uuid int64 541B 3,299B | dataset stringclasses 1
value | text stringlengths 1 4.29M |
|---|---|---|
1,314,259,993,563 | arxiv | \section{Introduction}
Broad Feshbach resonances in two-component atomic Fermi gases have made it possible to explore the crossover between Bardeen-Cooper-Schrieffer (BCS) superfluidity and Bose-Einstein condensation (BEC) \cite{Chin2010,Shin:2008,zwierlein-2006-442}. In the strongly interacting regime the difficulties in developing many-body theories for these systems has motivated the study of exact solutions to few-body problems as a means to gain insight into the many-body problem \cite{Busch:1998,Werner2006,Werner2006a,Kestner2007,Daily2010,Liu2009,Liu2010,Daily2010,Staferle2006}. Of particular interest has been the prediction \cite{Liu2010,Rakshit2012,Kaplan2011,Bhaduri2012} and observation \cite{Nascimbene2010,Ku2012} of universality in the strongly interacting regime of thermodynamic quantities, such as the energy and entropy. To date such calculations have been restricted to gases in non-rotating spherically symmetric harmonic traps.
In this work we consider the properties of a strongly interacting gas in a rotating trap. In particular we solve the two- \cite{Mulkerin2012} and three-body problems and calculate the virial expansion of the thermodynamic potential to third order, enabling the calculation of thermodynamic quantities. Interestingly, we find in the thermodynamic limit that the second- and third-order virial coefficients are universal, with respect to an external rotation and trapping frequencies. From this we show that thermodynamic quantities such as the energy and entropy are universal with respect to rotation through a simple rescaling of the Fermi energy.
In addition to these thermodynamic results we also examine the interplay between rotation and the emergence of itinerant ferromagnetism in strongly interacting ultra-cold Fermi gases. In the original work of Stoner \cite{stoner1938} it was proposed via a meanfield theory that a repulsive Fermi gas will always exhibit a ferromagnetic phase. Most recent experimental evidence \cite{Sanner2012} suggests there is no transition. Current theoretical work, Monte Carlo simulations and Tan relations \cite{Liu2010,Chang2011,Conduit2009a,Conduit2009} are contradictory. In this work we show that for the three-body problem, rotation suppresses the emergence of itinerant ferromagnetism. Additionally, in the thermodynamic limit itinerant ferromagnetism is suppressed for temperatures $T>10^{-7}T_{\text{F}}$ as the rotation frequency approaches the trapping frequency.
Our starting point is the wavefunction, $\psi(\mathbf{r}_1,\mathbf{r}_2,\ldots)$, of $N$ particles interacting in the $s$-wave channel at low energies. This satisfies the Bethe-Peierls boundary conditions
\begin{alignat}{1}
\lim_{r_{ij}\rightarrow 0}\partial(r_{ij}\psi)/\partial r_{ij} = -r_{ij} \psi /a
\label{eq:bethe-peierls},
\end{alignat}
where the interaction is parametrized by the scattering length $a$, and $r_{ij}=|\mathbf{r}_i-\mathbf{r}_j|$ is the separation of opposite-spin fermions.
Away from $r_{ij}=0$, the wavefunction of $N$ particles in a spherically symmetric rotating harmonic trap satisfies the non-interacting Schr\"odinger equation,
\begin{alignat}{1} \label{eq:rotHamiltonian}
\sum_{i=1}^{N} \left [ -\frac{\hbar^2}{2\mu}\mathbf{\nabla}_{i}^2 + \frac{1}{2}\mu\omega^2\mathbf{r}_i^2+ i\hbar\Omega_z\partial_{\phi_i}\right ] \psi=E\psi,
\end{alignat}
where $\mathbf{r}_i$ and $\mu$ are the position and mass of each particle and $\omega$ and $\Omega_z$ are the trapping and rotation frequencies, the latter assumed to be defined about the $z$ axis.
The center-of-mass Hamiltonian can be decoupled from Eq.~\eqref{eq:rotHamiltonian} and defines the rotating harmonic motion of a particle of mass $M=N\mu$ with energy spectrum
\begin{alignat}{1}
E_{\text{cm}}=(2n+l+3/2)\hbar\omega + m\hbar\Omega_z,
\label{eq:SingleParticleEnergies}
\end{alignat}
where $n,l,m$ label the usual harmonic oscillator eigenstates $R_{nl}Y_l^m$. The relative energy, $E_\text{rel}=E-E_\text{c.m.}$, incorporates the effects of the contact interaction but not the effects of the external rotation, since only $s$-wave states may interact. For two opposite-spin fermions the wavefunction in relative coordinates that satisfies the Bethe-Peierls boundary condition Eq.~\eqref{eq:bethe-peierls} is \cite{Busch:1998}
\begin{alignat}{1}
\psi^{\text{rel}}_{2b}(\mathbf{r};\nu)\propto\Gamma\left(-\nu\right)U\left(-\nu,3/2,r^2/d^2\right)\,\text{exp}\left(-r^2/d^2\right),
\end{alignat}
where $U$ is the confluent hypergeometric function of the second kind. A pseudo-quantum number, $\nu$, parametrizes the relative energy $E_\text{rel}=(2\nu+3/2)\hbar\omega$, and satisfies the relation
\begin{equation}\label{eq:Transcendental}
\frac{2\Gamma(-\nu)}{\Gamma(-\nu-1/2)}=\frac{d}{a},
\end{equation}
for harmonic oscillator length $d=\sqrt{\hbar/(\mu\omega)}$. In particular, for the unitary limit, where $a\rightarrow\pm\infty$, the relative energy spectrum simplifies to $E_\text{rel}=(2n+1/2)\hbar\omega$, where $n$ is any no-negative integer.
To find the energy spectrum of three interacting fermions in a rotating trap, we consider the configuration of two spin up fermions and one spin down, $\uparrow\downarrow\uparrow$, where two opposite spin particles interact at a point and form a pair, and the third moves relative to the pair. We define the center-of-mass coordinate of the three particles as $\smash{\mathbf{R}=(\mathbf{r}_1+\mathbf{r}_2+\mathbf{r}_3)/3}$, the relative coordinate between the interacting pair, $\smash{\mathbf{r}=\mathbf{r}_1-\mathbf{r}_2}$ and the relative coordinate between the third non-interacting particle and the center-of-mass of the pair as $\smash{\mathbf{\rho}=(2/\sqrt{3})[\mathbf{r}_3-(\mathbf{r}_1+\mathbf{r}_2)/2]}$. In this Jacobi coordinate system the center-of-mass Hamiltonian
decouples from the relative Hamiltonian
\begin{alignat}{1}
H_{\text{rel}}=
- \frac{\hbar^2}{\mu} \left(\nabla^{2}_{\mathbf{r}} + \nabla^{2}_{\mathbf{\rho}}\right) + \frac{1}{4}\mu\omega^2(\mathbf{r}^2 + \mathbf{\rho}^2) - i\hbar\Omega_z\partial_{\phi_\rho} \label{eq:jacobiHamil},
\end{alignat}
where $\mu/2$ is the reduced mass of the interacting pair. Like the two-body system, the angular momentum vanishes in the interacting pair but the third fermion can rotate around the pair and be affected by the external rotation, $\Omega_z$. This couples higher order angular momentum states to lower energies in the system.
In order to solve the relative Hamiltonian~\eqref{eq:jacobiHamil} we take
\begin{alignat}{1}
\Psi_{3b}^{\text{rel}}(\mathbf{r},\mathbf{\rho})=(1-P_{13})\sum_{n=0}^\infty c_n \psi_{2b}^{\text{rel}}(\mathbf{r};\nu_{nlm})R_{nm}(\mathbf{\rho})Y_{l}^{m}(\hat{\rho}),
\label{eq:ansatz}
\end{alignat}
as an ansatz for the wavefunction, where $P_{13}$ is an operator that exchanges the spin $\uparrow$ particles.
The eigenenergies of this system are
\begin{alignat}{1}
E_{\text{rel}}=\left[\left(2n+l+\frac{3}{2}\right)+\left(2\nu_{nlm}+\frac{3}{2}\right)\right]\hbar\omega+m\hbar\Omega_z \label{eq:3energies}.
\end{alignat}
The presence of the rotational term, $m\hbar\Omega_z$, in the eigenenergy spectrum shifts the non-rotating energy spectrum found in \cite{Liu2009}. To solve for the coefficients $c_n$ in Eq.~\eqref{eq:ansatz} we use the Bethe-Peierls boundary condition Eq.~\eqref{eq:bethe-peierls} and choose a set of quantum numbers $nlm$ and energy $E_{\text{rel}}$ to solve for a particular $\nu_{nlm}$ and scattering length $a$ \cite{Liu2009}.
\begin{figure}
\includegraphics{3particleswithrotation}
\caption{Energy spectrum of three interacting fermions with $l=2$ and $m=-2,0$ and $2$ for plots (a), (b) and (c) respectively with a rotation of $\Omega_z=0.9\omega$. We can see the shifting of the energy spectrum with the rotation included, in particular we can see the number of lower energy states increase for $m=-2$.}
\label{fig:spectrum}
\end{figure}
The relative energy spectrum can be found numerically for any scattering length, but in the unitary regime it is simpler to use the method of Werner and Castin \cite{Werner2006a} to obtain the energies using hyperspherical coordinates $(R,\alpha,\hat{r},\hat{\rho})$, where $R=\sqrt{(r^2+\rho^2)/2}$ is the hyperradius, $\alpha=\text{arctan}(r/\rho)$ is the first hyperangle and $\hat{r}$ and $\hat{\rho}$ are the direction of each Jacobi coordinate. Using this coordinate system and the ansatz for the wavefunction \cite{Werner2006a},
\begin{alignat}{1}
\Psi_{3b}^{\text{rel}}(\mathbf{r},\mathbf{\rho})=\frac{F(R)}{R^2}(1-P_{13})\frac{\varphi(\alpha)}{\sin(2\alpha)}Y_{l}^{m}(\hat{\rho}),
\label{eq:Fadeev}
\end{alignat}
the Hamiltonian, Eq.~\eqref{eq:jacobiHamil}, can be written as two decoupled Schr\"odinger equations,
\begin{alignat}{1}
-\frac{\hbar^2}{2m}\left(F''+\frac{1}{R}F'\right)+\left ( \frac{\hbar^2 s^{2}_{nl}}{2m R^2}+\frac{1}{2}m\omega^2 R^2 -m\hbar\Omega_z \right )F =E_{\text{rel}}F \label{eq:hyperradial},
\end{alignat}
and
\begin{alignat}{1}
-\varphi''(\alpha)+\frac{l(l+1)}{\cos^2(\alpha)}\varphi(\alpha)=s_{nl}^2\varphi(\alpha) \label{eq:hyperangular}.
\end{alignat}
For three fermions $s_{nl}^2$ is always positive and we can interpret the hyperradial Sch\"odinger equation \eqref{eq:hyperradial} as a particle moving in a two dimensional effective potential $\bigl( \hbar^2 s^{2}_{nl}/(2 m R^2) + \tfrac{1}{2} m \omega^2 R^2 \bigl)$ with energy spectrum \cite{Werner2006}
\begin{alignat}{1}
E_{\text{rel}}=(2q + s_{nl} + 1)\hbar\omega + m\hbar\Omega_z \label{eq:unitary},
\end{alignat}
where $q$ is a positive integer. The solutions to Eq.~\eqref{eq:hyperangular} must satisfy $\varphi(\pi/2)=0$ so that the ansatz Eq.~\eqref{eq:Fadeev} does not diverge. Hence,
\begin{alignat}{1}
\varphi(\alpha)\propto \cos(\alpha)^{l + 1}P^{(l + 1/2,-1/2)}_{(s_{nl} - l - 1)/2} \big [ -\cos(\alpha) \big ],
\label{eq:PhiExpansion}
\end{alignat}
where $P^{(\gamma,\beta)}_{n}(x)$ is the regular Jacobi polynomial \cite{abramowitz+stegun}. The eigenvalues $s_{nl}$ are determined from the Bethe-Peierls boundary condition \eqref{eq:bethe-peierls}, which
in hyperspherical coordinates reads
\begin{alignat}{1}
\varphi'(0)-(-1)^l \frac{4}{\sqrt{3}}\varphi \left ( \frac{\pi}{3} \right) =0
\label{eq:strongBethe}.
\end{alignat}
The procedure for solving Eq.~\eqref{eq:strongBethe} for values of $s_{nl}$ using the general solutions to the hyperangle equation is given in \cite{Liu2010}. It can be shown that the two spectra Eq.~\eqref{eq:unitary} and \eqref{eq:3energies} are the same in the unitary limit, where the solutions from Eq.~\eqref{eq:3energies} are found numerically.
\begin{figure}[t!]
\includegraphics{contours}
\caption
(a,b), Energy per particle $E/(NE_{\text{F}}^0)$ and (c,d), entropy per particle $S/(Nk_{\text{B}})$ for dimensionless rotation $\xi$ as a function of reduced temperature $T/T_{\text{F}}^0$, in the strongly (a,c), attractive and (b,d), repulsive regimes. For comparison we plot the temperature at which the second and third order virial expansions differ by $1\%$ for a rotation $\xi$, the dashed line. The hashed region in plots (a) and (c) indicate an unphysical solution to the virial expansion.}
\label{fig:contours}
\end{figure}
The energy spectrum of three interacting fermions is shown in Fig.~\eqref{fig:spectrum} for a rotation of $\Omega_z=0.9\omega$, a relative angular momentum of $l=2$ and axial angular momentum of $m=-2,0,2$. We see the energy levels for the three interacting fermions shift for each $m$ quantum number, lifting the $(2l+1)$ fold degeneracy.
To obtain the repulsive spectrum we omit the solution energy levels which are the lowest in each $l$-subspace lowest order $n=0$ bound state in each $l$ subspace, i.e. the $s_{0l}$ energy levels. The lowest energy in the repulsive regime is then the relative energy, which is dependent upon the rotation $\Omega_z$, plus the center-of-mass energy,
\begin{alignat}{1}
E_{gs}^{\uparrow\downarrow\uparrow}=(s_{nl} + 1)\hbar\omega + m\hbar\Omega_z + 1.5\hbar\omega.
\label{eq:intground}
\end{alignat}
This can be compared to the energy of three polarized fermions, which is the sum of the three lowest allowed energies:
\begin{alignat}{1}
E_{gs}^{\uparrow\uparrow\uparrow}=1.5\hbar\omega +2.5\hbar\omega + [1.5\hbar\omega+(\omega-\Omega_z)\hbar].
\label{eq:nonintground}
\end{alignat}
Comparing Eqs.~\eqref{eq:intground} and \eqref{eq:nonintground} it is possible to find a critical rotation, $\Omega_c$, for which the ground state energy of three repulsively interacting fermions becomes lower than three non-interacting polarized fermions. This is a rotation for which the three non-interacting fermions are unstable with respect to the three interacting fermion system. Choosing the lowest relative energy level in the $\smash{l=2}$ subspace, $\smash{s_{12}\simeq4.80}$, and a magnetic quantum number $\smash{m=-l}$, we find a critical rotation of $\smash{\Omega_c\simeq 0.8\omega}$. Since the ground state energy of three repulsively interacting fermions can be controlled by varying an external rotation, this indicates that an itinerant ferromagnetic transition, in the three-body system, only occurs for a rotation $\Omega_z<\Omega_c$.
The two- \cite{Mulkerin2012} and three-body solutions can now be used to calculate the many-body properties of strongly interacting rotating Fermi gases. This is done via a quantum virial expansion of the grand thermodynamic potential, $\Phi=-k_B T \ln \mathcal{Z}$, in terms of the fugacity $z$:
\begin{alignat}{1}
\Phi= & -k_B T Q_1 \frac{1}{2} \int_0^\infty \!\!\!\! \dd \epsilon \epsilon^2 \ln(1+z e^{-\epsilon}) \nonumber \\ & - k_B T Q_1 \left(z+\Delta b_2 z^2+\Delta b_3 z^3+\dots\right),
\label{eq:GTPOmega}
\end{alignat}
where
\begin{alignat}{1}
& \Delta b_2= \Delta Q_2/Q_1, \label{eq:Deltab2} \\ &
\Delta b_3= \Delta Q_3/Q_1 - \Delta Q_2, \label{eq:Deltab3}
\end{alignat}
and $\smash{\Delta Q_n = Q_n - Q_n^{(0)}}$ with $\smash{Q_N=\text{Tr}\left[\exp(-\mathcal{H}_N/k_B T)\right]}$ \cite{HuangBook}.
To first order for dimensionless rotation $\xi=\Omega_z/\omega$, $Q_1$ is given by
\begin{equation}
Q_1 = \left(\frac{k_{\text{B}}T}{\hbar\omega}\right)^3\frac{1}{1-\xi^2} +\dots
\label{eq:Q1}
\end{equation}
Following \cite{Mulkerin2012}, the second-order virial coefficient for a rotating trapped gas in the high temperature limit is
\begin{alignat}{1}
\Delta b_2^\text{att} & = \frac{1}{4} - \frac{\tilde{\omega}^2}{32} + \dots \label{eq:secondatt}\\
\Delta b_2^\text{rep} & = -\frac{1}{4} + \frac{\tilde{\omega}}{4} + \dots \label{eq:secondrep}
\end{alignat}
where $\tilde{\omega}=\hb\omega/k_{\text{B}} T$ is the reduced trapping frequency and $\smash{\tilde{\omega} \to 0}$ represents the thermodynamic limit. Extending the work of non-rotating systems in \cite{Liu2009} we find that $\Delta b_3^{att}$ is universal for any rotation and given by
\begin{alignat}{1}
\Delta b_3^\text{att} \simeq -0.06833960 + O(\tilde{\omega}^2). \label{eq:thirddatt}
\end{alignat}
For a repulsive gas the third-order virial coefficient to lowest order in $\tilde{\omega}$ is also universal,
\begin{alignat}{1}
\Delta b_3^\text{rep} \simeq 0.34976 +O(\tilde{\omega}). \label{eq:thirdrep}
\end{alignat}
Despite the rotational dependence of the two- and three-body eigenspectrums the second- and third-order virial coefficients are independent of rotation in the thermodynamic limit.
We are now able to calculate the total energy $E=-3\Phi$ and entropy $\smash{S=-\partial\Phi/\partial T}$ from the thermodynamic potential $\Phi$ of a strongly interacting gas \cite{HuangBook}. To determine the thermodynamic potential, the fugacity $\smash{z=\text{exp}(\mu/k_{\text{B}}T)}$ must be calculated from the total number of particles $\smash{N=-\pd\Phi/\pd\mu}$, where $\mu$ is the chemical potential.
In Fig.~\ref{fig:contours} we plot the energy, in units of $\smash{NE_{\text{F}}^0}$, where $\smash{E_{\text{F}}^0=(3N)^{1/3}\hbar\omega}$ is the non-rotating Fermi energy, and entropy per particle as functions of reduced temperature, $\smash{T_{\text{F}}^0=E_{\text{F}}^0/ k_{\text{B}}}$, and rotation using the virial expansion to third-order for a strongly repulsive, (b,d) and attractive, (a,c) Fermi gas. The hashed areas in plots (a) and (c) correspond to solutions of the energy and entropy that are unphysical. The dashed curve in Fig.~\ref{fig:contours} is the temperature at which the second and third order expansion differ by $1\%$, providing a conservative estimate of the temperature range of validity for the virial expansion. As the rotation is changed, for a fixed temperature, the energy (a,b) and entropy (c,d) change. Hence, the thermodynamic quantities appear not to be universal with respect to rotation.
From the three-body calculations Eqs.~\eqref{eq:intground} and \eqref{eq:nonintground} we see that itinerant ferromagnetism is suppressed for $\smash{\Omega_z\gtrsim0.8\omega}$. Fig.~\ref{fig:contours}(b) plots the total energy in the strongly repulsive regime. As $\smash{\xi \to 1}$ the validity of the solutions extends to $\smash{T \to 0}$. In this regime, by comparing the energy of the strongly interacting gas with the equivalent non-interacting polarized gas, we find that the itinerant ferromagnetic phase is suppressed for $T/T_{\text{F}}^0=10^{-7}$ \footnote{Due to numerical instabilities it is not possible to set $\xi=1$. Hence the limit $\smash{\xi\to 1}$ is evaluated at $\smash{\xi=1-10^{-7}}$, for which the virial expansion is valid for $\smash{T/T_{\text{F}}^0}>10^{-7}$}.
\begin{figure}[t]
\includegraphics{chemicalenergyentropy}
\caption{Chemical potential, (a), entropy per particle, (b) and energy per particle, (c) of a strongly attractive (solid), repulsive (dashed) and ideal Fermi gas (dotted) for dimensionless rotation $\xi=0, 0.5, 0.9$ and $0.99$ as a function of reduced temperature $\smash{T/T_{\text{F}}^\xi}$. }
\label{fig:energyentropy}
\end{figure}
Figure~\ref{fig:contours} demonstrates that for a given rotation frequency, $\xi$, there is a universal dependence of $\smash{E/(NE_{\text{F}}^{0})}$ and $\smash{S/(Nk_{\text{B}})}$ as a function of $T_{\text{F}}^{0}$, with respect to particle number and trapping frequency. However, through a simple rescaling of the Fermi energy [temperature] of the form $\smash{E_{\text{F}}^{\xi}=E_{\text{F}}^{0}(1-\xi^2)^{-1/3}}$ $\smash{\bigl[T_{\text{F}}^{\xi}=T_{\text{F}}^{0}(1-\xi^2)^{-1/3}\bigl]}$ it is possible to remove the rotational dependence observed in Fig.~\ref{fig:contours}. This generalized universality arises from the fact that under this rescaling the functional dependence of $\xi$ is removed from both the single particle cluster function, $Q_1$ [Eq.~\eqref{eq:Q1}], and the chemical potential. To emphasize the universal nature of the chemical potential with respect to rotation, $\mu/E_{\text{F}}^{\xi}$ is plotted in Fig.~\ref{fig:energyentropy}(a) as a function of $\smash{T/T_{\text{F}}^{\xi}}$ for various rotations, $\xi$, in the strongly attractive (solid), repulsive (dashed) and ideal (dotted) regimes. Hence, in conjunction with the fact that the second- [Eqs.(\ref{eq:secondatt},\ref{eq:secondrep})] and third-order [Eqs.(\ref{eq:thirddatt},\ref{eq:thirdrep})] virial coefficients are independent of rotation in the thermodynamic limit, the thermodynamic potential is independent of rotation. As a direct consequence the rescaled energy, $\smash{E/(NE_{\text{F}}^{\xi})}$, and entropy, $\smash{S/(Nk_{\text{B}})}$, are universal with respect to rotation as functions of $\smash{T/T_{\text{F}}^{\xi}}$,. This property is demonstrated in Figs.~\ref{fig:energyentropy}(b,c) which plot $\smash{E/(NE_{\text{F}}^{\xi})}$ (b) and $\smash{S/(Nk_{\text{B}})}$ (c) as a function of $\smash{T/T_{\text{F}}^{\xi}}$ for various values of $\xi$ in the strongly attractive (solid), repulsive (dashed) and ideal (dotted) regimes.
In conclusion we have examined the problem of three ultracold fermions in a harmonic trap subject to an external rotation. For the three-body problem we have demonstrated that rotation suppresses the transition to a ferromagnetic state. Explicitly we have shown that the three-body ferromagnetic state has a higher energy than the strongly interacting repulsive state for $\smash{\Omega_z\gtrsim0.8\omega}$ and is, consequently, unstable. Additionally, from the three-body solutions and the use of previous two-body results \cite{Mulkerin2012} we have calculated the equations of state using the virial expansion to third-order. Despite the rotational dependence of the two- and three-body eigenspectrums we have generalized the universal nature of strongly interacting fermions to include rotation by a simple rescaling of the Fermi energy and temperature. These results could be used as benchmarks in experiment to test the universal properties of strongly interacting rotating ultracold Fermi gases. \linebreak
H.M.Q. gratefully acknowledges the support of the ARC Centre of Excellence for Coherent X-ray Science.
|
1,314,259,993,564 | arxiv | \section{Introduction}
\noindent In this paper, we study the following fractional Kirchhoff-Schr\"{o}dinger-Poisson system involving a singular term.
\begin{align}\label{problem main}
\begin{split}
\left(a+b\int_{Q}\frac{(u(x)-u(y))^2}{|x-y|^{N+2s}}\right)(-\Delta)^{s} u+\phi u&=\lambda h(x)u^{-\gamma}+f(x,u)~\text{in}~\Omega,\\
(-\Delta)^{s}\phi&=u^{2}~\text{in}~\Omega,\\
u&>0~\text{in}~\Omega,\\
u&=\phi=0~\text{in}~\mathbb{R}^N\setminus\Omega,
\end{split}
\end{align}
where $\Omega$ is a smooth bounded domain in $\mathbb{R}^{N}$, $a, b\geq 0, a+b>0, \gamma> 0, \lambda>0$, $h\in L^{1}(\Omega)$, $h(x) >0$ a.e. in $\Omega$ and $f$ has some growth conditions.\\
In the recent time elliptic PDEs involving singularity has drawn interest to many researchers for both the local as well as the nonlocal operators. A noteworthy application of the fractional Laplacian operator can be found in \cite{valdinoci2009long} and the references therein. Further the application can be seen in the field of fluid dynamics, in particular to the study the thin boundary layer properties for viscous fluids \cite{thin bdry}, in probability theory to study the Levy process \cite{bertoin1996levy}, in finance \cite{cont2004financial}, in free boundary obstacle problems \cite{silvestre2008}. Another application of PDEs involving these type of nonlocal operator is in the field of image processing to find a clear image $u$ from a given noisy image $f$ \cite{bua,kinder}. Readers who are interested to know further details on applications of PDEs involving nonlocal operators can also refer to \cite{servadei2016book,dinezza2012}.\\
The following model problem of the type \eqref{problem main} was first introduced by Kirchhoff \cite{kirchhoff prob} as a generalization of the D'Alembert wave equation.
$$\rho\frac{\partial^2u}{\partial t^2}-\left(a+b\int_{0}^{l}\left|\frac{\partial u}{\partial x}\right|^2dx\right)\frac{\partial^2u}{\partial x^2}=g(x,u)$$
where $a, b, \rho$ are positive constants and $l$ is the changes in the length of the strings due to the vibrations. Recently, Fiscella and Valdinoci (\cite{valdinoci2014}, Appendix A), introduced the fractional Kirchhoff type problem considering the fractional length of the string for nonlocal measurements. A physical application of nonlocal Kirchhoff type problem can be found in \cite{pucci2016}.\\
The problem \eqref{problem main} is said to be degenerate if $a=0$ and $b>0$. Otherwise, if both $a>0$ and $b>0$, we say the problem \eqref{problem main} is non degenerate. For $a>0$ and $b=0$, the problem \eqref{problem main} reduces to Schr\"{o}dinger-Poisson system. For $b=\phi=0$, a vast amount of study to prove the existence, multiplicity and regularity of solutions to the problem of type \eqref{problem main} has been done involving both the local operator ($s=1$) as well as the nonlocal operator ($0<s<1$) with a singularity for both $0<\gamma<1$ and $\gamma>1$ and a power nonlinearity or an $L^1$ data or both. The literature is so vast that it is almost impossible to enlist all of them here. A few of such studies can be found in \cite{boccardo2010semilinear,canino2017nonlocal,crandall1977dirichlet,positivity,ghosh2018singular,giacomoni2009multiplicity,lazer1991singular,lei2015,liao2015,liu2013,oliva,saoudi2017critical,ghosh2018multiplicity,sun2013,sun2001,sun2014,yang2003} and the references therein.\\
For $b= f(x,u)=\phi(x)=0$, the problem \eqref{problem main}, reduces to a purely singular problem. In their celebrated article, Lazer and McKenna \cite{lazer1991singular}, have studied the purely singular problem involving the Laplacian operator, i.e. for $s=1$ and $b=f(x,u)=\phi(x)=0$. The authors in \cite{lazer1991singular}, has proved that the problem has a unique $C^1(\bar{\Omega})$ solution iff $0<\gamma<1$ and it has a $H_0^1(\Omega)$ solution iff $\gamma<3$. Later in \cite{sun2014}, the author proved that if $\gamma\geq3$, then the singular problem can not have $H_0^1(\Omega)$ solution.\\
Similar to the study with the Laplacian operator, Canino et al. \cite{canino2017nonlocal}, have studied the nonlocal PDE involving singularity. In \cite{canino2017nonlocal}, the authors considered the problem
\begin{align}\label{canino singular}
(-\Delta_p)^s u&=\lambda\frac{a(x)}{u^{\gamma}} ~\&~ u>0 ~\text{in}~\Omega,\nonumber\\
u&=0~\text{in}~\mathbb{R}^N\setminus\Omega.
\end{align}
The authors in \cite{canino2017nonlocal}, has guaranteed the existence of unique solution in $W_0^{s,p}(\Omega)$ for $0<\gamma\leq 1$ and in $W_{loc}^{s,p}(\Omega)$ for $\gamma>1.$ For $a(x)\equiv 1=\lambda$, Fang \cite{fang2014existence}, has proved the existence of a unique $C^{2, \alpha}(\Omega)$ solution for $0<\alpha<1$. One of the earliest study to show the existence of multiple solutions was made by Crandall et al. \cite{crandall1977dirichlet} involving the Laplacian operator. Further references on multiplicity involving local operator can be bound in \cite{giacomoni2009multiplicity} and the references therein. Recently, Saoudi et al. in \cite{ghosh2018multiplicity} has guaranteed the existence of at least two solutions by using min-max method with the help of modified Mountain Pass theorem involving fractional $p$-Laplacian operator. Saoudi in \cite{saoudi2017critical}, obtained two solutions involving fractional Laplacian operator For further references on the study of multiple solutions, refer \cite{giacomoni2009multiplicity,ghosh2018multiplicity} and the references therein.\\
Recently, the study of the Kirchhoff-Schr\"{o}dinger-Poisson system has drawn interest to many researchers. See for instance \cite{fiscella2019,fiscella2019-1,lei2015,fuyi2013,li2017,liao2016,liao2015,liu2013,sun2013,sun2001,zhang2016,jmp} and the references therein for a detailed study of existence, uniqueness and multiplicity of Kirchhoff type problem. Most of these studies, the authors used variational techniques, in particular min-max method, sub-super solution method, Nehari manifold method and Mountain Pass theorem to guarantee the existence and multiplicity of solutions.\\
In the recent past, for $a=0$ and $0<\gamma<1$, Fiscella \cite{fiscella2019} has obtained two distinct solution involving fractional Laplacian operator by variational technique. Later, for $a>0$, $0<\gamma<1$, Fiscella and Mishra \cite{fiscella2019-1} proved the multiplicity by Nehari manifold method. In \cite{liao2015,liu2013}, the authors studied the multiplicity of solutions involving singular nonlinearity. In \cite{fuyi2013}, Li and Zhang have studied the existence, uniqueness and multiplicity of solution(s) for Schr\"{o}dinger-Poisson system without compactness conditions. On the other hand, Zhang \cite{zhang2016} has studied for a Schr\"{o}dinger-Poisson system. For a detailed study on Schr\"{o}dinger-Poisson system, one can see \cite{antonio2008,antonio2013,ruiz2006} and there references therein. Liao et al. \cite{liao2016} has guaranteed the existence and uniqueness of solution for Kirchhoff type problem involving singularity. The author in \cite{sun2013}, provided a compatibility criterion to obtain the existence of solution for $b=\phi=0, \gamma>1$ and $f(x,t)=t^p, 0<p<1$. The author in \cite{sun2013}, proved that the problem \eqref{problem main} has a $H_{0}^{1}(\Omega)$ solution if and only if the following compatibility condition for the pair $(h, \gamma)$
\begin{equation}\label{compatibility local}
\int_{\Omega}h(x)|u_{0}|^{1-\gamma}<\infty~\text{for some}~ u_{0}\in H_{0}^{1}(\Omega)
\end{equation} holds true.\\
Recently, Zhang \cite{jmp}, proved a necessary and sufficient condition for the existence of solution for a Kirchhoff-Schr\"{o}dinger-Poisson system involving Laplacian operator with strong singularity.\\
In their pioneering work, Ambrosetti and Rabinowitz \cite{ambrosetti1973dual} has guaranteed the existence of infinitely many solutions to the problem of the type \eqref{problem main} for $a=s=1$, $b=\lambda=\phi=0$ by introducing the well known (AR) condition on $f$. In fact the (AR) condition has proved to be an important tool to obtain multiplicity of solutions. One can see \cite{binlin2015superlinear,positivity,wang infi} and the references therein for further details on infinitely many solutions. For $b=\phi=\lambda=0$, Binlin et al. \cite{binlin2015superlinear}, has proved the existence of infinitely many solutions for a superlinear data $f$. The study of Kirchhoff type problem to obtain infinitely many solution can be found in \cite{figi2015,li converge,zhao infi} and the references therein. For $\lambda=0$, Li et al. \cite{wang infi} guaranteed the existence of infinitely many solutions to the problem \eqref{problem main} for a sublinear data $f$. Recently, for $0<\gamma<1$, $b=\phi=0$, Ghosh and Choudhuri \cite{positivity}, guaranteed the existence of infinitely many solutions involving the fractional Laplacian operator. In all of these studies referred here that consists of infinitely many solutions, the authors used the symmetric Mountain Pass theorem under the crucial assumption that the data $f$ is odd. The authors in \cite{positivity}, assumed the following growth conditions on $f$.
\begin{itemize}
\item[(A1)] $f\in C(\Omega\times\mathbb{R}, \mathbb{R})$ and $\exists$ $\delta>0$ such that $\forall\,x\in\Omega$ and $|t|\leq\delta,$ $f(x,-t)=-f(x,t).$
\item[(A2)] $\lim\limits_{t\rightarrow0}\frac{f(x,t)}{t}=+\infty$ uniformly on $\Omega.$
\item[(A3)] There exists $r>0$ and $p\in(1-\gamma, 2)$ such that $\forall,\,x\in\Omega$ and $|t|\leq r$, $tf(x,t)\leq pF(x,t)$, where $F(x,t)=\int_{0}^{t}f(x,\tau) d\tau.$
\end{itemize}
Motivated from \cite{positivity,wang infi,sun2013}, we consider the fractional Kirchhoff-Schr\"{o}dinger-Poisson system \eqref{problem main} involving singularity. To the best of the author's knowledge, there in no study of infinitely many solutions for a fractional Kirchhoff-Schr\"{o}dinger-Poisson system involving a power nonlinearity and a singularity ($0<\gamma<1$) in the literature. On the other hand, the study of the problem \eqref{problem main} with strong singularity ($\gamma>1$) is comparatively challenging and hence can also be seen as a new addition to the literature involving nonlocal operator. One can expect the compatibility condition \eqref{compatibility local} as in \cite{sun2013}, for the existence of $X_0$ solution(s) to be
\begin{equation}\label{compatibility}
\int_{\Omega}h(x)|u_{0}|^{1-\gamma}<\infty~\text{for some}~ u_{0}\in X_0.
\end{equation}
The following two Theorems are the main results proved in this article.
\begin{theorem}\label{main thm1}
Assume $a, b\geq0, a+b>0$, $h\in L^{1}(\Omega), h>0$ a.e. in $\Omega$ and ($A1$)-($A3$) holds. Then for $0<\gamma<1$ and for any $\lambda\in(0,\Lambda)$, the problem \eqref{problem main} has a sequence of positive weak solutions ${u_n}\subset X_0\cap L^{\infty}(\Omega)$ such that $I(u_n) < 0$, $I(u_n)\rightarrow 0^{-}$ and $u_n \rightarrow 0$ in $X_0$. (See Section 2. for notations).
\end{theorem}
\begin{remark}
Note that in Theorem \ref{main thm1} there is no any restriction condition for $f$ in $t$ at infinity.
\end{remark}
\begin{theorem}\label{main theorem}
Assume $a, b\geq0, a+b>0$, $h\in L^{1}(\Omega), h>0$ a.e. in $\Omega$ and $f(x,u)=k(x)u^p$ such that $k\in L^{\infty}(\Omega)$ with $k>0$, $0<p<1$. Then, for $\gamma>1$ and for any $\lambda\in(0,\Lambda)$, the problem \eqref{problem main} has a weak solution in $X_0$ if and only if \eqref{compatibility} holds true.
\end{theorem}
\begin{remark}
If we assume $f(x,\cdot)\equiv0$, then the problem \eqref{problem main} possesses a unique solution.
\end{remark}
\noindent The paper is organised as follows. In Section 2, we will first give some mathematical formulation and define the space $X_0$. Moreover, we will discuss some preliminary properties of $\phi$ and prove that $\Lambda$ has a finite range. In the subsequent sections, Section 3 and Section 4, we will obtain the results as stated in Theorem \ref{main thm1} and Theorem \ref{main theorem} respectively.
\section{Mathematical formulations}
This section is devoted to give a few important results of fractional Sobolev spaces, embeddings, variational formulations and space setup. Let $\Omega$ be open bounded domain of $\mathbb{R}^N$ and $Q=\mathbb{R}^{2N}\setminus((\mathbb{R}^{N}\setminus\Omega)\times(\mathbb{R}^{N}\setminus\Omega))$. For $0<s<1$, the space ($X,\|.\|$), which is an intermediary Banach space between $H^1(\Omega)$ and $L^2(\Omega)$, is defined as
\begin{eqnarray}
X&=&\left\{u:\mathbb{R}^N\rightarrow\mathbb{R}~\text{is measurable}, u|_{\Omega}\in L^2(\Omega) ~\text{and}~\frac{|u(x)-u(y)|}{|x-y|^{\frac{N+2s}{2}}}\in L^{2}(Q)\right\}\nonumber
\end{eqnarray}
equipped with the norm
\begin{eqnarray}
\|u\|_X&=&\|u\|_{2}+[u]_2,\nonumber
\end{eqnarray}
where $[u]_2=\left(\int_{Q}\frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}dxdy\right)^{\frac{1}{2}}$ refers to the Gagliardo semi norm. Due to the zero Dirichlet boundary condition, it is natural to consider the space
$$X_0=\{u\in X:u=0 ~\text{a.e. in}~\mathbb{R}^N \setminus\Omega\},$$ endowed with the following Gagliardo norm on it.
\begin{eqnarray}
\|u\|&=&\left(\int_{Q}\frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}dxdy\right)^{\frac{1}{2}}.\nonumber
\end{eqnarray}
The space $(X_0, \|.\|)$ is a Hilbert space \cite{servadei2012mountain}. The best Sobolev constant is defined as
\begin{equation}\label{sobolev const}
S=\underset{u\in X_0\setminus\{0\}}{\inf}\cfrac{\int_{Q}\frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}dxdy}{\left(\int_\Omega|u|^{2_s^*}dx\right)^{\frac{2}{2_s^*}}}
\end{equation}
Let $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^N$. Then for every $q\in[1, 2_s^*]$, the space $X_0$ is continuously embedded in $L^q(\Omega)$ and for every $q\in[1, 2_s^*)$, the space $X_0$ is compactly embedded in $L^q(\Omega)$, where $2_s^*=\frac{2N}{N-2s}.$ Prior to define the weak solution to our problem, let us first consider the following problem
\begin{align}\label{poisson eqn}
(-\Delta)^{s}\phi=u^{2}~\text{in}~\Omega,\nonumber\\
\phi=0~\text{in}~\mathbb{R}^N\setminus\Omega.
\end{align}
In light of the Lax-Milgram theorem, for every $u\in X_0$, the problem \eqref{poisson eqn} has a unique solution $\phi_{u}\in X_0$ and we have the following Lemma consisting some properties of the solution $\phi_{u}$.
\begin{lemma}\label{phi prop}
For each solution $\phi_u\in X_0$ of \eqref{poisson eqn}, we have
\begin{enumerate}[label=(\roman*)]
\item $\|\phi_{u}\|^{2}=\int_{\Omega}\phi_{u}u^{2}dx=\int_{\Omega}|(-\Delta)^{s/2}\phi_{u}|^{2}dx\leq C_{\phi}\|u\|^{4}, ~\forall~u\in X_0$;
\item $\phi_{u}\geq 0$. Moreover, $\phi_{u}>0$ if $u\neq 0$;
\item for all $t\neq 0$, $\phi_{tu}=t^{2}\phi_{u}$;
\item $\|u_n-u\|\rightarrow 0$ implies that $\|\phi_{u_n}-\phi_u\|\rightarrow 0$ and $\int_{\Omega}\phi_{u_{n}}u_{n}vdx\rightarrow\int_{\Omega}\phi_{u}uvdx$, for any $v\in X_0$;
\item for any $u, v\in X_0$, we have $\int_{\Omega}(\phi_{u}u-\phi_{v}v)(u-v)dx\geq\frac{1}{2}\|\phi_{u}-\phi_{v}\|^{2}.$
\end{enumerate}
\end{lemma}
\noindent Now by replacing $\phi_u$ in place of $\phi$ in \eqref{problem main}, the problem \eqref{problem main} reduces to the following Dirichlet boundary value problem
\begin{align}\label{problem reduced}
\begin{split}
\left(a+b\int_{Q}\frac{(u(x)-u(y))^2}{|x-y|^{N+2s}}\right)(-\Delta)^{s} u+\phi_u u&=\lambda h(x)u^{-\gamma}+f(x,u)~\text{in}~\Omega,\\
u&>0~\text{in}~\Omega,\\
u&=0~\text{in}~\mathbb{R}^N\setminus\Omega,
\end{split}
\end{align}
We now define a weak solution to the problem \eqref{problem reduced}.
\begin{definition}\label{weak solution defn}
A function $u\in X_0$ is a weak solution to the problem \eqref{problem reduced}, if $u>0$ and
{\small \begin{align}\label{weak formulation}
&(a+b[u]^{2})\int_{Q}(-\Delta)^{s/2} u\cdot(-\Delta)^{s/2}\psi+\int_{\Omega}\phi_{u}u\psi -\lambda\int_{\Omega}h(x)u^{-\gamma}\psi -\int_{\Omega}f(x,u)\psi=0,
\end{align}}
for every $\psi\in X_0.$
\end{definition}
\noindent The associated energy functional to the problem \eqref{problem reduced} is defined as
{\small \begin{align}\label{functional}
I(u)=\frac{a}{2}\|u\|^{2}+\frac{b}{4}\|u\|^{4}&+\frac{1}{4}\int_{\Omega}\phi_{u}u^{2}-\frac{\lambda}{1-\gamma}\int_{\Omega}h(x)|u|^{1-\gamma}-\int_{\Omega}F(x,u),~ u\in X_0,
\end{align}}
\noindent where $F(x,u)=\int_{0}^{u}f(x,t)dt$. Observe that for $0<\gamma<1$, the term $\int_{\Omega}h(x)|u|^{1-\gamma}<\infty$ but the functional $I$ fails to be $C^1$. Therefore, by modifying the problem \eqref{problem main}, we will use the Kajikiya's Symmetric mountain pass theorem \cite{kajikiya2005critical} and a cut-off technique developed in \cite{clark1972variant} to obtain a $C^1$ functional to guarantee the existence of infinitely many solutions. On the other hand, for $\gamma>1$ the integral $\int_{\Omega}h(x)|u|^{1-\gamma}dx$ is not finite for $u \in X_0$. Therefore, the energy functional $I$ fails to be continuous and we cannot use the usual variational technique to guarantee the existence of solution. We will use arguments from \cite{sun2013} to obtain a weak solution to the problem \eqref{problem reduced}. Similar type of results can also be found in \cite{zhang2016}. We now state and prove the following Lemma to guarantee a finite range for $\Lambda$,which is defined as
$$\Lambda=\inf\{\lambda>0: ~\text{The problem \eqref{problem main} has no solution}\}.$$
\begin{lemma}\label{lambda finite}
Assume $a, b, \gamma>0$, ($A1$)-($A3$) and \eqref{compatibility} holds. Then $0\leq\Lambda<\infty$.
\end{lemma}
\begin{proof}
By definition, $\Lambda\geq 0$. Let $\phi_1>0$ be the first eigenfunction \cite{brasco2016second} corresponding to the first eigenvalue $\lambda_1$ for the fractional Laplacian operator. Then we have
\begin{align}
\begin{split}
(-\Delta)^s\phi_1&=\lambda_1\phi_1~\text{in}~\Omega\\
\phi_1&>0~\text{in}~\Omega\\
\phi_1&>0~\text{in}~\mathbb{R}^N\setminus\Omega.
\end{split}
\end{align}
Therefore, by putting $\phi_1$ as the test function in Definition \ref{weak solution defn}, we obtain
\begin{align}\label{lambda invalid}
\begin{split}
\lambda_1\int_{\Omega}(a+b\|u\|^2)u\phi_1dx&=\int_{\Omega}(a+b\|u\|^2)(-\Delta)^s\phi_1udx\\
&=\int_{\Omega}\left(\lambda h(x)u^{-\gamma}+f(x,u)-\phi_uu \right)\phi_1dx
\end{split}
\end{align}
At this stage, we choose $\tilde{\Lambda}>0$ such that
$$\tilde{\Lambda}h(x_0)t^{-\gamma}+f(x_0,t)>2\lambda_1t(a+bt^2)+\phi_tt$$ for all $t>0$ and for some $x_0\in\Omega$, which gives a contradiction to \eqref{lambda invalid}. Hence $\Lambda<\infty$.
\end{proof}
\noindent In the subsequent two sections, we establish the existence of solution(s).
\section{Existence of infinitely many solutions for $0<\gamma<1$.}
\noindent We begin this section with the definition of genus of a set.
\begin{definition}\label{genus} {(\bf{Genus})}
Let $X$ be a Banach space and $A\subset X$. A set $A$ is said to be symmetric if $u\in A$ implies $(-u)\in A$. Let $A$ be a closed, symmetric subset of $X$ such that $0\notin A$. We define a genus $\gamma(A)$ of $A$ by the smallest integer $k$ such that there exists an odd continuous mapping from $A$ to $\mathbb{R}^{k}\setminus\{0\}$. We define $\gamma(A)=\infty$, if no such $k$ exists.
\end{definition}
\noindent We now define the following family of sets, $$\Gamma_n=\{A_n\subset X: A_n~\text{is closed, symmetric and}~ 0\notin A_n~\text{such that the genus}~ \gamma(A_n)\geq n\}.$$
Further, we will use the following version of the symmetric Mountain Pass Theorem from \cite{kajikiya2005critical}.
\begin{theorem}\label{sym mountain}
Let $X$ be an infinite dimensional Banach space and $\tilde{I}\in C^1(X,\mathbb{R})$ satisfies the following
\begin{itemize}
\item[(i)] $\tilde{I}$ is even, bounded below, $\tilde{I}(0)=0$ and $\tilde{I}$ satifies the $(PS)_c$ condition.
\item[(ii)] For each $n\in\mathbb{N}$, there exists an $A_n\in\Gamma_n$ such that $\sup\limits_{u\in A_n}\tilde{I}(u)<0.$
\end{itemize}
Then for each $n\in\mathbb{N}$, $c_n=\inf\limits_{A\in \Gamma_n}\sup\limits_{u\in A}\tilde{I}(u)<0$ is a critical value of $\tilde{I}.$
\end{theorem}
\noindent We will modify the problem \eqref{problem reduced} to apply the symmetric Mountain Pass Theorem as follow
\begin{align}\label{main2}
\begin{split}
\left(a+b\int_{Q}\frac{(u(x)-u(y))^2}{|x-y|^{N+2s}}\right)(-\Delta)^{s} u+\phi_u u&=\lambda h(x)sign(u)|u|^{-\gamma}+f(x,u)~\text{in}~\Omega,\\
u&=0~\text{in}~\mathbb{R}^N\setminus\Omega,
\end{split}
\end{align}
\noindent The associated energy functional to the problem \eqref{main2} is defined as
\begin{align}\label{energy modified}
J(u)=\frac{a}{2}\|u\|^{2}+\frac{b}{4}\|u\|^{4}&+\frac{1}{4}\int_{\Omega}\phi_{u}u^{2}-\frac{\lambda}{1-\gamma}\int_{\Omega}h(x)|u|^{1-\gamma}-\int_{\Omega}F(x,u),~ u\in X_0,
\end{align}
where $F(x,u)=\int_{0}^{|u|}f(x,t)dt$. Observe that the functional $J$ is even by using the assumption ($A1$) and Lemma \ref{phi prop}(iii). We now define a weak solution to the modified problem \eqref{main2}.
\begin{definition}\label{weak modified}
A function $u\in X_0$ is a weak solution of \eqref{main2}, if $\phi |u|^{-\gamma}\in L^1(\Omega)$ and
{\small \begin{align}
&(a+b[u]^{2})\int_{Q}(-\Delta)^{s/2} u\cdot(-\Delta)^{s/2}\psi+\int_{\Omega}\phi_{u}u\psi -\int_{\Omega}\left(\lambda h(x)sign(u)|u|^{-\gamma}+f(x,u)\right)\psi=0,
\end{align}}
\end{definition}
\noindent for every $\psi\in X_0.$ Observe that if $u>0$ a.e. in $\Omega$, then weak solutions to the problem \eqref{main2} and to the problem \eqref{problem reduced} coincide. Therefore, it is sufficient to obtain a sequence of nonnegative weak solutions to the problem \eqref{problem reduced}. We now extend and modify $f(x,u)$ for $u$ outside a neighbourhood of $0$ by $\tilde{f}(x, u)$ as follow. We will follow \cite{clark1972variant} by considering a cut-off problem. Choose $l>0$ sufficiently small such that $0<l\leq\frac{1}{2}\min\{\delta, r\}$, where $\delta$ and $r$ are same as in the assumptions on $f$. We now define a $C^1$ function $\xi:\mathbb{R}\rightarrow\mathbb{R}^+$ such that $0\leq\xi(t)\leq1$ and
$$\xi(t)=\begin{cases}
1, ~\text{if}~ |t|\leq l\\
\xi ~\text{is decreassing, if}~ l\leq t\leq 2l\\
0,~\text{if}~ |t|\geq 2l.
\end{cases}$$
We now consider the following cut-off problem by defining $\tilde{f}(x, u)=f(x, u)\xi(u)$.
\begin{align}\label{main3}
\begin{split}
\left(a+b\int_{Q}\frac{(u(x)-u(y))^2}{|x-y|^{N+2s}}\right)(-\Delta)^{s} u+\phi_u u&=\lambda h(x)sign(u)|u|^{-\gamma}+\tilde{f}(x,u)~\text{in}~\Omega,\\
u&=0~\text{in}~\mathbb{R}^N\setminus\Omega,
\end{split}
\end{align}
\noindent The associated energy functional to the problem \eqref{main3} is defined as
\begin{align}\label{energy cutoff}
\tilde{I}(u)=\frac{a}{2}\|u\|^{2}+\frac{b}{4}\|u\|^{4}&+\frac{1}{4}\int_{\Omega}\phi_{u}u^{2}-\frac{\lambda}{1-\gamma}\int_{\Omega}h(x)|u|^{1-\gamma}-\int_{\Omega}\tilde{F}(x, u)dx, ~ u\in X_0.
\end{align}
\noindent We define a weak solution to the problem \eqref{main3} as follows.
\begin{definition}\label{weak cutoff}
A function $u\in X_0$ is a weak solution of \eqref{main3}, if $\phi |u|^{-\gamma}\in L^1(\Omega)$ and
{\small \begin{align}
&(a+b[u]^{2})\int_{Q}(-\Delta)^{s/2} u\cdot(-\Delta)^{s/2}\psi+\int_{\Omega}\phi_{u}u\psi -\int_{\Omega}(\lambda h(x)sign(u)|u|^{-\gamma}+\tilde{f}(x,u))\psi=0
\end{align}}
\end{definition}
\noindent for every $\psi\in X_0$. Again, if $\|u\|_{\infty}\leq l$ holds, then the weak solutions of \eqref{main3} and the weak solutions of \eqref{main2} coincide. We establish the existence result for the problem \eqref{main3}. Finally, we prove our main theorem by showing that the solutions to \eqref{main3} are positive and $\|u\|_{\infty}\leq l$.
\noindent We first prove the following Lemmas which are the hypotheses to the Symmetric mountain pass theorem.
\begin{lemma}\label{lemma ps}
The functional $\tilde{I}$ is bounded from below and satisfies $(PS)_c$ condition.
\end{lemma}
\begin{proof}
By the definition of $\xi$ and using the H\"{o}lder's inequality, we get
\begin{align*}
\tilde{I}(u)&\geq\frac{a}{2}\|u\|^{2}+\frac{b}{4}\|u\|^{4}+\frac{1}{4}\int_{\Omega}\phi_{u}u^{2}-C\|u\|^{1-\gamma}-C_1\\
&\geq\frac{a}{2}\|u\|^{2}+\frac{b}{4}\|u\|^{4}- C\|u\|^{1-\gamma}-C_1
\end{align*}
where, $C$, $C_1$ are nonnegative constants. Since $a, b>0$, this implies that $\tilde{I}$ is coercive and bounded from below in $X_0$. Let $\{u_n\}\subset X_0$ be a Palais Smale sequence for the functional $\tilde{I}$. Therefore, by using the coerciveness property of $\tilde{I}$ we have $\{u_n\}$ is bounded in $X_0$. Thus, we may assume that $\{u_n\}$ has a subsequence (still denoted by $\{u_n\}$) such that $u_n\rightharpoonup u$ in $X_0$. Therefore, we have
\begin{equation}\label{convergence weak}
\int_{Q}(-\Delta)^{s/2} u_n\cdot(-\Delta)^{s/2}\psi dxdy\longrightarrow\int_{Q}(-\Delta)^{s/2} u\cdot(-\Delta)^{s/2}\psi dxdy
\end{equation}
for all $\phi\in X_0.$
By the embedding result \cite{servadei2012mountain}, we can assume for every $q\in[1, 2_s^*)$
\begin{align}
u_n&\longrightarrow u ~\text{in}~ L^q(\Omega),\label{embed strong}\\
u_n(x)&\longrightarrow u(x) ~\text{a.e.}~ L^q(\Omega).\label{embed pointwise}
\end{align}
Therefore, from Lemma A.1 \cite{willem1997minimax}, we get that there exists $g\in L^q(\Omega)$ such that
\begin{equation}\label{appendeix A1}
|u_n(x)|\leq g(x) ~\text{a.e. in}~ \Omega, \forall\,n\in\mathbb{N}.
\end{equation}
Now on using \eqref{embed strong}, \eqref{embed pointwise}, \eqref{appendeix A1} and applying the Lebesgue dominated convergence theorem, we obtain
\begin{equation}\label{convergence f tilla}
\int_{\Omega}\tilde{f}(x,u_n)udx\rightarrow\int_{\Omega}\tilde{f}(x,u)udx ~\text{and}~ \int_{\Omega}\tilde{f}(x,u_n)u_ndx\rightarrow\int_{\Omega}\tilde{f}(x,u)udx.
\end{equation}
Moreover,
\begin{equation}\label{convergence phi}
\int_{\Omega}\phi_{u_n}u_nudx\rightarrow\int_{\Omega}\phi_{u}u^2dx ~\text{and}~ \int_{\Omega}\phi_{u_n}u_n^2dx\rightarrow\int_{\Omega}\phi_{u}u^2dx.
\end{equation}
Again, on using the H\"{o}lder's inequality and passing the limit $n\rightarrow\infty$, we get
\begin{align}
\begin{split}
\int_{\Omega}u_n^{1-\gamma}dx&\leq\int_{\Omega}u^{1-\gamma}dx+\int_{\Omega}|u_n-u|^{1-\gamma}dx\\
&\leq\int_{\Omega}u^{1-\gamma}dx+C\|u_n-u\|_{L^2(\Omega)}^{1-\gamma}\\
&=\int_{\Omega}u^{1-\gamma}dx +o(1).
\end{split}
\end{align}
Similarly, we have
\begin{align}
\begin{split}
\int_{\Omega}u^{1-\gamma}dx&\leq\int_{\Omega}u_n^{1-\gamma}dx+\int_{\Omega}|u_n-u|^{1-\gamma}dx\\
&\leq\int_{\Omega}u_n^{1-\gamma}dx+C\|u_n-u\|_{L^2(\Omega)}^{1-\gamma}\\
&=\int_{\Omega}u_n^{1-\gamma}dx +o(1).
\end{split}
\end{align}
Therefore,
\begin{equation}\label{convergence singular}
\int_{\Omega}u_n^{1-\gamma}dx=\int_{\Omega}u^{1-\gamma}dx+o(1).
\end{equation}
Since, $\{u_n\}$ is a Palais Smale sequence of $\tilde{I}$ therefore, by weak convergence, we gave
\begin{equation}\label{ps seq}
\langle\tilde{I}'(u_n)-\tilde{I}'(u), u_n-u\rangle=o(1)~\text{as}~n\rightarrow\infty.
\end{equation}
On the other hand,
\begin{align}
\langle\tilde{I}'(u_n)-\tilde{I}'(u), (u_n-u)\rangle&= (a+b[u_n]^{2})\langle u_n, (u_n-u)\rangle-(a+b[u]^{2})\langle u, (u_n-u)\rangle\nonumber\\
&+\int_{\Omega}[(\phi_{u_n}u_n-\phi_{u}u) -\lambda h(x)(sign(u_n)|u_n|^{-\gamma}- sign(u)|u|^{-\gamma})](u_n-u)\nonumber\\
&-\int_{\Omega}(\tilde{f}(x,u_n)-\tilde{f}(x,u))(u_n-u)
\end{align}
Now, on using \eqref{convergence f tilla}, \eqref{convergence phi} and \eqref{convergence singular} we get
\begin{align}\label{ps seq1}
&\langle\tilde{I}'(u_n)-\tilde{I}'(u), (u_n-u)\rangle= (a+b[u_n]^{2})\langle u_n, (u_n-u)\rangle-(a+b[u]^{2})\langle u, (u_n-u)\rangle+o(1)
\end{align}
as $n\rightarrow\infty$. Observe that
\begin{align}\label{ps seq2}
&(a+b[u_n]^{2})\langle u_n, (u_n-u)\rangle-(a+b[u]^{2})\langle u, (u_n-u)\rangle\nonumber\\
&=(a+b[u_n]^{2})[u_n-u]^2+b([u_n]^{2}-[u]^{2})\langle u, (u_n-u)\rangle.
\end{align}
Since, the sequence $(a+b[u_n]^{2})$ is bounded in $X_0$. Thus by using the definition of weak convergence, we get
\begin{align}\label{ps seq3}
b([u_n]^{2}-[u]^{2})\langle u, (u_n-u)\rangle=o(1)~\text{as}~n\rightarrow\infty.
\end{align}
Therefore, from \eqref{ps seq2} and \eqref{ps seq3}, we obtain
\begin{align}\label{ps seq4}
&(a+b[u_n]^{2})\langle u_n, (u_n-u)\rangle-(a+b[u]^{2})\langle u, (u_n-u)\rangle\geq a[u_n-u]^2~\text{as}~n\rightarrow\infty.
\end{align}
Finally, on using \eqref{ps seq}, \eqref{ps seq1} and \eqref{ps seq4}, we conclude that
\begin{align}
o(1)\geq\min\{a,1\}\|u_n-u\|^2+o(1)~\text{as}~n\rightarrow\infty.
\end{align}
Hence, $u_n\rightarrow u$ strongly in $X_0$ and this completes the proof.
\end{proof}
\begin{lemma}\label{lemma genus}
For any $n\in\mathbb{N}$, there exists a closed, symmetric subset $A_n\subset X_0$ with $0\notin A_n$ such that the genus $\gamma(A_n)\geq n$ and $\sup\limits_{u\in A_n}\tilde{I}(u)<0.$
\end{lemma}
\begin{proof}
We will first obtain the existence of a closed, symmetric subset $A_n$ of $X_0$ over every finite dimensional subspace such that $\gamma(A_n)\geq n.$ Let $X_k$ be a subspace of $X_0$ such that $\dim (X_k)=k.$ Since, every norm over a finite dimensional Banach space are equivalent then there exists a positive constant $M=M(k)$ such that $\|u\|\leq M\|u\|_{L^2(\Omega)}$ for all $u\in X_k.$\\
{\bf Claim:}
There exists a positive constant $R$ such that
\begin{equation}\label{claim genus}
\frac{1}{2}\int_{\Omega}|u|^2dx\geq\int_{\{|u|> l\}}|u|^2dx,~\forall\,u\in X_k ~\text{such that}~ \|u\|\leq R.
\end{equation}
We proof it by contradiction. Let $\{u_n\}$ be a sequence in $X_k\setminus\{0\}$ such that $u_n\rightarrow0$ in $X_0$ and
\begin{equation}\label{eq 3.24}
\frac{1}{2}\int_{\Omega}|u_n|^2dx<\int_{\{|u_n|> l\}}|u_n|^2dx.
\end{equation}
Choose, $v_n=\frac{u_n}{\|u_n\|_{L^2(\Omega)}}.$ Then \eqref{eq 3.24} reduces to
\begin{equation}\label{claim contra eqn}
\frac{1}{2}<\int_{\{|u_n|> l\}}|v_n|^2dx.
\end{equation}
Since, $X_k$ is finite dimensional and $\{v_n\}$ is bounded, we can assume $v_n\rightarrow v$ in $X_0$ upto a subsequence. Therefore, $v_n\rightarrow v$ also in $L^2(\Omega).$ Further observe that, $$m\{x\in\Omega: |u_n|>l\}\rightarrow0 ~\text{as}~ n\rightarrow\infty,$$ since $u_n\rightarrow0$ in $X_0$, where $m$ refers to the Lebesgue measure. This is a contradiction to the equation \eqref{claim contra eqn}. Hence, the claim is established. Again, from the assumption $(A2)$, one can choose $0<l\leq1$ sufficiently small such that, $$\tilde{F}(x,t)=F(x,t)\geq4\left(\frac{a}{2}+\frac{b}{4}+C_{\phi}\right)M^2t^2, ~\forall\, (x,t)\in\Omega\times[0,l].$$
Hence, for all $u\in X_k\setminus\{0\}$ such that $\|u\|\leq R$ and by using \eqref{claim genus}, we get
\begin{align*}
\tilde{I}(u)&\leq\frac{a}{2}\|u\|^2+\frac{b}{4}\|u\|^4+\int_{\Omega}\phi_uu^2dx-\frac{\lambda}{1-\gamma}\int_{\Omega}|h(x)||u|^{1-\gamma}dx-\int_{\{|u|\leq l\}}\tilde{F}(x, u)dx\\
&\leq\frac{a}{2}\|u\|^2+\frac{b}{4}\|u\|^4+C_{\phi}\|u\|^4-\frac{\lambda}{1-\gamma}\int_{\Omega}|h(x)||u|^{1-\gamma}dx-4\left(\frac{a}{2}+\frac{b}{4}+C_{\phi}\right)M^2\int_{\{|u|\leq l\}}|u|^2dx\\
&\leq\left(\frac{a}{2}+\frac{b}{4}+C_{\phi}\right)\|u\|^2-\frac{\lambda}{1-\gamma}\int_{\Omega}|h(x)||u|^{1-\gamma}dx\\
&\hspace{4.2cm}-4\left(\frac{a}{2}+\frac{b}{4}+C_{\phi}\right)M^2\left(\int_{\Omega}|u|^2dx-\int_{\{|u|> l\}}|u|^2dx\right)
\end{align*}
\begin{align*}
&\leq\left(\frac{a}{2}+\frac{b}{4}+C_{\phi}\right)\|u\|^2-\frac{\lambda}{1-\gamma}\int_{\Omega}|h(x)||u|^{1-\gamma}dx-2\left(\frac{a}{2}+\frac{b}{4}+C_{\phi}\right)M^2\int_{\Omega}|u|^2dx\\
&\leq-\left(\frac{a}{2}+\frac{b}{4}+C_{\phi}\right)\|u\|^2-\frac{\lambda}{1-\gamma}\int_{\Omega}|h(x)||u|^{1-\gamma}dx\\
&<0,~\text{for all}~u\in X_0~\text{such that}~\|u\|\leq\min\{1, R\}.
\end{align*}
\noindent We now choose, $0<\rho\leq\min\{1,R\}$ and $A_n=\{u\in X_n: \|u\|=\rho\}$. Thus $\Gamma_n\neq\phi$. This concludes that $A_n$ is symmetric, closed with $\gamma(A_n)\geq n$ such that $\sup\limits_{u\in A_n}\tilde{I}(u)<0.$
\end{proof}
\noindent We now state the following Lemmas which are essential to prove the boundedness of the solutions to the problem \eqref{main3}. The Lemma \ref{bounded l1} and Lemma \ref{bounded l2} are taken from \cite{brasco2016second} and a simple proof can be found in \cite{positivity}.
\begin{lemma}\label{bounded l1}
Let $g:\mathbb{R}\rightarrow\mathbb{R}$ be a convex $C^1$ function. Then for every $c, d, C, D\in\mathbb{R}$ with $C, D>0$ the following inequality holds.
\begin{equation}
(g(c)-g(d))(C-D)\leq (c-d)(Cg'(c)-Dg'(d))
\end{equation}
\end{lemma}
\begin{lemma}\label{bounded l2}
Let $\tilde{h}:\mathbb{R}\rightarrow\mathbb{R}$ be an increasing function, then for $c, d, \tau\in\mathbb{R}$ with $\tau\geq 0$ we have
\begin{equation}
[\tilde{H}(c)-\tilde{H}(d)]^2\leq (c-d)(\tilde{h}(c)-\tilde{h}(d))
\end{equation}
where, $\tilde{H}(t)=\int_0^t \sqrt{\tilde{h}'(\tau)}d\tau$, for $t\in\mathbb{R}.$
\end{lemma}
\noindent The following Lemma is based on the Moser iteration technique, which gives an uniform $L^{\infty}$ bound to the weak solutions of the problem \eqref{main3}.
\begin{lemma}\label{bounded}
Let $u\in X_0$ be a positive weak solution to the problem in \eqref{main3}, then $u\in L^{\infty}(\Omega).$
\end{lemma}
\begin{proof}
The proof is based on arguments as in \cite{positivity}. We will make use of the fact that
$$\int_{Q}\frac{(u(x)-u(y))(\psi(x)-\psi(y))}{|x-y|^{N+2s}}dxdy=C\int_{Q}(-\Delta)^{s/2} u\cdot(-\Delta)^{s/2}\psi dxdy,$$ for $\psi\in X_0.$ For every small $\epsilon>0,$ consider the smooth function
\begin{equation*}
g_{\epsilon}(t)=(\epsilon^2+t^2)^{\frac{1}{2}}
\end{equation*}
Note that the function $g_{\epsilon}$ is convex as well as Lipschitz. We choose $\psi=\tilde{\psi} g'_{\epsilon}(u)$ as the test function in \eqref{main3} for all positive $\tilde{\psi}\in C_c^{\infty}(\Omega)$. Now by taking $c=u(x), d=u(y), C=\psi(x)$ and $D=\psi(y)$ in Lemma \ref{bounded l1}, we get
\begin{align}\label{bound est 1}
(a+b\|u\|^2)&\int_{Q}\cfrac{(g_{\epsilon}(u(x))-g_{\epsilon}(u(y)))(\tilde{\psi}(x)-\tilde{\psi}(y))}{|x-y|^{N+2s}}dxdy\nonumber\\
&\leq\int_\Omega\left(|\lambda h(x)u^{-\gamma}+\tilde{f}(x, u)|-\phi_uu\right)|g'_{\epsilon}(u)|\tilde{\psi} dx\nonumber\\
&\leq\int_\Omega\left(|\lambda h(x) u^{-\gamma}+\tilde{f}(x, u)|\right)|g'_{\epsilon}(u)|\tilde{\psi} dx
\end{align}
Since, $g_{\epsilon}(t)\rightarrow|t|$ as $t\rightarrow0$, hence $|g'_{\epsilon}(t)|\leq1$ for all $t\geq0$. Therefore, on using the Fatou's Lemma and passing the limit $\epsilon\rightarrow0$ in \eqref{bound est 1}, we obtain
\begin{align}\label{bound est 2}
(a+b\|u\|^2)\int_{Q}\cfrac{(|u(x)|-|u(y)|)(\tilde{\psi}(x)
-\tilde{\psi}(y))}{|x-y|^{N+2s}}dxdy\leq\int_\Omega\left(|\lambda h(x) u^{-\gamma}+\tilde{f}(x, u)|\right)\tilde{\psi} dx
\end{align}
for all $\tilde{\psi}\in C_c^{\infty}(\Omega)$ with $\tilde{\psi}>0.$ The inequality \eqref{bound est 2} remains true for all $\tilde{\psi}\in X_0$ with $\tilde{\psi}\geq0.$ We define the {cut-off} function $u_k=\min\{(u-1)^+, k\}\in X_0$ for $k>0.$ Now for any given $\beta>0$ and $\delta>0$, we choose $\tilde{\psi}=(u_k+\delta)^{\beta}-\delta^{\beta}$ as the test function in \eqref{bound est 2} and get
\begin{align}\label{bound est 3}
(a+b\|u\|^2)&\int_{Q}\cfrac{(|u(x)|-|u(y)|)((u_k(x)+\delta)^{\beta}-(u_k(y)+\delta)^{\beta})}{|x-y|^{N+2s}}dxdy\nonumber\\
&\leq\int_\Omega\left(|\lambda h(x) u^{-\gamma}+\tilde{f}(x, u)|\right)\left((u_k+\delta)^{\beta}-\delta^{\beta}\right) dx
\end{align}
Now applying the Lemma \ref{bounded l2} to the function $\tilde{h}(u)=(u_k+\delta)^{\beta},$ we get
\begin{align}\label{bound est 4}
\begin{split}
&(a+b\|u\|^2)\int_{Q}\cfrac{|((u_k(x)+\delta)^{\frac{\beta+1}{2}}
-(u_k(y)+\delta)^{\frac{\beta+1}{2}})|^2}{|x-y|^{N+2s}}dxdy\\
&\leq\frac{(\beta+1)^2}{4\beta}(a+b\|u\|^2)\int_{Q}\cfrac{(|u(x)|-|u(y)|)((u_k(x)+\delta)^{\beta}
-(u_k(y)+\delta)^{\beta})}{|x-y|^{N+2s}}dxdy\\
&\leq\frac{(\beta+1)^2}{4\beta}\int_\Omega\left(|\lambda h(x) u^{-\gamma}+\tilde{f}(x, u)|\right)\left((u_k+\delta)^{\beta}-\delta^{\beta}\right) dx\\
&\leq\frac{(\beta+1)^2}{4\beta}\int_\Omega\left(|\lambda h(x) u^{-\gamma}|+|\tilde{f}(x, u)|\right)\left((u_k+\delta)^{\beta}-\delta^{\beta}\right) dx\\
&=\frac{(\beta+1)^2}{4\beta}\int_{\{u\geq1\}}\left(|\lambda h(x) u^{-\gamma}|+|\tilde{f}(x, u)|\right)\left((u_k+\delta)^{\beta}-\delta^{\beta}\right) dx\\
&\leq\frac{(\beta+1)^2}{4\beta}\int_{\{u\geq1\}}\left(|\lambda|\|h\|_{\infty}+(|c_1|+|c_2||u|^{p})\right)\left((u_k+\delta)^{\beta}-\delta^{\beta}\right) dx\\
&\leq C_1\frac{(\beta+1)^2}{4\beta}\int_{\{u\geq1\}}\left(1+|u|^{p}\right)\left((u_k+\delta)^{\beta}-\delta^{\beta}\right) dx\\
&\leq 2C_1\frac{(\beta+1)^2}{4\beta}\int_{\{u\geq1\}}|u|^{p}\left((u_k+\delta)^{\beta}-\delta^{\beta}\right) dx\\
&\leq C\frac{(\beta+1)^2}{4\beta}|u|_{2_s^*}^{p}|(u_k+\delta)^{\beta}|_q
\end{split}
\end{align}
where, $q=\frac{2_s^*}{2_s^*-p}$ and $C=\max\{1,|\lambda|\}.$ The rest of the proof is similar to the Lemma 2.7 in \cite{positivity} to obtain
\begin{equation}\label{bound est 11}
\|u_k\|_{\infty}\leq C\eta^{\frac{\eta}{(\eta-1)^2}}\left(|\Omega|^{1-\frac{1}{q}-\frac{2s}{N}} \right)^{\frac{\eta}{\eta-1}}\left(|(u-1)^+|_q+\delta|\Omega|^{\frac{1}{q}}\right)
\end{equation}
Now letting $k\rightarrow\infty$ in \eqref{bound est 11}, we have
\begin{equation}\label{bound est 12}
\|(u-1)^+\|_{\infty}\leq C\eta^{\frac{\eta}{(\eta-1)^2}}\left(|\Omega|^{1-\frac{1}{q}-\frac{2s}{N}} \right)^{\frac{\eta}{\eta-1}}\left(|(u-1)^+|_q+\delta|\Omega|^{\frac{1}{q}}\right)
\end{equation}
Hence, we conclude that $u\in L^{\infty}(\Omega).$
\end{proof}
\begin{proof}[{\bf Proof of Theorem \ref{main thm1}}]
By using the assumption $(A1)$ and the definition of $\xi$, we get the functional $\tilde{I}$ is even and $\tilde{I}(0)=0.$ Thus, on using Theorem \ref{sym mountain}, Lemma \ref{lemma ps} and Lemma \ref{lemma genus}, we conclude that $\tilde{I}$ has sequence of critical points $\{u_n\}$ such that $\tilde{I}(u_n)<0$ and $\tilde{I}(u_n)\rightarrow0^-$.\\
We now prove that the critical points of $\tilde{I}$ are nonnegative.\\
{\bf Claim:} Let $u_n$ be a critical point of $\tilde{I}$, then $u_n\geq0$ a.e. in $X_0$ for every $n\in\mathbb{N}$.
\begin{proof}
We first divide the domain as $\Omega= \Omega^+\cup\Omega^-$, where $\Omega^+=\{x\in X_0: u_n(x)\geq0 \}$ and $\Omega^-=\{x\in X_0: u_n(x)<0 \}$. We define $u_n=u_n^+-u_n^-$, where $u_n^+(x)=\max\{u_n(x), 0\}$ and $u_n^-(x)=\max\{-u_n(x), 0\}$. We proceed through a contradiction by taking $u_n<0$ a.e. in $\Omega$. Then on choosing, $\phi=u_n^-$ as the test function in the equation \eqref{weak cutoff} in association with the inequality $(a-b)(a^--b^-)\leq-(a^--b^-)^2$, we obtain
\begin{align*}
&\int_{\Omega}\left(\lambda h(x) \frac{sign(u_n)u_n^-}{|u_n|^{\gamma}}+\tilde{f}(x,u_n)u_n^-\right)dx\\
&=(a+b[u_n]^2)\int_{Q}\frac{(u_n(x)-u_n(y))(u_n^-(x)-u_n^-(y))}{|x-y|^{N+2s}}dxdy+\int_{\Omega}\phi_{u_n}u_nu_n^-dx\\
&=-(a+b\|u_n\|^2)\|u_n^-\|^2-\int_{\Omega}\phi_{u_n}\|u_n^-\|^2dx\\
&\Rightarrow\lambda\int_{\Omega^-}h(x)|u_n^-|^{1-\gamma}dx<0.
\end{align*}
Therefore, $|\Omega^-|=0$, which is a contradiction to the assumption $u_n<0$ a.e. in $\Omega$.
\end{proof}
\noindent We now prove $u_n\rightarrow0$ in $X_0.$ Indeed by the definition of $\tilde{I}$, we obtain
\begin{align*}
\frac{1}{p}\langle\tilde{I}^{'}(u_n), u_n\rangle-\tilde{I}(u_n)&=\frac{1}{p}\left[(a+b\|u_n\|^2)\|u_n\|^2+\int_{\Omega}\phi_{u_n}u_n^2-\int_{\Omega}\left(\lambda\frac{h(x)sign(u_n)u_n}{|u_n|^{\gamma}}+\tilde{f}(x,u_n)u_n\right)dx\right]\\
&-\left[\frac{a}{2}\|u_n\|^2+\frac{b}{4}\|u_n\|^4+\frac{1}{4}\int_{\Omega}\phi_{u_n}u_n^2-\int_{\Omega}\left(\frac{\lambda h(x)}{1-\gamma}|u_n|^{1-\gamma}+\tilde{F}(x,u_n)\right)dx\right]\\
&=a(\frac{1}{p}-\frac{1}{2})\|u_n\|^2+b(\frac{1}{p}-\frac{1}{4})\|u_n\|^4+(\frac{1}{p}-\frac{1}{4})\int_{\Omega}\phi_{u_n}u_n^2\\ &-\lambda(\frac{1}{p}-\frac{1}{1-\gamma})\int_{\Omega}h(x)|u_n|^{1-\gamma}dx+\frac{1}{p}\int_{\Omega}(p\tilde{F}(x,u_n)-\tilde{f}(x,u_n))dx\\
&\geq a(\frac{1}{p}-\frac{1}{2})\|u_n\|^2 +b(\frac{1}{p}-\frac{1}{4})\|u_n\|^4 +\lambda(\frac{1}{1-\gamma}-\frac{1}{p})\int_{\Omega}h(x)|u_n|^{1-\gamma}dx\\
&\geq(\frac{1}{p}-\frac{1}{2})\|u_n\|^2
\end{align*}
Therefore, by using the fact
\begin{align*}
&\frac{1}{p}\langle\tilde{I}^{'}(u_n), u_n\rangle-\tilde{I}(u_n)=o(1)\\
&\Rightarrow(\frac{1}{p}-\frac{1}{2})\|u_n\|^2\leq o(1),
\end{align*}
as $n\rightarrow\infty$. Since, $1-\gamma<p<2$, we conclude that $u_n\rightarrow0$ in $X_0$. Thus from the Lemma \ref{bounded}, we can obtain $\|u_n\|_{L^{\infty}(\Omega)}\leq l$ as $n\rightarrow\infty$, thanks to the Moser iteration method. Hence, the problem \eqref{main2} has infinitely many solutions. Moreover, by using $u_n\geq0$ and $\tilde{I}(u_n)<0$, we conclude that the problem \eqref{problem main} has infinitely many weak solutions in $X_0$. Thus Theorem \ref{main thm1} is proved.
\end{proof}
\section{Existence of solution for $\gamma>1$.}
This section is fully devoted to establish the existence of a weak solution to the problem \eqref{problem main} in $X_0$. Further, we will prove that for $k\equiv0$, the problem \eqref{problem main} possesses a unique solution. Let us define the following two subsets of $X_0$ similar to the Nehari manifold.
$$N_{1}=\{u\in X_0: (a+b[u]^{2})\|u\|^{2}+\int_{\Omega}\phi_{u}u^{2} -\lambda\int_{\Omega}h(x)|u|^{1-\gamma}-\int_{\Omega}k(x)|u|^{1+p}\geq0\},$$
$$N_{2}=\{u\in X_0: (a+b[u]^{2})\|u\|^{2}+\int_{\Omega}\phi_{u}u^{2} -\lambda\int_{\Omega}h(x)|u|^{1-\gamma}-\int_{\Omega}k(x)|u|^{1+p}=0\}.$$
We will show that the fractional Kirchhoff-Schr\"{o}dinger-Poisson system with a strong singularity has a weak solution in $N_2$. One can see that $N_2$ is not closed. We will prove that $N_1$ is closed in $X_0$ and the functional $I$ is coercive and bounded below on $N_1$. Further we will obtain a minimizing sequence $\{u_n\}$ of $c=\inf_{N_{1}}I$ such that $\{u_n\}$ converges to $u\in X_0$. Finally, we will show that $u\in N_2$ and hence $u$ is a weak solution to the problem \eqref{problem reduced}. We begin with the following Lemmas.
\begin{lemma}\label{l2.2}
Let $\int_{\Omega}\lambda h(x)|u|^{1-\gamma}<\infty$ for some $u\in X_0$. Then there exists a unique $t_0>0$ such that $t_0u\in N_{2}$ and $tu\in N_{1}$, for $t\geq t_0$, i.e. $N_{1}$, $N_{2}\neq\emptyset.$ Moreover, for $t\geq 0, \psi \in X_0$ the function $f$ defined as $\theta(t)=t(u+t\psi)$ is continuous on $[0, \infty)$.
\end{lemma}
\begin{proof}
Let $\int_{\Omega}\lambda h(x)|u|^{1-\gamma}<\infty$ for some $u\in X_0$. Now for $t>0$, we get
$$I(tu)= \frac{at^{2}}{2}\|u\|^{2}+\frac{bt^{4}}{4}\|u\|^{4} +\frac{t^{4}}{4}\int_{\Omega}\phi_{u}u^{2}-\frac{t^{1-\gamma}}{1-\gamma}\int_{\Omega}\lambda h(x)|u|^{1-\gamma}-\frac{t^{p+1}}{p+1}\int_{\Omega}k(x)|u|^{1+p}.$$
It is easy to see that $tu\in N_{1} \Leftrightarrow I'(tu) \geq 0$ and $tu\in N_{2} \Leftrightarrow I'(tu) = 0$. Also, since $0<p<1<\gamma$, $I(tu)\rightarrow+\infty$, if $t\rightarrow 0^+$ as well as $t\rightarrow +\infty$ and there exists a unique $t_0>0$ such that $I'(t_0u) = 0$, $I'(tu)\geq 0$, $t\geq t_0$ and $I(t_0u) = \min_{t\geq 0}I(tu)$. Therefore, $tu\in N_{1}$, $t_0u\in N_{2}$ for $t\geq t_0$, and $I(tu)\geq I(t_0u)$.\\
Again, observe that for $t, \psi\geq 0$, $\int_{\Omega}\lambda h(x)|u+t\psi|^{1-\gamma}<\infty$. Now, consider a nonnegative sequence $\{t_{n}\}$ such that $t_{n}\rightarrow t$ as $n\rightarrow\infty$. On using the arguments as above, there exists $\theta(t_{n}),\theta(t)\geq 0$ such that $\theta(t_{n})(u+t_{n}\psi), \theta(t)(u+t\psi)\in N_{2}$. Thus, we get
\begin{align}\label{2.1}
\begin{split}
a\theta^{1+\gamma}(t_{n})\| u+t_{n}\psi\|^{2}&+\theta^{3+\gamma}(t_{n})\left(b\| u+t_{n}\psi\|^{4}+\int_{\Omega}\phi_{u+t_{n}\psi}(u+t_{n}\psi)^{2}\right)\\&-\theta^{p+\gamma}(t_{n})\int_{\Omega}k(x)|u+t_{n}\psi|^{1+p}= \int_{\Omega}\lambda h(x)|u+t_{n}\psi|^{1-\gamma}
\end{split}
\end{align} and
\begin{align}\label{2.2}
\begin{split}
a\theta^{1+\gamma}(t)\|u+t\psi\|^{2}&+\theta^{3+\gamma}(t)\left(b\|u+t\psi\|^{4}+\int_{\Omega}\phi_{u+t\psi}(u+t\psi)^{2}\right)\\&-\theta^{p+\gamma}(t)\int_{\Omega}k(x)|u+t\psi|^{1+p}= \int_{\Omega}\lambda h(x)|u+t\psi|^{1-\gamma}.
\end{split}
\end{align}
Now for all $n \in \mathbb{N}$, we have $\lambda h(x)|u+t_{n}\psi|^{1-\gamma}\leq \lambda h(x)|u|^{1-\gamma}$.and for each $x\in\Omega$, we have the pointwise convergence $\lambda h(x)|u+t_{n}\psi|^{1-\gamma}\rightarrow \lambda h(x)|u+t\psi|^{1-\gamma}$ as $n\rightarrow\infty$. Therefore, by using Lebesgue's dominated convergence theorem, we get $\int_{\Omega}\lambda h(x)|u+t_{n}\psi|^{1-\gamma}\rightarrow\int_{\Omega}\lambda h(x)|u+t\psi|^{1-\gamma}$ as $n\rightarrow\infty$. Further, from \eqref{2.1}, one can see that the sequence $\{\theta(t_{n})\}$ is bounded. Therefore, it has a convergent subsequence. Let $\{\theta(t_{n_k})\}$ convergence to $s$. Then, on using \eqref{2.1} and \eqref{2.2} and the above arguments, we can conclude that $s=\theta(t)$. Hence $\theta$ is continuous.
\end{proof}
\noindent In the following Lemma, we establish that $N_1$ is closed in $X_0$ and the the functional $I$ is coercive and bounded below on $N_1$.
\begin{lemma}\label{l2.}
$N_1$ is closed in $X_0$ and for all for $u\in N_{1}$, there exists $C>0$ such that $\|u\|\geq C$. Moreover, the functional $I$ is coercive and bounded below on $N_1$.
\end{lemma}
\begin{proof}
We first show that $N_1$ is closed. Let $\{u_{n}\}\subset N_{1}$ be such that $u_{n} \rightarrow u$ in $X_0$. Since, $\{u_{n}\}\subset N_{1}$ and $\int_{\Omega}\lambda h(x)|u_{n}|^{1-\gamma}<\infty$, then $u_{n}(x)>0$ a.e. in $\Omega$ and then up to a subsequence, $u_{n}(x)\rightarrow u(x)$ a.e. in $\Omega$. Therefore, on applying the Fatou's lemma and then using Sobolev embedding, we get
\begin{align*}
\int_{\Omega}\lambda h(x)|u|^{1-\gamma}&\leq\lim_{n\rightarrow\infty}\inf\int_{\Omega}\lambda h(x)|u_{n}|^{1-\gamma}\\
&\leq\lim_{n\rightarrow\infty}\inf\left((a+b\|u_{n}\|^{2})\|u_{n}\|^{2}+\int_{\Omega}\phi_{u_{n}}u_{n}^{2}-\int_{\Omega}k(x)|u_{n}|^{1+p}\right)\\
&\leq(a+b\| u\|^{2})\| u\|^{2}+\int_{\Omega}\phi_{u}u^{2}-\int_{\Omega}k(x)|u|^{1+p}.
\end{align*}
Thus we have $u\in N_{1}$ and hence $N_{1}$ is closed in $X_0$. We prove the functional $I$ is bounded below on $N_1$ by using the method of contradiction. Suppose, there exists $\{u_{n}\}\subset N_{1}$ such that $u_{n}\rightarrow 0$ in $X_0$ as $n\rightarrow\infty$. Then, on using the Reverse H\"{o}lder inequality, we get
\begin{align*}
(a+b\|u_{n}\|^{2})\|u_{n}\|^{2}+\int_{\Omega}\phi_{{u}_{n}}u_{n}^{2}&\geq\int_{\Omega}\lambda h(x)|u_{n}|^{1-\gamma}+\int_{\Omega}k(x)|u_{n}|^{1+p}\\
&\geq\left(\int_{\Omega}|\lambda h(x)|^{\frac{1}{\gamma}}\right)^{\gamma}\left(\int_{\Omega}|u_{n}|\right)^{1-\gamma}\\
&\geq c\left(\int_{\Omega}|\lambda h(x)|^{\frac{1}{\gamma}}\right)^{\gamma}\| u_{n}\|^{1-\gamma}
\end{align*}
where, $c$ is a positive constant. This gives a contradiction, since $\gamma>1$. Therefore, there exists $C>0$ such that $\|u\|\geq C$ for all $u\in N_{1}$.\\
Since, $u\in N_{1}$ implies that $\int_{\Omega}\lambda h(x)|u|^{1-\gamma}\leq(a+b\|u\|^{2})\|u\|^{2} +\int_{\Omega}\phi_{u}u^{2}-\int_{\Omega}k(x)u^{1+p}<\infty$. Therefore, from the definition \eqref{functional} of $I$, we have \begin{align}\label{2.3}
I(u)&=\frac{a}{2}\|u\|^{2}+\frac{b}{4}\|u\|^{4}+\frac{1}{4}\int_{\Omega}\phi_{u}u^{2}-\frac{1}{1-\gamma}\int_{\Omega}\lambda h(x)|u|^{1-\gamma}-\frac{1}{p+1}\int_{\Omega}k(x)|u|^{1+p}\nonumber\\
&\geq\frac{a}{2}\| u\|^{2}+\frac{b}{4}\| u\|^{4}-c|k|_{\infty}\| u\|^{1+p}.
\end{align}
Now, since $0<p<1$ and $a+b\geq 0$, therefore by using the Lemma \ref{l2.2}, we get that $I$ is coercive and bounded below on $N_{1}.$
\end{proof}
\begin{lemma}\label{lemma min-seq}
Assume the compatibility condition \eqref{compatibility} holds. Then there exists a minimizing sequence $\{u_{n}\}\subset N_{1}$ of $c=\inf\limits_{N_{1}}I$, i.e. there exists $u\in N_{1}$ such that $I(u)=c$. Moreover, $u\in N_{2}$.
\end{lemma}
\begin{proof}
It is easy to see that the functional $I$, defined as in \eqref{functional} is lower semicontinuous. Since, $N_1 (\neq\emptyset)$ is closed, then by using Ekeland's variational principle for $c=\inf_{N_{1}}I$, we can extract a minimizing sequence $\{u_{n}\}\subset N_{1}$ such that
\begin{enumerate}[label=(\roman*)]
\item $I(u_{n}) \leq\ c+\frac{1}{n^{2}}$;
\item $I(u_{n}) \leq I(v)+\frac{1}{n}\| u_{n}-v\|$, $\forall\, v\in N_{1}.$
\end{enumerate}
Now, from the fact $I(|u|) =I(u)$, one can assume that $u_{n} >0$ a.e. in $\Omega$. By Lemma \ref{l2.}, on using the coerciveness of $I$, we have $\{u_{n}\}$ is bounded. Therefore, upto to a subsequence, we have
\begin{enumerate}[label=(\roman*)]
\item $u_{n}\rightharpoonup u$ weakly in $X_0$
\item $u_{n}\rightarrow u$ strongly in $L^{q}(\Omega)$ for $q\in [1, 2_s^*)$, $2_s^*=\frac{2N}{N-2s}$, and
\item $u_{n}(x) \rightarrow u(x)$ pointwise a. e. in $\Omega$.
\end{enumerate}
Since, $N_1$ is closed and $\gamma>1$, then from Fatou's lemma and $\{u_{n}\}\subset N_{1}$, we get $u>0$ a. e. in $\Omega$, $\int_{\Omega}\lambda h(x)|u|^{1-\gamma}<\infty$ and $u\in N_1$. On using Fatou's lemma and Lemma \ref{l2.2}, we have
\begin{align}\label{strong sub}
\inf_{N_{1}}I&=\liminf_{n\rightarrow\infty} I(u_{n})\nonumber\\
&=\liminf_{n\rightarrow\infty}\left(\frac{a}{2}\|u_{n}\|^{2}+\frac{b}{4}\|u_{n}\|^{4}+\frac{1}{4}\int_{\Omega}\phi_{u_{n}}u_{n}^{2}-\frac{1}{1-\gamma}\int_{\Omega}\lambda h(x)u_{n}^{1-\gamma}-\frac{1}{1+p}\int_{\Omega}k(x)u_{n}^{1+p}\right)\nonumber\\
&\geq\liminf_{n\rightarrow\infty}\frac{a}{2}\|u_{n}\|^{2}+\frac{b}{4}\liminf_{n\rightarrow\infty}\|u_{n}\|^{4}+\frac{1}{4}\phi_{u}u^{2}-\frac{1}{1-\gamma}\liminf_{n\rightarrow\infty}\int_{\Omega}\lambda h(x)u_{n}^{1-\gamma}-\frac{1}{1+p}\int_{\Omega}k(x){u}^{1+p}\nonumber\\
&\geq\frac{a}{2}\|u\|^{2}+\frac{b}{4}\|u\|^{4}+\frac{1}{4}\int_{\Omega}\phi_{u}u^{2}-\frac{1}{1-\gamma}\int_{\Omega}\lambda h(x)u^{1-\gamma}-\frac{1}{p+1}\int_{\Omega}k(x)u^{1+p}\nonumber\\
&=I(u)\geq I(t_0u)\nonumber\\
&\geq\inf_{N_{2}}I\geq\inf_{N_{1}}I.
\end{align}
Hence, $t_0=1$, i.e. $u\in N_2$ and hence $I(u)=c$. This completes the proof.
\end{proof}
\noindent{\bf Proof of Theorem \ref{main theorem}.}\\
Suppose $u$ is a solution to the problem \eqref{problem main}, then the compatibility condition \eqref{compatibility} must be true. We will prove the other part. Let \eqref{compatibility} be true. We first prove the following inequality which is essential to guarantee the existence of solution.
\begin{align}\label{1.7}
(a+b\|u\|^{2})\int_{\Omega}(-\Delta)^{s/2} u\cdot(-\Delta)^{s/2}\psi+\int_{\Omega}\phi_{u}u\psi -\int_{\Omega}k(x)u^{p}\psi\geq\int_{\Omega}\lambda h(x)u^{-\gamma}\psi.
\end{align}
for every nonnegative $\psi\in X_0$. We will divide the proof of \eqref{1.7} in two cases, i.e. either $\{u_{n}\}\subset N_{1}\setminus N_{2}$ or $\{u_{n}\}\subset N_{2}$.\\
{\bf Case 1.} {\it $\{u_{n}\}\subset N_{1}\setminus N_{2}$ for $n$ large enough.}\\
For a given nonnegative function $\psi\in X_0$, by $\{u_{n}\}\subset N_{1}\setminus N_{2}$, we derive that
\begin{equation*}
(a+b\| u_{n}\|^{2})\| u_{n}\|^{2}+\int_{\Omega}\phi_{u_{n}}u_{n}^{2}-\int_{\Omega}k(x)|u_{n}|^{1+p}>\int_{\Omega}\lambda h(x)u_{n}^{1-\gamma}\geq\int_{\Omega}\lambda h(x)(u_{n}+t\psi)^{1-\gamma},\ t\geq 0,
\end{equation*}
then by the continuity, there exists $t>0$ small enough such that
\begin{equation*}
(a+b\| u_{n}+t\psi\|^{2})\| u_{n}+t\psi\|^{2}+\int_{\Omega}\phi_{u_{n}+t\psi}(u_{n}+t\psi)^{2}-\int_{\Omega}k(x)(u_{n}+t\psi)^{1+p}\geq\int_{\Omega}\lambda h(x)(u_{n}+t\psi)^{1-\gamma},
\end{equation*}
that is $(u_{n}+t\psi)\in N_{1}$. Then, by (ii) of Ekeland's variational principle, we have
\begin{equation*}
\frac{1}{n}\| t\psi\|+I(u_{n}+t\psi)-I(u_{n})\geq 0.
\end{equation*}
That is,
\begin{align*}
\frac{\|t\psi\|}{n}&+\frac{a}{2}\left(\|u_{n}+t\psi\|^{2}-\|u_{n}\|^{2}\right)+\frac{b}{4}\left(\|u_{n}+t\psi\|^{4}-\|u_{n}\|^{4}\right)\\
&+\frac{1}{4}\int_{\Omega}\left(\phi_{u_{n}+t\psi}(u_{n}+t\psi)^{2}-\phi_{u_{n}}u_{n}^{2}\right)-\frac{1}{1+p}\int_{\Omega}k(x)\left((u_{n}+t\psi)^{1+p}-u_{n}^{1+p}\right)\\
&\geq\frac{1}{1-\gamma}\int_{\Omega}\lambda h(x)\left((u_{n}+t\psi)^{1-\gamma}-u_{n}^{1-\gamma}\right).
\end{align*}
Dividing by $t>0$ and by Fatou's lemma, we conclude that
\begin{align*}
\frac{1}{n}\|\psi\|+(a+b\|u_{n}\|^{2})\int_{\Omega}(-\Delta)^{s/2} u_{n}\cdot(-\Delta)^{s/2}\psi &+\int_{\Omega}\phi_{u_{n}}u_{n}\psi-\int_{\Omega}k(x)u_{n}^{p}\psi\\
&\geq\lim_{t\rightarrow 0} \inf\int_{\Omega}\frac{\lambda h(x)\left((u_{n}+t\psi)^{1-\gamma}-u_{n}^{1-\gamma}\right)}{(1-\gamma)t}\\
&\geq\int_{\Omega}\lambda h(x)u_{n}^{-\gamma}\psi,
\end{align*}
Now by Lemma \ref{lemma min-seq}, we have $I(u)=c$ for some $u\in N_2$. Therefore, by using \eqref{strong sub}, we obtain $\|u_n\|^2\rightarrow\|u\|^2$ for every $a>0, b\geq0$ and similarly, $\|u_n\|^4\rightarrow\|u\|^4$ for every $b>0$ with $a=0$. In both of the cases, $\|u_n\|\rightarrow\|u\|$ as $n\rightarrow\infty$. Thus, by applying Fatou's lemma once again, we get
\begin{equation*}
\left(a+b\| u\|^{2}\right)\int_{\Omega}(-\Delta)^{s/2} u\cdot(-\Delta)^{s/2}\psi+\int_{\Omega}\phi_{u}u\psi -\int_{\Omega}k(x)u^{p}\psi\geq\int_{\Omega}\lambda h(x)u^{-\gamma}\psi.
\end{equation*}
{\bf Case 2.} {\it There exists a subsequence of $\{u_{n}\}$ (still denoted by $\{u_{n}\}$) belonging to $N_{2}$.}\\
In this case, we can also show that \eqref{1.7} holds. For given nonnegative $\psi\in X_0$, for each $u_{n}\in N_{2}$ and $t\geq 0$,
\begin{equation*}
\int_{\Omega}\lambda h(x)(u_{n}+t\psi)^{1-\gamma}\leq\int_{\Omega}\lambda h(x)u_{n}^{1-\gamma}<\infty.
\end{equation*}
By Lemma \ref{l2.2}, there exists $t(u_{n}+t\psi)>0$ satisfying $t(u_{n}+t\psi)(u_{n}+t\psi)\in N_{2}$. For clarity, we denote $\theta_{n}(t)=t(u_{n}+t\psi)$, it is obvious that $\theta_{n}(0)=1$. By $\theta_{n}(t)(u_{n}+t\psi)\in N_{2}$, we have
\begin{align}\label{2.4}
a\theta_{n}^{2}(t)\|u_{n}+t\psi\|^{2}&+b\theta_{n}^{4}(t)\|u_{n}+t\psi\|^{4}+\theta_{n}^{4}(t)\int_{\Omega}(\phi_{u_{n}+t\psi}(u_{n}+t(\psi)^{2}\nonumber\\
&-\theta_{n}^{1-\gamma}(t)\int_{\Omega}\lambda h(x)(u_{n}+t\psi)^{1-\gamma}-\theta_{n}^{1+p}(t)\int_{\Omega}k(x)(u_{n}+t\psi)^{1+p}=0.
\end{align}
By Lemma \ref{l2.2}, for given $n$, $\theta_{n}$ is continuous on $[0, \infty)$. We denote $D_{+}\theta_{n}(0)$ the right lower Dini derivative of $\theta_{n}$ at zero. Next, we shall show that $\theta_{n}$ has uniform behavior at zero with respect to $n$, i.e., $|D_{+}\theta_{n}(0)|\leq{C}$ for suitable $C>0$ independent of $n$. By the definition of $D_{+}\theta_{n}(0)= \lim_{t\rightarrow 0^+}\inf\frac{\theta_{n}(t)-\theta_{n}(0)}{t}$, there exists a sequence $\{t_{k}\}$ with $t_{k}>0$ and $t_{k}\rightarrow 0$ as $k\rightarrow\infty$ such that
\begin{equation*}
D_{+}\theta_{n}(0)=\lim_{k\rightarrow\infty}\frac{\theta_{n}(t_{k})-\theta_{n}(0)}{t_{k}}.
\end{equation*}
By $u_{n}\in N_{2}$ and \eqref{2.4}, for $t>0$, we get that
\begin{align}\label{2.5}
0=\frac{1}{t}&\left[a\left(\theta_{n}^{2}(t)-1\right)\|u_{n}+t\psi\|^{2}+\left(\theta_{n}^{4}(t)-1\right)\left(b\|u_{n}+t\psi\|^{4}+\int_{\Omega}\phi_{u_{n}+t\psi}(u_{n}+t\psi)^{2}\right)\right.\nonumber\\
&-\left(\theta_{n}^{1-\gamma}(t)-1\right)\int_{\Omega}\lambda h(x)(u_{n}+t\psi)^{1-\gamma}-\left(\theta_{n}^{1+p}(t)-1\right)\int_{\Omega}k(x)(u_{n}+t\psi)^{1+p}\nonumber\\
&+a\left(\|u_{n}+t\psi\|^{2}-\|u_{n}\|^{2}\right)+b\left(\|u_{n}+t\psi\|^{4}-\|u_{n}\|^{4}\right)+\int_{\Omega}(\phi_{u_{n}+t\psi}(u_{n}+t\psi)^{2}-\phi_{u_{n}}u_{n}^{2})\nonumber\\
&\left.-\int_{\Omega}\lambda h(x)\left((u_{n}+t\psi)^{1-\gamma}-u_{n}^{1-\gamma}\right)-\int_{\Omega}k(x)\left((u_{n}+t\psi)^{1+p}-u_{n}^{1+p}\right)\right]\nonumber\\
&\geq\frac{\theta_{n}(t)-1}{t}\left[a(\theta_{n}(t)+1)\|u_{n}+t\psi\|^{2}+\frac{\theta_{n}^{4}(t)-1}{\theta_{n}(t)-1}\left(b\|u_{n}+t\psi\|^{4}+\int_{\Omega}\phi_{u_{n}+t\psi}(u_{n}+t\psi)^{2}\right)\right.\nonumber\\
&\left.-\frac{\theta_{n}^{1-\gamma}(t)-1}{\theta_{n}(t)-1}\int_{\Omega}\lambda h(x)(u_{n}+t\psi)^{1-\gamma}-\frac{\theta_{n}^{1+p}(t)-1}{\theta_{n}(t)-1}\int_{\Omega}k(x)(u_{n}+t\psi)^{1+p}\right]\nonumber\\
&+\frac{1}{t}\left[a(\|u_{n}+t\psi\|^{2}-\|u_{n}\|^{2})+b(\|u_{n}+t\psi\|^{4}-\|u_{n}\|^{4})+\int_{\Omega}(\phi_{u_{n}+t\psi}(u_{n}+t\psi)^{2}-\phi_{u_{n}}u_{n}^{2})\right.\nonumber\\
&\left.-\int_{\Omega}k(x)\left((u_{n}+t\psi)^{1+p}-u_{n}^{1+p}\right)\right],
\end{align}
replacing $t$ in \eqref{2.5} with $t_{k}$ and letting $k\rightarrow\infty$, we deduce that
\begin{align*}
0&\geq{D}_{+}\theta_{n}(0)\left[2a\|u_{n}\|^{2}+4\left(b\|u_{n}\|^{4}+\int_{\Omega}\phi_{u_{n}}u_{n}^{2}\right)-(1-\gamma)\int_{\Omega}\lambda h(x)u_{n}^{1-\gamma}-(1+p)\int_{\Omega}k(x)u_{n}^{1+p}\right]\\
&+(2a+4b\|u_{n}\|^{2})\int_{\Omega}(-\Delta)^{s/2}{u}_{n}\cdot(-\Delta)^{s/2}\psi+4\int_{\Omega}\phi_{u_{n}}u_{n}\psi-(p+1)\int_{\Omega}k(x)u_{n}^{p}\\
&=D_{+}\theta_{n}(0)\left[a(1-p)\|u_{n}\|^{2}+(3-p)\left(b\|u_{n}\|^{4}+\int_{\Omega}\phi_{u_{n}}u_{n}^{2}\right)+(p+\gamma)\int_{\Omega}\lambda h(x)u_{n}^{1+\gamma}\right]\\
&+(2a+4b\|u_{n}\|^{2})\int_{\Omega}(-\Delta)^{s/2}{u}_{n}\cdot(-\Delta)^{s/2}\psi+4\int_{\Omega}\phi_{u_{n}}u_{n}\psi-(p+1)\int_{\Omega}k(x)u_{n}^{p}\psi.
\end{align*}
By Lemma \ref{l2.2}, we get that
\begin{align*}
D_{+}\theta_{n}(0)(a(1-p)\alpha^{2}+(3-p)b\alpha^{4})&+(2a+4b\|u_{n}\|^{2})\int_{\Omega}(-\Delta)^{s/2} u_{n}\cdot(-\Delta)^{s/2}\psi\\
&+4\int_{\Omega}\phi_{u_{n}}u_{n}\psi-(p+1)\int_{\Omega}k(x)u_{n}^{p}\psi\leq 0.
\end{align*}
Since, $p\in(0,1)$, this implies that $D_{+}\theta_{n}(0)\neq+\infty$ and $D_{+}\theta_{n}(0)$ is bounded from above uniformly in $n$. That is $D_{+}f_{n}(0)\in[-\infty, +\infty$) and
\begin{equation}\label{2.6}
D_{+}\theta_{n}(0)\leq C_{1} ~\text{uniformly for}~ n
\end{equation}
for some $C_{1}>0.$\\
On the other hand, we can obtain the lower bound for $D_{+}\theta_{n}(0)$. If $D_{+}\theta_{n}(0)\geq 0$ for $n$ large, this gives the results. Otherwise, up to a subsequence, still denoted by $D_{+}\theta_{n}(0)$ such that $D_{+}\theta_{n}(0)$ are negative (possibly $-\infty$). Then by (ii) of Ekeland's variational principle, for $t>0$, we have
\begin{align*}
&\frac{(1-\theta_{n}(t))\|u_{n}\|+t\theta_{n}(t)\|\psi\|}{n}\\
&\geq\frac{\|u_{n}-\theta_{n}(t)(u_{n}+t\psi)\|}{n}\\
&\geq I(u_{n})-I(f_{n}(t)(u_{n}+t\psi))\\
&=\frac{a(\gamma+1)}{2(\gamma-1)}(\|u_{n}\|^{2}-\|u_{n}+t\psi\|^{2})+\frac{\gamma+3}{4(\gamma-1)}\left[b(\|u_{n}\|^{4}-\|u_{n}+t\psi\|^{4})+\int_{\Omega}(\phi_{u_{n}}u_{n}^{2}-\phi_{u_{n}+t\psi}(u_{n}+t\psi)^{2})\right]\\
&-\frac{p+\gamma}{(\gamma-1)(p+1)}\int_{\Omega}k(x)\left(u_{n}^{p+1}-(u_{n}+t\psi)^{p+1}\right)-\frac{a(\gamma+1)}{2(\gamma-1)}(f_{n}^{2}(t)-1)\|u_{n}+t\psi\|^{2}\\
&-\frac{\gamma+3}{4(\gamma-1)}(f_{n}^4(t)-1)\left(b\|u_{n}+t\psi\|^{4}+\int_{\Omega}(\phi_{u_{n}+t\psi}(u_{n}+t\psi)^{2})\right)\\
&+\frac{p+\gamma}{(\gamma-1)(p+1)}(f_{n}^{p+1}(t)-1)\int_{\Omega}k(x)(u_{n}+t\psi)^{p+1}.
\end{align*}
Then,
\begin{align}\label{2.7}
\frac{t\theta_{n}(t)}{n}\|\psi\|&\geq{I}(u_{n})-I(\theta_{n}(t)(u_{n}+t\psi))+\frac{\theta_{n}(t)-1}{n}\|u_{n}\|\nonumber\\
&=(\theta_{n}(t)-1)\left[\frac{\|u_{n}\|}{n}-\frac{a(\gamma+1)}{2(\gamma-1)}(\theta_{n}(t)+1)\|u_{n}+t\psi\|^{2}\right.\nonumber\\
&-\frac{\gamma+3}{4(\gamma-1)}\frac{\theta_{n}^{4}(t)-1}{\theta_{n}(t)-1}\left(b\|u_{n}+t\psi\|^{4}+\int_{\Omega}\phi_{u_{n}+t\psi}(u_{n}+t\psi)^{2}\right)\nonumber\\
&\left.+\frac{p+\gamma}{(\gamma-1)(p+1)}\frac{\theta_{n}^{p+1}(t)-1}{\theta_{n}(t)-1}\int_{\Omega}k(x)(u_{n}+t\psi)^{p+1}\right]+\frac{a(\gamma+1)}{2(\gamma-1)}(\|u_{n}\|^{2}-\|u_{n}+t\psi\|^{2})\nonumber\\
&+\frac{\gamma+3}{4(\gamma-1)}\left[b(\|u_{n}\|^{4}-\|u_{n}+t\psi\|^{4})+\int_{\Omega}(\phi_{u_{n}}u_{n}^{2}-\phi_{u_{n}+t\psi}(u_{n}+t\psi)^{2})\right]\nonumber\\
&-\frac{p+\gamma}{(\gamma-1)(p+1)}\int_{\Omega}k(x)\left(u_{n}^{p+1}-(u_{n}+t\psi)^{p+1}\right)
\end{align}
Then, replacing $t$ in \eqref{2.7}, dividing $t_{k}$ and letting $k\rightarrow\infty$, we deduce that
\begin{align}\label{2.8}
\frac{\|\psi\|}{n}&\geq{D}_{+}\theta_{n}(0)\left[\frac{\|u_{n}\|^{2}}{n}-\frac{a(\gamma+1)}{\gamma-1}\|u_{n}\|^{2}-\frac{\gamma+3}{\gamma-1}\left(b\|u_{n}\|^{4}+\int_{\Omega}\phi_{u_{n}}u_{n}^{2}\right)+\frac{p+\gamma}{\gamma-1}\int_{\Omega}k(x)u_{n}^{p+1}\right]\nonumber\\
&-\frac{a(\gamma+1)}{\gamma-1}\int_{\Omega}(-\Delta)^{s/2}{u}_{n}\cdot(-\Delta)^{s/2}\psi-\frac{\gamma+3}{\gamma-1}\left(b\|u_{n}\|^{2}\int_{\Omega}(-\Delta)^{s/2}{u}_{n}\cdot(-\Delta)^{s/2}\psi+\int_{\Omega}\phi_{u_{n}}u_{n}\psi\right)\nonumber\\
&+\frac{p+\gamma}{\gamma-1}\int_{\Omega}k(x)u_{n}^{p}\psi.
\end{align}
Since
\begin{align*}
&-\frac{a(\gamma+1)}{\gamma-1}\|u_{n}\|^{2}-\frac{\gamma+3}{\gamma-1}\left(b\|u_{n}\|^{4}+\int_{\Omega}\phi_{u_{n}}u_{n}^{2}\right)+\frac{p+\gamma}{\gamma-1}\int_{\Omega}k(x)u_{n}^{p+1}\\
&=-\frac{1}{\gamma-1}\left[a(1-p)\|u_{n}\|^{2}+(3-p)\left(b\|u_{n}\|^{2}+\int_{\Omega}\phi_{u_{n}}u_{n}^{2}\right)+(\gamma+p)\int_{\Omega}\lambda h(x)u_{n}^{1-\gamma}\right]\\
&\leq-\frac{a(1-p)}{\gamma-1}\|u_{n}\|^{2}\\
&\leq-\frac{a(1-p)}{\gamma-1}C^{2}.
\end{align*}
So, from \eqref{2.8}, we have
\begin{align}\label{2.9}
\frac{\|\psi\|}{n}&\geq{D}_{+}\theta_{n}(0)\left(\frac{\|u_{n}\|^{2}}{n}-\frac{(1-p)a\alpha^{2}}{\gamma-1}\right)-\frac{a(\gamma+1)}{\gamma-1}\int_{\Omega}(-\Delta)^{s/2}{u}_{n}\cdot(-\Delta)^{s/2}\psi\nonumber\\
&-\frac{\gamma+3}{\gamma-1}\left(b\|u_{n}\|^{2}\int_{\Omega}(-\Delta)^{s/2}{u}_{n}\cdot(-\Delta)^{s/2}\psi+\int_{\Omega}\phi_{u_{n}}u_{n}\psi\right)+\frac{p+\gamma}{\gamma-1}\int_{\Omega}k(x)u_{n}^{p}\psi.
\end{align}
We choose $n$ large enough such that $\frac{\|u_{n}\|^{2}}{n} -\frac{(1-p)aC^{2}}{\gamma-1} <0$, we know from \eqref{2.9} that $D_{+}\theta_{n}(0)\neq-\infty$ as $n\rightarrow\infty$. That is $D_{+}\theta_{n}(0)$ is bounded from below uniformly for $n$ large enough. Hence from \eqref{2.6}, we have
\begin{equation*}
|D_{+}\theta_{n}(0)|\leq{C} ~\text{for $n$ large enough},
\end{equation*}
for some $C>0$. Again, by (ii) of Ekeland's variation principle, we also have
\begin{align*}
&\frac{|\theta_{n}(t)-1|\|u_{n}\|+|t\theta_{n}(t)|\|\psi\|}{n}\\
&\geq\frac{\|u_{n}-\theta_{n}(t)(u_{n}+t\psi)\|}{n}\\
&\geq I(u_{n})-I(\theta_{n}(t)(u_{n}+t\psi))\\
&=\frac{a}{2}(\|u_{n}\|^{2}-\|u_{n}+t\psi\|^{2})+\frac{1}{4}\left[b(\|u_{n}\|^{4}-\|u_{n}+t\psi\|^{4})+\int_{\Omega}(\phi_{u_{n}}u_{n}^{2}-\phi_{u_{n}+t\psi}(u_{n}+t\psi)^{2})\right]\\
&+\frac{1}{\gamma-1}\int_{\Omega}\lambda h(x)(u_{n}^{1-\gamma}-(u_{n}+t\psi)^{1-\gamma})-\frac{1}{p+1}\int_{\Omega}k(x)(u_{n}^{p+1}-(u_{n}+t\psi)^{p+1})\\
&+\frac{a}{2}(1-\theta_{n}^{2}(t))\|u_{n}+t\psi\|^{2}+\frac{1}{4}(1-\theta_{n}^{4}(t))\left(b\|u_{n}+t\psi\|^{4}+\int_{\Omega}\phi_{u_{n}+t\psi}(u_{n}+t\psi)^{2}\right)\\
&+\frac{1}{\gamma-1}(1-\theta_{n}^{1-\gamma}(t))\int_{\Omega}\lambda h(x)(u_{n}+t\psi)^{1-\gamma}-\frac{1}{p+1}(1-\theta_{n}^{p+1}(t))\int_{\Omega}k(x)(u_{n}+t\psi)^{p+1}.
\end{align*}
The above inequality also holds for $t=t_{k}$, then dividing by $t_{k}>0$ and passing to the limit as $k\rightarrow\infty$, then by $u_{n}\in N_{2}$, we obtain that
\begin{align*}
\frac{|D_{+}\theta_{n}(0)|\|u_{n}\|}{n}+\frac{\|\psi\|}{n}&\geq-(a+b\|u_{n}\|^{2})\int_{\Omega}(-\Delta)^{s/2}{u}_{n}\cdot(-\Delta)^{s/2}\psi-\int_{\Omega}\phi_{u_{n}}u_{n}\psi+\int_{\Omega}k(x)u_{n}^{p}\psi\\
&+\lim_{k\rightarrow\infty}\inf\frac{1}{\gamma-1}\int_{\Omega}\frac{\lambda h(x)(u_{n}^{1-\gamma}-(u_{n}+t_{k}\psi)^{1-\gamma})}{t_{k}}.
\end{align*}
Again, proceeding to the similar arguments as in {\bf Case 1}, we can obtain $\|u_n\|\rightarrow\|u\|$ as $n\rightarrow\infty$. Hence, by using $|D_{+}f_{n}(0)|\leq C$, Fatou's lemma and the strong convergence, we obtain $\int_{\Omega}\lambda h(x)u^{-\gamma}\psi<\infty$ and
\begin{equation*}
(a+b\|u\|^{2})\int_{\Omega}(-\Delta)^{s/2}{u}\cdot(-\Delta)^{s/2}\psi+\int_{\Omega}\phi_{u}u\psi-\int_{\Omega}\lambda h(x)u^{-\gamma}\psi-\int_{\Omega}k(x)u^{p}\psi\geq 0.
\end{equation*}
Thus from {\bf Case 1} and {\bf Case 2}, we obtain that the above inequality holds for $\psi\in X_0$ with $\psi\geq 0$, that is the inequality \eqref{1.7} holds.\\
We now prove that $u$ is a weak solution to the system \eqref{problem reduced} by using$u\in N_2$ and the inequality \eqref{1.7}. For $t>0$ and $\psi\in X_0$, we define
$$\Omega_1=\{x\in\Omega:u(x)+t\psi(x)\geq0\}~\text{and}~\Omega_2=\{x\in\Omega:u(x)+t\psi(x)<0\}.$$
Now on using $\psi_t=(u+t\psi)^+$ as the test function in \eqref{1.7}, we get
{\small \begin{align*}
0&\leq(a+b\|u\|^{2})\int_{\Omega}(-\Delta)^{s/2}{u}\cdot(-\Delta)^{s/2}\psi_t+\int_{\Omega}\phi_{u}u\psi_t-\int_{\Omega}\lambda h(x)u^{-\gamma}\psi_t-\int_{\Omega}k(x)u^{p}\psi_t\\
&=(a+b\|u\|^{2})\int_{\Omega_{1}}(-\Delta)^{s/2}{u}\cdot(-\Delta)^{s/2}(u+t\psi)+\int_{\Omega_{1}}\phi_{u}u(u+t\psi)\\&\hspace{3cm}-\int_{\Omega_{1}}\lambda h(x)u^{-\gamma}(u+t\psi)-\int_{\Omega_{1}}k(x)u^{p}(u+t\psi)\\
&=(a+b\|u\|^{2})\int_{\Omega}(-\Delta)^{s/2}{u}\cdot(-\Delta)^{s/2}(u+t\psi)+\int_{\Omega}\phi_{u}u(u+t\psi)\\
&\hspace{3cm}-\int_{\Omega}\lambda h(x)u^{-\gamma}(u+t\psi) -\int_{\Omega}k(x)u^{p}(u+t\psi)\\
&-\left[(a+b\|u\|^{2})\int_{\Omega_{2}}(-\Delta)^{s/2}{u}\cdot(-\Delta)^{s/2}(u+t\psi)+\int_{\Omega_{2}}\phi_{u}u(u+t\psi)\right.\\
&\hspace{3cm}\left.-\int_{\Omega_{2}}\lambda h(x)u^{-\gamma}(u+t\psi) -\int_{\Omega_{2}}k(x)u^{p}(u+t\psi)\right]\\
&\leq t\left[(a+b\|u\|^{2})\int_{\Omega}(-\Delta)^{s/2}{u}\cdot(-\Delta)^{s/2}\psi+\int_{\Omega}\phi_{u}u\psi-\int_{\Omega}\lambda h(x)u^{-\gamma}\psi-\int_{\Omega}k(x)u^{p}\psi\right.\\
&\hspace{3cm}\left.-(a+b\|u\|^{2})\int_{\Omega_{2}}(-\Delta)^{s/2}{u}\cdot(-\Delta)^{s/2}\psi-\int_{\Omega_{2}}\phi_{u}u\psi\right]
\end{align*}}
Since, $u>0$ almost everywhere in $\Omega$ and the measure of $\Omega_{2}$ tends to zero as $t\rightarrow 0$, then dividing by $t>0$ and letting $t\rightarrow 0$, we obtain that
\begin{equation*}
(a+b\|u\|^{2})\int_{\Omega}(-\Delta)^{s/2}{u}\cdot(-\Delta)^{s/2}\psi+\int_{\Omega}\phi_{u}u\psi-\int_{\Omega}\lambda h(x)u^{-\gamma}\psi-\int_{\Omega}k(x)u^{p}\psi\geq 0, ~\psi\in X_0.
\end{equation*}
This inequality also holds for $-\psi$, so we have
\begin{equation*}
(a+b\|u\|^{2})\int_{\Omega}(-\Delta)^{s/2}{u}\cdot(-\Delta)^{s/2}\psi+\int_{\Omega}\phi_{u}u\psi-\int_{\Omega}\lambda h(x)u^{-\gamma}\psi-\int_{\Omega}k(x)u^{p}\psi=0, ~\psi\in X_0.
\end{equation*}
Thus, $u\in X_0$ is a solution of system \eqref{problem reduced}.\\
\noindent {\it\bf Uniqueness of solution.} Assume the compatibility condition \eqref{compatibility} holds. Let $u, v\in X_0$ be two weak solutions to system \eqref{problem reduced}. Then from Definition \ref{weak solution defn}, we have
\begin{equation}\label{2.10}
(a+b\|u\|^{2})\int_{\Omega}(-\Delta)^{s/2}{u}\cdot(-\Delta)^{s/2}(u-v)+\int_{\Omega}\phi_{u}u(u-v)=\int_{\Omega}\lambda h(x)u^{-\gamma}(u-v)+\int_{\Omega}f(x,u)(u-v).
\end{equation}
and
\begin{equation}\label{2.11}
(a+b\|v\|^{2})\int_{\Omega}(-\Delta)^{s/2}{v}\cdot(-\Delta)^{s/2}(u-v)+\int_{\Omega}\phi_{v}v(u-v)=\int_{\Omega}\lambda h(x)v^{-\gamma}(u-v)+\int_{\Omega}f(x,u)(u-v).
\end{equation}
On subtracting \eqref{2.11} from \eqref{2.10}, we get
\begin{align}\label{unique}
\begin{split}
&a\|u-v\|^{2}+b(\|u\|^{4}+\|v\|^{4}-\|u\|^{2}(u,v)-\|v\|^{2}(u,v))+\int_{\Omega}(\phi_{u}u-\phi_{v}v)(u-v)\\
&=\int_{\Omega}\lambda h(x)(u^{-\gamma}-v^{-\gamma})(u-v) +\int_{\Omega}(f(x,u)-f(x,v))(u-v)\\
&=\int_{\Omega}\lambda h(x)(u^{-\gamma}-v^{-\gamma})(u-v),~(\text{by using}~f(x;\cdot)\equiv0).
\end{split}
\end{align}
Now, on applying H\"{o}lder's inequality, we have
\begin{equation*}
\|u\|^{4}+\|v\|^{4}-\|u\|^{2}(u,v)-\|v\|^{2}(u,v)\geq(\|u\|-\|v\|)^2(\|u\|^2+\|u\|\|v\|+\|v\|^2)\geq 0.
\end{equation*}
and $\int_{\Omega}\lambda h(x)(u^{-\gamma}-v^{-\gamma})(u-v)\leq 0$, since $\gamma>0$. Therefore, on using $a, b\geq0$ with $a+b>0$ and from Lemma \ref{phi prop}, we deduce that $\|u-v\|^{2}\leq0$.
Hence, the solution to the system \eqref{problem reduced} is unique.
\begin{remark}
Observe that, if we replace $f$ by $-f$ in the problem \eqref{problem reduced}, then from \eqref{unique} one can conclude that the problem \eqref{problem reduced} possesses a unique solution.
\end{remark}
\begin{remark}
The existence of solution to the problem \eqref{problem reduced}, for $\gamma=1$ can be obtained as a limit of the following sequence of problems
\begin{align*}
\begin{split}
\left(a+b[u]^2\right)(-\Delta)^{s} u+\phi_u u&=\lambda\frac{h(x)}{(u+\frac{1}{n})}+f(x,u)~\text{in}~\Omega,\\
u&>0~\text{in}~\Omega,\\
u&=0~\text{in}~\mathbb{R}^N\setminus\Omega,
\end{split}
\end{align*}
\end{remark}
\section*{Acknowledgement}
The author thanks the Council of Scientific and Industrial Research (CSIR), India, for the financial assistantship received to carry out this research work (CSIR no. 09/983(0013)/2017-EMR-I). The author also thanks Prof. D. Choudhuri for the suggetions and discussions.
\bibliographystyle{plain}
|
1,314,259,993,565 | arxiv |
\section{Introduction}
The exclusive production of mesons was studied in the past
mostly close to the kinematical threshold.
The Tevatron is a first accelerator which opens
a possibility to study the (semi)exclusive production
of mesons at high energies. A similar program will be
carried out in the future at just being put into operation
LHC.
Here I briefly review several mechanisms of exclusive
meson production studied recently by the Cracow group
(the details can be found in \cite{SPT07,SS07,PST08,RSS08,SL08}).
In general, the mechanism of the reaction depends
on the quantum numbers of the meson and/or its internal
structure.
For heavy scalar mesons
(scalar quarkonia, scalar glueballs) the mechanism of
the production, shown in Fig.\ref{fig:scalar_diagram},
is exactly the same as for the diffractive Higgs boson
production extensively discussed in recent years
by the Durham group \cite{Durham}.
The dominant mechanism for the exclusive heavy
vector meson production is quite different.
Here there are two dominant processes shown in
Fig.\ref{fig:vector_diagram}. When going to lower
energies the mechanism of the meson production becoming
more complicated and usually there exist more mechanisms.
For illustration in Fig.\ref{fig:pion_pion_diagram}
I show a new mechanism of the glueball production
proposed recently in Ref.\cite{SL08}.
\begin{figure}
\includegraphics[height=.25\textheight]{QCDdiff.eps}
\caption{A sketch of the bare QCD mechanism of
exclusive heavy scalar meson production.
\label{fig:scalar_diagram}
}
\end{figure}
\begin{figure}
\includegraphics[height=.25\textheight]{vector_diagram.eps}
\caption{Two basic QED $\otimes$ QCD mechanisms of
exclusive heavy vector meson production.
\label{fig:vector_diagram}
}
\end{figure}
\begin{figure}
\includegraphics[height=.25\textheight]{pion_pion.eps}
\caption{A sketch of the bare QCD mechanism of
exclusive heavy scalar meson production.
\label{fig:pion_pion_diagram}
}
\end{figure}
\section{Some examples}
Recently we have calculated differential cross sections
in several reactions (including different mechanisms):
\begin{itemize}
\item $ p p \to p p \eta' $, $ p p \to p p \eta_c$
( IP IP + $\gamma \gamma$ )
\item $ p p \to p p \chi_c(0)$ (IP IP + $\gamma \gamma$ )
\item $ p p \to p p f_0(1500)$ (IP IP + $\pi^+ \pi^-$)
\item $ p p \to p p J/\psi$ (IP $\gamma$ + $\gamma$ IP)
\item $ p p \to p p \Upsilon$ ( IP $\gamma$ + $\gamma$ IP)
\item $ p p \to p p \pi^+ \pi^-$ ( (IP + IR) $\otimes$ (IP + IR))
\end{itemize}
The details of the formalism as well as a detailed analysis
of differential distribution in longitudinal and transverse
momenta can be found in the original papers
\cite{SPT07,SS07,PST08,RSS08,SL08}).
Here I wish to flash only some illustrative examples.
\subsection{Scalar meson production}
Let us start with the production of scalar particles.
In Ref.\cite{PST08} we discussed in detail the production
of scalar $\chi_c(0)$ meson. Here the dominant mechanism
is exactly the same as for celebrated recently
diffractive production of Higgs boson.
A study of this reaction can be an alternative
for inclusive searches for the Higgs boson at LHC.
In order to show the general features of the exclusive
diffractive production in Fig.\ref{fig:dsig_dxi_kl} I show
distribution in Feynman $x_F$ of $\chi_c(0^+)$ mesons
(the middle bump) as well as distributions of the
associated proton (left bump) and antiproton (right bump)
for W = 1960 GeV.
In this calculation the unintegrated gluon distribution
(UGDF) ala Kharzeev-Levin were used \cite{PST08}.
As discussed in Ref.\cite{PST08} the cross section strongly
depends on UGDF used.
A clear gaps in $x_F$ between the meson
and associated nucleons can be seen. The gaps in $x_F$
translate into gaps in rapidity. Generally the diffractive
production of single mesons can be characterized by
rapidity gaps. Can this be used as a criterion for
selecting appropriate events?
It would be useful for potential experiments to calculate
cross section for inclusive double diffraction of
$\chi_c(0)$ mesons to verify if the large rapidity gap
criterion can be used to pin down the exclusive channel.
\begin{figure}
\includegraphics[height=.3\textheight]{dsig_dxi_kl.eps}
\caption{Distribution in Feynman $x_F$ of $\chi_c(0)$
meson and associated proton (left bump) and antiproton (right bump) for the Tevatron energy.
\label{fig:dsig_dxi_kl}}
\end{figure}
It seems that the dominant mechanism of the glueball
production at high energies should be the same as for
the $\chi_c(0)$ meson. If there is appreciable gluonic
component in a meson the ladder gluons
could (should) strongly couple to the meson.
In Ref.\cite{SL08} we concentrated rather on the
intermediate and low energy regime. In addition to the QCD
diffractive mechanism we considered the two-pion fusion.
It seems that this new mechanism may dominate close to
threshold. This can be checked in the future with
the PANDA detector at the complex FAIR planned at
GSI Darmstadt. In the $p \bar p \to p \bar p f_0(1500)$
channel the pionic mechanism dominates close to threshold,
while the QCD diffraction takes over at larger energies.
In the $p \bar p \to n \bar n f_0(1500)$
channel the pion-exchange mechanism may dominate
in a broader range of energies.
\begin{figure}
\includegraphics[height=.3\textheight]{sig_w_f0_1500_ela.eps}
\includegraphics[height=.3\textheight]{sig_w_f0_1500_cex.eps}
\caption{Integrated cross section as a function of initial energy. The solid line corresponds to the pion-pion fusion whereas the dashed and dotted line to the diffractive QCDmechanism with different UGDFs.}
\end{figure}
\subsection{Vector meson production}
Because of their quantum numbers vector mesons cannot be
produced in a fusion of two pomerons (two gluonic ladders).
For heavy vector quarkonia the simplest possible mechanism
is a photon-pomeron or pomeron-photon fusion shown
in Fig.\ref{fig:vector_diagram}. For light vector mesons
the situation is slightly more complicated.
In Ref.\cite{SS07} we discussed several differential
distributions for exclusive $J/\Psi$ production
in a phenomenological model. In Ref.\cite{RSS08}
we performed a similar analysis for exclusive production
of $\Upsilon$ in the formalism of unintegrated
gluon distributions. In Fig. \ref{fig:dsig_dy_vector}
I show a compilation of the rapidity distributions
for $J/\Psi$, $\Psi'$, $\Upsilon$ and $\Upsilon'$
for both Tevatron (left panel) and LHC (right panel).
Both distributions obtained with bare amplitudes
(dashed lines) and
including absorption corrections (solid lines) are
shown.
More details about absorption corrections can be found
in Refs.\cite{SS07,RSS08}.
The transverse-momentum integrated rapidity distribution
is only mildly modified by the absorption corrections.
Much bigger effects can be seen at large transverse
momenta \cite{SS07,RSS08} where the cross section is
much smaller. We hope that in a near future the CDF
collaboration at the Tevatron will release the experimental
cross section for exclusive production of quarkonia
\cite{Albrow}.
\begin{figure}
\includegraphics[height=.3\textheight]{dsig_dy_Tev.eps}
\includegraphics[height=.3\textheight]{dsig_dy_LHC.eps}
\caption{Distribution in rapidity of
$J/\Psi$, $\Psi'$, $\Upsilon$, $\Upsilon'$ (from top to bottom) for Tevatron (left panel) and LHC (right panel).
The dashed line corresponds to calculation in
the Born approximation and the solid line includes
absorption corrections.
\label{fig:dsig_dy_vector}}
\end{figure}
\section{Conclusions}
I have briefly discussed some of our results
on exclusive production of mesons at high energies.
At present a direct comparison with experimental
data is not possible.
We expect some experimental data from the Tevatron soon.
In a more distant future one may hope for experimental
data from LHC.
\begin{theacknowledgments}
The collaboration with Wolfgang Sch\"afer, Roman Pasechnik,
Oleg Teryaev, Anna Cisek and Piotr Lebiedowicz
on the topics presented here is acknowledged.
\end{theacknowledgments}
|
1,314,259,993,566 | arxiv | \section{Introduction}
In 1976 Michael Atiyah
\cite{Atiyah:1976:elliptic}
introduced $L^2$-cohomology and $L^2$-Betti numbers
of manifolds with non-trivial fundamental group
and asked whether these numbers can be irrational.
For background on these concepts we refer to
the book~\cite{Lueck:2002:invariants}.
It was shown later that
Atiyah's question is equivalent to a purely analytic problem
on group von Neumann algebras
and we will exclusively work in this context.
Therefore no knowledge of cohomology theory and geometry is required
in the present paper, and in the remainder of this section
we recall a few basic facts from von Neumann algebra theory
which are necessary to state Atiyah's question.
Let $\Gamma$ be a finitely presented discrete group and $\IQ\Gamma$ its rational group
ring.
An element of this ring can be represented as a bounded left convolution
operator on the space $\ell_2(\Gamma)$ of square summable functions
on $\Gamma$. Denote this representation by $\lambda$.
The equivalent formulation of Atiyah's question in this
setting is:
\emph{
Let $\opT\in M_n(\IQ\Gamma)$ be a symmetric element. Is it possible that
the von Neumann dimension $\dim_{L(\Gamma)} \ker \lambda(\opT)$ is an irrational
number?
}
Here the \emph{von Neumann dimension} is to be understood in the setting
of the group von Neumann algebra, denoted $L(\Gamma)$,
which is the completion of the rational convolution operator algebra in the
weak operator topology. More precisely, one looks for matrices
of convolution operators, but for simplicity, in the present paper we will
work in $L(\Gamma)$ exclusively.
A symmetric element $\opT\in\IQ\Gamma$ gives rise to a selfadjoint
convolution operator $\lambda(\opT)$, all of whose spectral projections
lie in the von Neumann algebra $L(\Gamma)$.
On $L(\Gamma)$ there is a normal
faithful trace $\tau(\opT) = \langle \opT\delta_e,\delta_e\rangle$
and the value of $\tau(p)$ at projections $p\in L(\Gamma)$
is called the \emph{von Neumann dimension function}.
The von Neumann dimension of a closed subspace of $\ell_2(\Gamma)$
is the von Neumann dimension of the corresponding orthogonal
projection, if the latter happens to be an element of $L(\Gamma)$.
While rationality of kernel dimensions has been shown for many
examples, (see, e.g., \cite {Linnell:1993:division,LinnellSchick:2007:finite}),
recently Atiyah's question was answered affirmatively
by Tim Austin~\cite{Austin:2009:rational}
who constructed an uncountable family of groups and rational
convolution operators with distinct kernel dimensions,
which a fortiori must contain irrational numbers, even transcendental ones.
Subsequently more constructive answers were given in
\cite{Grabowski:2010:turing,PichotSchickZuk:2010:closed}.
A stronger variant of Atiyah's question had been solved earlier
\cite{GrigorchukLinnellSchickZuk:2000:Atiyah,
GrigorchukZuk:2001:lamplighter,DicksSchick:2002:spectral}.
In these examples, so-called lamplighter groups play a central role.
Although these are not finitely presented,
they are recursively presentable (see, e.g., \cite{Baumslag:2005:embedding})
and therefore by a theorem of Higman \cite{Higman:1961:subgroups}
can be embedded into finitely presented groups.
In the present paper, complementing the above results,
we pursue the ideas
of~\cite{DicksSchick:2002:spectral}
and
compute explicitly the von Neumann dimension of the kernel of the
``switch-walk-switch'' adjacency
operators on the free lamplighter groups $\CG_m\wr\IF_\f$
with respect to the canonical generators
and show that
they are irrational for any $\f\geq 2$ and $m>2\f-1$.
This provides another elementary explicit example of a rational convolution
operator with irrational kernel dimension.
The basic ingredient is Theorem~\ref{thm:lamplighterpercolation},
which generalises the methods of Dicks and
Schick~\cite{DicksSchick:2002:spectral} from the infinite cyclic group
to arbitrary discrete groups and makes a link to percolation theory,
thus providing a quite explicit description of the spectrum of
switch-walk-switch
transition operators on lamplighter groups as the union of the spectra of
all finite connected subgraphs of the Cayley graph.
In particular, the lamplighter kernel dimension equals the expected
normalised
kernel dimension of the percolation cluster.
The paper is organised as follows.
In Section~\ref{sec:lamplighter} we review
the necessary prerequisites about lamplighter groups and percolation
and state the main result.
In Section~\ref{sec:matchings} we recall the connection between
spectra and matchings of finite trees and compute the generating
function of the kernel dimensions of finite subtrees of
the Cayley graph of the free group. As a final step
we integrate this generating function
in Section~\ref{sec:parametrisation} and thus obtain the
dimensions we are interested in. Another example of a free
product of groups whose Cayley graph is not a tree is discussed in
Section~\ref{sec:freeproduct}.
\emph{Acknowledgements.}
We thank Slava Grigorchuk for explaining Atiyah's question,
Martin Widmer and Christiaan van de Woestijne for discussions about transcendental numbers,
and Mark van Hoeij for a hint to compute a certain abelian integral,
which ultimately led to the discovery of
the parametrisation \eqref{eq:parametrisation}.
Last but not least we thank three anonymous referees
for numerous remarks which helped to improve the presentation.
\section{Lamplighter groups and percolation}
\label{sec:lamplighter}
Let $G$ be a discrete group and fix a symmetic generating set $S$.
We denote by $\CX=\CX(G,S)$ the Cayley graph
of $G$ with respect to $S$ and in the rest of the paper
identify the rational group algebra
element $\opT=\sum_{s\in S}s$ with the corresponding convolution operator,
which coincides with the adjacency operator on $\CX$.
\subsection{Lamplighter groups}
The name \emph{lamplighter group} has been coined in recent years
to denote wreath products of the form $\Gamma=\CG_m\wr G$.
This is the semidirect product
$\LL\rtimes G$, where $\CG_m$ is the cyclic group of order $m$
and $\LL=\bigoplus_G \CG_m$ is the group
of \emph{configurations} $\eta:G\to \CG_m$ with finite support,
where we define $\supp \eta = \{x\in G : \eta(x) \ne \eCm\}$.
The group operation on $\LL$ is pointwise multiplication in $\CG_m$
and the natural
left action of $G$ on $\LL$ given by $L_g\eta(x) = \eta(g^{-1}x)$
induces the twisted group law on $\CG_m\wr G$
$$
(\eta,g)(\eta',g') = (\eta\cdot L_g\eta',gg')
$$
Certain random walks on $\CG_m\wr G$ can be interpreted as
a lamplighter walking around on $G$ and turning
on and off lamps. A pair $(\eta,g)$ encodes both
the position of the lamplighter as an element $g\in G$
and the states of the lamps as a function $\eta\in\LL$.
We will consider here the ``switch-walk-switch'' lamplighter adjacency operator
$$
\tilde{\opT} = \sum_{s\in S} EsE
$$
on the lamplighter group $\Gamma$ where $E=\frac{1}{m}\sum_{h\in\CG_m} h$
is the idempotent corresponding to the uniform distribution on the lamp group
$\CG_m$.
The \emph{underlying} convolution operator on $G$ is $\opT=\sum_{s\in S} s$.
Here we identify $\CG_m$ and $G$ with subgroups of $\Gamma$ via
the respective embeddings
$$
\begin{aligned}
\CG_m&\to \Gamma & \qquad\qquad G &\to\Gamma \\
h &\mapsto (\delta_e^h,e) & g &\mapsto (\iota, g)
\end{aligned}
$$
where $\iota$ is the neutral element of $\LL$ and
$$
\delta_g^h(x) =
\begin{cases}
h & x=g\\
\eCm & x\ne g
\end{cases}
.
$$
\subsection{Percolation clusters}
Let $\CX=(V,E)$ be a graph. We use the standard notation
``$x\in\CX$'' for vertices and $x\sim y$ for the neighbour relation.
Fix a parameter $0 < \mathsf{p} < 1$. In \emph{Bernoulli
site percolation} with parameter $\mathsf{p}$ on $\CX$, we have i.i.d.~Bernoulli random
variables $Y_x\,$, $x \in \CX\,$, sitting at the vertices of $\CX$, with
$$
\Prob_{\mathsf{p}}[Y_x=1] = \mathsf{p} ,\qquad \Prob_{\mathsf{p}}[Y_x=0] = q := 1-\mathsf{p}\,.
$$
We can realise those random variables on the probability space
$\Omega=\{0,1\}^\CX$ with a suitable probability measure $\Prob$.
Given $\omega\in\Omega$, denote by $\CX(\omega)$ the full subgraph
of $\CX$ induced on $\{x : Y_x(\omega)=1\}$ and for any vertex $x\in \CX$, denote by
$C_x(\omega)$ the connected component of $\CX(\omega)$ containing the vertex $x$,
which is called the \emph{percolation cluster} at $x$.
It is well known that for every connected graph
there is a critical parameter $\mathsf{p}_c$
such that for any vertex $x$ a phase transition occurs in the sense that
for $\mathsf{p}<\mathsf{p}_c$ the cluster $C_x$ is almost surely finite and for
$\mathsf{p}>\mathsf{p}_c$ it is infinite with positive probability.
In order to make use of this fact we recall a combinatorial interpretation
of criticality.
\begin{definition}
For a subset $\clA\subseteq \CX$ we denote its \emph{vertex boundary}
$$
d\clA = \{y\in \CX : y\not\in \clA, y\sim x \text{ for some $x\in \clA$}\}
.
$$
For $x\in \CX$, we denote
$$
\mathcal{A}_x=\{\clA\subseteq \CX: x\in \clA, \text{ $\clA$ finite and connected}\}
\cup\{\emptyset\}
$$
the set of finite, possibly empty, path-connected neighbourhoods of $x$.
These sets are sometimes called \emph{lattice animals}.
The boundary of the empty animal
is defined to be
the set $\{x\}$. We denote by $\mathcal{A}_x^*$ the set of animals at $x$ without
the empty animal.
\end{definition}
The probability of a fixed $\clA\in\mathcal{A}_x$ to occur as percolation
cluster at $x$ is
$$
\IP[C_x=\clA] = \mathsf{p}^{\abs{\clA}} q^{\abs{d\clA}}
;
$$
thus for $\mathsf{p}<\mathsf{p}_c$ we have
\begin{equation}
\label{equ:sump1-p=1}
\sum_{\clA\in \mathcal{A}_x} \mathsf{p}^{\abs{\clA}} q^{\abs{d\clA}} = 1
\end{equation}
because some $\clA\in \mathcal{A}_x$ occurs almost surely.
Now for a fixed animal $\clA$ consider the truncated operator
$$
\opT_\clA = P_\clA\opT P_\clA
$$
where $P_\clA$ is the orthogonal projection onto the finite dimensional
subspace $\{f\in \ell_2(\CX) : \supp f\subseteq \clA\}$.
We denote the random percolation adjacency operator by
$$
\opT_\omega = \opT_{C_e(\omega)},
$$
and by $\dim\ker \opT_\clA$ the dimension of the kernel of $\opT_F$
as a finite matrix, while $\frac{\dim\ker\opT_\clA}{\abs{\clA}}$ will
be the
von Neumann dimension of the kernel of $\opT_\clA$ regarded as an element
of the finite von Neumann algebra $M_{\abs{\clA}}(\IC)$ with
von Neumann trace $\frac{1}{\abs{\clA}}\Tr$.
Special care is needed for the empty animal, for which we define both
the cardinality of the boundary
and the von Neumann kernel dimension to be $1$.
Then we have the following relation between the spectrum of the lamplighter operator
$\tilde{\opT}$ and the spectra of $\opT_\clA$.
\begin{theorem}[{\cite{LehnerNeuhauserWoess:spectrum,Lehner:2009:eigenspaces}}]
\label{thm:lamplighterpercolation}
The spectral measure of the lamplighter adjacency operator $\tilde{\opT}$ of
order $m$ on a Cayley graph
$\CX$ is equal to the expected spectral measure of the random truncated
adjacency operator $\opT_\omega$ on the percolation clusters of $\CX$ with
percolation parameter $p=1/m$.
In addition, if $p<p_c$, then there is a one-to-one correspondence
between the eigenspaces of $\tilde{\opT}$ and the collection of eigenspaces
of $\opT_\omega$ and we have the formula
\begin{equation}
\label{eq:dimensionformula}
\dim_{L(\Gamma)} \ker \tilde{\opT} = \IE \frac{\dim \ker \opT_\omega}{\abs{C_x(\omega)}}
= \sum_{\clA\in\mathcal{A}_e} \frac{\dim \ker \opT_\clA}{\abs{\clA}}
p^{\abs{\clA}}q^{\abs{d\clA}}
.
\end{equation}
\end{theorem}
In general it is hard to evaluate formula~\eqref{eq:dimensionformula}
because one has to compute the kernels of the adjacency matrices of
all finite
clusters. Due to the recursive structure of the Cayley tree however
it is possible to compute an algebraic equation for the generating function
$$
\sum_{T\in \mathcal{A}_e^*} (\dim\ker \opT_\clA) x^{\abs{\clA}}
$$
on free groups and
to evaluate~\eqref{eq:dimensionformula} by integrating this function,
thus obtaining our main result, which concludes this section.
\begin{theorem}
Denote $g_1,g_2,\dots,g_\f$ the canonical generators
of the free group $\IF_\f$ and
consider the adjacency operator $\opT=\sum g_i+g_i^{-1}$
on its Cayley graph. Then the von Neumann dimension of the kernel of
the corresponding lamplighter operator
$\tilde{\opT}=E\opT E$ on $\CG_m\wr\IF_\f$
is the number
$$
\dim_{L(\Gamma)}\ker \tilde{\opT}
= 1-2p+\frac{(\tau(p)-1)(2-2\f+2\f{}\tau(p))}{\tau(p)^2}
$$
where $p=1/m$ and $\tau(p)$ is the unique positive solution of the equation
$t^{2\f-1}-t^{2\f-2}=p$. For $\f>1$, this is an irrational algebraic number,
e.g., for $\f=2$ and $m=4$, the dimension is
$$ -\frac56 - \frac{400}{3(766+258\sqrt{129})^{1/3}} +
\frac{2(766+258\sqrt{129})^{1/3}}{3} \approx 0.850971.
$$
\end{theorem}
\begin{remark}
Similar computations are possible in more general free product groups
$G_1*G_2*\dots*G_n$, where each factor $G_i$ is a finite group
whose Cayley graph possesses only cycles of length $\equiv 2\mod 6$,
like the cyclic groups $\IZ_2$, $\IZ_6$, $\IZ_{10}$, etc.
An example is briefly discussed in Section~\ref{sec:freeproduct}.
It should be noted however that our technique does not work
for nonzero eigenvalues, because in this case
it is more complicated to obtain the multiplicity of the eigenvalue.
Moreover,
in contrast to other approaches
(\cite{DicksSchick:2002:spectral,GrigorchukLinnellSchickZuk:2000:Atiyah}),
it is restricted to adjacency operators,
i.e., all group elements get the same weight.
\end{remark}
\section{Matchings, rooted trees, and generating functions}
\label{sec:matchings}
In this section we prepare the evaluation of the series
\eqref{eq:dimensionformula} by computing a generating function.
To this end let us recall some notations.
Let $G=(V,E)$ be a finite graph.
By \emph{characteristic polynomial} $\chi(G,x)$
(resp., \emph{spectrum, kernel dimension})
\emph{of a graph} we mean the characteristic polynomial (resp., spectrum,
kernel dimension) of its adjacency matrix.
A \emph{matching} of a finite graph is a set of disjoint edges,
i.e., every vertex occurs as an end point of at most one edge.
A \emph{perfect matching} is a matching which covers all the vertices
of the graph.
The \emph{matching polynomial} of a graph on $n$ vertices is the polynomial
$$
\sum_{j \geq 0} (-1)^j m(G,j)\, x^{n-2j},
$$
where $m(G,j)$ is the number of matchings of cardinality $j$.
It is well known (see, e.g., \cite{Godsil:1984:spectra,Cvetkovic:1988:recent})
that the characteristic polynomial of a finite tree
coincides with its matching polynomial.
Since the kernel dimension equals the multiplicity of eigenvalue zero,
which in turn is the degree of the polynomial $x^n\chi(G,x^{-1})$,
it follows immediately that the dimension of the kernel of a tree is given by
\begin{equation}\label{eq:kerdim}
\nu(T) = \dim \ker T = n - 2\mu(T),
\end{equation}
where $\mu(T)$ denotes the size of a matching of maximal cardinality in $T$.
In particular, $\dim \ker T = 0$ if and only if $T$ has a perfect matching.
As a first step to evaluate \eqref{eq:dimensionformula}
we have to determine the generating function
$$G(x) = \sum_{T\in \mathcal{A}_x^*} (\dim \ker T)\, x^{|T|}
= \sum_{T\in \mathcal{A}_x^*} (|T| - 2 \mu(T))\, x^{|T|},$$
where the sum is taken over all nonempty animals $T$, i.e.,
connected subgraphs of the
Cayley graph of the free group $\IF_\f{}$ that contain the unit element $e$.
To this end, we regard animals as rooted trees, with the root at $e$.
\begin{definition}
A \emph{ $k$-ary tree} is a planar rooted tree such that every vertex
has at most $k$ children. Hence every vertex has degree at most $k+1$, and the root has degree at most $k$.
A \emph{branch} of a $k$-ary tree is a rooted tree obtained
by splitting off a neighbor of the root together with its offspring.
Thus a $k$-ary tree can be defined recursively as
a rooted tree with an ordered collection of $k$ possibly empty
branches.
\end{definition}
Thus our animals are $k$-ary trees with $k=2\f-1$, with the single exception that the root vertex may have degree $k+1$ (but all branches are $k$-ary trees according to the above definition).
For reasons which will become apparent soon
we split the family of rooted trees into two groups,
following ideas similar to those employed in \cite{Wagner:2007:number}:
\begin{definition}
We say that a rooted tree is of \emph{type} $A$ if it has a maximum matching
that leaves the root uncovered. Otherwise $T$ is of type $B$.
\end{definition}
Suppose that $T$ is of type $A$. Then it has a maximum matching that does not
cover the root and is therefore a union of maximum matchings in the various
branches of $T$. Hence if $S_1,S_2,\cdots,S_k$ are the branches of $T$, we have
$$\mu(T) = \mu(S_1) + \mu(S_2) + \cdots + \mu(S_k).$$
We claim that in this case all $S_j$ are of type $B$.
For, suppose on the contrary that one of the branches, say $S_j$, is of type
$A$. Then we can choose a maximum matching in $S_j$ that does not cover the
root. Choose maximum matchings in all the other branches as well, and add the
edge between the roots of $T$ and $S_j$ to obtain a matching of cardinality
$$\mu(S_1) + \mu(S_2) + \cdots + \mu(S_k) + 1,$$
contradiction.
Conversely, if all branches of $T$ are of type $B$,
then $T$ is of type $A$: clearly, the
maximum cardinality of a matching that does not cover the root is $\mu(S_1) +
\mu(S_2) + \cdots + \mu(S_k)$, so it remains to show that there are no
matchings of greater cardinality that cover the root. Suppose that such a
matching contains the edge between the roots of $T$ and $S_j$. Then, since
$S_j$ is of type $B$, the remaining matching, restricted to $S_j$, can only
contain at most $\mu(S_j) - 1$ edges. Each of the other branches $S_i$ can only
contribute $\mu(S_i)$ edges, so that we obtain a total of
$$\mu(S_1) + \mu(S_2) + \cdots + \mu(S_k)$$
edges, as claimed. This proves the following fact:
\begin{lemma}\label{lem:rec}
\begin{enumerate}
\item Let $T$ be a rooted tree and $S_1,\ldots,S_k$ its branches,
then we have
$$\mu(T) = \sum_{i=1}^k \mu(S_i) + \begin{cases} 0 & \text{ if $T$ is of type $A$,} \\ 1 & \text{ otherwise.} \end{cases}$$
\item
A rooted tree $T$ is of type $A$ if and only if all its branches are of type $B$.
\end{enumerate}
\end{lemma}
The only part that was not explicitly proven above is the formula for $\mu(T)$
in the case that $T$ is of type $B$. This, however, is easy as well: Clearly
the cardinality of a matching is at most $\mu(S_1) + \cdots + \mu(S_k) + 1$
(the summand $1$ accounting for the edge that covers the root). On the other
hand, $\mu(T)$ must be strictly greater than $\mu(S_1) + \cdots + \mu(S_k)$,
since there are matchings of this cardinality that do not cover the root.
\begin{remark}
Consistently with Lemma~\ref{lem:rec}
we define the tree $T_1$ that only consists of a single vertex to be of
type $A$ with $\mu(T_1) = 0$ and
the empty tree $T_0$ to be of type $B$ with $\mu(T_0)= 0$.
This is important for the generating functions constructed below.
\end{remark}
Since we are interested in the parameter $\nu(T) = \dim \ker T = |T| - 2\mu(T)$
rather than $\mu(T)$ itself, we first translate the above formula to a
recursion for $\nu(T)$: since $|T| = |S_1| + \cdots + |S_k| + 1$, we have
$$\nu(T) = \sum_{i=1}^k \nu(S_i) + \begin{cases} 1 & \text{ if $T$ is of type
$A$,} \\ -1 & \text{ otherwise.} \end{cases}$$
Now let $\trees_k$, $\trees_{k,A}$, $\trees_{k,B}$ denote the set of all $k$-ary trees, $k$-ary trees of type $A$ and $k$-ary trees of type $B$ respectively.
We define the bivariate generating functions
$$
A:=A(u,x) = \sum_{T \in \trees_{k,A}} u^{\nu(T)} x^{|T|}
\qquad \text{and}
\qquad B:= B(u,x) = \sum_{T \in \trees_{k,B}} u^{\nu(T)} x^{|T|},
$$
the
summation being over $k$-ary trees in both cases (including the empty tree in
the case of $B$, and the one-vertex tree in the case of $A$).
Since any tree $T$ of type $A$ is a grafting of $k$ (possibly empty) branches $S_1,\ldots,S_k$ of type $B$ (which we write as $T = \bigvee_{i=1}^k S_i$),
we obtain
\begin{align*}
A(u,x)
&= \sum_{T\in \trees_{k,A}} u^{\nu(T)}\,x^{\abs{T}}\\
&= \sum_{S_1,\dots,S_k\in \trees_{k,B}} u^{\nu(\bigvee_{i=1}^k
S_i)}\,x^{\abs{\bigvee_{i=1}^k S_i}}\\
&= \sum_{S_1,\dots,S_k\in \trees_{k,B}} u^{1+\sum_{i=1}^k\nu(S_i)}\,x^{1+\sum_{i=1}^k\abs{S_i}}\\
&= ux \prod_{i=1}^k \sum_{S_i\in \trees_{k,B}} u^{\nu(S_i)}\,x^{\abs{S_i}}\\
&= ux B(u,x)^k
.
\end{align*}
Similarly, in the case of type $B$, we get the following equation
\begin{align*}
B(u,x)
&= \sum_{T\in \trees_{k,B}} u^{\nu(T)}\,x^{\abs{T}}\\
&= 1 + \sideset{}{^\prime}\sum_{S_1,\dots,S_k\in \trees_{k}} u^{\nu(\bigvee_{i=1}^k
S_i)}\,x^{\abs{\bigvee_{i=1}^k S_i}}\\
&= 1 + \sideset{}{^\prime}\sum_{S_1,\dots,S_k\in \trees_{k}} u^{-1+\sum_{i=1}^k\nu(S_i)}\,x^{\abs{\bigvee_{i=1}^k S_i}},\\
\intertext{where we took special care of the empty tree
and the remaining sum indicated by $\sum'$ runs over all $k$-tuples of trees such that
at least one of them is not of type $B$;
this means that we have to subtract the sum over $k$-tuples of type B trees
from the sum over $k$-tuples of arbitrary trees:}
B(u,x) &= 1 + \sum_{S_1,\dots,S_k\in \trees_{k}}
u^{-1+\sum_{i=1}^k\nu(S_i)}
\,
x^{1+\sum_{i=1}^k\abs{S_i}}
-
\sum_{S_1,\dots,S_k\in \trees_{k,B}}
u^{-1+\sum_{i=1}^k\nu(S_i)}
\,
x^{1+\sum_{i=1}^k\abs{S_i}}
\\
&= 1 + \frac{x}{u}
\biggl(
\prod_{i=1}^k \sum_{S_i\in \trees_{k}} u^{\nu(S_i)}\,x^{\abs{S_i}}
-
\prod_{i=1}^k \sum_{S_i\in \trees_{k,B}} u^{\nu(S_i)}\,x^{\abs{S_i}}
\biggr)\\
&= 1 + \frac{x}{u}
\biggl(
\bigl(A(u,x)+B(u,x)\bigr)^k - B(u,x)^k
\biggr)
.
\end{align*}
In conclusion, we have translated the recursive description into
the following two functional equations for $A(u,x)$ and $B(u,x)$:
\begin{equation}
\label{eq:ABequations}
\begin{aligned}
A(u,x) &= ux B(u,x)^k, \\
B(u,x) &= 1 + \frac{x}{u} ((A(u,x)+B(u,x))^k - B(u,x)^k). \\
\end{aligned}
\end{equation}
Finally, we obtain the following generating function for animals (the only
difference lying in the possibility that the root is allowed to have degree
$k+1 = 2\f{}$ as well and the empty tree is excluded this time):
$$F(u,x) = \sum_{\substack{ T \in\mathcal{A}_e^*}} u^{\nu(T)} x^{|T|} = ux B(u,x)^{k+1} + \frac{x}{u} ((A(u,x)+B(u,x))^{k+1} - (B(u,x))^{k+1}).$$
We are mainly interested in the derivative with respect to $u$, since
$$G(x) = \frac{\partial}{\partial u} F(u,x) \Big|_{u=1} = \sum_{T \in \mathcal{A}_e^*} \nu(T) x^{|T|}.$$
To save space, we will use the customary abbreviation $F_u$ etc.\ to denote
partial derivatives with respect to $u$. First note that
$$
A(u,x)+u^2B(u,x) = u^2+ux(A(u,x)+B(u,x))^k,
$$
and we can rewrite the identities~\eqref{eq:ABequations} as
\begin{equation}
\label{eq:BkABkequations}
\begin{aligned}
B(u,x)^k &= \frac{A(u,x)}{ux}, \\
(A(u,x)+B(u,x))^k &= \frac{A(u,x)+u^2(B(u,x)-1)}{ux}.
\end{aligned}
\end{equation}
Taking the derivative of the second identity at $u=1$ we obtain
$$
A_u(1,x) + B_u(1,x) = \frac{2(1-B(1,x))+x (A(1,x)+B(1,x))^k}{1-kx(A(1,x)+B(1,x))^{k-1}}.
$$
Using the identities~\eqref{eq:BkABkequations} we can express $F$ as
$$
F(u,x) = A(u,x)B(u,x)(1-\frac{1}{u^2}) + (A(u,x)+B(u,x))(\frac{A(u,x)}{u^2} + B(u,x)-1)
$$
and the derivative at $u=1$ is
\begin{align*}
F_u(1,x)
&= 2A(1,x)B(1,x) + (A_u(1,x)+B_u(1,x))(A(1,x)+B(1,x)-1) \\
&\phantom{=}+ (A(1,x)+B(1,x))(-2A(1,x) +A_u(1,x)+B_u(1,x)) \\
&= -2A(1,x)^2+(A_u(1,x)+B_u(1,x))(2A(1,x)+2B(1,x)-1)\\
&= -2A(1,x)^2
+ \frac{2(1-B(1,x))+x(A(1,x)+B(1,x))^k}{1-kx(A(1,x)+B(1,x))^{k-1}}
(2(A(1,x)+B(1,x))-1)\\
&= -2A(1,x)^2
+ \frac{(2(1-B(1,x))+A(1,x)+B(1,x)-1)(2(A(1,x)+B(1,x))-1)}{1-k(A(1,x)+B(1,x)-1)/(A(1,x)+B(1,x))},
\end{align*}
making use of~\eqref{eq:BkABkequations} in the last step once again. So we finally obtain
\begin{multline}\label{eq:G_in_terms_of_B}
G(x) = F_u(1,x) \\
= -2A(1,x)^2 + \frac{(A(1,x)+B(1,x))(A(1,x)-B(1,x)+1)(2A(1,x)+2B(1,x)-1)}{k-(k-1)(A(1,x)+B(1,x))}.$$
\end{multline}
\section{Parametrisation}
\label{sec:parametrisation}
Recall that we are considering percolation on a $(k+1)$-regular tree, where $p = \frac{1}{m} < \frac{1}{k}$ is the percolation probability, and $q = 1-p$. For an animal $T$ (i.e., a potential percolation cluster), the size of the boundary is $|dT| = 2 + (k-1)|T|$, as can be seen immediately by induction on $|T|$. In view of the identity~\eqref{eq:dimensionformula}, we are interested in the expression
\begin{equation}\label{eq:our_constant}
\begin{split}
C(p) &= q + \sum_{T\in\mathcal{A}_e^*} \frac{\dim \ker T}{|T|} p^{|T|}q^{2+(k-1)|T|} =
q + q^2 \sum_{T\in\mathcal{A}_e^*} \frac{\dim \ker T}{|T|} (pq^{k-1})^{|T|} \\
&= q + q^2 \int_0^{pq^{k-1}} \frac{G(x)}{x}\,dx,
\end{split}
\end{equation}
since it gives the von Neumann dimension of the kernel of the lamplighter operator $\tilde{\opT}$ on $\CG_m\wr\IF_\f$. The summand $q$ takes care of the ``empty'' animal, i.e., the possibility that the vertex $x$ is not actually in $\CX(\omega)$ (which happens with probability $q$).
In order to compute this integral, we determine a parametrisation of $G$; since
$G$ is a rational function of $x$, $A$ and $B$, we first find such a parametrisation
for the functions $A$ and $B$.
This is possible because the implicit equation~\eqref{eq:ABequations}
for $B$ defines an algebraic curve of genus zero.
Recall that $A = A(1,x)$ and $B = B(1,x)$ satisfy the equations
\begin{align*}
A &= xB^k, \\
B &= 1+x((A+B)^k-B^k).
\end{align*}
It turns out that the following parametrisation satisfies these two equations:
\begin{equation}
\label{eq:parametrisation}
\begin{aligned}
x &= (t-1)t^{k-1}(1+t^{k-1}-t^k)^{k-1}, \\
A &= \frac{t-1}{t(1+t^{k-1}-t^k)}, \\
B &= \frac{1}{t(1+t^{k-1}-t^k)}.
\end{aligned}
\end{equation}
This parametrisation was essentially obtained by ``guessing'', i.e., finding
the parametrisation in special cases, which was done with an algorithm
by M.~van~Hoeij~\cite {vanHoeij:1994:algorithm} in the
\verb|algcurves| package of the computer algebra system
\verb|Maple|${}^\mathrm{TM}$~\cite{Maple10},
and extrapolating to the general case. Once the parametrisation has
been found, however, it is easy to verify it directly.
The two equations determine the coefficients of the expansions of $A$ and $B$ at $x = 0$ uniquely, hence they define unique functions $A$ and $B$ that are analytic at $0$. The above parametrisation provides such an analytic solution in which $t = 1$ corresponds to $x = 0$. Furthermore, the interval $[1,t_0]$, where $t_0$ is the solution of $t^k-t^{k-1} = \frac{1}{k}$, maps to the interval $[0,x_0]$ with
$$x_0 = \frac1k \left(1 - \frac1k \right)^{k-1},$$
and the parametrisation is monotone on this interval. At $t = t_0$, it has a singularity (of square root type), which corresponds to the fact that $p = \frac1k$ is the critical percolation parameter and that $x_0$ is the radius of convergence and the smallest singularity of $A$ and $B$ (and thus in turn $G$). Therefore, the computation of~\eqref{eq:our_constant} amounts to integrating a rational function between $0$ and the unique solution $\tau(p)$ of
$$(t-1)t^{k-1}(1+t^{k-1}-t^k)^{k-1} = pq^{k-1}$$
inside the interval $[1,t_0]$. To show existence and uniqueness of $\tau(p)$, note again that the function $x(1-x)^{k-1}$ is strictly increasing on $[0,\frac{1}{k}]$ and thus maps this interval bijectively to $[0,x_0]$. Moreover, it follows that $\tau(p)$ is the unique positive solution of
$$\tau(p)^k-\tau(p)^{k-1} = p.$$
Plugging the parametrisations of $A$ and $B$ into~\eqref{eq:G_in_terms_of_B} yields
$$G(x) =\frac{(t-1)(t^{2k}(1-t)+t^{k-1}((2k+1)t^2-(4k+2)t+2k)+2)}{t^2(1+t^{k-1}-t^k)^2(1+kt^{k-1}-kt^k)}.$$
Together with
$$\frac{dx}{x} = \frac{(kt-k+1)(1+kt^{k-1}-kt^k)}{t(t-1)(1+t^{k-1}-t^k)}\,dt,$$
we finally end up with an integral which has a surprisingly simple antiderivative for arbitrary $k$. This antiderivative was also found by means of computer algebra, but can of course be checked directly to be an antiderivative:
\begin{align*}
C(p) &= q + q^2 \int_1^{\tau(p)} \frac{(kt-k+1)(t^{2k}(1-t)+t^{k-1}((2k+1)t^2-(4k+2)t+2k)+2)}{t^3(1+t^{k-1}-t^k)^3}\,dt \\
&= q + q^2 \frac{(t-1)(1-k+(k+1)t-t^{k+1})}{t^2(1+t^{k-1}-t^k)^2} \Big|_{t = \tau(p)} \\
&= q - p + \frac{(\tau(p)-1)(1-k+(k+1)\tau(p))}{\tau(p)^2},
\end{align*}
which shows that the constant $C(p)$ is always algebraic, since $p = \frac{1}{m}$ is rational in our context and $\tau(p)$ is a solution to an algebraic equation. In particular, for $k = 1$, one has $\tau(p) = 1 + p$, which yields
$$C(p) = 3 - 2p - \frac{2}{1+p}.$$
In general, however, $C(p)$ is not rational: take, for instance, $k = 3$ and $p = \frac14$, to obtain
$$C(p) = -\frac56 - \frac{400}{3(766+258\sqrt{129})^{1/3}} + \frac{2(766+258\sqrt{129})^{1/3}}{3} \approx 0.850971.$$
One can even easily prove the following:
\begin{proposition}
If $p = \frac1m$ for $m > k \geq 3$, then $C(p)$ is an irrational algebraic number.
\end{proposition}
\begin{proof}
Suppose that $C(p)$ is rational. Then
$$\frac{(\tau(p)-1)(1-k+(k+1)\tau(p))}{\tau(p)^2} = \frac{a}{b}$$
for some coprime integers $a,b$ with $b > 0$. Hence $\tau = \tau(p)$ is a root of the polynomial
$$P(t) = (b(k+1)-a)t^2-2bkt+b(k-1).$$
Divide by $g = \gcd(b(k+1)-a,-2bk,b(k-1))$ to obtain a primitive polynomial $\tilde{P}(t)$ (in the ring-theoretic sense, i.e., a polynomial whose coefficients have greatest common divisor $1$). It is easy to see that $g \leq 2$. Now note that $\tau$ is also a zero of
$$Q(t) = mt^k-mt^{k-1}-1.$$
If $\tau$ was rational, it would have to be of the form $\pm \frac{1}{r}$ (since the denominator has to divide the leading coefficient, while the numerator has to divide the constant coefficient), contradicting the fact that $\tau(p) > 1$. Hence $\tilde{P}$ is the minimal polynomial of $\tau$, and $Q(t)$ must be divisible by $\tilde{P}(t)$ in $\IZ[t]$ (by Gauss' lemma), which implies that the constant coefficient of $\tilde{P}$ must be $\pm 1$. But this is only possible if $b(k-1) = g = 2$, i.e., $k = 3$, $b = 1$, and $a$ must be even. But then
$$C(p) = q-p + \frac{a}{b} \geq q-p+2 = 1+2q > 1,$$
and we reach a contradiction.
\end{proof}
\section{A free product}\label{sec:freeproduct}
The method of the preceding sections is generally not applicable if the Cayley graph is not a tree; however, \eqref{eq:kerdim} remains true if all cycles of $T$ have length $\equiv 2 \mod 4$ (see for instance \cite[Theorem 2]{Borovicanin:2009:nullity}). Hence it is possible to apply the same techique if the free group $\IF_\f{}$ is replaced by special free products such as $\IZ_6 \ast \IZ_6$; the Cayley graph of this group has hexagons as its only cycles and therefore satisfies the aforementioned condition. Once again, one can distinguish between (rooted) animals with the property that there exists a maximum matching that does not cover the root (type A) and (rooted) animals for which this is not the case (type B) and derive recursions. In addition, one needs to take the size of the boundary of an animal into account, which is no longer uniquely determined by the size of an animal. Hence we consider the trivariate generating functions
$$A = A(u,x,y) = \sum_{\clA \text{ of type $A$}} u^{\nu(\clA)}
x^{|\clA|}y^{|d\clA|} \quad \text{and} \quad B = B(u,x,y) = \sum_{\clA \text{ of type $B$}} u^{\nu(\clA)} x^{|\clA|}y^{|d\clA|},$$
for which one obtains, after some lengthy calculations, functional equations in analogy to those in~\eqref{eq:ABequations} as well as an integral representation analogous to~\ref{eq:our_constant} for the von Neumann dimension of the kernel of the lamplighter operator $\tilde{\opT}$ on $\CG_m\wr(\IZ_6 \ast \IZ_6)$.
In order to determine the resulting integral, one can use the Risch-Trager algorithm, as implemented for example
in the computer algebra system
\texttt{FriCAS}, a fork of ~\cite{axiom},
Once again, we found that there exists an algebraic antiderivative, so that we obtain an algebraic von Neumann dimension for any $m \geq 3$ (the critical
percolation parameter is $p = 0.339303$ in this case, which is a zero of the polynomial $3p^5-2p^4-2p^3-2p^2-2p+1$). It is likely that it is also irrational for all $m \geq 3$, although we do not have a proof for this conjecture. Moreover, we conjecture that in fact the kernel dimension of the adjacency operator of an arbitrary free product of
cyclic groups $\IZ_{4k+2}$ is algebraic (and probably irrational), but the computations outlined above quickly become intractable by
the present method if more complicated examples are studied.
|
1,314,259,993,567 | arxiv | \section{Introduction}
\label{Sec1:Intro}
We consider optimal control problems with time-periodic pa\-ra\-bolic state equations.
Problems of this type often arise in different practical applications, e.g.,
in electromagnetics, chemistry, biology, or heat transfer, see also
\cite{LRW:Altmann:2013, LRW:AltmannStingelinTroeltzsch:2014,LRW:HouskaLogistImpeDiehl:2009}.
Moreover, optimal control problems are the subject matter of lots of different works, see, e.g.,
\cite{LRW:NeittaanmaekiSprekelsTiba:2006, LRW:HinzePinnauUlbrichUlbrich:2009,
LRW:Troeltzsch:2010, LRW:BorziSchulz:2012}
and the references therein.
The multiharmonic finite element method (MhFEM) is well adapted to the
class of parabolic time-periodic problems.
Within the framework of this method, the state functions are expanded
into Fourier series in time with coefficients depending on the spatial variables. In numerical computations,
these series are truncated and the Fourier coefficients are approximated by the
finite element method (FEM).
This scheme leads to the MhFEM (also called harmonic-balanced FEM), which
was successfully used for the simulation of electromagnetic devices
described by nonlinear eddy current problems with harmonic excitations,
see, e.g.,
\cite{PhD:YamadaBessho:1988,
LRW:BachingerKaltenbacherReitzinger:2002a,
PhD:BachingerLangerSchoeberl:2005,
PhD:BachingerLangerSchoeberl:2006,
PhD:CopelandLanger:2010} and the references therein.
Later, the MhFEM has been applied to linear time-periodic parabolic boundary value
and optimal control problems \cite{PhD:KollmannKolmbauer:2011,
LRW:KollmannKolmbauerLangerWolfmayrZulehner:2013, PhD:KrendlSimonciniZulehner:2012,
LRW:LangerWolfmayr:2013, LRW:Wolfmayr:2014}
and to linear time-periodic eddy current problems
and the corresponding optimal control problems
\cite{PhD:Kolmbauer:2012c, PhD:KolmbauerLanger:2012, PhD:KolmbauerLanger:2013}.
In the MhFEM setting, we are able to establish inf-sup and sup-sup conditions, which
provide existence and uniqueness of the solution to parabolic time-periodic problems.
For linear time-periodic parabolic problems, MhFEM
is a natural and very efficient numerical technology
based on the decoupling of computations related to different modes.
This paper is aimed to make a step towards the creation of fully
reliable error estimation methods for distributed time-periodic parabolic
optimal control problems.
We consider the multiharmonic finite element (MhFE) approximations of the
reduced optimality system,
and derive
guaranteed and fully computable bounds for the discretization errors.
For this purpose, we use the functional {\it a posteriori} error estimation
techniques earlier introduced by S.~Repin, see, e.g.,
the papers on parabolic problems
\cite{PhD:Repin:2002,
PhD:GaevskayaRepin:2005}
as well as on optimal control problems
\cite{PhD:GaevskayaHoppeRepin:2006,
PhD:GaevskayaHoppeRepin:2007},
the books
\cite{PhD:Repin:2008,
PhD:MaliNeittaanmaekiRepin:2014}
and the references therein.
In particular, our functional {\it a posteriori} error analysis
uses the techniques close to those suggested
in \cite{PhD:Repin:2002},
but the analysis contains essential changes due to the MhFEM setting.
In \cite{LRW:LangerRepinWolfmayr:2015}, the authors already derived
functional {\it a posteriori} error estimates for MhFE approximations
to parabolic time-periodic boundary value problems.
Similar results are now obtained for the MhFE approximations
to the state and co-state, which are the unique solutions of the
reduced optimality system.
It is worth mentioning
that these {\it a posteriori} error estimates for the state and co-state
immediately yield the corresponding {\it a posteriori} error estimates for the control.
In addition to these results, we deduce
fully computable estimates of the cost functional.
In fact, we generate a new formulation of the optimal control
problem, in which (unlike the original statement)
the state equations are accounted in terms of penalties.
It is proved that the modified cost functional attains its infimum at
the optimal control and the respective state function.
Therefore, in principle, we can use it as
an object of direct minimization, which value on each step provides a guaranteed
upper bound of the cost functional.
The paper is organized as follows:
In Section~\ref{Sec2:ParTimePerOCP}, we discuss
the time-periodic parabolic optimal control problem and the corresponding
optimality system.
The multiharmonic finite element discretization of this space-time weak formulation
is considered in Section~\ref{Sec3:MhFEApprox}.
Section~\ref{Sec4:FunctionalAPostErrorEstimates:OptSys} is devoted to
the derivation of functional {\it a posteriori} error
estimates for the optimality system associated with the multiharmonic setting.
In Section~\ref{Sec5:APosterioriErrorEstimation:OCP:CostFuncs},
we present new results related to
guaranteed and computable bounds of the cost functional.
In the final Section~\ref{Sec6:NumericalResults}, we discuss some implementation
issues and present the first numerical results.
\section{A Time-Periodic Parabolic Optimal Control Problem}
\label{Sec2:ParTimePerOCP}
Let $Q_T := \Omega \times (0,T)$ denote the space-time cylinder
and $\Sigma_T := \Gamma \times (0,T)$ its mantle boundary,
where the spatial domain $\Omega \subset \mathbb{R}^d$, $d=\{1,2,3\}$, is assumed to be
a bounded Lipschitz domain with boundary $\Gamma := \partial \Omega$,
and $(0,T)$ is a given time interval.
Let $\lambda>0$ be the regularization or cost parameter.
We consider the following parabolic time-periodic optimal control problem:
\begin{align}
\label{equation:minfunc:OCP}
\min_{y,u} \mathcal{J}(y,u)
:= \frac{1}{2} \int_0^T \int_{\Omega}
\left(y(\boldsymbol{x}, t) - y_d(\boldsymbol{x},t) \right)^2 d\boldsymbol{x}\,dt
+ \frac{\lambda}{2} \int_0^T \int_{\Omega} \left( u(\boldsymbol{x},t) \right)^2 d\boldsymbol{x}\,dt
\end{align}
subject to the parabolic time-periodic boundary value problem
\begin{align}
\left.
\label{equation:forwardpde:OCP}
\begin{aligned}
\sigma(\boldsymbol{x}) \, \partial_t y(\boldsymbol{x},t)
- \text{div} \, ( \nu(\boldsymbol{x}) \nabla y(\boldsymbol{x},t))
&= u(\boldsymbol{x},t) \hspace{1cm} &(\boldsymbol{x},t) \in Q_T, \\
y(\boldsymbol{x},t) &= 0 \hspace{1cm} &(\boldsymbol{x},t) \in \Sigma_T, \\
y(\boldsymbol{x},0) &= y(\boldsymbol{x},T) \hspace{1cm} &\boldsymbol{x} \in \overline{\Omega}.
\end{aligned}
\quad
\right \rbrace
\end{align}
Here, $y$ and $u$ are the state and control functions, respectively.
The coefficients $\sigma(\cdot)$ and $\nu(\cdot)$
are uniformly bounded and satisfy the conditions
\begin{align}
\label{assumptions:sigmaNu:sigmaStrictlyPositive}
0 < \underline{\sigma} \leq \sigma(\boldsymbol{x}) \leq \overline{\sigma}, \qquad
\text{and} \qquad
0 < \underline{\nu} \leq \nu(\boldsymbol{x}) \leq \overline{\nu}, \qquad
\boldsymbol{x} \in \Omega,
\end{align}
where $\underline{\sigma}$, $\overline{\sigma}$, $\underline{\nu}$ and, $\overline{\nu}$ are constants.
As usual, the cost functional (\ref{equation:minfunc:OCP}) contains
a penalty term weighted with a positive factor $\lambda$. This term restricts
(in the integral sense) values of the control function $u$.
We can reformulate the problem
(\ref{equation:minfunc:OCP})-(\ref{equation:forwardpde:OCP})
in an equivalent form.
For this purpose, we introduce the Lagrangian
\begin{align}
\label{equation:LagrangeFunctional}
\mathcal{L}(y,u,p):= \mathcal{J}(y,u) - \int_0^T \int_{\Omega} \big(\sigma \partial_t y
- \text{div} \, (\nu \nabla y) - u\big) p \, d\boldsymbol{x}\,dt,
\end{align}
which has a saddle point, see, e.g., \cite{LRW:HinzePinnauUlbrichUlbrich:2009, LRW:Troeltzsch:2010}
and the references therein.
The proper sets for which this saddle point problem is correctly stated are defined later.
Since the saddle point exists,
the corresponding solutions satisfy the system of necessary conditions
\begin{align}
\label{equation:optsysParabolicTimePeriodicOCP}
\begin{aligned}
\mathcal{L}_p (y,u,p) = 0, \qquad \qquad
\mathcal{L}_y (y,u,p) = 0, \qquad \qquad
\mathcal{L}_u (y,u,p) = 0.
\end{aligned}
\end{align}
Using the second condition, we can eliminate the control $u$ from the optimality system
(\ref{equation:optsysParabolicTimePeriodicOCP}), i.e.,
\begin{align}
\label{equation:eliminateControl}
u= - \lambda^{-1} p\; \;\mbox{ in}\;\; Q_T.
\end{align}
From (\ref{equation:eliminateControl}) it appears very natural to choose
$u$ and $p$ from the same space.
Moreover, we arrive at a reduced optimality system, written in its classical
formulation as
\begin{align}
\label{equation:KKTSysClassical}
\left.
\begin{aligned}
\sigma(\boldsymbol{x}) \, \partial_t y(\boldsymbol{x},t) -
\text{div} \, (\nu(\boldsymbol{x}) \nabla y(\boldsymbol{x},t))
&= - \lambda^{-1} p(\boldsymbol{x},t)
\quad &(\boldsymbol{x},t) \in Q_T, \\
y(\boldsymbol{x},t) &= 0
\quad &(\boldsymbol{x},t) \in \Sigma_T, \\
y(\boldsymbol{x},0) &= y(\boldsymbol{x},T)
\quad &\boldsymbol{x} \in \overline{\Omega},\\
- \sigma(\boldsymbol{x}) \, \partial_t p(\boldsymbol{x},t)
- \text{div} \, (\nu(\boldsymbol{x}) \nabla p(\boldsymbol{x},t))
&= y(\boldsymbol{x},t) - y_d(\boldsymbol{x},t)
\quad &(\boldsymbol{x},t) \in Q_T ,\\
p(\boldsymbol{x},t) &= 0
\quad &(\boldsymbol{x},t) \in \Sigma_T ,\\
p(\boldsymbol{x},T) &= p(\boldsymbol{x},0)
\quad &\boldsymbol{x} \in \overline{\Omega}.
\end{aligned}
\quad
\right \rbrace
\end{align}
In order to determine generalized solutions
of (\ref{equation:KKTSysClassical}),
we define the following
spaces (here and later on, the notation is similar to that was used in
\cite{LRW:Ladyzhenskaya:1973, LRW:LadyzhenskayaSolonnikovUralceva:1968}):
\begin{align*}
H^{1,0}(Q_T) &= \{v \in L^2(Q_T) : \nabla v \in [L^2(Q_T)]^d \}, \\
H^{0,1}(Q_T) &= \{v \in L^2(Q_T) : \partial_t v \in L^2(Q_T) \}, \\
H^{1,1}(Q_T) &= \{v \in L^2(Q_T) : \nabla v \in [L^2(Q_T)]^d, \partial_t v \in L^2(Q_T) \},
\end{align*}
which are endowed with the norms
\begin{align*}
\|v\|_{1,0} &:= \left(\int_{Q_T}
\left(v(\boldsymbol{x},t)^2 + |\nabla v(\boldsymbol{x},t)|^2 \right)
\, d\boldsymbol{x} \, dt \right)^{1/2}, \\
\|v\|_{0,1} &:= \left(\int_{Q_T} \left(v(\boldsymbol{x},t)^2 + |\partial_t v(\boldsymbol{x},t)|^2 \right) \, d\boldsymbol{x} \, dt \right)^{1/2}, \\
\|v\|_{1,1} &:= \left(\int_{Q_T} \left(v(\boldsymbol{x},t)^2 + |\nabla v(\boldsymbol{x},t)|^2
+ |\partial_t v(\boldsymbol{x},t)|^2 \right) \, d\boldsymbol{x} \, dt \right)^{1/2},
\end{align*}
respectively. Here,
$\nabla = \nabla_{\boldsymbol{x}}$ is the spatial gradient and $\partial_t$ denotes the generalized
derivative with respect to time.
Also we define subspaces of the above introduced spaces by putting subindex zero if the functions
satisfy the homogeneous Dirichlet condition on $\Sigma_T$ and subindex \textit{per} if they
satisfy the periodicity condition $v(\boldsymbol{x},0) = v(\boldsymbol{x},T)$.
All inner products and norms in $L^2$ related to the whole space-time domain $Q_T$
are denoted by $\langle \cdot, \cdot \rangle$ and $\| \cdot \|$, respectively.
If they are associated with the spatial domain $\Omega$, then we write
$\langle \cdot, \cdot \rangle_{\Omega}$ and $\| \cdot \|_{\Omega}$.
The symbols $\langle \cdot, \cdot \rangle_{1,\Omega}$ and $\| \cdot \|_{1,\Omega}$
denote the standard inner products and norms of the space $H^1(\Omega)$.
The functions used in our analysis are presented in
terms of Fourier series, e.g., for the function $v(\boldsymbol{x},t)$
is representation is
\begin{align}
\label{def:FourierAnsatz}
v(\boldsymbol{x},t) = v_0^c(\boldsymbol{x}) +
\sum_{k=1}^{\infty} \left(v_k^c(\boldsymbol{x}) \cos(k \omega t)
+ v_k^s(\boldsymbol{x}) \sin(k \omega t)\right).
\end{align}
Here,
\begin{align*}
\begin{aligned}
v_0^c(\boldsymbol{x}) &= \frac{1}{T} \int_0^T v(\boldsymbol{x},t) \,dt, \\
v_k^c(\boldsymbol{x}) = \frac{2}{T} \int_0^T v(\boldsymbol{x},t) \cos(k \omega t)\,&dt,
\hspace{0.3cm} \mbox{ and } \hspace{0.3cm}
v_k^s(\boldsymbol{x}) = \frac{2}{T} \int_0^T v(\boldsymbol{x},t) \sin(k \omega t)\,dt
\end{aligned}
\end{align*}
are the Fourier coefficients,
$T$ denotes the periodicity and $\omega = 2 \pi /T$ is the frequency.
Since the problem has time-periodical conditions,
these representations of the exact solution and respective
approximations are quite natural.
In what follows, we also use the spaces
\begin{align*}
H^{0,\frac{1}{2}}_{per}(Q_T) &= \{ v \in L^2(Q_T) : \big\| \partial^{1/2}_t v \big\| < \infty \}, \\
H^{1,\frac{1}{2}}_{per}(Q_T) &= \{ v \in H^{1,0}(Q_T) : \big\| \partial^{1/2}_t v \big\| < \infty \}, \\
H^{1,\frac{1}{2}}_{0,per}(Q_T) &= \{ v \in H^{1,\frac{1}{2}}_{per}(Q_T): v = 0 \mbox{ on } \Sigma_T \},
\end{align*}
where $\big\| \partial^{1/2}_t v \big\|$ is defined
in the Fourier space by the relation
\begin{align}
\label{definition:H01/2seminorm}
\big\| \partial^{1/2}_t v \big\| ^2 :=
|v|_{0,\frac{1}{2}}^2 :=
\frac{T}{2} \sum_{k=1}^{\infty} k \omega \|\boldsymbol{v}_k\|_{\Omega}^2.
\end{align}
Here, $\boldsymbol{v}_k = (v_k^c,v_k^s)^T$ for all $k \in \mathbb{N}$,
see also \cite{LRW:LangerWolfmayr:2013}.
These spaces can be considered as Hilbert spaces if we introduce the following
(equivalent) products:
\begin{align*}
\begin{aligned}
\langle \partial^{1/2}_t y, \partial^{1/2}_t v \rangle :=
\frac{T}{2} \sum_{k=1}^{\infty} k \omega \langle\boldsymbol{y}_k,\boldsymbol{v}_k\rangle_{\Omega},
\qquad \quad
\langle\sigma
\partial^{1/2}_t y, \partial^{1/2}_t v\rangle :=
\frac{T}{2} \sum_{k=1}^{\infty} k \omega \langle\sigma
\boldsymbol{y}_k,\boldsymbol{v}_k\rangle_{\Omega}.
\end{aligned}
\end{align*}
The above introduced spaces allow us to operate with a "symmetrized" formulation of the problem
(\ref{equation:KKTSysClassical}) presented by (\ref{problem:KKTSysSTVFAPost}).
The seminorm and the norm of the space $H^{1,\frac{1}{2}}_{per}(Q_T)$ are defined
by the relations
\begin{align*}
|v|_{1,\frac{1}{2}}^2
&:= \|\nabla v\| ^2 + \|\partial_t^{1/2} v\| ^2
= T \, \|\nabla y_0^c\|_{\Omega}^2
+ \frac{T}{2} \sum_{k=1}^{\infty} \left(k\omega \|\boldsymbol{v}_k\|_{\Omega}^2
+ \|\nabla \boldsymbol{v}_k\|_{\Omega}^2\right)
\qquad \mbox{ and } \\
\|v\|_{1,\frac{1}{2}}^2
&:= \|v\| ^2 + |v|_{1,\frac{1}{2}}^2
= T \, (\|y_0^c\|_{\Omega}^2 + \|\nabla y_0^c\|_{\Omega}^2)
+ \frac{T}{2} \sum_{k=1}^{\infty} \left((1+k\omega) \|\boldsymbol{v}_k\|_{\Omega}^2
+ \|\nabla \boldsymbol{v}_k\|_{\Omega}^2\right),
\end{align*}
respectively.
Using Fourier type series,
it is easy to define the function
"orthogonal" to $v$:
\begin{align*}
\begin{aligned}
v^{\perp}(\boldsymbol{x},t) &:=
\sum_{k=1}^{\infty}
\left(- v_k^c(\boldsymbol{x}) \sin(k \omega t)
+ v_k^s(\boldsymbol{x}) \cos(k \omega t)\right) \\
&= \sum_{k=1}^{\infty}
\underbrace{(v_k^s(\boldsymbol{x}),-v_k^c(\boldsymbol{x}))}_{=:(-\boldsymbol{v}_k^{\perp})^T} \cdot
\left( \begin{array}{l}
\cos(k \omega t) \\
\sin(k \omega t)
\end{array} \right).
\end{aligned}
\end{align*}
Obviously, $\|\boldsymbol{u}_k^\perp\|_{\Omega} = \|\boldsymbol{u}_k\|_{\Omega}$ and
we find that
\begin{align*}
\big\|\partial^{1/2}_t v^\perp\big\| ^2
= \frac{T}{2} \sum_{k=1}^\infty k \omega \|\boldsymbol{v}_k^\perp\|_{\Omega}^2
= \frac{T}{2} \sum_{k=1}^\infty k \omega \|\boldsymbol{v}_k\|_{\Omega}^2
= \big\|\partial^{1/2}_t v\big\| ^2 \qquad \forall \, v \in H^{0,\frac{1}{2}}_{per}(Q_T).
\end{align*}
Henceforth, we use the following subsidiary result (which
proof can be found in \cite{LRW:LangerRepinWolfmayr:2015}):
\begin{lemma}
\label{lemma:H11/2IdentiesAndOrthogonalities}
The identities
\begin{align}
\label{equation:H11/2identities}
\begin{aligned}
\langle \sigma \partial_t^{1/2} y,\partial_t^{1/2} v \rangle =
\langle \sigma \partial_t y,v^{\perp} \rangle \quad \mbox{ and } \quad
\langle \sigma \partial_t^{1/2} y,\partial_t^{1/2} v^{\perp} \rangle =
\langle \sigma \partial_t y,v \rangle
\end{aligned}
\end{align}
are valid
for all $y \in H^{0,1}_{per}(Q_T)$ and $v \in H^{0,\frac{1}{2}}_{per}(Q_T)$.
\end{lemma}
Also, we recall the orthogonality relations (see \cite{LRW:LangerWolfmayr:2013, LRW:Wolfmayr:2014})
\begin{align}
\left.
\label{equation:orthorelation}
\begin{aligned}
\langle \sigma \partial_t y,y \rangle = 0 \quad &\mbox{ and } \quad
\langle \sigma y^{\perp},y \rangle = 0 \qquad &&\forall \, y \in H^{0,1}_{per}(Q_T), \\
\langle \sigma \partial^{1/2}_t y,\partial^{1/2}_t y^{\perp} \rangle = 0 \quad &\mbox{ and } \quad
\langle \nu \nabla y, \nabla y^{\perp} \rangle = 0 \qquad &&\forall \, y \in H^{1,\frac{1}{2}}_{per}(Q_T),
\end{aligned}
\quad
\right \rbrace
\end{align}
and the identity
\begin{align}
\label{def:identityH01/2per}
\int_0^T \xi \, \partial_t^{1/2} v^\perp \, dt = - \int_0^T \partial_t^{1/2} \xi^\perp \, v \, dt
\qquad \forall \, \xi, v \in H^{0,\frac{1}{2}}_{per}(Q_T),
\end{align}
where
\begin{align}
\langle \xi,\partial_t^{1/2} v \rangle
:= \frac{T}{2} \sum_{k=1}^{\infty} (k \omega)^{1/2} \langle \boldsymbol{\xi}_k,\boldsymbol{v}_k \rangle_{\Omega}.
\end{align}
We note that for functions presented in terms of Fourier series the standard
Friedrichs inequality holds in $Q_T$. Indeed,
\begin{align}
\label{inequality:Friedrichs:FourierSpace}
\begin{aligned}
\|\nabla u\| ^2 &= \int_{Q_T} |\nabla u|^2 \, d\boldsymbol{x}\,dt
= T \, \|\nabla u_0^c\|_{\Omega}^2 + \frac{T}{2} \sum_{k=1}^\infty \|\nabla \boldsymbol{u}_k\|_{\Omega}^2 \\
&\geq \frac{1}{C_F^2} \left(T \, \|u_0^c\|_{\Omega}^2 + \frac{T}{2} \sum_{k=1}^\infty \|\boldsymbol{u}_k\|_{\Omega}^2 \right)
= \frac{1}{C_F^2} \|u\| ^2.
\end{aligned}
\end{align}
In order to derive the weak formulations of the equations
(\ref{equation:KKTSysClassical}), we multiply them
by the test functions $z, q \in H^{1,\frac{1}{2}}_{0,per}(Q_T)$,
integrate over $Q_T$, and apply
integration by parts with respect to the spatial variables
and time.
We arrive at
the following ``symmetric'' space-time weak
formulations of the reduced
optimality system (\ref{equation:KKTSysClassical}):
Given the desired state $y_d \in L^2(Q_T)$, find $y$ and $p$ from
$H^{1,\frac{1}{2}}_{0,per}(Q_T)$ such that
\begin{align}
\label{problem:KKTSysSTVFAPost}
\left.
\begin{aligned}
&\int_{Q_T} \Big( y\,z - \nu(\boldsymbol{x}) \nabla p \cdot \nabla z
+ \sigma(\boldsymbol{x}) \partial_t^{1/2} p \, \partial_t^{1/2} z^{\perp} \Big)\,d\boldsymbol{x}\,dt
= \int_{Q_T} y_d\,z\,d\boldsymbol{x}\,dt, \\
&\int_{Q_T} \Big( \nu(\boldsymbol{x}) \nabla y \cdot \nabla q
+ \sigma(\boldsymbol{x}) \partial_t^{1/2} y \, \partial_t^{1/2} q^{\perp}
+ \lambda^{-1} p\,q \Big)\,d\boldsymbol{x}\,dt = 0,
\end{aligned}
\quad
\right \rbrace
\end{align}
for all test functions $z, q \in H^{1,\frac{1}{2}}_{0,per}(Q_T)$.
We can represent (\ref{problem:KKTSysSTVFAPost}) in a somewhat different form.
For this purpose, it is convenient to introduce the bilinear form
\begin{align}
\label{definition:KKTSysSTVF}
\begin{aligned}
\mathcal{B}((y,p),(z,q)) = \int_{Q_T} &\Big( y\,z - \nu(\boldsymbol{x}) \nabla p \cdot \nabla z
+ \sigma(\boldsymbol{x}) \partial_t^{1/2} p \, \partial_t^{1/2} z^{\perp} \\
&+ \nu(\boldsymbol{x}) \nabla y \cdot \nabla q
+ \sigma(\boldsymbol{x}) \partial_t^{1/2} y \, \partial_t^{1/2} q^{\perp}
+ \lambda^{-1} p\,q \Big)\,d\boldsymbol{x}\,dt.
\end{aligned}
\end{align}
Then (\ref{problem:KKTSysSTVFAPost}) reads as
\begin{align*}
\mathcal{B}((y,p),(z,q)) = \langle (y_d,0),(z,q)\rangle \qquad \forall \, (z, q) \in (H^{1,\frac{1}{2}}_{0,per}(Q_T))^2.
\end{align*}
\section{Multiharmonic Finite Element Approximation}
\label{Sec3:MhFEApprox}
In order to solve the
optimal control problem (\ref{equation:minfunc:OCP})-(\ref{equation:forwardpde:OCP}) approximately,
we discretize the optimality system (\ref{problem:KKTSysSTVFAPost}) by
the MhFEM (see \cite{LRW:LangerWolfmayr:2013}).
Using the Fourier series ansatz (\ref{def:FourierAnsatz}) in
(\ref{problem:KKTSysSTVFAPost}) and exploiting
the orthogonality of $\cos(k \omega t)$ and $\sin(k \omega t)$,
we arrive at the following problem:
Find $\boldsymbol{y}_k, \boldsymbol{p}_k \in \mathbb{V} := V \times V = (H^1_0(\Omega))^2$ such that
\begin{align}
\label{equation:MultiAnsVFBlockk}
\left.
\begin{aligned}
&\int_{\Omega} \big( \boldsymbol{y}_k \cdot \boldsymbol{z}_k
- \nu(\boldsymbol{x}) \nabla \boldsymbol{p}_k \cdot \nabla \boldsymbol{z}_k
+ k \omega \, \sigma(\boldsymbol{x}) \boldsymbol{p}_k \cdot \boldsymbol{z}_k^{\perp} \big)\,d\boldsymbol{x}
= \int_{\Omega} \boldsymbol{y_d}_k \cdot \boldsymbol{z}_k\,d\boldsymbol{x}, \\
&\int_{\Omega} \big( \nu(\boldsymbol{x}) \nabla \boldsymbol{y}_k \cdot \nabla \boldsymbol{q}_k
+ k \omega \, \sigma(\boldsymbol{x}) \boldsymbol{y}_k \cdot \boldsymbol{q}_k^{\perp}
+ \lambda^{-1} \boldsymbol{p}_k \cdot \boldsymbol{q}_k \big)\,d\boldsymbol{x} = 0,
\end{aligned}
\quad
\right \rbrace
\end{align}
for all test functions $\boldsymbol{z}_k, \boldsymbol{q}_k \in \mathbb{V}$.
The system (\ref{equation:MultiAnsVFBlockk}) must be solved for every
mode $k \in \mathbb{N}$.
For $k = 0$, we obtain a reduced problem:
Find $y_0^c, p_0^c \in V$ such that
\begin{align}
\label{equation:MultiAnsVFBlock0}
\left.
\begin{aligned}
&\int_{\Omega} \big( y_0^c \cdot z_0^c - \nu(\boldsymbol{x}) \nabla p_0^c \cdot \nabla z_0^c \big)\,d\boldsymbol{x}
= \int_{\Omega} {y_d^c}_0 \cdot z_0^c\,d\boldsymbol{x}, \\
&\int_{\Omega} \big( \nu(\boldsymbol{x}) \nabla y_0^c \cdot \nabla q_0^c
+ \lambda^{-1} p_0^c \cdot q_0^c \big)\,d\boldsymbol{x} = 0,
\end{aligned}
\quad
\right \rbrace
\end{align}
for all test functions $z_0^c, q_0^c \in V$.
The problems (\ref{equation:MultiAnsVFBlockk}) and (\ref{equation:MultiAnsVFBlock0})
have unique solutions.
In order to solve these problems numerically, the Fourier series are truncated at a finite index $N$ and
the unknown Fourier coefficients
$ \boldsymbol{y}_k = (y_k^c, y_k^s)^T, \, \boldsymbol{p}_k = (p_k^c, p_k^s)^T \in \mathbb{V} $
are approximated by finite element (FE) functions
\begin{align*}
\boldsymbol{y}_{kh} = (y_{kh}^c, y_{kh}^s)^T, \, \boldsymbol{p}_{kh} = (p_{kh}^c, p_{kh}^s)^T
\in \mathbb{V}_h = V_h \times V_h \subset \mathbb{V},
\end{align*}
where $ V_h = \mbox{span} \{\varphi_1, \dots, \varphi_n\} $
with $\{\varphi_i(\boldsymbol{x}): i=1,2,\dots,n_h \}$
is a conforming FE space.
We denote by $h$ the usual discretization parameter such that $n = n_h = \mbox{dim} V_h = O(h^{-d})$.
In this work, we
use continuous, piecewise linear finite elements on
a regular triangulation $\mathcal{T}_h$ to construct
$V_h$ and its basis (see, e.g.,
\cite{LRW:Braess:2005, LRW:Ciarlet:1978, LRW:JungLanger:2013, LRW:Steinbach:2008}).
This leads to the following saddle point system
for every single mode $k=1,2,\dots,N$:
\begin{align}
\label{equation:MultiFESysBlockk}
\left( \begin{array}{cccc}
M_h & 0 & -K_{h,\nu} & k \omega M_{h,\sigma} \\
0 & M_h & -k \omega M_{h,\sigma} & -K_{h,\nu} \\
-K_{h,\nu} & -k \omega M_{h,\sigma} & -\lambda^{-1} M_h & 0 \\
k \omega M_{h,\sigma} & -K_{h,\nu} & 0 & -\lambda^{-1} M_h \end{array} \right) \left( \begin{array}{c}
\underline{y}_k^c \\
\underline{y}_k^s \\
\underline{p}_k^c \\
\underline{p}_k^s \end{array} \right) = \left( \begin{array}{c}
{\underline{y}_d^c}_k \\
{\underline{y}_d^s}_k \\
0 \\
0 \end{array} \right),
\end{align}
which has to be solved with respect to the nodal parameter vectors
\begin{align*}
\underline{y}_k^c = ( y_{k,i}^c)_{i=1,\dots,n}, \,
\underline{y}_k^s = ( y_{k,i}^s)_{i=1,\dots,n}, \,
\underline{p}_k^c = ( p_{k,i}^c)_{i=1,\dots,n}, \,
\underline{p}_k^s = ( p_{k,i}^s)_{i=1,\dots,n} \in \mathbb{R}^n
\end{align*}
of the FE approximations
$y _{kh}^c(\boldsymbol{x}) = \sum_{i=1}^n y_{k,i}^c \varphi_i(\boldsymbol{x})$
and
$y _{kh}^s(\boldsymbol{x}) = \sum_{i=1}^n y_{k,i}^s \varphi_i(\boldsymbol{x})$.
Similarly,
$p _{kh}^c(\boldsymbol{x}) = \sum_{i=1}^n p_{k,i}^c \varphi_i(\boldsymbol{x})$
and
$p _{kh}^s(\boldsymbol{x}) = \sum_{i=1}^n p_{k,i}^s \varphi_i(\boldsymbol{x})$.
The matrices $M_h$, $M_{h,\sigma}$, and $K_{h,\nu}$ denote the mass matrix,
the weighted mass matrix and the stiffness matrix, respectively. Their entries are
defined by the integrals
\begin{align*}
\begin{aligned}
M_h^{ij} = \int_{\Omega} \varphi_i \varphi_j \,d\boldsymbol{x}, \hspace{0.5cm}
M_{h,\sigma}^{ij} &= \int_{\Omega} \sigma \, \varphi_i \varphi_j \,d\boldsymbol{x}, \hspace{0.5cm}
K_{h,\nu}^{ij} &= \int_{\Omega} \nu \, \nabla \varphi_i \cdot \nabla \varphi_j \,d\boldsymbol{x}
\end{aligned}
\end{align*}
and the right hand side vectors have the form
\begin{align*}
\begin{aligned}
{\underline{y}_d^c}_k = \Big\lbrack \int_{\Omega} {y_d^c}_k \varphi_j \,d\boldsymbol{x}
\Big\rbrack_{j=1,\dots,n} \quad \mbox{and} \quad
{\underline{y}_d^s}_k = \Big\lbrack \int_{\Omega} {y_d^s}_k \varphi_j \,d\boldsymbol{x}
\Big\rbrack_{j=1,\dots,n}.
\end{aligned}
\end{align*}
For $k=0$, the problem (\ref{equation:MultiAnsVFBlock0}) generates a reduced
system of linear equations, i.e.,
\begin{align}
\label{equation:MultiFESysBlock0}
\left( \begin{array}{cc}
M_h & -K_{h,\nu} \\
-K_{h,\nu} & - \lambda^{-1} M_h \end{array} \right) \left( \begin{array}{c}
\underline{y}_0^c \\
\underline{p}_0^c \end{array} \right) = \left( \begin{array}{c}
{\underline{y}_d^c}_0 \\
0 \end{array} \right).
\end{align}
Fast and robust solvers for the systems
(\ref{equation:MultiFESysBlockk}) and (\ref{equation:MultiFESysBlock0})
can be found in
\cite{LRW:KollmannKolmbauerLangerWolfmayrZulehner:2013, LRW:KrausWolfmayr:2013,
LRW:LangerWolfmayr:2013, LRW:Wolfmayr:2014},
which we use in order to obtain the MhFE approximations
\begin{align}
\left.
\label{definition:MultiharmonicFEAnsatzStateANDAdjointState}
\begin{aligned}
y_{N h}(\boldsymbol{x},t) &= y_{0h}^c(\boldsymbol{x}) + \sum_{k=1}^N \left(y_{kh}^c(\boldsymbol{x}) \cos(k \omega t)
+ y_{kh}^s(\boldsymbol{x}) \sin(k \omega t)\right), \\
p_{N h}(\boldsymbol{x},t) &= p_{0h}^c(\boldsymbol{x}) + \sum_{k=1}^N \left(p_{kh}^c(\boldsymbol{x}) \cos(k \omega t)
+ p_{kh}^s(\boldsymbol{x}) \sin(k \omega t)\right).
\end{aligned}
\quad
\right \rbrace
\end{align}
\section{Functional A Posteriori Error Estimates for the Optimality System}
\label{Sec4:FunctionalAPostErrorEstimates:OptSys}
Now we are concerned with {\it a posteriori} estimates of the difference
between the exact solution $(y,p)$
and the respective finite element solution $(y_{N h},p_{N h})$.
First, we present the inf-sup and sup-sup conditions
for the bilinear form (\ref{definition:KKTSysSTVF}).
\begin{lemma}
\label{lemma:KKTSysSTVFinfsupsupsup}
For all $y,p \in H^{1,\frac{1}{2}}_{0,per}(Q_T)$,
the space-time bilinear form
$\mathcal{B}(\cdot,\cdot)$ defined by
(\ref{definition:KKTSysSTVF})
meets the following inf-sup and sup-sup conditions:
\begin{align}
\label{inequality:KKTSysSTVFinfsupsupsup}
\mu_{1} \|(y,p)\|_{1,\frac{1}{2}}
\leq
\sup_{0 \not= (z,q) \in (H^{1,\frac{1}{2}}_{0,per}(Q_T))^2}
\frac{\mathcal{B}((y,p),(z,q))}{\|(z,q)\|_{1,\frac{1}{2}}}
\leq
\mu_{2} \|(y,p)\|_{1,\frac{1}{2}},
\end{align}
where
$\mu_1 = \frac{\min\{\frac{1}{\sqrt{\lambda}},\underline{\nu},\underline{\sigma}\}
\min\{\sqrt{\lambda},\frac{1}{\sqrt{\lambda}}\}}{\sqrt{1+2\max\{\lambda,\frac{1}{\lambda}\}}}$
and
$\mu_{2} = \max\{1,\frac{1}{\lambda},\overline{\nu},\overline{\sigma}\}$
are positive constants.
\begin{proof}
Using the triangle and Cauchy-Schwarz inequalities, we obtain
\begin{align*}
\big|\mathcal{B}((y,p),(z,q))\big|
= &\, \Big| \int_0^T \int_\Omega \Big( y\,z - \nu(\boldsymbol{x}) \nabla p \cdot \nabla z
+ \sigma(\boldsymbol{x}) \partial_t^{1/2} p \, \partial_t^{1/2} z^{\perp} \\
&\qquad \qquad \, + \nu(\boldsymbol{x}) \nabla y \cdot \nabla q
+ \sigma(\boldsymbol{x}) \partial_t^{1/2} y \, \partial_t^{1/2} q^{\perp}
+ \lambda^{-1} p\,q \Big)\,d\boldsymbol{x}\,dt \Big| \\
\leq &\, \|y\| \|z\|
+ \overline{\nu} \, \|\nabla p\|\|\nabla z\|
+ \overline{\sigma} \, \big\|\partial^{1/2}_t p\big\| \big\|\partial^{1/2}_t z\big\| \\
&+ \overline{\nu} \, \|\nabla y\| \|\nabla q\|
+ \overline{\sigma} \, \big\|\partial^{1/2}_t y\big\| \big\|\partial^{1/2}_t q\big\|
+ \lambda^{-1} \, \|p\| \|q\| \\
\leq
&\, \mu_2 \|(y,p)\|_{1,\frac{1}{2}} \|(z,q)\|_{1,\frac{1}{2}},
\end{align*}
where
$\mu_2 := \max\{1,\frac{1}{\lambda},\overline{\nu},\overline{\sigma}\}$.
Thus, the right hand-side inequality in (\ref{inequality:KKTSysSTVFinfsupsupsup})
is proved.
In order to prove the left hand-side inequality, we select the test functions
\begin{align*}
(z,q) = (y - \frac{1}{\sqrt{\lambda}} p - \frac{1}{\sqrt{\lambda}} p^\perp,
p + \sqrt{\lambda} y - \sqrt{\lambda} y^\perp).
\end{align*}
Using the $\sigma$- and $\nu$-weighted orthogonality relations (\ref{equation:orthorelation}),
we obtain the relations
\begin{align*}
\mathcal{B}((y,p),(y,p))
=&\, \|y\|^2 + \lambda^{-1} \|p\|^2, \\
\mathcal{B}((y,p),(-\frac{1}{\sqrt{\lambda}} p, \sqrt{\lambda} y))
= &\, \frac{1}{\sqrt{\lambda}} \langle \nu \nabla p,\nabla p \rangle
+ \sqrt{\lambda} \langle \nu \nabla y,\nabla y \rangle, \\
\mathcal{B}((y,p),(-\frac{1}{\sqrt{\lambda}} p^\perp,-\sqrt{\lambda} y^\perp))
= &\, \frac{1}{\sqrt{\lambda}} \langle \sigma \partial_t^{1/2} p, \partial_t^{1/2} p \rangle
+ \sqrt{\lambda} \langle \sigma \partial_t^{1/2} y, \partial_t^{1/2} y \rangle,
\end{align*}
which lead to the estimate
\begin{align*}
\mathcal{B}((y,p)&,(y - \frac{1}{\sqrt{\lambda}} p - \frac{1}{\sqrt{\lambda}} p^\perp,
p + \sqrt{\lambda} y - \sqrt{\lambda} y^\perp)) \\
&\geq
\min\{\frac{1}{\sqrt{\lambda}},\underline{\nu},\underline{\sigma}\}
\min\{\sqrt{\lambda},\frac{1}{\sqrt{\lambda}}\}
\|(y,p)\|_{1,\frac{1}{2}}^2.
\end{align*}
Since
\begin{align*}
\|(z,q)\|_{1,\frac{1}{2}}^2
\leq \left(1+2\max\{\lambda,\frac{1}{\lambda}\}\right) \|(y,p)\|_{1,\frac{1}{2}}^2,
\end{align*}
we arrive at the estimate
\begin{align*}
\sup_{0 \not= (z,q) \in (H^{1,\frac{1}{2}}_{0,per}(Q_T))^2}
\frac{\mathcal{B}((y,p),(z,q))}{\|(z,q)\|_{1,\frac{1}{2}}}
&\geq \frac{\min\{\frac{1}{\sqrt{\lambda}},\underline{\nu},\underline{\sigma}\}
\min\{\sqrt{\lambda},\frac{1}{\sqrt{\lambda}}\} \|(y,p)\|_{1,\frac{1}{2}}^2}{
\sqrt{1+2\max\{\lambda,\frac{1}{\lambda}\}} \|(y,p)\|_{1,\frac{1}{2}}} \\
&= \mu_1 \, \|(y,p)\|_{1,\frac{1}{2}},
\end{align*}
where $\mu_1 = \frac{\min\{\frac{1}{\sqrt{\lambda}},\underline{\nu},\underline{\sigma}\}
\min\{\sqrt{\lambda},\frac{1}{\sqrt{\lambda}}\}}{\sqrt{1+2\max\{\lambda,\frac{1}{\lambda}\}}}$.
\end{proof}
\end{lemma}
In view of the Friedrichs inequality,
the norms $|\cdot|_{1,\frac{1}{2}}$ and
$\|\cdot\|_{1,\frac{1}{2}}$ are equivalent.
Therefore, Lemma \ref{lemma:KKTSysSTVFinfsupsupsup}
implies the following result:
\begin{lemma}
\label{lemma:KKTSysSTVFinfsupsupsup:Seminorm}
For all $y,p \in H^{1,\frac{1}{2}}_{0,per}(Q_T)$, the bilinear form
$\mathcal{B}(\cdot,\cdot)$ satisfies the inf-sup and sup-sup conditions
\begin{align}
\label{inequality:KKTSysSTVFinfsupsupsup:Seminorm}
\tilde \mu_1 |(y,p)|_{1,\frac{1}{2}}
\leq
\sup_{0 \not= (z,q) \in (H^{1,\frac{1}{2}}_{0,per}(Q_T))^2}
\frac{\mathcal{B}((y,p),(z,q))}{|(z,q)|_{1,\frac{1}{2}}}
\leq
\tilde \mu_2 |(y,p)|_{1,\frac{1}{2}},
\end{align}
where
$\tilde \mu_1 = \frac{\min\{\underline{\nu},\underline{\sigma}\} \min\{\lambda,\frac{1}{\lambda}\}}{\sqrt{2}} > 0$
and
$\tilde \mu_2 = \max\{1,\frac{1}{\lambda},\overline{\nu},\overline{\sigma}\} \max\{1,C_F^2+1\} > 0$, and
$C_F$ is the Friedrichs constant.
\begin{proof}
The right hand-side inequality in (\ref{inequality:KKTSysSTVFinfsupsupsup:Seminorm})
results from the triangle and Cauchy-Schwarz inequalities and, the
Friedrichs inequality (\ref{inequality:Friedrichs:FourierSpace}). Indeed,
\begin{align*}
\big|\mathcal{B}((y,p),(z,q))\big|
\leq \max\{1,\frac{1}{\lambda},\overline{\nu},&\,\overline{\sigma}\}
\Big((C_F^2+1) \|\nabla y\|^2 + \big\|\partial^{1/2}_t y\big\|^2 \\
&\qquad + (C_F^2+1) \|\nabla p\|^2 + \big\|\partial^{1/2}_t p\big\|^2\Big)^{1/2} \\
&\times \Big((C_F^2+1) \|\nabla z\|^2
+ \big\|\partial^{1/2}_t z\big\|^2 \\
&\qquad + (C_F^2+1) \|\nabla q\|^2 + \big\|\partial^{1/2}_t q\big\|^2 \Big)^{1/2} \\
\leq
\tilde \mu_2 |(y,p)|_{1,\frac{1}{2}} |&(z,q)|_{1,\frac{1}{2}},
\end{align*}
where $\tilde \mu_2 = \max\{1,\frac{1}{\lambda},\overline{\nu},\overline{\sigma}\} \max\{1,C_F^2+1\}$.
The left hand-side inequality in (\ref{inequality:KKTSysSTVFinfsupsupsup:Seminorm}) is
proved quite similarly to the previous case. We select
the test functions
\begin{align*}
(z,q) = (- \frac{1}{\sqrt{\lambda}} p - \frac{1}{\sqrt{\lambda}} p^\perp,
\sqrt{\lambda} y - \sqrt{\lambda} y^\perp),
\end{align*}
and use the $\sigma$- and $\nu$-weighted orthogonality relations
(\ref{equation:orthorelation}).
Then, we find that
\begin{align*}
\mathcal{B}((y,p),(- \frac{1}{\sqrt{\lambda}} p - \frac{1}{\sqrt{\lambda}} p^\perp,
\sqrt{\lambda} y - \sqrt{\lambda} y^\perp))
&\geq \min\{\underline{\nu},\underline{\sigma}\} \min\{\sqrt{\lambda},\frac{1}{\sqrt{\lambda}}\}
|(y,p)|_{1,\frac{1}{2}}^2.
\end{align*}
In view of the estimate
\begin{align*}
|(z,q)|_{1,\frac{1}{2}}^2
\leq 2 \max\{\lambda,\frac{1}{\lambda}\} |(y,p)|_{1,\frac{1}{2}}^2,
\end{align*}
we obtain
\begin{align*}
\sup_{0 \not= (z,q) \in (H^{1,\frac{1}{2}}_{0,per}(Q_T))^2}
\frac{\mathcal{B}((y,p),(z,q))}{\|(z,q)\|_{1,\frac{1}{2}}}
&\geq \frac{\min\{\underline{\nu},\underline{\sigma}\} \min\{\sqrt{\lambda},\frac{1}{\sqrt{\lambda}}\}
|(y,p)|_{1,\frac{1}{2}}^2}{
\sqrt{2} \max\{\sqrt{\lambda},\frac{1}{\sqrt{\lambda}}\} |(y,p)|_{1,\frac{1}{2}}} \\
&= \tilde \mu_1 \, |(y,p)|_{1,\frac{1}{2}},
\end{align*}
where
$\tilde \mu_1 = \frac{\min\{\underline{\nu},\underline{\sigma}\} \min\{\lambda,\frac{1}{\lambda}\}}{\sqrt{2}}$.
This justifies the left hand-side inequality in (\ref{inequality:KKTSysSTVFinfsupsupsup:Seminorm}).
\end{proof}
\end{lemma}
Let $(\eta,\zeta)$ be an approximation of $(y,p)$, which is
a bit more regular with respect to the time variable
than the exact solution $(y,p)$.
Namely, we assume that $\eta, \zeta \in H^{1,1}_{0,per}(Q_T)$
(this assumption is
of course true for the MhFE approximations
$y_{N h}$ and $p_{N h}$ defined in (\ref{definition:MultiharmonicFEAnsatzStateANDAdjointState})).
Our goal is to deduce a computable upper bound of the error
$e := (y,p) - (\eta,\zeta)$
in $H^{1,\frac{1}{2}}_{0,per}(Q_T) \times H^{1,\frac{1}{2}}_{0,per}(Q_T)$.
From (\ref{problem:KKTSysSTVFAPost}), it follows that the integral identity
\begin{align}
\left.
\label{problem:KKTSysSTVFAPost:Error}
\begin{aligned}
\int_{Q_T} &\Big( (y-\eta)\,z - \nu(\boldsymbol{x}) \nabla (p-\zeta) \cdot \nabla z
+ \sigma(\boldsymbol{x}) \partial_t^{1/2} (p-\zeta) \, \partial_t^{1/2} z^{\perp} \\
&+ \nu(\boldsymbol{x}) \nabla (y-\eta) \cdot \nabla q
+ \sigma(\boldsymbol{x}) \partial_t^{1/2} (y-\eta) \, \partial_t^{1/2} q^{\perp}
+ \lambda^{-1} (p-\zeta)\,q \Big)\,d\boldsymbol{x}\,dt \\
= &\int_{Q_T} \Big(y_d\,z
- \eta\,z + \nu(\boldsymbol{x}) \nabla \zeta \cdot \nabla z
- \sigma(\boldsymbol{x}) \partial_t^{1/2} \zeta \, \partial_t^{1/2} z^{\perp} \\
&- \nu(\boldsymbol{x}) \nabla \eta \cdot \nabla q
- \sigma(\boldsymbol{x}) \partial_t^{1/2} \eta \, \partial_t^{1/2} q^{\perp}
- \lambda^{-1} \zeta \,q \Big) \, d\boldsymbol{x}\,dt
\end{aligned}
\quad
\right \rbrace
\end{align}
holds for all $z, q \in H^{1,\frac{1}{2}}_{0,per}(Q_T)$.
The linear functional
\begin{align*}
\begin{aligned}
\mathcal{F}_{(\eta,\zeta)}(z,q)
= \int_{Q_T} &\Big(y_d\,z
- \eta\,z + \nu(\boldsymbol{x}) \nabla \zeta \cdot \nabla z
- \sigma(\boldsymbol{x}) \partial_t^{1/2} \zeta \, \partial_t^{1/2} z^{\perp} \\
&- \nu(\boldsymbol{x}) \nabla \eta \cdot \nabla q
- \sigma(\boldsymbol{x}) \partial_t^{1/2} \eta \, \partial_t^{1/2} q^{\perp}
- \lambda^{-1} \zeta \,q \Big) \, d\boldsymbol{x}\,dt
\end{aligned}
\end{align*}
is defined on $(z,q) \in H^{1,\frac{1}{2}}_{0,per}(Q_T) \times H^{1,\frac{1}{2}}_{0,per}(Q_T)$.
It can be viewed as a quantity measuring the accuracy of
(\ref{problem:KKTSysSTVFAPost:Error}) for any pair of test functions
$(z,q)$. Therefore, getting an upper bound of the error is reduced to the estimation of
\begin{align}
\label{inequality:supRHS:OCP}
\begin{aligned}
\sup_{0 \not= (z,q) \in (H^{1,\frac{1}{2}}_{0,per}(Q_T))^2}
\frac{\mathcal{F}_{(\eta,\zeta)}(z,q)}{\|(z,q)\|_{1,\frac{1}{2}}}
\qquad \text{ or } \quad
\sup_{0 \not= (z,q) \in (H^{1,\frac{1}{2}}_{0,per}(Q_T))^2}
\frac{\mathcal{F}_{(\eta,\zeta)}(z,q)}{|(z,q)|_{1,\frac{1}{2}}}.
\end{aligned}
\end{align}
We reconstruct $\mathcal{F}_{(\eta,\zeta)}$ using
the identity
\begin{align}
\label{equation:identityEtaH11/2}
\langle \sigma \partial_t^{1/2} \eta,\partial_t^{1/2} z^{\perp} \rangle
= \langle \sigma \partial_t \eta,z \rangle
\qquad \forall \, \eta \in H^{1,1}_{0,per}(Q_T) \quad \forall \, z \in H^{1,\frac{1}{2}}_{0,per}(Q_T).
\end{align}
Also, we use the identities
\begin{align*}
\int_\Omega \text{div} \, \boldsymbol{\rho} \, z \, d\boldsymbol{x}
= - \int_\Omega \boldsymbol{\rho} \cdot \nabla z \, d\boldsymbol{x}
\qquad \text{ and } \qquad
\int_\Omega \text{div} \, \boldsymbol{\tau} \, q \, d\boldsymbol{x}
= - \int_\Omega \boldsymbol{\tau} \cdot \nabla q \, d\boldsymbol{x},
\end{align*}
which hold for any $z,q \in H^1_0(\Omega)$ and any
\begin{align*}
\boldsymbol{\tau}, \boldsymbol{\rho} \in H(\text{div},Q_T) :=
\{\boldsymbol{\rho} \in [L^2(Q_T)]^d
: \text{div}_{\boldsymbol{x}} \, \boldsymbol{\rho}(\cdot,t) \in L^2(\Omega)
\text{ for a.e. } t \in (0,T)\}.
\end{align*}
For ease of notation, the index $\boldsymbol{x}$ in $\text{div}_{\boldsymbol{x}}$ will be henceforth omitted,
i.e., $\text{div} = \text{div}_{\boldsymbol{x}}$ denotes the generalized spatial divergence.
Using the Cauchy-Schwarz inequality yields the estimate
\begin{align*}
\mathcal{F}_{(\eta,\zeta)}(z,q&)
= \int_{Q_T} \Big(y_d\,z
- \eta\,z + \nu(\boldsymbol{x}) \nabla \zeta \cdot \nabla z
- \sigma(\boldsymbol{x}) \partial_t \zeta \, z
+ \text{div} \, \boldsymbol{\rho} \, z + \boldsymbol{\rho} \cdot \nabla z \\
&\qquad\, - \nu(\boldsymbol{x}) \nabla \eta \cdot \nabla q
- \sigma(\boldsymbol{x}) \partial_t \eta \, q
- \lambda^{-1} \zeta \,q
+ \text{div} \, \boldsymbol{\tau} \, q + \boldsymbol{\tau} \cdot \nabla q \Big) \, d\boldsymbol{x}\,dt \\
\leq &\,\|\mathcal{R}_1(\eta,\zeta,\boldsymbol{\tau})\|
\|q\| + \|\mathcal{R}_2(\eta,\boldsymbol{\tau})\| \|\nabla q\|
+ \|\mathcal{R}_3(\eta,\zeta,\boldsymbol{\rho})\| \|z\|
+ \|\mathcal{R}_4(\zeta,\boldsymbol{\rho})\| \|\nabla z\|,
\end{align*}
where
\begin{align}
\label{definition:R1R2R3R4}
\left.
\begin{aligned}
\mathcal{R}_1(\eta,\zeta,\boldsymbol{\tau})
&= \sigma \partial_t \eta + \lambda^{-1} \zeta - \text{div} \, \boldsymbol{\tau},
\qquad \,\,\,
\mathcal{R}_2(\eta,\boldsymbol{\tau}) = \boldsymbol{\tau}-\nu \nabla \eta, \\
\mathcal{R}_3(\eta,\zeta,\boldsymbol{\rho})
&= \sigma \partial_t \zeta + \eta - \text{div} \, \boldsymbol{\rho} - y_d,
\qquad
\mathcal{R}_4(\zeta,\boldsymbol{\rho}) = \boldsymbol{\rho} + \nu \nabla \zeta,
\end{aligned}
\quad
\right \rbrace
\end{align}
Applying (\ref{inequality:Friedrichs:FourierSpace}),
we find that
\begin{align*}
\mathcal{F}_{(\eta,\zeta)}(z,q)
&\leq \Big(C_F \, \|\mathcal{R}_1(\eta,\zeta,\boldsymbol{\tau})\|
+ \|\mathcal{R}_2(\eta,\boldsymbol{\tau})\|
\Big) \|\nabla q\| \\
&+ \Big(C_F \, \|\mathcal{R}_3(\eta,\zeta,\boldsymbol{\rho})\|
+ \|\mathcal{R}_4(\zeta,\boldsymbol{\rho})\|
\Big) \|\nabla z\|.
\end{align*}
Hence,
\begin{align*}
\sup_{0 \not= (z,q) \in (H^{1,\frac{1}{2}}_{0,per}(Q_T))^2}
\frac{\mathcal{F}_{(\eta,\zeta)}(z,q)}{|(z,q)|_{1,\frac{1}{2}}}
\leq &\, C_F \, \|\mathcal{R}_1(\eta,\zeta,\boldsymbol{\tau})\|
+ \|\mathcal{R}_2(\eta,\boldsymbol{\tau})\| \\
&+ C_F \, \|\mathcal{R}_3(\eta,\zeta,\boldsymbol{\rho})\|
+ \|\mathcal{R}_4(\zeta,\boldsymbol{\rho})\|
\end{align*}
and by the
inf-sup condition in (\ref{inequality:KKTSysSTVFinfsupsupsup:Seminorm}), we obtain
\begin{align*}
|e|_{1,\frac{1}{2}}
\leq \frac{1}{\tilde \mu_1}
\sup_{0 \not= (z,q) \in (H^{1,\frac{1}{2}}_{0,per}(Q_T))^2}
\frac{\mathcal{B}(e, (z,q))}{|(z,q)|_{1,\frac{1}{2}}}
= \frac{1}{\tilde \mu_1}
\sup_{0 \not= (z,q) \in (H^{1,\frac{1}{2}}_{0,per}(Q_T))^2}
\frac{\mathcal{F}_{(\eta,\zeta)}(z,q)}{|(z,q)|_{1,\frac{1}{2}}}.
\end{align*}
Thus, we arrive at the following result:
\begin{theorem}
\label{theorem:aposteriorEstimateH11/2Seminorm:OCP}
Let $\eta,\zeta \in H^{1,1}_{0,per}(Q_T)$,
$\boldsymbol{\tau}, \boldsymbol{\rho} \in H(\emph{div},Q_T)$,
and the bilinear form $\mathcal{B}(\cdot,\cdot)$
defined by (\ref{definition:KKTSysSTVF})
meets the inf-sup condition (\ref{inequality:KKTSysSTVFinfsupsupsup:Seminorm}). Then,
\begin{align}
\left.
\label{inequality:aposteriorEstimateH11/2Seminorm:OCP}
\begin{aligned}
|e|_{1,\frac{1}{2}} \leq \frac{1}{\tilde \mu_1}
& \big(C_F \, \|\mathcal{R}_1(\eta,\zeta,\boldsymbol{\tau})\|
+ \|\mathcal{R}_2(\eta,\boldsymbol{\tau})\| \\
&+ C_F \, \|\mathcal{R}_3(\eta,\zeta,\boldsymbol{\rho})\|
+ \|\mathcal{R}_4(\zeta,\boldsymbol{\rho})\|
\big)
=: \mathcal{M}^\oplus_{|\cdot|}(\eta,\zeta,\boldsymbol{\tau},\boldsymbol{\rho}),
\end{aligned}
\quad
\right \rbrace
\end{align}
where $e = (y,p) - (\eta, \zeta) \in (H^{1,\frac{1}{2}}_{0,per}(Q_T))^2$ and
$\tilde \mu_1 = \frac{\min\{\underline{\nu},\underline{\sigma}\} \min\{\lambda,\frac{1}{\lambda}\}}{\sqrt{2}}$.
\end{theorem}
\begin{remark}
For computational reasons,
it is useful to reformulate the majorants
in such a way that they are given by quadratic functionals,
see, e.g., \cite{PhD:GaevskayaHoppeRepin:2006}. This is done by introducing
parameters
$\alpha, \beta, \gamma > 0$ and applying Young's inequality.
For the error majorant
$\mathcal{M}^\oplus_{|\cdot|}(\eta,\zeta,\boldsymbol{\tau},\boldsymbol{\rho})$,
we have
\begin{align*}
\mathcal{M}^\oplus_{|\cdot|}(\eta,\zeta,\boldsymbol{\tau},\boldsymbol{\rho})^2
\leq &\,\, \mathcal{M}^\oplus_{|\cdot|}(\alpha,\beta,\gamma;\eta,\zeta,\boldsymbol{\tau},\boldsymbol{\rho})^2 \\
= &\, \frac{1}{\tilde \mu_1^2} \Big(C_F^2(1+\alpha)(1+\beta) \, \|\mathcal{R}_1(\eta,\zeta,\boldsymbol{\tau})\|^2
+ \frac{(1+\alpha)(1+\beta)}{\beta} \|\mathcal{R}_2(\eta,\boldsymbol{\tau})\|^2 \\
&+ C_F^2 \frac{(1+\alpha)(1+\gamma)}{\alpha} \, \|\mathcal{R}_3(\eta,\zeta,\boldsymbol{\rho})\|^2
+ \frac{(1+\alpha)(1+\gamma)}{\alpha \gamma} \|\mathcal{R}_4(\zeta,\boldsymbol{\rho})\|^2
\Big).
\end{align*}
\end{remark}
Finally, we note that the inf-sup condition
(\ref{inequality:KKTSysSTVFinfsupsupsup}) implies an estimate similar to
(\ref{inequality:aposteriorEstimateH11/2Seminorm:OCP})
for the error in terms of $\|\cdot\|_{1,\frac{1}{2}}$.
\begin{theorem}
\label{theorem:aposteriorEstimateH11/2Norm:OCP}
Let $\eta,\zeta \in H^{1,1}_{0,per}(Q_T)$,
$\boldsymbol{\tau},\boldsymbol{\rho} \in H(\emph{div},Q_T)$
and the bilinear form $\mathcal{B}(\cdot,\cdot)$
defined by (\ref{definition:KKTSysSTVF}) satisfies
(\ref{inequality:KKTSysSTVFinfsupsupsup}). Then,
\begin{align}
\left.
\label{inequality:aposteriorEstimateH11/2Nnorm:OCP}
\begin{aligned}
\|e\|_{1,\frac{1}{2}} \leq \frac{1}{\mu_{1}}
&\Big(\|\mathcal{R}_1(\eta,\zeta,\boldsymbol{\tau})\|^2
+ \|\mathcal{R}_2(\eta,\boldsymbol{\tau})\|^2 \\
&+ \|\mathcal{R}_3(\eta,\zeta,\boldsymbol{\rho})\|^2
+ \|\mathcal{R}_4(\zeta,\boldsymbol{\rho})\|^2
\Big)^{1/2}
=: \mathcal{M}^\oplus_{\|\cdot\|}(\eta,\zeta,\boldsymbol{\tau},\boldsymbol{\rho}),
\end{aligned}
\quad
\right \rbrace
\end{align}
where $e$ is defined in Theorem \ref{theorem:aposteriorEstimateH11/2Seminorm:OCP} and
$\mu_1 = \frac{\min\{\frac{1}{\sqrt{\lambda}},\underline{\nu},\underline{\sigma}\}
\min\{\sqrt{\lambda},\frac{1}{\sqrt{\lambda}}\}}{\sqrt{1+2\max\{\lambda,\frac{1}{\lambda}\}}}$.
\end{theorem}
\subsubsection*{The multiharmonic approximations}
Since the desired state $y_d$ belongs to $L^2$
it can be represented as a Fourier series.
Henceforth, we assume that the
approximations $\eta$ and $\zeta$ of
the exact state $y$ and the adjoint state $p$,
respectively,
are also represented in terms of truncated Fourier series
as well as the vector-valued functions
$\boldsymbol{\tau}$ and $\boldsymbol{\rho}$, i.e.,
\begin{align}
\label{definition:MhApproxEtaTau}
\left.
\begin{aligned}
\eta(\boldsymbol{x},t) &= \eta_0^c(\boldsymbol{x}) + \sum_{k=1}^N \left(\eta_k^c(\boldsymbol{x}) \cos(k \omega t)
+ \eta_k^s(\boldsymbol{x}) \sin(k \omega t)\right), \\
\boldsymbol{\tau}(\boldsymbol{x},t) &= \boldsymbol{\tau}_0^c(\boldsymbol{x}) + \sum_{k=1}^N \left(\boldsymbol{\tau}_k^c(\boldsymbol{x}) \cos(k \omega t)
+ \boldsymbol{\tau}_k^s(\boldsymbol{x}) \sin(k \omega t)\right),
\end{aligned}
\quad
\right \rbrace
\end{align}
where all the Fourier coefficients belong to the space $L^2(\Omega)$.
In this case,
\begin{align*}
\partial_t \eta(\boldsymbol{x},t) &= \sum_{k=1}^N \left(k \omega \, \eta_k^s(\boldsymbol{x}) \cos(k \omega t)
- k \omega \, \eta_k^c(\boldsymbol{x}) \sin(k \omega t)\right), \\
\nabla \eta(\boldsymbol{x},t) &= \nabla \eta_0^c(\boldsymbol{x})
+ \sum_{k=1}^N \left(\nabla \eta_k^c(\boldsymbol{x}) \, \cos(k \omega t)
+ \nabla \eta_k^s(\boldsymbol{x}) \, \sin(k \omega t)\right), \\
\text{div} \, \boldsymbol{\tau}(\boldsymbol{x},t) &= \text{div} \, \boldsymbol{\tau}_0^c(\boldsymbol{x})
+ \sum_{k=1}^N \left(\text{div} \, \boldsymbol{\tau}_k^c(\boldsymbol{x}) \, \cos(k \omega t)
+ \text{div} \, \boldsymbol{\tau}_k^s(\boldsymbol{x}) \, \sin(k \omega t)\right),
\end{align*}
and the $L^2(Q_T)$-norms of the functions
$\mathcal{R}_1$, $\mathcal{R}_2$, $\mathcal{R}_3$ and $\mathcal{R}_4$
defined in (\ref{definition:R1R2R3R4}) can be
represented in the form, which exposes each mode separately.
More precisely, we have
\begin{align*}
\|\mathcal{R}_1(\eta,\zeta,\boldsymbol{\tau})\|^2
= &\,T \|\lambda^{-1} \zeta_0^c - \text{div} \, \boldsymbol{\tau}_0^c\|_{\Omega}^2
+ \frac{T}{2} \sum_{k=1}^N
\|-k \omega \, \sigma \boldsymbol{\eta}_k^\perp + \lambda^{-1} \boldsymbol{\zeta}_k
- \text{\textbf{div}} \, \boldsymbol{\tau}_k \|_{\Omega}^2, \\
\|\mathcal{R}_2(\eta,\boldsymbol{\tau})\|^2
= &\,T \|\boldsymbol{\tau}_0^c - \nu \nabla \eta_0^c\|_{\Omega}^2 + \frac{T}{2} \sum_{k=1}^N
\|\boldsymbol{\tau}_k - \nu \nabla \boldsymbol{\eta}_k\|_{\Omega}^2, \\
\|\mathcal{R}_3(\eta,\zeta,\boldsymbol{\rho})\|^2
= &\,T \|\eta_0^c - \text{div} \, \boldsymbol{\rho}_0^c - {y_d^c}_0\|_{\Omega}^2
+ \frac{T}{2} \sum_{k=1}^N
\left(\|k \omega \, \sigma \zeta_k^s + \eta_k^c
- \text{div} \, \boldsymbol{\rho}_k^c - {y_d^c}_k\|_{\Omega}^2 \right. \\
&\left.+ \|-k \omega \, \sigma \zeta_k^c + \eta_k^s
- \text{div} \, \boldsymbol{\rho}_k^s - {y_d^s}_k\|_{\Omega}^2\right)
+ \frac{T}{2} \sum_{k=N+1}^\infty
\left(\|{y_d^c}_k\|_{\Omega}^2 + \|{y_d^s}_k\|_{\Omega}^2\right) \\
= &\,T \|\eta_0^c - \text{div} \, \boldsymbol{\rho}_0^c - {y_d^c}_0\|_{\Omega}^2
+ \frac{T}{2} \sum_{k=1}^N
\|-k \omega \, \sigma \boldsymbol{\zeta}_k^\perp + \boldsymbol{\eta}_k
- \text{\textbf{div}} \, \boldsymbol{\rho}_k - \boldsymbol{y_d}_k\|_{\Omega}^2 \\
&+ \frac{T}{2} \sum_{k=N+1}^\infty \|\boldsymbol{y_d}_k\|_{\Omega}^2, \\
\|\mathcal{R}_4(\zeta,\boldsymbol{\rho})\|^2
= &\,T \|\boldsymbol{\rho}_0^c + \nu \nabla \zeta_0^c\|_{\Omega}^2 + \frac{T}{2} \sum_{k=1}^N
\|\boldsymbol{\rho}_k + \nu \nabla \boldsymbol{\zeta}_k\|_{\Omega}^2,
\end{align*}
where $\text{\textbf{div}} \, \boldsymbol{\tau}_k
= (\text{div} \, \boldsymbol{\tau}_k^c,\text{div} \, \boldsymbol{\tau}_k^s)^T$ and
$\text{\textbf{div}} \, \boldsymbol{\rho}_k
= (\text{div} \, \boldsymbol{\rho}_k^c,\text{div} \, \boldsymbol{\rho}_k^s)^T$.
\begin{remark}
\label{remark:remainderterm}
Since $y_d$ is known, we can always compute the remainder term of truncation
\begin{align}
\label{def:remTerm}
\mathcal{E}_N :=
\frac{T}{2} \sum_{k=N+1}^\infty \|\boldsymbol{y_d}_k\|_{\Omega}^2
= \frac{T}{2} \sum_{k=N+1}^\infty \left(\|{y_d^c}_k\|_{\Omega}^2 + \|{y_d^s}_k\|_{\Omega}^2\right)
\end{align}
with any desired accuracy.
\end{remark}
It is important to outline that all the $L^2$-norms
of $\mathcal{R}_1$, $\mathcal{R}_2$, $\mathcal{R}_3$ and
$\mathcal{R}_4$ corresponding to every single mode $k=0,\dots,N$ are decoupled.
It is useful to introduce quantities related to each mode. For $k \geq 1$, we denote them by
\begin{align*}
{\mathcal{R}_1}_k(\boldsymbol{\eta}_k,\boldsymbol{\zeta}_k,\boldsymbol{\tau}_k)
&:= -k \omega \, \sigma \boldsymbol{\eta}_k^\perp + \lambda^{-1} \boldsymbol{\zeta}_k
- \text{\textbf{div}} \, \boldsymbol{\tau}_k
= ({\mathcal{R}_1}^c_k(\eta_k^s,\zeta_k^c,\boldsymbol{\tau}_k^c),
{\mathcal{R}_1}^s_k(\eta_k^c,\zeta_k^s,\boldsymbol{\tau}_k^s))^T \\
&= (k \omega \, \sigma \eta_k^s + \lambda^{-1} \zeta_k^c - \text{div} \, \boldsymbol{\tau}_k^c,
-k \omega \, \sigma \eta_k^c + \lambda^{-1} \zeta_k^s - \text{div} \, \boldsymbol{\tau}_k^s)^T,
\end{align*}
\begin{align*}
{\mathcal{R}_2}_k(\boldsymbol{\eta}_k,\boldsymbol{\tau}_k)
&:= \boldsymbol{\tau}_k - \nu \nabla \boldsymbol{\eta}_k
= ({\mathcal{R}_2}^c_k(\eta_k^c,\boldsymbol{\tau}_k^c),
{\mathcal{R}_2}^s_k(\eta_k^s,\boldsymbol{\tau}_k^s))^T \\
&= (\boldsymbol{\tau}_k^c - \nu \nabla \eta_k^c,
\boldsymbol{\tau}_k^s - \nu \nabla \eta_k^s)^T, \\
{\mathcal{R}_3}_k(\boldsymbol{\eta}_k,\boldsymbol{\zeta}_k,\boldsymbol{\rho}_k)
&:= -k \omega \, \sigma \boldsymbol{\zeta}_k^\perp + \boldsymbol{\eta}_k
- \text{\textbf{div}} \, \boldsymbol{\rho}_k - \boldsymbol{y_d}_k \\
&= ({\mathcal{R}_3}^c_k(\eta_k^c,\zeta_k^s,\boldsymbol{\rho}_k^c),
{\mathcal{R}_3}^s_k(\eta_k^s,\zeta_k^c,\boldsymbol{\rho}_k^s))^T \\
&= (k \omega \, \sigma \zeta_k^s + \eta_k^c - \text{div} \, \boldsymbol{\rho}_k^c - {y_d^c}_k,
-k \omega \, \sigma \zeta_k^c + \eta_k^s - \text{div} \, \boldsymbol{\rho}_k^s - {y_d^s}_k)^T, \\
\text{and} \hspace{1.9 cm}& \\
{\mathcal{R}_4}_k(\boldsymbol{\zeta}_k,\boldsymbol{\rho}_k)
&:= \boldsymbol{\rho}_k + \nu \nabla \boldsymbol{\zeta}_k
= ({\mathcal{R}_4}^c_k(\zeta_k^c,\boldsymbol{\rho}_k^c),
{\mathcal{R}_4}^s_k(\zeta_k^s,\boldsymbol{\rho}_k^s))^T \\
&= (\boldsymbol{\rho}_k^c + \nu \nabla \zeta_k^c,
\boldsymbol{\rho}_k^s + \nu \nabla \zeta_k^s)^T.
\end{align*}
For $k=0$, we have
\begin{align}
\label{definition:R1R2R3R4:Fourierk0}
\left.
\begin{aligned}
{\mathcal{R}_1}^c_0(\zeta_0^c,\boldsymbol{\tau}_0^c)
&:= \lambda^{-1} \zeta_0^c - \text{div} \, \boldsymbol{\tau}_0^c, \qquad \qquad \quad
{\mathcal{R}_2}^c_0(\eta_0^c,\boldsymbol{\tau}_0^c)
:= \boldsymbol{\tau}_0^c - \nu \nabla \eta_0^c, \\
{\mathcal{R}_3}^c_0(\eta_0^c,\boldsymbol{\rho}_0^c)
&:= \eta_0^c - \text{div} \, \boldsymbol{\rho}_0^c - {y_d^c}_0, \qquad \qquad
{\mathcal{R}_4}^c_0(\zeta_0^c,\boldsymbol{\rho}_0^c)
:= \boldsymbol{\rho}_0^c + \nu \nabla \zeta_0^c.
\end{aligned}
\right \rbrace
\end{align}
\begin{corollary}
The error majorants $\mathcal{M}^\oplus_{|\cdot|}(\eta,\zeta,\boldsymbol{\tau},\boldsymbol{\rho})$ and
$\mathcal{M}^\oplus_{\|\cdot\|}(\eta,\zeta,\boldsymbol{\tau},\boldsymbol{\rho})$
defined in (\ref{inequality:aposteriorEstimateH11/2Seminorm:OCP}) and
(\ref{inequality:aposteriorEstimateH11/2Nnorm:OCP}),
respectively,
can be represented
in somewhat new forms that contain quantities associated with the modes, namely,
\begin{align*}
\begin{aligned}
\mathcal{M}^\oplus_{|\cdot|}(\eta,\zeta,\boldsymbol{\tau},\boldsymbol{\rho})
= \frac{1}{\tilde \mu_1}
&\Big(C_F \, \big(T \|{\mathcal{R}_1}^c_0(\zeta_0^c,\boldsymbol{\tau}_0^c)\|_{\Omega}^2
+ \frac{T}{2} \sum_{k=1}^N
\|{\mathcal{R}_1}_k(\boldsymbol{\eta}_k,\boldsymbol{\zeta}_k,\boldsymbol{\tau}_k)\|_{\Omega}^2 \big)^{1/2} \\
&+ \big( T \|{\mathcal{R}_2}^c_0(\eta_0^c,\boldsymbol{\tau}_0^c)\|_{\Omega}^2
+ \frac{T}{2} \sum_{k=1}^N \|{\mathcal{R}_2}_k(\boldsymbol{\eta}_k,\boldsymbol{\tau}_k)\|_{\Omega}^2
\big)^{1/2} \\
&+ \,C_F \, \big(T \|{\mathcal{R}_3}^c_0(\eta_0^c,\boldsymbol{\rho}_0^c)\|_{\Omega}^2
+ \frac{T}{2} \sum_{k=1}^N
\|{\mathcal{R}_3}_k(\boldsymbol{\eta}_k,\boldsymbol{\zeta}_k,\boldsymbol{\rho}_k)\|_{\Omega}^2
+ \mathcal{E}_N
\big)^{1/2} \\
&+\, \big(T \|{\mathcal{R}_4}^c_0(\zeta_0^c,\boldsymbol{\rho}_0^c)\|_{\Omega}^2 + \frac{T}{2} \sum_{k=1}^N
\|{\mathcal{R}_4}_k(\boldsymbol{\zeta}_k,\boldsymbol{\rho}_k)\|_{\Omega}^2\big)^{1/2} \Big)
\end{aligned}
\end{align*}
and
\begin{align*}
\begin{aligned}
\mathcal{M}^\oplus_{\|\cdot\|}(\eta,\zeta,\boldsymbol{\rho}&,\boldsymbol{\tau})
= \frac{1}{\mu_{1}}
\Big(T \big(\|{\mathcal{R}_1}^c_0(\zeta_0^c,\boldsymbol{\tau}_0^c)\|_{\Omega}^2
+ \|{\mathcal{R}_2}^c_0(\eta_0^c,\boldsymbol{\tau}_0^c)\|_{\Omega}^2
+ \|{\mathcal{R}_3}^c_0(\eta_0^c,\boldsymbol{\rho}_0^c)\|_{\Omega}^2
\\
&+ \|{\mathcal{R}_4}^c_0(\zeta_0^c,\boldsymbol{\rho}_0^c)\|_{\Omega}^2 \big)
+ \frac{T}{2} \sum_{k=1}^N
\big(\|{\mathcal{R}_1}_k(\boldsymbol{\eta}_k,\boldsymbol{\zeta}_k,\boldsymbol{\tau}_k)\|_{\Omega}^2
+ \|{\mathcal{R}_2}_k(\boldsymbol{\eta}_k,\boldsymbol{\tau}_k)\|_{\Omega}^2 \\
&+ \|{\mathcal{R}_3}_k(\boldsymbol{\eta}_k,\boldsymbol{\zeta}_k,\boldsymbol{\rho}_k)\|_{\Omega}^2
+ \|{\mathcal{R}_4}_k(\boldsymbol{\zeta}_k,\boldsymbol{\rho}_k)\|_{\Omega}^2\big)
+ \mathcal{E}_N \Big)^{1/2}.
\end{aligned}
\end{align*}
\end{corollary}
\begin{remark}
Let $y_d$ has a multiharmonic representation, i.e.,
\begin{align}
\label{def:ydMultiharmonic}
y_d(\boldsymbol{x},t) = {y_d}_0^c(\boldsymbol{x}) + \sum_{k=1}^{N_d} \left(f_k^c(\boldsymbol{x}) \cos(k \omega t)
+ {y_d}_k^s(\boldsymbol{x}) \sin(k \omega t)\right),
\end{align}
where $N_d \in \mathbb{N}$.
If $N \geq N_d$, then
$(\eta,\zeta)$ is the exact solution of problem (\ref{problem:KKTSysSTVFAPost}) and
$(\boldsymbol{\tau},\boldsymbol{\rho})$ is the exact flux if and only if the error majorants
vanish, i.e.,
\begin{align*}
{\mathcal{R}_j}_0^c = 0 \qquad \text{and} \qquad
{\mathcal{R}_j}_k = 0 \qquad \forall \, k=1,\dots,N_d, \qquad \forall \, j \in \{1,2,3,4\}.
\end{align*}
Indeed, let the error majorants vanish. Then,
$\eta_0^c - \text{\emph{div}} \, \boldsymbol{\rho}_0^c = {y_d^c}_0$,
$\boldsymbol{\rho}_0^c = - \nu \nabla \zeta_0^c$,
$\lambda^{-1} \zeta_0^c - \text{\emph{div}} \, \boldsymbol{\tau}_0^c = 0$,
$\boldsymbol{\tau}_0^c = \nu \nabla \eta_0^c$
and we see that
\begin{align*}
&k \omega \, \sigma \eta_k^s + \lambda^{-1} \zeta_k^c - \text{\emph{div}} \, \boldsymbol{\tau}_k^c = 0, \qquad
-k \omega \, \sigma \eta_k^c + \lambda^{-1} \zeta_k^s - \text{\emph{div}} \, \boldsymbol{\tau}_k^s = 0, \\
&k \omega \, \sigma \zeta_k^s + \eta_k^c - \text{\emph{div}} \, \boldsymbol{\rho}_k^c = {y_d^c}_k,
\qquad \quad \, \,
-k \omega \, \sigma \zeta_k^c + \eta_k^s - \text{\emph{div}} \, \boldsymbol{\rho}_k^s = {y_d^s}_k, \\
&\boldsymbol{\tau}_k^c = \nu \nabla \eta_k^c, \qquad
\boldsymbol{\tau}_k^s = \nu \nabla \eta_k^s, \qquad
\boldsymbol{\rho}_k^c = - \nu \nabla \zeta_k^c, \qquad
\boldsymbol{\rho}_k^s = - \nu \nabla \zeta_k^s,
\end{align*}
for all $k=1,\dots,N_d$,
so that collecting the $N_d+1$ harmonics leads to
multiharmonic representations for $\eta$, $\zeta$, $\boldsymbol{\tau}$
and $\boldsymbol{\rho}$ of the form (\ref{def:ydMultiharmonic})
satisfying the equations
\begin{align*}
\sigma \partial_t \eta - \emph{div} \, \boldsymbol{\tau} + \lambda^{-1} \zeta = 0, \qquad
\boldsymbol{\tau} = \nu \nabla \eta, \qquad
\sigma \partial_t \zeta - \emph{div} \, \boldsymbol{\rho} + \eta = y_d, \qquad
\boldsymbol{\rho} = - \nu \nabla \zeta.
\end{align*}
Since $\eta$ and $\zeta$ also meet the boundary conditions,
we conclude that $\eta = y$ and $\zeta = p$.
\end{remark}
\section[A posteriori estimates for cost functionals]{Functional
A Posteriori Estimates for Cost Functionals of Parabolic Time-Periodic Optimal Control Problems}
\label{Sec5:APosterioriErrorEstimation:OCP:CostFuncs}
This section is aimed at deriving guaranteed and computable
upper bounds for the cost functional
and establishing their sharpness.
This is important because, in optimal control, we cannot in general compute the (exact)
cost functional since
the (exact) state function is unknown. By using {\it a posteriori} estimates for the state equation
we overcome this difficulty.
Similar results for elliptic optimal control problems can be found, e.g., in
\cite{PhD:GaevskayaHoppeRepin:2006, PhD:Repin:2008}.
Let $y=y(v)$ be the corresponding state to a control $v$.
The cost functional $\mathcal{J}(y(v),v)$
defined in (\ref{equation:minfunc:OCP}) has the form
\begin{align*}
\mathcal{J}(y(v),v)
= T \mathcal{J}_0(y_0^c(v_0^c),v_0^c)
+ \frac{T}{2} \sum_{k=1}^\infty \mathcal{J}_k(\boldsymbol{y}_k(\boldsymbol{v}_k),\boldsymbol{v}_k),
\end{align*}
where $\mathcal{J}_0(y_0^c(v_0^c),v_0^c)
= \frac{1}{2} \|y_0^c - {y_d^c}_0\|_{\Omega}^2
+ \frac{\lambda}{2} \|v_0^c\|_{\Omega}^2$
and
\begin{align*}
\mathcal{J}_k(\boldsymbol{y}_k(\boldsymbol{v}_k),\boldsymbol{v}_k)
= \frac{1}{2} \|\boldsymbol{y}_k-{\boldsymbol{y}_d}_k\|_{\Omega}^2
+ \frac{\lambda}{2} \|\boldsymbol{v}_k\|_{\Omega}^2.
\end{align*}
We wish to deduce majorants for the cost functional
$\mathcal{J}(y(u),u)$ of the exact control $u$ and corresponding state $y(u)$
by using some of the results presented in \cite{LRW:LangerRepinWolfmayr:2015},
which are obtained for the time-periodic boundary value problem (\ref{equation:forwardpde:OCP}).
In \cite{LRW:LangerRepinWolfmayr:2015},
the following functional {\it a posteriori} error estimate for problem
(\ref{equation:forwardpde:OCP}) has been proved:
\begin{align}
\label{inequality:aposteriorEstimateH11/2SeminormBVP}
|y(v) - \eta|_{1,\frac{1}{2}} \leq \frac{1}{\underline{\mu_1}}
\left(C_F \, \|\mathcal{R}_1(\eta,\boldsymbol{\tau},v)\|
+ \|\mathcal{R}_2(\eta,\boldsymbol{\tau})\| \right),
\end{align}
where
$\underline{\mu_{1}} = \frac{1}{\sqrt{2}} \min\{\underline{\nu},\underline{\sigma}\}$.
It holds for arbitrary functions $\eta \in H^{1,1}_{0,per}(Q_T)$ and
$\boldsymbol{\tau} \in H(\text{div},Q_T)$,
where
\begin{align*}
\mathcal{R}_1(\eta,\boldsymbol{\tau},v)
:= \sigma \partial_t \eta - \text{div} \, \boldsymbol{\tau} - v, \qquad
\qquad
\mathcal{R}_2(\eta,\boldsymbol{\tau})
:= \boldsymbol{\tau} - \nu \nabla \eta,
\end{align*}
and $v$ is a given function in $L^2(Q_T)$.
Now, adding and subtracting $\eta$ in the cost functional $\mathcal{J}(y(v),v)$ as well as
applying the triangle and Friedrichs inequalities yields the estimate
\begin{align*}
\mathcal{J}(y(v),v)
&\leq \frac{1}{2} \left(\|\eta - y_d\| + C_F \|\nabla y(v) - \nabla \eta\| \right)^2
+ \frac{\lambda}{2} \|v\|^2.
\end{align*}
Since
\begin{align*}
\|\nabla y(v) - \nabla \eta\|^2
\leq |y(v) - \eta|_{1,\frac{1}{2}}^2 = \|\nabla y(v) - \nabla \eta\|^2
+ \|\partial_t^{1/2} y(v) - \partial_t^{1/2} \eta\|^2,
\end{align*}
we conclude that
\begin{align*}
\mathcal{J}(y(v),v)
&\leq \frac{1}{2} \left(\|\eta - y_d\| + C_F |y(v) - \eta|_{1,\frac{1}{2}} \right)^2
+ \frac{\lambda}{2} \|v\|^2.
\end{align*}
Together with (\ref{inequality:aposteriorEstimateH11/2SeminormBVP})
this leads to the estimate
\begin{align*}
\mathcal{J}(y(v),v)
\leq \frac{1}{2} \left(\|\eta - y_d\|
+ \frac{C_F}{\underline{\mu_1}} \|\mathcal{R}_2(\eta,\boldsymbol{\tau})\|
+ \frac{C_F^2}{\underline{\mu_1}} \|\mathcal{R}_1(\eta,\boldsymbol{\tau},v)\| \right)^2
+ \frac{\lambda}{2} \|v\|^2.
\end{align*}
By introducing parameters $\alpha, \beta > 0$ and applying Young's inequality,
we can reformulate the estimate such that the
right-hand side is given by a quadratic functional
(the latter functional is more convenient from the computational point of view).
We have
\begin{align*}
\mathcal{J}(y(v),v)
\leq \mathcal{J}^\oplus(\alpha,\beta;\eta,\boldsymbol{\tau},v) \qquad \forall \, v \in L^2(Q_T),
\end{align*}
where
\begin{align}
\left.
\label{definition:majorantCostFunc}
\begin{aligned}
\mathcal{J}^\oplus(\alpha,\beta;\eta,\boldsymbol{\tau},v) :=
&\, \frac{1+\alpha}{2} \|\eta - y_d\|^2
+ \frac{(1+\alpha)(1+\beta) C_F^2 }{2 \alpha \underline{\mu_1}^2} \|\mathcal{R}_2(\eta,\boldsymbol{\tau})\|^2 \\
&+ \frac{(1+\alpha)(1+\beta) C_F^4}{2 \alpha \beta \underline{\mu_1}^2}
\|\mathcal{R}_1(\eta,\boldsymbol{\tau},v)\|^2
+ \frac{\lambda}{2} \|v\|^2.
\end{aligned}
\quad
\right \rbrace
\end{align}
The majorant (\ref{definition:majorantCostFunc}) provides a guaranteed
upper bound of the cost functional,
which can be computed for any approximate control and state functions.
Moreover, minimization of this functional with respect to
$\eta$, $\boldsymbol{\tau}$, $v$ and $\alpha, \beta > 0$
yields the exact value of the cost functional.
This important result is summarized in the following theorem:
\begin{theorem}
\label{theorem:majorantCostFuncInf}
The exact lower bound of the majorant $\mathcal{J}^\oplus$ defined in (\ref{definition:majorantCostFunc}) coincides
with the optimal value of the cost functional of problem (\ref{equation:minfunc:OCP})-(\ref{equation:forwardpde:OCP}),
i.e.,
\begin{align}
\label{definition:majorantCostFuncInf}
\inf_{\substack{\eta \in H^{1,1}_{0,per}(Q_T),\boldsymbol{\tau} \in H(\emph{div},Q_T) \\
v \in L^2(Q_T), \alpha, \beta > 0}}
\mathcal{J}^\oplus(\alpha,\beta;\eta,\boldsymbol{\tau},v) = \mathcal{J}(y(u),u).
\end{align}
\begin{proof}
The infimum of $\mathcal{J}^\oplus$ is attained for the optimal control $u$, the
corresponding state function $y(u)$ and the exact flux $(\nu \nabla y(u))$.
In this case, $\mathcal{R}_1$ and $\mathcal{R}_2$ vanish, and for $\alpha$ tending
to zero, values of $\mathcal{J}^\oplus$ tend to the exact value of the
cost functional.
\end{proof}
\end{theorem}
\begin{corollary}
From Theorem~\ref{theorem:majorantCostFuncInf}, we obtain
the following estimate:
\begin{align}
\label{estimate:majCostFunc}
\begin{aligned}
\mathcal{J}(y(u),u)
\leq \,\, &\mathcal{J}^\oplus(\alpha,\beta;\eta,\boldsymbol{\tau},v) \\
&\forall \, \eta \in H^{1,1}_{0,per}(Q_T), \,
\boldsymbol{\tau} \in H(\emph{div},Q_T), \,
v \in L^2(Q_T), \, \alpha, \beta > 0.
\end{aligned}
\end{align}
\end{corollary}
Now, it is easy to derive {\it a posteriori}
estimates for the cost functional in the setting of multiharmonic approximations.
Let $\eta$ be the MhFE approximation $y_{N h}$ to the state $y$.
Since the control $v$ can be chosen arbitrarily in (\ref{definition:majorantCostFunc}),
we choose a MhFE approximation
$u_{N h}$ for it as well. More precisely,
we can compute the MhFE approximation
$u_{N h}$ for the control from the MhFE approximation
$p_{N h}$ of the adjoint state, since $u_{N h} = - \lambda^{-1} p_{N h}$,
by solving the optimality system,
from which we obtain $y_{N h}$ as well.
We now apply (\ref{estimate:majCostFunc}) and select $\eta = y_{N h}$ and
$v = u_{N h}$.
Next, we need to make a suitable reconstruction of
$\boldsymbol{\tau}$, which can be done by different techniques,
see, e.g., \cite{PhD:Repin:2008, PhD:MaliNeittaanmaekiRepin:2014} and the references therein.
In the treatment of our approach it is natural to represent $\boldsymbol{\tau}$ in the form of
a multiharmonic function $\boldsymbol{\tau}_{N h}$. Then the majorant
\begin{align*}
\mathcal{J}^\oplus&(\alpha,\beta;y_{N h},
\boldsymbol{\tau}_{N h},u_{N h}) =
\frac{1+\alpha}{2} \|y_{N h} - y_d\|^2
+ \frac{(1+\alpha)(1+\beta) C_F^2 }{2 \alpha \underline{\mu_1}^2} \|\mathcal{R}_2(y_{N h},
\boldsymbol{\tau}_{N h})\|^2 \\
&\qquad \qquad \qquad \qquad \qquad \, + \frac{(1+\alpha)(1+\beta) C_F^4}{2 \alpha
\beta \underline{\mu_1}^2}
\|\mathcal{R}_1(y_{N h},
\boldsymbol{\tau}_{N h},u_{N h})\|^2
+ \frac{\lambda}{2} \|u_{N h}\|^2
\end{align*}
has a multiharmonic form
\begin{align}
\label{def:costfunc:multiharmonic}
\begin{aligned}
&\,\mathcal{J}^\oplus(\alpha,\beta;y_{N h},
\boldsymbol{\tau}_{N h},u_{N h})
= \, \frac{1+\alpha}{2} \Big(T \|y_{0 h}^c
- {y_d}_0^c\|_{\Omega}^2 \\
&\qquad \qquad \qquad \qquad \quad
+ \frac{T}{2} \sum_{k=1}^N
\left(\|y_{k h}^c - {y_d}_k^c\|_{\Omega}^2
+ \|y_{k h}^s - {y_d}_k^s\|_{\Omega}^2 \right) + \mathcal{E}_N
\Big) \\
&+ \frac{(1+\alpha)(1+\beta) C_F^2 }{2 \alpha \underline{\mu_1}^2}
\Big(T \|{\mathcal{R}_2}^c_0(y_{0 h}^c,
\boldsymbol{\tau}_{0 h}^c)\|_{\Omega}^2
\\
&\qquad \qquad \qquad \qquad \quad
+ \frac{T}{2} \sum_{k=1}^N
\left(\|{\mathcal{R}_2}^c_k(y_{k h}^c,
\boldsymbol{\tau}_{k h}^c)\|_{\Omega}^2
+ \|{\mathcal{R}_2}^s_k(y_{k h}^s,
\boldsymbol{\tau}_{k h}^s)\|_{\Omega}^2\right)\Big) \\
&+ \frac{(1+\alpha)(1+\beta) C_F^4}{2 \alpha \beta \underline{\mu_1}^2}
\Big(T \|{\mathcal{R}_1}^c_0(\boldsymbol{\tau}_{0 h}^c,u_{0 h}^c)\|_{\Omega}^2 \\
&\qquad \qquad \qquad
+ \frac{T}{2} \sum_{k=1}^N
\left(\|{\mathcal{R}_1}^c_k(y_{k h}^s,
\boldsymbol{\tau}_{k h}^c,u_{k h}^c)\|_{\Omega}^2
+ \|{\mathcal{R}_1}^s_k(y_{k h}^c,
\boldsymbol{\tau}_{k h}^s,u_{k h}^s)\|_{\Omega}^2\right)\Big) \\
&+ \frac{\lambda}{2} \Big(T \|u_{0 h}^c\|_{\Omega}^2
+ \frac{T}{2} \sum_{k=1}^N
\left(\|u_{k h}^c\|_{\Omega}^2
+ \|u_{k h}^s\|_{\Omega}^2\right)\Big),
\end{aligned}
\end{align}
where
${\mathcal{R}_1}^c_0(\boldsymbol{\tau}_{0 h}^c,u_{0 h}^c) = \text{div} \, \boldsymbol{\tau}_{0 h}^c + u_{0 h}^c$,
${\mathcal{R}_2}^c_0(y_{0 h}^c,\boldsymbol{\tau}_{0 h}^c) = \boldsymbol{\tau}_{0 h}^c - \nu \nabla y_{0 h}^c$,
\begin{align*}
{\mathcal{R}_1}_k(\boldsymbol{y}_{k h},
\boldsymbol{\tau}_{k h},\boldsymbol{u}_{k h})
&= k \omega \, \sigma \boldsymbol{y}_{k h}^\perp
+ \text{\textbf{div}} \, \boldsymbol{\tau}_{k h} + \boldsymbol{u}_{k h} \\
&= (-k \omega \, \sigma y_{k h}^s
+ \text{div} \, \boldsymbol{\tau}_{k h}^c + u_{k h}^c,
k \omega \, \sigma y_{k h}^c
+ \text{div} \, \boldsymbol{\tau}_{k h}^s + u_{k h}^s)^T \\
&= ({\mathcal{R}_1}^c_k(y_{k h}^s,
\boldsymbol{\tau}_{k h}^c,u_{k h}^c),
{\mathcal{R}_1}^s_k(y_{k h}^c,
\boldsymbol{\tau}_{k h}^s,u_{k h}^s))^T,
\end{align*}
and
\begin{align*}
{\mathcal{R}_2}_k(\boldsymbol{y}_{k h},\boldsymbol{\tau}_{k h})
&= \boldsymbol{\tau}_{k h} - \nu \nabla \boldsymbol{y}_{k h} \\
&= (\boldsymbol{\tau}_{k h}^c - \nu \nabla y_{k h}^c,
\boldsymbol{\tau}_{k h}^s - \nu \nabla y_{k h}^s)^T
\\
&= ({\mathcal{R}_2}^c_k(y_{k h}^c,
\boldsymbol{\tau}_{k h}^c),
{\mathcal{R}_2}^s_k(y_{k h}^s,
\boldsymbol{\tau}_{k h}^s))^T.
\end{align*}
Note that the
remainder term (\ref{def:remTerm}) remains
the same in (\ref{def:costfunc:multiharmonic}).
Since all the terms corresponding to every single mode $k$ in the majorant $\mathcal{J}^\oplus$ are decoupled,
we arrive at some majorants $\mathcal{J}_k^\oplus$, for which
we can, of course, introduce positive parameters $\alpha_k$ and
$\beta_k$ for every single mode $k$ as well. Then the majorant (\ref{def:costfunc:multiharmonic})
can be written as
\begin{align}
\label{def:costfunc:multiharmonic:new}
\begin{aligned}
\mathcal{J}^\oplus(\boldsymbol{\alpha}_{N+1},\boldsymbol{\beta}_{N};y_{N h},
\boldsymbol{\tau}_{N h},u_{N h})
= &\, T \, \mathcal{J}_0^\oplus(\alpha_0,\beta_0;y_{0 h}^c,\boldsymbol{\tau}_{0 h}^c,u_{0 h}^c) \\
&+ \frac{T}{2} \sum_{k=1}^N
\mathcal{J}_k^\oplus(\alpha_k,\beta_k; \boldsymbol{y}_{k h}, \boldsymbol{\tau}_{k h}, \boldsymbol{u}_{k h}) \\
&+ \frac{1+\alpha_{N+1}}{2} \, \mathcal{E}_N,
\end{aligned}
\end{align}
where $\boldsymbol{\alpha}_{N+1} = (\alpha_0,\dots,\alpha_{N+1})^T$,
$\boldsymbol{\beta}_N = (\beta_0,\dots,\beta_N)^T$, and
\begin{align}
\begin{aligned}
\label{def:Joplus0}
\mathcal{J}_0^\oplus(\alpha_0,\beta_0&;y_{0 h}^c,
\boldsymbol{\tau}_{0 h}^c,u_{0 h}^c)
= \frac{1+\alpha_0}{2} \|y_{0 h}^c
- {y_d}_0^c\|_{\Omega}^2 + \frac{\lambda}{2} \|u_{0 h}^c\|_{\Omega}^2 \\
&+ \frac{(1+\alpha_0)(1+\beta_0) C_F^2 }{2 \alpha_0 \underline{\mu_1}^2}
\|{\mathcal{R}_2}^c_0 \|_{\Omega}^2
+ \frac{(1+\alpha_0)(1+\beta_0) C_F^4}{2 \alpha_0 \beta_0 \underline{\mu_1}^2}
\|{\mathcal{R}_1}^c_0 \|_{\Omega}^2,
\end{aligned}
\end{align}
\begin{align}
\begin{aligned}
\label{def:Joplusk}
\mathcal{J}_k^\oplus(\alpha_k,\beta_k&; \boldsymbol{y}_{k h},
\boldsymbol{\tau}_{k h}, \boldsymbol{u}_{k h})
= \frac{1+\alpha_k}{2} \|\boldsymbol{y}_{k h}
- {\boldsymbol{y}_d}_k \|_{\Omega}^2
+ \frac{\lambda}{2} \|\boldsymbol{u}_{k h} \|_{\Omega}^2 \\
&+ \frac{(1+\alpha_k)(1+\beta_k) C_F^2 }{2 \alpha_k \underline{\mu_1}^2}
\|{\mathcal{R}_2}_k \|_{\Omega}^2
+ \frac{(1+\alpha_k)(1+\beta_k) C_F^4}{2 \alpha_k \beta_k \underline{\mu_1}^2}
\|{\mathcal{R}_1}_k \|_{\Omega}^2.
\end{aligned}
\end{align}
Next, we have to reconstruct the fluxes $\boldsymbol{\tau}_{0 h}^c$ and
$\boldsymbol{\tau}_{k h}$ for all $k=1,\dots,N$, which we denote by
\begin{align*}
\boldsymbol{\tau}_{k h}
= R_h^{\text{\tiny{flux}}}(\nu \nabla \boldsymbol{y}_{k h}).
\end{align*}
This can be done by various techniques.
In \cite{LRW:LangerRepinWolfmayr:2015},
we have used Raviart-Thomas elements of the lowest order
(see also
\cite{PhD:RaviartThomas:1977, LRW:BrezziFortin:1991, LRW:RobertsThomas:1991}),
in order to regularize the fluxes by a post-processing operator, which maps the $L^2$-functions
into $H(\text{div},\Omega)$.
Collecting all the fluxes corresponding to the
modes together yields
the reconstructed flux
\begin{align*}
\boldsymbol{\tau}_{N h}
= R_h^{\text{\tiny{flux}}}(\nu \nabla y_{N h}).
\end{align*}
After performing a simple minimization of the majorant
$\mathcal{J}^\oplus$
with respect to
$\boldsymbol{\alpha}_{N+1}$ and $\boldsymbol{\beta}_{N}$,
we finally arrive at the {\it a posteriori} estimate
\begin{align}
\begin{aligned}
\mathcal{J}(y(u),u)
\leq \mathcal{J}^\oplus(\boldsymbol{\bar{\alpha}}_{N+1},\boldsymbol{\bar{\beta}}_{N};
y_{N h},\boldsymbol{\tau}_{N h},u_{N h}),
\end{aligned}
\end{align}
where $\boldsymbol{\bar{\alpha}}_{N+1}$ and $\boldsymbol{\bar{\beta}}_{N}$
denote the optimized positive parameters.
It is worth outlining that the
majorant $\mathcal{J}^{\oplus}$ provides a guaranteed upper bound for the cost functional,
and, due to Theorem \ref{theorem:majorantCostFuncInf}, the infimum of the majorant
coincides with the optimal value of the cost functional.
\begin{remark}
In this work,
problems with constraints on the control or the state are not considered,
but inequality constraints imposed on the Fourier coefficients of the control
can easily be included into the MhFE approach, see
\cite{PhD:KollmannKolmbauer:2011},
and, hence, may be considered in the {\it a posteriori} error analysis of parabolic
time-periodic optimal control problems as well.
\end{remark}
\section{Numerical Results}
\label{Sec6:NumericalResults}
We compute and analyze the efficiency of the above derived
{\it a posteriori} estimates for different cases, namely,
\begin{enumerate}
\item[{1.}] the desired state is periodic and analytic in time, but not time-harmonic,
\item[{2.}] the desired state is analytic in time, but not time-periodic, and
\item[{3.}] the desired state is a non-smooth function in space and time.
\end{enumerate}
Note that convergence and other properties of numerical approximations generated
by the MhFEM have been studied in
\cite{LRW:KollmannKolmbauerLangerWolfmayrZulehner:2013, LRW:LangerWolfmayr:2013}
for the same three cases.
The optimal control problem
(\ref{equation:minfunc:OCP})-(\ref{equation:forwardpde:OCP}) is solved
on the computational domain
$\Omega = (0,1) \times (0,1)$
with the Friedrichs constant $C_F = 1/(\sqrt{2}\pi)$
using a uniform simplicial mesh
and standard continuous, piecewise linear finite elements.
The material coefficients are supposed to be
$\sigma = \nu = 1$.
In the first two examples, we choose the cost parameter $\lambda = 0.1$, and
in the third one, we choose $\lambda = 0.01$. Both choices are common.
Getting sharp error bounds requires an efficient construction of
$\eta$, $\zeta$, $\boldsymbol{\tau}$ and $\boldsymbol{\rho}$
in order to compute sharp guaranteed bounds from the majorants.
We choose MhFE approximations
(\ref{definition:MhApproxEtaTau}) for $\eta$ and $\boldsymbol{\tau}$ as well as for
$\zeta$ and $\boldsymbol{\rho}$.
In order to obtain
suitable fluxes $\boldsymbol{\tau}, \boldsymbol{\rho} \in H(\text{div},Q_T)$,
we reconstruct them
by the standard lowest-order Raviart-Thomas ($RT^0$-) extension of normal fluxes.
We refer the reader to \cite{LRW:LangerRepinWolfmayr:2015}, where the authors
have discussed this issue thoroughly.
In order to solve the saddle point systems (\ref{equation:MultiFESysBlockk}) for $k=1,\dots,N$
and (\ref{equation:MultiFESysBlock0}) for $k=0$,
we use the robust algebraic multilevel preconditioner of Kraus (see \cite{LRW:Kraus:2012})
for an inexact realization of the block-diagonal preconditioners
\begin{align}
\label{equation:preconditionerSigmaPWConst:OCP}
\mathcal{P}_k = \left( \begin{array}{cccc}
D & 0 & 0 & 0 \\
0 & D & 0 & 0 \\
0 & 0 & \lambda^{-1} D & 0 \\
0 & 0 & 0 & \lambda^{-1} D \end{array} \right),
\end{align}
where
$D = \sqrt{\lambda} K_{h,\nu} + k \omega \sqrt{\lambda} M_{h,\sigma} + M_h,$
and
\begin{align}
\label{equation:preconditionerSigmaPWConstkIs0:OCP}
\mathcal{P}_0 = \left( \begin{array}{cc}
M_h + \sqrt{\lambda} K_{h,\nu} & 0 \\
0 & \lambda^{-1} (M_h + \sqrt{\lambda} K_{h,\nu}) \end{array} \right),
\end{align}
in the minimal residual method, respectively.
The preconditioners (\ref{equation:preconditionerSigmaPWConst:OCP}) and
(\ref{equation:preconditionerSigmaPWConstkIs0:OCP})
were presented and discussed in
\cite{LRW:KollmannKolmbauerLangerWolfmayrZulehner:2013, LRW:LangerWolfmayr:2013,
LRW:Wolfmayr:2014}.
The numerical experiments were computed on grids of
mesh sizes 16$\times$16 to 256$\times$256.
The algorithms were implemented in C\texttt{++}, and
all computations were performed on an average class
computer with Intel(R) Xeon(R) CPU W3680 @ 3.33 GHz.
Note that the
presented CPU times $t^{\text{sec}}$ in seconds
include the computational times for computing the majorants,
which are very small in comparison to the
computational times of the solver.
In {\bf Example 1}, we set the desired state
\begin{align*}
y_d(\boldsymbol{x},t) = \frac{e^t \sin (t)}{10} \left(\left(12+4 \pi ^4\right) \sin ^2(t)-6 \cos ^2(t)-6 \sin (t) \cos (t)\right)
\sin (x_1 \pi) \sin (x_2 \pi),
\end{align*}
where $T=2 \pi / \omega$ and $\omega = 1$.
This function is time-periodic and analytic, but not time-harmonic.
Hence, the truncated Fourier series approximation of
$y_d$ has to be computed for
applying the MhFEM as presented in Section \ref{Sec3:MhFEApprox}.
For that, the Fourier coefficients of $y_d$ can be computed analytically,
and, then, they are approximated by the FEM.
Next the systems (\ref{equation:MultiFESysBlockk}) and (\ref{equation:MultiFESysBlock0})
are solved for all $k \in \{0,\dots,N\}$. We choose the truncation index $N=8$.
Finally, we reconstruct the fluxes by a $RT^0$-extension
and compute the corresponding majorants.
The exact state is given by
$ y(\boldsymbol{x},t) = e^t \sin(t)^3 \sin (x_1 \pi) \sin (x_2 \pi) $.
In Table~\ref{tab:Ex1:0}, we present the CPU times $t^{\text{sec}}$,
the majorants
\begin{align*}
\begin{aligned}
\mathcal{M}^{\oplus_0}_{|\cdot|}
= \sqrt{2} &\Big(C_F \, \big(\|{\mathcal{R}_1}^c_0
\|_{\Omega}
+ \|{\mathcal{R}_3}^c_0
\|_{\Omega}
\big)
+ \|{\mathcal{R}_2}^c_0
\|_{\Omega}
+ \|{\mathcal{R}_4}^c_0
\|_{\Omega} \Big)
\end{aligned}
\end{align*}
and $\mathcal{J}^{\oplus}_0$ as defined in (\ref{def:Joplus0}) as well as
the corresponding efficiency indices
\begin{align*}
I_{\text{eff}}^{\mathcal{M},0} =
\frac{\mathcal{M}^{\oplus_0}_{|\cdot|}}{|(y_0^c,p_0^c) - (\eta_0^c,\zeta_0^c)|_{1,\Omega}}
\end{align*}
and $I_{\text{eff}}^{\mathcal{J},0} = \mathcal{J}^{\oplus}_0 / \mathcal{J}_0$
obtained on grids of different mesh sizes.
\begin{table}[!ht]
\begin{center}
\begin{tabular}{|c|ccccc|}
\hline
grid & $t^{\text{sec}}$
& $\mathcal{M}^{\oplus_0}_{|\cdot|}$
& $I_{\text{eff}}^{\mathcal{M},0}$
& $\mathcal{J}^{\oplus}_0$
& $I_{\text{eff}}^{\mathcal{J},0}$ \\
\hline
$16 \times 16$ & 0.02 & 1.75e+01 & 2.50 & 1.26e+05 & 1.01 \\
$32 \times 32$ & 0.08 & 8.20e+00 & 2.20 & 1.27e+05 & 1.00 \\
$64 \times 64$ & 0.35 & 3.92e+00 & 2.05 & 1.27e+05 & 1.00 \\
$128 \times 128$ & 1.62 & 1.91e+00 & 1.98 & 1.27e+05 & 1.00 \\
$256 \times 256$ & 7.00 & 9.44e-01 & 1.94 & 1.27e+05 & 1.00 \\
\hline
\end{tabular}
\end{center}
\caption{The majorants $\mathcal{M}^{\oplus_0}_{|\cdot|}$ and $\mathcal{J}^{\oplus}_0$, and
their efficiency indices (Example 1).}
\label{tab:Ex1:0}
\end{table}
Moreover, Table~\ref{tab:Ex1:1} presents
the CPU times $t^{\text{sec}}$, the majorants
\begin{align*}
\begin{aligned}
\mathcal{M}^{\oplus_k}_{|\cdot|}
= \sqrt{2} \Big(C_F \,
\big(\|{\mathcal{R}_1}_k \|_{\Omega}
+ \|{\mathcal{R}_3}_k \|_{\Omega}
\big) +\, \|{\mathcal{R}_2}_k\|_{\Omega}
+ \|{\mathcal{R}_4}_k
\|_{\Omega} \Big)
\end{aligned}
\end{align*}
and $\mathcal{J}^{\oplus}_k$ as defined in (\ref{def:Joplusk})
for $k=1$ and, finally, the corresponding efficiency indices
\begin{align*}
I_{\text{eff}}^{\mathcal{M},k} =
\frac{\mathcal{M}^{\oplus_k}_{|\cdot|}}{|(\boldsymbol{y}_k,\boldsymbol{p}_k)-
(\boldsymbol{\eta}_k,\boldsymbol{\zeta}_k)|_{1,\Omega}}
\end{align*}
and $I_{\text{eff}}^{\mathcal{J},k} = \mathcal{J}^{\oplus}_k / \mathcal{J}_k$
obtained on grids of different mesh sizes.
Similar results are obtained for larger $k$ as well, which is illustrated
in Table~\ref{tab:Ex1:global}.
This table compares
the results for the modes $k \in \{0,\dots,8\}$
computed on the $256 \times 256$-mesh
and presents
the overall functional error estimates.
For that,
the remainder term
$\mathcal{E}_N$
is precomputed exactly,
see Remark \ref{remark:remainderterm}.
It can be observed that the values of the majorants
$\mathcal{M}^{\oplus_k}_{|\cdot|}$ and
$\mathcal{J}^{\oplus}_{k}$
decrease for increasing $k$,
but that the values of the efficiency indices
are all about the same,
which is a demonstration for
the \textit{robustness} of the method with respect to the modes.
Note that the overall efficiency index for $N=6$ is large ($I_{\text{eff}}^{\mathcal{M}} = 3.15$)
compared to the efficiency indices corresponding to the single modes. The reason for that is
that the remainder term $\mathcal{E}_6 = 640.25$ is still quite large, and hence, more modes are needed.
For $N=8$, the remainder term is $\mathcal{E}_8 =106.07$,
which leads to a much better overall efficiency index ($I_{\text{eff}}^{\mathcal{M}} = 1.69$).
The value for the cost functional is however sufficiently accurate with $N=6$.
This example demonstrates
that the {\it a posteriori} error estimates clearly show what amount of
modes would be sufficient for representing the solution with a desired accuracy.
\begin{table}[!ht]
\begin{center}
\begin{tabular}{|c|ccccc|}
\hline
grid
& $t^{\text{sec}}$
& $\mathcal{M}^{\oplus_1}_{|\cdot|}$
& $I_{\text{eff}}^{\mathcal{M},1}$
& $\mathcal{J}^{\oplus}_1$
& $I_{\text{eff}}^{\mathcal{J},1}$ \\
\hline
$16 \times 16$ & 0.02 & 3.40e+01 & 2.50 & 4.74e+05 & 1.00 \\
$32 \times 32$ & 0.09 & 1.59e+01 & 2.20 & 4.79e+05 & 1.00 \\
$64 \times 64$ & 0.36 & 7.63e+00 & 2.05 & 4.80e+05 & 1.00 \\
$128 \times 128$ & 1.62 & 3.72e+00 & 1.98 & 4.80e+05 & 1.00 \\
$256 \times 256$ & 6.99 & 1.84e+00 & 1.94 & 4.80e+05 & 1.00 \\
\hline
\end{tabular}
\end{center}
\caption{The majorants $\mathcal{M}^{\oplus_1}_{|\cdot|}$ and $\mathcal{J}^{\oplus}_1$, and
their efficiency indices (Example 1).}
\label{tab:Ex1:1}
\end{table}
\begin{table}[!ht]
\begin{center}
\begin{tabular}{|c|ccccc|}
\hline
& $t^{\text{sec}}$
& $\mathcal{M}^{\oplus}_{|\cdot|}$
& $I_{\text{eff}}^{\mathcal{M}}$
& $\mathcal{J}^{\oplus}$
& $I_{\text{eff}}^{\mathcal{J}}$ \\
\hline
$k=0$ & 7.00 & 9.44e-01 & 1.94 & 1.27e+05 & 1.00 \\
$k=1$ & 6.99 & 1.84e+00 & 1.94 & 4.80e+05 & 1.00 \\
$k=2$ & 7.02 & 1.18e+00 & 1.94 & 1.99e+05 & 1.00 \\
$k=3$ & 7.17 & 6.78e-01 & 1.94 & 6.74e+04 & 1.00 \\
$k=4$ & 6.95 & 2.35e-01 & 1.92 & 8.42e+03 & 1.00 \\
$k=5$ & 7.01 & 8.57e-02 & 1.94 & 1.13e+03 & 1.00 \\
$k=6$ & 6.70 & 4.03e-02 & 1.87 & 2.29e+02 & 1.00 \\
$k=7$ & 6.77 & 2.13e-02 & 2.12 & 6.37e+01 & 1.03 \\
$k=8$ & 6.87 & 1.25e-02 & 2.18 & 2.19e+01 & 1.04 \\
\hline
overall with $N=6$ & -- & 1.03e+01 & 3.15 & 3.17e+06 & 1.00 \\
overall with $N=8$ & -- & 6.13e+00 & 1.69 & 3.17e+06 & 1.00 \\
\hline
\end{tabular}
\end{center}
\caption{The overall majorants $\mathcal{M}^{\oplus}_{|\cdot|}$ and $\mathcal{J}^{\oplus}$,
and their parts computed on a $256 \times 256$-mesh (Example 1).}
\label{tab:Ex1:global}
\end{table}
In {\bf Example 2}, we set
\begin{align*}
y_d(\boldsymbol{x},t) = \frac{e^t}{10} \left(-2 \cos(t)+ (10 + 4 \pi^4) \sin (t) \right)
\sin (x_1 \pi) \sin (x_2 \pi),
\end{align*}
where $T=2 \pi / \omega$ with $\omega = 1$.
It is easy to see that this function is time-analytic, but not time-periodic.
As in the first example, we
compute the MhFE approximation
of the desired state and solve the systems
(\ref{equation:MultiFESysBlockk}) and (\ref{equation:MultiFESysBlock0})
for all $k \in \{0,\dots,N\}$ with first $N=6$ and second $N=8$ being the truncation index.
Finally, we compute also the solutions for $N=10$.
The exact state is given by $ y(\boldsymbol{x},t) = e^t \sin(t) \sin (x_1 \pi) \sin (x_2 \pi) $.
The results related to computational expenditures and efficiency indices are quite similar
to those for Example 1. Therefore, we present only numerical results in the form similar to
Table \ref{tab:Ex1:global} (see
Table~\ref{tab:Ex2:global}).
In this numerical experiment, we again
observe good and satisfying efficiency indices for $\mathcal{M}^{\oplus_k}_{|\cdot|}$.
The remainder terms for $N=6$, $N=8$ and $N=10$ are
$\mathcal{E}_6 = 44094.84$, $\mathcal{E}_8 = 19869.30$ and
$\mathcal{E}_{10} = 10597.20$, respectively.
The efficiency index for $\mathcal{M}^{\oplus}_{|\cdot|}$ with $N=10$ improves a lot compared to
the index with $N=6$.
Note that -- as in the first example -- the efficiency indices for the cost functional are approximately one.
This again demonstrates the accurateness of the majorants for the cost functional.
\begin{table}[!ht]
\begin{center}
\begin{tabular}{|c|ccccc|}
\hline
& $t^{\text{sec}}$ & $\mathcal{M}^{\oplus}_{|\cdot|}$
& $I_{\text{eff}}^{\mathcal{M}}$
& $\mathcal{J}^{\oplus}$
& $I_{\text{eff}}^{\mathcal{J}}$ \\
\hline
$k=0$ & 6.86 & 1.58e+00 & 1.95 & 3.56e+05 & 1.00 \\
$k=1$ & 6.89 & 2.83e+00 & 1.95 & 1.14e+06 & 1.00 \\
$k=2$ & 6.79 & 1.41e+00 & 1.93 & 2.85e+05 & 1.00 \\
$k=3$ & 6.86 & 6.73e-01 & 1.89 & 6.69e+04 & 1.00 \\
$k=4$ & 6.76 & 3.78e-01 & 1.85 & 2.19e+04 & 1.00 \\
$k=5$ & 6.93 & 2.37e-01 & 1.78 & 9.05e+03 & 1.00 \\
$k=6$ & 6.83 & 1.60e-01 & 1.71 & 4.38e+03 & 1.00 \\
$k=7$ & 6.73 & 1.12e-01 & 1.62 & 2.37e+03 & 1.00 \\
$k=8$ & 6.99 & 8.28e-02 & 1.54 & 1.40e+03 & 1.00 \\
$k=9$ & 6.83 & 6.28e-02 & 1.45 & 8.74e+02 & 1.00 \\
$k=10$ & 6.87 & 4.88e-02 & 1.37 & 5.75e+02 & 1.00 \\
\hline
overall with $N=6$ & -- & 6.94e+02 & 2.56 & 7.06e+06 & 1.00 \\
overall with $N=8$ & -- & 4.75e+02 & 1.75 & 7.06e+06 & 1.00 \\
overall with $N=10$ & -- & 3.55e+02 & 1.31 & 7.06e+06 & 1.00 \\
\hline
\end{tabular}
\end{center}
\caption{The overall majorants $\mathcal{M}^{\oplus}_{|\cdot|}$ and $\mathcal{J}^{\oplus}$,
and their parts computed on a $256 \times 256$-mesh (Example 2).}
\label{tab:Ex2:global}
\end{table}
In {\bf Example 3}, we set
\begin{align*}
y_d(\boldsymbol{x},t) =
\chi_{[\frac{1}{4},\frac{3}{4}]}(t) \,\chi_{[\frac{1}{2},1]^2}(\boldsymbol{x}),
\end{align*}
where $\chi$ denotes the characteristic function
in space and time.
Let $T=1$, then $\omega = 2 \pi$.
Again the coefficients of the Fourier
expansion associated with $y_d$
can be found analytically. They are
\begin{align*}
{y_d^c}_k(\boldsymbol{x}) = \frac{
\left(-\sin(\frac{k \pi}{2})
+ \sin(\frac{3 k \pi}{2}) \right)}{k \pi} \, \chi_{[\frac{1}{2},1]^2}(\boldsymbol{x}),
\end{align*}
and ${y_d^s}_k(\boldsymbol{x}) = 0$ for all $k \in \mathbb{N}$.
For $k=0$, ${y_d^c}_0(\boldsymbol{x}) = \chi_{[\frac{1}{2},1]^2}(\boldsymbol{x})/2$.
Since the exact solution cannot be
computed analytically, we compute its MhFE approximation on a finer mesh
($512 \times 512$-mesh).
Since the modes ${y_d^c}_k(\boldsymbol{x}) = 0$ for all even $k \in \mathbb{N}$,
it suffices to show the results for odd modes as well as for $k=0$.
Table \ref{tab:Ex3:global} presents the results with
truncation index $N=23$, since the results regarding the efficiency indices are similar
for higher modes.
The computational times presented include
the times for computing the
approximations of the exact modes on the finer mesh.
The numerical results again show
the efficiency of the majorants for both, the discretization error and the cost functional.
This is especially observed for the majorant related to the cost functional, which is
very close to the exact value (in spite of a really complicated $y_d$).
The majorant $\mathcal{M}^{\oplus}_{|\cdot|}$ exposes an overestimation
but anyway provides realistic estimates of the errors in the state and control functions
measured in terms of the combined error norm.
\begin{table}[!ht]
\begin{center}
\begin{tabular}{|c|ccccc|}
\hline
& $t^{\text{sec}}$
& $\mathcal{M}^{\oplus_k}_{|\cdot|}$
& $I_{\text{eff}}^{\mathcal{M}}$
& $\mathcal{J}^{\oplus_k}$
& $I_{\text{eff}}^{\mathcal{J}}$ \\
\hline
$k=0$ & 38.60
& 3.96e+02 & 3.64 & 8.20e+04 & 1.32 \\
$k=1$ & 38.88
& 4.80e+02 & 3.73 & 1.31e+05 & 1.30 \\
$k=3$ & 38.82
& 1.22e+02 & 2.52 & 1.35e+04 & 1.21 \\
$k=5$ & 38.98
& 5.58e+01 & 2.41 & 4.62e+03 & 1.14 \\
$k=7$ & 38.64
& 3.22e+00 & 2.45 & 2.28e+03 & 1.11 \\
$k=9$ & 39.06
& 2.12e+01 & 2.51 & 1.35e+03 & 1.08 \\
$k=11$ & 38.92
& 1.50e+01 & 2.55 & 8.89e+02 & 1.07 \\
$k=13$ & 39.13
& 1.14e+01 & 2.62 & 6.30e+02 & 1.06 \\
$k=15$ & 38.59
& 8.93e+00 & 2.63 & 4.68e+02 & 1.04 \\
$k=17$ & 38.58
& 7.36e+00 & 2.70 & 3.63e+02 & 1.04 \\
$k=19$ & 38.78
& 6.06e+00 & 2.69 & 2.88e+02 & 1.03 \\
$k=21$ & 38.72
& 5.27e+00 & 2.78 & 2.36e+02 & 1.03 \\
$k=23$ & 38.87
& 4.47e+00 & 2.74 & 1.96e+02 & 1.02 \\
\hline
\end{tabular}
\end{center}
\caption{The majorants and corresponding efficiency indices
computed on a $256 \times 256$-mesh (Example 3).}
\label{tab:Ex3:global}
\end{table}
\section{Conclusions and Outlook}
\label{Sec7:ConclusionsOutlook}
In \cite{LRW:LangerRepinWolfmayr:2015}, the authors derived functional-type
{\it a posteriori} error estimates for MhFE
approximations to linear parabolic time-periodic boundary value problems.
In this work, we extend this technique
to the derivation of {\it a posteriori} error estimates for MhFE
solutions of the corresponding distributed optimal control problem
which leads to additional challenges in the analysis.
The reduced optimality system is nothing but a coupled
parabolic time-periodic PDE system for the state and
the adjoint state. We are not only interested in computable
{\it a posteriori} error bounds for the state, the adjoint state
and the control, but also for the cost functional.
In case of linear time-periodic parabolic constraints,
the approximation via MhFE functions
leads to the decoupling of computations related to different modes.
Due to this feature of the MhFEM,
we can in principle use different meshes for different modes and
independently generate them by adaptive
finite element approximations of the respective Fourier coefficients.
To assure the quality of approximations constructed in this way,
we need
fully computable {\it a posteriori} estimates,
which provide guaranteed bounds of global
errors and reliable indicators of errors associated with the modes.
Then, by prescribing certain bounds, we can finally
filter out the Fourier coefficients, which are important for the
numerical solution of the problem.
This technology will lead to an
\emph{adaptive multiharmonic finite element method (AMhFEM)}
that will provide complete adaptivity in space and time.
The development and the analysis of such an AMhFEM
goes beyond the scope of this paper,
but will heavily be based on the results of this paper
as described above.
It is clear that the functional {\it a posteriori} estimates derived here for
time-harmonic parabolic optimal control problems can also be
obtained for distributed time-harmonic eddy current optimal control problems
as studied in
\cite{PhD:Kolmbauer:2012c, PhD:KolmbauerLanger:2012, PhD:KolmbauerLanger:2013}.
\section*{Acknowledgments}
The authors gratefully acknowledge the financial support by the Austrian Science Fund
(FWF) under the grants NST-0001 and W1214-N15 (project DK4), the Johannes Kepler University of Linz and
the Federal State of Upper Austria.
\bibliographystyle{abbrv}
|
1,314,259,993,568 | arxiv | \section{Introduction}
The goal of Text Simplification (TS) is to produce easily understandable texts that benefit disadvantaged readers such as children, dyslexic, or language learners. Simplifications are often generated by adapting a source text written for adult/native readers.
However, recent work in simplification has mostly addressed TS as a monolingual translation task, where individual sentences are "translated" into a simplified version \cite{zhu-etal-2010-monolingual,coster-kauchak-2011-simple,hwang-etal-2015-aligning}.
The main focus is put on either lexicographic replacements, paraphrasing, sentence splitting, or the dropping of words within a single sentence \cite{amancio-specia-2014-analysis}, which implies that the simplification of any input document will consist of roughly the same number of sentences.
While this approach is appropriate for sufficiently short source documents, longer articles become strenuous for disadvantaged readers. As can be seen in \Cref{tab:corpusstats}, articles in different corpora come with varying lengths of their respective source texts. When simplifications are generated via manual sentence-by-sentence translations, the simplified texts tend to have more sentences than the source documents. When alignments are constructed from a source and simplification text on the same topic instead, they exhibit a drastic length disparity.
Current simplification systems are, however, inherently limited in their ability to address the problem of joint simplification and summarization from much longer input documents.
Sentence-level alignments were traditionally seen as one way to circumvent certain problems in TS, namely:
\begin{enumerate}
\item Human feedback for judging simplification quality is more consistent for sentences, compared to longer samples, such as entire documents.
\item Metrics such as BLEU \cite{papineni-etal-2002-bleu} or SARI \cite{xu-etal-2016-optimizing} rely on (aligned) reference texts for automated evaluation.
\item Prior alignment of sentences limits the length of input samples, which is essential for algorithms with non-linear runtime, or length constraints.
\end{enumerate}
\begin{table}[t!]
\centering
\begin{tabular}{l@{\hspace{-0.8em}}r|r|r}
\hline
& \textbf{Aligned} & \multicolumn{2}{c}{\textbf{Avg.~\#Sentences}} \\
\textbf{Resource} & \textbf{Articles} & \textbf{Source} & \textbf{Simple}\\
\hline
Klexikon (Ours) & $2{,}898$ & $242.09$ & $32.51$ \\
\cite{hewett-stede-2021-automatically} & $978$ & $10.12$ & $43.54$ \\
\cite{battisti-etal-2020-corpus}$^*$ & $378$ & $45.29$ & $55.75$ \\
\hline
\cite{kauchak-2013-improving} & $59{,}775$ & $64.52$ & $8.46$ \\
\cite{xu-etal-2015-problems}$^*$ & $1{,}130$ & $49.59$ & $51.27$ \\
\end{tabular}
\caption
Corpus statistics for datasets with document alignments in German (top) and English (bottom). $^*$ indicates resources created by simplifying articles sentence-by-sentence. For \protect\cite{xu-etal-2015-problems,hewett-stede-2021-automatically}, we refer to the respective simplified corpora with simplification level 1.}
\label{tab:corpusstats}
\end{table}
In this work, we present remedies to the problem of missing document alignments, and argue that the inclusion of summarization into the broader context of Text Simplification is a necessary step towards end-to-end solutions for longer input texts. Specifically, it addresses the following problems:
\begin{enumerate}
\item Long-form documents can be compressed into significantly shorter summarized simplifications.
\item Document alignments provide context for models that are otherwise based on single sentence pairs.
\item The amount of accessible training data increases, which is especially important for languages other than English, where data is generally scarce.
\end{enumerate}
Simultaneously, TS offers interesting challenges to the summarization community, which hopefully facilitates exchange between the two fields:
On existing summarization datasets, simply taking the leading three sentences offers strikingly good results \cite{nallapati2017summarunner}, which may lead to systems learning specific extractive strategies instead of generalizing to broader textual relevance. Preliminary experiments show that our dataset poses a harder challenge for summarization systems, due to the additional simplification aspect.
Our proposed resource was obtained from semi-automated alignments between the German Wikipedia and the children's encyclopedia "Klexikon" \cite{schulte-dijk-2015-free}, written for children aged 8 to 13 years.\footnote{\url{https://klexikon.zum.de/wiki/Hilfe:Grunds\%C3\%A4tze}, accessed: 15.01.2022}
With almost $2{,}900$ articles, it is the largest non-English resource with document alignments.
\section{Related Work}
Related work can broadly be categorized into relevant simplification work, and associated works on resources for (German) summarization datasets.
\subsection{Text Simplification}
Previously mentioned work frequently deals with data aligned based on Simple Wikipedia \cite{zhu-etal-2010-monolingual,coster-kauchak-2011-simple,hwang-etal-2015-aligning}. The main differences between these approaches lie in their alignment strategies and underlying simplification model.
The only work on Simple Wikipedia that specifically introduces a document-aligned version is \cite{kauchak-2013-improving}, who investigates performance gains from supplementing language models with additional (non-simplified) texts. Importantly, it is not explicitly used for learning simplification.
\cite{hancke-etal-2012-readability} introduced a first German resource containing simplified texts based on unaligned articles from GEO and GEOlino, a German magazine similar to National Geographic, and its edition specifically for children.
They build a classification system that is able to classify between normal and simplified texts for several article categories.
A larger and improved version from the same source was collected by \cite{weiss-meurers-2018-modeling}, who also introduce a resource based on transcripts from German TV broadcasts (Tagesschau/Logo!), again without any alignment.
The first mention of an aligned corpus for German can be found in \cite{klaper-etal-2013-building}, who automatically align websites with their corresponding versions in accessible language. Their corpus contains a total of about 270 articles.
Most recently, \cite{battisti-etal-2020-corpus} collected a larger corpus, where 378 texts contain document alignments. Arguably, unaligned resources might still be helpful to facilitate pre-training of models.
In an attempt to circumvent data scarcity, \cite{mallinson-etal-2020-zero} employ multi-lingual pre-training, which they tested with a small, manually labeled German evaluation set.
To our knowledge, \cite{hewett-stede-2021-automatically} were the first to utilize alignments between Wikipedia and Klexikon, with an additional extension to MiniKlexikon, a secondary simplification level. Due to the further required alignments, the overall size of their data is about 10\% of our presented corpus. To avoid problems stemming from extreme length discrepancies, they also only extract introduction and abstracts for Wikipedia articles, which is something we explicitly encourage in our version. This also explains the different lengths while using the same document sources, as reported in \Cref{tab:corpusstats}~.
\subsection{Summarization}
\cite{parmanto2005access} are the first to explicitly explore summarization and simplification in a common context, albeit for the task of website accessibility.
Further work models summarization itself as a simplification technique, e.g.,
\cite{margarido2008automatic} investigated extractive summarization approaches and how they help disadvantaged readers. A similar experiment was conducted by \cite{smith-jonsson-2011-automatic} for Swedish texts, who find summarized texts to be more readable as well.
Also dealing with extractive summarizers, \cite{finegan2016sentence} look at simplifications in the biomedical and legal domain, but their findings indicate that altered sentences lead to fewer correctly answered questions by domain experts.
Simplification has also been suggested for multi-document summarization:
\cite{siddharthan-etal-2004-syntactic} select relevance exclusively over syntactically simplified sentences, whereas other works use simplification as an alternative to regular sentence selection \cite{vanderwende2006beyond,yih2007multi}.
Closest to a unified framework is the work by \cite{ma2017semantic}, who use the same neural encoder-decoder architecture for separate simplification and summarization tasks, which highlights the shared similarities in terms of shared model architectures and training.
To our knowledge, there exist few resources for German single-document summarization. \cite{nitsche2019abstractive} mention a (private) resource, provided by the German Press Agency (dpa), which uses headlines as target summaries. \cite{frefel-2020-summarization} generate a corpus based on German Wikipedia articles, and treat the overview paragraph at the beginning as the summary of the article. A similar approach including cross-lingual alignments between English and German has also been recently published \cite{fatima-strube-2021-novel}~.
\section{Text Simplification with Joint Summarization}
\begin{figure*}[ht]
\centering
\includegraphics[width=0.32\textwidth]{histogram_wiki}
\includegraphics[width=0.32\textwidth]{histogram_klexikon}
\includegraphics[width=0.32\textwidth]{histogram_ratios}
\caption{Histogram of our Klexikon dataset by number of sentences. Displayed are the distribution for source texts (left; bin width 50), simplified articles (center; bin width 5), and compression ratio of source over simplified lengths (right; bin width 2). Vertical lines represent median length (continuous orange), mean length (dashed black) and one standard deviation (dotted black).}
\label{fig:stats}
\end{figure*}
As previous work has shown, summarization in itself can already be considered a weaker form of simplification \cite{margarido2008automatic,smith-jonsson-2011-automatic}, although existing work never formalizes TS as a summarization problem.
Several points have to be addressed by both simplification and summarization components for a full end-to-end solution. In this section, we outline suggestions for a unified system design.
\subsection{Considerations for Simplification}
As previously stated, current simplification systems cannot generate significantly shorter output texts when simplifying individual sentences.
This is mainly due to the sentence-aligned training setup instead of training with the entire input document.
Further, this drops a sizable portion of the source text from training, since sentences are only considered when they align directly with a simplified part.
Several resources also lack a document alignment altogether, which completely precludes them from being used as a training resource for end-to-end systems.
Importantly, relevance of individual segments (sentences or paragraphs) has to be computed \emph{without knowledge about the output corpus}.
This can, for example, be achieved by pre-training strategies on monolingual corpora \cite{mallinson-etal-2020-zero}, but could otherwise be learned as an intermediate step in neural architectures.
This has been previously shown to work well for multi-document summarization \cite{liu-lapata-2019-hierarchical}, where paragraph relevance was learned across several documents.
Further, existing manually annotated corpora are frequently generating simplifications of short texts by "translating" sentence-by-sentence. This reinforces the bias towards equally long documents, which cannot be observed in post-aligned resources (i.e., where existing simplified texts were written independently on the same topic, cf. \Cref{tab:corpusstats}).
An amended assumption is that simplifications may only be \emph{up to a certain length}, due to varying attention spans of the target groups. This then requires additional "simplification" based on the length of the source document.
This could also be used as a parameter to model levels of difficulty, which is available for some resources, see the Newsela corpus \cite{xu-etal-2015-problems}.
Lastly, existing evaluation metrics strictly focus on sentence-level references \cite{xu-etal-2016-optimizing}. Extending system evaluations to document-level simplifications poses challenges that need to be overcome in order to collect both manual and automated feedback on the simplification quality.
\subsection{Considerations for Summarization}
For summarization, TS offers additional challenges not considered by current works.
Given a high enough compression rate, simplification can be seen as a special case of summarization. However, existing metrics, such as ROUGE \cite{lin-2004-rouge}, rely on the re-appearance of $n$-grams in the target summary (in our case, the simplification). This is not guaranteed, given that the simplification can appear in the form of lexicographic replacements.
It is thus unclear whether simplification should be considered a separate criterion or jointly modeled for the evaluation of summaries, specifically when considering other input factors as well \cite{ter2020makes}.
Additionally, the varying vocabulary and sentence structure pose a challenge to summarization systems, especially extractive approaches. See \Cref{sec:baseline} for experiments on our Klexikon corpus.
Previous work in that direction has mostly dealt with sentence-level lexicographic simplifications \cite{siddharthan-etal-2004-syntactic}, yet there are several other simplification operations to be considered \cite{amancio-specia-2014-analysis} in a joint end-to-end system.
\section{Klexikon Dataset}
We introduce a new dataset, loosely inspired in its construction by English Simple Wikipedia, to facilitate future research in joint simplification and summarization.
Specifically, we use the German children's encyclopedia "Klexikon" to obtain simplifications, and align them with reference articles from the German Wikipedia.
Compared to Simple Wikipedia, which can be freely edited, Klexikon specifically targets children between roughly the age of 8-13 as readers, and follows a strict reviewing procedure for individual articles, resulting in higher quality texts.
We only consider Wikipedia articles with a minimum length of 15 paragraphs, which helps to filter out disambiguation pages or stubs´. Additionally, this results in a clear contrast in overall article length between source and simplified texts (cf. \Cref{tab:corpusstats} and \Cref{fig:stats}).
The final dataset consists of $2{,}898$ article pairs, with Wikipedia documents having on average $8.94$ times more sentences compared to their Klexikon counterparts.
\subsection{Corpus Creation}
All manual steps during corpus creation were performed by the first author of this work.
We begin the extraction based on the list of all available articles from the Klexikon overview page in April 2021\footnote{\url{https://klexikon.zum.de/wiki/Kategorie:Klexikon-Artikel}, accessed 14.04.2021}~. At the time of experimentation, this returned 3{,}150 Klexikon articles, although more articles have been added since\footnote{In April 2022, the number of available articles has increased to 3{,}269.}.
\subsubsection{Document Alignment Strategy}
For the identification of matching articles between German Wikipedia and Klexikon, the following steps were performed:
\begin{enumerate}
\item Querying the MediaWiki Search API\footnote{\url{https://www.mediawiki.org/wiki/API:Search}, accessed: 14.04.2021} with the title of the Klexikon article.
2{,}861 articles, or around 90\%, have an entry with a directly matching heading on Wikipedia.
However, this may include disambiguation pages or stubs.
\item All remaining 289 unmatched articles are manually matched against the top five suggestions by the Wikimedia Search API.
If no candidate article is appropriate, the entry is dropped from the corpus.
\item Wikipedia articles with less than 15 paragraphs (108 articles) are again flagged and manually reviewed. Short Wikipedia entries may correspond to disambiguation pages (see next step), or are otherwise dropped because of their short length.
\item Disambiguation pages are replaced with a specific Wikipedia page, if it topically matches at least 66\% of the Klexikon paragraphs.
\footnote{For example, the Klexikon article for "Adler" (eagle) primarily talks about the animal, which is then chosen as the corresponding page in Wikipedia.}
\end{enumerate}
\subsubsection{Text Extraction}
The Klexikon website runs on the Wiki software, which makes text extraction across platforms very similar. For both websites, we extract all direct children elements of the main content block (div-class: \texttt{mw-parser-output}).
Of those, we only use text within \texttt{<p>} tags as the main paragraph content, and heading elements \texttt{<h1>}-\texttt{<h5>}.
This simultaneously discards non-textual contents, e.g., images, as well as malformed text elements, such as image captions or lists.
We note that the removal of lists can also remove valid content, but frequently suffers from inconsistent grammatical correctness; while some bullet lists are equivalent to a self-contained paragraph, more often than not, it simply contains enumerations.
Further limitations for summarization include the potential content split on Wikipedia. For example, in the Klexikon article about the city of \emph{Aarhus}, there is explicit information about the ARoS (Aarhus art museum); however, on Wikipedia, this information would be found in the article about the \emph{museum itself}, and not in the page about the city. For now, we defer these edge cases to future extensions including multi-document summarization/simplification.
To avoid encoding errors, we drop any character that appears less than 100 times in the corpus; more frequently appearing special characters are mapped to the closest latin character (e.g., \emph{\'{a}} to \emph{a}), with the exception of \emph{äöüß}, which are part of the standard German alphabet. In the absence of a close mapping (e.g., for Cyrillic letters), the character is dropped as well.
This assumes that foreign characters are irrelevant for simplified texts, which we can indeed observe from the utilized character set in Klexikon articles.
We process the raw text with spaCy's\footnote{\url{https://spacy.io}~, version 3.2} \texttt{de-core-news-md} model to separate sentences.
Our final data format maintains the following document representation:
\begin{enumerate}
\item Line-by-line sentence representations based on spaCy boundary detection,
\item Additional indication of separation of paragraphs (original \texttt{<p>} elements), and
\item Highlighted headings according to the indicated level (heading, subheading, etc.), available primarily for the Wikipedia documents.
\end{enumerate}
A statistical view of the corpus can be found in \Cref{tab:detailed}~.
\begin{table}[t]
\centering
\begin{tabular}{lr|r}
\hline
\textbf{} & \textbf{Wikipedia} & \textbf{Klexikon} \\
\hline
Documents & $2{,}898$ & $2{,}898$ \\
\hline
Average sentences & $242.09$ & $32.51$ \\
SD sentences & $227.39$ & $19.73$ \\
Median sentences & $162$ & $26$ \\
\hline
Average tokens & $5{,}442.83$ & $436.87$ \\
SD tokens & $5{,}093.82$ & $270.00$ \\
Median tokens & $3{,}705$ & $347$ \\
\end{tabular}
\caption{Corpus statistics of the Klexikon dataset. SD refers to one standard deviation.}
\label{tab:detailed}
\end{table}
\subsubsection{Sentence Alignments}
We also experimented with the creation of an automatically sentence-aligned variant of our data set.
Unfortunately, existing alignment algorithms from the TS community are not applicable here. CATS~\cite{stajner-etal-2018-cats} is one representative from the class of greedy alignment algorithms; these base their alignments on the assumption that a similar order of the content exists for both the source and simplification texts. This does not apply to our dataset, since texts have been written independently.
Algorithms with non-greedy alignment strategies exist~\cite{paetzold-etal-2017-massalign,jiang-etal-2020-neural}, but lack compatibility with German texts.
We instead experimented with alignments based on sentence embeddings from sentence-transformers~\cite{reimers-gurevych-2019-sentence}\footnote{\texttt{paraphrase-multilingual-mpnet-base-v2}, a multilingual variant also suitable for German texts.}, and selecting the most similar source sentence (or pair of sentences) for each Klexikon sentence.
However, sentence splitting and merging are impossible to model with this naive alignment strategy, but were frequently found to be the issue of sub-par alignments in a manual review of preliminary results. In particular, we also note that there were both cases of several relevant Wikipedia sentences for a single Klexikon sentence (highlighting the importance of a notion of "relevance"), as well as instances of long sentences from Wikipedia splitting into several (non-consecutive) sentences in the Klexikon text.
\subsection{Comparison to Existing Resources}
The only other two German datasets with document alignments are the recent resource by \cite{battisti-etal-2020-corpus}, as well as a smaller version of Klexikon data by \cite{hewett-stede-2021-automatically}. \cite{battisti-etal-2020-corpus} compiled documents from accessibility options on websites. Compared to our dataset, they potentially cover a more heterogeneous set of topics, but only provide alignments for a subset of articles.
As mentioned before, \cite{hewett-stede-2021-automatically} provide additional alignments to MiniKlexikon, and otherwise limit the maximum length of articles, which reduces the number of available alignments between all three resources to 295 documents. Even when considering only the equivalent Klexikon-Wikipedia alignments, there are less than $1{,}000$ documents, with additional constraints to the completeness of the Wikipedia texts.
Concerns raised about the quality of Wikipedia as a resource \cite{xu-etal-2015-problems} mention the problems with sentence alignment, inadequate simplifications, and poor generalization. Our version of the Klexikon dataset partially alleviates these issues:
\begin{enumerate}
\item We provide document and (automated) sentence alignments, which allows focusing on both summarization and simplification in a joint manner.
\item Articles for Klexikon are written following stricter guidelines both in their content structure, and we include stricter pre-processing criteria for the Wikipedia articles, resulting in a high-quality collection of text documents.
\item We provide sufficient training samples for potential neural approaches, by increasing the Klexikon-based resource to almost $2{,}900$ articles.
\end{enumerate}
\subsection{Baseline Performance}
\label{sec:baseline}
To quantify the quality of our automatically generated alignments, we investigate the dataset from both a summarization and simplification perspective.
\subsubsection{Summarization}
To verify the suitability of our corpus for \emph{summarization} purposes, we computed several baselines and compared them to the Klexikon articles as a presumable gold standard summary:
\begin{enumerate}
\item \textbf{Lead-$3$:} A baseline frequently used in news article summarization, which consists of the first three sentences. In our case, this corresponds to the first three sentences of the Wikipedia article.
\item \textbf{Lead-$k$:} A related baseline, taking all sentences of the overview section in the Wikipedia article.
\item \textbf{Full article:} The full Wikipedia article as a reference for the maximum possible vocabulary overlap (this corresponds to ROUGE-1 recall).
\item \textbf{ROUGE-2 oracle:} As an approximation of the upper limit for extractive summaries on this dataset, we select the sentence maximizing ROUGE-2 F1 scores for each sentence in the Klexikon article and
\item \textbf{Luhn:} A simple unsupervised baseline for extractive summaries can be generated by Luhn's algorithm~\cite{luhn-1958-automatic}. We use a target of 25 extracted sentences for each generated summary, which corresponds roughly to the median number of sentences in the Klexikon articles.
\item \textbf{LexRank S-T:} As a more sophisticated baseline, this approach supplies LexRank~\cite{erkan-radev-2004-lexrank} with embeddings extracted by \texttt{sentence-transformers}~\cite{reimers-gurevych-2019-sentence}\footnote{The same model as mentioned in footnote 7 is used.}~. The length is similarly limited to at most 25 extracted sentences.
\end{enumerate}
We use ROUGE \cite{lin-2004-rouge} to gauge summarization quality, which evaluates $n$-gram overlap between system outputs and gold references. In particular, we report F1 scores for ROUGE-1, ROUGE-2 and ROUGE-L.
Results in \Cref{tab:rouge} indicate that our dataset poses a significantly harder challenge compared to performance of baselines on standard summarization corpora, such as CNN/DailyMail \cite{nallapati2017summarunner}, where simple lead-3 baselines obtain extremely high ROUGE scores due to an overly pronounced lead bias.
On our dataset, lead-$3$ likely struggles with the very different output lengths and comparatively low recall scores; the opposite is true for the full article baseline, which does not summarize at all, and therefore scores poorly in terms of precision.
However, the full article baseline obtains a recall score of $77.3\%$ ROUGE-1, implying there is still a sizable vocabulary overlap between the Klexikon and Wikipedia articles.
With proper summarization methods, it is therefore possible to produce decent ROUGE scores, and another indicator of the corpus' suitability to summarization.
Best-suited as a baseline is lead-$k$, which is a decent approximation of the actual target article length. Even so, lead-$k$ is shorter than the corresponding Klexikon articles.
Based on these results, coupled with varying compression levels between articles (cf. \Cref{fig:stats}~), a high sensitivity to the overall input length seems to be required in order to generate appropriate summaries.
From the extractive summaries generated by unsupervised methods, it becomes obvious that content from sections outside the overview paragraph is beneficial in terms of ROUGE scores, which is a promising distinction from other summarization datasets, especially in German.
Finally, the ROUGE-2 oracle gives insights into the limitations of extractive summarization methods on this dataset. In particular, the differing expressiveness and vocabulary impacts the achievable ROUGE-2 and ROUGE-L scores. It should be noted, however, that the determination of output lengths seems to play a crucial role in the overall balance between precision and recall scores. Given that both unsupervised baselines work with informed choices of the expected summary length, their results should also be taken within the correct context.
\begin{table}
\centering
\begin{tabular}{lr|r|r}
\hline
\textbf{} & \textbf{R-1} & \textbf{R-2} & \textbf{R-L}\\
\hline
Lead-$3$ & $16.95$ & $3.77$ & $9.81$ \\
Lead-$k$ & $24.87$ & $5.10$ & $12.01$ \\
Full article & $16.81$ & $4.23$ & $6.95$ \\
\hline
ROUGE-2 oracle & $41.85$ & $10.68$ & $16.00$ \\
\hline
Luhn & $31.86$ & $5.55$ & $11.57$ \\
LexRank S-T & $33.90$ & $6.11$ & $12.86$ \\
\end{tabular}
\caption{Average ROUGE F1 for simple extractive baselines. $95\%$ confidence intervals for all scores differ by less than one point.}
\label{tab:rouge}
\end{table}
\subsubsection{Simplification}
We further provide different metrics to estimate the level of simplification present in the available documents.
For this, we compute Flesch reading-ease scores \cite{flesch1948new}, specifically an adjusted variation for German \cite{amstad1978verstandlich}.
In addition, we hypothesize that the average sentence length (in tokens), as well as the average number of characters per words are suitable proxies for simplification.
The latter is especially important for German, which is famous for its long compound words. In particular, we limit the word length calculation to "content word classes", i.e., nouns, verbs, adjectives, and adverbs only. \\
To cover lexicographic peculiarities in the data, we estimate the underlying vocabulary. Notably, the overall texts are quite different in lengths, so an absolute count of distinct tokens would heavily bias the results on Wikipedia.
Instead, we approximate this problem by looking at corpus-specific lemma coverage. By computing a corpus-specific list of the 1000 most frequently occurring lemmas, we are then able to compute what fraction of all used lemmas is contained in this top-1000 list.
A higher percentage likely points to fewer rare words used, and greater reliance on commonly understood words or an overall smaller vocabulary.
Indeed, we find a consistent pattern in our data (cf.~\Cref{tab:simpler}), where Klexikon data indicates simpler language on all our metrics, which confirms the suitability of our dataset for \emph{simplification} tasks.
We would like to point out the general consensus of the field that heuristics are only scratching the surface of representative readability judgments \cite{chall1958readability}, but still offer a chance for initial exploratory analysis of data suitability.
\begin{table}[t]
\centering
\begin{tabular}{lr|r}
\hline
\textbf{} & \textbf{Wikipedia} & \textbf{Klexikon} \\
\hline
Avg.~Flesch score & $40.1 \pm 7.3$ & $66.7 \pm 6.0$ \\
Avg.~sentence length & $22.7 \pm 2.6$ & $13.5 \pm 1.5$ \\
Avg.~word length & $8.7 \pm 4.0$ & $6.9 \pm 3.0$ \\
Share of top 1000 lemmas & $68.8$\% & $82.3$\% \\
\end{tabular}
\caption{Indicators of simplified target texts: averages for Flesch complexity scores (between 0 to 100; higher scores indicate simpler texts); average sentence length in tokens; average word length in characters (nouns, verbs, adjectives, adverbs); percentage share of occurrences of the top-1000 corpus-specific lemmas.}
\label{tab:simpler}
\end{table}
\section{Conclusion and Future Work}
In this work, we laid out basic requirements for a unified Text Simplification and Summarization framework.
Specifically, we also provided a document-aligned resource of German texts to facilitate future research in this area, and provide quantitative evidence of the suitability of our dataset.
We see the following points as the most critical issues for successful joint models:
\begin{inparaenum}[i)]
\item Learned sentence relevance and simplification in a joint setting. This can be potentially achieved by modeling sentence alignments similar to existing methods~ \cite{stajner-etal-2018-cats,jiang-etal-2020-neural}, but already during the training of an end-to-end system, instead of a separate pre-processing step.
\item Implementation of automated evaluation metrics that align both with human judgments of appropriateness for the summary, as well as simplification steps taken. ROUGE, based on $n$-grams, potentially suffers similar shortcomings to BLEU as an evaluation metric, since it fails to capture lexicographic simplifications. Existing simplification metrics, however, are unable to quantize the quality based on much longer source documents.
\item Extension of current abstractive summarization systems towards lexicographic simplification, potentially in the form of regularization during training.
\end{inparaenum}
\section{Bibliographical References}\label{reference}
\bibliographystyle{lrec2022-bib}
|
1,314,259,993,569 | arxiv | \section{Introduction}
Deep learning approaches to speech enhancement represent a significant leap in performance over previous approaches, such as the decision-directed (DD) approach \cite{1164453}. Multi-layer perceptrons (MLPs) were amongst the first artificial neural networks (ANNs) used for speech enhancement \cite{xu2017multi}. Recurrent neural networks (RNNs) employing long short-term memory (LSTM) cells provided a higher performance at the cost of parameter inefficiency and extensive training times \cite{chen2017long}. Convolutional neural networks (CNNs) were able to match the performance of LSTM networks, with fewer parameters and a reduction in training time \cite{Park2017}. LSTM networks were not outperformed until the introduction of residual \cite{he2016identity,wavenet} and densely connected \cite{huang_densely_2017,dense_SE} CNNs, as well as causal dilated convolutional units \cite{bai2018empirical}. A residual CNN (ResNet) aggregates layer outputs via a summation operation, which is given as input to deeper layers. A densely connected CNN (DenseNet) differs by aggregating layer outputs via a concatenation operation. Other ANNs that have been successfully applied to speech enhancement include generative adversarial networks (GANs) and encoder-decoder CNNs \cite{segan}.
Residual and dense aggregations of layer outputs have been found to benefit training. Residual links improve gradient flow during backpropagation \cite{he2016identity} and prevent the vanishing and exploding gradient problems \cite{279181}. This allows the training of very deep neural networks. Dense aggregations offer direct feature re-usage, as deeper layers have access to the outputs of shallower layers \cite{huang_densely_2017}. This allows a layer to explore a larger set of features during training. Despite the success of both ResNets and DenseNets, both aggregation types have drawbacks. For ResNets, information from the outputs of shallower layers can be lost after multiple summations with deeper layer outputs \cite{sparse_aggCNN}. This restricts feature re-usage and limits feature exploration during training. For DenseNets, while the concatenation of all previous layer outputs yields total feature re-usage, a significant number of parameters are unexploited due to large input sizes \cite{MLN}. This is exemplified in Figure \ref{fig:res_dense_block} (a), where the input size increases with the depth of the block.
Combining the benefits of both aggregation types has also been investigated. Mixed link networks (MLNs) are CNNs that employ both residual and dense aggregations \cite{MLN}. A network that employs densely connected residual blocks (DenseRNet) was able to outperform both ResNets and DenseNets on a speech recognition task \cite{DenseRNet}. As shown in \cite{MLN}, MLNs such as DenseRNets follow a similar dense aggregation strategy to DenseNets. For example, DenseRNet blocks have total feature re-usage between residual blocks. This indicates that current MLNs possess the same drawback inherent with DenseNets: too many parameters are allocated for feature re-usage in each block.
\begin{figure}[htp]
\centering
\centering
\includegraphics[scale=1.1]{input_size.pdf}%
\caption{Comparison of (a) the dense topology and (b) the proposed RDL topology. The number of input features to each convolutional unit is indicated. Given identical kernel and output sizes, more parameters are consumed for a larger input size. \textcircled{c} represents the concatenation operation.}
\label{fig:res_dense_block}
\end{figure}
In this paper, we propose a new CNN for speech enhancement that takes advantage of both aggregation types, without over-allocating parameters for feature re-usage. This is achieved by using a topology that differs from the chain-structure of MLNs, such as DenseRNet. The topology is a triangular lattice of convolutional units, as illustrated in Figure \ref{fig:model_block}. Local dense aggregations of convolutional unit outputs are formed strictly over the \textit{height} of the lattice. As can be seen by comparing Figure 1 (b) to Figure 1 (a), this reduces the maximum input size to a convolutional unit within a block. While RDL blocks do not allow for total feature re-usage, densely aggregating only a subset of previous outputs has been shown to be beneficial \cite{sparse_aggCNN}. Local residual and global dense links are also adopted, to improve intra block gradient flow, and inter block feature re-usage, respectively. We refer to the framework of applying residual and dense aggregations over a triangular lattice of convolutional units as a {\it residual-dense lattice} (RDL). Moreover, we show that the proposed RDL network (RDL-Net) is able to produce a higher speech enhancement performance than networks that employ residual and/or dense aggregations. An ablation study of RDL-Nets is also performed over multiple aggregation configurations. We also show that RDL-Nets outperform many state-of-the-art deep learning approaches to speech enhancement.
\section{Related works}\label{sec_rlw}
ANNs have been used for enhancing speech in both the time- and frequency-domain. In the time-domain, ANNs estimate clean speech frames from given noisy speech frames. A GAN was employed for speech enhancement in the time-domain (SEGAN), which used encoder-decoder CNNs for both the generator and discriminator \cite{segan}. A CNN employing non-causal dilated convolutional units and residual links was also used for speech enhancement in the time-domain (Wavenet) \cite{wavenet}.
In the frequency-domain, ANNs are employed to estimate either the clean speech magnitude spectra, a time-frequency mask, or the \textit{a priori} SNR from given noisy speech magnitude spectra. An MLP was used to estimate the clean speech log-power spectra (LPS) \cite{6932438}, with the framework later incorporating multi-objective learning, and ideal binary mask (IBM) post-processing (Xu2017) \cite{xu2017multi}. A DenseNet was also used to estimate the clean speech LPS in \cite{dense_SE}, and was able to outperform both MLP and LSTM networks in the same framework. Time-frequency masks, such as the ideal ratio mask (IRM), are applied as a suppression function to the noisy speech magnitude spectra. An LSTM network was used to estimate the IRM (LSTM-IRM) \cite{chen2017long}, which was able to generalise to unseen speakers. A GAN with a regularised loss function was also used to estimate the IRM (MMSE-GAN), and was able to outperform SEGAN \cite{MMSE-GAN}. This was outperformed by another GAN IRM estimator that used multiple objective measures during optimisation (Metric-GAN) \cite{metric-GAN}.
\textit{A priori} SNR estimates are used by minimum mean-square error (MMSE) estimators of the clean speech magnitude spectra \cite{1164453}. Recently, a deep learning approach to \textit{a priori} SNR estimation was proposed (Deep Xi) \cite{nicolson2019deep}. It used a residual LSTM network (ResLSTM) to estimate the \textit{a priori} SNR directly from noisy speech magnitude spectra. By estimating the \textit{a priori} SNR, different MMSE approaches can be used such as the MMSE log-spectral amplitude (MMSE-LSA) estimator \cite{1164550} and the square-root Wiener filter (SRWF) \cite{1455809}. RDL-Nets are examined within the Deep Xi framework, due to its flexibility of MMSE estimator choice.
\begin{figure}[t]
\centering
\centering
\includegraphics[scale=0.67]{model_block_compact.pdf}%
\caption{An RDL block with a length of 5 and height of 3. The two coloured triangles indicate the left and right halves of the lattice. Here, the kernel size is denoted by $k$.}
\label{fig:model_block}
\end{figure}
\begin{figure*}[ht]
\centering
\includegraphics[scale=0.83]{model_network_block.pdf}
\caption{The proposed RDL-Net for speech enhancement. The RDL-Net estimates the \textit{a priori} SNR from the given noisy speech magnitude spectrum. The estimated \textit{a priori} SNR is then used by an MMSE clean speech magnitude spectrum estimator.}
\label{fig:model_network}
\end{figure*}
\section{Residual-dense lattice networks} \label{seca}
The proposed RDL-Net is used to estimate the \textit{a priori} SNR from given noisy speech magnitude spectra, as shown in Figure \ref{fig:model_network}. The network consists of $B$ RDL blocks, and a sigmoidal fully-connected output layer, $\textbf{O}$. The block topology is a triangular lattice of convolutional units, as shown in Figure \ref{fig:model_block}. The location of each convolutional unit within the lattice, $C_{hl}$, is specified by a height and length co-ordinate $(h,l)$, where $h=1,2,...,H$, and, $l=1,2,...,L$. The number of convolutional units in each block is denoted by $N$, where $N$ is a square number, and $N\geq 4$. The height of the lattice is $H=\sqrt{N}$, and the length is $L=2\sqrt{N}-1$. The following notation is used to indicate in which section of the lattice a convolutional unit exists:
\begin{equation}
C_{hl}= \begin{cases}
C^{\lrtriangle}_{hl},& \text{if }h\leq l,~ l\leq H \\
C^{\lltriangle}_{hl},& \text{if }h \leq 2H-l,~ H<l\leq L \\
\emptyset,& \text{otherwise},\\
\end{cases}
\end{equation}
where $C^{\lrtriangle}_{hl}$ and $C^{\lltriangle}_{hl}$ are convolutional units that exist in the left and right triangles of the lattice, respectively. $\emptyset$ indicates that the convolutional unit does not exist.
\subsection{Convolutional units}
Each convolutional unit is a composite function, $f(\cdot)$, consisting of three operations, including layer normalisation \cite{ba2016layer}, followed by ReLU activation \cite{glorot_deep_2011}, and 1D causal dilated convolution \cite{bai2018empirical}. The output of a convolutional unit is given by $f(x_{hl},W_{hl})$, where $W_{hl}$ denotes the weights (and biases). Convolutional units within the RDL-Net are connected by both local and global links (i.e. intra and inter block links).
\subsection{Local dense aggregations}
The input to a convolutional unit in the left triangle of the lattice, $x^{\lrtriangle}_{hl}$, is the dense aggregation of the outputs at length $l-1$, and heights $h,h-1,...,1$:
\begin{equation}
\begin{aligned}
x^{\lrtriangle}_{hl} &= \begin{cases}
x_{11}, & \text{if } l=h=1\\
y_{h(l-1)}, & \text{if } C^{\lrtriangle}_{hl} ,~ l>1, h=1\\
[y_{h(l-1)},x_{(h-1)l}], & \text{if } C^{\lrtriangle}_{hl} ,~ l>h, h>1\\
x_{(h-1)l}, & \text{if } C^{\lrtriangle}_{hl} ,~ l=h, h>1,
\end{cases}
\end{aligned}
\end{equation}
where $[.]$ denotes the concatenation operation, and $y_{hl}$ is the local residual aggregation at $(h,l)$. The local dense aggregations in the left triangle of the lattice allow for multiple concise outputs to be progressively formed. The input to a convolutional unit in the right triangle of the lattice, $x^{\lltriangle}_{hl}$, is the dense aggregation of the outputs at length $l-1$, and heights $h,h+1,...,H$:
\begin{equation}
\begin{aligned}
x^{\lltriangle}_{hl} &= \begin{cases}
[y_{h(l-1)},y_{(h+1)(l-1)}], & \text{if } C^{\lltriangle}_{hl} ,~ h=2H-l\\
[y_{h(l-1)},x_{(h+1)l}], & \text{if } C^{\lltriangle}_{hl} ,~ h<2H-l.
\end{cases}
\end{aligned}
\end{equation}
In the right triangle of the lattice, the outputs are progressively amalgamated into a single output. By densely aggregating outputs over the height of the lattice, the input size to deeper convolutional units within the block is limited. This enables RDL-Nets to avoid the drawback associated with other densely connected residual networks: \textit{the over-allocation of parameters for feature re-usage}.
\subsection{Local residual aggregations}
To improve the flow of gradients over the length of the lattice, local residual links are adopted:
\begin{equation}
y_{hl}= \begin{cases}
f(x_{hl},W_{hl})+x_{h(l-1)}, & \text{if } C^{\lrtriangle}_{hl}\text{ or }C^{\lltriangle}_{hl},~l>h\\
f(x_{hl},W_{hl}), & \text{if } C^{\lrtriangle}_{hl}\text{ or }C^{\lltriangle}_{hl},~ l\leq h.\\
\end{cases}
\end{equation}
When the size of $y_{hl}$ and $x_{h(l-1)}$ are non-identical, the residual link is weighted so that $x_{h(l-1)}$ is the same size as $y_{hl}$. Local residual links also help to stabilise the training process~\cite{he2016identity}.
\subsection{Global dense aggregations}
Global dense links are adopted, to further enhance the propagation of information between RDL blocks:
\begin{equation}
x^{b+1}_{11}=[x^{b}_{11}, y^{b}_{1L}],
\end{equation}
where the superscript is added to the notation to indicate the block index, $b=1,2,...,B$. Utilising global dense links also enables feature re-usage between the RDL blocks.
\subsection{Implementation details}
The receptive field of an RDL block is controlled via the dilatation rate, $d=2^{h-1}$, and the kernel size, $k=2h-1$. However, this strategy can expend a large number of parameters. Hence, we alternate the kernel size of $k=2h-1$, with $k=1$ at each length, as depicted in Figure \ref{fig:model_block}. Moreover, we set the convolutional unit output size at each height to $m_{h}=\frac{m_1}{2^{h-1}}$, where $m_1$ is the output size at $h=1$. This policy ensures that a reduced number of parameters are used for feature re-usage. In this work, the total number of convolutional units for each RDL block was set to $N=16$ (hence, $H=4$ and $L=7$). The output size of the first level ($h=1$) was $m_{1}=64$. RDL-Nets with sizes of 0.53, 1.08, 1.48, 1.87, and 3.91 million parameters were formed by cascading 3, 6, 8, 10, and 18 blocks, respectively.
\section{Experiment setup} \label{secb}
\subsection{Network Configurations}
The aforementioned RDL-Net configurations and the following network configurations were tasked with estimating the \textit{a priori} SNR within the Deep Xi framework \cite{nicolson2019deep}. The estimated \textit{a priori} SNR is then used by MMSE approaches to speech enhancement.
\begin{description}
\item[ResNet:] Each residual block contained 2 causal dilated convolutional units with an output size of 64, and a kernel size of 3. For each block, $d$ was cycled from 1 to 8 (increasing by a power of 2). ResNets of sizes 0.53, 1.03, 1.53, and 2.03 million parameters were formed by cascading 20, 40, 60, and 80 residual blocks, respectively.
\item[DenseNet:] Each dense block contained 4 causal dilated convolutional units with an output size of 24, and a kernel size of 3. For each convolutional unit, $d$ was cycled from 1 to 8 (increasing by a power of 2). DenseNets of sizes 0.57, 0.97, 1.48, and 2.10 million parameters were formed by cascading 5, 7, 9, and 11 dense blocks, respectively.
\item[DenseRNet:] Each denseR block was composed of 4 densely connected residual blocks. Each residual block contained 2 causal dilated convolutional units with an output size of 24 and a kernel size of 3. For each residual block, $d$ was cycled from 1 to 8 (increasing by a power of 2). DenseRNets of sizes 0.60, 1.05, 1.44, and 2.02 million parameters were formed by cascading 2, 3, 4, and 6 denseR blocks, respectively.
\item[ResLSTM:] The cell size and number of residual blocks were 170 and 4, 188 and 5, and 200 and 6, for the ResLSTMs of sizes 1.02, 1.51, and 2.03 million parameters, respectively. This was the original network used in the Deep Xi framework \cite{nicolson2019deep}.
\end{description}
\subsection{Speech enhancement}
For each frame of noisy speech, the 257-point single-sided magnitude spectrum was computed, which included both the DC frequency component and the Nyquist frequency component, forming the input to each of the five previously described networks. The estimated \textit{a priori} SNR was used by an MMSE approach (MMSE-LSA estimator or SRWF approach) to estimate the clean speech magnitude spectrum. The short-time Fourier analysis, modification, and synthesis (AMS) framework was used to produce the final enhanced speech \cite{nicolson2019deep}. The Hamming window function was used for analysis and synthesis, with a frame length of 32 ms and a frame shift of 16 ms.
\subsection{Training set}
The \textit{train-clean-100} set from the Librispeech corpus \cite{panayotov2015librispeech}, the CSTR VCTK corpus (recordings from speakers $p232$ and $p257$ were excluded as they are used in Test Set 2) \cite{veaux2017cstr}, and the $si^*$ and $sx^*$ training sets from the TIMIT corpus \cite{garofolo1993darpa} were included in the training set ($73\,404$ clean speech recordings). $5\%$ of the clean speech recordings ($3\,667$) were randomly selected and used as the validation set. The $2\,382$ recordings adopted in \cite{nicolson2019deep} were used for the noise training set. All clean speech and noise recordings were single-channel, with a sampling frequency of 16 kHz. The noise corruption procedure for the training set is described in the next subsection.
\subsection{Training strategy}\label{sece}
Cross-entropy was used as the loss function. The \textit{Adam} algorithm \cite{kingma2014adam} with default hyper-parameters was used for stochastic gradient descent optimisation. A mini-batch size of $10$ noisy speech signals was used. The noisy speech signals were generated as follows: each clean speech recording selected for a mini-batch was mixed with a random section of a randomly selected noise recording at a randomly selected SNR level (-10 to 20 dB, in 1 dB increments). A total of 100 epochs were use to train all CNN architectures. A total of 10 epochs were used for the ResLSTM networks and the LSTM-IRM estimator \cite{chen2017long}, as each epoch required eight hours of training.
\subsection{Test sets}
The following two datasets were used for testing:
\begin{itemize}
\item {\textbf{Test set 1}:} The first test set was used to obtain the results in Figures~\ref{fig:abl-Curves} and~\ref{fig:Curves}, and Tables~\ref{table_ablstudy},~\ref{taba}, and~\ref{tabb}. The four noise sources included \textit{voice babble}, \textit{F16}, and \textit{factory} from the RSG-10 noise dataset \cite{steeneken1988description} and \textit{street music} (recording no. $26\,270$) from the Urban Sound dataset \cite{salamon2014dataset}. 10 clean speech recordings were randomly selected (without replacement) from the TSP speech corpus \cite{kabal2002tsp} for each of the four noise recordings. To generate the noisy speech, a random section of the noise recording was mixed with the clean speech at the following SNR levels: -5 to 15 dB, in 5 dB increments. This created a test set of 200 noisy speech signals. The noisy speech was single channel, with a sampling frequency of 16 kHz.
\item {\textbf{Test set 2}:} The second test set was used to obtain the results in Table \ref{table-stoa}. In order to make a direct comparison, the second test set is identical to those used in previous works. The test set included 824 clean speech recordings of two speakers from the Voice Bank corpus (393 from $p232$ and 431 from $p257$) \cite{6709856}. A total of 20 different conditions were used to create the noisy speech, including five noise types from the DEMAND dataset \cite{thiemann2013diverse}, and four SNR levels: 2.5, 7.5, 12.5, and 17.5 dB. This corresponds to approximately 20 different
sentences per condition for each speaker (824 noisy speech signals in the second test set).
\end{itemize}
\section{Results and discussion} \label{secc}
\subsection{Local and global aggregation study}\label{subsec:abls}
In this section we conduct an ablation study on the effects of two aggregation types used in the RDL-Net topology, including local residual links (LR) and global dense links (GD). To this end, four RDL-Net configurations are examined, as shown in Table~\ref{table_ablstudy}. The convergence of each configuration during training is also depicted in Figure~\ref{fig:abl-Curves}. The four configurations were formed using the aforementioned hyper-parameters, with 5 blocks. By adding either LR or GD to the baseline (no LR or GD), it can be seen that a lower validation error can be attained. While GD aggregations add more trainable parameters (0.23M) to the baseline, it achieved a lower validation error than LR (141.82 vs. 142.14). However, the GD configuration caused obvious fluctuations in the validation error during training. Utilising both LR and GD produced the lowest validation error, without the fluctuations in validation error exhibited by the GD configuration. This demonstrates that enhanced intra block gradient flow and inter block feature re-usage are both highly beneficial to the training of an RDL-Net.
\begin{table}[t]
\centering
\caption{Ablation study of local residual (LR) and global dense (GD) aggregations.}
\begin{tabular}{c|c|c|c|c}
\hline
\multicolumn{5}{c}{Different combinations of LR, GD} \\ [0.5ex]
\hline\hline
LR & \xmark & \cmark& \xmark & \cmark\\
GD & \xmark & \xmark & \cmark & \cmark\\ [0.5ex]
\hline\hline
\# params.& 0.39M & 0.53M & 0.62M & 0.86M \\\hline
Val. error & 146.37& 142.14& 141.82& 141.43 \\
\hline
\end{tabular}
\label{table_ablstudy}
\end{table}
\definecolor{darkgreen}{RGB}{0,128,0}
\begin{figure}[t]
\centering
\begin{tikzpicture}[scale=0.65]
\pgfplotstableread{abl_results.txt}
\datatable
\begin{axis}[grid=major,ymin=141.3,ymax=149,xmin=2,xmax=100,xlabel= Epoch,legend pos=outer north east,
ylabel= Validation error,
height=5.5cm, width=7cm]
\addplot[color=black, densely dotted, line width=1pt] table[y = ARF] from \datatable ;
\addlegendentry{Baseline}
\addplot[color=darkgreen, line width=1pt, densely dashdotted] table[y = RL-ARF] from \datatable ;
\addlegendentry{LR}
\addplot[color=blue, line width=1pt, densely dashed] table[y = GD-ARF] from \datatable ;
\addlegendentry{GD}
\addplot[color=red, line width=1pt] table[y = all] from \datatable ;
\addlegendentry{LR-GD}
\end{axis}
\end{tikzpicture}
\caption{Validation error attained by four RDL-Net configuration types: Baseline, LR, GD, and LR-GD.}
\label{fig:abl-Curves}
\end{figure}
\begin{table*}[ht]
\centering
\scriptsize
\setlength{\tabcolsep}{3.6pt}
\caption{Enhanced speech objective quality scores. The mean opinion score of the listening quality objective (MOS-LQO) was used as the metric, where the wideband perceptual evaluation of quality (Wideband PESQ) was the objective model used to obtain the MOS-LQO score \cite{itut2007application}. The tested conditions include clean speech mixed with real-world \textbf{non-stationary} (\textit{voice babble} and \textit{street music}) and \textbf{coloured} (\textit{F16} and \textit{factory}) noise sources at multiple SNR levels. The highest MOS-LQO score attained at each condition and for each parameter size is shown in boldface. The standard error (SE) over all conditions for each network is provided in the last column.}
\begin{tabular}{ll|lllll|lllll|lllll|lllll|l}
\toprule
\multirow{3}{*}{\begin{tabular}[c]{@{}c@{}}\textbf{Network}\end{tabular}} &
\multirow{3}{*}{\begin{tabular}[c]{@{}c@{}}\textbf{\# params.}\\$\pmb{\times 10^{6}}$\end{tabular}} &
\multicolumn{20}{c|}{\bf SNR level (dB)} & \multirow{3}{*}{\begin{tabular}[c]{@{}c@{}}\textbf{SE}\end{tabular}} \\
\cline{3-22}
& & \multicolumn{5}{c|}{\bf Voice babble} & \multicolumn{5}{c|}{\bf Street music} & \multicolumn{5}{c|}{\bf F16} & \multicolumn{5}{c|}{\bf Factory} & \\
\cline{3-22}
& & {\bf-5} & {\bf0} & {\bf5} & {\bf10} & {\bf15} & {\bf-5} & {\bf0} & {\bf5} & {\bf10} & {\bf15} & {\bf-5} & {\bf0} & {\bf5} & {\bf10} & {\bf15} & {\bf-5} & {\bf0} & {\bf5} & {\bf10} & {\bf15} & \\
\hline
Noisy speech & - & 1.04 & 1.07 & 1.15 & 1.35 & 1.71 & 1.04 & 1.06 & 1.11 & 1.27 & 1.58 & 1.03 & 1.06 & 1.11 & 1.25 & 1.52 & 1.04 & 1.04 & 1.09 & 1.24 & 1.54 & 0.016 \\
\midrule
ResLSTM & 1.02 & 1.07 & 1.20 & 1.51 & 1.96 & 2.49 & 1.10 & 1.23 & 1.50 & 1.92 & 2.47 & 1.13 & 1.28 & 1.49 & 1.80 & 2.33 & 1.09 & 1.22 & 1.54 & 1.86 & 2.31 & 0.035 \\
DenseNet & 0.97 & 1.08 & 1.20 & 1.54 & 2.01 & 2.50 & 1.09 & 1.24 & 1.51 & 1.86 & 2.36 & 1.14 & 1.35 & 1.64 & 2.05 & 2.45 & 1.07 & 1.25 & 1.55 & 1.97 & 2.43 & 0.037 \\
DenseRNet & 1.05 & 1.07 & 1.21 & 1.50 & 1.92 & 2.43 & 1.11 & 1.22 & 1.48 & 1.87 & 2.37 & 1.14 & 1.34 & 1.57 & 1.93 & 2.34 & 1.07 & 1.21 & 1.44 & 1.85 & 2.36 & 0.035 \\
ResNet & 1.03 & 1.08 & 1.22 & 1.58 & 2.10 & \textbf{2.67} & 1.11 & 1.30 & 1.58 & 2.04 & \textbf{2.53} & 1.19 & 1.44 & \textbf{1.78} & \textbf{2.17} & 2.58 & 1.11 & 1.29 & 1.60 & 2.03 & 2.46 & 0.040 \\
Prop. RDL-Net & 1.08 & \textbf{1.10} & \textbf{1.29} & \textbf{1.65} & \textbf{2.15} & 2.62 & \textbf{1.13} & \textbf{1.32} & \textbf{1.66} & \textbf{2.11} & \textbf{2.53} & \textbf{1.25} & \textbf{1.48} & 1.73 & \textbf{2.17} & \textbf{2.62} & \textbf{1.15} & \textbf{1.39} & \textbf{1.73} & \textbf{2.10} & \textbf{2.54} & 0.039 \\
\midrule
ResLSTM & 1.51 & 1.09 & 1.25 & 1.56 & 2.03 & 2.47 & 1.09 & 1.26 & 1.56 & 1.95 & 2.44 & 1.16 & 1.35 & 1.60 & 1.87 & 2.19 & 1.11 & 1.30 & 1.60 & 1.94 & 2.35 & 0.037 \\
DenseNet & 1.41 & 1.06 & 1.19 & 1.51 & 1.96 & 2.49 & 1.10 & 1.23 & 1.50 & 1.87 & 2.31 & 1.14 & 1.35 & 1.63 & 1.99 & 2.43 & 1.09 & 1.30 & 1.63 & 2.00 & 2.44 & 0.036 \\
DenseRNet & 1.37 & 1.07 & 1.22 & 1.54 & 2.00 & 2.50 & 1.12 & 1.29 & 1.59 & 1.99 & 2.45 & 1.20 & 1.41 & 1.71 & 2.10 & 2.53 & 1.06 & 1.24 & 1.57 & 2.00 & 2.45 & 0.038 \\
ResNet & 1.53 & 1.08 & 1.25 & 1.61 & 2.12 & 2.64 & 1.10 & 1.28 & 1.56 & 2.00 & 2.48 & 1.18 & 1.41 & 1.72 & 2.15 & \textbf{2.61} & 1.10 & 1.30 & 1.64 & 2.07 & 2.53 & 0.040\\
Prop. RDL-Net & 1.48 & \textbf{1.12} & \textbf{1.31} & \textbf{1.67} & \textbf{2.20} & \textbf{2.75} & \textbf{1.17} & \textbf{1.41} & \textbf{1.75} & \textbf{2.09} & \textbf{2.59} & \textbf{1.28} & \textbf{1.52} & \textbf{1.85} & \textbf{2.25} & \textbf{2.61} & \textbf{1.17} & \textbf{1.40} & \textbf{1.74} & \textbf{2.12} & \textbf{2.59} & 0.040 \\
\midrule
ResLSTM & 2.03 & 1.09 & 1.23 & 1.51 & 2.02 & 2.48 & 1.13 & 1.30 & 1.59 & 2.06 & 2.50 & 1.19 & 1.37 & 1.61 & 1.92 & 2.29 & 1.14 & 1.35 & 1.64 & 2.01 & 2.48 & 0.036 \\
DenseNet & 1.94 & 1.07 & 1.21 & 1.54 & 2.04 & 2.47 & 1.09 & 1.21 & 1.46 & 1.86 & 2.33 & 1.16 & 1.36 & 1.65 & 1.98 & 2.44 & 1.09 & 1.27 & 1.64 & 2.03 & 2.49 & 0.037 \\
DenseRNet & 2.02 & 1.08 & 1.20 & 1.48 & 1.83 & 2.24 & 1.10 & 1.21 & 1.42 & 1.77 & 2.23 & 1.19 & 1.37 & 1.60 & 1.93 & 2.29 & 1.06 & 1.18 & 1.42 & 1.81 & 2.29 & 0.033 \\
ResNet & 2.03 & \textbf{1.10} & 1.28 & 1.59 & 2.08 & 2.59 & 1.14 & 1.30 & 1.60 & 1.98 & 2.43 & 1.21 & 1.46 & 1.75 & 2.09 & 2.52 & 1.11 & 1.30 & 1.61 & 2.02 & 2.54 & 0.038\\
Prop. RDL-Net & 1.87 & \textbf{1.10} & \textbf{1.30} & \textbf{1.67} & \textbf{2.23} & \textbf{2.73} & \textbf{1.18} & \textbf{1.44} & \textbf{1.80} & \textbf{2.20} & \textbf{2.62} & \textbf{1.23} & \textbf{1.48} & \textbf{1.80} & \textbf{2.30} & \textbf{2.62} & \textbf{1.18} & \textbf{1.43} & \textbf{1.75} & \textbf{2.13} & \textbf{2.63} & 0.040 \\
\midrule
LSTM-IRM & 30.7 & 1.07 & 1.20 & 1.46 & 1.88 & 2.31 & 1.08 & 1.17 & 1.40 & 1.71 & 2.13 & 1.09 & 1.24 & 1.46 & 1.71 & 2.00 & 1.06 & 1.18 & 1.40 & 1.72 & 2.12 & 0.030 \\
Xu2017 & 19.1 & \textbf{1.18} & \textbf{1.43} & 1.78 & 2.20 & 2.66 & 1.15 & 1.33 & 1.58 & 1.94 & 2.35 & 1.17 & 1.43 & 1.80 & 2.25 & 2.65 & 1.09 & 1.23 & 1.47 & 1.87 & 2.34 & 0.040 \\
Prop. RDL-Net & 3.91 & 1.13 & 1.36 & \textbf{1.79} & \textbf{2.46} & \textbf{2.98} & \textbf{1.19} & \textbf{1.42} & \textbf{1.83} & \textbf{2.27} & \textbf{2.74} & \textbf{1.26} & \textbf{1.53} & \textbf{1.86} & \textbf{2.31} & \textbf{2.78} & \textbf{1.19} & \textbf{1.46} & \textbf{1.83} & \textbf{2.26} & \textbf{2.74} & 0.045 \\
\bottomrule
\end{tabular}
\label{taba}
\end{table*}
\begin{table*}[h!]
\centering
\scriptsize
\setlength{\tabcolsep}{3.6pt}
\caption{Enhanced speech objective intelligibility scores (in $\%$) as given by the short-time objective intelligibility (STOI) metric \cite{short_obj}. The tested conditions include clean speech mixed with real-world \textbf{non-stationary} (\textit{voice babble} and \textit{street music}) and \textbf{coloured} (\textit{F16} and \textit{factory}) noise sources at multiple SNR levels. The highest STOI score attained at each condition and for each parameter size is shown in boldface. The standard error (SE) over all conditions for each network is provided in the last column.}
\begin{tabular}{ll|lllll|lllll|lllll|lllll|l}
\toprule
\multirow{3}{*}{\begin{tabular}[c]{@{}c@{}}\textbf{Network}\end{tabular}} &
\multirow{3}{*}{\begin{tabular}[c]{@{}c@{}}\textbf{\# params.}\\$\pmb{\times 10^{6}}$\end{tabular}} &
\multicolumn{20}{c|}{\bf SNR level (dB)} & \multirow{3}{*}{\begin{tabular}[c]{@{}c@{}}\textbf{SE}\end{tabular}} \\
\cline{3-22}
& & \multicolumn{5}{c|}{\bf Voice babble} & \multicolumn{5}{c|}{\bf Street music} & \multicolumn{5}{c|}{\bf F16} & \multicolumn{5}{c|}{\bf Factory} & \\
\cline{3-22}
& & {\bf-5} & {\bf0} & {\bf5} & {\bf10} & {\bf15} & {\bf-5} & {\bf0} & {\bf5} & {\bf10} & {\bf15} & {\bf-5} & {\bf0} & {\bf5} & {\bf10} & {\bf15} & {\bf-5} & {\bf0} & {\bf5} & {\bf10} & {\bf15} & \\
\hline
Noisy speech & - & 60.2 & 72.4 & 83.0 & 90.7 & 95.5 & 59.0 & 70.9 & 81.9 & 90.3 & 95.6 & 60.4 & 71.8 & 82.4 & 90.5 & 95.7 & 57.8 & 69.9 & 80.9 & 89.2 & 94.5 & 0.010\\
\midrule
ResLSTM & 1.02 & 58.1 & 73.8 & 85.4 & 92.7 & 96.5 & 64.3 & 76.9 & 87.6 & 93.6 & 96.9 & 64.8 & 77.6 & 86.6 & 92.1 & 95.8 & 60.0 & 73.6 & 84.8 & 91.2 & 95.3 & 0.010 \\
DenseNet & 0.97 & 56.5 & 72.5 & 85.8 & 93.5 & 96.8 & 62.8 & 74.1 & 85.4 & 92.5 & 96.3 & 64.8 & 78.6 & 87.7 & 93.2 & 96.4 & 59.3 & 74.8 & 85.9 & 92.4 & 96.0 & 0.010\\
DenseRNet & 1.05 & 58.6 & 73.9 & 85.4 & 92.3 & 96.1 & 64.0 & 75.5 & 85.7 & 92.2 & 96.3 & 65.7 & 78.2 & 87.2 & 93.1 & 96.8 & 58.1 & 74.3 & 84.3 & 91.1 & 95.6 & 0.010 \\
ResNet & 1.03 & 59.8 & 75.4 & 87.4 & 94.0 & 97.0 & 65.4 & 77.2 & 88.0 & 94.0 & 97.2 & 68.0 & 80.3 & 88.7 & 94.1 & 97.1 & 62.5 & 76.9 & 86.7 & 92.9 & 96.4 & 0.009 \\
Prop. RDL-Net & 1.08 & \textbf{60.2} & \textbf{77.9} & \textbf{88.6} & \textbf{94.3} & \textbf{97.2} & \textbf{67.2} & \textbf{80.4} & \textbf{89.9} & \textbf{94.8} & \textbf{97.4} & \textbf{69.6} & \textbf{82.7} & \textbf{90.1} & \textbf{94.6} & \textbf{97.3} & \textbf{63.3} & \textbf{79.4} & \textbf{88.4} & \textbf{93.4} & \textbf{96.5} & 0.009 \\
\midrule
ResLSTM & 1.51 & 60.3 & 76.1 & 87.0 & 93.9 & 96.9 & 63.5 & 77.3 & 87.9 & 94.0 & 97.0 & 66.4 & 79.3 & 87.7 & 93.0 & 96.0 & 62.1 & 78.3 & 87.5 & 92.8 & 96.2 & 0.009 \\
DenseNet & 1.41 & 59.4 & 75.1 & 86.6 & 93.3 & 96.6 & 64.1 & 76.3 & 86.6 & 92.9 & 96.4 & 65.9 & 79.8 & 88.0 & 93.5 & 96.6 & 60.0 & 77.9 & 87.1 & 92.7 & 96.1 & 0.009 \\
DenseRNet & 1.37 &59.7 & 74.6 & 86.1 & 93.0 & 96.5 & 62.8 & 75.8 & 85.9 & 92.4 & 96.3 & 67.1 & 79.2 & 87.8 & 93.3 & 96.8 & 59.9 & 75.1 & 86.0 & 92.3 & 96.0 & 0.009 \\
ResNet & 1.53 & 60.9 & 76.5 & 87.9 & 94.0 & 97.1 & 66.0 & 77.9 & 88.2 & 93.8 & 97.0 & 67.9 & 80.9 & 89.3 & \textbf{94.3} & \textbf{97.3} & 63.2 & 78.3 & 87.8 & 93.1 & 96.5 & 0.009 \\
Prop. RDL-Net & 1.48 & \textbf{61.0} & \textbf{77.3} & \textbf{88.9} & \textbf{94.5} & \textbf{97.4} & \textbf{66.8} & \textbf{80.0} & \textbf{89.2} & \textbf{94.4} & \textbf{97.4} & \textbf{69.4} & \textbf{82.6} & \textbf{89.7} & \textbf{94.3} & 97.2 & \textbf{64.6} & \textbf{80.0} & \textbf{88.7} & \textbf{93.5} & \textbf{96.7} & 0.009 \\
\midrule
ResLSTM & 2.03 & 61.1 & 74.6 & 87.0 & 93.7 & 96.9 & 66.4 & 78.8 & 88.8 & 94.2 & 97.1 & 67.6 & 80.3 & 88.6 & 93.6 & 96.5 & 64.1 & 79.1 & 87.8 & 93.1 & 96.4 & 0.009 \\
DenseNet & 1.94 & 60.1 &75.3 & 86.9 & 93.9 & 96.9 & 64.8 & 77.1 & 86.9 & 93.2 & 96.8 & 66.7 & 79.9 & 88.3 & 93.3 & 96.5 & 60.4 & 77.2 & 87.4 & 92.8 & 96.3 & 0.009 \\
DenseRNet & 2.02 & 59.3 & 73.4 & 84.8 &92.0 & 95.8 & 64.0 & 74.8 & 84.4 & 91.2 & 95.5 & 66.5 & 77.8 & 86.4 & 92.4 & 96.1 & 58.6 & 73.4 & 84.1 & 91.2 & 95.4 & 0.009 \\
ResNet & 2.03 & \textbf{62.7} & 77.3 & 87.6 & 93.8 & 97.0 & 66.7 & 78.0 & 88.1 & 94.0 & 97.1 & 69.0 & 81.1 & 88.6 & 93.8 & 97.0 & 62.1 & 77.1 & 86.9 & 92.5 & 96.3 & 0.009 \\
Prop. RDL-Net & 1.87 & 61.5 & \textbf{77.8} & \textbf{89.0} & \textbf{94.7} & \textbf{97.4} & \textbf{68.5} & \textbf{81.2} & \textbf{90.1} & \textbf{94.8} & \textbf{97.4} & \textbf{69.3} & \textbf{82.5} & \textbf{90.4} & \textbf{94.9} & \textbf{97.4} & \textbf{64.8} & \textbf{80.6} & \textbf{88.6} & \textbf{93.5} & \textbf{96.6} & 0.009 \\
\midrule
LSTM-IRM & 30.7 & \textbf{64.2} & 78.5 & 88.0 & 93.5 & 96.5 & 66.1 & 77.4 & 86.6 & 92.6 & 96.0 & 67.3 & 79.1 & 87.3 & 92.5 & 95.8 & 62.3 & 76.7 & 86.6 & 92.5 & 95.9 & 0.009 \\
Xu2017 & 19.1 & 62.5 & 74.8 & 83.8 & 90.1 & 94.7 & 64.0 & 77.9 & 86.7 & 92.4 & 95.5 & 68.3 & 79.0 & 86.7 & 92.8 & 95.5 & 61.0 & 74.2 & 83.9 & 90.5 & 94.7 & 0.009 \\
Prop. RDL-Net & 3.91 & \textbf{64.2} & \textbf{80.82} & \textbf{89.0} & \textbf{94.7} & \textbf{97.4} & \textbf{71.6} & \textbf{82.9} & \textbf{90.7} & \textbf{95.0} & \textbf{97.5} & \textbf{72.6} & \textbf{83.9} & \textbf{91.0} & \textbf{95.4} & \textbf{97.8} & \textbf{67.1} & \textbf{81.7} & \textbf{89.5} & \textbf{93.9} & \textbf{96.8} & 0.008 \\
\bottomrule
\end{tabular}
\label{tabb}
\end{table*}
\begin{table*}[ht]
\centering
\caption{Comparison to recent deep learning approaches to speech enhancement using the second test set. As in previous works, the objective scores are averaged over all tested conditions. \textbf{CSIG}, \textbf{CBAK}, and \textbf{COVL} are mean opinion score (MOS) predictors of the signal distortion, background-noise intrusiveness, and overall signal quality, respectively \cite{4389058}. \textbf{PESQ} is the perceptual evaluation of speech quality measure \cite{4389058}. \textbf{STOI} is the short-time objective intelligibility measure (in \%) \cite{short_obj}. The highest scores attained for each measure are indicated in boldface.}
\begin{tabular}{@{}llllll@{}}
\toprule
\textbf{Method} & \textbf{CSIG} & \textbf{CBAK} & \textbf{COVL} & \textbf{PESQ} & \textbf{STOI} \\ \midrule
Noisy speech & 3.35 & 2.44 & 2.63 & 1.97 & 92 (91.5) \\
Wiener \cite{Wiener} & 3.23 & 2.68 & 2.67 & 2.22 & - \\
SEGAN \cite{segan} & 3.48 & 2.94 & 2.80 & 2.16 & 93 \\
Wavenet \cite{wavenet} & 3.62 & 3.23 & 2.98 & - & - \\
MMSE-GAN \cite{MMSE-GAN} & 3.80 & 3.12 & 3.14 & 2.53 & 93 \\
Deep Feature Loss \cite{deep_floss} & 3.86 & 3.33 & 3.22 & - & - \\
Metric-GAN \cite{metric-GAN} & 3.99 & 3.18 & 3.42 & 2.86 & - \\ \midrule
Proposed RDL-Net 1.87M (Deep Xi - MMSE-LSA) & 4.29 & 3.32 & 3.62 & 2.93 & 93 (93.4) \\
Proposed RDL-Net 1.87M (Deep Xi - SRWF) & 4.27 & 3.23 & 3.56 & 2.84 & 93 (93.5)\\
Proposed RDL-Net 3.91M (Deep Xi - MMSE-LSA) & \textbf{4.38} & \textbf{3.43} & \textbf{3.72} & \textbf{3.02} & \textbf{94} (\textbf{93.8}) \\
Proposed RDL-Net 3.91M (Deep Xi - SRWF) & 4.36 & 3.35 & 3.67 & 2.94 & \textbf{94} (\textbf{93.8}) \\ \bottomrule
\end{tabular}
\label{table-stoa}
\end{table*}
\subsection{Training and validation error}
The training and validation error curves for the RDL-Net, ResNet, DenseNet, and DenseRNet at a parameter sizes of approximately 2 million are shown in Figures~\ref{fig:Curves} (a) and (b), respectively. The RDL-Net was able to converge to a lower training and validation error than the other networks. This suggests that the proposed RDL-Net allocated an efficient number of parameters for feature re-usage. Conversely, the DenseNet and DenseRNet struggled at a parameter size of 2 million, indicating that too many parameters were wasted on feature re-usage.
\subsection{Parameter and computational efficiency}
The lowest validation error as a function of the number of parameters and computations for RDL-Nets, ResNets, DenseNets, and DenseRNets are shown in Figures\ref{fig:Curves} (c) and (d), respectively. RDL-Nets were able to achieve the same validation error as ResNets that employed significantly more parameters. For example, at a parameter size of 1 million, the RDL-Net attained the same lowest validation error as the ResNet with double the amount of parameters. A similar trend can be seen for the lowest validation error as a function of the number of FLOPs, (where FLOPs refers to the number of multiplication-addition operations during inference). For example, the RDL-Net that requires 2 million FLOPs achieved a lowest validation error similar to that of the ResNet that requires $4\times$ as many FLOPs.
\definecolor{darkgreen}{RGB}{0,128,0}
\begin{figure}[t]
\centering
\begin{tikzpicture}[scale=0.67]
\pgfplotstableread{error_results.txt}
\datatable
\begin{axis}[grid=major,ymin=140,ymax=148,xmin=2,xmax=100,xlabel= Epoch,
ylabel= Training error,
xlabel style={font=\normalsize}, xtick={20,40,60,80},
title=(a), title style={yshift=-3mm,},
ylabel style={font=\normalsize},yticklabel style={font=\normalsize},xticklabel style={font=\normalsize},
width=6.2cm, height=5.5cm
]
\addplot[line width=1pt, color=black, densely dotted] table[y = DenseRnet-tr] from \datatable ;
\addplot[line width=1pt, color=darkgreen, densely dashdotted] table[y = DenseNet-tr] from \datatable ;
\addplot[line width=1pt, color=blue, densely dashed] table[y = ResNet-tr] from \datatable ;
\addplot[line width=1pt, color=red] table[y = RDL-tr] from \datatable ;
\end{axis}
\end{tikzpicture}
\begin{tikzpicture}[scale=0.67]
\pgfplotstableread{error_results.txt}
\datatable
\begin{axis}[grid=major,ymin=140,ymax=148,xmin=2,xmax=100,xlabel= Epoch,
ylabel= Validation error,legend pos=north east,
xlabel style={font=\normalsize}, xtick={20,40,60,80},
ylabel style={font=\normalsize},yticklabel style={font=\normalsize},xticklabel style={font=\normalsize},
width=6.2cm, height=5.5cm, title=(b), title style={yshift=-3mm,},
]
\addplot[line width=1pt, color=black, densely dotted] table[y = DenseRnet-val] from \datatable ;
\addlegendentry{DenseRNet}
\addplot[line width=1pt, color=darkgreen, densely dashdotted] table[y = DenseNet-val] from \datatable ;
\addlegendentry{DenseNet}
\addplot[line width=1pt, color=blue, densely dashed] table[y = ResNet-val] from \datatable ;
\addlegendentry{ResNet}
\addplot[line width=1pt, color=red, mark repeat=15] table[y = RDL-val] from \datatable ;
\addlegendentry{RDL-Net}
\end{axis}
\end{tikzpicture}
\begin{tikzpicture}[scale=0.67]
\begin{axis}[grid=major,xlabel= \# parameters ($\times 10^{6}$),
ylabel= Lowest validation error,legend pos=north west,xlabel style={font=\normalsize}, ylabel style={font=\normalsize},yticklabel style={font=\normalsize},xticklabel style={font=\normalsize},
width=6.2cm, height=5.5cm,
title=(c), title style={yshift=-3mm,},
]
\addplot[color=black,mark=diamond*, densely dotted, mark options={solid}, line width=1pt] coordinates {
(0.6, 143.04)
(1.05, 142.82)
(1.37, 142.7)
(2.02, 142.51)};
\node [above,black] at (axis cs: 1.5, 142.7) {DenseRNet};
\addplot[color=darkgreen,mark=*, densely dashdotted, mark options={solid}, line width=1pt] coordinates {
(0.57, 143.25)
(0.97, 142.8)
(1.48, 142.51)
(2.1, 142.21)};
\node [above,darkgreen] at (axis cs: 1.5, 142) {DenseNet};
\addplot[color=blue,mark=square*,densely dashed, mark options={solid}, line width=1pt] coordinates {
(0.53, 142.19)
(1.02, 141.82)
(1.51, 141.42)
(2.03, 141.11)};
\node [right,blue] at (axis cs: 0.53, 142.29) {ResNet};
\addplot[color=red,mark=triangle*,mark options={solid}, line width=1pt] coordinates {
(0.53, 142.14)
(1.08, 141.03)
(1.48, 140.7)
(1.92, 140.5)};
\node [right,red] at (axis cs: 0.53, 140.8) {RDL-Net};
\draw[dashed,->,very thick] (axis cs:2, 141.11) -- (axis cs:1.03, 141.11);
\node [right,black] at (axis cs: 0.935, 141.26) {\textbf{$\approx $2$\pmb{\times}$ fewer params}};
\end{axis}
\end{tikzpicture}
\begin{tikzpicture}[scale=0.67]
\begin{axis}[grid=major,xlabel= \# FLOPs ($\times 10^{6}$),
ylabel= Lowest validation error,legend pos=north west,xlabel style={font=\normalsize}, ylabel style={font=\normalsize},yticklabel style={font=\normalsize},xticklabel style={font=\normalsize},
width=6.2cm, height=5.5cm, title=(d), title style={yshift=-3mm,},
]
\addplot[color=black,mark=diamond*, densely dotted, mark options={solid}, line width=1pt] coordinates {
(2.9, 143.04)
(4.2, 142.82)
(5.48, 142.7)
(8.06, 142.51)};
\node [above,black] at (axis cs: 7.5, 142.6) {DenseRNet};
\addplot[color=darkgreen,mark=*, densely dashdotted,mark options={solid}, line width=1pt] coordinates {
(2.26, 143.25)
(3.87, 142.8)
(5.9, 142.52)
(8.34, 142.21)};
\node [below,darkgreen] at (axis cs: 7.5, 142.23) {DenseNet};
\addplot[color=blue,mark=square*, densely dashed, mark options={solid}, line width=1pt] coordinates {
(2.48, 142.19)
(4.56, 141.82)
(6.59, 141.42)
(8.71, 141.11)};
\draw[dashed,->,very thick] (axis cs:8.71, 141.11) -- (axis cs:2.16, 141.11);
\node [right,blue] at (axis cs: 2.5, 142.25) {ResNet};
\node [right,black] at (axis cs: 2.2, 141.24) {\textbf{$\approx $4$\pmb{\times}$ fewer FLOPs}};
\addplot[color=red,mark=triangle*, mark options={solid}, line width=1pt] coordinates {
(1.23, 142.14)
(2.16, 141.03)
(2.9, 140.7)
(3.7, 140.5)};
\node [right,red] at (axis cs: 3.7, 140.54) {RDL-Net};
\end{axis}
\end{tikzpicture}
\caption{Training plots for RDL-Nets, ResNets, DenseNets and DenseRNets: \textbf{(a)} training error, \textbf{(b)} validation error, and lowest validation error as a function of the number of \textbf{(c)} parameters and \textbf{(d)} FLOPs.}
\label{fig:Curves}
\end{figure}
\subsection{Speech enhancement performance}
The enhanced speech objective quality scores attained by each of the networks in the Deep Xi framework are presented in Table \ref{taba}. Each network estimated the \textit{a priori} SNR for the MMSE-LSA estimator. It can be seen that RDL-Nets were able to achieve the highest objective quality scores for most of the tested conditions. The performance capability of RDL-Nets was demonstrated at a parameter size of 2 million for \textit{street music} at 10 dB, where the RDL-Net achieved a MOS-LQO improvement of 0.22 over the ResNet. Table~\ref{tabb} shows the objective intelligibility scores obtained by each of the networks. It can be seen that RDL-Nets were able to achieve the highest objective intelligibility scores for most of the tested conditions. RDL-Nets demonstrated its performance at a parameter size of 2 million for~\textit{factory} noise at 0 dB, attaining an STOI improvement of $3.5\%$ when compared to the equivalent ResNet. RDL-Nets in the Deep Xi framework were also able to produce enhanced speech with higher objective quality and intelligibility scores than two other widely known deep learning speech enhancement frameworks (LSTM-IRM and Xu2017) \cite{xu2017multi,chen2017long}.
We also compare RDL-Nets to recent deep learning approaches to speech enhancement. Here, RDL-Nets were used to estimate the \textit{a priori} SNR for the SRWF approach and the MMSE-LSA estimator. As shown in Table \ref{table-stoa}, RDL-Nets were able to attain the highest CSIG, CBAK, COVL, PESQ and STOI scores. The RDL-Net demonstrated an improvement of 0.39, 0.25, 0.3, and 0.16 over Metric-GAN for CSIG, CBAK, COVL, and PESQ, respectively. The RDL-Net also demonstrated an improvement of $1\%$ over MMSE-GAN for STOI. The enhanced speech produced by RDL-Net 3.91M is illustrated in Figure \ref{fig-spectogram} (d). It can be seen that the RDL-Net demonstrated superior noise suppression with little formant distortion. As illustrated in Figure \ref{fig-spectogram} (c), Deep Feature Loss over- and under-estimated multiple spectral components.\footnote{Enhanced speech recordings and additional results are available at: \url{https://github.com/nick-nikzad/RDL-SE}.}
\begin{figure} [h!]
\begin{center}
\includegraphics[scale=0.76]{spectro}
\caption{(a) Clean speech magnitude spectrogram ($|\textbf{S}|$) of female $p257$ uttering sentence $70$, ``The price cuts are really exciting''. (b) \textit{Crowd noise} mixed with (a) at an SNR level of 2.5 dB ($|\textbf{X}|$). Enhanced speech ($|\hat{\textbf{S}}|$) produced by (c) Deep Feature Loss and (d) RDL-Net 3.91M (Deep Xi-MMSE-LSA).}
\label{fig-spectogram}
\end{center}
\end{figure}
\section{Conclusion} \label{secd}
In this paper, we propose a novel convolutional neural network (CNN) for speech enhancement, called a residual-dense lattice (RDL) network. Unlike other CNNs that use both residual and dense aggregations, RDL-Nets take advantage of both aggregation types without over-allocating parameters for feature re-usage. This enables RDL-Nets to produce a higher speech enhancement performance than other networks, such as ResLSTM networks, ResNets, DenseNets, and DenseRNets. We also show that RDL-Nets are able to outperform many state-of-the-art deep learning approaches to speech enhancement. In future work, the RDL-Net topology will be investigated for speech separation, speech recognition, computer vision, and image denoising.
\bibliographystyle{aaai}
|
1,314,259,993,570 | arxiv | \section{Introduction}
\label{sec1}
The quantum approach that is commonly known as Bohmian mechanics\footnote{Within the field
of the quantum foundations, Bohmian mechanics is also widely known as the de Broglie-Bohm
interpretation.
However, in recent times the term Bohmian mechanics has become more widespread when it is
used in applications.
This will be the term also considered here.}
has been a source of controversy since its inception \cite{bohm:PR:1952-1,bohm:PR:1952-2},
formerly intended as a simple counter-proof to Von Neumann's theorem \cite{vonNeumann-bk:1932}
on the incompatibility between quantum mechanics and any possibility to complete this theory
with the introduction of local hidden variables.
Therefore, after having worked for a long time taking Bohmian mechanics as a
fundamental theoretic-analytical tool to explore, understand and describe different aspects
of quantum and optical phenomena, one learns to live with a series of recurrent questions
from colleagues and reviewers: Why should anyone be interested in Bohmian-related ``stuff''?
Which new physics does Bohmian mechanics add with respect to the other more conventional
quantum approaches?
Is it not redundant?
In addition, if the always appealing though controversial concept of hidden variable also
appears without having made explicit mention to it (or without having mentioned it at all),
things become even a bit more complicated.
All in all, the general trend seems to be smoothly changing towards what could be considered
to be, say, a more Bohmian-friendly attitude than it was ten or twenty years ago (not to say
earlier on).
Since the 1990s an increasing number of monographs have been published on the issue
\cite{holland-bk,bohm-hiley-bk,cushing-bk:1996,wyatt-bk,bacciagaluppi-valentini-bk,duerr-bk:2009,duerr-bk:2013,hughes-bk,chattaraj-bk,oriols-bk,sanz-bk-1,sanz-bk-2,bricmont-bk}.
These works describe and discuss the physical (and metaphysical) implications of Bohmian
mechanics, revisit the standard quantum formulation in terms of this approach or provide a
detailed account on its applications to different physical problems, which the interested
reader is kindly invited to consult (of course, bearing in mind that the list of works is far
from being complete, yet it serves to the purpose of illustration).
When facing such a flourishing landscape of new developments in the field, one feels compelled
to revisit the above questions, particularly the one about why any attention should be paid at
all to the Bohmian approach beyond the hidden-variable issue, i.e., beyond ontological questions
related to the completeness of quantum mechanics.
After a long and tough way, some conclusions have come up in that regard,
partly collected and discussed in previous works \cite{sanz:AJP:2012,sanz:foundphys:2015,sanz:FrontPhys:2019}.
Now, getting back to Bell's pedagogical view on Bohm's mechanics \cite{bell-bk},
the very first point that one should address is whether, keeping our feet on solid ground,
beyond metaphysical questions, this approach provides us with a natural scenario to think
the physics of quantum phenomena.
This does not mean to consider that Bohm's particle trajectory is the actual trajectory
followed by a real quantum particle\footnote{Note that the concept of trajectory needs not
be necessarily associated with the actual position of a particle or that of its center of mass.
Rather, it should be understood in a broad sense, i.e., as describing the evolution in time
of any type of degree of freedom (vibrations, rotations, etc.), which is a point often
neglected in discussions around Bohm's theory, where trajectories are immediately and uniquely
related to point-like (structureless) particles.
In this sense, Bohmian mechanics transcends the oversimplified framework that associates
the approach with a theory of motion for quantum particles; it can be applied to any aspect
of matter that is accounted for by Schr\"odinger's equation, although providing the
corresponding trajectories with the appropriate interpretation (i.e., in compliance with
the context considered).}.
Yet, the possibility to introduce this ``forbidden'' element in quantum mechanics allows us
to understand the evolution of quantum waves on formal and conceptual grounds analogous to
those used to describe the evolution of classical action in phase space, namely, the theory
of characteristics \cite{courant-hilbert-bk-2} and dynamical flows \cite{jordan-bk:1999}.
Of course, there are certain formal subtleties that generate necessary differences between
the classical and the quantum descriptions:
while space point dynamics is well-defined in the former, the latter precludes it in the same
terms, because it assumes mutual spatial coherence among different spatial points (nicely
evidenced by the Moyal-Wigner representation), which in turn implies a revision of the
laws of motion.
Nonetheless, this does not invalidate the existence of a common formal structure.
Stepping down from the formal level to, say, the level of our everyday experience, based on
real experiments with real quantum particles (including photons, whatever they might be),
several facts are worth noticing:
\begin{itemize}
\item[i)] The evolution of quantum particles takes place in real time.
Quantum particles cannot (or should not) be dissociated from the reality we live in (and
where they also live in).
In a typical diffraction or scattering experiment, for instance, pushing a trigger on they
start being launched; pushing the trigger off the flux of particles ends.
Now, in the meantime, each one of such particles has moved from wherever they were at a
$t_0$ to somewhere else at $t > t_0$ (relativistic issues are left aside for simplicity
and because they are not necessary at all in the discussion).
\item[ii)] The quantum theory is a statistical theory, where the so-called observables
correspond to statistical quantities.
Single events or realizations, e.g., the detection of one particle at a time $t$, do not
provide us with any relevant information about the process investigated; to obtain precise
(physically meaningful) information, a large number of events or realizations (detected
particles) is required, which involves a statistical analysis.
The probability distributions rendered by conventional quantum mechanics are directly
related to this large-numbers approach, which is the way how we can access and investigate
experimentally quantum systems.
In this regard, it is worth noting that the widespread conception that a full interference
pattern, for instance, is related with each single quantum particle is based on the
experimental performance previous to the advent of quantum mechanics, as it is inferred
from works at that time \cite{taylor:PCPS:1909,dempster:PhysRev:1927}.
After all, the atomistic understanding of matter was not solidly grounded.
But, more importantly, this view did not allow the settlement of solid conclusion about
the inherent statistical nature of quantum particles.
Only by the end of the 1970s and along the 1980s the refinement reached in the experimental
techniques enabled single-particle production/detection
\cite{davis:IEEEJQuantumElectron:1979,aspect:EPL:1986,kimura:OptCommun:1989},
which in turn gave complete sense to the statistical meaning of quantum observables, as
related to a collection of individually (detected) events.
This leads us directly to fact (iii) below.
\item[iii)] Even when it can be experimentally shown that there is no time-correlation
between the evolution of one particle ``identically'' prepared with respect to another
particle that precedes it, the two particles behave as if they shared some kind of
fundamental information.
Independently of their nature, all quantum particles exhibit the same behavior in
event-by-event experiments (photons \cite{weis:AJP:2008,weis:EJP:2010,padgett:AJP:2016},
electrons \cite{pozzi:AJP:1973,pozzi:AJP:1976,pozzi:AJP:2007,pozzi:EJP:2013,tonomura:ajp:1989,batelaan:NJP:2013},
atoms \cite{shimizu:pra:1992} or large macromolecular complexes
\cite{arndt:Nature:1999,arndt:NatCommun:2011,arndt:NNanotech:2012}).
This behavior has also been observed even in classical-type processes\footnote{Here, the
notion of ``classical-type'' applied to quantum or optical processes will be understood in
the context of wave descriptions that include a partial or even total lack of coherence (and
hence they are unable to display interference), regardless of how the latter arises.},
such as imaging produced by objects under extremely faint illumination conditions.
In these cases, the image becomes apparent after a rather long exposure time, once
the number of collected photons is relatively high, as it is shown and discussed in
\cite{rose:AdvBiolMedPhys:1957}, in the context of the limitations imposed on vision
by the quantum (granular) nature of light.
This is exactly the same problem that affect (the imaging of) interference patterns when
the source is very weak under total coherence conditions.
\end{itemize}
The above facts are relatively well known and hence they might seem natural to the reader
(even trivial).
However, they pave the way for event-to-event statistical descriptions of quantum systems
to the detriment of the single-particle approaches typically associated with Schr\"odinger's
equation.
It is precisely here where Bohmian mechanics comes into play: it gathers all the formal
elements to be consistent with quantum mechanics (it is actually quantum mechanics) without
the necessity to introduce any additional quantities or approximations.
Indeed it is an ideal candidate to investigate quantum dynamics in conformity with the above
three experimental facts: evolution in real time, observables arising from statistics
over individual events and uncorrelated realizations (events).
Leaving aside complications arising from computational implementations, we now have a
convenient tool to compare on equal (statistical) footing experiment and theory (detections
vs realizations).
This is precisely a legitimate argument to respond the everlasting criticism on the
additional physical content, which also leaves aside the hidden-variable issue,
because both concepts and formalism are well defined.
In fact, the role of the so-called quantum postulates is diminished.
So, what else could one wish?
Thus, so far, it is clear that, provided all sources of controversy are left aside
(at least in line with the renowned Copenhagian `shut up and calculate!'
\cite{mermin:PhysToday:1989}), nothing wrong is found in Bohmian mechanics, nor even
one needs to give further explanations on which new physics it provides us with.
Of course, there are some subtleties that make Bohmian mechanics different at an intuitive
level from the point of view of classical Newtonian mechanics, but this is also legitimate
for, after all, quantum mechanics itself is conceptually different from classical mechanics,
as stressed by the Bell inequalities \cite{bell:physics:1964,bell:RMP:1966}.
As mentioned by Hiley \cite{muser-hiley:SciAm:2013}, this difference used to be remarked by
Bohm by talking about Bohmian `non-mechanics', since this quantum approach {\it sensu
stricto} has little in common with mechanics.
Note, for instance, that the standard concept of force gets diluted
with the introduction of a quantum force mediated by Bohm's quantum potential.
Yet this is a hydrodynamic-like model that serves to the purpose, making more apparent
dynamical behaviors that go beyond our classical intuition, although they rule nature at
the microscopic and mesoscopic levels (with important implications on the macroscopic one).
Following the preceding discussion, the purpose here is to show and discuss a
``non-mechanical'' perspective of the Bohmian approach, that is, avoiding the traditional
ideas of quantum potential and quantum force, and trying to ground the description of quantum
phenomena on a ``non-observable'' (in the standard sense), namely, the quantum phase field
associated with the system state.
Typically, Bohmian mechanics includes discussions that turn around the concepts of Bohm's
quantum potential (or the forces generated by this potential) and how it rules the behavior
of the so-called Bohmian trajectories.
However, the quantum potential is only a measure of the curvature of the probability density
and, therefore, one feels compelled to find other alternative mechanisms responsible for the
dynamics exhibited by quantum systems.
The quantum phase and, more specifically, its local variations, though, have not been much
exploited in the literature, although they translate into a local velocity field that, in
principle, can be measured by means of the so-called weak measurements
\cite{aharonov:PRL:1988,sudarshan:PRD:1989,wiseman:NewJPhys:2007}.
When this velocity field, which is denoted as ``local'' because the system flow is determined
by its local value, is considered, a series of interesting properties emerge, which are not
proper of Bohmian mechanics, but of quantum mechanics in general, although they cannot be
easily perceived with other quantum formulations.
For instance, the so-called non-crossing rule in Bohmian mechanics is nothing but a
combination of the single-valuedness of the quantum phase (except for integer $2\pi$-jumps,
which are unnoticeable in the velocity field) and the dynamical domains determined by the
velocity field.
In order to illustrate these properties, a simple model of Gaussian diffraction and Young-type
interference are going to be analyzed in next sections, because of their interest not only
in quantum mechanics, but also in wave optics, which shares common theoretical grounds with
the former (despite the latter is typically regarded as a ``classical'' theory).
The work has thus been organized as follows.
Section~\ref{sec2} introduces and discusses some fundamental aspects of
Bohmian mechanics in the direction pointed out above, making emphasis on those formal aspects
that put the approach at the level of any other quantum representation rather than in those
that have traditionally associated it with a theory without observers, where trajectories are
(unfoundedly) related to paths followed by quantum particles.
Furthermore, on the analytical level, some particular aspects of the quantum potential are
illustrated by employing a simple diffraction model consisting of a free Gaussian wave
packet.
Section~\ref{sec3} is devoted to revisit and discuss some physical consequences related
to a Young-type interference.
To support the interest in the theory, particularly taking into account the discrete nature
of quantum phenomena, first the outcomes from a simple event-by-event Young-type experiment
are reported and discussed.
Then, Young-type fringes are analytically described in terms of a simple model consisting of
a coherent superposition of two Gaussian wave packets.
This models describes in a convenient manner the emergence of interference fringes along the
transverse direction (assuming the matter wave propagates forward with a fast speed, as it
is usually the case in slit and grating diffraction experiments).
More specifically, it will serve to show the main difference between the physics linked to
the quantum potential and the physics rendered by the velocity field.
Finally, the work concludes with a series of remarks summed up in Sec.~\ref{sec4}.
\section{A critical view on Bohmian mechanics: Concepts and formalism}
\label{sec2}
\subsection{Hidden variables {\em vs} experimental facts}
\label{sec21}
Since much has already been said in the literature about conceptual and formal aspects of
Bohmian mechanics, this section will be devoted to highlight other formal aspects, which
have not been so extensively considered.
Yet these aspects provide us with a different perspective of both Bohmian mechanics
itself and the quantum phenomena in general.
They will also be useful to get a better and broader understanding of the discussion in
Sec.~\ref{sec3}, at the same time that they in compliance with the statement made by Bohm
regarding his reformulation of quantum mechanics \cite{bohm:PR:1952-1}:
\begin{quotation}
\noindent
``[\ldots]\ the suggested interpretation provides a broader conceptual framework than the
usual interpretation, because it makes possible a precise and continuous description of all
processes, even at the quantum level.
This broader conceptual framework allows more general mathematical formulations of
the theory than those allowed by the usual interpretation.''
\end{quotation}
The above statement refers to interpretation, that is, how we should or could consider that
real particles behave in space and time.
Now, to put forth the question on a real-life context, consider the chip of a CCD
made of an array of pixels and connected to a screen where the detection of a photon in a
pixel translates into a scintillation on the screen.
Is there any good or deep reason preventing us from joining the scintillation (single photon
detection) with a specific source point at a previous time?
That is, can we establish a causal connection between the two points?
In principle, it seems there is no empirical evidence neither in favor nor against it.
However, assuming that we accept that such a link can be established, the next question is
whether the connection can be done by means of a smooth trajectory, more specifically, a
Bohmian trajectory.
This has been a central question in Bohmian mechanics since its beginning in the early 1950s.
Although the Bohmian approach prescribes a precise way to proceed, we have no way to
demonstrate that nature operates the same way; other alternative approaches could also be
formulated with a similar result, but without the need to consider a smooth causal connection.
This is the case, for instance, of the stochastic approaches proposed by Bohm and Vigier
\cite{bohm:pr:1954} (later on also considered by Bohm and Hiley \cite{bohm:PhysRep:1989}) or
by Nelson \cite{nelson:pr:1966}.
Nonetheless, it is clear that there is an appealing feature in Bohmian trajectories over
other types of trajectory-based approaches: it renders a fair reproduction of the detection
process, statistically speaking, at the same time that offers a precise description of the
system evolution (spatial diffusion, diffractive effects or whirlpool-type motion).
\subsection{Equations of motion and trajectories}
\label{sec22}
The standard starting point of Bohmian mechanics consists in recasting
Schr\"odinger's equation in the form of two coupled real-valued partial differential
equations \cite{bohm:PR:1952-1,holland-bk}.
This is achieved by writing the wave function (formerly given in the configuration
representation) in polar form, as
\begin{equation}
\Psi({\bf r},t)= A ({\bf r},t) e^{iS({\bf r},t)/\hbar} .
\label{eq1}
\end{equation}
This nonlinear transformation allows us to pass from the complex field variable
$\Psi$ to two real field variables, namely, an amplitude $A$ and a phase $S$.
This ansatz was formerly considered by Dirac \cite{dirac:PRSLA:1931}, in connection with the
existence of quantized singularities (magnetic monopoles), and by Pauli \cite{pauli-hbk-1},
in the context of quantum-classical correspondence.
After substitution into the time-dependent Schr\"odinger equation,
\begin{equation}
i\hbar\ \frac{\partial \Psi}{\partial t} = - \frac{\hbar^2}{2m} \nabla^2 \Psi + V \Psi ,
\label{eq2}
\end{equation}
and then proceeding with some simple algebraic manipulations, the real and imaginary parts
of the resulting equation give rise to two coupled partial differential equations:
\begin{subequations}
\begin{eqnarray}
\frac{\partial A^2}{\partial t}\ & +\ & \nabla \cdot \left( A^2 \frac{\nabla S}{m} \right) = 0 ,
\label{eq3a} \\
\frac{\partial S}{\partial t}\ & +\ & \frac{(\nabla S)^2}{2m} + V - \frac{\hbar^2}{2m}\frac{\nabla^2 A}{A} = 0 .
\label{eq3b}
\end{eqnarray}
\label{eq3}
\end{subequations}
The first equation, Eq.~(\ref{eq3a}), arising from the imaginary part of the Schr\"odinger
equation, deals with the spatial diffusion or dispersion of the probability density,
$\rho ({\bf r},t) = A^2 ({\bf r},t)$.
This is a continuity equation relating the evolution of the probability density in a
position (configuration) space with the vector quantity
\begin{equation}
{\bf J}({\bf r},t) = A^2 ({\bf r},t) \ \frac{\nabla S ({\bf r},t)}{m}
= \frac{1}{m}\ {\rm Re} \left\{ \Psi^* ({\bf r},t) \hat{\bf p} \Psi ({\bf r},t) \right\} ,
\label{eq4}
\end{equation}
where $\hat{\bf p} = - i\hbar \nabla$ is the expression of the momentum operator in the
configuration representation.
Equation~(\ref{eq4}) describes the quantum flux or current density, a well-known quantity
in quantum mechanics \cite{bohm-bk:QTh,schiff-bk}, introduced at the very beginning
of any elementary course on the subject.
As it can be noticed on the r.h.s.\ of the first equality in Eq.~(\ref{eq4}), the quantum
flux can be rewritten in a more compact form as
\begin{equation}
{\bf J}({\bf r},t) = \rho({\bf r},t) {\bf v}({\bf r},t) .
\label{eq4b}
\end{equation}
This expression not only makes emphasis on the causal relationship between the probability
density $\rho$ and its dispersion in terms of the quantum flux, but it also makes more
apparent the mechanism for such a dispersion, namely, the presence of an underlying
local velocity field,
\begin{equation}
{\bf v}({\bf r},t) = \frac{{\bf J}({\bf r},t)}{\rho({\bf r},t)}
= \frac{\nabla S ({\bf r},t)}{m} .
\label{eq5b}
\end{equation}
Physically, this vector field accounts for the density flow rate through the point ${\bf r}$
at a time $t$, i.e., its value changes locally following the variations of the quantum phase
$S$, unlike the average drift value obtained from the expectation value of the momentum,
$\langle \hat{\bf p} \rangle/m$.
The velocity field (\ref{eq5b}) is not a proper quantum observable in spite of its
connection to the usual momentum operator $\hat{\bf p}$.
Yet, it is going to play a fundamental role in the quantum dynamics, because of the
information that it provides on the concentration, expansion, diversion or rotation of the
quantum flow at each point.
The natural question that arises here is why this quantity is not mentioned at all
in any standard course on quantum mechanics, although it is well defined and provides extra
local information on the deformation of the probability density in the configuration space.
Actually, not only in standard quantum mechanics, but also in Bohmian mechanics this quantity
is often neglected in favor of other quantities, such as the so-called Bohm's quantum
potential, usually required to explain the behavior displayed by Bohmian trajectories.
To understand the above statement, let us get back to Eqs.~(\ref{eq3}).
The second differential equation, Eq.~(\ref{eq3b}), encoded in the real part of
Schr\"odinger's equation, keeps a formal resemblance with the classical Hamilton-Jacobi
equation \cite{goldstein-bk}.
This classical Hamilton-Jacobi equation arises from the so-called Hamiltonian analogy
between mechanics and optics \cite{bornwolf-bk}, which establishes a connection (analogy)
between the wavefronts of optics (surfaces of constant phase) and surfaces of constant
mechanical action.
Hence, in the same way that light rays are perpendicular to the wavefronts at each point,
the Newtonian trajectories are perpendicular to constant-action surfaces.
Following this analogy, Bohm considered Eq.~(\ref{eq3b}) to be a quantum version of
the Hamilton-Jacobi equation, thus postulating the existence of a quantum Jacobi law of
motion \cite{bohm:PR:1952-1,holland-bk}
\begin{equation}
\dot{\bf r} ({\bf r},t)= \frac{\nabla S ({\bf r},t)}{m} .
\label{eq5}
\end{equation}
This equation of motion is known as the guidance equation.
In agreement with the widespread Bohmian interpretation for this equation, it rules the
(quantum) way how particles moves, since integrating in time with the corresponding initial
conditions one obtains swarms of (Bohmian) trajectories.
Extending this idea further beyond, one might conclude that particle instantaneous
positions (e.g., the trajectory of an electron or a photon in an interference experiment)
are actually described by the solutions to (\ref{eq5}), thus becoming a sort of
hidden causal variables.
However, as seen above, this equation of motion is exactly the same as Eq.~(\ref{eq5b}),
which naturally follows from the standard formulation (from the continuity equation),
without any need to postulate anything, nor even the existence of hidden variables.
In classical mechanics there is a clear and direct connection between the trajectories
obtained from Jacobi's law, which describe the dynamical properties displayed by a
phase-space distribution function, and Newtonian trajectories, which account for the
evolution of individual systems.
That is, there is a one-to-one correspondence between the descriptors for ensembles and
for individuals, which is ultimately based on experience (statistics is the large-particle
limit of dynamics).
However, the same connection cannot be established for quantum systems.
Note that quantum mechanics, which is a statistical theory itself (with some
peculiar properties that make it different from classical statistics), lacks a quantum
counterpart for individuals, thus avoiding us to compare the Bohmian trajectories that
describe the dynamical properties of the probability density (in configuration space)
with the dynamics exhibited by individual particle trajectories obtained from a
singe-body dynamical law, i.e., the quantum analog to Newton's trajectories.
Bohmian trajectories have long been identified with such quantum Newtonian trajectories.
However, this is not based on solid empirical grounds, but on a weak conceptual inference:
because the statement holds for classical particles, it must also hold for quantum ones.
This is a very weak argumentation, because classical mechanics
and classical statistical mechanics are based on different formal grounds (phase space)
than quantum mechanics (positions or momenta, but not both at the same time, unless we
pay a price for it, as it happens ih the Wigner-Moyal representation).
Establishing a strong unique connection would require empirical evidence beyond
statistics-based experiments and reconsidering theoretical models in the direction of
de Broglie's former ideas of wave fields and particles both coexisting but being different
physical entities \cite{broglie-bk:1960}, the stochastic causal model \cite{bohm:pr:1954,bohm:PhysRep:1989}
or subquantum Brownian-type models \cite{furth:ZPhys:1933,comisar:PhysRev:1965,nelson:pr:1966}.
This is an important point, for instance, when using Eq.~(\ref{eq5}) in the interpretation and
understanding of single-photon experiments \cite{kocsis:Science:2011,steinberg:SciAdv:2016},
since the information provided by such experiments is indeed understandable in terms of our
actual theories of light; trajectories inferred from the experiments thus do not represent
the actual motion of real photons, but just the average expansion or contraction undergone
by the wave field corresponding in the large photon-number limit.
In order to switch from matter waves to light in this regard, it can easily be noted that
Eq.~(\ref{eq5}) shows a certain reminiscence of the optical ray equation, which describes
rays as lines always perpendicular to the surfaces of constant phase and would arise from
the aforementioned Hamiltonian analogy \cite{bornwolf-bk} (which, in turn, underlies the
derivation of Schr\"odinger's equation).
This is precisely the same conclusion found by de Broglie earlier on\footnote{In this
regard, the interested reader might like to consult the work by Drezet and Stock
\cite{drezet:AnnFondLdBroglie:2021} on an original manuscript sent by Bohm to de Broglie
in 1951, which predates the renowned 1952 papers and, according to the authors, seems to
be its origin.}
\cite{broglie:CompRend-2:1926}, which in the case of classical light (which does
not follow Schr\"odinger's equation, but Maxwell's equations) works very nicely
\cite{prosser:ijtp:1976-1,sanz:AnnPhysPhoton:2010,sanz:JRLR:2010}.
\subsection{Bohm's quantum potential}
\label{sec23}
Unlike classical particles, the motion displayed by quantum particles\footnote{Following
the preceding discussion, the meaning of ``particle'' here is as denoted above,
i.e., as an entity that serves us to keep track of the quantum density flux.} that follow
Eq.~(\ref{eq5}) is affected not only by the forces induced by $V({\bf r},t)$, but also
by an additional term, as seen in Eq.~(\ref{eq3b}).
This is the so-called Bohm's quantum potential,
\begin{equation}
Q({\bf r},t) = - \frac{\hbar^2}{2m} \frac{\nabla^2 A({\bf r},t)}{A({\bf r},t)}
= - \frac{\hbar^2}{4m} \left\{ \frac{\nabla^2 \rho({\bf r},t)}{\rho({\bf r},t)}
- \frac{1}{2} \left[ \frac{\nabla \rho({\bf r},t)}{\rho({\bf r},t)} \right]^2 \right\} .
\label{eq6}
\end{equation}
This contribution to the quantum Hamilton-Jacobi equation has little in common with usual
potential functions acting on physical systems.
Rather it is associated with the local curvature of the amplitude of the wave function,
undergoing important values in those regions where the amplitude becomes negligible, but
not its Laplacian.
This happens in nodes and nodal lines, where the quantum force acting on the particle
becomes very intense and so the changes in ${\bf v}$.
However, this is all related to the quantum state of the system itself and not to any
external interaction.
This is more apparent if we look at the right-hand side of the second equality, which is
explicitly written in terms of $\rho$.
In a Bohmian sense, $\rho$ describes the statistical distribution of independent realizations,
i.e., it is produced by the cumulative effect arising after, for instance, launching a large
number of independent photons or electrons (but all subjected to the same experiment), and
see how they start distributing spatially after a given (long enough) exposure time on the
corresponding detector \cite{tonomura:ajp:1989,weis:AJP:2008,padgett:AJP:2016,arndt:NNanotech:2012}.
How can independent realizations influence one another?
The above discussion leads us to a deeper question, namely, that of the reality of the wave
function as a physical field, beyond its usual conception as a probabilistic information
descriptor \cite{pusey:NaturePhys:2012}.
Quantum systems or, more strictly speaking, their seemingly random statistical distributions
would play the role of tracers that allow us to feel such a presence, in the same way, for
instance, that
iron powder allows us to make observable magnetic the line forces and, therefore, to detect
the presence of magnetic fields.
Of course, this does not provide any clue on the origin of this field or whether quantum
systems behave in the same way as Bohmian trajectories do (see discussion in next
section), but at least it seems there is a dynamical element that is totally neglected,
namely, the presence of an intrinsic velocity field.
This field does not require the presence (or existence) of a quantum potential, because it
is directly related to the phase (see below), hence removing the redundancy of describing
quantum effects in terms of $\rho$ and its curvature.
Furthermore, this quantity tells us that there is an underlying
statistical stream behavior not necessarily related with a quantum observable (although
its effects manifest through the topology displayed by $\rho$).
There is another important question regarding the quantum potential, more important to the
purpose here because of its dynamical implications, which is the fact
that this contribution to the quantum Hamilton-Jacobi equation arises from the kinetic
operator of Eq.~(\ref{eq2}).
Therefore, it is related to the diffusive part of the Schr\"odinger equation and not to
the action itself of an external potential function.
Therefore, even if it is used to explain in a sort of mechanistic way the
motion exhibited by quantum systems (within a Bohmian framework), it should be
interpreted as a kind of internal information conveyed to the system by its own
quantum state, which is continuously changing in time.
It is in this regard that the concept of ``mechanics'' is somehow dubious, because
the mechanism of the dynamical behaviors observed is partly due to the own system
(or, more strictly speaking, its quantum state).
\subsection{Dispersion of a localized Gaussian wave packet}
\label{sec24}
To illustrate the above facts in simple terms, consider the paradigm of localized quantum
system representing the free evolution of a particle of mass $m$, described by Gaussian
wave packet \cite{sanz:JPA:2008}.
In this case, although there are no
external forces acting on the particle, its states spreads out continuously in time,
first slowly and then, after undergoing an accelerating boost, linearly with time
\cite{sanz:AJP:2012}.
To better understand this behavior, consider that the particle is described by a normalized
one-dimensional wave packet centered at $x=0$,
\begin{equation}
\Psi(x) = \left( \frac{1}{2\pi\sigma_0^2} \right)^{1/4} e^{-x^2/4\sigma_0^2} ,
\label{eq8}
\end{equation}
with its width being $\sigma_0$.
The time-evolution of this wave packet is described by the time-dependent state
\begin{equation}
\Psi(x,t) = \left( \frac{1}{2\pi\tilde{\sigma}_t^2} \right)^{1/4}
e^{-x^2/4\sigma_0\tilde{\sigma}_t} ,
\label{eq9}
\end{equation}
with
\begin{eqnarray}
\tilde{\sigma}_t = \sigma_0 \left( 1 + \frac{\hbar t}{2m\sigma_0^2} \right) ,
\label{eq10}
\end{eqnarray}
with its (time-dependent) width being
\begin{equation}
\sigma_t = |\tilde{\sigma}_t| =
\sigma_0 \sqrt{1 + \left( \frac{\hbar t}{2m\sigma_0^2} \right)^2} .
\label{eq13}
\end{equation}
Because the initial momentum associated with the particle is zero, the center of this wave
packet remains at $x=0$ (there is no translational motion).
Yet, the expectation value of the energy is nonzero, but constant in time:
\begin{equation}
\bar{E} = \langle \hat{H} \rangle = \frac{\hbar^2}{8m\sigma_0^2} .
\label{eq11}
\end{equation}
This sort of average energy (\ref{eq11}) corresponds to the diffusive internal energy that
makes the wave packet to spread out with time, even if it can be recast as $\bar{E} = p_s^2/2m$,
in terms of the spreading momentum $p_s = \hbar/2\sigma_0$ \cite{sanz:JPA:2008}.
Nonetheless, we can still further investigate this contribution.
To that end, let us recast the wave packet (\ref{eq9}) in polar form and proceed with the
corresponding substitutions in Eq.~(\ref{eq3b}).
The kinetic contribution reads as
\begin{subequations}
\begin{equation}
K = \frac{1}{2m}\left(\frac{\partial S}{\partial x}\right)^2 = \frac{\hbar^2}{8m\sigma_0^2}
\left( \frac{\sigma_t^2 - \sigma_0^2}{\sigma_t^2} \right)
\frac{x^2}{\sigma_t^2} ,
\label{eq12a}
\end{equation}
while Bohm's quantum potential is
\begin{equation}
Q = \frac{\hbar^2}{8m\sigma_0^2} \frac{\sigma_0^2}{\sigma_t^2}
\left( 2 - \frac{x^2}{\sigma_t^2} \right) .
\label{eq12b}
\end{equation}
\label{eq12}
\end{subequations}
As it can be noticed from Eq.~(\ref{eq12a}), the lack of an initial momentum or the action
of an external potential makes more apparent how the quantum phase is responsible for the
generation of a dynamics, which eventually translates into the spreading of the wave packet.
This dynamics is null at $t=0$, but since the spreading factor, $\tilde{\sigma}_t$, in the
Gaussian state acquires a complex phase, it gradually induces the appearance of an internal
motion in the form of the spreading of the wave packet.
On the other hand, the quantum potential (\ref{eq12b}) is nonzero even at $t=0$, which is due
to its relationship with the curvature of the quantum state (either through its amplitude or
its density).
Because of their different origin, both quantities do not cancel each other or render
a constant value, but they keep evolving in time, spreading all over longer and longer
distances in configuration space, which explains the increasing spreading of free Gaussian
wave packets.
Following a standard Bohmian prescription and appealing to a typical description of dynamical
systems \cite{jordan-bk:1999}, it is seen that first the inverted parabola with $x$
corresponding to the quantum potential generates an unstable point (the kinetic term is zero),
which diverts trajectories towards positive and negative $x$ (with respect to $x=0$).
Then, the parabola describing the kinetic term starts gaining importance, which somehow helps
to counterbalance the action of the quantum potential, binding the motion.
Finally, at asymptotic times, the quantum potential becomes relatively flat compared to the
kinetic term (one term goes as $\sigma_t^{-2}$, while the other one goes as $\sigma_t^{-4}$),
which opens up gradually, as $x^2/t^2$, thus producing a linear spreading of the
trajectories (i.e., as $x/t$).
Nonetheless, their combined average, $\bar{K} + \bar{Q}$, when it is computed with respect
to the also time-dependent density $\rho$, i.e.,
\begin{equation}
\bar{K} + \bar{Q} = \int_{-\infty}^\infty \left[ \frac{1}{2m}\left(\frac{\partial S}{\partial x}\right)^2
- \frac{\hbar^2}{2m} \frac{1}{\rho^{1/2}} \frac{\partial^2 \rho^{1/2}}{\partial x^2} \right] \rho\ dx ,
\label{eq14}
\end{equation}
renders the constant value (\ref{eq11}), which
is consistent with the fact that this type of motion is energy-preserving, as it is expected
from a free particle.
Actually, it is interesting to note that in the long-time limit, we obtain
\begin{equation}
K + Q \approx \frac{1}{2} m \left( \frac{x}{t} \right)^2 ,
\label{eq15}
\end{equation}
which is consistent with the asymptotic behavior displayed by $K$ in the long-time limit,
as it was commented above.
Noticed that Eq.~(\ref{eq15}) resembles the usual expression for the kinetic energy of a
classical particle (with $v = x/t$), although its average is (\ref{eq11}), as it can readily
be shown by averaging over $\rho$.
After all, in the long-time limit, the dispersion undergone by the Gaussian, Eq.~(\ref{eq13}),
increases linearly with time as
\begin{equation}
\sigma_t \approx v_s t ,
\label{eq16}
\end{equation}
where $v_s = p_s/m$ \cite{sanz:JPA:2008}.
Actually, this expression not only provides the asymptotic spreading of a quantum wave
packet for a massive particle, but it also coincides with the divergence of a Gaussian
laser beam in paraxial optics \cite{sanz:ApplSci:2020} when the quantity $\hbar t/m$ is
substituted by $z/k$ (linear increase of the transverse dispersion with the longitudinal
coordinate $z$).
From a statistical viewpoint, the above discussion focuses around ensemble dynamics, which
is what $\rho$ eventually describes, namely the behavior of a large number of identically
distributed (non interacting) particles of mass $m$ in free space.
Following the standard Bohmian view (the one arising from his 1952 paper),
such particles follow well-defined trajectories, which are obtained after integrating
in time Eq.~(\ref{eq5}).
This is a legitimate way to understand such an equation.
However, there is an also legitimate but alternative way to understand it, namely, by
considering that such trajectories are just the streamlines that follow the flow described
by the local velocity field ${\bf v}$, as specified by Eq.~(\ref{eq5b}) and in compliance with
standard quantum mechanics.
In this case, irrespective of how real particles move (smoothly or randomly), Bohmian
trajectories only
reflect the local dynamics of the ensemble, just in the same way that a tiny floating particle
provides us with a clue on how a stream flows, but not on how each individual molecular
component of the stream behaves \cite{sanz:AJP:2012,sanz:JPhysConfSer:2012,sanz:FrontPhys:2019}.
This view is closer to Madelung's hydrodynamic formulation of the Schr\"odinger equation
\cite{madelung:ZPhys:1926}.
So, going to the point, after analytically integrating in time (\ref{eq5}) for the Gaussian
state (\ref{eq9}), the trajectories are found to follow the functional form
\begin{equation}
x(t) = \frac{\sigma_t}{\sigma_0}\ x(0) ,
\label{eq17}
\end{equation}
where $x(0)$ is the corresponding initial condition.
As it can be noticed, these trajectories diverge in compliance with the divergence undergone
by $\sigma_t$, being the effect more apparent as their initial condition is chosen further
and further away from the center of the wave packet.
At each time, it is possible to determine how the distribution described by $\rho$
behaves by only inspecting the behavior exhibited by a swarm of such trajectories,
which provides us clues on different dynamical regimes \cite{sanz:AJP:2012}.
\section{Young's two slits revisited}
\label{sec3}
To better appreciate the implications of the Bohmian formulation of quantum
mechanics and their reach in our understanding of quantum phenomena, now we are going
to focus on the analysis of Young's two-slit experiment, which, quoting Feynman
\cite{feynman-bk1}, ``has in it the heart of quantum mechanics''.
As it is shown, when the phenomenology of this experiment is revisited in terms of Bohmian
mechanics, a different perspective arises on what is going on, which challenges its
traditional Copenhagian explanation.
Furthermore, because this experiment stresses in the simplest manner the capability of
quantum systems to exhibit delocalization while keeping long-distance space correlations
(necessary to observe the well known interference fringes), by extension that analysis also
provides us with a new physical understanding of the concept of coherence, central not only
to quantum mechanics, but also to optics.
\subsection{Current experiments, old explanations}
\label{sec31}
The single-event-based picture of interference rendered by Bohmian mechanics is, perhaps,
better understood by revisiting the experiment and how it has been traditionally explained.
As it was mentioned in the introductory section, there is a number of interference
experiments performed with both photons and material particles, which show that whenever
the particle flux is faint enough (to the point that each single detection can be monitored
in real time), a random-like distribution of detected events is observed instead of the
well-defined interference fringes obtained in a standard high-flux experiment.
In such experiments, interference fringes start emerging gradually from among the
point-like distribution of detected events as time passes by
\cite{padgett:AJP:2016,rose:AdvBiolMedPhys:1957}.
This can easily be illustrated by considering a simple experiment, which, in the current
case, has been performed by the author in the teaching optics laboratory (an experiment
that our students routinely perform every year).
It is a Young-type experiment performed with a 631~nm wavelength laser ($< 5$~mW power)
illuminating a thin steel mask with two parallel narrow slits.
The slits have an average width of 0.145~mm and their center-to-center distance is 0.865~mm.
The light coming from the slits is made to converge with a 150~mm focal length lens on a
CCD consisting of a $1024 \times 768$ array of square pixels (the side of each pixel is
4.64~$\mu$m long).
\begin{figure}[!t]
\centering
\includegraphics[width=\textwidth]{fig1-sanz.png}
\caption{\label{fig1}
Snapshots illustrating the light distribution produced by two narrow slits (0.145~mm
wide) separated a distance of 0.865~mm when they are illuminated by a 631~nm wavelength
laser under high-intensity conditions (left) and low-intensity conditions (right).
The upper panels show the intensity distribution recorded by a CCD consisting of a
$1024 \times 768$ array of square pixels (with a side 4.64~$\mu$m long).
In both cases, the right-hand side gray-level scale in the upper panels denotes the
intensity registered by each pixel during the time the experiment has been run (few
seconds in both cases), which is proportional to the number of photons registered by
each pixel.
The lower panels show the intensity only along the transverse direction, that is,
integrating (summing) over the vertical direction in the upper plot in order to make
more apparent that the intensity can be assumed as a continuous distribution in the
high-intensity regime and as formed by discrete scintillations in the low-intensity
regime.}
\end{figure}
The upper panels of Fig.~\ref{fig1} show two snapshots that illustrate the result of
performing the experiment under high-intensity conditions (left) and under low-intensity
conditions (right).
In both cases, a very short exposure time has been considered in order to emphasize who,
in the high-intensity regime, the light interference fringes appear as a continuous
intensity distribution.
On the contrary, when the intensity is low enough, which is achieved by placing two
polarizers in front of the laser source and then making their transmission axes to be
nearly perpendicular one another, we observe a series of uncorrelated scintillations
that distribute randomly across the detector surface.
By zooming in the upper right panel (see inset), we can notice that the distribution is
rather sparse, thus offering no clue on any underlying interference-type structure.
In order to get a better quantitative idea, the lower panels show the transverse
intensity distribution, which has been obtained by integrating (summing) the intensity of
the upper panels along the vertical direction.
The $x$-axis labels the position of the capture pixels along this direction, while the
$y$-axis provides us with the relative intensity, proportional to the number of photons
collected in the corresponding pixels (remember the summation over the vertical pixels
for a given $x$-position) during the experiment performance time (for a better read of
the upper panels, the gray-level scale to their right represents the same).
Regarding the interpretation of the data shown, several comments are in order.
First, while the horizontal axis in the lower panels runs over all the 768~pixels, for
a better visualization in the upper panels only the region around the fringes has been
considered (the same regarding the vertical axis).
Hence, the upper and lower panels cannot be directly compared, which is the reason for the
mismatching when comparing the maxima and minima in both cases.
Second, the discrete numbers that appear, in particular, in the lower right panel does
not correspond to photon counts or, in other words, to number of photons per pixel, but to
a quantity proportional to CCD units of counts.
Yet it is clear by comparing the two lower panels that while in one case the pattern runs
smoothly along the transverse direction (left), the same does not happen in the low-intensity
regime, where a nearly uniform discretized accumulation.
Finally, even with the limitations of not having at hand a reliable single-photon source, but
a simple setup (after all, the experiment has been carried out by a theoretician), it still
serves to the purpose of illustrating the discreteness involved in the formation of
interference patterns.
This phenomenon can only be noticed with a very faint illumination of the slits, but is of
much relevance in the understanding of the two-slit experiment.
\begin{figure}[!t]
\centering
\includegraphics[width=0.9\textwidth]{fig2-sanz.png}
\caption{\label{fig2}
From (a) to (d), snapshots taken at subsequent times for the experiment of Fig.~\ref{fig1}
under low intensity conditions.
The upper frame shows the random-type distribution of scintillations for a few seconds.
Although there seems to be no correlation among those scintillations, as time proceeds it
is seen that photons arrive in a larger proportion to certain pixels, while avoid others,
thus giving rise to accumulations that increase beyond noise-type fluctuations.
Eventually these accumulations give rise to an intensity distribution mimicking the one
obtained under high-intensity conditions, which requires times of the order of several
minutes (note that, even so, the maxima are close to relative intensity values of 120,
while in the high-intensity regime (see Fig.~\ref{fig1}) they reached values up to 5,000).}
\end{figure}
In Fig.~\ref{fig2} the gradual appearance of the fringes is explicitly shown by beans of a
series of subsequent snapshots, each taken a larger exposure time.
In the upper panel of the figure there is a photograph for a short time, as in the right
panels of Fig.~\ref{fig1}.
As time proceeds, the sequence of fringes becomes more and more apparent, as it is shown in
the sequence from panels (a) to (d).
In this case, in order to focus on the region around the interference pattern, the intensity
in the 1D plots has been taken along a given range of the 2D photographs.
It can be noticed that these fringes are more apparent in the 2D plots than in the
representations of the relative intensity in terms of the transverse direction.
This is only an effect due to the summation over the vertical direction: since the pattern is
too faint, all other pixels exited with environmental noise photons are also going to contribute,
thus reducing the relative visibility of interference pattern.
This is the case in panels (a) and (b), for instance.
In the case of panels (c) and (d), the accumulation of photons in the regions around
interference maxima becomes more prominent than the background noise contribution.
The key question that arises now is that of the interpretation of the granular behavior involved
in the formation of interference fringes, as illustrated by the above experiment.
The traditional Copenhagian interpretation of the phenomenon is essentially based on the
pattern obtained in the high-intensity regime, that is, a continuous intensity distribution
that spreads all over some spatial region in the form of alternating light and dark spots
or fringes (regions with a high and low detection rates, respectively).
However, given that quantum systems consist of single, independent particles, the explanation
considers that, at some point the particle, understood a spatially localized system, becomes
and propagates as a wave before, while and after passing through the slits.
When it reaches the detector, the associated wave ``collapses'' and the particle acquires
again its corpuscular nature in the form of a spatially localized (single-event) detection
\cite{dempster:PhysRev:1927}.
Formally, this translates into the two different processes mentioned by Von Neumann to
describe the propagation and measurement of quantum systems \cite{vonNeumann-bk:1932}.
While the particle is not detected its evolution is unitary; when it is being detected, such
unitarity breaks down and it takes place a non-unitary irreversible ``collapse'' to a specific but previously indefinite spatial position.
This conception might seem odd and even uncomfortable, but it is what the experiment allows
us to know about the particle, even if we appeal to single-event experimental procedures, like
the one described above or all other that can be found in the literature; there is no way to go
further away and determine what is going on from the slits to the detector experimentally
without directly acting with the system, in which case interferences fly away.
However, it is also true that this is not the impression that one acquires when the particle
flux is weakened so much that the intensity distribution is finally reconstructed on an
event-to-event basis.
For some reason, many feel inclined to find a way to associated those individual, spatially
localized detections with the idea of a particle following a well-defined trajectory in space,
regardless of how this trajectory looks like, i.e., of which equation of motion describes it.
Therefore, even if cannot determine experimentally such trajectories, this ontic perspective
also seems to be a reasonable and legitimate explanation (neither better or worse than the
traditional Copenhagian collapse idea), which cannot be discarded (not, at least, with the
current experimental facts).
Apart from the single-event experiments mentioned so far, there are recent experimental
facts that are making us to reconsider our traditional conception of quantum systems
These changes are connected to the rather old concept of weak measurement
\cite{aharonov:PRL:1988,sudarshan:PRD:1989} and, more importantly, its experimental
implementation \cite{mir:NewJPhys:2007}.
Even though with its limitations regarding the interpretation of quantum phenomena (see below), this
technique has widened our view and understanding of quantum systems.
Contrary to a strong Von Neumann measurement (the usual measurement process in quantum mechanics),
a weak measurement only perturbs slightly the system, without making it to collapse, thus allowing
us to extract information on supplemental aspects of such a system at once \cite{lundeen:Nature:2011,kocsis:Science:2011}.
In the case of interference, both the probability density and the transverse flux, responsible for how
the former changes spatially with time [following Eq.~(\ref{eq4b})], can be determined within the same
experiment without requiring extra measurements, as it happens in quantum tomography, an also without
destroying the interference fringes.
With these two quantities, the local velocity field can be computed from Eq.~(\ref{eq5b})
at any time or, equivalently, any distance between slits and detector.
From here on, considering a series of initial conditions and integrating in time
(\ref{eq5}), one straightforwardly obtains the corresponding Bohmian trajectories,
as it is shown in \cite{kocsis:Science:2011} in the case of light (photons).
Of course, the information extracted from these experiments should be carefully considered,
avoiding conclusions that go beyond the experiment itself.
In the case we are dealing with, as it has been commented above (see Sec.~\ref{sec21}), there
is no empirical evidence on how to relate the inferred trajectories with the real motion of quantum particles.
These trajectories can be considered as streamlines accounting for the spatial dispersion of
ensembles, but not of individuals.
Note that although the average transverse flow, obtained from measurements over
many photons, behaves as specified by Eq.~(\ref{eq5b}) (or, to be more precise, in compliance
with the Maxwellian analog of this field \cite{sanz:AnnPhysPhoton:2010,sanz:ApplSci:2020}),
this does not mean that the detected photons follow Bohmian-type trajectories.
There are stochastic approaches, for instance, which also render the same average outcomes
\cite{bohm:pr:1954,bohm:PhysRep:1989,furth:ZPhys:1933,comisar:PhysRev:1965,nelson:pr:1966}.
Within this scenario, therefore, closer in spirit to Madelung's transformation of
Schr\"odinger's equation into a hydrodynamic form \cite{madelung:ZPhys:1926}, Bohmian
trajectories help us to understand the dynamics of this fluid when it reaches the two slits,
how it behaves after crossing them, giving rise to interference traits, or why interference
disappears if one of the slits is suddenly shut down (``observed'').
This brings in an alternative and very different picture of Young's two-slit experiment,
usually associated with the effects that follow the overlapping of the waves coming out from
each slit, and that has also overmagnified the role of the external observer in the removal
of the fringes.
\subsection{Dynamical role of the local velocity field}
\label{sec32}
Let us thus reconsider the problem from the Bohmian viewpoint.
To this end, a simple model based on the coherent superposition of two one-dimensional
Gaussian wave packets \cite{sanz:JPA:2008}.
In brief, to understand the model and its physical meaning, consider a screen with two slits
separated by a distance $d$ and both being parallel to the $y$-direction.
In this configuration, the $x$-axis cuts both slits in two symmetric halves and the $z$ axis
is perpendicular to the screen containing the slits.
If the slits are much wider along the $y$ direction than along the $x$ direction, and the
incident momentum, parallel to the $z$-axis, is relatively high, so that the angular spreading
by diffraction is negligible compared to the distance traveled along the $z$ axis, the wave
function of the system can be simplified by a product state, where interference takes place
along the (transverse) $x$ direction (further technical details on this modeling of diffraction
systems can be found in \cite{sanz:AOP:2015}).
To further simplify, the transmission function is assumed to be Gaussian, which produces two
diffracted Gaussian states at each slit.
In spite of its simplicity, this captures the essence of the phenomenon without any loss of
generality.
Accordingly, consider that the two diffracted waves are denoted by the coherent superposition
of two Gaussian wave packets,
\begin{equation}
\Psi(x,t) = \psi_-(x,t) + \psi_+(x,t) ,
\label{eq18}
\end{equation}
where each one of these wave packets has the same form as (\ref{eq9}) and subscripts
$\pm$ denote the position of their respective centers with respect to $x=0$, i.e., at
$x_\pm = \pm x_0$, with $x_0 = d/2$.
Recasting the wave packets in polar form, the following expressions for the probability
density and the quantum flux are readily obtained:
\begin{subequations}
\begin{eqnarray}
\rho (x,t)\ & =\ & \rho_+(x,t) + \rho_-(x,t) + 2 \sqrt{\rho_+(x,t) \rho_-(x,t)} \cos \varphi(x,t) ,
\label{eq19a} \\
J(x,t)\ & =\ & \frac{1}{m} \Bigg\{ \rho_+(x,t)\ \frac{\partial S_+(x,t)}{\partial x}
+ \rho_-(x,t)\ \frac{\partial S_-(x,t)}{\partial x}
\nonumber \\ & &
\quad
+ \sqrt{\rho_+(x,t) \rho_-(x,t)}\ \frac{\partial \left[S_+(x,t) + S_-(x,t)\right]}{\partial x} \cos \varphi (x,t) \Bigg\}
\nonumber \\ & &
\quad
+ \frac{\hbar}{2m}\ \sqrt{\rho_+(x,t) \rho_-(x,t)}\
\Bigg[ \frac{1}{\rho_+(x,t)}\frac{\partial \rho_+(x,t)}{\partial x}
\nonumber \\ & & \qquad \qquad \qquad
- \frac{1}{\rho_-(x,t)}\frac{\partial \rho_-(x,t)}{\partial x} \Bigg] \sin \varphi (x,t),
\label{eq19b}
\end{eqnarray}
\label{eq19}
\end{subequations}
with $\varphi (x,t) = [S_+(x,t) - S_-(x,t)]/\hbar$.
The superposition principle does not hold for any of these two magnitudes, since their
expressions involve the density and phase partial fields in a rather nonlinear fashion,
in particular, Eq.~(\ref{eq19b}).
From a dynamical point of view, this translates into an interesting property in the flux
that cannot be perceived in the probability density: at any time, it is zero at $x=0$.
This readily leads to an important physical consequence: the flux to the left of $x=0$
can never mix with the flux to the right \cite{sanz:JPA:2008}.
Accordingly, although the idea of constructive and destructive interference, based on
how the probability density is constructed, is formally correct, we find that the usual
explanation of the two-slits experiment is, to some extent, physically incorrect for it
neglects the dynamics of the probability density in terms of its flux.
When the latter is considered, the fact that the flows associated with each slit do not
mix implies that the left part of the interference pattern is always related to the left
slit, while the right part concerns to the right slit.
In other words, the flow makes distinguishable which part of the pattern is related with
each slit, even if there is no way to determine whether the same happens at an underlying
level with each individual real particle.
To above fact is better seen if Eqs.~(\ref{eq19}) are written explicitly in terms of the two
Gaussian wave packets and their parameters\footnote{For simplicity, the time-dependent
normalizing prefactor has been neglected, because it is dynamically irrelevant (it only
induces the gradual decrease of both quantities as they
spread out spatially).}:
\begin{subequations}
\begin{eqnarray}
\rho(x,t)\ & =\ & e^{-(x-x_+)^2/2\sigma_t^2} + e^{-(x-x_-)^2/2\sigma_t^2}
+ 2 e^{-(x^2 + x_0^2)/2\sigma_t^2} \cos (\kappa x) , \nonumber \\ & &
\label{eq20a} \\
J(x,t)\ & =\ & \frac{\hbar^2 t}{4m^2\sigma_0^2\sigma_t^2} \Big[ (x-x_+) e^{-(x-x_+)^2/2\sigma_t^2}
+ (x-x_-) e^{-(x-x_-)^2/2\sigma_t^2} \nonumber \\
& & \qquad \qquad \quad + 2 x e^{-(x^2 + x_0^2)/2\sigma_t^2} \cos (\kappa x) \Big] \nonumber \\
& & - \frac{\hbar x_0}{m\sigma_t^2}\ e^{-(x^2 + x_0^2)/2\sigma_t^2} \sin (\kappa x) ,
\label{eq20b}
\end{eqnarray}
\label{eq20}
\end{subequations}
with $\varphi = -\kappa x$ and $\kappa = \hbar t x_0/2m\sigma_0^2\sigma_t^2$.
Moreover, consider the timescale $\tau \equiv 2m\sigma_0^2/\hbar$, which provides us with a
measurement of the characteristic dispersion time associated with the wave packet and, hence,
different dynamical regimes characterizing its evolution \cite{sanz:JPA:2008,sanz:AJP:2012}.
As it can be noticed, for relative short times, $t \ll \tau$ ($\sigma_t \approx \sigma_0$),
when diffraction starts acting on each wave
packet but it is not enough to achieve their overlapping (an important value of $\rho$ in
the vicinity of $x=0$), Eq.~(\ref{eq20a}) describes two separate Gaussian distributions
\cite{sanz:AJP:2012,sanz-bk-2}.
In turn, the flux increases linearly with $x$ from negative to positive in both regions $x>0$
and $x<0$, where the sharp separation at $x=0$ removes any inconsistency.
This is a clear indication that, because both waves are present at the same time, there are
two dynamically separated spatial regions.
In the long-time limit, $t \gg \tau$ ($\sigma_t \approx \hbar t/2m\sigma_0$,
$\kappa \approx 2mx_0/\hbar t$), on the other hand, the probability density covers
long distances, $x \gg x_0$, and hence
\begin{subequations}
\begin{eqnarray}
\rho(x,t)\ & \approx\ & 2 e^{-2m\sigma_0^2 x^2/\hbar^2 t^2}
\Big[ \cosh (4m\sigma_0^2 x_0 x/\hbar^2 t^2) + \cos (2 m x_0 x/\hbar t) \Big] ,
\nonumber \\ & &
\label{eq21a} \\
J(x,t)\ & \approx\ & \frac{2x}{t}\ e^{-2m\sigma_0^2 x^2/\hbar^2 t^2}
\Big[ \cosh (4m\sigma_0^2 x_0 x/\hbar^2 t^2)
+ \cos (2 m x_0 x/\hbar t) \Big] \nonumber \\
& & - \frac{2x_0}{t}\ e^{-2m\sigma_0^2 x^2/\hbar^2 t^2}
\sinh (4m\sigma_0^2 x_0 x/\hbar^2 t^2) .
\label{eq21b}
\end{eqnarray}
\label{eq21}
\end{subequations}
The probability density, Eq.~(\ref{eq21a}), essentially consists of an oscillating function
modulated by a Gaussian prefactor, since the hyperbolic cosine grows spatially relatively slowly at a given time $t$ (like $x/t^2$, slower than the $x/t$ dependence of the Gaussian
prefactor or the cosine).
Accordingly, for a sufficiently large time, within the region of interest (ruled by the
argument of the Gaussian prefactor), the hyperbolic cosine can be assumed to be close to the unity, so that
Eqs.~(\ref{eq21}) can be conveniently further approximated as
\begin{subequations}
\begin{eqnarray}
\rho(x,t)\ & \approx\ & 4 e^{-2m\sigma_0^2 x^2/\hbar^2 t^2} \cos^2 (m x_0 x/\hbar t) ,
\label{eq21bba} \\
J(x,t)\ & \approx\ & \frac{4x}{t}\ e^{-2m\sigma_0^2 x^2/\hbar^2 t^2}
\cos^2 (m x_0 x/\hbar t) \nonumber \\
& & - \frac{2x_0}{t}\ e^{-2m\sigma_0^2 x^2/\hbar^2 t^2}
\sinh (4m\sigma_0^2 x_0 x/\hbar^2 t^2) .
\label{eq21bbb}
\end{eqnarray}
\label{eq21bb}
\end{subequations}
From Eq.~(\ref{eq21bba}) we notice that vanishing interference minima evolve in
time at a constant rate given by the expression
\begin{equation}
v_\nu^{\rm min} = \left(2\nu + 1\right) \frac{\pi\hbar}{md} ,
\label{eqcondmax}
\end{equation}
where $\nu = 0, \pm 1, \pm 2, \ldots$ denote the interferential order.
In turn, interference maxima (modulated by a Gaussian envelope) will also evolve
at the constant rate
\begin{equation}
v_\nu^{\rm max} = \frac{2\nu\pi\hbar}{m d} .
\label{eqcondmin}
\end{equation}
In both cases, the separation between adjacent interference minima and maxima remains
constant and depends on the inverse of the distance between the slits, as in the optical
Young experiment \cite{bornwolf-bk}.
Concerning Eq.~(\ref{eq21bbb}), despite the second term is negligible, it has not been
disregarded, because it plays a major role in the dynamics, as it will be seen later on.
Note that, whenever the probability density vanishes, this term does not.
\begin{figure}[!t]
\centering
\includegraphics[width=\textwidth]{fig3-sanz.png}
\caption{\label{fig3}
Time-evolution of the probability density (a) and Bohm's quantum potential (b) for a
coherent superposition of two Gaussian wave packets simulating Young's two-slit experiment
along the transverse coordinate ($x$).
In the color scale, lower values (nearly zero for the probability density and negative for
the velocity field) are denoted with blue, while the higher ones are represented with red;
note that, without any loss of generality, the quantum potential has been truncated both
from the bottom and also the top due to the high positive and negative values that it
reaches at some times and in some regions (in contrast with the nearly constant value that
it acquires along the regions associated with the maxima of the probability density).
The parameters considered in this simulation are $m = 1$, $\hbar = 1$, $\sigma_0 = 0.5$
and $x_0 = 5$ ($d = 10$), in arbitrary units.}
\end{figure}
The two regimes can be seen in the numerical simulation displayed in Fig.~\ref{fig3}(a),
which represents the evolution of a coherent superposition of two time-evolving Gaussian
wave packets separated a distance $d=10$ ($x_\pm = \pm 5$) and with a initial width
$\sigma_0 = 0.5$ (without loss of generality, $\hbar = 1$ and $m = 1$).
These regimes can be better appreciated in the density plot below: two nearly freely
propagating Gaussians at the beginning, for $t \lesssim 1$, and well-defined interference
fringes for $t \gtrsim 4$, with their minima located at $x_\nu(t) = \pm 0.1\pi t, \pm 0.3\pi t, \ldots$ (note that in the particular case $t=10$, the maxima are located at
$x = \pm \pi, \pm 3\pi, \ldots$
From a standard Bohmian perspective \cite{dewdney:NuovoCimB:1979,holland-bk,sanz:prb:2000,sanz:JPCM:2002},
where the quantum potential is a central quantity, we can see, by inspecting
Fig.~\ref{fig3}(b), that there is not much difference, because it actually measures the
curvature of the probability density, following the second expression of Eq.~(\ref{eq6}).
Accordingly, the structure displayed by the quantum potential is going to be pretty similar
to that of the probability density, substituting the interference maxima of the latter by
plateaus and the nodes by deep minima or ``canyons'' (because of the analogy with these
geological formations), which also satisfy the condition (\ref{eqcondmin}).
Typically, it is assumed that, because the quantum potential is nearly flat between two
adjacent canyons (negligible quantum force, $-\nabla Q \approx 0$), particles are going
to accumulate in such regions, giving rise to the interference maxima, while they avoid
staying at such canyons, where they are affected by the action of an intense quantum force.
No doubt, the idea is appealing.
However, not only it provides redundant information with respect to the probability
density (as mentioned above, it measures its curvature), but totally neglects the dynamical
role of the quantum phase, necessary to explain, for instance, in the renowned non-crossing
property satisfied by the Bohmian trajectories.
In order to provide a, say, non-redundant dynamical description to the emergence of the
interference pattern, let us get back to Eqs.~(\ref{eq21b}).
It consists of two contributions.
The first contribution is indeed the probability density multiplied by a prefactor $x/t$.
This prefactor is a (transverse) velocity that describes the overall spatial dispersion
(spreading) of the probability density.
Unlike the even parity displayed by the probability density (with respect to $x=0$), this
term has odd parity due to its additional dependence on $x$.
The second contribution, on the other hand, with an also odd parity, seems to play no role,
since it does not contain any information on interference and, moreover, decreases like
$t^{-1}$ all along $x$, thus becoming gradually less and less relevant.
Regarding the overall odd parity displayed by the flux, notice that it indicates that the
density is going to spread equally to both left and right.
Then, taking into account these features, how can the emergence of interference be explained
without relying again, as in the case of the quantum potential based explanation, on the
probability density?
To answer such a question, let us substitute Eqs.~(\ref{eq19}) into the second expression
of the equation of motion (\ref{eq5b}).
Thus, for the two wave packets, the latter equation reads as
\begin{eqnarray}
\dot{x}(x,t)\ & =\ & \frac{1}{m}
\Bigg\{ \frac{\rho_+(x,t)}{\rho(x,t)} \frac{\partial S_+(x,t)}{\partial x}
+ \frac{\rho_-(x,t)}{\rho(x,t)} \frac{\partial S_-(x,t)}{\partial x}
\nonumber \\ & & \quad
+ \frac{\sqrt{\rho_+(x,t) \rho_-(x,t)}}{\rho(x,t)}\ \frac{\partial \left[S_+(x,t) + S_-(x,t)\right]}{\partial x}\ \cos \varphi(x,t) \Bigg\} \nonumber \\
& & + \frac{\hbar}{m} \frac{\sqrt{\rho_+(x,t) \rho_-(x,t)}}{\rho(x,t)}
\Bigg[ \frac{1}{\rho_+(x,t)}\frac{\partial \rho_+(x,t)}{\partial x}
\nonumber \\ & & \qquad \qquad \qquad
- \frac{1}{\rho_-(x,t)} \frac{\partial \rho_-(x,t)}{\partial x} \Bigg] \sin \varphi(x,t) ,
\label{eq23}
\end{eqnarray}
where $\rho(x,t)$ is as given by Eq.~(\ref{eq19a}).
If we now reconsider the above limits, we find that, for short times, although the
probability density concentrates around either $x_+$ or $x_-$, Bohmian trajectories
associated with each wave packet are going to evolve seemingly like if the other wave
packet has no influence on them, i.e., like the trajectories related to a single Gaussian
wave packet problem \cite{sanz:AJP:2012,sanz:cpl:2007}.
This situation can be seen in Fig.~\ref{fig4}(a) up to $t \simeq 1$ (beyond this time the
trajectories closer to $x=0$ start undergoing deviations from the single wave-packet case).
However, this is only in appearance, as it can be noticed by inspecting Fig.~\ref{fig4}(b),
where the density plot represents the associated local velocity field, which changes very
abruptly at $x=0$, as expected according to the above discussion based on the quantum flux.
It is observed that, within the spatial domain of each slit, the flux associated to each
slit is the same and, therefore, the corresponding trajectories are expected to display
the same behavior.
\begin{figure}[t]
\centering
\includegraphics[width=\textwidth]{fig4-sanz.png}
\caption{\label{fig4}
Time-evolution of the probability density (a) and the local velocity field (b) for a
coherent superposition of two Gaussian wave packets simulating Young's two-slit experiment
along the transverse coordinate ($x$).
In the color scale, lower values (nearly zero for the probability density and negative for
the velocity field) are denoted with blue, while the higher ones are represented with red;
in the case of the velocity field, vanishing values appear with green (along the central
interference channel).
For a better visualization of the dynamics, two sets of Bohmian trajectories, each one
associated with one slit (Gaussian wave packet), are also on display (black solid lines).
The parameters considered in this simulation are $m = 1$, $\hbar = 1$, $\sigma_0 = 0.5$
and $x_0 = 5$ ($d = 10$), in arbitrary units.}
\end{figure}
As time increases and the probability density starts exhibiting interference maxima and
minima [see Fig.~\ref{fig4}(b)], the two swarms of Bohmian trajectories acquire a non-regular
distribution, loosing information about each particular slit and evolving along the
directions indicated by the maxima, while they undergo dramatic turns at nodal regions in
order to avoid them.
This is in agreement with the above quantum potential based argumentation.
But, why does this happen?
If Eqs.~(\ref{eq21bb}) are explicitly substituted into Eq.~(\ref{eq5b}), i.e., if the
long-time limit is considered in Eq.~(\ref{eq23}), this latter equation can no longer be
reduced to either one wave packet or the other, but needs to consider the full wave.
The equation of motion (\ref{eq5b}) reads now as
\begin{equation}
\dot{x} \approx \frac{x}{t} - \frac{x_0}{2t}\
\frac{\sinh (4m\sigma_0^2 x_0 x/\hbar^2 t^2)}{\cos^2 (m x_0 x/\hbar t)} .
\label{eq24}
\end{equation}
Following this expression for the local velocity it is clear that, out of the reach of the
nodes and at a given time, the flux increases linearly with the position, since the first
term on the r.h.s.\ is the dominant one.
This simply means that trajectories will move apart from $x=0$ either with positive velocity
in the positive half-plane or with negative velocity in the negative half-plane, as it is
observed in the case of a simple Gaussian wave packet \cite{sanz:AJP:2012}.
Actually, if we take into account that the maximum value of the cosine in the second term
becomes maximum for $m x_0 x/\hbar t = \nu \pi$, with $\nu = 0, \pm 1, \pm 2, \ldots$, on
average, the expression for the velocity between two neighboring nodes is given by
\begin{equation}
\dot{x} = \frac{\nu \pi\hbar}{m x_0} = \bar{v} \nu ,
\label{eq24b}
\end{equation}
where $\bar{v} \equiv \pi\hbar/m x_0 = 2\pi\hbar/md$.
This means that the velocity is a quantized quantity, showing a ladder-type structure, where
each step has nearly the same width and describes an interference channel, i.e., a region that
will accommodate an interference maximum of the probability density.
These structures are typical whenever diffraction channels appear regardless of whether they
have been produced by two slits \cite{luis:AOP:2015}, many slits \cite{sanz:AOP:2015} or
scattering with a metal surface \cite{sanz:EPL:2001}.
However, at the nodes, the numerator of the second term cancels out and the velocity acquires
a sudden change or kick either below or above the value indicated by the first term.
If the particle is in the positive half-plane, the kick is negative; if its in the negative
half-plane, then the kick is positive.
This translates into a reorientation of the trajectories, which instead of being pulled
apart, as in the case of the Gaussian wave packet, they are gradually redirected towards
inner interferential maxima.
This can be seen in Fig.~\ref{fig4}(b), where the effect of the kicks appears as a fast
twist in the trajectories each time they approach a nodal point, inducing the passage of the
trajectories from the region associated with an interference maximum to the immediately
nearby one (that is, the motion takes place in discrete jumps, one by one).
In the present example, this implies that central maxima eventually become more populated
than marginal ones, which, in turn, serves us to understand the evolution of the probability
density displayed in Fig.~\ref{fig3}(a).
This behavior, though, can readily be generalized to grating diffraction, thus providing an
extremely natural interpretation for the appearance of diffraction orders, Bragg's law or
the relationship between the beam size and the definition of diffractive features
\cite{sanz:AOP:2015}.
Analogously, the same can be directly translated to the realm of optics,
providing a more intuitive picture of interference and diffraction phenomena
\cite{sanz:AnnPhysPhoton:2010,sanz:JRLR:2010,luis:AOP:2015,sanz:ApplSci:2020}, which is
excellent agreement with the experimental findings reported by Steinberg and coworkers
regarding Young's two-slit experiment several years ago \cite{kocsis:Science:2011,sanz:PhysScr:2013,sanz:EPN:2013}.
Summing up, it is interesting to note that, although the standard or traditional Bohmian
view based on Bohm's quantum potential does not allow to explain the dynamical origin of
the renowned non-crossing rule among trajectories, the local velocity field provides a
rather complete understanding of the phenomenon.
Again, this does not mean that quantum particles cannot move in the most unexpected manners,
because that is, so far, a challenging unknown.
It simply means that the flux describing the evolution of the probability density,
which describes how the swarm of quantum particles behaves (distributes) spatially on average
at each time, exhibits a very precise dynamics, according to which it is indeed possible, to
some extent, to establish a well-defined separation between the regions covered by each slit
at a dynamical level.
None of the (tracer) Bohmian trajectories starting in one of the slits will ever cross the
region dominated by the other slit, and vice versa.
This is, therefore, a physical manifestation (or evidence) of the quantum phenomenon or
quantum resource that we call coherence.
As a consequence, two-slit experiments turn out to be equivalent to single slit experiments
coupled to short-range attractive walls \cite{sanz:JPA:2008}, where the presence of the
attractive well induces the appearance of long-living resonances near the wall (associated
with half the central maximum).
This picture is quite far from the usual one, although it resembles the typical reduction
in two-body classical scattering problems, where the two systems are substituted by a single
one acted by an effective central force.
Furthermore, there is another important related consequence.
By inspecting Eq.~(\ref{eq23}), it is noticed that in order to remove any trait of coherence,
not only the interference term must be somehow removed, as it is usually
mentioned when dealing with the removal of interference in the two-slit experiment.
The disappearance of such a term simply means that the flux does not
include any wavy term.
However, the information about the existence of two slits open at once still persists.
Therefore, the non-crossing rule is still preserved, i.e., the Bohmian trajectories leaving
each slit are not going to mix \cite{sanz:EPJD:2007,luis:AOP:2015}.
The fact that both slits still influence the dynamics means that there is coherence even if
there is no interference.
In order to remove all traits of coherence, it is important to also remove information about
the other slit \cite{sanz:CPL:2009-2}.
In the traditional picture of the two-slit experiment this actually happens when we decide
to include the action of an external observer (detector), which just removes the contribution
of one of the slits.
Within the more refined description provided by the theory of open quantum systems
\cite{breuer-bk:2002}, this removal is simply the effect of the different manner that the
system gets entangled with an environment when it crosses one slit or the other
\cite{sanz:FrontPhys:submit21}.
\section{Final remarks}
\label{sec4}
Quantum mechanics is supposed to be the mechanical theory of quanta, that is, of the
`bits' of matter (electrons, atoms, molecules, etc.) and radiation (photons).
However, what Bohm put forth was that quantum mechanics had more of a non-mechanical
theory, because of the important role played by the whole over the individual.
Formerly, it was proposed as a counterexample to Von Neumann's theorem on the impossibility
of hidden variables in quantum mechanics, although the fact that it introduces into this
theory a language pretty similar to that of the classical Hamilton-Jacobi formulation has
led to associate the corresponding trajectories with the actual motion displayed by real
quantum particles.
Thus, in the same way that a classical (interaction) potential function determines the
motion of a (classical) particle, in the quantum mechanical case it would be the combined
action of such a function plus the so-called Bohm's quantum potential the mechanism behind
the topology displayed by the trajectories pursued by quantum particles.
Of course, this potential acts on quantum particles even in the case of free motion,
where there is no external interaction ($V=0$).
In such a case, the bare action of Bohm's potential shows very nicely how it accounts
for pure quantum effects, such as interference, as it can be seen not only with a free
particle (a freely released wave packet), but also in slit diffraction problems
\cite{sanz:JPCM:2002}.
Much has been discussed in the literature about this potential, its properties and its
applications (see \cite{holland-bk} and references therein, for instance), yet it is nothing
but a measure of the local instantaneous curvature of the probability density, thus providing
us to some extent with redundant information (the same information already provided by the
probability density).
Nonetheless, recently it has received some attention as a magnitude that, in principle,
could be measured, particularly if instead of massive particles one considers light
\cite{umul:Optik:2020,hojman:arxiv:2021}, taking advantage of the one-to-one correspondence
between Schr\"odinger's equation and the paraxial Helmholtz equation.
Of course, this is not impossible, as it has also been the case of the transverse momentum
\cite{kocsis:Science:2011}, even though stricto sensu none of these quantities correspond to
quantum observables.
In order to avoid such redundancy, here the discussion has turned around the phase field
and, more specifically, the associated local velocity field, which allows us to establish a
connection between the probability density and the quantum flux, thus avoiding the extra
Bohmian postulate of a guidance condition.
The implications of the single valuedness of the quantum phase have long been discussed in
the literature to explain the non-crossing property exhibited by Bohmian trajectories in
the configurations space \cite{holland-bk}.
Unfortunately, the quantum phase only manifests through interference, thus providing little
clue on the dynamics displayed by the probability density.
The local velocity field, in turn, is a well-defined quantity with a precise physical
meaning, which can be experimentally determined through weak measurements, as shown in
\cite{kocsis:Science:2011}.
Accordingly, we have analyzed the dynamical information rendered by this quantity, which
allows us to understand and explain the time-evolution shown by the probability density at
each point of the configuration (in positions) space.
In analogy to classical hydrodynamic systems, this velocity field can be probed by
launching a series of tracers and let them to move accordingly, which provides us with
a more precise picture at a local level of the probability flux across the configuration
space in the form of probability-flow streamlines or trajectories.
These trajectories are the usual Bohmian trajectories, which here arise in a natural way,
without any need to introduce the concept of hidden variable.
It is in this way, used as tracers of the quantum dynamics, that Bohmian trajectories
constitute a remarkably beneficial tool to probe and understand quantum phenomena with
a language (that of dynamical systems) closer to our experience than abstract Hilbert
algebras ---perhaps more appropriate from a formal viewpoint, but totally useless to
understand what is going on in a real-lab experiment, where we know that we have something
that goes from somewhere to somewhere else, which can be acted and measured, etc.
With the purpose to illustrate the advantages of the local velocity field as a convenient
tool to analyze and explain quantum dynamics, Young-type interference has been studied.
Thus, while the usual Bohm's potential view provides an interpretation similar to the
Newtonian one (particles moving in regions with nearly constant potential values, while
avoiding others with strong, sudden changes), the velocity field provides us with a more
precise description of different dynamical regions and regimes.
Accordingly, it is seen that, the center of symmetry of the system, namely, the axis $x=0$
in our case, is a zero-flux line, which divides the configuration space into two dynamically
different regions.
Of course, the fact that the flux vanishes along this axis, and so the velocity field,
does not preclude the possibility that, at a ``subquantum'' level (i.e., at a level below the
equilibrium described by Schr\"odinger's equation), particles might randomly cross this
axis from one region to the other and vice versa, as it happens with chemicals (reactants
and products) under dynamical equilibrium conditions in a chemical reaction.
Yet this makes an important difference with respect to the traditional explanation
attributed to the two-slits experiments, where there is total indistinguishability.
Here, the distinguishability of dynamical domains gives rise to trajectories leaving one of
the slits that ``know'' of the existence of trajectories leaving the other slit.
These trajectories probe the dynamics associated with the flux, thus providing no clue on
how the motion of real particles might be, but only the resulting average (equilibrium)
motion.
The reveal how the velocity field changes locally at each time, undergoing fast and sudden
turns whenever there is a strong variation (similar to kicks), while moving nearly parallel
in those regions with (nearly) constant velocity.
The latter region happen to be quantized, i.e., the average velocity changes in units of
$\pi\hbar/m x_0 = 2\pi\hbar/md$ from one to the immediately neighboring one (both separated
by a kick).
Each one of these regions constitutes an interference channel, i.e., a region along which
trajectories tend to keep moving in a Newtonian sense.
These highly populated regions correspond to the interference maxima displayed by the
probability density.
If the mechanism to avoid regions with strong variations of the velocity field is clear,
which makes trajectories to get promoted from the outer interference channels to the
innermost ones, the same does not hold to explain why trajectories cannot cross the
$x=0$ axis, along which trajectories coming from both slits align.
The non-crossing here arises from having two different dynamical regimes well defined since
the very beginning.
Note here another deviation with respect to the standard view in terms of a simple
superposition relation, which holds true formally, but that cannot be accepted in dynamical
terms, since it will only appear provided both slits (both diffracted waves) are present
since the very beginning.
Accordingly, the concept of coherence acquires a different but totally unambiguous physical
meaning, in terms of the equivalence between this problem and that one of a single particle
colliding with an attractive potential wall \cite{sanz:JPA:2008}.
The attractive well happens to be relatively shallow, but with an extension beyond the two
wave packets (diffracted beams), which implies their mutual knowledge even if the probability
density is negligible in between.
Only if the information about the existence (presence) of one of the slits disappears (either
gradually or suddenly), trajectories from one domain will start crossing the trajectories from
the other domain \cite{luis:AOP:2015,sanz:EPJD:2007,sanz:CPL:2009-2}.
This situation thus describes a (partial or total) loss of coherence, which happens when
the system is strongly entangled with another environmental subsystem \cite{sanz:FrontPhys:submit21}.
At this point, based on the above discussion, one may still wonder whether real quantum
motion is still accessible.
As mentioned above, if it exists, it must be found at a subquantum level.
Nonetheless, the fact that the local value of the velocity field (transverse momentum) can be
experimentally determined opens new perspectives in our understanding of the quantum world.
Now we know that not only quantum particles distribute according to the usual probability
density even though they are totally uncorrelated, but also that in order to do it they
necessarily form currents.
This means that the usual Copenhagian view in terms of particle becoming a wave during the
experiment and then a particle again at the detector is currently getting blurred.
We have a precise quantum dynamical description of the average (equilibrium) behavior of
quantum systems where both their distribution (probability density) and spatial motion
(velocity field) can be determined without violating any of the fundamental principles
of quantum mechanics.
Because of the empirical impossibility to relate Bohmian trajectories with the actual
paths followed by real particles (in the sense that no experiment will be able to reveal
this very motion), one might wonder whether it provides or not a solution to the so-called
measurement problem \cite{zurek-bk}.
It is clear that, apart from point-like particles traveling along well-defined trajectories,
a proper description of such a problem requires including explicitly the presence of a second
agent or system, namely the detector.
When doing so, entanglement immediately arises \cite{giulini-bk,schlosshauer-bk:2007},
which is the physical mechanism behind the fact that we observe the system ``collapsing''
on any of the detector pointer states.
However, if we consider the simple experiment presented here, this is eventually equivalent to
make statistics over arrivals at certain spatial regions (pixels of a given finite dimension).
At this level, there is no need, therefore, to provide a better description of how each
photodetector state acts on or gets entangled with the system wave function, because each arrival
itself can be counted (registered).
The collection of these arrivals over time will provide us with a relatively fair picture of the
detection at a local level (pixel by pixel), which is equivalent to monitor in time the formation
of a full image over the whole scanning surface (e.g., a two-slit interference pattern with photons
or electrons, or, in the case of incoherent light, the appearance of a photograph).
In the standard quantum-mechanical approach the same is not possible, because the wave function
considered gives us the full solution (full image), even if later on some treatments (convolving
functions) are required in order to adapt such a solution to the finite-sized detection elements
(pixels, slits, etc.).
In this sense, and leaving aside other ontological connotations, a fully quantum-mechanical
trajectory-based approach proves to be more powerful than other standard quantum approaches,
since we are able to obtain first-principle theoretical descriptions of quantum phenomena
closer to real-life experiments without the need of extra treatments or elements (only the
necessary ones).
So, in conclusion, once the ``mysticism'' that usually accompanies Bohmian mechanics is removed,
whether this first-principle view can be considered of potential interest at a computational or
a fundamental level it is left to the reader's opinion.
\section*{Acknowledgements}
This work is partly based on the opening talk of the Journ\'ees Louis de Broglie
(November, 2019).
The author is much grateful to Thomas Durt and all the Organizers of the Journ\'ees for their
kind invitation to participate in this important meeting for those who think that other
understandings of quantum mechanics are possible.
He is also indebted to the Fondation Louis de Broglie for the kind attention received from
this institution.
Support to produce this work is also acknowledged to the Spanish Research Agency (AEI)
and the European Regional Development Fund (ERDF), under Grant No.\ FIS2016-76110-P.
|
1,314,259,993,571 | arxiv | \section{Doubly stochastic matrices and scaling}
Let $A = (a_{i,j})$ be an $n \times n$ matrix.
For $i \in \{1,\ldots, n\}$, the $i$th \emph{row sum}\index{row sum} of $A$ is
\[
\rowsum_i(A) = \sum_{j=1}^n a_{i,j}.
\]
For $j \in \{1,\ldots, n\}$, the $j$th \emph{column sum}\index{column sum} of $A$ is
\[
\colsum_j(A) = \sum_{i =1}^n a_{i,j}.
\]
The matrix $ A = (a_{i,j})$ is \emph{positive} if $a_{i,j}>0$ for all $i$ and $j$,
and \emph{nonnegative} if $a_{i,j} \geq 0$ for all $i$ and $j$.
The matrix $A = (a_{i,j})$ is \emph{row stochastic}\index{row stochastic}
if $A$ is nonnegative and $\rowsum_i(A) = 1$ for all $i \in \{1,\ldots, n\}$.
The matrix $A$ is \emph{column stochastic}\index{column stochastic}
if $A$ is nonnegative and $\colsum_j(A) = 1$ for all $j \in \{1,\ldots, n\}$.
The matrix $A$ is \emph{doubly stochastic}\index{doubly stochastic}
if it is both row and column stochastic.
Let $\diag(x_1,\ldots, x_n)$ denote the $n \times n$ diagonal matrix
whose $(i,i)$th coordinate is $x_i$ for all $i \in \{1,2,\ldots, n\}$.
The matrix $\diag(x_1,x_2,\ldots, x_n)$
is \emph{positive diagonal} if $x_i > 0$ for all $i$.
Let $A = (a_{i,j})$ be an $n \times n$ matrix.
The process of multiplying the rows of $A$ by scalars,
or, equivalently, multiplying $A$ on the left by a diagonal matrix $X$,
is called \emph{row-scaling},
and $X$ is called a \emph{row-scaling matrix}.
The process of multiplying the columns of $A$ by scalars,
or, equivalently, multiplying $A$ on the right by a diagonal matrix $Y$,
is called \emph{column-scaling},
and $Y$ is called a \emph{column-scaling matrix}.
If $X = \diag(x_1,x_2,\ldots, x_n)$ and $Y = \diag(y_1,y_2,\ldots, y_n)$, then
\[
XAY =
\left(\begin{matrix}
x_1a_{1,1} y_1 & x_1a_{1,2} y_2 & x_1a_{1,3} y_3 & \cdots & x_1a_{1,n} y_n \\
x_2a_{2,1} y_1 & x_2a_{2,2} y_2 &x_2a_{2,3} y_3 & \cdots & x_2a_{2,n} y_n \\
\vdots &&&& \vdots \\
x_n a_{n,1} y_1 & x_n a_{n,2} y_2 &x_n a_{n,3} y_3 & \cdots & x_n a_{n,n} y_n
\end{matrix}\right).
\]
Let $A = (a_{i,j})$ be an $n \times n$ matrix with positive row sums,
that is, $\rowsum_i(A) > 0$ for all $i \in \{1,\ldots, n\}$.
Let
\begin{equation} \label{Sinkhorn:rowsum}
X(A) = \diag\left( \frac{1}{\rowsum_1(A)}, \ldots, \frac{1}{\rowsum_n(A)} \right)
\end{equation}
and let
\[
\ensuremath{ \mathcal R}(A) = X(A) A.
\]
We have
\[
\ensuremath{ \mathcal R}(A)_{i,j} = \frac{a_{i,j}}{\rowsum_i(A)}
\]
and so
\[
\rowsum_i(\ensuremath{ \mathcal R}(A))
= \sum_{j=1}^n \ensuremath{ \mathcal R}(A)_{i,j}
= \sum_{j=1}^n\frac{a_{i,j}}{\rowsum_i(A)}
= \frac{\rowsum_i(A)}{\rowsum_i(A)} = 1
\]
for all $i \in \{1,\ldots, n \}$.
Therefore, $\ensuremath{ \mathcal R}(A)$ is a row stochastic matrix.
Similarly, if $A = (a_{i,j})$ is an $n \times n$ matrix with positive column sums and if
\begin{equation} \label{Sinkhorn:colsum}
Y(A) = \diag\left( \frac{1}{\colsum_1(A)}, \ldots, \frac{1}{\colsum_n(A)} \right)
\end{equation}
and
\[
\ensuremath{ \mathcal C}(A) = A Y(A),
\]
then
\[
\ensuremath{ \mathcal C}(A)_{i,j} = \frac{a_{i,j}}{\colsum_j (A)}
\]
and
\[
\colsum_j(\ensuremath{ \mathcal C}(A))
= \sum_{i=1}^n \ensuremath{ \mathcal C}(A)_{i,j} = \sum_{i=1}^n \frac{a_{i,j}}{\colsum_j(A)}
= \frac{\colsum_j(A)}{\colsum_j(A)} = 1
\]
for all $j \in \{1,\ldots, n\}$.
Therefore, $\ensuremath{ \mathcal C}(A)$ is a column stochastic matrix.
The following two theorems were stated by Sinkhorn~\cite{sink64},
and subsequently proved by
Brualdi, Parter, and Schneider~\cite{brua-part-schn66},
Djokovi\'{c}~\cite{djok70},
Knopp-Sinkhorn~\cite{sink-knop67},
Menon~\cite{meno67}, Letac~\cite{leta74}, and Tverberg~\cite{tver76}.
\begin{theorem} \label{Sinkhorn:theorem:Sinkhorn-1}
Let $ A = (a_{i,j})$ be a positive $n \times n$ matrix.
\begin{enumerate}
\item[(i)]
There exist positive diagonal $n\times n$ matrices $ X$ and $ Y$
such that $ X A Y$ is doubly stochastic.
\item[(ii)]
If $ X$, $ X'$, $ Y$, and $ Y'$ are positive diagonal $n\times n$ matrices such that
both $ X A Y$ and $ X' A Y'$ are doubly stochastic, then
$ X A Y = X' A Y'$ and there exists $\lambda > 0$
such that $ X' = \lambda X$ and $ Y' = \lambda^{-1} Y$.
\item[(iii)]
Let A\ be a positive symmetric $n \times n$ matrix. There exists a unique positive diagonal matrix X\
such that $X A X$ is doubly stochastic.
\end{enumerate}
\end{theorem}
The unique doubly stochastic matrix $XAY$ in Theorem~\ref{Sinkhorn:theorem:Sinkhorn-1}
is called the \emph{Sinkhorn limit} of A, and denoted $S(A)$.
\begin{theorem} \label{Sinkhorn:theorem:Sinkhorn-2}
Let A\ be a positive $n\times n$ matrix, and let $S(A)$ be the Sinkhorn limit of $A$.
Construct sequences of positive matrices $(A_{\ell})_{\ell=0}^{\infty}$
and $(A'_{\ell})_{\ell=0}^{\infty}$
and sequences of positive diagonal matrices $(X_{\ell})_{\ell=0}^{\infty}$
and $(Y_{\ell})_{\ell=0}^{\infty}$
as follows:
Let
\[
A_0 = A.
\]
Given the matrix $A_{\ell}$, let
\begin{equation} \label{Sinkhorn:X}
X_{\ell} = X(A_{\ell})
\end{equation}
be the row-scaling matrix of $A_{\ell}$ defined by~\eqref{Sinkhorn:rowsum}.
The matrix
\[
A'_{\ell} = \ensuremath{ \mathcal R}(A_{\ell}) = X_{\ell} A_{\ell}.
\]
is row stochastic.
Let
\begin{equation} \label{Sinkhorn:Y}
Y_{\ell} = Y(A'_{\ell})
\end{equation}
be the column-scaling matrix of $A'_{\ell}$ defined by~\eqref{Sinkhorn:colsum}, and let
\[
A_{\ell+1} = \ensuremath{ \mathcal C}(A'_{\ell}) = A'_{\ell}Y_{\ell}.
\]
The matrix $A_{\ell +1}$ is column stochastic.
The Sinkhorn limit is obtained by alternately row-scaling and column-scaling:
\[
S(A) = \lim_{\ell\rightarrow \infty} A_{\ell} = \lim_{\ell\rightarrow \infty} A'_{\ell}.
\]
\end{theorem}
It is an open problem to compute explicitly the Sinkhorn limit of a positive $n\times n$ matrix.
This is known for $2\times 2$ matrices (Nathanson~\cite{nath2019-184}).
The goal of this paper is the explicit computation of Sinkhorn limits
for certain $3 \times 3$ matrices.
\section{Sinkhorn limits of $3 \times 3$ symmetric matrices and
their doubly stochastic shapes}
Let $A$ and $B$ be positive $n\times n$ matrices.
We write $A \sim B$ if there exist $n\times n$ permutation matrices $P$ and $Q$
and $\lambda > 0$ such that
\[
B = \lambda PAQ.
\]
This is an equivalence relation. Moreover,
$A\sim B$ implies
\begin{equation} \label{Sinkhorn:PSQ}
S(B) = \lambda P S(A) Q.
\end{equation}
Thus, it suffices to determine the Sinkhorn limit of only one matrix in an equivalence class.
We shall compute the Sinkhorn limit of every symmetric positive $3\times 3$ matrix
whose set of coordinates consists of two distinct real numbers.
Let $A$ be such a matrix with coordinates $M$ and $N$ with $M \neq N$.
There are 9 coordinate positions in the matrix, and so exactly one
of the numbers $M$ and $N$ occurs at least five times.
Suppose that the coordinate $M$ occurs five or more times. Let $\lambda = 1/M$
and $K = N/M$.
The matrix $\lambda A$ has two distinct positive coordinates $1$ and $K$,
and $K$ occurs at most four times.
There are seven equivalence classes of such matrices with respect to permutations
and dilations.
The main result of this paper is the calculation of the
Sinkhorn limits of these matrices.
\begin{theorem}
Let $K >0$ and $K \neq 1$.
The matrices $A_1,\ldots, A_7$ below are a complete set
of representatives of the seven equivalence classes
of symmetric $3 \times 3$ matrices with coordinates 1 and $K$.
The matrix $S(A_i)$ gives the shape of the Sinkhorn limit of $A_i$ for $i = 1,\ldots, 7$.
The coordinates of the Sinkhorn limits as explicit functions of 1 and $K$ are computed
in Sections~\ref{Sinkhorn:section:A_1}--\ref{Sinkhorn:section:A_7}.
\begin{enumerate}
\item
\[
A_1 = \left(\begin{matrix}
K & 1 & 1 \\
1 & K & 1 \\
1 & 1 & K
\end{matrix}\right)
\qquad
S(A_1) = \left(\begin{matrix}
a & b & b \\
b & a & b \\
b & b & a
\end{matrix}\right)
\]
\item
\[
A_2 = \left(\begin{matrix}
K & 1 & 1 \\
1 & 1 & 1 \\
1 & 1 & 1
\end{matrix}\right)
\qquad
S(A_2) =
\left(\begin{matrix}
a & b & b \\
b & c & c \\
b & c & c
\end{matrix}\right)
\]
\item
\[
A_3 = \left(\begin{matrix}
1 & 1 & 1 \\
1 & K & K \\
1 & K & K
\end{matrix}\right)
\qquad
S(A_3) = \left(\begin{matrix}
a & b & b \\
b & c & c \\
b & c & c
\end{matrix}\right)
\]
\item
\[
A_4 = \left(\begin{matrix}
1 & K & K \\
K & 1 & 1 \\
K & 1 & 1
\end{matrix}\right)
\qquad
S(A_4) = \left(\begin{matrix}
a & b & b \\
b & c & c \\
b & c & c
\end{matrix}\right)
\]
\item
\[
A_5 = \left(\begin{matrix}
K & 1 & 1 \\
1 & K & 1 \\
1 & 1 & 1
\end{matrix}\right)
\qquad
S(A_5) = \left(\begin{matrix}
a & b & c \\
b & a & c \\
c & c & d
\end{matrix}\right)
\]
\item
\[
A_6 = \left(\begin{matrix}
K & K & 1 \\
K & 1 & 1 \\
1 & 1 & 1
\end{matrix}\right)
\qquad
S(A_6) = \left(\begin{matrix}
a & b &c \\
b & c & a \\
c & a & b
\end{matrix}\right)
\]
\item
\[
A_7 = \left(\begin{matrix}
K & K & 1 \\
K & 1 & 1 \\
1 & 1 & K
\end{matrix}\right)
\qquad
S(A_7) = \left(\begin{matrix}
a & b & c \\
b & d & e \\
c & e & f
\end{matrix}\right)
\]
\end{enumerate}
\end{theorem}
\section{The $MBN$ matrix}
Let $k$, $\ell$, and $n$ be positive integers such that
\[
n = k+\ell.
\]
Let $M$, $B$, and $N$ be positive real numbers.
Consider the $n \times n$ symmetric matrix
\begin{equation} \label{Sinkhorn:MBN}
A = \left(\begin{matrix}
M & M & \cdots & M & B & B & \cdots & B \\
M & M & \cdots & M & B & B & \cdots & B \\
\vdots &&& \vdots & \vdots &&& \vdots \\
M & M & \cdots & M & B & B & \cdots & B \\
B & B & \cdots & B & N & N & \cdots & N \\
B & B & \cdots & B & N & N & \cdots & N \\
\vdots &&& \vdots & \vdots &&& \vdots \\
B & B & \cdots & B & N & N & \cdots & N \\
\end{matrix}\right)
\end{equation}
in which the first $k$ rows are equal to
\[
(\underbrace{M,M,\ldots, M}_{k},\underbrace{ B, B, \ldots, B}_{\ell})
\]
and the last $\ell$ rows are equal to
\[
(\underbrace{B,B,\ldots, B}_{k}, \underbrace{ N,N, \ldots, N}_{\ell}).
\]
Let $X = \diag(x_1,x_2,x_3, \ldots, x_n ) $ be the unique positive $n \times n$ diagonal matrix
such that the alternate scaling limit $S(A) = X A X $ is doubly stochastic.
Thus, the matrix
\[
S(A)
= \left(\begin{matrix}
M x_1^2 & Mx_1x_2 & \cdots & M x_1x_k & B x_1x_{k+1} & B x_1x_{k+2} & \cdots & B x_1x_{n}\\
M x_2x_1 & Mx_2^2 & \cdots & M x_2x_k & B x_2x_{k+1} & B x_2x_{k+2} & \cdots & B x_2x_{n}\\
\vdots &&& \vdots & \vdots &&& \vdots \\
M x_k x_1 & Mx_kx_2 & \cdots & M x_k^2 & B x_kx_{k+1} & B x_kx_{k+2} & \cdots & B x_kx_{n}\\
B x_{k+1} x_1 & B x_{k+1} x_2 & \cdots & B x_{k+1}x_k & N x_{k+1}^2& N x_{k+1}x_{k+2} & \cdots & N x_{k+1}x_{n}\\
B x_{k+2} x_1 & B x_{k+2} x_2 & \cdots & B x_{k+2}x_k & N x_{k+2}x_{k+1}& N x_{k+2}^2 & \cdots & N x_{k+2}x_{n}\\
\vdots &&& \vdots & \vdots &&& \vdots \\
B x_n x_1 & B x_n x_2 & \cdots & B x_n x_k & N x_nx_{k+1}& N x_nx_{k+2} & \cdots & N x_{n}^2
\end{matrix}\right)
\]
satisfies
\[
x_i\left( M \sum_{j=1}^k x_j + B\sum_{j=k+1}^n x_j \right) = 1
\qquad \text{for $i = 1,2,\ldots k$}
\]
and
\[
x_i\left( B \sum_{j=1}^k x_j + N\sum_{j=k+1}^n x_j \right) = 1
\qquad \text{for $i = k+1, k+2, ,\ldots k+\ell$.}
\]
It follows that $x_i = x_1$ for $i = 1,2,\ldots k$ and $x_i = x_n$ for $i = k+1, k+2,\ldots k + \ell$.
Let $x_1 = x$ and $x_{k+1} = y$. Define the diagonal matrix
\[
X = \diag(\underbrace{x,x,\ldots, x}_{k}, \underbrace{ y,y, \ldots, y}_{\ell}).
\]
We obtain
\begin{align}
S(A) \label{Sinkhorn:MBN-limit}
& = \left(\begin{matrix}
M x^2& M x^2& \cdots & Mx^2& B xy & B xy & \cdots & B xy \\
M x^2& M x^2& \cdots & M x^2 & B xy & B xy& \cdots & B xy \\
\vdots &&& \vdots & \vdots &&& \vdots \\
M x^2 & Mx^2& \cdots & M x^2 & B xy & B xy & \cdots & B xy \\
B xy & B xy& \cdots & B xy & N y^2 & N y^2 & \cdots & N y^2 \\
B xy & B xy& \cdots & B xy & N y^2 & N y^2 & \cdots & N y^2 \\
\vdots &&& \vdots & \vdots &&& \vdots \\
B xy & B xy & \cdots & B xy & N y^2 & N y^2 & \cdots & N y^2
\end{matrix}\right) \\
& =
\left(\begin{matrix}
a & a & \cdots & a & b & b & \cdots & b \\
a & a & \cdots & a & b & b & \cdots & b \\
\vdots &&& \vdots & \vdots &&& \vdots \\
a & a & \cdots & a & b & b & \cdots & b \\
b & b & \cdots & b & c & c & \cdots & c \\
b & b & \cdots & b & c & c & \cdots & c \\
\vdots &&& \vdots & \vdots &&& \vdots \\
b & b & \cdots & b & c & c & \cdots & c \nonumber
\end{matrix}\right)
\end {align}
where
\begin{align} \label{Sinkhorn:MBN-a}
a & = Mx^2 \\
b & = Bxy = \frac{1-ka}{\ell} \label{Sinkhorn:MBN-b} \\
c & = Ny^2 = \frac{1-kb}{\ell} = \frac{\ell-k + k^2 a}{\ell^2}. \label{Sinkhorn:MBN-c}
\end{align}
Because $S(A)$ is row stochastic, we have
\begin{equation} \label{Sinkhorn:MBN-1}
x \left(kMx+ \ell B y \right) = 1
\end{equation}
and
\begin{equation} \label{Sinkhorn:MBN-2}
y \left( kBx + \ell N y \right) = 1.
\end{equation}
Equation~\eqref{Sinkhorn:MBN-1} gives
\[
y =\frac{1}{\ell B} \left( \frac{1}{x} -kMx \right).
\]
Inserting this into equation~\eqref{Sinkhorn:MBN-2} and rearranging gives
\begin{equation} \label{Sinkhorn:MBN-3}
k^2 M \left( MN - B^2 \right) x^4 - \left( 2k (MN - B^2) + nB^2 \right) x^2+ N = 0
\end{equation}
If $MN-B^2 = 0$, then
\[
x^2 = \frac{N}{nB^2} = \frac{1}{nM}
\]
and $M x^2 = a = b = c = 1/n$. Thus, $S(A)$ is the $n \times n$ doubly stochastic matrix
with every coordinate equal to $1/n$.
If $MN - B^2 \neq 0$, then~\eqref{Sinkhorn:MBN-3} is a quadratic equation in $x^2$.
Let
\[
L = \frac{MN}{B^2}.
\]
We obtain
\begin {align*}
x^2
& = \frac{ 2k (MN - B^2) + nB^2 \pm B\sqrt{ 4k\ell (MN-B^2) + n^2B^2}}{ 2k^2 M (MN-B^2)} \\
& = \frac{ 1}{kM} + \frac{ nB^2 \pm B\sqrt{ 4k\ell MN + (k-\ell)^2 B^2}}{ 2k^2 M (MN-B^2)} \\
& = \frac{ 1}{kM} + \frac{ n \pm \sqrt{ 4k\ell L + (k-\ell)^2 }}{ 2k^2 M( L-1)} \\
& = \frac{ 1}{kM} + \frac{ n \pm \sqrt{ n^2 + 4k\ell (L-1) }}{ 2k^2 M( L-1)}
\end {align*}
and
\begin {align*}
a = M x^2
& = \frac{ 1}{k} + \frac{ n \pm \sqrt{ n^2 + 4k\ell (L-1) }}{ 2k^2( L-1)}.
\end {align*}
Recall that $ka+\ell b = 1$ and so
\[
0 < a< \frac{1}{k}.
\]
If $MN > B^2$, then $L>1$ and
\[
\sqrt{ n^2 + 4k\ell (L-1) } > n > 0.
\]
The inequality $a < 1/k$ implies that
\begin{equation} \label{Sinkhorn:MBN-aa}
a = \frac{ 1}{k} + \frac{ n - \sqrt{ n^2 + 4k\ell (L-1) }}{ 2k^2( L-1)}.
\end{equation}
If $MN < B^2$, then $0 < L < 1$ and
\[
a = \frac{ 1}{k} - \frac{ n \pm \sqrt{ n^2 - 4k\ell (1-L) }}{ 2k^2( 1 - L)}.
\]
Because
\[
\frac{ n+ \sqrt{ n^2 - 4k\ell (1-L) }}{ 2k^2( 1 - L)} > \frac{1}{k}
\]
the inequality $a > 0$ implies~\eqref {Sinkhorn:MBN-aa}.
We have proved the following.
\begin{theorem} \label{Sinkhorn:theorem:MBN-1-1}
The Sinkhorn limit of the $MBN$ matrix~\eqref{Sinkhorn:MBN} is a
doubly stochastic matrix $S(A)$ with shape~\eqref{Sinkhorn:MBN-limit}.
If $L = MN/B^2 = 1$, then $a = b = c = 1/n$.
If $L \neq 1$, then equations~\eqref{Sinkhorn:MBN-aa},~\eqref{Sinkhorn:MBN-b}, and~\eqref{Sinkhorn:MBN-c} define the coordinates $a$, $b$, and $c$.
The matrix $S(A)$ depends only on the ratio $MN/B^2$.
\end{theorem}
For example, the matrices
\[
\left(\begin{matrix}
2 & 5 & 5 \\
5 & 3 & 3 \\
5 & 3 & 3
\end{matrix}\right),
\qquad
\left(\begin{matrix}
6 & 5 & 5 \\
5 & 1 & 1 \\
5 & 1 & 1
\end{matrix}\right),
\qquad
\left(\begin{matrix}
6/25 & 1 & 1 \\
1 & 1 & 1 \\
1 & 1 & 1
\end{matrix}\right)
\]
have the same Sinkhorn limit with
\begin{align*}
a & = - \frac{37}{38} + \frac{5\sqrt{73}}{38} = 0.1505\ldots \\
b & = \frac{75}{76} - \frac{5\sqrt{73}}{76} = 0.4247\ldots \\
c & = \frac{1}{152} + \frac{5\sqrt{73}}{152} = 0.2876\ldots.
\end{align*}
Let $\left( A^{(r)} \right)_{r=1}^{\infty}$ be a sequence of $MBN$ matrices
such that $\lim_{r\rightarrow \infty} MN/B^2 = \infty$.
Let
\[
S\left( A^{(r)} \right) =
\left(\begin{matrix}
a^{(r)} & a^{(r)} & \cdots & a^{(r)} & b^{(r)} & b^{(r)} & \cdots & b^{(r)} \\
a^{(r)} & a^{(r)} & \cdots & a^{(r)} & b^{(r)} & b^{(r)} & \cdots & b^{(r)} \\
\vdots &&& \vdots & \vdots &&& \vdots \\
a^{(r)} & a^{(r)} & \cdots & a^{(r)} & b^{(r)} & b^{(r)} & \cdots & b^{(r)} \\
b^{(r)} & b^{(r)} & \cdots & b^{(r)} & c^{(r)} & c^{(r)} & \cdots & c^{(r)} \\
b^{(r)} & b^{(r)} & \cdots & b^{(r)} & c^{(r)} & c^{(r)} & \cdots & c^{(r)} \\
\vdots &&& \vdots & \vdots &&& \vdots \\
b^{(r)} & b^{(r)} & \cdots & b^{(r)} & c^{(r)} & c^{(r)} & \cdots & c^{(r)} \\
\end{matrix}\right).
\]
We have
\[
\lim_{r \rightarrow \infty} a^{(r)} = \frac{1}{k}, \qquad
\lim_{r \rightarrow \infty} b^{(r)} = 0,
\qquad
\lim_{r \rightarrow \infty} c^{(r)} = \frac{1}{\ell}
\]
and
\[
\lim_{r\rightarrow \infty} S\left( A^{(r)} \right) =
\left(\begin{matrix}
1/k & 1/k & \cdots & 1/k & 0 & 0 & \cdots & 0 \\
1/k & 1/k & \cdots & 1/k & 0 & 0 & \cdots & 0 \\
\vdots &&& \vdots & \vdots &&& \vdots \\
1/k & 1/k & \cdots & 1/k & 0 & 0 & \cdots & 0 \\
0 & 0 & \cdots & 0 & 1/\ell & 1/\ell & \cdots & 1/\ell \\
0 & 0 & \cdots & 0 & 1/\ell & 1/\ell & \cdots & 1/\ell \\
\vdots &&& \vdots & \vdots &&& \vdots \\
0 & 0 & \cdots & 0 & 1/\ell & 1/\ell & \cdots & 1/\ell
\end{matrix}\right).
\]
Similarly, let $\left( A^{(r)} \right)_{r=1}^{\infty}$ be a sequence of $MBN$ matrices
such that $\lim_{r\rightarrow \infty} MN/B^2 = 0$.
It follows from~\eqref{Sinkhorn:MBN-a} that
\[
\lim_{r \rightarrow \infty} a^{(r)}
= \frac{ 1}{k} - \frac{ k+\ell - |k-\ell | }{ 2k^2},
\]
If $k \leq \ell$, then
\[
\lim_{r \rightarrow \infty} a^{(r)} = 0,
\qquad
\lim_{r \rightarrow \infty} b^{(r)} = \frac{1}{\ell},
\qquad
\lim_{r \rightarrow \infty} c^{(r)} = \frac{\ell-k}{\ell^2}.
\]
If $k > \ell$ , then
\[
\lim_{r \rightarrow \infty} a^{(r)} = \frac{k - \ell}{k^2},
\qquad
\lim_{r \rightarrow \infty} b^{(r)} = \frac{1}{k},
\qquad
\lim_{r \rightarrow \infty} c^{(r)} = 0.
\]
\section{The matrix $A_1$} \label{Sinkhorn:section:A_1}
The matrix
\[
A_1 = \left(\begin{matrix}
K & 1 & 1 \\
1 & K & 1 \\
1 & 1 & K
\end{matrix}\right)
\]
is the simplest. Just one row scaling or one column scaling produces
the doubly stochastic matrix
\[
S(A_1) = \left(\begin{matrix}
K/(K+2) & 1/(K+2) & 1/(K+2) \\
1/(K+2) & K/(K+2) & 1/(K+2) \\
1/(K+2) & 1/(K+2) & K/(K+2)
\end{matrix}\right)
\]
We have $S(A_1) = X A_1 X$, where
\[
X = \diag( \sqrt{1/(K+2)}, \sqrt{1/(K+2)}, \sqrt{1/(K+2)} ).
\]
We have the asymptotic limits
\[
\lim_{K\rightarrow \infty} S(A_1) =
\left(\begin{matrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{matrix}\right)
\qand
\lim_{K\rightarrow 0} S(A_1) =
\left(\begin{matrix}
0 & 1/2 & 1/2 \\
1/2 & 0 & 1/2 \\
1/2 & 1/2 & 0
\end{matrix}\right).
\]
\section{The matrices $A_2$, $A_3$, and $A_4$} \label{Sinkhorn:section:A_2}
These are $MBN$ matrices.
The matrix
\[
A_2 = \left(\begin{matrix} K & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{matrix}\right)
\]
is an $MBN$ matrix with $k=1$, $\ell = 2$, $M=K$, $B = N = 1$, and $L = K$.
The matrix
\[
A_3 = \left(\begin{matrix}
1 & 1 & 1 \\
1 & K & K \\
1 & K & K
\end{matrix}\right)
\]
is an $MBN$ matrix with $k=1$, $\ell = 2$, $M=B=1$, $N = K$, and $L = K$.
Both matrices satisfy $L = MN/B^2 = K \neq 1$, and so they have the same Sinkhorn limit
\[
S(A_2) = S(A_3) = \left(\begin{matrix}
a & b & b \\
b & c & c \\
b & c & c
\end{matrix}\right)
\]
with
\begin{align}
a & = \frac{2K+1 - \sqrt{8K + 1}}{2(K-1)} \label{Sinkhorn:A2-a} \\
b& = \frac{ -3 + \sqrt{8K + 1}}{4(K-1)} \label{Sinkhorn:A2-b} \\
c & = \frac{ 4K-1 - \sqrt{8K + 1}}{8(K-1)}. \label{Sinkhorn:A2-c}
\end{align}
We have the asymptotic limits
\[
\lim_{K\rightarrow \infty} S(A_2) =
\left(\begin{matrix}
1 & 0 & 0 \\
0 & 1/2 & 1/2 \\
0 & 1/2 & 1/2
\end{matrix}\right)
\qand
\lim_{K\rightarrow \infty} S(A_2) =
\left(\begin{matrix}
0 & 1/2 & 1/2 \\
1/2 & 1/4 & 1/4 \\
1/2 & 1/4 & 1/4
\end{matrix}\right).
\]
The matrix
\[
A_4 = \left(\begin{matrix}
1 & K & K \\
K & 1 & 1 \\
K & 1 & 1
\end{matrix}\right)
\]
is an $MBN$ matrix with $k=1$, $\ell = 2$, $M=N=1$, and $B = K$.
We have $L = MN/B^2 = 1/K^2 \neq 0$, and the Sinkhorn limit
\[
S(A_4) =
\left(\begin{matrix}
a & b & b \\
b & c & c \\
b & c & c
\end{matrix}\right)
\]
with
\begin {align*}
a & = \frac{-K^2 -2+ K \sqrt{K^2 + 8}}{2(K^2-1)} \\
b& = \frac{3K^2 - K \sqrt{K^2 + 8}}{4(K^2-1)} \\
c & = \frac{K^2 - 4 + K \sqrt{K^2 + 8}}{8(K^2-1)}.
\end {align*}
We have the asymptotic limits
\[
\lim_{K\rightarrow \infty} S(A_4) =
\left(\begin{matrix}
0 & 1/2 & 1/2 \\
1/2 & 1/4 & 1/4 \\
1/2 & 1/4 & 1/4
\end{matrix}\right)
\qand
\lim_{K\rightarrow 0} S(A_4) =
\left(\begin{matrix}
1 & 0 & 0 \\
0 & 1/2 & 1/2 \\
0 & 1/2 & 1/2
\end{matrix}\right).
\]
\section{The matrix $A_5$} \label{Sinkhorn:section:A_5}
The construction of the Sinkhorn limit of the $3 \times 3$ matrix
\[
A_5 = \left(\begin{matrix}
K & 1 & 1 \\
1 & K & 1 \\
1 & 1 & 1
\end{matrix}\right)
\]
requires only high school algebra.
There exists a unique positive diagonal matrix $X = \diag(x,y,z)$
such that $XA_5X$ is doubly stochastic and positive. We have
\[
S(A_5) = XA_5X = \left(\begin{matrix}
Kx^2 & xy & xz \\
xy & Ky^2 & yz \\
xz & yz & z^2
\end{matrix}\right)
\]
and so
\begin {align*}
Kx^2 + xy + xz & = 1 \\
xy + Ky^2 + yz & = 1 \\
xz + yz + z^2 & = 1
\end {align*}
We have
\[
z = \frac{1 - Kx^2 - xy }{x} = \frac{1 - xy - Ky^2 }{y}.
\]
Rearranging, we obtain
\begin{equation} \label{Sinkhorn:zC}
(y-x)((K-1)xy+1) = 0.
\end{equation}
Note that $0 < xy < 1$.
If $K > 1$, then $(K-1)xy+1 > 1$.
If $0 < K < 1$, then
\[
0 < (1-K)xy < 1-K < 1
\]
and $(K-1)xy+1 > 0$.
Therefore, $x=y$, and so
\begin{equation} \label{Sinkhorn:zC}
z = \frac{1 - (K+1)x^2}{x}
\end{equation}
\begin{equation} \label{Sinkhorn:zA}
(K+1)x^2 + xz = 1
\end{equation}
\begin{equation} \label{Sinkhorn:zB}
2xz + z^2 = 1.
\end{equation}
We obtain
\[
2\left( 1 - (K+1)x^2 \right) + \left(\frac{ 1 - (K+1)x^2}{x} \right)^2 = 1.
\]
Applying~\eqref{Sinkhorn:zC} and eliminating $xz$ from~\eqref{Sinkhorn:zA} and~\eqref{Sinkhorn:zB} gives
\[
\left( \frac{1 - (K+1)x^2}{x} \right)^2 = z^2 = 2(K+1)x^2 - 1.
\]
Therefore,
\[
(K^2-1)x^4 - (2K+1)x^2 + 1 = 0
\]
and so
\[
x^2 = \frac{2K+1 \pm \sqrt{4K+5}}{2(K^2 -1)}.
\]
The inequality $Kx^2 < 1$ implies
\[
x^2 = \frac{2K+1 - \sqrt{4K+5}}{2(K^2 -1)}
\]
and
\[
z^2 = \frac{K+2 - \sqrt{4K+5}}{K-1}.
\]
Thus, the Sinkhorn limit has the shape
\[
S(A_5) = \left(\begin{matrix}
a & b & c \\
b & a & c \\
c & c & d
\end{matrix}\right)
\]
where
\begin {align*}
a & = Kx^2 = \frac{K(2K+1 - \sqrt{4K+5})}{2(K^2 -1)} \\
b & = x^2 = \frac{2K+1 - \sqrt{4K+5}}{2(K^2 -1)} \\
c & = xz = \frac{ \sqrt{ 2K+7 - 3\sqrt{4K+5}}} {\sqrt{2}(K-1)} \\
d & = z^2 = \frac{K+2 - \sqrt{4K+5}}{K-1}.
\end {align*}
We have the asymptotic limits
\[
\lim_{K\rightarrow \infty} S(A_5) = \left(\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}\right)
\qand
\lim_{K\rightarrow 0} S(A_5) = \left(\begin{matrix}
0 & \frac{\sqrt{5}-1}{2} & \frac{3 - \sqrt{5}}{2} \\
\frac{\sqrt{5}-1}{2} & 0 & \frac{3 - \sqrt{5}}{2} \\
\frac{3 - \sqrt{5}}{2} & \frac{3 - \sqrt{5}}{2} & \sqrt{5}-2 \end{matrix}\right).
\]
\section{The matrix $A_6$} \label{Sinkhorn:section:A_6}
The construction of the Sinkhorn limit of the $3 \times 3$ matrix
\begin{equation} \label{Sinkhorn:KK}
A_6 = \left(\begin{matrix}
K & K & 1\\
K & 1 & 1 \\
1 & 1 & 1
\end{matrix}\right)
\end{equation}
also requires only high school algebra.
There exists a unique positive diagonal matrix
$X = \diag(x,y,z)$ such that
\[
S(A_6) = X A_6 X = \left(\begin{matrix}
Kx^2 & Kx y & x z \\
Kx y & y^2 & y z \\
x z & y z & z^2
\end{matrix}\right)
\]
is a doubly stochastic matrix, and so
\begin{align}
Kx^2 + Kx y + x z & =1 \label{Sinkhorn:A6-eqn1} \\
Kx y + y^2 + y z & = 1 \label{Sinkhorn:A6-eqn2} \\
x z + y z + z^2 & = 1. \label{Sinkhorn:A6-eqn3}
\end{align}
From~\eqref{Sinkhorn:A6-eqn1} and~\eqref{Sinkhorn:A6-eqn2} we obtain
\[
z = \frac{1}{x} -Kx - Ky = \frac{1}{y} -Kx - y
\]
and so
\begin{equation} \label{Sinkhorn:A6-eqn4}
x = \frac{y}{(K-1)y^2 + 1}
\end{equation}
and
\begin{equation} \label{Sinkhorn:A6-eqn5}
z = \frac{1}{y} - \frac{Ky}{(K-1)y^2 + 1} - y
= \frac{- (K-1)y^4 - 2y^2 + 1}{y( (K-1)y^2+1)}.
\end{equation}
Inserting~\eqref{Sinkhorn:A6-eqn4} and~\eqref{Sinkhorn:A6-eqn5} into~\eqref{Sinkhorn:A6-eqn3}
and simplifying, we obtain
\[
\left( (K-1)y^2 + 1 \right)^3 = K
\]
and so
\[
y^2 = \frac{K^{1/3} -1}{K-1} = \frac{1}{1+K^{1/3} + K^{2/3}}
\]
and
\[
y = \frac{1}{\sqrt{1+K^{1/3} + K^{2/3}}}.
\]
Inserting this into~\eqref{Sinkhorn:A6-eqn4} gives
\[
x = \frac{y}{K^{1/3}} = \frac{1}{ K^{1/3}\sqrt{1+K^{1/3} + K^{2/3}}}.
\]
and then~\eqref{Sinkhorn:A6-eqn5} gives
\[
z = K^{1/3}y = \frac{K^{1/3}}{ \sqrt{1+K^{1/3} + K^{2/3}}}.
\]
This determines the scaling matrix X.
The Sinkhorn limit is the circulant matrix
\[
S(A_6) = \left(\begin{matrix} a & b & c \\ b & c & a \\ c & a & b \end{matrix}\right)
\]
with
\begin {align*}
a & = Kx^2 = yz = \frac{K^{2/3} -K^{1/3} }{K-1} \\
b & = Kxy = z^2 = \frac{K - K^{2/3} }{K-1} \\
c & = xz = y^2 = \frac{K^{1/3} -1}{K-1}.
\end {align*}
The asymptotic limits are
\[
\lim_{K\rightarrow \infty} S(A_6) = \left(\begin{matrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{matrix}\right)
\qand
\lim_{K\rightarrow 0} S(A_6) = \left(\begin{matrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{matrix}\right).
\]
\section{The matrix $A_7$} \label{Sinkhorn:section:A_7}
Consider the symmetric $3\times 3$ matrix
\[
A_7= \left(\begin{matrix}
K & K & 1 \\
K & 1 & 1 \\
1 & 1 & K
\end{matrix}\right).
\]
There exists a unique positive diagonal matrix $X = \diag(x,y,z)$ such that
\[
S(A_7) = XA_7X = \left(\begin{matrix}
K x^2 & K xy & xz \\
Kxy & y^2 & yz \\
xz & yz & Kz^2
\end{matrix}\right)
\]
is doubly stochastic. Therefore,
\begin{align}
K x^2 + K xy+ xz & = 1 \label{Sinkhorn:A7-eqn1} \\
Kxy + y^2 + yz & = 1 \label{Sinkhorn:A7-eqn2} \\
xz + yz + K z^2 & = 1 \label{Sinkhorn:A7-eqn3}
\end{align}
Because equations~\eqref{Sinkhorn:A7-eqn1} and~\eqref{Sinkhorn:A6-eqn1}
are identical, and equations~\eqref{Sinkhorn:A7-eqn2} and~\eqref{Sinkhorn:A6-eqn2}
are identical, we obtain~\eqref{Sinkhorn:A6-eqn4} and~\eqref{Sinkhorn:A6-eqn5}.
Inserting these formulae for $x$ and $z$ into~\eqref{Sinkhorn:A7-eqn3} gives
the octic polynomial
\[
(K-1)^3y^8+3(K-1)^2y^6-(K-1)(2K-3)y^4-(4K-1)y^2+K = 0.
\]
By Theorem~\ref{Sinkhorn:theorem:Sinkhorn-1}, this polynomial has at least one solution $y \in (0,1)$.
If $K>1$, then, by Descartes's rule of signs, this polynomial has exactly two positive solutions.
If $0 < K < 1$, then this polynomial has one or three positive solutions.
For matrices of the form $A_7$, we do not have explicit formulae for the coordinates
of the Sinkhorn limit as functions of $K$. Computer calculations suggest
that the asymptotic limits of $S(A_7)$ as $K\rightarrow \infty$ and $K\rightarrow 0 $ are
\[
\left(\begin{matrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{matrix}\right)
\qand
\left(\begin{matrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{matrix}\right).
\]
\section{Gr\" obner bases and algebraic numbers} \label{Sinkhorn:section:Grobner}
I like solving problems using high school algebra.
However, it is important to note that the previous calculations are also easily
done using Gr\" obner bases.
For every $n \times n$ matrix $A = (a_{i,j})$ and diagonal matrix $X = \diag(x_1,\ldots, x_n)$,
we have the matrix
\[
XAX = \left(\begin{matrix} a_{i,j} x_i x_j \end{matrix}\right).
\]
If $A$ is positive and symmetric, then, by Theorems~\ref{Sinkhorn:theorem:Sinkhorn-1}
and~\ref{Sinkhorn:theorem:Sinkhorn-2}, the $n$ quadratic equations
\[
q_i = q_i(x_1,\ldots, x_n) = \sum_{j=1}^n a_{i,j} x_ix_j - 1 = 0 \qquad \qquad (i = 1,\ldots, n)
\]
have a unique positive solution, and the diagonal matrix
$X = \diag(x_1,\ldots, x_n)$ is the unique scaling matrix in the Sinkhorn limit $S(A) = XAX$.
Equivalently, $(x_1,\ldots, x_n)$ is the unique positive vector in the affine variety
of the ideal in $\ensuremath{\mathbf R}[x_1,\ldots, x_n]$ generated by the set of polynomials $\{q_1,\ldots, q_n\}$.
For each lexicographical ordering of the variables $x_1,\ldots, x_n$,
Maple (and other computer algebra programs) can compute a Gr\" obner basis for the ideal.
The Gr\" obner basis for this ideal shows that if the coordinates of the matrix $A = (a_{i,j})$
are rational numbers, then $x_1,\ldots, x_n$ are algebraic numbers of degrees
bounded in terms of $n$.
Here is an example. Let $n = 3$ and $X = \diag(x,y,z)$.
Consider the matrices
\[
A_7 = \left(\begin{matrix}
K & K & 1 \\
K & 1 & 1 \\
1 & 1 & K
\end{matrix}\right)
\qqand
XA_7X =
\left(\begin{matrix}
K x^2& Kxy & xz \\
Kxy & y^2 & yz \\
xz & xy & Kz^2
\end{matrix}\right).
\]
with $K>0$ and $K \neq 1$.
There exist unique positive real numbers $x,y,z$ that satisfy the quadratic
equations
\begin {align*}
K x^2 + K xy+ xz & = 1\\
K xy + y^2 + yz & = 1 \\
xz + yz + K z^2 & = 1.
\end {align*}
Equivalently, $(x,y,z)$ is the unique positive vector in the affine variety
$V(I)$, where $I$ is the ideal in $\ensuremath{\mathbf R}[x,y,z]$ generated by the polynomials
\begin {align*}
K x^2 + K xy+ xz & - 1\\
K xy + y^2 + yz & - 1 \\
xz + yz + K z^2 & - 1.
\end {align*}
Let $K=2$.
Using the Groebner package in Maple with the lexicographical order $(x,y,z)$, we obtain
the Gr\" obner basis
\begin {align*}
f_1(z) & = 4-28 z^2+62 z^4-57 z^6+18 z^8 \\
f_2(y,z) & = -17 z^3+39 z^5-18 z^7+2 y \\
f_3(x,z) & = -20 z+96 z^3-135 z^5+54 z^7+4 x.
\end {align*}
Applying Maple with the lexicographical order $(y,z,x)$, we obtain
the Gr\" obner basis
\begin {align*}
g_1(x) & = 2-17 x^2+22 x^4+48 x^6+36 x^8 \\
g_2(x,z) & = -103 x+378 x^3+624 x^5+396 x^7+14 z \\
g_3(x,y) & = 3 x-56 x^3-72 x^5-36 x^7+7 y.
\end {align*}
Applying Maple with the lexicographical order $(z,x,y)$, we obtain
the Gr\" obner basis
\begin {align*}
h_1(y) & = 2-7 y^2-y^4+3 y^6+y^8 \\
h_2(x,y) & = -4 y+2 y^5-3 y^3+y^7+6 x \\
h_3(y,z) & = -7 y+5 y^5+3 y^3+y^7+6 z.
\end {align*}
Thus, $x^2$, $y^2$, and $z^2$ are algebraic numbers of degree at most 4, and we have explicit polynomial representations of each variable $x$, $y$, $z$ in terms of the other two variables.
For arbitrary $K$, applying Maple with the lexicographical order $(y,z,x)$, we obtain
the Gr\" obner basis
\begin {align*}
h_1(y)
& = K- (4 K - 1) y^2 -(K-1)(2K-3) y^4+3 (K-1)^2 y^6+(K-1)^3 y^8 \\
h_2(x,y)
& = K (K+1) x -2 K y - (K-1)(2K-1) y^3+ 2 (K-1)^2 y^5+(K-1)^3 y^7 \\
h_3(y,z)
& = K (K+1) z - (K-1)^2 y -3(K-1) y^3+ (K-1)^2(K-3) y^5+( K-1)^3 y^7.
\end{align*}
For each of the 8 roots of $h_1(y)$, the polynomials $h_2(z,y)$ and $h_3(x,y)$
determine unique numbers $x$ and $z$.
Exactly one of the triples $(x,y,z)$ will be positive.
\section{Rationality and finite length} \label{Sinkhorn:section:finiteLength}
For what positive $n\times n$ matrices does the alternate scaling algorithm
converge in finitely many steps? This problem has been solved for $2 \times 2$ matrices
(Nathanson~\cite{nath2019-184}), but it is open for all dimensions $n \geq 3$.
In dimension 3, matrices equivalent to $A_1$ become doubly stochastic in one step,
that is, after one row or one column scaling.
Ekhad and Zeilberger~\cite{ekha-zeil19} computed a positive $3\times 3$ matrix
that becomes doubly stochastic in exactly two steps, and Nathanson~\cite{nath2019-186}
generalized this construction. It is not know if there exists a positive $3\times 3$ matrix
that becomes doubly stochastic in exactly $s$ steps for some $s \geq 3$.
Consider the matrix $A_2 = \left(\begin{matrix} K & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{matrix}\right)$ with parameter $K$.
If $K$ is a rational number, then every matrix generated by iterated row
and column scalings has rational coordinates.
If the Sinkhorn limit contains an irrational coordinate,
then the alternate scaling algorithm cannot terminate in finitely many steps.
Let $K$ be an integer, $K \geq 2$.
In Section~\ref{Sinkhorn:section:A_2} we proved that the Sinkhorn limit $S(A_2)$
has coordinates in the quadratic field
$\ensuremath{ \mathbf{Q} }(\sqrt{8K+1})$. For example, from~\eqref{Sinkhorn:A2-a},
the $(1,1)$ coordinate of $S(A_2)$ is
\[
\frac{2K+1-\sqrt{8K+1}}{2(K-1)}.
\]
This number is rational if and only if the odd integer $8K+1$ is the square of an odd integer, that is,
if and only if $8K+1 = (2r+1)^2$ for some positive integer $r$
and so $K = r(r+1)/2$ is a triangular number.
From~\eqref{Sinkhorn:A2-a},~\eqref{Sinkhorn:A2-b}, and~\eqref{Sinkhorn:A2-c},
we obtain
\begin {align*}
a & = \frac{r^2-r}{r^2+r-2} = \frac{r}{r+2} \\
b & = \frac{r -1}{r^2+r-2} = \frac{1}{r+2} \\
c & = \frac{r^2-1}{2(r^2+r-2)} = \frac{r+1}{2(r+2) }.
\end {align*}
Moreover, $S(A_2) = X A_2 X$, where $X = \diag (x,y,y)$
with $Kx^2 = a$ and $y^2 = c$. Thus,
\[
x = \sqrt{\frac{a}{K} } = \sqrt{ \frac{2}{ (r+1)(r+2)} }
\qqand
y = \sqrt{c} = \sqrt{ \frac{r+1}{2(r+2)} }.
\]
For example, if $K = 3$, then $r=2$ and
\[
A_2 = \left(\begin{matrix} 3 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{matrix}\right)
\qqand
X A_2 X = S(A_2 ) = \left(\begin{matrix} 1/2 & 1/4 & 1/4 \\ 1/4 & 3/8 & 3/8 \\ 1/4 & 3/8 & 3/8 \end{matrix}\right)
\]
where
\[
X = \diag( \sqrt{6}/6, \sqrt{6}/4, \sqrt{6}/4 ).
\]
Note that $A_2$ also has a scaling by rational matrices
\[
S(A_2) = X'A_2 Y'
\]
where
\[
X' = \diag( 1/6, 1/4,1/4) \qqand Y' = \diag(1, 3/2, 3/2).
\]
It is not known if there exists a triangular number $K$
for which the alternate scaling algorithm terminates in a finite number of steps.
\section{Open problems}
\begin{enumerate}
\item
Compute explicit formulas for the Sinkhorn limits of matrices of the form $A_7$.
More generally, compute explicit formulas for the Sinkhorn limits of all positive symmetric
$3\times 3$ matrices.
This is a central problem.
\item
Here is a special case.
Let $K, L, M$ and 1 be pairwise distinct positive numbers. Compute the Sinkhorn limits of the matrices
\[
\left(\begin{matrix}
K & 1 & 1 \\
1 & L & 1 \\
1 & 1 & 1
\end{matrix}\right)
\qqand
\left(\begin{matrix}
K & 1 & 1 \\
1 & L & 1 \\
1 & 1 & M
\end{matrix}\right).
\]
\item
For what positive $n\times n$ matrices does the alternate scaling algorithm
converge in finitely many steps? This is the problem discussed in Section~\ref{Sinkhorn:section:finiteLength}.
\item
It is not known what algebraic numbers appear as coordinates of Sinkhorn limits
of matrices with positive integral coordinates.
It would be interesting to have an example of an algebraic number in the unit interval
that is not a coordinate of the Sinkhorn limit of a positive integral matrix.
\item
Does every possible shape of a doubly stochastic $3\times 3$ matrix $A$ appear as the
nontrivial Sinkhorn limit of some $3\times 3$ matrix?
\item
Why does the shape of the Sinkhorn limit $S(A)$ seem to depend only on the shape of the matrix $A$
and not on the numerical values of the coordinates of $A$?
\item
Let A\ be a nonnegative $m\times n$ matrix.
Let $ \ensuremath{\mathbf r} = (r_1, r_2,\ldots, r_m) \in \ensuremath{\mathbf R}^m$ and let $ \ensuremath{\mathbf c} = (c_1, c_2, \ldots, c_n) \in \ensuremath{\mathbf R}^n$.
The matrix A\ is \emph{$ \ensuremath{\mathbf r}$-row stochastic}
if $\rowsum_i( A ) = r_i$ for all $i \in \{1,2,\ldots, m\}$.
The matrix A\ is \emph{$ \ensuremath{\mathbf c}$-column stochastic}
if $\colsum_j( A ) =c_j$ for all $j \in \{1,2,\ldots, n\}$.
The matrix $ A $ is \emph{$(\ensuremath{\mathbf r},\ensuremath{\mathbf c})$-stochastic}
if it is both $\ensuremath{\mathbf r}$-row stochastic and $\ensuremath{\mathbf c}$-column stochastic.
Let A\ be a positive matrix. Let $X$ be the $m\times m$ diagonal matrix whose
$i$th coordinate is $r_i/\rowsum_i(A)$, and let $Y$ be the $n \times n$ diagonal matrix whose
$j$th coordinate is $c_j/\colsum_j(A)$.
The matrix $XA$ is $\ensuremath{\mathbf r}$-row stochastic and the matrix
$AY$ is $\ensuremath{\mathbf c}$-column stochastic.
A simple modification of the alternate scaling algorithm produces
an $(\ensuremath{\mathbf r},\ensuremath{\mathbf c})$-stochastic Sinkhorn limit.
It is an open problem to compute explicit Sinkhorn limits in
the $(\ensuremath{\mathbf r},\ensuremath{\mathbf c})$-stochastic setting.
\item
It is a old problem in number theory to understand the continued fractions
of the cube roots of integers, and, in particular, to understand
the approximation of $\sqrt[3]{2}$ by rationals.
One coordinate of the Sinkhorn limit of the matrix $A_6$ with $K=2$ is $\sqrt[3]{2} - 1$.
The matrix $A_6$ with $K=2$ has rational coordinates, and so
the matrices constructed by the alternate scaling algorithm also have
rational coordinates, and generate explicit sequences of rational
approximations to $\sqrt[3]{2}$. The nature of these approximations
remains mysterious.
\end{enumerate}
\section{Notes}
The computational complexity of Sinkhorn's alternate scaling algorithm
is investigated in
Kalantari and Khachiyan~\cite{kala-khac93,kala-khac96},
Kalantari, Lari, Ricca, and Simeone~\cite{kala08},
Linial, Samorodnitsky and Wigderson~\cite{lini-samo-wigd98b}
and Allen-Zhu, Li, Oliveira, and Wigderson~\cite{alle-li-oliv-wigd17}.
An extension of matrix scaling to operator scaling began
with Gurvits~\cite{gurv04}, and is developed in
Garg, Gurvits, Oliveira, and Wigderson~\cite{garg-gurv-oliv-wigd16,garg-gurv-oliv-wigd17},
Gurvits~\cite{gurv15}, and Gurvits and Samorodnitsky~\cite{gurv-samo14}.
Motivating some of this recent work are the classical papers of Edmonds~\cite{edmo67}
and Valient~\cite{vali79a,vali79b}.
The literature on matrix scaling is vast. See the recent survey paper of
Idel~\cite{idel16}. For the early history of matrix scaling, see
Allen-Zhu, Li, Oliveira, and Wigderson~\cite[Section 1.1]{alle-li-oliv-wigd17}.
\emph{Acknowledgements.}
The alternate scaling algorithm was discussed in several lectures
in the New York Number Theory Seminar,
and I thank the participants for their useful remarks.
In particular, I thank David Newman for making the initial computations
that suggested some of the problems considered in this paper.
I also benefitted from a careful and thoughtful referee's report.
\def$'$} \def\cprime{$'$} \def\cprime{$'${$'$} \def$'$} \def\cprime{$'$} \def\cprime{$'${$'$} \def$'$} \def\cprime{$'$} \def\cprime{$'${$'$}
\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
\providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR }
\providecommand{\MRhref}[2]{%
\href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2}
}
\providecommand{\href}[2]{#2}
|
1,314,259,993,572 | arxiv | \section{Introduction}
For a set $V$ and a positive integer $r$, let $V^{(r)}$ be the family of all $r$-subsets of $V$. An $r$-uniform hypergraph or $r$-graph $G$ consists of a set $V(G)$ of vertices and a set $E(G) \subseteq V(G) ^{(r)}$ of edges. When $r=2$, an $r$-graph is a simple graph. When $r\ge 3$, an $r$-graph is often called a hypergraph. An edge $e=\{a_1, a_2, \ldots, a_r\}$ will be simply denoted by $a_1a_2 \ldots a_r$. Let $K^{(r)}_t$ denote the complete $r$-graph on $t$ vertices, that is the $r$-graph on $t$ vertices containing all possible edges. A complete $r$-graph on $t$ vertices is also called a clique with order $t$. A clique is said to be maximum if it has maximum cardinality. Let ${\mathbb N}$ be the set of all positive integers. For an integer $n \in {\mathbb N}$, let $[n]$ denote the set $\{1, 2, 3, \ldots, n\}$. Let $[n]^{(r)}$ represent the complete $r$-graph on the vertex set $[n]$.
For an $r$-graph $G=(V,E)$, denote the $(r-1)$-neighborhood of a vertex $i \in V$ by $E_i=\{A \in V^{(r-1)}: A \cup \{i\} \in E\}$. Similarly, denote the $(r-2)$-neighborhood of a pair of vertices $i,j \in V$ by $E_{ij}=\{B \in V^{(r-2)}: B \cup \{i,j\} \in E\}$. Denote the complement of $E_i$ by $E^c_i=\{A \in V^{(r-1)}: A \cup \{i\} \in V^{(r)} \backslash E\}$. Also, denote the complement of $E_{ij}$ by
$E^c_{ij}=\{B \in V^{(r-2)}: B \cup \{i,j\} \in V^{(r)} \backslash E\}$ and $E_{i\setminus j}=E_i\cap E^c_j.$
\begin{defi}
For an $r$-graph $G=([n], E(G))$ and a vector $\vec{x}=(x_1,\ldots,x_n) \in R^n$,
define
$$\lambda (G,\vec{x})=\sum_{i_1i_2 \cdots i_r \in E(G)}x_{i_1}x_{i_2}\ldots x_{i_r}.$$
Let $S=\{\vec{x}=(x_1,x_2,\ldots ,x_n): \sum_{i=1}^{n} x_i =1, x_i
\ge 0 {\rm \ for \ } i=1,2,\ldots , n \}$. The Lagrangian\footnote{Let us note that this use of the name Lagrangian is at odds with the tradition. Indeed, names as Laplacian, Hessian, Gramian, Grassmanian, etc., usually denote a structured object like matrix, operator, or manifold, and not just a single number.} of
$G$, denoted by $\lambda (G)$, is the maximum
of the above homogeneous function over the standard simplex $S$. Precisely,
$$\lambda (G) = \max \{\lambda (G, \vec{x}): \vec{x} \in S \}.$$
\end{defi}
The value $x_i$ is called the weight of the vertex $i$.
A vector $\vec{x}=(x_1, x_2, \ldots, x_n) \in R^n$ is called a feasible weighting for $G$ if
$\vec{x}\in S$. A vector $\vec{y}\in S$ is called an optimal weighting for $G$
if $\lambda (G, \vec{y})=\lambda(G)$.
The following fact is easily implied by the definition of the Lagrangian.
\begin{fact}\label{mono}
Let $G_1$, $G_2$ be $r$-uniform graphs and $G_1\subseteq G_2$. Then $\lambda (G_1) \le \lambda (G_2).$
\end{fact}
In \cite{MS}, Motzkin and Straus established a remarkable connection between the clique number and the Lagrangian of a graph.
\begin{theo} \cite{MS} \label{MStheo}
If $G$ is a 2-graph in which a maximum clique has order $t$ then
$\lambda(G)=\lambda(K^{(2)}_t)={1 \over 2}(1 - {1 \over t})$.
\end{theo}
The Motzkin-Straus result provides solutions to the optimization problem of a class of homogeneous multilinear functions over the standard simplex of the Euclidean space. The Motzkin-Straus result
and its extension were also successfully employed in optimization to provide heuristics for the
maximum clique problem (see \cite{B1,B2,B3,G9,PP15}). It is interesting to explore whether similar results holds for hypergraphs. The obvious generalization of Motzkin and Straus' result to hypergraphs is false because there are many examples of hypergraphs that do not achieve their
Lagrangian on any proper subhypergraph.
Lagrangians of hypergraphs has been proved to be a useful tool in hypergraph extremal problems. Applications of Lagrangian method can be found in \cite{FF,FR84,keevash,mubayi06,sidorenko89}. In most applications, an upper bound is needed. Frankl and F\"uredi \cite{FF} asked the following question. Given $r \ge 3$ and $m \in {\mathbb N}$ how large can the Lagrangian of an $r$-graph with $m$ edges be?
For distinct $A, B \in {\mathbb N}^{(r)}$ we say that $A$ is less than $B$ in the {\em colex ordering} if $max(A \triangle B) \in B$, where $A \triangle B=(A \setminus B)\cup (B \setminus A)$. For example, the first $t \choose r$ $r$-tuples in the colex ordering of ${\mathbb N}^{(r)}$ are the edges of $[t]^{(r)}$. The following conjecture of Frankl and F\"uredi (if it is true) proposes a solution to the question mentioned above.
\begin{con} \cite{FF} \label{conjecture} The $r$-graph with $m$ edges formed by taking the first $m$ sets in the colex ordering of ${\mathbb N}^{(r)}$ has the largest Lagrangian of all $r$-graphs with $m$ edges. In particular, the $r$-graph with $t \choose r$ edges and the largest Lagrangian is $[t]^{(r)}$.
\end{con}
This conjecture is true when $r=2$ by Theorem \ref{MStheo}. For the case $r=3$, Talbot in \cite{T} proved the following.
\begin{theo} \cite{T} \label{Tal} Let $m$ and $t$ be integers satisfying
${t-1 \choose 3} \le m \le {t-1 \choose 3} + {t-2 \choose 2} - (t-1).$
Then Conjecture \ref{conjecture} is true for $r=3$ and this value of $m$.
\end{theo}
Recently, in \cite{TPZZ}, using some different approaches, Conjecture \ref{conjecture} is confirmed for $r=3$ when the value of $m$ satisfying
${t-1 \choose 3} \le m \le {t-1 \choose 3} + {t-2 \choose 2} - \frac{1}{2}(t-1).$
Although the obvious generalization of Motzkin and Straus' result to hypergraphs is false as mentioned earlier, we attempt to explore the relationship between the Lagrangian of a hypergraph and the size of its maximum cliques for hypergraphs when the number of edges is in certain ranges. In \cite{PZ}, it is conjectured that the following Motzkin and Straus type results are true for hypergraphs.
\begin{con} \label{conjecture1} \cite{PZ}
Let $t$, $m$, and $r\ge 3$ be positive integers satisfying ${t-1 \choose r} \le m \le {t-1 \choose r} + {t-2 \choose r-1}$.
Let $G$ be an $r$-graph with $m$ edges and $G$ contain a clique of order $t-1$. Then $\lambda(G)=\lambda([t-1]^{(r)})$.
\end{con}
\begin{con} \label{conjecture2} \cite{PZ} Let $t$, $m$, and $r\ge 3$ be positive integers satisfying ${t-1 \choose r} \le m \le {t-1 \choose r} + {t-2 \choose r-1}$.
Let $G$ be an $r$-graph with $m$ edges without containing a clique of order $t-1$. Then $\lambda(G) < \lambda([t-1]^{(r)})$.
\end{con}
Note that the upper bound ${t-1 \choose r} + {t-2 \choose r-1}$ in Conjecture \ref{conjecture1} is the best possible (see \cite{PZ}). Conjecture \ref{conjecture1} is confirmed when $r=3$ in \cite{PZ}. Let $C_{r,m}$ denote the $r$-graph with $m$ edges formed by taking the first $m$ sets in the colex ordering of ${\mathbb N}^{(r)}$.
The following result was given in \cite{T}.
\begin{lemma} \cite{T} \label{LemmaTal7}
For any integers $m, t,$ and $r$ satisfying ${t-1 \choose r} \le m \le {t-1 \choose r} + {t-2 \choose r-1}$,
we have $\lambda(C_{r,m}) = \lambda([t-1]^{(r)})$.
\end{lemma}
In \cite{PTZ}, the following result is obtained for $r$-graphs.
\begin{theo} \cite{PTZ}\label{TheoremPTZ}
Let $t$,$m$ and $r$ be positive integers satisfying ${t-1 \choose r} \le m \le {t-1 \choose r} + {t-2 \choose r-1}-(2^{r-3}-1)({t-2 \choose r-2}-1)$. Let $G$ be an $r$-graph with $t$ vertices and $m$ edges and contain a clique of order $t-1$. Then $\lambda(G) = \lambda([t-1]^{(r)})$.
\end{theo}
In \cite{T}, the following result is also proved, which is the evidence for Conjecture \ref{conjecture} for
$r$-graphs $G$ on exactly $t$ vertices.
\begin{theo} \cite{T} \label{Talr} For any $r \ge 4$ there exists constants $\gamma_r$ and $\kappa_0(r)$ such that if $m$ satisfies
$${t-1 \choose r} \le m \le {t-1 \choose r} + {t-2 \choose r-1} - \gamma_r (t-1)^{r-2},$$
with $t \ge \kappa_0(r)$, let $G$ be an $r$-graph on $t$ vertices with $m$ edges, then $\lambda(G)\leq\lambda([t-1]^{(r)})$.
\end{theo}
The main result in this paper is Theorem \ref{mainresult} which is a accompany result of Theorem \ref{TheoremPTZ}.
\begin{theo} \label{mainresult} Let $m$, $t$, and $r\geq 4$ be integers satisfying
${t-1\choose r}\leq m \leq {t-1\choose r}+ {t-2\choose r-1}-[(2r-6)\times2^{r-1}+2^{r-3}+(r-4)(2r-7)-1]({t-2\choose r-2}-1).$ Let $G$ be an $r$-graph with $t$ vertices and $m$ edges and without containing a clique of order $t-1$. Then $\lambda(G) < \lambda([t-1]^{(r)})$.
\end{theo}
Theorem \ref{mainresult} and Theorem \ref{TheoremPTZ} give a Motzkin-Straus result for some $r$-graph. Combing Theorems \ref{TheoremPTZ} and \ref{mainresult}, we have the following result immediately.
\begin{coro} \label{improvement} Let $m$, $t$, and $r\geq 4$ be integers satisfying
${t-1\choose r}\leq m \leq {t-1\choose r}+ {t-2\choose r-1}-[(2r-6)\times2^{r-1}+2^{r-3}+(r-4)(2r-7)-1]({t-2\choose r-2}-1).$ Let $G$ be an $r$-graph with $t$ vertices and $m$ edges. Then $\lambda(G) \leq \lambda([t-1]^{(r)})$.
\end{coro}
Note that ${t-1\choose r}\leq m \leq {t-1\choose r}+ {t-2\choose r-1}-[(2r-6)\times2^{r-1}+2^{r-3}+(r-4)(2r-7)-1]({t-2\choose r-2}-1)$ implies the number of vertices $t$ should be sufficiently large such that ${t-2\choose r-1}\geq[(2r-6)\times2^{r-1}+2^{r-3}+(r-4)(2r-7)-1]({t-2\choose r-2}-1)$ in Theorem \ref{mainresult} and Corollary \ref{improvement}.
Theorem \ref{mainresult} and Corollary \ref{improvement} provide evidence for both Conjecture \ref{conjecture2} and Conjecture \ref{conjecture} respectively. The contribution of Corollary \ref{improvement} is that the method developed in the proof of Theorem \ref{mainresult} is simpler and different from that in Theorem \ref{Talr} in some ways. The upper bound in Corollary \ref{improvement} for the number of edges $m$ is more explicit and an improvement comparing to the bound in Theorem \ref{Talr}. The proof of Theorem \ref{mainresult} will be given in Section \ref{section1}. Further remarks and conclusions are in Section \ref{section3}.
\section{Proof of Theorem \ref{mainresult}}\label{section1}
We will impose one additional condition on any optimal weighting ${\vec x}=(x_1, x_2, \ldots, x_n)$ for an $r$-graph $G$:
\begin{eqnarray}
&&|\{i : x_i > 0 \}|{\rm \ is \ minimal, i.e. \ if} \ \vec y {\rm \ is \ a \ feasible \ weighting \ for \ } G {\rm \ satisfying }\nonumber \\
&&|\{i : y_i > 0 \}| < |\{i : x_i > 0 \}|, {\rm \ then \ } \lambda (G, {\vec y}) < \lambda(G) \label{conditionb}.
\end{eqnarray}
When the theory of Lagrange multipliers is applied to find the optimum of $\lambda(G, {\vec x})$, subject to $\sum_{i=1}^n x_i =1$, notice that $\lambda (E_i, {\vec x})$ corresponds to the partial derivative of $\lambda(G, \vec x)$ with respect to $x_i$. The following lemma gives some necessary conditions of an optimal weighting for $G$.
\begin{lemma} \cite{FR84} \label{LemmaTal5} Let $G=(V,E)$ be an $r$-graph on the vertex set $[n]$ and ${\vec x}=(x_1, x_2, \ldots, x_n)$ be an optimal weighting for $G$ with $k$ ($\le n$) non-zero weights $x_1$, $x_2$, $\cdots$, $x_k$ satisfying condition (\ref{conditionb}). Then for every $\{i, j\} \in [k]^{(2)}$, (a) $\lambda (E_i, {\vec x})=\lambda (E_j, \vec{x})=r\lambda(G)$, (b) there is an edge in $E$ containing both $i$ and $j$.
\end{lemma}
\begin{defi}
An $r$-graph $G=(V,E)$ on the vertex set $[n]$ is left-compressed if $j_1j_2\ldots j_r\in E$ implies $i_1i_2\ldots i_r\in E$ whenever $i_k \leq j_k, 1\leq k \leq r$. Equivalently, an $r$-graph $G=(V,E)$ on the vertex set $[n]$ is left-compressed if $E_{j\setminus i}=\emptyset$ for any $1\le i<j\le n$.
\end{defi}
\begin{remark}\label{r1} (a) In Lemma \ref{LemmaTal5}, part (a) implies that
$x_j\lambda(E_{ij}, {\vec x})+\lambda (E_{i\setminus j}, {\vec x})=x_i\lambda(E_{ij}, {\vec x})+\lambda (E_{j\setminus i}, {\vec x}).$
In particular, if $G$ is left-compressed, then
$(x_i-x_j)\lambda(E_{ij}, {\vec x})=\lambda (E_{i\setminus j}, {\vec x})$
for any $i, j$ satisfying $1\le i<j\le k$ since $E_{j\setminus i}=\emptyset$.
(b) If $G$ is left-compressed, then for any $i, j$ satisfying $1\le i<j\le k$,
\begin{equation}\label{enbhd}
x_i-x_j={\lambda (E_{i\setminus j}, {\vec x}) \over \lambda(E_{ij}, {\vec x})}
\end{equation}
holds. If $G$ is left-compressed and $E_{i\setminus j}=\emptyset$ for $i, j$ satisfying $1\le i<j\le k$, then $x_i=x_j$.
(c) By (\ref{enbhd}), if $G$ is left-compressed, then an optimal weighting ${\vec x}=(x_1, x_2, \ldots, x_n)$ for $G$ must satisfy
$x_1 \ge x_2 \ge \ldots \ge x_n \ge 0$.
\end{remark}
Denote $\lambda_{(m, t)}^{r}=\max \{ \lambda(G):$
$ G$ is an $r$-graph with $t$ vertices and $m$ edges $ \}$. The following Lemma is proved in \cite{T}.
\begin{lemma}\cite{T}\label{leftcom0}
There exists a left-compressed $r$-graph $G$ with $t$ vertices and $m$ edges such that
$\lambda(G)=\lambda_{(m,t)}^{r}$.
\end{lemma}
\begin{remark}\label{r2}
Since the only left-compressed $r$-graph with $t$ vertices and $m={t\choose r}$ edges is $[t]^{(r)}$. Hence by Lemma \ref{leftcom0} and Fact \ref{mono}, we have $\lambda_{(m,t)}^{r}\leq\lambda([t]^{(r)})$.
\end{remark}
Denote $\lambda_{(m, t-1, t)}^{r-}=\max \{ \lambda(G):$
$ G$ is an $r$-graph with $t$ vertices and $m$ edges not containing a clique of order $t-1 \}$. The following lemma implies that we only need to consider left-compressed $r$-graphs $G$ when we prove Theorem \ref{mainresult}.
\begin{lemma}\label{leftcom} Let $m$ and $t$ be integers satisfying
${t-1\choose r}\leq m \leq {t-1\choose r}+ {t-2\choose r-1}-[(2r-6)\times2^{r-1}+2^{r-3}+(r-4)(2r-7)-1]({t-2\choose r-2}-1).$
There exists a left-compressed $r$-graph $G$ on vertex set $[t]$ with $m$ edges without containing $[t-1]^{(r)}$ such that
$\lambda(G)=\lambda_{(m, t-1, t)}^{r-}$.
\end{lemma}
In the proof of Lemma \ref{leftcom}, we need to define some partial order relation.
An $r$-tuple $i_1 i_2\cdots i_r$ is called a descendant of an $r$-tuple $j_1j_2\cdots j_r$ if $i_s\le j_s$ for each $1\le s\le r$, and $i_1+i_2+\cdots +i_r < j_1+j_2+\cdots +j_r$. In this case, the $r$-tuple $j_1j_2\cdots j_r$ is called an ancestor of $i_1 i_2\cdots i_r$. The $r$-tuple $i_1i_2\cdots i_r$ is called a direct descendant of $j_1 j_2\cdots j_r$ if $i_1i_2\cdots i_r$ is a descendant of $j_1j_2\cdots j_r$ and $j_1+j_2+\cdots +j_r=i_1+i_2+\cdots +i_r +1$. We say that $i_1 i_2\cdots i_r$ has lower hierarchy than $j_1j_2\cdots j_r$ if $i_1 i_2\cdots i_r$ is a descendant of $j_1j_2\cdots j_r$. This is a partial order on the set of all $r$-tuples.
\noindent {\em Proof of Lemma \ref{leftcom}.} Let $G$ be an $r$-graph with $t$ vertices and $m$ edges without containing a clique of order $t-1$ such that $\lambda(G)=\lambda_{(m, t-1, t)}^{r-}$. We call $G$ an extremal $r$-graph for $m$, $t-1$ and $t$. Let ${\vec x}=(x_1, x_2, \ldots, x_t)$ be an optimal weighting of $G$. We can assume that $x_i\ge x_j$ when $i<j$ since otherwise we can just relabel the vertices of $G$ and obtain another extremal $r$-graph for $m$, $t-1$ and $t$ with an optimal weighting ${\vec x}=(x_1, x_2, \ldots, x_t)$ satisfying $x_i\ge x_j$ when $i<j$. Next we obtain a new $r$-graph $H$ from $G$ by performing the following:
\begin{enumerate}
\item If $(t-r)\ldots(t-1) \in E(G)$, then there is at least one $r$-tuple in $[t-1]^{(r)}\setminus E(G)$, we replacing $(t-r)\ldots(t-1)$ by this $r$-tuple;
\item If an edge in $G$ has a descendant other than $(t-r)\ldots(t-1)$ that is not in $E(G)$, then replace this edge by a descendant other than $(t-r)\ldots(t-1)$ with the lowest hierarchy. Repeat this until there is no such an edge.
\end{enumerate}
Then $H$ satisfies the following properties:
\begin{enumerate}
\item The number of edges in $H$ is the same as the number of edges in $G$.
\item $\lambda(G)=\lambda(G, {\vec x})\le \lambda(H, {\vec x})\le \lambda(H).$
\item $(t-r)\ldots(t-1) \notin E(H)$.
\item For any edge in $E(H)$, all its descendants other than $(t-r)\ldots(t-1)$ will be in $E(H)$.
\end{enumerate}
If $H$ is not left-compressed, then there is an ancestor of $(t-r)\ldots(t-1)$, says $e$, such that $e\in E(H)$. Hence $(t-r)\ldots(t-2)t$ and all the descendants of $(t-r)\ldots(t-2)t$ other than $(t-r)\ldots(t-1)$ will be in $E(H)$. Then
$$m\ge {t-1 \choose r}-1 + {t-2 \choose r-1}>{t-1\choose r}+ {t-2\choose r-1}-[(2r-6)\times2^{r-1}+2^{r-3}+(r-4)(2r-7)-1]({t-2\choose r-2}-1)$$ which is a contradiction. $H$ does not contain $[t-1]^{(r)}$ since $H$ does not contain $(t-r)\ldots(t-1)$. Clearly $H$ is on vertex set $[t]$. So we complete the proof of Lemma \ref{leftcom}\qed
In the rest of the paper we assume that $r\geq 4$ be an integer. In the following three lemmas, Lemma \ref{Lemmaeq} implies the maximum weight of $G$ should distribute 'uniform' on the $t$ vertices if $\lambda(G)\geq \lambda([t-1]^{(r)})$, and Lemma \ref{lemmac} implies $G$ contains most of the first ${t-2r+6 \choose r}$ edges in colex ordering of $N^{(r)}$ if $\lambda(G) \geq \lambda([t-1]^{(r)}),$ while Lemma \ref{lemmab} implies $G$ also contains most of the first ${t-2r+6 \choose r-1}$ edges containing $t-1$. Since $G$ is left-compressed, $G$ also contains most of the the first ${t-2r+6 \choose r-1}$ edges containing vertex $i$, where $t-2r+7\leq i\leq t-1.$ So $G$ contains most edges of $[t-1]^{(r)}$. Note that, in the proof of Lemma 2.6, whenever the lower bound of a product is greater than the upper bound, we take this to be the empty product.
\begin{lemma}\label{Lemmaeq} (a) Let $G$ be an $r$-graph on vertex set $[t]$. Let $\vec{x}=(x_{1},x_{2},\ldots ,x_{t})$ be an optimal weighting for $G$ satisfying $x_1 \ge x_2 \ge \ldots \ge x_t \ge 0$. Then
$x_1< x_{t-2r+3}+x_{t-2r+4}$ or $$\lambda(G)\le \frac{1}{r!}\frac{(t-r)^{r-1} \prod\limits_{i=t-r+2}^{t-2}i}{(t-r+1)^{r-2}(t-1)^{r-2}}<\frac{1}{r!}\frac{ \prod\limits_{i=t-r}^{t-1}i}{(t-1)^r}=\lambda([t-1]^{(r)}).$$
(b) Let $G$ be an $r$-graph on vertex set $[t]$. Let $\vec{x}=(x_{1},x_{2},\ldots ,x_{t})$ be an optimal weighting for $G$ satisfying $x_1 \ge x_2 \ge \ldots \ge x_t \ge 0$. Then
$x_1<2( x_{t-2r+4}+x_{t-2r+5})$ or
$$\lambda(G)\le \frac{1}{r!}\frac{(t-r)^{r-1} \prod\limits_{i=t-r+2}^{t-2}i}{(t-r+1)^{r-2}(t-1)^{r-2}}<\frac{1}{r!}\frac{ \prod\limits_{i=t-r}^{t-1}i}{(t-1)^r}=\lambda([t-1]^{(r)}).$$
\end{lemma}
{\em Proof.} (a) If $x_1\geq x_{t-2r+3}+x_{t-2r+4}$, then $rx_1+x_2+\cdots+x_{t-2r+2}\geq x_1+x_2+\cdots +x_{t-2r+4}+x_{t-3}+x_{t-2r+6}+x_{t-1}+x_{t}=1$. Recalling that $x_1 \ge x_2 \ge \ldots \ge x_{t-2r+2}$, we have $x_1\geq \frac{1}{t-r+1}$.
Using Lemma \ref{LemmaTal5}, we have $\lambda(G)=\frac{1}{r}\lambda(E_1,x).$ Note that $E_1$ is an $(r-1)$-graph with $t-1$ vertices and total weights at most $1-\frac{1}{t-r+1}$. Hence by Remark \ref{r2}(change the total weights 1 to $1-\frac{1}{t-r+1}$).
\begin{eqnarray}\label{eq234}
\lambda(G)&=&\frac{1}{r}\lambda(E_1,x)\leq\frac{1}{r}{t-1 \choose r-1}(\frac{1-\frac{1}{t-r+1}}{t-1})^{r-1}\nonumber\\
&=&\frac{1}{r!}\frac{(t-r)^{r-1} \prod\limits_{i=t-r+2}^{t-2}i}{(t-r+1)^{r-2}(t-1)^{r-2}}.
\end{eqnarray}
Next we prove
\begin{eqnarray}\label{eqa}
\frac{1}{r!}\frac{(t-r)^{r-1} \prod\limits_{i=t-r+2}^{t-2}i}{(t-r+1)^{r-2}(t-1)^{r-2}}
< \frac{1}{r!}\frac{ \prod\limits_{i=t-r}^{t-1}i}{(t-1)^r}
=\lambda([t-1]^{(r)}).
\end{eqnarray}
To show this, we only need to prove
\begin{eqnarray}\label{eqb}
(t-r)^{r-2}(t-1)<(t-r+1)^{r-1}.
\end{eqnarray}
If $t=r,r+1$, (\ref{eqb}) clearly holds. Assuming $t\geq r+2$, we prove this inequality by induction.
Now we suppose that (\ref{eqb}) holds for some $r\geq 4$, we will show it also holds for $r+1$.
Replacing $t$ by $t-1$ in (\ref{eqb}). We have
$$[t-(r+1)]^{r-2}(t-2)<(t-r)^{r-1}.$$
Multiplying $t-(r+1)$ to the above inequality, we have
$$[t-(r+1)]^{r-1}(t-2)<(t-r)^{r-1}[t-(r+1)].$$
Adding $[t-(r+1)]^{r-1}$ to the above inequality, we obtain
\begin{eqnarray}
[t-(r+1)]^{r-1}(t-1)&<&(t-r)^{r-1}[t-(r+1)]+[t-(r+1)]^{r-1}\nonumber\\
&=&(t-r)^{r}-(t-r)^{r-1}+[t-(r+1)]^{r-1}<(t-r)^{r}.
\end{eqnarray}
Hence (\ref{eqb}) also holds for $r+1$ and the induction is complete.
(b) If $x_1\geq 2( x_{t-2r+5}+x_{t-2r+6})$, then $x_1+x_2+\cdots+x_{t-2r+4}+(r-2)\frac{x_1}{2}\geq x_1+x_2+\cdots +x_{t-2r+4}+x_{t-3}+x_{t-2r+6}+x_{t-1}+x_{t}=1$. Recalling that $x_1 \ge x_2 \ge \ldots \ge x_{t-2r+4}$ and $r\geq 4$, we have $x_1\geq \frac{1}{t-2r+4+\frac{r-2}{2}}\geq \frac{1}{t-r+1}$. The rest of the proof is identical to that in part (a), we omit the computation details here.
\qed
\begin{lemma} \label{lemmab}
Let $G$ be a left-compressed $r$-graph on the vertex set $[t]$ without containing $[t-1]^{(r)}$, then $|[t-2r+6]^{(r-1)} \backslash E_{t-1}| \leq 2^{r-1}|E_{(t-1)t}|$ or $\lambda(G) < \lambda([t-1]^{(r)}).$
\end{lemma}
\noindent{\em Proof.}
Let ${\vec x}=(x_1, x_2, \ldots, x_t)$ be an optimal weighting for $G$. Since $G$ is left-compressed, by Remark \ref{r1}(a), $x_1\ge x_2 \ge \cdots \ge x_t \ge 0$. If $x_{t}=0$, then $\lambda(G)=\lambda(G, \vec{x}) < \lambda([t-1]^{(r)})$ since $G$ does not contain $[t-1]^{(r)}.$ So we assume that $x_{t}>0$.
Consider a new weighting for $G$, ${\vec y}=(y_1, y_2, \ldots, y_t)$ given by $y_i=x_i$ for $i\neq t-1, t$, $y_{t-1}=x_{t-1}+x_t$ and $y_t=0$. By Lemma \ref{LemmaTal5}(a), $\lambda(E_{t-1}, \vec{x})=\lambda(E_{t}, \vec{x})$, so
\begin{eqnarray}\label{eq10b}
\lambda(G,\vec {y})- \lambda(G,\vec {x})&=&x_{t}(\lambda(E_{t-1}, \vec{x})-x_{t}\lambda(E_{(t-1)t}, \vec{x})) \nonumber \\
&&-x_{t}(\lambda(E_{t}, \vec{x})-x_{t}\lambda(E_{(t-1)t}, \vec{x}))-x_{t-1}x_t\lambda(E_{(t-1)t}, \vec{x}))\nonumber \\
&=&x_{t}(\lambda(E_{t-1}, \vec{x})-\lambda(E_{t}, \vec{x}))-x_{t}^2\lambda(E_{(t-1)t}, \vec{x}) \nonumber\\
&=& -x_{t}^2\lambda(E_{(t-1)t}, \vec{x}).
\end{eqnarray}
Assume that $|[t-2r+6]^{(r-1)} \backslash E_{t-1}| > 2^{r-1}|E_{(t-1)t}|$. If $\lambda(G) < \lambda([t-1]^{(r)})$ we are done. Otherwise if $\lambda(G) \geq \lambda([t-1]^{(r)})$ we will show that there exists a set of edges $F\subset [t-1]^{(r)}\setminus E$ satisfying
\begin{equation}\label{eq11b}
\lambda(F,\vec {y})> x_{t}^2\lambda(E_{(t-1)t}, \vec{x}).
\end{equation}
Then using (\ref{eq10b}) and (\ref{eq11b}), the $r$-graph $G^{*}=([t], E^{*})$, where $E^{*}=E\cup F$, satisfies $\lambda(G^{*}, \vec {y})=\lambda(G, \vec {y})+\lambda(F, \vec {y})> \lambda(G, \vec{x})=\lambda(G)$. Since $\vec {y}$ has only $t-1$ positive weights, then $\lambda(G^{*}, \vec {y})\le \lambda([t-1]^{(r)})$, and consequently,
$\lambda(G)<\lambda([t-1]^{(r)}).$ This is a contradiction.
We now construct the set of edges $F$. Let $C=[t-2r+6]^{(r-1)} \setminus E_{t-1}$. Then by the assumption,
$\vert C\vert > 2^{r-1}|E_{(t-1)t}|$ and
$\lambda(C, \vec{x})\ge 2^{r-1}|E_{(t-1)t}|x_{t-3r+8}\ldots x_{t-2r+6}.$
Let $F$ consist of those edges in $[t-1]^{(r)}\setminus E$ containing the vertex $t-1$. Since $\lambda(G) \geq \lambda([t-1]^{(r)})$ then $x_{t-2r+3}> \frac{x_1}{2}$ by Lemma \ref{Lemmaeq}(a) and $x_{t-2r+4}\geq x_{t-2r+5} >\frac{x_1}{4}$ by Lemma \ref{Lemmaeq}(b). Hence
\begin{eqnarray}
\lambda(F,\vec {y}) &=&(x_{t-1}+x_{t})\lambda(C, \vec{x}) > 2x_{t}\cdot 2^{r-1}|E_{(t-1)t}|x_{t-3r+8}\ldots x_{t-2r+6} \nonumber \\
&\ge& x_{t}^2 |E_{(t-1)t}|(x_1)^{2} \ge x_{t}^2 \sum_{i_1\ldots i_{r-2}\in E_{(t-1)t}} x_{i_1}\ldots x_{i_2} = x_{t}^2 \lambda(E_{(t-1)t}, \vec{x}).
\end{eqnarray}
Hence $F$ satisfies (\ref{eq11b}). This proves Lemma \ref{lemmab}.\qed
\begin{lemma} \label{lemmac}
Let $G$ be a left-compressed $r$-graph on the vertex set $[t]$ without containing $[t-1]^{(r)}$, then $|[t-2r+6]^{(r)} \backslash E| \leq 2^{r-1}|E_{(t-1)t}|$ or $\lambda(G) < \lambda([t-1]^{(r)}).$
\end{lemma}
\noindent{\em Proof.}
Let ${\vec x}=(x_1, x_2, \ldots, x_t)$ be an optimal weighting for $G$. Since $G$ is left-compressed, by Remark \ref{r1}(a), $x_1\ge x_2 \ge \cdots \ge x_t \ge 0$. If $x_{t}=0$, then $\lambda(G) < \lambda([t-1]^{(r)})$ since $G$ does not contain $[t-1]^{(r)}.$ So we assume that $x_{t}>0$.
Consider a new weighting for $G$, ${\vec y}=(y_1, y_2, \ldots, y_t)$ given by $y_i=x_i$ for $i\neq t-1, t$, $y_{t-1}=x_{t-1}+x_t$ and $y_t=0$. By Lemma \ref{LemmaTal5}(a), $\lambda(E_{t-1}, \vec{x})=\lambda(E_{t}, \vec{x})$, similar to (4), we have
\begin{eqnarray}\label{eq10c}
\lambda(G,\vec {y})- \lambda(G,\vec {x})=-x_{t}^2\lambda(E_{(t-1)t}, \vec{x}).
\end{eqnarray}
Assume that $|[t-2r+6]^{(r)} \backslash E| > 2^{r-1}|E_{(t-1)t}|$. If $\lambda(G) < \lambda([t-1]^{(r)})$ we are done. Otherwise if $\lambda(G) \geq \lambda([t-1]^{(r)})$ we will show that there exists a set of edges $F\subset [t-2r+6]^{(4)}\setminus E$ satisfying
\begin{equation}\label{eq11c}
\lambda(F,\vec {y})> x_{t}^2\lambda(E_{(t-1)t}, \vec{x}).
\end{equation}
Then using (\ref{eq10c}) and (\ref{eq11c}), the $r$-graph $G^{*}=([t], E^{*})$, where $E^{*}=E\cup F$, satisfies $\lambda(G^{*}, \vec {y})=\lambda(G, \vec {y})+\lambda(F, \vec {y})> \lambda(G, \vec{x})=\lambda(G)$. Since $\vec {y}$ has only $t-1$ positive weights, then $\lambda(G^{*}, \vec {y})\le \lambda([t-1]^{(r)})$, and consequently,
$\lambda(G)<\lambda([t-1]^{(r)}).$ This is a contradiction.
We now construct the set of edges $F$. Let $C=[t-2r+6]^{(r)} \setminus E$. Then by the assumption,
$\vert C\vert > 2^{r-1}|E_{(t-1)t}|$ and
$\lambda(C, \vec{x})\ge 2^{r-1}|E_{(t-1)t}|x_{t-3r+7}\ldots x_{t-2r+6}.$
Let $F=C$. Since $\lambda(G) \geq \lambda([t-1]^{(r)})$ then $x_{t-2r+3}\geq \frac{x_1}{2}$ by Lemma \ref{Lemmaeq}(a) and $x_{t-2r+4}\geq x_{t-2r+5}>\frac{x_1}{4}$ by Lemma \ref{Lemmaeq}(b). Hence
\begin{eqnarray}
\lambda(F,\vec {y}) &=&\lambda(C, \vec{x}) > 2^{r-1}|E_{(t-1)t}|x_{t-3r+7}\ldots x_{t-2r+6} \ge x_{t}^2 |E_{(t-1)t}|(x_1)^{2} \nonumber \\
&\ge& x_{t}^2 \sum_{i_1\ldots i_{r-2}\in E_{(t-1)t}} x_{i_1}\ldots x_{i_{r-2}} = x_{t}^2 \lambda(E_{(t-1)t}, \vec{x}).
\end{eqnarray}
Hence $F$ satisfies (\ref{eq11c}). This proves Lemma \ref{lemmac}.\qed
Now we are ready to prove Theorem \ref{mainresult}.
\noindent{\em Proof of Theorem \ref{mainresult}.}
Let $m$ and $t$ be integers satisfying
${t-1\choose r}\leq m \leq {t-1\choose r}+ {t-2\choose r-1}-[(2r-6)\times2^{r-1}+2^{r-3}+(r-4)(2r-7)-1]({t-2\choose r-2}-1).$ Let $G$ be an $r$-graph with $t$ vertices and $m$ edges without containing a clique of order $t-1$ such that $\lambda(G)=\lambda_{(m, t-1, t)}^{r-}$. Then by Lemma \ref{leftcom}, we can assume that $G$ is left-compressed and does not contain $[t-1]^{(r)}$. Let ${\vec x}=(x_1, x_2, \ldots, x_t)$ be an optimal weighting for $G$. Since $G$ is left-compressed, by Remark \ref{r1}(a), $x_1\ge x_2 \ge \cdots \ge x_t \ge 0$. If $x_{t}=0$, then $\lambda(G) < \lambda([t-1]^{(r)})$ since $G$ does not contain $[t-1]^{(r)}$. So we assume that $x_{t}>0$.
If $\lambda(G) < \lambda([t-1]^{(r)})$ we are done. Otherwise $|[t-2r+6]^{(r-1)} \backslash E_{t-1}| \leq 2^{r-1}|E_{(t-1)t}|$ by Lemma \ref{lemmab}. Recalling that $G$ is left-compressed, we have $|[t-2r+6]^{(r-1)} \backslash E_{i}| \leq 2^{r-1}|E_{(t-1)t}|$ for $t-2r+7\leq i\leq t-1.$ We also have $|[t-2r+6]^{(4)} \backslash E| \leq 2^{r-1}|E_{(t-1)t}|$ by Lemma \ref{lemmac}. Note that $|E_{(t-1)t}|\leq{t-2\choose r-2}-1$, then
\begin{eqnarray}
|[t-1]^{(r)}\bigcap E|&\geq &|[t-2r+6]^{(r)} \bigcap E|+\sum\limits_{i=t-2r+7}^{t-1}|[t-2r+6]^{(r-1)} \bigcap E_{i}|\nonumber\\
&\geq & {t-2r+6 \choose r}-2^{r-1}|E_{(t-1)t}|+(2r-7)({t-2r+6 \choose r-1}-(2r-7)\times 2^{r-1}|E_{(t-1)t}|)\nonumber\\
&\geq & {t-2r+6 \choose r}+(2r-7){t-2r+6 \choose r-1}-(2r-6)\times2^{r-1}({t-2\choose r-2}-1).
\end{eqnarray}
Repeated using the equality ${m+1\choose n}={m\choose n}+{m\choose n-1}$ to the above inequality, we have
$$|[t-1]^{(r)}\bigcap E| \geq {t-1 \choose r}-[(2r-6)\times2^{r-1}+(r-4)(2r-7)]({t-2\choose r-2}-1).$$
So
$$0<|[t-1]^{(r)}\backslash E|\leq [(2r-6)\times2^{r-1}+(r-4)(2r-7)]({t-2\choose r-2}-1).$$
Since $G$ does not contain $[t-1]^{(r)}$.
Let $E^*=E\bigcup [t-1]^{(r)}$ and $G^*=([t], E^*)$. Denote the number of edges of $G^*$ by $m^*$, then ${t-1\choose r}\leq m^* \leq {t-1\choose r}+ {t-2\choose r-1}-2^{r-3}({t-2\choose r-2}-1).$ So $\lambda(G^*)=\lambda([t-1]^{(r)})$ by Theorem \ref{TheoremPTZ}.
Clearly, $\lambda(G^*,\vec{x})-\lambda(G,\vec{x})>0$ since $x_1\ge x_2 \ge \cdots \ge x_t > 0$ and $|[t-1]^{(r)}\backslash E|>0$. Hence $\lambda(G)=\lambda(G,\vec{x})<\lambda(G^*,\vec{x})\leq \lambda(G^*)=\lambda([t-1]^{(r)}).$ This completes the proof of Theorem \ref{mainresult}.\qed
\section{Remarks and conclusions}\label{section3}
We remark that, in the proof of Theorem \ref{Talr}, we see that $\gamma_r=2^{2^r}$ and $t\ge \kappa_0(r)$, where $\kappa_0(r)$ is a sufficiently large integer such that ${t-2 \choose r-1} \geq\gamma_r (t-1)^{r-2}=2^{2^r}(t-1)^{r-2}$ for $t\ge \kappa_0(r)$. In Corollary \ref{improvement}, we improve the upper bound of $m$ from ${t-1 \choose r} + {t-2 \choose r-1} - \gamma_r (t-1)^{r-2}$ to ${t-1\choose r}+ {t-2\choose r-1}-[(2r-6)\times2^{r-1}+2^{r-3}+(r-4)(2r-7)-1]({t-2\choose r-2}-1).$ Correspondingly, we improve the condition on $t$ from ${t-2 \choose r-1}\geq2^{2^r}(t-1)^{r-2}$ to ${t-2 \choose r-1}\geq[(2r-6)\times2^{r-1}+2^{r-3}+(r-4)(2r-7)-1]({t-2\choose r-2}-1).$
The method developed in the proof of Theorem \ref{mainresult} can also be used to deal with the case for $r=3$ ( see \cite{TPZZ} ). A natural question in the future study is how to prove similar results as Theorem \ref{mainresult} and Corollary \ref{improvement} without the restriction of the number of vertices. This will be considered in the future work.
\noindent{\bf Acknowledgments} This research is partially supported by National Natural Science Foundation of China (Grant No.61304021
).
|
1,314,259,993,573 | arxiv | \section{Introduction}
\label{prva}
A close-packed dimer model is a classical example of a lattice statistical mechanical model that can be solved exactly on regular planar lattices. It has emerged as a simplified version of the so called monomer-dimer model, that was introduced in 1937 by Fowler and Rushbroke \cite{Fowler} in their study of the liquid mixtures. Exact solution of the close-packed dimer model on the square lattice was given in the $60$-ies by Kasteleyn \cite{Kasteleyn1,Kasteleyn2}, and independently by Temperly and Fisher \cite{Temperley,Temperley1} and Fisher \cite{Fisher}. More recent solution on arbitrary planar bipartite graphs has been given by Kenyon et al \cite{Kenyon}. Although introduced as a simple model for adsorption of diatomic molecules on crystal surfaces, connections with other models in physics and chemistry have been established since then. It has been shown that the close-packed dimer model is equivalent to the two dimensional Ising model \cite{Fisher2}, and correspondence with many quantum theoretical field theory models has been recognized \cite{Nienhuis, Kondev, Iqbal, Jacobsen, Jacobsen1}. In graph theory the model is also referred to as perfect matchings and is closely related with other combinatorial objects such as domino tillings \cite{ Cohn} and spaning trees \cite{ Kenyon2,Temperley2}.
Besides the square lattice, the close-packed dimer model has also been studied on other translationally invariant lattices \cite{Elser,Fendley, Wu}, on some graphs without translational symmetry, such as self-similar graphs \cite{Harris, Dangeli}, and particularly on the Sierpinski gasket lattice and its generalizations \cite{Chang}.
In this paper we analyze close-packed dimer model on the subset of planar fractal lattices that belong to the family of the modified rectangular (MR) lattice introduced by Dhar \cite{Dhar}. We also consider the model on 4-simplex lattice which is a fractal lattice embedded in three dimensional space and whose graph is consequently non-planar. Fractal lattices are constructed iteratively, which makes them suitable for exact recursive treatment if the ramification number is low. Recursive enumerative method on these lattices can provide exact solutions for the close-packed dimer model, in addition to solutions obtained on translationally invariant planar lattices, which proved to be analytically tractable in that case. However, there are no exact solutions of the close-packed dimer model on three dimensional translationally invariant lattices, and result obtained on $4$-simplex fractal lattice in this paper, together with the results on other fractal lattices embedded in three dimensional space \cite{Dhar2, Chang}, can give valuable insights into the problem essence.
The paper is organized as follows. In section~\ref{druga} we shortly describe the closed-packed dimer model and fractal lattices relevant for this paper. In section~\ref{treca} we develop recursive method for the enumeration of all dimer configurations on MR family for scaling parameter $2\leq p\leq8$ and analyze the model to obtain the entropy. In the same section, we also apply the method on 4-simplex lattice. Discussion of the results and comparison with other lattices are given in section~\ref{cetvrta}.
\section{The model and lattices}
\label{druga}
We assume that dimer is a diatomic molecule, i.e. two monomer units bonded chemically. On a lattice it covers two adjacent lattice points. In close-packed dimer model each lattice site is occupied exactly once by a monomer, and each monomer is connected by a lattice bond with an adjacent monomer into a dimer. Neglecting any other interactions besides hard-core repulsion, partition function of this model is simply the total number of dimer configurations, whose logarithm determines the entropy.
\begin{figure
\begin{center}
\includegraphics[scale=0.95]{figure1.eps}
\end{center}
\caption{(a) Generator of order $r=5$ for MR lattice with $p=2$. Red dashed rectangles highlight generators of order $r=2$, $r=3$, and $r=4$ obtained in the subsequent stages of construction. (b) Generator of order $r=3$ for $p=3$ MR fractal, with smaller sub-generator (of order $r=2$) highlighted by red dashed rectangle. In both cases generator of order $r=1$ (initiator) is a unit square.}
\label{fig1}
\end{figure}
Lattices under consideration are MR family of fractals and $4$-simplex lattice.
Fractals from MR family are labeled with the scaling parameter $p$ (an integer, $2\leq p\leq \infty$). In iterative constructive procedure, structure obtained in the construction step $r$ is called $r$-th order generator and denoted by $G_r$. For each particular $p$, at the first step of construction ($r=1$) one has a graph consisting of four points forming a unit square. Then, $p$ unit squares are joined into the rectangle to obtain the generator of the second order. In the next step, $p$ rectangles are joined into the square, and the process should be repeated infinitely many times to obtain fractal lattice. In figure~\ref{fig1}$(a)$ generator of order $r=5$ for $p=2$ MR lattice is shown, while in figure~\ref{fig1}$(b)$ generator of order $r=3$ for $p=3$ member of MR family is shown. Generator of order $r$ for each fractal contains $N_{r}=4p^{\,r-1}$ lattice sites (it is also the number of monomers - twice the number of dimers, because of close-packing) and $N_{br}=\frac{3}{2}N_r-2=\frac{6}{p}p^{\,r}-2$ lattice bonds (edges). Fractal dimension is $d_f=\frac{\ln p^2}{\ln p}=2$ for each fractal from the family.
\begin{figure
\begin{center}
\includegraphics[scale=0.7]{figure2.eps}
\end{center}
\caption{First two steps of the iterative construction of $4$-simplex lattice. Fractal lattice is obtained in the limit $r\rightarrow\infty$.}
\label{fig2}
\end{figure}
Initiator of $4$-simplex lattice graph is a complete graph of four points. To obtain second order generator, four initiators are joined into two times larger structure, as shown in figure~\ref{fig2}, and the process should be repeated ad infinitum to obtain graph of $4$-simplex lattice. The number of lattice points in the $r$-th order generator is $N_r=4^r$, whereas the number of lattice bonds is $N_{br}=\frac{4}{2}N_r-2=2\cdot4^r-2$. Fractal dimension of the lattice is $d_f=\frac{\ln 4}{\ln 2}=2$.
It should be emphasized that for all considered lattices the number of lattice points in generators of any order is even, which is a necessary condition for a lattice to have close-packed dimer covering.
\section{Recursive enumerative method for dimer coverings on fractal lattices \label{treca}}
In this section we will develop the method for recursive enumeration of dimer configurations on aforementioned fractal lattices. Firstly, we establish the exact set of recurrent equations on MR lattices with $2\leq p\leq 8$ and analyze equations in order to determine the asymptotic form for the numbers of dimer coverings. Also, we numerically find the corresponding entropies per dimer in the thermodynamic limit. Although for some lattice models it was possible to find exact set of recurrence equations on the whole MR family \cite{dusa}, in this case we were not able to do so for the reason that would be explained in the appendix~A. Secondly, we apply the same method on $4$-simplex lattice and determine the entropy.
\subsection{Dimer coverings on MR fractals \label{dimeriMR}}
One close-packed dimer configuration on the $5$-th order generator of $p=2$ MR lattice is shown in figure~\ref{fig3}. In order to develop recurrence equation for the number of dimer coverings on MR lattice, we focus on the corner monomers of smaller generators, that is $G_4$, $G_3$, $G_2$, and $G_1$, as sub-generators of $G_5$ depicted in figure~\ref{fig3}. One can notice that the corner monomers of generators of any order form dimers either by the monomers on the same generator or the neighboring ones. We designate the corner monomers as black if their partner is on the same generator, and white if it is on the neighboring one. All generators have four corner monomers and, due to the parity, the only possible combinations are to have all four black, two black and two white, and all four white. Configurations with two black and two white along the diagonals are not possible on MR lattices of any $p$.
\begin{figure
\begin{center}
\includegraphics[scale=0.7]{figure3.eps}
\end{center}
\caption{One close-packed dimer configuration on generator $G_5$ of $p=2$ MR lattice. Dimer configurations on some sub-generators (rectangles or squares) are enclosed into blue dashed rectangles (squares), outlined on the sides and labeled as $f$, $g$, $h$ or $k$, depending on the type of the configuration.}
\label{fig3}
\end{figure}
According to the pairing of the corner monomers, for MR lattices of arbitrary $p$, we introduce four types of configurations on generators of any order $r$, namely $f$, $g$, $h$ and $k$ with the following meaning:
\begin{itemize}
\item $f$ - denotes each dimer configuration in which all four corner monomers are black. They form dimers with the 'internal' monomers, i.e. monomers on the same generator,
\item $g$ - denotes each dimer configuration in which two corner monomers are black and belong to the \textbf{different} \textbf{ sub-generators} of order $(r-1)$. These two black corner monomers form dimers with the monomers on the same $G_r$, while the other two white corner monomers form dimers with the monomers on the two neighboring $G_r$,
\item $h$ - denotes each dimer configuration in which two corner monomers are black and belong to the \textbf{same} \textbf{ sub-generator} of order $(r-1)$. As in type $g$, these two black corner monomers form dimers with the monomers on the same $G_r$, while the other two white corner monomers form dimers with the monomers on the two neighboring $G_r$,
\item $k$ - denotes each dimer configuration in which all four corner monomers are white. They form dimers with the 'external' monomers, i.e. monomers on the neighboring generators.
\end{itemize}
\begin{figure
\begin{center}
\includegraphics[scale=0.75]{figure4.eps}
\end{center}
\caption{ All types of dimer configurations on arbitrary generator $G_{r+1}$ of $p=2$ MR fractal, denoted as $f$, $g$, $h$ and $k$, and their composing configurations on generators $G_{r}$, from which recurrence equations (\ref{re1}) stem. }
\label{fig4}
\end{figure}
In figure~\ref{fig3}, all four types of configurations are enframed and schematically represented on the sides. In this schematic representation only corner monomers are shown with the internal structure of generators omitted, except that the two consecutive sub-generators are indicated by the dashed lines in order to easily distinguish between $g$ and $h$ configurations.
The numbers of configurations of each type on the $r$-th order generator are designated as $f_r$, $g_r$, $h_r$ and $k_r$. The total number of dimer configurations on $G_r$ is given by $f_r$, and can be determined through the system of recurrence equations that involve all four configurations. The closed system of the recurrence equations for $p=2$ is given as
\begin{eqnarray}
f_{r+1}&=&f_r^2+g_r^2\, , \nonumber\\
g_{r+1}&=&h_r^2\, , \nonumber \\
h_{r+1}&=&f_r g_r+ g_r k_r\, , \nonumber\\
k_{r+1}&=&g_r^2+k_r^2\, , \label{re1}
\end{eqnarray}
and can be inferred on the basis of figure~\ref{fig4}, where we illustrate how each configuration on the $(r+1)$-th order generator can be composed from the configurations on the two constituent $r$-th order generators.
Similarly, with the help of figure~\ref{fig5}, one can formulate recurrence equations for $p=3$ as
\begin{eqnarray}
f_{r+1}&=&f_r^3+2f_r g_r^2+g_r^2k_r\, , \nonumber\\
g_{r+1}&=&h_r^3\, , \nonumber \\
h_{r+1}&=&f_r^2 g_r+f_r g_r k_r+g_r^3+g_r k_r^2\, , \nonumber\\
k_{r+1}&=&f_r g_r^2+2g_r^2k_r+k_r^3\, . \label{re2}
\end{eqnarray}
Recurrence equations on fractals with $4\leq p\leq8$ are given in appendix~A. The initial conditions of these equations are the numbers of configurations on the unit square, and, for each $p$ they are given by: $f_1=2$ (both two possible), $g_1=1$ (only one of the two possible), $h_1=1$ (only one of the two possible) and $k_1=1$ (the only possible). By computer iteration of recursion relations~(\ref{re1}), and similarly ~(\ref{re2}), starting with the initial conditions, it is possible to obtain exact numbers of dimer configurations on generators of arbitrary order.
\begin{figure
\begin{center}
\includegraphics[scale=0.95]{figure5.eps}
\end{center}
\caption{Configurations $f$, $g$, $h$, and $k$ on generator $G_{r+1}$ of $p=3$ member of MR fractals and their composing parts on generators $G_r$. }
\label{fig5}
\end{figure}
These numbers grow very fast with the order $r$ of generator, and for illustration, in table~\ref{tab1} we give the numbers of close-packed dimer configurations on generators of order from $r=1$ to $r=5$ for MR fractals with $p=2$ and $p=3$.
\begin{table}
\caption{The numbers of closed-packed dimers on the first five generators for MR fractals labeled by $p=2$ and $p=3$.}
\centering
\label{tab1}
\vspace{2mm}
\begin{tabular}{|c|ccccc|}\hline
&$r=1$& $r=2$ & $r=3$ & $r=4$&$r=5$ \\ \hline\hline
$p=2$& 2 & 5& 26&757&575450\\
$p=3$&2 & 13& 2228&12266667328&1845787045627790291334622871552\\ \hline
\end{tabular}
\end{table}
Since the systems of difference equations given by~(\ref{re1}) and (\ref{re2}) are not solvable exactly, it is not possible to find exact expressions for $f_r$ as functions of $r$. Therefore, we analyze relations numerically and find asymptotic solutions. To make numerical analysis more tractable, we introduce new, rescaled variables defined as $x_r=g_r/f_r$, $y_r= h_r/f_r$ and $z_r= k_r/f_r$, whose recurrence equations can be obtained from their definitions and the equations~(\ref{re1}) and (\ref{re2}). New equations are
\begin{eqnarray}
f_{r+1}&=&f_r^2\left(1+x_r^2\right)\, , \nonumber\\
x_{r+1}&=&\frac{y_r^2}{1+x_r^2}\, , \nonumber \\
y_{r+1}&=&\frac{x_r\left(1+z_r\right)}{1+x_r^2}\, , \nonumber\\
y_{r+1}&=&\frac{x_r^2+z_r^2}{1+x_r^2}\, , \label{re1x}
\end{eqnarray}
for $p=2$, and
\begin{eqnarray}
f_{r+1}&=&f_r^3\left(1+2x_r^2+x_r^2z_r\right)\, , \nonumber\\
x_{r+1}&=&\frac{y_r^3}{1+2x_r^2+x_r^2z_r}\, , \nonumber \\
y_{r+1}&=&\frac{x_r\left(1+z_r+x_r^2+z_r^2\right)}{1+2x_r^2+x_r^2z_r}\, , \nonumber\\
y_{r+1}&=&\frac{x_r^2+2x_r^2z_r+z_r^3}{1+2x_r^2+x_r^2z_r}\, , \label{re2x}
\end{eqnarray}
for $p=3$.
Initial values of new variables are $x_1=y_1=z_1=\frac{1}{2}$.
For arbitrary $p$, the recurrence equation for the number $f_r$ of close-packed dimers as a function of rescaled variables can be written as
\begin{equation}\label{fpx}
f_{r+1}=f_r^p\left(1+\sum_{i}a_ix_i^{\alpha_i}z_i^{\beta_i}\right)\, ,
\end{equation}
where the coefficients $a_i$ and the exponents $\alpha_i$ and $\beta_i$ depend on $p$. First equation in systems~(\ref{re1x}) and (\ref{re2x}) are indeed of the given form. Iterating sequences $x$, $y$ and $z$, we find that for all considered $2\leq p\leq8$, elements $x_r$, $y_r$ and $z_r$ tend to zero very quickly with each iteration step $r$, implying that the equation~(\ref{fpx}) for $r\gg1$ has the form
\begin{equation}\label{fpas}
f_{r+1}\sim f_r^p\, .
\end{equation}
This further implies that $f_r$ asymptotically grows exponentially with $p^{\,r}$ i.e. $f_r\sim[const]^{p^{\,r}}$. Since the number of monomers in $G_r$ is given by the $N_r=\frac{4}{p}p^{\,r}$ it follows that $f_r$ exponentially grows with the number of monomers
\begin{equation}\label{fpas1}
f_{r}\sim\omega^{N_r}\, ,
\end{equation}
\begin{table}
\caption{Entropies per dimer $s_d$ of close-packed dimer model on MR fractals with $2\leq p\leq 8$. The last digits are rounded off.}
\centering
\label{tab2}
\vspace{2mm}
\begin{tabular}{c|ccccccc}
\hline\hline
${ p}$ &2 & 3 & 4 &5 \\ \hline
$s_d$ & 0.414750739 & 0.430188671 & 0.441389262 & 0.449006803\\ \hline\hline
${ p}$ &6 & 7 & 8 & - \\ \hline
$s_d$& 0.454290896 & 0.458114819 & 0.460997823 &-\\ \hline\hline
\end{tabular}
\end{table}
for $r\gg1$. Growth constant $\omega$ is defined through the relation $\ln\omega=\lim_{N_r\to\infty}\frac{\ln f_r}{N_r}$. To determine growth constant, we take logarithm of the equation (\ref{fpx}) and divide obtained equation with $N_{r+1}=4p^{\,r}$, after which we obtain
\begin{equation}\label{nizs}
\frac{\ln f_{r+1}}{N_{r+1}}=\frac{\ln f_r}{N_r}+\frac{1}{4p^{\,r}}\ln\left(1+\sum_{i}a_ix_i^{\alpha_i}z_i^{\beta_i}\right)\, .
\end{equation}
\begin{figure
\begin{center}
\includegraphics[scale=0.4]{figure6.eps}
\end{center}
\caption{Entropies per dimer of the close-packed dimer model on MR lattices with $2\leq p\leq8$ as functions of $1/p$. }
\label{fig7}
\end{figure}
This equation recursively defines sequence of numbers with the elements given by $s_r=\frac{\ln f_r}{N_r}$, so that $s_m=\lim_{r\to\infty} s_r=\ln\omega$ holds. For each $p$, the sequence converges very fast, so that for example for $p=3$, after only nine iterations more than twenty significant figures can be achieved, and for higher $p$ convergence is even faster. By numerical iteration, value of $s_m$ is determined for each $2\leq p\leq 8$. Limiting values $s_m$ are actually the entropies per monomer in the thermodynamic limit, as can be seen from the definition of the entropy through the number of dimer configurations $S=k_B\ln f_r$ and equation~(\ref{fpas1}). Setting $k_B=1$, it follows that $\lim_{N_r\to\infty} S/N_r=\ln \omega=s_m$. The number of dimers is just half of the number of monomers $N_r$, so that the entropy per dimer $s_d$ is twice the entropy per monomer. In table~\ref{tab2} we present entropies per dimer in the thermodynamic limit, calculated numerically from the numbers of dimer configurations on MR fractals with $2\leq p\leq 8$. Entropies $s_d$ as functions of $1/p$
are also presented graphically in figure~\ref{fig7}.
\subsection{Dimer coverings on $4$-simplex lattice \label{dimeri4sim}}
\begin{figure
\begin{center}
\includegraphics[scale=0.7]{figure7.eps}
\end{center}
\caption{Close-packed dimer configuration on the generator $G_3$ of $4$-simplex lattice. Possible types of configurations on sub-generators are enframed and schematically shown on the sides. }
\label{fig8}
\end{figure}
Recursive enumeration of dimer configurations on $4$-simplex lattice can be done in a similar manner as on MR lattices. In figure~\ref{fig8} one close-packed dimer configuration on the third order generator of $4$-simplex lattice is presented.
Again, corner monomers can be all four black - corresponding to $f$ configuration, two black and two white - corresponding to $g$ configuration and all four white - corresponding to $h$ configuration. All four corner vertices of $4$-simplex lattice are permutationally equivalent, so it is irrelevant which two monomers are black (white). Four constitutive sub-generators of $G_3$ in figure~\ref{fig8} are enframed, with the configurations classified according to the type they belong to, and schematically represented on the sides. Also, one configuration of type $h$ on $G_1$ is enframed, and schematically represented on the side. Recurrence equations for all three configurations, as illustrated in figure~\ref{fig9}, are given as
\begin{eqnarray}
f_{r+1}&=&f_r^4+4f_r g_r^3+3g_r^4\, , \nonumber\\
g_{r+1}&=&f_r^2g_r^2+2f_r g_r^3+g_r^2h_r^2+2g_r^3h_r+2g_r^4\, , \nonumber \\
h_{r+1}&=&h_r^4 + 4g_r^3h_r+3g_r^4\, , \label{resim}
\end{eqnarray}
with the initial conditions $f_1=3$ (all three possible), $g_1=1$ (one of six possible) and $h_1=1$ (the only possible).
\begin{figure
\begin{center}
\includegraphics[scale=0.95]{figure8.eps}
\end{center}
\caption{Configurations $f$, $g$ and $h$ on generator $G_{r+1}$ of $4$-simplex lattice and their composing parts on generators $G_r$, which determine corresponding terms in equations~(\ref{resim}). Multiplication by the factors $4$ or $2$ are due to the symmetrically related configurations. }
\label{fig9}
\end{figure}
Analysis of recurrence equations~(\ref{resim}) proceeds in a similar way as on MR lattices, and we find that the number of all close-packed dimer configurations grows with the number of monomers as
\begin{equation}\label{fsim}
f_{r}\sim\omega^{N_r}\, ,
\end{equation}
with the entropy per dimer $s_d=2\ln\omega=0.571832556 $.
\section{Discussion and conclusions} \label{cetvrta}
We have studied the close-packed dimer model on two types of fractal lattices, namely, lattices from MR family with the scaling parameter $2\leq p \leq 8$ embedded in two dimensional space, and $4$-simplex lattice embedded in three dimensional space. The asymptotic forms for the number of dimer configurations on these lattices have been determined. It is found that, on all considered fractals, the asymptotic form for number of dimer configurations is a pure exponential function of the number of monomers. In addition, microcanonical entropies per dimer in the thermodynamic limit are determined numerically. On MR lattices, entropy is an increasing function of fractal parameter $p$, as can be seen in table~\ref{tab2}. This deserves some insight into the geometry of the lattices. All lattices from this family are constructed iteratively through succession of generators, and all have the same fractal dimension. The ramification number of each lattice is two, and for all latices coordination number of each vertex is three (vertex degree), except for the corner vertices of the largest generator. But, the number of bonds and their distribution are different, and lattices with higher value of $p$ at each construction stage have larger number of bonds (caused by the larger number of vertices), so the configurational space is larger. In figure~\ref{fig7} entropies are shown as functions of inverse scaling parameter $p$, and an obvious question arises: is this sequence convergent when $p\rightarrow\infty$, and if so, what is the limit? We expect that there is a finite limiting value, and suppose that it is smaller than the value of entropy $s_{sq}=0.583121808$ obtained on the square lattice in \cite{Kasteleyn1}. This conclusion is justified by the fact that all MR lattices resemble to the square lattice since they can be obtained from it by deleting some bonds. Consequently, MR lattices have smaller number of bonds than the square lattice of equal size.
On the other hand, entropy obtained on $4$-simplex lattice is $s_{sim}=0.571832556$, and compared with the values in table~\ref{tab2}, we see that it is larger than any value on MR fractals considered. Coordination number of $4$-simplex lattice is four and thus greater than for MR lattices, but for MR fractals with $p\geq5$ at $r$-th construction stage there are more bonds than on $4$-simplex lattice at the same stage. However, MR lattices are highly anisotropic and their connectedness does not allow some configurations (in subsection~\ref{dimeriMR} configurations with two black and two white along the square (rectangle) diagonals were forbidden), resulting in smaller entropy.
Value $s_{sim}$ obtained here, should also be compared with the value of entropy of the same model studied on the Sierpinski gasket (SG) embedded in three dimensional space\cite{Chang}, which is $s_{3dSG}=0.857927798$, thus much larger than $s_{sim}$. These two lattices both have tetrahedral structure, but vertices of the neighboring tetrahedra are glued on $3d$ SG, so that the coordination number of this lattice is six.
Furthermore, entropy per dimer is larger on square lattice than on $4$-simplex lattice, although they both have the same coordination number.
To conclude, we can confirm that besides the coordination number, which is the most relevant lattice parameter that determines the number of close-packed dimer configurations, other lattice parameters and geometric constraints are important too. In favor of latter is the finding that on translationally invariant lattices the boundary effects play an important role. For example, on square lattice the entropy is the same for both open and periodic boundary conditions \cite{Kasteleyn1, Fisher}, whereas on hexagonal lattice it strongly depends on the boundary \cite{Grensing, Elser}. Extensions of the studies to account for cylindrical boundary conditions have also been done \cite{Yan, Li}. Similar problem has been encountered on other 'close-packed' models, namely Hamiltonian walk problem, where entropic exponent $\gamma$ depended on the boundary conditions\cite{Duplantier}. With respect to all these unresolved questions, we can say that additional studies should be conducted in order to specify all relevant parameters on fractal lattices as well as on translationally invariant ones, taking into account all metric properties of lattices.
Finally, we would like to mention that the dimer model considered here could be supplemented with the interaction weights and studied on fractal lattices. Interacting dimer model is extensive and physically more interesting, but also more difficult to approach.
\ack{This paper has been done as a part of the
work within the project No. 171015, funded by the Ministry of Education, Science and Technological Development of the Republic of Serbia.
}
|
1,314,259,993,574 | arxiv | \section{\label{sec1}Introduction}
The unique properties of the Majorana fermion behaving as its
own anti-particle and the existence of particle and
anti-particle in the same system are the subjects of intense
research interest in various branches of physics,
especially in quantum many-body systems since its proposal
by E. Majorana~\cite{majo1,wil,berni}. The recent studies
show that the concept and existence of Majorana, a zero energy
mode, like quasi-particles is very common in many branches of
physics. Interestingly, it appears as an emergent particle in
various condensed matter systems, such as Bogoliubov
quasi-particle in a one-dimensional
superconductor~\cite{ivanov, kraus, wimmer, sato}, semiconductor
quantum wire~\cite{stanescu2013, mourik, deng, rokin, das, finc, jafari}, proximity
induced topological
superconductor~\cite{kitaev, fu1, fu2, ali1, ali, sau, potter, ali2, fukui, sarkar3},
and the cold atoms trapped in one-dimension~\cite{zhang, jiang}.
Other than the exotic physics of this topological state, the absence
of decoherence in Majorana fermionic system makes it a prospective candidate
for application in the non-abelian quantum
computation~\cite{nayak, kitaev, sau2, ali3}.
The real world applications of these fermionic systems depend on
the stability of the Majorana fermions. A single spin polarized
fermion band system with spin orbit coupling and proximity induced
superconductor shows Majorana like modes in the presence of weak
fermionic interaction. However, it was shown that the interaction
weakens the stability of the Majorana fermion~\cite{suhas}.
Potter and Lee~\cite{potter} showed that the $p+ip$ superconductor
possesses localized Majorana particles in a rectangular system with
width less than the coherence length of the superconductor.
The helical liquid system is another candidate where the Majorana
like quasi-particles can exist. The helical liquid system generally
originates because of the quantum spin Hall effect in a system with
or without Landau levels. In this system, a coupling of the left
moving down spin with the right moving up spin at the edge of two
dimensional quantum hall systems gives rise to a quantized transport
process. In this phase, the spin and the momentum degrees of freedom are
coupled together without breaking the time reversal symmetry.
Various aspects of helical spin liquid is discussed in~\cite{sela,sarkar2,qi,qi2,dspms}.
The field theoretical calculation by Sela et al.~\cite{sela} shows
that the Majorana bound state in a helical liquid possesses a higher
degree of stability. In presence of the interaction, the scattering
processes between the two constituent fermion bands in the helical
liquid system stabilizes the Majorana bound state by opening a
gap~\cite{suhas,ali2,sela,sarkar2}. However, the strong interaction
may induce decoherence in the Majorana modes. They also considered
a highly anisotropic spin model with transverse and longitudinal
fields, and the system shows Majorana to Ising transition
(MI)~\cite{sela,sarkar2,dspms}. One of our coauthors showed using
RG calculation that a transition from a phase with Majorana edge
modes to the Ising phase exists in the helical liquid system for both
presence and absence of interaction~\cite{sarkar2}. However, a
systematic and accurate calculation of the phase boundary of the
MI quantum phase transition is still absent in the literature.
In this paper, we study the existence of Majorana fermion
modes in the interacting helical liquid system, and also propose
different criteria to characterize the MI quantum phase transition.
We will also analyze the entanglement spectrum (ES) to characterize
the topological aspects of the Majorana edge modes.
Sela et al.~\cite{sela} introduced a helical fermionic system, which
can be written in the field theoretical representation as
\begin{eqnarray}
H&=& H_0+\delta H+ H_{fw}+ H_{um},
\label{eq:h0}
\end{eqnarray}
where $H_0$ and $\delta H$ terms include the kinetic energy, single
potential energy, external magnetic field, and proximity induced
energy terms. $H_{fw}$ and $H_{um}$ represent the forward scattering
and the umklapp scattering terms respectively. The $H_0$ and $\delta H$
terms can be written as
\begin{eqnarray}
H_0&=&\int dx \left[\psi_{L \downarrow}^{\dagger}
(v_F i \partial_x - \mu)\psi_{L \downarrow} \right. \nonumber \\
& & \qquad + \left. \psi_{R \uparrow}^{\dagger}
(-v_F i \partial_x - \mu)\psi_{R \uparrow} \right],\nonumber \\
\delta H &=& \int dx \left[ B \psi_{L \downarrow}^{\dagger}\psi_{R \uparrow}
+ \Delta {\psi_{L \downarrow}} {\psi_{R \uparrow}} + h.c.\right],
\label{eq:h0p}
\end{eqnarray}
where $ \Psi $ are field operators, $ v_F $ and $ \mu $ are the
Fermi-velocity and chemical potential of the helical liquid. The
system is coupled to the magnetic field $ B $, and proximity
coupled to $ s $-wave superconducting gap $ \Delta $ which is
shown by additional terms in the Hamiltonian.
Both the scattering terms are given as
\begin{eqnarray}
\label{eq:scatter}
H_{fw} &=& \int dx \left[ g_2 \psi_{L \downarrow}^{\dagger} \psi_{L \downarrow}
\psi_{R \uparrow}^{\dagger} \psi_{R \uparrow} \right. \nonumber \\
& & + \left. \frac{g_4}{2} \left\{ \left( \psi_{L \downarrow}^{\dagger}
\psi_{L \downarrow}\right)^2 + \left( \psi_{R \uparrow}^{\dagger}
\psi_{R \uparrow}\right)^2 \right\} \right], \nonumber \\
H_{um} &=& g_u \int dx \left[ \psi_{L \downarrow}^{\dagger} \partial_x \psi_{L \downarrow}^{\dagger}
\psi_{R \uparrow} \partial_x \psi_{R \uparrow} + h.c.\right].
\end{eqnarray}
The conventional analytical expression for the umklapp scattering
term $H_{um}$ for the half filling~\cite{wu,sela,sarkar2,dspms}
is written in \eref{eq:scatter}. This analytical expression gives a
regularized theory using the lattice constant $a$ as an ultraviolet cut-off.
This complete Hamiltonian $ H + \delta H + H_{fw} + H_{um} $ can
be mapped to a spin-$1/2$ $XYZ$ model
Hamiltonian~\cite{sela, sarkar2, dspms} and can be written
as~\cite{sela}
\begin{equation}
H_i = \sum_{i} J^{\alpha} S_i^{\alpha} S_{i+1}^{\alpha}
- \left[ \mu + B (-1)^i \right] S_i^z ,
\label{sela:ham}
\end{equation}
where $\alpha\equiv x,y,z$ and $ J^\alpha $ are different
components of spin exchange interaction between neighboring
spins. $ \mu $ and $ B $ are longitudinal normal and staggered
magnetization applied externally. Here, we assume $J^x=J+\Delta$
and $ J^y=J-\Delta $ where $ \Delta $ is the superconducting gap.
\Eref{sela:ham} can be rewritten in terms of new variables
as
\begin{eqnarray}
\label{eq:spin_ham}
H &=& \frac{J}{2} \sum_{i} \left( S_{i}^{+} S_{i+1}^{-} + S_{i}^{-} S_{i+1}^{+}\right) +
J^{z} \sum_{i} S_{i}^{z} S_{i+1}^{z} \nonumber \\
& &+ \frac{\Delta}{2} \sum_{i} \left( S_{i}^{+} S_{i+1}^{+} +S_{i}^{-} S_{i+1}^{-}\right) -
\sum_{i} \left[ \mu + B(-1)^i \right] S_{i}^{z} . \nonumber \\
\end{eqnarray}
To have a better understanding of the Majorana modes, we map
this spin system to the spinless fermion model using Jordan-Wigner
transformation~\cite{fradkin} of \eref{eq:spin_ham}, and
it can be written in terms of spinless fermion as~\cite{katsura1}
\begin{eqnarray}
\label{eq:fermion_ham}
H &=& \frac{-J}{2} \sum_{i} \left( c_{i}^{\dagger} c_{i+1} + h.c.\right) \nonumber \\
& & + J^{z} \sum_{i} \left( c_{i}^{\dagger}c_i -\frac{1}{2}\right)
\left( c_{i+1}^{\dagger}c_{i+1} -\frac{1}{2}\right) \nonumber \\
& & + \frac{\Delta}{2} \sum_{i} \left( c_{i+1}^{\dagger} c_{i}^{\dagger} + h.c.\right) \nonumber \\
& &- \sum_{i} \left[ \mu + B(-1)^i \right] \left( c_{i}^{\dagger}c_i -\frac{1}{2}\right) .
\end{eqnarray}
For the sake of completeness, let us try to understand the results
in the limiting cases. This model is well studied in the limit of
$B=0$ and $ J^z=0 $, and \eref{eq:fermion_ham} reduces to
1D Kitaev model~\cite{kitaev,katsura2} which is given by
\begin{eqnarray}
\label{eq:kitaev_ham}
H &=& -\frac{J}{2} \sum_{i} \left( c_{i}^{\dagger} c_{i+1} + h.c.\right) \nonumber \\
& &+\frac{\Delta}{2} \sum_{i} \left( c_{i+1}^{\dagger} c_{i}^{\dagger} + h.c.\right) -
\mu \sum_{i} \left( c_{i}^{\dagger}c_i -\frac{1}{2}\right) .
\end{eqnarray}
Now let us consider a transformation $ {c_j} = \frac{1}{2} ( a_{2j-1} + i a_{2j} )$
and
$ c_j^{\dagger} = \frac{1}{2} ( a_{2j-1} - i a_{2j} )$. Here, $ a_j^{\dagger} $
and $ a_j $ are the creation and annihilation operators of $j^{\rm{th}}$ Majorana fermion.
We can write~\eref{eq:kitaev_ham} as
\begin{eqnarray}
\label{eq:reduced_kitaev_ham}
H&=&\frac{i}{2} \sum_{j} [ - \mu a_{2j-1}a_{2j} +\frac{1}{2} \left( J+\Delta\right) a_{2j}a_{2j+1}
\nonumber \\
& & \qquad +\frac{1}{2} \left( -J+\Delta\right) a_{2j-1} a_{2j+2} ] .
\end{eqnarray}
There are two conditions: first, when $J=\Delta=0$ and $\mu < 0$,
system shows trivial phase and two Majorana operators at each site
are paired together to form a ground state with occupation number $0$.
Secondly, for $J=\Delta>0 $ and $ \mu=0 $, the Majorana
operators from two neighboring sites are coupled together leaving two
unpaired Majorana operators at the two ends, and these two Majorana
modes are not coupled to the rest of the chain~\cite{kitaev,ali}.
The fermionic edge state formed with these two end operators has
occupation 0 or 1 with degenerate ground state, i.e., generating zero
energy excitation modes. However, the bulk properties of these systems
can be gapped.
In most of the papers~\cite{gergs,affleck_dmrg,ali2,haldane_dmrg,satoshi},
the Majorana zero modes (MZM) are characterized by exponential decay of
the lowest excitation gap with system size, and a large expectation value
of the creation operator of the fermion $\langle GS|c^{\dagger}|GS \rangle$
near the edges of the system. However, in the spin language there is no trivial relation
between the Majorana mode and spin raising operators. Therefore, in this
paper our main focus is to find the accurate phase boundary of the MI
transition, and for this purpose we focus on the lowest excitation gap
$ \Gamma $, derivative of longitudinal spin-spin correlations $C(1)$
between edge spin and its nearest neighbor spin, and spin density $ \rho_e $
of edge sites. In principle, the MZM do not couple with the bulk
states~\cite{kitaev,ali}; therefore, this correlation of the
edge spin should decay exponentially. We notice that a local magnetic
susceptibility $ \chi_i $ shows a discontinuity near the phase transition.
Fifth quantity is $ P(r) $ of edge spin, and it is similar to the spin
quadrupolar/spin-nematic order parameter~\cite{nematic1,nematic2,nematic3}
or superconducting order parameter of model in~\eref{eq:fermion_ham}.
The structure factor $S(q)$ gives us information about the phase boundary
and the bulk state. Based on the above quantities the MI transition
boundary is calculated in this paper. The bulk properties of the Majorana
state is rarely discussed in the literature, however we will try to
discuss those properties in this paper. In later part of this paper,
the ES of both states are also discussed to understand the topological
aspects of Majorana modes.
The Hamiltonian mentioned in~\eref{eq:spin_ham} is solved using the
Density matrix renormalization group (DMRG)~\cite{white-prl92, white-prb93}
and the exact diagonalization (ED) method. The DMRG method is a state of
the art numerical technique to solve the $1D$ interacting system, and
is based on the systematic truncation of irrelevant degrees of freedom
in the Hilbert space~\cite{white-prl92, white-prb93}. This numerical
method is best suited to calculate a few low lying excited states of
strongly interacting quantum systems accurately. To solve the interacting
Hamiltonian for ladder and chain with periodic boundary condition, the
DMRG method is further improved by modifying conventional DMRG
method~\cite{dd2016a} for zigzag chains~\cite{mk2010}, quasi one
dimension~\cite{mk2016} and higher dimensions~\cite{mk2012}. The left
and right block symmetry of DMRG algorithm for a XYZ-model of a
spin-$1/2$ chain in a staggered magnetic field (in \eref{eq:spin_ham})
is broken. Therefore, we use conventional unsymmetrized DMRG
algorithm~\cite{white-prl92, white-prb93} with open boundary condition.
In this model, the total $S^z$ is not conserved as $S^z$ does not commute
with the Hamiltonian in~\eref{eq:spin_ham}. As a result, the
superblock dimension is large. We keep $m \sim 500$ eigenvectors
corresponding to the highest eigenvalues of the density matrix to
maintain the desired accuracy of the results. The truncation error
of density matrix eigenvalues is less than $10^{-12}$. The energy
convergence is better than $0.001\%$ after five finite DMRG sweeps.
We go upto $N=200$ sites for the extrapolation of the transition points.
This paper is divided in to three sections. In~\sref{sec2},
we discuss our numerical results, and this is divided into eight
subsections. Results are discussed and compared with the existing
literatures in~\sref{sec3}.
\section{\label{sec2}Numerical results}
In this section, various criteria for the MI transition are discussed.
We start with a three dimensional phase diagram in $B$, $ \Delta $
and $ \mu $ parameter space for given $J^z =0$ and $0.5$. Thereafter,
various criteria of the MI such as lowest excitation energy $ \Gamma $,
edge spin correlation with its nearest neighbor $C(r=1)$, local
susceptibility at the site nearest neighbor to the edge $\chi_2$,
superconducting or spin-nematic order parameter of edge spin $P(r=1)$,
and structure factor $S(k)$ are studied. We show that all these
quantities show extrema at a transition parameter $B_m$ in $\Delta-B$
parameter space. However, all these extrema $B_m$ are extrapolated to
the same point in the thermodynamic limit, and this extrapolation is
done in the~\sref{Sec:H}. The ES is analyzed in~\sref{Sec:G}
to show the distinction between topological and the Ising phase.
\subsection{\label{Sec:A}Phase diagram}
\begin{figure}
\begin{center}
\includegraphics[width=3.3 in]{figure1.pdf}
\caption{MI phase boundary for a helical liquid mentioned
in~\eref{eq:spin_ham} in the parameter space of $\Delta $ and
$ \mu $ for $J^z = 0$. The color gradient represents the
critical magnetic field $B_c $ in the phase boundary.}
\label{fig1}
\end{center}
\end{figure}
A phase diagram of the model Hamiltonian in~\eref{eq:spin_ham},
is shown as a color gradient plot in~\fref{fig1}, where the
color gradient represents the critical value of $B_c$, and $X$-
and $Y$-axis are the $ \mu $ and the $ \Delta $ of the MI transition
points for a system size $N=100$ for $J^z=0$. The phase transition
in the $ \mu-\Delta $ parameter space shows that a finite $ \Delta_c $
is required to generate the Majorana modes in a finite system. The
$\mu$ favors the longitudinal degrees of freedom and tries to induce
the ferromagnetic order, although $B$ tries to align the nearest
spins in opposite directions to induce the Antiferromagnetic N\'eel
phase. In fact the $B$ and the $\mu$ both favor the Ising order,
whereas the $ \Delta $ breaks the parity symmetry and induces the
degeneracy in the system. It also induces the formation of
Cooper-pairs or magnon-pairs like excitations at the two neighboring
sites for the model Hamiltonian given by~\eref{eq:fermion_ham}
or~\eref{eq:spin_ham} respectively. As shown in~\fref{fig1},
the transition value $B_c$ increases with increasing
$ \mu_c $ for a fixed value of $ \Delta_c $,
and it increases with increasing $ \Delta_c $ for a given $ \mu_c $,
and the trends are similar for both $J^z=0$ and $0.5$. We notice
that the Majorana state occurs for $(B-\mu)^2 < \Delta^2$.
\subsection{\label{Sec:B}Excitation gap $ \Gamma $}
To characterize the Majorana modes in one dimension for the helical
system, the lowest excitation gap $ \Gamma $ is defined as
\begin{equation}
\Gamma=E_1(\Delta,\mu,B)-E_0(\Delta,\mu,B),
\label{eq:gap}
\end{equation}
where $ E_0 $ and $E_1$ are ground state and lowest excited state
of the Hamiltonian in~\eref{eq:spin_ham}. The $\Gamma-N$ is
plotted in log-linear scale in~\fref{fig2}.
The $\Gamma-N$ plot for $\mu = 0$, $ \Delta = 0.5 $, $J^z = 0.0$,
and for five values of $B$. For $B = 0.35$, $0.4$ and $0.45$,
$ \Gamma $ shows the exponential decay (Majorana regime), whereas
$ \Gamma $ goes linearly with $1/N$ for $B=0.5$ and $0.55$
(Ising phase) as shown in the~\fref{fig2}. The phase boundary
of MI transition is evaluated based on change in $\Gamma-N$ relation
from exponential to the power law. We also notice that the
contribution to the excitation gap $ \Gamma $ is uniformly
distributed in the Ising phase, whereas, in the Majorana state,
the major contribution comes from the edge as shown in Fig.~5 of~\cite{dspms}.
\begin{figure}
\begin{center}
\includegraphics[width=3.3 in]{figure2.pdf}
\caption{Lowest excitation gap $\Gamma$ (in~\eref{eq:gap})
vs. the system size $N$ for $\mu = J^z = 0$ and $\Delta=0.5$
with different values of $B$ chosen around the $B_c = 0.48$
(see~\fref{fig1}).}
\label{fig2}
\end{center}
\end{figure}
\subsection{\label{Sec:C}Correlation function $C(r=1)$ from the edge}
The longitudinal spin-spin correlation fluctuation $C(r)$ at
a distance $r$ from reference point $i$ is defined as
\begin{equation}
\label{spin_spin_corl}
C(r)=\left\langle S^z_iS^z_{i+r}\right\rangle -
\left\langle S^z_i\right\rangle \left\langle S^z_{i+r}\right\rangle,
\end{equation}
where $ \left\langle S^z_i\right\rangle$ and
$\left\langle S^z_{i+r}\right\rangle$ are spin densities at
the reference site $i$ and other site $i + r$. In~\fref{fig3},
the edge spin site $i = 1$ is considered as the reference spin.
The distance dependence of $C(r)$ for $\mu = 0$, $J^z=0$ and
$\Delta=0.5$ is shown in inset of~\fref{fig3}. It decreases
exponentially for $r \geq 2$, and effectively, only last two sites
are correlated. Therefore, $C(r=1)$ between nearest neighbors is
important. $C(r=1)$ first increases with $B$ in the Majorana state
and decreases afterwards in the Ising phase. The $dC(r=1)/dB$ is
plotted as a function of $B/B_c$ in the main~\fref{fig3} for
$\left( J^z=0\right. $, $\left. \Delta=0.5\right) $, and
$\left( J^z=0.5\right.$, $ \left. \Delta=1.5 \right) $ and $ \mu=0 $.
This minimum for the given value of parameters is also consistent
with the MI transition point calculated from energy degeneracy.
\begin{figure}
\begin{center}
\includegraphics[width=3.3 in]{figure3.pdf}
\caption{The derivative of longitudinal spin-spin correlation
between the spins corresponding to the edge bond $dC(r=1)/dB$
with the staggered magnetic field $B$ both for ($\Delta = 0.5$,
$\mu = J^z = 0$) and ($\Delta = 1.5$, $J^z = 0.5$, $ \mu =0 $).
The inset shows the distance dependence of $\ln |C(r)|$ for
$B = 0.3$, $0.48$ and $0.6$ and for $\Delta = 0.5$, $\mu = J^z = 0$.}
\label{fig3}
\end{center}
\end{figure}
\subsection{\label{Sec:D}Local magnetic susceptibility $\chi_i$}
The Majorana modes are confined to the edge of the system,
therefore we focus on the spin density of edge sites $i = 1$
and $2$ for $ \Delta=0.5 $, $ \mu=0 $ and $ J^z=0 $. The
spin density and the local staggered magnetic susceptibility
is defined as
\begin{eqnarray}
\rho_i=2\left\langle S^z_i \right\rangle, \\
\chi_i=\left|\frac{d\rho_i}{dB}\right|,
\label{eq:sus}
\end{eqnarray}
where, $ \left\langle S^z_i \right\rangle $ are the longitudinal
spin density at site $i$. The magnitude of spin density
$ \rho_i $ for site $i = 1, 2$ increases at both the sites with $B$,
it is positive at site $1$ and negative at site $2$. The magnitude
of $ \rho_i $ at site $2$ is lower than that is at site $1$. However,
both of them saturate with high staggered field, but $ \rho_1 $
continuously increases and there is no maxima for this function.
The $ \chi_2 $ as a function of $B/B_c$ are plotted in the main~\fref{fig4}
for ($J^z=0$, $\Delta=0.5$) and ($J^z=0.5$, $\Delta=1.5$)
and both for $ \mu=0 $. These two functions show a maxima near the
transition. The $ \rho_i $ for the whole system for $J^z=0$,
$\Delta=0.5$ and $\mu=0$ is shown in the inset of~\fref{fig4}.
We notice that the variation of $ \rho_i $ is confined to the edge
and the first few neighboring sites, and has constant value throughout
the rest of system. The $ \rho_i $ for $J^z=0.5$, $\Delta=1.5$ and
$\mu=0$ behaves in a similar manner.
\begin{figure}
\begin{center}
\includegraphics[width=3.3 in]{figure4.pdf}
\caption{Local staggered magnetic susceptibility (in~\eref{eq:sus})
of the second site nearest neighbor to the edge $\chi_2$ vs. $B/B_c$ both for
($\Delta = 0.5$, $\mu = J^z = 0$) and ($\Delta = 1.5$, $J^z = 0.5$,
$ \mu =0 $). In the inset, the spin density $\rho(r)$ for the
whole system for $B = 0.3$, $0.48$ and $0.6$ with
$\Delta = 0.5$, $\mu = J^z = 0$ is shown.}
\label{fig4}
\end{center}
\end{figure}
\subsection{\label{Sec:E}Quadrupolar order parameter $P(r=1)$}
The third term of the Hamiltonian in~\eref{eq:spin_ham} induces
the spin quadrupolar/spin-nematic order in the system. In this
phase $ \left\langle S^+\right\rangle $ vanishes, whereas
$\left\langle S^+S^+\right\rangle $ has non-zero value, and two magnon
pair formation is favored similar to superconductor system where
two electrons form Cooper pair. The spin distance dependent
quadrupolar order parameter is defined as
\begin{eqnarray}
P(r)&=& \left\langle S^x_iS^x_{i+r} \right\rangle -\left\langle
S^y_iS^y_{i+r} \right\rangle, \nonumber \\
&=& \left\langle S^+_iS^+_{i+r} + S^-_iS^-_{i+r} \right\rangle.
\label{eq:pr}
\end{eqnarray}
$P( r )$ is the difference in the $X$ and $Y$ component of correlation
$ \left\langle \vec{S}_i \cdot \vec{S}_{i+r} \right\rangle $.
For $r = 1$, this quantity is very similar to the spin quadrupolar
or superconducting order parameter of the spinless fermion model.
The $P(r = 1)$ is calculated as a function of $B$ for
($ \Delta = 0.5$, $J^z=0$) and ($\Delta = 1.5$, $J^z=0.5$) and both
for $ \mu=0 $, and we notice that when $ \Delta $ dominates over $B$,
the system goes from non-degenerate Ising phase to doubly degenerate
states which favors Majorana edge state and the bulk phase goes to
spin quadrupolar phase as $P(r) \neq 0$. The $P(r = 1)$ first
increases with $B$, and then it saturates with higher $B$. The
derivative of $P(r = 1)$ with $B$ for ($\Delta = 0.5$, $J^z=0$,
$\mu=0$) and ($\Delta = 1.5$, $J^z=0.5$, $\mu=0$) shows maxima at
$B= 0.48$ and $B=1.04$ respectively as shown in the main~\fref{fig5}.
The distance dependence of $P(r)$ as a function $r$
are shown for both the regime of the Majorana modes and the Ising
states in inset of~\fref{fig5}. The reference site $i$ is the
edge site of the chain. In the Majorana state P(r) shows long range
behavior, however, it decays exponentially in case of Ising phase.
\begin{figure}
\begin{center}
\includegraphics[width=3.3 in]{figure5.pdf}
\caption{ $ \frac{dP(r=1)}{dB} $ as a function of $\frac{B}{B_c}$ is shown
in the main figure for ($\Delta = 0.5$, $\mu = J^z = 0$) and ($\Delta = 1.5$,
$J^z = 0.5$, $ \mu =0 $). $P(r)$ (in~\eref{eq:pr}) is plotted for the whole system in
the inset for $B = 0.3$, $0.48$ and $0.6$ with $\Delta = 0.5$, $\mu = J^z = 0$.}
\label{fig5}
\end{center}
\end{figure}
\subsection{\label{Sec:F}Longitudinal structure factor $S(k)$}
The longitudinal structure factor $S(k)$ is the Fourier transformation
of $C(r)$ given in~\eref{spin_spin_corl}, and can be defined as
\begin{equation}
S(k)=\sum_r \left( \left\langle S^z_iS^z_{i+r}\right\rangle -
\left\langle S^z_i\right\rangle \left\langle S^z_{i+r}\right\rangle\right) e^{ikr}.
\end{equation}
Now let us define a quantity $ K_\rho $ in small $k$ limit defined as
\begin{equation}
K_\rho=\frac{S(k)}{k/\pi};k\rightarrow 0,
\label{eq:krho}
\end{equation}
where $ k=\frac{2\pi m}{N}; $ $ m=0,\pm 1,\pm 2,\ldots,\pm \frac{N}{2}$.
$ K_\rho $ is proportional to the Luttinger Liquid
parameter~\cite{luttinger1,luttinger2,luttinger3,luttinger4}. We take
the value of the function $ \frac{\pi S(k)}{k} $ at
$ k=\frac{2\pi}{N} \, \left( m=1\right) $. We calculate $ K_\rho $ for
the ($ \Delta =0.5$, $J^z = 0 $) and ($ \Delta =1.5$, $J^z = 0.5 $) as
a function of $ \mu $ and $B$. The derivative of $ K_\rho $ as a
function of $B$ shows maxima at $B=0.48$, $0.52$, and $0.7$ for
$\mu=0$, $0.2$ and $0.5$ at $J^z=0$ and $ \Delta=0.5 $. The derivative
of $ K_\rho $ with $B/B_c$ for ($\Delta =0.5$, $J^z = 0$) and
($\Delta =1.5$, $J^z = 0.5$) is shown for three different values of
$\mu$ in the main and in the inset of~\fref{fig6}. The maxima of
$\frac{dK_\rho}{dB} $ indicates the boundary between the Majorana and
the Ising state. The extrapolated value of the transition point is very
close to the transition point calculated from other criteria. It also
shows that the critical value of $B$ for the transition calculated from
$ \frac{dK_\rho}{dB} $ at a fixed $\Delta $ increases with increasing $ \mu $.
\begin{figure}
\begin{center}
\includegraphics[width=3.3 in]{figure6.pdf}
\caption{Derivative of $K_\rho$ (in~\eref{eq:krho}) with the
staggered magnetic field $B$, as a function of $ \frac{B}{B_c}$
is shown in the main figure for $\Delta = 0.5$, $J^z = 0$, and the
inset shows the same for $\Delta = 1.5$, $J^z = 0.5$ for $\mu=0$, $0.2$
and $0.5$.}
\label{fig6}
\end{center}
\end{figure}
\subsection{\label{Sec:G}Entanglement spectrum in Ising and Majorana state}
As the Majorana mode is topological in nature, there is no
well-defined order parameter. Therefore, direct measurement of
this property is not possible~\cite{entanglement1,entanglement2,entanglement3}.
Therefore, we study the ES of the reduced density matrix of the
system to indirectly study the topological aspect.
In this phase, all the states are either
doubly or multiply degenerate~\cite{entanglement1, entanglement2}.
The reduced density matrix of a system (half of the full system) can
be constructed in the GS of the full system by integrating out the
environment degrees of freedom. The eigenvalues of the reduced
density matrix are represented as $\lambda_i$. The Schmidt gap is defined as
$\Delta_S = \lambda_0 - \lambda_1$, where $\lambda_0$ and $\lambda_1$
are the largest and second largest eigenvalues of the reduced density
matrix. Topological phases are also characterized by $\Delta_S = 0$,
whereas it is finite in the trivial phase~\cite{entanglement2}.
In a non-topological state the largest eigenvalue is non-degenerate
and has mixed degenerate and non-degenerate eigenvalues~\cite{entanglement1, entanglement2}.
The ES of the reduced density matrix is analyzed for a chain of $N=96$ spins with PBC
in the deep Ising state, and in the deep Majorana state for $J^z=0$ and $0.5$. We plot the
Schmidt gap as a function of $ \frac{\Delta}{\Delta_c} $ for
$J^z =0$ and $J^z=0.5$ shown in the inset of~\fref{fig7} for
$\mu=0 $ and $ B=0.5 $. We notice that $\lambda_{n=0}$
is non-degenerate in the Ising state and Schmidt gap is finite
in this regime but goes to zero in the Majorana state as shown in
the inset of~\fref{fig7}. In the Ising phase many of $\lambda_n$
are non-degenerate for ($ J^z =0$, $\mu=0$, $B=0.5$,
$\Delta = 0.2 $) and ($J^z =0.5 $, $\mu=0.0$, $B=0.5$, $\Delta = 0.7$).
In this phase the ES is of mixed type. The spectrum
is shown as open and filled symbols in the main~\fref{fig7} for
$J^z=0$ and $0.5$ respectively. In the Majorana phase the $\lambda_{n=0}$
is triply degenerate, and for other higher $n$, these are either doubly
or multiply degenerate as shown in the main~\fref{fig7} for
($J^z =0$, $ \mu=0$, $B=0.5$, $\Delta = 0.9$) and ($J^z =0.5$, $\mu=0$,
$B=0.5$, $\Delta = 1.2$). The phase boundary of the system can be
characterized at the point where Schmidt gap goes to zero and the whole
spectrum becomes doubly or multiply degenerate. In the Majorana phase
first few largest $\lambda_n$ are independent of parameters.
\begin{figure}
\begin{center}
\includegraphics[width=3.3 in]{figure7.pdf}
\caption{The entanglement spectrum $\lambda_n$ is plotted for
$\mu=0$ and $B=0.5$ in both the Ising and Majorana regime for
$J^z =0$ and $J^z =0.5$ in the main figure. In the inset, the Schmidt gap
$ \Delta_S $ is shown as a function of $ \frac{\Delta}{\Delta_c} $ for
$B=0.5$, $ \mu=0 $ both for $J^z =0$ and $J^z =0.5$.}
\label{fig7}
\end{center}
\end{figure}
\subsection{\label{Sec:H}MI phase boundary}
In the last seven subsections, different criteria give us different
MI phase boundary in a finite system size and the phase boundary
$B_c$, calculated from different criteria have different finite size
dependence. For $\mu=0.2$, the finite size scaling of the $B_c$ for
four different criteria, i.e., ground state degeneracy,
$ K_\rho $, $P(r=1)$ and $C(r=1)$ are shown in the inset of~\fref{fig8}.
The ground state degeneracy and $C(r=1)$ have same finite size effect.
We notice that the extrapolation from all the criteria leads to same
$B_c=0.52$ in the thermodynamic limit. The effect of $ \mu $ on the MI
transition point $B_c$ is shown in the thermodynamic limit for
($J^z=0$, $ \Delta=0.5 $) and ($J^z=0.5$, $ \Delta=1.5 $) in the main~\fref{fig8}.
\begin{figure}
\begin{center}
\includegraphics[width=3.3 in]{figure8.pdf}
\caption{The effect of $\mu$ on MI transition in the thermodynamic
limit both for $J^z=0$, $ \Delta=0.5 $ and $J^z=0.5$, $ \Delta=1.5 $
are shown in the main figure. The finite size scaling of $B_c$ from
different criteria for $J^z =0$, $\Delta=0.5$ and $ \mu=0.2 $ is
shown in the inset.}
\label{fig8}
\end{center}
\end{figure}
\section{\label{sec3}Discussion}
We have studied the helical liquid system and mapped this
model into a $ XYZ $ spin-$1/2$ chain model. The Majorana-Ising
transition is characterized by calculating the lowest
excitation gap of the model Hamiltonian in~\eref{eq:spin_ham}
on a chain geometry. In the Majorana state, the system has
finite gap for a finite system which decays exponentially with
the system size $N$. The closing of the gap in the thermodynamic
limit is consistent with the study by Sela et al.~\cite{sela}.
Our aim of this paper is to explore various criteria to
characterize Majorana mode other than closing of the lowest excitation
gap, and the accurate determination of the MI transition boundary.
We have calculated various quantities, e.g., $\rho_i$, $C( r )$, $P( r )$,
$K_\rho$, $\Delta_S$. We have shown that the phase boundary
calculated from the various criteria are the same for given value
of $\Delta$ and $\mu$ in the thermodynamic limit.
We have shown that in strong repulsive interaction limit $J^z>0 $,
Majorana mode occurs at higher value of $ \Delta $ than the
non-interacting case $ \left( J^z=0\right) $, which is
consistent with the study of Gangadharaiah et al.~\cite{suhas},
where they have shown that the repulsive interaction weakens the
Majorana modes. The $J^z$ term of~\eref{eq:spin_ham} is
similar to the repulsive interaction term of the spinless fermion
model in~\eref{eq:fermion_ham}. In the mean field limit, $J^z$
term reduces to effective $ \mu $. We have noticed that the
Majorana state occurs for
$(\mu-B)^2 < \Delta^2$, therefore, any change in $\mu$ changes the
value of critical field $B_c$. The $P(r)$ is long range in the
Majorana state, whereas it decays exponentially in the
Ising phase as shown in the inset of~\fref{fig5}. In the Majorana
state, the bulk of system shows quadrupolar/or spin nematic phase
like behavior. The local magnetic susceptibility $ \chi $ of the
edge sites shows maxima near the phase boundary.
We have also studied the ES of this model, and it is shown that the
Ising phase has finite Schmidt gap $ \Delta_S $, and non-degenerate
eigenvalues are present in the spectrum of the reduced density matrix
of the system, whereas the topological aspect of Majorana state is
characterized by the doubly degenerate eigenvalues and zero Schmidt
gap~\cite{entanglement1,entanglement2}. We have also shown that ES
of the reduced density matrix of the ground state shows double and
multiple degeneracy in the Majorana state. We have also noticed
threefold degeneracy in the largest eigenvalues in this state in the
thermodynamic limit. Degeneracy of all the eigenvalues is very similar
to the study by Pollmann et al.~\cite{entanglement1} to distinguish
the topological and trivial phase of $ S=1$ system. The first few
eigenvalues of ES in the Majorana state is almost independent of
parameters as shown in~\fref{fig7}.
In conclusion, we have studied the helical liquid phase in one
dimensional system. This system shows the Majorana-Ising transition,
and the phase boundary is calculated using various criteria. The
topological aspect of the Majorana modes is studied for helical model,
and the closing of the Schmidt gap and degeneracy of full spectrum of
reduced density matrix can also be used to characterize the phase
boundary of Majorana-Ising transition. This model is one of the most
interesting and a general model for spin-$1/2$ systems. Our
study shows that an anisotropic spin-$1/2$ chain can be a good
candidate to observe the Majorana modes and MI transition. The local
experimental probe like neutron magnetic resonance can be used to
measure the local spin density at the edge of the sample.
\ack
MK thanks Z. G. Soos and Sumanta Tewari for their valuable comments.
MK thanks DST for a Ramanujan Fellowship SR/S2/RJN-69/2012 and funding
computation facility through SNB/MK/14-15/137. SS thanks the DST
(SERB, SR/S2/LOP-07/2012) fund and SNBNCBC for supporting the visit.
\section*{References}
|
1,314,259,993,575 | arxiv | \section{Introduction}
\label{INTRO}\label{sec:intro}
Understanding the power of quantum algorithms has been a central research goal over the last few decades.
One success story in this regard has been the discovery of powerful methods that establish limitations on quantum algorithms in the standard setting of \emph{query complexity}. This setting roughly asks, for a
specified function $f$, how many
bits of the input must be examined by any quantum algorithm that computes $f$ (see \cite{bw} for a survey of query complexity).
A fundamental topic of study in complexity theory is algorithms that are ``augmented'' with additional information, such as an untrusted witness provided by a powerful prover.
For example, the classical complexity class $\mathsf{NP}$ is defined this way.
In the quantum setting, if we go beyond standard query algorithms, and allow algorithms to receive a quantum state, the model becomes much richer, and we have very few techniques to establish lower bounds for these algorithms.
In this paper, we develop such techniques.
Our methods crucially use \emph{Laurent polynomials}, which are polynomials with positive and negative integer exponents.
We demonstrate the power of these lower bound techniques by proving optimal lower bounds for the \emph{approximate counting} problem, which captures the following task.
Given a nonempty finite set $S\subseteq [N] :=\left\{
1,\ldots ,N\right\} $, estimate its cardinality, $\left\vert S\right\vert $,
to within some constant (say, 2) multiplicative accuracy.
Approximate counting is a fundamental task with a rich
history in computer science. This includes the works of Stockmeyer~\cite%
{Sto85}, which showed that approximate counting is in the polynomial
hierarchy, and Sinclair and Jerrum~\cite{sinclairjerrum}, which showed the
equivalence between approximate counting and approximate sampling that
enabled the development of a whole new class of algorithms based on Markov chains.
Additionally, approximate counting precisely highlights the limitations of current lower bound techniques for the complexity class $\mathsf{QMA}$
(as we explain in \Cref{sec:QMAintro}).
Formally, we study the following decision version of the problem in this paper:
\begin{problem}[Approximate Counting]
In the $\mathsf{ApxCount}_{N,w}$ problem, our goal is to decide whether a
nonempty set $S\subseteq \lbrack N]$ satisfies $\left\vert S\right\vert \geq
2w$ (YES) or $\left\vert S\right\vert \leq w$ (NO), promised that one of
these is the case.
\end{problem}
In the query model, the algorithm is given a membership oracle for $S$: one that, for any $i\in \left[ N\right] $, returns whether $%
i\in S$. \ How many queries must we make, as a function of both $N$\ and $%
\left\vert S\right\vert $, to solve approximate counting with high
probability?
For classical randomized algorithms, it is easy to see that $\Theta (N/|S|)$\
membership queries are necessary and sufficient.
For quantum algorithms, which can query the membership oracle on superpositions
of inputs, Brassard et al.~\cite{bht:count,BHMT02} gave an algorithm
that makes only $O\bigl(\sqrt{N/|S|}\bigr)$\ queries.
It follows from the optimality of Grover's algorithm (i.e., the BBBV Theorem \cite{bbbv}) that this cannot be improved. Hence, the classical and quantum complexity of approximate counting with membership queries alone is completely understood.
In this paper, we study the complexity of approximate counting in models with untrusted and trusted quantum states.
\subsection{First result: QMA complexity of approximate counting}
\label{sec:QMAintro}
Our first result, presented in \Cref{sec:SBPQMA}, considers the standard Quantum Merlin--Arthur ($\mathsf{QMA}$) setting, in which the quantum algorithm receives an untrusted quantum state (called the witness).
This model is the quantum analogue of the classical complexity class $\mathsf{NP}$, and is of great
interest in quantum complexity theory. It captures natural problems about ground states of physical systems, properties of quantum circuits and channels, noncommutative constraint satisfaction problems, consistency of representations of quantum systems, and more~\cite{Boo13}.
In a $\mathsf{QMA}$ protocol, a skeptical verifier
(Arthur) receives a quantum witness state $|\psi \>$ from an all-powerful but
untrustworthy prover (Merlin), in support of
the claim that $f(x)=1$.
\ Arthur then needs to verify $|\psi \>$, via some
algorithm that satisfies the twin properties of \textit{completeness} and
\textit{soundness}. \ That is, if $f(x)=1$, then there must exist some $|\psi \>$\ that causes Arthur
to accept with high probability, while if $f(x)=0$, then every $|\psi \>$\ must cause Arthur to reject with high probability. \ We call such a protocol a $\mathsf{QMA}$ (Quantum Merlin--Arthur) protocol for computing $f$.
In the query complexity setting, there are two resources to consider: the length of the quantum witness, $m$, and
the number of queries, $T$, that Arthur makes to the membership oracle. \
A $\mathsf{QMA}$ protocol for $f$ is efficient if both $m$ and $T$
are $\mathrm{polylog}(N)$.
\para{The known lower bound technique for $\mathsf{QMA}$.}
Prior to our work, all known $\mathsf{QMA}$ lower bounds used the same proof technique.\footnote
There is one special case in which it is trivial to lower-bound $\mathsf{QMA}$ complexity.
Consider the $\mathsf{AND}_N$ function on $N$ bits that outputs 1 if and only if all $N$ bits equal $1$. For this function, since Merlin wants to convince Arthur that $f(x)=1$, intuitively there is nothing interesting that Merlin can say to Arthur other than ``$x$ is all ones'' since that is the only input with $f(x)=1$. Formally, Arthur can simply create the witness state that an honest Merlin would have sent on the all ones input, and hence Arthur does not need Merlin \cite{razshpilka}.
For such functions, $\mathsf{QMA}$ complexity is the same as standard quantum query complexity.
}
The technique establishes (and exploits) the complexity class containment
$\mathsf{QMA}\subseteq \mathsf{SBQP}$, where $\mathsf{SBQP}$\ is
a complexity class that models quantum algorithms with tiny acceptance and
rejection probabilities.
Specifically, we say that a function $f$ has $\mathsf{SBQP}$ query complexity at most $k$
if there exists a $k$-query quantum algorithm that
\begin{itemize}
\item outputs $1$ with probability $\geq \alpha $\ when $f(x)=1$, and
\item outputs $1$ with probability $\leq \alpha /2$\ when $f(x)=0$,
\end{itemize}
\noindent for some $\alpha $ that does not depend on the input (but may depend on the input size).
Note
that when $\alpha =2/3$, we recover standard quantum query complexity. \ But
$\alpha $ could be also be exponentially small, which makes $\mathsf{SBQP}$
algorithms very powerful.
Nevertheless, one can establish significant limitations on $\mathsf{SBQP}$
algorithms, by using a variation of the polynomial method of Beals et al.\
\cite{bbcmw}. \ If a function $f$ can be evaluated by an $\mathsf{SBQP}$
algorithm with $k$ queries, then there exists a real polynomial $p$ of
degree $2k$ such that $p(x)\in \lbrack 0,1]$ whenever $f(x)=0$ and $p(x)\geq
2$ whenever $f(x)=1$. \ The minimum degree of such a polynomial is also
called \emph{one-sided approximate degree}~\cite{BT15}.
The relationship between $\mathsf{SBQP}$ and $\mathsf{QMA}$ protocols is
very simple: if $f$ has a $\mathsf{QMA}$ protocol that receives an $m$-qubit
witness\ and makes $T$ queries, then it also has an $\mathsf{SBQP}$
algorithm that makes $O(mT)$ queries. \ This was essentially observed by Marriott and
Watrous~\cite[Remark 3.9]{marriott} and used by Aaronson \cite%
{aaronson_szk} to show an oracle relative to which $\mathsf{SZK} \not\subset \mathsf{QMA}$.
\para{Beyond the known lower bound technique for $\mathsf{QMA}$.}
Our goal is to find a new method of lower bounding $\mathsf{QMA}$, that does not go through $\mathsf{SBQP}$ complexity.
The natural way to formalize this quest is to find a problem that has an efficient
$\mathsf{SBQP}$ algorithm, and show that it
does not have an efficient $\mathsf{QMA}$ protocol.
A natural candidate for this is the $\mathsf{ApxCount}_{N,w}$ problem.
We know that $\mathsf{ApxCount}_{N,w}$ \textit{does} have a very simple $\mathsf{SBQP}$ algorithm of cost 1:
the algorithm picks an $i\in \left[ N\right] $\ uniformly at random, and accepts if and only if $i\in S$. \
Clearly the algorithm accepts with probability greater than $2w/N$ on yes inputs and with probability at most $w/N$ on no inputs.
Our first result establishes that $\mathsf{ApxCount}_{N,w}$
does \emph{not} have an efficient $\mathsf{QMA}$ protocol.
\begin{restatable}{theorem}{qmabound}
\label{thm:qmabound}
Consider a $\mathsf{QMA}$ protocol that solves $\mathsf{ApxCount}_{N,w}$. If the protocol receives a quantum witness of length $m$, and makes $T$ queries to the membership oracle for $S$, then
either $m = \Omega(w)$ or
$ T = \Omega\bigl(\sqrt{N/w}\bigr)$.
\end{restatable}
This lower bound proved in \Cref{sec:QMA} resolves the $\mathsf{QMA}$ complexity of $\mathsf{ApxCount}_{N,w}$, as (up to a $\log N$ factor) it matches the cost of two trivial $\mathsf{QMA}$ protocols.
In the first, Merlin sends $2w$ items claimed to be in $S$,
and Arthur picks a constant number of the items at random
and confirms they are all in $S$ with one membership query each.
This protocol has witness length $m=O(w\log N)$ (the number of bits needed to specify $2w$ elements out of $N$) and $T=O(1)$.
In the second protocol, Merlin does nothing, and Arthur solves the problem with $T=O\bigl(\sqrt{N/w}\bigr)$ quantum queries.
\setlength{\columnsep}{1.5em}
\setlength{\intextsep}{3ex}
\begin{wrapfigure}{r}{0.245\textwidth}
\centering
\begin{tikzpicture}[x=1cm,y=1cm]
\node (MA) at(2,0){$\cl{MA}$};
\node (QMA) at(1,1){$\cl{QMA}$};
\node (SBP) at(3,1){$\cl{SBP}$};
\node (SBQP) at(2,2){$\cl{SBQP}$};
\node (PP) at(2,3){$\cl{PP}$};
\node (AM) at(4,2){$\cl{AM}$};
\path[-] (MA) edge (QMA);
\path[-] (QMA) edge (SBQP);
\path[-] (MA) edge (SBP);
\path[-] (SBP) edge (SBQP);
\path[-] (SBP) edge (AM);
\path[-] (SBQP) edge (PP);
\end{tikzpicture}
\caption{Relationships between complexity classes.
An upward line indicates that a complexity class
is contained in the one above it relative to
all oracles.\label{fig:relations}\vspace{2ex}}
\end{wrapfigure}
\para{Oracle separation.}
Our result also yields new oracle separations.
The approximate counting problem is complete for the complexity class $\mathsf{SBP}$ \cite{BGM06}, which is sandwiched between $\mathsf{MA}$ (Merlin--Arthur) and $\mathsf{AM}$\ (Arthur--Merlin). \ The class $\mathsf{SBQP}$ (discussed above), first defined by Kuperberg \cite{kuperberg}, is a
quantum analogue of $\mathsf{SBP}$ that contains both $\mathsf{SBP}$ and $\mathsf{QMA}$.
By the usual connection between oracle separations and query complexity
lower bounds, \Cref{thm:qmabound} implies an oracle separation
between $\mathsf{SBP}$ and $\mathsf{QMA}$---i.e., there exists an oracle $A$
such that $\mathsf{SBP}^{A}\not\subset \mathsf{QMA}^{A}$
(see \Cref{cor:qma_separation}). Prior to our
work, it was known that there exist oracles $A,B$ such that $\mathsf{SBP}%
^{A}\not\subset \mathsf{MA}^{A}$~\cite{BGM06} and $\mathsf{AM}%
^{B}\not\subset \mathsf{QMA}^{B}$, which follows from $\mathsf{AM}^B \not\subset \mathsf{PP}^B$ \cite{vereshchagin},
but the relation
between $\mathsf{SBP}$\ and\ $\mathsf{QMA}$\ remained elusive.\footnote{%
It is interesting to note that in the non-relativized world, under plausible
derandomization assumptions~\cite{amnp}, we have $\mathsf{NP}=\mathsf{MA}=%
\mathsf{SBP}=\mathsf{AM}$. In this scenario, all these classes are equal,
and all are contained in $\mathsf{QMA}$.} \Cref{fig:relations}
shows the known inclusion relations among these
classes (all of which hold relative to all oracles).
Previous techniques were inherently
unable to establish this oracle separation for the reason stated above: all existing $\mathsf{QMA}$ lower bounds
intrinsically apply to $\mathsf{SBQP}$ as well. Since $\mathsf{SBP}$ is contained in $\mathsf{SBQP}$, prior techniques cannot establish $\mathsf{SBP}^{A}\not\subset \mathsf{QMA}^{A}$, or even $\mathsf{SBQP}^{A}\not\subset \mathsf{QMA}^{A}$, for any oracle $A$.
Our analysis also yields the first oracle with respect to
which $\mathsf{SBQP}$ is not closed under intersection.
\para{Proof overview.} \ To get around the
issue of $\mathsf{ApxCount}_{N,w}$\ being in $\mathsf{SBQP}$,\ we use a
clever strategy that was previously used by G\"{o}\"{o}s et al.~\cite%
{GLMWZ16}, and that was suggested to us by Thomas Watson (personal
communication).
Our strategy exploits a structural property of $\mathsf{QMA}$:
the fact that $\mathsf{QMA}$\ is closed under intersection,
but (at least relative to oracles, and as we'll show) $\mathsf{SBQP}$\ is not.
Given a function $f$, let $\mathsf{AND}_{2}\circ f$ be the
$\mathsf{AND}$ of two copies of $f$ on separate inputs.\footnote{Because we focus on lower bounds, for a promise problem $f$ (such as $\mathsf{ApxCount}_{N,w}$), we take the promise for $\mathsf{AND}_2 \circ f$ to be that both instances of $f$ must satisfy $f$'s promise. Then, any lower bound also applies to more relaxed definitions, such as only requiring one of the two instances to be in the promise.}
\ Then if $f$ has small $\mathsf{QMA}$ query complexity, it's not hard to see that $\mathsf{AND}%
_{2}\circ f$ does as well:\ Merlin simply sends witnesses corresponding to
both inputs; then Arthur checks both of them independently. \ While it's not
completely obvious, one can verify that a dishonest Merlin would gain
nothing by entangling the two witness states. \ Hence if $\mathsf{ApxCount}%
_{N,w}$ had an efficient $\mathsf{QMA}$ protocol, then so would $\mathsf{AND}%
_{2}\circ \mathsf{ApxCount}_{N,w}$, with the witness size and query
complexity increasing by only a constant factor.
By contrast, even though $\mathsf{ApxCount}_{N,w}$ does have an
efficient $\mathsf{SBQP}$ algorithm, we will show that $\mathsf{AND}%
_{2}\circ \mathsf{ApxCount}_{N,w}$ does not. \ This is the technical core of
our proof and proved in \Cref{sec:SBQP}.
\begin{restatable}{theorem}{sbqp}
\label{thm:sbqp}
Consider an $\mathsf{SBQP}$ algorithm for $\mathsf{AND}_2 \circ \mathsf{ApxCount}_{N,w}$ that makes $T$ queries to membership oracles for the two instances of $\mathsf{ApxCount}_{N,w}$. Then $T=\Omega\left(\min\bigl\{w,\sqrt{N/w}\bigr\}\right)$.
\end{restatable}
\Cref{thm:sbqp} is quantitatively optimal, as we'll exhibit
a matching $\mathsf{SBQP}$ upper bound. \
Combined with the
connection between $\mathsf{QMA}$ and $\mathsf{SBQP}$, \Cref{thm:sbqp}
immediately implies a $\mathsf{QMA}$ lower bound for $\mathsf{AND}_2 \circ \mathsf{ApxCount}_{N,w}$, and by extension $\mathsf{ApxCount}_{N,w}$ itself. However, this $\mathsf{QMA}$ lower bound is not quantitatively optimal. To obtain the optimal bound of \Cref{thm:qmabound},
we exploit additional analytic properties of
the $\mathsf{SBQP}$ protocols that are
derived from $\mathsf{QMA}$ protocols.
At a high level, the proof of \Cref{thm:sbqp} assumes that there's an
efficient $\mathsf{SBQP}$ algorithm for $\mathsf{AND}_{2}\circ \mathsf{%
ApxCount}_{N,w}$. \ This assumption yields a low-degree one-sided
approximating polynomial for the problem in $2N$ Boolean variables, where $N$
variables come from each $\mathsf{ApxCount}_{N,w}$\ instance. \ We then
symmetrize the polynomial (using the standard Minsky--Papert symmetrization argument~\cite{mp})
to obtain a bivariate polynomial in two variables $%
x$ and $y$ that represent the Hamming weight of the original instances.\footnote{%
The term ``symmetrization'' originally referred to the process of averaging a multivariate polynomial over permutations of its inputs to obtain a symmetric polynomial. More recently, authors have used ``symmetrization'' more generally to refer to any method for turning a multivariate polynomial into a univariate one in a degree non-increasing manner (see, e.g., \cite{sherstov2009halfspaces, sherstov2010halfspaces}). In this paper, we use the term ``symmetrization'' in this more general sense.
} \
This yields a polynomial $p(x,y)$ that for \emph{integer pairs} $x, y$ (also called lattice points) satisfies
$p(x,y)\in [0,1]$ when either $x \in \{0,\ldots,w\}$ and $y\in \{0,\ldots,w\} \cup \{2w,\ldots,N\}$, or
(symmetrically) $y \in \{0,\ldots,w\}$ and $x\in \{0,\ldots,w\} \cup \{2w,\ldots,N\}$. \ If
both $x \in \{2w,\ldots,N\}$ and $y \in \{2w,\ldots,N\}$, then $p(x,y)\geq 2$.
This polynomial $p$ is depicted in \Cref{fig:introL}.
\begin{figure}[tbp]
\centering
\begin{tikzpicture}[y=0.35cm, x=0.35cm]
\pgfmathsetmacro{\N}{20};
\pgfmathsetmacro{\w}{4};
\pgfmathsetmacro{\tw}{\w+\w};
\pgfmathsetmacro{\Np}{\N+1};
\fill[gray,opacity=.4] (\tw,0) -- (\tw,\w) -- (\N,\w) -- (\N,0);
\fill[gray,opacity=.4] (0,\tw) -- (\w,\tw) -- (\w,\N) -- (0,\N);
\fill[gray,opacity=.4] (0,0) -- (\w,0) -- (\w,\w) -- (0,\w);
\fill[gray,opacity=.1] (\tw,\tw) -- (\tw,\N) -- (\N,\N) -- (\N,\tw);
\draw[->] (0,0) -- coordinate (x axis mid) (\Np,0);
\draw[->] (0,0) -- coordinate (y axis mid) (0,\Np);
\draw (0,1pt) -- (0,-3pt) node[anchor=north] {0};
\draw (\w,1pt) -- (\w,-3pt) node[anchor=north] {$w$\vphantom{l}};
\draw (\tw,1pt) -- (\tw,-3pt) node[anchor=north] {$2w$};
\draw (\N,1pt) -- (\N,-3pt) node[anchor=north] {$N$};
\draw (1pt,0) -- (-3pt,0) node[anchor=east] {0};
\draw (1pt,\w) -- (-3pt,\w) node[anchor=east] {$w$};
\draw (1pt,\tw) -- (-3pt,\tw) node[anchor=east] {$2w$};
\draw (1pt,\N) -- (-3pt,\N) node[anchor=east] {$N$};
\node[below=0.8cm] at (x axis mid) {$x$};
\node[rotate=90, above=0.8cm] at (y axis mid) {$y$};
\node at (\w+\N/2, \w/2) {$p(x, y) \in [0,1]$};
\node[rotate=90] at (\w/2, \w+\N/2) {$p(x, y) \in [0,1]$};
\node at (\w/2, \w/2) [align=left]{$p(x, y)$ \\ $\in [0,1]$};
\node at (\w+\N/2, \w+\N/2) {$p(x, y) \geq 2$};
\draw[thick,domain=1:{\N/(2*\w)},smooth,variable=\t,blue] plot ({2*\w*\t},{2*\w/\t});
\draw[thick,domain=1:{\N/(2*\w)},smooth,variable=\t,blue] plot ({2*\w/\t},{2*\w*\t});
\end{tikzpicture}
\caption{The behavior of the (Minsky--Papert symmetrized) bivariate polynomial $p(x,y)$ at integer points $(x,y)$ in the proof of \Cref{thm:sbqp}. The polynomial $q$ obtained by erase-all-subscripts symmetrization is not depicted. We later restrict $q$ to a hyperbola similar to the one drawn in blue.}
\label{fig:introL}
\end{figure}
One difficulty is that we have a guarantee on the behavior of $p$ at lattice
points only, whereas the rest of our proof requires precise control over
the polynomial even at non-integer points. We ignore this issue for now and assume that $p(x,y)\geq 2$ for all
real values $x, y \in [2w, N]$, and $p(x, y) \in [0, 1]$
whenever $x \in [0, w]$ and $y \in [2w, N]$
or vice versa. We outline how we
address integrality issues one
paragraph hence.
The key remaining difficulty is that we want to lower-bound the degree of
a bivariate polynomial, but almost all known
lower bound techniques apply only to univariate polynomials.
To address this, we introduce a new technique
to reduce the number of variables (from $2$ to $1$)
in a degree-preserving way: we pass a \textit{hyperbola}
through the $xy$ plane (see \Cref{fig:introL}) and consider
the polynomial $p$ restricted to the hyperbola.\
Doing so gives us a new univariate \emph{Laurent} polynomial $\ell (t) = p(2wt, 2w/t)$,
whose positive and negative degree is at most $\deg(p)$. \
This Laurent polynomial has an additional symmetry,
which stems from the fact that $\mathsf{AND}%
_{2}\circ \mathsf{ApxCount}_{N,w}$ is the $\mathsf{AND}$ of two identical
problems (namely, $\mathsf{ApxCount}_{N,w}$). \ We leverage this symmetry to
view $\ell (t)$, a Laurent polynomial in $t$, as an ordinary univariate
polynomial $r$ in $t+1/t$ of degree $\deg(p)$.
We know that $r(2)=\ell(1)=p(2w, 2w)\geq 2$,
while for all $k \in [2.5, N/w+w/N]$, we know that
$r(k) \in [0, 1]$.
It then follows from classical
results in approximation theory that this univariate polynomial
must have degree $\Omega\bigl(\sqrt{N/w}\bigr)$.
\
Returning to integrality issues, to obtain a polynomial
whose behavior we can control at non-integer points,
we use a different symmetrization argument (dating
back at least to work of Shi \cite{shi})
that we call
\textquotedblleft erase-all-subscripts\textquotedblright\ symmetrization (see \cref{lem:eraseallsubscripts}).
This symmetrization yields a bivariate polynomial $q$ of the same degree as $p$ that is bounded in $[0,1]$ at all \emph{real-valued}
inputs in $[0, N] \times [0, N]$. However, while
we have more control over $q$'s values at non-integer
inputs relative to $p$,
we have
\emph{less} control over $q$'s values at integer
inputs relative to $p$, and
this introduces additional challenges. (These
challenges are not merely annoyances; they are
why the $\mathsf{SBQP}$ complexity of $\mathsf{AND}_2 \circ \mathsf{ApxCount}_{N,w}$
is $T=\Theta\bigl(\min\bigl\{w,\sqrt{N/w}\bigr\}\bigr)$,
and not $\Theta\bigl(\sqrt{N/w}\bigr)$).
Ultimately,
both types of symmetrization play an important role in our analysis, as we use $p$ to
bound $q$ when the polynomials have degree $o(w)$, using tools from approximation
theory and Chernoff bounds.
\subsection{Second result: Approximate counting with quantum samples}
\label{s:secondresultintro}
Our second result resolves the complexity of $\mathsf{ApxCount}_{N,w}$ in a different generalization of the quantum query model, in which the algorithm is given access to certain (trusted) quantum states.
\para{Quantum samples.} In practice, when
trying to estimate the size of a set $S\subseteq\left[ N\right] $, often we
can do more than make membership queries to $S$. \ At the least, often we
can efficiently generate nearly uniform \textit{samples} from $S$, for
instance by using Markov Chain Monte Carlo techniques. \ To give two
examples, if $S$ is the set of perfect matchings in a bipartite graph, or
the set of grid points in a high-dimensional convex body, then we can
efficiently sample $S$ using the seminal algorithms of Jerrum, Sinclair, and
Vigoda \cite{jsv} or of Dyer, Frieze, and Kannan \cite{dfk},\ respectively.
The natural quantum generalization of uniform sampling from a set $S$ is \emph{QSampling} $S$---a term coined in 2003 by Aharonov and Ta-Shma \cite{at}, and which means that we can approximately prepare the uniform superposition%
\begin{equation}
\left\vert S\right\rangle := \frac{1}{\sqrt{\left\vert S\right\vert }}%
\sum_{i\in S}\left\vert i\right\rangle
\end{equation}
via a polynomial-time quantum algorithm (where \textquotedblleft
polynomial\textquotedblright\ here means $\mathrm{polylog}(N)$). \ Because
we need to uncompute garbage, the ability to prepare $\left\vert
S\right\rangle $ as a coherent superposition is a more stringent requirement
than the ability to classically sample from $S$.
Indeed, Aharonov and Ta-Shma \cite{at}
showed that the ability to QSample lends considerable power:
all problems in the complexity class $\mathsf{SZK}$ (which
contains problems that are widely believed
be hard on average \cite{goldreich1993perfect, goldwasser1989knowledge, micciancio2003statistical, goldreich2000limits, peikert2008noninteractive})
can be efficiently reduced to the task of \emph{QSampling} some set
that can be \emph{classically} sampled in polynomial time.
To be clear, QSampling supposes that the algorithm
is given trusted copies of $|S\>$; unlike in the $\mathsf{QMA}$ setting, the state need not be ``verified'' by the algorithm.
On the other hand, Aharonov and Ta-Shma \cite{at},\ and Grover and Rudolph
\cite{groverrudolph}, observed that many interesting sets $S$\ can be
efficiently QSampled as well.\footnote{In particular, this holds for all sets $S$\ such that we
can approximately count not only $S$ itself, but also the restrictions of $S$
obtained by fixing bits of its elements.
So in
particular, the set of perfect matchings in a bipartite graph, and the set
of grid points in a convex body, can both be efficiently QSampled.
There
are other sets that can be QSampled but not because of this reduction. \ A
simple example would be a set $S$\ such that $\left\vert S\right\vert \geq%
\frac{N}{\mathrm{polylog}N}$: in that case we can efficiently prepare $%
\left\vert S\right\rangle $\ using postselection, but approximately counting
$S$'s restrictions might be hard.}
\para{QSampling via unitaries.}
In many applications (such as when $S$ is the set of perfect matchings in a bipartite graph or grid points in a convex body),
the reason an algorithm can QSample $S$ is because
it is possible to efficiently construct a quantum circuit
implementing a unitary operator $U$ that
prepares the state $|S\>$.
Access to this unitary $U$ potentially conveys
substantially more power than QSampling alone.
For example, access to $U$ conveys (in a black box manner)
the ability not only to QSample, but also to perform
reflections about $|S\>$: that is, to apply the unitary
transformation
\begin{equation}
\mathcal{R}_{S}:=\mathbbold{1}-2|S\>\langle S|,
\end{equation}%
which has eigenvalue $-1$ for $|S\>$ and eigenvalue $+1$ for all states
orthogonal to $|S\>$. \
More concretely, let $U$ be the unitary that performs the
map $U|0\>=|S\>$, for some canonical starting state $|0\>$. \ Since we know
the circuit $U$, we can also implement $U^{\dagger }$, by reversing the
order of all the gates and replacing all the gates with their adjoints. Then
$\mathcal{R}_{S}$ is simply
\begin{equation}
\mathcal{R}_{S}=\mathbbold{1}-2|S\>\langle S|=U\left( \mathbbold{1}%
-2|0\>\langle 0|\right) U^{\dagger }.
\end{equation}
Note that \textit{a priori}, QSamples and reflections about $|S\>$ could be
incomparable resources; it is not obvious how to simulate either one using
the other. \ On the other hand, it is known how to apply a quantum channel
that is $\varepsilon $-close to $\mathcal{R}_{S}$ (in the diamond norm)
using $\Theta (1/\varepsilon )$ copies of $|S\>$~\cite{lmr:pca,KLL+17}.
Access to a quantum circuit computing $U$ also permits an algorithm to efficiently apply $U$ on inputs
that do not produce the state $|S\>$, to construct
a controlled version of $U$, etc.
\para{Results.} As previously mentioned,
Aharonov and Ta-Shma \cite{at}
showed that the ability to QSample lends considerable power,
including the ability to efficiently solve $\mathsf{SZK}$-complete problems.
It is natural to ask just how much power the ability to QSample conveys.
In particular, can one extend the result of Aharonov and Ta-Shma \cite{at}
from any problem in $\mathsf{SZK}$ to any problem in $\mathsf{SBP}$?
Equivalently stated, can one
solve approximate counting efficiently,
using \textit{any} combination of $\mathrm{polylog}(N)$ queries and applications of a unitary $U$ that permits QSampling?\footnote{%
We thank Paul Burchard (personal communication) for bringing this question
to our attention.}
In this work, we show that the answer is no.
We begin by focusing on the slightly simplified
setting where the algorithm
is only permitted to perform membership queries,
QSamples, and reflections about the state $|S\>$.
\begin{restatable}{theorem}{main}
\label{thm:main}
Let $Q$ be a quantum algorithm that makes $T$ queries to the membership oracle for $S$,
and uses a total of $R$ copies of $|S\>$ and reflections about $|S\>$.
If $Q$ decides whether $\left\vert S\right\vert =w$\ or $\left\vert S\right\vert =2w$
with high probability, promised that one of those is the case, then either
\begin{equation}
T=\Omega\left( \sqrt{\frac{N}{w}}\right) \qquad \textrm{or} \qquad
R=\Omega\left( \min\left\{ w^{1/3},\sqrt{\frac{N}{w}}\right\} \right).
\end{equation}
\end{restatable}
This is proved in \Cref{sec:dualpoly}.
So if (for example) we set $w:=N^{3/5}$, then any quantum algorithm must
either query $S$, or use the state $\left\vert S\right\rangle $\ or
reflections about $\left\vert S\right\rangle $, at least $\Omega (N^{1/5})$\
times.
Put another way, \Cref{thm:main} means that unless $w$ is very small ($w \leq \mathrm{polylog}(N))$) or extremely large ($w\geq N/\mathrm{polylog}(N)$), the ability to QSample $S$, reflect about $|S\>$, and determine membership in $S$ is not sufficient to approximately count $S$ efficiently. Efficient quantum algorithms for approximate counting will have to leverage additional structure of $S$, beyond the ability to QSample, reflect about $|S\>$, and determine membership in $S$.
In \Cref{thm:unitary} of \Cref{sec:unitary}, we then strengthen \Cref{thm:main} to hold
not only against algorithms
that can QSample
and reflect about $|S\>$ (in addition
to performing membership queries to $S$),
but also against all algorithms that are given
access to
a specific unitary $U$ that
conveys the power to QSample
and reflect about $|S\>$.\footnote{To be precise,
the unitary $U$ to which the lower bound of \Cref{thm:unitary} applies maps a canonical starting state to $|S\>|S\>$.
As we explain in \Cref{sec:unitary}, such a unitary suffices to implement QSampling, reflections
about $|S\>$, etc.,
since the register containing the second
copy of $|S\>$ can simply be ignored.}
\medskip
Finally, we prove that the lower bounds in \Cref{thm:main} and \Cref{thm:unitary} are optimal. \ As
mentioned before, Brassard et al.~\cite{bht:count} gave a quantum algorithm
to solve the problem using $T=O(\sqrt{{N}/{w}})$\ queries alone, which
proves the optimality of the lower bound on the number of queries. On the other hand, it's easy to solve the problem using $O\left( \sqrt{w}%
\right) $\ copies of $\left\vert S\right\rangle $ alone, by simply measuring
each copy of $\left\vert S\right\rangle $\ in the computational basis and
then searching for birthday collisions. \ Alternately, we can solve the
problem using $O\bigl(\frac{N}{w}\bigr)$\ copies of $\left\vert
S\right\rangle $ alone, by projecting onto the state $|\psi \>=\frac{1}{%
\sqrt{N}}\left( \left\vert 1\right\rangle +\cdots +\left\vert N\right\rangle
\right) $ or its orthogonal complement. \ This measurement succeeds with
probability $|\langle S|\psi \>|^{2}=\frac{|S|}{N}$, so we can approximate $|S|$ by
simply counting how many measurements succeed.
In \Cref{UPPER} we improve on these algorithms by using samples \emph{and}
reflections, and thereby establish that \Cref{thm:main} and \Cref{thm:unitary} are tight.
\begin{restatable}{theorem}{alg}
\label{thm:alg}
There is a quantum algorithm that solves $\mathsf{ApxCount}_{N,w}$ with high probability using
$R$ copies of $|S\>$ and reflections about $|S\>$, where $R = O\left( \min \left\{ w^{1/3} , \sqrt{\frac{N}{w}} \right\} \right)$.
\end{restatable}
\para{The Laurent polynomial method.}In our
view, at least as interesting as \Cref{thm:main} is the technique used to
achieve it. \ In 1998, Beals et al.\ \cite{bbcmw}\ famously observed that, if
a quantum algorithm $Q$\ makes $T$ queries to an input $X$, then $Q$'s
acceptance probability can be written as a real multilinear polynomial in
the bits of $X$, of degree at most $2T$. \ And thus, crucially, if we want
to \textit{rule out} a fast quantum algorithm to compute some function $%
f(X) $, then it suffices to show that any real polynomial $p$\
that approximates $f$\ pointwise must have high degree. \ This general
transformation, from questions about quantum algorithms to questions about
polynomials, has been used to prove many results that were not known
otherwise at the time, including the quantum lower bound for the collision
problem \cite{aar:col,as}\ and the first direct product theorems for quantum
search \cite{aar:adv,ksw}.
In our case, even in the simpler model with only queries and samples (and no
reflections), the difficulty is that the quantum algorithm starts with many
copies of the state $\left\vert S\right\rangle $. \ As a consequence of
this---and specifically, of the ${1}/\sqrt{\left\vert S\right\vert }$\
normalizing factor in $\left\vert S\right\rangle $---when we write the
average acceptance probability of our algorithm as a function of $\left\vert
S\right\vert $, we find that we get a \textit{Laurent polynomial}: a
polynomial that can contain both positive and negative integer powers of $%
\left\vert S\right\vert $. \ The degree of this polynomial (the highest
power of $\left\vert S\right\vert $) encodes the sum of the number of
queries, the number of copies of $\left\vert S\right\rangle $, and the
number of uses of $\mathcal{R}_S$, while the \textquotedblleft
anti-degree\textquotedblright\ (the highest power of ${\left\vert
S\right\vert^{-1} }$) encodes the sum of the number of copies of $\left\vert
S\right\rangle $ and number of uses of $\mathcal{R}_S$.
This is described more precisely in \Cref{sec:Laurent}.
We're thus faced with the
task of lower-bounding the degree and the anti-degree of a Laurent
polynomial that's bounded in $[0,1]$ at integer points and that encodes the
approximate counting problem.
We then lower bound the degree of Laurent polynomials that approximate $%
\mathsf{ApxCount}_{N,w}$, showing that degree $\Omega \bigl(\min \bigl\{%
w^{1/3},\sqrt{{N}/{w}}\bigr\}\bigr)$ is necessary. \ We give two very different lower bound arguments. \ The first approach, which we call the
\textquotedblleft explosion argument,\textquotedblright\ is shorter but
yields suboptimal lower bounds, whereas the second approach using
\textquotedblleft dual polynomials\textquotedblright\ yields the optimal
lower bound.
There are two aspects of this that we find surprising: first, that Laurent
polynomials appear at all, and second, that they seem to appear in a
completely different way than they appear in our other result about $\mathsf{QMA}$ (\Cref{thm:sbqp}),
despite the close connection between the two statements. \
For \Cref{thm:main}, Laurent polynomials are needed just to describe the quantum
algorithm's acceptance probability, whereas for \Cref{thm:sbqp}, ordinary (bivariate)
polynomials sufficed to describe this probability; Laurent polynomials appeared only when we restricted a
bivariate polynomial to a hyperbola in the plane. \
In any case, the coincidence suggests that the \textquotedblleft Laurent polynomial
method\textquotedblright\ might be useful for other problems as well.\footnote{Since writing this, a third application of the Laurent polynomial method was discovered by the third author \cite{kretschmer}: a simple proof that the $\mathsf{AND}$-$\mathsf{OR}$ tree $\mathsf{AND}_m \circ \mathsf{OR}_n$ has approximate degree $\widetilde{\Omega}(\sqrt{mn})$.}
Before describing our techniques at a high level, observe that there are
\emph{rational} functions\footnote{A rational function of degree $d$ is of the form $\frac{p(x)}{q(x)}$, where $p$ and $q$ are both real polynomials of degree at most $d$.} of degree $O(\log (N/w))$ that approximate $%
\mathsf{ApxCount}_{N,w}$. \ This follows, for example, from Aaronson's $%
\mathsf{PostBQP}=\mathsf{PP}$ theorem \cite{aar:pp}, or alternately from
the classical result of Newman \cite{newman} that for any $k>0$, there is a
rational polynomial of degree $O(k)$ that pointwise approximates the sign
function on domain $[-n,-1]\cup \lbrack 1,n]$ to error $1-n^{-1/k}$. \ Thus,
our proof relies on the fact that Laurent polynomials are an extremely
special kind of rational function.
We also remark that in the randomized classical setting, the complexity of $\mathsf{ApxCount}_{N,w}$ with queries and uniform (classical) samples is easily characterized without such powerful techniques. \
Either $O(N/w)$ queries or $O(\sqrt{w})$ samples are sufficient, and furthermore either $\Omega(N/w)$ queries or $\Omega(\sqrt{w})$ samples are necessary. \
For completeness, we provide a sketch of these bounds in \Cref{sec:classical_samples_queries}.
\para{Overview of the explosion argument.}
Our first proof (in \Cref{sec:explosion}) uses
an \textquotedblleft explosion argument\textquotedblright\
that, as far as we know, is new in quantum query complexity. \ We separate
out the purely positive degree\footnote{%
Throughout this paper we allow any \textquotedblleft purely positive
degree\textquotedblright\ Laurent polynomial and any \textquotedblleft
purely negative degree\textquotedblright\ Laurent polynomial to include a
constant (degree zero) term.} and purely negative degree parts of our
Laurent polynomial as $q\left( \left\vert S\right\vert \right) =u\left(
\left\vert S\right\vert \right) +v({1}/{\left\vert S\right\vert })$, where $%
u $ and $v$ are ordinary polynomials. \ We then show that, if $u$ and $v$
both have low enough degree, namely $\deg \left( u\right) =o\bigl(\sqrt{{N}/{%
w}}\bigr)$ and $\deg \left( v\right) =o\left( w^{1/4}\right) $, then we get
\textquotedblleft unbounded growth\textquotedblright\ in their values. \
That is: for approximation theory reasons, either $u$ or $v$ must attain
large values, far outside of $\left[ 0,1\right] $, at some integer values of
$\left\vert S\right\vert $. \ But that means that, for $q$ itself to be
bounded in $\left[ 0,1\right] $\ (and thus represent a probability), the
other polynomial must \textit{also} attain large values. \ And that, in
turn, will force the first polynomial to attain even larger values, and so
on forever---thereby proving that these polynomials could not have existed.
\para{Overview of the method of dual polynomials.}
Our second argument (in \Cref{sec:dualpoly}) obtains the (optimal) lower bound stated
in \Cref{thm:main}, via a novel adaptation of the so-called \emph{method of
dual polynomials}.
With this method, to lower-bound the approximate degree of a Boolean
function $f$, one exhibits an explicit \emph{dual polynomial} $\psi $ for $f$%
, which is a dual solution to a certain linear program. Roughly speaking, a
dual polynomial $\psi $ is a function mapping the domain of $f$ to $\mathbb{R%
}$ that is (a) uncorrelated with any polynomial of degree at most $d$, and
(b) well-correlated with $f$.
Approximating a univariate function $g$ via low-degree Laurent polynomials
is also captured by a linear program, but the linear program is more
complicated because Laurent polynomials can have negative-degree terms. \ We
analyze the value of this linear program in two steps.
In Step 1, we transform the linear program so that it refers only to
ordinary polynomials rather than Laurent polynomials. \ Although simple,
this transformation is crucial, as it lets us bring techniques developed for
ordinary polynomials to bear on our goal of proving Laurent polynomial
degree lower bounds.
In Step 2, we explicitly construct an optimal dual witness to the
transformed linear program from Step 1. \ We\ do so by first identifying two
weaker dual witnesses: $\psi _{1}$, which witnesses that \emph{ordinary} (i.e., purely positive degree)
polynomials encoding approximate counting require degree at least $\Omega
\bigl( \sqrt{N/w}\bigr) $, and $\psi _{2}$, which witnesses that
purely negative degree polynomials
encoding approximate counting require degree $\Omega (w^{1/3})$. \ The first
witness is derived from prior work of Bun and Thaler \cite{bt13} (who refined earlier work of {\v{S}}palek \cite{spalek}), while the
second builds on a non-constructive argument of Zhandry \cite{zhandry}.
Finally, we show how to \textquotedblleft glue together\textquotedblright\ $%
\psi _{1}$ and $\psi _{2}$, to get a dual witness $\psi $ showing that any
general Laurent polynomial that encodes approximate counting must have
either positive degree $\Omega \bigl( \sqrt{N/w}\bigr) $ or negative degree
$\Omega (w^{1/3})$.
\para{Overview of the upper bound.}To recap, %
\Cref{thm:main} shows that any quantum algorithm for $\mathsf{ApxCount}%
_{N,w} $ needs either $\Theta (\sqrt{N/w})$ queries or $\Theta \bigl(\min %
\bigl\{w^{1/3},\sqrt{{N}/{w}}\bigr\}\bigr)$ samples and reflections. \ Since
we know from the work of Brassard, H{\o }yer, Tapp~\cite{bht:count} that the
problem can be solved with $O(\sqrt{N/w})$ queries alone, it remains only to
show the matching upper bound using samples and reflections, which we
describe in \Cref{sec:upper}.
First we describe a simple algorithm that uses $O(\sqrt{N/w})$ samples and
reflections. If we take one copy of $\left\vert S\right\rangle $, and
perform a projective measurement onto $|\psi \>=\frac{1}{\sqrt{N}}\left(
\left\vert 1\right\rangle +\cdots +\left\vert N\right\rangle \right) $ or
its orthogonal complement, the measurement will succeed with probability $|%
\langle S|\psi \>|^{2}=\left\vert S\right\vert /N$. \ Thus $O(N/w)$ repetitions of this will allow us to distinguish the probabilities $w/N$
and $2w/N$. We can improve this by using amplitude amplification~\cite{BHMT02} and only make $O(\sqrt{N/w})$ repetitions. \
Our second algorithm solves the problem with $O(w^{1/3})$ reflections and
samples and is based on the quantum collision-finding algorithm~\cite{BHT98}%
. \ We first use $O(w^{1/3})$ copies of $|S\>$ to learn $w^{1/3}$ distinct
elements in $S$. \ We now know a fraction of elements in $S$, and this
fraction is either $w^{-2/3}$ or $\frac{1}{2}w^{-2/3}$. \ We then use
amplitude amplification (or quantum counting) to distinguish these two cases,
which costs $O(w^{1/3})$ repetitions, where each repetition uses a
reflection about $|S\>$.
\section{Preliminaries}
\label{sec:prelim}
In this section we introduce some definitions and known facts about
polynomials and complexity classes.
\subsection{Approximation theory}
\label{s:approxprelim}
We will use several results from approximation theory,\ each of which has
previously been used (in some form) in other applications of the polynomial
method to quantum lower bounds. \ We start with the basic inequality of A.A.
Markov \cite{markov1890question}.
\begin{lemma}[Markov]
\label{markovlem}Let $p$\ be a real polynomial, and suppose that%
\begin{equation}
\max_{x,y\in\left[ a,b\right] }\left\vert p\left( x\right) -p\left( y\right)
\right\vert \leq H.
\end{equation}
Then for all $x\in\left[ a,b\right] $, we have
\begin{equation}
\left\vert p^{\prime}\left( x\right) \right\vert \leq\frac{H}{b-a}\deg\left(
p\right) ^{2},
\end{equation}
where $p^{\prime}(x)$ is the derivative of $p$ at $x$.
\end{lemma}
We'll also need a bound that was explicitly stated by Paturi \cite{paturi},
and which amounts to the fact that, among all degree-$d$
polynomials that are bounded within a given range, the Chebyshev polynomials
have the fastest growth outside that range.
\begin{lemma}[Paturi]
\label{paturilem}Let $p$\ be a real polynomial, and suppose that $\left\vert
p\left( x\right) \right\vert \leq1$\ for all $\left\vert x\right\vert \leq1$%
. \ Then for all $x\leq1+\mu$, we have%
\begin{equation}
\left\vert p\left( x\right) \right\vert \leq\exp\left( 2\deg\left( p\right)
\sqrt{2\mu+\mu^{2}}\right) .
\end{equation}
\end{lemma}
We now state a useful corollary of \Cref{paturilem}, which says (in effect)
that slightly shrinking the domain of a low-degree real polynomial can only
modestly shrink its range.
\begin{corollary}
\label{paturicor}Let $p$\ be a real polynomial of degree $d$, and suppose
that%
\begin{equation}
\max_{x,y\in\left[ a,b\right] }\left\vert p\left( x\right) -p\left( y\right)
\right\vert \geq H.
\end{equation}
Let $\varepsilon\leq\frac{1}{100d^{2}}$\ and $a^{\prime}:=a+\varepsilon%
\left( b-a\right) $. \ Then%
\begin{equation}
\max_{x,y\in\left[ a^{\prime},b\right] }\left\vert p\left( x\right) -p\left(
y\right) \right\vert \geq\frac{H}{2}.
\end{equation}
\end{corollary}
\begin{proof}
Suppose by contradiction that%
\begin{equation}
\left\vert p\left( x\right) -p\left( y\right) \right\vert <\frac{H}{2}%
\end{equation}
for all $x,y\in\left[ a^{\prime},b\right] $. \ By affine shifts, we can
assume without loss of generality that $\left\vert p\left( x\right)
\right\vert <\frac{H}{4}$\ for all $x\in\left[ a^{\prime},b\right] $. \ Then
by \Cref{paturilem}, for all $x\in\left[ a,b\right] $\ we have%
\begin{equation}
\left\vert p\left( x\right) \right\vert <\frac{H}{4}\cdot\exp\left(
2d\sqrt{2\left( \frac{1}{1-\varepsilon}-1\right) +\left( \frac
{1}{1-\varepsilon}-1\right) ^{2}}\right) \leq\frac{H}{2}.
\end{equation}
But this violates the hypothesis.
\end{proof}
We will also need a bound that relates the range of a low-degree polynomial
on a discrete set of points to its range on a continuous interval. \ The
following lemma generalizes a result due to Ehlich and Zeller \cite{ez} and
Rivlin and Cheney \cite{rc}, who were interested only in the case where the
discrete points are evenly spaced.
\begin{lemma}
\label{ezrclem}Let $p$\ be a real polynomial of degree at most $\sqrt{k}$,
and let $0=z_{1}<\cdots<z_{M}=k$\ be a list of points such that $%
z_{i+1}-z_{i}\leq1$\ for all $i$ (the simplest example being the integers $%
0,\ldots,k$). \ Suppose that%
\begin{equation}
\max_{x,y\in\left[ 0,k\right] }\left\vert p\left( x\right) -p\left( y\right)
\right\vert \geq H.
\end{equation}
Then%
\begin{equation}
\max_{i,j}\left\vert p\left( z_{i}\right) -p\left( z_{j}\right) \right\vert
\geq\frac{H}{2}.
\end{equation}
\end{lemma}
\begin{proof}
Suppose by contradiction that%
\begin{equation}
\left\vert p\left( z_{i}\right) -p\left( z_{j}\right) \right\vert
<\frac{H}{2}%
\end{equation}
for all $i,j$. \ By affine shifts, we can assume without loss of generality
that $\left\vert p\left( z_{i}\right) \right\vert <\frac{H}{4}$\ for all
$i$. \ Let%
\begin{equation}
c:=\max_{x\in\left[ 0,k\right] }\frac{\left\vert p\left( x\right)
\right\vert }{H/4}.
\end{equation}
If $c\leq1$, then the hypothesis clearly fails, so assume $c>1$. \ Suppose
that the maximum, $\left\vert p\left( x\right) \right\vert =\frac{cH}{4}$,
is achieved between $z_{i}$\ and $z_{i+1}$. \ Then by basic calculus, there
exists an $x^{\ast}\in\left[ z_{i},z_{i+1}\right] $\ such that%
\begin{equation}
\left\vert p^{\prime}\left( x^{\ast}\right) \right\vert >\frac{2\left(
c-1\right) }{z_{i+1}-z_{i}}\cdot\frac{H}{4}\geq\frac{\left( c-1\right)
H}{2}.
\end{equation}
So by \Cref{markovlem},%
\begin{equation}
\frac{\left( c-1\right) H}{2}<\frac{cH/4}{k}\deg\left( p\right) ^{2}.
\end{equation}
Solving for $c$, we find%
\begin{equation}
c<\frac{2k}{2k-\deg\left( p\right) ^{2}}\leq2.
\end{equation}
But if $c<2$, then $\max_{x\in\left[ 0,k\right] }\left\vert p\left(
x\right) \right\vert <\frac{H}{2}$, which violates the hypothesis.
\end{proof}
We also use a related inequality due to Coppersmith and Rivlin~\cite%
{coppersmith-rivlin} that bounds a polynomial on a continuous interval in
terms of a bound on a discrete set of points, but now with the weaker
assumption that the degree is at most $k$, rather than $\sqrt{k}$. \ This
gives a substantially weaker bound.
\begin{lemma}[Coppersmith and Rivlin]
\label{lem:coppersmith_rivlin} Let $p$ be a real polynomial of degree at
most $k$, and suppose that $\left\vert p(x)\right\vert \leq 1$ for all
integers $x\in \{0,1,\ldots ,k\}$. \ Then there exist universal constants $%
a,b$ such that for all $x\in \lbrack 0,k]$, we have
\begin{equation}
\left\vert p(x)\right\vert \leq a\cdot \exp \left( b\,\deg (p)^{2}/k\right) .
\end{equation}
\end{lemma}
\subsection{Symmetric polynomials}
\para{Univariate symmetrizations.}
Our starting point is the well-known \textit{symmetrization lemma} of Minsky
and Papert \cite{mp} (see also Beals et al.\ \cite{bbcmw}\ for its
application to quantum query complexity), by which we can often reduce
questions about multivariate polynomials to questions about univariate ones.
\begin{lemma}[Minsky--Papert symmetrization]
\label{symlem} Let $p:\left\{ 0,1\right\} ^{N}\rightarrow\mathbb{R}$\ be a
real multilinear polynomial of degree $d$, and let $q:\{0,1,\ldots,N\}\to
\mathbb{R}$ be defined as
\begin{equation}
q\left( k\right) :=\E_{\left\vert X\right\vert =k}\left[ p\left( X\right) %
\right] .
\end{equation}
Then $q$ can be written as a real polynomial in $k$ of degree at most $d$.
\end{lemma}
We now introduce a different, lesser known
notion of symmetrization,
which we call the \emph{erase-all-subscripts} symmetrization
for reasons to be explained shortly. This symmetrization previously appeared in \cite{shi} under the name ``linearization,'' and it is also equivalent to the noise operator used in analysis of Boolean functions \cite[Definition 2.46]{odonnell}.
\begin{lemma}[Erase-all-subscripts
symmetrization]
Let $p:\left\{ 0,1\right\} ^{N}\rightarrow\mathbb{R}$\ be a
real multilinear polynomial of degree $d$,
and for
any real number $k \in [0, 1]$,
let $M_{k}$ denote
the distribution over $\{0, 1\}^N$, wherein each coordinate is selected
independently to be 1 with probability $k$.
Let $q:[0, 1] \to
\mathbb{R}$ be defined as
\begin{equation}
q\left( k\right) :=\E_{X \sim M_{k}}\left[ p\left( X\right) %
\right] .
\end{equation}
Then $q$ can be written as a real polynomial in $k$ of degree at most $d$.
\label{lem:eraseallsubscripts}
\end{lemma}
\begin{proof} (see, for example, \cite[Proof of Theorem 3]{STT12}).
Given the multivariate polynomial expansion of $p$,
we can obtain $q$ easily just by ``erasing all the subscripts in each variable''. For example, if
$p(x_1, x_2, x_3)= 2 x_1 x_2 + x_2 x_3 + x_2$, we replace every $x_i$ with $k$ to obtain
$q(k)=2k\cdot k+ k\cdot k + k=3k^2 + k$.
This follows from linearity of expectation along with the fact that $M_k$
is defined to be the
product distribution wherein
each coordinate has expected value $k$.
\end{proof}
We highlight the following key difference between Minsky--Papert symmetrization
and the erase-all-subscripts symmetrization. Let $p:\left\{ 0,1\right\} ^{N}\rightarrow [0, 1]$
be a real multivariate polynomial whose evaluations at Boolean inputs are in
$[0, 1]$, i.e., for all $x\in\{0,1\}^n$, we have $p(x)\in[0,1]$.
If $q$ is the erase-all-subscripts symmetrization of $p$,
then $q$ takes values in $[0,1]$ at all \emph{real-valued}
inputs in $[0, 1]$: $q(k) \in [0,1]$ for all $k \in [0,1]$.
If $q$ is the Minsky--Papert symmetrization of $p$,
then it is only guaranteed to take values in $[0,1]$ at
\emph{integer-valued} inputs in $[0, N]$, i.e., $q(k)\in [0,1]$
is only guaranteed to hold at $k\in\{0,1,\ldots,N\}$.
This is the main reason we use erase-all-subscripts symmetrization
in this work.
\para{Bivariate symmetrizations.}
In this paper, it will be convenient
to consider bivariate versions of both
Minsky--Papert and erase-all-subscripts
symmetrization, and their applications to oracle separations. To this end,
define $X\in \left\{ 0,1\right\} ^{N}$, the \textquotedblleft characteristic
string\textquotedblright\ of the set $S\subseteq \left[ N\right] $, by $%
x_{i}=1$\ if $i\in S$\ and $x_{i}=0$\ otherwise. \ Let $\mathcal{O}_{S}$
denote the unitary that performs a membership query to $S$, defined as
\begin{equation}
\mathcal{O}_{S}\left\vert i\right\rangle \left\vert b\right\rangle
=(1-2bx_{i})\left\vert i\right\rangle \left\vert b\right\rangle
\end{equation}%
for any index $i\in \lbrack N]$ and bit $b\in \{0,1\}$.
Because we study oracle intersection problems, it is often convenient to
think of an algorithm as having access to \textit{two} oracles, wherein the
first bit in the oracle register selects the choice of oracle. \ As a
consequence, we need a slight generalization of a now well-established fact
in quantum complexity: that the acceptance probability of a quantum
algorithm with an oracle can be expressed as a polynomial in the bits of the
oracle string.
\begin{lemma}[Symmetrization with two oracles]
\label{lem:symmetrization} Let $Q^{\mathcal{O}_{S_{0}},\mathcal{O}_{S_{1}}}$
be a quantum algorithm that makes $T$ queries to a pair of membership
oracles for sets $S_{0},S_{1}\subseteq \lbrack N]$. \ Let $D_{\mu }$ denote
the distribution over subsets of $[N]$ wherein each element is selected
independently with probability $\frac{\mu }{N}$. \ Then there exist
bivariate real polynomials $q(s,t)$ and $p(x,y)$ of degree at most $2T$
satisfying:
\begin{align*}
\textrm{for all real numbers } s, t \in [0, N],& \quad
q(s,t)=\E_{\substack{ S_{0}\sim D_{s}, \\ S_{1}\sim D_{t}}}\left[ \Pr [Q^{%
\mathcal{O}_{S_{0}},\mathcal{O}_{S_{1}}}\text{ accepts}]\right], \textrm{ and} \\
\textrm{for all integers } x, y \in \{0, 1, \dots, N\},& \quad
p(x,y)=\E_{\substack{ |S_{0}|=x, \\ |S_{1}|=y}}\left[ \Pr [Q^{\mathcal{O}%
_{S_{0}},\mathcal{O}_{S_{1}}}\text{ accepts}]\right].
\end{align*}
\end{lemma}
\begin{proof}
Take $X = X_0|X_1$ to be the concatenation of the characteristic strings of the two oracles, and let $S \subseteq [2N]$ be such that $X$ is the characteristic string of $S$. Then, Lemma 4.2 of Beals et al.\ \cite{bbcmw} tells us that there is a real multilinear polynomial $r(X)$ of degree at most $2T$ in the bits of $X$ such that $r(X) = \Pr[ Q^{\mathcal{O}_S} \text{ accepts}]$.
Observe that $r$ has a meaningful probabilistic interpretation over arbitrary inputs in $[0, 1]$. A vector $X \in [0, 1]^{2N}$ of probabilities corresponds to a distribution over $\{0,1\}^{2N}$ wherein each bit is chosen from a Bernoulli distribution with the corresponding probability. Because $r$ is multilinear, $r$ in fact computes the expectation of the acceptance probability over this distribution. In particular, the polynomial
\begin{equation}
q(s, t) = r\bigg(\underbrace{\frac{s}{N},\ldots,\frac{s}{N}}_{N \text{ times}}, \underbrace{\frac{t}{N},\ldots,\frac{t}{N}}_{N \text{ times}}\bigg) = \E_{\substack{S_0 \sim D_s,\\S_1 \sim D_t}}\left[\Pr[ Q^{\mathcal{O}_{S_0},\mathcal{O}_{S_1}} \text{ accepts}] \right]
\end{equation}
corresponds to selecting $S_0 \sim D_s$ and $S_1 \sim D_t$. The total degree of $q$ is obviously
at most the degree of $r$, by the same reasoning as in the proof of \Cref{lem:eraseallsubscripts}.
To construct $p$, we apply the symmetrization lemma of Minsky and Papert \cite{mp} to symmetrize $r$, first with respect to $X_0$, then with respect to $X_1$:
\begin{equation}
p_0(x, X_1) = \E_{|S_0|=x} r(X_0,X_1) = \E_{|S_0|=x}\left[\Pr[ Q^{\mathcal{O}_{S_0},\mathcal{O}_{S_1}} \text{ accepts}] \right]
\end{equation}
\begin{equation}
p(x, y) = \E_{|S_1|=y} p_0(x,X_1) = \E_{\substack{|S_0|=x,\\|S_1|=y}}\left[\Pr[ Q^{\mathcal{O}_{S_0},\mathcal{O}_{S_1}} \text{ accepts}] \right]
\end{equation}
The degree of $p$ is at most the degree of $r$, due to \Cref{symlem}.
\end{proof}
We remark that, as a consequence of their definitions in \Cref{lem:symmetrization}, $p$ and $q$ satisfy:
\begin{equation}
q(s, t) = \E \left[ p(X, Y) \right],
\end{equation}
where $X$ and $Y$ are drawn from $N$-trial binomial distributions with means $s$ and $t$, respectively.
\para{Symmetric Laurent polynomials.}
Finally, we state a useful fact about Laurent polynomials:
\begin{lemma}[Symmetric Laurent polynomials]
\label{lem:symmetric} Let $\ell (x)$ be a real Laurent polynomial of positive
and negative degree $d$ that satisfies $\ell (x)=\ell (1/x)$. \ Then there exists a (ordinary) real
polynomial $q$ of degree $d$ such that $\ell (x)=q(x+1/x)$.
\end{lemma}
\begin{proof}
$\ell(x) = \ell(1/x)$ implies that the coefficients of the $x^i$ and $x^{-i}$ terms are equal for all $i$, as otherwise $\ell(x) - \ell(1/x)$ would not equal the zero polynomial. Thus, we may write $\ell(x) = \sum_{i=0}^d a_i \cdot (x^i + x^{-i})$ for some coefficients $a_i$. So, it suffices to show that $x^i + x^{-i}$ can be expressed as a polynomial in $x + 1/x$ for all $0 \le i \le d$.
We prove by induction on $i$. The case $i = 0$ corresponds to constant polynomials. For $i > 0$, by the binomial theorem, observe that $(x + 1/x)^i = x^i + x^{-i} + r(x)$ where $r$ is a degree $i - 1$ real Laurent polynomial satisfying $r(x) = r(1/x)$. By the induction assumption, $r$ can be expressed as a polynomial in $x + 1/x$, so we have $x^i + x^{-i} = (x + 1/x)^i - r(x)$ is expressed as a polynomial in $x + 1/x$.
\end{proof}
\subsection{Complexity classes}
\begin{definition}
\label{def:qma}
The complexity class $\mathsf{QMA}$ consists of the
languages $L$ for which there exists a quantum polynomial time
verifier $V$ with the following properties:
\begin{enumerate}
\item Completeness: if $x \in L$, then there exists a quantum witness state $|\psi\>$ on $\mathrm{poly}(|x|)$ qubits such that $\Pr\left[V(x, |\psi\>) \text{ accepts}\right] \ge \frac{2}{3}$.
\item Soundness: if $x \not\in L$, then for any quantum witness state $|\psi\>$ on $\mathrm{poly}(|x|)$ qubits, $\Pr\left[V(x, |\psi\>) \text{ accepts}\right] \le \frac{1}{3}$.
\end{enumerate}
\end{definition}
A quantum verifier that satisfies the above
promise for a particular language will be referred to as a $\mathsf{QMA}$
verifier or $\mathsf{QMA}$ protocol throughout.
Though $\mathsf{SBP}$ and $\mathsf{SBQP}$ can be defined in terms of
counting complexity functions, for our purposes it is easier to
work with the following equivalent definitions (see B\"ohler et al.\ \cite%
{BGM06}):
\begin{definition}
\label{def:sbp_sbqp} The complexity class $\mathsf{SBP}$ consists of the
languages $L$ for which there exists a probabilistic polynomial time
algorithm $M$ and a polynomial $\sigma$ with the following properties:
\begin{enumerate}
\item If $x \in L$, then $\Pr\left[M(x) \text{ accepts}\right] \ge
2^{-\sigma(|x|)}$.
\item If $x \not\in L$, then $\Pr\left[M(x) \text{ accepts}\right] \le
2^{-\sigma(|x|)}/2$.
\end{enumerate}
The complexity class $\mathsf{SBQP}$ is defined analogously, wherein the
classical algorithm is replaced with a quantum algorithm.
\end{definition}
A classical (respectively, quantum) algorithm that satisfies the above
promise for a particular language will be referred to as an $\mathsf{SBP}$
(respectively, $\mathsf{SBQP}$) algorithm throughout. \ Using these
definitions, a query complexity relation between $\mathsf{QMA}$
protocols and $\mathsf{SBQP}$ algorithms follows from the procedure of
Marriott and Watrous \cite{marriott}, which shows that one can exponentially
improve the soundness and completeness errors of a $\mathsf{QMA}$ protocol
without increasing the witness size. \
This relationship is now standard; see
for example \cite[Remark 6]{marriott} or \cite[Proposition 4.2]%
{sherstov2019vanishing} for a proof of the following lemma:
\begin{lemma}
\label{lem:guessing} Suppose there is a $\mathsf{QMA}$ protocol for some
problem that makes $T$ queries and receives an $m$-qubit witness. \ Then
there is a quantum query algorithm $Q$ for the same problem that makes $O(mT)$ queries, and satisfies the following:
\begin{enumerate}
\item If $x \in L$, then $\Pr\left[Q(x) \text{ accepts}\right] \ge
2^{-m}$.
\item If $x \not\in L$, then $\Pr\left[Q(x) \text{ accepts}\right] \le
2^{-10m}$.
\end{enumerate}
\end{lemma}
\section{QMA complexity of approximate counting}
\label{sec:SBPQMA}
This section establishes
an
optimal lower bound on the $\mathsf{QMA}$
complexity of approximate counting.
We first lower bound the $\mathsf{SBQP}$ complexity
of the $\mathsf{AND}_2 \circ \mathsf{ApxCount}_{N,w}$ problem (%
\Cref{thm:sbqp}).
This implies a $\mathsf{QMA}$ lower bound
for $\mathsf{ApxCount}_{N,w}$ via \Cref{lem:guessing},
but it is not quantitatively optimal.
We prove the optimal $\mathsf{QMA}$ lower bound
(\Cref{thm:qmabound}) via
\Cref{lem:betterforqma}, which
leverages additional properties
of
the $\mathsf{SBQP}$ protocol derived via
\Cref{lem:guessing}
from any $\mathsf{QMA}$ protocol with small
witness length. Finally, \Cref{cor:qma_separation}
describes new oracle separations that
are immediate consequences of \Cref{thm:qmabound} and \Cref{thm:sbqp}.
\subsection{Lower bound for \texorpdfstring{$\mathsf{SBQP}$}{SBQP} algorithms}
\label{sec:SBQP}
Our lower bound on the $\cl{SBQP}$ complexity of $\mathsf{AND}_2 \circ \mathsf{ApxCount}_{N,w}$ hinges
on the following theorem. \ The theorem
uses Laurent polynomials to prove a degree lower bound for bivariate
polynomials that satisfy an upper bound on an \textquotedblleft
L\textquotedblright -shaped pair of rectangles and a lower bound at a nearby
point:
\begin{theorem}
\label{thm:L} Let $0<w<32w<N$ and $M\geq 1$. \ Let $R_{1}=[4w,N]\times \lbrack 0,w/2]$ and
$R_{2}=[0,w/2]\times \lbrack 4w,N]$ be disjoint rectangles in the plane, and
let $L=R_{1}\cup R_{2}$. \ Let $p(x,y)$ be a real polynomial of degree $d$
with the following properties:
\begin{enumerate}
\item $p(4w, 4w) \ge 1.5 \cdot M$.
\item $0 \le p(x, y) \le 1$ for all $(x, y) \in L$.
\end{enumerate}
Then $d = \Omega(\sqrt{N/w} \cdot \log M)$.
\end{theorem}
\begin{figure}[tbp]
\centering
\begin{tikzpicture}[y=0.4cm, x=0.4cm]
\pgfmathsetmacro{\N}{20};
\pgfmathsetmacro{\w}{3};
\pgfmathsetmacro{\tw}{\w+\w};
\pgfmathsetmacro{\Np}{\N+1};
\draw[<-,thick] (\tw+0.5, \w/2) -- (\w+1,\w/2) node[anchor=east] {$R_1$};
\draw[<-,thick] (\w/2, \tw+0.5) -- (\w/2,\w+1) node[anchor=north] {$R_2$};
\fill[gray,opacity=.4] (\tw,0) -- (\tw,\w) -- (\N,\w) -- (\N,0);
\fill[gray,opacity=.4] (0,\tw) -- (\w,\tw) -- (\w,\N) -- (0,\N);
\draw[->] (0,0) -- coordinate (x axis mid) (\Np,0);
\draw[->] (0,0) -- coordinate (y axis mid) (0,\Np);
\draw[<-,thick] (\tw,\tw) -- (\tw+2,\tw+2) node[anchor=south west] {$t=1$};
\draw[<-,thick] (\w*4,\w) -- (\w*4,\w*2) node[anchor=south] {$t=8$};
\draw[<-,thick] (\N,4*\w*\w/\N) -- (\N+2,1+4*\w*\w/\N) node[anchor=west] {$t=\frac{N}{4w}$};
\draw[<-,thick] (1.5*\w,2*\w*4/3) -- (1.5*\w+2,2*\w*4/3+2) node[anchor=south west] {$(x=4wt,y=4w/t)$};
\draw (0,1pt) -- (0,-3pt) node[anchor=north] {0};
\draw (\w,1pt) -- (\w,-3pt) node[anchor=north] {$w/2$\vphantom{l}};
\draw (\tw,1pt) -- (\tw,-3pt) node[anchor=north] {$4w$};
\draw (\N,1pt) -- (\N,-3pt) node[anchor=north] {$N$};
\draw (1pt,0) -- (-3pt,0) node[anchor=east] {0};
\draw (1pt,\w) -- (-3pt,\w) node[anchor=east] {$w/2$};
\draw (1pt,\tw) -- (-3pt,\tw) node[anchor=east] {$4w$};
\draw (1pt,\N) -- (-3pt,\N) node[anchor=east] {$N$};
\node[below=0.8cm] at (x axis mid) {$x$};
\node[rotate=90, above=0.8cm] at (y axis mid) {$y$};
\node at (\w+\N/2, \w/2) {$0 \le p(x, y) \le 1$};
\node[rotate=90] at (\w/2, \w+\N/2) {$0 \le p(x, y) \le 1$};
\draw[thick,domain=1:{\N/(2*\w)},smooth,variable=\t,blue] plot ({2*\w*\t},{2*\w/\t});
\draw[thick,domain=1:{\N/(2*\w)},smooth,variable=\t,blue] plot ({2*\w/\t},{2*\w*\t});
\node at ({\w+\w},{\w+\w}) {\tiny\textbullet};
\node at (4*\w,\w) {\tiny\textbullet};
\node at (\N,4*\w*\w/\N) {\tiny\textbullet};
\end{tikzpicture}
\caption{Diagram of \Cref{thm:L} (not drawn to scale).}
\label{fig:L}
\end{figure}
\begin{proof}
Observe that if $p(x, y)$ satisfies the statement of the theorem, then so does $p(y, x)$. This is because the constraints in the statement of the theorem are symmetric in $x$ and $y$ (in particular, because $R_1$ and $R_2$ are mirror images of one another along the line $x = y$; see \Cref{fig:L}). As a result, we may assume without loss of generality that $p$ is symmetric, i.e.,\ $p(x, y) = p(y, x)$. Else, we may replace $p$ by $\frac{p(x, y) + p(y, x)}{2}$ because the set of polynomials that satisfy the inequalities in the statement of the theorem are closed under convex combinations.
Consider the hyperbolic parametric curve $(x = 4wt, y=4w/t)$ as it passes through $R_1$ (see \Cref{fig:L}). We can view the restriction of $p(x, y)$ to this curve as a Laurent polynomial $\ell(t) = p(4wt, 4w/t)$ of positive and negative degree $d$. The bound of $p(x,y)$ on all of $R_1$ implies that $|\ell(t)| \le 1$ when $t \in [8, \frac{N}{4w}]$ and that $\ell(1) \ge 1.5$ (see \Cref{fig:L}). Moreover, the condition that $p(x, y)$ is symmetric implies that $\ell(t) = \ell(1/t)$.
By \Cref{lem:symmetric} for symmetric Laurent polynomials, $\ell(t)$ can be viewed as a degree $d$ polynomial $q(t + 1/t)$. Under the transformation $s = t + 1/t$, $q$ satisfies $|q(s)| \le 1$ for $s \in [8 + 1/8, \frac{N}{4w} + \frac{4w}{N}]$ and $q(2) \ge 1.5 M$. Note that the length of the interval $[8 + 1/8, \frac{N}{4w} + \frac{4w}{N}]$ is $\Theta(N/w)$ because $w < N$. By an appropriate affine transformation of $q$, we can conclude from \Cref{paturilem} with $\mu = \Theta(w/N)$ that $d = \Omega(\sqrt{N/w} \cdot \log M)$.
\end{proof}
Why is \Cref{thm:L} useful? \ One may be tempted to apply this theorem
directly to the polynomial $p(x,y)$ obtained in \Cref{lem:symmetrization} to
conclude a degree lower bound (and thus a query complexity lower bound), as
the \textquotedblleft L\textquotedblright -shaped pair of rectangles $%
L=R_{1}\cup R_{2}$ correspond to \textquotedblleft no\textquotedblright\
instances of $\mathsf{AND}_{2}\circ \mathsf{ApxCount}_{N,w}$, while $(4w,4w)$
corresponds to a \textquotedblleft yes\textquotedblright\ instance. \
However, even though $p(x,y)$ is bounded at lattice points in $L$, it need
not be bounded along the entirety of $L$.\footnote{%
One can nevertheless use this intuition to obtain a nontrivial (though
suboptimal) lower bound by inspecting $p$ alone. Using the Markov brothers'
inequality (\Cref{markovlem}), if $\deg (p)=o(\sqrt{w})$, then the bounds on
$p(x,y)$ at lattice points in $L$ imply that $|p(x,y)|\leq 1+o_{w}(1)$ for
all $(x,y)\in L$. Thus, \Cref{thm:L} applies if $\deg (p)=o(\sqrt{w})$, so
overall we get a lower bound of $\Omega \left( \min \left\{ \sqrt{w},\sqrt{%
N/w}\right\} \right) $ for the $\mathsf{SBQP}$ query complexity of $\mathsf{%
AND}_{2}\circ \mathsf{ApxCount}_{N,w}$. See
\href{https://arxiv.org/abs/1902.02398}{arXiv:1902.02398} for details.} \
To obtain a lower bound, we
instead use the connection between the polynomials $p(x,y)$ and $q(s,t)$
from \Cref{lem:symmetrization}, and establish \Cref{thm:sbqp} from the
introduction, restated for convenience:
\sbqp*
\begin{proof}
Let $N > 32w$ (otherwise the theorem holds trivially). Since $Q$ is an $\mathsf{SBQP}$ algorithm, we may suppose that $Q$ accepts with probability at least $2\alpha$ on a ``yes'' instance and with probability at most $\alpha$ on a ``no'' instance (note that $\alpha$ may be exponentially small in $N$). Take $p(x,y)$ and $q(s, t)$ to be the symmetrized bivariate polynomials of degree at most $2T$ defined in \Cref{lem:symmetrization}. Define $L' = ([0, w] \times [0, w]) \cup ([0, w] \times [2w, N]) \cup ([2w, N] \times [0, w])$. The conditions on the acceptance probability of $Q$ for all $S_0, S_1$ that satisfy the $\mathsf{ApxCount}_{N,w}$ promise imply that $p(x, y)$ satisfies these corresponding conditions:
\begin{enumerate}
\item $1 \ge p(x, y) \ge 2\alpha$ for all $(x, y) \in \left([2w, N] \times [2w, N]\right) \cap \mathbb{Z}^2$.
\item $0 \le p(x, y) \le \alpha$ for all $(x, y) \in L' \cap \mathbb{Z}^2$.
\end{enumerate}
Our strategy is to show that if $T = o(w)$, then these conditions on $p$ imply that the polynomial $q(s, t) \cdot \frac{0.9}{\alpha}$ satisfies the statement of \Cref{thm:L} for all sufficiently large $w$. This in turn implies $T = \Omega(\sqrt{N/w})$. This allows us conclude that either $T = \Omega(w)$ or $T = \Omega(\sqrt{N/w})$, which proves the theorem.
Suppose $T = o(w)$, so that $p(x, y)$ and $q(s, t)$ both have degree $d=o(w)$. We begin by upper bounding $p(x, y)$ at the lattice points $(x, y)$ outside of $L'$.
We claim the following:
\begin{enumerate}[(a)]
\item $|p(x, y)| \le \alpha \cdot a \cdot \exp(bd^2 / w) \le \alpha \cdot a \cdot \exp(bd)$ whenever $(x, y) \in L'$ and either $x$ or $y$ is an integer, where $a$
and $b$ are the constants from \Cref{lem:coppersmith_rivlin}. This follows from \Cref{lem:coppersmith_rivlin} by fixing either $x$ or $y$ to be an integer and viewing the resulting restriction of $p(x, y)$ as a univariate polynomial in the other variable.
\item $|p(x, y)| \le \alpha \cdot a \cdot \exp(bd) \cdot \exp(2\sqrt{3}d) = \alpha \cdot a \cdot \exp((b + 2\sqrt{3})d)$ whenever $x \in [w, 2w]$, $y \in [0, w]$, and $y$ is an integer. This follows \Cref{paturilem}: consider the univariate polynomial $p(\cdot, y)$ on the intervals $[0, w]$ and $[2w, 3w]$, where it is bounded by (a).
\item $|p(x, y)| \le \alpha \cdot a \cdot \exp((b + 2\sqrt{3})d) \cdot a \cdot \exp(bd^2/w) \le \alpha \cdot a^2 \cdot \exp((2b + 2\sqrt{3})d)$ whenever $x \in [w, 2w]$ and $y \in [0, w]$. This follows from \Cref{lem:coppersmith_rivlin}: consider the univariate polynomial $p(x, \cdot)$ on the interval $[0, w]$, where it is bounded at integer points by (b).
\item $|p(x, y)| \le \alpha \cdot a^2 \cdot \exp((2b + 2\sqrt{3})d) \cdot \exp(4dy/w) = \alpha \cdot a^2 \cdot \exp((2b + 2\sqrt{3} + 4y/w)d)$ whenever $x \in [0, N]$, $y \in [w + 1, N]$, and $x$ is an integer. This follows from \Cref{paturilem}: consider the univariate polynomial $p(x, \cdot)$ on the interval $[0, w]$, where it is bounded by (a) when $x \in [0, w]$ or $x \in [2w, N]$, or bounded by (c) when $x \in [w, 2w]$.
By an affine shift, this corresponds to applying \Cref{paturilem} with $\mu = 2y/w - 2$, with the observation that $\sqrt{2\mu + \mu^2} < \mu + 2$.
\end{enumerate}
We now use this to upper bound $q(s, t)$ when $s \in [4w, N]$ and $t \in [0, w/2]$. Let $X$ and $Y$ be drawn from $N$-trial binomial distributions with means $s$ and $t$, respectively,
so that $q(s, t) = \E[p(X,Y)]$. Using the above bounds and basic probability, we have
\begin{align}
0 \leq q(s, t) = \E[p(X,Y)] &\le \alpha \cdot \bigg( \Pr[X \ge 2w, Y \le w] + \Pr[X \le 2w, Y \le w] \cdot a \cdot \exp\left(\left(b+2\sqrt{3}\right)d\right) \nonumber \\
& \quad + \sum_{y = w+1}^N \Pr[Y = y] \cdot a^2 \cdot \exp\left(\left(2b + 2\sqrt{3} + 4y/w\right)d\right) \bigg)\\
&\le \alpha \cdot \bigg(1 + \Pr[X \le 2w] \cdot a \cdot \exp\left(\left(b+2\sqrt{3}\right)d\right) \nonumber \\
& \quad + \sum_{y = w+1}^N \Pr[Y \ge y] \cdot a^2 \cdot \exp\left(\left(2b + 2\sqrt{3} + 4y/w\right)d\right)\bigg).
\end{align}
The probabilities above are easily bounded with a Chernoff bound:
\begin{align}
q(s, t)=\E[p(X,Y)] &\le \alpha \cdot \bigg( 1 + a \cdot \exp\left(\left(b+2\sqrt{3}\right)d - w/2\right) \nonumber\\
& \quad + \sum_{y = w+1}^N a^2 \cdot \exp\left(\left(2b + 2\sqrt{3} + 4y/w\right)d - y/6\right)\bigg).
\end{align}
Because $a$ and $b$ are universal constants from \Cref{lem:coppersmith_rivlin}, when $d = o(w)$, the first exponential term becomes arbitrarily small for all sufficiently large $w$. Moreover, for all sufficiently large $w$, the remaining sum becomes bounded by a geometric sum. For some constant $c$, we have
\begin{align*}
\sum_{y = w+1}^N a^2 \cdot \exp\left(\left(2b + 2\sqrt{3} + 4y/w\right)d - y/6\right) &\le \sum_{y = w+1}^\infty c \cdot \exp\left(-y/12\right)\\
&\le \frac{c}{1 - \exp(-1/12)} \cdot \exp(-w/12)\\
&= o_w(1).
\end{align*}
Thus we conclude that $0 \le q(s, t) \le \alpha \cdot (1 + o_w(1))$ when $s \in [4w, N]$ and $t \in [0, w/2]$ (i.e.,\ $(s, t) \in R_1$ in the statement of \Cref{thm:L}). By symmetry, we can conclude the same bound when $s \in [0, w/2]$ and $t \in [4w, N]$ (i.e.,\ $(s, t) \in R_2$ in the statement of \Cref{thm:L}).
Now, we lower bound $q(4w, 4w)$. Let $X$ and $Y$ be drawn from independent $N$-trial binomial distributions with mean $4w$, so that $q(4w, 4w) = \mathbb{E}\left[p(X,Y)\right]$. Then we have
\begin{align*}
\mathbb{E}\left[p(X,Y)\right] &\ge 2\alpha \cdot \Pr[X \ge 2w, Y \ge 2w]\\
&\ge 2\alpha \cdot \left(1 - \Pr[X \le 2w] - \Pr[Y \le 2w]\right)\\
&\ge 2\alpha \cdot \left(1 - 2\exp(-w/2)\right)\\
&\ge 2\alpha \cdot (1 - o_w(1))
\end{align*}
We conclude that $q(s, t) \cdot \frac{0.9}{\alpha}$ satisfies the statement of \Cref{thm:L} (with $M=1$) for all sufficiently large $w$.
\end{proof}
We remark that this lower bound is tight, i.e.,\ there exists an $\mathsf{%
SBQP}$ algorithm that makes $O\left( \min \left\{ w,\sqrt{N/w}\right\}
\right) $ queries. \ The $O(\sqrt{N/w})$ upper bound follows from the $%
\mathsf{BQP}$ algorithm of Brassard, H{\o }yer, and Tapp \cite{bht:count}. \
The $O(w)$ upper bound is in fact an $\mathsf{SBP}$ upper bound with the
following algorithmic interpretation: first, guess $w+1$ items randomly from
each of $S_{0}$ and $S_{1}$. \ Then, verify using the membership oracle that
the first $w+1$ items all belong to $S_{0}$ and that the latter $w+1$ items
all belong to $S_{1}$, accepting if and only if this is the case. \ Clearly,
this accepts with nonzero probability if and only if $\left\vert
S_{0}\right\vert \geq w+1$ and $\left\vert S_{1}\right\vert \geq w+1$.
\subsection{Lower bound for \texorpdfstring{$\mathsf{QMA}$}{QMA}}
\label{sec:QMA}
In this section, we establish the optimal $\mathsf{QMA}$ lower bound
(\Cref{thm:qmabound}).
We begin by quantitatively improving the $\mathsf{SBQP}$
lower bound for $\mathsf{AND}_2 \circ \mathsf{ApxCount}_{N,w}$ of \Cref{thm:sbqp},
under the stronger assumption that the parameter $\alpha$
in the $\mathsf{SBQP}$ protocol is not smaller than $2^{-w}$.
(In addition to a stronger
conclusion, this assumption also permits a considerably simpler
analysis than was required to prove \Cref{thm:sbqp}).
\begin{lemma}
\label{lem:betterforqma}
Consider any quantum query algorithm $Q^{\mathcal{O}_{S_{0}},\mathcal{O}_{S_{1}}}$
for $\mathsf{AND}_2 \circ \mathsf{ApxCount}_{N,w}$
that makes $T$ queries to the membership oracles
$\mathcal{O}_{S_{0}}$ and $\mathcal{O}_{S_{1}}$
for the two instances of $\mathsf{ApxCount}_{N,w}$ and
satisfies the following. For some $m=o(w)$, $\alpha=2^{-m}$, and $M \in [1, \alpha^{-1}]$:
\begin{enumerate}
\item If $x \in L$, then $\Pr\left[Q(x) \text{ accepts}\right] \ge
\alpha$.
\item If $x \not\in L$, then $\Pr\left[Q(x) \text{ accepts}\right] \le
\alpha/(2 M)$.
\end{enumerate}
Then $T = \Omega\left(\sqrt{N/w} \cdot \log M\right)$
\end{lemma}
\begin{proof}
As in the proof of \Cref{thm:sbqp}, define $L' = ([0, w] \times [0, w]) \cup ([0, w] \times [2w, N]) \cup ([2w, N] \times [0, w])$, and
take $p(x,y)$ and $q(s, t)$ to be the symmetrized bivariate polynomials of degree at most $2T$ defined in \Cref{lem:symmetrization}. $p(x, y)$ satisfies the following
properties.
\begin{enumerate}
\item[(a)] $1 \ge p(x, y) \ge \alpha$ for all $(x, y) \in \left([2w, N] \times [2w, N]\right) \cap \mathbb{Z}^2$.
\item[(b)] $0 \le p(x, y) \le \alpha/(1.5 M)$ for all $(x, y) \in L' \cap \mathbb{Z}^2$.
\item[(c)] $0 \leq p(x,y) \leq 1$ for all $(x, y) \in \left( [0,N] \times [0, N]\right) \cap \mathbb{Z}^2$.
\end{enumerate}
We use these properties to upper bound $q(s, t)$ when $s \in [4w, N]$ and $t \in [0, w/2]$. Let $X$ and $Y$ be drawn from $N$-trial binomial distributions with means $s$ and $t$, respectively,
so that $q(s, t) = \E[p(X,Y)]$. Using the above bounds and basic probability, we have
\begin{align*}
0 \leq q(s, t)=\E[p(X,Y)] &\le \alpha/(2M) \Pr[X \ge 2w, Y \le w] +
(1-\Pr[X \ge 2w, Y \le w])
\\
&\le \alpha/(2M) + 2^{-\Omega(w)} \leq (1+o(1)) \alpha/(2 M)
\end{align*}
Here, the first inequality holds by Properties (a)-(c) above,
while the second follows from a Chernoff Bound, and the third holds because
$\alpha/(2 M) \geq 2^{-o(w)}$.
Thus we conclude that $0 \le q(s, t) \le \alpha/(2M) \cdot (1 + o_w(1))$ when $s \in [4w, N]$ and $t \in [0, w/2]$ (i.e.,\ $(s, t) \in R_1$ in the statement of \Cref{thm:L}). By symmetry, we can conclude the same bound when $s \in [0, w/2]$ and $t \in [4w, N]$ (i.e.,\ $(s, t) \in R_2$ in the statement of \Cref{thm:L}).
Now, we lower bound $q(4w, 4w)$. Let $X$ and $Y$ be drawn from independent $N$-trial binomial distributions with mean $4w$, so that $q(4w, 4w) = \mathbb{E}\left[p(X,Y)\right]$. Then we have
\begin{align*}
\mathbb{E}\left[p(X,Y)\right] &\ge \alpha \cdot \Pr[X \ge 2w, Y \ge 2w]\\
&\ge \alpha \cdot \left(1 - \Pr[X \le 2w] - \Pr[Y \le 2w]\right)\\
&\ge \alpha \cdot \left(1 - 2\exp(-w/2)\right)\\
&\ge \alpha \cdot (1 - o_w(1))
\end{align*}
We conclude that $q(s, t) \cdot \frac{1.8 M}{\alpha}$ satisfies the statement of \Cref{thm:L} for all sufficiently large $w$.
Hence, $T = \Omega\left(\sqrt{N/w} \cdot \log M\right)$ as claimed.
\end{proof}
We now establish \Cref{thm:qmabound} from the introduction,
which quantitatively
lower bounds the $\mathsf{QMA}$ complexity of $\mathsf{ApxCount}_{N,w}$.
The analysis exploits
two key properties of the $\mathsf{SBQP}$ protocols that result
from applying \Cref{lem:guessing} to
a $\mathsf{QMA}$ protocol with witness length $m$:
(1) the
parameter $\alpha$ of the $\mathsf{SBQP}$
protocol is not too small (at least $2^{-m}$)
and (2) the multiplicative gap between acceptance probabilities when $f(x)=0$
vs. $f(x)=1$ is at least $2^m$, which may be much greater than $2$.
\qmabound*
\begin{proof}
Consider a $\mathsf{QMA}$
protocol for $\mathsf{ApxCount}_{N,w}$ with witness size $m$ and query cost $T$. If $m=\Omega(w)$, the theorem is vacuous, so suppose
that $m=o(w)$. Running the verifier, Arthur, a constant number of times with fresh witnesses to reduce the soundness and completeness errors, one obtains a verifier with soundness and completeness errors $1/6$ that receives an $O(m)$-length witness and makes $O(T)$ queries.
Repeating twice with two oracles and computing the $\mathsf{AND}$, one obtains a $\mathsf{QMA}$ verifier $V'^{\mathcal{O}_{S_0},\mathcal{O}_{S_1}}$ for $\mathsf{AND}_2 \circ \mathsf{ApxCount}_{N,w}$ with soundness and completeness errors $1/3$ that receives an $O(m)$-length witness and makes $O(T)$ queries.
Applying \Cref{lem:guessing} to $V'$, there exists a quantum query algorithm $Q^{\mathcal{O}_{S_0},\mathcal{O}_{S_1}}$ for $\mathsf{AND}_2 \circ \mathsf{ApxCount}_{N,w}$ that makes $O(m \cdot T)$ queries and satisfies the hypothesis of \Cref{lem:betterforqma} with $M=2^{-\Theta(m)}$. \Cref{thm:sbqp} tells us that $m \cdot T = \Omega\left(\sqrt{N/w} \cdot m\right)$.
Equivalently, $T = \Omega\left(\sqrt{N/w}\right)$.
\end{proof}
\Cref{thm:sbqp} also implies several oracle separations:
\begin{corollary}
\label{cor:qma_separation} There exists an oracle $A$ and a pair of
languages $L_0, L_1$ such that:
\begin{enumerate}
\item $L_0, L_1 \in \mathsf{SBP}^A$
\item $L_0 \cap L_1 \not\in \mathsf{SBQP}^A$.
\item $\mathsf{SBP}^A \not\subset \mathsf{QMA}^A$.
\end{enumerate}
\end{corollary}
\begin{proof}
For an arbitrary function $A: \{0,1\}^* \to \{0,1\}$ and $i \in \{0,1\}$, define $A_i^n = \{x \in \{0,1\}^n : A(i, x) = 1\}$. Define the unary language $L^A_i = \{1^n : |A_i^n| \ge 2^{n/2}\}$. Observe that as long as $A$ satisfies the promise $|A_i^n| \ge 2^{n/2}$ or $|A_i^n| \le 2^{n/2-1}$ for all $n \in \mathbb{N}$, then $L^A_i \in \mathsf{SBP}^A$. Intuitively, the oracles $A$ that satisfy this promise encode a pair of $\mathsf{ApxCount}_{N,w}$ instances $|A_0^n|$ and $|A_1^n|$ for every $n \in \mathbb{N}$ where $N = 2^n$ and $w = 2^{n/2 - 1}$.
\Cref{thm:sbqp} tells us that an $\mathsf{SBQP}$ algorithm $Q$ that makes $o(2^{n/4})$ queries fails to solve $\mathsf{AND}_2 \circ \mathsf{ApxCount}_{N,w}$ on \textit{some} pair $(S_0, S_1)$ that satisfies the promise. Thus, one can construct an $A$ such that $L_0, L_1 \in \mathsf{SBP}^A$ and $L_0 \cap L_1 \not\in \mathsf{SBQP}^A$, by choosing $(A_0^n, A_1^n)$ so as to diagonalize against all $\mathsf{SBQP}$ algorithms.
Because $\mathsf{QMA}^A$ is closed under intersection for any oracle $A$, and because $\mathsf{QMA}^A \subseteq \mathsf{SBQP}^A$ for any oracle $A$, it must be the case that either $L_0 \not\in \mathsf{QMA}^A$ or $L_1 \not\in \mathsf{QMA}^A$.
\end{proof}
\section{Approximate counting with quantum samples and reflections}
\label{sec:samplesreflections}
\subsection{The Laurent polynomial method}
\label{RESULT}\label{sec:lower}\label{sec:Laurent}
By using Minsky--Papert symmetrization (\Cref{symlem}), we now prove the key
fact that relates quantum algorithms, of the type we're considering, to real
Laurent polynomials in one variable. \ The following lemma generalizes the
connection between quantum algorithms and real polynomials established by
Beals et al.\ \cite{bbcmw}.
\begin{lemma}
\label{laurentlem} Let $Q$ be a quantum algorithm that makes $T$ queries to $%
\mathcal{O}_{S}$, uses $R_1$ copies of $\left\vert S\right\rangle $, and
makes $R_2$ uses of the unitary $\mathcal{R}_S$. \ Let $R:= R_1 + 2R_2$.
For $k\in\{1,\ldots,N\}$, let
\begin{equation}
q\left( k\right) :=\E_{\left\vert S\right\vert =k}\left[ \Pr\left[ Q^{%
\mathcal{O}_{S},\mathcal{R}_S}\left( \left\vert S\right\rangle ^{\otimes
R_1}\right) \text{ accepts}\right] \right] .
\end{equation}
Then $q$ can be written a univariate Laurent polynomial, with maximum
exponent at most $2T+R$\ and minimum exponent at least $-R$.
\end{lemma}
\begin{proof}
Let $|\psi_\mathrm{initial}\>$ denote the initial state of the algorithm,
which we can write as
\begin{align*}
|\psi_\mathrm{initial}\> &=
\left\vert S\right\rangle ^{\otimes R_1}
=\left(\frac{1}{\sqrt{|S|}} \sum_{i\in S}|i\>\right)^{\otimes R_1}
=\left(\frac{1}{\sqrt{|S|}} \sum_{i\in [N]}x_i|i\>\right)^{\otimes R_1}\\
&=\frac{1}{\left\vert S\right\vert ^{R_1/2}}\sum_{i_{1},\ldots,i_{R_1}\in\left[
N\right] }x_{i_{1}}\cdots x_{i_{R_1}}\left\vert i_{1},\ldots,i_{R_1}\right\rangle.
\end{align*}
Thus, each amplitude is a complex multilinear polynomial in $X=\left(
x_{1},\ldots,x_{N}\right) $ of degree $R_1$, divided by $\left\vert
S\right\vert ^{R_1/2}$.
Throughout the algorithm, each amplitude will remain a complex multilinear
polynomial in $X$ divided by some power of $|S|$.
Since $x_{i}^{2}=x_{i}$\ for all $i$, we can always
maintain multilinearity without loss of generality.
Like Beals et al.\ \cite{bbcmw}, we now consider how the polynomial degree of each amplitude and the power of $|S|$ in the denominator change as the algorithm progresses.
We have to handle 3 different kinds of unitaries that the quantum circuit may use:
the membership query oracle $\mathcal{O}_S$, unitaries independent of the input,
and the reflection unitary $\mathcal{R}_S$.
The first two cases are handled as in Beals et al.\
Since $\mathcal{O}_S$ is a unitary whose entries are degree-1 polynomials
in $X$, each use of this unitary increases a particular
amplitude's degree as a polynomial by $1$ and does not change the power of $|S|$ in the denominator. \
Second, input-independent unitary transformations only take linear
combinations of existing polynomials and hence do not increase the degree of the amplitudes or the power of $|S|$ in the denominator.
Finally, we consider the reflection unitary $\mathcal{R}_S = \mathbbold{1} - 2|S\>\<S|$.
The $(i,j)^\mathrm{th}$ entry of this operator is $\delta_{ij} - \frac{2x_ix_j}{|S|} = \frac{\delta_{ij}|S| - 2x_ix_j}{|S|}$, where $\delta_{ij}$ is the Kronecker delta function.
Since $|S| = \sum_i x_i$, this is a degree-2 polynomial divided by $|S|$.
Hence applying this unitary will increase the degree of the amplitudes by $2$ and increase
the power of $|S|$ in the denominator by $1$.
In conclusion, we start with each amplitude being a polynomial of degree $R_1$ divided by $|S|^{R_1/2}$. $T$ queries to the membership oracle will increase the degree of each amplitude by at most $T$ and leave the power of $|S|$ in the denominator unchanged. $R_2$ uses of the reflection unitary will increase the degree by at most $2R_2$ and the power of $|S|$ in the denominator by $R_2$. It follows that $Q$'s final state has the form%
\begin{equation}
\left\vert \psi_\mathrm{final}\right\rangle =\sum_z \alpha_{z}\left( X\right)
\left\vert z\right\rangle ,
\end{equation}
where each $\alpha_{z}\left( X\right) $\ is a complex multilinear polynomial
in $X$ of degree at most $R_1+2R_2+T = R+T$, divided by $\left\vert S\right\vert
^{R_1/2+R_2} = |S|^{R/2}$. \ Since $X$ itself is real-valued, it follows that the real and
imaginary parts of $\alpha_{z}\left( X\right) $, considered individually,
are real multilinear polynomials in $X$\ of degree at most $R+T$\ divided by
$\left\vert S\right\vert ^{R/2}$.
Hence, if we let%
\begin{equation}
p\left( X\right) :=\Pr\left[ Q^{\mathcal{O}_{S},\mathcal{R}_S}\left( \left\vert
S\right\rangle ^{\otimes R_1}\right) \text{ accepts}\right] ,
\end{equation}
then%
\begin{equation}
p\left( X\right) =\sum_{\text{accepting }z}\left\vert \alpha_{z}\left(
X\right) \right\vert ^{2}=\sum_{\text{accepting }z}\left( \operatorname{Re}%
^{2}\alpha_{z}\left( X\right) +\operatorname{Im}^{2}\alpha_{z}\left(
X\right) \right)
\end{equation}
is a real multilinear polynomial in $X$ of degree at most $2\left(
R+T\right) $, divided through (in every monomial) by $\left\vert S\right\vert
^{R}=\left\vert X\right\vert ^{R}$.
Now consider%
\begin{equation}
q\left( k\right) :=\E_{\left\vert X\right\vert =k}\left[
p\left( X\right) \right] .
\end{equation}
By \Cref{symlem}, this is a real univariate polynomial in $\left\vert
X\right\vert $ of degree at most $2\left( R+T\right) $, divided through (in
every monomial) by $\left\vert S\right\vert ^{R}=\left\vert X\right\vert ^{R}%
$. \ Or said another way, it's a real Laurent polynomial in $\left\vert
X\right\vert $, with maximum exponent at most $R+2T$\ and minimum exponent at
least $-R$.
\end{proof}
\subsection{Upper bounds}
\label{UPPER}\label{sec:upper}
Before proving our lower bounds on the degree of
Laurent polynomials approximating $\mathsf{ApxCount}_{N,w}$, we
establish some simpler \emph{upper bounds}.
We show upper bounds on Laurent polynomial degree and in the
queries, samples, and reflections model.
\para{Laurent polynomial degree of approximate counting.}
We now describe a \emph{purely negative} degree Laurent polynomial of degree $O(w^{1/3})$ for
approximate counting. \ This upper bound will serve as an important source
of intuition when we prove the (matching) lower bound of \Cref{thm:main}
(see \Cref{s:nextsubsection}). We are thankful to user \textquotedblleft
fedja\textquotedblright\ on MathOverflow for describing this construction.%
\footnote{%
See %
\url{https://mathoverflow.net/questions/302113/real-polynomial-bounded-at-inverse-integer-points}%
}
\begin{lemma}[fedja]
\label{fedjatight} For all $w$, there is a real polynomial $p$ of degree $O\left(w^{1/3}\right)$ such that:
\begin{enumerate}
\item $0 \le p(1/k) \le \frac{1}{3}$ for all $k \in [w]$.
\item $\frac{2}{3} \le p(1/k) \le 1$ for all integers $k \ge 2w$.
\item $0 \le p(1/k) \le 1$ for all $k \in \{w+1,w+2,\ldots,2w-1\}$.
\end{enumerate}
\end{lemma}
\begin{proof}
Assuming for simplicity that $w$ is a perfect cube, consider%
\begin{equation}
u\left( x\right) :=\left( 1-x\right) \left( 1-2x\right) \cdots\left(
1-w^{1/3}x\right) .
\end{equation}
Notice that $\deg\left( u\right) =w^{1/3}$\ and $u\left( \frac{1}%
{k}\right) =0$ for all $k\in\left[ w^{1/3}\right] $. \ Furthermore, we have
$u\left( x\right) \in [0,1]$\ for all $x\in\left[
0,\frac{1}{w^{1/3}}\right] $, and also $u\left( x\right) \in\left[
1-O\left( \frac{1}{w^{1/3}}\right) ,1\right] $\ for all $x\in\left[
0,\frac{1}{w}\right] $. \ Now, let $v$\ be the Chebyshev polynomial of degree
$w^{1/3}$, affinely adjusted so that $v\left( x\right)
\in [0,1]$\ for all $x\in\left[ 0,\frac{1}{w^{1/3}}\right] $ (rather
than in $[-1,1]$ for all all $\left\vert x\right\vert \leq1$), and with a large jump between
$\frac{1}{2w}$\ and $\frac{1}{w}$. \ Then the product, $p(x):=u\left( x\right) v\left( x\right) $, has degree $2w^{1/3}$\ and
satisfies all the requirements, except possibly that the constants $\frac{1}{3}$ and $\frac{2}{3}$ in the first two requirements may be off. Composing with a constant degree polynomial corrects this, and gives a polynomial of degree $O(w^{1/3})$ that satisfies all three requirements.
\end{proof}
Interestingly, if we restrict our attention to purely negative degree
Laurent polynomials, then a matching lower bound is not too hard to show. \
In the same MathOverflow post, user fedja also proves the following, which can also
be shown using earlier work of Zhandry \cite[Proof of Theorem 7.3]{zhandry}):
\begin{lemma
\label{fedjalem}Let $p$\ be a real polynomial, and suppose that $\left\vert
p\left( 1/k\right) \right\vert \leq1$ for all $k\in\left[ 2w\right] $, and
that $p\left( \frac{1}{w}\right) \leq\frac{1}{3}$\ while $p\left( \frac{1}{2w%
}\right) \geq\frac{2}{3}$. \ Then $\deg\left( p\right) =\Omega\left(
w^{1/3}\right) $.
\end{lemma}
\Cref{s:explosion} and \Cref{s:dualpoly} below take the considerable step of
extending \Cref{fedjalem} from purely negative degree Laurent
polynomials to general Laurent polynomials.
\para{Upper bounds in the queries, samples, and reflections model.}
Although we showed that there is a purely negative degree Laurent polynomial of degree
$O(w^{1/3})$ for $\mathsf{ApxCount}_{N,w}$, this does not imply the existence of a quantum algorithm
in the queries, samples, and reflections model with similar complexity.
We now show that our lower bounds in the queries, samples, and reflections model (in \Cref{thm:main}) are tight (up to constants). This is \Cref{thm:alg} in the introduction, restated here for convenience:
\alg*
\begin{proof}
We describe two quantum algorithms for this problem with the two stated complexities.
The first algorithm uses $O(w^{1/3})$ samples and reflections. This algorithm is reminiscent of the original collision finding algorithm of Brassard, H{\o}yer, and Tapp~\cite{BHT98}. We first use $O(w^{1/3})$ copies of $|S\>$ to learn a set $M\subset S$ of size $w^{1/3}$ by simply measuring copies of $|S\>$ in the computational basis. Now we know that the ratio $|S|/|M|$ is either $w^{2/3}$ or $2w^{2/3}$. Now consider running Grover's algorithm on the set $S$ where the elements in $M$ are considered the ``marked'' elements. Grover's algorithm alternates reflections about the uniform superposition over the set being searched, $S$, with an operator that reflects about the marked elements in $M$. The first reflection is simply $\mathcal{R}_S$, which we have access to. The second unitary can be constructed since we have an explicit description of the set $M$. Now Grover's algorithm can be used to distinguish whether the fraction of marked elements is $1/w^{2/3}$ or half of that, and the cost will be $O(w^{1/3})$.
The second algorithm uses $O(\sqrt{{N}/{w}})$ reflections only and no copies of $|S\>$.
Consider running the standard approximate counting algorithm~\cite{BHMT02} that uses membership queries to $S$ and distinguishes $|S|\leq w$ from $|S|\geq 2w$ using $O(\sqrt{N/w})$ membership queries.
Observe that this algorithm starts with the state $|\psi\> =\frac{1}{%
\sqrt{N}}\left( \left\vert 1\right\rangle +\cdots +\left\vert N\right\rangle
\right)$, which is in $\textrm{span}\{|S\>,|\bar{S}\>\}$, and only uses reflections about $|\psi\> $ and membership queries to $|S\>$ in the form of a unitary that maps $|i\>$ to $-|i\>$ when $i \in S$.
This means the state of the algorithm remains in $\textrm{span}\{|S\>,|\bar{S}\>\}$
at all times.
Within this subspace, a membership query to $S$ is the same as a reflection about $|S\>$.
Hence we can replace membership queries with the reflection operator to get an approximate counting algorithm that only uses $O(\sqrt{N/w})$ reflections and no copies of $|S\>$.
\end{proof}
Note that both the algorithms presented above generalize to the situation
where we want to distinguish $|S|=w$ from $|S|=(1+\varepsilon )w$. \ For the
first algorithm, we now pick a subset $M$ of size $w^{1/3}/\varepsilon ^{2/3}
$. Now we want to $(1+\varepsilon )$-approximate the fraction of marked
elements, which is either $1/(w\varepsilon )^{2/3}$ or $(1+\varepsilon )^{-1}
$ times that. This can be done with approximate counting~\cite[Theorem 15]%
{BHMT02}, and the cost will be $O\left( \frac{1}{\varepsilon }(w\varepsilon
)^{1/3}\right) =O\left( \frac{w^{1/3}}{\varepsilon ^{2/3}}\right) $.
The second algorithm is simpler to generalize, since we simply plug in
the query complexity of $\varepsilon$-approximate counting, which is
$O\Bigl( \frac{1}{\varepsilon }\sqrt{\frac{N}{w}}\Bigr) $.
\subsection{Lower bound using the explosion argument}
\label{s:explosion}\label{sec:explosion}
We now show a weaker version of \Cref{thm:main} using the explosion argument
described in the introduction. \ The difference between the following
theorem and \Cref{thm:main} is the exponent of $w$ in the lower bound.
\begin{theorem}
\label{thm:explosion}Let $Q$ be a quantum algorithm that makes $T$ queries
to the membership oracle for $S$, and uses a total of $R$ copies of $|S\>$
and reflections about $|S\>$. \ If $Q$ decides whether $\left\vert
S\right\vert =w$\ or $\left\vert S\right\vert =2w$ with success probability
at least $2/3$, promised that one of those is the case, then either
\begin{equation}
T=\Omega \left( \sqrt{\frac{N}{w}}\right) \qquad \text{or}\qquad R=\Omega
\left( \min \left\{ w^{1/4},\sqrt{\frac{N}{w}}\right\} \right) .
\end{equation}
\end{theorem}
\begin{proof}
Since we neglect multiplicative constants in our lower bounds, let us allow the algorithm to use up to $R$ copies of $|S\>$ and $R$ uses of $\mathcal{R}_S$.
Let%
\begin{equation}
q\left( k\right) :=\E_{\left\vert S\right\vert =k}\left[
\Pr\left[ Q^{\mathcal{O}_{S},\mathcal{R}_S}\left( \left\vert S\right\rangle ^{\otimes
R}\right) \text{ accepts}\right] \right] .
\end{equation}
Then by \Cref{laurentlem}, we can write $q$ as a Laurent polynomial:
\begin{equation}
q\left( k\right) =u\left( k\right) +v\left( 1/k\right) ,
\end{equation}
where $u$\ is a real polynomial in $k$ with $\deg\left( u\right) = O(T+R)$,
and $v$\ is a real polynomial in $1/k$\ with $\deg\left( v\right) = O(R)$.
\ So to prove the theorem, it suffices to show that either $\deg\left(
u\right) =\Omega\left( \sqrt{\frac{N}{w}}\right) $, or else $\deg\left(
v\right) =\Omega\left( w^{1/4}\right) $. \ To do so, we'll assume that
$\deg\left( u\right) =o\left( \sqrt{\frac{N}{w}}\right) $\ and
$\deg\left( v\right) =o\left( w^{1/4}\right) $, and derive a contradiction.
Our high-level strategy is as follows: we'll observe that, if approximate
counting is being successfully solved, then either $u$ or $v$ must attain a
large first derivative somewhere in its domain. \ By the approximation theory
lemmas that we proved in \Cref{s:approxprelim}, this will force that polynomial to have a large
range---even on a subset of integer (or inverse-integer) points. \ But the
sum, $u\left( k\right) +v\left( 1/k\right) $, is bounded in $\left[
0,1\right] $\ for all $k\in\left[ N\right] $. \ So if one polynomial has a
large range, then the other does too. \ But this forces the \textit{other}
polynomial to have a large derivative somewhere in its domain, and therefore
(by approximation theory) to have an even larger range, forcing the first
polynomial to have an even larger range to compensate, and so on. \ As long as
$\deg\left( u\right) $\ and $\deg\left( v\right) $ are both small enough,
this endless switching will force both $u$ and $v$ to attain
\textit{unboundedly }large values---with the fact that one polynomial is in
$k$, and the other is in $1/k$, crucial to achieving the desired
\textquotedblleft explosion.\textquotedblright\ \ Since $u$ and $v$ are
polynomials on compact sets, such unbounded growth is an obvious absurdity,
and this will give us the desired contradiction.
In more detail, we will study the following quantities.%
\begin{equation}%
\begin{tabular}
[c]{ll}%
$G_{u}:=\max_{x,y\in\left[ \sqrt{w},2w\right] }\left\vert u\left( x\right)
-u\left( y\right) \right\vert ~\ \ \ \ \ \ $ & $G_{v}:=\max_{x,y\in\left[
\frac{1}{N},\frac{1}{w}\right] }\left\vert v\left( x\right) -v\left(
y\right) \right\vert $\\
$\Delta_{u}:=\max_{x\in\left[ \sqrt{w},2w\right] }\left\vert u^{\prime
}\left( x\right) \right\vert $ & $\Delta_{v}:=\max_{x\in\left[ \frac{1}%
{N},\frac{1}{w}\right] }\left\vert v^{\prime}\left( x\right) \right\vert
$\\
$H_{u}:=\max_{x,y\in\left[ \sqrt{w},N\right] }\left\vert u\left( x\right)
-u\left( y\right) \right\vert $ & $H_{v}:=\max_{x,y\in\left[ \frac{1}%
{N},\frac{1}{\sqrt{w}}\right] }\left\vert v\left( x\right) -v\left(
y\right) \right\vert $\\
$I_{u}:=\max_{x,y\in\left[ w,N\right] }\left\vert u\left( x\right)
-u\left( y\right) \right\vert $ & $I_{v}:=\max_{x,y\in\left[ \frac{1}%
{2w},\frac{1}{\sqrt{w}}\right] }\left\vert v\left( x\right) -v\left(
y\right) \right\vert $\\
$L_{u}:=\max_{x,y\in\left\{ w,\ldots,N\right\} }\left\vert u\left(
x\right) -u\left( y\right) \right\vert $ & $L_{v}:=\max_{x,y\in\left\{
\sqrt{w},\ldots,2w\right\} }\left\vert v\left( \frac{1}{x}\right) -v\left(
\frac{1}{y}\right) \right\vert $%
\end{tabular}
\end{equation}
We have $0\leq q\left( k\right) \leq1$\ for all $k\in\left[ N\right] $,
since in those cases $q\left( k\right) $\ represents a probability. \ Since
$Q$ solves approximate counting, we also have $q\left( w\right) \leq\frac
{1}{3}$\ and $q\left( 2w\right) \geq\frac{2}{3}$. \ This means in particular
that either
\begin{enumerate}
\item[(i)] $u\left( 2w\right) -u\left( w\right) \geq\frac{1}{6}$, and
hence $G_{u}\geq\frac{1}{6}$, or else
\item[(ii)] $v\left( \frac{1}{2w}\right) -v\left( \frac{1}{w}\right)
\geq\frac{1}{6}$, and hence $G_{v}\geq\frac{1}{6}$.
\end{enumerate}
We will show that either case leads to a contradiction.
We have the following inequalities regarding $u$:%
\begin{equation}%
\begin{tabular}
[c]{ll}%
$G_{u}\geq L_{v}-1$ & by the boundedness of $q$\\
$\Delta_{u}\geq\frac{G_{u}}{2w}$ & by basic calculus\\
$H_{u}\geq\frac{\Delta_{u}\left( N-\sqrt{w}\right) }{\deg\left( u\right)
^{2}}$ & by \Cref{markovlem}\\
$I_{u}\geq\frac{H_{u}}{2}$ & by \Cref{paturicor}\\
$L_{u}\geq\frac{I_{u}}{2}$ & by \Cref{ezrclem}%
\end{tabular}
\end{equation}
Here the fourth inequality uses the fact that, setting $\varepsilon
:=\frac{\sqrt{w}}{N}$, we have $\deg\left( u\right) =o\left( \frac{1}%
{\sqrt{\varepsilon}}\right) $ (thereby satisfying the hypothesis of \Cref{paturicor}), while the fifth inequality uses the fact that $\deg\left(
u\right) =o\left( \sqrt{N}\right) $.
Meanwhile, we have the following inequalities regarding $v$:%
\begin{equation}%
\begin{tabular}
[c]{ll}%
$G_{v}\geq L_{u}-1$ & by the boundedness of $q$\\
$\Delta_{v}\geq G_{v}w$ & by basic calculus\\
$H_{v}\geq\frac{\Delta_{v}\left( \frac{1}{\sqrt{w}}-\frac{1}{N}\right)
}{\deg\left( v\right) ^{2}}$ & by \Cref{markovlem}\\
$I_{v}\geq\frac{H_{v}}{2}$ & by \Cref{paturicor}\\
$L_{v}\geq\frac{I_{v}}{2}$ & by \Cref{ezrclem}%
\end{tabular}
\end{equation}
Here the fourth inequality uses the fact that, setting $\varepsilon
:=\frac{1/2w}{1/\sqrt{w}}=\frac{1}{2\sqrt{w}}$, we have $\deg\left( v\right)
=o\left( \frac{1}{\sqrt{\varepsilon}}\right) $ (thereby satisfying the
hypothesis of \Cref{paturicor}). \ The fifth inequality uses the fact
that, if we set $V\left( x\right) :=v\left( x/w\right) $, then the
situation\ satisfies the hypothesis of \Cref{ezrclem}:\ we are interested
in the range of $V$ on the interval $\left[ \frac{1}{2},\sqrt{w}\right] $,
compared to its range on discrete points $\frac{w}{\sqrt{w}},\frac{w}{\sqrt
{w}+1},\ldots,\frac{w}{2w}$\ that are spaced at most $1$ apart from each
other; and we also have $\deg\left( V\right) =\deg\left( v\right)
=o\left( w^{1/4}\right) $.
All that remains is to show that, if we insert either $G_{u}\geq\frac{1}{6}%
$\ or $G_{v}\geq\frac{1}{6}$ into the coupled system of inequalities above,
then we get unbounded growth and the inequalities have no solution. \ Let us
collapse the two sets of inequalities to%
\begin{align*}
L_{u} & \geq\frac{1}{4}\frac{N-\sqrt{w}}{\deg\left( u\right) ^{2}}%
\frac{G_{u}}{2w}=\Omega\left( \frac{N}{w\deg\left( u\right) ^{2}}%
G_{u}\right) ,\\
L_{v} & \geq\frac{1}{4}\frac{\frac{1}{\sqrt{w}}-\frac{1}{N}}{\deg\left(
v\right) ^{2}}G_{v}w=\Omega\left( \frac{\sqrt{w}}{\deg\left( v\right)
^{2}}G_{v}\right) .
\end{align*}
Hence%
\begin{align*}
G_{u} & \geq L_{v}-1=\Omega\left( \frac{\sqrt{w}}{\deg\left( v\right)
^{2}}G_{v}\right) -1,\\
G_{v} & \geq L_{u}-1=\Omega\left( \frac{N}{w\deg\left( u\right) ^{2}%
}G_{u}\right) -1.
\end{align*}
By the assumption that $\deg\left( v\right) =o\left( w^{1/4}\right) $\ and
$\deg\left( u\right) =o\left( \sqrt{\frac{N}{w}}\right) $, we have
$\frac{\sqrt{w}}{\deg\left( v\right) ^{2}}\gg1$\ and $\frac{N}{w\deg\left(
u\right) ^{2}}\gg1$. \ Plugging in $G_{u}\geq\frac{1}{6}$\ or $G_{v}\geq
\frac{1}{6}$, this is enough to give us unbounded growth.
\end{proof}
\subsection{Lower bound using dual polynomials}
\label{s:dualpoly}\label{sec:dualpoly}
In this section we use the method of dual polynomials to
establish our main result, \Cref{thm:main}, restated for convenience:
\main*
Let $p(r)$ be a univariate Laurent polynomial of negative degree $D_{1}$ and
positive degree $D_{2}$. \ That is, let $p(r)$ be of the form
\begin{equation}
p(r)=a_{0}/r^{D_{1}}+a_{1}/r^{D_{1}-1}+\dots
+a_{D_{1}-1}/r+a_{D_{1}}+a_{D_{1}+1}\cdot r+\cdots +a_{D_{2}+D_{1}}\cdot
r^{D_{2}}.
\end{equation}
\Cref{thm:main} follows by combining the Laurent polynomial method (\Cref{laurentlem}) and the following
theorem.
\begin{theorem}
\label{mainthm}
Let $\varepsilon < 1$. Suppose that $p$ has negative degree $D_1$ and positive
degree $D_2$ and satisfies the following properties.
\begin{itemize}
\item $\left\vert p(w)-1\right\vert \leq \varepsilon $
\item $\left\vert p(2w)+1\right\vert \leq \varepsilon $
\item $\left\vert p(\ell )\right\vert \leq 1+\varepsilon \text{ for all }%
\ell \in \{1,2,\dots ,n\}$
\end{itemize}
Then either $D_1 \geq \Omega\left(w^{1/3}\right)$ or $D_2 \geq \Omega\left(\sqrt{N/w}%
\right)$.
\end{theorem}
In fact, our proof of \Cref{mainthm} will show that the lower bound holds
even if $\left\vert p(\ell )\right\vert \leq 1+\varepsilon $ only for $\ell
\in \{w^{1/3},w^{1/3}+1,\dots ,w\}\cup \{2w,2w+1,\dots ,N\}$. \ We refer to
a Laurent polynomial $p$ satisfying the three properties of \Cref{mainthm}
as an \emph{approximation for approximate counting}.
\paragraph{Proof of \Cref{mainthm}.}
Let $p$ be any Laurent polynomial satisfying the hypothesis of \Cref{mainthm}%
. \ We begin by transforming $p$ into a (standard) polynomial $q$ in a
straightforward manner. \ This transformation is captured in the following
lemma, whose proof is so simple that we omit it.
\begin{lemma}
\label{mainlem} If $p$ satisfies the properties of \Cref{mainthm}, then the
polynomial $q(r) = p(r) \cdot r^{D_1}= a_0 + a_1 r + \dots + a_{D_1 + D_2}
r^{D_1 + D_2}$ is a (standard) polynomial of degree at most $D_1 + D_2$, and
$q$ satisfies the following three properties.
\begin{itemize}
\item $\left\vert q(w)-w^{D_{1}}\right\vert \leq \varepsilon \cdot w^{D_{1}}$
\item $\left\vert q(2w)+(2w)^{D_{1}}\right\vert \leq \varepsilon \cdot
(2w)^{D_{1}}$
\item $\left\vert q(\ell )\right\vert \leq \left( 1+\varepsilon \right) \ell
^{D_{1}}\text{ for all }\ell \in \{1,2,\dots ,N\}$
\end{itemize}
\end{lemma}
We now turn to showing that, for any constant $\varepsilon < 1$, no
polynomial $q$ can satisfy the conditions of \Cref{mainlem} unless $D_1 \geq
\Omega(w^{1/3})$ or $D_2 \geq \Omega\left(\sqrt{N/w}\right)$.
Consider the following linear program. \ The variables of the linear program
are $\varepsilon $, and the $D_{2}+D_{1}+1$ coefficients of $q$.
\begin{equation}
\boxed{\begin{array}{ll} \text{minimize } & \varepsilon \\ \mbox{such that} & \\
&|q(w) - w^{D_1}| \leq \varepsilon \cdot w^{D_1}\\ &|q(2w) + (2w)^{D_1}| \leq \varepsilon
\cdot (2w)^{D_1}\\ & |q(\ell)| \leq (1+\varepsilon) \cdot \ell^{D_1} \text{ for all
} \ell \in \{1, 2, \dots, N\}\\ &\varepsilon \geq 0 \end{array}}
\end{equation}
Standard manipulations reveal the dual.
\begin{equation} \label{duallp}
\boxed{\begin{array}{ll} \text{maximize } \phi(w) \cdot w^{D_1} - \phi(2w)
\cdot (2w)^{D_1} - \sum_{\ell \in \{1, \dots, N\}, \ell \not\in \{w, 2w\}}
|\phi(\ell)| \cdot \ell^{D_1} \\ \mbox{such that} & \\ \sum_{\ell=1}^{N}
\phi(\ell) \cdot \ell^j =0 \text{ for } j= 0, 1, 2, \dots, D_1 + D_2 &\\
\sum_{\ell=1}^N |\phi(\ell)| \cdot \ell^{D_1} = 1\\ \phi \colon \mathbb{R} \to
\mathbb{R} \end{array}}
\end{equation}
\label{s:duallp}
\medskip\medskip
\Cref{mainthm} will follow if we can exhibit a solution $\phi $ to the dual
linear program achieving value $\varepsilon >0$, for some
setting of $D_{1}\geq \Omega (w^{1/3})$ and $D_{2}\geq \Omega \left( \sqrt{%
N/w}\right) $.\footnote{%
We will alternatively refer to such dual solutions $\phi $ as \emph{dual
witnesses}, since they act as a witness to the fact that any low-degree
Laurent polynomial $p$ approximating the approximate counting problem must
have large error.} \ We now turn to this task.
\subsubsection{Constructing the dual solution}
\label{duallpsec}
For a set $T\subseteq \{0,1,\dots ,N\}$, define
\begin{equation}
Q_{T}(t)=\prod_{i=0,1,\dots ,N,i\not\in T}(t-i).
\end{equation}%
Let $c>2$ be an integer constant that we will choose later (the bigger we choose $c$
to be, the better the objective value achieved by our final dual witness. \
But choosing a bigger $c$ will also lower the degrees $D_{1},D_{2}$ of
Laurent polynomials against which our lower bound will hold).
We now define two sets $T_{1}$ and $T_{2}$. \ The size of $T_{1}$ will be
\begin{equation}
d_{1}:=\lfloor \left( w/c\right) ^{1/3}\rfloor =\Theta \left( w^{1/3}\right)
\end{equation}%
and the size of $T_{2}$ will be $d_{2}$ for
\begin{equation}
d_{2}:=\lfloor \sqrt{N/(cw)}\rfloor =\Theta \left( \sqrt{N/w}\right) .
\end{equation}%
Let
\begin{equation}
T_1 = \left\{\lfloor w/(c i^2) \rfloor \colon i=1, 2, \dots, d_1\right\}
\end{equation}
and
\begin{equation}
T_2 = \left\{c \cdot i^2 \cdot w \colon i = 1, 2, \dots, d_2:=\sqrt{N/(cw)}%
\right\}.
\end{equation}
Finally, define
\begin{equation}
T = \{w, 2w\} \cup T_1 \cup T_2.
\end{equation}
At last, define $\Phi \colon \{0, 1, \dots, N\} \to \mathbb{R}$ via
\begin{equation} \label{phidef}
\Phi(t) = (-1)^t \cdot \binom{N}{t} \cdot Q_T(t).
\end{equation}
Our final dual solution $\phi $ will be a scaled version of $\Phi $. \
Specifically, $\Phi $ itself does not satisfy the second constraint of the
dual linear program, that $\sum_{\ell =1}^{N}|\Phi (\ell )|\cdot \ell
^{D_{1}}=1$. \ So letting
\begin{equation}
C=\sum_{\ell =1}^{N}|\Phi (\ell )|\cdot \ell ^{D_{1}}, \label{Cdef}
\end{equation}%
our final dual witness $\phi $ will be $\Phi /C$.
\para{The sizes of $T_{1}$ and $T_{2}$.} \ Clearly,
under the above definition of $T_{2}$, $|T_{2}|=d_{2}$ as claimed above. \
It is not as immediately evident that $|T_{1}|=d_{1}$: to establish this, we
must show that for distinct $i,j\in \{1,2,\dots ,d_{1}\}$, $\lfloor
w/(ci^{2})\rfloor \neq \lfloor w/(cj^{2})\rfloor $. \ This is handled in the
following easy lemma.
\begin{lemma}
\label{lem:distinct} Let $i\neq j$ be distinct numbers in $\{1,\dots ,d_{1}\}
$ and $c>2$ be a constant. \ Then as long as $d_{1}<\left( w/c\right) ^{1/3}$%
, it holds that $\lfloor w/(ci^{2})\rfloor \neq \lfloor w/(cj^{2})\rfloor $.
\end{lemma}
\begin{proof}
Assume without loss of generality that $i> j$. Then $w/(c j^2)- w/(c i^2)$ is clearly minimized when $i=d_1$ and $j=i-1$.
For the remainder of the proof, fix $i=d_1$.
In this case,
\begin{align}
w/(c j^2) - w/(c i^2) \geq
w/\left(c (i-1)^2\right) - w/\left(c i^2\right)
= \frac{w i^2 - w \left(i-1\right)^2}{c \cdot i^2 \cdot \left(i-1\right)^2} \notag \\
= \frac{w}{c} \cdot \frac{2i-1}{i^2\left(i-1\right)^2 }
\geq \frac{w}{c} \cdot \frac{2i-1}{i^4}
\geq \frac{w}{c i^3} \geq 1. \label{eq28}
\end{align}
Here, the final inequality holds because $i^3 = d_1^3 \leq w/c$.
\Cref{eq28} implies the lemma, as two numbers whose difference is at least 1 cannot have
the same integer floor.
\end{proof}
\Cref{lem:distinct} is false for $d_1 = \omega(w^{1/3})$, highlighting on a
technical level why one cannot choose $d_1$ larger than $\Theta(w^{1/3})$
without the entire construction and analysis of $\Phi$ breaking down.
\subsubsection{Intuition: ``gluing together'' two simpler dual solutions}
Before analyzing the dual witnesses
$\Phi$ and $\phi$ constructed in \Cref{phidef} and \Cref{Cdef},
in this subsection and the next, we provide detailed intuition
for why the definitions of $\Phi$ and $\phi$ are natural,
and briefly overview their analysis.
\label{sec:intuition}
\para{A dual witness for purely positive degree (i.e.,
approximate degree).} \ Suppose we were merely interested in showing an
approximate degree lower bound of $\Omega (\sqrt{N/w})$ for approximate
counting (i.e., a lower bound on the degree of traditional polynomials that
distinguish input $w$ from $2w$, and are bounded at all other integer inputs
in $1,\dots ,N$). \ This is equivalent to exhibiting a solution to the dual
linear program with $D_{1}=0$. \ A valid dual witness $\phi_1$ for this
simpler case is to also use \Cref{phidef}, but to set
\begin{equation} \label{simplerT1}
T= \{w, 2w\} \cup T_2,
\end{equation}
rather than $T=\{w, 2w\} \cup T_1 \cup T_2$.
We will explain intuition for why \Cref{simplerT1} is a valid dual
solution for the approximate degree of approximate counting in the next
subsection. \ For now, we wish to explain how this construction relates to
prior work. In \cite{bt13}, for any constant $\delta >0$, a dual
witness is given for the fact that the $(1-\delta )$-approximate degree
of $\mathsf{OR}$ is $\Omega (\sqrt{N})$. \ This dual witness \emph{nearly}
corresponds to the above, with $w=1$. \ Specifically, Bun and Thaler \cite{bt13} use the
set $T=\{0,1\}\cup \{ci^{2}\colon i=1,2,\dots ,\sqrt{N/c}\}$, and they show
that almost all of the \textquotedblleft mass\textquotedblright\ of this
dual witness is located on the inputs $0$ and $1$, i.e.,
\begin{equation}
\left\vert \Phi (0)\right\vert +\left\vert \Phi (1)\right\vert \geq
(1-\delta )\cdot \sum_{i=2}^{N}\left\vert \Phi (i)\right\vert .
\label{epseq}
\end{equation}%
Here, the bigger $c$ is chosen to be, the smaller the value of $\delta$
for which \Cref{epseq} holds.
In the case of $w=1$, our dual witness for approximate counting differs from
this only in that $\{0, 1\}$ is replaced with $\{1, 2\}$. This is
because, in order to show a lower bound for distinguishing input $w=1$ from
input $2w=2$, we want almost all of the mass to be on inputs $\{1, 2\}$
rather than $\{0, 1\}$ (this is what will ensure that the objective function
of the dual linear program is large).
For general $w$, we want most of the mass of $\psi $ to be concentrated on
inputs $w$ and $2w$. \ Accordingly, relative to the $w=1$ case, we
effectively multiply \emph{all} points in $T$ by $w$, and one can show that
this does not affect the calculation regarding concentration of mass.
\para{A dual witness for purely negative degree.} \
Now, suppose we were merely interested in showing that Laurent polynomials
of \emph{purely negative} degree require degree $\Omega (w^{1/3})$ to
approximate the approximate counting problem. \ This is equivalent to
exhibiting a solution to the dual linear program with $D_{2}=0$. Then a
valid dual witness $\phi_2 $ for this simpler case is to also use
\Cref{phidef}, but to set
\begin{equation} \label{simplerT2}
T= \{w, 2w\} \cup T_1.
\end{equation}
Again, we will give intuition for why this is a valid dual solution in the
next subsection (\Cref{s:nextsubsection}). \ For now, we wish to explain how
this construction relates to prior work. \ Essentially, the $\Omega \left(
w^{1/3}\right) $-degree lower bound for Laurent polynomials with \emph{only
negative} powers was proved by Zhandry \cite[Theorem 7.3]{zhandry}. \
Translating Zhandry's theorem into our setting is not entirely trivial, and
he did not explicitly construct a solution to our dual linear program. \
However (albeit with significant effort), one can translate his argument to
our setting to show that \Cref{simplerT2} gives a valid dual
solution to prove a lower bound against Laurent polynomials with only
negative powers.
\para{Gluing them together.} The above discussion
explains that the key ideas for constructing dual solutions $\phi _{1}$, $%
\phi _{2}$ witnessing degree lower bounds for Laurent polynomials of \emph{%
only negative} or \emph{only positive} powers were essentially already
known, or at least can be extracted from prior work with enough effort. \ In this work, we are interested in proving lower bounds for Laurent
polynomials with both positive and negative powers. \ Our dual solution $%
\Phi $ essentially just \textquotedblleft glues together\textquotedblright\
the dual solutions that can be derived from prior work. \ By this, we mean
that the set $T$ of integer points on which our $\Phi $ is nonzero is the
\emph{union} of the corresponding sets for $\phi _{1}$ and $\phi _{2}$
individually. \ Moreover, this union is nearly disjoint, as the only points
in the intersection of the two sets being unioned are $w$ and $2w$.
\para{Overview of the analysis.}
To show that we have constructed
a valid solution
to the dual linear program (\Cref{duallp}),
we
must establish
that (a) $\Phi$ is uncorrelated
with every polynomial of degree
at most $D_1 + D_2$
and (b) $\Phi$ is
well-correlated with
any function $g$ that evaluates
to $+1$ on input $w$, to $-1$
on input $2w$, and is bounded in $[-1, 1]$ elsewhere.
In (b), the correlation is taken
with respect to
an appropriate weighting
of the inputs, that on input $\ell \in [N]$ places
mass proportional to $\ell^{D_1}$.
The definition of $\Phi$
as a ``gluing together'' of $\phi_1$ and $\phi_2$
turns out, in a
straightforward manner, to ensure
that $\Phi$ is uncorrelated
with polynomials of degree at
$D_1 + D_2$.
All that remains is to
show that $\Phi$ is well-correlated
with $g$ under the appropriate
weighting of inputs.
This turns out to be technically demanding, but ultimately
can be understood as stemming from the fact that
$\phi_1$ and $\phi_2$ are individually
well-correlated with $g$ (albeit, in the case of $\phi_2$, under a
\emph{different} weighting of the inputs than the weighting that is relevant for $\Phi$).
\subsubsection{Intuition via complementary slackness}
\label{s:nextsubsection}We now attempt to lend some insight into why the
dual witnesses $\phi_1 $ and $\phi_2 $ for the purely positive degree and purely
negative degree take the form that they do. \ This section is deliberately
slightly imprecise in places, and builds on intuition that has been put
forth in prior works proving approximate degree lower bounds via dual
witnesses \cite{bt13, Tha16, bkt}.
Notice that $\phi_1 $ is precisely defined so that $\phi_1 (i)=0$ for any $%
i\not\in \{w,2w\}\cup T_{2}$, and similarly $\phi_2 (i)=0$ for any $i\not\in
\{w,2w\}\cup T_{1}$. \ The intuition for why this is reasonable comes from
complementary slackness, which states that an optimal dual witness should
equal $0$ except on inputs that correspond to primal constraints that are
\emph{made tight by an optimal primal solution}. \ By \textquotedblleft
constraints made tight by an optimal primal solution\textquotedblright , we
mean constraints that, for the optimal primal solution, hold with equality
rather than (strict) inequality.
Unpacking that statement, this means the following. \ Suppose that $q$ is an
optimal solution to the primal linear program of \Cref{s:duallp}, meaning it
minimizes the error $\varepsilon $ amongst all polynomials of the same same
degree. \ The constraints made tight by $q$ are precisely those inputs $\ell
$ at which $q$ hits its \textquotedblleft maximum error\textquotedblright\
(e.g., an input $\ell$ such that $|q(\ell)|=(1+\varepsilon )\cdot \ell ^{D_{1}}$).
\ We call these inputs \emph{maximum-error} inputs for $q$. \ Complementary
slackness says that there is an optimal solution to the dual linear program (\Cref{duallp})
that equals $0$ at all inputs that are not maximum-error inputs for $q$.
In both the purely positive degree case, and the purely negative degree
case, we know roughly what primal optimal solutions $q$ look like, and
moreover we know what roughly their maximum-error points look like. \ In the
first case, the maximum-error points are well-approximated by the points in $%
T_{2}$, and in the purely negative degree case, the maximum error points are
well-approximated by the points in $T_{1}$. \ Let us explain.
\para{Purely positive degree case.} \ Let $T_{d}$ be
the degree $d$ Chebyshev polynomial of the first kind. It can be seen that $%
P(\ell )=T_{\sqrt{N}}\left( 1+2/N-\ell /N\right) $ satisfies $P(1)\geq 2$,
while $|P(\ell )|\leq 1$ for $\ell =2,3,\dots ,N$. \ That is, up to scaling,
$P$ approximates the approximate counting problem for $w=1$, and its known
that its degree is within a constant factor of optimal.
It is known that the extreme points of $T_{d}$ are of the following form,
for $k=1,\dots ,d$:
\begin{equation}
\cos \left( \frac{(2k-1)}{2d}\pi \right) \approx 1-k^{2}/(2d^{2}),
\label{tester}
\end{equation}%
where the approximation uses the Taylor expansion of the cosine function
around $0$. \ \Cref{tester} means that the extreme points of $P$
are roughly those inputs $\ell $ such that $1+2/N-\ell /N\approx
1-k^{2}/(2d^{2})$, where $d=\sqrt{N}$. \ Such $\ell $ are roughly of the
form $\ell \approx c\cdot i^{2}$ for some constant $c$, as $i$ ranges from $1
$ up to $\Theta (N^{1/2})$.
More generally, when $w\geq 1$, an asymptotically optimal approximation for
distinguishing input $w$ from $2w$ is $P(\ell )=T_{\sqrt{N/w}}\left(
1+2w/N-\ell /(wN)\right) $. \ The extreme points of $P$ are roughly of the
form $\ell \approx c\cdot i^{2}\cdot w$ for some constant $c$, as $i$ ranges
from $1$ up to $\Theta (\sqrt{N/w})$, which is exactly the form of the
points in our set $T_{2}$.
\para{Purely negative degree case.} In \Cref{fedjatight}%
, we exhibited a simple, purely negative degree Laurent polynomial $p$
(i.e., $p(\ell )$ is a standard polynomial in $1/\ell $) with degree $%
D_{1}=w^{1/3}$ that solves the approximate counting problem (the
construction is due to MathOverflow user \textquotedblleft
fedja\textquotedblright ). \ Roughly speaking, $p$ can be written as a
product $p(\ell )=u(\ell )\cdot v(\ell )$, where $u(\ell )$ has the roots $%
\ell =1,2,\dots ,w^{1/3}$, and $v(\ell )$ is (an affine transformation) of a
Chebyshev polynomial of degree $w^{1/3}$, applied to $1/\ell $. \ One can
easily look at this construction and see that $p(\ell )$ outputs \emph{%
exactly} the correct value on inputs $\{1,2,\dots ,w^{1/3}\}$, so these are
not maximum error points for $p$. \ Moreover, the analysis of the maximum
error points for Chebyshev polynomials above can be applied to show that the
maximum error points of $p$ are roughly of the form $\ell $ such that $%
1/\ell =c\cdot i^{2}/w$ for some constant $c$, with $i$ ranging from $1$ up
to $\Theta (w^{1/3})$. \ This means that the extreme points are roughly of
the form $\ell \approx \frac{w}{ci^{2}}$, which is why our set $T_{1}$
consists of points of the form $\lfloor \frac{w}{ci^{2}}\rfloor $ (the
floors are required because we are proving lower bounds against polynomials
whose behavior is only constrained at integer inputs).
\subsubsection{Analysis of the dual solution \texorpdfstring{$\Phi$}{Phi}}
\begin{lemma}
\label{phdlem}Let $d_{1}=|T_{1}|$ and $d_{2}=|T_{2}|$. \ Then for any $%
j=0,1,\dots ,d_{1}+d_{2}$, it holds that
\begin{equation*}
\sum_{\ell =1}^{N}\Phi (\ell )\cdot \ell ^{j}=0.
\end{equation*}
\end{lemma}
\begin{proof}
A basic combinatorial fact is that for any polynomial $Q$ of degree at most $N-1$,
the following identity holds:
\begin{equation}
\label{keyidentity}
\sum_{\ell=0}^N \binom{N}{\ell} (-1)^\ell Q(\ell) = 0.
\end{equation}
Observe that for any $j \leq d_1 + d_2+1$,
\begin{equation} \label{isapolynomial}
Q_T(\ell) \cdot \ell^j \text{ is a polynomial in } \ell \text{ of degree at most } N-1.
\end{equation}
Furthermore, $\Phi(0)=0$, because $0 \not\in T$. Hence
\begin{equation} \label{nozeroterm}
\sum_{\ell=0}^N \binom{N}{\ell} (-1)^\ell Q_T(\ell) \cdot \ell^j= \sum_{\ell=1}^N \binom{N}{\ell} (-1)^\ell Q_T(\ell) \cdot \ell^j.
\end{equation}
Thus, we can calculate:
\begin{eqnarray*}
\sum_{\ell=1}^N \Phi(\ell) \cdot \ell^{j} = \sum_{\ell=1}^N (-1)^{\ell} \cdot \binom{N}{\ell} \cdot Q_T(\ell) \cdot \ell^j\\
= \sum_{\ell=0}^N (-1)^\ell \cdot \binom{N}{\ell} \cdot Q_T(\ell) \cdot \ell^j
=0.
\end{eqnarray*}
Here, the second equality follows from \Cref{nozeroterm}, while the third follows
from Equations \eqref{keyidentity} and \eqref{isapolynomial}.
\end{proof}
Let us turn to analyzing $\Phi $'s value on various inputs. \ Clearly the
following condition holds:
\begin{equation}
\Phi (\ell )=0~\text{for all }\ell \not\in T\text{.} \label{zeros}
\end{equation}%
Next, observe that for any $r\in T$,
\begin{equation*}
\left\vert \Phi (r)\right\vert =N!\cdot \frac{1}{\prod_{j\in T,j\neq r}|r-j|}%
.
\end{equation*}
Consider any quantity $c\cdot i^{2}\cdot w\in T_{2}$. \ Then
\begin{flalign} & \left|\Phi(c \cdot w \cdot i^2)\right|/\left|\Phi(w)\right| = \frac{\prod_{j \in T, j \neq w} |w- j|}{ \prod_{j \in T, j \neq c \cdot i^2 \cdot w} |w \cdot c \cdot i^2 - j|} \notag \\
& = \frac{\left|w-2w\right| \cdot \left(\prod_{j=1}^{d_2} \left|w-c \cdot j^2 \cdot w\right|\right) \cdot \left(\prod_{j=1}^{d_1} \left(w-\left\lfloor \frac{w}{cj^2}\right\rfloor\right)\right)}{\left|c \cdot i^2 \cdot w - w\right| \cdot \left|c \cdot i^2 \cdot w - 2w\right| \cdot \left( \prod_{j=1, j\neq i}^{d_2} \left|w \cdot c \cdot i^2 - w \cdot c \cdot j^2\right| \right)\cdot \left( \prod_{j=1}^{d_1} \left(w \cdot c \cdot i^2 - \left\lfloor\frac{w}{c \cdot j^2}\right\rfloor\right)\right)}\notag \\
& = \frac{c^{d_2} \cdot \left(\prod_{j=1}^{d_2} \left( j^2 - \frac{1}{c}\right)\right) \cdot \prod_{j=1}^{d_1} \left(w-\left\lfloor\frac{w}{c\cdot j^2}\right\rfloor\right)}{ \left(ci^2 -1\right)\cdot \left(ci^2 -2 \right) \cdot c^{d_2-1} \cdot \left( \prod_{j=1, j \neq i}^{d_2} \left|i^2 - j^2\right|\right) \cdot \left(\prod_{j=1}^{d_1} \left(w \cdot c \cdot i^2 - \left\lfloor \frac{w}{c \cdot j^2}\right\rfloor\right)\right)} \notag \\
& \leq
\frac{ c \cdot \left( \prod_{j=1}^{d_2} \left( j^2 - \frac{1}{c}\right)\right) \cdot \prod_{j=1}^{d_1} \left(w-\left\lfloor\frac{w}{c\cdot j^2}\right\rfloor\right)}{ \left(ci^2 -1\right)\cdot \left(ci^2 -2 \right) \cdot \left(\prod_{j=1, j \neq i}^{d_2} \left|i^2 - j^2\right|\right) \cdot \left(\prod_{j=1}^{d_1} \left(w \cdot c \cdot i^2 - \frac{w}{c \cdot j^2}\right)\right)} \label{firstlongequation}
\end{flalign}
Now, observe that
\begin{align}
\prod_{j=1}^{d_1} \left(w-\left\lfloor\frac{w}{c\cdot j^2}\right\rfloor\right)
\leq \prod_{j=1}^{d_1} \left(w - \frac{w}{cj^2} +1 \right) =
\prod_{j=1}^{d_1} w \cdot \left(1-\frac{1}{cj^2}\right) \cdot \left( 1+
\frac{1}{w \cdot \left(1-\frac{1}{cj^2}\right)}\right) \notag \\
\leq \prod_{j=1}^{d_1} w \cdot \left(1-\frac{1}{cj^2}\right) \left(1 + \frac{%
1}{(1-1/c) \cdot w}\right) \leq \left( \prod_{j=1}^{d_1} w \cdot \left(1-%
\frac{1}{cj^2}\right)\right) \cdot \left(1+o(1) \right). \label{alongequation}
\end{align}
Hence, we see that Expression \eqref{firstlongequation} is bounded by
\begin{align}
& \frac{c\cdot \left( \prod_{j=1}^{d_{2}}\left( j^{2}-\frac{1}{c}\right)
\right) \cdot \left( \prod_{j=1}^{d_{1}}\left( 1-\frac{1}{c\cdot j^{2}}%
\right) \right) \cdot (1+o(1))}{\left( ci^{2}-1\right) \cdot \left(
ci^{2}-2\right) \cdot \left( \prod_{j=1,j\neq i}^{d_{2}}\left\vert
i^{2}-j^{2}\right\vert \right) \cdot \left( \prod_{j=1}^{d_{1}}\left( c\cdot
i^{2}-\frac{1}{c\cdot j^{2}}\right) \right) } \notag \\
& \leq \frac{c\cdot \left( d_{2}!\right) ^{2}\cdot \left(
\prod_{j=1}^{d_{1}}\left( 1-\frac{1}{c\cdot j^{2}}\right) \right) \cdot
(1+o(1))}{\left( ci^{2}-1\right) \cdot \left( ci^{2}-2\right) \cdot \left(
\prod_{j=1,j\neq i}^{d_{2}}\left\vert i-j\right\vert \left\vert
i+j\right\vert \right) \cdot (c\cdot i^{2})^{d_{1}}\cdot \left(
\prod_{j=1}^{d_{1}}\left( 1-\frac{1}{c^{2}\cdot i^{2}\cdot j^{2}}\right)
\right) } \notag \\
& =\frac{c\cdot \left( d_{2}!\right) ^{2}\cdot 2i^{2}\cdot \left(
\prod_{j=1}^{d_{1}}\left( 1-\frac{1}{c\cdot j^{2}}\right) \right) \cdot
(1+o(1))}{\left( ci^{2}-1\right) \cdot \left( ci^{2}-2\right) \cdot \left(
d_{2}+i\right) !\left( d_{2}-i\right) !\cdot (c\cdot i^{2})^{d_{1}}\cdot
\left( \prod_{j=1}^{d_{1}}\left( 1-\frac{1}{c^{2}\cdot i^{2}\cdot j^{2}}%
\right) \right) } \notag \\
& \leq \frac{c\cdot 2i^{2}\cdot \left( d_{2}!\right) ^{2}\cdot (1+o(1))}{%
\left( ci^{2}-1\right) \left( ci^{2}-2\right) \cdot \left( d_{2}+i\right)
!\left( d_{2}-i\right) !\cdot (c\cdot i^{2})^{d_{1}}}\leq \frac{2\left(
1+o(1)\right) }{\left( 1-\frac{1}{c\cdot i^{2}}\right) \cdot (c\cdot
i^{2}-2)\cdot (c\cdot i^{2})^{d_{1}}}. \label{verylongequation}
\end{align}%
In the penultimate inequality, we used the fact that $\frac{(d_{2}!)^{2}}{%
(d_{2}+i)!(d_{2}-i)!}=\frac{\binom{2d_{2}}{d_{2}+i}}{\binom{2d_{2}}{d_{2}}}%
\leq 1$.
Next, consider any quantity $\left\lfloor \frac{w}{c\cdot i^{2}}\right\rfloor \in T_{1}$%
. \ Then
\begin{flalign} & \left|\Phi\left( \left\lfloor \frac{w}{c \cdot i^2}\right\rfloor\right)\right|/\left|\Phi(w)\right| \notag \\
& = \frac{|w-2w| \left(\prod_{j=1}^{d_2} |w - cj^2 w| \right) \left(\prod_{j=1}^{d_1} \left( w - \left\lfloor \frac{w}{cj^2}\right\rfloor\right)\right)}{ \left(w-\left\lfloor \frac{w}{c \cdot i^2}\right\rfloor\right) \cdot \left(2w - \left\lfloor \frac{w}{c \cdot i^2}\right\rfloor \right) \left(\prod_{j=1}^{d_2} \left(w \cdot c \cdot j^2 - \left\lfloor \frac{w}{c \cdot i^2}\right\rfloor\right)\right) \prod_{j=1, j\neq i}^{d_1} \left|\left\lfloor \frac{w}{c \cdot i^2}\right\rfloor - \left\lfloor \frac{w}{c \cdot j^2}\right\rfloor\right|} \notag \\
& \leq
\frac{|w-2w| \left(\prod_{j=1}^{d_2} |w - cj^2 w| \right) \left(\prod_{j=1}^{d_1} \left( w - \left\lfloor \frac{w}{cj^2}\right\rfloor\right)\right)}{ \left(w- \frac{w}{c \cdot i^2}\right) \cdot \left(2w - \frac{w}{c \cdot i^2} \right) \left(\prod_{j=1}^{d_2} \left(w \cdot c \cdot j^2 - \frac{w}{c \cdot i^2}\right)\right) \prod_{j=1, j\neq i}^{d_1} \left|\left\lfloor \frac{w}{c \cdot i^2}\right\rfloor - \left\lfloor \frac{w}{c \cdot j^2}\right\rfloor\right|} \notag \\
& \leq \frac{|w-2w| \left(\prod_{j=1}^{d_2} |w - cj^2 w| \right) \left(\prod_{j=1}^{d_1} \left( w - \frac{w}{cj^2}\right)\right) \cdot \left(1+o(1)\right)}{ \left(w- \frac{w}{c \cdot i^2}\right) \cdot \left(2w - \frac{w}{c \cdot i^2} \right) \left(\prod_{j=1}^{d_2} \left(w \cdot c \cdot j^2 - \frac{w}{c \cdot i^2}\right)\right) \prod_{j=1, j\neq i}^{d_1} \left|\left\lfloor \frac{w}{c \cdot i^2}\right\rfloor - \left\lfloor \frac{w}{c \cdot j^2}\right\rfloor\right|} \label{anotherlongequation}
\end{flalign}
Here, the final inequality used \Cref{alongequation}.
Let us consider the expression
$\prod_{j=1,j\neq i}^{d_{1}}\left\vert \left\lfloor \frac{%
w}{c\cdot i^{2}}\right\rfloor - \left\lfloor \frac{w}{c\cdot j^{2}}\right\rfloor \right\vert $%
. \
This quantity is \emph{at least}
\begin{align}
\prod_{j=1, j\neq i}^{d_1}\left( \left|\frac{w}{c \cdot i^2} - \frac{w}{c
\cdot j^2} \right| -1\right) = w^{d_1-1} \cdot \prod_{j=1, j \neq i}^{d_1}
\frac{\left|j^2 - i^2\right| - \frac{c i^2 j^2}{w}}{ci^2 j^2} \notag \\
= w^{d_1-1} \cdot \prod_{j=1, j \neq i}^{d_1} \frac{\left|j-i| \cdot
|j+i\right| - \frac{c i^2 j^2}{w}}{ci^2 j^2} \notag \\
= \left(\frac{w}{ci^2}\right)^{d_1-1} \cdot \prod_{j=1, j \neq i}^{d_1}
\frac{\left|j-i| \cdot |j+i\right| - \frac{c i^2 j^2}{w}}{j^2}
\label{aboveexp}
\end{align}
We claim that Expression \eqref{aboveexp} is at least
\begin{equation}
\left( \frac{w}{ci^{2}}\right) ^{d_{1}-1}\cdot \frac{1}{2}.
\label{provedintheappendix}
\end{equation}%
In the case that $c=2$ and $d_{1}$ is (at most) $w^{1/3}$, this is precisely
\cite[Claim 4]{zhandry}. \ We will ultimately take $c$ to be a constant
strictly greater than 2 and hence $d_{1}=\lfloor \left( w/c\right)
^{1/3}\rfloor $ is a constant factor smaller than $w^{1/3}$. \ The proof of
\cite[Claim 4]{zhandry} works with cosmetic changes in this case. \ For
completeness, we present a derivation of the claim in \Cref{app}.
\Cref{provedintheappendix} implies that Expression \eqref{anotherlongequation} is
at most:
\begin{align}
\frac{|w-2w| \left(\prod_{j=1}^{d_2} |w - cj^2 w| \right)
\left(\prod_{j=1}^{d_1} \left( w - \frac{w}{cj^2}\right)\right) \cdot
\left(1+o(1)\right)}{ \left(w- \frac{w}{c \cdot i^2}\right) \cdot \left(2w -
\frac{w}{c \cdot i^2} \right) \left(\prod_{j=1}^{d_2} \left(w \cdot c \cdot
j^2 - \frac{w}{c \cdot i^2}\right)\right) \left(\frac{w}{ci^2}%
\right)^{d_1-1} \cdot \frac{1}{2}} \notag \\
= \frac{2 \left(\prod_{j=1}^{d_2} |1 - cj^2| \right) \left(\prod_{j=1}^{d_1}
\left(1 - \frac{1}{cj^2}\right)\right) \cdot \left(1+o(1)\right)}{ \left(1-
\frac{1}{c \cdot i^2}\right) \cdot \left(2 - \frac{1}{c \cdot i^2} \right)
\left(\prod_{j=1}^{d_2} \left(c \cdot j^2 - \frac{1}{c \cdot i^2}%
\right)\right) \left(\frac{1}{ci^2}\right)^{d_1-1}} \notag \\
= \frac{2 \left(\prod_{j=1}^{d_2} (j^2-1/c) \right) \left(\prod_{j=1}^{d_1}
\left(1 - \frac{1}{cj^2}\right)\right) \cdot \left(1+o(1)\right)}{ \left(1-
\frac{1}{c \cdot i^2}\right) \cdot \left(2 - \frac{1}{c \cdot i^2} \right)
\left(\prod_{j=1}^{d_2} \left(j^2 - \frac{1}{c^2 \cdot i^2}\right)\right)
\left(\frac{1}{ci^2}\right)^{d_1-1}} \notag \\
\leq \frac{2 \left(1+o(1)\right)}{ \left(1- \frac{1}{c \cdot i^2}\right)
\cdot \left(2 - \frac{1}{c \cdot i^2} \right) \left(\frac{1}{ci^2}%
\right)^{d_1-1}} \leq 4 \cdot \left(ci^2\right)^{d_1-1}. \label{implication}
\end{align}
Summarizing Equations \eqref{verylongequation} and \eqref{implication}, we have shown
that: for any quantity $c \cdot i^2 \cdot w \in T_2$,
\begin{equation} \label{keyeq1}
\left|\Phi(c \cdot w \cdot i^2)\right|/\left|\Phi(w)\right| \leq \frac{2
\left(1+o(1)\right)}{\left(1-\frac{1}{c \cdot i^2}\right) \cdot (c \cdot i^2
- 2) \cdot (c\cdot i^2)^{d_1}}
\end{equation}
and for any quantity $\left\lfloor \frac{w}{c \cdot i^2} \right\rfloor \in T_1$,
\begin{equation} \label{keyeq2}
\left|\Phi\left( \left\lfloor \frac{w}{c \cdot i^2}\right\rfloor\right)\right|/\left|\Phi(w)%
\right| \leq 4\cdot \left(ci^2\right)^{d_1-1}.
\end{equation}
Let $\phi =\Phi /C$, where $C$ is as in \Cref{Cdef}. \ Let $%
D_{1}=d_{1}$ and $D_{2}=d_{2}$. \ \Cref{phdlem} implies that $\phi $ is a
feasible solution for the dual linear program of \Cref{duallpsec}. \ We now
show that, for any constant $\delta >0$, by choosing $c$ to be a
sufficiently large constant (that depends on $\delta $), we can ensure
that $\phi $ achieves objective value $1-2\delta $.
Let
\begin{eqnarray*}
A &=&|\Phi (w)|\cdot w^{D_{1}}, \\
B &=&|\Phi (2w)|\cdot (2w)^{D_{1}},
\end{eqnarray*}%
and
\begin{equation*}
E=\sum_{i=1}^{d_{1}}|\Phi (\lfloor w/ci^{2}\rfloor )|\cdot \left( \lfloor
w/ci^{2}\rfloor \right) ^{D_{1}}+\sum_{i=1}^{d_{2}}|\Phi (\lfloor w\cdot
ci^{2}\rfloor )|\cdot \left( w\cdot c\cdot i^{2}\right) ^{D_{1}}.
\end{equation*}%
By \Cref{zeros}, $C=A+B+E$.
Moreover, observe that $\text{sgn}(\Phi(w)) = - \text{sgn}(\Phi(2w))$, so
without loss of generality we may assume $\Phi(w) \geq 0$ and $\Phi(2w) \leq
0$ (if not, then replace $\Phi$ with $-\Phi$ throughout).
We now claim that by choosing $c$ to be a sufficiently large constant, we
can ensure that $E \leq \delta \cdot A$. To see this, observe that
Equations \eqref{keyeq1} and \eqref{keyeq2}, along with the fact that $%
D_1=d_1$ and $D_2=d_2$ implies that
\begin{align*}
E/A \leq \frac{1}{w^{D_1}} \left[ \left(\sum_{i=1}^{d_1} \left(\lfloor
w/ci^2\rfloor\right)^{D_1}\cdot 4 \cdot \left(ci^2\right)^{d_1-1} \right) +
\left(\sum_{i=1}^{d_2} \left(w \cdot c \cdot i^2\right)^{D_1} \frac{2
\left(1-\frac{1}{c \cdot i^2}\right)\left(1+o(1)\right)}{(c \cdot i^2 - 2)
\cdot (c\cdot i^2)^{d_1}}\right)\right] \\
\leq \frac{1}{w^{D_1}} \left[ \left(\sum_{i=1}^{d_1}
\left(w/ci^2\right)^{D_1} \cdot 4 \cdot \left(ci^2\right)^{d_1-1} \right) +
\left(\sum_{i=1}^{d_2} \left(w \cdot c \cdot i^2\right)^{D_1} \frac{2
\left(1-\frac{1}{c \cdot i^2}\right)\left(1+o(1)\right)}{(c \cdot i^2 - 2)
\cdot (c\cdot i^2)^{d_1}}\right) \right] \\
\leq 4 \left(\sum_{i=1}^{d_1} \frac{1}{c\cdot i^2}\right) +
\left(\sum_{i=1}^{d_2} \frac{2 \left(1+o(1)\right)}{\left(1-\frac{1}{c \cdot
i^2}\right) \left(c \cdot i^2 - 2\right)}\right) \\
\end{align*}
Since $\sum_{i=1}^{\infty}1/(ci^2) \leq \frac{\pi^2}{6c}$, we see that
choosing $c$ to be a sufficiently large constant depending on $\delta$
ensures that $E/A \leq \delta$ as desired.
Hence, $\phi$ achieves objective value at least
\begin{align*}
\phi(w) \cdot w^{D_1} - \phi(2w) \cdot (2w)^{D_1} - \sum_{\ell \in \{1,
\dots, N\}, \ell \not\in \{w, 2w\}} |\phi(\ell)| \cdot \ell^{D_1} \\
\geq \frac{A + B - E}{A+B+E} \geq \frac{(1-\delta) A + B}{%
(1+\delta)A + B} \geq 1-2\delta.
\end{align*}
\subsection{Approximate counting with classical samples}
\label{sec:classical_samples_queries}
For completeness, in this section, we sketch classical counterparts of \Cref{thm:main} and \Cref{thm:alg}. That is, we show tight bounds on classical randomized algorithms for $\mathsf{ApxCount}_{N,w}$ that make membership queries and have access to uniform random samples from the set being counted.
\begin{proposition}
There is a classical randomized algorithm that solves $\mathsf{ApxCount}_{N,w}$ with high probability using either $O(N/w)$ queries to the membership oracle for $S$, or else using $O(\sqrt{w})$ uniform samples from $S$.
\end{proposition}
\begin{proof}[Proof sketch]
By reducing approximate counting to the problem of estimating the mean of a biased coin, $O(N/w)$ queries are sufficient.
Alternatively, if we take $R$ samples, then the expected number of birthday collisions is $\binom{R}{2} \cdot \frac{1}{|S|}$ and the variance is $\binom{R}{2} \cdot \frac{1}{|S|}\left(1-\frac{1}{|S|}\right)$. So, taking $O(\sqrt{w})$ samples and computing the number of birthday collisions is sufficient to distinguish $|S| \le w$ from $|S| \ge 2w$ with $\frac{2}{3}$ success probability.
\end{proof}
\begin{proposition}
Let $M$ be a classical randomized algorithm that makes $T$ queries to the membership oracle for $S$,
and takes a total of $R$ uniform samples from $S$.
If $M$ decides whether $\left\vert S\right\vert =w$\ or $\left\vert S\right\vert =2w$
with high probability, promised that one of those is the case, then either
$T = \Omega(N/w)$ or $R = \Omega(\sqrt{w})$.
\end{proposition}
\begin{proof}[Proof sketch]
Note that without loss of generality, we may
assume that the algorithm first takes all of the samples it needs, and then queries random elements of $[N]$ that did not appear in the samples. Suppose the algorithm takes $R = o(\sqrt{w})$ samples and then makes $T = o(N/w)$ queries. Consider what happens when the algorithm tries to distinguish a random subset of size $w$ from a random subset of size $2w$ of $[N]$. By a union bound, the probability that the algorithm sees any collisions in the samples is $o(1)$, and the probability that the algorithm finds any additional elements of $S$ via queries is also $o(1)$. So, if the set has size either $w$ or $2w$, with $1 - o(1)$ probability, the algorithm's view of the samples is just a random subset of size $R$ of $[N]$ drawn without replacement, and the algorithm's view of the queries is just $T$ ``no'' answers to membership queries. Hence, the algorithm fails to distinguish random sets of size $w$ and size $2w$ with any constant probability of success.
\end{proof}
\subsection{Extending the lower bound to QSampling unitarily}
\label{sec:unitary}
So far in this section we have proved upper and lower bounds on the power of quantum algorithms for approximate counting that have
access to two resources (in addition to membership queries):
copies of $|S\>$, and the unitary transformation that reflects about $|S\>$.
The assumption of access to the reflection unitary is justified by the argument that, if we had access to a unitary that prepared $|S\>$, then it could be used to reflect about $|S\>$ as well.
Giving the algorithm access
to just the two resources above is an appealing model to use for upper bounds, since it does not assume anything about the method by which copies of $|S\>$ are prepared. This means algorithms
derived in this model work in many different settings.
For example, the algorithm may be able to QSample
because someone else simply handed the algorithm copies of $|S\>$, or perhaps several copies of $|S\>$ just happen to be stored in the algorithm's quantum memory as a side effect of the execution of some earlier quantum algorithm. The upper bound given
in \Cref{thm:alg} applies in any of these settings.
On the other hand, since only
permitting access to QSamples
and reflections about $|S\>$ ties the algorithm's hands,
lower bounds for this model
(e.g., \Cref{thm:main}) could be viewed
as weaker than is desirable.
In particular, our original
justification
for allowing access to reflections about $|S\>$
was that access to a unitary that prepared
the state $|S\>$ would in particular allow
such reflections to be done.
Given this justification, it is very natural to wonder whether
our lower bounds extend beyond just
QSamples and reflections, to
algorithms that are given access
to \emph{some} unitary process that
permits both QSampling and reflections about $|S\>$.
Note that
an algorithm with access to such a unitary could potentially exploit the unitary in ways other
than QSamples and reflections to learn information about $|S\>$.
For example, the algorithm could choose to run the unitary on inputs that do not produce $|S\>$.
More generally, given a quantum circuit that implements a unitary, it is possible to construct, in a completely black-box manner, the inverse of this unitary, and also a controlled version of the unitary.
The algorithm may choose to run the inverse on a state other than $|S\>$ to learn some additional information that is not captured by access
to QSamples and reflections alone.
In summary, in this section
we ask whether we can we extend the lower bound of \Cref{thm:main} to work in a model where the algorithm is given access to some unitary operator that conveys the power to both QSample and reflect about $|S\>$.\footnote{We thank Alexander Belov (personal communication) for raising this question.}
Via \cref{thm:finalthm} below, we explain that the answer is yes.
\medskip
It may seem convenient to assume that the unitary
transformation preparing $|S\>$
maps the all-zeros state to $|S\>$.
But this is not the most general method of preparing $|S\>$ by a unitary.
A unitary $U$ that maps the all-zeros state to $|S\>|\psi\>$ would also suffice to create copies of $|S\>$, since the register containing $|\psi\>$ can simply be ignored for the remainder of the computation.
More formally, assume $U$ behaves as
\begin{equation}
U|0^m\> = |S\>|\psi\>,
\end{equation}
where $|S\>|\psi\>$ is some $m$-qubit state.
Clearly we can use $U$ to create as many copies of $|S\>$ as we like,
which as a by-product also creates copies of $|\psi\>$.
This unitary also lets us reflect about $|S\>$.
To see how, first use this unitary to create a copy of $|\psi\>$, and then consider the action of the unitary $U(\mathbbold{1} - 2|0^m\>\<0^m|)U^\dagger$ on the state $|\phi\>|\psi\>$ for any state $|\phi\>$. We claim that this unitary acts as a reflection about $|S\>$ when restricted to the first register.
This establishes that any $U$ of this form subsumes the power of both QSamples and reflections about $|S\>$.
Let us also assume without loss of generality that $|S\>|\psi\>$ is orthogonal to $|0^m\>$ from now on. This can be achieved by adding an additional qubit to the input that is always negated by the unitary. That is, we could instead consider the map $(U \otimes X)|0^m\>|0\> = |S\>|\psi\>|1\>$, which is orthogonal to the starting state by construction, and only increases the value of $m$ by $1$.
Of course, the requirement that $U|0^m\> = |S\>|\psi\>$ does not fully specify $U$, as
it does not prescribe how $U$ behaves on other input states.
A reasonable prescription is that $U$ should behave ``trivially'' on other input states, so that it does not leak information about $S$ by its behavior on other states.
In tension with this prescription is the fact the rest of the unitary must depend on $S$, since the first column of the unitary contains $|S\>$, and the rest of the columns have to be orthogonal to this.
Alexander Belov (personal communication) brought to our attention a
very simple construction of such a unitary that leaks minimal additional information about $S$.
Consider the unitary $U$ that satisfies $U|0^m\> = |S\>|\psi\>$ and $U|S\>|\psi\> = |0^m\>$, with $U$ acting as identity outside $\mathrm{span}\{|0^m\>,|S\>|\psi\>\}$. $U$ is simply a reflection about the state $\frac{1}{\sqrt{2}}\bigl(|0^m\>-|S\>|\psi\>\bigr)$. This state is correctly normalized because we assumed that $|S\>|\psi\>$ is orthogonal to $|0^m\>$.
Clearly $U$ is now fully specified on the entire domain (once we have fixed $|\psi\>$) and it does not seem to leak any additional information about $S$.
In order to prove concrete lower bounds on the cost
of algorithms for approximate counting given access to $U$, we need to fix $|\psi\>$.
To answer the question posed in this section, we only need to establish that there exists \emph{some} choice of $|\psi\>$ for which our algorithms cannot be improved. (Note that we cannot hope to establish lower bounds for arbitrary $|\psi\>$, since $|\psi\>$ could just contain the answer to the problem we are solving.)
To this end we make the specific choice of $|\psi\> = |S\>$ and consider the unitary $V$ that acts as the unitary $U$ above with $|\psi\> = |S\>$.
In other words, $V$ maps $|0^m\>$ to $|S\>|S\>$, $|S\>|S\>$ to $|0^m\>$, and acts as identity on the rest of the space.
We also assume that $|0^m\>$ is orthogonal to $|S\>|S\>$.
In other words, $V$ simply reflects about the state $\frac{1}{\sqrt{2}}\bigl(|0^m\>-|S\>|S\>\bigr)$.
As previously discussed, granting an algorithm access to this unitary $V$ lends the algorithm at least as much power the ability to QSample and perform reflections about $|S\>$. How efficiently can we solve approximate counting with membership queries and uses of the unitary $V$?
We can use our Laurent polynomial method to establish optimal lower bounds in this model as well and we obtain lower bounds identical to \Cref{thm:main}.
\begin{theorem} \label{thm:finalthm} \label{thm:unitary}
Let $Q$ be a quantum algorithm that makes $T$ queries to the membership oracle for $S$,
and makes $R$ uses of the unitary $V$ defined above (and its inverse and controlled-$V$).
If $Q$ decides whether $\left\vert S\right\vert =w$\ or $\left\vert S\right\vert =2w$
with high probability, promised that one of those is the case, then either
\begin{equation}
T=\Omega\left( \sqrt{\frac{N}{w}}\right) \qquad \textrm{or} \qquad
R=\Omega\left( \min\left\{ w^{1/3},\sqrt{\frac{N}{w}}\right\} \right).
\end{equation}
\end{theorem}
\begin{proof}
We follow the same strategy as in
the proof of \Cref{thm:main}. Recall that $x \in \{0, 1\}^N$ denotes the indicator vector of the set $S$. We only need to show that such a quantum algorithm gives rise to a Laurent polynomial in $|S|:=\sum_{i=1}^n x_i$, with maximum exponent $O(T+R)$ and minimum exponent at least $-O(R)$ (as shown in \Cref{laurentlem} for the QSamples and reflections model).
We can prove this exactly the same way as \Cref{laurentlem} is established. Our quantum algorithm starts out from a canonical starting state that does not depend on the input and hence each entry of the starting state is a degree-$0$ polynomial.
Membership queries involve multiplication with an oracle whose entries are ordinary polynomials of degree at most $1$. The only thing that remains is understanding what the entries of the unitary $V$ look like. We claim that the entries of $V$ are given
by a polynomial of degree at most 2 in the entries of the input $x$,
with all coefficients of this degree-2 polynomial equal
to either a constant, or a constant multiple of $|S|^{-1}$.
To see this, note that $V$ is simply a reflection about the state
\begin{equation}
\frac{1}{\sqrt{2}}\bigl(|0^m\>-|S\>|S\>\bigr)
= \frac{1}{\sqrt{2}}\left(|0^m\>-\frac{1}{|S|}\Bigl(\sum_{i}x_i|i\>\Bigr)\Bigl(\sum_{j}x_j|j\>\Bigr)
\right).
\end{equation}
The coefficient in front of $|0^m\>$ is a degree-$0$ polynomial and the other nonzero coefficients are
a polynomial of degree at most 2 in the entries of the input $x$, with each coefficient of this polynomial equal to a constant multiple of $|S|^{-1}$.
Hence, each entry of the unitary $V$ is also a polynomial of degree at most 2 in the entries of the input $x$,
with each coefficient of this degree-2 polynomial equal
to either a constant, or a constant multiple of $|S|^{-1}$. The same also holds for controlled-$V$, since that unitary is just the direct sum of identity with $V$. $V$ is also self-inverse, so we do not need to account for that separately.
After the algorithm has made all the membership queries and uses of $V$, each amplitude of the final quantum state can be expressed as a polynomial of degree $O(T + R)$ in the input $x$, in which all coefficients are constant multiples of $|S|^{-R}$. The acceptance probability $p(x)$ of this algorithm will be a sum of squares of such polynomials.
Exactly
as in the proof of \Cref{thm:main},
\Cref{symlem} implies that there is a univariate polynomial
$q$ of degree at most $O(T+R)$, with coefficients that are multiples
of the coefficients of $p$, such that for all integers $k \in \{0, \dots, N\}$,
\begin{equation}
q\left( k\right) :=\E_{\left\vert X\right\vert =k}\left[
p\left( X\right) \right] .
\end{equation}
Since the coefficients of $p(X)$ are constant multiples of $|X|^{-2R}$,
$q$ is in fact a real Laurent polynomial in $k$, with maximum exponent at most $O(R+T)$\ and minimum exponent at
least $-2R$.
The theorem follows by a direct
application \Cref{mainthm} to $q$.
\end{proof}
\section{Discussion and open problems\label{OPEN}}
\label{sec:open}
\subsection{Approximate counting with QSamples and queries only}
\label{IMPROVE}If we consider the model where we only have membership
queries and samples (but no reflections), then the best upper bound we can
show is $O\left( \min \left\{ \sqrt{w},{\sqrt{{N}/{w}}}\right\} \right) $,
using the sampling algorithm that looks for birthday collisions, and the
quantum counting algorithm. It
would be interesting to improve the lower bound further in this case, but it
is clear that the Laurent polynomial approach cannot do so, since it hits a
limit at $w^{1/3}$. \ Hence a new approach is needed to tackle the model
without reflections.
We now give what we think is a viable path to solve this problem. \
Specifically, we observe that our problem---of lower-bounding the number of
copies of $\left\vert S\right\rangle $\ \textit{and} the number of queries
to $\mathcal{O}_{S}$\ needed for approximate counting of $S$---can be
reduced to a pure problem of lower-bounding the number of copies of $%
\left\vert S\right\rangle $. \ To do so, we use a hybrid argument, closely
analogous to an argument recently given by Zhandry \cite{zhandry:lightning}
in the context of quantum money.
Given a subset $S\subseteq\left[ L\right] $, let $\left\vert S\right\rangle $%
\ be a uniform superposition over $S$ elements. \ Then let%
\begin{equation}
\rho_{L,w,k}:=\E_{S\subseteq\left[ L\right] ~:~\left\vert S\right\vert =w}%
\left[ \left( \left\vert S\right\rangle \left\langle S\right\vert \right)
^{\otimes k}\right]
\end{equation}
be the mixed state obtained by first choosing $S$\ uniformly at random
subject to $\left\vert S\right\vert =w$, then taking $k$ copies of $%
\left\vert S\right\rangle $. \ Given two mixed states $\rho$\ and $\sigma$,
recall also that the \textit{trace distance}, $\left\Vert
\rho-\sigma\right\Vert _{\mathrm{tr}}$, is the maximum bias with which $%
\rho $\ can be distinguished from $\sigma$\ by a single-shot measurement.
\begin{theorem}
\label{hybridthm}Let $2w\leq L\leq N$. \ Suppose $\left\Vert
\rho_{L,w,k}-\rho_{L,2w,k}\right\Vert _{\mathrm{tr}}\leq\frac{1}{10}$. \
Then any quantum algorithm $Q$\ requires either $\Omega\left( \sqrt{\frac{N}{%
L}}\right) $\ queries to $\mathcal{O}_{S}$\ or else $\Omega\left( k\right) $%
\ copies of $\left\vert S\right\rangle $ to decide whether $\left\vert
S\right\vert =w$\ or $\left\vert S\right\vert =2w$ with success probability
at least $2/3$, promised that one of those is the case.
\end{theorem}
\begin{proof}
Choose a subset $S\subseteq\left[ N\right] $ uniformly at random, subject to
$\left\vert S\right\vert =w$\ or $\left\vert S\right\vert =2w$, and consider
$S$ to be fixed. \ Then suppose we choose $U\subseteq\left[ N\right]
$\ uniformly at random, subject to both $\left\vert U\right\vert =L$\ and
$S\subseteq U$. \ Consider the hybrid in which $Q$ is still given $R$\ copies
of the state $\left\vert S\right\rangle $, but now gets oracle access to
$\mathcal{O}_{U}$\ rather than $\mathcal{O}_{S}$. \ Then so long as $Q$ makes
$o\left( \sqrt{\frac{N}{L}}\right) $\ queries to its oracle, we claim that
$Q$ cannot distinguish this hybrid from the \textquotedblleft
true\textquotedblright\ situation (i.e., the one where $Q$\ queries
$\mathcal{O}_{S}$) with $\Omega\left( 1\right) $\ bias. \ This claim follows
almost immediately from the BBBV Theorem \cite{bbbv}. \ In effect, $Q$ is
searching the set $\left[ N\right] \setminus S$ for any elements of
$U\setminus S$\ (the \textquotedblleft marked items,\textquotedblright\ in
this context), of which there are $L-\left\vert S\right\vert $ scattered
uniformly at random. \ In such a case, we know that $\Omega\left( \sqrt
{\frac{N-\left\vert S\right\vert }{L-\left\vert S\right\vert }}\right)
=\Omega\left( \sqrt{\frac{N}{L}}\right) $\ quantum queries are needed to
detect the marked items with constant bias.
Next suppose we first choose $U\subseteq\left[ N\right] $ uniformly at
random, subject to $\left\vert U\right\vert =L$, and consider $U$ to be fixed.
\ We then choose $S\subseteq U$\ uniformly at random, subject to $\left\vert
S\right\vert =w$\ or $\left\vert S\right\vert =2w$. \ Note that this produces
a distribution over $\left( S,U\right) $\ pairs identical to the
distribution that we had above. \ In this case, however, since $U$ is fixed,
queries to $\mathcal{O}_{U}$\ are no longer relevant. \ The only way to decide
whether\ $\left\vert S\right\vert =w$\ or $\left\vert S\right\vert =2w$\ is by
using our copies of $\left\vert S\right\rangle $---of which, by assumption, we
need $\Omega\left( k\right) $\ to succeed with constant bias, even after
having fixed $U$.
\end{proof}
One might think that \Cref{hybridthm} would lead to immediate improvements
to our lower bound for the queries and samples model. \ In practice,
however, the best lower bounds that we currently have, even purely on the
number of copies of $\left\vert S\right\rangle $, come from the Laurent
polynomial method (\Cref{thm:main})! \ Having said that, we are optimistic
that one could obtain a lower bound that beats \Cref{thm:main}\ at least
when $w$\ is small, by combining \Cref{hybridthm} with a brute-force
computation of trace distance.
\subsection{Approximate counting to multiplicative factor
\texorpdfstring{$1+\varepsilon$}{1+eps}}
Throughout, we considered the task of approximating $\left\vert S\right\vert
$ to within a multiplicative factor of $2$. \ But suppose our task was to
distinguish the case $\left\vert S\right\vert \leq w$\ from the case $%
\left\vert S\right\vert \geq \left( 1+\varepsilon \right) w$; then what is
the optimal dependence on $\varepsilon $?
In the model with quantum membership queries only, the algorithm
of Brassard et al.~\cite[Theorem 15]{BHMT02}
makes $O\Bigl(\frac{1}{\varepsilon }\sqrt{\frac{N}{w}}\Bigr)$\ queries, which is optimal~\cite{nayak-wu}.
\ The algorithm uses amplitude amplification, the basic primitive of Grover's search algorithm
\cite{grover}. \ The original algorithm of Brassard et al.\ also used
quantum phase estimation, in effect \textit{combining} Grover's algorithm
with Shor's period-finding algorithm. \ However, one can remove the phase estimation, and adapt Grover search with an unknown number of marked items to get an approximate count of the number of marked items~\cite{aaronson-rall}.
One can also show without too much difficulty that in the
queries+QSamples model, the problem can be solved with
\begin{equation}
O\left( \min \left\{ \frac{\sqrt{w}}{\varepsilon^2 },\frac{1}{\varepsilon}\sqrt{\frac{N}{w}}\right\} \right)
\end{equation}
queries and copies of $\left\vert S\right\rangle $. \
As observed after \Cref{thm:alg}, the problem can also be solved with
\begin{equation}
O\left( \min \left\{ \frac{w^{1/3}}{\varepsilon ^{2/3}},\frac{1}{\varepsilon
}\sqrt{\frac{N}{w}}\right\} \right)
\end{equation}%
samples and reflections. \ On the lower bound side, what generalizations of %
\Cref{thm:main}\ can we prove that incorporate $\varepsilon $? \ We note
that the explosion argument doesn't automatically\ generalize; one would
need to modify something to continue getting growth in the polynomials $u$\
and $v$\ after the first iteration. \ The lower bound using dual polynomials
should generalize, but back-of-the-envelope calculations show that the lower
bound does not match the upper bound.
\subsection{Other questions}
\para{Non-oracular example of our result.} \
Is there any interesting real-world example of a class of sets for which
QSampling and membership testing are both efficient, but approximate
counting is not? \ (I.e., is there an interesting non-black-box setting that
appears to exhibit the behavior that this paper showed can occur in the
black-box setting?)
\para{The Laurent polynomial connection.} \ At
a deeper level, is there is any meaningful connection between our two uses
of Laurent polynomials? {\ And what other applications can be found for the
Laurent polynomial method?}
\section{Followup work}
Since this work was completed, Belovs and Rosmanis \cite{belovs} obtained essentially tight lower bounds on the complexity of approximate counting with access to membership queries, QSamples, reflections, and a unitary transformation that prepares the QSampling state, for all possible tradeoffs between these different resources. Additionally, they resolve the $\varepsilon$-dependence of approximate counting to multiplicative factor $1 + \varepsilon$. The techniques involved are quite different from ours: Belovs and Rosmanis use a generalized version of the quantum adversary bound that allows for multiple oracles, combined with tools from the representation theory of the symmetric group.
\section*{Acknowledgments}
We are grateful to many people: Paul Burchard, for suggesting the problem of
approximate counting with queries and QSamples;\ MathOverflow user
\textquotedblleft fedja\textquotedblright\ for letting us include
\Cref{fedjatight} and \Cref{fedjalem};\ Ashwin Nayak, for extremely helpful
discussions, and for suggesting the transformation of linear programs used
in our extension of the method of dual polynomials to the Laurent polynomial
setting; Thomas Watson, for suggesting the intersection approach to proving
an $\mathsf{SBP}$ vs. $\mathsf{QMA}$ oracle separation; and Patrick Rall,
for helpful feedback on writing. JT would particularly like to thank Ashwin
Nayak for his warm hospitality and deeply informative discussions during a visit to Waterloo.
|
1,314,259,993,576 | arxiv | \section{Related Work}
\label{sec:related_work}
\subsection{Traditional methods}
Many traditional methods for denoising such as BM3D \cite{dabov2006image}, non-local means \cite{buades2011non}, and weighted non-nuclear norm minimizaiton \cite{gu2014weighted} perform denoising by comparing the neighborhood of a pixel to other similar regions in the image. The advantage of learning-based methods is that they can also take advantage of examples from other images in the dataset beyond the input image to be denoised. Other methods such as total-variation denoising \cite{chambolle2004algorithm} enforce smoothness priors on the image which tend to lead to highly quantized results.
While most previous methods for denoising are designed for additive Gaussian noise; in the case of Poisson-Gaussian noise, a variance stabilizing transform \cite{makitalo2012optimal} is applied to approximately transform the noise to be Gaussian. However, these methods are designed explicitly for Poisson-Gaussian noise \cite{luisier2010image}.
\subsection{Deep learning methods}
At present, supervised deep learning methods for denoising \cite{zhang2017beyond,weigert2018content} typically far outperform traditional and self-supervised methods in terms of peak signal-to-noise ratio (PSNR). Most supervised methods apply a fully convolutional neural network \cite{long2015fully,ronneberger2015u} and simply regress to the clean image.
Recently, several approaches to self-supervised denoising have been developed.
Some methods \cite{NIPS2018_7587} use as a loss function Stein's Unbiased Risk Estimate (SURE) \cite{stein1981estimation,ramani2008monte}, which estimates the mean squared error (MSE) between a denoised image and the clean image without actually having access to the clean image. An analogous estimator for Poisson-Gaussian noise has been developed \cite{le2014unbiased}. However, these methods require \emph{a priori} knowledge of the noise level which is unrealistic in a practical setting. Our approach supports blind denoising and adaptively estimates the noise level at test time.
Lehtinen et al.~\cite{lehtinen2018noise2noise} introduced a highly successful approach to self-supervised denoising called Noise2Noise. In this approach, the network learns to transform one noisy instantiation of a clean image into another; under the MSE loss function, the network learns to output the expected value of the data which corresponds to the clean image. While this method can achieve results very close to a supervised method, it requires multiple, corresponding noisy images and thus is similarly limited in application in the live cell microscopy context.
An alternate approach to self-supervised denoising which does not require multiple noise instantiations of the same clean image is to learn a filter which predicts the center pixel of the receptive field based on the surrounding neighborhood of noisy pixels. By training such a filter to minimize the MSE to the noisy input, the resulting filter will theoretically output the clean value \cite{batson2019noise2self,krull2019noise2void}. Laine et al.~\cite{laine2019high} refer to a neural network built around this concept as a ``blindspot neural network.'' They improved upon the blindspot concept by extending it to a Bayesian context and introduced loss functions for pure Gaussian or Poisson noise, showing results very close to the supervised result when trained on synthetically noised data. However, their method requires a regularization term in the loss function which can't practically be tuned in the self-supervised setting; in our evaluation we found that the regularization strength indeed needs to be tuned for best results on different datasets. Our method avoids the need for regularization and outperforms the regularized version in our experiments.
Krull et al.~\cite{krull2019probabilistic} introduced Probabilistic Noise2Void (PN2V) which takes a non-parametric approach to modeling both the noise distribution and the network output; however, their approach requires paired clean and noisy images in order to calibrate the noise model. A recent follow-on work called PPN2V \cite{prakash2019fully} estimates the noise model using a Gaussian Mixture Model (GMM) in a fully unsupervised manner. Again, this approach involves several hyperparameters controlling the complexity of the noise model which need to be tuned, while ours does not. Additionally, in our experiments, we show that our approach outperforms PPN2V on several datasets.
\section{Self-supervised learning of denoising}
\label{sec:self_supervised}
The goal of denoising is to predict the values of a ``clean'' image $\mathbf{x} = (x_1,\ldots,x_n)$ given a ``noisy'' image $\mathbf{y}=(y_1,\ldots,y_n)$.
Let $\Omega_{y_i}$ denote the neighborhood of pixel $y_i$, which does not include $y_i$ itself. We make two assumptions that are critical to the setup of Noise2Void \cite{krull2019noise2void} and follow-on works: that the noise at each pixel is sampled independently, i.e. $p(y_i|x_1,\ldots,x_n)=p(y_i|x_i)$; and that each clean pixel is dependent on its neighborhood, a common assumption about natural images. The consequence of these assumptions is that $\Omega_{y_i}$ only gives information about $x_i$, not $y_i$. Therefore a network trained to predict $y_i$ given $\Omega_{y_i}$ using a mean squared error loss will in fact learn to predict $x_i$ \cite{krull2019noise2void,batson2019noise2self}.
In this work, we take a probabilistic approach rather than trying to regress to a single value. Following Laine et al.~\cite{laine2019high} and Krull et al.~\cite{krull2019probabilistic}, we can connect $y_i$ to its neighborhood $\Omega_{y_i}$ by marginalizing out the unknown clean value $x_i$:
\begin{equation}
\underbrace{p(y_i|\Omega_{y_i})}_{\text{Noisy observation}} = \int \underbrace{p(y_i|x_i)}_{\text{Noise model}}\underbrace{p(x_i|\Omega_{y_i})}_{\text{Clean prior}} dx_i.
\end{equation}
Since we only have access to observations of $y_i$ for training, this formulation allows us to fit a model for the clean data by minimizing the negative log likelihood of the noisy data, i.e. minimizing a loss function defined as
\begin{equation}
\label{eq:loss}
\mathcal{L}^{\text{marginal}} = \sum_i -\log p(y_i|\Omega_{y_i}).
\end{equation}
In the following we will drop the $\Omega_{y_i}$ to save space.
\subsection{Poisson-Gaussian noise}
In the case of Poisson-Gaussian noise, the noisy observation $y_i$ is sampled by first applying Poisson corruption to $x_i$ and then adding Gaussian noise which is independent of $x_i$. We have
\begin{equation}
\label{eq:poisson_gaussian_noise}
y_i = aP(x_i/a) + N(0,b)
\end{equation}
where $a>0$ is a scaling factor (related to the gain of the camera) and $b$ is the variance of the Gaussian noise component, which models other sources of noise such as electric and thermal noise \cite{foi2008practical}.
We apply the common approximation of the Poisson distribution as a Gaussian with equal mean and variance:
\begin{align}
\label{eq:pg_noise}
y_i &\approx aN(x_i/a,x_i/a) + N(0,b) \\
&= N(x_i,a x_i + b).
\end{align}
The noise model is then simply a Gaussian noise model whose variance is an affine transformation of the clean value. Note that in practice we allow $b$ to be negative; this models the effect of an offset or ``pedestal'' value in the imaging system \cite{foi2008practical}. This general formulation encompasses both pure Gaussian ($a=0$) and Poisson noise ($b=0$).
\subsection{Choice of prior}
In order to implement our loss function (Equation \ref{eq:loss}) we need to choose a form for the prior $p(x_i|\Omega_{y_i})$ that makes the integral tractable. One approach is to use the conjugate prior of the noise model $p(y_i|x_i)$, so that the integral can be computed analytically. For example, Laine et al.~\cite{laine2019high} model the prior $p(x_i|\Omega_{y_i})$ as a Gaussian, so that the marginal is also a Gaussian. Alternatively, Krull et al.~\cite{krull2019probabilistic} take a non-parametric approach and sample the prior.
In this work, similar to Laine et al.~\cite{laine2019high} we model the prior as a Gaussian with mean $\mu_i$ and variance $\sigma^2_i$. We replace the $ax$ term in Equation \ref{eq:pg_noise} with $a\mu$ to make the integral in Equation \ref{eq:loss} tractable; this approximation should be accurate as long as $\sigma^2_i$ is small. The marginal distribution of $y_i$ is then
\begin{equation}
p(y_i)=
\frac{1}{\sqrt{2\pi(a\mu_i+b+\sigma_i^2)}}\exp \left( -\frac{(y_i-\mu_i)^2}{2(a\mu_i+b+\sigma_i^2)}\right)
\end{equation}
and the corresponding loss function is
\begin{equation}
\mathcal{L}^{\text{marginal}} = \sum_i \left( \frac{(y_i-\mu_i)^2}{a\mu_i+b+\sigma_i^2}+\log(a\mu_i+b+\sigma_i^2) \right)
\end{equation}
\subsection{Posterior mean estimate}
At test time, $\mu_i$ is an estimate of the clean value $x_i$ based on $\Omega_{y_i}$, the neighborhood of noisy pixels around $y_i$. However, this estimate does not take into account the actual value of $y_i$ which potentially provides useful information about $x_i$.
Laine et al.~\cite{laine2019high} and Krull et al.~\cite{krull2019probabilistic} suggest to instead use the expected value of the posterior to maximize the PSNR of the resulting denoised image. In our case we have
\begin{equation}
\label{eq:pme}
\hat{x}_i =
\mathbb{E}[p(x_i|y_i)] =
\frac{y_i \sigma_i^2 + (a\mu_i+b) \mu_i}{a\mu_i + b + \sigma_i^2}.
\end{equation}
Intuitively, when the prior uncertainty is large relative to the noise estimate, the formula approaches the noisy value $y_i$; when the prior uncertainty is small relative to the noise estimate, the formula approaches the prior mean $\mu_i$.
\subsection{Blindspot neural network}
In our approach, $\mu_i$ and $\sigma_i^2$ are the outputs of a blindspot neural network \cite{laine2019high} and $a$ and $b$ are global parameters learned along with the network parameters.
The ``blind-spot neural network'' is constructed in such a way that the network cannot see input $y_i$ when outputting the parameters for $p(x_i)$. The blindspot effect can be achieved in multiple ways. Noise2Void ~\cite{krull2019noise2void} and Noise2Self \cite{batson2019noise2self} replace a random subset of pixels in each batch and mask out those pixels in the loss computation. Laine et al.~\cite{laine2019high} instead construct a fully convolutional neural network in such a way that the center of the receptive field is hidden from the neural network input. In our experiments we use the same blindspot neural network architecture as Laine et al.~\cite{laine2019high}.
\subsection{Regularization}
\input{table-loss}
\input{table-synthetic-params}
In a practical setting, the parameters $a$ and $b$ of the noise model are not known \emph{a priori}; instead, we need to estimate them from the data. However, an important issue arises when attempting to learn the noise parameters along with the network parameters: the network's prior uncertainty and noise estimate are essentially interchangeable without any effect on the loss function. In other words, the optimizer is free to increase $a$ and $b$ and decrease $\sigma_i^2$, or vice-versa, without any penalty. To combat this, we add a regularization term to the per-pixel loss which encourages the prior uncertainty to be small:
\begin{equation}
\mathcal{L}^{\text{regularized}} = \mathcal{L}^{\text{marginal}} + \lambda \sum_i |\sigma_i|.
\end{equation}
We found in our experiments that the choice of $\lambda$ strongly affects the results. When $\lambda$ is too high, the prior uncertainty is too small, and the results are blurry. When $\lambda$ is too low, the prior uncertainty is too high, and the network does not denoise at all. Unfortunately, in the self-supervised setting, it is not possible to determine the appropriate setting of $\lambda$ using a validation set, because we do not have ground truth ``clean'' images with which to evaluate a particular setting of $\lambda$.
\section{Learning an uncalibrated model}
\label{sec:uncalibrated}
This realization led us to adopt a different training strategy which defers the learning of the noise parameter models to test time.
In our uncalibrated model, we do not separate out the parameters of the noise model from the parameters of the prior. Instead, we learn a single variance value $\hat{\sigma_i}^2$ representing the total uncertainty of the network. Our uncalibrated loss function is then
\begin{equation}
\label{eq:uncalib_loss}
\mathcal{L}^{\text{uncalibrated}}=\sum_i
\left(
\frac{(y_i-\mu_i)^2}{\hat{\sigma_i}^2}+\log(\hat{\sigma_i}^2)
\right)
\end{equation}
At test time, however, we need to know the noise parameters $a$ and $b$ in order to compute $\sigma_i^2=\hat{\sigma_i}^2-a\mu_i-b$ and ultimately compute our posterior mean estimate $\hat{x}_i$.
If we had access to corresponding clean and noisy observations ($x_i$ and $y_i$, respectively) then we could fit a Poisson-Gaussian noise model to the data in order to learn $a$ and $b$. In other words, we would find
\begin{equation}
\label{eq:pg_fit}
a,b = {\arg \min}_{a,b} \sum_i \left( \frac{(y_i-x_i)^2}{ax_i+b}+\log(ax_i+b)\right).
\end{equation}
As we are in a self-supervised setting, however, we do not have access to clean data. Instead, we propose to use the prior mean $\mu_i$ as a stand-in for the actual clean value $x_i$. This bootstrapping approach is similar to that proposed by Prakash et al.~\cite{prakash2019fully}; however, they fit a general parametric noise model to the training data where as we propose to fit a Poisson-Gaussian model to each image in the test set.
Our approach is summarized in the following steps:
\begin{enumerate}
\item Train a blindspot neural network to model the noisy data by outputting a mean and variance value at each pixel, using the uncalibrated loss (Equation \ref{eq:uncalib_loss}).
\item For each test image:
\begin{enumerate}[label=\roman*.]
\item Run the blindspot neural network with the noisy image as input to obtain mean $\mu_i$ and total variance $\hat{\sigma_i}^2$ estimate at each pixel.
\item Determine the optimal noise parameters $a,b$ by fitting a Poisson-Gaussian distribution to the noisy and psuedo-clean images given by the mean values of the network output (Equation \ref{eq:pg_fit}).
\item Calculate the prior uncertainty at each pixel as $\sigma_i^2 = \max(0.0001,\hat{\sigma_i}^2-a\mu_i-b)$.
\item Use the noise parameters $a,b$ and the calculated prior uncertainties $\sigma_i^2$ to compute the denoised image as the posterior mean estimate (Equation \ref{eq:pme}).
\end{enumerate}
\end{enumerate}
We believe our approach has two theoretical advantages over the bootstrap method proposed by Prakash et al. \cite{prakash2019fully}. We can achieve a better fit to the data by training our system end-to-end, whereas Prakash et al. \cite{prakash2019fully} impose a fixed noise model during training by first estimating the noise parameters and then training the network. Second, we estimate noise parameters for each image separately at test time, whereas Prakash et al. \cite{prakash2019fully} estimate common noise parameters for all images first and fixes those parameters during training. Our approach allows for slight deviations in the noise parameters for each image, which might be more realistic for an actual microscope imaging system where the camera configuration slightly fluctuates between images.
\input{fig-hist-fmd}
\section{Experiments and Results}
\label{sec:experiments}
\subsection{Implementation details}
Our implementation uses the Keras library with Tensorflow backend. We use the same blindspot neural network architecture as Laine et al.~\cite{laine2019high}. We use the Adam optimizer \cite{kingma2014adam} with a learning rate of 0.0003 over 300 epochs, halving the learning rate when the validation loss plateaued. Each epoch consists of 50 batches of $128\times128$ crops from random images from the training set. For data augmentation we apply random rotation (in multiples of 90 degrees) and horizontal/vertical flipping.
To fit the Poisson-Gaussian noise parameters at test time, we apply Nelder-Mead optimization \cite{nelder1965simplex} with $(a=0.01,b=0)$ as the initialization point. We cut off data in the bottom 2\% and top 3\% of the noisy image's dynamic range before estimating the noise parameters.
\subsection{Datasets}
\subsubsection{Synthetic Data}
We generate a synthetic dataset using the ground truth images of the Confocal MICE dataset from the FMD benchmark \cite{zhang2019poisson} (described below). For training, we use the ground truth images from 19 of the 20 views and generate 50 noisy examples of each view by synthetically adding Poisson-Gaussian noise to the ground truth images using equation \ref{eq:poisson_gaussian_noise} where $a=1/\lambda$ and $b=(\sigma/255)^2$. For testing, we use the ground truth image from the 20th view and generate 50 noisy examples by synthetically adding Poisson-Gaussian noise in the same manner as during training. To ensure our method works for a wide range of noise levels, we train/test our method on all combinations $(\lambda,\sigma) \in \{0,10,20,30,40,50\}\times\{0,10,20,30,40,50\}$.
\input{table-synthetic-psnr}
\subsubsection{Real Data}
We evaluated our method on two datasets consisting of real microscope images captured with various imaging setups and types of samples. Testing on real data gives us a more accurate evaluation of our method's performance in contrast to training and testing on synthetically noised data, since real data is not guaranteed to follow the theoretical noise model.
The fluoresence microscopy denoising (FMD) benchmark \cite{zhang2019poisson} consists of a total of 12 datasets of images captured using either a confocal, two-photon, or widefield microscope. We used the same subset of datasets (Confocal Mice, Confocal Fish, and Two-Photon Mice) used to evaluate PN2V \cite{krull2019probabilistic} so that we could compare our results. Each dataset consists of 20 views of the sample with 50 noisy images per view. The 19th-view is withheld for testing, and the ground truth images are created by averaging the noisy images in each view. We trained a denoising model on the raw noisy images in each dataset separately.
Prakash et al.~\cite{prakash2019fully} evaluated PPN2V on three sequences from a confocal microscope, imaging Convallaria, Mouse Nuclei, and Mouse Actin. Each sequence consists of 100 noisy images and again the clean image is computed as the average of the noisy images. Whereas the FMD dataset provides 8-bit images clipped at 255, these images are 16-bit and thus are not clipped. Following their evaluation procedure, each method is trained on all 100 images and then tested on a crop of the same 100 images; this methodology is allowable in the self-supervised context since no label data is used during training.
\input{table-reg}
\subsection{Experiments}
In the following we will refer to the competing methods under consideration as
\begin{itemize}
\item \textbf{Regularized (Ours)}: Blindspot neural network trained using the regularized Poisson-Gaussian loss function (Equation \ref{eq:loss}) with regularization strength $\lambda$.
\item \textbf{Uncalibrated (Ours)}: Blindspot neural network trained using the uncalibrated loss function (Equation \ref{eq:uncalib_loss}) with noise parameter estimation done adaptively at test time (Section \ref{sec:uncalibrated}).
\item \textbf{N2V}: Noise2Void which uses the MSE loss function and random masking to create the blindspot effect \cite{krull2019noise2void}.
\item \textbf{PN2V}: Probabilistic Noise2Void -- same setup as N2V but uses a histogram noise model created from the ground truth data and a non-parametric prior \cite{krull2019probabilistic}.
\item \textbf{Bootstrap GMM} and \textbf{Bootstrap Histogram}: PPN2V training -- same setup as PN2V but models the noise distribution using either a GMM or histogram fit to the Noise2Void output \cite{prakash2019fully}.
\item \textbf{U-Net}: U-Net \cite{ronneberger2015u} trained for denoising in a supervised manner using MSE loss \cite{weigert2018content}.
\item \textbf{N2N}: Noise2Noise training using MSE loss \cite{lehtinen2018noise2noise}.
\end{itemize}
\input{table-results}
\input{fig-images-fmd-revised}
\input{fig-images}
\subsubsection{Noise parameter estimation}
We first evaluate whether our bootstrap approach to estimating the Poisson-Gaussian noise parameters is accurate in comparison to estimating the noise parameters using the actual ground truth clean values.
To evaluate our bootstrapping method, we compare the ground truth and estimated Poisson-Gaussian noise models fit for a test image in each dataset in the FMD benchmark \cite{zhang2019poisson}. Figure \ref{fig:hist-fmd} shows that the Poisson-Gaussian pdfs generated using our bootstrapping technique closely match that of the Poisson-Gaussian pdfs generated from the ground truth images.
We further evaluate our approach by comparing the loss and estimated Poisson-Gaussian noise parameters obtained when using actual ground truth data or the pseudo-clean data generated in our bootstrap method. Table \ref{table:loss} shows that bootstrapping can provide an accurate estimation of noise parameters and result in a loss similar to that obtained from using ground truth clean data. Here the loss value is
\begin{equation}
\label{eq:pg_fit}
\frac{1}{N} \sum_i \left( \frac{(y_i-x_i)^2}{ax_i+b}+\log(ax_i+b)\right)
\end{equation}
where $y_i$ is a pixel from the noisy image and $x_i$ is a corresponding pixel from either the ground truth clean image or the pseudo-clean image.
We perform a similar evaluation on our synthetic dataset where instead of having to estimate the true noise parameters from fitting a Poisson-Gaussian noise model with the ground truth clean image we readily have available the true noise parameters that correspond to the level of synthetically added Poisson-Gaussian noise. Table \ref{table:synthetic-params} shows the true noise parameters as well as the ones obtained using our uncalibrated method and the method described by Foi et al. in \cite{foi2008practical}. Our method tends to overestimate the $a$ parameter, whereas the estimate of the $b$ parameter is consistently accurate. This is probably because a majority of the pixels in the Confocal MICE images are dark and thus there are not many good samples for fitting the level of Poisson noise, whereas every pixel can be effectively used to estimate the Gaussian noise no matter the underlying brightness. Unlike the method of Foi et al.~\cite{foi2008practical} which obtains poor noise estimates most likely because of this, our method is still able to obtain a good estimate of the parameters by leveraging information from both the noisy image and our pseudo-clean image.
The effectiveness of fitting a Poisson-Gaussian noise model at test time is further evaluated in Table \ref{table:synthetic-psnr} which provides a comparison of peak signal-to-noise ratio (PSNR) on a subset of our synthetic dataset. Our method of estimating the noise parameters with our bootstrapping technique consistently improves the denoised results of the pseudo-clean image, but is ultimately bounded by the result obtained from using the pseudo-clean image along with the true noise parameters. Results for all noise parameter combinations are given in the supplemental material.
\subsubsection{Effect of regularization}
To highlight the difficulties of hyperparameter tuning in the self-supervised context, we trained our uncalibrated model and several regularized models on the FMD datasets. We tested a regularization strength of $\lambda=0.1,1,$ and $10$.
The results are shown in Table \ref{table:reg}. The test set PSNR of the regularized model varies greatly depending on the setting of $\lambda$, and indeed a different setting of $\lambda$ is optimal for each dataset. This indicates that hyperparameter tuning is critical for the regularized approach, but it is not actually possible in a self-supervised context.
In contrast, our uncalibrated method outperforms the regularized method at any setting of $\lambda$, and does not require any hyperparameters.
\subsubsection{Comparison to state-of-the-art}
Next we present the results of our performance evaluation on the FMD and PPN2V benchmark datasets. Table \ref{table:results} shows a comparison between our uncalibrated method and various competing methods, including self-supervised and supervised methods.
Between the fully unsupervised methods that do not require paired noisy images (our Uncalibrated method, N2V, Bootstrap GMM, and Bootstrap Histogram), our method outperforms the others on four out of six datasets. A comparison of denoising results on both benchmark datasets are shown in Figures \ref{fig:images-fmd} and \ref{fig:images}
\section{Conclusions and Future Work}
\label{sec:conclusions}
Noise is an unavoidable artifact of imaging systems, and for some applications such as live cell microscopy, denoising is a critical processing step to support quantitative and qualitative analysis. In this work, we have introduced a powerful new scheme for self-supervised learning of denoising which is appropriate for processing of low-light images. In contrast to the state-of-the-art, our model handles Poisson-Gaussian noise which is the standard noise model for most imaging systems including digital microscopes. In addition, we eliminate the need for loss function regularization in our method, thus making self-supervised denoising more practically applicable. Our evaluation on real datasets show that our method outperforms competing methods in terms of the standard PSNR metric on many datasets tested.
Our work opens up new avenues in live-cell imaging such as extreme low-light imaging over long periods of time. Future work lies in extending our model to other noise models appropriate to other imaging modalities, and exploring whether our uncalibrated method could be combined with a non-parametric prior \cite{krull2019probabilistic}.
\paragraph{Acknowledgments} This work was supported in part by NSF \#1659788, NIH \#1R15GM128166-01 and the UCCS Biofrontiers Center.
{\small
\bibliographystyle{ieee_fullname}
|
1,314,259,993,577 | arxiv | \section{Introduction}
The structure of nucleons has been studied extensively in experiments,
and nucleons also play a vital role as experimental probes. This makes
the controlled study of nucleons using lattice QCD a very attractive
goal. Some of the simplest matrix elements of interest include the
scalar and tensor charges, which control BSM contributions to neutron
beta decay, and the sigma terms, which control the sensitivity of dark
matter detection to WIMPs.
In order to make a significant impact, lattice calculations of nucleon
structure must become precision calculations, with full control over
all systematics. The traditional test observable is the isovector
axial charge $g_A$:
\begin{equation}
\left\langle p(P,s'\middle)|\bar u\gamma^\mu \gamma_5 d\middle|n(P,s)\right\rangle
\equiv g_A \bar u_p(P,s') \gamma^\mu \gamma_5 u_n(P,s).
\end{equation}
It is simple to compute, being an isovector forward matrix element,
and it is measured precisely in beta decay experiments\footnote{It
should be noted, however, that over time the experimental value has
drifted upward slightly; see the PDG's history plot.}, with the
latest PDG value~\cite{Tanabashi:2018oca} being
$g_A=1.2724(23)$. However, calculating it accurately has been
challenging: there is a long history of results being below the
experimental value. Because systematic effects in $g_A$ are
significant and it has been studied the most extensively, the axial
charge will be the exclusive focus of this review. This review will
also focus on recent results, particularly those presented at this
conference.
This review is organized as follows. Section~\ref{sec:methods} is a
brief overview of how matrix elements are computed. Excited-state
effects, which are a particularly challenging source of systematic
uncertainty, are discussed at length in
Section~\ref{sec:exc}. Finite-volume effects and dependence on the
pion mass are briefly reviewed in Sections~\ref{sec:volume} and
\ref{sec:chiral}. Finally, a summary and outlook is given in
Section~\ref{sec:outlook}.
\section{Methodology of matrix elements}\label{sec:methods}
The simplest approach for computing the forward hadronic matrix
element of operator $\mathcal{O}$ is to use a single interpolator
$\chi$ at zero momentum. One computes two-point and three-point
functions and performs a spectral decomposition,
\begin{gather}
\begin{aligned}
C_\text{2pt}(t) &\equiv \left\langle \chi(t) \chi^\dagger(0) \right\rangle & \qquad
C_\text{3pt}(\tau,T) &\equiv \left\langle \chi(T) \mathcal{O}(\tau) \chi^\dagger(0) \right\rangle \\
&= \sum_n |Z_n|^2 e^{-E_n t}, &
&= \sum_{n,n'} Z_{n'} Z_n^* \langle n'|\mathcal{O}|n\rangle
e^{-E_n\tau} e^{-E_{n'}(T-\tau)},
\end{aligned}
\end{gather}
where $Z_n\equiv\langle\Omega|\chi|n\rangle$ is the overlap of the
interpolator onto state $n$. In the limit where all time separations
$t$, $\tau$, and $T-\tau$ are large, the ground state dominates:
\begin{align}
C_\text{2pt}(t) &\to |Z_0|^2 e^{-E_0 t}\left(1 + O(e^{-\Delta E t})\right),\\
C_\text{3pt}(\tau,T) &\to |Z_0|^2 e^{-E_0 t}\left( \langle 0|\mathcal{O}|0\rangle
+ O(e^{-\Delta E \tau}) + O(e^{-\Delta E(T-\tau)} + O(e^{-\Delta E T}) \right),
\end{align}
where $\Delta E\equiv E_1-E_0$ is the energy gap to the first excited
state.
In the \textbf{ratio method}, the prefactors are cancelled to obtain
the ground-state matrix element:
\begin{equation}
R(\tau,T) \equiv \frac{C_\text{3pt}(\tau,T)}{C_\text{2pt}(T)}
\to \langle 0|\mathcal{O}|0\rangle + O(e^{-\Delta E\tau}) + O(e^{-\Delta E(T-\tau)})
+ O(e^{-\Delta E T}).
\end{equation}
For forward matrix elements, the excited-state contributions are
symmetric about $\tau=T/2$. They can be minimized by placing the
operator at the midpoint, yielding $R(\tfrac{T}{2},T) = \langle
0|\mathcal{O}|0\rangle + O(e^{-\Delta E T/2})$.
An alternative approach is the \textbf{summation
method}~\cite{Maiani:1987by,Gusken:1989ad}, which involves summing
over the operator insertion time $\tau$. The terms that are
independent of $\tau$ grow linearly with the source-sink separation
$T$ in the sum, whereas the terms that have an exponential dependence
on $\tau$ produce partial sums of geometric series. The derivative of
the sum yields the ground-state matrix element:
\begin{equation}
S(T) \equiv a\sum_\tau R(\tau,T),\quad
\frac{d}{dT}S(T) = \langle 0|\mathcal{O}|0\rangle + O(T e^{-\Delta E T}).
\end{equation}
There is some flexibility about the interval over which $\tau$ is
summed. One option is to use the ``interior'' region where
$\mathcal{O}$ is between the source and the sink, possibly excluding a
fixed number of points at each end so that the sum is from $\tau_0$ to
$T-\tau_0$. Another option is to sum over the whole lattice, which is
the method that was first used. In two talks at the Lattice 2010
conference~\cite{Capitani:2010sg,Bulava:2010ej}, it was pointed out
that contributions from excited states decay more rapidly than for the
ratio-midpoint method; this revived interest in the summation
method. In practice, the summation method has been found to produce a
larger statistical uncertainty than the ratio method, which can negate
some of its advantage; see Fig.~\ref{fig:ratio_summ}.
\begin{figure}
\centering
\includegraphics[width=0.7\textwidth]{plots/LHPC_gA_vs_Tsep}
\caption{Bare axial charge, determined using the ratio method
(purple circles) and summation method (green squares), versus
source-sink separation. This calculation was performed at the
physical pion mass with lattice spacing
$a=0.116$~fm~\cite{Hasan_inprep}. Note that statistics for
$T/a\geq 6$ are doubled compared with $T/a<6$.}
\label{fig:ratio_summ}
\end{figure}
In the last few years, there has been some use of methodologies based
on the Feynman-Hellmann theorem. This states that a matrix element in
a given state can be obtained from the derivative of the state's
energy with respect to a perturbation in the Lagrangian:
\begin{equation}\label{eq:FH}
\mathcal{L}(\lambda)\equiv \mathcal{L} + \lambda\mathcal{O}
\implies
\left.\frac{\partial}{\partial\lambda}E_n(\lambda)\right|_{\lambda=0}
= \langle n|\mathcal{O}|n\rangle.
\end{equation}
Discrete derivatives are sometimes used, particularly for the nucleon
sigma terms, where the theorem relates nucleon scalar matrix elements
to derivatives of the nucleon mass with respect to quark
masses. Evaluating the derivative of a two-point function exactly
leads directly to the summation method:
\begin{equation}
-\left.\frac{\partial}{\partial\lambda}\log C_\text{2pt}(t)\right|_{\lambda=0}
= S(t),
\end{equation}
where the sum is taken over all timeslices. In the large-$t$ limit,
the time derivative of this equation yields Eq.~\eqref{eq:FH} for the
ground state. This result appeared in the original summation-method
paper~\cite{Maiani:1987by} and has been rederived in recent
years~\cite{Chambers:2014qaa,Savage:2016kon,Bouchard:2016heu}.
\section{Excited-state contamination}\label{sec:exc}
Unwanted contributions from excited states decay exponentially and
will be highly suppressed if $\Delta E T \gg 1$. However, the
signal-to-noise problem~\cite{Lepage:1989hd} prevents calculations
from being performed at large $T$: this ratio decays as
$e^{-(E_0-\tfrac{3}{2}m_\pi)T}$. To be more concrete about the ``brute
force'' approach of simply using a large source-sink separation,
assume that we are using the ratio method in the asymptotic regime,
where the statistical errors and excited-state contributions scale as
\begin{equation}
\delta_\text{stat}\propto N^{-1/2}e^{(m_N-\tfrac{3}{2}m_\pi)T},\quad
\delta_\text{exc}\propto e^{-\Delta E T/2},
\end{equation}
respectively, where $N$ is the number of statistical
samples. Supposing that we want these two uncertainties to scale
together, i.e.\ $\delta_\text{exc}=\alpha\delta_\text{stat}\equiv
\delta$ for some constant $\alpha$, then as $\delta$ is decreased the
source-sink separation must be increased to suppress excited state
effects. An increase in statistics is also required, both to
compensate for the reduced signal-to-noise ratio, and also to meet the
smaller target statistical error; the required statistics are given by
\begin{equation}
N \propto \delta^{-\left(2+\frac{4m_N-6m_\pi}{\Delta E}\right)}.
\end{equation}
At the physical pion mass with $\Delta E=2m_\pi$ (see the next
subsection), the exponent is roughly $-13$, much larger than the $-2$
that is obtained when neglecting excited states. This situation could
be significantly improved if multilevel
methods~\cite{Ce:2016idq,Ce:2016ajy} or other
ideas~\cite{Detmold:2018eqd} are able to reduce the signal-to-noise
problem.
Because of the difficulty in going to large $T$, there has been much
effort spent on removing excited states from data at relatively small
source-sink separations: the two main approaches are improving the
interpolating operator and modeling the excited-state contributions.
\subsection{Theoretical expectations}
An approximation to the finite-volume spectrum consists of the energy
levels of any number of noninteracting stable hadrons,
$E=\sum_{i,j}\sqrt{m_i^2+p_j^2}$, where $p_j=\tfrac{2\pi}{L}n_j$ and
$n_j$ is a vector of integers. For a nucleon at rest, the leading
excitations are states with a nucleon and any number of
pions. Positive parity requires that for an odd number of pions there
must be some nonzero momenta. These noninteracting energy gaps are
shown in Fig.~\ref{fig:dE_free}. At the physical pion mass with $m_\pi
L=4$, the lowest excitation is $\Delta E=2m_\pi$ and there are eight
$N\pi$ and $N\pi\pi$ levels below $4m_\pi$. As the box size grows, the
spectrum becomes denser.
On the other hand, at heavier pion
masses the energies rapidly increase and there are soon just a few
levels below $\Delta E=1$~GeV. Going beyond the noninteracting
approximation, in Ref.~\cite{Hansen:2016qoz} finite-volume
quantization conditions were applied in the $N\pi$ sector using the
experimentally measured scattering phase shift. Significant deviations
from the noninteracting levels were found, particularly for energy
gaps between 0.4 and 0.8~GeV, however the general features of the
$N\pi$ spectrum were not significantly changed.
\begin{figure}
\centering
\includegraphics[width=0.495\textwidth]{plots/dE_free4}
\includegraphics[width=0.495\textwidth]{plots/dE_free_vs_mpi}
\caption{Energy gaps between noninteracting multiparticle excited
states states and the nucleon ground state. Left: as a function of
box size, at the physical pion mass. Right: versus pion mass, for
$m_\pi L=4$. At heavier pion masses one also expects additional
relatively low-lying states as resonances become stable; some
possibilities include $N^*(1440)$, $\Delta\pi$, and $N\sigma$.}
\label{fig:dE_free}
\end{figure}
Predicting excited-state contributions to two-point and three-point
functions requires knowing the spectrum $E_n$, the overlap factors
$Z_n$, and the operator matrix elements $\langle
n'|\mathcal{O}|n\rangle$. The key insight that allows for study using
chiral perturbation theory (ChPT) is that, if one assumes the smearing
size of the interpolator is small compared with $m_\pi^{-1}$, then at
leading order a single low-energy constant controls the coupling of a
local interpolator to both nucleon and nucleon-pion states, and it is
eliminated when forming ratios~\cite{Bar:2015zwa}.
The prediction from ChPT for the nucleon effective mass is a
percent-level excited-state contribution for $T\gtrsim
1$~fm~\cite{Tiburzi:2009zp,Bar:2015zwa,Tiburzi:2015tta}. On the other
hand, using the ratio method, an effect at the 10\% level is predicted
for $g_A$ at $T=1$~fm~\cite{Tiburzi:2015tta,Bar:2016uoj}. This effect
increases the effective lattice value of $g_A$, in contrast with most
numerical studies, which find that excited states decrease the
extracted value of $g_A$. At this conference, O.~Bär presented the
first study of observables at nonzero momentum transfer, namely the
form factors $G_A$ and $G_P$ of the axial current; a new tree-level
diagram was found to produce a very large excited-state effect in
$G_P$~\cite{Bar:2018akl}.
ChPT is not expected to produce accurate results for the contributions
from excited states with higher energies, particularly in the vicinity
of resonances; this means that it is only expected to be valid at
large source-sink separations where these contributions are
suppressed, namely $T\gtrsim 2$~fm for $g_A$ and similar
observables~\cite{Bar:2017kxh}. Deviations from ChPT for $N\pi$ states
in the resonance regime were modeled in Ref.~\cite{Hansen:2016qoz};
see Fig.~\ref{fig:gA_HM}. Under some model scenarios, the ratio-method
result for $g_A$ could be suppressed by excited states at short
source-sink separations and then rise to sit 1--2\% above the true
value for a wide range of larger source-sink separations. Clearly,
this corresponds to multiple excited states contributing with
different signs. This sort of scenario is particularly troublesome, as
it makes systematically improving a calculation by removing
excited-state effects quite difficult.
\begin{figure}
\centering
\includegraphics[width=0.6\textwidth]{plots/HM_ratio}
\caption{Model prediction of excited-state contributions to $g_A$
determined using the ratio method~\cite{Hansen:2016qoz}. The upper
pair of curves show the ChPT prediction, with the normalization of
states modified using the experimental $N\pi$ phase shift and
finite-volume quantization conditions. The other two pairs of curves show
different scenarios for modifications to the ChPT result in the
resonance regime. The figure is reproduced from
Ref.~\cite{Hansen:2016qoz} under the
{Creative Commons Attribution License}.
}
\label{fig:gA_HM}
\end{figure}
\subsection{Numerical studies}
Before discussing studies in the literature, it should be noted that
the available set of source-sink separations $T$ and source-operator
separations $\tau$ depends on how the three-point function is
computed.
\begin{compactitem}
\item The most common approach uses a \textbf{fixed
sink}. $C_\text{3pt}(\tau,T)$ is evaluated using a sequential
propagator through the sink, so that $T$, the sink momentum, and the
interpolating operators are fixed. All values of $\tau$ can be
obtained and any quark bilinear operator $\mathcal{O}(\tau)$ can be
used. The computational cost increases with every value of $T$.
\item If one instead uses a \textbf{fixed operator}, the sequential
propagator is evaluated through $\mathcal{O}(\tau)$, which is thus
fixed. All values of $T$ can be obtained, and the sink momentum and
interpolator can be varied. The computational cost increases with
each operator insertion and with each value of $\tau$. This approach
has been used recently in some variational studies by the CSSM
group~\cite{Owen:2012ts,Stokes:2018emx}.
\item Rather than fixing $\tau$, one can sum over it to obtain a
\textbf{summed operator} in the sequential propagator. In this case,
$T$ becomes the only relevant time separation and all values can be
obtained. The sink interpolator can be varied and the computational
cost increases with each operator insertion. This approach has been
used recently by CalLat~\cite{Bouchard:2016heu} and
NPLQCD~\cite{Savage:2016kon}.
\end{compactitem}
It is possible to replace the sequential propagator with a stochastic
one. This allows for increased flexibility but at the possible cost of
increased noise~\cite{Alexandrou:2013xon, Bali:2013gxx, Yang:2015zja,
Bali:2017mft, Gambhir_Lat18}.
\subsubsection{Improving the nucleon interpolator}
The usual\footnote{A common alternative is to replace $C\gamma_5$ with
$C\gamma_5 P_+$, where $P_+\equiv\tfrac{1}{2}(1+\gamma_0)$ is a
positive parity projector.} proton interpolator is
$\chi=\epsilon_{abc}(u^T_a C\gamma_5 d_b)u_c$, constructed using
smeared quark fields. Standard practice is to tune the smearing width
such that the nucleon effective mass reaches a plateau as early as
possible. It is well known from spectroscopy that the variational
method~\cite{Michael:1985ne,Luscher:1990ck}, where one finds a linear
combination of interpolating operators $\chi_\text{var}\equiv\sum_i
c_i \chi_i$ with optimized coefficients $c_i$, is a powerful
systematic approach for eliminating contributions from the
lowest-lying excited states to estimates of energy levels and matrix
elements~\cite{Blossier:2009kd,Bulava:2011yz}. In practice, its
effectiveness depends on the choice of interpolator basis
$\{\chi_i\}$.
A simple way to produce several interpolators is to vary the smearing
width; there are two recent studies\footnote{See
Refs.~\cite{Engel:2009nh,Owen:2012ts} for earlier studies.} that
used bases comprising interpolators with three different smearing
widths~\cite{Yoon:2016dij,Dragos:2016rtx}. Some results from
Ref.~\cite{Yoon:2016dij} are shown in
Fig.~\ref{fig:NME_variational}. The effective mass from the
variationally optimized interpolator lies very close to that of the
standard interpolator with the widest smearing. The optimized operator
and the widest smearing also both show little sign of excited-state
effects in the plateaus for the axial charge, but narrower smearings
do show clear signs (see e.g.\ Fig.~15 of
\cite{Yoon:2016dij}). Ref.~\cite{Dragos:2016rtx} also found that
narrower smearings suffer from larger excited-state effects. In that
study, the variationally optimized interpolator produced smaller
excited-state effects than the largest smearing. However, a still
larger smearing might be as good as the variational interpolator. One
should take two clear lessons from these studies. The first is that
tuning the smearing width is important, since it can have a
significant effect on excited states. The second is that when studying
a computationally more expensive alternative to the standard approach,
a fair comparison should be with a well-tuned operator; it is easy to
make the standard approach appear to be worse by using a too-narrow smearing.
\begin{figure}
\centering
\includegraphics[width=0.525\textwidth]{plots/NME_meff_V579}
\includegraphics[width=0.465\textwidth]{plots/NME_gA_S9_vs_var}
\caption{Comparison of analyses using a smeared standard
interpolator (denoted S$_i$ for smearing width $i$ in lattice
units) and using a variationally optimized interpolator (denoted
V$ijk$, using the basis
$\{\text{S}_i,\text{S}_j,\text{S}_k\}$). The calculation was
performed using an ensemble with $m_\pi=312$~MeV and
$a=0.081$~fm. Left: nucleon effective mass. Right: estimators for
bare $g_A$, using the interpolators S$_9$ (with four source-sink
separations), V357, and V579 (with source-sink separation
$12a$). The plots are reproduced from Ref.~\cite{Yoon:2016dij}.}
\label{fig:NME_variational}
\end{figure}
Beyond varying the smearing width, one can use different local
operator structures such as $\epsilon_{abc}(u_a^TCd_b)\gamma_5
u_c$. This approach has been used to add negative-parity interpolators
to the basis, which can be important for coupling to excited states in
moving frames~\cite{Stokes:2018emx}. Including covariant derivatives
or the chromomagnetic field strength in the interpolator allows for a
much larger basis~\cite{Edwards:2011jj,Dudek:2012ag}. In
Ref.~\cite{Egerer:2018xgu}\footnote{C.~Egerer presented preliminary
results at this conference.}, such a basis was used, employing the
distillation method to efficiently construct the correlators. In
general, a larger basis was more effective at removing excited-state
contributions, with the largest effect seen in the tensor charge. This
reference also reports that the Laplacian-Heaviside smearing used in
distillation produces smaller excited-state effects from the standard
operator than the more commonly used Wuppertal smearing, however it is
unclear whether either smearing was tuned.
\subsubsection{Fitting excited states}
A natural strategy for removing contributions from excited states is
to fit correlators [or derived quantities such as $R(\tau,T)$ or
$S(T)$] using a model that includes excited-state effects. By far the
most common model is based on a truncation of the spectral
decomposition to a small number of states (two or three). Generally
the two-point function provides the strongest constraints on the
energies $E_n$ and overlaps $Z_n$, and the three-point function serves
to determine the matrix elements $\langle n'|\mathcal{O}|n\rangle$.
Commonly, the energy gap $\Delta E$ is found to be between 0.5 and
1.0~GeV, which is usually greater than the expected lowest-lying
excitation energy shown in Fig.~\ref{fig:dE_free}.
\begin{figure}
\centering
\includegraphics[width=0.495\textwidth]{plots/gA_a09m220_CalLat}
\begin{minipage}[b]{0.495\textwidth}
\begin{center}
\includegraphics[width=\textwidth]{plots/gA_a09m220_PNDME}\\[-0.9ex]
{\scriptsize $(\tau-T/2)/a$}
\end{center}
\end{minipage}
\caption{Excited-state fits for bare $g_A$. Both calculations were
performed using the same HISQ ensemble with $a\approx 0.09$~fm and
$m_\pi\approx 220$~MeV. Left: two-state fit to data using domain
wall valence quarks and the summed operator method, with smeared
and point sinks (filled circles and open squares,
respectively). Right: three-state fit to data with clover valence
quarks and the fixed sink method, for three source-sink
separations. The left plot is adapted from
Ref.~\cite{Chang:2018uxx} and the right plot is reproduced from
Ref.~\cite{Gupta:2018qil} under the Creative Commons Attribution
License.}
\label{fig:gA_fits}
\end{figure}
Some recent examples are given in Refs.~\cite{Chang:2018uxx,
Gupta:2018qil}. In Ref.~\cite{Chang:2018uxx}, the extracted value of
$g_A$ was shown to be stable when the fit range is varied; similarly,
in Ref.~\cite{Gupta:2018qil} stability was shown with respect to
varying fit ranges and the number of states in the fit
model. Figure~\ref{fig:gA_fits} shows the preferred fits for $g_A$ on
the same ensemble (albeit with different valence quark action). In
both cases, the fits start with a minimum time separation of
$3a$. Given the pion mass and box size, there are more than ten
noninteracting excited energy levels with energy gap $\Delta E<1$~GeV;
for the first points included in the fit these are only suppressed by
$e^{-3a\Delta E}>0.25$. Clearly, it is difficult to associate the
``excited state'' in the fit with a single actual state.
At this conference, K.~Ottnad presented a different fitting
approach~\cite{Ottnad:2018fri} that does not determine the energy gap
$\Delta E$ from the two-point function. Instead, fits are performed to
the ratios $R(\tau,T)$; fitting six different observables
simultaneously is sufficient to constrain $\Delta E$. Interestingly,
in this case the fitted energy gap approaches the expected
lowest-lying noninteracting level as the minimum time separation
included in the fit, $t_\text{start}$, is increased.
In order for these fits to be trustworthy, ideally they would be
required to have good fit qualities ($p$-values). (This is not a
sufficient condition!) A strong test is given in
Ref.~\cite{Borsanyi:2014jba}: when the fit is repeated on many
ensembles, the distribution of fit qualities should be uniform, which
can be checked using a Kolmogorov-Smirnov (KS) test. More than half of
the fit qualities in Ref.~\cite{Chang:2018uxx} are below 0.2, and
hence the KS test indicates that they are not compatible with the
uniform distribution ($p<10^{-3}$). In contrast, the fits in
Ref.~\cite{Gupta:2018qil} are acceptable from this point of view
($p=0.26$). Some caution may be required, however, when fitting to
many variables, as the difficulty in inverting a large covariance
matrix can make it difficult to reliably estimate $\chi^2$ and the fit
quality.
\subsection{Outlook on excited states}
It is natural to ask why the picture in Fig.~\ref{fig:dE_free} of
low-lying $N\pi$ and $N\pi\pi$ states does not appear in the spectrum
of typical variational analyses or multi-state fits. One argument is
that the coupling of a multiparticle state to a local interpolator is
suppressed by the inverse lattice volume. However, this should be
compensated by the density of states so that in infinite volume the
interpolator will couple to continua of multiparticle states. In fact,
model predictions such as Fig.~\ref{fig:gA_HM} show a weak volume
dependence; this suggests that continuum spectral functions might
yield suitable fit models for excited states in large volumes.
The absence of multiparticle states is familiar from meson
spectroscopy. It has been found that a variational basis must include
nonlocal operators in order to identify the complete and correct
spectrum; see, e.g., \cite{Wilson:2015dqa}. Even though nucleon
structure only requires removing excited states and not obtaining
precise knowledge of them, it may still be necessary to include
nonlocal operators in order to benefit in practice from the proven
improved asymptotic approach to the ground state when using the
variational method~\cite{Blossier:2009kd,Bulava:2011yz}.
Given that state of the art nucleon structure calculations have not
reconciled their data with theoretical expectations for excited-state
effects, perhaps the safest approach is to analyze data in multiple
ways: ratio and summation methods, which don't make specific
assumptions about the spectrum of excitations, as well as fits that
can make use of shorter time separations. This was done in the
extensive excited-state study of Ref.~\cite{Dragos:2016rtx}, which
also included a variational setup, as well as in some recent
physical-pion-mass calculations by ETMC (see e.g.\
\cite{Alexandrou:2017hac}).
\section{Finite-volume effects}\label{sec:volume}
There is a long history of attributing a low value for $g_A$ computed
on the lattice to finite-volume effects. These effects were computed
in ChPT in Ref.~\cite{Beane:2004rf}; neglecting loops with $\Delta$
baryons, the leading contribution at large volume is
\begin{equation}\label{eq:gA_volume}
\frac{g_A(L)-g_A}{g_A} \sim \frac{m_\pi^2 g_A^2}{\pi^2F_\pi^2}
\sqrt{\frac{\pi}{2m_\pi L}} e^{-m_\pi L}.
\end{equation}
In addition to being exponentially suppressed at large $m_\pi L$, if
one fixes $m_\pi L$ and decreases $m_\pi$ this effect will also be
reduced. If this expression holds true, then a calculation with $m_\pi
L=3$ at the physical pion mass will have smaller finite-volume effects
than one with $m_\pi L=4$ at $m_\pi=300$~MeV.
\begin{figure}
\centering
\includegraphics[width=\textwidth]{plots/gA_mpiL_m}
\caption{Controlled studies of finite-volume effects in
$g_A$~\cite{Bali:2014nma, Gupta:2018qil, Chang:2018uxx,
Lauer_Lat18, Horsley:2013ayv, Bratt:2010jn, Green:2013hja,
Yamazaki:2008py}. Each study has two or more volumes at the same
pion mass, which is indicated in the legend. The largest volume
of each study is used for normalization, and a black dot is placed
over the symbol to indicate that the central value is fixed at
one. The horizontal axis contains the dependence on pion mass and
volume from Eq.~\protect\eqref{eq:gA_volume}.}
\label{fig:gA_volume}
\end{figure}
There have been several fully-controlled studies of finite-volume
effects in $g_A$, i.e., the same calculation performed on ensembles
that differ only by their volume. These are
summarized\footnote{Calculations where the lattice temporal extent was
varied together with the spatial volume are also included.} in
Fig.~\ref{fig:gA_volume}. For large values of $m_\pi L$ and small pion
masses, no effect is observed within uncertainties at the few percent
level. As the right hand side of Eq.~\eqref{eq:gA_volume} is
increased, the first significant effect is at the 5\% level in the
calculation by RQCD at $m_\pi=290$~MeV and $m_\pi L =3.4$. However, this
is a negative effect rather than the positive effect predicted by
Eq.~\eqref{eq:gA_volume}. Encouragingly, the physical pion mass with
$m_\pi L=3$ corresponds to 0.03 on the horizontal axis, where no
effect has been detected.
Global fits to a set of ensembles --- where the pion mass, lattice
spacing, and volume are all varied --- provide a different approach to
study finite-volume effects. The challenge is that any failure of the
fit function to accurately describe the dependence on the other
variables is a source of systematic uncertainty in the estimate of
finite-volume effects. This is especially true because most sets of
ensembles will tend to have larger values of $m_\pi L$ at larger pion
masses and on coarser lattice spacings. Using a global fit,
Ref.~\cite{Gupta:2018qil} assumed the volume dependence of $g_A$ has
the form $cm_\pi^2 e^{-m_\pi L}$ with $c$ a free parameter, and found
a $-0.9(5)\%$ effect at the physical pion mass with $m_\pi L=4$. A
similar approach was used by Ref.~\cite{Ottnad:2018fri}, and a similar
effect size was found. Finally, Ref.~\cite{Chang:2018uxx} assumed the
leading heavy baryon ChPT expression and allowed a higher-order term
proportional to $m_\pi^3$; this also produced a small effect at the
physical pion mass.
\section{Chiral extrapolation}\label{sec:chiral}
In heavy baryon ChPT, the pion mass dependence of the axial charge is
known to take the form
\begin{equation}
g_A(m_\pi) = g_0 - (g_0 + 2g_0^3)\left(\frac{m_\pi}{4\pi F_\pi}\right)^2\log\frac{m_\pi^2}{\mu^2} + c_1 m_\pi^2 + c_2 m_\pi^3 + O(m_\pi^4),
\end{equation}
where $g_0$ is the axial charge in the chiral limit and $c_{1,2}$ are
additional low-energy constants. Although the prefactor of the chiral
log is known, it is unclear how high of a pion mass can be reached
before the convergence of ChPT breaks down. As a result, recent
calculations that performed a chiral fit~\cite{Capitani:2017qpc,
Yamanaka:2018uud, Chang:2018uxx, Liang:2018pis, Gupta:2018qil,
Ottnad:2018fri}, including results presented by R.~Gupta and
K.~Ottnad at this conference, generally preferred to use a simple
polynomial dependence on $m_\pi^2$.
The issue of chiral extrapolation can, of course, be avoided by using
only lattice ensembles with near-physical pion masses. Results using
this approach were presented at this conference by
M.~Constantinou~\cite{Constantinou_Lat18},
Y.~Kuramashi~\cite{Shintani:2018ozy}, C.~Lauer~\cite{Lauer_Lat18},
Y.~Lin~\cite{Lin_Lat18}, and S.~Ohta~\cite{Ohta:2018zfp}.
\begin{SCfigure}
\includegraphics[width=0.7\textwidth]{plots/gA_results}
\caption{Recent calculations of the nucleon axial charge using $2$
flavours of dynamical sea quarks~\cite{Bali:2014nma,
Alexandrou:2017hac, Capitani:2017qpc} (green), $2+1$
flavours~\cite{Yamanaka:2018uud, Liang:2018pis, Ishikawa:2018rew,
Shintani:2018ozy, Ottnad:2018fri} (orange), and $2+1+1$
flavours~\cite{Bhattacharya:2016zcn, Chang:2018uxx, Gupta:2018qil,
Constantinou_Lat18} (blue). Published results are shown with a
filled symbol. The PDG value~\cite{Tanabashi:2018oca} is indicated
by the vertical band.}
\label{fig:gA_results}
\end{SCfigure}
Figure~\ref{fig:gA_results} shows some recent determinations of the
axial charge. It is encouraging that several collaborations are now
able to reproduce the experimental value, although there is still a
tendency for results to sit below experiment and no result is even
half a standard deviation above experiment. Given that the
experimental value is known, it may be useful for future calculations
to perform a blinded analysis.
\section{Summary and outlook}\label{sec:outlook}
Excited-state contamination remains a major focus of nucleon structure
calculations. A full variational study including nonlocal
interpolators that couple well to multiparticle states could help to
determine whether current methods are adequate. In contrast, no sign
of large finite-volume effect in $g_A$ has been observed in the
existing fully-controlled studies at low pion masses.
Discretization effects were not discussed in this review, in part
because they are not universal. In general they appear to be less
important than excited states, but they are nevertheless important for
controlling uncertainties and can have a significant impact on the
outcome such as in Ref.~\cite{Gupta:2018qil}.
It should be stressed that the results in Fig.~\ref{fig:gA_results}
have not been filtered based on any quality criteria. Such an
evaluation is necessary for obtaining a reliable ``lattice QCD
average'' of any observable. However, it may be particularly difficult
to set standards for controlling excited-state effects, since analysis
strategies vary significantly. A first community attempt was made in
Ref.~\cite{Lin:2017snn}, and some nucleon structure will also be
included in the next FLAG review.
Bringing simple observables like $g_A$ under precise control over all
sources of systematic uncertainty will be an important step toward
reliable calculations of more complex observables such as the proton
charge radius and parton distribution functions. Systematics in these
observables will require further study; in particular, finite-volume
effects could be important for form factors~\cite{Shintani:2018ozy}
and discretization effects might be significant for parton
distribution functions, which are generally not $O(a)$ improved.
\acknowledgments
I thank everyone who sent results in advance of my talk and who
replied to my questions: C.~Alexandrou, C.~C.~Chang, C.~Egerer,
R.~Gupta, J.~Liang, S.~Ohta, K.~Ottnad, F.~M.~Stokes, A.~Walker-Loud,
and T.~Yamazaki. I also thank my colleagues at DESY for their comments
on an early version of this talk and Karl Jansen for sending comments
on a draft of these proceedings.
|
1,314,259,993,578 | arxiv | \section{Introduction. Infrared Instabilities and Dynamical Chaos}
Recently, the nonperturbative treatment of the high multiplicity
scattering amplitudes which are not suppressed in the weak coupling
limit has attracted attention \cite{RT92,GNV92,Gong94,Hu95}. It is
quite natural in the connection with the exciting expectations that
the rate of the baryon-number violating electroweak processes,
non-perturbative in essence \cite{Hooft76}, might be significant at
ultra-high energies \cite{Ring90}. The semiclassical technique,
however, meets with difficulties in the study of $2 \to$ many
particles amplitude, since the {\it initial} state of few highly
energetic particles is not semiclassical at all (see, e.g.
\cite{Vol95}).
In the extreme non-perturbative classical treatment of the high energy
multiparticle amplitude, the question is the following: Does there
exist a mechanism for energy transfer from high frequency modes,
corresponding to two (or few) initial high energy particles, to low
frequency modes representing a multiparticle final states?
At first glance, the answer to this question, formulated in terms of
nonlinear dynamics, seems to be affirmative since the gauge field
equations are nonlinear. However, the studies of $(1+1)$-dimensional
Abelian Higgs model \cite{RT92} and $\lambda\varphi^4$-theory
\cite{GNV92} have shown no indication for a mechanism providing the
coupling between the initial high and the final low frequency modes.
Of course, the gauge field nonlinearities inherent in the non-Abelian
gauge theories and which are absent in the Abelian
models,\footnote{The nonlinearities of Abelian models due to the
Higgs-gauge fields and the Higgs self-coupling, as we see below, are
not important at the high energy gauge boson collisions.}
are essential and, in general, lead to the infrared instabilities.
One may say that for non-Abelian gauge theories the infrared
instabilities are not an exception, but rather a rule, and they are
intimately connected with i) the masslessness of the gauge field, ii)
its isospin (color) charge, and iii) the gyromagnetic ratio of the
gluon equal 2. Nonlinearity itself is not enough to furnish the above
mentioned coupling between fast and slow modes, as one can argue from
negative results of \cite{RT92,GNV92}.
From a more general point of view, the observed inability of the
nonlinearity alone to provide a mechanism for the formation of the
inelastic final states is intimately connected with the integrable
nature of the classical systems considered in \cite{RT92,GNV92}. It
is well known that non-Abelian gauge theories are nonintegrable in the
classical limit and exhibit dynamical chaos \cite{Mat81,Mul92} (see
also \cite{Biro94} for details and extended literature).
This dynamical stochasticity of the non-Abelian gauge fields together
with their mentioned instability are two possible sources of the
mechanism for the coupling between high and low frequency
modes.\footnote{It is not excluded that these two sources have a
common deep origin, though I am unable to prove this assertion on
theorem-like grounds.} At the same time, it is not superfluous to
recall the role of the Higgs condensate in the suppressing of the
chaos of the non-Abelian gauge fields \cite{MST81}.
In \cite{Hu95} we studied the collision of two SU(2) gauge field wave
packets. As we expected, based on our previous results \cite{Gong94},
the collisions of essentially non-Abelian initial configurations
trigger the decay of initial states into many low frequency modes with
dramatically different momentum distributions, whereas for Abelian
configurations wave packets pass through each other without
interaction.
Here I will present the study of the collisions of wave packets in the
SU(2) Higgs model where the fundamental excitations of the gauge-field
are massive.
\section{Collisions of Classical Wave Packets in SU(2) Higgs Model}
\subsection{SU(2) Higgs Model}
We briefly describe the spontaneously broken SU(2) model with an
isodoublet Higgs field $\Phi$. This model retains the most relevant
ingredients of the electroweak theory. The action of this model is
given by
\begin{eqnarray}
S &= &\int d^3xdt \left\{ -{1\over 2}{\rm tr}\left(
F_{\mu\nu}F^{\mu\nu}\right) + {1\over 2}{\rm tr}\left[({\cal D}_{\mu}
\Phi)^{\dagger}{\cal D}^{\mu}\Phi\right] \right . \nonumber \\
&&\left . \quad - \lambda \left[{1\over 2}{\rm tr} (\Phi^+\Phi) -v^2\right]^2
\right\}. \label{e2}
\end{eqnarray}
with ${\cal D}_{\mu} = \partial_{\mu}-ig A_{\mu}^a\tau^a/2$,
$F_{\mu\nu}\equiv F_{\mu\nu}^a{\tau^a\over 2} = {i\over g}[{\cal
D}_{\mu}, {\cal D}_{\nu}]$ and $\Phi = \phi^0 - i\tau^a\phi^a$;
$\tau^a(a=1,2,3)$ are Pauli matrices. $v$ is a vacuum expectation
value $(v.e.v.)$ of the neutral component of the scalar field.
By proper scaling transformations of the space-time coordinates and
fields, it is easy to see that the action (\ref{e2}) and the
corresponding equations of motion possesses only a single parameter
\begin{equation}
{\lambda\over g^2} = {M_H^2\over 8M_W^2}
\end{equation}
($M_H = 2v\sqrt{\lambda}\; {\rm and}\; M_W = {gv\over\sqrt{2}}$
are the tree masses of Higgs and gauge $W$-bosons respectively; $v =
174$ GeV.) However, in the simulation of the wave packet collisions,
initial conditions introduce extra physical parameters.
We work in the unitary gauge where only physical excitations appear:
\begin{eqnarray}
\Phi &= &\left( v+ {\rho\over\sqrt{2}}\right) U(\theta) \nonumber \\
A_{\mu} &= &U(\theta)W_{\mu} U^{-1}(\theta) + {i\over g}
\left( \partial_{\mu}U(\theta)\right) U^{-1}(\theta) \label{e3}
\end{eqnarray}
with $U(\theta) = \exp (i\tau^a\theta^a)$.
The real field $\rho$ describes the oscillations of the scalar field
about its $v.e.v$., and $W_{\mu}$ is the $W$-boson field:
\begin{equation}
(\partial_{\mu}\partial^{\mu}+M_H^2) \rho + 3\sqrt{2} \lambda v\rho^2
+ \lambda\rho^3 - {1\over 4} g^2W_{\mu}^aW^{a\mu}\rho^2 - {1\over
2\sqrt{2}} g^2vW_{\mu}^a W^{a\mu} = 0, \label{e4}
\end{equation}
\begin{equation}
[{\cal D}_{\mu},F^{\mu\nu}] + \left( M_W^2 + {1\over\sqrt{2}} g^2v\rho
+ {1\over 4}g^2\rho^2\right) W^{\nu} = 0. \label{e5}
\end{equation}
We emphasize that the gauge field acts as a source for Higgs
excitations in (\ref{e4}) (last term in (\ref{e4})). This permits us
to consider the $W$-field classically.
\subsection{Scattering of Wave Packets}
Our numerical study is based on the Hamiltonian formulation of lattice
SU(2) gauge theory \cite{KS75} (see \cite{Hu95,GongT} for details).
We work on an one-dimensional lattice with a size $L=Na$ ($N$ is the
number of lattice sites, $a$ is the lattice spacing).
To implement the temporal gauge $W_0^a=0$, most convenient in the
Hamiltonian formulation of the lattice gauge theory, we ``collide''
transverse $W$-bosons, for which the relation
$\partial_{\mu}W^{a\mu}=0$ holds.
The initial configuration is given by two well-separated right- and
left- moving Gaussian wave packets originally centered at $z_{R(L)}$
with average momenta ${\bf k}=(0,0,\bar k)$ and width $\Delta k\;
(\Delta k\ll \bar k)$:
\begin{eqnarray}
W^{c,\mu} &= &W_R^{c,\mu} + W_L^{c,\mu} \nonumber \\
W_{R(L)}^{c,\mu} &= &\delta^{\mu2} n_{R(L)}^c \psi(z-z_{R(L)},\pm t)
\label{e6}
\end{eqnarray}
with $n_{R(L)}^c$ the unit isospin vectors. To specify the profile
function $\psi(z,t)$ we take, for a right- moving wave packet centered
at $z=0$ at $t=0$,
\begin{equation}
\psi(z,t) = {1\over \sqrt{\pi^{3/2}\bar\omega\Delta k\sigma}}
\int_{-\infty}^{\infty} dk\; e^{-(k-\bar k)^2/2(\Delta k)^2}
\cos(\omega t-kz) \label{e7}
\end{equation}
with $\omega = (k^2+M_W^2)^{1/2}$, and the normalization is fixed by
requiring energy equal to $\bar\omega$ per cross-sectional area
$\sigma$.
From (\ref{e7}) we get the initial conditions for $\psi(z,0)$ and
${d\psi\over dt}(z,t)\big\vert_{t=0}$ which we don't give explicitly here.
The initial condition for the Higgs field is given by the vacuum solution
at $t=0$: $\phi^0=v,\; \phi^a=0,\; \dot\phi^0 = \dot\phi^a=0$. $\left(.
\equiv {d\over dt}\right)$. It is possible to see that the initial
conditions introduce three new dimensionless parameters $\bar k/v,
\;{\Delta k\over v}$ and ${\sigma v^2\over g^2}$ in addition to
${\lambda\over g^2}$ (or $M_H/M_W$) (we fix in the following $g=0.65$).
One more parameter appears in the initial conditions---the angle
$\theta_c$ between the relative orientation of isospins of two wave
packets. Non-zero $\theta_c$ corresponds to the essentially
non-Abelian configuration, $\theta_c=0$ (parallel isospins) gives the
initial pure Abelian configurations.
For the pure Yang-Mills wave packet collisions, the non-linearity is
due to the self-interaction of gauge fields. As it was established in
\cite{Hu95} for $\theta_c=0$ no indications of the final inelastic
states had a place: wave packets passed through each other without
interaction. On the contrary, for $\theta_c\not= 0$ collisions
resulted in strongly inelastic final states \cite{Hu95}. It is
remarkable that the inelastic patterns remain qualitatively similar
for $\theta_c$ as small as $\sim 10^{-12}$, clearly connecting these
phenomena with the dynamical chaos of the non-Abelian gauge fields.
For the Yang-Mills-Higgs system, the situation is more involved due to
the additional non-linearities induced by the gauge field-Higgs and
the Higgs self-couplings.
Figures 1 and 2 (top rows) show a few ``snapshots'' of the space-time
development of the colliding $W$-boson wave packets for parallel (Fig.
1) and orthogonal (Fig. 2) isospin orientations. The figures show the
absolute value of the scaled gauge field amplitude $\vert A\vert/v$.
For parallel isospin orientations, the result of the
``collisions'' is a slight distortion of the initial wave packets
showing no sign of the inelasticity. Decreasing of the energy
$(\bar k = \pi/25)$ shows a small inelasticity for $\theta_c =
0$.\footnote{This fact is easily explained by the consideration of the
tree diagrams in $WW$-scattering since the scalar exchanges are
decreased with energy.} As is seen from the top row
of Fig. 2, for $\theta_c=\pi/2$, final states are strongly inelastic.
\begin{figure}
\def\epsfsize#1#2{.6#1}
\centerline{\epsfbox{fig1.ps}}
\caption{Collision of two $W$-wave packets with parallel isospin
polarizations. We choose $M_H=M_W=0.126$, $\bar k = \pi/5$, $\Delta k
= \pi/100$, $g=0.65$, and $\sigma = 0.336$. This simulation, as well
as all others below, was performed on a lattice of length $L=2048$ and
lattice spacing $a=1$. The top row shows the space-time evolution of
the scaled gauge field amplitude $\vert A\vert/v$, the median row
exhibits the corresponding Fourier spectra of the gauge field energy
density, and the bottom row shows the space-time evolution of the
scaled Higgs field $\vert\Phi\vert^2/v^2$. The abscissae of top and
bottom rows are labelled in units of the lattice spacing, and the
abscissa of the median row is in units of $\sigma/1024$.}
\end{figure}
\begin{figure}
\def\epsfsize#1#2{.6#1}
\centerline{\epsfbox{fig2.ps}}
\caption{Same as for Figure 1, for orthogonal polarizations.}
\end{figure}
The difference between the two cases ($\theta_c=0,\;\pi/2)$ is more
striking by looking at the evolution of the absolute value of the
Fourier transform of the gauge-invariant energy density (scaled by
$v^2$) (median row in Figs. 1 and 2).
The bottom rows of Figs. 1 and 2 illustrate the time evolution of the
Higgs field excitations around its $v.e.v.\; v$ (scaled to unity).
Here we have plotted the $\vert\Phi\vert^2/v^2$ as a function of space
coordinate at three different times.
\section{Symmetry Restoration in WW collisions}
As is seen from Figures 1-2 (bottom rows), for not very large
$r=M_H/M_W$ the Higgs field oscillates not about its $v.e.v. \;v$ but
rather about zero. This suggests that the collisions of the gauge
boson wave packets, accompanied by energy transfer from gauge field to
Higgs field, lead to the restoration of the broken SU(2) symmetry.
This pehnomenon occurs for the large gauge fields amplitude (see
\cite{KL72,KP76}). Indeed, (\ref{e4}), describing the excitations of
the scalar field about the Higgs vacuum $\vert\Phi\vert = v$, has
another exact solution $\rho=-\sqrt{2}v$ $(\vert\Phi\vert = 0$, see
(\ref{e3})) with $W_{\mu}^c$ being arbitrary. In terms of the small
excitations $\chi=\rho+v/\sqrt{2}$ about $\Phi=0$, for (\ref{e4}) and
(\ref{e5}) we have:
\begin{eqnarray}
\left[ \partial_{\mu}\partial^{\mu} - {M_H^2\over 2} \left( 1 +
{g^2W^2\over 8\lambda v^2}\right)\right] \chi + \lambda\chi^3 &= &0
\label{e10} \\
\left[ {\cal D}_{\mu},F^{\mu\nu}\right] + {1\over 4} g^2\chi^2W^{\nu}
&= &0 \label{e11}
\end{eqnarray}
where $W^2(x) \equiv W_{\mu}^a(x) W^{a\mu}(x) = -(W_i^a(x))^2 <0$ for
our choice of the transverse $W$-wave packets. $W^2(x)$ is always
negative for time-like bosons.\footnote{We recall that the luminosity
of transverse $W$-bosons generated by the energetic fermions is much
higher than that of longitudinal ones and increases with energy
\cite{CG84}.}
Eq. (\ref{e11}) describes the massless $W$-boson. From (\ref{e10}) and
(\ref{e11}), we have the effective potential for $\chi$-excitations:
\begin{equation}
V(\chi,W^2) = -\lambda v^2(1-\eta) \chi^2 + {\lambda\over 4}\chi^4,
\label{e12}
\end{equation}
where we introduce $\eta = {g(W_i^a)^2\over 8\lambda v^2} = {1\over
r^2} \langle W_i^{a^2}\rangle$ as a parameter where the intensity
$(W_i^a)^2$ of the high frequency gauge pulses is replaced by its
space-time average $\langle W^2\rangle$. Depending on whether $\eta
>1$ or $\eta<1$, the potential (\ref{e12}) has two different {\it
stable} minima:
\begin{eqnarray}
\eta < 1:\quad \chi_{\rm min} &= &\pm \sqrt{2} v(1-\eta)^{1/2}, \quad
{\rm i.e.}\; \vert\phi\vert = v(1-\eta)^{1/2}, \label{e13} \\
\eta > 1: \quad \chi_{\rm min} &= &0, \quad {\rm i.e.}\; \Phi = 0.
\label{e14}
\end{eqnarray}
Stable excitations about these ``vacua'' have the following squared
masses:
\begin{eqnarray}
\tilde M_W^2 &= &M_W^2(1-\eta) \theta (1-\eta) \label{e15} \\
\tilde M_W^2 &= &{M_H^2\over 2} \vert 1-\eta\vert [1+\theta(1-\eta)].
\label{e16}
\end{eqnarray}
Thus for $\eta>1$, the symmetry is restored and the scalar
oscillations occur about the symmetrical state $\Phi=0$ with the zero
effective mass of the $W$-boson.
For $\eta<1$, the vacuum is changed gradually as $(1-\eta)^{1/2}$.
For this case, $\tilde r = \tilde M_H/\tilde M_W=r$. In Figure 3,
first column, the space development of the $WW$ collision is shown for
$\eta=1.32$ for different times. As seen from this figure, at time
$t\approx 300$ wave packets collide and then begin to separate. Just
about at this time one expects to observe the restoration of symmetry,
i.e. the oscillations of the scalar field $\phi$ about a new ground
state located below the ``old'' vacuum $\vert\phi\vert=v$. After the
separation of the wave packets $(t>300)$ the scalar field excitations
tend again to oscillate about ``old'' vacuum, i.e. the gauge symmetry
is broken again. The second column exhibits the space-time evolution
of the $\vert\phi\vert/v$. The third column shows the Higgs field
smoothed over 50 lattice sites in order to facilitate a comparison
with the definition (\ref{e12}) of the parameter $\eta$ in terms of
the averaged strength of the $W$-boson field.
\bigskip
\begin{figure}
\def\epsfsize#1#2{.6#1}
\centerline{\epsfbox{fig3.ps}}
\caption{Space-time development of symmetry restoration induced by two
colliding gauge wave packets with orthogonal isospins in the presence
of the Higgs vacuum condensate. First column shows the scaled gauge
field $\vert A\vert/v$ as a function of space coordinate $z$ five
chosen times. Second column demonstrates the corresponding space-time
evolution of the scaled Higgs field $\vert\Phi\vert/v$. Third column
shows the scaled Higgs field after smoothing over 50 lattice sites.
This simulation was done on a lattice of sites $n=2048$ and lattice
spacing $a=1$. The parameters were $\bar k=\pi/4$, $\Delta k=\pi/16$,
$M_H=M_W=0.15$, $g=0.65$, and $\sigma =1$.}
\end{figure}
\section{Concluding Remarks}
In the numerical studies of the collisions classical wave packets of
transversely polarized gauge bosons with non-parallel isospin
orientations in the broken SU(2) gauge theory we have found evidence
for the creation of final states with strongly ``inelastic'' events for
a wide range of the essential parameters of the problem.
We have observed and visualized the process of the SU(2) symmetry
res-toration in some finite space-time region as a result of the
collisions of the intense gauge pulses. At last but not least, it is
important to emphasize the observed correlation between the occurence
of the inelastic events (for non-parallel isospin configurations) and
the restoration of the symmetry in high energy collisions. Both these
phenomena require the same order of the amplitude of the initial
configurations.
\vfill
\eject
\subsection*{Acknowledgements}
It is my great pleasure to thank C. R. Hu, B. M\"uller and D. Sweet
for enlightening, stimulating discussions and collaboration. I am
grateful also to K. Rajagopal and R. Singleton for useful comments.
This work was supported in part by grant DE-FG02-96ER40945 from the
U.S. Department of Energy, and by the North Carolina Supercomputing
Center.
|
1,314,259,993,579 | arxiv | \section{Introduction}
\label{sec:intro}
Understanding the behaviour of stellar magnetic activity, and the nature of the underlying dynamo mechanism, are some of the most pressing challenges in solar and stellar physics. It is well known that the Sun displays an 11~yr sun spot cycle. Since the first detection of cyclic magnetic behaviour in solar-like stars (e.g. \citealt{wilson1978}), there has been great interest in determining which parameters, such as binarity, spin, or convective zone depth (and hence stellar type), are pivotal to both the duration and amplitude of magnetic activity cycles.
In a survey of stellar activity on 111 lower main-sequence stars, \cite{baliunas1995} used chromospheric Ca~\textsc{ii}~HK measurements as a proxy of the surface magnetic fields. They found that, of the stars with solar-like activity cycles, the measured activity cycle periods $P_{\mathrm{cyc}}$ ranged from 2.5~yr to the 25~yr maximum baseline of observations. They also found that G0--K5V type stars show changes in rotation and chromospheric activity on an evolutionary timescale, with stars of intermediate age showing moderate levels of activity and occasional smooth cycles, whereas young rapidly-rotating stars exhibit high average levels of activity and rarely display a smooth, cyclic variation.
In other work, \cite{saar1999} used a large and varied stellar sample (including evolved stars and cataclysmic variable secondaries) to explore the relationships between the length of the activity cycle $P_{\text{cyc}}$ and the stellar rotation period $P_{\text{rot}}$. They parameterized the relationships using the ratio of cycle and rotation frequencies ${\omega_{\text{cyc}}/\Omega~(=P_{\text{rot}}/P_{\text{cyc}})}$, as well as the inverse Rossby number ${\text{Ro}^{-1}~(\equiv 2 \tau_{\text{c}} \Omega}$, where $\tau_{\text{c}}$ is the convective turnover timescale). They found that stars with ages >0.1~Gyr lay on two nearly parallel branches, separated by a factor of $\sim6$ in ${\omega_{\text{c}}/\Omega}$, with both branches exhibiting increasing ${\omega_{\text{c}}/\Omega}$ with increasing $\text{Ro}^{-1}$. Furthermore, they found that, if the secondary stars in close binaries can be used as proxies for young, rapidly rotating single stars, the cycles of these stars populate a third `superactive' branch, that shows the opposite trend of \emph{decreasing} ${\omega_{\text{cyc}}/\Omega}$ with increasing $\text{Ro}^{-1}$.
Elsewhere, \cite{radick1998} found that the luminosity variation of young stars was anti-correlated with their chromospheric emission, in the sense that young stars are fainter near their activity maxima. This suggests that the long-term variability of young stars is spot-dominated, whereas older stars are faculae-dominated (\citealt{lockwood2007}). Such behaviour has been observed in a number of young single stars (e.g. \citealt{berdyugina2002,messina2002,jarvinen2005}) as well as in binary systems (e.g. \citealt{henry1995}). Indeed, magnetic activity cycles have been found in several systems using photometric techniques, with some systems appearing to show preferred longitudes for spot activity. The increase and corresponding decrease of spot activity on opposite stellar longitudes has been interpreted as a so-called `flip-flop' magnetic activity cycle (e.g. \citealt{berdyugina1998,berdyugina2005flipflop}). By tracking the number and position of spots on the stellar surface using Doppler imaging, such activity was also found on the RS~CVn star, II~Peg, by \cite{berdyugina1999}.
While the magnetic activity of single stars and detached binaries is reasonably well studied, studies of magnetic activity cycles on interacting binaries are critically lacking. Cataclysmic variables (CVs) are semi-detached binaries consisting of a (typically) lower main-sequence star transferring mass to a white dwarf (WD) primary via Roche-lobe overflow. These systems, with both rapid rotation and tidal distortion, provide a unique parameter regime to allow critical tests of stellar dynamo theories. In addition, CVs form the foundation of our understanding of a wide range of accretion driven phenomena, and in turn, the secondary stars are key to our understanding of the origin, evolution and behaviour of this class of interacting binary. The secondary star regulates the mass transfer history and is intimately tied in with the orbital angular momentum transport that determines the evolutionary timescales of the various accretion stages. In particular, magnetic braking is thought to drain angular momentum from the system, sustaining the mass transfer that causes CVs to evolve to shorter orbital periods. This has been a standard ingredient of compact binary evolution theory for several decades.
Furthermore, magnetic activity cycles in secondary stars have been invoked to explain the variations in orbital periods in interacting binaries caused by the \cite{applegate1992} mechanism. This causes angular momentum changes within the secondary star throughout the activity cycle to be transmitted to the orbital motion, resulting in cyclical orbital period variations. In addition, an increase in the number of magnetic flux tubes on the secondary star during a stellar maximum is thought to cause the star to expand (\citealt{richman1994}) and to result in enhanced mass transfer -- giving rise to an increased mass transfer rate through the disc and a corresponding increase in the system luminosity. Additional mass transfer also reduces the time required to build up sufficient material in the disc to trigger an outburst, resulting in shorter time intervals between consecutive outbursts (e.g. \citealt{bianchini1990}). On shorter timescales, starspots are thought to quench mass transfer from the secondary star as they pass the mass-losing `nozzle', resulting in the low-states observed in many CVs (see \citealt{livio1994,king1998,hessman2000}). Previous surface maps of the secondary stars in the CVs BV~Cen (\citealt{watson2007}) and AE~Aqr (\citealt{watson2006,hill2014}) show a dramatic increase in spot coverage on the side of the star facing the WD. This suggests that magnetic flux tubes are forced to emerge at preferred longitudes, as predicted by \cite{holzwarth2003b}, and is possibly related to the impact of tidal forces from the nearby compact object. If these particular spot distributions are confirmed to be long-lasting features, they would require explanation by stellar dynamo theory (e.g. \citealt{sokoloff2002,moss2002}), and would provide evidence for the impact of tidal forces on magnetic flux emergence. In addition, since the number of star spots should change dramatically over the course of an activity cycle, the density of spots around the mass transfer nozzle may also vary. This would provide an explanation for the extended high and low periods seen in polar type CVs such as AM Her (\citealt{hessman2000}).
Thus, the magnetic activity of CV secondary stars is crucial to the long and short term behaviour of these systems. Furthermore, it is clear that comparisons of the long-term magnetic activity across a range of stellar types, in different systems, are crucial to understanding the nature of the stellar dynamo, how it evolves, and what system parameters are most important in its operation. In light of this, we present a study of the long-term magnetic activity of the secondary star in the CV, AE~Aqr ($\ensuremath{P_{\text{rot}}}\xspace = 9.88$~h) by using Roche tomography to map the number, size, distribution and variability of starspots on the surface. This is the first time a CV secondary has been tracked with this type of a campaign, and given that CVs with both rapid-rotation and tidal distortion provide unique test-beds for dynamo theories, we can better understand what parameters are most important to the behaviour of the underlying dynamo mechanism, and the duration and amplitude of magnetic activity cycles
\section{Observations and reduction}
Simultaneous spectroscopic and photometric data of AE Aqr were taken in 2001, 2004, 2005 and 2006 (hereafter D01, D04, D05, D06), with spectroscopic data only in 2008 and 2009 (hereafter D08, D09a, D09b), where D9a and D09b were taken 9~d apart. As D01 and D09a~\&~D09b have previously been published in \cite{watson2006} and \cite{hill2014}, respectively, we refer the reader to these works for details of the reduction methods for both the spectroscopic and photometric data. Logs of the observations are shown in Tables~\ref{tab:obs}~and~\ref{tab:obsphotometry}.
\subsection{Spectroscopy}
\subsubsection{MIKE+Magellan Clay telescope}
For D04, D05 and D06, spectroscopic observations were carried out using the dual-beam Magellan Inamori Kyocera Echelle spectrograph (MIKE, \citealt{bernstein2003}) on the 6.5 m Magellan Clay telescope, situated at the Las Campanas Observatory in Chile. The standard set-up was used, allowing a wavelength coverage of 3330--5070 \r{A} in the blue arm and 4460--7270 \r{A} in the red arm, with significant wavelength overlap between adjacent orders. A slit width of 0.7 arcsec was used, providing a spectral resolution of around 38,100 ($\sim7.8$~kms\ensuremath{^{-1}}\xspace) and 31,500 ($\sim9.5$~kms\ensuremath{^{-1}}\xspace) in the blue and red channels, respectively. A Gaussian fit to several arc lamp lines gave a mean instrumental resolution of $\sim9$~kms\ensuremath{^{-1}}\xspace, which was adopted for use in Roche tomography in Section~\ref{sec:rochetomography}. Exposure times of 250~s (0.7~per~cent of the orbital period) were used in order to minimize velocity smearing of the data due to the orbital motion of the secondary star. ThAr lamp exposures were taken every 10 exposures for the purpose of wavelength calibration.
The data were reduced using the MIKE pipeline written in \textsc{python} by \cite{mikepipeline}. This automatically conducts bias subtraction, flat-fielding, blaze correction and wavelength calibration. The final output provides 1-D spectra split into orders, for both the blue and red arms. After reduction, it was found that each extracted order was not fully blaze-corrected, and so we applied an additional correction using a flux standard star. After flux calibration, the orders in the blue and red arms, respectively, were combined into continuous spectra by taking a variance-weighted mean across the spectral range. The blue spectra were then scaled to match the red spectra by optimally-subtracting the overlapping spectral regions (where the blue spectra was scaled and subtracted from the red spectra, with the optimal scaling factor being that which minimizes the residuals). Finally, a variance-weighted mean was made by combining the spectra from both arms, creating a single spectrum for each exposure.
\subsubsection{UVES+VLT}
For D08, spectroscopic observations were carried out using the Ultraviolet and Visual Echelle Spectrograph (UVES, \citealt{dekker2000}) on the 8.2-m UT2 of the VLT, situated on Cerro Paranal in Chile. UVES was used in the Dichroic-1/Standard setting (390+580 nm) mode, allowing a wavelength coverage of $3282-4563$~\r{A} in the blue arm and $4726-6835$~\r{A} in the red arm. A slit width of 0.9~arcsec was used, providing a spectral resolution of around 46,000 ($\sim6.5$~kms\ensuremath{^{-1}}\xspace) and 43,000 ($\sim7$~kms\ensuremath{^{-1}}\xspace) in the blue and red channels, respectively -- an instrumental resolution of 7~kms\ensuremath{^{-1}}\xspace was adopted for use in Roche tomography in Section~\ref{sec:rochetomography}. Exposure times of 230~s (0.65~per~cent of the orbital period) were used, with ThAr lamp exposures taken at the start and end of the night. The data were taken from the European Southern Observatory (ESO) data products archive after being reduced automatically using version 5.1.5 of the ESO/UVES pipeline. The final output consisted of 1-D spectra for both the red and blue arms.
\subsection{Photometry}
\label{sec:photometry}
Simultaneous photometry was carried out for D04, D05 and D06 using a Harris V-band filter on the Carnegie Institution's Henrietta Swope 1-m telescope, situated at the Las Campanas Observatory in Chile. The data were reduced using standard techniques. The master bias frame showed no ramp or large scale structure across the CCD, and so the bias level of each frame was removed by subtracting the median value of pixels in the overscan region. Pixel-to-pixel variations were corrected by dividing the target frames by a master flat-field taken at twilight. Optimal photometry was performed using the package \textsc{photom} \citep{eaton2009}, where three suitable comparison stars were identified using the catalogue of \cite{henden1995} to perform differential photometry. The light curves of D04, D05 and D06 are shown in Figure~\ref{fig:photometry}. Flaring activity is clearly evident over both slow and rapid timescales, with amplitudes of up to $\sim0.6$~mag. This most likely stems from accretion variability rather than the secondary star.
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{photometry2004.pdf}
\includegraphics[width=0.5\textwidth]{photometry2005.pdf}
\includegraphics[width=0.5\textwidth]{photometry2006.pdf}
\caption{The light curves of AE Aqr for D04 (top panel), D05 (middle) and D06 (bottom). The points are phase folded for clarity, and the typical uncertainties (not shown) are given in Table~\ref{tab:obsphotometry}. Rapid and frequent flaring is apparent in all plots, and is due to accretion variability.}
\label{fig:photometry}
\end{figure}
\begin{table*}
\caption{A log of the spectroscopic observations of AE Aqr. Columns 1-3 list the UT date, the start, and end times of observations, respectively. Column 4 lists the instrument and telescope used. Columns 5-8 show the exposure time, the number of spectra taken, the peak signal-to-noise ratio around the central wavelength of each spectrum (with the typical value in parentheses), and the phase coverage achieved. Column 9 gives the abbreviation used throughout the text to refer to that specific data set.}
\label{tab:obs}
\begin{tabular}{lcccccccc}
\toprule
\textsc{ut} date & \textsc{ut} start & \textsc{ut} end & Instrument & $T_{\mathrm{exp}}$~(s) & No. spectra & SNR & Phase coverage & Abbreviation \\
\midrule
2001 Aug 09 & 21:01 & 04:22 & UES+WHT & 200 & 88 & $22-44$ & $0.18-0.92$ & D01 \\
2001 Aug 10 & 20:49 & 04:37 & UES+WHT & 200 & 95 & $22-44$ & $0.59-0.37$ & \\
2004 Jul 09 & 02:42 & 09:18 & MIKE+Magellan & 250 & 69 & $69-129 ~(\sim96)$ & $0.46-0.81$ & D04 \\
2004 Jul 10 & 03:08 & 08:57 & MIKE+Magellan & 250 & 64 & $44-116 ~(\sim83)$ & $0.91-0.49$ & \\
2004 Jul 11 & 03:13 & 08:49 & MIKE+Magellan & 250 & 63 & $64-121 ~(\sim81)$ & $0.35-0.90$ & \\
2005 Aug 05 & 01:49 & 07:33 & MIKE+Magellan & 250 & 64 & $22-99 ~(\sim62)$ & $0.03-0.60$ & D05 \\
2005 Aug 06 & 02:49 & 07:29 & MIKE+Magellan & 250 & 57 & $77-131 ~(\sim98)$ & $0.56-0.02$ & \\
2006 Jul 04 & 03:36 & 07:50 & MIKE+Magellan & 250 & 48 & $45-100 ~(\sim95)$ & $0.13-0.55$ & D06 \\
2008 Aug 06 & 00:01 & 05:02 & UVES+VLT & 230 & 65 & $91-185 ~(\sim125)$ & $0.26-0.77$ & D08 \\
2008 Aug 07 & 00:08 & 05:13 & UVES+VLT & 230 & 66 & $119-176 ~(\sim149)$ & $0.70-0.22$ & \\
2009 Aug 27 & 23:49 & 05:11 & UVES+VLT & 230 & 57 & $81-150 ~(\sim121)$ & $0.34-0.89$ & D09a \\
2009 Aug 28 & 00:08 & 04:55 & UVES+VLT & 230 & 61 & $76-147 ~(\sim129)$ & $0.90-0.28$ & \\
2009 Sept 05 & 23:43 & 04:38 & UVES+VLT & 230 & 61 & $83-125 ~(\sim109)$ & $0.19-0.68$ & D09b \\
2009 Sept 06 & 00:34 & 05:20 & UVES+VLT & 230 & 60 & $107-158 ~(\sim136)$ & $0.71-0.18$ & \\
\bottomrule
\end{tabular}
\end{table*}
\begin{table*}
\caption{A log of the photometric observations taken of AE Aqr taken with the Henrietta Swope 1-m telescope. Columns 1-3 list the date, the start and end times of observations, respectively. Columns 4-6 give the exposure time, the number of exposures taken, and the typical uncertainty in the measured magnitude. Column 7 gives the abbreviation used throughout the text to refer to that specific data set.}
\label{tab:obsphotometry}
\begin{tabular}{lcccccc}
\toprule
\textsc{ut} date & \textsc{ut} start & \textsc{ut} end & $T_{\mathrm{exp}}$ (s) & No. exp. & $\sigma_{mag}$ & Abbreviation \\
\midrule
2004 Jul 09 & 02:42 & 09:22 & 10 & 416 & $\sim0.023$ & D04 \\
2004 Jul 10 & 02:54 & 09:02 & 10 & 384 & " & \\
2004 Jul 11 & 03:14 & 08:50 & 10 & 348 & " & \\
2005 Aug 05 & 01:55 & 07:33 & 20 & 290 & $\sim0.024$ & D05 \\
2005 Aug 06 & 02:56 & 07:32 & 15 & 272 & " & \\
2006 Jul 04 & 03:27 & 07:33 & 15 & 228 & $\sim0.019$ & D06 \\
\bottomrule
\end{tabular}
\end{table*}
\section{Ephemeris and Radial velocity curves}
\label{sec:ephemeris}
The analysis carried out in this section was completed for the sole purpose of determining a revised ephemeris in order to improve the quality of the Roche tomograms in Section~\ref{sec:surfacemaps}. New ephemerides were determined from the radial-velocity curves independently for each AE~Aqr data set, by cross-correlation with a spectral-type template star, following \cite{watson2006} and \cite{hill2014}. The details of the template star used for each data set are shown in Table~\ref{tab:templatestars}, where the systemic velocity was measured by a Gaussian fit to the least-squares deconvolution (LSD, see Section~\ref{sec:lsd}) line profile for each star (using a line-list where lines with a central depth shallower than 10~per~cent of the continuum were excluded).
\begin{table}
\centering
\begin{tabular}{llc}
\toprule
Data set & Star & $\gamma$ (kms\ensuremath{^{-1}}\xspace) \\
\midrule
D04 & HD 214759 & $1.9091\pm0.0014$ \\
D05, D06 & HD 24916 & $-5.144\pm0.005$ \\
D08 & HD 187760 & $-21.545\pm0.003$ \\
\bottomrule
\end{tabular}
\caption{Spectral-type templates used to calculate new ephemerides for AE Aqr. Columns 1-3 list the data for which the template star was used, the star's designation, and its systemic velocity.}
\label{tab:templatestars}
\end{table}
\begin{table*}
\centering
\begin{tabular}{lcccc}
\toprule
Data & T$_{0}$ (HJD) & $\gamma$ (kms\ensuremath{^{-1}}\xspace) & \ensuremath{K_{2}}\xspace (kms\ensuremath{^{-1}}\xspace) & \ensuremath{v\sin{i}}\xspace (kms\ensuremath{^{-1}}\xspace) \\
\midrule
D01 & $ 2452131.31345 \pm 0.00007 $ & $-59.0\pm1.0$ & $168.4\pm0.2$ & - \\
D04 & $ 2453195.440623 \pm 0.000013 $ & $-60.065\pm0.024$ & $169.82\pm0.03$ & 98.9 \\
D05 & $ 2453588.571407 \pm 0.000016 $ & $-59.33\pm0.04$ & $169.15\pm0.05$ & 99.2 \\
D06 & $ 2453921.60201 \pm 0.00005 $ & $-59.53\pm0.18$ & $165.85\pm0.21$ & 97.6 \\
D08 & $ 2454684.399364 \pm 0.000010 $ & $-61.652\pm0.018$ & $166.049\pm0.025$ & 102.8 \\
D09 \& D09b & $ 2455071.356125 \pm 0.000009 $ & $-62.072\pm0.017$ & $167.433\pm0.022$ & 100.1 \\
\bottomrule
\end{tabular}
\caption{The new ephemerides for each data set of AE Aqr based on the radial velocity analysis described in Section~\ref{sec:ephemeris}. Columns 1-4 list the data set, the ephemeris and associated statistical uncertainty, as calculated from the fit to the RV curves, the systemic velocity, and the radial velocity semi-amplitude. Only the ephemerides are adopted for the Roche tomography analysis.}
\label{tab:ephemeris}
\end{table*}
For this, we restricted ourselves to the spectral regions lying between $6000-6270$~\r{A} and $6320-6500$~\r{A}, as these contain strong absorption lines from the secondary star, and reduce the probability of introducing a continuum slope from the blue primary. Both the AE Aqr and K4V template spectra were normalized by dividing by a constant, and the continuum was fit using a third-order polynomial, and subtracted, thus preserving the line strength. The template spectrum was then artificially broadened (initially by 100~kms\ensuremath{^{-1}}\xspace) to account for the rotational velocity (\ensuremath{v\sin{i}}\xspace) of the secondary, multiplied by a constant, and subtracted from an averaged high-signal-to-noise orbitally-corrected AE Aqr spectrum. These latter three steps were repeated, artificially broadening the template spectrum in 0.1~kms\ensuremath{^{-1}}\xspace steps until the scatter in the residual spectrum was minimized. This typically took two to three iterations. Through the above process, a cross-correlation function (CCF) was calculated for each AE Aqr spectrum, and the peak of the CCF was found using a parabolic fit. A radial velocity (RV) curve was then derived by fitting a sinusoid through the CCF peaks, obtaining new zero-point ephemerides for each data set (shown in Table~\ref{tab:ephemeris}), with the orbital period fixed at $P_{\text{orb}} = 0.41165553$~d (from \citealt{casares1996}). All subsequent analysis of each data set have been phased with respect to these new ephemerides. Separate ephemerides were calculated for each data set as the RV curves are affected by systematics, and so combining all data to calculate a single global ephemeris may not be optimal. Furthermore, the scatter in the O--C values of all data is relatively small, with a standard deviation of 0.12~per~cent of the orbital period.
The RV measurements obtained from the cross-correlation method described above are relatively insensitive to the use of a poorly-matched template, or an incorrect amount of template broadening. Surface features such as irradiation or star spots, as well as the tidal distortion of the secondary are more likely to introduce systematic errors in RV measurements, if not properly accounted for (e.g. \citealt{davey1992}). In addition, no detailed attempt was made to determine the best-fitting spectral-type or binary parameter determination in this analysis, however, for completeness we include the systemic velocity ($\gamma$), the radial velocity semi-amplitude (\ensuremath{K_{2}}\xspace) and the projected rotational velocity (\ensuremath{v\sin{i}}\xspace) in Table~\ref{tab:ephemeris}.
Figure~\ref{fig:rvcurve} shows the measured radial velocities for each data set, the fitted sinusoid, and the residuals after subtracting the sinusoid. The inherent systematic biases are clearly evident as deviations from a perfect sinusoid, and as such, the binary parameters derived from this RV analysis have not been used in the subsequent analysis presented in this work.
The small variation in $\gamma$, \ensuremath{K_{2}}\xspace and \ensuremath{v\sin{i}}\xspace between data sets (see Table~\ref{tab:ephemeris}) may be due to instrumental offsets between observations, but the spread in values is most likely dominated by the systematic biasing of RV measurements due to surface features such as irradiation and starspots. Such features may alter the slope of the RV curve, changing \ensuremath{K_{2}}\xspace, and due to their non-uniform distribution, surface features may cause a shift in the measured $\gamma$. We note that the $1\sigma$ spread in ephemerides is~$\sim37$~s, suggesting the period was stable over the duration of all observations.
\begin{figure*}
\centering
\begin{minipage}{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{aeaqr2004_rv.pdf}
\includegraphics[width=\textwidth]{aeaqr2005_rv.pdf}
\end{minipage}\hfill
\begin{minipage}{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{aeaqr2006_rv.pdf}
\includegraphics[width=\textwidth]{aeaqr2008_rv.pdf}
\end{minipage}
\label{fig:rvcurve}
\caption{The radial velocity curves of AE Aqr for D04 (top left), D05 (bottom left), D06 (top right) and D08 (bottom right). The points are phase folded for clarity using the ephemerides in Table~\ref{tab:ephemeris}, and a least-squares sinusoid fit to the RV points (assuming a circular orbit) is shown as a solid line. The lower panels of each plot show the residuals after subtracting the fitted sinusoid, as well as the statistical uncertainies of the measured RVs.}
\end{figure*}
\begin{table}
\centering
\caption{Spectral regions excluded from analysis.}
\label{tab:wavemasked}
\begin{tabular}{cl}
\toprule
Masked region (\r{A}) & Comments \\
\midrule
$<4600$ & Noisy, He~\textsc{i} \& H emission \\
$4670 - 4700$ & He~\textsc{ii} emission \\
$4830 - 4885$ & H$\beta$ emission \\
$5850 - 5900$ & He~\textsc{i} emission \& Na~\textsc{i} doublet \\
$6270 - 6320$ & Tellurics \\
$6525 - 6600$ & H$\alpha$ emission \\
$6650 - 6700$ & He~\textsc{i} emission \\
$6864 - 6970$ & Tellurics \\
$7031 - 7093$ & He~\textsc{i} emission \\
$7158 - 7370$ & Tellurics \\
$7590 - 7705$ & Tellurics \\
\bottomrule
\end{tabular}
\end{table}
\section{Roche tomography}
\label{sec:rochetomography}
Roche tomography is a technique analogous to Doppler imaging (e.g. \citealt{vogt1983}), and is specifically designed to indirectly image the secondary stars in close binaries such as CVs (\citealt{rutten1994,rutten1996,schwope2004,watson2003,watson2006,watson2007,dunford2012,hill2014}), pre-CVs (e.g.~\citealt{parsons2015}) and X-ray binaries (e.g.~\citealt{shahbaz2014}). The technique assumes that the secondary is locked in synchronous rotation with a circularized orbit, and that the star is Roche-lobe filling. We refer the reader to the references above and the technical reviews of Roche tomography by \cite{watson2001} and \cite{dhillon2001} for a detailed description of the axioms and methodology.
\section{Least squares deconvolution}
\label{sec:lsd}
Least squares deconvolution (LSD) was applied to all spectra in the same manner as in \cite{watson2006}, producing mean line profiles with a substantially increased signal-to-noise ratio (SNR). LSD requires that the spectral continuum be flattened. However, the contribution to each spectrum from the accretion regions is unknown, and a constantly changing continuum slope due to, for example, flaring (see Figure~\ref{fig:photometry}) or the varying aspect of the accretion regions, means a master continuum fit to the data cannot be used. In addition, as the contribution of the secondary star to the total light of the system is constantly varying, normalizing the continuum by division would result in the photospheric absorption lines from the secondary star varying in relative strength from one exposure to the next. Hence, we are forced to subtract the continuum from each spectrum. This was achieved by fitting a spline to the data. As the spectral type of AE Aqr has been determined to lie in the range K3-K5V (\citealt{crawford1956,chincarini1981,tanzi1981,bruch1991}), we generated a stellar line list for a K4V type star ($T_{\mathrm{eff}} = 4750$~K and $\log{g} = 4.5$, the closest approximation available) using the Vienna Atomic Line Database (VALD, see \citealt{kupka2000}), adopting a detection limit of 0.2. The normalized line depths were scaled by a fit to the continuum of a K4V template star so each line's relative depth was correct for use with the continuum subtracted spectra.
Emission lines and telluric lines were masked in the spectra and line list -- the excluded spectral regions are detailed in Table~\ref{tab:wavemasked}. This meant that over the 4600--7700~\r{A} spectral range for D04, D05 and D06, 2354 lines were available over which to carry out LSD. Similarly, 1558 lines were available for D08 in the spectral range 4780--6810~\r{A}. After carrying out LSD, a small continuum slope was present in the LSD profiles. This was removed by masking out the line centre and subtracting a second-order polynomial which was fit to the continuum. Details for D01 and D09a~\&~D09b may be found in \cite{watson2001} and \cite{hill2014}, respectively.
The variable light contribution of the secondary means we cannot normalize the data in the usual way. Instead, we are forced to use relative line fluxes, requiring the spectra to be slit-loss corrected. For D04, D05 and D06 we used simultaneous photometry to monitor transparency and target brightness variations (see Section~\ref{sec:photometry}). We corrected for slit losses by dividing each LSD profile by the ratio of the flux in the spectrum (after integrating the spectrum over the photometric filter response function) to the corresponding photometric flux. The value of photometric flux used was the mean over the duration of the spectroscopic exposure. As we were unable to obtain simultaneous photometry for D08, we used the fits from Roche tomography to iteratively scale the LSD line profiles in the same manner as carried out for D09a~\&~D09b in~\cite{hill2014}. The resulting scaled profiles were visually inspected and found to be consistent.
The final LSD profiles, the computed fits, and the residuals after subtracting the fits from the LSD profiles, are trailed for each data set in Figures~\ref{fig:trails04}~to~\ref{fig:trails08}, where the orbital motion has been removed. Starspots and surface features are clearly visible as emission bumps moving through the profiles from negative to positive velocities as AE Aqr rotates. The variation in \ensuremath{v\sin{i}}\xspace due to the tidal-distortion is also clearly apparent. Furthermore, the residuals of D04, D05 and D08 (see Figures~\ref{fig:trails04},~\ref{fig:trails05}~and~\ref{fig:trails08}) show narrow emission features that are seen to lie outside the stellar absorption profile, and appear to move in anti-phase with respect to the secondary. Similar features were also seen in the trails of D09a~\&~D09b (see Figure~3 of \citealt{hill2014}), and as they are visible at all phases, they may be due to accretion material in the system. However, as this emission is weak, we did not carry out any further analysis. The residuals also show the relatively poor fit to the wings of the LSD profiles, resulting from adopting a spherical limb darkening law for a non-spherical object. We note that the SNR of D08 is significantly higher than for any other data set (see Table~\ref{tab:obs}), resulting in a relatively better fit with Roche tomography. This means that visually, the fits to the wings of the LSD profiles appear of lower quality, but the absolute level of the residuals is indeed lower than that of the other data sets.
\begin{figure*}
\centering
\includegraphics[width=0.6\textwidth,angle=270]{2004trails.pdf}
\caption{Trailed LSD profiles of AE Aqr for D04. The orbital motion has been removed assuming the binary parameters found in Section~\ref{sec:incsystemicmasses}, allowing individual starspot tracks across the profiles and the variation in \ensuremath{v\sin{i}}\xspace to be more clearly observed. Panels show (from left to right) the observed LSD profiles, the computed fits to the data using Roche tomography, and the residuals (increased by a factor of 10). Starspots and surface features appear bright in these panels, where a grey-scale of 1 corresponds to the maximum line depth in the reconstructed profiles. }
\label{fig:trails04}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[width=0.6\textwidth,angle=270]{2005trails.pdf}
\caption{The same as Figure~\ref{fig:trails04} but for D05.}
\label{fig:trails05}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[width=0.6\textwidth,angle=270]{2006trails.pdf}
\caption{The same as Figure~\ref{fig:trails04} but for D06.}
\label{fig:trails06}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[width=0.6\textwidth,angle=270]{2008trails.pdf}
\caption{The same as Figure~\ref{fig:trails04} but for D08. The fits to the wings of the LSD profiles appear relatively poor, but due to the higher SNR of the data, the absolute level of the residuals is actually lower (see text for discussion).}
\label{fig:trails08}
\end{figure*}
\section{System parameters}
\label{sec:systempars}
The system parameters (systemic velocity $\gamma$, orbital inclination $i$, primary star mass $M_{1}$ and secondary star mass $M_{2}$) of AE Aqr were determined using the standard methodology of Roche tomography (e.g. \citealt{watson2006,watson2007}). Adopting incorrect system parameters when carrying out Roche tomography reconstructions results in spurious artefacts in the final image. These artefacts are well characterised (see \citealt{watson2001}), and always increase the amount of structure (information content) of the map, decreasing the map entropy. We can constrain the binary parameters by carrying out map reconstructions for many pairs of component masses, fitting to the same $\chi^{2}$. This can be visualized as an entropy landscape, with an example shown in Figure~\ref{fig:entland}, where the optimal masses are the pair that produce the map of maximum entropy (least information content). Entropy landscapes are then repeated for different values of $i$ and $\gamma$, with the optimal set of parameters those which produce the map containing least structure (the map of maximum entropy).
The optimal system parameters are unique to each data set, as systematic effects may result in different optimal parameters between data sets. Hence, we do not adopt the mean values across all data sets for our analysis, as to do so may increase the number of artefacts reconstructed in the maps. This is further discussed in Section~\ref{sec:surfacemaps}.
\begin{figure}
\centering
\includegraphics[width=0.4\textwidth]{2005land.pdf}
\caption{The entropy landscape for AE Aqr using D05, assuming the parameters given in Table~\ref{tab:systemparameters}. Dark regions indicate masses for which no acceptable solution could be found. The cross marks the point of maximum entropy, corresponding to component masses of $M_{1} = 0.86~\mathrm{M}_{\sun}$ and $M_{2} = 0.56~\mathrm{M}_{\sun}$.}
\label{fig:entland}
\end{figure}
\subsection{Limb Darkening}
\label{sec:limbdarkening}
Following \cite{hill2014}, we adopted the four-parameter non-linear limb darkening model of \cite{claret2000}. The stellar parameters closest to that of a K4V star were adopted, which for the PHOENIX model atmosphere were $\log{g} = 4.5$ and $T_{\mathrm{eff}} = 4800$~K. The adopted coefficients for each data set are shown in Table~\ref{tab:limbcoeff}, where different (but very similar) values were used for each data set due to the different central wavelengths of the spectra. The treatment of limb darkening for D01 and D09a \& D09b may be found in the previously published work of \cite{watson2006} and \cite{hill2014}, respectively.
\begin{table}
\centering
\caption{Limb-darkening coefficients.}
\label{tab:limbcoeff}
\begin{tabular}{crrrr}
\toprule
Coefficient & D04 & D05 & D06 & D08 \\
\midrule
$a_{1}$ & 0.724 & 0.724 & 0.724 & 0.724 \\
$a_{2}$ & -0.759 & -0.768 & -0.764 & -0.764 \\
$a_{3}$ & 1.349 & 1.356 & 1.353 & 1.353 \\
$a_{4}$ & -0.415 & -0.411 & -0.412 & -0.412 \\
\bottomrule
\end{tabular}
\end{table}
\subsection{Systemic velocity, inclination and masses}
\label{sec:incsystemicmasses}
All data sets were fit independently. For each, we constructed a series of entropy landscapes for a range of orbital inclinations $i$ and systemic velocities $\gamma$. For given values of $i$ and $\gamma$ we selected the pair of masses that produced the map of maximum entropy. The results of this analysis are presented here.
\subsubsection{Systemic velocity}
\label{sec:systemic}
Figure~\ref{fig:systemic} shows the map entropy (after adopting optimal values of $M_{1}, M_{2}~\&~i$) as a function of systemic velocity, for each data set. Crosses mark the peak of the `entropy parabola', giving the optimal values of $\gamma$, as listed in Table~\ref{tab:systemparameters}. The measured values of $\gamma$ are consistent with that of previous work, falling within the uncertainties given by both \cite{welsh1995} and \cite{casares1996}. The significant difference between the $\gamma$ found here and that found by \cite{echevarria2008} stems from the uncertainties in the absolute radial velocities of the template stars used in the latter authors' analysis. The spread in $\gamma$ as measured by Roche tomography may be explained by instrumental offsets between different instruments, as well as for the same instrument over different observation periods. Values of $\gamma$ obtained by using entropy landscapes was found to be independent of the assumed inclination, as previously found by \cite{watson2003,watson2006} and \cite{hill2014}. In addition, the values obtained using the radial velocity curves are similar, although these will be biased, as discussed in section~\ref{sec:ephemeris}.
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{systemic_velocity.pdf}
\caption{The points show the maximum entropy obtained
in each data set as a function of systemic velocity for AE Aqr. The optimal inclination and masses were adopted for each data set, as found in Sections~\ref{sec:inclination}~\&~\ref{sec:masses}, respectively. The points are offset in the ordinate for clarity. Crosses mark the optimal value of $\gamma$, and solid lines are shown only as a visual aid.}
\label{fig:systemic}
\end{figure}
\subsubsection{Inclination}
\label{sec:inclination}
Figure~\ref{fig:inclination} shows the maximum entropy obtained as a function of inclination, for each data set, assuming the systemic velocities derived in Section~\ref{sec:systemic}. Crosses mark the maximum entropy obtained for a given data set, and the corresponding inclinations are listed in Table~\ref{tab:systemparameters}. All values of $i$ are consistent with previously published work by \cite{welsh1995} and \cite{casares1996}, although all values lie below that found by \cite{echevarria2008}. Furthermore, all values of $i$ lie between the previously determined inclinations of D01 ($i = 66\degr$, \citealt{watson2006}) and D09a~\&~D09b ($i = $50--51$\degr$, \citealt{hill2014}).
The consistency of $i =$~57--60$\degr$ across the four data sets is a reassuring result, as inclination is the worst constrained parameter when using Roche tomography. We do not have a clear explanation for the discrepancy between the inclinations found here and that of D01 and D09a~\&~D09b.
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{inclination.pdf}
\caption{The points show the maximum entropy obtained for each data set as a function of inclination, assuming the optimal values of $\gamma$ and masses, as found in Sections~\ref{sec:systemic}~\&~\ref{sec:masses}, respectively. Crosses mark the optimal value of $i$, and a solid line is shown as a visual aid.}
\label{fig:inclination}
\end{figure}
\subsubsection{Masses}
\label{sec:masses}
The component masses of AE Aqr were determined using entropy landscapes, with an example shown for D05 in Figure~\ref{fig:entland}. For each data set we assumed $i$ and $\gamma$, as derived in Sections~\ref{sec:systemic}~\&~\ref{sec:inclination}. Our derived masses shown in Table~\ref{tab:systemparameters} are consistent (within the uncertainties) of those found by \cite{echevarria2008}, \cite{welsh1995}, and \cite{casares1996}, once the masses have been adjusted to account for the change in inclination. The differences between the masses determined here, and those found in previous studies of AE Aqr using Roche tomography, are simply due to the use of slightly different inclination values. Indeed the mass ratios $q$ are in excellent agreement with previous work, and are typically 6~per~cent larger than those found by \cite{echevarria2008}, \cite{welsh1995}, and \cite{casares1996}. We note that the masses determined in this work are (in principle) the most reliable, as we correct for the systematic effects of surface features that may bias the measured RVs used to determine the system parameters in other work.
The target reduced~$\chi^{2}$ to which our data were fit was chosen as the point where the entropy of the reconstructed maps dramatically decreased when fits to a lower $\chi^{2}$ were performed. An increase in small scale structure contributes to a dramatic decrease in entropy, indicative of mapping noise in the Roche tomograms. Fitting to a higher reduced-$\chi^{2}$ caused fewer features to be mapped, and thus the system parameters were less well defined as more map pixels were assigned the default map value. Figure~\ref{fig:chivsentropy} shows how the map entropy depended on the aim $\chi^{2}$, where the adopted value is circled. The absolute value of $\chi^{2}$ is not a good reflection of the quality of fit, as a value above or below 1 indicates our error bars were systematically under or over estimated.
Assigning uncertainties to any of the derived system parameters ($i, \gamma, M_{1}, M_{2}$) is not trivial. As previously discussed in \cite{watson2001} and \cite{watson2006}, it would require using a Monte Carlo style technique combined with bootstrap resampling to generate synthetic data sets drawn from the same parent population as the observed data. Then, the same analysis carried out in this work would need to be applied to the hundreds of bootstrapped data sets, requiring an unfeasible amount of computation. Hence we do not assign strict uncertainties to our derived system parameters.
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{chi_entropy.pdf}
\caption{The reconstructed-map entropy as a function of reduced $\chi^{2}$ for the Roche tomograms of each data set (marked in top left of panels). The system parameters derived in Section~\ref{sec:incsystemicmasses} were adopted for the fits. The selected aim $\chi^{2}$ is circled in each plot, and is taken as the point where there is a dramatic decrease in map entropy, and a corresponding increase in small scale features in the reconstruction. The final reduced~$\chi^{2}$ for D04, D05, D06 and D08 are 0.95,~1.2,~1.3~\&~0.26, respectively, where a $\chi^{2}<1$ indicates that we overestimated the size of our propagated uncertainties. See Section~\ref{sec:masses} for further discussion.}
\label{fig:chivsentropy}
\end{figure}
\begin{table*}
\caption{System parameters. Columns 1-6 list the data set or paper from which the parameters were taken, the systemic velocity as measured by Roche tomography, the inclination, the mass of the primary star, the mass of the secondary star and the mass ratio. The significantly higher component masses found in D09a~\&~D09b are due to the lower inclination found in that study.}
\label{tab:systemparameters}
\begin{tabular}{cccccc}
\toprule
Author / Data & $\gamma$ (kms\ensuremath{^{-1}}\xspace) & $i$ (degrees) & $M_{1}$ (M$_{\sun}$) & $M_{2}$ (M$_{\sun}$) & $q = \sfrac{\mathrm{M}_{2}}{\mathrm{M}_{1}}$ \\
\midrule
D01 & $-63$ & 66 & 0.74 & 0.50 & 0.68 \\
D04 & $-60.6$ & 60 & 0.84 & 0.55 & 0.65 \\
D05 & $-60.7$ & 59 & 0.86 & 0.56 & 0.65 \\
D06 & $-62.4$ & 59 & 0.87 & 0.56 & 0.64 \\
D08 & $-61.4$ & 57 & 0.94 & 0.64 & 0.68 \\
D09a & $-64.7\pm 2.1$ & 50 & 1.20 & 0.81 & 0.68 \\
D09b & $-62.9\pm 1.0$ & 51 & 1.17 & 0.78 & 0.67 \\
\cite{echevarria2008} & -63 & $70\pm3$ & $0.63\pm0.05$ & $0.37\pm0.04$ & 0.60 \\
\cite{casares1996} & $-60.9\pm 2.4$ & $58\pm6$ & $0.79\pm0.16$ & $0.50\pm0.10$ & 0.63 \\
\cite{welsh1995} & $-63\pm 3$ & $54.9\pm7.2$ & $0.89\pm0.23$ & $0.57\pm0.15$ & 0.64 \\
\bottomrule
\end{tabular}
\end{table*}
\section{Surface maps}
\label{sec:surfacemaps}
Roche tomograms of the secondary star in AE Aqr were constructed for each data set using the system parameters derived in Section~\ref{sec:incsystemicmasses}. The corresponding fits to the data are shown in Figures~\ref{fig:trails04}~to~\ref{fig:trails08}, and the Roche tomograms are shown in Figures~\ref{fig:map04}~to~\ref{fig:map08}. For ease of comparison, the previously published Roche tomograms of D01 and D09a~\&~D09b are shown in Figures~\ref{fig:map01},~\ref{fig:map09a}~\&~\ref{fig:map09b}, respectively. These were previously analysed by \cite{watson2006} and \cite{hill2014}, and we highlight the relevant features here. In the analysis which follows, the map coordinates are defined such that $0\degr$ longitude is the centre of the back of the star, with increasing longitude towards the leading hemisphere, and with the L$_{1}$ point at $180\degr$. We note that, due to the inclination of the system combined with limitations in the technique, we only consider features mapped in the Northern hemisphere to be reliable. Hence, the Southern hemisphere is excluded from our analysis.
We have adopted the optimal system parameters determined for each data set rather than the mean values across all data sets. This is due to the fact that systematic effects between data sets may result in different optimal system parameters being determined. Hence, adopting the mean values may lead to an increase in the number of artefacts reconstructed in the maps. Nevertheless, we have assessed the impact of adopting the mean system parameters by additionally carrying out the analysis in this section for Roche tomograms reconstructed using the mean values of $i = 57.4\degr$, $M_{1} = 0.92$~M$_{\sun}$ and $M_{2} = 0.61$~M$_{\sun}$ (where the component masses were calculated from the mean of the constant $M_{(1,2)}\sin^{3}{i}$). We find that, for each data set, the spot features reconstructed using the mean parameters are not significantly different to those reconstructed using the optimal parameters. Furthermore, there are no significant differences in the fractional spot coverage as a function of longitude and latitude (see Section~\ref{sec:spotcover}) on the maps reconstructed using the mean and the optimal parameters. This shows the robustness of the surface features reconstructed against incorrect inclination and component masses. However, to obtain the same map entropy using the mean parameters, as compared to the optimal parameters, we were required to fit the data to a higher aim $\chi^{2}$. Thus, we adopted the optimal parameters for the spot analysis as they provide a better fit to the data.
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{2001map_label.pdf}
\caption{The Roche tomogram of AE Aqr using D01. Dark grey-scales indicate regions of reduced absorption-line strength that is due to either the presence of starspots, the impact of irradiation, or gravity darkening. The absolute grey-scales are relative and are not necessarily comparable between maps. The orbital phase is indicated above each panel. Roche tomograms are shown without limb darkening for clarity.}
\label{fig:map01}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{2004map_label.pdf}
\caption{The same as Figure~\ref{fig:map01} but for D04.}
\label{fig:map04}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{2005map_label.pdf}
\caption{The same as Figure~\ref{fig:map01} but for D05.}
\label{fig:map05}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{2006map_label.pdf}
\caption{The same as Figure~\ref{fig:map01} but for D06.}
\label{fig:map06}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{2008map_label.pdf}
\caption{The same as Figure~\ref{fig:map01} but for D08.}
\label{fig:map08}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{2009amap_label.pdf}
\caption{The same as Figure~\ref{fig:map01} but for D09a.}
\label{fig:map09a}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{2009bmap_label.pdf}
\caption{The same as Figure~\ref{fig:map01} but for D09b.}
\label{fig:map09b}
\end{figure}
\subsection{Global features}
\label{sec:global}
Spot features, both large and small, are clearly prevalent in all tomograms. Common to all maps are the dark regions around the L$_{1}$ point, primarily due to the time-averaged effects of irradiation (ionising weak metal lines preventing photon absorption), as well as gravity darkening. As both of these effects appear dark in the maps, we are unable to clearly distinguish between the two. Likewise, we are unable to disentangle any starspots that may inhabit the affected region. Nevertheless, the impact of gravity darkening on the fractional spot coverage, \ensuremath{f_{\mathrm{s}}}\xspace, was assessed, and is discussed in Section~\ref{sec:spotcover}.
The map of D01 (see Figure~\ref{fig:map01}) clearly shows a single large spot (labelled `A') extending 60--80$\degr$~latitude and 260--320$\degr$~longitude, with a second prominent spot centred at 50$\degr$~latitude and $180\degr$~longitude. Also clearly apparent is a spot extending from $\sim40\degr$ latitude down to the L$_{1}$ point, becoming indistinguishable in the irradiated region. The map of D04 (see Figure~\ref{fig:map04}) shows two separate large spots, with the largest (labelled `B') extending 40--75$\degr$ in latitude and 160--220$\degr$ in longitude, and the second largest (labelled `C') covering 60--70$\degr$~latitude and 310--340$\degr$~longitude. The map of D05 (see Figure~\ref{fig:map05}) has a single large spot (labelled `D'), extending 55--75$\degr$ in latitude and 140--210$\degr$ in longitude. The poor phase coverage of AE Aqr in D06 resulted in the reconstructed map having a relatively featureless leading hemisphere, with spots on the rest of the star becoming smeared out over the image, resulting in a lower contrast (see Figure~\ref{fig:map06}). Despite this, at least one, possibly two large spots (labelled `E') are evident above $\sim65\degr$ latitude, centred on $\sim180\degr$ longitude, with another prominent feature extending from $40\degr$ down to the L$_{1}$ point. The map of D08 (see Figure~\ref{fig:map08}) exhibits a single large spot (labelled `F') extending 65--85$\degr$ and 340--060$\degr$ in latitude and longitude, respectively. Finally, the maps of D09a and D09b (see Figures~\ref{fig:map09a}~and~\ref{fig:map09b}) show one large spot (labelled `G') spanning 65--90$\degr$~latitude and 340--050$\degr$~longitude.
The latitude of the largest spot in each map (labelled A, B, D, E, F, G) remains fairly constant over the 8~years between the first and last observations, ranging between 60--80$\degr$. However, the longitude of the dominant spot is not fixed. In D01, the dominant spot (A) lies $\sim290\degr$~longitude, whereas for D04, D05 and D06, the dominant spot (labelled B, D and E, respectively) lies $\sim180\degr$~longitude, although the position of spot E in D06 is less certain due to smeared features. In contrast, the largest spot in D08 (labelled F) and D09a~\&~D09b (labelled G) lies at $\sim15\degr$~longitude. Such monolithic spots are prevalent in many Doppler imaging studies of rapidly-rotating solar-type stars such as LQ~Hya \citep{donati1999}, and in CVs such as BV~Cen \citep{watson2007}. The high-latitude spots imaged here, and their possible evolution, are further discussed in Section~\ref{sec:discussion}.
Starspots are also prevalent at low to mid latitudes in all maps, and in order to make a more quantitive assessment of their properties and the underlying dynamo mechanism, we must consider their size and distribution across the stellar surface.
\subsection{Pixel intensity and spot filling factor} \label{sec:intensity}
To determine the spot coverage in the Roche tomograms, it was first necessary to define the pixel intensity of the immaculate photosphere as well as that of a totally spotted pixel. We do not adopt a two-temperature model when fitting with Roche tomography, where a spot filling factor is predetermined (e.g. \citealt{cameron1994}), as secondary stars in CVs are expected to exhibit large temperature differences due to irradiation by the primary. Our method of determining a totally spotted pixel was to simply select the lowest pixel intensity at the centre of the largest spot feature. The adopted value of a totally spotted pixel for each map is shown as a dotted line on the left side of the histograms of pixel intensities in Figure~\ref{fig:histogram}, where the brightest pixel is assigned an intensity of 100 and other pixels scaled linearly relative to this. Pixels with a lower intensity are present in the Roche tomograms, but these are confined to the region around the L$_{1}$ point, and as discussed above, are not likely to be due to a spot feature. Pixels on the Southern hemisphere are not included in the histograms or any of the analysis below for two reasons. Due to the inclination of the binary, a large portion of the surface is not visible, and so a substantial number of pixels on the Southern hemisphere are assigned the default map value. Furthermore, as RVs cannot constrain whether a feature is located in the Northern or Southern hemisphere, features may be mirrored about the equator, reducing their contrast as they are smeared over a larger area.
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{all_histogram.pdf}
\caption{Histograms of the pixel intensities in the Roche tomograms of AE~Aqr for each data set, where the pixel density is the fraction of the total number of pixels. Pixels on the Southern hemisphere (latitude $<0\degr$) were not included (see Section~\ref{sec:intensity} for details). The brightest pixel in each map was assigned an intensity of 100 and all other pixel intensities were scaled linearly against this. The definition of the pixel intensity representing the immaculate photosphere is shown as a dashed line on the right side of the histogram (a dotted line for D09b), and that representing a totally spotted pixel is shown as a dashed line on the left side (see text for details).}
\label{fig:histogram}
\end{figure}
The intensity of the immaculate photosphere was more difficult to define due to the growth of bright pixels -- an artefact known to affect maps that are not thresholded (e.g. \citealt{hatzes1992}). To assess the extent of bright pixel growth in our Roche tomograms, we carried out reconstructions of simulated maps with a random spot distribution, adding in varying levels of noise. We found that the brightest reconstructed pixel was between 4--15~per~cent higher in intensity than that of the original map, and that up to 12\% of pixels were classed as `bright' (typically $\sim1$~per~cent). Data with a higher SNR increased the number of bright pixels, however, the most dramatic increase was found when the data were fit to progressively lower reduced~$\chi^{2}$. Indeed, the maps with a large number of bright pixels were clearly overfit, exhibiting substantial reconstructed noise. Given that our Roche tomograms of AE~Aqr were fit to an aim reduced~$\chi^{2}$ that limited the amount of reconstructed noise (see Figure~\ref{fig:chivsentropy}), we assumed a percentage growth of bright pixels of 3\% of the total number of pixels in the map. This somewhat conservative estimate, given the results of our simulations, means we likely underestimate \ensuremath{f_{\mathrm{s}}}\xspace. Hence, we defined the immaculate photosphere as the lowest pixel intensity that includes 97\% of all pixels, and is shown in Figure~\ref{fig:histogram} as a dotted line on the right side of the histograms.
The histograms of pixel intensities in Figure~\ref{fig:histogram} show broad peaks, with long tails towards lower intensities. An idealised histogram would have a significantly bimodal distribution of pixel intensities, where spotted pixels and those representing the immaculate photosphere would be more clearly separated. The large number of intermediate pixel intensities found here may be explained by a lack of contrast in the maps, stemming from both a population of unresolved spots, as well as spots that have been smeared in latitude, increasing their areal coverage.
\subsection{Spot coverage as a function of longitude and latitude}
\label{sec:spotcover}
Figures~\ref{fig:spotcoverlong}~and~\ref{fig:spotcoverlat} show the fractional spot coverage \ensuremath{f_{\mathrm{s}}}\xspace as a function of longitude and latitude for each map, respectively. We calculate \ensuremath{f_{\mathrm{s}}}\xspace using Equation~\ref{eqn:fsc}, where $I$ is the pixel intensity, $I_{\mathrm{p}}$ is the intensity of the immaculate photosphere, and $I_{\mathrm{s}}$ is the intensity of a totally spotted pixel.
\begin{equation}
\ensuremath{f_{\mathrm{s}}}\xspace = \text{max}\left[0,\text{min}\left[1, \frac{I_{\mathrm{p}} - I}{I_{\mathrm{p}} - I_{\mathrm{s}}}\right]\right]
\label{eqn:fsc}
\end{equation}
The most prominent feature common to all plots is the large value of \ensuremath{f_{\mathrm{s}}}\xspace around $180\degr$ longitude in Figure~\ref{fig:spotcoverlong}, as well the increased coverage below $20\degr$ latitude in Figure~\ref{fig:spotcoverlat}. These regions include the features around the L$_{1}$ point which are dominated by the effects of irradiation and gravity darkening (as discussed above). The impact of these phenomena on the maps was assessed by simulating blank maps using the corresponding system parameters and limb darkening coefficients specific to each data set. Additionally, a gravity darkening coefficient of $\beta = 0.1$ was adopted, as this is representative of that measured for late-type secondary stars in close binaries \citep{djurasevic2003,djurasevic2006}. The simulated maps were used to create synthetic line profiles with the same orbital phases, exposure times and instrumental resolutions of the original data. These synthetic line profiles were then reconstructed in the same manner as the original data, and \ensuremath{f_{\mathrm{s}}}\xspace was calculated using the same definitions of a totally-spotted pixel and that representing the immaculate photosphere, as determined above. The value of \ensuremath{f_{\mathrm{s}}}\xspace of these synthetic maps was then subtracted from that of the original maps, effectively removing the systematic effects of inclination, phase sampling, and incorrect limb and gravity darkening in the spot coverage of the original maps. However, we cannot distinguish between a spot and the effects of gravity darkening or irradiation, as all three appear as dark regions in the maps. Hence, when we apply the correction described above, spots that are located in the regions most affected by irradiation and gravity darkening are not preserved (as the regions are made brighter, regardless of spots being present; see discussion in Section~\ref{sec:global}), and thus we may underestimate \ensuremath{f_{\mathrm{s}}}\xspace in those regions (see Figures~\ref{fig:spotcoverlong}~and~\ref{fig:spotcoverlat}).
\subsubsection{Longitude cover}
\label{sec:longcover}
The fractional spot coverage \ensuremath{f_{\mathrm{s}}}\xspace as a function of longitude varies significantly for each map (see Figure~\ref{fig:spotcoverlong}). Even after subtraction of the simulated maps there still remains a significant spot coverage around $180\degr$~longitude for D04, D05, D06 and D09a~\&~D09b. The map of D01 shows an increase in \ensuremath{f_{\mathrm{s}}}\xspace around $250\degr$~longitude, with the maps of D04 and D06 showing a significant increase in \ensuremath{f_{\mathrm{s}}}\xspace between 280--360$\degr$~longitude, and the maps of D08 and D09a~\&~D09b showing a larger \ensuremath{f_{\mathrm{s}}}\xspace over a broader range of 280--060$\degr$~longitude. The increase in \ensuremath{f_{\mathrm{s}}}\xspace between 0--120$\degr$~longitude for D06 is not real and results from poor phase coverage, leading to the pixels in this region being assigned a value similar to that of the default map (the mean of all map pixels).
The distributions of \ensuremath{f_{\mathrm{s}}}\xspace in longitude suggests the existence of two longitude regions, separated by $\sim180\degr$, with significantly higher \ensuremath{f_{\mathrm{s}}}\xspace in the maps of D04, D06 and D09a~\&~D09b. However, no such distributions are present in D01 and D05, with the high \ensuremath{f_{\mathrm{s}}}\xspace around $180\degr$ in D08 becoming much lower once the effects of irradiation and gravity darkening have been removed. Such `active longitudes' are observed in single stars such as LQ~Hya, AB~Dor and EK~Dra \citep{berdyugina2005} as well as in RS CVn binaries (e.g. \citealt{rodono2000}). In shorter period systems ($\ensuremath{P_{\text{rot}}}\xspace<1$~d), active regions are preferentially located at quadrature longitudes (e.g. \citealt{olah1994,heckert1998}), whereas longer period systems show no such preference. Indeed the RS CVn binaries HD~106225 \citep{strassmeier1994} and II~Peg \citep{rodono2000} show a migration of active longitudes with respect to the companion star, ascribed to differential rotation. However, the lack of clearly defined active longitudes in the maps of D01, D05 and D08 suggest that such active regions are not permanent features in AE~Aqr, or are at least not fixed with respect to the companion. We discuss possible explanations of the observed stellar activity in Section~\ref{sec:discussion}.
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{all_longitude_cover.pdf}
\caption{The fractional spot coverage \ensuremath{f_{\mathrm{s}}}\xspace as a function of longitude for the Northern hemisphere of AE~Aqr for each data set, where \ensuremath{f_{\mathrm{s}}}\xspace is normalized by the number of pixels within a $6\degr$~longitude bin. The solid line shows \ensuremath{f_{\mathrm{s}}}\xspace for the original map, and the dotted line shows \ensuremath{f_{\mathrm{s}}}\xspace after subtracting \ensuremath{f_{\mathrm{s}}}\xspace for the simulated map (removing the effects of irradiation and gravity darkening, see Section~\ref{sec:spotcover} for details). The shaded region for D06 indicates that features in this region are not reliable due to a lack of phase coverage.}
\label{fig:spotcoverlong}
\end{figure}
\subsubsection{Latitude cover}
\label{sec:latcover}
The fractional spot coverage \ensuremath{f_{\mathrm{s}}}\xspace as a function of latitude is shown in Figure~\ref{fig:spotcoverlat} for all maps. Latitudes above $60\degr$ show high \ensuremath{f_{\mathrm{s}}}\xspace due to the large high-latitude spots. The distribution of \ensuremath{f_{\mathrm{s}}}\xspace above $60\degr$~latitude is similar in D01, D08, and D09a~\&~D09b, with maximum \ensuremath{f_{\mathrm{s}}}\xspace at latitudes $>70\degr$. In comparison, \ensuremath{f_{\mathrm{s}}}\xspace peaks around $65\degr$~latitude for D04 and D06, with the flatter distribution of D05 exhibiting a peak that is consistent with D04 and D06.
At mid-latitudes of $\sim45\degr$ we see the apparent growth of \ensuremath{f_{\mathrm{s}}}\xspace over the 8~years of observations, with a peak becoming more pronounced in the plots of D08 and D09a~\&~D09b. This may be indicative of increasing spot coverage at this latitude, however, it could also be explained by a relative \emph{decrease} in the spot coverage at surrounding latitudes. Given our lack of an absolute spot filling factor, the total spot coverages between maps (given as a percentage in the top-left of each panel in Figure~\ref{fig:spotcoverlat}) are not necessarily comparable on an absolute scale, rather, they indicate the relative spot coverage for that particular map. However, given the measured spot coverage increases between D05 and D09b, we can be confident the increase of \ensuremath{f_{\mathrm{s}}}\xspace at $45\degr$~latitude is indeed due to an increase in spot coverage localised to this latitude.
A common feature to all maps, excluding that of D01, is the apparent lack of spots around $30\degr$~latitude. This is most obvious for D04, D05 and D08, as \ensuremath{f_{\mathrm{s}}}\xspace is lowest at this latitude, and becomes more pronounced for D06 and D09a~\&~D09b when the systematic effects of irradiation and gravity darkening are subtracted. The lower quality of data in D01 means features at lower latitudes are more smeared out compared with other maps in our sample, and so while this feature may exist, we cannot resolve it to the same degree as in the other maps.
At lower latitudes, we see a clear increase in \ensuremath{f_{\mathrm{s}}}\xspace around $20\degr$ for all maps. As previously discussed, the effects of irradiation and gravity darkening are significant at these latitudes, which makes it difficult to determine how \ensuremath{f_{\mathrm{s}}}\xspace varies between maps. However, even after these effects have been subtracted, we still see a persistently high-level of \ensuremath{f_{\mathrm{s}}}\xspace around $20\degr$~latitude, suggesting spots at this latitude are common.
The reliability of the reconstruction of small scale features in the maps was tested with reconstructions of simulated data sets. Test maps were created using the same parameters as those for D04, D05 and D08, with a large polar spot, bands of spots at $45\degr$ and $20\degr$ latitude, and gravity and limb darkening. The bands of spots each contained at least 17 individual spots, with sizes ranging between 5--10$\degr$ in latitude and longitude, separated by 10--20$\degr$ longitude. Trailed spectra were created using the same phases as the original data, and representative noise was added. Maps were reconstructed from the synthetic data in the same manner as that carried out for the original data. The resulting maps of all three simulated data sets show that the polar spot and the spot band at $45\degr$ are clearly recovered, with the spot band at $20\degr$ becoming moderately smeared in latitude, reducing the contrast. However, the latter spots are still clearly distinct from the reconstructed noise, and so we are confident that we can reliably reconstruct features of this size at these latitudes. Furthermore, the latitudinal spread in \ensuremath{f_{\mathrm{s}}}\xspace due to these smeared spots has been taken into account in the analysis presented here.
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{all_latitude_cover.pdf}
\caption{The fractional spot coverage \ensuremath{f_{\mathrm{s}}}\xspace as a function of latitude for the Northern hemisphere of AE~Aqr for each data set, normalized by the surface area at that latitude. The solid line shows \ensuremath{f_{\mathrm{s}}}\xspace for the original map, and the dotted line shows \ensuremath{f_{\mathrm{s}}}\xspace after subtracting \ensuremath{f_{\mathrm{s}}}\xspace for the simulated map (removing the effects of irradiation and gravity darkening, see Section~\ref{sec:spotcover} for details). The total spot coverage for each map is given as a percentage in the top left of each panel, where the value in parenthesis is the total spot coverage after subtracting the simulated map.}
\label{fig:spotcoverlat}
\end{figure*}
\section{Discussion}
\label{sec:discussion}
\subsection{Evolution of high-latitude spots}
\label{sec:highlatspots}
Several high-latitude spots are seen in the maps of AE~Aqr, and are labelled A-G in Figures~\ref{fig:map01}-\ref{fig:map09b}. There are several possible scenarios that may explain the behaviour of the dominant high-latitude spot; The largest spot in each map may be the same feature, with only its position changing between observations.
Alternatively, the largest spot may not be the same feature, but instead is evolving significantly, disappearing and appearing elsewhere between maps.
In the first scenario, if the largest spot in each map (labelled A, B, D, E, F, G) is the same feature, and is able to move position, then stable spots may have a lifetime of $\sim8$~yr. However, if the largest spot must remain in a fixed position, then spots B, D and E in Figures~\ref{fig:map04}--\ref{fig:map06} imply that a large, stable spot can live for $\sim4$~yr. The changing position of the dominant spot may be explained by the presence of differential rotation (DR) on the surface of the star, as measured by \cite{hill2014} using the maps of D09a~\&~D09b. By using the measured shear-rate, we have determined the longitude-shift for latitudes between 60--80$\degr$ over the intervals between observations. By comparing the shift in longitude due to DR with the observed change in longitude of the spot in question, we find that spot~A in D01 may have shifted to the position of spot~B in D04, and spot~E in D06 may have shifted to the position of spot~F in D08. Only the movement of the spots in these two cases may be explained by DR, with all other observed shifts being incompatible with this mechanism. It is possible that the spot~C in D04 grew in area \emph{and} rotated due to DR to the same position as spot~D in D05, however, the spot~B in D04 would then have to disappear completely in D05, which seems unlikely. Furthermore, the fact that spot~F in D08 is at the same location as spot~G in D09a~\&~D09b suggests DR is unlikely to be the cause of the observed shifts. Indeed DR may primarily affect smaller spots that have magnetic flux tubes anchored closer to the surface, whereas larger spots are less affected as their flux tubes may be anchored deeper in the stellar interior.
The second scenario requires the evolution of these dominant spots over a relatively short time period. Similar behaviour, observed in other systems, has been explained by a phenomenon known as the `flip-flop' activity cycle. In this scenario, active longitudes are present on the star. The `flip-flop' cycle occurs when the active longitude with the highest level of activity (i.e. the most spotted) switches to the opposite longitude, with cycles taking years to decades to occur (see \citealt{berdyugina2005} for a summary). The disappearance of spot~C in D04, as compared to D05, may imply the switching of dominant longitudes between observations of AE~Aqr. However, the lack of at least two clearly defined active longitudes in D01, D05 and D08 suggests that this type of cycle is not present in AE~Aqr. In any case, a robust detection of a `flip-flop' cycle in AE~Aqr would require a much shorter gap between observations than obtained for our sample in order to clearly track the emergence of spots at preferred longitudes.
\subsection{Magnetic flux tube dynamics in close binaries}
\label{sec:magfluxtube}
To understand why surface features are distributed as they are, and why they evolve in the observed manner, we can compare the Roche tomograms to numerical simulations of emerging magnetic flux tubes in close binary systems. \cite{holzwarth2003a} carried out such simulations, assuming that starspots are formed by erupting flux tubes that originate from the bottom of the stellar convective zone (by analogy with the Sun). Magnetic fields, believed to be amplified in the rotational shear layer (tacholine) near the base of the convection zone, are stored in the form of toroidal flux tubes in the convective overshoot layer \citep{schussler1994}. By studying the equilibrium and linear stability properties of these flux tubes, \cite{holzwarth2003a,holzwarth2003b} examined whether the influence of the companion star was able to trigger rising flux loops at preferred longitudes, since the presence of the companion breaks the rotational symmetry of the star.
The authors find that while the magnitudes of tidal effects are rather small, they nevertheless lead to the formation of clusters of flux tube eruptions at preferred longitudes on opposite sides of the star, a phenomenon resulting from the resonating action of tidal effects on rising flux tubes. Pertinently, the authors establish that the longitude distribution of spot clusters on the surface depends on the initial magnetic field strength and latitude of the flux tubes in the overshoot region, implying there is no preferred longitude in a globally fixed direction. Moreover, flux tubes that are perturbed at different latitudes in the convective overshot region, show a wide latitudinal range of emergence on the stellar surface, with considerable asymmetries appearing as highly peaked \ensuremath{f_{\mathrm{s}}}\xspace distributions or broad preferred longitudes. In a binary with $P_{\text{orb}} = 2$~d, the authors find it takes several months to years for a flux tube, perturbed from the convective overshoot region, to emerge on the stellar surface. Over this time, the Coriolis force acts on the internal gas flow and causes the poleward deflection of the tube \citep{schussler1992}, with the largest deflection for flux tubes starting at lower latitudes, and those starting $>60\degr$~latitude showing essentially no deflection. Such simulations are consistent with the large high-latitude spots found on AE~Aqr.
Clearly the behaviour of flux tubes in a binary system is complex. However, the results of the models by \cite{holzwarth2003b} may explain the varying size, distribution and evolution of the starspots imaged in AE~Aqr; Namely, the low value of \ensuremath{f_{\mathrm{s}}}\xspace around $30\degr$~latitude in the maps of AE~Aqr is consistent with the fact that some latitudes are avoided by the simulated erupting flux tubes. Furthermore, the variable peak of \ensuremath{f_{\mathrm{s}}}\xspace at high latitudes in AE~Aqr may be a redistribution of magnetic energy, changing the field strength of perturbed flux tubes and causing them to emerge at different latitudes, as well as shifting in longitude. In addition, flux tube eruption at high latitudes, due to flow instabilities, leads to spots emerging over a broad longitude region -- similar to the large high-latitude spots observed in AE~Aqr.
\section{A magnetic activity cycle?}
It is unclear what the dominant dynamo mechanism is in AE~Aqr. There is no clear evidence that we see a `flip-flop' cycle, especially given the lack of active longitudes in three maps of our sample. Indeed the most prominent evidence of an activity cycle is the increase in \ensuremath{f_{\mathrm{s}}}\xspace at $45\degr$~latitude over the course of the 8~years of observations, combined with the persistently high \ensuremath{f_{\mathrm{s}}}\xspace around $20\degr$~latitude. The growth in spot coverage around $45\degr$~latitude may be indicative of an emerging band of spots, forming part of an activity cycle similar to that seen on the Sun. Furthermore, the increase in \ensuremath{f_{\mathrm{s}}}\xspace around $20\degr$~latitude may be a second band of spots that form part of a previous cycle. In the case of the Sun, the latitude of emergence of flux tubes gradually moves towards the equator over the course of an activity cycle, taking $\sim11$~years, with little overlap between consecutive cycles of flux tube emergence. However, simulations by \cite{isik2011} show that stronger dynamo excitation may cause a larger overlap between consecutive cycles. Given that the high spot coverage in AE~Aqr suggests a strong dynamo excitation, the presence of two prominent bands of spots may be indicative of such an overlap between cycles. However, if such a cycle were to exist, we would expect to see the higher-latitude peak move towards lower latitudes over the course of our observations. Given that we do not clearly see this, any solar-like activity cycle must take place over a timescale longer than 8~years. \cite{saar1999}, in their study of a large sample of stars in a range of systems (including single stars and binaries), found several correlations between the duration of the magnetic activity cycle and the rotation period. Pertinently, using the fit to all stars in their sample, we estimate that AE~Aqr would have a magnetic activity cycle lasting $\sim16$~years. Furthermore, we estimate a longer cycle period of $\sim22$~years by using the fit to stars defined as `superactive' by \cite{saar2001}. If the correlations found for other systems are also true for AE~Aqr, then we may have observed less than half of an activity cycle.
This is the first time the number, size and distribution of starspots has been tracked in a CV secondary. While any specific interpretation of the long term behaviour of the imaged starspots is somewhat challenging, the presence and evolution of two distinct bands of spots may indicate an ongoing magnetic activity cycle in the secondary star in AE~Aqr. Hence, it is crucial we continue our study of its magnetic activity. Future maps would allow us to track the evolution of the large high-latitude spots to determine if their long term behaviour is periodic. Moreover, by tracking the evolution of the spot bands we may determine if they form part of a periodic activity cycle, and if so, the length of such a cycle could be measured, providing a unique insight into the behaviour of the stellar dynamo in an interacting binary. In addition, shorter, more intensive campaigns would allow the position of specific spot features to be tracked. This would allow further measurements of differential rotation as well as determining if meridional flows are present. The impact of tidal forces on magnetic flux tube emergence and possible quenching of mass transfer could then be assessed.
\section{Conclusions}
\label{sec:conclusions}
We have imaged starspots on the secondary star in AE~Aqr for 7~epochs, spread over 8~years. This is the first time such as study has been carried out for a secondary star in a CV and, in some cases, the number, size and distribution of spots varies significantly between maps. In particular, the changing positions of the large high-latitude spots cannot be explained by differential rotation, nor by the `flip-flop' activity cycle. At lower latitudes, we see the emergence of a band of spots around $45\degr$~latitude, as well as a persistently high spot coverage around $20\degr$~latitude. These bands may form part of an activity cycle similar to that seen in the Sun, where magnetic flux tubes emerge at progressively lower latitudes throughout a cycle. Furthermore, the complex distribution and behaviour of spots may be attributed to the impact of the companion WD on flux tube dynamics.
\section*{Acknowledgments}
We thank Tom Marsh for the use of his \textsc{molly} software package in this work, and VALD for the stellar line-lists used. We thank the staff at Carnegie Observatories for their assistance with the MIKE pipeline and for access to the Henrietta Swope Telescope. C.A.H. acknowledges the Queen's University Belfast Department of Education and Learning PhD scholarship, C.A.W. acknowledges support by STFC grant ST/L000709/1, and D.S. acknowledges support by STFC grant ST/L000733/1. This research has made use of NASA's Astrophysics Data System and the Ureka software package provided by Space Telescope Science Institute and Gemini Observatory.
\bibliographystyle{mn_new}
|
1,314,259,993,580 | arxiv | \section{Introduction}
\label{sec:introduction}
Deep Neural Networks (DNNs) and specifically Convolutional Neural Networks (CNNs) have recently attained significant success in image and video classification tasks. \behzad{They} are fundamental for state-of-the-art real-world applications running on embedded systems as well as data centers. \behzad{These neural networks learn a model} from a dataset in \behzad{their} training phase and make predictions on \behzad{new, previously-unseen} data in \behzad{their} classification phase.
However, their power-efficiency is inherently the primary concern due to the massive amount of data movement and computational power required. Thus, the scalability of CNNs for enterprise applications and \behzad{their deployment} in battery-limited scenarios, such as \behzad{in} drones and mobile devices, \behzad{are crucial concerns}.
Typically, hardware acceleration using Graphics Processing Units (GPUs)~\cite{zhang2018shufflenet}, \behzad{Field Programmable Gate Arrays (FPGAs)~\cite{sharma2016high,qiu2016going}}, or Application-Specific Integrated Circuits (ASICs)~\cite{sharma2018bit,jouppi2017datacenter,chen2016eyeriss} leads to a significant reduction \behzad{in CNN} power consumption~\cite{sze2017efficient}. Among \behzad{these}, FPGAs are rapidly becoming popular and are expected to be used in 33\% of modern data centers by 2020 \cite{top500}. This increase in the popularity of FPGAs is attributed to their power-efficiency compared to GPUs, their flexibility compared to ASICs, and \behzad{recent} advances in High-Level Synthesis (HLS) tools that significantly facilitate easier \behzad{mapping of applications on FPGAs}~\cite{putnam2014reconfigurable,vaishnav2018survey, salami2015hatch, salami2016accelerating,salami2017axledb,park2017scale,arcas2016hardware}. Hence, major companies, such as Amazon \cite{karandikar2018firesim} \behzad{(with EC2 F1 cloud)} and Microsoft~\cite{fowers2018configurable} \behzad{(with Brainwave project)}, have made large investments in FPGA-based CNN accelerators. However, recent studies show that FPGA-based accelerators are at least 10X less power-efficient compared to ASIC-based ones~\cite{boutros2018you,nurvitadhi2016accelerating,nurvitadhi2019compete}. In this paper, we aim to bridge this power-efficiency gap by \behzad{empirically understanding and} leveraging an effective undervolting technique for FPGA-based CNN accelerators.
\behzad{Power-efficiency} of state-of-the-art CNNs \behzad{generally} improves \behzad{via} architectural-level techniques, such as \behzad{quantization~\cite{zhu2019configurable} and pruning~\cite{molchanov2016pruning}}. These techniques do not \behzad{significantly} compromise \behzad{CNN} accuracy as they \behzad{exploit} the sparse nature of \behzad{CNN} applications~\cite{parashar2017scnn, albericio2016cnvlutin,zhang2016cambricon}. To further improve the power-efficiency of FPGA-based CNN accelerators, we propose to employ an orthogonal \behzad{hardware}-level approach\behzad{: undervolting (\textit{i.e.,} circuit supply voltage underscaling)} below the nominal/default level ($V_{nom}$), combined with the aforementioned architectural-level techniques. \behzad{FPGA vendors usually add a} voltage guardband to ensure the correct operation of FPGAs under the worst-case \behzad{circuit} and environmental conditions. However, these guardbands can be very conservative and unnecessary for state-of-the-art applications. Supply voltage underscaling below the nominal level \behzad{was} already shown \behzad{to provide} significant efficiency \behzad{improvements in} CPUs~\cite{papadimitriou2019adaptive}, GPUs~\cite{zou2018voltage,miller2012vrsync}, ASICs~\cite{chandramoorthy2019resilient}, and \behzad{DRAMs~\cite{chang2017understanding,koppula2019eden}}. This paper extends such studies \behzad{to FPGAs. Specifically, we study the classification phase of FPGA-based CNN accelerators}, as this phase can be repeatedly used in power-limited edge devices (unlike the training phase, which is \behzad{invoked much less frequently}). \behzad{Unlike} simulation-based approaches that may not be \fff{accurate enough}~\cite{zhang2018thundervolt, salamat2019workload}, our study is based on real off-the-shelf FPGA devices.
The extra voltage guardband can range between 12-35\% of the nominal supply voltage of modern CPUs~\cite{papadimitriou2019adaptive}, GPUs~\cite{zou2018voltage}, and DRAM \behzad{chips}~\cite{chang2017understanding}. Reducing the supply voltage in this guardband region does \behzad{\emph{not}} lead to \behzad{reliability issues under normal operating conditions,} and thus, eliminating this guardband \behzad{can} result in a significant power reduction for \behzad{a wide variety of} real-world applications. \textcolor{black}{We experimentally \behzad{demonstrate} a large voltage guardband for modern FPGAs\behzad{: an} average of 33\% with a slight variation across hardware platforms and software benchmarks. Eliminating this guardband leads to significant power-efficiency \behzad{($GOPs/W$) improvement}, on average, 2.6X, without any performance or reliability overheads.
With further undervolting, the power\behzad{-efficiency improves} by an extra 43\%, leading to a total \behzad{improvement} of more than 3X. This additional \behzad{gain} does not come for free, as we observe \behzad{exponentially}-increasing CNN accuracy loss below the guardband region. With further undervolting below this guardband, our experiments indicate that the \behzad{minimum} supply voltage at which the internal FPGA components could be functional ($V_{crash}$) is equal to, on average, 63\% of $V_{nom}$. Further reducing the supply voltage results in system crash.}
\textcolor{black}{We evaluate our undervolting technique on three identical samples of the Zynq-based ZCU102 platform \cite{zcu102}, a representative modern FPGA from Xilinx. However, we believe that our experimental observations are applicable to other FPGA platforms \behzad{as well}, perhaps with some minor differences. We previously showed \behzad{benefits of reduced-voltage operation} for on-chip memories on \behzad{different, }older FPGA \behzad{platforms}~\cite{salami2018comprehensive}. \behzad{Other works observed similar behavior} for different \behzad{types} of CPUs~\cite{papadimitriou2019adaptive}, GPUs~\cite{zou2018voltage}, and DRAM \behzad{chips}~\cite{chang2017understanding}. In this paper, we characterize the power dissipation of \behzad{FPGA-based CNN accelerators under reduced-voltage levels and} apply undervolting \behzad{to} improve the power-efficiency of such \behzad{accelerators}.}\footnote{ \textcolor{black}{\behzad{Our} exploration of the FPGA voltage behavior and the subsequent power-efficiency gain is applicable to any application.}}
\behzad{We experimentally evaluate the effects of reduced-voltage operation in} on-chip components of the FPGA platform, including Block RAMs (BRAMs) and internal FPGA \behzad{components,} containing Look-Up Tables (LUTs), Digital Signal Processors (DSPs), buffers, and routing resources.\footnote{These internal FPGA components share a single voltage rail in the studied FPGA platform. To our knowledge, \behzad{such voltage rail sharing} is a typical case for most modern FPGA platforms.} We \behzad{perform} our experiments on five state-of-the-art CNN image classification benchmarks, including VGGNet~\cite{miniV}, GoogleNet~\cite{miniG}, AlexNet~\cite{alex}, ResNet~\cite{he2016deep}, and Inception~\cite{miniG}. This enables us to experimentally study the workload-to-workload variation on the power-reliability trade-offs of FPGA-based CNN \behzad{accelerators. Specifically}, we extensively characterize the reliability behavior of the studied benchmarks below the guardband level and evaluate a frequency underscaling technique to prevent the accuracy loss in this voltage region. Our study also \behzad{examines} the effects of architectural quantization and pruning techniques with reduced-voltage FPGA operation. Finally, we experimentally evaluate the \behzad{effect} of environmental temperature variation on the power-reliability behavior of \behzad{FPGA-based CNN accelerators}.
\subsection{Contributions}
To our knowledge, for the first time, this paper experimentally studies the power-performance-accuracy characteristics of CNN accelerators with greatly reduced supply voltage capability implemented in real FPGAs. In summary, we achieve a total of more than 3X power-efficiency improvement for FPGA-based CNN accelerators. \behzad{We} gain insights into the reduced-voltage operation of such accelerators and, in turn, the \behzad{effect} of FPGA supply voltage on the power-reliability trade-off. \behzad{We make the following major contributions:}
\begin{itemize}
\item We characterize the power consumption of FPGA-based CNN accelerators across \behzad{different FPGA components}. We identify that the internal on-chip components, including processing elements, \behzad{contribute to} a vast majority of the total power consumption. \behzad{We reduce} this source of power consumption \behzad{via} our undervolting technique.
\item \textcolor{black}{We improve the power-efficiency of FPGA-based CNN accelerators \behzad{by} more than 3X\behzad{, measured across five state-of-the-art image classification benchmarks}. 2.6X of the power-efficiency gain is due to eliminating the voltage guardband\behzad{, which we measure} to be on average 33\%. An additional 43\% gain is due to further undervolting below the guardband\behzad{, which} comes at the cost of CNN accuracy loss.}
\item \behzad{We} characterize the reliability behavior of \behzad{FPGA}-based CNN accelerators when executed below the voltage guardband level and observe an exponential reduction \behzad{in} CNN accuracy \behzad{as voltage reduces}. We observe that workloads with more parameters, \textit{e.g.,} ResNet and Inception, are relatively more vulnerable to undervolting-related faults.
\item To prevent \behzad{CNN} accuracy loss below the voltage guardband level, we combine voltage underscaling with frequency underscaling. We experiment with a supply voltage lower than $V_{nom}$ and with operating frequency $F_{op} < F_{max}$. Our experiments \behzad{show} that the most \behzad{\emph{energy-efficient}} operating point is the one with the maximum frequency and minimum safe voltage, namely, $V_{min}$. However, lower voltage and lower frequency lead to better \behzad{\emph{power-efficiency}}.
\item We combine voltage underscaling with \behzad{the} existing CNN quantization and pruning techniques and study the power-reliability trade-off of \behzad{such} optimized FPGA-based CNN \behzad{accelerators}. We observe that these bit/parameter-size reduction techniques \behzad{(quantization and pruning)} slightly increase the vulnerability of a CNN to undervolting-related faults; but, they deliver significantly higher power-efficiency when integrated with our undervolting technique.
\item We study the effect of environmental temperature on the power-reliability trade-off of reduced-voltage FPGA-based CNN accelerators. \behzad{We} observe that temperature has a direct \behzad{effect on} the power consumption of such accelerators. However, at \behzad{very} low voltage levels, this \behzad{effect} is not noticeable.
\item We evaluate the effect of hardware platform variability by repeating \behzad{our} experiments on three identical samples of the Xilinx ZCU102 \behzad{FPGA} platform. We \behzad{find} large voltage \behzad{guardbands} in all platforms (an average of 33\%), \textit{i.e.,} $V_{min}= 0.67 * V_{nom}= 570mV$. \behzad{However, across three FPGAs, we observe a variation on $V_{min}$\textit{, i.e.,} $\Delta V_{min}=31mV$}. This variation can be due to process variation. \fff{Our results show that} the variation of \behzad{guardband} regions \behzad{across} different CNN workloads is insignificant.
\end{itemize}
\section{Background}
\label{sec:background}
In this section, we briefly introduce the most important concepts used in this paper, including the architecture of CNNs as well as the undervolting technique.
\subsection{Convolutional Neural Networks (CNNs)}
DNNs are a class of Machine Learning (ML) methods that are designed to classify unseen objects or entities using non-linear transformations applied to \behzad{input data} \cite{lecun2015deep}. DNNs are composed of biologically inspired neurons, interconnected to each other. Among different DNN models, multi-layer CNNs are a common type, which has recently shown acceptable success in \behzad{classification} tasks for real-world applications.
\subsubsection{Phases of a CNN: Training and Classification.}
\behzad{A CNN model encompasses} two \behzad{stages:} training and classification (inference). Training \behzad{learns a model from a set of training data. It} is an iterative, usually \behzad{a} single-time \behzad{(or relatively infrequently-executed) step, including} backward and forward phases. It \behzad{adjusts} the CNNs parameters, \textit{i.e.,} weights and biases, which determine the strength of the \behzad{connections} between different neurons \behzad{across} CNN layers. The training phase \behzad{minimizes} a loss function, which directly relates to \behzad{the accuracy of the neural network in the classification phase}. \behzad{In contrast,} inference is a post-training phase \behzad{that} aims to classify \behzad{unknown data, using the trained network model}. The inference phase is more \behzad{frequently executed in} edge devices with power-\behzad{constrained} environments. \behzad{The} target of this paper \behzad{is the} inference stage, similar to \behzad{many} existing efforts on the acceleration \behzad{of CNNs~\cite{guo2017survey,sze2017efficient, koppula2019eden}}.
\subsubsection{Internal Architecture of a CNN.}
\behzad{A CNN is} composed of multiple processing layers such as Convolution, Pooling, Fully-Connected, and SoftMax for feature extraction with various abstractions. Other customized layers can be used case by case for more optimized feature extraction, such as Batch \behzad{Normalization~\cite{nakahara2017batch}}. The functionality of each type of layer depends on the way \behzad{in which} the neurons are interconnected. Convolution layers generate a more profound abstraction of the input data, called a feature map. Following each Convolution layer, there is usually a Max/Avg Pooling layer to reduce the \behzad{dimensionality} of the feature map. Successive multiple Convolution and Pooling layers generate in-depth information from the input data. Afterward, Fully-Connected layers are typically applied for classification purposes. Finally, the SoftMax layer generates the class probabilities from the class scores in the output layer. Between layers, there are activation functions, such as Relu or Sigmoid, to \behzad{add} non-linear \fff{properties} to the network. \behzad{The} required computations of different layers are translated to matrix multiplication computations. Thus, \behzad{matrix} multiplication optimization techniques, such as FFT \behzad{or Strassen~\cite{lavin2016fast}}, can be applied to accelerate the \behzad{inference} implementation. Matrix multiplication is an ideal application to take advantage of parallel and data flow execution model \behzad{used in FPGA-based} hardware accelerators.
\subsubsection{Architectural Optimizations.}
To improve the power-efficiency of CNNs\behzad{, two} most \behzad{commonly-used} architectural-level techniques are quantization~\cite{zhou2017incremental} and pruning~\cite{molchanov2016pruning}.\footnote{\behzad{There are also other techniques, such as batching~\cite{shen2017escher}, loop unrolling~\cite{ma2017optimizing}, and memory compression~\cite{kim2015compression}.}} These \behzad{two} techniques rely on the \behzad{sparse nature} of CNNs, \behzad{\textit{i.e.,}} a vast majority of CNN computations are unnecessary. Quantization aims to reduce the complexity of \behzad{high-precision CNN} computation units by \behzad{substituting selected} floating-point parameters \behzad{with} low-precision fixed-point. \behzad{Pruning} aims to reduce the model size by eliminating \behzad{unnecessary} weight/neurons/connections of \behzad{a} CNN. These \fff{architectural} techniques are applicable to any underlying hardware. There are numerous extensions of quantization~\cite{zhou2017incremental,zhu2019configurable} and pruning~\cite{yazdani2018dark,he2017channel} techniques. In our experiments, we integrate typical \behzad{quantization~\cite{han2016eie} and pruning~\cite{han2015learning}} techniques with \behzad{our} proposed \behzad{hardware}-level undervolting technique to \behzad{further} improve the power-efficiency of FPGA-based CNN accelerators.
\subsection{Undervolting: Supply Voltage Underscaling Below the Nominal Voltage Level}
The total power consumption of any hardware \fff{substrate} is directly related to its supply voltage\behzad{: quadratically} and linearly with \behzad{dynamic} and static power, respectively. Thus, supply voltage underscaling toward the threshold voltage significantly reduces power consumption. Voltage underscaling is a common power-saving approach \behzad{as} manufacturing technology node \behzad{size reduces}. For instance, the $V_{nom}$ of Xilinx FPGAs is $1V$, $0.9V$, and $0.85V$ for \behzad{implementations in} 28nm, 20nm, and 16nm \behzad{technology nodes}, respectively. The aim of our undervolting technique \behzad{is to reduce the supply voltage below} the default $V_{nom}$.
However, circuit latency can increase substantially \behzad{ when supply voltage is reduced} below the guardband level, and in turn, timing faults can appear. These timing faults are manifested as bit-flips in memories or logic timing violations in \behzad{data} paths. They can potentially \behzad{cause} the application to produce wrong results, \behzad{leading to} reduced accuracy in CNNs\behzad{, or}, in the worst-case, they may cause system crashes. \behzad{There} are several approaches \behzad{to deal with undervolting faults}, such as preventing these faults by\behzad{: \textit{i)}} simultaneously decreasing the frequency~\cite{tang2019impact}\behzad{, which has an associated} performance degradation \behzad{cost}, \fff{\textit{ii)}} \behzad{fixing} the faults by using fault mitigation techniques, such as Error Correction Codes (ECCs) for memories~\cite{bacha2014using,salami2019evaluating} and Razor shadow latches for data paths~\cite{ernst2003razor}\behzad{, which comes at} the cost of extra hardware, or \behzad{\textit{iii)}} architectural improvements, such as additional iterations in CNN \behzad{training~\cite{zhang2018analyzing} that may incur \fff{hardware and/or software} adaptation costs}.
\behzad{There} are two approaches \behzad{to} undervolting studies\behzad{: \textit{i)}} simulation-based \behzad{studies~\cite{roelke2017pre,yalcin2016exploring,zhang2018thundervolt, swaminathan2017bravo}}, or \behzad{\textit{ii)}} direct implementation \behzad{or testing} on real hardware fabrics, mainly performed on CPUs, GPUs, ASICs, and \behzad{DRAMs~\cite{zou2018voltage,bacha2014using,papadimitriou2019adaptive, chang2017understanding, 8416495,koppula2019eden}}. The simulation-based approach requires less engineering \behzad{effort}. However, \fff{validation of} \behzad{simulation} results on real hardware is the primary concern \behzad{with such an approach}. \behzad{In contrast}, \behzad{the real hardware} \fff{evaluation} approach \fff{requires} substantial engineering \behzad{effort}, and it \behzad{is device-} and vendor-dependent. Such \behzad{a real hardware approach} leads to exact experimental results \behzad{and} it provides an opportunity to \behzad{study} device-dependent parameters, such as voltage guardbands and real power and reliability behavior of underlying hardware. In this paper, we follow the \behzad{real hardware} approach by evaluating \behzad{undervolting} on real modern off-the-shelf FPGA devices for state-of-the-art CNN workloads and benchmarks.
\section{Experimental Methodology}
\label{sec:methodology}
Figure~\ref{fig:overall} \behzad{depicts} the overall \behzad{methodological} flow of our experiments. In this section, we elaborate on its different aspects, including \behzad{our} implementation methodology, benchmarks, and \behzad{undervolting} methodology of \behzad{our} FPGA platform.
\begin{figure}[H]
\centering
\includegraphics[width=\linewidth]{Figures/overall.pdf}
\caption{Our overall methodology (for simplicity, we show a simplified block diagram of Xilinx DNNDK
\label{fig:overall}
\end{figure}
\begin{table*}[]
\caption{Evaluated CNN Benchmarks.}
\label{table:benchmarks}
\begin{tabular}{|l||l|r|r||r|r||r|r|}
\hline
\multicolumn{1}{|c||}{\textbf{CNN}} & \multicolumn{3}{c||}{\textbf{Dataset}} & \multicolumn{2}{c||}{\textbf{Parameters}} & \multicolumn{2}{c|}{\textbf{Inference Accuracy (\%)}} \\ \cline{2-8}
\multicolumn{1}{|c||}{\textbf{Model}} & \textbf{Name} & \multicolumn{1}{l|}{\textbf{Inputs}} & \multicolumn{1}{l||}{\textbf{Outputs}} & \multicolumn{1}{l|}{\textbf{\#Layers}} & \multicolumn{1}{l||}{\textbf{Size}} & \multicolumn{1}{l|}{\textbf{Literature}} & \multicolumn{1}{l|}{\textbf{Our \behzad{design} @Vnom}} \\ \hline \hline
\textbf{VGGNet} & Cifar-10 & 32*32 & 10 & 6& 8.7MB & 87\% \cite{miniV} & 86\% \\ \hline
\textbf{GoogleNet} & Cifar-10 & 32*32 & 10 & 21 & 6.6MB & 91\% \cite{miniG} & 91\% \\ \hline
\textbf{AlexNet} & Kaggle Dogs vs. Cats & 227*227 & 2 & 8 & 233.2MB & 96\% \cite{alex} & 92.5\% \\ \hline
\textbf{ResNet50} & ILSVRC2012 & 224*224 & 1000 & 50& 102.5MB & 76\% \cite{he2016deep} & 68.8\% \\ \hline
\textbf{Inception} & ILSVRC2012 & 224*224 & 1000 & 22& 107.3MB & 68.7\% \cite{miniG} & 65.1\% \\ \hline
\end{tabular}
\end{table*}
\subsection{CNN Model Development Platform}
\textcolor{black}{For \behzad{our} implementation, we leverage \fff{the} Deep Neural Network Development Kit (DNNDK)~\cite{dnndk}, a CNN framework from Xilinx. DNNDK is an integrated framework to facilitate CNN development and deployment on Deep learning \behzad{Processing} Units (DPUs). \behzad{In} this paper, we use DNNDK as \behzad{it is} a freely-available framework instead of a specialized custom design, to ensure that the results reported in this paper are reproducible and general-enough for state-of-the-art CNN implementations. Although we do not expect a significant difference by experimenting on DNNDK versus other DNN platforms, our future plan is to verify this by repeating the experiments on other platforms, such as DNNWeaver~\cite{sharma2016high}.} DNNDK provides a complete set of toolchains with compression, compilation, deployment, and profiling, for the mapping of CNN classification phases onto FPGAs integrated with hard CPU cores via a comprehensive and easy-to-use C/C++ programming interface.
Among the components of DNNDK, the DEep ComprEssioN Tool (DECENT) is responsible for quantization and pruning tasks. The quantization utility of \behzad{DECENT} can convert \behzad{a \fff{floating-point} CNN} model to a quantized model with the precision of at most INT8~\cite{han2016eie}. \behzad{The} pruning utility aims to minimize the model size by removing unnecessary connections of the CNN
\cite{han2015learning}. \behzad{We} perform our baseline evaluation on a model with INT8 precision and without any pruning optimization. However, in Section~\ref{subsec:architectural}, \behzad{we evaluate different configurations to provide} a more comprehensive analysis.
There are different sizes of soft DPUs provided by DNNDK with various hardware utilization \behzad{rates}~\cite{DPU}. Among them, B4096 is the \behzad{largest} model that utilizes a maximum \behzad{fraction} of BRAMs and DSPs, \textit{i.e.,} 24.3\% and 25.6\%, respectively, \behzad{resulting} in a peak performance of 4096 operations/cycle with a default DPU frequency of 333Mhz and DSP frequency of 666Mhz. In total, a maximum of three B4096 DPUs can be \behzad{used} in the hardware platform evaluated in this paper. Our experiments are based on the B4096 configuration to achieve peak performance.
\subsection{CNN Benchmarks}
We evaluate undervolting in FPGA-based CNN accelerators with \behzad{five} commonly-used image classification \behzad{benchmarks, shown in Table \ref{table:benchmarks}:} VGGNet \cite{miniV}, GoogleNet \cite{miniG}, AlexNet \cite{alex}, ResNet \cite{he2016deep}, and Inception \cite{miniG}. \behzad{To perform} a comprehensive analysis and \fff{study} workload-to-workload variation \fff{better,} we choose models \behzad{whose parameter sizes vary from} a few MBs, \textit{e.g.,} GoogleNet, to hundreds of MBs, \textit{e.g.,} AlexNet. \behzad{Our} benchmarks have \behzad{different numbers and types} of layers, as \behzad{shown} in Table~\ref{table:benchmarks}. \behzad{The} default activation function used \behzad{in} benchmarks is Relu.
\subsection{Undervolting}
In this section, we briefly explain the prototype FPGA platform and the associated voltage \behzad{control} setup.
\subsubsection{Prototype FPGA Platform.}
Our prototype is based on the Xilinx ZCU102 \behzad{FPGA} platform fabricated at a 16nm technology node. We choose this platform because \behzad{it} is \behzad{\textit{i)}} equipped \behzad{with} voltage underscaling capability\behzad{, \textit{ii)}} supported by DNNDK. We repeat experiments on three identical samples of ZCU102 to \behzad{study} the effect of \behzad{hardware} platform variability. ZCU102 is populated with the Zynq UltraScale+ XCZU9EG-2FFVB1156E MPSoC that combines a Processing System (PS) and user-Programmable Logic (PL) \behzad{in} the same device. The PS part features a quad-core \behzad{64-bit} \fff{ARM} Cortex-A53 and is mainly used for the host communication in DNNDK. The PL \behzad{part} has 32.1Mbit of BRAMs, 600K LUTs, and 2520 DSPs. For the CNN implementation, DPUs are mapped into the PL \behzad{side}. As mentioned earlier, \behzad{our} baseline hardware configuration employs three B4096 DPUs, the maximum possible number, leading to a maximum utilization \behzad{fraction} of more than 75\% for BRAMs and DSPs. \behzad{ZCU102} is equipped with an 8GB 64-bit DDR-4 off-chip memory. In our implementation, this memory contains input images and CNN parameters. \behzad{It is also} used for interfacing purposes with the host.
\subsubsection{Undervolting Methodology.}
Unfortunately, there is \behzad{no} voltage scaling standard for FPGAs. Different vendors have their unique voltage management methodologies. Moreover, there are some platforms without voltage scaling capability, such as \behzad{the} Xilinx Zedboard~\cite{zedboard}. Even a single vendor's different devices do not necessarily have the same voltage distribution model. \textcolor{black}{Although this \fff{non-standard} approach of vendors adds some constraints to experimental studies, such as the one conducted in this paper, we believe that, with minor changes, the methodology we explain below for ZCU102 can be applicable to other platforms, as, for instance, we previously studied for on-chip memories of older FPGA generations \cite{salami2018comprehensive}.}
Figure~\ref{fig:overall-voltage}, \behzad{adapted from \cite{zcu102},} depicts the voltage distribution model of ZCU102. \behzad{Here}, the voltage scaling capability is provided using an \behzad{on-board} voltage regulator that can convert an input voltage level of $12V$ into different voltage levels. The voltage level of the output lines, usually called voltage rails, \behzad{is} fully configurable and also addressable using the Power Management Bus (PMBus) \behzad{standard~\cite{pmbus}}. Each voltage rail feeds one or more components of the FPGA platform. ZCU102 is equipped with three voltage regulators, which in total provide 26 voltage rails accessible through the PMBus. In this paper, we focus on on-chip voltage \behzad{rails:} $V_{CCINT}$ and $V_{CCBRAM}$, as shown in \behzad{Figure~\ref{fig:overall-voltage}}. $V_{CCINT}$ is accessible with PMBus address 0x13 and $V_{nom}= 850mV$; it supplies multiple PL components\behzad{, including} DSPs, LUTs, buffers, and routing resources. $V_{CCBRAM}$ is accessible with PMBus \behzad{address} 0x14 and $V_{nom}= 850mV$; it supplies the BRAMs of the PL. \behzad{To} access these voltage rails for monitoring and regulation, we use a PMBus adapter and the provided API~\cite{maxtool}. Using a similar approach and different PMBus commands, we monitor the power consumption of each voltage rail as well as the on-chip temperature.
\begin{figure}[H]
\centering
\includegraphics[width=\linewidth]{Figures/voltage.pdf}
\caption{Voltage distribution on the Xilinx ZCU102 FPGA, adapted from~\protect\cite{zcu102}.}
\label{fig:overall-voltage}
\end{figure}
\section{Experimental Results}
\label{sec:results}
\behzad{We} present and \behzad{analyze our} experimental results \behzad{from reduced-voltage operation on FPGA boards}.
\behzad{These results are collected} at ambient \behzad{temperature.} Section~\ref{sec:temperature} presents further temperature analysis. \behzad{Each} result presented in this paper is the average of 10 experiments\fff{, in} order to account for any variation between different experiments; although, the variation \behzad{we} observed was negligible.
\subsection{Power Analysis of FPGA-based CNN Accelerators at the Nominal Voltage Level ($V_{nom}$)}
We measure the total on-chip power consumption of the baseline configuration to be \behzad{an average of $12.59W$ for benchmarks}, at \behzad{the} nominal voltage \behzad{level ($V_{nom}$)} and ambient temperature. \behzad{This value} includes the power consumption at on-chip voltage rails, including $V_{CCBRAM}$ and $V_{CCINT}$. We observe that internal FPGA components on \behzad{the $V_{CCINT}$ rail dissipate more than 99.9\% of} this on-chip power. \behzad{We believe} this observation \behzad{is due to} power-efficient BRAM \behzad{designs, using techniques like dynamic power gating~\cite{bramU}}, in modern Ultrascale+ FPGA platforms, including in the studied \behzad{ZCU102 FPGA}. Older generations of Xilinx FPGAs like \behzad{the} 7-series are not equipped with this capability \cite{bram7}\behzad{. Thus}, for \behzad{such} older devices, BRAM power consumption was the main source of \behzad{FPGA} power consumption, \behzad{as shown} in previous studies~\cite{salami2018comprehensive, salami2018fault, salami2019evaluating, ahmed2018automatic}. \behzad{For} the rest of the paper, as we study the power-reliability trade-off, we concentrate on $V_{CCINT}$ \behzad{due to its dominance in FPGA power consumption}.
\subsection{Overall Voltage Behavior}
\behzad{Our experiments reveal} that a \behzad{large} voltage guardband below $V_{nom}$ exists for $V_{CCINT}$, \behzad{as shown in Figure~\ref{subfig:guardband}} for three hardware platforms and five CNN benchmarks.
In the voltage guardband region, \behzad{as we reduce supply voltage} there is no performance or reliability degradation, and thus, \behzad{under} normal conditions, eliminating this \behzad{voltage guardband} can lead to significant \behzad{power savings} without any overhead. As Figure~\ref{subfig:guardband} shows, we measure the \behzad{average} guardband \behzad{amount to} be $850mV-570mV= 280mV$, with a slight variation \behzad{across} different \behzad{benchmarks}. In other words, we observe that $V_{min}= 570mV$ (on average) is the minimum safe voltage level of the accelerator, where there is no accuracy \behzad{loss.} \behzad{As we} further \behzad{undervolt} below $V_{min}$, we enter a region called \behzad{the} \textit{critical region} in which the reliability of the hardware and, in turn, the accuracy of the CNN starts to decrease significantly. As Figure~\ref{subfig:guardband} depicts, we measure the \behzad{average} critical voltage region \behzad{size, to be} $570mV-540mv= 30mV$, with a slight variation \behzad{across} different benchmarks. \behzad{As we further undervolt} below $V_{min}$, we reach a \behzad{point at} which the FPGA does not respond to \behzad{requests} and it is not \behzad{functional. This} point is called $V_{crash}$. We \behzad{find that} $V_{crash}= 540mV$ on average, with a slight variation \fff{across different hardware platforms.}
\begin{figure}[H]
\centering
\includegraphics[width=\linewidth]{Figures/guardband.png}
\caption{Voltage regions with a slight workload-to-workload variation (averaged across three hardware platforms).}
\label{subfig:guardband}
\end{figure}
\behzad{Figure~\ref{subfig:overallTradeoff} illustrates} the overall behavior \behzad{we} observe for \behzad{the} power-efficiency and CNN accuracy trade-off on our FPGA-based CNN accelerator. As we perform undervolting, \behzad{the FPGA enters} \fff{the} guardband region\behzad{, where} we observe no reliability degradation (\textit{i.e.,} CNN accuracy loss), and therefore, the power-efficiency comes with no cost. We observe this behavior until we reach the point $V_{min}$, \behzad{\textit{i.e.,}} minimum safe voltage level. With further undervolting, \fff{the FPGA enters} the critical \behzad{region, where} power-efficiency constantly increases\behzad{, but we} start to observe \behzad{fast}-increasing \behzad{CNN} accuracy loss. \behzad{When} we undervolt \behzad{down} to a specific point, called $V_{crash}$, the FPGA becomes non-functional and \behzad{starts to hang}. Sections~\ref{subsec:power} and \ref{subsec:resilience} provide more \fff{details on} the power-reliability trade-off. \textcolor{black}{\behzad{Our} demonstration is on three identical samples of Xilinx ZCU102. However, we believe that the overall voltage behavior, illustrated in Figure~\ref{subfig:overallTradeoff}, is reproducible for other FPGA platforms \behzad{as well}.
\begin{figure}[H]
\centering
\includegraphics[width=\linewidth]{Figures/overallTradeoff.png}
\caption{Overall voltage behavior observed for $V_{CCINT}$.}
\label{subfig:overallTradeoff}
\end{figure}
\subsection{Detailed Power-Efficiency Analysis}
\label{subsec:power}
Figure~\ref{fig:gops} presents the power-efficiency experimental results ($GOPs/W$) for five CNN workloads, averaged \behzad{across} three FPGA hardware platforms.
The power-efficiency gain at $V_{crash}$ is more than 3X \behzad{of that at} nominal voltage level, \textit{i.e.,} $V_{nom}$, for the same design of the given CNN accelerator. 2.6X of the gain in power-efficiency is the result of eliminating the voltage guardband without any \behzad{CNN} accuracy loss. \behzad{43\% further} power-efficiency \behzad{gain} is due to further undervolting in the critical \fff{region,} which has an associated CNN accuracy loss cost.
\begin{figure}[H]
\centering
\includegraphics[width=\linewidth]{Figures/gops.png}
\caption{Power-efficiency ($GOPs/W$) improvement via undervolting (averaged across three hardware platforms).}
\label{fig:gops}
\end{figure}
\begin{figure*}[hb]
\centering
\subfloat[VGGNet.]{\includegraphics[width=0.2\textwidth]{Figures/pv-1.png}\label{subfig:VGGNet}}
\subfloat[GoogleNet.]{\includegraphics[width=0.2\textwidth]{Figures/pv-2.png}\label{subfig:GoogleNet}}
\subfloat[AlexNet.]{\includegraphics[width=0.2\textwidth]{Figures/pv-3.png}\label{subfig:AlexNet}}
\subfloat[ResNet.]{\includegraphics[width=0.2\textwidth]{Figures/pv-4.png}\label{subfig:ResNet}}
\subfloat[Inception.]{\includegraphics[width=0.2\textwidth]{Figures/pv-5.png}\label{subfig:Inception}}
\caption{Effect of reduced supply voltage on the accuracy of CNN workloads (separately for three hardware platforms).}
($V_{nom}:\CIRCLE, V_{min}:\blacklozenge, V_{crash}:\bigstar)$
\label{fig:resilience}
\end{figure*}
\behzad{The} power-efficiency \behzad{gain via} undervolting \fff{until $V_{min}$} is not application-dependent, so it is useful for any application mapped onto the same FPGA. However, the reliability overhead in the critical \behzad{region} \fff{below $V_{min}$} is application-dependent due to different vulnerability levels of different applications/workloads.
\subsection{Detailed Reliability Analysis}
\label{subsec:resilience}
\textcolor{black}{As \behzad{we undervolt} until $V_{min}$, there is no reliability overhead. However, \behzad{as we further undervolt} below $V_{min}$, the reliability of the hardware is significantly affected due to the further increase \behzad{in} datapath delay. The effect of the reliability loss is fully application-dependent due to different \behzad{inherent} resilience levels of different applications. In this paper, we study this effect on several CNN workloads. Figure~\ref{fig:resilience} depicts \behzad{our} experimental results.} As \behzad{shown before}, as we reduce the supply voltage, \behzad{power}-efficiency improves. \behzad{When} we reduce the supply voltage below $V_{min}$, we observe that the accuracy of all benchmarks gradually \behzad{reduces}. With further undervolting, when the supply voltage reaches \behzad{an} average of $V_{crash}=540mV$ \behzad{across} different platforms and benchmarks, the accuracy of the benchmarks drops \behzad{greatly}, and the classifier behaves randomly. Our experiments show that benchmarks with more parameters, \textit{e.g.,} ResNet and Inception are relatively more vulnerable to undervolting faults below $V_{min}$. Also, as seen, there is a variation \behzad{of} $\Delta V_{min}=31mV$ and $\Delta V_{crash}=18mV$ \behzad{across} different \behzad{FPGAs}. This variation can be due to the process variation \behzad{across} different \behzad{FPGAs}.
\section{Frequency Underscaling}
\label{sec:mitigation}
As \behzad{shown} earlier, in the critical voltage \behzad{region} below the guardband, \behzad{CNN} classification accuracy dramatically decreases. In this section, we aim to overcome this \behzad{accuracy loss} by exploiting frequency underscaling. To be more precise, we aim to find a more energy-efficient \behzad{voltage} setting than the undervolted $V_{min}$\behzad{,} which also provides accurate results. \behzad{To this end}, for each supply \behzad{voltage} setting below \behzad{$V_{min}$,} we aim to identify the maximum frequency value $F_{max}$ with which the system does not experience any accuracy loss. When we find this frequency point, we \behzad{evaluate} the energy efficiency of the system. As we underscale the frequency of the system, the performance of the application \behzad{reduces}. Therefore, we \behzad{use} the $GOPs/J$ metric as it accommodates for both performance and energy consumption.
Table~\ref{table:frequency} summarizes the results of the frequency underscaling in the critical region. \behzad{These} experiments are based on frequency and voltage steps of $25Mhz$ and $5mV$, respectively. The column $V_{CCINT}$ corresponds to the supply voltage of \behzad{a given} setting. The column $F_{max}$ corresponds to the maximum frequency \behzad{at which \fff{there is} no accuracy loss}. The remaining columns: $GOPs$, $Power$, \behzad{$GOPS/W$,} $GOPS/J$ are normalized to the respective values of executing the system in the default setting \behzad{$V_{CCINT}=V_{min}=570mV, F_{max}=333Mhz$} which are the baseline settings \fff{of our accelerator}. Table~\ref{table:frequency} indicates that multiple voltage settings $V_{CCINT}$ map to the same operating Frequency \behzad{$F_{max}$: supply} voltages between $560mV$ to $545mV$ \behzad{require \fff{the} same} frequency of $F_{max}=250Mhz$. \behzad{This is} because the frequency step we use \behzad{is $25Mhz$}. Using smaller steps of frequency can \fff{lead to more spread-out} $F_{max}$ values.
\begin{table}[H]
\centering
\caption{Evaluation of frequency underscaling to prevent CNN accuracy loss in the critical voltage region (averaged across three hardware platforms). Best result with frequency underscaling in terms of each metric is marked in blue.}
\label{table:frequency}
\begin{tabular}{c|r||rrrr|}
\hline
\multicolumn{1}{|c}{\textbf{\begin{tabular}[c]{@{}c@{}}\small{$V_{CCINT}$}\\ \small{(mV)}\end{tabular}}} & \multicolumn{1}{c||}{\textbf{\begin{tabular}[c]{@{}c@{}}\small{$F_{max}$}\\ \small{(Mhz)}\end{tabular}}} & \multicolumn{1}{c}{\textbf{\begin{tabular}[c]{@{}c@{}}\small{$GOPs$}\\ \small{(Norm)}\end{tabular}}} & \multicolumn{1}{c}{\textbf{\begin{tabular}[c]{@{}c@{}}\small{$Power(W)$}\\ \small{(Norm)}\end{tabular}}} & \multicolumn{1}{c}{\textbf{\begin{tabular}[c]{@{}c@{}}\behzad{\small{$GOPs/W$}}\\ \behzad{\small{(Norm)}}\end{tabular}}} &
\multicolumn{1}{c|}{\textbf{\begin{tabular}[c]{@{}c@{}}\small{$GOPs/J$}\\ \small{(Norm)}\end{tabular}}} \\ \hline \hline
\multicolumn{1}{|c}{\textbf{570}} & 333 & \textcolor{blue}{\textbf{1.00}} & 1.00 & \behzad{1.00} & \textcolor{blue}{\textbf{1.00}} \\
\multicolumn{1}{|c}{\textbf{565}} & 300 & 0.94 & 0.97 & \behzad{0.97} & 0.87 \\
\multicolumn{1}{|c}{\textbf{560}} & 250 & 0.83 & 0.84 & \behzad{0.99} & 0.75 \\
\multicolumn{1}{|c}{\textbf{555}} & 250 & 0.83 & 0.78 & \behzad{1.06} & 0.80 \\
\multicolumn{1}{|c}{\textbf{550}} & 250 & 0.83 & 0.75 & \behzad{1.10} & 0.83 \\
\multicolumn{1}{|c}{\textbf{545}} & 250 & 0.83 & 0.74 & \behzad{1.12} & 0.84 \\
\multicolumn{1}{|c}{\textbf{540}} & 200 & 0.70 & \textcolor{blue}{\textbf{0.56}} & \textcolor{blue}{\textbf{1.25}} & 0.75 \\ \hline
\end{tabular}
\end{table}
\behzad{For} all the combinations of $(V_{i}, F_{i})$ \behzad{that} provide error-free results presented \fff{in} Table~\ref{table:frequency} in the critical region, \behzad{power} decreases with \behzad{decreasing} $V_{i} < V_{min}$ and $F_i < F_{max}$. \fff{This is} because we decrease both the supply voltage and the operating frequency. However, at the same time, this leads to decreasing the system performance. Consequently, the best voltage-frequency combination in terms of \emph{energy-efficiency} \fff{($GOPs/J$)} is the one with the highest frequency of $F_{max} = 333Mhz$, which also is our baseline. In other words, it is not worth to underscale the frequency and voltage to find a more energy-efficient optimal point. However, as a trade-off, the design is more \emph{power-efficient} \fff{(\textit{i.e.,} has higher $GOPs/W$) at lower voltage-frequency levels\fff{, up to 25\% at $V_{crash}=540mV$.}}
\section{Combining Undervolting with Architectural CNN Optimization Techniques}
\label{sec:qp}
\label{subsec:architectural}
In this section, we experimentally evaluate undervolting for \behzad{employing the CNN's} quantization and pruning techniques. \behzad{Via} experiments, we observe that \behzad{these} bit reduction techniques can deliver additional power-\behzad{efficiency} gains proportional to the quantization/pruning level. However, applying these techniques can slightly increase the vulnerability of CNNs to undervolting-related faults. \behzad{This section \fff{reports} results \fff{for} VGGNet as we observe similar results for other workloads.}
\subsection{Quantization}
\behzad{Our} baseline is optimized with INT8 \behzad{precision}. As shown in Table~\ref{table:benchmarks}, this precision does not incur any significant accuracy loss in comparison to baseline models \behzad{that use} floating-point precision. For further analysis of the \behzad{effect} of undervolting with lower precision models, we evaluate INT7, INT6, INT5, and INT4 precisions. \behzad{Using} DNNDK, we observe significant \behzad{accuracy} loss for INT3, INT2, and INT1 when executed at $V_{nom}$\behzad{. Thus}, we do not present them in this paper.
Figure~\ref{fig:quantization} shows results of different precisions \behzad{(INT8 to INT4)}. \behzad{We find \fff{that} \textit{i)} when operating at reduced-voltage levels, accuracy loss is relatively high due to lower precision; \textit{ii)} power-efficiency is proportional \fff{to} voltage as well as \fff{quantization} levels.} In conclusion, combining low-\behzad{precision} and low-voltage \behzad{operation} can significantly deliver higher power-efficiency. However, it comes at the cost of accuracy loss.
\begin{figure}[H]
\centering
\subfloat[\behzad{CNN} Accuracy.]{\includegraphics[width=0.25\textwidth]{Figures/qu-accuracy.png}\label{subfig:quantizatonA}}
\subfloat[Power-efficiency ($GOPs/W$).]{\includegraphics[width=0.25\textwidth]{Figures/qu-gops.png}\label{subfig:quantizationP}}
\caption{Effect of undervolting at different quantization levels for VGGNet (averaged across three hardware platforms).}
\label{fig:quantization}
\end{figure}
\subsection{Pruning}
Figure~\ref{fig:pruning} shows results \behzad{of pruned and baseline (without any pruning) models}. \behzad{We find that}
\behzad{undervolting}-related faults have a relatively more significant effect on the pruned model. However, \fff{this} comes with \behzad{higher power}-efficiency of the pruned model, as shown in Figure~\ref{subfig:pruningP}, due \behzad{to fewer} operations \behzad{in} the pruned model. \behzad{With} undervolting, \behzad{power} consumption reduces for both pruned and baseline models, \behzad{at} a similar rate. $V_{crash}$ is different for the pruned model. \behzad{Specifically}, the pruned version demonstrates a higher $V_{crash}$ \behzad{voltage} equal to $555mV$ in contrast to the baseline \behzad{$V_{crash}$} of $540mV$.
\begin{figure}[H]
\centering
\subfloat[\behzad{CNN} Accuracy.]{\includegraphics[width=0.25\textwidth]{Figures/pru-acuuracy.png}\label{subfig:pruningA}}
\subfloat[Power-efficiency ($GOPs/W$).]{\includegraphics[width=0.25\textwidth]{Figures/pru-gops.png}\label{subfig:pruningP}}
\caption{Effect of undervolting on pruned CNN models for VGGNet (averaged across three hardware platforms).}
\label{fig:pruning}
\end{figure}
\section{Effect of Environmental Temperature}
\label{sec:temperature}
The power consumption of a modern chip\fff{, including FPGAs,} also depends \fff{on} temperature. \behzad{Temperature} affects \behzad{static} power consumption. As the external temperature increases, the leakage current and, in turn, the leakage-induced static power increases~\cite{Borkar1999,Kaul2009,Kim2003,Huang2011}. \behzad{As technology node size reduces, a large \fff{fraction} of power consumption comes from the static power}. \behzad{Therefore,} temperature has a larger \behzad{effect} on the power consumption of \behzad{denser chips}~\cite{moradi2014side}. On the other \behzad{hand}, \behzad{temperature} can have a considerable \behzad{effect} on \behzad{circuit} latency~\cite{neshatpour2018enhancing,mottaghi2019aging}, \textit{i.e.,}, circuit latency \behzad{\emph{decreases}} as the temperature increases \behzad{in} contemporary technology nodes. \behzad{Therefore,} there are \behzad{fewer} undervolting\behzad{-related} faults at higher temperatures.
\behzad{To understand the combination of multiple effects mentioned above, we} study the effect of the environmental temperature \behzad{on} the power-reliability trade-off \behzad{of \fff{our} FPGA-based CNN accelerator under reduced-voltage operation}. \behzad{To this end}, we use GoogleNet as a benchmark and undervolt \behzad{$V_{CCINT}$}. We discuss the voltage behavior in both critical and guardband regions \behzad{at} different temperatures ranging from 34$^{\circ}$C to 52$^{\circ}$C degrees. \behzad{To} regulate the FPGA temperature, we control the fan speed using the PMBus interface. \behzad{We also use the same PMBus interface to monitor the on-board live temperature.} By doing so, we can test different ambient \behzad{temperatures} ranging from 34$^{\circ}$C to 52$^{\circ}$C degrees.\footnote{\behzad{[34$^{\circ}$C, 52$^{\circ}$C]} is the temperature range that we could generate using the fan speed. Experimenting \behzad{with} wider temperature ranges \behzad{requires} more facilities, which were not available \behzad{to us}.}
\subsection{Temperature Effect on Power Consumption}
\label{ssec:voltNetTempPower}
Figure~\ref{subfig:temperatureP} depicts the power consumption of \fff{our} \behzad{CNN} accelerator when executing GoogleNet \behzad{with} different $V_{CCINT}$ values \behzad{at different temperatures.} \behzad{Clearly,} temperature has a direct \behzad{effect on} power consumption. \behzad{As} temperature increases, power consumption proportionally increases. This is due to \behzad{increase in} static power when the chip heats up. \behzad{Dynamic} power consumption is \behzad{also} affected by \behzad{temperature}, but \behzad{this effect} is almost negligible. \behzad{Importantly,} we observe \behzad{that} \behzad{the effect of} temperature \behzad{on} power consumption \behzad{reduces} for lower voltages. For \behzad{example} power change from 34$^{\circ}$C to 52$^{\circ}$C \behzad{are 0.46\% and 0.15\%, respectively} at $V_{CCINT}=850mV$ and $V_{CCINT}=650mV$
\begin{figure}[H]
\centering
\includegraphics[width=\linewidth]{Figures/TempNew2.png}
\caption{Power consumption of our reduced-voltage CNN accelerator at temperature range of [34$^{\circ}$C, 52$^{\circ}$C], shown for GoogleNet (averaged across three hardware platforms).}
\label{subfig:temperatureP}
\end{figure}
\subsection{Temperature Effect on Reliability}
\behzad{Figure~\ref{subfig:temperatureA} shows} the \behzad{effect} \behzad{of temperature on} the accuracy of \behzad{our reduced-voltage CNN} accelerator. Our experiment demonstrates that \behzad{\textit{i)}} there is no \fff{noticeable} change in the \behzad{size} of \fff{the} guardband and critical regions, and \behzad{\textit{ii)}}
\behzad{higher} temperature at a particular voltage level \behzad{leads to higher} CNN accuracy. \behzad{This is because} at higher temperatures, there are \behzad{fewer} undervolting related errors due to decreased circuit latency, an artifact due \behzad{to} \fff{the} Inverse Thermal Dependence \behzad{(ITD) property} \fff{of} contemporary technology nodes~\cite{neshatpour2018enhancing,Uht2004}.
\begin{figure}[H]
\centering
\includegraphics[width=\linewidth]{Figures/TempNew1.png}
\caption{Accuracy of our reduced-voltage CNN accelerator at temperature range of [34$^{\circ}$C, 52$^{\circ}$C], shown for GoogleNet (averaged across three hardware platforms).}
\label{subfig:temperatureA}
\end{figure}
\subsection{Discussion}
In our setup, \behzad{considering the power-reliability trade-off discussed,} the optimal setting is at Temp=50$^{\circ}$ and $V_{CCINT}=565mV$, \behzad{\textit{i.e.,}} the minimum voltage level at which there is almost no accuracy loss due to the healing effect of high temperature. \behzad{However}, the disadvantage of operating at higher temperatures \behzad{is} the overall decrease in lifetime reliability. Below, we summarize our findings \behzad{on} temperature effects.
\begin{itemize}
\item There is a negligible change in the value of $V_{min}=570mV$ \behzad{across temperatures}, and thus, there is no significant change \behzad{in the} guardband region. However, the system crashes relatively earlier over temperature variation. We expect, though, that when the system \behzad{undergoes a} wider temperature \behzad{range,} there will be \behzad{a more noticeable change} in the $V_{min}$ and $V_{crash}$.
\item At any specific voltage \behzad{point} in \fff{either region}, \behzad{power} consumption directly increases as temperature increases, mainly due to the direct relation \behzad{of static power consumption} and temperature.
\item The \behzad{effect of temperature on power} consumption is significantly less at lower voltage levels, due to the relatively lower contribution \behzad{of static} power \behzad{to total power consumption}.
\item In the critical voltage region and at any specific voltage level, higher temperature leads to \behzad{higher CNN accuracy}. The power cost of the higher temperature in the critical voltage region is \behzad{relatively low}.
\end{itemize}
Consequently, a lower voltage can be applied at higher temperatures without causing significant accuracy loss \fff{at a small power cost}
\section{Related Work}
\label{sec:related}
\behzad{To our knowledge, this paper provides the first study evaluating the effect of reduced-voltage operation in FPGA-based CNN accelerators. In this section, we review related works \fff{on} \textit{i)} undervolting, \textit{ii)} power-efficient CNNs, and \textit{iii)} reliability of CNNs.}
\subsection{Undervolting}
Supply voltage underscaling below the nominal level is \fff{an} effective approach to improve the power-efficiency \behzad{of digital} circuits. There are two different approaches \behzad{to studying undervolting: simulation or real \fff{experiments}}.
\subsubsection{Simulation Studies.}
\behzad{This approach simulates hardware to study undervolting. It is convenient for early-stage studies as \behzad{it} does not require large engineering \fff{effort}. However, this approach lacks the information of real hardware, and thus, \behzad{validation of} results \behzad{is} the main concern.} Most of the existing simulation-based studies are for \behzad{CPUs}~\cite{roelke2017pre,yalcin2016exploring,swaminathan2017bravo,8416495} and specifically for \behzad{CPU components} \fff{such as caches~\cite{alameldeen2010adaptive,wilkerson2008trading,wilkerson2010reducing,yalcin2014exploiting,chishti2009improving}} and branch predictors~\cite{chatzidimitriou2019assessing}. \behzad{There are also studies for} ASIC CNN accelerators~\cite{reagen2016minerva,zhang2018thundervolt,andri2017yodann}. \behzad{Following this approach, studies on FPGA}-based designs are either \behzad{fully in} simulation~\cite{mottaghi2019aging} or emulation of FPGA netlists \behzad{on simulation} frameworks~\cite{khaleghi2019fpga,salamat2019workload}.
\subsubsection{Experimental Studies on Real Hardware.}
\label{subsec:relatedUndervolting}
Evaluating \behzad{undervolting} on real hardware is another approach that has recently been considered for multiple devices~\cite{gizopoulos2019modern, george2020exceeding}. \behzad{Doing so requires} relatively more engineering effort \behzad{as well as considering} physical constraints, such \behzad{as} non-standard device- and vendor-dependent voltage distribution \behzad{models. Yet,} the results produced are accurate and can be directly used in real-world applications.
\fff{Undervolting} of real hardware \behzad{is studied for various system components,} such as CPUs~\cite{bacha2013dynamic,papadimitriou2017harnessing,papadimitriou2017voltage,kaliorakis2018statistical,bertran2014voltage}, GPUs~\cite{zou2018voltage, leng2015gpu,leng2015safe}, ASICs~\cite{chandramoorthy2019resilient,pandey2019greentpu,kim2018matic}, \behzad{DRAMs~\cite{chang2017understanding, chang2018voltron,koppula2019eden},} and \behzad{Flash disks~\cite{cai2013threshold, cai2015read, cai2017error}}. \behzad{These studies focus} on voltage guardband analysis, fault characterization, and fault mitigation. \behzad{Undervolting on real} FPGAs is not thoroughly \behzad{investigated.} \behzad{Very recent works} on \fff{FPGA} undervolting are either accompanied with \behzad{frequency} underscaling~\cite{ahmed2018automatic,shen2019fast} that can diminish \behzad{performance, or are} limited \behzad{to} BRAMs~\cite{salami2018demo,salami2018comprehensive, salami2018fault, salami2019evaluating,salami2018aggressive}. This paper\behzad{, for the first time,} extends \behzad{real FPGA undervolting studies to} multiple on-chip components of modern FPGA fabrics and evaluates it in-detail on the power-accuracy trade-off of CNN applications.
\subsection{Power-efficient CNNs}
\behzad{Many works aim to improve CNN power-efficiency by optimizing \fff{the} CNN architecture as well as the underlying hardware.} \fff{In this paper, to achieve significant power-efficiency, we combine our hardware-level FPGA undervolting technique with architectural CNN optimization techniques, including quantization and pruning.}
\subsubsection{Architectural Techniques.}
This approach aims to reduce the parameter size of a CNN. The methods of this approach are independent of the underlying hardware, and in theory, they can be applied to any hardware\behzad{, including} hardware accelerators. The most common techniques are \fff{quantization}~\cite{zhou2017incremental, han2016eie,zhu2019configurable}, \fff{pruning}~\cite{molchanov2016pruning,yazdani2018dark,han2015learning}, batching~\cite{shen2017escher}, loop unrolling~\cite{ma2017optimizing}, and memory compression~\cite{deng2018permdnn,kim2015compression}. Among \behzad{these}, \behzad{quantization and pruning} have shown significant efficiency without \fff{significantly} compromising the CNN accuracy; hence, we \behzad{focus on} them in our experiments.
\subsubsection{Hardware-level Techniques.}
\behzad{An orthogonal approach to reducing CNN power is to optimize the underlying hardware.} \fff{To} this \behzad{end}, \behzad{since} traditional processor-based architectures are power-hungry and not suitable for CNNs, exploiting a dedicated hardware is the first approach. \behzad{Further} power \behzad{savings are possible with} low-level techniques, such as \behzad{undervolting.}
\begin{itemize}
\item Hardware Accelerators: Data-flow execution models using GPUs~\cite{khorasani2018register,hill2017deftnn}, FPGAs~\cite{sharma2016high,suda2016throughput,xiao2017exploring,ma2018optimizing,li2019rnn} and ASICs~\cite{jouppi2017datacenter,andri2016yodann,chen2014diannao,wang2019bit} are more efficient choices \behzad{for CNNs than} traditional CPUs. Among \behzad{these}, FPGAs are more flexible compared to ASICs and more efficient than GPUs. \behzad{Efficient} exploitation of the underlying hardware is fundamental \behzad{for power-efficiency}, using \behzad{techniques like resource partitioning~\cite{shen2017maximizing}} and \behzad{reuse~\cite{zhang2015optimizing,riera2018computation}. Our work uses} an industrial \behzad{tool~\cite{dnndk}} that inherently exploits these techniques.
\item Undervolting: \behzad{Undervolting \fff{has been shown to provide} significant power-efficiency benefit for CNNs \fff{when applied to} SRAMs~\cite{chandramoorthy2019resilient}, DRAMs~\cite{koppula2019eden}, ASICs~\cite{moons201714, zhang2018thundervolt, yang2017sram,chandramoorthy2019resilient,kim2018matic}, and heterogeneous systems~\cite{ cristal2018legato2,cristal2018legato1,salami2020legato}}.
\end{itemize}
\subsection{Reliability of CNNs}
Although CNNs are inherently resilient to some error rate \behzad{in data or underlying hardware}, \behzad{high enough error} rates can \behzad{cause} significant accuracy \behzad{loss}. \behzad{Error sources} can be harsh environments, process manufacturing defects, undervolting, ionizing particles, \behzad{noise in data,} among others. Hence, \behzad{CNN reliability is an active research area.} \behzad{Existing} studies \behzad{are based on fault} injection or \behzad{real errors.}
\subsubsection{Simulation-based Fault Injection.}
\behzad{These studies} inject randomly-generated faults \behzad{into} CNNs, \behzad{but they do not consider undervolting}~\cite{jha2019ml,salami2018resilience,reagen2018ares,li2017understanding,leng2020asymmetric, givaki2019resilience,jha2019kayotee,liu2017fault}. This approach provides an opportunity \behzad{for} comprehensive fault characterization \behzad{of} CNNs, such as the sensitivity of different layers, different location of faults, among others. However, \behzad{these works} do not consider \behzad{faults in real hardware\fff{, which potentially can lead to inaccurate analysis}}.
\subsubsection{Faults in Real Hardware.}
In real-world applications, such as IoT, airspace, and driver-less cars, CNNs can potentially experience different \behzad{types} of faults. \behzad{Various works evaluate CNN reliability on faulty real hardware, \textit{e.g.,} soft errors~\cite{libano2018selective, libano2020understanding,trindade2019assessment,brewer2019impact} and undervolting in ASICs~\cite{li2019chip,chandramoorthy2019resilient, whatmough201714, whatmough2018dnn, lee201916}. This approach requires significant engineering effort but can result in relatively more accurate results. None of these works study CNN reliability on undervolted FPGAs.}
\section{Summary and Future Work}
\label{sec:conclusion}
In this paper, we \behzad{experimentally} evaluated the effects of supply voltage underscaling below the nominal level on real FPGA-based CNN accelerators. We \behzad{showed that we could improve} the power-efficiency of \behzad{such} accelerators by more than 3X \behzad{via} \fff{undervolting}. 2.6X of the power-efficiency improvement comes from eliminating the voltage guardband (without compromising CNN accuracy), while the remaining 43\% improvement comes from undervolting further below the guardband \behzad{(which comes \fff{with} CNN accuracy loss). \fff{We conclude that undervolting can significantly improve the power-efficiency of FPGA-based neural network accelerators.}
As future work, we aim to develop \behzad{\textit{i)}} fault mitigation techniques for low-voltage regions even when the design operates at the maximum frequency ($F_{max}$), \behzad{\textit{ii)}} dynamic voltage adjustment techniques \behzad{considering} temperature, accuracy, power consumption, and performance trade-off. \textcolor{black}{We also aim to expand our experiments in \fff{hardware}\behzad{, by} evaluating more FPGAs\behzad{, as} well as in software\behzad{, by} repeating experiments on other CNN platforms like DNNWeaver~\cite{sharma2016high}. Finally, we \behzad{believe it is promising to} study potential security issues of FPGA-based CNN accelerators under reduced supply voltage levels.}
\section*{Acknowledgments}
We thank \behzad{the} anonymous \behzad{DSN2020} reviewers for their feedback and comments, as well as Dr. Long Wang, who helped us with \behzad{shepherding}. Also, we thank Dr. Konstantinos Parasyris for his in-depth review of the first version of this paper. The work done for this paper was partially supported by a HiPEAC Collaboration Grant funded by the H2020 HiPEAC Project under grant agreement \behzad{No.} 779656. \behzad{The} research leading to these results has received funding from the European Union’s Horizon 2020 Programme under the LEGaTO Project (www.legato-project.eu), grant agreement \behzad{No.} 780681. This work is supported in part by funding from the SRC and gifts from Intel, Microsoft and VMware to Onur Mutlu.
\bibliographystyle{plain
|
1,314,259,993,581 | arxiv |
\section{Introduction}
An emerging problem in Theoretical Computer Science and Data Science is
the low-rank approximation $\bZ\bZ^T\bA$ of a matrix $\bA\in\rmn$ by means of an orthonormal
basis $\bZ\in\real^{m\times k}$ \cite{Drineas2016,TCS-060}.
The ideal low-rank approximation consists
of the left singular vectors $\bU_k$ associated with the $k$ dominant singular values
$\sigma_1(\bA)\geq \cdots \geq \sigma_k(\bA)$ of $\bA$, because the low-rank approximation error in the two-norm
is minimal and equal to the first neglected singular value, $\|(\bI-\bU_k\bU_k^T)\bA\|_2=\sigma_{k+1}(\bA)$.
Low-rank approximation $\bZ$ can be determined with subspace iteration or a Krylov space
method \cite{G2014,MM2015}, with bounds for
$\|(\bI-\proj)\bA\|_{2,F}$ that contain $\sigma_{k+1}(\bA)$ as an additive
or multiplicative factor. Effort has been put into deriving bounds that not depend on the existence of the
singular value gap $\sigma_k(\bA)-\sigma_{k+1}(\bA)>0$.
A closely related problem in numerical linear algebra is the approximation of the dominant subspace
proper \cite{Saa80,Saa11},
that is, computing an orthonormal basis $\bZ\in \real^{m\times k}$ whose space is close to the
dominant subspace $\range(\bU_k)$. Closeness here means that the sine of the largest principal angle
between the two spaces, $\|\sin{\bTheta}(\bZ,\bU_k)\|_2=\|\bZ\bZ^T-\bU_k\bU_k^T\|_2$ is small.
For the dominant subspace $\bU_k$ to be well-defined, the associated singular values
must be separated from the remaining singular values, and there must be a gap
$\sigma_k(\bA)-\sigma_{k+1}(\bA)>0$, see
\cite{Ips99b,Ste73,STS90,Wedin1972,Wedin1983,ZKny13} which are all based on the
perturbation results for invariant subspaces of Hermitian matrices \cite{DaK69,DaK70}.
The purpose of our paper, following up on \cite{DMKI16},
is to establish a clear distinction between the mathematical problems of low-rank approximation, and approximation of dominant subspaces. In particular we show that low-rank approximations are well-defined and well-conditioned,
by deriving bounds for the low-rank approximation error $(\bI-\bZ\bZ^T)\bA$
in the two-norm, Frobenius norm, and more generally, any Schatten $p$-norm.
We establish relationships between the mathematical problems of dominant subspace computation
and of low-rank approximation.
\paragraph{Overview}
After setting the notation for the singular value decomposition (Section~\ref{s_sv}),
and reviewing Schatten $p$-norms (Section~\ref{s_schatten}) and angles between subspaces (Section~\ref{s_angles}),
we highlight the main results (Section~\ref{s_high}), followed by proofs for low-rank approximations
(Section~\ref{s_sensla}) and subspace angles (Section~\ref{s_bounds}, Appendix~\ref{s_app}).
\subsection{Singular Value Decomposition (SVD)}\label{s_sv}
Let the non-zero matrix $\bA\in\rmn$ have a full SVD $\bA=\bU\bSigma\bV^T$, where
$\bU\in\rmm$ and $\bV\in\rnn$ are orthogonal matrices, i.e.\footnote{The superscript~$T$
denotes the transpose.}
$$\bU\bU^T=\bU^T\bU=\bI_m, \qquad \bV\bV^T=\bV^T\bV=\bI_n,$$
and $\bSigma\in\rmn$ a diagonal matrix with diagonal elements
\begin{eqnarray}\label{e_svo}
\|\bA\|_2=\sigma_1(\bA)\geq \dots\geq\sigma_r(\bA)\geq 0,\qquad r\equiv\min\{m,n\}.
\end{eqnarray}
For $1\leq k\leq \rank(\bA)$, the respective leading $k$ columns of $\bU$ and $\bV$ are
$\bU_k\in\real^{m\times k}$ and $\bV_k\in\real^{m\times k}$. They
are orthonormal, $\bU_k^T\bU_k=\bI_k=\bV_k^T\bV_k$, and
are associated with the $k$ dominant singular values
$$\bSigma_k\equiv\diag\begin{pmatrix}\sigma_1(\bA) & \cdots & \sigma_k(\bA)\end{pmatrix}\in\real^{k\times k}.$$
Then
\begin{eqnarray}\label{e_bestrank}
\bA_k\ \equiv \ \bU_k\bSigma_k\bV_k^T \ = \ \bU_k\bU_k^T\bA
\end{eqnarray}
is a best rank-$k$ approximation of $\bA$, and satisfies in the two norm and in the Frobenius norm, respectively,
$$\|(\bI-\bU_k\bU_k^T)\bA\|_{2,F}\ =\ \|\bA-\bA_k\|_{2,F} \ = \ \min_{\rank(\bB)=k}{\|\bA-\bB\|_{2,F}}.$$
\paragraph{Projectors}
We construct orthogonal projectors to capture the target space, which is a dominant subspace of $\bA$.
\begin{definition}\label{d_proj}
A matrix $\proj\in\rmm$ is an {\rm orthogonal projector}, if it is idempotent and symmetric,
\begin{eqnarray}\label{e_dproj}
\proj^2\ =\ \proj \ = \ \proj^T.
\end{eqnarray}
\end{definition}
For $1\leq k\leq \rank(\bA)$,
the matrix $\bU_k\bU_k^T=\bA_k\bA_k^{\dagger}$ is the orthogonal projector onto the
$k$-dimensional dominant subspace $\range(\bU_k)=\range(\bA_k)$.
Here the pseudo inverse is $\bA_k^{\dagger}=\bV_k \bSigma_k^{-1}\bU_k^T$.
\subsection{Schatten $p$-norms}\label{s_schatten}
These are norms defined on the singular values of real and complex matrices, and thus special cases of symmetric
gauge functions.
We briefly review their properties, based on \cite[Chapter IV]{Bhatia1997} and \cite[Sections 3.4-3.5]{HoJ91}.
\begin{definition}\label{d_norms}
For integers $p\geq 1$, the {\rm Schatten $p$ norms} on $\rmn$ are
$$\spn{\bA}\ \equiv \ \sqrt[p]{\sigma_1(\bA)^p+\cdots+\sigma_r(\bA)^p}, \qquad
r\equiv\min\{m,n\}.$$
\end{definition}
\paragraph{Popular Schatten norms:}
\begin{description}
\item[$\qquad p=1:$\ ] Nuclear (trace) norm
$\ \|\bA\|_* \ =\ \sum_{j=1}^r{\sigma_j(\bA)}\ = \son{\bA}$.
\item[$\qquad p=2:$\ ] Frobenius norm
$\ \|\bA\|_F \ =\ \sqrt{\sum_{j=1}^r{\sigma_j(\bA)^2}}\ = \ \stn{\bA}$.
\item[$\qquad p=\infty:$\ ] Euclidean (operator) norm
$\ \|\bA\|_2\ =\ \sigma_1(\bA)\ = \ \sinf{\bA}$.
\end{description}
\bigskip
We will make ample use of the following properties.
\begin{lemma}
Let $\bA\in\rmn$, $\bB\in\real^{n\times \ell}$, and $\bC\in\real^{s\times m}$.
\begin{compactitem}
\item Unitary invariance:\\
If $\bQ_1\in\real^{s\times m}$ with $\bQ_1^T\bQ_1=\bI_m$ and
$\bQ_2\in\real^{\ell \times n}$ with $\bQ_2^T\bQ_2=\bI_n$, then
$$\spn{\bQ_1\bA\bQ_2^T}\ = \ \spn{\bA}.$$
\item Submultiplicativity: $\ \spn{\bA\bB}\ \leq \ \spn{\bA} \spn{\bB}$.
\item Strong submultiplicativity (symmetric norm):
$$\spn{\bC\bA\bB}\ \leq \ \sigma_1(\bC)\,\sigma_1(\bB)\,\spn{\bA} \ = \ \|\bC\|_2\,\|\bB\|_2\,\spn{\bA}.$$
\item Best rank-$k$ approximation:
$$\spn{(\bI-\bU_k\bU_k^T)\bA}\ =\ \spn{\bA-\bA_k}\ = \ \min_{\rank(\bB)=k}{\spn{\bA-\bB}}$$
\end{compactitem}
\end{lemma}
\subsection{Principal Angles between Subspaces}\label{s_angles}
We review the definition of angles between subspaces, and the connections between angles and projectors.
\begin{definition}[Section 6.4.3 in \cite{GovL13} and Section 2 in \cite{ZKny13}]
Let $\bZ\in\real^{m\times k}$ and $\hZ\in\real^{m\times \ell}$ with $\ell\geq k$
have orthonormal columns so that $\bZ^T\bZ=\bI_k$ and $\hZ^T\hZ=\bI_{\ell}$.
The singular values of $\bZ^T\hZ$ are the diagonal elements of the $k\times k$ diagonal matrix
$$\cos{\bTheta}(\bZ,\hZ)\equiv\diag\begin{pmatrix}\cos{\theta_1}&\cdots &\cos{\theta_k}\end{pmatrix},$$
where $\theta_j$ are the {\rm principal (canonical) angles} between $\range(\bZ)$ and $\range(\hZ)$.
\end{definition}
Next we show how to extract the principal angles between two subspaces of possibly different dimensions,
we make use of projectors.
\begin{lemma}\label{l_projbasis}
Let $\proj\equiv \bZ\bZ^T$ and $\projt\equiv \hZ\hZ^T$ be orthogonal projectors, where
$\bZ\in\real^{m\times k}$ and $\hZ\in\real^{m\times \ell}$ have orthonormal columns. Let $\ell\geq k$, and define
$$\sin{\bTheta}(\proj,\projt)=\sin{\bTheta}(\bZ,\hZ)\
\equiv\ \diag\begin{pmatrix}\sin{\theta_1}&\cdots &\sin{\theta_k}\end{pmatrix},$$
where $\theta_j$ are the $k$ principal angles between $\range(\bZ)$ and $\range(\hZ)$.
\begin{enumerate}
\item If $\rank(\hZ)=k=\rank(\bZ)$, then
\begin{eqnarray*}
\spn{\sin{\bTheta}(\bZ,\hZ)}&=&\spn{(\bI-\proj)\,\projt}=\spn{(\bI-\projt)\,\proj}.
\end{eqnarray*}
In particular
$$\|(\bI-\proj)\,\projt\|_2=\|\proj-\projt\|_2\leq 1$$
represents the distance between the subspaces $\range(\proj)$ and $\range(\projt)$.
\item If $\rank(\hZ)>k=\rank(\bZ)$, then
$$\spn{\sin{\bTheta}(\bZ,\hZ)}=\spn{(\bI-\projt)\,\proj}\ \leq \ \spn{(\bI-\proj)\,\projt}.$$
\end{enumerate}
\end{lemma}
\begin{proof}
The two-norm expressions follow from \cite[Section~2.5.3]{GovL13} and \cite[Section~2]{Wedin1983}.
The Schatten $p$-norm expressions follow from the CS decompositions in
\cite[Theorem 8.1]{PaigeWei94}, \cite[Section 2]{ZKny13}, and Section~\ref{s_app}.
\end{proof}
\subsection{Highlights of the Main Results}\label{s_high}
We present a brief overview of the main results: The well-conditioning of low-rank
approximations under additive perturbations in projector basis and the matrix (Section~\ref{s_low});
the well-conditioning of low-rank approximations under perturbations that change the matrix dimension
(Section~\ref{s_dimchange});
and the connection between low-rank approximation errors and angles between subspaces (Section~\ref{s_angle}).
Thus: Low-rank approximations are well-conditioned, and don't need a gap.
\subsubsection{Additive perturbations in the projector basis and the matrix}\label{s_low}
We show that the low-rank approximation error is insensitive to
additive rank-preserving perturbations in the projector basis (Theorem~\ref{t_a} and Corollary~\ref{c_a}),
and to additive perturbations in the matrix (Theorem~\ref{t_ba} and Corollary~\ref{c_ba}).
We start with perturbations in the projector basis.
\begin{mytheorem}[Additive rank-preserving perturbations in the projector basis]\label{t_a}
Let $\bA\in\rmn$; let $\bZ\in\real^{m\times\ell}$ be a projector basis with orthonormal columns
so that $\bZ^T\bZ=\bI_{\ell}$;
and let $\hZ\in\real^{m\times \ell}$ be its perturbation with
$$\epsilon_Z \equiv\ \|\hZ^{\dagger}\|_2\,\|\bZ-\hZ\|_2\ = \
\underbrace{\|\hZ\|_2\|\hZ^{\dagger}\|_2}_{\mathrm{Deviation~from} \atop \mathrm{orthonormality}}
\underbrace{\frac{\|\hZ-\bZ\|_2}{\|\hZ\|_2}.}_{\mathrm{Relative~distance} \atop \mathrm{from~exact~basis}}$$
\begin{compactenum}
\item If $\rank(\hZ)=\rank(\bZ)$ then
\begin{eqnarray*}
\spn{(\bI-\bZ\bZ^T)\bA} - \epsilon_Z\,\spn{\bA} & \leq &\spn{(\bI-\hZ\hZ^{\dagger})\bA}\\
&&\qquad\qquad \leq \spn{(\bI-\bZ\bZ^T)\bA}+ \epsilon_Z\,\spn{\bA}.
\end{eqnarray*}
\item If $\|\bZ-\hZ\|_2\leq 1/2$, then $\rank(\hZ)=\rank(\bZ)$ and $\epsilon_Z\leq 2\,\|\bZ-\hZ\|_2$.
\end{compactenum}
\end{mytheorem}
\begin{proof}
See Section~\ref{s_sensla}, and in particular Theorem~\ref{t_perturbla}.
\end{proof}
Theorem~\ref{t_a} bounds the change in the absolute approximation error
in terms of the additive perturbation $\epsilon_Z$ amplified by the norm of $\bA$.
The term $\epsilon_Z$ can also be written as the product of two factors:
(i) the two-norm condition number $\|\hZ\|_2\|\hZ^{\dagger}\|_2$ of the perturbed basis with regard to (left) inversion;
(ii) and relative two-norm distance between the bases.
The assumption here is that the perturbed vectors $\hZ$
are linearly independent, but not necessarily orthonormal. Hence the Moore
Penrose inverse replaces the transpose in the orthogonal projector,
and the condition number represents the deviation of $\hZ$ from orthonormality.
The special case $\|\bZ-\hZ\|_2\leq 1/2$ implies both that the perturbed projector
basis is well-conditioned and that it is close to the exact basis.
The lower bound in Theorem~\ref{t_a} simplifies when the columns of $\bZ$ are dominant singular vectors of~$\bA$. No singular value gap is required below, as we merely pick
the leading $k$ columns of $\bU$ from some SVD of $\bA$, and then perturb them.
\begin{mycorollary}[Rank-preserving perturbation of dominant singular vectors]\label{c_a}
Let $\bU_k\in\real^{m\times k}$ in (\ref{e_bestrank}) be $k$ dominant left singular
vectors of $\bA$. Let $\hU\in\real^{m\times k}$ be a perturbation of $\bU_k$
with $\rank(\hU)=k$ or $\|\bU_k-\hU\|_2\leq 1/2$;
and let $\epsilon_U \equiv\ \|\hU^{\dagger}\|_2\,\|\bU_k-\hU\|_2$.
Then
\begin{eqnarray*}
\spn{(\bI-\bU_k\bU_k^T)\bA} & \leq & \spn{(\bI-\hU\hU^{\dagger})\bA}\leq
\spn{(\bI-\bU_k\bU_k^T)\bA} + \epsilon_U\,\spn{\bA}.
\end{eqnarray*}
\end{mycorollary}
Next we consider perturbations in the matrix, with a bound that is completely general and
holds for any projector $\proj$ in any Schatten $p$-norm.
\begin{mytheorem}[Additive perturbations in the matrix]\label{t_ba}
Let $\bA$, $\bA+\bE\in\rmn$; and let $\proj\in\rmm$ be an orthogonal projector as in
(\ref{e_dproj}). Then
\begin{eqnarray*}
\bigg|\, \spn{(\bI-\proj)\, (\bA+\bE)}- \spn{(\bI-\proj)\bA}\, \bigg| \ \leq\ \spn{\bE}.
\end{eqnarray*}
\end{mytheorem}
\begin{proof} See Section~\ref{s_sensla}, and in particular Theorem~\ref{t_perturbm}.
\end{proof}
Theorem~\ref{t_ba} shows that the low-rank approximation error is well-conditioned, in the absolute sense,
to additive perturbations in the matrix.
Theorem~\ref{t_ba} also implies the following upper bound for
a low-rank approximation of $\bA$ by means of singular vectors of $\bA+\bE$.
Again, no singular value gap is required. We merely pick
the leading $k$ columns $\bU_k$ obtained from some SVD of $\bA$, and the
leading $k$ columns $\hU_k$ obtained from some SVD of $\bA+\bE$.
\begin{mycorollary}[Low-rank approximation from additive perturbation]\label{c_ba}
Let $\bU_k\in\real^{m\times k}$ in (\ref{e_bestrank}) be $k$ dominant left singular vectors
of $\bA$;
and let $\hU_k\in\real^{m\times k}$ be $k$ dominant left singular vectors of $\bA+\bE$.
Then
$$\|(\bI-\bU_k\bU_k^T)\bA\|_2 \ \leq\
\|(\bI-\hU_k \hU_k^T)\bA\|_{2}\, \ \leq\ \|(\bI-\bU_k\bU_k^T)\bA\|_2 + 2\|\bE\|_{2}.$$
\end{mycorollary}
\begin{proof}
See Section~\ref{s_sensla}, and in particular Corollary~\ref{c_cba}.
\end{proof}
Bounds with an additive dependence on $\bE$, like the two-norm bound above,
can be derived for other Schatten $p$-norms as well, and can then be combined
with bounds for $\bE$ in
\cite{Kundu2014,Achlioptas2007,Drineas2011}
where $\bA+\bE$ is obtained from element-wise sampling from $\bA$.
\subsubsection{Perturbation that change the matrix dimension}\label{s_dimchange}
We consider perturbations that can change the number of columns in~$\bA$ and
include, among others,
multiplicative perturbations of the form $\wA=\bA\bX$. However, our bounds
are completely general
and hold also in the absence of any relation between $\range(\bA)$ and $\range(\wA)$.
Presented below are bounds for the two-norm (Theorem~\ref{t_bb2}), the Frobenius norm (Theorem~\ref{t_bbF})
and general Schatten $p$-norms (Theorem~\ref{t_bbp}).
\begin{mytheorem}[Two-norm]\label{t_bb2}
Let $\bA\in\rmn$; $\wA\in\real^{m\times c}$; and $\proj\in\rmm$ an orthogonal projector
as in (\ref{e_dproj}) with $\rank(\proj)=c$.
Then
\begin{eqnarray}\label{e_bb21}
\bigg|\|(\bI-\proj)\,\bA\|_{2}^2- \|(\bI-\proj)\,\wA\|_{2}^2\bigg| \ \leq \ \|\bA\bA^T-\wA\wA^T\|_2.
\end{eqnarray}
If also $\rank(\wA)=c$ then
\begin{eqnarray}\label{e_bb22}
\|(\bI-\wA\wA^{\dagger})\,\bA\|_{2}^2 \ \leq \ \|\bA\bA^T-\wA\wA^T\|_2.
\end{eqnarray}
If $\wA_k\in\real^{m\times c}$ is a best rank -$k$ approximation of $\wA$ with $\rank(\wA_k)=k<c$ then
\begin{eqnarray}\label{e_bb23}
\|(\bI-\wA_k\wA_k^{\dagger})\,\bA\|_{2}^2 \ \leq\ \|\bA-\bA_k\|_2^2+ 2\,\|\bA\bA^T -\wA\wA^T\|_{2}.
\end{eqnarray}
\end{mytheorem}
\begin{proof}
See Section~\ref{s_sensla}, specifically
Theorem~\ref{t_perturbmm} for (\ref{e_bb21});
Theorem~\ref{t_lc} for (\ref{e_bb22}); and
and Theorem~\ref{t_lck} for (\ref{e_bb23}).
\end{proof}
The bounds~(\ref{e_bb23}) are identical to~\cite[Theorem 3]{DKM06}, while
(\ref{e_bb21}) and~(\ref{e_bb22}), though similar in spirit, are novel.
The bound (\ref{e_bb21}) holds for any orthogonal projector $\proj$,
in contrast to prior work which was limited to multiplicative perturbations $\wA=\bA\bX$
with bounds for $\|\bA\bA^T-\wA\wA^T\|_2$ for matrices $\bX$ that sample and rescale columns
\cite{DKM06a,Holodnak2015}, and other constructions of $\bX$ \cite{TCS-060}.
\begin{mytheorem}[Frobenius norm]\label{t_bbF}
Let $\bA\in\rmn$; $\wA\in\real^{m\times c}$; and $\proj\in\rmm$ an orthogonal projector
as in (\ref{e_dproj})
with $\rank(\proj)=c$.
Then
\begin{eqnarray}
\lefteqn{\bigg|\|(\bI-\proj)\,\wA\|_F^2 -\|(\bI-\proj)\,\bA\|_F^2\bigg| \ \leq}\label{e_bbF1}\\
& &\ \min\bigg\{ \|\bA\bA^T-\wA\wA^T\|_*, \quad \sqrt{m-c}\,\|\bA\bA^T-\wA\wA^T\|_F\bigg\}.\nonumber
\end{eqnarray}
If also $\rank(\wA)=c$ then
\begin{eqnarray}\label{e_bbF2}
\|(\bI-\wA\wA^{\dagger})\,\bA\|_{F}^2 \ \leq
\ \min\bigg\{ \|\bA\bA^T-\wA\wA^T\|_*, \ \sqrt{m-c}\,\|\bA\bA^T-\wA\wA^T\|_F\bigg\}.
\end{eqnarray}
If $\wA_k\in\real^{m\times c}$ is a best rank-$k$ approximation of $\wA$ with $\rank(\wA_k)=k<c$ then
\begin{eqnarray}
(\bI-\wA_k\wA_k^{\dagger})\,\bA\|_{F}^2 \ &\leq&\ \|\bA-\bA_k\|_F^2\label{e_bbF3}\\
&&+2\, \min\big\{ \|\bA\bA^T-\wA\wA^T\|_*, \, \sqrt{m-c}\,\|\bA\bA^T-\wA\wA^T\|_F\big\}.\nonumber
\end{eqnarray}
\end{mytheorem}
\begin{proof}
See Section~\ref{s_sensla}, specifically
Theorem~\ref{t_perturbmm} for (\ref{e_bbF1});
Theorem~\ref{t_lc} for (\ref{e_bbF2});
and Theorem~\ref{t_lck} for (\ref{e_bbF3}).
\end{proof}
The bound (\ref{e_bbF1}) holds for any $\proj$, and is the first one of its kind in this generality.
The bound (\ref{e_bbF3}) is similar to \cite[Theorem 2]{DKM06}, and weaker for smaller $k$ but tighter for larger $k$.
More generally, Theorem~\ref{t_bbF} relates the low-rank approximation error in the Frobenius norm
with the error $\bA\bA^T-\wA\wA^T$ in the trace norm, i.e. the Schatten one-norm.
This is a novel connection, and it should motivate further work into understanding the behaviour
of the trace norm, thereby complementing prior investigations into the two-norm and Frobenius norm.
\begin{mytheorem}[General Schatten $p$-norms]\label{t_bbp}
Let $\bA\in\rmn$; $\wA\in\real^{m\times c}$; and $\proj\in\rmm$ an orthogonal projector
as in (\ref{e_dproj})
with $\rank(\proj)=c$.
Then
\begin{eqnarray}
\lefteqn{\bigg|\spn{(\bI-\proj)\,\bA}^2- \spn{(\bI-\proj)\,\wA}^2\bigg| \ \leq\label{e_bbp1}}\\
& &\ \min\bigg\{ \spnt{\bA\bA^T-\wA\wA^T}, \quad \sqrt[p]{m-c}\,\spn{\bA\bA^T-\wA\wA^T}\bigg\}.
\nonumber
\end{eqnarray}
If $\rank(\wA)=c$ then
\begin{eqnarray}
\lefteqn{\spnt{(\bI-\wA\wA^{\dagger})\,\bA}^2 \ \leq}\label{e_bbp2}\\
& & \ \min\bigg\{ \spnt{\bA\bA^T-\wA\wA^T}, \quad \sqrt[p]{m-c}\,\spn{\bA\bA^T-\wA\wA^T}\bigg\}.\nonumber
\end{eqnarray}
If $\wA_k\in\real^{m\times c}$ is a best rank approximation of $\wA$ with $\rank(\wA_k)=k<c$ then
\begin{eqnarray}
\spnt{(\bI-\wA_k\wA_k^{\dagger})\,\bA}^2 \ &\leq&\ \spn{\bA-\bA_k}^2+\label{e_bbp3}\\
&&2\, \min\big\{ \spnt{\bA\bA^T-\wA\wA^T}, \, \sqrt[p]{m-c}\,\spn{\bA\bA^T-\wA\wA^T}\big\}.\nonumber
\end{eqnarray}
\end{mytheorem}
\begin{proof}
See Section~\ref{s_sensla}, specifically
Theorem~\ref{t_perturbmm} for (\ref{e_bbp1});
Theorem~\ref{t_lc} for (\ref{e_bbp2});
and Theorem~\ref{t_lck} for (\ref{e_bbp3}).
\end{proof}
Theorem~\ref{t_bbp} is new. To our knowledge, non-trivial bounds for
$\spn{\bA\bA^T-\wA\wA^T}$ for general~$p$ do no exist.
\subsubsection{Relations between low-rank approximation error and subspace angle}\label{s_angle}
For matrices $\bA$ with a singular value gap,
we bound the low-rank approximation error in terms of the subspace
angle (Theorem~\ref{t_c})
and discuss the tightness of the bounds (Remark~\ref{r_tc}).
The singular value gap is required for the dominant subspace to be well-defined,
but no assumptions on the accuracy of the low-rank approximation are required.
Assume that $\bA \in \rmn$ has a gap after the $k$th singular value,
\begin{equation*}
\|\bA\|_2=\sigma_1(\bA)\geq \cdots\geq \sigma_k(\bA)>
\sigma_{k+1}(\bA)\geq \cdots \geq \sigma_r(\bA)\geq 0,\qquad
r\equiv\min\{m,n\},
\end{equation*}
Below, we highlight the bounds from Section~\ref{s_bounds} for low-dimensional approximations,
compared to the dimension $m$ of the host space for $\range(\bA$).
\begin{mytheorem}\label{t_c}
Let $\proju\equiv \bA_k\bA_k^{\dagger}$ be the orthogonal projector onto the
dominant $k$-dimensional subspace of $\bA$; and
let $\proj\in\rmm$ be some orthogonal projector as in (\ref{e_dproj}) with
$k\leq \rank(\proj)<m-k$. Then
\begin{equation*}\label{e_tc1}
\sigma_k(\bA)\,\spn{\sin{\bTheta(\proj,\proju)}} \leq \spn{(\bI-\proj)\bA} \leq
\|\bA\|_2\,\spn{\sin{\bTheta(\proj,\proju)}}+\spn{\bA-\bA_k}.
\end{equation*}
\end{mytheorem}
\begin{proof} See Section~\ref{s_bounds}, and in particular Theorems~\ref{t_lau} and~\ref{t_lal1}.
\end{proof}
Theorem~\ref{t_c} shows that for dominant subspaces of sufficiently low dimension, the approximation error is
bounded by the product of the subspace angle with a dominant singular value.
The upper bound also contains the subdominant singular values.
\begin{myremark}[Tightness of Theorem~\ref{t_c}]\label{r_tc}
\begin{compactitem}
\item If $\rank(\bA)=k$, so that $\bA-\bA_k=\bzero$, then the tightness
depends on the spread of the non-zero singular values,
$$\sigma_k(\bA)\,\spn{\sin{\bTheta(\proj,\proju)}} \leq \spn{(\bI-\proj)\,\bA} \leq
\|\bA\|_2\,\spn{\sin{\bTheta(\proj,\proju)}}.$$
\item If $\rank(\bA)=k$ and $\sigma_1(\bA)= \dots =\sigma_k(\bA)$, then the bounds are tight,
and they are equal to
$$ \spn{(\bI-\proj)\,\bA} = \|\bA\|_2\,\spn{\sin{\bTheta(\proj,\proju)}}.$$
\item If $\range(\proj)=\range(\proju)$, so that $\sin{\bTheta(\proj,\proju)}=\bzero$,
then the upper bound is tight and equal to
$$\spn{(\bI-\proj)\,\bA} = \spn{\bA-\bA_k}.$$
\end{compactitem}
\end{myremark}
\section{Well-conditioning of low-rank approximations}\label{s_sensla}
We investigate the effect of additive, rank-preserving perturbations in the projector basis
on the orthogonal projector (Section~\ref{s_opperturb})
and on the low-rank approximation error (Section~\ref{s_lrperturb}); and the effect
on the low-rank approximation error of matrix perturbations, both additive and dimension changing
(Section~\ref{s_lrmperturb}). We also relate low-rank approximation
error and error matrix (Section~\ref{s_errmatrix}).
\subsection{Orthogonal projectors, and perturbations in the projector basis}\label{s_opperturb}
We show that orthogonal projectors and subspace angles are insensitive
to additive, rank-preserving perturbations in the projector basis (Theorem~\ref{t_perturbp})
if the perturbed projector basis is well-conditioned.
\begin{theorem}\label{t_perturbp}
Let $\bA\in\rmn$; let $\bZ\in\real^{m\times s}$ be a projector basis with orthonormal
columns so that $\bZ^T\bZ=\bI_s$; and let $\hZ\in\real^{m\times s}$ be its perturbation.
\begin{compactenum}
\item If $\rank(\hZ)=\rank(\bZ)$, then the {\rm distance} between $\range(\bZ)$ and $\range(\hZ)$ is
\begin{equation*}
\|\bZ\bZ^T-\hZ\hZ^{\dagger}\|_2=\|\sin{\bTheta(\bZ,\hZ)}\|_2\leq \epsilon_Z\equiv\ \|\hZ^{\dagger}\|_2\,\|\bZ-\hZ\|_2.
\end{equation*}
\item If $\|\bZ-\hZ\|_2\leq 1/2$, then $\rank(\hZ)=\rank(\bZ)$ and $\epsilon_Z\leq 2\,\|\bZ-\hZ\|_2$.
\end{compactenum}
\end{theorem}
\begin{proof}
\begin{compactenum}
\item The equality follows from Lemma~\ref{l_projbasis}.
The upper bound follows from \cite[Theorem 3.1]{Ste2011} and \cite[Lemma 20.12]{Higham2002},
but we provide a simpler proof for this context. Set $\hZ=\bZ+\bF$, and abbreviate
$\proj\equiv\bZ\bZ^T$ and $\projt\equiv\hZ\hZ^{\dagger}$.
Writing
$$(\bI-\proj)\projt=(\bI-\bZ\bZ^T)\hZ\hZ^{\dagger}=
(\bI-\bZ\bZ^T)(\bZ+\hZ-\bZ)\,\hZ^{\dagger}=(\bI-\proj)\bF\,\hZ^{\dagger}$$
gives
\begin{equation}\label{e_sin2}
\|\sin{\bTheta(\bZ,\hZ)}\|_2=\|(\bI-\proj)\projt\|_2\leq \|\hZ^{\dagger}\|_2\,\|\bF\|_2.
\end{equation}
\item To show $\rank(\hZ)=\rank(\bZ)$ in the special case $\|\bZ-\hZ\|_2\leq 1/2$, consider
the singular values $\sigma_j(\bZ)=1$ and $\sigma_j(\hZ)$, $1\leq j\leq s$.
The well-conditioning of singular values \cite[Corollary~8.6.2]{GovL13} implies
$$\left| 1-\sigma_j(\hZ)\right| = \left| \sigma_j(\bZ)-\sigma_j(\hZ)\right| \leq \|\bF\|_2\leq 1/2,
\qquad 1\leq j\leq s.$$
Thus $\min_{1\leq j\leq s}{\sigma_j(\hZ)}\geq 1/2>0$ and $\rank(\projt)=\rank(\hZ)=s=\rank(\proj)$.
Hence (\ref{e_sin2}) holds with
\begin{equation}\label{e_sin3}
\|\sin{\bTheta(\bZ,\hZ)}\|_2\leq \|\hZ^{\dagger}\|_2\|\bF\|_2\leq 2\|\bF\|_2.
\end{equation}
\end{compactenum}
\end{proof}
Note that $\epsilon_Z$ represents both, an absolute and a relative perturbation as
$$\epsilon_Z =\ \|\hZ^{\dagger}\|_2\,\|\bZ-\hZ\|_2\ = \ \|\hZ\|_2\|\hZ^{\dagger}\|_2\>
\frac{\|\hZ-\bZ\|_2}{\|\hZ\|_2}.$$
\subsection{Approximation errors, and perturbations in the projector basis}\label{s_lrperturb}
We show that low-rank approximation errors are insensitive to additive, rank-preserving perturbations
in the projector basis (Theorem~\ref{t_perturbla}), if the perturbed projector basis is well-conditioned.
\begin{theorem}\label{t_perturbla}
Let $\bA\in\rmn$; let $\bZ\in\real^{m\times s}$ be a projector basis with orthonormal
columns so that $\bZ^T\bZ=\bI_s$; and let $\hZ\in\real^{m\times s}$ be its perturbation with
$\epsilon_Z \equiv\ \|\hZ^{\dagger}\|_2\,\|\bZ-\hZ\|_2$.
\begin{compactenum}
\item If $\rank(\hZ)=\rank(\bZ)$ then
\begin{eqnarray*}
\lefteqn{\spn{(\bI-\bZ\bZ^*)\bA} - \epsilon_Z\,\spn{\bA} \leq \spn{(\bI-\hZ\hZ^{\dagger})\bA}}
\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\\
& \leq & \spn{(\bI-\bZ\bZ^*)\bA} + \epsilon_Z\,\spn{\bA}.
\end{eqnarray*}
\item If $\|\bZ-\hZ\|_2\leq 1/2$, then $\rank(\hZ)=\rank(\bZ)$ and $\epsilon_Z\leq 2\, \|\bZ-\hZ\|_2$.
\end{compactenum}
\end{theorem}
\begin{proof}
This is a straightforward consequence of Theorem~\ref{t_perturbp}.
Abbreviate $\proj\equiv \bZ\bZ^T$ and $\projt\equiv\hZ\hZ^{\dagger}$, and write
\begin{equation*}
(\bI-\projt)\bA = (\bI-\proj)\bA +(\proj-\projt)\bA.
\end{equation*}
Apply the triangle and reverse triangle inequalities, strong submultiplicativity, and
then bound the second summand with Theorem~\ref{t_perturbp},
\begin{equation*}
\spn{(\proj-\projt)\bA}\leq\|\hZ^{\dagger}\|_2\>\|\hZ-\bZ\|_2\spn{\bA} = \epsilon_Z\,\spn{\bA}.
\end{equation*}
\end{proof}
\subsection{Approximation errors, and perturbations in the matrix}\label{s_lrmperturb}
We show that low-rank approximation errors are insensitive to matrix perturbations that are additive
(Theorem~\ref{t_perturbm} and Corollary~\ref{c_cba}), and that are dimension changing
(Theorem~\ref{t_perturbmm})
\begin{theorem}[Additive perturbations]\label{t_perturbm}
Let $\bA,\bE\in\rmn$; and let $\proj\in\rmm$ be an orthogonal projector with $\proj^2=\proj=\proj^T$.
Then
\begin{equation*}
\spn{(\bI-\proj)\bA}- \spn{\bE} \leq
\spn{(\bI-\proj)(\bA+\bE)} \leq \spn{(\bI-\proj)\bA} + \spn{\bE}.
\end{equation*}
\end{theorem}
\begin{proof}
Apply the triangle and reverse triangle inequalities, and the fact that an orthogonal projector has at most unit norm,
$\|\bI-\proj\|_2\leq 1$.
\end{proof}
\begin{corollary}[Low-rank approximation from singular vectors of $\bA+\bE$]\label{c_cba}
Let $\bU_k\in\real^{m\times k}$ in (\ref{e_bestrank}) be $k$ dominant left singular vectors of $\bA$;
and let $\hU_k\in\real^{m\times k}$ be $k$ dominant left singular vectors of $\bA+\bE$.
Then
$$\|(\bI-\bU_k\bU_k^T)\bA\|_2 \ \leq\
\|(\bI-\hU_k \hU_k^T)\bA\|_{2}\, \ \leq\ \|(\bI-\bU_k\bU_k^T)\bA\|_2 + 2\|\bE\|_{2}.$$
\end{corollary}
\begin{proof}
Setting $\proj= \hU_k \hU_k^T$ in the upper bound of Theorem~\ref{t_perturbm} gives
\[\|(\bI-\hU_k \hU_k^T)\bA\|_{2}\, \ \leq\ \|(\bI-\hU_k\hU_k^T)\, (\bA+\bE)\|_{2} + \|\bE\|_{2}.\]
Express the approximation errors in terms of singular values,
$$\|(\bI-\hU_k \hU_k^T)\, (\bA+\bE)\|_2=\sigma_{k+1}(\bA+\bE), \qquad
\|(\bI-\bU_k\bU_k^T)\bA\|_2 =\sigma_{k+1}(\bA),$$
apply Weyl's theorem
\[|\sigma_{k+1}(\bA+\bE)-\sigma_{k+1}(\bA)| \leq \|\bE\|_2,\]
and combine the bounds.
\end{proof}
\begin{theorem}[Perturbations that change the number of columns]\label{t_perturbmm}
Let $\bA\in\rmn$; $\wA\in\real^{m\times c}$; let $\proj\in\rmm$ be an orthogonal projector
as in (\ref{e_dproj}) and $\rank(\proj)=s$; and let $p\geq 1$ an even integer. Then
\begin{compactenum}
\item Two norm $(p=\infty)$
$$\bigg|\|(\bI-\proj)\,\bA\|_{2}^2- \|(\bI-\proj)\wA\|_{2}^2\bigg| \ \leq \ \|\wA\wA^T-\bA\bA^T\|_2.$$
\item Schatten $p$ norm $(p$ even$)$
\begin{eqnarray*}
\lefteqn{\bigg|\spn{(\bI-\proj)\,\bA}^2- \spn{(\bI-\proj)\wA}^2\bigg| \ \leq}\\
& &\ \min\bigg\{ \spnt{\wA\wA^T-\bA\bA^T}, \quad \sqrt[p]{m-s}\,\spn{\wA\wA^T-\bA\bA^T}\bigg\}.
\end{eqnarray*}
\item Frobenius norm $(p=2)$
\begin{eqnarray*}
\lefteqn{\bigg|\|(\bI-\proj)\,\wA\|_F^2 -\|(\bI-\proj)\,\bA\|_F^2\bigg| \ \leq}\\
& &\ \min\bigg\{ \|\wA\wA^T-\bA\bA^T\|_*, \quad \sqrt{m-s}\,\|\wA\wA^T-\bA\bA^T\|_F\bigg\}.
\end{eqnarray*}
\end{compactenum}
\end{theorem}
\begin{proof}
The proof is motivated by that of \cite[Theorems 2 and 3]{DKM06}.
The bounds are obvious for $s=m$ where $\proj=\bI_m$, so assume $s<m$.
\paragraph{1. Two-norm}
The invariance of the two norm under transposition and the triangle inequality imply
\begin{eqnarray*}
\|(\bI-\proj)\wA\|_{2}^2&=&\|\wA^T(\bI-\proj)\|_{2}^2=\|(\bI-\proj)\wA\wA^T(\bI-\proj)\|_{2}\\
&=&\|(\bI-\proj)\bA\bA^T(\bI-\proj)\ +\ (\bI-\proj)\left(\wA\wA^T-\bA\bA^T\right)(\bI-\proj)\|_{2}\\
&\leq &\|(\bI-\proj)\bA\bA^T(\bI-\proj)\|_{2} +\|(\bI-\proj)\left(\wA\wA^T-\bA\bA^T\right)(\bI-\proj)\|_{2}.
\end{eqnarray*}
The first summand equals
$$\|(\bI-\proj)\bA\bA^T(\bI-\proj)\|_{2} =\|(\bI-\proj)\bA\|_{2}^2,$$
while the second one is bounded by submultiplicativity and $\|\bI-\proj\|_2\leq 1$,
\begin{eqnarray*}
\|(\bI-\proj)\left(\wA\wA^T-\bA\bA^T\right)(\bI-\proj)\|_{2}
&\leq& \|\bI-\proj\|_2^2\,\|\wA\wA^T-\bA\bA^T\|_{2}\\
&\leq& \|\wA\wA^T-\bA\bA^T\|_{2}.
\end{eqnarray*}
This gives the upper bound
$$\|(\bI-\proj)\wA\|_{2}^2-\|(\bI-\proj)\bA\|_{2}^2\leq \|\wA\wA^T-\bA\bA^T\|_{2}.$$
Apply the inverse triangle inequality to show the lower bound,
$$-\|\wA\wA^T-\bA\bA^T\|_{2}\leq \|(\bI-\proj)\wA\|_{2}^2-\|(\bI-\proj)\bA\|_{2}^2.$$
\paragraph{2. Schatten $p$-norm $(p$ even$)$}
The proof is similar to that of the two-norm, since an even Schatten $p$-norm
is a \textit{Q-norm} \cite[Definition IV.2.9]{Bhatia1997}, that is, it represents a quadratic gauge function.
This can be seen in terms of singular values,
$$\spn{\bM}^{p}=\sum_{j}{\left(\sigma_j(\bM)\right)^{p}}
=\sum_{j}{\left(\sigma_j(\bM\bM^T)\right)^{p/2}}=\spnt{\bM\bM^T}^{p/2}.$$
Hence
\begin{equation}\label{e_qnorm}
\spn{\bM}^2\ = \ \spnt{\bM\bM^T}.
\end{equation}
Abbreviate $\bM\equiv \wA\wA^T-\bA\bA^T$, and $\bB\equiv \bI-\proj$ where $\bB^T=\bB$ and $\|\bB\|_2=1$.
Since singular values do not change under transposition,
it follows from (\ref{e_qnorm}) and the triangle inequality that
\begin{eqnarray*}
\spn{\bB\,\wA}^2&=&\spn{\wA^T\,\bB}^2=\spnt{\bB\,\wA\wA^T\,\bB}
=\spnt{\bB\bA\bA^T\bB\ +\ \bB\,\bM\,\bB}\\
&\leq &\spnt{\bB\,\bA\bA^T\,\bB} +\spnt{\bB\,\bM\,\bB}.
\end{eqnarray*}
Apply (\ref{e_qnorm}) to the first summand,
$\ \spnt{\bB\,\bA\bA^T\,\bB} =\spn{\bB\,\bA}^2$,
and insert it into the above inequalities,
\begin{equation}\label{e_24}
\spn{\bB\,\wA}^2-\spn{\bB\,\bA}^2\leq\spnt{\bB\,\bM\,\bB}.
\end{equation}
\begin{compactenum}
\item First term in the minimum:
Bound (\ref{e_24}) with strong submultiplicativity and $\|\bB\|_2= 1$,
\begin{eqnarray*}
\spn{\bB\,\bM\,\bB}\leq \|\bB\|_2^2\,\spn{\bM}\leq \spn{\bM},
\end{eqnarray*}
which gives the upper bound
$$\spn{\bB\,\wA}^2-\spn{\bB\,\bA}^2\leq \spn{\bM}.$$
Apply the inverse triangle inequality to show the lower bound
$$-\spn{\bM}\leq \spn{\bB\,\wA}^2-\spn{\bB\,\bA}^2.$$
\item Second term in the minimum: From
$$\rank\left(\bB\,\bM\,\bB\right)\leq \rank(\bB)=\rank(\bI-\proj)=m-s>0$$
follows $\sigma_j(\bB)=1$, $\leq j\leq m-s$.
With the non-ascending singular value ordering in (\ref{e_svo}),
the Schatten $p$-norm needs to sum over only the largest $m-s$ singular values.
This, together with the singular value inequality \cite[(7.3.14)]{HoJ12}
$$\sigma_j(\bB\,\bM\,\bB)\leq \sigma_1(\bB)^2\,\sigma_j(\bM) = 1\cdot \,\sigma_1(\bM), \qquad 1\leq j\leq m-s, $$
gives
$$\spnt{\bB\,\bM\,\bB}^{p/2}=\sum_{j=1}^{m-s}{\left(\sigma_j(\bB\,\bM\,\bB)\right)^{p/2}}
\leq \sum_{j=1}^{m-s}{1\cdot\left(\sigma_j(\bM)\right)^{p/2}}.$$
At last apply the Cauchy-Schwartz inequality to the vectors of singular values
$$\sum_{j=1}^{m-s}{1\cdot \left(\sigma_j(\bM)\right)^{p/2}}\leq \sqrt{m-s}\,
\sqrt{\sum_{j=1}^{m-s}{\left(\sigma_j(\bM)\right)^p}}\leq \sqrt{m-s}\,\spn{\bM}^{p/2}.$$
Merging the last two sequences of inequalities gives
$$\spnt{\bB\,\bM\,\bB}^{p/2}\leq \sqrt{m-s}\,\spn{\bM}^{p/2}.$$
Thus $\spnt{\bB\,\bM\,\bB}\leq \sqrt[p]{m-s}\,\spn{\bM}$, which can now be substituted into~(\ref{e_24}).
\end{compactenum}
\paragraph{3. Frobenius norm}
This is the special case $p=2$ with $\stn{\bA}=\|\bA\|_F$ and $\son{\bA}=\|\bA\|_*$.
\end{proof}
\subsection{Approximation error, and error matrix}\label{s_errmatrix}
We generalize \cite[Theorems 2 and 3]{DKM06} to Schatten $p$-norms.
\begin{theorem}\label{t_lc}
Let $\bA\in\rmn$ with $\rank(\bA)\geq k$; $\bC\in\real^{m\times c}$ with $\rank(\bC)=c\geq k$;
and let $p\geq 1$ be an even integer. Then
\begin{compactenum}
\item Two-norm $(p=\infty)$
$$\|(\bI-\bC\bC^{\dagger})\,\bA\|_{2}^2 \ \leq\ \|\bA\bA^T -\bC\bC^T\|_{2}.$$
\item Schatten $p$-norm $(p$ even$)$
$$\spn{(\bI-\bC\bC^{\dagger})\,\bA}^2 \ \leq\
\ \min\bigg\{ \spnt{\bA\bA^T-\bC\bC^T}, \quad \sqrt[p]{m-c}\,\spn{\bA\bA^T-\bC\bC^T}\bigg\}.$$
\item Frobenius norm $(p=2)$
$$\|(\bI-\bC\bC^{\dagger})\,\bA\|_{F}^2 \ \leq\
\ \min\bigg\{ \|\bA\bA^T-\bC\bC^T\|_*, \quad \sqrt{m-c}\,\|\bA\bA^T-\bC\bC^T\|_F\bigg\}.$$
\end{compactenum}
\end{theorem}
\begin{proof}
This follows from Theorem~\ref{t_perturbmm} with $\proj=\bC\bC^{\dagger}$, $\rank(\proj)=c$, $\wA=\bC$, and
$$(\bI-\proj)\,\wA=(\bI-\bC\bC^{\dagger})\,\bC=\bC=\bC\bC^{\dagger}\bC=\bzero.$$
\end{proof}
Recall
\textit{Mirsky's Theorem} \cite[Corollary 7.4.9.3]{HoJ12}, an extension of the Hoffman-Wielandt theorem
to any unitarily invariant norm and, in particular, Schatten $p$-norms: for $\bA,\bH\in\rmn$, the singular values $\sigma_j(\bA\bA^T)$ and $\sigma_j(\bH\bH^T)$, $1\leq j\leq m$
are also eigenvalues and satisfy
\begin{equation}\label{e_M}
\sum_{j=1}^m{|\sigma_j(\bA\bA^T)-\sigma_j(\bH\bH^T)|^p}\leq \spn{\bA\bA^T-\bH\bH^T}^p.
\end{equation}
\begin{theorem}\label{t_lck}
Let $\bA\in\rmn$ with $\rank(\bA)\geq k$; let $\bC\in\real^{m\times c}$ with $\rank(\bC)=c\geq k$
and best rank-$k$ approximation $\bC_k$; and let $p\geq 1$ be an even integer. Then
\begin{compactenum}
\item Two-norm $(p=\infty)$
$$\|(\bI-\bC_k\bC_k^{\dagger})\,\bA\|_{2}^2 \ \leq\ \|\bA-\bA_k\|_2^2+ 2\,\|\bA\bA^T -\bC\bC^T\|_{2}.$$
\item Schatten $p$-norm $(p$ even$)$
\begin{eqnarray*}
\spn{(\bI-\bC_k\bC_k^{\dagger})\,\bA}^2 &\leq& \spn{\bA-\bA_k}^2+\\
&& 2\,\min\big\{ \spnt{\bA\bA^T-\bC\bC^T}, \, \sqrt[p]{m-c}\,\spn{\bA\bA^T-\bC\bC^T}\big\}.
\end{eqnarray*}
\item Frobenius norm $(p=2)$
\begin{eqnarray*}
\|(\bI-\bC_k\bC_k^{\dagger})\,\bA\|_{F}^2 \ &\leq&\ \|\bA-\bA_k\|_F^2+\\
&&2\, \min\big\{ \|\bA\bA^T-\bC\bC^T\|_*, \, \sqrt{m-c}\,\|\bA\bA^T-\bC\bC^T\|_F\big\}.
\end{eqnarray*}
\end{compactenum}
\end{theorem}
\begin{proof}
We first introduce some notation before proving the bounds.
\paragraph{0. Set up}
Partition $\bA=\bA_k+\bA_{\perp}$ and $\bC=\bC_k+\bC_{\perp}$ to distinguish the
respective best rank-$k$ approximations $\bA_k$ and $\bC_k$.
From $\bA_k\bA_{\perp}^T=\bzero$ and $\bC_k\bC_{\perp}^T=\bzero$ follows
\begin{equation}
\bA\bA^T=\bA_k\bA_k^T+\bA_{\perp}\bA_{\perp}^T, \qquad
\bC\bC^T=\bC_k\bC_k^T+\bC_{\perp}\bC_{\perp}^T.
\end{equation}
Since the relevant matrices are symmetric positive semi-definite,
eigenvalues are equal to singular values. The dominant ones are
$$\sigma_j(\bA_k\bA_k^T)=\sigma_j(\bA\bA^T)=\sigma_j(\bA)^2,\quad
\sigma_j(\bC_k\bC_k^T)=\sigma_j(\bC\bC^T)=\sigma_j(\bC)^2, \quad 1\leq j \leq k,$$
and the subdominant ones are, with $j \geq 1$,
$$\sigma_j(\bA_{\perp}\bA_{\perp}^T)=\sigma_{k+j}(\bA\bA^T) =\sigma_{k+j}(\bA)^2,\quad
\sigma_j(\bC_{\perp}\bC_{\perp}^T)=\sigma_{k+j}(\bC\bC^T)=\sigma_{k+j}(\bC)^2.$$
To apply Theorem~\ref{t_perturbmm}, set $\wA=\bC$,
$\proj=\bC_k\bC_k^{\dagger}$, $\rank(\proj_k)=k$, so that
$$(\bI-\proj)\,\wA=(\bI-\bC_k\bC_k^{\dagger})\,(\bC_k+\bC_{\perp})=\bC_{\perp}.$$
Thus
\begin{equation}\label{e_cperp}
\spn{(\bI-\proj)\,\wA}^2=\spn{\bC_{\perp}}^2.
\end{equation}
\paragraph{Two-norm}
Substituting (\ref{e_cperp}) into the two norm bound in Theorem~\ref{t_perturbmm} gives
\begin{equation}\label{e_two}
\|(\bI-\bC_k\bC_k^{\dagger})\,\bA\|_{2}^2 \leq \|\bC_{\perp}\|_2^2 + \|\bA\bA^T -\bC\bC^T\|_{2}.
\end{equation}
The above and Weyl's theorem imply
\begin{eqnarray*}
\|\bC_{\perp}\|_2^2&=&\|\bC_{\perp}\bC_{\perp}^T\|_2=\lambda_{k+1}(\bC\bC^T)\\
&\leq& \left|\lambda_{k+1}(\bC\bC^T)-\lambda_{k+1}(\bA\bA^T)\right|+\lambda_{1}(\bA_{\perp}\bA_{\perp}^T)\\
&\leq& \|\bA\bA^T-\bC\bC^T\|_2 +\|\bA_{\perp}\|_2^2.
\end{eqnarray*}
Substituting this into (\ref{e_two}) gives
$$\|(\bI-\bC_k\bC_k^{\dagger})\,\bA\|_{2}^2 \leq \|\bA-\bA_k\|_2^2 + 2\, \|\bA\bA^T -\bC\bC^T\|_{2}.$$
\paragraph{Schatten $p$-norm $(p$ even$)$}
Substituting (\ref{e_cperp}) into the Schatten-$p$ norm bound in Theorem~\ref{t_perturbmm} gives
\begin{eqnarray}\label{e_p}
\spn{(\bI-\bC_k\bC_k^{\dagger})\,\bA}^2 &\leq &\spn{\bC_{\perp}}^2+ \\
&&\min\big\{ \spnt{\bA\bA^T-\bC\bC^T}, \, \sqrt[p]{m-c}\,\spn{\bA\bA^T-\bC\bC^T}\big\}.\nonumber
\end{eqnarray}
From (\ref{e_24}) follows $\spn{\bC_{\perp}}^2=\spnt{\bC_{\perp}\bC_{\perp}^T}$.
For a column vector $\bx$, let
$$\|\bx\|_p=\sqrt[p]{\sum_j{|x_j|^{1/p}}}$$
be the ordinary vector $p$-norm, and put the singular values of $\bC_{\perp}\bC_{\perp}^T$
into the vector
$$\bc_{\perp}\equiv\begin{pmatrix} \sigma_{1}(\bC_{\perp}\bC_{\perp}^T) & \cdots &
\sigma_{m-k}(\bC_{\perp}\bC_{\perp}^T)\end{pmatrix}^T.$$
Move from matrix norm to vector norm,
$$\spnt{\bC_{\perp}\bC_{\perp}^T}^{p/2}=\sum_{j=1}^{m-k}{\sigma_j(\bC_{\perp}\bC_{\perp}^T)^{p/2}}
=\sum_{j=1}^{m-k}{c_j^{p/2}}=\spnt{\bc_{\perp}}^{p/2}.$$
Put the singular values of $\bA_{\perp}\bA_{\perp}^T$ into the vector
$$\ba_{\perp}\equiv\begin{pmatrix} \sigma_{1}(\bA_{\perp}\bA_{\perp}^T) & \cdots &
\sigma_{m-k}(\bA_{\perp}\bA_{\perp}^T)\end{pmatrix}^T,$$
and apply the triangle inequality in the vector norm
$$\spnt{\bC_{\perp}\bC_{\perp}^T}=\spnt{\bc_{\perp}}
\leq \spnt{\bc_{\perp}-\ba_{\perp}}+\spnt{\ba_{\perp}}.$$
Substituting the following expression
$$\spnt{\ba_{\perp}}^{p/2}=\sum_{j=1}^{m-k}{\sigma_j(\bA_{\perp}\bA_{\perp}^T)^{p/2}}
=\sum_{j=1}^{m-k}{\sigma_j(\bA_{\perp})^p}=\spn{\bA_{\perp}}^p$$
into the previous bound and applying (\ref{e_24}) again gives
\begin{equation}\label{e_sp}
\spn{\bC_{\perp}}^2=\spnt{\bC_{\perp}\bC_{\perp}^T}\leq \spnt{\bc_{\perp}-\ba_{\perp}}+\spn{\bA_{\perp}}^2.
\end{equation}
\begin{compactenum}
\item First term in the minimum in (\ref{e_p}): Apply Mirsky's Theorem (\ref{e_M}) to the first summand in (\ref{e_sp})
\begin{eqnarray*}
\spnt{\bc_{\perp}-\ba_{\perp}}^{p/2}&=&
\sum_{j=1}^{m-k}{\left|\sigma_{k+j}(\bC\bC^T) -\sigma_{k+j}(\bA\bA^T)\right|^{p/2}}\\
&\leq&\sum_{j=1}^{m}{\left|\sigma_j(\bC\bC^T) -\sigma_j(\bA\bA^T)\right|^{p/2}}
\leq \spnt{\bC\bC^T-\bA\bA^T}^{p/2}.
\end{eqnarray*}
Thus,
$$\spnt{\bc_{\perp}-\ba_{\perp}}\leq \spnt{\bC\bC^T-\bA\bA^T}.$$
Substitute this into (\ref{e_sp}), so that
\begin{equation*}
\spn{\bC_{\perp}}^2\leq \spn{\bA_{\perp}}^2 + \spnt{\bC\bC^T-\bA\bA^T},
\end{equation*}
and the result in turn into (\ref{e_p}) to obtain the first term in the minimum,
\begin{eqnarray*}
\spn{(\bI-\bC_k\bC_k^{\dagger})\,\bA}^2 \leq \spn{\bA_{\perp}}^2 + 2\, \spnt{\bC\bC^T-\bA\bA^T}.
\end{eqnarray*}
\item Second term in the minimum in (\ref{e_p}): Consider the first summand in (\ref{e_sp}), but apply
the Cauchy Schwartz inequality before Mirsky's Theorem~(\ref{e_M}),
\begin{eqnarray*}
\spnt{\bc_{\perp}-\ba_{\perp}}^{p/2}&=&
\sum_{j=1}^{m-k}{\left|\sigma_{k+j}(\bC\bC^T) -\sigma_{k+j}(\bA\bA^T)\right|^{p/2}}\\
&\leq& \sqrt{m-k}\>\sqrt{\sum_{j=1}^{m-k}{\left|\sigma_{k+j}(\bC\bC^T) -\sigma_{k+j}(\bA\bA^T)\right|^{p}}}\\
&\leq& \sqrt{m-k}\>\sqrt{\sum_{j=1}^{m}{\left|\sigma_{k+j}(\bC\bC^T) -\sigma_{k+j}(\bA\bA^T)\right|^{p}}}\\
&\leq &\sqrt{m-k}\>\spn{\bC\bC^T-\bA\bA^T}^{p/2}.
\end{eqnarray*}
Thus,
$$\spnt{\bc_{\perp}-\ba_{\perp}}\leq \sqrt[p]{m-k}\>\spn{\bC\bC^T-\bA\bA^T}.$$
Substitute this into (\ref{e_sp}), so that
\begin{equation*}
\spn{\bC_{\perp}}^2\leq \spn{\bA_{\perp}}^2 + \sqrt{m-k}\,\spn{\bC\bC^T-\bA\bA^T},
\end{equation*}
and the result in turn into (\ref{e_p}) to obtain the second term in the minimum,
\begin{eqnarray*}
\spn{(\bI-\bC_k\bC_k^{\dagger})\,\bA}^2 \leq \spn{\bA_{\perp}}^2 + 2\, \sqrt[p]{m-k}\>\spn{\bC\bC^T-\bA\bA^T}.
\end{eqnarray*}
\end{compactenum}
\paragraph{3. Frobenius norm}
This is the special case $p=2$ with $\stn{\bA}=\|\bA\|_F$ and $\son{\bA}=\|\bA\|_*$.
\end{proof}
\section{Approximation errors and angles between subspaces}\label{s_bounds}
We consider approximations where the rank of the orthogonal projector is at least as large as
the dimension of the
dominant subspace, and relate the low-rank approximation error to the
subspace angle between projector and target space.
After reviewing assumptions and notation (Section~\ref{s_ass}), we bound the low-rank approximation error
in terms of the subspace angle from below (Section~\ref{s_lower}) and from above (Section~\ref{s_upper}).
\subsection{Assumptions}\label{s_ass}
Given $\bA \in \rmn$ with a gap after the $k$th singular value,
\begin{equation*}
\|\bA\|_2=\sigma_1(\bA)\geq \cdots\geq \sigma_k(\bA)>
\sigma_{k+1}(\bA)\geq \cdots \geq \sigma_r(\bA)\geq 0,\qquad
r\equiv\min\{m,n\}.
\end{equation*}
Partition the full SVD $\bA=\bU\bSigma\bV^T$ in Section~\ref{s_sv} to distinguish between dominant
and subdominant parts,
$$\bU=\begin{pmatrix}\bU_k & \bU_{\perp}\end{pmatrix}, \qquad
\bV=\begin{pmatrix} \bV_k & \bV_{\perp}\end{pmatrix}, \qquad
\bSigma=\diag\begin{pmatrix}\bSigma_k & \bSigma_{\perp}\end{pmatrix},$$
where the dominant parts are
$$\bSigma_k\equiv\diag\begin{pmatrix}\sigma_1(\bA) & \cdots & \sigma_k(\bA)\end{pmatrix} \in\real^{k\times k},
\qquad \bU_k\in\real^{m\times k}, \qquad \bV_k\in\real^{n\times k},$$
and the subdominant ones
\begin{equation*}
\bSigma_{\perp}\in\real^{(m-k)\times (n-k)},
\qquad \bU_{\perp}\in\real^{m\times (m-k)}, \qquad \bV_{\perp}\in\real^{n\times (n-k)}.
\end{equation*}
Thus $\bA$ is a "direct sum"
$$\bA=\bA_k+\bA_{\perp}\qquad \text{where} \qquad
\bA_k\equiv \bU_k\bSigma_k\bV_k^T, \quad \bA_{\perp}\equiv \bU_{\perp}\bSigma_{\perp}\bV_{\perp}$$
and
\begin{equation}\label{e_orth}
\bA_{\perp}\bA_k^{\dagger} =\bzero=\bA_{\perp}\bA_k^T.
\end{equation}
The goal is to approximate the $k$-dimensional dominant left singular vector space,
\begin{equation}\label{e_proj}
\proju \equiv \bU_k\bU_k^T=\bA_k\bA_k^{\dagger}.
\end{equation}
To this end, let $\proj\in\rmm$ be an orthogonal projector as in (\ref{e_dproj}),
whose rank is at least as large as the dimension of the targeted subspace,
$$\rank(\proj)\geq \rank(\proju).$$
\subsection{Subspace angle as a lower bound for the approximation error}\label{s_lower}
We bound the low-rank approximation error from below by the subspace angle
and the $k$th singular value of $\bA$, in the two-norm and the Frobenius norm.
\begin{theorem}\label{t_lau}
With the assumptions in Section~\ref{s_ass},
\begin{equation*}
\|(\bI-\proj)\bA\|_{2,F} \geq \sigma_k(\bA)\,\|\sin{\bTheta}(\proj,\proju)\|_{2,F}.
\end{equation*}
\end{theorem}
\begin{proof}
From Lemma~\ref{l_projbasis}, (\ref{e_proj}), and (\ref{e_orth}) follows
\begin{eqnarray*}
\|\sin{\bTheta}(\proj,\proju)\|_{2,F}&=&\|(\bI-\proj)\,\proju\|_{2,F}=\|(\bI-\proj)\,\bA_k\bA_k^\dagger\|_{2,F}\\
&=& \|(\bI-\proj)\,(\bA_k+\bA_{\perp})\,\bA_k^\dagger\|_{2,F}= \|(\bI-\proj)\,\bA\bA_k^\dagger\|_{2,F}\\
& \leq &\|\bA_k^{\dagger}\|_2\, \|(\bI-\proj)\,\bA\|_{2,F}=\|(\bI-\proj)\bA\|_{2,F}/\sigma_k(\bA).
\end{eqnarray*}
\end{proof}
\subsection{Subspace angle as upper bound for the approximation error}\label{s_upper}
We present upper bounds for the low-rank approximation error in terms of the subspace
angle, the two norm (Theorem~\ref{t_lal1}) and Frobenius norm (Theorem~\ref{t_lal2}).
The bounds are guided by the following observation.
In the ideal case, where $\proj$ completely captures the target space, we have
$\range(\proj)=\range(\proju)=\range(\bA_k)$,
and
\begin{equation*}
\|\sin{\bTheta}(\proj,\proju)\|_{2,F}=0, \qquad
\|(\bI-\proj)\bA\|_{2,F}=\|\bA_{\perp}\|_{2,F}=\|\bSigma_{\perp}\|_{2,F},
\end{equation*}
thus suggesting an additive error in the general, non-ideal case.
\begin{theorem}[Two-norm]\label{t_lal1}
With the assumptions in Section~\ref{s_ass},
\begin{eqnarray*}
\|(\bI-\proj)\bA\|_2\leq \|\bA\|_2\,\|\sin{\bTheta(\proj,\proju)}\|_2
+\|\bA-\bA_k\|_2\,\|\cos{\bTheta(\bI-\proj,\bI-\proju)}\|_2.
\end{eqnarray*}
If also $k<\rank(\proj)+k<m$, then
\begin{eqnarray*}
\|(\bI-\proj)\bA\|_2\leq \|\bA\|_2\,\|\sin{\bTheta(\proj,\proju)}\|_2 +\|\bA-\bA_k\|_2.
\end{eqnarray*}
\end{theorem}
\begin{proof}
From $\bA=\bA_k+\bA_{\perp}$ and the triangle inequality follows
\begin{eqnarray*}
\|(\bI-\proj)\bA\|_2&\leq&\|(\bI-\proj)\bA_k\|_2+ \|(\bI-\proj)\bA_{\perp}\|_2.
\end{eqnarray*}
\paragraph{First summand}
Since $\rank(\proj)\geq \rank(\proju)$, Lemma~\ref{l_projbasis} implies
\begin{eqnarray*}
\|(\bI-\proj)\bA_k\|_2&\leq& \|(\bI-\proj)\,\bU_k\|_2\|\bSigma_k\|_2=\|\bA\|_2\,\|(\bI-\proj)\proju\|_2\\
&=&\|\bA\|_2\, \|\sin{\bTheta}(\proj,\proju)\|_2
\end{eqnarray*}
Substitute this into the previous bound to obtain
\begin{eqnarray}\label{e_5}
\|(\bI-\proj)\bA\|_2&\leq&\|\bA\|_2\,\|\sin{\bTheta}(\proj,\proju)\|_2+ \|(\bI-\proj)\bA_{\perp}\|_2.
\end{eqnarray}
\paragraph{Second summand}
Submultiplicativity implies
\begin{eqnarray*}
\|(\bI-\proj)\bA_{\perp}\|_2\leq
\|(\bI-\proj)\,\bU_{\perp}\|_2\,\|\bSigma_{\perp}\|_2=
\|\bA-\bA_k\|_2\>\|(\bI-\proj)\,\bU_{\perp}\|_2.
\end{eqnarray*}
Regarding the last factor, the full SVD of $\bA$ in Section~\ref{s_ass} implies
$$\range(\bU_{\perp})=\range(\bU_{\perp}\bU_{\perp}^T)=\range(\bU_k\bU_k^T)^{\perp}=\range(\proju)^{\perp}
=\range(\bI-\proju)$$
so that
\begin{equation*}
\|(\bI-\proj)\,\bU_{\perp}\|_2=\|(\bI-\proj)\,(\bI-\proju)\|_2=\|\cos{\bTheta(\bI-\proj,\bI-\proju)}\|_2.
\end{equation*}
Thus,
$$\|(\bI-\proj)\bA_{\perp}\|_2\leq\|\bA-\bA_k\|_2\,\|\cos{\bTheta(\bI-\proj,\bI-\proju)}\|_2.$$
Substitute this into (\ref{e_5}) to obtain the first bound.
\paragraph{Special case $\rank(\proj)+k<m$}
From Corollary~\ref{c_csnequal} follows with $\ell\equiv \rank(\proj)$
\begin{eqnarray*}
\|\cos{\bTheta}(\bI-\proj,\bI-\proju)\|_2=
\Big\|\begin{pmatrix}\bI_{m-(k+\ell)} & \\ & \cos{\bTheta}(\proj,\proju)\end{pmatrix}\Big\|_2=1.
\end{eqnarray*}
\end{proof}
\begin{theorem}[Frobenius norm]\label{t_lal2}
With the assumptions in Section~\ref{s_ass}
\begin{eqnarray*}
\|(\bI-\proj)\bA\|_2\leq \|\bA\|_2\,\|\sin{\bTheta(\proj,\proju)}\|_F+
\min\left\{\|\bA-\bA_k\|_2 \, \|\bGamma\|_F,\>\|\bA-\bA_k\|_F\,\|\bGamma\|_2\right\},
\end{eqnarray*}
where $\bGamma\equiv \cos{\bTheta(\bI-\proj,\bI-\proju)}$.
If also $k<\rank(\proj)+k<m$, then
\begin{eqnarray*}
\|(\bI-\proj)\bA\|_F&\leq& \|\bA\|_2\,\|\sin{\bTheta(\proj,\proju)}\|_F +\|\bA-\bA_k\|_F.
\end{eqnarray*}
\end{theorem}
\begin{proof}
With strong submultiplicativity, the analogue of (\ref{e_5}) is
\begin{eqnarray}\label{e_5b}
\|(\bI-\proj)\bA\|_F&\leq&\|\bA\|_2\,\|\sin{\bTheta}(\proj,\proju)\|_F+ \|(\bI-\proj)\bA_{\perp}\|_F.
\end{eqnarray}
There are two options to bound
$\|(\bI-\proj)\bA_{\perp}\|_F=\|(\bI-\proj)\,\bU_{\perp}\bSigma_{\perp}\|_F$, depending on which factor gets
the two norm. Either
\begin{eqnarray*}
\|(\bI-\proj)\,\bU_{\perp}\bSigma_{\perp}\|_F\leq \|(\bI-\proj)\,\bU_{\perp}\|_F\,\|\bSigma_{\perp}\|_2
=\|\bA-\bA_k\|_2\,\|(\bI-\proj)\,\bU_{\perp}\|_F
\end{eqnarray*}
or
\begin{eqnarray*}
\|(\bI-\proj)\,\bU_{\perp}\bSigma_{\perp}\|_F&\leq &\|(\bI-\proj)\,\bU_{\perp}\|_2\,\|\bSigma_{\perp}\|_F
=\|\bA-\bA_k\|_F\,\|(\bI-\proj)\,\bU_{\perp}\|_2.
\end{eqnarray*}
As in the proof of Theorem~\ref{t_lal1} one shows
\begin{equation*}
\|(\bI-\proj)\,\bU_{\perp}\|_{2,F}=\|\cos{\bTheta(\bI-\proj,\bI-\proju)}\|_{2,F},
\end{equation*}
as well as the expression for the special case $\rank(\proj)+k<m$.
\end{proof}
\section{CS Decompositions}\label{s_app}
We review expressions for the CS decompositions from
\cite[Theorem 8.1]{PaigeWei94} and \cite[Section 2]{ZKny13}.
We consider a subspace $\range(\bZ)$ of dimension $k$, and a subspace of $\range(\hZ)$ of dimension $\ell\geq k$,
whose dimensions sum up to less than the dimension of the host space.
Let $\begin{pmatrix} \bZ & \bZ_{\perp}\end{pmatrix},
\begin{pmatrix} \hZ & \hZ_{\perp}\end{pmatrix} \in\rmm$ be orthogonal matrices where
$\bZ\in\real^{m\times k}$ and $\hZ\in\real^{m\times \ell}$ with $\ell\geq k$.
Let
$$\begin{pmatrix} \bZ & \bZ_{\perp}\end{pmatrix}^T\,
\begin{pmatrix} \hZ & \hZ_{\perp}\end{pmatrix} =
\begin{pmatrix} \bZ^T\hZ & \bZ^T\hZ_{\perp} \\ \bZ_{\perp}^T\hZ & \bZ_{\perp}^T\hZ_{\perp}\end{pmatrix}=
\begin{pmatrix}\bQ_{11} & \\ &\bQ_{12}\end{pmatrix}\, \bD\,
\begin{pmatrix}\bQ_{21} & \\ & \bQ_{22}\end{pmatrix}$$
be a CS decomposition where $\bQ_{11}\in\real^{k\times k}$, $\bQ_{12}\in\real^{(m-k)\times (m-k)}$,
$\bQ_{21} \in\real^{\ell\times \ell}$ and $\bQ_{22}\in\real^{(m-\ell)\times (m-\ell)}$ are all orthogonal matrices.
\begin{theorem}\label{t_csnequal}
If $k<\ell<m-k$ then
\begin{equation*}
\bD \ = \ \begin{blockarray}{ccccccc}
r&s&\ell-(r+s)&m-(k+\ell)+r&s& k-(r+s)\\
\begin{block}{[ccc|ccc]c}
\bI_r & & &\bzero& & &r \\
&\bC & & &\bS&&s\\
& & \bzero & & &\bI_{k-(r+s)} &k-(s+r)\\
\BAhline
\bzero& & &-\bI_{m-(k+\ell)+r}&&&m-(k+\ell)+r\\
& \bS & & & -\bC &&s \\
&& \bI_{\ell-(r+s)} & & & \bzero& \ell-(r+s)\\
\end{block}
\end{blockarray}.
\end{equation*}
Here $\bC^2+\bS^2=\bI_s$ with
$$\bC=\diag\begin{pmatrix}\cos{\theta_1} & \cdots & \cos{\theta_s} \end{pmatrix}, \quad
\bS=\diag\begin{pmatrix}\sin{\theta_1} & \cdots & \sin{\theta_s}\end{pmatrix},$$
and
\begin{eqnarray*}
&&r= \dim\left(\range(\bZ)\cap\range(\hZ)\right), \quad
m-(k+\ell)+r=\dim\left(\range(\bZ_{\perp})\cap\range(\hZ_{\perp})\right)\\
&&\ell-(r+s) = \dim\left(\range(\bZ_{\perp})\cap\range(\hZ)\right), \
k-(r+s)=\dim\left(\range(\bZ)\cap\range(\hZ_{\perp})\right).
\end{eqnarray*}
\end{theorem}
\begin{corollary}\label{c_csnequal}
From Theorem~\ref{t_csnequal} follows
\begin{eqnarray*}\label{e_csequal1}
\|\sin{\bTheta}(\bZ,\hZ)\|_{2,F}&=&\|\bZ^T\hZ_{\perp}\|_{2,F}
=\Big\|\begin{pmatrix}\bS & \\ & \bI_{k-(r+s)}\end{pmatrix}\Big\|_{2,F}\\
\|\cos{\bTheta}(\bZ,\hZ)\|_{2,F}&=&\|\bZ^T\hZ\|_{2,F}=\Big\|\begin{pmatrix}\bI_r & \\ & \bC\end{pmatrix}\Big\|_{2,F}\\
\|\cos{\bTheta}(\bZ_{\perp},\hZ_{\perp})\|_{2,F}&=&\|\bZ_{\perp}^T\hZ_{\perp}\|_{2,F}=
\Big\|\begin{pmatrix}\bI_{m-(k+\ell)} & \\ & \cos{\bTheta}(\bZ,\hZ)\end{pmatrix}\Big\|_{2,F}.
\end{eqnarray*}
\end{corollary}
|
1,314,259,993,582 | arxiv | \section{Introduction}
The main asteroid belt is collisionally dominated with large asteroids' shapes, sizes and surface geology controlled by impacts. Studies of collisions help us to understand the evolution of the shape of the asteroid population and in turn the formation of our Solar system. These studies may involve laboratory experiments, computer modelling or observational programmes.
The evidence for collisions can be seen indirectly in main-belt asteroid families \citep{families}, asteroid satellites and binaries \citep{satellites}. It can also be seen directly in recently observed collisions \citep{snodgrass, jewitt, stevenson}.
There are three possible collisions observed to date.
In 2009 the 120 m diameter asteroid P/2010 A2 suffered a collision with a 6-9 m estimated diameter impactor \citep{snodgrass} (but see section 4.4).
In 2010 another asteroid, (596) Scheila (113 km diameter), was hit with a $\sim35$m diameter impactor \cite{jewitt}.
The most recent potential collision involved the object P/2012 F5 (Gibbs), which like others was originally identified as a potential main-belt comet \cite{stevenson}.
Events like the (596) Scheila collision should occur approximately every 5 years and collisions with asteroids $<$10m even more often \citep{bodewits}.
Several recent surveys are capable of detecting collisions and cratering events. For example, the Canada-France-Hawaii Telescope Legacy Survey was used to search for Main-Belt comets among 25240 objects in 2003-2009 \citep{cfhtls}, the Thousand Asteroid Lightcurve Survey (924 objects) was conducted with the Canada-France-Hawaii Telescope in September 2006 \citep{tals} and the Hawaii Trails project was conducted in 2009 (599 objects) \citep{htrails}. While none of the surveys mentioned above were specifically looking for main belt collisions, the methods used in search for main belt comets would have also revealed any collisional events.
There are also current surveys fully or partly dedicated to discovering Near Earth Asteroids, such as Pan-STARRS 1 \citep{pstarrs1}, the Lincoln Near-Earth Asteroid Research (LINEAR, responsible for discovery of P/2010 A2) \citep{linear}, the Catalina Sky Survey \citep{catalina} and the VST ATLAS survey \citep{atlas} that are all capable of detecting main-belt collisions.
Much work has been done in modelling the parameters (i.e. shape of debris, brightness, total ejected mass, impactor mass) of known observed collisions \citep{kleyna}, \citep{ishiguro}, \citep{HH2007}, \citep{HH2011} ; and hydrodynamic modelling of generalised collision \citep{benz}.
This work focuses solely on the magnitude change following an impact as it is most likely to be observable by optical telescopes. Rather than looking at a specific object in the main belt, the described model looks at what would be expected with generic asteroids.
\section{Model description}
\subsection{Cratering physics}
Our model is based on the work by \cite{HH2007}, who provide a summary of scaling laws that allows calculation of crater size using properties of the target and impactor, based on the results of impact experiments. These laws can also be used to calculate the evolution of the ejecta dispersal and consequently estimate the amount of material ejected and increase in brightness following a collision.
The decrease in magnitude of the target asteroid is going to depend on the amount of material that was ejected and whether it is optically thin or not.
At high impact speeds, transfer of the energy and momentum of the impactor into the target occurs over area on the order of impactor size, while the resulting crater usually exceeds this size by many times. It is therefore a reasonable approximation to assume that impact occurs as a point source. Using theoretical analyses of mechanics of crater formation, Holsapple and Housen showed that the crater and ejected material characteristics depend on the quantity $aU^{\mu} \delta^{\nu}$, where $a$ is the radius, $U$ is the normal velocity component of the impactor and $\delta$ is the density of the impactor; $\mu$ and $\nu$ are scaling exponents.
The scaling exponents depend on the material properties. Theoretical values of $\mu$ range from 1/3 to 2/3 \citep{numuvalues} and are a measure of the energy dissipation by material; a more porous material can dissipate energy more effectively and will have a lower value of this exponent. Experimentally determined values of $\mu$ are $\sim$0.55 for non-porous materials (e.g. rocks and wet soils), 0.41 for moderately porous materials (e.g. sand and cohesive soils) and 0.33 to 0.40 for highly porous materials \citep{numuvalues}. Experimental values for $\nu$ were found to be the same for all materials at around 0.4 \citep{numuvalues}. By selecting appropriate material scaling parameters for a given impact and inserting them into a general expression for the relationship between radii of involved objects and crater size, a reasonably accurate estimate of the crater size (as well as crater formation time and transient crater growth) can be made.
We now summarise how we use the previous studies in our calculations. Consider a spherical, non-rotating asteroid of radius $r$ following an impact from an object with radius $a$ at sub earth point. The general form of equation for crater size $R$ consists of strength and gravity term:
\begin{equation}
R=aK_{1}(\rm{gravity \:term}+\rm{strength\: term})^{-\frac{\mu}{2+\mu}}
\end{equation}
where
\begin{equation}
\rm{gravity\: term}= \frac{S_{g}a}{U^{2}}\big( \frac{\rho}{\delta_{grain}} \big)^{ \frac{2\nu}{\mu}}
\end{equation}
\begin{equation}
\rm{strength\: term} = \big(\frac{Y}{\rho U^{2}}\big)^{ \frac{2+\mu}{2} } \big( \frac{\delta_{grain}}{\rho}\big)^{ \frac{\nu(2+\mu)}{\mu}}
\end{equation}
Here $K_{1}$ is a scaling parameter (1.03, 1.17, 0.725 for sand/cohesive soil, wet soils/rock and highly porous material respectively; \citep{HH2007}), $Y$ is the average strength of the target material, $\rho$ is the grain density of target, $U = 5$ km s$^{-1}$~\citep{impactvel} is the normal velocity component, $\delta_{grain}$ is the grain density of the impactor, $S_{g}$ is the surface gravity of the target asteroid with mass $M$ and radius $r$, calculated as follows:
\begin{equation}
S_{g}=\frac{GM}{ r^{2}}
\end{equation}
Depending on the asteroid type, different values of bulk density (for calculation of target asteroid mass) and grain density (for calculation of ejected mass) are used. The values and their sources are summarised in Table \ref{tab:CSD}. Bulk densities of C- and S-type asteroids were taken from weighted averages of corresponding subclasses as summarised in Table 3 of \cite{carry}. Grain densities of C- and S-types are assumed to be the same as their most likely meteorite analogues \citep{porosity}. Density of D-type asteroids is approximated by bulk and grain densities of the Tagish lake meteorite \citep{Dbulkdensity, Dgraindensity}.
\begin{table}[ht]
\caption{Average bulk and grain density of asteroids depending on taxonomic type.}
\centering
\begin{tabular}{c c c }
\hline\hline
Asteroid type & Bulk density (kg m$^{-3}$) & Grain density (kg m$^{-3}$). \\ [0.5ex]
\hline
C & 1840 \citep{carry} & 2710 \citep{porosity} \\
S & 2640 \citep{carry} & 3700 \citep{porosity} \\
D & 1670 \citep{Dbulkdensity} & 2770 \citep{Dgraindensity} \\ [1ex]
\hline
\end{tabular}
\label{tab:CSD}
\end{table}
The range of material strengths used is presented in Table~\ref{tab:strength}. The strength value selected in this study was varied for each taxonomic type to explore the relationship between the type, strength and the corresponding magnitude change.
\begin{table}[ht]
\caption{Average material strength used in calculations for a selected scaling.}
\centering
\begin{tabular}{c c}
\hline\hline
Material & Average strength $Y$ (MPa) \\ [0.5ex]
\hline
Sand/cohesive soil & 0.18 \citep{H1993} \\
Wet soil/rock & 1.14 \citep{H1993} \\
Highly porous & 0.001 \citep{HH2007}\\ [1ex]
\hline
\end{tabular}
\label{tab:strength}
\end{table}
The crater radius $R$ calculated in this way has a corresponding mass $M_{\rm{crater}}$:
\begin{equation}
M_{\rm{crater}}=k_{\rm{crater}}\rho R^{3}
\end{equation}
where the scaling factor $k_{\rm{crater}}$ is taken to be 0.75 for cohesive soils, 0.8 for wet soils/rocks or 0.4 for highly porous material \citep{HH2011}.
As we are interested in the ejected mass, since it is only that which contributes to the observed magnitude change of the asteroid, the full crater mass will give an overestimate of brightness. The total crater volume is made up of a volume of ejected mass, a volume of the mass that is uplifted near the crater rim and a volume due to compaction. The fraction $k_{\rm{ejecta}}$ of the total crater mass that corresponds to ejected mass is of order 0.2-0.5 \citep{HH2011}.
\begin{equation}
M_{\rm{ejecta}}=k_{\rm{ejecta}}M_{\rm{crater}}
\end{equation}
Throughout this work we assume $k_{\rm{ejecta}}$ of 0.3 as being most appropriate to asteroids.
\subsection{Velocity shell model}
We consider the ejecta leaving the asteroid surface after the collision event. For simplicity, we assume that the debris expands spherically outwards from asteroid, the debris with each velocity $v_{n}$ forming a shell of radius $r_{s}$ (see Figure ~\ref{fig:velshell}). Effects from rotation of the target or impactor are beyond the scope of the current model. Impact experiments show that there is no significant correlation between velocity and mass of the particles \citep{massvel}. Therefore, each velocity shell is taken to have the same particle size distribution described below, in Section \ref{magnitude_change_calculation_section}. As our aim is to model observable brightening from Earth, we assume that the ejecta cloud is centred on the asteroid, as at early epochs the asteroid itself and the ejecta will be unresolved. We also assume that the brightness of the asteroid plus ejecta is measured through an aperture of fixed radius $r_{\rm{ap}}$ and centred on the asteroid. To obtain the amount of material that is visible in the aperture, and consequently the visible increase in asteroid magnitude, we need to look at the fraction of each debris shell that fits completely or partially in the aperture (ejecta visibility fraction $f_{\rm{vis}}$).
\vspace{0.5in}
\begin{figure}[ht!]
\begin{center}
\subfigure[Velocity shells.]{
\label{fig:shell}
\includegraphics[trim = 70mm 0.1mm 70mm 20mm, clip, width=0.45\textwidth]{concentric_shells_eye.pdf}
}
\subfigure[Spherical cap.]{
\label{fig:cap}
\includegraphics[trim = 50mm 190mm 80mm 30mm, clip, width=0.45\textwidth]{cap.pdf}
}
\end{center}
\caption{Velocity shell model: (a) Ejecta expands spherically outwards from the asteroid. The n$^{th}$ shell expands with velocity $v_{n}$. The portion of each shell that fits into the aperture $r_{ap}$ is marked in solid line. (b) Spherical cap indicates the portion of each shell that fits the aperture.}
\label{fig:velshell}
\end{figure}
\vspace{0.5in}
There are three possibilities for an individual shell at a given time $t$:
\begin{enumerate}
\item the entire shell fits within the aperture and hence $f_{\rm{vis}} = 1$
\item none of the shell is visible, i.e. the material is not above the surface of the asteroid as it has fallen back, $f_{\rm{vis}} = 0$
\item part of the shell fits into the aperture ($0<f_{\rm{vis}}<1$) (see Figure ~\ref{fig:velshell})
\end{enumerate}
In the latter case, the amount of material visible can be estimated by calculating the ratio of the surface area corresponding to the arc that fits into the aperture (spherical cap) to the total surface area of the sphere.
\begin{equation}
f_{\rm{vis}} = \frac{2(2 \pi r h)}{4\pi r^{2}}
\end{equation}
Here $h$ is the height of the cap and the extra factor of 2 accounts for near and far side material, $h=r-\sqrt{r^{2}-r_{\rm{ap}}^{2}}$ and $r_{\rm{ap}}$ is the aperture radius that equals the radius of the cap.
In the situation of no gravity acting on the debris, the radii at each velocity can be described straightforwardly by $r_{s} = vt$, however we need to consider the gravitational case.
For simplicity, we consider a single debris shell of radius $r_{s}$ and velocity $v$ over the asteroid of radius $r$. The acceleration due to the asteroid's gravity that the debris experience is inversely proportional to the square of shell radius ($\frac{1}{r_{s}^{2}}$) and also happens to be the second derivative of radius with respect to time.
\begin{equation}
\frac{d^{2}r_{s}}{dt^{2}}=Cr_{s}^{-2}
\end{equation}
$C$ is a constant of proportionality which we calculate by considering the boundary conditions. At the asteroid surface $r_{s} = r$ and the acceleration due to gravity is the surface gravity $S_{g}$. At the surface the original equation reduces to $Cr^{-2} = S_{g}$, which in turn gives $C= S_{g}r^{2}$. Therefore, the radius of the spherical shell of debris at velocity $v$ is governed by a second order differential equation of the form:
\begin{equation}
\frac{d^{2}r_{s}}{dt^{2}}=S_{g}r^{2}r_{s}^{-2}
\end{equation}
with the boundary conditions that at $t = 0$, $r_{s} = r$, $\frac{dr_{s}}{dt} = v_{\rm{initial}}$. Solving this equation will give the shell radii at all times $t$. The total mass within the aperture $M_{\rm{vis}}$ is the integral of all velocity shell fractions within the aperture i.e. the shell mass $M_{shell}$ multiplied by $f_{\rm{vis}}$
$$ M_{\rm{vis}} = \int_{v_{\rm{min}}}^{v_{\rm{max}}}M_{\rm{shell}}f_{\rm{vis}} {\rm{d}}v$$
In the code it is calculated by taking a sum of the individual contributions from each velocity shell, thus
$$M_{\rm{vis}} = \Sigma_{v{\rm{=}}v_{\rm{min}}}^{v_{\rm{max}}}M_{\rm{shell}}f_{\rm{vis}}$$
As the asteroid fragments will be ejected at a continuous range of velocities, increasing the number of shells considered improves the accuracy of calculations.
\subsection{Magnitude change calculation}
\label{magnitude_change_calculation_section}
At this stage we implicitly assume that the scattering function for the asteroid and the ejecta particles is the same and that the ejecta is optically thin throughout.
Assuming all the ejected particles contribute to reflected light with the same albedo as the target asteroid, the total visible surface area after the collision is $A_{c} = A_{a}+ A_{p}$ where $A_{a}$ is the area of target asteroid, $A_{p}$ is the area of ejected particles. The latter can be calculated assuming that the particle size distribution follows a power law of the following form \citep{ishiguro}:
\begin{equation}
N(a)da = \left\{ \begin{array}{rl}
N_{0} \big( \frac{a}{a_{0}}\big)^{q}da, &\mbox{ $a_{min}\leq a \leq a_{max}$} \\
0, &\mbox{ $a<a_{min}, a>a_{max}$}
\end{array} \right.
\end{equation}
Here $a_{min}=0.1$ $\mu$m is chosen to be the minimum particle radius because scattering is inefficient for particle sizes much smaller than the wavelength of the scattered light, $a_{max}$ is the maximum particle radius, $N_{0}$ is the reference dust abundance at reference size $a_{0}$ and $q=3.5$ is the assumed power law index \citep{jewittcent}.
The maximum particle radius $a_{max}$ is calculated using the expression derived from the power-law function of the ejection velocity of dust particles \citep{ishiguro}.
\begin{equation}
a_{max}= \sqrt[k]{ \frac{1}{V_{0}} \sqrt{ \frac{2GM}{r}} }
\end{equation}
Here $V_{0}$=80 m s$^{-1}$ is the reference ejection velocity of 1 $\mu$m particles, $k=1/4$ is the power index of size dependence of ejection velocity, $M$ and $r$ are mass and radius of the target respectively.
After integration we get cross-sectional area per unit mass of $\sigma_{a}$ (m$^{2}$ kg$^{-1}$), and the total cross-sectional area of particles around the asteroid $A_{p}$ are:
\begin{equation}
\sigma_{a}=-\frac{3}{4\rho_{grain}}\frac{ a_{max}^{-0.5}-a_{min}^{-0.5} }{ a_{max}^{0.5}-a_{min}^{0.5}}
\end{equation}
\begin{equation}
A_{p}= \sigma_{a}M_{\rm{vis}}
\end{equation}
The change in magnitude is going to be related to the ratio of the total area after collision $A_{c}$ and initial target asteroid area $A_{a}$:
\begin{equation}
\Delta m= - 2.5\log_{10}\frac{A_{c}}{A_{a}}
\end{equation}
Any ejecta in front of the asteroid will obscure a cross-sectional area equal to its own area. Given that we assume the same scattering function for asteroid and ejecta, this means ejecta directly in front of the asteroid does not contribute to the net brightness. Any ejecta behind the asteroid is clearly not visible. Therefore all ejected material along the line of sight of the asteroid should be ignored for brightness calculations. This is implemented by the subtraction of a pair of spherical caps with radius equal to that of the asteroid.
\section{Modelling results}
\subsection{Model setup}
We have looked at three taxonomic types that are common in the main belt - C, S and D types \citep{taxtypes}. The differences in the types are accounted for using their corresponding grain and bulk densities during calculation of the ejected mass and magnitude change.
In the simulation we varied the taxonomic type (C, S, D) and strength (sand/cohesive soil and highly porous) value to see if any patterns emerge. We have used a 50 by 50 point grid of target and impactor radii, ranging from 1-100 km and 1-100 m respectively and logarithmically spaced, and 500 velocity shells linearly spaced over interval $10^{-2}$ to $10^{2}$ m s$^{-1}$. The lowest initial velocity has to be above zero for the purposes of calculations and the amount of material being ejected at and beyond $10^{2}$ m s$^{-1}$ is a very small fraction of the total ejected mass and therefore considered negligible, so we take $10^2$ m s$^{-1}$ as a upper limit of the initial velocity range. Given that there is a range of target and impactor pairs that would result in catastrophic disruption (i.e. a dispersive shattering event leaving no remnant larger than 1/2 of the original mass of target asteroid \citep{greenberg2}) rather than in sub catastrophic collision, the results of our model for that region will not be physical. We have estimated the location of the disruption region by calculating the approximate impactor radius necessary for catastrophic disruption of a basalt target assuming average density of 2500 kg m$^{-3}$ and impact velocity of 5 km s$^{-1}$ \citep{benz}. This region is marked in black on the figures.
The code for this model was written in Python 2.7 (includes NumPy and SciPy packages), plotting of the output was performed in MatLab. The code requires a separate run for each pair of values for the taxonomic type and strength regime (i.e. C-type in sand/cohesive soil and C-type in highly porous regime would be two different code runs). Each run from impact until 7 days after impact with the above parameters takes approximately 2 hours on a single core of an Intel Core i5 2410M processor running at 2.3 GHz.
\subsection{Model Results}
Figure~\ref{fig:subfiguresCHPSAND} shows the resulting predicted change in magnitude post-collision for a C-type asteroid with both a highly porous and a sand/cohesive soil strength at two epochs after the impact (1 and 7 days). Figure~\ref{fig:subfiguresCHPSANDm-1} shows the calculated time it takes for the brightness increase to reach $-1.0$ magnitudes for C-types in the highly porous and the sand/cohesive soil strength regimes. This value has been chosen because the one sub-catastrophic asteroid collision confirmed so far between the known main-belt asteroid (596) Scheila and an impactor was observed to have a brightness increase of $\simeq-1$ magnitude~\citep{jewitt}. Conclusive attribution of a collisional nature to a smaller magnitude decrease may be difficult, due to natural variations in brightness caused by rotational modulation of the light curves and uncertainty in asteroidal absolute magnitudes~\citep{pravecbias}.
\begin{figure}[htp]
\begin{center}
\subfigure[highly porous regime, 1 day]{
\label{fig:CHP24}
\includegraphics[trim = {16mm} {50mm} {25mm} {60mm}, clip, width=0.45\textwidth]{C_hp_C_1day_new_density_500.pdf}
}
\subfigure[highly porous regime, 7 days]{
\label{fig:CHP1w}
\includegraphics[trim = {16mm} {50mm} {25mm} {60mm}, clip,width=0.45\textwidth]{C_hp_C_7days_new_density_500.pdf}
}
\subfigure[sand/cohesive soil regime, 1 day]{
\label{fig:CSAND24}
\includegraphics[trim = {16mm} {50mm} {25mm} {60mm}, clip, width=0.45\textwidth]{C_s_C_1day_new_density_500.pdf}
}
\subfigure[sand/cohesive soil regime, 7 days]{
\label{fig:CSAND1w}
\includegraphics[trim = {16mm} {50mm} {25mm} {60mm}, clip,width=0.45\textwidth]{C_s_C_7days_new_density_500.pdf}
}
\end{center}
\caption{
C-type impactor collision with C-type target: \ref{fig:CHP24}-\ref{fig:CHP1w} highly porous strength regime: magnitude change 1 and 7 days post-collision for a range of target and impactor radii. \ref{fig:CSAND24}-\ref{fig:CSAND1w} sand/cohesive soil strength regime: magnitude change 1 and 7 days post-collision for a range of target and impactor radii. Impactor radii range from 1 to 100 m, target radii: 1-100 km. Colour bar shows the magnitude change. Catastrophic disruption region is marked in black. Region where optical thickness of the debris is potentially significant is above the dashed line. Solid black lines indicate contours where magnitude change is -1, -3, -5, -7 and -9. }%
\label{fig:subfiguresCHPSAND}
\end{figure}
\begin{figure}[ht!]
\begin{center}
\subfigure[highly porous regime, 1 day]{
\label{fig:CHPm-124}
\includegraphics[trim = 15mm 50mm 23mm 60mm, clip,width=0.45\textwidth]{magfall_C_hp_C_1day_500.pdf}
}
\subfigure[highly porous regime, 7 days]{
\label{fig:CHPm-11w}
\includegraphics[trim = 15mm 50mm 23mm 60mm, clip,width=0.45\textwidth]{magfall_C_hp_C_7days_500.pdf}
}
\subfigure[sand/cohesive soil regime, 1 day]{
\label{fig:CSANDm-124}
\includegraphics[trim = 15mm 50mm 23mm 60mm, clip,width=0.45\textwidth]{magfall_C_s_C_1day_500.pdf}
}
\subfigure[sand/cohesive soil regime, 7 days]{
\label{fig:CSANDm-11w}
\includegraphics[trim = 15mm 50mm 23mm 60mm, clip,width=0.45\textwidth]{magfall_C_s_C_7days_500.pdf}
}
\end{center}
\caption{
Time taken for the magnitude change to fall to -1 magnitudes after a C-type impactor collision with C-type target. Figures \ref{fig:CHPm-124} and \ref{fig:CSANDm-124} show the impactor-target regime where this occurs within 1 day for highly porous and sand/cohesive soil strength regimes respectively. Figures \ref{fig:CHPm-11w}-\ref{fig:CSANDm-11w} show the same data but now extended to show where this occurs within 7 days of impact. For each figure the colour coding for this timescale is shown on the right, with the truncated region (timescale greater than 1 day or 7 days) shown in dark blue on the plot. The region where a catastrophic disruption would occur is marked in black. }%
\label{fig:subfiguresCHPSANDm-1}
\end{figure}
\begin{figure}[ht]
\centerline{\includegraphics[trim = 12mm 50mm 23mm 60mm, clip, width=0.6\textwidth]{stepsill.pdf}}
\caption{Magnitude change with time for three targets of radii 1325, 1600 and 1930 m after collision with 20 m radius impactor (all S-type, sand/cohesive soil regime). Steps are clearly visible due to quantised nature of the modelled velocity shells and occur at a different position for each target. }
\label{fig:steps}
\end{figure}
\clearpage
There are vertical structures in these plots that are not of physical significance, but are rather features of the model limitations. Three processes affect the total magnitude decrease post collision: the constant expansion of debris moving out of the aperture, material falling back on the surface of the target asteroid and therefore disappearing from observation; and material re-entering the aperture due to the gravitational attraction of the target asteroid. In an ideal simulation of the process, the change in magnitude as a function of time would be a smooth function, being the result of the interplay of all three processes. The function would be smooth because we expect the size and velocity distributions of the debris to be continuous. As velocity shells in this model are quantised, this leads to the creation of artefacts at certain values of target radius.
The effect of this can be seen clearly in Figure~\ref{fig:steps}, which shows the predicted brightness increase after an impact of a 20 m radius impactor onto 1325, 1600, 1930 m radius target (all S-type, sand/cohesive soil strength regime). For lower velocity shells that fall back onto the asteroid at early times these steps merge to give a continuous slope. Higher velocity shells never return as they have reached escape velocity, and thus never produce a sharp decline.
For a small number of shells with intermediate velocity, the return of the shell to the surface will be sufficiently distinct temporally from the other shells to produce a clearly identifiable step.
The velocity shells leaving the aperture do not produce a sharp decline, because the fraction of the shell in the aperture decreases as a continuous function of time after the impact.
The location of these steps is independent of the impactor radius and depends solely on the target radius due to the dependance on the surface gravity. These steps are clearly visible in Figures \ref{fig:subfiguresCHPSAND} and after as vertical structures that should not be interpreted as being significant.
One of the ways to minimise this and have a good resolution in the images is by using a large number of shells, however this increases the computation time. The 500 shells used in this study was selected as a compromise between resolution and run time. Figure \ref{fig:subfiguresCCsand_shells_comparison} shows a comparison between a plot using 500 shells and 5000 shells (20 hours run-time). The latter shows a reduction in vertical structures without substantial difference in output. Difference images show a discrepancy between the 500 shells and 5000 shells results being confined to the vertical structures only.
\begin{figure}[ht]
\begin{center}
\subfigure[7 days, 500 shells]{%
\label{fig:CCsand500_7d}
\includegraphics[trim = {16mm} {50mm} {25mm} {60mm}, clip,width=0.45\textwidth]{C_s_C_7days_new_density_500.pdf}
}
\subfigure[7 days, 5000 shells]{
\label{fig:CCsand5000_7d}
\includegraphics[trim = {16mm} {50mm} {25mm} {60mm}, clip,width=0.45\textwidth]{C_s_C_7days_new_density_5000.pdf}
}
\end{center}
\caption{
C-type impactor collision with C-type target, sand/cohesive soil strength regime: magnitude change 7 days post-collision for a range of target and impactor radii. Impactor radii range from 1 to 100 m, target radii: 1-100 km. Colour bar shows the magnitude change. Catastrophic disruption region is marked in black. Comparison of using (a) 500 and (b) 5000 velocity shells. The plot with 5000 shells shows a clear reduction in the vertical structures in comparison to 500 shells plot. Solid black lines indicate contours where magnitude change is -1, -3, -5, -7 and -9.}%
\label{fig:subfiguresCCsand_shells_comparison}
\end{figure}
We have chosen to only run the model up to 7 days post-collision, because a large proportion of the material (by surface area) would leave the aperture due to radiation pressure effects, which are not currently included in the model.
The time $t$ it takes for material to escape the aperture radius $r_{ap}$ is approximately equal to
\begin{equation}
t\approx\left(\frac{2r_{ap}}{a}\right)^{\frac{1}{2}}
\end{equation}
where $a$ is acceleration, that can be defined in terms of ratio between radiation pressure acceleration and the local gravity, $\beta$, and gravitation acceleration to the Sun at 1 AU $g_{\odot}$: $a=\beta g_{\odot}$.
In turn, $g_{\odot}\approx \frac{GM_{\odot}}{{R_{h}}^2}$, where $R_{h}$ is heliocentric distance.
This gives the approximate time for radiation pressure to remove material from the aperture as
\begin{equation}
t\approx\left({\frac{2r_{ap} {R_{h}}^2}{\beta GM_{\odot}}}\right)^\frac{1}{2}
\end{equation}
where $R_{h}$ is distance from the sun to the asteroid in question.
Assuming $\beta$ is on the order of $\approx 0.1$\citep{beta} for the small grains in the ejecta expected to dominate the scattering, $R_{h}=3$ AU (mid-main belt), the time $t$ is approximately 2 days. Therefore, we do not present any model results that go beyond 7 days post collision, as without the inclusion of the radiation pressure they do not reflect physical reality.
\subsection{Optical depth effects}
Throughout the modelling, all ejecta within the aperture and not behind the asteroid is treated as contributing to the brightness increase via the projected surface area. Effectively, it is assumed that optical depth $\tau$ is sufficiently low that the approximation $\tau\simeq1-e^{-\tau}$ is valid. (For $\tau <0.2$ this leads to a brightening overestimate of less than 10\%).
To estimate the potential optical depth effects, the ejecta cloud was divided into a series of concentric rings and the column density of each was calculated, then converted, using the particle size distribution, into the optical depth $\tau$. Any rings with $\tau < 0.02$ were ignored (this corresponds to an error of less than 1\%), and any cases where more than 10\% of ejecta mass was contained in rings with $\tau>0.02$ were considered to have their brightness potentially overestimated. We found that at one day after impact, ignoring opacity may lead to overestimating the brightness increase by up to twenty percent ($\sim 0.2$) magnitudes. The region where this may be relevant is outlined in shown in figures \ref{fig:CHP24}, \ref{fig:CSAND24}, \ref{fig:SSAND24} and \ref{fig:DSAND24} showing the brightness 1 day post collision. Clearly for all impacts apart from those on the largest asteroids, the correction to the calculated magnitude change will have little effect on detection and can be ignored. By one week after impact, we calculate that optical depth effects are negligible for all target impactor pairs.
\section{Discussion}
A collision between asteroids is a multi parameter problem requiring us to make some initial assumptions. This section is therefore divided into 4 parts.
First we describe an application of the model to the only known collision in the parameter space explored - (596) Scheila. In the second part we examine the effect of the different strength regimes in a collision between C-type asteroids (as the most numerous in the outer belt where (596) Scheila is situated). In the third part we keep the strength regime the same (sand/cohesive soil) and vary the taxonomic types to examine the effect it has on the magnitude change. We conclude with application of the model to predicting the type of observable collisions by currently active surveys such as Pan-STARRS 1 and the Catalina Sky Survey.
\subsection{Modelling the magnitude decrease in the (596) Scheila collision}
In December 2010 asteroid (596) Scheila was observed with the 0.68 meter Catalina Schmidt telescope to have increased in brightness by approximately 1.3 magnitudes in comparison to the previous month \citep{larson}. There have been multiple photometric observations of (596) Scheila reported as summarised in Table~\ref{tab:scheila_obs}, from which the impact is estimated to have occurred on December 3.5 UT, 2010 \citep{ishiguro_obs}.
All observations measured the total magnitude, i.e. from the asteroid and the surrounding debris. \cite{jewitt} used a 105,600 km aperture for the HST observations of (596) Scheila and this value is used for the calculations throughout this section. We have selected these observations for comparison with our model as giving the true magnitude decrease because they were taken by the same instrument and in the same observational circumstances.
\begin{table}[ht]
\caption{Summary of observed change in magnitude for (596) Scheila}
\centering
\begin{tabular}{c c c }
\hline\hline
Date of observation (UT) & Approximate time & Magnitude \\
& post-collision (days) & change \\[0.5ex]
\hline
2010 Dec. 3.4 \citep{larson} & 0& -1.3 \\
2010 Dec. 11.44-11.47 \citep{larson}& 7.9 & -1.1 \\
2010 Dec. 12 \citep{hsieh}& 9 & -0.86 \\
2010 Dec. 14-15 \citep{bodewits2}& 11 & -0.66 \\
2010 Dec. 27.9 \citep{jewitt} & 24.4 & -1.26 \\
2011 Jan. 4.9 \citep{jewitt} & 32.4 & -1.00 \\ [1ex]
\hline
\end{tabular}
\label{tab:scheila_obs}
\end{table}
A collision with a smaller asteroid is the most likely cause of this brightening and there have been a range of impactor diameters suggested in literature: 35 m \citep{jewitt}, 30-50 m \citep{ishiguro}, 30-90 m \citep{moreno}, $<$100 m \citep{bodewits}.
The impactor diameters above are calculated using the estimated mass of high speed ejecta and the relationship $m(v) \approx m_{i}$ for $v = 10^{-2} v_{i}$ \citep{HH2011} where $m(v)$ is the mass of ejecta above velocity $v$, $m_{i}$ is the impactor mass and $v_{i}$ the impactor velocity.
Scheila has semimajor axis a = $2.926$ AU, eccentricity $e = 0.164$, and inclination $i = 14.7^{\circ}$, which places it in the outer main belt. The most numerous asteroids in that region and therefore the most likely impactors are C-type \citep{taxtypes}, although (596) Scheila itself is a D-type \citep{dtype}. The taxonomic type determines the densities used in the model which affect the final output parameter of change in magnitude.
Figure ~\ref{fig:subfigures_scheila} shows the post-collision magnitude change during 60 days for a collision between Scheila and C-, D- and S-type impactors of diameters reported in the literature. We also used the observations from \cite{jewitt} to find the best fit size for each impactor type by matching the calculated magnitude change in our model to the observed lightcurve. The results are 59 m, 49 m and 65 m diameter impactors for S-, C- and D-types respectively.
As has been mentioned in the previous section, we believe that in its current state the model gives best description of reality if we limit the results to 7 days post-collision. However, in the case of Scheila collision, which is an important real observed collisional event, we lack a sufficient number of data points of good quality in the first 7 days to make a meaningful comparison. However, the large aperture used in this study includes a large portion of the ejecta even after being significantly affected by radiation pressure, allowing comparison with our model.
The model results are in approximate agreement with estimated impactor diameters by other authors. The differences most likely come from a variety of initial assumptions made by the different researchers, i.e. density, type of ejecta particle distribution, range of particle sizes and the estimated ejected mass. It is also clear from these results, that to the first order the taxonomic type of the impactor asteroid is not an important factor in diameter estimation.
\begin{figure}[ht!]
\begin{center}
\subfigure[C-type impactors]{
\label{fig:Cimp}
\includegraphics[trim = 18mm 50mm 25mm 60mm, clip,width=0.45\textwidth]{scheila_C_obs_bw.pdf}
}
\subfigure[S-type impactors]{
\label{fig:Simp}
\includegraphics[trim = 18mm 50mm 25mm 60mm, clip,width=0.45\textwidth]{scheila_S_obs_bw.pdf}
}\\
\subfigure[D-type impactors]{%
\label{fig:Dimp}
\includegraphics[trim = 18mm 50mm 25mm 60mm, clip,width=0.45\textwidth]{scheila_D_obs_bw.pdf}
}
\end{center}
\caption{
Magnitude change as a function of time following a collision of C-, S- and D-type impactors with a range of diameters with asteroid (596) Scheila. Individual points show reported magnitude change by various authors. Dot-dash line shows our fit to Jewitt et al.'s data. The best fit estimates of impactor diameter with our model are 59, 49 and 65m for C-, S- and D-type impactors respectively. }%
\label{fig:subfigures_scheila}
\end{figure}
\subsection{Effect of strength regime on post-collision magnitude change for C-type asteroids.}
We consider a collision where both impactor and the target are generic C-type asteroids. Figures~\ref{fig:CHP24} and \ref{fig:CHP1w} show the magnitude decrease post-collision in highly porous regime, while Figures~\ref{fig:CSAND24} and \ref{fig:CSAND1w} show the effect of changing the regime to the sand/cohesive soil.
Following the collision, a shock wave travels through the target asteroid and the outcome of the collision depends on how the stress wave propagates and is attenuated through the target. The cratering and amount of ejecta following an impact is determined by the target internal structure, porosity and strength. A porous asteroid would react to the impact differently than a more solid body and it is particularly important to consider impacts on these as most observable
main belt asteroids should have significant macroporosities, apart from the very largest. It has been found that highly porous objects such as (253) Mathilde are particularly good at attenuating stress waves due to energy going into the compaction of the pores \citep{porosity}.
An object with large macroporosity is most likely to be highly fractured or even a rubble pile. Dark and primitive asteroids are more likely to have significant porosity, however it is worth noting that it is not exclusive to C-types.
Figures~\ref{fig:CHP24}/\ref{fig:CHP1w} and \ref{fig:CSAND24}/\ref{fig:CSAND1w} are similar for both strength regimes at each point of time considered, however the sand/cohesive soil regime shows a larger magnitude decrease due to more material ejected, while in the highly porous regime some of the impact energy goes into compaction.
This tendency is also reflected in plots of amount of time taken for the magnitude change to decrease to $1.0$ magnitudes above the pre-impact brightness (Figure \ref{fig:subfiguresCHPSANDm-1}). The brightness increase caused by the ejecta reaches the assumed observable limit of approximately 1 magnitude faster for the highly porous case than the corresponding impactor-target pair undergoing a collision in the sand/cohesive soil regime. This is due to less ejecta being produced.
Recent research by \cite{carry} indicates that the density may be dependant on target size, rather than being a fixed quantity for each taxonomic type. However, the data is sparse in the size range used in this model and unavailable for the D-types. Therefore, our model makes a simplifying assumption that the bulk density stays fixed for all sizes considered and depends only on the taxonomic type.
\subsection{Effect of taxonomic type (C, D and S) on post-collision magnitude change in sand/cohesive soil regime}
It is important to understand the taxonomy effects on the collision lightcurve due to the established composition gradient across the main belt.
To investigate we kept the strength regime (sand/cohesive soil) constant and varied taxonomic type of both target and impactor (for S-, C- and D-types). Figures \ref{fig:subfiguresSSAND} and
\ref{fig:subfiguresDSAND} show the magnitude change 1 and 7 days post-collision for target-impactor pairs. Other combinations of target-impactor types are included in the Supplementary material (available online).
S-types impacting each other are more likely to occur in the inner belt, while D-types impacting each other are more likely in the Trojan population. Generally, both sets of plots show qualitatively similar results for magnitude change with time, with a high magnitude change in the day 1, followed by the decline over time, with C-types having the least magnitude increase at 7 days. From our results we observed that collisions between the lowest grain density and highest porosity C-types produce the least amount of magnitude decrease lasting for less time that it does for other taxonomic types. The S-types which have lower porosity and the highest considered grain density show the opposite - a larger magnitude change lasting longer. D-type collisions fall between these regimes.
Figure \ref{fig:taxtypes} shows a direct comparison between different combinations of target and impactor taxonomic types for a 20 km target and 50 m diameter impactor.
In a collision with an S-type target, the largest magnitude change is produced by a S-type impactor, then C-type and D-type producing the least magnitude change. The same pattern is followed by a C-type target being impacted by S-, C- and D-type asteroids. However, a D-type target deviates from this pattern, with an S-type impactor giving the largest change in magnitude, followed by D-types and C-types giving the least magnitude change. Overall, S-type impactors produce the brightest collision signature in all types of targets considered, due to their inherent high grain density.
\begin{figure}[ht!]
\begin{center}
\subfigure[1 day]{
\label{fig:SSAND24}
\includegraphics[trim = {16mm} {50mm} {25mm} {60mm}, clip, width=0.45\textwidth]{S_s_S_1day_new_density_500.pdf}
}
\subfigure[ 7 days]{
\label{fig:SSAND1w}
\includegraphics[trim = {16mm} {50mm} {25mm} {60mm}, clip,width=0.45\textwidth]{S_s_S_7days_new_density_500.pdf}
}
\end{center}
\caption{
S-type impactor collision with S-type target, sand/cohesive soil strength regime: magnitude change 1 and 7 days post-collision for a range of target and impactor radii. Catastrophic disruption region is marked in black. Region where optical thickness of the debris is potentially significant is above the dashed line. Solid black lines indicate contours where magnitude change is -1, -3, -5, -7 and -9. }%
\label{fig:subfiguresSSAND}
\end{figure}
\begin{figure}[ht!]
\begin{center}
\subfigure[1 day]{
\label{fig:DSAND24}
\includegraphics[trim = {16mm} {50mm} {25mm} {60mm}, clip, width=0.45\textwidth]{D_s_D_1day_new_density_500.pdf}
}
\subfigure[ 7 days]{
\label{fig:DSAND1w}
\includegraphics[trim = {16mm} {50mm} {25mm} {60mm}, clip,width=0.45\textwidth]{D_s_D_7days_new_density_500.pdf}
}
\end{center}
\caption{
D-type impactor collision with D-type target, Sand/cohesive soil strength regime: magnitude change 1 and 7 days post-collision for a range of target and impactor radii. Catastrophic disruption region is marked in black. Region where optical thickness of the debris is potentially significant is above the dashed line. Solid black lines indicate contours where magnitude change is -1, -3, -5, -7 and -9. }%
\label{fig:subfiguresDSAND}
\end{figure}
\begin{figure}[ht]
\centerline{\includegraphics[trim = 18mm 50mm 18mm 60mm, clip, width=0.7\textwidth]{taxtypes.pdf}}
\caption{The effect of varying taxonomic type in a collision of 20km target and 50m impactor on magnitude change as a function of time.}
\label{fig:taxtypes}
\end{figure}
\subsection{Detection from current sky surveys}
The catastrophic collision rate in
the main belt has been calculated from observationally constrained dynamical models as $\sim1$ per year at a diameter $D\sim 100$ m
~\citep{durda, greenberg, Bottke2005}. The general hope within the scientific community has been that
such collisions would be detected via the on-going wide-field NEO surveys.
However, several objects initially identified as potential collisional disruption events are now
suspected of being caused by rotational disruption due to YORP spinup {\it e.g.}
P/2010 A2 ~\citep{Agarwal2013}, P/2013 P5 ~\citep{Jewitt2013}. Additionally, a
recent study by ~\cite{Denneau2014} using $1.2$ years of Pan-STARRS 1 data found only one
plausible candidate of a collision event, and concluded that collisional
disruptions of 100-m scale asteroids may be extremely rare. Hence observing
much more frequent sub-catastrophic collisions may be a viable method for
constraining the overall collision rate.
To look at the likelihood of detecting these events, we first look at collisions with
100 km diameter targets similar to Scheila. According to \cite{Bottke2005}, the size of impactor that would disrupt a $D=100$ km asteroid
is $D\geq25$km and such a disruption would happen every $\sim10^{7}$ years in the main belt.
Using their CoDDEM model size distribution, a subcatastrophic collision impact by a $D\geq50$ m
impactor should occur once per $\simeq 25$ years, in rough agreement with the single such collision observed
since the start of modern surveys in the 1990's.
The current Pan-STARRS and Catalina surveys are complete down to an apparent magnitude $V\simeq 20$ (equivalent to $D\sim 2$ km in the main-belt), but are sensitive to asteroids down to $V\simeq 22$. The Bottke et al. disruption frequency at $D\sim 2$km is approximately one per 1000 years, requiring an impactor $D\simeq 60$ m. Scaling by the CoDDEM model size distribution predicts a collision between a $D\sim 2$km asteroid and a $D\geq10$m impactor should occur approximately once per year. As $\sim 50$\% of the asteroid belt is visible at any time, current surveys should be able to detect a sub-catastrophic impact once every couple of years.
So why are these collisions between smaller bodies not found? In reality, observational surveys are affected by
factors such as weather, moonlight and occasional losses due to detector gaps in the cameras. Perhaps most
importantly, current surveys have a cadence for most individual asteroids measured in weeks rather than days. Using a detection aperture of $\sim 1000$~km equivalent to $\sim 1-3$ arcsec as appropriate to surveys such as Pan-STARRS, our model predicts a brightening of $\geq-1$ magnitudes for $D\simeq10$m impacts on $D\sim 2$km for only 10--20 days. In reality, effects on the ejecta such as radiation pressure and Keplerian shear may shorten this timescale further. We can use the calculations presented by
\cite{Denneau2014} (their figure 5) to estimate the likelihood of detection. With a maximum magnitude change of $\sim2$ magnitudes
and resulting rate of dimming of $\geq0.1$ magnitudes/day, the probability of the Pan-STARRS 1 survey detecting
such a collision during the first 1.2 years of operation was $<0.05$. Therefore we conclude that observational losses together with the rapid dispersal of the ejecta can significantly decrease the possibility of photometric detection of sub-catastrophic impacts smaller that the (596) Scheila event, and
this implies a current low probability of detection using automated photometry software such as MOPS \citep{Denneau2013}. Finally, another additional factor may be inaccurate absolute magnitudes in current reference catalogues, as investigated by \cite{pravecbias}. They found that smaller asteroids were systematically fainter than expected from catalogues. Hence predicted magnitudes would be brighter than in reality, and any small brightness increase due to a collision could be masked by an incorrectly calculated residual magnitude.
Of course, during the first few days to weeks the ejecta may be visible as an
extended ejecta cloud around the target asteroid, and could be detected via direct manual
observation or software algorithms designed for comet comae detection, as in the case of (596) Scheila.
On the other hand, if the cadence of a survey is a week or less i.e. significantly less
than the decay timescale within the aperture, the chance of
photometrically detecting a small impact should be substantially higher. In Figure \ref{fig:visualmag}
we plot the predicted total $V$-band magnitudes for C-on-C and S-on-S collisions
1 day and 7 days after impact, for asteroids at opposition at $R_h=3.5$ AU. (Although S-types are predominantly found in the inner main belt, we calculate the predicted apparent magnitude at the likely maximum distance).
First, it is clear that for the larger (but less frequent) impacts, the total magnitude will be relatively bright and would produce saturated images in large aperture surveys such as Pan-STARRS. However it may be possible to deal with this situation by either fitting to the wings of asteroid image point-spread function, or by recognising that an expected asteroid in the field was rejected in software processing due to its increased brightness. For fainter impact events, it is clear that
Pan-STARRS with a limiting magnitude of $V\sim22$ is able to detect such impacts throughout
the asteroid belt, as long as the asteroid is observed soon after the collision. Importantly, the forthcoming ATLAS programme will have a nightly cadence
over the visible sky and have a limiting magnitude of $V\simeq20$ \citep{Tonry2011}. From comparing both Figure \ref{fig:visualmag} with the brightness increases presented earlier, we find that ATLAS would be able to detect almost all of our studied collisions as well, but here the survey cadence should not be an issue. Therefore we conclude that current and future high-cadence all sky surveys should be able to detect many more asteroid collisions at early epochs.
\begin{figure}[ht]
\begin{center}
\subfigure[C-type target impactor, 1 day]{
\label{fig:VCsC1}
\includegraphics[trim = 15mm 50mm 26mm 60mm, clip, width=0.45\textwidth]{observability_C_s_C_1day_new_density_500.pdf}
}
\subfigure[C-type target impactor, 7 days]{
\label{fig:VCsC7}
\includegraphics[trim = 15mm 50mm 26mm 60mm, clip,width=0.45\textwidth]{observability_C_s_C_7days_new_density_500.pdf}
}
\subfigure[S-type target impactor, 1 day]{
\label{fig:VSsS1}
\includegraphics[trim = 15mm 50mm 26mm 60mm, clip, width=0.45\textwidth]{observability_S_s_S_1day_new_density_500.pdf}
}
\subfigure[S-type target impactor, 7 days]{
\label{fig:VSsS7}
\includegraphics[trim = 15mm 50mm 26mm 60mm, clip,width=0.45\textwidth]{observability_S_s_S_7days_new_density_500.pdf}
}
\end{center}
\caption{
Visual magnitude following a collision. \ref{fig:VCsC1}-\ref{fig:VCsC7} show magnitude 1 and 7 days post-collision for C-type impactor and target in the sand/cohesive soil strength regime for a range of target and impactor radii. \ref{fig:VSsS1}-\ref{fig:VSsS7} show magnitude 1 and 7 days post-collision for S-type impactor and target in the sand/cohesive soil strength regime for a range of target and impactor radii. Impactor radii range from 1 to 100 m, target radii: 1-100 km. Colour bar shows the visual magnitude. Catastrophic disruption region is marked in black. Region where optical thickness of the debris is potentially significant is above the dashed line. }%
\label{fig:visualmag}
\end{figure}
\clearpage
\section{Summary}
The model presented in this paper predicts the brightness increases caused by
impacts of small asteroids on larger asteroids with radii $\geq 1$ km. We use the
scaling laws of \cite{HH2007} and \cite{HH2011} to estimate
the amount of ejected material. Our model separates the ejecta into discrete velocity shells,
and assuming an ejecta size distribution calculates the magnitude change post-collision by calculating the surface
cross-sectional area of the asteroid plus ejected material in a small photometric aperture. The scope of the model is limited to
non-rotating asteroids whose ejecta particles do not interact with each other, however by extending
the model results to the large apertures used for the reported ejects from the (596) Scheila collision,
we find an estimate for the impactor size of
$40-65$ m (depending on taxonomic type) similar to previous studies in the literature. The
model could be further improved by introducing particle light scattering laws, radiation pressure on the ejecta and the effect of
rotation of the target asteroid. However this will lead to a significant increase the complexity of the model and will be developed in future. We believe our results
are generic enough to be used to estimate the possibility of detection of such events by current automated surveys such as Pan-STARRS1 and the Catalina Sky Survey. The model estimates that within the parameter space examined (impactor radius $1-100$ m, target radius $1-100$ km, sub-catastrophic collisions) , a magnitude change of less than $-1$ is observable by an automated survey like Pan-STARRS 1 with effective aperture radii of 1000 km for only 10--20 days, which implies low probability of detection given the current low cadence for individual asteroids. However, detection may still be possible by direct manual observation or software algorithms designed for comet coma detection.
\section{Acknowledgements}
EMcL acknowledges support from the Astrophysics Research Centre, QUB. AF acknowledges support by STFC grant ST/L000709/1. EMcL and AF also thank Robert Jedicke for helpful comments during the preparation of this paper.
\clearpage
\section{Supplimentary material}
\begin{figure}[htp]
\begin{center}
\subfigure[1 day]{
\label{fig:SDsand1d}
\includegraphics[trim = {16mm} {50mm} {25mm} {60mm}, clip, width=0.45\textwidth]{S_s_D_1day_new_density_500.pdf}
}
\subfigure[ 7 days]{
\label{fig:SDsand7d}
\includegraphics[trim = {16mm} {50mm} {25mm} {60mm}, clip,width=0.45\textwidth]{S_s_D_7days_new_density_500.pdf}
}
\end{center}
\caption{
D-type impactor collision with S-type target, sand/cohesive soil regime: magnitude change 1 and 7 days post-collision for a range of target and impactor radii. Impactor radii range from 1 to 100 m, target radii: 1-100 km. Colour bar shows the magnitude change. Catastrophic disruption region is marked in black. Region where optical thickness of the debris is potentially significant is above the dashed line. Solid black lines indicate contours where magnitude change is -1, -3, -5, -7 and -9. }%
\label{fig:subfiguresSDsand}
\end{figure}
\begin{figure}[htp]
\begin{center}
\subfigure[1 day]{
\label{fig:SCsand1d}
\includegraphics[trim = {16mm} {50mm} {25mm} {60mm}, clip, width=0.45\textwidth]{S_s_C_1day_new_density_500.pdf}
}
\subfigure[ 7 days]{
\label{fig:SCsand7d}
\includegraphics[trim = {16mm} {50mm} {25mm} {60mm}, clip,width=0.45\textwidth]{S_s_C_7days_new_density_500.pdf}
}
\end{center}
\caption{
C-type impactor collision with S-type target, sand/cohesive soil regime: magnitude change 1 and 7 days post-collision for a range of target and impactor radii. Impactor radii range from 1 to 100 m, target radii: 1-100 km. Colour bar shows the magnitude change. Catastrophic disruption region is marked in black. Region where optical thickness of the debris is potentially significant is above the dashed line. Solid black lines indicate contours where magnitude change is -1, -3, -5, -7 and -9. }%
\label{fig:subfiguresSCsand}
\end{figure}
\begin{figure}[htp]
\begin{center}
\subfigure[1 day]{
\label{fig:CSsand1d}
\includegraphics[trim = {16mm} {50mm} {25mm} {60mm}, clip, width=0.45\textwidth]{C_s_S_1day_new_density_500.pdf}
}
\subfigure[ 7 days]{
\label{fig:CSsand7d}
\includegraphics[trim = {16mm} {50mm} {25mm} {60mm}, clip,width=0.45\textwidth]{C_s_S_7days_new_density_500.pdf}
}
\end{center}
\caption{
S-type impactor collision with C-type target, sand/cohesive soil regime: magnitude change 1 and 7 days post-collision for a range of target and impactor radii. Impactor radii range from 1 to 100 m, target radii: 1-100 km. Colour bar shows the magnitude change. Catastrophic disruption region is marked in black. Region where optical thickness of the debris is potentially significant is above the dashed line. Solid black lines indicate contours where magnitude change is -1, -3, -5, -7 and -9. }%
\label{fig:subfiguresCSsand}
\end{figure}
\begin{figure}[htp]
\begin{center}
\subfigure[1 day]{
\label{fig:CDsand1d}
\includegraphics[trim = {16mm} {50mm} {25mm} {60mm}, clip, width=0.45\textwidth]{C_s_D_1day_new_density_500.pdf}
}
\subfigure[ 7 days]{
\label{fig:CDsand7d}
\includegraphics[trim = {16mm} {50mm} {25mm} {60mm}, clip,width=0.45\textwidth]{C_s_D_7days_new_density_500.pdf}
}
\end{center}
\caption{
D-type impactor collision with C-type target, sand/cohesive soil regime: magnitude change 1 and 7 days post-collision for a range of target and impactor radii. Impactor radii range from 1 to 100 m, target radii: 1-100 km. Colour bar shows the magnitude change. Catastrophic disruption region is marked in black. Region where optical thickness of the debris is potentially significant is above the dashed line. Solid black lines indicate contours where magnitude change is -1, -3, -5, -7 and -9. }%
\label{fig:subfiguresCDsand}
\end{figure}
\begin{figure}[htp]
\begin{center}
\subfigure[1 day]{
\label{fig:DSsand1d}
\includegraphics[trim = {16mm} {50mm} {25mm} {60mm}, clip, width=0.45\textwidth]{D_s_S_1day_new_density_500.pdf}
}
\subfigure[ 7 days]{
\label{fig:DSsand7d}
\includegraphics[trim = {16mm} {50mm} {25mm} {60mm}, clip,width=0.45\textwidth]{D_s_S_7days_new_density_500.pdf}
}
\end{center}
\caption{
S-type impactor collision with D-type target, sand/cohesive soil regime: magnitude change 1 and 7 days post-collision for a range of target and impactor radii. Impactor radii range from 1 to 100 m, target radii: 1-100 km. Colour bar shows the magnitude change. Catastrophic disruption region is marked in black. Region where optical thickness of the debris is potentially significant is above the dashed line. Solid black lines indicate contours where magnitude change is -1, -3, -5, -7 and -9. }%
\label{fig:subfiguresDSsand}
\end{figure}
\begin{figure}[htp]
\begin{center}
\subfigure[1 day]{
\label{fig:DCsand1d}
\includegraphics[trim = {16mm} {50mm} {25mm} {60mm}, clip, width=0.45\textwidth]{D_s_C_1day_new_density_500.pdf}
}
\subfigure[ 7 days]{
\label{fig:DCsand7d}
\includegraphics[trim = {16mm} {50mm} {25mm} {60mm}, clip,width=0.45\textwidth]{D_s_C_7days_new_density_500.pdf}
}
\end{center}
\caption{
C-type impactor collision with D-type target, sand/cohesive soil regime: magnitude change 1 and 7 days post-collision for a range of target and impactor radii. Impactor radii range from 1 to 100 m, target radii: 1-100 km. Colour bar shows the magnitude change. Catastrophic disruption region is marked in black. Region where optical thickness of the debris is potentially significant is above the dashed line. Solid black lines indicate contours where magnitude change is -1, -3, -5, -7 and -9. }%
\label{fig:subfiguresDCsand}
\end{figure}
\clearpage
|
1,314,259,993,583 | arxiv | \section{Introduction}
As of 2019, the {\it {Kepler mission}} has discovered approximately ten circumbinary (CB)
planetary systems. All binary components define compact systems with orbital periods less than
$\sim 40$ days and a wide range of eccentricities and mass ratios. The planets surrounding them
also have a diversity of masses (between super-Earths to Jupiter masses) but they are all almost
coplanar with the binary. With the exception of Kepler 34b and Kepler 413b, all CB planets seem
characterized by small semimajor axis low eccentricities.
While the low inclinations suggest that these planets formed in a CB disc aligned with the
orbital plane of the central binary, it is well accepted that {\it{in situ}} formation so close
to the binary is unlikely due to the strong eccentricity excitation induced by the secondary
star \cite[e.g.][]{Lines2014,Meschiari2012}. However, as we move away from the binary, the
gravitational potential approaches that of a single star and planetary formation appears to be
easier, following usual core-accretion models. This suggest that CB planets could have formed
farther out, later migrated inward due to interaction with a primordial disc and finally stalled
near their current orbits by some mechanism (\cite{Dunhill2013}).
In a previous work \citep{Zoppetti2018}, we tested the possibility that the circumbinary
planets may have halted its inward migration due to a capture in a high order mean-motion
resonance (MMR) with the binary and, once the disc is dissipated, slowly escaped from the
commensurability due to tidal forces. We applied this hypothesis to Kepler-38, a very old system
in which captures in the 5/1 MMR had been reported with hydro-simulation \citep{Kley2014}.
Tidal interactions were modeled following \cite{Mignard1979} and incorporated to a N-body
integrator following the prescription detailed in \cite{Rodriguez2011}. We observed that while
the binary orbit shrinks due to tidal interactions, the planet seemed to increase its semimajor
axis, even after the system reached stationary solutions in the spin rates. We were unable to
explain these findings, which in principle could have been caused by a non-consistent treatment
of the tidal interactions between the different bodies of the dynamical system.
In this article we present and discuss a self-consistent tidal model for a multi-body system,
in which all tidal forces between pairs are computed adopting a weak-friction (Mignard-type)
model. While the model is general, we will focus primarily on the spin and orbital evolution
of the CB planet. To allow for a simpler comparison with our previous results, we will once
again employ Kepler-38 as a reference system \citep{Orosz2012}. However, we will also explore
a wider range of system parameters as well as different initial orbital elements and spin
rates.
This paper is organized as follows. In Section \ref{model} we present the model in two steps:
in Section \ref{sub1} we first discuss which tidal forces have a net effect onto the
dynamical evolution of an 3-extended-body system while in Section \ref{sub2} we show how
these forces are incorporated, self-consistently, into our tidal model. Section \ref{numsim}
presents a series of numerical integrations of the full spin and orbital equations of motion. We
concentrate on two different time-scales: the early dynamical evolution of the system before the
spins reached stationary solutions, and the subsequent long-term orbital evolution of the CB
planet in spin stationarity. In Section \ref{anali}, we construct analytical expressions for the
orbital and spin evolution, averaged over the orbital periods but retaining secular terms,
including those containing the difference between longitudes of pericenter. These allow us to
estimate the stationary spin rate of CB planets, as well as the direction and magnitude of the
orbital migration. We compare these predictions with full N-body simulations. Finally, Section
\ref{conclu} summarizes our main results and discusses their implications.
\section{The model}
\label{model}
Let us consider a binary system in which $m_0$ and $m_1$ are the masses of the stellar
components and $m_2$ is a circumbinary planet. We suppose that all the bodies lie in the same
orbital plane and their spins are perpendicular to it. We also assume that all the bodies are
extended masses with physical radii ${\cal R}_i$ and are deformable due to tidal effects between
them.
For the gravitational interactions between each pair of bodies we will be adopt the classical
weak-friction tidal model \citep{Mignard1979}. However, since now the tidal deformation of
each body will have to incorporate the gravitational potential generated by both of its
companions, we first need to address two issues: (i) which tidal deformation have a net effect
on the long-term dynamical evolution of the system and, (ii) how the different forces should be
incorporated into a self-consistent physical model. These questions are addressed in the next
two subsections.
\subsection{The Mignard forces revisited}
\label{sub1}
We begin considering our three-body system with two simplifications. First, we will neglect the
gravitational perturbations generated by $m_2$ on the other two bodies, as well as the effects
of $m_1$ on $m_2$. Second, only $m_0$ will be assumed to be an extended mass while $m_1$ and
$m_2$ will be taken as point masses. As a consequence of these approximations, the dynamics
of both $m_1$ and $m_2$ around $m_0$ will be defined by the point-mass approximation plus the
tidal deformation of $m_0$ generated solely by $m_1$. The role of $m_2$ is thus reduced to serve
as a tracker of the dynamical effect of the tidal bulge on any generic orbit in the
configuration plane.
\begin{figure}
\centering
\includegraphics[width=0.8\columnwidth,clip]{imagenes001.jpg}
\caption{A tidal lagged bulge generated on $m_0$ due to $m_1$ and its effect on a test body
$m_2$, from a $m_0$-centric coordinate frame.}
\label{fig:1}
\end{figure}
A schematics of this scenario is presented in Figure \ref{fig:1}, where ${\bf r_i}$ are the
$m_0$-centric position vectors of the other masses. Following Mignard (1979), the tidal bulge
of $m_0$ considered is displaced with respect to the instantaneous position of $m_1$ by a
constant time-lag $\Delta t_0$. We assume the lag is sufficiently small to expand the
gravitational potential $U$ generated by $m_0$ in anywhere in the space up to first-order in
$\Delta t_0$, such that
\begin{equation}
U({\bf {r}},{\bf {r}_1}) = U^{(0)}({\bf {r}},{\bf {r}_1}) + U^{(1)}({\bf {r}},{\bf {r}_1}) +
\mathcal{O}(\Delta t_0^2)
\label{eq1}
\end{equation}
where $U^{(0)}$ and $U^{(1)}$ are the expanded tidal potentials of order $\mathcal{O}(0)$ and
$\mathcal{O}(\Delta t_0)$, respectively. In particular, if we evaluate (\ref{eq1}) on the
position of $m_2$ (i.e. ${\bf {r}}={\bf {r}_2}$), we obtain
\begin{eqnarray}
\begin{split}
U^{(0)}({\bf {r}_2},{\bf {r}_1}) &= \frac{\mathcal{G} m_1 \mathcal{R}_0^5}{2 r_1^5 r_2^5}
k_{2,0} \bigg[ 3 {( {\bf{r}_2} \cdot {\bf{r}_1})}^2 - r_2^2 r_1^2 \bigg] \\
U^{(1)}({\bf {r}_2},{\bf {r}_1}) &= \frac{3 \mathcal{G} m_1 \mathcal{R}_0^5}{r_1^5 r_2^5}
k_{2,0} \Delta t_0 \bigg[ \frac{({\bf{r_1}} \cdot \dot{{\bf{r}_1}})} {2 r_1^2} [5
{({\bf{r}_2} \cdot {\bf{r}_1})}^2 - r_2^2 r_1^2 ] \\
& \hspace*{1.8cm} - ({\bf{r}_2} \cdot {\bf{r}_1}) [{\bf{r}_1} \cdot ({\bf{\Omega}}_0 \times
{\bf{r}_2}) + {\bf{r}_2} \cdot \dot{{\bf{r}_1}}] \bigg] \\
\end{split}
\label{eq2}
\end{eqnarray}
where $\mathcal{G}$ is the gravitational constant, ${\bf{\Omega}}_0$ is the spin vector of
$m_0$ and $k_{2,0}$ its the second degree Love number.
\begin{figure*}
\centering
\includegraphics[width=0.99\textwidth]{torque_medio.png}
\caption{Secular normalized torques of zero-order $|\langle {\bf{T}}^0({\bf{r}}_2) \rangle|$
(left column) and first-order $|\langle{\bf{T}}^{1}({\bf{r}}_2)\rangle|$ (right column),
computed on $m_2$ due to the tidal deformation on $m_0$ induced by $m_1$, plotted as a function
of the mean-motion ratio $n_1/n_2$. We considered $m_0 = 1$, $a_1 = 1$ and varied $a_2$ to
include orbits both interior and exterior to $m_1$. Upper panels correspond to circular orbits
($e_1=e_2=0$) while the lower panels assume eccentric orbits with $e_1=e_2=0.1$. Light
brown vertical lines highlight the location of some important mean-motion resonances. Note that
the first-order torques in the right panels are also normalized respect to the time-lag $\Delta
t_0$.}
\label{fig:torques}
\end{figure*}
The tidal force per unit mass ${\cal {\bf f}}$ generated by $m_0$ at a generic position vector
${\bf r}$ can be obtained as
\begin{equation}
{\bf f} = \nabla_{\bf {r}} (U^{(0)} + U^{(1)}) = {\bf f}^{(0)} + {\bf f}^{(1)}
\label{eq3}
\end{equation}
where explicit expressions evaluated on $m_2$ are given by
\begin{eqnarray}
\begin{split}
{\bf f}^{(0)} &= \frac{3 \mathcal{G} m_1 \mathcal{R}_0^5}{2
r_2^5 r_1^5} k_{2,0} \bigg[ 2 ( {\bf{r}_2} \cdot {\bf{r}_1}){\bf{r}_1}
+ \bigg({r_1}^2 -\frac{5}{r_2^2} {({\bf{r}_2} \cdot {\bf{r}_{1}})}^2 \bigg){\bf{r_2}} \bigg]
\\
{\bf f}^{(1)} &= \frac{3 \mathcal{G} m_1 \mathcal{R}_0^5}{r_2^5
r_1^5} k_{2,0} \Delta t_0 \bigg[ \frac{({\bf{r}_1} \cdot \dot{{\bf{r}_1}})}{r_1^2} [5
{\bf{r}_1} ({\bf{r}_2} \cdot {\bf{r}_1}) - {\bf{r}_2} {r_1}^2 ] \\
&- [{\bf{r}}_1 \cdot ({\bf{\Omega}}_0 \times {\bf{r}_2}) + {\bf{r_2}} \cdot \dot{{\bf{r}_1}} ]
{\bf{r}_1} - ({\bf{r}_1} \times {\bf{\Omega}}_0 + \dot{{\bf{r}_1}})({\bf{r}_2} \cdot
{\bf{r}_1})
\\
&+ \frac{5 {\bf{r}_2}}{r_2^2} \bigg[({\bf{r}_2} \cdot {\bf{r}_1})[{\bf{r}_1} \cdot
({\bf{\Omega}}_0 \times {\bf{r_2}}) + {\bf{r}_2} \cdot \dot{{\bf{r}_1}} ] \\
&- \frac{({\bf{r}_1} \cdot \dot{{\bf{r}_1}})}{2 r_1^2} [5 {({\bf{r}_2} \cdot {\bf{r}_{1}})}^2
- {r}^2_{2} {r_1}^2 ] \bigg] \bigg] . \\
\end{split}
\label{eq4}
\end{eqnarray}
Finally, the torques per unit mass can be calculated as ${\bf{T}}({\bf {r}},{\bf {r}_1}) \simeq
{\bf{r}} \times ( {\bf f}^{(0)}+{\bf f}^{(1)} ) = {\bf{T}}^{(0)}({\bf {r}},{\bf {r}_1}) +
{\bf{T}}^{(1)}({\bf {r}},{\bf {r}_1})$. As before, evaluating on the position of $m_2$ yields
\begin{eqnarray}
\begin{split}
{\bf{T}}^{(0)}({\bf {r}_2},{\bf {r}_1}) &= \frac{3 \mathcal{G} m_1 k_{2,0}
\mathcal{R}_0^5}{r_2^5
r_1^5} ({\bf{r}_2} \cdot {\bf{r}_1})({\bf{r}_2} \times {\bf{r}_1}) \\
{\bf{T}}^{(1)}({\bf {r}_2},{\bf {r}_1}) &= \frac{3 \mathcal{G} m_1 k_{2,0}
\mathcal{R}_0^5}{r_2^5 r_1^5}\Delta t_0 \bigg[ 5 \frac{({\bf{r}_1} \cdot
\dot{{\bf{r}}_1})}{r_1^2} ({\bf{r}_2} \cdot {\bf{r}_{1}})
({\bf{r}_{2}} \times {\bf{r}_1}) \\
&- [{\bf{r}_1} \cdot ({\bf{\Omega}_0} \times {\bf{r}_2}) + {\bf{r}_2} \cdot \dot{{\bf{r}}_1}]
({\bf{r}_2} \times {\bf{r}_1}) \\
&- ({\bf{r}_2} \cdot {\bf{r}_1}) [ ({\bf{r}_{2}} \cdot {\bf{\Omega}}_0){\bf{r}_{1}} -
({\bf{r}_{2}} \cdot {\bf{r}_1}){\bf{\Omega}_0} + {\bf{r}_2} \times \dot{{\bf{r}}_1} ] \bigg]
.\\
\end{split}
\label{eq5}
\end{eqnarray}
In the classical two-body tidal problem, the acceleration ${\bf f}$ and the torque ${\bf {T}}$
are computed on the position of the deforming body $m_1$. It is easy to see that in such a case,
the zero-order torque reduces to zero (i.e. ${\bf{T}}^{(0)} ({\bf{r}} = {\bf{r}_1},{\bf{r}_1}) =
{\bf{0}}$) and the only net contribution to the orbital and spin evolution (notwithstanding a
precession term) stems from the the first-order expressions ${\bf f}^{(1)}$ and ${\bf{T}}^{(1)}$
(see equations (5) and (6) of \cite{Mignard1979}). However, it is not immediately clear what
occurs if ${\bf r} \ne {\bf r_1}$. In other words, we wish to analyze what are the (long-term)
dynamical effects of a tidal bulge generated on $m_0$, due to the perturbing potential of $m_1$,
on the orbit of another body $m_2$.
To address this question, let the $m_0$-centric orbits of $m_i$ be characterized by semimajor
axis $a_i$, eccentricity $e_i$, mean longitude $\lambda_i$ and longitude of pericenter
$\varpi_i$. Furthermore, let $n_i$ denote the mean-motion (orbital frequency) of each body.
We then compute the net secular torques ($\langle\bf{T}^{(0)}({\bf {r}}_2,{\bf{r}}_1)\rangle$
and $\langle\bf{T}^{(1)}({\bf {r}}_2,{\bf{r}}_1)\rangle$) for different values of $a_2$,
assuming fixed values for $(a_1,e_1,e_2,\varpi_1,\varpi_2)$. The secular torques are calculated
averaging over the short-period terms associated to $\lambda_1$ and $\lambda_2$. Since we will
not restrict our analysis to non-resonant configurations between $m_1$ and $m_2$, we cannot
assume that both mean longitudes are necessarily independent. We thus substitute the classical
double averaging over $\lambda_i$ with a time average over time, such that
\begin{eqnarray}
\langle{\bf{T}}^{(i)}({\bf{r}_{2}},{\bf{r}_{1}})\rangle = \lim_{\tau \to \infty}
\frac{1}{\tau} \int_{0}^\tau {\bf{T}}^{(i)} ({\bf{r}_2}(t),{\bf{r}_1}(t)) \, dt ,
\label{eq6}
\end{eqnarray}
with $i=0,1$. This technique allows us to evaluate the net secular contribution in both resonant
and secular configurations of both bodies. In particular, the classical Mignard expressions
should be obtained assuming equal orbits (and orbital frequencies) for $m_1$ and $m_2$.
Results are shown in Figure \ref{fig:torques} for $m_0 = 1$, $a_1 = 1$, and $\varpi_1 =
\varpi_2$ and $e_1=e_2$. The values of the averaged torques are normalized with respect to $m_1$
and $\Delta t_0$, and plotted as function of the mean-motion ratio $n_1/n_2$. The left-hand
panels correspond to the zero-oder torque $\langle\bf{T}^{(0)}\rangle$, while the first-order
contributions $\langle\bf{T}^{(1)}\rangle$ are depicted in the right-hand graphs.
In the upper panels we analyze the circular case ($e_1=e_2=0$) and in the lower panel eccentric
orbits ($e_1=e_2=0.1$). Eccentricities are assume fixed throughout the numerical averaging. All
plots show distinct peaks, where the net torque is different from zero, overlaid with respect
to background values that decrease smoothly as $n_1/n_2 \rightarrow \infty$. This background
trend is a consequence of the numerical approximation employed to evaluate the time integral
(\ref{eq6}), which basically consisted in a discrete sum over a finite time interval equal to
500 orbital periods of the outermost body.
For circular orbits we observe that the only value of $a_2$ for which $m_2$ receives a non-zero
net torque corresponds to a $1/1$ MMR, that is in a coorbital position with the deforming body
$m_1$. In particular, the case in which the position of both masses coincide (i.e.
$\lambda_1=\lambda_2$) yields results analogous to those obtained from the 2-body tidal model.
Different initial values of the mean longitudes would, in principle, allow us to estimate both
torques in other coorbital configurations, such as that occurring for $m_2$ located in a
Trojan-like orbit with the other masses. Finally, although $|\langle {\bf{T}}^0({\bf{r}}_2)
\rangle|$ is different from zero in the $1/1$ MMR, its dynamical effect on the orbit reduces to
a tidal precession term (e.g. Correia et al. 2011) and does not contribute to any secular
changes in the orbits or spin rates.
The eccentric case, depicted in the lower frames, exhibits a richer diversity. Non-zero torques
are found in several mean-motion resonances and not only in the coorbital region. This seems to
imply that the tidal deformation generated by $m_1$ on $m_0$ should affect the dynamical
evolution of $m_2$ whenever there exists a commensurability relation between the orbital
frequencies. This is an important finding, indicating that tidal models for resonant bodies
could require the full tidal deformation on each body as generated by all the other bodies
of the system.
The numerical results described in Figure \ref{fig:torques} were confirmed introducing elliptic
expansions for the position and velocity vectors in the equations (\ref{eq4}) and (\ref{eq5}),
truncated up to fourth order in semimajor axis ratio and eccentricities, and integrating the
resulting expressions analytically.
In conclusion, in the absence of any mean-motion relation between $m_1$ and $m_2$, the only
tidal forces that need to be considered on $m_i$ are those stemming from the deformation that
$m_i$ generates on $m_j$ and $m_k$ (with $i \ne j \ne k$). Since the tidal deformation
generated on $m_j$ by $m_k$ may be neglected, a multi-body tidal model may be constructed
simply by adding the forces and torques between deformed-deforming pairs as given in equations
(5) and (6) of \cite{Mignard1979}. The effect of the tidal torques on resonant orbits will be
investigated in a forthcoming work.
\begin{figure}
\centering
\includegraphics[width=0.9\columnwidth,clip=true]{imagenes3.jpg}
\caption{Tidally interacting 3-body system. The tidal bulge generated on each body (filled
gray ellipsoids) are the sum of the deformations generated by each of its companions. ${\bf
R_i}$ denote the position vectors with respect to a generic inertial reference frame.}
\label{fig:3c}
\end{figure}
\subsection{The equations of motion}
\label{sub2}
Having identified the tidal forces affecting the long-term and secular dynamical evolution of
the system, we now discuss how they should be incorporated into the equations of motion of the
circumbinary system in a self-consistent manner.
We return to our full circumbinary system where now all bodies are considered extended and
gravitationally interacting. As shown in Figure \ref{fig:3c}, the equilibrium deformation of
body $m_i$ is the sum of two ellipsoids, each generated by the gravitational potential of the
other two masses. As shown in \cite{Folonier2017}, the sum of two ellipsoidal bulges can be
approximated by a single ellipsoidal bulge with its own flattening and orientation. However, as
discussed in Section \ref{sub1}, we only need to consider the direct distortion between pairs.
Let us denote by ${\bf R_i}$ the position vector of $m_i$ in an inertial reference frame.
Then, the complete equations of motion, including both tidal and point-mass terms, may be
expressed as:
\begin{eqnarray}
\begin{split}
m_0 {\bf \ddot{R}_0} &= \;\;\,
\frac{\mathcal{G}m_0 \, m_1}{|{\bf{\Delta_{10}}}|^3} {\bf \Delta_{10}}
+\frac{\mathcal{G}m_0 \, m_2}{|{\bf \Delta_{20}}|^3} {\bf \Delta_{20}} + \bf{F_0} \\
m_1 {\bf \ddot{R}_1} &=
-\frac{\mathcal{G}m_0 \, m_1}{|{\bf \Delta_{10}}|^3} {\bf \Delta_{10}}
+\frac{\mathcal{G}m_1 \, m_2}{|{\bf \Delta_{21}}|^3} {\bf \Delta_{21}} + \bf{F_1} \\
m_2 {\bf \ddot{R}_2} &=
-\frac{\mathcal{G}m_0 \, m_2}{|{\bf \Delta_{20}}|^3} {\bf \Delta_{20}}
-\frac{\mathcal{G}m_1 \, m_2}{|{\bf \Delta_{21}}|^3} {\bf \Delta_{21}} + \bf{F_2}
\label{eq7}
\end{split}
\end{eqnarray}
where for compactness we have denoted the relative position vectors as
\begin{equation}
{\bf \Delta_{ij}} \equiv {\bf R_i} - {\bf R_j} .
\label{eq8}
\end{equation}
In terms of the $m_0$-centric position vectors, these are simply given by ${\bf \Delta_{10}} =
{\bf r_{1}}$, ${\bf \Delta_{20}} = {\bf r_{2}}$ and ${\bf \Delta_{21}} = {\bf r_{2}} - {\bf
r_{1}}$. The last terms of the equations of motion are the complete tidal forces acting on each
mass. Following \cite{Ferraz-Mello2008}, considering the reacting forces, these may be
expressed by
\begin{eqnarray}
\begin{split}
\bf{F_0} &= \bf{F_{0,1}} + \bf{F_{0,2}} - \bf{F_{1,0}} - \bf{F_{2,0}} \\
\bf{F_1} &= \bf{F_{1,0}} + \bf{F_{1,2}} - \bf{F_{0,1}} - \bf{F_{2,1}} \\
\bf{F_2} &= \bf{F_{2,0}} + \bf{F_{2,1}} - \bf{F_{0,2}} - \bf{F_{1,2}} , \\
\end{split}
\label{eq9}
\end{eqnarray}
where ${\bf{F}_{i,j}}$ to the tidal force acting on $m_i$ due to the deformation in $m_j$. Note
that the positive contributions in $\bf{F_i}$ are the direct effect of the deformation of the
other bodies while the negative terms corresponds to the reaction of the force due to the
deformation of $m_i$. These have the form
\begin{eqnarray}
{\bf{F}_{i,j}} = -\frac{\mathcal{K}_{i,j}} {{|{\bf \Delta_{ij}}|}^{10}}
\bigg[ 2({\bf \Delta_{ij}} \cdot {\bf \dot{\Delta}_{ij}}) {\bf \Delta_{ij}}
+ {\bf \Delta_{ij}}^2 ( {\bf \Delta_{ij}} \times {\bf{\Omega}_j} + {\bf \dot{\Delta}_{ij}} )
\bigg]
\label{eq10}
\end{eqnarray}
(Mignard 1979), where $\mathcal{K}_{i,j}$ is a measure of the magnitude of the tidal force and
is given by
\begin{equation}
\mathcal{K}_{i,j} = 3 \mathcal{G}m_i^2 \mathcal{R}_j^5 k_{2,j}\Delta t_j.
\label{eq11}
\end{equation}
As before, ${\bf \Omega_j}$ is the spin angular velocity of $m_j$ and is assumed parallel to
the orbital angular momentum. We have neglected the tidal contributions which arise from the
zero-order potential since its effect is restricted to a precession of the pericenters and does
not introduce any secular changes in the spins, semimajor axes or eccentricities.
\subsection{The rotational dynamics}
While the orbital dynamics can be obtained solving the equations of motion (\ref{eq7}), the
time variation of the spins are deduced from the conservation of the total angular momentum
${\bf L_{\rm tot}}$. Since we assumed rotations perpendicular to the common orbital plane,
\begin{equation}
{\bf L_{\rm tot}} = {\bf L_{\rm orb}} + \sum_{i=0}^2 C_i {\bf \Omega_i} = const.,
\label{eq12}
\end{equation}
where $C_i$ is the principal moment of inertia of $m_i$. In turn, the orbital angular momentum
in the inertial reference frame is given by
\begin{equation}
{\bf L_{\rm orb}} = \sum_{i=0}^2 m_i ({\bf R_i} \times {\bf \dot{R}_i} ) .
\label{eq13}
\end{equation}
Differentiating this equation with respect to time and substituting expressions (\ref{eq7}) for
the accelerations ${\bf \ddot{R}_i}$, we obtain
\begin{eqnarray}
\begin{split}
{\bf \dot{L}_{\rm orb}} &=
{\bf \Delta_{10}} \times {\bf{F}_{1,0}} + {\bf \Delta_{20}} \times {\bf{F}_{2,0}} \\
&+ {\bf \Delta_{01}} \times {\bf{F}_{0,1}} + {\bf \Delta_{21}} \times {\bf{F}_{2,1}} \\
&+ {\bf \Delta_{02}} \times {\bf{F}_{0,2}} + {\bf \Delta_{12}} \times {\bf{F}_{1,2}} .\\
\end{split}
\label{eq14}
\end{eqnarray}
Furthermore, assuming that the variation in the spin angular momenta of the body $m_j$ is only
due to the terms in ${\bf {\dot L}_{\rm orb}}$ associated to its deformation, we obtain
\begin{equation}
C_j {\bf {\dot \Omega}_j} = - \sum_{i \ne j} {\bf \Delta_{ij}} \times {\bf{F}_{i,j}} .
\label{eq15}
\end{equation}
Note than in the limit where the physical radius of $m_j$ reduces to zero (i.e.
$\mathcal{R}_j=0$), the tidal terms ${\bf{F}_{i,j}}$ are also zero for all $i \ne j$, and
equation (\ref{eq15}) is automatically satisfied.
Finally, using expression (\ref{eq10}) for the tidal forces, the time evolution of the spin
vectors are given by
\begin{equation}
\frac{d{\bf \Omega_j}}{dt} = \frac{1}{C_j} \sum_{i \ne j}
\frac{{\cal K}_{i,j}}{|{\bf \Delta_{ij}}|^6} \bigg[ \frac{{\bf \Delta_{ij}} \times {\bf
{\dot \Delta}_{ij}}}{|{\bf \Delta_{ij}}|^2} - {\bf \Omega_j} \bigg] .
\label{eq16}
\end{equation}
Contrary to the 2-body case (e.g. Ferraz-Mello et al. 2008), the time derivative of the spin
is given by the sum of two distinct terms. Depending on the magnitudes of each tidal term, it
is not immediately obvious what would be the equilibrium rotational frequencies associated to
stationary solutions.
\section{Numerical simulations}
\label{numsim}
In order study the dynamical predictions of our model, we analyze the tidal evolution of a
3-body system consisting of a single planet around a binary star. The orbital and rotational
evolution will be followed solving the equations of motion (\ref{eq7}) for the orbit and
equation (\ref{eq16}) for each of the spins.
As before, we choose the Kepler-38 system as a test case, previously studied in
(\cite{Zoppetti2018}) using a simpler tidal model. Nominal values for system parameters and
initial orbital elements are detailed in Table \ref{tab1}. Stellar masses and radii were taken
from (\cite{Orosz2012}), while the value of $m_2$ was estimated from the semi-empirical
mass-radius from (\cite{Mills2017}). The orbital elements of the secondary star respect to
$m_0$ are those expected during the early stages of the system before tidal interactions had
time to act (see \cite{Zoppetti2018}), assuming tidal parameters and moments of inertia equal
to those given in the table.
The orbital parameters for the planet are similar to those presented by \cite{Orosz2012}, while
the value of $Q'_2$ is consistent with rocky bodies \citep{Ferraz-Mello2008}. However, it is
important to stress that there is little dynamical constraint on the values of the planetary
tidal parameters; the value adopted here is for illustrative purposes only. Finally, the
parameters highlighted with an asterisks were varied in our different simulations.
We will focus our attention on two different timescales: (i) an early stage (up to $\sim 1-2$
Gyr) characterized by the evolution of the rotation rates towards stationary solutions, and
(ii) the subsequent long-term dynamical orbital evolution of the system. In this second part we
will concentrate primarily on the orbital migration and eccentricity damping of the planet.
\subsection{Early dynamical evolution}
Figure \ref{fig:shortevo} shows the early rotational and orbital evolution of the binary stars
and the planet. Except for the spin rates, all initial conditions and system parameters were
taken equal to the nominal values of summarized in Table \ref{tab1}.
\begin{table}
\centering
\caption{Initial conditions for our reference numerical simulation, representing the primordial
Kepler 38 system \protect\citep{Orosz2012, Zoppetti2018}. Orbital elements are given in a
Jacobi reference frame. The parameters highlighted with an asterisk were varied in different
simulations as indicated in the text.}
\label{tab1}
\begin{tabular}{lccc}
\hline
body & $m_0$ & $m_1$ & $m_2$ \\
\hline
mass & $0.949 \, M_\odot$ & $0.249 \, M_\odot$ & $10 \, M_\oplus$ \\
radius & $0.84 \, R_\odot$ & $0.272 \, R_\odot$ & $4.35 \, R_\oplus$ \\
$C_i/(m_i\mathcal{R}_i^2)$ & $0.07$ & $0.25$ & $0.25$ \\
$Q'_i$ & $1 \times 10^6$ & $1 \times 10^6$ & $1 \times 10^1$(*) \\
$\Omega_i$ & $10 \, n_1$(*) & $10 \, n_1$(*) & $10 \, n_2$(*) \\
$a_i$ [AU] & & $0.15$ & $0.48$ \\
$e_i$ & & $0.15$ & $0.05$(*) \\
\hline
\end{tabular}
\end{table}
We begin our analysis with the binary components, shown in the left-hand plots of Figure
\ref{fig:shortevo}. The blue curves correspond to initial spin rates for both stellar
components equal to $\Omega_0 = \Omega_1 = n_1/10$ (i.e. slow rotators), while the black
curves show results where the star were considered initially fast rotators: $\Omega_0 = \Omega_1
= 10 n_1$. Regardless of the initial spin, both stars reach a pseudo-synchronization state in a
few Gyrs, with a final rotational frequency equal to the value predicted by 2-body tidal models:
$\Omega_0/n_1=\Omega_1/n_1=1+6 e_1^2$ (e.g. Ferraz-Mello et al. 2008).
If the stars were initially super-synchronous, the change in spin rates deliver angular momenta
to their orbit (eq. \ref{eq15}), increasing the semimajor axis $a_1$ and eccentricity
$e_1$. The opposite effect is observed if the stars were initially sub-synchronous: the orbit
delivers angular momenta to the stars to increase their spins, decreasing its semimajor axis and
eccentricity. Once the rotational stationary solution is attained, the subsequent dynamical
effect of the stellar tides acts to reduce the semimajor axis and damp the eccentricity until
the circularization is reached (\cite{Correia2016}, \cite{Hut1980}). Due to its small mass, the
presence of the CB planet has no noticeable influence on the tidal evolution of the binary.
\begin{figure}
\centering
\includegraphics[width=.99\columnwidth,clip=true]{short_evo.png}
\caption{Early tidal evolution of a circumbinary system. In all the panels, the black curve
represents the results of our reference simulation (Table \ref{tab1}). {\bf Left:} Dynamical
evolution of the binary, showing the spin rate (top), semimajor axis (middle) and eccentricity
(bottom panel). The results depicted in blue consider initially slow-rotating stars with
$\Omega_0/n_1=\Omega_1/n_1=0.1$ (at $t=0$), while those in black correspond to primordial fast
rotators $\Omega_0/n_1=\Omega_1/n_1=10$. {\bf Right:} Evolution of the planetary spin and
orbit. Black (respectively green) curves correspond to initial super-synchronous (respectively
sub-synchronous) planetary spin rates. Time variation of the semimajor axis $a_2$ and
eccentricity $e_2$ are practically equal in both cases (middle and bottom panels).}
\label{fig:shortevo}
\end{figure}
The right-hand panels of Figure \ref{fig:shortevo} show the dynamical evolution of the
planetary spin (top panel) and orbit (middle and bottom plots). As before we considered two
different initial spin rates: $\Omega_2/n_2 = 10$ is shown in black while $\Omega_2/n_2 = 0.1$
in green. The stellar spins were taken equal to the nominal values. We found no appreciable
change in the time evolution of the semimajor axis or eccentricity regardless of the initial
spins and, as seen in the middle and lower panels, both curves are practically
indistinguishable.
Concerning the evolution of the planetary spin, both initial conditions reach stationary values
much faster than the stars (typically in a few Myrs), although the equilibrium value is
sub-synchronous and significantly displaced with respect to the 2-body expectation (red
horizontal line). This behavior will be discussed in detail in section \ref{anali_spin} and
constitutes a new finding. Instead of the super-synchronous stationary solutions found in
classical tidal models for eccentric orbits, the interacting binary system leads to a stable
sub-synchronous state which does not change even after the stars themselves evolve towards their
rotational stationary spins.
\begin{figure*}
\centering
\includegraphics[width=.8\textwidth,clip=true]{long_evo.png}
\caption{Long-term orbital tidal evolution of our the planet in our Kepler-38-like system.
Except for the parameters inlaid in the left-hand plots, all parameters and initial
conditions were taken equal to those in Table \ref{tab1}. Light-tone curves for $a_2$ and
$e_2$ show osculating values while darker curves correspond to mean elements obtained from a
digital filter. The magenta curves in the lower panels are the result of a simulation
disregarding tidal interaction between the stars.}
\label{fig:planetevo}
\end{figure*}
\subsection{Long-term orbital evolution}
Figure \ref{fig:planetevo} shows three different long-term simulations, integrated over
timescales comparable with the estimated age of Kepler-38 system (\cite{Zoppetti2018}). All
system parameters and initial conditions were chosen equal to their nominal values (Table
\ref{tab1}) except for those described in the left-hand panels of each set. In all cases the
planetary spin reached a sub-synchronous stationary solution early in the simulation; thus
we concentrate on the orbital elements: semimajor axis $a_2$ in the left-hand plots,
eccentricity $e_2$ in the center graphs, and difference between longitudes of pericenter
$\Delta \varpi = \varpi_2 - \varpi_1$ in the right-hand graphs. Results after the application
of the low-pass filter are shown in darker curves for $a_2$ and $e_2$.
The black curves in the top panels correspond to the results of our reference simulation (see
Table \ref{tab1}) while in the cyan curves we consider a more dissipative planet with $Q'_2=1$.
The middle panels show results considering a more eccentric initial orbit $e_2(0)=0.1$, again
for the same two values of the tidal parameter. Finally, in the lower panel we analyze the case
in which the stars in the binary are not tidally interacting. This scenario correspond to
setting ${\bf{F}_{0,1}}={\bf{F}_{1,0}}=0$ (see eq. \ref{eq10}) in our code. Results with
non-tidally interacting stars are shown in magenta, while cyan curves repeat the results of our
simulation with tidal effects for the stars.
Independently of the adopted tidal parameter $Q'_2$, the planet is always observed to migrate
outwards, marking a second distinct difference with respect to expectations from classical
2-body tidal models. This result was already described in \cite{Zoppetti2018}, although in that
case we used a simpler and non-consistent tidal model. Lower values of $Q'_2$ (cyan curves
in the upper and middle panels) lead to more larger excursions in semimajor axis, ultimately
leading to scattering in a high-order MMR and temporary excitation of the eccentricity. The
magenta curve in the lower panels show that the outwards migration is not a consequence of
tidal effects in the stars, but seems to be independent of their tidal evolution.
The planetary eccentricity, on the other hand, always seems to decrease, as long as not
mean-motion resonances are encountered. For low initial values of $e_2$ (upper panels of
Figure \ref{fig:planetevo}) the planet and secondary star enter an aligned secular mode
(\cite{Michtchenko2004}) in which $\Delta \varpi$ librates around zero. The amplitude of
oscillation increases for larger initial eccentricities until $\Delta \varpi$ is observed to
circulate for $e_2(0)=0.1$. However, the libration/circulation is purely kinematic and the
difference in behavior is related to the amplitude of oscillation of the eccentricity.
Regardless, these results seem to indicate that an analytical model for the tidal evolution of
these type of systems must include terms involving the secular angle $\Delta \varpi$, even if
the tidal evolution timescales are much longer than those associated to the precession of
pericenters $\varpi_1$ and $\varpi_2$.
\section{Analytical secular model}
\label{anali}
In order to construct an analtical model from the equations of motion (\ref{eq7}) and
(\ref{eq16}), we first introduce a Jacobi reference frame for the position and velocity vectors
of the bodies. In terms of the inertial coordinates ${\bf R_i}$, the positions of the
masses in Jacobi coordinates are given by:
\begin{eqnarray}
\begin{split}
{\boldsymbol \rho_{\bf 0}} &= \frac{1}{\sigma_2} (m_0 \, {\bf R_0} + m_1 \, {\bf R_1} + m_2 \,
{\bf R_2}) \\
{\boldsymbol \rho_{\bf 1}} &= {\bf R_1} - {\bf R_0} \\
{\boldsymbol \rho_{\bf 2}} &= {\bf R_2} - \frac{1}{\sigma_1} (m_0 \, {\bf R_0} + m_1 \, {\bf
R_1}) ,\\
\end{split}
\label{eq17}
\end{eqnarray}
where
\begin{equation}
\sigma_i=\sum_{k=0}^i m_k .
\label{eq18}
\end{equation}
Analogous expressions relate the velocities vectors in both reference systems.
\subsection{Secular evolution of the planetary spin}
\label{anali_spin}
Expanding the position and velocity vectors in equation (\ref{eq16}) up to second order in
$\alpha = a_1/a_2$ and the eccentricities, and averaging with respect to both mean longitudes,
we finally obtain the rate of change of the rotational frequency of the planet as:
\begin{equation}
\left<\frac{d\Omega_2}{dt}\right> = \frac{1}{2 C_2 a_2^6}
\sum_{i,j,k=0}^2 A^{(s)}_{i,j,k} \, \alpha^i e_1^j e_2^k ,
\label{eq19}
\end{equation}
where the non-zero coefficients different are given by
\begin{eqnarray}
\begin{split}
A^{(s)}_{0,0,0} &= 2 ({\cal K}_{0,2}+{\cal K}_{1,2}) (n_2-\Omega_2) \\
A^{(s)}_{2,0,0} &= 6(\gamma_0^2{\cal K}_{0,2}+\gamma_1^2 {\cal K}_{1,2}) (4n_2-n_1-3\Omega_2)\\
A^{(s)}_{2,2,0} &= 3(\gamma_0^2{\cal K}_{0,2}+\gamma_1^2 {\cal K}_{1,2})(12n_2+n_1-9\Omega_2)\\
A^{(s)}_{0,0,2} &= 3 ({\cal K}_{0,2}+{\cal K}_{1,2}) (9 n_2-5\Omega_2) \\
A^{(s)}_{2,0,2} &= 12 (\gamma_0^2 {\cal K}_{0,2}+\gamma_1^2 {\cal K}_{1,2}) (44 n_2 - 7n_1 -
21\Omega_2) \\
A^{(s)}_{1,1,1} &= 9 (8 n_2 - 5 \Omega_2) (\gamma_0 {\cal K}_{0,2}+\gamma_1
{\cal K}_{1,2}) \cos(\Delta\varpi) ,\\
\end{split}
\label{eq20}
\end{eqnarray}
with
\begin{equation}
\gamma_0 = \frac{m_1}{\sigma_1} \hspace*{0.4cm} ; \hspace*{0.4cm}
\gamma_1 = -\frac{m_0}{\sigma_1} .
\label{eq21}
\end{equation}
The stationary spin rate $\big< \Omega_2 \big>_{\rm stat}$ predicted by this equation can be
easily calculated by equating expression (\ref{eq19}) to zero. The explicit form of the
equilibrium rotational frequency was found to be
\begin{eqnarray}
\begin{split}
\big< \Omega_2 \big>_{\rm stat} &= (1 + 6 e_2^2) n_2 - 6 \frac{\gamma_0^2
\gamma_1^2}{\gamma_0^2 + \gamma_1^2} \left( n_1-n_2 \right) \alpha^2 \\
&+ 3 \frac{\gamma_0^2 \gamma_1^2}{\gamma_0^2 + \gamma_1^2} \left( n_1 + 3n_2 \right) \alpha^2
e_1^2 \\
&- 3 \frac{\gamma_0^2 \gamma_1^2}{\gamma_0^2 + \gamma_1^2} \left( 13 n_1 - 41 n_2 \right)
\alpha^2 e_2^2 \\
&+ \frac{27}{2} \frac{\gamma_0 \gamma_1 (\gamma_0+\gamma_1)}{\gamma_0^2 + \gamma_1^2} n_2
\alpha e_1 e_2 \cos(\Delta \varpi) . \\
\end{split}
\label{eq22}
\end{eqnarray}
In the limit case in which the mass of one of the stars reduces to zero we recover the classical
2-body super-synchronous stationary solution $\big<\Omega_2 \big>_{\rm stat} = (1+6 e_2^2)
\, n_2$ (\cite{Ferraz-Mello2008},\cite{Correia2011}). On the other hand, we can observe that
for low binary and planetary eccentricities ($e_1,e_2 \to 0$), the CB planet stationary solution
is sub-synchronous by a factor that decreases proportional to $\alpha^2$ as we move outward
from the binary, and is maximum for equal-mass stars $m_1=m_0$.
\begin{figure}
\centering
\includegraphics[width=1.05\columnwidth,clip=true]{stat.png}
\caption{Stationary planetary spin $\big<\Omega_2 \big>_{\rm stat}/n_2$ as function of the
semimajor axis ratio $\alpha$ and mass of the secondary star (top), and as function of the
eccentricities (bottom). All other system parameters were chosen equal to those given in
Table \ref{tab1}. The nominal parameters for Kepler-38 are highlighted with a filled white
circle and marked as ``K38''. Dashed curve in the top graph corresponds to $\big<\Omega_2
\big>_{\rm stat}/n_2 = 1 + 6 e_2^2$.}
\label{fig:stat}
\end{figure}
Figure \ref{fig:stat} shows two color plots with the value of $\big<\Omega_2 \big>_{\rm stat}$
as a function of different system parameters and eccentricities (assumed constant). The top
frame shows the dependence of the equilibrium spin rate of the planet with the distance from
the binary system and the mass of the secondary star. Except for initial conditions very
close to the binary or $m_1/m_0 \lesssim 0.1$, the estimated value of $\big<\Omega_2 \big>_{\rm
stat}$ is always sub-synchronous with respect to the mean orbital frequency. The dashed black
curve corresponds to the equilibrium value of the spin as obtained from the 2-body problem,
i.e. $\big<\Omega_2 \big>_{\rm stat}/n_2 = 1 + 6 e_2^2$. Our model predicts lower values for
practically all values of the system parameters, at least for the nominal eccentricities. This
seems to imply that even a low-mass secondary, or even a large interior planet may counteract
the super-synchronous state deduced from the 2-body solution and lead to appreciable
differences in the rotational dynamics.
The dependence of $\big<\Omega_2 \big>_{\rm stat}$ with the eccentricities is analyzed in the
bottom frame of Figure \ref{fig:stat}. We note that the sub-synchronous equilibrium state is
only observed for low eccentricities of the planet, typically $e_2 \lesssim 0.1-0.15$, while
super-synchronous states may be attained form more eccentric planets. However, since we expect
tidal effects to damp the eccentricity, it appears that $\big<\Omega_2 \big>_{\rm stat} < n_2$
should probably the most common configuration in real-life systems. Finally, we observe little
sensitivity of the equilibrium spin with respect to the eccentricity of the binary.
In order to test the validity and precision of our analytical model, Figure \ref{fig:subsin}
shows four sets of different N-body simulations of the evolution of the planetary spin,
considering binaries with different mass ratios and planets in orbits with different initial
eccentricities. All results were digitally filtered to remove short-period variations.
The top right-hand panel uses initial conditions from Table \ref{tab1} while the bottom
right-hand panel considers a more eccentric CB planet. The left panels explore the case in
which the mass of the secondary star is smaller than the nominal value. In every case the black
curves correspond to initially super-synchronous planets while the green curves correspond to
initially sub-synchronous CB planets. In dashed yellow curve, we show the synchronization spin
predicted by our model (eq. \ref{eq22}) while in dashed red curve we compare with the
stationary 2-body solution.
In accordance with the initial simulations presented in the previous section, the planetary
spin reaches a stationary state rapidly, typically in about $10^5$ years, and our model
seems to reproduce the equilibrium behavior extremely well. In the case of low-massive
secondary star (left panels), the synchronization spin is very close to that predicted by the
2-body model. However, when we consider binaries with mass ratios similar to Kepler-38 system,
the synchronization spin is very different: sub-synchronous by an amount that can be very large
for binaries with stars of comparable mass. Since the gravitational interaction causes
long-term (secular) variations in the eccentricity of the planet, the value of $\Omega_2$ also
suffers periodic oscillations.
\begin{figure}
\centering
\includegraphics[width=1.07\columnwidth,clip=true]{subsin_osci.png}
\caption{N-body simulation of the spin evolution of fictitious CB planets, considering binaries
with different mass ratios (different columns) and different initial eccentricity for the
planets (different rows). In all the panels, the black curves correspond to the evolution of an
initially super-synchronous planet while the green curves represent the initially
sub-synchronous case. The dashed yellow curves are the stationary spins predicted by our model
(eq. \ref{eq22}) while the dashed red curves are the 2-body stationary solution.}
\label{fig:subsin}
\end{figure}
Finally, as can be observed from equation (\ref{eq22}), the stationary spin solution for the
CB planets is not a function of the planetary mass $m_2$ nor of the physical radii of the
bodies. Thus, if we assume that all currently known circumbinary planets have reached their
stationary spin, we can predict their current rotational period just from the stellar masses and
planetary orbits. As an example, considering its maximum possible eccentricity
\citep{Orosz2012} and that the planet is in an aligned secular mode \citep{Zoppetti2018}, we
estimate the rotation period of the planet in the Kepler-38 system in $P_{K38} \simeq 118$ days,
about a $12 \%$ higher than the one predicted by the 2-body synchronization model.
\subsection{Variational equations for the orbital evolution}
Having developed an analytical model for the rotational dynamics, we turn our attention to the
time evolution of the semimajor axis $a_2$ and eccentricity $e_2$. As before, we will focus on
the planetary orbit, although analogous expressions can be found also for the binary.
Following \cite{Beutler2005}, the variational equation for the semimajor axis in the
Jacobi reference frame may be written as
\begin{equation}
\frac{da_2}{dt} = \frac{2 a^2_2 }{\mathcal{G}\sigma_2} (\dot{\boldsymbol{\rho}}_{\bf 2} \cdot
\delta\bf{f_2})
\label{eq23}
\end{equation}
where $\delta{\bf{f}_2}$ is the total tidal force (per unit mass) affecting the 2-body motion
of the planet around the center of mass of $m_0$ and $m_1$, and has the form:
\begin{equation}
\delta{\bf{f}_2} = \frac{{\bf{F}_2}}{m_2}-\frac{1}{\sigma_1}({\bf{F}_0}+{\bf{F}_1}) .
\end{equation}
Substituting equation (\ref{eq9}) in order to express the total force in terms of the
individual two-body tidal interactions, we obtain
\begin{equation}
\delta{\bf{f}_2} = \frac{1}{\beta_2} \bigg[ ({\bf{F}_{2,0}} - {\bf{F}_{0,2}}) + ({\bf{F}_{2,1}}
- {\bf{F}_{1,2}}) \bigg] ,
\label{eq25}
\end{equation}
where
\begin{equation}
\beta_i = \frac{m_i\sigma_{i-1}}{\sigma_i}
\label{eq26}
\end{equation}
is the reduced-mass \cite[e.g.][]{Beauge2007}. An analogous reasoning leads to a similar
equation for the binary orbital evolution.
Expression (\ref{eq25}) shows that the total tidal force $\delta{\bf{f}}_2$ may be written in
terms of differences of the type $({\bf{F}_{2,j}} - {\bf{F}_{j,2}})$, where $j=0,1$. From
equations (\ref{eq10}), each of these differences may be explicitly written as
\begin{eqnarray}
\begin{split}
{\bf{F}_{2,j}}-{\bf{F}_{j,2}} &= -\frac{\mathcal{K}^{(+)}_j}{|{\bf{\Delta_{2j}}}|^{10}}
\bigg[ 2({\bf \Delta_{2j}} \cdot {\bf \dot{\Delta}_{2j}}) {\bf \Delta_{2j}} \\
&\hspace*{2cm} + {\bf \Delta_{2j}}^2 ( {\bf \Delta_{2j}} \times {\bf{\bar {\Omega}}^{(j)}_2}
+ {\bf \dot{\Delta}_{2j}} ) \bigg] \\
\end{split}
\label{eq27}
\end{eqnarray}
where we have defined
\begin{equation}
{\cal K}_{j}^{(+)}={\cal K}_{2,j}+{\cal K}_{j,2}
\label{eq28}
\end{equation}
and a new ``averaged'' rotational frequency
\begin{equation}
{\bf {\bar {\Omega}}^{(j)}_2} =
\frac{\mathcal{K}_{2,j}{\bf{\Omega}_j}+\mathcal{K}_{j,2}{\bf{\Omega}_2}}{{\cal K}_{2,j}+{\cal
K}_{j,2}}.
\label{eq29}
\end{equation}
Notice that expression (\ref{eq27}) has the same functional form as the tidal force in the
2-body problem (eq. \ref{eq10}) with a magnitude given by ${\cal K}_{j}^{(+)}$ and a
rotational frequency defined by ${\bf{\bar {\Omega}}^{(j)}_2}$. In the limit where $m_1
\rightarrow 0$ and ${\cal R}_1 \rightarrow 0$, the term in the tidal force associated to ${\cal
K}^{(+)}_1$ becomes negligible and we recover the same expression as found in the classical
2-body case.
Writing the tidal forces in terms of Jacobi coordinates through ${\bf \Delta_{2j}} =
\boldsymbol{\rho}_2 + \gamma_j \boldsymbol{\rho}_1$, substituting in the Gauss equation
(\ref{eq23}), expanding in power series of $\alpha$, $e_1$ and $e_2$ and, finally, averaging
over the mean longitudes, we obtain:
\begin{equation}
\bigg< \frac{da_2}{dt} \bigg> = \frac{n_2}{ {\cal G} m_2 \sigma_2 a_2^4}
\sum_{i=0}^4 \sum_{j,k=0}^2 \sum_{l=0}^1 A^{(a)}_{i,j,k,l} \, {\cal K}_{l}^{(+)}
\gamma_l^i \alpha^i e_1^j e_2^k ,
\label{eq30}
\end{equation}
where the non-zero coefficients are explicitly given by
\begin{eqnarray}
\begin{split}
A^{(a)}_{0,0,0,l} &= 2 \Big[ \bar{\Omega}^{(l)}_2 - n_2 \Big] \\
A^{(a)}_{2,0,0,l} &= 2 \Big[ 12 \bar{\Omega}^{(l)}_2 + 5 n_1 - 17 n_2 \Big] \\
A^{(a)}_{4,0,0,l} &= 20 \Big[ 6 \bar{\Omega}^{(l)}_2 + 5 n_1 - 11 n_2 \Big] \\
A^{(a)}_{2,2,0,l} &= \Big[ 36 \bar{\Omega}^{(l)}_2 - 5 n_1 - 51 n_2 \Big] \\
A^{(a)}_{4,2,0,l} &= 100 \Big[ 6 \bar{\Omega}^{(l)}_2 + n_1 - 11 n_2 \Big] \\
A^{(a)}_{1,1,1,l} &= 6 \Big[ 12 \bar{\Omega}^{(l)}_2 - 19 n_2 \Big] \cos(\Delta\varpi) \\
A^{(a)}_{3,1,1,l} &= \frac{25}{2} \Big[ 96 \bar{\Omega}^{(l)}_2 + 32 n_1 - 193 n_2 \Big]
\cos(\Delta\varpi) \\
A^{(a)}_{0,0,2,l} &= \Big[ 27 \bar{\Omega}^{(l)}_2 - 46 n_2 \Big] \\
A^{(a)}_{2,0,2,l} &= \Big[ 528 \bar{\Omega}^{(l)}_2 + 5 (44 n_1 - 227 n_2) \Big] \\
A^{(a)}_{4,0,2,l} &= 10 \Big[ 390 \bar{\Omega}^{(l)}_2 + (325 n_1 - 1008 n_2) \Big] .\\
\end{split}
\label{eq31}
\end{eqnarray}
Figure \ref{fig7} shows the normalized value of $\big< da_2/dt \big>$ in the $(\alpha,m_1/m_0)$
plane for three different values of the binary and planet eccentricities. For each value of
$m_1$ the physical radius of the star was modified following the empirical rule ${\cal R_1}
\simeq 0.9 m_1$. The nominal values are shown in the top panel, and the parameters corresponding
to Kepler-38 highlighted with a white circle. All initial conditions and physical parameters
leading to an inward orbital migration of the planet are colored in tones of blue, while those
leading to a secular increase of $a_2$ in tones of red. The limit between both is marked with a
white curve.
\begin{figure}
\centering
\includegraphics[width=1.0\columnwidth,clip=true]{dadt.png}
\caption{Normalized values of the secular rate of change of the planetary semimajor axis, as
function of the binary mass ratio and $\alpha$. Each panel shows results for different
eccentricities, assumed fixed for this calculation. Blue tones denote regions where the planet
experiences an inward orbital migration, while red tone identify regions where the migration is
outward. The primordial parameters of Kepler-38 are again highlighted in the top pannel with a
filled white circle and marked as ``K38''.}
\label{fig7}
\end{figure}
Although the plots show some quantitative differences as function of the eccentricities, in all
cases there seems to exist a lower value of $m_1/m_0$ above which the tidal interaction of the
system leads to an outward migration of the planet. The critical value of $m_1$ appears to be
larger for more eccentric binaries and lower for stars in almost circular orbits. As expected,
as $m_1 \rightarrow 0$ the migration is inwards, in accordance with known results for the
2-body case.
It is necessary to point out that our analytical model was obtained through a Legendre
expansion of the elliptic functions truncated at fourth-order of $\alpha$. Consequently, the
results shown here and in Figure \ref{fig:stat} are not expected to be accurate (or even valid)
for $\alpha \rightarrow 1$. We have nevertheless opted to include the complete range solely for
illustrative purposes.
The time variation of the eccentricity $e_2$ may be found from the orbital angular momentum
${\bf{L}}_2$ in the Jacobi reference frame. In the planar case, we have
\begin{equation}
L_2 = \beta_2 |(\boldsymbol{\rho}_{\bf 2} \times \dot{\boldsymbol{\rho}}_{\bf 2})| = \beta_2
\sqrt{\mathcal{G}\sigma_2 a_2 (1-e^2_2)} ,
\label{eq32}
\end{equation}
whose time derivative due to tidal forces leads to
\begin{equation}
\frac{1}{\beta_2}\dot{L}_2 = \frac{\mathcal{G}\sigma_2\beta_2}{2 L_2} \bigg( (1-e^2_2)
\frac{da_2}{dt} - a_2 \frac{de_2^2}{dt} \bigg) = |(\boldsymbol{\rho}_{\bf 2} \times
\delta{\bf{f}_2})| .
\label{eq33}
\end{equation}
Extracting the eccentricity term, we finally obtain:
\begin{equation}
\frac{d}{dt} (e^2_2) = \frac{1}{a_2} \bigg[(1-e^2_2)\frac{da_2}{dt} - \frac{2
L_2}{\mathcal{G}\sigma_2\beta_2} ({\boldsymbol\rho}_{\bf 2} \times \delta\bf{f_2}) \bigg] .
\label{eq34}
\end{equation}
Introducing elliptic expansions in a similar manner as done for (\ref{eq30}), and averaging
over short-period terms, we obtain:
\begin{equation}
\bigg< \frac{de_2^2}{dt} \bigg> = \frac{n_2}{4 {\cal G} m_2 \sigma_2 a_2^{5}}
\sum_{i=0}^4 \sum_{j,k=0}^2 \sum_{l=0}^1 A^{(e)}_{i,j,k,l} \, {\cal K}_{l}^{(+)}
\gamma_l^i \alpha^i e_1^j e_2^k
\label{eq35}
\end{equation}
where now the non-zero coefficients are given by
\begin{eqnarray}
\begin{split}
A^{(e)}_{0,0,2,l} &= 4 \Big[ 11 \bar{\Omega}^{(l)}_2 - 18 n_2 \Big] \\
A^{(e)}_{2,0,2,l} &= 20 \Big[ 36 \bar{\Omega}^{(l)}_2 + 15 n_1 - 74 n_2 \Big] \\
A^{(e)}_{4,0,2,l} &= 40 \Big[ 139 \bar{\Omega}^{(l)}_2 + 95 n_1 - 282 n_2 \Big] \\
A^{(e)}_{1,1,1,l} &= 2 \Big[ 39 \bar{\Omega}^{(l)}_2 - 54 n_2 \Big] \cos(\Delta\varpi) \\
A^{(e)}_{3,1,1,l} &= 10 \Big[ 102 \bar{\Omega}^{(l)}_2 + 34 n_1 - 185 n_2 \Big]
\cos(\Delta\varpi) .\\
\end{split}
\label{eq36}
\end{eqnarray}
Contrary to $da_2/dt$, we found that the eccentricity of the planet is always damped, at least
for the initial conditions and system parameters tested here.
\subsection{Comparisons with numerical integrations}
To test the accuracy of our analytical model, for given initial conditions we compare the
variation in planetary semimajor axis and eccentricity predicted by equations (\ref{eq30}) and
(\ref{eq35}) with the numerical results obtained using the original unaveraged equations
(\ref{eq23}) and (\ref{eq34}). We consider the nominal system parameters detailed in Table
\ref{tab1} but varied the planetary eccentricity and semimajor axis ratio $\alpha$. For each we
computed $da_2/dt$ and $de_2/dt$ as a function of the reduced mass
\begin{equation}
\tilde{\mu} = \frac{m_1}{m_0+m_1}
\label{37}
\end{equation}
by varying $m_1$. Due to the rapid rotational synchronization timescales, we consider
stationary spins for the stars and for the planet according to equation (\ref{eq22}).
\begin{figure*}
\centering
\includegraphics[width=.95\textwidth,clip=true]{dade_e115_alf4.png}
\caption{Time derivative of the semimajor axis (left panels) and eccentricity variation
(right panels) of a circumbinary planet at different distances from the binary: $\alpha=5/32$
(top panels) and $\alpha=5/16$ (bottom panels). Different colors are employed for different
eccentricities ($e_2=0.01$ in blue, $e_2=0.05$ in green and $e_2=0.1$ in red) and different type
of curves make reference to the calculation method: numerical (full line) and analytical (dashed
line).}
\label{fig:dade}
\end{figure*}
Results are shown in Figure (\ref{fig:dade}). In all the panels the colors represent different
planetary eccentricities ($e_2=0.01$ in blue, $e_2=0.05$ in green and $e_2=0.1$ in red) while
the type of curve makes reference to the calculation method (full line for numerical and dashed
line for analytical). Different rows correspond to different values of $\alpha$: the reference
value in the bottom panels ($\alpha=5/16$, see Table (\ref{tab1})) and half the nominal value
in top panels.
From the right panels we note that, as a result of the tidal interaction, the eccentricity of
the planet always decreases with a rate that seems weakly dependent on the secondary mass.
However, as in the 2-body case, $e_2$ decays more rapidly for eccentric planets. Thus, the
effect of tides on the eccentricity of circumbinary planets is very similar to that in the case
of bodies around single stars. In the absence of additional forces we expect the systems
to evolve towards quasi-circular orbits. Since our analytical model only included terms up
to second order in $e_i$, the accuracy decreases substantially for larger eccentricities,
leading to an relative error of the order of $20 \%$ for $e_2 \sim 0.1$. A more complete model,
perhaps including Mignard eccentricity functions (Mignard 1980) are necessary for more
eccentric orbits.
The rate of change of the semimajor axis (left-hand plots) shows a better agreement between our
model and the full unaveraged equations, leading to practically the same magnitude in the
derivatives even for moderate eccentricities. In particular, the values of the critical reduced
mass $\tilde{\mu}_{crit}$ associated to the limit between inward and outward migration is very
well reproduced.
Finally, Figure (\ref{fig:mucri}) shows the dependence of $\tilde{\mu}_{crit}$ as
function of $\alpha$ for different eccentricities. As before, calculations performed
with the unaveraged equations are plotted in continuous lines, while dashed curves
show results with the analytical model including terms up to fourth order in $\alpha$. To test
the necessity of such high orders, the dotted lines show analogous results, this time
truncating the expansions at third order in the semimajor-axis ratio. While the precision of
the fourth-order analytical model is very good up to $\alpha \sim 0.3$, the truncated version
shows a much smaller region of validity, reduced down to $\alpha \sim 0.1$. Thus, systems such
as Kepler-38 require a high-order model in order to reproduce the dynamics with a fair accuracy.
It is interesting to note that $\tilde{\mu}_{crit}$ increases for smaller values of $\alpha$.
In the limit when $\alpha \rightarrow 0$, we expect the system to behave as a planet orbiting a
single star of mass $m_0+m_1$ and all initial conditions should lead to an inward migration of
the semimajor axis.
\begin{figure}
\centering
\includegraphics[width=1.1\columnwidth,clip=true]{mucri_gato.png}
\caption{Critical value of $\tilde{\mu}$ above which tidal effects on the planet lead to
outwards orbital migration. Different colors represent different eccentricities for the planet
(same as in Figure (\ref{fig:dade})) and different types of curves refer to different
calculation method: numerical (continuous curves), analytical up to fourth order in $\alpha$
(dashed) and analytical up to third order in the semimajor-axis ratio (dotted line).}
\label{fig:mucri}
\end{figure}
\section{Summary and discussion}
\label{conclu}
In this work we present a model for treating the tides in a circumbinary system with one
planet, in which all bodies are assumed to be extended and tidally interacting. To built the
model, we consider a weak friction regime where the tidal forces can be approximated by the
classical expressions of \cite{Mignard1979} and proceed in two steps:
\begin{enumerate}
\item First, we revisited the Mignard theory and studied which tidal forces have a net effect
onto the dynamical evolution of the system. In the classical 2-body problem, where we are
computing the torques on the same body that exerts the deformation, the zero-order Mignard
torques have zero net secular effect. We found that this torques also has a null effect on
the third body, as long as there are no mean-motion resonances between $m_1$ and $m_2$.
Thus, in the non-resonant circumbinary problem, the only forces that should be taken into
account are those that are applied on the same body that exerts the deformation.
In a resonant case the zero-order torques may have important effects; their consequences will
be the focus of a forthcoming work.
\item Secondly, we incorporate the tidal forces in the gravitational equations of motion in a
self-consistent approach. Namely, we consider that each of the bodies is deformed by the other
two and there is a reaction force for each tidal force applied. As a result, we obtain the spin
evolution equation for the bodies and the orbital evolution equation for the planet.
\end{enumerate}
We have undertaken a series of numerical simulations, considering Kepler-38 system as a working
example, in order to compare the results of this model with our previous work
(\cite{Zoppetti2018}). We observed that in the short-timescales the dynamics is dominated by the
spin synchronization of the bodies: the planet, assumed a rocky body, synchronize very quickly
(in $\sim$ Myr) in a stationary spin lower than the orbital mean motion. On the other hand, the
stars exhibit super-synchronous spins in values predicted by the 2-body classical problem. The
subsequent orbital evolution of the binary is little affected by the planet and proceeds to a
decrease in the semimajor axis $a_1$ and eccentricity $e_1$.
The long-term orbital evolution of the planet is curiously different: as a result of the tidal
interaction the planet migrates outward and the direction of migration is not dependent on the
initial planetary eccentricity or the assumed planetary tidal parameter. Moreover, the outward
migration is also not an indirect effect of the migration of the binary, but observed even if
the tidal evolution of the stars is neglected.
During the tidal migration, the eccentricity of the planet oscillates around the force
eccentricity, which decreases as we move away from the binary (\cite{Leung2013}). For some
initial conditions, we found that the difference of pericenter angle $\Delta \varpi$ librates
around zero. Thus, when studying the secular tidal evolution of circumbinary planets, the usual
procedure of averaging over the longitudes of pericenters may not be accurate.
To better understand the numerical results, we constructed an analytical secular model
expanding the full spin and orbital equations of motion and averaging only over the mean
longitudes. Regarding the spins, the simplicity of the full equation, allows us to expand only
up to second-order in $\alpha$ and the eccentricities $e_1$ and $e_2$. The resulting
expressions showed a very good agreement with N-body simulations. We furthermore
obtained a simple equation estimating the stationary spin of CB planets that is not dependent on
the planetary mass. If we assume that their spins have reached their equilibrium state,
this allow us to predict the rotation period of almost all circumbinary systems requiring only
knowledge of the stellar masses and the orbital configuration of its members. Our analytical
approach was validated comparing the planetary stationary spin of the numerical simulation
with those predicted by our analytical equations.
Contrary to the spins, the analytical model for the orbital evolution required an expansion in
the semimajor-axis ratio up to fourth-order in $\alpha$. We maintained the eccentricities up to
second order; however, latter simulations showed that higher orders are probably needed in
systems with moderate-to-high eccentricities.
Regarding the eccentricity evolution, we found that the tidal forces on the CB planet always
seem to act circularizating its orbit. We observed a strong dependence on the eccentricities
but only a marginal dependence on the mass ratio of the stellar components. On the other hand,
the complex dependence of the planetary semimajor axis evolution with the mass of the stars is
reflected in the fact that the direction of migration depends on the binary mass ratio: for
binaries in which the secondary star is much less massive, even the case in which the secondary
companion is a planet, the tidal migration direction is inward. However, when the mass of
both stars are of the same order the planet migrates outward. The critical value of mass ratio
for which the direction of migration changes sign is dependent on the planetary eccentricity and
also on the position of the CB planet but can be predicted very accurately with our model.
The magnitude of the semimajor-axis variation is also very sensitive to the planetary
eccentricity and proximity to the binary, but mainly dominated by the amount of energy that is
dissipated in the planet due to tides. This quantity is very uncertain; however, the unexpected
outward tidal migration of CB planet seems to be only dependent on the stellar masses and
system configuration. A preliminary application of our model to other observed {\it Kepler}
systems seems to indicate that many systems could also have suffered an outward tidal
migration.
\begin{acknowledgements}
We wish to express our gratitude to IATE for an extensive use of their computing facilities,
without which this work would not have been possible. This research was funded by CONICET,
SECYT/UNC, FONCYT and FAPESP (Grant 2016/20189-9).
\end{acknowledgements}
\bibliographystyle{aa}
|
1,314,259,993,584 | arxiv | \section{Introduction}
Gas layers dramatically affect the flow boundary conditions in microfluidic systems, reducing drag by up to 75\%, \cite{lee2016superhydrophobic} but are frequently overlooked when one or more dimension is nanoscale due to difficulty in detecting them. Nanoscale gas layers, also known as surface nanobubbles, are extremely difficult to observe and characterize as they
are too small to be quantitatively analysed using optical
techniques which are typically used to study liquid-gas
interfaces. \cite{Alheshibri_Qian_Jehannin_Craig_2016} \par
Surface nanobubbles were first reported in the year 2000 \cite{Alheshibri_Qian_Jehannin_Craig_2016, Ishida_Inoue_Miyahara_Higashitani_2000} and have been controversial ever since. The high Laplace pressure inside bubbles with radius of curvature smaller than \SI{100}{\nano\meter} (\textit{e.g.} $\Delta$P $\approx$ \SI{29}{\atm} for r = \SI{50}{\nano\meter} and $\gamma = \SI{72}{\milli\newton\per\meter}$) indicates that they should have very short lifetimes (\textit{i.e.} on the order of \SI{100}{\micro\second} \cite{Lohse_Zhang_2015}), however they are routinely seen to be stable on much longer timescales. \cite{Zhang_Maeda_Craig_2006} Their unexpected stability is due to the fact that, despite having thickness in the range of a few tens to hundreds of nanometers, they have micrometric lateral size, which produces a flat interfacial shape with low radius of curvature. This reduces the internal Laplace pressure and allows the bubble to remain stable on the immersed surface for hours to days. For a complete review of the field, the reader is directed to the review by Lohse and Zhang. \cite{Lohse_Zhang_2015} \par
Here, we report a method to detect and map the presence of gas layers on structured hydrophobic surfaces covered with a thin layer of a hydrophobic oil. This type of surface is known as lubricant-infused surface (LIS) and has been the topic of intense research over the past decade due to their desirable properties introduced by the presence of the entrapped lubricant layer, \cite{Peppou-Chapman_Hong_Waterhouse_Neto_2020} such as anti-fouling, \cite{Epstein_Wong_Belisle_Boggs_Aizenberg_2012, Sunny_Vogel_Howell_Vu_Aizenberg_2014, Maccallum_Howell_Kim_Sun_Friedlander_Ranisau_Ahanotu_Lin_Vena_Hatton_2015, Sotiri_Overton_Waterhouse_Howell_2016, Ban_Lee_Choi_Li_Jun_2017, Al-Sharafi_Yilbas_Ali_2017, Ware_Smith-Palmer_Peppou-Chapman_Scarratt_Humphries_Balzer_Neto_2018, Wang_Zhao_Wu_Wang_Wu_Xue_2019} anti-icing, \cite{Kim_Wong_Alvarenga_Kreder_Adorno-Martinez_Aizenberg_2012, Kreder_Alvarenga_Kim_Aizenberg_2016, Subramanyam_Rykaczewski_Varanasi_2013, Yamazaki_Tenjimbayashi_Manabe_Moriya_Nakamura_Nakamura_Matsubayashi_Tsuge_Shiratori_2019} condensation enhancement \cite{Anand_Rykaczewski_Subramanyam_Beysens_Varanasi_2015, Al-Sharafi_Yilbas_Ali_2017, Preston_Lu_Song_Zhao_Wilke_Antao_Louis_Wang_2018, Sett_Sokalski_Boyina_Li_Rabbi_Auby_Foulkes_Mahvi_Barac_Bolton_2019} and drag reduction \cite{Solomon_Khalil_Varanasi_2014, Kim_Rothstein_2016, Rosenberg_Van_Buren_Fu_Smits_2016, Wang_Zhang_Liu_Zhou_2016, Fu_Arenas_Leonardi_Hultmark_2017, Asmolov_Nizkaya_Vinogradova_2018, Garcia-Cartagena_Arenas_An_Leonardi_2019, Lee_Kim_Choi_Yoon_Seo_2019}. \par
Generally, the \textit{L} in LIS is used interchangeably to indicate either \textit{liquid} \cite{Kim_Rothstein_2016} or \textit{lubricant}, \cite{Subramanyam_Rykaczewski_Varanasi_2013} as the most common liquids to impregnate surface structure are hydrophobic lubricants. \cite{Peppou-Chapman_Hong_Waterhouse_Neto_2020} Our recent insight showed that air and lubricant can both coexist within a hydrophobic surface structure and both act as lubricants, leading to drag reduction. \cite{Vega2021} Therefore, the distinction between \textit{liquid} and \textit{lubricant} is important as we showed that air is the fluid providing the greatest degree of lubrication when both are present. In this work, for clarity, the two lubricants will clearly be identified as oil (which could be any water-immiscible liquid lubricant) and as a gas layer. The term 'LIS' will be used to refer to a surface initially infused with a hydrophobic liquid lubricant before being submerged as we have done previously. \cite{Peppou_Chapman_AFM_Nanobubble_Mapping_2021,Peppou-Chapman_Hong_Waterhouse_Neto_2020} \par
Of particular interest is the ability of LIS to reduce interfacial drag. \cite{Schoenecker_Baier_Hardt_2014, Solomon_Khalil_Varanasi_2014, Kim_Rothstein_2016, Alinovi_Bottaro_2018, Ge_Holmgren_Kronbichler_Brandt_Kreiss_2018, Lee_Kim_Choi_Yoon_Seo_2019} Observed drag reduction \cite{Solomon_Khalil_Varanasi_2014, Kim_Rothstein_2016,Lee_Kim_Choi_Yoon_Seo_2019} is much higher than is expected by the interfacial slip model which predicts drag reduction only when the infused oil is less viscous than the flowing liquid. \cite{Vinogradova1999,Schoenecker_Baier_Hardt_2014,Alinovi_Bottaro_2018} Our recent work showed that the presence of isolated nanobubbles on silicone oil-infused Teflon wrinkled surfaces can quantitatively explain the observed drag reduction on LIS. \cite{Vega2021}
In this work we demonstrate that meniscus force mapping can be used to map hydrophobic oil and gas thickness simultaneously to reveal the pressence of nanobubbles on LIS. We describe AFM meniscus force measurements and how they can be used to detect and measure the thickness of a nanothin gas layer on top of a nanothin immiscible liquid layer, (\textit{i.e.} a nanobubble on a submerged LIS). To our knowledge, this is the first time two liquid/gas interfaces have been detected in a single AFM force-distance curve, and these force curves compiled to generate a time-resolved map of the spatial distribution of both phases. \par
\section{Materials and Methods}
\subsection{Sample Preparation}
Wrinkled Teflon surfaces were prepared as previously described \cite{Ware_Smith-Palmer_Peppou-Chapman_Scarratt_Humphries_Balzer_Neto_2018}.
Briefly, a shrinkable polystyrene substrate (Polyshrink\textsuperscript{TM}) was spin-coated with a thin layer ($\sim$\SI{40}{\nano\metre}) of Teflon AF (Chemours, 1.5\% in FC-40), and then annealed in an oven (France Etuves XFM020) at \SI{130}{\celsius}, inducing shrinking of the substrate and wrinkling of the top Teflon layer. \par
The as-produced wrinkles were infused by pipetting an excess of the lubricant (approx. \SI{200}{\micro\litre\per\centi\metre\squared}) of silicone oil (\SI{10}{\cSt}, Aldrich), spreading it, and then depleting the oil through repeated immersion through an air/water interface \cite{Peppou-Chapman_Neto_2021} or using a spin coater. \cite{Peppou-Chapman_Neto_2018} \par
\subsubsection*{Control over air content in working fluids}
Water with different air content was used in the experiments: Milli-Q water, used as produced, and gassed water. The procedure is described in \cite{Vega2021}. Briefly, the oxygen concentration in water was measured using a dissolved oxygen sensor (RCYACO, Model DO9100) and was used to estimate the air concentration in water. Milli-Q water as produced was air-saturated at atmospheric pressure (\SI{101}{\kilo\pascal}), and had an air content of $c_{air}\sim23.0\pm0.3$ \SI{}{\milli\gram\per\kilo\gram}. To produce gassed water, Milli-Q water was pressurized at \SI{203}{\kilo\pascal} to obtain an air content of $c_{air}\sim$ $44\pm4$ \SI{}{\milli\gram\per\kilo\gram}.
\subsection{Meniscus Force Measurements}
AFM meniscus force measurements were all performed using the force mapping feature on an MFP-3D (Asylum, Santa Clara, CA) using hydrophobized Multi-75 probes (k = \SIrange{1}{7}{\newton\per\meter}; Budget Sensors, Sofia, Bulgaria). The AFM probes are hydrophobized by depositing a thin layer of polydimethylsiloxane (PDMS) by chemical vapour deposition. The AFM probes are first cleaned using piranha solution, 3:1 sulfuric acid (98\%, Ajax) : hydrogen peroxide (30\%, Merck) for 5 minutes before being rinsed twice in Milli-Q water, once in toluene and dried under a gentle nitrogen flow. They are then placed in a glass staining jar with a small amount of uncured PDMS (Sylgard 184, Dow Corning) and placed in an oven at \SI{200}{\celsius} for 4 hours. After cooling, they are rinsed once more with toluene and dried under a gentle nitrogen flow. The procedure deposits \SIrange{1}{2}{\nano\meter} of PDMS (by ellipsometry). \cite{Peppou-Chapman_Neto_2021} \par
A custom-made sample holder is used to flood the samples with Milli-Q water \textit{in situ}. The cell consists of a superhydrophobic barrier with a small tubing through which water can be pumped, see our previous publication for details. \cite{Peppou-Chapman_Neto_2021} The custom cell was used for enhanced visibility and ease of use compared to the Asylum closed liquid cell when flooding a sample with water. This was important in previous work, but any underwater cell is sufficient to image nanobubbles using the technique described herein. \par
All data was analysed using Python 3 \cite{CS-R9526} using packages included in the Anaconda scientific computing distribution \cite{anaconda}. Raw force curve data was exported to ASCII format before being loaded in the Python script and analysed using the algorithm described herein. All code used in this work is available online. \cite{Peppou_Chapman_AFM_Nanobubble_Mapping_2021}
\section{Results and Discussion}
The detection of a thin gas layer sandwiched between the oil layer and the bulk water depends on detecting two features seen in AFM force-distance measurements. Oil layers are identified by a sharp negative deflection of the AFM cantilever, due meniscus formation (meniscus force measurements) \cite{Friedrich_Cappella_2020, Mate_Lorenz_Novotny_1990, Mate_Lorenz_Novotny_1989, Ally2010, Peppou-Chapman_Neto_2018, Scarratt_Zhu_Neto_2019} and gas layers are identified through the positive deflection of the cantilever. \cite{An_Tan_Ohl_2016, Tyrrell_Attard_2001, Walczyk_Schoenherr_2014a, Walczyk_Schoenherr_2014b, Wang_Zhao_Hu_Wang_Tai_Gao_Zhang_2017} Both these features are seen in the example force curve in \autoref{fig:Fig1_schematic}c, which was captured over an area identified as having both layers. Here, a positive deflection (\textit{i.e.} a force pushing the AFM tip away from the surface) is seen at larger separations and a negative deflection (\textit{i.e.} a force drawing the tip towards the surface) is seen a smaller separations, corresponding to the nanobubble and oil layer, respectively. \par
In this work, only extension curves were used and are presented, as the forces of interest occur as the tip moves towards the surface. Unless specified otherwise, all data was collected on a LIS composed of wrinkled Teflon infused with \SI{10}{\cSt} silicone oil. \cite{Ware_Smith-Palmer_Peppou-Chapman_Scarratt_Humphries_Balzer_Neto_2018, Vega2021} \par
\begin{figure}[H]
\centering
\includegraphics[width=0.9\linewidth]{Figure1_schematic.pdf}
\caption{Schematic showing an AFM tip approaching a nanobubble over a layer of oil: both the (a) oil thickness and (b) the nanobubble thickness (not to scale) are measured due to the opposite forces they exert on the AFM tip. c) Example experimentally obtained force curve from which the nanobubble thickness and oil thickness are extracted.}
\label{fig:Fig1_schematic}
\end{figure}
\subsection{Meniscus Force Measurements}
Meniscus force measurements are a subset of AFM force spectroscopy in which the dominant force on the cantilever is due to the formation of a liquid meniscus around the AFM tip. In contrast to the case where the AFM tip first encounters the solid surface (see \autoref{fig:Fig2_ex_FCs}a), when the AFM tip first contacts the air/liquid interface, a liquid meniscus forms around the tip and pulls down the cantilever towards the surface (see \autoref{fig:Fig1_schematic}a), causing a rapid negative deflection (see \autoref{fig:Fig2_ex_FCs}b). The distance between this so-called `jump-in' and the position where the AFM tip contacts the hard substrate underneath is the thickness of the liquid film, and can be revealed with nanoscale accuracy. \cite{Friedrich_Cappella_2020, Mate_Lorenz_Novotny_1990, Mate_Lorenz_Novotny_1989, Ally2010, Peppou-Chapman_Neto_2018, Scarratt_Zhu_Neto_2019} A detailed discussion on the accuracy of the technique is provided \cite{Friedrich_Cappella_2020, Mate_Lorenz_Novotny_1990, Mate_Lorenz_Novotny_1989}. \par
An advantage of this technique is that it is agnostic to the identity of the two phases. As long as the interfacial energy of the liquid is sufficiently high to observe a jump-in when the AFM tip touches it, the interface will be detected in the force curve. As a result, we have used this technique to map the thickness of a hydrophobic oil layer both in air \cite{Peppou-Chapman_Neto_2018,Tonelli_Peppou-Chapman_Ridi_Neto_2019} and underwater \cite{Peppou-Chapman_Neto_2021} as the water/oil interface has sufficient wetting contrast to produce the required jump-in. \autoref{fig:Fig2_ex_FCs}b shows the shape of a typical force curve of a thin oil film underwater where a sharp negative deflection is seen at about \SI{50}{\nano\meter} separation, before a sharp positive deflection when the tip contacts the underlying substrate.
\subsection{Force-Distance Curves on Surface Nanobubbles}
Due to their nanoscale dimensions, nanobubbles are not routinely studied using optical techniques typical in the study of larger gas bubbles. Instead, the nanoscale nature of AFM measurements has made it the technique of choice for the study of nanobubbles using both tapping mode and force-distance spectroscopy. \cite{Lohse_Zhang_2015,Alheshibri_Qian_Jehannin_Craig_2016} \par
In AFM force-distance measurements, nanobubbles are identified through a characteristic positive deflection (pushing the tip away from the substrate, see \autoref{fig:Fig1_schematic}b) due to deformation of the air/water interface. \cite{An_Tan_Ohl_2016, Walczyk_Schoenherr_2014b} \autoref{fig:Fig2_ex_FCs}c shows the shape of a typical force curve taken on a nanobubble, showing a section of force curve with a positive gradient between the zero force baseline and the hard contact point. Many publications have confirmed that this particular feature in the force curve shape is due to nanobubbles. \cite{An_Tan_Ohl_2016,Tyrrell_Attard_2001, Walczyk_Schoenherr_2014a, Walczyk_Schoenherr_2014b, Zhang_Maeda_Craig_2006, Wang_Zhao_Hu_Wang_Tai_Gao_Zhang_2017} \par
\subsection{Imaging Nanobubbles on LIS}
The two types of force curves described above combine when a gas layer is present on top of an oil layer underwater. In its approach towards the surface, the tip first deflects away due the deformation of the bubble and then towards the substrate due to meniscus formation when it contacts the oil layer, see \autoref{fig:Fig2_ex_FCs}d. The first point at which the positive deflection occurs is used to indicate the top of the gas layer and the first point of the negative deflection indicates the top of the oil layer. This interpretation may lead to small under- or overestimation of the layer thickness in certain cases. These effects are explored more in Section \ref{sec:limitations}. \par
\begin{figure}[H]
\centering
\includegraphics[width=\linewidth]{Figure2_example_FC_v2.pdf}
\caption{Example of the four types of force curves observed on silicone oil-infused Teflon LIS under Milli-Q water. Extension force curve as the AFM tip comes in contact with a) bare solid substrate under water; b) an oil film on the solid substrate under water; c) a gas layer on the solid substrate under water; d) a gas layer on an oil layer on the solid substrate under water.}
\label{fig:Fig2_ex_FCs}
\end{figure}
The process of imaging nanobubbles on LIS is similar to that of imaging the oil layer underwater, \cite{Peppou-Chapman_Neto_2021} with most of the difference consisting in the data analysis, see Section \ref{sec:data_anaysis}. \par
The four expected configurations of fluid layers and example force curves for an immersed LIS under static conditions are shown in \autoref{fig:Fig2_ex_FCs}: water on substrate, water on oil on substrate, water on gas on substrate, water on gas on oil on substrate. We do not observe a thick layer of oil on a nanobubble, and indeed a thick layer is not expected to be stable as the static pressure and Laplace pressure would destabilise the oil layer. \cite{Kreder_Daniel_Tetreault_Cao_Lemaire_Timonen_Aizenberg_2018} On the other hand, a nano-thin film ($\lesssim$ \SI{5}{\nano\meter}) of oil is expected to cloak the nanobubble due to the positive spreading parameter of oil at the water/air interface, but was not detectable with the current experimental set-up (see Section \ref{sec:limitations} for discussion of possible reasons). \par
The time required for the formation of nanobubbles is dependent on the gas content in the water. Nanobubbles appeared faster in water gassed with air (\textit{e.g.} for water air content $c_{air}\sim44\pm4$ \SI{}{\milli\gram\per\kilo\gram} they appeared instanteously, see \autoref{fig:Fig6_maps_cross_sec}e), while they appeared a few hours after immersion in plain Milli-Q water, which was saturated at atmospheric pressure ($c_{air}\sim23.0\pm0.3$ \SI{}{\milli\gram\per\kilo\gram}). \autoref{fig:Fig6_maps_cross_sec}a-d show the nucleation of a nanobubble on a surface through successive mapping of the same location. Each map took $\sim$35 minutes and the bubble appeared between successive maps. The surface was submerged for a total 2-3 hours with static conditions before the nanobubble appeared. \par
\subsection{AFM Probe Considerations}
\label{sec:AFM_probe}
The choice of tip shape and chemistry is important to achieving a clear image of both the oil and the gas layer. In AFM mapping techniques, a sharp tip is generally preferred as it gives maximum spatial resolution, but for meniscus force measurement a thinner tip decreases the force on the cantilever as the force is determined by the length of the contact line. \cite{Cappella_2017,Friedrich_Cappella_2020} As a result, there is a balance between spatial resolution and force resolution. If a tip of a known shape is used (\textit{e.g.} cylindrical), the length of the contact line is known and the surface tension of the fluid can be calculated. \cite{McGuiggan_Wallace_2006} \par
Similarly, cantilever spring constant is critical to successful meniscus force measurements. A low spring constant is needed for the cantilever to deflect due to meniscus formation and to deflect from the reaction force caused by nanobubble deformation. Additionally, a lower spring constant allows for a sharper tip to be used. However, the spring constant cannot be too low (\textit{e.g.} a contact mode probe), as the cantilever is not able to break free from the meniscus during retraction. Here, force modulation AFM probes with a spring constant of $\sim$ \SI{5}{\newton\per\meter} and a sharp tip were used, as these provide sufficiently large deflection values with good spatial resolution.\par
For imaging hydrophobic oil layers, tip chemistry determines both whether a meniscus is formed in the first place and the magnitude of the force exerted by this meniscus (as this is determined by the length of the contact line). The fluid layer that forms the meniscus should preferentially wet the tip for a meniscus to form. In air, this is trivial as almost all fluids wet the high surface energy silicon nitride which makes up most AFM tips. Underwater, this is less simple as a hydrophilic tip may be wet preferentially by the water and therefore a clear jump-in from a hydrophobic oil might not be apparent. Here, the tip was hydrophobized with a thin layer of PDMS to ensure a wettability contrast between the water and the thin oil layer. An unmodified tip was seen to work initially, but the image quality deteriorated quickly, leading to the loss of jump-in. \par
For imaging nanobubbles, the opposite is required. The thickness of a nanobubble is better imaged using a hydrophilic tip as the tip is less likely to penetrate into the bubble so all changes in deflection are due to nanobubble deformation. \cite{Walczyk_Schoenherr_2014b} A hydrophobic tip still shows deflection due to nanobubble deformation, however, it also shows signs of meniscus formation as the tip contacts the nanobubble. This aspect is discussed further in Section \ref{sec:limitations}.
\subsection{Data Analysis}
\label{sec:data_anaysis}
The key to using AFM meniscus force measurements for mapping of multiple layers was the automated data analysis with feature recognition to determine the points at which the tip contacts different interfaces. This section describes the logic used by our Python script to calculate the thickness of both the gas and oil layer from the collected force curves. \par
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{Figure3+4_flow_chart_ex_FC.pdf}
\caption{a) Flowchart showing the process used to characterise each force curve in the raw force map data. The inset in the red box is the logic used in the decision nodes with a red outline to determine whether the data between two features of a force curve is gas or lubricant. See GitHub \cite{Peppou_Chapman_AFM_Nanobubble_Mapping_2021} for code. b) Raw force curve data of cantilever deflection (y scale is in \SI{}{\meter}) as function of \textit{z}-displacement for an example force curve. c) The first derivative of b) (y scale is \SI{}{\meter\per\nano\meter}). d) The second derivative of b) (y scale is \SI{}{\meter\per\nano\meter\squared}). The orange vertical lines represent negative peaks picked from the second derivative. The red, green, and blue lines represent the location of the substrate, the top of the oil and the top of the nanobubble, respectively.}
\label{fig:Fig3+4_flow_FC}
\end{figure*}
For an underwater LIS there are four possible scenarios for the fluid layers each force curve may encounter (see \autoref{fig:Fig2_ex_FCs}):
\begin{enumerate}[label=(\alph*)]
\item Water on substrate;
\item Water on oil on substrate;
\item Water on gas on substrate;
\item Water on gas on oil on substrate;
\end{enumerate}
In our previous work, \cite{Peppou-Chapman_Neto_2018, Tonelli_Peppou-Chapman_Ridi_Neto_2019, Peppou-Chapman_Neto_2021} the script only detected an oil layer (\textit{i.e.} scenarios (a) and (b)), where only two key points needed to be detected: the jump-in and hard contact point. As a result of the added complexity added by the gas layer, a completely new detection algorithm was used here to detect all four of these scenarios. The features associated with the gas layer are less pronounced than in previous iterations, and so a lower threshold for detection of features needed to be implemented.\par
Here, the algorithm utilises automated peak finding to find areas of rapid changes in the gradient of the deflection and the fact that the gradient of the deflection is opposite signs when the tip is in contact with a nanobubble versus when in contact with a meniscus due to an oil layer. The general procedure used in the script is (see also \autoref{fig:Fig3+4_flow_FC}a):
\begin{itemize}
\item Convert piezo movement data to separation, and photo diode signal to deflection, using sensitivity of cantilever calculated from compliance region measured on a hard surface.
\item Find the hard-contact point by looking for the first point where the gradient changes substantially while moving towards larger separations from the turn around point.
\item Calculate the second derivative to the whole data and apply a smoothing algorithm to the y-data (deflection).
\item Use a peak finding algorithm to find negative peaks in the smoothed second derivative (orange lines in \autoref{fig:Fig3+4_flow_FC}b-d).
\item If no peaks are detected, there is no oil or gas (\textit{i.e.} case in \autoref{fig:Fig2_ex_FCs}a)
\item If one peak is detected, then the region between it and the hard contact point can either be oil or gas. Check if gas or oil using the algorithm outlined below. (\textit{i.e.} case in \autoref{fig:Fig2_ex_FCs}b,c).
\item If multiple peaks are detected, each segment between two peaks is checked to see if it is gas or not using the algorithm outlined below. The first segment identified as oil demarcates the boundary between the gas and oil (\textit{i.e.} case in \autoref{fig:Fig2_ex_FCs}d). If no oil is detected before reaching the hard contact point, then there is no oil present and the layer is entirely gas (i.e case in \autoref{fig:Fig2_ex_FCs}c).
\end{itemize}
To determine whether a segment between two of the negative peaks identified in the second derivative is gas or oil, the following algorithm is used (see also inset within the red box in \autoref{fig:Fig3+4_flow_FC}a):
\begin{itemize}
\item If 100\% of the first derivative is negative – the portion is oil. This is because the force acting on the cantilever is proportional to the length of the contact line and this force always increases (due the triangular shape of the tip) as the tip moves towards the surface – causing the deflection to become more negative.
\item If more than 60\% of the points are negative and the end is lower than the start by a threshold deflection (\SI{0.5}{\nano\meter} in this work, determined using trial and error), then portion is considered oil. Otherwise, it is gas.
\end{itemize}
The script then outputs the height of the hard contact point (absolute height from $z$-sensor data) and the thickness of the oil or gas layers found (calculated as the distance between relevant peaks in separation). \autoref{fig:Fig5_example_map}a shows a map with example forces curves and their corresponding location in the map. The force curves (\autoref{fig:Fig5_example_map}b-g) also show the features located by the algorithm described above (blue = start of gas layer, green = start of oil layer, and red = start of the substrate). \par
\subsubsection{Validation}
This analysis was validated by a manual review of analysed force curves to judge if the script had picked the correct locations for the start of gas and oil layers. Force curves from multiple maps were plotted with the vertical lines showing the interfaces as picked by the script as in \autoref{fig:Fig3+4_flow_FC}b and \autoref{fig:Fig5_example_map} and judged by eye if these were correct. A total of 494 force curves were selected at random and 93\% of them were judged to be correctly fitted. Potential reasons for incorrect fitting are discussed below in Section \ref{sec:limitations}. \par
\subsection{Data Visualisation}
As shown in \autoref{fig:Fig5_example_map}, the data is presented as multi-panel maps showing the spatial distribution of the different quantities measured. First, the sample topography is presented (extracted from the hard contact point of the force curves). Then maps of the oil thickness and the gas thickness are presented separately. Both oil and gas thickness are presented as non-linear contours where colours are evenly spaced at thickness of \SI{2}{\nano\meter} up to \SI{20}{\nano\meter} and then contours of \SI{150}{\nano\meter}, \SI{300}{\nano\meter} and >\SI{300}{\nano\meter} to give an indication of distribution of both thin layers and thick layers. This method of presentation was selected as it gives the ability to easily attribute the effect of topography on the distribution of either the liquid or the gas layer. \par
\begin{figure}[H]
\centering
\includegraphics[width=\linewidth]{Figure5_example_map.pdf}
\captionof{figure}{a) Map of a nanobubble on a LIS with (b-g) example force curves shown for points with different fluid layers. The map (a) is presented with three panels showing the topography of the underlying Teflon wrinkles (left), the thickness of the oil (middle) and the thickness of the gas (left); the units of all colour scales is \SI{}{\nano\meter}. Points 1 and 2 show an example of a nanobubble directly on the substrate. Points 3 and 4 show an example of a nanobubble on oil. Point 5 shows an example of the water directly contacting the substrate and point 6 shows an example of water contacting the oil. The vertical lines in the force curves show where the script has found the start of the different fluid layers, blue = start of gas layer, green = start of oil layer, and red = start of the substrate.}
\label{fig:Fig5_example_map}
\end{figure}
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{Figure6_Maps_crosssections.pdf}
\caption{AFM force maps showing oil and gas thickness (a, c, e, g) with cross sections (b, d, f, h) corresponding the cyan lines in the maps. The units of all colour scales is \SI{}{\nano\meter}. The marker symbols in the cross section represent points at which the height was measured and colors were added to help visualise the different layers. a) and b) are successive maps on the same area of the surface, showing the appearance of a nanobubble \textit{in situ}. The force curves were collected and analysed using the same parameters. e, f) example of a LIS submersed in gassed water ($c_{air}\sim44\pm4$ \SI{}{\milli\gram\per\kilo\gram}) imaged immediately after submersion. g, h) a superhydrophobic surface (the underlying TW substrate of the LIS without any silicone oil applied) showing a partially collapsed Cassie state. The non-zero measurements of oil thickness at multiple points on this map are a result of the feature detection algorithm not always being able to discern the difference between oil and gas or the particular force curve being noisy. See Section \ref{sec:limitations} for more information.}
\label{fig:Fig6_maps_cross_sec}
\end{figure*}
Cross sectional profiles of these maps can be extracted to visualise the shape of the interfaces, as shown in \autoref{fig:Fig6_maps_cross_sec}, in which color was added to the line profiles to highlight the different fluid layers. Each cross section corresponds the cyan line in the adjacent maps. As mentioned above, part (a) and part (b) in \autoref{fig:Fig6_maps_cross_sec} are sequential maps (taken approximately 35 mins apart) of the same surface before and after a nanobubble nucleates. \autoref{fig:Fig6_maps_cross_sec}e,f shows a cross sections from the LIS in \autoref{fig:Fig5_example_map} while \autoref{fig:Fig6_maps_cross_sec}g,h shows gas pockets on an immersed superhydrophobic surface (Teflon wrinkles with no oil present) in a partially-collapsed Cassie state. Despite the apparently high resolution of the contact line of the mapped interfaces, the local contact angle values can not be estimated, as discussed in Section \ref{sec:limitations}.
\subsection{Limitations and Sources of Error}
\label{sec:limitations}
There are several limitations and sources of error related to this technique which are somewhat accounted for by the mapping nature which allows individual errant pixels to be ignored. This section discusses limitations and sources of error in this technique. \par
The accuracy of the measured thickness depends on many factors and is not uniform for all film thicknesses. Very thin (<\SI{5}{\nano\meter}) films of either gas or oil are difficult to both detect and distinguish. The technique’s ability to detect a nanoscale film is proportional to the film's thickness as thicker films (either gas or oil) produces a greater deflection while films of just a few nanometers produce a deflection on the same order as measurement noise. As a result, regions of zero film thickness may contain undetected films of a few nanometers. \par
Automated feature detection exacerbates this limitation as there is a trade-off between sensitivity (\textit{i.e.} detecting smaller deflections) and avoiding false detection at large separations (\textit{e.g.} more than \SI{100}{\nano\meter} from the first interface as in \autoref{fig:Fig7_bad_Fcs}d). The noise present in the oil and gas thickness maps in \autoref{fig:Fig5_example_map} is the result of imperfect fitting that occurs due to higher feature detection sensitivity. Using a thicker tip would reduce this effect but, as mentioned above, would reduce lateral resolution. Slightly thicker films (up to tens of \SI{}{\nano\meter}) can be detected but with fewer data points to judge whether the feature in the force curve is due to oil or gas, the script is prone to mislabelling these. \autoref{fig:Fig7_bad_Fcs}a,b shows examples of two features at small separations with deflections $\sim$\SI{1}{\nano\meter} which are ambiguous and were identified as different features by the script, despite being adjacent pixels in the map. \par
The presence of a nano-thin layer of oil spread on the AFM tip cannot be excluded, which would increase the measured thickness of the oil layer. As with our previous publications, this does not seem to be an issue as there are multiple cases where no oil thickness is detected either under a nanobubble or elsewhere (\textit{e.g.} \autoref{fig:Fig2_ex_FCs}a,c). However, the lack of sensitivity to very thin films (<\SI{5}{\nano\meter}) described in the previous paragraph may hide the effects of such a film. \par
The thickness of the oil layer may be overestimated if long range van der Waals attraction between the tip and an interface draws the interface up to ‘meet’ the tip. \cite{Ally2010} In our previous publication, this effect was suppressed by sufficiently fast scan rate in the force curve \cite{Peppou-Chapman_Neto_2018}. As a result, here the fastest scan rate possible on our AFM (\SI{2}{\hertz}) was used to ensure this effect is minimized. This deformation will also effect the gas layer, with the air/water interface deforming to meet the tip, especially for a hydrophobic tip. \cite{Walczyk_Schoenherr_2014b} As a result of this deformation, Walczyk \& Sch{\"o}nherr \cite{Walczyk_Schoenherr_2014b} define the top of the a nanobubble measured with a hydrophobic tip to be where the force curve crosses zero deflection after the initial jump-in. Here potential long-range attraction of the air/water interface was ignored, due to the high scan rates used and because this correction would cause the calculated gas height to be heavily dependent on the quality of the baseline correction. Additionally, Walczyk \& Sch{\"o}nherr's definition of the top of a nanobubble assumes that the AFM tip only contacts the bubble with minimal contact line at zero deflection (\textit{i.e.} the only force on the cantilever is due to nanobubble deformation), which is impossible given the fact that a jump-in is seen, signifying meniscus formation and a non-trivial contact line. \par
\begin{figure}[H]
\centering
\includegraphics[width=\linewidth]{Figure7_bad_FCs.pdf}
\caption{Force curves showing the limitations of the automatic feature detection script presented in this work. Vertical lines indicate where the analysis script found an interface with blue = start of gas layer, green = start of oil layer, and red = start of the substrate. a,b) Two force curves obtained on adjacent locations, where the identity of the feature is ambiguous, with the script identifying in (a) a gas layer and in (b) a oil layer; c) an example of the script not detecting gas; d) an example of the script incorrectly detecting the gas/water interface; e) an example of the script incorrectly detecting the oil/gas interface instead of more gas; f) an example of a noisy force curve, likely due to vibrations. }
\label{fig:Fig7_bad_Fcs}
\end{figure}
There are limitations of the experimental setup which contribute noise in the maps. Force curves collected under water are particularly sensitive to vibrations, as they can be transmitted through the liquid to the cantilever. This means that a higher proportion of force curves are noisy throughout their entire range of motion, compared to the same force measurement in air, leading to incorrect feature detection (see \autoref{fig:Fig7_bad_Fcs}f). \par
The cross sections generated from the map cannot be used to quantify contact angle values as the interface is 3-dimensional and contributions from in-plane and out-of-plane features are impossible to account for. Additionally, the shape of the interface may be slightly deformed due van der Waals interactions not being consistent across the surface. \par
\section{Conclusion}
In summary, we have shown that a single AFM force-distance curve can capture the thickness of both a gas layer and an immiscible liquid layer underwater simultaneously. By mapping a surface with force curves and analysing the data automatically, the thickness of the gas and the immiscible liquid are both spatially resolved. This presents an exciting new technique to study systems such as the nucleation of nanobubbles on LIS. \cite{Vega2021} There are still many outstanding questions related to nanobubbles on LIS including the effects of substrate topography, the presence of surfactants or ion concentration and effect of static pressure. This force mapping technique will enable detailed studies of these parameters and could help in establishing whether nanobubbles are more stable on superhydrophobic or on LIS.
\printbibliography
\end{multicols}
\end{document}
\section{Introduction}
Gas layers dramatically affect the flow boundary conditions in microfluidic systems, reducing drag by up to 75\%, \cite{lee2016superhydrophobic} but are frequently overlooked when one or more dimension is nanoscale due to difficulty in detecting them. Nanoscale gas layers, also known as surface nanobubbles, are extremely difficult to observe and characterize as they
are too small to be quantitatively analysed using optical
techniques which are typically used to study liquid-gas
interfaces. \cite{Alheshibri_Qian_Jehannin_Craig_2016} \par
Surface nanobubbles were first reported in the year 2000 \cite{Alheshibri_Qian_Jehannin_Craig_2016, Ishida_Inoue_Miyahara_Higashitani_2000} and have been controversial ever since. The high Laplace pressure inside bubbles with radius of curvature smaller than \SI{100}{\nano\meter} (\textit{e.g.} $\Delta$P $\approx$ \SI{29}{\atm} for r = \SI{50}{\nano\meter} and $\gamma = \SI{72}{\milli\newton\per\meter}$) indicates that they should have very short lifetimes (\textit{i.e.} on the order of \SI{100}{\micro\second} \cite{Lohse_Zhang_2015}), however they are routinely seen to be stable on much longer timescales. \cite{Zhang_Maeda_Craig_2006} Their unexpected stability is due to the fact that, despite having thickness in the range of a few tens to hundreds of nanometers, they have micrometric lateral size, which produces a flat interfacial shape with low radius of curvature. This reduces the internal Laplace pressure and allows the bubble to remain stable on the immersed surface for hours to days. For a complete review of the field, the reader is directed to the review by Lohse and Zhang. \cite{Lohse_Zhang_2015} \par
Here, we report a method to detect and map the presence of gas layers on structured hydrophobic surfaces covered with a thin layer of a hydrophobic oil. This type of surface is known as lubricant-infused surface (LIS) and has been the topic of intense research over the past decade due to their desirable properties introduced by the presence of the entrapped lubricant layer, \cite{Peppou-Chapman_Hong_Waterhouse_Neto_2020} such as anti-fouling, \cite{Epstein_Wong_Belisle_Boggs_Aizenberg_2012, Sunny_Vogel_Howell_Vu_Aizenberg_2014, Maccallum_Howell_Kim_Sun_Friedlander_Ranisau_Ahanotu_Lin_Vena_Hatton_2015, Sotiri_Overton_Waterhouse_Howell_2016, Ban_Lee_Choi_Li_Jun_2017, Al-Sharafi_Yilbas_Ali_2017, Ware_Smith-Palmer_Peppou-Chapman_Scarratt_Humphries_Balzer_Neto_2018, Wang_Zhao_Wu_Wang_Wu_Xue_2019} anti-icing, \cite{Kim_Wong_Alvarenga_Kreder_Adorno-Martinez_Aizenberg_2012, Kreder_Alvarenga_Kim_Aizenberg_2016, Subramanyam_Rykaczewski_Varanasi_2013, Yamazaki_Tenjimbayashi_Manabe_Moriya_Nakamura_Nakamura_Matsubayashi_Tsuge_Shiratori_2019} condensation enhancement \cite{Anand_Rykaczewski_Subramanyam_Beysens_Varanasi_2015, Al-Sharafi_Yilbas_Ali_2017, Preston_Lu_Song_Zhao_Wilke_Antao_Louis_Wang_2018, Sett_Sokalski_Boyina_Li_Rabbi_Auby_Foulkes_Mahvi_Barac_Bolton_2019} and drag reduction \cite{Solomon_Khalil_Varanasi_2014, Kim_Rothstein_2016, Rosenberg_Van_Buren_Fu_Smits_2016, Wang_Zhang_Liu_Zhou_2016, Fu_Arenas_Leonardi_Hultmark_2017, Asmolov_Nizkaya_Vinogradova_2018, Garcia-Cartagena_Arenas_An_Leonardi_2019, Lee_Kim_Choi_Yoon_Seo_2019}. \par
Generally, the \textit{L} in LIS is used interchangeably to indicate either \textit{liquid} \cite{Kim_Rothstein_2016} or \textit{lubricant}, \cite{Subramanyam_Rykaczewski_Varanasi_2013} as the most common liquids to impregnate surface structure are hydrophobic lubricants. \cite{Peppou-Chapman_Hong_Waterhouse_Neto_2020} Our recent insight showed that air and lubricant can both coexist within a hydrophobic surface structure and both act as lubricants, leading to drag reduction. \cite{Vega2021} Therefore, the distinction between \textit{liquid} and \textit{lubricant} is important as we showed that air is the fluid providing the greatest degree of lubrication when both are present. In this work, for clarity, the two lubricants will clearly be identified as oil (which could be any water-immiscible liquid lubricant) and as a gas layer. The term 'LIS' will be used to refer to a surface initially infused with a hydrophobic liquid lubricant before being submerged as we have done previously. \cite{Peppou_Chapman_AFM_Nanobubble_Mapping_2021,Peppou-Chapman_Hong_Waterhouse_Neto_2020} \par
Of particular interest is the ability of LIS to reduce interfacial drag. \cite{Schoenecker_Baier_Hardt_2014, Solomon_Khalil_Varanasi_2014, Kim_Rothstein_2016, Alinovi_Bottaro_2018, Ge_Holmgren_Kronbichler_Brandt_Kreiss_2018, Lee_Kim_Choi_Yoon_Seo_2019} Observed drag reduction \cite{Solomon_Khalil_Varanasi_2014, Kim_Rothstein_2016,Lee_Kim_Choi_Yoon_Seo_2019} is much higher than is expected by the interfacial slip model which predicts drag reduction only when the infused oil is less viscous than the flowing liquid. \cite{Vinogradova1999,Schoenecker_Baier_Hardt_2014,Alinovi_Bottaro_2018} Our recent work showed that the presence of isolated nanobubbles on silicone oil-infused Teflon wrinkled surfaces can quantitatively explain the observed drag reduction on LIS. \cite{Vega2021}
In this work we demonstrate that meniscus force mapping can be used to map hydrophobic oil and gas thickness simultaneously to reveal the pressence of nanobubbles on LIS. We describe AFM meniscus force measurements and how they can be used to detect and measure the thickness of a nanothin gas layer on top of a nanothin immiscible liquid layer, (\textit{i.e.} a nanobubble on a submerged LIS). To our knowledge, this is the first time two liquid/gas interfaces have been detected in a single AFM force-distance curve, and these force curves compiled to generate a time-resolved map of the spatial distribution of both phases. \par
\section{Materials and Methods}
\subsection{Sample Preparation}
Wrinkled Teflon surfaces were prepared as previously described \cite{Ware_Smith-Palmer_Peppou-Chapman_Scarratt_Humphries_Balzer_Neto_2018}.
Briefly, a shrinkable polystyrene substrate (Polyshrink\textsuperscript{TM}) was spin-coated with a thin layer ($\sim$\SI{40}{\nano\metre}) of Teflon AF (Chemours, 1.5\% in FC-40), and then annealed in an oven (France Etuves XFM020) at \SI{130}{\celsius}, inducing shrinking of the substrate and wrinkling of the top Teflon layer. \par
The as-produced wrinkles were infused by pipetting an excess of the lubricant (approx. \SI{200}{\micro\litre\per\centi\metre\squared}) of silicone oil (\SI{10}{\cSt}, Aldrich), spreading it, and then depleting the oil through repeated immersion through an air/water interface \cite{Peppou-Chapman_Neto_2021} or using a spin coater. \cite{Peppou-Chapman_Neto_2018} \par
\subsubsection*{Control over air content in working fluids}
Water with different air content was used in the experiments: Milli-Q water, used as produced, and gassed water. The procedure is described in \cite{Vega2021}. Briefly, the oxygen concentration in water was measured using a dissolved oxygen sensor (RCYACO, Model DO9100) and was used to estimate the air concentration in water. Milli-Q water as produced was air-saturated at atmospheric pressure (\SI{101}{\kilo\pascal}), and had an air content of $c_{air}\sim23.0\pm0.3$ \SI{}{\milli\gram\per\kilo\gram}. To produce gassed water, Milli-Q water was pressurized at \SI{203}{\kilo\pascal} to obtain an air content of $c_{air}\sim$ $44\pm4$ \SI{}{\milli\gram\per\kilo\gram}.
\subsection{Meniscus Force Measurements}
AFM meniscus force measurements were all performed using the force mapping feature on an MFP-3D (Asylum, Santa Clara, CA) using hydrophobized Multi-75 probes (k = \SIrange{1}{7}{\newton\per\meter}; Budget Sensors, Sofia, Bulgaria). The AFM probes are hydrophobized by depositing a thin layer of polydimethylsiloxane (PDMS) by chemical vapour deposition. The AFM probes are first cleaned using piranha solution, 3:1 sulfuric acid (98\%, Ajax) : hydrogen peroxide (30\%, Merck) for 5 minutes before being rinsed twice in Milli-Q water, once in toluene and dried under a gentle nitrogen flow. They are then placed in a glass staining jar with a small amount of uncured PDMS (Sylgard 184, Dow Corning) and placed in an oven at \SI{200}{\celsius} for 4 hours. After cooling, they are rinsed once more with toluene and dried under a gentle nitrogen flow. The procedure deposits \SIrange{1}{2}{\nano\meter} of PDMS (by ellipsometry). \cite{Peppou-Chapman_Neto_2021} \par
A custom-made sample holder is used to flood the samples with Milli-Q water \textit{in situ}. The cell consists of a superhydrophobic barrier with a small tubing through which water can be pumped, see our previous publication for details. \cite{Peppou-Chapman_Neto_2021} The custom cell was used for enhanced visibility and ease of use compared to the Asylum closed liquid cell when flooding a sample with water. This was important in previous work, but any underwater cell is sufficient to image nanobubbles using the technique described herein. \par
All data was analysed using Python 3 \cite{CS-R9526} using packages included in the Anaconda scientific computing distribution \cite{anaconda}. Raw force curve data was exported to ASCII format before being loaded in the Python script and analysed using the algorithm described herein. All code used in this work is available online. \cite{Peppou_Chapman_AFM_Nanobubble_Mapping_2021}
\section{Results and Discussion}
The detection of a thin gas layer sandwiched between the oil layer and the bulk water depends on detecting two features seen in AFM force-distance measurements. Oil layers are identified by a sharp negative deflection of the AFM cantilever, due meniscus formation (meniscus force measurements) \cite{Friedrich_Cappella_2020, Mate_Lorenz_Novotny_1990, Mate_Lorenz_Novotny_1989, Ally2010, Peppou-Chapman_Neto_2018, Scarratt_Zhu_Neto_2019} and gas layers are identified through the positive deflection of the cantilever. \cite{An_Tan_Ohl_2016, Tyrrell_Attard_2001, Walczyk_Schoenherr_2014a, Walczyk_Schoenherr_2014b, Wang_Zhao_Hu_Wang_Tai_Gao_Zhang_2017} Both these features are seen in the example force curve in \autoref{fig:Fig1_schematic}c, which was captured over an area identified as having both layers. Here, a positive deflection (\textit{i.e.} a force pushing the AFM tip away from the surface) is seen at larger separations and a negative deflection (\textit{i.e.} a force drawing the tip towards the surface) is seen a smaller separations, corresponding to the nanobubble and oil layer, respectively. \par
In this work, only extension curves were used and are presented, as the forces of interest occur as the tip moves towards the surface. Unless specified otherwise, all data was collected on a LIS composed of wrinkled Teflon infused with \SI{10}{\cSt} silicone oil. \cite{Ware_Smith-Palmer_Peppou-Chapman_Scarratt_Humphries_Balzer_Neto_2018, Vega2021} \par
\begin{figure}[H]
\centering
\includegraphics[width=0.9\linewidth]{Figure1_schematic.pdf}
\caption{Schematic showing an AFM tip approaching a nanobubble over a layer of oil: both the (a) oil thickness and (b) the nanobubble thickness (not to scale) are measured due to the opposite forces they exert on the AFM tip. c) Example experimentally obtained force curve from which the nanobubble thickness and oil thickness are extracted.}
\label{fig:Fig1_schematic}
\end{figure}
\subsection{Meniscus Force Measurements}
Meniscus force measurements are a subset of AFM force spectroscopy in which the dominant force on the cantilever is due to the formation of a liquid meniscus around the AFM tip. In contrast to the case where the AFM tip first encounters the solid surface (see \autoref{fig:Fig2_ex_FCs}a), when the AFM tip first contacts the air/liquid interface, a liquid meniscus forms around the tip and pulls down the cantilever towards the surface (see \autoref{fig:Fig1_schematic}a), causing a rapid negative deflection (see \autoref{fig:Fig2_ex_FCs}b). The distance between this so-called `jump-in' and the position where the AFM tip contacts the hard substrate underneath is the thickness of the liquid film, and can be revealed with nanoscale accuracy. \cite{Friedrich_Cappella_2020, Mate_Lorenz_Novotny_1990, Mate_Lorenz_Novotny_1989, Ally2010, Peppou-Chapman_Neto_2018, Scarratt_Zhu_Neto_2019} A detailed discussion on the accuracy of the technique is provided \cite{Friedrich_Cappella_2020, Mate_Lorenz_Novotny_1990, Mate_Lorenz_Novotny_1989}. \par
An advantage of this technique is that it is agnostic to the identity of the two phases. As long as the interfacial energy of the liquid is sufficiently high to observe a jump-in when the AFM tip touches it, the interface will be detected in the force curve. As a result, we have used this technique to map the thickness of a hydrophobic oil layer both in air \cite{Peppou-Chapman_Neto_2018,Tonelli_Peppou-Chapman_Ridi_Neto_2019} and underwater \cite{Peppou-Chapman_Neto_2021} as the water/oil interface has sufficient wetting contrast to produce the required jump-in. \autoref{fig:Fig2_ex_FCs}b shows the shape of a typical force curve of a thin oil film underwater where a sharp negative deflection is seen at about \SI{50}{\nano\meter} separation, before a sharp positive deflection when the tip contacts the underlying substrate.
\subsection{Force-Distance Curves on Surface Nanobubbles}
Due to their nanoscale dimensions, nanobubbles are not routinely studied using optical techniques typical in the study of larger gas bubbles. Instead, the nanoscale nature of AFM measurements has made it the technique of choice for the study of nanobubbles using both tapping mode and force-distance spectroscopy. \cite{Lohse_Zhang_2015,Alheshibri_Qian_Jehannin_Craig_2016} \par
In AFM force-distance measurements, nanobubbles are identified through a characteristic positive deflection (pushing the tip away from the substrate, see \autoref{fig:Fig1_schematic}b) due to deformation of the air/water interface. \cite{An_Tan_Ohl_2016, Walczyk_Schoenherr_2014b} \autoref{fig:Fig2_ex_FCs}c shows the shape of a typical force curve taken on a nanobubble, showing a section of force curve with a positive gradient between the zero force baseline and the hard contact point. Many publications have confirmed that this particular feature in the force curve shape is due to nanobubbles. \cite{An_Tan_Ohl_2016,Tyrrell_Attard_2001, Walczyk_Schoenherr_2014a, Walczyk_Schoenherr_2014b, Zhang_Maeda_Craig_2006, Wang_Zhao_Hu_Wang_Tai_Gao_Zhang_2017} \par
\subsection{Imaging Nanobubbles on LIS}
The two types of force curves described above combine when a gas layer is present on top of an oil layer underwater. In its approach towards the surface, the tip first deflects away due the deformation of the bubble and then towards the substrate due to meniscus formation when it contacts the oil layer, see \autoref{fig:Fig2_ex_FCs}d. The first point at which the positive deflection occurs is used to indicate the top of the gas layer and the first point of the negative deflection indicates the top of the oil layer. This interpretation may lead to small under- or overestimation of the layer thickness in certain cases. These effects are explored more in Section \ref{sec:limitations}. \par
\begin{figure}[H]
\centering
\includegraphics[width=\linewidth]{Figure2_example_FC_v2.pdf}
\caption{Example of the four types of force curves observed on silicone oil-infused Teflon LIS under Milli-Q water. Extension force curve as the AFM tip comes in contact with a) bare solid substrate under water; b) an oil film on the solid substrate under water; c) a gas layer on the solid substrate under water; d) a gas layer on an oil layer on the solid substrate under water.}
\label{fig:Fig2_ex_FCs}
\end{figure}
The process of imaging nanobubbles on LIS is similar to that of imaging the oil layer underwater, \cite{Peppou-Chapman_Neto_2021} with most of the difference consisting in the data analysis, see Section \ref{sec:data_anaysis}. \par
The four expected configurations of fluid layers and example force curves for an immersed LIS under static conditions are shown in \autoref{fig:Fig2_ex_FCs}: water on substrate, water on oil on substrate, water on gas on substrate, water on gas on oil on substrate. We do not observe a thick layer of oil on a nanobubble, and indeed a thick layer is not expected to be stable as the static pressure and Laplace pressure would destabilise the oil layer. \cite{Kreder_Daniel_Tetreault_Cao_Lemaire_Timonen_Aizenberg_2018} On the other hand, a nano-thin film ($\lesssim$ \SI{5}{\nano\meter}) of oil is expected to cloak the nanobubble due to the positive spreading parameter of oil at the water/air interface, but was not detectable with the current experimental set-up (see Section \ref{sec:limitations} for discussion of possible reasons). \par
The time required for the formation of nanobubbles is dependent on the gas content in the water. Nanobubbles appeared faster in water gassed with air (\textit{e.g.} for water air content $c_{air}\sim44\pm4$ \SI{}{\milli\gram\per\kilo\gram} they appeared instanteously, see \autoref{fig:Fig6_maps_cross_sec}e), while they appeared a few hours after immersion in plain Milli-Q water, which was saturated at atmospheric pressure ($c_{air}\sim23.0\pm0.3$ \SI{}{\milli\gram\per\kilo\gram}). \autoref{fig:Fig6_maps_cross_sec}a-d show the nucleation of a nanobubble on a surface through successive mapping of the same location. Each map took $\sim$35 minutes and the bubble appeared between successive maps. The surface was submerged for a total 2-3 hours with static conditions before the nanobubble appeared. \par
\subsection{AFM Probe Considerations}
\label{sec:AFM_probe}
The choice of tip shape and chemistry is important to achieving a clear image of both the oil and the gas layer. In AFM mapping techniques, a sharp tip is generally preferred as it gives maximum spatial resolution, but for meniscus force measurement a thinner tip decreases the force on the cantilever as the force is determined by the length of the contact line. \cite{Cappella_2017,Friedrich_Cappella_2020} As a result, there is a balance between spatial resolution and force resolution. If a tip of a known shape is used (\textit{e.g.} cylindrical), the length of the contact line is known and the surface tension of the fluid can be calculated. \cite{McGuiggan_Wallace_2006} \par
Similarly, cantilever spring constant is critical to successful meniscus force measurements. A low spring constant is needed for the cantilever to deflect due to meniscus formation and to deflect from the reaction force caused by nanobubble deformation. Additionally, a lower spring constant allows for a sharper tip to be used. However, the spring constant cannot be too low (\textit{e.g.} a contact mode probe), as the cantilever is not able to break free from the meniscus during retraction. Here, force modulation AFM probes with a spring constant of $\sim$ \SI{5}{\newton\per\meter} and a sharp tip were used, as these provide sufficiently large deflection values with good spatial resolution.\par
For imaging hydrophobic oil layers, tip chemistry determines both whether a meniscus is formed in the first place and the magnitude of the force exerted by this meniscus (as this is determined by the length of the contact line). The fluid layer that forms the meniscus should preferentially wet the tip for a meniscus to form. In air, this is trivial as almost all fluids wet the high surface energy silicon nitride which makes up most AFM tips. Underwater, this is less simple as a hydrophilic tip may be wet preferentially by the water and therefore a clear jump-in from a hydrophobic oil might not be apparent. Here, the tip was hydrophobized with a thin layer of PDMS to ensure a wettability contrast between the water and the thin oil layer. An unmodified tip was seen to work initially, but the image quality deteriorated quickly, leading to the loss of jump-in. \par
For imaging nanobubbles, the opposite is required. The thickness of a nanobubble is better imaged using a hydrophilic tip as the tip is less likely to penetrate into the bubble so all changes in deflection are due to nanobubble deformation. \cite{Walczyk_Schoenherr_2014b} A hydrophobic tip still shows deflection due to nanobubble deformation, however, it also shows signs of meniscus formation as the tip contacts the nanobubble. This aspect is discussed further in Section \ref{sec:limitations}.
\subsection{Data Analysis}
\label{sec:data_anaysis}
The key to using AFM meniscus force measurements for mapping of multiple layers was the automated data analysis with feature recognition to determine the points at which the tip contacts different interfaces. This section describes the logic used by our Python script to calculate the thickness of both the gas and oil layer from the collected force curves. \par
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{Figure3+4_flow_chart_ex_FC.pdf}
\caption{a) Flowchart showing the process used to characterise each force curve in the raw force map data. The inset in the red box is the logic used in the decision nodes with a red outline to determine whether the data between two features of a force curve is gas or lubricant. See GitHub \cite{Peppou_Chapman_AFM_Nanobubble_Mapping_2021} for code. b) Raw force curve data of cantilever deflection (y scale is in \SI{}{\meter}) as function of \textit{z}-displacement for an example force curve. c) The first derivative of b) (y scale is \SI{}{\meter\per\nano\meter}). d) The second derivative of b) (y scale is \SI{}{\meter\per\nano\meter\squared}). The orange vertical lines represent negative peaks picked from the second derivative. The red, green, and blue lines represent the location of the substrate, the top of the oil and the top of the nanobubble, respectively.}
\label{fig:Fig3+4_flow_FC}
\end{figure*}
For an underwater LIS there are four possible scenarios for the fluid layers each force curve may encounter (see \autoref{fig:Fig2_ex_FCs}):
\begin{enumerate}[label=(\alph*)]
\item Water on substrate;
\item Water on oil on substrate;
\item Water on gas on substrate;
\item Water on gas on oil on substrate;
\end{enumerate}
In our previous work, \cite{Peppou-Chapman_Neto_2018, Tonelli_Peppou-Chapman_Ridi_Neto_2019, Peppou-Chapman_Neto_2021} the script only detected an oil layer (\textit{i.e.} scenarios (a) and (b)), where only two key points needed to be detected: the jump-in and hard contact point. As a result of the added complexity added by the gas layer, a completely new detection algorithm was used here to detect all four of these scenarios. The features associated with the gas layer are less pronounced than in previous iterations, and so a lower threshold for detection of features needed to be implemented.\par
Here, the algorithm utilises automated peak finding to find areas of rapid changes in the gradient of the deflection and the fact that the gradient of the deflection is opposite signs when the tip is in contact with a nanobubble versus when in contact with a meniscus due to an oil layer. The general procedure used in the script is (see also \autoref{fig:Fig3+4_flow_FC}a):
\begin{itemize}
\item Convert piezo movement data to separation, and photo diode signal to deflection, using sensitivity of cantilever calculated from compliance region measured on a hard surface.
\item Find the hard-contact point by looking for the first point where the gradient changes substantially while moving towards larger separations from the turn around point.
\item Calculate the second derivative to the whole data and apply a smoothing algorithm to the y-data (deflection).
\item Use a peak finding algorithm to find negative peaks in the smoothed second derivative (orange lines in \autoref{fig:Fig3+4_flow_FC}b-d).
\item If no peaks are detected, there is no oil or gas (\textit{i.e.} case in \autoref{fig:Fig2_ex_FCs}a)
\item If one peak is detected, then the region between it and the hard contact point can either be oil or gas. Check if gas or oil using the algorithm outlined below. (\textit{i.e.} case in \autoref{fig:Fig2_ex_FCs}b,c).
\item If multiple peaks are detected, each segment between two peaks is checked to see if it is gas or not using the algorithm outlined below. The first segment identified as oil demarcates the boundary between the gas and oil (\textit{i.e.} case in \autoref{fig:Fig2_ex_FCs}d). If no oil is detected before reaching the hard contact point, then there is no oil present and the layer is entirely gas (i.e case in \autoref{fig:Fig2_ex_FCs}c).
\end{itemize}
To determine whether a segment between two of the negative peaks identified in the second derivative is gas or oil, the following algorithm is used (see also inset within the red box in \autoref{fig:Fig3+4_flow_FC}a):
\begin{itemize}
\item If 100\% of the first derivative is negative – the portion is oil. This is because the force acting on the cantilever is proportional to the length of the contact line and this force always increases (due the triangular shape of the tip) as the tip moves towards the surface – causing the deflection to become more negative.
\item If more than 60\% of the points are negative and the end is lower than the start by a threshold deflection (\SI{0.5}{\nano\meter} in this work, determined using trial and error), then portion is considered oil. Otherwise, it is gas.
\end{itemize}
The script then outputs the height of the hard contact point (absolute height from $z$-sensor data) and the thickness of the oil or gas layers found (calculated as the distance between relevant peaks in separation). \autoref{fig:Fig5_example_map}a shows a map with example forces curves and their corresponding location in the map. The force curves (\autoref{fig:Fig5_example_map}b-g) also show the features located by the algorithm described above (blue = start of gas layer, green = start of oil layer, and red = start of the substrate). \par
\subsubsection{Validation}
This analysis was validated by a manual review of analysed force curves to judge if the script had picked the correct locations for the start of gas and oil layers. Force curves from multiple maps were plotted with the vertical lines showing the interfaces as picked by the script as in \autoref{fig:Fig3+4_flow_FC}b and \autoref{fig:Fig5_example_map} and judged by eye if these were correct. A total of 494 force curves were selected at random and 93\% of them were judged to be correctly fitted. Potential reasons for incorrect fitting are discussed below in Section \ref{sec:limitations}. \par
\subsection{Data Visualisation}
As shown in \autoref{fig:Fig5_example_map}, the data is presented as multi-panel maps showing the spatial distribution of the different quantities measured. First, the sample topography is presented (extracted from the hard contact point of the force curves). Then maps of the oil thickness and the gas thickness are presented separately. Both oil and gas thickness are presented as non-linear contours where colours are evenly spaced at thickness of \SI{2}{\nano\meter} up to \SI{20}{\nano\meter} and then contours of \SI{150}{\nano\meter}, \SI{300}{\nano\meter} and >\SI{300}{\nano\meter} to give an indication of distribution of both thin layers and thick layers. This method of presentation was selected as it gives the ability to easily attribute the effect of topography on the distribution of either the liquid or the gas layer. \par
\begin{figure}[H]
\centering
\includegraphics[width=\linewidth]{Figure5_example_map.pdf}
\captionof{figure}{a) Map of a nanobubble on a LIS with (b-g) example force curves shown for points with different fluid layers. The map (a) is presented with three panels showing the topography of the underlying Teflon wrinkles (left), the thickness of the oil (middle) and the thickness of the gas (left); the units of all colour scales is \SI{}{\nano\meter}. Points 1 and 2 show an example of a nanobubble directly on the substrate. Points 3 and 4 show an example of a nanobubble on oil. Point 5 shows an example of the water directly contacting the substrate and point 6 shows an example of water contacting the oil. The vertical lines in the force curves show where the script has found the start of the different fluid layers, blue = start of gas layer, green = start of oil layer, and red = start of the substrate.}
\label{fig:Fig5_example_map}
\end{figure}
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{Figure6_Maps_crosssections.pdf}
\caption{AFM force maps showing oil and gas thickness (a, c, e, g) with cross sections (b, d, f, h) corresponding the cyan lines in the maps. The units of all colour scales is \SI{}{\nano\meter}. The marker symbols in the cross section represent points at which the height was measured and colors were added to help visualise the different layers. a) and b) are successive maps on the same area of the surface, showing the appearance of a nanobubble \textit{in situ}. The force curves were collected and analysed using the same parameters. e, f) example of a LIS submersed in gassed water ($c_{air}\sim44\pm4$ \SI{}{\milli\gram\per\kilo\gram}) imaged immediately after submersion. g, h) a superhydrophobic surface (the underlying TW substrate of the LIS without any silicone oil applied) showing a partially collapsed Cassie state. The non-zero measurements of oil thickness at multiple points on this map are a result of the feature detection algorithm not always being able to discern the difference between oil and gas or the particular force curve being noisy. See Section \ref{sec:limitations} for more information.}
\label{fig:Fig6_maps_cross_sec}
\end{figure*}
Cross sectional profiles of these maps can be extracted to visualise the shape of the interfaces, as shown in \autoref{fig:Fig6_maps_cross_sec}, in which color was added to the line profiles to highlight the different fluid layers. Each cross section corresponds the cyan line in the adjacent maps. As mentioned above, part (a) and part (b) in \autoref{fig:Fig6_maps_cross_sec} are sequential maps (taken approximately 35 mins apart) of the same surface before and after a nanobubble nucleates. \autoref{fig:Fig6_maps_cross_sec}e,f shows a cross sections from the LIS in \autoref{fig:Fig5_example_map} while \autoref{fig:Fig6_maps_cross_sec}g,h shows gas pockets on an immersed superhydrophobic surface (Teflon wrinkles with no oil present) in a partially-collapsed Cassie state. Despite the apparently high resolution of the contact line of the mapped interfaces, the local contact angle values can not be estimated, as discussed in Section \ref{sec:limitations}.
\subsection{Limitations and Sources of Error}
\label{sec:limitations}
There are several limitations and sources of error related to this technique which are somewhat accounted for by the mapping nature which allows individual errant pixels to be ignored. This section discusses limitations and sources of error in this technique. \par
The accuracy of the measured thickness depends on many factors and is not uniform for all film thicknesses. Very thin (<\SI{5}{\nano\meter}) films of either gas or oil are difficult to both detect and distinguish. The technique’s ability to detect a nanoscale film is proportional to the film's thickness as thicker films (either gas or oil) produces a greater deflection while films of just a few nanometers produce a deflection on the same order as measurement noise. As a result, regions of zero film thickness may contain undetected films of a few nanometers. \par
Automated feature detection exacerbates this limitation as there is a trade-off between sensitivity (\textit{i.e.} detecting smaller deflections) and avoiding false detection at large separations (\textit{e.g.} more than \SI{100}{\nano\meter} from the first interface as in \autoref{fig:Fig7_bad_Fcs}d). The noise present in the oil and gas thickness maps in \autoref{fig:Fig5_example_map} is the result of imperfect fitting that occurs due to higher feature detection sensitivity. Using a thicker tip would reduce this effect but, as mentioned above, would reduce lateral resolution. Slightly thicker films (up to tens of \SI{}{\nano\meter}) can be detected but with fewer data points to judge whether the feature in the force curve is due to oil or gas, the script is prone to mislabelling these. \autoref{fig:Fig7_bad_Fcs}a,b shows examples of two features at small separations with deflections $\sim$\SI{1}{\nano\meter} which are ambiguous and were identified as different features by the script, despite being adjacent pixels in the map. \par
The presence of a nano-thin layer of oil spread on the AFM tip cannot be excluded, which would increase the measured thickness of the oil layer. As with our previous publications, this does not seem to be an issue as there are multiple cases where no oil thickness is detected either under a nanobubble or elsewhere (\textit{e.g.} \autoref{fig:Fig2_ex_FCs}a,c). However, the lack of sensitivity to very thin films (<\SI{5}{\nano\meter}) described in the previous paragraph may hide the effects of such a film. \par
The thickness of the oil layer may be overestimated if long range van der Waals attraction between the tip and an interface draws the interface up to ‘meet’ the tip. \cite{Ally2010} In our previous publication, this effect was suppressed by sufficiently fast scan rate in the force curve \cite{Peppou-Chapman_Neto_2018}. As a result, here the fastest scan rate possible on our AFM (\SI{2}{\hertz}) was used to ensure this effect is minimized. This deformation will also effect the gas layer, with the air/water interface deforming to meet the tip, especially for a hydrophobic tip. \cite{Walczyk_Schoenherr_2014b} As a result of this deformation, Walczyk \& Sch{\"o}nherr \cite{Walczyk_Schoenherr_2014b} define the top of the a nanobubble measured with a hydrophobic tip to be where the force curve crosses zero deflection after the initial jump-in. Here potential long-range attraction of the air/water interface was ignored, due to the high scan rates used and because this correction would cause the calculated gas height to be heavily dependent on the quality of the baseline correction. Additionally, Walczyk \& Sch{\"o}nherr's definition of the top of a nanobubble assumes that the AFM tip only contacts the bubble with minimal contact line at zero deflection (\textit{i.e.} the only force on the cantilever is due to nanobubble deformation), which is impossible given the fact that a jump-in is seen, signifying meniscus formation and a non-trivial contact line. \par
\begin{figure}[H]
\centering
\includegraphics[width=\linewidth]{Figure7_bad_FCs.pdf}
\caption{Force curves showing the limitations of the automatic feature detection script presented in this work. Vertical lines indicate where the analysis script found an interface with blue = start of gas layer, green = start of oil layer, and red = start of the substrate. a,b) Two force curves obtained on adjacent locations, where the identity of the feature is ambiguous, with the script identifying in (a) a gas layer and in (b) a oil layer; c) an example of the script not detecting gas; d) an example of the script incorrectly detecting the gas/water interface; e) an example of the script incorrectly detecting the oil/gas interface instead of more gas; f) an example of a noisy force curve, likely due to vibrations. }
\label{fig:Fig7_bad_Fcs}
\end{figure}
There are limitations of the experimental setup which contribute noise in the maps. Force curves collected under water are particularly sensitive to vibrations, as they can be transmitted through the liquid to the cantilever. This means that a higher proportion of force curves are noisy throughout their entire range of motion, compared to the same force measurement in air, leading to incorrect feature detection (see \autoref{fig:Fig7_bad_Fcs}f). \par
The cross sections generated from the map cannot be used to quantify contact angle values as the interface is 3-dimensional and contributions from in-plane and out-of-plane features are impossible to account for. Additionally, the shape of the interface may be slightly deformed due van der Waals interactions not being consistent across the surface. \par
\section{Conclusion}
In summary, we have shown that a single AFM force-distance curve can capture the thickness of both a gas layer and an immiscible liquid layer underwater simultaneously. By mapping a surface with force curves and analysing the data automatically, the thickness of the gas and the immiscible liquid are both spatially resolved. This presents an exciting new technique to study systems such as the nucleation of nanobubbles on LIS. \cite{Vega2021} There are still many outstanding questions related to nanobubbles on LIS including the effects of substrate topography, the presence of surfactants or ion concentration and effect of static pressure. This force mapping technique will enable detailed studies of these parameters and could help in establishing whether nanobubbles are more stable on superhydrophobic or on LIS.
\printbibliography
\end{multicols}
\end{document}
|
1,314,259,993,585 | arxiv | \section{Charged Lepton Flavor Violation and muon to electron conversion}
\label{sec:intro}
Within the Standard Model (SM), transitions in the lepton sector between charged and neutral particles preserve flavor, since the neutrinos are considered massless.
Even considering the discovery of neutrino oscillations, in the minimal extension of SM,
the predicted branching ratios of Charged Lepton Flavor Violation (CLFV)
processes in the muon sector are smaller than 10$^{-50}$.
No CLFV process has been observed yet, so any experimental detection
would be a clear signature of New Physics (NP) beyond the Standard Model.
One of the most promising process for probing CLFV is the coherent muon conversion
in the field of a nucleus, $\mu$~N~$\rightarrow$~e~N.
In this process the nucleus is left intact and the resulting electron has a monochromatic
energy slightly below the muon rest mass ($\sim$~104.96 MeV for Al), due to the nucleus recoil.
The Mu2e experiment \cite{tdr} is designed to improve the current limit on the conversion rate, R$_{\mu e}$, by 4
orders of magnitude over the SINDRUM II experiment \cite{sindrum}.
R$_{\mu e}$ is defined as the ratio between the number of electrons from the conversion process
and the number of captured muons:
\begin{equation*}
R_{\mu e} = \frac{\mu^- \thinspace N(Z,A) \rightarrow e^- \thinspace N(Z,A)}{\mu^- \thinspace N(Z,A) \rightarrow\nu_{\mu} \thinspace N(Z-1,A)}
\end{equation*}
where, in the Mu2e case, N(Z,A) is an Aluminum nucleus.
Many NP scenarios, like SUSY, Leptoquarks, Heavy Neutrinos, GUT, Extra Dimensions or Little Higgs,
predict significantly enhanced values for R$_{\mu e}$, allowing the detection of the process with the expected Mu2e sensitivity~\cite{marciano}.
A model independent description of the CLFV transitions, for NP models, is provided by
an effective Lagrangian~\cite{deGouvea} where the different processes are divided in dipole amplitudes
and contact term operators. The $\mu \to e \gamma$ decay is mainly sensitive to the
dipole amplitude, while $\mu \to e$ conversion and $\mu \to 3e$ receive contributions
also from the conctact interactions. It is possible to parametrise the interpolation between the
two amplitudes by means of two parameters~\cite{deGouvea}:
$\Lambda$, which sets the mass scale, and $\kappa$, which governs the ratio of the four fermion
to the dipole amplitude.
For $\kappa <\!\!\!< 1 (>\!\!\!> 1)$ the dipole-type (contact) operator dominates.
Figure~\ref{fig:degouvea} summarises the power of different searches to explore this parameter space~\cite{bob}.
\begin{figure}
\begin{center}
\includegraphics[width=0.8\columnwidth]{lambda_kappa.png}
\caption{\label{fig:degouvea}
Sensitivity of $\mu \to e \gamma$, $\mu \to e$ transition and $\mu \to 3e$ to the
scale of new physics $\Lambda$ as a function of the parameter $\kappa$ .
The shaded areas are excluded by present limits. On the left (right) side, the dipole (four-fermion) diagrams
are shown for the different processes.}
\end{center}
\end{figure}
Present experimental limits already excluded lepton flavour violation up to a mass scale up of $\Lambda < 700$~TeV.
The interpretation of an eventual direct observation of NP at LHC
will have to take into account precise measurements (or constraints) from MEG~\cite{meg} and Mu2e:
the comparison between these determinations will help pinning down the underlying theory.
\section{Muonic Aluminum atom}
When negative muons stop in the Aluminum target, they are captured in an atomic
excited state. They promptly fall to the ground state, then 39\% of them decay in
orbit, $\mu^- \to e^- \bar{\nu_e} \nu_{\mu}$ (DIO), while the remaining 61\% are captured on the nucleus. Low energy photons, neutrons and
protons are emitted in the nuclear capture process and constitutes an
environmental background that produces a ionisation dose and a neutron
fluency on the detection systems
as well as an accidental occupancy for the reconstruction program.
The kinematic limit for the muon decay in vacuum is at about 54 MeV,
but the nucleus recoil generates a long tail that has the
endpoint exactly at the conversion electron energy.
DIO electrons are an irreducible background that have to
be distinguished by the mono-energetic conversion electron (CE). The finite tracking
resolution and the positive reconstruction tail has a large effect
on the falling spectrum of the DIO background that translates in a
residual contamination in the signal region as shown in Fig~\ref{fig:tail}.
\begin{figure}[hbt]
\begin{center}
\includegraphics[width=0.45\columnwidth]{limit.pdf}
\caption{\label{fig:tail} Full simulation of DIO and CE events for an assumed $R_{\mu e}$ of 10$^{-16}$.}
\end{center}
\end{figure}
\section{The Mu2e experimental apparatus
The Mu2e apparatus consists of three superconductive solenoid magnets,
as shown in Figure~\ref{fig:mu2esetup}: the Production Solenoid (PS),
the Transport Solenoid (TS) and the Detector Solenoid (DS).
\begin{figure}[ht]
\begin{center}
\includegraphics[width=1\columnwidth]{mu2e_layout.pdf}
\caption{\label{fig:mu2esetup} Layout of the Mu2e experiment.}
\end{center}
\end{figure}
The proton beam interacts in the PS with a tungsten target, producing mostly pions and muons.
The gradient field in the PS increases from 2.5 to 4.6 T
in the same direction of the incoming beam and opposite to the
outgoing muon beam direction. This gradient field
works as a magnetic lens to focus charged particles into the transport
channel. The focused beam is constituted by muons, pions with
a small contamination of protons and antiprotons. When the beam passes through the S-shaped TS,
low momentum negative charged particles are selected and delivered to the Aluminum stopping
targets in the DS. Electrons from the $\mu$-conversion (CE) in the stopping target are captured
by the magnetic field in the DS and transported through the Straw Tube Tracker,
that reconstructs the CE trajectory and its momentum. The CE then strikes the Electromagnetic
Calorimeter, that provides independent measurements of the energy, the impact time and the position.
Both detectors operate in a 10$^{-4}$ Torr vacuum and in an uniform 1 T axial field.
A Cosmic Ray Veto (CRV) system covers the entire
DS and half of the TS, as shown in Figure~\ref{fig:calorimeter} (right).
Additional details on the Mu2e apparatus can be found in~\cite{tdr}.
\section{The Mu2e Tracker and Cosmic Ray Veto
The tracking detector is made of low
mass straw drift tubes oriented transversally to the
solenoid axis. The detector consists of about 21000 straw tubes arranged
in 18 stations, as shown in Figure~\ref{fig:tracker} (left). Each tube is of
5 mm in diameter and contains a 25~$\mu$m sense wire. The straw walls are
made of Mylar and have a thickness of 15~$\mu$m.
\begin{figure}[ht]
\begin{center}
\begin{tabular}{cc} \\
\includegraphics[width=0.46\columnwidth]{tracker.pdf} &
\includegraphics[width=0.50\columnwidth]{Tracker_panel.pdf} \\
\end{tabular}
\caption{\label{fig:tracker} (Left) Sketch of the Mu2e straw tracker system.
(right) Picture of the first prototype built for straw tube panel. }
\end{center}
\end{figure}
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.40\columnwidth]{helics.pdf}
\caption{\label{fig:helics} Transverse view of the Tracker active area. Only tracks emerging from the stopping target (yellow spot) with
momentum $p > 55$ MeV/c (green circles) leave hits in the straw tubes. Lower momentum tracks (black circles) leave the detectors undetected.}
\end{center}
\end{figure}
The gas used is a 80:20 mixture of Argon-CO$_2$. The tracker is around
3 m long and measures
the momenta of the charged particles from the reconstructed trajectories
using the hits detected in the straw. As shown in Figure~\ref{fig:helics}, a circular inner un-instrumented region inside the tracker
makes it insensitive to charged particle with momenta below 55 MeV$/$c.
Indeed its acceptance is optimised to identify $\sim~100$~MeV electrons.
Each straw tube is instrumented on both sides with TDCs to measure the particle crossing time and ADCs to measure the specific energy loss $d$E/$d$X used to separate electrons from highly ionizing particles. The Momentum resolution for 105 MeV electrons is expected to be better than 180 keV$/$c, enough to suppress background electrons coming from the decays of muons captured by Al nuclei and from DIO.
In Figure~\ref{fig:tracker}~(right),
an example of the first panel prototype built is shown.
One major background source for Mu2e is related to cosmic ray muons faking CEs when
interacting with the detector materials.
In order to reduce their contributions to below 0.1 event in the experiment lifetime,
the CRV system is required to get a vetoing efficiency of at least 99.99\%
for cosmic ray tracks while withstanding an intense radiation environment. The basic element
of the CRV is constituted by four staggered layers of scintillation bars,
each having two embedded Wavelength Shifting Fibres readout by means of
(2$\times$2)~mm$^2$ SiPM.
\section{The Mu2e Calorimeter}
The electromagnetic calorimeter \cite{miscetti} is a high granularity crystal-based calorimeter
needed to:
\begin{itemize}
\item identify conversion electrons
\item provide particle identification to suppress muons and pions faking conversion electrons
\item add trigger capabilities
\item add seed positioning and timing in the track reconstructions
\end{itemize}
It is composed of two annuli with inner and outer radii of 37.4 cm and 66 cm respectively, filled by
pure CsI scintillating crystals and is placed downstream
the tracker. Each annulus is composed of 674 crystals of ($34\times34\times200$)~mm$^3$ dimensions, each readout by two custom arrays of 2$\times3$ $6\times6$ mm$^2$ UV-extended Silicon Photomultiplier (SiPM); the SiPM are optimised to increase the quantum efficiency for 315 nm photons, the fast emission component of the scintillating process of CsI crystals.
The granularity and crystal dimensions have been optimised to maximise light collection for readout photosensor, time and energy resolutions and take under control particles pile-up.
Each crystal is wrapped with 150 $\mu$m Tyvek 4173D to maximise light collection.
In Figure~\ref{fig:calorimeter} (left) a drawing of these two annuli is shown.
Similarly to the tracker, the inner circular hole allows
electrons up to 55 MeV$/$c momenta to escape undetected.
In Figure~\ref{fig:calorimeter} (right) an exploded view of a single calorimeter annulus is shown.\\
It consists of an outer monolithyc Al cylinder that provides the main support for the crystals and integrates the feet and adjustment mechanism to park the detector on the rails inside the detector solenoid. The inner support is made of a Carbon Fiber cylinder that maximise passive material $X_0$.
The crystals are then sandwiched between two cover plates. A Carbon Fiber front plate also integrates thin wall Al pipes to flow the radioactive Fluorinert fluid to calibrate the response; a back plate made of PEEK with apertures in correspondence of each crystals where the Front End Electronics (FEE) and SiPM holders will be inserted.
The back plate houses also the Copper pipes where a coolant is flown to thermalise the photosensors and extract the power dissipated by both the FEE and the sensors.
The calorimeter has to operate in the hostile experimental environment with 1 $T$ magnetic field and a vacuum of $10^{-4}$ Torr, a maximum neutron fluence of $10^{12} \;n/$cm$^2$ in 3 years, a maximum ionising dose of 100 krad in the hottest region at lower radii of the calorimeter.
10 custom made crates are arranged on top of the outer cylinder and are connected to the cooling circuit.
The calorimeter particle identification provides a good separation between CE's
and muons, un-vetoed by the CRV, mimicking the signal.
The required muon rejection factor (200) is achieved with 95\% efficiency on the signal,
combining the time of flight difference between the tracker track and the calorimeter
cluster with the E$/$p ratio.
In order to satisfy these requirements, the calorimeter has to reach
an energy resolution of O($5 \%$), a time resolution less than 500~ps and
a position resolution better than 1~cm for 100~MeV electrons. The selected
crystals should also be radiation hard up to 100 krad. The photosensors are shielded
by the crystals themselves and should only sustain a fluency up to
$3 \times 10^{11}$~n/cm$^2$.
\begin{figure}
\begin{center}
\begin{tabular}{ll} \\
\includegraphics[width=0.45\columnwidth]{calo.pdf} &
\includegraphics[width=0.5\columnwidth]{exploded.pdf} \\
\end{tabular}
\caption{\label{fig:calorimeter} (Left) CAD drawings of the calorimeter disks.
Calorimeter innermost (outermost) radius is of 350 mm (600 mm).
Layout of the FEE and digitization crates is also shown. (Right) Exploded view of all the calorimeter parts.}
\end{center}
\end{figure}
\subsection{Calorimeter performances and prototyping}
A calorimeter prototype consisting of a $3 \times 3$ matrix of $30 \times 30 \times 200$ mm$^2$ un-doped CsI crystals wrapped with 150 $\mu$m Tyvek and read out by one $12 \times 12$ mm$^2$ SPL TSV SiPM by Hamamatsu
has been tested with an electron beam at the Beam Test Facility (BTF) in Frascati during
April 2015.
The results, described in \cite{btf}, are coherent with the ones predicted by the GEANT4 simulation ~\cite{geant} and are shown in Figure~\ref{fig:btf}:
\begin{itemize}
\item time resolution better than 150 $p$s for 100 MeV electrons. The timing resolution ranges from about 250 ps
at 22 MeV to about 120 ps in the energy range above 50 MeV.
\item energy resolution of $\sim7\% $ for 100 MeV electrons, dominated by the shower non-containment.
\end{itemize}
\begin{figure}
\begin{center}
\begin{tabular}{ccc} \\
\includegraphics[width=0.3\columnwidth]{eres.pdf} &
\includegraphics[width=0.3\columnwidth]{eres2.pdf}&
\includegraphics[width=0.3\columnwidth]{tres.pdf}\\
\end{tabular}
\caption{\label{fig:btf} BTF results:(left) Data-MC comparison for energy reconstruction of 100 MeV electrons; a typical fit to the data with a log-normal function is shown in red; left long tail is due to not full containment of the shower;
(center) energy resolution as a function of the reconstructed total energy; (right) time resolution as function of energy for different configurations of the matrix as function of energy.}
\end{center}
\end{figure}
We have built a Module-0 prototype composed of 51 CsI crystals from different vendors (Siccas, St. Gobain and Amcrys) instrumented with SiPM from 3 different companies (Hamamatsu, Sensl, Advansid) to test their quality and to test the current design technological performance. Figure ~\ref{fig:module0} shows the CAD drawing of the Module-0 and its actual realization. This Module-0 has been tested at the Frascati BTF and data analysis is underway.
A full scale mock-up of the mechanical structure is being built to test the assembly of the crystals, FEE electronics, cooling system and overall structure robustness : the Al outer ring, the inner Carbon Fiber cylinder, sections of the front and back plates, crate prototype have been manufactured.
A whole annulus will be assembled using a mixture of fake Iron crystals and a sample of preproduction CsI crystals. Figure~\ref{fig:mockup} shows the ongoing mock-up.
\begin{figure}
\begin{center}
\begin{tabular}{cc} \\
\includegraphics[width=0.6\columnwidth]{module0.pdf} &
\includegraphics[width=0.4\columnwidth]{module0pic.jpg} \\
\end{tabular}
\caption{\label{fig:module0} CAD drawing of the Module-0 (left) and its realization (right)}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\begin{tabular}{ccc} \\
\includegraphics[width=0.3\columnwidth]{outer.jpg} &
\includegraphics[width=0.3\columnwidth]{front.pdf} &
\includegraphics[width=0.3\columnwidth]{foot.jpg} \\
\end{tabular}
\caption{\label{fig:mockup} Full scale mock-up. Outer Al cylinder, front plate with source piping, foot with x-y adjustment.}
\end{center}
\end{figure}
\section{Conclusions and perspectives}
The Mu2e experiment design and construction proceeds well and
it is on schedule to be commissioned with beam for the end of 2021.
Its goal is to probe CLFV with a single event sensitivity of 2.5 $\times$
10$^{-16}$ or set an upper limit on the conversion rate $< 6 \times 10^{-17}$
at 90 \% C.L. improving of four orders of magnitude the
sensitivity of previuos measurements .
A Mu2e second phase is already planned with the goal of increasing the sensitivity
of an additional factor of 10.
The Calorimeter design is almost complete and the prototyping of the most delicate components is underway. The Module-0 construction showed a good coupling between photosensors and crystals and also the capability to cool down the SiPM and extract heat with the current cooling scheme.
We will start the construction of the final components in Year 2018 together with the opening of the tenders for crystals and SiPM purchase.
\acknowledgments
This work was supported by the EU Horizon 2020 Research and Innovation Programme under the Marie Sklodowska-Curie Grant Agreement No. 690835.
|
1,314,259,993,586 | arxiv | \section{Acknowledgement}
We would like to thank Burkhard D\"{u}nweg, Kostas Daoulas, Tristan Bereau, Cristina Greco, Horacio Vargas, Torsten Stuehn and Alexander Grosberg for stimulating discussions.
|
1,314,259,993,587 | arxiv | \section{Introduction}
Recently discovered\cite{top_disc} by the CDF and D0 Collaborations, the top quark is the least well understood fundamental particle. As a consequence of its enormous mass, around 175 GeV$/$c$^2$, this hefty particle has a lifetime shorter than the hadronization timescale.
With the top mass close to the scale of electroweak symmetry breaking there are hints of an intimate relationship between top and this mechanism--Higgs is most strongly coupled to the top quark. By studying top we are testing electroweak theory, as this quark may be sensitive to physics beyond the standard model (SM).
\section{Top Quark Production and Decay Modes}
At the Tevatron, a $p\overline{p}$ collider with a center of mass energy of $\sqrt{s} = 1.96$ TeV, top is predominantly pair produced via $q\overline{q}\rightarrow t\overline{t}$ 85\% of the time with the remaining fraction generated via $gg\rightarrow t\overline{t}$. The cross-section for $t\overline{t}$ production at the Tevatron, assuming $m_t = 175$ GeV$/$c$^2$, is\cite{top_theory} $6.7^{+0.7}_{-0.9}$ pb. By comparison, the total cross-section for producing top singly via electroweak processes is smaller by about a factor of two.
Top decays to a $W$ boson and a $b$ quark nearly 100\% of the time due to unitarity constraints on the CKM matrix. The experimental signature of a $t\overline{t}$ event includes two $b$ quark jets, the presence of multiple light-quark jets and/or a single high-$p_T$ charged lepton accompanied by significant missing transverse energy ($\mathbin{E\mkern - 11mu/_T}$) from an undetected neutrino. Top candidate events are frequently classified by the $W$ boson decays. The ``dilepton'' mode occurs when each of the two $W$s decay leptonically. Both $W$s decaying hadronically produce the ``all-hadronic'' mode; events with a mixture of leptonic and hadronic decays are dubbed ``lepton-plus-jets'' events.
\begin{figure}[t]
\centerline{\epsfxsize=3.5in\epsfbox{num_of_jets.eps}}
\caption{The top quark signal region of the lepton-plus-jets sample occupies the 3 and $\geq 4$ jet bins.\label{fig:num_of_jets}}
\end{figure}
\section{Top Quark Cross-section Measurements}
\begin{table}[hb]
\tbl{Top quark pair-production cross-section measurements.}
{\footnotesize
\begin{tabular}{@{}llc@{}}
\hline
{} &{} &{}\\[-1.5ex]
{} & \multicolumn{1}{c}{$\sigma_{t\overline{t}}$ [pb]} & $\int\mathcal{L}\mbox{ }dt$ [pb$^{-1}$] \\[1ex]
\hline
{} &{} &{}\\[-1.5ex]
Dilepton: Combined & $7.0^{+2.7}_{-2.3}\mbox{ (stat.) }^{+1.5}_{-1.4}\mbox{ (syst.)}$ & 200\\[1ex]
Dilepton: $\mathbin{E\mkern - 11mu/_T}$, Num. jets & $8.6^{+2.5}_{-2.4}\mbox{ (stat.) }\pm 1.1\mbox{ (syst.)}$ & 200\\[1ex]
\hline
{} &{} &{}\\[-1.5ex]
Lepton + Jets: Kinematic & $4.7\pm 1.6\mbox{ (stat.) }\pm 1.8\mbox{ (syst.)}$ & 193\\[1ex]
Lepton + Jets: Kinematic NN & $6.7\pm 1.1\mbox{ (stat.) }\pm 1.6\mbox{ (syst.)}$ & 193\\[1ex]
Lepton + Jets: Vertex Tag + Kinematic & $6.0\pm 1.6\mbox{ (stat.) }\pm 1.2\mbox{ (syst.)}$ & 162\\[1ex]
Lepton + Jets: Vertex Tag & $5.6^{+1.2}_{-1.1}\mbox{ (stat.) }^{+0.9}_{-0.6}\mbox{ (syst.)}$ & 162\\[1ex]
Lepton + Jets: Double Vertex Tag & $5.0^{+2.4}_{-1.9}\mbox{ (stat.) }^{+1.1}_{-0.8}\mbox{ (syst.)}$ & 162\\[1ex]
Lepton + Jets: Jet Probability Tag & $5.8^{+1.3}_{-1.2}\mbox{ (stat.) }\pm 1.3\mbox{ (syst.)}$ & 162\\[1ex]
Lepton + Jets: Soft Muon Tag & $5.2^{+2.9}_{-1.9}\mbox{ (stat.) }^{+1.3}_{-1.0}\mbox{ (syst.)}$ & 193\\[1ex]
\hline
{} &{} &{}\\[-1.5ex]
All Hadronic: Vertex Tag & $7.8\pm 2.5\mbox{ (stat.) }^{+4.7}_{-2.3}\mbox{ (syst.)}$ & 165\\[1ex]
\hline
\end{tabular}\label{tab:xsec_table} }
\vspace*{-13pt}
\end{table}
Conducting a top quark production cross-section measurement validates our top-enriched samples and could yield the first signs of new physics in the top quark sector.
The dilepton analyses use event selections of either two identified charged leptons ($e$ or $\mu$), at least two jets and large $\mathbin{E\mkern - 11mu/_T}$ or a single charged lepton in addition to a well isolated track.
Analyses in the lepton-plus-jets channel have a larger initial data sample than dilepton analyses, albeit one which contains a larger amount of background contamination. Tagging $b$-jets is a technique used to increase the fraction of top in the lepton-plus-jets sample. The tagger used by CDF is sensitive to displaced secondary vertices due to the relatively long-lived $b$ quarks, in addition to semi-leptonic decays of the $b$. The signal region for tagged lepton-plus-jets events is shown in Fig.~\ref{fig:num_of_jets}.
The $b$ tagging is also used by the all-hadronic analysis in addition to a trigger which requires four high $p_T$ jets and a large total transverse energy in the event.
Results of CDF's most recent cross-section measurements are summarized in Table~\ref{tab:xsec_table}. All results obtained thus far are consistent with the SM.\cite{cross_section}
\section{Top Quark Mass Measurements}
Through the correlation with other SM parameters, a measurement of the top mass puts constraints on the Higgs. A precise top mass measurement is difficult due to the level of understanding necessary concerning jet energies and the relationship between parton-level objects and detector observables. This measurement is also hampered by our ability to correctly assign jets to the parton-level objects in top decays.
\begin{table}[ht]
\tbl{Top quark mass measurements.}
{\footnotesize
\begin{tabular}{@{}llc@{}}
\hline
{} &{} &{}\\[-1.5ex]
{} & \multicolumn{1}{c}{$m_t$ [GeV$/$c$^2$]} & $\int\mathcal{L}\mbox{ }dt$ [pb$^{-1}$] \\[1ex]
\hline
{} &{} &{}\\[-1.5ex]
Dilepton: $\phi$ of $\nu$ & $170.0\pm 16.6\mbox{ (stat.) }\pm 7.4\mbox{ (syst.)}$ & 193\\[1ex]
Dilepton: $p_z$ $t\overline{t}$ & $176.5^{+17.2}_{-16.0}\mbox{ (stat.) }\pm 6.9\mbox{ (syst.)}$ & 193\\[1ex]
Dilepton: $\nu$ weighting & $168.1^{+11.0}_{-9.8}\mbox{ (stat.) }\pm 8.6\mbox{ (syst.)}$ & 200\\[1ex]
\hline
{} &{} &{}\\[-1.5ex]
Lepton + Jets: Multivariate & $179.6^{+6.4}_{-6.3}\mbox{ (stat.) }\pm 6.8\mbox{ (syst.)}$ & 162\\[1ex]
Lepton + Jets: $M_{\rm reco}$ & $177.2^{+4.9}_{-4.7}\mbox{ (stat.) }\pm 6.6\mbox{ (syst.)}$ & 162\\[1ex]
Lepton + Jets: DLM & $177.8^{+4.5}_{-5.0}\mbox{ (stat.) }\pm 6.2\mbox{ (syst.)}$ & 162\\[1ex]
\hline
\end{tabular}\label{tab:mass_table} }
\vspace*{-13pt}
\end{table}
Several methods are used by CDF to measure the top mass in the dilepton and lepton-plus-jets channels. Most of the analysis techniques use templates generated from Monte Carlo events in conjunction with likelihood fitting. The Dynamical Likelihood Method (DLM), however, takes into account all possible jet combinations in an event and the likelihood is multiplied event-by-event to derive the top quark mass using a maximum likelihood method. The advantage of the DLM method over the canonical template methods lies in the fact that the cross-section is used as a posterior probability whereas in the template methods it is used as a prior probability.
Results of the most recent top mass measurements are summarized in Table~\ref{tab:mass_table}.
\section{Single Top Quark Searches}
Investigating single top production is a great opportunity to study the charged-current weak interaction and to search for new physics thought to be exclusive to these channels. Single top production cross-sections are proportional to the CKM matrix element $V_{tb}$; a measurement of the cross-section provides a direct measurement of this quantity. CDF has searched for the $s$ and $t$-channel single top production modes which have theoretical cross-sections of 0.88 and 1.98 pb, respectively.\cite{single_top_theory}
The strategy for single top analyses is to search for $W$ decay products plus two or three jets. Two analyses were conducted in 162 pb$^{-1}$ of data to search for single top. A combined search using the scalar sum of the event transverse energy ($H_T$ distribution) was used for single top discovery and to measure $|V_{tb}|$. Separate $s$ and $t$-channel searches using the $Q\times\eta$ distribution\footnote{$Q$ is the charge of the lepton and $\eta$ of the light-quark jet.} were carried out to reveal any new physics.
The combined search sets a limit of $< 17.8\mbox { pb @ } 95\%\mbox{ CL}$. Limits from the $s$ and $t$-channel searches are $< 13.6\mbox { pb @ } 95\%\mbox{ CL}$ and $<10.1\mbox { pb @ } 95\%\mbox{ CL}$, respectively.\cite{single_top}
\section{Measurements of Top Quark Properties}
Now that sizable samples of top quark candidate events have been accumulated, we can proceed to measure various top quark properties.
The fraction of right-handed $W$ bosons from top decay is heavily suppressed in the SM. The charged-lepton $p_T$ and angular distributions for each of the three $W$ helicity states are very distinct--a feature exploited to make a helicity measurement using templates in likelihood fits to the CDF data. CDF uses both the charged-lepton $p_T$ and lepton angular distributions for extracting the fraction of longitudinal $W$s. The lepton $p_T$ analysis carries out a measurement in both the dilepton and lepton-plus-jets datasets measuring $F_0 = 0.27^{+0.35}_{-0.21}\mbox{ (stat.) }\pm0.17\mbox{ (syst.)}$, the angular distribution method uses the lepton-plus-jets dataset measuring $F_0 = 0.89^{+0.30}_{-0.34}\mbox{ (stat.) }\pm 0.17\mbox{ (syst.)}$. CDF has recently revisited the Run I data to make a measurement of the right-handed fraction,\cite{w_helicity} $F_+ < 0.18\mbox{ @ }95\%\mbox{ CL}$.
An analysis of top decay kinematics\cite{anom_kin} yields results consistent with the SM. CDF has measured several ratios of branching fractions: $BR(t\rightarrow\tau\nu b)/BR_{\rm SM}(t\rightarrow\tau\nu b) < 5.0\mbox{ @ }95\%\mbox{ CL}$ and $BR(t\rightarrow WB)/BR(t\rightarrow Wq) > 0.62\mbox{ @ }95\%\mbox{ CL}$. We have also measured the ratio of the cross-sections $\sigma_{\rm dilepton}/\sigma_{\rm lepton + jets} = 1.45^{+0.83}_{-0.55}\mbox{(stat. + syst.)}$.
\section{Outlook}
Experimentally, top quark physics is still in its infancy. While no unexpected results have been observed thus far, many opportunities for discovery still exist at CDF.
As CDF continues the trend of doubling its dataset each year, statistical and systematic uncertainties will be reduced greatly improving the current measurements.
|
1,314,259,993,588 | arxiv | \section{Sketch of Analysis}
To analyze the algorithm shown in Figure~\ref{alg:lazy}, first we decompose the regret into a number of terms, which are then bounded one by one.
Let $\widetilde{x}_{t+1}^a \sim P(.\,|\,x_t,a,\TTh_t)$, i.e. an imaginary next state sample assuming we take action $a$ in state $x_t$ when parameter is $\widetilde \theta_t$. Also let $\widetilde{x}_{t+1}\sim P(.\,|\,x_t,a_t,\TTh_t)$ and $x_{t+1}\sim P(.\,|\,x_t,a_t,\theta_*)$. By the average cost Bellman optimality equation \citep{bertsekas1995dynamic}, for a system parametrized by $\widetilde \theta_t$, we can write
\begin{align}
\label{eq:bellman}
J(\TTh_t) + h_t(x_t) &= \min_{a\in \cA} \left\{ \ell(x_t,a) + \EE{h_t(\widetilde{x}_{t+1}^a)\,|\,\cF_t, \TTh_t} \right\} \;.
\end{align}
Here $h_t(x) = h(x, \TTh_t)$ is the differential value function for a system with parameter $\TTh_t$. We assume there exists $H>0$ such that $h_t(x)\in [0,H]$ for any $x\in \cX$. Because the algorithm takes the optimal action with respect to parameter $\widetilde\theta_t$ and $a_t$ is the action at time $t$, the right-hand side of the above equation is minimized and thus
\begin{align}
\label{eq:bellman2}
J(\TTh_t) + h_t(x_t) = \ell(x_t,a_t) + \EE{h_t(\widetilde{x}_{t+1})\,|\,\cF_t, \TTh_t} \;.
\end{align}
The regret decomposes into two terms as shown in Lemma \ref{lemma:regret-decomposition}.
\begin{lemma}
\label{lemma:regret-decomposition}
We can decompose the regret as follows:
\begin{align*}
R_T &= \sum_{t=1}^T \EE{\ell(x_t,a_t) - J(\theta_*)}
\leq \, H \sum_{t=1}^T \EE{\one{A_t}}+\sum_{t=1}^T \EE{h_{t}(x_{t+1})-h_t(\widetilde x_{t+1})} + \, H \nonumber \\
\end{align*}
where $A_t$ denotes the event that the algorithm has changed its policy at time t.
\end{lemma}
The first term $H \sum_{t=1}^T \EE{\one{A_t}}$ is related to the sequential changes in the differential value functions, $h_{t+1} - h_t$.
We control this term by keeping the number of switches small; $h_{t+1} = h_t$ as long as the same parameter $\widetilde\theta_t$ is used.
Notice that under DS-PSRL, $\sum_{t=1}^T \one{A_t} \leq \log_2(T)$ always holds.
Thus, the first term can be bounded by $H \sum_{t=1}^T \EE{\one{A_t}} \leq H \log_2(T)$.
The second term $\sum_{t=1}^T \EE{h_{t}(x_{t+1})-h_t(\widetilde x_{t+1})}$ is related to how fast the posterior concentrates around the true parameter vector.
To simplify the exposition, we define
\begin{align*}
\Delta_t =& \, \int_\cX \Big( P(x\,|\,x_t,a_t,\theta_*) - P(x\,|\,x_t,a_t,\TTh_t) \Big) h_t(x) dx \nonumber \\
=& \, \EE{h_{t}(x_{t+1})-h_t(\widetilde x_{t+1}) \middle | x_t, a_t} \; .
\end{align*}
Recall that $\widetilde{x}_{t+1}\sim P(.\,|\,x_t,a_t,\TTh_t)$ while $x_{t+1}\sim P(.\,|\,x_t,a_t,\theta_*)$, thus, from the tower rule,
we have
\begin{align*}
\EE{\Delta_t }=\EE{h_{t}(x_{t+1})-h_t(\widetilde x_{t+1})} \; .
\end{align*}
The following two lemmas bound $\sum_{t=1}^T \EE{\Delta_t } $ under Assumption~\ref{ass:lipschitz}
and~\ref{ass:concentrating}.
\begin{lemma}
\label{lemma:delta1}
Under Assumption~\ref{ass:lipschitz}, let $m$ be the number of schedules up to time $T$, we can show:
\begin{align*}
\EE{\sum_{t=1}^T \Delta_t}
&\le C H \sqrt{ T \EE{ \sum_{j=1}^{m} M_{j} \abs{\theta_{*} - \TTh_{j}}^2 }} \;
\end{align*}
where $M_{j}$ is the number of steps in the $j$th episode.
\end{lemma}
\begin{lemma}
\label{lemma:delta2}
Given Assumption \ref{ass:concentrating} we can show:
\begin{align*}
\EE{ \sum_{j=1}^{m} M_{j} \abs{\theta_{*} - \TTh_{j}}^2 } \le 2 C' \log^2 T \;.
\end{align*}
\end{lemma}
Thus,
\begin{align*}
\EE{\sum_{t=1}^T \Delta_t} &\le C H \sqrt{ 2 C' T \log^2 T }= O(\sqrt{T } \log T) \;.
\end{align*}
Combining the above results, we have
\begin{align}
R_T \leq & \, H \log_2(T) + C H \sqrt{ 2 C' T \log^2 T } + H \nonumber = \, O(CH \sqrt{C' T } \log T) \; .
\end{align}
This concludes the proof.
\section{Proof of lemma \ref{lemma:regret-decomposition}}
\begin{proof}
For deterministic schedule,
\begin{align*}
\EE{J(\theta_*)} = \EE{ J(\TTh_{t}) } \;.
\end{align*}
Thus we can write
\begin{align*}
R_T &= \sum_{t=1}^T \EE{\ell(x_t,a_t) - J(\theta_*)}\\
&= \sum_{t=1}^T \EE{\ell(x_t,a_t) - J(\TTh_t)} \\
&= \sum_{t=1}^T \EE{h_t(x_t) - \EE{h_t(\widetilde x_{t+1})\,|\, \cF_t,\TTh_t}} \\
&= \sum_{t=1}^T \EE{h_t(x_t) - h_t(\widetilde x_{t+1})} \;.
\end{align*}
Thus, we can bound the regret using
\begin{align*}
R_T &= \EE{h_1(x_1) - h_{T+1}(x_{T+1})} \\
&+ \sum_{t=1}^T \EE{h_{t+1}(x_{t+1}) - h_t(\widetilde x_{t+1})}\\
&\le H + \sum_{t=1}^T \EE{h_{t+1}(x_{t+1}) - h_t(\widetilde x_{t+1})}\;,
\end{align*}
where the second inequality follows because $h_1(x_1)\le H$ and $-h_{T+1}(x_{T+1})\le 0$.
Let $A_t$ denote the event that the algorithm has changed its policy at time t. We can write
\begin{align*}
R_T - H
&\leq \sum_{t=1}^T \EE{h_{t+1}(x_{t+1}) - h_t(\widetilde x_{t+1})}\\
&= \sum_{t=1}^T \EE{h_{t+1}(x_{t+1}) - h_t(x_{t+1})} \\
& +\sum_{t=1}^T \EE{h_{t}(x_{t+1})-h_t(\widetilde x_{t+1})}\\
&\leq H \sum_{t=1}^T \EE{\one{A_t}}\\
&+\sum_{t=1}^T \EE{h_{t}(x_{t+1})-h_t(\widetilde x_{t+1})}\;.
\end{align*}
\end{proof}
\section{Proof of lemma \ref{lemma:delta1}}
\begin{proof}
By Cauchy-Schwarz inequality and Lipschitz dynamics assumption,
\begin{align*}
\Delta_t &\le \norm{P(.|x_t,a_t,\theta_*) - P(.|x_t,a_t,\TTh_t)}_1 \norm{h_t}_\infty \\
&\le C H \abs{\theta_{*} - \TTh_{t}} \;.
\end{align*}
Recall that $\TTh_t = \TTh_{\tau_t}$. Let $T_j$ be the length of episode $j$. Because we have $m$ episodes, we can write
\begin{align*}
\sum_{t=1}^T \Delta_t &\le \sqrt{T \sum_{t=1}^T \Delta_t^2}\\
&= C H \sqrt{ T \sum_{j=1}^{m} \sum_{s=1}^{T_j} \abs{\theta_{*} - \TTh_{j}}^2 } \\
&= C H \sqrt{ T \sum_{j=1}^{m} M_{j} \abs{\theta_{*} - \TTh_{j}}^2 } \,,
\end{align*}
where $M_{j}$ is the number of steps in the $j$th episode. Thus
\begin{align*}
\EE{\sum_{t=1}^T \Delta_t} &\le C H \EE{\sqrt{ T \sum_{j=1}^{m} M_{j} \abs{\theta_{*} - \TTh_{j}}^2 }} \\
&\le C H \sqrt{ T \EE{ \sum_{j=1}^{m} M_{j} \abs{\theta_{*} - \TTh_{j}}^2 }} \;.
\end{align*}
\end{proof}
\section{Proof of lemma \ref{lemma:delta2}}
\begin{proof}
Let $S = \EE{ \sum_{j=1}^{m} M_{j} \abs{\theta_{*} - \TTh_{j}}^2 }$.
Let $N_{j}$ be one plus the number of steps in the first $j$ episodes. So $N_{j} = N_{j-1} + M_{j}$ and $N_{0}=1$. We write
\begin{align*}
S &= \EE{ \sum_{j=1}^{m} N_{j-1} \abs{\theta_{*} - \TTh_{j}}^2 \frac{M_{j}}{N_{j-1}} } \\
& \stackrel{(a)}{\le} 2 \EE{ \sum_{j=1}^{m} N_{j-1} \abs{\theta_{*} - \TTh_{j}}^2 } \\
& \stackrel{(b)}{\le} 2 \log T \max_j \EE{N_{j-1} \abs{\theta_{*} - \TTh_{j}}^2 } \\
& \stackrel{(c)}{\le} 2 C' \log^2 T \;,
\end{align*}
where (a) follows from the fact that $M_{j}/N_{j-1}\le 2$ for all $j$,
(b) follows from
\[
\EE{ \sum_{j=1}^{m} N_{j-1} \abs{\theta_{*} - \TTh_{j}}^2 } \leq m \max_j \EE{N_{j-1} \abs{\theta_{*} - \TTh_{j}}^2 }
\]
and $m \le \log T$, and (c) follows from Assumption~\ref{ass:concentrating}.
\end{proof}
\section{Proof of lemma \ref{lemma:poi-lipschitz}}
\label{proof:poi-lipschitz}
\begin{proof}
To simplify the expositions, we use $p$ to denote $P(s=a|X)$ in this proof.
Notice that $z(\theta)=\frac{1-p} {1-p ^{1/ \theta}}$. Based on the definition of $\| \cdot\|_1$, we have
\small
\begin{align}
& \, \| P(\cdot|X,a,\theta) - P(\cdot|X,a,\theta') \|_1 \nonumber \\
= & \, \left| p^{\frac{1}{\theta}} - p^{\frac{1}{\theta'}} \right| +\sum_{s \neq a} \left | \frac{P(s | X)}{z(\theta)} - \frac{P(s | X)}{z(\theta')} \right | \nonumber \\
=& \, \left| p^{\frac{1}{\theta}} - p^{\frac{1}{\theta'}} \right| +\left | \frac{1-p ^{1/ \theta}}{1-p} - \frac{1-p ^{1/ \theta'}}{1-p}\right | \sum_{s \neq a} P(s | X) \nonumber \\
=& \, \left| p^{\frac{1}{\theta}} - p^{\frac{1}{\theta'}} \right| +\left | \frac{1-p ^{1/ \theta}}{1-p} - \frac{1-p ^{1/ \theta'}}{1-p}\right | (1-p) \nonumber \\
= &\, 2 \left| p^{\frac{1}{\theta}} - p^{\frac{1}{\theta'}} \right| .
\end{align}
\normalsize
We also define $h(\theta, p) \stackrel{\Delta}{=} p^{\frac{1}{\theta}}$. Based on calculus, we have
\small
\begin{align}
\frac{\partial h}{\partial \theta} (\theta, p) =& \, p^{\frac{1}{\theta}} \log \left(\frac{1}{p} \right) \frac{1}{\theta^2} \nonumber \\
\frac{\partial^2 h}{\partial \theta \partial p} (\theta, p) =& \, \frac{1}{\theta^2} p^{\frac{1}{\theta}-1} \left[\frac{1}{\theta} \log \left( \frac{1}{p} \right)-1 \right].
\end{align}
\normalsize
The first equation implies that $h$ is strictly increasing in $\theta$, and the second equation implies that
for all $\theta>0$, $\frac{\partial h}{\partial \theta} (\theta, p)$ is maximized by setting
$p=\exp(-\theta)$. This implies that for all $\theta>0$, we have
\[ 0<\frac{\partial h}{\partial \theta} (\theta, p) \leq \frac{\partial h}{\partial \theta} (\theta, \exp(-\theta)) = \frac{1}{e \theta}.\]
Hence, for all $\theta \geq 1$, we have
$0<\frac{\partial h}{\partial \theta} (\theta, p) \leq \frac{1}{e \theta} \leq \frac{1}{e}$. Consequently, $h(\theta, p)$ as a function of $\theta$ is
globally $\left( \frac{1}{e} \right)$-Lipschitz continuous for $\theta \geq 1$. So we have
\small
\[
\| P(\cdot|X,a,\theta) - P(\cdot|X,a,\theta') \|_1 = 2 \left| p^{\frac{1}{\theta}} - p^{\frac{1}{\theta'}} \right| \leq \frac{2}{e} |\theta -\theta'|.
\]
\normalsize
\end{proof}
\section{Posterior Concentration for POI Recommendation}
\label{appendix_concentration}
Recall that the parameter space $\Theta = \left \{ \theta_1, \ldots, \theta_K \right \}$ is a finite set, and
$\theta_*$ is the true parameter.
Notice that if $P(s_t=a_t | X_t) $ is close to $0$ or $1$, then the DS-PSRL will not learn much about
$\theta_*$ at time $t$, since in such cases $P(s_t | X_t, a_t, \theta)$'s are roughly the same for all $\theta \in \Theta$.
Hence, to derive the concentration result, we make the following simplifying assumption:
\[
\Delta_P \leq P(s | X) \leq 1-\Delta_P \quad \forall (X,s)
\]
for some $\Delta_P \in (0, 0.5)$. Moreover, we assume that all the elements in $\Theta$ are distinct, and define
\[
\Delta_{\theta} \stackrel{\Delta}{=} \min_{\theta \in \Theta, \theta \neq \theta_*} |\theta - \theta_*|
\]
as the minimum gap between $\theta_*$ and another $\theta \neq \theta_*$.
To simplify the exposition, we also define
\begin{align}
B \stackrel{\Delta}{=} &\, 2 \max \left \{
\max_{\theta \in \Theta} \max_{p \in [\Delta_P, 1-\Delta_P] } \left | \log \left( \frac{p^{1/\theta}}{p^{1/\theta_*}} \right) \right| , \right. \nonumber \\
& \left.
\max_{\theta \in \Theta} \max_{p \in [\Delta_P, 1-\Delta_P] } \left | \log \left( \frac{1- p^{1/\theta}}{1- p^{1/\theta_*}} \right) \right|
\right \} \nonumber \\
c_0 \stackrel{\Delta}{=} & \, \frac{ \min \left \{ \ln \left(\frac{1}{\Delta_P}\right) \Delta_P, \ln \left( \frac{1}{1-\Delta_P} \right) (1-\Delta_P) \right \}}{(\max_{\theta \in \Theta} \theta)^2}
\nonumber \\
\kappa \stackrel{\Delta}{=} & \, \left( \max_{\theta \in \Theta} \theta - \min_{\theta \in \Theta} \theta \right)^2. \nonumber
\end{align}
Then we have the following lemma about the concentrating posterior of this problem:
\begin{lemma}
(Concentration)
\label{lemma:poi-concentration}
Assume that $\theta_t$ is sampled from $P_t$ at time step $t$,
then
under the above assumptions, for any $t>2$, we have
\begin{align*}
\EE{(\theta_t -\theta_*)^2 } & \leq \frac{3}{e c_0^2 t} \frac{1-P_0(\theta_*)}{P_0(\theta_*)} \times \\
&\exp \left \{
- c_0^2 \Delta_\theta^2 t +
\sqrt{2 B^2 t \ln \left( K \kappa t^2 \right)}
\right \} + \frac{1}{t^2},
\end{align*}
where $B$, $c_0$, and $\kappa$ are constants defined above.
Note that they only depend on $\Delta_P$ and $\Theta$
\end{lemma}
Notice that Lemma~\ref{lemma:poi-concentration} implies that
\begin{align*}
t \EE{(\theta_t -\theta_*)^2 } \leq & O \left(
\exp \left \{
- c_0^2 \Delta_\theta^2 t +
\sqrt{2 B^2 t \ln \left( K \kappa t^2 \right)}
\right \} \right ) + \frac{1}{t} = O(1)
\end{align*}
for any $t>2$.
This directly implies that
$ \max_j \EE{ N_{j-1} \abs{\theta_{*} - \TTh_{j}}^2 } =O(1)$. Q.E.D.
\subsection{Proof of lemma \ref{lemma:poi-concentration}}
\begin{proof}
We use $P_0$ to denote the prior over $\theta$, and use $P_t$ to denote the posterior distribution over $\theta$ at the end of time $t$.
Note that by Bayes rule, we have
\small
\[
P_t(\theta) \propto P_0(\theta) \prod_{\tau=1}^{t} P(s_\tau | X_\tau, a_\tau, \theta) \quad \forall t \, \text{ and } \forall \theta \in \Theta .
\]
\normalsize
We also define the posterior log-likelihood of $\theta$ at time $t$ as
\small
\[
\Lambda_t (\theta) = \log \left \{ \frac{P_t(\theta)}{P_t(\theta_*)} \right \}=
\log \left \{ \frac{P_0(\theta)}{P_0(\theta_*)} \prod_{\tau=1}^t \left[ \frac{P(s_\tau | X_\tau, a_\tau, \theta)}{P(s_\tau | X_\tau, a_\tau, \theta_*)} \right] \right \}
\]
\normalsize
\noindent
for all $t$ and all $\theta \in \Theta$.
Notice that $P_t (\theta) \leq \exp \left[ \Lambda_t (\theta) \right]$ always holds, and $\Lambda_t (\theta_*)=0$ by definition.
We also define $p_t \stackrel{\Delta}{=} P(s_t=a_t | X_t) $ to simplify the exposition.
Note that by definition, we have
\small
\[
P(s_t | X_t, a_t, \theta) = \left \{
\begin{array}{ll}
p_t^{1/\theta} & \text{if $s_t=a_t$} \\
\frac{P(s_t | X_t)}{1-p_t}(1-p_t^{1/\theta}) & \text{otherwise}
\end{array}
\right.
\]
\normalsize
Define the indicator $z_t = \mathbf{1} \left \{ s_t = a_t \right \}$, then we have
\small
\[
\log \left \{ \frac{P(s_t | X_t, a_t, \theta)}{P(s_t | X_t, a_t, \theta_*)} \right \} = z_t \log \left[ \frac{p_t^{1/\theta}}{p_t^{1/\theta_*}}\right] +
(1-z_t) \log \left[ \frac{1-p_t^{1/\theta}}{1-p_t^{1/\theta_*}}\right]
\]
\normalsize
Since $p_t$ is $\cF_{t-1}$-adaptive, we have
\small
\begin{align}
& \, \EE{\log \left \{ \frac{P(s_t | X_t, a_t, \theta)}{P(s_t | X_t, a_t, \theta_*)} \right \} \middle | \cF_{t-1} , \theta_* } \nonumber \\
=& \,
p_t^{1/\theta_*} \log \left[ \frac{p_t^{1/\theta}}{p_t^{1/\theta_*}}\right] +
(1- p_t^{1/\theta_*}) \log \left[ \frac{1-p_t^{1/\theta}}{1-p_t^{1/\theta_*}}\right] \nonumber \\
=& \, - \mathrm{D_{KL}} \left( p_t^{1/\theta_*} \| p_t^{1/\theta} \right)
\leq \, - 2 \left( p_t^{1/\theta_*} - p_t^{1/\theta} \right)^2 , \nonumber
\end{align}
\normalsize
where the last inequality follows from Pinsker's inequality. Notice that function $h(x)=p_t^x$ is a strictly convex function of $x$, and
$\frac{d h}{dx} (x)= p_t^x \ln(p_t)$, we have
\small
\[
p_t^{1/\theta}-p_t^{1/\theta_*} \geq \ln(p_t) p_t^{1/\theta_*} (1/\theta - 1/\theta_*)=\ln(1/p_t) p_t^{1/\theta_*} \frac{(\theta-\theta_*)}{\theta \theta_*}
\]
\normalsize
Similarly, we have
$p_t^{1/\theta_*}-p_t^{1/\theta} \geq \ln(1/p_t) p_t^{1/\theta} \frac{(\theta_*-\theta)}{\theta \theta_*}$. Consequently, we have
\small
\begin{align}
\left | p_t^{1/\theta}-p_t^{1/\theta_*} \right | \geq & \, \ln(1/p_t) \min \left \{ p_t^{1/\theta_*} , p_t^{1/\theta} \right \} \frac{|\theta-\theta_*|}{\theta \theta_*}
\nonumber \\
\geq & \, \ln(1/p_t) p_t \frac{|\theta-\theta_*|}{\theta \theta_*}, \nonumber
\end{align}
\normalsize
where the last inequality follows from the fact $\theta, \theta_* \in [1, \infty)$. Since function $\ln(1/x)x$ is concave on $[0,1]$ and
$p_t \in [\Delta_P, 1-\Delta_P]$, we have $ \ln(1/p_t) p_t \geq \min \left \{ \ln(1/\Delta_P) \Delta_P, \ln(1/(1-\Delta_P)) (1-\Delta_P) \right \}$. Define
\small
\begin{equation}
c_0 \stackrel{\Delta}{=} \frac{\min \left \{ \ln \left(1/\Delta_P \right) \Delta_P, \ln \left( 1/(1-\Delta_P) \right) (1-\Delta_P) \right \} }{(\max_{\theta \in \Theta} \theta)^2},
\end{equation}
\normalsize
then we have $\left | p_t^{1/\theta}-p_t^{1/\theta_*} \right | \geq c_0 |\theta-\theta_*|$. Hence we have
\small
\[ - \mathrm{D_{KL}} \left( p_t^{1/\theta_*} \| p_t^{1/\theta} \right) \leq - 2 c_0^2 (\theta-\theta_*)^2. \]
\normalsize
Furthermore, we define
\small
\begin{align}
\xi_t(\theta) \stackrel{\Delta}{=}& \,
\log \left \{ \frac{P(s_t | X_t, a_t, \theta)}{P(s_t | X_t, a_t, \theta_*)} \right \} \nonumber \\
-& \,
\EE{\log \left \{ \frac{P(s_t | X_t, a_t, \theta)}{P(s_t | X_t, a_t, \theta_*)} \right \} \middle | \cF_{t-1} , \theta_* }.
\end{align}
\normalsize
Obviously, by definition,
$
\EE{\xi_t(\theta) \middle | \cF_{t-1} , \theta_* }=0
$.
We also define
\small
\begin{align}
B \stackrel{\Delta}{=} & \, 2 \max \left \{
\max_{\theta \in \Theta} \max_{p \in [\Delta_P, 1-\Delta_P] } \left | \log \left( \frac{p^{1/\theta}}{p^{1/\theta_*}} \right) \right| , \right . \nonumber \\
& \left .
\max_{\theta \in \Theta} \max_{p \in [\Delta_P, 1-\Delta_P] } \left | \log \left( \frac{1- p^{1/\theta}}{1- p^{1/\theta_*}} \right) \right|
\right \},
\end{align}
\normalsize
then $ \left | \xi_t(\theta) \right | \leq B$ always holds. This allows us to use Azuma's inequality.
Specifically, for any $\theta \in \Theta$, any $t$, and any $\delta \in (0,1)$, we have
$
\sum_{\tau=1}^t \xi_\tau(\theta) \leq \sqrt{2 B^2 t \ln \left( K/\delta \right)}
$ with probability at least $1-\delta/K$. Taking a union bound over $\theta \in \Theta$, we have
\small
\begin{align}
\label{eqn:lemma6_conc1}
\sum_{\tau=1}^t \xi_\tau(\theta) \leq \sqrt{2 B^2 t \ln \left( K/\delta \right)} \quad \forall \theta \in \Theta
\end{align}
\normalsize
with probability at least $1-\delta$.
Consequently, we have
\small
\begin{align}
\Lambda_t (\theta) =& \, \log \left \{ \frac{P_0(\theta)}{P_0(\theta_*)} \right \} \nonumber \\
+& \, \sum_{\tau=1}^t \left \{
z_\tau \log \left[ \frac{p_\tau^{1/\theta}}{p_\tau^{1/\theta_*}}\right] + (1-z_\tau) \log \left[ \frac{1-p_\tau^{1/\theta}}{1-p_\tau^{1/\theta_*}}\right]
\right \} \nonumber \\
=& \, \log \left \{ \frac{P_0(\theta)}{P_0(\theta_*)} \right \} - \sum_{\tau=1}^t \mathrm{D_{KL}} \left( p_\tau^{1/\theta_*} \| p_\tau^{1/\theta} \right) +
\sum_{\tau=1}^t \xi_\tau(\theta) \nonumber \\
\leq & \, \log \left \{ \frac{P_0(\theta)}{P_0(\theta_*)} \right \} - 2 c_0^2 (\theta-\theta_*)^2 t +
\sum_{\tau=1}^t \xi_\tau(\theta)
\end{align}
\normalsize
Combining the above inequality with equation~\ref{eqn:lemma6_conc1}, we have
\small
\[
\Lambda_t (\theta) \leq \log \left \{ \frac{P_0(\theta)}{P_0(\theta_*)} \right \} - 2 c_0^2 (\theta-\theta_*)^2 t +
\sqrt{2 B^2 t \ln \left( K/\delta \right)} \quad \forall \theta \in \Theta
\]
\normalsize
with probability at least $1-\delta$. Hence, we have
\small
\begin{align}
\label{eqn:lemma6_conc2}
P_t (\theta) \leq & \, \exp \left[ \Lambda_t (\theta) \right] \\
\leq & \, \frac{P_0(\theta)}{P_0(\theta_*)} \exp \left \{
- 2 c_0^2 (\theta-\theta_*)^2 t +
\sqrt{2 B^2 t \ln \left( K/\delta \right)}
\right \} \nonumber
\end{align}
\normalsize
for all $\theta \in \Theta$
with probability at least $1-\delta$.
Thus, for any $\cF_{t-1}$ s.t. the above inequality holds, we have
\small
\begin{align}
& \EE{(\theta_t -\theta_*)^2 \middle | \cF_{t-1}, \theta_*} = \sum_{\theta \neq \theta_*} P_t (\theta) (\theta-\theta_*)^2 \nonumber \\
\leq & \sum_{\theta \neq \theta_*}
\frac{P_0(\theta)}{P_0(\theta_*)} \exp \left \{
- 2 c_0^2 (\theta-\theta_*)^2 (t-1) \right. \nonumber \\
+ & \,
\left.
\sqrt{2 B^2 (t-1) \ln \left( K/\delta \right)}
\right \} (\theta-\theta_*)^2
\end{align}
\normalsize
For $t>2$, we have
\small
\[
\exp \left \{
- c_0^2 (\theta-\theta_*)^2 (t-2)
\right \} (\theta-\theta_*)^2 \leq \frac{1}{e c_0^2 (t-2)} \leq \frac{3}{e c_0^2 t},
\]
\normalsize
where the last inequality follows from the fact that $t-2 \geq \frac{t}{3}$.
Hence we have
\small
\begin{align}
& \, \EE{(\theta_t -\theta_*)^2 \middle | \cF_{t-1}, \theta_*} \nonumber \\
\leq & \,
\frac{3}{e c_0^2 t} \sum_{\theta \neq \theta_*}
\frac{P_0(\theta)}{P_0(\theta_*)} \exp \left \{
- c_0^2 (\theta-\theta_*)^2 t +
\sqrt{2 B^2 t \ln \left( K/\delta \right)}
\right \} \nonumber \\
\leq & \,
\frac{3}{e c_0^2 t} \sum_{\theta \neq \theta_*}
\frac{P_0(\theta)}{P_0(\theta_*)} \exp \left \{
- c_0^2 \Delta_\theta^2 t +
\sqrt{2 B^2 t \ln \left( K/\delta \right)}
\right \} \nonumber \\
=& \, \frac{3}{e c_0^2 t} \frac{1-P_0(\theta_*)}{P_0(\theta_*)}
\exp \left \{
- c_0^2 \Delta_\theta^2 t +
\sqrt{2 B^2 t \ln \left( K/\delta \right)}
\right \}, \nonumber
\end{align}
\normalsize
where the second inequality follows from $(\theta-\theta_*)^2 \geq \Delta_\theta^2$.
For $\cF_{t-1}$ s.t. inequality~\ref{eqn:lemma6_conc2} does not hold, we use the naive bound
$$(\theta_t -\theta_*)^2 \leq \kappa \stackrel{\Delta}{=} \left( \max_{\theta \in \Theta} \theta - \min_{\theta \in \Theta} \theta \right)^2.$$
Since inequality~\ref{eqn:lemma6_conc2} holds with probability at least $1-\delta$, we have
\small
\begin{align}
& \, \EE{(\theta_t -\theta_*)^2 \middle | \theta_*} \\
\leq & \, \frac{3}{e c_0^2 t} \frac{1-P_0(\theta_*)}{P_0(\theta_*)}
\exp \left \{
- c_0^2 \Delta_\theta^2 t +
\sqrt{2 B^2 t \ln \left( K/\delta \right)}
\right \} + \delta \kappa. \nonumber
\end{align}
\normalsize
Finally, by choosing $\delta = \frac{1}{\kappa t^2}$ and taking an expectation over $\theta_*$, we have
\small
\begin{align}
& \, \EE{(\theta_t -\theta_*)^2 } \\
\leq & \, \frac{3}{e c_0^2 t} \frac{1-P_0(\theta_*)}{P_0(\theta_*)}
\exp \left \{
- c_0^2 \Delta_\theta^2 t +
\sqrt{2 B^2 t \ln \left( K \kappa t^2 \right)}
\right \} + \frac{1}{t^2}. \nonumber
\end{align}
\normalsize
\end{proof}
\section{The Proposed Algorithm: Deterministic Schedule PSRL}
\begin{figure}[h]
\begin{center}
\framebox{\parbox{8cm}{
\begin{algorithmic}
\STATE {\bf Inputs}: $P_1$, the prior distribution of $\theta_*$.
\STATE $L \leftarrow 1$.
\FOR{$t\gets 1,2,\dots$}
\IF{$t = L $}
\STATE Sample $\TTh_{t}\sim P_t$.
\STATE $L \leftarrow 2L$.
\ELSE
\STATE $\TTh_{t} \leftarrow \TTh_{t-1}$.
\ENDIF
\STATE Calculate near-optimal action $a_t \leftarrow \pi^*(x_t, \TTh_t)$.
\STATE Execute action $a_t$ and observe the new state $x_{t+1}$.
\STATE Update $P_t$ with $(x_t,a_t,x_{t+1})$ to obtain $P_{t+1}$.
\ENDFOR
\end{algorithmic}
}}
\end{center}
\caption{The DS-PSRL algorithm with deterministic schedule of policy updates.}
\label{alg:lazy}
\end{figure}
In this section, we propose a PSRL algorithm with a deterministic policy update schedule, shown in Figure~\ref{alg:lazy}. The algorithm changes the policy in an exponentially rare fashion; if the length of the current episode is $L$, the next episode would be $2L$. This switching policy ensures that the total number of switches is $O(\log T)$. We also note that, when sampling a new parameter $\widetilde \theta_t$, the algorithm finds the optimal policy assuming that the sampled parameter is the true parameter of the system. Any planning algorithm can be used to compute this optimal policy \citep{sutton1998introduction}. In our analysis, we assume that we have access to the exact optimal policy, although it can be shown that this computation need not be exact and a near optimal policy suffices (see \citep{Abbasi-Yadkori-Szepesvari-2015}).
To measure the performance of our algorithm we use Bayes regret $R_T$ defined in Equation~\ref{eqn:bayes_regret}. The slower the regret grows, the closer is the performance of the learner to that of an optimal policy. If the growth rate of $R_T$ is sublinear ($R_T = o(T))$, the average loss per time step will converge to the optimal average loss as $T$ gets large, and in this sense we can say that the algorithm is asymptotically optimal. Our main result shows that,
under certain conditions, the construction of such asymptotically optimal policies
can be reduced to efficiently sampling from the posterior of $\theta_*$ and
solving classical (non-Bayesian) optimal control problems.
First we state our assumptions. We assume that MDP is weakly communicating. This is a standard assumption and under this assumption, the optimal average loss satisfies the Bellman equation. Further, we assume that the dynamics are parametrized by a scalar parameter and satisfy a smoothness condition.
\begin{ass}[Lipschitz Dynamics]
\label{ass:lipschitz}
There exist a constant $C$ such that for any state $x$ and action $a$ and parameters $\theta,\theta'\in \Theta \subseteq \Re$,
\[
\norm{P(.|x,a,\theta) - P(.|x,a,\theta')}_1 \le C \abs{\theta-\theta'} \;.
\]
\end{ass}
We also make a concentrating posterior assumption, which states that the variance of the difference between the true parameter and the sampled parameter gets smaller as more samples are gathered.
\begin{ass}[Concentrating Posterior]
\label{ass:concentrating}
Let $N_{j}$ be one plus the number of steps in the first $j$ episodes.
Let $\theta_j$ be sampled from the posterior at the current episode $j$. Then there exists a constant $C'$ such that
\[
\max_{j} \EE{ N_{j-1} \abs{\theta_{*} - \TTh_{j}}^2 } \le C' \log T \;.
\]
\end{ass}
The \ref{ass:concentrating} assumption simply says the variance of posterior decreases given more data. In other words, we assume that the problem is learnable and not a degenerate case. \ref{ass:concentrating} was actually shown to hold for two general categories of problems, finite MDPs and linearly parametrized problems with Gaussian noise \cite{Abbasi-Yadkori-Szepesvari-2015}. In addition, in this paper we prove how this assumption is satisfied for a large class of practical problems, such as smoothly parametrized sequential recommendation systems in Section \ref{sec:poi}.
Now we are ready to state the main theorem. We show a sketch of the analysis in the next section. More details are in the appendix.
\begin{thm}
\label{thm:main}
Under Assumption~\ref{ass:lipschitz} and \ref{ass:concentrating},
the regret of the DS-PSRL algorithm is bounded as
\[
R_T = \widetilde{O}(C \sqrt{C' T}),
\]
where the $\widetilde{O}$ notation hides logarithmic factors.
\end{thm}
Notice that the regret bound in Theorem~\ref{thm:main} does not directly depend on $S$ or $A$.
Moreover, notice that the regret bound is smaller if the Lipschitz constant $C$ is smaller or the posterior concentrates faster (i.e. $C'$ is smaller).
\section{Experiments}
In this section we compare through simulations the performance of DS-PSRL algorithm with the latest PSRL algorithm called Thompson Sampling with dynamic episodes (TSDE) \cite{Ouyang2017}. We experimented with the RiverSwim environment \cite{STREHL20081309}, which was the domain used to show how TSDE outperforms all known existing algorithms in \cite{Ouyang2017}. The RiverSwim example models an agent swimming in a river who can choose to swim either left or right. The MDP consists of $K$ states arranged in a chain with the agent starting in the leftmost state ($s = 1$). If the agent decides to move left i.e with the river current then he is always successful but if he decides to move right he might `fail' with some probability. The reward function is given by: $r(s, a) = 5$ if $s = 1$, $a = \text{left}$; $r(s, a) = 10000$ if $s = K$, $a = \text{right}$; and $r(s, a) = 0$ otherwise.
\subsection{Scalar Parametrization}
For scalar parametrization a scalar value defines the transition dynamics of the whole MDP. We did two types of experiments, In the first experiment the transition dynamics (or fail probability) were the same for all states for a given scalar value. In the second experiment we allowed for a single scalar value to define different fail probabilities for different states. We assumed two probabilities of failure, a high probability $P_1$ and a low probability $P_2$. We assumed we have two scalar values $\{\theta_1, \theta_2\}$. We compared with an algorithm that switches every time-step, which we otherwise call t-mod-1, with TSDE and DS-PSRL algorithms. We assumed the true model of the world was $\theta_*=\theta_2$ and that the agent starts in the left-most state.
In the first experiment, $\theta_1$ sets $P_1$ to be the fail probability for all states and $\theta_2$ sets $P_2$ to be the fail probability for all states. For $\theta_1$ the optimal policy was to go left for the states closer to left and right for the states closer to right. For $\theta_2$ the optimal policy was to always go right. The results are shown in Figure \ref{fig:plot-exp1}, where all schedules are quickly learning to optimize the reward.
In the second experiment, $\theta_1$ sets $P_1$ to be the fail probability for all states. And $\theta_2$ sets $P_1$ for the first few states on the left-end, and $P_2$ for the remaining. The optimal policies were similar to the first experiment. However the transition dynamics are the same for states closer to the left-end, while the polices are contradicting.
For $\theta_1$ the optimal policy is to go left and for $\theta_2$ the optimal policy is to go right for states closer to the left-end. This leads to oscillating behavior when uncertainty about the true $\theta$ is high and policy switching is done frequently. The results are shown in Figure \ref{fig:plot-exp2-50} where t-mod-1 and TSDE underperform significantly. Nonetheless, when the policy is switched after multiple interactions, the agent is likely to end up in parts of the space where it becomes easy to identify the true model of the world. The second experiment is an example where multi-step exploration is necessary.
\begin{figure*}[ht!]
\begin{center}
%
\subfigure[]{%
\label{fig:plot-exp1}
\includegraphics[width=0.39\textwidth]{plot-reward-exp1-prm1-50}
}%
\subfigure[]{%
\label{fig:plot-exp2-50}
\includegraphics[width=0.39\textwidth]{plot-reward-exp2-prm1-50}
}%
\end{center}
\caption{%
When multi-step exploration is necessary DS-PSRL outperforms.}%
\label{fig:subfigures}
\end{figure*}
\subsection{Multiple Parameters}
Even though our theoretical analysis does not account for the case with multiple parameters, we tested empirically our algorithm with multiple parameters. We assumed a Dirichlet prior for every state and action pair. The initial parameters of the priors were set to one (uniform) for the non-zero transition probabilities of the RiverSwim problem and zero otherwise. Updating the posterior in this case is equivalent to updating the parameters after every transition. We did not compare with the t-mod-1 schedule, due to the computational cost of sampling and solving an MDP every time-step. Unlike the scalar case we cannot define a small finite number of values, for which we can pre-compute the MDP policies. The ground truth model used was $\theta_2$ from the second scalar experiment. Our results are shown in Figures \ref{fig:plot-exp2-43-15} and \ref{fig:plot-exp2-58-20}. DS-PSRL performs better than TSDE as we increase the number of parameters.
\begin{figure*}[ht!]
\begin{center}
%
\subfigure[]{%
\label{fig:plot-exp2-43-15}
\includegraphics[width=0.33\textwidth]{plot-reward-exp2-prm43-15}
}%
\subfigure[]{%
\label{fig:plot-exp2-58-20}
\includegraphics[width=0.33\textwidth]{plot-reward-exp2-prm58-20}
}%
\subfigure[The LQ problem.]{%
\label{fig:plot-LQ}
\includegraphics[width=0.33\textwidth]{plot-LQ}
}%
\end{center}
\caption{%
Multiple parameters (a,b) and continuous domain (c).}%
\label{fig:subfigures}
\end{figure*}
\subsection{Continuous Domains}
In a final experiment we tested the ability of DS-PSRL algorithm in continuous state and action domains. Specifically, we implemented the discrete infinite horizon linear quadratic (LQ) problem in \cite{Abbasi-Yadkori-Szepesvari-2015, pmlr-v19-abbasi-yadkori11a}:
$$x_{t+1} = A_*x_t + B_*u_t + w_{t+1} \text{ and } c_t = x^T_t Qx_t + u^T_t Ru_t,$$
where $t=0,1,...,u_t \in R^d$ is the control at time $t$, $x_t \in R^n$ is the state at time $t$, $c_t \in R$ is the cost at time $t$, $w_{t+1}$ is the `noise', $A_* \in R^{n \times n}$ and $B_* \in R^{n \times d}$ are unknown matrices while $Q \in R^{n \times n}$ and $R \in R^{d \times d}$ are known (positive definite) matrices. The problem is to design a controller based on past observations to minimize the average expected cost. Uncertainty is modeled as a multivariate normal distribution. In our experiment we set $n=2$ and $d=2$.
We compared DS-PSRL with t-mod-1 and a recent TSDE algorithm for learning-based control of unknown linear
systems with Thompson Sampling \cite{quyang-TSDE-LQ}. This version of TSDE uses two dynamic conditions. The first condition is the same as in the discrete case, which activates when episodes increase by one from the previous episode. The second condition activates when the determinant of the sample covariance matrix is less than half of the previous value. All algorithms learn quickly the optimal $A_*$ and $B_*$ as shown in Figure \ref{fig:plot-LQ}. The fact that switching every time-step works well indicates that this problem does not require multi-step exploration.
\section{Introduction}
Thompson sampling \citep{Thompson1933}, or posterior sampling for reinforcement learning (PSRL), is a conceptually simple approach to deal with unknown MDPs \citep{Strens:2000:BFR:645529.658114,Osband-Russo-VanRoy-2013}. PSRL begins with a prior distribution over the MDP model parameters (transitions and/or rewards) and typically works in episodes. At the start of each episode, an MDP model is sampled from the posterior belief and the agent follows the policy that is optimal for that sampled MDP until the end of the episode. The posterior is updated at the end of every episode based on the observed actions, states, and rewards. A special case of MDP under which PSRL has been recently extensively studied is MDP with state resetting, either explicitly or implicitly. Specifically, in \citep{Osband-Russo-VanRoy-2013,Osband-VanRoy-2014} the considered MDPs are assumed to have fixed-length episodes, and at the end of each episode the MDP's state is reset according to a fixed state distribution. In \citep{Gopalan-Mannor-2015}, there is an assumption that the environment is ergodic and that there exists a recurrent state under any policy. Both
approaches have developed variants of PSRL algorithms under their respective assumptions, as well as state-of-the-art regret bounds, Bayesian in \citep{Osband-Russo-VanRoy-2013,Osband-VanRoy-2014} and Frequentist in \citep{Gopalan-Mannor-2015}.
However, many real-world problems are of a continuing and non-resetting nature. These include sequential recommendations and other common examples found in controlled mechanical systems (e.g., control of manufacturing robots), and process optimization (e.g., controlling a queuing system), where `resets' are rare or unnatural. Many of these real world examples could easily be parametrized with a scalar parameter, where each value of the parameter could specify a complete model. These type of domains do not have the luxury of state resetting, and the agent needs to learn to act, without necessarily revisiting states. Extensions of the PSRL algorithms to general MDPs without state resetting has so far produced non-practical algorithms and in some cases buggy theoretical analysis. This is due to the difficulty of analyzing regret under policy switching schedules that depend on various dynamic statistics produced by the true underlying model (e.g., doubling the visitations of state and action pairs and uncertainty reduction of the parameters). Next we summarize the literature for this general case PSRL.
The earliest such general case was analyzed as Bayes regret in a `lazy' PSRL algorithm~\citep{Abbasi-Yadkori-Szepesvari-2015}. In this approach a new model is sampled, and a new policy is computed from it, every time the uncertainty over the underlying model is sufficiently reduced; however, the corresponding analysis was shown to contain a gap~\citep{Osband-VanRoy-2016}.
A recent general case PSRL algorithm with Bayes regret analysis was proposed in \citep{Ouyang2017}. At the beginning of each episode, the algorithm generates a sample from the posterior distribution over the unknown model parameters. It then follows the optimal stationary policy for the sampled model for the rest of the episode. The duration of each episode is dynamically determined by two stopping criteria. A new episode starts either when the length of the current episode exceeds the previous length by one, or when the number of visits to any state-action pair is doubled. They establish $ \widetilde{O}(HS\sqrt{AT})$ bounds on expected regret under a Bayesian setting, where $S$ and $A$ are the sizes of the state and action spaces, $T$ is time, and $H$ is the bound of the span, and $\widetilde{O}$ notation hides logarithmic factors. However, despite the state-of-the-art regret analysis, the algorithm is not well suited for large and continuous state and action spaces due to the requirement to count state and action visitations for all state-action pairs.
In another recent work \citep{Agrawal2017}, the authors present a general case PSRL algorithm that achieves near-optimal worst-case regret bounds when the underlying Markov decision process is communicating with a finite, though unknown, diameter. Their main result is a high probability regret upper bound of $\widetilde{O}(D\sqrt{SAT})$ for any communicating MDP with $S$ states, $A$ actions and diameter $D$, when $T \geq S^5A$. Despite the nice form of the regret bound, this algorithm suffers from similar practicality issues as the algorithm in \citep{Ouyang2017}. The epochs are computed based on doubling the visitations of state and action pairs, which implies tabular representations. In addition it employs a stricter assumption than previous work of a fully communicating MDP with some unknown diameter. Finally, in order for the bound to be true $T \geq S^5A$, which would be impractical for large scale problems.
Both of the above two recent state-of-the-art algorithms \citep{Ouyang2017,Agrawal2017}, do not use generalization, in that they learn separate parameters for each state-action pair. In such non-parametrized case, there are several other modern reinforcement learning algorithms, such as UCRL2 \citep{jaksch2010near}, REGAL \citep{bartlett2009regal}, and R-max \citep{brafman2002r}, which learn MDPs using the well-known `optimism under uncertainty' principle. In these approaches a confidence interval is maintained for each state-action pair, and observing a particular state transition and reward provides information for only that state and action. Such approaches are inefficient in cases where the whole structure of the MDP can be determined with a scalar parameter.
Despite the elegant regret bounds for the general case PSRL algorithms developed in \citep{Ouyang2017,Agrawal2017}, both of them focus on tabular reinforcement learning and hence are sample inefficient for many practical problems with exponentially large or even continuous state/action spaces. On the other hand, in many practical RL problems, the MDPs are parametrized in the sense that system dynamics and reward/loss functions are assumed to lie in a known parametrized low-dimensional manifold~\citep{Gopalan-Mannor-2015}. Such model parametrization (i.e. model generalization) allows researchers to develop sample efficient algorithms for large-scale RL problems. Our paper belongs to this line of research. Specifically, we propose a novel general case PSRL algorithm, referred to as DS-PSRL, that exploits model parametrization (generalization). We prove an $\widetilde{O}(\sqrt{T})$ Bayes regret bound for DS-PSRL, assuming we can model every MDP with a single smooth parameter.
DS-PSRL also has lower computational and space complexities than algorithms proposed in \citep{Ouyang2017,Agrawal2017}. In the case of \citep{Ouyang2017} the number of policy switches in the first $T$ steps is $K_T =O \left( \sqrt{2SAT log(T)} \right)$; on the other hand, DS-PSRL adopts a deterministic schedule and its number of policy switches is $K_T \leq \log(T)$. Since the major computational burden of PSRL algorithms is to solve a sampled MDP at each policy switch, DS-PSRL is computationally more efficient than the algorithm proposed in \citep{Ouyang2017}. As to the space complexity, both algorithms proposed in \citep{Ouyang2017,Agrawal2017} need to store counts of state and action visitations. In contrast, DS-PSRL uses a model independent schedule and as a result does not need to store such statistics.
In the rest of the paper we will describe the DS-PSRL algorithm, and derive a state-of-the-art Bayes regret analysis. We will demonstrate and compare our algorithm with state-of-the-art on standard problems from the literature. Finally, we will show how the assumptions of our algorithm satisfy a sensible parametrization for a large class of problems in sequential recommendations.
\section{Application to Sequential Recommendations}
\label{sec:poi}
With `sequential recommendations' we refer to the problem where a system recommends various `items' to a person over time to achieve a long-term objective. One example is a recommendation system at a website that recommends various offers. Another example is a tutorial recommendation system, where the sequence of tutorials is important in advancing the user from novice to expert over time. Finally, consider a points of interest recommendation (POI) system, where the system recommends various locations for a person to visit in a city, or attractions in a theme park. Personalized sequential recommendations are not sufficiently discussed in the literature and are practically non-existent in the industry. This is due to the increased difficulty in accurately modeling long-term user behaviors and non-myopic decision making. Part of the difficulty arises from the fact that there may not be a previous sequential recommendation system deployed for data collection, otherwise known as the cold start problem.
Fortunately, there is an abundance of sequential data in the real world. These data is usually `passive' in that they do not include past recommendations. A practical approach that learns from passive data was proposed in \cite{Theocharous:2017:IPI:3030024.3040983}. The idea is to first learn a model from passive data that predicts the next activity given the history of activities. This can be thought of as the `no-recommendation' or passive model. To create actions for recommending the various activities, the authors perturb the passive model. Each perturbed model increases the probability of following the recommendations, by a different amount. This leads to a set of models, each one with a different `propensity to listen'. In effect, they used the single `propensity to listen' parameter to turn a passive model into a set of active models. When there are multiple model one can use online algorithms, such as posterior sampling for Reinforcement learning (PSRL) to identify the best model for a new user \citep{Strens:2000:BFR:645529.658114,Osband-Russo-VanRoy-2013}. In fact, the algorithm used in \cite{Theocharous:2017:IPI:3030024.3040983} was a deterministic schedule PSRL algorithm. However, there was no theoretical analysis.
The perturbation function used was the following:
\begin{equation}
\label{eq:poi-dynamics}
P(s|X,a,\theta) =
\begin{cases}
P(s|X) ^{1/ \theta}, & \text{if } a = s\\
P(s|X)/z(\theta), & \text{otherwise}
\end{cases}
\end{equation}
where $s=$ is a POI, $X=(s_1, s_2 \dots s_t)$ a history of POIs, and $z(\theta)=\frac{\sum_{s \neq a} P(s|X)} {1-P(s=a|X) ^{1/ \theta}}$ is a normalizing factor. Here we show how this model satisfies both assumptions of our regret analysis.
\paragraph{Lipschitz Dynamics} We first prove that the dynamics are Lipschitz continuous:
\begin{lemma}
\label{lemma:poi-lipschitz}
(Lipschitz Continuity) Assume the dynamics are given by Equation \ref{eq:poi-dynamics}. Then for all
$\theta, \theta' \geq 1$ and all $X$ and $a$, we have
\[
\| P(\cdot|X,a,\theta) - P(\cdot|X,a,\theta') \|_1 \leq \frac{2}{e} |\theta -\theta'|.
\]
\end{lemma}
Please refer to Appendix~\ref{proof:poi-lipschitz} for the proof of this lemma.
\paragraph{Concentrating Posterior} As is detailed in Appendix~\ref{appendix_concentration} (see Lemma~\ref{lemma:poi-concentration}), we can also show that Assumption~\ref{ass:concentrating} holds in this POI recommendation example. Specifically, we can show that under mild technical conditions, we have
\[
\max_j \EE{ N_{j-1} \abs{\theta_{*} - \TTh_{j}}^2 } =O(1)
\]
\section{Problem Formulation}
We consider the reinforcement learning problem in a parametrized Markov decision process (MDP) $(\cX, \cA, \ell, P^{\theta_*} )$ where $\cX$ is the state space, $\cA$ is the action space, $\ell:\cX\times\cA\ra\real$ is the instantaneous loss function, and $P^{\theta_*}$ is an MDP transition model parametrized by $\theta_*$. We assume that the learner knows $\cX$, $\cA$, $\ell$, and the mapping from the parameter $\theta_*$ to the transition model $P^{\theta_*}$, but does not know $\theta_*$. Instead, the learner has a prior belief $P_0$ on $\theta_*$ at time $t=0$, before it starts to interact with the MDP. We also use $\Theta$ to denote the support of the prior belief $P_0$. Note that in this paper, we do not assume $\cX$ or $\cA$ to be finite; they can be infinite or even continuous.
For any time $t=1, 2, \ldots$, let $x_t\in\cX$ be the state at time $t$ and $a_t\in\cA$ be the action at time $t$. Our goal is to develop an algorithm (controller) that adaptively selects an action $a_t$ at every time step $t$ based on prior information and past observations to minimize the long-run Bayes average loss
\[
\EE{ \limsup_{n\ra\infty} \frac1n\sum_{t=1}^n \ell(x_t,a_t)} .
\]
Similarly as existing literature \citep{Osband-Russo-VanRoy-2013, Ouyang2017}, we measure the performance of such an algorithm using Bayes regret:
\begin{equation}
R_T = \EE{ \sum_{t=1}^T \left( \ell(x_t,a_t) - J^{\theta_*}_{\pi^*} \right)} , \label{eqn:bayes_regret}
\end{equation}
where $J^{\theta_*}_{\pi^*}$ is the average loss of running the optimal policy under the true model $\theta_*$.
Note that under the mild `weakly communicating' assumption, $J^{\theta_*}_{\pi^*}$ is independent of the initial state.
The Bayes regret analysis of PSRL relies on the key observation that at each stopping time $\tau$
the true MDP model $\theta_*$ and the sampled model $\TTh_\tau$ are identically distributed \citep{Ouyang2017}. This fact allows to relate quantities that depend on the true, but unknown, MDP $\theta_*$, to those of the sampled MDP $\TTh_\tau$ that is fully observed by the agent. This is formalized by the following Lemma \ref{lemma:psrl}.
\begin{lemma}
\label{lemma:psrl}
(Posterior Sampling \citep{Ouyang2017}).
Let $(\cF_s)_{s=1}^\infty$ be a filtration ($\cF_s$ can be thought of as the historic information until current time $s$) and let $\tau$ be an almost surely finite $\cF_s$-stopping time.
Then, for any
measurable function $g$,
\begin{equation}
\EE{g(\theta_*)|\cF_{\tau} } = \EE{g(\TTh_\tau)|\cF_{\tau}} \;.
\label{eq:psrl-lemma}
\end{equation}
Additionally, the above implies that $\EE{g(\theta_*)} = \EE{g(\TTh_\tau)}$ through the tower property.
\end{lemma}
\section{Summary and Conclusions}
We proposed a practical general case PSRL algorithm, called DS-PSRL with provable guarantees. The algorithm has similar regret to state-of-the-art. However, our result is more generally applicable to continuous state-action problems; when dynamics of the system is parametrized by a scalar, our regret is independent of the number of states. In addition, our algorithm is practical. The algorithm provides for generalization, and uses a deterministic policy switching schedule of logarithmic order, which is independent from the true model of the world. This leads to efficiency in sample, space and time complexities. We demonstrated empirically how the algorithm outperforms state-of-the-art PSRL algorithms. Finally, we showed how the assumptions satisfy a sensible parametrization for a large class of problems in sequential recommendations.
|
1,314,259,993,589 | arxiv | \section{Introduction}
The recent development of the Internet of Things (IoT) promises to connect billions of smart devices. The dynamic underground environment is a frontier of IoT, where wireless sensors are employed to monitor soil status for designing climate-smart agriculture systems, studying forest dynamics, and detecting underground pipeline leakage \cite{vuran2018internet}. Different from remote sensing or ground penetration radar, underground sensors have unprecedented advantages in providing accurate real-time in-situ sensing.
The main challenge of wireless underground communications resides in penetrating through the dense and lossy soil medium, due to which the communication range is very limited and the power consumption is huge. Wireless channels of electromagnetic (EM) waves \cite{dong2013environment} and magnetic induction (MI) \cite{guo2016increasing} have been extensively studied for underground communications. The carrier frequency of EM waves are several hundred MHz and, thus, the data rate is high, while the MI uses low frequency to create long wavelength to reduce the propagation loss, which results in low data rate. Although designing an efficient wireless network in lossy soil medium is already challenging, its dynamic change further increases the complexity.
Existing works consider a static underground environment, where dielectric parameters of soil do not change with time. In reality, due to precipitation, the water content of soil changes frequently. Since the relative permittivity of water is around 81, which is much larger than the dry soil's (around 2), the water content significantly affects the soil's dielectric parameters. The same effect can be observed for electric conductivity. Although the works in \cite{vuran2018internet,dong2013environment,guo2015text} consider various water content and conductivity of soil, there is no discussion on the continuous change of soil dielectric parameters. On one hand, the high conductivity of soil reduces the communication range, while the low conductivity of soil allows long-range communication. If we design wireless underground sensor networks by considering the worst case, i.e., the highest conductivity, the lifetime of networks can be very short due to the high power consumption. On the other hand, if the design considers the best case, i.e., lowest conductivity, the connectivity and packet loss can be dramatically increased when the precipitation is high, which increase the soil conductivity in a short period. Therefore, we need optimal communication policies for wireless sensors in dynamic underground environments, which can be adaptive to environmental changes.
Reinforcement learning (RL) is an effective approach to solve dynamic problems. The dynamic environment is modeled as Hidden Markov Models and optimal actions can be derived using Bellman's equation. In \cite{ku2015data}, RL is used for energy harvesting networks to optimal use of the harvested energy to communicate. In \cite{li2018reinforcement}, RL is used to schedule wireless power transfer for wireless sensors based on their battery level. In \cite{wang2018deep}, deep RL is used for dynamic multichannel access in wireless networks. Although not listed here, RL has shown its unprecedented advantages in dealing with dynamic changes in wireless communications.
In this paper, we propose a data-driven model to capture the dynamic change of dielectric parameters of soil, upon which we develop an adaptive model to efficiently use sensor's energy and reduce the number of packet loss. We use the underground soil dielectric data from Nevada Climate Change Portal \cite{dascalu2014overview}, which is collected by underground sensors at Snake Range West Montane in 2017, to develop a Markov Decision Process to capture the dynamic change. From the data, we notice that some measurements are missing, which not only shows the low reliability of the system but also motivates this work. We study the impact of the soil parameter change on wireless channels and derive states based on the change of channel path loss. Then, we use RL to obtain the optimal transmission policies at each state. We show that the delay and the packet loss are controllable and the system is more reliable and efficient than transmission policies considering static environments. To the best of our knowledge, this is the first paper that investigates the optimal transmission policies for wireless underground sensor networks considering the dynamic change of the soil medium.
The rest of this paper is organized as follows. In Section II, we introduce the system model, including the dynamic environment, channel, states, actions, and the reward model. After that, we derive the optimal policies by using reinforcement learning in Section III. In Section IV, we present the numerical simulations and insights on the optimal policies. Finally, this paper is concluded in Section V.
\section{System Model}
In this section, we present the system model to capture key factors that can affect the system performance. First, we introduce the communication protocol to support the developed optimal transmission policy.
\begin{figure}[t]
\centering
\includegraphics[width=0.25\textwidth]{fig/sys}
\vspace{-5pt}
\caption{Illustration of the proposed wireless underground sensor network considering dynamic environmental changes.}
\vspace{-10pt}
\label{fig:sys}
\end{figure}
\subsection{Communication Protocol}
In the underground sensor networks, there are multiple sensors buried in soil. In this paper, we focus on the performance of a single wireless sensor to show the improvement of the proposed adaptive transmission policy. We consider that the sensor $S_1$ is buried underground with depth $d_{ug}$. The soil's permittivity is $\epsilon_1$ and conductivity is $\sigma_1$. The sensor also has a packet queue, which can save packets when the channel status is not good. The number of packets in the queue is denoted by $Q_1$ and $0 \leq Q_1\leq N_q $. The data packet is generated periodically since the sensor samples the environmental parameters in a periodical way (e.g., every one or ten minutes). Also, the sensor directly communicates with a basestation (BS) without rerouting packets. Note that, $\epsilon_1$, $\sigma_1$, and $Q_i$ are time-dependent.
In this paper, we consider framed transmission. The sensing and transmission period is $T$ and it is divided into time slots. Since the communication takes much shorter time than the period $T$, e.g, less than 1 second compared with 10 minutes. In each time slot only one packet can be transmitted. The sensor is allocated $t_{max}$ time slots and it can decide the number of transmitted packets, which ranges from 0 (no transmission) to $t_{max}$ packets. However, due to the harsh channel and limited transmission power, not all the transmissions are successful. If the sensor does not receive an acknowledgment from BS, it saves the packet in its queue, which will be retransmitted in the next period. An illustration of the protocol is shown in Fig.~\ref{fig:sys}. Since the BS is above ground and it only sends limited amount of data to underground sensors, we focus on the uplink communication, through which sensors report sensed data to the BS.
\subsection{Effect of $\epsilon_1$ and $\sigma_1$}
The underground wireless channel is not static due to the dynamic change of underground soil conductivity and permittivity. According to \cite{dong2013environment}, considering the soil-air interface the uplink path loss between sensors and BS can be written as
\begin{align}
\label{equ:received_power}
-10\log\frac{{\tilde P}_r}{P_t}=-10\log\frac{G_tG_r}{L_{ug}(d_{ug})L_{ag}(d_{ag})L(R)},
\end{align}
where ${\tilde P}_r$ is the received power, $P_t$ is the transmission power and $L_{ug}(d_{ug})$, $L_{ag}(d_{ag})$, and $L(R)$ are the underground propagation loss, aboveground propagation loss, and reflection loss on the boundary, respectively, which are given in \cite{dong2013environment,vuran2010channel}. $L_{ug}(d_{ug})$ and $L(R)$ are functions of the propagation constant of soil, which is
\begin{align}
k_s= j2 \pi f \sqrt{\mu_0 \left(\epsilon_1-j\frac{\sigma_1} {2\pi f}\right)}
\end{align}
where $j=\sqrt{-1}$, $f$ is the carrier frequency, $\mu_0$ is the permeability, $\epsilon_s$ is the permittivity of the soil, and $\sigma_s$ is the conductivity of the soil. Although \eqref{equ:received_power} captures the dominant components of the received power, the noise is neglected. Here, we use a more comprehensive model and the received power can be updated as $P_r={\tilde P}_r+ \eta$, where $\eta$ is the noise power.
\begin{figure}[t]
\centering
\includegraphics[width=0.4\textwidth]{fig/snake_permittivity}
\vspace{-5pt}
\caption{Soil permittivity at Snake Range West Montane in 2017. Data are downloaded from Nevada Climate Change Portal \cite{dascalu2014overview}.}
\vspace{-10pt}
\label{fig:snake_permittivity}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=0.35\textwidth]{fig/pathloss}
\vspace{-5pt}
\caption{Dynamic path loss in 2017 (left) and zoom-in in July and August (right). $d=9.5$~cm, $f=300$~MHz, $d_{ag}=20$~m, $G_t=G_r=5$~dB.}
\vspace{-10pt}
\label{fig:pathloss}
\end{figure}
The received power is affected by $\epsilon_1$ and $\sigma_1$, which varies with time. In Fig. \ref{fig:snake_permittivity}, we show the soil permittivity measured every 10 minutes at Snake Range West Montane in 2017 with depth 9.5~cm. The data are collected from Nevada Climate Change Portal \cite{dascalu2014overview} which is based on a real-time underground sensing platform at multiple locations in the state of Nevada. The data show that $\epsilon_1$ changes significantly during a year, e.g., the value in summer is much larger than that in winter. Moreover, as shown in Fig.~\ref{fig:snake_permittivity}, the day-to-day change is also considerable. Although not shown here, the soil conductivity also demonstrates drastically change during the year and it also has a day-to-day changing pattern. Also, we notice the change of conductivity and permittivity are not correlated. Since the wireless channel between underground sensors and above ground BS is a function of $\epsilon_1$ and $\sigma_1$, we can expect a significant change of the channel.
In Fig.~\ref{fig:pathloss}, the change of path loss by considering dynamic $\epsilon_1$ and $\sigma_1$ using \eqref{equ:received_power} is shown. As we can see, the path loss can change as high as 25~dB and also there are small daily changes. Moreover, the path loss may increase significantly during a couple of days. If we consider a static environment, such a large change of path loss can be ignored, which is too optimistic. Next, we train a Hidden Markov Model to capture the dynamic change of path loss.
\subsection{State Model}
Our state model consists of two parts, namely, the dynamic wireless channel due to environment changes and the sensor's packet queue.
\subsubsection{Wireless Channel}
Instead of modeling the dynamic change of $\epsilon_1$ and $\sigma_1$ individually, we consider their effects jointly by looking at the wireless channel model in \eqref{equ:received_power}. By substituting the continuous time series $\epsilon_1$ and $\sigma_1$ into \eqref{equ:received_power}, we obtain the channel path loss. Then, we employ the Hidden Markov Model and Gaussian emission model \cite{ku2015data} to derive the states for Markov Decision Process. Assume that there are $N_g$ Gaussian distributions ${\mathcal {N}}_n(u_n, \Sigma_n),~n=1,2,\cdots, N_g$ and $N_g$ states. Each distribution is associated with a state. Each path loss measured at the sampling time is a variable generated from a state with the associated Gaussian distribution. In this way, a time-series of the path loss is captured by using the finite state Markov model and the parameter of Gaussian distributions can be obtained by using the Expectation-Maximization algorithm. Note that in Fig.~\ref{fig:snake_permittivity}, some of the measured $\epsilon_1$ are negative which is mainly due to the sensor's low accuracy when the temperature is low. To avoid misleading results, we consider all the negative $\epsilon_1$ as 1 which can reflect the physics better. Also, due to unreliable wireless transmission, some of data are missing. Although the portion is very small, this can affect the training result. We use linear filling to obtain substitutions for the missing data.
We consider that there are 15 states, which can reflect the change well. Although using a model with more states can obtain the optimal policy that can improve the system performance, it also increases the computation burden. In addition, we notice that simply train the states by using path loss values cannot fully capture the dynamics. For example, two samples may have the same path loss, but for the first one the next sample is increasing, while for the second one the next sample is decreasing. For the optimal transmission, the sensor should transmit more in the first case since the path loss is increasing, while for the second case the sensor should save packets if it has space since the path loss is decreasing and if it transmits in the future, rather than now, it can have better probability to successfully transmit packets. Thus, in the training we also consider the change of path loss. For each training input, it has the path loss, as well as the difference between the current path loss and the previous one. In this way, depends on the path loss is increasing or decreasing, the sensor may make different decisions.
In Fig.~\ref{fig:state_pathloss}, the mean value and change of path loss for each state is shown. As we can see from the figure, the change of path loss between two samples is small and most are around 0. The transition probability is given in Fig.~\ref{fig:transition_prob}. The probability of changing from one state to another is low and most of the times the path loss remains in its current state. The probability of state transition can be written as $P(c'|c)$, where $c$ is the current channel path loss and $c'$ is the next channel path loss.
\begin{figure}[t]
\centering
\includegraphics[width=0.4\textwidth]{fig/state_pathloss}
\vspace{-5pt}
\caption{Mean path loss and change of path loss for each state.}
\vspace{-10pt}
\label{fig:state_pathloss}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=0.4\textwidth]{fig/transition_prob}
\vspace{-5pt}
\caption{Transition probability of states.}
\vspace{-10pt}
\label{fig:transition_prob}
\end{figure}
\subsubsection{Packet Queue}
The sensor measures the information of interest every $T$ and generate a data packet. Traditional solution is transmitting the packet immediately after it is created if it is allowed to access the wireless channel. However, it is not always good for the sensor to transmit its data packet due to the harsh wireless channel.
In this paper, we consider the sensor's queue is first-in-first-out. When the packet is generated, it is able to make a decision to transmit that packet or save it in its queue based on its observation of the surrounding environment and the number of packets in its packet queue. If the sensor generates a new packet, but its queue is full, it has to drop the oldest packet. If it sends a packet but cannot receive an acknowledgment, it saves the packet in its queue and it will be retransmitted in next period $T$. Since we consider the sensor communicates directly with the BS, there is not relaying packets.
The queue can save up to $N_q$ packets. The unsuccessful transmission does not drop a packet since the packet is saved and will be retransmitted. Only if the queue is full and the oldest packet is dropped.
The state transition probability for the queue with $t_{max}$ scheduled transmissions can be written as
\begin{align}
\label{equ:queue}
&P_a(Q_1'=q_1-t_{max}+1+N_u | Q_1=q_1)=\\
&\begin{cases}
& \binom{t_{max}}{N_u}[1-(1-P_e)^{P_L}]^{N_u}(1-P_e)^{P_L(t_{max}-N_u)}\\
&~~~~~~~~~~~~~~~~~\text{ when } 0\leq p \leq t_{max}\\
& 0,~~~~~~~~~~~~~~~ \text{otherwise}
\end{cases}
\end{align}
where $P_L$ is the packet length, $N_u$ is the unsuccessfully transmitted packet number, $P_e$ is the bit-error-rate (BER), and $a$ is the transmission action that is taken at the current state.
\subsection{Wireless Communication Policies}
The action policy consists of choosing the modulation scheme and determining the transmitting packet number. The modulation schemes are MPSK. We consider the allocated time slot number for BPSK is $N_{pmax}=t_{max}$. Using the same time, this can accommodate $N_{pmax}=2t_{max}$ packets for QPSK and $N_{pmax}=3t_{max}$ for 8PSK. As a result, the allocated time interval is a constant, but the maximum number of transmitted packets are different, which depends on the modulation scheme.
The BER for MPSK modulation can be written as \cite{lu1999m}
\begin{align}
P_{e}\approx \frac{2}{\max (\log_2 M,2)}\sum_{i=1}^{\max (M/4,1)}Q\left(\sqrt{\frac{2P_t l_p}{\eta}}\sin\frac{(2i-1)\pi}{M}\right),
\end{align}
where $Q(x)=(1/\sqrt{2\pi})\int_{x}^{\infty} e^{-t^2/2}dt$ and $M$ can be 2, 4, or 8. Also, we consider the sensor transmission power is a constant.
\subsection{Rewards Model}
Wireless communication consumes a large portion of a sensor's energy. To efficiently use energy, a sensor prefers to transmit when the channel status is good, while to save the packet in its queue when the path loss is high. Our reward model consists of two key components, namely, the successfully transmitted packet and the unsuccessfully transmitted packet, both of which are scaled by the overall transmission energy. In general, the reward is defined as
\begin{align}
\label{equ:reward}
{\mathcal R}_a(s) = \frac{\left[N_t-\alpha_1 N_u-\alpha_2(q_1-N_t+1)\right]\log_2(M)}{t_{sym}P_{t}(N_t+N_u)}
\end{align}
where $N_t$ is the successfully transmitted packet number, $t_{sym}$ the symbol time, $\alpha_1$ is the coefficient to scale dropped packets number, and $\alpha_2$ is the coefficient to scale the packet number in the queue. If $N_t>N_u$, the sensor receives a positive reward. If $N_t = N_u$ or there is no packet being transmitted, the reward is 0. If $N_t<N_u$, the sensor transmits when BER is large and it receives negative rewards.
The $\alpha_1 N_u$ in \eqref{equ:reward} penalizes the unsuccessfully transmitted packets. In this way, if the channel path loss is high, the sensor can try to use low level modulation. Although the unsuccessfully transmitted packets have been counted in the denominator, the $\alpha_1 N_u$ provides more flexibility. If $\alpha_1$ is large, the sensor is more cautious to transmit a packet and more likely to save the packet in its queue.
By using $\alpha_2(q_1-N_t+1)$, the sensor tends to send more packets to reduce the number of packets in its queue. In other words, when the channel path loss is small, the more transmitted packets, the more rewards. This is also equivalent to control the delay, i.e., a long queue causes more packets being delayed.
\section{Optimal Policy using Reinforcement Learning}
Based on the developed state, action, and reward model, we provide the RL algorithm in this section. The Bellman's equation for the expected value function can be written as
\begin{align}
V_{\pi^{*}}(s)=\max _{a \in \mathcal{A}}\left({\mathcal R}_{a}(s)+\lambda \sum_{s' \in \mathcal{S}} P_{a}\left(s' | s\right) V_{\pi^{*}}\left(s'\right)\right)
\end{align}
where $V_{\pi^{*}}(s)$ is value function at state $s$ using the optimal policy $\pi^{*}$, $a \in \mathcal{A}$ is one of the possible actions, $s'$ is the next state, $P_{a}\left(s' | s\right)$ is the transition probability from $s$ to $s'$ using action $a$, $\mathcal{S}$ contains all the states, and $\lambda$ is a discount scalar. The transition probability can be expressed as
\begin{align}
P_{a}\left(s' | s\right) = P(c'|c)\cdot P_a(Q_1'|Q_1)
\end{align}
where $P(c'|c)$ is trained from the data and is shown in Fig. \ref{fig:transition_prob}, and $P(Q_1'|Q_1)$ is the queue transition probability, which is given in \eqref{equ:queue}. Then, the value iteration can be employed to obtain the optimal policy \cite{sutton2018reinforcement}, i.e.,
\begin{align}
&V_{k+1}^a(c,q_1) = {\mathcal R}_a(c,q_1)+\lambda \sum_{(c',q_1') \in \mathcal{S}}P_a (c',q_1'|c,q_1)V_k (c',q_1')\\
&V_{k+1}(c,q_1) = \max_{a \in \mathcal{A}} V_{k+1}^a(c,q_1)
\end{align}
where the subscript $k$ denotes the iteration round.
To gain more insights, we focus on the value function and the reward model to see how the modulation scheme, queue length, and the environmental change affect each other and we try to understand how the wireless sensor responds to the change.
Intuitively, there is a tradeoff between the delay and the number of dropped packets. If the queue is short, more packets are dropped which reduces the reward, while if the queue is long, the delay increases which also reduces the reward. However, this is not true since the performance is not only determined by the queue but also the channel. Next, we show that, when $q_1\leq N_{pmax}$, where $N_{pmax}$ is the maximum number of packets that can be transmitted in interval $T$, $V_{\pi^{*}}(s)$ is a constant, while when $q_1>N_{pmax}$, $V_{\pi^{*}}(s)$ is an monotonically decreasing function. Here, we assume the path loss is small which results in a small BER. Consider the values functions $V_{k+1}^a(c,q_1)$ and $V_{k+1}^{a'}(c,q_1+1)$, the latter has one more packet in its queue than the former and they have different actions, i.e., $a$ and $a'$. First, we assume the queue length is small and the channel does not change. To compare the values, we have
\begin{align}
&V_{k+1}^a(c,q_1)-V_{k+1}^{a'}(c,q_1+1)={\mathcal R}_a(c,q_1)-{\mathcal R}_{a'}(c,q_1+1)\nonumber \\
&\label{equ:value1}+\lambda P_a(c,q_1'|c,q_1)V_k(c,q_1')-\lambda P_{a'}(c,q_1^{\prime\prime}|c,q_1+1)V_k(c,q_1^{\prime\prime}).
\end{align}
Since we consider $q_1$ is small and the channel status is good, the sensor can empty its queue with probability 1. Thus, $q_1'=q_1^{\prime\prime}$ and $\lambda P_a(c,q_1'|c,q_1)V_k(c,q_1')-\lambda P_{a'}(c,q_1^{\prime\prime}|c,q_1+1)V_k(c,q_1^{\prime\prime})=0$. Also, since BER is small, $N_u=0$, and $q_1-N_t+1=0$, \eqref{equ:reward} can be simplified to ${\mathcal R}_a(c,q_1)={\mathcal R}_{a'}(c,q_1+1)=\log_2(M)/(t_{sym}P_t)$. Thus, when $q_1\leq N_{pmax}$, \eqref{equ:value1} is equivalent to 0 and the value functions are constants, which are not affected by the queue length.
When $q_1> N_{pmax}$, the sensor cannot transmit all the packets in its queue and there are some packets left after transmission. In this case, the longer the queue, the smaller the reward. In \eqref{equ:value1}, ${\mathcal R}_a(c,q_1)$ is larger than ${\mathcal R}_{a'}(c,q_1+1)$ since $q_1-N_t+1<q_1-N_t+2$. Then, we can apply the induction method. When $q_1'$ and $q_1^{\prime\prime}$ are smaller than $N_{pmax}$, $V_k(c,q_1')=V_k(c,q_1^{\prime\prime})$. Since ${\mathcal R}_a(c,q_1)$ is larger, $V_{k+1}^a(c,q_1)-V_{k+1}^{a'}(c,q_1+1)>0$. When, $q_1'=N_{pmax}$ and $q_1^{\prime\prime}=N_{pmax}+1$, $V_k(c,q_1')>V_k(c,q_1^{\prime\prime})$ which can be proved by using the conclusion when $q_1'$ and $q_1^{\prime\prime}$ are smaller than $N_{pmax}$. As a result, $V_{k+1}^a(c,q_1)-V_{k+1}^{a'}(c,q_1+1)>0$, which is still valid. Similarly, we can prove that it is valid when $q_1'$ and $q_1^{\prime\prime}$ both are larger than $N_{pmax}$. Here, we find that a long queue cannot guarantee a larger expected value, since it creates larger delay. When path loss is small, the long queue does not help and the optimal strategy is sense-then-transmit. If the packet number in the queue is larger than $N_{pmax}$, the expected value reduces.
When the channel path loss is high, the state transition probability is not 1 and $N_u$ can be larger than 0. In this case, when $q_1<N_{pmax}$ the expected value is not a constant since neither the rewards or value functions are equivalent. With more transmission errors, the long queue can provide more space to save packets to avoid unsuccessful transmissions with high path loss. Also, depends on the tolerance of delay, we can adjust $\alpha_2$ to provide more flexibility.
\section{Numerical Simulations}
In this section, we numerically evaluate the proposed solution. Here, we compare with two baseline models, the sense-then-transmit using BPSK modulation and 8PSK modulation. Note that, there is no queue for the baseline models since we consider the traditional wireless underground sensor communication model without considering dynamic environmental change. The simulation parameters are given in Table \ref{tab:parameter}.
\begin{table}[t]
\renewcommand{\arraystretch}{1.3}
\caption{Simulation Parameters}
\label{tab:parameter}
\centering
\begin{tabular}{c|c||c|c}
\hline
Symbol & Value &Symbol & Value\\
\hline
\hline
$\eta$ & -100~dBm& $N_{q}$ &150 \\
\hline
$t_{max}$ & 15 &$t_{sym}$&1/60000\\
\hline
$P_L$ & 1000 &$\alpha_2$&0.1\\
\hline
$\alpha_1$ & 1 &$\lambda$&0.1\\
\hline
\end{tabular}
\end{table}
First, we show how the expected rewards change at different states. The state is composed of queue length and channel state. The transmission power is 20 dBm to provide small BER. Note $N_{pmax}= 3t_{max}$ since the 8PSK is used when the signal-to-noise ratio (SNR) is high. As shown in Fig. \ref{fig:reward}, when the queue length is smaller than $N_{pmax}$, the expected reward is a constant, while when the queue length becomes larger, the expected reward decreases. This shows that when SNR is high, i.e., equivalently the path loss is small, there is no need to use long queues since the expected reward is no larger than that with one packet in the queue. This is consistent with the discussion in the previous section.
\begin{figure}[t]
\centering
\includegraphics[width=0.35\textwidth]{fig/reward}
\vspace{-5pt}
\caption{Expected reward at different states.}
\vspace{-10pt}
\label{fig:reward}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=0.35\textwidth]{fig/dropped}
\vspace{-5pt}
\caption{The number of dropped packets.}
\vspace{-10pt}
\label{fig:dropped}
\end{figure}
The number of dropped packets is shown in Fig.~\ref{fig:dropped}. The transmission power is increased from 0.001 W to 0.1 W. The sense-then-transmit using BPSK and 8PSK do not have queues. The 8PSK suffers from higher BER, and thus its dropped packet number is large. The BPSK is more reliable. However, there are some extreme scenarios where the path loss is very high. Even using the BPSK modulation, we cannot avoid dropping packets. We notice that the RL solution can significantly reduce the number of dropped packets, e.g., when the transmission power is slightly higher than 0.01 W, the dropped packets number becomes 0. The RL solution can observe the environment and save packets when the channel has high path loss. It waits until the channel becomes better to transmit the packets. Therefore, it enjoys smaller dropped packets number. One may argue that the benefits of using RL arises from the queue. However, even we provide queues to the baseline models, it requires a strategy to transmit the packets in the queue. Depending on the intelligence level of the transmission strategy, the performances can be drastically different. Currently, there is no existing research efforts on this topic and we only consider the simple baseline models.
\begin{figure}[t]
\centering
\includegraphics[width=0.35\textwidth]{fig/energy}
\vspace{-5pt}
\caption{Effective energy used to successfully transmit a packet.}
\vspace{-10pt}
\label{fig:energy}
\end{figure}
In Fig.~\ref{fig:energy}, we show the ratio of the overall energy used over the successfully transmitted packet number, i.e., $[{t_{sym}P_{t}(N_t+N_u)}]/N_t$. Note that, the smaller the ratio, the better the energy is used since a unit energy can successfully transmit more packets. From the figure, we can see that with low transmission power, RL solution has slightly higher ratio. Because the RL solution prefers to not transmit packets when transmission power is low and the channel is not good. If the transmission fails, it saves the packet and retransmit it. This increases the overall transmission number since the low SNR causes most of the transmissions failing. As the transmission power increases, the SNR becomes large and the RL solution can successfully transmit more packets. Moreover, since the SNR is large, the RL solution uses 8PSK modulation and therefore its performance converges to the 8PSK model.
\begin{figure}[t]
\centering
\includegraphics[width=0.35\textwidth]{fig/queue}
\vspace{-5pt}
\caption{Effect of queue length of the number of dropped packets.}
\vspace{-10pt}
\label{fig:queue}
\end{figure}
In Fig.~\ref{fig:queue}, we show the effect of $N_q$. The transmission power is 0.01 W. The maximum transmit packet number in an interval $T$ is $t_{max}=\lceil 0.1 N_q \rceil$. As we can see in the figure, when $N_q$ is small, the number of dropped packets is large, since the SNR is not high enough to avoid errors and the queue is not long enough to save packets. As the queue length becomes longer, the dropped packets number reduces and it is even smaller than the BPSK baseline model.
\begin{figure}[t]
\centering
\includegraphics[width=0.4\textwidth]{fig/queueL}
\vspace{-5pt}
\caption{The status of queue length.}
\vspace{-10pt}
\label{fig:queueL}
\end{figure}
In Fig.~\ref{fig:queueL}, we show an example of the packet number in the queue. The transmission power is 0.001 W and all other parameters are given in Table~\ref{tab:parameter}. The results suggest that when the path loss is high, the sensor tries to save packets and only transmit the minimum number of packet, i.e., one packet, to accommodate the new packet. Refer back to Fig.~\ref{fig:pathloss}, the path loss from March to June and from September to October are large and that is when the sensor's queue is almost full.
\section{Conclusion}
The dynamic change of dielectric parameters in underground environments poses significant challenges in designing reliable wireless sensor networks. The network performance can easily be deteriorated by the dynamic change. In this paper, we propose a data-driven model for adaptive wireless communication in underground using reinforcement learning. We provide a systematic model to capture the change of dielectric parameters and relate the change to wireless channels. We consider the sensor has a queue, which can be leveraged to avoid unsuccessful transmissions. The optimal communication policies are derived and evaluated. The results show that the proposed solution can significantly reduce the number of dropped packets with a controllable delay and reasonable energy consumption. Our future work will consider not only the underground environmental change but also the RF environmental change such as interference and noise, as well as the sensor's battery. Also, instead of using a model-based solution, we will investigate the model-free solution using Q-learning to make the system more adaptive.
\bibliographystyle{IEEEtran}
|
1,314,259,993,590 | arxiv | \section{Introduction}\label{S0}
In its simplest form the \emph{light bulb lemma} \cite{Ga} asserts that if a surface $R$ in the 4-manifold $M$ has a geometrically dual sphere $G$, then one can perform the \emph{crossing change} of Figure 1 (Figure 2.1 in \cite{Ga}) via an isotopy of $R$, provided there is a path $\alpha$ from $y$ to $z=R\cap G$ that is disjoint from the tube $B$. Recall that a geometrically dual sphere is an embedded sphere $G$ with trivial normal bundle that intersects $R$ once and transversely. This paper investigates what happens when such path $\alpha$ must cross $B$, i.e. is \emph{self-referential}. It leads to the discovery of homotopic, concordant but non isotopic discs with common geometrically dual spheres, thereby exhibiting new phenomena not seen for spheres in a large class of manifolds. It also leads to the discovery of knotted 3-balls in certain 4-manifolds.
\begin{figure}[ht]
\includegraphics[scale=0.80]{Figure1.eps}
\caption{}
\end{figure}
Perhaps the simplest example is shown in Figure 2. Here $V=S^2\times D^2\natural S^1\times B^3 := W\times [-1,1]$ where $W$ is a solid torus with an open 3-ball removed. Let $G$ denote the 2-sphere component of $\partial W_0$, where $W_0=W\times 0$. Let $D_0$ be a vertical disc in the $S^2 \times D^2$ factor and $P$ a round 2-sphere centered in $W_0$ that projects to a disc in $W_0$ disjoint from $D_0$. See Figure 2 a). Note that $D_0\cap W_0$ (resp. $P \cap W_0$) is an arc (resp. a circle). Let $D_1$ be obtained by tubing the disc $D_0$ to the 2-sphere $P$, such that the projection of $D_1$ to $W_0$ is as in Figure 2 b). Here $D_1\cap W_0$ is an arc and the shading indicates projections from the past and future to $W_0$. Note that $D_0$ and $D_1$ have the common geometrically dual sphere $G$. If we could apply the light bulb lemma to $D_1$ near where the tube links the sphere, then $D_1$ is isotopic to $D_0 \rel \partial$.
Here is the idea for showing that $D_0$ and $D_1$ are non isotopic $\rel \partial$. Let $I_0$ denote the arc $D_0\cap W_0$ oriented to point into $G$ and $\Emb(I,V; I_0)$ the space of proper arc embeddings based at $I_0$ that coincide with $I_0$ near $\partial I_0$. Then $D_0, D_1$ naturally correspond to loops $\alpha_0, \alpha_1$ in $\Emb(I,V; I_0)$ where $\alpha_0$ is the constant loop. Using methods from Dax \cite{Da} we will show that $\alpha_1$ is not homotopic to $\alpha_0$ in $\Emb(I,V; I_0)$ and hence $D_1$ is not isotopic to $D_0 \rel \partial$.
\setlength{\tabcolsep}{60pt}
\begin{figure}
\centering
\begin{tabular}{ c c }
$\includegraphics[width=3.5in]{Figure2.eps}$
\end{tabular}
\caption[(a) X; (b) Y]{\label{Figure2}
\begin{tabular}[t]{ @{} r @{\ } l @{}}
A Self-Referential Disc
\end{tabular}}
\end{figure}
\begin{remarks} \label{failure remarks} i) Let $M$ be a 4-manifold such that $\pi_1(M)$ has no 2-torsion. Theorem 1.2 \cite{Ga} shows that if two homotopic 2-spheres $A_0, A_1\subset M$ have a common geometrically dual sphere $G$ and coincide near $G$, then they are ambiently isotopic fixing a neighborhood of $G$ pointwise. Since the isotopy is supported in a disc in the domain, I initially thought that Theorem 1.2 proved that properly homotopic discs with geometrically dual spheres are properly isotopic. However, the proof of Theorem 1.2 uses that $A_0$ is a sphere as opposed to a disc in one crucial spot. See Remark \ref{key point}.
ii) On the other hand, there is nothing new when $G\subset S^2\times S^1\subset \partial M$, for filling this component with a $S^2 \times D^2$ reduces to the study of isotopy classes of spheres with geometrically dual spheres. That was solved for spheres in 4-manifolds $M$ such that $\pi_1(M)$ has no 2-torsion in \cite{Ga} and in general 4-manifolds by Schneiderman and Teichner \cite{ST}.
iii) Hannah Schwartz \cite{Sch} showed that there exist manifolds with 2-torsion in their fundamental groups supporting homotopic spheres with a common geometric dual that are not isotopic, in fact not even concordant. Rob Schneiderman and Peter Teichner \cite{ST} identified the Freedman - Quinn (FQ) concordance invariant \cite{FQ} as the exact obstruction and showed that concordance implies isotopy.
iv) Note that $D_1$ is concordant to $D_0$, thus their difference is not detected by the FQ invariant. A secondary obstruction to isotoping one sphere to another is the \emph{km} invariant of Stong \cite{St} which is only defined when FQ=0. See \cite{KM} for a modern exposition. The Stong invariant does not detect that $D_1$ is not isotopic to $D_0$. First, one can attempt to transform the isotopy problem for discs to one for spheres by attaching a 0-framed 2-handle to $V$ along $\partial D_0$ and extending $D_0$ and $D_1$ to spheres, but then these spheres become isotopic by \cite{Ga}. Secondly, $km=0$ when the spheres have a common geometrically dual sphere. \end{remarks}
We now define our obstruction generally and introduce the work of Dax before stating our main results.
\vskip 8pt
\begin{construction} (An obstruction to isotopy) \label{obstruction} Let $D_0$ be a properly embedded disc in the 4-manifold $M$. View $D_0$ as $I\times I$ with $I_0$ denoting $I\times 1/2$ and $ \mathcal{F}_0$ this product foliation. If $D$ is another properly embedded disc that coincides with $D_0$ near $\partial D_0$, then $D$ gives rise to a canonical element $[\phi_{D_0}(D)]\in \pi_1(\Emb(I,M;I_0))$, where $\Emb(I,M;I_0)$ is the space of smooth embeddings of $I$ based at $I_0$. To see this, view $ D=I\times I$ where this foliation $\mathcal{F}$ coincides with $\mathcal{F}_0$ near $\partial D_0$. Use $D_0$ to inform how to modify $\mathcal{F}$ to a loop $\phi_{D_0}(D)$ in $\Emb(I,M;I_0)$ based at $ I_0$. (See Definition \ref{novel} for more details.) Since $[\phi_{D_0}(D_0)] = [1_{I_0}]$, where $1_{I_0}$ is the constant map to $I_0$, and $\Diff(D^2\fix \partial)$ is connected \cite{Sm3}, the class $[\phi_{D_0}(D)] \in \pi_1(\Emb(I,M;I_0))$ is well defined and gives an obstruction to isotoping $D$ to $D_0 \rel \partial D_0$. \end{construction}
Let $f_0: N^n\to M^m$ be an embedding where $N$ and $M$ are closed manifolds. In 1972 Jean-Pierre Dax showed \cite{Da} that $\pi_k(\Maps(N,M), \Emb(N,M), f_0)$ is isomorphic to a certain bordism group when $2\le k\le 2m-3n-3$. While stated very abstractly, the case $N=I$ and $M$ a 4-manifold can be restated with a strikingly elegant formulation. This paper gives that reformulation a self contained exposition. See \S 3. Let $\pi_1^D(\Emb(I,M; I_0))$ denote the subgroup of $ \pi_1(\Emb(I,M; I_0))$ represented by loops that are inessential in $\Maps(I,M:I_0)$. The following result is a slightly stronger version of the restated Theorem A \cite{Da} p.345 for $N=I$ and $M$ a 4-manifold.
\begin{theorem} (Dax Isomorphism Theorem) \label{dax} Let $I_0$ be an oriented properly embedded $[0,1]$ in the oriented 4-manifold $M$. Then
i) There is a homomorphism $d_3:\pi_3(M, x_0)\to \mathbb Z[\pi_1(M)\setminus 1]$ with image $D(I_0)$, called the \emph{Dax kernal}.
ii) $\pi_1^D(Emb(I,M;I_0))$ is canonically isomorphic to $\mathbb Z[\pi_1(M)\setminus 1]/D(I_0)$ and generated by $\{\tau_g|g\neq 1, g\in \pi_1(M)\} $. \end{theorem}
\begin{remark} The $\tau_g$'s arise from a spinning construction of Ryan Budney. See Definition \ref{spin}.\end{remark}
Thus Construction \ref{obstruction} together with the Dax isomorphism theorem gives a concrete obstruction to isotoping one embedded disc to another $\rel \partial$.
\begin{corollary} Let $D_0$ be a properly embedded disc in the oriented 4-manifold and $\mathcal{D}$ be the isotopy classes of embedded discs homotopic $\rel \partial$ to $D_0$, then there is a canonical function $\phi_{D_0}: \mathcal{D} \to \mathbb Z[\pi_1(M)\setminus 1]/D(I_0)$ such that if $D$ is a embedded disc homotopic rel $\partial$ to $D_0$, then $\phi_{D_0}([D])\neq 0$ implies $D$ is not isotopic to $D_0\rel\partial$.\end{corollary}
Note that $\phi_{D_0}$ is a function of $D_0$.
\vskip 8pt
In the setting of properly embedded discs with a common dual sphere, the methods of \cite{Ga} show that $\phi_{D_0}$ is a homomorphism whose image contains a particular subgroup and also proves the converse when $\pi_1(M)=1$.
\begin{theorem} \label{main} Let $M$ be a compact 4-manifold and $D_0$ a properly embedded 2-disc with a geometrically dual sphere $G\subset \partial M$. Let $\mathcal{D}$ be the isotopy classes of embedded discs homotopic $\rel \partial$ to $D_0$.
i) If $\pi_1(M)=1$, then $\mathcal{D}=[D_0]$, i.e. if $D_0$ and $D_1$ are homotopic rel $\partial$, then they are isotopic $\rel \partial$.
ii) In general, $\mathcal{D}$ is an abelian group with zero element $[D_0]$. There is a homomorphism $\phi_{D_0}:\mathcal{D}\to \mathbb Z[\pi_1(M)\setminus 1]/D(I_0)\cong \pi_1^D(\Emb(I,M; I_0))$. It maps onto the subgroup generated by elements of the form $g+g^{-1}$ and $\hat \lambda$, where $\hat\lambda^2=1$. \end{theorem}
\begin{remarks} i) We shall see in \S 4 that for $M=S^2\times D^2\natural S^1\times B^3$ the Dax kernal is trivial and the disc $D_1$ of Figure 2 maps to $t+t^{-1}$, thus $D_0$ and $D_1$ are not isotopic rel $\partial$.
ii) $\mathcal{D}$ is a torsor when there is a dual sphere. Fixing the element $[D_0]$ turns it into a group with identity $[D_0]$. $ \mathbb Z[\pi_1(M)\setminus 1]$ acts on $\mathcal{D}$ by adding self-referential tubes and $\mathbb Z[T_2]$ acts on $\mathcal{D}$ by adding double tubes, where $T_2$ is the set of non trivial 2-torsion elements. See \S 4. \end{remarks}
As an application we show the existence of knotted 3-balls in 4-manifolds.
\begin{theorem}\label{knotted ball} If $V=S^2\times D^2\natural S^1\times B^3$ and $B_0= x_0\times B^3$, then there exists a properly embedded 3-ball $B_1\subset V$ such that $B_1$ is properly homotopic but not properly isotopic to $B_0$. See Figure 3.\end{theorem}
Here is the idea of the proof. An extension of Hannah Schwartz' Lemma 2.3 \cite{Sch} to discs implies that there is a diffeomorphism $\phi:V\to V$ fixing a neighborhood of $\partial V$ pointwise and homotopic to $\id$ rel $\partial$ such that $\phi(D_0)=D_1$. Let $B_0$ denote the 3-ball $x_0\times B^3$ in the $S^1\times B^3$ factor of $V$ and $B_1:=\phi(B_0)$. If $B_1$ is isotopic to $B_0$, then since $B_1$ is disjoint from $D_1$, $D_1$ can be isotoped into the $S^2\times D^2$ factor of $V$. Theorem 10.4 \cite{Ga} implies that $D_1$ is isotopic to $D_0$ rel $\partial$, a contradiction. $B_1$ is obtained from $B_0$ by embedded surgery as described in more detail in \S 5. See Figure 3. \vskip 10 pt
\setlength{\tabcolsep}{40pt}
\begin{figure}
\centering
\begin{tabular}{ c c }
$\includegraphics[width=5in]{Figure3,1.eps}$ \end{tabular}
\caption[(a) X; (b) Y]{\label{fig:Fig1}\begin{tabular}[t]{ @{} r @{\ } l @{}}
A Knotted 3-Ball
\end{tabular}}
\end{figure}
This paper is organized as follows. Basic definitions will be given in \S1. Section \S2 will describe to what extent the methods of \cite{Ga} extend to discs. In particular we will show that if $D_0$ and $D_1$ are homotopic and have a common dual sphere, then $D_1$ can be put into a \emph{self-referential form} with respect to $D_0$. This is the analogue of the normal form of \cite{Ga} except that in addition to double tubes, $D_1$ can have finitely many self-referential discs. Theorem \ref{main} i) will also be proved. The Dax isomorphism theorem \cite{Da} will be stated and proved in \S3.
A slightly sharper version of Theorem \ref{main} ii) will be proved in \S4. Applications to knotted 3-balls in 4-manifolds and further questions will be given in \S5.
\begin{acknowledgements} We thank Hannah Schwartz for helpful conversations and Ryan Budney for his comments and for teaching me about the modern theory of embedding spaces.
We thank the Max Planck Institute in Bonn and the Banff International Research Station for the opportunity to present the main results at workshops respectively in September and November 2019. Much of this paper was written while a member of the Institute for Advanced Study. \end{acknowledgements}
\section{Basic Definitions}
We say that $G$ is a \emph{dual sphere} for the properly embedded disc $D\subset M$ if $G\subset \partial M$ and $D$ intersects $G$ exactly once and transversely. It would be more proper to call such a $G$ a \emph{geometrically dual boundary} sphere to distinguish it from geometrically dual spheres intersecting $D$ at an interior point. A \emph{geometric dual sphere} is one with trivial normal bundle that intersects a given surface exactly once and transversely. Trivial normal bundle is automatic here since $G$ is an embedded homologically non trivial sphere in an orientable 3-manifold. Unless said otherwise all dual spheres for discs lie in the boundary of the 4-manifold.
If $S_0$ and $S_1$ are oriented surfaces, then we say that they are tubed \emph{coherently} if the tubing creates an oriented surface whose orientation agrees with that of $S_0$ and $S_1$.
This paper works in the smooth category. All manifolds are orientable.
\section{Self-Referential Form}
Let $D_0$ be a properly embedded disc with dual sphere $G\subset \partial M$. In this section we show that if $D_1$ is an embedded disc with $\partial D_0=\partial D_1$ and $D_1$ is homotopic $\rel \partial$ to $D_0$, then $D_1$ can be isotoped to a \emph{self-referential form}, i.e. $D_1$ looks like $D_0$ except for finitely many double tubes representing distinct non trivial 2-torsion elements of $\pi_1(M)$ and self-referential discs.
\setlength{\tabcolsep}{60pt}
\begin{figure}
\centering
\begin{tabular}{ c c }
$\includegraphics[width=3.5in]{FigureB,1.eps}$
\end{tabular}
\caption[(a) X; (b) Y]{\label{FigureB,1}
\begin{tabular}[t]{ @{} r @{\ } l @{}}
A Self-Referential Disc
\end{tabular}}
\end{figure}
\begin{definition} Let $S_0$ be a properly embedded oriented surface in the 4-manifold $M, B\subset \inte(M)$ an oriented embedded 3-ball with $B\cap S_0=\emptyset$ and $\partial B=P$. Let $\tau:[0,1]\to M$ be an embedded path from $\inte(S_0)$ to $P$ such that $\tau(0)=\tau\cap S_0, \tau(1)=\tau \cap P$ and $\inte(\tau)$ intersects $B$ exactly once and transversely. Let $S_1$ be obtained from $S_0$ by tubing $S_0$ to $P$ along $\tau$. We say that $S_1$ is obtained from $S_0$ by attaching a \emph{self-referential disc}. See Figure \ref{FigureB,1}.\end{definition}
\begin{remarks} i) The disc $D_1$ in Figure 2 is obtained by attaching a self-referential disc to the disc $D_0$.
ii) A priori to define the tubing, $\tau$ should be a framed embedded path as in Definition 5.4 \cite{Ga}. Up to isotopy supported in $N(\tau)$ there are four isotopy classes, exactly two of which are coherent with the orientations of $S_0$ and $P$. These two, as do the non coherent ones, differ by the non trivial element of $\pi_1(SO(3))$ on the $B^3$ normal fibers of $N(\tau)$ as one traverses $\tau$. Since $\tau$ attaches to a sphere, the two choices give isotopic $S_1$'s. Thus $S_1$ depends only on $\tau$ and coherence/noncoherence. Equivalently, we can fix the orientation of the sphere one way or the other and then insist that the attachment be coherent. \end{remarks}
\begin{definition} Now assume that $D_0\subset M$ is a properly embedded oriented disc with dual sphere $G$. Let $B\subset\inte(M)$ an oriented 3-ball with $\partial B=P$ and $B\cap D_0=\emptyset$. Let $\tau_0$ be an embedded arc from $\inte(D_0)$ to $\inte(B)$ intersecting $B\cup D_0$ only at its endpoints. Think of it as being very short and view $D_0\cup \tau_0\cup B$ as the basepoint for $\pi_1(M)$. Associated to $g\in \pi_1(M)$ and $\sigma\in \pm$ construct $D_1$ by attaching a self-referential disc as follows. Let $\tau_1 $ be a path from $B$ to $\inte(D_0)\setminus \tau_0$ such that $\tau_1(0)=\tau_0(1), \tau_1\cap(D_0\cup\tau_0\cup B)=\partial \tau_1$ and $\tau_1$ represents the class $g$. Use $\tau=\tau_0 * \tau_1$ to construct $ D_1$ where $\sigma$ determines whether or not the attachment is coherent. See Figure 4.
Given $\sigma_1 g_1, \cdots, \sigma_n g_n$ construct a disc $D_1$ by attaching $n$ self-referential discs to $D_0$ by starting with $n$ adjacent copies of $\tau_0 \cup B$ and then attaching $n$ self-referential discs as above. \end{definition}
\begin{remark} Since $D_0$ has a dual sphere the inclusion $M\setminus(D_0\cup \tau_0\cup B)\to M$ induces a $\pi_1$-isomorphism. Thus once $B$ and $\tau_0$ are chosen, if $D_1$ is obtained by attaching one self-referential disc, then $D_1$ is determined up to isotopy by $\sigma$ and $g$. In a similar manner, if $D_1$ is obtained by attaching n self-referential discs, then once the $n$ adjacent copies of $\tau_0 \cup B$ are chosen it is determined up to isotopy by $\sigma_1 g_1, \cdots, \sigma_n g_n$. \end{remark}
The statement of \emph{self-referential form} given in Defintion \ref{sr form} below is quite technical, so for now we give the following informal one. Starting with $D_0$ construct the normal form analogue of Definition 5.23 and Figure 5.10 \cite{Ga} and then attach self-referential discs to obtain $D_1$. The actual definition includes some constraints and keeps track of certain orientations. The following is the main result of this section.
\begin{theorem} \label{srf} Let $D_0, D_1$ be properly embedded discs in the 4-manifold $M$ that coincide near their boundaries and have a geometrically dual sphere $G\subset \partial M$. If $D_0$ and $D_1$ are homotopic $\rel \partial$, then $D_1$ can be isotoped rel $\partial$ to self-referential form with respect to $D_0$.\end{theorem}
Before embarking on the proof we recall the following result which is a rewording of Theorems 1.2 and 1.3 \cite{Ga}.
\begin{theorem} \label{light bulb} Let $M$ be a 4-manifold such that the embedded spheres $R_0$ and $R_1$ have a common geometrically dual sphere $G$ and coincide near $G$. If $R_1$ and $R_0$ are homotopic and $\pi_1(M)$ has no 2-torsion, then they are ambiently isotopic fixing $N(G)$ pointwise. In general $R_1$ can be ambiently isotoped fixing $N(G)$ pointwise to be in normal form with respect to $R_0$. \end{theorem}
\begin{remarks} \label{key point} i) As mentioned in the introduction, since the isotopy fixes $N(G)$ pointwise, I originally thought that this theorem is a result about properly homotopic discs with dual spheres, which seems to contradict the main result of this paper.
ii) The key point is this. In the proof of Theorem \ref{light bulb} the dual sphere is repeatedly used to enable various geometric operations. When $R_1$ is a sphere, $\partial N(G)=S^2\times S^1$. Therefore, if $z=R_1\cap G$, then through each point of $\partial N(z)\cap R_1$ there is a distinct dual sphere. On the other hand, when $D_1$ is a disc we assume that $G\subset \partial M$ and so $N(G)=G\times I$. Here there may only be an interval $[a,b]\subset \partial D_1$ with the property that for $\theta\in [a,b]$, $D_1$ has a distinct dual sphere through $\theta$. For example, consider the disc $D_1$ of Figure 2. For most of the proof of Theorem \ref{light bulb} an interval suffices, but near the end, at one crucial spot, we require the whole circle. See the second paragraph preceding Lemma 8.1 \cite{Ga} where it is stated ``We can further assume that $q_1\in\partial D_0$." Note that when $G\subset S^2\times S^1\subset \partial M$, each point of $\partial D_0$ sees its own dual sphere, so the proofs of \cite{Ga} and \cite{ST} apply to discs without modification.
iii) There is the temptation to push $G$ to $G'\subset\inte(M)$ and use $G'$ as a dual sphere; however, an argument along the lines of \cite{Ga} requires that $D_1$ be $G'$-inessential, a condition automatic for spheres but not for discs.\end{remarks}
\begin{definition} Parametrize $\partial D_0=\partial D_1$ by $[0,2\pi]/{\sim}$ and $N(G)\cap \partial M$ as $G\times [\pi/2,3\pi/2]$ so that $\partial D_0\cap (G\times \theta)= \theta$. Call $[\pi/2,3\pi/2]\subset \partial D_0$ the \emph{approach interval}. \end{definition}
The proof of Theorem \ref{light bulb} extends essentially directly to the proof of Theorem \ref{srf} until the third paragraph of \S 8. We now elaborate on this extension and then state a result that summarizes what survives for discs.
\vskip 8pt
\noindent\emph{Section 2}: The extension is direct. In particular, the light bulb lemma goes through unchanged.
\vskip 8pt
\noindent\emph{Section 3}: Not relevant.\vskip 8pt
\noindent\emph{Section 4}: Smale's theorem implies that embedded discs that are homotopic $\rel \partial$ are properly regularly homotopic $\rel \partial$. \vskip 8pt
\setlength{\tabcolsep}{60pt}
\begin{figure}
\centering
\begin{tabular}{ c c }
$\includegraphics[width=1in]{FigureB,2.eps}$
\end{tabular}
\caption[(a) X; (b) Y]{\label{FigureB,2}
\begin{tabular}[t]{ @{} r @{\ } l @{}}
Tubed Surface
\end{tabular}}
\end{figure}
\noindent\emph{Section 5}: 1) Definition of \emph{tubed surface}. Recall that a tubed surface $\mathcal{A}$ is the data for constructing an embedded surface in $M$. In the end of the proof of our Theorem \ref{srf} above the associated surface $A_1$ will be our $D_0$ and the realization $A$ will be our $D_1$. While stated for closed surfaces, the definition of a tubed surface applies to compact surfaces with boundary. For us, $A_0$ is a disc with $\partial A_0$ parametrized by $[0,2\pi]/{\sim}$ where $[\pi/2,3\pi/2]$ is the approach interval, $z_0=\pi\in \partial A_0$ and $f(z_0)=z=A_1\cap G$. In the closed surface setting we can assume that the $\sigma, \alpha, \beta, \gamma$ tube guide curves approach $z_0\in A_0$ radially. In the disc setting these curves approach $[\pi/2,3\pi/2]\subset\partial A_0$ transversely and intersect $N(\partial A_0)$ in distinct arcs. See Figure \ref{FigureB,2}. That figure shows $\partial A_0$ together with the tube guide curves in a small neighborhood of the approach interval, which is shown in green.
2) Construction of the realization $ A$. The construction is essentially the same. Here a tube guide curve $\kappa$ connecting to $\theta\in \partial A_0$ corresponds to a tube paralleling $f(\kappa)\subset A_1$ that connects to a parallel copy of $G\times \theta$ pushed slightly into $\inte(M)$.
3) Tube sliding moves. With one exception all the moves yield isotopic realizations as before. In the disc setting, the \emph{reordering move} between tube guide curves $\kappa_j, \kappa_k$ requires that the relevant component between their endpoints lies in the approach interval.
4) Finger and tube locus free Whitney moves. Same as before.
5) Theorem 5.21. The proof is the same as before, in particular reordering is not used.
6) Lemma 5.25. The proof holds since one can permute pairs $(\beta_i, \gamma_i), (\beta_j, \gamma_j)$ that are adjacent in the approach interval.
Summary: Except for a restricted reordering move, all the results of Section 5 directly hold.
\vskip 8pt
\noindent\emph{Section 6}: Direct analogues of all the results of this section hold for discs. Here are some additional remarks.
1) Lemma 6.1 holds tautologically since $D_0$ and $D_1$ are homtopic $\rel \partial$.
\begin{notation}\label{sign convention} Sign Convention: We continue to adopt the orientation convention on $\beta_i, \lambda_i$ and $\gamma_i$ as in that section. As in 6.3 \cite{Ga} the tube guide curve $\alpha$ corresponds to a sphere $P(\alpha)$ obtained by connecting oppositely oriented copies of $G$ by a tube that parallels $f(\alpha)$. Orient $\alpha $ so that the copy giving $-[G]$ (resp. $[G])$ is at the negative (resp. positive) end of $f(\alpha)$. \end{notation}
2) If $\pi:\tilde M\to M$ is the universal covering map, then the components of $\pi^{-1}(D_1\cup G)$ are in natural 1-1 correspondence with elements of $\pi_1(M, z)$ and the components of $\pi^{-1}(G)$ freely generate a $\mathbb Z[\pi_1(M)]$ submodule of $H_2(\tilde M)$, thus the algebra of \S 6 extends to the disc case.
3) In our context the associated surface $A_1$ in the statement of Proposition 6.9 is a disc. The proof is a direct translation.\vskip8pt
\noindent\emph{Section 7:} The statement and proof of the crossing change lemma hold as before.\vskip 8pt
\noindent\emph{Section 8}: The proof holds as before, until the second to last sentence of the third paragraph. That sentence ``We can further assume that $q_1\in \partial D_0$." requires that the approach interval is the whole circle. \vskip 8pt
\setlength{\tabcolsep}{60pt}
\begin{figure}
\centering
\begin{tabular}{ c c }
$\includegraphics[width=1.5in]{FigureB,3.eps}$
\end{tabular}
\caption[(a) X; (b) Y]{\label{FigureB,3}
\begin{tabular}[t]{ @{} r @{\ } l @{}}
Sector Form
\end{tabular}}
\end{figure}
Putting this all together we have the following result.
\begin{proposition} (Sector Form) \label{whats left} Let $D_0, D_1$ be properly embedded discs in the 4-manifold $M$ such that $D_0$ and $D_1$ coincide near their boundaries and have the dual sphere $G\subset \partial M$. Then there exists a tubed surface $\mathcal{A}$ with underlying surface $A_0$ parametrized as the unit disc in $\mathbb R^2$, with $f(A_0)=D_0$ and with realization $A$ isotopic $\rel \partial$ to $D_1$.
$\mathcal{A}$ has data $(\alpha_1,(p_1,q_1), \tau_1), \cdots, (\alpha_r,(p_r, q_r), \tau_r),(\beta_0, \gamma_0, \lambda_0), (\beta_1, \gamma_1,\lambda_1), \cdots, (\beta_n, \gamma_n, \lambda_n))$.
Each each of these data sets lie in distinct sectors of $A_0$. This means that there exists linearly ordered $a_0=\pi/2, a_1, \cdots, a_{r+n+1}=3\pi/2 \subset \partial A_0$ such that $(\alpha_i, (p_i,q_i))\subset$ the sector defined by $(a_{i-1},a_i, 0)$ and
$(\beta_j,\gamma_j)$ lies in the sector defined by $(a_{r+j}, a_{r+j+1},0)$ with $\beta_j\cap \gamma_j=\emptyset$. \emph{See Figure \ref{FigureB,3}}. \end{proposition}
\begin{lemma} \label{normal permutation} The data of the various sectors can be permuted without changing the isotopy class of the realization. \end{lemma}
\begin{proof} Using the tube sliding operations any two adjacent pairs $(\alpha_i,(p_i,q_i), \tau_i)$, $(\beta_j, \gamma_j, \lambda_j)$, i.e. two of one type or one of each type, in the approach interval can be permuted, but we cannot \emph{permute} data within a given sector, i.e. the $\beta_i$ and $\gamma_i$ curves. \end{proof}
\begin{definition} A tubed surface $\mathcal{A}$ with data as in Proposition \ref{whats left} is said to be in \emph{sector form}. Let $\mathcal{A}$ be a tubed surface in sector form. Let $\lambda$ be a framed embedded path in $M$ with disjoint embedded tube guide curves $\beta$ and $\gamma \subset A_0$, all oriented with the above sign convention. We denote the pair $(\beta,\gamma)$ as $+ (\beta,\gamma)$ (resp. $- (\beta,\gamma))$ if $\beta$ appears before (resp. after) $ \gamma$ in the approach interval. Call an embedded $\alpha $ curve $+$ (resp. $-$) if the negative (resp. positive) end of $\alpha$ appears before the positive (resp. negative) end in the approach interval. \end{definition}
\begin{definition}\label{sr form} We say that the tubed surface $\mathcal{A}$ is in \emph{self-referential form} with data $(\lambda_1, \lambda_2, \cdots, \lambda_n, \sigma_1 g_1, \cdots, \sigma_k g_k)$ if
a) The immersion $f:A_0 \to M$ is a proper embedding with $f(A_0)=A_1$ a 2-disc with dual sphere $G\subset \partial M$.
b) The paths $\beta_1, \gamma_1, \cdots, \beta_n, \gamma_n, \sigma_1\alpha_1, \cdots, \sigma_k\alpha_k$ are embedded and linearly arrayed along the approach interval, where $\sigma_i\in \pm$ and $+\alpha_i$ (resp. $-\alpha_i$) denotes that its negative (resp. positive) end is closer to $\pi/2$ than its positive end. The point $q_i$ associated to $\alpha_i$ lies in the half disc bounded by $\alpha_i$ and the approach interval.
c) The framed embedded paths $\lambda_1, \lambda_2, \cdots, \lambda_n$ represent distinct nontrivial 2-torsion elements of $\pi_1(M)$.
d) Each $g_i$ represents a non trivial element of $\pi_1(M, z_0)$ and no $i,j$ is $\sigma_i g_i=-\sigma_j g_j$.
We say that the disc $D_1$ is in \emph{self-referential form} with data $(\lambda_1, \lambda_2, \cdots, \lambda_n, \sigma_1 g_1, \cdots, \sigma_k g_k)$ with respect to the disc $D_0$ if $D_1$ is the realization of the tubed surface $ \mathcal{A} $ with this data where $A_1=D_0$. \end{definition}
We now show the key connection between the formal definition and the earlier one for self-referential form.
\begin{lemma} \label{alpha to srf} If $D_1$ is in self-referential form with respect to $D_0$ with data $(\lambda_1, \lambda_2, \cdots, \lambda_n,\newline \sigma_1 g_1, \cdots, \sigma_k g_k)$ and $D'_0$ is in self-referential form with respect to $D_0$ with data $(\lambda_1, \lambda_2, \cdots, \lambda_n)$, then $D_1$ is isotopic to the surface obtained from $D'_0$ by attaching the self-referential discs associated to the data $(\sigma_1 g_1, \cdots, \sigma_k g_k)$.\end{lemma}
\begin{proof} Since $q_1$ lies to the approach interval side of $\alpha_1$ sliding the sphere $P(\alpha_1)$ off of $D_0$ entangles the tube connecting $D_0$ to $P(\alpha_1)$ to create a self-referential disc of the type claimed. See Figures \ref{Figure4,3} to \ref{Figure4,5}. The result follows by induction on the number of $\alpha$ curves.\end{proof}
\setlength{\tabcolsep}{60pt}
\begin{figure}
\centering
\begin{tabular}{ c c }
$\includegraphics[width=5in]{FigureB,4.eps}$
\end{tabular}
\caption[(a) X; (b) Y]{\label{FigureB,4}
\begin{tabular}[t]{ @{} r @{\ } l @{}}
$D_g+D_{-g}=D_0$
\end{tabular}}
\end{figure}
\begin{lemma}\label{alpha cancellation} An embedded surface $T$ with dual sphere $G$ is isotopic to the surface $T'$ obtained from $T$ by tubing self-referential discs of type $g, -g$.\end{lemma}
\begin{proof} See Figure \ref{FigureB,4}. Figure \ref{FigureB,4} a) shows $T$ with self-referential discs of type $g, -g$. The green dot denotes intersection with a geometrically dual sphere, which is on $\partial T$, when $T$ is a disc. Two applications of the light bulb lemma enable the isotopy to Figure \ref{FigureB,4} b). Figure \ref{FigureB,4} c) is after sliding one of the tubes. Since the spheres now cancel, that surface is isotopic to $T$ itself.\end{proof}
\begin{definition} We say that the embedded surface $T$ is obtained from the embedded surface $S$ by \emph{tubing a sphere $P$ along $\tau$}, if $P$ bounds a 3-ball disjoint from $S$ and $T$ is obtained by tubing $S$ and $P$ along a framed embedded path $\tau$.\end{definition}
\begin{lemma} \label{stabilization} Let $S$ be an embedded surface with dual sphere $G$. If the surface $T$ is obtained from $S$ by tubing a sphere $P$ along $\tau$, then $T$ is isotopic to a surface obtained from $S$ by attaching finitely many self-referential discs.\end{lemma}
\begin{proof} If $P=\partial B$ and $|B\cap \tau|=k$, then squeeze $B$ into two balls $B_1, B_2$ so that $|\tau\cap B_1|=1, |\tau\cap B_2|=k-1$ and $(\partial \tau\cap B)\subset B_2\setminus B_1$. If $P_i=\partial B_i$, then we can further assume that $P_1$ is connected to $P_2$ by a tube $ \tau_1$ disjoint from $\tau$. Use $\tau$ to slide $\tau_1$ off of $P_2$ so that now $\tau_1$ connects $P_1$ with $S$. Here we abused notation by identifying the framed embedded path $\tau$ with its corresponding tube. By construction $\tau_1$ will link $P_1$ exactly once. Next, use the light bulb lemma to unlink $\tau_2$ from $P_1$ and $\tau_1$ from $P_2$. The result follows by induction on $k$. \end{proof}
\begin{lemma} \label{alpha stabilization} Let $\mathcal{A}$ be a tubed surface in sector form containing a sector $J$ with data $(\alpha_i,(p_i,q_i), \tau_i)$. There exists another tubed surface $\mathcal{A}'$ with isotopic realizations whose data agrees with that of $\mathcal{A}$ except that the $(\alpha_i,(p_i,q_i), \tau_i)$ data has been deleted and the sector $J$ has been subdivided into finitely many sectors each of which contains data of the form $(\sigma_s\alpha_s,(p_s,q_s), \tau_s))$ where $\alpha_s$ is embedded and $q_s$ lies in the halfdisc bounded by $\alpha_s$ and the approach interval.\end{lemma}
\begin{proof} By the crossing change Lemma 7.1 \cite{Ga} we can assume that $\alpha_i$ is monotonically increasing. Sliding $P(\alpha_i)$ off of $A_1$ as in the proof of Lemma \ref{alpha to srf} we obtain an unknotted 2-sphere $P_i$, which is entangled with $ \tau_i$. If $S$ denotes the realization of the tubed surface $\mathcal{A}$ with the data $(\alpha_i,(p_i,q_i), \tau_i)$ deleted, it follows that the realization $A$ of $\mathcal{A}$ is obtained by tubing $S$ to the sphere $P_i$. By Lemma \ref{stabilization} $A$ is isotopic to a surface obtained by adding self-referential discs to $S$. The proof of that lemma further shows that they can be attached in subsectors of $J$ without the self-referential discs linking with other parts of $A$. Finally, reverse the proof of Lemma \ref{alpha to srf} to obtain the desired $\mathcal{A}'$ satisfying all but possibly the last conclusion. If a $q_s$ lies outside the halfdisc bounded by $\alpha_s$ and the approach interval, then deleting the data $(\sigma_s\alpha_s,(p_s,q_s), \tau_s)$ does not change the isotopy class of the realization, \end{proof}
The next result follows from Lemmas \ref{alpha cancellation} and \ref{alpha stabilization}.
\begin{corollary} \label{alpha addition} Let $\mathcal{A}$ be a tubed surface in sector form. Given the data $(\alpha_s,(p_s,q_s), \tau_s)$ there exists a tubed surface $\mathcal{A}'$ in sector form with realization isotopic to that of $\mathcal{A}$ such that the data of $\mathcal{A}'$ consists of the data from the sectors of $\mathcal{A}$ plus another sector with data $(\alpha_s,(p_s,q_s), \tau_s)$ together with other sectors having data only involving $\alpha $ curves.\qed\end{corollary}
\noindent\emph{Proof of the Self-referential Form Theorem}. By Proposition \ref{whats left} we can assume that $\mathcal{A}$ is in sector form.
0) By Lemma \ref{normal permutation} the data of the various sectors can be permuted.
i) Elimination of the $(\beta_0, \gamma_0, \lambda_0)$ data can be done as in Remark 8.2 \cite{Ga}. This might create additional data of the form $(\alpha_s,(p_s,q_s), \tau_s)$.
ii) We can further assume that the $\lambda_i$'s represent distinct non trivial 2-torsion elements since the methods of \S 6 \cite{Ga} enable the exchange of a pair of double tubes representing the same 2-torsion element for a pair of single tubes. Again, this might create data of the form $(\alpha_s,(p_s,q_s), \tau_s)$.
iii) The modification of the $\beta_i, \gamma_i$ curves to embedded tube guide curves can be done as in the two paragraphs after Remark 8.2 \cite{Ga}. This might require that $\mathcal{A}$ has particular sectors of the form $(\alpha_s,(p_s,q_s),\tau_s)$ in order to invert the operation of \S 6 \cite{Ga}. We can create such sectors by Lemma \ref{alpha addition} at the cost of creating other sectors with data of the form $(\alpha_t,(p_t,q_t), \tau_t)$. Also, the modification may create other sectors of this type.
iv) To reverse the ordering of the tube guide curves in $(\gamma_i,\beta_i, \lambda_i)$ where $\lambda_i$ represents 2-torsion, modify $\mathcal{A}$ to create two new sectors with data of the form $(\beta_i, \gamma_i, \lambda_i), (\beta_i,\gamma_i, \lambda_i)$ at the cost of adding sectors with $(\alpha_s,(p_s,q_s), \tau_s)$ type data. Then cancel the $(\gamma_i,\beta_i, \lambda_i), (\beta_i, \gamma_i, \lambda_i)$ pairs at the possible cost of additional type $(\alpha_s,(p_s,q_s), \tau_s)$ sectors.
v) Apply Lemma \ref{alpha stabilization} to each sector with $(\alpha_s,(p_s,q_s), \tau_s)$ data.\qed.
\vskip 10 pt
If $\pi_1(M)=1$, then the self-referential form data is trivial, thus, we have proved the following, stated as Theorem \ref{main} i) in the introduction.
\begin{theorem} \label{pi trivial} Let $D_0, D_1$ be properly embedded discs in the 4-manifold that coincide near their boundaries and have the common dual sphere $G\subset \partial M$. If $M$ is simply connected, then $D_1$ is homotopic to $D_0 \rel \partial $ if and only if it is isotopic $\rel \partial$. \end{theorem}
\section{The Dax Isomorphism Theorem}
Let $f_0: N^n\to M^m$ be an embedding where $N$ and $M$ are closed manifolds. In 1972 J. P. Dax showed that $\pi_k(\Maps(N,M), \Emb(N,M), f_0)$ is isomorphic to a certain bordism group when $2\le k\le 2m-3n-3$. See Theorem A and Theorem 1.1 \cite{Da}. While both the statement and proof are expressed in the very abstract and general style of the day, our case of interest is a strikingly clean and beautiful geometric result with an elementary proof. Using different language and in part different methods we exposit this result when $N=I:=[0,1]$ and $f_0: I\to M^4$ is a proper embedding with image $I_0$. Again, unless stated otherwise, all maps and spaces are smooth and in this section manifolds are oriented. Standard spaces are standardly oriented.
\begin{definition} Define the \emph{Dax group} $\pi_1^D(\Emb(I,M; I_0))$ to be the subgroup of $\pi_1(\Emb(I,M; I_0))$ consisting of classes represented by loops in $\Emb(I,M;I_0)$ that are homotopically trivial in $\pi_1(\Maps(I,M; I_0))$. Here $\Emb(I,M;I_0)$ (resp. $\Maps(I,M;I_0)$) is the based space of proper embeddings (resp. proper continuous maps) that coincide with $I_0$ near $\partial I_0$. Here we abuse notation by identifying the interval $I_0$ with the embedding $f_0:I\to I_0$. \end{definition}
The following definition is a special case of the \emph{spinning} operation of Ryan Budney \cite{Bu}, see Figure \ref{FigureD,1}. That figure shows the projection of a 4-ball $B\subset M$ to a 3-ball $\hat B$. Our path $\alpha_{t}$, which is constant near $t=.5$, intersects $B$ (resp. $\hat B$) in arcs $\sigma$ and $\tau$ (resp. $\sigma$ and a point). It is modified to one where $\sigma$ spins about the point. What follows is a slightly more formal definition.
\begin{definition}\label{spin} Let $\alpha_t:L\to M, t\in [0,1]$ be a path in $\Emb(L,M)$ where $L$ is an oriented 1-manifold and $M$ an oriented 4-manifold. Assume that $\alpha_t$ is constant for $t\in [.45, .55]$. Let $B\subset M$ be parametrized by $[-2,2]\times [-2,2]\times [-1,1]\times [-1,1]$. With respect to local coordinates assume that $B\cap L=\sigma\cup\tau$ where $\tau = (0,0,0,-s), s\in [-1,1]$, $\sigma=\{-1,0,s,0\}, s\in [-1,1]$ and both are oriented from the $s=-1$ to the $s=+1$ end. We modify $\alpha$ to $\gamma$ so that $\alpha_t(s)=\gamma_t(s)$ unless $t\in [.45,55]$ and $\alpha_{.5}(s)\in \sigma$. Within $t\in [.45,55]$, keeping endpoints fixed and staying within the 2-sphere $Q\subset [-2,2]\times [-2,2]\times [-1,1]\times 0=\hat B$, swing $\sigma$ around $\tau$ by first going around the negative $y$-side and then back along the positive $y$-side of $Q$. This can be done so that $\gamma_t$ is a smooth loop. See Figure \ref{FigureD,1}. We say that $\gamma$ is obtained by \emph{spinning} $\alpha$. Note that Lk($\tau$,Q)=+1, where (motion of $\sigma$, orientation of $\sigma$) orients $Q$, in this case the standard orientation. If in local coordinates $\lambda$ denotes the straight path from $(-1,0,0,0)$ to $(0,0,0,0)$, then we say that $\gamma$ is obtained from $\alpha$ by $\lambda$-\emph{spinning}.\end{definition}
\begin{remarks} i) The inverse $\tau^{-1}$ of $\tau$ corresponds to going around $Q$ the other way, thereby reversing the orientation of $Q$ and hence the linking number.
ii) Up to homotopy in $\Emb(L,M;L_0)$, $\lambda$-spinning depends only on the path homotopy class of $\lambda$ and the linking number.
\end{remarks}
\setlength{\tabcolsep}{60pt}
\begin{figure}
\centering
\begin{tabular}{ c c }
$\includegraphics[width=5in]{FigureD,1,2.eps}$
\end{tabular}
\caption[(a) X; (b) Y]{\label{FigureD,1}
\begin{tabular}[t]{ @{} r @{\ } l @{}}
Obtaining $\gamma$ by $\lambda$-spinnning $\alpha$ \\
\end{tabular}}
\end{figure}
\begin{notation} Let $I_0$ be a properly embedded $[0,1]$ in the 4-manifold $M$ and let $1_{I_0}$ denote the identity element in $\pi_1^D(\Emb(I,M;I_0))$. Let $p<q\in I_0$ and $g\in \pi_1(M,I_0)$ where $I_0$ is viewed as the basepoint, then denote by $\tau_g\in \pi_1^D(\Emb(I,M;I_0))$ the loop obtained by spinning $1_{I_0}$ using a path $\lambda$ from $p$ to $q$ representing $g$. Let $\tau_{-g}$ denote $\tau_g^{-1}$. \end{notation}
\begin{remarks} i) Spinning can be viewed as the arc pushing map that defines the barbell map of \cite{BG}. Reversing the orientation of $\lambda$ changes a spin to its inverse up to homotopy in $\Emb(L,M)$. See Theorem 6.6 \cite{BG}. Do not confuse $\tau_{-g}=\tau_g^{-1}$ with $\tau_{g^{-1}}$.
ii) Modifying the orientation preserving parametrization of $B$, e.g., by an element of $\pi_1(SO(3))$ as one moves along $\lambda$, does not change the path homotopy class of $\gamma$. See Remark 6.4 i) \cite{BG}.
iii) The homotopy class of $\gamma$ is independent of the representative of $\lambda$. In particular $\tau_g$ is well defined up to homotopy in $\Emb(I,M;I_0)$ and represents an element of $\pi_1^D(\Emb(I,M;I_0))$. If $g=1\in \pi_1(M,I_0)$, then $\tau_g = 1_{I_0}\in \pi_1^D(\Emb(I,M;I_0))$.
\end{remarks}
\begin{lemma} Spinning commutes up to homotopy in $\Emb(I,M; I_0)$.\end{lemma}
\begin{proof} After an isotopy we can assume that the support of the spins are disjoint.\end{proof}
\begin{theorem} (Dax Isomorphism Theorem) Let $I_0$ be an oriented properly embedded $[0,1]$ in the oriented 4-manifold $M$. Then
i) There is a homomorphism $d_3:\pi_3(M, x_0)\to \mathbb Z[\pi_1(M)\setminus 1]$ with image $D(I_0)$, called the \emph{Dax kernal}.
ii) $\pi_1^D(Emb(I,M;I_0))$ is generated by $\{\tau_g|g\neq 1, g\in \pi_1(M)\} $ and canonically isomorphic to $\mathbb Z[\pi_1(M)\setminus 1]/D(I_0)$. \end{theorem}
\noindent\emph{Proof}.
Let $\alpha=\alpha_t, t\in I$ represent an element of $\pi_1^D(\Emb(I,M;I_0))$. Being in the Dax group, there exists a homotopy $\alpha_{t,u} \in \Maps(I,M;I_0) $ such that $ \alpha_{t,u}$ equals $1_{I_0}$ for $u$ near $0$ and $\alpha_{t,u}$ equals $\alpha_t $ for $u$ near 1.
\vskip 8pt
\noindent\emph{Step 1}: Define $d(\alpha_{t,u})\in \mathbb Z[\pi_1(M)\setminus 1]$.
\vskip 8pt
As in \cite{Da} define $F_0:I\times I^2 \to M\times I^2$ by $F_0(s,t,u)=(\alpha_{t,u}(s),t, u)$. As in Chapter III \cite{Da} we can assume that $F_0$ is \emph{parfait}, in particular is an immersion, has finitely many double points and no triple points. Furthermore, $F_0$ is self transverse at the double points which we can assume occur at distinct values of the last factor. The results in Chapter III are stated for closed manifolds but apply to manifolds with boundary since the support of the modification occurs away from the boundary. See also Chapter VI \cite{Da} which mentions the bounded case.
\setlength{\tabcolsep}{60pt}
\begin{figure}
\centering
\begin{tabular}{ c c }
$\includegraphics[width=5in]{Figure3,2.eps}$
\end{tabular}
\caption[(a) X; (b) Y]{\label{Figure3,2}
\begin{tabular}[t]{ @{} r @{\ } l @{}}
\end{tabular}}
\end{figure}
Assign a generator $\sigma_x g_x\in \mathbb Z[\pi_1(M)]$ to each double point $x$ as follows. Suppose $x=\alpha_{t,u}(p)=\alpha_{t,u}(q)$, where $p<q$. Let $g_x\in \pi_1(M,I_0)$ be represented by $\alpha_{t,u}|[0,p]*\alpha_{t,u}|[q,1]$. See Figure \ref{Figure3,2}. Note that $I_0$ functions as the basepoint. Let $\sigma_x$ be the self intersection number obtained by comparing the orientation of $DF_0(T_{p,t,u}(I^3))\oplus DF_0(T_{q,t,u}(I^3))$ with that of $T_x(M\times I^2)$. If $x_1, \cdots, x_n$ are the double points with $g_{x_i}\neq 1$, then define $d(\alpha_{t,u})=\sum_{i=1}^n \sigma_{x_i} g_{x_i}$.
\vskip 8pt
The next two steps show that modulo $D(I_0)$, different choices of $\alpha_{t,u}$ give the same $d$ value.
\vskip 8pt
\noindent\emph{Step 2}: If $\alpha^0_{t,u}$ is properly homotopic to $\alpha^1_{t,u}$, then $d(\alpha^0_{t,u})=d(\alpha^1_{t,u})$.
\vskip 8pt
\noindent\emph{Proof}. By properly homotopic we mean that there exists $\alpha^v_{t,u}, v\in I$ such that each $ \alpha^v_{t,u}\in \Maps(I,M, I_0)$, $\alpha^v_{t,1}$, $v\in I$ is a homotopy in $\Emb(I,M, I_0)$ from $\alpha^0_{t,1}$ to $\alpha^1_{t,1}$ and $ \alpha^v_{t,u}$ equals $1_{I_0}$ for $u$ near $0$ and $v\in I$.
Suppose that we have two homotopies $F_0, F_1$ as in Step 1, that are homotopic rel $\partial$. Then we can interpolate by maps $F_v$ and combine them to a map $F:(I\times I\times I)\times I\to (M\times I\times I)\times I$, such that $F(s,t,u,v)=(\alpha^v_{t,u}(s),t,u,v)$. Again, we can assume that $F$ is parfait and hence away from finitely many singularities $F$ is a self transverse immersion without triple points. The double points form a 1-manifold whose endpoints in the interior of $M\times I^3$ occur at singularities. The local form of a singularity, p.332 \cite{Da}, implies that a double point $x$ sufficiently close to a singular point has $g_x= 1$. Indeed, since each $\alpha^v_{t,u}$ is path homotopic to $I_0$, if $x=\alpha^v_{t,u}(r)=\alpha^v_{t,u}(s)$, then $g_x=1$ when the loop $\alpha^v_{t,u}|[r,s]$ is homotopically trivial. Here, that loop is homotopically trivial since its diameter converges to $0$ as $x$ approaches the singular point. Finally, use the other double curves to equate the $d$ values coming from $F_0$ and $F_1$. \qed
\vskip 8pt
If $\pi_3(M)\neq 0$, then there will be non homotopic null homotopies of $\alpha_t$ in $\Maps(I,M; I_0)$ which may lead to different values of $d(\alpha_{t,u})$. The Dax kernal keeps track of this indetermanency. Call an $\alpha_{t,u}$ a \emph{kernal map} if for all $u$ close to either 0 or 1, $\alpha_{t,u}=1_{I_0}$. In a natural way, up to homotopy supported away from $\partial I^3$ there is a natural isomorphism between kernal maps and $\pi_3(M, x_0)$, where $x_0=I_0(1/2)$ and the addition of kernal maps is given by concatenation.
\begin{definition} Define $d_3:\pi_3(M,x_0)\to \mathbb Z[\pi_1(M)\setminus 1]$ as follows. Represent $a\in \pi_3(M,x_0)$ as a kernal map $\alpha_{t,u}$. Now define $d(a)=d(\alpha_{t,u})\in \mathbb Z[\pi_1(M)\setminus 1]$ as in Step 1. Define $D(I_0)=d_3(\pi_3(M,x_0))$. When $I_0$ is clear from context, we will write $D(I_0)$ as $D$. \end{definition}
\noindent\emph{Step 3}: $d_3:\pi_3(M)\to \mathbb Z[\pi_1(M)\setminus 1]$ is a homomorphism as is $d: \pi_1^D(\Emb(I,M;I_0))\to \mathbb Z[\pi_1(M)\setminus 1]/D$ where $d(\alpha_t):=d(\alpha_{t,u})$ for some $\alpha_{t,u}$.
\vskip 8pt
\noindent\emph{Proof}. The proof of Step 2 shows that $d_3:\pi_3(M)\to \mathbb Z[\pi_1(M)\setminus 1]$ is well defined. Its additivity with respect to concatenation shows that it is a homomorphism. If $\alpha^0_{t,u}$, $\alpha^1_{t,u}$ are two null homotopies of $\alpha_t$ in $\Maps(I,M;I_0)$, then after concatenating with a kernal map we obtain a new null homotopy whose $d$ value differs by an element of $D$. It follows that $d: \pi_1^D(\Emb(I,M;I_0))\to \mathbb Z[\pi_1(M)\setminus 1]/D$ is well defined.
To show that $d$ is a homomorphism first observe that $d(1_{I_0})=0$. By concatenating $F_0$'s for $\alpha$ and $\beta$ we see that $d(\alpha*\beta)=d(\alpha)+d(\beta)$.\qed
\vskip 8pt
\noindent\emph{Step 4}: If $[\alpha]\in \pi_1^D(\Emb(I,M;I_0))$ and without cancellation $d(\alpha_{t,u}) = \sigma_{x_1} g_{x_1} + \cdots + \sigma_{x_n} g_{x_n}$, then $\alpha$ is homotopic to the compositions of spin maps $\tau_{\sigma_{x_1} g_{x_1}}, \cdots, \tau_{\sigma_{x_n} g_{x_n}}$.
\vskip 8pt
\noindent\emph{Proof}. Let $F_0:I\times I\times I \to M\times I^2$ as in Step 1. We prove Step 3 by induction on the number of double points. Assume for the moment Step 3 is true if $F_0$ has $\le k$ double points where $k\ge 1$. If $F_0$ has $k+1$ double points, then by changing coordinates we can assume that one occurs at $x=F_0(p,\frac{1}{2},\frac{1}{2})=F_0(q,\frac{1}{2}, \frac{1}{2})$ where $p<q$ and the others occur at $F_0(s,t,u)$ where $u>3/4$. Thus, $F_0|I\times I\times 5/8$ is homotopic to a spin map
$\tau$ and there is a homotopy $G_0$ from $1_{I_0}$ to $\tau^{-1}* \alpha$ with $k$ double points of the same group ring types as $F_0|I\times I\times [5/8,1]$ and hence the result follows by induction.
\vskip 8pt
We now consider the case that there is a single double point. By modifying the homotopy rel $\partial$ we can assume that with respect to local coordinates on $M\times I\times I$ and local variables $-\epsilon\le s',t',u' \le \epsilon$;
\vskip 8pt
$F(q+s',t'+\frac{1}{2},u'+\frac{1}{2})=(0,0,0,-s',t'+\frac{1}{2},u'+\frac{1}{2})$,
$F(p+s',t'+\frac{1}{2},u'+\frac{1}{2})=(u', t', s',0,t'+\frac{1}{2},u'+\frac{1}{2})$ if $\sigma_x=+1$,
$F(p+s',t'+\frac{1}{2},u'+\frac{1}{2})=(u', -t', s',0, t'+\frac{1}{2},u'+\frac{1}{2})$ if $\sigma_x=-1$.
\vskip 8pt
Thus, the passage from $\alpha_{t,\frac{1}{2}-\epsilon}$ to $\alpha_{t,\frac{1}{2}+\epsilon}$ changes $1_{I_0}$ to $\tau_{ \sigma_x g_x}$, where $g_x$ is the loop $\phi_0*\phi_1$ where $\phi_0$ (resp. $ \phi_1$) is the arc $F_0(p,\frac{1}{2},w), 0\le w \le \frac{1}{2}\ $(resp. $F_0(q,\frac{1}{2},1-w), \frac{1}{2}\le w\le 1)$ which is homotopic to the loop $g_x$. \qed
\vskip 8pt
\noindent\emph{Step 5}: $d$ is canonical; i.e. if $\alpha$ is a composition of $\tau_{\sigma_1 g_1}, \cdots, \tau_{\sigma_n g_n}$, with all $g_i\neq 1$, then there exists $\alpha_{t,u}$ with $d(\alpha_{t,u})=\sigma_1 g_1 +\cdots +\sigma_n g_n$.
\vskip 8pt
\noindent\emph{Proof}. The local functions defined in Step 4 show how to construct a homotopy $F_0$ from $1_{I_0}$ to $\alpha$ whose double points evaluate to $\sigma_1 g_1, \cdots, \sigma_n g_n.$ \qed
\vskip 8pt
\noindent\emph{Step 6}: $d: \pi_1^D(\Emb(I,M;I_0))\to \mathbb Z[\pi_1(M)\setminus 1]/D$ is an isomorphism.
\vskip 8pt
\noindent\emph{Proof}. Step 3 and 5 show that $d$ is a surjective homomorphism. We now prove injectivity. If $\alpha\in \pi_1^D(\Emb(I,M;I_0))$ and $d(\alpha_{u,t})\in \mathcal{D}$ then by concatenating with a kernal map we can assume that $d(\alpha_{u,t})=0$. It follows from Step 4 that $\alpha$ is homotopic to a composit of spin maps $\tau_{\sigma_{x_1} g_{x_1}}, \cdots, \tau_{\sigma_{x_n} g_{x_n}}$ whose sum is equal to 0 in $\mathbb Z[\pi_1(M)\setminus 1]$. Since spin maps commute it follows that $\alpha$ is homotopic to
$1_{I_0}$. This completes the proof of the Dax isomorphism theorem. \qed
\vskip 8pt
\begin{theorem} Let M be a 4-manifold such that $\pi_3(M)=0$, then $\pi_1^D(Emb(I,M;I_0))$ is freely generated by $\{\tau_g|g\neq 1, g\in \pi_1(M)\} $ and canonically isomorphic to $\mathbb Z[\pi_1(M)\setminus 1]$.\qed\end{theorem}
\begin{theorem} If $M=S^1\times B^3\natural S^2\times D^2$, then $\pi_1^D(Emb(I,M;I_0))$ is isomorphic to $\mathbb Z[\mathbb Z\setminus 1]$ and is freely generated by $\{\tau_g|g\neq 1, g\in \pi_1(M)\} $. (Here $\pi_1(M)$ is expressed multiplicatively.) \end{theorem}
\begin{proof} $\pi_3(M)$ as a $\mathbb Z[\pi_1]$ module is generated by the Hopf map of $S^3$ to a 2-sphere $ Q$ and Whitehead products of conjugates of $\pi_2(Q)$. Once given $ I_0, Q$ can be chosen disjoint from $I_0$ and hence any element of $\pi_3(M)$ has support in a simply connected subcomplex. \end{proof}
\begin{theorem}\label{connect sum} If $M=S^1\times B^3\# S^2\times D^2$, then $\pi_1^D(Emb(I,M;I_0))$ is isomorphic to $\mathbb Z[\mathbb N]$ and is freely generated by $\{\tau_g|g\ge 1\}$.\end{theorem}
\begin{proof} Here the Dax kernal $\neq 0$. The various $\pi_1(M)$ conjugates in $\pi_3(M)$ of the separating $S^3$ give, up to sign, the relations $g^i=g^{-i}$ in $\mathbb Z[\pi_1(M)\setminus 1]$.\end{proof}
\begin{remarks} i) Theorem \ref{dax} is stronger than the one given in \cite{Da} in that we identified generators of $\pi_1^D(\Emb(I,M;I_0))$. Working with these commuting elements enables us to avoid a parametrized double point elimination argument and the need to modify $F_0$ to eliminate double points $x$ with $g_x=1$. Also, we have a natural isomorphism of $\pi_1^D(\Emb(I,M;I_0))$ with a computable quotient of the group ring as opposed to one arising from an abstract bundle cobordism construction.
ii) The ordering of $I_0$ enables us to unambiguously define $\sigma_x$ and $g_x$.
iii) We note that the Dax group $\pi_1^D(\Emb(S^1,M; S^1_0))$, has an extra relation from being able to cancel double points of $F_0$ by going around the $S^1$. Dax computed the case $M=S^1\times S^3$, P. 369 \cite{Da}. See also \cite{AS} and \cite{BG} for the case $M=S^1\times S^3$. \end{remarks}
\begin{question} What is the relation between the Dax kernal and the six dimensional self intersection invariant? \end{question}
\begin{remark} Schneiderman and Teichner \cite{ST} show that for an oriented six dimensional manifold $P$ the self intersection invariant $\mu_3:\pi_1(P)\to \mathbb Z[\pi_1(P)]/<g+g^{-1},1>$ specializes to a map $\mu_3:\pi_3(N)\to \mathcal{F}_2 T_N$, when $P=N\times I$ and where $T_N$ is the vector space with basis the non trivial torsion elements of $\pi_1(N)$ and $\mathcal{F}_2$ is the field with two elements. Our setting is both similar and different in that we are looking at an \emph{ordered} self intersection of mapped \emph{3-balls with fixed boundary} into $M\times I\times I$. As indicated in Theorem \ref{connect sum} the Dax kernal can be nontrivial, e.g. in manifolds with $\pi_1(M)=\mathbb Z$.\end{remark}
\begin{remarks} i) Syunji Moriya \cite{Mo} shows that for certain simply connected 4-manifolds M, $\pi_1(\Emb(S^1, M))\cong H_2(M,\mathbb Z)$.
ii) See Danica Kosanovic's thesis \cite{Ko1} and paper \cite{Ko2} for results on $\Emb(I,M)$ for general manifolds $M$. \end{remarks}
\section{From Discs to Paths}
\begin{definition} Let $D_0$ be a properly embedded disc in $M$ with dual sphere $G$. Let $\mathcal{D}$ be the set of isotopy classes $\rel \partial$ of discs homotopic $\rel \partial$ to $D_0$. If $D_1, D_2\in \mathcal{D}$, then define $D_1+D_2=D_3$ so that $D_3$ is the realization of a tubed surface whose sector form data is the concatenation of that of $D_1$ and $D_2$. This means that if $D_1$ (resp. $D_2)$ has $n_1$ (resp. $n_2$) sectors with data then $D_3$ has $n_1+n_2$ sectors with the corresponding data. \end{definition}
\begin{proposition} $\mathcal{D}$ is an abelian group with unit $[D_0]$ under the operation $+$.\end{proposition}
\begin{proof} We need to show that $D_3$ is independent of the choice of representatives of $D_1$ and $D_2$, the other conditions being immediate. In particular, by Lemma \ref{normal permutation} $D_3$ is independent of the concatenation order and hence $\mathcal{D}$ is abelian. We can assume that $D_1$ coincides with $D_0$ near their boundaries, so an isotopy of $D_1$ to $D_1'$ can be chosen to be supported away from some neighborhood of $\partial D_0$. Since the data of $D_2$, except for its framed embedded paths, can be isotoped within their sectors to be very close to $\partial D_0$, we see that the isotopy of $D_1$ can be chosen to avoid it. While the framed embedded paths associated to $D_2$ may get moved during the ambient isotopy of $D_1$ to $D_1'$, the light bulb lemma enables them to isotope back to their original positions without introducing intersections with $D_1'$. \end{proof}
\begin{remark} $\mathcal{D}$ is a torsor, where $\mathbb Z[\pi_1(M)\setminus 1]$ and $\mathbb Z[T_2]$ act on $\mathcal{D}$. Here $T_2$ is the set of non trivial 2-torsion elements. The former acts by attaching the appropriate self-referential discs and the latter by attaching the appropriate double tubes. \end{remark}
\begin{notation} If $\lambda$ is a framed embedded path with endpoints in $D_0$ representing a nontrivial 2-torsion element of $\pi_1(M)$, then let $\hat\lambda$ denote this element and let $D_\lambda$ denote the realization of the self-referential form tubed surface whose data consists exactly of $(\lambda)$. If $1\neq g\in \pi_1(M)$, then let $D_g$ (resp. $D_{-g})$ denote the realization of the self-referential form tubed surface whose data only consists exactly of $(+g)$ (resp. $(-g)$.\end{notation}
\begin{remark} Since an element of $\mathcal{D}$ can be put into self-referential form it follows that the $D_g$'s and $D_\lambda$'s are generators of $\mathcal{D}$.\end{remark}
\begin{definition} \label{novel} Let $D_0$ be a properly embedded disc in the 4-manifold $M$, not necessarily with a dual sphere. View $D_0$ as $I\times I$ with $I_0$ denoting $I\times 1/2$ and $ \mathcal{F}_0$ this product foliation. If $D$ is another properly embedded disc that agrees with $D_0$ along $\partial D_0$, then $D$ gives rise to an element $[\phi_{D_0}(D)]\in \pi_1(\Emb(I,M;I_0))$, where $\Emb(I,M;I_0)$ is the space of smooth embeddings of $I$ based at $I_0$. To construct $\phi_{D_0}(D)$, first isotope $D$ to coincide with $D_0$ near $\partial D_0$. Next view $ D=I\times I$ where this foliation $\mathcal{F}$ coincides with $\mathcal{F}_0$ near $\partial D_0$. Use $D_0$ to inform how to modify $\mathcal{F}$ to a loop $\phi_{D_0}(D)$ in $\Emb(I,M;I_0)$ based at $ I_0$. To do this first define $\beta\in \Emb(I,M)$ as follows. For $t\in [0,1/4], \beta_t$ traces $I\times(1/2-2t)$ using $\mathcal{F}_0$; for $t\in [1/4,3/4], \beta_t$ traces $I\times(2t-.5)$ using $\mathcal{F}$; and for $t\in [3/4,1], \beta_t$ traces $I\times(1.5-2t)$ using $\mathcal{F}_0$. Naturally modify the ends of each $\beta_t$ to coincide with $I_0$ near $ \beta_t(0)$ and $\beta_t(1)$ to obtain $\phi_{D_0}(D)$ with $[\phi_{D_0}(D)]$ denoting the corresponding class in $\pi_1(\Emb(I,M;I_0))$.\end{definition}
\begin{remark} For the sake of exposition, $D_0$ was parametrized as a disc with corners. The definition is readily modified to the smooth setting.\end{remark}
Since $\Diff(D^2\fix \partial)$ is connected \cite{Sm3} it follows that $\phi_{D_0}$ is well defined and depends only on $D_0$ and $I_0$. If $\mathcal{D}$ is the set of isotopy classes of discs homotopic to $D_0 \rel \partial$, then together with the Dax isomorphism theorem we obtain the following result.
\begin{theorem} Let $D_0$ be a properly embedded disc in the oriented 4-manifold, $I_0$ an oriented properly embedded arc in $D_0$ and $\mathcal{D}$ be the isotopy classes of embedded discs homotopic $\rel \partial$ to $D_0$, then there is a canonical function $\phi_{D_0}: \mathcal{D} \to \mathbb Z[\pi_1(M)\setminus 1]/D(I_0)$ such that if $D$ is a embedded disc homotopic rel $\partial$ to $D_0$, then $\phi_{D_0}([D])\neq 0$ implies $D$ is not isotopic to $D_0\rel\partial$.\end{theorem}
We have more algebraic structure when $D_0$ has a dual sphere. The following is a sharper form of Theorem \ref{main} ii) of the introduction.
\begin{theorem} \label{main ii} Let $D_0\subset M$ be a properly embedded disc with the dual sphere $G$ and $\mathcal{D}$ the isotopy classes of discs homotopic to $D_0\rel \partial D_0$. Then $\mathcal{D}$ is an abelian group with zero element $[D_0]$ and there exists a natural homomorphism $\phi_{D_0}:\mathcal{D}\to \mathbb Z[\pi_1(M)\setminus 1]/D(I_0)\cong \pi_1^D(\Emb(I,M; I_0))$, where $D(I_0)$ is the Dax kernal, such that the generators of $\mathcal{D}$ are mapped as follows.
i) $\phi_{D_0}([D_\lambda])=\hat \lambda$
ii) $\phi_{D_0}([D_g])=g+g^{-1}$.\end{theorem}
\setlength{\tabcolsep}{60pt}
\begin{figure}
\centering
\begin{tabular}{ c c }
$\includegraphics[width=5in]{Figure4,1.eps}$
\end{tabular}
\caption[(a) X; (b) Y]{\label{Figure4,1}
\begin{tabular}[t]{ @{} r @{\ } l @{}}
\end{tabular}}
\end{figure}
\begin{proof} We first set the local picture. View $N(D_0\cup G)$ as the manifold with corners $J\times [-1,1]$, where $J=H\setminus \inte(B)$, where $B$ is an open 3-ball and $H$ is a half 3-ball with $\partial H = \partial_e H\cup \partial_i H$, the \emph{external} and \emph{internal boundaries}. Also, $\partial M\cap J\times [-1,1]=(\partial_e H\cup \partial B)\times [-1,1]\cup J\times \{-1,1\}$. Here $G_t:=\partial B\times t$ and $N(G)\cap \partial M=G\times [-1,1]$. $D_0$ is a vertical disc in $J\times [-1,1]$ with $I_t:=D_0\cap J\times t$, where $I_0$ is an arc from $\partial_e H\times 0$ to $G:=G_0$. See Figure \ref{Figure4,1} a). Figure \ref{Figure4,1} b) shows a one dimension lower version. In that figure $G$ is a circle and $D_0$ is a disc. $\partial M$ is the union of $G\times [-1,1]$ and the shaded face which is the analogue of $\partial_e(H)\times [-1,1]$ and the top and bottom faces.
We now define $\phi_{D_0}$ from this point of view. If $D$ is a properly embedded disc that coincides with $D_0$ near $\partial D$, then the $I_t$ fibering of $D_0$ induces $\phi_{D_0}(D)\in\pi_1^D(\Emb(I,M;I_0))$ as follows. It first induces a map $\phi_{D_0}':[-1,1]\to (\Maps: [-1,1]\to \Emb(I,M))$. The projection of $I_t$ to $I_0$ then informs how to close up to a loop and modify the ends to coincide with $I_0$ to obtain a well defined element of $\pi_1^D(\Emb(I,M;I_0))$. It is a homomorphism since by construction $\phi_{D_0}([D_0])]=[1_{I_0}]$. Since addition is given by concatenation of sector forms it follows that $\phi_{D_0}([D_1]+[D_2])=\phi_{D_0}([D_1])+\phi_{D_0}([D_2])$.
\setlength{\tabcolsep}{60pt}
\begin{figure}
\centering
\begin{tabular}{ c c }
$\includegraphics[width=1.5in]{Figure4,2.eps}$
\end{tabular}
\caption[(a) X; (b) Y]{\label{Figure4,2}
\begin{tabular}[t]{ @{} r @{\ } l @{}}
Orientations on $D_0$ and $G$
\end{tabular}}
\end{figure}
\setlength{\tabcolsep}{60pt}
\begin{figure}
\centering
\begin{tabular}{ c c }
$\includegraphics[width=5in]{Figure4,3.eps}$
\end{tabular}
\caption[(a) X; (b) Y]{\label{Figure4,3}
\begin{tabular}[t]{ @{} r @{\ } l @{}}
Orientation on $P(\alpha)$
\end{tabular}}
\end{figure}
\setlength{\tabcolsep}{60pt}
\begin{figure}
\centering
\begin{tabular}{ c c }
$\includegraphics[width=4in]{Figure4,4.eps}$
\end{tabular}
\caption[(a) X; (b) Y]{\label{Figure4,4}
\begin{tabular}[t]{ @{} r @{\ } l @{}}
Isotoping to a self-referential disc I
\end{tabular}}
\end{figure}
\setlength{\tabcolsep}{60pt}
\begin{figure}
\centering
\begin{tabular}{ c c }
$\includegraphics[width=1.75in]{Figure4,5.eps}$
\end{tabular}
\caption[(a) X; (b) Y]{\label{Figure4,5}
\begin{tabular}[t]{ @{} r @{\ } l @{}}
Isotoping to a self-referential disc II
\end{tabular}}
\end{figure}
We now show ii). Given $D_g\in \mathcal{D}$, represent $\phi_{D_0}(D_g)$ as $\alpha_t$ a loop in $\Emb(I,M;I_0)$. As in \S 3 we construct a homotopy $\alpha_{t,u}$ in $ \Maps(I,M;I_0)$ from $ \alpha_t $ to $1_{I_0}$ and then compute $d(\alpha_{t,u})$. To compute the required intersection numbers we need to establish and keep track of orientations. First $J\times [-1,1]$ has the standard orientation $(\epsilon_1, \epsilon_2, \epsilon_3, \epsilon_4)$ induced from $\mathbb R^3\times \mathbb R$. Figure \ref{Figure4,2} shows our orientations on $D_0$ and $G$ as seen from $J\times 0$. Here $T_z(D_0)$ is oriented by $(\epsilon_2,\epsilon_4)$ and $T_z(G)$ is oriented by $(\epsilon_3,\epsilon_1)$. Note that $<D_0,G>_z=1$. Recall that $D_g$ is obtained by coherently tubing $D_0$ with the oriented sphere $P(\alpha)$ along a path $\tau$ representing $g$, so to know the orientation on $D_g$ it remains to know the orientation of $P(\alpha)$ which is shown in Figure \ref{Figure4,3}. The numbers next to the vectors indicate which goes first. Recall that $P(\alpha)$ is obtained by tubing two copies of $G$, say $G_{ - .5}$ and $G_{.5} $ where the orientation of $G\times -.5$ (resp. $G\times +.5$) disagrees (resp. agrees) with that of $G$. Figure \ref{Figure4,4} a) shows the projection of $P(\alpha)\cup D_0\cup \tau$ to $J\times 0$; the solid line indicating intersection with the present and shading indicates projection from either the past or future. Here $J_t; t<0, t=0$ or $t>0$ refers to the past, present or future. The orientation shown is that of the projection of the disc from the future. Figure \ref{Figure4,4} b) is another projection after an isotopy of $P(\alpha)\cup \tau$. To obtain the full picture of this $D_g$ we coherently connect $D_0$ to this isotoped $P(\alpha)$ by the tube $T_\tau$ that follows the isotoped $\tau$. See Figure \ref{Figure4,5}.
\setlength{\tabcolsep}{60pt}
\begin{figure}
\centering
\begin{tabular}{ c c }
$\includegraphics[width=4in]{Figure4,6.eps}$
\end{tabular}
\caption[(a) X; (b) Y]{\label{Figure4,6}
\begin{tabular}[t]{ @{} r @{\ } l @{}}
Computing the Intersection Numbers
\end{tabular}}
\end{figure}
We now describe $\alpha_{t,u}$. The passage from the original $D_g$ to the above one induces a homotopy of $\alpha_{t,0}$ to $\alpha_{t,1/4}$. Here is a description of the loop $\alpha_{t,1/4}, t\in [-1,1]$. Starting at $\alpha_{-1,1/4} =I_0$, keeping neighborhood of $\partial I_0$ fixed, $\alpha_{t,1/4} $ sweeps out along $T_\tau$ staying slightly in the past, then remaining slightly in the past continues across $P(\alpha)$ to reach $\alpha_{1/2,1/4}$, the dark line in Figure \ref{Figure4,5} which is totally in the present. It then sweeps back across $P(\alpha)$ staying slightly in the future and then back across $T_\tau$ before returning to $I_0=\alpha_{1,1/4}$. Our homotopy $\alpha_{t,u}$ will have the feature that for all $u, \alpha_{1/2,u}\cap J\times [-1,1]\subset J\times 0$. If $D_g(u)$ denotes the image of $\alpha_{t,u}, t\in [-1,1]$, then Figure \ref{Figure4,5} shows the projection of $D_g(1/4)$ to $J\times 0$. We now homotope $D_g(1/4)$ to $D_g(3/8)$ as shown in Figure \ref{Figure4,6} a). Here we abuse notation by conflating the domain with the image. While the embedded part of $D_g(u)$ now becomes immersed, the homotopy induces a homotopy of $\alpha_{t,1/4}$ to $\alpha_{t, 3/8}$ as loops in $\Emb(I,M; I_0)$. Figure \ref{Figure4,6} b), (resp. Figure \ref{Figure4,6} c)) shows the result of a further homotopy to $\alpha_{t,9/16}$ (resp. $\alpha_{t, 3/4}$) this time as loops in $\Maps(I,M;I_0)$. $\alpha_{t,u}$ fails to be a loop in $\Emb(I,M;I_0)$ when $u=1/2$ and $5/8$. This can be done so that at $u=1/2$ (resp. $u=5/8)$ there is a single self-intersection when $t=1/2$ and $s=a_0$ and $s=b_0$ (resp. $t=1/2$ and $s=a_1$ and $s=b_1$.) Note that the loop $\alpha_{t,3/4}$ is homotopic in loops $\Emb(I,M;I_0)$ to $1_{I_0}$. Use this homotopy to complete the construction of $\alpha_{t,u}$.
We now compute the self-intersection values. Recall that $I_0$ is oriented to point into $G$. Following the rules of \S 3, since $b_0<a_0$ the group element to this self-intersection is $g^{-1}$. With notation as in \S 3 we now compute the sign of the self-intersection by comparing $DF_{0_{b_0, 1/2, 1/2}}(T_{b_0,1/2,1/2}(I^3))\oplus DF_{0_{a_0,1/2, 1/2}}(T_{a_0,1/2,1/2}(I^3))$ with that of $T_{x_1,1/2,1/2}(M\times I^2)$ where $x_1=\alpha(1/2,1/2)(a_0)=\alpha(1/2,1/2)(b_0)$. Parametrized as in \S 3 we have $DF_{0_{b_0,1/2,1/2}}(\partial/\partial s, \partial/ \partial t, \partial/\partial u)= (\epsilon_1, \epsilon_5, \epsilon_6)$ and $DF_{0_{a_0, 1/2,1/2}} (\partial/\partial s, \partial/\partial t, \partial/\partial u)= (\epsilon_3, \epsilon_4+\epsilon_5, \epsilon_2 + \epsilon_6)$ which as a 6-vector is equivalent to $(\epsilon_1, \epsilon_5,\epsilon_6, \epsilon_3, \epsilon_4, \epsilon_2)$ which is equivalent to the standard basis, hence the self-intersection number is $+1$. Since $a_1<b_1$, A similar calculation shows that at the second self-intersection the group element is g and the 6-tuple of vectors is $(\epsilon_3, \epsilon_4+\epsilon_5, \epsilon_2+\epsilon_6, -\epsilon_1, \epsilon_5, \epsilon_6)$ which is equivalent to $(\epsilon_3, \epsilon_4, \epsilon_2, -\epsilon_1, \epsilon_5, \epsilon_6)$ which also gives the standard basis. Therefore, $\phi(D_g)=d(\alpha_{t,u})=g+g^{-1}$.
\setlength{\tabcolsep}{60pt}
\begin{figure}
\centering
\begin{tabular}{ c c }
$\includegraphics[width=5in]{Figure4,7.eps}$
\end{tabular}
\caption[(a) X; (b) Y]{\label{Figure4,7}
\begin{tabular}[t]{ @{} r @{\ } l @{}}
Two double tubes equals one single tube
\end{tabular}}
\end{figure}
We now show i) by proving that $2\phi_{D_0}(D_\lambda)=\phi_{D_0}(2D_\lambda) =2\hat\lambda$. Figure \ref{Figure4,7} a) shows a tubed surface with self-referential form data $(\lambda, \lambda)$. Figure \ref{Figure4,7} b) shows the result of applying the operation of \S 6 \cite{Ga} to this tubed surface. Tube sliding moves allow for the $q$ point to $\alpha_2$ to be placed to either side of $\alpha_1$ and vice versa. Note that the orientations on the $\alpha$ curves are determined by the sign convention. As in \S 2, deleting the data corresponding to the $\alpha_2$ curve does not change the realization since it's $q$ point lies on the far side of the approach interval. What's left is a tubed surface of Figure \ref{Figure4,7} c) with self-referential form data $(+\hat \lambda)$ whose realization is $D_{\hat\lambda}$. By ii), $\phi_{D_0}(D_{\hat\lambda})= 2\hat \lambda$.\end{proof}
\begin{corollary} \label{knotted disc} Let $M=S^2\times B^2\natural S^1\times B^3, D_0$ be the standard 2-disc as in Figure 2 and $g$ be a generator of $\pi_1(M)$. Then the discs $D_{g^i}$, $i \in \mathbb N$ are pairwise not properly isotopic. On the other hand each $D_{g^i}$ is concordant to $D_0$.\end{corollary}
\begin{proof} By Theorem \ref{connect sum}, the Dax kernal $D(I_0)=0$. It follows that if $i\neq j$, then $D_{g^i}$ is not isotopic to $D_{g^j}$ since $g^i+g^{-i}\neq g^j+g^{-j}$. Since each $D_{g^i}$ differs from $D_0$ by a ribbon 3-disc, they are concordant. See Figure 2 in the introduction. \end{proof}
\section{Applications and Questions}
As an application we give examples of knotted 3-balls in 4-manifolds with boundary. See \cite{BG} and \cite{Wa} for constructions in closed manifolds. As a prototype we state a result for $M=S^2\times D^2\natural S^1\times B^3$ and indicate a generalization to other manifolds.
\begin{theorem}\label{knotted ball two} If $M=S^2\times D^2\natural S^1\times B^3$ and $\Delta _0= x_0\times B^3$ in the $S^1\times B^3$ factor, then there exist infinitely many 3-balls properly homotopic to $\Delta_0$, but not pairwise properly isotopic.\end{theorem}
\begin{remark} The following result is a straight forward extension of Hannah Schwartz' Lemma 2.3 \cite{Sch} for spheres with dual spheres to discs with dual spheres, with a somewhat different proof. \end{remark}
\begin{lemma} \label{schwartz} Let $D_0\subset N$ be a properly embedded 2-disc with dual sphere $G$. If $D_1$ is a properly embedded 2-disc that coincides with $D_0$ near $\partial D_0$ and $D_1$ is homotopic $\rel \partial$ to $D_0$, then there exists a diffeomorphism $\psi:(N,D_0)\to (N, D_1)$. If $D_1$ is homotopic $\rel \partial$ to $D_0$, then $\psi$ can be chosen to fix a neighborhood of $\partial N$ pointwise. If $D_0$ is concordant to $D_1$, then $\psi$ can also be chosen to be homotopic to $\id \rel \partial$.\end{lemma}
\begin{proof} Let $G\times [-\epsilon, \epsilon]$ be a product neighborhood of $G\subset \partial N$. Let $N_1=N\cup_{G\times [-\epsilon,\epsilon]} B^3\times [-\epsilon,\epsilon]$. Then $N$ is obtained from $N_1$ by removing a neighborhood of the arc $\kappa=0\times [-\epsilon,\epsilon]$. Any loop $\gamma \in \Emb(I,N_1;\kappa)$ whose time 1 map preserves the framing of $T(\kappa)$ induces $\psi_1:(N_1,\kappa)\to (N_1,\kappa)$ fixing $\partial N_1\cup N(\kappa)$ pointwise and hence a map $ \psi_\gamma:N\to N$ fixing $\partial N$ pointwise, otherwise it induces a diffeomorphism that twists the boundary. Such a diffeomorphism is called an \emph{arc pushing map}.
Since $D_0, D_1$ coincide near $N(\partial D_0)$, we can extend slightly to discs $E_1, E_0$ in $N_1$, which coincide in $N_1\setminus N$ with $\partial E_0\subset \kappa\cup \partial N_1$. Let $\gamma$ be the arc pushing map which first deformation retracts $E_0$ to a small neighborhood of $\partial E_0$ and then expands along $E_1$. If $D_1$ is homotopic to $D_0$ such an isotopy can be constructed to preserve the normal framing of $\kappa$ and hence induce a diffeomorphism $\psi_\gamma:(N,D_0)\to (N,D_1)$ which fixies $N(\partial N)$ pointwise.
If $\hat\psi_\gamma:N_1\times I\to N_1\times I$ is the map induced from suspending the ambient isotopy induced from $ \gamma$, then $ \kappa$ tracks out a properly embedded disc. If $D_1$ is concordant to $D_0$, then this disc is isotopic $\rel \partial$ to $\kappa\times I$, in which case $\psi_\gamma$ is homotopic to $\id \rel \partial$.\end{proof}
\begin{remark} It suffices that $D_1$ and $D_0$ induce the same framing on their boundaries to enable $\psi$ to fix $\partial N$ pointwise.\end{remark}
\noindent\emph{Proof of Theorem \ref{knotted ball two}}. Let $g$ be a generator of $\pi_1(M)$. Let $D_i$ be the disc $D_{g^i}$ of Theorem \ref{knotted disc}. By that result all these $D_i$'s are homotopic, in fact concordant, yet pairwise not isotopic $\rel \partial$. Apply the lemma to obtain $\psi_i:M\to M$ a diffeomorphism, properly homotopic to $\id$ fixing $N(\partial M)$ pointwise such that $\psi_i(D_0)=D_i$. Let $\Delta_i=\psi_i(\Delta_0)$. Since $\Delta_0\cap D_0=\emptyset$ it follows that for all $i$, $\Delta_i\cap D_i=\emptyset$. If $\Delta_i$ is properly isotopic to $\Delta_j, i\neq j$, then the corresponding ambient isotopy takes $D_i$ to $D'_i$ with $D'_i\cap \Delta_j=\emptyset$. Now $M\setminus \inte(N(\Delta_0))$ is diffeomorphic to $S^2\times D^2$ and hence so is $M\setminus \inte(N(\Delta_j))$. Since $\Delta_i'$ is properly homotopic to $\Delta_j$ in $M$. $D_i'$ is homotopic $\rel \partial$ to $D_j$ in this $S^2\times D^2$. By Theorem 10.4 \cite{Ga}, $D'_i $ is isotopic $\rel \partial$ to $ D_j$, which is a contradiction.\qed
\begin{remark} In a somewhat similar manner we obtain knotted 3-balls in some manifolds of the form $W=M\natural S^1\times B^3$ where $D_0\subset M$ has a dual sphere $G\subset M$. Here $\pi_1(W)=\pi_1(M)*\mathbb Z$. Let $t$ denote a generator of $\mathbb Z$. We require that the subgroup of $\mathbb Z[\pi_1(W)\setminus 1]$ generated by $t^n+t^{-n}, n\in \mathbb N$ is not contained in the subgroup generated by $\mathbb Z[\pi_1(M)]+D(I_0)$. For example, manifolds $W$, where $M$ of the form $S^2\times D^2\natural Y$ and $\pi_3(Y)=0$. Define $\Delta_0=x_0\times B^3$ and let $D_1$ be obtained by attaching self-referential discs to $D_0$ so that $\phi_{D_0}(D_1)\notin \mathbb Z[\pi_1(M)]+D(I_0)$. Now modify $\Delta_0$ to $\Delta_1$ by embedded surgery so that $\Delta_1\cap D_1=\emptyset$ and $\Delta_1$ is homotopic $\rel \partial$ to $\Delta_0$. If $\Delta_1$ can be isotoped to $\Delta_0$, then $D_1$ can be isotoped into $M$. Since $D_1$ is homotopic to $D_0$ in $W$, a homotopy can be constructed to be supported in $M$. This can be seen be remembering that $\pi_2(W)=H_2(\tilde W)$, so a 2-sphere in $\tilde W$ homologically trivial in $\tilde W$ is homologically trivial in $\tilde W \setminus \pi^{-1}(\Delta_0)$, where $\pi$ is the covering projection. It follows that $\phi_{D_0}(D_1)\in \mathbb Z[\pi_1(M)]+D(I_0)$ a contradiction.
Note that the analogous construction does not work for $V=S^2\times D^2\# S^1\times B^3$ for the standard $D_0$ which lies in the $S^2\times D^2$ factor, since for this $D_0$ homotopy implies isotopy. That is because the separating 3-sphere can be used to disentangle a single self-referential disc. Also multiple self-referential discs can be disentangled using the separating 3-sphere and the light bulb lemma. \end{remark}
\vskip 10pt
We conclude with a problem and two questions.
\begin{problem} Complete the isotopy classification of properly embedded discs in 4-manifolds with dual spheres. \end{problem}
The following question specializes this problem to 4-manifolds without 2-torsion in their fundamental groups?
\begin{questions} Let $D_0\subset M$ be a properly embedded disc with dual sphere $G$ such that $\pi_1(M)$ has no 2-torsion. Let $\mathcal{D}$ be the isotopy classes of embedded discs homotopic to $D \rel \partial$. Let $\phi_{D_0}:\mathcal{D}\to \mathbb Z[\pi_1(M, z)\setminus 1]/D\cong\Emb(I,M;I_0)$ be the canonical homomorphism. What is $\ker \phi_{D_0}$? In particular, if $M=S^2\times D^2\natural S^1\times B^3$, is $D_g$ isotopic $\rel \partial $ to $D_{g^{-1}}$? \end{questions}
|
1,314,259,993,591 | arxiv | \section{Introduction}
Protostellar jets are spectacular signposts of star formation. They are
highly collimated structures consisting of a chain of knots and bow shocks,
emanating from young protostellar objects, propagating away at highly
supersonic speeds. Since they are believed to be launched from accretion
disks around protostars, they are not only the fossil records of accretion
history of the protostars but also are expected to play an important role in
facilitating the accretion process. Studying their dynamics and evolution
also allows us to probe disk evolution and potentially planetary formation
in the disks.
In low-mass star formation, the protostellar phase with active accretion can
be divided into the Class 0 and Class I phases
\citep{Mckee2007,Evans2009,Kennicutt2012}. The Class 0 phase starts when a
protostar is first formed at the center with a mass of $\sim 10^{-3}
$M_\odot$$ and a radius of $\sim 2\, R_\odot$
\citep{Larson1969,Masunaga2000}. In this phase, the protostar is deeply
embedded in a large cold dust envelope and actively accreting material from
it. This phase will end in $\sim 10^5$ yrs when the cold envelope decreases
to $\sim$ 10\% of its original amount \citep{Evans2009}. This phase is then
followed by the Class I phase, which will end in $\sim 5\times 10^5$ yrs
when most of the envelope material has been consumed and the accretion
almost comes to an end \citep{Evans2009}. Then the protostar enters the
Class II phase or T-Tauri phase, and becomes a pre-main-sequence star
visible in the optical. Note that some protostars could already become
pre-main-sequence stars earlier during the Class I phase.
\begin{figure}[htb]
\centering
\includegraphics[angle=0, width=\textwidth]
{HH212_Jet_Review.pdf}
\caption{HH 212: (a) H$_2$ map of the jet in parsec scale adopted from
\citet{Reipurth2019}. (b) H$_2$ map of the inner jet adopted from
\citet{McCaughrean2002}. (c) A composite image of the jet within $\sim$
10000 au of the central source, with H$_2$ map (blue) from
\citet{McCaughrean2002} and SiO (green) and CO (red) maps from
\citet{Lee2015}. (d) The jet in SiO (green) within $\sim$ 1000 au of the central source
and the accretion disk in dust continuum at 850 $\mu$m{}
(orange), adopted from \citep{Lee2017Jet}. (e) shows the SiO jet (green)
within $\sim$ 100 au of the central source with the dusty accretion disk
(orange), adopted from \citep{Lee2017Jet}. The blue and red arrows show the
disk rotation.
\label{fig:HH212_Jet}}
\end{figure}
Jets are commonly seen in the Class 0 to the early Class II phase when the
accretion process is active. Their velocities scale with protostellar mass
and thus increase with the evolutionary phase, increasing from $\sim$ 100
km s$^{-1}${} in the early phase to a few 100 km s$^{-1}${} in the late phase
\citep{Hartigan2005,Anglada2007,Hartigan2011}. Therefore, they can
propagate for a long distance into the ISM, producing spectacular
parsec-scale Herbig-Haro objects \citep{Reipurth2001} even in the Class 0
phase \cite[][see also Fig.~\ref{fig:HH212_Jet} for the Class 0 jet HH
212]{Reipurth2019}. When these jets propagate into the ISM, they push and
sweep up the surrounding material, forming molecular outflows around the jet
axis \citep{Lee2000,Arce2007}, perturbing the ISM and thus potentially
reducing the star-formation efficiency \citep{Bally2016}. Interestingly,
jets are also detected around brown dwarfs
\cite[e.g.,][]{Whelan2005,Riaz2017}, intermediate-mass protostars
\cite[e.g.,][]{Zapata2010,Reiter2017,Takahashi2019}, and high-mass
protostars
\citep{Reipurth2001,Zapata2006,Carrasco2010,Ellerbroek2013,Caratti2015},
suggesting that the low-mass star formation scenario may apply to these
objects.
Current observations also show an evolution in the gas content of the jets.
In the Class 0 phase, the jets are mainly detected in molecular gas, e.g.,
CO, SiO, and SO at (sub)millimeter wavelengths and H$_2$ in infrared
\citep{Arce2007,Frank2014}. In the Class I and Class II phases, they are
mainly detected in atomic and ionized gas, e.g., O, H$\alpha$, and S II
\citep{Reipurth2001,Bally2016}, appearing as Herbig-Haro flows at optical
and infrared wavelengths. At the base near the launching points, the jets
are ionized and thus radiate free-free emission at centimeter wavelengths
\citep{Anglada2018}. Therefore, different telescopes are used to study
different regions of the jets in different evolutionary phases.
Recent comprehensive reviews of the jets have been presented by
\citet{Frank2014} and \citet{Bally2016}, and a further review on radio jets
by \citet{Anglada2018}. Here I will focus on molecular jets seen in the
early phase of low-mass star formation, probing the initial phase of the jet
formation and accretion process in the first $10^{5}$ yrs in the Class 0
phase. Without suffering from dust extinction effects, molecular lines at
(sub)millimeter wavelengths allow us to probe the jets close to the
innermost regions, where the sources are deeply embedded and where the jets
are launched and collimated (see Fig.~\ref{fig:HH212_Jet}). The jets are
believed to be launched from accretion disks around protostars through
magneto-centrifugal forces \citep{Shu2000,Konigl2000}, and are thus expected
to be magnetized and rotating. In this case, the jets can also solve the
angular momentum problem in the innermost edges of the disks by carrying away
angular momentum from there to allow disk material to fall onto the central
protostars. I will review current exciting and revolutionary results on
molecular jets, especially those obtained with unprecedented angular
resolution, velocity resolution, and sensitivity using ALMA. Together with
previous detailed studies of jets in their later phase, we can set strong
constraints on jet launching and collimation. With a detailed study of jet
physical properties (e.g., mass-loss rate, velocity, rotation, radius,
wiggle, molecular content, shock formation, periodical variation, magnetic
field, etc), we can also probe accretion process, binary formation, disk
instability and evolution, and potentially planetary formation in the disks.
\section{Observed properties of molecular jets}
Protostellar jets in the Class 0 phase are mainly molecular. They can be
traced by high-velocity CO emission
\citep{Gueth1999,Lee2007HH212,Santiago2009,Hirano2010,Plunkett2015}, which
allows us to derive the density and thus the mass-loss rate in the jets.
Their knots and bow shocks can be traced by shock tracers, e.g., H$_2$
\citep{McCaughrean1994,Zinnecker1998}, SiO
\citep{Gueth1998,Hirano2006,Palau2006,Codella2007,Lee2007HH212,Codella2014,Podio2016,Bjerkeli2019},
and SO \citep{Lee2007HH212,Lee2010HH211,Codella2014}. More importantly, SiO
is a dense shock tracer, tracing uniquely the jets within $\sim 10^4$ au
of the central sources down to the bases where the density is high, allowing
us to probe the jet launching and collimation regions.
\begin{table}
\small
\centering
\caption{Molecular jets from low-mass Class 0 protostars detected in both SiO and CO}
\label{tab:jetsource}
\begin{tabular}{llllllllll}
\hline Source & $D$ & $L_{\mathrm{bol}}$ & $M_\ast$ & $r_d$ & $v_j$ & $\dot{M}_j$ & $L_j$ & $\dot{M}_{\mathrm{acc}}$ & Refs. \\
& (pc) & ($L_\odot$) & ($M_\odot$) & (au) & (km/s) & ($M_\odot$ \ yr$^{-1}$)& $L_\odot$ & ($M_\odot$ \ yr$^{-1}$) \\ \hline\hline
IRAS 04166+2706 & 140 & 0.4 & & ? & 61 & $0.7\times10^{-6}$ & 0.21 & & 1 \\
B335 & 100 & 0.7 & 0.05 & $<5$ &160 & $0.1\times10^{-6}$ & 0.21 & $1.2\times10^{-6}$ & 2\\
NGC1333 IRAS4A2 & 320 & 3.5 & 0.11 & $<65$ & 100 & & & & 3 \\
L1157 & 250 & 3.6 & 0.04 & $<50$ & 112 & $0.8\times10^{-6}$ & 0.82 & $7.1\times10^{-6}$ & 4\\
HH 211 & 320 & 4.7 & 0.08 & 16 & 100 & $1.1\times10^{-6}$ & 1.1 & $4.7\times10^{-6}$ & 5\\
L1448 C & 320 & 8.4 & 0.08 & $50?$ &160 & $2.4\times10^{-6}$ & 5.0 & $11\times10^{-6}$ & 6 \\
HH 212 & 400 & 9.0 & 0.25 & 44 &135 & $1.3\times10^{-6}$ & 1.9 & $2.8\times10^{-6}$ & 7\\
\hline
\multicolumn{10}{p{11cm}}{
Here $D$ is distance, $L_{\mathrm{bol}}$ is bolometric luminosity, $M_\ast$ is mass of central protostar,
$r_d$ is disk radius, $v_j$ is jet velocity, $\dot{M}_j$ is mass-loss rate in the jet, $L_j$ is mechanical
luminosity of the jet, and $\dot{M}_{\mathrm{acc}}$ is accretion rate.
References: (1) \citet{Santiago2009}, \citet{Tafalla2017}
(2) \citet{Yen2010}, \citet{Green2013}, \citet{Bjerkeli2019}
(3) \citet{Choi2006,Choi2010,Choi2011}
(4) \citet{Green2013}, \citet{Kwon2015}, \citet{Podio2016}, \citet{Maury2019}
(5) \citet{Froebrich2005}, \citet{Lee2018HH211}, \citet{Jhan2016}
(6) \citet{Hirano2010}, \citet{Green2013}, \citet{Maury2019}
(7) \citet{Zinnecker1998}, \citet{Lee2017Disk,Lee2017Jet}
} \\
\end{tabular}
\end{table}
Table~\ref{tab:jetsource} lists the properties of a few well-studied
molecular jets from low-mass Class 0 protostars detected in both SiO and CO at
(sub)millimeter wavelengths. Note that although the jets listed here are
bipolar, there are also monopolar jets, e.g., in NGC1333-IRAS2A
\citep{Codella2014}. This table also lists the mechanical luminosity of
the jets calculated from the following equation
\begin{equation} L_j =
\frac{1}{2}\dot{M}_j v_j^2 \approx 0.82 \ \frac{\dot{M}_j}{10^{-6}
$M_\odot$ \ \textrm{yr}^{-1}} \left(\frac{v_j}{100 \; \textrm{km s}^{-1}}\right)^2 L_\odot
\end{equation}
\noindent and the accretion rate onto the protostars estimated from the
following equation
\begin{equation}
\dot{M}_{\mathrm{acc}} \sim
\frac{(L_{\mathrm{bol}}+L_j)R_\ast}{G M_\ast} \approx 6.44 \times 10^{-8} \
\frac{L_{\mathrm{bol}}+L_j}{L_\odot} \frac{$M_\odot$}{M_\ast} \ $M_\odot$ \
\textrm{yr}^{-1}
\end{equation}
\noindent assuming that both the bolometric luminosity and the jet's mechanical
luminosity come from accretion. Here we assume the radius of the central
protostars is $R_\ast \sim 2 \, R_\odot$ \citep{Stahler1988}. Rotating disks have
been detected or suggested in most of these sources, as required in current
magneto-centrifugal models of jet launching \citep{Shu2000,Konigl2000}.
Although the sample is small and the measurements have significant
uncertainties, we still can see some trends in comparison to the jets in the
later phase of star formation. The mass of the protostars is $0.05$--$0.25
$M_\odot$$, smaller than that in the Class I and II phases
\cite[see, e.g.,][]{Simon2000,Yen2017}. Thus, the mass in most of the protostars
here might be too small to have deuterium burning \citep{Stahler1988}. The
jet velocity is 61--160 km s$^{-1}${}, much smaller than that of the Class I and II
jets. The mass-loss rate is (0.7-- 2.4)$\times10^{-6} $M_\odot$ \mathrm{\
yr}^{-1}$ (except for B335 with a much smaller value), and is much larger
than those in the Class I and II phases. The jets have a mechanical
luminosity about 20--50\% of the bolometric luminosity. The accretion rate
is a few times the mass-loss rate. The disks and candidate disks mostly
have a radius $r_d <$ 50 au, smaller than those in the Class I phase, as
discussed in \citet{Maury2019}.
\begin{figure}[htb]
\centering
\includegraphics[angle=270, width=\textwidth]
{Macc_Mjet.pdf}
\caption{Observed mass-loss rate in the jets $\dot{M}_{j}$ versus accretion rate
$\dot{M}_{\mathrm{acc}}$ of our jet sources (blue stars)
compared with those in different classes of YSOs associated with jets and outflows
presented in \citet{Ellerbroek2013}.
The dashed lines indicate $\frac{\dot{M}_{j}}{\dot{M}_{\mathrm{acc}}}$ = 0.01 and 0.1.
\label{fig:MaccMjet}}
\end{figure}
The ratio of the mass-loss rate to the accretion rate has been used to probe
the mass-ejection efficiency that is then compared to the current
jet-launching models. Figure \ref{fig:MaccMjet} shows the mass-loss rate
versus the accretion rate for the Class 0 jets here (as marked with blue
stars) in comparison to those found in different classes of young stellar
objects (YSOs) associated with jets and outflows presented in
\citet{Ellerbroek2013}. As can be seen, our data points follow roughly the
trend found before, which shows the mass-loss rate and accretion rate
decreasing from the Class 0 to Class II (T-Tauri) phase from $10^{-6}$ to
$10^{-10} $M_\odot$ \mathrm{\ yr}^{-1}$ and from $10^{-5}$ to $10^{-9}
$M_\odot$ \mathrm{\ yr}^{-1}$, respectively. Our data points here are in
the higher end with higher mass-loss rate and accretion rate, because the
jet sources here are younger. A linear fit to our data points results in a
ratio $\frac{\dot{M}_{j}}{\dot{M}_{\mathrm{acc}}}\sim 0.19$, higher than
$\sim$ 0.12 obtained by fitting to all the Class 0/I data points. However,
since the data distribution is rather scattered and our sample is small,
further work with more accurate measurements are needed to check this.
Nonetheless, both ratios are consistent with magneto-centrifugal
jet-launching models. Note that in current jet-launching models, there will
be wide-angle wind components around the jets, thus the mass-loss rate
estimated here could be a lower limit of the true value, and so could be the
ratio.
With the ratio, we can estimate the magnetic lever arm parameter
\citep{Shu2000,Pudritz2007}, which is defined as
\begin{equation}
\lambda \equiv \left(\frac{r_A}{r_0}\right)^2 \approx \frac{\dot{M}_{\mathrm{acc}}}{\dot{M}_j}
\end{equation}
\noindent where $r_0$ is the launching radius of the jets in the disks and
$r_A$ is the Alfv{\'e}n radius along the streamline launched from $r_0$.
This parameter determines the extracted angular momentum and poloidal
acceleration, assuming a conservation of angular momentum and energy along
the streamline. For example, the poloidal acceleration determines the
terminal velocity the jet can achieve. Let $v_{\mathrm{kep}}$ be the
Keplerian velocity at the launching point, then $v_j \sim
\sqrt{2\lambda-3}\; v_{\mathrm{kep}}$, according to the current
magneto-centrifugal jet-launching models \citep{Shu2000,Pudritz2007}. Thus,
with a given protostellar mass and jet velocity, we can derive the launching
radius with
\begin{equation}
r_0 \sim (2\lambda-3)\; \frac{GM_\ast }{v_j^2}
\label{eq:ro_l}
\end{equation}
\noindent Based on Table~\ref{tab:jetsource} using the 6 sources with both
mass and jet velocity measured, the jet sources have a mean mass of $M_\ast
\sim 0.10 $M_\odot$$ and the jets have a mean velocity of $v_j \sim$ 130
km s$^{-1}${}. Then with a mean $\lambda \sim 1/0.19 \sim 5$ for our jet sources,
the mean jet launching radius is $\sim$ 0.04 au, which is $\sim$ $4 \, R_\ast$.
Further observations of a larger sample with better measurements of all
related quantities are needed to refine this.
\section{Jet rotation} \label{sec:jetrotation}
In current jet-launching models, the jets are launched by
magneto-centrifugal force from the innermost parts of the disks, and are
thus expected to be rotating and magnetized. In this way, angular momentum
can be carried away by the jets from the innermost parts of the disks,
allowing material there to actually fall onto the central protostars.
Therefore, measuring jet rotation is an important task to confirm these
models and the role of the jets in removing the angular momentum from the
disks. The measured amount of angular momentum in the jets can also help us
differentiate between the two competing models of jet launching, namely, the
X-wind model \citep{Shu2000} and the disk-wind model \citep{Konigl2000},
without spatially resolving the launching zones on the $\sim$ 0.05 au scale.
These two models predict different amounts of angular momentum to be carried
away by the jets. In particular, the jets in the X-wind model are launched
from the innermost edge of the disks and thus carry only a small amount of
specific angular momentum of $\lesssim 10$ au km s$^{-1}${} \citep{Shu2000}, while
the jets in the disk-wind model are launched from a range of radii from the
innermost edge out to $\sim$ 1 au, and thus can carry a larger amount of
specific angular momentum up to a few 100 au km s$^{-1}${} \citep{Pudritz2007}.
Previously, jet rotation was tentatively detected in T-Tauri jets in
optical [OI], [NII], and [SII] lines using the STIS instrument aboard the
{\it Hubble Space Telescope}, with a velocity sampling of $\sim$ 25 km s$^{-1}${}
pixel$^{-1}$ and a spatial sampling of $\sim$ 7 au (or \arcsa{0}{05} in
Taurus) pixel$^{-1}$ \citep{Bacciotti2002,Coffey2007}. The jets were found
to have a specific angular momentum up to a few 100 au km s$^{-1}${}, suggesting a
launching radius of 0.2 to 1 au \citep{Coffey2007}. Similar measurements
were also made towards a few Class 0 jets, e.g., HH 212 \cite[with a spatial
resolution of $\sim$ 140 au and a velocity resolution of $\sim$ 1
km s$^{-1}${},][]{Lee2008}, HH 211 \cite[with a spatial resolution of $\sim$ 77 au
and a velocity resolution of $\sim$ 0.35 km s$^{-1}${},][]{Lee2009}, and NGC 1333
IRAS 4A \cite[with a spatial resolution of $\sim$ 480 au and a velocity
resolution of $\sim$ 0.67 km s$^{-1}${},][]{Choi2011} in SiO at radio wavelengths,
finding a launching radius of 0.03 to 2 au. Tentative jet rotation was also
reported in the Class I jet HH 26 \cite[with a spatial sampling of $\sim$ 60
au and a velocity resolution of $\sim$ 34 km s$^{-1}${},][]{Chrysostomou2008} in
H$_2$, suggesting a launching radius of 2 to 4 au. Tentative jet rotation
was also reported in SO in the intermediate-mass source Ori-S6
\citep{Zapata2010}, suggesting a launching radius of $\sim$ 50 au. However,
all of these measurements are based on shock emission (e.g., SiO, SO, H$_2$,
and [OI]) and thus could be uncertain if the shock structures and
kinematics of the jets are not spatially resolved, for example, the
rotation in T-Tauri jets was later found to be false \citep{Coffey2012}. In
addition, asymmetric shock structure and jet precession can also produce a
velocity gradient mimicking a jet rotation.
Since the jets are highly collimated and narrow, a spatial resolution of
better than 10 au is needed to spatially resolve them. In addition, since
the rotation speed is small in the jets, a velocity resolution of $\sim$ 1
km/s is also needed to detect it. More importantly, we need to zoom in to
the innermost parts of the jets, where the (internal) shocks have not yet
developed significantly, where the jets are not yet interacting
significantly with the surrounding and cavity material, and where the jet
precession effect is not significant, so that the rotation signature can be
preserved and the velocity gradient across the jet axis can be attributed to
jet rotation. Moreover, since the jet radius decreases towards the central
source as discussed later, we also need to avoid zooming in too close to the
central source where the jets become unresolvable with current instruments.
\begin{figure}[htb]
\centering
\includegraphics[angle=270, width=\textwidth]
{jetrot_HH212.pdf}
\caption{ALMA results of the SiO jet in HH 212 within 100 au of the central source, adopted
from \citet{Lee2017Jet}.
The orange image shows the dusty disk observed at 850 $\mu$m{} \citep{Lee2017Disk}.
In (a), the green image shows the SiO jet. In (b), blueshifted emission and
redshifted emission are plotted separately to show the jet rotation around the jet axis.
The blue and red arrows show the disk rotation.
\label{fig:jetrot}}
\end{figure}
Thanks to the powerful Atacama Large Millimeter/submillimeter Array (ALMA)
with an unprecedented combination of high spatial resolution of $\sim$ 8 au
and high velocity resolution of $\sim$ 1 km s$^{-1}${}, \citet{Lee2017Jet} reported
a more reliable detection of jet rotation in the Class 0 jet HH 212 in SiO.
This jet is almost in the plane of the sky, allowing us to measure the jet
rotation without being affected by projection effects. In this jet, six
knots (N1, N2, N3 and S1, S2, S3) were detected in SiO within 100 au of the
central source, with three on each side, as shown in Figure
\ref{fig:jetrot}a. As shown in Fig.~\ref{fig:jetrot}b, the blueshifted
emission and redshifted emission are on the opposite sides of the jet axis,
showing a consistent velocity gradient with the same velocity sense as the
disk rotation, indicating that the velocity gradient is due to jet rotation.
Note that although the inner knots are less resolved, they still show a hint
of velocity gradient with the same velocity sense as the outer knots. This
measurement strongly supports the role of the jet in carrying away the
angular momentum from the disk. Based on the position-velocity diagrams
across the knots in the jet (Fig.~\ref{fig:jetpvHH212}), the specific
angular momentum in the knots is estimated to be $\sim$ 10 au km s$^{-1}${}, but
could be smaller because the knots are not well resolved spatially.
\begin{figure}[htb]
\centering
\includegraphics[angle=270, width=\textwidth]
{pvHH212.pdf}
\caption{Position-velocity (PV) diagrams cut across the knots (N1-N3 and S1-S3) in the HH 212 jet,
adopted from \citet{Lee2017Jet}.
The horizontal dashed lines indicate the peak (central)
position of the knots. The vertical dashed lines indicate roughly
the systemic (mean) velocities for the northern and southern jet components.
The red contours mark the 7$\sigma$ detections in knots N2, N3 and S3.
The solid lines mark the linear velocity structures across the knots
due to the jet rotation.
In (c), the blue ellipse is a tilted elliptical PV
structure expected for a rotating and expanding ring (see text).
\label{fig:jetpvHH212}}
\end{figure}
In the framework of magneto-centrifugal jet-launching models, the jet is the
central core of a magnetized wind. The launching radius at the foot point
of the jet in the accretion disk can be derived from the observed specific
angular momentum and the observed velocity of the jet at a large distance
from the central source, assuming a conservation of energy and angular
momentum along the field line. In addition, the wind can be assumed to have
enough energy to climb out of the potential well of the central star easily,
so that the kinetic energy of the wind is substantially greater than the
gravitational binding energy at the launching surface. With these
assumptions, \citet{Anderson2003} has derived a useful relation between the
angular momentum at large distance and the jet launching radius ($r_0$) at
the foot point of the field line in the disk:
\begin{equation}
r_0 \approx 0.7 \, \textrm{au} \;
\left(\frac{l_j} {100\, \textrm{au} \;\textrm{km s}^{-1}}\right)^{2/3}
\left(\frac{v_j} {100\, \textrm{km s$^{-1}$}}\right)^{-4/3}
\left(\frac{M_\ast} {1\,M_\odot}\right)^{1/3}
\label{eq:Anderson}
\end{equation}
\noindent where $l_j$ and $v_j$ are the specific angular momentum and
velocity of the jet at large distance. In the case of HH 212 where $l_j$ is
small, we need to add two correction terms to the above solution as in the following
\citep{Lee2017Jet}
\begin{equation}
r_0 \approx 0.7 \, \textrm{au} \;
\left(\frac{l_j} {100\, \textrm{au} \ \textrm{km s}^{-1}}\right)^{2/3}
\left(\frac{v_j} {100\, \textrm{km s$^{-1}$}}\right)^{-4/3}
\left(\frac{M_\ast} {1\, M_\odot}\right)^{1/3} \Big[1-\frac{2}{3}\eta + \frac{1}{9}
\eta^2\Big] \label{eq:jradius}
\end{equation}
\noindent
with
\begin{equation}
\eta = \frac{3}{2^{2/3}}(\frac{GM_\ast}{v_j l_j})^{2/3} \approx
0.38 \;\left(\frac{l_j} {100\, \textrm{au} \ \textrm{km s}^{-1}}\right)^{-2/3}
\left(\frac{v_j} {100\, \textrm{km s}^{-1}}\right)^{-2/3}
\left(\frac{M_\ast} {1\, M_\odot}\right)^{2/3} \,.
\end{equation}
The two correction terms can improve the accuracy of the launching
radius estimate in the case where the dimensionless parameter $\eta$ is not
much smaller than unity, particularly when the specific angular momentum
$l_j$ is relatively small, as is true for HH 212. Then with $l_j \sim 10 $
au km s$^{-1}${}, $v_j \sim$ 135 km s$^{-1}${}, and $M_\ast \sim 0.25 $M_\odot$$ (see
Table \ref{tab:jetsource}), the launching radius of the HH 212 jet is
estimated to be $\sim$ 0.04 au, consistent with current models of jet
launching. This results in a magnetic lever arm parameter of $\lambda \sim
3.2$ (see Eq. \ref{eq:ro_l}), indicating that the jet has a specific
angular momentum about 3 times that in the disk at the launching radius.
Since the mass-loss rate in the jet is estimated to be $\sim
1.3\times10^{-6} $M_\odot$$ (see Table~\ref{tab:jetsource}), the angular
momentum flux would be $1.3\times10^{-5} $M_\odot$$ au km s$^{-1}${} yr$^{-1}$.
Fig.~\ref{fig:jmodHH212} shows a possible scenario of the jet launching in
HH 212 from the accretion disk. In the innermost part of the disk, part of
the disk material is ejected, forming a bipolar jet carrying the angular
momentum away, allowing the disk material to fall onto the central protostar
through a funnel flow \citep{Shu2000}.
\begin{figure}[htb]
\centering
\includegraphics[width=\textwidth]
{disk_jet_press4.pdf}
\caption{Cartoon showing a possible jet launching scenario for HH 212.
In the innermost part of the disk, part of the
disk material is ejected, forming a bipolar jet carrying the angular momentum
away, allowing the disk material to fall onto the central protostar through a
funnel flow.
\label{fig:jmodHH212}}
\end{figure}
Recent ALMA observations at an unprecedented high resolution of $\sim$ 3 au
have detected a small SiO jet in B335 extending out from the central source
along the outflow axis \cite[][see also
Fig.~\ref{fig:BjerkeliB335}]{Bjerkeli2019}. This jet is located within
$\sim$ 4 au of the central source and has a radius of $\lesssim$ 1 au. No
clear rotation is detected in this jet, likely because the jet is not
resolved at current resolution. In addition, since the central source
seems to have a smaller protostellar mass of only $\sim 0.05 $M_\odot$$
\citep{Bjerkeli2019}, the jet rotation is expected to be smaller than that
seen in HH 212 and is thus more difficult to detect. The SiO line in this
jet has a linewidth of $\sim$ 13 km s$^{-1}${} \citep{Bjerkeli2019}. Assuming that
this linewidth is mainly due to jet rotation, then the specific angular
momentum of the jet is $\lesssim$ 6.5 au km s$^{-1}${}. If this is the case, then
the jet launching radius would be $\lesssim$ 0.02 au, assuming a jet
velocity of 160 km s$^{-1}${} (see Table~\ref{tab:jetsource}). Further
observations are needed to check this.
\begin{figure}[htb]
\centering
\includegraphics[width=0.9\textwidth]
{Bjerkeli2019_F6.jpg}
\caption{SiO maps of the jet in B335 obtained with ALMA at $\sim$ 3 au resolution
\citep{Bjerkeli2019}.
Blue and red contours show the blueshifted and redshifted emission maps
of the jet in SiO J=5-4.
\label{fig:BjerkeliB335}}
\end{figure}
\section{Jet radius near the central source: collimation} \label{sec:jetradius}
With the unprecedented angular resolutions of ALMA, we can now measure the
radius of molecular jets within 100 au of the central sources in order to
probe the collimation zones, as done before for T-Tauri jets in the optical.
Figure \ref{fig:jetradius} shows the radii of the jets in the two Class 0
sources HH 212 \cite[assumed to be half of the Gaussian widths reported
in][]{Lee2017Jet} and B335 \cite[measured from the SiO maps shown in
Fig.~\ref{fig:BjerkeliB335} adopted from][]{Bjerkeli2019}
in comparison to those in the two T-Tauri sources, RW Aur \citep{Woitas2002}
and DG Tau \citep{Agra-Amboage2011}, previously measured from the
high-velocity emission maps in the optical. As seen from the figure, the
Class 0 jets seem to be narrower and have a smaller expansion rate in jet
radius with distance than the T-Tauri jets. However, further work with
a larger sample is needed to confirm it. Interestingly, for both types of
jets, the jet radius could be increasing roughly parabolically with the
distance (i.e., with the square of the distance), as guided by the gray
shaded area in the figure, and thus roughly consistent with that predicted
in current magneto-centrifugal models of jet launching
\citep{Shu2000,Konigl2000}. In these models, the jets are collimated
internally by their own toroidal magnetic fields and expand roughly
parabolically with the distance. If the jet radius indeed expands roughly
parabolically with the distance, then the jets must have launched from the
innermost parts of the disks within a radius much less than 1 au of the
central sources, as guided by the gray shaded area.
\begin{figure}[htb]
\centering
\includegraphics[angle=270, width=\textwidth]
{jet_radius.pdf}
\caption{Observed jet radius versus the distance from the central source.
Here, DG Tau data points are adopted from \citet{Agra-Amboage2011},
RW Aur from \citet{Woitas2002}, HH 212
from \citet{Lee2017Jet}, and B335 from
\citet{Bjerkeli2019}. Note that in HH 212,
the radius of knot N3 in HH 212 has
been revised to be $\sim$ 6 au here after excluding the emission in the west
around $-3$ km s$^{-1}${} (see Fig.~\ref{fig:jetpvHH212}c, marked with a ``?''),
which could result from an interaction with the material in the cavity.
The gray shaded area is to guide the readers that the jet radius could be
increasing roughly parabolically with the distance.
The black dotted curve outlines the radius of a hollow cone
in an X-wind launched at a radius of 0.04 au.
The cyan dotted curve indicates the streamline within which the X-wind
contains 30\% of the total mass-loss rate.
The orange triangles indicate the radius of the SiO knots in HH 212, assuming
the SiO emission comes from a ring (see text).
\label{fig:jetradius}}
\end{figure}
\section{Are the jets hollow?}
In current magneto-centrifugal jet-launching models, there is a narrow
hollow cone in the jet center along the jet axis because of an intrinsic
expansion of the magneto-centrifugal winds coming from the disks. This
hollow cone is also supported by an axially opened stellar magnetic field.
However, there will be no hollow cone if the jets turn out to be stellar
winds launched by stellar magnetic field. Therefore, it is critical to
check for the existence of a hollow cone in the jets in order to determine
the jet-launching models.
Here we can compare the observed radius of the HH 212 molecular jet to the
predicted radius of the hollow cone in the X-wind model \citep{Shu2000}.
Since the launching radius of the jet is estimated to be $\sim$ 0.04 au, we
calculate the predicted radius of the hollow cone for this launching radius,
as shown as the black dotted curve in Fig.~\ref{fig:jetradius}. For
comparison, we also plot the streamline (the cyan dotted curve) within which
the X-wind contains 30\% of the total mass-loss rate. As can be seen, the
radius of the knots of HH 212 is larger than that of the hollow cone,
indicating that we can not rule out the existence of a hollow cone in this
jet. In addition, the radius of the knots is smaller than the cyan
streamline, supporting that the jet traces the inner core of the wind.
We can also check the existence of the hollow cone by investigating the
kinematics. The outer SiO knots (i.e., N2, S3, and N3) are roughly
resolved. The position-velocity (PV) diagrams across them show a roughly
linear PV structure (see Fig.~\ref{fig:jetpvHH212}), suggesting that they
could actually be rotating rings, as expected if there is a hollow cone in
the jet. For knot N3, the PV diagram even shows a tilted elliptical PV
structure (blue ellipse in Fig.~\ref{fig:jetpvHH212}c), as expected for a
rotating ring with an expansion \citep{Lee2018Dwind}. Although the inner
knots (i.e., N1, S1, and S2) are not resolved, knots N1 and S2 also seem to
show the similar linear PV structure, suggesting that they could also be
rotating rings. As a result, the kinematics across the knots in the HH 212
jet is consistent with a presence of a hollow cone in the jet. If this is
the case, the radius of the knot is better assumed to be the radius of the
ring rather than the Gaussian radius. Assuming that the SiO emission of each knot
comes from a ring, we can estimate the ring radius using the two ends of the
linear velocity structures in the PV diagrams. As can be seen, the ring
radius of the knots (marked as orange triangles) is slightly larger than
that of the hollow cone, supporting that the SiO knots come from the
innermost core of the wind and the jet is hollow.
Since knot N3 shows an expansion in the PV diagram, we can also check if its
expansion velocity can be consistent with the intrinsic expansion velocity
of a magneto-centrifugal wind. A rough fit (blue ellipse in Figure
\ref{fig:jetpvHH212}c) to the PV structure suggests an expansion velocity of
$\sim$ 3 km s$^{-1}${} in this knot. Since the jet has a velocity of $\sim$ 135
km s$^{-1}${}, this expansion velocity would require a streamline with a very small angle of
$\sim$ 1.3$^\circ${} to the jet axis, and thus consistent with a streamline
in the innermost core of the wind around the possible hollow cone.
\section{Shock formation and sideways ejection}
Studying the formation of the knots and bow shocks in the molecular jets
allows us not only to further constrain the origins of the jets but also to
understand the jet chemistry. Since the molecular jets, e.g., HH 211
\citep{Gueth1999,Lee2010HH211}, HH 212 \citep{Lee2008,Lee2015}, and L1448 C
\citep{Hirano2010}, appear to be continuous in the inner parts, the jet
ejection at the base should be continuous. Some of them show roughly equal
spacings in between knots and bow shocks, indicating that the knots and bow
shocks are produced by quasi-periodical variations in ejections. Since the
knots and bow shocks are well detected in, e.g., SiO and H$_2$, they trace
strong shocks, as in the optical jets \citep{Ray2007,Hartigan2011,Bally2016}.
Therefore, they can not merely trace quasi-periodical
ejections of enhanced jet densities, which alone would not produce shocks
\citep{Frank2014}. There must also be a quasi-periodical variation in
ejection velocity, so that a shock can be formed as the fast jet material
catches up with the slow jet material. A simple form of velocity variation
is a sinusoidal variation, presumably induced by an orbital motion of the
perturbations. An eccentric orbit can even cause an enhanced accretion rate
and thus mass-ejection rate near periastrons \citep{Reipurth2000,Benisty2013}.
Many hydrodynamical simulations have been performed to study the formation
of knots and bow shocks with a periodical sinusoidal variation of ejection
velocity \cite[e.g.,][]{Raga1990,Stone1993,Suttner1997,Lee2001}. In the
body of the jet, the fast material catches up with the slower material,
forming an internal shock (which consists of a forward shock, a backward
shock, and an internal working surface in between). This internal shock is
first seen as a knot. As it propagates down along the jet axis, because of
sideways ejection in the shock, it expands laterally and grows to a wider
knot and then a bow shock and then an internal shell of post-shock gas
closing back to the central source \citep{Lee2001}. Each bow shock will
show a spur velocity structure in the PV diagram along the jet axis
\citep{Lee2001,Ostriker2001}, because the gas velocity decreases rapidly
away from the bow tip. If the amplitude of the velocity variation is
larger, the resulting bow shock is bigger.
A periodical variation in jet velocity has been suggested in the jet in
L1448 C \citep{Hirano2010}. In particular, each SiO knot has its higher
velocity in the upstream (closer to the jet source) side and lower velocity
in the downstream side. The opposite velocity gradient is always seen in
the faint emission between the knots. Such a velocity pattern is likely to
be formed by a periodical variation in the ejection velocity, as seen in the
simulations \citep{Stone1993,Suttner1997}. In HH 211 (a jet close to the
plane of the sky), the innermost pair of jetlike structures BK1 and RK1,
which consists of a chain of smaller subknots, show a velocity range
decreasing with distance \citep{Lee2010HH211}, also as expected for the
knots being formed by a periodical variation in velocity
\citep{Suttner1997}. The two knots further away are believed to be newly
formed internal shocks, each showing a pair of backward and forward shocks
expanding longitudinally with time \citep{Jhan2016}. In these jets, a
velocity variation of 20--30 km s$^{-1}${} has been suggested to produce the knots
that have a spacing corresponding to a period of a few ten yrs
\citep{Hirano2010,Jhan2016}.
\begin{figure}[htb]
\centering
\includegraphics[width=0.4\textwidth]
{Tafalla_f1.pdf}
\includegraphics[width=0.4\textwidth]
{Tafalla_f5.pdf}
\caption{Molecular jet in IRAS 04166+2706:
(Left) A chain of knots are detected in CO along the jet axis \citep{Santiago2009}.
(Right) A schematic diagram showing the sideways ejection in
knots B6 and R6, as concluded from
a detailed ALMA kinematic study \citep{Tafalla2017}.
\label{fig:sideways}}
\end{figure}
A clear example showing the sideways ejection of the internal shocks is the
molecular jet in IRAS 04166+2706. It consists of at least 7 pairs of knots
seen in SiO and CO, with the width increasing with distance from the central
source \cite[][see also Fig.~\ref{fig:sideways} left]{Santiago2009}. The
velocity field along the jet axis shows a sawtooth pattern for each knot.
This pattern, together with a systematic widening of the knots with distance
to the central source, is consistent with them tracing the
laterally-expanding internal shocks viewed at a high inclination angle to
the plane of the sky \citep{Stone1993,Suttner1997}. At a large distance
from the central source, the knots, B6 and R6, have grown to possess
bow-like structures. A detailed kinematic study of this pair of knots with
ALMA confirmed that they are internal bow shocks where material is being
ejected laterally away from the jet axis \citep{Tafalla2017}, as shown in
Fig.~\ref{fig:sideways}.
Sideways ejection is also detected in other jets. In HH 212, the sideways
ejection is clearly seen in the two spatially resolved internal bow-like
knots SK4 and SK5 in CO and SiO \citep{Lee2015}. Since the HH 212 jet is
almost in the plane of the sky, the knots are associated with arclike
velocity patterns \citep{Stone1993}, instead of a sawtooth velocity pattern
seen in a highly inclined jet. Nested internal shells are also
seen in CO, with each extending from a H$_2$ bow shock back to the central
source \citep{Lee2015}, as seen in the simulations. Sideways ejection is
also detected in CARMA-7. In this source, a chain of small CO bow shocks is
seen along the jet axis, with each bow shock associated with a spur velocity
structure in the PV diagram along the jet axis \cite[][see also
Fig.~\ref{fig:pvC7}]{Plunkett2015}, consistent with laterally-expanding
internal bow shocks. The spur structure is slightly tilted because of a
small inclination of the jet to the plane of the sky.
\begin{figure}[htb]
\centering
\includegraphics[width=0.6\textwidth]
{pvCARMA7.jpg}
\caption{Position-velocity diagram of the CO jet in CARMA-7 along the jet axis \citep{Plunkett2015}.
Labels B1 to B11 and R1 to R11 mark the positions of the internal knots and bow shocks.
\label{fig:pvC7}}
\end{figure}
\section{Periodical variations and their possible origins}
As discussed earlier, some of the molecular jets show roughly equal spacings
in between knots and bow shocks that can be due to quasi-periodical
variations in ejections, allowing us to probe quasi-periodical perturbations
of the underlying accretion process in the disks. In addition, the rotating
disks associated with some molecular jets are now resolved and found to have
a radius smaller than 50 au (see Table~\ref{tab:jetsource}), providing a
strong constraint on the possible perturbations in the disks. Therefore, we
can now further constrain the origins of the quasi-periodical variations in
ejections.
The period of variation can be obtained by dividing the knot or bow shock
spacing by the jet velocity. Previously in the optical jets in the later
phase, at least two periods of variations, one short and one long, have been
found to operate at the same time in one single jet. For example,
\citet{Raga2002} found two periods of 270 yr and 1400 yr in HH 34, and two
periods of 60 yr and 950 yr in HH 111. Similarly in the molecular jet HH
212 in the early phase, \citet{Zinnecker1998} also found two periods, one
for the inner knots with a spacing of $\sim$ 1700 au and the other for the
prominent bow shocks with a spacing of $\sim$ 17200 au (see Figure
\ref{fig:spacingHH212}). Adopting a mean jet velocity of $\sim$ 135 km s$^{-1}${}
(see Table 1), these spacings correspond to periods of 60 yrs and 605 yrs,
respectively. In recent high-resolution ALMA observations, an additional
knot spacing of $\sim$ 30 au (see Fig.~\ref{fig:jetradius}) with a
corresponding period of $\sim$ 1 yr is also detected near the jet source.
Such a short period of variation was also detected before in, e.g., the DG
Tau jet which has a period of $\sim$ 2.5 yr \citep{Agra-Amboage2011}. It is
believed that such a short period of variation will be found in more jets
near the central sources when more high-resolution observations are
obtained.
\begin{table}
\small
\centering
\caption{Periodical variations in well-defined molecular jets}
\label{tab:jetperiod}
\begin{tabular}{lllll}
\hline Source & Spacing & Period & a & References \\
& (au) & ($yr$) & (au) & \\ \hline\hline
IRAS 04166+2706 & 1065 & 83 & ? & 1 \\
& 5330 & 415 & ? & 1 \\
NGC1333 IRAS4A2 & 1600 & 76 & 9 & 2 \\
L1157 & 15000 & 660 & 26 & 3 \\% 60" , Kwon et al, 2015, 0.04 Ms if edge-on, Podio et al 73 degree to the line of sight
HH 211 & 256 & 12 & 2 & 4 \\
& 800 & 38 & 5 & 4 \\
& 4300 & 204 & 15 & 4 \\
L1448 C& 680 & 20 & 3 & 5 \\
& 8000 & 240 & 17 & 5 \\
HH 212 & 30 & 1 & 0.6 & 6 \\
& 1700 & 60 & 10 & 6 \\
& 17200 & 605 & 45 & 6 \\
\hline
\multicolumn{5}{p{11cm}}{
Here $a$ is the corresponding orbital radius of the perturbation defined by Eq. \ref{eq:binsep}.
References:
(1) \citet{Santiago2009},
(2) \citet{Choi2011}
(3) \citet{Kwon2015}
(4) \citet{McCaughrean1994}, \citet{Lee2010HH211},
(5) \citet{Hirano2010}
(6) \citet{Zinnecker1998}, \citet{Lee2017Jet}
} \\
\end{tabular}
\end{table}
Table \ref{tab:jetperiod} lists the periods of variations estimated from six
well-defined molecular jets. These periods range from 1 yr to 660 yrs and
can be broadly divided into 3 different groups: periods of a few yrs
associated with knots near the jet sources, periods of a few ten yrs
associated with knots and small bow shocks in the inner parts of the jets,
and periods of a few hundred yrs associated with prominent bow shocks (see
Fig.~\ref{fig:spacingHH212} for HH 212 and Fig.~\ref{fig:spacingHH211}
for HH 211). Recent ALMA observations of a jet in the Class 0 protostar
CARMA-7 show a clear chain of small CO bow shocks along the jet axis, with a
spacing of $\sim$ 680 au \citep{Plunkett2015}, or a period of $\sim$ 32 yrs
assuming a velocity of 100 km s$^{-1}${}, and thus belongs to the period group of a
few ten yrs. These periods could come from periodical perturbations of the
underlying accretion in the disks on the orbital time scales. In order to
study this possibility, we derive the corresponding orbital radii for these
periods with the following equation from the Kepler's third law of orbital motion:
\begin{equation}
a = \left(\frac{G M_\ast T^2}{4 \pi^2}\right)^{1/3} = \left(\frac{M_\ast}{M_\odot}\right)^{1/3}
\left(\frac{T}{\textrm {yr}}\right)^{2/3} \, \textrm{au}
\label{eq:binsep}
\end{equation}
\noindent where $T$ is the period and $M_\ast$ is mass of the central protostar.
\begin{figure}[htb]
\centering
\includegraphics[angle=180, width=\textwidth]
{H2_CF.pdf}
\caption{Spacings between knots and major bow shocks in HH 212 seen in H$_2$ \citep{Zinnecker1998}.
Red lines show the spacing for the inner knots and black lines for the prominent bow shocks.
\label{fig:spacingHH212}}
\end{figure}
\begin{figure}[htb]
\centering
\includegraphics[angle=270, width=\textwidth]
{jet_spacing.pdf}
\caption{Spacing between major bow shocks in HH 211 seen in H$_2$ \citep{Hirano2010}.
\label{fig:spacingHH211}}
\end{figure}
As can be seen from Table \ref{tab:jetperiod}, periods of a few hundred yrs
correspond to orbital radii of a few tens of au. Interestingly in HH 212
and HH 211, the implied orbital radii of the perturbations are similar to
the radii of their rotating disks, which have a radius of $\sim$ 44 and 16
au (see Table \ref{tab:jetsource}), respectively. For other jet sources,
their disks are not resolved yet and further observations are needed to
check this possibility. Therefore, it seems that the longest period of
perturbations in the disks can come from the outer edge of the disk,
probably induced by gravitational instability (GI) powered by envelope
accretion. Such instability has been detected in the embedded Class I disk
in HH 111, which shows a pair of spiral arms extending from the outer edge
of the disk to the inner disk where the Toomre Q parameter is of the order
of unity \citep{Lee2019HH111}. An accretion shock is also detected around
that disk in SO, indicative of an active accretion from the envelope
\citep{Lee2016HH111}. In simulations, protostellar disks could be
periodically gravitationally unstable, forming spiral arms in the disks and
thus producing enhanced accretion rates onto the protostars periodically
\citep{Tomida2017}. These periodical enhanced accretion rates can then
cause periodical major ejections (such as FU Ori-like bursts) of the jets,
producing the prominent bow shocks in the jets. A recent study of a
few FU Ori-type objects have detected arms and fragmented structures that
can be attributed to gravitationally unstable disks \citep{Takami2018}, also
supporting this possibility. In this scenario, older protostars, which
have larger masses and disks, would have periods of a few thousand yrs. For
example, HH 111, which has a protostellar mass of $\sim 1.5 $M_\odot$$ and
a disk radius of $\sim$ 160 au \citep{Lee2019HH111}, would have a period of
$\sim$ 1650 yrs. HH 111 has a length of more than 10 pc
\citep{Reipurth2001}, and deeper observations in H$_2$ are needed to check
for this period.
The next group of periods has a mean of $\sim$ 50 yr, with a mean
corresponding orbital radius of $\sim$ 6 au. This radius could be the
location where the deadzone is located with negligible ionization. One
possible disturbance source is a close binary companion, which could evolve
from a secondary fragmentation in a second core collapse
\citep{Machida2008}. The possibility of a binary companion was also
suggested in, e.g., HH 212 \citep{Lee2007HH212}, L1448 C \citep{Hirano2010},
and HH 211 \citep{Lee2010HH211} because of the wiggle in their jets. Other
origins could be stellar magnetic cycles or global magnetospheric
relaxations of the star-disk system \citep{Frank2014}. Alternatively,
perhaps GI can penetrate to the inner parts of the disks, forming dense
(protoplanetary) clumps in spiral arms, causing enhanced accretion rates onto
the protostars \citep{Vorobyov2005}.
The shortest period is $\sim$ 1 yr in HH 212, corresponding to an orbital
radius of $\sim$ 0.6 au. Similarly in the DG Tau jet, which is driven by a
T-Tauri star with a mass of $\sim 0.7 $M_\odot$$, the corresponding
orbital radius for its 2.5 yr period is $\sim$ 1.5 au
\citep{Agra-Amboage2011}. It is not clear what origin can introduce this
perturbation on this small orbital scale. It could be due to a periodical
turn on of magneto-rotational instability (MRI) \citep{Balbus2006} in the
innermost parts of the disks where the temperature is high (1000 K) and the
ionization is sufficient \citep{Audard2014}. It could also be related to
the phase transition from dust-rich to dust-free region in the disks,
because the jets are likely launched from dust-free regions in the innermost
parts of the disks, as discussed later.
\section{Wiggles and binaries?}
Two types of wiggles are detected in protostellar jets in the early phase,
as in the later phase. One is point-symmetric (i.e., S-shaped) wiggles that
could be due to precession of the jets \cite[e.g.,][]{Raga1993}, which in
turn could be due to precession of the accretion disks because of tidal
interactions in noncoplanar binary systems \cite[see, e.g.,][]{Terquem1999}.
The other is reflection-symmetric (i.e., C-shaped or W-type) wiggles
\citep{Fendt1998,Masciadri2002,Lee2010HH211,Moraghan2016} that could be due to orbital
motion of the jet sources around a binary companion. Both types of wiggles
can take place simultaneously in single jets \citep{Raga2009}.
Unlike most of the optical jets, molecular jets are seen on both sides,
allowing us to better determine the types of the wiggles and derive the
binary properties. Combining with the total mass derived from other
methods, we can also derive the mass of each binary component. In addition, the
periods of these wiggles could be compared to those of the ejection
variations in producing the knots and bow shocks in the jets, providing
further constraints on the potential binary formation in the central regions
of star formation in the early phase. In particular, formation of a close
binary can start in the early phase before the collapse of a second core
and even during the protostellar phase because of angular momentum
redistribution at the center of the system \citep{Machida2008}.
The young source L1157 has a S-shaped outflow and thus a precessing jet was
proposed to produce its outflow morphology \cite[][see also Figure
\ref{fig:L1157Prec} Left]{Gueth1996,Bachiller2001,Takami2011}. Recently,
\citet{Kwon2015} proposed that there could be two precessing jets, each
along one side of the outflow cavity walls, to produce the outflow
morphology. However, \citet{Podio2016} detected only a single jet in SiO
and CO within 200 au of the central source propagating along the symmetry
axis of the outflow lobes (see Fig.~\ref{fig:L1157Prec} Middle and Right),
confirming the previously proposed one precessing jet model
\citep{Gueth1996,Bachiller2001}. The precession period is $\sim$ 1640 yrs,
about twice that estimated from the bow shock separation (see Table
\ref{tab:jetperiod}).
\begin{figure}[htb]
\centering
\includegraphics[width=0.45\textwidth]
{Podio2016_F1left.jpg}
\includegraphics[width=0.45\textwidth]
{Podio2016_F1right.jpg}
\caption{L1157 jet and outflow adopted from \citet{Podio2016}. (Left)
A precessing jet model plotted on the Spitzer/IRAC 8 $\mu$m{} map of the outflow.
(Middle) Blueshifted and redshifted CO emission of the jet.
(Right) Blueshifted and redshifted SiO emission of the jet.
\label{fig:L1157Prec}}
\end{figure}
One mechanism to produce a precessing jet is disk precession induced by
tidal interaction with a binary companion in a non-coplanar orbit
\cite[e.g.,][]{Terquem1999}. The jet source has a low-mass accretion disk
(with an outer radius $r_d$), which is precessing as a result of the
perturbations due to the companion. Let the jet source have a mass $M_j$
and a companion a mass $M_c$ and the binary has a separation $a$ in a
circular orbit. The ratio of precession period $\tau_p$ to orbital period
$\tau_o$ can then be given by
\begin{equation}
\frac{\tau_p}{\tau_o}
=\frac{32}{15 \cos \alpha} \left(\frac{a}{r_d}\right)^{3/2} \left(1+\frac{M_c}{M_j}\right)^{1/2} \left(\frac{M_j}{M_c}\right)
\end{equation}
\noindent where $\alpha$ is the half-opening angle of the resulting
precession cone \citep{Terquem1999,Anglada2007,Raga2009}.
\citet{Terquem1999} suggested that the tidal truncation of the accretion
disk will lead to a ratio $a/r_d=2-4$. The jet precession in L1157 could be
due to this disk precession. Taking a mean ratio of $a/r_d \sim 3$ and
assuming $M_C \sim M_J$, we have $\frac{\tau_p}{\tau_o} \sim 15$ and thus
the orbital period $\tau_o \sim$ 110 yrs. With Equation \ref{eq:binsep},
the binary separation is estimated to be $\sim$ 8 au, and thus the disk
radius is $\sim$ 2.7 au. Further work is needed to check this possibility.
Another possible mechanism is an asymmetric envelope accretion onto the
disk. Simulations have shown that a large misalignment between the magnetic
axis and rotation axis in the star-forming core can produce a flattened
infalling envelope and a rotating disk at the center. However, the major
axis of the flattened infalling envelope will be misaligned with the major
axis of the disk \citep{Hirano2019}, as also seen in the observations of HH
211 \citep{Lee2019HH211}. In this case, the envelope accretion onto the
disk will be asymmetric, causing the disk and the jet to precess
\citep{Hirano2019}. Indeed, the jet axis in HH 211 has already been
found to have precessed by $\sim$ 3$^\circ${} in the past
\citep{Eisloffel2003}, supporting this scenario.
On the other hand, reflection-symmetric wiggle has been suggested in the jets
HH 211 \citep{Lee2010HH211,Moraghan2016} and HH 111 \citep{Noriega2011}, and could be due
to an orbital motion of the jet sources. As discussed in
\citet{Masciadri2002} and \citet{Lee2010HH211}, we can derive the period
$P_o$, velocity $v_o$, and radius $R_o$ of the orbit of the jet source
around the companion from three measurable quantities: the jet velocity
$v_j$, the half-opening angle $\kappa$ and the periodic length (i.e.,
wavelength) $\Delta z$ of the wiggle, with the following equations:
\begin{equation}
P_o = \frac{\Delta z}{v_j} \;,\hspace{1cm} v_o \approx \kappa \; v_j \;,
\hspace{1cm}R_o \approx \frac{\kappa \Delta z}{2 \pi}
\end{equation}
\noindent Again, assuming the jet source has a mass $M_j$ and the
companion a mass $M_c$, and $M_j=m M_c$, then the binary separation
would be $a=(1+m)R_o$.
From Kepler's third law of orbital motion,
the total mass of the binary would be
\begin{eqnarray}
M_t \approx 9.5\times 10^{-4}\ (1+m)^3
\times \left(\frac{\eqt{v}{j}}{100 \;\textrm{km s}^{-1}{}}\right)^2 \left(\frac{\kappa}{1^\circ}\right)^3
\frac{\triangle z}{100 \, \mathrm{au}}\, M_\odot
\end{eqnarray}
\noindent In HH 211, with $\Delta z \sim 1750$ au, $\kappa \sim 0.0094$ (or
0.54$^\circ$), $v_j \sim 110$ km s$^{-1}${}, and $M_t \sim 0.08 $M_\odot$$
\citep{Lee2019HH211}, we have $P_o \sim$ 83 yrs, $v_o \sim 0.94$ km s$^{-1}${},
$R_o \sim 2.6$ au, $m \sim 2.1$, and $a \sim 8.2$ au. The mass of the jet
source would be $\sim 0.053 $M_\odot$$. The orbital period is $\sim$ 2
times that generates the knot separation, which is $\sim$ 38 yrs (see Table
\ref{tab:jetperiod}).
It is possible that the orbit is eccentric and two perturbations could
generate two knots per orbit. This could occur either through the
periastron passage to the companion and a close approach to the inner edge
of the circumbinary disk, or two close approaches to the inner edge of the
circumbinary disk per eccentric orbit
\citep{Moraghan2016}. Since the binary separation here is
about half of the disk radius, observations at higher resolution are needed
to resolve the binary and check this possibility. On the other hand, in HH
111, with $\Delta z \sim 86400$ au, $\kappa \sim 0.013$ (or 0.74$^\circ${}),
and $v_j\sim$ 240 km s$^{-1}${} \citep{Noriega2011}, and a total mass of $\sim 1.5
$M_\odot$$ \citep{Lee2010HH111,Lee2016HH111}, we have $P_o \sim$ 1710 yrs,
$v_o \sim 3.1$ km s$^{-1}${}, and $R_o \sim 178$ au, $m\sim -0.1$, and $a \sim 163$
au. Thus, within the uncertainty, the jet source has almost no mass,
inconsistent with it driving a powerful jet longer than 10 pc
\citep{Reipurth2001}. In addition, the binary separation would be similar
to the disk radius, which is $\sim$ 160 au \citep{Lee2019HH111}, but no
binary companion was detected there around the edge of the disk
\citep{Lee2019HH111}. Thus, further work is needed to check the
reflection-symmetric wiggle in the HH 111 jet.
\section{Magnetic fields in the jets}
For high-mass protostars, synchrotron radiation have been detected in a few
protostellar jets, e.g., HH 80-81 \citep{Marti1993} and W3(OH)
\citep{Wilner1999}, at centimeter wavelengths, indicating a presence of
magnetic fields in the jets. Magnetic field morphology can thus be derived
from the polarization pattern of the synchrotron radiation. So far,
polarization has only been detected in HH 80-81 \citep{Carrasco2010}. It was
detected at a spatial resolution of $\sim$ 20,000 au towards the jetlike
structures at a large ($\gtrsim$ 30,000 au) distance from the central
protostar, where the underlying jet interacts with the ambient material
\citep{Rodriguez-K2017}. The implied magnetic fields there are mainly
poloidal, with a field strength of $\sim$ 0.2 mG. There is also a hint of
toroidal fields near the edges of the jetlike structures. However, more
sensitive observations are needed to detect the polarized emission and thus
the field morphology in the underlying jet itself, which is much narrower with
a radius of $\sim$ 2000 au \citep{Rodriguez-K2017}. Recent observations in
W3(H$_2$O) \citep{Goddi2017} also detected linear and circular polarizations
in the water masers at the center of the synchrotron jet within tens to
hundreds of au from the central source. They suggested that the magnetic
field could evolve from having a dominant component parallel to the outflow
velocity in the pre-shock gas, with field strengths of a few tens of mG, to
being mainly dominated by the perpendicular component of a few hundred of mG
in the post-shock gas where the H$_2$O masers are excited.
For low-mass protostars, the jets are less energetic and thus do not have
strong enough synchrotron radiation and water masers for polarization
detection with current instruments. Fortunately, when they are young, they
have a high content of molecular gas because of high mass-loss rate
\citep{Glassgold1991,Shang2006,Cabrit2007,Lee2007HH212,Hirano2010}, allowing
us to map their magnetic fields using the linear polarization in molecular
lines with the so-called Goldreich--Kylafis (GK) effect
\citep{Goldreich1981,Goldreich1982}. In the presence of a magnetic field, a
molecular rotational level splits into magnetic sublevels, producing a line
polarization with its orientation either parallel or perpendicular to the
magnetic field. Recent successful detection of this GK effect in the SiO line
in HH 211 confirmed this method of mapping the field morphology in the jets
from the low-mass protostars \citep{Lee2018Bjet}. In the future, sensitive
observations of the Zeeman effect in molecular lines will allow us to derive the
strength of the magnetic fields \citep{Cazzoli2017}.
\begin{figure}[htb]
\centering
\includegraphics[angle=270, width=\textwidth]
{jet_SiO_H2_Review.pdf}
\caption{SiO J=8-7 line polarization detection in the HH 211 jet \citep{Lee2018Bjet}.
The asterisk marks the position of the central driving source.
Gray image shows the outflow and the outer part of the jet in H$_2$ adopted
from \citet{Hirano2006}. Blue image
and red image show the blueshifted and redshifted jet components in SiO, respectively.
Yellow line segments indicate the polarization orientations seen in SiO.
\label{fig:Bjet}}
\end{figure}
Figure \ref{fig:Bjet} shows the SiO line polarization in J=8-7 transition
detected in the HH 211 jet within a few hundred au of the protostar. The
jet is best seen in SiO in the J=8-7 transition. At this transition, the
optical depth is close to 1 \citep{Lee2009} and the collision rate is lower
than the radiative transition rate for a typical jet density of
$10^{6}$--$10^{7}$ cm$^{-3}${}, both optimal for polarization from the GK effect
\citep{Goldreich1981,Goldreich1982}. As can be seen, the polarization
orientations are all roughly parallel to the jet axis. The implied magnetic
fields could be either mainly poloidal or mainly toroidal and additional
polarization observations in another SiO transition line are needed to
resolve this ambiguity in the field morphology, as done before in a CO outflow
\citep{Ching2016}. It could be mainly toroidal, as suggested in current
jet-launching models, in order to collimate the jet at large distances. The
field strength (projected on the plane of the sky) was estimated to be
$\sim$ 15 mG \citep{Lee2018Bjet}. This field strength is about 2 times the
toroidal field strength expected from a typical X-wind model \citep{Shu1995}
for a low-mass protostellar jet like HH 211, which is reasonable considering
a shock compression in the jet and all the uncertainties in the
measurements.
\defn_{\textrm{\scriptsize H}_2}{n_{\textrm{\scriptsize H}_2}}
The jets may have poloidal fields as well. In HH 212, the jet is found to be
wiggling, but with the amplitude of the wiggle being saturated at some
distance, inconsistent with a typical jet precession that has an amplitude
increasing linearly with the distance. The saturation in the amplitude may
suggest a current-driven kink instability in the jet \citep{Cerqueira2001},
see, e.g., Fig.~3 in both \citet{Cerqueira2001} and \citet{Mizuno2014}.
For the kink instability to take place, we have the Kruskal--Shafranov
criterion $|B_p/B_\phi| < \lambda/ 2 \pi A_m$ \cite[e.g.,][]{Bateman1978},
where $A_m$ is the maximum displacement and $\lambda$ is the wiggle
wavelength. In HH 212, with $A_m \sim$ \arcsa{0}{1} and $\lambda \sim$
\arcsa{5}{6}, we have $|B_p/B_\phi| < 9$. Therefore, this kink instability,
if in action, could be initiated in the central part of the jet, where the
magnetic field is dominated by the poloidal field \citep{Pudritz2012}. This
poloidal field could serve as a ``backbone" to stabilize the jet
\citep{Ouyed2003}. The toroidal field dominates only near the jet edges in
order to collimate the jet.
\section{Origin of molecular gas in the jets}
Unlike those in the Class I and Class II phases, the jets in the Class 0
phase appear to be mainly molecular and well detected in CO, SiO, and SO.
Note that near the base of the jets, free-free emission from ionic gas can
also be detected in the Class 0 phase
\citep{Rodriguez1997,Reipurth2002,Reipurth2004,Rodriguez2014,Tobin2016Per}. It is possible that the
high molecular content in the Class 0 phase is due to the high mass-loss
rate and thus the fast formation rate of the molecular gas. Since the
mass-loss rate in the jets is high ($\sim 10^{-6} $M_\odot$$ yr$^{-1}$),
molecules such as CO, SiO, and SO could have formed via gas-phase
reactions in an initially atomic jet close to the launching point within 0.1
au \citep{Glassgold1991}. The abundances of SiO and SO in the gas phase are
found to be highly enhanced in the jets as compared to those in the
quiescent molecular clouds, even close to within 30 au of the central
sources where the dynamical timescale is $<$ 1 yr
\citep{Lee2017Jet,Lee2018Dwind,Bjerkeli2019}.
As discussed earlier, based on the ejection efficiency, jet rotation, and
the expansion of jet radius near the jet sources, the molecular jets in the
Class 0 phase should be launched within $\sim$ 0.1 au of the jet sources.
Most of these jet sources have a bolometric luminosity $\gtrsim 1
$L_\odot$$, suggesting that the dust sublimation radius in their accretion
disks is $\gtrsim$ 0.1 au \citep{Millan2007}, which is outside the inferred
jet launching radius. In the case of HH 212, \citet{Tabone2017} proposed
that the SiO jet could be the innermost part of a disk wind launched at
$\sim$ 0.05--0.2 au, by modeling the PV structures of the SiO emission
across the jet axis (see their Figs.~2e and 2f). Since their model PV
structures show emission peaks at the two high-velocity ends instead of at the
low velocity on the jet axis, the major part of the jet must still be
launched significantly closer than 0.2 au. In any case, since the jet
source in HH 212 has a bolometric luminosity of $9 $L_\odot$$, the
sublimation radius should be $\sim$ 0.2 au \citep{Millan2007}, and thus
still larger than the launching radius.
It is likely that elements such as Si, S, C, and O, are already released
from the grains into the gas phase at the base of the jets. Since the jets
are well collimated with a high mass-loss rate of $\sim 10^{-6}
$M_\odot$ \mathrm{\ yr}^{-1}$ (see Table \ref{tab:jetsource}), SiO, CO, and SO can
form quickly within $\sim$ 0.1 au because of the high density in the jet
\citep{Glassgold1991}. The jets are bright in SiO J=8-7, which has a high
critical density of $\gtrsim 10^7$ cm$^{-3}${}, further supporting the high density in
the jets.
In addition, the Si$^+$ recombination and SiO
formation are expected to be faster than the photodissociation caused by
possible far-ultraviolet radiation of the central protostar
\citep{Cabrit2012}. Since these molecules are fully released from the dust
grains at the base, the observed abundances in the jets are higher
as compared to those in the quiescent molecular clouds.
For the jet sources with a bolometric luminosity $< 1 $L_\odot$$, e.g.,
B335 and IRAS 04166+2706 (see Table \ref{tab:jetsource}), the dust
sublimation radius is $<$ 0.1 au \citep{Millan2007}. However, it is still
possible that their jets are launched from a dust-free zone, because their
jet launching radius could be smaller, as discussed earlier. In B335, since
the mass-loss rate is only $\sim$ $10^{-7} $M_\odot$ \mathrm{\ yr}^{-1}$, further
work is needed to check if the density in the jet is high enough to form the
molecules in the gas phase. In this source, the SiO J=5-4 emission is only
detected within a few au of the jet source, probably because the mass-loss
rate is low.
In case the jets are launched from dusty zones, SiO abundance in the jets
can be enhanced as a consequence of grain sputtering or grain-grain
collisions in the shocks releasing Si-bearing material into the gas phase,
which reacts rapidly with O-bearing species (e.g., O$_2$ and OH ) to form
SiO \citep{Schilke1997,Caselli1997}. Another possible explanation is that
the SiO molecules existed on the grain mantles and are released into the gas
phase by means of shocks as suggested by \cite{Gusdorf2008}. Similarly,
sulfur can be released from dust grains in the form of H$_2$S in the shocks
and that is then oxidized to SO \citep{Bachiller2001}. Alternatively, the
SO molecules might be abundant on dust grains and directly released from
grains in the shocks \citep{Jimenez2005,Podio2015}.
\section{Disk winds around the jets?}
As discussed earlier, protostellar jets can carry angular momentum away from
the innermost region of the disks, allowing the disk material to feed the
central protostars. However, other mechanisms are also needed to transfer
angular momentum outward within the disks or carry it away from the disks,
so that the disk material can be transported to the innermost region from
the outer region of the disks. For example, MRI can turn on in the inner
part of the disks where the temperature is high ($\gtrsim$ 1000 K)
\citep{Audard2014} to transfer angular momentum outward. GI can also be
excited in the outer part of the disks to transfer angular momentum outward,
as suggested in many simulations \citep{Bate1998,Tomida2017} and by the
detections of a pair of spirals in the embedded disk in HH 111
\citep{Lee2019HH111} and probably in the older disk in Elias 2-26
\citep{Perez2016}. Besides, low-velocity extended tenuous disk winds can be
present as well to carry angular momentum away from the disks. In
particular, previous observations have shown that wide-angle radial winds
are needed to drive molecular outflows
\citep{Lee2000,Lee2001,Hirano2010,Arce2013}, especially in the later phase
of star formation \citep{Lee2005L43}.
In the two popular magneto-centrifugal jet-launching models from the disks,
e.g., the X-wind model and the disk-wind model, the jets are merely the
central cores of the winds. Therefore, in addition to the jets, the
models also predict wide-angle tenuous winds around the jets. In the X-wind
model, the wide-angle wind comes from the same disk radius as the jet within
$\sim$ 0.05 au of the central source. This model has been proposed to
produce the molecular outflows and the collimated jets simultaneously
\citep{Shang2006}. On the other hand, in the disk-wind model, an extended
(wide-angle) disk wind comes from a range of radii up to $\sim$ 20 au
around the central jet, extracting angular momentum from the disks at larger
radii. The detections of poorly collimated and low-velocity rotating
molecular outflows near the disks in a few older sources, including CB 26
(Class II) \citep{Launhardt2009}, TMC1A (Class I) \citep{Bjerkeli2016}, and
HH 30 (Class II) \citep{Louvet2018}, strongly support this possibility. For
example, in TMC1A, a low-velocity and poorly collimated CO outflow was
proposed to be driven by an extended disk wind coming from the disk surface
with a radius up to 25 au from the central source. A poorly collimated and
low-velocity rotating molecular outflow was also detected in the high-mass
source Orion BN/KL Source I \citep{Greenhill2013,Hirota2017} and it can also
be driven by an extended disk wind coming from the disk surface with a
radius up to $\sim$ 20 au from the central source. Nonetheless, further
works are still needed to check if the rotating outflows can also be rotating
envelope material swept up by inner winds, either the wide-angle components
of X-winds or the inner disk winds.
\begin{figure}[htb]
\centering
\includegraphics[width=0.6\textwidth]
{Tabone2017_F3.jpg}
\caption{Schematic view of the inner 180 au region of the HH212 system, showing
the SiO jet and a possible SO disk wind around the jet \citep{Tabone2017}.
Here CB means centrifugal barrier and COMs means complex organic molecules
in the disk atmosphere.
\label{fig:TaboneDwind}}
\end{figure}
\begin{figure}[htb]
\centering
\includegraphics[angle=270, width=0.8\textwidth]
{so_cont_mom1.pdf}
\caption{SO rotating outflow from the disk around the SiO jet
in HH 212 \citep{Lee2018Dwind}.
Magenta contours show the SiO jet.
Gray contours show the continuum map of the disk at 850 $\mu$m{}.
White contours show the disk atmosphere detected in CH$_2$DOH.
Color maps show the intensity-weighted velocity of the SO outflow.
The red and blue arrows show the rotation of the disk.
\label{fig:HH212SO}}
\end{figure}
In the early phase, disk winds could also be present around the molecular
jets, e.g, in HH 212 \citep{Tabone2017,Lee2018Dwind} and HH 211
\citep{Lee2018HH211}. For example, in HH 212, by modeling a rotating SO
outflow near the disk around the jet, \citet{Tabone2017} suggested that the
outflow can trace a wind coming from the disk at a radius up to 40 au to the
outer edge of the disk, as shown in the schematic diagram in Figure
\ref{fig:TaboneDwind}. The disk wind could be massive and can carry away
$\sim$ 50\% of the incoming accretion flow \citep{Tabone2017}. This SO
outflow was later observed and resolved at higher angular resolution, and
found to extend out to only $\sim$ 70 au above and below the disk midplane
\cite[][see also Fig.~\ref{fig:HH212SO}]{Lee2018Dwind}. It has a specific
angular momentum of $\lesssim$ 40 au km s$^{-1}${}, indicating that it could trace
a disk wind launched at a radius up to $\sim$ 7 au \citep{Lee2018Dwind}.
However, further observations are needed to check if the rotating SO outflow
could also trace the material in the rotating disk atmosphere being pushed
up and out by an inner wind launched at the innermost disk, e.g., a
wide-angle radial wind component surrounding the fast jet as in the X-wind
model.
\section{Summaries and conclusions}
Recent results of a small sample of molecular jets at high spatial and
velocity resolution have provided strong constraints on jet launching and
collimation. The mean ratio of the mass-loss rate in the jet to the
accretion rate is $\sim$ 0.2, as expected in magneto-centrifugal
jet-launching models. This ratio implies a magnetic lever arm parameter of
$\sim$ 5 and a jet launching radius of $\sim$ 0.04 au. More importantly, a
clear rotation is detected in the HH 212 jet within 100 au of the central source,
with a rotation sense the same as that of the disk, confirming the role of
the jet in removing angular momentum from the disk. The specific angular
momentum of the jet is $\sim$ 10 au km s$^{-1}${}, consistent with the jet being
launched from the innermost edge of the disk at $\sim$ 0.04 au. The jet is
expanding roughly with the square of the distance from the central source,
roughly consistent with that predicted in magneto-centrifugal jet-launching
models, where the jet is collimated internally by its own toroidal magnetic
field. There is also a hint of a hollow cone in this jet, because the knots
in this jet show a linear velocity structure that can come from a ring.
Assuming that the knots are indeed rings, then the knots have a radius
slightly larger than that of a hollow cone with a launching radius of $\sim$
0.04 au, consistent with the jet tracing the innermost core of the wind in
the magneto-centrifugal jet-launching models.
The knots and bow shocks in the jets likely trace the internal shocks
produced by quasi-periodical variations in ejection velocity, which in turn
is induced by quasi-periodical perturbations of the accretion in the disks.
Sideways ejections are detected in the knots and bow shocks. Nested
internal shells are also detected, with each extending from a bow shock back
to the central source. Up to 3 periodical variations are detected in the
jets, with one having a period of a few yrs, one with a period of a few ten
yrs, and one with a period of a few hundred yrs. The one with a period of a
few hundred yr could be induced by gravitational instability powered by
envelope accretion. Others could be due to binary companions, either
stellar or planetary, magneto-rotational instabilities, gravitational
instability penetrating to the inner disks, etc.
Two types of wiggles, point-symmetric and reflection-symmetric, are detected
in the jet trajectories. The point-symmetric wiggle could be due to jet
precession, which in turn could be due to disk precession. Disk precession can
be induced by tidal interactions in noncoplanar binary systems. It can also
be induced by asymmetric envelope accretion, when there is a misalignment
between the flattened envelope and the disk because of a misalignment
between the magnetic axis and rotation axis in the star-forming core. On
the other hand, the reflection-symmetric wiggle could be due to an orbital
motion of the jet sources around the binary companions.
High sensitivity observations with ALMA can detect SiO line polarization in
molecular jets due to the Goldreich--Kylafis effect, allowing us to map the
magnetic field morphology in the jets. Recently, SiO line polarization has
been detected toward a well-defined jet within a few hundred au of the
central source, with an orientation parallel to the jet axis, indicating
that the magnetic field there could be either mainly toroidal or mainly
poloidal. Additional polarization observations in SiO in different line
transitions are needed to resolve the ambiguity in the field morphology.
Moreover, current-driven kink instability might have been detected in one of
the jets, suggesting a possible presence of a poloidal field in the jet core
as well.
Being launched within a radius $<$ 0.1 au of the central sources, molecular
jets are likely launched from dust-free regions. The high content of
molecular gas in the jets likely arise because the mass-loss rate is high $\sim
10^{-6} $M_\odot$ \mathrm{\ yr}^{-1}$, so that molecules can form quickly in the
jets. Current observations show that a SiO jet can be detected down to within
$\sim$ 3 au of the central source, where the dynamical age is less than a
month, strongly supporting this possibility.
Rotating molecular outflows are detected around a few molecular jets,
suggestive of a presence of extended disk winds around the jets. The
extended disk winds are expected to carry angular momentum away from the
disks, allowing the disk material to be transported to the inner parts from
the outer parts. However, more works are needed to check if the rotating
outflows can also be rotating envelope material or disk atmosphere swept up
by inner winds, either the wide-angle components of the X-winds or the inner
disk winds.
In summary, recent observations of a small sample of molecular jets have
opened up opportunities for us to detect the jet rotation, to resolve the
jet structure and search for a hollow cone, to map the magnetic field, to
study the periodical variation in ejection and thus the periodical
perturbations in disk accretion, etc. Future systematic observations with a
large sample of molecular jets at high spatial and velocity resolution with
ALMA are expected to lead to a breakthrough understanding in the study of jets.
In addition, sensitive observations of the Zeeman effect in molecular lines will
allow us to derive the strength of the magnetic fields in the jets.
\begin{acknowledgements}
C.-F.L. acknowledges grants from the Ministry of Science and Technology of
Taiwan (MoST 107-2119-M-001-040-MY3) and the Academia Sinica (Investigator
Award). I thank the referee Bo Reipurth for his carefully reading my manuscript
and for his useful comments and suggestions. I also thank Anthony Moraghan
for his helpful suggestions on English grammar.
\end{acknowledgements}
\bibliographystyle{spbasic}
|
1,314,259,993,592 | arxiv | \section{Introduction}
Due to the downscaling of modern nanoelectronic devices the surface-to-volume ratio increases continuously and the surface becomes increasingly important as an additional conductance channel for charge transport. To assess the influence of this surface channel on the device performance or even be able to use it as a functional unit, a reliable value for the two-dimensional surface conductivity has to be known. However, the determination of the surface conductivity from electrical four-point measurements is quite a challenging task, as the main difficulty is to separate the 2D conductance at the surface from the conductance through other channels, e.g. the bulk and the space charge layer.
Often indirect measurement methods are used for the separation of the 2D conductance at the surface, but these methods have special requirements on the material and the preparation of the sample under study. For example, one method for separating the surface conductivity is based on the comparison of measurements before and after quenching the surface states by adsorption of atoms or molecules \cite{HasegawaA,HasegawaB,Petersen,Hasegawa1,Wolkow}. The adsorption species has to be chosen specifically for the material under study and for the quenched system several conditions have to be carefully confirmed. First, all of the surface states have to be quenched and, secondly, the conductivity of the near-surface space charge region has to remain unchanged under the influence of the adsorbed surface layer. Thirdly, no additional surface conductance has to be induced by the adsorbed layer.
If one of these conditions is not fulfilled, the experiments based on the difference method can result in underestimated values for the surface conductivity.
Here, we present a generic N-layer conductance model, free of such requirements, for describing the measured four-point resistance on samples consisting of a surface channel, a space charge region due to the near-surface band-bending and a semi-infinite bulk.
No special sample preparation is necessary and the model can directly be applied to the raw data, which in combination with a calculation of the conductivity profile in the space charge region permits to extract the value for the surface conductivity from distance-dependent four-point measurements.
First, we compare a very simple model often used to describe measurements on samples with mixed 2D-3D conduction channels, the parallel-circuit model, to the N-layer model, and point out that the application of the parallel-circuit model is very limited, as there are already significant deviations if the N-layer model is reduced to the simplest case of a 3-layer model ($N=3$).
Secondly, we apply the N-layer model to different distance-dependent four-point measurements from the literature obtained with a multi-tip scanning tunneling microscope on the semiconductors Ge(100) and Si(100) with different types and concentrations of doping, and determine values for the surface conductivity of these materials.
The analytical derivation of the N-layer model is shown in detail in the appendix.
\section{Mixed 2D-3D conduction channels}
\begin{figure}[t!]
\centering
\includegraphics[width=0.435\textwidth]{./rpr4pe.pdf}
\includegraphics[width=0.445\textwidth]{./surfcurr.pdf}
\caption{(Color online) (a) Calculated four-point resistance of the Si(111)-($7\times7$) surface with a bulk conductivity of $\sigma_B = 0.14\,\mathrm{S/m}$ and a surface conductivity of $\sigma_{S} = 5.14 \cdot 10^{-6}\,\mathrm{S/\square}$ as a function of the equidistant probe distance $s$ and with the ratio $\sigma_{SC}/\sigma_B$ between the conductivities of the space charge layer and the bulk as additional parameter (colored curves). The orange curve located between the two limiting cases of pure 2D and pure 3D conductance (dotted blue and red curves) is based on measurements \cite{Just}, while the magenta, green, blue and red curves correspond to variations of the ratio $\sigma_{SC}/\sigma_B$ over several orders of magnitude. The black curve results from the description by the parallel-circuit model without considering an additional space charge layer between surface and bulk. In the inset, the equidistant linear tip arrangement with the outer current-injecting tips and the inner voltage-measuring tips is shown.
(b) Calculated percentage of surface current $I_{\mathrm{surf}}$ as function of the ratios $\sigma_S\,z_S^{-1}/\sigma_B$ between the surface conductivity and the bulk ($z_S = 3\,\mathrm{\AA}$), and $\sigma_{SC}/\sigma_B$ between the conductivity of the space charge layer and the bulk. The colored points correspond to the position of the curves in (a). Inside the region marked by the two dotted lines the parallel-circuit model can be applied for describing the four-point resistance on the surface with an error of less than 10\%. }
\label{fig1}
\vspace{-2ex}
\end{figure}
\begin{figure*}[t!]
\centering
\includegraphics[height=0.345\textwidth]{./jx3-3.pdf}
\includegraphics[height=0.345\textwidth]{./jx1-3.pdf}
\includegraphics[height=0.345\textwidth]{./jx2-3.pdf}
\caption{(Color online) Color plots of the absolute value of the in-line component of the current density $\mathbf{j}(x,y,z)$ in the xz-plane as a function of depth $z$ into the sample and lateral distance $x$ along the tip positioning line. The current density is calculated from the 3-layer model for a distance $3s = 150\,\mu\mathrm{m}$ of the current-injecting tips, and for a sample with a bulk conductivity $\sigma_B = 0.14\,\mathrm{S/m}$, a surface conductivity $\sigma_{S} = 5.14 \cdot 10^{-6}\,\mathrm{S/\square}$ and an average thickness $z_2 = 2.5 \,\mathrm{\mu m}$ of the intermediate space charge layer. The average conductivity of the intermediate space charge layer is varied in the three cases (a) - (c) showing the significant influence of the space charge region on the vertical current distribution in the sample. According to the 3-layer model the red dashed lines indicate the interfaces between the surface, the space charge layer and the bulk. The black dotted vertical lines mark the position of the current-injecting tips on the surface.
(a) In the case of a very low conducting space charge layer with $\sigma_{SC} \ll \sigma_B$ ($\sigma_{SC} = 2.5 \cdot 10^{-4}\,\mathrm{S/m}$) the majority of the current flows through the surface even if the bulk is highly conductive, as the space charge region acts as a blockade for the injection into the bulk and an enhanced 2D transport can be observed.
(b) If $\sigma_{SC} = \sigma_B$, there is effectively no space charge region and the current flow through the bulk takes place according to the bulk conductivity. In this case the four-point resistance on the surface can be approximated by the parallel-circuit model.
(c) If the space charge layer is highly conductive with $\sigma_{SC} \gg \sigma_B$ ($\sigma_{SC} = 2.5 \cdot 10^{2}\,\mathrm{S/m}$), the current flows not only through the surface, but also equally through the space charge layer, while the current in the bulk is again reduced. }
\label{fig3}
\end{figure*}
For pure 2D or pure 3D charge transport, there exist simple analytic relations between the measured four-point resistance and the conductivity. For an equidistant probe setup with a distance $s$ between the tips, the following equations are obtained for a 2D sheet and a 3D half-space \cite{Schroder}, respectively
\begin{equation}\label{eq:1}
R^{4p}_{2D}=\frac{\ln 2}{\pi \sigma_{2D}},\quad \mathrm{and}\quad R^{4p}_{3D}=\frac{1}{2 \pi \sigma_{3D}} \cdot s^{-1}
\end{equation}
with the 2D surface conductivity $\sigma_{2D}$ and the 3D bulk conductivity $\sigma_{3D}$. The equation for the 2D case shows a constant four-point resistance, independent of the probe spacing, while the conductance through a 3D channel depends on the spacing $s$. Due to this characteristic probe-spacing dependency, it is possible to distinguish between 2D and 3D channels from distance-dependent four-point measurements.
However, if a sample consists of a mixed 2D-3D geometry, e.g. a conducting sheet on a conducting substrate, these two equations cannot be applied any more. Often, a simple approximation of a parallel-circuit consisting of the four-point resistance of the surface and the bulk according to Eq.(\ref{eq:1}) is used \cite{Perkins,Wells3}
\begin{equation}\label{eq:2}
R^{4p}_{\parallel}(s) = \left(\frac{1}{R^{4p}_{2D}} + \frac{1}{R^{4p}_{3D}(s)}\right)^{-1}\,\mathrm{,}
\end{equation}
but this approach has restrictions and shortcomings, as it can be seen in the following.
In the parallel-circuit model a complete separation of the surface conductance channel and the bulk is assumed. The splitting of the injection current between the surface and the bulk only takes place at the injection points and depends on the ratio of the four-point resistances of the two individual layers. However, the two-point resistance, and not the four-point resistance, should determine, which amount of current flows through the surface channel and which part through the bulk \cite{Polley}. Therefore, the exact current path through the sample depends also on details of the injection, e.g. the tip diameter, which are not included in the parallel-circuit model.
The most important point, however, is the fact that in the approximation of the parallel-circuit model the current is injected equally into the surface channel and the bulk, and any influence of a possible near-surface space charge region, which particularly exists in semiconductors, is neglected. But especially this space charge region has a significant influence on the charge transport through the sample, as it will be discussed in the following.
A different approach presented in \cite{Durand} uses an approximation for the surface current to solve the current continuity equations for 2D and 3D resulting in a combination of both 2D and 3D conduction channels. This approach removes the artificial separation between surface and bulk and uses a real injection geometry with extended tips, but it takes only into account a two-layer structure consisting of the surface and the bulk, so that the results are very similar to the parallel-circuit model. Any additional conductivity distribution between the surface and the bulk caused by a space charge region is neglected, which is also the major restriction in the parallel-circuit model. For this reason, the model can only be applied, if no near-surface band-bending occurs and a sharp transition between surface and bulk exists.
Another approach published in \cite{Szymonski2} attempts to describe the deviation from a pure 3D conductance behavior caused by an additional 2D channel with an expansion of distance-dependent terms, and introduces an effective conductivity consisting of the bulk conductivity and a value for the deviation from the pure 3D case. However, although this model may also be able to treat deviations caused by a near-surface space charge region, it is not suitable to determine a value for the surface conductivity, as the deviations from the pure 3D conductance are only indicated by one numerical value, which cannot be easily interpreted as a physical quantity.
In \cite{Wells2} a computational method is described using no longer an analytical model for the four-point resistance but a finite element calculation for approximating the different conduction channels in the sample. In this case, also the near-surface space-charge layer between the surface channel and the 3D bulk can be taken into account. However, as the surface channel has only a depth of several \AA, while the space-charge layer may be extended up to several $\mu m$, very different length scales are involved, so that the finite element calculation of the complete sample geometry can be very sophisticated and computationally time consuming.
The best way to point out the important role of the space charge region, which is especially important for semiconductors, and the limited applicability of a two-layer model, like the parallel-circuit model, is a comparison of the four-point resistance with the lowest N-layer model including the influence of the space charge region, i.e. the 3-layer model. Apart from the surface layer and the bulk region this 3-layer model uses only one additional layer to approximate the space charge region, but despite this quite rough approximation it is able to describe four-point resistance measurement values much better than the parallel-circuit model and was successfully applied to determine the surface conductivity of the Si(111)-($7\times7)$ surface \cite{Just}.
In Fig. \ref{fig1} (a) the calculated distance-dependent four-point resistance for the Si(111)-($7\times7)$ surface on an n-doped substrate ($700\,\mathrm{\Omega cm}$) is shown (orange line) located between the two limiting cases of pure surface conductance (dotted blue line) and pure bulk conductance (dotted red line). The calculation is based on the 3-layer model with parameters obtained in \cite{Just} and assumes an equidistant linear tip configuration with a tip spacing $s$. Using the same parameters for surface and bulk conductivity the four-point resistance expected from the parallel-circuit model according to Eq. \ref{eq:2} is plotted as solid black line, which exhibits a very strong deviation from the curve based on the 3-layer model. The major reason for this behavior is the absence of the additional space charge layer between surface and bulk in the parallel-circuit model. In the case of the Si(111)-($7\times7)$ surface on an n-doped Si substrate with $\sigma_B = 0.14\,\mathrm{S/m}$ the ratio between the average conductivity of the space charge region $\sigma_{SC}$ and the bulk can be estimated as $\sigma_{SC}/\sigma_B = 0.002$ \cite{Just}. For smaller values of this ratio, the deviation of the 3-layer model from the parallel-circuit model increases and the calculated four-point resistance approaches the 2D case (magenta curve). On the other hand, if the ratio becomes larger, the deviation between the two models decreases (green and blue curves). But only if the ratio $\sigma_{SC}/\sigma_B$ is close to 1 (red curve), the deviation between both models is so small, that the parallel-circuit model can be used as approximation without a large error. This error is smallest, if the near-surface space charge region vanishes completely, and in this case the parallel-circuit model is a suitable simple approach to approximate the four-point resistance of a two-layer structure consisting of a 2D and a 3D conduction channel.
The significant influence of the space charge region can also be deduced from the amount of current flowing through the surface compared to the totally injected current. In Fig. \ref{fig1} (b) the calculated percentage of surface current is shown in dependence of the conductivity ratios between space charge layer and bulk $\sigma_{SC}/\sigma_B$ and the surface and bulk $\sigma_S\,z_S^{-1}/\sigma_B$ (thickness of surface layer $z_S \approx 3\,\mathrm{\AA}$) for a constant tip distance of $s = 50\,\mu\mathrm{m}$. The calculation is again based on the 3-layer model and on parameters obtained in \cite{Just} for the measurements of the Si(111)-($7\times7$) surface.
For a vanishing space charge layer, i.e. $\sigma_{SC}/\sigma_B \approx 1$, the amount of surface current approximately only depends on the ratio $\sigma_S\,z_S^{-1}/\sigma_B$ and increases with an increasing ratio. However, if the influence of the space charge layer becomes larger, i.e. if the ratio $\sigma_{SC}/\sigma_B$ deviates from 1, the contour lines in the plot get distorted, so that for large ratios the amount of surface current is reduced and for small ratios enhanced.
The reason for this behavior is that the conductivity of the space charge layer controls the current injection into the bulk below.
If the near-surface band-bending leads to a depletion zone or an inversion zone so that the average conductivity in the space charge region is significantly reduced compared to the bulk, then this region behaves as a blocking region preventing the injected current to flow through the bulk, even if it has a very high conductivity. This results in an enhanced surface domination of charge transport, which cannot be considered in the parallel-circuit model.
In Fig. \ref{fig3} this behavior is visualized by the depth-dependent current density inside the sample. The absolute value of the in-line component of the current density $\mathbf{j}(x,y,z)$ in the xz-plane is plotted as function of depth $z$ into the sample and lateral distance $x$ along the tip positioning line. The calculation is based on the 3-layer model with the same parameters as used in Fig. \ref{fig1}(b) and a distance of $3s = 150\,\mu\mathrm{m}$ for the current injecting tips. For the first case in Fig. \ref{fig3}(a), a very low conducting space charge layer with $\sigma_{SC} \ll \sigma_B$ (thickness $z_{SC} = 2.5\, \mathrm{\mu m}$) is used for the calculation, and the result shows that the majority of the current flows through the surface layer (thickness $z_S = 3\, \mathrm{\AA}$), whereas only a very small amount of current is injected through the space charge layer into the bulk. The current density inside the bulk material is one order of magnitude lower than in the case of a vanishing near-surface band-bending, where the space charge layer coincides with the bulk ($\sigma_{SC} \approx \sigma_B$), which is depicted in Fig. \ref{fig3} (b).
On the other hand, if an accumulation zone is formed near the surface with a high conductivity compared to the bulk, this region can act as an additional conductance channel totally surpassing the current flow through the bulk and also reducing the current through the surface states. In this case shown in Fig. \ref{fig3} (c), where $\sigma_{SC} \gg \sigma_B$, the current flow through the bulk is again reduced by an order of magnitude, while not only transport through the surface states but also through the space charge region is now preferred equally. As the space charge layer has a finite thickness, the current transport may seem to be purely 2-dimensional for larger probe spacings and the usage of the parallel-circuit model for the four-point resistance on such a system would result in a largely overestimated value for the surface conductivity.
In conclusion, the parallel-circuit model has only a very limited applicability within a certain range of conductivity parameters, where the space charge region does not play a significant role for the current transport. In Fig. \ref{fig1}(b) the dotted lines indicate the region, inside which the parallel-circuit model can be applied to four-point resistance measurements with an error of less than 10\%. Inside this region, the contour lines of the color plot are approximately perpendicular to the x-axis indicating that the surface current is nearly independent of the ratio $\sigma_{SC}/\sigma_B$, which is an essential requirement for the application of the parallel-circuit model. For comparison, the four colored points indicate the positions of the resistance curves from Fig. \ref{fig1} (a). Only the red curve, which is very close to the parallel-circuit model, is located inside the dotted region, while the orange curve representing a measurement of the Si(111)-($7\times7$) surface on an n-doped substrate is clearly outside the region.
Although the 3-layer model is obviously better suitable to describe measurement data over a wider range of conductivity parameters than the parallel-circuit model, it still has a basic restriction: the very rough description of the space charge region by only a single layer. Especially for semiconductors, which can have a very strong band-bending near the surface, this can be a major drawback. For this reason, the 3-layer model should be refined by introducing more layers resulting in an N-layer model, which is discussed in the following section.
\section{The N-layer model}
The 3-layer model offers only a rough approximation of the space charge region described by only a single layer with an average conductivity and average thickness. However, the conductivity profile in this region can exhibit a very strong dependence on the z-position, and, especially, if an inversion layer is formed in the near-surface region, the description by a single layer is not sufficient any more. Therefore, we try to approximate the space charge region by more than one layer and present an N-layer model for charge transport consisting of a thin surface layer, $N-2$ layers for the near-surface space charge region, and a semi-infinite bulk. Such a multi-layer model was first proposed by Schumann and Gardner \cite{Schumann, Schumann2, Gardner} and primarily applied to the method of spreading resistance measurements \cite{Leong, Berkowitz, Vandervorst}, but also extended to four-point measurements \cite{Wang} for determining individual sheet conductivities. However, as far as we know, it has not yet been used for obtaining the conductivity of surface states of semiconductors in combination with a calculated conductivity profile of the space charge region as input.
A detailed description and mathematical derivation of the N-layer model is shown in the appendix \ref{appendix}.
In the following section, the application of the N-layer model is demonstrated and it is used to obtain values for the surface conductivity of the Ge(100) and Si(100) surfaces.
\section{Application of the N-layer model}
The advantage of the N-layer model is that it can be used for evaluation of all distance-dependent four-probe resistance measurements without the need of any special sample preparation before the measurement, e.g. in order to quench the surface states \cite{Hasegawa1,HasegawaA,HasegawaB,Petersen}, or special measurement conditions, e.g. varying the temperature \cite{WellsPRL,Tanikawa,Yoo}. For this reason, we apply the N-layer model to already published data of the semiconductor surfaces Ge(100) and Si(100), which were described previously by either pure 2D or pure 3D conductance, but not by a mixed transport channel. In combination with the N-layer model, it is now possible to take into account simultaneously the current transport through the 2D surface and through the 3D bulk both influenced by the presence of the near-surface space charge layer, and to determine values for the surface conductivities of the materials from these measurements.
\subsection{Germanium(100)}
\begin{figure}[t!]
\centering
\includegraphics[width=0.44\textwidth]{./ge-0-3ohmcm.pdf}
\vspace{1ex}
\includegraphics[width=0.47\textwidth]{./ge-0-3ohmcmLeit.pdf}
\caption{(Color online) (a) Four-point resistance of a p-doped Ge(100) sample (nominal bulk resistivity $(0.1 - 0.5) \,\Omega\mathrm{cm}$) as function of probe distance $s$ between the inner voltage-measuring tips \cite{Wojtaszek}. Different colored data points correspond to different distances $D$ in the symmetric linear tip configuration shown in the inset. The solid lines represent one single fit to all data points using the N-layer model for charge transport, which results in a value for the surface conductivity of $\sigma_{S} = (2.9 \,\pm\,0.6) \cdot 10^{-4}\,\mathrm{S/\square}$ and for the bulk resistivity of $\rho_B = (0.22\,\pm\,0.01) \, \mathrm{\Omega cm}$. The dotted lines indicate the expected four-point resistances for a vanishing surface conductance channel, i.e. $\sigma_S = 0 $, taking into account only the space charge region and the bulk. (b) The calculated conductivity profile of the space charge layer as function of the depth $z$ into the sample starting from the surface. This profile is approximated with $N = 20$ layers and used as input for the N-layer model. The band diagram in the inset shows the surface pinning of the Fermi level $E_F$ (red) located $0.11\,\mathrm{eV}$ above the valence band edge and the resulting near-surface band-bending of the conduction band $E_C$ (green) and the valence band $E_V$ (blue).}
\label{fig5}
\end{figure}
Distance-dependent four-point transport measurements on the Ge(100) surface were published by Woj\-taszek \textit{et al.} \cite{Wojtaszek}. They used a room-temperature, ultra-high vacuum multi-tip STM and carried out four-point resistance measurements on Ge(100) substrates with different bulk doping concentration and type. A symmetric linear probe configuration was used, where the outer current-injecting tips have a distance $D$ and the inner voltage-measuring tips are separated by the distance $s$. The complete setup is symmetric with respect to the centre plane of the tip positioning line.
In Fig. \ref{fig5}(a), the experimental data for a p-type Ga-doped sample with a nominal bulk resistivity of $0.1 - 0.5\, \Omega\mathrm{cm}$ are shown \cite{Wojtaszek}. The measured four-point resistance is plotted as a function of the spacing $s$ between the voltage-measuring tips and with the distance $D$ between the current-injecting tips as additional parameter. In the framework of the publication \cite{Wojtaszek}, these data were described by a pure 3D conductance channel. However, it was mentioned that there were some systematic deviations from the 3D model, which increasingly appear, if the voltage-measuring tips approach the positions of the current-injecting tips, i.e. $s/D \ge 0.7$, but the origin of these deviations could not be explained quantitatively.
In fact, for the symmetric linear tip configuration, it is particularly the region with a ratio $s/D$ close to 1, where the setup is most sensitive to surface transport and a possible surface conductance channel would have the most influence on the measured four-point resistance. So, it is reasonable to assume that the observed deviations are caused by an additional 2D conductance channel through the surface states of the Ge(100)-(2$\times$1) surface, which cannot be considered by the pure 3D model.
\begin{figure}[t!]
\centering
\includegraphics[width=0.45\textwidth]{./ge-45ohmcm.pdf}
\includegraphics[width=0.47\textwidth]{./ge-45ohmcmLeit.pdf}
\caption{(Color online) (a) Four-point resistance of an n-type doped, almost intrinsic Ge(100) sample (nominal bulk resistivity $\sim 45\,\Omega\mathrm{cm}$) as function of probe distance s between the inner voltage-measuring tips \cite{Wojtaszek}. Different colored data points correspond to different distances $D$ in the symmetric linear tip configuration (inset in Fig. \ref{fig5} (a)). The solid lines represent a single fit to all data points using the N-layer model for charge transport ($N = 20$), which results in a value for the surface conductivity of $\sigma_{S} = (3.4\,\pm\,0.2) \cdot 10^{-4}\,\mathrm{S/\square}$ and for the bulk resistivity of $\rho_B = (45\,\pm\,22)\,\mathrm{\Omega cm}$. The dotted lines correspond to the expected four-point resistances without any surface channel ($\sigma_S = 0$) taking into account only the bulk and the space charge region. (b) Calculated conductivity profile of the space charge region as function of the depth $z$ from the surface (red line). The approximated profile (green line) is used as input for the N-layer model. In the upper inset, the complete range of the conductivity profile of the space charge region exhibiting a shape of an inversion layer is shown. The lower inset depicts the surface pinning of the Fermi level $E_F$ (red) and the induced near-surface band-bending of the conduction band $E_C$ (green) and the valence band $E_V$ (blue).}
\label{fig6}
\end{figure}
In order to describe this additional 2D transport channel more quantitatively, we evaluate the existing data with the N-layer model. First, the near-surface band-bending of the p-type Ge(100) sample is calculated by solving Poisson's equation and using a Fermi level pinning at the surface of $\sim 0.11\,\mathrm{eV}$ above the valence band \cite{Tsipas,Tsipas2,Broqvist}. Fig \ref{fig5} (b) shows the resulting depth-dependent conductivity profile of the space charge region consisting of a near-surface accumulation layer. This conductivity profile is approximated by a step function of $(N-2)$ steps ($N = 20$) determining the values for $\sigma_n$ and $z_n$ to be used as input for the N-layer model (details in the appendix). For the symmetric linear tip setup the four-point resistance according to the N-layer model can be expressed as function of $s$ and $D$ by the equation
\begin{align}
R^{4p}(s,D) = & \frac{2}{I} \int_0^\infty \left[ a_0(k) + a_1(k) \right] \cdot \left[J_0\left(k\cdot\frac{D-s}{2}\right) \right. \nonumber \\[1ex]
& \left. - J_0\left(k\cdot\frac{D+s}{2}\right)\right] \, \mathrm{d}k\, \mathrm{,} \label{r4psymm}
\end{align}
which is fitted to the measurement data resulting in the colored solid curves shown in Fig. \ref{fig5} (a). All four curves for the different values for the distance $D$ correspond to only a single fit with the surface conductivity $\sigma_S$ and the bulk conductivity $\sigma_B$ confined close to the range of the nominal values as free parameters. As the conductivity profile of the space charge region also depends on the bulk conductivity, an iterative fitting process is applied, which includes the calculation of the space charge region and the fit to the data in each iteration. For values of $\sigma_S = (2.9\,\pm\,0.6) \cdot 10^{-4} \mathrm{S}/\square$ and $\sigma_B = (460\,\pm\,11) \mathrm{S}/\mathrm{m}$ the iterative process converges and the best fit is obtained describing the data very precisely throughout the complete measurement range without any systematic deviations. A further advantage is the resulting single value for each of the parameters $\sigma_S$ and $\sigma_B$, which is sufficient to describe precisely all four resistance curves for the different distances $D$. In the case of a pure 3D model, as it is used for the fitting process in \cite{Wojtaszek}, it is not possible to model all four data sets with only one value for the bulk conductivity $\sigma_B$. The 3D fit has to be applied separately to each curve resulting in different values for $\sigma_B$ spreading by a relative deviation of $\sim$ 25\%. However, the measured bulk conductivity should not change during the variation of the tip configuration by the distance $D$ on the same substrate.
This reveals that, even if the transport in the sample is mostly 3D dominated due to the highly conductive bulk and the weak accumulation zone near the surface, a description of the data by a pure 3D model is not sufficient and an additional 2D channel has to be taken into account.
For validating the results for the additional surface conductance channel and ensuring that the observed amount of two-dimensional conductance is not merely caused by the near-surface accumulation layer, the dotted colored curves in Fig. \ref{fig5} (a) correspond to the expected four-point resistance for a vanishing surface channel. In these curves, only the bulk conductivity and the conductivity profile of the space charge region according to Fig. \ref{fig5} (b) are taken into account, while the value for the surface conductivity $\sigma_S$ is set to zero. The clearly visible deviation of the dotted curves from the measurement data verifies that an additional 2D surface conductance channel is necessary for describing the measured four-point resistance, and, therefore, proves the existence of conducting surface states.
Fig. \ref{fig6} (a) shows similar distance-dependent four-point resistance measurements on an n-type doped, almost intrinsic Ge(100) sample with a nominal bulk resistivity of $\sim 45\,\Omega\mathrm{cm}$ \cite{Wojtaszek}. As the measurement data show an enhanced two-dimensional character of conductance, a pure 2D model was used in \cite{Wojtaszek}, which was justified by the presence of a near-surface inversion layer totally preventing the current to be injected into the bulk and acting as a 2D channel, which confines the current close to the surface. However, any possible presence of an additional 2D surface channel caused by surface states was neglected. In this case, a further disentanglement between the conductivity of the near-surface p-type part of the inversion layer and the surface conductivity would be required.
So, we try again to describe the measurement data with the N-layer model. The calculated conductivity profile of the space charge region shows the expected inversion layer depicted in Fig. \ref{fig6} (b). For the calculation, the transition region between p-type and n-type of conduction has not been taken into account and only the absolute value of the conductivity is considered, but, as the majority of the current flows through the near-surface p-type part of the inversion layer and through the surface channel, this approximation should be suitable in the present case. The conductivity profile is described by a step function (green line) and used in combination with the N-layer model for a fit to the data according to Eq. \ref{r4psymm}. In Fig. \ref{fig6} (a), the two solid curves result from a single fit with the parameters $\sigma_S = (3.4\,\pm\,0.2)\cdot 10^{-4}\,\mathrm{S}/\square$ and $\sigma_B = (2.2\,\pm\,1.1)\,\mathrm{S}/\mathrm{m}$ and describe the data very precisely. For verification, the dotted lines shown in Fig. \ref{fig6} (a) again represent the expected four-point resistance without any additional surface channel ($\sigma_S = 0$). The very strong deviation from the measurement data indicates clearly that the observed transport behavior cannot only be caused by the enhanced conductivity close to the surface due to the inversion layer, but that there has to be an additional surface conductance channel also on the n-type sample.
If the results for the p-type and n-type Ge(100) samples are compared, the values for the obtained surface conductivity coincide within the error limits. This is expected, as the surface states should not be influenced by the doping type of the substrate. Thus, this is another confirmation that really the conductivity of the surface states was determined. By combining the results of the p- and n-type sample, a more precise value for the surface conductivity of the Ge(100)-(2$\times$1) surface of $\sigma_{S,Ge(100))} = (3.1\,\pm\,0.6) \cdot 10^{-4}\,\mathrm{S}/\square$ can be obtained.
\subsection{Silicon(100)}
\begin{figure}[t!]
\centering
\includegraphics[width=0.41\textwidth]{./si100.pdf}
\includegraphics[width=0.425\textwidth]{./si100_ndop_leitf.pdf}
\includegraphics[width=0.425\textwidth]{./si100_pdop_leitf.pdf}
\caption{(Color online) (a) Four-point resistance of a p-doped (red data points) and an n-doped (blue data points) Si(100)-(2$\times$1) sample (nominal bulk resistivity $(1-10)\,\mathrm{\Omega cm}$) as function of the equidistant probe distance $s$ reproduced from \cite{Polley}. Fits to the data (solid lines) based on the N-layer model result in a surface conductivity of $\sigma_{S} = (1.9\,\pm\,1.4) \cdot 10^{-4}\,\mathrm{S/\square}$ and in a bulk resistivity of $\rho_B = (7.5\,\pm\,0.9) \,\mathrm{\Omega cm}$ for the p-doped case, and in $\sigma_{S} = (1.6\,\pm\,0.4) \cdot 10^{-4}\,\mathrm{S/\square}$ and $\rho_B = (10\,\pm\,7.5)\,\mathrm{\Omega cm}$, respectively, for the n-doped sample. The inset shows the equidistant tip configuration. (b),(c) Calculated conductivity profiles of the space charge region for the p- and n-doped samples (red curves). The approximation by $N = 20$ layers (green curves) is used for the N-layer model. In the insets, the near-surface band-bending of the conduction band $E_C$ (green) and the valence band $E_V$ (blue) caused by the surface pinning of the Fermi level $E_F$ (red) due to the surface states located $\approx 0.31\, eV$ above the valence band edge is shown.}
\label{fig7}
\vspace{-0.7cm}
\end{figure}
Distance-dependent four-point resistance measurements on p-type and n-type doped Si(100) substrates were carried out by Polley \textit{et al.} \cite{Polley}. For the measurements, a room temperature, ultra-high vacuum multi-tip STM was used with a linear equidistant tip configuration with spacing $s$ between adjacent tips. The current was injected by the outer tips and the potential drop between the inner tips was measured. In Fig. \ref{fig7} (a), the measured four-point resistance is shown as a function of the tip distance $s$ for an n-type (blue points) and a p-type (red points) Si(100) substrate both with a nominal bulk resistivity of $(1-10)\,\Omega\mathrm{cm}$. Although the bulk doping concentrations of p- and n-type sample are similar, the observed transport behavior is completely different. In the p-type case, a 3D conduction channel is more dominant, while in the n-type case the majority of current flows through a 2D transport channel. Again, this was explained by the presence of an inversion layer in the n-type sample preventing the current to flow through the bulk. So, the measured data were described in \cite{Polley} by a pure 3D conductance model for the p-type substrate and by a pure 2D model in the n-type case. However, this approach cannot consider any possible mixed 2D-3D conductance channels through the space charge region and the bulk in both samples, and, especially, neglects the two-dimensional surface state, which should be present on the Si(100)-(2$\times$1) surface \cite{Martensson}.
For refining the description of the measured data on the Si(100) substrates and for determining a value for the conductivity of the Si(100)-(2$\times$1) surface state, we use the N-layer model. Fig. \ref{fig7} (b) and (c) show the corresponding conductivity profiles of the space charge region for the p-type and n-type Si(100) substrates, respectively. For the calculation, a Fermi level pinning of the surface states of $\sim 0.31\,\mathrm{eV}$ above the valence band is used \cite{Yoo,Martensson,Himpsel2}. In the p-type case, a depletion zone is formed close to the surface, while in the n-type case an inversion layer separates the bulk from the near-surface region. Again, the pn-transition is not considered for the inversion layer, as the n-type bulk does not contribute significantly to current transport. The approximation of the conductivity profiles (green curves) is used as input for fitting the respective measurement data in Fig. \ref{fig7} (a) according to Eq. \ref{r4p-linear}. The results are depicted as solid curves in Fig. \ref{fig7} (a) and correspond to fitparameters for the surface conductivity of $\sigma_S = (1.9\,\pm\,1.4) \cdot 10^{-4}\,\mathrm{S}/\square$ and for the bulk conductivity, which is confined to the range of the nominal value, of $\sigma_B = (13.3\,\pm\,1.7) \,\mathrm{S}/\mathrm{m}$ for the p-type sample, and to values of $\sigma_S = (1.6\,\pm\,0.4) \cdot 10^{-4}\,\mathrm{S}/\square$ and $\sigma_B = (10\,\pm\,7.5) \,\mathrm{S}/\mathrm{m}$ in the n-type case.
As the four-point resistance measurement for the p-type sample in the chosen tip distance range is not very surface sensitive, the determined value for the surface conductivity has quite a large error, even if the curve fits quite well to the data. The fitted curve for the n-type substrate shows some larger deviations due to a larger spread and a slight increasing behavior of the data, which might be caused by tip positioning errors or influence of the sample edges. However, the obtained value for the surface conductivity is more precise, as the transport behavior in the n-type sample is now more dominated by the near-surface region. So, as both values are still consistent within the error limits, the value resulting from the n-type sample can describe the conductivity of the Si(100)-(2$\times$1) surface state more precisely as $\sigma_{S,Si(100)} = (1.6\,\pm\,0.4) \cdot 10^{-4} \,\mathrm{S}/\square$.
\section{Conclusion}
\begin{table}[b]
\begin{ruledtabular}
\centering
\begin{tabular}{ l l }
&\\[-1.5ex]
Surface reconstruction & Surface conductivity $\sigma_S$\\[1ex]
\hline\\[-1ex]
Si(100)-(2$\times$1) & $(1.6\,\pm\,0.4) \cdot 10^{-4}\,\mathrm{S}/\square$\\
Ge(100)-(2$\times$1) & $(3.1\,\pm\,0.6) \cdot 10^{-4}\,\mathrm{S}/\square$\\
Si(111)-(7$\times$7) & $(8.6\,\pm\,1.9) \cdot 10^{-6}\,\mathrm{S}/\square$ \cite{Just}\\
Bi/Si(111)-($\sqrt{3}\times\sqrt{3}$)R$30^{\circ}$ & $(1.4\,\pm\,0.1) \cdot 10^{-4}\,\mathrm{S}/\square$ \cite{Just}\\
Ag/Si(111)-($\sqrt{3}\times\sqrt{3}$)R$30^{\circ}$ & $(3.1\,\pm\,0.4) \cdot 10^{-3}\,\mathrm{S}/\square$ \cite{Luepke}\\[0.5ex]
\end{tabular}
\caption{Values for the surface conductivity of different reconstructed and passivated surfaces of silicon and germanium.}
\label{table1}
\end{ruledtabular}
\end{table}
In conclusion, we applied an analytically derived N-layer model for current transport through multiple layers of different conductivity including the calculation of the near-surface band-bending to interpret distance-dependent four-point resistance measurements on semiconductor surfaces.
First, the important role of the space charge region for the current distribution in the sample was discussed and it was shown that already the lowest case of the N-layer model, i.e. the 3-layer model, can describe measured four-point resistance data much better than the often used parallel-circuit model, which completely neglects the space charge region. The derivation of the N-layer model and its usage for multi-probe distance-dependent four-point resistance measurements on surfaces was presented. Finally, the N-layer model was used for describing published distance-dependent four-point measurements on Ge(100) and Si(100) surfaces and values for the conductivities of the surface states of these materials could be determined as summarized in Tab. \ref{table1}. For comparison, values for the surface conductivities of differently reconstructed and passivated Si(111) surfaces are also listed. In total, the presented method is quite generic and can easily be used for many other materials to determine values for the surface conductivity.
|
1,314,259,993,593 | arxiv | \section{Introduction}
\PARstart{C}{admium} Zinc Telluride (CZT) is rapidly coming of age as a detector material for photons from a few keV to a few MeV. It offers superior spatial and energy resolution compared to scintillators; and it is more economical and compact compared to Ge, particularly considering that CZT does not require cryogenic cooling.
\\
CZT is a compound semiconductor, with a band gap between 1.5 and 2.2 eV, depending on the Zn/Cd fraction. This band gap allows room-temperature operation. Its high average Z of 50 and high density contribute to effective stopping of photons. One of its limitations is a poor hole mobility $(\mu_h\:\tau_h=\rm(0.2-5)\cdot10^{-5}cm^2/V)$ and trapping. Advanced electrode designs, including pixilation,
crossed strips, and steering electrodes, mitigate this by virtue of the ``small pixel effect'' \cite{Barret95,Luke95}.
The best energy resolutions are achieved by combining small-pixel detector designs with
corrections for any residual dependence of the induced signals on the depth of the interaction
(DOI)
of the primary photons. In practice, the DOI can be estimated by measuring the
timing delay between cathode rise and anode rise, \cite{Kalem02,Zhang04}, or
from the anode to cathode signal ratio (e.g.\ \cite{Kraw04}).
Since the use of small anode contacts leaves gaps to which charge can drift and escape collection \cite{bolotnikov},
steering grids biased somewhat below anode potential have been used to steer the charges to the anodes.
Figure \ref{potential} shows the results from a 3-D detector simulation and illustrates the potential distribution
in such a detector when the grid is biased at -300~V relative to the pixels.\begin{figure}
\vspace{-0.4cm}
\centering
\includegraphics[width=8cm]{potential.eps}
\vspace*{-0.6cm}
\caption{Potential distribution of a central pixel in a
0.5~cm thick detector with pixel pitch of 0.24 mm, pixel width of
0.16 mm and a steering grid width of 0.016 mm from a 3-D detector
simulation. While the anode pixels were held at ground, the cathode
was biased at -500~V and the steering grid at -300~V. The 3-D Possion
solver was developed at Washington University by S.\ Komarov.}
\label{potential}
\end{figure}
CZT detectors with steering grids encounter two difficulties. First, the surface resistivity of CZT is much lower than the
bulk resistivity ($\sim$10$^9$ Ohm cm) and the grid bias voltage gives rise to currents between
the steering grids and the pixels. The noise associated with these currents may deteriorate
the energy resolution of the detectors.
Second, for grid bias voltage much lower than the cathode bias voltage, a considerable
fraction of electric field lines inside the detector connect to the steering grids.
Thus, biased in this way steering grids tend to collect some
of the charge generated in the detectors and tend to reduce the detection efficiency of
the detectors. Here we explore a novel approach that uses steering grids that are
isolated from the CZT bulk material by a thin isolation layer (Fig.\ \ref{Layer}).
In the following we briefly outline our technique of fabricating the detectors;
subsequently we describe our measurement equipment and the results from testing the
detectors with flood illumination and with a collimated X-ray beam.
In the following, all energy resolutions are full width half maximum (FWHM)
resolutions.
\begin{figure} \vspace{-0.2cm}
\centering
\includegraphics[width=5.5cm]{Layer.eps}
\caption{Sketch of a CZT detector with planar cathode contact, pixelated anode contacts
and a steering grid isolated from the CZT substrate by a high-resistivity isolation layer.
We show schematically that the steering grid can steer electrons generated in the
volume below adjacent pixels to the pixels.}
\label{Layer}
\end{figure}
\section{Results}
\subsection{Detector fabrication}
The studies use modified Horizontal Bridgeman CZT from the company Orbotech
\cite{Orbotech}.
We fabricated several detectors. On some we used
the In pixels deposited by the Orbotech. On others we deposited the pixels
ourselves using e-beam evaporation through a mask.
The results shown in the following are from detector ``D1''
with pixels deposited
by Orbotech and an isolated steering grid deposited in our laboratory.
The detectors have a volume of 2.0$\times$2.0$\times$0.5~cm$^3$ and are
contacted with a planar In cathode and 8$\times$8 In pixels with a pitch of 0.25~cm and a pixel width of 0.16~cm.
After evaluating the performance of the detectors, the isolated steering grids were
deposited using standard photolithographic techniques.
We used the photoresist S1813 and the developer Cd-30 from the company Shipley \cite{shipley}.
We optimized the processing parameters based on a series of empirical tests
varying the prebake and softbake temperatures and durations,
the exposure time, and the development time.
The optimization has been carried through at Washington University in St. Louis and
at Fisk University.
After protecting the pre-deposited pixels with photoresist,
first a 150 nm thick isolation layer, and subsequently a 200 nm thick grid were deposited at
$\sim$10$^{-7}$ Torr with an electron beam evaporator. We used Al$_2$O$_3$ as
isolation material because of its very high resistivity ($>10^{14}$ Ohm cm), and excellent
mechanical properties. Almost any metal with good sticking properties could be used
for the steering grid. We chose Ti because of its low price and relative ease
with which it can be deposited.
The thickness of the isolation layer has still to be optimized.
The width of the steering grid is approximately 0.02~cm.
Figure \ref{GridSmallBWN} shows one of our detectors with a steering grid.
\begin{figure}
\centering
\includegraphics[width=4.5cm]{Stack40-43_4xpixesl_J10-small.eps}
\caption{Pixelated 2.0$\times$2.0$\times$0.5~cm$^3$ CZT detector with steering grid. A high-resistivity Al$_2$O$_3$
film isolates the steering grid from the CZT substrate.}
\label{GridSmallBWN}
\end{figure}
\subsection{Experimental Setup}
Several setups have been used to test the performance of the detector.
At Washington University the pixel performance of three of the
central pixels are measured by using flood illumination with
a $^{137}$Cs source.
The setup reads out four channels (the three
central anode pixels plus the cathode, all other
anode pixels are grounded) that are AC coupled to fast Amptek 250 amplifiers
followed by a second amplifier stage. The amplified signals are digitized by a
500 MHz oscilloscope and transferred to a PC via Ethernet. The time resolved
readout enables us to measure the drift time of electrons through the detector with a
resolution of 10 ns. Each event gets a time stamp. A fit of an exponential to
a histogram of the times between successive events after deadtime correction,
is used to calculate
the detection rate. Care was taken to locate the source always at
the same location above the detector, so that the measured rates can be
used to estimate the detection efficiency of the detectors.
The detector is mounted by using gold plated
pogo-pins to contact the anode, the steering grid and the cathode.
The cathode is negatively biased, and the anode pixels are held at ground.
The steering grid was biased at -30~V, -60~V, $-120$~V and -200~V.
The electronic noise of our test set-up has been measured before each detector
evaluation and lies between 5~keV and 10~keV. We used a Keithley picoammeter
to measure cathode-pixel, steering grid-pixel, and pixel-pixel IV curves.
A more detailed description of the test-equipment as well as previous
results on Orbotech detectors have been given in \cite{Kraw04,Perk03,Jung05}.
At the University of California in San Diego the detector response has been
studied with a collimated gamma-ray source. The 7.5 cm thick
tungsten collimator has a tapered hole. The hole diameter is 0.02 cm
at the source and 0.05 cm at the detector end of the collimator.
The CZT cathode is 0.10 cm away from the collimator.
The collimator position is controlled by a x-y stage with 1$\mu$m
accuracy. Eight channels can be read out, the information obtained is the
pulse height of the signals. The FWHM noise of anode channels
lies between 5.75~keV to 8.1~keV for cathode biases between $-100$~V and
$-1000$~V and grid bias between $-30$~V and $-120$~V. The FWHM noise of the cathode
channel lies between 7.2~keV and 41~keV.
More information on the experimental set-up as well as previous results
can be found in \cite{Kalem02}.
\subsection{Results}
In the following, we report on the results obtained with detector D1.
Figure \ref{IVCurveGridPixel} shows the current between the isolated steering grid and
several anode pixels as function of the grid bias (pixels and cathode grounded).
For a grid-bias of -60~V grid-pixel currents per pixel below 0.25 nA were observed.
Comparing these results to pixel-pixel IV measurements of detectors without
a steering grid, we observe that the Al$_2$O$_3$ layer results in a substantial current suppression.
Figure \ref{EnergyResolution} shows the 662~keV energy resolution of the three central
pixels for flood-illuminating the detector with a $^{137}$Cs radioactive source.
Before depositing the steering grid, the three pixels gave energy resolutions
of 2.10\%, 2.03\% and 1.85\%. Biasing the steering grid at -30~V, the performance
of the same three pixels improved to 1.69\%, 1.49\% and 1.39\%, respectively.
The energy resolutions changed little for steering grid voltages between 0~V and
-200~V. For biases $<$-200~V, the detector performance deteriorated significantly.
Before and after deposition of the steering grid, we obtained detection rates
that are identical within the statistical errors. Mantaining the same detection
efficiency while substantially improving the energy resolution of the
detectors is a very encouraging result.
\begin{figure}
\centering
\includegraphics[width=8cm]{IV.eps}
\vspace*{-0.6cm}
\caption{The currents between the steering grid and
three central pixels as function of relative grid bias for detector D1.}
\label{IVCurveGridPixel}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=8cm]{EnergyResolution.eps}
\vspace*{-0.6cm}
\caption{Energy resolutions of the three central pixels of detector D1.
The horizontal lines show the energy resolutions before deposition of the steering grid (``No SG''),
and the data points give the energy resolutions with the isolated steering grid at different bias
voltages (``SG'').}
\label{EnergyResolution}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[width=8cm]{A1vsA2_2.eps}
\vspace*{-0.3cm}
\caption{Scatter plot of anode 1 versus anode 2 signals at two different
positions of the collimated $^{137}$Cs beam: a)
at the center of anode 1 (0 mm, 0mm) and
b) in the middle between anode 1 and anode 2. The different colors correspond
to different sharing ratios.
Ref for a sharing ratio between 0-0.04 and 0.96-1.0, blue for 0.04-0.1 and 0.9-0.96
and green for 0.1-0.9.}
\label{A1vsA2}
\end{figure}
In order to understand the detector performance for energy depositions
in the detector volume below adjacent pixels, we scanned
the detector D1 with a collimated 0.05 cm diameter $^{137}$Cs source.
The scan with a step size of 0.125 cm started at the center of
one pixel and ended at the center of an adjacent pixel. The detector
pixels were held at ground, the cathode was biased at -500~V, and
the steering grid at -60~V. The electronic noise for the anode
and cathode channels was 7~keV and 20~keV, respectively.
Here we show first results from the scans of the detector with
an isolated steering grid and their initial interpretation.
Note that we obtained qualitatively very similar results for detectors with
and without steering grid. More detailed studies aiming
at studying the effect of the steering grid are in preparation.
Figure \ref{A1vsA2} shows that the pixel signals depend strongly on the
position of the radioactive beam. Illuminating the center of ``pixel 1''
charge sharing is negligible and almost all events are only detected
by pixel 1 (Fig. \ref{A1vsA2}a). The small fraction (2\%) of photo-effect
events (around channel 3040) detected with pixel 2 are most likely
662~keV photons that made it through the collimator walls and hit
pixel 2 directly.
In the following, we call the ratio of the charge collected with pixel
1 divided by
the charge collected by pixels~1 and 2 the sharing ratio.
We measured the percentage of events with a sharing ratio between
$>$10\% and $<$90\%
using a cut on the sum of the two pixel signals ($>1500$ channels).
With the collimator centered at pixel 1 this fraction of
events with substantial charge sharing was $\sim$10\%, at the position
(0 mm, 0.5 mm)
$\sim$25\% and in the middle between the pixels $\sim$40\% (Fig. \ref{A1vsA2}b).
We examined the same data in more detail. The sum of the anode signals
of pixels~1 and 2 was plotted as a function of the cathode signal
for the collimated beam pointing at the center of pixel 1
(Fig.~\ref{SumAnodevsCathode}a) and at the location
between pixels 1 and 2 (Fig.~\ref{SumAnodevsCathode}b) . Only signals which exceed a threshold of 120 channels were
included in the analysis. The different colors in Fig. \ref{SumAnodevsCathode}
show events with different sharing ratios (red: 0-4\% and
96-100\%, blue: 4\%-10\% and 90\%-96\% and in green 10\%-90\%).
When the collimator was pointed at the region between the two pixels,
three different populations contribute to the photo-peak event line.
The events with approximately no charge sharing (red) (in the following called line 1)
exhibit a similar anode to cathode dependency as the events
taken when the collimated beam pointed at the center of pixel 1 (Fig. \ref{Anode1AndSum}).
Most events with sharing ratios between 4\%-10\% and 90\%-96\%
exhibit relatively small cathode signals and the summed anode signals
lie above the ones with no charge sharing on a line, called ``line 2'' in the
following.
For the events with charge sharing ratios between 10\% and 90\%,
the photo-peak line (``line 3'') is shifted to lower summed anode values.
In the following, we refer to the three photopeak lines in Fig.
\ref{SumAnodevsCathode} as lines 1, 2 and 3.
Line 3 shows a strong dependency on the depth of interaction.
A possible explanation is lost charge in the gap/steering grid region.
The charge loss depends
on the depth of interaction \cite{Kalem02}.
After energy deposition in the detector the electrons undergo diffusion.
Neglecting the electron charge, the diffusion can be described by the Fick's equation $D\nabla^{2}M = \delta M/\delta t$. (D is the diffusivity, for CZT $\sim$26 cm$^{2}$/s) (\cite{Kalem02}). The one dimensional solution for the
concentration M(x,t) at
position x and time t for a delta function initial concentration M$_{0}\delta(x_{0},0)$ at position $x_{0}$
is $M(x,t)=M_{0}/\sqrt{4\pi Dt}\exp{(-(x-x_{0})^{2}/(4Dt))}$.
If one makes a rough estimate, that all events at line 3 interact
in the middle of the gap and that a region of 18$\mu$m exactly between
the pixels, steals charge
from the signal, line 1 can be reproduced.
To fully understand this effect, further investigations are needed.
Detailed simulations and measurements are planned.
Events with small sharing ratios (4\%-10\% and 90\%-96\%) are mostly
found in line 2. These are probably events interacting
near the anode side and near the
pixel edges. In this area, a signal can be induced in the adjacent
pixel that increases the summed signal.
Because the different lines are dependent on the sharing ratio,
this is an effect which can be corrected for.
In Fig. \ref{correct} the spectra for the collimator position at
anode 1 and for the collimator position at the gap between
anode 1 and anode 2 is shown. Both spectra are corrected for the
effect mentioned above. The obtained energy resolutions are
1.7\% at anode 1 and 2.1\% at the gap.
\\
One remainig question is the behaviour of the summed anode signal vs. cathode
signal as a function of different grid biases. For biases
equal to the pixel voltage and for a grid-bias of -30~V
the number of charge sharing events is strongly suppressed compared
to a grid-bias of -60~V.
The number of events in the region
between 2560 channels and 3200 channels (the photopeak range)
are 1062, 1846 and 2910 for bias 0~V, -30~V and -60~V respectively.
The steering grid works effectively and increases the number of
detected photopeak events by a factor of almost 3, for events
in the gap between pixels.
\begin{figure}
\centering
\includegraphics[width=8cm]{W66_SumAnodevsCathode2.eps}
\vspace*{-0.3cm}
\caption{Scatter plot of the sum of the signals at anodes 1 and 2 versus the
cathode signal. Only signals exceeding the threshold of 120 channels were
included in the summation. The two plots relate to different positions
of the collimator equipped with a $^{137}$Cs source: a)
at the center of pixel 1, b) in the middle of the gap between anodes 1 and 2.}
\label{SumAnodevsCathode}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=8cm]{W66_Anode1AndSum_Gap.eps}
\caption{For the collimator position at the center of
the gap, the summed anode signals are shown as a function of the cathode
signal for events with sharing ratios between 0\%-4\% and 96\%-100\% (red markers). Overlayed in black, the anode~1 signal distribution
is shown for the same collimator position. }
\label{Anode1AndSum}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=8cm]{Correct.eps}
\caption{Spectra for two different collimator positions. The black, solid line shows the spectra for the collimator centered at anode 1 (1.7\% energy resolution), the red, dashed line
for the collimator at the pixel gap (2.1\% energy resolution). Both spectra
include corrections for three
different sharing ratios: a) 0\%-4\% and 96\%-100\%, b) 4\%-10\% and 90\%-96\% and c) 10\%-90\%.}
\label{correct}
\end{figure}
\section{Conclusion}
We fabricated pixelated CZT detectors with steering grids. As an innovation,
we used photolithography and thin film deposition methods to fabricate detectors
with steering grids that are isolated from the CZT substrates by thin-film Al$_2$O$_3$ layers.
We presented first results obtained from testing the detectors at 662~keV.
The results are extremely encouraging: the Al$_2$O$_3$ layer seems to
reduce the grid-pixel currents and the steering grid seems to improve the
energy resolution of the detector by a factor of 1.3.
While the grid biased at -30~V did not reduce the detection efficiency, when
using only the information from one pixel. Using the information
from adjacent pixels by summing their signals, the steering grid even
increases the number of events reconstructed in the photopeak.
With the aim to achieve a good understanding of the performance of the detector
with and without steering grids, we have started to scan the detector response
with a collimated X-ray beam. The results show a very complex behavior of
the detector for primary energy depositions in the volume below adjacent pixels. We have shown, that the sharing ratio can be used to identify different
classes of events. It should be possible to use the sharing ratio
to correct inter-pixel events, even when the detector is flood illuminated.
We still have to test this method.
We plan to achieve further progress by complementing the measurements
with detailed 3-D models of the detector.
Our future work will concentrate on a systematic study of the dependence
of the detector response on the steering grid voltage, the dependence of the
grid-pixel currents on the thickness of the Al$_2$O$_3$ isolation layer,
and the relative performance of different isolation materials.
\subsection*{Acknowledgements}
This work has been supported by NASA under contracts NNG04WC17G and NNG04GD70G, and the NSF/HRD grant no.\ 0420516 (CREST) and by DOE National Nuclear Security Administration of Nonproliferation Research and Engineering NA-22 under grants DE-FG07-04ID14555 and DE-FG52-05NA27036.
We thank S.\ Komarov and L.\ Sobotka for the joint work on CZT detectors
and for access to their 3-D detector simulation code. We thank Orbotech, especially Y.\ Raab, A.\ Shani and U.\ El-Hanany for fruitful discussions.
|
1,314,259,993,594 | arxiv | \section{Introduction}
The determination of hadronic parameters, in particular meson-baryon coupling
constants, requires some information about the physics at large distances.
Therefore one has to employ some specific nonperturbative method to obtain
detailed predictions. Among the various nonperturbative methods,
QCD sum rules \cite{R1} have proved to be very useful to extract the low-lying
hadron masses and coupling constants. This method is a framework which connects
hadronic parameters with QCD parameters. It is based on short-distance
operator product expansion (OPE) in the deep Euclidean region of vacuum-vacuum
correlation function in terms of quark and gluon condensates. Further
progress has been achieved by an alternative method \cite{R2} known as the
QCD sum rules on the light cone.
The light cone QCD sum rules method is based on OPE on the light cone
which is an expansion over the twist of the operators rather than
the dimensions of the operators as in the traditional QCD sum rules approaches. In this expansion,
the main contribution comes from the lowest twist operators. The matrix elements of
non-local composite operators between meson and vacuum states define the
meson wave functions. The light-cone QCD sum rules have been employed to study
hadronic properties and these application can be found in \cite{R3}-\cite{R10}
and references therein.
In this work we study the $p\pi^{+}\rightarrow\Delta^{++}$ transition and
we employ the light-cone QCD sum rules approach to calculate the
$\Delta\pi N$ coupling $g_{\Delta\pi N}$.
We note that this coupling constant was investigated by using the traditional
QCD sum rules method in the soft pion limit \cite{R11}.
In the spirit of the philosophy
of the light cone QCD sum rules we study the time-ordered two-point
correlation function of the interpolating field for $N$ and $\Delta$
with pion. We use a dispersion relation that relates the hadronic
spectral density to the correlation function and thus write the
correlator in terms of phenomenological hadronic spectrum. Since the
interpolating fields are written in terms of quark fields, we also
calculate this correlator directly from QCD on the light cone. We consider
the structure $q_{\mu}$ and match these two descriptions. In order
to calculate the coupling constant $g_{\Delta\pi N}$, we invoke
the double Boorel transformation so that the excited states and the
continuum contributions can be safely seperated out.
We consider the two-point correlation function with pion
\begin{eqnarray}
\Pi_{\mu}(p,q)=\int d^{4}x~e^{ipx}~<0|T\{\eta^{\Delta}_{\mu}(x)\bar{\eta}^{N}(0)\}|\pi^{+}(q)>
\label{E1}
\end{eqnarray}
with $p$ and ${\eta}^{\Delta}_{\mu}$ the four-momentum and the interpolating
current of delta, ${\eta}^{N}$ the interpolating current of nucleon, and $q$
the four-momentum of pion. The interpolating currents for delta
and nucleon \cite{R12} are
\newpage
\begin{eqnarray}
{\eta}^{\Delta}_{\mu}&=&\epsilon_{abc}(u_{a}^{T}C\gamma_{\mu}u_{b})u_{c}
\label{E2} \\
{\eta}^{N}&=&\epsilon_{abc}(u_{a}^{T}C\gamma_{\mu}u_{b})\gamma_{5}\gamma^{\mu}d_{c} \label{E3}
\end{eqnarray}
where a,b,c are the color indices, $C=i\gamma_{2}\gamma_{0}$ is the charge conjugation matrix,
T denotes transpose, and $u$ and $d$ are up and down quark fields respectively.
In order to construct the phenomenological side of the correlator we use
the following Lagrangian density
\begin{eqnarray}
{\cal{L}}=g_{\Delta\pi N}\bar{N}\Delta_{\mu}\partial^{\mu}\pi \label{E4}
\end{eqnarray}
which defines the $\Delta\pi N$ coupling constant $g_{\Delta\pi N}$ and
where $N$, $\Delta_{\mu}$ and $\pi$ are the nucleon, delta and pion fields.
Thus, at the phenomenological level the correlator can be saturated
by delta and nucleon states as
\begin{eqnarray}
\Pi_{\mu}(p,q)=
\frac{<0|\eta^{\Delta}_{\mu}|\Delta><\Delta|N\pi><N|\bar{\eta}^{N}|0>}
{(p^{2}-m_{\Delta}^{2})(p^{\prime}~^{2}-m_{N}^{2})}+...
\label{E5}
\end{eqnarray}
with the contributions from the higher states and the continuum denoted by dots. The overlapping
amplitudes of the interpolating currents with delta and nucleon states are
\begin{eqnarray}
<N|\bar{\eta}^{N}|0>=\lambda_{N}\bar{u}^{N}(p^{\prime}) \label{E6} \\
<0|\eta^{\Delta}_{\mu}|\Delta>=\lambda_{\Delta}u^{\Delta}_{\mu}(p) \label{E7}
\end{eqnarray}
where $u^{\Delta}_{\mu}$ is the Rarita-Schwinger spinor
and the matrix element $<\Delta|N\pi>$ is given as
\begin{eqnarray}
<\Delta|N\pi>=-g_{\Delta\pi N}~q^{\nu}~\bar{u}^{\Delta}_{\mu}(p)u^{N}(p^{\prime})~~~~~~~.
\label{E07} \end{eqnarray}
Substitution of Eqs.(6,7,8) into Eq. (5) results in
\begin{eqnarray}
\Pi_{\mu}(p,q)=-\frac{\lambda_{N}\lambda_{\Delta}g_{\Delta\pi N}}
{(p^{2}-m_{\Delta}^{2})(p^{\prime}~^{2}-m_{N}^{2})}
&[&g_{\mu\nu}-\frac{1}{3}\gamma_{\mu}\gamma_{\nu}-\frac{2p_{\mu}p_{\nu}}{3m_{\Delta}^{2}}
+\frac{\gamma_{\mu}p_{\nu}-\gamma_{\nu}p_{\mu}}{3m_{\Delta}}]
\nonumber \\
&&~~~~~\times(\rlap/p+m_{\Delta})q^{\nu}(\rlap/p^{\prime}+m_{N})+... \label{E9}
\end{eqnarray}
We then consider the theoretical part of the correlator Eq. (1), for which
we obtain
\begin{eqnarray}
\Pi_{\mu}(p,q)=\int d^{4}x~e^{ipx}~[-2(S(\gamma_{\nu}C)S^{T}(C\gamma_{\mu})
{\cal{M}}(\gamma^{\nu}\gamma_{5})
-{\cal{M}}(\gamma^{\nu}\gamma_{5})tr((C\gamma_{\mu})S(\gamma_{\nu}C)S^{T})]
\label{E10}
\end{eqnarray}
with
\begin{eqnarray}
{\cal{M}}&=&\gamma_{5}<0|\bar{d}_{a}(0)\gamma_{5}u_{a}(x)|\pi>
-\gamma_{5}\gamma_{\lambda}<0|\bar{d}_{a}(0)\gamma_{5}\gamma^{\lambda}u_{a}(x)|\pi>
\nonumber \\
&&+\frac{1}{2}\sigma_{\alpha\beta}<0|\bar{d}_{a}(0)\sigma^{\alpha\beta}u_{a}(x)|\pi>~.
\label{Eq11}
\end{eqnarray}
The full light quark propagator in Eq. (10) with both perturbative term and
contribution from vacuum fields is given as
\begin{eqnarray}
iS(x)=&&i\frac{\rlap/x}{2\pi^{2}x^{4}}-\frac{<\bar{q} q>}{12}
-\frac{x^{2}}{192}<\bar{q} g_{s}~\sigma\cdot G~ q> \nonumber \\
&&-i\frac{g_{s}}{16\pi^2}\int_{0}^{1}du~\{\frac{\rlap/x}{x^2}\sigma\cdot
G(ux)
-4iu\frac{x_{\mu}}{x^{2}}G^{\mu\nu}(ux)\gamma_{\nu}\}+... \label{E12}
\end{eqnarray}
The matrix elements of the nonlocal operators between the vacuum and pion
state defines the two particle pion wave functions, and up to twist four the
Dirac components of these wave functions can be written as \cite{R13}
\begin{eqnarray}
<0|\bar{d}(0)\gamma_{\mu}\gamma_{5}u(x)|\pi^+>&=&if_{\pi}q_{\mu}\int_{0}^{1}du~e^{-iqux}
(\varphi_{\pi}(u)+x^{2}g_{1}(u)) \nonumber \\
&&+f_{\pi}(x_{\mu}-\frac{x^{2}q_{\mu}}{q^{2}}) \int_{0}^{1}du~e^{-iqux}g_{2}(u)~, \label{E13} \\
<0|\bar{d}(0)i\gamma_{5}u(x)|\pi^+>&=&\frac{f_{\pi}m_{\pi}^{2}}{m_{u}+m_{d}}
\int_{0}^{1}du~e^{-iqux}\varphi_{P}(u)~~~, \label{E14} \\
<0|\bar{d}(0)\sigma^{\mu\nu}\gamma_{5}u(x)|\pi^+>&=&(q_{\mu}x_{\nu}-q_{\nu}x_{\mu})
\frac{if_{\pi}m_{\pi}^{2}}{6(m_{u}+m_{d})}
\int_{0}^{1}du~e^{-iqux}\varphi_{\sigma}(u)~. \label{E15}
\end{eqnarray}
We further define
\begin{eqnarray}
G_{2}(u)=-\int_{0}^{u}g_{2}(v)dv, ~~~~G_2(1)=G_2(0)=0 \label{E16}
\end{eqnarray}
and
\begin{eqnarray}
g_{3}(u)=g_{1}(u)+G_{2}(u)~~~~~~~~~~. \label{E17}
\end{eqnarray}
From the theqretical part of the correlator given in Eq. (10),
we consider the structure $q_{\mu}$, and for this structure after
Fourier transformation over $x$ and double Borel transformations with respect to
variables $p_{1}^{2}=p^{2}$ and $p_{2}^{2}=(p-q)^{2}$ we finally obtain
\begin{eqnarray}
\Pi^{theor.}&=&f_{\pi}\mu_{\pi}<\bar{q} q>[~2M^{2}f_{0}(s_{0}/M^{2})
-\frac{1}{2}m_{0}^{2}~]~\varphi_{P}(u_{0})u_{0} \nonumber \\
&&+\frac{f_{\pi}}{\pi^{2}}~[-\frac{1}{2}M^{6}f_{2}(s_{0}/M^{2})
+\frac{1}{24}g_{s}^{2}<G^{2}>M^{2}f_{0}(s_{0}/M^{2})~]~
\varphi_{\pi}(u_{0}) \nonumber \\
&&+\frac{f_{\pi}}{\pi^{2}}~[-\frac{1}{6}M^{6}f_{2}(s_{0}/M^{2})
-\frac{1}{24}g_{s}^{2}<G^{2}>M^{2}f_{0}(s_{0}/M^{2})~]~
u_{0}\varphi_{\pi}^{\prime}(u_{0}) \nonumber \\
&&+\frac{f_{\pi}}{\pi^{2}}~\frac{1}{6}~g_{s}^{2}<G^{2}>g_{2}(u_{0})~
\frac{f_{\pi}}{\pi^{2}}~f_{1}(s_{0}/M^{2})g_{3}(u_{0})
\nonumber \\
&&+\frac{f_{\pi}}{\pi^{2}}~[~2M^{4}f_{1}(s_{0}/M^{2})
+\frac{1}{6}g_{s}^{2}<G^{2}>]~u_{0}~g_{3}^{\prime}(u_{0})
\nonumber \\
&&+\frac{1}{18}f_{\pi}\mu_{\pi}<\bar{q} q>
[-8M^{2}f_{0}(s_{0}/M^{2})\varphi_{\sigma}(u_{0})
+(-4M^{2}f_{0}(s_{0}/M^{2})+m_{0}^{2})u_{0}\varphi_{\sigma}^{\prime}(u_{0})]
\nonumber \\
&& \label{E18}
\end{eqnarray}
where the function
\begin{eqnarray}
f_{n}(s_{0}/M^{2})=1-e^{-s_{0}/M^{2}}
{\sum_{k=0}^{n}}\frac{(s_{0}/M^{2})^{k}}{k!} \nonumber
\end{eqnarray}
is the factor used to subtract the continuum, which is modeled by
the dispersion integral in the region $s_{1},~s_{2}\geq s_{0}$, $s_{0}$
being the continuum threshold,
$\mu_{\pi}=\frac{m_{\pi}^{2}}{m_u+m_d}$, and
\begin{eqnarray}
u_{0}=\frac{M_{1}^{2}}{M_{1}^{2}+M_{2}^{2}},~~~~
M^{2}=\frac{M_{1}^{2}M_{2}^{2}}{M_{1}^{2}+M_{2}^{2}} \nonumber
\end{eqnarray}
with $M_{1}^{2}$ and $M_{2}^{2}$ are the Borel parameters, and
$\varphi^{\prime}(u_{0})=\frac{d\varphi}{du}\mid_{u_{0}}$.
Performing double Borel transformation over the variables $p_{1}^{2}=p^{2}$
and $p_{2}^{2}=(p-q)^{2}$ on the phenomenological part in Eq. (9), and then
equating the result obtained for the Lorentz structure $g_{\mu\nu}$ part
to that theoretical result given in Eq. (18) we finally obtain the sum rule
for the coupling constant $g_{\Delta\pi N}$
\begin{eqnarray}
g_{\Delta\pi N}\lambda_{N}\lambda_{\Delta}=
\frac{2}{(m_{N}+m_{\Delta})^{2}}e^{\frac{m_{\Delta}^{2}}{M_{1}^{2}}}
e^{\frac{m_{N}^{2}}{M_{2}^{2}}}\Pi^{theor.}~~. \label{E19}
\end{eqnarray}
We note that this sum rule depends on the value of the wave functions
at a specific point, which are much better known than the whole wave
functions \cite{R14}.
\section{Numerical results and discussion}
The various parameters we adopt are
$m_{0}^{2}=0.8~ GeV^{2}$,
$<g_{s}^{2}G^{2}>=0.474~GeV^{4}$,
$f_{\pi}=0.132~GeV$,
$\mu_{\pi}=1.65~GeV$,
$<\bar{q}q>=-(0.225~GeV)^{3}$,
$s_{0}=3~GeV^{2}$.
In our calculation of the theoretical part we use the two particle
pion wave functions based on the QCD sum rule approach given in
\cite{R14} as
\begin{eqnarray}
\varphi_{\pi}(u,\mu)&=&
6u\bar{u}~[~1+a_{2}(\mu)C_{2}^{3/2}(2u-1)+a_{4}C_{4}^{3/2}(2u-1)~]~, \nonumber \\
\varphi_{\sigma}(u,\mu)&=&
6u\bar{u}~\big [~1+C_{2}\frac{3}{2}~[~5(u-\bar{u})^{2}-1~]
+C_{4}\frac{15}{8}~[~21(u-\bar{u})^{4}-14(u-\bar{u})^{2}+1~]~\big ]~, \nonumber \\
\varphi_{P}(u,\mu)&=&
1+B_{2}\frac{1}{2}~[~3(u-\bar{u})^{2}-1~]
+B_{4}\frac{1}{8}~[~35(u-\bar{u})^{4}-30(u-\bar{u})^{2}+3~]~, \nonumber \\
g_{1}(u,\mu)&=&
\frac{5}{2}\delta^{2}(\mu)u^{2}\bar{u}^{2}+\frac{1}{2}\epsilon (\mu)
~[~u\bar{u}(2+13u\bar{u})+10u^{3}\ln u~(2-3u+\frac{6}{5}u^{2})
\nonumber \\
&&+10\bar{u}^{3}\ln \bar{u}~(2-3\bar{u}+\frac{6}{5}\bar{u}^{2})~]~, \nonumber \\
g_{2}(u,\mu)&=&\frac{10}{3}\delta^{2}u\bar{u}(u\bar{u})~, \nonumber \\
G_{2}(u,\mu)&=&\frac{5}{3}\delta^{2}u^{2}\bar{u}^{2}~,
\label{E20}
\end{eqnarray}
where $\bar{u}=1-u$, $C_{2}^{3/2}$ and $C_{4}^{3/2}$ are the Gegenbauer
polynomials defined as
\begin{eqnarray}
C_{2}^{3/2}(2u-1)&=&\frac{3}{2}~[~5(2u-1)^{2}+1~]~, \nonumber \\
C_{4}^{3/2}(2u-1)&=&\frac{15}{8}~[~21(2u-1)^{4}-14(2u-1)^{2}+1~]~,
\label{E21}
\end{eqnarray}
and $a_{2}=2/3$, $a_{4}=0.43$ corresponding to the normalization point
$\mu=0.5~GeV$. The remaining parameters are taken from the QCD sum rule
estimates of Ref. \cite{R15} as $\delta^{2}(\mu=1~GeV)=0.2$, and
$\epsilon (\mu=1~GeV)=0.5$ and from those of Refs.\cite{R3,R13} as
$B_{2}=0.48$, $B_{4}=1.51$, $C_{2}=0.10$, $C_{4}=0.07$.
Since the mass difference of $N$ and $\Delta$ is not very significant,
we let $M_{1}^{2}=M_{2}^{2}=2M^{2}$ from which it follows that $u_{0}=1/2$.
We study the dependence of the sum rule of Eq. (19) on the continuum threshold
$s_{0}$ and on the Borel parameter $M^{2}$. We find that the sum rule is
very stable with resonable variations of $s_{0}$ and $M^{2}$ as can be seen
in Fig. 1. In order to determine the value of the coupling constant
$g_{\Delta\pi N}$ from the sum rule Eq. (19), the residues
$\lambda_{\Delta}$ and $\lambda_{N}$ of the hadronic currents are needed.
We use the following values that are obtained from the corresponding sum rules for
$\Delta$ and $N$ which are \cite{R11}
\begin{eqnarray}
|\lambda_{N}|^{2}~2(2\pi)^{4}~e^{-\frac{m_{N}^{2}}{M^{2}}}&=&
M^{6}f_{2}(s_{0}^{N}/M^{2})+b~M^{2}f_{0}(s_{0}^{N}/M^{2})+\frac{4}{3}a^{2}~,
\label{E22} \\
|\lambda_{\Delta}|^{2}~5(2\pi)^{4}~e^{-\frac{m_{\Delta}^{2}}{M^{2}}}&=&
M^{6}f_{2}(s_{0}^{\Delta}/M^{2})
-\frac{25}{18}~b~M^{2}f_{0}(s_{0}^{\Delta}/M^{2})+\frac{20}{3}a^{2}~,
\label{E23}
\end{eqnarray}
where
\begin{eqnarray}
a&=&-2\pi^{2}<\bar{q}q>=0.5~GeV^{3}~, \nonumber \\
b&=&\frac{\alpha_{s}}{\pi}<G^{2}>=0.12~GeV^{4}~, ~~and \nonumber \\
s_{0}^{\Delta}&=&(m_{\Delta}+0.5)^{2}~,~~~s_{0}^{N}=(m_{N}+0.5)^{2}~.
\nonumber
\end{eqnarray}
We substitute these values of the residues into the sum rule in Eq. (19),
and further study the dependence of $g_{\Delta\pi N}$ on the continuum
threshold $s_{0}$ and on the Borel parameter $M^{2}$. This dependence
is shown in Fig. 2, examination of which indicates that it is quite
stable with resonable variations of $s_{0}$ and $M^{2}$.
We thus obtain the coupling constant $g_{\Delta\pi N}$ as
\begin{eqnarray}
g_{\Delta\pi N}=(14.5\pm 1.5)~ GeV^{-1} \label{E24}
\end{eqnarray}
The uncertainty we included is due to the variation of the continuum
threshold and the Borel parameter.
Finally, we consider the Lagrangian density in Eq. (4) and obtain the
expression for the decay width as
\begin{eqnarray}
\Gamma (\Delta^{++}\rightarrow \pi^{+}p)=\frac{1}{6}\frac{g_{\Delta\pi N}^{2}}{4\pi}
\frac{(m_{N}+m_{\Delta})^{2}-m_{\pi}^{2}}{m_{\Delta}^{2}}~p_{\pi}^{3}
\label{E25}
\end{eqnarray}
where the pion momentum $p_{\pi}$ is
\begin{eqnarray}
p_{\pi}=\sqrt{(\frac{m_{\pi}^{2}+m_{\Delta}^{2}-m_{p}^{2}}{2m_{\Delta}})^{2}-m_{\pi}^{2}}~~.
\nonumber
\end{eqnarray}
We use the experimental result \cite{R16} for the decay width and obtain the coupling constant
as
\begin{eqnarray}
g_{\Delta\pi N} =(15.2\pm 0.1)~~~GeV^{-1} \nonumber
\end{eqnarray}
which indicates that our value in Eq. (24) is in satisfactory agreement with the experimental
result.
In summary, we calculated $\Delta\pi N$ coupling constant
$g_{\Delta\pi N}$ using light-cone QCD sum rules. Our result is consistent with
the results obtained using traditional QCD sum rules \cite{R11} and it
is in good agreement with the value of the coupling constant deduced from the
experimental decay rate of $\Delta^{++}$ baryon.
\vspace{0.5cm}
{\bf Acknowledgment}\\
We thank to T. M. Aliev for suggesting us this problem and
his guidance during the course of our work.
\pagebreak
|
1,314,259,993,595 | arxiv | \section{Introduction}
\noindent In this paper, we consider only simple and finite graphs. We use $V(G)$ for the vertex set and $E(G)$ for the edge set of a graph $G$. The neighborhood, $N_G(v)$ or shortly $N(v)$ of a vertex $v$ of $G$ is the set of all vertices adjacent to $v$ in $G$. For further graph-theoretic terminology and notation, we refer Bondy and Murty \cite{MR2368647} and Hammack $et$ $al.$\cite{MR2817074}.
\par A Kotzig array \cite{kotzig1971magic}, $KA(a,b)$ is an $a\times b$ array in which every row contains each of the integers $0,1,\cdots, b-1$ exactly once and the sum of the entries in each column is the same constant $c = \frac{a(b - 1)}{2}$. The following theorem can be obtained from \cite{MR3013201}.
\par In 2003, Miller {\cite{MR1999203}} introduced the concept of distance magic labeling of a graph $G$, which is injective function $l : V(G)\rightarrow \{1,2,\cdots,|V(G)|\}$ such that for any $u$ of $G$, the weight $w_G(u)$ of $u$ is a constant $\mu$, where $w_G(u)$ is the sum of labels of all neighbors of $u$. A graph $G$ that admits such a labeling $l$ is called a distance magic graph, or shortly, a $dmg$.
\par The following results provide some necessary condition for distance magicness of regular graphs.
\begin{theorem}\label{oddregular}
\textnormal{\cite{MR1999203,Rao2004,vilfred}} No $r$-regular graph with $r$-odd can be a distance magic graph.
\end{theorem}
\begin{theorem}\label{2mode4}
\textnormal{\cite{MR2259699}} Let $EIT(a, r)$ be an equalized tournament with an even number $a$ of teams and $r\equiv 2 \mod 4$. Then $a\equiv 0 \mod 4$.
\end{theorem}
\noindent A simple argument on parity, can prove that an odd regular graph is not distance magic \cite{MR1999203} and therefore, $G=K_{3,3}$ given in Figure \ref{Fig5.1}, is not a distance magic. That is, it is not possible to find a labeling from $V(G)$ to $\{1,2,...,6\}$, with all vertices have same constant weight. But the weights of all vertices, are a unique constant, when the existing set $\{1,2,\cdots,6\}$ is replaced by certain sets $S_1,S_2$ and $S_3$ with $|S_1|=|S_2|=|S_3|=6$ (see Figure 1).
\begin{figure}[ht]
\centering
\includegraphics[width=120mm]{Fig5_1}
\caption{$S_1,$ $S_2$ and $S_3$-magic graph $G$}
\label{Fig5.1}
\end{figure}
\par Motivated by the above fact, Godinho et al. \cite{MR3743955} defined the concept of $S$-magic labeling of a graph and introduce certain measure namely, the distance magic index to predict the nature of distance of magicness of a graph. \begin{definition}
\textnormal{\cite{MR3743955}} Let $G$ be a graph on $n$ vertices and let $S$ be a set of any $n$ positive integers. Then $G$ is said to be \textit{$S$-magic} if there exists a bijection $l$ from $V(G)$ to $S$ such that for any vertex $u$ of $G$, $\sum_{v\in N(u)} l(v)$ is a constant $\mu'$ and $\mu'$ is the $S$-magic constant.
\end{definition}
\begin{definition}
\textnormal{\cite{MR3743955}} Let $G$ be a graph on $n$ vertices and let $S$ be a set of $n$ positive integers with the largest number $\eta(S)$. Define $i(G)$ to be the infimum of all $\eta(S)$'s, where the infimum runs overall $S$ for which $G$ admits $S$-magic labeling. Then the \textit{distance magic index} of $G$, $\theta(G)=i(G)-n$.
\end{definition}
In 2015, Godinho et al. \cite{godinho2015s} defined the concept of fractional domination number and its relationship with $S$-magic graphs.
\begin{definition}\cite{godinho2015s}
Let $G = (V,E)$ be a graph without isolated vertices. A function $g : V \rightarrow [0, 1]$ is called a total dominating function of $G$ if $g(N(v)) = \sum_{u \in N(v)} g(u) \ge 1$ for all $v \in V$. The fractional total domination number $\gamma_{ft}(G)$ is defined to be $min\{|g| : g$ is a total dominating function of $G\}$, where $|g| = \sum_{v \in V} g(v)$.
\end{definition}
\begin{theorem}\textnormal{{\cite{MR1999203,kotlar}}} Let $1 \le n_1 \le n_2 \le n_3 \le n_4$ and $s_{i} = \sum_{i = 1}^{4} n_i$. Suppose $G$ is either $K_{n_1,n_2}$ or $K_{n_1,n_2,n_3}$ or $K_{n_1,n_2,n_3, n_4}$. There exists a distance magic labeling of $G$ if and only if the following conditions hold.
\begin{enumerate}
\item[$(a)$] $n_2 \ge 2$,
\item[$(b)$] $n(n + 1) \equiv 0$ (mod $2p)$, where $n = s_{p} = |V (K_{n_{1},,...,n_{p}})|$, and
\item[$(c)$] $\displaystyle \sum_{j=1}^{s_{i}} (n - j + 1) \ge \frac{n(n+1)i}{2p}$ for $1 \le i \le p$,
\end{enumerate}
where $p=2$ if $G$ is complete bipartite graph, and $p=3$ if $G$ is complete tripartite graph, and $p=4$ if $G$ is complete four-partite graph.
\end{theorem}
\begin{lemma} \textnormal{ \cite{godinho2015s})}
The complete r-partite graph $G = K_{m_1,m_2,...,n_r}$ is $S$-magic with labeling f if and only if the sum of the labels of all the vertices in any two partite sets are equal.
\end{lemma}
\begin{theorem} \textnormal{\cite{MR3743955}}
Let $G$ be the complete bipartite graph $K_{n_{1}, n_{2}}$, where $2 \le n_{1} \le n_{2}$ and $n_{1} + n_{2} = n$. Then\\
$\displaystyle \theta(G)$ = $\begin{cases}
0 & \text{ if } n(n+1) \ge 2n_{2}(n_{2} + 1) \textnormal{\it ~and~} n\equiv 0 \text{ or } 3 (\text{mod~} 4)\\
1 & \text{ if } n(n+1) \ge 2n_{2}(n_{2} + 1) \textnormal{\it ~and~} n\equiv 1 \text{ or } 2 (\text{mod~} 4)\\
\bigl\lceil \frac{|n(n+1) - 2n_{2}(n_{2} + 1)|}{2n_
{1}} \bigr\rceil & \text{ if } n(n+1) < 2n_{2}(n_{2} + 1).
\end{cases}$
\end{theorem}
\begin{theorem} \textnormal{\cite{godinho2015s}}
If a graph $G$ admits an $S$-magic labeling $f$ with magic constant $k$, then $k = \frac{\alpha}{\gamma_{ft}(G)}$, where $\alpha = \sum_{i \in S}i$.
\end{theorem}
\begin{corollary} \textnormal{\cite{godinho2015s}}
If $G$ is $S$-magic, then the smallest $S$-magic constant corresponds to the $S$-magic labeling for which $ \sum_{i \in S}i$ is minimum.
\end{corollary}
\par Let $G$ be a graph on $n$ vertices. It is clear that $G$ is distance magic if and only if $\theta(G) = 0$. On the other hand, if $G$ is not $S$-magic, for any $S$ with $|V(G)|$ positive integers, then $\theta(G)$ is not finite. Godinho et al.\cite{MR3743955} identified a parameter for a general graph $G$, defined by $g(x) = \frac{1}{2}[(2\delta(n+x)-\delta^2+\delta)-\Delta(\Delta+1)]$, which can be effectively used to determine the distance magic index of general graphs. From \cite{MR3743955}, one can see that if $g(0)<0$, then $G$ is not a distance magic graph and hence $\theta(G)\geq 1$.
The following result gives a lower bound for $\theta(G)$.
\begin{theorem}\label{godinho}
\textnormal{\cite{MR3743955}} If $G$ is a graph of order $n$ such that $g(0)<0$, then $\theta(G)\geq \biggl\lceil \frac{|g(0)|}{\delta}\biggr\rceil$.
\end{theorem}
\par However, if $g(0)\geq 0$, there is no result on $G$ to argue whether for that graph, $\theta(G)=0$ or $\theta(G) \geq 1$. Note that if $G$ is an $r$-regular graph with $n$ vertices, then $g(0)>0$ and hence Theorem \ref{godinho} does predict any possible $\theta(G)$.
\section{Distance magic index of complete-tripartite graphs}
In this section, we prove the distance magicness and the magic index of the complete tripartite graph $K_{n_1, n_2, n_3}$, where $V=\bigcup_{i=1}^3 V_i$ and $|V_i|=n_i$ so that $n_1 + n_2 + n_3 = |V|=n$ (say) and the edge set $E =\{uv: u\in V_i,v\in V_j, 1\leq i,j\leq 3 \textnormal{~and~} i\ne j\}$ with $|E |=n_1n_2 + n_2n_3 + n_1n_3$. Also when $l$ is a mapping from the vertex set of $K_{n_1, n_2, n_3}$ to $S$, where $S$ is set of positive integers with $|S|=|V(K_{n_1, n_2, n_3})|$, we say label set of $V_i$ is $\{l(v) : v\in V_i\}$, which is denoted by $L_i$ and we use $s(L_i)$ to denote the sum of labels of all the elements in $L_i$, where $i=1,2,3$. Without loss of generality, one can restrict $2 \le n_{1} \le n_{2} \le n_{3}$. For any $i<j$, we denote $\zeta_i^j = \sum_{\gamma=i}^{j} \gamma$. Now, by using the above notation and terminology, we first rewrite the third condition in Theorem 1.3\cite{MR1999203, kotlar} for the complete tripartite graph $K_{n_1, n_2, n_3}$
\begin{equation}\label{1a}
\zeta_{n - n_1 + 1}^{n} = \sum_{j=1}^{n_{1}} (n-j+1) = \frac{n_{1}}{2} (2n - n_{1} + 1) \ge \frac{n(n+1)}{6} = \frac{\zeta_1^{n}}{3}
.\end{equation}
and,
\begin{equation*}
\zeta_{n_3 + 1}^{n} = \sum_{j=1}^{n_{1}+n_{2}} (n-j+1) = \frac{n_{1}+n_{2}}{2} (2n - n_{1} - n_{2} + 1) \ge \frac{n(n+1)}{3} = \frac{2\zeta_1^{n}}{3}
.\end{equation*}
Since,
\begin{equation}\label{2a}
\frac{n_{1}+n_{2}}{2} (2n - n_{1} - n_{2} + 1) + \frac{n_{3}(n_{3}+1)}{2} = \frac{\zeta_1^{n}}{3} + \frac{2\zeta_1^{n}}{3} = \frac{n(n+1)}{2} = \zeta_1^{n}
.\end{equation}
we get,
\begin{equation}\label{3a}
3\zeta_1^{n_3} \le \zeta_1^{n}
.\end{equation}
From the above equations \eqref{1a} and \eqref{3a}, one can write the following theorem.
\begin{theorem}
$K_{n_{1}, n_{2}, n_{3}}$ is distance magic if and only if the following conditions hold.
\begin{enumerate}
\item[$1)$] $n_{2} \ge 2$,
\item[$2)$] $2\zeta_1^n \equiv 0\mod 6$,
\item[$3)$] $3\zeta_{n-n_1+1}^{n} \ge \zeta_1^{n}$ and,
\item[$4)$] $3\zeta_{1}^{n_3} \le \zeta_1^{n}$.
\end{enumerate}
\end{theorem}
Note that Theorem [3.1] is a rewritten from Theorem [1.3] for the particular case of complete tripartite graphs, $K_{n_{1}, n_{2}, n_{3}}$. Therefore, from above theorem, one can classify $K_{n_{1}, n_{2}, n_{3}}$ into the following five cases.
\begin{enumerate}
\item[$i)$] $3\zeta_{n-n_1+1}^{n} \ge \zeta_1^{n} \ge 3\zeta_{1}^{n_3} \text{ and }2\zeta_1^{n} \equiv 0 \text{ mod } 6$.
\item[$ii)$] $\zeta_1^{n} > 3\zeta_{n-n_1+1}^{n}$ and $\zeta_1^{n} \ge 3\zeta_{1}^{n_3} $.
\item[$iii)$] $3\zeta_{n-n_1+1}^{n} < \zeta_1^{n} < 3\zeta_{1}^{n_3} $.
\item[$iv)$] $3\zeta_{n-n_1+1}^{n} \ge \zeta_1^{n} \ge 3\zeta_{1}^{n_3} \text{ and } 2\zeta_1^{n} \equiv 2 \text{ mod } 6$.
\item[$v)$] $\zeta_1^{n} \le 3\zeta_{n-n_1+1}^{n}$ and $\zeta_1^{n} < 3\zeta_{1}^{n_3} $.
\end{enumerate}
The resulting theorem determines the distance magic index of the complete tripartite graphs for some of the cases mentioned above.
\begin{lemma}
Let $K_{n_{1}, n_{2}, n_3}$ be a complete tripartite graph such that $$3\zeta_{n-n_1+1}^{n} \ge \zeta_1^{n} \ge 3\zeta_{1}^{n_3} \text{ and } 2\zeta_1^{n} \equiv 2 \mod 6.$$ Then $K_{n_1 + 1, n_2, n_3}$ is a distance magic graph and consequently, $\theta(K_{n_1 + 1, n_2, n_3}) = 0$.
\end{lemma}
\begin{proof} Consider, $6\zeta_{n-n_1+1}^{n+1} = 3(n_1 + 1)(2(n+1) - (n_1+1) + 1)$,
\begin{eqnarray*}
3(n_1 + 1)(2(n+1) - (n_1+1) + 1) &=& 3n_1(2n - n_1 + 1) - 6n_1 + 6n\\ &\ge& n(n+1) - 6n_1 + 6n.
\end{eqnarray*}
Since $n_1 \le \frac{n}{3}$. Hence,
\begin{eqnarray*}
3(n_1 + 1)(2(n+1) - (n_1+1) + 1) &\ge& n(n+1) + 6n - 6n_1\\ &\ge& n(n+1) + 4n\\ &=& n^2 + 5n \\ &>& n^2 + 3n + 2\\ &=& (n+1)(n+2) = 2\zeta_{1}^{n+1}.
\end{eqnarray*}
Now consider $6\zeta_{1}^{n_3} = 3n_3(n_3+1)$,
\begin{eqnarray*}
3n_3(n_3 + 1) \le n(n+1) \le (n+1)(n+2) = 2\zeta_{1}^{n+1}.
\end{eqnarray*}
\noindent Therefore,
\begin{eqnarray*}
2\zeta_{1}^{n+1} = (n+1)(n+2) \cong n(n+1) + 2(n+1) \text{ mod } 6 \cong 2n + 4 \text{ mod } 6.
\end{eqnarray*}
Since $2\zeta_{1}^{n} \cong 2$ mod $6$, $2n \cong 2$ mod $6$, we have,
\begin{equation*}
2\zeta_{1}^{n+1} \cong 0 \text{ mod } 6
.\end{equation*}
Hence by Theorem 3.1, $K_{n_1 + 1, n_2, n_3}$ is a distance magic graph.
\end{proof}
\begin{theorem}
Let G be the complete tripartite graph $K_{n_{1}, n_{2}, n_{3}}$. Then,
\begin{enumerate}
\item[$(i)$] If $3\zeta_{n-n_1+1}^{n} \ge \zeta_1^{n} \ge 3\zeta_{1}^{n_3} \text{ and }2\zeta_1^{n} \equiv 0 \text{ mod } 6$, then $\theta(G)=0$.
\item[$(ii)$] If $\zeta_1^{n} > 3\zeta_{n-n_1+1}^{n}$ and $\zeta_1^{n} \ge 3\zeta_{1}^{n_3} $, then
\begin{equation*}
\theta(G) = \begin{cases}
\biggl\lceil \frac{3\zeta_{1}^{n-n_1} - 2\zeta_{1}^{n}}{2n_{1}} \biggr\rceil & \text{ if } n - n_{1} \equiv 0 \text{ or } 3 \text{ (mod } 4)\\
\biggl\lceil \frac{3\zeta_{1}^{n-n_1} - 2\zeta_{1}^{n} + 1}{2n_{1}}\biggr \rceil & \text{ if } n - n_{1} \equiv 1 \text{ or } 2 \text{ (mod } 4). \end{cases} .\end{equation*}
\item[$(iii)$] If
$3\zeta_{n-n_1+1}^{n} < \zeta_1^{n} < 3\zeta_{1}^{n_3}$, then
\begin{equation*}
\theta(G) \ge \biggl\lceil \frac{\zeta_{1}^{n_3} - \zeta_{n-n_1+1}^{n}}{n - n_{3}} \biggr\rceil.
\end{equation*}
\item[$(iv)$] If $3\zeta_{n-n_1+1}^{n} \ge \zeta_1^{n} \ge 3\zeta_{1}^{n_3} \text{ and } 2\zeta_1^{n} \equiv 2 \text{ mod } 6$,
\begin{equation*}
1 \le \theta(G) \le n+1.
\end{equation*}
\item[$(v)$] If $\zeta_1^{n} \le 3\zeta_{n-n_1+1}^{n}$ and $\zeta_1^{n} < 3\zeta_{1}^{n_3},$ then
\begin{equation*}
\theta(G) = \biggl\lceil \frac{\zeta_{1}^{n_3} - \zeta_{n-n_1+1}^{n}}{n_{1}} \biggr\rceil, \text{ if } \frac{\zeta_{1}^{n_3} - \zeta_{n-n_1+1}^{n}}{n_{1}} \ge \theta(H)
\end{equation*}
\begin{equation*}
\theta(G) \le \theta(H) + 1, \text{ if } \frac{\zeta_{1}^{n_3} - \zeta_{n-n_1+1}^{n}}{n_{1}} < \theta(H),\textnormal{~~~~~~~~}
\end{equation*}\\
where $H \cong K_{n_{2}, n_{3}}$.
\end{enumerate}
\end{theorem}
\begin{proof}
Let $G=K_{n_{1}, n_{2}, n_{3}}$ be a complete tripartite graph, where $2 \le n_{1} \le n_{2}\le n_{3}$ and $V =V_1\cup V_2\cup V_3$ with $|V_i|=n_i$, and $|V | =n_1+n_2+n_3=n$~(say) and with the edge set $E =\{uv: u\in V_i,v\in V_j, 1\leq i,j\leq 3 \textnormal{~and~} i\ne j\}$. Note that $|E|=n_1n_2 + n_2n_3 + n_1n_3$. For any vertex $v\in V_i$, since the neighborhood of $v$ is $V_j\cup V_k$, the degree of $v$ is $n_j+n_k$, where $i,j,k$ are three distinct integers and $1 \leq i,j,k\leq 3$.\\
\noindent \textbf{Proof of case $(i)$} \\ Suppose that $3\zeta_{n-n_1+1}^{n} \ge \zeta_1^{n} \ge 3\zeta_{1}^{n_3} \text{ and}2\zeta_1^{n} \equiv 0 \mod 6$. By using Theorem 3.1, we have $\theta(G) = 0$.
\noindent \textbf{Proof of case $(ii)$} \\ Let $L_{i}$ denote the set of labels of the partition $V_{i}$, where $i=1,2,3$.
Let the label sets of partitions $V_{1}$, $V_{2}$ and $V_{3}$ be $L_{1} = \{n - n_{1} + 1, \cdots, n\}$, $L_{2} = \{n_{3} + 1,\cdots, n - n_{1}\}$ and $L_{3} = \{1,\cdots, n_{3}\}$ respectively. Let $H$ be a subgraph of $G$ such that $H \cong K_{n_{2},n_{3}}$. Consider,
\begin{eqnarray*}
\frac{(n - n_{1})(n - n_{1} + 1)}{2} = \frac{n^{2} + n + n_{1}^{2} - 2nn_{1} - n_{1}}{2}\\
= \frac{n^{2} + n}{2} - \frac{n_{1}(2n - n_{1} + 1)}{2}.
\end{eqnarray*}
Since $3n_{1}(2n - n_{1} +1) < n(n+1)$ and $3n_{3}(n_{3} + 1) \le n(n+1)$,
\begin{eqnarray}\label{6}
\frac{(n - n_{1})(n - n_{1} + 1)}{2} > \frac{n^{2} + n}{3} \ge n_{3}(n_{3} + 1).
\end{eqnarray}
Hence by Theorem 1.4,\\
\begin{equation}\label{7}
\theta(H) = \begin{cases}
0 & \text{if} n - n_{1} \equiv 0 \text{or} 3 (\mod 4)\\
1 & \text{ if } n - n_{1} \equiv 1 \text{ or } 2 (\mod 4).\\
\end{cases} \\
\end{equation}
\vskip .2cm \noindent Hence by Theorem 1.4 \cite{MR3743955}, the graph $H \equiv K_{n_{2},n_{3}}$ is $S$-magic with $S = \{1, 2,\cdots, n-n_{1}\}$ if $\theta(H) = 0$ and with $S = \{1, 2,\cdots, n-n_{1}-1, n-n_{1}+1\}$ if $\theta(H) = 1$. Hence from this,
\begin{eqnarray}\label{8}
h = \begin{cases} \frac{(n-n_{1})(n-n_{1}+1)}{4}, & \textnormal{if} n-n_{1} \equiv 0 \textnormal{ or} 3 (\mod 4)\\
\frac{(n-n_{1})(n-n_{1}+1)+2}{4}, & \textnormal{if} n-n_{1} \equiv 1 \text{or} 2 (\mod 4).
\end{cases}
\end{eqnarray}
where $h$ is the magic constant of the subgraph $H \cong K_{n_{2}, n_{3}}$. Note that $h$ is an integer.\newline
\noindent \textbf{Subcase 1} Let $n - n_{1} \equiv 0$ or $3(\mod 4)$. Then from \eqref{7}, $\theta(H) = 0$. Thus $L_{1} = \{n - n_{1} + 1,\cdots, n\}$, $L_{2}$ is the set of labels of $V_{2}$ in $H$ and $L_{3}$ is the set of all labels of $V_{3}$ in $H$. Now for the graph $G$ to be $S$-magic, $s(L_{1}) = h$. But $s(L_{1}) = \frac{n_{1}}{2}(2n - n_{1} + 1)$. Therefore, let
\begin{eqnarray*}
\centering
\lambda &=& h - s(L_1) = \frac{(n-n_1)(n-n_1+1)}{4} - \frac{n_1}{2}(2n - n_1 + 1).\\
\textnormal{Then, }\mbox{~~~~~~~~~} \lambda &=& h - \{s(L_1) + s(L_2) + s(L_3)\} + s(L_2) + s(L_3)\\
&=& 3h - \frac{n(n+1)}{2}.
\end{eqnarray*}
\begin{equation}\label{9}
\therefore \lambda = \frac{6h - n(n+1)}{2} = \frac{3(n-n_{1})(n-n_{1}+1)-2n(n+1)}{4}
.\end{equation}
From equation \eqref{6}, $\lambda > 0$. Hence, let $\lambda = n_{1}q + r$, where $q \ge 0$ and $0 \le r < n_{1}$. Now, increase each label of $L_{1}$ by $q$ and keep $L_{2}$ and $L_{3}$. Then, $L_{1}^{'} = \{n - n_{1} + q + 1,\cdots, n + q\}$, $L_{2}^{'} = L_{2}$ and $L_{3}^{'} = L_{3}$.
\vskip .2cm \noindent Therefore, $$s(L_{1}^{'}) = s(L_{1}) + n_{1}q.$$
\noindent {\bf (i)} Suppose $r=0$. We can label the partitions $V_{1}$, $V_{2}$ and $V_{3}$ with the labels $L_{1}^{'}$, $L_{2}^{'}$ and $L_{3}^{'}$ respectively. From equation \eqref{8} and \eqref{9}, it is clear that, $s(L_{1}^{'}) = s(L_{2}^{'}) = s(L_{3}^{'})$. Hence by Lemma 1.1, $G$ is $S$-magic. Since the highest label is $n+q$, the magic index, $\theta(G) = q$.\\
{\bf (ii)} Suppose $r>0$, We replace the label $n + q - r + 1 \in L_{1}^{'}$ with the label $n + q - r + 1 + r = n + q + 1$. Hence the resulting labels are $L_{1}^{''} = \{n - n_{1} + q + 1,\cdots, n + q - r, n + q - r +2,\cdots, n + q + 1\}$, $L_{2}^{''} = L_{2}^{'}$ and $L_{3}^{''} = L_{3}^{'}$. Now $s(L_{1}^{''}) = s(L_{1}^{'}) + r = s(L_{1}) + n_{1}q + r$, $s(L_{2}^{''})=s(L_{2})$ and $s(L_{3}^{''}) = s(L_{3})$. We label the partitions $V_{1}$, $V_{2}$ and $V_{3}$ with the labels $L_{1}^{''}$, $L_{2}^{''}$ and $L_{3}^{''}$ respectively.
\vskip .2cm \noindent From equation \eqref{8} and \eqref{9}, it is clear that $s(L_{1}^{''}) = s(L_{2}^{''}) =s(L_{3}^{''})$. Hence by Lemma 1.1, $G$ is $S$-magic. Since the highest label is $n + q + 1$, the magic index, $\theta(G) = q + 1$. Therefore,
\begin{equation*}
\theta(G) = \biggl\lceil \frac{6h - n(n+1)}{2n_{1}} \biggr\rceil = \biggl\lceil \frac{3(n - n_{1})(n - n_{1} + 1) - 2n(n+1)}{4n_{1}} \biggr\rceil
.\end{equation*}
Since the subgraph $H$ is $S$-magic with $\theta(H) = 0$, by Corollary 1.1, $h$ is the minimum magic constant for the subgraph $H$. Therefore, $\theta(G)$ is minimum as it is a linear function of $h$. \newline
\noindent \textbf{Subcase 2}. Let $n - n_{1} \equiv 1 $ or $2$ (mod $4)$. Hence from \eqref{7}, $\theta(H) = 1$. So, take $L_{1}^{'} = \{n-n_{1}, n-n_{1}+2,\cdots, n\}$, $L_{2}^{'}$ is the set of labels of $V_{2}$ in $H$ and $L_{3}^{'}$ is the set of labels of $V_{3}$ in $H$. Now if $G$ is $S$-magic, then by Lemma 1.1, we must have $s(L_{1}^{'}) = s(L_{2}^{'}) = s(L_{3}^{'})$. But $s(L_{1}^{'}) = s(L_{1}) - 1 = \frac{n_{1}}{2}(2n - n_{1} +1) - 1$. Hence, let
\begin{eqnarray*}
\lambda &=& h - s(L_{1}) + 1 = \frac{(n-n_{1})(n-n_{1}+1)+2}{4} - \bigl(\frac{n_{1}}{2}(2n - n_{1} + 1) -1\bigr) \\
\implies \lambda &=& h - s(L_{1}) + 1 - s(L_{2}) - s(L_{3}) + s(L_{2}) + s(L_{3}) \\ &=& 3h - \frac{n(n+1)}{2}.\\
\textnormal{Thus,~~} \lambda &=& \frac{6h - n(n+1)}{2} = \frac{3(n-n_{1})(n-n_{1}+1)-2n(n+1) + 6}{4}
\end{eqnarray*}
\\ \noindent As in the previous subcase, from equation \eqref{6}, $\lambda > 0$. Hence, let $\lambda - 1 = n_{1}q + r - 1$, where $q \ge 0$ and $0 \le r < n_{1}$.
\vskip .2cm \noindent First, let us consider $q > 1$. Now increase each label of $L_{1}^{'}$ by $q$ and keep $L_{2}^{'}$ and $L_{3}^{'}$ unchanged, we get $L_{1}^{''} = \{n - n_{1} + q, n - n_{1} + 2 + q,\cdots, n + q\}$, $L_{2}^{''} = L_{2}$ and $L_{3}^{''} = L_{3}$. Therefore, $s(L_{1}^{''}) = s(L_{1}) + n_{1}q - 1$. If $r = 0$, we can label the partitions $V_{1}$, $V_{2}$ and $V_{3}$ with the labels $L_{1}^{''}$, $L_{2}^{''}$ and $L_{3}^{''}$ respectively. From equation \eqref{8} and \eqref{9}, we see that $s(L_{1}^{''}) = s(L_{2}^{''}) = s(L_{3}^{''})$. Hence by Lemma 1.1, $G$ is $S$-magic. Since the highest label is $n + q$, the magic index, $\theta(G) = q$. Since $\lambda - 1 = n_{1}(q - 1) + n_{1} - 1$, we have the distance magic index, $$\theta(G) = q = \biggl\lceil \frac{\lambda - 1}{n_{1}} \biggr\rceil.$$
\vskip .2cm \noindent {\bf (i)} Suppose $r$ is $1$. Then replace the label $n - n_{1} + q \in L_{1}^{''}$ with the label $n - n_{1} + q +1$. Hence the resulting labels are $L_{1}^{'''} = \{n - n_{1} + q + 1, n - n_{1} + q + 2, \dots, n + q\}$, $L_{2}^{'''} = L_{2}$ and $L_{3}^{'''} = L_{3}$. Therefore, $s(L_{1}^{'''}) = s(L_{1}) + n_{1}q$, $s(L_{2}^{'''}) = s(L_{2})$ and $s(L_{3}^{'''}) = s(L_{3})$. We label the partitions $V_{1}$, $V_{2}$ and $V_{3}$ with the labels $L_{1}^{'''}$, $L_{2}^{'''}$ and $L_{3}^{'''}$ respectively. From equation \eqref{8} and \eqref{9}, we see that $s(L_{1}^{'''}) = s(L_{2}^{'''}) = s(L_{3}^{'''})$. Hence, by Lemma 1.1, $G$ is $S$-magic. Since the highest label is $n + q$, the distance magic index $$\theta(G) = q = \biggl\lceil \frac{\lambda - 1}{n_{1}} \biggr\rceil.$$
\vskip .2cm \noindent {\bf (ii)} Suppose $r>1$. Then replace the label $n - n_{1} + q \in L_{1}^{''}$ with the label $n - n_{1} + q +1$ and $n + q - r + 2 \in L_{1}^{''}$ with the label $n + q - r + 2 + r - 1 = n + q + 1$. Hence the resulting labels are $L_{1}^{''''} = \{n - n_{1} + 1 + q, n - n_{1} + 2 + q, \dots, n + q - r + 1, n + q - r + 3, ..., n + q + 1\}$, $L_{2}^{''''} = L_{2}^{'}$ and $L_{3}^{''''} = L_{3}^{'}$. Now $s(L_{1}^{''''}) = s(L_{1}^{'}) + r = s(L_{1}) + n_{1}q + r - 1$, $s(L_{2}^{''''}) = s(L_{2})$ and $s(L_{3}^{''''}) = s(L_{3})$. We label the partitions $V_{1}$, $V_{2}$ and $V_{3}$ with the labels sets $L_{1}^{''''}$, $L_{2}^{''''}$ and $L_{3}^{''''}$ respectively. From equation \eqref{8} and \eqref{9}, we see that $s(L_{1}^{''''}) = s(L_{2}^{''''}) = s(L_{3}^{''''})$. Hence, by Lemma 1.1, $G$ is $S$-magic. Since the highest label is $n + q + 1$, the distance magic index $\theta(G) = q + 1$. Since $\lambda - 1 = n_{1}q + r - 1$ and $r > 1$, $$\theta(G) = q + 1 = \biggl\lceil \frac{\lambda - 1}{n_{1}} \biggr\rceil.$$
\noindent Now, if we take $q = 1$, then $\lambda - 1 = n_{1} + r - 1$. Hence, we can not increase $n - n_{1} \in L_{1}^{'}$ by 1 as $n - n_{1} + 1 \in L_{2}^{'} \cup L_{3}^{'}$. Hence, increase each label of $L_{1}^{'}-\{n-n_1\}$ by 1, and keep $L_{2}^{'}$ and $L_{3}^{'}$ unchanged and hence, we get, $L_{1}^{'''} = \{n - n_{1}, n - n_{1} + 3,\cdots, n + 1 \}$, $L_{2}^{'''} = L_{2}$ and $L_{3}^{'''} = L_{3}$. Therefore $$s(L_{1}^{'''}) = s(L_{1}) + n_{1} - 1.$$
\vskip .2cm \noindent {\bf (i)} Suppose $r=0$. Then replace the label $n + 1 \in L_{1}^{'''}$ with $n + 2$. Hence the resulting labels are $L_{1}^{''''} = \{n - n_{1}, n - n_{1} + 3,\cdots, n, n + 2\}$, $L_{2}^{''''} = L_{2}$ and $L_{3}^{''''} = L_{3}$. Therefore, $s(L_{1}^{''''}) = s(L_{1}) + n_{1}$, $s(L_{2}^{''''}) = s(L_{2})$ and $s(L_{3}^{''''}) = s(L_{3})$. We label the partitions $V_{1}$, $V_{2}$ and $V_{3}$ with the labels $L_{1}^{''''}$, $L_{2}^{''''}$ and $L_{3}^{''''}$, respectively. From equation \eqref{8} and equation \eqref{9}, it is clear that, $s(L_{1}^{''''}) = s(L_{2}^{''''}) = s(L_{3}^{''''})$. Hence by Lemma 1.1, $G$ is $S$-magic. Since the highest label is $n + 2 = n + q + 1$, the magic index, $\theta(G) = 2 = q+1$. Since $\lambda - 1 = n_{1} + r - 1$, we have $\theta(G) = \lceil \frac{\lambda - 1}{n_{1}} \rceil$.\\
\vskip .2cm \noindent {\bf (ii)} Suppose $r = 1$. Then replace the label $n - n_{1} \in L_{1}^{'''}$ with $n - n_{1} + 2$. Hence the resulting labels are $L_{1}^{''''} = \{n - n_{1} + 2, n - n_{1} + 3,\cdots, n + 1\}$, $L_{2}^{''''} = L_{2}$ and $L_{3}^{''''} = L_{3}$. Therefore, $s(L_{1}^{''''}) = s(L_{1}) + n_{1}$, $s(L_{2}^{''''}) = s(L_{2})$ and $s(L_{3}^{''''}) = s(L_{3})$. We label the partitions $V_{1}$, $V_{2}$ and $V_{3}$ with the label set $L_{1}^{''''}$, $L_{2}^{''''}$ and $L_{3}^{''''}$, respectively. From \eqref{8} and \eqref{9} it is clear that, $s(L_{1}^{''''}) = s(L_{2}^{''''}) = s(L_{3}^{''''})$. Hence by lemma 1.1, $G$ is $S$-magic. Since the highest label is $n + 1 = n + q $, the magic index, $\theta(G) = 1 = q$. Since $\lambda - 1 = n_{1}$, $\theta(G) = \lceil \frac{\lambda - 1}{n_{1}} \rceil$.\\
\vskip .2cm \noindent {\bf (iii)} Suppose that $r > 1$. We replace the label $n - n_{1} \in L_{1}^{'''}$ with the label $n - n_{1} + 2$ and $n - r + 3 \in L_{1}^{'''}$ with $n - r + 3 + r - 1 = n + 2$. Hence the resulting label sets are $L_{1}^{''''} = \{n - n_{1} + 2, n - n_{1} + 3,\cdots, n - r + 2, n - r + 4,\cdots, n +2\}$, $L_{2}^{''''} = L_{2}$ and $L_{3}^{''''} = L_{3}$. Now $s(L_{1}^{''''}) = s(L_{1}) + n_{1} + r$, $s(L_{2}^{''''}) = s(L_{2})$ and $s(L_{3}^{''''}) = s(L_{3})$. We label the partitions $V_{1}$, $V_{2}$ and $V_{3}$ with the labels $L_{1}^{''''}$, $L_{2}^{''''}$ and $L_{3}^{''''}$ respectively. From \eqref{8} and \eqref{9} it is clear that, $s(L_{1}^{''''}) = s(L_{2}^{''''}) = s(L_{3}^{''''})$. Hence by lemma 1.1, $G$ is $S$-magic. Since the highest label is $n + 2 = n + q + 1$, the magic index, $\theta(G) = 2 = q+1$. Since $\lambda - 1 = n_{1} + r - 1$, $\theta(G) = \lceil \frac{\lambda - 1}{n_{1}} \rceil$.\\
\noindent Now, take $q = 0$. Then $\lambda = r$. Since $\lambda > 0$, we have $r > 0$. If $r = 1$, we replace $n \in L_{1}^{'}$ with $n + 1$. Hence the resulting labels are $L_{1}^{v} = \{n - n_{1}, n - n_{1} + 2,\cdots, n - 1, n + 1\}$, $L_{2}^{v} = L_{2}$ and $L_{3}^{v} = L_{3}$. Therefore, $s(L_{1}^{v}) = s(L_{1}) + 1$, $s(L_{2}^{v}) = s(L_{2})$ and $s(L_{3}^{v}) = s(L_{3})$. We label the partitions $V_{1}$, $V_{2}$ and $V_{3}$ with the label sets $L_{1}^{v}$, $L_{1}^{v}$ and $L_{1}^{v}$, respectively. From \eqref{8} and \eqref{9}, it is clear that $s(L_{1}^{v}) = s(L_{3}^{v}) = S(L_{2}^{v})$. Hence by Lemma 1.1, $G$ is $S$-magic. Since the highest label is $n + 1$, the magic index, $\theta(G) = 1 = q + 1$. Since $\lambda - 1 = r - 1$, $\theta(G) = \lceil \frac{\lambda - 1}{n_{1}} \rceil$.
even
If $r > 1$. we replace the label $n - r + 1 \in L_{1}^{'}$ with $n - r + 1 + r = n + 1$. Hence the resulting labels are $L_{1}^{v} = \{n - n_{1}, n - n_{1} + 2,\cdots,n - r, n - r + 2, \cdots, n + 1\}$, $L_{2}^{v} = L_{2}$ and $L_{3}^{v} = L_{3}$. Therefore, $s(L_{1}^{v}) = s(L_{1}) + r$, $s(L_{2}^{v}) = s(L_{2})$ and $s(L_{3}^{v}) = s(L_{3})$. We label the partitions $V_{1}$, $V_{2}$ and $V_{3}$ with the label sets $L_{1}^{v}$, $L_{1}^{v}$ and $L_{1}^{v}$, respectively. From \eqref{8} and \eqref{9}, one can see that $s(L_{1}^{v}) = s(L_{3}^{v}) = s(L_{2}^{v})$. Hence by Lemma 1.1, $G$ is $S$-magic. Since the highest label is $n + 1$, the magic index, $\theta(G) = 1 = q + 1$. Since $\lambda - 1 = r - 1$, we have $\theta(G) = \bigl\lceil \frac{\lambda - 1}{n_{1}} \bigr\rceil$.
\begin{equation*}
\therefore \theta(G) = \biggl\lceil \frac{6h - n(n+1) - 2}{2n_{1}} \biggr\rceil = \biggl\lceil \frac{3(n - n_{1})(n - n_{1} + 1) - 2n(n+1) + 2}{4n_{1}} \biggr\rceil
.\end{equation*}
Since the subgraph $H$ is $S$-magic with $\theta(H) = 1$, by Corollary 2.7.1, $h$ is the minimum magic constant for the subgraph $H$. Therefore, $\theta(G)$ is minimum as it is a linear function of $h$.\\
\noindent \textbf{Proof of case} (iii). \\ Assume that $3\zeta_{n-n_1+1}^{n} < \zeta_1^{n} < 3\zeta_{1}^{n_3}$. Then
\begin{equation*}
\frac{n_{1}(2n - n_{1} + 1)}{2} < \frac{n_{3}(n_{3} + 1)}{2}
.\end{equation*}
Hence, from equation (4), $g(0) < 0$. Therefore, by Lemma 1.2,
\begin{equation}
\theta(G) \ge \biggl\lceil \frac{|n_{1}(2n - n_{1} + 1) - n_{3}(n_{3}+1)|}{2(n - n_{3})} \biggr\rceil.
\end{equation}
\begin{example} Consider the complete tripartite graph $K_{5,6,7}$. Now, by the case $(i)$ of Theorem 2.2. its distance magic index $0$ with magic constant $k = 114$ and it is illustrated in Figure 1.
\begin{figure}[ht]
\centering
\includegraphics[scale=0.1]{k5,6,7.png}
\caption{Example of case $(i)$, $G \cong K_{5, 6, 7}$, $\theta(G) = 0$, $k = 114$}
\end{figure}
On the other hand, Figure 2 gives an $S$-magic labeling of the complete tripartite graph $K_{3,8,9}$ and its distance magic index $7$ with magic constant $k = 154$. Note that by using case $(ii)$ of Theorem 2.2, one can say that the magic index of $K_{3, 8, 9}$ cannot be $0, 1, 2, 3, 4, 5$ and $6$,
\begin{figure}[ht]
\centering
\includegraphics[scale=0.1]{k3,8,9.png}
\caption{Example of case $(ii)$, $G \cong K_{3, 8, 9}$, $\theta(G) = 7$, $k = 154$}
\end{figure}\\
\end{example}
\noindent \textbf{Proof of case $(iv)$}\\
Here, $3\zeta_{n-n_1+1}^{n} \ge \zeta_1^{n} \ge 3\zeta_{1}^{n_3} \text{ and } 2\zeta_1^{n} \equiv 2 \text{ mod } 6$. Hence by Theorem 1.3, $\theta(G) \ge 1$. By Lemma 3.1, one can see that $\theta(K_{n_1 + 1, n_2, n_3}) = 0$.
Thus, by removing the vertex with maximum label, say $u$, from $V_1$ and adding that label to any other vertex, say $u'$, of $V_1$ such that $u + u' > n$ will make $K_{n_1, n_2, n_3}$, $S$-magic. Since $u \le n + 1$, and $u' \le n$, the highest label, $u + u' \le 2n +1$. Therefore, $\theta(G) \le 2n + 1 - n = n +1$. Hence,
$1 \le \theta(G) \le n + 1.$
\newline \noindent But we have to prove the existence of vertices $u$ and $u'$ in $V_1$ such that $u + u' > n$.
Suppose $u + u' \le n$. Let us consider $S(L_1)$ of graph $K_{n_1+1,n_2,n_3}$
\begin{eqnarray*}
s(L_1) &<& (n_1)u + u' \le (n_1-1)u + n\\
&<& (n_1-1)\frac{n}{2} + n =
\frac{n(n_1 + 1)}{2}\\
&\le& \frac{n(n+3)}{6} < \frac{(n+1)(n+2)}{6}.
\end{eqnarray*}
Which is a contradiction, Since $s(L_1) = \frac{(n+1)(n+2)}{6}$.\\
\textbf{Proof of case $(v)$}.\\
In this case, we have, $\zeta_1^{n} \le 3\zeta_{n-n_1+1}^{n}$ and $\zeta_1^{n} < 3\zeta_{1}^{n_3} $. Suppose that the partitions $V_1$, $V_2$ and $V_3$ are labelled by the set of labels $L_{1} = \{n,\cdots, n - n_{1} + 1\}$, $L_{2} = \{n - n_{1},\cdots, n_{3} + 1\}$ and $L_{3} = \{n_{3},\cdots, 1\}$, respectively. Let $H \cong K_{n_{2}, n_{3}}$. Consider,
\begin{eqnarray*}
\frac{(n-n_{1})(n - n_{1} + 1)}{2} &=& \frac{n^2 + n}{2} - \frac{n_{1}(2n - n_{1} + 1)}{2}\\
&\le& \frac{n^2 + n}{3} \\ &<& n_{3}(n_{3} + 1).
\end{eqnarray*}
Hence, by using Theorem 1.4, we have
\begin{equation}
\theta(H) = \biggl\lceil \frac{|(n - n_{1})(n - n_{1} + 1) - 2n_{3}(n_{3}+1)|}{2n_{2}} \biggr\rceil
.\end{equation}
Let $$\frac{|(n - n_{1})(n - n_{1} + 1) - 2n_{3}(n_{3}+1)|}{2} = n_{2}\mu + r, \quad 0 \le r < n_{2}$$
Hence $H$ is $S$-magic with $L_{3}^{'} = L_{3}$ and\\
$L_{2}^{'} =$ $\begin{cases}\{n_{3} + 1 + \mu, \cdots, n - n_{1} + \mu\}, & \text{ if } r = 0 \\
\{n_{3} + 1 + \mu, \cdots, n - n_{1} + \mu - r, n - n_{1} + \mu - r + 2,\cdots, n - n_{1} + \mu + 1\}, & \text{ if } r > 0.\end{cases}$
Let $L_{1} ^{'} = \{n - n_{1} + \mu + 2, \cdots, n + \mu + 1\}$. Implies, $s(L_{1}^{'}) = s(L_{1}) + n_even{1}(\mu+1)$.\\
{\bf Subcase 1}. Suppose $s(L_{1}^{'}) \le s(L_{3})$. Let $s(L_{3}) - s(L_{1}^{'}) = n_{1}\lambda + r_{1}$ where, $0 \le r_{1} < n_{1}$. This implies, $s(L_{3}) - s(L_{1}) = n_{1}\lambda + n_{1}(\mu+1) + r_{1}$. Now, increase each label of $L_{1}^{'}$ by $\lambda$ and hence, we get $L_{1}^{''} = \{n - n_{1} + \mu + \lambda + 2,\cdots, n + \mu + \lambda + 1\}$. If $r_{1} = 0$, we can label the vertices of the partition $V_{1}$, $V_{2}$ and $V_{3}$ by the label set $L_{1}^{''}$, $L_{2}^{'}$ and $L_{3}$, respectively. Since $s(L_{1}^{''}) = s(L_{2}^{'}) = s(L_{3})$, by Lemma 1.1, the graph $G$ is $S$-magic. Since the highest label is $n + \mu + \lambda + 1$, the magic index $\theta(G) = \mu + \lambda + 1$. If $r_{1} > 0$, replace $n + \mu + \lambda + 2 - r_{1} \in L_{1}^{''}$, with $n + \mu + \lambda + 2$. Thus the resulting label set $L_{1}^{'''} = \{n - n_{1} + \mu + \lambda + 2, \cdots, n + \mu + \lambda - r_{1}+1, n + \mu + \lambda + 3 - r_{1}, \cdots, n + \mu + \lambda + 2\}$. Hence, $s(L_{1}^{'''}) = s(L_{1}^{'}) + n_{1}\lambda + r_{1} = s(L_{2}^{'}) = s(L_{3})$. Therefore, by Lemma 1.1, we label the partitions $V_{1}$, $V_{2}$ and $V_{3}$ respectively by $L_{1}^{'''}$, $L_{2}^{'}$ and $L_{3}$ in such a way that the graph $G$ is an $S$-magic. Since the highest label is $n + \mu + \lambda + 2$, the magic index, $\theta(G) = \mu + \lambda + 2$. Therefore,
\begin{equation*}
\theta(G) = \biggl\lceil \frac{2h - n_{1}(2n-n_1+1)}{2n_{1}} \biggr\rceil = \biggl\lceil \frac{n_{3}(n_{3} + 1) - n_1(2n-n_1 + 1)}{2n_{1}} \biggr\rceil
.\end{equation*}
{\bf Subcase 2}. Suppose $s(L_{1}^{'}) > s(L_3)$. Hence the maximum highest possible label, is $n + \mu + 1$. Therefore, $\theta(G) \le \mu + 1 \le \theta(H) + 1$ and hence $\theta(G) \le \theta(H) + 1$.
\end{proof}
\par Now, we discuss the distance magic index of complete multi-partite graphs and some related graphs. We denote $K(a,b)$ as the complete multi-partite graph with $b$ partitions and each partition having $a$ vertices in it, where $a\geq 1, b\geq 1$. Note that for a graph $G$, the graph $mG$ is the disjoint union of $m$ copies of the graph $G$. Recall that the lexicographic product~\cite{MR2817074} of two graphs $G$ and $H$ is a graph $G \circ H$ with the vertex set $V(G)\times V(H)$ and two vertices $(u,v)$ and $(u',v')$ are adjacent if and only if $u$ is adjacent to $u'$ in $G$, or $u=u'$ and $v$ is adjacent to $v'$ in $H$.
\begin{theorem}
Let $h \geq 1.$ A Kotzig array $KA(2h, b)$ exists for any $b$;
$KA(2h + 1, b)$ exists if and only if $b$ is odd.
\end{theorem}
In 2022, Froncek et al.\cite{AVPrajeesh} introduced the notion of the lifted quasi-Kotzig arrays and used them to construct the quasimagic rectangles as a generalization of magic rectangles.
\begin{definition} \cite{AVPrajeesh}
For an odd $a$ and even $b$, a quasi-magic rectangle $QMR(a,b:d)$ is an $a\times b$ array with the entries $1,2,\dots,d-1,d+1,\dots, ab+1$, each appearing exactly once, such that the sum of each row is equal to a constant $\rho$ and the sum of each column is equal to a constant $\sigma$.\end{definition}
They proved the existence of $QMR(a,b:d)$ for all possible values of $a$ and $b$ and $d=ab/2+1$. When $\gcd(a,b)=1$, and $d=ab/2+1$, they obtained a characterization for the existence of $QMR(a,b:d)$.
\begin{theorem}\textnormal{\cite{AVPrajeesh}}\label{thm:main}
{There exist quasimagic rectangles $QMR(a,2t:a t+1)$ for all odd $a$ and $t \geq 1$ with exceptions of $QMR(a,2:a+1)$ for $a \equiv 1(\bmod~4)$. In addition, when $\gcd(a, 2 t)=1$, then the necessary condition is also sufficient.}
\end{theorem}
Note that it is certainly, one of the tools to determine the distance magic index of certain partite graphs.
\section{Quasimagic rectangle: A tool to determine the distance magic index}
\par In this section, the distance magic indices of the graphs such as $mK(a,b)$, $m(C_{b}\circ\overline{K}_a)$ and $G\circ \overline{K}_a$, are computed, where $G$ is an arbitrary regular graph. These graphs already appear in the literature in the articles \cite{MR2848089,MR1999203,MR2533199}. These articles mainly discuss about the distance magicness of the above mentioned graphs. The existing results of distance magicness of these graphs, are tabulated below for a quick reference. \\
\noindent Case 1: The distance magicness of $K(a,b)$ is given in Table \ref{Table1}.
\begin{table}[h]
\centering
\begin{tabular}{|c|c|c|}
\hline
values of $a$ & odd $b$'s & even $b$'s\\
\hline
even & $dmg$ \cite{MR1999203} & $dmg$ \cite{MR1999203} \\
\hline
odd & $dmg$ \cite{MR1999203} & not~ $dmg$ \cite{MR1999203} \\
\hline
\end{tabular}
\caption{}
\label{Table1}
\end{table}
\\ \noindent Case 2: The distance magicness of $mK(a,b),m>1$ is given in Table \ref{Table2}.
\begin{table}[htbp]
\centering
\begin{tabular}{|c|c|c|c|}
\hline values of $m$ & values of $a$ & odd $b$'s & even $b$'s\\ \hline
odd & even & $dmg$ \cite{MR2533199} &$dmg$\cite{MR2533199}\\ \hline
even & even & $dmg$ \cite{MR2533199}& $dmg$ \cite{MR2533199} \\ \hline
odd & odd &$dmg$ \cite{MR2533199} & not $dmg$ \cite{MR2533199} \\ \hline
even & odd & not dmg if $b=4t+1$ \cite{MR2848089}, not dmg if $b=4t+3$ \cite{MR2533199}
& not $dmg$ \cite{MR2533199} \\ \hline
\end{tabular}
\caption{}
\label{Table2}
\end{table}
\newline
Case 3: The distance magicness of $m(C_b\circ\overline{K}_a)$ is given in Table \ref{Table3}.
\begin{table}[htbp]
\begin{center}
{\footnotesize \begin{tabular}{|c|c|c|c|}
\hline
values of $m$ & values of $a$ & even $b$'s & odd $b$'s\\
\hline
odd & {even} & $dmg$ \cite{MR2533199}& $dmg$ \cite{MR2533199} \\\hline
even & even & $dmg$ \cite{MR2533199} & $dmg$ \cite{MR2533199} \\
\hline
odd & odd & $dmg$ if $b=4t$, {not $dmg$} if $b=4t+2$ \cite{MR2533199}& {$dmg$} \cite{MR2533199} \\ \hline
even & odd & $dmg$ if $b=4t$, {not $dmg$} if $b=4t+2$ \cite{MR2533199} & {not $dmg$} \cite{MR2533199} \\
\hline
\end{tabular}}
\end{center}
\caption{}
\label{Table3}
\end{table}
\newline
Case 4: If $G$ is an $r$-regular graph on $b$ vertices, the distance magicness of $G\circ\overline{K}_a$ is given in Table \ref{Table4}.
\begin{table}[h]
\begin{center}
{\footnotesize \begin{tabular}{ |c|c|c|c|c|}
\hline
values of $r$
& values of $a$ & even $b$'s, $b=4t$ & even $b$'s, $b=4t+2$& odd $b$'s\\
\hline
even, $r=4s$ & odd & {not yet solved}& { not yet solved} & $dmg$ \cite{MR2848089}\\\hline
even, $r=4s+2$ & odd & { not yet solved } & {not $dmg$} \cite{MR2848089} & $dmg$ \cite{MR2848089}\\hline
odd & & {not $dmg$}& {not $dmg$}& no such graphs\\\hline
even & even & $dmg$\cite{MR1999203} & $dmg$ \cite{MR1999203}
& $dmg$ \cite{MR1999203}
\\\hline
odd &even& $dmg$ \cite{MR1999203}& $dmg$ \cite{MR1999203}& no such graphs \\
\hline
\end{tabular}}
\end{center}
\caption{}
\label{Table4}
\end{table}
\begin{theorem}\label{index1hnp} For the graph $K(a,b)$, \\
$\theta(K(a,b)) =$
$\begin{cases}
0 & \text{for $a$ even
or $a$ and $b$ both odd} \\
1 & \text{for $a$ odd and $b$ even except when $a \equiv 1 \bmod 4$ and $b=2$}.
\end{cases}$
\end{theorem}
\begin{proof}
If $a$ is even or $a$ and $b$ are both odd, then from Table 1, $\theta(K(a,b))=0$. Now, if $a$ is odd and $b$ is even except when $a \equiv 1 \mod 4$ and $b=2$, then again from Table 1, $\theta(K(a,b))\neq 0$.
\par Now, if $a$ is odd and $b$ is even, then by using Theorem 2.4, construct quasimagic rectangle $QMR(a,b)$ from the set $S = \{1,2,\cdots,\frac{ab}{2},\frac{ab}{2}+2,\cdots,ab+1\}$. Then, label $i^{th}$ partition of $K(a,b)$ by $i^{\textnormal{th}}$ column of $QMR(a,b)$, where $1\leq i \leq b$. Thus, if $\rho$ is the column sum of a $QMR(a,b)$, then $K(a,b)$ is $S$-magic graph with magic constant $\rho(b-1)$ and therefore, $\theta(K(a,b))$ is 1.
\end{proof}
\begin{theorem}\label{multi6}
Let $a>1,b>1,m> 1$. Then\\
$\theta(mK(a,b))$ =
$\begin{cases}
0 & \text{for} a \text{is even $(or)$}
~mab \text{~is odd}\\
1 & \text{otherwise.}
\end{cases}$
\end{theorem}
\begin{proof}
From Table 2, it can be observed that when $m$ is even, and $a$ is odd or $m,a$ are odd and $b$ is even; we have $\theta(mK(a,b))\neq 0$. For the remaining cases $\theta(mK(a,b))= 0$.
\par For all the cases, when $mK(a,b)$ is not distance magic, by using Theorem 2.4, construct a $QMR(a,mb)$ with column sums $\rho$.
Now, use it to label $m$ copies of the graph $K(a,b)$ by choosing $mb$ columns of $QMR(a,mb)$ to the $(mb)$-partitions of $mK(a,b)$. Hence, an $S$-magic labeling of $mK(a,b)$ with $\mu' =\rho(b-1)$ is obtained. Therefore, $\theta(mK(a,b))=1.$
\end{proof}
\begin{theorem}\label{thm14}
Let $m\geq 1, a>1$ and $b\geq3$. Then\\
\begin{equation*}
\theta(m(C_{b}\circ \overline{K}_a) =
\begin{cases}
0 & \text{if $a$ is~even $(or)$ $mab$ is odd, $(or)$ $a$ is odd, and $b\equiv 0 \textnormal{~mod~} 4$},\\
1 & otherwise.
\end{cases}
\end{equation*}
\end{theorem}
\begin{proof}
If $a$ is even or $mab$ is odd, or $a$ is odd and $b \equiv 0 \bmod 4$, then from Table 3, $\theta(G)=0.$
Now the remaining cases are given below. \\
Case 1. $a$ is odd, $m$ is even, $b\equiv 2 \bmod 4.$\\
Case 2. $a$ is odd, $m$ is odd, $b\equiv 2 \bmod 4.$\\
Case 3. $a$ is odd, $m$ is even, $b$ is odd.
\par Now, for all these cases, a $QMR(a,mb)$ is constructed by using Theorem 2.4, with individual column-sum $\rho$ and individual row-sum $\sigma$. The first $b$ columns of $QMR(a,mb)$ is used to label the first copy of $m(C_{b}\circ \overline{K}_a)$, by labeling the $i^{th}$ layer of $K_{a}$ of $C_{b}\circ \overline{K}_a$ using the $i^{th}$ column of $QMR(a,mb)$. Similarly, the next $b$ columns are used to label the vertices of the second copy $C_{b}\circ \overline{K}_a$. If we proceed like this, one can label all copies of $m(C_{b}\circ \overline{K}_a)$, using the column of $QMR(a,mb)$. Thus, $m(C_{b}\circ \overline{K}_a)$ is $S$-magic with $\mu' = 2\rho$ and therefore, $\theta(m(C_{b}\circ \overline{K}_a))=1$.
\end{proof}
\begin{theorem}\label{thm15}
Let $a\ge 1$, and $G$ be an $r$-regular graph on $b$ vertices. Then
\begin{equation*}
\theta(G\circ \overline{K}_a) =
\begin{cases}
1 & \text{for $a$ and $r$ are odd except when $a \equiv 1 \bmod 4$ and $b=2$} \\
1 & \text{for $a$ is odd, $r\equiv 2 \bmod 4$, $b \equiv 2\bmod 4$},\\
0 & \text{for $a$ is even $(or)$ $a$ and $b$ are odd, and $r$ is even}.
\end{cases}
\end{equation*}
\end{theorem}
\begin{proof}
Let $v_{1},v_2,\cdots,v_{b}$ be the vertices of $G$. Now, for any $j\in\{1,2,\cdots,b\}$, let $V_{j} = \{v_{j}^{i}: i\in\{1,2,\cdots, a\}\}$ be set the vertices of $G\circ \overline{K}_a$ that replace the vertex $v_{j}$ of $G$. In fact, the vertex set of $G\circ \overline{K}_a$ is $\displaystyle \bigcup_{j=1}^{b}V_{j}$ and $G\circ \overline{K}_a$ is a $(ra)$-regular graph on $ab$ vertices. \par Now consider the following cases:\\ Case 1. $a$ is even. From Table 4, we get $\theta(G\circ \overline{K}_a)=0$.\\
Case 2. $a$ is odd. When $r$ is even, and $b$ is odd, from Table 4, we have
$\theta(G\circ \overline{K}_a)=0$. On the other hand, if $r\equiv 2 \bmod 4$, and $b \equiv 2 \bmod 4$, then from Table 4, $\theta(G\circ \overline{K}_a)\neq 0$. Finally, if $r$ is odd, then $b$ is even, and by Theorem \ref{oddregular}, $\theta(G\circ \overline{K}_a)\neq0$.\\
\par Now, for both the cases, when $\theta(G\circ \overline{K}_a)\neq0$, construct $QMR(a,b)$ using Theorem 2.4. Use the $j^{th}$ column of $QMR(a,b)$ to label the set of vertices, $V_{j}$, for all $j \in \{1,2,\cdots,b\}$. Hence, an $S$-magic labeling of $G\circ \overline{K}_a$, with $\mu' = r\rho$ is obtained. Therefore, $\theta(G\circ \overline{K}_a)=1.$
\end{proof}
\begin{example}
A quasimagic rectangle $QMR(3,10)$ is given below. Now, Figure \ref{Fig5.3} illustrates how one can use this $QMR$ to give $S$-magic labeling of the $9$-regular graph $G \circ \overline{K}_3$, where $S$ is the set of all entries of this $QMR$. \vskip 0.12cm
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}
\hline
22 & 23 & 5 & 20 & 1 & 10 & 12 & 6 & 30 & 31 \\\hline
17 & 18 & 19 & 3 & 26 & 27 & 8 & 13 & 14 & 15 \\\hline
9 & 7 & 24 & 25 & 21 & 11 & 28 & 29 & 4 & 2 \\\hline
\end{tabular}
\end{center}
\vskip .2cm \noindent One can see that the column sums are 48 and row sums are 160 and moreover, $\theta(G\circ \overline{K}_3)=1.$
\begin{figure}[ht]
\centering
\includegraphics[width=120mm]{Fig5_3}
\caption{An $S$-magic labeling of $G \circ \overline{K}_3$ using $QMR(3,10)$}
\label{Fig5.3}
\end{figure}
\end{example}
\section{Conclusion and scope}
In this paper, the distance magic index of complete tri-partite graphs $K_{n_1,n_2,n_3}$ for all possible values of $n_1,n_2$ and $n_3$, are determined and Table 5 gives the summary. However, the problem of determining the distance magic index of complete $r$-partite graphs $K_{n_1,n_2,\cdots, n_r}$ is still open, where $r>3$. In last section, the quasimagic rectangle $QMR$ is identified as useful tool to determine the distance magic indices of certain regular graphs such as disjoint union of $m$ copies of $K(a,b)$, disjoint union of $m$ copies of $C_{b}\circ \overline{K}_a$ and $G \circ {K}_a$, where $G$ is an arbitrary regular graph.
|
1,314,259,993,596 | arxiv | \section{Introduction} \label{sec:introduction}
In recent years, machine learning, especially deep learning, has been applied to various domains (\textit{e.g.}, computer vision, speech recognition, and video analytics).
Emerging \textit{Intelligent Applications} (IAs) such as image classification based on deep Convolutional Neural Networks (CNNs)~\cite{Krizhevsky2017}, traffic flow prediction based on deep Recurrent Neural Networks (RNNs)~\cite{WangZhumei2020}, and game development based on deep Generative Adversarial Networks (GANs)~\cite{Kim2020}, are demonstrating superior performance in terms of accuracy and latency.
Such performance, however, requires tremendous computation and network resources to deal with the increasing size of Machine Learning (ML)/Deep Learning (DL) models and the proliferation of vast amounts of training data~\cite{MayerJ2020}.
Cloud computing is indisputably attractive to IA developers as the predominating high-performance computing paradigm~\cite{10.1145/3364684}.
Typically, cloud providers offer services like Infrastructure-as-a-Service (IaaS), Platform-as-a-Service (PaaS), and Software-as-a-Service (SaaS) to facilitate application implementation, where resources like high-performance computation, massive elastic storage, and reliable network services are allocated according to user requirements.
Intuitively, mainstream IAs are deployed on the Cloud to leverage centralized resources for computationally-intensive Artificial Intelligence (AI) tasks like data processing, ML/DL model training, and inference.
For instance, the distributed training of AlphaGo~\cite{Silver2016} is a typical \textit{`Cloud Intelligence'} (CI) representative.
However, novel challenges to CI emerge when modern IAs rapidly proliferate and are required to be in production in practice, where \textit{high end-to-end service latency}, \textit{high network bandwidth overhead}, and \textit{severe privacy leakage threat} are among the most critical ones~\cite{zhou2019edge}.
Instead of concentrating on the Cloud, increasing efforts attempt to exploit heterogeneous resources distributed at the network Edge to address such issues.
For example, some IAs offload DL tasks to edge servers (\textit{e.g.}, Nvidia Jetson TX2 Board)~\cite{Daniel2019} for privacy preservation and timely responses.
Such an edge offloading of relatively simple AI tasks, or \textit{`Edge Intelligence'} (EI)~\cite{10.1145/3409977,10.1145/3429945}, manages to alleviate the controversy between broadened requirements of modern IAs and the conventional CI paradigm.
The rapid development of EI and corresponding prototypes demonstrates that, due to edge devices' heterogeneous resource constraints, the Cloud is still critical to modern production level IAs with multi-faceted performance requirements~\cite{zhou2019edge}.
Increasing IA developers start to focus on efficiently leveraging edge resources under cloud coordination to collaboratively conduct AI tasks with optimized performance~\cite{Song2018,Abdelzaher2020}, or \textit{`Edge-Cloud Collaborative Intelligence'} (ECCI).
ECCI relies on pivotal interdisciplinary technologies of cloud and edge computing (supporting ECCI infrastructure and runtime), and ML/DL-based AI (introducing rich IA workloads).
Existing ECCI applications (\textit{e.g.}, HOLMES~\cite{10.1145/3394486.3403212} for healthcare,
EdgeRec~\cite{10.1145/3340531.3412700} for E-commerce, SurveilEdge~\cite{Wang2020} for urban surveillance, and general solutions like CLIO~\cite{10.1145/3372224.3419215} and SPINN~\cite{Laskaridis2020}) are individually developed and deployed by either academic researchers or industrial communities, where both the application design and system implementation are highly \textit{developer-dependent} and \textit{scenario-specific}.
For example, SurveilEdge~\cite{Wang2020} is a typical ECCI application for real-time intelligent urban surveillance video query.
In its prototypical implementation, the developers depend on relatively higher edge computation capabilities (\textit{i.e.}, X86 PCs) to support system scaling without subtly designing an ECC infrastructure management scheme.
For the ease of implementation, they hard-code the load balancing policy with the video query workload for latency reduction.
Additionally, to achieve intelligent video query, the entire solution is specifically designed to support CNN training and inference workloads, where dedicated service links (\textit{e.g.}, message service links) among all application components are individually configured to achieve edge-cloud collaborations.
Without impacting the application performance, such developer-dependent design and implementation, however, are impeding others to migrate the application to general ECC infrastructures (\textit{e.g.}, resource-constrained Industrial IoTs) or pursue customizable performance optimizations (\textit{e.g.}, joint optimization of latency and bandwidth consumption).
Moreover, if others want to adopt SurveilEdge (or other existing applications) as the backbone of other applications, driven by different DL models and deployed at different infrastructures, corresponding DL runtimes and different ECC services have to be designed and implemented by the adopters themselves thoroughly.
Such a non-generic manner is severely hindering the proliferation of production level ECCI applications.
Therefore, \textit{for the cost-efficient implementation of high-performance production level ECCI applications, it is necessary to construct a unified platform handling both ever-increasing edge and cloud resources and emerging IA workloads with increasing scale and complexity.}
Particularly, to construct such a platform, the following \textbf{four challenges} need to be explicitly addressed:
\textbf{Support for unified management of hierarchical and heterogeneous infrastructures.}
The efficient implementation of ECCI applications requires unified management of not only infrastructures offered by traditional centralized cloud providers but also heterogeneous computation, storage, and network resources geographically dispersed at the edge.
The development and deployment of ECCI application components on edge devices are extremely inefficient due to the lack of a unified platform.
Furthermore, it is infeasible to directly migrate IaaS and PaaS technologies in cloud computing to the management of inherently distributed edge resources~\cite{Bagchi2019}.
\textbf{Support for user-transparent ECC services.}
ECCI application developers require services providing user-transparent edge-edge and edge-cloud collaborations.
In most cases, components of existing ECCI applications are independently deployed on edge nodes, only interacting through services deployed on the Cloud.
Such a manner increases both bandwidth cost and response latency.
Few existing edge services (\textit{e.g.}, Dapr~\cite{dapr}) can improve edge autonomy and application performance to a certain extent.
However, due to the lack of links between edge and cloud services, they cannot provide user-transparent collaborative services to developers.
\textbf{Support for complex IA workloads.}
Efficient ECCI application implementations require comprehensive system-level supports to complex IA workloads like ML/DL model training and inference, which cannot be provided by existing cloud and edge computing platforms.
For instance, in edge computing systems for IoT data processing, the message-driven communication solution for transmitting KB-level sensor data cannot effectively handle the transmission of DL models as large as hundreds of MBs.
Moreover, most existing distributed ML/DL solutions are designed for datacenter networks with high bandwidth and low transmission latency.
Such methods are inefficient in ECC systems with inherent constraints like prolonged and unstable End-to-End (E2E) communication latency.
\textbf{Support for unified optimization of ECCI applications.}
Unified performance optimization mechanisms are important to efficient ECCI application implementations.
For most existing edge computing applications, the efficiency of resource utilization highly depends on the developer's design, where effective optimizations require a profound understanding of system architectures and optimization theories~\cite{Harchol2019}.
For existing ECCI applications, except for the multi-component development and cross-device deployment of inherently complex IA workloads, the developers also have to deal with the overall performance optimization across ECC infrastructures by themselves, not to mention the difficulties in application debugging, monitoring, and profiling caused by the distributed and heterogeneous environment.
Such a requirement is quite challenging to not only developers of emerging ECCI applications, but also those who want to migrate existing IAs to ECC infrastructures.
\section{ECCI Application Patterns} \label{sec:apppattern}
Currently, there exists no commonly-accepted abstraction of general ECCI application patterns, which are critical to improving the efficiency of ECCI application development and deployment.
As the foundation of the unified platform, considering the subject of different application tasks, we extract \textbf{four} common patterns, \textit{i.e.}, \textit{ECC processing}, \textit{ECC training}, \textit{ECC inference}, and \textit{hybrid collaboration}.
\textbf{\textit{ECC Processing}} of data is the basis of various ECCI applications.
Collaborative data processing applications are often built as \textit{pipelines} or \textit{Directed Acyclic Graphs} (DAGs).
For example, the Steel framework~\cite{Noghabi2018} deploys a streaming analytic pipeline of different data processing tasks (\textit{e.g.}, filtering, anomaly detection, and storage) for ECC IoT anomaly detection applications.
\textbf{\textit{ECC Training}} refers to conducting ML/DL model training based on edge-cloud collaborations.
Unlike ECC processing, ECC training requires complex interactive and iterative data and control flows between edges and the Cloud (\textit{e.g.}, training data, model, and hyper-parameter exchanges).
For instance, Federated Learning (FL) is a typical ECC training application, which conducts ML training across devices to protect data privacy (\textit{e.g.}, Gboard Mobile Keyboard~\cite{Google2017} and Apple QuickType Keyboard~\cite{Apple2019}), and to bridge data silos (\textit{e.g.}, model training for bank fraud detection~\cite{WeBank2019}).
\textbf{\textit{ECC Inference}} focuses on ML/DL model inference, where only forward propagation is conducted.
Generally, ECC inference is achieved through either intra-model or inter-model collaborations.
In intra-model solutions, a single DL model is decomposed by layers into two parts (\textit{i.e.}, neural network partitioning) deployed at edges and the Cloud respectively for collaborative inference (\textit{e.g.}, Neurosurgeon~\cite{neurosurgeon}, SPINN~\cite{Laskaridis2020}, and JointDNN~\cite{jointdnn}).
In inter-model ones, however, multiple DL models with different functionality or performance are deployed at edges and the Cloud respectively for collaborative inference (\textit{e.g.}, VideoEdge~\cite{Hung2018} and SurveilEdge~\cite{Wang2020}).
\textbf{\textit{Hybrid Collaboration}} combines at least two of three ECCI application patterns above or integrates additional CI/EI capabilities into ECCI applications.
For example, ShadowTutor~\cite{Chung2020} enables robust HD video semantic segmentation with significant throughput improvement and network data transmission reduction.
Here, cloud servers conduct both the inference of the heavy and general `teacher' model and the training of the lightweight `student' model.
Mobile edge devices conduct the 'student' model inference.
\section{ECCI Platform Design Principles} \label{sec:designprinciple}
In this article, we aim to construct a unified platform for the efficient development and deployment of ECCI applications.
It is required to provide efficient management of heterogeneous ECC infrastructures, user-transparent ECC services, and customizable performance optimizations, supporting scalable, reliable, and robust ECCI application development and deployment.
The desired platform should be treated as \textit{ECCI-as-a-Service} (ECCIaaS), similar to the concept of Machine Learning-as-a-Service (MLaaS).
Particularly, we extract \textbf{five} essential design principles as follows.
\textit{\textbf{Principle One:} an instance of ECCI application should be an integrated entity that can be managed in a scalable manner.}
This principle requires the unified management of typical edge and cloud infrastructures, including hardware nodes like edge gateways, clusters like Kubernetes~\cite{kubernetes}, virtual machines, and third-party cloud services like Azure IoT Hub~\cite{iothub}.
Any operation of ECCI applications (\textit{e.g.}, deployment and monitoring) should be carried out on large-scale collaborative infrastructures organized as a unity.
ECCI applications should be able to provide continuously available services when the infrastructures are scaled or upgraded.
\textit{\textbf{Principle Two:} ECCI application components at edges and the Cloud should be able to operate in both collaborative and autonomous manners.}
Unlike the datacenter network on the Cloud, the edge-cloud network has limited capabilities (\textit{e.g.}, bandwidth), and may perform unstably.
While supporting collaborations with the Cloud, edges should be able to cache data and provide partial services autonomously to mitigate the impact of network partitioning.
\textit{\textbf{Principle Three:} orchestration is essential to ECCI applications.}
Except for edge-cloud separations, modularized ECCI application components have specific requirements of computation and storage resources, as well as deployment locations.
Moreover, there can be multiple applications co-located at the same infrastructure.
Therefore, component orchestration is necessary to ensure that all applications' resource and user requirements can be satisfied.
\textit{\textbf{Principle Four:} provide in-app control of ECCI applications.}
In most cases, offloading computation to edges may not directly improve application performance.
Here, in-app control optimization has been demonstrated to be effective in various aspects like bandwidth saving~\cite{nigade2020} and E2E latency reduction~\cite{Ren2020}, which should be seriously considered for application performance enhancement.
\textit{\textbf{Principle Five:} support as many types of ECCI application workloads as possible.}
ECCI application scenarios are ever-increasing, such as federated model training and ECC model inference.
It is essential for the platform to support common application patterns and services, facilitating efficient development and deployment of a broadened spectrum of ECCI applications.
\section{Application-Centric ECCI Platform} \label{sec:ace}
Driven by all principles above, the explicit design of our Application-Centric ECCI (ACE) platform is as follows.
\begin{figure*}[htbp]
\centering
\includegraphics[width=0.85\textwidth]{figure/design.pdf}
\caption{The general architecture of ACE.}
\label{fig:design}
\end{figure*}
\subsection{Overview} \label{subsec:aceoverview}
We illustrate the general ECCI application development and deployment procedure based on ACE in Figure~\ref{fig:design}.
For application developers, this procedure comprises \textbf{three} major phases, \textit{i.e.}, \textit{user registration}, \textit{application development}, and \textit{application deployment}.
In the \textit{user registration} phase, any ECCI application developer can register at ACE as a \textit{platform user}.
The user first requests the registration of an ECC infrastructure at ACE, and registers all his/her edge and cloud \textit{nodes} to form an infrastructure according to operational instructions replied by ACE (see Part~\ref{subsubsec:infraorg}).
Here, a \textit{node} can be either a physical device or a virtual service like an edge gateway, a cloud server, a private or public cloud, \textit{etc.}
The user can also select to deploy different resource-level services based on service components provided by ACE on the infrastructure, which can be shared among all his/her ECCI applications (see Part~\ref{subsubsec:resourceservice}).
Then, in the \textit{application development} phase, the user implements applications in a modularized manner.
Specifically, for each application, different components are separated according to user-defined business logic or functionalities.
Meanwhile, requested by ACE, the user deliberately decouples application control flows with workload flows for collaboration optimization and component reuse (see Part~\ref{subsubsec:reusedev}).
All components are then implemented using the ACE Software Development Kits (SDKs), and encapsulated into containers that can be deployed on edge or cloud according to components' resource and user requirements.
For each application, the user constructs a \textit{topology file} describing component relations and resource and user requirements of each component.
All component images and corresponding topology files are then submitted to ACE.
Finally, in the \textit{application deployment} phase, ACE determines a \textit{deployment plan} for all components of a specific application according to the topology file, guaranteeing that all resource and user requirements are satisfied (see Part~\ref{subsubsec:depauto}).
According to the plan, the application can be deployed on the user's ECC infrastructure through ACE.
All deployed applications are continuously monitored by ACE for maintenance, and corresponding users can upgrade, monitor, and remove their applications at any time.
To achieve the procedure above, we construct our ACE platform in a hierarchical manner with \textbf{three} layers, \textit{i.e.}, \textit{platform layer}, \textit{resource layer}, and \textit{application layer}.
The general architecture of ACE is illustrated in Figure~\ref{fig:design}.
Details of each layer are as follows.
\subsection{Platform Layer} \label{subsec:aceplatformlayer}
This layer manages the ACE platform, all registered users, and users' nodes and applications.
It also offers platform-level services for users and their applications.
\subsubsection{Platform Management} \label{subsubsec:platformmanagement}
Our platform-layer manager comprises \textit{controller}, \textit{orchestrator}, \textit{API server}, \textit{Pub/Sub service}, \textit{monitoring service}, and \textit{user interfaces}:
\textbf{Controller}
manages platform users, their nodes and applications, \textit{e.g.}, registers and deletes users, shields failed nodes, and controls node component deployment.
\textbf{Orchestrator}
determines a deployment plan for all components of each application based on the topology file (see Part~\ref{subsubsec:depauto}), ensuring resource (\textit{e.g.}, computing) and user (\textit{e.g.}, location) requirements of all components are satisfied.
\textbf{API Server}
provides uniform APIs for querying and manipulating the status of ACE entities (\textit{e.g.}, users, nodes, applications) to other platform manager components (\textit{e.g.}, orchestrator, controller).
\textbf{Pub/Sub Service}
provides a bi-directional data transmission channel between ACE and user nodes and applications (\textit{e.g.}, delivering deployment instructions from the controller to user nodes).
\textbf{Monitoring Service}
collects the status, performance metrics, and runtime logs of ACE, user nodes and applications.
\textbf{User Interfaces}
enhances ACE's user-friendliness with Command Line Interface (CLI) and web-based dashboard.
For example, the dashboard with a `drag-and-drop' visual application editor can be used for handy application development.
\subsubsection{Platform-level Services} \label{subsubsec:platformservice}
Platform-level services are not ACE's internal features.
They can be implemented as requested to improve the efficiency of ECCI application development and deployment based on ACE.
Following are two typical examples:
\textbf{Image Registry}
hosts ACE-provided images (\textit{e.g.}, controller, orchestrator), generic runtime images (\textit{e.g.}, Python runtime), and user-provided customized application images.
\textbf{Validation Testbed}
allows users to develop, debug, and monitor ECCI applications on an SDN-based application validation testbed.
For example, the impact of edge-cloud channel dynamics (\textit{e.g.}, bandwidth, delay, jitter) on the testbed can help users understand the actual performance of an ECCI application in real-world networks.
\subsection{Resource Layer} \label{subsec:aceresourcelayer}
This layer manages the ECC infrastructure of each user.
It also provides resource-level services shared among applications deployed on the same infrastructure.
\begin{figure*}[htbp]
\centering
\includegraphics[width=0.85\textwidth]{figure/services.pdf}
\caption{Illustration of ACE provided resource-level services.}
\label{fig:services}
\end{figure*}
\subsubsection{Infrastructure Organization} \label{subsubsec:infraorg}
Considering \textit{Principles One} and \textit{Two}, ACE requires all nodes of each user to be organized as several \textit{Edge Clouds} (ECs) and one \textit{Central Cloud} (CC) to host scalable ECCI applications, and to enable autonomous operations of edge components.
For a specific user, all his/her edge nodes are divided into several groups according to the user's preferences (\textit{e.g.}, in terms of nodes' geographical locations or resources).
Each group is organized as an EC, serving all end nodes (\textit{e.g.}, IoT sensors and cameras) that can access the EC through Local Area Network (LAN).
All cloud nodes of the user are organized as a single CC, and it can interact with each EC through Wide Area Network (WAN).
For each EC and the CC, internal nodes are organized as a cluster (similar to Kubernetes ideally, or a node pool for simple implementation).
Treating each EC and the CC as a resource-level operational unit allows ACE to effectively manage the infrastructure and deploy applications on such an infrastructure, without considering the explicit management of potentially massive underlay nodes.
Moreover, when there is no cloud coordination caused by either CC or edge-cloud communication failure, each EC as a cluster remains (partially) functional, enabling local area collaborations among edge components based on corresponding edge services.
Receiving the user's registration request, ACE assigns a unique infrastructure ID to the user, and establishes a node information record for infrastructure organization.
Meanwhile, ACE assigns a unique second layer ID affiliated to the infrastructure ID to each EC and the CC claimed by the user, where corresponding node registration instructions are generated automatically.
Replied from ACE, such instructions are executed by the user on nodes.
An agent is deployed on each node, informing ACE of the node information and the EC or CC the node belongs.
ACE assigns a unique third layer ID affiliated to the EC or CC's ID to each node.
The agent is also used for application deployment and application status collection.
\subsubsection{Resource-level Services} \label{subsubsec:resourceservice}
For ECCI applications with the typical patterns discussed in Section~\ref{sec:apppattern}, essential services like small/big packet communication and data caching/storage are commonly required~\cite{Harchol2019,MongaRS2019,LiL2020}.
In a specific ECC infrastructure, existing services supporting ECCI applications are conventionally deployed on both ECs and the CC, serving EC and CC clients (\textit{i.e.}, application components) respectively to ensure the autonomy of ECs.
Each service is accessible to all applications deployed on the same infrastructure.
However, due to the lack of links between edge and cloud services, conventional services require application developers to handle complex edge-cloud interactions.
Treating conventional message service for small packet communication as an example, as shown in Figure~\ref{fig:services}, for edge-cloud unicast communications, all EC clients have to directly access the message service at CC (\textit{i.e.}, \circled{1}) to communicate with CC clients.
Here, the developer has to handle the CC message service authorization to each EC client individually, which is quite expensive for large-scale ECCI applications.
Considering \textit{Principle Five}, to facilitate efficient application development, ACE prefers to provide E2E resource-level services with unified interfaces to EC and CC clients, respectively.
Therefore, \textit{long-lasting links} between services on ECs and the CC need to be established.
Some conventional services support the direct establishment of such links (\textit{e.g.}, service bridging for the message service).
Specifically, as shown in Figure~\ref{fig:services}, ACE implements a \textit{resource-level message service}, where the long-lasting link between EC and CC message services (\textit{i.e.}, \circled{2}) is established using MQTT topic-bridging~\cite{roger_mosquitto}.
Here, edge-cloud interactions are conducted by ACE provided SDK, and each client only needs to interact with its local service with a dedicated interface.
For other services, directly establishing long-lasting links is expensive or even infeasible.
For example, the link between edge and cloud file services could be established using file synchronization, which induces additional requirements on network condition, computation, and access authorization.
Instead, ACE uses the resource-level message service to establish long-lasting links for other services.
ACE implements a \textit{resource-level file service}, whose control flow (\textit{e.g.}, \circled{3},\circled{4}) is separated from the data flow and handled by the resource-level message service.
Furthermore, the proverbial object storage service is used to handle the data flow (\textit{e.g.}, \circled{5},\circled{6}) for transmission simplification.
Note that, as shown in Figure~\ref{fig:services}, three types of links are used in resource-level services, \textit{i.e.}, ad-hoc links (grey) for repetitive interactions, ad-hoc links (orange) for one-off interactions, and long-lasting links (green) established once the service is deployed.
Besides, resource-level services should provide basic operations for applications through their lifecycle (\textit{e.g.}, temporary storage for intermittent models and data, and permanent storage for final trained models in the file service).
\subsection{Application Layer} \label{subsec:aceapplicationlayer}
This layer supports user applications through the entire lifecycle.
\subsubsection{ACE Supported ECCI Application Lifecycle.} \label{subsubsec:applife}
As a unified platform, ACE supports each application through its entire lifecycle (\textit{i.e.}, designing, coding, building, testing, deploying, and monitoring).
For designing, ACE provides a standard specification (\textit{i.e.}, the topology file) to achieve modularized development for ACE-organized ECC infrastructures.
For coding, ACE provides the SDKs with access to resource-level services for application components and the user interface to access the essential Integrated Development Environment (IDE).
For building, ACE provides the image registry for efficient image management and distribution.
For testing, ACE provides the validation testbed for application verification and evaluation.
For deployment, ACE provides the orchestrator and the controller for automatic deployment.
For monitoring, ACE provides the monitoring service collecting the status of application components and nodes where they are deployed.
Such supports from ACE enable users to develop and deploy basic ECCI applications efficiently.
For applications with specific performance requirements (\textit{e.g.}, the minimal E2E latency), or with advanced architectures (\textit{e.g.}, large-scale components with complex topology), ACE provides \textbf{two} extra supports, \textit{i.e.}, \textit{reusable development} and \textit{deployment automation}.
\subsubsection{Reusable Development.} \label{subsubsec:reusedev}
Considering \textit{Principles Four} and \textit{Five}, ACE requires developers to decouple and separate control and workload planes of all application components.
The control plane conducts in-app control operations, component monitoring, and policy execution (\textit{e.g.} decide the best partition point for intra-model inference solutions~\cite{neurosurgeon,jointdnn}).
The workload plane conducts computation, storage, and transmission instructed by the control plane (\textit{e.g.}, deep feature compression module ~\cite{ChoiB18} or hybrid collaboration pipeline for data processing and inference tasks~\cite{colliflow}).
Such a separation allows ACE to construct a reusable in-app controller, enabling developers to concentrate on implementing ECCI workloads and efficiently contribute to the ACE based ECCI ecosystem.
For the reusable in-app controller, ACE constructs a series of general in-app control operations (\textit{e.g.}, start, filter, aggregate, and terminate), component monitoring operations, and a basic control policy.
Determined by the ECC infrastructure, the controller is constructed at the resource level in an ECC manner (see Part~\ref{subsubsec:infraorg}).
The CC controller conducts global coordination related operations, and the EC controller coordinates components within the EC.
Resource-level services support interactions between CC and EC controllers.
For applications with specific performance requirements, developers can inherit the general in-app controller and override optimization methods as advanced control policies (\textit{e.g.}, the rate control based optimal edge-cloud bandwidth allocation ~\cite{AlvarB21}).
\begin{figure*}[!htbp]
\centering
\includegraphics[width=0.85\textwidth]{figure/vq_illustration.pdf}
\caption{ACE based intelligent video query workflow.}
\label{fig:video_query}
\end{figure*}
\subsubsection{Deployment Automation.} \label{subsubsec:depauto}
Considering \textit{Principle One}, ACE needs to support efficient application deployment regardless of the topology complexity and the infrastructure scale.
To achieve this, ACE constructs an automatic application deployment method only requiring the application topology file containing information like application specification, component clarifications, parameters, relations, and deployment requirements.
Such a manner prevents users from handling complex component-infrastructure mapping.
Specifically, to deploy an application, the user submits the topology file through the user interface to ACE, and triggers the orchestration process.
According to component deployment requirements, the ACE platform-layer orchestrator binds each component with specific nodes in the infrastructure and resource-level services required, generating the deployment plan.
When the user triggers the deployment process, the ACE platform-layer controller generates the instruction to deploy each component instance on the corresponding node according to the deployment plan, and sends the instruction to the node agent for execution.
Note that users can manage applications (\textit{e.g.}, update and delete) by modifying the topology file.
For example, for updates, submitting an updated topology file, the user can trigger a thorough update, \textit{i.e.}, ACE deletes the previous application and repeats the entire deployment process.
An incremental update can also be triggered, \textit{i.e.}, ACE automatically deploys updated components according to the new topology file.
\section{How It Works: Intelligent Video Queries using ACE} \label{sec:casestudy}
To validate our platform in supporting efficient and high-performing ECCI application development and deployment, we first present the entire development and deployment process of an intelligent video query application based on ACE, then compare the performance of the application implemented with ACE, CI, and EI, respectively.
\subsection{Application Development and Deployment} \label{subsec:appdevdep}
Video query~\cite{Wang2020,reducto} is one of killer ECCI applications.
To fulfil latency-sensitive user-specific video query requests (\textit{e.g.}, query about the existence of a type of objects in video streams from a geographic area), it generally uses edge and cloud resources to retrieve targeted objects from the video streams with a proper tradeoff between query accuracy and response latency under practical edge-cloud bandwidth limitations.
Based on Subsection~\ref{subsec:aceoverview}, we developed and deployed a video query application (based on ~\cite{Wang2020}) using ACE.
\subsubsection{User Registration.} \label{subsubsec:caseuserreg}
As an ACE user, we first mounted all our nodes and conducted the organization of our ECC infrastructure instructed by ACE.
Our infrastructure comprised a CC (one node, \textit{i.e.}, a GPU workstation), and three ECs (each with four nodes, \textit{i.e.}, an X86 mini PC and three Raspberry Pis).
Node details are in Figure~\ref{fig:video_query}.
For each EC, all edge nodes connected to an individual 100Mbps WLAN.
Each EC connected to CC through WAN (campus network) with software-limited bandwidth (\textit{i.e.}, 20Mbps uplink and 40Mbps downlink) and one-way delay (\textit{i.e.}, 0ms and 50ms as ideal and practical networks, respectively).
Let each Raspberry Pi receive the real-time video stream from a surveillance camera.
We deployed the resource-level message service on the infrastructure.
\subsubsection{Application Development.} \label{subsubsec:caseappdev}
Our application~\cite{ace2022} aimed at fulfilling user-specific video query requests accurately and rapidly through edge-cloud collaborations under practical network limitations (\textit{i.e.}, bandwidth and delay).
We developed the following components:
Data Generator (DG) providing the real-time video stream to the edge node,
Object Detector (OD) rapidly extracting video frame crops potentially containing moving objects from the video stream,
Edge Object Classifier (EOC) conducting lightweight query-specific binary object classification,
Cloud Object Classifier (COC) conducting accurate multi-class object classification,
In-app Controller (IC) executing the control policy,
and Result Storage (RS) saving all query results.
OD on edge nodes was implemented using frame differencing~\cite{Wang2020} (\textit{i.e.}, cropping regions with salient pixel differences across frames) instead of accurate but complex object detector like YOLOv3~\cite{yolov3} for rapid crop extraction on resource-limited edge nodes.
COC on CC was a ResNet152~\cite{resnet} pre-trained on ImageNet ILSVRC15~\cite{ILSVRC15} (4.49\% Top-5 error rate).
EOC was a MobileNetV2~\cite{mobilenetv2} rapidly trained on-the-fly by CC whenever there were user-specific queries.
To form its query-specific training set, video frame crops containing different classes of objects were extracted on CC by a YOLOv3 pre-trained on COCO~\cite{coco} (57.9\% mAP measured at 0.5 IOU) from historical video data uploaded by cameras at (or nearby) the queried area at leisure time, then labelled by COC.
The trained EOC (training details are in ~\cite{Wang2020}) was then deployed on edge nodes in the queried area.
We used real video clips from Youtube Live~\cite{youtubelive} ($30$ fps, $1920 \times 1080$ resolution, various durations) as historical video data and real-time video streams to query.
For a motorcycle query task, EOC's training set had 14,000 crops extracted from clips (170 hours total duration) from 14 surveillance cameras at or nearby the queried area (\textit{i.e.}, historical video data).
Another 6433 `motorcycle' and 68749 `non-motorcycle' crops were extracted as EOC's test set, where EOC achieved 11.06\% error rate under 80\% object identification confidence, tending to be less accurate than COC.
Another three video clips with $5$ minutes duration were used as real-time video streams.
Each node in the three ECs had one of the three clips.
The video query workflow after EOC's deployment is shown in Figure \ref{fig:video_query}.
For each edge node receiving the real-time video stream from DG (\textit{i.e.}, \circled{1}), OD selected three consecutive frames with a specific interval (\textit{e.g.}, 0.5 seconds), and rapidly extracted crops potentially containing moving objects.
Such crops were classified by EOC (\textit{i.e.}, \circled{2},\circled{3}), and the results were used by IC for crop scheduling based on the \textit{Basic Policy} (BP) (\textit{i.e.}, \circled{4},\circled{5}).
If the object identification confidence of a crop was above $80\%$, a targeted object was identified (predicted as positive due to the lack of ground truth of the real-time video), and its metadata were sent to RS (\textit{i.e.}, \circled{3},\circled{6},\circled{7}).
If the confidence was below $10\%$, the crop was dropped.
Otherwise the crop was sent to COC (\textit{i.e.}, \circled{3},\circled{6},\circled{8}).
If the Top-5 classification results of the crop on COC contained the targeted label, a targeted object was identified (\textit{i.e.}, predicted as positive), and its metadata was sent to RS (\textit{i.e.}, \circled{8},\circled{7}).
Since BP may induce queue backlog at EOC and frequent reprocessing at COC, we constructed an \textit{Advanced Policy} (AP) (\textit{i.e.}, \circled{4},\circled{10}) based on BP as a customized IC to further reduce E2E Inference Latency (EIL).
AP collected and estimated EILs of EOC (\textit{i.e.}, \circled{5},\circled{4}) and COC (\textit{i.e.}, \circled{9},\circled{11},\circled{4}) to guide crop uploading from OD (\textit{i.e.}, load balancing~\cite{Wang2020}, always sent to the one with a lower estimated EIL, \circled{2}, \circled{6},\circled{8}).
Then, AP reduced crops uploaded from EOC to COC by shrinking the identification confidence thresholds when either EOC's or COC's EIL got deteriorated.
\begin{figure}[tbp]
\centering
\includegraphics[width=0.85\columnwidth]{figure/deployment.pdf}
\caption{Automatic application deployment.}
\label{fig:deployment}
\end{figure}
\subsubsection{Application Deployment.} \label{subsubsec:caseappdep}
As shown in Figure~\ref{fig:deployment}, we submitted a topology file to ACE, which was an extended YAML file containing meta information of both the application and all components.
We illustrate the deployment of component OD as an instance.
Receiving the topology file, as Step \circled{1}, the orchestrator determined the node(s) (\textit{e.g.}, Raspberry Pi `ec-1-rpi1' on edge cloud `EC-1') satisfying all requirements of OD (\textit{i.e.}, `connections' implying OD's dependencies with components Local In-app Controller (LIC), EOC, and COC, `resources' implying CPU and memory required by OD, and `labels' implying that OD should be deployed on edge nodes connected to cameras).
Such decisions were recorded in the deployment plan (\textit{i.e.}, `instances'), a topology replica modified by the orchestrator.
Note that, to manage nodes in an EC as a cluster, ACE can delegate node-level orchestration to the EC.
Receiving the deployment plan, as Step \circled{2}, the controller transformed information of OD instances into specific deployment instructions in a standard Docker-compose YAML file, which was distributed to the node agent (\textit{e.g.}, the container engine at `ec-1-rpi1') for OD deployment.
\subsection{Impact of Implementation Paradigm on Intelligent Application Performance} \label{subsec:caseeva}
We compared the performance of our application implemented with different paradigms.
For CI, COC was deployed on CC.
For EI, EOCs were deployed on ECs.
For ECCI, based on ACE, two versions of the application with BP (ACE) and AP (ACE+) were deployed.
Different system loads were simulated by varying the sampling interval of frame differencing in OD from 0.5 to 0.1 seconds.
Since all comparatives used the same OD, we compared their video query performance using their object classification performance.
Particularly, we used \textit{F1-score}~\cite{f1score}~\footnote{
Since real-time video streams to query were not labelled, we classified all crops extracted by OD during the entire query task with COC after the task was finished, and treated COC's predicted labels as the query ground truth for F1-score calculation.}, \textit{edge-cloud BandWidth Consumption} (BWC), and \textit{E2E Inference Latency} (EIL)~\footnote{ Time from a crop is transmitted by OD to its predicted label is given by EOC or COC.} as evaluation metrics.
We conducted the motorcycle query task under different system load and edge-cloud network delay (\textit{i.e.}, 0ms for ideal and 50ms for practical) settings.
Results are illustrated in Figure~\ref{fig:exps}.
When the system load increases, F1-scores of CI and EI basically remain the same, where CI using COC only and EI using EOC only achieve the highest and lowest F1-scores under all system loads, respectively.
ACE and ACE+ using COC and EOC collaboratively manage to achieve F1-scores slightly lower than CI but significantly higher than EI.
Unlike EI, in ACE and ACE+, many crops that cannot be confidently classified by EOCs (with a confidence from 10\% to 80\% and dropped by EI) are uploaded to COC.
Compared with CI, few crops are dropped by EOCs (with a confidence below 10\%) in ACE and ACE+.
Besides, the higher the system load, the better ACE+ performs compared with ACE.
Under higher system loads, more crops are directly uploaded from ODs to COC by IC with AP for load balancing in ACE+, reducing crops dropped by IC with BP in ACE.
Furthermore, when the system load increases, ACE+ achieves higher F1-scores under practical than ideal network delay.
In ACE+, under practical network delay, fewer crops are uploaded from EOCs to COC to avoid higher EILs by shrinking the confidence thresholds, and more are from ODs to COC for load balancing.
When the system load increases, BWCs of all except for EI increase.
ACE and ACE+ induce significantly lower BWCs than CI since considerable objects are identified by EOCs.
Furthermore, the higher the system load, the higher BWCs of ACE+ compared with ACE.
In ACE+, some crops (increase with system load) are directly uploaded by IC with AP for load balancing, where, however, only some of them are uploaded by IC with BP in ACE (with identification confidence from 10\% to 80\%), inducing higher BWCs.
When the system load is low, CI induces the lowest EIL under different network delay settings benefiting from COC's fast processing (\textit{i.e.}, the inference time of COC is about 32.3ms on CC, and that of EOC on edge node is above 44ms).
When the system load increases, different from EI, ACE, and ACE+, CI's EIL increases significantly due to the large queue backlog aggregated from all ODs (normal in large-scale edge-cloud systems).
Besides, the practical network delay also enlarges CI's EIL more obviously (significantly higher than the 50ms network delay).
Compared to CI, EILs of EI, ACE, and ACE+ are not obviously impacted by both system load (\textit{i.e.}, low queue backlog at EOCs) and network delay (no/low uploading).
ACE's EIL is slightly higher than EI since EOCs manage to identify most objects, and only a few crops are uploaded to COC.
Furthermore, the higher the system load, the lower EIL of ACE+ compared with ACE.
Some crops (increase with system load) are directly uploaded to COC for load balancing by IC with AP in ACE+.
Compared with CI and EI, ACE-based video query manages to better fulfill query requests accurately and rapidly with efficient bandwidth consumption.
ACE also facilitates developers for customized optimization (\textit{i.e.}, EIL reduction with customized AP).
\begin{figure}[tbp]
\centering
\includegraphics[width=0.85\columnwidth]{figure/results.pdf}
\caption{Intelligent video query performance.}
\label{fig:exps}
\end{figure}
\section{Future of ACE} \label{sec:discussion}
As a prototype for cost-efficient ECCI application development and deployment, ACE is still in its infancy.
The construction of ACE reveals fundamental challenges to address and sheds light on the vision of an ACE-based ECCI ecosystem deserving explorations.
\subsection{Challenges} \label{subsec:challenge}
\textit{Agile ECCI application orchestration} is critical, but challenging, to improve the performance of ACE-based applications.
ACE's platform-layer orchestrator manages to allocate application components to proper nodes satisfying basic (node-level) resource and user (\textit{i.e.}, edge/cloud deployment) requirements.
However, fine-grained orchestration under more explicit constraints is still hard to achieve, which is significant to fully infrastructure utilization.
Furthermore, ACE's static application orchestration cannot adjust to application or infrastructure changes. A dynamic orchestrator is also necessary.
\textit{Resource contention prevention} has to be further investigated to ensure the performance of ECCI applications co-located at the same infrastructure.
Currently, ACE manages to achieve component-level resource isolation through containerization, and support inter-component resource allocation optimization through the customized in-app controller, where, however, application-level resource isolation and allocation is still an open issue.
Critical resources like edge-cloud bandwidth should be allocated appropriately to co-located applications under ACE's coordination.
It is also promising to integrate the serverless technology~\cite{Castro2019} for elastic resource allocation that cannot be directly achieved by container-based solutions.
\textit{Security} is another critical issue.
ACE now contains no security module, where state-of-the-art encryption and authentication techniques can be directly integrated for fundamental secrecy.
The actual challenge, however, is access control.
In our design, an ACE user has full access to his/her infrastructure and ECCI applications, where no user collaboration is currently supported.
For specific applications (\textit{e.g.}, federated learning) that have to be developed and deployed by multiple users collaboratively, ACE is required to provide a fine-grained access control mechanism.
It needs to ensure that each collaborator has limited access to the shared application and infrastructure without jeopardizing others' privacy.
\subsection{Vision} \label{subsec:vision}
\textit{ACE demonstrates the potential in supporting closed-loop DevOps of ECCI applications.}
ACE manages to facilitate the cost-efficient development and deployment of ECCI applications effectively.
Taking a step further, we believe it is viable to integrate proper operation and maintenance modules into ACE, aiming at the close loop of continuous ECCI application development, deployment, monitoring, delivering, and testing.
Such full DevOps supports will enable ACE to act as the foundation of the approaching ECCI ecosystem.
\textit{ACE is promising in promoting a broad spectrum of production level ECCI applications.}
ECCI applications, especially high-performing ones, are difficult to design, develop, and deploy, which hinders such a paradigm from contributing to the rapidly expanding IA market.
ACE manages to provide supports along the entire ECCI application lifecycle, facilitating general users to conduct unified and user-friendly application development and deployment.
Besides, ACE can also ease the migration of existing IAs based on CI and EI to ECCI applications satisfying specific practical requirements.
\section{Conclusion} \label{sec:conclusion}
ML/DL-based IAs with harsher practical requirements cast challenges on conventional CI implementations.
The emerging ECCI paradigm can support proliferating IAs that, however, are currently developed and deployed individually without generality.
We envision systematic designs of a unified platform for cost-efficient development and deployment of high-performing ECCI applications, guiding us to construct the ACE platform handling heterogeneous resources and IA workloads.
Our initial experience shows that ACE manages to help developers and operators along the entire lifecycle of ECCI applications, where customizable optimizations can be conducted efficiently.
Further research is still required, and we discuss both the challenges and visions of the newborn ACE.
\begin{acks}
This work was supported in part
by the National Key Research and Development Program of China under Grant 2020YFA0713900;
the National Natural Science Foundation of China under Grants 61772410, 61802298, 62172329, U1811461, U21A6005, 11690011;
the China Postdoctoral Science Foundation under Grants 2020T130513, 2019M663726;
and the Alan Turing Institute.
\end{acks}
\bibliographystyle{ACM-Reference-Format}
|
1,314,259,993,597 | arxiv | \section{Introduction}
The discovery of the Higgs boson at the Large Hadron Collider (LHC) has completed the standard model particle spectrum. The most important task is now to find physics beyond the standard model. Precision study of the Higgs boson is a powerful tool for this purpose. The International Linear Collider (ILC)\,\cite{Aus0} is an ideal machine to carry out the precision Higgs measurements.
The motivation of our study is to find new physics effects in $h\gamma\gamma$ and $h \gamma Z$ couplings. Since these couplings appear only at the loop level in the standard model, they are potentially very sensitive to new physics and being studied at the LHC. As one example, the expected deviations on the $e^+e^- \to h \gamma$ cross section and the $h \to \gamma \gamma$ branching ratio in the Inert Doublet Model~\cite{Aus2} are shown in \Figref{fig:1}, which suggests that depending on model parameters the deviations can be as large as 100$\%$.
\begin{figure}[ht]
\centering \includegraphics[width=0.4\columnwidth]{idm_graph.pdf}
\caption{
\label{fig:1}
The relative deviations of the $e^+ e^- \to h\gamma$ cross section and the $h \xrightarrow{} \gamma\gamma$ branching ratio with respect to the Standard Model values~\cite{Aus2}}
\end{figure}
A usual method to measure $h\gamma\gamma$ and $h \gamma Z$ couplings is to use decay branching ratios of $h \to \gamma \gamma / \gamma Z$. It is, however, very challenging to measure the $h \to \gamma Z$ branching ratio even at the HL-LHC: only a 3$\sigma$ significance is expected. As a complementary method we study these couplings in a production process at the ILC, $e^+e^- \to h \gamma$ (see \Figref{fig:6}).
\begin{figure}[ht]
\centering \includegraphics[width=0.4\columnwidth]{fynman.pdf}
\caption{
\label{fig:6}
The Feynman diagram of $e^+ e^- \to h \gamma$
}
\end{figure}
In addition, the $h \gamma Z$ coupling appears in an s-channel photon exchange diagram for the leading single Higgs production process: $e^+ e^- \to h Z$ in the effective field theory framework. It is hence necessary to know the contribution from this diagram.
Furthermore, it turns out that the anomalous $h \gamma Z$, $h \gamma \gamma$, $hZZ$, and $hWW$ couplings come from a common set of a few dimension-6 operators, hence the measurement of the $h \gamma Z$ coupling using $e^+e^- \to h \gamma$ has a potential to provide one very useful constraint on those operators.
In section 2, we introduce our theoretical framework. In section 3, we describe our experimental method and simulation framework. In section 4, we present our event selection and analysis result. Section 5 gives our plan for further study.
\section{Theoretical Framework}
In this analysis, we use the effective Lagrangian shown in \Equref{Equ:0} to include new physics contributions to the $e^+ e^- \xrightarrow{} h \gamma $ cross section model-independently,
\begin{eqnarray}
{\cal{L}} _ {h \gamma } = {\cal{L}} _ { \mathrm { SM } } + \frac { \zeta _ { A Z } } { v } A _ { \mu \nu } Z ^ { \mu \nu } h + \frac { \zeta _ { A } } { 2 v } A _ { \mu \nu } A ^ { \mu \nu } h,
\label{Equ:0}
\end{eqnarray}
where $\zeta_{AZ}$ and $\zeta_A$ terms represent respectively effective $h \gamma Z $ and $h \gamma \gamma $ couplings from new physics. $A_{\mu \nu}$, and $ Z_{\mu \nu}$ are field strength tensors. $v$ is the vacuum expectation value. The first term is the Standard Model Lagrangian. The three terms contribute to $e^+e^- \to h \gamma$ process via the Feynman diagrams shown in \Figref{fig:5}, where the first SM diagram represents several loop induced diagrams as shown in \Figref{fig:7}.
\begin{figure}[ht]
\centering \includegraphics[width=0.9\columnwidth]{lagrangian.pdf}
\caption{
\label{fig:5}
Diagrams arising from each of the three terms of \Equref{Equ:0}, respectively. }
\end{figure}
\begin{figure}[ht]
\centering \includegraphics[width=0.9\columnwidth]{dyagram2.pdf}
\caption{
\label{fig:7}
The loop induced Feynman diagrams in the Standard Model for $e^+e^- \to h \gamma$~\cite{Aus2}}
\end{figure}
The cross section normalized to SM can be written as \Equref{Equ:1} and \Equref{Equ:2}~\cite{Aus3}.
The SM cross sections at $\sqrt{s}$ = 250 GeV are calculated as shown in \Tabref{tbl:1}. The cross sections including effective $h \gamma Z / h\gamma \gamma$ couplings from new physics are calculated as in $P(e^-,e^+)=(+100\%,-100\%)$) , up to interference term.
\begin{eqnarray}
\frac { \sigma _ { \gamma H } } { \sigma _ { S M } } = 1 - 201 \zeta _ { A } - 273 \zeta _ { A Z }
\label{Equ:1}
\end{eqnarray}
\begin{eqnarray}
\frac { \sigma _ { \gamma H } } { \sigma _ { S M } } = 1 + 492 \zeta _ { A } - 311 \zeta _ { A Z }
\label{Equ:2}
\end{eqnarray}
\begin{table}[htbp]
\begin{center}
\caption{SM cross sections for different beam polarizations ($\sqrt{s}$ = 250 GeV )}
\label{tbl:1}
\begin{tabular}{|c|c|c|c|}
\hline
\multicolumn{1}{|c}{$P_{e^-}$ } & \multicolumn{1}{|c|}{$P_{e^+}$} & \multicolumn{1}{c|}{$\sigma_{SM}$[fb]} \\
\hline
-100$\%$ & +100$\%$ & 0.35\\
+100$\%$ & -100$\%$ & 0.016\\
-80$\%$ & +30$\%$ & 0.20\\
\hline
\end{tabular}
\end{center}
\end{table}
\section{Experimental Method and Simulation Framework}
\subsection{Experimental Method}
In order to determine both $\zeta_{AZ}$ and $\zeta_{A}$, we need two measurements. There are two strategies:\\
1. to measure the cross sections of $e^+ e^- \xrightarrow{} \gamma h$ for two different beam polarizations, and\\
2. to use the measurement of the $h \xrightarrow{} \gamma\gamma$ branching ratio at the LHC to constrain $\zeta_A$ and determine $\zeta_{AZ}$ by just measuring $e^+e^- \to h \gamma$ cross section for one single polarization.\\
\subsection{Simulation framework}
We use fully-simulated Monte-Carlo (MC) samples produced with the ILD detector model~\cite{Aus4}. For event generation, we use Physsim~\cite{Aus7} for signal, and Whizard~\cite{Aus11} for background processes. For detector simulation, we use Mokka~\cite{Aus5}, which is based on Geant4~\cite{Aus12}, and for event reconstruction, we use Marlin in iLCSoft~\cite{Aus6}, where particle flow analysis (PFA) is done with PandoraPFA~\cite{Aus13} and flavor tagging is done with LCFI+~\cite{Aus10}. The analysis is carried out at $\sqrt{s}$=250 GeV, assuming an integrated luminosity of 2000 fb$^{-1}$.
\section{Event Selection and Results}
\subsection{Event Selection}
The signal channel studied in this paper is $e^+ e^- \to h \gamma $, followed by $h \to b \bar{b}$. In the final states of the signal events, we expect one isolated monochromatic photon with an energy of $E_{\gamma}=\sqrt{s}/{2}\left( 1- {\left(m_h/\sqrt{s}\right)}^2 \right) = 93$~GeV, where $m_h$ is the Higgs mass, and two $b$ jets with an invariant mass consistent with the Higgs mass. The enegy resolution of the electromagnetic calorimeter is given by $\sigma_E=0.16 \times \sqrt{E}$~(GeV), where the photon energy $E$ is in units of GeV~\cite{Aus4}. The energy resolution for the isolated photon is thus 1.5~GeV.
The main background would be $e^+e^-\to \gamma q\bar{q}$, dominantly coming from $e^+e^- \to \gamma Z$.
As pre-selection, we start with identifying one isolated photon with an energy greater than 50 GeV. Sometimes, the reconstruction software PandoraPFA splits calorimetric clusters created by a single high energy photon into several objects. Such split clusters fall within a narrow cone ($\cos \theta_{cone}$=0.998, where $\cos \theta_{cone}$ is cone angle), and are considered as a single object in the following analysis.The particles other than the photon are clustered into two jets using the Durham algorithm~\cite{Aus15}.
In the final selection, the first cut requires two $b$ jets. \Figref{fig:8} shows the distribution normalized to unity of the larger $b$-likeliness value among the two jets for signal and background events. We require this larger $b$-likeliness to be greater than 0.77 to suppress the light flavor $\gamma q\bar{q} $ events. The cut value is optimized to maximize the signal significance defined as
\begin{eqnarray}
\text {significance} = \frac { N _ { S } } { \sqrt { N _ { S } + N _ { B } } },
\label{Equ:7}
\end{eqnarray}
where $N_S$ and $N_B$ are the numbers of signal and background events, respectively.
\begin{figure}[ht]
\centering \includegraphics[width=0.6\columnwidth]{htobb.pdf}
\caption{
\label{fig:8}
The distribution of $b$-likeliness for signal and background events with unit normalization}
\end{figure}
\if 0
\begin{figure}[ht]
\centering \includegraphics[width=0.5\columnwidth]{htobb2.pdf}
\caption{
\label{fig:9}
}
\end{figure}
\fi
The second cut requires small enough missing energy. \Figref{fig:10} shows the missing energy distribution. We cut the missing energy at 35~GeV.
\begin{figure}[ht]
\centering \includegraphics[width=0.6\columnwidth]{emis.pdf}
\caption{
\label{fig:10}
Normalized missing energy distribution for signal and background events}
\end{figure}
\if 0
\begin{figure}[ht]
\centering \includegraphics[width=0.4\columnwidth]{emis2}
\caption{
\label{fig:11}
}
\end{figure}
\fi
To finalize our event selection, we use a multivariate analysis method, the BDT algorithm as implemented in the TMVA package~\cite{Aus8}. It is trained using five input variables: the 2-jet invariant mass, the energy of the isolated photon, its polar angle, the smaller angle between the photon and a jet, and the angle between the two jets. \Figref{fig:12} illustrates these input variables. \Figref{fig:14} shows the distributions of each input variable for signal and background events. The blue histograms are for signal events, and the red histograms are for background events. Our final cut requires the BDT output to be greater than 0.0126.
\begin{figure}[h]
\centering \includegraphics[width=0.7\columnwidth]{input.pdf}
\caption{
\label{fig:12}
Input variables for TMVA}
\end{figure}
\begin{figure}[h]
\centering \includegraphics[width=0.7\columnwidth]{tmvapara.pdf}
\caption{
\label{fig:14}
Distributions of TMVA input variables for signal and background events. }
\end{figure}
\Figref{fig:15} shows the distribution of $m(b\bar{b})$ after all the other cuts for the signal and background events normalized to an integrated luminosity of 2000 fb$^{-1}$ for the left-handed beam polarization. The remaining background events are dominated by 2-fermion processes.
\begin{figure}[h]
\centering \includegraphics[width=0.7\columnwidth]{20190214_cutsall.pdf}
\caption{
\label{fig:15}
The distribution of $m(b\bar{b})$ after all the other cuts for the signal and background events normalized to 2000 fb$^{-1}$}
\end{figure}
\clearpage
\subsection{Result}
\Tabref{tbl:bins} gives the number of signal and background events, as well as the signal significance after each cut. The significance is defined by \Equref{Equ:7}.
After all the cuts, the signal significance is expected to be 0.53$\sigma$, for the SM signal process $e^+e^-\to h \gamma$ followed by $h\to b\bar{b}$ decay.
The 95 $\%$ confidence level upper limit for the cross section at $\sqrt{s}=250$ of $e^+ e^- \to h \gamma$ is calculated using \Equref{Equ:4} to be $\sigma_{h\gamma}^{CL95} < 1.08$ fb, for 2000 fb$^{-1}$ GeV and left handed beam polarizations.
\begin{table}[htbp]
\begin{center}
\caption{The cut table}
\label{tbl:bins}
\begin{tabular}{|c|c|c|c|}
\hline
\multicolumn{1}{|c}{ } & \multicolumn{1}{|c|}{Signal} & \multicolumn{1}{c|}{background} & \multicolumn{1}{c|}{Significance} \\
\hline
Expected &237 & 3.14$\times 10^8$ & 0.01\\
Pre selection &222 & 6.54$\times10^7$ &0.02 \\
$b_{tag}\geq$0.8 &200 & 4.96$\times10^6$ &0.09\\
$E_{mis}\leq$35 & 182 & 4.30$\times10^6$ &0.09 \\
$mvabdt \geq$ 0.0126 &75 &1.98$\times10^4$ &0.53\\
\hline
\end{tabular}
\end{center}
\end{table}
\begin{eqnarray}
\sigma = \frac { 1.64 } { \text { significance } } \sigma _ { S M }
\label{Equ:4}
\end{eqnarray}
From this upper limit and \Equref{Equ:1} and \Tabref{tbl:1}, we have
\begin{eqnarray}
3.09 > \frac { \sigma _ { \gamma H } } { \sigma _ { S M } } = 1 - 201 \zeta _ { A } - 273 \zeta _ { A Z } > 0
~(\mbox{assuming}~ \zeta _ { A } = 0).
\label{Equ:5}
\end{eqnarray}
We can then set the bound on the parameter $\zeta_{AZ}$:
\begin{eqnarray}
- 0.0077 > \zeta _ { A Z } > 0.0037.
\label{Equ:6}
\end{eqnarray}
\section{Further Study}
We are planning to improve the analysis by adding the $h \xrightarrow{} WW^*$ channel. The branching ratio of this channel is around 21$\%$ corresponding to about 50 event for 2000 fb$^{-1}$. The main background we expect in this channel would be $e^+e^-\to W^+W^-$ with a hard ISR photon, while in the $h\to b\bar{b}$ channel, the main background is $e^+e^-\to b\bar{b}$ with a hard ISR photon, which is significantly enhanced due to the radiative return to $Z$-pole. We would hence expect a higher signal to background ratio in the $h \to WW^*$ channel.
After $h\to WW^*$ channel is completed, the experimental bound on $\zeta_{AZ}$ will be translated into a bound on Dimension-6 operators.
\section*{Acknowledgements}
We would like to thank the LCC generator working group and the ILD software working group for providing the simulation and reconstruction tools and producing the Monte Carlo samples used in this study.
This work has benefited from computing services provided by the ILC Virtual Organization, supported by the national resource providers of the EGI Federation and the Open Science GRID.
|
1,314,259,993,598 | arxiv | \section{Introduction}
Distributed Energy Resources (DER), such as rooftop solar and wind turbine, have seen widespread proliferation among consumers in recent years\cite{IEA}. In addition, the advent of Smart Grid (SG) technologies, Advanced Metering Infrastructures (AMI), and home energy management systems, have added flexibility in energy generation/consumption for consumers. This, in turn, has allowed traditionally passive consumers to become actively involved in energy trading by sharing the excess energy generated at their premise to either grid or other buyers \cite{timilsina2021reinforcement,Parag2016prosumer_era}. These active consumers with energy production capabilities have been referred to as \textit{prosumers} \cite{Parag2016prosumer_era}, as a portmanteau of ``producers'' and ``consumers''. The role of prosumers in energy market has been recognized to some extent with the adoption of incentive schemes like \textit{Feed-in-Tariff} (FiT) mechanism \cite{tushar2020peer, tushar2018peer}. FiT allows prosumers to sell excess energy to the grid and buy from grid when required \cite{tushar2018peer}. However, existing energy trading modalities offer limited benefits to participating prosumers. This is due to the minimal prices at which energy is purchased by grid, as well as the low limits on the amount of energy that can be purchased \cite{tushar2020peer,tushar2018peer,Parag2016prosumer_era}.
\subsection{Literature Review and Motivation}
{\em Peer-to-peer (P2P) energy trading} is a recently proposed decentralized modality for energy sharing aiming at solving limitations of centralized techniques. This modality has been gaining significant traction recently \cite{tushar2018peer,tushar2020peer}.
Specifically, P2P energy trading allows prosumers to trade energy among each other at a negotiated price with or without the involvement of the grid \cite{tushar2020peer}. It generates better monetary incentives for prosumers compared to existing mechanisms while also reducing their grid dependency \cite{tushar2018peer}.
Additionally, increased local energy generation/consumption resulting from P2P trading leads to the minimization of overall system energy loss while providing an effective way to achieve demand side management\cite{Zhu2013SmartMicrogrids}. Benefits extend also to the grid operator, by providing
savings in investments that would have been otherwise required to develop/maintain transmission infrastructure in a centralized power distribution architecture \cite{Parag2016prosumer_era,tushar2020peer}.
P2P energy trading has received attention from the research community in recent years. The works in \cite{tushar2018transforming,tushar2019grid} present game theoretic approaches in a P2P setting, while a greedy rule-based P2P mechanism to assign energy among prosumers is proposed in \cite{azim2019feasibility} that includes mid-market pricing. Similarly, the physical aspects of P2P energy trading, such as power loss minimization and voltage regulation, have been explored in \cite{nasimifar2019peer,paudel2020peer}.
These works, however, largely overlook the user behavior in designing their solutions. As established in \cite{Parag2016prosumer_era,tushar2018peer,timilsina2021reinforcement}, accommodating the user behavioral modeling in P2P energy trading ensures sustained participation from prosumers while incentivizing their contribution. In fact, the papers \cite{tushar2018transforming,tushar2019grid} consider prosumers to be actively involved and fully compliant with the system as rational decision-makers. First concern with this assumption is that the continuous online presence of participating prosumers with the system might not always be possible in real-world application. Secondly, research on user behavioral models and decision making \cite{gigerenzer2002bounded,agosto2002bounded} have found users to have \textit{bounded rationality}.
Therefore, requiring constant active participation overwhelms the users and incentivizes non-rational decisions \cite{earl2016bounded}. In the worst case, it might even result in users opting to terminate their participation altogether \cite{agosto2002bounded,timilsina2021reinforcement}.
In that light, the works in \cite{timilsina2021reinforcement,agate2020enabling} incorporates bounded rationality and user preferences into P2P energy trading. However, it requires continuous human participation and assumes a simplistic linear model for user perception. Conversely, the authors of \cite{tushar2018peer} limit their focus on coalition formation in game theoretic setting and do not explicitly consider user behavioral modeling.
As a result, a prosumer-centric P2P energy trading model, that effectively incorporates the prosumers' decision-making behavior and their perceived loss/gain value from trading, is still lacking in the existing literature. Such a trading modality is expected to require minimal active participation from users while also ensuring their sustained involvement through the adoption of user behavioral modeling.
To this end, the framework of \textit{Prospect Theory} (PT) \cite{kahneman2013prospect} can be used to model the non-rational user behavior in the face of uncertain decision-making. It is often regarded as fairly accurate mathematical representation of human behavior
\cite{kahneman2013prospect,el2016prospect, el2017managing}.
Recently, there has been few efforts in integrating PT in energy related applications as well to capture the irrationality of users \cite{saad2016toward,el2017managing,wang2020prospect,yao2021distributed}. In relation to P2P energy trading, the authors in \cite{yao2021distributed} have proposed a PT-based distributed energy trading model to optimize trading decisions for prosumers in a competitive market.
Although these papers model the user behavior in some ways, they require active participation from users and also assume that such behavior (e.g., the parameters of PT) is homogeneous for all the users. Social science studies, such as the one conducted in Italy \cite{contu2016modeling} to investigate the social acceptance of nuclear energy using an online survey, show that users exhibit significant heterogeneity in their preferences for the sources of energy. Neuroscience studies have also stressed the heterogeneity of humans in reference to PT parameters \cite{fox2009prospect}. Not capturing such heterogeneity provides little benefits in terms of user behavioral modeling.
\subsection{Paper Contributions}
In this paper, we design a PT-based optimization framework for prosumer-centric P2P energy trading as shown in Fig. \ref{fig:system_overview_PT}. The framework aims at matching energy production and consumption (step $1$ in Fig. \ref{fig:system_overview_PT}) to maximize the perceived utility of individual buyers while taking into account the intrinsic heterogeneity of human perception. Given that the optimization problem is non-linear and non-convex, we further devise a \textit{Differential Evolution}-based \cite{storn1997differential} metaheuristic algorithm called $DEbATE$ to solve the problem (\textit{energy allocation}, step $2$). In order to ensure minimal active participation of prosumers, we employ a Reinforcement Learning (RL) framework, called $PQR$, in tandem with $DEbATE$ to automate the pricing mechanism for sellers (\textit{pricing mechanism}, step $3$). In doing so, $PQR$ learns the selling price for each sellers using a PT-based risk-sensitive Q-learning algorithm \cite{shen2014risk}. The output of the algorithms is then returned to the prosumers for executing the physical energy transactions (step $4$).
Using real datasets for energy production and consumption, paired with recent survey data for PT perception modeling, results show that $DEbATE$ performs $25\%$ higher in buyer's perception and $7\%$ higher in seller's reward compared to state-of-the-art approach.
The major contributions of the paper are the following:
\begin{itemize}
\item We develop a PT-inspired optimization framework for P2P energy trading;
\item We design a metaheuristic algorithm $DEbATE$ to solve the non-linear energy allocation problem;
\item We design dynamic pricing mechanism with $PQR$ algorithm using risk-sensitive Q-learning approach;
\item Experiments using real data show the superiority of proposed approach compared to the state-of-the-art;
\end{itemize}
\begin{figure}[!thb]
\includegraphics[width=.96\linewidth]{images/P2P_trading.png}
\caption{P2P Energy Trading System Overview.}\label{fig:system_overview_PT}
\end{figure}
\section{System Model and Problem Formulation}\label{chap:system_prob}
We consider a P2P energy trading system as shown in Fig. \ref{fig:system_overview_PT}. The system consists of prosumers that can exchange energy among each other through an existing distribution network. The grid serves as backup for prosumers to either buy or sell energy, if the local energy trading is insufficient or not possible. Let $P$ be the set of all prosumers participating in the P2P energy market. We refer to $B_t \subset P$ as the set of \textit{Buyers}, i.e. the set of prosumers that have higher self-consumption than generation at a timeslot $t$, and consumers without energy generation capabilities. Similarly, $S_t \subset P$ is the set of \textit{Sellers}, i.e., prosumers that have excess generation at a timeslot $t$. For simplicity of notation, we drop the subscript $t$ in the following.
We model the perceived loss and gain of prosumers using the \textit{prospect theory} (PT) value function to capture user perception on gains and losses. Specifically, consider the excess energy generation of seller $i \in S$ be $r_i$ and demand of buyer $j \in B$ be $w_j$. Then, let $x_{ij} \in [0,1]$ represent the fraction of $w_j$ that a buyer $j$ is willing to buy from seller $i$ at $\rho_{i}$ price per $kWh$ amount of energy. There is an {\em energy loss} during the physical energy transfer through wires \cite{Zhu2013SmartMicrogrids}, which depends on the wire-length between $i$ and $j$ and directly proportional to the amount of energy exchanged. The loss is modeled as a fraction $l_{ij} \in [0,1]$ of the energy exchanged.
Assume $\rho_{gs},\rho_{gb}$ be the energy selling and purchasing prices from the grid. We adopt a modified PT value function to model realistic user perception in an energy market \cite{kahneman2013prospect}. The function quantifies perceived utility of humans towards gain and loss based on degree of deviation from a reference point. Particularly, in our problem, it captures the difference of total actual buying cost $y_{j}$ from the buyer's desired total reference cost $\rho_{j}w_j$ where $\rho_j$ is the {\em reference price} of buyer $j$ for purchasing energy. This utility function is formulated as
\begin{equation} \label{eq:PT_val_buyers}
v(y_j)=
\begin{cases}
k_{+,j}(\rho_{j}w_j - y_j)^{\zeta_{+,j}}, & y_j < \rho_{j}w_j\\
-k_{-,j}(y_j - \rho_{j}w_j)^{\zeta_{-,j}}, & y_j \geq \rho_{j}w_j
\end{cases}
\end{equation}
where $k_{+,.}, k_{-.},\zeta_{+,.}, \zeta_{-,.}$ are the parameters that control the degree of loss-aversion and risk-sensitivity. These parameters are found to be highly heterogeneous and vary from person to person based on factors like gender and age group
\cite{balavz2013testing,rieger2017estimating}.
$y_j$ is the total actual cost of buying energy for $j^{th}$ buyer s.t.
$$ y_j = \sum_{i \in S} \rho_{i}x_{ij}w_j + \rho_{gs}(1-\sum_i x_{ij})w_j$$
Note that, similar to the PT value function in \cite{kahneman2013prospect}, the utility function in Eq. \eqref{eq:PT_val_buyers} is concave in the gain domain (i.e. case $y_j < \rho_{j}w_j$) while convex in loss domain (i.e. case $y_j \geq \rho_{j}w_j$).
The problem of matching demand and production of heterogeneous prosumers is formalized as follows.
\begin{subequations}\label{obj_func_pt}
\begin{align}
{\text{maximize}}& &&f(y):\sum_{j \in B} v(y_j) \tag{\ref{obj_func_pt}}\\
\mbox{s.t.}& && \sum_{j \in B} (1+l_{ij}) x_{ij}w_j \leq r_i, &&&& \forall i\label{const2_pt}\\
&&& \sum_{i \in S} x_{ij} \leq 1, &&&& \forall j\label{const3_pt}\\
&&& x_{ij} = 0 \text{, if } l_{ij}\geq l_{max},&&&& \forall i\label{const4_pt}\\
&&& \rho_{gb} \leq \rho_{i},\rho_{j} \leq \rho_{gs},&&&& \forall i\label{const5_pt}\\
&&& x_{ij} \in [0, 1], &&&&\forall i,j\label{const6_pt}
\end{align}
\end{subequations}
The problem maximizes the sum of perceived utility for buyers in Eq. \eqref{obj_func_pt}. Constraint in Eq. \eqref{const2_pt} prevents the problem from exceeding the amount of energy being sold by each sellers while incorporating the losses in electric lines. The constraint in Eq. \eqref{const3_pt} ensures that the energy demand for each buyers is not exceeded, while constraint \eqref{const4_pt} limits the loss between sellers and buyers to be within the loss threshold $l_{max}$. Finally, the constraint \eqref{const5_pt} limits upper and lower bound for energy price to the selling and buying price of the grid.
It is to be noted that the problem in Eq. \eqref{obj_func_pt} is non-linear, non-convex
optimization problem. Hence, we propose a heuristic based on Differential Evolution Algorithm (DEA) \cite{storn1997differential} described in the following section. Additionally, in the above problem, the selling price is considered as a fixed amount for a trading period. However, the reference price $\rho_j$ of buyer $j$ is a personal value which may under- or over-estimate the competitiveness of market. In order to maximize the sellers' perceived objectives through prospect theory, we resort to the risk-sensitive Q-learning algorithm \cite{shen2014risk}.
\begin{algorithm}[!hpbt]
\SetAlgoLined
\footnotesize
\caption{DEbATE}\label{alg:DEBATE}
\footnotesize
\SetKwInOut{Input}{Input}
\SetKwInOut{Output}{Output}
\Input{set of buyers $B$, sellers $S$, fitness function $f(.)$, max iterations $G_{max}$, population size $NP$, crossover probability $CR$, differential weight $F$}
\Output{best identified feasible solution $\mathbf{x^*}$}
Update set of buyers $B$ and sellers $S$, $count = 0$\;
Generate initial population $\mathcal{X} = \{\mathbf{x_k} |\mbox{ } k = 1, \dots, NP\}$\;
\While{$count < G_{max}$}{
\For{each $\mathbf{x_k} \in \mathcal{X}$}{
Choose $3$ different vectors $\{\mathbf{x_a}, \mathbf{x_b}, \mathbf{x_c}\}\in \mathcal{X}$ at random and $R \sim U(1,|S| \times |B|)$\;
Create mutated solution $\mathbf{\Bar{x}_k} = \mathbf{x_k}$\;
\tcc{\textbf{Mutation and Crossover}}
\For{each $i \in |S|$, $j \in |B|$}{
Select $u \sim U(0,1)$ \;
\uIf{$u < CR || (i \times j) == R$}{
$\Bar{x}_{ij}^{(k)} = x_{ij}^{(a)} + F \times (x_{ij}^{(b)} - x_{ij}^{(c)})$\;
$\Bar x_{ij}^{(k)} = \min(1,\max(0,\Bar x_{ij}^{(k)}))$
}
}
\tcc{\textbf{Check Constraints}}
$\forall i,j$, \lIf{$l_{ij}\geq l_{max}$}{$\Bar{x}_{ij}=0$}
$\forall i$, \lIf{$\sum_j (1+l_{ij})\Bar x_{ij}w_j > r_i$}{$\Bar{x}_{ij} = \frac{\Bar{x}_{ij}r_{i}} {\sum_{\hat{j}} \Bar (1+l_{i\hat{j}})\Bar{x}_{i\hat{j}}w_{\hat{j}}} $}
$\forall j$, \lIf{$\sum_i \Bar x_{ij} > 1$}{ $\Bar{x}_{ij} = \frac{\Bar{x}_{ij}} {\sum_{\hat{i}} \Bar x_{\hat{i}j}} $}
\tcc{\textbf{Compare fitness}}
\lIf{$f(\mathbf{\Bar{x}_k}) > f(\mathbf{x_k})$}{$\mathcal{X} = (\mathcal{X} \setminus \{ \mathbf{x_k} \}) \cup \{\mathbf{\Bar{x}_k}\}$}
}
count = count++\;
}
\tcc{\textbf{Find the best solution to execute trading}}
Let $\mathbf{x^*} = \arg \max\limits_{\mathbf{x_k} \in \mathcal{X}} f(\mathbf{x_k})$\;
Execute transactions for each prosumers to $\mathbf{x^*}$ \;
\end{algorithm}
\section{The DEbATE and PQR Heuristics} \label{chap_fut_heuristic}
In this section, we describe the \textit{Differential Evolution-based Algorithm for Trading Energy (DEbATE)} (Alg. \ref{alg:DEBATE}), designed for the problem presented in Section \ref{chap:system_prob}, and the \textit{Pricing mechanism with Q-learning and Risk-sensitivity (PQR)}, designed to dynamically adjust the sellers' prices.
\subsection{DEbATE}
$DEbATE$ is executed at each trading period (e.g., 12 hours) to solve the non-linear optimization problem in Eq. \eqref{obj_func_pt}. It uses differential evolution to determine an optimal amount of energy to be traded between prosumers that maximizes the perceived utility of buyers.
\textit{DEbATE} initially updates the list of buyers ($B$) and sellers ($S$) based on the expected production and consumption for current trading period. These can be predicted accurately with recent approaches \cite{kong2017short,casella2022dissecting}.
The differential evolution-based optimization begins on line $2$ where an {\em initial population} $\mathcal{X}$ is generated with population size of $NP$. An element $\mathbf{x_k} \in \mathcal{X}$, with $k=1,2,\dots,NP$ is a {\em candidate solution} vector of variables $x_{ij}$ representing the amount of energy to be traded between each seller $i$ and buyer $j$ . These variables correspond to the decision variables of our optimization problem.
The $while-$loop (line $3-19$) is the differential evolution loop that aims at finding solution to the non-linear optimization problem with Eq. \eqref{obj_func_pt} as the fitness function. The loop is executed for $G_{max}$ iterations.
At each iteration, for each candidate solution $\mathbf{x}_k \in \mathcal{X}$, the algorithm creates a {\em mutated solution} $\mathbf{\bar{x}_k}$. Initially, $\mathbf{\bar{x}_k} = \mathbf{x_k}$.
The mutated solution is subsequently updated through mutation and crossover with $3$ random candidates $\mathbf{x}_a, \mathbf{x}_b, \mathbf{x}_c \in \mathcal{X}$ (line $5$). A value $R \in [1,|S|\times |B|]$ is selected at random. $R$ will be used in the following $for-$loop to ensure a minimum mutation. The for loop in line $7$ iterates over the components (dimensions in evolutionary terms) of $\mathbf{\bar{x}_k}$.
During each iteration, a value $u \in [0,1]$ is sampled at random as mutation probability (line $8$). Subsequently, a mutation occurs for the component $ij$ of $\mathbf{\bar{x}_k}$ with crossover probability $CR$ (line $9$). The mutation occurs irrespective of the probability if $(i \times j) = R$ (to ensure at least one minimum mutation). A mutation is executed by combining the corresponding component of $\mathbf{x_a}$, $\mathbf{x_b}$, and $\mathbf{x_c}$ with the differential weight parameter $F \in [0,2]$ as in line $10$.
The mutated component $\mathbf{\bar{x}_{ij}^{(k)}}$ is clipped to ensure that it falls within $[0,1]$ as minimum and maximum threshold to satisfy constraint Eq. \eqref{const6_pt} in line $11$ of the algorithm.
After the mutated solution is finalized, it is checked, and adjusted if needed, to meet the constraints in Eqs. \eqref{const2_pt}-\eqref{const4_pt} of the optimization problem. Specifically, line $13$ ensures that no exchange occurs (i.e., $\mathbf{\bar{x}_{ij}^{(k)}} = 0$) between users having a loss higher than $l_{max}$. Lines $14-15$ ensure that the production of a seller and the demand of each buyer are not exceeded, respectively.
Finally, in line $16$, the fitness function $f(.)$ of the mutated solution $\mathbf{\Bar{x}_k}$ is compared against the original candidate solution $\mathbf{x_k}$. If $f(\mathbf{\Bar{x}_k}) > f(\mathbf{{x}_k}) $, then $\mathbf{\Bar{x}_k}$ replaces $\mathbf{{x}_k}$ in the set of candidate solutions $\mathcal{X}$.
At the end of the while loop, $DEbATE$ selects the best solution $\mathbf{{x}^*}$ in $\mathcal{X}$ (line $20)$ and executes the transactions accordingly (line $21$).
In the following theorem \ref{Theo:complexity}, we show that the $DEbATE$ has polynomial complexity and hence, computationally efficient.
\begin{theorem}\label{Theo:complexity}
The complexity of the $DEbATE$ algorithm is $O(G_{max} \times NP \times |S||B|)$.
\end{theorem}
\begin{proof}
The complexity is dominated by the $while$ loop (lines $3-19$), which is executed $G_{max}$ times. Within this loop, the $for-$loop (lines $4-17$) does $|\mathcal{X}| = NP$ total iterations. In each iteration, the inner $for-$loop (lines $7-12$) iterates over the sets $S$ and $B$, and only contains constant operations. Similarly, checking the constraints (lines $13-15$) requires to iterate over the same sets. Finally, calculating the function $f(.)$ (line $16$) has cost $|B|$. Overall, the complexity is $O(G_{max} \times NP \times (|S||B| + 3|S||B| + |B|)) = O(G_{max} \times NP \times |S||B|)$
\end{proof}
\begin{algorithm}[!hpbt]
\SetAlgoLined
\footnotesize
\caption{PQR}\label{alg:PQRs}
\footnotesize
\tcc{\textbf{Pricing with Risk-sensitive Q-learning}}
Collect transaction information for each prosumers from $DEbATE$ (Alg. \ref{alg:DEBATE}) for current timestep $t$\;
\For{each $i \in S$}{
Select an action, $a \in \{+\delta,-\delta,0\}$ based on exploration and exploitation \;
$s=\rho_i;s_{new} = s+a; R_i = (\rho_i+a) \sum\limits_{j \in B} x_{ij}$\;
Update $Q(s,a)$ as in Eq. \eqref{eq:PT_Q_update}\
$\rho_i = s_{new}$\;
Send information on updated price $\rho_i$ to seller $i$\;
}
\end{algorithm}
\subsection{PQR}
After determining the solution to the energy allocation problem in $DEbATE$, the selling price for sellers is then updated through the $PQR$ algorithm. In order to learn the optimal selling price dynamically over time, we model the sellers as independent learning agents. Note that, to preserve the privacy and avoid the conflict between prosumers, these agents do not have access to information about other sellers or buyers. The state space in the Q-learning formulation consists of the prices between the grid buying ($\rho_{gb}$) and selling ($\rho_{gs}$), discretized by a step size, $\delta$, i.e., $\rho_i \in \{\rho_{gb},\ \rho_{gb}+\delta,\ \rho_{gb}+2\delta,\ ...,\ \rho_{gb} + \big(\frac{\rho_{gs}-\rho_{gb}}{\delta}-1\big)\delta,\ \rho_{gs}\}.$
The action space consists of a price increasing action, price decreasing action, and no change action, i.e. $a \in \{+\delta,-\delta,0\}$, where $\delta$ is the amount by which price is increased or decreased. Seller $i$ reward function is the total revenue generated at the current trading period i.e. $R_i = (\rho_{i}+a)\sum_{j \in B} x_{ij}w_j$. For updating Q-values, we modify the approach proposed in \cite{shen2014risk} by considering the following Q-learning update rule that includes the PT-based perceived utility of sellers.
\begin{equation} \label{eq:PT_Q_update}
Q^{(new)}(s,a) = Q^{(old)}(s,a) +\alpha v(y_i)
\end{equation}
\begin{equation} \label{eq:PT_val_sellers}
v(y_i)=
\begin{cases}
k_{+,i}(y_i)^{\zeta_{+,i}}, & y_i > 0\\
-k_{-,i}(-y_i)^{\zeta_{-,i}}, & y_i \leq 0
\end{cases}
\end{equation}
where, $y_i = R_i + \gamma \max_a Q(s_{new},a) - Q(s,a)$ is the Temporal Difference (TD) error of $i^{th}$ seller for current iteration, and $v(y_i)$ is transformation of TD error to capture each seller's personalized perceived utility on loss and gain. $\alpha$ refers to the learning rate for updating Q-values in Eq. \eqref{eq:PT_Q_update}.
The action is selected based on an \textit{$\epsilon$-greedy} exploration-exploitation strategy \cite{sutton2018reinforcement}. Specifically, $\epsilon$ refers to the probability of exploration and it is initially set to $1$. It is then decreased over time using an \textit{$\epsilon-$decay} value, as the system learns the optimal policy.
Based on the selected action, the new selling price, reward, and Q-value are updated as per Eqs. \eqref{eq:PT_Q_update} and \eqref{eq:PT_val_sellers}. Updated selling price is then sent to the respective seller $i$ for next trading period.
The system runs both $DEbATE$ and $PQR$ sequentially at every trading period. Input of $DEbATE$ is updated based on the prices computed by $PQR$. $PQR$ then takes as input the reward from executing energy transactions by $DEbATE$.
\section{Experimental Results}
\subsection{Experimental Setup}
\begin{figure}[htbp]
\centering
\includegraphics[width=0.5\linewidth]{images/de_obj_val_plot.png}
\caption{Normalized objective value vs. number of iterations.}
\label{fig:de_obj_val}
\end{figure}
\begin{figure*}[!th]
\minipage{0.33\linewidth}
\includegraphics[width=.99\linewidth]{images/cumulative_obj_val_15_15_plot.png}%
\caption{Buyers' perceived values.}\label{fig:obj_val_PT}
\endminipage\hfill
\minipage{0.33\linewidth}%
\includegraphics[width=.99\linewidth]{images/cumulative_reward_q_15_15_plot.png}
\caption{Sellers' cumulative reward.}\label{fig:cum_reward_PT}
\endminipage\hfill
\minipage{0.33\linewidth}%
\includegraphics[width=0.99\linewidth]{images/ind_price_5_5_plot.png}
\caption{Individual prices.}\label{fig:ind_price_PT}
\endminipage
\end{figure*}
In this section, we evaluate the performance of \textit{DEbATE} and \textit{PQR}, hereafter jointly referred as $DEbATE-PQR$, against a recent state-of-the-art approach referred to as \textit{Rule} \cite{azim2019feasibility}. $Rule$ allocates energy using a greedy heuristic that assigns cheapest sellers to buyers based on their registration order in the system, while final price of each transaction follows mid-market pricing, i.e., mid value of seller's and buyer's asking price. We consider a system with $40$ prosumers, split evenly as buyers and sellers. This is considered a representative number of prosumers in a microgrid or set of houses supplied by a single distribution transformer. We use a realistic dataset for buyers' energy consumption obtained from \cite{Energy_dataset}. Similarly, we consider sellers equipped with $4kW$ rooftop solar located in Lexington, Kentucky, USA.
The energy generated is estimated using NREL's PVWatts Calculator \cite{Solar_dataset} given the solar irradiance in Lexington and size of solar panels.
Losses are assigned uniformly at random from set $\{1\%, 2\%, 3\%, 4\%\}$ and maximum loss threshold $L_{max} = 2.5\%$.
We assume that prosumers complete a survey before joining the system to estimate their individual prospect theory parameters, similar to \cite{rieger2017estimating,balavz2013testing,fox2009prospect}, and use realistic prospect theory parameters determined by them.
Specifically, we sample the risk-averting parameter for gains $(\zeta_+) \in [0.60,0.88]$, the risk-seeking parameter for losses $(\zeta_-) \in [0.52, 1.0]$, the loss-aversion parameters for gain and loss $(k_+),(k_-) \in [2.10,2.61]$ for each individual prosumers. The grid energy buying price is set to $\rho_{gb} = \$ 0.06$ and the selling price to $\rho_{gs} = \$ 0.12$. The reference price for each sellers is initially randomly sampled from range $[0.09, 0.12]$. It is then updated using $PQR$ at each iteration. The reference price for each buyer is selected in the range $[0.06,0.10]$ and considered static for the duration of experiments, which is $365$ days. The parameters for $PQR$ algorithm are set as follows: learning rate $\alpha = 10^{-4}$, step size for discretizing state space $\delta =\$ 0.001$, and $\epsilon-$decay $=0.965$.
\subsection{Results}
We consider several experimental scenarios and performance metrics, as discussed in the following.
\textbf{Experimental Scenario 1:}
We first run experiments to study the convergence of \textit{DEbATE}. We considered different system size by scaling the number of sellers and buyers. Fig. \ref{fig:de_obj_val} shows the normalized objective value as a function of the number of iterations using a population size $NP=20$. The plot averaged over 10 runs shows that $10,000$ iterations are sufficient for the algorithm to converge in the considered settings. As a result, in the following scenarios we set $G_{max} = 10,000$ and the population size $NP=20$.
\textbf{Experimental Scenario 2:} In the second experimental scenario we study the performance of the considered approaches over time. Two performance metrics are considered, namely the buyers' objective value and the sellers' cumulative reward. These are represented in Figs. \ref{fig:obj_val_PT} and \ref{fig:cum_reward_PT}, respectively, with a moving average of $10$ days. In this experiments we consider $15$ buyers and $15$ sellers. The benefits of $DEbATE-PQR$ over $Rule$ are more prominent from April through October, when the energy demand and production is higher. The greedy nature of $Rule$ penalizes the quality of the resulting matching, significantly reducing the buyers' perceived value. Note that, the buyers' objective values are negative because they are paying higher prices than their reference purchase price. Therefore, transactions are seen as loss from a prospect theory perspective. Nevertheless, our approach optimizes the energy assignment to maximize the buyers perceived value. Additionally, our approach is able to generate higher rewards than $Rule$ by dynamically learning the prices for sellers through the $PQR$ algorithm. The the sellers' reward decreases after mid-september for both the approaches due to the reduced energy production during winter.
\begin{figure*}[tbhp]
\minipage{0.5\linewidth}
\includegraphics[width=.99\linewidth]{images/obj_vals_vs_size.png}%
\caption{Obj. values for buyer vs. network size.}\label{fig:obj_size_PT}
\endminipage\hfill
\minipage{0.5\linewidth}%
\includegraphics[width=.99\linewidth]{images/reward_vs_size.png}
\caption{Total rewards for sellers vs. network size.}\label{fig:tot_reward_PT}
\endminipage
\end{figure*}
We further study the performance over time by considering the evolution of average and individual sellers' prices. We consider a smaller system of $5$ sellers and $5$ buyers for ease of representation of the results.
Fig. \ref{fig:ind_price_PT} shows the individual prices. $DEbATE-PQR$ is able to learn and adjust the price over time to improve the buyers' perceived value considering their competitiveness. The competitiveness is a function of a buyer's reference price, their production, and their location in the system (e.g., loss w.r.t. sellers). As a result, our approach is able to improve the perception of both buyers and sellers while ensuring the competitiveness of the market.
\textbf{Experimental Scenario 3:} In this scenario we test the scalability with respect to the system size. Specifically, we increase the system proportionately from $5$ sellers and $5$ buyers to $20$ sellers and $20$ buyers.
Figs. \ref{fig:obj_size_PT}-\ref{fig:tot_reward_PT} show the buyers' total perceived value and the sellers' reward, respectively, over a year.
By considering the loss-averse and risk-seeking PT-value functions, $DEbATE-PQR$ achieves an increasing advantage as the system size increases compared to $Rule$, for both sellers and buyers. As a numerical example, $DEbATE-PQR$ achieves as much as $26\%$ increase in buyers' perceived value while ensuring $7\%$ profit improvement for sellers.
\section{Concluding Remarks}
In this paper,
we bring together the concept of perceived utility from behavioral economics and reinforcement learning into the P2P energy trading scene. Unlike existing literature,
we propose an automated and dynamic P2P energy trading problem that maximizes the perceived value for buyers while simultaneously learning the optimal selling price. Given the non-linear and non-convex nature of the problem, we propose a novel differential evolution-based metaheuristic algorithm, called $DEbATE$. $DEbATE$ is paired with a prospect theory enhanced Q-learning algorithm, called $PQR$, to adjust the selling price over time. Results show the advantages of the proposed approaches with respect to a state of the art solution using real energy consumption and production data.
\section*{Acknowledgment}
This work is
supported by the NSF grant EPCN-1936131 and NSF CAREER grant CPS-1943035.
\appendices
\bibliographystyle{IEEEtran}
|
1,314,259,993,599 | arxiv | \section{Introduction}
The aerosolization of biomatter caused by flushing toilets has long been known to be a potential source of transmission of infectious microorganisms~\cite{Darlow1959,Gerba1975}. Toilet flushing can generate large quantities of microbe-containing aerosols~\cite{Johnson2013} depending on the design and water pressure or flushing energy of the toilet~\cite{Bound1966,Johnson2013Aerosol,Lai2018}. A variety of different pathogens which are found in stagnant water or in waste products (e.g., urine, feces, and vomit) can get dispersed widely via such aerosolization, including the legionella bacterium responsible for causing Legionnaire's disease~\cite{Hamilton2018,Couturier2020}, the Ebola virus~\cite{Lin2017}, the norovirus which causes severe gastroenteritis (food poisoning)~\cite{Caul1994,Marks2000}, and the Middle East Respiratory Syndrome coronavirus (MERS-CoV)~\cite{Zhou2017}. Such airborne dispersion is suspected to have played a key role in the outbreak of viral gastroenteritis aboard a cruise ship, where infection was twice as prevalent among passengers who used shared toilets compared to those who had private bathrooms~\cite{Ho1989}. Similarly, transmission of norovirus via aerosolized droplets was linked to the occurrence of vomiting or diarrhea within an aircraft restroom~\cite{Widdowson2005}, as passengers and crew who got infected were more likely to have visited restrooms than those that were not infected. The participants in the study reported that all of the restroom surfaces appeared to be clean, which indicates that infection is likely to have occurred via bioaerosols suspended within the restroom.
In more controlled studies investigating toilet-generated aerosols, Barker \& Bloomfield~\cite{Barker2000} isolated salmonella bacteria from air samples collected after flushing. Bacteria and viruses could be isolated from settle plates for up to an hour to 90 minutes after flushing~\cite{Barker2005,Best2012}, suggesting that the microorganisms were present in aerosolized droplets and droplet nuclei. An experimental study in a hospital-based setting measured bioaerosol generation when fecal matter was flushed by patients~\cite{Knowlton2018}. A significant increase in bioaerosols was observed right after flushing, and the droplets remained detectable for up to 30 minutes afterwards. Notably, flushing does not remove all of the microorganisms which may be present in the bowl. In various studies where the toilet bowl was seeded with microorganisms, sequential flushes led to a drop in microbe count, however, some residual microbes remained in the bowl even after up to 24 flushes~\cite{Gerba1975,Barker2000,Barker2005,Johnson2017,Aithinne2019}. In some cases, residual microbial contamination was shown to persist in biofilm formed within the toilet bowl for several days to weeks~\cite{Barker2000}.
In an effort to reduce aerosol dispersal, certain studies conducted measurements with the toilet seat lid closed~\cite{Barker2005,Best2012}. Closing the lid led to a decrease, but not a complete absence of bacteria recovered from air samples. This suggests that smaller aerosolized droplets were able to escape through the gap between the seat and the lid. In addition to the experiment-based studies mentioned here, numerical simulations have been used recently to investigate the ejection of aerosolized particles from toilets and urinals, specifically in the context of COVID-19 transmission~\cite{Li2020,Wang2020}.
The issue of aerosolization is particularly acute for viruses compared to bacteria, given their different response to levels of relative-humidity (RH). High RH levels result in slower evaporation of aerosolized droplets, whereas lower levels accelerate the phenomenon, leading to the formation of extremely small droplet nuclei which can remain airborne for long periods of time\blue{ and can deposit deep into the lungs~\cite{Mallik2020,Wang2020Motion}}. Various studies have indicated that the viability of bacteria decreases at low RH levels~\cite{Won1966,Lin2020}, which makes them less likely to retain their infectivity in droplet nuclei form. On the other hand, viruses exhibit lowest viability at intermediate RH levels, and retain their viability at either low or high RH values~\cite{Songer1967,Benbough1971,Schaffer1976,Donaldson1976,Lin2020}, making them more likely to remain intact in droplet nuclei which can stay suspended from hours to days. Viruses are also more likely to aerosolize easily, as indicated by Lee at al.~\cite{Lee2016} who used wastewater sludge (both synthetic and real) to demonstrate that when viruses were seeded into the sludge, $94\%$ stayed mobile in the liquid phase while only a small fraction adhered to the solid biomatter or to the surfaces of the toilet. This suggests that the presence of solid biomatter, which is more difficult to aerosolize, might not reduce the potential for virus transmission since they are more likely to get aerosolized with the liquid phase.
Apart from gastrointestinal diseases, viruses associated with respiratory illnesses have also been detected in patients' stool and urine samples. For instance, the SARS-CoV (Severe Acute Respiratory Syndrome Coronavirus) responsible for the SARS outbreak of 2003 was found in patients' urine and stool specimens for longer than 4 weeks~\cite{Xu2005}. Similarly, recent studies have confirmed the presence of SARS-CoV-2 (the virus associated with COVID-19) viral RNA in patients' stool samples~\cite{Xiao2020,Wu2020Lancet,Chen2020,Xiao2020,Zhang2020Emerging,Foladori2020,Gupta2020}, even if they did not experience gastrointestinal symptoms and regardless of the severity of their respiratory symptoms~\cite{Wu2020Lancet,Chen2020,Zhang2020,Ling2020}. Surprisingly, viral RNA could be detected in feces for several days to weeks after it was no longer detectable in respiratory samples from nasal and oral swabs~\cite{Xiao2020,Wu2020Lancet,Chen2020,Gupta2020}. Moreover, Wu et al.~\cite{Wu2020} recovered large quantities of viral RNA from urban wastewater treatment facilities. The levels detected were several orders of magnitude higher than would be expected for the number of clinically confirmed cases in the region, which suggests that there was a high prevalence of asymptomatic and undetected cases.
Although enveloped viruses like SARS-CoV-2 are susceptible to the acids and bile salts found in digestive juices, it has been shown that they can survive when engulfed within mucus produced by the digestive system. Hirose et al.~\cite{Hirose2017} demonstrated that influenza viruses could be protected from degradation by simulated digestive juices using both artificial and natural mucus. This might help explain why recent studies have been able to isolate viable SARS-CoV-2 virus particles (i.e., those able to infect new cells) that remained intact when passing through the digestive and urinary systems, albeit in smaller quantities compared to respiratory fluids~\cite{Jones2020}. Wang et al.~\cite{Wang2020JAMA} detected live virus in feces from patients who did not have diarrhea, and Xiao et al.~\cite{Xiao2020Viable} demonstrated the infectivity of intact virions isolated from a patient's stool samples. In urine specimens, SARS-CoV-2 RNA is found less frequently than in fecal and respiratory samples~\cite{Peng2020,Ling2020,Xiao2020}. However, Sun et al.~\cite{Sun2020} managed to isolate the virus from a severely infected patient's urine, and showed that these virions were capable of infecting new susceptible cells. As with fecal samples, viral RNA has been found in urine even after the virus is no longer detectable in respiratory swabs~\cite{Ling2020}.
These findings suggest that the aerosolization of biomatter could play a potential role in the transmission of SARS-CoV-2, which is known to remain viable in aerosol form~\cite{vanDoremalen2020,Fears2020}. Environmental samples taken by Ding et al.~\cite{Ding2020} in a hospital designated specifically for COVID-19 patients indicated high prevalence of the virus within bathrooms used by the patients, both on surfaces and in air samples. The authors hypothesized that aerosolized fecal matter may have dispersed the virus within the bathroom, since viral samples were not detected on surfaces in the patients' rooms.
Given the potential role of aerosolized biomatter in spreading a wide variety of gastrointestinal and respiratory illnesses, we investigate droplet generation from toilets and urinals in a public restroom operating under normal ventilation condition. We examine the size, number, and various heights to which the droplets rise when generated by the flushing water. The main aim is to better understand the risk of infection transmission that the droplets pose in public restrooms, since these relatively confined locations often experience heavy foot traffic. The experimental methodology is described in Section~\ref{sec:methods} followed by results and discussion in Section~\ref{sec:results} and conclusion in Section~\ref{sec:conclusion}.
\section{Methods}
\label{sec:methods}
The flush-generated aerosol measurements were recorded in a medium sized restroom on the university campus, consisting of 3 bathroom cubicles, 6 urinals, and 3 sinks. The restroom was deep cleaned and closed twenty four hours prior to conducting the experiments, with the ventilation system operating normally to remove any aerosols generated during cleaning. \blue{The temperature and relative humidity within the restroom were measured to be $21\degree C$ and $52\%$, respectively.} For the measurements reported here, one particular toilet and one urinal were selected, both equipped with flushometer type flushing systems. The urinal used 3.8 liters of water per flush whereas the toilet used 4.8 liters per flush.
The size and concentration of aerosols generated by flushing were measured using a handheld particle counter (9306-V2 - TSI Incorporated). \blue{The sensor's size resolution is less than $15\%$, which is indicative of the uncertainty in the measured particle diameter. More specifically, the resolution is specified as the ratio of standard deviation to the mean size of the particles being sampled. The counting efficiency of the sensor is $50\%$ at $0.3\mu m$ and $100\%$ for particles larger than $0.45\mu m$. These values denote the ratio of particle numbers measured by the counter to those measured using a reference instrument. Handheld counters with comparable specifications have been used for estimating the likelihood of aerosol transmission in typical public spaces~\cite{Somsen2020}.}
The \blue{particle} counter was positioned at various heights close to the toilet and the urinal as shown in Figure~\ref{fig:Schematics}.
\begin{figure*}[ht!]
\centering
\begin{subfigure}{0.38\linewidth}
\centering
\includegraphics[width=\linewidth]{toiletSchematic.pdf}
\caption{\label{fig:toiletSchemat}}
\end{subfigure}
\begin{subfigure}{0.379\linewidth}
\centering
\includegraphics[width=\linewidth]{urinalSchematic.pdf}
\caption{\label{fig:urinalSchemat}}
\end{subfigure}
\caption{\label{fig:Schematics} Measurement locations where the aerosol sensor was placed for (\subref{fig:toiletSchemat}) the toilet and (\subref{fig:urinalSchemat}) the urinal. Measurements for the toilet were taken at heights of $0.43m$ from the ground ($1ft \ 5in$), $1.22m$ ($4ft$), and $1.52m$ ($5ft$), whereas those for the urinal were taken at $0.53m$ ($1ft \ 9in$), $0.97m$ ($3ft \ 2in$), and $1.22m$ ($4ft$).}
\end{figure*}
Measurements for the toilet were taken at 3 different heights, at approximately $0.43m$ from the ground ($1ft \ 5in$), $1.22m$ ($4ft$), and $1.52m$ ($5ft$), with the toilet seat raised up. The lowest level corresponds to the distance between the ground and the toilet seat, and represents the scenario where the particle counter was placed level with the seat. Measurements for the urinal were taken at \blue{3 different heights, at approximately $0.53m$ from the ground ($1ft \ 9in$), $0.97m$ ($3ft \ 2in$), and $1.22m$ ($4ft$)}. \blue{The particle counter's intake probe was oriented parallel to the floor and perpendicular to the back wall, with the inlet pointing in the direction of the flushing water. The probe was centered laterally for both toilet and urinal measurements. The placement and orientation were selected to be representative of a person breathing in when flushing the toilet/urinal after use, since different choices were observed have a notable impact on the measured droplet count. The probe inlet was positioned $5cm$ inside the rim of the toilet, and it was placed $5cm$ outside the edge of the urinal, as depicted in Figure~\ref{fig:Schematics}.} In addition to measurements taken during normal operation of the toilet, aerosol measurements were recorded after a large flat plate was placed over the toilet opening, to assess the impact of flushing with the lid closed. The use of a separate cover was necessary since public restrooms in the United States often do not come equipped with toilet seat lids.
The particle counter drew air samples at a volume flow rate of 2.83 liters per minute \blue{(0.1 Cubic Feet per Minute - CFM)}, and measured aerosol concentrations in six different size ranges, namely, (0.3 to 0.5)$\mu m$, (0.5 to 1.0)$\mu m$, (1.0 to 3.0)$\mu m$, (3.0 to 5.0)$\mu m$, (5.0 to 10.0)$\mu m$, and (10.0 to 25.0)$\mu m$. For the tests reported here, air samples were recorded at a sampling frequency of $1Hz$ for a total of 300 seconds at each of the levels depicted in Figure~\ref{fig:Schematics}. \blue{We note that although it is feasible to compute droplet concentration at a given measurement location, it is difficult to determine overall characteristic droplet production rates for the toilet or urinal, since the measured values depend on both the location and orientation of the probe.} During the 300-second sampling, the toilet and urinal were flushed manually 5 different times at the 30, 90, 150, 210, and 270 second mark, with the flushing handle held down for five consecutive seconds. The data obtained from the three different scenarios, i.e., toilet flushing, covered toilet flushing, and urinal flushing, were analyzed to determine the increase in aerosol concentration. The behavior of droplets of different sizes, the heights that they rose to, and the impact of covering the toilet are discussed in detail in Section~\ref{sec:results}.
\section{Results and Discussion}
\label{sec:results}
The measurements from the particle counter were analyzed to determine the extent of aerosolization, and the various heights to which the droplets rise after flushing. Figure~\ref{fig:ToiletTime} shows the time-variation of the total number of particles recorded by the sensor from measurements for the uncovered toilet.
\begin{figure*}[ht]
\centering
\begin{subfigure}{0.49\linewidth}
\centering
\includegraphics[width=\linewidth]{toilet_timeSeries_plot1.pdf}
\caption{\label{fig:T_pt3topt5andpt5to1}}
\end{subfigure}
\begin{subfigure}{0.49\linewidth}
\centering
\includegraphics[width=\linewidth]{toilet_timeSeries_plot2.pdf}
\caption{\label{fig:T_1to3and3to5}}
\end{subfigure}
\begin{subfigure}{0.49\linewidth}
\centering
\includegraphics[width=\linewidth]{toilet_timeSeries_plot3.pdf}
\caption{\label{fig:T_5to10and10to25}}
\end{subfigure}
\caption{\label{fig:ToiletTime}Particle-count from the toilet-flushing test, measured at a height of $0.43m$ ($1ft 5in$). The time series plots are shown for particles in various size ranges: (\subref{fig:T_pt3topt5andpt5to1}) (0.3 to 0.5)$\mu m$ - black, and (0.5 to 1)$\mu m$ - blue; (\subref{fig:T_1to3and3to5}) (1 to 3)$\mu m$ - black, and (3 to 5)$\mu m$ - blue; (\subref{fig:T_5to10and10to25}) (5 to 10)$\mu m$ - black, and (10 to 25)$\mu m$ - blue. The black curves in (\subref{fig:T_pt3topt5andpt5to1}) and (\subref{fig:T_1to3and3to5}) correspond to the left vertical axes, whereas the blue curves correspond to the right vertical axes. The dashed gray lines indicate the instances when the flushing handle was depressed and held down for 5 seconds.}
\end{figure*}
The data plotted has been smoothed using a moving average window of size 4 to reduce noise levels. Figure~\ref{fig:T_pt3topt5andpt5to1} depicts the time series for particles of size (0.3 to 0.5)$\mu m$ and (0.5 to 1)$\mu m$, whereas the size groups (1 to 3)$\mu m$ and (3 to 5)$\mu m$ are shown in Figure~\ref{fig:T_1to3and3to5}, and size groups (5 to 10)$\mu m$ and (10 to 25)$\mu m$ are shown in Figure~\ref{fig:T_5to10and10to25}. We observe a noticeable increase in particle count for all of the size ranges a few seconds after flushing. This indicates that flushing the toilet generates droplets \blue{in significant numbers}, which can be detected at seat-level for up to 30 seconds after initiating the flush.
In Figure~\ref{fig:T_pt3topt5andpt5to1} we observe a large variation in the measured levels of the smallest particles, i.e., those smaller than $1 \mu m$. These particles are highly susceptible to flow disturbances in the ambient environment due to their low mass, which may account for the high variability. The time series for particles larger than 1$\mu m$ (Figures~\ref{fig:T_1to3and3to5} and~\ref{fig:T_5to10and10to25}) exhibit distinctive surges in particle count after each flushing event. Importantly, the total number of droplets generated in the smaller size ranges is considerably larger than that generated in the larger ranges, even though the surges appear to be less prominent for the smaller droplets. We note that for the smallest aerosols (i.e., those smaller than 1$\mu m$), ambient levels in the restroom were relatively high prior to starting the experiment ($\sim O(3000)$). Thus, in these size ranges the flush-generated droplets comprise a small fraction of the total particle count. On the other hand, ambient levels for particle sizes larger than 1$\mu m$ were negligible in the restroom ($\sim O(1)$ to $O(10)$), resulting in the distinctive surges observed after flushing.
Similar plots depicting the time-variation of droplet counts for the covered toilet test and the urinal-flushing test are shown in Figure~\ref{fig:CovToiletTime} and Figure~\ref{fig:UrinalTime}, respectively.
\begin{figure*}[ht!]
\centering
\begin{subfigure}{0.49\linewidth}
\centering
\includegraphics[width=\linewidth]{covToilet_timeSeries_plot1.pdf}
\caption{\label{fig:TC_pt3topt5andpt5to1}}
\end{subfigure}
\begin{subfigure}{0.49\linewidth}
\centering
\includegraphics[width=\linewidth]{covToilet_timeSeries_plot2.pdf}
\caption{\label{fig:TC_1to3and3to5}}
\end{subfigure}
\begin{subfigure}{0.49\linewidth}
\centering
\includegraphics[width=\linewidth]{covToilet_timeSeries_plot3.pdf}
\caption{\label{fig:TC_5to10and10to25}}
\end{subfigure}
\caption{\label{fig:CovToiletTime}Particle-count from the flushing test when the toilet was covered using a large flat plate. Measurements taken at a height of $0.43m$ ($1ft 5in$). The time series plots are shown for particles in various size ranges: (\subref{fig:TC_pt3topt5andpt5to1}) (0.3 to 0.5)$\mu m$ - black, and (0.5 to 1)$\mu m$ - blue; (\subref{fig:TC_1to3and3to5}) (1 to 3)$\mu m$ - black, and (3 to 5)$\mu m$ - blue; (\subref{fig:TC_5to10and10to25}) (5 to 10)$\mu m$ - black, and (10 to 25)$\mu m$ - blue. The black curves in (\subref{fig:TC_pt3topt5andpt5to1}) and (\subref{fig:TC_1to3and3to5}) correspond to the left vertical axes, whereas the blue curves correspond to the right vertical axes. The dashed gray lines indicate the instances when the flushing handle was depressed and held down for 5 seconds.}
\end{figure*}
\begin{figure*}[ht]
\centering
\begin{subfigure}{0.49\linewidth}
\centering
\includegraphics[width=\linewidth]{urinal_timeSeries_plot1.pdf}
\caption{\label{fig:U_pt3topt5andpt5to1}}
\end{subfigure}
\begin{subfigure}{0.48\linewidth}
\centering
\includegraphics[width=\linewidth]{urinal_timeSeries_plot2.pdf}
\caption{\label{fig:U_1to3and3to5}}
\end{subfigure}
\begin{subfigure}{0.48\linewidth}
\centering
\includegraphics[width=\linewidth]{urinal_timeSeries_plot3.pdf}
\caption{\label{fig:U_5to10and10to25}}
\end{subfigure}
\caption{\label{fig:UrinalTime}Particle-count from the urinal-flushing test, measured at a height of $0.53m$ ($1ft 9in$). The time series plots are shown for particles in various size ranges: (\subref{fig:U_pt3topt5andpt5to1}) (0.3 to 0.5)$\mu m$ - black, and (0.5 to 1)$\mu m$ - blue; (\subref{fig:U_1to3and3to5}) (1 to 3)$\mu m$ - black, and (3 to 5)$\mu m$ - blue; (\subref{fig:U_5to10and10to25}) (5 to 10)$\mu m$ - black, and (10 to 25)$\mu m$ - blue. The black curves in (\subref{fig:U_pt3topt5andpt5to1}) and (\subref{fig:U_1to3and3to5}) correspond to the left vertical axes, whereas the blue curves correspond to the right vertical axes. The dashed gray lines indicate the instances when the flush was activated using the proximity sensor.}
\end{figure*}
For the covered toilet, the plots display a large variation in the number of the smallest droplets in Figure~\ref{fig:TC_pt3topt5andpt5to1}, and comparatively small surges relative to ambient levels due to the background count being high. Importantly, the observed peak values of the surges are lower for the covered toilet compared to the uncovered tests. This is evident in Figure~\ref{fig:TC_1to3and3to5}, where the peak values are approximately 35 droplets on average for the (1 to 3)$\mu m$ range, and 3 droplets for the (3 to 5)$\mu m$ range. The same numbers for the uncovered toilet are approximately 50 droplets and 5 droplets, respectively, in Figure~\ref{fig:T_1to3and3to5}. Notably, there is a significant reduction in the number of droplets larger than 5$\mu m$ for the covered toilet (Figure~\ref{fig:TC_5to10and10to25}) compared to the uncovered toilet (Figure~\ref{fig:T_5to10and10to25}). This indicates that the covering helps to reduce the dispersion of flush-generated droplets, especially those larger than 5$\mu m$, but it does not completely contain the escape of droplets smaller than 5$\mu m$.
The data from the urinal-flushing tests in Figure~\ref{fig:UrinalTime} indicate a large number of droplets generated in all size ranges observed; the post-flush surges are much more pronounced than those for the toilet-flushing tests, even for droplets smaller than 1$\mu m$ (Figure~\ref{fig:U_pt3topt5andpt5to1}). This may be related to the closer proximity of the sensor to the water drain in the urinal, compared to the toilet-flushing tests for which the sensor was placed at the outer edge of the toilet bowl. \blue{We observe that there is no consistent increasing or decreasing trend in either the peaks or the baseline levels with subsequent flushes in the time series plots. The same holds true for data from the toilet-flushing tests in Figures~\ref{fig:ToiletTime} and \ref{fig:CovToiletTime}. Thus, any short term changes in temperature and RH at the measurement location due to flushing do not have a noticeable impact on the droplet count.} \blue{Furthermore, while the smallest droplets will remain suspended for longer than $300s$, the time series plots indicate that droplet counts at the sensor location return to ambient levels within approximately half a minute. Nonetheless, as these droplets move past the particle counter they become part of the ambient environment, leading to a measurable increase in background levels as demonstrated later in this section.}
To compare the increase in droplet concentration for the three different scenarios at various measurement heights, the time series data were examined manually to identify the time delay between flush initiation and the observed rise in particle count, as well as the total time span for which the particle counts remained elevated. The corresponding values are provided in Table~\ref{tab:startAndSpan}.
\begin{table}[htp]
\caption{\label{tab:startAndSpan} Average time delay between flush initiation and the observed rise in particle count. The last column indicates the average time taken for the particle count to return to ambient levels.}
\blue{
\begin{ruledtabular}
\begin{tabular}{llcc}
&Height &Time Delay [s] &Time Span [s] \\ \hline
\multirow{3}{*}{Toilet}
&$0.43m$ ($1ft 5in$) & 10 & 20 \\
&$1.22m$ ($4ft$) & 10 & 20 \\
&$1.52m$ ($5ft$) & 10 & 20 \\
\hline
\multirow{2}{*}{Covered Toilet}
&$0.43m$ ($1ft 5in$) & 0 & 20 \\
&$1.22m$ ($4ft$) & 5 & 20 \\
\hline
\multirow{2}{*}{Urinal}
&$0.53m$ ($1ft 9in$) & 0 & 15 \\
&$0.97m$ ($3ft 2in$) & 5 & 15 \\
&$1.22m$ ($4ft$) & 6 & 20
\end{tabular}
\end{ruledtabular}
}
\end{table}
We note that the time delay between flush initiation and the measured surge for the uncovered toilet at seat-level was 10 seconds, whereas that for the covered toilet was 0 seconds. \blue{Furthermore, the delay was smaller for the covered toilet at a height of $1.22m$ ($5s$ versus $10s$), suggesting that the aerosols were forced through gaps in between the seat and the plate for the covered toilet}. In both cases, the droplet counts remained elevated for a further 20 seconds after first detection of the surge. \blue{For the covered toilet and the urinal, we observe a consistent increase in time delay with increasing height, which also corresponds to increasing distance from the flushing water, but the observed delay remained nearly constant for the uncovered toilet at $10s$}. \blue{We remark that the time delay and detection duration are expected to be influenced strongly} by the placement of the sensor, the \blue{fixture} geometry, the flushing mechanism, as well as the water volume \blue{and pressure}.
The number of droplets produced during the flushes were determined by numerically integrating the sections comprising the `surge' segments in the unfiltered time series. More specifically, within each 1-minute window associated with a particular flush, the start of the surge was identified using the time-delay values specified in Table~\ref{tab:startAndSpan}. Starting at this time the area under the particle-count curve was computed numerically up until the end of the surge, the corresponding time span for which is also specified in Table~\ref{tab:startAndSpan}. \blue{The average surge count was determined by dividing this area by the corresponding time span.} The area under the remaining parts of the curve, i.e., the segments lying outside the surge but within the 1-minute time window, was determined similarly to obtain the average ambient droplet count. This ambient count was subtracted from the surge count to yield the average number of flush-generated droplets measured per second by the particle counter. \blue{The resulting values from the 4 different full-minute flush measurements were averaged to obtain the increase in droplet count per second, and the standard deviation was computed to determine the uncertainty.} The resulting data for the flushing toilet is depicted graphically in Figure~\ref{fig:UnToiletDelta}, and the corresponding numerical values are provided in Table~\ref{tab:ToiletTable}.
\begin{figure}[h!]
\centering
\includegraphics[width=\linewidth]{bar_toilet.pdf}
\caption{\label{fig:UnToiletDelta}Average increase in the number of droplets measured per second after flushing the toilet. \blue{The error bars indicate the standard deviation of the measured increase from multiple flushes.} Each bar cluster corresponds to particles in a given size range, and indicates how the droplet count varies with measurement height. The corresponding values are provided in Table~\ref{tab:ToiletTable}.}
\end{figure}
\begin{table}[htp]
\caption{\label{tab:ToiletTable}Numerical values for the average increase in droplet count per second from the toilet-flushing tests\blueTwo{, with the standard deviation provided in parentheses}. The data corresponds to the bar graphs shown in Figure~\ref{fig:UnToiletDelta}.}
\blue{
\begin{ruledtabular}
\begin{tabular}{lccc}
Height & (0.3 to 0.5)$\mu m$ &(0.5 to 1)$\mu m$ &(1 to 3)$\mu m$ \\ \hline
$0.43m$ & 186 ($\pm 25$) & 51 ($\pm 20$) & 17 ($\pm 3$) \\
$1.22m$ & 27 ($\pm 24$) & 14 ($\pm 7$) & 7 ($\pm 2$) \\
$1.52m$ & 29 ($\pm 5$) & 13 ($\pm 5$) & 5 ($\pm 2$)
\end{tabular}
\end{ruledtabular}
}
\end{table}
We note that droplets larger than 3$\mu m$ were excluded from this analysis since very few droplets in these size ranges were detected at the higher locations, which made it difficult to distinguish between the measured values and \blue{background} noise.
The bar graphs in Figure~\ref{fig:UnToiletDelta} indicate that a significant number of droplets smaller than $0.5 \mu m$ were generated by the flushing toilet. If these droplets contain infectious microorganisms from aerosolized biomatter, they can pose a significant transmission risk since they remain suspended for long periods of time. For instance, in a poorly ventilated location where gravitational settling is the only means of removing suspended particles, the Stokes settling time for a spherical water droplet of size $0.5 \mu m$ from a height of $1.52m$ ($5ft$) would be approximately 56 hours, or more than 2 days. Apart from the smallest aerosols, comparatively larger aerosols also pose a risk in poorly ventilated areas even though they experience stronger gravitational settling. They often undergo rapid evaporation in the ambient environment and the resulting decreases in size and mass, or the eventual formation of droplet nuclei, can allow microbes to remain suspended for several hours\blue{~\cite{Wells1934,Duguid1946,Basu2020}}.
\blue{In Figure~\ref{fig:UnToiletDelta}, we observe a large variation for aerosols in the size range (0.3 to 0.5)$\mu m$. This may be attributed to the small droplets' high sensitivity to ambient flow fluctuations, as well as to the sensor's limited counting efficiency in this range. Notably, droplets smaller than $3\mu m$ are detectable in significant numbers even at a height of $1.52m$ ($5ft$). We observe a consistent decline in droplet count with increasing height; there is a significant drop in droplet count going from seat-level to $1.22m$, and a very small decrease with a further move up to $1.52m$ for droplets larger than $0.5\mu m$. The smallest aerosols exhibit some variation in the trend, which is likely due to the sensor limitations mentioned above. The observed decrease in droplet count with increasing measurement height} is expected, since the {droplet concentration is highest when the probe is placed closer to the flushing water, and it decreases at farther locations} due to dispersal of the droplets over a wider area. We remark that gravitational forces \blue{are not expected to} play a dominant role in the observed behavior, given the extremely small mass of the aerosols \blue{being considered here.} \blueTwo{Rather, it is aerodynamic drag that dominates.} {The Stokes settling speed for the largest aerosol being considered, i.e., a $3\mu m$ droplet, is approximately $0.00027m/s$. This amounts to a settling time of $1589s$ from a height of $0.43m$, and even longer for the smaller droplets. Thus, the effects of gravitational forces are not dominant at the time scales being considered ($\sim O(10s)$).} \blueTwo{Finally, the monotonic decrease in particle count with increasing particle size is similar to the trend observed by Johnson et al.~\cite{Johnson2013Aerosol} for various toilet designs and flushing mechanisms.}
\blue{The data collected after flushing the covered toilet and the urinal were also processed in a similar manner to determine the corresponding increases in droplet count per second. The results for the covered toilet} are presented in Figure~\ref{fig:CovToiletDelta} and Table~\ref{tab:CovToiletTable}, whereas those from \blue{flushing the urinal} are presented in Figure~\ref{fig:UrinalDelta} and Table~\ref{tab:UrinalTable}. Results from measurements at $1.52m$ height were not included in the analysis for the covered toilet, since \blue{it was difficult to discern droplet counts from background noise due to the extremely low measured values}. This indicates that the covering plate prevented the aerosols from rising upward \blue{and instead deflected them to lower levels, also resulting in shorter time delays compared to the uncovered toilet (Table~\ref{tab:startAndSpan})}. Over the long term however, these aerosols could rise up with updrafts created by the ventilation system or by \blue{the movement of people in the restroom}.
We observe a large number of aerosolized droplets smaller than 1$\mu m$ in Figure~\ref{fig:CovToiletDelta}, and an appreciable number of droplets in the (1 to 3)$\mu m$ range. \blue{This suggests that while the covering is able to suppress the dispersion of droplets to some extent, it does not eliminate them completely. Thus, although a toilet lid may appear to be a straightforward solution for reducing aerosol dispersal, other alternatives may need to be evaluated when designing public restrooms, such as modifying the fixture design, water pressure, vent placement, airflow rate, or even employing a liquid `curtain' incorporated into the fixture~\cite{Wu2020Curtain}.}
\begin{figure}[h!]
\centering
\includegraphics[width=\linewidth]{bar_coveredToilet.pdf}
\caption{\label{fig:CovToiletDelta}Average increase in number of droplets measured per second from flushing the covered toilet. \blue{The error bars indicate the standard deviation of the measured increase from multiple flushes.} Each bar cluster corresponds to particles in a given size range, and indicates how the droplet count varies with measurement height. The corresponding values are provided in Table~\ref{tab:CovToiletTable}.}
\end{figure}
\begin{table}[htp]
\caption{\label{tab:CovToiletTable}Numerical values for the average increase in droplet count per second from the covered toilet-flushing tests\blueTwo{, with the standard deviation provided in parentheses}. The data corresponds to the bar graphs shown in Figure~\ref{fig:CovToiletDelta}.}
\blue{
\begin{ruledtabular}
\begin{tabular}{lccc}
Height & (0.3 to 0.5)$\mu m$ &(0.5 to 1)$\mu m$ &(1 to 3)$\mu m$ \\ \hline
$0.43m$ & 147 ($\pm 47$) & 35 ($\pm 9$) & 9 ($\pm 2$) \\
$1.22m$ & 80 ($\pm 36$) & 27 ($\pm 7$) & 7 ($\pm 3$)
\end{tabular}
\end{ruledtabular}
}
\end{table}
\blue{The bars in Figure~\ref{fig:CovToiletDelta} display a consistent decline in droplet count with increasing height, similar to the trend observed for the uncovered toilet. One unexpected observation is the occurrence of higher droplet counts for the covered toilet at $1.22m$, compared to analogous measurements for the uncovered toilet in Table~\ref{tab:ToiletTable}. We remark that this does not indicate that the covering led to an increase in droplet count, but rather that the aerosols were redirected in higher concentrations to the position where the counter was located, after being forced through gaps between the seat and the cover. Examining the data from the the urinal-flushing tests in Figure~\ref{fig:UrinalDelta}, we observe a similar decline in droplet count with increasing height as for the other two cases. A large number of droplets were detected in the (0.3 to 0.5)$\mu m$ size range (approximately 300 droplets per second on average) at the lowest measurement level, which can be attributed to the close proximity of the sensor to the flushing water. Moreover, a significant number of droplets reached heights of up to $1.22m$ ($4ft$) from the ground, similar to the toilet-flushing tests.}
\begin{figure}[h]
\centering
\includegraphics[width=\linewidth]{bar_urinal.pdf}
\caption{\label{fig:UrinalDelta}Average increase in number of droplets measured per second from flushing the urinal. \blue{The error bars indicate the standard deviation of the measured increase from multiple flushes.} Each bar cluster corresponds to particles in a given size range, and indicates how the droplet count varies with measurement height. The corresponding values are provided in Table~\ref{tab:UrinalTable}.}
\end{figure}
\begin{table}[htp]
\caption{\label{tab:UrinalTable}Numerical values for the average increase in droplet count per second from the urinal-flushing tests\blueTwo{, with the standard deviation provided in parentheses}. The data corresponds to the bar graphs shown in Figure~\ref{fig:UrinalDelta}.}
\blue{
\begin{ruledtabular}
\begin{tabular}{lccc}
Height & (0.3 to 0.5)$\mu m$ &(0.5 to 1)$\mu m$ &(1 to 3)$\mu m$ \\ \hline
$0.53m$ & 315 ($\pm 209$) & 80 ($\pm 47$) & 17 ($\pm 8$) \\
$0.97m$ & 46 ($\pm 23$) & 14 ($\pm 5$) & 8 ($\pm 2$) \\
$1.22m$ & 34 ($\pm 12$) & 10 ($\pm 2$) & 5 ($\pm 2$)
\end{tabular}
\end{ruledtabular}
}
\end{table}
\blue{We remark that the total number of droplets generated in each flushing test described here can range in the tens of thousands. The numbers reported here indicate average droplet count per second, for cases where the time span for each surge varies from $15s$ to $20s$ (Table~\ref{tab:startAndSpan}). Thus, an average count of 50 droplets per second for one size range would amount to a total of 750 to 1000 droplets at one particular measurement location. Considering that similar measurements could be made all around the periphery of the fixtures, and that droplets are generated in several different size ranges, the overall total count would likely end up being significantly higher.} \blueTwo{Furthermore, droplet generation and accumulation depend on a variety of factors, such as the design of the toilet fixtures, the water pressure, the ventilation positioning, airflow, temperature, and RH, to name a few. The aim of the present work is not to present detailed characterizations of the influence of these factors on droplet dynamics, but instead to highlight the occurrence of aerosol generation and accumulation within public restrooms. These observations can help stimulate further studies to investigate steps to mitigate the issues involved.} {We further note that while the results presented here are restricted to specific measurement heights, there is a high likelihood of the aerosols getting dispersed throughout the room over time due to updrafts created by the ventilation system or by the movement of people.}
In addition to the flush-generated aerosol measurements, ambient aerosol levels were measured prior to starting the experiments and again after completing all of the tests. After approximately 3 hours of tests involving over 100 flushes, there was a substantial increase in the measured aerosol levels in the ambient environment. The corresponding data is presented in Figure~\ref{fig:BackgroundChange} and Table~\ref{tab:BackgroundValues}.
\begin{figure}[h]
\centering
\includegraphics[width=\linewidth]{background_timeSeries_plot.pdf}
\caption{\label{fig:BackgroundChange} Particle-count from ambient measurements within the restroom. The plot indicates the time-variation of particles in two different size ranges, (0.3 to 0.5)$\mu m$ - black, and (0.5 to 1)$\mu m$ - blue. The black curves correspond to the left vertical axis, whereas the blue curves correspond to the right vertical axis. The dashed lines indicate initial background readings before conducting any flushing tests, whereas the solid lines indicate measurements taken at the conclusion of all tests, approximately 3 hours and 100 flushes later.}
\end{figure}
\begin{table}[htp]
\caption{\label{tab:BackgroundValues} Average values for the background measurements shown in Figure \ref{fig:BackgroundChange}. Additionally, average measurements for the (1 to 3)$\mu m$ size group are also provided below. The `Before' column indicates the average ambient levels measured within a 5-minute time window before conducting any flushing experiments, and the `After' column indicates similar measurements taken after concluding all the experiments.}
\begin{ruledtabular}
\begin{tabular}{crrr}
Particle Size Group & Before & After & Percent Change \\ \hline
0.3 to 0.5 $\mu$m & 2537 & 4301 & 69.5\% \\
0.5 to 1 $\mu$m & 201 & 621 & 209\% \\
1 to 3 $\mu$m & 8 & 12 & 50\%
\end{tabular}
\end{ruledtabular}
\end{table}
There was a $69.5\%$ increase in measured levels for particles of size (0.3 to 0.5)$\mu m$, a $209\%$ increase for the (0.5 to 1)$\mu m$ particles, and a $50\%$ increase for the (1 to 3)$\mu m$ particles. Particles larger than 3$\mu m$ were excluded from the analysis due to the the impact of background noise on the extremely low measured values. The results point to significant accumulation of flush-generated aerosolized droplets within the restroom over time, which indicates that the ventilation system was not effective in removing them from the enclosed space, although there was no perceptible lack of airflow within the restroom\blue{; the room was equipped with two vents rated at volume flow rates of $7.5 m^3/min$ (265 CFM) and $5.66 m^3/min$ (200 CFM)}. Furthermore, a comparison with ambient levels outside the restroom (a few meters away from the closed restroom door, but within the same building) indicated that the levels of droplets smaller than $1\mu m$ were more than 10 times higher within the restroom compared to ambient levels outside the restroom. This was unexpected since the restroom had been closed off for more than 24 hours after deep cleaning, with the ventilation system operating normally. \blue{While it is difficult to ascertain the exact source of the droplets that contributed to high background levels within the restroom, it is likely that they were generated during the cleaning operation. There were no other readily apparent sources, since both locations, i.e., inside and outside the restroom, employed the same centralized air-conditioning system, and the RH and temperature were maintained at comparable levels \blueTwo{(Figure~\ref{fig:RH_temp})}.} These observations \blue{further} highlight the importance of employing adequate ventilation in enclosed spaces \blue{to extract suspended droplets effectively, in order to reduce} the chances of infection transmission via aerosolized droplets.
\begin{figure}[h]
\centering
\includegraphics[width=\linewidth]{RHtemp_plot.pdf}
\caption{\label{fig:RH_temp} \blueTwo{Relative humidity (black) and temperature measurements (blue) inside and outside the restroom. Solid lines indicate measurements taken inside the restroom, whereas dashed lines correspond to measurements outside the restroom, a few meters away from the closed restroom door.}}
\end{figure}
The results presented here indicate that although the likelihood of infection for respiratory illnesses via bioaerosols may be low compared to the risk posed by respiratory droplets (since virions are detected in larger quantities in respiratory samples), it presents a viable transmission route especially in public restrooms which often experience heavy foot-traffic within a relatively confined area. As demonstrated here, multiple flush-use over time can lead to an accumulation of potentially infectious aerosols, which poses a measurable risk considering the large number of individuals who may visit a public restroom and subsequently disperse into the broader community. Moreover, apart from flush-generated bioaerosols, the accumulation of respiratory aerosols also poses a concern in public restrooms if adequate ventilation is not available. \blue{Overall, the results presented here highlight the crucial need for ensuring effective aerosol removal capability in high density and frequently visited public spaces.}
\section{Conclusion}
\label{sec:conclusion}
The aerosolization of biomatter from flushing toilets is known to play a potential role in spreading a wide variety of gastrointestinal and respiratory illnesses. To better understand the risk of infection transmission that such droplets may pose in confined spaces, this paper investigates droplet-generation by flushing toilets and urinals in a public restroom operating under normal ventilation condition. The measurements were conducted inside a medium-sized public restroom, with a particle counter placed at various heights to determine the size and number of droplets generated upon flushing. The results indicate that both toilets and urinals generate large quantities of droplets smaller than 3$\mu m$ in size, which can pose a significant transmission risk if they contain infectious microorganisms from aerosolized biomatter. The droplets were detected at heights of up to $1.52m$ ($5ft$) for 20 seconds or longer after initiating the flush. Owing to their small size, these droplets can remain suspended for long periods of time, as is demonstrated in the present study via ambient measurements taken before and after conducting the experiments. When a large flat plate was used to cover the toilet opening, it led to a decrease in droplet dispersion but not a complete absence of the measured aerosols. This indicates that installing toilet seat lids in public restrooms may help reduce droplet dispersal to some extent, but it may not sufficiently address the risk posed by the smallest aerosolized droplets. Ambient aerosol levels measured before and after conducting the experiments indicated a substantial increase in particle count, pointing to significant accumulation of flush-generated aerosols within the restroom over time. This indicates that the ventilation system was not effective in removing the aerosols, although there was no perceptible lack of airflow within the restroom. Importantly, this suggests that multiple flush-use over time can lead to the accumulation of high levels of potentially infectious aerosols within public restrooms, which poses an elevated risk of airborne disease transmission. In addition to flush-generated bioaerosols, the accumulation of respiratory aerosols also poses a concern in public restrooms in the absence of adequate ventilation. Overall, the results presented here indicate that ensuring adequate ventilation in public restrooms is essential, since these relatively confined areas often experience heavy foot traffic and could pose a risk for widespread community transmission of various gastrointestinal and respiratory illnesses.
\section*{Data Availability}
The data that support the findings of this study are available from the corresponding author upon reasonable request.
|
1,314,259,993,600 | arxiv | \section{Introduction}
One of the oldest mathematical problems is to solve a linear system, that is to find a solution $\*x$ satisfying $\*A\*x = \*b$ given an $n \times n$ matrix $\*A$ and a $n$-dimensional vector $\*b$ as input.
In the RealRAM model this can be done in $O(n^{\omega})$ time, where $\omega \leq 2.3728$ \cite{Gall14a} is the matrix multiplication constant.
Much faster algorithms exist for approximately solving linear systems when $\*A$ is the Laplacian of undirected graphs.
Indeed recent breakthroughs showed that it can be done in nearly linear time \cite{SpielmanT14,CohenKMPPRX14}.
Kyng and Zhang \cite{KyngZ17} further showed that if such solvers can be extended to nearly linear time solvers for some classes slightly larger than undirected Laplacians, we can also solve general linear systems in nearly linear time.
In this paper we are interested in the space complexity of this problem.
For general linear systems Ta-shma gave a quantum algorithm using logarithmic space \cite{Ta-Shma13}.
For undirected Laplacian, Doron et al. showed that it has a probabilistic logarithmic-space algorithm \cite{DoronGT17} and hence a deterministic $O(\log^{3/2} n)$-space algorithm by a well-known space-efficient derandomization result \cite{SaksZ95}.
This was improved later to $\tilde{O}(\log n)$ by Murtagh et al \cite{MurtaghRSV17}.
\subsection{Our results}
We prove a space hardness version of Kyng and Zhang's results \cite{KyngZ17}, showing space hardness of approximate linear system solvers for some classes slightly larger than undirected Laplacians, namely multi-commodity Laplacians, 2D Truss Stiffness Matrices, and Total Variation Matrices.
\begin{theorem}\label{thm:hard}
Suppose that for multi-commodity Laplacians, 2-D Truss Stiffness Matrices, or Total Variation Matrices, the linear system $\*A\*x = \*b$ can be approximately solved in (nearly) logarithmic space with logarithmic dependence on condition number $\kappa$ and accuracy $\e^{-1}$ (even if it only works in expectation or with high probability), then any linear systems with polynomially bounded integers and condition number can be solved in (nearly) logarithmic space with high accuracy (in expectation or with high probability, respectively).
\end{theorem}
This shows that if the probabilistic logspace solver for undirected Laplacian in \cite{DoronGT17}, or the deterministic $\tilde{O}(\log n)$-space one in \cite{MurtaghRSV17}, can be extended to solve any of these three slightly larger subclasses of linear systems, we would have a surprising result that all linear systems can be approximately solved in probabilistic logspace or in deterministic $\tilde{O}(\log n)$-space.
Pessimistically speaking the theorem means that it is very hard to get space efficient algorithms for these subclasses, as it is as difficult as solving all linear systems in a space efficient way.
On the bright side, we actually prove that any progress on solving these subclasses using less space will immediately translate into similar progress for solving all linear system using less space.
Kyng and Zhang \cite{KyngZ17} proved their results via reductions from approximate solvers of general linear systems to those of three subclasses.
In this paper we prove Theorem \ref{thm:hard} by proving that their reductions can be carried out in a space efficient manner.
Indeed we prove a much stronger result that their reductions can be implemented in $\mathsf{TC}^0$ circuits,
which are constant-depth polynomial-size unbounded-fan in circuits with $\mathsf{MAJORITY}$ and $\neg$ gates.
It shows that these reductions are actually highly parallelizable.
We denote $\+G$ as the class of all matrices with integer valued entries.
In the context of solving linear systems, an all-zero row or column can be trivially handled, so we can assume without loss of generality that matrices in $\+G$ has at least one non-zero entry in every row and column.
For 2-commodity matrices $\+{MC}_2$, we have two set of variables $X$ and $Y$ of the same size, and the equations are scalings of $x_i - x_j = 0$, $y_i - y_j = 0$, and $x_i - y_i - (x_j - y_j) = 0$, where $x_i, x_j \in X$ and $y_i, y_j \in Y$.
This generalizes undirected Laplacians, as the incidence matrices of undirected Lapacians only have equations of the form $x_i - x_j = 0$ for $x_i, x_j \in X$.
Our main technical result proves that the reduction from $\+G$ to $\+{MC}_2$ in \cite{KyngZ17} is $\mathsf{TC}^0$-computable.
\begin{theorem}\label{thm:main}
There is a $\mathsf{TC}^0$-reduction from approximately solving $\+G$ to approximately solving $\+{MC}_2$.
\end{theorem}
In \cite{KyngZ17} it is shown that the matrix produced by this reduction is also a 2D Truss Stiffness Matrix as well as a Total Variation Matrix, therefore Theorem \ref{thm:main} also works for these classes.
Also note that this reduction is a Karp-style reduction, i.e. it requires only one linear system solve and uses the solution in a black-box way.
That is why Theorem \ref{thm:hard} still applies if the solver only works in expectation or with high probability.
We also show $\mathsf{TC}^0$-computability of the reductions in \cite{KyngZ17} to some more restrictive subclasses of $\+{MC}_2$, including $\+{MC}^>_2$, the exact class we have to solve when we use Interior Point Methods for 2-commodity problems, as explained in the Kyng and Zhang's full paper \cite{KyngZ17full}.
They also showed that the promise problem of deciding if a vector is in the image of a matrix or $\e$-far from the image can be directly reduced to approximately solving linear systems.
Combining with the above results, this shows that the promise problem can be reduced to approximately solving the above-mentioned subclasses in $\mathsf{TC}^0$.
\section{Simplified reductions in $\mathsf{TC}^0$: the easy parts}
Throughout this paper we use the sign-magnitude representation to encode a $w$-bit signed integer $z \in [-2^{w+1}+1, 2^{w+1}-1]$ into a sign bit, $0$ for positive and $1$ for negative, and $w$ bits for $|z|$ in binary. Note that in this method, $0$ has two representations and we accept both.
For simplicity we first prove the special case for the reduction from exact solver of $\+G$ to exact solver of $\+{MC}_2$.
\begin{theorem}
Given an $m \times n$ $w$-bit matrix $\*A \in \+G$ and a vector $\*b$ as input, we can compute in $\mathsf{TC}^0$ an $O(nmw \log n) \times O(nmw \log n)$ $O(w \log n)$-bit matrix $\*A' \in \+{MC}_2$ and a vector $\*b'$ such that:
\begin{itemize}
\item $\*A\*x = \*b$ has a solution if and only if $\*A'\*x' = \*b'$ has a solution;
\item if $\*A'\*x' = \*b'$ has a solution $\*x'$, then we can compute $\*x$ in $\mathsf{TC}^0$ from $\*x'$ so that $\*A\*x = \*b$.
\end{itemize}
\end{theorem}
As in \cite{KyngZ17}, this reduction is split into several steps using the following classes of matrices.
\begin{definition}[\cite{KyngZ17}]
\begin{itemize}
\item Let $\+G_z \subset \+G$ denote the class of matrices with integer valued entries such that every row has zero row sum;
\item Let $\+G_{z,2} \subset \+G_z$ denote the class of matrices with integer valued entries such that every row has zero row sum, and for each row the sum of positive coefficients is a power of $2$.
\end{itemize}
\end{definition}
\begin{lemma}\label{lem:reduc}
There are $\mathsf{TC}^0$-reductions for exact solvers of the following classes:
\begin{enumerate}[(i)]
\item\label{red:1} from $\+G$ to $\+G_z$;
\item\label{red:2} from $\+G_z$ to $\+G_{z,2}$;
\item\label{red:3} from $\+G_{z,2}$ to $\+{MC}_2$.
\end{enumerate}
\end{lemma}
Lemma \ref{lem:reduc} (\ref{red:3}) is the main reduction in this paper (same as in \cite{KyngZ17}), which will be proved in the next section. In the remaining of this section we prove Lemma \ref{lem:reduc} (\ref{red:1}) and (\ref{red:2}).
\paragraph*{From $\+G$ to $\+G_z$}
\begin{proof}[Proof sketch]
Given a matrix $\*A \in \+G$, we can define a matrix $\*A'$ with one more column by $\*A' = \begin{pmatrix}
\*A & -\*A\*1
\end{pmatrix}$.
Obviously $\*A' \in \+G_z$, and $\*A \*x = \*b$ has a solution if and only if $\*A' \*x' = \*b$ has a solution.
We can also recover $\*x \in \mathbb{R}^n$ from $\*x' \in \mathbb{R}^{n+1}$ by taking the first $n$ rows of $\*x'$ and minus each of them by $\*x'_{n+1}$.
The following results about additions imply that $\*A'$ can be calculated in $\mathsf{TC}^0$, and we can recover $\*x'$ from $\*x$ in $\mathsf{AC}^0$ (for simplicity we ignore the precision problem here).
\end{proof}
\begin{fact}\label{fat:add}
\begin{itemize}
\item Addition of $2$ $w$-bit numbers has $\mathsf{AC}^0$ circuit of size $\poly(w)$ (c.f. \cite{CloteK02});\footnote{$\mathsf{AC}^0$ circuits are constant-depth polynomial-size unbounded-fan in circuits with $\wedge$, $\vee$, and $\neg$ gates.}
\item Addition of $n$ $w$-bit numbers has $\mathsf{TC}^0$ circuit of size $\poly(n,w)$ \cite{ImmermanL89}.
\end{itemize}
\end{fact}
\paragraph*{From $\+G_z$ to $\+G_{z,2}$}
\begin{proof}[Proof sketch]
Given an $m \times n$ $w$-bit signed-integer matrix $\*A' \in \+G_z$, we just need to add two more columns to make the sum of positive (and negative) entries in each row to the closet power of $2$. This can be done in $\mathsf{TC}^0$ in the following way. For each row $1 \leq i \leq m$, we calculate the sum of positive entries $s_i$ by checking the sign bit then do the iterated addition in $\mathsf{TC}^0$ by Fact \ref{fat:add}.
We then take $s = \max s_i$, which can be computed in $\mathsf{AC}^0$ given $s_i$'s.
$s$ has at most $O(w \log n)$ bits so given $s$ by searching brute-forcely in $\mathsf{AC}^0$ we can find the minimum $k$ s.t. $2^{k} \geq s$.
Therefore by Fact \ref{fat:add} we can calculate $a_i = 2^{k} - s_i$ in $\mathsf{AC}^0$ given $s_i$.
Then for each row $i$ we add $a_i$ and $-a_i$ to the last two columns of $\*A'$ to get $\*A''$. Additionally we need to add a new row to $\*A''$ (and to $\*b''$ accordingly) to zero out the last two variables we just added: set the last two entries into $1$ and $-1$, and set all other entries $0$, and also add a $0$ entry to $\*b'$ to get $\*b''$.
Obviously we have $\*A'' \in \+G_{z,2}$, and $\*A'\*x' = \*b'$ has a solution iff $\*A''\*x'' = \*b''$ has a solution. We can easily recover $\*x'$ from $\*x''$ by taking the first $n$ rows.
\end{proof}
For the next section we need the following definition.
\begin{definition}
We say a matrix $\*A \in \+G_{z}$ is \emph{$w$-bounded} if for each row the sum of positive coefficients is at most $2^w$. Note that in such matrix every entry is a $w$-bit signed integer.
\end{definition}
Note that the reduction from $\+G$ to $\+G_z$ reduces an $m \times n$ $w$-bit matrix $\*A$ into an $m \times (n+1)$ $O(w \log n)$-bounded matrix $\*A'$, then the reduction from $\+G_z$ to $\+G_{z,2}$ reduces $\*A'$ into an $(m+1) \times (n+3)$ $O(w \log n)$-bounded matrix $\*A''$.
\section{Simplified main reduction in $\mathsf{TC}^0$}
The main reduction in \cite{KyngZ17} uses the pair-and-replace scheme to transform a linear system in $\+G_{z,2}$ to a $2$-commodity equation system. The main idea is to use $\+{MC}_2\mathtt{Gadget}$s consisting of $\+{MC}_2$ equations to replace pairs of variables in the original linear system round-by-round according to the bit representation of their coefficients.
A simplified version of the gadget, implicitly given in \cite{KyngZ17}, is as follows.
\begin{definition}[Simplified $\+{MC}_2\mathtt{Gadget}$]
Define $\+{MC}_2\mathtt{Gadget}(t, t', j_1, j_2)$ to be the following set of 2-commodity linear equations representing ``$2x_t = x_{j_1} + x_{j_2}$'':
\begin{align*}
x_t - x_{t'+1} &= 0 \\
x_{t'+2} - x_{j_2} &= 0\\
y_{t'+1} - y_{t'+3} &= 0 \\
y_{t'+4} - y_{t'+2} &= 0 \\
x_{t'+3} - x_{j_1} &= 0\\
x_t - x_{t'+4} &= 0\\
x_{t'+4} - y_{t'+4} - (x_{t'+3} - y_{t'+3}) &= 0\\
x_{t'+1} - y_{t'+1} - (x_{t'+2} - y_{t'+2}) &= 0.
\end{align*}
For convenience we use an extra parameter $t'$ to keep track of new variables that are only used in the gadgets.
\end{definition}
Correctness of this gadget can be easily verified by summing up all the equations in it.
We now present a simplified reduction from exactly solving $\+G_{z,2}$ to exactly solving $\+{MC}_2$ in Algorithm \ref{alg:reduc1}. We use $\*A_i$ to denote the $i$-th row of $\*A$.
\begin{algorithm}[ht]
\DontPrintSemicolon
\SetKwInOut{Input}{Input}\SetKwInOut{Output}{Output}
\SetKwFunction{Gadget}{$\+{MC}_2\mathtt{Gadget}$}
\Input{a $w$-bounded $m \times n$ matrix $\*A \in \+G_{z,2}$ and $\*c \in \mathbb{R}^n$.}
\Output{a $w$-bit $m' \times n'$ matrix $\*B \in \+{MC}_2$ and $\*d \in \mathbb{R}^n$.}
\BlankLine
$m' \leftarrow m$, $n' \leftarrow n$\;
$n_g \leftarrow 0$ \tcp*{\# of new variables used only in the $\+{MC}_2\mathtt{Gadget}$s}
\For{$i \leftarrow 1$ \KwTo $m$}{
\For{$s \in \{+, -\}$}{
\For{$k \leftarrow 1$ \KwTo $w$}{
\While{strictly more than 1 entry in $\*A_i$ has sign $s$ and the $k$-th bit being $1$}{
Let $j_1,j_2$ be the first and second indices of such entries in $\*A_i$\;
In $\*A_i$, replace $2^k(x_{j_1} + x_{j_2})$ by $2^{k+1}x_{n'+1}$ (thus adding one column to the right of $\*A$)\; \label{alg:ln:replace}
Add the coefficients of \Gadget{$n'+ n_g +1$, $n'+n_g+1$, $j_1$, $j_2$} to $\*C$\;\label{alg:ln:gadget}
$n' \leftarrow n' + 1$\;\label{alg:ln:nnv}
$n_g \leftarrow n_g + 4$\;
$m' \leftarrow m' + 8$\;
}
}
}
}
$n' \leftarrow 2 \times (n' + n_g)$\;\label{alg:ln:double}
Stack $\*C$ in the bottom of $\*A$ and fill in $0$'s to get $\*B$\;
Add $m' - m$ $0$'s under $\*c$ to get $\*d$\;
\caption{Simplified $\textsc{Reduce} \+G_{z,2}\textsc{To}\+{MC}_2$}\label{alg:reduc1}
\end{algorithm}
Note that in $\+{MC}_2$ we have two input variables sets $X$, $Y$ of the same size. That is why we have to multiply $n'$ by 2 at last in the reduction. But here we will only use variables in $Y$ in the $\+{MC}_2\mathtt{Gadget}$s, and all the other $Y$ variables are unused.
For convenience we arrange the variables in the following way.
\begin{remark}[Arrangement of variables]\label{rmk:v}
In $\*B$ we put all the variables in $X$ before those in $Y$.
More importantly, we put those $X$ variables that are only used in the gadgets behind all those $x_{n'+1}$ in Line \ref{alg:ln:replace}.
Equivalently it can be viewed as the following process.
We first run Algorithm \ref{alg:reduc1} virtually before Line \ref{alg:ln:double} to get $n'$, which is the number of $X$ variables ignoring those only used in the gadgets.
Then we run it again on the original input, starting with $n_g$ being this value, and in Line \ref{alg:ln:gadget} we use $\+{MC}_2\mathtt{Gadget}(n'+1, n_g, j_1, j_2)$ instead.
\end{remark}
We give an example showing how the reduction works under this arrangement.
\begin{example}\label{exp}
We show how the reduction runs on $3x_1 + 5x_2 + x_3+7x_4 -16x_5 =0$:
\begin{align*}
& 00011 x_1 + 00101 x_2 + 00001 x_3 + 00111x_4 - 10000x_5 = 0 \\
\xrightarrow{x_1 + x_2 - 2x_6 = 0}~& 00010x_1 + 00100x_2 + 00001x_3 + 00111x_4 + 00010x_6 -10000 x_5 =0\\
\xrightarrow{x_3 + x_4 - 2x_7 = 0}~& 00010x_1 + 00100x_2 + 00110x_4 + 00010x_6 + 00010x_7 -10000 x_5 =0\\
\xrightarrow{x_1 + x_4 - 2x_8 = 0}~& 00100x_2 + 00100x_4 + 00010x_6 + 00010x_7 + 00100x_8 -10000 x_5 =0\\
\xrightarrow{x_6 + x_7 - 2x_9 = 0}~& 00100x_2 + 00100x_4 + 00100x_8 + 00100x_9 -10000 x_5 =0\\
\xrightarrow{x_2 + x_4 - 2x_{10} = 0}~& 01000x_{10} + 00100x_8 + 00100x_9 -10000 x_5 =0\\
\xrightarrow{x_8 + x_9 - 2x_{11} = 0}~& 01000x_{10} + 01000x_{11} -10000 x_5 =0\\
\xrightarrow{x_{10} + x_{11} - 2x_{12} = 0}~& 10000x_{12} -10000 x_5 =0.
\end{align*}
We are only eliminating the positive coefficient variables in this example for simplicity.
In the first round we use new variables (and the corresponding gadgets) $x_6$ and $x_7$ to eliminate the first bit, getting the equation $2x_1 + 4 x_2 + 6 x_4 + 2x_6 + 2x_7 = 0$.
These two generated variables are then eliminated in the second round by $x_9$, in addition to $x_8$ for the second bit.
In the third round we use $x_{10}$ and $x_{11}$.
Finally in the fourth round we use $x_{12}$ and get
the equation after the reduction $16 x_{12} - 16 x_5 = 0$, with
$\+{MC}_2\mathtt{Gadget}$s representing $2x_6 = x_1 + x_2$, $2x_7 = x_3 + x_4$, $2x_8 = x_1 + x_4$, $2x_9 = x_6 + x_7$, $2x_{10} = x_2 + x_4$, $2x_{11} = x_8 + x_9$, and $2x_{12} = x_{10} + x_{11}$.
\end{example}
Correctness of this simplified reduction follows easily by correctness of the gadget, as our transformation preserves the original solution. Moreover, given a solution $\*x^*$ such that $\*B\*x^* = \*d$ where $\*B$ and $\*d$ are obtained from running Algorithm \ref{alg:reduc1} on input $\*A$ and $\*c$, we can easily get the solution to the original equation system $\*A\*x = \*c$ by simply taking the first $n$ elements in $\*x^*$. It is also easy to get $\*d$ from $\*c$ if we can calculate $m'$ in $\mathsf{TC}^0$.
In the remaining of this section we are going to prove that Algorithm \ref{alg:reduc1} can be implemented in $\mathsf{TC}^0$.
For $1 \leq i \leq m$, $1 \leq k \leq w$, we define
\begin{align*}
\mtt{Len}^+_{i}(\*A) &= \text{ log of sum of positive coefficients in $\*A_i$},\\
\mtt{CountBit}^+_{i,k}(\*A) &= \# \text{ of positive coefficient variables in the original $\*A_i$ with the $k$-th bit $1$}, \\
\mtt{NumGadget}^+_{i,k}(\*A) &= \# \text{ of $\+{MC}_2\textsc{Gadget}$s used for $\*A_i$ to eliminate the $k$-th bit}\\
&\phantom{= \# }\text{ of positive coefficient variables},
\end{align*}
and similarly $\mtt{Len}^-_i(\*A)$, $\mtt{CountBit}^-_{i,k}(\*A)$, $\mtt{NumGadget}^-_{i,k}(\*A)$ for the negative coefficient variables.
\begin{example}
For the above example, we have $\mtt{Len}^+_i(\*A) = 5$,
and the following values for each $k$.
\begin{table}[h]\caption{Values of $\mtt{CountBit}^+_{i,k}(\*A)$ and $\mtt{NumGadget}^+_{i,k}(\*A)$ for Example \ref{exp}}
\centering
\begin{tabular}{|c|c|c|c|c|c|}
\hline
$k$ & $1$ & $2$ & $3$ & $4$ & $5$ \\\hline
$\mtt{CountBit}^+_{i,k}(\*A)$ & $4$ & $2$ & $2$ & $0$ & $0$ \\\hline
$\mtt{NumGadget}^+_{i,k}(\*A)$ & $2$ & $2$ & $2$ & $1$ & $0$ \\\hline
\end{tabular}
\end{table}
\end{example}
We have the following simple but crucial properties for these values.
\begin{claim}\label{clm:obs}
\begin{enumerate}[(i)]
\item $\mtt{Len}^s_i(\*A) \leq w$ for all $s \in \{+, -\}$, $1 \leq i \leq m$;
\item $\mtt{CountBit}^s_{i,k}(\*A) \leq n$ for all $s \in \{+, -\}$, $1 \leq i \leq m$, $1 \leq k \leq w$;
\item For all $s \in \{+, -\}$, $1 \leq i \leq m$,
\[
\mtt{NumGadget}^s_{i,k}(\*A) =
\begin{cases}
2^{-(k+1)} \sum_{k' = 1}^k 2^{k'}\mtt{CountBit}^s_{i,k'}(\*A) & \text{ for } 1 \leq k \leq \mtt{Len}^s_i(\*A)-1, \\
0 & \text{ for } \mtt{Len}^s_i(\*A) \leq k \leq w,
\end{cases}
\]
thus $\mtt{NumGadget}^s_{i,k}(\*A) \leq O(n)$;
\item $m' = m + 8 \sum_{i = 1}^m \sum_{s \in \{+, -\}} \sum_{k = 1}^w \mtt{NumGadget}^s_{i,k}(\*A) \leq O(nmw)$;
\item $n' = 2n + 10\sum_{i = 1}^m \sum_{s \in \{+, -\}} \sum_{k = 1}^w \mtt{NumGadget}^s_{i,k}(\*A) \leq O(nmw)$.
\end{enumerate}
\end{claim}
\begin{proof}
(i) and (ii) are trivial by definition.
(iv) and (v) are straightforward from Algorithm \ref{alg:reduc1} and (iii).
For (iii), note that in each round for $k$ we will eliminate all the variables generated in the previous round for $k-1$ by construction (ignoring those variables that are only used in the gadgets), therefore we have $\mtt{NumGadget}^s_{i,1}(\*A) = \mtt{CountBit}^s_{i,1}(\*A)/2$ and
$\mtt{NumGadget}^s_{i,k}(\*A) = (\mtt{CountBit}^s_{i,k}(\*A) + \mtt{NumGadget}^s_{i, k-1}(\*A))/2$ for $2\leq k \leq \mtt{Len}^s_i(\*A) -1$.
Here we rely on the property that the sum of positive (and negative) coefficients in each row is a power of $2$ to ensure that $\mtt{NumGadget}^s_{i,k}(\*A)$ as calculated in this way are always integers.
Then by induction we get the formula.
\end{proof}
Note that in (iii) $2^{-(k+1)}$ and $2^{k'}$ are just right and left shifts, which can be easily implemented in the circuit model.
Combining Claim \ref{clm:obs} and Fact \ref{fat:add}, we can see that all of these values can be computed in $\mathsf{TC}^0$ for all $i, k, s$, i.e. the depths of the $\mathsf{TC}$ circuits are absolute constants independent of $i$, $k$, and $s$.
\begin{lemma}[Informal]\label{lem:aux}
\begin{itemize}
\item $\mtt{Len}^s_i$, $\mtt{CountBit}^s_{i,k}$, and $\mtt{NumGadget}^s_{i,k}$ have $\mathsf{TC}^0$ circuits of size $\poly(n,w)$ for all $i, k, s$;
\item $n', m'$ has $\mathsf{TC}^0$ circuits of size $\poly(n,m,w)$.
\end{itemize}
\end{lemma}
Now we can prove that the reduction from $\+G_{z,2}$ to $\+{MC}_2$ can be done in $\mathsf{TC}^0$. We represent the input matrix $\*A$ explicitly by giving all its entries in sign-magnitude form, and the output matrix $\*B$ implicitly by outputting the entry of $\*B$ in row $i$ column $j$ with $i$,$j$ as additional inputs.
\begin{theorem}
There is a $\mathsf{TC}^0$ circuit family $\{C_{n,m,w}\}_{n,m,w \in \mathbb{N}}$ of size $\poly(n,m,w)$ with $(O(nmw) + O(\log nmw))$-bit input and $O(w)$-bit output such that:
\begin{itemize}
\item the first $O(nmw)$ bits of input represent a $w$-bounded $m \times n$ matrix $\*A \in \+G_{z,2}$ in sign-magnitude form, the next $O(\log nmw)$ bits of input represent an index $i$, and the last $O(\log nmw)$ bits represent another index $j$;
\item for $i \leq m'$ and $j \leq n'$ where $m', n' \leq O(nmw)$ is calculated as in Algorithm \ref{alg:reduc1} on $\*A$, the output represents the entry of $\*B$ (in sign-magnitude form) in row $i$ column $j$ as calculated by Algorithm \ref{alg:reduc1} on $\*A$; otherwise it represents $0$.
\end{itemize}
\end{theorem}
\begin{proof}
Consider the $i$-th row of $\*B$, we need to know the equation it corresponds to. Because we put those equations of $\+{MC}_2\mathtt{Gadget}$s behind the modified equations of $\*A$, we have two cases:
\begin{itemize}
\item $i \leq m$: in this case, the equation is $\*A_i$ after the transformation, which has the form $C (x_{j_+} - x_{j_-}) = \*c_i$ with $C = \underbrace{100\cdots 0}_{\mtt{Len}^+_i(\*A)\text{ bits}}$ in binary since our transformation does not change the sum of positive (and negative) coefficients in $\*A$. What remains is to calculate the indices $j_+$ and $j_-$ in $\mathsf{TC}^0$.
Let $\mtt{SumNumG}^s_i(\*A) = \sum_{k=1}^w \mtt{NumGadget}^s_{i,k}$ for all $s \in \{+,-\}$ and $1 \leq i \leq m$. For each $s \in \{+, -\}$, there are two cases:
\begin{itemize}
\item No new variable is added, i.e. $\mtt{SumNumG}^s_i(\*A) = 0$: there is only one entry in the original $\*A_i$ with the corresponding sign thus $j_s$ is the index of this entry. We search for this entry to get its index, which can be implemented in $\mathsf{AC}^0$ because there are $n$ possibilities.
\item Some new variables are added: then $j_s$ is the index of the last added new variable, ignoring those only in $\+{MC}_2\mathtt{Gadget}$s. It can be calculated by
\begin{align*}
j_s = \begin{cases}
n + \mtt{SumNumG}^+_i(\*A) + \sum_{i'=1}^{i-1}\sum_{s'\in\{+,-\}}\mtt{SumNumG}^{s'}_{i'}(\*A) &\text{ if } s = +, \\
n + \sum_{i'=1}^{i}\sum_{s'\in\{+,-\}}\mtt{SumNumG}^{s'}_{i'}(\*A) &\text{ if } s = -.
\end{cases}
\end{align*}
\end{itemize}
$\mtt{SumNumG}^s_i \in \mathsf{TC}^0$ by Fact \ref{fat:add} and Lemma \ref{lem:aux} so both $j_+, j_-$ can be calculated in $\mathsf{TC}^0$ by Fact \ref{fat:add}. Therefore we can calculate the entries in row $i$ in $\mathsf{TC}^0$ for $i \leq m$.
\item $i > m$: in this case, this equation is in an $\+{MC}_2\mathtt{Gadget}(t, t', j_1, j_2)$ gadget for some $t, t', j_1, j_2$. We need to calculate $t, t', j_1$, and $j_2$ in $\mathsf{TC}^0$ then it is easy to recover the equation from the offset in the gadget.
We can first calculate in $\mathsf{AC}^0$ the gadget's index $ind = \lfloor \frac{i-m-1}{8}\rfloor+1$ as $\lfloor \frac{\cdot}{8}\rfloor$ is just ignoring the three least significant bits.
By Algorithm \ref{alg:reduc1} and Remark \ref{rmk:v}, we know $t = n + ind$ thus it is $\mathsf{AC}^0$-computable,
and we can also calculate $t' = n + \sum_{i'=1}^m\sum_{s' \in \{+,-\}} \mtt{SumNumG}^{s'}_{i'}(\*A) + 4 (ind - 1)$ in $\mathsf{TC}^0$ by Fact \ref{fat:add}.
Then we can calculate the number $i'$, $k'$, sign $s' \in \{+,-\}$, and number $\ell'$ such that this gadget is the $\ell'$-th gadget we created to eliminate the $k'$-th bit of $\*A_{i'}$ for variables with sign $s'$. More specifically, we want to find the minimum $i'$, $k'$, $s'$ (where we treat `$+$' $<$ `$-$') such that
\begin{align*}
\mtt{PrefixSum}^+_{i',k'}(\*A)
< ind \leq \mtt{PrefixSum}^+_{i',k'}(\*A) + \mtt{NumGadget}^+_{i',k'} & \text{ if } s' = +,\\
\mtt{PrefixSum}^-_{i',k'}(\*A) < ind \leq \mtt{PrefixSum}^-_{i',k'}(\*A) + \mtt{NumGadget}^-_{i',k'} & \text{ if } s' = -,
\end{align*}
where
\begin{align*}
\mtt{PrefixSum}^+_{i',k'}(\*A) &= \sum_{i''=1}^{i'-1}\sum_{s''\in\{+,-\}}\mtt{SumNumG}^{s''}_{i''}(\*A)
+\sum_{k''=1}^{k'-1}\sum_{s''\in\{+,-\}}\mtt{NumGadget}^{s''}_{i',k''}(\*A),\\
\mtt{PrefixSum}^-_{i',k'}(\*A) &= \mtt{PrefixSum}^+_{i',k'}(\*A) + \mtt{NumGadget}^+_{i',k'}.
\end{align*}
There are $O(mw)$ possible choices for $i'$, $k'$, $s'$ and each condition is a prefix sum of $\mtt{NumGadget}^s_{i,k}$'s, therefore
this can be done in $\mathsf{TC}^0$ by a parallel comparison of $ind$ to the prefix sums of $\mtt{NumGadget}^s_{i,k}$'s. After getting $i'$, $k'$, and $s'$, we can get $\ell' = ind - \mtt{PrefixSum}^{s'}_{i',k'}$ by Fact \ref{fat:add}.
What remains is to calculate $j_1$ and $j_2$ from $i'$, $k'$, $s'$, and $\ell'$. Note that when eliminating the $k'$-th bit of $\*A_{i'}$ for variables with sign $s'$, we first eliminate in pairs those variables in the original $\*A_{i'}$ that have $1$ in the $k'$-th bit before the reduction (we call them \emph{original pairs}), then eliminate in pairs those generated in the previous round for $i'$, $k'-1$, and $s'$. The number of original pairs is given by $p = \lfloor \mtt{CountBit}^{s'}_{i',k'}(\*A) / 2\rfloor$, computable in $\mathsf{TC}^0$. There are two cases:
\begin{itemize}
\item $\ell' \leq p$: $j_1$ and $j_2$ is the indices of variables in the $\ell'$-th original pairs, which are the indices of the $(2\ell'-1)$-th and $2\ell'$-th variables in $\*A_{i'}$ that have $1$ in the $k'$-th bit. Similarly as above, this can be done by a simple parallel comparison to prefix sums of the $k'$-th bits of variables in $\*A_{i'}$.
\item $\ell' > p$: then $j_1$ and $j_2$ are the $(2(\ell'-p)-1)$-th and $2(\ell'-p)$-th new variables generated in the previous round, therefore $j_1 = \mtt{PrefixSum}^{s'}_{i', k'-1}(\*A) + 2(\ell'-p)-1$ and $j_2 = j_1 + 1$.
\end{itemize}
However the second case above only works for even $\mtt{CountBit}^{s'}_{i',k'}(\*A)$. When it is odd, $\ell' = p+1$ corresponds to a pair with the last original variable in this round and the first generated variables in the previous round, thus $j_1$ can be calculated as in the first case and $j_2 = \mtt{PrefixSum}^{s'}_{i', k'-1}(\*A) + 1$. Then for $\ell' > p+1$, we have $j_1 = \mtt{PrefixSum}^{s'}_{i', k'-1}(\*A) + 2(\ell'-p)-2$ and $j_2 = j_1 + 1$.
\end{itemize}
In conclusion, given $i$ as input, we can first check if $i \leq m'$ in $\mathsf{TC}^0$ by Lemma \ref{lem:aux} and if so compute coefficients of $\*B_i$ in $\mathsf{TC}^0$, therefore with $j$ as input, we check if $j \leq n'$ in $\mathsf{TC}^0$ and if so compute the entry of $\*B$ in row $i$ column $j$ in $\mathsf{TC}^0$.
\end{proof}
\section{Genearlization to approximate solvers, and more restrictive classes}
Now we are going to generalize the above results of reductions for exact solvers into those for approximate solvers, thus proving Theorem \ref{thm:main}. First we need the following result showing the power of $\mathsf{TC}^0$.
\begin{fact}\label{fat:tc0}
Division and iterated multiplication are in $\mathsf{DLOGTIME}$-uniform $\mathsf{TC}^0$ \cite{HesseAB02}. Moreover, we can approximate in $\mathsf{TC}^0$ functions represented by sufficiently nice power series, such as $\log$, $\exp$, and $x^{1/k}$ \cite{ReifT92,MacielT99,HesseAB02}.
\end{fact}
\begin{proof}[Proof sketch of Theorem \ref{thm:main}]
Based on our proofs on the simplified reductions, we are going to prove that the original reductions from $\+G$ to $\+{MC}_2$ in \cite{KyngZ17} can be computed in $\mathsf{TC}^0$.
In the context of approximate solvers, the error $\e$ that solvers are required to achieve is also part of the instance.
By Fact \ref{fat:tc0}, all the errors in the reduced instances, as defined in the reductions in Kyng and Zhang's full paper \cite{KyngZ17full}, can be computed in $\mathsf{TC}^0$ given the size, condition number, and bit-complexity of the original matrix as parameters.
\begin{itemize}
\item \textbf{From $\+G$ to $\+G_z$}: this reduction remains the same, but now knowing the accuracy we can recover $\*x'$ from $\*x$ using fix-point arithmetic in $\mathsf{TC}^0$.
\item \textbf{From $\+G_z$ to $\+G_{z,2}$}: the only difference (besides the calculation of accuracy) is that in the original reduction in \cite[Section~7.2]{KyngZ17full}, in the last row of the reduced matrix $\*A'$ the last two entries are set to be $w$ and $-w$ for some value $w$, instead of $1$ and $-1$ as in our simplified version. However $w$ is computable in $\mathsf{TC}^0$ by Fact \ref{fat:tc0} so the reduction is still computable in $\mathsf{TC}^0$.
\item \textbf{From $\+G_{z,2}$ to $\+{MC}_2$}: for approximate solvers we will have to use the original $\+{MC}_2\mathtt{Gadget}$ in \cite[Section~4]{KyngZ17full} consisting of ten $2$-commodity equations instead of eight and with additional variables.
So we need to modify the corresponding numbers in our calculation, and in particular the gedget's index becomes $ind = \lfloor \frac{i -m - 1}{10} \rfloor +1$, which is still $\mathsf{TC}^0$-computable.
Besides, the equations in the $\+{MC}_2\mathtt{Gadget}$ for eliminating the $k$-th bit of variables with sign $s$ in $\*A_i$ will be multiplied by a factor $-s \cdot 2^k \cdot w_i$, where $w_i = \sqrt{10 \sum_{s' \in \{+,-\}} \mtt{SumNumG}^s_i(\*A)}$, as specified in Algorithm 1 of \cite{KyngZ17full}.
These factors can be calculated with desired accuracy in $\mathsf{TC}^0$ by Fact \ref{fat:tc0}.
Therefore the reduction is still computable in $\mathsf{TC}^0$.
\end{itemize}
Additionally in the second and third reduction, when recovering $\*x'$ from $\*x$ we need to check if the original matrix $\*A$ and vector $\*b$ satisfy $\*A^\top \*b = \*0$ and simply return $\*0$ if so. This can be done in $\mathsf{TC}^0$ by Fact \ref{fat:add}.
In conclusion, we can reduce the problem of approximately solving equations in $\+G$ to approximately solving equations in $\+{MC}_2$ in $\mathsf{TC}^0$.
\end{proof}
In \cite{KyngZ17} they also considered some more restrictive subclasses of $\+{MC}_2$.
Intuitively, the set of \emph{strict $2$-commodity matrices} $\+{MC}^{>0}_2 \subset \+{MC}_2$ is the class of 2-commodity equations $\*A$ such that for every pair $i, j$, equation $x_i - x_j = 0 \in \*A$ iff equation $y_i - y_j = 0 \in \*A$ iff equation $x_i - y_i - (x_j - y_j) = 0 \in \*A$.
The set of strict $2$-commodity matrices with integer entries is denoted by $\+{MC}^{>0}_{2,\mathbb{Z}}$.
They showed that reductions from approximately solving $\+{MC}_2$ to approximately solving $\+{MC}^{>0}_2$, and from $\+{MC}^{>0}_2$ to $\+{MC}^{>0}_{2,\mathbb{Z}}$.
We are going to show that these reductions can be computed in $\mathsf{TC}^0$.
\paragraph*{From $\+{MC}_2$ to $\+{MC}^{>0}_2$}
\begin{proof}[Proof sketch]
The reduction, as defined in \cite[Section~5.1]{KyngZ17full}, runs by checking for each pair $i,j$ that are involved in some equation in $\*A$ if any of the three types of equations is missing and add it if so.
The added equations will be multiplied by a factor that is computable in $\mathsf{TC}^0$. Obviously the resulting equation systems is in $\+{MC}^{>0}_2$. It is easy to see that the number of added equations for each pair $i,j$ can be computed in $\mathsf{AC}^0$, thus all the prefix sums of these numbers can be calculated in $\mathsf{TC}^0$ simultaneously, and so we can determine the equations in the reduced instance in $\mathsf{TC}^0$.
\end{proof}
\paragraph*{From $\+{MC}^{>0}_2$ to $\+{MC}^{>0}_{2,\mathbb{Z}}$}
\begin{proof}[Proof sketch]
In \cite[Section~6]{KyngZ17full} it is done by scaling up all the numbers in the matrix and the vector by a factor of $2^k$, where $k$ is computable in $\mathsf{TC}^0$ by Fact \ref{fat:tc0}, then take the ceiling function on entries of the matrix to convert them into integer entries, which also can be done in $\mathsf{TC}^0$.
\end{proof}
|
1,314,259,993,601 | arxiv | \section{Formalism}
\subsection{Transport}
Our method generalises the purely classical
Boltzmann-Green-function theory
of Stanton and Wilkins.\cite{swI,swII,swt}
We start with the one-body transport equation.
For simplicity consider a uniform conductor.
The semiclassical one-electron Boltzmann equation is
\begin{eqnarray}
{\left[
{\partial\over {\partial t}}
- {{e{\bf E}}\over {\hbar}}
{\bbox \cdot}
{\partial\over {\partial {\bf k}}}
\right]} f_{s,{\bf k}}(t)
=&& {\cal C}_{s,{\bf k}}[f(t)],
\label{eq1}
\end{eqnarray}
\noindent
where the time-dependent distribution $f_{s,{\bf k}}(t)$
is labelled by band-and-spin index $s$ and wave vector
${\bf k}$; the local driving field (uniform here)
is ${\bf E}$. The collision operator ${\cal C}_{s,{\bf k}}[f]$
carries, at the semiclassical level, the effects of microscopic
scattering from all sources.
In a degenerate system, ${\cal C}$ is inherently nonlinear in $f$.
Normalisation to the density $n$ is such that
$n = \Sigma_{s,{\bf k}} f_{s,{\bf k}}/\Omega $
in a system of volume $\Omega$.
In the steady state, Eq. (\ref{eq1}) maps
the equilibrium Fermi-Dirac distribution $f^{\rm eq}$
to $f$. Introducing the difference function
$g \equiv f - f^{\rm eq}$, we have
\begin{eqnarray}
- {{e{\bf E}}\over {\hbar}}
{\bbox \cdot}
{{\partial g_{s,{\bf k}}}\over {\partial {\bf k}}}
=&&
{{e{\bf E}}\over {\hbar}}
{\bbox \cdot}
{{\partial f^{\rm eq}_{s,{\bf k}}}\over {\partial {\bf k}}}
+ {\cal C}_{s,{\bf k}}[f^{\rm eq} + g].
\label{eq2}
\end{eqnarray}
\subsection{Steady-State Fluctuations}
The equilibrium density-density fluctuation for free electrons is
$\Delta f^{\rm eq} \equiv k_BT \partial f^{\rm eq}/\partial \varepsilon_F$,
where $k_BT$ is the thermal energy and $\varepsilon_F$
is the chemical potential.
Eq. (\ref{eq2}), linearised, maps $\Delta f^{\rm eq}$
adiabatically to its nonequilibrium counterpart. Solving
for the fluctuation difference $\Delta g$ according to
\begin{eqnarray}
- {{e{\bf E}}\over {\hbar}}
{\bbox \cdot}
{{\partial \Delta g_{s,{\bf k}}}\over {\partial {\bf k}}}
=&&
{{e{\bf E}}\over {\hbar}}
{\bbox \cdot}
{{\partial \Delta f^{\rm eq}_{s,{\bf k}}}\over {\partial {\bf k}}}
\cr
{\left. \right.} \cr
&& +
\sum_{s',{\bf k'}}
{ {\delta {\cal C}_{s,{\bf k}}}\over {\delta f_{s',{\bf k'}}} }
{\Bigl(
\Delta f^{\rm eq}_{s',{\bf k'}} + \Delta g_{s',{\bf k'}}
\Bigr)},
\label{eq3}
\end{eqnarray}
\noindent
the complete steady-state fluctuation
can be constructed as $\Delta f \equiv \Delta f^{\rm eq} + \Delta g$.
The effect of degeneracy on $\Delta g$ is explicit,
since $\Delta f^{\rm eq} = f^{\rm eq}(1 - f^{\rm eq})$.
Moreover, the leading right-hand term of Eq. (\ref{eq2})
for $g$ itself contains
{~$\partial f^{\rm eq}/\partial {\bf k} \propto \Delta f^{\rm eq}$}. Thus
{\em the very nature of the equilibrium state imposes
a connection between the nonequilibrium one- and two-particle
structures}, $g$ {\em and} $\Delta f$.
\cite{fg}
\subsection{Dynamic Fluctuations}
These are obtained from the linearised form of Eq. (\ref{eq1}) by
calculating its retarded resolvent $R(t)$,
\cite{ks}
with which one then constructs
the transient current autocorrelation $C_{xx}(t)$:
\begin{eqnarray}
R_{s s'; {\bf k} {\bf k'}}(t - t') \equiv&&
\theta(t - t')
{ {\delta f_{s,{\bf k}}(t)}\over {\delta f_{s',{\bf k'}}(t')} }
\label{eq4}
\end{eqnarray}
\noindent
and
\begin{eqnarray}
C_{xx}(t) \equiv&& {e^2\over l^2}
\sum_{s,{\bf k}} \sum_{s',{\bf k'}}
(v_x)_{s,{\bf k}}
{\Bigl(
R_{s s'; {\bf k} {\bf k'}}(t)
\Bigr.} \cr
&& -
{\Bigl.
R_{s s'; {\bf k} {\bf k'}}(\infty)
\Bigr)}
(v_x)_{s',{\bf k'}} \Delta f_{s',{\bf k'}},
\label{eq5}
\end{eqnarray}
\noindent
where $(v_x)_{s,{\bf k}}$ is the group velocity
in the direction of $-{\bf E}$,
and $l$ is the sample length. Eq. (\ref{eq5}) extends
the classical definition of Stanton and Wilkins\cite{swI}
to degenerate systems, with the following interpretation.
At $t = 0$ a spontaneous fluctuation $v'_x \Delta f'$
perturbs the flux in its steady state. Its subsequent fate is
determined by the propagator $R(t)$.
After removal of the steady-state asymptote at $t \to \infty$,
the product
$(-ev_x/l) R (-ev'_x/l) \Delta f'$
gives the dynamical
current-current correlation.
Our premise is that fluctuations in the electron gas
are induced almost instantaneously compared with their
relaxation rate. This allows a two-step calculation.
First, compute the strength of the spontaneous
fluctuations in the steady state
(with the underlying equilibrium statistics included manifestly).
Second, derive their semiclassical dynamics
and hence the noise spectrum.
This prescription satisfies the fluctuation-dissipation theorem.
\cite{fg}
\section{Consequences}
The simplest form of our theory is based on the Drude model
in a single parabolic conduction band, for which
${\cal C} \equiv - g/\tau$; at room temperature
the inelastic collision time $\tau$ is of order $10^{-13}{\rm s}$.
The noise spectral density is
\cite{swI,gc}
\begin{eqnarray}
S(\omega)
\equiv&& 4 \int^{\infty}_0\!\! C_{xx}(t) \cos(\omega t) dt
\cr
{\left. \right.} \cr
=&& {{4Gk_BT}\over {1 + \omega^2 \tau^2}}
{\left[
1 + {\left( {{\Delta n}\over n} \right)}
{{m^* \mu^2 E^2}\over {k_BT}}
\right]},
\label{eq6}
\end{eqnarray}
\noindent
in which $G$ is the sample conductance,
$m^*$ the effective mass, $\mu = e\tau/m^*$ the mobility, and
$\Delta n = 2\Sigma_{\bf k} \Delta f_{\bf k}/\Omega$.
For $E = 0$ in the static limit, $S(0)$ is
the Johnson-Nyquist noise. At high fields
the ``hot-electron'' term $\propto E^2$
in Eq. (\ref{eq6}) is quite strongly suppressed by degeneracy;
while a classical system has $\Delta n/n = 1$,
a $\nu$-dimensional degenerate system
has, instead, $\Delta n/n = \nu k_BT/2\varepsilon_F \ll 1$.
Equations (\ref{eq3}), (\ref{eq5}), and (\ref{eq6})
show $S$ to be a linear functional of $\Delta f^{\rm eq}$. Insofar as
Coulomb and exchange interactions modify the
the free-particle form of the equilibrium
density-density fluctuation, it follows that
{\em thermal noise carries a signature of
the internal correlations of the electron gas}.
Apart from its physical significance,
this has immediate practical implications.
\subsection{Device Noise}
An important consequence for microwave technology
is reduction of thermal noise in a two-dimensional (2D)
electron gas,\cite{gc}
confined at the heterojunction of a
high-electron-mobility transistor (HEMT).
\cite{wv}
Basically, the occupancy within a HEMT is
$f^{\rm eq}_{\bf k} =
\{ 1 + \exp[(\varepsilon_{\bf k} + \varepsilon_0(n)
- \varepsilon_F)/k_BT] \}^{-1}$,
where $\varepsilon_{\bf k}$ is the 2D band energy and $\varepsilon_0(n)$
is the ground-state energy in the heterojunction quantum well.
The density governs $\varepsilon_0$ because
the mean-field electronic potential in the well
is self-consistent. As a result $\Delta f^{\rm eq}$ is renormalised
to $\gamma {\Delta f}^{\rm eq}$ by the suppression factor
\cite{gc}
$\gamma(n) = [1 + (d\varepsilon_0/dn)\Delta n/k_BT]^{-1}$.
The observable noise is therefore reduced by
as much as 65\% for sheet densities
of $10^{12} {\rm cm}^{-2}$, typical in conductive channels.
Under normal conditions this means that the effective noise
temperature in a HEMT is
{\em 100K}, not the 300K of a bulk conductor.
Such suppression is unique to self-consistently quantised systems.
Circuit-theoretical arguments show that, in production
devices, it should lead to
an extrinsic noise figure which is
half that of bulk field-effect transistors,
of otherwise similar circuit performance.\cite{gc}
This agrees well with real device comparisons.\cite{wrb}
\subsection{Mesoscopic Noise}
Sample lengths approaching the mean free path $\lambda$
take us into the mesoscopic regime.
\cite{l+djb}
Even for an embedded
slice of long, spatially uniform conductor,
the propagator $R(t)$ acquires spatial structure
below $\lambda$, satisfying
\begin{eqnarray}
&&{\left[
{\partial\over {\partial t}}
+ {\bf v}_{s,{\bf k}}
{\bbox \cdot}
{\partial\over {\partial {\bf r}}}
- {{e{\bf E}}\over {\hbar}}
{\bbox \cdot}
{\partial\over {\partial {\bf k}}}
\right]}
R_{s s'; {\bf k} {\bf k'}}({\bf r} - {\bf r'}, t - t')
\cr
{\left. \right.} \cr
&& {~} =
\Omega \delta_{{\bf k} {\bf k'}} \delta_{s s'}
\delta({\bf r} - {\bf r'}) \delta(t - t')
\cr
{\left. \right.} \cr
&& {~~~} + \sum_{s'',{\bf k''}}
{ {\delta {\cal C}_{s,{\bf k}}}\over {\delta f_{s'',{\bf k''}}} }
R_{s'' s'; {\bf k''} {\bf k'}}({\bf r} - {\bf r'}, t - t').
\label{eq7}
\end{eqnarray}
\noindent
The sums in Eq. (\ref{eq5}) for $C_{xx}$ now include
integrals over the mesoscopic slice, while
the macroscopically homogeneous propagator of Eq. (\ref{eq4})
re-emerges as $\int\!d^\nu r R/\Omega$.
Using Eqs. (\ref{eq7}), (\ref{eq5}), and (\ref{eq6})
we have computed the thermal-noise spectrum
for an embedded uniform 1D wire, in the Drude model
previously studied by Stanton\cite{swt} within the
classical limit $k_BT \gg \varepsilon_F$.
The example, although suggestive, is artificial not least because
``wire'' and ``leads'' are operationally indistinguishable.
We examine the degenerate limit, for which $\lambda = \tau v_F$
in terms of the Fermi velocity. At zero frequency the
equilibrium noise is given by
\begin{equation}
S^{\rm eq}(0) = 4Gk_BT
{\left[
1 - {\lambda\over l}(1 - e^{-l/\lambda})
\right]},
\label{eq8}
\end{equation}
\noindent
where $l$ is the length of the wire segment, much smaller
than the enclosing 1D ``volume'' $\Omega$. In the limit $l \ll \lambda$
Eq. (\ref{eq8}) reduces to $S^{\rm eq}(0) = (4e^2/\pi \hbar)k_BT$.
Since this is the ballistic regime we note that the thermal noise
scales with $e^2/\hbar$,
the universal unit of conductance.\cite{l+djb}
On the other hand the nonequilibrium ballistic noise is
\begin{equation}
S(0) = {{4e^2}\over {\pi \hbar}}k_BT
{\left[
1 + {\left( 1 + {{\mu E}\over v_F} \right)}
\exp{\left( -{{2v_F}\over {\mu E}} \right)}
\right]}.
\label{eq9}
\end{equation}
\noindent
The expression is nonperturbative in $E$
owing to nonanalyticity of the solutions to
Eqs. (\ref{eq2}) and (\ref{eq3})
for a uniform nonequilibrium system.
\cite{bw,swt}
Our exact, if simple, model calculation shows
that one cannot take for granted
the existence of an expansion for the fluctuations
near the equilibrium state, in powers of the field.
At large fields the ballistic noise becomes linear:
\cite{swt}
\begin{eqnarray}
S(0) =&& {{4e^2}\over {\pi \hbar}}k_BT
{\left( {{\mu E}\over v_F} \right)} + {\cal O}(E^{-2})
\cr
{\left. \right.} \cr
\rightarrow&& 4Gk_BT{\left( {{eV}\over {4\varepsilon_F}} \right)},
\label{eq10}
\end{eqnarray}
\noindent
where $V = El$ is the voltage across the wire
(still with $l \ll \lambda$),
and we have reinstated the Johnson-Nyquist normalisation.
Two remarks can be made on this equation:
$(a)$ the linear dependence on $E$ is kinematic, reflecting the
shift of the centroid of the fluctuation distribution
in $k$-space by $eE\tau/\hbar$. In contrast,
the quadratic dependence of the macroscopic noise in
Eq. (\ref{eq6}) is dissipative, and sensitive to
thermal broadening of $\Delta f_k$ over the bulk.
$(b)$ The second form of Eq. (\ref{eq10})
is equivalent to $S(0) = 2eI(\Delta n/n)$,
where $I = GV$ is the current through the wire.
Thus, ballistic thermal noise in the 1D Drude approximation
is given by the classical shot-noise formula
\cite{l+djb,swt}
attenuated by the degeneracy, a result valid for all
densities and temperatures.
While thermal noise must go to zero with the temperature,
this is not the case for true shot noise, whose non-thermal
origin is the random transit of individual carriers
across the sample.
It is beyond our present scope to discuss applications of the
Boltzmann-Green-function formalism to diffusive mesoscopic shot noise,
\cite{l+djb,smd,sbkpr}
which we are actively investigating.
\section{Summary}
We have outlined a semiclassical framework for calculating thermal
fluctuations in metallic electron systems far from equilibrium.
Our approach also describes how Coulomb and exchange correlations,
present at equilibrium, appear in the
nonequilibrium current noise. The consequences
for device design are exemplified by the physics of
correlation-induced noise suppression
in heterojunction field-effect transistors.
The same formalism can be applied to
diffusive mesoscopic noise. An illustrative 1D model provides
evidence that mesoscopic noise may not always have a perturbation
expansion at low fields, if the underlying distributions
are nonanalytic in the equilibrium limit.
At high fields the model recovers the shot-noise-like behaviour of
ballistic thermal noise.\cite{swt}
For a metallic wire we find that this is
attenuated in proportion to the degeneracy of the system.
|
1,314,259,993,602 | arxiv | \section{Introduction}
\label{sec:introduction}
Observations of cosmological perturbations on quasilinear scales, with the cosmic microwave background (CMB) and large scale structure (LSS), have resulted in precise measurements of fundamental cosmological parameters.
However, quasilinear scales contain only a small fraction of the theoretically accessible information.
On smaller scales, traditional $N$-point correlation function analysis becomes sub-optimal, and inference methods such as forward modelling \cite{Seljak:2017rmr,2013MNRAS.432..894J} or machine learning \cite{Ravanbakhsh:2017bbi,Villaescusa-Navarro:2021pkb,Lazanu:2021tdl,Villaescusa-Navarro:2021cni,Hortua:2021vvj} can provide stronger constraints.
Here, we will focus on inference of the important primordial physics parameter $f_{NL}$, which arises in multi-field models of inflation \cite{Linde_1997, Dvali_2004, Bartolo_2004, Biagetti_2019}.
In such models, the primordial Bardeen potential $\Phi({\bf x})$ can be parameterized as:
\begin{equation}
\Phi({\bf x}) = \Phi_G({\bf x}) + f_{NL}(\Phi_G({\bf x})^2 - \langle \Phi_G^2 \rangle) \label{eq:pbs_fnl}
\end{equation}
where $\Phi_G$ is a Gaussian field and $f_{NL}$ quantifies the level of non-Gaussianity. Constraining $f_{NL}<1$ is a major goal of upcoming galaxy surveys \cite{Alvarez:2014vva}.
Several statistical approaches for estimating $f_{NL}$ in LSS have been proposed, including the squeezed bispectrum \cite{MoradinezhadDizgah:2020whw} and the scale-dependent power spectrum approach \cite{Dalal:2007cu, Slosar:2008hx} together with the idea of sample variance cancellation using a variety of probes \cite{Seljak:2008xr, Smith:2018bpn, Munchmeyer:2018eey, Giri:2020pkk}.
In this paper, we will propose a neural network (NN) based approach to estimating $f_{NL}$.
We will show that our NN-based analysis obtains significantly smaller error bars than an analysis based on large-scale matter and halo fields.
This is perhaps unsuprising since a neural network with enough capacity, trained on enough simulations, should give {\em statistically} optimal parameter constraints.
However, {\em robustness} is a central challenge for neural networks.
As is widely appreciated, simulations on small scales suffer from baryonic feedback uncertainties and are not reliable at the sub-percent level accuracy required to tighten parameter bounds.
For example, an NN trained to measure $\sigma_8$ on one set of simulations will likely give incorrect results on different simulations or on real data \cite{Villanueva-Domingo:2022rvn,Villaescusa-Navarro:2020rxg}.
This problem is not unique to NN-based methods.
In particular, the average dark matter halo density $\bar{n}_h$ is statistically a precise probe of $\sigma_8$, but it is not robust because $\bar n_h$ is sensitive to uncertain local physics.
However, {\em anisotropy} in the halo field can be used to place robust constraints on $f_{NL}$.
A famous result \cite{Dalal:2007cu} states that if $f_{NL}\ne 0$, the halo bias $b_h(k)$ contains a characteristic $1/k^2$ term, which cannot be induced by local physics.
The idea of this paper is to replace the halo field by an NN-based local estimate $\pi({\bf x})$ of $\sigma_8$.
As in the halo case, $\bar\pi$ is not robust as an absolute measurement of $\sigma_8$, but the bias $b_\pi(k)$ contains a term proportional to $f_{NL}/k^2$, which can be used to place robust constraints on $f_{NL}$.
We will show that this approach combines the statistical power of neural networks with the robustness of the traditional halo-based analysis.
\section{Formalism}
\label{sec:formalism}
In an $f_{NL}$ cosmology, the halo bias $b_h(k)$ contains a term proportional to $f_{NL}/k^2$ \cite{Dalal:2007cu}.
In the next few paragraphs, we review the derivation, in a language which will generalize to NN-based observables.
If $f_{NL}\ne 0$, the locally observed amplitude of short-wavelength modes is a function $\sigma_8^{\rm loc}({\bf x})$ of position.
On large scales ${\bf k}_L \rightarrow 0$, anisotropy in $\sigma_8^{\rm loc}$ is related to the potential $\Phi$ by:
\begin{equation}
\frac{\sigma_8^{\rm loc}({\bf k}_L)}{\bar\sigma_8} = 2 f_{NL} \, \Phi({\bf k}_L)
\label{eq:pbs_sigma8_loc}
\end{equation}
The coupling between large and small scales described by Eq.\ (\ref{eq:pbs_sigma8_loc}) arises from the term $f_{NL} \Phi_G^2$ in Eq.\ (\ref{eq:pbs_fnl}), which mixes scales.
For a formal derivation of (\ref{eq:pbs_sigma8_loc}), see \cite{Slosar_2008, Biagetti_2019}.
On large scales, the local halo abundance $\delta_h$ is sensitive to both $\sigma_8^{\rm loc}$ and the matter overdensity $\delta_m$:
\begin{equation}
\delta_h({\bf x}) = b^G_h \delta_m({\bf x})
+ \frac{1}{2} b^{NG}_h \log\left( \frac{\sigma_8^{\rm loc}({\bf x})}{\bar\sigma_8} \right)
+ (\mbox{noise})
\end{equation}
The first term $b^G_h \delta_m$ is the usual (Gaussian) halo bias, and the second term $b^{NG}_h \log(\sigma_8^{\rm loc})$ is a non-Gaussian bias term.
The coefficient $b^{NG}_h$ is the derivative of the halo density $\bar n_h$ with respect to the cosmological parameter $\sigma_8$ \cite{Slosar_2008,Baldauf_2011,Desjacques:2016bnm,Biagetti_2017}:
\begin{equation}
b^{NG}_h = 2 \frac{\partial \log \bar n_h}{\partial\log\sigma_8}
\label{eq:bh_ng}
\end{equation}
(Sometimes the approximation $b^{NG}_h \approx 2 \delta_c (b^G_h-1)$ is used, but we will not use it in this paper.)
On linear scales, the density field $\delta_m$ and potential $\Phi$ are related via the Fourier-space Poisson equation $\delta_m({\bf k}) = \alpha({\bf k},z)\Phi({\bf k})$ \cite{Dodelson:2003ft}. Here, $\alpha(k,z)$ is given by %
\begin{equation}
\alpha(k,z) \equiv \frac{2k^2 T(k) D(z)}{3\Omega_m H_0^2}
\label{eq:alpha_def}
\end{equation}
Combining Eqs.\ (\ref{eq:pbs_sigma8_loc})--(\ref{eq:alpha_def}), we derive the NG halo bias:
\begin{equation}
\delta_h({\bf k}_L) = b_h(k_L) \delta_m({\bf k}_L) + (\mbox{Poisson noise})
\end{equation}
where:
\begin{equation}
b_h(k) = b^G_h + b^{NG}_h \frac{f_{NL}}{\alpha(k,z)}
\end{equation}
In this paper, we generalize the preceding results as follows.
First we note that they do not depend on any specific properties of halos, other than the local halo abundance $n_h({\bf x})$ being sensitive to $\sigma_8^{\rm loc}({\bf x})$.
Any field $\pi({\bf x})$ which is derived from the nonlinear density field, in a reasonably local way, should have the same property.
We propose constructing a field $\pi({\bf x})$ using a neural network trained to maximize sensitivity to $\sigma_8^{\rm loc}$.
We assume that our input data consists of the nonlinear density field $\delta_m({\bf x})$ as a 3-d pixelized map at $\sim$2 Mpc resolution without noise.
Thus, our constraints on $f_{NL}$ should be interpreted as information content in principle, given complete information at a specified resolution.
In future work, we plan to apply our approach to simulated galaxy catalogs, which are more representative of real data.
We define a field $\pi({\bf x})$ by applying a CNN to $\delta_m({\bf x})$ (Figure \ref{fig:schematic_cnn}).
The CNN has a small receptive field ($\sim$18 $h^{-1}\, $Mpc), so that the output field $\pi({\bf x})$ is fairly local in the input field $\delta_m({\bf x})$.
As described in \S\ref{sec:pipeline}, we train the CNN so that $\pi({\bf x})$ is an estimate of $\sigma_8^{\rm loc}({\bf x})$ with low statistical noise.
Following the logic above for halos, we make the following predictions for the behavior of $\pi({\bf x})$ on large scales.
First, we predict that the matter-$\pi$ and $\pi$-$\pi$ power spectra are given by:
\begin{align}
P_{m\pi}(k) &= b_\pi(k) P_{mm}(k) \label{eq:P_m_pi} \\
P_{\pi\pi}(k) &= b_\pi(k)^2 P_{mm}(k) + N_{\pi\pi} \label{eq:P_pi_pi}
\end{align}
where the linear bias $b_\pi(k)$ is the sum of Gaussian (constant) and non-Gaussian terms:
\begin{equation}
b_\pi(k) = b^G_\pi + b^{NG}_\pi \frac{f_{NL}}{\alpha(k,z)}
\label{eq:bpi_general}
\end{equation}
We also predict that the non-Gaussian bias $b^{NG}_\pi$ is related to the $\sigma_8$ dependence of the mean $\pi$-field:
\begin{equation}
b^{NG}_\pi = 2 \frac{\partial\bar\pi}{\partial\log\sigma_8}
\label{eq:bngpi_general}
\end{equation}
Finally, we predict that the noise $N_{\pi\pi}$ defined in (\ref{eq:P_pi_pi}) is constant in $k$.
In \S\ref{sec:mcmc}, we will verify these predictions and show that they lead to strong constraints on $f_{NL}$.
\begin{figure*}
\includegraphics[width=0.80\linewidth]{plots/figure1.pdf}
\caption{A 2D schematic representation of our CNN architecture.
The input is the nonlinear 3-d density field $\delta_m({\bf x})$ from an $N$-body simulation, and we train the network so that the output 3-d field $\pi({\bf x})$ is an estimate of $\sigma_8$.
The total receptive field size is $(9 \times 9 \times 9)$ voxels, equivalent to $(18\ h^{-1} \mbox{Mpc})^3$.
Each convolution except the last is followed by a ReLU activation function.
Each grey square represents a logical array of size $(512\times 512 \times 512)$ with periodic boundary conditions.
However, as an implementation detail to reduce GPU memory usage, we divide the simulation volume into slightly overlapping subvolumes which can be processed independently.
}
\label{fig:schematic_cnn}
\end{figure*}
\section{Neural Network}
\label{sec:pipeline}
\vskip5pt
\noindent
{\bf Architecture:}
Our neural network uses a fully convolutional, sliding-window architecture with a total of 16433 parameters (Figure \ref{fig:schematic_cnn}).
The network takes the 3D matter density field $\delta_m({\bf x})$ from an $N$-body simulation, and produces an output field $\pi({\bf x})$ with the same resolution as the input.
We use small convolution kernels, including several layers with $(1\times 1\times 1)$ kernels, so that the total receptive field of the network will be small (18 $h^{-1}\,$Mpc).
The size of the receptive field limits the scales which the neural network can use for estimating $\pi$ and thus enforces locality.
We leave systematic exploration of neural network architecture to future work.
\vskip5pt
\noindent
{\bf Simulations:}
We want to train the network so that its output field $\pi({\bf x})$ is an optimal estimate of $\sigma_8$.
To do this, we need a training set of $N$-body simulations with multiple values of $\sigma_8$.
We use the \verb|s8_p| and \verb|s8_m| datasets from the \texttt{Quijote} simulations \cite{Villaescusa-Navarro:2019bje}, with $\sigma_8=0.849$ and 0.819 respectively.
The remaining cosmological parameters are $\Omega_{\rm m}=0.3175$, $\Omega_{\rm b}=0.049$, $h=0.6711$, $n_s=0.9624$, and $w=-1$.
Each dataset contains 400 collisionless simulations.
Each simulation has $512^3$ particles and volume $(1\ h^{-1} \mbox{Gpc})^3$. For each simulation, we inpaint particles from the $z=0$ snapshot on a $512^3$ 3D mesh using the Cloud-in-Cell algorithm implemented in \texttt{nbodykit}. This produces a voxelized 3D matter density field $\delta_m({\bf x})$, which we save to disk for neural network training. We require only two values of $\sigma_8$ in the training data rather than a continuum, because the variance of $\pi({\bf x})$ per receptive field is much larger than the difference between the two $\sigma_8$ values.
\vskip5pt
\noindent
{\bf Loss function and optimizer:}
For each simulation, let $\pi({\bf x})$ be the CNN output, and let $\sigma_8^{\rm true}$ be the value of $\sigma_8$ in the simulation. We define the loss function:
\begin{equation}
\mathcal{L} = \Bigg[
\Bigg(
\frac{1}{N_{\rm voxels}} \sum_{{\rm voxels}\ {\bf x}} \pi({\bf x})
\Bigg) - \sigma_8^{\rm true}
\Bigg]^2
\label{eq:loss}
\end{equation}
Intuitively, minimizing $\mathcal{L}$ should produce an output field $\pi({\bf x})$ which is an optimal estimate of $\sigma_8$, by minimizing the difference between $\sigma_8^{\rm true}$ and the spatially averaged $\pi$-field.
We use the Adam optimizer \cite{2014arXiv1412.6980K} with a learning rate of $(5 \times 10^{-5})$ to minimize the loss function. The learning rate is reduced by a factor of 0.7 whenever the loss fails to register any improvement for 5 successive epochs of training. The architecture is implemented in \texttt{PyTorch} \cite{https://doi.org/10.48550/arxiv.1912.01703} and uses \texttt{PyTorch-lightning} \cite{falcon2019pytorch} for high level interfacing with mixed precision training \cite{https://doi.org/10.48550/arxiv.1710.03740}.
We find that the overall normalization $W$ and additive bias $b$ of the NN are slow to converge, so as a final training step, we fix all parameters except $(W,b)$, and minimize the loss (\ref{eq:loss}).
This minimization can be done exactly in a single epoch, since the loss is a quadratic function of $(W,b)$.
\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{plots/figure2.pdf}
\caption{Estimating $\sigma_8$ using the neural network from Figure\ \ref{fig:schematic_cnn}.
{\em Top panel.} Histogrammed NN estimates $\bar\pi = V_{\rm box}^{-1} \int_{{\bf x}} \pi({\bf x})$ on a test set of simulations with $\sigma_8=0.819$ (green) and $\sigma_8=0.849$ (blue).
{\em Bottom panel.} Histogrammed halo counts from the same test set, showing worse statistical separation between $\sigma_8$ values than the NN.}
\label{fig:sigma8_histograms}
\end{figure}
\comment{
\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{plots/projection_plot_10_percent.eps}
\caption{2D slice of the $\ln(\delta_m)$ and $\ln(\pi)$ fields projected over 50$ \ h^{-1}\mathrm{Mpc}$~ along the z-axis. Large-scale correlations between the fields is visually evident.}
\label{fig:my_label}
\end{figure}
}
\vskip5pt
\noindent
{\bf Validation:}
Our NN has been trained so that the spatially averaged $\pi$ field $\bar\pi = V_{\rm box}^{-1} \int_{{\bf x}} \pi({\bf x})$ is an estimate of $\sigma_8$ with lowest possible noise.
In the top panel of Figure \ref{fig:sigma8_histograms}, we verify this statement, by evaluating $\bar\pi$
on a test set of 100+100 simulations with $\sigma_8 \in \{ 0.819, 0.849 \}$.
We see that the network recovers the correct value of $\sigma_8$, and that the NN obtains better statistical separation between $\sigma_8$ values than counting halos (bottom panel).
\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{plots/figure3.pdf}
\caption{\emph{Top panel.} Bias model (\ref{eq:bpi_specific}) for the neural network output field $\pi({\bf x})$, compared to the empirical bias $P_{m\pi}(k)/P_{mm}(k)$ from simulation, for $f_{NL} \in \{0,250\}$. \emph{Bottom panel.} Power spectrum of the residual field $\epsilon({\bf k}) = \pi({\bf k}) - b_\pi(k) \delta_m({\bf k})$ compared to a best-fit constant $N_\pi$.
Throughout this figure, best-fit model parameters $(b^G_\pi, N_\pi)$ are obtained from the MCMC pipeline described in \S\ref{sec:mcmc}, with $k_{\rm max}=0.014$ $h\, \mathrm{Mpc}^{-1}$ (shown as the shaded region).}
\label{fig:conjecture_plot}
\end{figure}
\section{Estimating $f_{NL}$}
\label{sec:mcmc}
In this section, we apply our neural network to simulations with $f_{NL}\ne 0$.
We use a test set of 10 $N$-body simulations with $\sigma_8=0.834$, $f_{NL}=250$, and non-Gaussian initial conditions generated using the Zeldovich approximation.
\vskip5pt
\noindent
{\bf Power spectra:}
We next verify the predictions in Eqs.\ (\ref{eq:P_m_pi})--(\ref{eq:bngpi_general}) for the large-scale power spectra $P_{m\pi}$, $P_{\pi\pi}$ in an $f_{NL}$ cosmology.
First, we note that since $\bar\pi = \sigma_8$ (Fig.\ \ref{fig:sigma8_histograms}), our prediction (\ref{eq:bngpi_general}) for the non-Gaussian bias $b^{NG}_\pi$ is:
\begin{equation}
b^{NG}_\pi = 2 \frac{\partial\bar\pi}{\partial\log\sigma_8} = 2 \sigma_8 \label{eq:bngpi_specific}
\end{equation}
and so our prediction (\ref{eq:bpi_general}) for the total bias $b_\pi(k)$ is:
\begin{equation}
b_\pi(k) = b_\pi^G + 2 \sigma_8 \frac{f_{NL}}{\alpha(k,z)} \label{eq:bpi_specific}
\end{equation}
In the the top panel of Fig.\ \ref{fig:conjecture_plot}, we compare the bias model (\ref{eq:bpi_specific}) to the empirical bias obtained from cross-correlating $\pi$ and $\delta_m$ in $k$-bins, and find good agreement.
In the bottom panel of Fig. \ref{fig:conjecture_plot}, we verify the prediction that the noise power spectrum $N_\pi$ defined in Eq.\ (\ref{eq:P_pi_pi}) is constant in $k$, by plotting the power spectrum of the residual field $\epsilon({\bf k}) = \pi({\bf k}) - b_\pi(k) \delta_m({\bf k})$.
We emphasize that simulations with $f_{NL}\ne 0$ were never seen during the training process.
The predictions in Eqs.\ (\ref{eq:P_m_pi})--(\ref{eq:bngpi_general}) for power spectra in an $f_{NL}$ cosmology (in particular the prediction $b_\pi^{NG} = 2\sigma_8$) are based entirely on the NN response to varying $\sigma_8$, and general considerations of locality.
Therefore, the verification of these predictions is a strong test of our formalism.
\vskip5pt
\noindent
{\bf MCMC pipeline:}
Now that our model for the power spectra $P_{m\pi}$, $P_{\pi\pi}$ has been verified, we develop an MCMC pipeline that combines large-scale modes of $\pi({\bf k})$ and $\delta_m({\bf k})$ in a joint analysis.
\begin{figure*}
\begin{subfigure}%
\centering
\includegraphics[width=0.49\linewidth]{plots/figure4a.pdf}
\label{fnl_constraint_halo}
\end{subfigure}
\begin{subfigure}%
\centering
\includegraphics[width=0.49\linewidth]{plots/figure4b.pdf}
\label{fig:fnl_constraint_pi}
\end{subfigure}
\caption{MCMC posteriors on $f_{NL}$ and nuisance parameters (either $(b^G_h,N_{hh})$ or $(b^G_\pi,N_{\pi\pi})$) from joint analysis of 100 Quijote simulations with $f_{NL} = 0$.
\textit{Left.} Traditional halo based analysis using large-scale modes of the matter field $\delta_m({\bf k})$ and halo field $\delta_h({\bf k})$.
\textit{Right.} Neural network based analysis using $\delta_m({\bf k})$ and the NN output field $\pi({\bf k})$. The neural network reduces the error bar on $f_{NL}$ by a factor $\sim$3.5.}
\label{fig:fnl_constraint}
\end{figure*}
\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{plots/figure5.pdf}
\caption{MCMC posterior on $f_{NL}$ from joint analysis of 10 $N$-body simulations with $f_{NL}=250$, using either the matter+halo fields (red), or matter+$\pi$ fields (black), where $\pi({\bf x})$ is the NN output field.
The 1-d $f_{NL}$ likelihoods are marginalized over nuisance parameters (either $(b^G_h,N_{hh})$ or $(b^G_\pi,N_{\pi\pi})$).}
\label{fig:fnl250_posterior}
\end{figure}
\comment{
\vskip5pt
\noindent
{\bf Model:} For $N$-body simulations with Gaussian initial condition, the field $\pi_i$ on large scales can be expressed using leading-order perturbative bias \footnote{Higher-order and non-local contributions are negligible for case in hand.} expansion
\begin{equation}
\pi_i = b_\pi \delta + \epsilon_i
\end{equation}
where $\epsilon_i$ is random gaussian noise uncorrelated with $\delta$ with power spectrum defined by
\begin{equation}
\langle \epsilon_i(k) \epsilon_i(k`) \rangle = (2\pi)^{3} \delta^3(k-k`)N_{\pi_i\pi_i}
\end{equation}
The bias expansion has a simple physical interpretation under the \textit{peak-background split} formalism where the bias is
\begin{equation}
b_{\pi_i} = \frac{\partial \bar{\pi}_i}{\partial \delta_l}
\end{equation}
where $\delta_l$ is large-wavelength matter mode such that $\delta = \delta_l + \delta_s$. \\
\noindent In an $f_{nl}\neq 0$ universe, the expansion becomes
\begin{equation}
\pi_i = \Bigg(b_{\pi_i} + \frac{2\beta_{\pi_i} f_{nl}}{\alpha}\Bigg)\delta_m + \epsilon_i
\end{equation}
where $\beta_{{\pi}_i}$ is the non-Gaussian bias of the $\pi$ field given under PBS by
\begin{equation}
\beta_{\pi_i} = \frac{\partial \ln \bar{\pi}_i }{\partial \ln \sigma_8}
\end{equation}
\utkarsh{add remaining details in appendix??}
}
The Gaussian likelihood for our data vector $\mathcal{D}=[\delta_m$, $\pi]$ given model parameters $\Theta = (f_{NL}, b^G_\pi, N_{\pi\pi})$ is:
\begin{equation}
\mathcal{L}(\Theta|\mathcal{D}) \propto \prod_{k}
\frac{1}{\sqrt{\mbox{Det}\, C(k)}}
\exp(-\frac{{\mathcal D}({\bf k})^{\dagger} C(k)^{-1} {\mathcal D}({\bf k})}{2V}) \nonumber
\end{equation}
where the $2\times2$ covariance matrix $C(k)$ is:
\begin{align}
C(k) &=
\begin{bmatrix}
P_{mm}(k) & P_{m\pi}(k)\\
P_{m\pi}(k) & P_{\pi \pi}(k) %
\end{bmatrix} \nonumber \\
&=
\begin{bmatrix}
P_{mm}(k) & b_\pi(k) P_{mm}(k)\\
b_\pi(k) P_{mm}(k) & b_\pi(k)^2 P_{mm}(k) + N_{\pi\pi}
\end{bmatrix}
\end{align}
with $b_\pi(k)$ given by Eq.\ (\ref{eq:bpi_specific}).
We truncate the likelihood at $k_{\rm max}=0.014$ $h\, \mathrm{Mpc^{-1}}$. The posterior is defined using flat priors over a reasonable range for the model parameters $\Theta$. We sample the posterior using affine-invariant sampling implemented in \texttt{emcee}\cite{Foreman_Mackey_2013} to obtain constraints on $f_{NL}$.
To compare the neural network to a traditional halo based analysis, we also run our MCMC pipeline using the halo field $\delta_h({\bf k})$ instead of the NN-derived field $\pi({\bf k})$. The only change is that we replace $b^{NG}_\pi = 2 \sigma_8$ by the non-Gaussian halo bias $b^{NG}_h$, which we measure in simulations using Eq.\ (\ref{eq:bh_ng}).
\vskip5pt
\noindent \textbf{MCMC results:}
We begin by analysing $N$-body simulations with Gaussian initial conditions \textit{i}.\textit{e}.\ $f_{NL}=0$. %
We jointly analyze 100 \texttt{fiducial} Quijote simulations by multiplying together their posteriors before sampling.
In the right panel of Fig.\ \ref{fig:fnl_constraint}, we show $f_{NL}$ constraints from a joint analysis of the large-scale matter density $\delta_m({\bf k})$ and the NN-derived field $\pi({\bf x})$.
In the left panel, we show a similar analysis using $\delta_m({\bf k})$ and the {\em halo} field $\delta_h({\bf k})$.
In both cases, the result is consistent with $f_{NL}=0$ as expected.
However, the neural network gives an $f_{NL}$ error which is 3.5 times better than the halo based analysis!
In the left panel of Figure \ref{fig:fnl_constraint}, we used a single halo field consisting of all halos with $\ge 20$ particles ($M_{\rm min} = 1.3 \times 10^{13}$ $h^{-1}\,M_\odot$).
We checked that if narrow halo mass bins are used with optimal weighting, the Fisher forecasted error $\sigma(f_{NL})$ is only 25\% better than the single-bin case.
In Fig.\ \ref{fig:fnl250_posterior}, we show $f_{NL}$ constraints from a joint MCMC analysis of 10 simulations with $f_{NL}=250$.
We can see that for these non-Gaussian simulations, the correct value of $f_{NL}$ is recovered, and the NN improvement over halos is just as good as in the $f_{NL}=0$ case. %
\vskip5pt
\noindent \textbf{Robustness:}
To frame the issue of robustness concretely, imagine that the small-scale astrophysics in the real universe is slightly different from the training set.
How will our constraints on $f_{NL}$ be affected?
Since the parameters $(b^G_\pi, b_\pi^{NG}, N_\pi)$ are sensitive to small-scale physics, their values will differ slightly from the training set.
For $b^G_\pi$ and $N_\pi$, this is harmless since we marginalize these parameters in our MCMC anyway.
For $b^{NG}_\pi$, we note that in the formalism from \S\ref{sec:formalism}, the parameters $b^{NG}_\pi$ and $f_{NL}$ only appear in the combination $(b^{NG}_\pi f_{NL})$.
Therefore, a small change in $b^{NG}_\pi$ is equivalent to a change in the normalization of $f_{NL}$ -- it cannot ``fake'' a detection of nonzero $f_{NL}$.
Physically, this is because local physics cannot generate a term in the bias $b_\pi(k)$ proportional to $1/k^2$.
This is qualitatively similar to the familar case of halo counts.
Our method does depend on having a rough estimate for $b^{NG}_\pi$ based on training data.
As a check, we estimated the cosmological parameter dependence of $b^{NG}_\pi$ using the Quijote ``latin hypercube'' simulations, which vary cosmological parameters over wide ranges.
We find that $\bar\pi$ is well modelled by a quadratic polynomial in $(\Omega_m,\Omega_b,h,n_s,\sigma_8)$.
Using this quadratic model, we find that if cosmological parameters are varied within Planck+BAO $2\sigma$ errors \cite{Planck:2018vyg}, the change in $b^{NG}_\pi = 2 (\partial\bar\pi/\partial\log\sigma_8)$ is $\le 1$\%.
In future work, we hope to extend this analysis to study dependence of $b^{NG}_\pi$ on subgrid physics.
\section{Conclusion and Outlook}
\label{sec:conclusion}
In this letter we have demonstrated that the statistical power of neural networks can be combined with
the idea of $(1/k^2)$ non-Gaussian bias
to arrive at a robust measurement of $f_{NL}$ from the matter distribution.
Unlike forward modelling approaches which are difficult at strongly non-linear scales, our approach can use information from very small scales and still remain robust.
Our main next step will be to quantify to what extent the method presented here can improve $f_{NL}$ constraints from realistic galaxy surveys,
rather than the matter field.
Machine learning based $\sigma_8$ constraints from simulated galaxy distributions have been examined in \cite{Ntampaka_2020} (using a halo occupation distribution), \cite{Villanueva-Domingo:2022rvn} (using the hydrodynamic CAMELS simulations) and \cite{Perez:2022nlv} (using a semi-analytic galaxy formation model). In particular, \cite{Villanueva-Domingo:2022rvn} highlighted the problem that different baryonic subgrid models lead to inconsistent results, which is precisely the issue our method is designed to overcome for $f_{NL}$. Recently, \cite{Valogiannis:2021chp} introduced a non-linear estimator based on the Wavelet Scattering Transform (WST) and even applied it to BOSS data \cite{Valogiannis:2022xwu} to extract cosmological parameters including $\sigma_8$. The WST behaves similarly to a neural network and thus the claimed improvements in $\sigma_8$ suggest that our $f_{NL}$ method could also work well for galaxies. We will investigate this question in detail in upcoming work.
A straightforward generalization of our method is to other scale-dependent biases, such as those induced by the trispectrum $g_{NL}$ parameter \cite{Smith_2012}, neutrino masses \cite{Chiang:2017vuk}, and isocurvature perturbations \cite{Barreira:2019qdl}.
More generally, our approach of using a neural network as a local probe
may generalize to other observables which are large-scale modulations of local non-Gaussian fields or cross-correlations. This is a common setup in cosmology, often exploited for quadratic estimators. \\
\section*{Acknowledgements}
Part of this work was performed at the Aspen Center for Physics, which is supported by National Science Foundation grant PHY-1607611.
MM acknowledges support from DOE grant DE-SC0022342.
KMS was supported by an NSERC Discovery Grant and a CIFAR fellowship.
Research at Perimeter Institute is supported in part by the Government of Canada through the Department of Innovation, Science and Economic Development Canada and by the Province of Ontario through the Ministry of Colleges and Universities.
Perimeter Institutes’s HPC system ``Symmetry'' was used to perform some of the analysis presented in the letter. We have extensively used several python libraries including \texttt{numpy}\cite{harris2020array}, \texttt{matplotlib}\cite{Hunter:2007}, \texttt{CLASS}\cite{Blas_2011}, \texttt{getdist}\cite{Lewis:2019xzd} and \texttt{SciencePlots}\cite{SciencePlots}.
|
1,314,259,993,603 | arxiv | \section{Introduction}
The notion of $p$-adic modular forms was introduced by Serre in the study of congruences between modular forms. It is well-known that to get a better spectral theory of the $U_p$-operator, one should consider the subspace of \textit{overconvergent} modular forms, on which $U_p$ acts completely continuously. In this short note, we will show that Hecke operators away from $p$ also have a better convergence when acting on overconvergent modular forms. As a consequence, we deduce that the action of the (big) Hecke algebra $\mathbb{T}$ naturally extends to an action of the rigid functions on its generic fiber (denoted by $\mathbb{T}^{\mathrm{rig}}$ by some people). Since having a Hodge-Tate-Sen weight $0$ is a Zariski-closed property on $\Spec\mathbb{T}^{\mathrm{rig}}$, the density of classical points implies directly that
\begin{thm}[Corollary \ref{HTS}]
The two dimensional semi-simple Galois representation associated to an overconvergent eigenform of weight $k\in\mathbb{Z}$ has Hodge-Tate-Sen weights $0,k-1$.
\end{thm}
This confirms a conjecture of Gouv\^{e}a \cite[Conjecture 4]{Gou88}. We remark that this result was recently obtained by myself in \cite{Pan20} and by Sean Howe independently in \cite{Howe20} (when $k\neq 1$), by relating overconvergent modular forms with completed cohomology. Our method here is more straightforward. Hopefully it will be clear to the readers that the argument can be easily generalized to other contexts.
This note is organized as follows. We will first introduce a class of actions of algebras on a $p$-adic Banach space called \textit{locally analytic action} and give several (simple) examples. Then using fake-Hasse invariants introduced by Scholze \cite{Sch15}, we show that the action of the Hecke algebra on the space of overconvergent modular forms (with fixed radius) is locally analytic. As suggested by Matthew Emerton, this also reproves a result of Calegari-Emerton. At the end, we also discuss a similar phenomenon in the context of locally analytic vectors of completed cohomology.
\subsection*{Acknowledgement}
I would like to thank Matthew Emerton for his comments on an earlier draft of this note and the anonymous referee for many helpful comments on this paper.
\section{Locally analytic action}
\begin{defn}
Let $W$ be a $p$-adic Banach space over $\mathbb{Q}_p$. A continuous linear operator $T\in\End(W)$ is called \textit{locally analytic} if there exists a monic polynomial $f(X)\in\mathbb{Z}_p[X]$ such that $f(T)(W^o)\subset pW^o$, where $W^o$ denotes the unit ball of $W$.
\end{defn}
Note that for a locally analytic operator $T$, if $W^o$ is $T$-stable, then the image of $T$ in $\End(W^o/pW^o)$ generates a finite $\mathbb{F}_p$-algebra.
\begin{exa}
Suppose $W$ is a finite dimensional vector space over $\mathbb{Q}_p$. Then any linear operator of norm $\leq 1$ is locally analytic by considering its characteristic polynomial.
\end{exa}
\begin{exa}
Suppose $W=\mathbb{Q}_p\langle X\rangle$, the ($p$-adic) completion of $\mathbb{Q}_p[X]$ with respect to the unit ball $\mathbb{Z}_p[X]$. Let $T\in\End(\mathbb{Q}_p\langle X\rangle)$ be the translation $X\mapsto X+1$. It is locally analytic because $(T^p-1)\cdot F(X)=F(X+p)-F(X)\in p\mathbb{Z}_p[X]$ for any $F(X)\in\mathbb{Z}_p[X]$.
\end{exa}
Recall that an operator $T$ on $W$ is called topologically nilpotent if $\displaystyle \lim_{n\to\infty}T^n\cdot v=0$ for any $v\in W$, i.e. the sequence $\{T^n\}_{n\geq 0}$ converges to zero in the space of linear operators on $W$ with respect to the \textit{weak topology}.
\begin{prop} \label{tnun}
Let $W$ be a $p$-adic Banach space over $\mathbb{Q}_p$. Suppose $T\in\End(W^o)$ is topologically nilpotent. The following are equivalent.
\begin{enumerate}
\item $T$ is locally analytic;
\item $T^n(W^o)\subseteq pW^o$ for some $n\geq 1$, i.e. $T^n\cdot v$ converges to $0$ uniformly for all $v\in W^o$;
\item The sequence $\{T^n\}_{n\geq 0}$ converges to zero in $\End(W^o)$ with respect to the $p$-adic topology (equivalently the norm topology).
\end{enumerate}
\end{prop}
\begin{proof}
(2) and (3) are clearly equivalently. (2) implies (1) by taking $f(T)=T^n$ in the definition of locally analytic operators. It remains to show that (1) implies (2). Suppose that $f(T)(W^o)\subset pW^o$ for some monic polynomial $f(X)\in\mathbb{Z}_p[X]$. Write $f(X)\mod p =X^k g(X)$ with $g(X)\in\mathbb{F}_p[X]$ and $g(0)\neq 0$. Then $g(T)$ is invertible on $W^o/pW^o$ as $T$ is topologically nilpotent. Hence $T^k=f(T)=0$ viewed as elements in $\End(W^o/p)$.
\end{proof}
We can also generalize this notion to representations of algebras.
\begin{defn}
Suppose $A$ is a ring and $W$ is a $p$-adic Banach space equipped with an $A$-module structure. We say the action of $A$ on $W$ is \textit{locally analytic} if there exists an $A$-stable open and bounded lattice $\mathcal{L}\subseteq W$ such that the image of $A\to \End(\mathcal{L}/p^n\mathcal{L})$ is finite for any $n\geq 1$. If this happens,
the image of $A\to \End(\mathcal{L}'/p^n\mathcal{L}')$ is finite for any $n$ and any $A$-stable open and bounded lattice $\mathcal{L}'\subseteq W$.
\end{defn}
In some cases, we only need to consider the image of $A\to \End(\mathcal{L}/p\mathcal{L})$.
\begin{lem}\label{noela}
Suppose $A$ is a Noetherian ring and $W$ is a $p$-adic Banach space equipped with an $A$-module structure. The action of $A$ on $W$ is locally analytic if there exists an $A$-stable open and bounded lattice $\mathcal{L}\subseteq W$ such that the image of $A\to \End(\mathcal{L}/p\mathcal{L})$ is finite.
\end{lem}
\begin{proof}
Let $I_n$ be the kernel of $A\to \End(\mathcal{L}/p^n\mathcal{L})$. Clearly $I_1^n\subseteq I_n$. Hence it is enough to show that $A/I_1^n$ is finite. The assumption implies that $A/I_1$ is finite. Since $A$ is Noetherian, we have $I_1^n/I_1^{n+1}$ is also finite. Our claim follows as $A/I_1^n$ is filtered by $I_1^k/I_1^{k+1}$, $k=0,\cdots,n-1$.
\end{proof}
The following Proposition explains our choice of the notion ``locally analytic".
\begin{prop} \label{lapLg}
Suppose $G$ is a compact $p$-adic Lie group and $W$ is a continuous $p$-adic Banach space representation of $G$. Then the following are equivalent.
\begin{enumerate}
\item there exists a $G$-stable open and bounded lattice $\mathcal{L}\subseteq W$ such that $\mathcal{L}/p\mathcal{L}$ is fixed by some open subgroup $G'$ of $G$;
\item the induced action of $\mathbb{Z}_p[G]$ (the group algebra of $G$) on $W$ is locally analytic;
\item the induced action of $\mathbb{Z}_p[[G]]$ (the Iwasawa algebra) on $W$ is locally analytic;
\item $W$ is an analytic representation of some open subgroup $G'$ of $G$. In particular, $W$ is a locally analytic representation of $G$ in the usual sense.
\end{enumerate}
\end{prop}
\begin{proof}
Note that $\mathbb{Z}_p[[G]]$ is Noetherian, cf. \cite[Theorem 33.4, Theorem 27.1]{Sch11}. Hence by Lemma \ref{noela},
part (3) follows from (1) by noting that the action of $\mathbb{Z}_p[[G]]$ on $\mathcal{L}/p\mathcal{L}$ factors through $\mathbb{F}_p[G/G']$ for some open normal subgroup of $G$. Part (3) implies (2) because $\mathbb{Z}_p[G]\subseteq\mathbb{Z}_p[[G]]$. To see (2) implies (1), we may assume $G$ is pro-p by replacing it by an open subgroup. Then $g-1$ is topologically nilpotent on $W$ for any $g\in G$. The same argument of Proposition \ref{tnun} shows that $(g-1)^{p^n}(\mathcal{L})\subseteq p\mathcal{L}$ for some $n>0$ and some open bounded $G$-stable lattice $\mathcal{L}$ and any $g\in G$. Hence $G^{p^n}$ fixes $\mathcal{L}/p\mathcal{L}$. Part (1) follows as $G^{p^n}$ contains an open subgroup of $G$, cf. Theorem 27.1 and Remark 26.9 of \cite{Sch11}.
It remains to to prove the equivalence between (1) and (4). This is well-known. Part (4) follows from (1) by considering the Mahler coefficients and invoking Amice's theorem, cf. \cite[IV.4]{CD14}. Now assume (4). There is a $G'$-equivariant isomorphism $W\cong (W\widehat\otimes_{\mathbb{Q}_p} \mathscr{C}^{an}(G',\mathbb{Q}_p))^{G'}$ where $G'$ acts on the right hand side via the right translation action on $\mathscr{C}^{an}(G',\mathbb{Q}_p)$, the space of $\mathbb{Q}_p$-valued analytic functions on $G'$, cf. \cite[2.1]{Pan20}. Part (1) follows by noting that $\mathscr{C}^{\mathrm{an}}(G',\mathbb{Q}_p)^{o}/p$ is fixed by an open subgroup $G'$ by \cite[Lemma 2.1.2.]{Pan20}.
\end{proof}
\begin{exa} \label{kex}
Suppose $G=\mathbb{Z}_p^k$ and $W$ is a $\mathbb{Q}_p$-Banach space representation of $G$. Then $\mathbb{Z}_p[[G]]\cong \mathbb{Z}_p[[T_1,\ldots,T_k]]$ where $T_i=g_i-1$ and $g_1,\ldots, g_k$ form a basis of $G$. Now suppose the action of $\mathbb{Z}_p[[G]]$ on $W$ is locally analytic. It follows from the previous discussion that there exists $n>0$ such that $T_i^n(W^o)\subseteq pW^o$, $i=1,\ldots,k$, or equivalently $\frac{T_i^n}{p}$ has norm $\leq 1$. Hence the action of $\mathbb{Z}_p[[T_1,\ldots,T_k]]$ on $W$ can be extended to $\mathbb{Q}_p\langle \frac{T_1^n}{p},\ldots, \frac{T_k^n}{p} ,T_1,\ldots,T_k\rangle$.
Geometrically, the generic fiber of $\mathbb{Z}_p[[T_1,\ldots,T_k]]$ is an open ball and the rigid analytic space associated to $\mathbb{Q}_p\langle \frac{T_1^n}{p},\ldots, \frac{T_k^n}{p} ,T_1,\ldots,T_k\rangle$ corresponds to a \textit{closed} polydisc inside. Roughly speaking, this means the spectrum of $W$ is in a bounded region with radius strictly less than $1$.
\end{exa}
\begin{rem}
Proposition \ref{lapLg} shows that one can extend the notion of locally analytic representations to general topological groups. More precisely, a $\mathbb{Q}_p$-Banach space representation $W$ of a topological group $G$ is called \textit{locally analytic} if the action of $\mathbb{Z}_p[G]$ on $W$ is locally analytic in our sense. When $G=G_{K}$, the local Galois group of a finite extension $K$ of $\mathbb{Q}_p$, these locally analytic representations show up naturally in the recent development of Sen's theory. For example, one can show that for a locally analytic representation $W$ of $G_{K}$, there is a natural isomorphism
\[W\widehat\otimes_{\mathbb{Q}_p} \overbar{K}\cong (W\widehat\otimes_{\mathbb{Q}_p} \overbar{K})^{H_{K},\Gamma_{K,n}-\mathrm{an}}\widehat\otimes_{K_n} \overbar{K}\]
for some $n>0$. Here $H_K=\Gal(\overbar{K}/K(\mu_{p^\infty}))$, $K_n=K(\mu_{p^n})$, $\Gamma_{K,n}=\Gal(K(\mu_{p^\infty})/K_n)$, and the superscript $H_K$ denotes taking the $H_K$-invariants and ``$\Gamma_{K,n}-\mathrm{an}$'' denotes taking the $\Gamma_{K,n}$-analytic vectors. See \cite[Theorem 3.3.3]{CJER22} for a relative version of this result. We remark that when $G=G_K$, the equivalence between parts (1) and (3) of Lemma \ref{noela} still holds, even though $\mathbb{Z}_p[[G_K]]$ is not Noetherian. This is a consequence of the local class field theory: for any finite extension $L$ of $\mathbb{Q}_p$, the dimension of $\Hom(G_L,\mathbb{F}_p)$ is finite.
\end{rem}
\section{Fake-Hasse invariants} \label{FHi}
In order to study the Hecke action on overconvergent modular forms, we need fake-Hasse invariants and strange formal integral models of the modular curve constructed by Scholze in Chapter 4 of \cite{Sch15}. That the Hecke action is locally analytic will be a formal consequence of the existence of these Hecke-invariant sections.
Our setup is as follows. Let $C=\mathbb{C}_p$ the $p$-adic completion of $\overbar\mathbb{Q}_p$ with ring of integers $\mathcal{O}_C$. For a sufficiently small open compact subgroup $K$ of $\mathrm{GL}_2(\mathbb{A}_f)$, we denote by $X^*_{K,C}$ the complete adelic modular curve over $C$ of level $K$ and by $\mathcal{X}^*_{K}$ its associated rigid analytic space. We will always assume $K$ is sufficiently small so that $X^*_{K,C}$ is a variety. If we choose an isomorphism between $C$ and $\mathbb{C}$, the non-cusp points of $X^*_{K,C}(C)$ are given by the usual double cosets $\mathrm{GL}_2(\mathbb{Q})\setminus (\mathbb{C}-\mathbb{R})\times \mathrm{GL}_2(\mathbb{A}_f)/K$.
On $\mathcal{X}^*_{K}$, we have the usual automorphic line bundle $\omega_{K^pK_p}$. Fix an open compact subgroup $K^p\subseteq\mathrm{GL}_2(\mathbb{A}_f^p)$ contained in the level-$N$-congruence subgroup for some $N\geq 3$ prime to $p$. For a sufficiently small open subgroup $K_p\subseteq \mathrm{GL}_2(\mathbb{Q}_p)$, in the proof of Theorem 4.3.1. of \cite{Sch15}, Scholze constructed
\begin{itemize}
\item a formal integral model $\mathfrak{X}^*_{K^pK_p}$ of $\mathcal{X}^*_{K^pK_p}$ together with an affine open cover $\mathfrak{V}_{K_p,1},\mathfrak{V}_{K_p,2}$;
\item an ample line bundle $\omega^{\mathrm{int}}_{K^pK_p}$ on $\mathfrak{X}^*_{K^pK_p}$ whose generic fiber is $\omega_{K^pK_p}$;
\end{itemize}
Moreover fix $n\geq 1$. For a sufficiently small open subgroup $K_p\subseteq \mathrm{GL}_2(\mathbb{Q}_p)$, there are
\begin{itemize}
\item global sections $\bar{s}_{n,1},\bar{s}_{n,2}\in H^0(\mathfrak{X}^*_{K^pK_p},\omega^{\mathrm{int}}_{K^pK_p}/p^n)$ (fake-Hasse invariants) such that ${\mathfrak{V}_{K_p,i}}$ is the locus where $\bar{s}_{n,i}$ is invertible for $i=1,2$. In particular, $\bar{s}_{n,1},\bar{s}_{n,2}$ generate $\omega^{\mathrm{int}}_{K^pK_p}/p^n$.
\end{itemize}
All $\mathfrak{X}^*_{K^pK_p},\mathfrak{V}_{K_p,1},\mathfrak{V}_{K_p,2}$ and $\omega^{\mathrm{int}}_{K^pK_p}$ are functorial in $K^pK_p$, hence $\mathrm{GL}_2(\mathbb{A}_f^p)$ acts on the tower of $(\mathfrak{X}^*_{K^pK_p},\omega^{\mathrm{int}}_{K^pK_p})$. Both sections $\bar{s}_{n,1},\bar{s}_{n,2}$ are invariant under this action.
We briefly recall Scholze's construction. Scholze proved that when the level at $p$ varies, the inverse limit $\displaystyle \varprojlim_{K_p\subseteq \mathrm{GL}_2(\mathbb{Q}_p)}\mathcal{X}^*_{K^pK_p}$ exists as a perfectoid space, which will be denoted by $\mathcal{X}^*_{K^p}$. Moreover, there is the so-called Hodge-Tate period morphism
\[\pi_{\mathrm{HT}}:\mathcal{X}^*_{K^p}\to {\mathscr{F}\!\ell}\]
defined via the position of the Hodge-Tate filtration on the first cohomology of the universal elliptic curve (on the non-cusp points). Here ${\mathscr{F}\!\ell}$ ($\cong\mathbb{P}^1$) denotes the associated adic space of the flag variety of $\mathrm{GL}_2/C$. The pull-back of the tautological ample line bundle $\omega_{{\mathscr{F}\!\ell}}$ on ${\mathscr{F}\!\ell}$ along $\pi_{\mathrm{HT}}$ is canonically identified with the pull-back of $\omega_{K^pK_p}$ to $\mathcal{X}^*_{K^p}$ (up to a Tate twist). Note that $\Gamma({\mathscr{F}\!\ell},\omega_{\mathscr{F}\!\ell})$ has a canonical basis $f_1,f_2$, whose pull-back to $\mathcal{X}^*_{K^p}$ will be denoted by $e_1,e_2$. Let $U_1,U_2\subseteq \mathcal{X}^*_{K^p}$ be the open subsets defined by $\|e_2/e_1\|\leq 1$ and $\|e_1/e_2\|\leq 1$ respectively. Hence $e_i$ is an invertible section on $U_i$ for $i=1,2$. Scholze proved that $U_1$ and $U_2$ are affinoid perfectoid and are the preimages of some affinoid open subsets $V_{K_p,1},V_{K_p,2}$ of $\mathcal{X}^*_{K^pK_p}$ for sufficiently small $K_p$. Fix $n\geq 1$. For a sufficiently small subgroup $K_p$ and $i=1,2$, we may find
\begin{itemize}
\item $s_{n,i}\in \Gamma(V_{K_p,i},\omega_{K^pK_p})$ such that $\displaystyle \|1-\frac{s_{n,i}}{e_i}\|\leq \|p^n\|$;
\item $x_{n,i}\in \Gamma(V_{K_p,i},\mathcal{O}_{\mathcal{X}^*_{K^pK_p}})$ such that $\displaystyle \|\frac{e_{3-i}}{e_i}-x_{n,i}\|\leq \|p^n\|$.
\end{itemize}
This is possible because the natural map
\[\varinjlim_{K_p}\Gamma(V_{K_p,i},\mathcal{O}_{\mathcal{X}^*_{K^pK_p}})\to \Gamma(U_{i},\mathcal{O}_{\mathcal{X}^*_{K^p}})\]
has dense images. The formal model $\mathfrak{X}^*_{K^pK_p}$ is obtained by glueing $\mathfrak{V}_{K_p,1}:=\Spf \mathcal{O}^+(V_{K_p,1})$ and $\mathfrak{V}_{K_p,2}:=\Spf \mathcal{O}^+(V_{K_p,2})$ along $\Spf \mathcal{O}^+(V_{K_p,1}\cap V_{K_p,2})$. The integral line bundle $\omega^{\mathrm{int}}_{K^pK_p}$ is defined by requiring $s_{n,i}$ being invertible on $\mathfrak{V}_i$ for $i=1,2$. This does not depend on $n$ and the choice of $s_{n,i}$. For the fake-Hasse invariants, observe that $s_{n,1}\mod p^n$ and $s_{n,2}x_{n,2}\mod p^n$ glue a global section $\bar{s}_{n,1}\in H^0(\mathfrak{X}^*_{K^pK_p},\omega^{\mathrm{int}}_{K^pK_p}/p^n)$ by our choice of $s_{n,i},x_{n,i}$. Similarly one can construct $\bar{s}_{n,2}$. We remark that $\bar{s}_{n,1},\bar{s}_{n,2}$ are independent of the choice of $s_{n,i},x_{n,i}$ because $\bar{s}_{n,i}$ may be viewed as $e_i\mod p^n$. Thus $\bar{s}_{n,1},\bar{s}_{n,2}$ are fixed by the action of $\mathrm{GL}_2(\mathbb{A}^p_f)$.
Let $\mathbb{T}=\mathbb{T}_{K^p}=\mathbb{Z}_p[\mathrm{GL}_2(\mathbb{A}_f^p)//K^p]$ be the abstract Hecke algebra of $K^p$-biinvariant compactly supported functions on $\mathrm{GL}_2(\mathbb{A}_f^p)$, where the Haar measure gives $K^p$ measure $1$. Let $K_p$ be a sufficiently small subgroup of $\mathrm{GL}_2(\mathbb{Q}_p)$ so that $\mathfrak{X}^*_{K^pK_p}$ and $\mathfrak{V}_1,\mathfrak{V}_2$ are defined. (We drop some subscripts $K_p$ from the notations.) It follows from the functorial properties of $\mathfrak{V}_1,\mathfrak{V}_2$ that $H^0(\mathfrak{V}_i,(\omega^{\mathrm{int}}_{K^pK_p})^{\otimes k}),i=1,2$ and $k\in \mathbb{Z}$ admits a natural action of $\mathbb{T}$. Denote by $V_i=V_{K_p,i}\subseteq\mathcal{X}^*_{K^pK_p}$ the generic fiber of $\mathfrak{V}_i$. Then $H^0(V_i,\omega_{K^pK_p}^{\otimes k})$ is a $p$-adic Banach space with unit ball $H^0(\mathfrak{V}_i,(\omega^{\mathrm{int}}_{K^pK_p})^{\otimes k})$. Our main result here is
\begin{thm} \label{mT}
For $i=1,2$ and $k\in\mathbb{Z}$, the Hecke action of $\mathbb{T}$ on $H^0(V_i,\omega_{K^pK_p}^{\otimes k})$ is locally analytic.
\end{thm}
\begin{rem}
We will relate $H^0(V_i,\omega_{K^pK_p}^{\otimes k})$ with classical overconvergent modular forms later in the next section. See the proof of Corollary \ref{HTS}.
\end{rem}
Since $\mathfrak{V}_i$ is affine, $H^0(\mathfrak{V}_i,(\omega^{\mathrm{int}}_{K^pK_p})^{\otimes k})/p^n=H^0(\mathfrak{V}_i,(\omega^{\mathrm{int}}_{K^pK_p})^{\otimes k}/p^n)$. It follows from the construction of $\omega^{\mathrm{int}}_{K^pK_p}$ that if $K'_p$ is an open subgroup of $K_p$, the pull-back map
\[H^0(\mathfrak{V}_i,(\omega^{\mathrm{int}}_{K^pK_p})^{\otimes k}/p^n)\to H^0(\mathfrak{V}_{K_p',i},(\omega^{\mathrm{int}}_{K^pK'_p})^{\otimes k}/p^n)\]
is injective as $\bar{s}_{1,i}^k$ generates both $(\omega^{\mathrm{int}}_{K^pK_p})^{\otimes k}/p$ and $(\omega^{\mathrm{int}}_{K^pK'_p})^{\otimes k}/p$ on $\mathfrak{V}_i$ and $\mathfrak{V}_{K'_p,i}$ respectively. Hence for a fixed $n$, we are free to replace $K_p$ by a smaller subgroup. In particular, we may assume $\bar{s}_{n,i}$ exists. Note that $\bar{s}_{n,i}$ is an invertible section on $\mathfrak{V}_i$ and commutes with the Hecke actions. There are $\mathbb{T}$-equivariant isomorphisms:
\[H^0(\mathfrak{V}_i,\mathcal{O}_{\mathfrak{X}^*_{K^pK_p}}/p^n)\stackrel{\times \bar{s}_{n,i}^{k}}{\to}H^0(\mathfrak{V}_i,(\omega^{\mathrm{int}}_{K^pK_p})^{\otimes k}/p^n).\]
Hence $H^0(\mathfrak{V}_i,(\omega^{\mathrm{int}}_{K^pK_p})^{\otimes k})/p^n$ is independent of $k$ as a Hecke module. Thus it suffices to prove Theorem \ref{mT} for $k=0$ and we have the following corollary.
\begin{cor} \label{cor1}
The Hecke actions of $\mathbb{T}$ on
\[(\prod_{k\in\mathbb{Z}} H^0(\mathfrak{V}_i,(\omega^{\mathrm{int}}_{K^pK_p})^{\otimes k}))\otimes_{\mathbb{Z}_p}\mathbb{Q}_p,\,i=1,2\]
are locally analytic.
\end{cor}
\begin{proof}[Proof of Theorem \ref{mT}]
Fix $n\geq 1$. By definition, we need to show that the image of
\[\mathbb{T}\to\End\left(H^0(\mathfrak{V}_i,\mathcal{O}_{\mathfrak{X}^*_{K^pK_p}}/p^n)\right)\]
is finite. By shrinking $K_p$ if necessary, we may assume $\bar{s}_{n,i}$ exists.
Since ${\mathfrak{V}_i}$ is the locus where $\bar{s}_{n,i}$ is invertible, we may write
\[H^0(\mathfrak{V}_i,\mathcal{O}_{\mathfrak{X}^*_{K^pK_p}}/p^n)=\varinjlim_{\times\bar{s}_{n,i}}H^0(\mathfrak{X}^*_{K^pK_p},(\omega^{\mathrm{int}}_{K^pK_p})^{\otimes k}/p^n).\]
Hence it suffices to show the image of
\[\mathbb{T}\to\End\left(H^0(\mathfrak{X}^*_{K^pK_p},(\omega^{\mathrm{int}}_{K^pK_p})^{\otimes k}/p^n)\right)\]
is finite and the kernel stabilizes when $k$ is sufficiently large. For the finiteness, by the ampleness of $\omega^{\mathrm{int}}_{K^pK_p}$, when $k$ is sufficiently large, we have
\[H^0\left(\mathfrak{X}^*_{K^pK_p},(\omega^{\mathrm{int}}_{K^pK_p})^{\otimes k}/p^n\right)=H^0\left(\mathfrak{X}^*_{K^pK_p},(\omega^{\mathrm{int}}_{K^pK_p})^{\otimes k}\right)/p^n.\]
Since $H^0\left(\mathfrak{X}^*_{K^pK_p},(\omega^{\mathrm{int}}_{K^pK_p})^{\otimes k}\right)$ is $p$-torsion free, it is enough to show that the image of
\[\mathbb{T}\to\End_{\mathcal{O}_C}\left(H^0(\mathfrak{X}^*_{K^pK_p},(\omega^{\mathrm{int}}_{K^pK_p})^{\otimes k})\right)\subseteq\End_C\left(H^0(\mathcal{X}^*_{K^pK_p},\omega_{K^pK_p}^{\otimes k})\right)\]
is a finite $\mathbb{Z}_p$-module. Indeed, the properness of $\mathfrak{X}^*_{K^pK_p}$ implies that $H^0(\mathfrak{X}^*_{K^pK_p},(\omega^{\mathrm{int}}_{K^pK_p})^{\otimes k})$ is a finite $\mathcal{O}_C$-module and $H^0(\mathcal{X}^*_{K^pK_p},\omega_{K^pK_p}^{\otimes k})$ is a finite dimensional $C$-vector space. Our claim is clear as $\mathcal{X}^*_{K^pK_p}$, the sheaf $\omega_{K^pK_p}$ and Hecke actions are all defined over $\mathbb{Q}_p$.
To see that the kernel of $\mathbb{T}\to\End\left(H^0(\mathfrak{X}^*_{K^pK_p},(\omega^{\mathrm{int}}_{K^pK_p})^{\otimes k}/p^n)\right)$ stabilizes, consider the exact sequence
\[0\to (\omega^{\mathrm{int}}_{K^pK_p})^{\otimes k-1}/p^n\stackrel{(\bar{s}_{n,1},\bar{s}_{n,2})}{\longrightarrow} (\omega^{\mathrm{int}}_{K^pK_p})^{\otimes k}/p^n\oplus (\omega^{\mathrm{int}}_{K^pK_p})^{\otimes k}/p^n
\stackrel{(\bar{s}_{n,2},-\bar{s}_{n,1})}{\longrightarrow}
(\omega^{\mathrm{int}}_{K^pK_p})^{\otimes k+1}/p^n\to 0.\]
(This essentially comes from the non-split sequence $0\to \mathcal{O}(-1)\to\mathcal{O}^{\oplus 2}\to\mathcal{O}(1)\to 0$ on $\mathbb{P}^1$.)
When $k$ is sufficiently large, taking global sections of this exact sequence remains exact as $\omega^{\mathrm{int}}_{K^pK_p}$ is ample. Thus the Hecke action of $\mathbb{T}$ on $H^0(\mathfrak{X}^*_{K^pK_p},(\omega^{\mathrm{int}}_{K^pK_p})^{\otimes k+1}/p^n)$ factors through $H^0(\mathfrak{X}^*_{K^pK_p},(\omega^{\mathrm{int}}_{K^pK_p})^{\otimes k}/p^n)^{\oplus 2}$, which proves the claim.
\end{proof}
\section{Hodge-Tate-Sen weights}
In this section, we study Galois representations attached to eigenforms in $H^0(V_i,\omega_{K^pK_p}^{\otimes k})$. Let me introduce some (standard) notation first. For simplicity, from now on we assume $K^p\subset\mathrm{GL}_2(\mathbb{A}_f^p)$ is of the form $\prod_{l\neq p}K_l$. Let $S$ be a finite set of rational primes containing $p$ such that $K_l\cong \mathrm{GL}_2(\mathbb{Z}_l)$ for $l\notin S$. Denote by $\mathbb{T}_S=\mathbb{Z}_p[\mathrm{GL}_2(\mathbb{A}_f^S)//\prod_{l\notin S}K_l]\subset \mathbb{T}$ the subalgebra generated by spherical Hecke operators. Consider the image $\mathbb{T}_{i,1}$ of $\mathbb{T}_S\to \End(H^0(\mathfrak{V}_i,\mathcal{O}_{\mathfrak{X}^*_{K^pK_p}}/p))$. By Theorem \ref{mT}, this is a finite $\mathbb{F}_p$-algebra. Moreover, by Corollary 5.11 of \cite{Sch15}, there is a continuous $2$-dimensional determinant $D$ of $G_{\mathbb{Q},S}$ valued in $\mathbb{T}_{i,1}$ in the sense of Chenevier \cite{Che14} satisfying the following property: for any $l\notin S$, the characteristic polynomial of $D(\Frob_l)$ is
\[X^2-l^{-1}T_lX+l^{-1}S_l.\]
Here $G_{\mathbb{Q},S}$ denotes the Galois group of the maximal extension of $\mathbb{Q}$ unramified outside of $S$ and infinity, $\Frob_l\in G_{\mathbb{Q},S}$ denotes a geometric Frobenius element at $l$ and $T_l,S_l$ denote the usual Hecke operators
\[[K_l\begin{pmatrix}l&0\\0&1\end{pmatrix}K_l],\,[K_l\begin{pmatrix}l&0\\0&l\end{pmatrix}K_l].\]
Let $\mathbb{F}$ be a finite field so that all residue fields of $\mathbb{T}_{i,1}$ can be embedded into $\mathbb{F}$. Fix an embedding of $W(\mathbb{F})[\frac{1}{p}]$ into $\overbar\mathbb{Q}_p$, or equivalently an embedding $\mathbb{F}\to \mathcal{O}_C/p$. Then $\mathbb{T}_{i,1}\otimes_{\mathbb{F}_p}\mathbb{F}$ acts on $H^0(\mathfrak{V}_i,\mathcal{O}_{\mathfrak{X}^*_{K^pK_p}}/p)$ and we denote by $\mathbb{T}_{i}$ its image in $ \End(H^0(\mathfrak{V}_i,\mathcal{O}_{\mathfrak{X}^*_{K^pK_p}}/p))$. Finally, for any maximal ideal $\mathfrak{m}$ of $\mathbb{T}_i$, we have a continuous $2$-dimensional determinant $D_{\mathfrak{m}}$ of $G_{\mathbb{Q},S}$ valued in $\mathbb{T}_{i}/\mathfrak{m}=\mathbb{F}$. Let $R^{\mathrm{ps}}_{\mathfrak{m}}$ be the universal formal $W(\mathbb{F})$-algebra parametrizing all liftings of $D_{\mathfrak{m}}$. This is a noetherian ring. Denote the product over all $\mathfrak{m}$ by
\[R^{\mathrm{ps}}=\prod_{\mathfrak{m}\in\Spec\mathbb{T}_i}R^{\mathrm{ps}}_{\mathfrak{m}}.\]
Now for any $k\in\mathbb{Z},n>0$, by Corollary 5.1.11 of \cite{Sch15}, there is a lifting of $\prod_{\mathfrak{m}\in\Spec\mathbb{T}_i}D_{\mathfrak{m}}$ valued in the image of $\mathbb{T}_S\otimes_{\mathbb{Z}_p}W(\mathbb{F})\to\End(H^0(\mathfrak{X}^*_{K^pK_p},(\omega^{\mathrm{int}}_{K^pK_p})^{\otimes k}/p^n))$. By the universal property, this image receives a map from $R^{\mathrm{ps}}$. Hence we obtain an action of $R^{\mathrm{ps}}$ on $H^0(V_i,\omega_{K^pK_p}^{\otimes k})$ factoring through the Hecke action. In particular, by Corollary \ref{cor1},
\begin{cor}
The action of $R^{\mathrm{ps}}$ on $(\prod_{k\in\mathbb{Z}} H^0(\mathfrak{V}_i,(\omega^{\mathrm{int}}_{K^pK_p})^{\otimes k}))\otimes_{\mathbb{Z}_p}\mathbb{Q}_p$
is locally analytic.
\end{cor}
Concretely, since each $R^{\mathrm{ps}}_{\mathfrak{m}}$ is a noetherian local formal $W(\mathbb{F})$-algebra, it can be written as a quotient of $W(\mathbb{F})[[x_1,\ldots,x_g]]$ for some $g$. As explained in Example \ref{kex}, there exists an integer $n>0$ such that $\frac{x_j^n}{p}$ has norm $\leq 1$ acting on $(\prod_{k\in\mathbb{Z}} H^0(\mathfrak{V}_i,(\omega^{\mathrm{int}}_{K^pK_p})^{\otimes k}))\otimes_{\mathbb{Z}_p}\mathbb{Q}_p$ for any $j=1,\ldots,g$. Therefore, let $E\subset\overbar\mathbb{Q}_p$ be a finite extension of $W(\mathbb{F})[\frac{1}{p}]$ containing a $n$-th root of $p$ and fix such a root $p^{1/n}\in E$. We can extend the action of $W(\mathbb{F})[[x_1,\ldots,x_g]]$ to an $E$-linear action of $E\langle\frac{x_1}{p^{1/n}},\ldots,\frac{x_g}{p^{1/n}}\rangle$ on $(\prod_{k\in\mathbb{Z}} H^0(\mathfrak{V}_i,(\omega^{\mathrm{int}}_{K^pK_p})^{\otimes k}))\otimes_{\mathbb{Z}_p}\mathbb{Q}_p$. Recall that geometrically, the generic fiber of $W(\mathbb{F})[[x_1,\ldots,x_g]]$ is an open ball and $E\langle\frac{x_1}{p^{1/n}},\ldots, \frac{x_g}{p^{1/n}}\rangle$ corresponds to a \textit{closed} polydisc inside. This means the spectrum of $H^0(V_i,\omega_{K^pK_p}^{\otimes k})$ is in a bounded region with radius strictly less than $1$.
We make such a choice for each $\mathfrak{m}$. As a consequence, the action of $R^{\mathrm{ps}}$ can be extended to an action of a topologically finitely generated Banach $E$-algebra. We denote its image in
\[\End((\prod_{k\in\mathbb{Z}} H^0(\mathfrak{V}_i,(\omega^{\mathrm{int}}_{K^pK_p})^{\otimes k}))\otimes_{\mathbb{Z}_p}\mathbb{Q}_p)\]
by $\mathcal{R}$. There is a natural map $R^{\mathrm{ps}}\to\mathcal{R}$. Hence we have a $2$-dimensional determinant $D_{\mathcal{R}}$ of $G_{\mathbb{Q},S}$ valued in $\mathcal{R}$ which is continuous with respect to the $p$-adic topology on $\mathcal{R}$. The whole point of showing that the Hecke action is locally analytic is to improve the continuity of the determinant on $R^{\mathrm{ps}}$ from the $\mathrm{rad}(R^{\mathrm{ps}})$-adic topology to a $p$-adic topology.
The Hecke action on $H^0(\mathcal{X}^*_{K^pK_p},\omega_{K^pK_p}^{\otimes k}),k\geq 0$ extends naturally to $\mathcal{R}$. In fact, the image of $\mathbb{T}_S\otimes E$ in $\End(H^0(\mathcal{X}^*_{K^pK_p},\omega_{K^pK_p}^{\otimes k}))$ agrees with the image of $\mathcal{R}$. In particular, the action of $\mathcal{R}$ on $H^0(\mathcal{X}^*_{K^pK_p},\omega_{K^pK_p}^{\otimes k})$ is semi-simple.
\begin{lem} \label{dense}
The kernel of
\[\mathcal{R}\to \End(\prod_{k\geq 0}H^0(\mathcal{X}^*_{K^pK_p},\omega_{K^pK_p}^{\otimes k}))\]
is trivial.
\end{lem}
\begin{proof}
This is a standard application of fake Hasse invariants. See the proof of Theorem 4.4.1. of \cite{Sch15}. We give a sketch here. Suppose $f\in\mathcal{R}$ is a non-zero element in the kernel of the above map. We may assume it has norm $\leq 1$ acting on $\prod_{k\in\mathbb{Z}} H^0(\mathfrak{V}_i,(\omega^{\mathrm{int}}_{K^pK_p})^{\otimes k})$ and its image in $\End (\prod_{k\in\mathbb{Z}} H^0(\mathfrak{V}_i,(\omega^{\mathrm{int}}_{K^pK_p})^{\otimes k}/p))$ is non-zero. Now since $R^{\mathrm{ps}}\otimes_{W(\mathbb{F})} E\to\mathcal{R}$ has dense image, the action of $f$ on $H^0(\mathfrak{V}_i,(\omega^{\mathrm{int}}_{K^pK_p})^{\otimes k}/p)$ commutes with $\bar{s}_{1,i}^{n}$ if $n$ is sufficiently divisible by $p$. Indeed, since $\omega^{\mathrm{int}}_{K^pK_p}$ is ample, $\bar{s}_{1,i}^{l}$ lifts to a global section $s_1\in H^0(\mathfrak{X}^*_{K^pK_p},(\omega^{\mathrm{int}}_{K^pK_p})^{\otimes l})$ for $l$ large enough. Hence $(g-1)\cdot s_1\in pH^0(\mathfrak{X}^*_{K^pK_p},(\omega^{\mathrm{int}}_{K^pK_p})^{\otimes l})$ for any $g\in\mathrm{GL}_2(\mathbb{A}_f^S)$. Thus $(g-1)\cdot s_1^{p^k} \in p^{k+1}H^0(\mathfrak{X}^*_{K^pK_p},(\omega^{\mathrm{int}}_{K^pK_p})^{\otimes lp^k})$ for $k\geq0$. In particular,
\[T(s_1^{p^k}x)-s_1^{p^k} T(x)\in p^{k+1} H^0(\mathfrak{V}_i,(\omega^{\mathrm{int}}_{K^pK_p})^{\otimes m+lp^k})\]
for $T\in\mathbb{T}_S$ and $x\in H^0(\mathfrak{V}_i,(\omega^{\mathrm{int}}_{K^pK_p})^{\otimes m})$. By continuity, this also holds for $T\in R^{\mathrm{ps}}$. If we write $f=\frac{f'}{p^k}+pf''$, where $f'\in R^{\mathrm{ps}}\otimes \mathcal{O}_E$ and $f''\in \mathcal{R}$ with norm $\leq 1$ acting on $\prod_{k\in\mathbb{Z}} H^0(\mathfrak{V}_i,(\omega^{\mathrm{int}}_{K^pK_p})^{\otimes k})$. It follows that $\frac{f'}{p^k}$ and $f$ commute with $(\bar{s}_{1,i})^{lp^k}$. This means $f$ acts non-trivially on $H^0(\mathfrak{X}^*_{K^pK_p},(\omega^{\mathrm{int}}_{K^pK_p})^{\otimes k}/p)$ for some sufficiently large $k$ by the same argument as in the proof of Corollary \ref{cor1}. In this case, $H^0(\mathfrak{X}^*_{K^pK_p},(\omega^{\mathrm{int}}_{K^pK_p})^{\otimes k}/p)=H^0(\mathfrak{X}^*_{K^pK_p},(\omega^{\mathrm{int}}_{K^pK_p})^{\otimes k})/p$ by the ampleness of $\omega^{\mathrm{int}}_{K^pK_p}$. But this contradicts our assumption on $f$.
\end{proof}
Recall that there is a determinant $D_{\mathcal{R}}$ of $G_{\mathbb{Q},S}$ valued in $\mathcal{R}$. Since $\mathcal{R}$ is over a characteristic zero field, one can also view this as a function $T:G_{\mathbb{Q},S}\to\mathcal{R}$, which behaves like the trace of a two-dimensional representation, i.e. a pseudo-representation. For any non-zero $E$-algebra homomorphism $\lambda:\mathcal{R}\to\overbar\mathbb{Q}_p$, we can associate a two-dimensional semi-simple continuous representation $\rho_\lambda:G_{\mathbb{Q},S}\to\mathrm{GL}_2(\overbar\mathbb{Q}_p)$, well-defined up to conjugation, whose trace is given by $\lambda\circ T$. Moreover, if $\lambda$ arises from an eigenform in $H^0(\mathcal{X}^*_{K^pK_p},\omega_{K^pK_p}^{\otimes k})$, then by Faltings's result \cite{Fa87}, $\rho_\lambda|_{G_{\mathbb{Q}_p}}$ has Hodge-Tate weights $0,k-1$. Our convention is that the cyclotomic character has Hodge-Tate weight $-1$. The density result \ref{dense} has the following consequence.
\begin{thm} \label{mtS}
For any $\lambda:\mathcal{R}\to\overbar\mathbb{Q}_p$, one of the Hodge-Tate-Sen weights of $\rho_\lambda|_{G_{\mathbb{Q}_p}}$ is $0$, i.e. $(\rho_\lambda\otimes_{\mathbb{Q}_p}C)^{G_{\mathbb{Q}_p}}\neq 0$.
\end{thm}
\begin{proof}
Recall that given a continuous representation of $G_{\mathbb{Q}_p}\to\mathrm{GL}_n(\overbar\mathbb{Q}_p)$, Sen constructs a monic polynomial $P_{\mathrm{Sen},\rho}$ of degree $n$ with coefficients in $\overbar\mathbb{Q}_p\otimes_{\mathbb{Q}_p}\mathbb{Q}_p(\mu_{p^\infty})$. It is called the Sen polynomial of $\rho$ and only depends on the semi-simplification of $\rho$. Its roots are called the Hodge-Tate-Sen weights of $\rho$ (or up to a sign depending on the normalization). Moreover, Sen shows that this polynomial varies analytically in family. See \cite{Sen88,Sen93} and also Th\'eor\`eme 5.1.4. of \cite{BC08}. We are going to apply Sen's theory in our context.
First, suppose that there exists a continuous Galois representation $\rho_{\mathcal{R}}:G_{\mathbb{Q},S}\to\mathrm{GL}_2(\mathcal{R})$ whose trace is $T$. Then by Sen's result, we can find a polynomial $P_{\mathrm{Sen},\rho_{\mathcal{R}}}$ with coefficients in $\mathcal{R}\otimes_{\mathbb{Q}_p}\mathbb{Q}_p(\mu_{p^\infty})$, such that for any $\lambda:\mathcal{R}\to\overbar\mathbb{Q}_p$, the Sen polynomial of $\rho_{\lambda}$ is given by $\lambda( P_{\mathrm{Sen},\rho_{\mathcal{R}}})$. By Lemma \ref{dense} and Faltings's result, the constant term of $P_{\mathrm{Sen},\rho_{\mathcal{R}}}$ vanishes as it vanishes after composing with any $\lambda$ arisen from an eigenform in $H^0(\mathcal{X}^*_{K^pK_p},\omega_{K^pK_p}^{\otimes k})$. (Implicitly we using $\mathbb{Q}_p(\mu_p^\infty)$ is flat over $\mathbb{Q}_p$.) This immediately implies our claim.
In general, we may assume $\mathcal{R}$ is an integral domain. We are going to use the following lemma, whose proof will be given later.
\begin{lem} \label{uSen}
Assume $\mathcal{R}$ is normal. There exists a polynomial $P\in \mathcal{R}\otimes_{\mathbb{Q}_p}\mathbb{Q}_p(\mu_{p^\infty})[X]$ such that for any $\lambda:\mathcal{R}\to\overbar\mathbb{Q}_p$, the Sen polynomial of $\rho_{\lambda}$ is $\lambda( P)$.
\end{lem}
Let $\mathcal{R}'$ be the normal closure of $\mathcal{R}$ in its fraction field. Note that $\mathcal{R}$ is a quotient of products of $E\langle x_1,\ldots,x_k\rangle$. Hence it is excellent because the Tate algebra $E\langle x_1,\ldots,x_k\rangle$ is excellent by the weak Jacobian condition (\cite[Theorem102]{Mat2}). In particular, $\mathcal{R}$ is a Nagata ring and $\mathcal{R}'$ is a finite $\mathcal{R}$-algebra. Thus $\mathcal{R}'$ is a Banach $E$-algebra.
Now consider the pseudo-representation $G_{\mathbb{Q},S}\stackrel{T}{\to} \mathcal{R}\to\mathcal{R}'$. Note that by the going-up property of integral extension, any $\lambda:\mathcal{R}\to\overbar\mathbb{Q}_p$ can be extended to a map $\lambda':\mathcal{R}'\to\overbar\mathbb{Q}_p$ and $\rho_{\lambda}\cong\rho_{\lambda'}$. In particular, it is enough to show that $\rho_{\lambda'}$ has a Hodge-Tate-Sen weight zero for any $\lambda':\mathcal{R}'\to\overbar\mathbb{Q}_p$. Applying the previous lemma to $\mathcal{R}'$, we get a universal Sen polynomial $P$ with coefficients in $\mathcal{R}'\otimes_{\mathbb{Q}_p}\mathbb{Q}_p(\mu_{p^\infty})$. Again it suffices to show the constant term of $P$ vanishes. Write the constant term of $P$ as $\sum_{i=1}^l a_i\otimes b_i$ with $a_i\in\mathcal{R}',b_i\in \mathbb{Q}_p(\mu_p^\infty)$ and $b_i$ are linearly independent over $\mathbb{Q}_p$. If one of $a_i$ is non-zero, say $a_1$, we can find a monic polynomial $Q(X)\in \mathcal{R}[X]$ with constant term $Q(0)\neq 0$ and $Q(a_1)=0$. By Lemma \ref{dense}, there exists a $\lambda:\mathcal{R}\to\overbar\mathbb{Q}_p$ arisen from an eigenform in $H^0(\mathcal{X}^*_{K^pK_p},\omega_{K^pK_p}^{\otimes k})$ and $\lambda(Q(0))\neq 0$. Let $\lambda':\mathcal{R}'\to\overbar\mathbb{Q}_p$ be a map extending $\lambda$. By Faltings's result, $\lambda'(a_1)=0$. But $0=\lambda'(Q(a_1))=\lambda(Q(0))\neq 0$. Contradiction. Thus we prove $P(0)=0$.
\end{proof}
\begin{proof}[Proof of Lemma \ref{uSen}]
First let me recall some standard constructions in the theory of pseudo-representations. Fix a complex conjugation $\sigma^*\in G_{\mathbb{Q},S}$. Our pseudo-representation $T$ is odd in the sense that $T(\sigma^*)=0$. For any $\sigma,\tau\in G_{\mathbb{Q},S}$, let
\begin{itemize}
\item $a(\sigma)=\frac{T(\sigma^*\sigma)+T(\sigma)}{2}$;
\item $d(\sigma)=T(\sigma)-a(\sigma)$;
\item $x(\sigma,\tau)=a(\sigma\tau)-a(\sigma)a(\tau)$.
\end{itemize}
We denote by $\mathcal{I}$ the ideal of $\mathcal{R}$ generated by all $x(\sigma,\tau)$. It is called the ideal of reducibility as $\rho_\lambda$ is reducible if and only if $\lambda(\mathcal{I})=0$. If $\mathcal{I}$ is generated by some $x(\sigma_0,\tau_0)\neq 0$, then
\[\sigma\in G_{\mathbb{Q},S}\mapsto \begin{pmatrix} a(\sigma) & \frac{x(\sigma,\tau_0)}{x(\sigma_0,\tau_0)}\\ x(\sigma_0,\sigma) & d(\sigma)\end{pmatrix}\]
defines a representation $G_{\mathbb{Q},S}\to\mathrm{GL}_2(\mathcal{R})$ whose trace is $T$. In this case, our claim follows from Sen's result directly.
In general, $\mathcal{I}$ might not even be principal. Here is a sketch of our strategy. $\mathcal{X}:=\Spm \mathcal{R}$ is viewed as an affinoid rigid analytic variety. Consider the blowup $\tilde{\mathcal{X}}$ of $\mathcal{X}$ along the ideal sheaf defined by $\mathcal{I}$. Then $\mathcal{I}$ becomes an invertible sheaf on $\tilde{\mathcal{X}}$ and we can apply the previous construction and glue a polynomial on $\tilde{\mathcal{X}}$ interpolating Sen polynomial at each point. Now the normal assumption guarantees that the coefficients of this polynomial actually belong to $\mathcal{R}$. This gives the polynomial we are looking for. Since everything is relatively simple here, the blowup process will be replaced by the explicit construction below. But it seems helpful to keep this blowup picture in mind.
If $\mathcal{I}= 0$, then $a,d$ are characters and our claim is clear. So we may assume $\mathcal{I}\neq 0$ from now on. Let $x_1=x(\sigma_1,\tau_1),\ldots, x_r=x(\sigma_r,\tau_r)$ be a set of non-zero generators of $\mathcal{I}$. Denote by $\mathcal{R}^+$ the unit ball of $\mathcal{R}$ and by $\mathcal{K}$ the fraction field of $\mathcal{R}$. For each $i\in\{1,\ldots,r\}$, we define $\mathcal{R}_i^+$ as the $p$-adic completion of $\mathcal{R}^+[\frac{x_1}{x_i},\ldots,\frac{x_r}{x_i}]\subset \mathcal{K}$ and $\mathcal{R}_i=\mathcal{R}_i^+[\frac{1}{p}]$. Consider the pseudo-representation $G_{\mathbb{Q},S}\stackrel{T}{\to} \mathcal{R}\to\mathcal{R}_i$. The ideal of reducibility in this case is generated by $x_i$. Hence we have a polynomial $P_i\in\mathcal{R}_i\otimes_{\mathbb{Q}_p}\mathbb{Q}_p(\mu_{p^\infty})[X]$ interpolating Sen polynomial at each point of $\Spm \mathcal{R}_i$.
Denote by $\mathcal{Y}_i\subset\Spm\mathcal{R}_i$ the open subset defined by $x_i\neq 0$ and by $\mathcal{X}_i\subset\mathcal{X}$ the open subset defined by $x_i\neq 0,\|x_j\|\leq \|x_i\|,j=1,\ldots,r$. It is easy to see that $\mathcal{Y}_i$ maps isomorphically onto $\mathcal{X}_i$ under the natural map $\pi_i:\Spm \mathcal{R}_i\to \mathcal{X}$. Hence we may view $P_i|_{\mathcal{Y}_i}$ as a polynomial on $\mathcal{X}_i$. Clearly, it interpolates Sen polynomial at each point in $\mathcal{X}_i$. Hence we can glue all $P_i$ and get a polynomial $P$ on $\mathcal{X}':=\mathcal{X}\setminus V(\mathcal{I})$, the locus of irreducible representations. (Here we are using $\mathcal{R}$ is reduced.) Since $\mathcal{R}$ is normal and the coefficients of $P$ are bounded functions, by Bartenwerfer's result \cite[\S 3]{Bar76}, the coefficients of $P$ can be extended to functions defined everywhere on $\mathcal{X}$, i.e. $P\in\mathcal{R}\otimes_{\mathbb{Q}_p}\mathbb{Q}_p(\mu_{p^\infty})[X]$.
We claim this polynomial $P$ interpolates the Sen polynomial at each point in $\mathcal{X}$. By construction, this is true for points in $\mathcal{X}'$. It remains to verify points in $V(\mathcal{I})$. Let $\lambda:\mathcal{R}\to \overbar\mathbb{Q}_p$ be a non-zero map whose kernel contains $\mathcal{I}$. Note that there exists $i\in\{1,\ldots,r\}$ such that $\lambda$ can be extended to a map $\lambda':\mathcal{R}[\frac{x_1}{x_i},\ldots,\frac{x_r}{x_i}]\to \overbar\mathbb{Q}_p$. This is because the usual blowup (in algebraic geometry) of $\Spec \mathcal{R}$ along $\mathcal{I}$ maps surjectively onto $\Spec\mathcal{R}$. Fix an integer $n$ so that $\lambda'(\mathcal{R}^+[\frac{p^nx_1}{x_i},\ldots,\frac{p^nx_r}{x_i}])$ is contained in the ring of integers of $\overbar\mathbb{Q}_p$. We define $\mathcal{R}'^+_i$ as the $p$-adic completion of $\mathcal{R}^+[\frac{p^nx_1}{x_i},\ldots,\frac{p^nx_r}{x_i}]$ and $\mathcal{R}'_i=\mathcal{R}'^+_i[\frac{1}{p}]$. Then $\lambda'$ extends to $\mathcal{R}'_i$ naturally. Again there is a polynomial $P_i'\in\mathcal{R}'_i\otimes_{\mathbb{Q}_p}\mathbb{Q}_p(\mu_{p^\infty})[X]$ interpolating Sen polynomial at each point of $\Spm \mathcal{R}'_i$. It suffices to prove that $P$, considered as an element of $(\mathcal{R}'_i)^{\mathrm{red}}\otimes_{\mathbb{Q}_p}\mathbb{Q}_p(\mu_{p^\infty})[X]$, agrees with $P'_i$. Clearly this is true for points in the irreducible locus $\Spm\mathcal{R}'_i\setminus V(x_i)$. But this also implies points in $V(x_i)$ as $x_i$ is not a zero-divisor in $\mathcal{R}'_i$ by the flatness of $\mathcal{R}'^+_i$ over $\mathcal{R}^+[\frac{p^nx_1}{x_i},\ldots,\frac{p^nx_r}{x_i}]\subset\mathcal{K}$. (Note that $\mathcal{R}'^+_i$ is the $p$-adic completion of the noetherian ring $\mathcal{R}^+[\frac{p^nx_1}{x_i},\ldots,\frac{p^nx_r}{x_i}]$.) This finishes the proof.
\end{proof}
\begin{rem}
In fact, the normal assumption in Lemma \ref{uSen} can be waived here because the local universal deformation ring at $p$ of a pseudo-representation is normal by \cite[Theorem 1.4]{BIP21}.
\end{rem}
\begin{cor} \label{HTS}
The two dimensional semi-simple Galois representation associated to an overconvergent eigenform of weight $k\in\mathbb{Z}$ has Hodge-Tate-Sen weights $0,k-1$.
\end{cor}
\begin{proof}
We use the (generalized) notion of overconvergent modular forms introduced in \cite[Definition 5.2.5]{Pan20}. See also the discussion there for its relation with classical overconvergent modular forms. For an open compact subgroup $K_p$ of $\mathrm{GL}_2(\mathbb{Q}_p)$, denote by $\mathcal{V}_{K_p}$ the set of open subsets $V\subseteq \mathcal{X}^*_{K^pK_p}$ such that
\begin{itemize}
\item $\pi_{K_p}^{-1}(V)=\pi_{\mathrm{HT}}^{-1}(V_\infty)$ for some open neighborhood $V_\infty$ of $\infty\in\mathbb{P}^1={\mathscr{F}\!\ell}$.
\end{itemize}
Here $\pi_{K_p}:\mathcal{X}^*_{K^p}\to \mathcal{X}^*_{K^pK_p}$ denotes the projection map and $\pi_{\mathrm{HT}}:\mathcal{X}^*_{K^p}\to {\mathscr{F}\!\ell}$ is the Hodge-Tate period morphism, cf. the discussion in the beginning of Section \ref{FHi}. For example $V_{K_p,2}$ introduced in Section \ref{FHi} is an element of $\mathcal{V}_{K_p}$ if $K_p$ is sufficiently small. Open sets in $\mathcal{V}_{K_p}$ form a projective system by inclusions. If $K'_p\subseteq K_p$ is an open subgroup, there is a natural map $\mathcal{V}_{K_p}\to \mathcal{V}_{K'_p}$ induced by taking the preimages. The space of overconvergent modular forms of weight $k$ is defined as
\[M^{\dagger}_k(K^p):=\varinjlim_{K_p\to 1}\varinjlim_{V\in\mathcal{V}_{K_p}} H^0(V,\omega^k_{K^pK_p}).\]
(This is equivalent with \cite[Definition 5.2.5]{Pan20} by \cite[Proposition 5.2.6, Lemma 5.2.9]{Pan20}.)
Fix a $V\in \mathcal{V}_{K_p}$. The Hecke operators away from $p$ acts on $H^0(V,\omega^k_{K^pK_p})$. An (non-zero) eigenvector of $\mathbb{T}_S$ in $M^{\dagger}_k(K^p)$ is called an overconvergent eigenform of weight $k$. We remark that elements of form $\begin{pmatrix} p^l & 0\\ 0 & 1\end{pmatrix}\in\mathrm{GL}_2(\mathbb{Q}_p)$ act on $M^{\dagger}_k(K^p)$.
Let $M_2\subseteq M^\dagger_k(K^p)$ be the image of $ \varinjlim_{K_p\to 1} H^0(V_{K_p,2},\omega^k_{K^pK_p})\to M^\dagger_k(K^p)$. We claim that
\[M^{\dagger}_k(K^p)=\bigcup_{n\in\mathbb{Z}} \begin{pmatrix} p^n& 0\\ 0 & 1\end{pmatrix}\cdot M_2.\]
This implies the corollary. Indeed, the action of $\begin{pmatrix} p^n& 0\\ 0 & 1\end{pmatrix}$ commutes with the action of the Hecke algebra. Hence our assertion follows from Theorem \ref{mtS} because the (usual) determinant of the Galois representation associated to an overconvergent eigenform of weight $k$ has Hodge-Tate weight $k-1$. To prove the claim, we first note that $\pi_{K_p}^{-1}(V_{K_p,2})=\pi_{\mathrm{HT}}^{-1}(U)$ for some open subset $U$ of ${\mathscr{F}\!\ell}$ containing $\infty$, which is independent of $K_p$. For any open neighborhood $V_\infty$ of $\infty$, $ \begin{pmatrix} p^n& 0\\ 0 & 1\end{pmatrix}\cdot U\subseteq V_\infty$ for some $n$. This implies that given $V\in\mathcal{V}_{K_p}$, there exist a sufficiently small open subgroup $K'\subseteq \mathrm{GL}_2(\mathbb{Q}_p)$ and integer $n$ such that
\begin{itemize}
\item $gK'g^{-1}\subseteq K_p$, where $g= \begin{pmatrix} p^n& 0\\ 0 & 1\end{pmatrix}$;
\item under the isomorphism $\varphi:\mathcal{X}^*_{K^pK'}\stackrel{\sim}{\to}\mathcal{X}^*_{K^pgK'g^{-1}}$ induced by $g^{-1}$, we have $\varphi(V_{K',2})\subseteq \pi^{-1}(V)$ where $\pi:\mathcal{X}^*_{K^pgK'g^{-1}}\to \mathcal{X}^*_{K^pK_p}$ denotes the projection map.
\end{itemize}
Thus the map $H^0(V,\omega_{K^pK_p}^k)\to M^\dagger_k(K^p)$ factors through $g\cdot H^0(V_{K',2},\omega_{K^pK'}^k)$ and our claim follows.
\end{proof}
\section{A result of Calegari-Emerton}
Matthew Emerton pointed out the following consequence of Corollary \ref{cor1}, which reproves a result of Calegari-Emerton \cite[Theorem 2.2]{CE04} and can be viewed as some evidence towards a question of Buzzard \cite[Question 4.4]{Bu05} asking whether for a fixed level, all Hecke eigenvalues of arbitrary weights lie in a \textit{finite} extension of $\mathbb{Q}_p$. We denote by $\overbar\mathbb{Z}_p$ the ring of integers of $\overbar\mathbb{Q}_p$ and by $\mathfrak{m}$ its maximal ideal.
\begin{thm}
Let $S$ be a finite set of rational primes containing $p$ and $K=\prod_{l}K_l$ be an open compact subgroup of $\mathrm{GL}_2(\mathbb{A}_f)$ with $K_l\cong\mathrm{GL}_2(\mathbb{Z}_l)$ for $l\notin S$. There exists a rational number $\kappa=\kappa(K,p)$ such that for any $\lambda:\mathbb{T}_S\to\overbar\mathbb{Z}_p$ appearing in $H^0(V_i,\omega_{K^pK_p}^{\otimes k})$ and $\lambda':\mathbb{T}_S\to\overbar\mathbb{Z}_p$ appearing in $H^0(V_i,\omega_{K^pK_p}^{\otimes k'})$ for some integers $k,k'$, ($\lambda,\lambda'$ may come from classical forms for example,) if $\lambda\equiv \lambda' \mod \mathfrak{m}$, then
\[\lambda\equiv \lambda' \mod p^{\kappa}\overbar\mathbb{Z}_p.\]
\end{thm}
\begin{proof}
Clear as the action of $\mathbb{T}_S$ on $H^0(V_i,\omega_{K^pK_p}^{\otimes k})$ is locally analytic.
\end{proof}
\section{Hecke action on locally analytic vectors of admissible representations}
In this last section, we provide another example of locally analytic Hecke actions: the case of the Hecke algebra acting on the locally analytic vectors in the completed cohomology. In fact we will prove this result in a more general setup. Suppose
\begin{itemize}
\item $G$ is a finite-dimensional $p$-adic Lie group;
\item $W$ is an \textit{admissible} Banach space representation of $G$. Recall that this means that for any open compact subgroup $K$ of $G$ and any open bounded $K$-stable lattice $\mathcal{L}\subseteq W$, the $\mathbb{F}_p$-dimension of $(\mathcal{L}/p\mathcal{L})^{K}$ is finite.
\item $A$ is a ring and $W$ is equipped with an $A$-module structure which commutes with $G$.
\end{itemize}
For simplicity, we also assume the following:
\begin{itemize}
\item $W^o$ is $A[K]$-stable for some open subgroup $K$ of $G$.
\end{itemize}
One Typical example to keep in mind is that $W$ is Emerton's completed cohomology introduced in \cite{Eme06} for arithmetic quotients of symmetric spaces and $A$ is the Hecke algebra. If these arithmetic quotients are Shimura varieties defined over a number field $F$, one can also take $A=\mathbb{Z}_p[G_F]$.
Let $K$ be an open subgroup of $G$ sufficiently small so that $W^o$ is $K$-stable and it makes sense to talk about analytic functions on it, cf. Theorem 27.1 of \cite{Sch11}. We denote by $W^{K-\mathrm{an}}\subseteq W$ the subspace of $K$-analytic vectors. It is a $\mathbb{Q}_p$-Banach space and an $A$-module.
\begin{thm} \label{ocla}
The action of $A$ on $W^{K-\mathrm{an}}$ is locally analytic.
\end{thm}
\begin{proof}
We denote by $\mathscr{C}^{\mathrm{an}}(K,\mathbb{Q}_p)$ the space of $\mathbb{Q}_p$-valued analytic functions on $K$ with the unit open ball $\mathscr{C}^{\mathrm{an}}(K,\mathbb{Q}_p)^o$ . Fix $n\geq 1$. Then
\[W^{K-\mathrm{an},o}=(W^o\widehat\otimes_{\mathbb{Z}_p} \mathscr{C}^{\mathrm{an}}(K,\mathbb{Q}_p)^o)^K,\]
cf. \cite[2.1]{Pan20}. The completed tensor product is $p$-torsion free. Hence there is an inclusion
\[W^{K-\mathrm{an},o}/p^n\subseteq (W^o\otimes_{\mathbb{Z}_p} \mathscr{C}^{\mathrm{an}}(K,\mathbb{Q}_p)^o/p^n)^K.\]
Note that $ \mathscr{C}^{\mathrm{an}}(K,\mathbb{Q}_p)^o/p^n$ is fixed by some open subgroup $K'$ of $K$: when $n=1$, this is \cite[Lemma 2.1.2.]{Pan20}. The same argument works for any $n$. Hence
\[ W^{K-\mathrm{an},o}/p^n\subseteq (W^o\otimes_{\mathbb{Z}_p} \mathscr{C}^{\mathrm{an}}(K,\mathbb{Q}_p)^o/p^n)^{K'}=(W^o/p^n)^{K'}\otimes_{\mathbb{Z}_p/p^n} \mathscr{C}^{\mathrm{an}}(K,\mathbb{Q}_p)^o/p^n.\]
(Implicitly we use that $\mathscr{C}^{\mathrm{an}}(K,\mathbb{Q}_p)^o/p^n$ is flat over $\mathbb{Z}_p/p^n$.) Note that all the maps are $A$-equivariant. Thus the image of $A\to\End(W^{K-\mathrm{an},o}/p^n)$ factors through the image of $A\to \End((W^o/p^n)^{K'})$. But $(W^o/p^n)^{K'}$ is finite by the admissibility. By definition, this means that the action of $A$ on $W^{K-\mathrm{an}}$ is locally analytic.
\end{proof}
\begin{rem}
When $W$ is the completed cohomology of modular curves and $A=\mathbb{Z}_p[G_{\mathbb{Q}_p}]$, this result implies that Sen's theory can be applied to $W^{K-\mathrm{an}}$. For example it follows that the Sen operator acts on $W^{K-\mathrm{an}}\widehat\otimes_{\mathbb{Q}_p} C$, cf. \cite[Remark 5.1.16]{Pan20}.
\end{rem}
\bibliographystyle{amsalpha}
|
1,314,259,993,604 | arxiv | \section{Introduction} \label{sec:Intro}
Defining off-shell amplitudes in closed string field theory requires selecting a set of string vertices $\mathcal{V}_{g,n}$ with $2g-2+n > 0$~\cite{Zwiebach:1992ie,Erler:2019loq}. These are subsets of the moduli spaces $\widehat{\mathcal{P}}_{g,n}$ of compact Riemann surfaces of genus $g$ and $n$ punctures with a choice of local coordinates (defined up to global phases) around each puncture. String vertices ought to satisfy the geometric master equation in order to define a consistent quantum theory~\cite{Sen:1994kx,Sen:1993kb,Sonoda:1989wa}.
There have been few proposals in the past for how to explicitly specify string vertices $\mathcal{V}_{g,n}$. The oldest, and probably the most well-known, is the one that uses the minimal area metrics on Riemann surfaces~\cite{Zwiebach:1992ie,Zwiebach:1990nh}. Using such metrics there is a simple prescription for how to specify string vertices that solves the geometric master equation~\cite{Zwiebach:1992ie}. The minimal area metrics for higher genus surfaces, however, are not known explicitly and still lack rigorous proof of existence. Nonetheless, one may expect that these will soon follow in the light of the recent discoveries~\cite{Headrick:2018ncs,Headrick:2018dlw,Naseer:2019zau}.
Another proposal for string vertices $\mathcal{V}_{g,n}$ that utilizes the fact that the Riemann surfaces considered for $\mathcal{V}_{g,n}$ admit hyperbolic metrics (of constant negative Gaussian curvature $K=-1$) was recently made by Moosavian and Pius~\cite{Moosavian:2017qsp,Moosavian:2017sev}. This interesting approach seems particularly promising considering the rigorously established existence of hyperbolic metrics and the recent developments in evaluating integrals over the moduli spaces of Riemann surfaces using the associated Teichm\"uller spaces~\cite{mcshane1998simple,mirzakhani2007weil,mirzakhani2007simple,Eynard:2007fi,ellegard1,ellegard2,Dijkgraaf:2018vnm}. However, it has been shown that these string vertices solve the geometric master equation only to the first approximation and they require a correction at each order of approximation. It is not known that such corrected string vertices always exist.
Although they are intriguing in their own rights, we see that two proposals for string vertices above suffer from either missing the proof of existence or failing to satisfy the geometric master equation exactly, therefore falling short of providing a consistent string field theory. In order to have a consistent string field theory we must guarantee that the string vertices exist on the moduli spaces of Riemann surfaces while exactly satisfying the geometric master equation. \emph{Hyperbolic string vertices} by Costello and Zwiebach simultaneously achieved both of these conditions recently~\cite{Costello:2019fuh}. To that end, the authors considered Riemann surfaces endowed with hyperbolic metric with \emph{geodesic boundaries} of length $L$, for $0<L\leq2$ arcsinh(1), and with systole\footnote{Systole on a bordered surface is defined as the length of the shortest closed geodesic that is \emph{not} a boundary component.} greater than or equal to $L$. Then they specified the string vertices by attaching flat semi-infinite cylinders of circumference $L$ at each boundary component to such surfaces. By the existence of hyperbolic metrics on Riemann surfaces of genus $g$ and $n$ boundaries with $2g-2+n > 0$, it was argued that this construction is always possible. Furthermore, it has been shown that the resulting string vertices exactly satisfy the geometric master equation by the virtue of the collar theorems of hyperbolic geometry~\cite{buser2010geometry}. We are going to call the closed bosonic string field theory hyperbolic string vertices define \emph{hyperbolic string field theory}.
Beyond establishing the first rigorous, explicit, and exact construction for the string vertices, using the hyperbolic string vertices also seems promising from the perspective of the aforementioned developments in computing integrals over the moduli spaces of Riemann surfaces by exploiting the underlying hyperbolic geometry, just like in the case of the vertices of Moosavian and Pius. One might imagine (or hope) similar methods can be applied to evaluate the string amplitudes to arbitrary orders and provide a useful handle for the computations in hyperbolic string field theory as a result.
A natural first step in this direction would be to compute the off-shell three-string amplitudes using the hyperbolic three-string vertex $\mathcal{V}_{0,3}$, which is constructed by grafting three flat semi-infinite cylinders to the three-holed sphere (or \emph{pair of pants}) equipped with a hyperbolic metric, since $\mathcal{V}_{0,3}$ contains just a single surface. For the sake of generality, we are going to leave the circumferences of the grafted cylinders arbitrary for this vertex, even though only the case of equal circumferences is needed for $\mathcal{V}_{0,3}$~\cite{Costello:2019fuh}. This \emph{generalized hyperbolic three-string vertex} is of interest in the hyperbolic string field theory in the long run on account of the well-known pants decomposition of Riemann surfaces~\cite{buser2010geometry}. For brevity, we will also denote this generalized vertex as hyperbolic three-string vertex without making a distinction.
In order to perform the computations mentioned above using the operator formalism of conformal field theory (CFT), one needs to obtain the explicit expressions of the local coordinates around the punctures for the hyperbolic three-string vertex~\cite{Erler:2019loq}. In this paper, we find these local coordinates, investigate their various limits, and derive the associated conservation laws by following the procedure in~\cite{rastelli2001tachyon}.
In principle, the local coordinates for the hyperbolic three-string vertex can be obtained by the following procedure. First, recall that the hyperbolic metric on the three-holed sphere with geodesic boundaries of lengths $L_i$ ($i=1,2,3$) is unique up to isometry~\cite{buser2010geometry}. So we can simply write down this hyperbolic metric as
\begin{equation} \label{eq:intrometric}
ds^2 = e^{\varphi(z,\bar{z})} |dz|^2,
\end{equation}
on the Riemann sphere minus three disjoint simply connected regions, or \emph{holes}, \emph{unique} up to PSL(2,$\mathbb{C}$) transformations, whose boundaries are geodesics of given lengths $L_i$. From this point of view, one can obtain the local coordinates by finding how punctured unit disks conformally map onto these simply connected regions, since a semi-infinite cylinder is conformal to a punctured unit disk and it canonically introduces the local coordinates~\cite{Costello:2019fuh}. Note that such conformal transformations exist by the Riemann mapping theorem.
Therefore, we see that the problem of finding the local coordinates for the hyperbolic three-string vertex is a two-step procedure:
\begin{enumerate}
\item Find an explicit description of the union of three disjoint simply connected regions on the Riemann sphere whose complement is endowed with a hyperbolic metric~\eqref{eq:intrometric} and boundary components are geodesics of lengths $L_i$,
\item Find the conformal transformations from punctured unit disks to the aforementioned simply connected regions.
\end{enumerate}
The first step clearly involves solving a complicated boundary value problem for a partial differential equation, \emph{Liouville's equation}, and getting an exact answer is a hard endeavor in general. Luckily, it is known that the solutions for such boundary value problem can be related to a monodromy problem of a particular second-order linear ordinary differential equation with regular singularities, or a \emph{Fuchsian equation}, on the complex plane~\cite{hadasz2003polyakov,hadasz2004classical}. Exploiting this relation, which we review and expand in sections~\ref{sec:Fuchsian} and~\ref{sec:Monodromy}, we find the explicit description of the hyperbolic metric~(\ref{eq:intrometric}) and of the three holes on the Riemann sphere, up to PSL(2,$\mathbb{C}$) transformations.
Furthermore, the second step becomes trivial after we find such explicit description as we argue in section~\ref{sec:Monodromy}. In the end, for the hyperbolic three-string vertex whose grafted flat cylinders have the circumferences
\begin{equation} \label{eq:circum}
L_i \equiv 2 \pi \lambda_i,
\end{equation}
we obtain the following local coordinates $w_i(z)$ around the punctures at $z=0,1,\infty$ respectively:\footnote{We fix the locations of the punctures to $z=0,1,\infty$ using PSL(2,$\mathbb{C}$) transformations without loss of generality. So indices and numbers $i,j=1,2,3$ appearing on the objects denote the punctures $z=0,1,\infty$ respectively, unless otherwise stated. This shall be obvious from the context and we are not going to report it every time. If there is no label on an object, it should be understood that it has the same value for each puncture.}
\begin{subequations} \label{eq:lc}
\begin{align} \label{eq:lc1}
w_1(z) &= \frac{1}{N_1} \exp\left( \frac{v(\lambda_1,\lambda_2, \lambda_3)}{\lambda_1} \right) z (1-z)^{-\frac{\lambda_2}{\lambda_1}} \nonumber \\
&\qquad \qquad \times \left[ \frac{_2F_1 \left(\frac{1}{2}(1+i \lambda_1 -i \lambda_2 +i \lambda_3), \, \frac{1}{2}(1+i \lambda_1 -i \lambda_2 -i \lambda_3);\, 1+ i \lambda_1;\, z\right)}{_2F_1 \left(\frac{1}{2}(1-i \lambda_1 +i \lambda_2 -i \lambda_3),\,\frac{1}{2}(1-i \lambda_1 +i \lambda_2 +i \lambda_3);\, 1- i \lambda_1;\, z\right)} \right]^{\frac{1}{i \lambda_1}} \nonumber \\
&= \frac{1}{N_1} \exp\left( \frac{v(\lambda_1,\lambda_2, \lambda_3)}{\lambda_1} \right) \left(z + \frac{1+ \lambda_{1}^2 + \lambda_{2}^2 - \lambda_{3}^2}{2(1+\lambda_{1}^2)} z^2 + \mathcal{O}(z^3)\right) ,\\
\label{eq:lc2}
\quad \;\; w_2(z) &= \frac{1}{N_2} \exp\left( \frac{v(\lambda_2,\lambda_1, \lambda_3)}{\lambda_2} \right)(1-z) z^{-\frac{\lambda_1}{\lambda_2}} \nonumber\\
&\qquad \qquad \times
\left[ \frac{_2F_1 \left(\frac{1}{2}(1+i \lambda_2 -i \lambda_1 +i \lambda_3),\,\frac{1}{2}(1+i \lambda_2 -i \lambda_1 -i \lambda_3);\, 1+ i \lambda_2;\, 1-z\right)}{_2F_1 \left(\frac{1}{2}(1-i \lambda_2 +i \lambda_1 -i \lambda_3),\,\frac{1}{2}(1-i \lambda_2 +i \lambda_1 +i \lambda_3);\, 1- i \lambda_2;\, 1-z\right)} \right]^{\frac{1}{i \lambda_2}}\nonumber\\
&= \frac{1}{N_2} \exp\left( \frac{v(\lambda_2,\lambda_1, \lambda_3)}{\lambda_2} \right)\left((1-z) + \frac{1+ \lambda_{2}^2 + \lambda_{1}^2 - \lambda_{3}^2}{2(1+\lambda_{2}^2)} (1-z) ^2 + \mathcal{O}((1-z) ^3)\right), \\
\label{eq:lc3}
w_3(z) &= \frac{1}{N_3} \exp\left( \frac{v(\lambda_3,\lambda_2, \lambda_1)}{\lambda_3} \right)\left(\frac{1}{z}\right) \left(1-\frac{1}{z}\right)^{-\frac{\lambda_2}{\lambda_3}} \nonumber\\
&\qquad \qquad \times
\left[ \frac{_2F_1 \left(\frac{1}{2}(1+i \lambda_3 -i \lambda_2 +i \lambda_1),\,\frac{1}{2}(1+i \lambda_3 -i \lambda_2 -i \lambda_1);\, 1+ i \lambda_3;\, \frac{1}{z}\right)}{_2F_1 \left(\frac{1}{2}(1-i \lambda_3 +i \lambda_2 -i \lambda_1),\,\frac{1}{2}(1-i \lambda_3 +i \lambda_2 +i \lambda_1);\, 1- i \lambda_3;\, \frac{1}{z}\right)} \right]^{\frac{1}{i \lambda_3}} \nonumber \\
&= \frac{1}{N_3} \exp\left( \frac{v(\lambda_3,\lambda_2, \lambda_1)}{\lambda_3} \right) \left(\frac{1}{z} + \frac{1+ \lambda_{3}^2 + \lambda_{2}^2 - \lambda_{1}^2}{2(1+\lambda_{3}^2)}\frac{1}{z^2} + \mathcal{O}\left(\frac{1}{z^3} \right)\right).
\end{align}
\end{subequations}
Here ${_2}F_1(a,b;c;z)$ is the ordinary hypergeometric function
\begin{equation} \label{eq:hypergeometric}
{_2}F_1(a,b;c;z) = 1 + \frac{ab}{c} \frac{z}{1!} + \frac{a(a+1)b(b+1)}{c(c+1)} \frac{z^2}{2!} + \cdots,
\end{equation}
and the function $v\left(\lambda_{1}, \lambda_{2}, \lambda_{3}\right) $ is given in terms of the gamma function $\Gamma(z)$ as
\begin{align}
v\left(\lambda_{1}, \lambda_{2}, \lambda_{3}\right) &\equiv \frac{1}{2i} \text{Log}\left[\frac{\Gamma\left(-i \lambda_{1}\right)^{2}}{\Gamma\left(i \lambda_{1}\right)^{2}} \frac{\gamma\left(\frac{1}{2}(1+i \lambda_{1}+i \lambda_{2}+i \lambda_{3})\right) \gamma\left(\frac{1}{2}(1+i \lambda_{1}-i \lambda_{2}+i \lambda_{3})\right)}{\gamma\left(\frac{1}{2}(1-i \lambda_{1}-i \lambda_{2}+i \lambda_{3})\right) \gamma\left(\frac{1}{2}(1-i \lambda_{1}+i \lambda_{2}+i \lambda_{3})\right)} \right],\nonumber \\
\gamma(x) &\equiv \frac{\Gamma(1-x)}{\Gamma(x)}.
\end{align}
The factors $N_i$ above will be called \emph{scale factors}. They are fixed by integer $\tilde{l}_i \in \mathbb{Z}$ via
\begin{equation}
N_i = \exp\left[ \frac{\pi}{\lambda_i} \left(\tilde{l}_i + \frac{1}{2}\right) \right].
\end{equation}
Tilde on the integers $\tilde{l}_i$ comes from our construction. As we will see, among the sets of integers $l_i$, only specific ones, $l_i=\tilde{l}_i$, would give the correct scale factor. The integers $\tilde{l}_i$ can be determined for a given set of $\lambda_i$'s in principle. Even though we couldn't find a closed-form expression for the integers $\tilde{l}_i$'s for arbitrary $\lambda_i$'s, one can still easily find them by investigating numerical plots of the local coordinates. In certain symmetric situations it is possible to find the integers $\tilde{l}_i$'s without resorting the plots. For example, when $0 \leq \lambda_i = \lambda \leq 10$ for the grafted cylinders, we find $\tilde{l}_i=\tilde{l}=-1$. This result is anticipated to hold for all values of $\lambda_i = \lambda$.
Since we have the explicit expressions of the local coordinates for the hyperbolic three-vertex~\eqref{eq:lc}, we can check their consistency with the other local coordinates in the literature by investigating their limiting behaviors~\cite{sonoda1990covariant,Moosavian:2017qsp,Zwiebach:1988qp}. For instance, as argued in~\cite{Costello:2019fuh}, this vertex must produce the three-string vertex obtained from the minimal area metric as all $\lambda_i \to \infty$ at the same rate, which we are going to denote as the \emph{minimal area limit}. In the light of this fact, we consider the minimal area limit of the coordinates~\eqref{eq:lc} and show that the local coordinates for the minimal area three-string vertex and those for the hyperbolic three-string vertices with $\lambda \to \infty$ match perturbatively to the order $\mathcal{O}(z^{10})$ in section~\ref{sec:Limits}. We then discuss the possibility of extending our argument to all orders in $z$.
Moreover, the hyperbolic three-string vertex must reduce to the three-string vertex considered by Moosavian and Pius~\cite{Moosavian:2017qsp} as $\lambda_i \to 0$ after a suitable modification, since the geodesic boundaries become cusps in this regime and this is exactly what is considered there. We argue that this limiting behavior indeed holds in section~\ref{sec:Limits}. Lastly, we consider the situation $\lambda_{2} = \lambda_1 + \lambda_3$ with $\lambda_i \to \infty$ for which the geometry resembles the light-cone vertex~\cite{Zwiebach:1988qp}. We show that the hyperbolic three-string vertex reduces to the light-cone vertex in this limit, in accord with our expectations.
Having an explicit expression for the local coordinates~\eqref{eq:lc} also means that it is possible to derive the conservation laws for the hyperbolic three-string vertex in the spirit of~\cite{rastelli2001tachyon}, which we do in section~\ref{sec:Conservation}. Again, we can investigate various limits of these conservation laws. Especially we observe that all of our expressions in section~\ref{sec:Conservation} reduces to their respective counterparts in~\cite{rastelli2001tachyon} in the minimal area limit. This is consistent, since the open string Witten vertex and its closed string analog must generate the same conservation laws. It is known that conservation laws provide systematic and easily implementable procedure for computations in the cubic open string field theory, especially for the level truncation~\cite{Gaiotto:2002wy}, and we hope that these expressions will accomplish the same in the hyperbolic string field theory in the future.
As a sample computation using the local coordinates~\eqref{eq:lc}, we calculate the $t^3$ term in the closed string tachyon potential $V$ with $t$ is the zero-momentum tachyonic field in the case of $\lambda_i=\lambda$. Remember this is the case that appears in the string action. We find ($\alpha'=2$)
\begin{align} \label{eq:tachyon}
&\kappa^2 V = -t^2 + \frac{1}{3} \frac{t^3}{r^6} + \dots = -t^2 +\frac{1}{3} \exp\left[\frac{ 6v(\lambda,\lambda,\lambda) + 3\pi}{\lambda} \right] t^3 + \cdots.
\end{align}
Here $\kappa$ is the closed string coupling constant and $r$ is the mapping radius of the local coordinates, whose inserted expression is derived in section~\ref{sec:Monodromy}. Note that this calculation is exactly like in~\cite{kostelecky1990collective,belopolsky1995off,yang2005closed}, the only difference being the mapping radii we used for the expression above.
In order to get a sense of its value, let us set $\lambda= L_{\ast}/(2 \pi) = \text{arcsinh}(1)/\pi \approx 0.28055$, which is the largest value of $\lambda$ for which the hyperbolic vertices solves the geometric master equation~\cite{Costello:2019fuh}. Substituting this value and evaluating, we obtain the closed string tachyon potential $V$ in the hyperbolic string field theory is given by
\begin{equation} \label{eq:tp}
\kappa^2 V \approx -t^2 + \left( 1.62187 \times 10^8\right) t^3 + \cdots.
\end{equation}
The coefficient for the $t^3$ term is quite large compared to the corresponding one in the minimal area three-string vertex, which is approximately equal to $1.602$~\cite{kostelecky1990collective,belopolsky1995off,yang2005closed}. However, this coefficient in fact has the expected order of magnitude. We can see this by considering the coefficient obtained from the minimal area three-string vertex with stubs of length $\pi$, which roughly \emph{looks like} a hyperbolic three-string vertex geometrically. The coefficient for the case with stubs is easily obtained by observing that adding stubs scales mapping radii by $e^{-\pi}$, and in turn multiplies the no-stub coefficient by $e^{6 \pi}$ by the first equality in~\eqref{eq:tachyon}. This gives approximately $e^{6\pi} \cdot 1.602 \approx 2.460 \times 10^8$, which is close to the value given in~\eqref{eq:tp}.
The outline of the paper is as follows. In section~\ref{sec:Fuchsian} we introduce the boundary value problem for the hyperbolic metric with geodesic boundaries of fixed lengths on the Riemann sphere minus three holes and its relation to Fuchsian equations. In section~\ref{sec:Monodromy} we consider the relevant monodromy problem in order to find the explicit description of the holes on the Riemann sphere. The results of these two sections are well-established in the literature~\cite{hadasz2003polyakov,hadasz2004classical}, but we provide a self-contained discussion where we emphasize and investigate the resulting hyperbolic geometry in more detail. Additionally, we construct the local coordinates around the punctures for the hyperbolic three-string vertex in section~\ref{sec:Monodromy} and later in section~\ref{sec:Limits} we investigate their various limits. Lastly, we obtain the conservation laws associated with the hyperbolic three-string vertex in section~\ref{sec:Conservation}. We conclude the paper and discuss the possible future directions in section~\ref{sec:Conc}.
\section{Liouville's equation on a three-holed sphere} \label{sec:Fuchsian}
In this section, we describe the problem of finding an explicit description of the hyperbolic metric on the three-holed sphere with geodesic boundaries of lengths $L_i$ on the Riemann sphere, which will help us obtain the shapes and locations of the geodesic boundaries and the local coordinates later on. As we mentioned briefly, this is equivalent to solving Liouville's equation with specified boundary conditions on the Riemann sphere minus three holes. This problem is hard by itself, so instead we introduce a stress-energy tensor (in the sense of Liouville theory) and consider its associated Fuchsian equation, which we define below. The properties of this equation is investigated. Most importantly, we show that its multi-valued solutions can be related to hyperbolic metrics. The results of this section are well-known in the literature in the context of Liouville theory and the uniformization problem~\cite{hadasz2003polyakov,hadasz2004classical,hadasz2006liouville,cantini2001proof,cantini2002liouville,cantini2003polyakov,zograf1988liouville,takhtajan2003hyperbolic,Seiberg:1990eb,Bilal:1987cq,Hadasz:2005gk,Teschner:2003at,hempel1988uniformization}, but we are going to provide a self-contained review that focuses on the issues relevant to us.
As noted above, our first goal is to solve \emph{Liouville's equation}
\begin{equation} \label{eq:LiouvilleEq}
\partial \bar{\partial} \varphi(z,\bar{z}) = \tfrac{1}{2} e^{\varphi(z,\bar{z}) },
\end{equation}
on the three-holed sphere $X$ whose boundaries are chosen to be geodesics of lengths $L_i$ of the metric~(\ref{eq:intrometric}). It can be easily seen that satisfying Liouville equation is equivalent to the metric~(\ref{eq:intrometric}) having constant negative curvature $K=-1$. We will call $e^{\varphi(z,\bar{z}) }$ the \emph{conformal factor} and take $\varphi(z,\bar{z}) \in \mathbb{R}$ always to define a real metric. Like we mentioned before, we can think the surface $X$ endowed with the metric~(\ref{eq:intrometric}) as the Riemann sphere $\widehat{\mathbb{C}} = \mathbb{C} \cup \{\infty\}$ with three disjoint simply connected regions taken out and this understanding will be implicit. So $(z, \bar{z})$ will denote the complex coordinates on $X \subset \widehat{\mathbb{C}}$.
Solving the boundary value problem described above directly is non-trivial and we won't attempt to do that. Instead, we are going to relate this problem to solving a more manageable linear ordinary differential equation. In order to do that, let the factor $\varphi$ denote a solution of Liouville's equation~(\ref{eq:LiouvilleEq}) and define the (holomorphic) \emph{stress-energy tensor} associated with $\varphi$ as follows~\cite{Seiberg:1990eb}:
\begin{equation} \label{eq:stress-energy}
T_{\varphi}(z) \equiv -\tfrac{1}{2}(\partial \varphi)^2 + \partial^2 \varphi = -2 e^{\frac{\varphi}{2}} \partial^2 e^{-\frac{\varphi}{2}}.
\end{equation}
Observe that we only wrote the dependence on $z$, and not on $\bar{z}$, of the stress-energy tensor since it can be shown that $T_{\varphi}$ is holomorphic, $\bar{\partial} T_{\varphi} = 0$, using Liouville's equation~\eqref{eq:LiouvilleEq}. Furthermore, the converse of this statement holds as well: If $T_{\varphi} = T_{\varphi}(z)$ is holomorphic, then the factor $\varphi$ defined by $(\ref{eq:stress-energy})$ solves the Liouville's equation. Lastly, we note that $T_{\varphi}(z)$ is the (classical) stress-energy tensor in the context of Liouville theory and it transforms under conformal transformation $z \to \tilde{z}(z)$ as follows~\cite{Seiberg:1990eb}:
\begin{align} \label{eq:transofT}
T_{\varphi}(z) = \left(\frac{\partial \tilde{z}}{\partial z} \right)^2 \widetilde{T}_{\varphi}(\tilde{z}) + \{ \tilde{z},z \}.
\end{align}
Here tilde on the stress-energy tensor indicates that it is written in the $\tilde{z}$ coordinates and $\{\cdot, \cdot\}$ is the Schwarzian derivative:
\begin{equation}
\{ \tilde{z},z\} \equiv
\frac{\partial^3\tilde{z}}{\partial \tilde{z}} - \frac{3}{2} \left(\frac{\partial^2 \tilde{z}}{\partial \tilde{z}} \right)^2.
\end{equation}
We can similarly define the anti-holomorphic stress-energy tensor $\overline{T_{\varphi}}$ by replacing $\partial \to \bar{\partial}$ in~\eqref{eq:stress-energy}.
Now consider the following second-order linear ordinary differential equation constructed with the stress-energy tensor $T_{\varphi}(z)$ above~\cite{hadasz2003polyakov,hadasz2004classical}:
\begin{equation} \label{eq:Fuchsian}
\partial^2 \psi(z) + \tfrac{1}{2} T_{\varphi}(z) \psi (z) = 0.
\end{equation}
We will call this the holomorphic \emph{Fuchsian equation} associated with $T_{\varphi}(z)$. The reason for the name \emph{Fuchsian} will be justified in section~\ref{sec:Monodromy} when we show that the relevant $T_{\varphi}(z)$ contains at most double poles, so that the equation~(\ref{eq:Fuchsian}) has only regular singularities (i.e. \emph{Fuchsian}). Similarly, we can define the anti-holomorphic Fuchsian equation associated with $\overline{T_{\varphi}}(\bar{z})$. Considering~\eqref{eq:transofT}, in order to make the equation~(\ref{eq:Fuchsian}) conformal invariant, we are going to take the object $\psi(z)$ transforms as a conformal primary of dimension $(-\frac{1}{2},0)$. That is, we demand
\begin{equation} \label{eq:transofpsi}
\tilde{\psi} (\tilde{z}) = \left( \frac{\partial \tilde{z}}{\partial z}\right)^{\frac{1}{2}} \psi(z),
\end{equation}
under conformal transformation $z \to \tilde{z}(z)$.
Now suppose we have solved the Fuchsian equation and found two linearly independent, not necessarily single-valued, complex-valued solutions $\psi^{+}(z)$ and $\psi^{-}(z)$. We are going to always assume these solutions are normalized appropriately, in the sense that their Wronskian $W(\psi^{-},\psi^{+})$ is equal to one:
\begin{equation} \label{eq:Wronskian}
W(\psi^{-},\psi^{+}) \equiv (\partial \psi^{+}) \psi^{-} -\psi^{+} (\partial \psi^{+}) = 1.
\end{equation}
Now define the ratio $A(z)$ of these solutions and observe that we have the relations
\begin{equation} \label{eq:ratio}
A(z) \equiv \frac{\psi^{+}(z)}{\psi^{-}(z)} \quad \iff \quad \psi^{+}(z) = \frac{A(z)}{\sqrt{\partial A(z)}}, \quad \psi^{-}(z) = \frac{1}{\sqrt{\partial A(z)}}.
\end{equation}
From this, we immediately see the stress-energy tensor can be written as follows:
\begin{align} \label{eq:TasSch}
T_{\varphi}(z) = - 2 \frac{\partial^2 \psi^{-}}{\psi^{-}} &= -2 (\partial A)^{\frac{1}{2}} \partial^2 (\partial A)^{-\frac{1}{2}} = (\partial A)^{\frac{1}{2}} \partial \left((\partial A)^{-\frac{3}{2}} \partial^2 A\right) \nonumber \\
&= \frac{\partial^3A(z)}{\partial A(z)} - \frac{3}{2} \left(\frac{\partial^2 A(z)}{\partial A(z)} \right)^2 \equiv \{ A(z),z\} \implies T_{\varphi}(z) = \{ A(z),z\}.
\end{align}
In general, it is highly non-trivial to find the function $A(z)$ for a given $T_{\varphi}(z)$ satisfying (\ref{eq:TasSch}) above. However, if we know the solutions to the Fuchsian equation (\ref{eq:Fuchsian}), we see that $A(z)$ is determined by~\eqref{eq:ratio} up to M\"{o}bius transformations. That is one utility of the Fuchsian equation. Moreover, given $A(z)$ satisfying $T_{\varphi}(z) = \{A(z),z\}$, we can find the normalized solutions for the Fuchsian equation from~(\ref{eq:ratio}) as well. Note that $A(z)$ is a scalar under conformal transformations as can be seen from~\eqref{eq:transofpsi} and~\eqref{eq:ratio}.
Also we can see that putting the stress-energy tensor $T_{\varphi}(z)$ in the form (\ref{eq:TasSch}) and knowing such $A(z)$ is advantageous on the account of the transformation property of the stress-energy tensor~\eqref{eq:transofT}. The relation (\ref{eq:TasSch}), combined with the transformation property of the Schwarzian derivative and the stress-energy tensor, allows us to find the explicit expression of the stress-energy tensor $T_{\varphi}$ in other coordinates. We will see the benefit of this observations in the next section.
Another utility of the Fuchsian equation (\ref{eq:Fuchsian}) can be understood as follows. We can easily see that $\psi = e^{-\frac{\varphi(z,\bar{z})}{2}}$ solves (\ref{eq:Fuchsian}) using the second equality in~\eqref{eq:stress-energy}. This solution of the Fuchsian equation is real and single-valued because the metric (\ref{eq:intrometric}) itself is real and single-valued. It is important to observe that such factor solves the Fuchsian equation, because this allows us to relate the linearly independent, normalized solutions $\psi^{\pm}(z)$ of the Fuchsian equation to the hyperbolic metric (\ref{eq:intrometric}). In other words, knowing $\psi^{\pm}(z)$ would suffice to construct the metric.
Before we do that more precisely, we should first describe the multi-valuedness of the solutions $\psi^{\pm}(z)$. For our purposes, it is going to be sufficient to assume that the multi-valuedness of the solutions $\psi^{\pm}(z)$ are described by SL(2,$\mathbb{R}$) transformations, in the sense that when we go around any point $z=u \in \widehat{\mathbb{C}}$ by $(z-u)\to e^{2 \pi i} (z-u)$ the solutions are taken to be transforming as follows:
\begin{equation}
\begin{bmatrix}
\psi^{+} \\ \psi^{-}
\end{bmatrix} \; \to \;
\begin{bmatrix}
a & b \\ c & d
\end{bmatrix}
\begin{bmatrix}
\psi^{+} \\ \psi^{-}
\end{bmatrix} \quad \text{where} \quad a,b,c,d \in \mathbb{R}, \quad ad-bc=1,
\end{equation}
unless otherwise stated. That is, we assume the values that the functions $\psi^{\pm}(z)$ attain at a given point are related by SL(2,$\mathbb{R}$) transformations like above. From this, it is easy to see that the solution $e^{-\frac{\varphi(z,\bar{z})}{2}}$ of~\eqref{eq:Fuchsian} is given by the following linear combination of $\psi^{+}(z)$ and $\psi^{-}(z)$:
\begin{equation} \label{eq:FtoL}
e^{-\frac{\varphi(z,\bar{z})}{2}} = C \frac{i}{2} ( \overline{\psi^{-}(z)} \psi^{+}(z) - \overline{\psi^{+}(z)} \psi^{-}(z) ),
\end{equation}
since this is the unique real linear combination of the solutions $\psi^{\pm}(z)$ that is invariant under SL(2,$\mathbb{R}$) transformations (i.e. single-valued). As usual, the bar over the solutions denotes the complex conjugation. Here $C$ is a real constant, which turns out to be $C=\pm1$, as we will show it shortly. With this, the following metric has constant negative curvature $K= -1$:
\begin{equation} \label{eq:firstmetric}
ds^2 = e^{\varphi(z,\bar{z})} |dz|^2 = \cfrac{-4 |dz|^2}{( \overline{\psi^{-}(z)} \psi^{+}(z) - \overline{\psi^{+}(z)} \psi^{-}(z) )^2}.
\end{equation}
Note that a version of these expressions appears in the context of Liouville theory~\cite{Seiberg:1990eb}. There, the solutions $\psi^{\pm}(z)$ are interpreted as spin-$1/2$ representations of SL(2,$\mathbb{R}$) and their physical meaning is discussed.
The main takeaway from the discussion in the previous paragraphs is that the hyperbolic metric on a three-holed sphere $X$ can be related to the solutions of the Fuchsian equation using a suitable $T_{\varphi}(z)$. From the expression in (\ref{eq:stress-energy}), it might seem that finding $T_{\varphi}(z)$ as a function of $z$ is as hard as finding the explicit form of the metric (\ref{eq:intrometric}). However, as we will see in section \ref{sec:Monodromy}, $T_{\varphi}(z)$ can be found without knowing the metric. Then we can deduce the form of the hyperbolic metric by solving the associated Fuchsian equation through the relation~\eqref{eq:firstmetric}, which will eventually lead us to the local coordinates.\footnote{These relations hold for other hyperbolic Riemann surfaces with geodesic boundaries as well. But we will restrict our discussion to three-holed sphere, since it is the simplest case to perform these computations explicitly.}
Before we conclude this section, we need to show $C = \pm 1$ as we claimed. It is clear that not every value of a priori unfixed $C \in \mathbb{R}$ can define a hyperbolic metric with $K=-1$, so we need to choose the right value(s). This is essentially the reflection of the fact that the Fuchsian equation is linear: Every scaling of $e^{-\frac{\varphi(z,\bar{z})}{2}}$ is also a solution of (\ref{eq:Fuchsian}), even though the scaled ones don't define a hyperbolic metric with $K=-1$ because the Liouville's equation~(\ref{eq:LiouvilleEq}) is non-linear.
We can fix such $C$ once and for all as follows. First note that the conformal factor
\begin{equation} \label{eq:conformalfactor}
e^{\varphi(z,\bar{z})} = \frac{\lambda^2 |\partial f (z)|^2}{|f(z)|^2 \sin^2(\lambda \log|f(z)|)} = \frac{|\partial(\lambda \log(f(z)))|^2}{\sin^2(\lambda \log|f(z)|)}.
\end{equation}
always defines a (possibly singular) hyperbolic metric with $K=-1$, or equivalently, $\varphi$ above solves the Liouville's equation (\ref{eq:LiouvilleEq}) for an arbitrary holomorphic function $f(z)$ and an arbitrary $\lambda \in \mathbb{R}_{\geq 0}$, as one can check by explicit calculation. Now take the function $f(z)$ to be equal to
\begin{equation} \label{eq:HolChoice}
f(z) = A(z)^{\frac{1}{i \lambda}} = \left( \frac{\psi^{+}(z)}{\psi^{-}(z)} \right)^{\frac{1}{i \lambda}}.
\end{equation}
We will denote the right-hand side as the \emph{scaled ratio}. After substituting this expression into~(\ref{eq:conformalfactor}) we exactly get the metric~(\ref{eq:firstmetric}). This shows $C =\pm1$. For us, the equivalence between~(\ref{eq:firstmetric}) and (\ref{eq:conformalfactor}), with the choice (\ref{eq:HolChoice}), is going to be extremely useful and we will use both forms interchangeably in our arguments.
In summary, we have seen that we can relate the hyperbolic metric on a three-holed sphere $X$ to the solutions of the Fuchsian equation (\ref{eq:Fuchsian}) through (\ref{eq:firstmetric}). Not only this will provide us a solution to the Liouville's equation (\ref{eq:LiouvilleEq}), but, more importantly, it will be also used to make the boundaries of $X$ geodesics of the metric~\eqref{eq:intrometric}. After all, that's the whole reason we are taking this detour into Fuchsian equations. We have already seen that the conformal factor~(\ref{eq:conformalfactor}) always defines a (possibly singular) hyperbolic metric for any given $f(z)$, but the boundaries of $X$ are going to be geodesics only when we relate it to a particular set of solutions for the Fuchsian equation through the relation (\ref{eq:HolChoice}), as we shall see. In the next section, we are going to focus on the three-punctured sphere $\widetilde{X} = \mathbb{C} \setminus \{0,1,\infty\}$, rather than a three-holed sphere $X$, since it is simpler to deal with initially. Then we will cut open appropriate holes around the punctures in $\widetilde{X}$ to return back to $X \subset \widehat{\mathbb{C}}$ and graft flat semi-infinite cylinders to these holes to construct the local coordinates for the hyperbolic three-string vertex.
\section{A monodromy problem of Fuchsian equation} \label{sec:Monodromy}
In this section we find the hyperbolic metric on a three-holed sphere $X$ by investigating a certain monodromy problem of the Fuchsian equation (\ref{eq:Fuchsian}) on the three punctured sphere $\widetilde{X}$ and construct the local coordinates for the hyperbolic three-string vertex. First, we describe the relevant monodromy problem and solve the Fuchsian equation on $\widetilde{X}$ accordingly. Then we find the explicit form of the (singular) hyperbolic metric on $\widetilde{X}$ by the relations given in section~\ref{sec:Fuchsian}. The resulting geometry looks like three semi-infinite series of hyperbolic cylinders, attached where they flare up, connected to each other while keeping the curvature constant and negative.
Next, we cut these hyperbolic cylinders out from the geometry appropriately, which leave us with a three-holed sphere $X$. This procedure doesn't change the hyperbolic metric, so at the end we obtain an explicit description of the hyperbolic metric with geodesic boundaries on a three-holed sphere. Moreover, we describe the holes on the Riemann sphere explicitly by investigating the simple closed geodesics of this hyperbolic metric. After that, grafting flat semi-infinite cylinders needed for the construction of the local coordinates amounts to simple conformal transformations of the punctured unit disks to these holes.
Most of the results from this section (except for subsection \ref{sec:Local}) are from~\cite{hadasz2003polyakov,hadasz2004classical}, for which we provide a detailed summary. However, we elaborate the geometric picture coming from the hyperbolic metric in more detail and prove some important results necessary for the explicit construction of the local coordinates.
\subsection{Description of the monodromy problem}
Consider the three-punctured sphere $\widetilde{X} = \mathbb{C} \setminus \{0,1,\infty\}$ and suppose that the solutions of the Fuchsian equation (\ref{eq:Fuchsian}) have hyperbolic SL(2,$\mathbb{R}$) monodromy around each puncture. That is, as we go around a puncture by $(z-z_j) \to e^{2 \pi i }(z-z_j)$, we demand that the solutions for the Fuchsian equation $\psi^{\pm}(z)$ change as,
\begin{equation} \label{eq:SL2R}
\begin{bmatrix}
\psi^{+} \\ \psi^{-}
\end{bmatrix} \; \to \;
M^j
\begin{bmatrix}
\psi^{+} \\ \psi^{-}
\end{bmatrix} \quad \text{where} \quad M^j \in \text{SL(2,}\mathbb{R}), \quad |\text{Tr} M^j|>2.
\end{equation}
Note that the condition on the trace makes the matrix $M^j$ a hyperbolic element of $\text{SL(2,}\mathbb{R})$ and that's why we say we have a hyperbolic monodromies around the puncture $z=z_j$. Realizing this structure for the solutions to the Fuchsian equation and finding them is our \emph{monodromy problem}. This problem is first considered in~\cite{hadasz2004classical} in the context of Liouville theory. We will call a puncture \emph{hyperbolic singularity} if the solutions of the Fuchsian equation have a hyperbolic SL(2,$\mathbb{R}$) monodromy around it.
In order to solve the monodromy problem, we need to first determine appropriate $T_{\varphi}(z)$ as a function of $z$ (if exists) so that the solutions of the Fuchsian equation can realize these monodromies around the punctures. Then declaring that particular $T_{\varphi}(z)$ to be equal to (\ref{eq:stress-energy}) coming from Liouville theory and using the reasoning in section \ref{sec:Fuchsian} we can extract the possibly singular hyperbolic conformal factor on $\widetilde{X}$ with the solutions that realize these monodromies. As explained above the equation ~\eqref{eq:FtoL}, this metric is going to be single-valued by $\text{SL(2,}\mathbb{R})$ monodromies and it will eventually lead us to the hyperbolic metric with geodesic boundaries on a three-holed sphere $X$.
Before we do that, let us investigate an individual hyperbolic singularity. We begin by picking a puncture, say $z=0$, and choosing a normalized basis of solution $\psi_{1}^{\pm}(z)$ for which the monodromy around $z=0$ is diagonal as follows:
\begin{equation} \label{eq:DiagMon}
\begin{bmatrix}
\psi_1^{+} \\ \psi_1^{-}
\end{bmatrix} \; \to \;
\begin{bmatrix}
-e^{-\pi \lambda_1} & 0 \\ 0 & -e^{\pi \lambda_1}
\end{bmatrix}
\begin{bmatrix}
\psi_1^{+} \\ \psi_1^{-}
\end{bmatrix}
\iff
\psi_{1}^{\pm}(z) = \frac{e^{\pm \frac{i v_1}{2}}}{\sqrt{i \lambda_1}} z^{\frac{1\pm i \lambda_1}{2}} (1+ \mathcal{O}(z)).
\end{equation}
The solutions always can be put into this form around $z=0$ since hyperbolic elements of SL(2,$\mathbb{R}$) can be diagonalized by conjugation, which amounts to performing a SL(2,$\mathbb{R}$) change of basis of the solutions. Here $\lambda_1 \in \mathbb{R}$ will be called the \emph{geodesic radius} associated with the $z=0$ puncture and the reason for its name will be apparent shortly. Without loss of generality we will take $\lambda_1>0$. Note that the Wronskian of these solutions is equal to $1$ thanks to the factor $1/\sqrt{i \lambda_1}$ in front. Furthermore, we also included the factors $e^{\pm \frac{i v_1}{2}}$, with $v_1 \in \mathbb{C}$, to account for the multiplicative constant that is not fixed by the Wronskian condition~\eqref{eq:Wronskian}. As we shall see, the constant $v_1$ will be fixed below by demanding SL(2,$\mathbb{R}$) monodromies around each puncture.
Using~\eqref{eq:DiagMon}, we can write the scaled ratio associated with the puncture $z=0$, as in~\eqref{eq:HolChoice},
\begin{equation} \label{eq:ScaledRatio}
\rho_1(z) \equiv \left( \frac{\psi_1^+(z)}{\psi_1^-(z)} \right)^{\frac{1}{i \lambda_1}} = e^{\frac{v_1}{\lambda_1}}(z + \mathcal{O}(z^2)).
\end{equation}
Note that this series expansion converges only on the open unit disk $D_1 = \{z \in \mathbb{C} \; | \; |z|<1 \}$, around the puncture $z=0$, since outside $D_1$ the scaled ratio $\rho_1(z)$ is multi-valued by the solutions $\psi_{1}^{\pm}(z)$ having a non-diagonal monodromy around the punctures at $z=1, \infty$. We can analytically continue the scaled ratio defined above outside the disk $D_1$, but inevitably this will require us to choose a branch for which $\rho_1(z)$ is continuous across $\partial D_1$ except at the punctures/branch cuts. We will choose the branch cut $\widetilde{L}_1$ of $\rho_1(z)$ to extend from 1 to $\infty$ along the real axis and take this to be the principal branch of $\rho_1(z)$. Thus, we conclude that the scaled ratio $\rho_1(z)$ can be defined analytically on the set
\begin{equation} \label{eq:s1}
S_1 = \mathbb{C} \setminus \widetilde{L}_1,
\end{equation}
with the expansion (\ref{eq:ScaledRatio}). When we mention the scaled ratio, we will consider the principal branch implicitly henceforth, unless otherwise stated. Lastly, note that the scaled ratio is an analytic scalar under conformal transformations, just like the ratio $A(z)$ in~\eqref{eq:ratio}.
Now by performing the conformal transformation $z \to \rho_1 = \rho_1(z)$ on $S_1$ and using the equation (\ref{eq:TasSch}) along with the properties of the Schwarzian derivative we see
\begin{equation} \label{eq:Tsect3}
T_{\varphi}(z) =\{ A(z), z\}= \left\{ \frac{\psi_1^+(z)}{\psi_1^-(z)}, z\right\} =\{\rho_1(z)^{i \lambda}, z \} = (\partial \rho_1)^2 \{ \rho_1^{i \lambda_1}, \rho_1\}+ \{\rho_1,z \}.
\end{equation}
Comparing the final form with~\eqref{eq:transofT} we read that the stress-energy tensor $\widetilde{T_{\varphi}}(\rho_1)$ in the $\rho_1$-plane takes the following form:
\begin{equation} \label{eq:closetopuncture}
\widetilde{T_{\varphi}}(\rho_1) = \{ \rho_1^{i \lambda_1}, \rho_1\} = \frac{\Delta_1}{\rho_1^2} \quad \text{where} \quad \Delta_1 \equiv \frac{1}{2} + \frac{\lambda_1^2}{2}.
\end{equation}
Here the real number $\Delta_1$ will be called the \emph{weight}. As a result of this, the Fuchsian equation in the $\rho_1$-plane takes a very simple form and we can easily obtain its solutions:
\begin{equation}
\frac{\partial^2 \tilde{\psi}(\rho_1)}{\partial \rho_1^2} + \frac{\Delta_1}{2 \rho_1^2} \tilde{\psi}(\rho_1) =0
\implies \widetilde{\psi}_1^{\pm} (\rho_1)= \frac{\rho_1^{\frac{1 \pm i \lambda_1}{2}}}{\sqrt{i \lambda_1}}.
\end{equation}
Here, $\widetilde{\psi}_1^{\pm}(z)$ are normalized solutions that are chosen to have diagonal monodromy around the puncture $z=0$, or equivalently $\rho_1 = 0$. Here we set the phase factor not fixed by Wronskian equal to one for convenience.\footnote{Considering this factor just adds a phase shift for the sine that appears in~\eqref{eq:met}, which would be unimportant for our considerations in this subsection.} Note that the scaled ratio of these two solutions is simply
\begin{equation}
\left( \frac{ \tilde{\psi}^{+}_1(z)}{ \tilde{\psi}^{-}_1(z)} \right)^{\frac{1}{i\lambda_1}} = \rho_1.
\end{equation}
As a result, the hyperbolic metric that the Fuchsian equation produces in the $\rho_1$-and $z$-plane are simply given by, using the relation (\ref{eq:conformalfactor}) with the choice $f(\rho_1)=\rho_1$,
\begin{equation} \label{eq:met}
ds^2 = \frac{\lambda_{1}^2 }{|\rho_1|^2 \sin^2(\lambda_1 \log|\rho_1|)} |d\rho_1|^2 = \frac{\lambda_{1}^2 |\partial \rho_1 (z)|^2}{|\rho_1(z)|^2 \sin^2(\lambda_1 \log|\rho_1(z)|)} |dz|^2.
\end{equation}
There are two important things we should notice here. First, the metric takes the form of a series of hyperbolic cylinders that are attached to each other where they flare up in the $\rho_1$-plane, and by the expansion (\ref{eq:ScaledRatio}), when we are sufficiently close to $z=0$ in the $z$-plane. We will explain this fact, along with the closed geodesics/singularities of this metric in more detail after we obtain the explicit form for $\rho_1(z)$.
Secondly, the $z$-plane metric is smooth (except for the singularities) not only over $S_1$ but across the branch cut $\widetilde{L}_1$ as well. The reason is simply that we demanded SL(2,$\mathbb{R}$) monodromy around each puncture and we know that the metric above is invariant under the monodromies of that kind by the equivalent form in (\ref{eq:firstmetric}). So we can use the metric above in the entirety of the $z$-plane minus punctures as long as we guarantee the SL(2,$\mathbb{R}$) monodromies around all punctures simultaneously.
Since we are also demanding hyperbolic SL(2,$\mathbb{R}$) monodromies for the remaining punctures, two facts above hold for them without too much modification. We just have to change $\rho_1$ with appropriate $\rho_j$. Moreover, these produce the same hyperbolic metric when we pullback them to the $z$-plane from any $\rho_j$-plane. This can be easily seen by noticing the fact that the appropriate SL(2,$\mathbb{R}$) change of basis of solutions $\psi_1^{\pm}(z)$ can diagonalize the monodromy around another puncture, by the fact that hyperbolic elements in SL(2,$\mathbb{R}$) are conjugate to a diagonal matrix. Such transformations of the solutions don't affect the metric as we argued before.
In conclusion, we see the motivation behind using the Fuchsian equation with correct monodromy structure in more detail from these comments. Even though any choice of holomorphic function works in (\ref{eq:conformalfactor}) to define a hyperbolic metric, using the scaled ratio coming from the Fuchsian equation with the monodromy data above will guarantee to generate the hyperbolic metric (\ref{eq:met}) on the $z$-plane where three series of attached hyperbolic cylinders connected to each other with hyperbolic pair of pants (i.e. three-holed sphere endowed with a hyperbolic metric), as shown in figure \ref{fig:CylinderSketch}. Moreover, it is easy to see from figure \ref{fig:CylinderSketch} that one can obtain a description of the hyperbolic pair of pants by taking out the hyperbolic cylinders and considering the remaining connected region only. This justifies why we considered this particular monodromy problem of Fuchsian equation on $\widetilde{X}$: It is a natural starting point to generate the hyperbolic metric with geodesic boundaries on $X$.
\begin{figure}[!t]
\centering
\fd{6cm}{fig1a.pdf}
\fd{9cm}{fig1b.pdf}
\caption{Sketch of the hyperbolic metric described by the Fuchsian equation with three hyperbolic singularities. The smooth hyperbolic pair of pants with geodesic boundaries is going to connect the hyperbolic cylinders, as we shall see more explicitly.}
\label{fig:CylinderSketch}
\end{figure}
\subsection{Solution to the monodromy problem}
Before we describe the hyperbolic pair of pants, we are going to get an explicit expression for the hyperbolic metric on the three-punctured sphere $\widetilde{X}$ resulting from three hyperbolic singularities. First, we solve the monodromy problem. That is, we find $T_{\varphi}(z)$ for which the solutions of the Fuchsian equation can realize the monodromy structure described in~\eqref{eq:SL2R}. Then we solve the resulting Fuchsian equation with these prescribed monodromies and proceed to construct the metric by finding the scaled ratio.
In order to find $T_{\varphi}(z)$ as a function of $z$, observe that when we are close to the puncture $z=0$, i.e. $\rho_1 = 0$, the stress-energy tensor in~\eqref{eq:Tsect3} takes the form
\begin{equation}
T_{\varphi}(z) = (\partial \rho_1)^2 \frac{\Delta_1}{\rho_1^2} + \{\rho_1,z \} = (e^{\frac{v_1}{\lambda_1}} + \dots)^2 \frac{\Delta_1}{(e^{\frac{v_1}{\lambda_1}} z + \dots)^2} + \{e^{\frac{v_1}{\lambda_1}} z + \dots,z \}
= \frac{\Delta_1}{z^2} + \mathcal{O}(\frac{1}{z}),
\end{equation}
using (\ref{eq:ScaledRatio}) and (\ref{eq:closetopuncture}). From this, we see that $T_{\varphi}(z)$ must have at most double poles of residues $\Delta_1,\Delta_2$ and $\Delta_3$ at $z=0,1,\infty,$ respectively in order to have a hyperbolic singularity. One can easily show that the unique $T_{\varphi}(z)$ that has such structure is
\begin{equation} \label{eq:3T}
T_{\varphi}(z) = \frac{\Delta_1}{z^2} + \frac{\Delta_2}{(z-1)^2} + \frac{\Delta_3 - \Delta_1 - \Delta_2}{z(z-1)},
\end{equation}
with $\Delta_i = (1+\lambda_i^2)/2$. Clearly we have at most double poles at $z=0,1$ with appropriate residues. Using the inversion map $z \to \tilde{z} =1/z$, along with $\{\tilde{z},z\}=0$, we can easily see that we have the correct structure at infinity, i.e. $\tilde{z} = 0$, as well:
\begin{equation}
\widetilde{T_{\varphi}}(\tilde{z}) = \frac{1}{\tilde{z}^4} \left[\Delta_1 \tilde{z}^2 + \Delta_2\tilde{z}^2 + (\Delta_3 - \Delta_1 - \Delta_2)\tilde{z}^2 + \mathcal{O}(\tilde{z}^3) \right]
= \frac{\Delta_3}{\tilde{z}^2} +\mathcal{O}(\frac{1}{\tilde{z}}).
\end{equation}
The stress-energy tensor $T_{\varphi}(z)$ in (\ref{eq:3T}) solves the monodromy problem. In order to see that, first observe the Fuchsian equation in this case takes the form
\begin{equation} \label{eq:Fuchplugged}
\partial^2 \psi(z) + \frac{1}{2} \left[\frac{\Delta_1}{z^2} + \frac{\Delta_2}{(z-1)^2} + \frac{\Delta_3 - \Delta_1 - \Delta_2}{z(z-1)}\right] \psi (z) = 0.
\end{equation}
This is the hypergeometric equation, written in the so-called $Q$-form. The solutions of this equation and their properties are well tabulated (see Schwarz's function in \cite{hypergeometric}, also \cite{hadasz2004classical,Bilal:1987cq}). They are, with proper normalization and assignment of diagonal monodromy around $z=0$,
\begin{subequations}
\begin{align}
\psi_1^{\pm}(z) &= \frac{e^{\pm \frac{i v(\lambda_1, \lambda_2,\lambda_3)}{2}}}{\sqrt{i \lambda_1}} z^{\frac{1 \pm i \lambda_1}{2}} (1-z)^{\frac{1 \mp i \lambda_2}{2}} \nonumber \\ &\quad \times {_2}F_1 \left(\frac{1 \pm i \lambda_1 \mp i \lambda_2 \pm i \lambda_3}{2},\frac{1 \pm i \lambda_1 \mp i \lambda_2 \mp i \lambda_3}{2}; 1 \pm i \lambda_1; z\right).
\end{align}
Here ${_2}F_1(a,b;c;z)$ is the ordinary hypergeometric function~\eqref{eq:hypergeometric}. Using the transformation properties of these solutions (\ref{eq:transofpsi}) and appropriately exchanging punctures, we can also find the normalized solutions having a diagonal monodromy around $z=1$ and $z=\infty$. They are, respectively,
\begin{align}
\psi_2^{\pm}(z) &= i\frac{ e^{\pm \frac{i v(\lambda_2, \lambda_1,\lambda_3)}{2}}}{\sqrt{i \lambda_2}} (1-z)^{\frac{1 \pm i \lambda_2}{2}} z^{\frac{1 \mp i \lambda_1}{2}} \nonumber \\ &\quad\times {_2}F_1 \left(\frac{1 \pm i \lambda_2 \mp i \lambda_1 \pm i \lambda_3}{2},\frac{1 \pm i \lambda_2 \mp i \lambda_1 \mp i \lambda_3}{2}; 1 \pm i \lambda_2; 1-z\right), \\
\psi_3^{\pm}(z) &= (iz)\frac{e^{\pm \frac{i v(\lambda_3, \lambda_2,\lambda_1)}{2}}}{\sqrt{i \lambda_3}} \left(\frac{1}{z}\right)^{\frac{1 \pm i \lambda_3}{2}} \left(1-\frac{1}{z}\right)^{\frac{1 \mp i \lambda_2}{2}} \nonumber\\ &\quad \times {_2}F_1 \left(\frac{1 \pm i \lambda_3 \mp i \lambda_2 \pm i \lambda_1}{2},\frac{1 \pm i \lambda_3 \mp i \lambda_2 \mp i \lambda_1}{2}; 1 \pm i \lambda_3; \frac{1}{z}\right).
\end{align}
\end{subequations}
We should emphasize again that the constant $v(\lambda_1,\lambda_2, \lambda_3) = v_1$ above is not fixed by the Wronskian and we will determine it below by demanding hyperbolic SL(2,$\mathbb{R}$) monodromies around all punctures. We will call this \emph{compatibility} of monodromies. Notice that compatibility is not guaranteed a priori. This is because when we demand a SL(2,$\mathbb{R}$) monodromy around a puncture, the monodromies around remaining punctures are elements of SL(2,$\mathbb{C}$), rather than SL(2,$\mathbb{R}$), in general.\footnote{It can still have unit determinant without loss of generality if one assumes appropriately normalized solutions in the sense of~\eqref{eq:Wronskian}.} So, actually, in order to solve the monodromy problem completely, we must show that the compatibility is achievable for the Fuchsian equation~\eqref{eq:Fuchplugged}.
In order to ensure compatibility, first observe that we have some SL(2,$\mathbb{C}$) monodromy around $z=1$ if we use the basis $\psi_1^{\pm}(z)$. That is, as $(1-z) \to e^{2 \pi i}(1-z)$, we have
\begin{equation}
\begin{bmatrix}
\psi_1^{+} \\ \psi_1^{-}
\end{bmatrix} \; \to \;
M_1^2
\begin{bmatrix}
\psi_1^{+} \\ \psi_1^{-}
\end{bmatrix} \quad \text{where} \quad M_1^2 \in SL(2,\mathbb{C}).
\end{equation}
Here, and throughout, we are going to denote the monodromy of the solutions $\psi_i^{\pm}(z)$ around the puncture $z=z_j$ as $M_i^j$. In order to have hyperbolic $\text{SL(2,}\mathbb{R})$ monodromy around $z=1$ while simultaneously having hyperbolic $\text{SL(2,}\mathbb{R})$ monodromy around $z=0$, we have to make sure that $M_1^2 \in \text{SL(2,}\mathbb{R})$ and $|\text{Tr} M_1^2|>2$ by adjusting $v_1$ appropriately. To that end, first observe that we have a diagonal hyperbolic $\text{SL(2,}\mathbb{R})$ monodromy around $z=1$ if we use the basis $\psi_2^{\pm}(z)$:
\begin{equation}
\begin{bmatrix}
\psi_2^{+} \\ \psi_2^{-}
\end{bmatrix} \; \to \;
M_2^2
\begin{bmatrix}
\psi_2^{+} \\ \psi_2^{-}
\end{bmatrix}
=
\begin{bmatrix}
-e^{-\pi \lambda_2} & 0 \\ 0 & -e^{\pi \lambda_2}
\end{bmatrix}
\begin{bmatrix}
\psi_2^{+} \\ \psi_2^{-}
\end{bmatrix}.
\end{equation}
Secondly, notice that two basis $\psi_1^{\pm}(z)$ and $\psi_2^{\pm}(z)$ are related via the connection formulas for the hypergeometric function~(see section 2.9 in~\cite{hypergeometric}, also~\cite{hadasz2004classical,Bilal:1987cq})
\begin{equation} \label{eq:connection}
\begin{bmatrix}
\psi_1^{+} \\ \psi_1^{-}
\end{bmatrix}
=
S
\begin{bmatrix}
\psi_2^{+} \\ \psi_2^{-}
\end{bmatrix}
=
\sqrt{\lambda_1 \lambda_2} \begin{bmatrix}
e^{i \frac{v_1-v_2}{2}} \; g_{-}& e^{i \frac{v_1+v_2}{2}} \; g_{+} \\ -e^{-i \frac{v_1+v_2}{2}} \; \overline{g_{+}}& -e^{-i \frac{v_1-v_2}{2}} \; \overline{g_{-}}
\end{bmatrix}
\begin{bmatrix}
\psi_2^{+} \\ \psi_2^{-}
\end{bmatrix},
\end{equation}
here $v_2 = v(\lambda_2, \lambda_1,\lambda_3)$ and the functions $g_{\pm}$ are given by
\begin{equation} \label{eq:gpm}
g_{\pm}=\frac{\Gamma\left(i \lambda_{1}\right) \Gamma\left(\pm i \lambda_{2}\right)}{\Gamma\left(\frac{1+i \lambda_{1} \pm i \lambda_{2}+i \lambda_{3}}{2}\right) \Gamma\left(\frac{1+i \lambda_{1} \pm i \lambda_{2}-i \lambda_{3}}{2}\right)}.
\end{equation}
Using them, we observe the monodromies in two basis are related by the following conjugation:
\begin{equation} \label{eq:conj}
\begin{aligned}
M_1^2 =& S M_2^2 S^{-1} =\lambda_{1} \lambda_{2} \left[\begin{array}{cc}
\mathrm{e}^{-\pi \lambda_{2}}\left|g_{-}\right|^{2}-\mathrm{e}^{\pi \lambda_{2}}\left|g_{+}\right|^{2} & -\left(\mathrm{e}^{\pi \lambda_{2}}-\mathrm{e}^{-\pi \lambda_{2}}\right) \mathrm{e}^{i v_{1}} g_{+} g_{-} \\[10pt]
\left(\mathrm{e}^{\pi \lambda_{2}}-\mathrm{e}^{-\pi \lambda_{2}}\right) \mathrm{e}^{-i v_{1}} \overline{g_{+} g_{-}} & \mathrm{e}^{\pi \lambda_{2}}\left|g_{-}\right|^{2}-\mathrm{e}^{-\pi \lambda_{2}}\left|g_{+}\right|^{2}
\end{array}\right].
\end{aligned}
\end{equation}
Here we used the fact
\begin{equation} \label{eq:identity}
|g_+|^2 - |g_-|^2 = \frac{1}{\lambda_1 \lambda_2},
\end{equation}
which can be derived from the expression~\eqref{eq:gpm}.
Now it is a simple calculation using~\eqref{eq:conj} and~\eqref{eq:identity} to check that $\det M_1^2 =1$. Therefore in order to have $M_1^2\in$ SL(2$, \mathbb{R}$) it is enough to make sure the entries of $M_1^2$ are real. That means we have
\begin{align} \label{eq:v}
\mathrm{e}^{2 i v\left(\lambda_{1}, \lambda_{2}, \lambda_{3}\right)} =\frac{\overline{g_{+} g_{-}}}{g_{+} g_{-}} = \frac{\Gamma\left(-i \lambda_{1}\right)^{2}}{\Gamma \left(i \lambda_{1}\right)^{2}} \frac{\gamma\left(\frac{1+i \lambda_{1}+i \lambda_{2}+i \lambda_{3}}{2}\right) \gamma\left(\frac{1+i \lambda_{1}-i \lambda_{2}+i \lambda_{3}}{2}\right)}{\gamma\left(\frac{1-i \lambda_{1}-i \lambda_{2}+i \lambda_{3}}{2}\right) \gamma\left(\frac{1-i \lambda_{1}+i \lambda_{2}+i \lambda_{3}}{2}\right)} ,
\end{align}
with the function $\gamma(x)$ defined as
\begin{equation}
\gamma(x) \equiv \frac{\Gamma(x)}{\Gamma(1-x)}.
\end{equation}
The equality~\eqref{eq:v} fixes the exponent $v(\lambda_1,\lambda_2, \lambda_{3})=v_1$, but in a rather complicated way, and shows that it is real. Moreover, we can also easily observe Tr$M_1^2 = -2 \cosh(\pi \lambda_2)$ using~\eqref{eq:conj} and~\eqref{eq:identity}, which unsurprisingly shows the monodromy is still hyperbolic. Thus, we conclude that we can have hyperbolic SL(2$, \mathbb{R}$) monodromy around $z=1$ while having a hyperbolic SL(2$, \mathbb{R}$) monodromy around $z=0$.
Note that that guaranteeing a hyperbolic SL(2$, \mathbb{R}$) monodromies around $z=0,1$ simultaneously with the correct choice of $v(\lambda_1,\lambda_2, \lambda_{3})$ would be sufficient for guaranteeing a hyperbolic SL(2$, \mathbb{R}$) monodromy around $z=\infty$ as well, which is the only remaining point where we have a nontrivial monodromy around. This is because we can imagine a contour that surrounds both $z=0$ and $z=1$ whose associated monodromy would be a product of two hyperbolic SL(2$, \mathbb{R}$) matrices, which is another SL(2$, \mathbb{R}$) matrix. Furthermore, this monodromy would be clearly hyperbolic by construction. As a result, the solutions would have the desired monodromy structure around $z=\infty$ as well when we think this contour to surround $z=\infty$ instead. So we conclude that the solutions of the Fuchsian equation (\ref{eq:Fuchplugged}), with the right choice of $v(\lambda_1,\lambda_2, \lambda_{3})$, can realize hyperbolic SL(2$, \mathbb{R}$) monodromies around each puncture and they are compatible. We solved the monodromy problem.
Finally, we can list the scaled ratios $\rho_i = (\psi_i^+(z)/\psi_i^-(z))^{1/i \lambda_i}$ associated with each puncture. They are:
\begin{align}
\rho_1(z) &= e^{\frac{v(\lambda_1,\lambda_2, \lambda_3)}{\lambda_1}} z (1-z)^{-\frac{\lambda_2}{\lambda_1}}
\left[ \frac{_2F_1 \left(\frac{1+i \lambda_1 -i \lambda_2 +i \lambda_3}{2},\frac{1+i \lambda_1 -i \lambda_2 -i \lambda_3}{2}; 1+ i \lambda_1; z\right)}{_2F_1 \left(\frac{1-i \lambda_1 +i \lambda_2 -i \lambda_3}{2},\frac{1-i \lambda_1 +i \lambda_2 +i \lambda_3}{2}; 1- i \lambda_1; z\right)} \right]^{\frac{1}{i \lambda_1}}, \nonumber\\
\rho_2(z) &= e^{\frac{v(\lambda_2,\lambda_1, \lambda_3)}{\lambda_2}} (1-z) z^{-\frac{\lambda_1}{\lambda_2}} ,
\left[ \frac{_2F_1 \left(\frac{1+i \lambda_2 -i \lambda_1 +i \lambda_3}{2},\frac{1+i \lambda_2 -i \lambda_1 -i \lambda_3}{2}; 1+ i \lambda_2; 1-z\right)}{_2F_1 \left(\frac{1-i \lambda_2 +i \lambda_1 -i \lambda_3}{2},\frac{1-i \lambda_2 +i \lambda_1 +i \lambda_3}{2}; 1- i \lambda_2; 1-z\right)} \right]^{\frac{1}{i \lambda_2}},\nonumber \\
\rho_3(z) &= e^{\frac{v(\lambda_3,\lambda_2, \lambda_1)}{\lambda_3}} \left(\frac{1}{z}\right) \left(1-\frac{1}{z}\right)^{-\frac{\lambda_2}{\lambda_3}}
\left[ \frac{_2F_1 \left(\frac{1+i \lambda_3 -i \lambda_2 +i \lambda_1}{2},\frac{1+i \lambda_3 -i \lambda_2 -i \lambda_1}{2}; 1+ i \lambda_3; \frac{1}{z}\right)}{_2F_1 \left(\frac{1-i \lambda_3 +i \lambda_2 -i \lambda_1}{2},\frac{1-i \lambda_3 +i \lambda_2 +i \lambda_1}{2}; 1- i \lambda_3; \frac{1}{z}\right)} \right]^{\frac{1}{i \lambda_3}}. \label{eq:scaledratio}
\end{align}
From above it is clear that $\rho_2(z)$ and $\rho_3(z)$ can be obtained from $\rho_1(z)$ by exchanging punctures, as well as their associated $\lambda_j$'s, $(1) \leftrightarrow (2)$ and $(1) \leftrightarrow (3)$ respectively while keeping the remaining puncture fixed. Moreover, one can also show that the scaled ratio associated with the fixed puncture remains invariant (up to a sign) under this exchange, either by reasoning through our construction above or by checking it directly using the identities for hypergeometric functions~\cite{hypergeometric}. In any case, we see that the set of three scaled ratios given above would be invariant (up to a sign) under the permutation group $S_3$ acting on the positions and the parameters of the punctures. This fact will eventually lead us to a similar symmetry for the local coordinates of the hyperbolic three-string vertex.
As we already argued in the previous subsection, these scaled ratios will define the following single-valued, singular, hyperbolic metric on the whole three-punctured sphere (\ref{eq:met}):
\begin{equation} \label{eq:yetanothermetric}
ds^2 = \frac{\lambda_{j}^2 |\partial \rho_j (z)|^2}{|\rho_j(z)|^2 \sin^2(\lambda_j \log|\rho_j(z)|)} |dz|^2
= \frac{\lambda_{j}^2 }{|\rho_j|^2 \sin^2(\lambda_j \log|\rho_j|)} |d\rho_j|^2,
\end{equation}
for which we have three semi-infinite series of attached hyperbolic cylinders connected to each other. Again, each $j=1,2,3$ defines the same metric.
\subsection{The resulting geometry on the three-punctured sphere}
Before we construct the local coordinates, we should understand the geometry of (\ref{eq:yetanothermetric}) better and show that it looks exactly like in figure \ref{fig:CylinderSketch} as we have claimed. In order to do that, focus on the set $S_1$, which was the complex plane with a cut from $1$ to $\infty$ (see~\eqref{eq:s1}). This will be mapped to the set $\rho_1(S_1)$ in the $\rho_1$-plane.\footnote{It can be shown that this map is invertible, see \cite{hypergeometric}. So this mapping would be bijective.} The rough sketch of these regions, based on numerics, but \emph{not} on scale, is given in figure \ref{fig:rhoSketch} and \ref{fig:zSketch}. We will consider and explain this geometry on the $\rho_1$-plane for now, but geometries on the other $\rho_j$-planes are analogous.
\begin{figure}[!pht]
\centering
\fd{14cm}{fig2.pdf}
\caption{The rough sketch of the geometry on the $\rho_1$-plane. The meaning of the curves are explained in the text. The coloring conventions for the curves will be the same for all figures in this subsection. Dashed curves indicate the line singularities. \vspace{1cm}}
\label{fig:rhoSketch}
\centering
\fd{14cm}{fig3.pdf}
\caption{The corresponding geometry on the $z$-plane after we pullback the metric~(\ref{eq:yetanothermetric}) from the $\rho_1$-plane above. Note that the gray region would be endowed with the hyperbolic metric with geodesic boundaries $\Gamma_i$.}
\label{fig:zSketch}
\end{figure}
As we mentioned previously, the metric on the $\rho_1$-plane (\ref{eq:yetanothermetric}) takes the form of the hyperbolic metric of series of attached hyperbolic cylinders. Indeed, we see that the line singularities (where the metric blow up on a curve) and the simple closed geodesics surrounding the origin $\rho_1=0$ of the hyperbolic metric~\eqref{eq:yetanothermetric} are located at
\begin{equation}
\text{Line singularities:} \; |\rho_1| = e^{\frac{\pi l_1}{\lambda_1}}, \qquad
\text{Simple Closed Geodesics:} \; |\rho_1| = e^{\frac{\pi}{\lambda_1} \left(l_1+\frac{1}{2}\right)},
\end{equation}
where $ l_1 \in \mathbb{Z}$. We can see these by noting that the sine in the denominator of the metric~\eqref{eq:yetanothermetric} is equal to zero in the case of line singularity by $\sin(\pi l_1) = 0$ and one in the case of simple closed geodesics by $\sin\left(\pi l_1+ \pi/2 \right ) = 1$, which makes the metric~\eqref{eq:yetanothermetric} blow up and minimize respectively.
Notice that the line singularities and simple closed geodesics form alternating, exponentially separated circles around the origin on the $\rho_1$-plane, as shown in figure \ref{fig:rhoSketch} with green and purple respectively; except for the geodesic colored with magenta which will turn out to be special. Additionally, it is clear that every simple geodesic surrounding the origin has the length $2 \pi \lambda_1$ by the metric (\ref{eq:yetanothermetric}), which justifies the name \emph{geodesic radii} for $\lambda_j$. Obviously we can pullback these curves to the $z$-plane with a cut from 0 to $\infty$, which will result in closed, simple geodesics/line singularities around the puncture $z=0$ by $\rho_1(0)=0$. These are shown in figure~\ref{fig:zSketch} correspondingly.
Observe that the lines just above/below the branch cut of $\rho_1(z)$, denoted as $\widetilde{L}_1^{\pm}$ and shown in figure \ref{fig:zSketch}, are mapped to the red/blue curves $\rho_1 (\widetilde{L}_1^{\pm})$ in $\rho_1(S_1)$. These curves are shown in figure \ref{fig:rhoSketch}. They are symmetric with respect to the real axis on the $\rho_1$-plane by the choice of the principal branch for the scaled ratio. The set $S_1$ is mapped between $\rho_1(\widetilde{L}_1^+)$ and $\rho_1(\widetilde{L}_1^-)$, which is the shaded region in figure \ref{fig:rhoSketch}. Moreover, if we identify the two curves $\rho_1 (\widetilde{L}_1^{\pm})$, the whole $z$-plane minus the punctures maps to the region between them. But, in any case, we indicated where the punctures $z=1$ and $z= \infty$ are heuristically getting mapped to in figure \ref{fig:rhoSketch}: $z=1$ is mapped to the right-side infinity and $z= \infty$ is mapped to the left-side infinity.
Now observing figure \ref{fig:rhoSketch}, we see that some simple closed geodesics/line singularities don't intersect $\rho_1(\widetilde{L}^{\pm}_1)$. As a result, we see $\exists \, \tilde{l}_1 \in \mathbb{Z}$ such that the geodesic at $|\rho_1| = e^{\frac{\pi}{\lambda_j}\left( \tilde{l}_1+\frac{1}{2}\right)}$ does not intersect $\rho_1(\widetilde{L}^{\pm}_1)$ and surrounds \emph{all} the closed simple geodesics/line singularities that do not intersect $\rho_1(\widetilde{L}^{\pm}_1)$ (i.e. those with $l_1 \leq \tilde{l}_1$). The closed geodesic with $l_1 = \tilde{l}_1$ is shown with magenta instead of purple in figure~\ref{fig:rhoSketch} in order to differentiate it from the others. At this stage nothing prevents us to having a line singularity that surrounds this geodesic and doesn't intersect $\rho_1(\widetilde{L}^{\pm}_1)$, but this turns out not to be the case as we will prove it shortly. We just assume this is the case for now.
We can pullback the geodesic with $l_1=\tilde{l}_1$ described above to the $z$-plane, which we denote it by $\Gamma_1$. Defining the closed geodesics homotopic to the puncture $z=0$ as \emph{separating} geodesics of $z=0$, we see the simple closed geodesic $\Gamma_1$ would be the separating geodesic farthest away from $z=0$ by construction. So we will call $\Gamma_1$ as \emph{the most-distant separating} geodesic of $z=0$. This geodesic is shown in figure \ref{fig:zSketch} with magenta as well.
From this, we see that the simply-connected region $H_1$ on the $z$-plane surrounded by $\Gamma_1$ contains every geodesic/line singularity with $l_1 \leq \tilde{l}_1$. Furthermore, as $l_1 \to - \infty$, the geodesics/line singularities get closer to the puncture. So we conclude that the geometry on $H_1$ looks like a series of semi-infinite hyperbolic cylinders attached at where they flare up, like shown in figure~\ref{fig:CylinderSketch}. The places where they flare up are the line singularities of the metric.
We can repeat the same procedure for the other punctures and obtain their most-distant separating geodesics $\Gamma_j$, associated simply-connected regions $H_j$, and integers $\tilde{l}_j$. Note that $\Gamma_i \cap \Gamma_j = \emptyset$ for $i \neq j$, by $\Gamma_i$'s being simple geodesics of the same metric. Hence, the resulting geometry on the $z$-plane would indeed look like in figure \ref{fig:zSketch}. Again, the most-distant separating geodesics $\Gamma_j$ are shown with different colors. In this figure, we also see there are alternating closed curves around each puncture representing the simple closed geodesics/line singularities surrounding them. These can be related to the geodesics/line singularities that intersect $\rho_1 (\widetilde{L}_1^{\pm})$ on the $\rho_1$-plane (hence their colors), but this wouldn't be necessary for our purposes.
Now let us inspect how the most-distant separating geodesics of the punctures $z=1$ and $z=\infty$, denoted as $\Gamma_2$ and $\Gamma_3$ respectively, look like on $\rho_1(S_1)$. In order to do that, let us call the line singularity with $l_1 = \tilde{l}_1 + 1$ to be the \emph{first line singularity} of $z=0$. Clearly, the first line singularity encloses $\rho_1(\Gamma_1)$ and is enclosed by every other line singularity that encloses $\rho_1(\Gamma_1)$ on the $\rho_1$-plane, hence the name \emph{first}. Moreover, it is clear that the first line singularity intersects with the curves $\rho_1(\widetilde{L}_1^{\pm})$ by definition. We define the \emph{first geodesic} of a puncture in similar fashion.
Now, we will find the shortest geodesic that is enclosed by the first line singularity of $z=0$ and stretches between the curves $\rho_1(\widetilde{L}_1^{\pm})$ for both right/left of the origin $\rho_1 =0$ on $\rho_1(S_1)$, which we will call $\rho_1(\Omega_2)$ and $\rho_1(\Omega_3)$ respectively. Clearly, $\rho_1(\Omega_2)$ and $\rho_1(\Omega_3)$ can be made shorter by eliminating any self intersections, so we will consider the simple geodesics without loss of generality. Moreover, $\rho_1(\Omega_2)$ and $\rho_1(\Omega_3)$ can be made shorter by making them intersect $\rho_1(\widetilde{L}_1^{\pm})$ perpendicularly, which we will also take to be the case.
There might be multiple curves satisfying the definition for $\rho_1(\Omega_2)$ and $\rho_1(\Omega_3)$ above. However, this cannot be the case since their pullbacks on the $z$-plane would correspond to closed simple geodesics without a line singularity between them around the punctures $z=0$ and $z=\infty$, and we know that this can't happen as we saw above. So $\rho_1(\Omega_2)$ and $\rho_1(\Omega_3)$ are unique for the left and right side. This is shown in figure~\ref{fig:rhoSketch}. Additionally, this argument shows that $\Omega_2$ and $\Omega_3$ are the the most-distant separating geodesics for the punctures $z=1$ and $z=\infty$ respectively, i.e. $\Gamma_2 = \Omega_2$ and $\Gamma_3=\Omega_3$, since there are no geodesics that surround them and separate from the other punctures.
Keeping this in mind, we can now demonstrate that the there is no line singularity that surrounds the geodesic $\rho_1(\Gamma_1)$ and doesn't intersect $\rho_1(\widetilde{L}^{\pm}_1)$ on the $\rho_1$-plane, which we only assumed previously. For the sake of contradiction, suppose there is one and call it $\rho_1(\Lambda_1)$, which is shown in figure~\ref{fig:argument}. Then it is clear by above that the geodesic $\rho_1(\Gamma_2)$ around the puncture $z=1$ would be a piece of the first geodesic of $z=0$. Now going to the $\rho_2$-plane after we pullback this geometry to the $z$-plane, we see that $\Lambda_1$ maps to a piece of a line singularity $\rho_2(\Lambda_1)$ on the $\rho_2$-plane stretching between $\rho_2(\widetilde{L}_2^{\pm})$, while $\Gamma_2$ maps to a circle around the origin and doesn't intersect $\rho_2(\widetilde{L}_2^{\pm})$. Similar to the arguments above, we can always find a simple geodesic $\rho_2(\Omega_1)$ between these two, but this leads to a contradiction with the fact that $\Gamma_1$ being the most-distant separating geodesic of $z=0$ since the separating geodesic $\Omega_1$ would be enclosing $\Gamma_1$. Clearly this argument can be repeated for other punctures, so what we have assumed regarding having a line singularity that surrounds the most-distant separating geodesic and doesn't intersect $\rho_1(\widetilde{L}^{\pm}_1)$ was justified.
\begin{figure}[!t]
\centering
\fd{7.5cm}{fig4a.pdf}
\fd{7.5cm}{fig4b.pdf}
\caption{The illustration of the geometry described in the argument above on the $\rho_1$-plane (left) and $\rho_2$-plane (right). Here the line singularity $\Lambda_1$ is shown with dark green, while the geodesic $\Omega_1$ is shown with brown.}
\label{fig:argument}
\centering
\end{figure}
In order to complete our construction, we now need to find the integers $\tilde{l}_j$. For that, first notice the following inequality is satisfied:
\begin{equation} \label{eq:ineq}
|\rho_j(z)| = \exp \left[\frac{\pi}{\lambda_j}\left( \tilde{l}_j+\frac{1}{2}\right)\right] \leq \min_{z \in \widetilde{L}_j} |\rho_j(z)|,
\end{equation}
with $\widetilde{L}_j$ denoting the branch cut of the function $\rho_j(z)$. This inequality is evident since we demanded above that the geodesic $\rho_j(\Gamma_j)$, located at $|\rho_j| = e^{\frac{\pi}{\lambda_j}\left( \tilde{l}_j+\frac{1}{2}\right)}$, is not intersecting the curves $\rho_j(\widetilde{L}^{\pm}_1)$ on the $\rho_j$-plane. Note that $|\rho_j(z)|$ would be single-valued on the branch cut $\widetilde{L}_j$ because of the choice of the principal branch. From~\eqref{eq:ineq} and noting that $\tilde{l}_j$ is the greatest integer that satisfies it by definition, we can write a prescription for $\tilde{l}_j$ as follows:
\begin{align} \label{eq:ltilde}
\tilde{l}_j = \left\lfloor \frac{\lambda_j}{\pi} \log \min_{z \in \widetilde{L}_j} |\rho_j(z)| - \frac{1}{2}\right\rfloor.
\end{align}
Here $\lfloor \cdot \rfloor: \mathbb{R} \to \mathbb{Z}$ denotes the floor function. We couldn't be able to find an explicit expression for this in terms of $\lambda_j$'s. However, determining the \emph{exact} values of the integers $\tilde{l}_j$ numerically for given $\lambda_j$ is trivial by the expression above and using the scaled ratios~\eqref{eq:scaledratio}.
Although it is hard to find an expression for $\tilde{l}_j$ in terms of arbitrary $\lambda_j$'s, we can still make some progress for the case where two of the $\lambda_j$'s are equal by exploiting the permutation symmetry. In order to do that, suppose we want to find $\tilde{l}_1$ in the case of $\lambda_2=\lambda_3 = \lambda$. Now recall that three scaled ratios~\eqref{eq:scaledratio} are invariant under the permutations of the punctures and their associated geodesic radii up to a sign. Specifically, in the case where we exchange the punctures at $z=1,\infty$ while keeping $z=0$ fixed, which is implemented by the conformal transformation $z \to \frac{z}{z-1}$, we get the following relation for $\rho_1(z)$ on $S_1$
\begin{equation}
\rho_1(z) = -\rho_1 \left(\frac{z}{z-1}\right).
\end{equation}
Note that it was essential to take $\lambda_2 = \lambda_3$ to establish this relation.
Clearly, $z=2$ is the fixed point of the transformation $z \to \frac{z}{z-1}$. One consequence of this is that $|\rho_1(z)|$ when restricted to the branch cut $\widetilde{L}_1$ is symmetric around $z=2$ when we apply the transformation $z \to \frac{z}{z-1}$. Then using this fact and analyticity of $\rho_1(z)$, it can be shown that the point $z=2$ would be where $|\rho_1(z)|$ attains its global minimum on the branch cut.
\begin{figure}[!t]
\centering
\fd{12cm}{fig5-eps-converted-to.pdf}
\caption{The plot of $\mathcal{L}(\lambda,\lambda,\lambda)$ as a function of $\lambda$. Note that the floor of this function gives $-1$ in the range shown. An analytic proof for $\tilde{l}_1 = -1$ for any $\lambda>0$ would require better understanding of $\mathcal{L}(\lambda,\lambda, \lambda) $, especially for the large values of $\lambda$.}
\label{fig:Length}
\end{figure}
So we see that permutation symmetry of the situation $\lambda_2=\lambda_3 = \lambda$ allow us to find the global minimum of $|\rho_1(z)|$ on its branch cut $\widetilde{L}_1$, which is at $z=2$. Now define the following function and notice
\begin{equation}
\mathcal{L}(\lambda_1,\lambda, \lambda) \equiv \frac{\lambda_1}{\pi} \log |\rho_1(2)| - \frac{1}{2} \implies \tilde{l}_1 = \lfloor \mathcal{L}(\lambda_1,\lambda, \lambda) \rfloor,
\end{equation}
using (\ref{eq:ltilde}). This expression is certainly more manageable then what has been given in~(\ref{eq:ltilde}). Obviously, we can get similar expressions for the other punctures when the remaining punctures has equal $\lambda_j$'s. As an example for what we have discussed so far, we plotted $\mathcal{L}(\lambda,\lambda, \lambda)$ in figure~\ref{fig:Length}. This suggests $\tilde{l}_1=-1$, and by symmetry $\tilde{l}_2 = \tilde{l}_3=-1$, for $0 < \lambda < 10$.
In summary, we see the geometry of the metric~(\ref{eq:yetanothermetric}) is indeed given by figure \ref{fig:CylinderSketch}. Remember the metric (\ref{eq:yetanothermetric}) was on the three-punctured sphere $\widetilde{X}$, but, clearly, we can now obtain the hyperbolic metric with geodesic boundaries on a three-holed sphere by restricting to the region $X=\widehat{\mathbb{C}} \setminus (H_1 \cup H_2 \cup H_3)$, which is shaded gray in figure \ref{fig:zSketch}. On the $\rho_j$-plane this corresponds to the region between the most-distant geodesics with the curves $\rho_j(L^{\pm}_j)$ are identified, which is also shaded gray in figure \ref{fig:rhoSketch}. Note that $X$ is still endowed with $K=-1$ metric (\ref{eq:yetanothermetric}), but now free from singularities, and it is clear by the construction that its boundaries $\Gamma_j$ are geodesics. In other words, we performed a \emph{surgery} where we amputated the hyperbolic cylinders around the hyperbolic singularities and left with the geodesic boundaries instead while keeping everything the same. Four examples of such region on the $z$-plane are shown in figure~\ref{fig:example}. In the next subsection, we are going to graft flat semi-infinite cylinders into the places of amputated hyperbolic cylinders in order to construct the local coordinates explicitly.
\begin{figure}[!t]
\centering
\fd{7.5cm}{lambda028-eps-converted-to.pdf}
\fd{7.5cm}{lambda075-eps-converted-to.pdf}
\fd{7.5cm}{lambda15-eps-converted-to.pdf}
\fd{7.5cm}{lambda2-eps-converted-to.pdf}
\caption{Four examples for the pants diagram region $X$ on the $z$-plane in the case of equal geodesic radii. Punctures are located at $z=0,1,\infty$, and indicated by black dots. We only show the most-distant separating geodesics $\Gamma_j$ because we performed a surgery and take out everything surrounded by them. The region remaining $X$ is endowed with the hyperbolic metric (\ref{eq:yetanothermetric}) and $\Gamma_j$ are its geodesics by construction. Note that the geodesics $\Gamma_1$ and $\Gamma_2$ in the critical case $\lambda =$ arcsinh(1)$/\pi \approx 0.28$ are so small that they haven't rendered in the top-left figure.}
\label{fig:example}
\end{figure}
\subsection{Local coordinates} \label{sec:Local}
In this subsection, we describe how to construct the local coordinates around the punctures for the hyperbolic three-string vertex by attaching flat semi-infinite cylinders of radius $\lambda_j$ at each geodesic boundary component of $X$. First, note that when we perform the surgery described above to obtain the geodesic boundaries, we essentially take out the disk
\begin{equation}
\rho_j(H_j) = \left\{\rho_j \in \mathbb{C} \; | \; |\rho_j| < \exp \left[ \frac{\pi}{\lambda_j} \left(\tilde{l}_j+\frac{1}{2} \right) \right]\right\} ,
\end{equation}
from the $\rho_j$-plane for each $j=1,2,3$. Now imagine we have a punctured unit disk $0 < |w_j| \leq 1$ with the metric
\begin{equation} \label{eq:cylmetric}
ds^2 = \lambda_j^2 \frac{ |dw_j|^2}{|w_j|^2},
\end{equation}
which describes a flat semi-infinite cylinder ($K=0$) of radius $\lambda_j$. We can map this punctured unit disk into the hole $\rho_j(H_j)$ on the $\rho_j$-plane with a simple scaling:
\begin{equation} \label{eq:scale}
\rho_j = \exp \left[ \frac{\pi}{\lambda_j} \left(\tilde{l}_j+\frac{1}{2} \right) \right] w_j = N_j w_j, \quad N_j \equiv \exp \left[ \frac{\pi}{\lambda_j} \left(\tilde{l}_j+\frac{1}{2} \right) \right].
\end{equation}
We will call $N_j$ the \emph{scale factor}. Above we haven't considered the overall rotations of the punctured unit disk, $w_j \to e^{i \theta} w_j$, while we are mapping to $\rho_j(H_j)$, since such global phase factors are not relevant in closed string field theory.
Clearly, the flat metric (\ref{eq:cylmetric}) does not change under this scaling. Furthermore, the flat metric~(\ref{eq:cylmetric}) and the hyperbolic metric (\ref{eq:yetanothermetric}) for the pair of pants as well as their first derivatives match at the circular seams $\rho_j(\Gamma_j)$ of radius $\lambda_j$. As a result, we fill the regions $\rho_j(H_j)$ with flat semi-infinite cylinders and discontinuity first appears in the curvature as we desire. Note that the metric we obtain after grafting these semi-infinite flat cylinders is a Thurston metric on the three-punctured sphere~\cite{Costello:2019fuh}.
Now we can pullback these filled $\rho_j(H_j)$ to the otherwise empty holes $H_j$ on the $z$-plane with the maps $\rho_j(z)$ to construct the local coordinates around the punctures $z=0,1,\infty$ describing three semi-infinite flat cylinders grafted on to the hyperbolic pair of pants on $\widehat{\mathbb{C}}$. Thus, from~\eqref{eq:scale}, we see that the local coordinates $w_j$ around the punctures $w_j=0$ are given by
\begin{equation} \label{eq:LocalCoord}
w_j = \exp \left[- \frac{\pi}{\lambda_j} \left(\tilde{l}_j+\frac{1}{2} \right) \right] \rho_j(z) = N_j^{-1} \rho_j(z),
\end{equation}
with $|w_j| \leq 1$. This yields the local coordinates~\eqref{eq:lc} using~\eqref{eq:scaledratio}. Equivalently, we can write $z = \rho_j^{-1}\left( N_j w_j\right)$ on the coordinate patches $H_j$ with the punctures are located at $z=z_j$. Note that $|w_j|=1$ maps to $\partial H_j = \Gamma_j$ by construction. Obviously, we can get the anti-holomorphic local coordinates $w_j(\bar{z})$ in similar fashion. Moreover, we see that they satisfy $w_j(\bar{z}) = \overline{w_j(z)}$ from~\eqref{eq:lc}, up to possible overall phase ambiguity. This shows all the coefficients in the expansions of $w_j(z)$ in $z$ can be chosen to be real.
As can be seen from~\eqref{eq:lc}, and alluded before, the local coordinates are invariant under permutations of the punctures and their associated $\lambda_j$. Adding the scale factor $N_j$ doesn't spoil this symmetry, since its value is getting permuted as well. Moreover, when we take all $\lambda_j = \lambda$ equal (recall this is the version that appears in the string action), this vertex becomes cyclic in the technical sense~\cite{sonoda1990covariant}. These results are certainly consistent with what is expected form the geometry of the hyperbolic pair of pants with three grafted flat cylinders.
As a final note, the mapping radius $r_j = \left| \frac{dz}{dw_j} \right|_{w_j=0}$ for this local coordinates can be easily read from~\eqref{eq:lc}, and they are
\begin{equation}
w_j = e^{-\frac{\pi (\tilde{l}_j+\frac{1}{2})}{\lambda_j}} e^{\frac{v_j}{\lambda_j}} (z-z_j) + \dots \implies r_j = \exp\left[\frac{\pi (\tilde{l}_j+\frac{1}{2})}{\lambda_j} - \frac{v_j}{\lambda_j}\right] = N_j \exp\left[- \frac{v_j}{\lambda_j}\right].
\end{equation}
Remember both $v_j$ and $N_j$ depends on the circumferences of the grafted cylinders as can be seen from~\eqref{eq:v} and~\eqref{eq:scale}.
\section{Limits of the hyperbolic three-string vertex} \label{sec:Limits}
In this section, we investigate various limits of the local coordinates (\ref{eq:LocalCoord}) to check that they are consistent with the literature~\cite{Moosavian:2017qsp,sonoda1990covariant, Zwiebach:1988qp}. We show that it is possible to produce the minimal area three-string vertex, Kleinian vertex, and the light-cone vertex as different limits of the hyperbolic three-string vertex.
\subsection{Minimal area three-string vertex} \label{sec:Witten}
In order to produce the minimal area three-string vertex from the hyperbolic three-sting vertex, we set the lengths of the boundary components of $X$ the same, $\lambda_1=\lambda_{2}=\lambda_{3} =\lambda$, and take $\lambda \to \infty$. Since the lengths of the boundaries of the pair of pants get larger at the same rate while the area of the pair of pants remains constant by the Gauss-Bonnet Theorem in this limit, the pair of pants shrinks and it becomes like a ribbon graph of vanishing width. As a result, after grafting the flat cylinders and rescaling their circumferences, we get the three-vertex obtained from the minimal area metric~\cite{Costello:2019fuh}. Therefore, we see that this is indeed the correct limit to generate the minimal area three-string vertex and we will call it \emph{minimal area limit}. Note that this limiting behavior is also evident from the examples given in figure \ref{fig:example}. We seem to get the usual representation of the minimal area three-string vertex as $\lambda$ gets larger~\cite{Erler:2019loq}.
In order to consider the minimal area limit explicitly, first notice that we have
\begin{equation}
\lim_{\lambda \to \infty} \exp \left[\frac{v(\lambda,\lambda,\lambda)}{\lambda} \right] = \frac{3\sqrt{3}}{4}.
\end{equation}
This can be obtained from the expression~(\ref{eq:v}) for the function $v(\lambda,\lambda,\lambda)$ and evaluating its limit in Mathematica.
Next, we need to find the limiting value of $N=N_1=N_2=N_3$ in the minimal area limit. Already from figure~\ref{fig:Length} and~\eqref{eq:scale} it can be visually argued that $N \to 1$ as $\lambda \to \infty$, but here we are going to provide an additional heuristic argument why this expectation is correct in case figure~\ref{fig:Length} is misleading in large $\lambda$. To that end, we should first understand the minimal area limit of the hyperbolic metric (\ref{eq:yetanothermetric}). This metric certainly diverges in the minimal area limit, but since we are going to rescale our cylinders at the end, this overall divergence is not a problem. Ignoring this divergence, indicated by $\sim$, the hyperbolic metric, now formally on the ribbon graph of vanishing width, takes the following form in the minimal area limit:
\begin{equation}
ds^2 \sim \frac{ |\partial \rho_i (z)|^2}{|\rho_i(z)|^2 \sin^2(\infty \log|\rho_i(z)|)} |dz|^2.
\end{equation}
Above $\infty$ in the denominator indicates infinite oscillations of the metric as $\lambda \to \infty$ except for when $|\rho_i(z)|=1$. But note that if we have such infinite oscillations, the metric would certainly be ill-defined. The only time it is well-defined is when we have $|\rho_i(z)|=1$, which produces just a divergence and that is acceptable as we mentioned. Thus, we conclude that the shape of the ribbon graph of vanishing width is described by $|\rho_i(z)|=1$ in the minimal area limit, because this is the only time we have a meaningful limit of the geometry. Now note that this ribbon graph at $|\rho_i(z)|=1$ can be thought as the union of $\Gamma_i$, which is described by $|\rho_i(z)|=N$, in the minimal area limit by shrinking hyperbolic pair of pants. This gives
\begin{equation}
\lim_{\lambda \to \infty} N = \lim_{\lambda \to \infty} \exp\left[\frac{\pi}{\lambda} \left(\tilde{l}+\frac{1}{2}\right) \right]= 1.
\end{equation}
So our expectation above was indeed correct.
Using the two limits we argued above, we see that the local coordinate around $z=0$~\eqref{eq:lc1} has the following expansion in the minimal area limit:\footnote{We also checked the similar results hold for other punctures. We omit reporting them to avoid repetition.}
\begin{align} \label{eq:WittenExpansion}
w_1 = \frac{3 \sqrt{3}}{4}z &+ \frac{3 \sqrt{3}}{8}z^2 + \frac{27 \sqrt{3} }{64}z^3 + \frac{57 \sqrt{3} }{128}z^4 + \frac{231 \sqrt{3} }{512}z^5 + \frac{459 \sqrt{3} }{1024} z^6+ \frac{7275 \sqrt{3} }{16384}z^7 \nonumber \\
&+ \frac{14493 \sqrt{3} }{32768} z^8 + \frac{58077 \sqrt{3} }{131072}z^9 + \frac{116565 \sqrt{3} }{262144}z^{10} + \mathcal{O}(z^{11}).
\end{align}
We obtained this expression by expanding (\ref{eq:lc1}) in $z$ first, then taking the minimal area limit. One can easily observe that the local coordinates around $z=0$ of the minimal area three-string vertex, as given in equation (2.19) of~\cite{sonoda1990covariant} with $a=1$,
\begin{equation} \label{eq:minimalvertex}
z_1 = i\frac{\left(1-\frac{i \sqrt{3} z}{z-2}\right)^{3/2}-\left(1+\frac{i \sqrt{3} z}{z-2}\right)^{3/2}}{\left(1-\frac{i \sqrt{3}
z}{z-2}\right)^{3/2}+\left(1+\frac{i \sqrt{3} z}{z-2}\right)^{3/2}},
\end{equation}
also has the same expansion (\ref{eq:WittenExpansion}) after an unimportant phase rotation $z_1 \to -z_1$. So, unsurprisingly, these local coordinates match in the minimal area limit.
Comparison was perturbative in $z$ above, however, we think this limiting behavior holds for all orders in $z$. That is $w_1=-z_1$ exactly in the minimal area limit. The best way to show this would be by finding an appropriate asymptotic formula for the hypergeometric function when $\lambda$ is large to generate the expression (\ref{eq:minimalvertex}), similar to the cases given in~\cite{Watson}. In any case, this perturbative analysis would be sufficient for our purposes. In conclusion, we see that the hyperbolic three-string vertex reduces to the minimal are three-string vertex in the limit $\lambda \to \infty$.
\subsection{Kleinian vertex}
Now we consider the opposite limit for which $\lambda_j=\lambda \to 0$. Clearly, the grafted flat cylinders disappears in this limit\footnote{Since this is the case, this naive limit of the local coordinates~\eqref{eq:LocalCoord} seems actually ill-defined. We are going to comment on this point below.} and instead we are left with a purely hyperbolic metric on the three-punctured sphere. So the local coordinates for the hyperbolic three-string vertex is expected to be related to the Kleinian vertex of~\cite{sonoda1990covariant} in this limit, whose local coordinates $z_i$ are given by
\begin{equation} \label{eq:KleinianVertex}
z_1 = e^{i \pi \tau(z)}, \qquad z_2 = e^{-i \pi/\tau(z)}, \qquad z_3 = e^{-i \pi/(\tau(z)\pm 1)},
\end{equation}
around the punctures $z=0,1,\infty$ respectively, since it involves the same hyperbolic geometry in its construction which emphasized more recently in~\cite{Moosavian:2017qsp,Moosavian:2017sev}. Here the function $\tau(z)$ is the inverse of the modular $\lambda$-function, which is equal to~\cite{hypergeometric}
\begin{equation} \label{eq:tau}
\tau(z) = i \frac{{_2}F_1(\frac{1}{2},\frac{1}{2},1,1-z)}{{_2}F_1(\frac{1}{2},\frac{1}{2},1,z)} = -\frac{i}{\pi} \log\left( \frac{z}{16} \right) + \mathcal{O}(z).
\end{equation}
We will denote the limit $\lambda_j=\lambda \to 0$ as the \emph{Kleinian limit}.
In order to argue for this limit, first notice that the function $\tau(z)$ satisfies the following equality~\cite{hempel1988uniformization}
\begin{equation}
\{\tau,z\} = \frac{1}{2z^2} + \frac{1}{2(z-1)^2} - \frac{1}{2z(z-1)},
\end{equation}
But recall from~\eqref{eq:Tsect3} and~\eqref{eq:3T} we also have
\begin{equation}
\lim_{\lambda \to 0} \{\rho_j^{i \lambda},z\} = \lim_{\lambda \to 0} T_{\varphi}(z) = \frac{1}{2z^2} + \frac{1}{2(z-1)^2} - \frac{1}{2z(z-1)} ,
\end{equation}
So from these two we immediately conclude
\begin{equation} \label{eq:logroh}
\{\tau,z\} = \lim_{\lambda \to 0} \{\rho_j^{i \lambda},z\} = \{\lim_{\lambda \to 0} \log(\rho_j) ,z\} \implies \lim_{\lambda \to 0} \log(\rho_j(z)) = \frac{a \tau(z) + b}{c \tau(z) + d}.
\end{equation}
Above we moved the limit inside the Schwarzian derivative and used the fact that two equal Schwarzian derivatives must be related to each other by a PGL(2,$\mathbb{C}$) transformation. So here $a,b,c,d \in \mathbb{C}$ and $ad-bc \neq 0$. Note that we can easily determine these constants by expanding both sides of~\eqref{eq:logroh} to leading order in $z$.
Take $z=0$ for instance. We already know the expansion of $\log(\rho_1(z))$ around $z=0$ from~(\ref{eq:ScaledRatio}). In the Kleinian limit this would then yield
\begin{equation}
\lim_{\lambda \to 0} \log(\rho_1(z)) = \log\left( \frac{z}{16} \right) + \mathcal{O}(z),
\end{equation}
using properties of Gamma functions for the limit of the function $v(\lambda,\lambda,\lambda)$. Comparing this to~\eqref{eq:tau}, we see the constants above get fixed and we obtain the following in the Kleinian limit:
\begin{equation}
a = i \pi, \quad b=c=0, \quad d=1 \implies \lim_{\lambda \to 0} \rho_1(z) = e^{i \pi \tau(z)} =z_1.
\end{equation}
Note that we can repeat the same procedure for other punctures and similarly obtain $\lim_{\lambda \to 0} \rho_j = z_j$ up to an unimportant phase factor. We explicitly checked this is indeed the case.
Now observe the scale factor $N=e^{-\frac{\pi}{2\lambda}}$ that relates the actual local coordinates $w_j$ to $\rho_j$ by $N w_j =\rho_j$ approaches to zero as $\lambda \to 0$, which is essentially a consequence of shrinking grafted cylinders. So in order to get a well-defined limit, it is necessary to place a \emph{cut-off} on the scale factor $N$ which we can do it as follows. We know $\lambda \to 0$ would make the length of the boundary geodesics $L$ smaller. So, as we take this limit, we choose some value of $L = \epsilon \ll 1$ that we put in $N$ and keep using it for any $L < \epsilon $. In other words, we take $N\approx e^{-\frac{\pi^2}{\epsilon }}$ for sufficiently small $L = 2 \pi \lambda \geq 0$ instead of what is given before. Note that this procedure essentially mirrors what is done in~\cite{Moosavian:2017qsp,Moosavian:2017sev}.
With this cut-off in place, we now have $N w_j = \rho_j = z_j$ in the Kleinian limit. Like in~\cite{Moosavian:2017qsp,Moosavian:2017sev}, we will multiply the original local coordinates $z_j$ for the Kleinian vertex given in (\ref{eq:KleinianVertex}) by $N^{-1}$ and define a new set of local coordinates $z_j' \equiv N^{-1} z_j$ in order to use the standard plumbing parameters. With this, we get $\lim_{\lambda \to 0} w_j = z_j'$ and see the scaled local coordinates for the Kleinian vertex matches what we find from the Kleinian limit of the hyperbolic three-string vertex as anticipated.
\subsection{Light-cone vertex}
Lastly, consider the situation
\begin{equation}
\lambda_1 = r \lambda, \qquad \lambda_2 = \lambda, \qquad \lambda_3 = (1-r) \lambda,
\end{equation}
for $0<r<1$ and take $\lambda \to \infty$. Having $\lambda_{2}=\lambda_{1}+\lambda_{3}$ while all of them being large, this limit should produce the local coordinates for the light-cone vertex~\cite{Zwiebach:1988qp}, by using similar geometric reasoning given in subsection \ref{sec:Witten}. Thus, we are going to call this limit the \emph{light-cone limit}.
In order to understand this limit better, first note that the restriction $\lambda_{2}=\lambda_{1}+\lambda_{3}$ always makes one of the first two arguments of the hypergeometric function appearing in local coordinates~\eqref{eq:lc} independent of $\lambda$ and finite as $\lambda \to \infty$. This is crucial because then a generic term in the expansion of these hypergeometric function around the puncture $z=z_j$ takes the following form:
\begin{equation}
\text{term} \sim \frac{\# \lambda^n + \dots }{\# \lambda^n + \dots} (z-z_j)^n.
\end{equation}
Here $\#$ denotes some numbers while dots denote the lower order terms in $\lambda$. The important point here is that since one of the arguments of the hypergeometric function is independent of $\lambda$, the same power of $\lambda$ appears in the numerator and the denominator of the coefficient of $(z-z_j)^n$ in its expansion. Therefore, these coefficients remain finite as we take $\lambda \to \infty$.
On the other hand, observe that the ratio of hypergeometric functions is raised to the power $1/i\lambda$ in~\eqref{eq:lc} and this exponent approaches to $0$ in the light-cone limit. But as we have just argued, the expansion of the hypergeometric functions remains finite in this limit. So we conclude that the part depending on the hypergeometric functions must completely drop out. The resulting limit gives, after taking the limits of prefactors like in subsection~\ref{sec:Witten},
\begin{subequations}
\begin{align}
w_1(z) &= r^{-1} (r-1)^{\frac{r-1}{r}}z (1-z)^{-\frac{\lambda_{2}}{\lambda_{1}}},\\
w_2(z) &= r^{r} (r-1)^{1-r} (z-1) z^{-\frac{\lambda_{1}}{\lambda_{2}}}, \\
w_3(z) &= r^{\frac{r}{r-1}} (r-1)^{-1} (z-1)^{-\frac{\lambda_{2}}{\lambda_{3}}} z^{\frac{\lambda_{1}}{\lambda_{3}}}.
\end{align}
\end{subequations}
From this, it is clear that if we relate the lengths of strings $2 \pi \lambda_j = L_j$ to the light-cone momenta $p_j^+$ in the usual fashion after an infinite rescaling, i.e. $p_j^+ \sim L_j$, and include the appropriate signs for the incoming/outgoing momenta, we arrive the light-cone vertex given in~\cite{Zwiebach:1988qp} up to an unimportant phase ambiguity. Therefore, the hyperbolic three-string vertex indeed reduces to the light-cone vertex in the light-cone limit in accord with our geometric expectation.
\section{Conservation laws for the hyperbolic three-string vertex} \label{sec:Conservation}
In this section we derive the conservation laws associated with the hyperbolic three-string vertex in the spirit of~\cite{rastelli2001tachyon}. Let us denote the hyperbolic three-string vertex with the geodesic boundaries of length $L=2 \pi \lambda$ as $\bra{V_{0,3}(\lambda)}$. This should be thought as an element of three-string dual Fock space, so it takes 3 states in Fock space and maps to a complex number. For simplicity of reporting, we set all the boundary lengths equal and report the holomorphic Virasoro conservation laws, but arguments here can be extended trivially to the cases with unequal lengths; ghosts and current conservation laws; and/or anti-holomorphic analogues.
First, let us put the punctures at $z=\sqrt{3},0, - \sqrt{3}$ in order to be consistent with~\cite{rastelli2001tachyon} and report the expansions for $z$ in terms of the local coordinates~$w_j$:
\begin{align}
f_1(w_1) &= \sqrt{3} + 2 \sqrt{3} N e^{-\frac{v}{\lambda }} w_1 + 3 \sqrt{3} N^2 e^{-\frac{2 v}{\lambda }} w_1^2 \nonumber
+\frac{\sqrt{3} \left(31 \lambda ^2+139\right)}{8 \left(\lambda ^2+4\right)}N^3 e^{-\frac{3 v}{\lambda }} w_1^3 + \mathcal{O}(w_1^4), \\
f_2(w_2) &= \frac{1}{2} \sqrt{3} N e^{-\frac{v}{\lambda }} w_2 -\frac{5 \sqrt{3} \left(\lambda ^2+1\right) N^3 e^{-\frac{3 v}{\lambda }}}{32 \left(\lambda ^2+4\right)} w_2^3 + \mathcal{O}(w_2^5),\nonumber\\
f_3(w_3) &= -\sqrt{3} + 2 \sqrt{3} N e^{-\frac{v}{\lambda }} w_3 - 3 \sqrt{3} N^2 e^{-\frac{2 v}{\lambda }} w_3^2
+\frac{\sqrt{3} \left(31 \lambda ^2+139\right)}{8 \left(\lambda ^2+4\right)} N^3 e^{-\frac{3 v}{\lambda }} w_3^3 + \mathcal{O}(w_3^4).
\end{align}
These are based on inverting~\eqref{eq:lc} respectively after performing the global conformal transformation
\begin{equation} \label{eq:move}
z \to - \frac{z-\sqrt{3}}{z+\sqrt{3}},
\end{equation}
that makes the monodromies around $z=\sqrt{3},0, -\sqrt{3}$ non-trivial. Here we will refer functions from the $w_j$-plane to the $z$-plane as $f_j$, $f_j(w_j) = z$. Like before, here $N=e^{-\frac{\pi}{2 \lambda}}$ and $v=v_1=v_2=v_3$. The global phase of the local coordinates $w_j$ are not important as usual, so we used this freedom to put $f_j$'s into rather symmetric form shown above. We are going to work perturbatively in $w_i$ below.
Notice that when we consider the minimal area limit, these expressions reduce to the one given in (2.11) of~\cite{rastelli2001tachyon}. This limiting behavior is expected, since the vertex given there, open string Witten vertex, when considered in the entirety of the complex plane becomes the closed string minimal area three-vertex, and we know from the previous sections that's what the hyperbolic three-string vertex approaches in the minimal area limit. So it shouldn't be too surprising that the identities we will write below reduces to their counterparts given in~\cite{rastelli2001tachyon} in the minimal area limit.
As an example of conservation laws, we derive the Virasoro conservation laws by which we mean the identities of the type, for $k>0$,
\begin{equation} \label{eq:Form}
\bra{V_{0,3}(\lambda)} L_{-k}^{(2)} = \bra{V_{0,3}(L)} \left[A^k(\lambda) \cdot c + \sum_{n\geq 0 } a_n^k(\lambda) L_n^{(1)}+ \sum_{n\geq 0 } c_n^k(\lambda) L_n^{(2)}+ \sum_{n\geq 0 } d_n^k(\lambda) L_n^{(3)}\right].
\end{equation}
Here $A^k, a_n^k, c_n^k, d_n^k$ are some functions of $\lambda$ that we are going to explicitly derive, $L_n$ are Virasoro generators, and the superscript denotes the slot that they apply in $\bra{V_{0,3}(\lambda)}$. By cyclicity of the hyperbolic three-vertex similar identities holds as we permute $(1) \to (2), (2) \to (3), (3) \to (1)$. So it would be sufficient to report the form above. The idea here is to exchange the negatively-moded Virasoro charges with the positively-moded ones plus the central term.
Now let $v(z)$ be a vector field holomorphic everywhere except for the punctures.\footnote{Not to be confused with $v$ appearing in the local coordinates.} That is, it changes as $v(z) \to \tilde{v}({\tilde{z}}) = (\partial \tilde{z}) v(z)$ under $z \to \tilde{z}$. Note that $v(z)$ should be regular at $z=\infty$ by its definition, so we must ensure $z^{-2}v(z)$ is finite as $z \to \infty$ by the inversion map $z \to \tilde{z} = 1/z$.
It is important to note that the object $v(z) T(z) dz$ is almost a 1-form, where $T(z)$ is stress-energy tensor.\footnote{In this section $T(z)$ will denote the stress-energy tensor of an arbitrary CFT with central charge $c$, not to be confused with the stress-energy tensor $T_{\varphi}(z)$ we previously considered.} Under $z \to \tilde{z}$ it transforms as,
\begin{equation} \label{eq:formtransform}
v(z) T(z) dz = \tilde{T}(\tilde{z}) \tilde{v}(\tilde{z}) d \tilde{z} - \frac{c}{12} \{z, \tilde{z}\} \tilde{v}(\tilde{z}) d \tilde{z}.
\end{equation}
As we see above, we have an extra contribution from the central term. Nonetheless, we can integrate this object on the complex plane on contours and use the usual properties of the complex integration, as long as we keep track of this additional term under the change of coordinates.
In order to derive the Virasoro conservation laws, the following equality is crucial:
\begin{equation}
\bra{V_{0,3}(\lambda)} \oint_{\mathcal{C}} \text{d} z \; v(z) T(z) = 0.
\end{equation}
Here, $\mathcal{C}$ is a contour that surrounds the three punctures, oriented counterclockwise. This is a shorthand notation for the vanishing of the correlator of the integral $\oint_{\mathcal{C}} v(z) T(z) dz$ with any three vertex operator placed at the punctures. Note that this correlator vanishes because we can push the contour to shrink around $z=\infty$ by the inversion map. In this case, the central charge term does not contribute since the Schwarzian derivative of the inversion map is zero.
Now we can deform the contour $\mathcal{C}$ to separate it into positively oriented, disjoint contours $\mathcal{C}_i$ around each punctures and write down the expression above in terms of the local coordinates as follows:
\begin{equation} \label{eq:conservation}
\bra{V_{0,3}(\lambda)} \sum_{i=1}^{3} \oint_{\mathcal{C}_i} \text{d} w_i \; v^{(i)}(w_i) \left[T^{(i)}(w_i) - \frac{c}{12} \{f_i(w_i), w_i\}\right] = 0,
\end{equation}
using the transformation property of $v(z) T(z) dz$ given in~\eqref{eq:formtransform}. Here $v^{(i)}(w_i)$ denotes the components of the vector field $v(z)\frac{\partial}{\partial z}$ in the local coordinates $w_i$ and similarly for the stress-energy tensor.
We will clearly need to find $\{f_i(w_i), w_i\} $ because of~\eqref{eq:conservation}. This is easy to do:
\begin{align}
\{f_i(w_i), w_i\} &= -\frac{15 \left(\lambda ^2+1\right)}{8 \left(\lambda
^2+4\right)} N^2 e^{-\frac{2 v}{\lambda }} + \frac{135 \left(\lambda ^2+1\right) \left(3 \lambda ^4+19 \lambda ^2-2\right)}{64 \left(\lambda ^2+4\right)^2 \left(\lambda ^2+16\right)} N^4 e^{-\frac{4 v}{\lambda }} w_i^2 + \cdots.
\end{align}
This is the same for each puncture because of the cyclicity, which we explicitly checked. Unsurprisingly, in minimal area limit we arrive the expression given in equation (3.8) of~\cite{rastelli2001tachyon}. By this expansion it is easy to see this term only appears if we have odd-powered poles around a puncture by (\ref{eq:conservation}).
Now remember
\begin{equation}
L_{-k}^{(i)} = \oint_{\mathcal{C}_i} \frac{\text{d} w_i}{2\pi i} w_i^{-k+1} T^{(i)}(w_i),
\end{equation}
so we need a vector field that behaves like $v^{(2)} \sim w_2^{-k+1}$ for $k>0$ around the puncture $(2)$ while behaves like $v^{(1)} \sim w_1$ and $v^{(3)} \sim w_3$ around the other punctures in order to put the Virasoro generators in the form given in (\ref{eq:Form}). Additionally, we have to ensure the regularity at infinity.
For $k=1$ case, all of these can be achieved with the following globally defined holomorphic vector field:
\begin{equation}
v_1(z) = -\frac{N e^{-\frac{v}{\lambda }}}{2 \sqrt{3}} \left(z^2-3\right).
\end{equation}
Normalization is chosen to get the convention in~(\ref{eq:Form}) and in the minimal area limit this reduces to one given in (3.10) of~\cite{rastelli2001tachyon}. This has the following expansion in the local coordinates $w_i$
\begin{align}
v_1^{(1)}(w_1) &= -Ne^{-\frac{v}{\lambda }}w_1 + \frac{1}{2} N^2 e^{-\frac{2 v}{\lambda }} w_1^2 - \frac{5 \left(\lambda ^2+1\right) }{8 \left(\lambda
^2+4\right)} N^3 e^{-\frac{3 v}{\lambda }} w_1^3 + \mathcal{O}(w_1^4), \nonumber\\
v_1^{(2)}(w_2) &= 1 + \frac{\left(11 \lambda ^2-1\right)}{16 \left(\lambda
^2+4\right)} N^2 e^{-\frac{2 v}{\lambda }}w_2^2 + \frac{5 \left(-8 \lambda ^6-6 \lambda ^4+3 \lambda ^2+1\right) }{128 \left(\lambda ^2+4\right)^2 \left(\lambda ^2+16\right)} N^4 e^{-\frac{4 v}{\lambda
}} w_2^4 + \mathcal{O}(w_2^6),\nonumber\\
v_1^{(3)}(w_3) &= Ne^{-\frac{v}{\lambda }}w_3 + \frac{1}{2} N^2 e^{-\frac{2 v}{\lambda }} w_3^2 + \frac{5 \left(\lambda ^2+1\right) }{8 \left(\lambda
^2+4\right)} N^3 e^{-\frac{3 v}{\lambda }} w_3^3 + \mathcal{O}(w_1^4).
\end{align}
Unsurprisingly, these reduce to the equation (3.11) of \cite{rastelli2001tachyon} in the minimal area limit. After substituting this into (\ref{eq:conservation}), each integration amounts to doing the replacement $w_i^n \to L_{n-1}^{(i)}$ by the residue theorem. Therefore we get
\small
\begin{alignat}{2}
0 &= \bra{V_{0,3}(\lambda)} \left( -Ne^{-\frac{v}{\lambda }}L_0 + \frac{1}{2} N^2 e^{-\frac{2 v}{\lambda }} L_1 - \frac{5 \left(\lambda ^2+1\right) N^3 e^{-\frac{3 v}{\lambda }}}{8 \left(\lambda
^2+4\right)} L_2 + \frac{5 \left(\lambda ^2+1\right) N^4 e^{-\frac{4 v}{\lambda }}}{32 \left(\lambda ^2+4\right)}L_3 + \dots \right)^{(1)}\nonumber\\
&+\bra{V_{0,3}(\lambda)} \left(L_{-1} + \frac{\left(11 \lambda ^2-1\right) N^2 e^{-\frac{2 v}{\lambda }}}{16 \left(\lambda
^2+4\right)} L_1 + \frac{5 \left(-8 \lambda ^6-6 \lambda ^4+3 \lambda ^2+1\right) N^4 e^{-\frac{4 v}{\lambda
}}}{128 \left(\lambda ^2+4\right)^2 \left(\lambda ^2+16\right)} L_3 + \dots \right)^{(2)}\nonumber\\
&+\bra{V_{0,3}(\lambda)} \left( Ne^{-\frac{v}{\lambda }}L_0 + \frac{1}{2} N^2 e^{-\frac{2 v}{\lambda }} L_1 + \frac{5 \left(\lambda ^2+1\right) N^3 e^{-\frac{3 v}{\lambda }}}{8 \left(\lambda
^2+4\right)} L_2+ \frac{5 \left(\lambda ^2+1\right) N^4 e^{-\frac{4 v}{\lambda }}}{32 \left(\lambda ^2+4\right)}L_3 + \dots \right)^{(3)}.
\end{alignat}
\normalsize
Again, this reduces to (3.12) of~\cite{rastelli2001tachyon} in the minimal area limit. Note that this doesn't have any central charge contribution since the vector $v_1(z)$ does not have a pole around the punctures.
We can continue to generate identities of the form (\ref{eq:Form}) by using the following vector fields:
\begin{align}
v_2(z) &= -\frac{N^2 e^{-\frac{2v}{\lambda }}}{4} \frac{z^2-3}{z},\\
v_3(z) &= -\frac{\sqrt{3 }N^3 e^{-\frac{3v}{\lambda }}}{8} \frac{z^2-3}{z^2}-
\frac{3 (3+7\lambda^2) }{16(4+\lambda^2)} N^2 e^{-\frac{2v}{\lambda }} v_1(z).
\end{align}
They produce the following identities respectively,
\small
\begin{alignat}{2}
0 &= \bra{V_{0,3}(\lambda)} \left( -\frac{1}{2} N^2 e^{-\frac{2 v}{\lambda }} L_0
+\frac{5}{4} N^3 e^{-\frac{3 v}{\lambda }} L_1
-\frac{3 \left(7 \lambda ^2+23\right) }{16 \left(\lambda
^2+4\right)} N^4 e^{-\frac{4 v}{\lambda }} L_2 + \dots \right)^{(1)}\nonumber\\
&+\bra{V_{0,3}(\lambda)} \left( L_{-2} + \frac{5 \left(\lambda ^2+1\right) }{32 \left(\lambda
^2+4\right)} N^2 e^{-\frac{2 v}{\lambda }} c + \frac{\left(4 \lambda ^2+1\right)}{4 \left(\lambda
^2+4\right)} N^2 e^{-\frac{2 v}{\lambda }} L_0 + \dots \right)^{(2)}\nonumber\\
&+\bra{V_{0,3}(\lambda)} \left( -\frac{1}{2} N^2 e^{-\frac{2 v}{\lambda }} L_0
-\frac{5}{4} N^3 e^{-\frac{3 v}{\lambda }} L_1
-\frac{3 \left(7 \lambda ^2+23\right)}{16 \left(\lambda
^2+4\right)} N^4 e^{-\frac{4 v}{\lambda }} L_2+ \dots \right)^{(3)}, \\
0 &= \bra{V_{0,3}(\lambda)} \left(\frac{\left(17 \lambda ^2-7\right) }{16 \left(\lambda
^2+4\right)} N^3 e^{-\frac{3 v}{\lambda }} L_0 + \frac{15 \left(\lambda ^2+9\right) }{32 \left(\lambda
^2+4\right)} N^4 e^{-\frac{4 v}{\lambda }} L_1 + \dots \right)^{(1)}\nonumber\\
&+\bra{V_{0,3}(\lambda)} \left(L_{-3} -\frac{15 \left(\lambda ^2+9\right) \left(2 \lambda ^2-1\right) \left(4 \lambda ^2+1\right) }{128 \left(\lambda ^2+4\right)^2 \left(\lambda ^2+16\right)} N^4 e^{-\frac{4 v}{\lambda
}} L_1 + \dots \right)^{(2)} \nonumber \\
&+\bra{V_{0,3}(\lambda)} \left(-\frac{\left(17 \lambda ^2-7\right) }{16 \left(\lambda
^2+4\right)} N^3 e^{-\frac{3 v}{\lambda }}L_0 + \frac{15 \left(\lambda ^2+9\right) }{32 \left(\lambda
^2+4\right)} N^4 e^{-\frac{4 v}{\lambda }} L_1 + \dots \right)^{(3)}.
\end{alignat}
\normalsize
We explicitly checked these reduces to their counterparts in~\cite{rastelli2001tachyon} in the minimal area limit. Note that we can continue generating similar identities for $L_{-k}$ recursively by using vector fields $v_k(z) \sim (z^2-3)z^{-k+1}$ and appropriately subtracting previous ones. Doing this allows us to put the identities in the form (\ref{eq:Form}) for which only a single negatively-moded Virasoro generator appears in the left-hand side.
\section{Remarks and open questions} \label{sec:Conc}
In this paper, we constructed the local coordinates for the hyperbolic three-string vertex first described in~\cite{Costello:2019fuh} and investigated its various limits explicitly. We calculated the $t^3$ term in the closed string tachyon potential and developed the conservation laws associated with such vertex in the spirit of~\cite{rastelli2001tachyon}. We conclude by providing some final remarks and highlighting possible future directions relevant to us:
\begin{enumerate}
\item Since we now know the local coordinates for the hyperbolic three-string vertex, it is possible to construct the Feynman diagrams by identifying them as
\begin{equation}
w_j w_j' = \exp \left[ -\frac{2 \pi s}{L_j} + i \theta\right] \quad \text{with} \quad s \in \mathbb{R}_{\geq 0}, \quad \theta \in [0,2 \pi)
\end{equation}
using the local coordinates $w_j$ and $w_j'$ associated to boundaries of equal length on not-necessarily-distinct pair of pants. Making this identification corresponds to having a finite flat cylinder of circumference $L_j$ and length $s$ with a twist $\theta$ stretching between not-necessarily-distinct pair of pants and it has the natural interpretation of the string propagator.
As usual, we must consider every possible value of $(s,\theta)_A$ when we are computing the string amplitudes. Here we added an index $A$ to indicate there are generally more than one propagator in the Feynman diagrams. It would be interesting the study the Feynman regions these diagrams cover in the moduli spaces of Riemann surfaces of genus $g$ and $n$ punctures $\mathcal{M}_{g,n}$ to see if they provide a piece of a section over the bundle $\widehat{\mathcal{P}}_{g,n} \to \mathcal{M}_{g,n}$ or not. The simplest Feynman regions to study would be for four-string scattering or string tadpole interaction. Note that with the metric we constructed on the hyperbolic pair of pants, it is possible to describe a Thurston metric of~\cite{Costello:2019fuh} explicitly on Riemann surfaces.
\item The local coordinates (\ref{eq:LocalCoord}) we constructed in this paper also can be used for the open-closed hyperbolic string vertices without moduli~\cite{Cho:2019anu}. There are two additional vertices without moduli on top of the sphere with three closed string punctures in this situation. They are disk with three open string punctures and disk with one open string puncture and one closed string puncture. Note that if we cut open the hyperbolic three-closed string vertex along a geodesics connecting all punctures for the former and one puncture connecting back to itself for the latter, we generate these additional cases exactly. From this construction it is clear that these would carry hyperbolic metric with appropriately grafted flat strip/cylinder parts and would be the same as what is constructed in~\cite{Cho:2019anu}. So we can still use the local coordinates (\ref{eq:LocalCoord}) for these additional cases.
\item The primary method we used in this paper, that is relating Liouville's equation on a specified domain to a monodromy problem, can be generalized to construct the local coordinates of the classical ($g=0$) hyperbolic $n$-string vertices in principle. For this, instead of (\ref{eq:3T}) we should take the stress-energy tensor of Liouville's equation to be
\begin{equation} \label{eq:Tconc}
T_{\varphi}(z) = \sum_{i=1}^n \left[ \frac{\Delta_i}{(z-z_i)^2} + \frac{c_i}{(z-z_i)}\right] ,
\end{equation}
with punctures positioned at $z=z_i$ and use this stress-energy tensor to generate the local coordinates. Here $c_i \in \mathbb{C}$ are so-called \emph{accessory parameters}~\cite{hadasz2003polyakov}.
There are two important problems with this approach. First, after we fixed the positions of three punctures by PSL(2,$\mathbb{C}$) symmetry, assigned prescribed weights at all punctures, and demanded regularity at infinity, we would still have $n-3$ unfixed $c_i$ parameters functions of $n-3$ unfixed positions $z_i$, the usual moduli for the $n$-punctured sphere. It is argued that such accessory parameters can be fixed in terms of moduli using the action of Liouville theory so that the metric associated to $T_{\varphi}(z)$ is smooth and hyperbolic~\cite{hadasz2003polyakov}, which goes under the name \emph{Polyakov Conjecture}. Computing these parameters exactly is not known, so this is the first problem. However, some numerical results are available in the case of vanishing $L_i$, see~\cite{hadasz2006liouville}.
Secondly, even if we find the correct $c_i$, guaranteeing the correct monodromy structure for the resulting Fuchsian equation with $n$ regular singularities is impractical. This is because of the lack of analogous formulas given in (\ref{eq:connection}) for the solutions to the Fuchsian equation associated with~\eqref{eq:Tconc}. It seems to us this is not the direction one should pursue if their goal is to do practical computations.
\item It would seem more promising to evaluate all higher elementary string interactions by exploiting the pants decomposition of the (marked) Riemann surfaces and their associated Teichm\"uller spaces~\cite{buser2010geometry}. The idea would be to decompose the contribution from a given Riemann surface as sums of products of cubic interaction of appropriate string fields dictated by a given pair of pants decomposition of such Riemann surface and to use an appropriate region in Teichm\"uller space to perform the moduli integration, similar to what is suggested in~\cite{Moosavian:2017qsp,Moosavian:2017sev}. Both of these steps need further study. Related to this idea, it may be possible to form a recursion relations in the similar vein of~\cite{mirzakhani2007weil,Eynard:2007fi,ellegard1,ellegard2}.
From the possibility of using pants decomposition we see the relevance of the hyperbolic three-string vertex with \emph{unequal} $L_i$ we considered so far. After the pants decomposition, we would only need to use the hyperbolic three-string vertex of arbitrary $L_i$ to compute CFT correlators and the rest of the computation would presumably just involve combining them together in correct fashion. We leave investigating this to a future work.
\end{enumerate}
\acknowledgments
The author would like to thank Barton Zwiebach for suggesting this problem and his guidance in the writing process. This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of High Energy Physics of U.S. Department of Energy under grant Contract Number DE-SC0012567.
\section{Introduction} \label{sec:Intro}
Defining off-shell amplitudes in closed string field theory requires selecting a set of string vertices $\mathcal{V}_{g,n}$ with $2g-2+n > 0$~\cite{Zwiebach:1992ie,Erler:2019loq}. These are subsets of the moduli spaces $\widehat{\mathcal{P}}_{g,n}$ of compact Riemann surfaces of genus $g$ and $n$ punctures with a choice of local coordinates (defined up to global phases) around each puncture. String vertices ought to satisfy the geometric master equation in order to define a consistent quantum theory~\cite{Sen:1994kx,Sen:1993kb,Sonoda:1989wa}.
There have been few proposals in the past for how to explicitly specify string vertices $\mathcal{V}_{g,n}$. The oldest, and probably the most well-known, is the one that uses the minimal area metrics on Riemann surfaces~\cite{Zwiebach:1992ie,Zwiebach:1990nh}. Using such metrics there is a simple prescription for how to specify string vertices that solves the geometric master equation~\cite{Zwiebach:1992ie}. The minimal area metrics for higher genus surfaces, however, are not known explicitly and still lack rigorous proof of existence. Nonetheless, one may expect that these will soon follow in the light of the recent discoveries~\cite{Headrick:2018ncs,Headrick:2018dlw,Naseer:2019zau}.
Another proposal for string vertices $\mathcal{V}_{g,n}$ that utilizes the fact that the Riemann surfaces considered for $\mathcal{V}_{g,n}$ admit hyperbolic metrics (of constant negative Gaussian curvature $K=-1$) was recently made by Moosavian and Pius~\cite{Moosavian:2017qsp,Moosavian:2017sev}. This interesting approach seems particularly promising considering the rigorously established existence of hyperbolic metrics and the recent developments in evaluating integrals over the moduli spaces of Riemann surfaces using the associated Teichm\"uller spaces~\cite{mcshane1998simple,mirzakhani2007weil,mirzakhani2007simple,Eynard:2007fi,ellegard1,ellegard2,Dijkgraaf:2018vnm}. However, it has been shown that these string vertices solve the geometric master equation only to the first approximation and they require a correction at each order of approximation. It is not known that such corrected string vertices always exist.
Although they are intriguing in their own rights, we see that two proposals for string vertices above suffer from either missing the proof of existence or failing to satisfy the geometric master equation exactly, therefore falling short of providing a consistent string field theory. In order to have a consistent string field theory we must guarantee that the string vertices exist on the moduli spaces of Riemann surfaces while exactly satisfying the geometric master equation. \emph{Hyperbolic string vertices} by Costello and Zwiebach simultaneously achieved both of these conditions recently~\cite{Costello:2019fuh}. To that end, the authors considered Riemann surfaces endowed with hyperbolic metric with \emph{geodesic boundaries} of length $L$, for $0<L\leq2$ arcsinh(1), and with systole\footnote{Systole on a bordered surface is defined as the length of the shortest closed geodesic that is \emph{not} a boundary component.} greater than or equal to $L$. Then they specified the string vertices by attaching flat semi-infinite cylinders of circumference $L$ at each boundary component to such surfaces. By the existence of hyperbolic metrics on Riemann surfaces of genus $g$ and $n$ boundaries with $2g-2+n > 0$, it was argued that this construction is always possible. Furthermore, it has been shown that the resulting string vertices exactly satisfy the geometric master equation by the virtue of the collar theorems of hyperbolic geometry~\cite{buser2010geometry}. We are going to call the closed bosonic string field theory hyperbolic string vertices define \emph{hyperbolic string field theory}.
Beyond establishing the first rigorous, explicit, and exact construction for the string vertices, using the hyperbolic string vertices also seems promising from the perspective of the aforementioned developments in computing integrals over the moduli spaces of Riemann surfaces by exploiting the underlying hyperbolic geometry, just like in the case of the vertices of Moosavian and Pius. One might imagine (or hope) similar methods can be applied to evaluate the string amplitudes to arbitrary orders and provide a useful handle for the computations in hyperbolic string field theory as a result.
A natural first step in this direction would be to compute the off-shell three-string amplitudes using the hyperbolic three-string vertex $\mathcal{V}_{0,3}$, which is constructed by grafting three flat semi-infinite cylinders to the three-holed sphere (or \emph{pair of pants}) equipped with a hyperbolic metric, since $\mathcal{V}_{0,3}$ contains just a single surface. For the sake of generality, we are going to leave the circumferences of the grafted cylinders arbitrary for this vertex, even though only the case of equal circumferences is needed for $\mathcal{V}_{0,3}$~\cite{Costello:2019fuh}. This \emph{generalized hyperbolic three-string vertex} is of interest in the hyperbolic string field theory in the long run on account of the well-known pants decomposition of Riemann surfaces~\cite{buser2010geometry}. For brevity, we will also denote this generalized vertex as hyperbolic three-string vertex without making a distinction.
In order to perform the computations mentioned above using the operator formalism of conformal field theory (CFT), one needs to obtain the explicit expressions of the local coordinates around the punctures for the hyperbolic three-string vertex~\cite{Erler:2019loq}. In this paper, we find these local coordinates, investigate their various limits, and derive the associated conservation laws by following the procedure in~\cite{rastelli2001tachyon}.
In principle, the local coordinates for the hyperbolic three-string vertex can be obtained by the following procedure. First, recall that the hyperbolic metric on the three-holed sphere with geodesic boundaries of lengths $L_i$ ($i=1,2,3$) is unique up to isometry~\cite{buser2010geometry}. So we can simply write down this hyperbolic metric as
\begin{equation} \label{eq:intrometric}
ds^2 = e^{\varphi(z,\bar{z})} |dz|^2,
\end{equation}
on the Riemann sphere minus three disjoint simply connected regions, or \emph{holes}, \emph{unique} up to PSL(2,$\mathbb{C}$) transformations, whose boundaries are geodesics of given lengths $L_i$. From this point of view, one can obtain the local coordinates by finding how punctured unit disks conformally map onto these simply connected regions, since a semi-infinite cylinder is conformal to a punctured unit disk and it canonically introduces the local coordinates~\cite{Costello:2019fuh}. Note that such conformal transformations exist by the Riemann mapping theorem.
Therefore, we see that the problem of finding the local coordinates for the hyperbolic three-string vertex is a two-step procedure:
\begin{enumerate}
\item Find an explicit description of the union of three disjoint simply connected regions on the Riemann sphere whose complement is endowed with a hyperbolic metric~\eqref{eq:intrometric} and boundary components are geodesics of lengths $L_i$,
\item Find the conformal transformations from punctured unit disks to the aforementioned simply connected regions.
\end{enumerate}
The first step clearly involves solving a complicated boundary value problem for a partial differential equation, \emph{Liouville's equation}, and getting an exact answer is a hard endeavor in general. Luckily, it is known that the solutions for such boundary value problem can be related to a monodromy problem of a particular second-order linear ordinary differential equation with regular singularities, or a \emph{Fuchsian equation}, on the complex plane~\cite{hadasz2003polyakov,hadasz2004classical}. Exploiting this relation, which we review and expand in sections~\ref{sec:Fuchsian} and~\ref{sec:Monodromy}, we find the explicit description of the hyperbolic metric~(\ref{eq:intrometric}) and of the three holes on the Riemann sphere, up to PSL(2,$\mathbb{C}$) transformations.
Furthermore, the second step becomes trivial after we find such explicit description as we argue in section~\ref{sec:Monodromy}. In the end, for the hyperbolic three-string vertex whose grafted flat cylinders have the circumferences
\begin{equation} \label{eq:circum}
L_i \equiv 2 \pi \lambda_i,
\end{equation}
we obtain the following local coordinates $w_i(z)$ around the punctures at $z=0,1,\infty$ respectively:\footnote{We fix the locations of the punctures to $z=0,1,\infty$ using PSL(2,$\mathbb{C}$) transformations without loss of generality. So indices and numbers $i,j=1,2,3$ appearing on the objects denote the punctures $z=0,1,\infty$ respectively, unless otherwise stated. This shall be obvious from the context and we are not going to report it every time. If there is no label on an object, it should be understood that it has the same value for each puncture.}
\begin{subequations} \label{eq:lc}
\begin{align} \label{eq:lc1}
w_1(z) &= \frac{1}{N_1} \exp\left( \frac{v(\lambda_1,\lambda_2, \lambda_3)}{\lambda_1} \right) z (1-z)^{-\frac{\lambda_2}{\lambda_1}} \nonumber \\
&\qquad \qquad \times \left[ \frac{_2F_1 \left(\frac{1}{2}(1+i \lambda_1 -i \lambda_2 +i \lambda_3), \, \frac{1}{2}(1+i \lambda_1 -i \lambda_2 -i \lambda_3);\, 1+ i \lambda_1;\, z\right)}{_2F_1 \left(\frac{1}{2}(1-i \lambda_1 +i \lambda_2 -i \lambda_3),\,\frac{1}{2}(1-i \lambda_1 +i \lambda_2 +i \lambda_3);\, 1- i \lambda_1;\, z\right)} \right]^{\frac{1}{i \lambda_1}} \nonumber \\
&= \frac{1}{N_1} \exp\left( \frac{v(\lambda_1,\lambda_2, \lambda_3)}{\lambda_1} \right) \left(z + \frac{1+ \lambda_{1}^2 + \lambda_{2}^2 - \lambda_{3}^2}{2(1+\lambda_{1}^2)} z^2 + \mathcal{O}(z^3)\right) ,\\
\label{eq:lc2}
\quad \;\; w_2(z) &= \frac{1}{N_2} \exp\left( \frac{v(\lambda_2,\lambda_1, \lambda_3)}{\lambda_2} \right)(1-z) z^{-\frac{\lambda_1}{\lambda_2}} \nonumber\\
&\qquad \qquad \times
\left[ \frac{_2F_1 \left(\frac{1}{2}(1+i \lambda_2 -i \lambda_1 +i \lambda_3),\,\frac{1}{2}(1+i \lambda_2 -i \lambda_1 -i \lambda_3);\, 1+ i \lambda_2;\, 1-z\right)}{_2F_1 \left(\frac{1}{2}(1-i \lambda_2 +i \lambda_1 -i \lambda_3),\,\frac{1}{2}(1-i \lambda_2 +i \lambda_1 +i \lambda_3);\, 1- i \lambda_2;\, 1-z\right)} \right]^{\frac{1}{i \lambda_2}}\nonumber\\
&= \frac{1}{N_2} \exp\left( \frac{v(\lambda_2,\lambda_1, \lambda_3)}{\lambda_2} \right)\left((1-z) + \frac{1+ \lambda_{2}^2 + \lambda_{1}^2 - \lambda_{3}^2}{2(1+\lambda_{2}^2)} (1-z) ^2 + \mathcal{O}((1-z) ^3)\right), \\
\label{eq:lc3}
w_3(z) &= \frac{1}{N_3} \exp\left( \frac{v(\lambda_3,\lambda_2, \lambda_1)}{\lambda_3} \right)\left(\frac{1}{z}\right) \left(1-\frac{1}{z}\right)^{-\frac{\lambda_2}{\lambda_3}} \nonumber\\
&\qquad \qquad \times
\left[ \frac{_2F_1 \left(\frac{1}{2}(1+i \lambda_3 -i \lambda_2 +i \lambda_1),\,\frac{1}{2}(1+i \lambda_3 -i \lambda_2 -i \lambda_1);\, 1+ i \lambda_3;\, \frac{1}{z}\right)}{_2F_1 \left(\frac{1}{2}(1-i \lambda_3 +i \lambda_2 -i \lambda_1),\,\frac{1}{2}(1-i \lambda_3 +i \lambda_2 +i \lambda_1);\, 1- i \lambda_3;\, \frac{1}{z}\right)} \right]^{\frac{1}{i \lambda_3}} \nonumber \\
&= \frac{1}{N_3} \exp\left( \frac{v(\lambda_3,\lambda_2, \lambda_1)}{\lambda_3} \right) \left(\frac{1}{z} + \frac{1+ \lambda_{3}^2 + \lambda_{2}^2 - \lambda_{1}^2}{2(1+\lambda_{3}^2)}\frac{1}{z^2} + \mathcal{O}\left(\frac{1}{z^3} \right)\right).
\end{align}
\end{subequations}
Here ${_2}F_1(a,b;c;z)$ is the ordinary hypergeometric function
\begin{equation} \label{eq:hypergeometric}
{_2}F_1(a,b;c;z) = 1 + \frac{ab}{c} \frac{z}{1!} + \frac{a(a+1)b(b+1)}{c(c+1)} \frac{z^2}{2!} + \cdots,
\end{equation}
and the function $v\left(\lambda_{1}, \lambda_{2}, \lambda_{3}\right) $ is given in terms of the gamma function $\Gamma(z)$ as
\begin{align}
v\left(\lambda_{1}, \lambda_{2}, \lambda_{3}\right) &\equiv \frac{1}{2i} \text{Log}\left[\frac{\Gamma\left(-i \lambda_{1}\right)^{2}}{\Gamma\left(i \lambda_{1}\right)^{2}} \frac{\gamma\left(\frac{1}{2}(1+i \lambda_{1}+i \lambda_{2}+i \lambda_{3})\right) \gamma\left(\frac{1}{2}(1+i \lambda_{1}-i \lambda_{2}+i \lambda_{3})\right)}{\gamma\left(\frac{1}{2}(1-i \lambda_{1}-i \lambda_{2}+i \lambda_{3})\right) \gamma\left(\frac{1}{2}(1-i \lambda_{1}+i \lambda_{2}+i \lambda_{3})\right)} \right],\nonumber \\
\gamma(x) &\equiv \frac{\Gamma(1-x)}{\Gamma(x)}.
\end{align}
The factors $N_i$ above will be called \emph{scale factors}. They are fixed by integer $\tilde{l}_i \in \mathbb{Z}$ via
\begin{equation}
N_i = \exp\left[ \frac{\pi}{\lambda_i} \left(\tilde{l}_i + \frac{1}{2}\right) \right].
\end{equation}
Tilde on the integers $\tilde{l}_i$ comes from our construction. As we will see, among the sets of integers $l_i$, only specific ones, $l_i=\tilde{l}_i$, would give the correct scale factor. The integers $\tilde{l}_i$ can be determined for a given set of $\lambda_i$'s in principle. Even though we couldn't find a closed-form expression for the integers $\tilde{l}_i$'s for arbitrary $\lambda_i$'s, one can still easily find them by investigating numerical plots of the local coordinates. In certain symmetric situations it is possible to find the integers $\tilde{l}_i$'s without resorting the plots. For example, when $0 \leq \lambda_i = \lambda \leq 10$ for the grafted cylinders, we find $\tilde{l}_i=\tilde{l}=-1$. This result is anticipated to hold for all values of $\lambda_i = \lambda$.
Since we have the explicit expressions of the local coordinates for the hyperbolic three-vertex~\eqref{eq:lc}, we can check their consistency with the other local coordinates in the literature by investigating their limiting behaviors~\cite{sonoda1990covariant,Moosavian:2017qsp,Zwiebach:1988qp}. For instance, as argued in~\cite{Costello:2019fuh}, this vertex must produce the three-string vertex obtained from the minimal area metric as all $\lambda_i \to \infty$ at the same rate, which we are going to denote as the \emph{minimal area limit}. In the light of this fact, we consider the minimal area limit of the coordinates~\eqref{eq:lc} and show that the local coordinates for the minimal area three-string vertex and those for the hyperbolic three-string vertices with $\lambda \to \infty$ match perturbatively to the order $\mathcal{O}(z^{10})$ in section~\ref{sec:Limits}. We then discuss the possibility of extending our argument to all orders in $z$.
Moreover, the hyperbolic three-string vertex must reduce to the three-string vertex considered by Moosavian and Pius~\cite{Moosavian:2017qsp} as $\lambda_i \to 0$ after a suitable modification, since the geodesic boundaries become cusps in this regime and this is exactly what is considered there. We argue that this limiting behavior indeed holds in section~\ref{sec:Limits}. Lastly, we consider the situation $\lambda_{2} = \lambda_1 + \lambda_3$ with $\lambda_i \to \infty$ for which the geometry resembles the light-cone vertex~\cite{Zwiebach:1988qp}. We show that the hyperbolic three-string vertex reduces to the light-cone vertex in this limit, in accord with our expectations.
Having an explicit expression for the local coordinates~\eqref{eq:lc} also means that it is possible to derive the conservation laws for the hyperbolic three-string vertex in the spirit of~\cite{rastelli2001tachyon}, which we do in section~\ref{sec:Conservation}. Again, we can investigate various limits of these conservation laws. Especially we observe that all of our expressions in section~\ref{sec:Conservation} reduces to their respective counterparts in~\cite{rastelli2001tachyon} in the minimal area limit. This is consistent, since the open string Witten vertex and its closed string analog must generate the same conservation laws. It is known that conservation laws provide systematic and easily implementable procedure for computations in the cubic open string field theory, especially for the level truncation~\cite{Gaiotto:2002wy}, and we hope that these expressions will accomplish the same in the hyperbolic string field theory in the future.
As a sample computation using the local coordinates~\eqref{eq:lc}, we calculate the $t^3$ term in the closed string tachyon potential $V$ with $t$ is the zero-momentum tachyonic field in the case of $\lambda_i=\lambda$. Remember this is the case that appears in the string action. We find ($\alpha'=2$)
\begin{align} \label{eq:tachyon}
&\kappa^2 V = -t^2 + \frac{1}{3} \frac{t^3}{r^6} + \dots = -t^2 +\frac{1}{3} \exp\left[\frac{ 6v(\lambda,\lambda,\lambda) + 3\pi}{\lambda} \right] t^3 + \cdots.
\end{align}
Here $\kappa$ is the closed string coupling constant and $r$ is the mapping radius of the local coordinates, whose inserted expression is derived in section~\ref{sec:Monodromy}. Note that this calculation is exactly like in~\cite{kostelecky1990collective,belopolsky1995off,yang2005closed}, the only difference being the mapping radii we used for the expression above.
In order to get a sense of its value, let us set $\lambda= L_{\ast}/(2 \pi) = \text{arcsinh}(1)/\pi \approx 0.28055$, which is the largest value of $\lambda$ for which the hyperbolic vertices solves the geometric master equation~\cite{Costello:2019fuh}. Substituting this value and evaluating, we obtain the closed string tachyon potential $V$ in the hyperbolic string field theory is given by
\begin{equation} \label{eq:tp}
\kappa^2 V \approx -t^2 + \left( 1.62187 \times 10^8\right) t^3 + \cdots.
\end{equation}
The coefficient for the $t^3$ term is quite large compared to the corresponding one in the minimal area three-string vertex, which is approximately equal to $1.602$~\cite{kostelecky1990collective,belopolsky1995off,yang2005closed}. However, this coefficient in fact has the expected order of magnitude. We can see this by considering the coefficient obtained from the minimal area three-string vertex with stubs of length $\pi$, which roughly \emph{looks like} a hyperbolic three-string vertex geometrically. The coefficient for the case with stubs is easily obtained by observing that adding stubs scales mapping radii by $e^{-\pi}$, and in turn multiplies the no-stub coefficient by $e^{6 \pi}$ by the first equality in~\eqref{eq:tachyon}. This gives approximately $e^{6\pi} \cdot 1.602 \approx 2.460 \times 10^8$, which is close to the value given in~\eqref{eq:tp}.
The outline of the paper is as follows. In section~\ref{sec:Fuchsian} we introduce the boundary value problem for the hyperbolic metric with geodesic boundaries of fixed lengths on the Riemann sphere minus three holes and its relation to Fuchsian equations. In section~\ref{sec:Monodromy} we consider the relevant monodromy problem in order to find the explicit description of the holes on the Riemann sphere. The results of these two sections are well-established in the literature~\cite{hadasz2003polyakov,hadasz2004classical}, but we provide a self-contained discussion where we emphasize and investigate the resulting hyperbolic geometry in more detail. Additionally, we construct the local coordinates around the punctures for the hyperbolic three-string vertex in section~\ref{sec:Monodromy} and later in section~\ref{sec:Limits} we investigate their various limits. Lastly, we obtain the conservation laws associated with the hyperbolic three-string vertex in section~\ref{sec:Conservation}. We conclude the paper and discuss the possible future directions in section~\ref{sec:Conc}.
\section{Liouville's equation on a three-holed sphere} \label{sec:Fuchsian}
In this section, we describe the problem of finding an explicit description of the hyperbolic metric on the three-holed sphere with geodesic boundaries of lengths $L_i$ on the Riemann sphere, which will help us obtain the shapes and locations of the geodesic boundaries and the local coordinates later on. As we mentioned briefly, this is equivalent to solving Liouville's equation with specified boundary conditions on the Riemann sphere minus three holes. This problem is hard by itself, so instead we introduce a stress-energy tensor (in the sense of Liouville theory) and consider its associated Fuchsian equation, which we define below. The properties of this equation is investigated. Most importantly, we show that its multi-valued solutions can be related to hyperbolic metrics. The results of this section are well-known in the literature in the context of Liouville theory and the uniformization problem~\cite{hadasz2003polyakov,hadasz2004classical,hadasz2006liouville,cantini2001proof,cantini2002liouville,cantini2003polyakov,zograf1988liouville,takhtajan2003hyperbolic,Seiberg:1990eb,Bilal:1987cq,Hadasz:2005gk,Teschner:2003at,hempel1988uniformization}, but we are going to provide a self-contained review that focuses on the issues relevant to us.
As noted above, our first goal is to solve \emph{Liouville's equation}
\begin{equation} \label{eq:LiouvilleEq}
\partial \bar{\partial} \varphi(z,\bar{z}) = \tfrac{1}{2} e^{\varphi(z,\bar{z}) },
\end{equation}
on the three-holed sphere $X$ whose boundaries are chosen to be geodesics of lengths $L_i$ of the metric~(\ref{eq:intrometric}). It can be easily seen that satisfying Liouville equation is equivalent to the metric~(\ref{eq:intrometric}) having constant negative curvature $K=-1$. We will call $e^{\varphi(z,\bar{z}) }$ the \emph{conformal factor} and take $\varphi(z,\bar{z}) \in \mathbb{R}$ always to define a real metric. Like we mentioned before, we can think the surface $X$ endowed with the metric~(\ref{eq:intrometric}) as the Riemann sphere $\widehat{\mathbb{C}} = \mathbb{C} \cup \{\infty\}$ with three disjoint simply connected regions taken out and this understanding will be implicit. So $(z, \bar{z})$ will denote the complex coordinates on $X \subset \widehat{\mathbb{C}}$.
Solving the boundary value problem described above directly is non-trivial and we won't attempt to do that. Instead, we are going to relate this problem to solving a more manageable linear ordinary differential equation. In order to do that, let the factor $\varphi$ denote a solution of Liouville's equation~(\ref{eq:LiouvilleEq}) and define the (holomorphic) \emph{stress-energy tensor} associated with $\varphi$ as follows~\cite{Seiberg:1990eb}:
\begin{equation} \label{eq:stress-energy}
T_{\varphi}(z) \equiv -\tfrac{1}{2}(\partial \varphi)^2 + \partial^2 \varphi = -2 e^{\frac{\varphi}{2}} \partial^2 e^{-\frac{\varphi}{2}}.
\end{equation}
Observe that we only wrote the dependence on $z$, and not on $\bar{z}$, of the stress-energy tensor since it can be shown that $T_{\varphi}$ is holomorphic, $\bar{\partial} T_{\varphi} = 0$, using Liouville's equation~\eqref{eq:LiouvilleEq}. Furthermore, the converse of this statement holds as well: If $T_{\varphi} = T_{\varphi}(z)$ is holomorphic, then the factor $\varphi$ defined by $(\ref{eq:stress-energy})$ solves the Liouville's equation. Lastly, we note that $T_{\varphi}(z)$ is the (classical) stress-energy tensor in the context of Liouville theory and it transforms under conformal transformation $z \to \tilde{z}(z)$ as follows~\cite{Seiberg:1990eb}:
\begin{align} \label{eq:transofT}
T_{\varphi}(z) = \left(\frac{\partial \tilde{z}}{\partial z} \right)^2 \widetilde{T}_{\varphi}(\tilde{z}) + \{ \tilde{z},z \}.
\end{align}
Here tilde on the stress-energy tensor indicates that it is written in the $\tilde{z}$ coordinates and $\{\cdot, \cdot\}$ is the Schwarzian derivative:
\begin{equation}
\{ \tilde{z},z\} \equiv
\frac{\partial^3\tilde{z}}{\partial \tilde{z}} - \frac{3}{2} \left(\frac{\partial^2 \tilde{z}}{\partial \tilde{z}} \right)^2.
\end{equation}
We can similarly define the anti-holomorphic stress-energy tensor $\overline{T_{\varphi}}$ by replacing $\partial \to \bar{\partial}$ in~\eqref{eq:stress-energy}.
Now consider the following second-order linear ordinary differential equation constructed with the stress-energy tensor $T_{\varphi}(z)$ above~\cite{hadasz2003polyakov,hadasz2004classical}:
\begin{equation} \label{eq:Fuchsian}
\partial^2 \psi(z) + \tfrac{1}{2} T_{\varphi}(z) \psi (z) = 0.
\end{equation}
We will call this the holomorphic \emph{Fuchsian equation} associated with $T_{\varphi}(z)$. The reason for the name \emph{Fuchsian} will be justified in section~\ref{sec:Monodromy} when we show that the relevant $T_{\varphi}(z)$ contains at most double poles, so that the equation~(\ref{eq:Fuchsian}) has only regular singularities (i.e. \emph{Fuchsian}). Similarly, we can define the anti-holomorphic Fuchsian equation associated with $\overline{T_{\varphi}}(\bar{z})$. Considering~\eqref{eq:transofT}, in order to make the equation~(\ref{eq:Fuchsian}) conformal invariant, we are going to take the object $\psi(z)$ transforms as a conformal primary of dimension $(-\frac{1}{2},0)$. That is, we demand
\begin{equation} \label{eq:transofpsi}
\tilde{\psi} (\tilde{z}) = \left( \frac{\partial \tilde{z}}{\partial z}\right)^{\frac{1}{2}} \psi(z),
\end{equation}
under conformal transformation $z \to \tilde{z}(z)$.
Now suppose we have solved the Fuchsian equation and found two linearly independent, not necessarily single-valued, complex-valued solutions $\psi^{+}(z)$ and $\psi^{-}(z)$. We are going to always assume these solutions are normalized appropriately, in the sense that their Wronskian $W(\psi^{-},\psi^{+})$ is equal to one:
\begin{equation} \label{eq:Wronskian}
W(\psi^{-},\psi^{+}) \equiv (\partial \psi^{+}) \psi^{-} -\psi^{+} (\partial \psi^{+}) = 1.
\end{equation}
Now define the ratio $A(z)$ of these solutions and observe that we have the relations
\begin{equation} \label{eq:ratio}
A(z) \equiv \frac{\psi^{+}(z)}{\psi^{-}(z)} \quad \iff \quad \psi^{+}(z) = \frac{A(z)}{\sqrt{\partial A(z)}}, \quad \psi^{-}(z) = \frac{1}{\sqrt{\partial A(z)}}.
\end{equation}
From this, we immediately see the stress-energy tensor can be written as follows:
\begin{align} \label{eq:TasSch}
T_{\varphi}(z) = - 2 \frac{\partial^2 \psi^{-}}{\psi^{-}} &= -2 (\partial A)^{\frac{1}{2}} \partial^2 (\partial A)^{-\frac{1}{2}} = (\partial A)^{\frac{1}{2}} \partial \left((\partial A)^{-\frac{3}{2}} \partial^2 A\right) \nonumber \\
&= \frac{\partial^3A(z)}{\partial A(z)} - \frac{3}{2} \left(\frac{\partial^2 A(z)}{\partial A(z)} \right)^2 \equiv \{ A(z),z\} \implies T_{\varphi}(z) = \{ A(z),z\}.
\end{align}
In general, it is highly non-trivial to find the function $A(z)$ for a given $T_{\varphi}(z)$ satisfying (\ref{eq:TasSch}) above. However, if we know the solutions to the Fuchsian equation (\ref{eq:Fuchsian}), we see that $A(z)$ is determined by~\eqref{eq:ratio} up to M\"{o}bius transformations. That is one utility of the Fuchsian equation. Moreover, given $A(z)$ satisfying $T_{\varphi}(z) = \{A(z),z\}$, we can find the normalized solutions for the Fuchsian equation from~(\ref{eq:ratio}) as well. Note that $A(z)$ is a scalar under conformal transformations as can be seen from~\eqref{eq:transofpsi} and~\eqref{eq:ratio}.
Also we can see that putting the stress-energy tensor $T_{\varphi}(z)$ in the form (\ref{eq:TasSch}) and knowing such $A(z)$ is advantageous on the account of the transformation property of the stress-energy tensor~\eqref{eq:transofT}. The relation (\ref{eq:TasSch}), combined with the transformation property of the Schwarzian derivative and the stress-energy tensor, allows us to find the explicit expression of the stress-energy tensor $T_{\varphi}$ in other coordinates. We will see the benefit of this observations in the next section.
Another utility of the Fuchsian equation (\ref{eq:Fuchsian}) can be understood as follows. We can easily see that $\psi = e^{-\frac{\varphi(z,\bar{z})}{2}}$ solves (\ref{eq:Fuchsian}) using the second equality in~\eqref{eq:stress-energy}. This solution of the Fuchsian equation is real and single-valued because the metric (\ref{eq:intrometric}) itself is real and single-valued. It is important to observe that such factor solves the Fuchsian equation, because this allows us to relate the linearly independent, normalized solutions $\psi^{\pm}(z)$ of the Fuchsian equation to the hyperbolic metric (\ref{eq:intrometric}). In other words, knowing $\psi^{\pm}(z)$ would suffice to construct the metric.
Before we do that more precisely, we should first describe the multi-valuedness of the solutions $\psi^{\pm}(z)$. For our purposes, it is going to be sufficient to assume that the multi-valuedness of the solutions $\psi^{\pm}(z)$ are described by SL(2,$\mathbb{R}$) transformations, in the sense that when we go around any point $z=u \in \widehat{\mathbb{C}}$ by $(z-u)\to e^{2 \pi i} (z-u)$ the solutions are taken to be transforming as follows:
\begin{equation}
\begin{bmatrix}
\psi^{+} \\ \psi^{-}
\end{bmatrix} \; \to \;
\begin{bmatrix}
a & b \\ c & d
\end{bmatrix}
\begin{bmatrix}
\psi^{+} \\ \psi^{-}
\end{bmatrix} \quad \text{where} \quad a,b,c,d \in \mathbb{R}, \quad ad-bc=1,
\end{equation}
unless otherwise stated. That is, we assume the values that the functions $\psi^{\pm}(z)$ attain at a given point are related by SL(2,$\mathbb{R}$) transformations like above. From this, it is easy to see that the solution $e^{-\frac{\varphi(z,\bar{z})}{2}}$ of~\eqref{eq:Fuchsian} is given by the following linear combination of $\psi^{+}(z)$ and $\psi^{-}(z)$:
\begin{equation} \label{eq:FtoL}
e^{-\frac{\varphi(z,\bar{z})}{2}} = C \frac{i}{2} ( \overline{\psi^{-}(z)} \psi^{+}(z) - \overline{\psi^{+}(z)} \psi^{-}(z) ),
\end{equation}
since this is the unique real linear combination of the solutions $\psi^{\pm}(z)$ that is invariant under SL(2,$\mathbb{R}$) transformations (i.e. single-valued). As usual, the bar over the solutions denotes the complex conjugation. Here $C$ is a real constant, which turns out to be $C=\pm1$, as we will show it shortly. With this, the following metric has constant negative curvature $K= -1$:
\begin{equation} \label{eq:firstmetric}
ds^2 = e^{\varphi(z,\bar{z})} |dz|^2 = \cfrac{-4 |dz|^2}{( \overline{\psi^{-}(z)} \psi^{+}(z) - \overline{\psi^{+}(z)} \psi^{-}(z) )^2}.
\end{equation}
Note that a version of these expressions appears in the context of Liouville theory~\cite{Seiberg:1990eb}. There, the solutions $\psi^{\pm}(z)$ are interpreted as spin-$1/2$ representations of SL(2,$\mathbb{R}$) and their physical meaning is discussed.
The main takeaway from the discussion in the previous paragraphs is that the hyperbolic metric on a three-holed sphere $X$ can be related to the solutions of the Fuchsian equation using a suitable $T_{\varphi}(z)$. From the expression in (\ref{eq:stress-energy}), it might seem that finding $T_{\varphi}(z)$ as a function of $z$ is as hard as finding the explicit form of the metric (\ref{eq:intrometric}). However, as we will see in section \ref{sec:Monodromy}, $T_{\varphi}(z)$ can be found without knowing the metric. Then we can deduce the form of the hyperbolic metric by solving the associated Fuchsian equation through the relation~\eqref{eq:firstmetric}, which will eventually lead us to the local coordinates.\footnote{These relations hold for other hyperbolic Riemann surfaces with geodesic boundaries as well. But we will restrict our discussion to three-holed sphere, since it is the simplest case to perform these computations explicitly.}
Before we conclude this section, we need to show $C = \pm 1$ as we claimed. It is clear that not every value of a priori unfixed $C \in \mathbb{R}$ can define a hyperbolic metric with $K=-1$, so we need to choose the right value(s). This is essentially the reflection of the fact that the Fuchsian equation is linear: Every scaling of $e^{-\frac{\varphi(z,\bar{z})}{2}}$ is also a solution of (\ref{eq:Fuchsian}), even though the scaled ones don't define a hyperbolic metric with $K=-1$ because the Liouville's equation~(\ref{eq:LiouvilleEq}) is non-linear.
We can fix such $C$ once and for all as follows. First note that the conformal factor
\begin{equation} \label{eq:conformalfactor}
e^{\varphi(z,\bar{z})} = \frac{\lambda^2 |\partial f (z)|^2}{|f(z)|^2 \sin^2(\lambda \log|f(z)|)} = \frac{|\partial(\lambda \log(f(z)))|^2}{\sin^2(\lambda \log|f(z)|)}.
\end{equation}
always defines a (possibly singular) hyperbolic metric with $K=-1$, or equivalently, $\varphi$ above solves the Liouville's equation (\ref{eq:LiouvilleEq}) for an arbitrary holomorphic function $f(z)$ and an arbitrary $\lambda \in \mathbb{R}_{\geq 0}$, as one can check by explicit calculation. Now take the function $f(z)$ to be equal to
\begin{equation} \label{eq:HolChoice}
f(z) = A(z)^{\frac{1}{i \lambda}} = \left( \frac{\psi^{+}(z)}{\psi^{-}(z)} \right)^{\frac{1}{i \lambda}}.
\end{equation}
We will denote the right-hand side as the \emph{scaled ratio}. After substituting this expression into~(\ref{eq:conformalfactor}) we exactly get the metric~(\ref{eq:firstmetric}). This shows $C =\pm1$. For us, the equivalence between~(\ref{eq:firstmetric}) and (\ref{eq:conformalfactor}), with the choice (\ref{eq:HolChoice}), is going to be extremely useful and we will use both forms interchangeably in our arguments.
In summary, we have seen that we can relate the hyperbolic metric on a three-holed sphere $X$ to the solutions of the Fuchsian equation (\ref{eq:Fuchsian}) through (\ref{eq:firstmetric}). Not only this will provide us a solution to the Liouville's equation (\ref{eq:LiouvilleEq}), but, more importantly, it will be also used to make the boundaries of $X$ geodesics of the metric~\eqref{eq:intrometric}. After all, that's the whole reason we are taking this detour into Fuchsian equations. We have already seen that the conformal factor~(\ref{eq:conformalfactor}) always defines a (possibly singular) hyperbolic metric for any given $f(z)$, but the boundaries of $X$ are going to be geodesics only when we relate it to a particular set of solutions for the Fuchsian equation through the relation (\ref{eq:HolChoice}), as we shall see. In the next section, we are going to focus on the three-punctured sphere $\widetilde{X} = \mathbb{C} \setminus \{0,1,\infty\}$, rather than a three-holed sphere $X$, since it is simpler to deal with initially. Then we will cut open appropriate holes around the punctures in $\widetilde{X}$ to return back to $X \subset \widehat{\mathbb{C}}$ and graft flat semi-infinite cylinders to these holes to construct the local coordinates for the hyperbolic three-string vertex.
\section{A monodromy problem of Fuchsian equation} \label{sec:Monodromy}
In this section we find the hyperbolic metric on a three-holed sphere $X$ by investigating a certain monodromy problem of the Fuchsian equation (\ref{eq:Fuchsian}) on the three punctured sphere $\widetilde{X}$ and construct the local coordinates for the hyperbolic three-string vertex. First, we describe the relevant monodromy problem and solve the Fuchsian equation on $\widetilde{X}$ accordingly. Then we find the explicit form of the (singular) hyperbolic metric on $\widetilde{X}$ by the relations given in section~\ref{sec:Fuchsian}. The resulting geometry looks like three semi-infinite series of hyperbolic cylinders, attached where they flare up, connected to each other while keeping the curvature constant and negative.
Next, we cut these hyperbolic cylinders out from the geometry appropriately, which leave us with a three-holed sphere $X$. This procedure doesn't change the hyperbolic metric, so at the end we obtain an explicit description of the hyperbolic metric with geodesic boundaries on a three-holed sphere. Moreover, we describe the holes on the Riemann sphere explicitly by investigating the simple closed geodesics of this hyperbolic metric. After that, grafting flat semi-infinite cylinders needed for the construction of the local coordinates amounts to simple conformal transformations of the punctured unit disks to these holes.
Most of the results from this section (except for subsection \ref{sec:Local}) are from~\cite{hadasz2003polyakov,hadasz2004classical}, for which we provide a detailed summary. However, we elaborate the geometric picture coming from the hyperbolic metric in more detail and prove some important results necessary for the explicit construction of the local coordinates.
\subsection{Description of the monodromy problem}
Consider the three-punctured sphere $\widetilde{X} = \mathbb{C} \setminus \{0,1,\infty\}$ and suppose that the solutions of the Fuchsian equation (\ref{eq:Fuchsian}) have hyperbolic SL(2,$\mathbb{R}$) monodromy around each puncture. That is, as we go around a puncture by $(z-z_j) \to e^{2 \pi i }(z-z_j)$, we demand that the solutions for the Fuchsian equation $\psi^{\pm}(z)$ change as,
\begin{equation} \label{eq:SL2R}
\begin{bmatrix}
\psi^{+} \\ \psi^{-}
\end{bmatrix} \; \to \;
M^j
\begin{bmatrix}
\psi^{+} \\ \psi^{-}
\end{bmatrix} \quad \text{where} \quad M^j \in \text{SL(2,}\mathbb{R}), \quad |\text{Tr} M^j|>2.
\end{equation}
Note that the condition on the trace makes the matrix $M^j$ a hyperbolic element of $\text{SL(2,}\mathbb{R})$ and that's why we say we have a hyperbolic monodromies around the puncture $z=z_j$. Realizing this structure for the solutions to the Fuchsian equation and finding them is our \emph{monodromy problem}. This problem is first considered in~\cite{hadasz2004classical} in the context of Liouville theory. We will call a puncture \emph{hyperbolic singularity} if the solutions of the Fuchsian equation have a hyperbolic SL(2,$\mathbb{R}$) monodromy around it.
In order to solve the monodromy problem, we need to first determine appropriate $T_{\varphi}(z)$ as a function of $z$ (if exists) so that the solutions of the Fuchsian equation can realize these monodromies around the punctures. Then declaring that particular $T_{\varphi}(z)$ to be equal to (\ref{eq:stress-energy}) coming from Liouville theory and using the reasoning in section \ref{sec:Fuchsian} we can extract the possibly singular hyperbolic conformal factor on $\widetilde{X}$ with the solutions that realize these monodromies. As explained above the equation ~\eqref{eq:FtoL}, this metric is going to be single-valued by $\text{SL(2,}\mathbb{R})$ monodromies and it will eventually lead us to the hyperbolic metric with geodesic boundaries on a three-holed sphere $X$.
Before we do that, let us investigate an individual hyperbolic singularity. We begin by picking a puncture, say $z=0$, and choosing a normalized basis of solution $\psi_{1}^{\pm}(z)$ for which the monodromy around $z=0$ is diagonal as follows:
\begin{equation} \label{eq:DiagMon}
\begin{bmatrix}
\psi_1^{+} \\ \psi_1^{-}
\end{bmatrix} \; \to \;
\begin{bmatrix}
-e^{-\pi \lambda_1} & 0 \\ 0 & -e^{\pi \lambda_1}
\end{bmatrix}
\begin{bmatrix}
\psi_1^{+} \\ \psi_1^{-}
\end{bmatrix}
\iff
\psi_{1}^{\pm}(z) = \frac{e^{\pm \frac{i v_1}{2}}}{\sqrt{i \lambda_1}} z^{\frac{1\pm i \lambda_1}{2}} (1+ \mathcal{O}(z)).
\end{equation}
The solutions always can be put into this form around $z=0$ since hyperbolic elements of SL(2,$\mathbb{R}$) can be diagonalized by conjugation, which amounts to performing a SL(2,$\mathbb{R}$) change of basis of the solutions. Here $\lambda_1 \in \mathbb{R}$ will be called the \emph{geodesic radius} associated with the $z=0$ puncture and the reason for its name will be apparent shortly. Without loss of generality we will take $\lambda_1>0$. Note that the Wronskian of these solutions is equal to $1$ thanks to the factor $1/\sqrt{i \lambda_1}$ in front. Furthermore, we also included the factors $e^{\pm \frac{i v_1}{2}}$, with $v_1 \in \mathbb{C}$, to account for the multiplicative constant that is not fixed by the Wronskian condition~\eqref{eq:Wronskian}. As we shall see, the constant $v_1$ will be fixed below by demanding SL(2,$\mathbb{R}$) monodromies around each puncture.
Using~\eqref{eq:DiagMon}, we can write the scaled ratio associated with the puncture $z=0$, as in~\eqref{eq:HolChoice},
\begin{equation} \label{eq:ScaledRatio}
\rho_1(z) \equiv \left( \frac{\psi_1^+(z)}{\psi_1^-(z)} \right)^{\frac{1}{i \lambda_1}} = e^{\frac{v_1}{\lambda_1}}(z + \mathcal{O}(z^2)).
\end{equation}
Note that this series expansion converges only on the open unit disk $D_1 = \{z \in \mathbb{C} \; | \; |z|<1 \}$, around the puncture $z=0$, since outside $D_1$ the scaled ratio $\rho_1(z)$ is multi-valued by the solutions $\psi_{1}^{\pm}(z)$ having a non-diagonal monodromy around the punctures at $z=1, \infty$. We can analytically continue the scaled ratio defined above outside the disk $D_1$, but inevitably this will require us to choose a branch for which $\rho_1(z)$ is continuous across $\partial D_1$ except at the punctures/branch cuts. We will choose the branch cut $\widetilde{L}_1$ of $\rho_1(z)$ to extend from 1 to $\infty$ along the real axis and take this to be the principal branch of $\rho_1(z)$. Thus, we conclude that the scaled ratio $\rho_1(z)$ can be defined analytically on the set
\begin{equation} \label{eq:s1}
S_1 = \mathbb{C} \setminus \widetilde{L}_1,
\end{equation}
with the expansion (\ref{eq:ScaledRatio}). When we mention the scaled ratio, we will consider the principal branch implicitly henceforth, unless otherwise stated. Lastly, note that the scaled ratio is an analytic scalar under conformal transformations, just like the ratio $A(z)$ in~\eqref{eq:ratio}.
Now by performing the conformal transformation $z \to \rho_1 = \rho_1(z)$ on $S_1$ and using the equation (\ref{eq:TasSch}) along with the properties of the Schwarzian derivative we see
\begin{equation} \label{eq:Tsect3}
T_{\varphi}(z) =\{ A(z), z\}= \left\{ \frac{\psi_1^+(z)}{\psi_1^-(z)}, z\right\} =\{\rho_1(z)^{i \lambda}, z \} = (\partial \rho_1)^2 \{ \rho_1^{i \lambda_1}, \rho_1\}+ \{\rho_1,z \}.
\end{equation}
Comparing the final form with~\eqref{eq:transofT} we read that the stress-energy tensor $\widetilde{T_{\varphi}}(\rho_1)$ in the $\rho_1$-plane takes the following form:
\begin{equation} \label{eq:closetopuncture}
\widetilde{T_{\varphi}}(\rho_1) = \{ \rho_1^{i \lambda_1}, \rho_1\} = \frac{\Delta_1}{\rho_1^2} \quad \text{where} \quad \Delta_1 \equiv \frac{1}{2} + \frac{\lambda_1^2}{2}.
\end{equation}
Here the real number $\Delta_1$ will be called the \emph{weight}. As a result of this, the Fuchsian equation in the $\rho_1$-plane takes a very simple form and we can easily obtain its solutions:
\begin{equation}
\frac{\partial^2 \tilde{\psi}(\rho_1)}{\partial \rho_1^2} + \frac{\Delta_1}{2 \rho_1^2} \tilde{\psi}(\rho_1) =0
\implies \widetilde{\psi}_1^{\pm} (\rho_1)= \frac{\rho_1^{\frac{1 \pm i \lambda_1}{2}}}{\sqrt{i \lambda_1}}.
\end{equation}
Here, $\widetilde{\psi}_1^{\pm}(z)$ are normalized solutions that are chosen to have diagonal monodromy around the puncture $z=0$, or equivalently $\rho_1 = 0$. Here we set the phase factor not fixed by Wronskian equal to one for convenience.\footnote{Considering this factor just adds a phase shift for the sine that appears in~\eqref{eq:met}, which would be unimportant for our considerations in this subsection.} Note that the scaled ratio of these two solutions is simply
\begin{equation}
\left( \frac{ \tilde{\psi}^{+}_1(z)}{ \tilde{\psi}^{-}_1(z)} \right)^{\frac{1}{i\lambda_1}} = \rho_1.
\end{equation}
As a result, the hyperbolic metric that the Fuchsian equation produces in the $\rho_1$-and $z$-plane are simply given by, using the relation (\ref{eq:conformalfactor}) with the choice $f(\rho_1)=\rho_1$,
\begin{equation} \label{eq:met}
ds^2 = \frac{\lambda_{1}^2 }{|\rho_1|^2 \sin^2(\lambda_1 \log|\rho_1|)} |d\rho_1|^2 = \frac{\lambda_{1}^2 |\partial \rho_1 (z)|^2}{|\rho_1(z)|^2 \sin^2(\lambda_1 \log|\rho_1(z)|)} |dz|^2.
\end{equation}
There are two important things we should notice here. First, the metric takes the form of a series of hyperbolic cylinders that are attached to each other where they flare up in the $\rho_1$-plane, and by the expansion (\ref{eq:ScaledRatio}), when we are sufficiently close to $z=0$ in the $z$-plane. We will explain this fact, along with the closed geodesics/singularities of this metric in more detail after we obtain the explicit form for $\rho_1(z)$.
Secondly, the $z$-plane metric is smooth (except for the singularities) not only over $S_1$ but across the branch cut $\widetilde{L}_1$ as well. The reason is simply that we demanded SL(2,$\mathbb{R}$) monodromy around each puncture and we know that the metric above is invariant under the monodromies of that kind by the equivalent form in (\ref{eq:firstmetric}). So we can use the metric above in the entirety of the $z$-plane minus punctures as long as we guarantee the SL(2,$\mathbb{R}$) monodromies around all punctures simultaneously.
Since we are also demanding hyperbolic SL(2,$\mathbb{R}$) monodromies for the remaining punctures, two facts above hold for them without too much modification. We just have to change $\rho_1$ with appropriate $\rho_j$. Moreover, these produce the same hyperbolic metric when we pullback them to the $z$-plane from any $\rho_j$-plane. This can be easily seen by noticing the fact that the appropriate SL(2,$\mathbb{R}$) change of basis of solutions $\psi_1^{\pm}(z)$ can diagonalize the monodromy around another puncture, by the fact that hyperbolic elements in SL(2,$\mathbb{R}$) are conjugate to a diagonal matrix. Such transformations of the solutions don't affect the metric as we argued before.
In conclusion, we see the motivation behind using the Fuchsian equation with correct monodromy structure in more detail from these comments. Even though any choice of holomorphic function works in (\ref{eq:conformalfactor}) to define a hyperbolic metric, using the scaled ratio coming from the Fuchsian equation with the monodromy data above will guarantee to generate the hyperbolic metric (\ref{eq:met}) on the $z$-plane where three series of attached hyperbolic cylinders connected to each other with hyperbolic pair of pants (i.e. three-holed sphere endowed with a hyperbolic metric), as shown in figure \ref{fig:CylinderSketch}. Moreover, it is easy to see from figure \ref{fig:CylinderSketch} that one can obtain a description of the hyperbolic pair of pants by taking out the hyperbolic cylinders and considering the remaining connected region only. This justifies why we considered this particular monodromy problem of Fuchsian equation on $\widetilde{X}$: It is a natural starting point to generate the hyperbolic metric with geodesic boundaries on $X$.
\begin{figure}[!t]
\centering
\fd{6cm}{fig1a.pdf}
\fd{9cm}{fig1b.pdf}
\caption{Sketch of the hyperbolic metric described by the Fuchsian equation with three hyperbolic singularities. The smooth hyperbolic pair of pants with geodesic boundaries is going to connect the hyperbolic cylinders, as we shall see more explicitly.}
\label{fig:CylinderSketch}
\end{figure}
\subsection{Solution to the monodromy problem}
Before we describe the hyperbolic pair of pants, we are going to get an explicit expression for the hyperbolic metric on the three-punctured sphere $\widetilde{X}$ resulting from three hyperbolic singularities. First, we solve the monodromy problem. That is, we find $T_{\varphi}(z)$ for which the solutions of the Fuchsian equation can realize the monodromy structure described in~\eqref{eq:SL2R}. Then we solve the resulting Fuchsian equation with these prescribed monodromies and proceed to construct the metric by finding the scaled ratio.
In order to find $T_{\varphi}(z)$ as a function of $z$, observe that when we are close to the puncture $z=0$, i.e. $\rho_1 = 0$, the stress-energy tensor in~\eqref{eq:Tsect3} takes the form
\begin{equation}
T_{\varphi}(z) = (\partial \rho_1)^2 \frac{\Delta_1}{\rho_1^2} + \{\rho_1,z \} = (e^{\frac{v_1}{\lambda_1}} + \dots)^2 \frac{\Delta_1}{(e^{\frac{v_1}{\lambda_1}} z + \dots)^2} + \{e^{\frac{v_1}{\lambda_1}} z + \dots,z \}
= \frac{\Delta_1}{z^2} + \mathcal{O}(\frac{1}{z}),
\end{equation}
using (\ref{eq:ScaledRatio}) and (\ref{eq:closetopuncture}). From this, we see that $T_{\varphi}(z)$ must have at most double poles of residues $\Delta_1,\Delta_2$ and $\Delta_3$ at $z=0,1,\infty,$ respectively in order to have a hyperbolic singularity. One can easily show that the unique $T_{\varphi}(z)$ that has such structure is
\begin{equation} \label{eq:3T}
T_{\varphi}(z) = \frac{\Delta_1}{z^2} + \frac{\Delta_2}{(z-1)^2} + \frac{\Delta_3 - \Delta_1 - \Delta_2}{z(z-1)},
\end{equation}
with $\Delta_i = (1+\lambda_i^2)/2$. Clearly we have at most double poles at $z=0,1$ with appropriate residues. Using the inversion map $z \to \tilde{z} =1/z$, along with $\{\tilde{z},z\}=0$, we can easily see that we have the correct structure at infinity, i.e. $\tilde{z} = 0$, as well:
\begin{equation}
\widetilde{T_{\varphi}}(\tilde{z}) = \frac{1}{\tilde{z}^4} \left[\Delta_1 \tilde{z}^2 + \Delta_2\tilde{z}^2 + (\Delta_3 - \Delta_1 - \Delta_2)\tilde{z}^2 + \mathcal{O}(\tilde{z}^3) \right]
= \frac{\Delta_3}{\tilde{z}^2} +\mathcal{O}(\frac{1}{\tilde{z}}).
\end{equation}
The stress-energy tensor $T_{\varphi}(z)$ in (\ref{eq:3T}) solves the monodromy problem. In order to see that, first observe the Fuchsian equation in this case takes the form
\begin{equation} \label{eq:Fuchplugged}
\partial^2 \psi(z) + \frac{1}{2} \left[\frac{\Delta_1}{z^2} + \frac{\Delta_2}{(z-1)^2} + \frac{\Delta_3 - \Delta_1 - \Delta_2}{z(z-1)}\right] \psi (z) = 0.
\end{equation}
This is the hypergeometric equation, written in the so-called $Q$-form. The solutions of this equation and their properties are well tabulated (see Schwarz's function in \cite{hypergeometric}, also \cite{hadasz2004classical,Bilal:1987cq}). They are, with proper normalization and assignment of diagonal monodromy around $z=0$,
\begin{subequations}
\begin{align}
\psi_1^{\pm}(z) &= \frac{e^{\pm \frac{i v(\lambda_1, \lambda_2,\lambda_3)}{2}}}{\sqrt{i \lambda_1}} z^{\frac{1 \pm i \lambda_1}{2}} (1-z)^{\frac{1 \mp i \lambda_2}{2}} \nonumber \\ &\quad \times {_2}F_1 \left(\frac{1 \pm i \lambda_1 \mp i \lambda_2 \pm i \lambda_3}{2},\frac{1 \pm i \lambda_1 \mp i \lambda_2 \mp i \lambda_3}{2}; 1 \pm i \lambda_1; z\right).
\end{align}
Here ${_2}F_1(a,b;c;z)$ is the ordinary hypergeometric function~\eqref{eq:hypergeometric}. Using the transformation properties of these solutions (\ref{eq:transofpsi}) and appropriately exchanging punctures, we can also find the normalized solutions having a diagonal monodromy around $z=1$ and $z=\infty$. They are, respectively,
\begin{align}
\psi_2^{\pm}(z) &= i\frac{ e^{\pm \frac{i v(\lambda_2, \lambda_1,\lambda_3)}{2}}}{\sqrt{i \lambda_2}} (1-z)^{\frac{1 \pm i \lambda_2}{2}} z^{\frac{1 \mp i \lambda_1}{2}} \nonumber \\ &\quad\times {_2}F_1 \left(\frac{1 \pm i \lambda_2 \mp i \lambda_1 \pm i \lambda_3}{2},\frac{1 \pm i \lambda_2 \mp i \lambda_1 \mp i \lambda_3}{2}; 1 \pm i \lambda_2; 1-z\right), \\
\psi_3^{\pm}(z) &= (iz)\frac{e^{\pm \frac{i v(\lambda_3, \lambda_2,\lambda_1)}{2}}}{\sqrt{i \lambda_3}} \left(\frac{1}{z}\right)^{\frac{1 \pm i \lambda_3}{2}} \left(1-\frac{1}{z}\right)^{\frac{1 \mp i \lambda_2}{2}} \nonumber\\ &\quad \times {_2}F_1 \left(\frac{1 \pm i \lambda_3 \mp i \lambda_2 \pm i \lambda_1}{2},\frac{1 \pm i \lambda_3 \mp i \lambda_2 \mp i \lambda_1}{2}; 1 \pm i \lambda_3; \frac{1}{z}\right).
\end{align}
\end{subequations}
We should emphasize again that the constant $v(\lambda_1,\lambda_2, \lambda_3) = v_1$ above is not fixed by the Wronskian and we will determine it below by demanding hyperbolic SL(2,$\mathbb{R}$) monodromies around all punctures. We will call this \emph{compatibility} of monodromies. Notice that compatibility is not guaranteed a priori. This is because when we demand a SL(2,$\mathbb{R}$) monodromy around a puncture, the monodromies around remaining punctures are elements of SL(2,$\mathbb{C}$), rather than SL(2,$\mathbb{R}$), in general.\footnote{It can still have unit determinant without loss of generality if one assumes appropriately normalized solutions in the sense of~\eqref{eq:Wronskian}.} So, actually, in order to solve the monodromy problem completely, we must show that the compatibility is achievable for the Fuchsian equation~\eqref{eq:Fuchplugged}.
In order to ensure compatibility, first observe that we have some SL(2,$\mathbb{C}$) monodromy around $z=1$ if we use the basis $\psi_1^{\pm}(z)$. That is, as $(1-z) \to e^{2 \pi i}(1-z)$, we have
\begin{equation}
\begin{bmatrix}
\psi_1^{+} \\ \psi_1^{-}
\end{bmatrix} \; \to \;
M_1^2
\begin{bmatrix}
\psi_1^{+} \\ \psi_1^{-}
\end{bmatrix} \quad \text{where} \quad M_1^2 \in SL(2,\mathbb{C}).
\end{equation}
Here, and throughout, we are going to denote the monodromy of the solutions $\psi_i^{\pm}(z)$ around the puncture $z=z_j$ as $M_i^j$. In order to have hyperbolic $\text{SL(2,}\mathbb{R})$ monodromy around $z=1$ while simultaneously having hyperbolic $\text{SL(2,}\mathbb{R})$ monodromy around $z=0$, we have to make sure that $M_1^2 \in \text{SL(2,}\mathbb{R})$ and $|\text{Tr} M_1^2|>2$ by adjusting $v_1$ appropriately. To that end, first observe that we have a diagonal hyperbolic $\text{SL(2,}\mathbb{R})$ monodromy around $z=1$ if we use the basis $\psi_2^{\pm}(z)$:
\begin{equation}
\begin{bmatrix}
\psi_2^{+} \\ \psi_2^{-}
\end{bmatrix} \; \to \;
M_2^2
\begin{bmatrix}
\psi_2^{+} \\ \psi_2^{-}
\end{bmatrix}
=
\begin{bmatrix}
-e^{-\pi \lambda_2} & 0 \\ 0 & -e^{\pi \lambda_2}
\end{bmatrix}
\begin{bmatrix}
\psi_2^{+} \\ \psi_2^{-}
\end{bmatrix}.
\end{equation}
Secondly, notice that two basis $\psi_1^{\pm}(z)$ and $\psi_2^{\pm}(z)$ are related via the connection formulas for the hypergeometric function~(see section 2.9 in~\cite{hypergeometric}, also~\cite{hadasz2004classical,Bilal:1987cq})
\begin{equation} \label{eq:connection}
\begin{bmatrix}
\psi_1^{+} \\ \psi_1^{-}
\end{bmatrix}
=
S
\begin{bmatrix}
\psi_2^{+} \\ \psi_2^{-}
\end{bmatrix}
=
\sqrt{\lambda_1 \lambda_2} \begin{bmatrix}
e^{i \frac{v_1-v_2}{2}} \; g_{-}& e^{i \frac{v_1+v_2}{2}} \; g_{+} \\ -e^{-i \frac{v_1+v_2}{2}} \; \overline{g_{+}}& -e^{-i \frac{v_1-v_2}{2}} \; \overline{g_{-}}
\end{bmatrix}
\begin{bmatrix}
\psi_2^{+} \\ \psi_2^{-}
\end{bmatrix},
\end{equation}
here $v_2 = v(\lambda_2, \lambda_1,\lambda_3)$ and the functions $g_{\pm}$ are given by
\begin{equation} \label{eq:gpm}
g_{\pm}=\frac{\Gamma\left(i \lambda_{1}\right) \Gamma\left(\pm i \lambda_{2}\right)}{\Gamma\left(\frac{1+i \lambda_{1} \pm i \lambda_{2}+i \lambda_{3}}{2}\right) \Gamma\left(\frac{1+i \lambda_{1} \pm i \lambda_{2}-i \lambda_{3}}{2}\right)}.
\end{equation}
Using them, we observe the monodromies in two basis are related by the following conjugation:
\begin{equation} \label{eq:conj}
\begin{aligned}
M_1^2 =& S M_2^2 S^{-1} =\lambda_{1} \lambda_{2} \left[\begin{array}{cc}
\mathrm{e}^{-\pi \lambda_{2}}\left|g_{-}\right|^{2}-\mathrm{e}^{\pi \lambda_{2}}\left|g_{+}\right|^{2} & -\left(\mathrm{e}^{\pi \lambda_{2}}-\mathrm{e}^{-\pi \lambda_{2}}\right) \mathrm{e}^{i v_{1}} g_{+} g_{-} \\[10pt]
\left(\mathrm{e}^{\pi \lambda_{2}}-\mathrm{e}^{-\pi \lambda_{2}}\right) \mathrm{e}^{-i v_{1}} \overline{g_{+} g_{-}} & \mathrm{e}^{\pi \lambda_{2}}\left|g_{-}\right|^{2}-\mathrm{e}^{-\pi \lambda_{2}}\left|g_{+}\right|^{2}
\end{array}\right].
\end{aligned}
\end{equation}
Here we used the fact
\begin{equation} \label{eq:identity}
|g_+|^2 - |g_-|^2 = \frac{1}{\lambda_1 \lambda_2},
\end{equation}
which can be derived from the expression~\eqref{eq:gpm}.
Now it is a simple calculation using~\eqref{eq:conj} and~\eqref{eq:identity} to check that $\det M_1^2 =1$. Therefore in order to have $M_1^2\in$ SL(2$, \mathbb{R}$) it is enough to make sure the entries of $M_1^2$ are real. That means we have
\begin{align} \label{eq:v}
\mathrm{e}^{2 i v\left(\lambda_{1}, \lambda_{2}, \lambda_{3}\right)} =\frac{\overline{g_{+} g_{-}}}{g_{+} g_{-}} = \frac{\Gamma\left(-i \lambda_{1}\right)^{2}}{\Gamma \left(i \lambda_{1}\right)^{2}} \frac{\gamma\left(\frac{1+i \lambda_{1}+i \lambda_{2}+i \lambda_{3}}{2}\right) \gamma\left(\frac{1+i \lambda_{1}-i \lambda_{2}+i \lambda_{3}}{2}\right)}{\gamma\left(\frac{1-i \lambda_{1}-i \lambda_{2}+i \lambda_{3}}{2}\right) \gamma\left(\frac{1-i \lambda_{1}+i \lambda_{2}+i \lambda_{3}}{2}\right)} ,
\end{align}
with the function $\gamma(x)$ defined as
\begin{equation}
\gamma(x) \equiv \frac{\Gamma(x)}{\Gamma(1-x)}.
\end{equation}
The equality~\eqref{eq:v} fixes the exponent $v(\lambda_1,\lambda_2, \lambda_{3})=v_1$, but in a rather complicated way, and shows that it is real. Moreover, we can also easily observe Tr$M_1^2 = -2 \cosh(\pi \lambda_2)$ using~\eqref{eq:conj} and~\eqref{eq:identity}, which unsurprisingly shows the monodromy is still hyperbolic. Thus, we conclude that we can have hyperbolic SL(2$, \mathbb{R}$) monodromy around $z=1$ while having a hyperbolic SL(2$, \mathbb{R}$) monodromy around $z=0$.
Note that that guaranteeing a hyperbolic SL(2$, \mathbb{R}$) monodromies around $z=0,1$ simultaneously with the correct choice of $v(\lambda_1,\lambda_2, \lambda_{3})$ would be sufficient for guaranteeing a hyperbolic SL(2$, \mathbb{R}$) monodromy around $z=\infty$ as well, which is the only remaining point where we have a nontrivial monodromy around. This is because we can imagine a contour that surrounds both $z=0$ and $z=1$ whose associated monodromy would be a product of two hyperbolic SL(2$, \mathbb{R}$) matrices, which is another SL(2$, \mathbb{R}$) matrix. Furthermore, this monodromy would be clearly hyperbolic by construction. As a result, the solutions would have the desired monodromy structure around $z=\infty$ as well when we think this contour to surround $z=\infty$ instead. So we conclude that the solutions of the Fuchsian equation (\ref{eq:Fuchplugged}), with the right choice of $v(\lambda_1,\lambda_2, \lambda_{3})$, can realize hyperbolic SL(2$, \mathbb{R}$) monodromies around each puncture and they are compatible. We solved the monodromy problem.
Finally, we can list the scaled ratios $\rho_i = (\psi_i^+(z)/\psi_i^-(z))^{1/i \lambda_i}$ associated with each puncture. They are:
\begin{align}
\rho_1(z) &= e^{\frac{v(\lambda_1,\lambda_2, \lambda_3)}{\lambda_1}} z (1-z)^{-\frac{\lambda_2}{\lambda_1}}
\left[ \frac{_2F_1 \left(\frac{1+i \lambda_1 -i \lambda_2 +i \lambda_3}{2},\frac{1+i \lambda_1 -i \lambda_2 -i \lambda_3}{2}; 1+ i \lambda_1; z\right)}{_2F_1 \left(\frac{1-i \lambda_1 +i \lambda_2 -i \lambda_3}{2},\frac{1-i \lambda_1 +i \lambda_2 +i \lambda_3}{2}; 1- i \lambda_1; z\right)} \right]^{\frac{1}{i \lambda_1}}, \nonumber\\
\rho_2(z) &= e^{\frac{v(\lambda_2,\lambda_1, \lambda_3)}{\lambda_2}} (1-z) z^{-\frac{\lambda_1}{\lambda_2}} ,
\left[ \frac{_2F_1 \left(\frac{1+i \lambda_2 -i \lambda_1 +i \lambda_3}{2},\frac{1+i \lambda_2 -i \lambda_1 -i \lambda_3}{2}; 1+ i \lambda_2; 1-z\right)}{_2F_1 \left(\frac{1-i \lambda_2 +i \lambda_1 -i \lambda_3}{2},\frac{1-i \lambda_2 +i \lambda_1 +i \lambda_3}{2}; 1- i \lambda_2; 1-z\right)} \right]^{\frac{1}{i \lambda_2}},\nonumber \\
\rho_3(z) &= e^{\frac{v(\lambda_3,\lambda_2, \lambda_1)}{\lambda_3}} \left(\frac{1}{z}\right) \left(1-\frac{1}{z}\right)^{-\frac{\lambda_2}{\lambda_3}}
\left[ \frac{_2F_1 \left(\frac{1+i \lambda_3 -i \lambda_2 +i \lambda_1}{2},\frac{1+i \lambda_3 -i \lambda_2 -i \lambda_1}{2}; 1+ i \lambda_3; \frac{1}{z}\right)}{_2F_1 \left(\frac{1-i \lambda_3 +i \lambda_2 -i \lambda_1}{2},\frac{1-i \lambda_3 +i \lambda_2 +i \lambda_1}{2}; 1- i \lambda_3; \frac{1}{z}\right)} \right]^{\frac{1}{i \lambda_3}}. \label{eq:scaledratio}
\end{align}
From above it is clear that $\rho_2(z)$ and $\rho_3(z)$ can be obtained from $\rho_1(z)$ by exchanging punctures, as well as their associated $\lambda_j$'s, $(1) \leftrightarrow (2)$ and $(1) \leftrightarrow (3)$ respectively while keeping the remaining puncture fixed. Moreover, one can also show that the scaled ratio associated with the fixed puncture remains invariant (up to a sign) under this exchange, either by reasoning through our construction above or by checking it directly using the identities for hypergeometric functions~\cite{hypergeometric}. In any case, we see that the set of three scaled ratios given above would be invariant (up to a sign) under the permutation group $S_3$ acting on the positions and the parameters of the punctures. This fact will eventually lead us to a similar symmetry for the local coordinates of the hyperbolic three-string vertex.
As we already argued in the previous subsection, these scaled ratios will define the following single-valued, singular, hyperbolic metric on the whole three-punctured sphere (\ref{eq:met}):
\begin{equation} \label{eq:yetanothermetric}
ds^2 = \frac{\lambda_{j}^2 |\partial \rho_j (z)|^2}{|\rho_j(z)|^2 \sin^2(\lambda_j \log|\rho_j(z)|)} |dz|^2
= \frac{\lambda_{j}^2 }{|\rho_j|^2 \sin^2(\lambda_j \log|\rho_j|)} |d\rho_j|^2,
\end{equation}
for which we have three semi-infinite series of attached hyperbolic cylinders connected to each other. Again, each $j=1,2,3$ defines the same metric.
\subsection{The resulting geometry on the three-punctured sphere}
Before we construct the local coordinates, we should understand the geometry of (\ref{eq:yetanothermetric}) better and show that it looks exactly like in figure \ref{fig:CylinderSketch} as we have claimed. In order to do that, focus on the set $S_1$, which was the complex plane with a cut from $1$ to $\infty$ (see~\eqref{eq:s1}). This will be mapped to the set $\rho_1(S_1)$ in the $\rho_1$-plane.\footnote{It can be shown that this map is invertible, see \cite{hypergeometric}. So this mapping would be bijective.} The rough sketch of these regions, based on numerics, but \emph{not} on scale, is given in figure \ref{fig:rhoSketch} and \ref{fig:zSketch}. We will consider and explain this geometry on the $\rho_1$-plane for now, but geometries on the other $\rho_j$-planes are analogous.
\begin{figure}[!pht]
\centering
\fd{14cm}{fig2.pdf}
\caption{The rough sketch of the geometry on the $\rho_1$-plane. The meaning of the curves are explained in the text. The coloring conventions for the curves will be the same for all figures in this subsection. Dashed curves indicate the line singularities. \vspace{1cm}}
\label{fig:rhoSketch}
\centering
\fd{14cm}{fig3.pdf}
\caption{The corresponding geometry on the $z$-plane after we pullback the metric~(\ref{eq:yetanothermetric}) from the $\rho_1$-plane above. Note that the gray region would be endowed with the hyperbolic metric with geodesic boundaries $\Gamma_i$.}
\label{fig:zSketch}
\end{figure}
As we mentioned previously, the metric on the $\rho_1$-plane (\ref{eq:yetanothermetric}) takes the form of the hyperbolic metric of series of attached hyperbolic cylinders. Indeed, we see that the line singularities (where the metric blow up on a curve) and the simple closed geodesics surrounding the origin $\rho_1=0$ of the hyperbolic metric~\eqref{eq:yetanothermetric} are located at
\begin{equation}
\text{Line singularities:} \; |\rho_1| = e^{\frac{\pi l_1}{\lambda_1}}, \qquad
\text{Simple Closed Geodesics:} \; |\rho_1| = e^{\frac{\pi}{\lambda_1} \left(l_1+\frac{1}{2}\right)},
\end{equation}
where $ l_1 \in \mathbb{Z}$. We can see these by noting that the sine in the denominator of the metric~\eqref{eq:yetanothermetric} is equal to zero in the case of line singularity by $\sin(\pi l_1) = 0$ and one in the case of simple closed geodesics by $\sin\left(\pi l_1+ \pi/2 \right ) = 1$, which makes the metric~\eqref{eq:yetanothermetric} blow up and minimize respectively.
Notice that the line singularities and simple closed geodesics form alternating, exponentially separated circles around the origin on the $\rho_1$-plane, as shown in figure \ref{fig:rhoSketch} with green and purple respectively; except for the geodesic colored with magenta which will turn out to be special. Additionally, it is clear that every simple geodesic surrounding the origin has the length $2 \pi \lambda_1$ by the metric (\ref{eq:yetanothermetric}), which justifies the name \emph{geodesic radii} for $\lambda_j$. Obviously we can pullback these curves to the $z$-plane with a cut from 0 to $\infty$, which will result in closed, simple geodesics/line singularities around the puncture $z=0$ by $\rho_1(0)=0$. These are shown in figure~\ref{fig:zSketch} correspondingly.
Observe that the lines just above/below the branch cut of $\rho_1(z)$, denoted as $\widetilde{L}_1^{\pm}$ and shown in figure \ref{fig:zSketch}, are mapped to the red/blue curves $\rho_1 (\widetilde{L}_1^{\pm})$ in $\rho_1(S_1)$. These curves are shown in figure \ref{fig:rhoSketch}. They are symmetric with respect to the real axis on the $\rho_1$-plane by the choice of the principal branch for the scaled ratio. The set $S_1$ is mapped between $\rho_1(\widetilde{L}_1^+)$ and $\rho_1(\widetilde{L}_1^-)$, which is the shaded region in figure \ref{fig:rhoSketch}. Moreover, if we identify the two curves $\rho_1 (\widetilde{L}_1^{\pm})$, the whole $z$-plane minus the punctures maps to the region between them. But, in any case, we indicated where the punctures $z=1$ and $z= \infty$ are heuristically getting mapped to in figure \ref{fig:rhoSketch}: $z=1$ is mapped to the right-side infinity and $z= \infty$ is mapped to the left-side infinity.
Now observing figure \ref{fig:rhoSketch}, we see that some simple closed geodesics/line singularities don't intersect $\rho_1(\widetilde{L}^{\pm}_1)$. As a result, we see $\exists \, \tilde{l}_1 \in \mathbb{Z}$ such that the geodesic at $|\rho_1| = e^{\frac{\pi}{\lambda_j}\left( \tilde{l}_1+\frac{1}{2}\right)}$ does not intersect $\rho_1(\widetilde{L}^{\pm}_1)$ and surrounds \emph{all} the closed simple geodesics/line singularities that do not intersect $\rho_1(\widetilde{L}^{\pm}_1)$ (i.e. those with $l_1 \leq \tilde{l}_1$). The closed geodesic with $l_1 = \tilde{l}_1$ is shown with magenta instead of purple in figure~\ref{fig:rhoSketch} in order to differentiate it from the others. At this stage nothing prevents us to having a line singularity that surrounds this geodesic and doesn't intersect $\rho_1(\widetilde{L}^{\pm}_1)$, but this turns out not to be the case as we will prove it shortly. We just assume this is the case for now.
We can pullback the geodesic with $l_1=\tilde{l}_1$ described above to the $z$-plane, which we denote it by $\Gamma_1$. Defining the closed geodesics homotopic to the puncture $z=0$ as \emph{separating} geodesics of $z=0$, we see the simple closed geodesic $\Gamma_1$ would be the separating geodesic farthest away from $z=0$ by construction. So we will call $\Gamma_1$ as \emph{the most-distant separating} geodesic of $z=0$. This geodesic is shown in figure \ref{fig:zSketch} with magenta as well.
From this, we see that the simply-connected region $H_1$ on the $z$-plane surrounded by $\Gamma_1$ contains every geodesic/line singularity with $l_1 \leq \tilde{l}_1$. Furthermore, as $l_1 \to - \infty$, the geodesics/line singularities get closer to the puncture. So we conclude that the geometry on $H_1$ looks like a series of semi-infinite hyperbolic cylinders attached at where they flare up, like shown in figure~\ref{fig:CylinderSketch}. The places where they flare up are the line singularities of the metric.
We can repeat the same procedure for the other punctures and obtain their most-distant separating geodesics $\Gamma_j$, associated simply-connected regions $H_j$, and integers $\tilde{l}_j$. Note that $\Gamma_i \cap \Gamma_j = \emptyset$ for $i \neq j$, by $\Gamma_i$'s being simple geodesics of the same metric. Hence, the resulting geometry on the $z$-plane would indeed look like in figure \ref{fig:zSketch}. Again, the most-distant separating geodesics $\Gamma_j$ are shown with different colors. In this figure, we also see there are alternating closed curves around each puncture representing the simple closed geodesics/line singularities surrounding them. These can be related to the geodesics/line singularities that intersect $\rho_1 (\widetilde{L}_1^{\pm})$ on the $\rho_1$-plane (hence their colors), but this wouldn't be necessary for our purposes.
Now let us inspect how the most-distant separating geodesics of the punctures $z=1$ and $z=\infty$, denoted as $\Gamma_2$ and $\Gamma_3$ respectively, look like on $\rho_1(S_1)$. In order to do that, let us call the line singularity with $l_1 = \tilde{l}_1 + 1$ to be the \emph{first line singularity} of $z=0$. Clearly, the first line singularity encloses $\rho_1(\Gamma_1)$ and is enclosed by every other line singularity that encloses $\rho_1(\Gamma_1)$ on the $\rho_1$-plane, hence the name \emph{first}. Moreover, it is clear that the first line singularity intersects with the curves $\rho_1(\widetilde{L}_1^{\pm})$ by definition. We define the \emph{first geodesic} of a puncture in similar fashion.
Now, we will find the shortest geodesic that is enclosed by the first line singularity of $z=0$ and stretches between the curves $\rho_1(\widetilde{L}_1^{\pm})$ for both right/left of the origin $\rho_1 =0$ on $\rho_1(S_1)$, which we will call $\rho_1(\Omega_2)$ and $\rho_1(\Omega_3)$ respectively. Clearly, $\rho_1(\Omega_2)$ and $\rho_1(\Omega_3)$ can be made shorter by eliminating any self intersections, so we will consider the simple geodesics without loss of generality. Moreover, $\rho_1(\Omega_2)$ and $\rho_1(\Omega_3)$ can be made shorter by making them intersect $\rho_1(\widetilde{L}_1^{\pm})$ perpendicularly, which we will also take to be the case.
There might be multiple curves satisfying the definition for $\rho_1(\Omega_2)$ and $\rho_1(\Omega_3)$ above. However, this cannot be the case since their pullbacks on the $z$-plane would correspond to closed simple geodesics without a line singularity between them around the punctures $z=0$ and $z=\infty$, and we know that this can't happen as we saw above. So $\rho_1(\Omega_2)$ and $\rho_1(\Omega_3)$ are unique for the left and right side. This is shown in figure~\ref{fig:rhoSketch}. Additionally, this argument shows that $\Omega_2$ and $\Omega_3$ are the the most-distant separating geodesics for the punctures $z=1$ and $z=\infty$ respectively, i.e. $\Gamma_2 = \Omega_2$ and $\Gamma_3=\Omega_3$, since there are no geodesics that surround them and separate from the other punctures.
Keeping this in mind, we can now demonstrate that the there is no line singularity that surrounds the geodesic $\rho_1(\Gamma_1)$ and doesn't intersect $\rho_1(\widetilde{L}^{\pm}_1)$ on the $\rho_1$-plane, which we only assumed previously. For the sake of contradiction, suppose there is one and call it $\rho_1(\Lambda_1)$, which is shown in figure~\ref{fig:argument}. Then it is clear by above that the geodesic $\rho_1(\Gamma_2)$ around the puncture $z=1$ would be a piece of the first geodesic of $z=0$. Now going to the $\rho_2$-plane after we pullback this geometry to the $z$-plane, we see that $\Lambda_1$ maps to a piece of a line singularity $\rho_2(\Lambda_1)$ on the $\rho_2$-plane stretching between $\rho_2(\widetilde{L}_2^{\pm})$, while $\Gamma_2$ maps to a circle around the origin and doesn't intersect $\rho_2(\widetilde{L}_2^{\pm})$. Similar to the arguments above, we can always find a simple geodesic $\rho_2(\Omega_1)$ between these two, but this leads to a contradiction with the fact that $\Gamma_1$ being the most-distant separating geodesic of $z=0$ since the separating geodesic $\Omega_1$ would be enclosing $\Gamma_1$. Clearly this argument can be repeated for other punctures, so what we have assumed regarding having a line singularity that surrounds the most-distant separating geodesic and doesn't intersect $\rho_1(\widetilde{L}^{\pm}_1)$ was justified.
\begin{figure}[!t]
\centering
\fd{7.5cm}{fig4a.pdf}
\fd{7.5cm}{fig4b.pdf}
\caption{The illustration of the geometry described in the argument above on the $\rho_1$-plane (left) and $\rho_2$-plane (right). Here the line singularity $\Lambda_1$ is shown with dark green, while the geodesic $\Omega_1$ is shown with brown.}
\label{fig:argument}
\centering
\end{figure}
In order to complete our construction, we now need to find the integers $\tilde{l}_j$. For that, first notice the following inequality is satisfied:
\begin{equation} \label{eq:ineq}
|\rho_j(z)| = \exp \left[\frac{\pi}{\lambda_j}\left( \tilde{l}_j+\frac{1}{2}\right)\right] \leq \min_{z \in \widetilde{L}_j} |\rho_j(z)|,
\end{equation}
with $\widetilde{L}_j$ denoting the branch cut of the function $\rho_j(z)$. This inequality is evident since we demanded above that the geodesic $\rho_j(\Gamma_j)$, located at $|\rho_j| = e^{\frac{\pi}{\lambda_j}\left( \tilde{l}_j+\frac{1}{2}\right)}$, is not intersecting the curves $\rho_j(\widetilde{L}^{\pm}_1)$ on the $\rho_j$-plane. Note that $|\rho_j(z)|$ would be single-valued on the branch cut $\widetilde{L}_j$ because of the choice of the principal branch. From~\eqref{eq:ineq} and noting that $\tilde{l}_j$ is the greatest integer that satisfies it by definition, we can write a prescription for $\tilde{l}_j$ as follows:
\begin{align} \label{eq:ltilde}
\tilde{l}_j = \left\lfloor \frac{\lambda_j}{\pi} \log \min_{z \in \widetilde{L}_j} |\rho_j(z)| - \frac{1}{2}\right\rfloor.
\end{align}
Here $\lfloor \cdot \rfloor: \mathbb{R} \to \mathbb{Z}$ denotes the floor function. We couldn't be able to find an explicit expression for this in terms of $\lambda_j$'s. However, determining the \emph{exact} values of the integers $\tilde{l}_j$ numerically for given $\lambda_j$ is trivial by the expression above and using the scaled ratios~\eqref{eq:scaledratio}.
Although it is hard to find an expression for $\tilde{l}_j$ in terms of arbitrary $\lambda_j$'s, we can still make some progress for the case where two of the $\lambda_j$'s are equal by exploiting the permutation symmetry. In order to do that, suppose we want to find $\tilde{l}_1$ in the case of $\lambda_2=\lambda_3 = \lambda$. Now recall that three scaled ratios~\eqref{eq:scaledratio} are invariant under the permutations of the punctures and their associated geodesic radii up to a sign. Specifically, in the case where we exchange the punctures at $z=1,\infty$ while keeping $z=0$ fixed, which is implemented by the conformal transformation $z \to \frac{z}{z-1}$, we get the following relation for $\rho_1(z)$ on $S_1$
\begin{equation}
\rho_1(z) = -\rho_1 \left(\frac{z}{z-1}\right).
\end{equation}
Note that it was essential to take $\lambda_2 = \lambda_3$ to establish this relation.
Clearly, $z=2$ is the fixed point of the transformation $z \to \frac{z}{z-1}$. One consequence of this is that $|\rho_1(z)|$ when restricted to the branch cut $\widetilde{L}_1$ is symmetric around $z=2$ when we apply the transformation $z \to \frac{z}{z-1}$. Then using this fact and analyticity of $\rho_1(z)$, it can be shown that the point $z=2$ would be where $|\rho_1(z)|$ attains its global minimum on the branch cut.
\begin{figure}[!t]
\centering
\fd{12cm}{fig5-eps-converted-to.pdf}
\caption{The plot of $\mathcal{L}(\lambda,\lambda,\lambda)$ as a function of $\lambda$. Note that the floor of this function gives $-1$ in the range shown. An analytic proof for $\tilde{l}_1 = -1$ for any $\lambda>0$ would require better understanding of $\mathcal{L}(\lambda,\lambda, \lambda) $, especially for the large values of $\lambda$.}
\label{fig:Length}
\end{figure}
So we see that permutation symmetry of the situation $\lambda_2=\lambda_3 = \lambda$ allow us to find the global minimum of $|\rho_1(z)|$ on its branch cut $\widetilde{L}_1$, which is at $z=2$. Now define the following function and notice
\begin{equation}
\mathcal{L}(\lambda_1,\lambda, \lambda) \equiv \frac{\lambda_1}{\pi} \log |\rho_1(2)| - \frac{1}{2} \implies \tilde{l}_1 = \lfloor \mathcal{L}(\lambda_1,\lambda, \lambda) \rfloor,
\end{equation}
using (\ref{eq:ltilde}). This expression is certainly more manageable then what has been given in~(\ref{eq:ltilde}). Obviously, we can get similar expressions for the other punctures when the remaining punctures has equal $\lambda_j$'s. As an example for what we have discussed so far, we plotted $\mathcal{L}(\lambda,\lambda, \lambda)$ in figure~\ref{fig:Length}. This suggests $\tilde{l}_1=-1$, and by symmetry $\tilde{l}_2 = \tilde{l}_3=-1$, for $0 < \lambda < 10$.
In summary, we see the geometry of the metric~(\ref{eq:yetanothermetric}) is indeed given by figure \ref{fig:CylinderSketch}. Remember the metric (\ref{eq:yetanothermetric}) was on the three-punctured sphere $\widetilde{X}$, but, clearly, we can now obtain the hyperbolic metric with geodesic boundaries on a three-holed sphere by restricting to the region $X=\widehat{\mathbb{C}} \setminus (H_1 \cup H_2 \cup H_3)$, which is shaded gray in figure \ref{fig:zSketch}. On the $\rho_j$-plane this corresponds to the region between the most-distant geodesics with the curves $\rho_j(L^{\pm}_j)$ are identified, which is also shaded gray in figure \ref{fig:rhoSketch}. Note that $X$ is still endowed with $K=-1$ metric (\ref{eq:yetanothermetric}), but now free from singularities, and it is clear by the construction that its boundaries $\Gamma_j$ are geodesics. In other words, we performed a \emph{surgery} where we amputated the hyperbolic cylinders around the hyperbolic singularities and left with the geodesic boundaries instead while keeping everything the same. Four examples of such region on the $z$-plane are shown in figure~\ref{fig:example}. In the next subsection, we are going to graft flat semi-infinite cylinders into the places of amputated hyperbolic cylinders in order to construct the local coordinates explicitly.
\begin{figure}[!t]
\centering
\fd{7.5cm}{lambda028-eps-converted-to.pdf}
\fd{7.5cm}{lambda075-eps-converted-to.pdf}
\fd{7.5cm}{lambda15-eps-converted-to.pdf}
\fd{7.5cm}{lambda2-eps-converted-to.pdf}
\caption{Four examples for the pants diagram region $X$ on the $z$-plane in the case of equal geodesic radii. Punctures are located at $z=0,1,\infty$, and indicated by black dots. We only show the most-distant separating geodesics $\Gamma_j$ because we performed a surgery and take out everything surrounded by them. The region remaining $X$ is endowed with the hyperbolic metric (\ref{eq:yetanothermetric}) and $\Gamma_j$ are its geodesics by construction. Note that the geodesics $\Gamma_1$ and $\Gamma_2$ in the critical case $\lambda =$ arcsinh(1)$/\pi \approx 0.28$ are so small that they haven't rendered in the top-left figure.}
\label{fig:example}
\end{figure}
\subsection{Local coordinates} \label{sec:Local}
In this subsection, we describe how to construct the local coordinates around the punctures for the hyperbolic three-string vertex by attaching flat semi-infinite cylinders of radius $\lambda_j$ at each geodesic boundary component of $X$. First, note that when we perform the surgery described above to obtain the geodesic boundaries, we essentially take out the disk
\begin{equation}
\rho_j(H_j) = \left\{\rho_j \in \mathbb{C} \; | \; |\rho_j| < \exp \left[ \frac{\pi}{\lambda_j} \left(\tilde{l}_j+\frac{1}{2} \right) \right]\right\} ,
\end{equation}
from the $\rho_j$-plane for each $j=1,2,3$. Now imagine we have a punctured unit disk $0 < |w_j| \leq 1$ with the metric
\begin{equation} \label{eq:cylmetric}
ds^2 = \lambda_j^2 \frac{ |dw_j|^2}{|w_j|^2},
\end{equation}
which describes a flat semi-infinite cylinder ($K=0$) of radius $\lambda_j$. We can map this punctured unit disk into the hole $\rho_j(H_j)$ on the $\rho_j$-plane with a simple scaling:
\begin{equation} \label{eq:scale}
\rho_j = \exp \left[ \frac{\pi}{\lambda_j} \left(\tilde{l}_j+\frac{1}{2} \right) \right] w_j = N_j w_j, \quad N_j \equiv \exp \left[ \frac{\pi}{\lambda_j} \left(\tilde{l}_j+\frac{1}{2} \right) \right].
\end{equation}
We will call $N_j$ the \emph{scale factor}. Above we haven't considered the overall rotations of the punctured unit disk, $w_j \to e^{i \theta} w_j$, while we are mapping to $\rho_j(H_j)$, since such global phase factors are not relevant in closed string field theory.
Clearly, the flat metric (\ref{eq:cylmetric}) does not change under this scaling. Furthermore, the flat metric~(\ref{eq:cylmetric}) and the hyperbolic metric (\ref{eq:yetanothermetric}) for the pair of pants as well as their first derivatives match at the circular seams $\rho_j(\Gamma_j)$ of radius $\lambda_j$. As a result, we fill the regions $\rho_j(H_j)$ with flat semi-infinite cylinders and discontinuity first appears in the curvature as we desire. Note that the metric we obtain after grafting these semi-infinite flat cylinders is a Thurston metric on the three-punctured sphere~\cite{Costello:2019fuh}.
Now we can pullback these filled $\rho_j(H_j)$ to the otherwise empty holes $H_j$ on the $z$-plane with the maps $\rho_j(z)$ to construct the local coordinates around the punctures $z=0,1,\infty$ describing three semi-infinite flat cylinders grafted on to the hyperbolic pair of pants on $\widehat{\mathbb{C}}$. Thus, from~\eqref{eq:scale}, we see that the local coordinates $w_j$ around the punctures $w_j=0$ are given by
\begin{equation} \label{eq:LocalCoord}
w_j = \exp \left[- \frac{\pi}{\lambda_j} \left(\tilde{l}_j+\frac{1}{2} \right) \right] \rho_j(z) = N_j^{-1} \rho_j(z),
\end{equation}
with $|w_j| \leq 1$. This yields the local coordinates~\eqref{eq:lc} using~\eqref{eq:scaledratio}. Equivalently, we can write $z = \rho_j^{-1}\left( N_j w_j\right)$ on the coordinate patches $H_j$ with the punctures are located at $z=z_j$. Note that $|w_j|=1$ maps to $\partial H_j = \Gamma_j$ by construction. Obviously, we can get the anti-holomorphic local coordinates $w_j(\bar{z})$ in similar fashion. Moreover, we see that they satisfy $w_j(\bar{z}) = \overline{w_j(z)}$ from~\eqref{eq:lc}, up to possible overall phase ambiguity. This shows all the coefficients in the expansions of $w_j(z)$ in $z$ can be chosen to be real.
As can be seen from~\eqref{eq:lc}, and alluded before, the local coordinates are invariant under permutations of the punctures and their associated $\lambda_j$. Adding the scale factor $N_j$ doesn't spoil this symmetry, since its value is getting permuted as well. Moreover, when we take all $\lambda_j = \lambda$ equal (recall this is the version that appears in the string action), this vertex becomes cyclic in the technical sense~\cite{sonoda1990covariant}. These results are certainly consistent with what is expected form the geometry of the hyperbolic pair of pants with three grafted flat cylinders.
As a final note, the mapping radius $r_j = \left| \frac{dz}{dw_j} \right|_{w_j=0}$ for this local coordinates can be easily read from~\eqref{eq:lc}, and they are
\begin{equation}
w_j = e^{-\frac{\pi (\tilde{l}_j+\frac{1}{2})}{\lambda_j}} e^{\frac{v_j}{\lambda_j}} (z-z_j) + \dots \implies r_j = \exp\left[\frac{\pi (\tilde{l}_j+\frac{1}{2})}{\lambda_j} - \frac{v_j}{\lambda_j}\right] = N_j \exp\left[- \frac{v_j}{\lambda_j}\right].
\end{equation}
Remember both $v_j$ and $N_j$ depends on the circumferences of the grafted cylinders as can be seen from~\eqref{eq:v} and~\eqref{eq:scale}.
\section{Limits of the hyperbolic three-string vertex} \label{sec:Limits}
In this section, we investigate various limits of the local coordinates (\ref{eq:LocalCoord}) to check that they are consistent with the literature~\cite{Moosavian:2017qsp,sonoda1990covariant, Zwiebach:1988qp}. We show that it is possible to produce the minimal area three-string vertex, Kleinian vertex, and the light-cone vertex as different limits of the hyperbolic three-string vertex.
\subsection{Minimal area three-string vertex} \label{sec:Witten}
In order to produce the minimal area three-string vertex from the hyperbolic three-sting vertex, we set the lengths of the boundary components of $X$ the same, $\lambda_1=\lambda_{2}=\lambda_{3} =\lambda$, and take $\lambda \to \infty$. Since the lengths of the boundaries of the pair of pants get larger at the same rate while the area of the pair of pants remains constant by the Gauss-Bonnet Theorem in this limit, the pair of pants shrinks and it becomes like a ribbon graph of vanishing width. As a result, after grafting the flat cylinders and rescaling their circumferences, we get the three-vertex obtained from the minimal area metric~\cite{Costello:2019fuh}. Therefore, we see that this is indeed the correct limit to generate the minimal area three-string vertex and we will call it \emph{minimal area limit}. Note that this limiting behavior is also evident from the examples given in figure \ref{fig:example}. We seem to get the usual representation of the minimal area three-string vertex as $\lambda$ gets larger~\cite{Erler:2019loq}.
In order to consider the minimal area limit explicitly, first notice that we have
\begin{equation}
\lim_{\lambda \to \infty} \exp \left[\frac{v(\lambda,\lambda,\lambda)}{\lambda} \right] = \frac{3\sqrt{3}}{4}.
\end{equation}
This can be obtained from the expression~(\ref{eq:v}) for the function $v(\lambda,\lambda,\lambda)$ and evaluating its limit in Mathematica.
Next, we need to find the limiting value of $N=N_1=N_2=N_3$ in the minimal area limit. Already from figure~\ref{fig:Length} and~\eqref{eq:scale} it can be visually argued that $N \to 1$ as $\lambda \to \infty$, but here we are going to provide an additional heuristic argument why this expectation is correct in case figure~\ref{fig:Length} is misleading in large $\lambda$. To that end, we should first understand the minimal area limit of the hyperbolic metric (\ref{eq:yetanothermetric}). This metric certainly diverges in the minimal area limit, but since we are going to rescale our cylinders at the end, this overall divergence is not a problem. Ignoring this divergence, indicated by $\sim$, the hyperbolic metric, now formally on the ribbon graph of vanishing width, takes the following form in the minimal area limit:
\begin{equation}
ds^2 \sim \frac{ |\partial \rho_i (z)|^2}{|\rho_i(z)|^2 \sin^2(\infty \log|\rho_i(z)|)} |dz|^2.
\end{equation}
Above $\infty$ in the denominator indicates infinite oscillations of the metric as $\lambda \to \infty$ except for when $|\rho_i(z)|=1$. But note that if we have such infinite oscillations, the metric would certainly be ill-defined. The only time it is well-defined is when we have $|\rho_i(z)|=1$, which produces just a divergence and that is acceptable as we mentioned. Thus, we conclude that the shape of the ribbon graph of vanishing width is described by $|\rho_i(z)|=1$ in the minimal area limit, because this is the only time we have a meaningful limit of the geometry. Now note that this ribbon graph at $|\rho_i(z)|=1$ can be thought as the union of $\Gamma_i$, which is described by $|\rho_i(z)|=N$, in the minimal area limit by shrinking hyperbolic pair of pants. This gives
\begin{equation}
\lim_{\lambda \to \infty} N = \lim_{\lambda \to \infty} \exp\left[\frac{\pi}{\lambda} \left(\tilde{l}+\frac{1}{2}\right) \right]= 1.
\end{equation}
So our expectation above was indeed correct.
Using the two limits we argued above, we see that the local coordinate around $z=0$~\eqref{eq:lc1} has the following expansion in the minimal area limit:\footnote{We also checked the similar results hold for other punctures. We omit reporting them to avoid repetition.}
\begin{align} \label{eq:WittenExpansion}
w_1 = \frac{3 \sqrt{3}}{4}z &+ \frac{3 \sqrt{3}}{8}z^2 + \frac{27 \sqrt{3} }{64}z^3 + \frac{57 \sqrt{3} }{128}z^4 + \frac{231 \sqrt{3} }{512}z^5 + \frac{459 \sqrt{3} }{1024} z^6+ \frac{7275 \sqrt{3} }{16384}z^7 \nonumber \\
&+ \frac{14493 \sqrt{3} }{32768} z^8 + \frac{58077 \sqrt{3} }{131072}z^9 + \frac{116565 \sqrt{3} }{262144}z^{10} + \mathcal{O}(z^{11}).
\end{align}
We obtained this expression by expanding (\ref{eq:lc1}) in $z$ first, then taking the minimal area limit. One can easily observe that the local coordinates around $z=0$ of the minimal area three-string vertex, as given in equation (2.19) of~\cite{sonoda1990covariant} with $a=1$,
\begin{equation} \label{eq:minimalvertex}
z_1 = i\frac{\left(1-\frac{i \sqrt{3} z}{z-2}\right)^{3/2}-\left(1+\frac{i \sqrt{3} z}{z-2}\right)^{3/2}}{\left(1-\frac{i \sqrt{3}
z}{z-2}\right)^{3/2}+\left(1+\frac{i \sqrt{3} z}{z-2}\right)^{3/2}},
\end{equation}
also has the same expansion (\ref{eq:WittenExpansion}) after an unimportant phase rotation $z_1 \to -z_1$. So, unsurprisingly, these local coordinates match in the minimal area limit.
Comparison was perturbative in $z$ above, however, we think this limiting behavior holds for all orders in $z$. That is $w_1=-z_1$ exactly in the minimal area limit. The best way to show this would be by finding an appropriate asymptotic formula for the hypergeometric function when $\lambda$ is large to generate the expression (\ref{eq:minimalvertex}), similar to the cases given in~\cite{Watson}. In any case, this perturbative analysis would be sufficient for our purposes. In conclusion, we see that the hyperbolic three-string vertex reduces to the minimal are three-string vertex in the limit $\lambda \to \infty$.
\subsection{Kleinian vertex}
Now we consider the opposite limit for which $\lambda_j=\lambda \to 0$. Clearly, the grafted flat cylinders disappears in this limit\footnote{Since this is the case, this naive limit of the local coordinates~\eqref{eq:LocalCoord} seems actually ill-defined. We are going to comment on this point below.} and instead we are left with a purely hyperbolic metric on the three-punctured sphere. So the local coordinates for the hyperbolic three-string vertex is expected to be related to the Kleinian vertex of~\cite{sonoda1990covariant} in this limit, whose local coordinates $z_i$ are given by
\begin{equation} \label{eq:KleinianVertex}
z_1 = e^{i \pi \tau(z)}, \qquad z_2 = e^{-i \pi/\tau(z)}, \qquad z_3 = e^{-i \pi/(\tau(z)\pm 1)},
\end{equation}
around the punctures $z=0,1,\infty$ respectively, since it involves the same hyperbolic geometry in its construction which emphasized more recently in~\cite{Moosavian:2017qsp,Moosavian:2017sev}. Here the function $\tau(z)$ is the inverse of the modular $\lambda$-function, which is equal to~\cite{hypergeometric}
\begin{equation} \label{eq:tau}
\tau(z) = i \frac{{_2}F_1(\frac{1}{2},\frac{1}{2},1,1-z)}{{_2}F_1(\frac{1}{2},\frac{1}{2},1,z)} = -\frac{i}{\pi} \log\left( \frac{z}{16} \right) + \mathcal{O}(z).
\end{equation}
We will denote the limit $\lambda_j=\lambda \to 0$ as the \emph{Kleinian limit}.
In order to argue for this limit, first notice that the function $\tau(z)$ satisfies the following equality~\cite{hempel1988uniformization}
\begin{equation}
\{\tau,z\} = \frac{1}{2z^2} + \frac{1}{2(z-1)^2} - \frac{1}{2z(z-1)},
\end{equation}
But recall from~\eqref{eq:Tsect3} and~\eqref{eq:3T} we also have
\begin{equation}
\lim_{\lambda \to 0} \{\rho_j^{i \lambda},z\} = \lim_{\lambda \to 0} T_{\varphi}(z) = \frac{1}{2z^2} + \frac{1}{2(z-1)^2} - \frac{1}{2z(z-1)} ,
\end{equation}
So from these two we immediately conclude
\begin{equation} \label{eq:logroh}
\{\tau,z\} = \lim_{\lambda \to 0} \{\rho_j^{i \lambda},z\} = \{\lim_{\lambda \to 0} \log(\rho_j) ,z\} \implies \lim_{\lambda \to 0} \log(\rho_j(z)) = \frac{a \tau(z) + b}{c \tau(z) + d}.
\end{equation}
Above we moved the limit inside the Schwarzian derivative and used the fact that two equal Schwarzian derivatives must be related to each other by a PGL(2,$\mathbb{C}$) transformation. So here $a,b,c,d \in \mathbb{C}$ and $ad-bc \neq 0$. Note that we can easily determine these constants by expanding both sides of~\eqref{eq:logroh} to leading order in $z$.
Take $z=0$ for instance. We already know the expansion of $\log(\rho_1(z))$ around $z=0$ from~(\ref{eq:ScaledRatio}). In the Kleinian limit this would then yield
\begin{equation}
\lim_{\lambda \to 0} \log(\rho_1(z)) = \log\left( \frac{z}{16} \right) + \mathcal{O}(z),
\end{equation}
using properties of Gamma functions for the limit of the function $v(\lambda,\lambda,\lambda)$. Comparing this to~\eqref{eq:tau}, we see the constants above get fixed and we obtain the following in the Kleinian limit:
\begin{equation}
a = i \pi, \quad b=c=0, \quad d=1 \implies \lim_{\lambda \to 0} \rho_1(z) = e^{i \pi \tau(z)} =z_1.
\end{equation}
Note that we can repeat the same procedure for other punctures and similarly obtain $\lim_{\lambda \to 0} \rho_j = z_j$ up to an unimportant phase factor. We explicitly checked this is indeed the case.
Now observe the scale factor $N=e^{-\frac{\pi}{2\lambda}}$ that relates the actual local coordinates $w_j$ to $\rho_j$ by $N w_j =\rho_j$ approaches to zero as $\lambda \to 0$, which is essentially a consequence of shrinking grafted cylinders. So in order to get a well-defined limit, it is necessary to place a \emph{cut-off} on the scale factor $N$ which we can do it as follows. We know $\lambda \to 0$ would make the length of the boundary geodesics $L$ smaller. So, as we take this limit, we choose some value of $L = \epsilon \ll 1$ that we put in $N$ and keep using it for any $L < \epsilon $. In other words, we take $N\approx e^{-\frac{\pi^2}{\epsilon }}$ for sufficiently small $L = 2 \pi \lambda \geq 0$ instead of what is given before. Note that this procedure essentially mirrors what is done in~\cite{Moosavian:2017qsp,Moosavian:2017sev}.
With this cut-off in place, we now have $N w_j = \rho_j = z_j$ in the Kleinian limit. Like in~\cite{Moosavian:2017qsp,Moosavian:2017sev}, we will multiply the original local coordinates $z_j$ for the Kleinian vertex given in (\ref{eq:KleinianVertex}) by $N^{-1}$ and define a new set of local coordinates $z_j' \equiv N^{-1} z_j$ in order to use the standard plumbing parameters. With this, we get $\lim_{\lambda \to 0} w_j = z_j'$ and see the scaled local coordinates for the Kleinian vertex matches what we find from the Kleinian limit of the hyperbolic three-string vertex as anticipated.
\subsection{Light-cone vertex}
Lastly, consider the situation
\begin{equation}
\lambda_1 = r \lambda, \qquad \lambda_2 = \lambda, \qquad \lambda_3 = (1-r) \lambda,
\end{equation}
for $0<r<1$ and take $\lambda \to \infty$. Having $\lambda_{2}=\lambda_{1}+\lambda_{3}$ while all of them being large, this limit should produce the local coordinates for the light-cone vertex~\cite{Zwiebach:1988qp}, by using similar geometric reasoning given in subsection \ref{sec:Witten}. Thus, we are going to call this limit the \emph{light-cone limit}.
In order to understand this limit better, first note that the restriction $\lambda_{2}=\lambda_{1}+\lambda_{3}$ always makes one of the first two arguments of the hypergeometric function appearing in local coordinates~\eqref{eq:lc} independent of $\lambda$ and finite as $\lambda \to \infty$. This is crucial because then a generic term in the expansion of these hypergeometric function around the puncture $z=z_j$ takes the following form:
\begin{equation}
\text{term} \sim \frac{\# \lambda^n + \dots }{\# \lambda^n + \dots} (z-z_j)^n.
\end{equation}
Here $\#$ denotes some numbers while dots denote the lower order terms in $\lambda$. The important point here is that since one of the arguments of the hypergeometric function is independent of $\lambda$, the same power of $\lambda$ appears in the numerator and the denominator of the coefficient of $(z-z_j)^n$ in its expansion. Therefore, these coefficients remain finite as we take $\lambda \to \infty$.
On the other hand, observe that the ratio of hypergeometric functions is raised to the power $1/i\lambda$ in~\eqref{eq:lc} and this exponent approaches to $0$ in the light-cone limit. But as we have just argued, the expansion of the hypergeometric functions remains finite in this limit. So we conclude that the part depending on the hypergeometric functions must completely drop out. The resulting limit gives, after taking the limits of prefactors like in subsection~\ref{sec:Witten},
\begin{subequations}
\begin{align}
w_1(z) &= r^{-1} (r-1)^{\frac{r-1}{r}}z (1-z)^{-\frac{\lambda_{2}}{\lambda_{1}}},\\
w_2(z) &= r^{r} (r-1)^{1-r} (z-1) z^{-\frac{\lambda_{1}}{\lambda_{2}}}, \\
w_3(z) &= r^{\frac{r}{r-1}} (r-1)^{-1} (z-1)^{-\frac{\lambda_{2}}{\lambda_{3}}} z^{\frac{\lambda_{1}}{\lambda_{3}}}.
\end{align}
\end{subequations}
From this, it is clear that if we relate the lengths of strings $2 \pi \lambda_j = L_j$ to the light-cone momenta $p_j^+$ in the usual fashion after an infinite rescaling, i.e. $p_j^+ \sim L_j$, and include the appropriate signs for the incoming/outgoing momenta, we arrive the light-cone vertex given in~\cite{Zwiebach:1988qp} up to an unimportant phase ambiguity. Therefore, the hyperbolic three-string vertex indeed reduces to the light-cone vertex in the light-cone limit in accord with our geometric expectation.
\section{Conservation laws for the hyperbolic three-string vertex} \label{sec:Conservation}
In this section we derive the conservation laws associated with the hyperbolic three-string vertex in the spirit of~\cite{rastelli2001tachyon}. Let us denote the hyperbolic three-string vertex with the geodesic boundaries of length $L=2 \pi \lambda$ as $\bra{V_{0,3}(\lambda)}$. This should be thought as an element of three-string dual Fock space, so it takes 3 states in Fock space and maps to a complex number. For simplicity of reporting, we set all the boundary lengths equal and report the holomorphic Virasoro conservation laws, but arguments here can be extended trivially to the cases with unequal lengths; ghosts and current conservation laws; and/or anti-holomorphic analogues.
First, let us put the punctures at $z=\sqrt{3},0, - \sqrt{3}$ in order to be consistent with~\cite{rastelli2001tachyon} and report the expansions for $z$ in terms of the local coordinates~$w_j$:
\begin{align}
f_1(w_1) &= \sqrt{3} + 2 \sqrt{3} N e^{-\frac{v}{\lambda }} w_1 + 3 \sqrt{3} N^2 e^{-\frac{2 v}{\lambda }} w_1^2 \nonumber
+\frac{\sqrt{3} \left(31 \lambda ^2+139\right)}{8 \left(\lambda ^2+4\right)}N^3 e^{-\frac{3 v}{\lambda }} w_1^3 + \mathcal{O}(w_1^4), \\
f_2(w_2) &= \frac{1}{2} \sqrt{3} N e^{-\frac{v}{\lambda }} w_2 -\frac{5 \sqrt{3} \left(\lambda ^2+1\right) N^3 e^{-\frac{3 v}{\lambda }}}{32 \left(\lambda ^2+4\right)} w_2^3 + \mathcal{O}(w_2^5),\nonumber\\
f_3(w_3) &= -\sqrt{3} + 2 \sqrt{3} N e^{-\frac{v}{\lambda }} w_3 - 3 \sqrt{3} N^2 e^{-\frac{2 v}{\lambda }} w_3^2
+\frac{\sqrt{3} \left(31 \lambda ^2+139\right)}{8 \left(\lambda ^2+4\right)} N^3 e^{-\frac{3 v}{\lambda }} w_3^3 + \mathcal{O}(w_3^4).
\end{align}
These are based on inverting~\eqref{eq:lc} respectively after performing the global conformal transformation
\begin{equation} \label{eq:move}
z \to - \frac{z-\sqrt{3}}{z+\sqrt{3}},
\end{equation}
that makes the monodromies around $z=\sqrt{3},0, -\sqrt{3}$ non-trivial. Here we will refer functions from the $w_j$-plane to the $z$-plane as $f_j$, $f_j(w_j) = z$. Like before, here $N=e^{-\frac{\pi}{2 \lambda}}$ and $v=v_1=v_2=v_3$. The global phase of the local coordinates $w_j$ are not important as usual, so we used this freedom to put $f_j$'s into rather symmetric form shown above. We are going to work perturbatively in $w_i$ below.
Notice that when we consider the minimal area limit, these expressions reduce to the one given in (2.11) of~\cite{rastelli2001tachyon}. This limiting behavior is expected, since the vertex given there, open string Witten vertex, when considered in the entirety of the complex plane becomes the closed string minimal area three-vertex, and we know from the previous sections that's what the hyperbolic three-string vertex approaches in the minimal area limit. So it shouldn't be too surprising that the identities we will write below reduces to their counterparts given in~\cite{rastelli2001tachyon} in the minimal area limit.
As an example of conservation laws, we derive the Virasoro conservation laws by which we mean the identities of the type, for $k>0$,
\begin{equation} \label{eq:Form}
\bra{V_{0,3}(\lambda)} L_{-k}^{(2)} = \bra{V_{0,3}(L)} \left[A^k(\lambda) \cdot c + \sum_{n\geq 0 } a_n^k(\lambda) L_n^{(1)}+ \sum_{n\geq 0 } c_n^k(\lambda) L_n^{(2)}+ \sum_{n\geq 0 } d_n^k(\lambda) L_n^{(3)}\right].
\end{equation}
Here $A^k, a_n^k, c_n^k, d_n^k$ are some functions of $\lambda$ that we are going to explicitly derive, $L_n$ are Virasoro generators, and the superscript denotes the slot that they apply in $\bra{V_{0,3}(\lambda)}$. By cyclicity of the hyperbolic three-vertex similar identities holds as we permute $(1) \to (2), (2) \to (3), (3) \to (1)$. So it would be sufficient to report the form above. The idea here is to exchange the negatively-moded Virasoro charges with the positively-moded ones plus the central term.
Now let $v(z)$ be a vector field holomorphic everywhere except for the punctures.\footnote{Not to be confused with $v$ appearing in the local coordinates.} That is, it changes as $v(z) \to \tilde{v}({\tilde{z}}) = (\partial \tilde{z}) v(z)$ under $z \to \tilde{z}$. Note that $v(z)$ should be regular at $z=\infty$ by its definition, so we must ensure $z^{-2}v(z)$ is finite as $z \to \infty$ by the inversion map $z \to \tilde{z} = 1/z$.
It is important to note that the object $v(z) T(z) dz$ is almost a 1-form, where $T(z)$ is stress-energy tensor.\footnote{In this section $T(z)$ will denote the stress-energy tensor of an arbitrary CFT with central charge $c$, not to be confused with the stress-energy tensor $T_{\varphi}(z)$ we previously considered.} Under $z \to \tilde{z}$ it transforms as,
\begin{equation} \label{eq:formtransform}
v(z) T(z) dz = \tilde{T}(\tilde{z}) \tilde{v}(\tilde{z}) d \tilde{z} - \frac{c}{12} \{z, \tilde{z}\} \tilde{v}(\tilde{z}) d \tilde{z}.
\end{equation}
As we see above, we have an extra contribution from the central term. Nonetheless, we can integrate this object on the complex plane on contours and use the usual properties of the complex integration, as long as we keep track of this additional term under the change of coordinates.
In order to derive the Virasoro conservation laws, the following equality is crucial:
\begin{equation}
\bra{V_{0,3}(\lambda)} \oint_{\mathcal{C}} \text{d} z \; v(z) T(z) = 0.
\end{equation}
Here, $\mathcal{C}$ is a contour that surrounds the three punctures, oriented counterclockwise. This is a shorthand notation for the vanishing of the correlator of the integral $\oint_{\mathcal{C}} v(z) T(z) dz$ with any three vertex operator placed at the punctures. Note that this correlator vanishes because we can push the contour to shrink around $z=\infty$ by the inversion map. In this case, the central charge term does not contribute since the Schwarzian derivative of the inversion map is zero.
Now we can deform the contour $\mathcal{C}$ to separate it into positively oriented, disjoint contours $\mathcal{C}_i$ around each punctures and write down the expression above in terms of the local coordinates as follows:
\begin{equation} \label{eq:conservation}
\bra{V_{0,3}(\lambda)} \sum_{i=1}^{3} \oint_{\mathcal{C}_i} \text{d} w_i \; v^{(i)}(w_i) \left[T^{(i)}(w_i) - \frac{c}{12} \{f_i(w_i), w_i\}\right] = 0,
\end{equation}
using the transformation property of $v(z) T(z) dz$ given in~\eqref{eq:formtransform}. Here $v^{(i)}(w_i)$ denotes the components of the vector field $v(z)\frac{\partial}{\partial z}$ in the local coordinates $w_i$ and similarly for the stress-energy tensor.
We will clearly need to find $\{f_i(w_i), w_i\} $ because of~\eqref{eq:conservation}. This is easy to do:
\begin{align}
\{f_i(w_i), w_i\} &= -\frac{15 \left(\lambda ^2+1\right)}{8 \left(\lambda
^2+4\right)} N^2 e^{-\frac{2 v}{\lambda }} + \frac{135 \left(\lambda ^2+1\right) \left(3 \lambda ^4+19 \lambda ^2-2\right)}{64 \left(\lambda ^2+4\right)^2 \left(\lambda ^2+16\right)} N^4 e^{-\frac{4 v}{\lambda }} w_i^2 + \cdots.
\end{align}
This is the same for each puncture because of the cyclicity, which we explicitly checked. Unsurprisingly, in minimal area limit we arrive the expression given in equation (3.8) of~\cite{rastelli2001tachyon}. By this expansion it is easy to see this term only appears if we have odd-powered poles around a puncture by (\ref{eq:conservation}).
Now remember
\begin{equation}
L_{-k}^{(i)} = \oint_{\mathcal{C}_i} \frac{\text{d} w_i}{2\pi i} w_i^{-k+1} T^{(i)}(w_i),
\end{equation}
so we need a vector field that behaves like $v^{(2)} \sim w_2^{-k+1}$ for $k>0$ around the puncture $(2)$ while behaves like $v^{(1)} \sim w_1$ and $v^{(3)} \sim w_3$ around the other punctures in order to put the Virasoro generators in the form given in (\ref{eq:Form}). Additionally, we have to ensure the regularity at infinity.
For $k=1$ case, all of these can be achieved with the following globally defined holomorphic vector field:
\begin{equation}
v_1(z) = -\frac{N e^{-\frac{v}{\lambda }}}{2 \sqrt{3}} \left(z^2-3\right).
\end{equation}
Normalization is chosen to get the convention in~(\ref{eq:Form}) and in the minimal area limit this reduces to one given in (3.10) of~\cite{rastelli2001tachyon}. This has the following expansion in the local coordinates $w_i$
\begin{align}
v_1^{(1)}(w_1) &= -Ne^{-\frac{v}{\lambda }}w_1 + \frac{1}{2} N^2 e^{-\frac{2 v}{\lambda }} w_1^2 - \frac{5 \left(\lambda ^2+1\right) }{8 \left(\lambda
^2+4\right)} N^3 e^{-\frac{3 v}{\lambda }} w_1^3 + \mathcal{O}(w_1^4), \nonumber\\
v_1^{(2)}(w_2) &= 1 + \frac{\left(11 \lambda ^2-1\right)}{16 \left(\lambda
^2+4\right)} N^2 e^{-\frac{2 v}{\lambda }}w_2^2 + \frac{5 \left(-8 \lambda ^6-6 \lambda ^4+3 \lambda ^2+1\right) }{128 \left(\lambda ^2+4\right)^2 \left(\lambda ^2+16\right)} N^4 e^{-\frac{4 v}{\lambda
}} w_2^4 + \mathcal{O}(w_2^6),\nonumber\\
v_1^{(3)}(w_3) &= Ne^{-\frac{v}{\lambda }}w_3 + \frac{1}{2} N^2 e^{-\frac{2 v}{\lambda }} w_3^2 + \frac{5 \left(\lambda ^2+1\right) }{8 \left(\lambda
^2+4\right)} N^3 e^{-\frac{3 v}{\lambda }} w_3^3 + \mathcal{O}(w_1^4).
\end{align}
Unsurprisingly, these reduce to the equation (3.11) of \cite{rastelli2001tachyon} in the minimal area limit. After substituting this into (\ref{eq:conservation}), each integration amounts to doing the replacement $w_i^n \to L_{n-1}^{(i)}$ by the residue theorem. Therefore we get
\small
\begin{alignat}{2}
0 &= \bra{V_{0,3}(\lambda)} \left( -Ne^{-\frac{v}{\lambda }}L_0 + \frac{1}{2} N^2 e^{-\frac{2 v}{\lambda }} L_1 - \frac{5 \left(\lambda ^2+1\right) N^3 e^{-\frac{3 v}{\lambda }}}{8 \left(\lambda
^2+4\right)} L_2 + \frac{5 \left(\lambda ^2+1\right) N^4 e^{-\frac{4 v}{\lambda }}}{32 \left(\lambda ^2+4\right)}L_3 + \dots \right)^{(1)}\nonumber\\
&+\bra{V_{0,3}(\lambda)} \left(L_{-1} + \frac{\left(11 \lambda ^2-1\right) N^2 e^{-\frac{2 v}{\lambda }}}{16 \left(\lambda
^2+4\right)} L_1 + \frac{5 \left(-8 \lambda ^6-6 \lambda ^4+3 \lambda ^2+1\right) N^4 e^{-\frac{4 v}{\lambda
}}}{128 \left(\lambda ^2+4\right)^2 \left(\lambda ^2+16\right)} L_3 + \dots \right)^{(2)}\nonumber\\
&+\bra{V_{0,3}(\lambda)} \left( Ne^{-\frac{v}{\lambda }}L_0 + \frac{1}{2} N^2 e^{-\frac{2 v}{\lambda }} L_1 + \frac{5 \left(\lambda ^2+1\right) N^3 e^{-\frac{3 v}{\lambda }}}{8 \left(\lambda
^2+4\right)} L_2+ \frac{5 \left(\lambda ^2+1\right) N^4 e^{-\frac{4 v}{\lambda }}}{32 \left(\lambda ^2+4\right)}L_3 + \dots \right)^{(3)}.
\end{alignat}
\normalsize
Again, this reduces to (3.12) of~\cite{rastelli2001tachyon} in the minimal area limit. Note that this doesn't have any central charge contribution since the vector $v_1(z)$ does not have a pole around the punctures.
We can continue to generate identities of the form (\ref{eq:Form}) by using the following vector fields:
\begin{align}
v_2(z) &= -\frac{N^2 e^{-\frac{2v}{\lambda }}}{4} \frac{z^2-3}{z},\\
v_3(z) &= -\frac{\sqrt{3 }N^3 e^{-\frac{3v}{\lambda }}}{8} \frac{z^2-3}{z^2}-
\frac{3 (3+7\lambda^2) }{16(4+\lambda^2)} N^2 e^{-\frac{2v}{\lambda }} v_1(z).
\end{align}
They produce the following identities respectively,
\small
\begin{alignat}{2}
0 &= \bra{V_{0,3}(\lambda)} \left( -\frac{1}{2} N^2 e^{-\frac{2 v}{\lambda }} L_0
+\frac{5}{4} N^3 e^{-\frac{3 v}{\lambda }} L_1
-\frac{3 \left(7 \lambda ^2+23\right) }{16 \left(\lambda
^2+4\right)} N^4 e^{-\frac{4 v}{\lambda }} L_2 + \dots \right)^{(1)}\nonumber\\
&+\bra{V_{0,3}(\lambda)} \left( L_{-2} + \frac{5 \left(\lambda ^2+1\right) }{32 \left(\lambda
^2+4\right)} N^2 e^{-\frac{2 v}{\lambda }} c + \frac{\left(4 \lambda ^2+1\right)}{4 \left(\lambda
^2+4\right)} N^2 e^{-\frac{2 v}{\lambda }} L_0 + \dots \right)^{(2)}\nonumber\\
&+\bra{V_{0,3}(\lambda)} \left( -\frac{1}{2} N^2 e^{-\frac{2 v}{\lambda }} L_0
-\frac{5}{4} N^3 e^{-\frac{3 v}{\lambda }} L_1
-\frac{3 \left(7 \lambda ^2+23\right)}{16 \left(\lambda
^2+4\right)} N^4 e^{-\frac{4 v}{\lambda }} L_2+ \dots \right)^{(3)}, \\
0 &= \bra{V_{0,3}(\lambda)} \left(\frac{\left(17 \lambda ^2-7\right) }{16 \left(\lambda
^2+4\right)} N^3 e^{-\frac{3 v}{\lambda }} L_0 + \frac{15 \left(\lambda ^2+9\right) }{32 \left(\lambda
^2+4\right)} N^4 e^{-\frac{4 v}{\lambda }} L_1 + \dots \right)^{(1)}\nonumber\\
&+\bra{V_{0,3}(\lambda)} \left(L_{-3} -\frac{15 \left(\lambda ^2+9\right) \left(2 \lambda ^2-1\right) \left(4 \lambda ^2+1\right) }{128 \left(\lambda ^2+4\right)^2 \left(\lambda ^2+16\right)} N^4 e^{-\frac{4 v}{\lambda
}} L_1 + \dots \right)^{(2)} \nonumber \\
&+\bra{V_{0,3}(\lambda)} \left(-\frac{\left(17 \lambda ^2-7\right) }{16 \left(\lambda
^2+4\right)} N^3 e^{-\frac{3 v}{\lambda }}L_0 + \frac{15 \left(\lambda ^2+9\right) }{32 \left(\lambda
^2+4\right)} N^4 e^{-\frac{4 v}{\lambda }} L_1 + \dots \right)^{(3)}.
\end{alignat}
\normalsize
We explicitly checked these reduces to their counterparts in~\cite{rastelli2001tachyon} in the minimal area limit. Note that we can continue generating similar identities for $L_{-k}$ recursively by using vector fields $v_k(z) \sim (z^2-3)z^{-k+1}$ and appropriately subtracting previous ones. Doing this allows us to put the identities in the form (\ref{eq:Form}) for which only a single negatively-moded Virasoro generator appears in the left-hand side.
\section{Remarks and open questions} \label{sec:Conc}
In this paper, we constructed the local coordinates for the hyperbolic three-string vertex first described in~\cite{Costello:2019fuh} and investigated its various limits explicitly. We calculated the $t^3$ term in the closed string tachyon potential and developed the conservation laws associated with such vertex in the spirit of~\cite{rastelli2001tachyon}. We conclude by providing some final remarks and highlighting possible future directions relevant to us:
\begin{enumerate}
\item Since we now know the local coordinates for the hyperbolic three-string vertex, it is possible to construct the Feynman diagrams by identifying them as
\begin{equation}
w_j w_j' = \exp \left[ -\frac{2 \pi s}{L_j} + i \theta\right] \quad \text{with} \quad s \in \mathbb{R}_{\geq 0}, \quad \theta \in [0,2 \pi)
\end{equation}
using the local coordinates $w_j$ and $w_j'$ associated to boundaries of equal length on not-necessarily-distinct pair of pants. Making this identification corresponds to having a finite flat cylinder of circumference $L_j$ and length $s$ with a twist $\theta$ stretching between not-necessarily-distinct pair of pants and it has the natural interpretation of the string propagator.
As usual, we must consider every possible value of $(s,\theta)_A$ when we are computing the string amplitudes. Here we added an index $A$ to indicate there are generally more than one propagator in the Feynman diagrams. It would be interesting the study the Feynman regions these diagrams cover in the moduli spaces of Riemann surfaces of genus $g$ and $n$ punctures $\mathcal{M}_{g,n}$ to see if they provide a piece of a section over the bundle $\widehat{\mathcal{P}}_{g,n} \to \mathcal{M}_{g,n}$ or not. The simplest Feynman regions to study would be for four-string scattering or string tadpole interaction. Note that with the metric we constructed on the hyperbolic pair of pants, it is possible to describe a Thurston metric of~\cite{Costello:2019fuh} explicitly on Riemann surfaces.
\item The local coordinates (\ref{eq:LocalCoord}) we constructed in this paper also can be used for the open-closed hyperbolic string vertices without moduli~\cite{Cho:2019anu}. There are two additional vertices without moduli on top of the sphere with three closed string punctures in this situation. They are disk with three open string punctures and disk with one open string puncture and one closed string puncture. Note that if we cut open the hyperbolic three-closed string vertex along a geodesics connecting all punctures for the former and one puncture connecting back to itself for the latter, we generate these additional cases exactly. From this construction it is clear that these would carry hyperbolic metric with appropriately grafted flat strip/cylinder parts and would be the same as what is constructed in~\cite{Cho:2019anu}. So we can still use the local coordinates (\ref{eq:LocalCoord}) for these additional cases.
\item The primary method we used in this paper, that is relating Liouville's equation on a specified domain to a monodromy problem, can be generalized to construct the local coordinates of the classical ($g=0$) hyperbolic $n$-string vertices in principle. For this, instead of (\ref{eq:3T}) we should take the stress-energy tensor of Liouville's equation to be
\begin{equation} \label{eq:Tconc}
T_{\varphi}(z) = \sum_{i=1}^n \left[ \frac{\Delta_i}{(z-z_i)^2} + \frac{c_i}{(z-z_i)}\right] ,
\end{equation}
with punctures positioned at $z=z_i$ and use this stress-energy tensor to generate the local coordinates. Here $c_i \in \mathbb{C}$ are so-called \emph{accessory parameters}~\cite{hadasz2003polyakov}.
There are two important problems with this approach. First, after we fixed the positions of three punctures by PSL(2,$\mathbb{C}$) symmetry, assigned prescribed weights at all punctures, and demanded regularity at infinity, we would still have $n-3$ unfixed $c_i$ parameters functions of $n-3$ unfixed positions $z_i$, the usual moduli for the $n$-punctured sphere. It is argued that such accessory parameters can be fixed in terms of moduli using the action of Liouville theory so that the metric associated to $T_{\varphi}(z)$ is smooth and hyperbolic~\cite{hadasz2003polyakov}, which goes under the name \emph{Polyakov Conjecture}. Computing these parameters exactly is not known, so this is the first problem. However, some numerical results are available in the case of vanishing $L_i$, see~\cite{hadasz2006liouville}.
Secondly, even if we find the correct $c_i$, guaranteeing the correct monodromy structure for the resulting Fuchsian equation with $n$ regular singularities is impractical. This is because of the lack of analogous formulas given in (\ref{eq:connection}) for the solutions to the Fuchsian equation associated with~\eqref{eq:Tconc}. It seems to us this is not the direction one should pursue if their goal is to do practical computations.
\item It would seem more promising to evaluate all higher elementary string interactions by exploiting the pants decomposition of the (marked) Riemann surfaces and their associated Teichm\"uller spaces~\cite{buser2010geometry}. The idea would be to decompose the contribution from a given Riemann surface as sums of products of cubic interaction of appropriate string fields dictated by a given pair of pants decomposition of such Riemann surface and to use an appropriate region in Teichm\"uller space to perform the moduli integration, similar to what is suggested in~\cite{Moosavian:2017qsp,Moosavian:2017sev}. Both of these steps need further study. Related to this idea, it may be possible to form a recursion relations in the similar vein of~\cite{mirzakhani2007weil,Eynard:2007fi,ellegard1,ellegard2}.
From the possibility of using pants decomposition we see the relevance of the hyperbolic three-string vertex with \emph{unequal} $L_i$ we considered so far. After the pants decomposition, we would only need to use the hyperbolic three-string vertex of arbitrary $L_i$ to compute CFT correlators and the rest of the computation would presumably just involve combining them together in correct fashion. We leave investigating this to a future work.
\end{enumerate}
\acknowledgments
The author would like to thank Barton Zwiebach for suggesting this problem and his guidance in the writing process. This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of High Energy Physics of U.S. Department of Energy under grant Contract Number DE-SC0012567.
|
1,314,259,993,605 | arxiv | \section{Introduction}
According to many textbooks, a hallmark of turbulence is its unpredictability \cite{tritton1988,tennekes1972}. Here we address this issue using experimental data from a turbulent soap film. The starting point is Shannon's information theory \cite{shannon1964,cover1991,brillouin1962}, where in Neil Gershenfeld's words, ``...information is what you don't already know" \cite{gershenfeld2000}. Our experiment conveys information about the physical state of the system. The amount of previously unknowable information is our measure of unpredictability.
Our objective is to quantify the prediction of turbulent velocity fluctuations and in the process characterize turbulence. We will measure both the limits on making predictions and how much we need to know to do so \cite{crutchfield2012}. This approach parallels the use of Lyapunov exponents to characterize the sensitivity to initial conditions (unpredictability) of chaotic systems \cite{baker1996}. The main finding is a transition in our ability to predict, corresponding to the emergence of a cascade.
The turbulent cascade envisioned by Richardson and described mathematically by Kolmogorov is the prevalent picture of turbulence \cite{davidson2004}. In this picture, energy (or enstrophy in two dimensions) is transported across scales from some injection scale until it reaches a dissipative scale and the cascade ends. This cascade exists in both three dimensional (3D) and two dimensional (2D) turbulence, such as the one studied here. We argue that a cascade influences prediction, as discussed below.
The central quantity in information theory is the entropy density $h$ \cite{cover1991}. It is the information we receive per measurement (which in this case means a single velocity value). Think of this as the number of yes/no questions needed on average to determine the next measurement (not necessarily an integer) \cite{shannon1964}. This number can be reduced if the data is not random and structureless \cite{cover1991}. By looking at all previous data, we reduce the unpredictability to $h$. Of course, knowing the value of $h$ does not tell us how to make a prediction, only the limits on our ability to do so.
While $h$ is the amount of information that we don't know, we could also ask how much we do know. This is the excess entropy $E$, which is the information about correlations in the system \cite{schurmann1996,crutchfield2003}. It is the reduction of unpredictability. Accurate prediction requires an amount of information at least equal to $E$ \cite{ellison2009}. Although $E$ further characterizes our ability to predict, we still must decide how to do so.
Now we must decide how to make a prediction. There are many options, but our choice is to make a statistical model with a set of states and the probabilities to transition between them. For a binary system with $0s$ and $1s$, this will look like the schematic in Fig. \ref{digraph2}. There is more than one way to define which states to use and potential benefits from choosing them cleverly.
\begin{figure}[h!]
\centering
\includegraphics[scale = 0.33]{digraph2.eps}
\caption{A schematic of a binary system's states and transition probabilities between them (``digraph" \cite{ellison2009}). The transition probabilities are the conditional probabilities $p(0|1)$, $etc.$ In this notation, $p(0|1)$ means the probability of 0 given 1.}
\label{digraph2}
\end{figure}
Starting with the states that are present in the data $U$, that set is then reduced by combining those states which statistically lead to the same future \cite{crutchfield2012}. This makes the connection with prediction clear. The information contained in these optimally predictive ``causal states" $S$ is the statistical complexity $C$ of Crutchfield \cite{shalizi2001,crutchfield2012,ellison2009}. It is defined so that it is zero for laminar flows and zero also for completely random velocity fluctuations. In both limits the system's prior behavior tells one nothing about velocity fluctuations to follow. It is known that $C \ge E$, but the reasons why are not always clear \cite{crutchfield2012,ellison2009}. More details on $h$, $E$ and $C$ can be found in the Appendices \ref{appendix:data}-\ref{appendix:C}.
This study focuses on predicting the spatial variations of turbulence. A prediction in space means that given the velocity $u$ at a point $x$, one anticipates the velocity at some other point $r$ away. Prediction is normally associated with time \cite{aurell1996,leith1972}, but there are several reasons for considering the spatial alternative.
We know that the temporal and spatial features of turbulence are distinct. The fundamental work of Kolmogorov dealt only with the spatial structure of turbulence \cite{kolmogorov1941,davidson2004}. Kraichnan and others have also shown that many of the essential features of turbulence are retained if one throws away temporal correlations but keeps spatial ones \cite{kraichnan1994, shraiman2000, falkovich2001}. Thus, a treatment of spatial prediction is arguably of more fundamental interest than temporal prediction, at least for turbulence.
For a specific application, consider airplane flight. The typical cruise speed of a Boeing 747 is $V \simeq 250$ m/s \cite{boeing}. Contrast this with the rms velocity fluctuations $\sigma$ of ``strong" atmospheric turbulence $\sigma \simeq 7$ m/s \cite{mcminn1997}. Since $\sigma/V \simeq 0.03$ is small, one must use Taylor's frozen turbulence hypothesis when discussing the turbulence the airplane encounters \cite{kellay2002,tennekes1972}. In other words, an airplane flies fast enough to sample only the spatial variations of turbulence. There is not enough time for the turbulent velocity field to evolve temporally.
We have previously used $h$ to characterize two-dimensional (2D) turbulence in a flowing soap film as a function of Reynolds number $Re$ \cite{cerbus2013}. Here we use $C$ and $E$ to go beyond this and fully characterize the prediction of turbulent velocity fluctuations. This leads to the following conclusions. (1) The presence of correlated velocity fluctuations $reduces$ the amount of information $C$ needed to predict. However, those same velocity correlations increase $E$. Thus, $C$ and $E$ may be used as an indicator of the presence of a turbulent cascade. (2) 2D turbulence becomes increasingly easy to predict as $Re$ increases. While this study is on an experimental 2D system, the arguments apply for 3D turbulence as well. Moreover, no specific assumptions about the data are made. Thus, this study serves as an experimental test bed for these tools, which can be used generally for other complex systems.
\section{Example}
\label{sec:example}
As a simple illustration of these ideas, consider a coin flipping experiment where each subsequent flip will be the same as the previous one with probability $P \in [0,1]$ \cite{quax2013}. This is the statistical model for, $e.g.$, correlated random walks \cite{codling2008}. The statistical evolution of this system will look like Fig. \ref{digraph2} but with, $e.g.$, $p(0|0) = P$.
If $P = 0.5$ we have the usual fair coin toss experiment, with $h = 1$ and $C = E = 0$, since this system is maximally uncertain but statistically simple to predict with no information being shared between the past and future. In this fully random case ($P = 0.5$) both 0 and 1 predict the same future, so they are combined into a single causal state. Of course, with only one causal state, $C = 0$ automatically (see Eq. \ref{eq:C}).
Consider now a slight deviation of $P$ from 0.5. Now $C = 1$ since we will always need to know 1 bit of information (the previous flip) to predict the future. We can also calculate $h$ and $E$ (see Appendices \ref{appendix:h} and \ref{appendix:E}), which are plotted together with $C$ vs. $P$ in Fig. \ref{simple_example}. Since $P > 0.5$ means more predictable, it is clear that $h$ should decrease with increasing $P$, while $E$ should increase.
\begin{figure}[h!]
\centering
\includegraphics[scale = 0.33]{simple_example.eps}
\caption{Plot of the fundamental quantities $h$ $(\bigcirc)$, $E$ $(\square)$ and $C$ $(\triangle)$ for the simple example given here. Although $h$ and $E$ are continuous functions of $P$, $C$ is not.}
\label{simple_example}
\end{figure}
This example highlights the difference between $E$ and $C$, the crypticity $\chi \equiv C - E$ \cite{ellison2009,mahoney2011}. Here $C = E + h$, which is a unique feature of this system being first-order Markovian \cite{crutchfield2003}. The extra information needed to predict beyond $E$ is due to the randomness still intrinsic in the causal states themselves. There are many examples for which $C \ne E$ \cite{crutchfield2009,ellison2009}, but this is not always so.
An important lesson we learn from this example is that $h$, $E$ and $C$ were all necessary to characterize this system's behavior. For $P$ only slightly different from 0.5, $h$ and $E$ will still suggest a nearly random system, much like a slightly biased coin. The fact that $C$ is large and not 0 (its random value), shows that there are important correlations not present in a simple biased coin system. The system is both unpredictable (large $h$) and difficult to predict (large $C$). A similar result will be found for the low Reynolds number flow in Sec. \ref{sec:results}.
\section{Experimental setup}
Now consider a turbulent soap film, which is a good approximation to 2D turbulence since the film is only several $\mu$m thick \cite{kellay2002,boffetta2012}. The soap solution is a mixture of Dawn (2$\%$) detergent soap and water with 4 $\mu$m particles added for laser doppler velocimetry (LDV) measurements. Figure \ref{setup} contains a diagram of the experimental setup as well as thickness fluctuations visualized through thin film interference using a monochromatic light source. The thickness fluctuations act as a surrogate for velocity fluctuations \cite{kellay2002,boffetta2012}.
\begin{figure}[h!]
\hspace{-1.5em}
\includegraphics[scale = 0.19]{setup_bw.eps}
\caption{Left: Experimental setup showing the reservoirs ($TR$, $BR$), pump ($P$), valve ($V$), comb ($C$), blades ($LB$, $RB$), LDV and weight ($W$). Middle: Fluctuations in film thickness from turbulent velocity fluctuations with smooth walls and a comb. Right: Thickness fluctuations with smooth and rough walls.}
\label{setup}
\end{figure}
The soap film is suspended between two vertical blades. Nylon fishing wire connects the blades to the nozzle above and the weight below. The nozzle is connected by tubes to a valve and a top reservoir which is constantly replenished by a pump that brings the spent soap solution back up to the top reservoir. The flow is gravity-driven. Typical centerline speeds $\overline{u}$ are several hundred cm/s with rms fluctuations $u'$ ranging roughly from 1 to 30 cm/s. The channel width $w$ is usually several cm. The Reynolds number $Re = u'w/\nu$, where $\nu$ is the kinematic viscosity, thus ranges from 10 to 10,000.
Turbulence is generated using several different protocols. We can (1) insert a row of rods (comb) perpendicular to the film, (2) replace on or both smooth walls with rough walls (saw blades) with the comb removed and possibly a rod inserted near the top \cite{kellay2012}, or (3) use a comb with smooth walls as in (1) but now very near the top of the soap film where the flow is still quite slow. The comb teeth are $\sim 1$ mm in diameter and several mm apart. The saw blade teeth are $\sim 2$ mm tall and wide.
When protocol (1) is used we almost always observe the direct enstrophy cascade \cite{kellay2002,boffetta2012}. If procedure (2) is used, we can observe an inverse energy cascade \cite{kellay2002,boffetta2012,kellay2012}, although this depends sensitively on the flux and $w$. When protocol (3) is used, we see no cascade at all.
The type of cascade is identified by calculating the one-dimensional velocity energy spectrum $\mathcal{E}(k)$, where $\frac{1}{2}u'^2 = \int_0^{\infty} \mathcal{E}(k) dk.$ For the enstrophy cascade, $\mathcal{E}(k) \propto k^{-3}$ and for the energy cascade $\mathcal{E}(k) \propto k^{-5/3}$ \cite{kellay2002,boffetta2012}. A number of measurements were taken above the blades where the flow is slower. For protocol (3), $\mathcal{E}(k)$ is flat and so apparently there is no cascade, although the flow is not laminar ($u' \neq 0$). See Fig. \ref{spectra} for some representative spectra. In Fig. \ref{Re_E&C} the data for $Re < 100$ have a flat $\mathcal{E}(k)$.
\begin{figure}[h!]
\hspace{-1.5em}
\includegraphics[scale = 0.36]{spectra.eps}
\caption{Representative one-dimensional energy spectra in a log-log plot of $\mathcal{E}(k)$ vs. $k$. The enstrophy cascade ($\triangle$) has a slope close to -3 while the energy cascade ($\square$) has a slope close to -5/3. The flat curve ($\bigcirc$) has no cascade.}
\label{spectra}
\end{figure}
In all cases, we measure the longitudinal (streamwise) velocity component at the horizontal center of the channel. The data rate is $\simeq$ 5000 Hz and the time series typically had more than $10^6$ data points. For this system the time series is really a spatial series by virtue of Taylor's frozen turbulence hypothesis \cite{davidson2004,kellay2002,boffetta2012,tennekes1972}. This means that the spatial variations are swept through the LDV's measuring point by the mean flow so quickly that it is as if the LDV were scanning a frozen-in-time velocity field. This distinction between spatial and temporal is essential, as discussed above and in Ref. \cite{cerbus2013}.
\section{Results}
\label{sec:results}
The quantities $C$, $E$ and $h$ are plotted vs. $Re$ in Fig. \ref{Re_E&C}. The data are roughly divided in $Re$ into no-cascade (flat $\mathcal{E}(k)$ for $Re < 100$) and cascade (power law $\mathcal{E}(k)$ for $Re > 100$) regimes. Although $C$ and $E$ intersect at finite $Re \simeq 7000$ in Fig. \ref{Re_E&C}, this meeting point depends on the data analysis. In order to calculate probabilities from continuous data, one must bin the measurements. For different binning protocols we find a different meeting point. However, the $Re$-dependent behavior of $h$, $E$ and $C$ discussed below is the same. See Appendices \ref{appendix:data} and \ref{appendix:C} for more details on the treatment of the data.
\begin{figure}[h!]
\includegraphics[scale = 0.35]{Re_4h_E_C_binary_chi2.eps}
\caption{The statistical complexity $C$ ($\square$), excess entropy $E$ ($\bigcirc$) and entropy density $h$ ($\triangle$) as functions of $Re$ for binarized ($A = 2$) data (see Appendix \ref{appendix:data} for details on binning). We plot $h$ on a different scale for better visibility. The maximum value of $h$ here is $\log_2 2 = 1$, which the no-cascade data for $Re < 100$ approach very closely. Here $L = 10$ and we used our MATLAB program with the $\chi^2$ test to calculate $C$ (see Appendix \ref{appendix:C} for details). The lines are not fits to the data but are meant to suggest the behavior of $C$ and $E$ as functions of $Re$. For the cascade region, $C$ and $h$ are decreasing functions of $Re$ while $E$ increases. The vertical line separates the data according to whether there is a cascade or not.}
\label{Re_E&C}
\end{figure}
\subsection{Cascade Turbulence}
Now consider the behavior of $h$, $E$ and $C$ in the ``cascade regime" of Fig. \ref{Re_E&C}, $Re > 100$. At these values of $Re$, $\mathcal{E}(k)$ shows power law scaling as in Fig. \ref{spectra}. Both energy and enstrophy cascade data are present. We see from Fig. \ref{Re_E&C} that the unpredictability ($h$) is decreasing, the amount of information needed to predict ($C$) is also decreasing, while information about correlations ($E$) is increasing (all logarithmically). The opposite trend in $Re$ for $E$ and $C$ is noteworthy. It is surprising that the behavior of $h$, $E$ and $C$ for $Re > 100$ does not depend on which cascade is present, only on whether or not there is a cascade at all.
The increase of $E$ with $Re$ can be understood from the traditional view that as $Re$ increases, the ``inertial range" of correlated scales broadens \cite{davidson2004}. The increase in correlations across spatial scales is reflected by an increase in $E$. We can go further to suggest a connection between $E$ and the broadness of the inertial range. Dimensional arguments suggest that the turbulent degrees of freedom go as $N \propto Re$ for the enstrophy cascade and $N \propto Re^{3/2}$ for the inverse energy cascade. In the 3D energy cascade, $N \propto Re^{9/4}$ \cite{landau1987}. Thus the behavior $E \propto \log_2 Re$ in Fig. \ref{Re_E&C} indicates that $E$ is a logarithmic measure of the extent of the inertial range.
An interpretation of the behavior of $C$ is also suggested by the traditional picture of 2D turbulence \cite{kellay2002,boffetta2012}. As $Re$ grows, the inertial range broadens, and more of the velocity fluctuations come under the governance of the cascade. Thus, the randomness $h$ will decrease, and because the cascade's structure is dominating, our prediction cost $C$ decreases. This is the result of the general principle that patterns help us to predict \cite{shalizi2001}. Here the pattern is the cascade's structure.
Although turbulence has traditionally been thought of as unpredictable \cite{tritton1988,tennekes1972}, with $h$, $E$ and $C$ we see that the spatial predictability of (2D) turbulence is increasing with $Re$ in its fullest sense: we can predict further and more easily. This is in stark contrast to turbulence's increasing temporal unpredictability with $Re$, at least as evidence by numerical work \cite{aurell1996,leith1972}. This reiterates the important difference between time and space in turbulence, which is of fundamental interest and practical importance (recall the airplane).
\subsection{Transition to Cascade Turbulence}
Next consider the region of Fig. \ref{Re_E&C} labeled ``no-cascade". The absence of a cascade is evidenced by a lack of power law scaling in $\mathcal{E}(k)$ as in Fig. \ref{spectra}. Here $h$, $E$ and $C$ are relatively constant with respect to $Re$. It is notable that $h$ is very near to the random (white noise) value of $\log_2 2 = 1$, which is nothing like laminar flow where $h = 0$. When a cascade emerges at $Re \simeq 100$, all three quantities begin to change noticeably. This change in behavior is decidedly different from the laminar to turbulent transition which only involves the onset of fluctuations \cite{tritton1988,landau1987}.
The fluctuations of pre-cascade turbulence are apparently difficult to predict ($C$ is large in Fig. \ref{Re_E&C}). Moreover, the wide separation between $E$ and $C$ is surprising. We emphasize that $C$, $E$ and $h$ have made a clear distinction between simply unsteady velocity fluctuations and cascade turbulence. It is natural that tools designed to quantify randomness and order should be able to detect this transition.
Simulations of 3D turbulence have shown that statistics of the velocity derivatives are gaussian (or sub-gaussian) up until a small value of the Reynolds number \cite{schumacher2007,schumacher2014}. Below this value of Reynolds number, there is a ``regime which is a complex time-dependent flow rather than a turbulent one." They observe a transition similar to the one described here. Their transition is evidenced primarily by non-gaussian velocity derivative statistics. Recall that nongaussian statistics are a general feature of fully developed turbulence \cite{sreenivasan1997}.
We also resort here to a more traditional tool from turbulence, the correlation function $c(r) \equiv \langle u(x)u(x+r) \rangle_x / u'^2$ plotted in Fig. \ref{correlation} \cite{davidson2004}. $c(r)$ has typically been thought of as a tool for determining the range of length scales over which $u$ is correlated. $c(r)$ is telling us that for small $Re \le 100$, the range of scales over which $u$ is correlated is very small.
Figures \ref{spectra} and \ref{correlation} both indicate that for $Re \le 100$ the data is like white noise. The values of $h \simeq 1$ and $E \simeq 0$ in Fig. \ref{Re_E&C} reinforce this interpretation. On the other hand, if the fluctuations were truly like white noise, then $C$ should also be zero in this regime, which it is not. Recall that in the simple example from Sec. \ref{sec:example}, $C$ is large when $h$ and $E$ are close to their random values. The data are nearly random but have an explicit albeit short dependence on the past which drives $C$ from zero to its maximum value. If we were to only look at $h$ (or $E$), we would miss that there is nontrivial (non-random) behavior for low $Re$.
We have yet to understand why self-similar turbulence emerges from this ``complex, time-dependent flow" \cite{schumacher2007}. One sees from another nonlinear system, Rayleigh-Benard convection, that there is a lot to be learned at modest levels of excitation \cite{kadanoff2001}.
\begin{figure}[h!]
\includegraphics[scale = 0.36]{correlation.eps}
\caption{The velocity autocorrelation function $c(r)$ plotted $vs.$ $r$ for several values of $Re$. For small $Re$, $c(r)$ quickly decays to zero, indicating little correlation in the velocity $u$. For larger $Re$, where Fig. \ref{spectra} indicates spatial structure, there is a wider range of correlated scales. The $Re = 300$ curve has a longer correlation length $L$ than the higher $Re = 6000$ curve presumably because this lower $Re$ curve corresponds to an inverse energy cascade. The inverse energy cascade is supposed to involve larger length scales than the enstrophy cascade \cite{kellay2002,boffetta2012}.}
\label{correlation}
\end{figure}
The traditional approaches to the laminar-turbulent transition deal with instabilities of the laminar flow \cite{tritton1988,brandstater1983}. Whether it is the quasi-periodicity of Landau \cite{landau1987} or the nonperiodicity of Ruelle and Takens \cite{ruelle1971}, none of these approaches deal with the development of a cascade \cite{swinney1978}. And yet a cascade is always present in ``fully-developed turbulence" \cite{kolmogorov1941,davidson2004}. How does this cascade emerge? New approaches and models are necessary to understand how cascade behavior develops out of a ``complex, time-dependent flow" \cite{schumacher2007}. Since this development is clearly visible in Fig. \ref{Re_E&C}, an information theory approach seems promising.
\begin{figure}[h!]
\includegraphics[scale = 0.35]{Re_pred_eff_binary_chi2.eps}
\caption{The predictive efficiency $E/C$ plotted vs. $Re$ using the same data as in Fig. \ref{Re_E&C} as well as a quaternary partition $A = 4$ with partition walls placed symmetrically with respect to the mean (see Appendix \ref{appendix:data} for details on binning). We used $L = 10$ for both partitions (see Appendix \ref{appendix:C}). Here we find that $E/C$ is increasing only after a cascade develops.}
\label{pred_eff}
\end{figure}
We suggest an information-theoretic indicator of a cascade. Based on the above arguments, large $E$ and $1/C$ should both indicate a well-developed cascade. With that in mind, we can also consider the ``predictive efficiency" $E/C$ \cite{wiesner2012}, which is an increasing function of $Re$, as shown in Fig. \ref{pred_eff} for two different binning protocols. The ratio $E/C$ tells us the fraction of the information needed to predict $C$ that is due to correlations $E$. It is nearly zero when no cascade is present and grows smoothly after one has emerged. This shows that $E/C$ is a nice tool for studying the transition to cascade turbulence.
Besides this cascade transition, the laminar to fluctuation transition is also of interest. Unfortunately, we are not able to access a truly laminar regime with our apparatus. For laminar flow and this geometry, $h = E = C = 0$ \cite{crutchfield2012}. Looking at Fig. \ref{Re_E&C}, and with the reasonable assumption that $h$ and $C$ are continuous functions of $Re$, one expects a local maximum in $C$ and $h$ at some low value of $Re$. This maximum would correspond to a special transition in the evolution of the flow between laminar and turbulent behavior. The observation of this maximum requires a different experimental setup.
\section{Conclusion}
The approach here is not limited to incompressible Navier-Stoke's turbulence. In fact it is useful for any nonlinear system, even those for which one does not know the equations of motion. When we think of turbulence in terms of information and prediction, we can make new distinctions and draw new insights. We have been able to highlight a cascade transition and have seen that spatially, turbulence is becoming easier to predict statistically as $Re$ increases. As for our airplane, Figs. \ref{Re_E&C} and \ref{pred_eff} bring bittersweet news. Although its passengers will certainly experience a rougher flight as $Re$ increases, at least they won't be as surprised.
We would like to thank D. P. Feldman for explaining several concepts to us and for making his excellent lecture notes available online. C. J. Ellison was kind enough to explain some of the finer points of the formalism to us. We are also indebted to M. Bandi for providing numerous suggestions and insights. The criticisms and suggestions from several referees have also been beneficial. This work is supported by NSF Grant No. 1044105 and by the Okinawa Institute of Science and Technology (OIST). R.T.C. is also supported by a Mellon Fellowship through the University of Pittsburgh.
\begin{appendix}
\section{Data}
\label{appendix:data}
The approach used here is data driven. We are given a data stream and use it to say something about the system that made it. The main assumption is that the system is stationary \cite{cover1991,crutchfield2012}. We don't appeal to the Navier-Stoke's equation or any of Kolmogorov's universality assumptions \cite{davidson2004,kolmogorov1941}. This method is generally applicable to many types of systems.
The formalism is now introduced. In the discussion that follows an uppercase $U$ denotes the data (the random variable, the message) with possible velocity values $\mathcal{U}$ and the lowercase $u$ denotes a particular member of that set. We can also consider groups of length $L$ denoted by the set $\mathcal{U}^L$ and its particular members $u^L$. We are interested in treating a group because of the correlations that may exist between its members. Overhead arrows indicate a direction in the 1D data set relative to an arbitrary reference point $x$. For example, $\overrightarrow{U^L}$ refers to any block of data of size $L$ taken to the right of $x$. For example, if $L = 3$, then a particular block $\overrightarrow{u^3}$ is as below
\begin{equation*}
...u_{x-\Delta x},u_{x},\overrightarrow{u_{x+\Delta x},u_{x+2\Delta x},u_{x+3\Delta x}},u_{x+4\Delta x},...
\end{equation*}
where $\Delta x$ is the spatial resolution. If no $L$ is mentioned, the block is (semi-)infinite.
Let $U$ be a velocity component in the soap film, which is characterized by the experimental probability distribution $P(U)$. The focus is on the information shared between different directions $\overleftarrow{U}$ and $\overrightarrow{U}$ relative to the arbitrary point $x$ \cite{crutchfield1997,crutchfield2012}. If we had data with explicit time dependence, we would talk about the past, future and present \cite{crutchfield2012}.
In order to use this formalism with turbulence, the continuous experimental data must be converted to symbols \cite{daw2002}. A partition is defined which assigns data values in specific ranges to unique symbols \cite{daw2002, schurmann1996}. This is usually referred to as binning the data. All experiments of continuous systems do this because of limited resolution $\epsilon$. There are numerous previous studies where even binarizing a turbulent velocity signal has given more insight than traditional techniques \cite{daw2002,palmer2000,lehrman2001,cerbus2013}.
In this work we primarily use a binary partition (alphabet size $A = 2$) with the single partition wall located at the mean velocity. This smaller alphabet allows us to use a larger $L$ with confidence and so cover a wider range of length scales in our analysis. Just as with $h$ in Ref. \cite{cerbus2013}, we have found that the general behavior of $C$ and $E$ with respect to $Re$ is independent of the partition size; partitions of sizes $A = 4$, $8$ gave similar results. Here the choice was made to use the same alphabet size $A$ for all $Re$. This was done so that all data, if random, would have the same maximum value of $h = \log_2A$. Thus, all data are treated at the same level of description. Of course, there are alternative choices for setting the partition size.
\section{Entropy density $h$}
\label{appendix:h}
We have already spoken of the entropy density $h$ as a measure of unpredictability. The definition of entropy we are most familiar with is \cite{cover1991,shannon1964}
\begin{equation}
H(U) = -\sum_{u \in U} p(u) \log_2 p(u),
\end{equation}
with units of ``bits". This is the unpredictability of single data points given no immediate knowledge of any previous data points. An example of this would be estimating the unpredictability of letters in the English language based solely on the frequency of the letters and not on words.
Consider two examples. First look at a random string of 1s and 0s where $p(0) = p(1) = 0.5$. Here $H = 1$ is the maximum possible value. Next consider a periodic string such as ``...0101...". Here again $p(0) = p(1) = 0.5$, and so here also $H = 1$. However, something is wrong since a periodic string should be perfectly predictable.
Since this definition of unpredictability misses any structure or correlations extending across scales, it is generalized to the block entropies \cite{schurmann1996,crutchfield2003}
\begin{equation}
H_L = H(U^L) = -\sum_{u^L \in U^L} p(u^L) \log_2 p(u^L).
\end{equation}
This is the unpredictability of blocks of data. Of course, if we want to go back to looking at the unpredictability of a single data point, we can manipulate the $H_L$. The unpredictability of a single data point knowing $L$ immediately previous data points is
\begin{equation}
h_L = H_{L+1} - H_L.
\label{eq:hL}
\end{equation}
The $L$-dependence is inconvenient, but if we make $L$ large enough $h_L$ will become $L$-independent (for most systems) \cite{schurmann1996,crutchfield2003}. We are now ready to introduce the entropy density
\begin{equation}
h = \lim_{L \rightarrow \infty} h_L = H(\overrightarrow{U^1} | \overleftarrow{U})
\label{eq:h}
\end{equation}
with an equivalent definition in terms of the conditional entropy \cite{cover1991}. This says explicitly how unpredictable a single data point is given all previous ones.
To further develop intuition for how $h$ is associated with unpredictability, recall the Lyapunov exponents \cite{baker1996}. If a system is chaotic, its largest Lyapunov exponent $\lambda$ is greater than 0 \cite{baker1996}. If our measurement has a resolution of $\epsilon$ and we enforce a tolerance of $\Delta$, then our system is typically predictable up to a distance of $\frac{\log_2(\Delta/\epsilon)}{\lambda}$. Consider an information approach to the same problem. We choose to (or are forced to) have a particular partition size $\epsilon$. This will correspond to $A = \frac{\max(U) - \min(U)}{\epsilon}$. Our maximum possible uncertainty in bits is $\log_2 A$. It will take $n = \frac{\log_2 A}{h}$ steps into the future to add up to this uncertainty and beyond this our data stream is unpredictable.
We estimate $h$ using the limit of $h_L$ from Eq. \ref{eq:hL} in Eq. \ref{eq:h}, as discussed in Ref. \cite{cerbus2013} and elsewhere \cite{schurmann1996,crutchfield2003}. The undersampling bias in the $H(U^L)$ is corrected using Grassberger's method \cite{schurmann1996}, although this did not affect the value of $h$ very much. The $h_L$ typically reached $h$ at $L \simeq 10$.
\section{Excess entropy $E$}
\label{appendix:E}
While $h$ tells us about the unpredictability of $\overrightarrow{U^1}$ given $\overleftarrow{U}$, we may also want to know how much we actually learned about $\overrightarrow{U}$ from $\overleftarrow{U}$. This is the excess entropy $E$. It is in some sense the opposite of unpredictability. $E$ doesn't ask how much information we get from $\overrightarrow{U}$ upon measuring, but how much we don't get. We already know it. Stated mathematically \cite{schurmann1996,crutchfield2003}:
\begin{equation}
E = H(\overrightarrow{U}) - H(\overrightarrow{U} | \overleftarrow{U}) \equiv I(\overrightarrow{U} ; \overleftarrow{U})
\end{equation}
where $I(\overrightarrow{U} ; \overleftarrow{U})$ is the mutual information shared between $\overrightarrow{U}$ and $\overleftarrow{U}$ \cite{cover1991}.
This $E$ is the information we got from $\overleftarrow{U}$ that reduces unpredictability. However, just like $h$, this is a statistical statement that doesn't tell us how to use that information. $E$ does provide us with a lower bound on the amount of information needed to make predictions, since we need to account for all correlations. No matter how it's done, $E$ bits will be necessary \cite{crutchfield2003}, otherwise we ignore some structure in the system.
An alternative expression is used to estimate $E$ \cite{crutchfield2003}:
\begin{equation}
E = \sum_{L = 1}^{\infty} (h_L - h)
\end{equation}
This calculation uses essentially the same quantities involved in estimating $h$. It turns out that for many chaotic systems, $h_L - h \propto 2^{-\gamma L}$ ($\gamma$ is some constant independent of $L$) \cite{crutchfield2003}. This empirical relationship has been shown to improve the estimation of $E$ \cite{crutchfield2003}. This expression will be used when possible.
\section{Crutchfield complexity $C$}
\label{appendix:C}
We now come to prediction using a statistical model. We must determine a set of special states called causal states $S$ \cite{crutchfield2012}. These will make up a minimal representation of our system for predictive purposes. In other words, we are trying to build the simplest possible statistical model of our data. For more details see Ref. \cite{shalizi2001}. There Shalizi $et$ $al.$ show that within the information theory framework, the approach described below is maximally predictive with a minimal amount of information needed.
A statistical model consists of a set of states and the transition probabilities between them. To determine $S$ consider all unique blocks of data $U^L$. One would like to make $L$ large to capture as many correlations as possible, but the finite amount of data means only finite $L$ can be statistically reliable. For our data, $L \simeq 10$ is a good compromise. This $L$ is also chosen because it is the value of $L$ at which $h_L$ typically reached $h$.
We now calculate the conditional probability $p(\overrightarrow{U}^L|\overleftarrow{u}^L)$ that any particular block $\overleftarrow{u}^L$ will give rise to any other block of the same length. If the conditional probability distributions conditioned on two blocks are the same, they are indistinguishable from a statistically predictive point of view. Thus block 1 and block 2 are equivalent, $u^L_1 \sim u^L_2$, if $p(\overrightarrow{U}^L|\overleftarrow{u}^L_1) = p(\overrightarrow{U}^L|\overleftarrow{u}^L_2)$. This process incorporates pattern recognition by construction, which is why $C$ was originally introduced as a complexity quantifier \cite{feldman1998,crutchfield2012}.
All equivalent blocks are then combined and organized into a set of predictive causal states $S$. For example, suppose there are only three states $u_1$, $u_2$, and $u_3$ (forget about $L$ here). If $p(\overrightarrow{U}|\overleftarrow{u}_1) = p(\overrightarrow{U}|\overleftarrow{u}_2) \ne p(\overrightarrow{U}|\overleftarrow{u}_3)$, then $u_1 \sim u_2 \nsim u_3$ and we have two causal states $s_1 = (u_1, u_2)$ and $s_2 = (u_3)$. Refer back to the example in Sec. \ref{sec:example}. It is apparent that if $P = 0.5$ (or 1) there is only one causal state, but if $P \neq 0.5$ (or 1), there are two causal states.
The Shannon information (entropy) contained in $S$ is the statistical complexity \cite{crutchfield2012,crutchfield1989}
\begin{equation}
C = H[S] = - \sum_{s} p(s) \log_2 p(s).
\label{eq:C}
\end{equation}
This is the total amount of information needed to statistically reproduce the data, as we shall soon see.
Here is how this prediction work in practice: we find the causal states $S$ as just described and so we also have the transition probabilities between the states $S$. Start out in some state $u$ belonging to a particular $s$. Determine the next $s'$ statistically using the known transition probabilities $p(s'|s)$ (the $'$ means the next step). Then determine a particular $u'$ belonging to this $s'$ according to $p(u'|s')$. This is symbolically represented by
\begin{equation*}
u \xrightarrow{u \in s} s \xrightarrow{p(s'|s)} s' \xrightarrow{p(u'|s')} u'.
\end{equation*}
Then repeat. In this way the data is reproduced in a statistical sense. In summary, we can write down the probability of any $u$ starting from any other $u$. This is statistical prediction.
We needed to know an amount of information $C = H[S]$ to carry out the above prediction program. That is, we need to ask (on average) $C$ ``yes" or ``no" questions in order to find the current state of the system, and then predict from there. By design, this connects with the system's predictability, since organizing the message's parts into causal states will affect the value of $C$.
We can appreciate the distinction between $C$ and $h$ by considering an unbiased coin flip. The system is maximally unpredictable with $h = 1$, since one has no clue as to what will come next. In contrast, $C = 0$ since no information is needed for statistical prediction. There is only one causal state. This may strike the readers as strange, since random data is supposedly impossible to predict. This is only true if we insist on a prediction that has absolute certainty. Here we are predicting statistically.
When actually handling real data to identify $S$, one must deal with imperfections. These may be due to external noise or the finiteness of the amount of data. Regardless of the origin, one must set some sensible threshold to determine if two conditional probability distributions are the same, since they will never be identical. An example of some conditional probability distributions is shown in Fig. \ref{cond_pdf}. Two of the distributions are similar, indicating that the two states belong to the same causal state. The third distribution is entirely different. The task is to choose a sensible metric to make this distinction objectively.
\begin{figure}[h!]
\hspace{-1.5em}
\includegraphics[scale = 0.35]{cond_pdf.eps}
\caption{An example of three conditional pdfs used to determine the causal states. The data used here is binarized turbulence data with $L = 5$ (giving a total of 32 possible states) and $Re = 3300$ ($\bigcirc = 00001$, $+ = 00011$, $\triangle = 00010$). The horizontal axis features all the possible future states while the vertical axis is the conditional probability that given a certain past state, any of those possible future states will occur. Here the distribution for states $\bigcirc$ and $\square$ appear similar while that for state $\triangle$ is quite different.}
\label{cond_pdf}
\end{figure}
We wrote a MATLAB program that uses a $\chi^2$ test to compare conditional probability distributions \cite{lehmann2005}. We use a 0.95 confidence level, but the results are not sensitive to this choice. Results from our method are in good agreement with another frequently used algorithm \cite{shalizi2003,shalizi_website}. In the end, of course, the choice has an element of subjectivity to it.
Note that alternative expressions for $h$ and $E$ are \cite{crutchfield2012,crutchfield2009}
\begin{equation}
h = H[\overrightarrow{U^1} | \overleftarrow{S}]
\label{eq:hS}
\end{equation}
and
\begin{equation}
E = I[\overrightarrow{S} ; \overleftarrow{S}] = H[\overrightarrow{S}] - H[\overrightarrow{S} ; \overleftarrow{S}] = C - H[\overrightarrow{S} ; \overleftarrow{S}].
\label{eq:ES}
\end{equation}
Equations \ref{eq:hS} and \ref{eq:ES} say that the causal states serve as a sufficient representation. Equation \ref{eq:hS} also serves as a check on our determination of $S$ by comparing $h$ calculated with Eq. \ref{eq:hS} with our previous method from Eqs. \ref{eq:hL} and \ref{eq:h}. From Eq. \ref{eq:ES} we see that $C$ may be different from $E$. Actually, it can be shown that $C \ge E$. The difference between these two has various interpretations.
The interpretation of Crutchfield and coworkers is that a system may have some ``hidden" information, or crypticity $\chi = C - E$ \cite{crutchfield2009,mahoney2011}. Despite looking at the infinite $\overleftarrow{U}$, we missed out on the need to have an extra amount of information $\chi$ for prediction. Wiesner and coworkers have interpreted $\chi$ as the information erased at each step in the system's evolution \cite{wiesner2012}. If we were to simulate this process on a computer, $k_B T \chi$ (where $k_B$ = Boltzmann's constant and $T$ is the computer's temperature) would be the minimum thermodynamic cost. This is an extension of Landauer's work on computation. He was the first to suggest that the erasure of information has a thermodynamic cost \cite{landauer1996}.
\end{appendix}
|
1,314,259,993,606 | arxiv | \section{Introduction}
Amodal segmentation aims to infer the amodal mask, including both the visible region and the possible invisible region of the target object.
Different from semantic segmentation or traditional instance segmentation, amodal segmentation
is designed to exploit the amodal perception capability \cite{zhu2017semantic}, where human could infer and perceive the whole semantic concept of the target object mainly according to the partially visible region of the target object.
Simulating the amodal perception is quite challenging
Lots of efforts have been done for the amodal segmentation, which could be roughly divided into two groups. Methods in the first group directly estimate both the visible and the amodal regions from the images \cite{qi2019amodal, follmann2019learning, zhu2017semantic}, while methods in the second group use inferred depth order information to help the amodal mask prediction \cite{zhang2019learning}.
However, all of them learn the mapping relationship from the feature corresponding to the whole view to the amodal mask.
This processing brings explicitly ambiguity that the same image appearances of occlusion may require different predictions.
\begin{figure*}
\centering
\includegraphics[width=0.9\textwidth, height=4cm]{figures/teaser2.pdf}
\caption{Performance comparison between ours and Mask R-CNN.
We overlay different objects on top of the target green vegetables with Adobe Photoshop. Our method could estimate the almost same invisible regions, given different occlusions. For the Mask R-CNN using features corresponding to both occlusions and visible regions, the results are different.
}
\label{fig:teaser}
\end{figure*}
Motivated by the behavior of human leveraging only visible regions and memorizing the category-specific shape prior for amodal segmentation, we propose a solution that estimates the visible region of the target object and leverages the visible region and shape prior for amodal segmentation.
Without the shape prior of an object, the amodal mask inferred by human might be in an arbitrary shape. Similarly, the estimated amodal mask might also suffer from the lack of shape prior, resulting in the arbitrary edges.
Thus, our model leads to more robust results.
As shown in Fig. \ref{fig:teaser}, although the target (green vegetable) is occluded by different objects, our method could perceive the almost same occluded regions, like human ignoring the different occlusion contexts such as banana, apples, or cabbage. However, the existing baseline estimates the different occluded regions.
In particular, our proposed method consists of a coarse mask segmentation module, a visible mask segmentation module, and an amodal mask segmentation module.
In the coarse mask segmentation module, we utilize the backbone of Mask R-CNN \cite{he2017mask} with an amodal mask head and a visible mask head to predict the coarse amodal and visible mask respectively.
In the visible mask segmentation module, we propose to leverage the amodal mask to refine the visible mask and a reclassification regularizer to alleviate the misleading effect of occlusion in classification. Specifically, we use the coarse amodal mask as the attention multiplying with the feature of region-of-interest for more accurate visible mask estimation. The coarse amodal mask alleviates the effect of the background and contains more information than the coarse visible mask, which provides a cue for the visible mask.
Further, for the reclassification regularizer, we propose to apply the feature of the visible region instead of the whole view to for classification, which alleviates the influence of the occlusion and background.
In the amodal mask segmentation module, we propose to use the feature of the visible region and the shape prior to refine the amodal mask.
Unlike the diversity of the visible mask, the amodal mask has the inherent category-specific shape prior.
To encode the shape prior, we design an auto-encoder for amodal mask encoding, and a codebook is used as the memory obtained from the K-Means of amodal ground-truth mask embeddings. We utilize the shape prior for amodal mask prediction in two aspects: Firstly, we utilize the shape prior to refine the amodal mask.
Secondly, in the inference, after the final amodal mask prediction, we use the shape prior to post-process the score of proposed boxes to filter out the proposals with a low-quality amodal mask.
As shown in Fig. \ref{fig:teaser}, although the target (green vegetable) is occluded by different objects, our method could perceive the almost same occluded regions with different occlusion contexts such as banana, apples, or cabbage. However, the existing baseline estimates the different occluded regions with features corresponding to different occlusions.
The contributions of this paper could be summarized as followings: (1) inspired by the behavior of human's amodal perception, we propose a novel amodal segmentation model;
(2) a cross-task attention based refinement strategy is proposed, where the amodal mask and the visible mask is used as attention to refine each other for improving the performance of amodal segmentation; (3) this is the first work that proposes to utilize the shape prior knowledge in the amodal segmentation, and two ways to use shape prior are discussed; (4) experimental results on three public datasets show our method outperforms the existing state-of-the-art methods.
\begin{figure*}[h]
\centering
\includegraphics[width=0.9\textwidth, height=6.1cm]{figures/network9.pdf}
\caption{The overview of our approach.}
\label{fig:overview}
\end{figure*}
\section{Related Work}
\subsection{Visual Occlusion Learning}
Occlusion is inevitable in a large amount of visual task regardless of detection \cite{huang2020nms,xuangeng2020one}, segmentation \cite{fu2019stacked,huang2019ccnet} or inpainting \cite{ren2019structureflow,yu2019free,kar2015amodal} tasks.
Some researches about occlusion propose interesting and innovative methods or framework to solve the occlusion problem.
In \cite{yang2019embodied}, Yang \emph{et al.} propose a strategy to see the information behind the occlusion by moving the position of the camera.
The BANet\cite{chen2020banet} computes the similarity of the pixels in the boundary region of instance to recognize whether these pixels belong to the occlusion or not.
Some other works tend to explicitly remove the influence of occlusion, the conditional random fields (CRFs) is applied in \cite{John2006layout} to represent the occlusion probability of object parts.
The recent work \cite{huang2020nms,xuangeng2020one} deals with the occlusion problem in human crowd detection which is another task with heavy occlusion.
The visible region is used to guide the full region of human in \cite{huang2020nms} which indicates that a pair of boxes is generated by a single proposal.
Besides, in \cite{xuangeng2020one}, Xuan \emph{et al.} applied a similar measure which takes advantage of a proposal corresponding to multiple predictions to avoid the ambiguous regression.
In 3D area, many methods \cite{zhang2020object}, \cite{ramamonjisoa2020predicting,cheng2019occlusion} about solving the negative influence of occlusion are utilized for 3D keypoint, mesh and depth estimation.
\subsection{Amodal Segmentation}
A large number of researches has been accomplished on instance segmentation \cite{bai2017deep,ren2017end,xu2019explicit,neven2019instance,chen2019hybrid}, based on classic detection framework such as Faster-RCNN \cite{ren2015faster} or YOLO \cite{redmon2016you}.
Mask R-CNN \cite{he2017mask}, one of the most representative two-stage approach, uses a mask head added in the Faster-RCNN to process the aggregated feature sampled by the ROIAlign module.
In order to utilize the multiple scale information, the FPN \cite{lin2017feature} is proposed to detect instances on different scales.
Further, the Path aggregation network \cite{liu2018path} is proposed to boost information flow in feature hierarchy.
Besides, some other works localize the position of instances by the center point of each instance \cite{xie2019polarmask} instead of the boundary box or iteratively deform an initial contour to the boundary box \cite{peng2020deep}.
As a newly developing direction of instance segmentation,
The earliest work about amodal segmentation is proposed by \cite{li2016amodal}, which utilizing generated data by overlapping instances on other instances to train and test their method. They extend the boxes of each instance and refine the heatmap.
And in \cite{zhu2017semantic}, the SharpMask \cite{pinheiro2015learning} which predicts the object mask from coarse to fine is provided as the baseline model.
Recently, several amodal segmentation datasets are released to help for the amodal segmentation research.
The ORCNN \cite{follmann2019learning} uses a visible mask head and an amodal mask head to directly predict the visible mask and amodal mask respectively, obtaining the occlusion mask by subtracts the visible mask from the amodal mask.
The SLN \cite{zhang2019learning} modeling a depth order representation to infer the amodal mask.
The method proposed in \cite{qi2019amodal} utilizes an occlusion classifier to recognize whether an instance is occluded and ensemble the feature from box and class head by multi-level coding.
\section{Method}
\subsection{Problem Formulation}
Given an image $\mathbf{I}$, amodal segmentation aims to estimate the amodal mask $\mathbf{M}_a$ as well as the visible mask $\mathbf{M}_v$ for a region-of-interest (ROI).
The visible mask $\mathbf{M}_v$ could be estimated directly from the image. The amodal mask $\mathbf{M}_a$ consists of both the visible part and the invisible part. The most challenging part of amodal segmentation is to estimate the invisible region based on the visible region and without being affected by the occlusions.
The human amodal perception means the ability to infer the global instance according to the partial observation.
When inferring the invisible region, human utilizes the feature of the visible region and the shape prior of the object.
Inspired by this, we formulate the amodal segmentation as learning a nonlinear mapping function that maps the ROI feature $\mathbf{F}$ of an image $\mathbf{I}$ to the visible mask $\mathbf{M}_v$ and the amodal mask $\mathbf{M}_a$ with the regularization of shape prior.
As shown in Fig. \ref{fig:overview}, our proposed method consists of a coarse mask segmentation module, a visible mask segmentation module, and an amodal mask segmentation module.
\subsection{The Coarse Mask Segmentation Module}
The coarse mask segmentation module aims to extract the target visual features and predict the coarse amodal mask $\mathbf{M}_a^c$ and the coarse visible mask $\mathbf{M}_v^c$.
Following the existing works \cite{zhang2019learning, follmann2019learning, qi2019amodal}, we also employ a ResNet50 \cite{he2016deep} based FPN \cite{lin2017feature} as the backbone to extract the visual feature of the region-of-interest $\mathbf{F}$ containing the target object.
Specifically, in this module, both the visible mask head $f_v$ and the amodal mask head $f_a$ take the feature $\mathbf{F}$ as input.
The amodal mask head and the visible mask head have the same network structure that consists of 4 convolution layers and 1 deconvolution layer with different parameters.
There are four loss terms in this module, including a coarse amodal mask loss $\mathcal{L}_{BCE}(\mathbf{M}_a^c, \mathbf{M}_a^g)$, a coarse visible mask loss $\mathcal{L}_{BCE}(\mathbf{M}_v^c, \mathbf{M}_v^g)$, a classification loss $\mathcal{L}_{\text{cls}}$ and an object bounding box regression loss $\mathcal{L}_{\text{reg}}$.The $\mathbf{M}_a^g$ and $\mathbf{M}_v^g$ are the amodal and visible ground-truth mask. Both of the $\mathcal{L}_{cls}$ and $\mathcal{L}_{reg}$ are the same as the loss function of the class and box head in Mask R-CNN \cite{he2017mask}. The $\mathcal{L}_{BCE}(\cdot, \cdot)$ is the binary cross-entropy loss.
\subsection{The Visible Mask Segmentation Module}
The visible segmentation module aims to further refine the visible mask, via the amodal mask and the reclassification regularizer.
Because the amodal mask contains the visible region, we use it as the attention to multiple with the ROI feature $\mathbf{F}$ for the visible mask refinement. This operation enhances the capability to distinguish occlusion and target instance of the visible mask head. Further, it also alleviates the effect of background features for visible mask refinement.
The loss term of visible mask refinement is
\begin{equation}
\begin{aligned}
\mathcal{L}_v^r & = \frac{1}{N}\begin{matrix} \sum_i^N \end{matrix}
\mathcal{L}_{BCE}(f_v(\mathbf{F}_i \cdot \mathbf{M}_{a,i}^c), \mathbf{M}_{v,i}^g),
\end{aligned}
\end{equation}
where $N$ is the number of predicted instances. We denote the refined visible mask of $i^{th}$ instance as $\mathbf{M}^r_{v,i}$, which is the output of the visible mask head $f_v$ in this term.
The reclassification regularizer aims to classify each instance by processing the feature of the visible region, which avoids the misleading effect of the feature corresponding to the occlusion and background. The input of the reclassification regularizer is the feature of refined visible region regarded as $\mathbf{F}\cdot \mathbf{M}_v^r$. The reclassification regularizer $f_{rc}$ composes of two fully connected layers. In the inference, we obtain the class score of each instance by multiplying the score of the class head and reclassification regularizer.
The loss term of the reclassification regularizer is
\begin{equation}
\begin{aligned}
\mathcal{L}_{rc} = \frac{\lambda_{rc}}{N}\begin{matrix} \sum_i^N \end{matrix}\mathcal{L}_{CE}(f_{rc}(\mathbf{F}_i \cdot \mathbf{M}_{v,i}^r), y_i),
\end{aligned}
\end{equation}
where $y_i$ is the class label of $i^{th}$ instance. $\mathcal{L}_{CE}(\cdot,\cdot)$ is the cross-entropy loss. The the hyper-parameter $\lambda_{rc}=0.25$.
\subsubsection{Feature matching}
Feature matching is commonly used in network model acceleration and compression \cite{Ba2013Do, Li_2017_CVPR}, where a compact student model mimics the feature maps extracted from a large teacher model to improve its accuracy.
In this work, we also employ the feature matching strategy for the visible mask head to reduce the gaps between feature maps extracted in the coarse mask prediction and the refined mask prediction.
Since the visible mask head has the same network structure and the same parameters in predicting the coarse and refined mask, and the only difference is the inputs.
The feature matching loss between the coarse visible mask prediction and refined visible mask prediction helps the network concentrate more on visible region appearance for visible mask segmentation and alleviate the effect of the background.
In our implementation, we use feature in the last two convolution layers of the visible mask head to measure the feature matching loss. We denote the loss function of the feature matching of the visible mask head as $ L_{vfm}$, and use a subscript ($j$) on the visible mask head $f_v$ to denote the feature maps in the $i^{\text{th}}$ layer. Here $j=4,5$, which means we only use features in the higher level convolution layers. Then the feature matching loss of the visible mask head is
\begin{equation}
\begin{aligned}
\mathcal{L}_{vfm} = \frac{1}{N\cdot S}\begin{matrix}\sum_{i,j}^{N,S}\end{matrix}
\lambda_j\mathcal{L}_S(f^{(j)}_v(\mathbf{F}_i),f^{(j)}_v(\mathbf{F}_i\cdot \mathbf{M}_{a,i}^c)),
\end{aligned}
\end{equation}
where $N,S$ is the number of instances and the number of convolution layers of the visible mask head respectively.
And we apply cosine similarity $\mathcal{L}_S$ \cite{zhu2019deformable} in feature matching. We set the hyper-parameters $\lambda_4=0.01$, $\lambda_5=0.05$ and $\lambda_j=0 (j\in\{1,2,3\})$.
\subsection{The Amodal Mask Segmentation Module}
The amodal mask segmentation module is designed to refine the coarse amodal mask by using the feature of the visible region and the shape prior. Inferring the amodal mask from the visible region appearance helps our model alleviate the misleading effect of the occlusion feature. Besides, different from the visible mask, for each category, the shapes of amodal masks are explicitly more stable, without the effect of arbitrary occlusions.
The feature of the visible region can be obtained by using the refined visible mask $\mathbf{M}_v^r$ from the visible mask segmentation module as visible attention.
\begin{figure}[t]
\centering
\includegraphics[height=4cm, width=0.48\textwidth]{figures/ret2.pdf}
\caption{The illumination of the shape prior post-process. The predicted amodal mask of box B gets a higher IoU value than the predicted amodal mask of box A. But the class score of box B (0.97) is lower than box A (0.99), which results in the suppression of box B in NMS. The shape prior post-process multiplies the shape prior similarity with the class score to refine the score of object. The score of A is $0.99\times0.69 \approx 0.68$ while the score of B is $0.97\times 0.91 \approx 0.88$. The box B is remained while the box A is suppressed by NMS.}
\label{fig:shape prior}
\end{figure}
In the pre-training phase, to model the shape prior, we collect the amodal ground-truth masks in the training set and use an auto-encoder to obtain the embedding of each amodal mask. Then, for each category, we partition the embeddings into $K$ clusters via K-Means clustering and use the centers of these clusters as a codebook. Thus the codebook could memorize the embeddings of category-specific shape prior. This codebook is used to further refine the amodal mask.
\begin{table*}[h]
\centering
\resizebox{\textwidth}{14mm}{
\begin{tabular}{l|ccccc|cc|ccccc|cc|ccccc|cc}
\toprule
\multirow{3}{*}{Methods} & \multicolumn{7}{c|}{D2SA} & \multicolumn{7}{c|}{KINS} & \multicolumn{7}{c}{COCOA cls} \\ \cline{2-22}
& \multicolumn{5}{c|}{Amodal} & \multicolumn{2}{c|}{Visible} & \multicolumn{5}{c|}{Amodal} & \multicolumn{2}{c|}{Visible} & \multicolumn{5}{c|}{Amodal} & \multicolumn{2}{c}{Visible} \\ \cline{2-22}
& AP & AP50 & AP75 & AR & \begin{tabular}[c]{@{}c@{}}AP\\ (Occluded)\end{tabular} & AP & AR & AP & AP50 & AP75 & AR & \begin{tabular}[c]{@{}c@{}}AP\\ (Occluded)\end{tabular} & AP & AR & AP & AP50 & AP75 & AR & \begin{tabular}[c]{@{}c@{}}AP\\ (Occluded)\end{tabular} & AP & AR \\ \hline
Mask RCNN & 63.57 & 83.85 & 68.02 & 65.18 & NA & 68.98 & 70.11 & 30.01 & 54.53 & 30.11 & 19.42 & NA & 28.00 & 19.23 & 33.67 & 56.50 & 35.78 & 34.18 & NA & 30.10 & 31.52 \\
Mask RCNN(C8) & 64.85 & 84.05 & 70.72 & 65.61 & NA & 69.81 & 70.54 & 30.71 & 54.36 & 31.47 & 19.77 & NA & 28.74 & 19.38 & 34.72 & \textbf{57.50} & 36.93 & 35.45 & NA & 31.89 & 32.88 \\
ORCNN & 64.22 & 83.55 & 69.12 & 65.25 & 45.27 & 69.67 & 70.46 & 30.64 & 54.21 & 31.29 & 19.66 & 34.23 & 28.77 & 20.01 & 28.03 & 53.68 & 25.36 & 29.83 & 17.40 & 30.80 & 32.23 \\
SLN & 25.10 & 30.80 & 29.40 & 19.20 & NA & NA & NA & 6.60 & 10.70 & 6.90 & 6.10 & NA & NA & NA & 14.40 & 23.60 & 15.80 & 17.10 & NA & NA & NA \\
Our method & \textbf{70.27} & \textbf{85.11} & \textbf{75.81} & \textbf{69.17} & \textbf{51.17} & \textbf{72.28} & \textbf{71.85} & \textbf{32.08} & \textbf{55.37} & \textbf{33.34} & \textbf{20.90} & \textbf{37.40} & \textbf{29.88} & \textbf{19.88} & \textbf{35.41} & 56.03 & \textbf{38.67} & \textbf{37.11} & \textbf{22.17} & \textbf{34.58} & \textbf{36.42} \\
\bottomrule
\end{tabular}
}
\caption{The comparison on the D2SA dataset, the KINS dataset, and the COCOA cls dataset. Because some methods only output the amodal mask prediction, the AP (Occluded) and visible mask prediction performance of them are unavailable (NA).}
\label{tab:exp_comparison}
\end{table*}
In the training phase, for a predicted coarse amodal mask $\mathbf{M}_a^c$, we feed it into the encoder of the pre-trained auto-encoder to obtain the embedding.
We use an L2 distance to find $k$ nearest embeddings in the category-specific codebook according to the predicted category. This is denoted as shape prior search to obtain the shape prior embeddings.
Then we feed these $k$ nearest embeddings into the decoder to get decoded shape prior masks $\mathbf{M}_{sp}^k=f_{sp}(\mathbf{M}_a^c)$ ($\mathbf{M}_a^c$$\in$$\mathbb{R}^{H\times W}$, $\mathbf{M}_{sp}^k$$\in$$\mathbb{R}^{k\times H\times W}$). The $f_{sp}$ denotes the operation using an auto-encoder with category-specific codebook for shape prior search.
From this operation, we obtain the category-specific shape prior masks $\mathbf{M}_{sp}^k$ which are the most similar to the coarse amodal mask in the shape prior.
Then, we concatenate the feature of the visible region $\mathbf{F}\cdot\mathbf{M}_v^r$ and the $k$ nearest shape prior amodal masks $\mathbf{M}_{sp}^k$ as the input of the amodal mask head for amodal mask refinement. This process is to imitate the perception of human that infers the amodal mask objects by focusing on the appearance at the visible region and using the shape prior knowledge. The total loss term of this processing can be denoted by
\begin{equation}
\begin{aligned}
\mathcal{L}_a^r = \frac{1}{N} \begin{matrix}\sum_i^N\end{matrix}
\mathcal{L}_{CE}(f_a(cat(\mathbf{F}_i\cdot \mathbf{M}_{v,i}^r, \mathbf{M}_{sp,i}^k)), \mathbf{M}_{a,i}^g),
\end{aligned}
\end{equation}
where the $f_a$ is the amodal mask head whose output is $\mathbf{M}_a^r$. The $cat(\cdot,\cdot)$ is the matrix concatenate operation. In our implementation, we set $k=16$.
Besides, similar to the equation (3) of the visible segmentation module, we also apply feature matching to the amodal mask head to enhance the capacity of focusing on the visible region. The loss function of feature matching in the amodal mask head is
\begin{equation}
\begin{aligned}
\mathcal{L}_{afm} = \frac{1}{N\cdot S} \begin{matrix}\sum_{i,j}^{N,S}\end{matrix}\lambda_j\mathcal{L}_S(f^{(j)}_a(\mathbf{F}_i),f^{(j)}_a(\mathbf{F}_i\cdot \mathbf{M}_{v,i}^r)).
\end{aligned}
\end{equation}
In the inference phase, the shape prior can also be used to help the network further improve amodal segmentation.
As shown in Fig. \ref{fig:shape prior}, we use the difference between the refined amodal mask and its nearest counterpart in the codebook $\|\mathbf{M}_a^r - \mathbf{M}_{sp}^k\|_2$ as a measurement to rank the scores of bounding box proposals, and filter out the proposals with larger differences. Here the difference is measured by an L1 distance. Here we set $k=1$.
If the predicted amodal mask shape of an instance is explicitly different from the nearest shape prior in the memory, it should be treated as a low-quality prediction even with a high class score.
\begin{figure*}
\centering
\includegraphics[width=0.85\textwidth, height=8cm]{figures/comparsion_exp2.pdf}
\caption{The columns from left to right are the images, the ground-truth amodal masks, estimations of Mask R-CNN, ORCNN and ours, respectively.}
\label{fig:exp_examples}
\end{figure*}
\subsection{The Implementation Details}
Our model could predict the amodal mask and the visible mask.
In the visible mask segmentation module and amodal mask segmentation module, we use the coarse amodal mask and refined visible mask as attention. However, if directly using the predicted masks in the warm-up phase, these inaccurate predictions might destroy the following parts. Thus, we design a weighting operation, where each instance is assigned an amodal weight and a visible weight to measure the weight of an instance in optimization.
This operation can be found in supplementary. The final loss function is
\begin{equation}
\mathcal{L} = \mathcal{L}_{cls} + \mathcal{L}_{reg} + \mathcal{L}_a^c + \mathcal{L}_v^c + \mathcal{L}_a^r + \mathcal{L}_v^r + \mathcal{L}_{rc} + \mathcal{L}_{afm} + \mathcal{L}_{vfm}.
\end{equation}
Stochastic Gradient Descent (SGD) \cite{zinkevich2010parallelized} with weight decay is used for optimization in the training.
\begin{table*}[h]
\centering
\resizebox{0.93\textwidth}{14mm}{
\begin{tabular}{lccccl|ccc|cc|ccc|cc}
\toprule
\multirow{2}{*}{} & \multicolumn{5}{c|}{\multirow{2}{*}{Ablation Study}} & \multicolumn{5}{c|}{D2SA} & \multicolumn{5}{c}{KINS} \\ \cline{7-16}
& \multicolumn{5}{c|}{} & \multicolumn{3}{c|}{Amodal} & \multicolumn{2}{c|}{Visible} & \multicolumn{3}{c|}{Amodal} & \multicolumn{2}{c}{Visible} \\ \hline
& \begin{tabular}[c]{@{}c@{}}Visible\\ Attention\end{tabular} & Reclass & \begin{tabular}[c]{@{}c@{}}Shape Prior\\ Refinement\end{tabular} & \begin{tabular}[c]{@{}c@{}}Shape Prior\\ Post-process\end{tabular} & \multicolumn{1}{c|}{\begin{tabular}[c]{@{}c@{}}Feature\\ Matching\end{tabular}} & AP & AR & \begin{tabular}[c]{@{}c@{}}AP\\ (Occluded)\end{tabular} & \begin{tabular}[c]{@{}c@{}}Visible\\ AP\end{tabular} & \begin{tabular}[c]{@{}c@{}}Visible\\ AR\end{tabular} & AP & AR & \begin{tabular}[c]{@{}c@{}}AP\\ (Occluded)\end{tabular} & AP & AR \\ \hline
\textbf{1} & & \checkmark & \checkmark & \checkmark & & 66.67 & 65.68 & 46.40 & 70.50 & 70.24 & 31.61 & 20.16 & 36.84 & 29.44 & 19.61 \\
\textbf{2} & \checkmark & & \checkmark & \checkmark & \multicolumn{1}{c|}{} & 69.02 & 68.09 & 50.31 & 71.66 & 70.83 & 31.65 & 20.55 & 36.93 & 29.25 & 19.81 \\
\textbf{3} & \checkmark & \checkmark & & \checkmark & & 68.84 & 67.42 & 49.08 & 71.53 & 70.70 & 31.70 & 20.33 & 37.25 & 29.45 & 19.63 \\
\textbf{4} & \checkmark & \checkmark & \checkmark & & & 68.15 & 68.30 & 50.03 & 70.71 & 70.43 & 31.87 & 20.46 & 37.48 & 29.53 & 19.70 \\
\textbf{5} & \checkmark & \checkmark & \checkmark & \checkmark & & 69.98 & 68.87 & 51.01 & 71.92 & 71.15 & 31.94 & 20.60 & 37.55 & 29.61 & \textbf{19.88} \\
\textbf{6} & \checkmark & \checkmark & \checkmark & \checkmark & \multicolumn{1}{c|}{\checkmark} & \textbf{70.27} & \textbf{69.17} & \textbf{51.17} & \textbf{72.28} & \textbf{71.85} & \textbf{32.08} & \textbf{20.90} & \textbf{37.57} & \textbf{29.88} & \textbf{19.88} \\
\bottomrule
\end{tabular}
}
\caption{The ablation studies results on the D2SA dataset and the KINS dataset.}
\label{tab:exp_ablation}
\end{table*}
\section{Experiments}
\subsection{Experimental Setting}
We implement our proposed model based on Detectron2 \cite{wu2019detectron2} on the PyTorch framework.
The main parameter setting is: For the D2SA dataset, batch size(2), learning rate (0.005), and the number of iteration (70000). For the KINS dataset, batch size(1), learning rate (0.0025), and the number of iteration (48000). For the COCOA cls dataset, batch size (2), learning rate (0.0005), and the number of iteration (10000).
\textbf{Datasets.} We evaluate the model performance for amodal segmentation on three
datasets: the D2SA (D2S amodal) \cite{follmann2019learning}, the KINS dataset \cite{qi2019amodal}, the COCOA cls dataset \cite{zhu2017semantic}.
The D2SA dataset is built based on the D2S(Densely Segmented Supermarket) dataset with 60 categories of instances. It contains 2000 images in the training set and 3600 images in the validation set, where the annotations of the amodal mask are generated by overlapping one to another.
The KINS dataset is built based on the KITTI dataset \cite{geiger2012we}. It consists of 7474 images in the training set and 7517 images in the validation set. Different from the D2SA dataset, its amodal ground-truth is manually annotated.
There are 7 categories about the autonomous driving task in the KINS dataset.
The COCOA cls dataset \cite{zhu2017semantic} is built based on the COCO dataset \cite{lin2014microsoft}. It consists of 2476 images in the training set and 1223 images in the validation set. There are 80 categories in this dataset.
\textbf{Metrics.}
Following the \cite{zhu2017semantic, zhang2019learning}, we use the mean average precision (AP) and mean average recall (AR) to evaluate performances. The AP (Occluded) is computed on the instances whose occlusion rate is larger than 15\%.
We use the evaluation api of the COCO dataset \cite{lin2014microsoft} for fair comparisons.
\textbf{Baselines.}
We use following state-of-the-art methods for comparison.
(1) Mask R-CNN\cite{he2017mask} predicts the amodal masks by a mask head consisting of 4 convolution layers and 1 deconvolution layer. We train two Mask R-CNN baselines predicting amodal mask and visible mask respectively since the Mask R-CNN cannot predict both of them simultaneously. The Mask R-CNN (C8) means that the mask head has 8 convolution layers.
(2) ORCNN \cite{follmann2019learning} uses an amodal mask head and a visible mask head to infer the amodal masks and visible maks. The invisible mask is obtained by abstracting the visible mask from the amodal mask.
(3) SLN \cite{zhang2019learning} claims the importance of depth information in amodal segmentation. It uses a semantics-aware distance map to predict the amodal mask by utilizing depth order information.
\subsection{Performance Comparison}
We compare our model with all comparative methods on the datasets mentioned, and the performance comparisons are shown on Table \ref{tab:exp_comparison}.
We can see that our model always outperforms other methods. Compared with Mask R-CNN and Mask R-CNN (C8), the improvement resulting from directly adding the depth of the mask head is explicitly lower than the improvement achieved by our model. Our method gets better performance mainly due to our reasonable design of the network rather than the expansion of the network.
We also show some qualitative results estimated by Mask R-CNN, ORCNN, and our method in Fig .\ref{fig:exp_examples}.
We can see that our method can segment more accurately than other methods, owning to the help of the attention on the visible region and the shape prior. For the 1st and 2nd row, the predictions of our method are not misled by the feature of occlusions such as the glass bottle and cabbage. For the 3rd and 4th row, our method keeps robust even the occlusion rate is large.
\subsection{Ablation Studies}
We conduct the ablation studies on both the D2SA dataset and the KINS dataset. All the results are shown on Table \ref{tab:exp_ablation}.
\subsubsection{The Effect of Visible Attention.}
To evaluate the effect of visible attention in refining the amodal mask, we design the baseline refining the amodal mask without using the visible mask as attention. The input of the amodal mask head for predicting refined amodal mask is the concatenation of ROI feature $\mathbf{F}$ and shape prior $\mathbf{M}_{sp}^k$. Experimental results are shown at the 1st and 5th rows on Table \ref{tab:exp_ablation}.
\begin{table*}[h!]
\centering
\resizebox{0.9\textwidth}{12mm}{
\begin{tabular}{l|ccccc|cc|ccccc|cc}
\toprule
\multirow{3}{*}{Types of Utilizing Attention} & \multicolumn{7}{c|}{D2SA} & \multicolumn{7}{c}{KINS} \\ \cline{2-15}
& \multicolumn{5}{c|}{Amodal} & \multicolumn{2}{c|}{Visible} & \multicolumn{5}{c|}{Amodal} & \multicolumn{2}{c}{Visible} \\ \cline{2-15}
& AP & AP50 & AP75 & AR & \begin{tabular}[c]{@{}c@{}}AP\\ (Occluded)\end{tabular} & AP & AR & AP & AP50 & AP75 & AR & \begin{tabular}[c]{@{}c@{}}AP\\ (Occluded)\end{tabular} & AP & AR \\ \hline
(a) Both Self Attention & 64.55 & 82.53 & 68.96 & 65.62 & 44.31 & 70.50 & 71.02 & 31.18 & 54.24 & 32.08 & 20.15 & 36.89 & 29.27 & 19.76 \\
(b) Only Visible Attention & 66.98 & 83.99 & 72.87 & 67.78 & 47.53 & 70.92 & 71.50 & 31.32 & \textbf{54.79} & 32.23 & 20.25 & 36.98 & 29.40 & 19.81 \\
(c) Cross Attention & 67.11 & 83.90 & 72.81 & 67.70 & 47.75 & 71.79 & 71.77 & 31.57 & 54.58 & 32.54 & 20.61 & 37.13 & 29.65 & \textbf{20.09} \\
(d) Ours & \textbf{67.33} & \textbf{84.10} & \textbf{72.97} & \textbf{68.06} & \textbf{47.91} & \textbf{71.88} & \textbf{72.43} & \textbf{31.69} & 54.52 & \textbf{32.96} & \textbf{20.75} & \textbf{37.30} & \textbf{29.68} & 20.01 \\
\bottomrule
\end{tabular}
}
\caption{The experimental results of the attention analysis on the D2SA dataset and the KINS dataset.}
\label{tab:exp_cross-task}
\end{table*}
\subsubsection{The Effect of Reclassification Regularizer.}
To validate the effectiveness of the reclassification regularizer. We conduct the experiments at the 2nd and 5th rows on Table \ref{tab:exp_ablation}.
The experimental results show the importance of the reclassification regularizer.
\subsubsection{The Effect of the Shape Prior Refinement.}
To investigate the effect of shape prior in refinement, we plan to train the baseline at the 3rd without shape prior refinement. We only utilize the feature of the refined visible region to refine the amodal mask.
The gap between the results at the 3rd row the 5th row shows the importance of shape prior knowledge, which agrees with the usage of shape prior in the human's amodal perception.
\begin{figure*}[t
\begin{center}
\begin{tabular}{ccc}
\includegraphics[height=3.8cm,width=0.28\textwidth]{figures/kmeans_1.png} &
\includegraphics[height=3.8cm,width=0.28\textwidth]{figures/kmeans_5.png} &
\includegraphics[height=3.8cm,width=0.28\textwidth]{figures/kmeans_6.png}\\
(a) Carrot & (b) Bottle & (c) Cucumber \\
\end{tabular}
\end{center}
\caption{Visualization of shape prior embeddings in shape codebook. We use t-SNE \cite{maaten2008visualizing} to visualize the latent space of each category in the codebook. This proves that the learned shape prior has certain interpretability.}
\label{fig:exp_kmeans}
\end{figure*}
\subsubsection{The Effect of the Shape Prior Post-process.}
To investigate the effect of shape prior post-process, we plan to remove the shape prior post-process based on the baseline at the 5th row. The result is shown at the 4th row. Compared with the results at the 5th row, the gap shows that the utilizing of shape prior post-process can achieve improvement.
\subsubsection{The Effect of Feature Matching.}
To validate the effect of feature matching in our method, we train a baseline via removing the feature matching on both the amodal and visible mask head. The result is shown in the 5th row on Table \ref{tab:exp_ablation}. Compared with our model at 6th row, the feature matching could achieve further better performance, which indicates that the feature matching could further help the amodal mask head learn the feature extracted from the visible regions.
\subsection{Attention Analysis}
To evaluate the effect of amodal attention and visible attention in our model, we propose 4 different types to utilize amodal and visible attention. The details are shown in supplementary. We conduct the experiments as shown on Table \ref{tab:exp_cross-task} to compare different ways to use amodal and visible attention. The shape prior and reclassification regularizer are removed in this section.
\textbf{(a) Both Self-Attention} uses the coarse amodal mask and coarse visible mask as attention to refine amodal mask and visible mask respectively. The input of the amodal mask head and the visible mask head in attention-based refinement are the feature of the amodal region $\mathbf{F}\cdot\mathbf{M}_a^c$ and the feature of the visible region $\mathbf{F}\cdot\mathbf{M}_v^c$ respectively.
\textbf{(b) Only Visible Attention} uses the coarse visible mask as attention to refine the visible mask prediction, which indicates $\mathbf{M}_v^r = f_v(\mathbf{F}\cdot\mathbf{M}_v^c)$. Then, the refined visible mask $\mathbf{M}_v^r$ is utilized as attention to refine amodal mask prediction. The refined amodal mask is obtained by $\mathbf{M}_a^r = f_a(\mathbf{F}\cdot\mathbf{M}_v^r)$.
\textbf{(c) Cross Attention} uses the coarse amodal mask and coarse visible mask as attention to refine the visible and amodal mask prediction respectively. The formula is $\mathbf{M}_a^r=f_a(\mathbf{F}\cdot\mathbf{M}_v^c)$ and $\mathbf{M}_v^r=f_v(\mathbf{F}\cdot\mathbf{M}_a^c)$.
\textbf{(d) Ours} uses the coarse amodal mask as attention to refine the visible mask, $\mathbf{M}_v^r = f_v(\mathbf{F}\cdot\mathbf{M}_a^c)$. Then, using the refined visible mask as attention to refine the amodal mask, $\mathbf{M}_a^r = f_a(\mathbf{F}\cdot\mathbf{M}_v^r)$.
We can observe that utilizing the visible mask as attention to refine amodal mask (b,c,d) achieves explicitly better performance on amodal mask prediction than using the coarse amodal mask (a). This indicates that applying visible attention for amodal mask prediction is more reasonable than applying amodal attention. Using the amodal mask as attention (c,d) to refine visible mask can get better performance on visible mask prediction than using the coarse visible mask (a,b). This indicates the effect of applying amodal attention in visible mask prediction. Using the refined visible mask as attention to refine amodal mask (d) has slight improvement than using the coarse visible mask (c). This result shows that using more accurate visible attention can obtain improvement in amodal mask prediction.
\subsection{Visualization of Shape Prior in the Codebook}
We also show some category-specific shape prior clusters such as carrot, bottle and cucumber, via t-SNE \cite{maaten2008visualizing} in Fig. \ref{fig:exp_kmeans}.
For each category, we partition the latent feature into 1024 clusters via K-Means, since we need to use many redundant shape prior items to store the various changes of rotation.
Thus, there exist a huge number of shape prior in each category in Fig. \ref{fig:exp_kmeans} (a)-(c).
In particular, Fig. \ref{fig:exp_kmeans} (c) represents the learned codebook of cucumber, just like a circle. As we know, cucumber is a formable object with little shape changes, while the 6 amodal masks reflect its rotation changing process. This visualization shows that the category-specific shape prior has certain interpretability.
\section{Conclusion}
In this amodal segmentation work, we propose a novel model to mimic the human amodal perception using the shape prior to imagine the invisible regions mainly based on the feature of visible regions.
However, almost existing methods use the appearance of the whole region-of-interest to infer the amodal masks, which is against the human amodal perception. And this strategy brings the ambiguity that the same appearance of occlusion may require different predictions.
To simulate the imagination from visible region and shape prior,
we use the visible mask as attention to focus on the visible regions and build a codebook to store the collected amodal shape prior embeddings for refinement and post-process.
The experimental results indicate our method outperforms other state-of-the-art methods.
\section{Acknowledgment}
The work was supported by National Key R\&D Program of China (2018AAA0100704), NSFC \#61932020, Science and Technology Commission of Shanghai Municipality (Grant No. 20ZR1436000) and ShanghaiTech-Megvii Joint Lab.
|
1,314,259,993,607 | arxiv | \section{Introduction}
\label{sect-Introduction}
\setcounter{equation}{0}
In this paper we are interested in the following {\it mean-field-type} stochastic control problem,
on a given filtered probability space $(\Omega, {\cal F}, \mathbb{P};\mathbb{F}=\{{\cal F}_t\}_{t\ge 0})$:
\begin{eqnarray}
\label{SDE-0}
\left\{\begin{array}{lll}
dX_t= \mathbb{E}\{b(t, \varphi_{\cdot\wedge t},\mathbb{E}[X_{ t}|{\cal G}_t], u)\}|_{\varphi=X, u=u_t}dt+
\mathbb{E}\{\sigma(t, \varphi_{\cdot\wedge t}, \mathbb{E}[X_{t}|{\cal G}_t], u)\}|_{\varphi=X, u=u_t}dB_t, \\
X_0=x,
\end{array}\right.
\end{eqnarray}
where $B$ is an $\mathbb{F}$-Brownian motion, $b$ and $\sigma$ are measurable functions satisfying reasonable conditions,
$\varphi_{\cdot \wedge t}$ and $X_{\cdot\wedge t}$ denote the continuous function and process, respectively, ``stopped" at $t$;
$\mathbb{G}\stackrel{\triangle}{=}\{{\cal G}_t\}_{t\ge0}$ is a given filtration that could involve the information of $X$ itself, and
$u=\{u_t:t\ge 0\}$ is the ``control process", assumed to be adapted to a filtration $\mathbb{H}=\{{\cal H}_t\}_{t\ge0}$, where
${\cal H}_t\subseteq {\cal F}^X_t\vee{\cal G}_t$, $t\ge 0$. We note that if ${\cal G}_t=\{\emptyset, \Omega\}$, for all $t\ge 0$ (i.e., the
conditional expectation in (\ref{SDE-0}) becomes expectation), ${\cal H}_t={\cal F}^X_t$, and coefficients are ``Markovian"
(i.e., $\varphi_{\cdot\wedge t}=\varphi_t$), then the problem becomes a stochastic control problem with McKean-Vlasov dynamics
and/or a Mean-field game (see, for example, \cite{CD1,CD2,CD3} in its ``forward" form, and \cite{BDL, BDLP, BLP} in
its ``backward" form). On the other hand, when $\mathbb{G}$ is a given filtration, this is the so-called {\it conditional
mean-field SDE} (CMFSDE for short) studied in \cite{CZ}. We note that in that case the conditioning is essentially
``open-looped".
The problem that this paper is particularly focusing on is when ${\cal G}_t={\cal F}^Y_t$, $t\ge 0$, where $Y$ is an
``observation process" of the dynamics of $X$, i.e., the case when the pair $(X, Y)$ forms a ``close-looped"
or ``coupled" CMFSDE.
More precisely, we shall consider the following partially observed controlled dynamics (assuming $b=0$ for
notational simplicity):
\begin{eqnarray}
\label{dynamics}
\left\{\begin{array}{lll}
\displaystyle dX_t=
\mathbb{E}\{\sigma(t, \varphi_{\cdot\wedge t}, \mathbb{E}[X_{t}|{\cal F}^Y_t], u)\}|_{\varphi=X, u=u_t}dB^1_t;\medskip \\
\displaystyle dY_t=h(t, X_{ t})dt +\hat \sigma d B^2_t; \qquad X_0=x, ~Y_0=0.
\end{array}\right.
\end{eqnarray}
Here $X$ is the ``signal" process that can only be observed through $Y$, $(B^1, B^2)$ is a standard Brownian motion,
and $\hat \sigma$ is a constant. We should
note that in SDEs (\ref{dynamics}) the conditioning filtration $\mathbb{F}^Y$ now depends on $X$ itself, therefore it is much
more convoluted than the CMFSDE we have seen in the literature. Furthermore, the path-dependent nature of the
coefficients makes the SDE essentially {\it non-Markovian}.
Such form of CMFSDEs, to the best of our knowledge,
has not been explored fully in the literature.
Our study of the CMFSDE (\ref{dynamics}) is strongly motivated by the following variation of the mean-field game in a finance
context, which would result in a type of stochastic control problem involving a controlled dynamics of such a form. Consider a firm whose {\it fundamental value}, under the risk neutral measure $\mathbb{P}^0$ with zero interest, evolves
as the following SDE with ``stochastic volatility" $\sigma=\sigma(t,\omega)$, $(t,\omega)\in[0,\infty)\times\Omega$:
\begin{eqnarray}
\label{fvalue}
X_t=x+\int_0^t \sigma(s, \cdot) dB^1_s, \quad t\ge 0,
\end{eqnarray}
where $B^1$ is the intrinsic noise from inside the firm. We assume that such fundamental value process cannot be observed directly, but can be observed through a stochastic dynamics (e.g., its stock value) via an SDE:
\begin{eqnarray}
\label{stock}
Y_t=\int_0^t h(s, X_{s})ds+B^2_t, \quad t\ge 0,
\end{eqnarray}
where $B^2$ is the noise from the market, which we assume is independent of $B^1$ (this is by no means
necessary, we can certainly consider the filtering problem with correlated noises).
Now let us assume that the volatility $\sigma$ in (\ref{fvalue}) is affected by the actions of a large number of investors,
and all can only
make decisions based on the information from the process $Y$. Therefore, similar to \cite{CD2} (or \cite{HMC}) we begin by considering
$N$ individual investors, and assume that $i$-th investor's
private state dynamics is of the form:
\begin{eqnarray}
\label{playeri}
dU^i_t=\sigma^i(t, U^i_{\cdot\wedge t}, \bar \nu^N_t, \alpha^i_t)dB^{1,i}_t, \qquad t\ge 0, \quad 1\le i\le N,
\end{eqnarray}
where $B^{1,i}$'s are independent Brownian motions, and $\bar\nu^N_t$ denotes the empirical conditional distribution of
$U=(U^1, \cdots, U^N)$, given the (common) observation $Y=\{Y_t:t\ge0\}$, that is, $\bar\nu^N_t\stackrel{\triangle}{=}\frac1N\sum_{j=1}^N \delta_{\mathbb{E}[U^j_t|{\cal F}^Y_t]}$, where $\delta_x$ denotes the Dirac measure at $x$. More precisely, the notation in (\ref{playeri}) means (see, e.g., \cite{CD2}),
\begin{eqnarray}
\label{empirical}
\sigma^i(t, U^i_{\cdot\wedge t}, \bar \nu^N_t, \alpha^i_t)&\stackrel{\triangle}{=}&\int_{\mathbb{R}}\tilde\sigma^i(t, U^i_{\cdot\wedge t}, y, \alpha^i_t)\bar\nu^N_t(dy) \nonumber\\
&=& \frac1N\sum_{j=1}^N\int_{\mathbb{R}}\tilde\sigma^i(t, U^i_{\cdot\wedge t}, y, \alpha^i_t)\delta_{\mathbb{E}[U^j_t|{\cal F}^Y_t]}(dy)\\
&=& \frac1N\sum_{j=1}^N\tilde\sigma^i(t, U^i_{\cdot\wedge t},\mathbb{E}[U^j_t|{\cal F}^Y_t], \alpha^i_t). \nonumber
\end{eqnarray}
Here, $\tilde\sigma^i$'s are the functions defined on appropriate (Euclidean) spaces.
We now assume that each investor chooses an individual strategy to minimize
the cost; the cost functional of the $i$-th agent is of the form:
\begin{eqnarray}
\label{costi}
J^i(\alpha^i)\stackrel{\triangle}{=}\mathbb{E}\Big\{\Phi^i(U^i_T)+\int_0^TL^i(t,U^i_{\cdot\wedge t}, \bar\nu^N_t, \alpha^i_t)dt\Big\},\qquad 1\le i\le N,
\end{eqnarray}
Following the argument of Lasry and Lions \cite{LasryLions} (see also \cite{CD2, CD3, CD5, CZ, HMC}), if we assume that the game is {\it symmetric}, i.e., $\tilde\sigma^i=\tilde\sigma,\ L^{i}$ and $\Phi^{i}=\Phi$ are independent of $i$, and that the number of investors $N$ converges to $+\infty$, then under suitable technical conditions, one could
find (approximate) Nash equilibriums through a limiting dynamics, and assign
a representative investor the unified strategy $\alpha$, determined by a {\it conditional} McKean-Vlasov type SDE
\begin{eqnarray}
\label{McKeanV}
dX_t=\sigma(t, X_{\cdot\wedge t}, \mu_t, \alpha_t)dB^1_t, \quad t\ge 0,
\end{eqnarray}
where $\mu$ is the conditional distribution of $X_t$ given ${\cal F}^Y_t$, and
$$ \sigma(t, X_{\cdot\wedge t}, \mu_t, u_t)\stackrel{\triangle}{=}\int \sigma(t, X_{\cdot\wedge t}, y, u_t)\mu_t(dy)=\mathbb{E}\{\sigma(t, \varphi_{\cdot\wedge t}, \mathbb{E}[X_t|{\cal F}^Y_t], u)\}|_{\varphi
=X, u=u_t}.
$$
Furthermore, the value function becomes, with similar notations,
\begin{eqnarray}
\label{cost0}
V(x)=\inf_\alpha J(\alpha)\stackrel{\triangle}{=}\mathbb{E}\Big\{\Phi(X_T)+\int_0^TL(t,X_{\cdot\wedge t}, \mu_t, \alpha_t)dt\Big\}.
\end{eqnarray}
We note that (\ref{McKeanV}) and (\ref{cost0}), together with (\ref{stock}), form a stochastic control problem involving
CMFSDE dynamics and partial observations, as we are proposing.
The main objective of this paper is two-fold: We shall first study the exact meaning as well as the well-posedness of the dynamics, and then investigate the Stochastic Maximum Principle for the corresponding stochastic control problem. For the wellposedness of (\ref{dynamics}) we shall use a scheme that combines the idea of \cite{CD1} and the techniques of nonlinear filtering, and prove the existence and uniqueness of the solution to SDE (\ref{McKeanV}) via Schauder's fixed point theorem on $\mathscr{P}_2(\Omega)$, the space of probability measures with finite second moment, endowed with the 2-Wasserstein metric. We note that the
important elements in this argument include the so-called {\it reference probability space} that is often seen in the nonlinear
filtering theory and the Kallianpur-Striebel formula (cf. e.g., \cite{Bens, Zeit}), which enable us to define the solution mapping.
Our next task is to prove Pontryagin's Maximum Principle for our stochastic control problem. The main idea is similar to
earlier works of the first two authors (\cite{BLP, Li}), with some significant modifications. In particular, since in the present case
the control problem can only be carried out in a weak form, due to the lack of strong solution of CMFSDE, the existence of
the common reference probability space is essential. Consequently, extra efforts are needed to overcome the complexity caused
by the change of probability measures, which, together with the path-dependent nature of the underlying dynamic system,
makes even the first order adjoint equation more complicated than the traditional ones. To the best of our knowledge, the resulting mean-field backward SDE is new.
The paper is organized as follows. In Section 2 we provide all the necessary preparations, including some known
facts of nonlinear filtering. In Sections 3 and 4 we prove the well-posedness of the partially observable dynamics. In Section 5 we
introduce the stochastic control problem, and in Section 6 we study the variational equations and give some important
estimates. Finally, in Section 7 we prove the Pontryagin maximum principle.
\section{Preliminaries}
\setcounter{equation}{0}
Throughout this paper we consider the {\it canonical space} $(\Omega, {\cal F})$, where $\Omega\stackrel{\triangle}{=} \mathbb{C}_0([0,\infty);\mathbb{R}^{2d})= \{\omega\in \mathbb{C}([0,\infty);\mathbb{R}^{2d}): \omega_0 = {\bf 0}\}$, and ${\cal F}$ be its topological $\sigma$-field.
Let $\mathbb{F}=\{{\cal F}_t\}_{t\ge 0}$ be the natural filtration on $\Omega$, that is, for each $t\ge 0$,
${\cal F}_t$ is the topological $\sigma$-field of the space $\Omega_t\stackrel{\triangle}{=} \{\omega(\cdot\wedge t): \omega\in\Omega\}$.
For simplicity, throughout this paper we assume $d=1$, and that all the processes are 1-dimensional, although
the higher dimensional cases can be argued similarly without substantial difficulties.
Furthermore, we let $\mathscr{P}(\Omega)$ denote the space
of all probability measures on $(\Omega, {\cal F})$, and for each $\mathbb{P}\in \mathscr{P}(\Omega)$, we assume that $\mathbb{F}$ is $\mathbb{P}$-augmented so that the filtered probability space $(\Omega, {\cal F}, \mathbb{P}; \mathbb{F})$
satisfies the {\it usual hypotheses}.
Next, for given $T>0$ we denote $\mathbb{C}_T=\mathbb{C}([0,T])$ endowed by the supremum norm $\|\cdot\|_{\mathbb{C}_T}$, and let $\mathscr{B}(\mathbb{C}_T)$ be its topological $\sigma$-field. Consider now the space of all probability measures on $(\mathbb{C}_T, \mathscr{B}(\mathbb{C}_T))$, denoted by $\mathscr{P}(\mathbb{C}_T)$, and for $p\ge 1$ we let $\mathscr{P}_p(\mathbb{C}_T)
\subseteq \mathscr{P}(\mathbb{C}_T)$ be those that have finite $p$-th moment. We recall that the {\it $p$-Wasserstein metric} on
$\mathscr{P}_p(\mathbb{C}_T)$ is defined as a mapping
$W_p:\mathscr{P}_p(\mathbb{C}_T)\times\mathscr{P}_p(\mathbb{C}_T)\mapsto \mathbb{R}_+$ such that, for all $\mu, \nu\in\mathscr{P}_p(\mathbb{C}_T)$,
\begin{eqnarray}
\label{Wp-metric}
W_p(\mu,\nu) \stackrel{\triangle}{=} \inf\{(\int_{\mathbb{C}^2_T}\|x-y\|^p_{\mathbb{C}_T}\pi(dx,dy))^{\frac1p}: \pi\in \mathscr{P}_p(\mathbb{C}^2_T)~\mbox{with marginals $\mu$ and $\nu$}\}.
\end{eqnarray}
In this paper we shall use the 2-Wasserstein metric $W_2$, and abbreviate $(\mathscr{P}_2(\mathbb{C}_T), W_2)$ by $\mathscr{P}_2(\mathbb{C}_T)$. Since $\mathbb{C}_T$ is a separable Banach space, it is known that $\mathscr{P}_2(\mathbb{C}_T)$ is a separable and complete metric space. Furthermore, it is known that (cf. e.g., \cite{Villani}), for $\mu_n,\mu\in{\mathscr{P}}_2(\mathbb{C}_T)$,
\begin{eqnarray}\begin{array}{lcl}
\label{Wasserstein}
\lim_{n\to \infty}W_2(\mu_n,\mu)= 0 & \Longleftrightarrow&\, \, \mu_n\mathop{\buildrel w\over\rightarrow} \mu ~\mbox{ in $\mathscr{P}_2(\mathbb{C}_T)$ and, as } N\rightarrow +\infty,\\
& & ~ \displaystyle\sup_n \int_{\Omega} \|\varphi\|^2_{\mathbb{C}_T}I\{\|\varphi\|_{\mathbb{C}_T}\ge N\}\mu_n(d\varphi)\rightarrow 0.
\end{array}
\end{eqnarray}
Next, for any $\mathbb{P}\in\mathscr{P}(\Omega)$, $p,q\ge 1$, any sub-filtration $\mathbb{G}\subseteq \mathbb{F}$, and any Banach space $\mathbb{X}$, we denote
$L^p(\mathbb{P}; \mathbb{X})$ to be all $\mathbb{X}$-valued $L^p$-random variables under $\mathbb{P}$. In particular, we denote by $L^p(\mathbb{P};\mathbb{R})$ to be
all real valued $L^p$-random variables under $\mathbb{P}$. Further, we denote by $L^p_{\mathbb{G}}(\mathbb{P}; L^q([0,T]))$ the $L^p$-space of all $\mathbb{G}$-adapted processes $\eta$, such that
\begin{eqnarray}
\label{LpqPnorm}
\| \eta\|_{p,q,\mathbb{P}}\stackrel{\triangle}{=} \Big\{\mathbb{E}^{\mathbb{P}}\Big[\int_0^T|\eta_t|^qdt\Big]^{p/q}\Big\}^{1/p}<\infty.
\end{eqnarray}
If $p=q$, we simply write $L^p_\mathbb{G}(\mathbb{P};[0,T])\stackrel{\triangle}{=} L^p_\mathbb{G}(\mathbb{P}; L^p([0,T]))$. Finally,
we define $L^{\infty-}_{\mathbb{G}}(\mathbb{P};[0,T])\stackrel{\triangle}{=} \bigcap_{p>1} L^p_{\mathbb{G}}(\mathbb{P};[0,T])$ and $\mathscr{L}^{\infty-}_\mathbb{G}(\mathbb{P}; \mathbb{C}_T)
\stackrel{\triangle}{=} \bigcap_{p>1} L^p_\mathbb{G}(\mathbb{P};\mathbb{C}_T)$, where $L^p_\mathbb{G}(\mathbb{P}; \mathbb{C}_T)$ is the space of all continuous, $\mathbb{F}$-adapted,
processes $\xi=\{\xi_t\}$ such that $\|\xi\|_{\mathbb{C}_T}\in L^p(\mathbb{P};\mathbb{R})$.
We will often drop ``$\mathbb{P}$" from the subscript/superscript when the context is clear.
We now give a more precise description of
the SDEs (\ref{dynamics}), in terms of the
standard McKean-Vlasov SDE. Again we consider only the case $b=0$, and we assume further that $\hat\sigma=1$\ in (\ref{dynamics}) for simplicity.
We begin by introducing some notations. Let $X$ be the state process and $Y$ the observation process, defined on $(\Omega, {\cal F}, \mathbb{P})$, for some $\mathbb{P}\in\mathscr{P}(\Omega)$. We denote the ``filtered" state process by $U^{X|Y}_t=\mathbb{E}^\mathbb{P} [X_t|{\cal F}^Y_t]$, $t\ge0$. Since (as we show in Lemma \ref{contiU} below) the process
$U^{X|Y}$ is continuous, we denote its law
under $\mathbb{P} $ on $\mathbb{C}_T$ by $\mu^{X|Y}=\mathbb{P} \circ [U^{X|Y}]^{-1}\in\mathscr{P}(\mathbb{C}_T)$. Next, let $P_t(\varphi)=\varphi(t)$,
$\varphi\in \mathbb{C}_T$, $t\ge0$, be the projection mapping, and define $\mu^{X|Y}_t=\mu^{X|Y}\circ {P_t}^{-1}$. Then, for any $\varphi\in \mathbb{C}_T$, and
$u\in \mathbb{R}$,
we can write
$$ \mathbb{E} [\sigma(t, \varphi_{\cdot\wedge t}, \mathbb{E} [X_{t}|{\cal F}^Y_t], u)]=\int\sigma(t, \varphi_{\cdot\wedge t}, y, u)\mu^{X|Y}_t(dy)
\stackrel{\triangle}{=}\sigma(t, \varphi_{\cdot\wedge t}, \mu^{X|Y}_t, u).
$$
We should note that since the dynamics $X$ is non-observable, the decision of the controller
can only be made based on the information observed from the process $Y$. Therefore, it is reasonable
to assume that the control process $u$ is $\mathbb{F}^Y=\{{\cal F}^Y_t\}_{t\ge 0}$ adapted (or progressively measurable).
We should remark that, for a given such control, it is by no means clear that the state-observation SDEs will
have a strong solution on a prescribed probability space, as we shall see from our well-posedness result in the next sections. We therefore consider a ``weak formulation" which we now describe. Consider the pairs $( \mathbb{P}, u)$, where $\mathbb{P}\in\mathscr{P}(\Omega)$, $u \in L^2_{\mathbb{F}}(\mathbb{P};[0,T])$, such that the following SDEs are well-defined:
\begin{eqnarray}
\label{controlsys}
X_t&=&x+\int_0^t\mathbb{E}^{\mathbb{P}}[\sigma(s, \varphi_{\cdot\wedge s}, \mathbb{E}^{\mathbb{P}}[X_s|{\cal F}^Y_s], z)]\Big|_{\varphi=X, z=u_s}
dB^1_s\\
&=&
x+\int_0^t\int_{\mathbb{R}}\sigma(s, X_{\cdot\wedge s}, y, u_s)\mu_s(dy)dB^1_s=x+\int_0^t\sigma(s, X_{\cdot\wedge s}, \mu_s, u_s)dB^1_s,
\nonumber\\
\label{observation}
Y_t&=&\int_0^th(s, X_{ s})ds + B^2_t, \qquad t \ge 0,
\end{eqnarray}
where $(B^1,B^2)$ is a standard 2-$d$ Brownian motion under $\mathbb{P}$, and $\mu_t(\cdot)\stackrel{\triangle}{=} \mathbb{P}\circ \mathbb{E}^{\mathbb{P}}[X_t|{\cal F}^Y_t]^{-1}(\cdot)$ is the distribution, under $\mathbb{P}$, of the conditional expectation of
$X_t$, given ${\cal F}^Y_t$. We note that we {\it do not} require that the solution to (\ref{controlsys}) and (\ref{observation}) (or probability
$\mathbb{P}$ for given $u$) be unique(!).
Now let $U$ be a convex subset of $\mathbb{R}^k$. For simplicity, assume $k=1$.
\begin{defn}
\label{admissible}
A pair $(\mathbb{P}, u)\in \mathscr{P}(\Omega)\times L^2_{\mathbb{F}}(\mathbb{P};[0,T])$ is called an ``admissible control" if
{\rm (i)} $u_t\in U$, for all $t\in[0, T]$, and $B=(B^1,B^2)$ is a $(\mathbb{F},\mathbb{P})$-Brownian motion;
\medskip
{\rm (ii)} There exist processes $(X, Y)\in L^2_{\mathbb{F}}(\mathbb{P}; [0, T])$ satisfying SDEs (\ref{controlsys}) and (\ref{observation}); and
\medskip
{\rm (iii)} $u\in L^{\infty-}_{\mathbb{F}^Y}(\mathbb{P}; [0,T])$.
\hfill \vrule width.25cm height.25cm depth0cm\smallskip
\end{defn}
We shall denote the set of all admissible controls by $\mathscr{U}_{ad}$. For simplicity, we often write $u\in\mathscr{U}_{ad}$, and
denote the associated probability measure(s) $\mathbb{P}$ by $\mathbb{P}^u$, for $u\in\mathscr{U}_{ad}$.
\begin{rem}
\label{rem1}
{\rm As we will shall see later, under our standing assumptions to every control $u\in\mathscr{U}_{ad}$ there is only one probability measure $\mathbb{P}^{u}$ associated.
We should note, however, that
unlike the traditional filtering problem, the main difficulty of SDE (\ref{controlsys})-(\ref{observation}) lies in
the mutual dependence between the solution pair $X^u$ and $Y$, via
the law of conditional expectation
$\mu^u_t=\mathbb{P}^u\circ \mathbb{E}^{\mathbb{P}^u}[X^u_t|{\cal F}^Y_t]^{-1}$ in the coefficients.
Moreover, the requirement that $u$ is $\mathbb{F}^Y$-adapted
adds an additional seemingly ``circular" nature to the problem. Thus, the well-posedness of the problem is far from obvious, and will be the main subject of \S3. \hfill \vrule width.25cm height.25cm depth0cm\smallskip}
\end{rem}
We note that under the weak formulation the state-observation processes $(X^u, Y)$ are often defined on different probability spaces. To facilitate our discussion we shall designate a common space on which all the controlled dynamics can be evaluated. In light of the nonlinear filtering theory, we make the following assumption.
\begin{assum}
\label{Assump2}
There exists a probability measure $\mathbb{Q}^0$ on $(\Omega, {\cal F})$, such that, under $\mathbb{Q}^0$, $(B^1, Y)$ is a 2-dimensional Brownian motion, where
$Y$ is the observation process.
\hfill \vrule width.25cm height.25cm depth0cm\smallskip
\end{assum}
We note that the probability measure $\mathbb{Q}^0$ is commonly known as the ``reference probability measure" in nonlinear filtering theory. The existence of such measure can be argued once the existence of the weak solution of (\ref{controlsys})-(\ref{observation}) is known.
Indeed, suppose that $u\in \mathscr{U}_{ad}$ and $\mathbb{P}^u\in \mathscr{P}(\Omega)$ is the associated probability such that the SDEs (\ref{controlsys}) and (\ref{observation}) have a solution $(X^u,Y)$ on $(\Omega, {\cal F}, \mathbb{P}^u)$.
Consider the following SDE:
\begin{eqnarray}
\label{barL}
\bar L_t=1-\int_0^t h(s, X^u_{s}) \bar L_sdB^2_s=1+\int_0^t \bar L_s dZ^u_s,
\end{eqnarray}
where $Z^u_t=-\int_0^t h(s, X^u_{ s})dB^2_s$. We denote its solution by $\bar L^u$. Then, under appropriate conditions on $h$,
both $Z^u$ and $\bar L^u$ are $\mathbb{P}^u$-martingales, and $\bar L^u$ is the stochastic exponential:
\begin{eqnarray}
\label{barLexp}
\bar L^u_t=\exp\Big\{Z^u_t-\frac12 \langle Z^u\rangle_t\Big\}=\exp\Big\{-\int_0^th(s, X^u_{s})dB^2_s-\frac12 \int_0^t
|h(s, X^u_{s})|^2ds\Big\}.
\end{eqnarray}
Thus, the Girsanov Theorem suggests that $d\mathbb{Q}^0=\bar{L}_T^{u}d\mathbb{P}^{u}$ defines a new probability measure $\mathbb{Q}^0$ under which
$(B^1,Y)$ is a Brownian motion, hence a ``reference measure".
The essence of Assumption \ref{Assump2} is, therefore, to assign a {\it prior distribution} on the observation process $Y$ {\it before} the well-posedness of the control system is established. In fact, with such an assumption one can begin by assuming that $(B^1, Y)$
is the canonical process (i.e., $(B^1_t, Y_t)(\omega)=\omega(t)$, $\omega\in\Omega$) and $\mathbb{Q}^0$ the Wiener measure on $(\Omega, {\cal F})$, and then
proceed to prove the existence of the weak solution of the system (\ref{controlsys}) and (\ref{observation}). This
scheme will be carried out in details in \S3.
Continuing with our control problem, for any $u\in\mathscr{U}_{ad}$, we define the {\it cost functional} by
\begin{eqnarray}
\label{cost}
J(t, x; u)&\stackrel{\triangle}{=}& \mathbb{E}^{\mathbb{Q}^0}\Big\{\int_t^T f(s, X^u_{\cdot\wedge s}, \mu^u_s, u_s)ds+\Phi(X^u_T, \mu^u_T)\Big\}\nonumber\\
&=&\mathbb{E}^{\mathbb{Q}^0}\Big\{\int_t^T\mathbb{E}^{\mathbb{P}^u}[f(s, \varphi_{\cdot\wedge s}, \mathbb{E}^{\mathbb{P}^u}[X^u_s|{\cal F}^Y_s], u)]\Big|_{\varphi=X^u, u=u_s}ds\\
&&+\mathbb{E}^{\mathbb{P}^u}[\Phi(x,\mathbb{E}^{\mathbb{P}^u}[X^u_T|{\cal F}^Y_T])]\Big|_{x=X^u_T}\Big\},\nonumber
\end{eqnarray}
and we denote the value function as
\begin{eqnarray}
\label{value}
V(t, x)\stackrel{\triangle}{=} \inf_{u\in\mathscr{U}_{ad}} J(t, x; u).
\end{eqnarray}
We shall make use of the following {\it Standing Assumptions} on the coefficients.
\begin{assum}
\label{Assum1}
{\rm (i)} The mappings $(t, \varphi,x, y, z)\mapsto \sigma(t, \varphi_{\cdot\wedge t},y, z)$, $h(t,x)$, $f(t, \varphi_{\cdot\wedge t}, y, z)$, and $\Phi(x, y)$ are bounded and continuous, for $(t,\varphi,x,y, z)\in [0,T]\times\mathbb{C}_T\times\mathbb{R}\times \mathbb{R}\times U$;
\smallskip
{\rm (ii)} The partial derivatives $\partial_y\sigma$, $\partial_z\sigma$, $\partial_y f$, $\partial_z f$, $\partial_x h$, $\partial_x\Phi$, $\partial_y\Phi$ are bounded and continuous, for $(\varphi, x, y,z)\in \mathbb{C}_T\times \mathbb{R}\times\mathbb{R}\times U$, uniformly in $t\in[0,T]$;
\smallskip
{\rm (iii)} The mappings $\varphi\mapsto \sigma(t, \varphi_{\cdot\wedge t}, y, z), f(t, \varphi_{\cdot\wedge t}, y, z)$, as functionals from $\mathbb{C}_T$ to $\mathbb{R}$, are
Fr\'echet differentiable. Furthermore, there exists a family of measures $\{\ell(t, \cdot)\}|_{t\in[0,T]}$, satisfying $0\le \int_0^T\ell(t,ds)\le C$, for all $t\in[0,T]$, such that both derivatives, denoted by $D_\varphi\sigma=D_\varphi\sigma(t, \varphi_{\cdot\wedge t}, y, z)$ and $D_\varphi f=D_\varphi f(t, \varphi_{\cdot\wedge t}, y, z)$, respectively, satisfy
\begin{eqnarray}
\label{DsiDf}
|D_\varphi\sigma(t, \varphi_{\cdot\wedge t}, y, z)(\psi)|+|D_\varphi f(t, \varphi_{\cdot\wedge t}, y, z)(\psi)|\le \int_0^T|\psi(s)|\ell(t, ds), \ \ \psi\in\mathbb{C}_T,
\end{eqnarray}
uniformly in $(t, \varphi, y,z)$;
\smallskip
{\rm (iv)} The mapping $y\mapsto y\partial_y\sigma(t, \varphi_{\cdot\wedge t}, y, z)$ is uniformly bounded, uniformly in $(t, \varphi,z)$;
\smallskip
{\rm (v)} The mapping $x\mapsto x\partial_x h(t, x)$ is bounded, uniformly in $(t,x)\in[0,T]\times\mathbb{R}$;
\smallskip
{\rm (vi)} The mappings $x\mapsto x h(t,x), x^2\partial_x h(t,x)$ are bounded, uniformly in $(t,x)\in[0,T]\times\mathbb{R}$.
\hfill \vrule width.25cm height.25cm depth0cm\smallskip
\end{assum}
We note that some of the assumptions above are merely
technical and can be improved, but we prefer not to dwell on such technicalities and focus on the
main ideas instead.
\begin{rem}
\label{remark3}
{\rm Note that if $(t, \varphi, y, z)\mapsto\phi(t, \varphi_{\cdot\wedge t}, y, z)$ is a function defined on $[0,T]\times\mathbb{C}_T\times \mathbb{R}\times \mathbb{R}$
satisfying Assumption \ref{Assum1}-(i), (ii), then for any $\mu\in\mathscr{P}_2(\mathbb{C}_T)$, we can define a function on the space $[0,T]\times\Omega\times\mathbb{C}_T\times \mathscr{P}_2(\mathbb{C}_T)\times U$:
\begin{eqnarray}
\label{barphi}
\bar\phi(t,\omega, \varphi_{\cdot\wedge t}, \mu_t, z)\stackrel{\triangle}{=} \int_\mathbb{R}\phi(t, \varphi_{\cdot\wedge t}, y, z)\mu_t(dy),
\end{eqnarray}
where $\mu_t=\mu\circ P_t^{-1}$ and $P_t(\varphi)\stackrel{\triangle}{=}\varphi(t)$, $(t, \varphi)\in [0,T]\times\mathbb{C}_T$. Then, $\bar \phi$ must satisfy
the following Lipschitz condition:
\begin{eqnarray}
\label{Lip}
|\bar\phi(t, \varphi^1_{\cdot\wedge t}, \mu^1_t, z^1)-\bar\phi(t, \varphi^2_{\cdot\wedge t}, \mu^2_t, z^2)|\le K\Big\{\|\varphi^1-
\varphi^2\|_{\mathbb{C}_t}+ W_2(\mu^1, \mu^2)+|z^1-z^2|\Big\},
\end{eqnarray}
where $\|\cdot\|_{\mathbb{C}_t}$ is the sup-norm on $\mathbb{C}([0,t])$ and $W_2(\cdot, \cdot)$ is the 2-Wasserstein metric.
\hfill \vrule width.25cm height.25cm depth0cm\smallskip}
\end{rem}
\begin{rem}
\label{remark6}
{\rm
The Fr\'echet derivatives $D_\varphi\sigma$ and $D_\varphi f$ by definition belong to $\mathbb{C}_T^*\stackrel{\triangle}{=} \mathscr{M}[0,T]$, the
space of all finite signed Borel measures on $[0,T]$, endowed with the total variation norm $|\cdot|_{TV}$ (with a slight abuse
of notation, we still denote it by $|\cdot|$). Thus the Assumption \ref{Assum1}-(iii)
amounts to saying that, as measures,
\begin{eqnarray}
\label{dominate}
|D_\varphi\sigma(t, \varphi_{\cdot\wedge t}, y, z)(ds)|+|D_\varphi f(t, \varphi_{\cdot\wedge t}, y, z)(ds)|\le \ell(t, ds), \quad \forall (t, \varphi, y, z).
\end{eqnarray}
This inequality will be crucial in our discussion in Section 7.
\hfill \vrule width.25cm height.25cm depth0cm\smallskip}
\end{rem}
To end this section we recall some basic facts in nonlinear filtering theory, adapted to our situation.
We begin by considering the inverse Girsanov kernel of $\bar L^u$ defined by (\ref{barLexp}):
\begin{eqnarray}
\label{Lexp}
L^u_t\stackrel{\triangle}{=} [\bar L^u_t]^{-1}=\exp\Big\{\int_0^th(s, X^u_{s})dY_s-\frac12 \int_0^t
|h(s, X^u_{s})|^2ds\Big\}, \ \ t\in [0,T].
\end{eqnarray}
Then $L^u$ is a $\mathbb{Q}^0$-martingale, $d\mathbb{P}^u=L^u_Td\mathbb{Q}^0$, and $L^u$
satisfies the following SDE on $(\Omega, {\cal F}, \mathbb{Q}^0)$:
\begin{eqnarray}
\label{L}
L_t=1+\int_0^t h(s, X_{s}) L_sdY_s, \qquad t\in[0,T].
\end{eqnarray}
Let us now denote $L=L^u$ for simplicity. An important ingredient that we are going to use frequently is the SDEs known as the {\it Kushner-Stratonovic} or {\it Fujisaki-Kallianpur-Kunita} (FKK) equation for the ``normalized conditional probability". Let us denote
\begin{eqnarray}
\label{S}
S_t\stackrel{\triangle}{=} \mathbb{E}^{\mathbb{Q}^0}[L_tX_t|{\cal F}^Y_t], \quad S^{0}_t\stackrel{\triangle}{=} \mathbb{E}^{\mathbb{Q}^0}[L_t|{\cal F}^Y_t], \quad t\ge 0.
\end{eqnarray}
Since under $\mathbb{Q}^0$ the process $(B^1,Y)$ is a Brownian motion, the $\sigma$-field ${\cal F}^Y_{t, T}$ and ${\cal F}^Y_t\vee {\cal F}^{B^1}_t$
are independent, where ${\cal F}^Y_{t, T}\stackrel{\triangle}{=} \sigma\{Y_r-Y_t: t\le r\le T\}$.
It is standard to show that (in light of (\ref{L})) $S$ and $S^0$ satisfy the following SDEs:
\begin{eqnarray}
\label{Sn0}
S^{0}_t=1+\int_0^t\mathbb{E}^{\mathbb{Q}^0}[ h(s, X_{s})L_s|{\cal F}^Y_s]dY_s, \quad t\ge 0.
\end{eqnarray}
and
\begin{eqnarray}
\label{SDES}
S_t=x+\int_0^t \mathbb{E}^{\mathbb{Q}^0}[L_sX_sh(s, X_{s})|{\cal F}^Y_s]dY_s, \quad t\ge 0.
\end{eqnarray}
Furthermore, let $U_t\stackrel{\triangle}{=} \mathbb{E}^{\mathbb{P}^u}[X_t|{\cal F}^Y_t]$, $t\ge 0$. Then, by the Bayes formula (also known as the Kallianpur-Striebel formula, see, e.g., \cite{Bens}) we have
\begin{eqnarray}
\label{Bayes}
U_t=\frac{\mathbb{E}^{\mathbb{Q}^0}[L_tX_t|{\cal F}^Y_t]}{\mathbb{E}^{\mathbb{Q}^0}[L_t|{\cal F}^Y_t]}=
\frac{S_t}{S^0_t}, \quad t\ge 0, \quad \mathbb{Q}^0\mbox{-a.s.}
\end{eqnarray}
A simple application of It\^o's formula and some direct computation then lead to the following FKK equation:
\begin{eqnarray}
\label{FKK}
dU_t&=&\Big\{\mathbb{E}^{\mathbb{P}^u}[X_th(t, X_{t})|{\cal F}^Y_t]-\mathbb{E}^{\mathbb{P}^u}[X_t|{\cal F}^Y_t]\mathbb{E}^{\mathbb{P}^u}
[h(t,X_{ t})|{\cal F}^Y_t]\Big\}dY_t\\
&&+\Big\{\mathbb{E}^{\mathbb{P}^u}[X_t|{\cal F}^Y_t]\big\{\mathbb{E}^{\mathbb{P}^u}[h(t,X_{ t})|{\cal F}^Y_t]\big\}^2-\mathbb{E}^{\mathbb{P}^u}[X_th(t,X_{ t})|{\cal F}^Y_t]\mathbb{E}^{\mathbb{P}^u}[h(t,X_{t})|{\cal F}^Y_t]\Big\}dt.\nonumber
\end{eqnarray}
In fact, one can easily show that
\begin{eqnarray}
\label{ZakaivsFKK}
S_t&=&U_t\exp\Big\{\int_0^t\mathbb{E}^{\mathbb{P}^u}[h(s,X_{s})|{\cal F}^Y_s]dY_s-\frac12\int_0^t\mathbb{E}^{\mathbb{P}^u}[h(s,X_{ s})|{\cal F}^Y_s]^2ds\Big\}.
\end{eqnarray}
\section{Well-posedness of the State-Observation Dynamics}
\setcounter{equation}{0}
In this and next sections we investigate the well-posedness of the controlled state-observation system (\ref{controlsys}) and (\ref{observation}). More precisely, we shall argue that the admissible control set $\mathscr{U}_{ad}$, defined by Definition \ref{admissible}, is not empty. We first note that, for a fixed $\mathbb{P}\in\mathscr{P}(\Omega)$ and
$u\in L^{\infty-}_{\mathbb{F}^Y}(\mathbb{P}, [0,T])$, if we define
\begin{eqnarray}
\label{sigma}
\phi^u(t,\omega, \varphi_{\cdot\wedge t}, \mu_t)\stackrel{\triangle}{=} \int_\mathbb{R}\phi(t, \varphi_{\cdot\wedge t}, y, u_t(\omega))\mu_t(dy),
\end{eqnarray}
where $\phi=b, \sigma$, then we can write the control-observation system (\ref{controlsys}) and (\ref{observation}) as
a slightly more generic form (denoting $b^u=b$ and $\sigma^u=\sigma$ for simplicity):
\begin{eqnarray}
\label{SDE}
\left\{\begin{array}{lll}
\displaystyle X_t=x+\int_0^tb(s,\cdot, X_{\cdot\wedge s}, \mu^{X|Y}_s)ds+\int_0^t\sigma(s,\cdot, X_{\cdot\wedge s}, \mu^{X|Y}_s)dB^1_s;\medskip\\
\displaystyle Y_t=\int_0^t h(s, X_{s})ds+B^2_t,
\end{array}\right. \qquad t\ge 0,
\end{eqnarray}
where $B=(B^1,B^2)$ is a $\mathbb{P}$-Brownian motion, and $\mu^{X|Y}_t=\mathbb{P}\circ [\mathbb{E}^{\mathbb{P}}[X_t|{\cal F}^Y_t]]^{-1}$. Our task is to prove the well-posedness of SDE (\ref{SDE}) in a {\it weak} sense (i.e., including the existence of
the probability measure $\mathbb{P}$(!)). In light of Remark \ref{remark3},
we shall assume that the coefficients $b$ and $\sigma$
in (\ref{SDE}) satisfy the following assumptions that are slightly weaker than Assumption \ref{Assum1}, but sufficient for our
purpose in this section.
\begin{assum}
\label{Assum2}
The coefficients $b, \sigma: [0, T]\times \mathbb{C}_T\times \mathscr{P}_2(\mathbb{C}_T)\mapsto \mathbb{R}$ enjoy the following properties:
{\rm(i)} For fixed $(\varphi, \mu)\in \mathbb{C}_T\times \mathscr{P}_2(\mathbb{C}_T)$, the mapping $(t,\omega)\mapsto (b, \sigma)(t,\omega, \varphi, \mu)$ is an $\mathbb{F}$-progressively
measurable process;
{\rm(ii)} For fixed $t\in [0,T]$, and $\mathbb{Q}^0$-a.e. $\omega\in\Omega$, there exists $K>0$, independent of $(t,\omega)$, such that for all
$(\varphi^1,\mu^1), (\varphi^2,\mu^2)\in \mathbb{C}_T\times \mathscr{P}_2(\mathbb{C}_T)$, it holds that
\begin{eqnarray}
\label{Lip2}
|\phi(t,\omega, \varphi^1_{\cdot\wedge t}, \mu^1_t) -\phi(t,\omega, \varphi^2_{\cdot\wedge t}, \mu^2_t)|\le K(\sup_{t\in[0,T]}|\varphi^1_t-\varphi^2_t|+W_2(\mu^1, \mu^2)),
\end{eqnarray}
for $\phi=b, \sigma$, respectively.
\hfill \vrule width.25cm height.25cm depth0cm\smallskip
\end{assum}
In the rest of the section we shall still assume $b=0$, as it does not add extra difficulties.
Now assume that $(X,Y)$ satisfies (\ref{SDE}) under $\mathbb{P}$, and
let us denote
$U^{X|Y}_t\stackrel{\triangle}{=} \mathbb{E}^{\mathbb{P}}[X_t|{\cal F}^Y_t]$, $t\ge0$. (We note that $U^{X|Y}$ should be understood as the ``optional
projection" of $X$ onto $\mathbb{F}^Y$!) We first check that $U^{X|Y}$ is indeed a continuous process.
\begin{lem}
\label{contiU}
Assume that Assumption \ref{Assum1} holds. Then $U^{X|Y}$ admits a continuous version.
\end{lem}
{\it Proof.} First note that $\mathbb{P}\sim \mathbb{Q}^0$, and $X$ has continuous paths, $\mathbb{P}$-a.s. By Bayes formula (\ref{Bayes}) we can write
$ U^{X|Y}_t=\frac{\mathbb{E}^{\mathbb{Q}^0}[L_tX_t|{\cal F}^Y_t]}{\mathbb{E}^{\mathbb{Q}^0}[L_t|{\cal F}^Y_t]}=\frac{S_t}{S^0_t}$, where $S^0$ and $S$ satisfy
(\ref{Sn0}) and (\ref{SDES}), respectively, and $L$ satisfies (\ref{L}). Clearly, the representations (\ref{Sn0}) and (\ref{SDES}) indicate that both $S^0$ and $S$ have continuous paths, thus $U^{X|Y}$ must have a continuous version.
\hfill \vrule width.25cm height.25cm depth0cm\smallskip
\medskip
We now define $\mu^{X|Y}(\cdot)=\mathbb{P}\circ [U^{X|Y}]^{-1}(\cdot)$, and $\mu^{X|Y}_t(\cdot)=\mathbb{P}\circ [U^{X|Y}_t]^{-1}(\cdot)$, for any $t\ge0$.
Lemma \ref{contiU} then implies that
$\mu^{X|Y}\in\mathscr{P}_2(\mathbb{C}_T)$, justifying the definition of SDE (\ref{SDE}). In what follows when the context is clear, we shall omit ``$X|Y$" from the superscript.
We note that the special circular nature of SDE (\ref{SDE}) between its solution and
its law of the conditional expectation (whence the underlying probability) makes it necessary to specify the meaning of a solution.
We have the following definition.
\begin{defn}[Weak Solution]
\label{sol}
An eight-tuple $(\Omega, {\cal F}, \mathbb{P}, \mathbb{F}, X, Y, B^1,B^2)$ is called a solution
to the filtering equation (\ref{SDE}) if
{\rm(i)} $(\Omega, {\cal F})$ is the canonical space, $\mathbb{P}\in \mathscr{P}(\Omega)$, and $\mathbb{F}$ is the canonical filtration;
{\rm(ii)} $(B^1, B^2)$ is a 2-dimensional $\mathbb{F}$-Brownian motion under $\mathbb{P}$;
{\rm(iii)} $(X, Y)$ is an $\mathbb{F}$-adapted continuous process such that (\ref{SDE}) holds for all $t\in[0,T]$, $\mathbb{P}$-almost surely.
\end{defn}
To prove the well-posedness we shall use a generalized version of the Schauder Fixed Point Theorem (see Cauty \cite{Cauty}, or
a recent generalization
in \cite{Cauty2}). To this end we consider the
following subset of $\mathscr{P}_2(\mathbb{C}_T)$:
\begin{eqnarray}
\label{cE}
\mathscr{E}\stackrel{\triangle}{=} \Big\{\mu\in \mathscr{P}_2(\mathbb{C}_T)
\big|\sup_{t\in [0,T]}\int_{\mathbb{R}}|y|^4\mu_t(dy)<\infty\Big\}.
\end{eqnarray}
In the above $\mu_t=\mu\circ {P_t}^{-1}\in\mathscr{P}_2(\mathbb{R})$, and $P_t(\varphi)=\varphi(t)$, $\varphi\in \Omega$, is the projection mapping.
Clearly, $\mathscr{E}$ is a convex subset of $\mathscr{P}_2(\mathbb{C}_T)$.
We now construct a mapping $\mathscr{T}:\mathscr{E}\mapsto \mathscr{E}$, whose fixed point, if exists, would give a solution to the SDE (\ref{SDE}).
We shall begin with
the reference probability space $(\Omega, {\cal F}, \mathbb{Q}^0)$, thanks to Assumption \ref{Assump2}, then $(B^1, Y)$ is a $\mathbb{Q}^0$-Brownian motion.
We may assume without loss of generality that $(B^1,Y)$ is the canonical process,
and $\mathbb{Q}^0$ is the Wiener measure.
For any $\mu\in \mathscr{E}$ we consider the SDE on the space $(\Omega, {\cal F}, \mathbb{Q}^0)$:
\begin{eqnarray}
\label{fileq2}
X_t=x+\int_0^t \sigma(s, \cdot, X_{\cdot\wedge s}, \mu_s)dB^1_s, \quad t\ge 0.
\end{eqnarray}
Note that as the distribution $\mu$ is given, (\ref{fileq2}) is an ``open-loop" SDE with ``functional Lipschitz" coefficient, thanks to
Assumption \ref{Assum2}. Thus,
there exists a unique (strong) solution to (\ref{fileq2}), which we denote by $X=X^\mu$.
Now, using $X^\mu$ we define the process $L^\mu=\{L^\mu_t\}_{t\ge0}$ as in (\ref{Lexp}) on probability space $(\Omega, {\cal F}, \mathbb{Q}^0)$,
and then we define the probability $d\mathbb{P}^\mu\stackrel{\triangle}{=} L^\mu_Td\mathbb{Q}^0$. By the Kallianpur-Striebel formula (\ref{Bayes}) we can define a process
\begin{eqnarray}
\label{KS}
U^\mu_t\stackrel{\triangle}{=}\mathbb{E}^{\mathbb{P}^\mu}[X^\mu_t|{\cal F}^Y_t]=\frac{\mathbb{E}^{\mathbb{Q}^0}[L^\mu_tX^\mu_t|{\cal F}^Y_t]}{\mathbb{E}^{\mathbb{Q}^0}[L^\mu_t|{\cal F}^Y_t]}=\frac{S^\mu_t}{S^{\mu,0}_t}, \quad t\ge0,
\end{eqnarray}
where $S^\mu_t\stackrel{\triangle}{=} \mathbb{E}^{\mathbb{Q}^0}[L^\mu_tX^\mu_t|{\cal F}^Y_T]$, $S^{\mu,0}_t\stackrel{\triangle}{=} \mathbb{E}^{\mathbb{Q}^0}[L^\mu_t|{\cal F}^Y_T]$, $t\ge 0$,
and then we denote
\begin{eqnarray}
\label{Tnu}
\mathscr{T}(\mu)\stackrel{\triangle}{=} \nu^\mu= \mathbb{P}^\mu\circ[U^\mu]^{-1}\in\mathscr{P}(\mathbb{C}_T).
\end{eqnarray}
Our task is to show that the solution mapping $\mathscr{T}: \mu\mapsto \nu^\mu$ satisfies the desired assumptions for
Schauder's Fixed Point Theorem.
\begin{thm}
\label{compact}
The solution mapping $\mathscr{T}:\mathscr{E}\to \mathscr{P}_2(\mathbb{C}_T)$ enjoys the following properties:
(1) $\mathscr{T}(\mathscr{E})\subseteq \mathscr{E}$;
(2) $\mathscr{T}(\mathscr{E})$ is compact under 2-Wasserstein metric.
(3) $\mathscr{T}:(\mathscr{E},W_1(\cdot,\cdot))\rightarrow (\mathscr{P}_2(\mathbb{C}_T),W_2(\cdot,\cdot))$ is continuous, i.e., whenever $\mu,\mu^n\in \mathscr{E},\, n\ge 1,$ is such that $W_1(\mu^n,\mu)\rightarrow 0$, we have that $W_2(\mathscr{T}(\mu^n),\mathscr{T}(\mu))\rightarrow 0.$
\end{thm}
We remark that an immediate consequence of (3) is that $\mathscr{T}:\mathscr{E}\rightarrow \mathscr{P}_2(\mathbb{C}_T)$ is continuous under both the 1- and the 2-Wasserstein metrics. Moreover, the compactness of $\mathscr{T}(\mathscr{E})$ under the 2-Wasserstein metric stated in (2) implies that in the 1-Wasserstein metric.
{\it Proof.} (1) Given $\mu\in \mathscr{E}$ we need only show that
\begin{eqnarray}
\label{moment4}
\sup_{t\in[0,T]}\int_{\mathbb{R}}|y|^4\nu^\mu_t(dy)<\infty.
\end{eqnarray}
To see this we note that for $t\in[0,T]$, by Jensen's inequality,
\begin{eqnarray*}
\int_{\mathbb{R}}|y|^4\nu^\mu_t(dy)=\int_{\mathbb{R}}|y|^4\mathbb{P}^\mu\circ [U^\mu]^{-1}(dy)=\mathbb{E}^{\mathbb{P}^\mu}[|\mathbb{E}^{\mathbb{P}^\mu}[X^\mu_t|{\cal F}^Y_t]|^4]
\le \mathbb{E}^{\mathbb{P}^\mu}[|X^\mu_t|^4].
\end{eqnarray*}
Since under $\mathbb{Q}^0$, $B^1$ is also a Brownian motion, it is standard to argue that, as $X^\mu$ is the solution to
the SDE (\ref{fileq2}), it holds that
\begin{eqnarray}
\label{momentQ0}
\sup_{0\le t\le T} \mathbb{E}^{\mathbb{Q}^0}[|X^\mu_t|^{2n}] \le C(1+|x|^{2n}), \qquad \mbox{for all } n\in\mathbb{N}.
\end{eqnarray}
Furthermore, noting that the process $L^\mu$ is an
$L^2$-martingale under $\mathbb{Q}^0$, we have
\begin{eqnarray*}
\sup_{0\le t\le T}\int_{\mathbb{R}^d}|y|^4\nu^\mu_t(dy)
&\le& \sup_{0\le t\le T}\mathbb{E}^{\mathbb{P}^\mu}\Big[|X^\mu_t|^4\Big]
=\sup_{0\le t\le T}\mathbb{E}^{\mathbb{Q}^0}\Big[L^\mu_T|X^\mu_t|^4\Big]\\
&\le&\big(\mathbb{E}^{\mathbb{Q}^0}[|L^\mu_T|^2]\big)^{\frac12}\sup_{0\le t\le T}\mathbb{E}^{\mathbb{Q}^0}\Big[|X^\mu_t|^8\Big]^{\frac12}
<\infty,
\end{eqnarray*}
thanks to (\ref{momentQ0}). In other words, $\nu^\mu=\mathscr{T}(\mu)\in\mathscr{E}$, proving (1).
(2) We shall prove that for any sequence $\{\mu^n_t\}\subseteq \mathscr{E}$, there exists a subsequence,
denoted by $\{\mu^{n}_t\}$ itself, such that $\lim_{n\to\infty}\mathscr{T}(\mu^{n})= \nu$ in 2-Wasserstein metric, for some
$\nu\in \mathscr{T}(\mathscr{E})$.
In light of the equivalence relation (\ref{Wasserstein}), we shall first argue that the family $\{\mathscr{T}(\mu^n)\}_{n\ge1}$ is tight. To this end, recall that
\begin{eqnarray}
\label{Un}
U^n_t=\mathbb{E}^{\mathbb{P}^n}[X^n_t|{\cal F}^Y_t]=\frac{S^n_t}{S^{n,0}_t},
\end{eqnarray}
where $S^n_t\stackrel{\triangle}{=} \mathbb{E}^{\mathbb{Q}^0}[L^n_tX^n_t|{\cal F}^Y_t]$,
$S^{n,0}_t\stackrel{\triangle}{=} \mathbb{E}^{\mathbb{Q}^0}[L^n_t|{\cal F}^Y_t]$, $t\ge 0$, and $d\mathbb{P}^n\stackrel{\triangle}{=} L^n_Td\mathbb{Q}^0$.
It then follows from the FKK equation (\ref{FKK}) that
\begin{eqnarray}
\label{FKK1}
dU^n_t&=&\big\{\mathbb{E}^{\mathbb{P}^n}[X^n_th(t, X^n_{t})|{\cal F}^Y_t]-\mathbb{E}^{\mathbb{P}^n}[X^n_t|{\cal F}^Y_t]\mathbb{E}^{\mathbb{P}^n}[h(t,X^n_{ t})
|{\cal F}^Y_t]\big\}dY_t\\
&&+\big\{\mathbb{E}^{\mathbb{P}^n}[X^n_t|{\cal F}^Y_t](\mathbb{E}^{\mathbb{P}^n}[h(t,X^n_{ t})|{\cal F}^Y_t])^2-\mathbb{E}^{\mathbb{P}^n}[X^n_th(t, X^n_{ t})|{\cal F}^Y_t]\mathbb{E}^{\mathbb{P}^n}[h(t, X^n_{ t})|{\cal F}^Y_t]\big\}dt. \nonumber
\end{eqnarray}
Now denote $B^{2,n}_t\stackrel{\triangle}{=} Y_t-\int_0^th(s,X^n_{\cdot\wedge s})ds$. Then $(B^1, B^{2,n})$ is a 2-dimensional
standard $\mathbb{P}^n$-Brownian motion. Furthermore, since $h$ is bounded, so is
$\mathbb{E}^{\mathbb{P}^n}[h(t, X^n_{\cdot\wedge t})|{\cal F}^Y_t]$.
We thus have the following estimate:
$$\begin{array}{lll}
\mathbb{E}^{\mathbb{P}^n}[|U^n_t-U^n_s|^4]&\le& C\mathbb{E}^{\mathbb{P}^n}\Big[\Big(\int_s^t\mathbb{E}^{\mathbb{P}^n}[|X^n_s|^2|{\cal F}^Y_s]ds\Big)^2
\Big]\nonumber\\
&\le& C\mathbb{E}^{\mathbb{P}^n}\Big[\sup_{0\le s\le T}\big|\mathbb{E}^{\mathbb{P}^n}[|X^n_s|^2|{\cal F}^Y_s]|^2\Big]|t-s|^2\nonumber\\
\end{array}$$
\begin{eqnarray} \label{est1}
&\le& C\mathbb{E}^{\mathbb{P}^n}\Big[\sup_{0\le s\le T}\big|\mathbb{E}^{\mathbb{P}^n}[\sup_{0\le r\le T}|X^n_r|^2|{\cal F}^Y_s]|^2\Big]|t-s|^2\\
&\le & C\mathbb{E}^{\mathbb{P}^n}\Big[\sup_{0\le s\le T}|X^n_s|^4\Big]|t-s|^2\le C|t-s|^2. \nonumber
\end{eqnarray}
Thus, as $U^n_0=x$, $n\geq 1$, the sequence of continuous processes $\{U^n\}$ is relatively compact (cf. e.g., Ethier-Kurtz \cite{EK}).
Therefore, the sequence of their laws $\{\mathscr{T}(\mu^n)\stackrel{\triangle}{=} \mathbb{P}^n\circ [U^n]^{-1}, n\ge1\}\subseteq \mathscr{P}(\mathbb{C}_T)$ is tight. Consequently,
we can find a subsequence, we may assume itself, that converges weakly to a limit $\nu\in\mathscr{P}_2(\mathbb{C}_T)$.
Furthermore, for each $n\ge 1$, we apply the Jensen, Burkholder-Davis-Gundy, and H\"older inequalities
to get, with $\nu^n\stackrel{\triangle}{=}\mathscr{T}(\mu^n)$,
\begin{eqnarray}
\label{estnu}
\int_{\mathbb{C}_T}\|\varphi\|^4_{\mathbb{C}_T}\nu^n(d\varphi)&=&\mathbb{E}^{\mathbb{P}^n}[\|U^n\|^4_{\mathbb{C}_T}]=\mathbb{E}^{\mathbb{P}^n}[\sup_{0\le t\le T}|\mathbb{E}^{\mathbb{P}^n}[X_t^n
|{\cal F}_t^n]|^4]\nonumber\\
&\le&
\mathbb{E}^{\mathbb{P}^n}\Big[\sup_{0\le t\le T} \mathbb{E}^{\mathbb{P}^n}\big[\sup_{0\le r\le T}|X_r^n||{\cal F}_t^n\big]^4\Big]\\
&\le& C\Big[\mathbb{E}^{\mathbb{P}^n}\big[\sup_{0\le r\le T}|X_r^n|^{6}\big]\Big]^{2/3}=C\Big[\mathbb{E}^{\mathbb{Q}_0}\big[L_T^n\sup_{0\le r\le T}|X_r^n|^{6}\big]\Big]^{2/3}\nonumber\\
&\le& C\Big[\mathbb{E}^{\mathbb{Q}_0}[(L_T^n)^4]\Big]^{1/6}\Big[\mathbb{E}^{\mathbb{Q}_0}[\sup_{0\le r\le T}|X_r^n|^8]\Big]^{1/2}<+\infty. \nonumber
\end{eqnarray}
But noting that $h$ is bounded, one deduces from (\ref{momentQ0}) that
\begin{eqnarray}
\label{uniformbdd}
\sup_{n\ge 1}\int_{\mathbb{C}_T}\|\varphi\|^4_{\mathbb{C}_T}\nu^n(d\varphi)<\infty,
\end{eqnarray}
and, thus,
\centerline{$\displaystyle\sup_{n\ge 1}\int_{\mathbb{C}_T}\|\varphi\|^2_{\mathbb{C}_T}I\{|\varphi\|_{\mathbb{C}_T}\ge N \}\nu^n(d\varphi)\rightarrow 0,
\mbox{ as }N\rightarrow +\infty.$}
\smallskip
\noindent This, together with the fact that $\nu^n=\mathscr{T}(\mu^n)\mathop{\buildrel w\over\rightarrow} \nu$, implies that $W_2(\nu^n, \nu)\to 0$, and $\nu\in \mathscr{E}$, as $n\to\infty$, where $W_2(\cdot,\cdot)$ is the 2-Wasserstein metric on $\mathscr{P}_2(\mathbb{C}_T)$. This proves (2).
(3) We now check that the mapping $\mathscr{T}:(\mathscr{E},W_1(\cdot,\cdot))\rightarrow (\mathscr{P}_2(\mathbb{C}_T),W_2(\cdot,\cdot))$ is continuous. To this end,
for each $\mu\in\mathscr{E}$, we consider the following SDE on the probability
space $(\Omega, {\cal F}, \mathbb{Q}^0)$:
\begin{eqnarray}
\label{XBLQ0}
\left\{\begin{array}{lll}
dX_t=\sigma(t, X_{\cdot\wedge t}, \mu_t)dB^1_t, \quad & X_0=x;\\
dB^2_t=dY_t-h(t, X_{t})dt, & B^2_0=0; \\
dL_t=h(t, X_{ t})L_t dY_t, & L_0=1.
\end{array}\right.
\end{eqnarray}
Now let $\{\mu^n\}\subseteq \mathscr{E}$ be any sequence such that $\mu^n\to \mu$, as $n\to\infty$, in the 1-Wasserstein metric, and denote by $(X^n, B^{n,2}, L^n)$ the corresponding solutions to (\ref{XBLQ0}). Define
$$\sigma^n(t, \omega_{\cdot\wedge t})\stackrel{\triangle}{=} \sigma(t, \omega_{\cdot\wedge t}, \mu^n_t), \quad (t, \omega)\in [0,T]\times \Omega.
$$
Then by Assumption \ref{Assum2}-(ii), the $\sigma^n$'s are functional Lipschitz deterministic functions, with Lipschitz constant independent
of $n$. This and standard SDE arguments lead to that, as $n\to\infty$,
\begin{eqnarray}
\label{converge}
\mathbb{E}^{\mathbb{Q}^0}\Big\{\sup_{0\le t\le T}|X^n_t-X_t|^p+\sup_{0\le t\le T}|L^n_t- L_t|^p\Big\}\to 0, \quad \mbox{in $L^p(\mathbb{Q}^0)$, \quad
$p\ge 1$.}
\end{eqnarray}
We deduce that $U^n_t=\mathbb{E}^{\mathbb{P}^n}[X^n_t|{\cal F}^Y_t]=S^n_t/S^{n,0}_t$ converges in probability under $\mathbb{Q}^0$ to
$\frac{\mathbb{E}^{\mathbb{Q}^0}[L_tX_t|{\cal F}^Y_t]}{\mathbb{E}^{\mathbb{Q}^0}[L_t|{\cal F}_t^Y]}=\mathbb{E}^{\mathbb{P}}[X_t|{\cal F}^Y_t]$, where $d\mathbb{P}\stackrel{\triangle}{=} L_T
d\mathbb{Q}^0$.
Now for any $\psi\in \mathbb{C}_b(\mathbb{R})$, letting $n\to\infty$ we have
\begin{eqnarray}
\label{lawconverge}
\mathop{\langle} \psi, \mathscr{T}(\mu^n)_t\mathop{\rangle}&=& \mathbb{E}^{\mathbb{P}^n}\big[\psi(\mathbb{E}^{\mathbb{P}^n}[X^n_t|{\cal F}^Y_t])\big]= \mathbb{E}^{\mathbb{Q}^0}\big[L^n_T
\psi(\mathbb{E}^{\mathbb{P}^n}[X^n_t|{\cal F}^Y_t])\big]\nonumber\\
&\longrightarrow& \mathbb{E}^{\mathbb{Q}^0}\big[L_T\psi(\mathbb{E}^{\mathbb{P}}[X_t|{\cal F}^Y_t])\big]=
\mathbb{E}^{\mathbb{P}}\big[\psi(\mathbb{E}^{\mathbb{P}}[X_t|{\cal F}^Y_t])\big]\\
&=& \mathop{\langle} \psi, \mathbb{P}\circ [\mathbb{E}^{\mathbb{P}}[X_t|{\cal F}^Y_t]]^{-1}\mathop{\rangle}, \qquad \mbox{\rm as~} n\to\infty. \nonumber
\end{eqnarray}
This implies that $\nu_t=\mathbb{P}\circ [\mathbb{E}^{\mathbb{P}}[X_t|{\cal F}^Y_t]]^{-1}=\mathscr{T}(\mu)_t$, for all $t\in[0,T]$. With the same argument one shows that,
for any $0\le t_1<t_2<\cdots <t_k<\infty$,
$$\mathscr{T}(\mu^n)_{t_1, \cdots, t_k}\stackrel{\triangle}{=} \mathbb{P}\circ\big(\mathbb{E}^{\mathbb{P}}[X^n_{t_1}|{\cal F}^Y_{t_1}], \cdots, \mathbb{E}^{\mathbb{P}}[X^n_{t_k}|{\cal F}^Y_{t_k}])^{-1} \stackrel{d}\longrightarrow \nu_{t_1, \cdots, t_k}, \quad \mbox{as $n\to \infty$. }
$$
That is, the finite dimensional distributions of $\mathscr{T}(\mu^n)$ converge to those of $\nu$, and as $\{\mathscr{T}(\mu^n)\}_{n\ge
1}$ is tight by part (2), we conclude that $\mathscr{T}(\mu^n)\mathop{\buildrel w\over\rightarrow} \nu$ in $\mathscr{P}(\mathbb{C}_T)$. This, together with
(\ref{estnu}), further shows that $W_2(\mathscr{T}(\mu^n), \mathscr{T}(\mu))\to 0$, as $n \to\infty$, proving the continuity of $\mathscr{T}$, whence (3).
The proof is now complete.
\hfill \vrule width.25cm height.25cm depth0cm\smallskip
As a consequence of Theorem \ref{compact}, we have the following existence result for SDE (\ref{SDE}).
\begin{prop}
\label{existence} Let Assumption \ref{Assum2} hold. Then SDE (\ref{SDE}) has at least one solution in the sense of Definition \ref{sol}.
\end{prop}
{\it Proof.}
The proof follows from Theorem \ref{compact} and a generalization of the Schauder Fixed Point Theorem by Cauty (see \cite{Cauty}, or a recent generalization
\cite{Cauty2}). To do this we must check:
(i) $\mathscr{E}$ is a convex subset of a Hausdorff topological linear space, (ii) $\mathscr{T}$ is continuous and $\mathscr{T}(\mathscr{E})\subseteq \mathscr{E}$; and (iii) $\mathscr{T}(\mathscr{E})\subset K$, for some compact $K$
in $\mathscr{P}_2(\mathbb{C}_T)$.
To imbed ${\mathscr{E}}$ into a Hausdorff topological linear space, we borrow the argument of Li-Min \cite{LiMin}.
Let $\mathscr{M}_1(\mathbb{C}_T)$ be the space of all bounded signed Borel measures $\nu(\cdot)$ on $\mathbb{C}_T$ such that $|\int_{\mathbb{C}_T}\|\varphi\|_{\mathbb{C}_T}\nu(d\varphi)|<+\infty$, endowed with the norm:
$$
\|\nu\|_1:=\sup\Big\{\Big|\int_{\mathbb{C}_T}hd\nu\Big|\ :\ h\in\mbox{Lip}_1(\mathbb{C}_T),~|h(0)|\le 1 \Big\}. \footnote{$Lip_1(\mathbb{C}_T)$ denotes the set of all real-valued Lipschitz functions over $\mathbb{C}_T$ with Lipschitz constant 1.}$$
Clearly $(\mathscr{M}_1(\mathbb{C}_T),\|\cdot\|_1)$ is a normed (hence Hausdorff topological) linear space. Since $\mathscr{P}_2(\mathbb{C}_T)\subset\mathscr{P}_1(\mathbb{C}_T)\subset\mathscr{M}_1(\mathbb{C}_T)$, and by the Kantorovich-Rubinstein formula,
\begin{eqnarray*}
W_1(\nu^1,\nu^2)=\sup\Big\{\Big|\int_{\mathbb{C}_T}hd(\nu^1-\nu^2)\Big|\ :\ h\in\mbox{Lip}_1(\mathbb{C}_T),~|h(0)|\le 1 \Big\}
=\|\nu^1-\nu^2\|_1,
\end{eqnarray*}
for all $\nu^1,\nu^2\in {\cal P}_1(\mathbb{C}_T)$, the
topology generated by the norm $\|\cdot\|_1$ on $\mathscr{P}_2(\mathbb{C}_T)$ coincides with the one generated by the 1-Wasserstein metric on $\mathscr{P}_2(\mathbb{C}_T)$. Thus, $\mathscr{E}\subset\mathscr{P}_2(\mathbb{C}_T)$ is a convex subset of $\mathscr{M}_1(\mathbb{C}_T)$, proving (i). Further, note
that
$\mathscr{T}:\mathscr{E}\rightarrow \mathscr{P}_2(\mathbb{C}_T)$ is continuous under the 1-Wasserstein metric, hence also under the $\|\cdot\|_1$-norm, verifying (ii). Finally, since $\mathscr{T}(\mathscr{E})\subset\mathscr{E}$, and $\mathscr{E}$ is compact under the 2-Wasserstein metric, hence also under the $\|\cdot\|_1$-norm, proving (iii). We can now apply Cauty's theorem to conclude the existence of a fixed point $\nu\in\mathscr{E}\subset\mathscr{P}_2(\mathbb{C}_T)$ such that $\mathscr{T}(\nu) =\nu.$
We note that the existence of the fixed point $\mu$ amounts to saying that SDE (\ref{XBLQ0}) has
a solution on the probability space $(\Omega, {\cal F}, \mathbb{Q}^0)$, with $\mu=\mu^{X|Y}=\mathbb{P}\circ [U]^{-1}$, and $U_t=\mathbb{E}^{\mathbb{P}}[X_t|{\cal F}^Y_t]$, $t\ge 0$, where $d\mathbb{P}=L_Td\mathbb{Q}^0$ by construction. But this in turn defines a solution of (\ref{SDE}) on the probability space $(\Omega, {\cal F}, \mathbb{P})$, thanks to the Girsanov transformation. However, since under $\mathbb{P}$,
$(B^1, B^2)$ constructed in (\ref{XBLQ0}) is a Brownian motion, $(\Omega, {\cal F}, \mathbb{P}, X, Y, B^1, B^2)$ defines a
(weak) solution of SDE (\ref{SDE}).
\hfill \vrule width.25cm height.25cm depth0cm\smallskip
\section{Uniqueness}
\setcounter{equation}{0}
In this section we investigate the uniqueness of the solution to SDE (\ref{SDE}).
We note that the general uniqueness for the weak solution for this problem is quite difficult, we will content ourselves with a version that is relatively more amendable.
To begin with, and let $\mathbb{Q}^0$ be the reference probability measure
under which $(B^1, Y)$ is a Brownian motion. For each $u\in L^{\infty-}_{\mathbb{F}^Y}(\mathbb{Q}^0,[0,T])$, consider the SDE on $(\Omega, {\cal F}, \mathbb{Q}^0)$:
\begin{eqnarray}
\label{XBLQ1}
\left\{\begin{array}{lll}
dX^u_t=\sigma(t, X^u_{\cdot\wedge t}, \mu^{X^u|Y}_t, u_t)dB^1_t, \quad &X^u_0=x;\medskip\\
dB^2_t=dY_t-h(t, X^u_{ t})dt, & B^2_0=0; \medskip\\
dL^u_t= h(t, X^u_{ t})L^u_t dY_t, & L_0=1,
\end{array}\right.
\end{eqnarray}
where $\mu_t^{X^{u}|Y}:=\mathbb{P}^{u}\circ[\mathbb{E}^{\mathbb{P}^{u}}[X^u_t|{\cal F}_t^Y]]^{-1}$, and $d\mathbb{P}^{u}:=L_T^{u}d\mathbb{Q}^0$.
We shall argue that, under Assumption \ref{Assum1}, the solution of the SDE (\ref{XBLQ1}) is pathwisely unique.
\begin{rem}
\label{remarkuniq} {\rm It should be clear that if $u\in L^{\infty-}_{\mathbb{F}^Y}(\mathbb{Q}^0,[0,T])$, and $(X^u, B^2, L^u)$ is a solution to (\ref{XBLQ1}) under $\mathbb{Q}^0$, then $u\in L^{\infty-}_{\mathbb{F}^Y}(\mathbb{P}^u,[0,T])$ (since
$\frac{d\mathbb{P}^u}{d\mathbb{Q}^0}\in L^p(\Omega)$ for all $p>1$, thanks to Assumption \ref{Assum1}),
and the process $(X^u, Y, B^1, B^2)$ is a solution to (\ref{controlsys}) and (\ref{observation}) on the probability space $(\Omega, {\cal F}, \mathbb{P}^u, \mathbb{F})$ in the sense of Definition \ref{sol}, where $\mathbb{F}:=\mathbb{F}^{B^1, Y}$.
Conversely, if $(\Omega,{\cal F},\mathbb{P}^u,\mathbb{F},B^1,B^2,X,Y)$ is a weak solution of (\ref{controlsys})-(\ref{observation}), then following the argument of \S2.2, we see that $d\mathbb{Q}^0=[L^{u}_T]^{-1}d\mathbb{P}^u$ defines a reference measure, where $L^u$ is defined by (\ref{barL}) or (\ref{barLexp}), and $(X, B^2, [L^u]^{-1})$ will be a solution of (\ref{XBLQ1}) with respect to the $\mathbb{Q}^0$-Brownian motion $(B^1,Y)$.
In what follows we shall call the solution to (\ref{XBLQ1}) the $\mathbb{Q}^0$-dynamics of the system (\ref{controlsys}) and (\ref{observation}).}
\hfill \vrule width.25cm height.25cm depth0cm\smallskip
\end{rem}
Bearing Remark \ref{remarkuniq} in mind, let us first try to establish a result in the spirit of the Yamada-Watanabe Theorem: {\it the pathwise uniqueness of $(\ref{XBLQ1})$ implies the uniqueness in law for the original SDEs (\ref{controlsys}) and (\ref{observation})}. To do this, we begin by noting that, given the ``regular" nature of the canonical space $\Omega$, a process $u\in L^{\infty-}_{\mathbb{F}^Y}(\mathbb{P}^u,[0,T])$ amounts to saying that (cf. e.g., \cite{SV,yong-zhou}) there exists a
progressively measurable functional ${\bf u}:[0,T]\times \mathbb{C}_T\mapsto U$ such that $u_t(\omega)={\bf u}(t, Y_{\cdot\wedge t}(\omega))$, $dtd\mathbb{P}^u$-a.s., such that $u$ has all the finite moments under $\mathbb{P}^u$ (hence also true under $\mathbb{Q}^0\sim
\mathbb{P}^u$!).
We have the following Proposition.
\begin{prop}
\label{YamaWata}
Assume that Assumption \ref{Assum1} is in force, and that the pathwise uniqueness holds for SDE (\ref{XBLQ1}). Let ${\bf u}:[0,T]\times \Omega\mapsto U$ be a given progressively measurable functional, and $(\Omega,{\cal F},\mathbb{P}^{i},\mathbb{F},B^{1,i},B^{2,i},X^{i},Y^{i})$, $i=1,2$, be two (weak) solutions of (\ref{controlsys})- (\ref{observation}) corresponding to the controls $u^{i}={\bf u}(\cdot, Y^{i})$, $i=1,2$, respectively. Then, it holds that
$$\mathbb{P}^{1}\circ[(B^{1,1},B^{2,1},X^{1},Y^{1})]^{-1}=\mathbb{P}^{2}\circ[(B^{1,2},B^{2,2},X^{2},Y^{2})]^{-1}.$$
\end{prop}
{\it Proof.} Following the argument of \S2.2, we define $d\mathbb{Q}^{0,i}=[L^{i}_T]^{-1}d\mathbb{P}^{i}$, where $L^{i}=[\bar{L}^{i}]^{-1}$ and $\bar{L}^{i}$ is the unique solution of the SDE (\ref{barL}) with respect to $(X^i, B^{1,i}, Y^i)$, $i=1,2$.
Then, as the $\mathbb{Q}^{0,i}$-dynamics, $(X^i, B^{2,i},L^{i})$ satisfies (\ref{XBLQ1}), $i=1,2$, $\mathbb{Q}^{0,i}$-a.s.
In particular, we recall (\ref{KS}) that
$$U_t^{X^{i}|Y^{i}}=\mathbb{E}^{\mathbb{P}^{i}}[X_t^{i}|{\cal F}^{Y^{i}}_t]= \frac{\mathbb{E}^{\mathbb{Q}^{0,i}}[L_t^{i}X_t^{i}|{\cal F}^{Y^{i}}_t]}{\mathbb{E}^{\mathbb{Q}^{0,i}}\left[L_t^{i}|{\cal F}^{Y^{i}}_t\right]},\qquad \mathbb{Q}^{0,i}\mbox{-a.s.},\, t\in[0,T].
$$
Thus, there exist two progressively measurable functionals $\Phi^{i}:[0,T]\times \Omega\mapsto \mathbb{R}$ such that $U_t^{X^{i}|Y^{i}}=\Phi^{i}(t,Y^{i}_{\cdot\wedge t})$, $dtd\mathbb{Q}^{0,i}$-a.s., $i=1,2$. We now consider an intermediate SDE on $(\Omega, {\cal F}, \mathbb{Q}^{0,2})$:
\begin{eqnarray}
\label{hatX2}
\left\{\begin{array}{lll}
d\widehat{X}^{2}_t=\sigma(t,\widehat{X}^{2}_{\cdot\wedge t},\Phi^{1}(t,Y^{2}_{\cdot\wedge t}),{\bf u}(t, Y^{2}_{\cdot\wedge t}))dB^{1,2}_t,\quad &\widehat{X}^{2}_0=x; \medskip \\
d\widehat{L}^{2}_t=h(t,\widehat{X}^{2}_t)\widehat{L}_t^{2}dY^{2}_t, &\widehat{L}^{2}_0=1,
\end{array}\right.
\qquad t\in[0,T].
\end{eqnarray}
Clearly, comparing to (\ref{XBLQ1}) for $\mathbb{Q}^{0,1}$-dynamics $(X^1, B^{2,1}, L^1)$, this SDE has the same coefficient $\widehat{\sigma}(t, \omega, \varphi_{\cdot\wedge t}):=\sigma(t,\varphi_{\cdot\wedge t},\Phi^1(t,\omega^2_{\cdot\wedge t}),{\bf u}(t,\omega^2_{\cdot\wedge t}))$, and
$h(t,x)\ell$, which is jointly measurable, uniformly Lipschitz in $\varphi$ with linear growth (in $\ell$),
uniformly in $(t,\omega, \varphi, \ell)$, thanks to Assumption \ref{Assum1}, except that it is
driven by the $\mathbb{Q}^{0,2}$-Brownian motion $(B^{1,2}, Y^2)$. Thus, by the classical SDE theory (cf. e.g., \cite{IW}) we know that
there exists a (unique) measurable functional $\Psi:\mathbb{C}_T\times\mathbb{C}_T\rightarrow\mathbb{C}_T\times\mathbb{C}_T$ such that $(X^1,L^1)=\Psi(B^{1,1},Y^1),\, \mathbb{Q}^{0,1}$-a.s., and
$(\widehat{X}^2,\widehat{L}^2)=\Psi(B^{1,2},Y^2),\, \mathbb{Q}^{0,2}$-a.s.
Since
$\mathbb{Q}^{0,1}\circ (B^{1,1},Y^1)^{-1}=\mathbb{Q}^{0,2}\circ (B^{1,2},Y^2)^{-1}=\mathbb{Q}^0$, the Wiener measure on $(\Omega, {\cal F})$, we deduce that
\begin{eqnarray}
\label{Q01=Q02}
\mathbb{Q}^{0,1}\circ (B^{1,1},Y^1,X^1,L^1)^{-1}=\mathbb{Q}^{0,2}\circ (B^{1,2},Y^2,\widehat{X}^2,\widehat{L}^{2})^{-1}.
\end{eqnarray}
We now claim that $(\widehat{X}^2, B^{2,2}, \widehat{L}^2)$ coincides with the $\mathbb{Q}^{0,2}$-dynamics of (\ref{controlsys})-(\ref{observation}).
Indeed, it suffices to argue that in SDE (\ref{hatX2}),
\begin{eqnarray}
\label{hatmu}
\Phi^1(t,Y^2_{\cdot\wedge t})=\mathbb{E}^{\widehat{\mathbb{P}}^{2}}[\widehat{X}_t^2|\mathbb{F}_t^{Y^2}]=U_t^{\widehat{X}^2|Y^2},\qquad \mathbb{Q}^{0,2}\mbox{-a.s.},
\end{eqnarray}
where $d\widehat{\mathbb{P}}^2:=\widehat{L}^2d\mathbb{Q}^{0,2}$. To see this, we note that,
for all $t\in[0,T]$ and any bounded Borel measurable function $f:\mathbb{C}_T\rightarrow \mathbb{R}$, it follows from (\ref{Q01=Q02}) and the
definition of $U_t^{X|Y}$ that
\begin{eqnarray*}
&&\mathbb{E}^{\widehat{\mathbb{P}}^{2}}[f(Y^2_{\cdot\wedge t})\Phi^1(t,Y^2_{\cdot\wedge t})]=\mathbb{E}^{\mathbb{Q}^{0,2}}[\widehat{L}_t^2 f(Y^2_{\cdot\wedge t})\Phi^1(t,Y^2_{\cdot\wedge t})]=\mathbb{E}^{\mathbb{Q}^{0,1}}[L_t^1 f(Y^1_{\cdot\wedge t})\Phi^1(t,Y^1_{\cdot\wedge t})]\\
&=&\mathbb{E}^{\mathbb{P}^{1}}[f(Y^1_{\cdot\wedge t})U_t^{X^1|Y^1}] =\mathbb{E}^{\mathbb{P}^{1}}[f(Y^1_{\cdot\wedge t})X_t^1]=\mathbb{E}^{\mathbb{Q}^{0,1}}[L_t^1f(Y^1_{\cdot\wedge t})X^1_t]=\mathbb{E}^{\mathbb{Q}^{0,2}}[\widehat{L}_t^2f(Y^2_{\cdot\wedge t})\widehat{X}^2_t]\\
&=&\mathbb{E}^{\widehat{\mathbb{P}}^{2}}[f(Y^2_{\cdot\wedge t})\widehat{X}_t^2]=\mathbb{E}^{\widehat{\mathbb{P}}^{2}}[f(Y^2_{\cdot\wedge t})U_t^{\widehat{X}^2|Y^2}],
\end{eqnarray*}
proving (\ref{hatmu}), whence the claim.
Now, by pathwise uniqueness of SDE (\ref{XBLQ1}), we conclude that $(X^2,L^2)=(\widehat{X}^2,\widehat{L}^2)$, $\mathbb{Q}^{0,2}$-a.s.
Thus (\ref{Q01=Q02}) implies that $ \mathbb{Q}^{0,1}\circ [(B^{1,1},Y^1,X^1,L^1)]^{-1}
=\mathbb{Q}^{0,2}\circ [(B^{1,2},Y^2,X^2,L^{2})]^{-1}$, and consequently,
$\mathbb{Q}^{0,1}\circ [(B^{1,1},B^{2,1},X^1,Y^1)]^{-1}=\mathbb{Q}^{0,2}\circ [(B^{1,2},B^{2,2},X^2,Y^2)]^{-1}$. This proves
the uniqueness in law for the system (\ref{controlsys})-(\ref{observation}).
\hfill \vrule width.25cm height.25cm depth0cm\smallskip
We now turn our attention to the main result of this section: the pathwise uniqueness of (\ref{XBLQ1}). We shall establish
some fundamental estimates which will
be useful in our future discussions. Since all controlled dynamics are constructed via the reference probability
space $(\Omega, {\cal F}, \mathbb{Q}^0)$, we shall consider only their $\mathbb{Q}^0$-dynamics, namely the solution to (\ref{XBLQ1}).
Recall the space $L^p(\mathbb{Q}^0; L^2([0,T]))$, $p>1$, and the norm $\|\cdot\|_{p,2,\mathbb{Q}^0}$
defined by (\ref{LpqPnorm}).
We have the following important result.
\begin{prop}
\label{est1}
Assume that Assumption \ref{Assum1} is in force. Let $u, v\in \mathscr{U}_{ad}$ be given. Then, for any $p>2$, there exists a constant $C_p>0$, such that the following estimates hold:
\begin{eqnarray}
\label{7.25}
&&{\rm (i)}\ \mathbb{E}^{\mathbb{Q}^0}\Big[\sup\limits_{0\leq s\leq T}(|X_s^{{u}}-X_s^{{v}}|^2+
|L_s^{{u}}-L_s^{{v}}|^2+|X_s^{{u}}L_s^{{u}}
-X_s^{{v}} L_s^{{v}}|^2)\Big]\leq C\|u-v\|^{2}_{2,2,\mathbb{Q}^0};\ \ \ \ \ \\
\label{Xpest}
&&{\rm (ii)}\ \mathbb{E}^{\mathbb{Q}^0}\Big[\sup\limits_{0\leq s\leq T}|X_s^{{u}}-X_s^{{v}}|^p\Big]\leq C_p\|u-v\|^{p}_{p,2,\mathbb{Q}^0}.
\end{eqnarray}
\end{prop}
{\it Proof.} We split the proof into several steps. Throughout this proof we let $C>0$ be a generic constant, depending only
on the bounds and Lipschitz constants of the coefficients and the time duration $T>0$, and it is allowed to vary from line to line.
\medskip
{\it Step 1} ({\it Estimate for $X$}). First let us denote, for any $u\in \mathscr{U}_{ad}$,
\begin{eqnarray}
\label{sigmamuu}
&& \sigma^u(t, \varphi_{\cdot\wedge t}, \mu^u_t) \stackrel{\triangle}{=} \int_{\mathbb{R}}\sigma(t, \varphi_{\cdot\wedge t}, y, u_t)\mu^u_t(dy), ~~(t, \varphi)\in [0,T]\times \mathbb{C}_T,
\end{eqnarray}
and
$ \mu^u_t \stackrel{\triangle}{=} \mu^{X^u|Y}\circ P^{-1}_t=\mathbb{P}^u\circ (\mathbb{E}^{\mathbb{P}^u}[X^u_t|{\cal F}^Y_t])^{-1}$, $t\ge 0$. Then, we have
\begin{eqnarray}
\label{siuv}
&&|\sigma^u(t, X^u_{\cdot\wedge t}, \mu^u_t)-\sigma^v(t, X^v_{\cdot\wedge t}, \mu^v_t)|\nonumber\\
&=& \Big|\int_{\mathbb{R}}\sigma(t, X^u_{\cdot\wedge t}, y, u_t)\mu^u_t(dy)-\int_{\mathbb{R}}\sigma(t, X^v_{\cdot\wedge t}, y, v_t)\mu^v_t(dy)\Big|\\
&\le & C\Big\{ |u_t-v_t|+\sup_{0\le s\le t}|X^u_s-X^v_s|+\Big|\int_{\mathbb{R}}\sigma(t, X^v_{\cdot\wedge t}, y, v_t)[\mu^u_t(dy)-
\mu^v_t(dy)]\Big|\Big\}.\nonumber
\end{eqnarray}
Next, let us denote $S^u_t=\mathbb{E}^{\mathbb{Q}^0}[L^u_tX^u_t|{\cal F}^Y_t]$ and $S^{u,0}_t=\mathbb{E}^{\mathbb{Q}^0}[L^u_t|{\cal F}^Y_t]$, and define
$S^{v}_t$, $S^{v,0}_t$ in a similar way. By (\ref{Bayes}) and the fact that $d\mathbb{P}^u=L^u_Td\mathbb{Q}^0$, we see that
\begin{eqnarray}
\label{I1I2}
&&\Big|\int_{\mathbb{R}}\sigma(t, X^v_{\cdot\wedge t}, y, v_t)[\mu^u(dy)-\mu^v(dy)]\Big|\nonumber\\
&=& \Big|\mathbb{E}^u[\sigma(t, \varphi_{\cdot\wedge t},\mathbb{E}^u[X^u_t|{\cal F}^Y_t], u)]-\mathbb{E}^v[\sigma(t, \varphi_{\cdot\wedge t},\mathbb{E}^v[X^v_t|{\cal F}^Y_t], u)]\big|_{\varphi=X^v, u=v_t}\Big|\\
&=& \Big|\mathbb{E}^{\mathbb{Q}^0}\Big\{L^u_t \sigma\big(t, \varphi_{\cdot\wedge t},\frac{\mathbb{E}^{\mathbb{Q}^0}[L^u_tX^u_t|{\cal F}^Y_t]}{\mathbb{E}^{\mathbb{Q}^0}[L^u_t|{\cal F}^Y_t]}, u\big)
-L^v_t \sigma\big(t, \varphi_{\cdot\wedge t},\frac{\mathbb{E}^{\mathbb{Q}^0}[L^v_tX^v_t|{\cal F}^Y_t]}{\mathbb{E}^{\mathbb{Q}^0}[L^v_t|{\cal F}^Y_t]}, u\big)\Big\}\big|_{\varphi=X^v, u=v_t}\Big|\nonumber\\
&\le & I_1+I_2, \nonumber
\end{eqnarray}
where (noting the definition of $S^u$, $S^{u,0}$ and the fact that they are both $\mathbb{F}^Y$-adapted)
\begin{eqnarray*}
\label{I1}
I_1&=&\Big|\mathbb{E}^{\mathbb{Q}^0}\Big\{L^u_t \sigma\big(t, \varphi_{\cdot\wedge t},\frac{S^u_t}{S^{u,0}_t}, u\big)
-L^v_t \sigma\big(t, \varphi_{\cdot\wedge t},\frac{S^u_t}{S^{v,0}_t}, u\big)\Big\}\big|_{\varphi=X^v, u=v_t}\Big|\\
&=&\Big|\mathbb{E}^{\mathbb{Q}^0}\Big\{S^{u,0}_t \sigma\big(t, \varphi_{\cdot\wedge t},\frac{S^u_t}{S^{u,0}_t}, u\big)
-S^{v,0}_t \sigma\big(t, \varphi_{\cdot\wedge t},\frac{S^u_t}{S^{v,0}_t}, u\big)\Big\}\big|_{\varphi=X^v, u=v_t}\Big|;
\end{eqnarray*}
and
\begin{eqnarray*}
I_2&=&\Big|\mathbb{E}^{\mathbb{Q}^0}\Big\{L^v_t \big[\sigma\big(t, \varphi_{\cdot\wedge t},\frac{S^u_t}{S^{v,0}_t}, u\big)
-\sigma\big(t, \varphi_{\cdot\wedge t},\frac{S^v_t}{S^{v,0}_t}, u\big)\big]\Big\}\big|_{\varphi=X^v, u=v_t}\Big|\\
&=&\Big|\mathbb{E}^{\mathbb{Q}^0}\Big\{S^{v,0}_t \big[\sigma\big(t, \varphi_{\cdot\wedge t},\frac{S^u_t}{S^{v,0}_t}, u\big)
-\sigma\big(t, \varphi_{\cdot\wedge t},\frac{S^v_t}{S^{v,0}_t}, u\big)\big]\Big\}\big|_{\varphi=X^v, u=v_t}\Big|.
\end{eqnarray*}
Clearly, we have
\begin{eqnarray}
\label{I2est}
I_2\le C\mathbb{E}^{\mathbb{Q}^0}\Big\{S^{v,0}_t\frac{|S^{u}_t-S^{v}_t|}{S^{v,0}_t}\Big\}\le C\mathbb{E}^{\mathbb{Q}^0}\left[|L^u_tX^u_t-L^v_tX^v_t|\right].
\end{eqnarray}
To estimate $I_1$, we write $\hat \sigma(t,\omega,\varphi_{\cdot\wedge t}, y,z)=y\sigma\big(t,\varphi_{\cdot\wedge t}, \frac{S^u_t(\omega)}{y}, z\big)$. Since
\begin{eqnarray}
\label{pahatsi}
\partial_y\hat \sigma(t,\omega, \varphi_{\cdot\wedge t}, y,z)=\sigma\Big(t,\varphi_{\cdot\wedge t}, \frac{S^u_t(\omega)}{y}, z\Big)-\frac{S^u_t(\omega)}{y}\partial_y\sigma
\Big(t,\varphi_{\cdot\wedge t}, \frac{S^u_t(\omega)}{y}, z\Big),
\end{eqnarray}
we see that $y\mapsto \partial_y\hat\sigma(t,\varphi_{\cdot\wedge t}, y, z) $ is uniformly bounded thanks to Assumption
\ref{Assum1}-(iv). Thus we have
\begin{eqnarray}
\label{I1est}
I_1
\le C\|\partial_y\hat\sigma\|_{\infty}\mathbb{E}^{\mathbb{Q}^0}|S^{u,0}_t-S^{v,0}_t|\le C\mathbb{E}^{\mathbb{Q}^0}|L^u_t-L^v_t|.
\end{eqnarray}
Now note that (\ref{XBLQ1}) implies that
$
X_t^{{u}}-X_t^{{v}}=\int_0^t[\sigma^u(s, X^u_{\cdot\wedge s}, \mu^{u}_s)-\sigma^v(s, X^v_{\cdot\wedge s}, \mu^{v}_s)]dB^1_s$.
Combining (\ref{siuv})--(\ref{I1est}), we see that
\begin{eqnarray}
\label{7.27}
\mathbb{E}^{\mathbb{Q}^0} \Big[\sup_{0\leq s\leq t}|X_s^{u}-X_s^{v}|^p\Big]&\le&C\mathbb{E}^{\mathbb{Q}^0}\Big\{\Big[\int_0^t[\sup_{r\in[0,s]}|X_r^{u}-X_r^{v}|^2+|u_s-v_s|^2\\
&&\qquad+ (\mathbb{E}^{\mathbb{Q}^0}|L_s^{u}-L_s^{v}|)^2+(\mathbb{E}^{\mathbb{Q}^0}|L_s^{u} X_s^{u}-L_s^{v}X_s^v|)^2]ds\Big]^{p/2}\Big\}.\nonumber
\end{eqnarray}
Applying the Gronwall inequality we obtain that
\begin{eqnarray}
\label{7.28}
\mathbb{E}^{\mathbb{Q}^0} \Big[\sup_{0\leq s\leq t}|X_s^{u}-X_s^{v}|^p\Big]&\leq& C\mathbb{E}^{\mathbb{Q}^0}\Big\{\Big[\int_0^t\big[|u_s-v_s|^2+\mathbb{E}^{\mathbb{Q}^0}[|L_s^{u}-L_s^{v}|^2]
\nonumber\\
&& +\mathbb{E}^{\mathbb{Q}^0}[|L_s^{u}X_s^{u}-L_s^{v} X_s^v|^2]\big]ds\Big]^{p/2}\Big\}.
\end{eqnarray}
\medskip
{\it Step 2} ({\it Estimate for $L$}). We first note that, for $t\in[0,T]$,
\begin{eqnarray}
\label{Lh}
&&|L_t^{u}h(t, X_{t}^{u})-L_t^{v} h(t, X_{ t}^{v})| =\Big|L_t^{u}h\Big(t, \frac{L^u_tX_{t}^u}{L^u_t}\Big)-L_t^{v} h\Big(t, \frac{L^v_tX_{t}^v}{L^v_t}\Big)\Big|\nonumber\\
&\le& \Big|L_t^{u}h\Big(t, \frac{L^u_tX_{t}^u}{L^u_t}\Big)-L_t^{u} h\Big(t, \frac{L^v_t X_{t}^v}{L^u_t}\Big)\Big|+
\Big|L_t^{u}h\Big(t, \frac{L^v_tX_{t}^v}{L^u_t}\Big)-L_t^{v} h\Big(t, \frac{L^v_tX_{t}^v}{L^v_t}\Big)\Big|
\\
&\le &C |L_t^{u}X_{ t}^{u}-L_t^{v}X_{t}^{v}|+\Big|L_t^{u}h\Big(t, \frac{L^v_tX_{t}^v}{L^u_t}\Big)-L_t^{v} h\Big(t, \frac{L^v_tX_{t}^v}{L^v_t}\Big)\Big|.\nonumber
\end{eqnarray}
To estimate the second term above we define, as before, $\hat h(t,\omega, x)\stackrel{\triangle}{=} xh\big(t, \frac{ L^v_t(\omega)X^v_t(\omega)}x\big)$. Then, similar to (\ref{pahatsi}), one shows that $x\mapsto \partial_x\hat h(t,\omega, x)$ is uniformly bounded, thanks to Assumption
\ref{Assum1}-(v). Consequently, we have
\begin{eqnarray}
\label{pahath}
\Big|L_t^{u}h\Big(t, \frac{L^v_tX_{t}^v}{L^u_t}\Big)-L_t^{v} h\Big(t, \frac{L^v_tX_{t}^v}{L^v_t}\Big)\Big|\le
\|\partial_x\hat h\|_\infty |L_t^{u}-L_t^{v}|.
\end{eqnarray}
Now, combining (\ref{Lh}) and (\ref{pahath}) we obtain
\begin{eqnarray}
\label{7.29}
|L_t^{u}h(t, X_{t}^{u})-L_t^{v} h(t, X_{ t}^{v})|
\leq C(|L_t^{u}-L_t^{v}|+|L_t^{u}X_{ t}^{u}-L_t^{v}X_{t}^{v}|).
\end{eqnarray}
Therefore, noting that $L_t^{{u}}=1+\int_0^t h(s, X_{ s}^{{u}})L_s^{{u}}dY_s$, we deduce from (\ref{7.29}) and Gronwall's
inequality that
\begin{equation}
\label{7.31}
\mathbb{E}^{\mathbb{Q}^0}[\sup_{0\leq s\leq t}|L_s^{u}-L_s^{v}|^2]\leq
C\mathbb{E}^{\mathbb{Q}^0}[\int_0^t|L_s^{u} X_{ s}^{u}-L_s^{v}X_{s}^{v}|^2ds],\ \mathbb{Q}^0\mbox{-a.s.},\ 0\leq t\leq T.
\end{equation}
\bigskip
{\it Step 3} ({\it Estimate for $L_tX_t$}). It is clear from (\ref{7.28}) and (\ref{7.31}) that it suffices to find the estimate of
$L^u_tX^u_t-L^v_tX^v_t$ in terms of $u-v$. To see this we note that
\begin{eqnarray}
\label{LX}
L_t^{u} X_t^{u}=x+\int_0^t L_s^{u} X_s^{u}h(s,X_{s}^{u})dY_s+\int_0^t L_s^{u}\mathbb{E}^{\mathbb{P}^u}[\sigma (s, \varphi_{\cdot\wedge s},
\mathbb{E}^{\mathbb{P}^u}[X_s^{u}|\mathcal{F}_s^Y], v)]\big|_{\varphi=X^{u}\atop v=u_s}dB_s^1.
\end{eqnarray}
Now define $\tilde h(t,x)\stackrel{\triangle}{=} xh(t,x)$. Then it is easily seen that as $h$ satisfies Assumption \ref{Assum1}-(vi),
$\tilde h$ satisfies Assumption \ref{Assum1}-(v). Thus, similar to (\ref{7.29}) we have
\begin{eqnarray}
\label{7.32}
|L_s^u X_s^u h(s, X_{s}^u)-L_s^v X_s^{v}h(s, X_{s}^{v})|&=&|L_s^u\tilde h(s,X_{s}^u)- L_{s}^{v}\tilde h(s,X_s^{v})|\nonumber\\
&\leq& C(|L_s^{u}-L_s^{v}|+|L_s^u X_{ s}^{u}-L_s^v X_{s}^{u}|).
\end{eqnarray}
On the other hand, for any $u\in \mathscr{U}_{ad}$, recalling (\ref{sigmamuu}) for the notations $\sigma^u$ and $\mu^u$,
we have,
\begin{eqnarray*}
\label{7.33}
\Delta^{u,v}_t &\stackrel{\triangle}{=}& \Big|L_s^u \mathbb{E}^{\mathbb{P}^u}[\sigma (s, \varphi_{\cdot\wedge s}, \mathbb{E}^{\mathbb{P}^u}[X_s^{u}|\mathcal{F}_s^Y], z)]\big|_{\varphi=X^{u};\atop z=u_s}
-L_s^v\mathbb{E}^{\mathbb{P}^v} [\sigma (s, \varphi_{\cdot\wedge s}, \mathbb{E}^{\mathbb{P}^v}[X_s^{v}|\mathcal{F}_s^Y], z)]\big|_{\varphi=X^{v}\atop z=v_s}\Big| \nonumber\medskip \\
&=& \big|L_t^u \sigma^u (t, X^{u}_{\cdot\wedge t}, \mu^u_t)
-L_t^v \sigma^v (t, X^{v}_{\cdot\wedge t}, \mu^v_t)\big|.
\end{eqnarray*}
Then, following a similar argument as in Step 1 we have
\begin{eqnarray*}
\label{Deltauv}
\Delta^{u,v}_t
&\leq& CL_t^{v}(\mathbb{E}^{\mathbb{Q}^0}[|L_t^{u}-L_t^{v}|]+\mathbb{E}^{\mathbb{Q}^0}[|X_t^{u} L_t^{u}-X_t^{v}L_t^v|]) \\
&& +C(|L_t^{u}-L_t^{v}|+|L_t^u X_t^{u}-L_t^v X_t^{v}|)+CL_t^v |u_t-v_t|.\nonumber
\end{eqnarray*}
Squaring both sides above and then taking the expectations we easily deduce that
\begin{equation}
\label{7.34}
\mathbb{E}^{\mathbb{Q}^0}[|\Delta^{u,v}_t|^2]\leq C(\mathbb{E}^{\mathbb{Q}^0}[|L_s^u-L_s^{v}|^2]+\mathbb{E}^{\mathbb{Q}^0}[|X_t^{u} L_t^{u}-X_t^v L_t^{v}|^2])+C\mathbb{E}^{\mathbb{Q}^0}[(L_t^{v})^2 |u_t-v_t|^2].
\end{equation}
Now, combining (\ref{LX})-- (\ref{7.34}), for $p>2$ we can find $C_p>0$ such that
\begin{eqnarray}
\label{7.35}
&&\mathbb{E}^{\mathbb{Q}^0}\Big[\sup_{0\leq s\leq t}|L_s^uX_s^u -L_s^vX_s^v |^2\Big] \nonumber\\
&\leq& C\mathbb{E}^{\mathbb{Q}^0}\Big[\int_0^t |L_s^u X_s^u h(s,X_{ s}^{u})-L_s^v X_s^v h(s,X_{ s}^v)|^2ds\Big]+C\mathbb{E}^{\mathbb{Q}^0}\int_0^t|\Delta^{u,v}_s|^2ds\\
&\leq& C_p\Big\{\mathbb{E}^{\mathbb{Q}^0}\Big[\Big(\int_0^t |u_s-v_s|^2ds\Big)^{p/2}\Big]\Big\}^{2/p}+C\mathbb{E}^{\mathbb{Q}^0}\int_0^t|L_s^{u}-L_s^{v}|^2ds \nonumber\\
&& +C\mathbb{E}^{\mathbb{Q}^0}\int_0^t|L_s^u X_s^{u}-L_s^v X_s^{v}|^2ds. \nonumber
\end{eqnarray}
Hence, applying Gronwall's inequality we obtain
\begin{eqnarray}
\label{7.36}
\mathbb{E}^{\mathbb{Q}^0}\Big[\sup_{0\leq s\leq t} |L_s^uX_s^u -L_s^vX_s^v |^2\Big]\leq C_p\|u-v\|^2_{p,2,\mathbb{Q}^0}+
C\mathbb{E}^{\mathbb{Q}^0}\int_{0}^{t}|L_s^u-L_s^v|^2ds.
\end{eqnarray}
Combining (\ref{7.36}) with (\ref{7.31}) and applying the Gronwall inequality again, we conclude that
\begin{eqnarray}
\label{7.37}
\mathbb{E}^{\mathbb{Q}^0}\Big\{\sup_{0\leq s\leq t}|L_s^u-L_s^v|^2\Big\}\leq C_p\|u-v\|^2_{p,2,\mathbb{Q}^0}.
\end{eqnarray}
This, together with (\ref{7.28}) and (\ref{7.36}), implies (\ref{7.25}). (\ref{Xpest}) then follows easily from (\ref{7.25}) and
(\ref{7.27}), proving the proposition.
\hfill \vrule width.25cm height.25cm depth0cm\smallskip
A direct consequence of Proposition \ref{est1} is the following uniqueness result.
\begin{cor}
\label{uniqueness}
Assume that Assumption \ref{Assum1} holds. Then the solution to SDE (\ref{XBLQ1}) is pathwisely unique.
\end{cor}
{\it Proof.} Setting $u=v$ in Proposition \ref{est1} we obtain the result.
\hfill \vrule width.25cm height.25cm depth0cm\smallskip
\section{A Stochastic Control Problem with Partial Observation}
\setcounter{equation}{0}
We are now ready to study the stochastic control problem with partial observation. We first note that
in theory for each $(\mathbb{P}^u, u)\in\mathscr{U}_{ad}$ our state-observation dynamics $(X^u, Y^u)$ lives on probability space
$(\Omega, {\cal F}, \mathbb{P}^u)$, which varies with control $u$. We shall consider their $\mathbb{Q}^0$-dynamics so that
our analysis can be carried out on a common probability space, thanks to Assumption \ref{Assump2}.
Therefore, in what follows, for each $(\mathbb{P}^u, u)\in \mathscr{U}_{ad}$ we consider only the $\mathbb{Q}^0$-dynamics $(X^u, Y, L^u)$, which
satisfies the following SDE:
\begin{eqnarray}
\label{7.18a}
\left\{
\begin{array}{lll}
\displaystyle dX_t^u=\sigma^u(t, X^u_{\cdot\wedge t},\mu^u_t)dB_t^1,\quad & X^u_0=x; \medskip \\
\displaystyle dB^{2,u}_t=dY_t-h(t, X^u_{t})dt, & B^{2, u}_0=0; \medskip\\
\displaystyle dL^u_t=h(t, X^u_t)L^u_tdY_t, & L^u_0=1,\quad t\geq 0,
\end{array}
\right.
\end{eqnarray}
where $(B^1, Y)$ is a $\mathbb{Q}^0$-Brownian motion, $d\mathbb{P}^u=L_T^ud\mathbb{Q}^0$, and $\mu_t^{X^{u}|Y}=\mathbb{P}^{u}\circ[\mathbb{E}^{\mathbb{P}^{u}}[X_t|{\cal F}_t^Y]]^{-1}$. For simplicity, we denote $\mathbb{E}^u[\cdot]\stackrel{\triangle}{=} \mathbb{E}^{\mathbb{P}^u}[\cdot]$ and
$\mathbb{E}^0[\cdot]\stackrel{\triangle}{=} \mathbb{E}^{\mathbb{Q}^0}[\cdot]$.
\begin{rem}{\rm
A convenient and practical way to identify admissible control is to simply consider the space $L^{\infty-}_{\mathbb{F}^Y}(\mathbb{Q}^0;[0,T])$
(cf. Definition \ref{admissible}),
which is independently well-defined, thanks to Assumption \ref{Assump2}. It is easy to check that, under Assumption \ref{Assum1},
$u\in L^{\infty-}_{\mathbb{F}^Y}(\mathbb{Q}^0;[0,T])$ if and only if $u\in L^{\infty-}_{\mathbb{F}^Y}(\mathbb{P}^u;[0,T])$.
Therefore in what follows by $u\in\mathscr{U}_{ad}$ we mean that $u\in L^{\infty-}_{\mathbb{F}^Y}(\mathbb{Q}^0;[0,T])$.
\hfill \vrule width.25cm height.25cm depth0cm\smallskip}
\end{rem}
We recall that for $u\in\mathscr{U}_{ad}$ and $\mu\in\mathscr{P}_2(\mathbb{C}_T)$, the coefficient $\sigma^u$ in (\ref{7.18a}) is defined
by (\ref{sigmamuu}). Thus we can write the cost functional as
\begin{equation}
\label{7.20}
J(u)\stackrel{\triangle}{=} \mathbb{E}^0\Big\{ \Phi(X^u_T, \mu^u_T)+
\int_0^T f^u(s, X_s^u, \mu^u_s)ds\Big\}.
\end{equation}
An admissible control $u^*\in\mathscr{U}_{ad}$ is said to be optimal if
\begin{equation}\label{7.21}
J(u^*)=\inf_{u\in\mathscr{U}_{ad}}J(u).
\end{equation}
We remark that the cost functional $J(\cdot)$ involves the law of the conditional expectation of the solution in a nonlinear way.
Therefore, such a control problem is intrinsically ``{\it time-inconsistent}" and, thus, the dynamic programming approach in general does not apply.
For this reason, we shall consider only the necessary condition of the optimal solution, that is, Pontryagin's
Maximum Principle.
To this end, we let $u^*\in\mathscr{U}_{ad}$ be
an {\it optimal control}, and consider the convex variations of $u^*$:
\begin{equation}
\label{7.22}
u_t^{\theta, v}:= u^*_t+\theta(v_t-u^*_t),\quad t\in[0,T], \quad 0<\theta<1,\quad v\in\mathscr{U}_{ad}.
\end{equation}
Here, we assume that $u^*, v\in L^{\infty-}_{\mathbb{F}^Y}(\mathbb{Q}^0; [0,T])$. Since $U$ is convex, $u^{\theta, v}_t\in U$, for all $t\in[0,T]$, $ v\in\mathscr{U}_{ad}$, and $\theta\in(0,1)$. We denote $(X^{\theta, v}, Y, L^{\theta, v})$ to be the corresponding $\mathbb{Q}^0$-dynamics that satisfies (\ref{7.18a}), with control $u^{\theta, v}$. Applying Proposition \ref{est1} ((\ref{7.25}) and (\ref{Xpest})) and noting that $Y$ is a Brownian motion under $\mathbb{Q}^0$, we get, for $p>2$,
\begin{eqnarray}
\label{limXth}
\lim_{\theta\to0} \mathbb{E}^0\Big[\sup_{0\leq t\leq T}|X_t^{\theta, v}-X_t^{u^*}|^2\Big]&\le& C_p \lim_{\theta\to0}\| u^{\theta, v}-u^*\|^2_{p,2,\mathbb{Q}^0}
=0; \\
\label{limLth}
\lim_{\theta\to0} \mathbb{E}^0\Big[\sup_{0\leq t\leq T}|L_t^{\theta, v}-L_t^{u^*}|^2\Big]&=&0.
\end{eqnarray}
In the rest of the section we shall derive, heuristically, the ``variational equations" which play a fundamental role
in the study of Maximum Principle. The complete proof will be given in the next section.
For notational simplicity we shall denote $u=u^*$, the optimal control, from now on, bearing in mind that all
discussions will be carried out for the $\mathbb{Q}^0$-dynamics, therefore on the same probability space.
Now for $u^1,u^2\in\mathscr{U}_{ad}$, let $(X^1, L^1)$ and $(X^2, L^2)$ denote the corresponding solutions of (\ref{7.18a}). We define
$\delta X=\delta X^{1,2}=\delta X^{u^1,u^2}\stackrel{\triangle}{=} X^{u^1}-X^{u^2}$ and $\delta L=\delta L^{1,2}=\delta L^{u^1,u^2}\stackrel{\triangle}{=} L^{u^1}-L^{u^2}$, and will
often drop the superscript ``$^{1,2}$" if the context is clear. Then $\delta X$ and $\delta L$ satisfy the equations:
\begin{eqnarray}
\label{deltaX}
\left\{\begin{array}{lll}
\displaystyle \delta X_t = \int_0^t [\sigma^{u^1}(s, X^1_{\cdot\wedge s}, \mu^1_s)-\sigma^{u^2}(s, X^2_{\cdot\wedge s}, \mu^2_s)]dB^1_s;\medskip\\
\displaystyle \delta L_t=\int_0^t [L^1_sh(s, X^1_s)-L^2_sh(s, X^2_s)]dY_s.
\end{array}\right.
\end{eqnarray}
As before, let $U^i_t\stackrel{\triangle}{=} \mathbb{E}^{u^i}[X^i_t|{\cal F}^Y_t]$ and $\mu^i_t=\mathbb{P}^{u^i}\circ [U^i_t]^{-1}$, $t\ge0$, $i=1,2$. We can easily check that
\begin{eqnarray}
\label{deltasigma}
&& \sigma^{u^1}(t, X^1_{\cdot\wedge t}, \mu^1_t)-\sigma^{u^2}(t, X^2_{\cdot\wedge t}, \mu^2_t) \nonumber\\
&=& \mathbb{E}^0\Big\{L^1_t \sigma(t, \varphi^1_{\cdot\wedge t}, U^1_t, z^1)-L^2_t \sigma(t, \varphi^2_{\cdot\wedge t}, U^2_t, z^2)\Big\}\Big|_{\varphi^1=X^1, \varphi^2=X^2;z^1=u^1_t, z^2=u^2_t}\nonumber\\
&=& \mathbb{E}^0\Big\{\delta L^{1,2}_t \sigma(t, \varphi^1_{\cdot\wedge t},U^1_t, z^1)\\
&&+L^2_t \Big[\int_0^1D_\varphi\sigma(t, \varphi^2_{\cdot\wedge t}+\lambda (\varphi^1_{\cdot\wedge t}-\varphi^2_{\cdot\wedge t}), U^1_t, z^1)(\varphi^1_{\cdot\wedge t}-\varphi^2_{\cdot\wedge t})d\lambda\nonumber\\
&&+\int_0^1\partial_y\sigma(t, \varphi^2_{\cdot\wedge t}, U^2_t+\lambda(U^1_t-U^2_t), z^1)d\lambda \cdot(U^1_t-U^2_t)\nonumber\\
&&+\int_0^1\partial_z\sigma(t, \varphi^2_{\cdot\wedge t}, U^2_t, z^2+\lambda(z^1-z^2))d\lambda \cdot(z^1-z^2)\Big]\Big\}\Big|_{\varphi^1=X^1, \varphi^2=X^2; z^1=u^1_t, z^2=u^2_t}. \nonumber
\end{eqnarray}
Now let $u^1=u^{\theta, v}$ and $u^2=u^*=u$, and denote
$$ \delta_\theta X\stackrel{\triangle}{=} \delta_\theta X^{u, v}=\frac{X^{\theta, v}-X^u}{\theta}, \quad \delta_\theta L\stackrel{\triangle}{=} \delta_\theta L^{u, v}=\frac{L^{\theta, v}-L^u}{\theta}, \quad
\delta_\theta U\stackrel{\triangle}{=} \delta_\theta U^{u,v}=\frac{U^{\theta, v}-U^u}{\theta}.
$$
Combining (\ref{deltaX}) and (\ref{deltasigma}) we have
\begin{eqnarray}
\label{dthX}
\delta_\theta X_t
&=& \int_0^t\Big\{\mathbb{E}^0\{\delta_\theta L_s \cdot\sigma(s, \varphi^1_{\cdot\wedge s},U^{\theta, v}_s, z^1)\}\Big|_{\varphi^1=X^{\theta, v}, z^1=u^{\theta, v}_s}+
[D\sigma]^{\theta, u, v}_s(\delta_\theta X_{\cdot\wedge s})\medskip\nonumber\\
&&\quad+\mathbb{E}^0\{B^{\theta,u,v}(s, \varphi^2_{\cdot\wedge s}, z^1) \delta_\theta U_s\}
\Big|_{\varphi^2=X^{u}; \atop z^1=u^{\theta,v}_s}
+C^{\theta, u,v}_\sigma(s)(v_s-u_s)\Big\}dB^1_s,
\end{eqnarray}
where
\begin{eqnarray}
\label{ABC}
&& [D\sigma]^{\theta, u, v}_t(\psi)=
\mathbb{E}^0\Big\{L^u_t\int_0^1D_\varphi\sigma(t, \varphi^2_{\cdot\wedge t}+\lambda (\varphi^1_{\cdot\wedge t}-\varphi^2_{\cdot\wedge t}), U^{\theta, v}_t, z^1)(\psi)d\lambda\Big\}
\Big|_{\varphi^1=X^{\theta,v}, \varphi^2=X^u, \atop
z^1=u^{\theta, v}_t\ \ \ \ \ \ \ },\nonumber\\
&&B^{\theta, u,v}(t, \varphi^2_{\cdot\wedge t}, z^1) =L^u_t\int_0^1\partial_y\sigma(t, \varphi^2_{\cdot\wedge t}, U^u_t+\lambda(U^{\theta,v}_t-U^u_t), z^1)d\lambda,
\\
&&C^{\theta, u,v}_\sigma(t) =\mathbb{E}^0\Big\{L^u_t \int_0^1\partial_z\sigma(t, \varphi^2_{\cdot\wedge t}, U^u_t, z^2+\lambda(z^1-z^2))d\lambda \Big\}\Big|_{\varphi^2=X^u; z^1=u^{\theta, v}_t, z^2=u_t}. \nonumber
\end{eqnarray}
Here the integral involving the Fr\'echet derivative $D_\varphi \sigma$ is in the sense of Bochner.
Noting that $U^{\theta, v}_t=\frac{\mathbb{E}^0[L^{\theta,v}_tX^{\theta,v}_t|{\cal F}^Y_t]}{\mathbb{E}^0[L^{\theta,v}_t|{\cal F}^Y_t]}$ and $U^u_t=\frac{\mathbb{E}^0[L^u_tX^u_t|{\cal F}^Y_t]}{\mathbb{E}^0[L^u_t|{\cal F}^Y_t]}$, we can easily check that
\begin{eqnarray}
\label{deltaU}
\delta_\theta U_t&=&\frac{\mathbb{E}^0[L^u_t|{\cal F}^Y_t]\mathbb{E}^0[L^{\theta,v}_tX^{\theta,v}_t|{\cal F}^Y_t]-\mathbb{E}^0[L^{\theta,v}_t|{\cal F}^Y_t]\mathbb{E}^0[L^u_tX^u_t|{\cal F}^Y_t]}{\theta\mathbb{E}^0[L^{\theta,v}_t|{\cal F}^Y_t]\mathbb{E}^0[L^u_t|{\cal F}^Y_t]}\\
&=& \frac{\mathbb{E}^0[L^u_t|{\cal F}^Y_t]\mathbb{E}^0[\delta_\theta L_tX^{\theta,v}_t+L^u_t\delta_\theta X_t|{\cal F}^Y_t]-\mathbb{E}^0[\delta_\theta L_t|{\cal F}^Y_t]
\mathbb{E}^0[L^u_tX^u_t|{\cal F}^Y_t]}{\mathbb{E}^0[L^{\theta,v}_t|{\cal F}^Y_t]\mathbb{E}^0[L^u_t|{\cal F}^Y_t]}\nonumber\\
&=&\frac{\mathbb{E}^0[\delta_\theta L_tX^{\theta,v}_t+L^u_t\delta_\theta X_t|{\cal F}^Y_t]}{\mathbb{E}^0[L^{\theta,v}_t|{\cal F}^Y_t]}-
\frac{\mathbb{E}^0[\delta_\theta L_t|{\cal F}^Y_t]}{\mathbb{E}^0[L^{\theta,v}_t|{\cal F}^Y_t]}U^u_t.\nonumber
\end{eqnarray}
Now, sending $\theta\to 0$, and assuming that
\begin{eqnarray}
\label{KRuv}
K_t=K^{u,v}_t\stackrel{\triangle}{=} \lim_{\theta\to 0} \delta_\theta X^{u,v}_t; \qquad R_t=R^{u,v}_t\stackrel{\triangle}{=}\lim_{\theta\to 0}\delta_\theta L^{u,v}_t
\end{eqnarray}
both exist in $L^2(\mathbb{Q}^0)$, then it follows from (\ref{deltaX})-(\ref{deltaU}) we have, at least formally,
\begin{eqnarray}
\label{Keq0}
K_t&=&\int_{0}^{t}\Big\{\mathbb{E}^0[R_s\sigma(s,\varphi_{\cdot\wedge s}, U^u_s, z)]\Big|_{\varphi=X^u,z=u_s}+ [D\sigma]^{u, v}_s(K_{
\cdot\wedge s})
\nonumber\\
& &+\mathbb{E}^0\Big[B^{u,v}(s, \varphi_{\cdot\wedge s}, z)\Big(\frac{\mathbb{E}^0[R_sX^u_s+L^u_sK_s|{\cal F}^Y_s]}{\mathbb{E}^0[L^{u}_s|{\cal F}^Y_s]}-
\frac{\mathbb{E}^0[R_s|{\cal F}^Y_s]}{\mathbb{E}^0[L^{u}_s|{\cal F}^Y_s]}U^u_s\Big)\Big]\Big|_{\varphi=X^u;\atop z=u_s}\\
& &+ C^{u,v}_\sigma(s)(v_s-u_s)\Big\}dB_s^1, \nonumber
\end{eqnarray}
where
\begin{eqnarray}
\label{DBC}
[D\sigma]^{u, v}_t(\psi)&\stackrel{\triangle}{=}&\mathbb{E}^0\{L^u_t D_\varphi\sigma(t, \varphi_{\cdot\wedge t}, U^{u}_t, z)( \psi) \}
\Big|_{\varphi=X^u; z=u_t},\nonumber\\
B^{u,v}(t, \varphi_{\cdot\wedge t}, z) &\stackrel{\triangle}{=}&L^u_t\partial_y\sigma(t, \varphi_{\cdot\wedge t}, U^u_t, z), \\
C^{u,v}_\sigma(t) &\stackrel{\triangle}{=}&\mathbb{E}^0\Big\{L^u_t \partial_z\sigma(t, \varphi_{\cdot\wedge t}, U^u_t, z)\Big\}\Big|_{\varphi=X^u; z=u_t}. \nonumber
\end{eqnarray}
Observing also that $U^u_t$ is ${\cal F}^Y_t$-measurable, we have
\begin{eqnarray}
\label{Buv}
&&\mathbb{E}^0\Big[B^{u,v}(s, \varphi_{\cdot\wedge s}, z)\Big(\frac{\mathbb{E}^0[R_sX^u_s+L^u_sK_s|{\cal F}^Y_s]}{\mathbb{E}^0[L^{u}_s|{\cal F}^Y_s]}-
\frac{\mathbb{E}^0[R_s|{\cal F}^Y_s]}{\mathbb{E}^0[L^{u}_s|{\cal F}^Y_s]}U^u_s\Big)\Big]\Big|_{\varphi=X^u;\atop z=u_s}\nonumber\\
&=&\mathbb{E}^u\Big[\partial_y\sigma(s, \varphi_{\cdot\wedge s}, U^u_s, z)\mathbb{E}^u\{(L^u_s)^{-1}R_s[X^u_s-U^u_s]+K_s|{\cal F}^Y_s\}
\Big]\Big|_{\varphi=X^u;\atop z=u_s}\\
&=&\mathbb{E}^u\Big[(L^u_s)^{-1}\partial_y\sigma(s, \varphi_{\cdot\wedge s}, U^u_s, z)\{R_s[X^u_s-U^u_s]+L^u_sK_s\}
\Big]\Big|_{\varphi=X^u;\atop z=u_s}\nonumber\\
&=&\mathbb{E}^0\Big[\partial_y\sigma(s, \varphi_{\cdot\wedge s}, U^u_s, z)(R_sX^u_s+L^u_sK_s)-U^u_s\partial_y\sigma(s, \varphi_{\cdot\wedge s}, U^u_s, z)R_s\Big]\Big|_{\varphi=X^u;\atop z=u_s}.
\nonumber
\end{eqnarray}
Consequently, if we define
\begin{eqnarray}
\label{Psi}
\Psi(t, \varphi_{\cdot\wedge t}, x, y,z)\stackrel{\triangle}{=}\sigma(t, \varphi_{\cdot\wedge t}, y,z)+\partial_y\sigma(t, \varphi_{\cdot\wedge t}, y,z)(x-y),
\end{eqnarray}
then we can rewrite (\ref{Keq0}) as
\begin{eqnarray}
\label{Keq}
K_t&=&\int_{0}^{t}\Big\{\mathbb{E}^0\Big[\Psi(s,\varphi_{\cdot\wedge s}, X^u_s, U^u_s,z)R_s
+\partial_y\sigma(s,\varphi_{\cdot\wedge s},U^u_s, z)L_s^uK_s\Big]\Big|_{\varphi=X^{u}; z=u_{s}}\\
&& \quad + [D\sigma]^{u, v}_s(K_{\cdot\wedge s})+C^{u,v}_\sigma(s)(v_s-u_s)\Big\}dB_s^1.\nonumber
\end{eqnarray}
Similarly, we can formally write down the SDE for $R$:
\begin{eqnarray}
\label{Req}
R_t =\int_{0}^{t}[R_sh(s,X_{s}^u)+L_s^u\partial_xh(s,X_{s}^u)K_{s}]dY_s,
\quad t\geq0.
\end{eqnarray}
The following theorem is regarding the well-posedness of the SDEs (\ref{Keq}) and (\ref{Req}).
\begin{thm}\label{Theorem 5.2}
Assume that Assumption \ref{Assum1} is in force, and let $u, v\in L^{\infty-}_{\mathbb{F}^Y}(\mathbb{Q}^0; [0,T])$ be given. Then,
there is a unique solution $(K, R)\in \mathscr{L}^{\infty-}_{\mathbb{F}}(\mathbb{Q}^0;\mathbb{C}_T^2)$ to SDEs (\ref{Keq}) and (\ref{Req}).
\end{thm}
{\it Proof.} Let $u, v\in L^{\infty-}_{\mathbb{F}^Y}(\mathbb{Q}^0; [0,T])$ be given. We define $F^1_t(K, R)$ and $F_t^2(K, R)$, $t\in [0,T]$,
to be the right hand side of (\ref{Keq}) and (\ref{Req}), respectively.
We first observe that $F_t^1(0,0)=\int_0^t C^{u,v}_\sigma(s)(v_s-u_s)dB^1_s$, and $F_t^2(0,0)\equiv 0$, $t\in[0,T]$. Then, for any $p>2$, it holds that
\begin{eqnarray}
\label{F1est}
\mathbb{E}^u\Big[\sup_{0\le s\le t}|F_s^1(0,0)|^p\Big]\le C_p\mathbb{E}^u\Big[\Big(\int_0^t|v_s-u_s|^2ds\Big)^{p/2}\Big], \qquad t\in [0,T].
\end{eqnarray}
Now let $(K^i,R^i)\in \mathscr{L}^{\infty-}_\mathbb{F}(\mathbb{Q}^0;\mathbb{C}_T)$, $i=1,2$. We define
$\widetilde K^i\stackrel{\triangle}{=} F_1(K^i, R^i)$, $\widetilde R^i\stackrel{\triangle}{=} F_1(K^i, R^i)$, $i=1,2$, and
$\bar K\stackrel{\triangle}{=} K^1-K^2$, $\bar R\stackrel{\triangle}{=} R^1-R^2$, $\hat K\stackrel{\triangle}{=} \widetilde K^1-\widetilde K^2$, and $\hat R\stackrel{\triangle}{=}\widetilde R^1-\widetilde R^2$.
Then, noting that $\sigma$, $\partial_y\sigma$, $y\partial_y\sigma$, and $\partial_z\sigma$ are all bounded, thanks to Assumption \ref{Assum1}, we see that
\begin{eqnarray*}
\label{Psiest}
|\Psi(t, \varphi_{\cdot\wedge t}, x,y,z)|\le C(1+|x|), \quad (t, x,y,z)\in [0,T]\times \mathbb{R}^3, ~\varphi\in\mathbb{C}_T,
\end{eqnarray*}
where, and in what follows, $C>0$ is some generic constant which is allowed to vary from line to line. It then follows that
\begin{eqnarray}
\label{Psiest2}
&&\Big|\mathbb{E}^0[\Psi(t, \varphi_{\cdot\wedge t}, X^u_t, U^u_t,z)\bar R_s+\partial_y\sigma(t, \varphi_{\cdot\wedge t}, U^u_t, z)L^u_t\bar K_t]\Big|
\nonumber\\
&& \le C\mathbb{E}^0[(1+|X^u_t|)|\bar R_t|+|L^u_t \bar K_t|] \le C\Big[ \mathbb{E}^0[|\bar K_t|^2+|\bar R_t|^2]\Big]^{1/2}.
\end{eqnarray}
Furthermore, since $D_\varphi\sigma$ is also bounded, we have
$|[D\sigma]^{u,v}_t(\psi)|\le C\sup_{0\le s\le t}|\psi(s)|$, for $\psi\in\mathbb{C}_T$.
Then from the definition of $\hat K$ and (\ref{Psiest2}) we have, for any $p\ge 2$ and $t\in [0,T]$,
\begin{eqnarray}
\label{hatK}
\mathbb{E}^0\Big[\sup_{0\le s\le t}|\hat K_s|^{2p}\Big] \le C_p\int_0^t\Big(\mathbb{E}^0[|\bar R_s|^2+|\bar K_s|^2]\Big)^pds
+C_p\int_0^t\mathbb{E}^0\Big[\sup_{0\le r\le s}|\bar K_r|^{2p}\Big]ds.
\end{eqnarray}
On the other hand, the boundedness of $h$ and $\partial_x h$ implies that, recalling the definition of $\hat R$, for $p\ge 2$ and
$t\in [0,T]$,
\begin{eqnarray}
\label{hatR}
\Big(\mathbb{E}^0\Big[\sup_{s\le t}|\hat R_s|^p\Big)^2&\le & C_p\int_0^t \mathbb{E}^0[|\bar R_s|^p]^2ds+C_p\int_0^t\mathbb{E}^0[|L_s^u\bar K_s|^p]^2ds\\
&\le & C_p\int_0^t (\mathbb{E}^0[|\bar R_s|^p])^2ds+C_p\int_0^t\mathbb{E}^0[|\bar K_s|^{2p}]ds. \nonumber
\end{eqnarray}
Combining (\ref{hatK}) and (\ref{hatR}) we have, for $t\in[0,T]$,
\begin{eqnarray*}
\mathbb{E}^0\Big[\sup_{0\le s\le t}|\hat K_s|^{2p}\Big]+\Big(\mathbb{E}^0\Big[\sup_{0\le s\le t}|\hat R_s|^p]\Big)^2\le
C_p\int_0^t\Big(\mathbb{E}^0\Big[\sup_{0\le r\le s}|\bar K_r|^{2p}\Big]+\Big(\mathbb{E}^0\Big[\sup_{0\le r\le s}|\bar R_r|^p\Big]^2\Big)ds.
\end{eqnarray*}
This, together with (\ref{F1est}), enables us to apply standard SDE arguments to deduce that there is a unique solution
$(K, R)\in \mathscr{L}^{\infty-}_{\mathbb{F}}(\mathbb{P};\mathbb{C}_T)$ of (\ref{Keq}) and (\ref{Req}), such that for all $p\ge 2$, it holds that
\begin{eqnarray}
\label{KRest}
\mathbb{E}^0\big[\|K\|^{2p}_{\mathbb{C}_T}\big]+\mathbb{E}^0\big[\|R\|^{2p}_{\mathbb{C}_T}\big]\le C_p\|v_s-u_s\|^2_{p,2,\mathbb{Q}^0}.
\end{eqnarray}
We leave it to the interested reader, and this completes the proof.
\hfill \vrule width.25cm height.25cm depth0cm\smallskip
\section{Variational Equations}
\setcounter{equation}{0}
In this section we validate the heuristic arguments in the previous section and derive the variational equation
of the optimal trajectory rigorously.
Recall the processes $\delta_\theta X=\delta_\theta X^{u,v}$, $\delta_\theta L=\delta_\theta L^{u,v}$, and $(K,R)$ defined in the previous
section. Denote
\begin{equation}\label{7.44}
\eta_t^\theta\stackrel{\triangle}{=}\delta_\theta X_t-K_t, \qquad
\tilde{\eta}_t^\theta\stackrel{\triangle}{=} \delta_\theta L_t-R_t,\qquad t\in[0,T].
\end{equation}
Our main purpose of this section is to prove the following result.
\begin{prop}
\label{lemma7.2}
Let $(\mathbb{P}^u, u)=(\mathbb{P}^{u^*}, u^*)\in \mathscr{U}_{ad}$ be an optimal control, $(X^u, L^u)$ be the
corresponding solution of (\ref{7.18a}), and let $U^u_t=\mathbb{E}^u[X^u_t|{\cal F}^Y_t]$, $t\ge 0$.
For any $v\in\mathscr{U}_{ad}$, let $(K, R) = (K^{u,v}, R^{u,v})$ be the solution of the linear equations (\ref{Keq}) and (\ref{Req}).
Then, for all $p>1$, it holds that
\begin{equation}\label{7.42}
\lim_{\theta\to0}\mathbb{E}^0[\|\eta^\theta\|^p_{\mathbb{C}_T}]= \lim_{\theta\rightarrow 0}\mathbb{E}^0\Big[\sup_{s\in[0,T]} \Big|\frac{X_s^{\theta, v}-X_s^u}{\theta}-K_s\Big|^p\Big]=0;
\end{equation}
\begin{equation}\label{7.43}
\lim_{\theta\to0}\mathbb{E}^0[\|\tilde\eta^\theta\|^p_{\mathbb{C}_T}]= \lim\limits_{\theta\rightarrow 0}\mathbb{E}^0\Big[\sup_{s\in[0,T]}\Big|\frac{L_s^{\theta, v}-L_s^u}{\theta}-R_s\Big|^p\Big]=0.
\end{equation}
\end{prop}
The proof of Proposition \ref{lemma7.2} is quite lengthy, we shall split it into two parts.
\medskip
[{\it Proof of (\ref{7.43})}]. This part is relatively easy.
We note that with
a direct calculation using the equations (\ref{deltaX}) and (\ref{Req}) it is readily seen that $\tilde \eta^\theta$ satisfies the
following SDE:
\begin{eqnarray}
\label{7.45}
\tilde{\eta}_t^\theta&=& \int_{0}^{t}\tilde{\eta}_r^{\theta}h(r,X_{ r}^{\theta,v})dY_r+\int_{0}^{t}L_r^u\int_{0}^{1}\partial_xh(r,X_{ r}^u+\lambda\theta(\eta_{ r}^{\theta}+K_{ r}))\eta_{ r}^{\theta }d\lambda d Y_r\nonumber\\
&&+I_t^{1,\theta}+I_t^{2,\theta},
\end{eqnarray}
where
\begin{eqnarray*}
I_t^{1,\theta}&=& \int_{0}^{t}R_r(h(r,X_{ r}^{\theta, v})-h(r,X_{ r}^u))dY_r;\\
I_t^{2,\theta}&=&\int_{0}^{t}L_r^u\int_{0}^{1}\partial_xh(r,X_{ r}^u+\lambda\theta(\eta_{ r}^{\theta}+K_{ r}))K_{ r} d\lambda dY_r-\int_{0}^{t}L_r^u\partial_xh(r,X_{ r}^u)K_{ r}dY_r.
\end{eqnarray*}
We claim that, for all $p>1$,
\begin{equation}\label{7.46}
\lim\limits_{\theta\rightarrow0}\mathbb{E}^u[\sup_{t\in[0,T]}|I_{t}^{1,\theta}|^p]=0,\ \ \
\lim_{\theta\rightarrow0}\mathbb{E}^u[\sup_{t\in[0,T]}|I_{t}^{2,\theta}|^p]=0.
\end{equation}
Indeed, note that $dY_t=dB^2_t-h(t,X^u_t)dt$, and $B^2$ is a $\mathbb{P}^u$-Brownian motion.
Proposition \ref{est1}, together with the bounded and continuity of $h$ and $\partial_x h$, leads to that, for all $p\ge 2$,
\begin{eqnarray*}
\mathbb{E}^u\Big\{\sup_{t\in[0,T]}|I^{1, \theta}_t|^p\Big\}&=&\mathbb{E}^0\Big\{L^u_T\sup_{t\in[0,T]}\Big|\int_0^tR_s[h(s,X^{\theta,v}_s)-h(s,X^u_s)]dY_s\Big|^p
\Big\}\\
&\le &2\mathbb{E}^u\Big\{\sup_{t\in[0,T]}\Big|\int_0^t R_s[h(s,X^{\theta,v}_s)-h(s,X^u_s)]dB^2_s\Big|^p\Big\}\\
&&+2\mathbb{E}^0\Big\{L^u_T\sup_{t\in[0,T]}\Big|\int_0^tR_s[h(s,X^{\theta, v}_s)-h(s,X^u_s)]h(s,X^u_s)ds\Big|^p\Big\}\\
&\le & C_p\mathbb{E}^0\Big\{L^u_T\int_0^T R^p_s(|X^{\theta,v}_s-X^u_s|^p\wedge 1)ds\Big\}\\
&\le & C_p\Big\{\negthinspace\mathbb{E}^0\big[(L^u_T)^3\big]\negthinspace\Big\}^{\frac13}
\Big\{\mathbb{E}^0\big[\negthinspace\neg\sup_{s\in[0,T]}|R_s|^{3p}\big]\negthinspace\Big\}^{\frac13}
\Big\{\mathbb{E}^0 \big[\negthinspace\neg\sup_{s\in[0,T]}(|X^{\theta,v}_s-X^u_s|^2\negthinspace\wedge \neg1)\big]\negthinspace\Big\}^{\frac13}\\
&\le & C_p\|u-u^{\theta,v}\|_{p,2,\mathbb{Q}^0}^{\frac23}\le C|\theta|^{\frac23},
\end{eqnarray*}
where we used the following estimate for
any function $f\in L^\infty(\mathbb{R})$ bounded by $C_0\ge 1$:
\begin{eqnarray}
\label{3pest}
|f(x)-f(x')|^{3p}\le (2C_0 (|f(x)-f(x')|\wedge 1))^{3p}\le (2C_0)^{3p}(|f(x)-f(x')|^2\wedge 1),~ \forall p\ge 2.
\end{eqnarray}
Similarly, we have
\begin{eqnarray*}
&&\mathbb{E}^u\Big\{\sup_{t\in[0,T]}|I^{2, \theta}_t|^p\Big\}\\
&=& \mathbb{E}^0\Big\{L^u_T\sup_{t\in[0,T]}\Big|\int_0^tL^u_rK_r\Big[\int_0^1\big[
\partial_xh(r, X^u_r+\lambda\theta(\eta^\theta_r+K_r))-\partial_xh(r,X^u_r)]d\lambda\Big]dY_r\Big|^p \Big\}\\
&\le& C_p \mathbb{E}^0\Big\{L^u_T\int_0^T|L^u_r|^p|K_r|^p\Big[\int_0^1\big|
\partial_xh(r, X^u_r+\lambda\theta(\eta^\theta_r+K_r))-\partial_xh(r,X^u_r)\big|d\lambda\Big]^pdr\Big\}\\
&\le& C_p\mathbb{E}^0\Big\{\int_0^T\Big[\int_0^1\big[|\partial_xh(r, X^u_r+\lambda\theta(\eta^\theta_r+K_r))-\partial_xh(r,X^u_r)|^2\wedge 1\big]d\lambda\Big] dr\Big\}^{1/3}.
\end{eqnarray*}
Here in the above the second inequality follows from (\ref{3pest}) applied to $\partial_x h$, the H\"{o}lder inequality, and the fact that $L^{u},K\in \mathscr{L}^{\infty-}_{\mathbb{F}}(\mathbb{Q}^0;\mathbb{C}_T)$ (see Theorem \ref{Theorem 5.2}), and the last inequality follows from the $L^p$-estimate (\ref{KRest}).
Now, from (\ref{7.25}), (\ref{Keq}), and (\ref{Req})
we see that
$$ \mathbb{E}^0\Big\{\sup_{t\in[0,T]}\big(|\eta^\theta_t|^2+|K_t|^2\big)\Big\}\le C, \qquad \theta\in(0,1).
$$
Hence, since $\theta [\|\eta^{\theta}\|_{\mathbb{C}_T}+\|K\|_{\mathbb{C}_T}]\to 0$, in probability $\mathbb{Q}^0$,
as $\theta\to0$, the continuity of $\partial_x h$ and the Bounded Convergence Theorem then imply (\ref{7.46}),
proving the claim. Recalling (\ref{7.45}), we see that (\ref{7.43}) follows from (\ref{7.46}), provided
(\ref{7.42}) holds, which we now substantiate.
[{\it Proof of (\ref{7.42})}]. This part is more involved. We first rewrite (\ref{dthX}) as follows
\begin{eqnarray}
\label{dthX1}
\delta_\theta X_t
&=& \int_0^t\Big\{\mathbb{E}^0\{(\tilde\eta^\theta_s+R_s)\sigma(s, \varphi_{\cdot\wedge s},U^{\theta,v}_s, z)\}\Big|_{\varphi=X^{\theta,v}, \atop
z=u^{\theta,v}_s}+
[D\sigma]^{\theta, u, v}_s(\eta^\theta_{\cdot\wedge s}+K_{\cdot\wedge s})\medskip\nonumber\\
&&\quad+\mathbb{E}^0\{B^{\theta,u,v}(s, \varphi_{\cdot\wedge s}, z) \delta_\theta U_s\}
\Big|_{\varphi=X^{u}; \atop z=u^{\theta,v}_s}
+C^{\theta, u,v}_\sigma(s)(v_s-u_s)\Big\}dB^1_s.
\end{eqnarray}
Here $[D\sigma]^{\theta, u,v}$, $B^{\theta,u,v}$, and $C^{\theta, u,v}$ are defined by (\ref{ABC}). Furthermore, in light of
(\ref{deltaU}), we can also write:
\begin{eqnarray}
\label{deltaU1}
\delta_\theta U_t
=\frac{\mathbb{E}^0[(\tilde\eta^\theta_t+R_t)X^{\theta,v}_t+L^u_t(\eta^\theta_t+K_t)|{\cal F}^Y_t]}{\mathbb{E}^0[L^{\theta,v}_t|{\cal F}^Y_t]}-
\frac{\mathbb{E}^0[(\tilde\eta^\theta_t+R_t)|{\cal F}^Y_t]}{\mathbb{E}^0[L^{\theta,v}_t|{\cal F}^Y_t]}U^u_t.\nonumber
\end{eqnarray}
Plugging this into (\ref{dthX1}) we have
\begin{eqnarray}
\label{dthX2}
\delta_\theta X_t
&=& \int_0^t\Big\{\mathbb{E}^0\{\tilde\eta^\theta_s\sigma(s, \varphi_{\cdot\wedge s},U^{\theta,v}_s, z)\}\Big|_{\varphi=X^{\theta,v}, \atop
z=u^{\theta,v}_s}+
[D\sigma]^{\theta, u, v}_s(\eta^\theta_{\cdot\wedge s})\medskip\nonumber\\
&&\quad+\mathbb{E}^0\Big\{B^{\theta,u,v}(s, \varphi_{\cdot\wedge s}, z) \Big[\frac{\mathbb{E}^0[\tilde\eta^\theta_sX^{\theta,v}_s+L^u_s\eta^\theta_s|{\cal F}^Y_s]}{\mathbb{E}^0[L^{\theta,v}_s|{\cal F}^Y_s]}-
\frac{\mathbb{E}^0[\tilde\eta^\theta_s|{\cal F}^Y_s]}{\mathbb{E}^0[L^{\theta,v}_s|{\cal F}^Y_s]}U^u_s\Big]\Big\}
\Big|_{\varphi=X^{u}; \atop z=u^{\theta,v}_s}\Big\}dB^1_s\nonumber\\
&&+ \int_0^t\Big\{\mathbb{E}^0\{R_s\sigma(s, \varphi_{\cdot\wedge s},U^{\theta}_s, z)\}\Big|_{\varphi=X^{\theta,v}, \atop
z=u^{\theta,v}_s}+ [D\sigma]^{\theta, u, v}_s(K_{\cdot\wedge s})\medskip\nonumber\\
&&\quad+\mathbb{E}^0\Big\{B^{\theta,u,v}(s, \varphi_{\cdot\wedge s}, z) \Big[\frac{\mathbb{E}^0[R_sX^{\theta,v}_s+L^u_sK_s|{\cal F}^Y_s]}{\mathbb{E}^0[L^{\theta,v}_s|{\cal F}^Y_s]}-
\frac{\mathbb{E}^0[R_s|{\cal F}^Y_t]}{\mathbb{E}^0[L^{\theta,v}_s|{\cal F}^Y_s]}U^u_s\Big]\Big\}
\Big|_{\varphi=X^{\theta,v}; \atop z=u^{\theta,v}_s}\nonumber\\%\nonumber
&&\quad+C^{\theta, u,v}_\sigma(s)(v_s-u_s)\Big\}dB^1_s.\nonumber
\end{eqnarray}
Now, recalling (\ref{Keq}) (or more conveniently, (\ref{Keq0})) we have
\begin{eqnarray}
\label{eta}
\eta_{t}^{\theta}&=& \delta_\theta X_t- K_t=\int_0^t\Big\{\mathbb{E}^0\{\tilde\eta^\theta_s\sigma(s, \varphi_{\cdot\wedge s},U^{\theta,v}_s, z)\}\Big|_{\varphi=X^{\theta,v}, \atop
z=u^{\theta,v}_s}+
[D\sigma]^{\theta, u, v}_s(\eta^\theta_{\cdot\wedge s})\medskip\nonumber\\
&&+\mathbb{E}^0\Big\{B^{\theta,u,v}(s, \varphi_{\cdot\wedge s}, z) \Big[\frac{\mathbb{E}^0[\tilde\eta^\theta_sX^{\theta,v}_s+L^u_s\eta^\theta_s|{\cal F}^Y_s]}{\mathbb{E}^0[L^{\theta,v}_s|{\cal F}^Y_s]}-
\frac{\mathbb{E}^0[\tilde\eta^\theta_s|{\cal F}^Y_s]}{\mathbb{E}^0[L^{\theta,v}_s|{\cal F}^Y_s]}U^u_s\Big]\Big\}
\Big|_{\varphi=X^{u}; \atop z=u^{\theta,v}_s}\Big\}dB^1_s\nonumber\\
&&+ I^{3, \theta, 1}_t+ I^{3, \theta, 2}_t+ I^{3, \theta, 3}_t+ I^{3, \theta, 4}_t,
\end{eqnarray}
where, for $t\in[0,T]$,
\begin{eqnarray}
\label{I3th1}
I^{3,\theta,1}_t&\stackrel{\triangle}{=}&\int_{0}^{t}\mathbb{E}^0\big\{R_s\big[\sigma(s,\varphi^1_{\cdot\wedge s}, U^{\theta,v}_s, z^1)-\sigma(s,\varphi^2_{\cdot\wedge s}, U^u_s, z^2)\big]\big\}\Big|_{\varphi^1=X^{\theta,v},z^1=u^{\theta,v}_s\atop \varphi^2=X^u,z^2=u_s}dB^1_s;\nonumber\\
I^{3,\theta,2}_t&\stackrel{\triangle}{=}&\int_{0}^{t}\mathbb{E}^0\big\{[D\sigma]^{\theta,u, v}_s(K_{\cdot\wedge s})-
[D\sigma]^{u, v}_s(K_{\cdot\wedge s})\big\}dB^1_s;\\
\nonumber
\end{eqnarray}
\begin{eqnarray}
I^{3,\theta,3}_t &\stackrel{\triangle}{=}&\int_{0}^{t}\Big\{\mathbb{E}^0\Big\{B^{\theta,u,v}(s, \varphi_{\cdot\wedge s}, z) \Big(\frac{\mathbb{E}^0[R_sX^{\theta,v}_s+L^u_sK_s|{\cal F}^Y_s]}{\mathbb{E}^0[L^{\theta,v}_s|{\cal F}^Y_s]}-
\frac{\mathbb{E}^0[R_s|{\cal F}^Y_t]}{\mathbb{E}^0[L^{\theta,v}_s|{\cal F}^Y_s]}U^u_s\Big)\Big\}\Big|_{\varphi=X^{u}; \atop z=u^{\theta,v}_s}\nonumber\\
&&-\mathbb{E}^0\Big\{B^{u,v}(s, \varphi_{\cdot\wedge s}, z)\Big(\frac{\mathbb{E}^0[R_sX^u_s+L^u_sK_s|{\cal F}^Y_s]}{\mathbb{E}^0[L^{u}_s|{\cal F}^Y_s]}-
\frac{\mathbb{E}^0[R_s|{\cal F}^Y_s]}{\mathbb{E}^0[L^{u}_s|{\cal F}^Y_s]}U^u_s\Big)\Big\}\Big|_{\varphi=X^u;\atop z=u_s} \Big\}dB^1_s \nonumber\\
I^{3,\theta,4}_t &\stackrel{\triangle}{=}&\int_{0}^{t}\mathbb{E}^0[C^{\theta, u,v}_\sigma(s)(v_s-u_s)- C^{u,v}_\sigma(s)(v_s-u_s)]dB_s^1. \nonumber
\end{eqnarray}
We have the following lemma.
\begin{lem}
\label{I3thconv}
Suppose that Assumption \ref{Assum1} holds. Then, for all $p>1$,
\begin{eqnarray}
\label{I3conv}
\lim_{\theta\to0}\mathbb{E}^0\Big\{\sup_{0\le t\le T}|I^{3,\theta,i}_t|^p\Big\}=0, \qquad i=1,\cdots, 4.
\end{eqnarray}
\end{lem}
{\it Proof.}
We first recall that $U^{\theta,v}_s\stackrel{\triangle}{=} \mathbb{E}^{\theta,v}[X_{s}^{\theta,v}|{\cal F}_{s}^{Y}]$ and $U^u_s\stackrel{\triangle}{=}\mathbb{E}^{u}[X_{s}^{u}|\mathcal{F}_{s}^{Y}]$.
Using the Kallianpur-Strieble formula we have
\begin{eqnarray}
\label{J12th}
\mathbb{E}^0\int_0^T|U^{\theta,v}_s-U^u_s|^pds&\le&C_p\Big\{\mathbb{E}^{0}\int_0^T\Big|\frac{\mathbb{E}^{0}[L^{\theta,v}_sX_{s}^{\theta,v}|{\cal F}_{s}^{Y}]}{\mathbb{E}^0[L^{\theta,v}_s|{\cal F}^Y_s]}-\frac{\mathbb{E}^{0}[L^u_sX_{s}^{u}|\mathcal{F}_{s}^{Y}]}{\mathbb{E}^0[L^{\theta,v}_s|{\cal F}^Y_s]}\Big|^pds\nonumber\\
&&+\mathbb{E}^{0}\int_0^T\Big|\frac{\mathbb{E}^{0}[L^u_sX_{s}^{u}|\mathcal{F}_{s}^{Y}]}{\mathbb{E}^0[L^{\theta,v}_s|{\cal F}^Y_s]}-
\frac{\mathbb{E}^{0}[L^u_sX_{s}^{u}|\mathcal{F}_{s}^{Y}]}{\mathbb{E}^0[L^u_s|{\cal F}^Y_s]}\Big|^pds\Big\}\\
&\stackrel{\triangle}{=}& C_p\{J^{1}_\theta+J^{2}_\theta\}. \nonumber
\end{eqnarray}
We now estimate $J^1_\theta$ and $J^2_\theta$ respectively.
First note that, for any $p>1$, we can find a constant $C_p>0$ such that for any $\theta \in(0,1)$ and $u\in \mathscr{U}_{ad}$,
$$ \mathbb{E}^0[(L^{\theta,v}_s)^p]+\mathbb{E}^0[(L^{\theta,v}_s)^{-p}]+\mathbb{E}^0[(L^u_s)^p] \le C_p.
$$
Thus, applying the H\"older and Jensen inequalities as well as Proposition \ref{est1}, we have, for any $p>1$,
and $\theta\in(0,1)$,
\begin{eqnarray}
\label{dL1}
&&\mathbb{E}^0\int_0^T\Big|\frac{\mathbb{E}^0[L^{\theta,v}_sX^{\theta,v}_s|{\cal F}^Y_s]-\mathbb{E}^0[L^u_sX^u_s|{\cal F}^Y_s]}{\mathbb{E}^0[L^{\theta,v}_s|{\cal F}^Y_s]}\Big|^pds \le
\int_0^T\mathbb{E}^0\Big\{\frac{|L^{\theta,v}_sX^{\theta,v}_s-L^u_sX^u_s|^p}{\mathbb{E}^0[
L^{\theta,v}_s|{\cal F}^Y_s]^p}\Big\}ds\nonumber\\
&\le& \int_0^T\Big\{\{\mathbb{E}^0|L^{\theta,v}_sX^{\theta,v}_s-L^u_sX^u_s|^2\}^{1/2}\cdot \Big\{\mathbb{E}^0\Big[\frac{|L^{\theta,v}_sX^{\theta,v}_s-L^u_sX^u_s|^{2p-2}}{\mathbb{E}^0[L^{\theta,v}_s|{\cal F}^Y_s]^{2p}}\Big]\Big\}^{1/2}\Big\}ds\\
&\le& \int_0^t\{\mathbb{E}^0|L^{\theta,v}_sX^{\theta,v}_s-L^u_sX^u_s|^2\}^{1/2}\cdot \Big\{\mathbb{E}^0[|L^{\theta,v}_sX^{\theta,v}_s-L^u_sX^u_s|^{2p-2}]
\mathbb{E}^0\big[[L^{\theta,v}_s]^{-2p}|{\cal F}^Y_s\big]\Big\}^{1/2}ds\nonumber\\
&\le& C_p\theta\|u-v\|_{2,2,\mathbb{Q}^0}.\nonumber
\end{eqnarray}
Similarly, one can also argue that, for any $p>1$, the following estimates hold:
\begin{eqnarray}
\label{dL2}
\mathbb{E}^0\int_0^T\Big| \frac1{\mathbb{E}^0[L^{\theta,v}_s|{\cal F}^Y_s]}-\frac1{\mathbb{E}^0[L^u_s|{\cal F}^Y_s]}
\Big|^pds \le C_p\theta\|u-v\|_{2,2,\mathbb{Q}^0}, \quad\theta\in(0,1).
\end{eqnarray}
Clearly, (\ref{dL1}) and (\ref{dL2}) imply that $J^1_\theta+J^2_\theta\le C_p\theta\|u-v\|_{2,2,\mathbb{Q}^0}$, for some constant $C_p>0$, depending only
on $p$, the Lipschitz constant of the coefficients, and $T$. Therefore we have
\begin{eqnarray}
\label{Uth}
\mathbb{E}^0\int_0^T|U^{\theta,v}_s-U^u_s|^pds \le C_p\theta\|u-v\|_{2,2,\mathbb{Q}^0}\to 0, \qquad \mbox{as $\theta\to0$.}
\end{eqnarray}
We can now prove (\ref{I3conv}) for $i=1,\cdots, 4$. First, by Burkholder-Davis-Gundy inequality we have
\begin{eqnarray*}
\label{I3th1}
\mathbb{E}^0[\sup_{0\le t\le T}|I^{3,\theta,1}_t|^2]\le C\int_0^T\mathbb{E}^0\Big|\mathbb{E}^0\big\{R_s\big[\sigma(s,\varphi^1_{\cdot\wedge s}, U^{\theta,v}_s, z^1)-\sigma(s,\varphi^2_{\cdot\wedge s}, U^u_s, z^2)\big]\big\}\Big|_{\varphi^1=X^{\theta,v},z^1=u^{\theta,v}_s\atop \varphi^2=X^u,z^2=u_s}\Big|^2ds.
\end{eqnarray*}
Since $\sigma$ is bounded and Lipschitz continuous in $(\varphi, y, z)$, it follows from Proposition \ref{est1} and (\ref{Uth})
that $\lim_{\theta\to0} \mathbb{E}^0[\sup_{0\le t\le T}| I^{3,\theta, 1}_t|^2]=0$.
By the similar arguments using the continuity of $D_\varphi\sigma$ and that of $\partial_z\sigma$, respectively, it is not hard to show that,
for all $p>1$,
$$\lim_{\theta\to 0}\mathbb{E}^0[\sup_{0\le t\le T}| I^{3,\theta, 2}_t|^p]=0; \qquad \lim_{\theta\to0}\mathbb{E}^0[\sup_{0\le t\le T}| I^{3,\theta, 4}_t|^p]=0.
$$
It remains to prove the convergence of $I^{3,\theta,3}$. To this end, we note that, for any $p>1$,
\begin{eqnarray}
\label{est2}
\mathbb{E}^0\Big[\sup_{s\in[0,T]}\big(|R_s|^p+|K_s|^p\big)\Big]&\le& C_p,
\end{eqnarray}
and by (\ref{Uth}) we have, for $p>1$,
\begin{eqnarray}
\label{B3th}
\lim_{\theta\to0}\mathbb{E}^0\int_{0}^{T}\Big|\mathbb{E}^{0}\big\{\big|B^{\theta,u,v}(s,\varphi_{\cdot\wedge s},z)
-B^{u,v}(s, \varphi_{\cdot\wedge s}, z^1)\big|^2\big\}\big|_{\varphi=X^{u},z=u^{\theta,v}_s\atop
z^1=u_s\ \ \ \ \ \ \ }\Big|^pds=0.
\end{eqnarray}
This, together with (\ref{dL2}), (\ref{Uth}), an estimate similar to (\ref{dL1}), and Proposition \ref{est1}, yields that
$\lim_{\theta\to 0}\mathbb{E}^0[\sup_{0\le t\le T}| I^{3,\theta, 3}_t|^2]=0$,
proving the lemma.
\hfill \vrule width.25cm height.25cm depth0cm\smallskip
We now continue the proof of (\ref{7.42}). First we rewrite (\ref{eta}) as
\begin{eqnarray}
\label{eta1}
\eta_{t}^{\theta}&=&\int_0^t\Big\{\mathbb{E}^0\{\tilde\eta^\theta_s\sigma(s, \varphi_{\cdot\wedge s},U^{\theta,v}_s, z)\}\Big|_{\varphi=X^{\theta,v}, \atop
z=u^{\theta,v}_s}+
[D\sigma]^{\theta, u, v}_s(\eta^\theta_{\cdot\wedge s})\medskip\nonumber\\
&&+\mathbb{E}^0\Big\{B^{\theta,u,v}(s, \varphi_{\cdot\wedge s}, z) \Big[\frac{\mathbb{E}^0[\tilde\eta^\theta_sX^{\theta,v}_s+L^u_s\eta^\theta_s|{\cal F}^Y_s]}{\mathbb{E}^0[L^{u}_s|{\cal F}^Y_s]}-
\frac{\mathbb{E}^0[\tilde\eta^\theta_s|{\cal F}^Y_s]}{\mathbb{E}^0[L^{u}_s|{\cal F}^Y_s]}U^u_s\Big]\Big\}
\Big|_{\varphi=X^{u}; \atop z=u^{\theta,v}_s}\Big\}dB^1_s\nonumber\\
&&+I^{3,\theta,0}_t+\sum_{i=1}^4 I^{3, \theta, i}_t,
\end{eqnarray}
where
\begin{eqnarray*}
I^{3,\theta,0}_t&\stackrel{\triangle}{=}&\int_0^t\mathbb{E}^0\Big\{B^{\theta,u,v}(s, \varphi_{\cdot\wedge s}, z) \Big[\frac{\mathbb{E}^0[\tilde\eta^\theta_sX^{\theta,v}_s+L^u_s\eta^\theta_s|{\cal F}^Y_s]}{\mathbb{E}^0[L^{\theta,v}_s|{\cal F}^Y_s]}-
\frac{\mathbb{E}^0[\tilde\eta^\theta_s|{\cal F}^Y_s]}{\mathbb{E}^0[L^{\theta,v}_s|{\cal F}^Y_s]}U^u_s\Big]\Big\}
\Big|_{\varphi=X^{u}; \atop z=u^{\theta,v}_s}\nonumber\\
&&-B^{\theta,u,v}(s, \varphi_{\cdot\wedge s}, z) \Big[\frac{\mathbb{E}^0[\tilde\eta^\theta_sX^{\theta,v}_s+L^u_s\eta^\theta_s|{\cal F}^Y_s]}{\mathbb{E}^0[L^{u}_s|{\cal F}^Y_s]}-
\frac{\mathbb{E}^0[\tilde\eta^\theta_s|{\cal F}^Y_s]}{\mathbb{E}^0[L^{u}_s|{\cal F}^Y_s]}U^u_s\Big]\Big\}
\Big|_{\varphi=X^{u}; \atop z=u^{\theta,v}_s}\Big\}dB^1_s\nonumber\\
\end{eqnarray*}
We note that with the same argument as before one shows that $\lim_{\theta\to0}\mathbb{E}^0[\sup_{0\le t\le T}|I^{3,\theta,0}_t|^2]=0$. On the other hand, similar to (\ref{Buv}) one can argue that
\begin{eqnarray*}
\label{Buv1}
&&\mathbb{E}^0\Big[B^{\theta,u,v}(s, \varphi_{\cdot\wedge s}, z)\Big(\frac{\mathbb{E}^0[\tilde\eta^\theta_sX^{\theta,v}_s+L^u_s\eta^\theta_s|{\cal F}^Y_s]}{\mathbb{E}^0[L^{u}_s|{\cal F}^Y_s]}-
\frac{\mathbb{E}^0[\tilde\eta^\theta_s|{\cal F}^Y_s]}{\mathbb{E}^0[L^{u}_s|{\cal F}^Y_s]}U^u_s\Big)\Big]\Big|_{\varphi=X^u;\atop z=u^{\theta,v}_s}\nonumber\\
&=&\mathbb{E}^0\Big[\int_0^1\partial_y\sigma(s, \varphi_{\cdot\wedge s}, U^u_s+\lambda (U^{\theta,v}_s-U^u_s), z)d\lambda \cdot (\tilde\eta^\theta_sX^{\theta,v}_s+L^u_s\eta^\theta_s-U^u_s
\tilde\eta^\theta_s)\Big]\Big|_{\varphi=X^u;\atop z=u^{\theta,v}_s}.
\end{eqnarray*}
Consequently, we have
\begin{eqnarray*}
\label{eta2}
\eta_{t}^{\theta}&=&\int_0^t\Big\{\mathbb{E}^0\{\alpha^{1,\theta}_s(\varphi^1_{\cdot\wedge s},\varphi^2_{\cdot\wedge s}, z)\tilde\eta^\theta_s\}\Big|_{\varphi^1=X^{\theta,v},\varphi^2=X^{u},\atop z=u^{\theta,v}_s\ \ \ \ \ \ \ \ }+\mathbb{E}^0\{\alpha^{2,\theta}_s(\varphi^2_{\cdot\wedge s}, z)\tilde\eta^\theta_s\}\Big|_{\varphi^2=X^{u},\atop z=u^{\theta,v}_s}\Big\}dB^1_s\\
& &+\int_0^t\Big\{\mathbb{E}^0\{\beta^\theta_s(\varphi^2_{\cdot\wedge s}, z)\eta^\theta_s\}\Big|_{\varphi^2=X^u \atop z=u^{\theta,v}_s}+[D\sigma]^{\theta, u, v}_s(\eta^\theta_{\cdot\wedge s})\Big\}dB^1_s+I^{3,\theta}_t,
\end{eqnarray*}
where $I^{3,\theta}_t=\sum_{i=0}^4 I^{3, \theta, i}_t$, and
\begin{eqnarray*}
&&\alpha^{1,\theta}_s(\varphi^1_{\cdot\wedge s},\varphi^2_{\cdot\wedge s}, z)\stackrel{\triangle}{=}\int_0^1D_\varphi\sigma(s,\varphi^2_{\cdot\wedge s}+\lambda(\varphi^1_{\cdot\wedge s}-\varphi^2_{\cdot\wedge s}),U^{\theta,v}_s,z)(\varphi^1_{\cdot\wedge s}-\varphi^2_{\cdot\wedge s})d\lambda,\\
&&\alpha^{2,\theta}_s(\varphi^2_{\cdot\wedge s}, z)\stackrel{\triangle}{=} \sigma(s,\varphi^2_{\cdot\wedge s},U^{\theta,v}_s,z)+\int_0^1\partial_y\sigma(s,\varphi^2_{\cdot\wedge s},U_s^u+\lambda(U^{\theta,v}_s-U_s^u),z)d\lambda(U^{\theta,v}_s-U_s^u);\\
&&\beta^\theta_s(\varphi^2_{\cdot\wedge s}, z)\stackrel{\triangle}{=} L_s^u\int_0^1 \partial_y\sigma(s,\varphi^2_{\cdot\wedge s},U_s^u+\lambda(U^{\theta,v}_s-U_s^u),z)d\lambda.
\end{eqnarray*}
\noindent Notice that
$$|\alpha^{1,\theta}_s(\varphi^1_{\cdot\wedge s},\varphi^2_{\cdot\wedge s}, z)| +|\alpha^{2,\theta}_s(\varphi^2_{\cdot\wedge s}, z)|\le C(1+|\varphi^1_{\cdot\wedge s}|+|\varphi^2_{\cdot\wedge s}|+|U_s^{\theta, v}|+|U^u_s|),\ |\beta_s^\theta(\varphi_{\cdot\wedge s}, z)|\le CL^u_s.
$$
Now by the Burkholder and Cauchy-Schwartz inequalities we have, for all $p\ge 2$, $t\in[0,T]$,
\begin{eqnarray*}
\mathbb{E}^0\Big[\sup_{s\in[0,t]}|\eta^\theta_s|^{2p}\Big]\le C_p \Big\{\mathbb{E}^0[\|I^{3,\theta}\|^{2p}_{\mathbb{C}_T}]+\mathbb{E}^0\Big\{\Big[\int_0^t\Big(\mathbb{E}^0[|\eta^\theta_s|^2
+|\tilde\eta^\theta_s|^2]+\sup_{r\in[0,s]}|\eta^\theta_s|^2\Big)ds\Big]^p\Big\}\Big\}, \end{eqnarray*}
and from Gronwall's inequality one has
\begin{eqnarray}
\label{etaest}
\mathbb{E}^0\Big[\sup_{s\in[0,t]}|\eta^\theta_s|^{2p}\Big]\le C_p\Big\{\mathbb{E}^0\Big[\|I^{3,\theta}\|^{2p}_{\mathbb{C}_T}+\int_0^t\Big(\mathbb{E}^0[|\tilde\eta^\theta_s|^p]
\Big)^2ds\Big\}, \quad t\in[0,T].
\end{eqnarray}
On the other hand, setting $I^\theta_t \stackrel{\triangle}{=} I^{1, \theta}_t+I^{2,\theta}_t$, $t\in[0,T]$, we have from (\ref{7.45}) that, for $p\ge 2$,
\begin{eqnarray*}
\mathbb{E}^0\Big[\sup_{s\in[0,t]}|\tilde\eta^\theta_s|^p\Big]\le C_p\Big\{\mathbb{E}^0[\|I^\theta\|^p_{\mathbb{C}_T}]+\int_0^t\mathbb{E}^0[|\tilde\eta^\theta_s|^p]ds
+\int_0^t\big(\mathbb{E}^0[|\eta^\theta_s|^{2p}]\big)^{1/2}ds\Big\}, ~t\in[0,T].
\end{eqnarray*}
Then Gronwall's inequality leads to that
\begin{eqnarray}
\label{tetaest}
\big(\mathbb{E}^0\Big[\sup_{s\in[0,t]}|\tilde\eta^\theta_s|^p\Big]\Big)^2\le C_p\Big\{\big(\mathbb{E}^0\|I^\theta\|^p_{\mathbb{C}_T}\big)^2+
\int_0^t\mathbb{E}^0[|\eta^\theta_s|^{2p}]ds\Big\}, \quad t\in[0,T].
\end{eqnarray}
Combining (\ref{etaest}), (\ref{tetaest}), applying (\ref{7.46}) and Lemma \ref{I3thconv} as well as the Gronwall inequality, we can easily
deduce (\ref{7.42}) by sending $\theta\to0$. Consequently, (\ref{7.43}) holds as well.
\hfill \vrule width.25cm height.25cm depth0cm\smallskip
From Proposition 6.1, (\ref{deltaU}) and the above development we also obtain the following corollary.
\begin{cor}
We assume that Assumption \ref{Assum1} holds. Then, for all $p>1$,
$$\lim_{\theta\to 0}\mathbb{E}^0[\|\delta_\theta U-\overline{V}\|_{\mathbb{C}_T}^p]=\lim_{\theta\to 0}\mathbb{E}^0[\sup_{0\le s\le T}|\frac{U_s^{\theta,v}-U_s^u}{\theta}-\overline{V}_s|^p]=0,
$$
where
$$\overline{V}_t\stackrel{\triangle}{=} \frac{\mathbb{E}^0[R_tX_t^u+L_t^uK_t|{\cal{F}}_t^Y]}{\mathbb{E}^0[L_t^u|{\cal{F}}_t^Y]}-\frac{\mathbb{E}^0[R_t|{\cal{F}}_t^Y]}{\mathbb{E}^0[L_t^u|{\cal{F}}_t^Y]}U_t^u,\ t\in [0, T]. $$
\end{cor}
\section{ Stochastic Maximum Principle}
\setcounter{equation}{0}
We are now ready to study the Stochastic Maximum Principle. The main task will be to determine the appropriate {\it adjoint equation}, which we expect to be a backward stochastic differential equation of Mean-field type. We begin with
a simple analysis. Suppose that $u=u^*$ is an optimal control, and for any $ v\in\mathscr{U}_{ad}$, we define $u^{\theta, v}$
by (\ref{7.22}). Then we have
\begin{eqnarray}
\label{7.23}
0&\le &\frac{J(u^{\theta, v})-J(u)}{\theta}\nonumber\\
&=&\frac1\theta ~\mathbb{E}^0\Big\{\mathbb{E}^0[L^{\theta, v}_T\Phi(x, U^{\theta, v}_T)]|_{x=X^\theta_T}-\mathbb{E}^0[L^u_T\Phi(x, U^u_T)]|_{x=X^u_T}\\
&&+\int_0^T\big[\mathbb{E}^0[L^{\theta, v}_sf(s, \varphi_{\cdot\wedge s}, U^{\theta, v}_s, z)]|_{\varphi=X^{\theta, v},\atop z=u^{\theta,v}_s}-\mathbb{E}^0[L^u_sf(s, \varphi_{\cdot\wedge s}, U^u_s, z)]|_{\varphi=X^u,\atop z=u_s}\big]ds\Big\}.\nonumber
\end{eqnarray}
Now, repeating the same analysis as that in Proposition \ref{est1}, then sending $\theta\to 0$, it follows from
Propositions \ref{est1}, \ref{lemma7.2} and the continuity of the functions $\Phi$ and $f$ that
\begin{eqnarray}
\label{terminal}
0&\leq& \mathbb{E}^0[K_{T}\xi]+\mathbb{E}^0[R_T\Theta]
+ \mathbb{E}^{0}\Big\{\int_0^T\Big\{\mathbb{E}^0[R_sf(s,\varphi_{\cdot\wedge s}, U^u_s,z)]|_{\varphi=X^{u}, z=u_s}\nonumber\\
&&\qquad+\mathbb{E}^0[\partial_yf(s,\varphi_{\cdot\wedge s}, U^u_s, z)(X^u_s-U^u_s)R_s+L^u_sK_s]|_{\varphi=X^{u}, z=u_s}\\
&&\qquad+\mathbb{E}^0[L^u_s D_\varphi f(s,\varphi_{\cdot\wedge s}, U^u_s,z)(\psi_{\cdot\wedge s})]|_{\varphi=X^{u}, z=u_s,\psi=K}\nonumber\\
&&\qquad +\mathbb{E}^0[L^u_s \partial_z f(s,\varphi_{\cdot\wedge s}, U^u_s,z)]|_{\varphi=X^{u}, z=u_s}(v_s-u_s)\Big\}ds\Big\},\nonumber
\end{eqnarray}
where
\begin{eqnarray}
\label{pTQT}
\xi&\stackrel{\triangle}{=}&\mathbb{E}^{0}[L_{T}^{u}\partial_x\Phi(x,U^u_T)]|_{x=X_{T}^{u}}+L^u_T\mathbb{E}^0[\partial_y \Phi(X_{T}^{u}, y)]|_{y=U^u_T},\nonumber\\
\Theta&\stackrel{\triangle}{=}&\mathbb{E}^{0}[\Phi(X_{T}^{u}, y)]\big|_{y=U^u_T}+(X^u_T-U^u_T)\mathbb{E}^0[\partial_y \Phi(X_{T}^{u}, y)]|_{y=U^u_T}.
\end{eqnarray}
We now consider the adjoint equations that take the following form of backward SDEs on the reference space
$(\Omega, {\cal F}, \mathbb{Q}^0)$:
\begin{eqnarray}
\label{BSDE}
\left\{\begin{array}{lll}dp_t= -\alpha_tdt+d\Gamma_t+q_tdB^1_t+\widetilde q_t dY_t, \qquad p_T=\xi,\\
dQ_t= -\beta_tdt+d\Sigma_t+M_tdB^1_t+\widetilde M_t dY_t, \qquad Q_T=\Theta.
\end{array}\right.
\end{eqnarray}
Here the coefficients $\alpha, \beta$ as well as the two bounded
variation processes $\Gamma$ and $\Sigma$ are to be determined. Applying It\^o's formula and recalling the variational equations
(\ref{Keq}) and (\ref{Req}), we can easily derive (denote $U^u_t=\mathbb{E}^{u}[X_{t}^{u}|{\mathcal{F}}_{t}^{Y}]$,
$t\in[0,T]$)
\begin{eqnarray}
\label{pKQR}
&& \mathbb{E}^0[\xi K_T]+\mathbb{E}^0[\Theta R_T] \nonumber\\
&=&\int_0^T\Big\{-\mathbb{E}^0[K_s\alpha_s]-\mathbb{E}^0[R_s\beta_s]+\mathbb{E}^0\Big[q_s\mathbb{E}^0[R_s\sigma(s, \varphi_{\cdot\wedge s}, U^u_s, z)]\big|_{\varphi=X^u, z=u_s}\Big]
\nonumber\\
&&+\mathbb{E}^{0}\Big[q_{s}\mathbb{E}^0\Big[\partial_y\sigma(s,\varphi_{z\cdot\wedge s}, U^{u}_s,z)[(X^{u}_s-U^u_s)R_s+L^u_sK_s]\Big]
\Big|_{\varphi=X^u, z=u_s}\Big]\\
&&+\mathbb{E}^{0}\big[q_s[D\sigma]^{u,v}_s(K_{\cdot\wedge s}) +q_s C^{u,v}_\sigma(s)(v_s-u_s)+\widetilde M_sR_sh(s, X^u_s)
+\widetilde M_{s}K_sL^u_s\partial_xh(s, X^u_s)]\Big\}ds \nonumber\\
&&+\mathbb{E}^0\Big\{\int_0^T[K_sd\Gamma_s+R_sd\Sigma_s]\Big\},\nonumber
\end{eqnarray}
where $[D\sigma]^{u,v}$ and $C^{u,v}$ are defined by (\ref{DBC}).
By Fubini's Theorem we see that
\begin{eqnarray}
\label{Fubini}
\left\{\begin{array}{llll}
\mathbb{E}^0\big[q_s\mathbb{E}^0[R_s\sigma(s, \varphi{\cdot\wedge s}, U^u_s, z)]\big|_{\varphi=X^u, z=u_s}\big]=
\mathbb{E}^{0}\big[R_s\mathbb{E}^0[q_{s}\sigma(s, X_{\cdot\wedge s}, y, u_s)]\big|_{y=U^{u}_s}\big]; \medskip\\
\mathbb{E}^{0}\Big[q_{s}\mathbb{E}^0\Big[\partial_y\sigma(s,\varphi_{z\cdot\wedge s}, U^{u}_s,z)[(X^{u}_s-U^u_s)R_s+L^u_sK_s]\Big]\Big|_{\varphi=X^u, z=u_s}\Big]\\
\qquad=\mathbb{E}^{0}\Big[\mathbb{E}^0\big[q_s\partial_y\sigma(s,X_{z\cdot\wedge s}, y,u_s)]\big|_{y=U^u_s}[(X^{u}_s-U^u_s)R_s+L^u_sK_s]\Big].
\end{array}\right.
\end{eqnarray}
Furthermore, in light of definition of $[D\sigma]^{u,v}$ ((\ref{DBC})), if we denote, for fixed $(t,\varphi, z)$,
\begin{eqnarray}
\label{mu0}
\mu^0_\sigma(t, \varphi_{\cdot\wedge t}, z)(\cdot)\stackrel{\triangle}{=} \mathbb{E}^0[L^u_t D_\varphi\sigma(t, \varphi_{\cdot\wedge t}, U^u_t, z)](\cdot)\in \mathscr{M}[0,T],
\end{eqnarray}
where $\mathscr{M}[0,T]$ denotes all the Borel measures on $[0,T]$, then we can write
\begin{eqnarray}
\label{DsiK}
[D\sigma]^{u,v}_t(K_{\cdot\wedge t})=\mathbb{E}^0\big[L^u_t D_\varphi\sigma(t, \varphi_{\cdot\wedge t}, U^u_t, z)(\psi)]\big|_{\varphi=X^u, z=u_t, \atop
\psi=K_{\cdot\wedge t}\ \ \ \ }=\int_0^tK_r\mu^0_\sigma(r, X^u_{\cdot\wedge r}, u_r)(dr).
\end{eqnarray}
Let us now argue that a similar Fubini Theorem argument holds for the random measure $\mu^0_\sigma(t, X^u_{\cdot\wedge t}, u_t)
(\cdot)$.
First, for a given process $q\in L^2_\mathbb{F}(\mathbb{Q}^0; [0,T])$, consider the following finite variation (FV) process (in fact, under Assumption \ref{Assum1}, {\it integrable variation} (IV) process):
\begin{eqnarray}
\label{A}
A^\sigma_t\stackrel{\triangle}{=} \int_0^T\int_0^{t\wedge s}q_s\mu^0_\sigma(s, X_{\cdot\wedge s}^u, u_s)(dr)ds, \qquad t\in[0,T].
\end{eqnarray}
It is easy to check, as a (randomized) signed measure on $[0,T]$, it holds $\mathbb{Q}^0$-almost surely that $dA^\sigma_t=\int_t^Tq_s\mu^0_\sigma
(s, X^u_{\cdot\wedge s}, u_s)(dt)ds$. We note that being a ``raw FV" process, the process $A^\sigma$ is not $\mathbb{F}$-adapted. We now consider its {\it dual predictable projection}:
\begin{eqnarray}
\label{dpA}
^p\negthinspace\Big(\int_t^Tq_s\mu^0_\sigma(s, X^u_{\cdot\wedge s}, u_s)(dt)ds\Big)\stackrel{\triangle}{=} d[^p\negthinspace\neg A^\sigma_t], \quad t\in[0,T].
\end{eqnarray}
We remark that $d[^p\negthinspace\neg A_t]$ is a predicable random measure that can be formally understood as
\begin{eqnarray*}
\label{dpA1}
d[ ^p\negthinspace\neg A^\sigma_t]=\mathbb{E}^0[dA^\sigma_t|{\cal F}_{t-}]=\mathbb{E}^0\Big[\int_t^Tq_s\mu^0_\sigma(s, X^u_{\cdot\wedge s}, u_s)(dt)ds\Big|{\cal F}_{t-}\Big], \quad t\in[0,T].
\end{eqnarray*}
Using the definition of dual predicable projection and (\ref{DsiK}), we see that, for the continuous process $K\in
L^2_\mathbb{F}(\mathbb{Q}^0;\mathbb{C}_T)$,
\begin{eqnarray}
\label{Duvsi}
\int_0^T\mathbb{E}^0[q_s[D\sigma]^{u,v}_s(K_{\cdot\wedge s})]ds&=&\int_0^T\mathbb{E}^0\Big[q_s\int_0^sK_r\mu^0_\sigma(r,
X^u_{\cdot\wedge r}, u_r)(dr)\Big]ds\nonumber\\
&=& \mathbb{E}^0\Big[\int_0^TK_r dA^\sigma_r\Big]=\mathbb{E}^0\Big[\int_0^T K_r d[^p\negthinspace\neg A^\sigma_r]\Big]\\
&=& \mathbb{E}^0\Big[\int_0^TK_r \, ^p\negthinspace\Big(\int_r^Tq_s\mu^0_\sigma(s, X^u_{\cdot\wedge s}, u_s)(dr)ds \Big)\Big].
\nonumber
\end{eqnarray}
Similarly, we denote $A^f_t \stackrel{\triangle}{=} \int_0^T\int_0^{t\wedge s}\mu^0_f(s, X^u_{\cdot\wedge s}, u_s)(dr)ds$, $t\in[0,T]$;
and denote its dual predicable projection by $^p\negthinspace\Big(\int_t^T\mu^0_f(s, X^u_{\cdot\wedge s}, u_s)(dt)ds\Big) = d[^p\negthinspace\neg A^f_t]$, $t\in[0,T]$.
We now plug (\ref{Fubini}) and (\ref{Duvsi}) into (\ref{pKQR}) to get:
\begin{eqnarray}
\label{pKQR1}
&& \mathbb{E}^0[\xi K_T]+\mathbb{E}^0[\Theta R_T] \nonumber\\
&=&\mathbb{E}^0\Big\{\int_0^T\Big\{K_s\Big[-\alpha_s+L^u_s\mathbb{E}^0\big[q_{s}\partial_y\sigma(s,X^u_{\cdot\wedge s}, y,u_s)\big]\big|_{y=U^{u}_s}
+M_{s}L^u_s\partial_xh(s, X^u_s)\Big]\nonumber\\
&& +R_s\Big[-\beta_s+\mathbb{E}^0[q_{s}\sigma(s, X_{\cdot\wedge s}, y, u_s)]\big|_{y=U^{u}_s}+\widetilde M_sh(s, X^u_s)\Big] +q_s C^{u,v}_s(v_s-u_s)\nonumber\\
&& +R_s\mathbb{E}^0\big[q_s\partial_y\sigma(s,X^u_{\cdot\wedge s}, y,u_s)\big]\big|_{y=U^{u}_s}(X^{u}_s-U^u_s)\Big\}ds+\int_0^TK_s d[^p\negthinspace A^\sigma_s]\Big\}\\
&&+\mathbb{E}^0\Big\{\int_0^T[K_sd\Gamma_s+R_sd\Sigma_s]\Big\},\nonumber\\
&=& \mathbb{E}^0\Big\{\int_0^T[-K_s\hat \alpha_s-R_s\hat\beta_s+q_s C^{u,v}_\sigma(s)(v_s-u_s)]ds+K_s d[^p\negthinspace A^\sigma_s]+[K_sd\Gamma_s+R_sd\Sigma_s]\Big\},\nonumber
\end{eqnarray}
where
\begin{eqnarray}
\label{alpha}
\left\{\begin{array}{llllll}
\hat\alpha_t \stackrel{\triangle}{=} \alpha_t-L^u_t\mathbb{E}^0\big[q_t\partial_y\sigma(t,X^u_{\cdot\wedge t}, y,u_t)\big]\big|_{y=U^{u}_t}
-\widetilde M_tL^u_t\partial_xh(t, X^u_t); \medskip\\
\hat\beta_t\stackrel{\triangle}{=} \beta_t-\mathbb{E}^0[q_{t}\sigma(t, X_{\cdot\wedge t}, y, u_t)]\big|_{y=U^{u}_t}-\widetilde M_th(t, X^u_t)\\
\qquad-\mathbb{E}^0\big[q_t\partial_y\sigma(t,X^u_{\cdot\wedge t}, y,u_t)\big|_{y=U^{u}_t}(X^{u}_t-U^u_t).
\end{array}\right.
\end{eqnarray}
Combining (\ref{terminal}) and (\ref{pKQR1}) and using the processes $dA^\sigma$, $dA^f$ and their dual predicable projections,
we have
\begin{eqnarray}
\label{terminal1}
0&\leq&
\mathbb{E}^0\Big\{\int_0^T[-K_s\hat \alpha_s-R_s\hat\beta_s+q_s C^{u,v}_\sigma(s)(v_s-u_s)]ds+\int_0^TK_sd[^p\negthinspace\neg A^\sigma_s]\Big\}\\
&&+\mathbb{E}^0\Big\{\int_0^T\Big[R_s\big[\mathbb{E}^0[f(s, X_{\cdot\wedge s}, y, u_s)]\big|_{y=U^{u}_s}+\mathbb{E}^0\big[\partial_y f(s,X^u_{\cdot\wedge s}, y,u_s)\big]\big|_{y=U^{u}_s}(X^{u}_s-U^u_s)\big]\nonumber\\
&&+L^u_sK_s\mathbb{E}^0\big[\partial_y f(s,X^u_{\cdot\wedge s}, y,u_s)\big]\big|_{y=U^{u}_s}+C^{u,v}_f(s)
(v_s-u_s)\Big]ds+\int_0^TK_s d[^p\negthinspace\neg A^f_s]\Big\}\nonumber\\
&&+\mathbb{E}^0\Big\{\int_0^T[K_sd\Gamma_s+R_sd\Sigma_s]\Big\},\nonumber
\end{eqnarray}
where
$C^{u,v}_f(s)\stackrel{\triangle}{=} \mathbb{E}^0[L^u_s \partial_z f(s,\varphi_{\cdot\wedge s}, U^u_s,z)]|_{\varphi=X^{u}, z=u_s}$.
Now, if we set $\Sigma_t=0$, and
\begin{eqnarray}
\label{abcd}
\hat\alpha_t&=& L^u_t\mathbb{E}^0\big[\partial_y f(t,X^u_{\cdot\wedge t}, y,u_t)\big]\big|_{y=U^{u}_t}\nonumber\\
\hat\beta_t&=& \mathbb{E}^0[f(t, X_{\cdot\wedge t}, y, u_t)]\big|_{y=U^{u}_t}+\mathbb{E}^0\big[\partial_y f(t,X^u_{\cdot\wedge t}, y,u_t)\big]\big|_{y=U^{u}_t}(X^{u}_t-U^u_t)\\
d\Gamma_t&=&-d[^p\negthinspace\neg A^\sigma_t] -d[^p\negthinspace\neg A^f_t], \nonumber
\end{eqnarray}
then (\ref{terminal1}) becomes
\begin{eqnarray}
\label{SMP0}
0&\le& \mathbb{E}^0\Big\{\int_0^T [q_tC^{u,v}_\sigma(s)+C^{u,v}_f(s)](v_s-u_s)ds\Big\}, \quad v\in\mathscr{U}_{ad}.
\end{eqnarray}
From this we should be able to derive the maximum principle, provided that the adjoint equation (\ref{BSDE}) with coefficients
$\alpha$, $\beta$, and
$\Gamma$ determined by (\ref{alpha}) and (\ref{abcd}) is well-defined.
\begin{rem}
\label{hDsi}
{\rm 1) We remark that the process $\Gamma$ in (\ref{abcd}) should be considered as a mapping from the space $L^2_\mathbb{F}([0,T]\times\Omega)\times L^2_\mathbb{F}(\Omega;\mathbb{C}_T)\times
L^2_\mathbb{F}([0,T]\times\Omega; U)$ to $\mathscr{M}_\mathbb{F}([0,T])$, the space of all the random measures on $[0,T]$, such that
(i) $(t,\omega)\mapsto \mu(t, \omega, A)$ is $\mathbb{F}$-progressively measurable, for all $A\in\mathscr{B}([0,T])$;
(ii) $\mu(t,\omega, \cdot)\in\mathscr{M}([0,T])$ is a finite Borel measure on $[0, T]$.
\medskip
2) Assumption \ref{Assum1}-(iii) implies that the random measure $\mathbb{D}_\sigma[q,X^u, u](t, dt)$ satisfies the following estimate:
for any $q\in L^2_\mathbb{F}([0,T]\times \Omega)$ and $u\in\mathscr{U}_{ad}$,
\begin{eqnarray}
\label{hDsiest}
\mathbb{E}^0\Big[\int_0^T|d\,^p\negthinspace\neg A^\sigma_t|\Big]&=&
\mathbb{E}^0\Big\{\int_0^T\Big|\,^p\negthinspace\Big(\int_t^Tq_s\mu^0_\sigma(s, X^u_{\cdot\wedge s}, u_s)(dt)ds\Big) \Big|\Big\}\nonumber\\
&\le& \mathbb{E}^0\Big\{\int_0^T\int_0^s|q_s||\mu^0_\sigma(s, X^u_{\cdot\wedge s }, u_s)(dt)|ds\Big\}\\
&\le&\mathbb{E}^0\Big\{\int_0^T|q_s|\int_0^s\ell(s,dt)ds\Big\}\le C\mathbb{E}^0\Big\{\int_0^T|q_s|ds\Big\}\le C\|q\|_{2,2,\mathbb{Q}^0}.\nonumber
\end{eqnarray}
The same estimate holds for $\mathbb{D}_f[X^u, u](t, dt)$ as well.
\medskip
3) Clearly, the processes $A^\sigma$ and $A^f$ are originated from the Fr\'echet derivatives of $\sigma$ and $f$, respectively, with respect to
the path $\varphi_{\cdot\wedge t}$. If $\sigma$ and $f$ are of Markovian type, then they will be absolutely
continuous with respect to the Lebesgue measure.
\hfill \vrule width.25cm height.25cm depth0cm\smallskip}
\end{rem}
We shall now validate all the arguments presented above. To begin with, we note that the choice of $\alpha$, $\beta$, and
$\Gamma$ via by (\ref{alpha}) and (\ref{abcd}), together with the terminal condition $(\xi, \Theta)$ by (\ref{pTQT}), amounts to saying
that the processes $(p, q, \tilde q)$ and $(Q, M, \tilde M)$ solve the BSDE:
\begin{eqnarray}
\label{BSDE1}
\left\{\begin{array}{lllllll}
dp_t=-L^u_t\Big\{\mathbb{E}^0\big[\partial_y f(t,X^u_{\cdot\wedge t}, y,u_t)\big]\big|_{y=U^{u}_t}+\mathbb{E}^0\big[q_t\partial_y\sigma(t,X^u_{\cdot\wedge t}, y,u_t)\big]\big|_{y=U^{u}_t}\medskip\\
\qquad\quad+\widetilde M_t\partial_xh(t, X^u_t)\Big\}dt-d\,^p\negthinspace\neg A^\sigma_t - d\,^p\negthinspace\neg A^f_t+q_tdB^1_t+\widetilde q_tdY_t\medskip\\
dQ_t=-\Big\{\mathbb{E}^0[q_{t}\sigma(t, X^u_{\cdot\wedge t}, y, u_t)]\big|_{y=U^{u}_t}-\widetilde M_th(t, X^u_t)\medskip\\
\qquad\quad+\mathbb{E}^0\big[q_t\partial_y\sigma(t,X^u_{\cdot\wedge t}, y,u_t)]\big|_{y=U^{u}_t}(X^{u}_t-U^u_t)\medskip\\
\qquad\quad+\mathbb{E}^0[f(t, X_{\cdot\wedge t}, y, u_t)]\big|_{y=U^{u}_t}+\mathbb{E}^0\big[\partial_y f(t,X^u_{\cdot\wedge t}, y,u_t)\big]\big|_{y=U^{u}_t}(X^{u}_t-U^u_t)\Big\}dt\medskip\\
\qquad\quad+M_tdB^1_t+\widetilde M_tdY_t,\medskip\\
p_T=\xi, \quad Q_T=\Theta.
\end{array}\right.
\end{eqnarray}
Now if we denote $\eta=(p, Q)^T$, $W=(B^1, Y)^T$, $\Xi=\Big[\begin{array}{ll}q&\tilde q\\ M&\tilde M\end{array}\Big]$, then we can rewrite
(\ref{BSDE1}) in a more abstract (vector) form:
\begin{eqnarray}
\label{BSDE2}
\left\{\begin{array}{ll}
d\eta_t=-\{A_t+\mathbb{E}^0[G_t\Xi_tg(t,y)]\big|_{y=U^u_t}+H_t\Xi_t h_t\}dt-\Gamma(\Xi)(t,dt)-\Gamma_0(t,dt)+\Xi_tdW_t, \medskip\\
\eta_T=\Upsilon,
\end{array}\right.
\end{eqnarray}
where $\Upsilon\in L^2_{\mathbb{F}^W_T}(\Omega;\mathbb{Q}^0)$;
$A, G, H$ and $h$ are bounded, vector or matrix-valued $\mathbb{F}^W$-adapted processes with appropriate dimensions,
$g$ is an $\mathbb{R}^2$-valued progressively measurable random field, and $U$ is an $\mathbb{F}^Y$-adapted process. Moreover,
the $\mathbb{R}^2$-valued finite variation processes $\Gamma(\Xi)(t,dt)$ and $\Gamma_0(t,dt)$ take the form:
\begin{eqnarray}
\label{Gamma}
\Gamma(\Xi)(t,dt)=~^p\negthinspace\Big(\int_t^T\Xi_r\mu^1_r(dt)dr\Big), \quad \Gamma_0(t, dt)=~^p\negthinspace\Big(\int_t^T\mu^2_r(dt)dr\Big),
\end{eqnarray}
where $r\mapsto \mu^i_r(\cdot)$, $i=1,2$, are $\mathscr{M}[0,T]$-valued measurable random processes satisfying, as
measures with respect to the total variation norm,
\begin{eqnarray}
\label{muest}
|\mu^1_r(dt)|+|\mu^2_r(dt)|\le \ell(r,dt), \quad r\in[0,T], ~\mathbb{Q}^0\mbox{a.s.}
\end{eqnarray}
We note that $\Gamma(\Xi)(dt)$ and $\Gamma_0(dt)$ are representing $d[^p\negthinspace A^\sigma_t]$ and $[^p\negthinspace A^f_t]$
in (\ref{BSDE1}), respectively, and can be substantiated by (\ref{A}) and (\ref{dpA}). Furthermore, by Assumption
\ref{Assum1}, they both satisfy (\ref{muest}). To the best of our knowledge,
BSDE (\ref{BSDE2}) is beyond all the existing frameworks of BSDEs, and we shall give a brief proof for its well-posedness.
\begin{thm}
\label{wellposed}
Assume that the Assumption \ref{Assum1} is in force. Then, the BSDE (\ref{BSDE2}) has a unique solution $(\eta, \Xi)$.
\end{thm}
{\it Proof.} The proof is more or less standard, we shall only point out a key estimate. For any given $\widetilde{\Xi}^i\in L^2_{\mathbb{F}^W}([0, T]\times\Omega;\mathbb{R}^4)$, obviously we have a unique solution $(\eta^i, \Xi^i)$ of (\ref{BSDE2}), $i=1, 2$, respectively, i.e.,
$$
\label{BSDE3}
\left\{\begin{array}{ll}
d\eta^i_t=-\{A_t+\mathbb{E}^0[G_t\widetilde{\Xi}^i_tg(t,y)]\big|_{y=U^u_t}+H_t\widetilde{\Xi}^i_t h_t\}dt-\Gamma(\widetilde{\Xi}^i)(t,dt)-\Gamma_0(t,dt)+\Xi^i_tdW_t, \medskip\\
\eta_T^i=\Upsilon.
\end{array}\right.
$$
We define $\widehat{\xi}=\xi^1-\xi^2$, $\xi^i=\eta^i, \Xi^i$, $i=1, 2$, respectively. $\widehat{\widetilde{\Xi}}=\widetilde{\Xi}^1-\widetilde{\Xi}^2$. Noting the linearity of BSDE (\ref{BSDE2}) we see that $\widehat{\eta}$ satisfies:
\begin{eqnarray}
\label{BSDE3}
\widehat{\eta}_t=\int_t^T\Big\{\mathbb{E}^0[G_s\widehat{\widetilde{\Xi}}_s g(s,y)]\big|_{y=U^u_s}+H_s\widehat{\widetilde{\Xi}}_s h_s\Big\}ds+\int_t^T\Gamma(\widehat{\widetilde{\Xi}})(s,ds)-M^T_t,
\end{eqnarray}
where $M^T_t\stackrel{\triangle}{=} \int_t^T\widehat{\Xi}_sdW_s$. Therefore,
\begin{eqnarray*}
|\widehat{\eta}_t+M_t^T|^2\le 2\Big\{\Big|\int_t^T\Big\{\mathbb{E}^0[G_s\widehat{\widetilde{\Xi}}_s g(s,y)]\big|_{y=U^u_s}+H_s\widehat{\widetilde{\Xi}}_s h_s\Big\}ds\Big|^2+\Big|\int_t^T\Gamma(\widehat{\widetilde{\Xi}})(s,ds)\Big|^2\Big\}.
\end{eqnarray*}
Taking expectation on both sides above and noting that
$ \mathbb{E}^0[\widehat{\eta}_t M^T_t]=0$ and
$$\mathbb{E}^0\Big\{\Big|\int_t^T\Big\{\mathbb{E}^0[G_s\widehat{\widetilde{\Xi}}_sg(s,y)]\big|_{y=U^u_s}+H_s\widehat{\widetilde{\Xi}}_s h_s\Big\}ds\Big|^2\Big\}\le C(T-t)\mathbb{E}^0\Big[\int_t^T|\widehat{\widetilde{\Xi}}_s|^2ds\Big],$$
we have
\begin{eqnarray}
\label{BSDEest}
\mathbb{E}^0[|\widehat{\eta}_t|^2] +\mathbb{E}^0\Big[\int_t^T|\widehat{\Xi}_s|^2ds\Big]\le C(T-t)\mathbb{E}^0\Big[\int_t^T|\widehat{\widetilde{\Xi}}_s|^2ds\Big] +\mathbb{E}^0\Big\{\Big|\int_t^T\Gamma(\widehat{\widetilde{\Xi}})(s,ds)\Big|^2\Big\}.
\end{eqnarray}
To estimate the term involving $\Gamma(\widehat{\widetilde{\Xi}})$ we note that (recall (\ref{Gamma})) if a square-integrable process $V$ is increasing and continuous,
then so is its dual predictable projection $^pV$. Thus, by the definition of $^pV$ we have
\begin{eqnarray*}
&&\mathbb{E}^0\Big[\Big|\int_t^T d[^pV_s]\Big|^2\Big]=2 \mathbb{E}^0\Big[\int_t^T(^pV_s-{}^pV_t)d[^pV_s]\Big]=
2 \mathbb{E}^0\Big[\int_t^T(^pV_s-{}^pV_t)dV_s\Big]\\
&\le& 2\mathbb{E}^0[(^pV_T-{}^pV_t)(V_T-V_t)]\le 2\Big(\mathbb{E}^0\Big[\Big|\int_t^T d[^pV_s]\Big|^2\Big]\Big)^{1/2}
\Big(\mathbb{E}^0\Big[\Big|\int_t^T dV_s\Big|^2\Big]\Big)^{1/2}.
\end{eqnarray*}
That is,
\begin{eqnarray}
\label{dpAest}
\mathbb{E}^0\Big[\Big|\int_t^T d[^pV_s]\Big|^2\Big]\le 4\mathbb{E}^0\Big[\Big|\int_t^T dV_s\Big|^2\Big].
\end{eqnarray}
Applying this to $V_t\stackrel{\triangle}{=} \int_0^T\int_0^{t\wedge r}|\widehat{\widetilde{\Xi}}_r||\mu^1_r(ds)|dr$, $t\in[0,T]$, we have
\begin{eqnarray*}
\mathbb{E}^0\Big[\Big|\int_t^T \Gamma(\widehat{\widetilde{\Xi}})(s,ds)\Big|^2\Big]
&\le&\mathbb{E}^0\Big[\Big|\int_t^T{}^p\Big(\int_s^T |\widehat{\widetilde{\Xi}}_r||\mu^1_r(ds)|dr\Big)\Big|^2\Big]\le 4\mathbb{E}^0\Big[\Big|\int_t^T\int_s^T |\widehat{\widetilde{\Xi}}_r| |\mu^1_r(ds)|dr\Big|^2\Big]\\
&\le&4\mathbb{E}^0\Big[\Big|\int_t^T \int_s^T |\widehat{\widetilde{\Xi}}_r|\ell(r,ds)dr\Big|^2\Big\}\\
&\le& C \mathbb{E}^0\Big[\Big|\int_t^T |\widehat{\widetilde{\Xi}}_r|dr\Big|^2\Big\}\le C(T-t)\mathbb{E}^0\Big[\int_0^T|\widehat{\widetilde{\Xi}}_s|^2ds\Big],
\end{eqnarray*}
and therefore (\ref{BSDEest}) becomes
\begin{eqnarray}
\label{BSDEest1}
\mathbb{E}^0[|\widehat{\eta}_t|^2] +\mathbb{E}^0\Big[\int_t^T|\widehat{\Xi}_s|^2ds\Big]\le C(T-t)\mathbb{E}^0\Big[\int_t^T|\widehat{\widetilde{\Xi}}_s|^2ds\Big] .
\end{eqnarray}
With this estimate, and following the standard argument one shows that BSDE (\ref{BSDE1}) is well-posed on $[T-\delta, T]$ for some (uniform)
$\delta>0$. Iterating the argument one can then obtain the well-posedness on $[0,T]$. We leave the details to the interested reader.
\hfill \vrule width.25cm height.25cm depth0cm\smallskip
We are now ready to prove the main result of this paper. Let us define the {\it Hamiltonian}:
for $(\varphi, \mu)\in \mathbb{C}_T\times \mathscr{P}(\mathbb{C}_T)$, and $k: [0, T]\times \Omega\rightarrow \mathbb{R}$ adapted process, $(t, \omega, z)\in[0,T]\times \Omega\times \mathbb{R}$,
\begin{eqnarray}
\label{Hamilton}
\mathscr{H}(t, \omega, \varphi_{\cdot\wedge t}, \mu, z; k)\stackrel{\triangle}{=} k_t(\omega)\cdot \sigma(t, \varphi_{\cdot\wedge t}, \mu,z)+f(t, \varphi_{\cdot\wedge t}, \mu, z).
\end{eqnarray}
We have the following theorem.
\begin{thm}[Stochastic Maximum Principle]
\label{theorem7.5}
Assume that the Assumptions \ref{Assum1} and \ref{Assum2} hold. Assume further that the mapping $z\mapsto \mathscr{H}(t, \varphi_{\cdot\wedge t}, \mu, z)$ is
convex. Let $u=u^*\in\mathscr{U}_{ad}$ be an optimal control and $X^u$ the corresponding trajectory. Then,
for $dt\times d\mathbb{Q}^0$-a.e. $(t,\omega)\in[0,T]\times\Omega$ it holds that
\begin{eqnarray}
\label{7.57}
\mathscr{H}(t, \omega, X_{\cdot\wedge t}^u, \mu^u_t, u_t; q_t)=\inf_{v\in {U}}\mathscr{H}(t, \omega, X_{\cdot\wedge t}^u, \mu^u_t, v; q_t),
\end{eqnarray}
where $(p,q, \tilde q)$ and $(Q, M, \tilde M)$ constitute the unique solution of the BSDE (\ref{BSDE1}).
\end{thm}
{\it Proof.} We first recall from (\ref{DBC}) that
\begin{eqnarray*}
C^{u,v}_f(t)&=& \mathbb{E}^0[L^u_t \partial_z f(t,\varphi_{\cdot\wedge t}, U^u_t,z)]|_{\varphi=X^{u}, z=u_t}=\partial_zf(t,X^u_{\cdot\wedge t}, \mu^u_t,u_t);\\
C^{u,v}_\sigma(t) &=&\mathbb{E}^0\Big\{L^u_t \partial_z\sigma(t, \varphi_{\cdot\wedge t}, U^u_t, z) \Big]\Big\}\Big|_{\varphi=X^u; z=u_t}=\partial_z\sigma(t,X^u_{\cdot\wedge t}, \mu^u_t,u_t).
\end{eqnarray*}
Then (\ref{SMP0}) implies that
\begin{eqnarray}
\label{7.55}
0&\le& \mathbb{E}^0\Big[\int_0^T [q_tC^{u,v}_\sigma(t)+C^{u,v}_f(t)](v_t-u_t) dt\Big]\\
&=&\mathbb{E}^0\Big[\int_0^T\partial_z\mathscr{H}(t, \omega, X_{\cdot\wedge t}^u, \mu^u_t, u_t; q_t)(v_t-u_t) dt\Big]. \nonumber
\end{eqnarray}
Therefore for $dt\times d\mathbb{Q}^0$-a.e. $(t,\omega)\in[0,T]\times\Omega$, and any $v\in U$, it holds that
\begin{eqnarray}
\label{7.56}
\partial_z\mathscr{H}(t, \omega, X_{\cdot\wedge t}^u, \mu^u_t, u_t; q_t)(v-u_t)\ge 0.
\end{eqnarray}
Now, for any $v\in {U}$, one has, $dt\times d\mathbb{Q}^0$-a.e. on $ [0,T]\times\Omega$,
\begin{eqnarray*}
&&\mathscr{H}(t, \omega, X_{\cdot\wedge t}^u, \mu^u_t, v; q_t)-\mathscr{H}(t, \omega, X_{\cdot\wedge t}^u, \mu^u_t, u_t; q_t)\\
&=&\int_0^1 \partial_z\mathscr{H}(t, \omega, X_{\cdot\wedge t}^u, \mu^u_t, u_t+\lambda(v-u_t); q_t)(v-u_t)d\lambda\\
&=&\int_0^1\Big[\partial_z\mathscr{H}(t, \omega, X_{\cdot\wedge t}^u, \mu^u_t, u_t+\lambda(v-u_t); q_t)-\partial_z\mathscr{H}(t, \omega, X_{\cdot\wedge t}^u, \mu^u_t, u_t; q_t)\Big](v-u_t)
d\lambda\\
&& +\partial_z\mathscr{H}(t, \omega, X_{\cdot\wedge t}^u, \mu^u_t, u_t; q_t)(v-u_t)\geq 0,.
\end{eqnarray*}
Here the first integral on the right hand side above is nonnegative due to the convexity of $\mathscr{H}$ in variable $z$, and the last term is non-negative because of (\ref{7.56}). The identity (\ref{7.57}) now follows immediately.
\hfill \vrule width.25cm height.25cm depth0cm\smallskip
\begin{rem}
{\rm In stochastic control literature the inequality (\ref{7.55}) is sometimes referred to as {\it Stochastic Maximum Principle in integral form}, which
in many applications is useful, as it does not require the convexity assumption on the Hamiltonian $\mathscr{H}$.
\hfill \vrule width.25cm height.25cm depth0cm\smallskip}
\end{rem}
\bigskip
\noindent{\bf Acknowledgment.} We would like to thank the anonymous referee for his/her very careful reading of the manuscript
and many incisive and constructive questions and suggestions, which helped us to make the paper a much better product.
|
1,314,259,993,608 | arxiv | \section{Introduction}
Given a domain $\Omega \subset \Cb^d$ let $\Aut(\Omega)$ denote the biholomorphism group of $\Omega$. When $\Omega$ is bounded, H. Cartan proved that $\Aut(\Omega)$ is a Lie group (with possibly infinitely many connected components) and acts properly on $\Omega$.
An old theorem of Wong-Rosay~\cite{W1977, R1979} states that if $\Omega \subset \Cb^d$ is a bounded domain with $C^2$ boundary and $\Aut(\Omega)$ acts co-compactly on $\Omega$, then $\Omega$ is biholomorphic to the unit ball. According to Wong~\cite[p. 257]{W1977}, Yau suggested that the co-compactness condition could be replaced by the assumption that $\Omega$ covers a finite volume manifold. More precisely:
\begin{conjecture}[Yau]
Let $\Omega \subset \Cb^d$ ($d \geq 2$) be a bounded pseudoconvex domain whose boundary is $C^2$. Assume that $\Omega$ has a (open) quotient of finite-volume (in the sense of K{\"a}hler-Einstein volume). Then $\Omega$ is biholomorphic to the unit ball in $\Cb^d$.
\end{conjecture}
Considering bounded domains that cover finite volume open manifolds seems more natural than studying those that cover compact manifolds. For instance, it is well known that $\Tc_{g}$, the Teichm{\"u}ller space of hyperbolic surfaces with genus $g$, is biholomorphic to a bounded domain and has a finite volume quotient. Further, Griffiths constructed the following examples.
\begin{theorem}\cite[Theorem I, Proposition 8.12]{G1971} Suppose $V$ is an irreducible, smooth, quasi-projective algebraic variety over the complex numbers. For any $x \in V$ there exists a Zariski neighborhood $U$ of $x$ such that $\wt{U}$, the universal cover of $U$, is biholomorphic to a bounded pseudoconvex domain in $\Cb^d$. Moreover, the Kobayashi-Eisenman volume of $U$ is finite.
\end{theorem}
In this paper we answer Yau's question:
\begin{theorem}\label{thm:main} Suppose $\Omega \subset \Cb^d$ is a bounded pseudoconvex domain with $C^2$ boundary and $\Gamma \leq \Aut(\Omega)$ is a discrete group acting freely on $\Omega$. If $\Gamma \backslash \Omega$ has finite volume with respect to either the Bergman volume, the K{\"a}hler-Einstein volume, or the Kobayashi-Eisenman volume, then $\Omega$ is biholomorphic to the unit ball.
\end{theorem}
\begin{remark} Recently, Liu and Wu~\cite{LW2018}, see Theorem~\ref{thm:LW} below, established the above theorem with the additional assumptions that \begin{enumerate}
\item $d=2$ and $\Omega$ is convex, or
\item $d>2$, $\Omega$ is convex, and $\Gamma$ is irreducible.
\end{enumerate}
\end{remark}
It is well known that Teichm{\"u}ller spaces admit a finite volume quotient and so Theorem~\ref{thm:main} provides a new proof of the following result.
\begin{corollary}[{Yau~\cite[p. 328]{Y2011}}]\label{cor:teich}
Let $\Tc_{g}$ denote the Teichm{\"u}ller space of hyperbolic surfaces with genus $g$. If $g \geq 2$, then $\Tc_{g}$ is not biholomorphic to a bounded domain with $C^2$ boundary.
\end{corollary}
\begin{remark} \ \begin{enumerate}
\item A theorem of Bers~\cite{B1960} says that $\Tc_{g}$ is biholomorphic to a bounded domain.
\item Recently, Gupta and Seshadri~\cite[Theorem 1.2]{GS2017} provided a proof of Corollary~\ref{cor:teich} which relies on the ergodicity of the Teichm{\"u}ller geodesic flow.
\end{enumerate}
\end{remark}
If $\Omega \subset \Cb^d$ is a bounded domain, $\Gamma \leq \Aut(\Omega)$ is a discrete group acting freely on $\Omega$, and $\Gamma \backslash \Omega$ is a quasi-projective variety, then a result of Griffiths implies that $\Gamma \backslash \Omega$ has finite volume with respect to the Kobayashi-Eisenman volume (see Proposition 8.12 and the discussion following Question 8.13 in~\cite{G1971}). So we have the following corollary of Theorem~\ref{thm:main}.
\begin{corollary}\label{cor:quasi_proj} Suppose $\Omega \subset \Cb^d$ is a bounded pseudoconvex domain with $C^2$ boundary and $\Gamma \leq \Aut(\Omega)$ is a discrete group acting freely on $\Omega$. If $\Gamma \backslash \Omega$ is a quasi-projective variety, then $\Omega$ is biholomorphic to the unit ball.
\end{corollary}
The proof of Theorem~\ref{thm:main} uses the Levi form of the boundary and hence does not easily generalize to domains whose boundaries have less than $C^2$ regularity. However, by assuming our domain is convex we can lower the required regularity to $C^{1,\epsilon}$ for any $\epsilon > 0$.
\begin{theorem}\label{thm:main_convex} Suppose $\Omega \subset \Cb^d$ is a bounded convex domain with $C^{1,\epsilon}$ boundary and $\Gamma \leq \Aut(\Omega)$ is a discrete group acting freely on $\Omega$. If $\Gamma \backslash \Omega$ has finite volume with respect to either the Bergman volume, the K{\"a}hler-Einstein volume, or the Kobayashi-Eisenman volume, then $\Omega$ is biholomorphic to the unit ball.
\end{theorem}
\begin{remark} \ \begin{enumerate}
\item The proof will use a recent result of Liu and Wu~\cite{LW2018}, see Theorem~\ref{thm:LW} below, and a recent result in~\cite{Z2017convex}, see Theorem~\ref{thm:Z} below.
\item It is conjectured that a bounded convex domain with a finite volume quotient (with no assumptions on the regularity of $\partial\Omega$) must be a bounded symmetric domain, see for instance~\cite[Conjecture 1.12]{LW2018}.
\item Using Theorem~\ref{thm:main_convex}, the hypothesis of Corollary~\ref{cor:quasi_proj} can be modified to assume that $\Omega$ is a bounded convex domain with $C^{1,\epsilon}$ boundary. Theorem~\ref{thm:main_convex} also can be used to show that $\Tc_g$ $(g \geq 2)$ is not biholomorphic to a convex domain with $C^{1,\epsilon}$ boundary, however a recent result of Markovic~\cite{M2017} implies that $\Tc_{g}$ is not biholomorphic any convex domain when $g \geq 2$ (with no regularity assumptions on the boundary of the convex domain).
\end{enumerate}
\end{remark}
\subsection{Outline of the proofs} We will use a theorem of Wong and Rosay to prove Theorem~\ref{thm:main}.
\begin{theorem}[Wong-Rosay Ball Theorem~\cite{W1977, R1979}]\label{thm:WR} Suppose $\Omega \subset \Cb^d$ is a bounded domain. Assume that $\partial \Omega$ is $C^2$ and strongly pseudoconvex in a neighborhood of $\xi \in \partial \Omega$. If there exists some $z_0 \in \Omega$ and a sequence $\varphi_n \in \Aut(\Omega)$ such that $\varphi_n(z_0) \rightarrow \xi$, then $\Omega$ is biholomorphic to the unit ball.
\end{theorem}
When $\Omega$ is a bounded domain with $C^2$ boundary, then there exists some $\xi \in \partial \Omega$ which is strongly pseudoconvex (see Observation~\ref{obs:strong_pc_exist} below). If $\Aut(\Omega)$ acts co-compactly on $\Omega$ then it is easy to show that there exists some $z_0 \in \Omega$ and a sequence $\varphi_n \in \Aut(\Omega)$ such that $\varphi_n(z_0) \rightarrow \xi$. So one has the following Corollary to Theorem~\ref{thm:WR}:
\begin{corollary} Suppose $\Omega \subset \Cb^d$ is a bounded domain with $C^2$ boundary. If $\Aut(\Omega)$ acts co-compactly on $\Omega$, then $\Omega$ is biholomorphic to the unit ball.
\end{corollary}
In the case when $\Omega$ only admits a finite volume quotient, finding $z_0 \in \Omega$ and a sequence $\varphi_n \in \Aut(\Omega)$ such that $\varphi_n(z_0)$ converges to a certain boundary point $\xi \in \partial \Omega$ is much harder. We accomplish this task by considering the behavior of the Bergman distance and in particular the shape of horospheres near a strongly pseudoconvex point. The squeezing function also plays an important role in understanding the complex geometry of $\Omega$.
For convex domains, there are precise estimates for the Kobayashi distance and so in the proof of Theorem~\ref{thm:main_convex} we consider horospheres with respect to the Kobayashi distance (instead of the Bergman distance). Since the hypothesis of Theorem~\ref{thm:main_convex} only assumes $\partial \Omega$ has $C^{1,\epsilon}$ boundary there is no hope of using the Wong-Rosay Ball Theorem. Instead we reduce to two recent results about the automorphism group of convex domains. Before stating these results we need a few definitions:
\begin{enumerate}
\item Given a bounded domain $\Omega \subset \Cb^d$ let $\Aut_0(\Omega)$ denote the connected component of the identity in $\Aut(\Omega)$.
\item When $\Omega \subset \Cb^d$ is a bounded domain, the \emph{limit set of $\Omega$}, denoted $\Lc(\Omega)$ is the set of points $x \in \partial \Omega$ where there exists some $z \in \Omega$ and a sequence $\varphi_n \in \Aut(\Omega)$ such that $\varphi_n(z) \rightarrow x$.
\item Given a convex domain $\Omega \subset \Cb^d$ with $C^1$ boundary and $x \in \partial \Omega$, let $T_{x}^{\Cb} \partial \Omega \subset \Cb^d$ be the complex affine hyperplane tangent to $\partial\Omega$ at $x$. Then the \emph{closed complex face of $x$ in $\partial \Omega$} is the set $T_{x}^{\Cb} \partial \Omega \cap \partial \Omega$.
\end{enumerate}
Liu and Wu recently proved the following rigidity result.
\begin{theorem}[{Liu-Wu~\cite{LW2018}\label{thm:LW}}] Suppose $\Omega \subset \Cb^d$ is a bounded convex domain, $\Gamma \leq \Aut(\Omega)$ is a discrete group acting freely on $\Omega$, and $\Gamma \backslash \Omega$ has finite volume with respect to either the Bergman volume, the K{\"a}hler-Einstein volume, or the Kobayashi-Eisenman volume. If either:
\begin{enumerate}
\item $\Gamma \leq \Aut_0(\Omega)$,
\item $\Aut_0(\Omega) \neq 1$ and $\Gamma$ is irreducible,
\item $\Omega$ has $C^1$ boundary and $\Gamma$ is irreducible,
\item $d=2$ and $\Aut_0(\Omega) \neq 1$, or
\item $d=2$ and $\Omega$ has $C^1$ boundary,
\end{enumerate}
then $\Omega$ is biholomorphic to a bounded symmetric domain.
\end{theorem}
\begin{remark} By the so-called rescaling method, part (3) (respectively part (5)) is a consequence of part (2) (respectively part (4)). Also, by a result of Mok and Tsai~\cite{MT1992}: if $\Omega$ is a bounded symmetric domain which is convex and has $C^1$ boundary, then $\Omega$ is biholomorphic to the unit ball. \end{remark}
We recently proved the following result.
\begin{theorem}\cite{Z2017convex}\label{thm:Z} Suppose $\Omega \subset \Cb^d$ is a bounded convex domain with $C^{1,\epsilon}$ boundary. If $\Lc(\Omega)$ intersects at least two different closed complex faces of $\partial \Omega$, then \begin{enumerate}
\item $\Aut(\Omega)$ has finitely many components,
\item there exists a compact normal subgroup $N \leq \Aut_0(\Omega)$ such that $\Aut_0(\Omega)/N$ is a non-compact simple Lie group with real rank one.
\end{enumerate}
\end{theorem}
Hence to prove Theorem~\ref{thm:main_convex} it is enough to show that $\Lc(\Omega)$ intersects at least two different closed complex faces of $\partial \Omega$.
\subsection*{Acknowledgements} This material is based upon work supported by the National Science Foundation under grant DMS-1760233.
\section{Preliminaries}
\subsection{Notations:}
Suppose $\Omega \subset \Cb^d$ is a bounded pseudoconvex domain, then
\begin{enumerate}
\item Let $k_\Omega: \Omega \times \Cb^d \rightarrow \Rb_{ \geq 0}$ denote the infinitesimal Kobayashi metric, $K_\Omega: \Omega \times \Omega \rightarrow \Rb_{\geq 0}$ denote the Kobayashi distance on $\Omega$, and $\Vol_{K}$ denote the Kobayashi-Eisenman volume form,
\item Let $g_B$ denote the Bergman metric on $\Omega$, $B_\Omega$ denote the Bergman distance on $\Omega$, and $\Vol_{B}$ denote the Riemannian volume form associated to $g_B$. We will also let $b_\Omega:\Omega \times \Cb^d \rightarrow \Rb$ denote the norm associated to $g_B$, that is
\begin{align*}
b_\Omega(x;v) = \sqrt{g_B(v,v)}
\end{align*}
when $v \in T_x \Omega$.
\item Let $g_{KE}$ denote the K{\"a}hler-Einstein metric on $\Omega$ with Ricci curvature $-1$ constructed by Cheng-Yau~\cite{CY1980} when $\Omega$ has $C^2$ boundary and Mok-Yau~\cite{MY1983} in general. And let $\Vol_{KE}$ denote the Riemannian volume form associated to $g_{KE}$.
\end{enumerate}
Throughout the paper $\norm{ \cdot}$ will denote the standard Euclidean norm on $\Cb^d$. Given $z_0 \in \Cb^d$ and $r >0$ define
\begin{align*}
\Bb_d(z_0;r) = \{ z\in \Cb^d : \norm{z-z_0} < r\}.
\end{align*}
Finally, given a domain $\Omega \subset \Cb^d$ and $z \in \Omega$ define
\begin{align*}
\delta_\Omega(z) = \inf\{ \norm{w -z} : w \in \partial \Omega\}.
\end{align*}
\subsection{The squeezing function}\label{sec:squeeze}
Given a domain $\Omega \subset \Cb^d$ let $s_\Omega : \Omega \rightarrow (0,1]$ be the \emph{squeezing function on $\Omega$}, that is
\begin{align*}
s_\Omega(z) = \sup\{ r : & \text{ there exists an one-to-one holomorphic map } \\
& f: \Omega \rightarrow \Bb_d(0;1) \text{ with } f(z)=0 \text{ and } \Bb_d(0;r) \subset f(\Omega) \}.
\end{align*}
In this section we recall a result of Sai-Kee Yeung.
\begin{theorem}\label{thm:squeeze}\cite[Theorem 2]{Y2009} Suppose $s > 0$ and $d > 0$. Then there exists $C, \delta, \epsilon, \kappa >0$ such that: if $\Omega \subset \Cb^d$, $z_0 \in \Omega$, $s_\Omega(z_0) > s$, and
\begin{align*}
B_\epsilon = \{ z \in \Omega : B_\Omega(z_0,z) \leq \epsilon\},
\end{align*}
then \begin{enumerate}
\item $B_\epsilon \Subset \Omega$,
\item $g_B$, $g_{KE}$, and $k_\Omega$ are all $C$-bi-Lipschitz on $B_\epsilon$,
\item the sectional curvature of $g_B$ is bounded in absolute value by $\kappa$ on $B_\epsilon$,
\item the injectivity radius of $g_B$ is bounded below by $\delta$ on $B_\epsilon$, and
\item if $\Vol$ denotes either the Bergman volume, the K{\"a}hler-Einstein volume, or the Kobayashi-Eisenman volume, then
\begin{align*}
\Vol\Big( \{ z \in \Omega : B_\Omega(z, z_0) \leq r\}\Big) \geq r^{2d}/C
\end{align*}
for all $r \in [0,\epsilon]$.
\end{enumerate}
\end{theorem}
Parts (1)-(4) follow from~\cite[Theorem 2]{Y2009}. In~\cite[Theorem 2]{Y2009} it is assumed that $s_\Omega(z) > s$ for all $z \in \Omega$, however all the arguments are local in nature and can be easily modified to prove parts (1)-(4) in the above Theorem. Part (2) also follows from the proof of~\cite[Theorem 7.2]{LSY2004}.
Part (5) is a consequence of the definition and part (2): since $s_\Omega(z_0) > s$ we can assume that $z_0=0$ and
\begin{align*}
\Bb_d(0;s) \subset \Omega \subset \Bb_d(0;1).
\end{align*}
Then
\begin{align*}
k_{\Bb_d(0;1)} \leq k_\Omega \leq k_{\Bb_d(0;s)}
\end{align*}
on $\Bb_d(0;s)$. Then, from the well known explicit description of the Kobayashi metric on the ball and part (2), we see that there exists $C_1 > 0$ such that $g_B$, $g_{KE}$, and $k_\Omega$ are all $C_1$-bi-Lipschitz to the Euclidean metric on $\Bb_d(0;s/2)$. So we can find $C, \epsilon > 0$ such that: if $\Vol$ denotes either the Bergman volume or the K{\"a}hler-Einstein volume, then
\begin{align*}
\Vol\Big( \{ z \in \Omega : B_\Omega(z, z_0) \leq r\}\Big) \geq r^{2d}/C
\end{align*}
for all $r \in [0,\epsilon]$. Next let $\Vol_{K}$ denote the Kobayashi-Eisenman volume on $\Omega$ and $\Vol_{\Bb_d(0;1)}$ denote the Kobayashi-Eisenman volume on $\Bb_d(0;1)$. Then by definition
\begin{align*}
\Vol_{K}(A) \geq \Vol_{\Bb_d(0;1)}(A)
\end{align*}
for all subsets $A\subset \Omega$. So from the well known explicit description of the Kobayashi-Eisenman volume for the ball, part (2), and by possibly modifying $C,\epsilon$ we can also assume that
\begin{align*}
\Vol_K\Big( \{ z \in \Omega : B_\Omega(z, z_0) \leq r\}\Big) \geq r^{2d}/C
\end{align*}
for all $r \in [0,\epsilon]$.
\subsection{Invariant metrics near a strongly pseudoconvex point}
We will use the following well known facts about invariant metrics near a strongly pseudoconvex point.
\begin{theorem}\label{thm:local} Suppose $\Omega \subset \Cb^d$ is a bounded domain. Assume that $\partial \Omega$ is $C^2$ and strongly pseudoconvex in a neighborhood of $\xi \in \partial \Omega$. Then there exists an neighborhood $U$ of $\xi$ in $\overline{\Omega}$ and $C > 0$ such that:
\begin{enumerate}
\item $k_\Omega$ and $g_B$ are $C$-bi-Lipshitz to each other on $U \cap \Omega$,
\item $k_\Omega(x;v) \geq C^{-1}\norm{v}\delta_\Omega(x)^{-1/2}$ for all $x \in U \cap \Omega$ and $v \in \Cb^d$, and
\item $g_B$ has negative sectional curvature on $U \cap \Omega$.
\end{enumerate}
\end{theorem}
\begin{proof} Fix open neighborhoods $V_2 \Subset V_1$ of $\xi$ such that there exist a holomorphic embedding $\varphi: V_1 \rightarrow \Cb^d$ with $\varphi(V_2 \cap \Omega)$ a convex domain which is strongly convex near $\varphi(\xi)$.
By~\cite[Theorem 2.1]{FR1987} there exists a neighborhood $V_3$ of $\xi$ such that $V_3 \Subset V_2$ and
\begin{align*}
k_\Omega(x;v) \leq k_{V_2 \cap \Omega}(x;v) \leq 2 k_\Omega(x;v)
\end{align*}
for all $x \in V_3$ and $v \in \Cb^d$ (notice that the first inequality is by definition). Further, by~\cite[Theorem 1]{DFH1984} there exists $C_0 > 1$ such that
\begin{align*}
\frac{1}{C_0} b_{\Omega \cap V_2}(x;v) \leq b_\Omega(x;v) \leq C_0b_{\Omega \cap V_2}(x;v)
\end{align*}
for all $x \in V_3$ and $v \in \Cb^d$.
Now since $V_2 \cap \Omega$ is biholomorphic to a convex domain, a result of Frankel~\cite{F1991} implies that $b_{\Omega \cap V_2}$ and $k_{\Omega \cap V_2}$ are $C_1$-bi-Lipschitz to each other for some $C_1 >1$. So we see that $k_\Omega$ and $g_B$ are $C$-bi-Lipshitz to each other on $V_3 \cap \Omega$ for some $C>1$.
Given a domain $\Oc \subset \Cb^d$, $x \in \Oc$, and nonzero $v \in \Cb^d$ define
\begin{align*}
\delta_{\Oc}(x;v) = \inf\{ \norm{y-x} : y \in \partial \Omega \cap (x+\Cb v)\}.
\end{align*}
Since $\Cc=\varphi(V_2 \cap \Omega)$ is convex, a result of Graham~\cite{G1990,G1991} says that
\begin{align*}
\frac{\norm{v}}{2\delta_{\Cc}(x;v)} \leq k_{\Cc}(x;v) \leq \frac{\norm{v}}{\delta_{\Cc}(x;v)}
\end{align*}
for all $x \in \Cc$ and $v \in \Cb^d$. Then, since $\Cc$ is strongly convex at $\varphi(\xi)$, there exists a neighborhood $W$ of $\varphi(\xi)$ and some $C_2 > 0$ such that
\begin{align*}
C_2 \frac{\norm{v}}{\delta_{\Cc}(x)^{1/2}} \leq k_{\Cc}(x;v)
\end{align*}
for all $x \in W$ and $v \in \Cb^d$. Since $V_2 \Subset V_1$, the map $\varphi:V_1 \rightarrow \Cb^d$ is bi-Lipschitz on $V_2$, so by possibly shrinking $V_3$ and increasing $C$ we can assume that
\begin{align*}
\frac{1}{C} \frac{\norm{v}}{\delta_{\Omega}(x)^{1/2}} \leq k_{\Omega}(x;v)
\end{align*}
for all $x \in V_3$ and $v \in \Cb^d$.
Finally, part (3) follows from~\cite[Theorem 1]{KY1996}.
\end{proof}
\subsection{Completeness of the Bergman metric}
We will use the following fact about the Bergman metric:
\begin{theorem}[Ohsawa~\cite{O1981}] If $\Omega \subset \Cb^d$ is a bounded pseudoconvex domain with $C^1$ boundary, then the Bergman metric is a complete Riemannian metric on $\Omega$.
\end{theorem}
\begin{remark} It is also known that the Bergman metric is complete on the more general class of hyperconvex domains, see~\cite{H1999} and~\cite{BP1998b}.\end{remark}
\subsection{A local version of E. Cartan's fixed point theorem}
E. Cartan showed that a compact group $G$ acting by isometries on $(X,g)$ a complete simply connected Riemannian manifold with non-positive sectional curvature always has a fixed point. One proof, see for instance~\cite[p. 21]{E1996}, uses the following lemma: if $K \subset X$ is compact, then the function
\begin{align*}
f(x) = \sup\{ d(x,k) : k \in K\}
\end{align*}
has a unique minimum in $X$. In this section we observe a local version of this lemma which will allow us to show that a certain compact subgroup has a fixed point in the proof of Theorem~\ref{thm:main}.
Given a complete Riemannian manifold $(X,g)$, $x_0 \in X$, and $R >0$ let $B_{(X,g)}(x_0,R)$ denote the open metric ball of radius $R$ centered at $x_0$.
\begin{proposition}\label{prop:COM} Suppose $(X,g)$ is a complete Riemannian manifold, $x_0 \in X$, $R>0$, the metric $g$ has non-positive sectional curvature on $B_{(X,g)}(x_0,8R)$, and $g$ has injectivity radius at least $16R$ at each point in $B_{(X,g)}(x_0,8R)$. If $K \subset B_{(X,g)}(x_0,R)$ is compact, then the function
\begin{align*}
f(x) = \sup\{ d(x,k) : k \in K\}
\end{align*}
has a unique minimum in $X$.
\end{proposition}
The following proof is nearly identical to the proof of the Lemma on p. 21 in~\cite{E1996}, but we provide the details for the reader's convenience.
\begin{proof}
When $\alpha \in [0,4]$, every two points in $B_{(X,g)}(x_0,\alpha R)$ are joined by a unique geodesic and this geodesic is contained in $B_{(X,g)}(x_0,2 \alpha R)$.
Since $g$ is non-positively curved on $B_{(X,g)}(x_0, 8R)$ and has injectivity radius at least $16R$ at each point in $B_{(X,g)}(x_0,8R)$ the Rauch comparison theorem implies (see \cite[p. 73]{H2001}): if $\Tc$ is a geodesic triangle contained in $B_{(X,g)}(x_0,4R)$ with side lengths $a,b,c$ then
\begin{align}
\label{eq:law_of_cosines}
a^2 + b^2 -2ab \cos \theta \leq c^2
\end{align}
where $\theta$ is the angle at the vertex opposite to the side of length $c$.
Since $f$ is a proper continuous function there exists at least one minimum. Since $f(x) > R$ when $x \in X \setminus B_{(X,g)}(x_0,2R)$ and $f(x_0) \leq R$ any minimum of $f$ is in $B_{(X,g)}(x_0,2R)$.
Suppose for a contradiction that there exists two distinct minimum points $x,y$ of $f$. Let $\sigma:[0,T] \rightarrow X$ denote the unique geodesic with $\sigma(0)=x$ and $\sigma(T)=y$. Let $m = \sigma(T/2)$. Then consider some $k \in K$ and let $\gamma: [0,S] \rightarrow X$ denote the unique geodesic in $X$ with $\gamma(0)=m$ and $\gamma(S) = k$. Since
\begin{align*}
\angle_m(-\sigma^\prime(T/2), \gamma^\prime(0)) + \angle_m(\sigma^\prime(T/2), \gamma^\prime(0))=\pi,
\end{align*}
by relabelling $x,y$ we can assume that $\theta:=\angle_m(\sigma^\prime(T/2), \gamma^\prime(0)) \geq \pi/2$. Then Equation~\eqref{eq:law_of_cosines} implies that
\begin{align*}
d(y,k)^2 \geq d(y,m)^2 + d(m,k)^2 - 2d(y,m)d(m,k)\cos \theta > d(m,k)^2.
\end{align*}
So $d(m,k) < d(y,k) \leq f(y)$. Since $k \in K$ was arbitrary and $K$ is compact, we then have $f(m) < f(y)$ which is a contradiction.
\end{proof}
\section{An estimate for the Bergman distance}
\begin{theorem}\label{thm:GP} Suppose $\Omega \subset \Cb^d$ is a bounded pseudoconvex domain. Assume that $\partial \Omega$ is $C^2$ and strongly pseudoconvex in a neighborhood of $\xi \in \partial \Omega$. If $z_0 \in \Omega$ and $\epsilon_0 > 0$, then there exists $\epsilon \in (0,\epsilon_0)$ and $R >0$ such that
\begin{align*}
B_\Omega(z,w) \geq B_\Omega(z,z_0)+B_\Omega(z_0,w) -R
\end{align*}
for all $z,w \in \Omega$ with $\norm{z-\xi} < \epsilon$ and $\norm{w-\xi} > \epsilon$.
\end{theorem}
\begin{remark} This says that a point $z$ near $\xi$ and point $w$ far away from $\xi$ can be joined by a path that passes through $z_0$ and is length minimizing up to an error of $R$. \end{remark}
The following argument is based on the proof of ~\cite[Lemma 36]{K2005b} which establishes a similar estimate for the Kobayashi distance.
\begin{proof} By Theorem~\ref{thm:local} there exists an neighborhood $U$ of $\xi$ and some $C > 1$ such that
\begin{align*}
\frac{1}{C} k_\Omega(x;v) \leq b_\Omega(x;v) \leq C k_\Omega(x;v)
\end{align*}
and
\begin{align*}
\frac{1}{C} \frac{\norm{v}}{\delta_\Omega(x)^{1/2}} \leq b_\Omega(x;v)
\end{align*}
for all $x \in \Omega \cap U$ and $v \in \Cb^d$.
By definition
\begin{align*}
k_\Omega(x;v) \leq \frac{\norm{v}}{\delta_\Omega(x)}
\end{align*}
and so
\begin{align*}
b_\Omega(x;v) \leq C\frac{\norm{v}}{\delta_\Omega(x)}
\end{align*}
for $x \in U \cap \Omega$ and $v \in \Cb^d$. Then since $\partial \Omega$ is $C^2$ near $\xi$ one can consider parametrizations of inward pointing normal lines to show that there exists $\alpha, \beta > 0$ and a neighborhood $V \subset U$ of $\xi$ such that
\begin{align*}
B_\Omega(z_0, z) \leq \alpha + \beta \log \frac{1}{\delta_\Omega(z)}
\end{align*}
for all $z \in V \cap \Omega$.
Now fix $\epsilon \in (0,\epsilon_0)$ such that
\begin{align*}
\{ z \in \Cb^d : \norm{z-\xi} < 2\epsilon \} \subset V.
\end{align*}
Consider points $z,w \in \Omega$ with $\norm{z-\xi} < \epsilon$ and $\norm{w-\xi} > \epsilon$. Let $\sigma: [0,T] \rightarrow \Omega$ be a geodesic (with respect to the Bergman distance) joining $z$ and $w$. Define
\begin{align*}
T_0 = \max\left\{ t \in [0,T] : \sigma([0,t]) \subset \overline{\Bb_d(z; \epsilon)} \right\}.
\end{align*}
Then let $\tau \in [0,T_0]$ be such that
\begin{align*}
\delta_\Omega(\sigma(\tau)) = \max \{ \delta_\Omega(\sigma(t)) : t \in [0,T_0]\}.
\end{align*}
Now for $t \in [0,T_0]$ we have
\begin{align*}
\abs{t-\tau} &= B_\Omega(\sigma(t), \sigma(\tau)) \leq B_\Omega(\sigma(t), z_0)+B_\Omega(z_0, \sigma(\tau))\\
& \leq 2\alpha + \beta \log \frac{1}{\delta_\Omega(\sigma(t))\delta_\Omega(\sigma(\tau))}
\end{align*}
So
\begin{align*}
\delta_\Omega(\sigma(t)) \leq \sqrt{\delta_\Omega(\sigma(t))\delta_\Omega(\sigma(\tau))} \leq
\exp \left( \frac{ -\abs{t-\tau}+2\alpha}{2\beta} \right).
\end{align*}
Now fix $M > 0$ such that
\begin{align*}
\int_{M}^{\infty} \exp \left( \frac{ -r+2\alpha}{4\beta} \right) dr < \epsilon/(4C).
\end{align*}
Then
\begin{align*}
\epsilon = & \norm{\sigma(0)-\sigma(T_0)} \leq \int_0^{T_0} \norm{\sigma^\prime(t)} dt \leq C \int_0^{T_0} \delta_\Omega(\sigma(t))^{1/2} dt
\end{align*}
since $b_\Omega(\sigma(t); \sigma^\prime(t))=1$. Then
\begin{align*}
\epsilon & \leq C \int_{[0,T_0] \cap (\tau-M, \tau +M)} \delta_\Omega(\sigma(t))^{1/2} dt + C \int_{[0,T_0] \cap (\tau-M, \tau +M)^c} \delta_\Omega(\sigma(t))^{1/2} dt \\
& \leq 2CM \delta_\Omega(\sigma(\tau))^{1/2} + 2C\int_{M}^\infty \exp \left( \frac{ -r+2\alpha}{4\beta} \right) dr \\
& \leq 2CM \delta_\Omega(\sigma(\tau))^{1/2} + \epsilon/2.
\end{align*}
So
\begin{align*}
\delta_\Omega(\sigma(\tau))^{1/2} \geq \epsilon/(4CM).
\end{align*}
Then
\begin{align*}
B_\Omega(z,w) &=B_\Omega(z,\sigma(\tau))+B_\Omega(\sigma(\tau),w) \geq B_\Omega(z,z_0)+B_\Omega(z_0,w) - 2B_\Omega(z_0,\sigma(\tau)) \\
& \geq B_\Omega(z,z_0)+B_\Omega(z_0,w) - R
\end{align*}
where
\begin{align*}
R = 2\alpha + 4\beta \log \frac{4CM}{\epsilon}.
\end{align*}
Notice that $R$ does not depend on $z$ or $w$, so the proof is complete.
\end{proof}
\section{Proof of Theorem~\ref{thm:main}}
For the rest of the section suppose that $\Omega \subset \Cb^d$ is a bounded pseudoconvex domain with $C^{2}$ boundary and $\Gamma \leq \Aut(\Omega)$ is a discrete group acting freely on $\Omega$. Further assume that $\Vol(\Gamma \backslash \Omega) < +\infty$ where $\Vol$ is either the Bergman volume, the K{\"a}hler-Einstein volume, or the Kobayashi-Eisenman volume.
By replacing $\Omega$ with an affine translate we can assume that $0 \in \Omega$, $\Omega \subset \Bb_d(0;1)$, and $(1,0,\dots,0) \in \partial \Omega$. Then, since $\partial \Omega$ is $C^2$, there exists some $r\in (0,1)$ such that
\begin{align*}
\Bb_d( (r,0,\dots,0); 1-r) \subset \Omega \subset \Bb_d(0;1).
\end{align*}
\begin{observation}\label{obs:strong_pc_exist} $\xi=(1,0,\dots,0)$ is a strongly pseudoconvex point of $\partial \Omega$.
\end{observation}
\begin{proof}
This is a simple consequence of the fact that $(1,0,\dots, 0) \in \partial \Omega$ and $\Omega \subset \Bb_d(0;1)$, see for instance~\cite[Lemma 4.1]{GS2017}.
\end{proof}
\begin{observation} Let $w_t = (t,0,\dots,0) \in \Cb^d$. Then there exists some $s_0 > 0$ such that $s_\Omega(w_t) \geq s_0$ for $t \in [r,1)$.
\end{observation}
\begin{proof}
For $t \in [r,1)$ consider the transformation
\begin{align*}
\varphi(z_1,\dots, z_d) = \left( \frac{z_1-t}{tz_1-1}, \frac{(1-t^2)^{1/2}}{tz_1-1}z_2, \dots, \frac{(1-t^2)^{1/2}}{tz_1-1}z_d \right).
\end{align*}
Then $\varphi \in \Aut(\Bb_d(0;1))$ and $\varphi(0)=w_t$. We claim that
\begin{align*}
\varphi( \Bb_d(0;s_0)) \subset \Omega
\end{align*}
where
\begin{align*}
s_0 = \frac{1-r}{12\sqrt{d}}.
\end{align*}
Suppose $z \in \Bb_d(0;s_0)$. Then $\norm{z} \leq 1/2$ and so
\begin{align*}
\abs{tz_1-1} \geq 1/2.
\end{align*}
Then
\begin{align*}
\abs{\frac{z_1-t}{tz_1-1}-r}^2
& = \abs{ (t-r)+\frac{(1-t^2)z_1}{tz_1-1}}^2 \leq (t-r)^2 + \frac{2(t-r)(1-t^2)}{\abs{tz_1-1}} \abs{z_1} + \frac{(1-t^2)^2}{\abs{tz_1-1}^2} \abs{z_1}^2 \\
& \leq (t-r)^2 + 4(1-t)\abs{z_1}+ 16 (1-t)\abs{z_1}^2 \\
& \leq (t-r)^2 + 4 (1-t)\abs{z_1} + 8(1-t)\abs{z_1}\\
& \leq (t-r)^2 + 12(1-t) \abs{z_1}.
\end{align*}
We also have
\begin{align*}
(t-r)^2 -(1-r)^2 = (2r-1-t)(1-t) \leq (r-1)(1-t)
\end{align*}
and
\begin{align*}
\abs{\frac{(1-t^2)^{1/2}}{tz_1-1}z_i}^2 \leq 8(1-t)\abs{z_i}^2 \leq 4(1-t)\abs{z_i}.
\end{align*}
So
\begin{align*}
\norm{\varphi(z)-w_r}^2
& \leq (1-r)^2 +(r-1)(1-t) + 12(1-t)(\abs{z_1}+\dots+\abs{z_d})\\
& \leq (1-r)^2 +(r-1)(1-t) + 12\sqrt{d}(1-t) \norm{z} \\
& < (1-r)^2.
\end{align*}
So $\varphi(z) \in \Bb_d(w_r;1-r) \subset \Omega$. Since $z \in \Bb_d(0;s_0)$ was arbitrary, we then have
\begin{align*}
\varphi(\Bb_d(0;s_0)) \subset \Omega.
\end{align*}
Then $\varphi^{-1}(w_t) = 0$ and
\begin{align*}
\Bb_d(0;s_0) \subset \varphi^{-1}(\Omega) \subset \Bb_d(0;1),
\end{align*}
so $s_\Omega(w_t) \geq s_0$.
\end{proof}
Then fix a sequence $r_n \nearrow 1$ and consider the points $y_n=(r_n,0,\dots,0) \in \Omega$. For each $n \in \Nb$ define
\begin{align*}
\delta_n = \min_{\gamma \in \Gamma \setminus \{1\}} B_\Omega(y_n, \gamma y_n).
\end{align*}
Then the quotient map $\pi : \Omega \rightarrow \Gamma \backslash \Omega$ restricts to an embedding on
\begin{align*}
B_n = \{ z \in \Omega : B_\Omega(z, y_n) < \delta_n/2\}.
\end{align*}
Further, by Theorem~\ref{thm:squeeze} there exists some $C,\epsilon_0 > 0$ such that
\begin{align*}
{ \rm Vol}(\pi(B_n)) \geq C\min\{\epsilon_0^{2d}, \delta_n^{2d}\}.
\end{align*}
After passing to a subsequence we can assume that
\begin{align*}
\lim_{n \rightarrow \infty} \delta_n = \delta \in \Rb_{\geq 0} \cup \{ \infty\}.
\end{align*}
\noindent \textbf{Case 1:} $\delta \neq 0$. Since $\Vol(\Gamma \backslash \Omega) < \infty$, the set $\{ \pi(y_n) : n \in \Nb\}$ must be relatively compact in $\Gamma \backslash \Omega$. So for each $n$, there exist some $\gamma_n \in \Gamma$ such that the set $\{ \gamma_n y_n : n \in \Nb\}$ is relatively compact in $\Omega$. Then we can pass to a subsequence such that $\gamma_n y_n \rightarrow y \in \Omega$. Then $\gamma_n^{-1}y \rightarrow \xi$. So $\Omega$ is biholomorphic to the ball by Theorem~\ref{thm:WR}. \\
\noindent \textbf{Case 2:} $\delta=0$. Pick $\gamma_n \in \Gamma$ such that
\begin{align*}
B_\Omega(\gamma_n y_n, y_n) = \delta_n.
\end{align*}
\noindent \textbf{Case 2(a):} The set $\{ \gamma_1, \gamma_2, \dots\}$ is infinite. Since $\Gamma$ is discrete, by passing to a subsequence we can suppose that $\gamma_n \rightarrow \infty$ in $\Aut(\Omega)$. Fix some $z_0 \in \Omega$. By passing to another subsequence we can assume that $\gamma_n^{-1} z_0 \rightarrow \eta \in \partial \Omega$. Since $(\Omega, B_\Omega)$ is a complete proper metric space we must have
\begin{align*}
B_\Omega(z_0, \gamma_n^{-1} z_0) \rightarrow \infty.
\end{align*}
We claim that $\eta= \xi$. Suppose not, then by Theorem~\ref{thm:GP} there exists $R > 0$ such that
\begin{align*}
B_\Omega(\gamma_n^{-1}z_0, z_0) &+ B_\Omega(z_0, y_n) - B_\Omega(\gamma_n^{-1} z_0, y_n) \leq R.
\end{align*}
However
\begin{align*}
B_\Omega(\gamma_n^{-1}z_0, z_0) &+ B_\Omega(z_0, y_n) - B_\Omega(\gamma_n^{-1} z_0, y_n) \\
& = B_\Omega(\gamma_n^{-1}z_0, z_0) + B_\Omega(z_0, y_n) - B_\Omega(z_0, \gamma_n y_n) \\
& \geq B_\Omega(\gamma_n^{-1}z_0, z_0) - B_\Omega(\gamma_n y_n, y_n) \rightarrow \infty.
\end{align*}
So we have a contradiction and hence $\xi = \eta$. So $\Omega$ is biholomorphic to the unit ball by Theorem~\ref{thm:WR}.\\
\noindent \textbf{Case 2(b):} The set $\{ \gamma_1, \gamma_2, \dots\}$ is finite. By passing to a subsequence we can suppose that $\gamma_n = \gamma$ in for all $n \in \Nb$. Fix some $z_0 \in \Omega$ and consider the functions
\begin{align*}
b_n(z) = B_\Omega(z,y_n)-B_\Omega(y_n,z_0).
\end{align*}
Since $b_n(z_0)=0$ and each $b_n$ is 1-Lipschitz (with respect to the Bergman distance) we can pass to a subsequence such that $b_n \rightarrow b$ locally uniformly. Then
\begin{align*}
b(\gamma^{-1}z) = \lim_{n \rightarrow \infty} B_\Omega(\gamma^{-1} z,y_n)-B_\Omega(y_n,z_0) = \lim_{n \rightarrow \infty} B_\Omega(z,\gamma y_n)-B_\Omega(y_n,z_0) =b(z)
\end{align*}
since
\begin{align*}
\abs{ B_\Omega(z,\gamma y_n)-B_\Omega(z, y_n)} \leq B_\Omega(y_n, \gamma y_n) \rightarrow 0.
\end{align*}
So
\begin{align*}
b(\gamma^{-n} z_0) = b(z_0)=0
\end{align*}
for all $n \in \Nb$.
\begin{observation} For any $t \in \Rb$
\begin{align*}
\overline{b^{-1}\Big((-\infty, t]\Big)}^{\Euc} \cap \partial \Omega = \{ \xi \}.
\end{align*}
\end{observation}
\begin{proof} Suppose $w_m \in b^{-1}\Big((-\infty, t]\Big)$ and $w_m \rightarrow \eta \in \partial \Omega$. If $\eta \neq \xi$, then Theorem~\ref{thm:GP} implies that there exists $R >0$ such that
\begin{align*}
B_\Omega(w_m, z_0) &+ B_\Omega(z_0, y_n) - B_\Omega(w_m, y_n) \leq R.
\end{align*}
Then
\begin{align*}
b(w_m) = \lim_{n \rightarrow \infty} B_\Omega(w_m, y_n)-B_\Omega(z_0, y_n) \geq B_\Omega(w_m, z_0)-R.
\end{align*}
However $B_\Omega(w_m, z_0) \rightarrow \infty$ since $B_\Omega$ is a proper metric on $\Omega$. So we have a contradiction.
\end{proof}
Using the previous observation, if $\gamma^{-n} z_0$ is unbounded in $\Omega$, then there exists $n_k \rightarrow \infty$ such that $\gamma^{-n_k} z_0 \rightarrow \xi$. Hence, in this case, $\Omega$ is biholomorphic to the unit ball by Theorem~\ref{thm:WR}.
It remains to consider the case where the sequence $\gamma^{-n} z_0$ is bounded in $\Omega$. Since $\Gamma$ is discrete and acts properly on $\Omega$, in this case
\begin{align*}
M:={\rm order}(\gamma) < \infty.
\end{align*}
We claim that $\gamma$ has a fixed point in $\Omega$. First, notice that
\begin{align*}
K_\Omega(\gamma^m y_n, y_n) \leq (M-1)\delta_n
\end{align*}
for all $m \in \Zb$. By Theorem~\ref{thm:squeeze} there exists some $\tau> 0$ such that the injectivity radius of $g_\Omega$ is bounded below by $\tau$ on each $U_n = \{ z \in \Omega : B_\Omega(z_0,y_n) \leq \tau\}$. By Theorem~\ref{thm:local}, $g_B$ is negatively curved on $U_n$ when $n$ is large. Then since $\delta_n \rightarrow 0$, Proposition~\ref{prop:COM} implies that when $n$ is large the function
\begin{align*}
f_n(x) = \sup\{ B_\Omega(\gamma^m y_n, x) : m=0,1,\dots, M-1\}
\end{align*}
has a unique minimum $c_n$ in $\Omega$. Since
\begin{align*}
\gamma \left\{ y_n, \gamma y_n, \gamma^2 y_n, \dots, \gamma^{M-1} y_n\right\} = \left\{ y_n, \gamma y_n, \gamma^2 y_n, \dots, \gamma^{M-1} y_n\right\},
\end{align*}
we then have $\gamma c_n = c_n$. So $\gamma$ has a fixed point in $\Omega$. Since $\Gamma$ acts freely on $\Omega$, we have a contradiction.
\section{The convex case}
Before starting the proof of Theorem~\ref{thm:main_convex} we will recall some results about convex domains.
As in Section~\ref{sec:squeeze}, let $s_\Omega: \Omega \rightarrow (0,1]$ denote the squeezing function on a bounded domain $\Omega \subset \Cb^d$.
\begin{theorem}\cite{F1991, KZ2016, NA2017} For any $d>0$ there exists some $s=s(d) > 0$ such that: if $\Omega \subset \Cb^d$ is a bounded convex domain, then $s_\Omega(z) \geq s$ for all $z \in \Omega$.
\end{theorem}
We will also need the following facts about the Kobayashi distance.
\begin{proposition}\label{prop:convex_complete} Suppose $\Omega \subset \Cb^d$ is a bounded convex domain. Then the metric space $(\Omega, K_\Omega)$ is proper and Cauchy complete.
\end{proposition}
For a proof of Proposition~\ref{prop:convex_complete} see for instance~\cite[Proposition 2.3.45]{A1989}.
\begin{theorem}\label{thm:GP_convex}\cite[Theorem 4.1]{Z2017} Suppose $\Omega \subset \Cb^d$ is a bounded convex domain with $C^{1,\epsilon}$ boundary. If $\xi, \eta \in \partial \Omega$ and $T_{\xi}^{\Cb} \partial \Omega \neq T_\eta^{\Cb} \partial \Omega$, then
\begin{align*}
\limsup_{x \rightarrow \xi, y \rightarrow \eta} K_\Omega(x, z_0) + K_\Omega(z_0, y) - K_\Omega(x, y) < \infty
\end{align*}
for some (hence any) $z_0 \in \Omega$.
\end{theorem}
\begin{remark} This says that a point $x$ near $\xi$ and point $y$ near $\eta$ can be joined by a path that passes through $z_0$ and is length minimizing up to a bounded error. \end{remark}
\subsection{Proof of Theorem~\ref{thm:main_convex}}
For the rest of the section suppose that $\Omega \subset \Cb^d$ is a bounded convex domain with $C^{1,\epsilon}$ boundary and $\Gamma \leq \Aut(\Omega)$ is a discrete group acting freely on $\Omega$. Further assume that $\Vol(\Gamma \backslash \Omega) < +\infty$ where $\Vol$ is either the Bergman volume, the K{\"a}hler-Einstein volume, or the Kobayashi-Eisenman volume.
Using Theorem~\ref{thm:LW} and Theorem~\ref{thm:Z} it is enough to show that $\Lc(\Omega)$ intersects at least two different closed complex faces of $\partial \Omega$.
\begin{lemma} If $\xi \in \partial \Omega$, then $\Lc(\Omega) \cap T_{\xi}^{\Cb} \partial \Omega \neq \emptyset$. \end{lemma}
The proof of the Lemma is nearly identical to the proof of Theorem~\ref{thm:main}, but we provide the complete argument for the reader's convenience.
\begin{proof} By replacing $\Omega$ with an affine translate, we may assume that $\xi = (1,0,\dots,0)$ and $0 \in \Omega$. Then fix a sequence $r_n \nearrow 1$ and consider the points $y_n=(r_n,0,\dots,0) \in \Omega$. For each $n \in \Nb$ define
\begin{align*}
\delta_n = \min_{\gamma \in \Gamma \setminus\{1\}} K_\Omega(y_n, \gamma y_n).
\end{align*}
Now for each $n \in \Nb$ the quotient map $\pi : \Omega \rightarrow \Gamma \backslash \Omega$ restricts to an embedding on
\begin{align*}
B_n = \{ z \in \Omega : K_\Omega(z, y_n) < \delta_n/2\}.
\end{align*}
Further, by Theorem~\ref{thm:squeeze} there exists some $C,\epsilon_0 > 0$ such that
\begin{align*}
{ \rm Vol}(\pi(B_n)) \geq C\min\{\epsilon_0^{2d}, \delta_n^{2d}\}.
\end{align*}
After passing to a subsequence we can assume that
\begin{align*}
\lim_{n \rightarrow \infty} \delta_n = \delta \in \Rb_{\geq 0} \cup \{ \infty\}.
\end{align*}
\noindent \textbf{Case 1:} $\delta \neq 0$. Since $\Vol(\Gamma \backslash \Omega) < \infty$, the set $\{ \pi(y_n) : n \in \Nb\}$ must be relatively compact in $\Gamma \backslash \Omega$. So for each $n$, there exist some $\gamma_n \in \Gamma$ such that the set $\{ \gamma_n y_n : n \in \Nb\}$ is relatively compact in $\Omega$. Then we can pass to a subsequence such that $\gamma_n y_n \rightarrow y \in \Omega$. Then $\gamma_n^{-1}y \rightarrow \xi$. So $\xi \in \Lc(\Omega)$. \\
\noindent \textbf{Case 2:} $\delta=0$. Then pick $\gamma_n \in \Gamma$ such that
\begin{align*}
K_\Omega(\gamma_n y_n, y_n) = \delta_n.
\end{align*}
\noindent \textbf{Case 2(a):} The set $\{ \gamma_1, \gamma_2, \dots\}$ is infinite. Since $\Gamma$ is discrete, by passing to a subsequence we can suppose that $\gamma_n \rightarrow \infty$ in $\Aut(\Omega)$. Fix some $z_0 \in \Omega$. By passing to another subsequence we can assume that $\gamma_n^{-1} z_0 \rightarrow \eta \in \partial \Omega$. Since $(\Omega, K_\Omega)$ is a complete proper metric space we must have
\begin{align*}
K_\Omega(z_0, \gamma_n^{-1} z_0) \rightarrow \infty.
\end{align*}
We claim that $\eta \in T_\xi^{\Cb} \partial \Omega$. Suppose not, then by Theorem~\ref{thm:GP_convex} there exists $R > 0$ such that
\begin{align*}
K_\Omega(\gamma_n^{-1}z_0, z_0) &+ K_\Omega(z_0, y_n) - K_\Omega(\gamma_n^{-1} z_0, y_n) \leq R.
\end{align*}
However
\begin{align*}
K_\Omega(\gamma_n^{-1}z_0, z_0) &+ K_\Omega(z_0, y_n) - K_\Omega(\gamma_n^{-1} z_0, y_n) \\
& = K_\Omega(\gamma_n^{-1}z_0, z_0) + K_\Omega(z_0, y_n) - K_\Omega(z_0, \gamma_n y_n) \\
& \geq K_\Omega(\gamma_n^{-1}z_0, z_0) - K_\Omega(\gamma_n y_n, y_n) \rightarrow \infty.
\end{align*}
So we have a contradiction and hence $\eta \in T_\xi^{\Cb} \partial \Omega$. \\
\noindent \textbf{Case 2(b):} The set $\{ \gamma_1, \gamma_2, \dots\}$ is finite. By passing to a subsequence we can suppose that $\gamma_n = \gamma$ in for all $n \in \Nb$.
Fix some $z_0 \in \Omega$. If the set $\{ \gamma^n(z_0) : n \in \Nb\}$ is relatively compact in $\Omega$, then $\gamma$ has a fixed point in $\Omega$ (see for instance~\cite[Theorem 5.1]{Z2017}). So, since $\Gamma$ acts freely on $\Omega$, the set $\{ \gamma^n(z_0) : n \in \Nb\}$ must be unbounded in $\Omega$.
Next consider the functions
\begin{align*}
b_n(z) = K_\Omega(z,y_n)-K_\Omega(y_n,z_0).
\end{align*}
Since $b_n(z_0)=0$ and each $b_n$ is 1-Lipschitz (with respect to the Kobayashi distance) we can pass to a subsequence such that $b_n \rightarrow b$ locally uniformly. Then
\begin{align*}
b(\gamma^{-1}z) = \lim_{n \rightarrow \infty} K_\Omega(\gamma^{-1} z,y_n)-K_\Omega(y_n,z_0) = \lim_{n \rightarrow \infty} K_\Omega(z,\gamma y_n)-K_\Omega(y_n,z_0) =b(z)
\end{align*}
since
\begin{align*}
\abs{ K_\Omega(z,\gamma y_n)-K_\Omega(z, y_n)} \leq K_\Omega(y_n, \gamma y_n) \rightarrow 0.
\end{align*}
So
\begin{align*}
b(\gamma^{-n} z_0) = b(z_0)=0
\end{align*}
for all $n \in \Nb$.
\begin{observation} For any $t \in \Rb$
\begin{align*}
\overline{b^{-1}\Big((-\infty, t]\Big)}^{\Euc} \cap \partial \Omega \subset T_{\xi}^{\Cb} \partial \Omega.
\end{align*}
\end{observation}
\begin{proof} Suppose $w_m \in b^{-1}\Big((-\infty, t]\Big)$ and $w_m \rightarrow \eta \in \partial \Omega$. If $\eta \notin T_\xi^{\Cb} \partial \Omega$, then Theorem~\ref{thm:GP_convex} implies that there exists $R >0$ such that
\begin{align*}
K_\Omega(w_m, z_0) &+ K_\Omega(z_0, y_n) - K_\Omega(w_m, y_n) \leq R.
\end{align*}
Then
\begin{align*}
b(w_m) = \lim_{n \rightarrow \infty} K_\Omega(w_m, y_n)-K_\Omega(z_0, y_n) \geq K_\Omega(w_m, z_0)-R.
\end{align*}
However $K_\Omega(w_m, z_0) \rightarrow \infty$ since $K_\Omega$ is a proper metric on $\Omega$. So we have a contradiction.
\end{proof}
Using the previous observation, there exists $n_k \rightarrow \infty$ such that
\begin{align*}
\lim_{k \rightarrow \infty} d_{\Euc}\left( \gamma^{-n_k} z_0, T_{\xi}^{\Cb} \partial \Omega \right)=0.
\end{align*}
So $\Lc(\Omega) \cap T_{\xi}^{\Cb} \partial \Omega \neq \emptyset$.
\end{proof}
\bibliographystyle{alpha}
|
1,314,259,993,609 | arxiv | \section{...}
\renewcommand{\theequation}{\Alph{app}.\arabic{equation}}
\newcommand{\sect}[1]{ \section{#1} \setcounter{equation}{0} }
\begin{document}
\begin{titlepage}
\title{Effective Action for the \\
Chiral Quark--Meson Model\thanks{Supported by the Deutsche
Forschungsgemeinschaft}}
\author{{\sc D.--U. Jungnickel\thanks{Email:
D.Jungnickel@thphys.uni-heidelberg.de}} \\
\\ and \\ \\
{\sc C. Wetterich\thanks{Email: C.Wetterich@thphys.uni-heidelberg.de}}
\\ \\ \\
{\em Institut f\"ur Theoretische Physik} \\
{\em Universit\"at Heidelberg} \\
{\em Philosophenweg 16} \\
{\em 69120 Heidelberg, Germany}}
\date{}
\maketitle
\begin{picture}(5,2.5)(-350,-450)
\put(12,-115){HD--THEP--95--7}
\put(12,-148){\today}
\end{picture}
\thispagestyle{empty}
\begin{abstract}
The scale dependence of an effective average action for mesons and
quarks is described by a nonperturbative flow equation.
The running couplings lead to spontaneous chiral symmetry
breaking. We argue that for
strong Yukawa coupling between quarks and mesons the low momentum
physics is essentially determined by infrared fixed points. This
allows us to establish relations between various parameters related
to the meson potential. The results for $f_\pi$ and
$\VEV{\olpsi\psi}$ are not very sensitive to the poorly
known details of the quark--meson effective action at scales where
the mesonic bound states form. For realistic constituent quark
masses we find $f_\pi$ around $100\,{\rm MeV}$.
\end{abstract}
\end{titlepage}
\sect{Introduction}
Quantum chromodynamics as the theory of strong interactions and its
symmetries are well tested both for high and low momenta. For momenta
$q^2\gta(2\,{\rm GeV})^2$ asymptotic freedom \cite{GW73-1} permits the use of
perturbation theory for a quark--gluon description with small gauge
coupling $g_s$. The long distance behavior for $q^2\lta(300\,{\rm MeV})^2$
can partially be described by chiral perturbation theory
\cite{Wei79-1,GL82-1}. Here the picture is based on a nonlinear or linear
$\si$--model \cite{GML60-1} for the pseudo--scalar mesons. The latter
can also be extended to describe in addition
scalar, vector and pseudo--vector mesons. Such
effective models
incorporate the chiral symmetries of QCD and use several
free couplings to parameterize the unknown strong interaction
dynamics. The parameters are determined phenomenologically
\cite{GL82-1,RRY85-1}, but
on a more fundamental level the question arises how they can be
related to the parameters of short distance QCD, i.e. the strong fine
structure constant $\alpha_s=\frac{g_s^2}{4\pi}$ and the current quark
masses. For example, one may ask how the most prominent quantity of
the mesonic picture, namely the pion decay constant $f_\pi$ which
measures the strength of spontaneous chiral symmetry breaking, can be
computed from $\alpha_s$ or vice versa.
Important progress in this question has been achieved by numerical
simulations in lattice gauge theories \cite{Lue}. Serious difficulties
in this approach remain, however, related to the treatment of
dynamical quarks and chiral symmetry.
There is also a vast amount of literature on various analytical
attempts to attack this problem. Examples can be found in \cite{Bij95-1}.
In this paper we employ a new
analytical method based on nonperturbative flow equations for scale
dependent effective couplings. These couplings parameterize the
effective average
action $\Gm_k$ \cite{Wet91-1} which is a type of coarse grained free
energy. It includes the effects of all quantum fluctuations with
momenta larger than an infrared cutoff $\sim k$. In the limit where
the average scale $k$ tends to zero $\Gm_{k\ra0}$ becomes therefore
the usual effective action, i.e. the generating functional of $1PI$
Green functions \cite{Wet93-2}. The scale dependence of $\Gm_k$ can be
described by an exact nonperturbative evolution equation
\cite{Wet93-1,Wet93-2}
\be
\prlt \Gm_k [\vp] =
\hal\mathop{\rm Tr}\left\{\left(
\Gm_k^{(2)}[\vp]+R_k\right)^{-1}
\frac{\prl R_k}{\prl t}\right\}
\label{ERGE}
\ee
where $t=\ln(k/\La)$ with $\La$ some suitable high momentum scale. The
trace represents here a momentum integration as well as a summation
over internal indices and we note the appearance on the right hand
side of the {\em exact} inverse propagator $\Gm_k^{(2)}$ as given by
the second
functional variation of $\Gm_k$ with respect to the field variables
$\vp$. The function $R_k(q)$ parameterizes the detailed form of the
infrared cutoff or the averaging procedure. With the
choice\footnote{$Z_{\vp,k}$ is an appropriate wave function
renormalization constant which will be specified later.}
\be
R_k(q)=\frac{Z_{\vp,k}q^2 e^{-q^2/k^2}}{1- e^{-q^2/k^2}}
\label{Rk(q)}
\ee
we observe that the momentum integration in eq. (\ref{ERGE}) is both
infrared and ultraviolet finite. For fluctuations with small momenta
$q^2\ll k^2$ the infrared cutoff $R_k\sim Z_{\vp,k}k^2$ acts like an
additional mass term in the propagator, whereas for $q^2\gg k^2$ it is
ineffective. The only difference between the flow equation
(\ref{ERGE}) and the $k$--derivative of a one--loop expression with
infrared cutoff $R_k$ concerns the appearance of $\Gm_k^{(2)}$ instead
of the second functional derivative of the {\em classical} action. This
turns eq. (\ref{ERGE}) into an exact equation, but also transmutes it
into a complicated functional differential equation which can
only be solved approximately by truncating the most
general form of $\Gm_k$. As it should be, eq. (\ref{ERGE}) can be
shown \cite{BAM93-1} to be equivalent to earlier forms of the exact
renormalization group equation \cite{WH73-1}. It may be interpreted as
a differential form of the Schwinger--Dyson equations
\cite{Dys49-1}. The difficult part is, however, not so much the
establishment of an exact flow equation but rather the finding of a
suitable nonperturbative truncation scheme which allows to solve the
differential equation. Then the flow equation can be integrated from
some short distance scale $\La$, where $\Gm_\La$ can be taken as the
classical action, to $k\ra0$ thus solving the model approximately.
Within the formalism of exact flow equations for the average action it
is possible to change the relevant degrees of freedom
\cite{EW94-1}. The idea is now to start from the exact flow equations
for quarks and gluons for $k>k_\vp$, and to use a similar exact
flow equation for quarks and mesons for $k<k_\vp$. The transition at
the scale $k_\vp$ ($600-700\,{\rm MeV}$) between the two pictures can be
encoded into an exact identity \cite{EW94-1} which replaces
multi--quark interactions in the quark--gluon picture by mesonic
interactions in the quark--meson picture. We emphasize that in the
quark--meson description the quarks remain important degrees of freedom as
long as $k$ is larger than a typical constituent quark mass
$m_q\simeq300\,{\rm MeV}$. We have therefore to deal with an effective
quark--meson model, where the mesons are described by a linear
$\si$--model with Yukawa coupling $h$ to the quarks. A first attempt
to describe the transition to a quark--meson model within a
QCD--inspired model with four--quark interactions \cite{EW94-1} has
been very encouraging. Spontaneous chiral symmetry breaking was
observed for low $k$, with a chiral condensate of the right order of
magnitude. In the following, the formalism has been generalized to QCD
\cite{Wet95-1}, with a method where the gluonic fluctuations with
$q^2>k^2$ are integrated out subsequently as $k$ is lowered.
So far, the treatment of
the quark--meson model for scales $k<k_\vp$ has been very rough,
however, since only quark fluctuations were included in
ref. \cite{EW94-1}. It is the purpose of this paper to present a
systematic study of the scale dependence of the quark--meson effective
action, including both quark and meson fluctuations.
Our main tool are nonperturbative flow equations which describe the
change of shape of the effective meson potential and the running of
the Yukawa coupling $h(k)$. In the perturbative limit these equations
reproduce the running of the couplings \cite{CGS93-1,BHJ94-1} in a
$U_L(N)\times U_R(N)$ model, which has been investigated in the context
of dynamical top quark condensation \cite{BHL91-1}. In our context $N$
stands for the number of quark flavors. The most important
nonperturbative ingredient in the flow equations will turn out to be
the appearance of effective mass threshold functions which account for
the decoupling of modes with mass larger than $k$. The solution of our
approximate flow equations allows us to express typical low momentum
quantities like $f_\pi$ in terms of the ``initial values'' for the
quark--meson model at the scale $k_\vp$, as for example the meson mass
term $\olm^2(k_\vp)$ or the wave function renormalization constant
$Z_\vp(k_\vp)$.
Not too surprisingly, the effective quark--meson Yukawa coupling $h$
will turn out to be rather strong. This can easily be seen by noting
that for $k\ra0$ this coupling is related to the ratio of a
constituent quark mass $m_q$ to $f_\pi=93\,{\rm MeV}$, namely
\be
h(k=0)=\frac{2m_q}{f_\pi}\simeq6.5\; .
\label{ValueOfHr}
\ee
For larger $k$ the Yukawa coupling must be even stronger, with a
typical nonperturbative initial value $h^2(k_\vp)/16\pi^2\gta1$.
The presence
of a strong coupling has important consequences for the predictive
power of the quark--meson model. Generically, the system of flow
equations exhibits (partial) infrared fixed points in the absence of a
mass scale. Due to the large Yukawa interaction the couplings are
driven very fast
towards these fixed points and the system ``looses its memory'' on the
detailed form of the initial values at $k_\vp$. Despite the fact that
the running is finally stopped by the formation of the chiral
condensate this infrared stability implies that the low momentum
quantities essentially depend only on one ``relevant'' parameter at
the scale $k_\vp$, i.e. the ratio $\olm^2(k_\vp)/k_\vp^2$. Using the
value (\ref{ValueOfHr}) for $h(0)$ and $h(k_\vp)$ between $12$ and
$100$ we find in a simplified model
\be
f_\pi\simeq(83-100)\,{\rm MeV}
\ee
in good agreement with the observed value $f_\pi=93\,{\rm MeV}$.
An estimate of the error as well
as a more complete treatment including more accurately
the effects of the chiral
anomaly and the strange quark mass is postponed to future work.
Besides the exciting prospect of computing $f_\pi$ and other
parameters of the low momentum meson interactions from QCD our
approach seems also capable to deal with other issues. Once the
parameters at the transition scale $k_\vp$ are fixed
either by a QCD--computation or by fitting
low momentum observational data,
it is straightforward to study the quark--meson
system at nonvanishing temperature. With methods described in ref.
\cite{TW93-1} the temperature dependence of $f_\pi$ or
$\VEV{\olpsi\psi}$ can be investigated. For $T\lta k_\vp/2\pi$ a study
within the effective quark--meson model with $T$--independent initial
values at $k_\vp$ should be sufficient, whereas for larger temperatures
the $T$--dependence of initial parameters like $\olm^2(k_\vp)$ starts
to become an important effect. One may therefore hope to gain new
insight into the nature of the chiral phase transition in QCD
\cite{PW84-1}. Another interesting issue concerns the use of
quark--meson models to describe hadronization in high energy scattering
experiments involving quarks or gluons \cite{KG-1}. Here our approach
may help to compute the phenomenological parameters used in those
models. Finally, the scalar--fermion models have been extensively
studied in the large--$N_c$ limit
\cite{BHJ94-1,BHL91-1}, and
our nonperturbative flow equations may help to access smaller values
of $N_c$.
Our paper is structured as follows: in section
\ref{TheChiralQuarkMesonModel} we give a brief phenomenological
introduction to the chiral quark--meson model with $N$ flavors. The
scale dependence of the effective meson potential is then described in
section \ref{ScaleDependenceOfTheEffectiveMesonPotential} and the
scalar wave function renormalization can be found in section
\ref{ScalarAnomalousDimension}. In section \ref{EvolutionEquationForH}
we derive the $\beta$--function for the running Yukawa coupling
between quarks and mesons as well as the quark wave function
renormalization. Section \ref{TheO(4)SymmetricSigmaModel} is
devoted to a short discussion of the chiral anomaly and the
presentation of two simplified models with two quark flavors. The
first model is based on the symmetry $U_L(2)\times U_R(2)$ and
neglects the effects of the chiral anomaly, whereas the second one
based on $O(4)$ neglects all scalars whose
masses obtain contributions from the chiral anomaly. In section
\ref{Results} we discuss in detail
the infrared stability properties for models with strong Yukawa
couplings and the consequences for the ``prediction'' of
$f_\pi$. Section \ref{Discussion} finally contains our quantitative
estimates for $f_\pi$ and the chiral condensate
$\VEV{\olpsi\psi}_0$. Conclusions are drawn in section
\ref{Conclusions}.
\sect{The chiral quark--meson model}
\label{TheChiralQuarkMesonModel}
We describe the low--energy degrees of freedom of QCD by an effective
action for quarks and mesons. We concentrate in this paper on
pseudo--scalar and scalar
mesons $\vp$ which transform in the $(\ol{\bf N},{\bf N})$
representation of the flavor symmetry group $SU_L(N)\times SU_R(N)$
for $N$ flavors. We consider the chiral limit where the
current quark masses are neglected. In its simplest form the
effective action $\Gm_k$ for quarks and mesons contains kinetic
terms, a potential for the scalar fields and a Yukawa coupling
between quarks and mesons:
\bea
\Gm_k &=&\ds{\int d^4 x\left\{
Z_\vp (k) \prl_\mu \vp^*_{ab} \prl^\mu \vp^{ab}+
U_k(\vp,\vp^\dagger ) \right.}\nnn
&+& \ds{\left. iZ_\psi (k)
\olpsi^a \gm^\mu \prl_\mu \psi_a +
\olh (k) \olpsi^a \left[
\frac{1+\ol{\gm}}{2}\vp_a^{\;\;b}-
\frac{1-\ol{\gm}}{2}{(\vp^\dagger )}_a^{\;\;b} \right]
\psi_b \right\} }.
\label{Truncation}
\eea
Our Euclidean conventions ($\olh (k)$ is real) are specified in appendix
\ref{LinearSigmaModel}. The scalar potential is assumed to be a
function of the invariants
\bea
\rho &=& \ds{\mathop{\rm tr}\left(\vp^\dagger \vp \right) }\nnn
\tau_2 &=& \ds{\frac{N}{N-1} \mathop{\rm tr}\left(\vp^\dagger \vp \right)^2 -
\frac{1}{N-1} \rho^2 }\nnn
\xi &=& \ds{
\det\vp +\det\vp^\dagger }
\label{BasicInvariants}
\eea
where we neglect the dependence on additional higher order invariants
present for $N\geq 3$ (cf. appendix \ref{ScalarMassSpectrum}).
Spontaneous chiral symmetry breaking with
a residual vector--like $SU(N)$ flavor symmetry occurs if the
potential has a minimum for $\si_0\neq0$
\be
\vp_0 = \left(
\ba{cccc}
\si_0 & & & \\
& \si_0 & & \\
& & \ddots & \\
& & & \si_0
\ea\right)\; ,\;\;\;
\rho_0=N\abs{\si_0}^2 \; .
\label{Minimum}
\ee
In this case we consider a quartic approximation for the potential
\be
U_k = -\olmu^2 (k) \rho +
\hal \olla_1 (k) \rho^2 + \frac{N-1}{4}
\olla_2 (k) \tau_2 -\hal \olnu(k)\xi
\label{QuarticPotentialSSB}
\ee
where $\olmu^2(k)$ is related to the $k$--dependent minimum value
$\rho_0 (k)$ by
\be
\olmu^2(k) =
\olla_1(k) \rho_0(k) - \frac{|\olnu(k)|}{2}
\left(\frac{\rho_0 (k)}{N}\right)^{\frac{N-2}{2}} \; .
\ee
Without loss of generality we will restrict ourselves
to positive $\olnu$. Up to an irrelevant constant we can also write
\be
U_k=\hal \olla_1 (k)
\left(\rho -\rho_0 (k)\right)^2 +
\frac{N-1}{4}\olla_2 (k) \tau_2 -
\hal \olnu(k)\xi
+\hal\olnu(k)
\left(\frac{\rho_0(k)}{N}\right)^{\frac{N-2}{2}}
\rho \; .
\ee
On the other hand, at short distance scales spontaneous chiral
symmetry breaking is not yet visible and $U_k$ is in the symmetric
regime ($\si_0=0$), where
we use the parameterization
\be
U_k = \olm^2 (k) \rho +
\hal \olla_1 (k) \rho^2 + \frac{N-1}{4}
\olla_2 (k) \tau_2 -\hal \olnu(k)\xi \; .
\label{PotentialSymRegime}
\ee
With these approximations our model can be described in terms of the
renormalized couplings
\bea
h(k) &=& \ds{
Z_\vp^{-1/2}(k) Z_\psi^{-1}(k) \olh (k) }\nnn
\la_{1,2}(k) &=& \ds{
Z_\vp^{-2}(k) \olla_{1,2}(k) }
\label{RenCoupl}
\eea
and either the mass term
\be
m^2 (k)=Z_\vp^{-1}(k) \olm^2 (k)
\label{RenMass}
\ee
or the location of the potential minimum
\be
\rho_R (k)=Z_\vp (k) \rho_0 (k) \; .
\label{RenRho}
\ee
The dimension of $\olnu$ depends on $N$ and the renormalized
coupling is
\be
\nu_R (k)=Z_\vp^{-\frac{N}{2}}(k) \olnu(k) \; .
\ee
For $\olnu=0$ the model has an additional axial $U_A (1)$ symmetry
which, however, is broken in QCD through the axial anomaly.
The quark--meson model is supposed to be obtained as an
effective model at some scale $k_\vp$, say $k_\vp\simeq600\,{\rm MeV}$.
It should be derivable from QCD by integrating out the gluonic
degrees of freedom and converting nonlocal four--quark interactions
into an effective quark--meson theory by the change of variables
described in \cite{EW94-1}. This gives a direct relation between
$\vp$ and a suitably defined \cite{EW94-1} composite
quark bilinear operator $\Oc_a^{\;\; b}=\VEV{\olpsi^b \psi_a}$
according to
\be
\Oc_a^{\;\; b}=\frac{2\olm^2 (k_\vp) Z_\psi (k_\vp)}
{\olh (k_\vp)}\vp_a^{\;\; b} \; .
\ee
The aim of this paper is to follow the evolution of $\Gm_k$ from the
``initial value'' at $k=k_\vp$ to $k=0$. The effective action
$\Gm =\Gm_{k=0}$ then describes the 1PI Green functions for the
collective meson fields or quark bilinears. In
particular, the chiral condensate is related to the vacuum
expectation value of $\vp$ corresponding to the minimum of the
effective potential $U=U_{k=0}$ through\footnote{We note that
(\ref{Codensate}) defines the quark condensate at the scale $k_\vp$
which may be different from the scale of usual chiral perturbation
theory estimates.}
\be
\VEV{\olpsi\psi}_0 =
\frac{2\olm^2 (k_\vp) Z_\psi (k_\vp)}
{\olh (k_\vp)} \si_0
\label{Codensate}
\ee
with\footnote{Neglecting fermion masses the phase of $\si_0$ is
arbitrary and we employ here a real and positive
$\si_0$. Correspondingly, $\VEV{\olpsi\psi}_0$ stands only for the
magnitude of the chiral condensate.}
\be
\si_0 = \left(
\frac{\rho_0 (k=0)}{N}\right)^{1/2} \; .
\ee
For $\si_0$ different from zero the chiral symmetry is spontaneously
broken and the spectrum of scalars contains $N^2 -1$ Goldstone bosons
corresponding to the pions. Their interactions are described by the
nonlinear $\si$--model. Neglecting the explicit $SU_V(N)$
breaking through quark masses the pion decay constant
$f_\pi$ is given in our conventions by
\be
f_\pi = 2\si_R
\ee
with renormalized expectation value
\be
\si_R = Z_\vp^{1/2} (k=0) \si_0 =
\left(\frac{Z_\vp (0)\rho_0 (0)}{N}\right)^{1/2} \; .
\ee
In our normalization the
experimental value reads $f_\pi =93\,{\rm MeV}$ or
\be
\si_R=46.5\,{\rm MeV}\; .
\ee
Another interesting quantity in our picture is the renormalized quark
mass ($h\equiv h(0)$)
\be
m_q = h \si_R
\ee
This corresponds to a constituent mass generated by chiral symmetry
breaking. (We remind that we consider the approximation of vanishing
current quark masses here.) A typical value should be around
$300\,{\rm MeV}$ and the renormalized Yukawa coupling therefore be relatively
large, $h\approx 6.5$.
The scalar spectrum is discussed in detail in appendix
\ref{ScalarMassSpectrum}.
For $\si_R\neq 0$ the meson sector contains besides the $N^2-1$
massless Goldstone bosons the $\si$--field (radial mode). With
$\la_1 =\la_1 (0)$ and $\nu_R=\nu_R(0)$ its mass is given by
\be
m_\si^2 = 2N\la_1 \si_R^2 -\hal\nu_R(N-2)\si_R^{N-2} \; .
\ee
One meson acquires a mass through the chiral anomaly. For the
realistic case of $N=3$ this can be identified with the
$\eta^\prime$ meson whereas the $\eta$ meson remains massless in the
chiral limit as one of the Goldstone bosons. For $N=2$
we are left
with the three pions as massless degrees of freedom
whereas the $K$--mesons and the $\eta$--meson are absent from the
spectrum. We will also for $N=2$ associate
the anomalously massive meson with the
$\eta^\prime$ meson. In our model its mass is given by
\be
m_{\eta^\prime}^2 = \frac{N}{2}\nu_R \si_R^{N-2} \; .
\ee
The remaining $N^2 -1$ massive scalar fields in the adjoint
representation of the diagonal flavor symmetry group $SU(N)$ have
mass (for $\la_2 =\la_2 (0)$)
\be
m_a^2 = N\la_2 \si_R^2 + \nu_R\si_R^{N-2} \; .
\ee
The neutral component can be associated with the $a_0$
meson with mass $983\,{\rm MeV}$. For realistic meson masses
($m_{\eta^\prime} = 958\,{\rm MeV}$) the couplings would
be
\bea
\nu_R &\simeq& \left( 958\,{\rm MeV}\right)^2\;\;\;
{\rm for}\;\;\; N=2 \nnn
\nu_R &\simeq& 13158\,{\rm MeV}\;\;\;
{\rm for}\;\;\; N=3
\label{ValuesForNyR}
\eea
\bea
\la_2 &\simeq& 11 \;\;\;
{\rm for}\;\;\; N=2 \nnn
\la_2 &\simeq& 55 \;\;\;
{\rm for}\;\;\; N=3 \; .
\eea
One should, however, notice that the values for $\nu_R$ and $\la_2$
are rather sensitive to the precise association of
$m_a$ or $m_{\eta^\prime}$ with known particle masses
and should therefore only be taken as a rough estimate. In particular,
for $N=3$ the effects of a nonzero strange quark mass have to be
incorporated for a more realistic estimate.
\sect{Scale dependence of the effective meson potential}
\label{ScaleDependenceOfTheEffectiveMesonPotential}
The meson degrees of freedom can be
introduced \cite{EW94-1} at
some short distance scale $k_\vp$ by
inserting the identity
\bea
1 &=& \ds{{\rm const} \int\Dc\si_A \Dc\si_H
\exp\left\{ -\hal \left[\left(\si_A^\dagger -
K_A^\dagger \tilde{G}-\Oc^\dagger [\psi]\tilde{G}\right)
\tilde{G}^{-1}\left(\si_A -\tilde{G}K_A -
\tilde{G}\Oc [\psi]\right)\right.\right. }\nnn
&+& \ds{\left.\left.\left(\si_H^\dagger -
K_H^\dagger \tilde{G}-\Oc^{(5)\dagger} [\psi]\tilde{G}\right)
\tilde{G}^{-1}\left(\si_H -\tilde{G}K_H -
\tilde{G}\Oc^{(5)} [\psi]\right)\right]\right\} }
\label{identity}
\eea
into the functional integral for the effective
average action for quarks. Here $K_{A,H}$ are sources for the
collective fields and correspond to the antihermitian and hermitian
parts\footnote{The fields $\ol{\si}_{A,H}$ associated to $K_{A,H}$ by
a Legendre transformation obey $\ol{\si}_A =-\frac{i}{2}(\vp
-\vp^\dagger )$, $\ol{\si}_H =\hal (\vp
+\vp^\dagger )$.} of
$\vp$. They are associated to the fermion bilinear operators $\Oc
[\psi]$,
$\Oc^{(5)}[\psi ]$ whose Fourier components read
\bea
\Oc_{\;\; b}^a (q) $=$ \ds{
-i\int\frac{d^4 p}{(2\pi)^4} g(-p,p+q)
\olpsi^a (p)\psi_b (p+q) }\nnn
\Oc_{\;\;\;\;\;\; b}^{(5)a} (q) $=$ \ds{
-\int\frac{d^4 p}{(2\pi)^4} g(-p,p+q)
\olpsi^a (p)\ol{\gm}\psi_b (p+q) }\; .
\eea
The wave function renormalization $g(-p,p+q)$ and the propagator
$\tilde{G}(q)$ are chosen such that the four--quark interaction
contained in (\ref{identity}) cancels the dominant part of the
QCD--induced nonlocal four--quark interaction in the
effective average action
formulated only for quarks. As a result, the introduction of
collective fields by (\ref{identity}) replaces the dominant part of
the
four--quark interaction by terms quadratic and linear
in the meson field. The resulting effective quark--meson
interactions are more general than those of the model described
in the last section.
The momentum dependence of the kinetic terms and the Yukawa couplings
can be described by an extended truncation of the effective average
action which, for general space--time dimensions $d$, is given
by
\bea
\Gm_k &=& \ds{ \int d^d x\;
U_k (\vp ,\vp^\dagger )}\nnn
&+& \ds{
\iddq\left\{
Z_{\vp ,k}(q) q^2 \mathop{\rm tr}\left(
\vp^\dagger (q)\vp (q)\right) +
Z_{\psi ,k}(q)\olpsi(q)\gm^\mu q_\mu \psi (q) \right. }\nnn
&+& \ds{ \left.
\iddq
\olh _k (-q,q-p) \olpsi(q) \left(
\frac{1+\ol{\gm}}{2}\vp (p)-
\frac{1-\ol{\gm}}{2}\vp^\dagger (-p) \right)
\psi (q-p) \right\} \; .}
\label{EffActAnsatz}
\eea
At the scale $k_\vp$ the average potential is
then purely quadratic
\be
U_{k_\vp} = \olm^2\mathop{\rm tr}\left(\vp^\dagger\vp\right)
\label{InitialU}
\ee
and the inverse scalar propagator is related to $\tilde{G}(q)$ in eq.
(\ref{identity}) by
\bea
\tilde{G}^{-1}(q) &=& 2\olm^2 +2\ol{Z}_\vp (q) q^2 \nnn
\ol{Z}_\vp (q) &\equiv& Z_{\vp ,k_\vp}(q) \; .
\eea
The initial value of the Yukawa coupling corresponds to the ``quark
wave function in the meson'' in eq. (\ref{identity}), i.e.
\be
\olh _{k_\vp}(-q,q-p) = g(-q,q-p)
\ee
which can be normalized with $\olh_{k_\vp}(0,0)=g(0,0)=1$.
The propagator $\tilde{G}$ and the wave function $g(-q,q-p)$ should be
optimized for a most complete elimination of terms quartic in the
quark fields. Neglecting the remaining $\psi^4$ terms and terms of
higher order in $\psi$ (e.g, $\psi^6$) one arrives at the initial
value for $\Gm_{k_\vp}$. In the present paper we often do not want to
keep the complete momentum dependence of $Z_{\psi ,k}$, $Z_{\vp ,k}$
and $\olh _k$. Useful definitions of the initial values of the
parameters of the model in section
\ref{TheChiralQuarkMesonModel} are then
\bea
Z_\psi (k_\vp ) &=& \ds{\left.
Z_{\psi ,k_\vp}(q)\right|_{q^2 =0} }\nnn
Z_\vp (k_\vp ) &=& \ds{ \left.\frac{1}{2q^2}\left(
\tilde{G}^{-1}(q)-\tilde{G}^{-1}(0)\right)
\right|_{q^2 =k_\vp^2} }\nnn
\olm^2 (k_\vp ) &=& \ds{
\hal \tilde{G}^{-1}(0) }\nnn
\olh (k_\vp ) &=& \ds{\left.
\olh _{k_\vp}(-q,q)\right|_{q^2 =0}\equiv 1 }\; .
\label{InitialValues}
\eea
Although the results of \cite{EW94-1} should only be considered as
rough estimates it seems convenient to use them as a guide for the
choice of initial values of the various couplings. The values found
for the transition scale $k_\vp$ and the scalar mass at this scale are
\cite{EW94-1} $k_\vp=630\,{\rm MeV}$, $\olm(k_\vp)=120\,{\rm MeV}$. The
$q^2$--dependence of $\tilde{G}$ was not computed very reliably in ref
\cite{EW94-1}. Large--$N_c$ estimates use a rather weak
$q^2$--dependence \cite{BHJ94-1}.
As a typical guess we consider here, somewhat
arbitrarily, that $\tilde{G}^{-1}(q^2=k_\vp^2)$ exceeds
$\tilde{G}^{-1}(q^2=0)$ by $15\%$. This leads to
$Z_\vp(k_\vp)=0.15\frac{\olm^2(k_\vp)}{k_\vp^2}\simeq\frac{1}{180}$. With
$Z_\psi(k_\vp)=1$, $\olh(k_\vp)=1$ this corresponds to a large
renormalized Yukawa coupling of $h^2(k_\vp)=180$. We will see later
(section \ref{Results}) that for strong initial Yukawa couplings the
decisive parameter is the ratio
\be
\teps_0=Z_\psi^2(k_\vp)\frac{\olm^2(k_\vp)}{k_\vp^2}\; .
\ee
The values of \cite{EW94-1} correspond to $\teps_0=0.036$.
Both, $Z_\psi$ and
$\olm$, may be somewhat lower than the values from \cite{EW94-1}
and we will often use typical values $\olm Z_\psi(k_\vp)=89\,{\rm MeV}
(63\,{\rm MeV})$ for which
$\teps_0\equiv\olm^2Z_\psi^2(k_\vp)/k_\vp^2\simeq0.02(0.01)$ and
$h^2(k_\vp)\simeq330(660)$ if the same assumption on the momentum
dependence of $\tilde{G}^{-1}(q)$ is made as above. The dependence of our
results on the choice of initial values will be discussed in detail in
section \ref{Results}.
We note that the use of the identity (\ref{identity})
does not lead to anomalous
$U_A(1)$ violating meson interactions
and (\ref{EffActAnsatz}) conserves the axial $U_A(1)$ symmetry.
Consequently the solution of the flow equations for the
$k$--dependence of the average potential also conserves this
symmetry. The formalism has therefore to be extended to incorporate
anomalous fermion interactions of the type
$\mathop{\rm Det}\left(\olpsi^a\psi_b\right)$ into the mesonic picture.
This issue may be addressed for the time being by introducing a term
$-\hal\olnu\xi$ into $U_k$ of (\ref{InitialU}) as a phenomenologically
determined coupling. We leave this for future work and
concentrate here on the $U_A(1)$ conserving case $\olnu=0$ and later
(section \ref{TheO(4)SymmetricSigmaModel}) on the opposite extreme
$\olnu\ra\infty$ for $N=2$.
We also observe that a generalization of the
formalism of \cite{EW94-1} may lead to nonvanishing meson
self--interactions $\olla_1$, $\olla_2$ at the scale $k_\vp$
as predicted by large--$N_c$ results \cite{BHJ94-1}.
At the scale $k_\vp$ the effective potential (\ref{InitialU})
has its minimum at the origin. As a result of quantum
fluctuations with momenta $q^2<k_\vp^2$ one expects that the potential
changes its shape and ends up at $k=0$ with a minimum for $\rho>0$,
resulting in a spontaneous breaking of chiral symmetry.
The aim of this paper is to derive flow equations for the
$k$--dependence of $Z_\psi$, $Z_\vp$, $U_k$ and $\olh$ and
to compute the observable quantities at $k=0$ described in the
last section from the initial values (\ref{InitialValues}).
Solving the flow equations numerically we find that chiral symmetry
breaking indeed occurs as demonstrated in figure \ref{Fig1}.
\begin{figure}
\unitlength1.0cm
\begin{picture}(13.,9.)
\put(.7,5.){\bf $\ds{\frac{m,\si_R}{\,{\rm MeV}}}$}
\put(8.,0.5){\bf $k/\,{\rm MeV}$}
\put(6.,3.5){\bf $\si_R$}
\put(11.5,5.){\bf $m$}
\put(-0.8,-11.5){
\epsfysize=22.cm
\epsffile{Fig1.ps}
}
\end{picture}
\caption{\footnotesize Evolution of the
renormalized mass $m$ in the symmetric regime (dashed line) and the
vacuum expectation value $\si_R$ of the scalar field in the SSB regime
(solid line) as functions of $k$ for the $U_L(2)\times U_R(2)$
model. Initial values are
$\la_1(k_\vp)=\la_2(k_\vp)=0$ for
$k_\vp=630\,{\rm MeV}$ with $h^2(k_\vp)=300$ and
$\teps_0=0.01$.}
\label{Fig1}
\end{figure}
We begin with the evolution equation for the effective average
potential $U_k$. Except for the $k$--dependence of $\olnu$ the evolution
equation for the
potential can be obtained (c.f. appendix
\ref{LinearSigmaModel})
by studying a constant scalar field
configuration which is real and
diagonal
\be
\vp_{ab}=\vp_a \delta_{ab}=
\widehat{m}_a \delta_{ab}
\ee
We evaluate the exact evolution equation (\ref{ERGE}) for this
configuration and insert the truncation (\ref{Truncation}) into the
right hand side. The flow equation has a fermionic and a bosonic
contribution. The bosonic part follows by neglecting for a moment
the quarks \cite{Wet93-1,Wet93-2}:
\bea
\ds{\prlt U_k} &=& \ds{
\hal\iddq\prlt R_k(q)
\left\{\sum_a \left[ \frac{1}{Z_\vp P+M_{Ra}^2}+
\frac{1}{Z_\vp P+M_{Ia}^2}\right]\right. }\nnn
&+& \ds{ \left.\sum_{a\neq b}\left[
\frac{1}{Z_\vp P+(M_{Rab}^+)^2}+
\frac{1}{Z_\vp P+(M_{Rab}^-)^2}\right.\right.}\nnn
&&\;\;\;\;\, \ds{\left.\left. +
\frac{1}{Z_\vp P+(M_{Iab}^+)^2}+
\frac{1}{Z_\vp P+(M_{Iab}^-)^2}\right]\right\} } \; .
\label{UkEvol}
\eea
We observe the appearance of the (massless) inverse average propagator
\be
P(q)=q^2+Z_\vp^{-1}R_k(q)=\frac{q^2}{1-e^{-\frac{q^2}{k^2}}}
\ee
which incorporates the infrared cutoff function $R_k$ (\ref{Rk(q)}).
The dependence of the various mass eigenvalues on $\vp_a$ can be found
in appendix \ref{ScalarMassSpectrum}. We restrict the discussion here
to the approximation $\olnu=0$ which
corresponds to neglecting the mass difference between the pseudo--scalar
pion triplet and the $\eta^\prime$ singlet. The mass eigenvalues on
the right hand side of (\ref{UkEvol}) are then
given by (\ref{MIa})---(\ref{MRac}).
In order to express the eigenvalues
$\widehat{m}_a^2$ in terms of $\rho$ and $\tau_2$ we consider a
particular configuration $\vp$ where $N-1$ eigenvalues are equal
to $\widehat{m}_1^2$ such that
\be\ba{rcl}
\rho &=& \ds{
(N-1)\widehat{m}_1^2 +\hat{m}_N^2 } \nnn
\mathop{\rm tr}\phi^2 &=& \ds{\frac{N-1}{N}\left(
\widehat{m}_1^2-\widehat{m}_N^2 \right)^2\; ,\;\;\;
\phi=\vp^\dagger\vp-\frac{1}{N}\rho }
\ea\ee
or
\be\ba{rcccl}
\vp_1^2 &=& \widehat{m}_1^2 &=&
\ds{\frac{1}{N}\left(\rho +\sqrt{\tau_2}\right)}\nnn
\vp_N^2 &=& \widehat{m}_N^2 &=&
\ds{\frac{1}{N}\left(\rho -(N-1)\sqrt{\tau_2}\right)}\; .
\label{configuration}
\ea\ee
This defines the right hand side of the evolution equation
(\ref{UkEvol}) as a function of $\rho$ and $\tau_2$. In the symmetric
regime the evolution of the couplings $\olla_1$, $\olla_2$ and
the mass term $\olm^2$ can now be extracted by suitable
differentiation of eq. (\ref{UkEvol}) with respect to
$\rho$ and $\tau_2$, evaluated for $\rho=\tau_2=0$. One finds for the
bosonic contributions
\begin{eqnarray}
\ds{\prlt\olm^2} &=& \ds{
-\hal\iddq\frac{\prl R_k}{\prl t}
\frac{2(N^2+1)\olla_1 +(N^2-1)\olla_2}
{(Z_\vp P+\olm^2)^2} }
\label{BosEvolMSymReg}\\[2mm]
\ds{\prlt\olla_1} &=& \ds{
\iddq\frac{\prl R_k}{\prl t}
\frac{2(N^2+4)\olla_1^2 +2(N^2-1)\olla_1 \olla_2+
(N^2-1)\olla_2^2}
{(Z_\vp P+\olm^2)^3} }
\label{BosEvolLa1SymReg}\\[2mm]
\ds{\prlt\olla_2} &=& \ds{
\iddq\frac{\prl R_k}{\prl t}
\frac{12\olla_1 \olla_2 +2(N^2-3)\olla_2^2}
{(Z_\vp P+\olm^2)^3} }\; .
\label{BosEvolLa2SymReg}
\end{eqnarray}
If in the course of the evolution towards smaller values of $k$ the
mass term $\olm^2$ becomes negative we should switch to the
couplings appropriate to the regime with spontaneous symmetry
breaking (SSB regime). There we define
\bea
\olla_1 &=& U_k^{\prpr}(\rho_0,\tau_2 =0) \nnn
\olla_2 &=& \ds{\frac{4}{N-1}\frac{\prl U_k}{\prl\tau_2}
(\rho_0,\tau_2 =0)}
\label{DefLambdas}
\eea
where $\rho_0$ corresponds to the $k$--dependent minimum of the
potential. We use that
\be
U_k^\prime (\rho_0)=0
\ee
is valid for all $k$ and therefore obtain
\be
\prlt\rho_0 =
-\frac{1}{\olla_1}\prlt
U_k^\prime (\rho_0) \; .
\label{DefRhoEvenN}
\ee
The evolution equations for $\rho_0$, $\olla_1$ and $\olla_2$ follow
directly from the definitions (\ref{DefLambdas}),
(\ref{DefRhoEvenN}). For $\olnu =0$ they read
\be
\prlt\rho_0 =
\hal\iddq\frac{\prl R_k}{\prl t}\left\{
\frac{N^2}{(Z_\vp P)^2}+\frac{3}{(Z_\vp P+2\olla_1
\rho_0)^2}+
\frac{(N^2-1)\left(1+\frac{\olla_2}{\olla_1}\right)}
{(Z_\vp P+\olla_2 \rho_0)^2}\right\}
\label{BosEvolRhoSSB}
\ee
\be
\prlt\olla_1 =
\iddq\frac{\prl R_k}{\prl t}\left\{
\frac{N^2 \olla_1^2}{(Z_\vp P)^3}+
\frac{9\olla_1^2}{(Z_\vp P+2\olla_1 \rho_0)^3}+
\frac{(N^2-1)\left(\olla_1+\olla_2\right)^2}
{(Z_\vp P+\olla_2 \rho_0)^3}\right\}
\label{BosEvolLa1SSB}
\ee
\bea
\ds{\prlt\olla_2} &=& \ds{
\iddq\frac{\prl R_k}{\prl t}\left\{
\frac{N^2}{4}\frac{\olla_2^2}{(Z_\vp P)^3} \right. }\nnn
&+& \ds{
\frac{9(N^2-4)}{4}\frac{\olla_2^2}
{(Z_\vp P+\olla_2 \rho_0)^3}
+ \frac{N^2\olla_2}{4\rho_0}\left[
\frac{1}{(Z_\vp P+\olla_2 \rho_0)^2}-
\frac{1}{(Z_\vp P)^2}\right] }\nnn
&+& \ds{\left.
\frac{3\olla_2 (\frac{1}{4}\olla_2+\olla_1)}
{\rho_0 (\hal\olla_2 -\olla_1)}\left[
\frac{1}{(Z_\vp P+2\olla_1 \rho_0)^2}-
\frac{1}{(Z_\vp P+\olla_2 \rho_0)^2}\right]\right\} }\; .
\label{BosEvolLambda2SSB}
\eea
It is straightforward to check that in the limits $\olm^2 \ra 0$,
$\rho_0 \ra 0$ the flow equations for $\olla_1$, $\olla_2$ coincide
in the symmetric and SSB regime. We also note that the evolution
equation for $\olla_2$ depends on the precise definition of this
coupling. This issue is shortly addressed in
appendix \ref{EvolEquLa2} where we
also give an alternative formulation of
eq. (\ref{BosEvolLambda2SSB}).
Next we turn to the fermionic contribution to the evolution equation
for the effective average potential which we denote by $\prl
U_{kF}/\prl t$. Using the general formulae of \cite{Wet90-1} it can
be computed without additional effort for the extended ansatz
(\ref{EffActAnsatz})
where we keep the momentum dependence of
$Z_{\psi,k}$ and part of the momentum dependence of the Yukawa
coupling with $\olh_k(q)\equiv\olh(-q,q)$. With $P_F$ given in
appendix \ref{InfraresCutoffFermions} and setting for a
moment $Z_{\psi,k} (q)=1$ one obtains:
\be
\prlt U_{kF} =
-2^{\frac{d}{2}-1}N_c \iddq\sum_{a=1}^N
\frac{\prl P_F(q)}{\prl t}\left( P_F (q)+m_a^2 (q)\right)^{-1} \; .
\ee
Here $m_a^2 (q)$ are the $N$ real nonnegative eigenvalues of the
$N\times N$ matrix $\olh_k (q)\olh_k^* (q)\vp^\dagger\vp$ with momentum
dependent Yukawa couplings defined by (\ref{SYuk}). Here we have taken
into account the $N_c$ colors of the quarks. For a given value of
$q$ we may use the identity (with $\prl /\prl t$ acting only on $P_F$,
$m^2 ={\rm diag}(m_a^2)$ and $\mathop{\rm tr}$ taken in flavor space)
\bea
\ds{\sum_{a=1}^N \frac{\prl P_F}{\prl t}
\left( P_F +m_a^2\right)^{-1}} &=& \ds{
\prlt\mathop{\rm tr}\ln\left( P_F+m^2\right) =
\prlt\ln\det\left( P_F+m^2\right)}\nnn
&=& \ds{\prlt
\ln\det\left( P_F+\abs{\olh_k (q)}^2\vp^\dagger\vp\right) }\nnn
&=& \ds{
\frac{\prl P_F}{\prl t}
\mathop{\rm tr}\left( P_F+\abs{\olh_k (q)}^2\vp^\dagger\vp\right)^{-1} }\; .
\eea
This gives an expression for general $\vp$
\be
\prlt U_{kF} =
-2^{\frac{d}{2}-1}N_c
\iddq\frac{\prl P_F}{\prl t}
\mathop{\rm tr}\left( P_F+\abs{\olh_k (q)}^2\vp^\dagger\vp\right)^{-1} \; .
\label{UFEvol}
\ee
Using the particular configuration with $N-1$ equal eigenvalues and
the relation (\ref{configuration}) one finally obtains
\bea
\ds{\prlt U_{kF} } &=& \ds{
-2^{\frac{d}{2}-1}N_c
\iddq\frac{\prl P_F}{\prl t}
\left\{ (N-1)\left( P_F +
\frac{1}{N}\abs{\olh_k}^2(\rho +\sqrt{\tau_2})\right)^{-1}
\right. }\nnn
&+& \ds{\left.
\left( P_F +
\frac{1}{N}\abs{\olh_k}^2(\rho -(N-1)\sqrt{\tau_2})\right)^{-1}
\right\} }\; .
\label{UFEvolN=2}
\eea
We emphasize that the Yukawa couplings in (\ref{EffActAnsatz})
conserve the axial $U_A(1)$ symmetry. The fermionic contribution to
$\prl U_{kF}/\prl t$ is therefore independent of $\xi$. The fermionic
wave function renormalization $Z_{\psi,k} (q)$ is easily restored if
we replace the function $P_F (q)$ in (\ref{UFEvolN=2}) by
$Z_{\psi,k}^2 (q)P_F(q)$ and note that the partial derivative
$\widehat{\prlt}(Z_{\psi,k}^2 P_F)$ only acts on the
pieces related to the infrared cutoff $R_k$. Within the truncation
(\ref{EffActAnsatz}) the fermionic contribution to the evolution
equation (\ref{UFEvol}) is then exact. Eq. (\ref{UFEvolN=2}) gives
the exact result for $N=2$ whereas for
$N>2$ one has an additional dependence on invariants $\tau_i$, $i\geq
3$, defined in appendix \ref{ScalarMassSpectrum}, which can be
extracted from (\ref{UFEvol}). The derivatives of
$\prlt U_{kF}$ with respect to $\rho$ and $\tau_2$ can be written in
a suggestive form as
\begin{eqnarray}
\ds{\prlt U_{kF}^\prime} &=& \ds{
-2^{\frac{d}{2}-1} \frac{N_c}{N}
\iddq\abs{\olh_k}^2 }
\label{UFprime} \\[2mm]
&\times& \ds{
\widehat{\prlt} \left\{
\frac{N-1}{Z_{\psi,k}^2 P_F +
\frac{1}{N}\abs{\olh_k}^2(\rho +\sqrt{\tau_2})} +
\frac{1}{Z_{\psi,k}^2 P_F +
\frac{1}{N}\abs{\olh_k}^2(\rho -(N-1)\sqrt{\tau_2})}
\right\} } \nnn
\ds{\prlt\frac{\prl U_{kF}}{\prl\tau_2}} &=& \ds{
-2^{\frac{d}{2}-2} N_c\frac{N-1}{N} \frac{1}{\sqrt{\tau_2}}
\iddq\abs{\olh_k}^2 }
\label{UFtau2}\\[2mm]
&\times& \ds{
\widehat{\prlt} \left\{
\frac{1}{Z_{\psi,k}^2 P_F +
\frac{1}{N}\abs{\olh_k}^2(\rho +\sqrt{\tau_2})} -
\frac{1}{Z_{\psi,k}^2 P_F +
\frac{1}{N}\abs{\olh_k}^2(\rho -(N-1)\sqrt{\tau_2})}
\right\} } \nonumber
\end{eqnarray}
with the formal definition
(cf. appendix \ref{InfraresCutoffFermions})
\be
\ds{\widehat{\prlt}}
\equiv \ds{
\frac{1}{Z_{\vp,k}}\frac{\prl R_k}{\prl t}
\frac{\prl}{\prl P} }
+ \ds{
\frac{2}{Z_{\psi,k}} \frac{P_F}{1+r_F}
\frac{\prl \left[Z_{\psi,k} r_F\right]}{\prl t}
\frac{\prl}{\prl P_F} }\; .
\ee
We may now combine the bosonic and fermionic contributions to the
running of the renormalized couplings. Here we restrict the
discussion again to momentum independent $Z_\vp$, $Z_\psi$ and (real)
$\olh$, i.e. we replace similarly to (\ref{InitialValues})
$Z_{\vp,k}(q)\ra Z_\vp (k)$, $Z_{\psi,k}(q)\ra Z_\psi (k)$ and
$\olh_k(q)\ra\olh (k)=\olh^*(k)$. For arbitrary $d$ it is
convenient to introduce dimensionless couplings in analogy to
(\ref{RenCoupl})---(\ref{RenRho}):
\bea
h^2 &=& \ds{
Z_\vp ^{-1} Z_\psi^{-2} k^{d-4} \olh^2 } \nnn
\la_{1,2} &=& \ds{
Z_\vp^{-2} k^{d-4} \olla_{1,2} } \nnn
\kappa &=& \ds{
k^{2-d} \rho_R\; =\; Z_\vp k^{2-d} \rho_0 }\nnn
\eps &=& \ds{k^{-2} m^2\; =\; Z_\vp^{-1} k^{-2} \olm^2 }
\label{DimensionlessCouplings}
\eea
and to define the anomalous dimensions for the scalar field,
$\eta_\vp$, and the fermion field, $\eta_\psi$, by
\be
\eta_\vp=-\prlt\ln Z_{\vp,k} \; ,\;\;\;
\eta_\psi=-\prlt\ln Z_{\psi,k}\; .
\label{AnoDimensions}
\ee
We also use dimensionless integrals
\ben
l_n^d (w;\eta_\vp) &=& \ds{
l_n^d (w) - \eta_\vp\hat{l}_n^d (w)
}\nnn
&=& \ds{
\frac{n}{4} v_d^{-1}k^{2n-d} \iddq
\left(\frac{1}{Z_\vp} \frac{\prl R_k(q)}{\prl t}\right)
\left( P+w k^2\right)^{-(n+1)} }\nnn
&=& \ds{
-\hal k^{2n-d} \int_0^{\infty} dx x^{\frac{d}{2}-1}
\widehat{\prlt} \left( P+w k^2\right)^{-n} }\\[2mm]
\ds{ l_{n_1,n_2}^d(w_1,w_2;\eta_\vp) }
&=& \ds{
\l_{n_1,n_2}^d(w_1,w_2)
-\eta_\vp \hat{l}_{n_1,n_2}^d(w_1,w_2)}\nnn
&=& \ds{
-\hal k^{2(n_1+n_2)-d}
\int_0^\infty dx\, x^{\frac{d}{2}-1}
\widehat{\prlt} \left\{\left(
P+w_1k^2\right)^{-n_1}
\left( P+w_2k^2\right)^{-n_2} \right\} \nonumber
}
\een
where the part $\sim\eta_\vp\hat{l}_n^d(w)$ arises from the
$t$--derivative acting on $Z_\vp$ within $R_k$ (cf. (\ref{Rk(q)})) and
\be
v_d^{-1} = 2^{d+1} \pi^\frac{d}{2} \Gm\left(\frac{d}{2}\right) \; .
\ee
The ``fermionic integrals''
$l_n^{(F)d} (w;\eta_\psi)=l_n^{(F)d} (w)-
\eta_\psi\check{l}_n^{(F)d} (w)$
are defined analogously,
with $P$ replaced by $P_F$. Combining
(\ref{BosEvolMSymReg})---(\ref{BosEvolLa2SymReg}) with
(\ref{UFprime}), (\ref{UFtau2}) we obtain the evolution equations for
the symmetric regime:
\begin{eqnarray}
\ds{ \frac{\prl\eps}{\prl t} } &=& \ds{
-(2-\eta_\vp )\eps -2v_d\left\{
[2(N^2+1)\la_1+(N^2-1)\la_2] l_1^d (\eps;\eta_\vp)
\right. }\nnn
&-& \ds{\left.
2^\frac{d}{2}N_c h^2
l_1^{(F)d}(\eta_\psi) \right\}
\label{FlowOfEpsilon} }\\[2mm]
\ds{\frac{\prl\la_1}{\prl t} } &=& \ds{
(d-4+2\eta_\vp )\la_1
+ 2v_d \left\{
[2(N^2+4)\la_1^2+(N^2-1)\la_2(2\la_1+\la_2)]l_2^d(\eps;\eta_\vp)
\right. }\nnn
&-& \ds{ \left.
2^\frac{d}{2} \frac{N_c}{N} h^4
l_{2}^{(F)d} (\eta_\psi)\right\} }\\[2mm]
\ds{\frac{\prl\la_2}{\prl t} } &=& \ds{
(d-4+2\eta_\vp )\la_2
+ 2v_d \left\{\left[
12\la_1\la_2 +2(N^2-3)\la_2^2\right]
l_2^d (\eps;\eta_\vp) \right. }\nnn
&-& \ds{ \left.
2^{\frac{d}{2}+1} \frac{N_c}{N} h^4
l_{2}^{(F)d}(\eta_\psi)\right\}\; . }
\een
Similarly, the evolution equations for the SSB regime read
\ben
\ds{ \frac{\prl\kappa}{\prl t} } &=& \ds{
(2-d-\eta_\vp )\kappa + 2v_d \left\{
N^2l_1^d(\eta_\vp)
+3l_1^d (2\la_1 \kappa;\eta_\vp) \right. }\nnn
&+& \ds{ \left.
(N^2-1)\left[ 1+\frac{\la_2}{\la_1}\right]
l_1^d (\la_2\kappa;\eta_\vp)-2^\frac{d}{2}N_c \frac{h^2}{\la_1}
l_{1}^{(F)d} (\frac{1}{N}h^2 \kappa;\eta_\psi)
\right\}
\label{FlowOfKappa} }\\[2mm]
\ds{\frac{\prl\la_1}{\prl t} } &=& \ds{
(d-4+2\eta_\vp )\la_1 +2v_d \left\{
N^2\la_1^2 l_2^d(\eta_\vp)
+9\la_1^2 l_2^d (2\la_1\kappa;\eta_\vp)
\right.}\nnn
&+& \ds{\left.
(N^2-1)\left[\la_1 +\la_2\right]^2
l_2^d (\la_2\kappa;\eta_\vp)-2^\frac{d}{2}\frac{N_c}{N}h^4
l_{2}^{(F)d} (\frac{1}{N}h^2\kappa;\eta_\psi)
\right\} }\\[2mm]
\ds{\frac{\prl\la_2}{\prl t} } &=& \ds{
(d-4+2\eta_\vp )\la_2 +2v_d \left\{
\frac{N^2}{4}\la_2^2 l_2^d(\eta_\vp) +
\frac{9}{4}(N^2-4)\la_2^2 l_2^d (\la_2\kappa;\eta_\vp)
\right. }\nnn
&-& \ds{
\hal N^2 \la_2^2
l_{1,1}^d(0,\la_2\kappa;\eta_\vp) +
3[\la_2+4\la_1]\la_2
l_{1,1}^d(2\la_1\kappa,\la_2\kappa;\eta_\vp)
\label{SSBFlowOfLambda2}
}\\[2mm]
&-& \ds{\left.
2^{\frac{d}{2}+1}\frac{N_c}{N}h^4
l_{2}^{(F)d} (\frac{1}{N}h^2\kappa;\eta_\psi)\right\}
}\nonumber
\een
where we have defined
\be
l_n^d(\eta_\vp)\equiv l_n^d(0;\eta_\vp)\; ,\;\;\;
l_{n}^{(F)d}(\eta_\psi)\equiv l_{n}^{(F)d}(0;\eta_\psi) \; .
\ee
No explicit dependence on $k$ appears in this scaling form of the flow
equations. In the limit $\eps,\kappa\ra 0$ one recovers for both
regimes to leading order in the coupling constants the known
\cite{CGS93-1,BHJ94-1} perturbative one--loop beta functions for
$\la_1$ and $\la_2$.
The system of flow equations
(\ref{FlowOfEpsilon})--(\ref{SSBFlowOfLambda2}) is the central piece
of this work. For the quark--meson model ($d=4$) we fix the initial
conditions at $k=k_\vp$ ($t=0$) with $\la_1(k_\vp)=\la_2(k_\vp)=0$,
$\eps(k_\vp)=Z_\vp^{-1}(k_\vp)k_\vp^{-2}\olm^2$. The solution of the
evolution equations should then reveal the phenomenon of spontaneous
chiral symmetry breaking for $k\ra 0$ ($t\ra -\infty$). More precisely,
we expect for some scale $k_s>0$ that the mass term vanishes, i.e.
$\eps(k_s)=0$. Subsequently, for $k<k_s$ we follow the evolution of
the minimum of the potential using the system
(\ref{FlowOfKappa})--(\ref{SSBFlowOfLambda2}), with initial condition
$\kappa(k_s)=0$. (The couplings $\la_1$ and $\la_2$ are continuous at
$k_s$.) For sufficiently small values of $k$ the minimum of the
potential will not move anymore, i.e.
\bea
\ds{\lim_{k\ra 0} Z_\vp(k) }
&=& \ds{Z_\vp}\nnn
\ds{ \lim_{k\ra 0}\rho_R(k) }
&=& \ds{ \lim_{k\ra 0}Z_\vp(k)\rho_0(k)
=\rho_R=Z_\vp \rho_0 }\nnn
&=& \ds{ N\si_R^2 = NZ_\vp \si_0^2
=\frac{N}{4}f_\pi^2 }\nnn
\ds{ \lim_{k\ra 0}\kappa(k)}
&\ra& \ds{
\rho_R k^{-2} \; .}
\eea
Supplementing the flow equations for $\kappa$, $\la_1$ and $\la_2$ by
the one for the Yukawa coupling $h^2$ and inserting the anomalous
dimensions --- these quantities will be computed in the next two
sections --- we can now study how the shape of the average potential
changes as $k$ is lowered. We have integrated the flow equations
numerically for $d=4$ and $N_c=3$ from
$t_i=0$ corresponding to $k=k_\vp$
to $t_f=\ln(m_\pi/k_\vp)$. This endpoint of the numerical integration
simulates the pion mass threshold which is neglected in our
approximation of vanishing quark masses. We ignore here all dependence
of the threshold functions on the anomalous dimensions $\eta_\vp$ and
$\eta_\psi$. This approximations will be justified in section
\ref{Results}. We use first $k_\vp=630\,{\rm MeV}$, $\teps_0=0.01$ and
$Z_\psi(k_\vp)=1$, $h^2(k_\vp)=Z_\vp^{-1}(k_\vp)=300$. For the initial
values of $\la_1$ and $\la_2$ we employ two different sets of boundary
conditions. One corresponds to the approximations used in
\cite{EW94-1}
\be
\la_1(k_\vp) = \la_2(k_\vp)=0 \; .
\label{EWBoundaryConditions}
\ee
The other set is obtained in the large--$N_c$ limit of the
$U_L(N)\times U_R(N)$ model \cite{BHJ94-1} and reads
\bea
\la_1(k_\vp) &=& \ds{\frac{2}{N}h^2(k_\vp)} \nnn
\la_2(k_\vp) &=& \ds{\frac{4}{N}h^2(k_\vp)} \; .
\label{LNBoundaryConditions}
\eea
\begin{figure}
\unitlength1.0cm
\begin{picture}(13.,9.)
\put(1.,5.){\bf $\la_{1/2}$}
\put(8.,0.5){\bf $k/\,{\rm MeV}$}
\put(-0.8,-11.5){
\epsfysize=22.cm
\epsffile{Fig2.ps}
}
\end{picture}
\caption{\footnotesize Flow of $\la_1$ (solid
lines) and $\la_2$ (dashed lines) with $k$,
for the $U_L(2)\times U_R(2)$ model with two sets
of initial conditions at $k_\vp=630\,{\rm MeV}$ with $h^2(k_\vp)=300$ and
$\teps_0=0.01$.}
\label{Fig2}
\end{figure}
For the numerical investigations we will concentrate in the present
work on $N=2$.
In fig. \ref{Fig1} we have plotted the renormalized mass
$m$ as a function of $k$.
Starting from a very large value
$m(k_\vp)=1091\,{\rm MeV}$ the mass rapidly decreases and reaches zero for
$k_s\simeq 450\,{\rm MeV}$. For $k<k_s$ the minimum of the potential occurs
for $\rho_0>0$ and we have to use the flow equations for the SSB
regime. In fig. \ref{Fig1} we also show the $k$--dependence of the
location of the minimum, $\si_R(k)=\left(\rho_R(k)/N\right)^\hal$,
and see how it stabilizes
for $k\lta250\,{\rm MeV}$. The final result for $\si_R$ is rather
insensitive to the exact choice of the endpoint $k_f=m_\pi$.
In fig. \ref{Fig2} we display the $k$--dependence of the quartic
couplings $\la_1$ and $\la_2$ for the two sets of boundary
conditions. We observe that the result for $k=k_f$ does not depend
strongly on the initial values. This is a first manifestation of the
infrared stability mentioned in the introduction. We will explain the
origin of this behavior in more detail later (section \ref{Results}),
since for an understanding we first need to discuss the anomalous
dimensions and the running of $h^2$.
\sect{Scalar anomalous dimension}
\label{ScalarAnomalousDimension}
A computation of $f_\pi$ in terms of the four--quark interaction at
the scale $k_\vp$ requires the ratio $Z_\vp(0)/Z_\vp(k_\vp)$.
An evaluation of this ratio is the subject of this section. We will
begin with the determination of the flow equation for the
momentum dependent scalar wave function
renormalization $Z_{\vp ,k} (q)$ or, equivalently, the scalar
anomalous dimension $\eta_{\vp ,k} (q)\equiv -Z_{\vp ,k}^{-1}(q)\prl_t
Z_{\vp ,k}(q)$.
For this purpose we have to consider
a spatially varying scalar field configuration. We choose a nondiagonal
distortion of the constant vacuum configuration (\ref{Minimum}):
\be
\vp_{ab}(x) = \vp\dt_{ab} +
\left[\dt\vp e^{-iQx} + \dt\vp^* e^{iQx}\right] \Si_{ab}
=\vp_{ab}^*(x)
\label{ConfAnDi}
\ee
with
\be
\Si_{ab}=\dt_{a1}\dt_{b2}-\dt_{a2}\dt_{b1}
\ee
or, in momentum space (with $\dt(q_1,q_2)=(2\pi)^d\dt(q_1-q_2)$),
\bea
\ds{\vp_{ab}(q) }
&=& \ds{ \vp\dt(q,0)\dt_{ab}+
\left[\dt\vp\,\dt(q,Q) + \dt\vp^*\,\dt(q,-Q)\right]\Si_{ab} }\nnn
&\equiv& \ds{\vp\dt(q,0)\dt_{ab} + \Dt(q,Q)\Si_{ab} }
\eea
Expanding around $\vp$ at the potential minimum, we observe that
$\dt\vp$ corresponds to an excitation of the
massless charged pion $\pi^\pm$.
We keep the discussion here for general $\vp$.
If we supplement the scalar configuration by a fermionic background
\be
\psi_\al = \olpsi_\al =0
\ee
$\Gm^{(2)}_k$ is easily seen to be
block--diagonal. It decays into matrices acting in scalar and fermion
subspaces, $\Gm_{Sk}^{(2)}$ and $\Gm_{Fk}^{(2)}$,
respectively. Hence, we can read off from (\ref{ERGE})
\bea
\ds{
\frac{\prl}{\prl t} Z_{\vp ,k}(Q) }
&=& \ds{
\frac{1}{4Q^2}\left(
\lim_{\dt\vp ,\dt\vp^* \ra 0}
\frac{\prl}{\prl (\dt\vp \dt\vp^*)}
\left\{\hal \mathop{\rm Tr} \left[\left(\Gm^{(2)}_{Sk}+R_k\right)^{-1}
\frac{\prl R_k}{\prl t}\right] \right.\right. } \nnn
&-& \ds{ \left.\left.
\mathop{\rm Tr}\left[\left(\Gm^{(2)}_{Fk}+R_{Fk}\right)^{-1}
\frac{\prl R_{Fk}}{\prl t}\right]\right\}
- \left( Q\ra 0\right) \right) }
\label{Zvpfull}
\eea
The right hand side may be
evaluated by expanding the traces in powers of $\dt\vp$ and
$\dt\vp^*$ up to order $\dt\vp\dt\vp^*$
and subtracting all $Q$--independent
terms. The flow equations for the renormalization constant $Z_\vp (k)$
or equivalently $\eta_\vp (k)$ are now determined as
\be
\frac{\prl}{\prl t} Z_\vp(k) =
\lim_{Q^2 \ra 0}
\frac{\prl}{\prl t} Z_{\vp ,k}(Q) \; ,\;\;\;
\eta_\vp (k)=\lim_{Q^2 \ra 0}\eta_{\vp ,k}(Q)
=-\prlt\ln Z_\vp(k) \;.
\ee
In order to evaluate the right hand side we split
\be
\Gm_{k}^{(2)} = \Gm_{k,0}^{(2)} +
\Dt \Gm_{k}^{(2)}
\ee
such that all $\dt\vp$, $\dt\vp^*$ dependence is contained in $\Dt
\Gm_{k}^{(2)}$. We may then expand
\bea
\ds{ \mathop{\rm Tr}\left[\left(\Gm_{k}^{(2)}+R_k\right)^{-1}
\frac{\prl R_k}{\prl t}\right] }
&=& \ds{
\mathop{\rm Tr}\left[\left(\Gm_{k,0}^{(2)}+R_k\right)^{-1}
\frac{\prl R_k}{\prl t}\right] }\nnn
&+& \ds{
\mathop{\rm Tr}\left[\widehat{\frac{\prl}{\prl t}}
\left\{
\left(\Gm_{k,0}^{(2)}+R_k\right)^{-1}
\Dt \Gm_{k}^{(2)} \right\}\right]} \\[2mm]
&-& \ds{
\hal\mathop{\rm Tr}\left[\widehat{\frac{\prl}{\prl t}}
\left\{
\left(\Gm_{k,0}^{(2)}+R_k\right)^{-1}
\Dt \Gm_{k}^{(2)}
\left(\Gm_{k,0}^{(2)}+R_k\right)^{-1}
\Dt \Gm_{k}^{(2)} \right\}\right] }\nnn
&+& \ds{ {\cal O}(\Dt^3) }
\label{TraceExp}
\eea
and compute the right hand side of (\ref{Zvpfull}) from the first terms.
Details of the calculation can be found in appendix
\ref{ScalarWaveFunctionRenormalization}. Taking the limit $Q^2\ra 0$
and neglecting all momentum
dependence of the Yukawa coupling and wave function
renormalizations we find for the SSB regime
\bea
\ds{ \eta_\vp} &=& \ds{
8\frac{v_d}{d}\kappa\left\{
2\la_1^2 m_{2,2}^d(0,2\kappa\la_1;\eta_\vp)+
\frac{N^2-2}{4}\la_2^2 m_{2,2}^d(0,\kappa\la_2;\eta_\vp)
\right\} }\nnn
&+& \ds{
2^{\frac{d}{2}+2} \frac{v_d}{d} N_c h^2
m_4^{(F)d} (\frac{1}{N}\kappa h^2; \eta_\psi ) }
\label{EtaVpBR}
\eea
and for the symmetric regime
\be
\eta_\vp =2^{\frac{d}{2}+2} \frac{v_d}{d} N_c h^2
m_4^{(F)d} (0; \eta_\psi )\; .
\label{EtaVpSR}
\ee
Here we have defined the threshold functions
\bea
\ds{m_{n_1,n_2}^d (w_1,w_2;\eta_\vp) } &\equiv& \ds{
m_{n_1,n_2}^d (w_1,w_2) - \eta_\vp
\hat{m}_{n_1,n_2}^d (w_1,w_2) }\nnn
&& \ds{ \hspace{-1cm}
= -\hal k^{2(n_1+n_2-1)-d}
\int_0^\infty dx\, x^{\frac{d}{2}}
\widehat{\frac{\prl}{\prl t}} \left\{
\frac{\dot{P} (x)}
{[P(x)+k^2 w_1]^{n_1} }
\frac{\dot{P} (x)}
{[P(x)+k^2 w_2]^{n_2}} \right\} }
\label{mn1n2d}
\eea
and
\bea
\ds{m_4^{(F)d} (w; \eta_\psi )}
&=& \ds{
m_4^{(F)d} (w)-\eta_\psi \check{m}_4^{(F)d} (w) }\nnn
&=& \ds{
-\hal k^{4-d}
\int_0^\infty dx\, x^{\frac{d}{2}+1}
\widehat{\prlt} \left(
\frac{\prl}{\prl x}
\frac{1+r_F(x)}{P_F(x)+k^2w}\right)^2 }
\label{m4Fd}
\eea
where
\be
x= q^2\; ,\;\; P(x)\equiv P(q)\; ,\;\;
\dot{P}(x)\equiv\frac{\prl}{\prl x}P(x)\; ,\;\;
\widehat{\prlt}\dot{P}\equiv\frac{\prl}{\prl x}\widehat{\prlt}P\; .
\ee
{}From the anomalous dimension $\eta_\vp(k)$ the wave function
renormalization can be obtained by numerical integration of
(\ref{AnoDimensions}). We have plotted the result in fig.
\ref{Fig3} for $k_\vp=630\,{\rm MeV}$, $\teps_0=0.01$,
$\la_1(k_\vp)=\la_2(k_\vp)=0$ and two values of $h^2(k_\vp)=300$ and
$10^4$.
\begin{figure}
\unitlength1.0cm
\begin{picture}(13.,9.)
\put(1.5,5.){\bf $Z_\vp$}
\put(8.,0.5){\bf $k/\,{\rm MeV}$}
\put(-0.8,-11.5){
\epsfysize=22.cm
\epsffile{Fig3.ps}
}
\end{picture}
\caption{\footnotesize Evolution of $Z_\vp$ with $k$, for the
$U_L(2)\times U_R(2)$ model with
$k_\vp=630\,{\rm MeV}$, $\teps_0=0.01$,
$\la_1(k_\vp)=\la_2(k_\vp)=0$ and
two values $h^2(k_\vp)=300$ (solid line) and $h^2(k_\vp)=10^4$ (dashed
line).}
\label{Fig3}
\end{figure}
We note that $Z_\vp$ increases strongly in the symmetric regime for
$k>450\,{\rm MeV}$ and stabilizes for low values of $k$. Again, the final
value at $k=k_f$ does not depend very much on the initial conditions
for $h^2$.
\sect{Evolution equation for the Yukawa coupling and fermion anomalous
dimension}
\label{EvolutionEquationForH}
To determine the evolution equation for the Yukawa coupling and the
fermionic wave function renormalization constant we will turn to a
field configuration ($\vp=\vp^*$)
\bea
\ds{\vp_{ab}(x)} &=& \ds{\vp\dt_{ab}}\nnn
\ds{\psi_a^{\hat{\al}}(x)} &=& \ds{\psi_a^{\hat{\al}}e^{-iQx}}\nnn
\ds{\olpsi^a_{\hat{\al}}(x)} &=& \ds{\olpsi^a_{\hat{\al}}e^{iQx}} \; .
\eea
Furthermore, we will approximate the momentum dependence of the Yukawa
coupling by $\olh_k(-q,q^\prime)\simeq\olh_k(\frac{q+q^\prime}{2})$.
This amounts to neglecting its dependence on the external scalar
momentum $\frac{q-q^\prime}{2}$ in the Yukawa vertex in
(\ref{EffActAnsatz}). Accordingly, the renormalized Yukawa coupling is
defined via
\be
h_k(q)=k^{\frac{d}{2}-2}Z_{\psi,k}^{-1}(q)
Z_{\vp,k}^{-\hal}(0)\olh_k(q)\; .
\label{YukRen}
\ee
The matrix of second functional derivatives of $\Gm_k$ simplifies
considerably for the above configuration. Omitting spinor indices
one finds
\be
\frac{\dt^2\Gm_k}
{\dt\olpsi^a (q)\dt\psi_b (q^\prime )}
=\left( Z_{\psi ,k}(q)\slash{q}
+\olh_k(q)\vp\olgm \right)
(2\pi)^d\dt(q-q^\prime)\dt_a^b \; .
\label{FFF}
\ee
The derivation of the flow equations for $\olh$ and $Z_\psi$
follows similar lines as for the
scalar anomalous dimension discussed in section
\ref{ScalarAnomalousDimension}. For details of the calculation we
refer to appendix \ref{FermionWaveFunctionRenormalization}.
Neglecting the effects of the chiral anomaly ($\olnu=0$) as well as
the momentum dependence of the wave function renormalizations and the
Yukawa coupling we find in the limit $Q\ra0$
\bea
\ds{ \prlt h^2}
&=& \ds{ \left[ d-4+2\eta_\psi +\eta_\vp\right] h^2
-\frac{4}{N}v_d h^4 \left\{
N^2l_{1,1}^{(FB)d} (\frac{1}{N}\kappa h^2,\eps;
\eta_\psi,\eta_\vp) \right. }\nnn
&-& \ds{\left.
(N^2-1)l_{1,1}^{(FB)d} (\frac{1}{N}\kappa h^2,\eps+\kappa\la_2;
\eta_\psi,\eta_\vp)
-l_{1,1}^{(FB)d} (\frac{1}{N}\kappa h^2,\eps+2\kappa\la_1;
\eta_\psi,\eta_\vp)
\right\} }
\label{RunningOfh2}
\eea
where
\bea
\ds{ l_{n_1,n_2}^{(FB)d}(w_1,w_2;\eta_\psi,\eta_\vp) }
&=& \ds{
l_{n_1,n_2}^{(FB)d}(w_1,w_2)
-\eta_\psi \check{l}_{n_1,n_2}^{(FB)d}(w_1,w_2)
-\eta_\vp \hat{l}_{n_1,n_2}^{(FB)d}(w_1,w_2) }\nnn
&& \ds{ \hspace{-2cm}
= -\hal k^{2(n_1+n_2)-d}
\int_0^\infty dx\, x^{\frac{d}{2}-1}
\widehat{\prlt}\left\{
\frac{1}{[P_F(x)+k^2w_1]^{n_1} [P(x)+k^2w_2]^{n_2} } \right\} }\; .
\label{ln1n2FBd}
\eea
Similarly, the fermionic anomalous dimension reads
\bea
\ds{\eta_\psi }
&=& \ds{
\frac{4}{N}\frac{v_d}{d} h^2 \left\{
N^2m_{1,2}^{(FB)d}(\frac{1}{N}h^2\kappa,\eps;\eta_\psi,\eta_\vp)
+m_{1,2}^{(FB)d}(\frac{1}{N}h^2\kappa,\eps+2\kappa\la_1;
\eta_\psi,\eta_\vp) \right. }\nnn
&+& \ds{ \left.
(N^2-1)m_{1,2}^{(FB)d}(\frac{1}{N}h^2\kappa,\eps+\kappa\la_2;
\eta_\psi,\eta_\vp) \right\} }
\label{EtaPsi}
\eea
with
\bea
\ds{m_{n_1,n_2}^{(FB)d}(w_1,w_2; \eta_\psi,\eta_\vp)}
&=& \ds{
m_{n_1,n_2}^{(FB)d}(w_1,w_2)
-\eta_\psi \check{m}_{n_1,n_2}^{(FB)d}(w_1,w_2)
-\eta_\vp \hat{m}_{n_1,n_2}^{(FB)d}(w_1,w_2) }\nnn
&& \ds{ \hspace{-2.5cm}
= -\hal k^{2(n_1+n_2-1)-d}
\int_0^\infty dx\, x^{\frac{d}{2}}
\widehat{\prlt}\left\{
\frac{1+r_F(x)}{[P_F(x)+k^2w_1]^{n_1}}
\frac{\dot{P}(x)}{[P(x)+k^2w_2]^{n_2}} \right\} \; . }
\label{mn1n2FBd}
\eea
In summary, the equations (\ref{EtaVpBR}) and
(\ref{EtaPsi}) constitute a linear system for $\eta_\vp$ and
$\eta_\psi$ with solution
\bea
\ds{\eta_\vp} &=& \ds{
\frac{A_1(1+A_6)-A_3A_4}{(1+A_2)(1+A_6)-A_3A_5} }\nnn
\ds{\eta_\psi} &=& \ds{
\frac{A_4(1+A_2)-A_1A_5}{(1+A_2)(1+A_6)-A_3A_5} }
\label{EtaSolution}
\eea
where $A_1,\ldots,A_6$ are defined by writing (\ref{EtaVpBR})
and (\ref{EtaPsi}) in an obvious notation as
\bea
\ds{\eta_\vp} &=& \ds{ A_1-A_2\eta_\vp -A_3\eta_\psi }\nnn
\ds{\eta_\psi} &=& \ds{
A_4-A_5\eta_\vp-A_6\eta_\psi} \; .
\label{DefinitionOfAs}
\eea
We note that in four dimensions the integrals
\be
l_{1,1}^{(FB)4}(0,0)=m_4^{(F)4}(0)=m_{1,2}^{(FB)4}(0,0)=1
\ee
are independent of the particular choice of the infrared cutoff. We
therefore find in the limit of small masses $\kappa$, $\eps$
in both
regimes to leading order in the coupling constants the known
\cite{CGS93-1,BHJ94-1} perturbative one--loop results for both
anomalous dimensions:
\bea
\ds{\eta_\vp} &=& \ds{
\frac{N_c}{8\pi^2}h^2 }\nnn
\ds{\eta_\psi} &=& \ds{
\frac{N}{16\pi^2}h^2 } \; .
\eea
This in turn yields the correct perturbative one--loop result
\be
\prlt h^2 =(2\eta_\psi+\eta_\vp)h^2=
\frac{N+N_c}{8\pi^2}h^4 \; .
\ee
We observe that for large $h^2$ the running of the Yukawa coupling is
very fast due to the term $\sim h^4$ in the flow equation
(\ref{RunningOfh2}). This explains why different large values of
$h^2(k_\vp)$ lead to very similar results for $h_R^2=h^2(0)$. We
demonstrate this in fig. \ref{Fig4} where two different initial values
$h^2(k_\vp)=300$ and $h^2(k_\vp)=10^4$ are compared.
\begin{figure}
\unitlength1.0cm
\begin{picture}(13.,9.)
\put(1.5,5.){\bf $h^2$}
\put(8.,0.5){\bf $k/\,{\rm MeV}$}
\put(-0.8,-11.5){
\epsfysize=22.cm
\epsffile{Fig4.ps}
}
\end{picture}
\caption{\footnotesize Dependence of $h^2$ on $k$,
for the $U_L(2)\times U_R(2)$ model with two different initial
values, $h^2(k_\vp)=300$ (solid line)
and $h^2(k_\vp)=10^4$ (dashed line). We use $k_\vp=630\,{\rm MeV}$,
$\teps_0=0.01$ and $\la_1(k_\vp)=\la_2(k_\vp)=0$.}
\label{Fig4}
\end{figure}
Fig. \ref{Fig5} shows the corresponding evolution of $Z_\psi(k)$.
\begin{figure}
\unitlength1.0cm
\begin{picture}(13.,9.)
\put(1.5,5.){\bf $Z_\psi$}
\put(8.,0.5){\bf $k/\,{\rm MeV}$}
\put(-0.8,-11.5){
\epsfysize=22.cm
\epsffile{Fig5.ps}
}
\end{picture}
\caption{\footnotesize Running of $Z_\psi$ with
$k$, for the $U_L(2)\times U_R(2)$ model with
two different initial values $h^2(k_\vp)=300$ (solid line),
$h^2(k_\vp)=10^4$ (dashed line), and $k_\vp=630\,{\rm MeV}$,
$\teps_0=0.01$, $\la_1(k_\vp)=\la_2(k_\vp)=0$.}
\label{Fig5}
\end{figure}
\sect{The chiral anomaly}
\label{TheO(4)SymmetricSigmaModel}
So far we have considered the somewhat unrealistic limit $\olnu\ra 0$
where the effects of the chiral anomaly are neglected.
In view of the large value of $\nu_R$
(\ref{ValuesForNyR}) as compared
to $k_\vp\simeq630\,{\rm MeV}$, however, it appears that the opposite
limit, $\olnu\ra\infty$, should be closer to reality.
For $N=2$ it is straightforward to take the effects of the chiral
anomaly in this limit into account. To see this we
notice that the complex $({\bf 2},{\bf 2})$ representation $\vp$ of
the global symmetry group $SU_L(2)\times SU_R(2)\simeq O(4)$
decomposes into two irreducible real vector representations of $O(4)$
(cf. the discussion at the end of appendix \ref{ScalarMassSpectrum}):
\be
\vp=\hal\left( \si-i\eta^\prime\right) +\hal\left(
a^k+i\pi^k\right)\tau_k \; .
\ee
By taking $\olnu\ra\infty$ while keeping
$\olm^2-\hal\olnu$ (or
$\olmu^2+\hal\olnu$) fixed,
$m_a$ and $m_{\eta^\prime}$
diverge and the four corresponding mesons decouple.
Hence, we are left with the real
vector representation $\vec{\phi}=(\si,\pi_1,\pi_2,\pi_3)$ of
$O(4)$. Its potential reads in the symmetric regime
\be
U_k=\hal(\olm^2-\hal\olnu)\phi_a\phi^a
+\frac{1}{8}\la_1\left(\phi_a\phi^a\right)^2
\ee
and similarly in the SSB regime.
We therefore end up with the $O(4)$ symmetric linear sigma model
coupled to fermions. The flow equations in the
symmetric regime are given by \cite{Wet93-1,BW93-1}
\bea
\ds{\frac{\prl\eps}{\prl t} }
&=& \ds{
-(2-\eta_\vp)\eps-2v_d\left\{6\la_1
l_1^d(\eps;\eta_\vp)
-2^{\frac{d}{2}}N_c h^2 l_1^{(F)d}(\eta_\psi)\right\} } \nnn
\ds{\frac{\prl\la_1}{\prl t} }
&=& \ds{
(d-4+2\eta_\vp)\la_1+2v_d\left\{
12\la_1^2 l_2^d(\eps;\eta_\vp)
-2^{\frac{d}{2}-1}N_c h^4l_2^{(F)d}(\eta_\psi) \right\} }\nnn
\ds{\frac{\prl h^2}{\prl t}}
&=& \ds{
(d-4+2\eta_\psi+\eta_\vp)h^2
-4v_dh^4l_{1,1}^{(FB)d}(0,\eps;\eta_\psi,\eta_\vp) }\nnn
\eta_\vp &=& \ds{
2^{\frac{d}{2}+2}\frac{v_d}{d}N_c h^2
m_4^{(F)d}(0;\eta_\psi) }\nnn
\eta_\psi &=& \ds{
8\frac{v_d}{d}h^2
m_{1,2}^{(FB)d}(0,\eps;\eta_\psi,\eta_\vp) }
\eea
where $\eps$ is defined here by $\eps=Z_\vp^{-1}k^{-2}(\olm^2-\hal\olnu)$.
For the SSB regime we find
\ben
\ds{\frac{\prl\kappa}{\prl t} }
&=& \ds{
(2-d-\eta_\vp)\kappa+2v_d\left\{
3l_1^d(\eta_\vp)+3l_1^d(2\la_1\kappa;\eta_\vp)
-2^{\frac{d}{2}}N_c \frac{h^2}{\la_1}
l_1^{(F)d}(\hal h^2\kappa;\eta_\psi)\right\} }\nnn
\ds{\frac{\prl\la_1}{\prl t} }
&=& \ds{
(d-4+2\eta_\vp)\la_1 }\nnn
&+& \ds{
2v_d\left\{
3\la_1^2 l_2^d(\eta_\vp)
+9\la_1^2 l_2^d(2\la_1\kappa;\eta_\vp)
-2^{\frac{d}{2}-1}N_c h^4
l_2^{(F)d}(\hal h^2\kappa;\eta_\psi) \right\} }\nnn
\ds{\frac{\prl h^2}{\prl t}}
&=& \ds{
(d-4+2\eta_\psi+\eta_\vp)h^2 }\\[2mm]
&-& \ds{
2v_d h^4 \left\{
3l_{1,1}^{(FB)d}(\hal h^2\kappa,0;\eta_\psi,\eta_\vp)
-l_{1,1}^{(FB)d}(\hal h^2\kappa,2\la_1\kappa;
\eta_\psi,\eta_\vp)\right\} }\nnn
\eta_\vp &=& \ds{
4\frac{v_d}{d}\left\{
4\kappa\la_1^2 m_{2,2}^d(0,2\la_1\kappa;\eta_\vp)
+2^{\frac{d}{2}}N_c h^2
m_4^{(F)d}(\hal h^2\kappa;\eta_\psi) \right\} }\nnn
\eta_\psi &=& \ds{
2\frac{v_d}{d}h^2 \left\{
3m_{1,2}^{(FB)d}(\hal h^2\kappa,0;\eta_\psi,\eta_\vp)
+m_{1,2}^{(FB)d}(\hal h^2\kappa,2\la_1\kappa;
\eta_\psi,\eta_\vp)\right\} }\; . \nonumber
\een
The difference between the results of the $O(4)$ model
($\olnu\ra\infty$) and the $U_L(2)\times U_R(2)$ model ($\olnu=0$)
can be taken as a measure for the uncertainty due to the rough
treatment of the anomaly in the present work. Here the $U_L(2)\times
U_R(2)$ model exhibits the effects of additional scalar degrees of
freedom beyond the pions (and the $\si$--mode). Since the additional
modes are relatively heavy, the $O(4)$ model should be closer to
reality. The best model with $N=3$, where the strange quark mass and
the chiral anomaly are properly taken into account, is expected to
deviate from the $O(4)$ model in the same direction as the
$U_L(2)\times U_R(2)$ model.
Since there are three light
quark flavors in nature with masses
smaller than $k_\vp$ one might naively expect the case $N=3$ to
correspond to the
most realistic description of the real world.
However, we have
neglected quark masses and in particular the strange quark mass in this
work. In the $SU_L(3)\times SU_R(3)\times U_V(1)$ model the four
$K$--mesons will therefore appear as massless
Goldstone degrees of freedom which will unnaturally
drive the evolution of all parameters even at scales much lower than
their physical masses of approximately $500\,{\rm MeV}$. The same holds, of
course, for the three pions. The effects of their physical masses
$m_\pi\simeq140\,{\rm MeV}$ can, however, be mimicked by stopping the
running for $k_f=m_\pi$. We therefore expect the case $N=2$ to yield
more realistic results than $N=3$ as long as the strange quark mass is
neglected. In addition, we note that for $N=3$ the scalar self coupling
$\la_1$ turns negative for positive but small values of $\olm^2$. This
happens despite the fact that $\la_1$ has acquired first a large
positive value due to the strong initial Yukawa
coupling. The cause is a large value of $\la_2$ which
can drive $\la_1$ negative when
the scalar loop contributions to the running of $\la_1$ become
numerically important around the scale $k_s$. We interpret this
quartic instability of the
truncation (\ref{PotentialSymRegime}) of the effective potential as a
signal for a first order phase transition in the mass parameter
even within the $U_L(3)\times U_R(3)$ model without chiral anomaly. A
proper treatment of the case $N=3$ therefore requires a more general
truncation of the effective potential \cite{BTW95-1}. We will leave
this problem for future
work. However, we would like to point out that the
inclusion of the cubic (for $N=3$)
$\xi$--term into the potential will change the phase transition to
first order anyway. One expects that the vacuum
expectation value of $\vp$ jumps discontinuously to a finite value
already for a large value of $\eps$ such that the scalar
contributions to
the evolution of $\la_1$ never become strong enough to turn it
negative.
The difference between the $O(4)$ model and the $U_L(2)\times U_R(2)$
model is exemplified in fig. \ref{Fig6}.
\begin{figure}
\unitlength1.0cm
\begin{picture}(13.,9.)
\put(1.,5.){\bf $\ds{\frac{f_\pi}{\,{\rm MeV}}}$}
\put(8.,0.5){\bf $k_\vp$}
\put(8.,4.){\bf $O(4)$}
\put(7.,6.5){\bf $U_L(2)\times U_R(2)$}
\put(-0.8,-11.5){
\epsfysize=22.cm
\epsffile{Fig6.ps}
}
\end{picture}
\caption{\footnotesize The pion decay constant
$f_\pi$ as a function
of $k_\vp$ for $\la_1(k_\vp)=\la_2(k_\vp)=0$,
$h^2(k_\vp)=300$ and $\teps_0=0.01$.}
\label{Fig6}
\end{figure}
There we show the ``prediction'' of $f_\pi$ as a function of the
initial scale $k_\vp$. The difference between the two models is not
very large. We observe for not too small values of $k_\vp$ a linear
behavior $f_\pi\sim k_\vp$. For fixed initial values of the
dimensionless
parameters like $\teps=\olm^2Z_\psi^2/k_\vp^2$, $h^2$ etc., this
proportionality follows on pure dimensional grounds if no other mass
scale is present. The bending of the curves for small $k_\vp$ is
therefore purely a consequence of the additional infrared cutoff
$k_f=m_\pi$ which obviously plays a negligible role for a realistic
size of $k_\vp$. Looking at fig. \ref{Fig6} the reader may prematurely
conclude that the predictive power for $f_\pi$ is severely limited by
the arbitrariness of the choice of the scale $k_\vp$ from where on the
meson description is used. We should emphasize that for a full
treatment along the lines of \cite{EW94-1,Wet95-1} this is
actually not the case: If one lowers the transition scale $k_\vp$ more
fluctuations are included in the momentum range $q^2>k_\vp^2$ where a
quark--gluon picture is used. As a result, the pole like structure
in the effective four--quark interaction becomes stronger and $\olm^2$
therefore decreases. This is the same effect as found in the
quark--meson picture used for the fluctuations with $q^2<k_\vp^2$: the
mass term $\olm^2$ decreases with smaller $k$ as a result of the
Yukawa coupling to the quarks. In the limit where the pole like
structure dominates the evolution of $\tilde{G}(0)=\frac{1}{2\olm^2}$
in the quark gluon picture the running of $\olm^2$ is identical in
both pictures for $k$ larger or smaller than $k_\vp$. The initial
value $\olm^2(k_\vp)$ as a function of
$k_\vp$ follows therefore the same renormalization group trajectory as
given by the flow equation (\ref{BosEvolMSymReg}). In this ideal case
the choice of the transition scale $k_\vp$ does not affect the
results, since the initial conditions move on trajectories of constant
physics. In practice, this ideal scenario will often not be fully
realized, since different types of fluctuations are included in the
quark--gluon and the quark--meson description. The dependence of the
results on $k_\vp$ can then be used as a quantitative check of the
reliability of the employed truncations for the effective action.
\sect{Infrared stability and predictive power for $f_\pi$}
\label{Results}
Comparing the results for $f_\pi$ from fig. \ref{Fig6} with the
experimental value $f_\pi=93\,{\rm MeV}$ we find a surprisingly good
agreement for $k_\vp=630\,{\rm MeV}$ as infered from ref. \cite{EW94-1}. The
question arises to what extent this result depends on the particular
choice of initial values at the scale $k_\vp$. In principle, the
values of the parameters of the quark--meson system at the scale
$k_\vp$ can be computed from QCD \cite{EW94-1,Wet95-1}. In practice,
however, many quantities will not be available with high accuracy,
since one has to deal with a problem involving strong interactions. If
$f_\pi$ would depend very sensitively on such quantities, a
computation of $f_\pi$ with satisfactory precision
would be extremely difficult. We will argue in this section that
for small enough $Z_\vp$ the opposite situation is realized. In this
event the prediction for $f_\pi$ turns out to be almost independent
of the initial values of many couplings. The reason is that a small
$Z_\vp$ corresponds to a strong Yukawa interaction.
The large value of $h$
induces then a very fast running of almost all couplings towards
values which are determined by an infrared attractive behavior. More
precisely, the ratios $\la_1/h^2$ and $\la_2/h^2$ are determined by
infrared fixed points of the type first found in the electroweak
standard model \cite{Wet81-1}. This explains the insensitivity with respect
to the initial values as demonstrated in fig. \ref{Fig2}. The Yukawa
coupling itself is also strongly renormalized and predicted to be in
the vicinity of the upper bound of the relevant infrared
interval\footnote{The infrared fixed point for $h$ is $h_*=0$ if no
infrared cutoff is present. Due to a finite amount of running from
$k_\vp$ to $k_s$ this translates into an infrared interval.}
\cite{Hill81-1}. Here the upper bound of the infrared interval is
essentially determined by the scale $k_s$ where spontaneous symmetry
breaking sets in and the effective quark masses constitute an
infrared cutoff. This can clearly be observed in fig. \ref{Fig4} where
also the insensitivity with respect to the initial value $h(k_\vp)$
becomes apparent. The only relevant parameter will turn out to be the
ratio
\be
\teps_0=\frac{\eps(k_\vp)}{h^2(k_\vp)}=
\frac{\olm^2(k_\vp)Z_\psi^2(k_\vp)}{k_\vp^2} \; .
\ee
This value determines $k_s$ and $f_\pi$ as well as all other
couplings at the scale $k_f$.
For small enough $Z_\vp$ one starts in a regime where
$\eps=m^2/k^2=\olm^2Z_\vp^{-1}k^{-2}$ is large.
For large $\eps$ the scalar fluctuations
are suppressed by inverse powers of $\eps$ appearing in the threshold
functions. Then the scalar fluctuations can be neglected and only
quark fluctuations drive the flow of the couplings. This is the
approximation used in ref. \cite{EW94-1} which remains valid as long
as $\eps\gg1$. The Yukawa coupling $\olh$ is normalized according to
(\ref{InitialValues}) as
$\olh(0)\equiv\olh_{k_\vp}(0,0)=1$. Consequently, the initial value
for the renormalized Yukawa coupling is
\be
h_0^2\equiv
h^2(k_\vp)=\frac{1}{Z_\vp(k_\vp)Z_\psi^2(k_\vp)}\; .
\ee
We use here a normalization of the fermion kinetic term such that
$Z_\psi(k_\vp)=1$. For small $Z_\vp(k_\vp)$ we therefore start with a
strong Yukawa coupling. In the limit $\eps\gg1$ the flow equations
simplify considerably.
If we define $\teps\equiv\eps/h^2$ and
$\tla_i\equiv\la_i/h^2$ we find
\bea
\ds{\frac{\prl\teps}{\prl t} } &=& \ds{
-(d-2)\teps+2^{\frac{d}{2}+1}v_dN_c }\nnn
\ds{\frac{\prl\tla_1}{\prl t} } &=& \ds{
2^{\frac{d}{2}+1}v_dN_ch^2
\left[\frac{2}{d}\tla_1-\frac{1}{N}\right] }\nnn
\ds{\frac{\prl\tla_2}{\prl t} } &=& \ds{
2^{\frac{d}{2}+2}v_dN_ch^2
\left[\frac{1}{d}\tla_2-\frac{1}{N}\right] }\nnn
\ds{\frac{\prl h^2}{\prl t} } &=& \ds{
(d-4)h^2+2^{\frac{d}{2}+2}\frac{v_d}{d}N_ch^4 }\; .
\label{ReducedSystem}
\eea
In the following we will specialize to the case $d=4$.\footnote{The
system (\ref{ReducedSystem}) remains solvable for general $d$.}
As a first observation we notice that the $\tla_1$--$\tla_2$ system
exhibits an infrared fixed point given by
\be
\tla_{1*}=\hal\tla_{2*}=\frac{2}{N}\; .
\label{FixedPoint}
\ee
This fixed point corresponds exactly to the large--$N_c$ estimate of
\cite{BHJ94-1}. The explicit solution of the differential equations
(\ref{ReducedSystem}) reads
\bea
\ds{\teps(t) } &=& \ds{
4v_4N_c\left[ 1-e^{-2t}\right]+\teps_0e^{-2t} }\nnn
\ds{\tla_1(t)} &=& \ds{
\frac{\frac{\la_{10}}{h_0^2}-8\frac{N_c}{N}v_4h_0^2 t}
{1-4N_cv_4h_0^2t} }\nnn
\ds{\tla_2(t)} &=& \ds{
\frac{\frac{\la_{20}}{h_0^2}-16\frac{N_c}{N}v_4h_0^2 t}
{1-4N_cv_4h_0^2t} }\nnn
h^2(t) &=& \ds{
\frac{h_0^2}{1-4N_cv_4h_0^2t} }
\eea
with $\teps_0\equiv\teps(t=0)$, etc.
The system crosses into the SSB regime when $\teps(t_s)=0$,
corresponding to a scale
\be
t_s=\hal\ln\left[1-\frac{\teps_0}{4N_cv_4}\right]\; ,\;\;\;
k_s^2=\left(1-\frac{8\pi^2}{3}\teps_0\right)k_\vp^2 \; .
\label{ttilde}
\ee
Here we have used $v_4=(32\pi^2)^{-1}$ and $N_c=3$ in the last
expression. Around the scale $k_s$
our approximation (large $\eps$) breaks down.
Nevertheless, it becomes apparent already at this stage that values of
$\teps_0$ substantially larger than $0.04$ are incompatible with
chiral symmetry breaking, since $\teps$ would remain positive for all
$k$ in this case.
For large $\teps_0$ the effect of the quark fluctuations is
simply not strong enough to turn the scalar mass term negative.
We notice that $\teps_0\ll1$
for $\olm(k_\vp)\sim\Oc(100\,{\rm MeV})$ and $Z_\psi(k_\vp)\lta 1$.
If we furthermore assume $-4N_cv_4h_0^2t_s\gg 1$ or
\be
\teps_0\gg 4N_cv_4\left[
1-\exp\{-\frac{1}{2N_cv_4h_0^2}\}\right] \; ,
\label{tepsCondition}
\ee
we find for not too large $\la_{10}/h_0^2$, $\la_{20}/h_0^2$
\be
\tla_1(t_s)\simeq\tla_{1*}\; ,\;\;\;
\tla_2(t_s)\simeq\tla_{2*}\; ,\;\;\;
h^2(t_s)\simeq\left[2N_cv_4\ln\frac{4N_cv_4}
{4N_cv_4-\teps_0}\right]^{-1} \; .
\label{FixedPointH2}
\ee
This result may be interpreted as follows:
Even though (\ref{ttilde}) should only give an approximate
estimate for the
entering point into the SSB regime this is sufficient to imply
that $\tla_1$ and $\tla_2$ approximately reach their fixed
point long before $\eps$ goes through zero provided
(\ref{tepsCondition}) is
fulfilled. Furthermore $h^2$ becomes
asymptotically independent of $h_0^2$ and is only a function of
$\teps_0$. We therefore conclude that for small $Z_\vp(k_\vp)$
the system is governed by an
infrared fixed point of $\tla_1$ and $\tla_2$ in the symmetric regime
and looses almost all its
information on the initial values at $k=k_\vp$. Infrared quantities
of the SSB regime like $f_\pi$ or meson masses will therefore
merely depend on $\teps_0$ for a given $k_\vp$.
It is straightforward to see that the same analysis holds for the
$O(4)$ model discussed in section \ref{TheO(4)SymmetricSigmaModel}.
The approximate flow equations for $\teps$, $\tla_1$ and $h^2$ are the
same as (\ref{ReducedSystem}). The difference between the
$U_L(2)\times U_R(2)$ and the $O(4)$ model arises principally from
the behavior of the running couplings around the scale $k_s$.
For larger values of $Z_\vp(k_\vp)$ or smaller values of $h(k_\vp)$
the attraction of the infrared fixed points becomes weaker. As a
consequence, the dependence of $f_\pi$ on the initial values of the
couplings becomes more important as demonstrated in fig. \ref{Fig7}.
\begin{figure}
\unitlength1.0cm
\begin{picture}(13.,9.)
\put(1.,5.){\bf $\ds{\frac{f_\pi}{\,{\rm MeV}}}$}
\put(8.,0.5){\bf $h_0$}
\put(-0.8,-11.5){
\epsfysize=22.cm
\epsffile{Fig7.ps}
}
\end{picture}
\caption{\footnotesize The
$O(4)$ model pion decay constant $f_\pi$
as a function of $h_0\equiv h(k_\vp)$ for $k_\vp=630\,{\rm MeV}$,
$\teps_0=0.02$ and
$\la_1(k_\vp)=0$ (solid line) as well as $\la_1(k_\vp)=h_0^2$
(dashed line).}
\label{Fig7}
\end{figure}
We conclude that for $h(k_\vp)$ substantially smaller than ten it will
become more and more difficult to obtain an accurate prediction for
$f_\pi$. On the other hand, fig. \ref{Fig7} clearly shows the
approximate independence of $f_\pi$ on $h^2(k_\vp)$ or $\la_1(k_\vp)$
if $h^2(k_\vp)$ exceeds 300.
An additional aspect of strong Yukawa couplings concerns the error in
$f_\pi$ due to the truncations of the quark--meson effective
action. The effects of truncations in the scalar sector are diminished
by the fact that scalar fluctuations are subdominant in the region of
very large Yukawa couplings. A similar argument justifies the
approximation of
neglecting the terms proportional to the anomalous dimensions in
the threshold functions. For the $m$--type functions this
approximation is valid, since
$A_2,A_3,A_5,A_6\ll 1$ in (\ref{EtaSolution}),
(\ref{DefinitionOfAs}). For $l$-type functions one might be
worried that $\eta_\vp$ and $\eta_\psi$ are large for
the initial part of the running in the symmetric regime due to large
values of $h^2$. However, because of the
large values of $\eps$ in this range of scales the contributions from
the $l$--type functions can be neglected altogether. For smaller
values of $\eps$, i.e. for scales closer to $k_s$, the anomalous
dimensions are expected to be already small (as indicated by
figs. (\ref{Fig3}) and (\ref{Fig4})).
\sect{Computation of $f_\pi$ for strong Yukawa coupling}
\label{Discussion}
In the last section we have shown that for large enough Yukawa
couplings, say $h^2(k_\vp)>200$, the value of $f_\pi$ only depends on
the parameter $\teps(k_\vp)\equiv\teps_0$ for given $k_\vp$. We
demonstrate this quantitatively in fig. \ref{Fig8}
\begin{figure}
\unitlength1.0cm
\begin{picture}(13.,9.)
\put(1.,5.){\bf $\ds{\frac{f_\pi}{\,{\rm MeV}}}$}
\put(8.,0.5){\bf $\tilde{\epsilon}_0$}
\put(10.5,5.5){\bf $O(4)$}
\put(5.5,2.5){\bf $U_L(2)\times U_R(2)$}
\put(-0.8,-11.5){
\epsfysize=22.cm
\epsffile{Fig8.ps}
}
\end{picture}
\caption{\footnotesize The pion decay constant
$f_\pi$ as a function
of $\teps_0$ for $k_\vp=630\,{\rm MeV}$, $\la_1(k_\vp)=\la_2(k_\vp)=0$ and
$h^2(k_\vp)=300$ (solid line) as well as $h^2(k_\vp)=10^4$ (dashed line).}
\label{Fig8}
\end{figure}
where we plot $f_\pi$ as a function of $\teps_0$, for both the
$O(4)$ and the $U_L(2)\times U_R(2)$ model as well as for two
different large initial values of $h^2$. The ``prediction'' for
$f_\pi$ is rather insensitive to $h^2(k_\vp)$ for moderate values of
$\teps_0$.\footnote{The decrease of $f_\pi$ for $h^2(k_\vp)=10^4$
observed for unnaturally small values of $\teps_0$ results
simply from the fact that the system enters almost immediately into
the SSB regime, having no ``time'' for $Z_\vp$ to grow much beyond
$Z_\vp(k_\vp)$.}
For the $O(4)$ model we also observe an extended plateau where
$f_\pi$ is not very sensitive to $\teps_0$ either.
For this plateau the value
of $f_\pi$ comes out between $80$ and $100\,{\rm MeV}$ which fits very well
with the experimental value of $93\,{\rm MeV}$. Also the renormalized Yukawa
coupling $h_R=h(k_f)$, or, equivalently, the constituent quark mass
$m_q=\hal h_Rf_\pi$ depends essentially only on $\teps_0$. We show
this in fig. \ref{Fig9},
\begin{figure}
\unitlength1.0cm
\begin{picture}(13.,9.)
\put(1.,5.){\bf $h_R$}
\put(8.,0.5){\bf $\tilde{\epsilon}_0$}
\put(8.,3.8){\bf $O(4)$}
\put(2.7,2.){\bf $U_L(2)\times U_R(2)$}
\put(-0.8,-11.5){
\epsfysize=22.cm
\epsffile{Fig9.ps}
}
\end{picture}
\caption{\footnotesize The renormalized Yukawa
coupling $h_R$ as a function of $\teps_0$ for $k_\vp=630\,{\rm MeV}$,
$\la_1(k_\vp)=\la_2(k_\vp)=0$ and $h^2(k_\vp)=300$ (solid lines),
$h^2(k_\vp)=10^4$ (dashed lines).}
\label{Fig9}
\end{figure}
again for two different large values of $h^2(k_\vp)$. Since both $h_R$
and $f_\pi$ are functions of only one parameter $\teps_0$,
there arises a
quantitative relation between those two quantities. Consider first the
$O(4)$ model: For a constituent quark mass of $300\,{\rm MeV}$ or
$h_R\simeq6.5$ we read off from fig. \ref{Fig9} that
$\teps_0\simeq0.02$. Inserting this into the plot of fig. \ref{Fig8}
one obtains
\be
f_\pi\simeq92\,{\rm MeV} \; .
\label{ResultFpi}
\ee
This value was obtained for $h^2(k_\vp)=300$ and
$\la_1(k_\vp)=\la_2(k_\vp)=0$, but it turns out to be not very
different for $h^2(k_\vp)=10^4$ or different initial values of
$\la_1$ and $\la_2$. A similar procedure gives for the $U_L(2)\times
U_R(2)$ model a value $\teps_0\simeq0.008$ and in turn
$f_\pi\simeq126\,{\rm MeV}$. Repeating this procedure for $k_\vp=700\,{\rm MeV}$ we
can infer from table \ref{tab1}
\begin{table}
\begin{center}
\begin{tabular}{|c|c|c||c|c|c||c|c|c|} \hline
\multicolumn{2}{|c|}{ } & $\frac{m_q}{\,{\rm MeV}}$\rule[-3mm]{0mm}{8mm} &
\multicolumn{3}{c||}{300} &
\multicolumn{3}{c|}{350} \\ \cline{3-9}
\multicolumn{2}{|c|}{ } &
$h^2(k_\vp)$ & 64 & 300 & $10^4$ &
106 & 300 & $10^4$ \\ \hline
& $\frac{k_\vp}{\,{\rm MeV}}\rule[-3mm]{0mm}{8mm} $
& & & & & & & \\ \hline\hline
$O(4)$ & $630$ & & 143.9 & 91.7 & 83.5 & 124.6 & 99.9 & 91.0 \\ \cline{2-9}
& $700$ & & 159.7 & 101.5 & 92.5
& 138.2 & 110.7 & 100.7\\ \hline\hline
$U_L(2)\times U_R(2)$
& $630$ & & - & 125.7 & 138.3 & - & - & 138.9 \\ \cline{2-9}
& $700$ & & - & 139.6 & 153.5 & - & - & 154.2 \\ \hline
\end{tabular}
\caption{\footnotesize $f_\pi$ in $\,{\rm MeV}$ for various initial
conditions at
$k_\vp$ and two values of the constituent quark mass.}
\label{tab1}
\end{center}
\end{table}
a guess of the uncertainty in $f_\pi$ that can be expected within the
quark--meson model for large Yukawa couplings, as
represented in the table by the values $h^2(k_\vp)=300$ and $10^4$. On
the other hand, a given
value of $m_q$ also implies a minimal value $h_{\rm min}(k_\vp)$
such that the evolution of $h(k)$ can reach the value
$h_R=2m_q/f_\pi$ at all.
We assume here that the result for $\olm^2(k_\vp)$ of ref. \cite{EW94-1}
should not be off by more than a factor of four. We can therefore
conclude that for $k_\vp$ in the range $(630-700)\,{\rm MeV}$ there exists a
lower bound on $\teps_0$, i.e. $\teps_0\gta0.01$.
This in turn amounts for the $O(4)$ model to $h_{\rm min}(k_\vp)\simeq
6.2,8.0,10.3$ for $m_q=250,300,350\,{\rm MeV}$,
respectively. The corresponding values in
table \ref{tab1} give an estimate for
the maximal deviation of $f_\pi$ from its value for strong Yukawa
coupling. For the $U_L(2)\times
U_R(2)$ model a value $h^2(k_\vp)=300$ is already near the lower limit
of what is compatible with realistic values for $m_q$ and $\teps_0$
(cf. figure \ref{Fig9}).
We can also invert these relations and look for the optimum value of
$\teps_0$ and $h^2(k_\vp)$ for fixed $f_\pi=93\,{\rm MeV}$ and $m_q=300\,{\rm MeV}$.
One obtains for the $O(4)$ model and $k_\vp=630\,{\rm MeV}$
\bea
\ds{\teps_0} &\simeq& \ds{0.02}\nnn
\ds{h^2(k_\vp)} &\simeq& \ds{280} \; .
\label{OptimalValues}
\eea
Within the simple QCD inspired model of ref. \cite{EW94-1} the
transition scale was found as $k_\vp=630\,{\rm MeV}$ and the mass term at
$k_\vp$ gave $\teps_0=0.036$. Comparing with fig. \ref{Fig8} we find
that this value of $\teps_0$ is actually too large to induce
spontaneous symmetry breaking if the meson fluctuations are taken into
account. Given the simplified character of the model considered in
\cite{EW94-1}, however, we find the agreement with the order of
magnitude of (\ref{OptimalValues}) very encouraging. On the other
hand, the estimate of $Z_\vp(k_\vp)\simeq0.85$
appears to be very inaccurate for
the model of \cite{EW94-1} and far away from the small values of
$Z_\vp(k_\vp)$ for which the values (\ref{ResultFpi}),
(\ref{OptimalValues}) were obtained.
We finally compute the chiral condensate from (\ref{Codensate}) with
$\olh(k_\vp)=1$ as
\be
\VEV{\olpsi\psi}_0=\teps_0 Z_\vp^{-\hal}(m_\pi)
Z_\psi^{-1}(k_\vp)f_\pi k_\vp^2 \; .
\ee
Extracting $Z_\vp(m_\pi)$ from section \ref{ScalarAnomalousDimension}
(fig. \ref{Fig3}) we can use this value for a check of the
self--consistency of our scenario. We observe that three different
small quantities, $\teps_0$, $Z_\vp(m_\pi)$ and $f_\pi/k_\vp$ enter
here, and it is by far not trivial that a reasonable value of the
chiral condensate can be obtained. We normalize the condensate at
$k_\vp$ with $Z_\psi(k_\vp)=1$. The result for $\VEV{\olpsi\psi}_0$ as
a function of $\teps_0$ is plotted in fig. \ref{Fig10}.
\begin{figure}
\unitlength1.0cm
\begin{picture}(13.,9.)
\put(0.3,5.){\bf $\ds{\frac{\abs{\VEV{\olpsi\psi}_0}^{\frac{1}{3}}}{\,{\rm MeV}}}$}
\put(8.,0.5){\bf $\teps_0$}
\put(5.7,3.){\bf $U_L(2)\times U_R(2)$}
\put(12.,3.){\bf $O(4)$}
\put(-0.8,-11.5){
\epsfysize=22.cm
\epsffile{Fig10.ps}
}
\end{picture}
\caption{\footnotesize The quark condensate as
a function of $\teps_0$ for $k_\vp=630\,{\rm MeV}$,
$\la_1(k_\vp)=\la_2(k_\vp)=0$ and $h^2(k_\vp)=300$ (solid line) as
well as $h^2(k_\vp)=10^4$ (dashed line).}
\label{Fig10}
\end{figure}
For $k_\vp=630\,{\rm MeV}$, $\teps_0=0.02$ and $h^2(k_\vp)=300$
we obtain in the $O(4)$ model
\be
\abs{\VEV{\olpsi\psi}_0}^{\frac{1}{3}} \simeq 163\,{\rm MeV} \; .
\ee
We may compare this value with a typical value infered from chiral
perturbation theory \cite{GL82-1}
\be
\abs{\VEV{\olpsi\psi}_{\rm CPT}}^{\frac{1}{3}}(1\,{\rm GeV})
\simeq (225\pm25)\,{\rm MeV} \; .
\ee
This value can be scaled down to $k_\vp=630\,{\rm MeV}$ by
exploiting $k\frac{\prl}{\prl k}\VEV{\olpsi\psi}_{\rm
CPT}(k)m_q(k)=0$.
We use here the
three--loop $\beta$--function of QCD ($\ol{\rm MS}$ scheme) for the
running quark mass \cite{JM95-1}
\be
\frac{k}{m_q(k)}\frac{\prl}{\prl k}m_q(k)=
-\left[\gm_1\frac{\alpha_s}{\pi}
+\gm_2\left(\frac{\alpha_s}{\pi}\right)^2
+\gm_3\left(\frac{\alpha_s}{\pi}\right)^3\right]
\ee
with
\bea
\ds{\gm_1} &=& \ds{\frac{3}{2}C_F\; ;\;\;\;
\gm_2=\frac{C_F}{48}\left[
97N_c+9C_F-10N\right]}\nnn
\ds{\gm_3} &=& \ds{
\frac{C_F}{32}\left[
\frac{11413}{108}N_c^2-\frac{129}{4}N_cC_F
-\left(\frac{278}{27}+24\zeta(3)\right)NN_c
+\frac{129}{2}C_F^2
\right. }\nnn
&-& \ds{\left.
\left(23-24\zeta(3)\right)NC_F-\frac{35}{27}N^2\right] }
\eea
and $C_F=\frac{N_c^2-1}{2N_c}$.
This yields
\be
\frac{m_q(k_\vp)}{m_q(1\,{\rm GeV})}\simeq 1.72
\label{BetaFunctionForMq}
\ee
for $k_\vp=630\,{\rm MeV}$ and therefore
\be
\abs{\VEV{\olpsi\psi}_{\rm CPT}}^{\frac{1}{3}}(k_\vp)\simeq
(188\pm21)\,{\rm MeV} \; .
\ee
This is in satisfactory agreement with our estimate.
\sect{Conclusions}
\label{Conclusions}
We have presented here an effective quark--meson model which is
supposed to describe the strong interaction dynamics between
(constituent) quarks, scalar and pseudo--scalar mesons at momentum
scales smaller than $k_\vp\simeq630\,{\rm MeV}$. The effective average action
$\Gm_k$ for this model depends on a scale $k$ which plays the role of
an infrared cutoff. The scale dependence of the average action obeys
an exact nonperturbative evolution equation. Using a truncation for
the general form of $\Gm_k$ this results in approximate flow equations
for the meson potential and kinetic term as well as the quark kinetic
term and the quark--meson Yukawa coupling $h$. The initial values of
these parameters at the scale $k_\vp$ can, in principle, be computed
\cite{EW94-1,Wet95-1} from evolution equations for QCD which are valid
for scales larger than $k_\vp$. Following the flow equations from
$k_\vp$ to lower scales $k$ one recovers for $k=0$ the effective
action, i.e. the generating functional for the $1PI$ Green functions
for the mesons. In particular, the standard nonlinear $\si$--model
framework of chiral perturbation theory obtains if the expectation
value of the meson field is kept fixed at a nonvanishing vacuum
expectation value.
We have solved the flow equations numerically and observe how the
minimum of the meson potential turns from $\si_0=0$ at high scales to
a nonzero value $\abs{\si_0}>0$ for small $k$. The nonvanishing
expectation value $\si_0$ indicates spontaneous chiral symmetry
breaking. Our numerical solution allows us to compute the mass scales
characteristic for chiral symmetry breaking, i.e. the pion decay
constant $f_\pi$, the constituent quark mass $m_q$ and the chiral
condensate $\VEV{\olpsi\psi}_0$. These quantities are computed as
functions of the initial values for the scalar mass term
$\olm^2(k_\vp)$ and wave function renormalization $Z_\vp(k_\vp)$. The
latter is related to the value of the renormalized Yukawa coupling at
$k_\vp$ by $h^2(k_\vp)=Z_\vp^{-1}(k_\vp)$ once we normalize the
quark wave function with
$Z_\psi(k_\vp)=1$.
The large ratio between the constituent quark mass $m_q$ and $f_\pi$
necessitates a large value of the renormalized Yukawa coupling
$h_R^2=h^2(k=0)$ according to
\be
h_R^2=\frac{4m_q^2}{f_\pi^2}\simeq 50 \; .
\ee
Since $h^2(k)$ decreases rapidly with decreasing $k$ we conclude that
at the scale $k_\vp$ we have to deal with a strong Yukawa
coupling. Our investigation therefore concentrates on large initial
values $h^2(k_\vp)\gta200$. The most crucial observation of the
present work is that strong Yukawa couplings imply a very fast running
of almost all couplings towards values determined by infrared fixed
points or corresponding infrared intervals. The quark--meson model for
small scales $k$ therefore looses its memory of the exact initial
values of most of the couplings at the scale $k_\vp$. In consequence,
strong Yukawa couplings greatly enhance the chances for a reliable
estimate of $f_\pi$! We find that $f_\pi$ as well as $m_q$ and
$\VEV{\olpsi\psi}_0$ depend only on one ``relevant'' initial value,
namely $\teps_0\sim\olm^2(k_\vp)/k_\vp^2$. The value of $h_R$ may then
be used to fix the required initial value of $\teps_0$ and therefore
to determine $f_\pi$ and $\VEV{\olpsi\psi}_0$. For the simplified
$O(4)$ model discussed in the present paper we obtain
$f_\pi\simeq92\,{\rm MeV}$ and
$\abs{\VEV{\olpsi\psi}_0}^{\frac{1}{3}}\simeq163\,{\rm MeV}$. The comparison
with the experimental result $f_\pi=93\,{\rm MeV}$ and the estimate from
chiral perturbation theory
$\VEV{\olpsi\psi}_{\rm CPT}(k_\vp)=(188\pm21)\,{\rm MeV}$ is very encouraging!
Despite the success of the simplified computation of the present work
our results have partly the character of a feasibility study. Several
steps should considerably improve the accuracy of our computation of
$f_\pi$. First, one should consider the case $N=3$ with a realistic
value of $\olnu$ for the chiral anomaly. This investigation is already
prepared in the appendices of the present paper where the central
results are exhibited for arbitrary $N$ and $\olnu$. Second, the
explicit chiral symmetry breaking due to current quark masses, in
particular the strange quark mass, should be included. Third, the
dependence of the Yukawa coupling and the quark kinetic term on the
quark momentum can be incorporated. This will provide information
on the quark wave function inside the mesons \cite{EW94-1}.
Furthermore, we have not included residual gluon effects for $k<k_\vp$
in the present
work. This would not be necessary if the gluons could completely be
integrated out for the determination of the effective quark--meson
action at the scale $k_\vp$. Since the latter seems to be a quite
difficult task, one may rather use the proposal of ref. \cite{Wet95-1}
and integrate out only the gluons with momenta $q^2>k_\vp^2$. The
remaining gluon fluctuations with momenta $q^2<k_\vp^2$ give then
additional contributions to the flow equations in the quark--meson
model. In the context of a computation of $f_\pi$
the most important effect seems to be a residual gluonic contribution to
$\eta_\psi$ and the effective quark meson vertex and therefore
to the running of the Yukawa coupling. This can be taken into account
by generalizing the average action to include gauge bosons
\cite{RW93-1} and using the formalism of \cite{Wet95-1} to integrate
them out consecutively. A great part of the gluon fluctuation effects
is already included in the contributions from the effective
four--fermion interactions for $k>k_\vp$ or from the quark--meson
interactions considered in the present paper for $k<k_\vp$. The
corrections from residual gluon fluctuations can be found from the
explicit formulae in \cite{Wet95-1}. Since
in addition the confinement scale is
below the constituent quark mass one may
hope that the complicated effects of
gluon condensation do not have a very important influence on
the determination of $f_\pi$.
Finally, a computation of $f_\pi$ within QCD, i.e. as a function of
$\alpha_s(1\,{\rm GeV})$ and the quark masses, necessitates a
reliable computation of $\teps_0$ and $h^2(k_\vp)$
within the QCD framework for the effective average action
proposed in \cite{Wet95-1}. Thus, the road to an analytical computation
of $f_\pi(\alpha_s(1\,{\rm GeV}))$ is still long. Our results should
encourage to go it.
\newpage
\noindent
{\LARGE Appendices}
|
1,314,259,993,610 | arxiv | \section*{Introduction}
\noindent
The ground field $\bbk$ is algebraically closed and of characteristic $0$.
In 1976, Vinberg et al{.} classified the irreducible representations of simple algebraic groups with
polynomial rings of invariants~\cite{KPV76}. Such representations are sometimes called {\it coregular}.
The most important class of coregular representations of reductive groups is provided by the $\theta$-groups introduced and studied in depth by Vinberg, see~\cite{V76}.
Since then
the classification of coregular representations of semisimple groups has attracted much attention. The
{\bf reducible} coregular representations of {\bf simple} groups have been classified independently by
Schwarz~\cite{gerry1} and Adamovich--Golovina~\cite{ag79}, while the {\bf irreducible} coregular representations of {\bf semisimple} groups are classified by Littelmann~\cite{litt}.
For several decades,
only rings of invariants of representations of {\bf reductive} groups were considered. However, invariants
of non-reductive groups are also very important in Representation Theory.
Let $S$ be an algebraic group
with $\es=\Lie S$. The invariants of $S$ in the symmetric algebra $\gS (\es)$ of $\es$
(=\,{\it symmetric invariants\/} of $\es$ or $S$) help us to understand
the coadjoint action $(S:\es^*)$ and in particular, coadjoint orbits, as well as representation theory of $S$.
Several classes of non-reductive groups $S$ such that $\gS (\es)^S$ is a polynomial ring have been
found recently, see e.g. \cite{J,trio,coadj,Y}. A quest for this type of groups continues.
Hopefully, one can find interesting properties of $S$ and its representations under the assumption that
the ring $\gS (\es)^S$ is polynomial.
A natural class of non-reductive groups, which is still tractable, is given by a semi-direct product
construction, see Section~\ref{sect:kos-th} for details. In~\cite{Y}, the following problem has been proposed:
to classify all representations $V$ of simple algebraic groups $G$ such that
the ring of symmetric invariants of the semi-direct product $\q=\g\ltimes V$ is polynomial (in other words,
the coadjoint representation of $\q$ is coregular). It is easily seen that if $\q$ has this property,
then $\bbk[V^*]^G$ is a polynomial ring, too. Therefore, the suitable representations $(G,V)$ have to
be extracted from the lists of \cite{gerry1,ag79}.
Some natural representations of $G=SL_n$ are studied in~\cite{Y}. Those considerations imply that the
$SL_n$-case is very difficult. For this reason, we take here the other end and classify such representations
$(G,V)$ for the {\it exceptional} algebraic groups $G$. In a forthcoming article, we provide such a
classification for the representations of the orthogonal and symplectic groups. To a great extent, our classification results rely on the theory developed by the second author in \cite{Y16}.
{\sl \un{ Notation}.}
Let $S$ act on an irreducible affine variety $X$. Then $\bbk[X]^S$
is the algebra of $S$-invariant regular functions on $X$ and $\bbk(X)^S$
is the field of $S$-invariant rational functions. If $\bbk[X]^S$ is finitely generated, then
$X\md S:=\spe \bbk[X]^S$. Whenever $\bbk[X]^S$ is a graded polynomial ring,
the elements of any set of algebraically independent homogeneous generators
will be referred to as {\it basic invariants\/}. If $V$ is an $S$-module and $v\in V$, then
$\es_v=\{\zeta\in\es\mid \zeta{\cdot}v=0\}$ is the {\it stabiliser\/} of $v$ in $\es$ and
$S_v=\{s\in S\mid s{\cdot}v=v\}$ is the {\it isotropy group\/} of $v$ in $S$.
In explicit examples of Section~\ref{sect:rank} and in Table~\ref{table-ex1},
we identify the representations $V$ of semisimple groups with their highest weights, using the {\it multiplicative\/} notation and the
Vinberg--Onishchik numbering of the fundamental weights~\cite{VO}. For instance, if $\varpi_1,\dots,\varpi_n$ are the fundamental weights of a simple algebraic group $G$, then $V=\varpi_i^2+2\varpi_{j}$ stands for the direct sum of three simple $G$-modules, with highest weights $2\varpi_i$ (once) and $\varpi_{j}$ (twice). If $H\subset G$ is semisimple and we are describing the restriction of $V$ to $H$ (i.e., $V\vert_H$), then the fundamental weights of
$H$ are denoted by $\tvp_i$. Write `$\mathrm{{1}\!\! 1}$' for the trivial one-dimensional representation.
\section{Preliminaries on the coadjoint representations}
\label{sect:prelim}
\noindent
Let $S$\/ be an affine algebraic group with Lie algebra $\es$. The symmetric algebra
$\gS (\es)$ over $\bbk$ is identified with the graded algebra of polynomial functions on $\es^*$ and we
also write $\bbk[\es^*]$ for it.
The {\it index of}\/ $\es$, $\ind\es$, is the minimal codimension of $S$-orbits in $\es^*$. Equivalently,
$\ind\es=\min_{\xi\in\q^*} \dim \es_\xi$. By Rosenlicht's theorem~\cite[2.3]{VP}, one also has
$\ind\es=\trdeg\bbk(\es^*)^S$. The ``magic number'' associated with $\es$ is $b(\es)=(\dim\es+\ind\es)/2$. Since the coadjoint orbits are even-dimensional, the magic number is an integer. If $\es$ is reductive, then
$\ind\es=\rk\es$ and $b(\es)$ equals the dimension of a Borel subalgebra. The Poisson bracket $\{\ ,\ \}$ in
$\bbk[\es^*]$ is defined on the elements of degree $1$ (i.e., on $\es$) by $\{x,y\}:=[x,y]$.
The {\it centre\/} of the Poisson algebra $\gS(\es)$ is $\gS(\es)^\es=\{F\in \gS(\es)\mid \{F,x\}=0 \ \ \forall x\in\es\}$. If $S^o$ is the identity component of $S$, then $\gS(\es)^\es=\gS(\es)^{S^o}$.
The set of $S$-{\it regular\/} elements of $\es^*$ is
$\es^*_{\sf reg}=\{\eta\in\es^*\mid \dim S{\cdot}\eta\ge \dim S{\cdot}\eta' \text{ for all }\eta'\in\es^*\}$.
We say that $\es$ has the {\sl codim}--$2$ property if $\codim (\es^*\setminus\es^*_{\sf reg})\ge 2$.
The following useful result appears in~\cite[Theorem\,1.2]{coadj}:
\\[.7ex]
\hbox to \textwidth{\ $(\blacklozenge)$ \hfil
\parbox{400pt}{\it
Suppose that $S$ is connected, $\es$ has the {\sl codim}--$2$ property, and there are homogeneous algebraically
independent $f_1,\dots,f_l\in \bbk[\es^*]^S$ such that $l=\ind\es$ and $\sum_{i=1}^l \deg f_i=b(\es)$.
Then $\bbk[\es^*]^S=\bbk[f_1,\dots,f_l]$ and $(\textsl{d}f_1)_\xi,\dots,(\textsl{d}f_l)_\xi$ are linearly
independent if and only if $\xi\in\es^*_{\sf reg}$.}
\hfil}
\noindent
More generally, one can define the set of $S$-regular elements for any $S$-action on an irreducible variety $X$; that is, $X_{\sf reg}=\{x\in X\mid \dim S{\cdot}x\ge \dim S{\cdot}x' \text{ for all } x'\in X\}$.
We say that the action $(S:X)$ {has a generic stabiliser}, if there exists
a dense open subset $\Omega\subset X$ such that all stabilisers $\es_x$, $x\in \Omega$, are
$S$-conjugate. Then {\bf any} subalgebra $\es_x$, $x\in\Omega$, is called a {\it generic stabiliser}
(=\,\textsf{g.s.}). The points of $\Omega$ are said to be $S$-generic (or, just generic if the group is clear from the context).
Likewise, one defines a {\it generic isotropy group} (=\,$\gig$),
which is a subgroup of $S$. By~\cite[\S\,4]{Ri}, $(S:X)$ has a generic stabiliser if and only if it has a generic isotropy group. It is also shown therein that $\gig$ always exists if $S$ is reductive and $X$ is smooth.
If $H$ is a generic isotropy group for $(S:X)$ and $\h=\Lie H$, then we write $H=\gig(S:X)$ and
$\h=\textsf{g.s.}(S:X)$.
A systematic treatment of generic stabilisers in the context of reductive group
actions can be found in \cite[\S 7]{VP}.
Note that if a generic stabiliser for $(S:X)$ exists, then any $S$-generic point is $S$-regular, but not vice versa.
Recall that $f\in \bbk[X]$ is called a {\it semi-invariant} of $S$ if
$S{\cdot}f\subset \bbk f$. A semi-invariant $f$ is {\it proper} if
$f\not\in \bbk[X]^S$.
\section{On the coadjoint representations of semi-direct products}
\label{sect:kos-th}
In this section, we gather some results on the coadjoint representation that are specific for
semi-direct products. In particular, we recall the necessary theory from \cite{Y16}.
\\ \indent
Let $G\subset GL(V)$ be a connected algebraic group with $\g=\Lie G$.
The vector space $\g\oplus V$ has a natural structure of Lie algebra, the {\it semi-direct product
of\/ $\g$ and $V$}.
Explicitly, if $x,x'\in \g$ and $v,v'\in V$, then
\[
[(x,v), (x',v')]=([x,x'], x{\cdot}v'-x'{\cdot}v) \ .
\]
This Lie algebra is denoted by $\es=\g\ltimes V$, and $V\simeq \{(0,v)\mid v\in V\}$
is an abelian ideal of $\es$.
The corresponding connected algebraic group $S$ is the semi-direct
product of $G$ and the commutative unipotent group $\exp(V)\simeq V$.
The group $S$ can be identified with $G\times V$, the product being given by
\[
(s,v)(s',v')= (ss', (s')^{-1}{\cdot}v+v'), \ \text{ where } \ s,s'\in G .
\]
In particular, $(s,v)^{-1}=(s^{-1}, -s{\cdot}v)$. Then $\exp(V)$ can be identified with
$1\ltimes V:=\{(1,v)\mid v\in V\} \subset G\ltimes V$.
If $G$ is reductive, then the subgroup $1\ltimes V$ is the unipotent radical of $S$, also denoted
by $R_u(S)$.
There is a general formula for the index of $\es=\g\ltimes V$, which is due to M.~Ra\"is \cite{rais}. Namely, there is a dense open subset $\Omega\subset V^*$ such that $\ind\es=\trdeg\bbk(V^*)^G+\ind\g_\xi$ for any $\xi\in\Omega$.
In particular, if a generic stabiliser for $(G:V^*)$ exists, then one can take $\g_\xi$ to be a generic stabiliser.
\begin{rmk} \label{rem:useful}
There are some useful observations related to the symmetric invariants of the semi-direct product
$\es=\g\ltimes V$:
\begin{itemize}
\item[\sf (i)] \ The decomposition $\es^*=\g^*\oplus V^*$ yields a bi-grading of
$\bbk[\es^*]^S$~\cite[Theorem\,2.3(i)]{coadj}. If ${\boldsymbol{H}}$ is a bi-homogenous $S$-invariant, then
$\deg_{\g}\! {\boldsymbol{H}}$ and $\deg_{V}\! {\boldsymbol{H}}$ stand for the corresponding degrees;
\item[\sf (ii)] \ The algebra $\bbk[V^*]^G$ is contained in $\bbk[\es^*]^S$. Moreover, a minimal generating
system for $\bbk[V^*]^G$ is a part of a minimal generating system of
$\bbk[\es^*]^S$~\cite[Sect.\,2\,(A)]{coadj}. Therefore, if $\bbk[\es^*]^S$ is a polynomial ring, then so is
$\bbk[V^*]^G$.
\end{itemize}
\end{rmk}
\begin{prop}[Prop.\,3.11 in \cite{Y16}] \label{non-red}
Let $G$ be a connected algebraic group acting on a finite-dimensional vector space $V$. Suppose that
$G$ has no proper semi-invariants in\/ $\bbk[\es^*]^{1\ltimes V}$ and\/ $\bbk[\es^*]^S$ is a polynomial
ring in $\ind \es$ variables.
For generic $\xi\in V^*$, we then have
\begin{itemize}
\item the restriction map
$\psi: \bbk[\es^*]^S \to \bbk[\g^*{\times}\{\xi\}]^{G_\xi{\ltimes}V}\simeq \gS(\g_\xi)^{G_\xi}$ is surjective;
\item $\gS(\g_\xi)^{G_\xi}$ coincides with $\gS(\g_\xi)^{\g_\xi}$;
\item $\gS(\g_\xi)^{G_\xi}$ is a polynomial ring in $\ind\g_\xi$ variables.
\end{itemize}
\end{prop}
\noindent
Note that $G$ is not assumed to be reductive and $G_\xi$ is not assumed to be connected in the above proposition! We mention also that there are
isomorphisms
$\bbk[\g^*{\times}\{\xi\}]^{G_\xi{\ltimes}V}\simeq \bbk[\mathfrak g_x]^{G_x}\simeq \gS(\g_\xi)^{G_\xi}$ for any $\xi\in V^*$, see \cite[Lemma~2.5]{Y16}.
From now on, $G$ is supposed to be reductive. The action $(G:V)$ is said to be {\it stable\/} if the union of closed $G$-orbits is dense in $V$. Then a generic stabiliser $\mathsf{g.s.}(G:V)$
is necessarily reductive.
Consider the following assumptions on $G$ and $V$:
\begin{itemize}
\item[($\diamondsuit$)] \
the action of $(G:V^*)$ is stable, $\bbk[V^*]^G$ is a polynomial ring, $\bbk[\g^*_\xi]^{G_\xi}$ is a polynomial
ring for generic $\xi\in V^*$, and $G$ has no proper semi-invariants in $\bbk[V^*]$.
\end{itemize}
The following result of the second author was excluded from the final text of
\cite{Y16}. .
\begin{thm} \label{V-rank-1}
Suppose that $G$ and $V$ satisfy condition $(\diamondsuit)$ and $V^*\md G=\mathbb A^1$, i.e.,
$\bbk[V^*]^G=\bbk[F]$ for some homogeneous $F$. Let $H$ be a generic isotropy group for $(G:V^*)$
and $\h=\Lie H$.
Assume further that $D=\{x\in V^*\mid F(x)=0\}$ contains an open $G$-orbit, say $G{\cdot} y$,
$\ind\g_y=\ind\h=:\ell$, and $\gS(\g_y)^{G_y}$ is a polynomial ring in $\ell$ variables
with the same degrees of generators as $\gS(\h)^{H}$.
Then $\bbk[\es^*]^S$ is a polynomial ring in $\ind\es=\ell+1$ variables.
\end{thm}
\begin{proof}
If $\ell=0$, then $\bbk[\es^*]^S=\bbk[F]$ and we have nothing to do.
Assume that $\ell\ge 1$.
Let $\{{\boldsymbol{H}}_i\mid 1\le i\le \ell\}$ be bi-homogeneous $S$-invariants chosen as
in \cite[Lemma\,3.5(i)]{Y16}.
Assume that $\deg_{\g} {\boldsymbol{H}}_i\le \deg_{\g} {\boldsymbol{H}}_j$ if $i<j$. We will show that these polynomials
can be modified in such a way that the new set
satisfies the conditions of \cite[Lemma\,3.5(ii)]{Y16} and therefore freely generates
$\bbk[\es^*]^S$ over $\bbk[F]$.
Notice that $F$ is an irreducible polynomial,
because $(\diamondsuit)$ includes also the absence of proper semi-invariants.
Thereby $\bbk[D]^G=\bbk$ and a non-trivial relation over $\bbk[D]^G$ among
$\tilde {\boldsymbol{H}}_i={{\boldsymbol{H}}_i}\vert_{\g^*{\times} D}$
gives also a non-trivial relation among $\tilde{\bf h}_i={{\boldsymbol{H}}_i}\vert_{\g^*{\times}\{y\}}$. Recall that $\tilde{\bf h}_i\in \gS(\mathfrak g_y)^{G_y}$
by \cite[Lemma~2.5]{Y16}.
Assume that
$\tilde{\bf h}_1,\ldots,\tilde{\bf h}_j$ are algebraically independent
if $j=d$ and dependent for $j=d+1$. Then $\tilde{\bf h}_{d+1}$ is not among the generators of
$\gS(\g_y)^{G_y}$ and it can be expressed as a polynomial
$R(\tilde{\bf h}_1,\ldots,\tilde{\bf h}_d)$. Then also
$\tilde {\boldsymbol{H}}_{d+1}-R(\tilde {\boldsymbol{H}}_1,\ldots,\tilde {\boldsymbol{H}}_d)=0$ and we can replace
${\boldsymbol{H}}_{d+1}$ by the bi-homogeneous part of $\frac{1}{F}({\boldsymbol{H}}_{d+1}-R({\boldsymbol{H}}_1,\ldots,{\boldsymbol{H}}_d))$
of bi-degree $(\deg_{\g} {\boldsymbol{H}}_{d+1},\deg_{V} {\boldsymbol{H}}_{d+1}-\deg F)$.
Clearly, $\sum_i\deg {\boldsymbol{H}}_i$ is decreasing and therefore the process will end up at some stage and bring
a new set $\{{\boldsymbol{H}}_i\}$ satisfying the conditions of \cite[Lemma\,3.5(ii)]{Y16}.
\end{proof}
\begin{rmk} \label{rem:deg-V}
Although Theorem~\ref{V-rank-1} asserts that $\bbk[\es^*]^S$ is a polynomial ring under certain conditions,
it does not say anything about the (bi)degrees of the generators ${\boldsymbol{H}}_i$. Finding these degrees is not an
easy task. Let us say a few words about it.
Suppose that $G$ is semisimple, $\bbk[\mathfrak s^*]^S$ is a polynomial ring, and
$\mathfrak s$ has the {\sl codim}--$2$ property. As is mentioned above,
$\ind\es={\rm tr.deg}\,\bbk(V^*)^G+\ind\g_\xi$ with $\xi\in V^*$ generic.
Here ${\rm tr.deg}\,\bbk(V^*)^G=\dim V^*\md G$ and
$\bbk[\mathfrak s^*]^S=\bbk[{\boldsymbol{H}}_1,\ldots,{\boldsymbol{H}}_\ell, F_1,\ldots,F_r]$, where $\ell=\ind\g_\xi$,
$r= \dim V^*\md G$, and all generators are bi-homogeneous.
The generators $F_j$ are elements of $\gS (V)$.
The $\mathfrak g$-degrees of the polynomials ${\boldsymbol{H}}_i$ are the degrees of basic invariants in
$\gS (\g_\xi)^{G_\xi}$.
Furthermore, $\sum_{i=1}^{\ell} \deg_V {\boldsymbol{H}}_i + \sum_{j=1}^r \deg F_j = \dim V$, see
\cite[Section~2]{Y} for a detailed explanation. Thus, the only open problem is how to determine the
$V$-degrees of the ${\boldsymbol{H}}_i$. In particular, the problem simplifies considerably, if $\ell$ is small.
\end{rmk}
\section{The classification and table}
\label{sect:rank}
\noindent
In this section, $G$ is an {\it \bfseries exceptional} algebraic group, i.e., $G$ is a simple algebraic group of
one of the types $\mathsf E_6, \mathsf E_7, \mathsf E_8,
\mathsf F_4, \mathsf G_2$. We classify the (finite-dimensional
rational) representations $(G:V)$ such that the symmetric invariants of $\es=\g\ltimes V$ form a polynomial ring. This will be referred to as property ({\sf FA}) for $\es$.
We also say that $\es$ (or just the action $(G:V)$) is {\it good} (resp. {\it bad}), if ({\sf FA}) is
(resp. is not) satisfied for $\es$.
\\ \indent To distinguish exceptional groups and Lie algebras, we write, say, $\GR{E}{7}$ for the group and
$\eus E_7$ for the respective algebra; while the corresponding Dynkin type is referred to as $\mathsf E_7$.
\begin{ex} \label{ex:adjoint}
If $G$ is arbitrary semisimple, then $\g\ltimes\g$ always has ({\sf FA})~\cite{takiff}.
Therefore we exclude the adjoint representations from our further consideration.
\end{ex}
If $\bbk[\es^*]^S$ is a polynomial ring, then so is $\bbk[V^*]^G$ (Remark~\ref{rem:useful}(ii)). For this
reason, we have to examine all representations of $G$ with polynomial rings of invariants.
If $(G:V)$ is a representation of an exceptional algebraic group such that $V\ne \g$ and $\bbk[V]^G$ is
a polynomial ring, then $V$ or $V^*$ is contained in Table~\ref{table-ex1}.
This can be extracted from the classifications in \cite{ag79} or \cite{gerry1}. Furthermore, the algebras
$\bbk[V]^G$ and $\bbk[V^*]^G$ (as well as
$\gS(\g\ltimes V)^{G\ltimes V}$ and $\gS(\g\ltimes V^*)^{G\ltimes V^*}$) are isomorphic, so that it is
enough to keep track of only $V$ or $V^*$.
As in~\cite{ag79,alela1,Y16},
we use the Vinberg-Onishchik numbering of fundamental weights $\varpi_i$. Here $H$ is a generic isotropy
group for $(G:V^*)$ and column ({\sf FA}) refers to the presence of property that $\bbk[\es^*]^S$ is a
polynomial ring, where $\es=\g\ltimes V$. We write $q(V\md G)$ for the sum of degrees of the basic invariants in $\bbk[V^*]^G$.
In Table~\ref{table-ex1}, the group $H$ is always reductive. Since $G$ is semisimple, this implies that
the action $(G:V^*)$ is stable in all cases, see~\cite[Theorem~7.15]{VP}. The fact that $G$ is semisimple
means also that $G$, as well as $G\ltimes V$, has only trivial characters and therefore has no
proper semi-invariants.
\begin{table}[ht]
\caption{Representations of the exceptional groups with polynomial ring $\bbk[V^*]^G$} \label{table-ex1}
\begin{center}
\begin{tabular}{>{\sf}c<{}|>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}ccc}
{\tencyr\cyracc N0}& G & V & \dim V & \dim V^*\md G & q(V\md G) & H & $\ind\es$ & ({\sf FA}) & Ref. \\ \hline \hline
1a & \GR{G}{2} & \varpi_1 & 7 & 1 & 2 & \GR{A}{2} & 3 & $+$ & Eq.~\eqref{eq:chain2}\\
1b & & 2\varpi_1 & 14 & 3 & 6 & \GR{A}{1} & 4 & + & \cite[Ex.\,4.8]{Y16} \\
1c & & 3\varpi_1 & 21 & 7 & 15 & \{1\} & 7 & + & Example~\ref{ex:h=0}\\[.5ex] \hline
2a & \GR{F}{4} & \varpi_1 & 26 & 2 & 5 & \GR{D}{4} & 6 & $-$ & Example~\ref{ex:F4-fi1} \\
2b & & 2\varpi_1 & 52 & 8 & 22 & \GR{A}{2} & 10 & $+$ & Eq.~\eqref{eq:chain2} \\
\hline
3a & \GR{E}{6} & \varpi_1 & 27 & 1 & 3 & \GR{F}{4} & 5 & + & Example~\ref{ex:E6-fi1} \\
3b & & \varpi_1+\varpi_5 & 54 & 4 & 12 & \GR{D}{4} & 8 & $-$ & Example~\ref{ex:E6-mnogo-fi1}\\
3c & & 2\varpi_1 & 54 & 4 & 12 & \GR{D}{4} & 8 & $-$ & Example~\ref{ex:E6-mnogo-fi1} \\
3d & & 3\varpi_1 & 81 & 11 & 36 & \GR{A}{2} & 13 & $+$ & Theorem~\ref{thm:E6-good}\\
3e & & 2\varpi_1+\varpi_5 & 81 & 11 & 36 & \GR{A}{2} & 13 & $+$ & Theorem~\ref{thm:E6-good} \\
\hline
4a & \GR{E}{7} & \varpi_1 & 56 & 1 & 4 & \GR{E}{6} & 7 & $-$ &Theorem~\ref{thm:E7-bad}\\
4b & & 2\varpi_1 & 112 & 7 & 28 & \GR{D}{4} & 11 & $-$ & Example~\ref{ex:E7-mnogo-fi1}\\ \hline
\end{tabular}
\end{center}
\end{table}
We provide below necessary explanations.
\begin{ex} \label{ex:E6-fi1}
Consider item {\sf 3a} in the table. Here $V^*\md G=\mathbb A^1$, i.e.,
$\bbk[V^*]^G=\bbk[F]$ for some homogeneous $F$. The divisor $D=\{\xi\in V^*\mid F(\xi)=0\}$ contains a
dense $G$-orbit,
say $G{\cdot}\eta$, whose stabiliser is the semi-direct product
$\g_\eta=\mathfrak{so}_9\ltimes \varpi_4$. A generic isotropy group for
$(G:V^*)$ is the exceptional group $\GR{F}{4}$ and $\g_\eta$ is a $\BZ_2$-contraction of $\eus F_4$.
By~\cite[Theorem\,4.7]{coadj}, $\gS(\g_\eta)^{G_\eta}$ is a polynomial ring whose degrees
of basic invariants are the same as those for $\eus F_{4}$. Therefore, using Theorem~\ref{V-rank-1}, we
obtain that $\bbk[\es^*]^S$ is a polynomial ring.
\end{ex}
\begin{ex} \label{ex:h=0}
If $\h=0$, then $\bbk[\es^*]^S\simeq \bbk[V^*]^G$~\cite[Theorem\,6.4]{p05} (cf.~\cite[Example\,3.1]{Y16}). Therefore item~{\sf 1c} is a good case.
\end{ex}
\begin{ex} \label{ex:F4-fi1}
The semi-direct product in {\sf 2a} is a $\BZ_2$-contraction of $\eus E_6$. This is one of the four bad
$\BZ_2$-contractions of simple Lie algebras, i.e., $\bbk[\es^*]^S$ is not a polynomial ring here,
see~\cite[Section 6.1]{Y16}.
\end{ex}
If $G$ is semisimple, $V$ is a reducible $G$-module, say $V=V_1\oplus V_2$,
then there is a trick that allows us to relate the polynomiality problem for the symmetric invariants of $\es=\g\ltimes V$ to a smaller semi-direct product. The precise statement is as follows.
\begin{prop} \label{prop:trick}
With $\es=\g\ltimes (V_1\oplus V_2)$ as above,
let $H$ be a generic isotropy group for $(G:V^*_1)$ and $\mathfrak h=\Lie H$. If\/ $\bbk[\es^*]^S$ is a polynomial ring, then so is\/
$\bbk[\tilde\q^*]^{\tilde Q}$, where $\tilde\q=\h\ltimes (V_2\vert_H)$.
\end{prop}
\begin{proof}
Consider the (non-reductive) semi-direct product $\q=\g\ltimes V_2$. Then
$\es=\q\ltimes V_1$. It is assumed that the unipotent radical of $Q$, $1\ltimes V_2$, acts trivially on $V_1$.
If $\xi\in V^*_1$ is generic for $(G:V^*_1)$, then it is also generic for $(Q:V^*_1)$. If $G_\xi=H$, then the corresponding isotropy group in $Q$
is $Q_\xi=H\ltimes V_2$ and $\q_\xi=\h\ltimes V_2\simeq \tilde \q$, where $V_2$ is considered as
$H$-module.
Since $G$ is semisimple, all hypotheses of Proposition~\ref{non-red} are satisfied for $Q$ in place of $G$
and $V=V_1$. Therefore $\gS(\q_\xi)^{Q_\xi} =\gS(\q_\xi)^{\q_\xi}$
is a polynomial ring.
\end{proof}
\begin{rmk}
It is easily seen that the passage from $(G:V=V_1\oplus V_2)$ to $(H:V_2)$, where $H=\gig(G:V_1)$, preserves generic isotropy groups. For generic stabilisers, this appears already in \cite[\S\,3]{alela1}.
\end{rmk}
One can use Proposition~\ref{prop:trick} as a tool for proving that $\bbk[\es^*]^S$ is not a polynomial ring.
\begin{ex} \label{ex:E6-mnogo-fi1}
In case~{\sf 3b}, we take $Q=\GR{E}{6}\ltimes \varpi_1$ and $V_1=\varpi_5$. Then
$\es=(\eus E_6\ltimes\varpi_1)\ltimes\varpi_5\simeq \eus E_6\ltimes(\varpi_1+\varpi_5)$.
If $\xi\in V^*_1=\varpi_1$ is generic, then $\g_\xi\simeq \eus F_4$ and
$\varpi_1\vert_{\GR{F}{4}}=\tilde\varpi_1+\mathrm{{1}\!\! 1}$ \cite{alela1}. Therefore, $\q_\xi$ is isomorphic to the semi-direct product
related to item 2a, modulo a one-dimensional centre. Therefore, $\gS(\q_\xi)^{Q_\xi}$ is not
a polynomial ring (see Example~\ref{ex:F4-fi1}), and we conclude, using Proposition~\ref{prop:trick}, that
$\bbk[\es^*]^S$ is not a polynomial ring, too.
\\ \indent
In case~{\sf 3c}, we take the same $Q$ and $V_1=\varpi_1$. The rest is more or less the same, since
$\GR{F}{4}\ltimes (\varpi_1\vert_{\GR{F}{4}})$ is again a generic isotropy subgroup for $(Q:V_1^*)$.
\end{ex}
\begin{ex} \label{ex:E7-mnogo-fi1}
In case~{\sf 4b}, we take $Q=\GR{E}{7} \ltimes \varpi_1$ and $V_1=\varpi_1$.
Note that $\varpi_1$ is a symplectic $\GR{E}{7}$-module. If $\xi\in V_1^*$ is generic, then the corresponding stabiliser in $\eus E_7$
is isomorphic to $\eus E_6$~\cite{H,alela1}. Thereby
$\q_\xi=\eus E_6\ltimes (\varpi_1\vert_{\eus E_6})=\eus E_6\ltimes (\tilde\varpi_1+\tilde\varpi_5+2\mathrm{{1}\!\! 1})$ \cite{alela1}.
Hence $\q_\xi$ represents item~{\sf 3b} (modulo a two-dimensional centre) and we have already demonstrated in the previous example that here $\bbk[\q^*_\xi]^{Q_\xi}$ is not
a polynomial ring!
\end{ex}
The output of Examples~\ref{ex:E6-mnogo-fi1} and \ref{ex:E7-mnogo-fi1} is that there is the tree of
reductions to a ``root'' bad semi-direct product:
\[
\xymatrix{ & (\GR{E}{6}, 2\varpi_1)\ar[dr] & \\
(\GR{E}{7}, 2\varpi_1)\ar[r] & (\GR{E}{6}, \varpi_1+\varpi_5)\ar[r] & \text{\framebox{$(\GR{F}{4},\varpi_1)$}},
}\]
and therefore all these items represent bad semi-direct products.
Using Proposition~\ref{prop:trick} in
a similar fashion, one obtains another tree of reductions:
\beq \label{eq:chain2}
\xymatrix{
(\GR{E}{6}, 2\varpi_1+\varpi_5)\ar[dr] & & & \\
(\GR{E}{6}, 3\varpi_1)\ar[r] & (\GR{F}{4},2\varpi_1)\ar[r] & (\GR{D}{4},\varpi_1+\varpi_3+\varpi_4)\ar[r] & (\GR{G}{2},\varpi_1).
}
\eeq
Some details for the passage from $\eus E_6$ to $\mathfrak{so}_8$ and then to $\eus G_2$ are given below in the proof of Theorem~\ref{thm:E6-good}.
Note that the representations occurring in~\eqref{eq:chain2} have one and the same generic isotropy group,
namely $SL_3$. As we will shortly see, tree~\eqref{eq:chain2} consists actually of good cases. Here
our strategy is to prove that both ``crown'' $\GR{E}{6}$-cases are good. To this end, we need some
properties of the representation $(\GR{G}{2}, \varpi_1)$ related to the ``root'' case.
\begin{lm} \label{lm:G2-case}
Let $v_1$ be a highest weight vector in the $\GR{G}{2}$-module $\varpi_1$ and
$Q:=(\GR{G}{2})_{v_1}$ the respective isotropy group. Then
{\sf (i)} $\q=\Lie Q$ has the {\sl codim}--$2$ property and {\sf (ii)} the coadjoint representation of $Q$
has a polynomial ring of invariants whose degrees of basic invariants are $2,3$.
\end{lm}
\begin{proof} A generic isotropy group for $(\GR{G}{2}:\varpi_1)$ is connected and isomorphic to
$SL_3$~\cite{H} and $\q$ is a contraction of $\mathfrak{sl}_3$ (See \cite[Ch.\,7, \S\,2]{t41} for Lie algebra contractions).
We also have $\q=\el{\ltimes}\n$, where $\el=\mathfrak{sl}_2$ and
the nilpotent radical $\n$ is a 5-dimensional $\BZ$-graded non-abelian Lie algebra of the form
\[
\n(1)\oplus\n(2)\oplus\n(3)=\bbk^2_{I}\oplus \bbk\oplus \bbk^2_{II} .
\]
Here $\bbk^2_{I}$ and $\bbk^2_{II}$ are standard $\tri$-modules and $\bbk$ is the trivial $\tri$-module.
Let $\{a_1,b_1\}$ be a basis for $\bbk^2_{I}$, $\{a_2,b_2\}$ a basis for $\bbk^2_{II}$, and $\{u\}$ a basis
for $\bbk$. W.l.o.g., we may assume that $[a_1,b_1]=u$,
$[a_1,u]=a_2$, and $[b_1,u]=b_2$.
{\sf (i)} \ Since $\q$ is a contraction of $\mathfrak{sl}_3$ and $\ind \mathfrak{sl}_3=2$, we have
$\ind\q\ge 2$. On the other hand, if $0\ne \xi\in\n(3)^*\subset \n^*\subset \q^*$, then $\dim\q_\xi=2$.
Hence $\ind\q=2$ and $\n(3)^*\setminus\{0\}\subset \q^*_{\sf reg}$. Since $\dim\n(3)=2$, the last property readily implies that
$\q^*\setminus \q^*_{\sf reg}$ cannot contain divisors.
{\sf (ii)} \ It is easily seen that
\[
{\mathbf h}_1=2a_1b_2-2b_1a_2+u^2 \in\gS(\n) \subset \gS(\q)
\]
is a $\q$-invariant. There is also another invariant of
degree three. Let $\{e,h,f\}$ be a standard basis of $\tri$ (i.e., $[e,f]=h, [h,e]=2e, [h,f]=-2f$). We assume that
$[e,a_1]=0$ and $[f,a_1]=b_1$, which implies $[h,a_1]=a_1$ and $[h,b_1]=-b_1$. Then
\[
{\mathbf h}_2=b_2^2e+a_2b_2h-a_2^2f+u(a_1b_2-a_2b_1)+\frac{1}{3}u^3
\]
is an $\tri$-invariant and in addition, the following Poisson bracket can be computed:
\[
\{a_1,{\mathbf h}_2\}=a_2b_2(-a_1)-a_2^2(-b_1)+a_2(a_1b_2-a_2b_1)-ua_2u+u^2a_2=0 .
\]
Since $\el=\tri$ and $a_1$ generate $\q$ as Lie algebra, ${\bf h}_2$ is also a $\q$-invariant. The
polynomials ${\mathbf h}_1$ and ${\mathbf h}_2$ are algebraically independent, because
${\mathbf h}_1\in {\gS}(\n)$ and ${\bf h}_2\not\in {\gS}(\n)$.
By {\sf (i)}, $\q$ has the {\sl codim}--$2$ property. Since $\dim\q=8$, $\ind\q=2$, and $b(\q)=5$,
we have $\deg {\mathbf h}_1+\deg {\mathbf h}_2=b(\q)$.
Therefore, ${\bf h}_1$ and ${\bf h}_2$ freely generate ${\gS}(\q)^{Q}$, see $(\blacklozenge)$ in
Section~\ref{sect:prelim}.
\end{proof}
\begin{rmk}
In this case, $(\GR{G}{2})_{v_1}$ is a so-called {\it truncated parabolic subgroup}.
Symmetric invariants of truncated (bi)parabolics
were intensively studied by Fauquant-Millet and Joseph, see e.g.
\cite{FMJ, J}.
Let $\rr_{\sf tr}$ be a truncated (bi)parabolic in type {\sf A} or {\sf C}. Then
${\eus S}(\rr_{\sf tr})^{\rr_{\sf tr}}$ is a polynomial ring in $\ind\rr_{\sf tr}$ homogeneous generators and
the sum of their degrees is equal to $b(\rr_{\sf tr})$.
The same properties hold for many truncated (bi)parabolics in other types~\cite{J}. It is very probable that
a sufficient condition of~\cite{J} is satisfied for $(\eus G_2)_{v_1}$. However, we prefer to keep the explicit construction of generators.
\end{rmk}
\begin{thm} \label{thm:E6-good}
If\/ $\es =\eus E_6\ltimes (3\varpi_1)$ or $\es =\eus E_6\ltimes(2\varpi_1+\varpi_5)$, then $\bbk[\es^*]^S$ is a polynomial ring.
\end{thm}
\begin{proof}
Here $G=\GR{E}{6}$, $\g=\eus E_6$, and $V=3\varpi_1$ or $2\varpi_1+\varpi_5$.
Accordingly, $V^*=3\varpi_5$ or $2\varpi_5+\varpi_1$. In both cases, $\ind\es=13$, $V^*\md G\simeq \mathbb A^{11}$, and a
generic isotropy group for $(G:V^*)$ is $SL_3$. By \cite[Theorem\,2.8 and Lemma\,3.5({\sf i})]{Y16}, there
are bi-homogeneous irreducible polynomials $F_1,F_2\in \bbk[\es^*]^S$ such that $\deg_{\g}F_1=2$, $\deg_{\g}F_2=3$, and
their restrictions to $\g^*\times\{\xi\}$, where $\xi\in V^*$ is $G$-generic, are the basic invariants of
$SL_3$. Here, for $\g^*\times\{\xi\} \subset \es^*$, we use the isomorphism
\beq \label{eq:canon-iso}
\bbk[\g^*\times\{\xi\}]^{G_\xi\times V}\simeq \gS(\g_\xi)^{G_\xi} ,
\eeq
see \cite[Lemma~2.5]{Y16}.
By \cite[Lemma\,3.5({\sf ii})]{Y16}, $F_1$ and $F_2$ generate $\bbk[\es^*]^S$ over $\bbk[V^*]^G$ if and only if
the restrictions $F_1\vert_{\g^*\times D}$ and $F_2\vert_{\g^*\times D}$ are algebraically
independent for any $G$-stable homogeneous divisor $D\subset V^*$. Let us prove that this is really the case.
\\ \textbullet\quad
If there is a non-trivial relation for the restrictions of $F_1$ and $F_2$ to $\g^*\times D$, then this also yields
a non-trivial relation for the subsequent restrictions of $F_1$ and $F_2$ to $\g^*\times\{\eta\}$, where
$\eta$ is $G$-generic in $D$.
\\ \textbullet\quad
If $G_\eta\simeq SL_3$, i.e., $\eta$ is also $G$-generic in $V^*$, then $F_1\vert_{\g^*\times\{\eta\}}$ and $F_2\vert_{\g^*\times \{\eta\}}$
are nonzero elements of $\gS(\mathfrak{sl}_3)^{SL_3}$ of degrees $2$ and $3$, respectively.
The invariants of degree $2$ and $3$ in $\gS(\mathfrak{sl}_3)^{SL_3}$ are uniquely determined, up to a scalar factor, and they are algebraically independent. Hence $F_1\vert_{\g^*\times D}$ and $F_2\vert_{\g^*\times D}$ are algebraically independent for such divisors.
\\ \textbullet\quad
However, it can happen that a divisor $D$ contains no ``globally'' $G$-generic points. To circumvent this difficulty, consider three projections from $V^*$ to its simple constituents, and their (non-trivial!) restrictions to $D$:
\[ \xymatrix{
& D\ar[dl]_{p_1}\ar[d]^{p_2}\ar[dr]^{p_3} & \\
\varpi_5 & \varpi_5 & \varpi_1 \text{ or } \varpi_5
}\]
For $\eta=y_1+y_2+y_3\in D\subset V^*$, we have
$p_i(\eta)=y_i$. Since $D$ is a divisor, at least two of the $p_i$'s are dominant.
Without loss of generality, we may assume that $p_1$ and $p_2$ are dominant, whereas
$p_3$ is not and then $\ov{p_3(D)}$ is a divisor in $p_3(V^*)$. (For, if all $p_i$'s are dominant, then $D$ contains a globally generic point.)
Therefore, we can take $y_1,y_2$ to be generic in the $p_i(V^*)$.
Then $y_1+y_2$ is $G$-generic in $2\varpi_5$, $G_{y_1+y_2}=Spin_8$, and
$\varpi_1\vert_{Spin_8}\simeq \varpi_5\vert_{Spin_8}\simeq \tvp_1+\tvp_3+\tvp_4+3\mathrm{{1}\!\! 1}$.
Here $G_\eta=(Spin_8)_{y_3}$.
Having obtained a $Spin_8$-stable divisor $\ov{p_3(D)}$ in the $Spin_8$-module $ \tvp_1+\tvp_3+\tvp_4+3\mathrm{{1}\!\! 1}$, we consider $\tilde V:=\tvp_1+\tvp_3+\tvp_4$ and
$\tilde D:=\ov{p_3(D)}\cap \tilde V$. If $\tilde D=
\tilde V$, then $(Spin_8)_{y_3}=SL_3$ for some $y_3$, i.e., again $G_\eta=SL_3$.
\\ \textbullet\quad
If $\tilde D$ is a divisor in $\tilde V$, then we
can play the same game with $Spin_8$ and $\tilde D$. Let $y_3=x_1+x_3+x_4\in \tilde D$, where
$x_i\in\tvp_i$. Again, at least two of the projections $\tilde p_i:\tilde V\to \tvp_i$ ($i=1,3,4$) are dominant.
Without loss of generality, we may assume that $\tilde p_1$ and $\tilde p_3$ are dominant and then
$x_1$ and $x_3$ are generic elements. Then
$(Spin_8)_{x_1+x_3}\simeq \GR{G}{2}$, $(\GR{G}{2})_{x_4}=(Spin_8)_{y_3}$,
and $\tvp_4\vert_{\GR{G}{2}}\simeq \hat \varpi_1+\mathrm{{1}\!\! 1}$.
The structure of $\GR{G}{2}$-orbits in $\hat \varpi_1$ shows that either $x_4$ is $\GR{G}{2}$-generic and then
$(\GR{G}{2})_{x_4}\simeq SL_3$ or $x_4$ is a highest weight vector and
$(\GR{G}{2})_{x_4}\simeq (\GR{G}{2})_{v_1}$, cf. Lemma~~\ref{lm:G2-case}.
Thus, for any $G$-stable divisor $D\subset V^*$, there is $\eta\in D$ such that
$G_\eta=SL_3$ or $(\GR{G}{2})_{v_1}$, and in both cases $\gS(\g_\eta)^{G_\eta}$ is generated by
algebraically independent invariants of degree $2$ and $3$ (see Lemma~\ref{lm:G2-case} for the latter).
It follows that $F_1\vert_{\g^*\times D}$ and $F_2\vert_{\g^*\times D}$ are algebraically
independent, and we are done.
\end{proof}
Combining Proposition~\ref{prop:trick}, Theorem~\ref{thm:E6-good}, and tree~\eqref{eq:chain2}, we conclude that
cases {\sf 3d, 3e, 2b}, and {\sf 1a} are good. (Note also that we have found one good case related to a representation of the classical Lie algebra $\eus D_4$.)
Thus, it remains to handle only the semi-direct product
$\eus E_7\ltimes \varpi_1$ (case {\sf 4a}).
\begin{thm} \label{thm:E7-bad}
If\/ $\es =\eus E_7\ltimes \varpi_1$, then $\bbk[\es^*]^S$ is not a polynomial ring.
\end{thm}
\begin{proof}
Here $G=\GR{E}{7}$, $\g=\eus E_7$, and $\bbk[V^*]^G=\bbk[F]$. By~\cite{H}, a generic isotropy group for
$(G:V^*)$ is
connected and isomorphic to $\GR{E}{6}$, and the null-cone
$D=\{\xi\in V^*\mid F(\xi)=0\}$ contains a dense $G$-orbit, say $G{\cdot}y$, whose isotropy group
$G_y$ is connected and isomorphic to $\GR{F}{4}\ltimes\varpi_1$.
Assume that $\bbk[\es^*]^S$ is polynomial.
By \cite[Theorem\,2.8 and Lemma\,3.5({\sf i})]{Y16}, since $\ind\es=7$ and $\gig$ is
$\GR{E}{6}$, there are bi-homo\-ge\-ne\-ous polynomials $H_1,\dots,H_6$
such that $\bbk[\es^*]^S=\bbk[F,{\boldsymbol{H}}_1,\dots,{\boldsymbol{H}}_6]$ and the ${\boldsymbol{H}}_i\vert_{\g^*\times\{\xi\}}$'s yield the basic invariants of $\eus E_6$ for a generic $\xi$. Here we again use Proposition~\ref{non-red} and isomorphism~\eqref{eq:canon-iso}.
By \cite[Lemma~3.5(ii)]{Y16}, the restrictions ${\boldsymbol{H}}_i\vert_{\g^*\times D}$, $i=1,\dots,6$ remain algebraically independent.
On the other hand, let us consider further restrictions ${\boldsymbol{H}}_i\vert_{\g^*\times \{y'\}}$, $i=1,\dots,6$, where $y'$ belongs to the dense $G$-orbit in $D$. Recall that $\g_{y'}\simeq \eus F_{4}\ltimes \varpi_1$ and the latter is a bad $\BZ_2$-contraction of $\eus E_{6}$, see Example~\ref{ex:F4-fi1}. Moreover, the algebra of symmetric invariants of $\eus F_{4}\ltimes \varpi_1$ does not have algebraically independent invariants whose degrees are the same as the degrees of basic invariants of $\eus E_{6}$~\cite[Section~6.1]{Y16}. This implies that ${\boldsymbol{H}}_i\vert_{\g^*\times \{y'\}}$, $i=1,\dots,6$ must be algebraically dependant for any $y'\in G{\cdot}y$.
Let $L({\boldsymbol{H}}_1\vert_{\g^*\times \{y'\}}, \dots, {\boldsymbol{H}}_6\vert_{\g^*\times \{y'\}})=0$ be a polynomial relation
for {\bf some} $y'$. Since the ${\boldsymbol{H}}_i$'s are $G$-invariant, the relation with the same coefficients holds
for {\bf all} $y'\in G{\cdot}y$. Hence, this dependance can be lifted to $\g^*\times G{\cdot}y$ and then
carried over to $\g^*\times D$. This contradiction shows that the ring $\bbk[\es^*]^S$ cannot be polynomial.
\end{proof}
Summarising, we obtain the main result of the article:
\begin{thm} \label{thm:main}
Let $G$ be an exceptional algebraic group, $V$ a (finite-dimensional rational) $G$-module, and
$\es=\g\ltimes V$. Then $\bbk[\es^*]^S$ is a polynomial ring if and only if one of the following two conditions is satisfied: {\sf (i)} $V=\g$; {\sf (ii)} $V$ or $V^*$ represents one of the seven good cases in Table~\ref{table-ex1}.
\\ Moreover, by Ra\"is' formula (see Section~\ref{sect:kos-th}), one has
$\trdeg \bbk[\es^*]^S=\ind\es=\dim V^*\!\md G +\rk H$.
\end{thm}
\noindent
In the good cases, we neither construct generators nor give their degrees. The reason is that the main results of \cite{Y16} as well as Theorem~\ref{V-rank-1} are purely existence theorems. Yet, as explained in Remark~\ref{rem:deg-V}, a great deal of information on the degrees is available. If $\ell$ is small, then using ad hoc methods one can determine all the degrees.
Let us see how this can be done for an example with $\ell=2$.
Take item~{\sf 1a}. Since $\mathfrak h=\mathfrak{sl}_3$, we have
$\bbk[\mathfrak s^*]^S=\bbk[V^*]^G[\boldsymbol{H}_1,\boldsymbol{H}_2]$ with $\deg_{\g}\boldsymbol{H}_1=2$, $\deg_{\g} \boldsymbol{H}_2=3$.
Set $a_i=\deg_V \boldsymbol{H}_i$. It can easily be seen that $\es$ has the {\sl codim}--$2$ property.
The discussion in Remark~\ref{rem:deg-V} shows that $a_1+a_2=5$.
Following the same strategy as in \cite[Prop.\,3.10]{Y16}, one can produce an $S$-invariant
of bi-degree $(2,2)$. This implies that $a_1=2$ and $a_2=3$.
|
1,314,259,993,611 | arxiv | \section{INTRODUCTION}
\label{INTRODUCTION}
The prevalence of more affordable high-speed internet and high-resolution cameras on mobile devices has enabled widespread live-streaming practices on the platforms like Tiktok, Twitch, and Youtube. Streamers share a wide range of live videos regarding pan entertainment~\cite{lu2018you}, goods selling~\cite{lu2018you,hu2020enhancing}, education content~\cite{lu2018you,chen2021towards,chen2021learning,fonseca2021knowledge,hammad2021towards}, gaming~\cite{tang2016meerkat,woodcock2019affective}, and intangible culture heritage~\cite{lu2019feel}. The ubiquity of live-streaming also attracts people with different abilities to engage, such as people with disabilities. Prior research discussed the enjoyable experiences and challenges of video game streamers with physical disabilities, ADHD (Attention Deficit Hyperactivity Disorder), and dyslexia on Twitch~\cite{johnson2019inclusion,anderson2021gamer}. \rv{Although} live-streaming is a highly visually-demanding activity (e.g., scrolling comments, filming, and filter), people who are blind or have low vision (i.e., BLV) were recently reported to be active content creators on Tiktok~\cite{alastair2021sheffield, torres2020blind,lasker2020teens,brown2020tiktok}. As online platforms that support live-streaming (e.g., Youtube, TikTok, and Douyin) often employ algorithms to curate, select, and present contents to viewers, content creators' activities on such algorithm-driven platforms tend to affect how the algorithms curate such contents ~\cite{karizat2021algorithmic}. However, little is known \rv{about how} streamers with visual impairments perceive the effects of such algorithms and how their practices on live-streaming platforms \rv{may be shaped by} the algorithms.
Moreover, recent work uncovered that the algorithms of video-sharing platforms were perceived to suppress the content from marginalized groups, such as people of color, the LGBT+ community, and content creators of lower socio-economic status~\cite{simpson2021you,karizat2021algorithmic}. The videos related to LGBT+ social identities were reported to be removed by Tiktok algorithms~\cite{simpson2021you}. Some anecdotes revealed that Tiktok prevented the content of disabled users (e.g., people with facial disfigurements, down's syndrome, or autism) from showing in able-bodied people's feeds~\cite{linessocial,trewin2019considerations, bozdag2013bias,robertson2019tiktok,biddle2020invisible}. \st{This raises the question of \textit{whether and to what extent BLV streamers' content is subjected to the manipulation~(e.g., removal, suppression) of the algorithm adopted by video-sharing and live-streaming platforms. How do BLV content creators perceive the algorithms? What strategies do they employ to mitigate the barriers created by the algorithms?}} \rv{This raises the questions of \textit{whether BLV content creators \textbf{perceive} there are any algorithmic challenges in relation to their status of visual impairments. If yes, what kinds of \textbf{perceived} algorithmic challenges that BLV content creators encounter? What strategies do they employ to mitigate the \textbf{perceived} barriers created by the algorithms?}}
To understand BLV content creators' perceptions of algorithms and to figure out whether they encountered similar \rv{perceived} algorithmic bias as other types of disabled people did, we must understand the challenges they encounter and the mitigation strategies they employ when interacting with the live-streaming ecosystem.
Motivated to understand BLV content creators' perceptions and challenges of the algorithms adopted by live-streaming platforms, we conducted semi-structured interviews with BLV streamers (N=19) of Douyin (i.e., Tiktok in China).
We chose to study Douyin due to its popularity in China~\cite{lu2018you}. A survey conducted by Shenzhen Accessibility Research Association indicated that although no screen readers were perfectly compatible with Douyin, twenty-six percent of BLV respondents still kept using Douyin frequently ~\cite{zijie2019}. In Douyin, content creators not only stream but also post and share short videos. Their live-streams or videos are promoted to users' For You Page~(FYP) by Douyin's algorithms. Thus, content creators on Douyin tend to rely on algorithms to make their content visible to expected audiences heavily.
Our findings uncovered BLV content creators' perceived factors contributing to their disadvantages under the algorithmic evaluation of BLV streamers' content, such as visual effects of filming and filters, interaction with viewers, video editing, misperceptions, and trolls. Moreover, they also suffered from perceived algorithmic biases regarding content suppression. For example, participants believed that the platform did not actively promote their content to the sighted audience \rv{(e.g., participant 16 noted:“It feels like being locked in a cage, since my content can hardly reach sighted viewers.”).} \rv{Also, they perceived that Douyin's algorithms} tended to limit the popularity of BLV streamers' content within the BLV community. BLV streamers developed mitigation strategies, such as hiding BLV identities, tagging geo-locations at downtown areas which had more potential sighted viewers, creating counter-intuitive feelings in their content, actively interacting with sighted users and providing mutual support within the BLV community. Based on the findings, we discussed the design considerations to make live-streaming platforms more accessible and inclusive for the BLV community.
In sum, we made the following contributions in this work: i) We identified BLV content creators' perceptions of how Douyin's algorithms operated in relation to BLV status; ii) We uncovered the perceived algorithmic challenges that BLV streamers encountered as well as their mitigation strategies; iii) We presented design considerations to make a more inclusive and fair live-streaming ecosystem for BLV streamers.
\section{RELATED WORK}
\label{RELATED WORK}
We first review the prior work in practices and populations on platforms that supported live-streaming, followed by a review of algorithmic biases and stereotypes for marginalized groups.
\subsection{The Practices and Populations on Platforms that Supported Live-streaming}
\label{The Practices and Populations on Platforms that supported live-streaming}
Due to the surge of live-streaming, \st{there was} an increasing number of studies \st{investigating} investigated users' live-streaming practices on social media platforms. Prior research indicated that people from a variety of professions actively streamed online, such as professional streamers~\cite{woodcock2019affective, lu2018you} , practitioners safeguarding Intangible Cultural Heritage (ICH)~\cite{lu2019feel} , programmers~\cite{chen2021towards,hammad2021towards} , university instructors~\cite{chen2021learning,hammad2021towards} , students preparing for specific tests~\cite{wang2021study} , community leaders and reporters~\cite{dougherty2011live} . For instance, Chen et al.\cite{chen2021towards} indicated that programmers chose live-streaming to share programming knowledge and skills. Some studies also examined a wide range of live-streaming
practices, including pan entertainment~\cite{lu2018you} , goods selling~\cite{lu2018you,hu2020enhancing} , online education~\cite{lu2018you,chen2021towards,chen2021learning,fonseca2021knowledge,hammad2021towards} , personal story sharing~\cite{lu2018you} , intangible culture promotion~\cite{lu2018you,lu2019feel}, eating~\cite{anjani2020people}, outdoor activities~(e.g. traveling, adventures)~\cite{lu2019vicariously,tang2016meerkat}, and gaming~\cite{tang2016meerkat,woodcock2019affective}. \st{Lu et al. found that streamers who promoted ICH-related content leveraged various streaming structures to showcase ICH practices, such as questions and answers, expert interviews, learners' live performances, and tutorials regarding basic knowledge.} \rv{ Lu et al.\cite{lu2019feel} found that streamers who promoted ICH-related content leveraged various streaming structures to showcase ICH practices. For example, ICH streamers held ``Questions and Answers'' sessions, interviewed experts, asked learners to perform, and delivered tutorials on basic knowledge.} Work by Lu et al.~\cite{lu2019vicariously} analyzed outdoor live-streaming in China.~\st{finding that in addition to broadcasting outdoor activities or travel in different environments~(e.g., hiking, fishing, and hunting), streamers also shared their spontaneous and unpredictable interaction with strangers (e.g., passersby on the street and the Didi~(Uber) drivers).} \rv{This work showed that in livestreams, streamers broadcasted outdoor activities or traveled in different environments (e.g., hiking, fishing, and hunting). In addition, they shared spontaneous and unpredictable interactions with strangers (e.g., passersby on streets and Didi (Uber) drivers).}
Some prior work presented that people with disabilities actively streamed. Anderson and Johnson~\cite{anderson2021gamer} studied how people with physical disabilities streamed for changing disabled viewers' negative mindsets about disabilities. In addition, work by Johnson~\cite{johnson2019inclusion} presented the enjoyable experiences and challenges of video game streamers with physical disabilities, ADHD (Attention Deficit Hyperactivity Disorder), and dyslexia on Twitch. Moreover, some prior work focused on economic and employment opportunities of live-streaming for disabled people ~\cite{johnson2019inclusion, johnson2020disability,qu2020internet}. Recently, Jun et al.~\cite{10.1145/3476038} explored the motivations, practices, and challenges of streamers with visual impairments on Youtube and Twitch.
Although this work uncovered streamers' motivations (e.g., achieving personal goals, delivering messages, overcoming limitations of recorded videos, getting inspired \rv{by} other streamers) and challenges (e.g.\rv{,} multitasking, screen reader related issues), it did not investigate BLV streamers' experiences with content-curation algorithms adopted by the platforms. However, recent research suggested that users from marginalized groups perceived live-streaming platforms
may have algorithmic biases against minority \rv{communities}, such as race and ethnicity minorities and the LGBT community~\cite{karizat2021algorithmic}.
Therefore, in this work, we take a first step to investigate BLV streamers' experiences with content-curation algorithms adopted by live-streaming platforms.
\subsection{Algorithmic Biases for Marginalized Groups on Video-sharing platforms}
\label{Algorithmic Biases for Marginalized Groups on Video-sharing Platforms}
Content creators of video-sharing platforms were aware that algorithms were a crucial factor, which influenced whether their content could be well-known among passersby ~\cite{bucher2017algorithmic, trewin2019considerations,pedersen2019my,wu2019agent}. Prior research ~\cite{simpson2021you, linessocial,trewin2019considerations} has presented how algorithms excluded the videos created by people from marginalized groups, such as people of color and those who came from LGBTQ groups or lower social-economic class. For example, several studies indicated algorithmic bias for sexual orientation or other gender issues. Simpson et al.~\cite{simpson2021you} described that \rv{users from the LGBT+ community believed that} the Tiktok algorithms censored or removed the content created by \rv{the LGBT+ group}. Haimson et al.~\cite{haimson2021tumblr} \rv{reported that users perceived that} the algorithms of Tumblr sometimes oppressed the content from trans through classifying them into adult content. Also, recently, some anecdotes~\cite{linessocial,trewin2019considerations, bozdag2013bias,robertson2019tiktok,biddle2020invisible,biddle2021tiktok,melonio2020tiktok} revealed that in the process of content evaluation, Tiktok limited the popularity of videos created by people with, for example, disabilities, facial disfigurements, Down syndrome, or autism. \rv{Tiktok prevented their} content from reaching non-disabled people's "For you" feed.
The \rv{perceived} algorithmic biases and suppression enabled content creators to come up with some coping strategies. For example, Simpson et al.~\cite{simpson2021you} showed that content creators recapped or reposted the content removed by the Tiktok algorithms to push back against the \rv{perceived} suppression. \rv{According to work by Karizat~\cite{karizat2021algorithmic},} content creators \rv{believed that the Tiktok algorithms }suppressed \rv{ the content related to marginalized }social identities. \rv{Content creators} resisted the Tiktok algorithms through individual actions~(e.g., sharing the content perceived to be suppressed ), collective actions~(e.g., collectively commenting and liking the videos silenced by algorithms), and altering the performance of content.
Despite increasing attention to algorithmic fairness for marginalized groups, whether BLV users perceive their contents are impacted unjustly by the algorithms of live-stream platforms remains unclear, which motivated our work.
\input{table1}
\section{METHOD}
\label{METHOD}
To better understand BLV streamers' motivations, practices, challenges, and coping strategies, we conducted \rv{an IRB-approved} semi-structured interview \rv{study} with 19 BLV streamers, who had live-streamed over ten times and over half an hour at least three times in the last three months on Douyin.
\subsection{Participants}
\label{Participants}
Table 1 shows participants' demographic information, including age, gender, vision condition, and occupation. Eleven~(N=11) participants were totally blind, and eight~(N=8) were low vision. Seven were female, and twelve were male. Participants were between 18 and 42 years old~($M=31, SD=4$).
\rv{It was worth noting that most of our participants were not tech-savvy, who mainly utilized the screen readers and Voice-over to access live-streaming platforms.}
Our participants were recruited by sending direct messages on Douyin or sending messages to their WeChat posted in their profiles on Douyin or snowball sampling. We searched on Douyin, using BLV-related keywords~(e.g., blind, visual impairment, promote accessibility, etc.), and sent messages directly to the eligible streamers. We contacted 82 streamers directly in total, and 6 were willing to accept our interview. Through snowball sampling, we recruited another 13 participants. It was worthy to note that the BLV participants we recruited had diverse educational levels, ranging from high school degree or less to bachelor's degree.
We watched and engaged in their live-streaming two weeks before the interviews to build rapport and trust with our participants. We also watched their short videos, ``like'', and left comments on the videos. At the same time, their live-streamings and short videos were recorded and saved.
\subsection{Procedure}
\label{Procedure}
We observed their live-streaming practices~(e.g., topics chosen, hashtags added, interacting approaches, etc.). Meanwhile, we checked their profiles to gain more useful information the streamers disclosed. This process helped us prepare for semi-structured interviews.
We conducted the interviews between June 22 and August 12 in 2021. All interviews lasted approximately 50 minutes to 3 hours via WeChat voice call, and participants were provided with 100 CNY \rv{(about 16 US dollars)} after interviewing. All interviews were conducted in the participants' native language~(i.e., Mandarin Chinese) and audio recorded.
We aimed to understand participants' perceptions of algorithms employed by living-stream platforms. We first their general understanding of algorithms. We then asked them how algorithms worked on living-stream platforms. We further asked about their perceptions of the factors related to the algorithmic evaluation of content, the perceived algorithm-related challenges they experienced, and their coping strategies.
\subsection{Data Analysis}
\label{Data Analysis}
Our data included the audio recordings of the interviews, recordings of their live-streaming, short videos they posted, and research field notes. We first transcribed the audio recordings verbatim. Then two native Mandarin-speaking authors coded the transcripts independently using an open-coding approach~\cite{corbin2014basics}. In the regular weekly meetings, the two coders discussed the codes (e.g., definitions and example quotes) and resolved conflicts to consolidate the code book. If there were any unresolved conflicts, the rest of the authors joined the discussion to gain a consensus. Afterward, the research team performed affinity diagramming~\cite{hartson2012ux} to group the codes into clusters based on their semantic similarity, and we identified the themes of the clusters. These themes and codes, along with representative quotes, form the structure of our findings, which will be reported in the next section.
\section{RESULTS}
\label{RESULTS}
\input{picture_findings}
\rv{The interviewed streamers streamed various contents, such as sharing their daily lives, sharing their professional lives, answering questions about visual impairments, explaining how to use assistive technologies (e.g., screen readers), and socializing (Figure 1) For example, P2 chose to share her daily life with her guide dog through livestreams and short videos to dispel the general public's misunderstanding of guide dogs. One such example was as follows, \textit{``Guide dogs were assistance dogs for people with visual impairments and were well trained. They (guide dogs) would not make any trouble for others.''}}
Next, we present our findings on \rv{three} themes: 1) \textit{Perceptions of Factors Contributing To Disadvantages Under Algorithmic Evaluation}; 2) \textit{Perceived Algorithmic Suppression};3) \textit{Mitigation Strategies.} \rv{ Figure~\ref{fig:schematic} shows how these themes and sub-themes are connected with each other.}
\subsection{Perceptions of Factors Contributing to Disadvantages Under Algorithmic Evaluation}
\label{Challenges with Douyin Algorithmic suppression and Mitigation Strategies}
Participants perceived the algorithms to own a great deal of power, assessing whether their content could pass the “examination” from Douyin.
For instance, P17 expressed her perceptions of which factors of a user's content played essential roles in influencing the content evaluation from the algorithms, which was informed by the several articles on new media and word of mouth from Douyin users.
\begin{quote}
\emph{``There were many metrics in terms of content evaluation, such as the completion rate of your videos, the time users spent on your live room, the number of video views, the `likes' and comments, as well as the virtual gifts you received. It was difficult for us to fit into these rules due to our visual disability.''} (P17)
\end{quote}
Participants felt that accessibility challenges with Douyin's user interfaces resulted in disadvantages under algorithmic evaluation. The accessibility challenges were related to \textit{difficulties in ensuring visual effect filters}, \textit{obstacles to keeping active in livestreams}, and \textit{barriers to video editing}.
\subsubsection{\textbf{Difficulties in Ensuring Visual Effect of Filming and Filters}}
\label{BLV Content Creators' perceptions of how Douyin's algorithms work}
Participants described that they encountered great difficulty ensuring the visual quality of the content they created. Due to visual impairments, they could not set a proper camera angle that captured the scenes they intended to show. As P2 said, \textit{``I wanted to film the whole body of my guide dog, yet only captured his head in the live-streaming.''} P15 noted that he was often unaware that he had been off the shooting range: \textit{``It was inevitable to move my body during live-streaming, which resulted in only showing my shoulders rather than my upper body in the camera. However, I did not notice it at the moment due to my sight loss.''}
\st{While prior work showed that Instagram users with low vision could apply filters to photos to achieve a specific aesthetic effect, it did not mention the usage of visual filters in live-streaming among BLV streamers. Our work filled this gap and revealed that BLV streamers, especially those who were blind, were unable to apply visual filters to short videos and live-streaming.} Our participants, especially those who were blind, expressed that they were unable to apply visual filters to short videos and live-streaming. The reason for this frustration was they had no idea about the effects that the filters would achieve. P3 explained, \textit{``It seems that visual filters uniquely belong to sighted users. I can not imagine what will happen to my appearance after turning a filter on.''} As a result, image management was also challenging for BLV streamers: \textit{``We could not dress up as sighted people did, and some of \rv{people with visual impairments}'s eyes may not look pretty or even normal to sighted people.''}~(P6)
\textbf{Participants often had difficulties creating visually-engaging content and felt that their content was not valued by Douyin's content promotion algorithms.} They believed that Douyin's algorithms assessed the aesthetic of content by the conventional beauty standards of sighted people. As P15 explained, \textit{``The slogan in Douyin's advertisement reflected the value of Douyin's algorithms: to explore beautiful things. ''
In addition, P6 described that BLV content creators, especially those who were blind, often could not shoot their faces from a proper position, and felt this could be an important reason why some potential viewers may skip his livestreams.
\begin{quote}
\emph{``The BLV content creators could not see.~(When using cell phones to shoot,) the camera may capture part of his or her face, even only one eye or the nose…some of BLV content creators were getting too close to the camera that the viewers could only see his or her face. \st{Those viewers may quickly swipe after seeing such images.}~\rv{One of my sighted friends told me, `Several short videos created by people with visual impairments randomly appeared on my For You Page. The visual effects were awful, because I could only see a nose or a forehead rather than a face. Therefore, I quickly skipped those videos.'}''}
\end{quote}
Participants also thought that their shortcomings in managing presented images made their content disadvantaged under the algorithms. As P19 noted,
\begin{quote}
\emph{``Given that \rv{people with visual impairments}'s visual status, we could not dress up as sighted people did, and we had no idea how to use visual filters and the effects of filters. Some viewers thought we were not good-looking, then left our live room or closed our videos.~\rv{For example, I was randomly paired with a sighted user (by Douyin algorithms), when he noticed my looking, he just said, `Oh, how ugly!' and left the live-stream immediately.}''}
\end{quote}
As a result, participants perceived that the algorithms would think BLV users' content was not worthy of being promoted. As P15 said, \textit{`` Some audiences were not willing to stay and would quickly swipe our videos, not even mention leaving comments or clicking "Like". (We believe)This led to a low completion rate as well as a low score from the algorithms.''}
\subsubsection{\textbf{\st{Interaction with Viewers}~\rv{Obstacles to Keeping Active in Livestreams}}}
\label{BLV Content Creators' perceptions of how Douyin's algorithms work}
Unlike sighted streamers who can instantaneously view and respond to onscreen text comments, BLV streamers could not immediately know the messages that their viewers typed in the comment area because they needed to suspend the live-streaming and touch the screen to listen to what the viewers said with the help of a screen reader. Also, participants noted that compared to sighted streamers who could skim through comments, it was inevitable for BLV streamers to spend a significant amount of time listening to the entire comments read by the screen reader sequentially. As P9 said, \textit{`` It is inefficient to interact with the audience since the time lag exists in every step of reading comments.''} Additionally, participants expressed that it was challenging to read every comment from viewers during live-streaming because the list of comments kept automatically scrolling in real-time when the audience typed new comments. P6 stated, \textit{``I need to touch the comments one by one. Each time I listen to a comment, I can not touch the following one since the list of comments has scrolled quickly to the bottom for showing the newest messages.''} Participants expressed the concern that such challenges with reading and responding to the comments resulted in the misunderstanding from the audience. As P3 said, \textit{``They may have thought that I intentionally ignored their comments.''}
\textbf{Participants believed that the barrier to actively and timely interacting with the audience resulted in disadvantages under Douyin's algorithmic metric.} Their' comments highlighted beliefs that Douyin's algorithms valued streamers who could keep talking during live-streaming sessions. As P18 said,
\begin{quote}
\emph{``\rv{We believe that} the algorithms could detect the streamers' voices in the live room. You needed to keep talking or singing in order to make a good impression on the algorithms. Otherwise, the algorithms may oppress your live-streaming.''}
\end{quote}
Such evaluation metrics from the algorithms were perceived as challenging by participants. For example, P18, who streamed to sing songs, found it challenging to keep singing due to the visual impairment: \textit{``It was impossible to sing while looking at the lyrics because I could not see. And I often forgot the lyrics, which induced me to stop singing. Therefore, the algorithms must dislike my live-streaming. ''}
In addition, participants hold the opinions that the algorithms valued those live-streams where streamers actively interacted with viewers because active interaction could contribute to longer viewership duration. However, BLV users could hardly interact with the audience as soon as possible, which resulted in misunderstandings among viewers. As P18 explained,
\begin{quote}
\emph{``I needed to touch the messages in the comment area to know what happened. Given that I could not keep touching the screen reader during the whole session, it was tricky to say hello to viewers quickly or reply to their comments immediately. \st{They may think I ignored them and left my live-streaming.}~\rv{Sometimes I received complaints from viewers, they made comments like `why was this streamer ignoring audiences? it was impossible not to see these messages!' When I noticed their comments (by touching the screen) and was about to reply, I found they had left.}''}
\end{quote}
This was echoed by P6, who could not show timely appreciation for the audience that gave virtual gifts to his live-streaming. \textit{``They~(the audience) must think I was rude, feeling uncomfortable, and leaving my live room, which led the algorithms to perceive my live-streaming as not attractive.''}
\subsubsection{\textbf{Barriers to Video Editing}}
\label{BLV Content Creators' perceptions of how Douyin's algorithms work}
Participants described that it was difficult to edit the video they shot due to the complex and visually-demanding interfaces of video editing software and the compatibility problems of screen readers: \textit{``the screen readers worked awfully on the video-editing application associated with Douyin...many functions could not be read, such as inserting music, adjusting the volume and locating a particular spot in progress bar...''}~(P14) Moreover, in addition to inaccessible user interfaces of video editing tools~\cite{Seo2020UnderstandingTC}, our participants reported a lack of description of their video's content, which offered them little control over the editing process, such as feeling hesitant about the segments that they should clip out.
\textbf{Participants believed that the inaccessibility of video editing interface led to the difficulties in creating content that was valued by Douyin's algorithms.} Participants felt that the algorithms appreciated the short videos that were well-edited according to Douyin's template. But such video creations relied highly on one's visual ability. For instance, P15, who was low vision, noted that the prevalent videos on the platform were generally following Douyin's templates: \textit{``Those videos were elaborate. In the first three seconds, you had to show affluent information like the template. You also needed to switch the scene precisely according to the rhythm of the background music.''}
P15 went on to express the difficulty of meeting such video editing standards.\textit{``Creating such videos required you to edit through looking at the screen and manipulating the complex interfaces, which was highly visually-demanding. ''}
This perception of the challenge to fit into the Douyin video template was echoed by P17. Basic video editing skills could hardly be performed due to visual disability. For instance, clipping invalid frames from the beginning of short videos. \textit{``The first three seconds were crucial. Once the video could not show attractive or informative content in the beginning few seconds, viewers may skip my video. As a result, the completion rate would be terrible.''}
In addition to accessibility challenges with Douyin's user interfaces, participants felt that toxic online behaviors against BLV content creators resulted in disadvantages under algorithmic evaluation.
\subsubsection{\textbf{Misperceptions and Trolls}}
\label{BLV Content Creators' perceptions of how Douyin's algorithms work}
Participants shared negative experiences about being trolled due to the misperceptions of viewers: \textit{``They doubted whether I was really blind since they did not know the existence of assistive technology, thinking it was impossible for us to use smartphones, not to mention streaming.''}~(P3) \textbf{Participants took internet trolls as another negative impact on algorithmic evaluation.} They believed that Douyin's algorithms would rate their content as detrimental to the community due to being maliciously reported by trolls: \textit{`` My live-streaming and videos were reported as defrauding by trolls, who thought I pretended to blind in order to obtain money. Being reported must result in ruining algorithms' impression on my content''}~(P13) Participants also \st{assumed}~\rv{expressed} that trolls \st{might} intentionally clicked ``not interested'' for their videos, which contributed to negative input into the algorithms' feedback loop. \rv{P8 noted,``some trolls made comments below my videos, said that `Why their (people with visual impairments) videos still show on my For You Page? I have already clicked `not interested' for many many times! I really do not want to see them angling for sympathy.'''} In some cases, participants noted that some viewers made outrageous comments
to disturb their live-streaming. Such a trolling behavior resulted in losing viewers of \rv{people with visual impairments}'s live-streaming: \textit{``They kept calling me a fraud and said I was consuming the sympathy of Douyin users to attract followers. Other viewers \st{may} feel annoyed with the conflicts of those emotional comments and choose to leave the live room, ~\rv{for instance, one of my viewers said, `I feel so sorry, I have to leave because I cannot stand such a bitter quarrel.'}... (I believe)the algorithms themselves treated this short length of viewership as low attractiveness''}~(P5).
\subsection{Perceived Algorithmic Suppression}
\label{Challenges with Douyin Algorithmic suppression and Mitigation Strategies}
Participants suffered from two main perceived algorithmic suppression.
\textbf{Failure in Reaching Sighted Audiences}.
Participants generally believed that their content was mainly ``locked'' within the BLV community by the algorithms and was barely amplified to the sighted audience. For example, P7, a blind massage shop owner who wanted to promote the business through Douyin, noticed that most viewers were BLV users, which did not meet his expectations because he looked forward to attracting sighted customers. As he explained, \textit{``Whenever I observed the video view counts provided by the system, I found few sighted users while most were BLV users~\rv{[based on his experience that BLV viewers' IDs often contain terms relating to `A person with visual impairments' or `a blind person')}. Also, the comments were mainly from \rv{people with visual impairments}.''} The perception of BLV streamers' content not reaching sighted users was echoed by P18, who often chatted with sighted streamers at their live-streaming sessions. As she said, \textit{``When they first met me~(at their live room), most of them felt surprised and said, `wow, you are the first BLV user we encountered these years! We never saw any content created by BLV community members before.'''} P18 further noted, \textit{``They~(sighted streamers) later told me that they eventually saw the BLV-related content only if they intentionally searched the keywords at Douyin, such as blind streamers.''}
Participants' comments spoke to a belief that the algorithms automatically detected their content and account interactions. The algorithms were perceived to mark the account network from followers or viewers who re-posted, liked, or left comments. As a result, they believed that once the algorithms found users with similar networks and behaviors, those users would be confined and categorized as a group with a dedicated user profile. BLV users, under such circumstances, could hardly send the content to massive sighted viewers. For instance, P14, a grocery store owner who wanted to promote goods to sighted people through Douyin while mostly followed by BLV users, elaborated as follows:
\begin{quote}
\emph{``My online business plan failed...one of my videos were viewed and reposted by several BLV users, which induced more BLV users to follow my account. Then the algorithms may consistently amplify my content to BLV users. As a result, this process kept looping, which enhanced the algorithms' belief that it made the right decision to recommend my content only within the BLV community.''}
\end{quote}
Participants' remarks expressed a perception that the more BLV-related information an account had, the more likely the account was identified as and clustered into the BLV user group. Hence, BLV users may find it more challenging to promote content to sighted audiences.
\textbf{Suppressed Popularity within BLV Community}.
Participants' comments highlighted a belief that Douyin's algorithms limited the popularity of BLV-related content within the BLV community. For example, P16, a public-spirited blind streamer who aimed at popularizing how to manipulate current assistive technology for BLV users through Douyin, perceived that the algorithms suppressed his live-streaming. As he explained, \textit{``It was last December, the number of my viewers dropped suddenly.''} P19 went on to note that he and other streamers found they could not purchase the Dou+ service, which could help boost the popularity of live-streaming.
\begin{quote}
\emph{}{``A pop-up window showed that the themes of our content were not suitable to be boosted. You know, once you could not buy such a service, then the popularity of the account was permanently suppressed.''}
\end{quote}
Other participants expressed that not only their live-streaming but also the short videos they created were not valued by the algorithm. P7, a blind content creator who took videos to promote available assistive technology for \rv{people with visual impairments}, shared an example of how Douyin held up his video. As he said,
\begin{quote}
\emph{``I had a video about how to quit the screen readers, I tried to upload it to Douyin several times, but the system just told me I failed to upload the video because the platform was overloaded. However, when I uploaded a non-BLV video, it went through''}
\end{quote}
This perception was echoed by P6, a user with low vision, who observed that compared to sighted users' videos, the window that showed the ``not interested button'' popped up at BLV content creators' videos with higher frequency. As he explained,
\begin{quote}
\emph{``One of my BLV classmates also noticed that such an indicative window seldom showed up in sighted users' content while it often popped up on BLV users' videos. The algorithms may not value BLV-related videos and thus provided the~(not interested) button to suppress our videos further.''}
\end{quote}
Participants' comments about the ``not interested button'' expressed a perception that even for BLV audiences, Douyin's content promotion algorithms may consider the BLV-related videos they viewed to be of low quality and not worthy of being promoted within the BLV community. It is worth noting that participants also perceived that sighted people most likely encountered such a ``not interested button'' when they saw BLV-related content on their For You Page. Participants believed that if sighted viewers clicked the button, the algorithm would likely not recommend their live-streams/videos to sighted populations (as Section 4.1.4
reported).
\subsection{Mitigation Strategies}
\label{Mitigation Strategies: Reaching Sighted Users}
Our findings show that BLV participants took various actions to mitigate the perceived algorithmic suppression. Participants adopted the following strategies related to their \textbf{BLV status}:~\textit{Hiding BLV Identity}, \textit{Creating Counter-intuitive Content for Sighted Viewers}, ~\textit{Leveraging Occupational Convenience to Attract Sighted Followers}, and \textit{Collectively Ignoring The ``Not Interested'' Button on BLV-related Videos}.
\subsubsection{\textbf{Hiding BLV Identity}} Participants noted that the hashtags of videos played a significant role in algorithmic user profile grouping and automatic viewership setting. Thus, participants intentionally avoided using hashtags that include BLV-related information, such as blind people, screen readers, etc. For example, P17, who intended to promote his massage shop to sighted people, shared his strategies.
\begin{quote}
\emph{``I used to set the hashtags as blind massage, which attracted many BLV users to view my videos. It would make my situation worse if I kept using this hashtag. Thus, I decided to rename the hashtag as healthy massage, keeping a healthy lifestyle, or osteopath. It worked well!''}
\end{quote}
In addition to not using BLV-related hashtags, some participants chose to hide their BLV identity completely. They created new accounts without showing any BLV-related information because they believed intuitive connections with BLV identity led the algorithms to confine their content within the BLV community. They detailed their strategies: \textit{avoid disclosing BLV identity by user name}, \textit{wearing sunglasses to cover their eyes}, \textit{avoiding following or being followed by BLV users}, and \textit{
never posting BLV-related content}. As P6 described, \textit{``It felt like creating an account of a sighted user.''} P6 further described that he only posted non-BLV-related content, such as what happened around the city he lived in, and kept following nearby sighted users, expressing how this strategy worked. \textit{``Some of them followed me back. When my sighted followers reached around 200, I found that some of my videos were viewed more than thirty thousand times by sighted people!''} P6 went on to share his experience about reassuring his video was promoted to nearby sighted users. \textit{``I encountered some unacquainted sighted people when I hung out. They said, 'wow, I know you, you are that [user name] on Douyin. I often watch your videos.' ''} Participants noted that they used the new accounts and the prior BLV-related accounts simultaneously since they perceived maintaining two accounts as a way to balance the tension of promoting content to sighted users and sustaining BLV identity. As P19 said, \textit{``I do not mean to repel the BLV community since I am a member of it. So I use two accounts, one for following and interacting with BLV users, and another for promoting my content to sighted audiences. ''}
\subsubsection{\textbf{Creating Counter-intuitive Content for Sighted Viewers}} Participants described that it was helpful to promote the counter-intuitive content that challenged Chinese sighted users' stereotype about \rv{people with visual impairments} because most of the Chinese sighted people were unfamiliar with the life of the BLV community. For instance, P7 noted that as long as the BLV users posted some videos about independently completing the tasks that sighted people perceive as impossible for a BLV person, such videos would most likely become prevalent within the sighted user community. As he explained,
\begin{quote}
\emph{``It was a stereotype of mainstream society that \rv{people with visual impairments} could not independently do many things. I witnessed some BLV users' videos that sighted people widely reposted. Those videos showcased how \rv{people with visual impairments} cooked, made coffee, surfed the internet, or built computers by themselves, which introduced strong contrast for sighted users.''}
\end{quote}
P7 went on to describe how such content was reached to the sighted community. \textit{``Even if the algorithms do not actively recommend such content to sighted people, one sighted user may feel inconceivable from his or her random browse. Then the content will be shared and reposted one by one...eventually become widespread.''}
\subsubsection{\textbf{Leveraging Occupational Convenience to Attract Sighted Followers.}} In addition, some participants described how they leveraged occupational convenience to attract sighted followers. P19, a blind massage shop owner, shared how he promoted his Douyin account to more sighted viewers by providing discounts to customers who were willing to follow and further promote the account. He elaborated on it: \textit{``Customers got a 20\% off for next visit if they successfully recommended more than three new followers.''} Other BLV streamers employed similar strategies to attract sighted audiences. P6 reported that \textit{``Some BLV masseurs may stream how they provide massage services to sighted clients...combining with tagging geolocation at the nearby downtown. It was effective to attract sighted followers while promoting business.'' }
\subsubsection{\st{\textbf{Providing Mutual Support within BLV User Community}}} \st{Consistent with prior work, BLV users mutually followed BLV accounts, liked and commented on BLV-related content to resist the perceived algorithmic suppression within the BLV community. It was worth noting that participants also described some mutual support strategies that were not discussed by previous research. Participants with more followers described how they helped the less popular users. For instance, P16, a popular BLV content creator in the BLV Douyin community, combined the video content from himself and other users, and the mixed video would be posted on Douyin. As P16 said, \textit{``As my channel has more subscribers, the mixed content posted on my channel with the same theme could help promote others' accounts.''} Similarly, P19 noted that he created a WeChat group that included all of the BLV streamers he knew, which could help them promote their content within the group. He explained, \textit{``There were more than four hundred people with visual impairments within the group. I thought it would more or less help attract some attention for those with few followers.''}}
\subsubsection{\textbf{Collectively Ignoring The ``Not Interested'' Button on BLV-related Videos}} Different from prior work~\cite{simpson2021you}, in which LGBTQ TikTok users used the ``not interested'' feature to block the promoted contents that did not align with their identities, our participants noted that they chose to collectively ignore the pop-up window containing the ``not interested'' button, which frequently appeared on BLV users' short videos. They perceived clicking the ``not interested'' button as a way of unconsciously being the conspirator with the algorithms. As P6 explained, \textit{``The algorithms do not like our content; I can not click that button to enhance its value of suppressing BLV content.''} P18 stated that avoiding clicking the “not interested” button was a way to support BLV users. As she said,
\begin{quote}
\emph{``It was challenging for BLV users to create videos on Douyin. I would absolutely not click that button even though I was not too fond of that video since it may discourage their passion for creating content. Instead, I would view the whole videos if they showed up on my For You Page to boost the completion rate.''}
\end{quote}
As reported in section 4.1.4, participants believed that sighted people who encountered and clicked such a button would prevent their streaming from being recommended to sighted viewers. Therefore, ignoring the ``not interested'' button was also perceived as a manner to \textit{``avoid making a bad situation worse''} (P6) and resist the perceived biases against amplifying their contents to sighted viewers.
\subsubsection{\st{\textbf{Leveraging Trending Topics within BLV Community}}} \st{Participants reported that even if their content was not amplified within the BLV community, it was still practical to spread their content through leveraging trending topics within their community. For example, P16 shared that setting the hashtags as the name of White Cane Safety Day led his video to be viewed by more than 100 thousand times. \textit{``The view counts were surprising. I just used that hashtag and made a video to express my festival greetings for the people with visual impairments.''} P7 noted that videos criticizing how people with visual impairments were treated unfairly in daily life had wider dissemination.}
\begin{quote}
\emph{``\st{People with visual impairments are mostly sensitive about the unfairness of the mainstream society, such as a rejection of boarding experience from an airline company because of one's visual impairment. Other BLV users, feeling strong engagement and compassion, will repost or share the content again and again.}''}
\end{quote}
Moreover, participants also adopted the following strategies \st{\textit{unrelated to BLV status}}~\rv{that \textit{may not uniquely belong to BLV community}}. ~\rv{Although other types of streamers may apply these tactics to mitigate the misalignments between expectations and algorithmic decisions, our findings provided affluent details about how these strategies worked within the BLV community.}
\subsubsection{\textbf{Tagging Geolocation at Downtown Areas that Had More Potential Target Viewers}} Participants perceived setting geolocations at the downtown areas as an effective way to resist perceived content suppression from the algorithms. They perceived that the platform seemed to have independent algorithms for geolocation and user profile based on behaviors and interactions patterns. Thus, BLV users could amplify their videos to the sighted users using geolocation tags. P17 reported that he tagged the geolocation associated with videos at a nearby supermarket, finding it useful, \textit{``The number of viewers increased obviously.''} Similarly, P6 said that he tended to tag the location at a middle school located in the downtown area. As he expressed,
\textit{``This strategy worked, even though the algorithms generally recommended my videos within the BLV community. Given the algorithms about recommending videos to nearby users mainly relied on geolocation, tagging the location could more or less 'open a small window' to help amplify my content to some sighted people.''}
\subsubsection{\textbf{Actively Interacting with Target Users}} Participants described that they actively participated in sighted users' live-streaming since they believed that attracting sighted followers was a straightforward way to promote their content to sighted audiences. P9 expressed that he often took the initiative to click like, write lots of comments, and send gifts to sighted users' live-streaming, which led to attention from sighted streamers. As she said,
\begin{quote}
\emph{``They remembered me because of my activeness. After knowing I was a blind user, they tried to help promote my account by introducing me at their live-streaming, advocating for sighted audiences to follow me, and allowing me to join their live chatting without charging gifts. Some of them even proactively joined my live-streaming for the whole session.''}
\end{quote}
Participants also shared their strategies on how they engaged with experienced sighted streamers' live room to learn Douyin's algorithmic impacts. For example, P18, a blind streamer, reported how she learned to play with Douyin's algorithms from those sighted streamers to sell her craft to sighted viewers. As she said,
\begin{quote}
\emph{``I learned a ton. I realized that I needed to keep posting craft-related videos rather than a variety of content...it could persuade the algorithms to tag me as a craft content creator, then precisely promote my live-streaming and short videos for those who felt interested in it.''}
\end{quote}
In some cases, participants indicated that actively participating in ``PK''~(i.e., a live video feature in Douyin that enabled two streamers to interact in real-time) with random sighted streamers could effectively attract sighted followers. For instance, P9 expressed that he found all streamers that the system randomly assigned were sighted users. He further noted how he seized the chance to attract those sighted users to follow him. As he explained,
\begin{quote}
\emph{``If the sighted streamers did not end the `PK' after knowing I was blind, it meant that they would like to chat with blind people. They often asked questions about the life of blind people, such as how I used the smartphone...I patiently answered their questions for making a friendly impression, which resulted in being followed by them afterward.''}
\end{quote}
\subsubsection{\textbf{\rv{Sticking to Trending Topics.}}} \rv{Participants reported that even if their content was not amplified within the BLV community, it was still practical to spread their content through leveraging trending topics within their community. For example, P16 shared that setting the hashtags as the name of White Cane Safety Day (which was set in an effort to celebrate the achievement of people with visual impairments) led his video to be viewed by more than 100 thousand times. \textit{``The view counts were surprising. I just used that hashtag and made a video to express my festival greetings for the people with visual impairments.''} P7 noted that videos criticizing how \rv{people with visual impairments} were treated unfairly in daily life had wider dissemination.}
\begin{quote}
\emph{``\rv{People with visual impairments are mostly sensitive about the unfairness from the mainstream society, such as a rejection of boarding experience from an airline company because of one's visual impairment. Other BLV users, feeling strong engagement and compassion, will repost or share the content again and again.}''}
\end{quote}
\subsubsection{\textbf{\rv{Peer Support}}} \rv{Consistent with prior work~\cite{karizat2021algorithmic}, BLV users followed each other on the platform, liked and commented on BLV-related content to resist the perceived algorithmic suppression within the BLV community. Moreover, participants also described mutual support strategies that were not discussed by previous research~\cite{karizat2021algorithmic, wu2019agent, simpson2021you}. Participants with more followers described how they helped the less popular BLV streamers. For instance, P16, a popular BLV streamer, combined his video content with other BLV streamers' video content to create mixed videos and posted them on Douyin. As P16 said, \textit{``As my channel has more subscribers, the mixed content posted on my channel with the same theme could help promote others' accounts.''} Similarly, P19 noted that he created a WeChat group that included all of the BLV streamers he knew to help them promote their content within the group. He explained, \textit{``There were more than four hundred \rv{people with visual impairments} within the group. I thought it would be more or less helpful to attract some attention for those with fewer followers.''}}
\subsubsection{\textbf{Directing More Comments From other Platforms}} Some participants described that the alternative strategy they employed was to have as many comments as possible. For example, P16 was a blind streamer who often answered accessibility-related questions from \rv{people with visual impairments} on Wechat; To direct more comments to his videos, he chose to transfer the Q\&A from WeChat to the comment area of a particular video. As he said,
\begin{quote}
\emph{``I hoped people use the comment area under the video as a place for chatting, where they enjoy more interactive chats and contribute more conversations. Moreover, other BLV users interested in the topic will join the conversations and feed more comments. ''}
\end{quote}
\subsubsection{\textbf{Negotiating with Platform's Authority}} Some participants described that they actively negotiated with the Douyin authority, hoping that Douyin could help address the challenges regarding suppressed popularity within the BLV community. P19 expressed how he reached out to Douyin customer support. As he explained, \textit{``I complained to them that I could not purchase the Dou+ service; it was unfair. I asked them whether they could ask algorithms engineers about what happened to my account.''} Similarly, P7 stated that he contacted Douyin customer service and complained: \textit{``The video could not be uploaded for several times, I did not buy the reason of platform overload. It must be held up by your algorithms!''} However, the customer service did not respond to participants' complaints.\newline
\section{DISCUSSION}
\label{DISCUSSION}
While prior work ~\cite{10.1145/3476038} described BLV streamer's motivations (e.g., achieving personal goals, delivering messages, overcoming limitations of recorded videos, getting inspired by other streamers) and accessibility challenges (e.g., multitasking, screen reader related issues), our findings extended this line of research ~\cite{10.1145/3476038} by uncovering factors contributing to disadvantages under algorithmic evaluation, and the perceived challenges with algorithmic suppression and mitigation strategies. Based on the findings, we present the following design considerations for building a more accessible and inclusive live-streaming ecosystem.
\subsection{Design for Inclusive Computer Vision Technology}
\label{How to Help BLV people Ensure Proper Shooting Angles?}
One common recurring difficulty was adjusting the shooting angles properly to ensure that a BLV streamer's face and body were within the camera frame when streaming due to visual impairment. Current face-tracking technology shows the promise to address this challenge, which could detect and locate objects or faces in nearly real-time~\cite{wagner2009multiple,lei2011real,naik2015robust,ataer2016object} on low-end devices like mobile phones~\cite{chen2009streaming}. However, BLV content creators' streaming behaviors complicated the utility of face-tracking technology. Some BLV streamers did not want to show their eyes and thus always wore sunglasses, which made it challenging to track their face since current face-tracking technologies rely on capturing the whole parts of a human face ~\cite{anwarul2020comprehensive,kaur2020facial}. Also, some BLV participants streamed in nearly total darkness because they were not sensitive to the indoor light conditions due to visual impairments, which deteriorated the accuracy of face-tracking technologies ~\cite{rahman2020face}. We suggest that algorithm designers may train the face recognition model with as many BLV streamers' videos as possible.
Additionally, when applying face-recognition technology in BLV users' live-streaming, it is essential to take into consideration how to avoid leaking private information. For example, the faces of the BLV streamer's family members or the photos of faces~\cite{gurari2019vizwiz} in the streamer's room might be accidentally captured. Prior research indicated that privacy was a core value in system design~\cite{friedman2013value} and \rv{people with visual impairments} were concerned about their privacy when using technologies~\cite{10.5555/3361476.3361478,10.5555/3235866.3235879,10.1145/3234695.3236342}. To protect users' privacy, Zhou et al. developed a technology to blur irrelevant people's faces during live-streaming~\cite{9218980}. However, many questions remain unresolved. For example, what visual content, in addition to faces, do BLV streamers perceive as private? Would failures of blurring different privacy-sensitive content cause equal concerns to BLV streamers?
\subsection{Design for Inclusive Virtual Identity Management}
\label{How to Better Help BLV Streamers Do Image Management without Compromising Viewers' Experiences?}
Some participants with eyeball atrophy or who were not good at dressing up also mentioned that they encountered barriers in image management. They worried that their appearance would leave a negative impression on the audience. Hence, some of them tended to wear sunglasses and even turned off the camera directly during streaming. This raises an interesting question of how to help BLV streamers better manage images without compromising viewers' experiences? A possible solution could be to explore the use of virtual idols on short-video platforms that support live-streaming~\cite{lu2021more,black2012virtual,freeman2020my,yin2018vocaloid}. \rv{A virtual idol is a 2D or 3D animated virtual avatar with the voice of a human, which could be operated by individuals or agencies. An increasing number of streamers already started to use virtual avatars in their live-streams, such as voice actors, amateur and professional musicians, and artists~\cite{lu2021more}. One streamer named CodeMico on Twitch used virtual avatars to achieve visual effects, such as blowing herself up and flying into space in the live-streaming, that would otherwise be impossible to perform by humans~\cite{virtual2022code}. Moreover, live-streams with virtual avatars had a broad audience in China~\cite{bilibili2020vtuber,lu2021more}. By June 2021, over 30,000 active streamers conducted live-streams with virtual avatars on Bilibili and gained over 560 million Danmu (i.e., scrolling comments)~\cite{bilibili2020vtuber}. Recently, some content creators with disabilities also began to use virtual avatars. For example, a content creator with paralysis recorded his videos with virtual avatars rather than his real appearance. He commented that without showing his appearance, he could avoid being questioned for sad-fishing with his disability~\cite{disablity2021virtual}. Thus, we believe that} if streamers or video creators with disabilities do not want to show their faces, they could use virtual avatars. However, it remains an open challenge of how to inform BLV streamers about the visual appearances of, and more importantly, the audience's reactions to, their virtual avatars. Towards this direction, designers should consider providing explicit audio descriptions of the virtual avatars' features such as gender, skin tone, hairstyles, makeup, accessories, clothing, and ability status~\cite{bennett2021s} for BLV streamers. It is also essential to \rv{inform BLV streamers promptly about their viewers' reactions and preferences of their avatars}.
In addition, when it comes to enabling BLV streamers to create and craft their virtual avatars, designing BLV user-friendly audio instructions and avatar building interfaces remains unclear. Thus, it is imperative for designers to consider lowering the burden of designing avatars while ensuring design quality. Furthermore, given that some BLV content creators preferred not to show their BLV identities in live-streaming or videos, a potential tension between BLV identity and avatar's appearance may emerge, which calls into questions: Is the BLV streamer willing to show his or her BLV identity through a virtual avatar? What are the viewers' reactions to the avatar that discloses BLV identity? How should BLV streamers balance maintaining disability identity and keeping the avatar attractive? It is worth investigating different stakeholders' perceptions of the avatar design of BLV streamers.
\subsection{Design for Inclusive Content Creation Experience}
\label{Design for Inclusive Content Creation Experience}
\textbf{\textit{Accessible Filters Editing Experience.}} \st{As discussed in Section 4.1.1, BLV streamers expressed that filters on Douyin were inaccessible.} ~\rv{While prior work~\cite{10.1145/3173574.3173650} showed that Instagram users with low vision could apply filters to photos to achieve a specific aesthetic effect, it did not mention the usage of visual filters in live-streaming among BLV streamers. Our work filled this gap and revealed that BLV streamers, especially those who were blind, were unable to apply visual filters to short videos and live-streaming.} For instance, some functional buttons of filters were not labeled, such as the percentage of luminance. Participants also emphasized that they could not picture the visual effects the filters may achieve.
Not only is it essential to ensure all UI buttons have been labeled correctly, but it is also important to provide audio instructions to help BLV streamers develop a prior idea of the functioning of the filters during live streaming.
However, as people's perceptions of the aesthetic of visual content are personal and subjective~\cite{10.1145/3134756}, it can be challenging for BLV streamers to judge whether a visual filter would meet their expectations. Thus, it is important for screen reader designers to investigate ways to make computer-generated descriptions of filters match their expectations of such filters.
Additionally, designers of screen readers need to personalize audio descriptions to different kinds of visually impaired users in order to improve intelligibility. For instance, an audio description comprehensible to people with non-congenital visual impairments may be incomprehensible to people with congenital visual impairments. Similarly, designers also need to personalize assistive editing features for streamers with differing levels of vision impairment. Unlike low-vision people, participants with total blindness noted that navigating the filter interfaces and locating the buttons were challenging. One possible solution is to use voice commands or gesture inputs (e.g., tapping, swiping, or even eyelid gestures) for navigation~\cite{Li2020iWink,10.1145/3308558.3314136,Fan2020Eyelid,Li2020iWink,Fan2021Eyelid,Li2017BrailleSketch}.
\textbf{\textit{Accessible Video Editing Experience.}} Another challenge for assistive editing design is to provide audio descriptions on mobile editing tools (e.g., Jianying, a software that allows users to edit videos on mobile phones and imports videos to Douyin). Although previous works introduced algorithms to generate video descriptions~\cite{Johnson2016DenseCapFC, Krishna2017DenseCaptioningEI,wang2021toward}, such algorithms were designed or trained to recognize videos taken by sighted people. The quality and content of videos taken by \rv{people with visual impairments}, however, may differ from those taken by sighted users. For example, a video shot by a BLV streamer may result in inaccurate or even absurd descriptions for current algorithms due to factors, such as jitters or being out of focus. Therefore, common video quality issues should be carefully considered when designing algorithms to generate audio descriptions of videos shot by BLV streamers. Designers should consider collecting diverse sets of videos shot by BLV streamers in different contexts and training such algorithms on those videos.
Further, as automatic generated audio descriptions may still contain errors, post-verification is needed. For example, previous research mentioned that \rv{people with visual impairments} tend to ask sighted friends or family members to verify the accuracy of computer-generated descriptions of photo, but this also imposes a burden on those sighted people~\cite{10.1145/2998181.2998364, 10.1145/3134756}. This issue may be exacerbated because of the extra efforts required to evaluate video descriptions due to longer length. Therefore, future work should investigate more effective way to validate the quality of video descriptions.
\subsection{Design For Inclusive Algorithmic Experiences}
\label{How to Design More Inclusive Algorithmic Experiences For BLV Content Creators?}
Our finding regarding BLV content creators’ perceived algorithmic barriers mirrored Karizat et al.'s identity strainer theory~\cite{karizat2021algorithmic}, which expressed that the algorithms acted as a strainer to marginalize a variety of social identities, including ability status. However, in ~\cite{karizat2021algorithmic}, there was only one participant who mentioned his or her observation that Tiktok valued the content of able-bodied streamers but oppressed the content created by people with disabilities. We extended Karizat et al. 's work~\cite{karizat2021algorithmic} by providing an in-depth investigation of BLV content creators' perceived challenges from Douyin's algorithms. Participants believed that they were vulnerable under the evaluation metric of the algorithms due to the harmful trolling behaviors and the lack of visual ability to ensure content’s visual appearance, timely interaction with audiences during streaming, and quality of video editing.
Participants perceived that content created by BLV content creators was seldom amplified to sighted audiences. Instead, they believed that algorithms limited the popularity of their content within the BLV community.
Participants adopted BLV status-related strategies to mitigate perceived algorithmic biases, including 1) hiding BLV identity; 2) creating counter-intuitive content; 3) leveraging occupational convenience in massage shops to attract sighted followers; ; and 4) collectively ignoring the “Not Interested” button on BLV-related videos. Among these strategies, “hiding BLV identity” may lead to conflicts with BLV identities. For instance, in order to “trick” Douyin's algorithms to promote their streaming contents to more sighted viewers, participants avoided disclosing any BLV-related information in their accounts or shied away from any engagement with any BLV-related content and community. Indeed, this tactic may help them reach a wider audience of sighted people, but it would unfortunately force them to hide themselves in non-BLV algorithmic identities~\cite{doi:10.1177/0263276411424420}. Such a strategy is contrary to those in previous works ~\cite{karizat2021algorithmic,simpson2021you}, in which marginalized users tamed the algorithms to make their algorithmic identities~\cite{doi:10.1177/0263276411424420} and their self-identities further closer together. Future research may explore whether content creators from other marginalized groups may use their own status-related mitigation strategies. It may also be fruitful to explore whether there exist tensions when trying to preserve identities and tame algorithms in order reach a certain targeted audience simultaneously. Researchers could also explore how \rv{people with visual impairments} may resolve these tensions?
Participants also adopted strategies that may not uniquely belong to the BLV community to mitigate perceived algorithmic biases, including: 1) tagging locations at popular places; 2) directing more comments from other platforms; 3) actively interacting with target users; 4)peer support; 5) sticking to trending topics and 6) negotiating with the Douyin authority. Since these mitigation strategies are non-BLV status-related (e.g., tagging location and direct comments), they may be applicable to other marginalized content creators (e.g., people of color, people of LGBTQ community, people with other types of disability, or those who are socioeconomically disadvantaged) who want to promote their contents to a broader community but perceive their contents being “locked” within a confined group.
Next, we present design suggestions for creating inclusive algorithmic experiences for BLV content creators.
\textbf{Accessible Explanation Interface.} Prior research~\cite{10.1145/3173574.3173860} defined algorithmic experience~(AX) as ``an analytic tool for approaching a user-centered perspective on algorithms, how users perceive them, and how to design better experiences with them”, which included five themes for improving AX in the context of social media: algorithmic profiling transparency, algorithmic profiling management, algorithmic awareness, algorithmic user-control, and selective algorithmic remembering. We suggest that algorithmic profiling management of AX
may be applied to create better AX for BLV content creators, which enables them to modify and tune their profile created by algorithms~(i.e., how algorithms think about them). For example, BLV content creators, who want to promote their
live-streams or videos to more sighted audiences, could adjust the factors contributing to failure in reaching sighted audiences in their algorithmic profiles, making such profiles aligned with their expectations. However, this raises an open question of how to effectively and sufficiently communicate complex internal operations of algorithms to \rv{people with visual impairments}. Current research on Human-Centered AI leveraged interactive visualizations to allow stakeholders to explore the data and audit the algorithms synchronously and asynchronously~\cite{10.1145/3290605.3300789,Cheng2021SolicitingSF,Fan2020VisTA,Fan2022HAC}. However, such visually-demanding interfaces are inaccessible to \rv{people with visual impairments}. Therefore, it is worth exploring ways to design accessible explainable AI interfaces for \rv{people with visual impairments}, such as combining the potentials of accessible visualization~\cite{Lundgard2019SociotechnicalCF} and dynamic shape-shifting user interfaces~\cite{10.1145/3173574.3173865}.
\textbf{Accessible Participatory algorithms Design.} Our study unpacked insights into BLV streamers' perceived algorithmic biases against their BLV identity, which can be difficult to be effectively incorporated into algorithmic design~\cite{10.1145/3274463}. Towards this challenge, participatory algorithm design may be a promising direction to explore~\cite{10.1145/3359283}. Participatory algorithm design involves stakeholders building the computational models that represent their motivations, values, and goals. However, it remains an open question of how to leverage such a participatory algorithm design. For example, how
to ensure the whole model training process accessible to BLV participants? How to deal with the different values and opinions of participants with different levels of visual impairments (e.g., blind and low-vision participants)? How to take the social-economical conditions of BLV participants into consideration to improve the diversity of the design of the participatory algorithms?
\textbf{Social Awareness.} While it is necessary to address the algorithmic challenges by leveraging technical solutions, \textbf{socio-technical solutions are also crucial}. For example, the misperceptions and trolling behaviors from sighted audiences may negatively influence the input into the video-sharing platform’s algorithms feedback loop, enabling algorithms to oppress content created by \rv{people with visual impairments}. Policymakers should make continued effort to mitigate the general public’s stigma towards people with disabilities ~\cite{li2021choose}, addressing the knowledge gap ~\cite{li2021choose} about disability from mainstream society, and strengthening the inclusive awareness of the public. It is also imperative for video-sharing platforms~(e.g., Douyin) to create an inclusive online community for content creators with BLV or other disabilities. Platforms may help promote the content regarding mitigating misperceptions about \rv{people with visual impairments} or promoting accessibility knowledge to a broader audience, outlining respect to people with different ability levels as the community rules, combating the toxic trolling behaviors towards BLV streamers. In addition, participants noted a lack of understanding about how algorithms work to influence content within the less-educated BLV content creators or those located in rural areas. Informed by~\cite{10.1145/3173574.3173860} ’s framework for AX, we suggest that in the future, video-sharing platforms~(e.g., Douyin) should strive for improving the algorithmic awareness of BLV content creators who are vulnerable in education or social-economic status, telling them explicitly how algorithms operate and what sorts of user behaviors affect the feedback loop of the algorithms. Furthermore, it is also necessary for algorithm designers to consider and respect the diverse needs of people with different abilities instead of just focusing on the algorithmic performance.
\textbf{Algorithmic Identity.} The perceived suppression from the algorithms shaped participants’ behavior to break the feedback loop~\cite{10.1145/2702123.2702174} to promote their content to their target audiences, for example, by manipulating the input to the Douyin's algorithms individually or collectively. While prior work~\cite{karizat2021algorithmic, wu2019agent} noted that influencing the input of the feedback loop had the potential to tilt the algorithms, we found that \textbf{a tension emerged when the BLV users wanted to promote their content to both the sighted and BVI community because the strategies they employed for reaching these two communities were paradoxical at times}. For example, the strategies that participants utilized to promote content to sighted people, such as hiding BLV identity, increasing view and like count of sighted people, attracting sighted followers, resulting in continuously influencing the input into algorithms, which might gradually enhance the algorithms’ belief that their content should be recommended to more sighted audiences. This strategy worked at the expense of abandoning the opportunities to make their content show at BLV users’ FYP~(for you page). This tension was echoed by P9, noting \textit{ ``You could only choose one side~(sighted or BLV community) under the current rules of the algorithms. ”} The aspect of algorithmic user-control in Alvarado and Waern’s AX framework~\cite{10.1145/3173574.3173860} showed the potential to address this tension, which empowered users to control the algorithms directly. For instance, algorithm designers can design a feature that enables BLV content creators to craft and switch between different algorithmic profiles~(i.e., promoting content to different categories of viewers ) to click a switch button on their profile pages directly and conveniently.
\section{LIMITATIONS AND FUTURE WORK}
\label{LIMITATIONS AND FUTURE WORK}
In this work, we studied the practices and challenges of BLV streamers when using user interfaces to create and stream contents that were affected by both the algorithms of Douyin and their corresponding mitigation strategies in China. Live-streaming platforms, such as Douyin, Kuaishou, and Tiktok, adopt similar content promotion algorithms. Thus, we expect that many of the findings might be applicable to such similar live-streaming platforms. However, different live-streaming platforms also adopt some unique content promotion algorithms that may affect their user experiences. Moreover, the culture and audience differences exist between different platforms. For example, while Douyin mainly has Chinese-speaking audiences, Tiktok has mostly English-speaking audiences. More research is warranted to investigate whether and to what extent the perceived algorithmic biases and other practices and challenges differ between different platforms for BLV streamers.
Our work uncovered BLV streamer's \textbf{perceived} biases of Douyin content promotion algorithms. Future work should adopt more objective approaches, such as analyzing promotion algorithms themselves or the promoted streams, to study whether such perceived biases objectively exist.~\rv{Moreover, future work should also investigate how BLV content creators audit algorithms (i.e., developing and testing hypotheses about observed problematic algorithmic behaviors), and what the ramifications of algorithmic biases, if any, may be.}
Furthermore, our study examined the live-streaming practices and challenges of people who are blind or have low vision. It remains unknown whether and to what extent live-streaming platforms are accessible to streamers with other forms of disabilities, such as hearing or motor disabilities, and whether these streamers perceive biases in the content promotion algorithms. To this end, it is imperative to investigate the accessibility of the user interfaces and the experiences of the content promotion algorithms of live-streaming platforms for streamers with different disabilities. Such efforts, along with ours, would collectively help identify issues to be addressed to make live-streaming ecosystems a more inclusive place for people with different disabilities.
\section{CONCLUSION}
\label{CONCLUSION}
In this paper, we have presented a qualitative study of BLV Streamers' perspectives on algorithms of the live-streaming platform, Douyin. We presented the findings from a semi-structured interview with nineteen BLV streamers.
We identified the BLV content creators' perceptions of factors (i.e., challenges with user interface and toxic online behaviors against BLV content creators) that have negative impacts on algorithmic evaluation of their content. We then identified the perceived algorithmic barriers of BLV users as content created by people \rv{with visual impairments} was not amplified to sighted audiences, and the popularity of BLV related content was limited within the BLV community. Followed by the perceived algorithmic challenges, we identified the mitigation strategies people \rv{with visual impairments} employed to make their content reach target audiences, sighted people~(i.e., hiding BLV identities, tagging geolocation at downtown area, creating counter-intuitive content, leveraging BLV-related occupational convenience and actively interacting with sighted users) or BLV users~(i.e., providing mutual support within the BLV community, collectively ignoring the ``not interested'' button on BLV-related video, directing more comments from other platforms, leveraging the trending topic within the BLV community and negotiating with Douyin authority). These findings contributed to our understanding of how BLV streamers perceived they were marginalized and excluded by the Douyin ecosystem, and informed the design considerations for future research in designing a more accessible Douyin interface, and creating more inclusive and equitable algorithm experience.
|
1,314,259,993,612 | arxiv | \section{Introduction}
According to Binney and Tremaine [1], the process of transformation of the universe may be divided into four main eras, which are vacuum energy era (or, Planck era), radiation era $\&$ matter dominated era and the dark energy era (or, de Sitter era). The universe goes through a stage of early inflation in the vacuum energy era (Planck era), that brings the universe from the Planck size $l_{p} = 1.62 \times 10^{-35} m$ to a 'macroscopic' size $a \sim 10^{-6} m$ in a lowest possible fraction of a second [2, 3]. After that, the universe moves into the radiation era and then, the matter-dominated era, whenever the temperature drops down below 103 K approximately [4]. Finally, in the dark energy era, the universe goes through a phase transition of late inflation [5]. The singularity problems are solved in the early inflation [2] whereas late inflation is necessary to explain accelerating expansion of the universe [6-14]. The cosmological results and data sets like Atacama Cosmology Telescope (ACT) [15], Planck 2015 results- XIII [16], are also suggested about late time inflation of the universe and allows the researchers to determine cosmological parameters such as the Hubble constant $H$ and the deceleration parameter $q$. Studying the constraints given by the data from CMBR (Cosmic Microwave Background Radiation) investigation [17, 18], WMAP (Wilkinson Microwave Anisotropy Probe) [19], observations of clusters of galaxies at low red-shift [20, 21] etc., it can be deduced that the universe is dominated by some mysterious components. This mystical component of the energy is called dark energy which has negative pressure and positive energy density (giving negative EoS parameter). In the energy budget, it is believed that about $73 \%$ of our universe is Dark energy, about $23 \%$ is occupied by Dark matter and the usual baryonic matter occupies about $4 \%$. Therefore, the study of the aspect of dark energy has become an interesting topics in the field of fundamental physics [22-28]. Einstein's cosmological constant $\Lambda$ is the best match for dark energy and physically, it corresponds to the quantum vacuum energy. The cosmological model with $\Lambda$ and cold dark matter (CDM) is usually called the $\Lambda$CDM model.\\
\\
In 1919, Einstein geometrizes gravitation in his theory of general relativity which is treated as a basis for a model of the universe. After that, many cosmologists and astrophysicists attempted to study gravitation in different contexts. For the illustration both gravitation and electromagnetism, Weyl [29] tried to formulate a novel gauge theory containing metric tensor. But, due to the non-integrability condition, it does not get importance in the cosmological society. In order to eliminate the non-integrability condition in the Weyl's geometry, Lyra [30] introduced a gauge function into the structure less manifold and recommended an adjustment in it. These customized theories of relativity are termed as the alternate theories of the gravitation or modified theories of the gravitation. Some significant customized theories of gravitation are Brans-Dicke theory [31], Scalar-tensor theories [32], Nordvedt [33], Saez and Ballester [34], Vector-tensor theory [35], Weyl's theory [36], F(R) gravity [37], mimetic gravity [38], mimetic F(R) gravity [39], Lyra geometry [40] etc. These customized theories of gravitation may be used to study the accelerating expansion of the universe. Amongst these customized theories of gravitation, here, the Lyra geometry is discussed. Soleng [41] shows that the gauge vector $\phi_{i}$ in Lyra's geometry will play either the role of creation field Hoyle [42] (equal to Hoyle's creation field [42]) or cosmological constant. Sen [43], Sen and Dunn [44], Rosen [45] are some well-known researchers who have studied scalar-tensor theory of gravitation on the basis of Lyra geometry. Halford [46, 47] recommended that the cosmological constant in the general theory of relativity and the constant displacement vector field $\phi_{i}$ in Lyra's geometry perform the same role and the scalar-tensor treatment in the framework of Lyra's geometry expect the identical result, within observational limits as predicted by Einstein's theory of relativity. Bhamra [48], Karade and Borikar [49], Rahaman [50], Khadekar and Nagpure [51], Rahaman et al. [52], Casana et al. [53, 54], Mohanty et al. [55, 56], Shchigolev [57], Mahanta et al. [58], Darabi et al. [59], Mollah et al. [60], Mollah and Priyokumar [61] etc. are the some authors who studied Einstein's field equations of gravitation in the framework of Lyra's geometry and obtained their solutions effectively under different circumstances.\\
\\
The equation of state in relativity and cosmology, which is nothing but the relationship among combined matter, temperature, pressure, energy and energy density for any region of space, plays an important role. Many researchers like Ivanov [62], Sharma and Maharaj [63], Thirukkanesh and Maharaj [64], Feroze and Siddiqui [65], Varela et al. [66] etc. studied cosmological models with linear and non-linear equation of state. For the study of dark energy and general relativistic dynamics in different cosmological models, the quadratic equation of state plays an important role. Dark energy universe with different equations of state were already discussed by various authors like, Bamba et al [25], Nojiri and Odintsov [67, 68], Nojiri et al. [69], Nojiri and Odintsov [70] and Capozziello et al. [71]. The general form of quadratic equation of state
\[p = p_{0} + \alpha \rho + \beta \rho^{2} \ , \]
where $p_{0} , \alpha , \beta$ are the parameters, is nothing but the first term of the Taylor's expansion of an equation of state of the form $p = p(\rho)$ about $\rho = 0$.\\
\\
Considering a non-linear equation of state (EoS) in the form of a quadratic equation $p = p_{0} + \alpha \rho + \beta \rho^{2}$, Ananda $ \&$ Bruni [72] investigated the general relativistic dynamics of Robertson-Walker models. They have shown that in general relativistic theory setting, the anisotropic behaviour at the singularity obtained in the Brane Scenario can be reproduced. In the general theory of relativity, they have also discussed the anisotropic homogeneous and inhomogeneous cosmological models with the consideration of quadratic equation of state of the form
\[p = \alpha \rho + \frac{\rho^{2}}{\rho_{c}} \ ,\]
and attempted to isotropize the model universe in the initial stages when the initial singularity is approached. In this paper, we have taken the quadratic equation of state of the form
\[p = \alpha \rho^{2} - \rho \ ,\]
is considered, where $\alpha \neq 0$ is a constant quantity, but we can take $p_{0} = 0$ to make our calculations simpler. This will not affect the quadratic nature of the equation of state.\\
\\
Considering an equation of state in quadratic form as $\frac{p}{c^{2}} = -\frac{4\rho^{2}}{3\rho_{p}} + \frac{\rho}{3} - \frac{4\rho_{\Lambda}}{3}$ , Chavanis [73] investigated a four-dimensional Friedmann-Lemaitre-Roberston-Walker (FLRW) cosmological model unifying radiation, vacuum energy and dark energy. Again, by using a quadratic form of equation of state, Chavanis [74] formulated a cosmological model that describes the early inflation, the intermediate decelerating expansion, and the late-time accelerating expansion of the universe.\\
\\
Many authors like Maharaj et al. [75]; Rahaman, F., et al. [76], Feroze and Siddiqui [65] have studied cosmological models on the basis of equation of state in quadratic form under different circumstances. Recently, V. U. M. Rao et al. [77], Reddy et al. [78], Adhav et al. [79] studied Kaluza-Klein Space-time cosmological models with a quadratic equation of state in general and modified theories of relativity.\\
\\
Inspired by the above studies, here, we have investigated a homogeneous and anisotropic Bianchi type III cosmological model universe with a quadratic equation of state in Lyra Geometry setting and find out the realistic solutions.\\
\\
This paper has been put into order as follows: In Sec. 2, we have formulated the problem where the physical and kinematical parameters are defined. In this section, exact solutions of the field equations are also obtained and the graphs of some of the parameters are shown as well. Sec. 3 consists of the physical and geometrical aspects of the derived model. Concluding remarks are given in Sec. 4. In Sec. 5, an acknowledgement to the funding authority finds a place.
\section{Formulation of problem}
Let us consider the homogeneous and anisotropic space-time metric described by the Bianchi type III line element in the form
\begin{equation}
ds^{2}=A^{2}dx^{2}+B^{2}e^{-2{\alpha}x}dy^{2}+C^{2}dz^{2}-dt^{2}
\end{equation}
where $A$, $B$ and $C$ are the scale factors, which are functions of cosmic time $t$ only and $\alpha \neq 0$ is a constant.\\
\\
The Einstein field equations in Lyra Geometry (as taken by Sen [43]; Sen and Dunn [44]) in geometric units ($c = G = 1$) is given by
\begin{equation}
R_{ij}-\frac{1}{2}Rg_{ij}+\frac{3}{2}\phi_{i}\phi_{j}-\frac{3}{4}g_{ij}\phi^k\phi_k=-8\pi T_{ij}
\end{equation}
and the energy momentum tensor $T_{ij}$ is taken as
\begin{equation}
T^{i}_{j}=(\rho + p) u^{i}u_{j}+pg^{i}_{j}
\end{equation}
where, $u^{i} = (0, 0, 0, 1)$ is the four velocity vector, $\phi_{i}=(0, 0, 0, \beta(t))$ is the displacement vector; $R_{ij}$ is the Ricci tensor; $R$ is the Ricci scalar, $\rho$ is the energy density and $p$ is the pressure so that
\begin{equation}
g_{ij}u^{i}u^{j}=1
\end{equation}
\\
Therefore, we have
\begin{equation}
T^{1}_{1}=T^{2}_{2}=T^{3}_{3} = p ~~ ; ~~ T^{4}_{4} = - \rho ~~~~ \textrm{and} ~~~~ T^{i}_{j}=0 ~~~ \textrm{for all} ~~ i\neq 0
\end{equation}
\\
Also, the general quadratic form of the Equation of state (EoS) for the matter of distribution is given by
\[p=p(\rho)= a\rho^{2} + b\rho + c\]
where $a \neq 0$, $b$ and $c$ are constants.\\
\\
From the field equations (2), the continuity equation is given by
\begin{equation}
\dot{\rho} + \frac{3}{2}\beta \dot{\beta} + \left[(\rho + p) + \frac{3}{2}\beta^{2}\right] \left(\frac{\dot{A}}{A}+\frac{\dot{B}}{B}+\frac{\dot{C}}{C}\right)= 0
\end{equation}
\\
Again, by the use of the energy conservation equation $T^{i}_{j;i}$ , the continuity equation for matter can be written as
\begin{equation}
\left(R^{i}_{j}-\frac{1}{2}g^{i}_{j}R\right)_{;i} + \frac{3}{2}\left( \phi^{i}\phi_{j} \right)_{;i}-\frac{3}{4}\left( g^{i}_{j}\phi^{k}\phi_{k} \right)_{;i} = 0
\end{equation}
\\
Simplifying equation (7) we have
\begin{equation}
\frac{3}{2}\phi_{j}\left[ \frac{\partial\phi^{i}}{\partial x^{i}} + \phi^{l}\Gamma^{i}_{li} \right] + \frac{3}{2}\phi^{i}\left[ \frac{\partial\phi_{j}}{\partial x^{i}} - \phi_{l}\Gamma^{l}_{ij} \right] - \frac{3}{4}g^{i}_{j}\phi_{k}\left[ \frac{\partial\phi^{k}}{\partial x^{i}} + \phi^{l}\Gamma^{k}_{lj} \right] - \frac{3}{4}g^{i}_{j}\phi^{k}\left[ \frac{\partial\phi_{k}}{\partial x^{i}} - \phi_{l}\Gamma^{l}_{kj} \right] = 0
\end{equation}
\\
This equation (8) identically satisfied for $j = 1, 2, 3$.\\
\\
But, for $j = 4$, this equation reduces to
\begin{equation}
\frac{3}{2}\beta \dot{\beta} + \frac{3}{2}\beta^{2} \left(\frac{\dot{A}}{A}+\frac{\dot{B}}{B}+\frac{\dot{C}}{C}\right)= 0
\end{equation}
\\
Therefore, the continuity equation (6) reduces to
\begin{equation}
\dot{\rho} + (\rho + p) \left(\frac{\dot{A}}{A}+\frac{\dot{B}}{B}+\frac{\dot{C}}{C}\right)= 0
\end{equation}
\\
Therefore, in a comoving coordinate system, the Bianchi type III space-time metric (1) for the energy momentum tensor (3), the Einstein's field equations (2) reduces to
\begin{equation}
\frac{\ddot{B}}{B}+\frac{\ddot{C}}{C}+\frac{\dot{B}\dot{C}}{BC}+\frac{3}{4}\beta^{2}=-8 \pi p
\end{equation}
\begin{equation}
\frac{\ddot{A}}{A}+\frac{\ddot{C}}{C}+\frac{\dot{A}\dot{C}}{AC}+\frac{3}{4}\beta^{2}=-8 \pi p
\end{equation}
\begin{equation}
\frac{\ddot{A}}{A}+\frac{\ddot{B}}{B}+\frac{\dot{A}\dot{B}}{AB}-\frac{\alpha^{2}}{A^{2}}+\frac{3}{4}\beta^{2}=-8 \pi p
\end{equation}
\begin{equation}
\frac{\dot{A}\dot{B}}{AB}+\frac{\dot{A}\dot{C}}{AC}+\frac{\dot{B}\dot{C}}{BC}-\frac{\alpha^{2}}{A^{2}}-\frac{3}{4}\beta^{2}=-8 \pi \rho
\end{equation}
\begin{equation}
\alpha \left( \frac{\dot{A}}{A} - \frac{\dot{B}}{B} \right) = 0
\end{equation}
where the overhead dot denote derivatives with respect to the cosmic time $t$.\\
\\
Now, for the metric (1) , important physical quantities like Volume $V$, average Scale factor $R$ , Expansion Scalar $\theta$ , Hubble's parameter $H$ , Shear Scalar $\sigma$ , Anisotropy Parameter $\Delta$ and Deceleration parameter $q$ are defined as
\begin{equation}
V = R^{3} = ABCe^{-\alpha x}
\end{equation}
\begin{equation}
\theta = \frac{\dot{A}}{A}+\frac{\dot{B}}{B}+\frac{\dot{C}}{C}
\end{equation}
\begin{equation}
H = \frac{1}{3}\theta = \frac{1}{3}\left(\frac{\dot{A}}{A}+\frac{\dot{B}}{B}+\frac{\dot{C}}{C}\right)
\end{equation}
\begin{equation}
\sigma^{2} = \frac{1}{3}\left( \frac{\dot{A}^{2}}{A^{2}} + \frac{\dot{B}^{2}}{B^{2}} + \frac{\dot{C}^{2}}{C^{2}}
- \frac{\dot{A}\dot{B}}{AB} - \frac{\dot{A}\dot{C}}{AC} - \frac{\dot{B}\dot{C}}{BC}\right)
\end{equation}
\begin{equation}
\Delta = \frac{1}{3H^{2}}\left( \frac{\dot{A}^{2}}{A^{2}} + \frac{\dot{B}^{2}}{B^{2}} + \frac{\dot{C}^{2}}{C^{2}} \right) -1
\end{equation}
and
\begin{equation}
q = 3 \frac{d}{dt}\left(\frac{1}{\theta}\right) - 1
\end{equation}
\\
Since $\alpha \neq 0$ , so from (15), we have
\[ \frac{\dot{A}}{A} = \frac{\dot{B}}{B} \]
\\
Integrating it, we get
\begin{equation}
A = kB
\end{equation}
where $k$ is a constant. Without loss of generality, we may choose $k=0$ then we have
\begin{equation}
A = B
\end{equation}
\\
Therefore, the equations (11)-(14) reduces to
\begin{equation}
\frac{\ddot{B}}{B}+\frac{\ddot{C}}{C}+\frac{\dot{B}\dot{C}}{BC}+\frac{3}{4}\beta^{2}=-8 \pi p
\end{equation}
\begin{equation}
2\frac{\ddot{B}}{B}+\frac{\dot{B}^{2}}{B^{2}}-\frac{\alpha^{2}}{B^{2}}+\frac{3}{4}\beta^{2}=-8 \pi p
\end{equation}
\begin{equation}
\frac{\dot{B}^{2}}{B^{2}}+2\frac{\dot{B}\dot{C}}{BC}-\frac{\alpha^{2}}{B^{2}}-\frac{3}{4}\beta^{2}= 8 \pi \rho
\end{equation}
\\
The equations (24) - (26) represents a system of three independent simultaneous equations involving five unknown parameters viz. $B$, $C$, $\beta$, $\rho$ and $p$. So, in order to find exact solution of the above system, it is required two more physical conditions involving these parameters. These two conditions are taken as follows-\\
i) choosing $b = -1$ and $c = 0$ in general form of quadratic equation of state, we have considered the equation of state in the form
\begin{equation}
p = a\rho^{2} - \rho
\end{equation}
where $a$ is a constant and
ii) we assume that the expansion scalar $\theta$ is proportional to the shear tensor $\sigma^{1}_{1}$ so that we get
\begin{equation}
BC = A^{n}
\end{equation}
where $n$ is a constant.\\
\\
So, from equations (23) - (26) and (28), it may be obtained
\begin{equation}
n\frac{\ddot{B}}{B} + (n-1)^{2}\frac{\dot{B}^{2}}{B^{2}} + \frac{3}{4}\beta^{2} = -8 \pi p
\end{equation}
\begin{equation}
2\frac{\ddot{B}}{B} + \frac{\dot{B}^{2}}{B^{2}} - \frac{\alpha^{2}}{B^{2}} + \frac{3}{4}\beta^{2} = -8 \pi p
\end{equation}
\begin{equation}
(2n-1)\frac{\dot{B}^{2}}{B^{2}} - \frac{\alpha^{2}}{B^{2}} - \frac{3}{4}\beta^{2} = 8 \pi \rho
\end{equation}
\\
Subtracting (30) from (29) and adding (30) $\&$ (31), we have
\begin{equation}
(n-2)\frac{\ddot{B}}{B} + (n^{2}-2n)\frac{\dot{B}^{2}}{B^{2}} + \frac{\alpha^{2}}{B^{2}} = 0
\end{equation}
and
\begin{equation}
\frac{\ddot{B}}{B} + n\frac{\dot{B}^{2}}{B^{2}} - \frac{\alpha^{2}}{B^{2}} = 4 \pi (\rho - p)
\end{equation}
From equations (32) and (33), it can be obtained that
\begin{equation}
\frac{\ddot{B}}{B} + n\frac{\dot{B}^{2}}{B^{2}} = \frac{4 \pi}{n-1} (\rho - p)
\end{equation}
\\
The equations (16) and (34) will give us
\begin{equation}
\dot{V} = \sqrt{m \rho V^{2} + k_{1}}
\end{equation}
where $k_{1}$ and $m = \frac{8 \pi (n+1)}{n-1}$ are integrating constants.\\
\\
Integrating equation (35), we have
\begin{equation}
\int \frac{dV}{\sqrt{m \rho V^{2}} + k_{1}} = t + k_{2}
\end{equation}
where $k_{2}$ is an integrating constant that represent a shift of cosmic time $t$. Therefore, it can be chosen as zero.\\
\\
Equations (10) and (27) will give us
\begin{equation}
\rho = (a logV)^{-1}
\end{equation}
\\
Now, if we take $k_{1} = k_{1} = 0$, then from equations (36), the volume $V$ is obtained as
\begin{equation}
V = exp \left[\left(\frac{9m}{4a}t^{2}\right)^{\frac{1}{3}}\right]
\end{equation}
\\
So, the scale factors $A$, $B$ and $C$ are obtained from equations (16), (23), (28) and (38) as
\begin{equation}
A = B = exp \left[\frac{1}{n+1}\left(\frac{9m}{4a}t^{2}\right)^{\frac{1}{3}}\right] e ^{\frac{\alpha x}{n+1}}
\end{equation}
and
\begin{equation}
C = exp \left[\frac{n-1}{n+1}\left(\frac{9m}{4a}t^{2}\right)^{\frac{1}{3}}\right] e ^{\frac{(n-1)\alpha x}{n+1}}
\end{equation}
\\
Using equations (39) and (40) in equation (1), the geometry of the model universe is given by
\begin{equation}
\begin{split}
ds^{2}=&exp \left[\frac{2}{n+1}\left(\frac{9m}{4a}t^{2}\right)^{\frac{1}{3}}\right] e ^{\frac{2 \alpha x}{n+1}}\left[dx^{2}+e^{-2{\alpha}x}dy^{2}\right]\\
&+exp \left[\frac{2(n-1)}{n+1}\left(\frac{9m}{4a}t^{2}\right)^{\frac{1}{3}}\right] e ^{\frac{2(n-1)\alpha x}{n+1}}dz^{2}-dt^{2}
\end{split}
\end{equation}
\\
The use of equation (38) in Equation (37), the energy density $\rho$ is obtained as
\begin{equation}
\rho = \frac{1}{a} \left(\frac{9m}{4a}\right)^{-\frac{1}{3}}t^{-\frac{2}{3}}
\end{equation}
\\
Therefore, from equation (27), the pressure $p$ can be obtained as
\begin{equation}
p = \frac{1}{a} \left(\frac{9m}{4a}\right)^{-\frac{1}{3}}t^{-\frac{2}{3}}\left[ \left(\frac{9m}{4a}\right)^{-\frac{1}{3}}t^{-\frac{2}{3}} -1 \right]
\end{equation}
\\
From equation (31), the displacement vector $\beta$ is given by
\begin{equation}
\frac{3}{4}\beta^{2} = \frac{n-2}{n-1}\frac{4\pi}{a} \left(\frac{9m}{4a}\right)^{-\frac{1}{3}}t^{-\frac{2}{3}}\left[ \frac{2}{n+1} - \left(\frac{9m}{4a}\right)^{-\frac{1}{3}}t^{-\frac{2}{3}} \right]
\end{equation}
\\
For the model universe (41), the other physical and geometrical properties like expansion scalar $\theta$, Hubble's expansion factor $H$, shear scalar $\sigma$, anisotropy parameter $\Delta$ and deceleration parameter $q$ can be easily obtained from equations (17)-(21).
\begin{equation}
\theta = \frac{2}{3} \left(\frac{9m}{4a}\right)^{\frac{1}{3}}t^{-\frac{2}{3}}
\end{equation}
\begin{equation}
H = \frac{2}{9} \left(\frac{9m}{4a}\right)^{\frac{1}{3}}t^{-\frac{2}{3}}
\end{equation}
\begin{equation}
\sigma^{2} = \frac{4(n-2)^{2}}{27(n+1)^{2}} \left(\frac{9m}{4a}\right)^{\frac{2}{3}}t^{-\frac{2}{3}}
\end{equation}
\begin{equation}
\Delta = \frac{2(n-2)^{2}}{(n+1)^{2}} = \textrm{constant} \neq 0 ~~ \textrm{for} ~~ n \neq 2 ~~ \textrm{and} ~~ n \neq -1
\end{equation}
\begin{equation}
q = \frac{3}{2} \left(\frac{9m}{4a}\right)^{-\frac{1}{3}}t^{-\frac{2}{3}} -1
\end{equation}
It is well known that the different values of the parameters will give rise different graph, so the variations of some parameters are shown, by taking particular values of the integrating constants as $a = 10$ and $n = 10$ so that $m = 30.730159$, in the following Fig. 1-6.
\begin{figure}[H]
\centerline{\psfig{file=fig1.eps,width=3in}}
\vspace*{8pt}
\caption{The variation of Expansion Scalar $\theta$ vs Time $t$, when $a$ = 10 and $n$ = 10 so
that $m$ = 30.730159.\label{Fig. 1.}}
\centerline{\psfig{file=fig2.eps,width=3in}}
\vspace*{8pt}
\caption{The variation of Volume $V$ vs Time $t$, when $a$ = 10 and $n$ = 10 so that
$m$ = 30.730159.\label{Fig. 2.}}
\centerline{\psfig{file=fig3.eps,width=3in}}
\vspace*{8pt}
\caption{The variation of Deceleration Parameter $q$ vs Time $t$, when $a$ = 10 and
$n$ = 10 so that $m$ = 30.730159.\label{Fig. 3.}}
\end{figure}
\begin{figure}[H]
\centerline{\psfig{file=fig4.eps,width=3in}}
\vspace*{8pt}
\caption{The variation of Energy Density $\rho$ vs Time $t$, when when $a$ = 10 and $n$ = 10
so that $m$ = 30.730159.\label{Fig. 4.}}
\centerline{\psfig{file=fig5.eps,width=3in}}
\vspace*{8pt}
\caption{The variation of Pressure $p$ vs Time $t$, when when $a$ = 10 and $n$ = 10 so
that $m$ = 30.730159.\label{Fig. 5.}}
\centerline{\psfig{file=fig6.eps,width=3in}}
\vspace*{8pt}
\caption{The variation of Displacement Vector $\beta$ vs Time $t$, when $a$ = 10 and $n$ = 10
so that $m$ = 30.730159.\label{Fig. 6.}}
\end{figure}
\section{Physical and Geometrical Properties of the Model Universe}
The evolution of expansion scalar $\theta$ has been shown in Fig. 1 corresponding to the equation (45) and it is observed that the expansion scalar $\theta$ starts with infinite value at initial epoch of cosmic time $t = 0$. But, as time $t$ progresses, it decreases and becomes constant after some finite time that explains the Big-Bang scenario. From equation (38) and Fig. 2, it is seen that the value of volume $V$ of the model universe is an increasing function of cosmic time $t$ and increases from zero at initial epoch of time to infinite volume whenever $t \rightarrow \infty$ , so our model represents an expanding universe.\\
\\
Now, from equation (49) and Fig. 3, it is clear that initially the value of deceleration parameter $q$ is negative and tends to $-1$ at infinite time i.e. the values of $q$ is in the range $-1 \leq q \leq 0$ implying that the values of $q$ satisfies the present observational data like Riess et al. [6] and Perlmutter et al. [7]. Also, from the expression (46) for Hubble's expansion factor $H$ it has been observed that $dH/dt$ is negative, so our model universe expands with an accelerated rate.\\
\\
Since the present day universe is isotropic, it is important to observe whether our models evolve into an isotropic or an anisotropic one. In order to investigate the isotropy of the model universe, here, we have considered the simple anisotropy parameter $\Delta$. From equation (48), it has been observed that the anisotropy parameter $\Delta$ is independent of cosmic time $t$ and $\Delta = 0$ for $n = 2$ but whenever $n \neq 2$ and $n \neq 1$ then $\Delta \neq 0$ , which shows that our model universe is isotropic throughout the evolution whenever $n = 2$ but for $n \neq 2$ and $n \neq 1$, the model remains anisotropic throughout the evolution.\\
\\
Again, it is known that the energy conditions for Bianchi type III model is energy density $\rho$ is positive i.e. $\rho > 0$. From Fig. 4 of equation (42), it is seen that the energy $\rho$ is always positive. Also, the Fig. 5 showing the evolution of pressure $p$ depicts that initially when $t \rightarrow \infty$ then $p$ is positive but as the time progresses $p$ changes sign from positive to negative. Therefore, our model universe trespasses through the transition from matter dominated period to inflationary period.\\
\\
From equation (44), the displacement vector $\beta$ is found to be positive which increases rapidly at initial epoch of time but with the increase of time it decreases and at infinite time the displacement vector $\beta$ becomes a small positive constant. The variation of the parameter 'displacement vector $\beta$' is shown in Fig. 6. Also, from equation (47) it can be seen that the shear scalar $\sigma$ is a decreasing function of cosmic time $t$ and vanishes as $t \rightarrow \infty$ . So, our model represents a shear free dark energy cosmological model universe for large values of cosmic time $t$.
\section{Conclusion}
Investigating a homogeneous and anisotropic space-time described by Bianchi type III metric in presence of perfect fluid in Lyra geometry setting under the assumption of quadratic equation of state (EoS), exact solutions of the Einstein's field equations have been obtained. Here, we have got a model universe which is expanding with acceleration that also passes through the transition from matter dominated period to inflationary period. The displacement vector $\beta$ and shear scalar $\rho$ becomes zero as $t \rightarrow \infty$. So, our model represents a shear free dark energy cosmological model universe for large values of cosmic time $t$. \\
\section{Acknowledgements}
The authors are very thankful to the UGC, India for funding under the sanction Order No. F. 5-332/2014-15/MRP/NERO/2386 to carry out this work successfully.
\\
|
1,314,259,993,613 | arxiv | \section{Introduction}
It has been shown that certain computational tasks, such as search \cite{Grover:1996rk} and factorization \cite{Shor:1994:AQC:1398518.1399018}, can be performed faster by a quantum computer than a classical one. This speed up over the classical information scheme can be attributed to hallmark properties of quantum mechanics, including superposition and entanglement. It is concievable, then, that continuing to add or tweak the features of the computation model might bring further improvements. The $\mathcal O( \sqrt N)$ scaling of Grover's search algorithm, for example, has been found to be optimal in the quantum model \cite{doi:10.1137/S0097539796300933}, whose dynamics are linear, as described by the Schrodinger Equation. If we instead assume non-linear dynamics, however, randomly walking particles can conditionally search with even better scalings \cite{PhysRevA.89.012312}, depending on the non-linearity, even approaching constant time scaling for certain models.
To make practical use of the speedups provided these approximations to the quantum model, we need to confirm their physical significance. The Gross-Pitaevskii Equation (GPE) \cite{Gross1961}, for instance, is a non-linear approximation which is particularly appropriate for describing the dynamics of Bose-Einstein condensates \cite{Rogel_Salazar_2013}. If we accept and apply the GPE to quantum random walks on the complete graph, search has been shown to conditionally scale like $\mathcal O(N^{1/4})$ \cite{Meyer_2013}.
In order for the GPE to be a reasonal approximatin to the Schrodinger Equation, though, a number of assumptions must be satisfied. Assuming many particles and short-ranged interactions is relatively benign, and enforcing permutation symmetry is simply a matter of restricting to identical Bosons. The more challenging assumptions to motivate physically are the mean field approximation,
\begin{equation}
\rho_2 \left(t,x_1,x_2;x_1'x_2' \right) \approx \rho_1 \left(t,x_1;x_1' \right) \otimes \rho_1 \left(t,x_2;x_2' \right),
\end{equation}
and approximate purity of the single-party reduced state,
\begin{equation}
\rho_1 \left(t,x;x' \right) \approx \psi \left(t,x \right) \psi^* \left( t,x' \right),
\end{equation}
where $\rho_2$ and $\rho_1$ are the continuous space one- and two-party reduced states respectively, noting that the permutation symmetry of the overall state makes the choice of specific parties labels irrelevant. These are strong restrictions to make to the overall Hilbert space, but are properties natural to Bose-Einstein condensates \cite{Rogel_Salazar_2013}, which make them a common physical medium for the continuous-space GPE.
To extend this analysis to quantum random walks, the GPE, along with its requisite assumptions, must be translated to a discrete space description. In particular, the mean field approximation and pure single-party marginal assumptions become
\begin{align}\label{MFA}
\rho_2 &\approx \rho_1 \otimes \rho_1 \\
\rho_1 &\approx \ketbra{\psi}{\psi},
\end{align}
where the states are now described by finite-dimensional vectors and density matrices. The aim of this paper is to examine the validity of these assumptions, particularly the mean field approximation (\ref{MFA}), for quantum random walks of permutation-invariant particles on the complete graph. Some entanglement properties of such states have been examined previously \cite{RobinReuvers:2019xew}, but we provide a more targeted approach.
At first glance it seems as if the symmetries associated to the permutation-invariance and the complete graph may be enough to guarantee (\ref{MFA}) because it is, in some sense, a measure of entanglement. Certainly if $\rho_2$ is entangled then (\ref{MFA}) will not be satisfied, and potentially the converse is true; the less entangled the state is, the greater the agreement with (\ref{MFA}). The monogamy of entanglement then suggests that the symmetric sharing of pairwise entanglement in the overall state will decay with the number of particles, $n$. Permutation-invariance alone in multi-qubit states bounds the pairwise Concurrence \cite{PhysRevLett.80.2245} by $\mathcal C_{i,j} \leq 2/n$ \cite{PhysRevA.62.050302}.
In this paper we begin by fully expressing the symmetries of permutation-invariance and the complete graph, then finding the resulting Young diagram basis for the symmetrized states. We then measure the validity of the mean field approximation on those states through the matrix Fidelity. We perform exact fidelity calculations on a subset of single basis elements, then bound the fidelity for a larger subset of states.
\section{Party-Site Symmetry}
Consider $n$ particles evolving on a complete graph of $d$ sites. To describe the state of such a system, we can index the site position of particle $j$ by $i_j$, and therefore the overall state can be described by $n$ qu$d$its, or, a vector in $\mathbb C_d^{\otimes n}$,
\begin{equation}\label{ogPSS}
\ket{\psi} = \sum_{i_1 = 1}^{d} \ldots \sum_{i_n=1}^{d} a_{i_1 \ldots i_n} \ket{i_1 \ldots i_n}.
\end{equation}
To this state we then want to enforce constraints which reflect the symmetries of both:
\begin{itemize}
\item \emph{Party:} Because the particles in question are identical, we want to enforce that any permutation of their labels leaves the state unchanged.
\item \emph{Site:} Because the particles are walking on a \emph{complete} graph, the sites themselves of that graph are identical, and any permutation of their labels would leave the graph, and therefore the state, unchanged.
\end{itemize}
We can formalize these combined symmetries by defining that a state,
\begin{equation}
\ket{\psi} = \sum_{i_1 = 1}^{d} \ldots \sum_{i_n=1}^{d} a_{i_1 \ldots i_n} \ket{i_1 \ldots i_n},
\end{equation}
is party-site symmetric (PSS) if
\begin{eqnarray}\label{Usym}
U_{\mu} \ket{\psi} &=& \ket{\psi} \quad \forall \quad \mu \in S_n \\ \label{Vsym}
V_{\nu}^{\otimes n} \ket{\psi} &=& \ket{\psi} \quad \forall \quad \nu \in S_d,
\end{eqnarray}
where $U_{\mu}$ is the unitary representation of $\mu$, which permutes the party labels,
\begin{equation}
U_{\mu} \ket{i_1 \ldots i_N} = \ket{ \mu \left( i_1 \ldots i_N \right)},
\end{equation}
while $V_{\nu}$ is the unitary representation of $\nu$, which permutes the basis (site) labels,
\begin{equation}
V_{\nu} \ket i = \ket{\nu(i)}.
\end{equation}
The individual symmetries associated to party (\ref{Usym}) and site (\ref{Vsym}) will be referred to as $U$ and $V$ symmetries respectively. For PSS states we expect that many of the coefficients, $a_{i_1 \ldots i_n}$, are constrained to be equal by the $U$ and $V$ symmetries, leaving some much smaller basis for the subspace. The $U$ symmetry implies that the ordering of $i_1$-$i_n$ does not matter. The $V$ symmetry then implies that the collective index values themselves can be freely permuted. Given these constraints, the only actually distinguishing feature of a given $a_{i_1 \ldots i_n}$, and the elements it is grouped with, is the partitioning of shared indices. For example, $a_{1,2,2,3,4,2,3}=a_{2,2,2,3,3,1,4}=a_{i_1,i_1,i_1,i_1,i_2,i_2,i_3,i_4}$ would be grouped under the label, $a_{3,2,1,1}$, where now the subscripts denote the number of parties who share a given index. One can recognize that such a grouping and labeling can be expressed in a young diagram,
\begin{widetext}
\begin{center}
\ytableausetup{centertableaux}
\begin{ytableau}
\; & & & \\
\; & \\
\\
\end{ytableau}
$=\left \{ \mu^{\vphantom{U^U}} \left( \nu(1^{\vphantom{U^U}}),\nu(2),\nu(2),\nu(3),\nu(4),\nu(2),\nu(3) \right)^{\vphantom{U^U}} \; \; \middle | \; \; \mu \in S_n, \; \nu \in S_d \right \}$.
\end{center}
\end{widetext}
In general, the number of rows in a Young diagram indicates that each of the elements in the set has that many distinct indices. The number of blocks in a row indicates how many parties share that index. Naturally, the total number of blocks is $n$, and there can be at most $d$ rows in a Young diagram. With the interpretation of Young diagrams established, we then find that they serve as an orthonormal basis for pure PSS states,
\begin{equation}\label{PSS}
\ket{\psi} = \sum_{y \in \mathcal Y(n,d)} a_y \ket y,
\end{equation}
where $\mathcal Y(n,d)$ is the set of Young diagrams with $n$ blocks and at most $d$ rows, and $\ket y$ is a normalized equal superposition of computational basis elements belonging to the set described by the Young diagram, $y$.
In evaluating the validity of the mean field approximation for a PSS state we will have to perform the partial trace on (\ref{PSS}) to find $\rho_1$ and $\rho_2$. Thankfully, the $U$ and $V$ symmetries greatly simplify that process. Consider the reduced state, $\rho_k$, obtained by tracing out the last $n-k$ parties,
\begin{widetext}
\begin{align}
\rho_k =& \text{Tr}_{\bar k} \left( \ketbra{\psi}{\psi} \right) \\
=& \sum_{l_{k+1} \ldots l_{N}} \; \sum_{i_{1} \ldots i_k} \; \sum_{j_{1} \ldots j_k} a_{i_1 \ldots i_k l_{k+1} \ldots l_N} \ketbra{i_1 \ldots i_k}{j_1 \ldots j_k} a_{j_1 \ldots j_k l_{k+1} \ldots l_N}^*.
\end{align}
\end{widetext}
Now consider $V_{\nu^{-1}}$ for some $\nu \in S_d$, acting on $\rho_k$,
\begin{widetext}
\begin{eqnarray}
V_{\nu^{-1}} \rho_k V_{\nu^{-1}}^{\dagger} &=& \sum_{i, j, l} a_{\nu \left(i_1\right) \ldots \nu \left(i_k\right) l_{k+1} \ldots l_N} \ketbra{i_1 \ldots i_k}{j_1 \ldots j_k} a_{\nu \left(j_1\right) \ldots \nu \left(j_k\right) l_{k+1} \ldots l_N}^* \quad \quad \quad \\
&=& \sum_{i, j, l} a_{\nu \left(i_1\right) \ldots \nu \left(i_k\right) \nu \left(l_{k+1}\right) \ldots \nu \left(l_N\right)} \ketbra{i_1 \ldots i_k}{j_1 \ldots j_k} a_{\nu \left(j_1\right) \ldots \nu \left(j_k\right) \nu \left(l_{k+1}\right) \ldots \nu \left(l_N\right)}^* \quad \quad \quad \\
&=& \sum_{i, j, l} a_{i_1 \ldots i_k l_{k+1} \ldots l_N} \ketbra{i_1 \ldots i_k}{j_1 \ldots j_k} a_{j_1 \ldots j_k l_{k+1} \ldots l_N}^* \\
&=& \rho_k.
\end{eqnarray}
\end{widetext}
Likewise consider $U_{\mu^{-1}}$ for $\mu \in S_k$. We can also extend $\mu \otimes \mathbb 1_{n-k} \in S_n$ as the permutation which acts on the first $k$ parties by $\mu$ and leaves the traced over parties fixed. Now examine
\begin{widetext}
\begin{eqnarray}
U_{\mu^{-1}} \rho_k U_{\mu^{-1}}^{\dagger} &=& \sum_{i, j, l} a_{\mu \left(i_1 \ldots i_k\right) l_{k+1} \ldots l_N} \ketbra{i_1 \ldots i_k}{j_1 \ldots j_k} a_{\mu \left(j_1 \ldots j_k\right) l_{k+1} \ldots l_N}^* \quad \quad \quad \\
&=& \sum_{i, j, l} a_{\mu \otimes \mathbb 1_{N-k} \left(i_1 \ldots i_k l_{k+1} \ldots l_N \right)} \ketbra{i_1 \ldots i_k}{j_1 \ldots j_k} a_{\mu \otimes \mathbb 1_{N_k} \left(j_1 \ldots j_k l_{k+1} \ldots l_N \right)}^* \quad \quad \quad \\
&=& \sum_{i, j, l} a_{i_1 \ldots i_k l_{k+1} \ldots l_N} \ketbra{i_1 \ldots i_k}{j_1 \ldots j_k} a_{j_1 \ldots j_k l_{k+1} \ldots l_N}^* \\
&=& \rho_k.
\end{eqnarray}
\end{widetext}
Even more interesting is that acting on only one side by $U$ would likewise leave $\rho_k$ invariant because $\mu$ can be freely extended to $\mu \otimes \mathbb 1_{N-k}$ for either the bra or the ket individually. This is not true for the $V$ symmetry, where absorbing $\nu$ into the sum in $l$ has to affect both the bra and the ket simultaneously. Altogether then we have
\begin{eqnarray}
U \rho_k = \rho_k U = \rho_k \\
V \rho_k V^{\dagger} = \rho_k.
\end{eqnarray}
These symmetries allow us to greatly constrain the elements of $\rho_1$ and $\rho_2$, leaving us with a fairly simple parametrization of the two matrices. Starting with $\rho_1$, the $V$ symmetry implies that $\left \{ \rho_1 \right \}_{i,j} = \left \{ \rho_1 \right \}_{\nu(i), \nu(j)}$ for any $\nu \in S_d$. This then equates all the diagonal elements as $\left \{ \rho_1 \right \}_{i,i} = \frac 1d$ and the off diagonal elements as $\left \{ \rho_1 \right \}_{i,j} = A$ for all $i \neq j$. Since the $V$ symmetry equates $\left \{ \rho_1 \right \}_{i,j}= \left \{ \rho_1 \right \}_{j,i}$, the hermiticity of $\rho_1$ then implies that $A \in \mathbb R$. The same application of the $U$ and $V$ symmetries along with hermiticity constrain the following elements of $\rho_2$, in which it is implied that $i$, $j$, $k$, and $l$ are distinct,
\begin{align}
B_1 =& \left \{ \rho_2 \right \}_{i j, k l} \\
B_2 =&\left \{ \rho_2 \right \}_{i i, k l} \\
B_3 =&\left \{ \rho_2 \right \}_{i j, i l} = \left \{ \rho_2 \right \}_{i j, l i} = \left \{ \rho_2 \right \}_{j i, i l} = \left \{ \rho_2 \right \}_{j i, l i} \\
B_4 =&\left \{ \rho_2 \right \}_{i i, k k} \\
B_5 =&\left \{ \rho_2 \right \}_{i i, i l} = \left \{ \rho_2 \right \}_{i i, l i} \\
C_1 =&\left \{ \rho_2 \right \}_{i j, i j} = \left \{ \rho_2 \right \}_{i j, j i} \\
C_2 =&\left \{ \rho_2 \right \}_{i i, i i}
\end{align}
where $B_1$, $B_3$, $B_4$, $C_1$, and $C_2$ are real, while $B_2$ and $B_5$ are complex. Notably, we can relate $A$ to the parameters of $\rho_2$ by
\begin{align}
A &= \left \{ \rho_1 \right \}_{i,j} \\
&= \left \{ \text{Tr}_2 \left ( \rho_2 \right ) \right \}_{i,j}\\
&= \sum_{k=1}^d \left \{ \rho_2 \right \}_{ik,jk} \\
&= (d-2)B_3 + B_5 + B_5^* \\
&= (d-2)B_3 + 2 \Re \left (B_5 \right ).
\end{align}
We can also use the normalization of $\rho_2$ to find $dC_1+d(d-1)C_2=1$.
\section{Exact Fidelity Calculations}
Having examined $\rho_1$ and $\rho_2$ for PSS states, we must now choose a metric by which to measure the agreement with (\ref{MFA}). We have chosen to use the matrix Fidelity \cite{doi:10.1080/09500349414552171},
\begin{equation}
F(A,B) = \left[ \text{Tr} \sqrt{\sqrt A B \sqrt A} \right]^2.
\end{equation}
The fidelity is a common choice in quantum information theory as a generalization of the pure state inner product to mixed states. Applied to the mean field approximation, let us label
\begin{equation}
F(\ket \psi) = \left [ \text{Tr} \sqrt M \right]^2,
\end{equation}
where $M = \sqrt{\rho_1 \otimes \rho_1} \rho_2 \sqrt{\rho_1 \otimes \rho_1}$. Unfortunately, for the most general PSS state (\ref{PSS}), the resulting $M$ matrix is analytically challenging to diagonalize or find the trace of its square root. In some simple cases, however, the fidelity can be determined exactly. Consider the following set of PSS states described by single, rectangular Young diagram basis elements,
\ytableausetup{centertableaux,boxsize=6mm}
\[y(k) = \quad \quad
\begin{ytableau}
\tikznode{a3}{~}\; & & \none[\dots] & & \\
\; & \none & \none & \none & \\
\none[\vdots] & \none & \none & \none & \none[\vdots]\\
\; & \none & \none & \none & \\
\tikznode{a1}{~} & & \none[\dots] & & \tikznode{a2}{~}
\end{ytableau}.
\]
\tikz[overlay,remember picture]{%
\draw[decorate,decoration={brace},thick] ([yshift=-3.5mm,xshift=3mm]a2.south east) --
([yshift=-3.5mm,xshift=-3mm]a1.south west) node[midway,below]{$\frac nk$};
\draw[decorate,decoration={brace},thick] ([yshift=-2.5mm,xshift=-4mm]a1.south west) --
([yshift=4.5mm,xshift=-3.2mm]a3.north west) node[midway,left]{$k$};
}
\vspace{5mm}
\noindent Expressed as a state in the computational basis,
\begin{equation}
\ket{y(k)} = \mathcal A_k^{- \frac 12} \sum_{\mu \in S_n} \; \sum_{i_1 < \ldots < i_k} U_{\mu} \bigotimes_{j=1}^k \ket{i_j}^{\otimes \frac nk},
\end{equation}
where $\mathcal A_k$ is a normalization constant equal to the number of computational basis elements present in $\ket{y(k)}$. It evaluates to
\begin{equation}
\mathcal A_k = \binom{d}{k} \frac{n!}{{\left[ \left(\frac nk \right) !\right]}^k} = \left(\frac{k!(d-k)! \left[ \left( \frac nk \right)! \right]^k}{d! \, n!} \right)^{-1}.
\end{equation}
The first step in calculating $F( \ket{y(k)})$ is the determination of the components of $\rho_2$ and $\rho_1$. Consider first
\begin{align}
\rho_1 &= \mathcal A_k^{-1} \sum_{i, k_2 \ldots k_n \in y} \sum_{j, k_2 \ldots k_n \in y} \ketbra ij \\
&=\mathcal A_k^{-1} \sum_{i,j} \mathcal N^{(y)}_{i,j} \ketbra ij,
\end{align}
where $\mathcal N_{i,j}^{(y)}$ is the number of strings, $(k_2 \ldots k_n)$, for which both $(i \, k_2 \ldots k_n)$ and $(j \, k_2 \ldots k_n)$ are contained in the set associated to $y$. We can analogously define $\mathcal N_{ij,kl}^{(y)}$ such that
\begin{equation}
\rho_2 = \mathcal A_k^{-1} \sum_{i,j,k,l} \mathcal N_{ij,kl}^{(y)} \ketbra{ij}{kl}.
\end{equation}
Determining each of the $\mathcal N$ for the family of $y(k)$ is a simple counting/combinatorics exercise. The results are the following, where it is assumed that $i$, $j$, $k$, and $l$ are distinct,
\begin{align}
\mathcal N_{i,j}^{(y(k))} &= \delta(k-n) \frac{(d-2)!}{(d-n-1)!} \\
\mathcal N_{ii,ii}^{(y(k))} &= \begin{cases} 0 & k=n \\ \binom{d-1}{k-1} \frac{(n-2)!}{ \left( \frac nk-2 \right)! \left[ \left( \frac nk \right)! \right ]^{k-1}} & k\neq n \end{cases} \\
\mathcal N_{ij,ij}^{(y(k))} &= \binom{d-2}{k-2} \frac{(n-2)!}{\left[ \left( \frac nk-1 \right)! \right]^2 \left[ \left( \frac nk \right)! \right ]^{k-2}} \\
\mathcal N_{ii,jj}^{(y(k))} &= \delta \left(k-\frac n2 \right) \binom{d-2}{\frac n2-1} \frac{(n-2)!}{ 2^{\frac n2-1}} \\
\mathcal N_{ij,ik}^{(y(k))} &= \delta(k-n) \frac{(d-3)!}{(d-n-1)!} \\
\mathcal N_{ij,kl}^{(y(k))} &= \delta(k-n) \frac{(d-4)!}{(d-n-2)!},
\end{align}
while $\mathcal N_{ii,ij}^{(y(k))}=\mathcal N_{ii,jk}^{(y(k))}=0$. Dividing by $\mathcal A_k$ then finally gives the components of each reduced density matrix,
\begin{align}
A &= \delta(k-n) \frac{d-n}{d(d-1)} \\
\left \{ \rho_2 \right \}_{ii,ii} &= \frac{n-k}{d \, k(n-1)} \\
\left \{ \rho_2 \right \}_{ij,ij} &= \frac{n(k-1)}{d (d-1) k(n-1)} \\
B_4 &= \delta \left(k- \frac n2 \right) \frac{d- \frac n2}{d(d-1)(n-1)} \\
B_3 &= \delta(k-n) \frac{d-n}{d(d-1)(d-2)} \\
B_1 &= \delta(k-n) \frac{(d-n)(d-n-1)}{d(d-1)(d-2)(d-3)} \\
B_2 &=B_5=0
\end{align}
From here there are three major cases to consider: $k<n/2$, $k=n/2$, and $k=n$. Starting with the $k<n/2$ case we have $\rho_1= d^{-1} \mathbb 1_d$ and
\begin{align}
\rho_2 =& \frac{1}{d \, k (n-1)} \biggr[ (n-k) \sum_i \ketbra{ii}{ii} \\ \notag &+ \frac{n(k-1)}{d-1} \sum_{i \neq j} \ketbra{ij}{ij} + \ketbra{ij}{ji} \biggr],
\end{align}
which leads to
\begin{align}
M =& \frac{1}{d^3 k (n-1)} \biggr[ (n-k) \sum_i \ketbra{ii}{ii} \\ \notag &+ \frac{n(k-1)}{d-1} \sum_{i \neq j} \ketbra{ij}{ij} + \ketbra{ij}{ji} \biggr].
\end{align}
Given that we will be tracing this matrix after finding its square root, we can jointly reorder the rows and columns together. Doing so yields the convenient representation,
\begin{equation}
M = \frac{n(k-1)}{d^4k(n-1)} \left[ \left( \begin{array}{c c} 1 & 1 \\ 1 & 1 \end{array} \right)^{\oplus d(d-1)/2} \oplus \frac{d(n-k)}{n(k-1)} \, \mathbb 1_d \right],
\end{equation}
which we be easily diagonalized and square rooted,
\begin{align}
\sqrt M =& \sqrt{\frac{n(k-1)}{d^4k(n-1)}} \biggr[ \left( \begin{array}{c c} \sqrt 2 & 0 \\ 0 & 0 \end{array} \right)^{\oplus d(d-1)/2} \\ \notag &\oplus \sqrt{\frac{d(n-k)}{n(k-1)}} \, \mathbb 1_d \biggr].
\end{align}
And finally we can find $F( \ket{y(k<n/2)})$,
\begin{align}
F = \frac{ \left( (d-1) \sqrt{n(k-1)}+ \sqrt{2d(n-k)} \right)^2}{2d^2k(n-1)}.
\end{align}
As $d \to \infty$, this simplifies to
\begin{equation}
F( \ket{y(k<n/2)}) = \frac{n(k-1)}{2k(n-1)}.
\end{equation}
Moving to the $k=n/2$ case, the same analysis arrives at
\begin{equation}
F = \frac{\left( \sqrt{2+2d-n} + (d+1) \sqrt{(d-1)(n-2)} \right)^2}{2d^3(n-1)},
\end{equation}
which, as $d \to \infty$, evaluates to
\begin{equation}
F( \ket{y(k=n/2)}) = \frac{n-2}{2(n-1)}.
\end{equation}
This leaves only the $k=n$ case, which has two notable limits; $n=d$ and $n \ll d$. For $n=d$ we find that
\begin{equation}
M = \frac{2}{d^3(d-1)} \left( \begin{array}{c c} 1 & 0 \\ 0 & 0 \end{array} \right)^{\oplus d(d-1)/2}.
\end{equation}
This makes determining the fidelity rather straightforward,
\begin{equation}
F( \ket{y(k=n=d)}) = \frac{d-1}{2d},
\end{equation}
which is equal to $1/2$ as $d \to \infty$. In the $n \ll d$ case, we actually have that in the large $d$ limit,
\begin{equation}
\rho_1 \approx \frac 1d \sum_{i,j} \ketbra ij,
\end{equation}
which is pure, and therefore $\rho_2 = \rho_1 \otimes \rho_1$ and $F( \ket{y(k=n \ll d)}) = 1$.
The results of these calculations are somewhat surprising given our intuitions regarding monogamy constraints on the symmetric sharing of entanglement. We had expected a heuristic connection between entanglement in the state and violation of the mean field approximation. This notion was only partially correct though, as the mean field approximation is a stronger assumption than the separability of $\rho_2$. Recall that a mixed state is separable if
\begin{equation}
\rho = \sum_i^r p_i \rho_i^{(1)} \otimes \rho_i^{(2)},
\end{equation}
for some decomposition, in which $r$ is unbounded. The mean field approximation, however, demands $r=1$, which is therefore only true for a subset of separable states. So it is then unsurprising that we were able to find PSS states for which $F(\ket \psi)$ was not close to 1, as the entanglement decaying with $n$ due to the symmetry implies that the state merely approaches a separable one, not one for which the mean field approximation is a necessarily good one.
The example of rectangular Young diagrams raises an important intuition regarding the validity of the mean field approximation. The results of this section can be summarized as larger $k$ leading to better agreement with the mean field approximation. Physically, small $k$ corresponds to more compact grouping of the particles. Therefore we are led to believe that the more spread out the particles are, the better the mean field approximation gets. This notion is given further context in the next section, where we conclude that the \emph{only} way to get good agreement with the mean field approximation is to have isolated particles. This intuition does give hope to the use of the Gross-Pitaevskii Equation in quantum search. The initial state for that algorithm is the uniform superposition,
\begin{equation}
\ket{\psi_0} = \left( \sum_{i=1}^d \ket i \right)^{\otimes n},
\end{equation}
which we know is approximately the $\ket{y(k=n)}$ state in the $n \ll d$ limit, and approaches perfect agreement with the mean field approximation. Left to evolve, we would expect that the particles would stay mostly spread because that is both entropically and energetically favored.
\section{Bounded Fidelity Analysis}
The previous section introduced the intuition that isolated particles make for better agreement with the mean field approximation. In this section we will confirm that notion by proving that good fidelity is \emph{impossible} without isolated particles. The following theorem, whose proof can be found in the Appendix, will be instrumental in that endeavor,
\begin{thm} \label{fidbound}
In the limits that $1 \ll d$ and $n \sqrt n \ll d$, if a PSS state, $\ket{\psi}$, has $A \leq \mathcal O(d^{-2})$, then $F(\ket{\psi}) \leq 1/2$.
\end{thm}
This theorem allows us to identify any PSS state with $A \leq \mathcal O(d^{-2})$ as one for which the mean field approximation is not valid. In finding sets of PSS states with such $A$, it will be important to establish notation which allows us to describe an arbitrary Young diagram, $y$, and its corresponding state vector, $\ket y$. First, as before, label the number of rows as $k^{(y)}$, but now denote the number of distinct row lengths as $p^{(y)}$. Denote the length of the $q^{\text {th}}$ distinct row from the bottom as $M_q^{(y)}$. Denote the number of rows of length $M_q^{(y)}$ as $l_q^{(y)}$. These labels are constrained by $p^{(y)}<k^{(y)}<d$ and $\sum_{q=1}^{p^{(y)}} l_q^{(y)}M_q^{(y)} =n$. Finally, this notation allows us to determine the normalization coefficient, $\mathcal A_y$, for a single Young diagram basis element,
\begin{equation}
\mathcal A_y = \frac{d! \, n!}{(d-k^{(y)})! \Pi_y},
\end{equation}
where
\begin{equation}
\Pi_y = \prod_{q=1}^{p^{(y)}} l_q! \left[M_q^{(y)}! \right]^{l_q^{(y)}}.
\end{equation}
To confirm the intuition of the previous section, that isolated particles are required for good fidelity, let us start by considering the fidelity for \emph{single} basis element states. In particular, let us examine Young diagrams which contain \emph{no} isolated particles, and denote the set of such Young diagrams as $\mathcal Y_>$,
\begin{equation}
\mathcal Y_> = \left \{ y \in \mathcal Y(n,d) \middle | M_1^{(y)} \geq 2 \right \},
\end{equation}
for example,
\ytableausetup{centertableaux,boxsize=6mm}
\[y \in \mathcal Y_> = \quad \quad
\begin{ytableau}
\; & & \none[\dots] & & & & \none[\dots] & & \\
\; & \none & \none & \none & \\
\none[\vdots] & \none & \none & \none & \none[\vdots] & \none & \none[\iddots] \\
\; & \none & \none & \none & \\
\tikznode{a4}{~} & & \none[\dots] & & \tikznode{a5}{~}
\end{ytableau}.
\]
\tikz[overlay,remember picture]{%
\draw[decorate,decoration={brace},thick] ([yshift=-3.5mm,xshift=3mm]a5.south east) --
([yshift=-3.5mm,xshift=-3mm]a4.south west) node[midway,below]{$\geq 2$};
}
\vspace{5mm}
We can then confirm that $A$ for any $\ket y$ such that $y \in \mathcal Y_>$ obeys $A \leq \mathcal O(d^{-2})$, and therefore, by Theorem \ref{fidbound}, $F(\ket y) \leq 1/2$. To see this, start by performing the partial trace on $\ket y$ to find $\rho_1$, which amounts to finding $\mathcal N_{i,j}^{(y)}$. Obviously, if $M_1^{(y)}=1$, there will be a contribution to $ \mathcal N_{i,j}^{(y)}$ which is proportional to $l_1^{(y)}$. But for $M_1^{(y)} \geq 2$, the only way to contribute to $\mathcal N_{i,j^{(y)}}$ is if $i$ and $j$ are in row blocks $q$ and $q+1$, and $M_q^{(y)}+1=M_{q+1}^{(y)}$. To add some intuition to that statement, we can only add to $ \mathcal N_{i,j}^{(y)}$ if, after removing a single block from $y$, there are at least two places (one for $i$ and one for $j$) to put that block back to return to $y$. All that remains is to sum over the possible arrangements and selections of the remaining indices which construct an element in $y$,
\begin{align}
\mathcal N_{i,j}^{(y)} =& \frac{(d-2)!(n-1)!}{(d-k^{(y)})! \Pi_y} \biggr(\delta(M_1^{(y)}-1) l_1^{(y)}(d-k^{(y)}) \\ \notag &+ 2\sum_{q=2}^{p^{(y)}} \Delta^{(y)}(q,1) l_q^{(y)}l_{q-1}^{(y)}M_q^{(y)} \biggr),
\end{align}
where $\Delta^{(y)}(q,r) = \delta \left( M_q^{(y)}-M_{q-1}^{(y)}-r \right)$. From here we can determine $A$, and find the following bounds,
\begin{align}
A =& \frac{2}{d^2n} \sum_{q=2}^{p^{(y)}} \Delta^{(y)}(q,1) l_q^{(y)} l_{q-1}^{(y)} M_q^{(y)} \\
\leq& \frac{2}{d^2n} \sum_{q=2}^{p^{(y)}} l_q^{(y)} l_{q-1}^{(y)} M_q^{(y)} \\
<& \frac{2k^{(y)}}{d^2n} \sum_{q=2}^{p^{(y)}} l_q^{(y)} M_q^{(y)} \\
<& \frac{2k^{(y)}}{d^2}.
\end{align}
So indeed, $A \leq \mathcal O(d^{-2})$ so long as $k^{(y)} = \mathcal O(1)$, and therefore $F(\ket{y}) \leq 1/2$ for $y \in \mathcal Y_>$.
Now let us broaden the picture by considering superpositions of basis elements with no isolated particles,
\begin{equation}
\ket{\psi_>} = \sum_{y \in \mathcal Y_>} a_{y} \ket{y}.
\end{equation}
From this state, tracing down to $\rho_1$ gives two components of $A$,
\begin{equation}
A= \sum_y A_>^{(y)} + \sum_{y \neq z} A_\times^{(y,z)},
\end{equation}
where, before defining them formally, the components can be described as $A_>^{(y)}$ being the contribution from the single $M_1^{(y)}>1$ basis elements, and $A_\times^{(y,z)}$ being the cross terms from different elements. Now, in more detail, we can start with the familiar term,
\begin{align}
A_>^{(y)} = \frac{2 \left|a_y \right|^2}{d(d-1)n} \sum_{q=2}^{p^{(y)}} \Delta^{(y)}(q,1)l_q^{(y)}l_{q-1}^{(y)}M_q^{(y)} .
\end{align}
The new term in the calculation is the cross term, $A_\times^{(y,z)}$, but notably not all cross terms are going to appear in the partial trace. Two Young diagrams, $y$ and $z$, will only contribute to $A_\times^{(y,z)}$ if $\ket{i k_2 \ldots k_n}$ is in $\ket y$ and $\ket{j k_2 \ldots k_n}$ is in $\ket z$ or vice versa. This then implies that Young diagrams, $y$ and $z$, differ by only one block placement. Possibly a clearer way to describe this is that removing a single block from $y$ and from $z$ will arrive at the same Young diagram, or, more precisely, they are connected to a common vertex with $n-1$ blocks in Young's lattice \cite{SUTER2002233}. This leads us to define the `compatibility function', $G(y,z)$, which evaluates to 1 if $y$ and $z$ are connected to a common vertex with $n-1$ blocks in Young's lattice, and evaluates to 0 otherwise. The size of $A_\times^{(y,z)}$ will then depend on the number permutations of the remaining $k_2 \ldots k_n$ which are consistent with $y$ and $z$. To quantify this, let $m_1^{(y)}$ indicate the row cluster from which the block is taken in $y$ and moved to the row cluster $m_2^{(y)}$ in $y$ to create $z$. We can analogously define $m_1^{(z)}$ and $m_2^{(z)}$. We could then choose which diagram, $y$ or $z$, to index the sum over the $k_2 \ldots k_n$. Rather than committing to one, we will do both simultaneously, relying on the following identity for compatible $y$ and $z$,
\begin{align}
\frac{l_{m_1}^{(y)}l_{m_2}^{(y)}M_{m_1}^{(y)}}{\Pi_y} &= \frac{l_{m_1}^{(z)}l_{m_2}^{(z)}M_{m_1}^{(z)}}{\Pi_z} \\ &= \sqrt{\frac{l_{m_1}^{(y)}l_{m_2}^{(y)}M_{m_1}^{(y)}}{\Pi_y}\frac{l_{m_1}^{(z)}l_{m_2}^{(z)}M_{m_1}^{(z)}}{\Pi_z}}.
\end{align}
From here we can finally determine
\begin{align}
A_\times^{(y,z)} = \frac{a_y^{\vphantom *}a_z^*}{d(d-1)n}G(y,z) \sqrt{l_{m_1}^{(y)}l_{m_2}^{(y)}M_{m_1}^{(y)}l_{m_1}^{(z)}l_{m_2}^{(z)}M_{m_1}^{(z)}}.
\end{align}
We can then bound the sums over these terms, starting with that over $A_>^{(y)}$
\begin{align}
\sum_y A_>^{(y)} &= \frac{2}{d(d-1)n} \sum_y \left | a_y \right |^2 \sum_{q=2}^{p^{(y)}} \Delta^{(y)}(q,1)l_q^{(y)}l_{q-1}^{(y)}M_q^{(y)} \\
&< \frac{2}{d(d-1)} \sum_y \left | a_y \right|^2 \sum_{q=1}^{p^{(y)}} l_q^{(y)} \\
&\leq \frac{n}{d(d-1)} \sum_y \left|a_y \right|^2 \\
&= \frac{n}{d(d-1)},
\end{align}
which is clearly still $\leq \mathcal O(d^{-2})$ so long as $n \ll d$. We can then turn our attention to bounding the sum over $A_\times^{(y,z)}$,
\begin{align}
\sum_{y \neq z} A_\times^{(y,z)} &= \frac{1}{d(d-1)n} \sum_{y \neq z} a_y^{\vphantom *}a_z^*G(y,z) \\ \notag & \hspace{10mm} \times \sqrt{l_{m_1}^{(y)}l_{m_2}^{(y)}M_{m_1}^{(y)}l_{m_1}^{(z)}l_{m_2}^{(z)}M_{m_1}^{(z)}} \\
&< \frac{n}{2d(d-1)} \sum_{y \neq z} a_y^{\vphantom *}a_z^*G(y,z) \\
&= \frac{n}{4d(d-1)} \sum_{y \neq z} \left(a_y^{\vphantom *} a_z^* + a_z^{\vphantom *}a_y^* \right) G(y,z) \\
&\leq \frac{n}{2d(d-1)} \sum_{y \neq z} \left|a_y \right| \left | a_z \right|G(y,z) \\
&\leq \frac{n}{d(d-1)} \sum_y \left|a_y \right| \sum_{z \leq y} \left|a_z \right| G(y,z),
\end{align}
where $z \leq y$ if $ \left|a_z \right| \leq \left|a_y \right|$. Continuing on,
\begin{align}
\sum_{y \neq z} A_\times^{(y,z)} &\leq \frac{n}{d(d-1)}\sum_y \left|a_y\right|^2 \sum_{z \leq y} G(y,z) \\
&< \frac{n}{d(d-1)} \sqrt{\frac n2} \sum_y \left| a_y \right|^2 \\
&= \frac{n}{d(d-1)} \sqrt{\frac n2},
\end{align}
which is likewise $\leq \mathcal O(d^{-2})$ so long as $n \sqrt n \ll d$. These two together imply that $A \leq \mathcal O(d^{-2})$, and therefore, by Theorem \ref{fidbound}, $F\left( \ket{\psi_>} \right) \leq 1/2$.
\section{Discussion}
The cumulative conclusion of the work of this paper is that isolate particles are \emph{required} for good agreement with the mean field approximation. Exactly how fidelity increases with the number of isolated particles, however, remains unknown. Ideally we would be able to bound or approximate the fidelity as a function of the ratio of isolated particles to non-isolated. To do so, though, new techniques will need to be developed to evaluate the trace of $\sqrt M$ for PSS states with $A> \mathcal O(d^{-2})$.
We have found that the mean field approximation is not in general appropriate for identical particles on the complete graph. It then follows that the Gross-Pitaevskii is not in general a good approximation to the Schrodinger Equation for such systems, potentially unless the particles of the system remain relatively spread throughout the dynamics. This analysis provides interesting context to the use of non-linear dynamics for quantum algorithms, but is merely an initial characterization. Search algorithms, for example, require a marked site or set of marked sites, which breaks the site symmetry. This analysis would then need to be repeated for states with bipartite site symmetry, whose basis elements would consist of pairs of Young diagrams.
This work was supported, in part, by NSF grant PHY-1620846.
|
1,314,259,993,614 | arxiv | \section{Introduction}
A successful model of formation and evolution of galaxy clusters must
explain, among other properties, the observed morphology segregation
found in nearby rich clusters. The first detected difference between
galaxy populations was the strong gradient in morphological type with
radial distance, producing a concentrated spatial distribution of
early-type in comparison with the sparse distribution of late-type
galaxies. This effect was associated to the well discussed
morphology-density relation, $T-\Sigma$ (Dressler 1980), or to the
alternative explanation of morphology-radius relation $T-R$ (Sanrom\`a
\& Salvador-Sol\'e 1990, Whitmore, Gilmore \& Jones 1993). These two
relations allow to attribute the morphology of galaxies to local or to
global cluster properties, respectively. The T-$\Sigma$ relation is
observed in both open and compact clusters in the local universe.
However, this is apparently not the case at intermediate-redshift.
Dressler et al. (1997) find that the T-$\Sigma$ relation is
qualitatively similar in compact clusters at intermediate-redshifts,
but completely absent in the open clusters at a similar epoch. This
result suggest that morphological segregation occurs hierarchically
over time, i.e. groups that make up the irregular, open clusters at
intermediate-redshifts have not undergone significant morphological
segregation to establish a T-$\Sigma$ relation. However, by the
present epoch, the groups that make up the local open clusters would
have had sufficient time to establish such correlation. If this
hypothesis is correct, clusters at high redshifts should show little
or no morphological segregation.
Another segregation in nearby clusters, reported by many authors (Moss
and Dickens 1977, Tully \& Shaya 1984, Huchra 1985, Dressler \&
Shectman 1988, Sodr\'e et al. 1989, Bird et al. 1994, Andreon 1994,
Biviano et al. 1997, Colless \& Dunn 1996, Fadda et al. 1996,
Girardi et al. 1996, Scodeggio et al. 1995, Andreon \& Davoust 1997
and Andreon 1996), is identified as a kinematical segregation,
characterized by a early-type population with a velocity dispersion
always lower than the velocity dispersion of the late-type population.
That effect is commonly explained as a consequence of the different
virialization state of both populations, where the observed behavior
of late-type is attributed to the possibility that they have been
accreted by the cluster more recently, after the collapse and violent
relaxation of the initial population of early-type galaxies which now
constitute the cluster core. We can also mention, continuing with the
detected kinematical segregation in clusters, the less studied
luminosity segregation (Biviano et al. 1992, Fusco-Fermiano \& Menci
1998, Kashikawa et al. 1998), which seems to affect preferentially the
more luminous early-type galaxies. Biviano et al. (1992) found that
this type of galaxies are located in the center of clusters and they
have velocities with respect to the mean cluster velocity lower than
other less massive members, as could be expected if they are affected
by dynamical friction. Recently, another type of galaxy segregation
was detected by Ram\'{\i}rez and de Souza (1998, hereafter RdS98).
They analyzed kinematically the early- and late-type galaxies in 18
nearby rich clusters, and they concluded that inside 1.0 \ifmmode{h^{-1}\,\hbox{Mpc }}\else{$h^{-1}$\thinspace Mpc }\fi
elliptical galaxies present orbits more eccentric than spiral
galaxies, and only well outside the fudicial radius r$_{200}$ the
orbits of spiral galaxies become more radial. A possible explanation
for it is related to the stability of the morphological shapes of
galaxies as they plunge towards the central regions. In this case
objects with roughly circular orbits, will not have their morphologies
seriously affected because they avoid the cluster center where the
probability of occurring a strong interaction will be higher.
However, those with more eccentric orbits will cross the densest
cluster regions and will experience on average a stronger
environmental influence and a higher encounter rate. Furthermore, if
along the cluster life-time there are significant orbital changes of
their members, high redshifts clusters could show a different level of
orbital-segregation than the nearby population.
In this paper we present a kinematical analysis of clusters at
intermediate redshifts. The goal of this analysis is to detect how
strong the orbital-segregation between early- and late-type galaxies
at this epoch is. The results are compared with those for a nearby
clusters sample. In section 2 we present the distribution function we
used to analyze the line-of-sight velocities of a cluster having a
velocity field with anisotropy. In section 3, it is defined two sample of
clusters and we define the morphology classification we
use. In section 4 we study the morphological segregation in the
intermediate-redshift sample. In section 5 we use the average
deviation of the line-of-sight velocity normalized to the velocity
dispersion to trace the orbit distributions of early- and late-type
galaxies. In section 6 we summarize our main conclusions.
\section{Distribution of velocities and the average deviation}
In this section we present a brief summary of the analytical analysis,
completely presented in RdS98, of the velocity distribution of systems
with velocity anisotropies.
Let us assume that for a given morphological class the velocity
distribution function is Gaussian, having however different
dispersions along the radial ($\sigma_R$) and transversal directions
($\sigma_\perp$). The behavior of the velocity distribution of this
system can therefore be described by the anisotropy parameter $\eta =
\sigma_R/\sigma_\perp$. A large value of $\eta$ for a given
population means that its members are crossing the cluster with an
almost radial orbit, and therefore are more sensitive to suffer
gravitational encounters with objects in the dense central regions.
On the contrary, objects with lower anisotropy parameter tend to have
a more circular orbit with small penetration in the dense core region.
Therefore, for a given morphological class the anisotropy parameter
should allow us to connect the efficiency of the environment
perturbations with the related kinematical orbital behavior.
The line-of-sight velocity distribution in a Gaussian velocity field
with a fixed anisotropy can be expressed as:
\begin{equation} F(u;\eta ) = \frac{1}{(2\pi)^{1/2}}
\int^1_0 \frac{1}{\Theta(\omega,\eta)}
e^{-\frac{u^2}{2\Theta(\omega,\eta)^2}}
d\omega
\end{equation}
\noindent where $\omega = \cos \theta$, $u = v_z/\sigma$ and $v_z$ is
directed along the line-of-sight. The velocity dispersion $\sigma$ can
be expressed in the form $\sigma = \sqrt{(\sigma_R^2 +
2\sigma_\perp^2)/3}$ and the term $\Theta(\omega,\eta)=
\sqrt{\frac{3(1-\omega^2 +\eta^2\omega^2)}{2+\eta^2}}$ represents a
correction of the velocity dispersion due to the presence of the
anisotropic field. In particular, for $\eta=1$ we retrieve the
expected Gaussian shape for an isotropic velocity field. Then, even
if the radial and transversal distributions are assumed to be
Gaussian, the observed distribution along the line-of-sight, when the
anisotropy parameter is different from one, is not Gaussian.
Although in the general case the line-of-sight velocity density
distribution can be estimated only by numerical methods, it is
interesting to note that the expressions for the moments can be solved
exactly. The distribution $F(u;\eta)$ is symmetric by construction,
resulting that the first centered moment is obviously zero. The
second moment corresponds to the variance of the distribution that
remains independent of the anisotropic parameter. Therefore, we may
expect that two populations responding to the same gravitational
potential can
present different orbital shapes, but their velocity dispersions
remain constant. Then, to detect anisotropies is better to work
with those expectation values that are function of the $\eta$
parameter. The more interesting of them is the average or mean
deviation (Kendall, Stuart and Ord, 1987) of the line-of-sight
velocity normalized to the velocity dispersion
\begin{equation} |u| = \sum_{i=1}^N |(v_z)_i|/\sigma
\end{equation}
\noindent The predicted value of this statistical
parameter, using $F(u;\eta)$, can be estimated by the expressions,
for $\eta < 1$
\[ |u| = \sqrt{\frac{6}{\pi (\eta^2+2)}} \biggl( \frac{\eta}{2} +
\frac{1}{2} \frac{1}{ \sqrt{1-\eta^2}} sin^{-1}(\sqrt{1-\eta^2} )
\biggr) \]
and for $\eta > 1$
\[ |u| = \sqrt{\frac{6}{\pi (\eta^2+2)}} \biggl( \frac{\eta}{2} +
\frac{1}{2} \frac{1}{ \sqrt{\eta^2 - 1}} ln(\sqrt{\eta^2 - 1} + \eta )
\biggr), \]
\noindent the behavior of this function is presented in
Figure 1 and the limits it reaches are the following:
\vspace{0.5cm}
(i) population with radial orbits
~~($\eta \gg 1) \longrightarrow |u| \simeq\sqrt{3/2\pi} \simeq 0.69$
\vspace{0.3cm}
(ii) population with purely circular orbits
~~($\eta = 0) \longrightarrow |u| =\sqrt{3\pi }/4 \simeq 0.77$
\vspace{0.3cm}
(iii) population with isotropic orbits
~~($\eta = 1) \longrightarrow |u| = \sqrt{2/\pi}\simeq 0.80$
\vspace{0.5cm}
From Figure 1 we can observe that the average deviation as a function
of the anisotropic parameter $\eta$ has a maximum at $\eta=1$. The
function is bi-valuated between $|u|$= 0.77 to 0.80, producing an
indetermination when we try to derive the value of $\eta$. As a
consequence we cannot distinguish between a purely circular model
($\eta = 0$) and one having a radial contribution slightly higher than
the isotropic case ($\eta \leq 2.4$). However, the region of
$|u| < 0.77$ or $\eta > 2.4$ is free from this problem, and as a
consequence the models of highly radial orbits can be easily
discriminated. Therefore, the average deviation is easy to
measured in real clusters, using the equation (2), and a direct comparison
with the expected values for each orbit family could be done.
\subsection{Comparison with other methods}
We must remark that our simplified treatment of the velocity
distribution produces the degeneracy in the average deviation value when
it is a function of the anisotropy parameter.
This is a consequence of
two assumptions we made, the first is the velocity
distribution as a Gaussian with constant anisotropy, and the second
is related to the density distribution which is only restricted to a
spherical symmetrical distribution. In fact, the distribution function
is not necessarily unique because the
density distribution, $\rho(\psi,r)$, could be originated from many
gravitational potential.
A usual method to fix the degeneracy is by the use of a complete set
of defined functions which follow the Jeans's equations and the Poisson's
equations in a self-consistent way. The unknown functions which
well define the system | i.e. spatial density distribution,
velocity dispersion profiles along the radial and tangential direction,
and the mass profile | could be obtained by fitting and inverting
the observed functions, i.e. projected velocity dispersion profiles
and surface brightness profile.
It is usually assumed a given profile for the anisotropy
together with a M/L distribution. In fact,
most of the works dealing with
these self-consistent distribution functions assume that M/L is constant
and the anisotropy is a function of the radius alone
(e.g. Binney \& Mamon 1982, The \& White 1986, Merritt 1987).
However, the solutions given by inverse problem are limited to systems with
high number of measured velocities | i.e. few
clusters or synthetic clusters produced by adding others |
since the errors or incompleteness in the
data will be amplified when going from data space to model space.
Then, in most of the cases the results tend to be noisy,
unless some objective smoothness condition is placed on the
solution. An approach is to replace the data by ad hoc
fitting functions for which the inversions can be carried out exactly,
or to use smooth functions that are generated from the data using
nonparametric algorithm (e.g. Qian et al. 1995).
For axisymmetric system Merritt (1996) concluded that
they are able to construct the gravitational potential and the kinematic
by the use of the full two-dimensional set of velocity,
line-of-sight velocity distribution (LOSVD) along
the radius. Binney et al. (1990) pioneered this method using only few
cuts along the radius. Both methods are very useful
when we have enough resolution in the shape of the LOSVD at different radius,
because it depends on the high order moments of the distribution. The
nonparametric fits of the LOSVD also enables
to detect the variation of the anisotropy without making
any hypotheses about the ratio M/L, although it is necessary to guess an
initial form of the gravitational potential.
When we compare our method with the models cited above,
we must remember that we are assuming
the anisotropy constant along the radius.
This difference produces that the
variability of the LOSVD along the radius is not due to a variability
in the anisotropy, instead it must be related to the mass
distribution and how well the galaxies trace the mass.
Furthermore, we must have in mind that we are
studing the global average deviation, which represents the
behavior of the integrated average deviation along the projected radius,
and we are integrating along the inner region of the cluster together
with a part of the outer region.
Nevertheless, if we want to get an idea of the behavior of the average
deviation along the radius, we can compare our model to these of
variable anisotropy as for example the models of Merritt (1987),
Dejonghe (1987), Gerhard (1993) and Merritt \& Gebhardt (1996).
They used different density profiles
and different gravitational potential to obtain that almost always in
the tangentially-anisotropic systems, i.e. $\sim \eta=0$, the LOSVD
tends to have a peaked shape in the center and evolves to a flat-topped
shape in the outer regions.
In the radial-anisotropic systems, $\eta > 2.4$, they have the LOSVD
shapes slightly flat-topped in the center an evolve to more peaked
shapes in the outer regions. These behavior must produce very different
variation in the average deviation along the radius, specially in the
outer region, and these variations must be stronger in the
tangentially-anisotropic systems than in the radial-anisotropic systems.
The center and the outer radius
in these cases are scaled by the core radius and they represent
a very central and a very external region within a galaxy cluster.
On other hand, if we compare our simple model with models
that assume the function distribution of quasi-separable form
(Gerhard 1991) and these ones that expand the distribution
function in the standard Hermite polynomials (Gerhard 1993),
we are approximately restricted to the terms zero-order Gaussian
and first-order odd of such distribution functions.
The goodness of the simplicity of our method is achieved
at the cost of having to determine not exactly which family of
orbits are more common when the studied systems have circular,
isotropic or mildly radial orbits, however, the highly radial family
are easily identified. This characteristic enable us to detect segregation
in the velocity anisotropy of our sample, because early and late-type
galaxy populations have anisotropies that are not constrained to the
degenerated region of the average deviation curve.
Also, we must remark that we are using the average
deviation to get the anisotropy of the system, which corresponds
to a first order moment of the distribution. This is
more robust than higher order moments and it was always
disconsidering in the past mainly due to the difficulty of
dealing with it analytically.
\section{The sample selection: intermediate-redshifts and nearby clusters}
Our basic sample contains all clusters observed by the Canadian
Network for Observational Cosmology, CNOC1 (Carlberg et al 1996). It
contains $\sim$2600 galaxies having Gunn $r$ magnitudes, $g-r$ colors,
and redshifts. They were selected in the fields of 16 high luminosity
X-ray clusters spread from redshift 0.18 to 0.55. This is a
relatively homogeneous sample of clusters that is guaranteed to be at
least partially virialized on the basis of their X-ray emission.
Galaxies of all colors above a k-corrected Gunn $r$ absolute magnitude
$M_r^k$ of $-18.5$ are considered in the calculations. Together with
the CNOC1 sample we use data for 12 nearby rich clusters, the same
sample analyzed in detail in RdS98. This nearby sample consists of
clusters with $z < 0.055$ having at least 65 members, within 2.5 \ifmmode{h^{-1}\,\hbox{Mpc }}\else{$h^{-1}$\thinspace Mpc }\fi
from the cluster center, morphologically classified as elliptical,
spiral or lenticular galaxies. Clusters with obvious substructures
were discarded, since in those cases the velocity distribution could
result from a complex association of several small groups.
Because of the non-homogeneity of the spatial distribution in the
CNOC1 sample (i.e. galaxy members selected along elongated strips), we
decided to work with data inside the fiducial radius r$_{200}$. This
radius | defined as the radius where the mean interior density is 200
times the critical density of the universe | is expected to contain
the bulk of the virialized cluster mass. To derive r$_{200}$ from the
observational virial radius, $r_v$ (which is largely fixed by the
outer boundary of the sample), we assume that $M(r)\propto r$. This
gives
\begin{equation}
r_{200} = \frac{\sqrt{3} \sigma}{10 H_o (1 +
z) \times (1 + \Omega_o z )^{1/2}}
\end{equation}
\noindent which is completely independent of the observational virial
radius (Carlberg, Yee \& Ellingson 1997). Here $\sigma$ is the global
velocity dispersion of each cluster. A Hubble constant of H$_0$ = 100
\ifmmode{\,\hbox{km}\,s^{-1}}\else {\rm\,km\,s$^{-1}$}\fi \ifmmode{h^{-1}\,\hbox{Mpc }}\else{$h^{-1}$\thinspace Mpc }\fi$^{-1}$ and $\Omega_0 = 0.2$ are adopted. An iterative
procedure was applied to select the members inside the r$_{200}$
region of each cluster. First an initial value of the velocity
dispersion was assumed, using values from Carlberg et al. (1996) for
the CNOC1 data and from RdS98 in the case of the nearby sample. The
r$_{200}$ was estimated and galaxies inside this radius were selected,
calculating a new velocity dispersion. The procedure was stopped if
the value for the velocity dispersion of the selected galaxies in two
subsequent iterations did not differ by more than 50 \ifmmode{\,\hbox{km}\,s^{-1}}\else {\rm\,km\,s$^{-1}$}\fi. No more
than 3 iteration were necessary for each of the clusters.
The properties of both samples are presented in Table 1, where the
clusters are ordered from high to low redshifts. Column (2) presents
the number of galaxies inside r$_{200}$ and column (3) shows r$_{200}$
in \ifmmode{h^{-1}\,\hbox{Mpc }}\else{$h^{-1}$\thinspace Mpc }\fi. The mean redshifts and the velocity dispersions (columns 4
and 5), and all other means and dispersions appearing in this paper,
were estimated using the bi-weighted estimators described in Beers et
al. (1990), using the ROSTAT\footnote{Version obtained from the ST-ECF
Astronomical Software Library ftp://ecf.hq.eso.org/pub/swlib} program,
which contains the versions of statistical routines tested by T.
Beers, K. Flynn, and K. Gebhardt for robust estimation of simple
statistics. Also, all the errors bars are at the 68\% confidence
level, and they were obtained via a bootstrap resampling procedure
with 1000 iterations. Column (6) presents the cluster concentration
parameter as was defined by Butcher and Oemler (1978), i.e. the
logarithm of the ratio between the radius that contains 60\% and 20\%
of the cluster population, $C = log_{10}(r_{60}/r_{20})$. Finally, in
column (7) clusters with obvious substructure are marked with a letter
{\it a} and those that do not cover the whole region inside r$_{200}$ are
marked with a {\it b}, all these clusters were not considered in the
final sample.
\subsection{Morphological classification}
The use of ground-based imaging for measuring morphology of
high-redshift galaxies is discussed in Schade et al. (1996, hereafter S96)
where
details are given of the procedure, including convolution with the
point-spread function. Although it is difficult to perform Hubble type
morphological classification even at moderate redshifts, it is
possible to define roughly 3 galaxy classes based on the fractional
bulge luminosity (B/T) after fitting two-component models
(de Vaucouleurs $R^{1/4}$ bulge and exponential disk). The data quality
here is lower than that used in S96 but the redshifts
are lower, on average on that a similar level of morphological
discrimination is feasible. The correspondence between the ratio B/T
and morphological type was established by using the following recipe:
(i) disk-dominated profiles, i.e. with $0.0 \leq$ B/T $\leq 0.4$,
were defined as late-type galaxies, (ii) profiles with $0.4 <$ B/T $<
0.7$ were defined as intermediate-type galaxies, and (iii)
bulge-dominated profiles, i.e. $0.7 \leq$ B/T $\leq 1.0$ were defined
as early-type galaxies.
The distribution of the rest frame color index (U-V)$_0$ of each
morphological class in the CNOC1 cluster galaxies are presented in
Figure 2. The difference between the early- and late-type color
distributions shows that morphological cuts that we have defined
provides some level of real discrimination between galaxy types since
the profile fits are independent of color. As expected from local
samples of galaxies, objects classified as early-type galaxies from
the B/T indicator are redder than those classified as late-type
galaxies. In fact, the early-type galaxies have a mean color index
(U-V)$_0$ = 1.84 $\pm$ 0.01 with a bi-weighted dispersion of 0.17.
This distribution present a small color scatter, as usually found in
early-type galaxies at intermediate-redshifts. However, the late-type
present two peaks: a blue distribution centered on (U-V)$_o$ = 1.06
$\pm$ 0.03 with a bi-weighted distribution of 0.21, and a red
distribution centered on 1.82 $\pm$ 0.01 (bi-weighted dispersion of
0.21). Then, the population of these late-type galaxies could be
further separated into two groups: a red disk-dominated galaxies, and
a blue disk-dominated galaxies, which could represent the very
late-type (i.e. Sd or Irr). This result shows that in the late-type
case, when the B/T ratio is considered, we are not separating the
galaxies by their star-formation activity. Instead, we are separating
them by how well they are represented by either an exponential and/or
a de Vaucouleurs profile, and this is not related with their internal
kinematical state, because virialized systems could present both types
of profiles. A comparison of the late-type color distribution with
Figure 6 of S96 indicates a significant difference in color between
disks in clusters and the fields. It is seems that the bluer peak of
the disk-dominated galaxies in clusters correspond to the field
population. In another side, the bulge-dominated galaxies in clusters
seems to be clean of the star forming galaxies, as Im, Sbc observed at
the field. But, the last result is accentuated by the fact that from
the final 716 galaxies morphologically classified, inside the
$r_{200}$ in the 9 selected clusters, 15 were excluded as peculiar
galaxies (with $B/T > 0.5$ and color index in the rest-frame $(U-V)_0
< 1.4$), as is suggested in S96. Other 39 were also discarded because
their low quality profile fitting.
\section{Morphological segregation}
We combine the two samples of clusters, in normalized co-ordinates to
construct two ensemble working samples, called ec-nearby and
ec-cnoc. Figure 3 presents the projected positions of the galaxies
inside the r$_{200}$ radius of the ec-nearby and ec-cnoc clusters. To
combine the clusters we adopt the same procedure of Carlberg, Yee \&
Ellingson (1997) and Ram\'{\i}rez \& de Souza (1998). The brightest
cluster galaxies were used as the nominal centers of each cluster on
the sky for the CNOC1 clusters. The galaxy velocities were normalized
to the velocity dispersion about the cluster mean, and the projected
radii were normalized to the empirically determined r$_{200}$ (see
section 3). This procedure diminishes substructure and asphericity to
a level where the galaxies can be treated as if they were in a
spherical distribution, preserving the radial dependence of the
kinematical properties of the samples, and clearing off the effects
due to eventually existing local substructures. Before any comparison
between both ensembles, it is important to clarify how the different
morphological populations will be referred. Due to the selection of
only elliptical, lenticular and spiral galaxies in the nearby sample,
hereafter the definition of early-type (late-type) galaxy will mean:
elliptical (spiral) galaxy in the nearby sample and bulge-dominated
(disk-dominated) galaxy at intermediate-redshift. Then, the early-type
galaxies not include the lenticular galaxies in the nearby sample, nor
the intermediate B/T class galaxies at intermediate-redshift. Also,
we must to note that the late-type definition in the nearby clusters
excluding the irregular galaxies.
\subsection{Projected position distribution}
From the Figure 3 we readily noted that the spatial distributions of
the late-type galaxies are broader than those for the early-type
galaxies, in both samples. This morphological segregation had already
been detected in the nearby sample and seems to be also present in the
CNOC1 data. A useful quantitative parameter to compare the spatial
distribution of each morphological class is the concentration C,
listed and defined in Table 1. We found that late-type galaxies tend
to present a mean concentration lower than that for early-type
galaxies. The typical values for the late-type galaxy populations
are: C$_{S-nearby} = 0.40$ and C$_{S-cnoc} = 0.31$. For the more
concentrated early-type galaxies the values are C$_{E-nearby}=0.59$
and C$_{E-cnoc}= 0.42$, and for the intermediate-type it is
C$_{S0-nearby} =0.50$ and C$_{S0-cnoc}= 0.41$. Although it is clear
that the concentration parameter is higher for the early-type galaxies
in both samples the effect is less pronounced in the case of the
intermediate-redshift clusters.
Figure 4 shows the morphology-radius relation that galaxies separated
by the B/T criteria follow. This figure shows that the
bulge-dominated galaxies are centrally concentrated while the
disk-dominated population prefers the periphery. This simply shows the
morphology-density (Dressler 1980) or morphology-radius (Whitmore,
Gilmore, \& Jones 1993) relation over a wide range in cluster-centric
distance. The field galaxies included in the CNOC1 survey are also
included in the plot.
\subsection{Kinematical segregation}
Figure 5 presents a summary of several cluster properties as a
function of $z/(1+z)$. For low values of redshifts $z/(1+z)
\rightarrow z$, and it is proportional to the look-back time ($\tau$)
when $\Omega_0 \rightarrow 0$. In fact, $\tau = z/(1+z) H_o^{-1}$ Gyr
when $\Omega_0 = 0$. We prefer this variable because the clusters
with redshifts lower than 0.1 lay more separated in a plot with this
scale than in a linear z scale, which is also labeled in the upper
axis.
Figure 5a shows the distribution of the radius r$_{200}$ in \ifmmode{h^{-1}\,\hbox{Mpc }}\else{$h^{-1}$\thinspace Mpc }\fi. The
dotted and dashed lines represent the r$_{200}$ of the Coma and Virgo
clusters, respectively. None of the CNOC1 clusters is as massive as
Coma and three of them have significantly low masses. Figures 5b
and 5c show the ratio between the velocity dispersion of late- and
early-type galaxies, $\sigma_S/\sigma_E$, and the ratio between
intermediate and early-type galaxies, $\sigma_{S0}/\sigma_E$,
respectively. As can be seen late-type galaxies at low and
intermediate-redshifts show similar values, while the intermediate
galaxies display values from $\sigma_{S0}/\sigma_E \sim 0.50$ to
$\sim$1.8 in both samples. It has previously been suggested that
late-type galaxies are just now being captured while early-type
galaxies may have been in the cluster since the initial formation,
representing the relaxed and virialized population, respectively (see
references in section 1). Then, a value of $\sqrt 2$ must be expected
(dot-and-dashed line in the Figure 5b and 5c), however in most
clusters the ratio is lower. Actually, the mean values are
$\sigma_S/\sigma_E = 1.14 \pm 0.07$ (bi-weighted dispersion of 0.18)
for the nearby sample and $\sigma_S/\sigma_E =1.17 \pm 0.06$
(bi-weighted dispersion of 0.18) for the CNOC1 data. The velocity
dispersion of the late-type galaxy population is approximately 15\%
higher than the velocity dispersion for the early-type galaxies.
These results show the existence of a kinematical segregation in both
samples | nearby and intermediate-redshift | where the population of
early-type galaxies always have a lower velocity dispersion than the
population of late-type galaxies. The similar values along the
redshift suggest that no-evolution for this type of segregation is
present.
Figure 5d shows the distribution of the concentration as a function of
redshifts, where the dotted line represents the Coma cluster
concentration, and the dashed line correspond to the Virgo cluster
concentration, and open diamonds show the concentrations of nine
Dressler et al. (1997) (hereafter D97) clusters. D97 measured the
concentration of 10 clusters at intermediate-redshifts, using galaxies
well inside the central region that cover $\sim$ 0.5-0.8 \ifmmode{h^{-1}\,\hbox{Mpc }}\else{$h^{-1}$\thinspace Mpc }\fi
(i.e. inside the corresponding $r_{200}$ if they have $\sigma < 1200$
\ifmmode{\,\hbox{km}\,s^{-1}}\else {\rm\,km\,s$^{-1}$}\fi). They re-evaluated the T-$\Sigma$ relation, using these
clusters, finding that the more concentrated clusters are the most
affected by the T-$\Sigma$ relation. We observe that most of the
nearby clusters have similar concentration values | i.e. concentration
around 0.42-0.52 | and they are more concentrated than the
intermediate-redshift cluster. The CNOC1 clusters appear with
concentrations similar to the low concentration cluster of D97 sample.
Then, in agreement with the conclusions of D97 we must expect a lower
morphological segregation within the CNOC1 clusters.
\subsection{Lenticular galaxies evolution}
Finally, in the last two plots of Figure 5 we detected evolution
comparing the numbers of late- and intermediate-type to early-type
galaxies. In this case we are based in the fact that early-type
galaxies must correspond to the oldest population in the cluster |
i.e. if they were formed by merging, that must have happened at the
beginning of the cluster formation, or if they were formed in a
monolithic way they were formed all together not after $z = 2$ | and
we are also assuming that their numbers kept constant along the
cluster life-time. In D97 the ratio of lenticular to elliptical
galaxies is cited as an indicator of evolution, and they suggest this
ratio decreases with redshift. This is the same tendency found in our
samples, as Figure 5f shows. Although, when we compare our data
(filled circles in Figure 5f), with those of D97, where a visual
morphological classification with HST images was used (open diamonds
in Figure 5f), we observe an underestimation of our number of
intermediate-type galaxies. This is not only an effect of the
difference between morphological classification methods, because the
nearby sample | where the classical visual morphology classification
was also used | follow the same tendency of the CNOC1
intermediate-redshifts clusters. Nevertheless, in all the cases the
number of intermediate-type galaxies is significantly lower at higher
redshifts. In fact, the mean value of $N_{S0}/N_E$ is $1.46 \pm 0.34$
for the nearby sample and $0.43 \pm 0.04$ for the CNOC1 sample.
Unfortunately, the physical meaning of this evolution is still in
discussion and the problem with the misidentification between
elliptical and lenticular galaxies is always present. Interesting is
to note that when the late-type galaxies are compared with the
early-type galaxies the ratio $N_{S}/N_E$ (Figure 5e) is almost
constant, and they have similar values of that of D97 sample.
\section{Segregation detected in the average deviation of the
velocity distribution}
The average deviation $|u|$ defined in section 2, is useful to connect
the line-of-sight velocity distribution of a population with the
family of orbits they represent. RdS98 analyzed the behavior of this
parameter for a sample of 18 nearby clusters, and concluded that
elliptical galaxies have more eccentric orbits than spiral galaxies.
In this section we investigate if the results found for the nearby
cluster sample also apply for the CNOC1 sample.
The velocity average deviation normalized by the velocity dispersion,
$|u|$, was calculated for each population class as follow:
\[ |u| = \frac{1}{N} \sum_{i=1}^N |v_i-v_{cl}|/\sigma_{cl} \]
\noindent where N is the number of galaxies of each morphological
class, $v_i$ is the line-of-sight velocity, $v_{cl}$ and $\sigma_{cl}$
are the mean velocity and velocity dispersion of the cluster,
and they were calculated with the bi-weighted estimators (Beers et al 1990).
Table 2 presents the average deviation and other kinematical properties of all
morphological classes in the nearby and CNOC1 clusters. Column (1)
shows the name of the cluster, in the case of Virgo the first entry
(Virgo-all) corresponds to the whole cluster, considering as a part of the cluster
all the substructures detected by Binggeli, Popescu \& Tammann (1993), inside $r_{200}$,
the second entry (Virgo-A) corresponds to the main cluster only, part A.
The columns (2), (3) and (4) present
the following properties for the late-type galaxies: number of
galaxies inside the r$_{200}$ radius, velocity dispersion normalized
by the cluster velocity dispersion, and average deviation also
normalized by the velocity dispersion with their errors to the 68\%
confidence level. Columns (5), (6) and (7) show the same properties
described above for intermediate-type galaxies and columns (8), (9)
and (10) for early-type galaxies. When less than 6 galaxies were
available the dispersion and the average deviation were not
calculated. In Figure 6 we plot the distribution of the average
deviation for the late-, intermediate-, and early-type galaxies as
a function of the same variable $z/(1+z)$ used in Figure 5. The
dotted lines show the expected value of $|u| = 0.77$, which represents
the highest value bellow which an eccentric orbit can be unambiguously
identified, as was described in section 2. For clusters below the dotted
we should have that the radial velocity dispersion is larger than the
transversal velocity dispersion, $\sigma_R > \sigma_\perp$, under the
assumption of Gaussian velocity field adopted in the present study.
There is a possible bias toward higher values of average deviation due
to substructures within clusters, because an artificial enhancement is
introduced in the projected velocity distribution.
This bias could be more important in the intermediate redshift sample,
where substructures are hardly detected.
Cen (1997) using n-body simulations suggested that the presence of
substructures modifies the velocity distribution in a complex way, but
the final velocity dispersion is slightly affected. He estimated
variations of the final velocity dispersion of only 5\% and 9\%
within 0.5 and 1.0 \ifmmode{h^{-1}\,\hbox{Mpc }}\else{$h^{-1}$\thinspace Mpc }\fi, respectively. Another study is presented
in Bird et al. (1996), using observational data, where they noted that
the existence of substructures is an important factor to determine the
dynamical parameters, but the effect is reduced when the data is
restricted to a region inside the virial radius, as we made.
It is worthwhile to mention that variations
on the velocity dispersion due to substructures could increase or reduce
the velocity dispersions (Bird, 1994), a fact which is
not consistent with spirals always presenting an slightly larger
dispersion than ellipticals. Then, the effect produced by substructures
seems to be weak enough to allow the segregation be detected.
However, to test how the average deviation could be modified by
substructures we applied the method to the Virgo cluster,
taking into account the substructures detected by Binggeli,
Popescu \& Tammann 1993, (BPT93).
First we worked with the main body of Virgo, called part A
in BPT93, then the same method was applied including
the part B and the clouds W and W'. All samples were
limited to the region inside the r$_{200}$ radius.
The results are presented in the two Virgo's entry in the
Table 2. Both samples show
very similar results and differences between
spiral, lenticular and elliptical populations are detected
on spite of the substructures.
\subsection{Late and early-type galaxies}
Figure 6 shows the segregation at low redshifts previously detected by
RdS98 (see their Figure 4), but now considering galaxies inside
r$_{200}$ radius instead of the limiting radius of 1.0 \ifmmode{h^{-1}\,\hbox{Mpc }}\else{$h^{-1}$\thinspace Mpc }\fi and 2.5
\ifmmode{h^{-1}\,\hbox{Mpc }}\else{$h^{-1}$\thinspace Mpc }\fi adopted in that analysis. The segregation at low redshift shows
that elliptical galaxies have $|u|$ values corresponding to highly
eccentric orbits, while the spiral galaxies have orbits that are
nearly isotropic, circular or slightly radial. Although in general
the population of early-type galaxies in the CNOC1 clusters follow the
same tendency of those of the nearby clusters, having also lower
average deviation values, the clear segregation detected for clusters
at $z < 0.1$ is not so obvious at higher redshifts. This result could
indicate a real evolution of the orbit shapes inside the $r_{200}$
radius, where the early-type population only recently got more
eccentric in their orbits, and the population of late-type galaxies
kept in circular or isotropic orbits. However, the distribution of
average deviations of early-type galaxies in the CNOC1 sample have
cases as ms1008-12 with orbits as eccentric as the nearby elliptical galaxies.
We must note that the intermediate-type galaxies in the nearby sample
follow the same behavior as the early-type population. However, at
intermediate redshift the small number of intermediate-type galaxies,
producing a behavior not well defined in their kinematical parameters.
To further quantify the degree of significance of the differences
between the late-type and early-type galaxies along the redshift, we have
applied statistical tests. The main goal is to verify if the
observed distributions of the average deviations in Figure 6, could
come from the same parent distribution. The same two-sampling tests
were applied to all clusters. The tests applied were developed
in IRAF/STSDAS ({\em twosampt} and {\em kolmov}). The results are
presented in Table 3, where columns show: (1) the two population
being compared, (2) the sample and the morphological criteria, (3)
the number of clusters used, and (4) the mean, the error and the
dispersion of the average deviation distribution of each sample.
The four last columns present the probability, expressed in percentage,
of the two compared
population belong to the same distribution, using the: (5) K-S
Test, (6) Gehan's Generalized Wilcoxon Test, (7) the Logrank Test
and the (8) Peto \& Prentice Generalized Wilcoxon Test. The
statistical tests show a significant difference between the average
deviation distribution of the local spiral and elliptical galaxies
at the 99\% of confidence level. In the case of bulge- and disk-dominated
galaxies at intermediate-redshifts the distributions of the average
deviation are different at the 67-89\% of confidence level,
depending of the statistical test. Then, the difference in the
velocity anisotropies seems to exist but is weaker than those for
the nearby galaxies.
\subsection{Red and blue galaxies}
Carlberg et al. (1997) compared the properties of the CNOC1 galaxies
separating them by colors. They applied the Jeans equation of stellar
hydrodynamical equilibrium to the red and blue subsamples, and they
found that both populations give separately statistically identical
cluster mass profiles. This is strong evidence that the CNOC1
clusters are effectively systems in equilibrium. Motivated by this
work we have also repeated the analysis separating the galaxies by
colors. Unfortunately we do not have access to homogeneous data on
colors for the nearby clusters. A comparison between nearby and
intermediate-redshift clusters requires complete data color for both
samples with the same set of filters. Therefore the comparison between
red and blue galaxies is applied to the CNOC1 data only.
The criteria to separate the red and blue galaxies was the following:
(i) galaxies with (U-V)$_0 \leq 1.4$ were defined as blue galaxies,
associated to the late-type galaxies, (ii) galaxies with (U-V)$_0 >
1.4$ were defined as red galaxies, associated to the early-type
population. Applying our orbital analysis to these two subsamples we
find that the average deviation of both populations, in all clusters,
have very different distributions (different at the 99\% confidence
level). The last line in Table 3 shows this result and Table 4 shows
in detail the properties of these populations. In this case we are
working with only 7 of the 9 clusters because ms1455+22 and ms0440+02
have less than 6 blue galaxies, and no-statistical results are
possible with this number of objects. The mean average deviation of
the blue population is $|u| = 1.02 \pm 0.09$ (dispersion of 0.21) and
for the red population is $|u| = 0.70 \pm 0.02$ (dispersion of
0.06). Therefore, similarly as occurred with nearby ellipticals, the
red galaxies observed at intermediate-redshift also have more
eccentric orbits than the blue galaxies. In addition, we observe that
the mean velocity dispersion of the red and blue galaxy populations |
calculated with the velocity dispersion of each population normalized
to the velocity dispersions of the respectively cluster | are 0.96
$\pm$ 0.04 and 1.29 $\pm$ 0.09, respectively. That is, the velocity
dispersion of blue cluster galaxies is about 30\% higher than for red
galaxies. This quantity is larger than the difference between
elliptical and spiral at low redshift and this one of disk- and
bulge-dominated galaxies at intermediate-redshift.
Before any consideration about the notable difference between red and
blue galaxy population inside $r_{200}$, we must mention two important
characteristic of these subsamples. First, the difference detected
here is not in contradiction with the conclusion of Carlberg et
al.(1997) about red and blue galaxies in the CNOC1 survey being
systems in equilibrium with the gravitational potential of their own
clusters. As was explained in RdS98, the global parameters as velocity
dispersions are independent of the existence of any difference in the
velocity anisotropy. And second, unfortunately up to the limit of
$M_r^k=-18.5$ mag, about 85\% of the cluster galaxies inside r$_{200}$
fall into the red subsample. This effect produce a limiting factor in
the precise numerical agreement because there are relatively few blue
galaxies in clusters, meaning that the mean velocity dispersion and
average deviation are less accurately measured than those for red
galaxies. On spite of the few galaxies, the behavior of the blue
population in all the clusters is always in the same direction (see
Table 4). If the blue galaxies are galaxies rich in gas they could
represent the star formation activity of the cluster. This activity
could be due to an internal perturbation (e.g. early stage of the gas
consumption), external perturbations (e.g. tidal effect by near
companions or recent merging), or due a mix of these two (e.g.
internal response to the galactic harassment). Define how all of
these perturbations will affect the observed stellar distribution is a
hard task. However, the orbital properties can produce measurable
properties compatible with the star formation activity. The objects
with very eccentric orbits reach the densest regions of the cluster
and are more prompt to suffer perturbations from other
galaxies. Although the high relative velocities of the galaxies in the
cluster center makes merging unprovable, the accumulated effect of
these perturbations could remove the gas content of the galaxies,
dimming the star formation rates (e.g. Fujita et al. 1999). In this
cases, the blue galaxies in clusters tends to appear redder after some
crossing times due to the velocity anisotropy. This could be the
case of the A2390 cluster studied by Abraham et al. (1996). In fact,
they found that the blue galaxies in this cluster are being altered by
the cluster environment, such that some of the their members are
likely leaving the blue population to join the red population. Then,
if the other clusters in the CNOC1 survey have velocity anisotropy, as
A2390 cluster, our finding that red galaxies have more eccentric
orbits than blue galaxies is consistent with a modification of the
star formation activity by an orbital-segregation.
\section{Conclusion}
We presented a kinematical analysis of clusters at
intermediate-redshifts. They were analyzed looking for the existence
of an orbital-segregation, in the sense that galaxies with different
morphological types have different orbit families, as was previously
found for ellipticals and spirals in nearby clusters. The velocity
anisotropies of each morphological population was estimated by the use
of the average deviation of the line-of-sight velocity, which is
associate to the orbital shapes, assuming a Gaussian distribution
having different dispersions along the radial and the transversal
directions.
The differences in the average deviation of velocity distributions of
elliptical, lenticular and spiral galaxies detected inside the 2.5
\ifmmode{h^{-1}\,\hbox{Mpc }}\else{$h^{-1}$\thinspace Mpc }\fi in nearby rich clusters (RdS98) is stronger inside the more
interesting in physical meaning $r_{200}$ radius. These differences
correspond to different anisotropies and it corresponds to an stronger
orbital-segregation. In this case the early-type galaxies, inside the
$r_{200}$ radius, have orbits more eccentrics than the late-type
galaxies, and the intermediate-type galaxies share an intermediate
orbital family.
When the procedure applied to the nearby clusters is applied to a
sample of intermediate-redshifts cluster, as the CNOC1 survey, an
orbital-segregation is again detected. However, this time the
differences is between the bulge-dominated galaxies and the
disk-dominated galaxies, and the orbital-segregation is less strong
than in the nearby sample. Moreover, when the orbits of the red and
blue galaxies of the intermediate-redshift sample are compared the
strongest orbital segregation is found. This result suggests the orbital
segregation seems to modify more efficiently the star formation
activity than the internal shape of the galaxies.
Then, our results impose a further restriction on the plausible models of
cluster of galaxies formation. Along the covered redshifts the models
must reproduce the observed velocity field
anisotropy of each morphological type. In this case the early-type
galaxies are evolving very fast from z=0.4 to 0, and late-type
galaxies remain with their orbits without obvious evolution. However
we cannot discard the possibility that this could be an effect of
comparing different population when we associate the elliptical
galaxies in nearby clusters to the bulge-dominated galaxies at
intermediate-redshifts. Unfortunately, in the case of
intermediate-type galaxies, their behavior at higher redshifts is not
so clear as in the nearby cluster, and we have | besides the always
present problem of misidentification | the problem of the dramatic
decrease in number.
Finally, some interesting question are opened:
Does the orbital-segregation represent the tendency of early-type
galaxies to preserve the initial conditions of an anisotropic
collapse? In effect, this type of collapse studied with high
resolution n-body simulations and semi-analytical models forms
clusters with an anisotropic velocity distribution of their members.
These clusters are different from those obtained in a classical
spherical collapse, without taking into account the anisotropy in
large-scale initial conditions. Cosmological n-body simulations have
shown that, in the most scenarios, the matter in filaments collapses
into dark matter halos, and these flow along filaments towards the
potential minima, where they form galaxy clusters. The collapse is
therefore a highly anisotropic process (West, Villumsen \& Dekel 1991,
Tormen 1997, Tormen, Diaferio \& Syer 1998; Ghigna et al. 1998;
Dubinski 1998; Splinter et al. 1997). For example, Tormen (1997)
studying cluster formation with n-body simulations found that more
massive satellites move along slightly more eccentric orbits, which
penetrate deeper in the cluster. Also, he found that the shape and
orientation of the final cluster and its velocity ellipsoid are
strongly correlated with each other and with the infall pattern of
merging satellites, suggesting that cluster alignments is related to
the anisotropy in the velocity. In addition, the properties of dark
matter halos within rich clusters studied by Ghigna et al. (1998)
present an orbital distribution close to isotropic,
however the circular orbits are rare and the radial orbits are
common. Although, the interesting results cited above are more
representative of the dark-matter distribution, few has been done to
compare with real clusters. Only some observations of distribution
of structure in X-ray clusters (West, Jones \& Forman 1995) seem
indicate that anisotropic collapse is common in also in real clusters.
Nevertheless, why only elliptical galaxies are able to preserve the
initial conditions? And why the youngest sample shows
very eccentric orbits only for red galaxies? Could this be the result
of a morphological transformation within an anisotropic collapse? In
this case, how the strength of this transformation will depend of the
galaxy orbit?. That question, must be studied and clusters are higher
redshift must be studied.
\acknowledgments
We thank to the members of the CNOC team who providing us with
the data, in special we thank the comments from Ray G. Carlberg.
Also AR thanks Claudia Mendes de Oliveira who helped her with
comment that led to substantial improvements in the presentation.
Also we gratefully acknowledge financial support
from FAPESP (AR Post-Doc fellowship grant No 1998/014345-9)
and FAPESP (RES grant No 1995/7008-76). And RES acknowledge
support for this work provided by the National Science
Foundation through a Gemini Fellowship from the Association of
Universities for Research in Astronomy, Inc., under NSF cooperative
agreement AST-8947990.
\clearpage
|
1,314,259,993,615 | arxiv | \section{Introduction}
Post-asymptotic giant branch (post-AGB) stars are luminous
(10$^3$-10$^4$~L$_{\sun}$) evolved stars with initial masses in the
range 0.8-8~M$_{\sun}$ \citep*[see][for a general review]{vw03}. At
the end of the AGB phase, mass-loss rates can peak at over $10^{-4}$
M$_{\sun}$~yr$^{-1}$ before dropping dramatically, as the star enters
its post-AGB evolution \citep*[e.g.][]{s83}, creating detached envelopes
of gas and dust. These dusty circumstellar envelopes (CSEs) are then
visible at optical and near-infrared wavelengths as proto-planetary
nebulae (PPN; \citealt*{k93}). A seemingly ubiquitous feature of PPN is
their lack of spherical symmetry, with many having a bipolar or
point-symmetric structure. Notable and well-studied examples are the
Egg Nebula (AFGL 2688; \citealt{sah98}) and the Red Rectangle (AFGL 915;
\citealt{coh04}). Optical and near-infrared surveys of PPN have shown
that in all cases where a CSE is detected then it appears
asymmetric in some way (e.g. \citealt*{ueta00};
\citealt{gchy01}). Possible mechanisms for the shaping of PPN usually
involve interaction of the mass-losing star with a binary companion,
and have been reviewed by \citet*{bal02}.
Imaging polarimetry is a differential imaging technique, which is
well-suited to the study of CSEs surrounding post-AGB stars. The
technique discriminates between the faint but polarized scattered
light from the PPN and any bright unpolarized emission from the
central star. This enables the imaging of circumstellar material that
would normally be lost under the wings of the stellar point spread
function (PSF), thereby obtaining information on the dust distribution
close to the central source. Imaging polarimetric surveys of post-AGB
stars using the UK Infrared Telescope have detected scattered light
from PPN around 34 stars, and all of these PPN were found to be
axisymmetric in some way (\citealt{gchy01}; \citealt*{g05}). Higher spatial
resolution polarimetry using the Hubble Space Telescope ({\it HST}) has
enabled more detailed studies of the morphology of
PPN, as well as providing constraints on dust grain properties in
these systems, and has revealed point-symmetries, jets and multi-lobed
structures \citep*[e.g.][]{umm05,shk03}.
In this paper, we examine \textit{IRAS} 19306+1407 (GLMP 923), which
has \textit{IRAS} colours typical of a post-AGB star with a cold CSE
\citep{olflhhs93}. Radio and millimetre surveys for molecular emission
have failed to detect OH or H$_2$O masers \citep*{l89} or CO emission
\citep{alk86,lfom91}. However, the object shows a number of dust
spectral features. \citet*{hvk00} present \textit{ISO} spectroscopy
showing emission features at 6.3, 7.8 and 10.7 ~$\umu$m, with a
``probable'' feature at 3.3 ~$\umu$m, and compare these features to
the unidentified infrared (UIR) bands at 3.3, 6.2 and 7.7 ~$\umu$m,
commonly attributed to polycyclic aromatic hydrocarbon (PAH) molecules
\citep*{atb89}. Given that the mid-infrared spectral features are
similar to those seen in hot carbon-rich PN, \citet{hvk00} suggest
that the object is a young PN. A further analysis of the {\it ISO}
data by \citet{hkpw04} confirms the presence of UIR features at 3.3,
6.2, and 7.7~$\umu$m, with the addition of the 8.6 and 11.2~$\umu$m
features. These authors also mention the presence of silicate emission
at 11, 19 and 23~$\umu$m, raising the possibility that \textit{IRAS}
19306+1407 may have a mixed CSE chemistry.
Optical spectroscopy shows a broadened H$\alpha$ emission line with
line width of $\sim$2300~km~s$^{-1}$ indicating a fast outflow
\citep*{sc04}, as well as H$\beta$ and [N{\small II}] emission,
leading \citet*{kh05} to suggest a spectral type of approximately B0
for the star. A number of H$_2$ emission lines are seen in the
$K$-band, with line ratios suggesting a mix of radiative and shock
excitation \citep*{kh05}. Imaging through a narrow-band H$_2$ filter,
centred on the 2.122~$\umu$m line, shows that the H$_2$ emission has a
ring-like structure with evidence for bipolar lobes extending
perpendicular to the ring \citep*{vhk04}.
We present the first near-infrared polarimetric images of the dusty
CSE of {\it IRAS} 19306+1407, showing the structure of the envelope in
scattered light. We also present new sub-millimetre photometry and
archived {\it HST} images. The observations are interpreted using
2-dimensional (axisymmetric) light scattering and radiation transport
models.
\section{Observations and Results}
\begin{table}
\caption{Summary of photometry for \textit{IRAS}
19306+1407 for \textit{HST} (using Vega zero points), UKIRT and SCUBA observations, including
integration time (Int.) and the
extent (Size) of the semi-major and minor axes of the aperture used in
photometry.
The PA angle of photometry aperture is equal to 18\degr (E of N).}
\label{table1}
\begin{tabular*}{\columnwidth}{@{}lccc@{\extracolsep{2pt}}c}
\hline
Band & Magnitude & Flux & Int. & Size \\
& & (mJy) & (s) & (arcsec $\times$ arcsec) \\
\hline
{\it F606W}$^{\rmn{a}}$ & $13.81 \pm 0.03$ & $9.5 \pm 0.3 $ & 300 & $3.2 \times 2.0$\\
{\it F814W}$^{\rmn{b}}$ & $12.45 \pm 0.02$ & $26.1 \pm 0.5 $ & 50 & $3.2 \times 1.9$\\
{\it J}$^{\rmn{c}}$ & $11.18 \pm 0.04$ & $51.5 \pm 0.8$ & 237.6 & $3.9 \times 2.4$ \\
{\it K}$^{\rmn{d}}$ & $10.29 \pm 0.12$ & $48.4 \pm 2.2$ & 237.6 & $3.9 \times 2.4$\\
{\it 450W}$^{\rmn{e}}$ & - & $49.9 \pm 38.7$ & 1334$^\dag$ & - \\
{\it 850W}$^{\rmn{f}}$ & - & $14.1 \pm 3.7$ & 1334$^\dag$ & - \\
\hline
\end{tabular*}
{\footnotesize Notes: central wavelengths at $^{\rmn{a}}0.5888\umu$m (Broad {\it V}),
$^{\rmn{b}}0.8115\umu$m (Johnson {\it I}),
$^{\rmn{c}}1.25\umu$m,
$^{\rmn{d}}2.2\umu$m,
$^{\rmn{e}}450\umu$m and
$^{\rmn{f}}850\umu$m;
and $^\dag$inclusive of observational overheads.}
\end{table}
\subsection{Imaging polarimetry observations and results}
\begin{centering}
\begin{figure*}
\vspace{15cm}
\caption{{\bf High resolution images are available at http://star-www.herts.ac.uk/$\sim$klowe/}.
The \textit{J-} and \textit{K-}band observations are
displayed at the top and bottom of the figure respectively. These
images have been scaled logarithmically. The total intensity (I) is
displayed in sub-figures (a) and (c) with overlaid polarization
vectors (pol).
Sub-figures (a) and (c) are scaled between 20 and 13
mag~arcsec$^{-2}$. The lowest outer contour levels are 19 and 18
mag~arcsec$^{-2}$ and separated by 1~mag~arcsec$^{-2}$ for (a) and (c)
respectively. The polarized flux (IP) images (b) and (d) are scaled
between 20 to 16 mag~arcsec$^{-2}$ and 19 to 16 mag~arcsec$^{-2}$
respectively. The lowest outer contours are 19 (b) and 18 (d)
mag~arcsec$^{-2}$ and separated by 0.5 mag~arcsec$^{-2}$.
}
\label{fig1}
\end{figure*}
\end{centering}
Polarimetric imaging at {\it J}- and {\it K}-band of
\textit{IRAS} 19306+1407 was obtained at the 3.8-m United Kingdom
Infrared Telescope (UKIRT) on Mauna Kea, Hawai'i, using the UKIRT 1-5
micron imager spectrometer (UIST) in conjunction with the infrared
half-waveplate module (IRPOL2). A pixel scale of 0.12 arcsec was used
and observations were made on 2003 June 8 with an average seeing of
0.5 arcsec. The total integration time for each filter was 237.6
seconds, comprising 24 exposures of 9.9 seconds each (see
Table~\ref{table1}). Linear polarimetry was obtained by observing at
four half-waveplate angles of 0$\degr$, 22.5$\degr$, 45$\degr$ and
67.5$\degr$. The data reduction was carried out using
\textsc{starlink}\footnote{Available from {\tt www.starlink.ac.uk}}
applications. A bad pixel mask was created using \textsc{oracdr} and
chopped to 512 by 512 pixels. The standard subtraction of dark frames
and flat fielding were carried out by \textsc{ccdpack}. A 3D cube
consisting of the \textit{I, Q} and \textit{U} Stokes images, was
produced using \textsc{polka} from the \textsc{polpack} suite, and
this was then used to derive the per cent polarization, polarized flux
and polarization angle. A more detailed description of dual-beam
polarimetry and the data reduction techniques is given by \citet*{bg99}.
Photometric standards, FS 147 ({\it J}) and FS 141 ({\it K}), were
used to flux calibrate the data giving $J$=$11.18 \pm 0.04$ and
$K$=$10.29 \pm 0.12$.
For these observations, the focal plane polarimetry mask was removed,
so that a 512 by 512 pixel sub-array of the UIST detector could be
used. This enabled faster read-out times and exposures of less than 1
second, so that observations of bright sources could be made without
the risk of saturation. This configuration of UIST resulted in the
overlapping of the \textit{o}- and \textit{e}-beams produced by the
Wollaston prism and a final analysis area of 20 by 60~arcsec. The
Wollaston prism splits each star into an \textit{e}- and
\textit{o}-component separated by 20 arcsec, so that any star in the
field lying more than 10 arcsec along the prism dispersion axis from
the target will only have one component in the analysis area. Since
both \textit{e}- and \textit{o}-beams are required to correctly
calculate the Stokes intensities \textit{I}, \textit{Q} and
\textit{U}, these offset stars appear as highly polarized artefacts in
the reduced data, and they are marked as such on Fig. \ref{fig1}.
As the prism dispersion varies slightly with wavelength, this
results in an apparent shift of the artefact stars between the
\textit{J-} and \textit{K}-filters.
\label{jkpolres}
The \textit{J}- and \textit{K}-band polarimetric results are shown in
Fig.~\ref{fig1}. The total intensity images are shown in
Fig.~\ref{fig1} (a) and (c), superimposed with polarization
vectors, and show the centrally peaked nature of the source. The
object is clearly extended, relative to the 0.5~arcsec seeing FWHM,
with faint emission detected out to a radius of approximately
3~arcsec. The lowest contour in both filters is 3 times the sky noise
and in the $I_{\rm J}$ image, shows that the faint emission is
elongated in a NNE/SSW direction. Details of contour levels are given
in the Figure caption. It is possible that a similar extension is
present in the $I_{\rm K}$ image, but confusion due to the presence
of the artefact stars makes this uncertain.
The polarized flux, produced by light scattering from dust grains, is
shown in Fig. \ref{fig1} (b) and (d). In both filters, the central
region appears elongated along a PA 136$\degr$ East of North, with two
bright shoulders of emission either side of the star. At {\it J}
(IP$_{\rm J}$ image) this structure is embedded within fainter more
extended emission orientated at 18$\degr$ East of North, seen in the
lowest three contours (the lowest contour is at 1.5 times the sky
noise). This faint extension is not as apparent in the $K$-band
polarized flux image (IP$_{\rm K}$), which is approximately
1~mag~arcsec$^{-2}$ shallower than the $J$-band data.
The NW shoulder
is brighter than the SE shoulder, particularly apparent in the
IP$_{\rm J}$ image. Similar morphology has been observed in polarised
flux in a number of other PPN. \citet{gchy01} found bright arc-like
structures on either side of the star in {\it IRAS} 17436+5003 as well as
shoulder-like features in {\it IRAS} 19500-1709 and more ring-like
features in {\it IRAS} 22223+4327 and 22272+5435.
They
interpreted these structures in terms of scattering from the inner
surfaces of a detached axisymmetric shell, with an equatorial density enhancement,
and classified these objects
as ``shell-type'' objects.
The arcs in {\it IRAS} 17436+5003 were
later fully resolved in mid-infrared imaging of thermal emission from
the dust \citep*{gy03} and successfully modelled using an axisymmetric
dust distribution based on that of \citet*{kw85}.
Further evidence for
arcs and shoulders is seen in polarized flux images of {\it IRAS}
06530-0213, 07430+1115 and 19374+2359 \citep*{g05} and was interpreted
using light-scattering in a Kahn \& West density distribution.
We therefore interpret the polarized flux shoulders seen around
{\it IRAS} 19306+1407 in the same way, and suggest that they result
from increased scattering at the inner boundary of a detached shell
with an equatorial dust density enhancement.
The polarization vectors shown in Fig. 1 (a) and (c) are binned over
0.36 $\times$ 0.36 arcsec ($3\times 3$ pixels) and have a
signal-to-noise threshold of 2 in per cent polarization. The vector
pattern appears approximately centro-symmetric in both filters,
indicating isotropic illumination by a central source. The maximum
per cent polarization is $15 \pm 6$ and $10 \pm 4$ at \textit{J}- and
\textit{K}-bands respectively (Table~\ref{table2}). These values are
lower limits to the intrinsic polarization, since in these
observations it has not been possible to correct for dilution of the
polarized flux by the unpolarized light from the central star.
\begin{table}
\caption{Summary of polarimetric results of \textit{IRAS}
19306+1407 for each band, detailing the maximum polarization,
integrated polarization and the position angle (E of N) of
the major and minor axis of the nebula in polarized flux. }
\label{table2}
\begin{tabular*}{\columnwidth}{ccccc}
\hline
Band & Max. Pol. & Integrated Pol.$^{\dagger}$ & PA$_{\rmn{major}}$ & PA$_{\rmn{minor}}$ \\
& (per cent) & (per cent) & ($\degr$) & ($\degr$) \\
\hline
{\it J} & $15 \pm 6$ & $1.7\pm0.1$ & 18 & 136 \\
{\it K} & $10 \pm 4$ & $1.3\pm0.1$ & 18 & 136 \\
\hline
\end{tabular*}
$^{\dagger}$ - The integrated polarization over the source with
apertures of radii of 1.7- and 1.4-arcsec for {\it J-} and {\it K-}band respectively.
\end{table}
\subsection{\textit{Hubble Space Telescope} observations and results}
\label{hst}
We have obtained archived {\it HST} images for {\it IRAS}
19306+1407\footnote{Based on observations made with the NASA/ESA
Hubble Space Telescope, obtained from the data archive at the Space
Telescope Institute. STScI is operated by the association of
Universities for Research in Astronomy, Inc. under the NASA contract
NAS 5-26555.} observed on 2003 September 8 (proposal ID: 9463). The
observations were obtained with the Advanced Camera for Surveys (ACS),
in conjunction with the High Resolution Channel (HRC), using {\it
F814W-} and {\it F606W}-filters with pivotal wavelengths of 5888 and
8115~\AA~respectively. The images were reduced using the On-the-Fly
Reprocessing of {\it HST} Data ({\sc otfr}), which produces a
cosmic-ray cleaned, calibrated, geometrically corrected mosaic image.
Aperture photometry was performed using \textsc{gaia}, using the Vega
zero points\footnote{{\tt
http://www.stsci.edu/hst/acs/analysis/zeropoints}}, and obtained
magnitudes of $13.81 \pm 0.03$ and $12.45 \pm 0.02$ for {\it F606W} and {\it F814W}
respectively (Table \ref{table1}).
The reduced {\it F606W} and {\it F814W} images are shown in Fig. 2 (a)
and (b). Fig. 2 (c) shows the {\it F606W} image superimposed with
contours of {\it J}-band polarized flux from Fig. 1 (b). The object is
clearly bipolar in the {\it F606W} image, and the curved edges of
bipolar cavities, extending for 3 to 4 arcsec from the source, can be
seen. The orientation of the bipolar axis, at PA 18 deg, is aligned
with the {\it J}-band elongation in total and polarized intensity seen
in Fig. 1 (a) and (b). The bipolar structure appears to be surrounded
by a faint, more spherically symmetric halo, seen in both {\em HST}
filters, and this corresponds in extent to the outer contours in
Fig. 1 (a) and (c). The polarized flux shoulders, at PA 136 deg, are
not perpendicular to the major axis of the nebula and this is clearly
seen in Fig. 2 (c). This non-orthogonality in the two axes will be
discussed further in Section 4.
The southern bipolar lobe appears to be the brighter of the two in both
{\em HST} filters, which could indicate that the major axis is slightly
inclined to the plane of the sky.
\begin{figure}
\begin{center}
\vspace{7.5cm}
\caption{
{\bf High resolution images are available at http://star-www.herts.ac.uk/$\sim$klowe/}.
The {\it HST} ACS images, scaled logarithmically, of {\it IRAS} 19306+1407.
(a) {\it F606W} (5888{\AA}) scaled between 22 and 13 mag~arcsec$^{-2}$ with
the angle of the major axis indicated by the arrow.
(b) {\it F814W} (8115{\AA}) scaled between 22 and 11 mag~arcsec$^{-2}$.
(c) {\it F606W} image, scaled as above, and {\it J-}band polarized flux contours.
The lowest contour level is 19 mag~arcsec$^{-2}$ and subsequent contours are
separated by 1 mag~arcsec$^{-2}$.}
\label{fig2}
\end{center}
\end{figure}
\subsection{Sub-millimetre observations and results}
Observations were made on 2005 January 8 using the Sub-millimetre
Common User Bolometer Array (SCUBA) at the 15~m James Clerk Maxwell
Telescope (JCMT) on Mauna Kea, Hawai'i. The SCUBA observations were
made simultaneously at 450 and 850~$\umu$m in photometry mode using
a jiggle pattern.
The 450 and 850~$\umu$m photometry data were reduced
using the \textsc{surf} package within the \textsc{starlink} suite.
The sky opacity was corrected using the Caltech Sub-millimetre
Observatory (CSO) tau relationship\footnote{Using the revised 2000
October 25 relations}. Flux calibration was performed using Mars,
inclusive of a maximum $\pm$5 per cent error due to the orientation of
Mars' poles relative to the Earth and Sun. \textit{IRAS} 19306+1407
was detected at 450- and 850-$\umu$m at $>1\sigma$ and $>3\sigma$
respectively inclusive of calibration errors. The fluxes obtained
(Table \ref{table1}) for $F_{450}$ and $F_{850}$ are $49.9 \pm 38.7$
mJy and $14.1 \pm 3.7$ mJy within a beam size of 7.5 and 14~arcsec
respectively.
\section{Modelling the CSE}
\label{mod:cse}
\subsection{Model details}
To investigate the dusty CSE around {\it IRAS} 19306+1407, we use
modified versions of the \citet{m89} axisymmetric light scattering
(ALS) code to produce Stokes {\it I, Q, U} images and the axisymmetric
radiative transfer (DART) code \citep{er90} to model the SED. Both
codes have previously been used to model the CSEs of post-AGB stars.
\citet*{gy03} used DART to simulate multi-wavelength mid-infrared
imaging observations of {\it IRAS} 17436+5003, in which an
axisymmetric shell was resolved. To simulate the axisymmetry, these
authors used a simple dust density formulation from \citet*{kw85}
which was found to successfully reproduce all of the axisymmetric
features, including the offset location of the brightness peaks seen
in the data, which was found to be due to the inclination of the
system to the plane of the sky. \citet*{g05} have used the ALS code to
produce generic light scattering models of PPN at varying optical
depth and also find that a Kahn \& West density model provides a good
representation of the observations with a minimum number of model
parameters. It is important that the dust density model uses a minimum
number of parameters whilst achieving an adaptable axisymmetric geometry, so
that there is a better chance of each parameter being observationally
well constrained.
More complex dust density formulae have
been used \citep[e.g.][]{mubs02}, which incorporate the presence of AGB
and superwind mass loss histories, but require more
parameters (twice as many in the case of \citealt{mubs02}).
These models result in morphologies that are
qualitatively similar to our simpler models, but are
unlikely to be well constrained by our observations.
In both the ALS
and DART models we therefore use a simpler density profile from \citet{kw85}
to model an axisymmetric shell, whilst recognising its limited ability to
reproduce more complex morphologies:
\begin{equation}
\label{density1}
\rho(r,\theta) = \rho_0\left(\frac{r}{r_{\rmn{in}}}\right)^{-\beta}
\left(1 + \epsilon
\sin^{\gamma}\theta\right),
\end{equation}
where $\rho_0$ is the density at the pole ($\theta = 0\degr$) at the
inner radius, $r_{\rmn{in}}$, and $\beta$ specifies the radial density
distribution. The azimuthal density distribution is determined by
parameters $\epsilon$ and $\gamma$, which specify the equator-to-pole
density ratio ($1+\epsilon$) and the degree of equatorial enhancement,
respectively. An increase to $\gamma$ flattens the density
distribution, creating a more toroidal structure.
All parameters in Equation 1 are optimized in the model, apart from
$\beta$, which is fixed at a value of 2 due to a limitation of the
DART code, corresponding to constant mass-loss rate and expansion velocity
for the AGB wind. The ALS density profile includes an extra
parameter, that restricts the axisymmetry to within a radius,
r$_{\rmn{{\sc sw}}}$, modifying Equation \ref{density1} to:
\begin{equation}
\rho(r,\theta)=\rho_{0}\left(\frac{r}{r_{\rmn{in}}}\right)^{-\beta}
\quad \rmn{when}\quad r>r_{\rmn{{\sc sw}}}.
\end{equation}
A power law size distribution is used with spherical grains of radius
$a$, between a minimum and maximum grain size of $a_{\rmn{min}}$
and $a_{\rmn{max}}$ respectively, and a power-law index, $q$,:
\begin{equation}
n(a) \propto a^{-q}\quad \rmn{for} \quad a_{\rmn{min}} \leq a \leq a_{\rmn{max}}.
\end{equation}
The inclination of the symmetry axis to the plane of the sky is not
known. As mentioned in Section~\ref{hst}, the southern bipolar lobe
appears slightly brighter than the northern one in {\em HST} imaging
(Fig~2), which could indicate a small inclination to the plane of the
sky. Although the near-infrared images appear consistent with zero
inclination (e.g. they are similar to edge-on axisymmetric shell
models shown in \citealt{g05}), we consider the inclination angle to
be a free parameter and allow it to vary in steps of 10 deg.
The overall chemistry of the system is uncertain. The results from
\citet{hvk00} suggest a C-rich nature based on emission features
consistent with C-rich PNe. \citet{hkpw04} re-evaluated the
mid-infrared spectra and classified {\it IRAS} 19306+1407 as ``UIR
features coupled with emission from crystalline silicates'' suggesting
a dual chemistry nature. The dust species that have been considered in
our models are amorphous carbon (amC), silicon carbide (SiC) and
Ossenkopf cold silicates, and we have obtained the optical constants
from \citet{poyh93}, \citet{p88} and \citet{ohm92} respectively.
We ran a total of over 150 ALS and over 300 DART models to create a
model grid for the free physical parameters (Table~\ref{table3}). The
minimum and maximum grain sizes were investigated from 0.005 to
1~\micron, with a variable grain size spacing typically 0.005 to
0.02~\micron. The grain size power law index was varied between 3.0
to 6.0 at increments of 0.5. The radial density fall off exponent is
fixed at $\beta=2$ and cannot not be varied. The bin widths for the
CSE parameters, common to both models are 1, 2, 10$\degr$ and
0.1$\times$10$^{-2}$ for the equator-to-pole contrast ($\epsilon$),
equatorial density enhancement ($\gamma$), inclination angle
($\theta$) and the ratio of the inner-to-outer radii ($r_{\rm
in}/r_{\rm out}$) respectively. The stellar temperature, $T_*$, was
investigated using a series of Kurucz models \footnote{{ \tt
http://kurucz.harvard.edu/grids.html}} with solar metalicities and
temperatures separated by 1000~K.
The ALS code is used to determine the best-fitting envelope parameters
based on the morphology, azimuthal profiles in polarized flux and
radial profiles of the percentage polarization and total intensities.
The ALS code is additionally used to
constrain the dust grain size by generating polarization information.
The ALS estimate of the grain size is an important input to the DART
calculations, which would otherwise suffer from a degeneracy between
grain size and outer CSE radius, both of which strongly influence the
long-wavelength tail of the SED. The optical depths at 0.55, 1.2 and
2.2~$\umu$m are also derived from the ALS model and subsequently
inserted into the DART model. The DART model fits to the SED are used
to constrain the temperature of the central star, inner-to-outer and
stellar-to-inner radii ratios. The two codes were used to iteratively
produce a convergent model.
\begin{centering}
\begin{figure*}
\vspace{16cm}
\caption{{\bf High resolution images are available at http://star-www.herts.ac.uk/$\sim$klowe/}.
The 1.2- and 2.2-$\umu$m smoothed model images of
\textit{IRAS} 19306+1407 are displayed at the top and bottom of the
figure respectively. These images are rotated to a PA of 136\degr to
mimic the observed data. As with the observed images they have been
scaled logarithmically. The total intensity (I) is displayed in
sub-figures (a) and (c) with overlaid polarization vectors (pol) and
polarized flux is shown in (b) and (d). The model images have been
normalised at the same levels as the observed images: (a) and (c) are
scaled between 20 and 13 mag~arcsec$^{-2}$ with lowest outer contour
levels at 19 and 18 mag~arcsec$^{-2}$, respectively, separated by
1~mag~arcsec$^{-2}$; (b) and (d) are scaled between 20 to 16
mag~arcsec$^{-2}$ and 19 to 16 mag~arcsec$^{-2}$ respectively with
lowest outer contours at 19 (b) and 18 (d) mag~arcsec$^{-2}$,
separated by 0.5 mag~arcsec$^{-2}$.
}
\label{fig3}
\end{figure*}
\end{centering}
\begin{figure*}
\vspace{15cm}
\caption{{\bf High resolution images are available at http://star-www.herts.ac.uk/$\sim$klowe/}.
{\bf Left:} Azimuthally averaged radial profiles of the
normalised total intensity. {\bf Centre:} Azimuthally averaged radial
profiles of the per cent polarization. {\bf Right:} Radially averaged
azimuthal profiles of the normalised polarized intensity. In all
cases, the {\it J-} and {\it K-}band data are displayed as squares and
triangles respectively, with 3$\sigma$ error bars, and the 1.2- and
2.2-$\umu$m smoothed model data are displayed as solid and dashed
curves respectively. }
\label{fig4}
\end{figure*}
\begin{table}
\caption{The CSE and dust grain parameters
for the best-fitting ALS and DART models for \textit{IRAS} 19306+1407.}
\label{table3}
\label{table4}
\label{table7}
\begin{tabular*}{\columnwidth}{@{}lcl}
\hline
Parameter & Value & Description \\
\hline
\multicolumn{3}{@{}l}{\textbf{Dust grain parameters}} \\
\multicolumn{3}{@{}l}{Ossenkopf} \\
Cold Silicates$^1$ & 1.0 $\pm$ 0.01 & Number fraction \\
$a_{\rmn{min}}$ ($\umu$m) & 0.10 $\pm$ 0.01 & Minimum grain radius \\
$a_{\rmn{max}}$ ($\umu$m) & 0.40 $\pm$ 0.01 & Maximum grain radius \\
$q$ & 3.5 $\pm$ 0.5 & Grain size power law index \\
\noalign{\smallskip}
\multicolumn{3}{@{}l}{\textbf{Common envelope model parameters}} \\
$\beta^{\dagger}$ & 2 & Radial density fall off \\
$\epsilon$ & 6 $\pm$ 1 & Equator-to-pole density contrast \\
$\gamma$ & 5 $\pm$ 2 & Equatorial density enhancement \\
$\theta$ (deg) & 0 $\pm$ 10 & Inclination angle (from equator) \\
$r_{\rmn{in}}/r_{\rmn{out}}$ (10$^{-2}$) & 7 $\pm$ 1 & Inner-to-outer radius ratio\\
\noalign{\smallskip}
\multicolumn{3}{@{}l}{\textbf{ALS model parameters}} \\
$\tau_{1.2}^{\ddagger}$ ($\times$10$^{-1}$) & 6.78 $\pm$ 0.05 & Optical depth at 1.2 $\umu$m \\
$\tau_{2.2}^{\ddagger}$ ($\times$10$^{-1}$) & 1.13 $\pm$ 0.01 & Optical depth at 2.2 $\umu$m \\
$r_{\rmn{{\sc sw}}}/r_{\rmn{in}}$ & 2.0 $\pm$ 0.5 & Super-wind \\
& & to inner radius ratio\\
\noalign{\smallskip}
\multicolumn{3}{@{}l}{\textbf{DART model parameters}} \\
$T_{\star}$ (10$^3$~K) & 21 $\pm$ 1 & Effective Stellar Temperature \\
$r_{\star}/r_{\rmn{in}}$ (10$^{-5}$) & 1.4 $\pm$ 0.2 & Stellar-to-inner radius ratio \\
$A^{\rmn{CSE}}_{V_{\mbox{ }}}$ (mag) & 2.0 $\pm$ 0.1 & Equatorial optical extinction \\
\hline
\end{tabular*}
$^1$\citet{ohm92}. $^{\dagger}$This variable is fixed in our model code and cannot be varied.
$^{\ddagger}$The optical depth is an output of the ALS model.
\end{table}
\subsection{Model results}
\subsubsection{ALS model}
Before the raw model images can be compared with the polarimetric
observations, they must be smoothed to mimic the effect of the
atmosphere and telescope. We find that a simple Gaussian filter is
unable to reproduce the wings of the PSF effectively, which is
essential since the PSF wings have a critical effect on the percentage
polarization in the envelope where the intensity is low, at $r \ga
r_{\rmn{in}}$. To obtain a more realistic fit we use a Moffat filter
profile:
\begin{equation}
M(r) \propto \left[1 + \left(\frac{r}{\alpha_{\rmn{mof}}}\right)^2\right]
^{-\beta_{\rmn{mof}}},
\end{equation}
where $r$ is radius from the source and $\alpha_{\rmn{mof}}$ and
$\beta_{\rmn{mof}}$ are fitting parameters \citep*{mof69}. The Moffat
parameters were calculated by fitting to the PSF of a bright field
star (Table~\ref{table5}) and their uncertainties were estimated by
examining the fit to the remaining field stars. The filter was then
applied to the raw ({\it I, Q } and {\it U}) model images, which were
then combined to obtain polarized flux and per cent polarization
values.
The resulting best-fitting smoothed model is shown in Fig.~\ref{fig3}
and the parameters used are displayed in Table~\ref{table4}. The model
reproduces the centrosymmetric polarization pattern and the observed
degrees of polarization in the {\it J}- and {\it K}-bands. The
polarized flux images show the shoulders seen in the observations, due
to the enhanced scattering at the inner edges of the axisymmetric
shell, where the dust density is greatest. In Fig.~1, the observed
polarized flux images show a peak of emission at the location of the
star. Any mis-alignment of the bright, centrally-peaked images during
the data reduction stages will lead to a residual polarization at this
location. Since the polarized flux peak is narrower than the seeing
disc size, we cannot treat it as significant. We do not see polarized
emission from the location of the star in the model images, since
forward-scattered light (i.e. scattering angles close to zero) is
strongly depolarized. Higher spatial resolution observations will be
required to investigate the polarization within 0.2 arcsec of the
star. If there is significant polarized emission from this region,
then an additional dust component would be required in the model.
The fit
was assessed by comparing the full grid of ALS models to the
polarimetric observations. In particular, the radial and azimuthal
profiles of the smoothed model images and the observations were
compared and the profiles for the best-fit model are shown in
Fig.~\ref{fig4}. The total intensity image radial profile fit
(Fig.~\ref{fig4} left) provides a check on the level of smoothing, and
shows an excellent fit to the observed intensity profile at both
wavelengths. The fit to the radial distribution of per cent
polarization (Fig.~\ref{fig4} centre) allows us to constrain the dust
grain parameters and optical depth. The maximum degree of polarization
produced by the model is very sensitive to the grain size distribution
so we consider that the grain size is well constrained. The radial
distribution of per cent polarization depends strongly on the optical
depth (and hence the dust density), since this determines the surface
brightness of the CSE relative to the unpolarized light from the
smoothed PSF. We determine an optical depth of 0.68 and 0.11 at $J$
and $K$ respectively, so that the CSE is optically thin in the
near-infrared. The axisymmetry parameters, $\epsilon$ and $\gamma$ are
determined by comparing azimuthal polarized flux profiles to the data
(Fig.~\ref{fig4} right). The best fit gives an equator-to-pole density
contrast of 7.
\begin{table}
\caption{The Moffat filter profile parameters, $\alpha_{\rmn{mof}}$ and
$\beta_{\rmn{mof}}$, for a bright field star at {\it J } \& {\it K}.}
\label{table5}
\begin{center}
\begin{tabular}{ccc}
\hline
Band & $\alpha_{\rmn{mof}}$ & $\beta_{\rmn{mof}}$ \\
\hline
{\it J} & $3.95 \pm 0.06$ & $2.4 \pm 0.2 $ \\
{\it K} & $3.03 \pm 0.02$ & $2.2 \pm 0.3 $ \\
\hline
\end{tabular}
\end{center}
\end{table}
\subsubsection{DART model}
\label{mod:DARTmodel}
\begin{table}
\caption{Photometric values for {\it IRAS}
19306+1407 collated from the literature:
(1) \citet{hvk00}; (2) \citet{metal03};
(3) {\it MSX} Bands \citep{epkmcwecg03}, and
(4) \citet{iras88}.}
\label{table6}
\begin{center}
\begin{tabular*}{\columnwidth}{cccc}
\hline
Band & Central & Flux density & Reference \\
& wavelength & & \\
& ($\umu$m) & (Jy) & \\
\hline
{\it V} & 0.55 & $ 7.40\times10^{-3} $ & (1) \\
{\it R} & 0.44 & $ 2.21\times10^{-2}$ & (2) \\
{\it MSX A} & 8.28 & 1.16 & (3) \\
{\it IRAS} 12$\umu$m & 12.0 & 3.58 & (4)\\
{\it MSX C} & 12.13 & 3.65 & (3) \\
{\it MSX D} & 14.65 & 9.12 & (3) \\
{\it MSX E} & 21.34 & 46.27 & (3) \\
{\it IRAS} 25$\umu$m & 25.0 & 58.65 & (4) \\
{\it IRAS} 60$\umu$m & 60.0 & 31.83 & (4) \\
{\it IRAS} 100$\umu$m & 100.0 & 10.03 & (4) \\
\hline
\end{tabular*}
\end{center}
\end{table}
\begin{figure}
\vspace{8cm}
\caption{
{\bf High resolution images are available at http://star-www.herts.ac.uk/$\sim$klowe/}.
The observed SED and best model fits for
\textit{IRAS} 19306+1407.
The dash line is the model fit and the
solid black line is the model fit with
interstellar reddening applied.
References: (1) this paper, (2) \citet{hvk00},
(3) \citet{ummpd03}, (4) \citet{metal03},
(5) \citet{epkmcwecg03} and (6) \citet{iras88}.}
\label{fig5}
\end{figure}
The SED of \textit{IRAS} 19306+1407 is plotted in Fig.~\ref{fig5}
using published photometry and spectroscopy from a variety of sources,
including this paper, and covering wavelengths from the $V$-band
through to the sub-millimetre. The photometric values are listed in
Table \ref{table6}. The double-peaked nature of the SED is immediately
evident, consisting of a reddened stellar peak around $1.6$~$\umu$m
and a broad thermal dust peak between 30 and 40~$\umu$m due to the
CSE. Double-peaked SEDs are typical of post-AGB stars with optically
thin detached CSEs \citep*{vdvhg89}.
Our best-fitting model is shown in Fig.~5, both with and without
correction for interstellar extinction (see below). Previous
attempts to model the SED using amorphous carbon dust and a cooler F/G
type star, were found not to provide sufficient flux in the dust peak
\citep{hvk00}. We have treated the stellar temperature as a free
parameter and determined a best-fitting value of 21,000 K, typical of
a B1 type star. This is consistent within errors with the
observationally determined spectral type of B0: \citep{vhk04,kh05},
where the colon denotes an uncertainty in the 0
(Hrivnak, private communication).
An optical extinction of $A_{\rmn{V}} = 2.0$~$\pm$~0.1~mag, through
the CSE in the equatorial direction, was determined from the model
fit. We have investigated the effect of inclination of the
nebula axis and have determined that the SED
is consistent with a value of $0\degr \pm 10\degr$. The extinction through
the CSE along our line of sight is, therefore, also $A_{\rmn{V}} = 2.0$.
{\it IRAS} 19306+1407 lies close to the galactic plane, $l=
50.30\degr$ and $b= -2.48\degr$, and the SED will be affected by
interstellar extinction. The extinction through the Galaxy at this
point is estimated to be $A_V =$~5.1~$\pm$~0.2~mag. This value was
obtained from the {\it IRAS} dust reddening and extinction
service\footnote{{\tt
http://irsa.ipac.caltech.edu/applications/DUST}}, based on the data
and technique in \citet*{sfd98}.
To correct the emergent model flux for interstellar extinction, we
apply a reddening model developed by \citet*{ccm89} which gives the
extinction, $A_{\lambda}$, at every wavelength between 0.1 and
3.3~$\umu$m for a given $A_{\rmn{V}}$ and extinction ratio,
$R_{\rmn{V}}$. The extinction at shorter and longer wavelengths has
been extrapolated. The DART model flux, $F_{\rmn{DART}}$, is then
modified to give the flux after correction for interstellar
extinction, $F_{\lambda}$:
\begin{equation}
\label{sedism}
F_{\lambda} = F_{\rmn{DART}} \times 10^{-\frac{A_{\lambda}}{2.5}},
\end{equation}
Assuming a standard value of $R_V$=3.1 for the ISM, then a fit to the
SED shortward of 6~$\umu$m gives a value of 4.2~$\pm$~0.1~mag for
interstellar extinction (solid curve in Fig.~\ref{fig5}). The total
extinction to the star is, therefore, 6.2~$\pm$~0.2~mag. This is
consistent with the observed $J$-$K$ colours. Assuming an extinction
ratio ($R_V$) of 3.1, and an intrinsic colour excess of $E(J-K)_0 =
-0.09$ for a \textsc{B1I} star, gives $A_{\rm V}=6.4 \pm 0.7$~mag.
The model parameters used in DART are presented in Table~\ref{table7}.
\subsubsection{Distance estimate and derived parameters}
The interstellar extinction can be used to estimate the distance of
the post-AGB star. Using \citet{j05}, based on extinction towards
open clusters, gives an estimated extinction of $1.58 \pm 0.04$~mag
kpc$^{-1}$. A visual extinction of 4.2~$\pm$~0.1~mag suggests a
distance of $2.7 \pm 0.1$~kpc, which we now adopt as our assumed
distance from this point onwards.
Using this distance estimate gives values for $r_{\rmn{in}}$ and
$r_{\rmn{out}}$ of $1.9 \pm 0.1 \times$10$^{14}$ and $2.7 \pm 0.1
\times$10$^{15}$~m respectively. Multiplying $r_{\rmn{in}}$ by
$r_{\star}/r_{\rmn{in}}$ gives a stellar radius, $R_{\star}$, of $3.8
\pm 0.6$~R$_{\sun}$.
The stellar luminosity, $L_{\star}$, is obtained by calculating the
integrated flux under the model SED, giving values of $1800 \pm 140$
and $4500 \pm 340 $~L$_{\sun}$, with and without interstellar
reddening applied respectively, for the assumed distance. Post-AGB
stellar evolution models suggest a lower limit of 2500~L$_{\sun}$ for
the central star of a PN \citep{s83}, which means that {\it IRAS}
19306+1407 must be at least 2.0~kpc away to satisfy this criterion.
To calculate the time scales of mass loss, $r_{\rmn{in}}$ and
$r_{\rmn{out}}$ are divided by the AGB wind speed. Only the H$_2$ and
H$\alpha$ kinematic information are available for {\it IRAS}
19306+1407. These speeds arise from the shocks and fast winds in the
post-AGB phase, and are not a true reflection of the AGB envelope
expansion speed, therefore we have assumed a typical speed of
15~km~s$^{-1}$ from \citet{nklbl98}. The age of the CSE is then $5700
\pm 160 $~yrs, became detached $400 \pm 10$~yrs and the mass loss
lasted $5300 \pm 160 $~yrs.
The number density of dust grains, $N_0$, at $r_{\rmn{in}}$ is
calculated from the optical depth, the extinction cross section of the
dust and the CSE thickness. The optical depth at 1.2~$\umu$m is $0.678
\pm 0.005$, giving a value of $N_0$ = $6.1 \pm 3.0
\times$10$^{-3}$~m$^{-3}$. Using $N_0$ and integrating the dust
density distribution gives the total dust mass ($M_{\rmn{d}}$), and
assuming a dust grain bulk density of 3$\times$10$^3$~kg~m$^{-3}$,
gives a value of $8.9 \pm 5.0 \times$10$^{-4}$~M$_{\sun}$.
The gas-to-dust ratio for this object is unknown and we have adopted a
value of 200 from \citet{hh05}. The total mass of the CSE is then
$1.8 \pm 1.0 \times$10$^{-1}$~M$_{\sun}$ with an average mass-loss
rate ($\dot{M}$) of $3.4 \pm 2.1
\times$10$^{-5}$~M$_{\sun}$~yr$^{-1}$.
The derived parameters given in this section are summarized in
Table~\ref{table8}.
\section{Discussion}
\subsection{CSE geometry}
The polarimetric observations, shown in Fig.~1, have been interpreted
in terms of an axisymmetric shell with an equatorial density enhancement,
which is optically thin in the
near-infrared. The shell model successfully reproduces the observed
SED from the V band to the sub-millimetre. As a further check on the
validity of the model, the ALS code was run at the central wavelength
of the {\it F606W} filter to simulate the {\it HST} observations shown
in Fig.~2 (a). The results are shown in Fig.~6 and we find that the
bipolar structure is reproduced, inclusive of the flattened contours
in the centre of the {\it HST} image. A single axisymmetric shell
model, based on the simple \citet*{kw85} density distribution, can
account for the morphology of this object over a wide range of
wavelengths. The transition from bipolar nebula in the optical to
limb-brightened shell in the near-IR is due to the variation in
optical depth through the envelope with wavelength. At the wavelength
of the {\em HST} observations, the CSE is optically thick along the
equatorial direction and so light is preferentially funnelled along
the polar axes before scattering into our line of sight, creating the
bipolar lobes. The fact that the general appearance and extent of the
lobes is reproduced by the model indicates that the density structure
of the shell, in particular the equator-to-pole density contrast of 7,
is reasonable. At near-infrared wavelengths, where the shell is
optically thin along the equator, light is mainly scattered at the
inner boundary in the equatorial plane, where the dust density is
greatest, creating the shoulders seen in polarized flux in our
observations.
Since our model calculations are limited to axisymmetric geometries,
one aspect of the observations that we have not been able to account
for is the non-orthogonality of the polarized flux shoulders, at PA
136~deg, and the major axis of the nebula, at PA 18~deg, illustrated
in Fig.~2. A similar `twist' has been detected in the mid-infrared
images of {\it IRAS} 17456+5003, which has a curving polar axis
\citep*{gy03}, and which was also modelled with an axisymmetric dust
shell. A further similarity between the two objects is the unequal
brightness of the polarized flux shoulders \citep[see][]{gchy01}. In
the context of our model, these are due to scattering at the inner
edge of the axisymmetric shell, so that the scattering optical depth
is greater on one side of the shell than the other. Assuming that the
dust properties are the same throughout the shell, then this suggests
that there is a greater concentration of dust in the brighter
shoulder. Further evidence for asymmetric dust distributions around
post-AGB stars is seen in mid-infrared images of {\it IRAS} 07134+1005
\citep{d98} and {\it IRAS} 21282+5050 \citep{m93}. \citet*{gy03}
discuss possible causes for these asymmetries and conclude that they
may arise due to interaction of the mass-losing star with a binary
companion, although exactly how this happens is not clear.
\citet{vhk04} imaged {\it IRAS} 19306+1407 using a narrowband H$_2$
filter (2.12~$\umu$m) and a narrowband {\it K} continuum filter
(2.26~$\umu$m), to investigate the molecular hydrogen emission. Their
continuum subtracted H$_2$ image (their Fig.~2 left) shows a broken
ring with limb-brightened edges, which appears cospatial with the
central dust structure, at PA 136, seen in our polarized flux images.
The ring can also be seen in their 2.26~$\umu$m continuum image, so
that they have resolved the dust structure that we see in polarized
flux. The similarity between the polarized flux and H$_2$ images
suggests that the scattered light and molecular emission originate in
the same region. \citet{vhk04} also detect faint extended H$_2$
emission lobes, extending from the ring, corresponding to the extended
bipolar structure seen in the HST images (Fig.~\ref{fig2}), oriented
PA 18$\degr$. It appears that the same axis twist seen in the
scattered light images may be present in H$_2$ emission. \citet{vhk04}
suggest that the H$_2$ ring seen in their images collimates the
H$_2$-emitting bipolar lobes.
\subsection{Estimation of the dust mass from our sub-mm observations}
\label{estdustsub}
The mass of dust in the CSE, $M_{\rmn{d}}$, can be estimated from the
{\it IRAS} 100~$\umu$m flux, $F_{100}$, and the SCUBA 850~$\umu$m
flux. We have used the method stated in \citet*{gby02} to calculate
an estimate of the dust mass from our observations. The dust
temperature is estimated to be 146~$\pm$~21~K, using Wien's
displacement law, with the peak dust emission at $35 \pm 5$~$\umu$m.
The 850~$\umu$m flux value given in Table~\ref{table1} and $F_{100}=
10.03 \pm 1.30$~Jy, gives an emissivity index of 1.3~$\pm$~0.1. The
assumed density for a silicate dust grain is
3$\times$10$^3$~kg~$\rmn{m}^{-3}$. The total dust mass in the CSE,
using the assumed distance, is then $4.3 \pm 0.7
\times$10$^{-4}$~$\rmn{M_{\sun}}$, which is a factor of $\sim 2$ less
than the value obtained from our radiative transfer model. The
difference may arise from the simple assumptions inherent in the
sub-millimetre estimate, particularly that of an isothermal CSE. The
bulk of dust in the envelope will be cooler than 146~K (the maximum
and minimum dust temperatures in the DART model are 130 and 40~K
respectively), and will radiate on the long wavelength tail of the
SED. An isothermal temperature of 100~K would result in a dust mass of
$7.2 \pm 1.7 \times$10$^{-4}$~$\rmn{M_{\sun}}$. Given these
approximations, we consider that the two results are comparable but
that the more rigorous model calculations from DART and ALS provide a
realistic value for the dust mass in the CSE.
\subsection{CSE chemistry}
\label{dis:geochem}
\begin{centering}
\begin{figure}
\vspace{8cm}
\caption{{\bf High resolution images are available at http://star-www.herts.ac.uk/$\sim$klowe/}.
A comparison of the {\it F606W} {\em HST} image and the raw
model image from ALS at the central wavelength of the {\it F606W}
filter. The model image has been rotated parallel to the long axis (PA
= 18$\degr$) to match the {\it HST} image. The contours are spaced at
an interval of 1~mag~arcsec$^{-2}$ from the peak value.}
\label{fig6}
\end{figure}
\end{centering}
We have modelled {\em IRAS} 19306+1407 using a silicate dust model,
with grain sizes between 0.1 and 0.4~$\umu$m, which reproduces the
shell-like morphology in the near-infrared, the observed degrees of
polarization and the SED. However, we also find that a purely C-rich
chemistry (amorphous carbon) using larger grains, typically
$>$0.6~$\umu$m, can reproduce the observed polarization \citep{lg05}
and fit the overall shape of the SED, although this produces a poor
fit at $<$1~$\umu$m after interstellar reddening is applied. Amorphous
carbon also does not reproduce the shape of the SED between 10 and
20~$\umu$m. We have investigated the possibility that silicon carbide
could fit the 10-20~$\umu$m region, but find that it provides too much
flux at 11-12~$\umu$m and was in general a poor fit to the SED. These
regions are modelled more effectively using Ossenkopf cold silicates.
As mentioned in Section~3.1, the simultaneous presence of emission
from PAHs and crystalline silicates \citep{hvk00,hkpw04} suggests that
the CSE has a mixed chemistry (both O- and C-rich). Our simple
investigations of mixes of carbon and silicate dust in the CSE, show
that amorphous carbon significantly dominates the SED at less than 1
per cent abundance. This suggests that if the 10-20~$\umu$m fits
require silicate grains, then they must be the dominant dust
component. However our models do not allow us to segregate the O- and
C-rich material to have, for example, a region of silicate grains
close to the star with a largely C-rich outflow at larger radii. Such
a configuration has been proposed to explain observations of mixed
chemistry objects \citep{mol02} in which the crystalline emission
comes from cool silicates trapped in stable circumstellar or
circumbinary discs. \citet{mat04} have shown that in the mixed
chemistry post-AGB object {\em IRAS} 16279-4757 the carbon-rich dust,
traced by PAH emission, is located in a low-density outflow, while the
continuum emission is concentrated toward the centre. Although our
single component model, based on silicate grains, is reasonably
successful in reproducing the observations, it is almost certain
that the chemistry of {\em IRAS} 19307+1407 involves both O- and
C-rich material, perhaps spatially segregated and with more than one
size distribution.
\begin{table}
\caption{The derived model parameters at the assumed distance of
2.7~kpc obtained from ALS$^{\dagger}$ and DART$^{\ddagger}$ models.}
\label{table8}
\begin{tabular*}{\columnwidth}{l@{\extracolsep{\fill}}rll}
\hline
Parameter & Value & Units & Description \\
\hline
$R_{\star}$ & 3.8 $\pm$ 0.6 & R$_{\sun}$ & Stellar Radius$^{\ddagger}$ \\
$r_{\rmn{in}}$ & 1.9 $\pm$ 0.1 & (10$^{14}$) m & Inner Radius$^{\dagger \ddagger}$ \\
$r_{\rmn{{\sc sw}}}$ & 3.8 $\pm$ 1.0 & (10$^{14}$) m & Super-wind Radius$^{\dagger}$\\
$r_{\rmn{out}}$ & 2.7 $\pm$ 0.1 & (10$^{15}$) m & Outer Radius$^{\dagger \ddagger}$ \\
$L^{\diamond}_{\star}$ & 4500 $\pm$ 340 & L$_{\sun}$ & Stellar Luminosity$^{\ddagger}$ \\
$N_0$ & 6.1 $\pm$ 3.0 & (10$^{-3}$) m$^{-3}$ & Number density at $r_{\rmn{in}}^{\dagger}$ \\
$M_{\rmn{d}}$ & 8.9 $\pm$ 5.0 & (10$^{-4}$) M$_{\sun}$ & Total mass of Dust$^{\dagger}$ \\
$A_{\rmn{V}}$ & 4.2 $\pm$ 0.1 & mag & Interstellar extinction$^{\ddagger}$ \\
$T_{\rmn{max}}$ & 130 $\pm$ 30 & K & Temperature at $r_{\rmn{in}}^{\ddagger}$ \\
$T_{\rmn{min}}$ & 40 $\pm$ 20 & K & Temperature at $r_{\rmn{out}}^{\ddagger}$ \\
\hline
\end{tabular*}
$^{\diamond}$The apparent luminosity of the star,
with applied interstellar reddening, is 1800~$\pm$~140~L$_{\sun}$.
\end{table}
\section{Conclusion}
We present near-infrared polarimetric images of the dusty CSE of {\it
IRAS} 19306+1407, in conjunction with new submillimetre photometry and
archived {\it HST} images. The polarization vectors show a
centrosymmetric structure with a maximum polarization of $15 \pm 6$
and $10 \pm 4$ per cent for {\it J-} and {\it K-}band respectively.
The polarized flux shows a very faint elongated distribution at PA
18$\degr$ with two bright scattering shoulders at PA 136$\degr$. The
object is clearly bipolar in archived {\em HST} images, with the bipolar
axis also at PA 18$\degr$
We model the polarimetric data using an axisymmetric light scattering
code and a dust model based on sub-micron sized silicate grains. The
observed polarization features are well described by a simple
axisymmetric shell geometry, with an equator-to-pole density contrast
of 7. The same shell model is used to fit the SED of {\it IRAS}
19306+1407 from optical to sub-millimetre wavelengths using an
axisymmetric radiation transport code, to constrain the stellar
temperature and radius, the optical depth of the CSE and the mass of
dust in the CSE. We find that a B-type stellar spectrum, with
T$_*$=21,000~K, best describes the SED, confirming previous
suggestions that the object is an early PN.
The models give a value for the CSE and interstellar extinction of
2.0~$\pm$~0.1~mag and 4.2~$\pm$~0.1~mag respectively. We estimate a
distance, from the interstellar extinction, of 2.7~$\pm$~0.1~kpc and
use this value to derive parameters from our models.
The polarimetric imaging shows deviations from axisymmetry that are
beyond the scope of our model calculations. There appears to be a
greater concentration of dust on one side of the star than the other,
plus the axisymmetric shell is not aligned with the larger-scale
bipolar axis, clearly seen in archive {\em HST} images. Similar
features are seen in other post-AGB objects and may result from
interaction of the mass-losing star with a binary companion. Further
evidence for a binary nature is provided by the probable mixed
chemistry nature of this object.
\section{Acknowledgements}
Krispian Lowe
is supported by a PPARC studentship.
The United Kingdom Infrared Telescope is operated by the
Joint Astronomy Centre on behalf
of the U.K. Particle Physics and Astronomy Council (PPARC).
The James Clerk Maxwell Telescope is operated by
The Joint Astronomy Centre on behalf of the
Particle Physics and Astronomy Research Council
of the United Kingdom, the Netherlands Organisation
for Scientific Research, and the National Research Council of Canada
(Program ID: S04BU09).
Based on observations made with the NASA/ESA Hubble Space Telescope,
obtained from the data archive at the Space Telescope Institute.
STScI is operated by the association of Universities for
Research in Astronomy, Inc. under the NASA contract NAS 5-26555.
All model calculations were run on the HiPerSpace Computing
Facility at University College London.
Thank you to Kim Clube for discussions on DART modelling.
A thank you to Bruce Hrivnak for the private communication
on \textit{IRAS} 19306+1407.
VizieR catalogue access tool and {\sc SIMBAD} database, {\sc CDS},
Strasbourg, France.
This research has made use of NASA's Astrophysics Data System.
|
1,314,259,993,616 | arxiv | \section{Introduction}
In Hamiltonian systems, important results (e.g. KAM theorem, Aubry-Mather theory and Nekhoroshev theorem) assume that the orbits have a monotonic frequency profile, known as twist condition \cite{meiss1992,lichtenberg,lochak1992}. However, many physical systems of physical importance may not satisfy that requirement, e.g., laboratory and atmospheric zonal flows \cite{diego2000} and magnetic field lines in tokamaks \cite{morrison2000,oda1995,petrisor2003}. Those systems, called nontwist, differ fundamentally from the twist ones. The degeneracies present in the frequency profile of nontwist systems originate twin island chains, whose separatrices can change their topology in a global bifurcation called reconnection \cite{wurm2005}.
Hamiltonian flow investigation has an intrinsic difficulty due to phase space dimension. For example, time-independent Hamiltonians with two degrees of freedom have a four-dimensional phase space. Fortunately, its dynamical universal behavior is equivalent to two-dimensional area-preserving maps, which reduces the dimensionality \cite{lichtenberg}. So, as in nontwist Hamiltonian flows, nontwist area-preserving maps violate the twist condition in, at least, one orbit. The so-called standard nontwist map captures the universal behavior of nontwist systems with a single orbit that violates the twist condition, called shearless invariant curve \cite{diego1996}. It has a typical phase space of quasi-integrable system: there are invariant curves (shearless included) and periodic orbits are surrounded by resonant islands. Small perturbations give rise to chaotic orbits around the saddle points, but for strong enough perturbations, the chaotic orbits spread out through phase space.
The transport in nontwist area-preserving maps has great importance due to its applications, as in fusion plasmas \cite{caldas2012} and fluids \cite{diego2000}. The chaotic regions are bounded by the invariant curves, acting like transport barriers. Global transport occurs when the last invariant curve is broken. Numerical investigations indicate that shearless curves are among the last invariant tori to break up \cite{diego1996}. However, even after their breakup, {\color{black}an effective transport barrier} still persists due to stickiness effect \cite{szezech2009}.
A nontwist area-preserving map model, proposed by Horton \textit{et al}., has been used to describe particle trajectories in tokamaks due to electric field drift, in order to understand the plasma transport in those devices \cite{horton1985,horton1998}. If the plasma has nonmonotonic profiles, {\color{black}such as magnetic and electric fields}, this model implies phase space with properties of nontwist systems \cite{horton1998,osorio2021}.
The emergence of multiple shearless curves in phase space is a topic under investigation in nontwist systems. One example appears in the standard nontwist map: the so-called secondary shearless curves arise in phase space after an odd-period orbit collision, and their breakup has different properties from the central shearless curve \cite{fuchss2006}. In fact, these bifurcations are so general that, locally, they can happen even in twist systems \cite{dullin2000,abud2012}. Moreover, recent works have found more than one shearless curves in Horton's map model \cite{grime,osorio2021}.
In this work, we derive an area-preserving nontwist map from Horton's map model \cite{horton1998}, named Biquadratic Nontwist Map. This map has a fourth degree polynomial twist function that violates the twist condition in three regions. The map presents four isochronous islands and three shearless curves. The reconnection scenarios of main resonances are presented in this paper, as well as the bifurcation scenario of the shearless curve. In addition, the map has the same shearless bifurcation scenario as obtained in the Hamiltonian flow from which it was derived \cite{grime}.
This paper is organized as follows. We derive the Biquadratic Nontwist Map from Horton's map model in Sec. 2. Some analytical results concerning symmetries and fixed points collision and reconnections are presented in Sec. 3. The shearless bifurcations in the map are shown in Sec. 4. Conclusions are presented in the last section.
\section{Derivation of the Biquadratic Nontwist Map}
We can derive the Biquadratic Nontwist Map (BNM) from a model for particle trajectories due to $\mathbf{E}\times\mathbf{B}$ drift, called Horton's map model \cite{horton1998}. Given a test particle in the plasma, its motion is subjected to the plasma electric and magnetic fields $\mathbf{E}$ and $\mathbf{B}$, respectively. Filtering out the gyromotion around the magnetic field lines, and the toroidal curvature, the particle motion is described by the differential equation
\begin{equation}
\label{eq:horton}
\dfrac{d\mathbf{x}}{dt} = v_{||}\dfrac{\mathbf{B}}{B} + \dfrac{\mathbf{E\times B}}{B^2}
\end{equation}
where $\mathbf{x}=(r,\theta,\varphi)$ is the position in local cylindrical coordinates and $v_{||}$ is the toroidal particle velocity. Waves are present at the plasma edge and lead to plasma transport. Those waves are represented by a fluctuating electric field $\mathbf{E}=-\nabla \Phi$, with potential given by
\begin{equation}
\Phi(\theta,\varphi,t)=\sum_{p=1}^{\infty} \phi_p\cos{(M\theta - L\varphi - p\omega_0t + \alpha_p)}
\end{equation}
where $M$ and $L$ stand for the spatial modes of oscillations, and the angular frequencies are multiples of $\omega_0$ \cite{horton1998}. Writing Eq. \eqref{eq:horton} in components, introducing action-angle variables $(I,\Psi)$ given by $I=(r/a^*)^2$ and $\Psi=M\theta - L\varphi$ and setting $\phi_p=\phi$ and $\alpha_p=0$ for all $p$, the differential equation \eqref{eq:horton} yields the area-preserving map
\begin{subequations}
\label{eq:horton.map.appox}
\begin{align}
\label{eq:horton.map.appox.a}
I_{n+1} &= I_n + \dfrac{4\pi M\phi}{{a^*}^2B\omega_0}\sin{(\Psi_n)}\\[0.3cm]
\label{eq:horton.map.appox.b}
\Psi_{n+1} &= \Psi_n + \dfrac{2\pi v_{||}}{\omega_0 R}\dfrac{\left[ M-Lq(I_{n+1}) \right]}{q(I_{n+1})}\qquad {\color{black} (\mathrm{mod}\ 2\pi)}
\end{align}
\end{subequations}
\noindent where the constant $B$ is related to the magnetic field, and $a^*$ and $R$ are geometrical constants \cite{horton1998}. The safety factor $q$ is a nonmonotonic function of the action coordinate and represents the spatial dependence of the magnetic field.
The aim is to obtain a map valid in the region near the minimum of the safety factor profile, also the location of the shearless transport barrier. In this situation, expanding the safety factor profile in the vicinity of a local minimum at $I=I_\mathrm{m}$, and considering up to second order terms, we obtain the $q(I)$ profile
\begin{equation}\label{eq:q.profile.map}
q(I) = q_\mathrm{m} + \dfrac{q^{''}_\mathrm{m}}{2}(I-I_\mathrm{m})^2,
\end{equation}
wherein $q_\mathrm{m}$ and $q^{''}_\mathrm{m}$ stand for the value of the safety factor and its second derivative at the minimum of the profile. Applying this profile on Eq. \eqref{eq:horton.map.appox.b}, we obtain
\begin{align}\label{eq:boquadratic}
\dfrac{M - Lq}{q} &= \dfrac{\delta}{q_\mathrm{m}}\dfrac{1-\dfrac{Lq_\mathrm{m}^{''}}{2\delta}(I-I_\mathrm{m})^2}{1+\dfrac{q_\mathrm{m}^{''}}{2q_\mathrm{m}}(I-I_\mathrm{m})^2}\\
&= \dfrac{\delta}{q_\mathrm{m}} \dfrac{1-y^2}{1+\epsilon y^2} \approx \dfrac{\delta}{q_\mathrm{m}}\left(1-(1+\epsilon)y^2 + \epsilon(1+\epsilon)y^4\right)
\end{align}
where $\delta=M -Lq_\mathrm{m}$, $\epsilon=\delta/(Lq_\mathrm{m})$ and $y=\sqrt{(Lq_\mathrm{m}^{''})/(2\delta)} \ (I-I_\mathrm{m})$. Defining the variable $x=\Psi/(2\pi)$, and the constants
\begin{equation}
\label{eq:qnm.params}
a = \dfrac{v_{||}\delta}{Rq_\mathrm{m}\omega_0}, \qquad b = -\dfrac{4\pi M\phi}{a'^2B\omega_0}\left( \dfrac{Lq^{''}_\mathrm{m}}{2\delta} \right)^{1/2},
\end{equation}
we obtain the map
\begin{subequations}
\label{eq:map}
\begin{align}
x_{n+1} &= x_n + a \left[ 1-(1+\epsilon)y_{n+1}^2 + \epsilon(1+\epsilon) y_{n+1}^4 \right]\\[0.3cm]
y_{n+1} &= y_n - b \sin{(2\pi x_n)}
\end{align}
\end{subequations}
that has a biquadratic polynomial, which can be factorized (provided the roots are real) as
\begin{equation}
a \left[ 1-(1+\epsilon)y^2 + \epsilon(1+\epsilon) y^4 \right] = a\epsilon(1+\epsilon)rs(1-y^2/r)(1-y^2/s)
\end{equation}
where $r$ and $s$ are the roots in the $y^2$ variable. Finally, defining $y'=y/\sqrt{r}$, $b'=b\sqrt{r}$, $a'=a\epsilon(1+\epsilon)rs$, $\epsilon'=r/s$, we obtain the Biquadratic Nontwist Map in the form:
\begin{subequations}
\label{eq:qnm}
\begin{align}
x_{n+1} &= x_n + a \left( 1-y_{n+1}^2 \right)\left( 1-\epsilon y_{n+1}^2 \right)\ {\color{black} (\mathrm{mod}\ 1)}\\[0.3cm]
y_{n+1} &= y_n - b \sin{(2\pi x_n)}.
\end{align}
\end{subequations}
\noindent where we omitted the primes in $y$, $\epsilon$, $a$ and $b$, for simplicity of notation.
The map \eqref{eq:qnm} models the particle drift motion in a tokamak plasma near the minimum of the safety factor profile \eqref{eq:q.profile.map}. It is a three-parameter family of nontwist area-preserving maps in $(x,y)$ variables, where $x\in [0,1)$ and $y\in \mathbb{R}$ are the angle and action variables, respectively. The parameters ${\color{black}a \in [0,1]}$ and ${\color{black}\epsilon \in \mathbb{R}^+}$ modulate the twist function of the map, and {\color{black}$b \in [0,1]$} is the perturbation parameter. In the limit $\epsilon\to 0$, the Biquadratic Nontwist Map \eqref{eq:qnm} reduces to the standard nontwist map \cite{diego1996}.
The BNM has the twist function
\begin{equation}\label{eq:twist.function}
\omega(y)=a\left( 1-y^2 \right)\left( 1-\epsilon y^2 \right).
\end{equation}
The twist condition of an area-preserving map reads \cite{lichtenberg}
\begin{equation}
\label{eq:twist.condition}
\dfrac{\partial x_{n+1}}{\partial y_n} = \dfrac{\partial \omega}{\partial y} \neq 0, \ \forall (x,y).
\end{equation}
Applying the definition \eqref{eq:twist.condition} to the twist function of the Biquadratic Nontwist Map \eqref{eq:twist.function}, it violates the twist condition, for $b=0$, at
\begin{equation*}
y = 0 \ \text{and} \ y = \pm \sqrt{\dfrac{1+\epsilon}{2\epsilon}}.
\end{equation*}
In the integrable limit ($b\to 0$), the map has three shearless curves, $C_1, C_2 \ \text{and} \ C_3$, defined by
\begin{align}
&C_1: \ y=b \sin{(2\pi x)}, \\[0.1cm]
&C_{2,3}: \ y= \pm \sqrt{\dfrac{1+\epsilon}{2\epsilon}} + b \sin{(2\pi x).}
\end{align}
Figure \ref{fig:twist.function} shows the twist function of the standard (dashed line) and biquadratic (continuous line) nontwist maps. The three extrema present in BNM are marked in red, blue and green. The red point, representing the central shearless curve $C_1$, is common to both maps, but the biquadratic map has two other shearless points, corresponding to $C_{2,3}$. The Standard Nontwist map also has scenarios with more than one shearless curves, but they are consequences of bifurcations in periodic orbits \cite{wurm2005}. In contrast, the BNM has three shearless curves even in the integrable limit, for $b=0$.
\begin{figure}[htb]
\centering
\includegraphics[width=0.5\textwidth]{images/Figure1.png}
\caption{Twist function of BNM [Eq. \eqref{eq:twist.function}] for the parameters $a=1$ and $\epsilon = 0$ (dashed line) and $\epsilon=0.6$ (filled line). There are three points violating the twist condition marked in red, blue and green.}
\label{fig:twist.function}
\end{figure}
For $b \neq 0$, the map is nonintegrable, and the shearless curves are calculated numerically by finding the extrema in rotation number profile. For a regular (nonchaotic) orbit with initial condition $(x_0,y_0)$, we define its rotation number $\Omega$ by the limit
\begin{equation}
\Omega(x_0,y_0) = \lim_{n\to \infty} \dfrac{x_n - x_0}{n},
\end{equation}
{\color{black}wherein the modulus operation is not applied.} If the initial condition belongs to a chaotic orbit, this limit does not exist, and we cannot define its rotation number.
In addition, a similar map, called quartic nontwist map, was proposed in Ref. \cite{wurm2012} to study the influence of symmetries in the shearless breakup. The quartic nontwist map has a fourth degree polynomial twist function {\color{black}equivalent to Eq. \eqref{eq:twist.function}, but considers $\epsilon < 0$. Therefore, the quartic nontwist map has only one shearless point and} its dynamical behavior departs from the Biquadratic Nontwist Map introduced in this article.
\section{Some results concerning the Biquadratic Nontwist Map}
Simple nontwist area-preserving maps, like the standard nontwist map, have spatial and time-reversal symmetries that make some numerical analysis tractable, like the search for periodic orbits \cite{diego1996,petrisor2001}. The Biquadratic Nontwist Map (BNM) has the same spatial symmetry as the standard nontwist map \cite{diego1996}. Let $M$ be the BNM and $S$ the transformation
\begin{equation}\label{eq:symmetry.transformation}
S(x,y) = \left ( x + 1/2, \ -y \right ),
\end{equation}
\noindent the map $M$ is invariant under $S$, so $M = S^{-1}MS$. Another property of the BNM, analogous to the SNT, is the time reversal symmetry \cite{diego1996}. We can decompose the map \eqref{eq:qnm} as a product of two involutions
\begin{equation}
M = R_1R_0
\end{equation}
\noindent where
\begin{subequations}
\label{eq:involutions}
\begin{align}
R_0(x,y) &= \left ( -x, \ y - b \sin{(2\pi x)} \right),\\
R_1(x,y) &= \left (-x + a (1 - y^2)(1 - \epsilon y^2), \ y \right ).
\end{align}
\end{subequations}
Each involution \eqref{eq:involutions} has an invariant set of points, defined by
\begin{equation}
\mathcal{I}_j = \left\{\mathbf{z} \ | \ R_j\mathbf{z} = \mathbf{z}\right\}, \ j=0,1,
\end{equation}
\noindent which are one-dimensional sets called symmetry sets of the map. The set $\mathcal{I}_0$ is formed by the union $\mathcal{S}_1\cup \mathcal{S}_2$, and $\mathcal{I}_1=\mathcal{S}_3\cup\mathcal{S}_4$, where $\mathcal{S}_i$ is the $i$-th symmetry line given by:
\begin{subequations}
\begin{align}
\mathcal{S}_1 &= \left\{ \ (x,y) \ | \ x = 0 \ \right\},\\
\mathcal{S}_2 &= \left\{ \ (x,y) \ | \ x = 1/2 \ \right\},\\
\mathcal{S}_3 &= \left\{ \ (x,y) \ | \ x = a (1-y^2)(1-\epsilon y^2)/2 \ \right\},\\
\mathcal{S}_4 &= \left\{ \ (x,y) \ | \ x = a (1-y^2)(1-\epsilon y^2)/2 + 1/2 \ \right\}.
\end{align}
\end{subequations}
\subsection{Fixed Points}
The Biquadratic Nontwist Map has eight fixed points. Using the notation $\mathbf{z}=(x,y)$, those points are:
\begin{subequations}
\label{eq:fixed.points}
\begin{align}
\mathbf{z}_1^{\pm}&=(0,\pm 1), \hspace{1cm} \mathbf{z}_2^{\pm}=\left( 0,\pm \dfrac{1}{\sqrt{\epsilon}} \right), \\[0.1cm]
\mathbf{z}_3^{\pm}&=\left(\dfrac{1}{2}, \ \pm 1\right), \hspace{0.5cm} \mathbf{z}_4^{\pm}=\left(\dfrac{1}{2}, \ \pm \dfrac{1}{\sqrt{\epsilon}}\right).
\end{align}
\end{subequations}
Four of the fixed points in Eq. \eqref{eq:fixed.points}, $\mathbf{z}_{1,3}^{\pm}$, are equivalent to those in the standard nontwist map \cite{diego1996}. The rest of them are introduced by the new term in the twist function, controlled by the parameter $\epsilon$. For small $\epsilon$, those points go to infinity, and we recover the phase space of the standard nontwist map. Figure \ref{fig:fixed.points.coordinate} displays the $y$ coordinate of the fixed points in BNM. In the critical value $\epsilon=1$, the fixed points collide in a bifurcation [see Fig. \ref{fig:homoclinic.bifurcation}b].
\begin{figure}[htb]
\centering
\includegraphics[width=.5\textwidth]{images/Figure2.png}
\caption{The fixed points position is controlled by the parameter $\epsilon$. Plot of $y$ coordinate of the fixed points: $\mathbf{z}_{1,3}^{+}$ (magenta dashed), $\mathbf{z}_{1,3}^{-}$ (green dashed), $\mathbf{z}_{2,4}^{+}$ (gold) and $\mathbf{z}_{2,4}^{-}$ (cyan) as a function of the parameter $\epsilon$. For $\epsilon=1$ the points $\mathbf{z}_{1}^\pm$ and $\mathbf{z}_{3}^\pm$ collide with $\mathbf{z}_{2}^\pm$ and $\mathbf{z}_{4}^\pm$.}
\label{fig:fixed.points.coordinate}
\end{figure}
The perturbation in the map generates primary resonances in the fixed points. As a result, the phase space contains four isochronous islands, shown in Figure \ref{fig:symmetry.lines}, together with the symmetry lines. In Figure \ref{fig:symmetry.lines}a, the phase space of the standard nontwist map is plotted for $a = 0.3$ and $b = 0.05$. It contains two resonances and four fixed points. Using the same parameters $a$ and $b$, and $\epsilon=0.4$, the Biquadratic Nontwist Map shows its four resonances, marked in magenta, cyan, green and gold [Figure \ref{fig:symmetry.lines}b]. We observe that symmetric fixed points in the same symmetry line have opposite stability, like all periodic orbits in the even scenario on standard nontwist map \cite{diego1996}. The rotation number profile for Figure \ref{fig:symmetry.lines}b is plotted in Figure \ref{fig:rot.profile}, using the initial condition $x_0=0.25$. We see the four plateaus in the rotation profile, corresponding to the isochronous islands, and the three extreme points related to the shearless curves in the system.
\begin{figure}[htb]
\centering
\includegraphics[width=.75\textwidth]{images/Figure3.png}
\caption{\label{fig:symmetry.lines}Comparison between (a) standard nontwist map and (b) Biquadratic Nontwist Map. The phase space of the standard nontwist map is plotted for parameters $a = 0.3$ and $b = 0.05$. For the biquadratic map, those parameters are the same and $\epsilon=0.4$. The symmetry lines $\mathcal{S}_3$ and $\mathcal{S}_4$ are drawn in gray.}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=.5\textwidth]{images/Figure4.png}
\caption{Rotation number profile of the map in Figure \ref{fig:symmetry.lines}b calculated for $x_0=0.25$. The three extreme points, one maximum (in red) and two minimum (in blue and green) give the $y$ initial conditions for the shealess curves. The four isochronous islands appear as four plateaus in the profile.}
\label{fig:rot.profile}
\end{figure}
The stability of a fixed point is determined by the eigenvalues of the tangent map evaluated at that point \cite{lichtenberg}. For area-preserving maps, {\color{black}these eigenvalues are a pair $\{ \lambda, 1 / \lambda \}$}, and if they are real (complex), the point is unstable (stable) \cite{lichtenberg}. For area-preserving maps, one way to write the criterion for the stability of a fixed point $\mathbf{z}$ is by its residue
\begin{equation}
R = \dfrac{1}{4}\left[ 2 - \mathrm{Tr}\left(J(\mathbf{z}) \right) \right],
\end{equation}
where $\mathrm{Tr}\left(J(\mathbf{z})\right)$ is the trace of the Jacobian matrix at the fixed point \cite{greene1968}. If $0<R<1$ the periodic orbit is elliptic (stable), if $R<0$ or $R>1$ it is hyperbolic (unstable) and it is parabolic in the critical values $R=0$ and $R=1$ \cite{greene1968}. For the map \eqref{eq:qnm}, the residues of the fixed points are
\begin{subequations}
\label{eq:residue.fixed.points}
\begin{align}
R\left(\mathbf{z}_1^{\pm}\right) &= \pm \pi ab(\epsilon -1),\\
R\left(\mathbf{z}_2^{\pm}\right) &= \mp \pi ab(\epsilon -1)/\sqrt{\epsilon},\\
R\left(\mathbf{z}_3^{\pm}\right) &= \mp \pi ab(\epsilon -1),\\
R\left(\mathbf{z}_4^{\pm}\right) &= \pm \pi ab(\epsilon -1)/\sqrt{\epsilon}
\end{align}
\end{subequations}
It is easy to verify that the stability of symmetric fixed points in the same symmetry line is opposite. For example, $\mathbf{z}_2^{+}$ is hyperbolic and $\mathbf{z}_2^{-}$ is an elliptic fixed point, both belong to symmetry lines $\mathcal{S}_2$ and $\mathcal{S}_4$. As we see, the parameter $\epsilon$ controls, together with $a$ and $b$, the stability of the fixed points. Considering $a,b \in [0,1]$, by the residue criterion, a change of stability occurs for $\epsilon=1$. As seen in Figure \ref{fig:homoclinic.bifurcation}, for $0< \epsilon <1$, $\mathbf{z}_{1,4}^{+}$ and $\mathbf{z}_{2,3}^{-}$ are hyperbolic; $\mathbf{z}_{2,3}^{+}$ and $\mathbf{z}_{1,4}^{-}$ are elliptic [Figure \ref{fig:homoclinic.bifurcation}a]. Otherwise, if $\epsilon>1$, $\mathbf{z}_{1,4}^{+}$ and $\mathbf{z}_{2,3}^{-}$ are elliptic; $\mathbf{z}_{2,3}^{+}$ and $\mathbf{z}_{1,4}^{-}$ are hyperbolic, Figure \ref{fig:homoclinic.bifurcation}c. In the critical value $\epsilon=1$, the fixed points collide [Figure \ref{fig:homoclinic.bifurcation}b] and the residues of all fixed points are zero, then they are all parabolic.
\begin{figure}
\centering
\includegraphics[width=0.99\textwidth]{images/Figure5.png}
\caption{\label{fig:homoclinic.bifurcation}Same as Figure \ref{fig:symmetry.lines}b, for parameters $a=0.1$, $b=0.01$ and (a) $\epsilon=0.6$, (b) $\epsilon=1.0$ and (c) $\epsilon =1.9$. The $\epsilon$ parameter controls the fixed points position and there is a bifurcation for $\epsilon=1$, where they collide.}
\end{figure}
Similar results, with four isochronous islands and three shearless curves, have also been obtained for a map derived in a model of particle trajectory in tokamaks with finite Larmor radius \cite{diego2012,fonseca2014}. However, the latter map has not the symmetries of the Biquadratic Nontwist Map introduced in this work.
\FloatBarrier
\subsection{Separatrix reconnection}
In this section, we investigate the separatrix reconnection for the Biquadratic Nontwist Map. In the standard nontwist map, which violates the twist condition in one point, there are more than one (usually, two) orbits with the same rotation number \cite{diego1996}. In contrast, the Biquadratic Nontwist Map has three extrema in the twist function, allowing four isochonous island chains. Those orbits may undergo a global bifurcation process, namely, the reconnection of separatrices, that changes the topology of invariant manifolds of the corresponding hyperbolic orbits \cite{wurm2005}. In the standard nontwist map, those reconnections have different properties depending if the periodic orbit has odd or even period \cite{diego1996}. For the Biquadratic Nontwist Map, we also have the same standard odd and even scenarios. We will focus the discussion on the reconnection process of the fixed points.
The Biquadratic Nontwist Map (BNM) has four primary resonances related to the fixed points given by Eq. \eqref{eq:fixed.points}. The hyperbolic manifolds of each resonance may reconnect to an adjacent island, so there are two possible reconnections of separatrices. One of them involves the hyperbolic points $\mathbf{z}_1^+ = (0,1)$ and $\mathbf{z}_3^- = (1/2,-1)$, displayed in Figure \ref{fig:reconnection1}, where $b$ is the control parameter. The hyperbolic manifolds of those fixed points have heteroclinic topology in Figure \ref{fig:reconnection1}a. The reconnection of separatrix is shown in Figure \ref{fig:reconnection1}b and a bifurcation changes its topology to homoclinic configuration, Figure \ref{fig:reconnection1}c. The appearance of meandering orbits (orbits that are not graphs over the $x$-axis) \cite{van1988,simo1998} is a consequence of that topology changing.
\begin{figure}[htb]
\centering
\includegraphics[width=.99\textwidth]{images/Figure6.png}
\caption{\label{fig:reconnection1}Separatrix reconnection of fixed points $\mathbf{z}_1^+ = (0,1)$ e $\mathbf{z}_3^- = (1/2,-1)$ in Biquadratic Nontwist Map. The parameters used are: $a=0.02$, $\epsilon=0.1$ and $b=$ (a) $0.0533$, (b) $0.0821003$ and (c) $0.133$. The reconnection occurs in (b), changing the topology of separatrices from heteroclinic (a) to homoclinic (c).}
\end{figure}
{\color{black}Considering the $x$ variable $\mathrm{mod}\ 1$, in Figure \ref{fig:reconnection1}a, the hyperbolic manifolds have homoclinic topology, because the fixed points on $x=0$ and $x=1$ are the same. Otherwise, if $x$ has an unlimited range, those fixed points are different and, by consequence, the separatrix has heteroclinic topology. The literature, and this paper, assumes the second convention \cite{wurm2005,diego1997}.}
Given $a$ and $\epsilon$, there is an analytical procedure, outlined in \ref{sec.appendixA}, that returns the approximate critical value of the parameter $b$ for which the bifurcation occurs. Applying this method, for the previously mentioned reconnection, we obtain the critical parameter
\begin{equation}
\label{eq:parameter.reconnection1}
b_1^* = \dfrac{4\pi a}{3}\left( 1 - \epsilon/5 \right),
\end{equation}
which agrees with the $b$ critical value in Figure \ref{fig:reconnection1}b.
The relation above is an approximation, valid for small values of $a$ and $\epsilon$. In the limit $\epsilon \to 0$, we recover the result for the standard nontwist map \cite{diego1996}.
Another possible reconnection is between the two pairs of islands close to the shearless curves $C_{2,3}$. The hyperbolic points involved are: $\mathbf{z}_1^+ = (0,1)$ and $\mathbf{z}_4^+ = (1/2,1/\sqrt{\epsilon})$; and $\mathbf{z}_3^- = (1/2,-1)$ and $\mathbf{z}_2^- = (0,-1/\sqrt{\epsilon})$. All the primary resonances are involved in this bifurcation. The islands reconnect, in pairs, in the same previous scenario: heteroclinic topology [Figure \ref{fig:reconnection2}a], reconnection of separatrices [Figure \ref{fig:reconnection2}b] and homoclinic topology with meander formation [Figure \ref{fig:reconnection2}c]. The analytical procedure described in \ref{sec.appendixA} results in the relation
\begin{equation}
\label{eq:parameter.reconnection2}
b_2^* = 2\pi a \dfrac{( 1 - 5\epsilon + 5\epsilon^{3/2} - \epsilon^{5/2})}{15\epsilon^{3/2}}
\end{equation}
for the critical $b$ value at the reconnection. Again, for small values of $a$ and $\epsilon$, this analytical relation agrees with numerical results [Figure \ref{fig:reconnection2}b].
\begin{figure}[htb]
\centering
\includegraphics[width=.99\textwidth]{images/Figure7.png}
\caption{\label{fig:reconnection2}Same as Figure \ref{fig:reconnection1}, using parameters $\epsilon=0.4$, $b=0.05$, $a=$ (a) $0.3$, (b) $0.18444$ and (c) $0.1$. The scenario is similar to Figure \ref{fig:reconnection1}, but involves different pairs of isochronous islands.}
\end{figure}
Scenarios with four isochronous island chains and three shearless curves have also been reported in atypical periodic orbit configurations in the standard nontwist map \cite{wurm2005}. The so-called inner and outer periodic orbits reconnect in the two forms present in Figures \ref{fig:reconnection1} and \ref{fig:reconnection2}. Although the standard nontwist map has the same scenario reported for the Biquadratic Nontwist Map, the multiple twin island chains and shearless curves are localized and come from bifurcations derived from the perturbations in the map. {\color{black}In contrast}, in the Biquadratic Nontwist Map the multiple shearless curves are related to the twist function.
\section{Shearless bifurcations}
The shearless curves in the Biquadratic Nontwist Map may be broken by the perturbation. The breakup of shearless curve in nontwist maps is the subject of many studies in literature \cite{diego1996,diego1997,wurm2012,wurm2005,shinohara1997,shinohara1998,howard1995}. For the Biquadratic Nontwist Map, there may be situations in which one or two of the shearless curves are broken, but the remnant shearless curve(s) still prevent global transport.
In Figure \ref{fig:shearless.bifuraction}a, the perturbation in the map has broken the shearless curves $C_{2,3}$ and we see just one shearless curve, $C_1$, in the phase space. In this particular example, the fixed points have collided and have parabolic stability. However, {\color{black}on changing parameter $a$}, the blue and green shearless curves reappear, as seen in Figure \ref{fig:shearless.bifuraction}b. The scenario of that shearless bifurcation is shown in Figure \ref{fig:shearless.bifuraction.scenario}. In the boundary of the chaotic region, there are secondary resonances: a pair of twin isochronous island chains, in pink and orange, Figure \ref{fig:shearless.bifuraction.scenario}a. The orange chain goes away from the chaos and the shearless curve $C_{2}$ emerges from that process, Figure \ref{fig:shearless.bifuraction.scenario}b. Due to the symmetry of the map, the blue and green shearless curves emerge concomitantly for the same critical parameter.
The scenario of shearless bifurcation displayed in Figure \ref{fig:shearless.bifuraction.scenario} was reported in a different system, with similar characteristics. In Ref. \cite{grime}, shearless bifurcations are analyzed in a Hamiltonian flow {\color{black}related to} Horton's model. More than one shearless curve appears, and the scenario of the shearless bifurcation is the same as the one reported in Figure \ref{fig:shearless.bifuraction.scenario}. In fact, we conjecture that the Biquadratic Nontwist Map captures the essential features of the shearless bifurcations present in other nontwist systems.
\begin{figure}[htb]
\centering
\includegraphics[width=.75\textwidth]{images/Figure8.png}
\caption{\label{fig:shearless.bifuraction}The Biquadratic Nontwist Map features a bifurcation in the shearless curves. In (a) we observe one shearless curve in phase space and two chaotic regions on top and bottom. Varying the parameter $a$, (b) the blue and green shearless curves shows up at the boundary of the chaotic regions. The parameters used are $\epsilon =1.0$, $b=0.16$, (a) $a=0.325$ and (b) $a=0.358$.}
\end{figure}
\begin{figure}[htb]
\centering
\includegraphics[width=.75\textwidth]{images/Figure9.png}
\caption{\label{fig:shearless.bifuraction.scenario}Emergence scenario of the shearless curves $C_{2,3}$. (a) There is a pair of twin islands, one in pink and the other in orange, (b) that leaves the chaotic region, simultaneously with the blue shearless curve emerges. The parameters used are $\epsilon=1.0$, $b=0.16$ and $a=$ (a) $0.325$, (b) $0.328$ and (c) $0.331$.}
\end{figure}
\FloatBarrier
\section{Conclusion}
In this paper, we derived an area-preserving nontwist map from a Hamiltonian model for particle trajectories in plasmas, named Biquadratic Nontwist Map. It has a fourth degree polynomial function, which implies the presence of three shearless curves and four main resonances in phase space. The map has symmetry properties similar to the standard nontwist map, that enable simplifications in some numerical problems. Although derived from a plasma model, the map captures the behavior of a broader range of nontwist systems with multiple shearless curves.
We reported reconnection scenarios, involving the main resonances, similar to those found in other nontwist map, and used analytical techniques involving integrable Hamiltonian flows to find its critical parameters. The results obtained agree with the map for a certain range of the parameters, when the chaos has not spread over the phase space.
Finally, we found shearless bifurcations in the Biquadratic Nontwist Map, with a scenario identical to the one found in more complex nontwist systems. The results in this paper suggest a relation between secondary twin island chains in the boundary of chaotic regions and the emergence of new shearless curves in phase space. So, it can be used as a model for these bifurcations in the shearless curve.
\section*{Ackowledgments}
The authors thank the financial support from the Brazilian Federal Agencies (CNPq) under Grant Nos. 407299/2018-1, 302665/2017-0, 403120/2021-7, and 301019/2019-3; the São Paulo Research Foundation (FAPESP, Brazil) under Grant Nos. 2018/03211-6 and 2022/04251-7; and support from Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) under Grants No. 88887.710886/2022-00, 88887.522886/2020-00, 88881.143103/2017-01 and Comité Français d'Evaluation de la Coopération Universitaire et Scientifique avec le Brésil (COFECUB) under Grant No. 40273QA-Ph908/18.
YE enjoyed the hospitality of the grupo controle de oscilações at USP.
|
1,314,259,993,617 | arxiv | \section{Introduction}
In strongly correlated materials\cite{RPP2017,Giustino_2021} various energy scales compete in defining the ground state.
Perturbing their balance, e.g., through pressure or doping, may induce a multitude of different
long-range orders or trigger metal-insulator transitions.\cite{imada}
Particularly rich are phenomena involving the orbital degrees of freedom.\cite{Tokura462,doi:10.1021/acs.chemrev.0c00579}
Their behavior is extremely sensitive to the local atomic environment that controls
hybridizations,\cite{jmt_radialKI} crystal-fields,\cite{Bethe1929} and their degeneracy.
Therefore, the advent of epitaxial growth of ultra-thin films and hetero-structures has
unlocked vast possibilities\cite{Tokura2003,doi:10.1146/annurev-matsci-070813-113248} to explore orbital physics and electronic anisotropies\cite{Benjamin_2D3D} in general.
Indeed, substrate strain, interfacial or surface reconstruction, and varying
surface terminations cause distortions, rotations, or even the severing of coordination polyhedra. These structural changes invariably affect the electronic, magnetic, and orbital state, often leading to properties absent in bulk samples of the same material.
Here, we study the extreme case of a monolayer of the perovskite transition-metal oxide SrVO$_3$ deposited on a SrTiO$_3$ substrate.
A moderately correlated paramagnetic metal in the bulk,\cite{PhysRevB.58.4372,PhysRevLett.93.156402} SrVO$_3$
is known to undergo a metal-to-insulator transition (MIT) in ultra-thin films on SrTiO$_3$ below a critical thickness of 2-3 unit-cells.\cite{PhysRevLett.104.147601,Kobayashi2017}
This Mott insulator\cite{PhysRevLett.104.147601,PhysRevLett.114.246401} has further been suggested\cite{PhysRevLett.114.246401} as active material in a Mott transistor,\cite{Motttransistor} where a gate voltage is used to switch between the insulator (OFF) and a metallic (ON) state.
Besides the characterization of the charge state, however, little is known about ordered phases or long-range fluctuations in ultra-thin films. Bulk Mott insulators typically order antiferromagnetically (AF) at low enough temperatures, while doping them may lead to various types of fluctuations and symmetry-broken phases, e.g., superconductivity or charge-order.
Here, we find that the SrVO$_3$ monolayer
realizes more than five distinct phases, see \fref{fig:phasediagram}, including (in)commensurate antiferromagnetism (AF), ferromagnetism (FM), as well as stripe (s) and checkerboard (c) orbital-order (OO).
Some of these regimes can be explored by chemical doping or an applied gate voltage. Additionally, we evidence a qualitatively different phase diagram for the two possible choices of surface termination, in which the top is formed by either a VO$_2$, or a SrO layer (\fref{fig:phasediagram}a, \ref{fig:phasediagram}b, respectively).
Ultimately, we link the character of dominant fluctuations to the orbital degrees of freedom, that are tuned through the (total) filling $n$ and the crystal-field splitting $\Delta_{\rm cfs}$ between the two-fold degenerate $d_{xz}$, $d_{yz}$ orbitals and the $d_{xy}$ orbital residing in the film's plane.
With these key ingredients being common to a multitude of correlated oxides, our study of the SrVO$_3$ monolayer anticipates rich phase diagrams in transition-metal oxide ultra-thin films.
\begin{figure*}
\centering
\subfloat[VO$_2$-terminated monolayer]{
\includegraphics[width=0.47\textwidth]{phasediagram_vo2.pdf}
}
\quad
\subfloat[SrO-terminated monolayer]{
\includegraphics[width=0.47\textwidth]{phasediagram_sro.pdf}
}
\caption{{\bf Phase diagrams.} (a) VO$_2$ and (b) SrO terminated monolayers of SrVO$_3$\ on a SrTiO$_3$\ substrate (see insets) realize various phases as a function of the number of electrons per site in the low-energy $t_{2g}$ ($n$; lower $x$-axis) or gate voltage ($V_G$; upper $x$-axis):
antiferromagnetism (AF: red), ferromagnetism (FM: blue), incommensurate magnetism (iM: blueish), checkerboard orbital order (cOO: green), stripe orbital order (sOO: turquoise).
The thus colored domes indicate the formation of long-range order
within dynamical mean-field theory (DMFT).
The ``$+$''--signs indicate points for which many-body calculations were performed (black signs represent non-divergent, white signs divergent DMFT susceptibilities). Based on the
dominant susceptibilities, the color domes have been drawn as a guide to the eye.
\label{fig:phasediagram}}
\end{figure*}
The outline of the paper is as follows: Section
\ref{Sec:Method} briefly discusses the methods employed: density functional theory (DFT) and dynamical mean-field theory (DMFT).
Section \ref{sec:DFT} presents the quantitative differences between the two different surface terminations on the one-particle level.
Section \ref{sec:Mott} presents a many-body discussion of the Mott insulating state found in the stoichiometric systems, while Section \ref{Sec:DMFT} presents the electronic structure trends upon doping the SrVO$_3$ monolayers.
Section \ref{Sec:DMFTsusc} characterizes magnetic and orbital fluctuations
and associated ordering instabilities on the basis of DMFT susceptibilities.
Section \ref{Sec:Discussion} puts our findings into perspective.
Finally, Section \ref{Sec:Conclusion} summarizes the results and provides an outlook.
\section{Method}%
\label{Sec:Method}
DFT calculations are performed with the WIEN2k package \cite{wien2k,doi:10.1063/1.5143061}
using the PBE\cite{PhysRevLett.77.3865} exchange-correlation potential.
We construct systems with one half of a unit cell of SrVO$_3$~(VO$_2$ termination) and one unit cell of SrVO$_3$~(SrO termination) on top of a substrate of six unit cells of SrTiO$_3$\ surrounded by sufficient vacuum of around $10$\AA\ in z-direction
(see insets in Fig.~\ref{fig:phasediagram}). In both setups the transition between SrTiO$_3$~and SrVO$_3$~consists of a TiO$_2$ - SrO - VO$_2$ interface, consistent with experiment \cite{C6CP07691B,PhysRevB.100.155114}, while at the bottom the SrTiO$_3$~substrate is terminated via SrO to vacuum.
Since experimentally SrVO$_3$\ is locked to the SrTiO$_3$\ substrate\cite{PhysRevMaterials.3.115001}
we initialize the in-plane lattice constant
with the PBE-optimized value for bulk SrTiO$_3$\
$a_{\mathrm{SrTiO}_3}=3.95$\AA.
To treat the surface properly the two unit cells of SrTiO$_3$\ furthest away from SrVO$_3$\ are then constrained to $a_{\mathrm{SrTiO}_3}$, simulating the transition to the SrTiO$_3$ bulk, while all other internal atomic positions are fully relaxed.
We then perform Wannier projections
onto maximally localized V-t$_{\mathrm{2g}}$ orbitals, using the WIEN2Wannier \cite{wien2wannier} interface to Wannier90 \cite{wannier90}, see Fig.~\ref{Fig:dft}.
These Wannier
Hamiltonians are supplemented with an effective SU(2)-symmetric Kanamori
interaction of $U=5$eV, $J=0.75$eV, $U'=3.5$eV
(similar to Ref.~\onlinecite{PhysRevLett.114.246401}),
for which we perform dynamical mean-field theory (DMFT) \cite{bible,doi:10.1080/00018730701619647}
calculations at various temperatures.
For the undoped bulk, this setup yields the correct mass enhancement and spectra qualitatively congruent with photoemission spectroscopy\cite{pavarini:176403, Mo2003}. These values can be thought of as a lower boundary since in ultrathin films, interaction parameters increase slightly\cite{PhysRevLett.114.246401} with respect to their bulk values.
The Hamiltonians are kept constant under doping.
The analytic continuations to real frequencies are performed with the maximum entropy method\cite{maxent} used in the \verb=ana_cont= library\cite{PhysRevLett.122.127601,kaufmann2021anacont}.
The DMFT self-consistency cycle as well as the sampling of the two-particle Green's function is done by continuous-time quantum Monte Carlo simulations in the hybridization expansion \cite{Werner2006,Gull2011a} using w2dynamics \cite{w2dynamics} with worm sampling \cite{Gunacker15}.
Momentum-dependent DMFT susceptibilities are calculated from the local vertex,
using the AbinitioD$\Gamma$A \cite{Anna_ADGA,CPC_ADGA} program package.
\begin{figure*}[!t]
\centering
\subfloat[VO$_2$-terminated monolayer: $\Delta_{\rm cfs}<0$]{
\includegraphics[width=0.47\textwidth]{bandstructure_dos_vo2.pdf}
}
\quad
\subfloat[SrO-terminated monolayer: $\Delta_{\rm cfs}>0$]{
\includegraphics[width=0.47\textwidth]{bandstructure_dos_sro.pdf}
}
\caption{{\bf Band structures and density of states.} (a) VO$_2$ and (b) SrO terminated monolayer. Left: DFT band structure along a momentum path through the Brillouin zone (black dots) overlain with the
t$_{\mathrm{2g}}$-orbital projected Wannier dispersion (red lines). Right: the resulting
density of states (DOS) clearly shows the (quasi--) 2D character of the $xy$-orbital
(blue) and the 1D character of the locally degenerate $xz/yz$-orbitals (green). The local orbital energy levels are marked as dashed horizontal lines in the DOS.
We find a crystal field splitting of $\Delta_{\rm cfs} = -0.252$eV for the VO$_2$ terminated layer and $\Delta_{\rm cfs} = +0.126$eV for the SrO terminated monolayer.}
\label{Fig:dft}
\end{figure*}
\section{Results}
\subsection{Surface termination, band structure and density of states}
\label{sec:DFT}
Both, SrO and VO$_2$-terminated films result in similarly looking DFT band structures whose relevant orbitals around the Fermi level are of vanadium $t_{2g}$ character, see Fig.~\ref{Fig:dft}.
The densities of states (DOS) of these orbitals showcase the abrupt surface termination of the sample: While the $xy$-projection (blue line) keeps its two-dimensional character (as in bulk SrVO$_3$) the (locally degenerate) $xz$- and $yz$-projections (green lines) now become one-dimensional and, concomitantly, display a strongly reduced bandwidth. Consequently a van-Hove singularity emerges, which, at zero doping, is in close proximity to the Fermi level. Indeed, in the VO$_2$ (SrO) terminated system this singularity is situated slightly below (above) the Fermi level.
On top of this dimensionality reduction we find the crystal-field splitting (cfs)
\begin{equation}
\Delta_{\rm cfs} = E_{xz/yz}-E_{xy},
\end{equation}
to have opposite signs for the two different setups:
The {\it positive} cfs of the SrO terminated monolayer, $\Delta_{\rm cfs}=+0.13$eV, is a direct result
of the tensile strain caused by the (in-plane) lattice mismatch between SrVO$_3$ and SrTiO$_3$ ($a_{\mathrm{SrVO}_3} < a_{\mathrm{SrTiO}_3}$).
The in-plane expansion triggers a structural compression in the perpendicular direction, so as to keep the volume approximately constant.
This structural anisotropy translates into an electronic anisotropy\cite{Benjamin_2D3D}:
The evident breaking of the cubic symmetry of SrVO$_3$ lifts the three-fold $t_{2g}$ degeneracy, making the $xy$-orbital energetically favorable.
The same effects take place in the VO$_2$ terminated monolayer as well. There, however, the geometric distortion gets overcompensated by the missing SrO layer: severing the apical oxygen of the transition metal coordination octahedron results in a reversed, {\it negative} $\Delta_{\rm cfs}=-0.25$eV. The $xz/yz$ orbitals have their lobes pointing in the $z$-direction, towards the lobes of the oxygen $p_x/p_y$ orbitals. The missing overlap to the absent apical oxygen leads to less electrostatic repulsion, thus lowering the energy required to occupy these states.
Another contributing factor is the abrupt termination to vacuum, removing any restriction in the positive $z$-direction for the structural relaxation:
The VO$_2$ (SrO) terminated system results in a concave (convex) final termination, i.e., the last VO$_2$ (SrO) layer bends inwards (outwards).
For both systems we find almost identical $xy$ orbitals with a nearest-neighbor hopping $t_{xy}\sim-230$meV, next-nearest-neighbor hopping $t^\prime_{xy}\sim-70$meV and bandwidth W$_{xy}\sim2.1$eV.
The $xz$ and $yz$ orbitals of both systems, on the other hand, can be described purely by nearest-neighbor hopping along the $x$ or $y$ direction, respectively: The VO$_2$-terminated monolayer allows for a large hopping amplitude ($t_{xz/yz}\sim-300$meV), resulting in a slightly larger bandwidth $W_{xz/yz}=1.2$eV in Fig.~\ref{Fig:dft}a, compared to only $W_{xz/yz}=0.95$eV ($t_{xz/yz}\sim-200$meV) for the SrO-terminated monolayer in Fig.~\ref{Fig:dft}b.
Recent experiments\cite{PhysRevLett.119.086801,Gabel2021} suggest
ultra-thin SrVO$_3$ films to be VO$_2$-terminated. However, for reasons of stability (surface oxidization, surface protection, etc.), a SrO-boundary could be preferable. This might be achievable either implicitly via a SrTiO$_3$ capping-layer (see Sec.~\ref{Sec:Conclusion}) or explicitly via deposition of SrO on top of the VO$_2$ surface. The latter has in fact been achieved
for LaNiO$_3$ films on LaAlO$_3$ substrate, where both LaO and NiO$_2$ terminations are possible through an ablation of La$_2$O$_3$ and NiO, respectively\cite{Golalikhani2018}.
\subsection{Stoichiometric Mott insulator}
\label{sec:Mott}
Next, we analyze the electronic structure for the stoichiometric samples ($n=1$)
on the many-body level, using DMFT at room temperature $T=290$K:
While in the VO$_2$-terminated monolayer (\fref{fig:mott}a) the out-of-plane $xz/yz$-orbitals realize a {\it quarter-filled} Mott insulator with a gap of $0.5$eV, in the SrO terminated monolayer (\fref{fig:mott}c) the in-plane ${xy}$-orbital hosts an essentially {\it half-filled} canonical Mott insulator with a gap of $1.2$eV.
This difference can be traced back to the bare crystal-fields.
Indeed, DMFT amplifies the effect of the DFT cfs for both terminations, leading to the depletion of the energetically higher lying orbital(s), i.e., the $xy$ and the $xz/yz$ orbital(s) for the VO$_2$ and SrO termination, respectively.
This correlation-enhanced orbital polarization\cite{poter_v2o3} leads to an effectively reduced orbital-degeneracy. As a consequence, charge (inter-orbital) fluctuations are suppressed and the critical interaction for reaching the Mott state diminishes \cite{PhysRevB.54.R11026,PhysRevB.70.205116,pavarini:176403}: The Coulomb interaction is large enough to open a Mott gap in the SrO (VO$_2$) terminated monolayer with a single (two-fold) degenerate lowest orbital, while three-fold orbitally degenerate bulk SrVO$_3$ is a stable metal.
Let us note that the evidenced orbital polarization persists when including charge self-consistency, which only yields minor corrections because charge is only redistributed between orbitals, not between sites\cite{PhysRevB.94.155131}.
However, for both terminations the insulating behavior is actually driven by a {\it combination} of
the crystal-field splitting\cite{PhysRevLett.114.246401}
and the reduced band-widths\cite{PhysRevLett.104.147601}.
Whereas the crystal-field splitting is essential for the bilayer system \cite{PhysRevLett.114.246401}, we find that the bandwidth reduction alone is sufficient to drive the monolayers insulating.
We illustrate this in \fref{fig:mott}b and \fref{fig:mott}d where we take the original Hamiltonians and set the cfs artificially to zero by shifting the local orbital energies. Both systems remain insulating in DMFT. Unswayed by the cfs, however, the Mott gaps turn out smaller and orbital occupations (in both cases: $n_{\mathrm{xy},\sigma}>n_{\mathrm{xz/yz},\sigma}$) only reflect the asymmetry of the orbitals' DOS. Let us note here that if we instead keep the cfs unchanged and adjust the $xz/yz$-bandwidths such that $W_{xz/yz} = W_{xy}$ both systems remain firmly metallic.
To investigate the stability of the insulating state further, we perform DMFT calculations for various intra-orbital interaction strengths $U$. While keeping the Hund's coupling $J$ fixed to $0.75$eV,\footnote{Note that, in contrast to $U$, $J$ is hardly screened so that there is much less uncertainty and ambiguity than for $U$.} we adjust the inter-orbital interaction strength $U^\prime$ according to spherical symmetry ($U^\prime=U-2J$)\cite{PhysRevB.90.165105}.
Fig.~\ref{fig:stability} shows the orbital occupations depending on the interaction $U$:
Starting from our standard value $U=5$eV (vertical dashed line), going to larger interaction strengths simply stabilizes the insulating solution further, while also increasing the orbital polarization slightly. Smaller interaction strengths on the other hand, reduce the orbital polarization until, eventually, the insulating solution can no longer be stabilized.
This metal-to-insulator transition is, as expected within DMFT, of first order (hysteresis or coexistence regime marked in gray in \fref{fig:stability}) and manifests itself by a sudden drop of the orbital polarization.
The Mott insulating state is stable down to $U=4.5$eV ($U=4.1$eV) for the VO$_2$ (SrO) terminated monolayer.
The stability of the stoichiometric Mott insulating solution is in particular important when doping away from it, see Sec.~\ref{Sec:DMFT}. As long as the stoichiometric sample is insulating, we expect that any
variation of the interaction will have no qualitative impact on the DMFT phase diagram. A smaller on-site repulsion will merely lead to weaker orbital polarizations and shifted boundaries in the phase diagrams, \fref{fig:phasediagram}a,b.
On top of the Mott physics discussed here, weak localization through disorder may play an additional role in the insulating behavior of transport properties\cite{PhysRevB.100.155114}.
However, the suppression of the one-particle spectra for thin films\cite{PhysRevLett.104.147601}, magneto-transport results for SrVO$_3$ thin films on an LSAT substrate\cite{https://doi.org/10.1002/admi.201300126} as well as SrVO$_3$/SrTiO$_3$ supperlattices\cite{Wang2020} argue against a dominant weak localization scenario for the insulator. Similar observations have been made for CaVO$_3$ thin films on SrTiO$_3$ substrate\cite{doi:10.1063/1.4798963}.
\begin{figure}[!t!h]
\centering
\includegraphics[width=0.48\textwidth]{mottorigin_sro_vo2.pdf}
\caption{{\bf Mott insulating ground state.} DMFT spectral functions $A(\omega)$ for (a) the VO$_2$ terminated and (c) the SrO terminated structure at $U=5$eV and room temperature ($T=290$K). In both cases a wide Mott gap forms which is accompanied by a strong orbital polarization.
Removing the crystal-field splitting, the reduced bandwidth alone results in a Mott insulator (b,d) with a slightly smaller band gap.
}
\label{fig:mott}
\end{figure}
\begin{figure}[!t!h]
\centering
\includegraphics[width=0.46\textwidth]{mottstability_sro_vo2.pdf}
\caption{{\bf Stability of Mott insulating state.}
Spin-dependent orbital occupation $n_{i\sigma}$ vs.~intra-band interaction strength $U$ at room temperature $T=290$K; Hund's coupling $J=0.75$eV; inter-band interaction $U^\prime = U-2J$. (a) The VO$_2$ terminated monolayer is effectively a two-orbital quarter-filled system ($n_{xz/yz,\sigma}\sim 0.25$) while (b) the SrO terminated monolayer becomes effectively a half-filled one-orbital system ($n_{xy,\sigma}\sim 0.5$) at large enough interaction strengths. Both lead to a Mott localization of carriers which can be upheld even if we reduce the interaction. The transition to the metallic solution is accompanied by a tight hysteresis after which the orbital polarization drops rapidly. The calculations under doping in the next figure are performed for $U=5$eV (vertical, dashed black line).}
\label{fig:stability}
\end{figure}
\subsection{Behavior under doping}
\label{Sec:DMFT}
\begin{figure}[!t]
\includegraphics[width=0.47\textwidth]{orbitaloccupation_sro_vo2.pdf}
\caption{{\bf System trends with doping.} Spin-dependent orbital occupation at $T=290$K, resolved for the $i=xy$ and $xz$/$yz$-orbitals.
(a) The VO$_2$ terminated system around nominal filling ($n = 1$)
can be effectively described by two quarter-filled (xz/yz) orbitals.
(b) The SrO terminated system at and below nominal filling can effectively be described by a single half-filled (xy) orbital.
The former explains the tendency for checkerboard orbital ordering, while the latter promotes antiferromagnetism in Fig.~\ref{fig:phasediagram}.
The shaded areas around undoped SrVO$_3$\ represent the doping levels
at which we find these checkerboard and antiferromagnetic orderings in DMFT at this temperature.
Inset: Semi-empirical condition where stripe orbital ordering emerges: If we frustrate the local site enough, we find a transition from checkerboard to stripe orbital-order (indicated by a change in background color)}
\label{fig:occupation}
\end{figure}
We now discuss the electronic structure of the doped monolayers in their
non-symmetry broken phases. From the information of orbital occupations and degeneracies, we motivate possible ordering instabilities (that will then be quantitatively assessed in Sec. \ref{Sec:DMFTsusc}).
First, the stoichiometric insulating state in \fref{fig:mott} and the various orbital occupations in \fref{fig:occupation}, indicate that
our systems are asymmetrical with respect to doping with electrons ($n>1$) or holes ($n<1$).
The VO$_2$ terminated monolayer (\fref{fig:occupation}a) somewhat upholds its orbital polarization when holes are introduced to the system. Such orbital occupations of the bipartite lattice
system make the system prone to a staggered, checkerboard orbital ordering (cOO) \cite{Held1998,PhysRevB.58.R567}, as two orbitals may now be occupied alternately on neighboring lattice sites. This is energetically favorable, since a nearest-neighbor hopping then results in a state where different orbitals are occupied so that a (virtual) hopping process costs only $U'=U-2J$ instead of $U$.
Doping with more holes, see \fref{fig:occupation}a, the VO$_2$-terminated monolayer
quickly moves away from quarter-filling by redistributing electrons from the $xz/yz$-orbitals into the $xy$-orbital,
As we shall see in Sec.~\ref{Sec:DMFTsusc}, before OO is fully suppressed
upon hole doping the ordering vector changes to stripe orbital ordering (sOO)
above a particular filling of the so far auxiliary $xy$-orbital.
Electron doping on the other hand maintains the quarter-filled state for longer, where for fillings up to $n\sim 1.2$ the additional electrons solely occupy the $xy$-orbital. Only above $n\sim 1.3$ we see a coincidental increase of all orbital occupations,
again disfavoring orbital order.
The SrO-terminated monolayer (\fref{fig:occupation}b) is even more asymmetrical: Introducing holes does not affect the effective one-band description of the system and the sparsely filled $xz/yz$ orbitals remain almost depleted.
More interesting is the electron-doped side, where the multi-orbital
character is promoted.
In this electron-doped regime, the Hund's coupling $J$ will promote a parallel spin alignment of the electrons in the three orbitals. It is natural to expect that the hopping transfers this local spin alignment into an FM order on the lattice, but other orders such as OO may emerge here as well \cite{Tokura2000}.
For strong Coulomb interactions and in an insulting state, these competing phases can be understood by superexchange as in the classical Kugel-Khomskii spin-orbital models \cite{KHOMSKII1973763}.
These phases have also been found in early DMFT calculations for a two-band model \cite{Held1998} and an oversimplified Stoner criterion predicts FM order of
the $m$-fold degenerate Hubbard model for
$A(0)\left(U+(m-1)J\right)\geq 1$\cite{Fazekas_book}.
At extremely large dopings around $n\sim 1.5$, the physics changes once again: The system now consists of three quite equally filled orbitals where the $xz/yz$ orbitals approach quarter-filling. Similarly to stoichiometric filling in the VO$_2$ terminated monolayer, such degenerate quarter-filled orbitals may lead again to orbital ordering.
\subsection{DMFT susceptibilities}
\label{Sec:DMFTsusc}
We now put the above analysis of potential ordered phases on firm footing:
For the prevailing magnetic and orbital orders,
\fref{fig:suscq}a and \fref{fig:suscq}b displays the relevant DMFT susceptibilities at temperatures above the respective instabilities.
Maxima in the shown susceptibilities indicate type and $\mathbf{Q}$-vector
of the dominant fluctuations.
\fref{fig:suscq}c illustrates, for selected examples, the critical behavior of the (inverse) susceptibilities and (inverse) correlation lengths
emerging when said maxima turn into instabilities.
Magnetic instabilities occur where the static susceptibility in the magnetic channel
\begin{equation}
\chi_m(\vek{Q}
= g^2 \sum_{ij,ll^\prime}e^{i\mathbf{Q}(\mathbf{R}_i-\mathbf{R}_j)}\int d\tau
\left\langle T_\tau S^z_{il}(\tau)S^z_{jl^\prime}(0) \right\rangle
\label{eq:suscmagn}
\end{equation}
diverges at a critical temperature, indicated by the intercept of $\chi^{-1}(\mathbf{Q})$ with the temperature axis in \fref{fig:suscq}c.
In \eref{eq:suscmagn}, $g$ is the Land\'e factor; $i,j$ are indices for the lattice sites ${\mathbf R}_{i(j)}$; $l,l^\prime$ are
orbital indices. The $z$-component of the spin operator $S_{il}^z=(n_{il \uparrow}-n_{il \downarrow})/2$ is expressed in terms of the
number operator $n_{il\sigma}$ for an electron on site $i$ in orbital $l$ with spin $\sigma$.
Ferromagnetism (FM) and antiferromagnetism (AF) correspond to the usual ordering vectors $\vek{Q}=(0,0)$ and $\vek{Q}=(\pi,\pi)$, respectively. The incommensurate magnetism (iM), that we find for the VO$_2$-termination, corresponds to an ordering vector $\vek{Q}$ with fixed length $\left|\vek{Q}\right| = \delta \geq 0$, see \fref{fig:suscq}a and \fref{fig:incomm}.
We now assess the instabilities resulting in the phase diagrams, \fref{fig:phasediagram}a,b.
{\it Antiferromagnetic} (AF) order from super-exchange is facilitated by effectively half-filled orbitals. For the $d^1$ configuration of SrVO$_3$, only the SrO-terminated monolayer provides this favorable condition. Indeed, there, the positive crystal-field
realizes a half-filled, Mott insulating $xy$-orbital that then hosts AF order,
see the diverging susceptibility in \fref{fig:suscq}(c).
Note that AF order was also predicted for a SrO-terminated SrRuO$_3$ monolayer on SrTiO$_3$ around nominal stoichiometry.\cite{Liang_SRO}
There, the $d^4$ configuration results in an essentially fully occupied $xy$-orbital, and the staggered moment is instead carried by half-filled $xz/xz$ orbitals.
In both cases, doping with either electrons or holes suppresses the AF state.
Doping the SrVO$_3$ monolayer with either termination towards their respective van-Hove singularities, i.e., hole (electron) doping for the VO$_2$ (SrO) terminated sample (see \fref{Fig:dft}) results in a strongly increased spectral density around the Fermi level
within DMFT.
Concomitantly doping
generates an orbital configuration where all involved orbitals are close to equally filled, promoting energy minimization through Hund's exchange $J$ and therefore a parallel alignment of the involved spins.
This situation, leading to {\it ferromagnetism} (FM), is found around $n=0.7$ in the VO$_2$-terminated and near $n=1.3$ in the SrO-terminated sample. In both cases, ferromagnetism is hosted by the degenerate $xz$ and $yz$-orbitals.
Subleading non-local AF fluctuations are, however, still present in the $xy$-orbital of the SrO-terminated system. Indeed, an antiferromagentic stripe pattern, $\vek{Q}=(0,\pi)$, and symmetrically related at $\vek{Q}=(\pi,0)$, appears, indicated by additional local maxima in the susceptibility, see \fref{fig:suscq}b. Quite notably, in the absence of Hund's rule coupling FM fluctuations are strongly suppressed and said frustrated AF spin-fluctuations would be on par with them (additional data, not shown).
Moreover, we also find incommensurate magnetic order in the VO$_2$-terminated system around $n=1.3$ in the $xz/yz$-orbitals (iM at $n=1.3$ in \fref{fig:suscq}a). There, instead of a specific ordering vector $\vek{Q}$ the magnetic susceptibility is maximal for all vectors $\vek{Q}$ with origin $(0,0$) and a length of $\delta = \left|\vek{Q}\right| \geq 0$, i.e., roughly a circle in the $\mathbf{q}$-plane. Upon lowering temperature, $\delta$ increases
and in close vicinity to the ordered phase anisotropy develops; the maximum susceptibility within the circle is found at $\vek{Q}=(\pm\delta,\pm\delta)$, see \fref{fig:incomm}.
These clear maxima suggest that a kind of frustrated ferromagnetism develops where the $xy$-orbital disturbs the alignment of the $xz/yz$-orbitals. Doping beyond $n=1.3$ further increases $\delta$ (data not shown). Let us note here that throughout the phase diagram we did not find any magnetic instabilities supported by Fermi surface nesting.
\begin{figure}[!t]
\includegraphics[width=0.47\textwidth]{susceptibilities_criticality_sro_vo2.pdf}
\caption{{\bf DMFT susceptibilities and criticality.} Momentum-dependence of the susceptibility $\chi_r(\vek{q})$ for (a) the VO$_2$ terminated and (b) the SrO terminated SrVO$_3$ monolayer in the vicinity of the respective phase transitions.
The dominating component can be either found directly in the magnetic channel (r=`m', Eq.\eqref{eq:suscmagn}) or be obtained via a linear orbital combination of the density channel (r=`d', Eq.\eqref{eq:suscoo}).
(c) Temperature-dependence of the inverse DMFT susceptibility $\chi_r^{-1}(\vek{Q})$ (first diverging $r$, $\vek{Q}$ at selected dopings; circles, left axis) for selected points from (a) and (b); lines are linear fits.
Intersections with the $T$-axis denote the transition temperature for the respective order.
On the secondary (right) axis the corresponding inverse correlation lengths $\xi_r^{-1}$ are shown (squares, right axis); dashed lines are fits to mean-field behavior.}
\label{fig:suscq}
\end{figure}
\begin{figure}[!t]
\includegraphics[width=0.48\textwidth]{incommensurate_magnetism.pdf}
\caption{{\bf Incommensurate magnetism (iM).} Left: magnetic susceptibility $\chi_m(q)$ along the high-symmetry path $\Gamma\rightarrow \mathrm{M}$ for the VO$_2$-terminated monolayer at $n=1.3$ at various temperatures. Right: magnetic susceptibility $\chi_m(q)$ for $T=170$K in the planar Brillouin zone. The susceptibility is maximal roughly on a circle centered around $\mathbf{q}=(0,0)$. Upon lowering temperature the maximum of the susceptibility moves to larger $\mathbf{q}$-vectors. Close to the transition temperature we find anisotropy on this circle where the susceptibility is clearly maximal (purple) for $\mathbf{q} = \left(\pm\delta,\pm\delta\right)$.}
\label{fig:incomm}
\end{figure}
\medskip
The other prevalent type of instabilities we find are of {\it orbital-ordering} type between the degenerate $xz/yz$ orbitals and can be monitored in the density channel
\begin{eqnarray}
\label{eq:suscoo}
\chi_d^{xz/yz}(\vek{Q}) & = & \sum_{ij \sigma \sigma'}e^{i\mathbf{Q}(\mathbf{R}_i-\mathbf{R}_j)}\int d\tau \times\\
&\times&\left\langle T_\tau
(n_{i\, \mathrm{xz}\, \sigma}-n_{i\, \mathrm{yz} \,\sigma})(\tau)(n_{j\, \mathrm{xz}\, \sigma'}-n_{j\, \mathrm{yz}\, \sigma'})(0)
\right\rangle\nonumber.
\end{eqnarray}
Towards quarter-filling ($n_{i,\sigma}=0.25$) of the $xz$/$yz$-orbitals, i.e., at and around stoichiometric filling in the VO$_2$-terminated monolayer and around $n\sim 1.5$ in the SrO-terminated setup, the wave-vector of critical fluctuations is firmly $\vek{Q}=(\pi,\pi)$. As previously alluded to, this leads to a {\it checker-board orbital-order} (cOO), consistent with model expectations \cite{Held1998,PhysRevB.58.R567}.
We note that the $xy$-orbital does not participate in the ordering, as signaled
by susceptibility enhancements being confined to components of the other two orbitals.
The $xy$ orbital can also be passive at larger fillings or valencies:
With one electron more, a t$_{2g}^2$ OO---with $xy$-orbitals near half-filling and one electron alternatingly in the $xz$ and the $yz$-orbital---can occur in YVO$_3$ (LaVO$_3$) if Y (La) ions are partially replaced by Ca (Sr) \cite{PhysRevX.5.011037}, cf.\ Refs.~\onlinecite{Khaliullin2001,Ren2003}.
An OO with all three $t_{2g}$-orbitals participating on the other hand is highly frustrated for a cubic lattice \cite{Khaliullin2000}.
Due to the strong asymmetry around nominal filling in our monolayers, we also find a strong asymmetry of the corresponding cOO-dome in \fref{fig:phasediagram}a, where the cOO transition temperature even increases upon electron-doping. If we move too far away from ideal quarter-filling, the ordering temperature is suppressed rapidly.
Despite this suppression of cOO we find an additional emerging ordering for $n\sim0.9$ in \fref{fig:phasediagram}a. The corresponding ordering can again be described via Eq.~\eqref{eq:suscoo}
with, however, a characteristic vector
$\vek{Q}=(0,\pi)$ (and $\vek{Q}=(\pi,0)$ related via symmetry), describing {\it stripe orbital-ordering} (sOO).
The cOO-to-sOO transition under hole-doping is not realized by a continuous move of the ordering vector from $\vek{Q}=(\pi,\pi)$ to $\vek{Q}=(0,\pi)$.
Instead, increased hole-doping suppresses cOO while simultaneously promoting sOO.
We conjecture that this transition can be ascribed to the `auxiliary' $xy$-orbital,
that does not contribute to the susceptibility enhancements of either fluctuations.
Illustrated in the inset of \fref{fig:occupation} we find a semi-empirical condition
that links the preference for stripe over checkerboard orbital order to the filling of the $xy$-orbital:
$n_{\mathrm{xy},\sigma} = (0.25-n_{\mathrm{xz/yz},\sigma})/2)$ below which checkerboard ordering and above which stripe ordering is preferred by the system. Effectively, enough $d_{xy}$ occupation frustrates the local site enough for stripe ordering to be energetically favorable. As for the electron-doped side, the orbital-ordering domes in both systems disappear when doping too far away from quarter-filling.
\section{Discussion}
\label{Sec:Discussion}
Mapping out all these different magnetic and orbital-ordering instabilities and their corresponding critical temperatures yields the phase diagrams in \fref{fig:phasediagram}.
Naturally, the DMFT long-range order exhibits mean-field criticality (Gaussian fluctuations), i.e., the critical exponents are $\gamma=1$ for the susceptibility $\chi$, and $\nu=0.5=\gamma/2$ for the correlation length $\xi$, see \fref{fig:suscq}.
We note that in strictly two dimensions, due to the Mermin-Wagner theorem \cite{Mermin1966}, long-range order can only set in at $T=0$. As a spatial mean-field theory, DMFT does not verify this constraint, while it is captured in, e.g., D$\Gamma$A\cite{Rohringer2011,Schaefer2020}.
In an experimental setting, however, perturbations by disorder (oxygen vacancies, etc.), spin-orbit coupling, single-ion anisotropy, surface adatoms, as well as tunneling into the substrate render strict 2Dness obsolete. This allows for phase transitions at finite temperatures.
Still, non-local correlations beyond DMFT\cite{RevModPhys.90.025003,Tomczak2017review} are expected to attenuate the tendency for long-range order with respect to our DMFT solution.
Non-local fluctuations may even drive pseudogaps, as shown for antiferromagnetic fluctuations in the one-band Hubbard model\cite{Schaefer2015-2,Schaefer2015-3,Gukelberger2016,Schaefer2020}.
In \fref{fig:phasediagram} we further indicate that instead of chemical doping, the phase diagram can be perused by applying a gate voltage $V_G$. We note that the estimated values of $V_G$ measure the necessary potential directly in the monolayer, not the truly external one applied at a certain distance through a dielectric medium. Realizing the whole phase diagram with a sheet carrier change of 0.5
electrons/unit cell ($3\cdot10^{14}$ electrons/cm$^2$) in a single device
might be challenging\cite{RevModPhys.78.1185}. Ionic electrolytes as dielectrics, however, allow for such a large amount of induced charges thanks to the large capacitance
of polarized ions\cite{Katase3979}. Experimentally, the phase diagram \fref{fig:phasediagram} can hence be realized either by doping or via gate voltage, or a combination of both.
Finally, let us comment on an important difference between our semi-{\it ab initio} calculations and typical model setups.
In multi-orbital (Hubbard) {\it models}, orbital complexity is often simplified, in the sense that the hopping of each orbital is considered to be {\it isotropic} vis-à-vis all neighboring atoms. Many model studies even employ semi-circular densities of states, corresponding to the Bethe lattice with infinite coordination number.
Such simplifications allowed distilling essential behaviors
of, e.g., correlation enhancements of crystal-fields\cite{PhysRevB.78.045115},
the influence of Hund's physics on the Mott transition \cite{PhysRevLett.107.256401}, or correlations in band-insulators \cite{PhysRevB.80.155116,NGCS}.
Indeed, qualitative {\it spectral properties} of 3D systems without broken symmetries,
are mainly controlled
by the orbitals' filling and the kinetic energy they mediate.
Near symmetry-broken phases, however, the associated fluctuations
in {\it physical susceptibilities} are strongly dependent on electronic-structure details. In fact, van-Hove singularities, nesting, or Kohn points may well be the cause of an instability.
In our context, besides anisotropies, e.g., induced by the tetragonal distortion\cite{Benjamin_2D3D}
and the obvious geometric restriction,
a crucial ingredient is the {\it per se} 2D nature of the $t_{2g}$-orbitals:
In perfectly 3D cubic perovskites (e.g., bulk SrVO$_3$) each transition-metal $t_{2g}$ orbital only hybridizes with four of the six oxygen ions of the coordination octahedron.
Therefore, both in bulk and ultra thin films, the in-plane $d_{xy}$ DOS is 2D-like.
The loss of hopping along the $z$-direction in ultra-thin films
further reduces the effective dimensionality, leading to the 1D-like DOS of the $d_{xz,yz}$ orbitals, see \fref{Fig:dft}. %
Our analysis suggests, that even for qualitative phase diagrams of ultra-thin oxide films or heterostructures, both, the crystal geometry and the orbital structure have to be accounted for.
\section{Conclusion and outlook}
\label{Sec:Conclusion}
\begin{figure}[!t]
\includegraphics[width=0.4\textwidth]{orbitaloccupation_nxy_vs_nxz_square.pdf}
\caption{{\bf Global phase diagram of SrVO$_3$ monolayers on SrTiO$_3$.} DMFT orbital occupations of each setup are mapped into a $n_{xy}$ vs.\ $n_{xz} + n_{yz}$ graph. Due to the reduced bandwidth in the monolayer on a SrTiO$_3$ substrate, any orbital occupation repartitioning
of the nominal $n=1$ filling (orange line) realizes a Mott insulator. SrO-terminated (black) and VO$_2$-terminated (gray) monolayers and their dominant non-local fluctuations (shaded background; colors identical to \fref{fig:phasediagram}, but lighter) are discussed in this publication. As an outlook we showcase the effect of embedding a SrVO$_3$ monolayer in a SrTiO$_3$ sandwich (navy blue, dashed): The crystal-field and orbital structure is somewhat intermediate to the two uncapped monolayers. The computation of respective instabilities is left for future work.
}
\label{fig:orbmap}
\end{figure}
We have studied a single layer of SrVO$_3$ on a SrTiO$_3$ substrate, using DFT and DMFT, considering both possible surface terminations, VO$_2$ and SrO, to vacuum.
We demonstrated that stoichiometric samples ($n=1$) are
bandwidth-controlled Mott insulators:
Depending on the surface termination, SrO or VO$_2$, the monolayer is an effectively half-filled one-orbital or a quarter-filled two-orbital Mott insulator.
We showed this orbital polarization to derive from the crystal-field splitting
having opposite signs for the two terminations and to be significantly enhanced by electronic correlations.
Electron or hole-doping reveals multi-orbital effects:
For the SrO-termination, AF-fluctuation are dominant around nominal filling. Doping with electrons populates the $xz/yz$-orbitals; they order ferromagnetically ($n\sim1.3$) or realize checkerboard orbital orbital order ($n\sim1.5$).
For the VO$_2$ termination checkerboard $xz/yz$ orbital-order already dominates around nominal filling. Doping then instead promotes the $xy$-orbital which acts as a mediator for ferromagnetism and stripe orbital-order on the hole-doped side and incommensurate magnetism
on the electron-doped side.
While the change in magnetic fluctuations and orders could be observed in neutron experiments, experimentally evidencing the orbital fluctuations is only possible indirectly: the staggered pattern of $xz$ and $yz$-orbitals will result in a dynamic (potentially static) alternation of the bond-length in the $x$ and $y$ direction, possibly detectable in future x-ray measurements.
In all, the orbital polarization is the essential driver of the phase diagram of the SrVO$_3$ monolayer on SrTiO$_3$.
We therefore summarize our results in Fig.~\ref{fig:orbmap} in form of
an orbital occupation map.
The considered surface terminations each realize, under doping, a characteristic trajectory in the $n_{xy}$ vs.\ $n_{xz}+n_{yz}$ space.
As an outlook, we include a third possibility---a SrVO$_3$ monolayer with SrTiO$_3$ on both sides: At nominal filling this sandwich is, again, a Mott insulator. However, owing to the symmetric embedding, the crystal-field is
minute (but positive).
The computation of ordering instabilities of capped ultra-thin films, in which quantum confinement effects could be studied in a more controlled fashion, is left for future work.
A different future avenue are even more realistic setups of the current geometries, e.g., atomic position relaxation with the inclusion of correlation effects and recent advances\cite{PhysRevLett.112.146401, PhysRevB.102.245104} which allow for calculations of forces and phonons within DFT+DMFT to test the dynamical stability against superstructure formations.
\begin{acknowledgments}
We thank R. Claessen, M.\ Fuchs, J. Gabel, A.\ Galler, G.\ Sangiovanni, M. Sing, P.\ Thunstr\"om and Z.\ Zhong for fruitful discussions.
The authors acknowledge support from the Austrian Science Fund (FWF) through grants P 30819, P 30997, P 32044, and P 30213. Calculations were performed in part on the Vienna Scientific Cluster (VSC).
\end{acknowledgments}
|
1,314,259,993,618 | arxiv | \section{Introduction}
The transverse field Ising antiferromagnet on the triangular lattice, with Hamiltonian,
\begin{equation}
H_{\rm Ising} = J_{1}\sum_{\langle \vec{R} \vec{R}'\rangle }\sigma^z_{\vec{R}} \sigma^z_{\vec{R}'} -\Gamma \sum_{\vec{R}}\sigma^x_{\vec{R}} -B \sum_{\vec{R}} \sigma^z_{\vec{R}}\; ,
\label{Hising}
\end{equation}
where $\vec{\sigma}_{\vec{R}}$ are Pauli matrices representing $S=1/2$ moments on sites $\vec{R}$ of the triangular lattice, $\langle \vec{R} \vec{R}'\rangle$ denote the nearest neighbour links of the triangular lattice, $J_1>0$ is the antiferromagnetic exchange among easy-axis components of the $S=1/2$ moments (a factor of $\frac{1}{4}$, appropriate for
$S=1/2$ moments, has been absorbed in the definition of $J_1$), and $B$ and $\Gamma$ are components of the external magnetic field along the easy axis $\hat{z}$ and transverse direction $\hat{x}$ respectively (a factor of $\frac{g\mu_B}{2}$, appropriate for $S=1/2$ moments, has been absorbed in the definition of these field components), provides perhaps the simplest example of a quantum ``order-by-disorder''~\cite{Villain,Moessner_Sondhi} effect, whereby a classical spin liquid develops long-range magnetic order upon the introduction of terms in the Hamiltonian that induce quantum fluctuations.
When $\Gamma=0$, the zero temperature classical Ising antiferromagnet at $B=0$
has a macroscopic degeneracy of minimum exchange-energy configurations on the triangular lattice. These are in correspondence with all dimer covers of the dual honeycomb lattice,
implying that the entropy-density remains nonzero in this classical zero temperature limit.\cite{Wannier,Houtappel} At non-zero temperature, thermal fluctuations of $\sigma^{z}$ allow for defects that take the system out of the minimum exchange-energy dimer subspace.
The Ising spins remain in a paramagnetic state all the way down to $T=0$,\cite{Wannier,Houtappel} albeit with a diverging correlation length\cite{Stephenson} at the three-sublattice wavevector ${\bf Q}$. This provides a simple example of classical spin
liquid behaviour, with the $T=0$ limit characterized by power-law spin correlations
at the three-sublattice wavevector ${\bf Q}$.
A transverse field $\Gamma$ that couples to $\sigma^x$ induces quantum fluctuations of
the Ising spins $\sigma^z$, and would ordinarily be expected to further reduce any residual ordering tendency of the Ising spins. However, in reality, these quantum fluctuations immediately stabilize a ground-state
with long-range three-sublattice order of $\sigma^z$ for any nonzero $\Gamma$. In contrast to the {\em ferrimagnetic} three-sublattice order exhibited by the classical Ising antiferromagnet with ferromagnetic further neighbour couplings\cite{Landau}, the $\Gamma >0$ ground state is characterized by {\em antiferromagnetic} three-sublattice order,\cite{Isakov_Moessner} {\em i.e.}, the modulation of $\langle \sigma^z\rangle$ at wavevector ${\mathbf Q}$
is not accompanied by any net ferromagnetic moment. At $T=0$, this three-sublattice ordered persists up to a critical value $\Gamma_{c}\approx 1.7$\cite{Isakov_Moessner} (in units of $J_1$), beyond which the system becomes a quantum paramagnet in which
the spins are polarized in the $\hat{x}$ direction.\cite{Moessner_Sondhi_Chandra,Moessner_Sondhi,Isakov_Moessner} When the system is heated to nonzero temperatures
above this three-sublattice ordered ground state, the three-sublattice order
melts via an intermediate-temperature phase characterized by power-law order: $\langle \sigma^z(\vec{R}) \sigma^z(0)\rangle \sim \cos({\mathbf Q}\cdot \vec{R}) /|\vec{R}|^{\eta(T)}$ for $T \in (T_{1},T_{2})$, with a temperature-dependent power-law exponent $\eta(T)$
that is expected\cite{Jose_Kadanoff_Kirkpatrick_Nelson} to increase from $\eta(T_1)=1/9$ to $\eta(T_2)=1/4$.\cite{Isakov_Moessner}
A recent field-theoretical analysis\cite{Damle} predicts that the {\em ferromagnetic easy-axis susceptibility} $\chi_{u}(B)$ to the uniform longitudinal field $B$ along the easy-axis diverges at small $B$ in
a large portion of such power-law ordered phases associated with the two-step melting
of three-sublattice order in frustrated easy-axis antiferromagnets with triangular lattice symmetry: $\chi_{u}(B) \sim |B|^{-\frac{4 - 18 \eta}{4-9\eta}}$ for $\eta(T) \in (1/9,2/9)$.
For the specific case of the transverse field Ising antiferromagnet on the triangular lattice, this is a rather counter-intuitive prediction:
The Ising spins in Eq.~\eqref{Hising} have no ferromagnetic couplings, and ferromagnetic correlations remain short-ranged in the low-temperature phase with long-range three-sublattice
ordered phase. Yet, the prediction is for the uniform easy-axis susceptibility to start diverging once this three-sublattice order melts partially due to thermal fluctuations.
Our goal here is to test this general prediction using the test-bed provided by the transverse
field Ising antiferromagnet (Eq.~\eqref{Hising}) on the triangular lattice. In order to do this, we need to obtain an accurate characterization of the long-distance form of the correlations of
the easy-axis magnetization density as well as correlations of the three-sublattice
order parameter for this model. These can be used to obtain
the finite-size easy-axis susceptibility $\chi_{u}(L)$ of the Ising spins, as well as the
value of $\eta(T)$ for a range of temperatures in the power-law ordered phase.
If the easy-axis susceptibility is indeed singular as predicted, then standard finite-size scaling arguments imply that $\chi_{u}(L) \sim L^{2-9 \eta}$
for $\eta(T) \in (1/9,2/9)$ in the power-law ordered phase. In this paper, we test this form of the prediction using
a newly developed quantum-cluster algorithm\cite{Biswas_Rakala_Damle} that provides an efficient tool
for performing Quantum Monte Carlo simulations of frustrated transverse field
Ising models within the Stochastic Series Expansion\cite{Melko,Sandvik_PRE} framework.
The rest of this paper is organized as follows: In Section.~\ref{PhasesandTransitions}, we discuss the antiferromagnetic nature of the three-sublattice order induced by the transverse field and contrast it with the ferrimagnetic three-sublattice ordered phase established by additional ferromagnetic couplings.
We also review the standard Landau theory framework used for describing this kind of
long-range order, and use it to discuss the possible theoretical scenarios for the phase transition between these two phases.
In Section.~\ref{Methods} we provide a brief sketch of the actual
computational method used to obtain our numerical results. In Section.~\ref{Results}, we
summarize our results for the uniform magnetization density as well as the
three-sublattice order parameter and compare them with the field-theoretical
predictions alluded to earlier.
\section{Phases and transitions}
\label{PhasesandTransitions}
The \emph{antiferromagnetic} (with no net easy-axis magnetic moment) three-sublattice
order exhibited by the $\Gamma > 0$ ground state of $H_{\rm Ising}$ can be thought
of in terms of the following useful caricature: Ising spins on one spontaneously chosen
sublattice (out of the three sublattices corresponding to the natural tripartite decomposition of the triangular lattice) freeze into the $\lvert \sigma^{x}=+1\rangle$ state. Equivalently,
one may think of them as fluctuating freely between the $\lvert \sigma^{z}=+1\rangle$ and $\lvert \sigma^{x}=-1\rangle$ states due to the effects of quantum fluctuations. On the other two sublattices of the triangular
lattice, the system orders antiferromagnetically, with spins on one sublattice pointing up
along the $\hat{z}$ axis, and spins on the other sublattice pointing down. This is also
the picture for the long-range ordered phase that persists up to the lower-critical
temperature $T_1(\Gamma)$ that marks the onset of the power-law ordered intermediate
phase associated with the two-step melting of three-sublattice order.
On incorporating an additional next-neighbour ferromagnetic coupling $J_{2}<0$, the antiferromagnetic three-sublattice order of the low-temperature phase gives way to \emph{ferrimagnetic} (with net easy axis moment) three-sublattice order beyond a
non-zero threshold value $J_{2c}$.\cite{Biswas_Rakala_Damle} This is because
the classical ($\Gamma=0$) model with $J_{2} < 0$ is known to develop ferrimagnetic
three-sublattice order beyond a $T=0$ threshold at which $\sigma^z$ have power-law
correlators $\langle \sigma^z(\vec{R})\sigma^z(0) \rangle \sim \cos({\mathbf Q} \cdot \vec{R})/|R|^{\eta}$ with $\eta=1/9$.\cite{Nienhuis_Hilhorst_Blotte}
This ferrimagnetic three-sublattice order can be understood in terms of
the following caricature: The system spontaneously chooses one sublattice on which the
spins all point along the $+\hat{z}$ direction ($-\hat{z}$ direction), while the spins
on the other two sublattices all point along the $-\hat{z}$ direction ($+\hat{z}$ direction).
With this picture of the low temperature phases in mind, we focus our
attention on the uniform easy axis magnetization $m$ and the complex three-sublattice order parameter $\psi$, defined as
\begin{align}
\label{orderparameters}
&m=\frac{1}{L^{2}}\sum_{\vec{R}}\sigma^{z}_{\vec{R}} \\
&\psi=\frac{1}{L^{2}}\sum_{\vec{R}}\sigma^{z}_{\vec{R}}\exp(i\mathbf{Q} \cdot \vec{R})
\end{align}
where $\mathbf{Q}$
is the three-sublattice ordering wave vector ($(2\pi/3,2\pi/3)$ in the standard basis) and $\vec{R}$ represents the coordinates
of triangular lattice sites. In the standard Landau-Ginzburg approach\cite{Domany_Schick_Walker_Griffiths,Domany_Schick,Alexander} to thermal
(nonzero temperature) phase transitions involving such three-sublattice ordered states, the physics of three-sublattice ordering is represented in terms of a classical order parameter field $\psi_{\mathrm{cl}}$, which may be identified with the static (Matsubara frequency $\omega_n = 0$) part of the $\psi$ operator defined above:
\begin{equation}
\psi_{\mathrm{cl}}=\frac{1}{\beta}\int_0^\beta d\tau \psi(\tau)
\label{OPDEFN}
\end{equation}
Here, we used the usual notation for the imaginary-time analog of Heisenberg operators, $\mathcal{O}(\tau)=e^{\tau H_{\rm{TFIM}}}\mathcal{O}e^{-\tau H_{\mathrm{TFIM}}}$, corresponding to any Schr\"{o}dinger operator $\mathcal{O}$.
In this Landau-Ginzburg framework, the free energy is written as an integral over
a coarse-grained free-energy density $\mathcal{F}(\psi_{\mathrm{cl}})$ that admits
an expansion in powers and gradients of a coarse-grained order-parameter field $\psi_{{\mathrm{cl}}}(\vec{r})$ which may be thought of as a local version of the order parameter
defined in Eq.~\ref{OPDEFN}. Keeping various
low-order terms consistent with the action of various symmetries of the microscopic
Hamiltonian, one writes:
\begin{align}
\mathcal{F}(\psi_{\mathrm{cl}})=\kappa \lvert \nabla \psi_{\mathrm{cl}} \rvert^2 + r\lvert \psi_{\mathrm{cl}}\rvert^{2}+u_4\lvert \psi_{\mathrm{cl}}\rvert^{4} \nonumber\\
+ u_6\lvert \psi_{\mathrm{cl}}\rvert^{6} + \lambda_{6}\lvert \psi_{\mathrm{cl}}\rvert^{6}\cos(6\theta) \nonumber \\
+\lambda_{12}\lvert \psi_{\mathrm {cl}}\rvert^{12}\cos(12\theta)+\ldots
\label{LGW}
\end{align}
where, $\theta(\vec{r})$ is the phase of the complex order parameter field $\psi_{cl}(\vec{r})$.
As usual, one assumes that the coefficients of various terms in this phenomenological free-energy are smooth functions of microscopic parameters. In this approach, three-sublattice ordering corresponds to $r<0$. The sign of $\lambda_{6}$ determines the nature of three-sublattice ordering : $\lambda_{6}>0$ favors antiferromagnetic ordering with the phase $\theta$ pinned at $(2n+1)\pi/6$ ($n = 0,1\dots 5$), while $\lambda_{6}<0$ favors ferrimagnetic ordering with the phase pinned at $(2n)\pi/6$ ($n=0,1\dots 5$). The $\lambda_{12}$ term is not expected to be important except when $\lambda_6$ is driven to the vicinity of zero by the competition between further-neighbour ferromagnetic couplings ( in the microscopic Hamiltonian ) that favour ferrimagnetic three-sublattice ordering, and other effects (such as quantum fluctuations induced by a transverse field) that favour antiferromagnetic three-sublattice ordering.
If fluctuations of $\theta$, the phase of the order parameter, play a dominant role in driving
the transition to a paramagnetic high-temperature state, one expects a phase-only
description to capture the long-wavelength properties near such a transition. In other words, one then expects that $|\psi_{\mathrm{cl}}|$, the amplitude of the order parameter, remains
nonzero near the transition (corresponding to $r<0$), and the physics of the transition
is controlled by the interplay between the effective phase-stiffness $\kappa |\psi_{\mathrm{cl}}|^2$ and the six-fold anisotropy
$\lambda_6$. This gives rise to the expectation\cite{Domany_Schick_Walker_Griffiths,Domany_Schick,Alexander} of critical behaviour in the universality
class of the six-state clock model\cite{Jose_Kadanoff_Kirkpatrick_Nelson,Challa_Landau,Cardy} of statistical mechanics.
As is well-known, two-dimensional six-state clock models represent an
unusual example of a system which can display a variety of critical behaviours, each
of which is a generic possibility that can be realized for a range of microscopic parameters.\cite{Cardy} Of particular interest in the present context is the possibility of a two-step
melting transition, whereby the low-temperature phase with long-range order in
$\exp(i\theta)$ is separated from a high-temperature paramagnetic phase by an intermediate
phase with power-law order in $\exp(i\theta)$. As is well known, this
power-law ordered phase is controlled by a line of Gaussian fixed points\cite{Jose_Kadanoff_Kirkpatrick_Nelson} with
effective free-energy density $\frac{1}{4\pi g}\int d^{2}r(\nabla \theta)^{2}$.
For $g\in(1/9,1/4)$, the six-fold anisotropy $\lambda_6$ and the vorticity
in the $xy$ field $\theta$ are both irrelevant perturbations of this fixed-point free-energy
density, which controls the long-wavelength behaviour of order parameter correlations
in the intermediate power-law ordered phase. The continuously varying power-law
exponent $\eta(T)$ for order parameter correlations, which serves as a ``universal coordinate'' that locates a given microscopic system within this power-law ordered phase, is set by the coupling constant $g$ via the relation $\eta(T) = g(T)$.
The Landau-Ginzburg theory also sheds light on the nature of the low temperature transition between the two kinds of three-sublattice ordered phases, modeled by $\lambda_6$ going through zero smoothly and changing
sign. Since both phases have long-range three-sublattice order, fluctuations of $\lvert \psi_{\mathrm{cl}}\rvert$ may again be neglected in the vicinity of this transition. With the amplitude $|\psi_{\mathrm{cl}}|$ remaining essentially constant across this transition, the physics
of the transition is again controlled by the phase $\theta$ of the three-sublattice order parameter. Minimizing the free-energy density $\mathcal{F}$ yields a spatially uniform configuration with a particular optimal value $\theta^{*}$ for this phase variable. When $\lambda_{12}<0$, $\theta^{*}$ takes on the
values $(2n+1)\pi/6$ ($(2n)\pi/6$) with $n=0,1\dots 5$ when $\lambda_{6}>0$ ($\lambda_{6}<0$ ). When $\lambda_6=0$, all values $\theta^*=m\pi/6$ $(m = 0,1\dots 11$) minimize the free-energy. Clearly, this corresponds to a first-order transition between
ferrimagnetic and antiferromagnetic three-sublattice ordered states, with both kinds of
three-sublattice order coexisting at the transition point.
If, on the other hand, $\lambda_{12}>0$, we obtain
\begin{equation}
\theta^{*}=\begin{cases}
\frac{2n\pi}{6} & \text{if } \lambda_{6} <-4\lambda_{12}\lvert\psi_{\rm cl}\rvert^{6}\\
\frac{2n\pi}{6}+ \frac{1}{6}\arccos(-\lambda_{6}/4\lambda_{12}\lvert\psi_{\rm cl}\rvert^{6}) & \text{if } \lvert \lambda_{6}\rvert <4\lambda_{12}\lvert\psi_{\rm cl}\rvert^{6}\\
\frac{(2n+1)\pi}{6} & \text{if } \lambda_{6} >4\lambda_{12}\lvert\psi_{\rm cl}\rvert^{6}\\
\end{cases}
\end{equation}
where $n=0,1\dots 5$ represents the six-fold degeneracy of the minima in each case.
In this case, as $\lvert \lambda_{6} \rvert$ becomes small and $\lambda_6$ goes through zero, $\theta^{*}$ switches continuously from the antiferromagnetic phase to the ferrimagnetic phase via an intermediate mixed phase that is established for $\lvert \lambda_{6} \rvert<4\lambda_{12}\lvert\psi_{\rm cl}\rvert^{6}$. In what follows, we will confront these two
quite different scenarios with data obtained in the vicinity of the transition between
antiferromagnetic and ferrimagnetic three-sublattice order in the low-temperature
state of $H_{\rm Ising}$ with an additional ferromagnetic second-neighbour coupling $J_2$ between the Ising spins.
\begin{figure}[t]
\includegraphics[width=\columnwidth]{plateau.pdf}
\caption{\label{plateau} The uniform easy-axis susceptibility $\chi_{u}$ of $H_{\rm Ising}$ on $L\times L$ triangular lattices, when plotted vs $1/L$ for a sequence of sizes, clearly saturates to a finite value in the limit of large $L$. Note the slow crossover to this thermodynamic limit,
with samples of linear size as large as $L^{*}=40$ not yet in the asymptotic large-$L$ regime. This behaviour demonstrates that the low temperature phase is indeed antiferromagnetic. However, the slow crossover indicates the presence of a proximate phase
with net magnetic moment along the easy-axis, suggesting that $H_{\rm Ising}$ could
be driven into a ferrimagnetic ground state for relatively small values of an additional second-neighbour ferromagnetic coupling $J_2$. All other
temperature and energy scales are measured in units of $J_1$ which is set to unity.}
\end{figure}
\begin{figure}[t]
\includegraphics[width=\columnwidth]{plateaufit.pdf}
\caption{\label{plateaufit} The uniform easy-axis susceptibility of $H_{\rm Ising}$ on $L \times L$ triangular lattices, now scaled
by the number of sites $L^2$, is fit reasonably well to the single parameter form $kL^{-2}$ with $k=15.18(8)$ for the largest four sizes studied here. This analysis also confirms that the low temperature phase of $H_{\rm Ising}$ is indeed antiferromagnetic, {\em i.e.} with no net easy-axis moment. All other
temperature and energy scales are measured in units of $J_1$ which is set to unity.}
\end{figure}
\begin{figure}[t]
\includegraphics[width=\columnwidth]{j2=0cross.pdf}
\caption{\label{cross1} Lower and upper transition temperatures $T_1$ and $T_2$, which mark the boundaries of the power-law ordered phase associated with the two-step melting of antiferromagnetic three-sublattice order, are obtained by plotting $\chi_{\mathbf{Q}}L^{\frac{1}{9}-2}$ and $\chi_{\mathbf{Q}}L^{\frac{1}{4}-2}$ versus $T$ for different values of $L$ and identifying
the temperatures at which curves corresponding to different $L$ cross. This
gives $T_{1}=0.198(5)$ and $T_{2}=0.414(5)$ when $\Gamma=0.8$. All other
temperature and energy scales are measured in units of $J_1$ which is set to unity.}
\end{figure}
\section{Methods}
\label{Methods}
Our numerical work uses the Stochastic Series Expansion (SSE) framework \cite{Melko,Sandvik_PRE,Sandvik_JPHYSA,Syljuasen_Sandvik,Sandvik_PRB} to compute equilibrium averages $\langle \dots \rangle$ for transverse field Ising models at nonzero temperature. For models with geometric frustration,
which results in a macroscopic degeneracy of minimally frustrated classical configurations
(with minimum Ising-exchange energy), it is important that the computational
method correctly captures the interplay between this macroscopic degeneracy,
and the disordering effects of classical and quantum fluctuations. In the present case, this interplay is expected to be crucial to the establishment of antiferromagnetic three-sublattice order in the low temperature phase,
as well as its two-step melting.\cite{Isakov_Moessner,Moessner_Sondhi_Chandra,Moessner_Sondhi}
Therefore, to obtain reliable results, we use the recently developed quantum cluster algorithm\cite{Biswas_Rakala_Damle} that works within the SSE framework to provide an
efficient way of sampling the partition function for such frustrated transverse field
Ising models. In this cluster algorithm, which works in the $\sigma^z$ basis, the diagonal Ising exchange part of $H_{\rm{Ising}}$ in Eq.~\eqref{Ising} is written as $\mathcal{H}_{diag}=\sum_{\triangle}\mathcal{H_{\triangle}}$, where $H_{\triangle}$ are operators living on elementary triangular plaquettes $\triangle$. This furnishes the algorithm local information
that enables it to distinguish between minimally
frustrated plaquettes and fully frustrated plaquettes of higher Ising-exchange energy. The transverse field part of the Hamiltonian is represented as single-site operators as
in the original SSE approach.\cite{Sandvik_PRE} The plaquette representation of $\mathcal{H}_{diag}$ facilitates the construction of ``space-time clusters'' with a broad
distribution of cluster sizes, allowing the algorithm to efficiently sample
the configuration space of SSE operator strings at low temperature.
Using this approach, we study $H_{\rm Ising}$ on $L\times L$ triangular lattice with periodic boundary conditions, with $L$ ranging from $L=24$ to $96$. We compute the static susceptibilities corresponding to the order parameters defined in Eq.~\eqref{orderparameters}. These susceptibilities are defined as
\begin{align}
&\chi_{u}=\frac{L^{2}}{\beta}\langle\lvert \int_{0}^{\beta} d\tau m(\tau)\rvert^{2}\rangle \label{chi0}\\
&\chi_{{\mathbf{Q}}}=\frac{L^{2}}{\beta}\langle\lvert \int_{0}^{\beta} d\tau \psi(\tau)\rvert^{2}\rangle
\label{chi}
\end{align}
Additionally, we compute the static susceptibility $\chi^{xx}_{{\mathbf{Q}}}$ to a transverse field (along $\hat{x}$) oscillating at wavevector ${\mathbf Q}$, defined as
\begin{equation}
\chi^{xx}_{{\mathbf{Q}}}=\frac{L^{2}}{\beta}\langle\lvert \int_{0}^{\beta} d\tau \sigma^{x}_{\mathbf{Q}}(\tau)\rvert^{2}\rangle
\label{chix}
\end{equation}
where $\sigma^{x}_{\mathbf{Q}}$ is given by
\begin{equation}
\sigma^{x}_{\mathbf{Q}}=\frac{1}{L^{2}}\sum_{\vec{R}}\sigma^{x}_{\vec{R}}\exp(i\mathbf{Q} \cdot \vec{R})
\end{equation}
\section{Results}
\label{Results}
We begin by revisiting the phase diagram obtained in previous work\cite{Isakov_Moessner} for the case with no next-nearest neighbour coupling ($J_{2}=0$).
From their results, we note that the low temperature order persists up to
the highest temperature when $\Gamma$ is in the vicinity of $\Gamma = 0.8$.
Therefore, we set the transverse field to this value in most of our work and study
the three-sublattice ordering of the low temperature phase, as well as its
two-step melting.
As expected, we find that the order parameter susceptibility $\chi_{{\mathbf{Q}}}$ scales
with the volume of the system at low enough temperature, confirming the presence of long-range three-sublattice order in the low temperature phase. Since this is entirely
consistent with earlier results,\cite{Isakov_Moessner} we do not display this explicitly here.
Since our focus in what follows will be an unusual singular behaviour in
the ferromagnetic susceptibility $\chi_{u}$ to a {\em uniform} field along the easy-axis,
we find it useful to first study the same quantity deep in the low-temperature
ordered state. From Fig.~\ref{plateau} and Fig.~\ref{plateaufit}, which display the $L$ dependence of $\chi_{u}$ and $\chi_{u}/L^2$ deep in the low-temperature ordered state, we see that the three-sublattice ordering in the low temperature phase is not accompanied by
any net moment along the easy-axis.
This confirms earlier results\cite{Isakov_Moessner}
that have identified the antiferromagnetic nature of the three-sublattice ordering
at low temperature. However, the approach to the
thermodynamic limit is seen to involve a slow crossover, suggesting the presence of
a proximate phase with a net easy-axis moment. This is consistent with the fact
that a relatively small value of second-neighbour ferromagnetic exchange $J_2 < 0$ is
sufficient to access a nearby state with ferrimagnetic three-sublattice
ordering at low temperature.\cite{Biswas_Rakala_Damle}
In the power-law ordered phase associated with the two-step melting of three-sublattice order, the static susceptibility $\chi_{\mathbf{Q}}$, defined in Eq.~\eqref{chi} for a finite size $L \times L$ system, is expected to scale as
\begin{equation}
\chi_{\mathbf{Q}}\sim L^{2-\eta(T)}
\label{chiQdecay}
\end{equation}
From the renormalization group picture (summarized in the previous section) of
this power-law ordered phase, it is also clear that $\eta(T)$ ranges from
$\eta(T_1)=1/9$ at the lower phase boundary $T_1(\Gamma)$ of the power-law phase, to $\eta(T_2)=1/4$ at the upper phase boundary $T_2(\Gamma)$.
To locate these upper and lower transition temperatures for $\Gamma=0.8$, we plot $\chi_{\mathbf{Q}}L^{\frac{1}{9}-2}$ and $\chi_{\mathbf{Q}}L^{\frac{1}{4}-2}$ for various sizes $L$ as a function of temperature and identify the temperature at which curves corresponding to the
different sizes all cross. This is shown in Fig.~\ref{cross1}. The location of transitions obtained in this way are consistent
with those obtained earlier in Ref.~\onlinecite{Isakov_Moessner}.
\begin{figure}[t]
\includegraphics[width=\columnwidth]{scalingj20.pdf}
\caption{\label{scaling1} Quantum Monte Carlo data for the static susceptibility $\chi_{\mathbf{Q}}$ of $H_{\rm Ising}$ at
wavevector ${\mathbf{Q}} $ on $L\times L$ triangular lattices collapses onto a universal
scaling form when $\chi_{\mathbf{Q}}(t,L)L^{\frac{1}{4}-2}$ for different $L$ and temperatures $T$ (in the vicinity of the upper transition temperature $T_2$) are plotted as a function of
the scaling variable defined in Eq.~{\protect{\eqref{finitesize}}} in the main text. All other
temperature and energy scales are measured in units of $J_1$ which is set to unity.}
\end{figure}
\begin{figure}[t]
\includegraphics[width=\columnwidth]{j2=01cross.pdf}
\caption{\label{cross2} The ferrimagnetic three-sublattice order that characterizes
the ground state in the presence of a second-neighbour ferromagnetic interaction $J_{2}=-0.1$ also melts in a two-step manner. Upper and lower transition temperatures $T_1$ and $T_2$, that demarcate the boundaries of the power law ordered phase associated with this two-step
melting, are obtained by plotting $\chi_{\mathbf{Q}}L^{\frac{1}{9}-2}$ and $\chi_{\mathbf{Q}}L^{\frac{1}{4}-2}$ versus $T$ for different values of $L$ and identifying
the temperatures at which curves corresponding to different $L$ cross. This
gives $T_{1}=0.440(6)$ and $T_{2}=0.851(8)$ when $\Gamma=0.8$. All other
temperature and energy scales are measured in units of $J_1$ which is set to unity.}
\end{figure}
\begin{figure}[t]
\includegraphics[width=\columnwidth]{scalingj2=01.pdf}
\caption{\label{scaling2} Quantum Monte Carlo data for the static susceptibility $\chi_{\mathbf{Q}}$ of $H_{\rm Ising}$ with $J_2=-0.1$ at
wavevector ${\mathbf{Q}} $ on $L\times L$ triangular lattices also collapses onto a universal scaling form when $\chi_{\mathbf{Q}}(t,L)L^{\frac{1}{4}-2}$ for different $L$ and temperatures $T$ (in the vicinity of the upper transition temperature $T_2$) are plotted as a function of
the scaling variable defined in Eq.~{\protect{\eqref{finitesize}}} in the main text. All other
temperature and energy scales are measured in units of $J_1$ which is set to unity.}
\end{figure}
Since the upper (lower) transitions out of the power-law ordered phase correspond
to vorticity (six-fold anisotropy) in $\theta$ becoming relevant, we expect
these transitions to be of the Kosterlitz-Thouless (inverted Kosterlitz-Thouless) type.
To confirm that this is indeed the case, we perform fits of our Quantum Monte Carlo data
in the vicinity of the upper phase boundary to the finite-size scaling form predicted by Kosterlitz-Thouless theory.\cite{Challa_Landau} This scaling form follows from
the following argument: Above $T_2(\Gamma)$, order parameter correlations decay
exponentially, with a correlation length $\xi$ given by\cite{Kosterlitz}
\begin{equation}
\xi \sim \exp(at^{-1/2}) \; ,
\label{xi}
\end{equation}
where $t=(T-T_{2})/T_{2} $ is the reduced temperature.
This Kosterlitz-Thouless form of the correlation length, Eq.~\eqref{xi}, in conjunction
with the standard finite size scaling ansatz $\chi_{\mathbf{Q}}(t,L)=L^{2-\eta_2}f(\xi/L)$ gives
the finite-size scaling form\cite{Challa_Landau}
\begin{equation}
\chi_{\mathbf{Q}}(t,L)L^{\frac{1}{4}-2}=f(L^{-1}\exp(at^{-1/2})) \ ;,
\label{finitesize}
\end{equation}
where we have used $\eta_2=1/4$, and $f$ is the finite-size scaling function that
we expect our data to collapse onto. In practice, we use
$T_{2}$ obtained from Fig.~\ref{cross1}, and attempt a finite-size
scaling collapse with a single adjustable parameter $a$. This is shown in Fig.~\ref{scaling1}.
\begin{figure}[t]
\includegraphics[width=\columnwidth]{powerlawj20.pdf}
\caption{\label{powerlaw1} $\chi_{\mathbf{Q}}$ and $\chi_{u}$ fit rather well
to power-law forms $k_{1}L^{2-\eta}$ and $k_{2}L^{2-9\eta}$ respectively for three different values of temperature in the intermediate power-law ordered phase associated with the melting of antiferromagnetic three-sublattice order when $J_{2}=0.0$, $\Gamma=0.8$. All other
temperature and energy scales are measured in units of $J_1$ which is set to unity.}
\end{figure}
\begin{figure}[t]
\includegraphics[width=\columnwidth]{powerlawj201.pdf}
\caption{\label{powerlaw2} $\chi_{\mathbf{Q}}$ and $\chi_{u}$ fit rather well
to power-law forms $k_{1}L^{2-\eta}$ and $k_{2}L^{2-9\eta}$ respectively for three different values of temperature in the intermediate power-law ordered phase associated with the melting of ferrimagnetic three-sublattice order when $J_{2}=-0.1$, $\Gamma=0.8$. All other
temperature and energy scales are measured in units of $J_1$ which is set to unity.}
\end{figure}
\begin{figure}[t]
\includegraphics[width=\columnwidth]{last.pdf}
\caption{\label{transversepowerlaw} $\chi^{xx}_{\mathbf{Q}}$ fits the power-law
form $k_3 L^{2-4\eta}$ for three different values of temperature in the intermediate power-law ordered phase associated with the melting of antiferromagnetic as well as ferrimagnetic three-sublattice order. All other temperature and
energy scales are measured in units of $J_1$ which is set to unity.}
\end{figure}
\begin{figure}[t]
\includegraphics[width=\columnwidth]{transition.pdf}
\caption{\label{transition} Histograms of $\chi_{u}/L^{2}$ show a characteristic two-peak structure suggestive of a first order transition. All other
temperature and energy scales are measured in units of $J_1$ which is set to unity.}
\end{figure}
When ferromagnetic second-neighbour interactions $J_{2}<0$ of sufficient magnitude
are present, one expects the ground state ordering pattern to change to ferrimagnetic
three-sublattice order.\cite{Nienhuis_Hilhorst_Blotte} In recent work,\cite{Biswas_Rakala_Damle}, the threshold value of $J_{2}$ corresponding to this onset of ferrimagnetism was estimated to be roughly $J_{2c} \approx -0.03$. With a view towards comparing the
melting behaviour of this ferrimagnetic three-sublattice order with the two-step
melting of antiferromagnetic three-sublattice order, we also study the effect of
thermal fluctuations at $J_2 =-0.1$, {\em i.e.} deep in this ferrimagnetic three-sublattice ordered state. We find that long-range order is again lost via a two-step melting process,
with an intermediate power-law ordered phase. The locations of the upper and lower
transitions that demarcate the extent of the power-law ordered phase are obtained
as before. This is displayed in Fig.~\ref{cross2}.
Above $T_2$, the static order parameter susceptibility again collapses quite
convincingly on to the Kosterlitz-Thouless finite-size scaling form. This is shown in Fig.~\ref{scaling2}.
With these preliminaries out of the way, we are now in a position to study in a unified way the behaviour
of the uniform easy-axis susceptibility $\chi_{u}$ in the power-law ordered phase associated with the
two-step melting of antiferromagnetic three-sublattice order as well as ferrimagnetic
three-sublattice order. As mentioned earlier, our goal is
to test a recent prediction\cite{Damle} that $\chi_{u}$ provides a
thermodynamic signature of two step melting due to the presence of a singular
$B$ dependence: $\chi_{u} (B) \sim |B|^{-\frac{4 - 18 \eta}{4-9\eta}}$ for $\eta(T) \in (1/9,2/9)$.
Here, we test this via the equivalent prediction\cite{Damle} for the finite-size
susceptibility $\chi_{u}(L)$ of an $L \times L$ sample when $B=0$: $\chi_{u}(L) \sim L^{2-9\eta}$ for $\eta(T) \in (1/9,2/9)$. In Landau theory terms, this singularity in $\chi_u$ is a direct consequence of a symmetry-allowed coupling of the form $m_{\mathrm{cl}} \lvert \psi_{\mathrm{cl}} \rvert^3 \cos(3\theta)$ between the static component $m_{\mathrm{cl}}(\vec{r})$ of the uniform magnetization density and the order parameter field $\psi_{\mathrm{cl}}$. In the power-law ordered
phase, this coupling is predicted\cite{Damle} to cause $m_{\mathrm{cl}}$ to have the same power-law correlations as $\cos(3 \theta)$, leading to a singular $\chi_{u}$ {\em independent}
of whether the low temperature ordered state is ferrimagnetic or antiferromagnetic.
Thus, while the predicted effect is particularly counter-intuitive for the antiferromagnetic
case, {\em i.e} with $J_2=0$ for the system under consideration, the underlying
mechanism is expected to be the same at $J_2 = -0.1$ as well.
As is clear from Fig.~\ref{powerlaw1}, simultaneous fits of $\chi_{\mathbf{Q}}$ to the form $k_{1}L^{2-\eta}$ and $\chi_{u}$ to the form $k_{2}L^{2-9\eta}$ work rather well at three different
points in the power-law ordered phase associated with the two-step melting of antiferromagnetic three-sublattice order. This can be compared to similar fits in Fig.~\ref{powerlaw2} for
the same quantities in the power-law ordered phase associated with the two-step
melting of ferrimagnetic three-sublattice order.
As is clear from these results, the uniform susceptibility does indeed provide
a thermodynamic signature of the power-law ordered phase, independent of
the ferri/antiferromagnetic nature of the low-temperature three-sublattice ordered phase,
exactly as predicted by the effective field theoretical arguments of Ref.~\onlinecite{Damle}.
A similar argument, which identifies $\beta^{-1} \int_0^{\beta}\sigma^{x}_{{\mathbf{Q}}}(\tau)$ with $\psi^2_{\mathrm{cl}}$ on symmetry grounds, immediately predicts that $\chi^{xx}_{{\mathbf{Q}}} \sim L^{2-4\eta}$ throughout the power-law ordered phase. As is clear from
Fig.~\ref{transversepowerlaw}, our data for $\chi^{xx}_{{\mathbf{Q}}} $ is seen to be completely consistent
with this prediction as well.
Finally, we comment on the nature of the transition between the antiferromagnetic and ferrimagnetic three sublattice ordered states. In previous work which studied\cite{Biswas_Rakala_Damle}
relatively small samples at moderately low temperatures in the vicinity of this transition, the phase of the estimator for the three-sublattice order parameter, as measured in the
Quantum Monte Carlo simulations, was seen to be distributed more or less uniformly in the interval
$(0,2\pi)$. If this behaviour were to persist to larger sizes, it would be indicative
of a power-law ordered phase that interpolates between the antiferromagnetic
and ferrimagnetic three-sublattice ordered phases at nonzero temperature. However, from the Landau
theory considerations of Sec.~\ref{PhasesandTransitions}, we see that the
two generic possibilities for this phase transition are first-order behaviour,
or an intermediate mixed-phase. An intervening power-law ordered phase
can, in this picture, only arise in the fine-tuned limiting case where $\lambda_{12}$
and higher order anisotropies are all absent. With this in mind, we measure the
histogram of the estimator for $\chi_{u}/L^{2}$ to look for signals of phase coexistence
in the transition region.
These histograms are shown in Fig.~\ref{transition}. The two-peak nature
of these histograms suggests that the transition is in fact of a weakly first-order type.
This is consistent with the fact that the $L$-dependence of $\chi_{\mathbf{Q}}$ is certainly
not a power-law, and the fact that Binder ratios of the estimator of $\chi_{\mathbf{Q}}$ also
do not show a clear crossing (indicative of a second-order transition),\cite{Binder_ZPHYSB,Binder_PRL} nor do they stick (as they would in a power-law ordered phase).~\cite{Challa_Landau} However, we do not see any indications
of non-monotonic Binder ratios\cite{Binder_Landau} of the type expected in the vicinity of first order transitions.
Thus, while our data is suggestive of a weakly first-order transition, more work is needed
to clarify the precise nature of this transition.
\section{Discussion}
Thus, we have obtained fairly convincing evidence for a
singular uniform easy-axis susceptibility $\chi_u(B)$ in the power-law ordered
phase associated with the two-step melting of antiferromagnetic three-sublattice order
in triangular lattice transverse-field Ising antiferromagnets. This (at-first-sight) counter-intuitive thermodynamic signature of two-step melting is already of
some general interest, since the transverse-field
Ising antiferromagnet on the triangular lattice is a paradigmatic example of the interplay
between quantum fluctuations and frustrated classical interactions.
Of course, this thermodynamic signature of two-step melting would be of much greater
interest and direct experimental relevance if the model Hamiltonian $H_{\rm Ising}$ were to emerge as
a good description of magnetic exchange interactions in some frustrated magnet.
In this context, it should
be noted that a closely related model Hamiltonian, the one-dimensional transverse
field Ising chain, does serve as a good starting point for the theoretical
description of an interesting quantum phase transition in the magnetic material
Columbite.\cite{Kinross_etal,Lee_etal,Coldea_etal} Columbite can be thought of as a triangular array of one dimensional
chains of magnetic moments, with strong intra-chain coupling between the moments
and weak inter-chain couplings. This hierarchy of exchange-couplings allows
for a theoretical description in terms of the quantum critical properties of the one-dimensional transverse-field Ising chain. It is possible that other materials, with somewhat different exchange pathways but the same strong easy-axis anisotropy, may have much stronger exchange couplings within a triangular plane, and much
weaker couplings between planes. For such a material, the model Hamiltonian $H_{\rm Ising}$ could in the same way serve as a good theoretical description, and our results on this
thermodynamic signature of two-step melting could then be of direct experimental relevance. We hope that our results provide some motivation for exploring this
possibility.
\section{Acknowledgements}
\label{Acknowledgements}
Our computational work was made possible by the computational resources
of the Department of Theoretical Physics of the Tata Institute of Fundamental
Research, as well as by computational resources funded by
DST (India) grant DST-SR/S2/RJN-25/2006. The analysis of our Monte Carlo
data was greatly facilitated by the general-purpose file-handling and data-analysis scripts
developed by Geet Rakala.
|
1,314,259,993,619 | arxiv | \section{Introduction}
In the seminal paper~\cite{So70} Solovay proved that in the Levy model
(after collapsing an inaccessible) every definable set is measurable and has
the Baire property.
In~\cite{Sh:176} Shelah showed that the inaccessible is necessary for
measurability, but the Baire property of every definable set can be obtained by
a forcing $P$ without the use of an inaccessible (i.e.\ in ZFC). This forcing
$P$ is constructed by amalgamation of universally meager forcings $Q$.
So every $Q$ adds
a co-meager set of generics and has many automorphisms,
and the forcing $P$ has
similar properties to the Levy collapse. The property of $Q$ that implies
that $Q$ can be amalgamated is called ``sweetness'' (a strong version of ccc).
One can ask about other ideals than Lebesgue-null and meager (or their defining
forcings, random and Cohen), and classify such ideals (or forcings). This was
done e.g.\ in Sweet {\&} Sour~\cite{RoSh:672} --- sweet forcings are close to
Cohen, sour ones close to random.
However, there is a completely different construction that brings measurability
of all definable sets: Instead iterating basic forcings $Q$ that have many
automorphisms and add a measure 1 set of generics, we use a $Q$ that adds only
a null set of generics (a single one in our case, and this real remains the
only generic over $V_i$ even in the final limit). So $Q$ has to be very
non-homogeneous. Instead of having many
automorphisms in $Q$, we assume that the skeleton of the iteration has many
automorphisms (so in particular a non-wellfounded iterations has to be used).
We use the word Saccharinity for this concept: a construction that achieves the
same effect as an (amalgamation of) sweet forcings, but using entirely
different means.
As a first instance of this concept we construct an ideal $I$ on the reals and a
forcing $P$ making every projective (or: definable) set measurable modulo $I$
(without using amalgamation or an inaccessible). $I$ will be defined from a
tree forcing in the same way that the Marczewski ideal is defined from Sacks
forcing. $I$ is not a ccc ideal.
\subsection*{Annotated contents}\nopagebreak
\begin{list}{}{\setlength{\leftmargin}{0.5cm}\addtolength{\leftmargin}{\labelwidth}}
\item[Section~\ref{sec:Q}, p. \pageref{sec:Q}:]
We define a class of tree forcings
with ``lim-sup norm''.
The forcing conditions are subtrees of a basic tree
that satisfy ``along every branch, many nodes have many successors''.
\item[Section~\ref{sec:nwi}, p. \pageref{sec:nwi}:]
We introduce a general construction to iterate such lim-sup tree-forcings along
non-wellfounded total orders. It turns out that the limit is proper,
$\omega\ho$-bounding and has other nice properties similar to the properties of
the lim-sup tree-forcings itself.
\item[Section~\ref{sec:myI}, p. \pageref{sec:myI}:]
We define (with respect to a lim-sup tree-forcing $Q$)
the ideals $\mathbb{I}$ and $\myI$ (the $<2^\al0$-closure
of $\mathbb{I}$). These ideals will generally not be ccc.
We define what we mean by
``$X$ is measurable with respect to $\myI$''
and formulate the main theorem: Assuming CH and a Ramsey property
for $Q$ (see Section~\ref{sec:tree}), we can force
that all definable sets are measurable.
\item[Section~\ref{sec:order}, p. \pageref{sec:order}:]
Assuming CH, we construct an order
$I$ which has
many automorphisms and a certain
cofinal sequence $(j_\alpha)_{\alpha\in\om2}$.
We show that $P$ (the non-wellfounded iteration
of $Q$ along the order $I$) forces that $2^\al0=\al2$, that
$\myI$ is nontrivial, that for every definable set
$X$ ``locally'' either
all or no $\n\eta_{j_\delta}$ are in $X$
and that the set $\{\n\eta_{j_\delta}:\, \delta\in\om2\}$ is of measure
1 in the set $\{\n \eta_i:\, i\in I\}$.
\item[Section~\ref{sec:tree}, p. \pageref{sec:tree}:]
We assume a certain Ramsey property for $Q$.
We show that \mbox{$\{\eta_i:\, i\in I\}$}
is of measure 1 (with respect to $\myI$). Together with
the result of the last section this proves the main theorem.
\end{list}
\section{finitely splitting lim-sup tree-forcings}\label{sec:Q}
We will define a class of tree forcings with ``lim-sup norm''. Such trees (and
generalizations) are investigated in~\cite{RoSh:470} under the name
$\mathbb{Q}^\text{tree}_0$, see Definition 1.3.5 there. Let us first introduce some
notation around trees on $\omega\hko$.
\begin{Def}
Let $T\subseteq \omega\ho$ be a tree
(i.e.\ $T$ is closed under initial segments), let
$s,t\in \omega\hko$, $A \subseteq T$.
\begin{itemize}
\item $\langle \rangle$ denotes the empty sequence.
\item $s\preceq t$ means that $s$ is a restriction of $t$ (or equivalently that $s\subseteq t$).
\item $t$ is immediate successor of $s$
if $t\succeq s$ and $\length(t)=\length(s)+1$.
\item $\SUCC_T(t)$ is the set of immediate successors of $t$ in a tree $T$.
If the tree $T$ is clear from the context we will also
write $\SUCC(t)$.
\item $s$ is compatible with $t$ ($s\ensuremath{\parallel} t$), if
$s\preceq t$ or $t\preceq s$.
Otherwise, $s$ and
$t$ are incompatible ($s\ensuremath{\perp} t$).
\item $T^{[t]}\ensuremath{\coloneqq} \{s\in T:\, s\ensuremath{\parallel} t\}$.\quad (So $T^{[t]}$ is a tree.)
\item $T\restriction n\ensuremath{\coloneqq}\{t\in T:\, \length(t)<n\}$.
\item $A\subseteq T$ is a chain if $s\ensuremath{\parallel} t$ for all $s, t\in A$.
\item $b\subseteq T$ is a branch if it is a maximal
chain.\\If there exists a $t\in b$ with length $n$ then
this $t$ is unique and denoted by $b\restriction n$.
\item $A\subseteq T$ is an antichain if $s\ensuremath{\perp} t$ for all
$s\neq t\in A$.
\item $A\subseteq T$ is a front if it is an antichain
and every branch $b$ meets $A$ (i.e.\ $\card{b\cap A}=1$).
\item $t\preceq A$ stands for: ``$t\preceq s$ for some $s\in A$''.
\item $T_\text{cldn}^A\ensuremath{\coloneqq}\{t\in \omega\hko:\, t\preceq A\}$.\\
(We will use this notion
only for finite antichains $A$. Then $T_\text{cldn}^A$ is a finite tree.)
\item If $A$ and $A'$ are antichains, then $A'$ is stronger than
$A$ if for each $t\in A'$ there is a $s\in A$ such that
$s\preceq t$ (cf.\ Figure~\ref{fig:fronts}).
\item If $A$ and $A'$ are antichains then $A'$ is purely stronger than
$A$ if it is stronger and
for each $s\in A$ there is a $t\in A'$ such that
$s\preceq t$ (cf.\ Figure~\ref{fig:fronts}).
\end{itemize}
\end{Def}
\begin{figure}[tb]
\begin{center}
\scalebox{0.4}{\input{fronts.pstex_t}}
\end{center}
\caption{\label{fig:fronts} $F'$ is stronger than $F$, $F''$ is purely stronger
than $F$.}
\end{figure}
We are only interested in finitely splitting trees
(i.e.\ $\SUCC(t)$ is finite for all $t\in T$).
Then all fronts are finite.
Note that being a front is stronger than being a maximal antichain.
For example, $\{0^n1:\, n\in\omega\}$ is a maximal antichain in
$2\hko$, but not a front.
\begin{uAsm}
Assume $\Tmax$ and $\mu$ satisfy the following:
\begin{itemize}
\item $\Tmax$ is a finitely splitting tree.
\item $\mu$ assigns a non-negative real to every subset
of $\SUCC_{\Tmax}(t)$ for every $t\in \Tmax$.
\item $\mu$ is monotone, i.e.\ if $A\subseteq B$ then $\mu(A)\leq \mu(B)$.
\item The measure of singletons is smaller than 1, i.e.\ $\mu(\{s\})<1$.
\item For all branches $b$ in $\Tmax$, $\limsup_{n\rightarrow \infty}
(\mu(\SUCC(b\restriction n)))=\infty$.
\end{itemize}
\end{uAsm}
Such a $\Tmax$ and $\mu$ define a lim-sup tree-forcing $Q$ in the
following way:
\begin{Def}\label{def:Q}
\begin{itemize}
\item
If $T$ is a subtree of $\Tmax$ and $t\in T$,
then $\mu_T(t)$ is defined as the measure of the $T$-successors of $t$,
i.e.\ $\mu_T(t)\ensuremath{\coloneqq}\mu(\SUCC_T(t))$.
\item
$Q$ consists of all subtrees $T$ of $\Tmax$
(ordered by inclusion)
such that along every branch $b$ in $T$,
$\limsup(\mu_T(b\restriction n ))=\infty$
\end{itemize}
\end{Def}
So $\Tmax$ itself is the weakest element of $Q$.
For example, Sacks forcing is such a forcing:
Set $\Tmax\ensuremath{\coloneqq} 2\hko$, and for $t\in \Tmax$ and $A\subseteq \SUCC(t)$ set
\[
\mu(A)\ensuremath{\coloneqq}\begin{cases}\length(t)&\text{if
}\card{A}=2,\\0&\text{otherwise.}\end{cases}
\]
Then a subtree $T$ of $2\hko$ is in $Q$ iff along every branch there
are infinitely many splitting nodes.\footnote{This example is ``discrete''
in the following sense: For a node $s\in T$ there is an $A\subset \SUCC(s)$ such
that $\mu(A)$ is large but $\mu(B)<1$ for every $B\subsetneq A$.
In this paper, we will be interested in ``finer'' norms. In particular
we will require the Ramsey property defined in \ref{def:ramseyprop}.}
\begin{Def}
A (finite or infinite) subtree $T$ of $\Tmax$ is $n$-dense if
there is a front $F$ in $T$ such that
$\mu_T(t)>n$ for every $t\in F$.
\end{Def}
\begin{Lem}
\begin{enumerate}
\item A subtree $T$ of $\Tmax$ is in $Q$ iff $T$ is $n$-dense
for every $n\in\mathbb{N}$.
\item ``$T\in Q$'' and ``$T\leq_Q S$'' are Borel statements, and ``$S\ensuremath{\perp} T $'' is $\mP11$
(in the real parameters $\Tmax$ and $\mu$).
\end{enumerate}
\end{Lem}
\begin{proof}
(1) $\rightarrow$: If $D_n\ensuremath{\coloneqq}\{s\in T:\, \mu_T(s)>n\}$ meets
every branch, then
\[F_n\ensuremath{\coloneqq}\{s\in D:\, (\forall s'\precneqq s)\, s'\notin D\}\]
is a front.\\
$\leftarrow$:
If $b$ is a branch, then $b$ meets every $F_n$, i.e.\ $\mu(b\restriction m)>n$
for some $m$.
Since $\mu(b\restriction m)$ is finite,
$\limsup(\mu_T(b \restriction n)))$ has to be infinite.
(2)
Since $\Tmax$ is finitely splitting, ``$F$ is a front'' is equivalent to
``$F$ is a finite maximal antichain''.
\end{proof}
A finite antichain $A$ can be seen as an approximation of a tree:
$A$ approximates $T$ means that $A$ is a front in $T$.
If $A'$ is purely stronger than $A$, then $A'$ gives more
information about the tree $T$ that is approximated
(i.e.\ every tree approximated by $A'$ is also approximated by $A$).
We will usually identify a finite antichain $F$ and the corresponding
finite tree $T_\text{cldn}^F$.
\begin{Def}\label{def:nsplitting}
\begin{itemize}
\item
A finite antichain $F$ is $n$-dense
if $T_\text{cldn}^F$ is $n$-dense.
\item $\bar F=(F_n)_{n\in \omega}$ is a front-sequence, if
$F_{n+1}$ is $n$-dense and purely stronger than $F_n$.
\item A front-sequence $\bar F$ and a tree $T\in Q$ correspond to each other
if $F_n$ is a front in $T$ for all $n$.
\end{itemize}
\end{Def}
\begin{Facts}
\begin{itemize}
\item
If $F$ is $n$-dense and $F'$ is purely stronger than $F$,
then $F'$ is $n$-dense as well. (This is not true if
$F'$ is just stronger than $F$.)
\item
If $T\in Q$ then there is a front-sequence corresponding to $T$.
\item
If $\bar F$ is a front-sequence
then there exists exactly one $T\in Q$ corresponding to $\bar F$,
which we call $\lim(\bar F)$. It is the tree
\[\lim(\bar F)\ensuremath{\coloneqq}\{t\in \Tmax:\, (\exists i\in\omega)\, t\preceq F_i\},\]
or equivalently
\[\lim(\bar F)\ensuremath{\coloneqq}\{t\in \Tmax:\, (\forall i\in\omega)\, (\exists s\in F_i)\, t\ensuremath{\parallel} s\}.\]
\end{itemize}
\end{Facts}
\begin{Lem}\label{lem:Qisproper}
Assume that $Q$ is a
finitely splitting lim-sup tree-forcing.
\begin{enumerate}
\item
If $T\in Q$ and $t\in T$ then $T^{[t]}\in Q$.
\item\label{item:basicQunion}
The finite union of elements of $Q$ is in $Q$.\footnote{$Q$ is generally
not closed under countable unions.}
\item The generic filter on $Q$ is determined by a real
$\n \eta$ defined by $\ensuremath{\Vdash}_Q\{\n \eta\}=\bigcap_{T\in G_Q}\lim(T)$.
It is forced that $\n \eta\notin V$ and that $T\in G_Q$ iff
$\n\eta\in\lim(T)$.\\
For every $T\in Q$ and $t\in T$ it is compatible with $T$ that $t\prec \n\eta$
(i.e.\ $T\not\ensuremath{\Vdash} t\not\prec\n\eta$).
\item (Fusion)
If $(T_i)_{i\in \omega}$ is a decreasing sequence in $Q$ and
$\bar F$ is a front-sequence such that
$F_i$ is a front in $T_i$ for all $i$,
then $\lim(\bar F)\leq_Q T_i$.
\item (Pure decision)
If $D\subseteq Q$ is dense, $T\in Q$ and $F$ is a front of $T$,
then there is an $S\leq T$ such that $F$ is a front of $S$
and for every $t\in F$, $S^{[t]}\in D$.
\item $Q$ is proper and $\omega\ho$-bounding.
\end{enumerate}
\end{Lem}
\begin{proof}[Sketch of proof]
(1) and (2) and (4) are clear. (1) and (2) imply (5).
(3): Let $G$ be $Q$-generic over $V$, and define
$X\ensuremath{\coloneqq} \bigcap_{T\in G}\lim(T)$.
$\lim(\Tmax)$ is compact and therefore satisfies the finite
intersection property. So $X$ is nonempty.
For every $T\in G$ and $n\in\omega$ there is exactly one
$t\in T$ of length $n$ such that $T^{[t]}\in G$.
So $X$ has at most one element.
If $r\in V$, then the set of trees $S\in Q^V$ such that
$r\notin \lim(S)$ is dense: If $r$ is a branch of
$T\in Q$ then pick an $m$ such that $\mu_T(r \restriction m)>2$.
Since singletons have measure less than 1,
$r(m)$ has at least two immediate successors in $T$,
and one of them (we call it $t$) is not an initial segment
of $r$. So $S\ensuremath{\coloneqq} T^{[t]}$ forces that $\n\eta\neq r$.
Assume towards a contradiction that $\n\eta\in\lim(T)$ for
some $T\in Q^V\setminus G$. Then this is forced by some $S\in G$.
In particular $S$ can not be a subtree of $T$.
So pick an $s\in S\setminus T$. Then
$S^{[s]}\leq S$ forces that $\n\eta\notin\lim(T)$, a contradiction.
If $T\in Q$ and $t\in T$ then $T^{[t]}$ forces that $t\prec \eta$.
(4) and (5) imply that
$Q$ is $\omega\ho$-bounding and satisfies a version of Axiom A
(with fronts as indices instead of natural numbers).%
\footnote{In the formulation of fusion and pure decision
we could use the classical Axiom
A version as well: Define
$F^T_n$ to be the minimal $n$-dense front, i.e.\
\[F^T_n\ensuremath{\coloneqq}\{t\in T:\, \mu_T(t)>n\,\&\, (\forall s\precneqq t)\,\mu_T(s)\leq n\},\]
and define $T\leq_n S$ by $T\leq S$ and $F^T_n=F^S_n$. It
should be clear how to formulate fusion and pure decision for this notions, and
that this proves Axiom A for $Q$. In \ref{lem:Qisproper} we implicitly
use a different notion:
$T\leq_A S$, meaning $T\leq S$ and $A$
is a front in $T$ and $S$.
The reason is that this notion will be generalized
for the non-wellfounded iteration.}
Se we get properness.
(We will prove a more general case
in \ref{thm:fusioncor}.)
\end{proof}
So a front can be seen as a finite set of (pairwise incompatible) possibilities
for initial segments of the generic real $\n \eta$. In the next section we
will generalize this to sequences of generic reals instead of a single one. We
will call sets of finite sequences of initial segments of these reals
approximations.
Later we will need the following fact:
\begin{Lem}\label{lem:treewithoutoldbranch}
If $S\in Q$ and the forcing $R$ adds a new real $\n r\in 2\ho$,
then
$R$ forces that
there is a $T\leq_Q S$ such that
$\lim(T)\cap V=\emptyset$, and moreover $\lim(T)\cap V$
remains empty in every extension of the universe.
\end{Lem}
\begin{proof}
Assume $S$ corresponds to the front-sequence $\bar F$.
Without loss of generality we can assume that
along every branch in $S$ there is exactly one split
between $F_{n-1}$ and $F_n$
and this split has measure $>n$.
\begin{figure}[tb]
\begin{center}
\scalebox{0.7}{\input{tree1b.pstex_t}}
\end{center}
\caption{\label{fig:tree1} An example for $S$ and its subtree $\protect\n T$ (bold) when $\protect\n r(0)=0$.}
\end{figure}
We define a name of a sequence of finite antichains $(\n F'_n)$
the following way (cf.\ Figure~\ref{fig:tree1}):
If $n$ is even, we ``take all splits'', i.e.\
$\n F'_n$ is the set of nodes in $F_n$ that
are compatible with $\n F'_{n-1}$.
If $n$ is odd, then we add no splittings at all:
for every $s\in \n F'_{n-1}$ we put exactly one successor $t\in F_n$ of $s$
into $\n F'_n$, namely the one continuing the $\n r(\frac{n-1}{2})$-th
successor of the (unique) splitting node over $s$.
It is clear that the sequence $(\n F'_n)$ defines a subtree $\n T$ of $S$.
Assume $V'$ is an arbitrary extension of $V$ containing an
$R$-generic filter $G$ over $V$. If
$\eta\in \lim(\n T[G])\cap V$, then
$\n r[G]$ can be decoded in $V$ using $S$ and $\eta$. This is a contradiction
to $\ensuremath{\Vdash}_R \n r\notin V$.
\end{proof}
We will also need the following family of definable dense subsets
of $Q$:
\begin{Def}\label{def:spl}
Fix a recursive bijection $\psi$ from $\omega$ to $2\hko$.
Assume that $f:\omega\fnto\omega$ is strictly increasing and
that $A\subseteq
\omega$.
\begin{itemize}
\item For $g\in 2\ho$, define $A^{\psi}_g\ensuremath{\coloneqq} \{n\in\omega:\, \psi(n)\prec g\}$.
\item $Q^f_A$ is the set of all $T\in Q$ such that for all splitting nodes $t\in T$,
$\length(t)$ is in the interval $[f(n),f(n+1)-1]$ for some $n\in A$.
\item $T\in Q$ has full splitting with respect to $f$
if for all $n\in\omega$ and $t\in T$ of length $f(n+1)$ there
is an $s\preceq t$ of length at least $f(n)$ such that
$\mu_T(s)>n$.
\item
$D^\text{spl}_f$ is the set of all $T\in Q$ such that
either $T\in Q^f_{A^{\psi}_g}$ for some $g\in 2\ho$
or $T \ensuremath{\perp}_Q S$ for all $g\in 2\ho$ and $S\in Q^f_{A^{\psi}_g}$.
\end{itemize}
\end{Def}
Of course the notions $Q^f_{A}$ and $D^\text{spl}_f$ depend
on the forcing $Q$ (i.e.\ on $\Tmax$ and $\mu$), so maybe it
would be more exact to write $Q^f_{A}[\Tmax,\mu]$ etc. However,
we always assume that the $Q$ is understood. The same
applies to other notation in this paper (e.g. \ref{def:IstrongI} or
\ref{def:positive}).
\begin{Lem}\label{lem:dspl}
Assume that $f:\omega\fnto\omega$ is strictly increasing and $A,B\subseteq \omega$.
\begin{enumerate}
\item If $g\neq g'$, then $A^{\psi}_g\cap A^{\psi}_{g'}$ is finite.
\item\label{item:dspl1}
If $A$ is finite then $Q^f_A=\emptyset$.\quad $Q^f_\omega=Q$.
\item\label{item:dspl2}
$Q^f_A\cap Q^f_B= Q^f_{A \cap B}$.\quad
If $A\subseteq B$, then $Q^f_A\subseteq Q^f_B$.
\item\label{item:dspl3}
If $T\leq_Q S$ and $S\in Q^f_A$ then $T\in Q^f_A$.
\item\label{item:dspl4}
For every $T\in Q$ there is a strictly increasing
$f$ such that $T$ has full splitting with respect to $f$.
\item\label{item:dspl5}
If $T\in Q$ has full splitting with respect to $f$
and $\card{A}=\al0$ then there is an $S\leq_Q T$
such that $S\in Q^f_A$.
\item\label{item:dspl6}
$D^\text{spl}_f$ is an (absolute definition of an) open dense subset of $Q$
(using the parameters $f$, $\Tmax$ and $\mu$).\footnote{$X_0\ensuremath{\coloneqq}
\{A^{\psi}_g:\, g\in 2\ho\}$
is an almost disjoint family, but not
maximal. So of course $Q(X_0)\ensuremath{\coloneqq} \bigcup_{A\in X_0} Q^f_A\subset Q$ is not
dense. We add the incompatible conditions to get the dense set
$D^\text{spl}_f$. One could ask whether $Q(X)$ is dense for a m.a.d. family
$X$. The following holds:
\begin{enumerate}
\item For every $f$ there is a m.a.d. family $X$ such that $Q(X)$
is not dense.
\item (CH) For every $f$ there is a m.a.d. family $X$
such that $Q(X)$ is dense.
\end{enumerate}
Proof: Fix $f$.
A node $s\in\Tmax$ has level $m$ if $f(m)\leq \length(s)<f(m+1)$. $S\in Q$ has
unique splitting if $S$ has at most one splitting point of level $n$ for all
$n\in\omega$. For every $T\in Q$ there is an $S\leq_Q T$ with unique
splitting.
For (a), fix a $T\in Q$ with unique splitting.
Set $Y\ensuremath{\coloneqq}\{A\in[\omega]^\al0:\, (\forall S\leq_Q T)\,S\notin Q^f_A\}$.
$Y$ is open dense in $([\omega]^\al0,\subseteq)$, therefore there is
a m.a.d. $X\subseteq Y$.
For (b), list $Q$ as $(T_\alpha)_{\alpha\in\om1}$, and build
$B_\alpha\in[\omega]^\al0$ by induction on $\alpha\in\om1$: Find an $S\leq_Q
T_\alpha$ with unique splitting. If some $S'\leq_Q S$ is in $Q^f_{B_\beta}$
($\beta<\alpha$) (or equivalently in $Q^f_{\bigcup_{i\in l} B_{\beta_i}}$
for some $l\in\omega$, $\beta_0,\dots,\beta_{l-1}<\alpha$),
then just pick any almost disjoint $B_\alpha$.
Otherwise
enumerate $(B_\beta)_{\beta\in\alpha}$ as $(C_n)_{n\in\omega}$,
and construct $B_\alpha$ and $S'\leq_Q S$ inductively:
At stage $n$, add a split of $S$ to
$S'$ whose level is not in $\bigcup_{m\leq n} C_m$, and use some
bookkeeping to guarantee that $S'\in Q$. Let $B_\alpha$
be the set of splitting-levels of $S'$.}
\item\label{item:dspl7}
In any extension $V'$ of $V$ the following holds:
If $r\in 2\ho\setminus V$ and
$S\in Q^f_{A^{\psi}_{r}}$, then
$T\ensuremath{\perp}_Q S$ for all $T\in V\cap D^\text{spl}_f$.
\end{enumerate}
\end{Lem}
\begin{proof}
(1)--(4) and (6) are clear.
(5):
Let $T$ be an element of $Q$.
Assume we already constructed $f(n)$. Let $N$ be the maximum
of $\mu_T(t)$ for $t\in T\restriction f(n)$.
There is an $N+n+1$-dense front $F$ in $T$.
Let $f(n+1)$ be the maximum of $\{\length(t):\, t\in F\}$.
(7):
``$T$ is incompatible with all $S\in Q^f_{A^{\psi}_g}$'' is absolute, since it is equivalent to
\[(\forall g\in 2^\omega)\, (\forall S\subseteq \Tmax)\ \left[S\notin Q^f_{A^{\psi}_g}\ \vee\
T\ensuremath{\perp}_Q S\right],\]
which is a $\mP11$ statement.
(8):
Let $r\in 2\ho\setminus V$
and $T\in V\cap D^\text{spl}_f$.
If $T\in Q^f_{A^{\psi}_g}$ for some $g\in 2^\omega\cap V$,
then $g\neq r$, so $A^{\psi}_g\cap A^{\psi}_r$ is finite and
$Q^f_{A^{\psi}_{r}}\cap Q^f_{A^{\psi}_{g}}$ is empty.
If on the other hand $T$ is incompatible with all $S\in Q^f_{A^{\psi}_g}$ in $V$
then this holds in $V'$ as well.
\end{proof}
Assume $f'(n)\geq f(n)$ for all $n\in\omega$. Define $h(n)$ by induction:
$h(n+1)\ensuremath{\coloneqq} f'(h(n)+1)$.
If $T$ has full splitting with respect to $f$, then $T$ has full splitting with respect to $h$:
$h(n)\leq f(h(n))$, since $f$ is strictly increasing.
$f(h(n)+1)\leq f'(h(n)+1)=h(n+1)$, and there
are $h(n)$-dense splits between
the levels $f(h(n))$ and $f(h(n)+1)$. So there are $n$-dense splits between the levels
$h(n)$ and $h(n+1)$. So we get:
\begin{Lem}\label{lem:fullsplitinV}
If $V'$ is an $\omega\ho$-bounding extension of $V$
and $T\in Q^{V'}$, then there is a strictly increasing $h\in V$ such that
(in $V'$) $T$ has full splitting with respect to $h$.
\end{Lem}
\section{A non-wellfounded Iteration}\label{sec:nwi}
In this section we introduce a general construction to iterate lim-sup
tree-forcings $Q_i$
(as defined in the last section) along non-wellfounded linear orders $I$. It
turns out that the limit $P$ is proper, $\omega\ho$-bounding and has other nice
properties similar to the properties of $Q_i$ itself. If $I$ is wellfounded,
then $P$ is equivalent to the usual countable support iteration of
(the evaluations of the definitions) $Q_i$.
There is quite some literature about non-wellfounded iteration.
E.g. Jech and Groszek~\cite{jech-groszek} investigated the wellfounded
but non-linear iteration of Sacks forcings. Building on this,
Kanovei~\cite{kanovei_illfounded_sacks} and
Groszek~\cite{MR1295981} develop non-wellfounded iterations of
Sacks. The description of the construction there is very close
to what we do here, but the realization is quite different.
Brendle~\cite{brendle_mad_fam_it} presented finite-support non-wellfounded
iteration constructions (extracted from Shelah's method of iterations along
smooth templates~\cite{Sh:700}).
\begin{Def}
Let $I^\infty$ be a linear order containing a maximal element $\infty$. For
$i\in I^\infty$ we set \mbox{$I_{<i}\ensuremath{\coloneqq} \{j\in I:\, j<i\}$} and
$I\ensuremath{\coloneqq} I_{<\infty}$.
\end{Def}
For every $i\in I$ we fix a finitely splitting lim-sup tree-forcing $Q_i$
(to be more exact, fix a pair $\Tmax^i,\mu^i$).
In the application of this paper, each $Q_i$ will be the same forcing $Q$.
\begin{Def}\label{def:precondition}(Pre-condition)
We call $p$ a pre-condition, if $p$ is a function, the domain of $p$
is a countable\footnote{this includes finite and empty.} subset of $I$,
and for each $i\in\dom(p)$,
$p(i)$ consists of the following:\\
\phantom{xxx}$\pD p i$, a countable subset of $\dom(p)\cap I_{<i}$, and\\
\phantom{xxx}a (definition of a) Borel function $\pB p i:
\left(\omega^\omega\right)^{\pD p i}\fnto Q_i$
\end{Def}
The idea is that we calculate the
condition $\pB p i\in Q_i$ using
countably many generic reals
$(\eta_j)_{j\in\pD p i}$
that have already been produced at stage $i$.
We will only be interested in $B$'s that are
continuous (on a certain Borel set),
i.e.\ if we want to know $\pB p i$ up to some finite height
we only have to know $(\eta_i\restriction m)_{i\in u}$
for some finite $u$ and $m\in\omega$.
Moreover, we will assume that we will have ``wellfounded continuity
parts'':
\begin{figure}[tb]
\begin{center}
\scalebox{0.4}{\input{approx.pstex_t}}
\end{center}
\caption{\label{fig:approx} An approximation $\mathfrak{g}$:
$u=\{i,j\}$, $\Tmax^i=2\hko$, $\Tmax^j=3\hko$.\protect\\
$\text{Pos}(\mathfrak{g})=\text{Pos}_{\leq j}(\mathfrak{g})=\{(a_i^1,b_j^0),(a_i^1,b_j^1),(a_i^0,b_j^2),(a_i^0,b_j^3),(a_0^1,b_j^4)\}$.}
\end{figure}
\begin{Def}\label{def:approximation} (Approximation)
\begin{itemize}
\item\label{item:approx} $\mathfrak{g}$ is an approximation, if
$\mathfrak{g}$ is a function with finite domain
$u\subseteq I$ of the following form:
Let $i_0$ be the smallest element of $u$.
$\text{Pos}_{<i_0}(\mathfrak{g})\ensuremath{\coloneqq}\{\emptyset\}$, and
(by induction on $i\in u$):
$\mathfrak{g}(i)$ is a function from $\text{Pos}_{<i}(\mathfrak{g})$
to finite antichains in $\Tmax^i$, and\\
$\text{Pos}_{\leq i}(\mathfrak{g})\ensuremath{\coloneqq}\{\bar a\cup (i,a_i):\,
\bar a\in \text{Pos}_{<i}(\mathfrak{g}),\, a_i\in \mathfrak{g}(i)(\bar a)\}$.
\item
For any $i\in I^\infty$, $\text{Pos}_{< i}(\mathfrak{g})$ is defined as
$\text{Pos}_{\leq j}(\mathfrak{g})$, where $j=\max(\dom(\mathfrak{g})\cap I_{<i})$
(or as $\{\emptyset\}$, if $\dom(\mathfrak{g})\cap I_{<i}$ is empty).
$\text{Pos}(\mathfrak{g})\ensuremath{\coloneqq}\text{Pos}_{<\infty}(\mathfrak{g})$.
$\text{Pos}(\mathfrak{g})$ is called the set of possibilities of $\mathfrak{g}$.
\item\label{item:incr_def_of_approx}
If $i\notin \dom(\mathfrak{g})$ or $\bar a\notin \text{Pos}_{<i}(\mathfrak{g})$
we set $\mathfrak{g}(i)(\bar a)\ensuremath{\coloneqq}\{\langle\rangle\}$
(i.e.\ the front in $\Tmax$ consisting only of the root.
This corresponds to ``no information'').
\item
Let $\mathfrak{g}$ be an approximation,
$J\subset I$, and $\bar \eta=(\eta_i)_{i\in J}$ a
sequence of reals.
Then $\bar \eta$ is compatible with
$ \mathfrak{g} $, if there is an $\bar a\in \text{Pos}(\mathfrak{g})$
such that $a_i\prec \eta_i$ for all
$i\in \dom(\mathfrak{g})\cap J$.
If in addition $J\supseteq \dom(\mathfrak{g})$, then this $\bar a$ is uniquely defined,
and we set \mbox{$\mathfrak{g}(i)(\bar \eta)\ensuremath{\coloneqq}\mathfrak{g}(i)(\bar a\restriction I_{<i})$}.
In the same way we
define $\bar b$ to be compatible with $\mathfrak{g}$ for
$\bar b=(b_i)_{i\in J}$ a sequence of elements of $\omega\hko$.
Then $\mathfrak{g}(i)(\bar b)\ensuremath{\coloneqq}\mathfrak{g}(i)(\bar a\restriction I_{<i})$
is well-defined if $J\supseteq \dom(\mathfrak{g})$ and the elements of
$\bar b$ have sufficient length.
\item
If $\mathfrak{g}$ and $\mathfrak{g}'$ are both approximations, then
$\mathfrak{g}'$ is stronger than $\mathfrak{g}$ if $\dom(\mathfrak{g}')\supseteq \dom(\mathfrak{g})$
and for all
$\bar b\in \text{Pos}(\mathfrak{g}')$
there is an $\bar a\in \text{Pos}(\mathfrak{g})$ such that
$\bar b\succeq \bar a$ (i.e.\ $b_i \succeq a_i$ for all $i\in \dom(\mathfrak{g})$).
In this case $\bar a$ is uniquely defined and called
$\bar b\restriction \mathfrak{g}$.
\item
$\mathfrak{g}'$ is purely stronger than $\mathfrak{g}$
if $\mathfrak{g}'$ is stronger than $\mathfrak{g}$ and
for all $i\in\dom(\mathfrak{g})$
and $\bar b\in \text{Pos}_{<i }(\mathfrak{g}')$
the front $\mathfrak{g}'(i)(\bar b)$ is purely stronger
than $\mathfrak{g}(i)(\bar b\restriction \mathfrak{g})$.
\item
For $u\subseteq \dom(\mathfrak{g})$,
$\maxlength_{u}(\mathfrak{g})$ is
$\max(\{\length(a_i):\, i\in u,\bar a\in\text{Pos}(\mathfrak{g})\})$.\\
$\maxlength(\mathfrak{g})$ is $\maxlength_{\dom(\mathfrak{g})}(\mathfrak{g})$.
$\minlength(\mathfrak{g})$ is defined analogously.
\item $\mathfrak{g}$ is $n$-dense for
$i\in I$, if $i\in \dom(\mathfrak{g})$
and for all $\bar a\in \text{Pos}_{<i}(\mathfrak{g})$, $\mathfrak{g}(i)(\bar a)$
is $n$-dense for $Q_i$ (see \ref{def:nsplitting}).
\item For all
$\bar a=(a_i)_{i\in u}$
such that $a_i\in \Tmax^i$ there is a (unique) approximation
$\bar g$ such that $\text{Pos}(\mathfrak{g})=\{\bar a\}$. We will call this
approximation $\bar a$ as well.
\end{itemize}
\end{Def}
So if $\mathfrak{g}'$ is stronger than $\mathfrak{g}$,
then
all $\bar \eta$ compatible with $\mathfrak{g}'$ are compatible with $\mathfrak{g}$.
Clearly ``stronger'' and
``purely stronger''
are transitive.
\begin{Def} (Approximation to $p$)
Let $p$ be a pre-condition.
\begin{itemize}
\item
$\mathfrak{g}$ approximates $p$, or: $\mathfrak{g}$ is a $p$-approximation,
if $\dom(\mathfrak{g})\subseteq \dom(p)$ and $\mathfrak{g}$ is an
approximation with the following property: If $i\in \dom(\mathfrak{g})$, $\bar a\in
\text{Pos}_{<i}(\mathfrak{g})$, and $(\eta_j)_{j\in\pD p i}$ is compatible with $\bar a$,
then $\mathfrak{g}(i)(\bar a)$ is a front in $\pB p i(\bar \eta)$.
\item
$\mathfrak{g}$ is an indirect approximation to $p$ witnessed by $\mathfrak{g}'$,
if $\mathfrak{g}'$ approximates $p$ and $\mathfrak{g}'$ is purely stronger than $\mathfrak{g}$.
\end{itemize}
\end{Def}
\begin{Fact}
If $\mathfrak{g}$ (indirectly) approximates $p$, $\mathfrak{g}'$ approximates $p$,
and $\mathfrak{g}'$ is stronger than $\mathfrak{g}$, then $\mathfrak{g}'$ is purely
stronger than $\mathfrak{g}$.
\end{Fact}
Now we can define the forcing $P$, the non-wellfounded
countable support limit along $I$:
\begin{Def} (The nwf-iteration $P=\nwflim_I(Q_i)$)
\begin{itemize}
\item $p\in P$ means:\\
$p$ is a pre-condition, and
for all $u\subseteq \dom(p)$, $i\in u$ and $n\in \omega$
there is a $p$-approximation $\mathfrak{g}$ such that $\dom(\mathfrak{g})\supseteq u$, $\mathfrak{g}$ is
$n$-dense for $i$, and $\minlength_{u}(\mathfrak{g})>n$.
\item For $p,q\in P$, $q\leq p$ means:\\
for all $p$-approximations $\mathfrak{g}$
there is a $q$-approximation $\mathfrak{h}$ which is stronger than $\mathfrak{g}$
(so in particular, $\dom (q)\supseteq \dom(p)$).
\item $q\leq_{\mathfrak{g}} p$ if $q\leq p$ and $\mathfrak{g}$ indirectly approximates $p$
and $q$.
\end{itemize}
\end{Def}
As we will see in
\ref{thm:fusioncor}(\ref{item:fusiocorCSI}), $P$ is just the usual
countable support iteration of $Q_i$ in case that $I$ is
well-ordered.
\begin{Facts}
\begin{itemize}
\item
$\leq$ is transitive, and for a fixed approximation $\mathfrak{g}$
the relation $\leq_{\mathfrak{g}}$ is transitive as well. Also, if $\mathfrak{g}'$ is
purely stronger than $\mathfrak{g}$ then $\leq_{\mathfrak{g}'}$ implies $\leq_{\mathfrak{g}}$.
\item
For every $p\in P$, the approximations of $p$ are directed:
If $\mathfrak{g}$ and $\mathfrak{g}'$ both (indirectly)
approximate $p$, then there is a $\mathfrak{g}''$ approximating $p$ that is
(purely) stronger than both $\mathfrak{g}$ and $\mathfrak{g}'$ (and every
$p$-approximation $\mathfrak{g}''$ has this property if
$\dom(\mathfrak{g}'')\supseteq \dom(\mathfrak{g})\cup\dom(\mathfrak{g}')$ and if
$\minlength_{\dom(\mathfrak{g})\cup\dom(\mathfrak{g}')}(\mathfrak{g}'')$
is large enough).
\end{itemize}
\end{Facts}
So in particular for every $p\in P$ there is an approximating sequence:
\begin{Def}
An approximating sequence for $p\in P$
is a sequence $(\mathfrak{g}_n)_{n\in\omega}$ of approximations of $p$
such that $\mathfrak{g}_{n+1}$ is purely stronger than $\mathfrak{g}_n$, $\mathfrak{g}_{n+1}$ is
$n$-dense for each $i\in\dom(\mathfrak{g}_n)$, and
\mbox{$\dom(p)=\bigcup \dom(\mathfrak{g}_n)$}.
\end{Def}
An approximating sequence contains all
relevant information about $p$.
In particular, $\mathfrak{g}$ is an indirect approximation to $p$ iff
there is an $n$ such that $\mathfrak{g}_n$ is purely stronger than $\mathfrak{g}$.
So if $p$ and $q$ both have the approximating sequence $(\mathfrak{g}_n)_{n\in\omega}$,
then $p\leq q$ and $q\leq p$ (and $p\leq_\mathfrak{g} q$ if $\mathfrak{g}$ indirectly
approximates $p$ or $q$).
Approximating sequences provide an equivalent definition for $P$:
\begin{Def} (Alternative definition of the nwf-iteration $P$)
Define the p.o. $P'$ as follows:
$\bar{\mathfrak{g}}\in P'$ iff $\bar{\mathfrak{g}}$ is a sequence of approximations
$(\mathfrak{g}_n)_{n\in\omega}$ such that
$\mathfrak{g}_{n+1}$ is purely stronger than $\mathfrak{g}_n$ and
$n$-dense for every $i\in\dom(\mathfrak{g}_n)$.
$\bar{\mathfrak{h}}<\bar{\mathfrak{g}}$ iff for every $n$ there is an $m$ such that
$\mathfrak{h}_m$ is stronger than $\mathfrak{g}_n$.
\end{Def}
\begin{Lem}\label{lem:sequofapprox}
There is a dense embedding%
\footnote{$\phi$ is even an isomorphism modulo $=^*$, where
$p=^*q$ if $q\leq p$ and $q\leq p$.}
$\phi:\, P'\fnto P$.
\end{Lem}
\begin{proof}
Given a sequence $\bar{\mathfrak{g}}\in P'$, define
$p=\phi(\bar{\mathfrak{g}})$ the following way:
$\dom(p)=\bigcup \dom(\mathfrak{g}_n)$. For $i\in \dom(p)$,
set $\pD p i\ensuremath{\coloneqq} \dom(p)\cap I_{<i}$.
Define $T= \pB p i(\bar \eta)$ as follows:
If $\bar \eta$ is compatible with all
$\mathfrak{g}_n$, then let $T$ be
$\{t\in \Tmax^i:\, (\exists n\in\omega)\, t\preceq \mathfrak{g}_n(i)(\bar\eta)\}$.
Otherwise, let $n$ be maximal such that $\bar \eta$ is compatible
with $\mathfrak{g}_n$, and let $T$ be $\{t\in \Tmax^i:\,
(\exists s\in \mathfrak{g}_n(i)(\bar \eta))\, t\ensuremath{\parallel} s\}$. Clearly,
$\pB p i\in Q_i$, $\pB p i$ is
a Borel function, and each $\mathfrak{g}_n$ approximates $p$.
Therefore $(\mathfrak{g}_n)_{n\in\omega}$ is an approximating sequence for $p\in P$.
It is clear that $\phi$ preserves the order.
Let $\psi$ map $p\in P$ to any approximating sequence for $p$.
$\psi: P\fnto P'$ preserves order as well and
$\phi(\psi(p))=^* p$. Therefore $\phi$ is a dense embedding.
\end{proof}
To summarize: Every $p\in P$ corresponds to a
purely increasing sequence $(\mathfrak{g}_n)$ of approximations such that
$\bigcup\dom(\mathfrak{g}_n)=\dom(p)$ and $\mathfrak{g}_{n+1}$ is $n$-dense for
$\dom(\mathfrak{g}_n)$.
The approximating sequences immediately prove a version of fusion:
\begin{Lem}\label{lem:fusion}
(Fusion)
Assume that $(p_n)_{n\in\omega}$
is a sequence of conditions,
$(\mathfrak{g}_n)_{n\in\omega}$ a sequence of approximations, and $i_n\in\dom(\mathfrak{g}_n)$
such that:\\
\phantom{xxx}$p_{n+1}\leq_{\mathfrak{g}_n} p_n$,\\
\phantom{xxx}$\mathfrak{g}_{n+1}$ is purely stronger than $\mathfrak{g}_n$ and $n$-dense for $i_n$,\\
\phantom{xxx}$(i_n)_{n\in\omega}$ covers $\bigcup \dom(p_n)$ infinitely
often.\\
Then there is a condition $p_\omega$ such that
$p_\omega\leq_{\mathfrak{g}_n} p_n$
for all $n$.
\end{Lem}
\begin{proof}
We already know that the sequence $(\mathfrak{g}_n)_{n\in\omega}$ of approximations
defines a condition $p_\omega$
such that each $\mathfrak{g}_n$ approximates $p_\omega$.
If $\mathfrak{h}$ approximates $p_n$, then some $\mathfrak{g}_m$
is stronger than $\mathfrak{h}$. $\mathfrak{g}_m$ approximates
$p_\omega$, so $p_\omega\leq p_n$.
\end{proof}
\begin{Notes}\label{notes:conditions}
\begin{enumerate}
\item\label{tmp4_11} If $\mathfrak{g}$ indirectly approximates $p$,
then there is a $q=^*p $ such that $\mathfrak{g}$ approximates $q$.
(Just let $q$ correspond to an approximating sequence of
$p$ starting with $\mathfrak{g}_0=\mathfrak{g}$.)
\item
It doesn't matter whether the $\mathfrak{g}_n$ in an approximating sequence
are approximations to $p$ or just indirect approximations.
\item
It doesn't matter whether $\mathfrak{g}_{n+1}$ proves $n$-density
for every $i\in\dom(\mathfrak{g}_n)$ or for just some $i_n$,
provided that the sequence $(i_n)_{n\in\omega}$ covers $\bigcup\dom(\mathfrak{g}_n)$
infinitely often.
\item
In Definition~\ref{def:precondition}
of pre-condition, instead of requiring $\pB p i$
to be a function into $Q$, we could have defined
$\pB p i$ to be a function to subtrees of
$\Tmax$. The additional ``$n$-dense'' requirements on a condition
guarantee $\pB p i(\bar \eta)\in Q$
anyway (for generic sequences $\bar \eta$).
\item
Every approximation $\mathfrak{g}$ can be interpreted as
a condition in $P$, by
\[\pB \mathfrak{g} i(\bar \eta)\ensuremath{\coloneqq} \{t:\, t\ensuremath{\parallel} \mathfrak{g}(i)(\bar \eta)\}
\text{ for }i\in\dom(\mathfrak{g}).\]
(Recall that $\mathfrak{g}(i)(\bar \eta)\ensuremath{\coloneqq} \{\langle\rangle\}$
if $\bar \eta$ is incompatible with $\mathfrak{g}$.)
Then $\mathfrak{g}$ approximates itself.
\item\label{tmp3_4}
For any approximation $\mathfrak{g}$ and $u\subseteq I$ finite
we can assume $u\subseteq \dom\mathfrak{g}$: Just set
\mbox{$\mathfrak{g}(i)=\{\langle\rangle\}$} for $i\notin\dom\mathfrak{g}$.
This is consistent with the notational convention in
\ref{def:approximation}.
\item\label{ijewqwqtqwt}
If $\mathfrak{g}$ and $\mathfrak{h}$ are approximations, we can assume without
loss of generality that $\dom(\mathfrak{g})=\dom(\mathfrak{h})$.
\item
For any $U\subseteq I$ countable and $p\in P$
we can assume without loss of generality
that \mbox{$\dom(p)\supseteq U$}. This is clear if
$p$ is interpreted as a sequence of Borel-functions
(just set \mbox{$\pB p i=\Tmax^i$} for \mbox{$i\notin\dom(p)$}).
If
$p$ is interpreted as sequence $(\mathfrak{g}_n)_{n\in\omega}$ of
approximations, we have to set $\mathfrak{g}_n(i)$ to
be $\Tmax^i\restriction k(n)$ for some sufficiently large $k(n)$.
(Using $\{\langle \rangle\}$ does not work here, since it
does not prove $n$-density.)
\end{enumerate}
\end{Notes}
\begin{figure}[tb]
\begin{center}
\scalebox{0.4}{\input{conjunctions.pstex_t}}
\end{center}
\caption{\label{fig:conjunction}}
\end{figure}
We now investigate conjunctions of antichains (cf.\
Figure~\ref{fig:conjunction}), of approximations and of a condition and an
approximation.
\begin{Def}
\begin{itemize}
\item
Let $F$ and $F'$ be finite antichains in $\Tmax^i$. Set
\[A\ensuremath{\coloneqq} \{t\in F:\, (\exists s\in F')\, t\succeq s\}\cup \{s\in F':\, (\exists t\in F)\, s\succeq t\}.\]
Either $A$ is empty (then $F$ and $F'$ are
called incompatible, and $F\wedge F'$ is undefined),
or $A$ is a nonempty antichain
(then $F$ and $F'$ are compatible, and $F\wedge F'\ensuremath{\coloneqq} A$).
\item
Now we define $h=\mathfrak{g}\wedge\mathfrak{g}'$
for two approximations $\mathfrak{g}$ and $\mathfrak{g}'$.
Without loss of generality\footnote{According to \ref{notes:conditions}(\ref{ijewqwqtqwt}).}
$\dom(\mathfrak{g})=\dom(\mathfrak{g}')=u$.
Let $\mathfrak{h}$ be the approximation
with domain $u$ defined inductively
for all $i\in u$: For $\bar t\in \text{Pos}_{<i}(\mathfrak{h})$ set
\[ \mathfrak{h}(i)(\bar t)\ensuremath{\coloneqq} \mathfrak{g}(i)(\bar t\restriction \dom(\mathfrak{g}))\wedge
\mathfrak{g}'(i)(\bar t\restriction \dom(\mathfrak{g}')),\text{ if defined}.\]
If at any $i\in u$ this $\mathfrak{h}(i)$ is undefined, then $\mathfrak{g}$, $\mathfrak{g}'$
are called incompatible and $\mathfrak{g}\wedge\mathfrak{g}'$ is undefined.
Otherwise set $\mathfrak{g}\wedge\mathfrak{g}'\ensuremath{\coloneqq} \mathfrak{h}$.
\end{itemize}
\end{Def}
\begin{Facts}
\begin{itemize}
\item If there is an approximation stronger than both $\mathfrak{h}$ and $\mathfrak{g}$,
then $\mathfrak{h}\wedge \mathfrak{g} $ is defined; and if $\mathfrak{h}\wedge \mathfrak{g} $ is defined then
it is the weakest approximation stronger than both $\mathfrak{h}$ and $\mathfrak{g}$.
\item If $\mathfrak{g}$ approximates $p$,
$\mathfrak{h}$ approximates $q$, and $\mathfrak{h}$, $\mathfrak{g}$ are incompatible,
then $p\ensuremath{\perp}_Q q$.
\item
If $\mathfrak{g}'$ is stronger than $\mathfrak{g}$, then $\mathfrak{h}\wedge \mathfrak{g}'$
(if defined) is stronger than $\mathfrak{h}\wedge \mathfrak{g}$.
\end{itemize}
\end{Facts}
\begin{uNotes}
\begin{itemize}
\item
It is possible that $\mathfrak{h}\wedge \mathfrak{g}$ is defined but
$\mathfrak{h}\wedge \mathfrak{g}'$ is not for some $\mathfrak{g}'$ stronger than $\mathfrak{g}$.
\item
If $\mathfrak{g}'$ is purely stronger than $\mathfrak{g}$, and $\mathfrak{h}\wedge \mathfrak{g}'$
is defined, then $\mathfrak{h}\wedge\mathfrak{g}'$ does {\em not} have to be
purely stronger than $\mathfrak{h}\wedge \mathfrak{g}$ (see Figure~\ref{fig:conjunction}).
\item
However, if for $u\ensuremath{\coloneqq} \dom(\mathfrak{h})\cap\dom(\mathfrak{g})$,
$\minlength_u(\mathfrak{g})>\maxlength_u(\mathfrak{h})$,
if $\mathfrak{g}\wedge \mathfrak{h}$ exists and if $\mathfrak{g}'$ is purely stronger
than $\mathfrak{g}$, then
$\mathfrak{g}'\wedge \mathfrak{h}$ exists and is purely stronger
than $\mathfrak{g}\wedge \mathfrak{h}$.
\end{itemize}
\end{uNotes}
It is possible to define $q\wedge p$ for $p\ensuremath{\parallel} q\in P$
(using some kind of Cantor Bendixon algorithm). However,
we will only use the version $p\wedge \mathfrak{h}$ for an approximation
$\mathfrak{h}$: Assume that $p$ corresponds to the sequence $(\mathfrak{g}_n)$ of
approximations. If for some $n$, $\mathfrak{h}$ and $\mathfrak{g}_n$
are incompatible, then there
is no $q\leq p,\mathfrak{h}$. On the other hand,
if $\mathfrak{h}$ is compatible with all $\mathfrak{g}_n$, then for $n$
large enough $\mathfrak{g}_{n+1}\wedge \mathfrak{h}$
is purely stronger than $\mathfrak{g}_n\wedge \mathfrak{h}$ and
$n$-dense for all $i\in\dom(\mathfrak{g}_n)$. So
$(\mathfrak{g}_n\wedge \mathfrak{h})_{n\in\omega}$ is an approximating sequence
corresponding to some $p\wedge \mathfrak{h}\in P$.
\begin{Def}
Let $p$ correspond to the approximating sequence $\bar \mathfrak{g}$.
$p$ is compatible with $\mathfrak{h}$
if
$\mathfrak{h}$ and $\mathfrak{g}_n$ are compatible
for all $n$. In this case
$p\wedge \mathfrak{h}\in P$ is defined by the approximating
sequence $(\mathfrak{g}_n\wedge \mathfrak{h})_{n\geq m}$ for some suitable $m$.
\end{Def}
Then we get the following:
\begin{Facts}
Let $p\in P$ be compatible to the approximation $\mathfrak{h}$.
\begin{itemize}
\item $p\wedge \mathfrak{h}\in P$, $p\wedge\mathfrak{h}\leq p$. If $\mathfrak{h}$
(indirectly) approximates $p$, then $p\wedge\mathfrak{h}=^*p$.
\item If $\mathfrak{g}$ indirectly approximates $p$ and the domain
and length of $\mathfrak{g}$ is large enough,
then $\mathfrak{g}\wedge \mathfrak{h}$ indirectly approximates
$p\wedge\mathfrak{h}$.
\item If $q\leq p$ and $\mathfrak{g}$ is compatible to $q$, then
$\mathfrak{g}$ is compatible to $p$ and $q\wedge \mathfrak{g}\leq p\wedge\mathfrak{g}$.
\item If $\mathfrak{h}$ is stronger than $\mathfrak{g}$ and $\mathfrak{h}$ is compatible
to $p$, then $\mathfrak{g}$ is compatible to $p$ and
$p\wedge \mathfrak{h}\leq p\wedge \mathfrak{g}$.
\item If $q\leq p$ and $\mathfrak{g}$ indirectly approximates $q$
then $\mathfrak{g}$ is compatible with $p$ and $q\leq p\wedge \mathfrak{g}$.
\end{itemize}
\end{Facts}
An important special case is that of a sub-approximation:
\begin{Def}
$\mathfrak{h}$ is
sub-approximation of $\mathfrak{g}$ if $\text{Pos}(\mathfrak{h})\subseteq \text{Pos}(\mathfrak{g})$.
\end{Def}
In particular a sub-approximation of $\mathfrak{g}$ is stronger than $\mathfrak{g}$.
\begin{Lem} (Sub-approximation)
Assume that $\mathfrak{g}$ indirectly approximates $p$ and that $\mathfrak{h}$ is a
sub-approximation of $\mathfrak{g}$.
Then $\mathfrak{h}$ is compatible
with $p$.
If $q\leq p$ and $\mathfrak{h}$ approximates $q$,
then there is an $r\leq_\mathfrak{g} p$
such that $r\wedge h=^* q$.
\end{Lem}
\begin{proof}
We will define $r$ so that it is identical to $q$ ``below $\mathfrak{h}$''
and identical to $p$ otherwise.
More exactly:
Without loss of generality we can assume that $p$
corresponds to $(\mathfrak{g}_n)_{n\in\omega}$,
\mbox{$\mathfrak{g}_0=\mathfrak{g}$}, $q$ corresponds to $(\mathfrak{h}_n)_{n\in\omega}$, $\mathfrak{h}_0=\mathfrak{h}$,
$\mathfrak{h}_n$ is stronger than $\mathfrak{g}_n$, and $\dom(\mathfrak{h}_n)=\dom(\mathfrak{g}_n)=u_n$.
Let $\af_n$ be an
approximation with domain $u_n$, $i_0$ minimal in $u_n$. So
$\text{Pos}_{<i_0}(\af_n)=\{\emptyset\}$. By induction on $i\in u_n$, define for
all $\bar a\in\text{Pos}_{<i}(\af_n)$
\[\af_n(i)(\bar a)\ensuremath{\coloneqq}\begin{cases}
\mathfrak{g}_n(i)(\bar a\restriction \mathfrak{g}_n)&
\text{if }\bar a\text{ is incompatible with }\mathfrak{h}_n,\\
\mathfrak{h}_n(i)(\bar a) \cup \{t\in \mathfrak{g}_n(i)(\bar a\restriction \mathfrak{g}_n):\, t\ensuremath{\perp}
\mathfrak{h}_n(i)(\bar a)\}&
\text{otherwise.}
\end{cases}\]
It is clear that the possibilities of
$\af_n$ follow $\mathfrak{h}_n$ up to some $i\in\dom \mathfrak{g}_n$
and from then on follow $\mathfrak{g}_n$.
To be more exact: $\bar a\in \text{Pos}(\af_n)$ iff $\bar a\restriction
\mathfrak{g}_n\in\text{Pos}(\mathfrak{g}_n)$ and
for some $i\in\dom(\mathfrak{g}_n)\cup\{\infty\}$, $\bar a\restriction I_{<i}$
is in $\text{Pos}(\mathfrak{h}_n)$ and either $i=\infty$ or $a_i \ensuremath{\perp}
\mathfrak{h}_n(i)(\bar a)$. From this it follows that
$\af_n$ is purely stronger than $\mathfrak{g}_n$, and that the
$\af_n$ are an approximating sequence
(converging to some $r\leq p$).
So $r\leq_\mathfrak{g} p$,
$\af_n\wedge \mathfrak{h}=\mathfrak{h}_n$, and $r\wedge \mathfrak{h}=^* q$.
\end{proof}
A special case of a sub-approximation is a
singleton:
\begin{Def}
Assume that $\mathfrak{g}$ indirectly approximates $p$ and $\bar a\in \text{Pos}(\mathfrak{g})$.
Then the approximation $\bar a$ is compatible with $p$.
Define $p^{[\bar a]}$ to be $p\wedge \bar a$.
\end{Def}
So we get:
\begin{Cor}\label{cor:restrtoa}(Specialization and pure decision)
Assume that $\mathfrak{g}$ indirectly approximates $p$ and that $\bar a\in\text{Pos}(\mathfrak{g})$.
\begin{enumerate}
\item $p^{[\bar a]}\in P$, $p^{[\bar a]}\leq p$ and
$\bar a$ indirectly approximates $p^{[\bar a]}$.
If $q\leq p$ and $\bar a$ indirectly approximates
$q$, then $q\leq p^{[\bar a]}$.
\item If $q\leq_{\mathfrak{g}}p$, then
$q^{[\bar a]}\leq p^{[\bar a]}$.
\item\label{item:rbetween} If $q\leq p^{[\bar a]}$
then there is a $r\leq_{\mathfrak{g}} p$ such that $r^{[\bar a]}=^*q$.
\item\label{item:specialipredense}
The set
$\{p^{[\bar a]}:\, \bar a\in\text{Pos}(\mathfrak{g})\}$ is predense below $p$.
\item\label{item:singletondense}
We can densely determine $\n\eta_i$ up to $n$,
i.e.\ for all $i\in I$, $n\in \omega$ the following set is dense:
$\{p\in P:\, (\exists a\in\omega^n)\, (i,a)\text{ approximates }p\}$.
\item\label{item:puredecision} (Pure decision)
If $D\subseteq P$ is open dense, and $\mathfrak{g}$ indirectly approximates
$p$, then
there is an $r\leq_{\mathfrak{g}} p$ such that
$r^{[\bar a]}\in D$ for all $\bar a\in\text{Pos}(\mathfrak{g})$.
\end{enumerate}
\end{Cor}
\begin{proof}
(1)--(3) follow from the last Facts.
(\ref{item:specialipredense}) If $\mathfrak{g}$ indirectly approximates $p$ and $q\leq
p$, then there is a $\mathfrak{h}$ stronger than $\mathfrak{g}$ approximating $q$. Let $\bar
b\in \text{Pos}(\mathfrak{h})$ and $\bar a=\bar b\restriction \mathfrak{g}\in\text{Pos}(\mathfrak{g})$. Then $q^{[\bar
b]}\leq q,p^{[\bar a]}$.
(\ref{item:singletondense})
Let $\mathfrak{h}$ approximates $p$ such that $\minlength_{\{i\}}(\mathfrak{h})>n$.
Let $\bar a\in\text{Pos}(\mathfrak{h})$. Then $(i,a_i)$ indirectly
approximates $p^{[\bar a]}\leq p$.
By \ref{notes:conditions}(\ref{tmp4_11}) we can find a
$q=^* p$ such that $a_i$ approximates $q$.
(\ref{item:puredecision})
Let $\text{Pos}(\mathfrak{g})=\{{\bar a}_0,\dots,{\bar a}_l \}$.
Pick $q_0\leq p^{[{\bar a}_0]}$ in $D$, and
$r_0\leq_{\mathfrak{g}}p$ as in (\ref{item:rbetween}).
So $r_0^{[{\bar a}_0]}\in D$.
Pick $q_1\leq r_0^{[{\bar a}_1]}$ in $D$
and $r_1\leq_{\mathfrak{g}}r_0$ as above, etc.
Then $r_l$ has the required property.
\end{proof}
We now list some trivial properties of $P$ regarding restriction:
\begin{Def} For $i\in I^\infty$ we define $P_i\ensuremath{\coloneqq} P_{<i}\ensuremath{\coloneqq}
\{p\in P:\,\dom(p)\subseteq I_{<i}\}$,
and analogously for $P_{\leq i}$.
\end{Def}
\begin{Facts}\label{facts:restriction}
(Restriction) Assume $p,q\in P$ and $i,j\in I^\infty$.
\begin{itemize}
\item
If $\dom(q)\supseteq \dom(p)$, $q\restriction \dom(p)=p$ and
$\mathfrak{g}$ approximates $\mathfrak{g}$,
then $q\leq_{\mathfrak{g}} p$.
\item $P=P_\infty$,
$ p\restriction I_{<i}\in P_{<i}$ and $p\leq p\restriction I_{<i}$.
\item If $p'\leq p$ then $p'\restriction I_{<i}\leq p\restriction I_{<i}$.
If $p\in P_{<i}$ then $p\restriction I_{<i}=p$.
\item Let $q\in I_{<i}$, $q\leq p\restriction I_{<i}$. Define
$q\wedge p\ensuremath{\coloneqq} q\cup p\restriction I_{\geq i}$. Then
$q\wedge p\in P$ is the weakest condition stronger than both
$q$ and $p$.
\item $p\restriction I_{<i}$
is a reduction of $p$ (i.e.\ $r'\in P_{<i}$ and $r'\leq p\restriction
I_{<i}$ implies $r' \ensuremath{\parallel} p$).
\item If $i\leq j$ then $P_{<i}<\hspace{-0.5ex}\cdot P_{<j}$ (i.e.\ $P_{<i}$ is a complete subforcing of $P_{<j}$).
\item If $p\restriction I_{<i} \ensuremath{\parallel} q\restriction I_{<i}$
and $\dom(p)\cap \dom(q)\subseteq I_{<i}$, then $p\ensuremath{\parallel} q$.
\item Similar facts hold for $P_{\leq i}$. E.g. if
$i<j$, then $P_{\leq i}<\hspace{-0.5ex}\cdot P_{<j}$.
\end{itemize}
\end{Facts}
\begin{Def}
Assume that $i\leq j\in I^\infty$ and that $G_j$ is a
$P_{<j}$-generic filter over $V$.
\begin{itemize}
\item
The filter induced by the complete embedding $P_{<i}<\hspace{-0.5ex}\cdot P_{<j}$
is called $G_i$, and $V[G_i]_{P_{<i}}$ is called $V_i$.
The canonical $Q_i$-generic filter over $V_i$ is called $G(i)$.
Analogously we define $V_{\leq i}$ and $G_{\leq i}$
(which will turn out to be $V_i[G(i)]$ and $G_i\ast G(i)$).
\item
In $V_j$ or $V_{\leq i}$
we define $\eta_i$ to be the union of all
$t\in\omega\hko$ such that
$(i,t)$ is an approximation of $p$
for some $p\in G_j$ (or $G_{\leq i}$).
\end{itemize}
\end{Def}
\begin{Lem}
Let $i,j,G_j$ be as above, $p\in G_j$, and set $\bar \eta=(\eta_l)_{l\leq j}$.
\begin{enumerate}
\item $\eta_i$ is a well-defined real.
\item If $\mathfrak{g}$ indirectly approximates $p$, then $\bar \eta$ is compatible with $\mathfrak{g}$.
\item $\{\eta_i\}=\bigcap \{\lim\pB p i(\bar \eta):\, p\in G_j,\,i\in\dom(p)\}$
\item If $p\ensuremath{\Vdash} (\forall i\in\dom(q))\, \eta_i\in\lim(\pB q i(\eta))$,
then $q\leq p$.
\item $q\in G$ iff
$\eta_i\in\lim(\pB q i(\bar \eta))$ for all $i\in\dom(q)$.
\end{enumerate}
\end{Lem}
\begin{proof}
(1) By \ref{cor:restrtoa}(\ref{item:singletondense}),
the set of conditions $q$ such that for some $t$ of length $n$
$(i,t)$ is an approximation of $q$ is dense.
Therefore $\eta_i$ is infinite. Also,
If $s\ensuremath{\perp} t$, $(i,t)$ is an approximation of $q$,
and $(i,s)$ is an approximation of $q'$,
then $q$ and $q$ are incompatible.
This shows that $\eta_i$ is indeed a real.
(2) According to \ref{cor:restrtoa}(\ref{item:specialipredense}), the set
$\{p^{[\bar a]}:\, \bar a\in\text{Pos}(\mathfrak{g})\}$ is
predense below $p$. Let $\bar a$ be such that
$p^{[\bar a]}\in G$.
Any $q\in G$ that is stronger than $p^{[\bar a]}$ and
decides $\n \eta_i$ up to the length of $a_i$ forces that
$\n \eta_i\supset a_i$. So $\bar \eta$ is compatible
with $\bar a$ and therefore with $\mathfrak{g}$.
(3) Let $n\in \omega$. We have to show that
$\eta_i\restriction n\in \pB p i(\bar \eta)$.
First pick an approximation $\mathfrak{g}$ of $p$
with $\minlength_{\{i\}}(\mathfrak{g})\geq n$.
We already know that $\bar \eta$ is compatible with $\mathfrak{g}$,
in particular $\eta_i$ is compatible with
$\mathfrak{g}(i)(\bar \eta)$.
And $\mathfrak{g}(i)(\bar \eta)$ is a front in
$\pB p i(\bar \eta)$, since $\mathfrak{g}$ approximates $p$.
(4) Let $\mathfrak{g}$ approximate $p$ and $\mathfrak{h}$ approximate $q$ such that
$\dom(\mathfrak{h})\supseteq \dom(\mathfrak{g})$ and the length of $\mathfrak{h}$ is sufficiently
large on $\dom(\mathfrak{g})$.
Then $\mathfrak{h}$ must be stronger than $\mathfrak{g}$.
(5) follows from (4).
\end{proof}
As immediate consequence of fusion and pure decision we get:
\begin{Thm}\label{thm:fusioncor}
\begin{enumerate}
\item\label{item:omegaomegabounding} $P$ is $\omega^\omega$-bounding.
For every $p$ and $P$-name $\n\tau$ for an $\omega$-sequence of ordinals
there is a $p'\leq p$ such that $p'$ reads $\n\tau$ continuously.
\item\label{item:nonewreals}
Assume the cofinality\footnote{We always mean
the ``upwards cofinality'', i.e.\ the
minimal size of an upwards cofinal subset.
$A\subset I$ is upwards cofinal if for every $i\in I$
there is an $a\in A$ such that $a\geq i$.}
of $I$ is $\geq\al1$, $G$ is $P$-generic over $V$
and $r\in\mathbb{R}^{V[G]}$. Then there is an $i\in I$
such that $r\in\mathbb{R}^{V_i}$.
\item $P$ is proper.
\item $P$ forces that $\n\eta_i$ is a $Q_i$-generic real over $V_i$.
\item
If $I=I_1+I_2$, then $\nwflim_I(Q_i)\cong\nwflim_{I_1}(Q_i)\ast \nwflim_{I_2}(Q_i)$.
\item\label{item:fusiocorCSI} If
$I=\Sigma_{\beta\in\epsilon} J_\beta$ is the concatenation of the orders
$J_\beta$ along the ordinal $\epsilon$,
then $\nwflim_I(Q_i)$ is equivalent to the countable support limit
$(P_\beta,\n Q'_\beta)_{\beta\in\epsilon}$
of the forcings $\n Q'_\beta\ensuremath{\coloneqq}\nwflim_{J_\beta}(Q_i)$.
\item If $I$ is well-founded, then $\nwflim_I(Q_i)$ is the
countable support limit of the $Q_i$.
\end{enumerate}
\end{Thm}
\begin{proof}
(1)
Fix for every countable
subset $J$ of $I$ an enumeration $\{j_m:\,m\in \omega\}$,
and denote $\{j_m:\, m\in n\}$ by $\text{first}(n,J)$.
Assume $\n\tau$ is a name of a real and $p\in P$.
We have to show that there is a $p_\omega\leq p$ and an
$f\in \omega\ho$
such that $p_\omega\ensuremath{\Vdash} \n\tau(n)<f(n)$.
Let $p_0\leq p$, $f(0)\in\omega$
be such that $p_0\ensuremath{\Vdash} \n\tau(0)=f(0)$,
and let $\mathfrak{g}_0$ approximate $p_0$.
Assume that $\mathfrak{g}_n$ and $p_n$ are already defined.
We define $p_{n+1}\leq_{\mathfrak{g}_n} p_n $, $f(n)$ and
$\mathfrak{g}_{n+1}$
the following way: Let $p_{n+1}\leq_{\mathfrak{g}_n} p_n $
be such that
$p^{[\bar a]}_{n+1}$ decides $\n\tau(n)$
for every $\bar a\in \text{Pos}(\mathfrak{g}_n)$,
see \ref{cor:restrtoa}(\ref{item:puredecision}).
Let $f(n)$ be the maximum of the possible values for
$\n\tau(n)$. Let $\mathfrak{g}_{n+1}$ be a
$p_{n+1}$-approximation
stronger than $\mathfrak{g}_n$ which is $n$-dense at every
$i\in \text{first}(n,\dom(p_1))\cup\dots\cup \text{first}(n,\dom(p_n))$.
Then the sequence $(p_n)_{n\in\omega}$
satisfies the conditions for fusion \ref{lem:fusion}
so there is a $p_\omega\leq p_n$. Clearly, $p_\omega\ensuremath{\Vdash} \n\tau(n)\leq f(n)$.
(2) The $p_\omega$ above completely determines $\n\tau$,
so if $p_\omega\in P_{<i}$, then $p_\omega \ensuremath{\Vdash}_P \n\tau\in V_i$.
(3) is very similar to the above: Assume that $N\esm H(\chi)$ and
$p_0\in N$.
Let $\{D_m:\, m\in\omega\}$ enumerate the dense sets in $N$.
Assume $p_n$, $\mathfrak{g}_n\in N$ are already defined.
Find (in $N$) $p_{n+1}\leq_{\mathfrak{g}_n} p_n$ such that
$p^{[\bar a]}_{n+1}\in D_n$ for all $\bar a\in\text{Pos}(\mathfrak{g}_n)$,
and pick $\mathfrak{g}_{n+1}\in N$ big enough.
Then we can (in $V$) fuse this sequence into a $p_\omega\in P$.
Note that $\dom(p_\omega)\subseteq N\cap I$.
If $G$ is $P$-generic over $V$ and $p_\omega\in G$,
then $p_n\in G$ and $\{p^{[\bar a]}_n:\, \bar a\in\text{Pos}(\mathfrak{g}_n)\}$
is predense below $p_n$, so some $p^{[\bar a]}_n\in G$,
and $p^{[\bar a]}_n\in D_n\cap N$.
(4)
is a special case of (5): Set $I_1\ensuremath{\coloneqq} I_{<i}$ and
$I_2\ensuremath{\coloneqq} \{i\}$. So $\n\eta_i$ is $V_i$-generic in
$V_{\leq i}$ and therefore in $V_\infty$ as well.
(5)
Set $P\ensuremath{\coloneqq} \nwflim_I(Q_i)$, $P_1\ensuremath{\coloneqq} \nwflim_{I_1}(Q_i)$, $P_2\ensuremath{\coloneqq} \nwflim_{I_2}(Q_i)$.
There is a natural map $\phi: p\mapsto (p_1,\n p_2)$
from $P$ to $P_1\ast P_2$:
$p_1\ensuremath{\coloneqq} p\restriction I_1$, and $p_2$ is defined by
$\dom(p_2)\ensuremath{\coloneqq}\dom(p)\setminus I_1$ and
$\pB {p_2} i(\bar\eta)\ensuremath{\coloneqq}
\pB p i ((\n\eta_i)_{(i\in I_1)}^\frown\bar\eta)$.
It is clear that $\phi$ preserves $\leq$.
We claim that it is dense and preserves $\ensuremath{\perp}$.
Assume $\phi(p)=(p_1,\n p_2)$, $\phi(q)=(q_1,\n q_2)$, and
$(r_1,\n r_2)\leq (p_1,\n p_2),(q_1,\n q_2)$.
We have to find a $r'\leq_P p,q$ such that $\phi(r')\leq (r_1,\n r_2)$.
$r_1$ forces that $\n p_2$, $\n q_2$ and $\n r_2$
correspond to approximating sequences
$(\n \mathfrak{g}^p_n)$, $(\n \mathfrak{g}^q_n)$ and $(\n \mathfrak{g}^r_n)$.
As in (1) we can find an $r'_1\leq r_1$ with an approximating
sequence $(\mathfrak{h}_n)$ such that
$\mathfrak{h}_n$ decides $\n \mathfrak{g}^i_n$ (for $i\in\{p,q,r\}$)
in a way such that $\n \mathfrak{g}^r_n$ is stronger than both
$\n \mathfrak{g}^p_n$ and $\n \mathfrak{g}^q_n$. Then we can concatenate
$(\mathfrak{h}_n)$ with the $(\n \mathfrak{g}^r_n)$ to an approximating sequence
to some $r'\in P$. Then $r'\leq p,q$ and $\phi(r')\leq (r_1,\n r_2)$.
(6)
By induction on $\epsilon$. The successor step follows from (5).
Let $\cf(\epsilon)>\omega$. Then
the nwf-limit as well as the cs-limit are just the unions of
the smaller limits, and therefore equal by induction.
If $\cf(\epsilon)=\omega$, then the nwf-limit as well as the cs-limit
are the full inverse limits of the iteration system, and
therefore again equal by induction.
(7) follows from (6).
\end{proof}
We will also use the following fact:
\begin{Lem}\label{lem:lem35}
If $q\restriction I_i$ reads $\n S \in Q$ continuously and $q\ensuremath{\Vdash} \n\eta_i\in\n
S$, then $q\restriction I_i$ forces that $\pB q i (\bar{\n \eta}) \leq_Q \n S$.
\end{Lem}
\begin{proof}
Assume otherwise. Then there is an approximation $\mathfrak{g}$ of $p\ensuremath{\coloneqq}
q\restriction I_i$, an $\bar a\in\text{Pos}(\mathfrak{g})$ and a
$t\in\Tmax^i$ such that
$p^{[\bar a]}$ forces $t$ to be in $\pB q i (\bar{\n \eta})$ but not in $\n S$.
Let $\bar a^+$ be $\bar a^\frown t$. Then $\bar a^+$ is a possible
value of some approximation of $q$, and $q^{[\bar a^+]}$ forces
that $\n\eta_i\notin \n S$, a contradiction.
\end{proof}
\begin{uRem}
The iteration technique defined here also works for larger classes of
forcings, e.g. for the tree forcings $\mathbb{Q}^\text{tree}_0$ of~\cite{RoSh:470}
mentioned already. If we assume additional properties such as bigness and
halving, we could also use lim-inf forcings. It is also possible to
extend the construction to non-total orders, or to allow $\Tmax^i,\mu^i$
to be $P_{<i}$-names.
\end{uRem}
\section{The ideal $\myI$}\label{sec:myI}
In this section we define (to a lim-sup forcing $Q$ as in section \ref{sec:Q})
ideals $\mathbb{I}$ and $\myI$ (the $<2^\al0$-closure of $\mathbb{I}$). These
ideals will generally not be ccc. We define what we mean by ``$X$ is
measurable with respect to $\myI$''. The main theorem of this paper is that
for certain $Q$ we can force measurability for all definable $X$ without
using an inaccessible or amalgamation of forcings.
\begin{Def}\label{def:IstrongI}
\begin{itemize}
\item
The ideal $\mathbb{I}$ on the reals is defined by:
$X\in \mathbb{I}$ if for all $S\in Q$ there is a $T\leq S$
such that $X\cap\lim(T)=\emptyset$.
\item
$\myI$ is the $<2^\al0$-closure of $\mathbb{I}$.
\item
$X$ is null if $X\in\myI$; $X$ is positive if $X\notin \myI$;
$X$ has measure 1 if $2\ho\setminus X \in \myI$.
\end{itemize}
\end{Def}
\begin{uNotes}
\begin{itemize}
\item
Of course the notion null etc.\ depend on the forcing $Q$,
so it might be more exact to use notation
such as $Q$-null or $(\Tmax,\mu)$-null. In this paper
this is not necessary, since we will always use a fixed $Q$.
\item
We use the words ``null'', ``positive'' and ``measure 1'' although the ideals
$\mathbb{I}$ and $\myI$ are not related to a measure
(they are not even ccc).
\item
If CH holds, then $\myI=\mathbb{I}$.
\item
$\mathbb{I}$ is always nontrivial (i.e.\ $V\notin \mathbb{I}$),
but this is not clear for $\myI$.
\end{itemize}
\end{uNotes}
$F: Q\fnto Q$ is a witness for $X\in \mathbb{I}$
if $F(S)\leq S$ and $X\cap\lim(F(S))=\emptyset$
for all $S\in Q$.
So every $X\in \mathbb{I}$ is contained in a set $\bigcap
\{\omega\ho\setminus\lim(F(S)):\, S\in Q\}$.\footnote{Note that this is not a
countable intersection.}
\begin{Lem}\label{lem:strongIissigma}
$\mathbb{I}$ is a non-trivial $\sigma$-ideal.
\end{Lem}
\begin{proof}
This follows from fusion:
Assume $X_i\in \mathbb{I}$ ($i\in\omega$) and $S=S_0\in Q$.
Pick any front $F_0\in S_0$, so $S_0=\bigcup_{t\in F_0} S_0^{[t]}$.
For each $t\in F_0$ pick an
$S_{1,t}\leq S_0^{[t]}$ such that $\lim(S_{1,t})\cap X_1=\emptyset$.
Set $S_1\ensuremath{\coloneqq} \bigcup_{t\in F_0} S_{1,t}$. So $S_1\in Q$,
and $F_0$ is a front in $S_1$.
Pick a $1$-dense front $F_1$ in $S_1$ (purely) stronger than $F_0$.
Iterate the construction. Fusion produces a
$T< S$ such that $\lim(T)\cap X_i=\emptyset$ for all $i\in \mathbb{N}$.
\end{proof}
For example, if $Q$ is Sacks forcing,
then $\mathbb{I}$ is called Marczewski ideal.
$X\in\mathbb{I}$ iff in every
perfect set $A$ there is a perfect subset $A'$ of $A$ such that
$A'\cap X=\emptyset$. So if $X$ is Borel
(or if $X$ has the perfect set property,
e.g. $X$ is $\mS11$),
then $X\in\mathbb{I}$ iff $X$ is countable.
$\mathbb{I}$ is not a ccc ideal:
For $A\subseteq \omega$, set
\[X_A\ensuremath{\coloneqq}\{f\in 2\ho:\, (\forall n\notin
A)\,f(n)=0\}.\]
Clearly $X_A\cap X_B=X_{A\cap B}$, and $\card{X_A}=2^\card{A}$. So if
$\{A_i:\, i\in 2^\al0\}$ is an almost disjoint family, then
$\{X_{A_i}\}$ is a family of closed sets not in $\mathbb{I}$ such that
$X_{A_i}\cap X_{A_j}$ is finite
for $i\neq j$.
For a Borel ccc ideal $I$, ``$X\subseteq \mathbb{R}$ is measurable''
can be defined by ``there is a Borel
set $A$ such that $A\Delta X\in I$''.
(Usually the basis of the ideal is simpler, e.g.
one can use open sets instead of Borel sets for meager, or $G_\delta$ sets for
Lebesgue-null.) Equivalently, $X$ is measurable iff for every $I$-positive
Borel set $A$ there is an $I$-positive Borel set $B\subseteq A$ such that
either $B\cap X\in I$ or $B\setminus X\in I$.
For non-ccc ideals that do not live on the Borel sets,
this second notion is usually the right one to to generalize.
In our case we use the following definition of measurability:
\begin{Def}\label{def:positive}
\begin{itemize}
\item
$X\subseteq \mathbb{R}$ is strongly measurable if for every
$T\in Q$ there is an $S\leq_Q T$ such that either
$\lim(S)\cap X\in \mathbb{I}$ or $\lim(S)\setminus X\in \mathbb{I}$.
\item $X\subseteq \mathbb{R}$ is measurable if for every
$T\in Q$ there is an $S\leq_Q T$ such that either
$\lim(S)\cap X\in \myI$ or $\lim(S)\setminus X\in \myI$.
\end{itemize}
\end{Def}
Again, it would be more exact (but it is not necessary here) to say
``$(\Tmax,\mu)$-measurable'' instead of ``measurable''.
Clearly strong measurability implies measurability.
\begin{Lem}
The family of (strongly) measurable sets contains the
Borel sets and is closed under complements and
countable unions.
\end{Lem}
\begin{proof}
Closure under complement is trivial.
Every closed set is strongly measurable: Let $X=\lim(T')$ be closed
and $T\in Q$. If there is a $t\in T\setminus T'$ then
$S\ensuremath{\coloneqq} T^{[t]}$ satisfies $\lim(S)\cap X=\emptyset$.
Otherwise $T\subseteq T'$ and $S\ensuremath{\coloneqq} T$ satisfies
$\lim(S)\setminus X=\emptyset$.
Assume that $(X_i)_{i\in\omega}$ is a sequence of measurable sets
and that $T\in Q$. If for some $i\in\omega$ there is an $S\leq T$
such that $\lim(S_i)\setminus X_i\in\myI$ then the same
obviously holds for $\bigcup_{i\in\omega} X_i$. So assume that
for all $i\in\omega$ and $T'\leq T$ there is an $S\leq T'$ such
that $\lim(S)\cap X_i\in\myI$.
Now repeat the proof of \ref{lem:strongIissigma}.
The same proof shows that the {\em strongly} measurable sets
are closed under countable unions as well.
\end{proof}
$\myI$ could be trivial. However, if it is ``everywhere nontrivial'',
then $\myI$ and $\mathbb{I}$ are the same on Borel sets:
\begin{Lem}\label{lem:IandstrongI}
Assume that $\lim(S)\notin\myI$ for all $S\in Q$. Then $\myI$ and $\mathbb{I}$
agree on strongly measurable sets. I.e.\ if $X$ is strongly measurable and
$X\in \myI$, then $X\in\mathbb{I}$.
\end{Lem}
In particular, in this case every Borel set is in $\myI$
iff it is in $\mathbb{I}$.
\begin{proof}
For every $T\in Q$ there is an $S\leq_Q T$ such
that $\lim(S)\cap X\in \mathbb{I}$:
Otherwise $\lim(S)\setminus
X\in\mathbb{I}\subseteq \myI$,
a contradiction to $X\in\myI$ and $\lim(S)\notin\myI$.
So by the definition of $\mathbb{I}$ there is a $S'\leq_Q S\leq_Q T$
such that that $ \lim(S')\cap X=\emptyset$. So $X\in\mathbb{I}$.
\end{proof}
\begin{Fact}
For Borel codes $B$, the statement ``$B\in\mathbb{I}$'' is $\mP12$ and therefore
invariant under forcing.
\end{Fact}
On the other hand, since $\mathbb{I}$ is not a Borel ideal (i.e.\ not
every $X\in\mathbb{I}$ is contained in a Borel set $B\in\mathbb{I}$),
there is no reason why $X\in\mathbb{I}$ should be upwards absolute
between universes.
The main theorem of this paper states that (using an artificially sweet forcing
$P$) we can force all definable sets to be measurable:
\begin{Thm}\label{thm:defaremeas}
Assume CH and that $Q$
satisfies the Ramsey property \ref{def:ramseyprop}.
Then there is a proper, $\al2$-cc, $\omega\ho$-bounding p.o. $P$
forcing that
every set of reals which is (first-order) definable using a parameter
in $L(\mathbb{R})$ is measurable.
\end{Thm}
$P$ will be the non-wellfounded iteration of $Q$ along an
order $I$ defined in the next section.
We already know the following (use \ref{lem:treewithoutoldbranch}):
\begin{Lem}\label{lem:nooldreal}
If $I$ has cofinality $\geq \al1$
and $i\in I$ then $\ensuremath{\Vdash}_P V_i\in \mathbb{I}$.
\end{Lem}
\begin{proof}
If $T\in V[G_P]$ then $T\in V_j$ for some $i<j<\infty$
because of \ref{thm:fusioncor}(\ref{item:nonewreals}).
So in $V_{\leq j}$ there is an $S<T$ such that
$\lim(S)\cap V_i=\emptyset$ (in $V_{\leq j}$ and $V[G_P]$ as well,
according to \ref{lem:treewithoutoldbranch}).
\end{proof}
For later reference, we will reformulate
the definition of $\mathbb{I}$:
If $S\in Q$, $X\subseteq Q$, $T\in X$ and $T'\leq_Q S,T$, then $\lim(T')\cap
(2^\omega\setminus \bigcup_{T\in X}\lim(T))
\subseteq \lim(T')\setminus \lim(T)=\emptyset$.
So we get:
\begin{Lem}\label{lem:predenseiscoi}
If $X\subseteq Q$ is predense then
$\bigcup_{T\in X}\lim(T)$ is of measure 1 with respect to
$\mathbb{I}$.
\end{Lem}
\section{An order with many automorphisms}\label{sec:order}
In this section we assume CH. We will construct an
order $I$ and define
$P$ to be the nwf-limit of $Q$ along $I$.
$I$ is
$\om2$-like,\footnote{$I$ is $\om2$-like if
$\card{I_{<i}}<\al2$ for all $i\in I$ and $\card{I}=\al2$.}
has a cofinal sequence $j_\alpha$ ($\alpha\in\om2$) and
many automorphisms.
We show that these properties imply that $P$ forces the following:
\begin{itemize}
\item $2^\al0=\al2$,
\item $\myI$ is nontrivial (and moreover $\lim(S)\notin \myI$ for all $S\in Q$),
\item for every definable set $X$,
``locally'' either all or no $\eta_{j_\delta}$ are in $X$ and
\item $\{\eta_{j_\delta}:\, \delta\in\om2\}$
is of measure 1 in $\{\eta_i:\, i\in I\}$.
\end{itemize}
In the next section it will be shown that
the set $\{\eta_i:\, i\in I\}$ is of measure 1, which will
finish the proof of the main theorem.
\begin{Lem}\label{lem:card}
Assume CH holds and $I$ is $\om2$-like.
Then
\begin{enumerate}
\item $P$ has the $\al2$-cc (and therefore preserves all cofinalities).
\item
$P_{<i}\ensuremath{\Vdash}$ CH
for each $i\in I$ and
$P\ensuremath{\Vdash} 2^\al0=\al2$.
\end{enumerate}
\end{Lem}
\begin{proof}
(1)
If $\card{I_{<i}}\leq 2^{\al0}$ then $\card{P_{<i}}\leq 2^{\al0}$:
There are at most $\card{I_{<i}}^{\al0}\leq 2^{\al0}$ may countable
subsets of $\card{I_{<i}}$. For each $p\in P_{<i}$ with a fixed
domain and each $j\in \dom(p)$
there are $2^{\al0}$ many possibilities for
$\pD p j$
and $2^{\al0}$ many possibilities for the Borel definition $\pB p j$.
If CH holds, then the usual delta system lemma applies: If $A\subseteq P$ is a
maximal antichain of size $\al2$ then without loss of generality the domains of
$p\in A$ form a delta system
(i.e.\ there is a countable $x\subseteq I$
such that
$\dom(p_1)\cap\dom(p_2)=x$
for all $p_1\neq p_2\in A$).
Since $I$ is $\om2$-like, $x$ cannot be cofinal.
Let $i$ be an upper bound of $x$. Without loss of generality
$p_1\restriction I_{<i}=p_2\restriction I_{<i}$
for $p_1\neq p_2\in A$
(since there are only $\al1$ many elements of $P_{<i}$).
But then $p_1\ensuremath{\parallel} p_2$ by \ref{facts:restriction}.
Proper and $\al2$-cc imply preservation of all
cofinalities and cardinalities.
(2)
Let $G$ be $P$-generic over $V$.
Then the reals in $V[G_P]$ are the union of the reals
in $V[G_{P_{<i}}]$, and every real in $V[G_{P_{<i}}]$
is decided by (i.e.\ read continuously from)
a condition $p\in P_{<i}$.
Since \mbox{$\card{P_{<i}}=(2^\al0)^V=\al1$}, there are at most
$\al2$ many reals in $V[G_{P_{<i}}]$.
And $\eta_{i}\notin V[G_{P_{i}}]$, so in particular $\eta_{i_1}\neq
\eta_{i_2}$ for $i_1\neq i_2$.
\end{proof}
The following is well known:
\begin{Lem}
If CH holds, then there is
an $\al1$ saturated\footnote{A linear order $\tilde I$
is $\al1$ saturated if there are no gaps of type
$(a,b)$, where $a,b\in \{1,\al0,\pm \infty \}$.}
linear order $\tilde I$ of size $\al1$, and
all such orders are isomorphic.
\end{Lem}
\begin{proof}
Induction of length $\om1$: Assume at stage $\alpha$ we
have a linear order $L_\alpha$ of size $\om1=2^\al0$.
List all the ($\om1$ many) gaps of the types $(a,b)$,
where $a,b\in \{1,\al0,\pm \infty \}$,
and add points to fill these gaps. At limits, take the union.
Then at stage $\om1$ we get a saturated order.
Uniqueness is proven by the standard back and forth argument.
\end{proof}
\newcommand{\mathfrak S}{\mathfrak S}
\begin{Def}
Let $\mathfrak S$ be the set of $0<\alpha<\om2$
such that $\cf(\alpha)\in\{1,\om1\}$.
Note that $\mathfrak S\subseteq\om2$ is stationary.
\end{Def}
We will now define the order $I$ along which we iterate.
(We do this assuming CH.)
Given $\tilde I$ as above, let $I$
be the following order:\\
$\underbrace{\tilde I}_0+
\underbrace{\{j_1\}+\tilde I}_1+\cdots+
\underbrace{\tilde I}_\omega+
\underbrace{\{j_{\omega+1}\}+\tilde I}_{\omega+1}+\cdots+
\underbrace{\{j_{\omega_1}\}+\tilde I}_{\omega_1}+\cdots
$\\
So at stages $\alpha\in \mathfrak S$, we add $\{i\}+\tilde I$,
in other stages we add just $\tilde I$.
\begin{Facts}\label{facts:cofinalsequ}
\begin{itemize}
\item
$I$ is $\om2$-like,
\item
$(j_\alpha)_{\alpha \in \mathfrak S}$ is an
increasing (and therefore cofinal) continuous sequence in $I$, and
\item
every $j_\alpha$ has cofinality $\al1$ in $I$.
\end{itemize}
\end{Facts}
Continuous
means that
$j_\delta=\sup(j_\alpha: \, \alpha\in\mathfrak S,\alpha < \delta)$
whenever
$\delta=\sup(\mathfrak S\cap \delta )\in \mathfrak S$ (which is
equivalent to $\cf(\delta)=\om1$).
\begin{uNote}
We could just as well define $j_\alpha$ for
$\alpha$ with cofinality $\om1$ only, or for all
$\alpha\in\om2$ (and require continuity for points of cofinality
$\om1$ only). All these versions are equivalent by simple
relabeling, cf.\ the beginning of the proof of \ref{lem:onlyetaalphamatter}.
\end{uNote}
\begin{Def}
We set $Q_i=Q$ for all $i\in I$ and let $P$ be the
nwf-iteration of $Q_i$ along $I$.
We will use the notation $I_\alpha$, $P_\alpha$, $V_\alpha$ and $\eta_\alpha$
for $I_{<j_\alpha}$, $P_{<j_\alpha}$, $V_{j_\alpha}$ and $\eta_{j_\alpha}$.
We set \mbox{$V_\om2\ensuremath{\coloneqq} V_\infty=V[G_P]$}.
\end{Def}
\begin{Lem} (CH) Let $S_0\subseteq \mathfrak S$ be stationary.
$P$ forces the following:
\begin{enumerate}
\item $\{\eta_{\delta}:\, \delta\in S\}\notin\myI$ for every
stationary $S\subseteq \mathfrak S $, and
\item
$\{\eta_{\delta}:\, \delta\in S_0\}\cap \lim(T_0)\notin\myI$
for every $T_0\in Q$.
\end{enumerate}
\end{Lem}
This lemma implies that in $V_\om2$ the assumption of
Lemma~\ref{lem:IandstrongI} is satisfied (i.e.\ that $\myI$ is
``everywhere nontrivial'').
Note that this lemma holds for all $I$ satisfying \ref{facts:cofinalsequ}.
\begin{proof}
(1)
Assume otherwise, i.e.\ there are $P$-names
$\n F_\zeta$ $(\zeta\in\om1)$ for functions from
$Q$ to $Q$ and $\n S$ for a stationary set
such that $p_0\in P$ forces\\
\centerline{$\n F_\zeta(T)\leq T$
and $(\forall \delta\in S)\, (\exists \zeta\in\om1)\,(\forall T\in Q)\,
\eta_{\delta}\notin\lim(\n F_\zeta(T))$.}
$P$ forces that
for each $\alpha\in \mathfrak S $ there is a $\beta \in \mathfrak S $ such that
$\n F_\zeta(T)\in Q^{V_{\beta}}$
for all $T\in Q^{V_{\alpha}}$ and $\zeta\in\om1$.
We need something slightly stronger:
For every name $\n T$ for an element of
$Q^{V_{\alpha}}$ and $\zeta\in \om1$ there is a maximal antichain
$A\subset P$ such that for every $q\in A$ there is a
$P$-name $\n{T}'_q$
such that $q$ forces $\n F_\zeta(\n T)=\n{T}'_q$ and $q$ continuously
reads $\n{T}'_q$. So if
$q\in G_P$ and
$\beta$ is bigger than $\dom(q)$,\footnote{More formally: if
$j_\beta>i$ for all $i\in\dom(q)$.}
then
$V_{\beta}$ not only contains $T'_q=F_\zeta(T)$, but also knows
that $T'_q$ will be $F_\zeta(T)$ in $V_\om2$.
Define $f^-(\alpha)$ to be the smallest $\beta$
which is bigger than $\dom(q)$ for every $a\in A$,
where $A$ is an antichain for some $\n T$ and $\zeta\in \om1$
as above. $P$ is $\al2$-cc, every $q\in A$ has countable domain,
and there are only $\al1$ many reals in $V_\alpha$.
So $f^-(\alpha)<\om1$, and we can define $f(\alpha)$
to be the smallest $\beta\in \mathfrak S$ that is larger or equal
to $\max(\alpha,f^-(\alpha))$.
If $\cf(\alpha)=\om1$, then $f(\alpha)$ is the supremum of
$\{f(\gamma):\, \gamma\in\mathfrak S\cap \alpha\}$,
since the reals in $V_\alpha$ are the union of
the reals in $V_\gamma$. So $f$ is continuous.
Then $P$ forces the following:
Since $\n S$ is stationary,
there is a $\beta\in \n S$ such that $f(\beta)=(\beta)$.
$V_{\beta}$ can calculate every $F_\zeta$, and
$\n F_\zeta''Q$ is dense in $Q$.
Since $\n \eta_{\beta}$ is a $Q$-generic real over
$V_{\beta}$, there is
(for every $\zeta\in\om1$)
a $T\in Q^{V_{\beta}}$ such that $\n \eta_{\beta}\in\lim(\n F_\zeta(T))$,
a contradiction.
(2): We can assume that $T_0\in V$.
Again, choose names $\n F_\zeta$
as above, and assume that $p_0\in P$ forces that\\
\centerline{$\n F_\zeta(T)\leq T$
and $(\forall \delta\in S_0)\, (\exists \zeta\in\om1)\,(\forall T\in Q)\,
\eta_{\delta}\notin\lim(\n F_\zeta(T))\cap\lim(T_0)$.}
Define $f$ as above, so there is a $\beta>\dom(p)$ such that
$\beta\in S_0$ and $f(\beta)=\beta$. So the same argument
proves that $p_0$ forces that $\n \eta_{\beta}\notin \lim(T_0)$,
a contradiction.
\end{proof}
We also get the following:
\begin{Lem}\label{lem:onlyetaalphamatter} (CH)
For every $C\subseteq \om2$ club,
$P$ forces the following:
\[
\{\n \eta_i:\, i\in I\}\setminus \{\n \eta_\alpha:\,
\alpha\in\mathfrak S\cap C\}\in\myI.
\]
\end{Lem}
Again, this lemma applies to all $I$ satisfying \ref{facts:cofinalsequ}.
\begin{proof}
We can assume that $C=\om2$, since
we can just relabel the sequence $\{j_\alpha:\, \alpha\in \mathfrak S\cap C\}$:
Set $j'_\alpha\ensuremath{\coloneqq} j_\beta$, where $\beta$ is the $\alpha$-th element of
$C\cap \mathfrak S$. Then $(j'_\alpha)_{\alpha\in\mathfrak S}$
satisfies \ref{facts:cofinalsequ} as well.
Recall Definition~\ref{def:spl} of $Q^f_{A^{\psi}_r}$ and $D^\text{spl}_f$ (for $f:\omega \fnto \omega$
increasing and $r\in 2\ho$). Enumerate all
increasing $f:\omega \fnto \omega$ in $V$
as $f_\zeta$ ($\zeta\in\om1$). (CH holds in $V$.)
\begin{figure}[tb]
\begin{center}
\scalebox{0.5}{\input{ordtree.pstex_t}}
\end{center}
\caption{\label{fig:ordtree}}
\end{figure}
{\bf Claim:}
In $V$, we can find $P_\alpha$-names $\n T_\alpha^\zeta$
($\zeta\in\om1$, $\alpha<\om2$ successor) for elements of $Q$
such that the following is forced by $P$ (cf.\ Figure~\ref{fig:ordtree}):
\begin{enumerate}
\item
The set
$\{\n T_{\alpha+1}^\zeta:\, \alpha<\om2\}\subseteq Q$ is
dense
for all $\zeta\in\om1$.
\item
$\n T_\alpha^\zeta\in D^\text{spl}_{f_\zeta}$ (in $V_\alpha$ or
equivalently in $V_\om2)$.\footnote{Recall \ref{def:spl}
and \ref{lem:dspl}.}
\item
If $\beta<\alpha$ is a successor, then
$\n T_\alpha^\zeta$ has no branch in $V_{\beta}$, and
for all $i<j_\alpha$
there is a $\zeta_0$ such that
$\n T_\alpha^\zeta$ has no branch in $V_i$
for all $\zeta \geq \zeta_0$.
\end{enumerate}
Proof of the claim:
Pick for all $\alpha+1$ a function $\phi_{\alpha+1}:
\om1\fnto I_{j_{\alpha+1}}\setminus I_{j_\alpha}$
which is increasing and cofinal. Also pick an
enumeration $(\n S_{\alpha+1})_{\alpha\in \om2}$
such that $\n S_{\alpha}$ is an $P_\alpha$-name
and $P$ forces that $Q=\{\n S_{\alpha+1}:\, \alpha\in \om2\}$.
(This is possible since $P$ forces that $Q^{V_\om2}=\bigcup Q^{V_\alpha}$,
cf.\ \ref{thm:fusioncor}(\ref{item:nonewreals}).)
To find $\n T_\alpha^\zeta$ ($\alpha$ successor)
note that $P_\alpha$ forces that we can perform
the following construction in $V_\alpha$:
First pick an $S'\leq \n S_\alpha$
such that $S'\in D^\text{spl}_{f_\zeta}$ (cf.\
\ref{lem:dspl}(\ref{item:dspl6})). $\cf(j_\alpha)=\al1$, so $S'\in V_i$
for some $i<j_\alpha$. Pick some $i'$ bigger than
$\max(i,\phi_{\alpha}(\zeta))$ and smaller than $j_\alpha$.
There is a real $r\in V_\alpha\setminus V_{i'}$
(e.g. $\eta_{i'}$). Therefore there is a $T^\zeta_\alpha<S'$
such that $\lim(T^\zeta_\alpha)\cap V_{i'}=\emptyset$
(in $V_\alpha$ and $V_\om2$ as well, cf.\ \ref{lem:treewithoutoldbranch}).
Let $\n T_\alpha^\zeta$ be a $P_\alpha$-name for $T^\zeta_\alpha$.
The $\n T_\alpha^\zeta$ constructed this way satisfy the claim:
(1): $\n T_\alpha^\zeta\leq \n S_\alpha$, (2):
$D^\text{spl}_{f_\zeta}$ is open dense and absolute, (3):
pick $\zeta_0$ such that $\phi_\alpha(\zeta_0)>i$.
This ends the proof of the claim.
From now on assume $G$ is $P$-generic over $V$.
We work in $V_\om2$ and set $T_\alpha^\zeta\ensuremath{\coloneqq} \n T_\alpha^\zeta[G]$.
So if $i\in I$ then the sequence
$(T_{\alpha+1}^\zeta)_{j_{\alpha+1}<i,\zeta\in\om1}$ is in $V_i$.
For all $\zeta\in\om1$, $X_\zeta\ensuremath{\coloneqq}
\bigcup_{\alpha+1<\om2} \lim(T_{\alpha+1}^\zeta)$ is of measure 1
with respect to $\mathbb{I}$
(cf.\ \ref{lem:predenseiscoi}).
So the set $Y\ensuremath{\coloneqq} \bigcap_{\zeta\in\om1} X_\zeta$ is
of measure 1.
It is enough to show that
\[\{\eta_i:\, i\in I\}\setminus \{\eta_\alpha:\, \alpha\in\mathfrak S\}
\cap Y=\emptyset.\]
Assume towards a contradiction that some $\eta_i$ is in $Y$ and
$\eta_i\neq \eta_\alpha$ for all $\alpha\in\mathfrak S$.
Let $\alpha\in \mathfrak S$ be minimal such that $\eta_i\in V_{\alpha}$ (i.e.\
$i<j_{\alpha}$, cf.\ Figure~\ref{fig:ordtree}).
So $\alpha$ is a successor (but not necessarily
a successor of a $\beta\in\mathfrak S$), and $i>j_\beta$ for all
$\beta\in \mathfrak S\cap\alpha$.
So according to (3) there is a $\zeta_0$ such that $\eta_i\notin
\lim(T_{\gamma+1}^\zeta)$ for all $\zeta>\zeta_0$ and all $\gamma+1\geq
\alpha$.
So we know the following:
$\eta_i\in Y$, i.e.\
\[\eta_i\in\bigcup_{\gamma+1<\om2} \lim(T_{\gamma+1}^\zeta)\quad\text{ for all }
\zeta\in\om1.\]
But
\[\eta_i\notin \bigcup_{\alpha\leq \gamma+1<\om2}
\lim(T_{\gamma+1}^\zeta)\quad\text{ for all }\zeta\geq \zeta_0.\]
Therefore
\[\eta_i\in\bigcup_{\gamma+1<\alpha} \lim(T_{\gamma+1}^\zeta)\quad\text{ for all }
\zeta\geq \zeta_0.\]
Recall that $V_i$ sees the sequence
$(T_{\gamma+1}^\zeta)_{\gamma+1<\alpha,\zeta\in\om1}$.
So in $V_i$, some $T\in Q$ forces that for all $\zeta>\zeta_0$
there is a successor $\beta(\zeta)<\alpha$ such that
$\n\eta_i\in \lim(T_{\beta(\zeta)}^\zeta)$.
In $V_i$, $T$ has full splitting for some $f_\zeta\in V$,
$\zeta>\zeta_0$ (see \ref{lem:dspl}(\ref{item:dspl4}), \ref{lem:fullsplitinV} and \ref{thm:fusioncor}(\ref{item:omegaomegabounding})).
Let $r$ be a real in
$V_{i}\setminus \bigcup_{\gamma+1<\alpha}{V_{\gamma+1}}$.
Pick in $V_i$ a $T'\leq T$ such that $T'\in Q^{f_\zeta}_{A^{\psi}_r}$
(cf.\ \ref{lem:dspl}(\ref{item:dspl5}))
and $T'$ decides $\beta(\zeta)$.
Then $T'$ forces that $\eta_i\in T'\cap T_{\beta(\zeta)}^\zeta$,
a contradiction to $T'\ensuremath{\perp} T_{\beta(\zeta)}^\zeta\in V_i$
(because of (2), either $T_{\beta(\zeta)}^\zeta$ is in $Q^{f_\zeta}_{A^{\psi}_s}$
for some old real $s$, or incompatible to all $Q^{f_\zeta}_{A^{\psi}_s}$).
\end{proof}
\begin{figure}[tb]
\begin{center}
\scalebox{0.65}{\input{automorph.pstex_t}}
\end{center}
\caption{\label{fig:auto}An automorphism $f$.}
\end{figure}
We call $f$ an automorphism if it is a $<$-preserving
bijection from $I$ to $I$.
If $f: I\fnto I$ is an automorphism, then $f$ defines an automorphism of $P$
in a natural way as well (provided of course that $f(i)=j$ implies $Q_i=Q_j$,
but in our case all the $Q_i$
are the same). Also, $f$ defines a map on all $P$-names, and we have:
$p\ensuremath{\Vdash} \varphi(\n \tau)$ iff $f(p)\ensuremath{\Vdash} \varphi(f\n\tau)$.
If $\ensuremath{\Vdash}_P\n x\in V_{i}$, then there is a $V_i$-name $\n \tau $
such that $\ensuremath{\Vdash}_P \n x=\n\tau$.
If $f\restriction I_{<i}$ is the identity, then $f(\n \tau)=\n \tau$.
So in this case $p\ensuremath{\Vdash} \phi(\n \tau)$ iff $f(p)\ensuremath{\Vdash} \phi(\n\tau)$.
Also, if $f\restriction \dom(p)\cap I_{<i}$
is the identity
then $\pB p i(\bar \eta)=\pB{f(p)}{f(i)}(\bar \eta)$.
\begin{Lem}\label{lem:manyaut}
The following holds for $I$ (see Figure~\ref{fig:auto}):
If $\alpha<\beta<\gamma<\delta\in \mathfrak S $, $A\subseteq I_{\beta}$ countable,
$B\subseteq I\setminus I_{\beta}$ countable, then there is an automorphism $f$
of $I$
such that $f\restriction (I_\alpha\cup A)$ is the identity,
$f(j_\beta)=j_\gamma$ and $f'' B>j_{\delta}$.
\end{Lem}
\begin{proof}
For every $i<j\in I$, $I_{<i}$ and $\{k:\, i<k<j\}$ are
isomorphic and also isomorphic to $\tilde I$ (since they are all $\al1$
saturated
linear orders of size $\al1$). If
$A\subset I$ countable, then there are $i<A<j$, and for all
such $i,j$, $\{k:\, i<k<A\}$ and $\{k:\, A<k<j\}$ are again
isomorphic to $\tilde I$.
Also, $I_{>i}$ is isomorphic to $I$ (since
$\om2\setminus \alpha$ is isomorphic to $\om2$).
So assume $\alpha<\beta<\gamma\in \mathfrak S$, $A<i<j_\beta$ countable, $i>j_\alpha$.
Then $I_{j_\beta}\setminus I_{\leq i}\cong I_{j_\gamma} \setminus I_{\leq i}\cong \tilde I$.
Also, if $B\subset I$ is countable, $\delta\in \mathfrak S$ and
$B>j_\beta$, then there is an $j\beta<i<B$,
and $I_i\cong I_{j_\delta}\cong \tilde I$, $I\setminus I_{\leq i}\cong
I\setminus
I_{\leq j_\delta}\cong I$. Now combine these automorphisms.
\end{proof}
\begin{Lem}\label{lem:oneetalalleta}
For $\beta\in\om2$ set $Y_\beta\ensuremath{\coloneqq} \{\n \eta_{\gamma}:\, \gamma\in\mathfrak S,\,\gamma>\beta\}$.
$P$ forces the following:
If $X$ is a set of reals defined with a parameter
$x\in \bigcup_{i\in I} V_i$,
and if $T\in Q$, then
there is an $S \leq T$ and a $\beta\in\om2$ such that
either $\lim(S)\cap X\cap Y_\beta=\emptyset$ or
$(\lim(S)\setminus X)\cap Y_\beta=\emptyset$.
\end{Lem}
This lemma holds for all $I$ satisfying \ref{facts:cofinalsequ}
and \ref{lem:manyaut}.
Note that every real in $V_\om2$ is in $\bigcup_{i\in I} V_i$.
We will see in the next section that (using additional assumptions)
$Y_\beta$ is a measure 1 set. Then this lemma implies that
$X$ is measurable, i.e.\ the main theorem~\ref{thm:defaremeas}.
Because of \ref{lem:onlyetaalphamatter},
it will be enough to show that
the set $\{\eta_i:\, i\in I\}$ is of measure 1.
\begin{proof}
Assume $\n X=\{r:\, \varphi(r,\n x)\}$ and fix some $\n T$.
Some $p_0$ forces that $\n x$ and $\n T$ are in $V_{\alpha}$, so
without loss of generality
${\n x}, \n T$ are $P_{\alpha}$ names and $\dom(p_0)\subset I_{\alpha}$.
Pick a $p_1\leq p_0$, $p_1\in P_\alpha$ such that $p_1$ continuously
reads $\n T$.
Fix some $\beta>\alpha$.
Then $p_2\ensuremath{\coloneqq} p_1\cup \{(j_\beta,\n T)\}$
is an element of $P_{\leq j_\beta}$ (since $\n T$ is read continuously).
Let $p\leq p_2$ decide $\varphi(\n\eta_{{\beta}},\n x)$.
Without loss of generality
$p\ensuremath{\Vdash} \varphi(\n\eta_{{\beta}},\n x)$.
$p\restriction I_\beta$
forces that \mbox{$\n S\ensuremath{\coloneqq} \pB p {j_\beta}(\bar{\n \eta})\leq_Q \n T$}
(since $p\leq p_2$).
Assume towards a contradiction that for some $q\leq p$, $\gamma\in \mathfrak S $ and
$\gamma>\beta$
\[q\ensuremath{\Vdash} \n\eta_{\gamma}\in\lim (\n S)\ \& \ \lnot \varphi(\n
\eta_{\gamma},\n x).\]
Note that $q\restriction I_\gamma$ reads $\n S$ continuously and forces
that $\pB q {j_\gamma} (\bar{\n \eta})\leq_Q \n S$
(cf.\ \ref{lem:lem35}).
Set $A\ensuremath{\coloneqq} \dom(p)\cap I_\beta$ and
$B\ensuremath{\coloneqq} \dom(p)\cap I_{>j_\beta}$.
Let $j_{\delta}$ be bigger than $\dom(q)$, and
let $f$ be an automorphism of $I$ such that
$f\restriction (I_\alpha\cup A)$ is the identity,
$f(j_\beta) = j_{\gamma}$ and $f'' B>\dom(q)$
(cf.\ \ref{lem:manyaut} or Figure~\ref{fig:auto}).
$\dom(f(p))\cap \dom(q)\subseteq A\cup \{j_\gamma\}$.
$f(p)\restriction A=p\restriction A\geq q\restriction A$,
and $q\restriction I_\gamma$ forces that
\[\pB {f(p)} {j_{\gamma}}(\bar{\n \eta})=\pB p {j_\beta}(\bar{\n \eta})=\n S
\geq_Q \pB q {j_\beta} (\bar{\n \eta}).\]
So $f(p)$ and $q$ are compatible, a contradiction to
$f(p)\ensuremath{\Vdash} \varphi(\n\eta_{{\gamma}},\n x)$.
\end{proof}
\begin{uRem} For $\alpha$ with cofinality $\omega$, we could define $I_\alpha$
to $\bigcup_{\beta<\alpha} I_\beta$, and $P_\alpha$ to be the nwf-limit of the
$P_\beta$ ($\beta<\alpha$). Then according to Theorem~\ref{thm:fusioncor},
$P$ is the c.s. limit of the iteration $(P_\alpha, R_\alpha)$, where
$R_\alpha$ is $\nwflim_{\tilde I} Q$ if $\alpha\notin \mathfrak S$, and
$\nwflim_{\{i\}+\tilde I} Q$ otherwise. However, we will not need this fact in
this paper.
\end{uRem}
\section{A very non-homogeneous tree}\label{sec:tree}
For the proof of Theorem~\ref{thm:defaremeas} it remains to be shown that
$\{\eta_i:\, i\in I \}$ is of measure 1.
For this we will need a certain Ramsey property for $Q$.
\begin{Def}
A subtree $T$ of $\Tmax$ is called $(n,r)$-meager
if
$\mu_T(t)<r$
for all $t\in T$ with length at least $n$.
\end{Def}
\begin{Lem}
If $T$ is meager for some $(n,r)$, then $\lim(T)\in \mathbb{I}$.
\end{Lem}
\begin{proof}
For any $S\in Q$ there is an $s\in S$ of length at least $n$ such that
$\mu_S(s)>r$. So there is an immediate successors $t$ of $s$ in $S$ such that
$t\notin T$. Then $\lim(S^{[t]})\cap \lim(T)=\emptyset$.
\end{proof}
\begin{Def}
Let $M,N$ be natural numbers. $N\rightarrow M$ means: If\\
\phantom{xxx}$s_1,\dots,s_M\in \Tmax$ such that $\length(s_i)>N$,\\
\phantom{xxx}$t\in \Tmax$ such that $s_i\ensuremath{\perp} t$ for $1\leq i\leq M$,\\
\phantom{xxx}$A\subseteq \SUCC(t)$ such that $\mu(A)>N$,\\
\phantom{xxx}$f_i:A\fnto \Tmax^{[s_i]}$ for $1\leq i\leq M$,\\
then there is a $B\subseteq A$ such that\\
\phantom{xxx}$\mu(B)>M$ and\\
\phantom{xxx}$\{s\in \Tmax:\, (\exists i\leq M)\, (\exists t\in B)\, s\preceq f_i(t)\}$ is $(N,1/M)$-meager.
\end{Def}
\begin{Def}\label{def:ramseyprop}
A lim-sup tree-forcing $Q$ is strongly non-homogeneous if
$\mu$ is sub-additive\footnote{$\mu(A\cup B)\leq \mu(A)+\mu(B)$.}
and for all $M$ there is an $N$ such that $N\rightarrow M$.
\end{Def}
There are many similar notions of bigness, see e.g. \cite[2.2]{RoSh:470}.
\begin{Lem}
There is a forcing $Q$ that is strongly non-homogeneous.
\end{Lem}
\begin{proof}
First note that it is enough to show that
for each $M$ there is an $N$ such that $N\rightarrow^- M$,
where $N\rightarrow^- M$ is defined as above but
with just one $s$ and $f$ instead of $M$ many.
To see this, just set $K_0\ensuremath{\coloneqq} M^2$ and
find $K_i$ such that $K_{l+1}\rightarrow^- K_l$.
Then $K_M \rightarrow M$. (Here we use that $\mu$
is sub-additive, since we need that the union of $m$
many $(n,r)$-meager trees is $(n,r\cdot m)$-meager.)
We will construct $\Tmax$ and $\mu$ by induction.
We define $s\lhd t$ by: $\length(s)<\length(t)$
or $\length(s)=\length(t)$ and $s$ is lexicographically smaller
than $t$.
Fix some $t\in\omega\hko$.
Assume that we already decided which $s\lhd t$ will
be elements of $\Tmax$ as that we already defined
the set of successors of all these $s$ as well as the
measure of their subsets. We
have to define $\SUCC(t)$ and the measure on it.
Let $m_t$ be the number of nodes $s\lhd t$ already defined (including the
successors of $s$ for the already defined for $s\lhd t$).
Set $M_t\ensuremath{\coloneqq} (2 m_t)^{m_t}$ .
Then we define $\SUCC(t)$ to be of size ${M_t}^{m_t}$.\footnote{
We can e.g. set $\SUCC(t)\ensuremath{\coloneqq} \{t^\frown k:\, 0\leq k<M_t\}$.}
For $A\subseteq \SUCC(t)$ we set $\mu(A)\ensuremath{\coloneqq}
\log_{M_t}(\card{A}/{M_t}+1)$.
Then $0\leq \mu(A)<m_t$,
$\mu(A)=0$ iff $A=\emptyset$, and $\mu$ is strictly monotonous and
sub-additive.\footnote{Since the function $g(x)\ensuremath{\coloneqq}\log_{M_t}(x+1)$
is concave and satisfies $g(0)=0$.} If $A,B\subseteq \SUCC(t)$ and
$\card{B}\geq \card{A}/{M_t}$, then $\mu(B)>\mu(A)-1$.
If $\card{B}\leq m_t$ then $\mu(B)<1/{m_t}$.
If $\mu(\SUCC(t))>M$, then $m_t>M$.
Now fix an arbitrary $M\in\omega$. There is an $N_0$ such that
$\mu(A)<1/M$
for all $s$ with $\length(s)>N_0$ and all $A\subseteq \SUCC(s)$
with $\card{A}<m_s$.
(Just
note that $m_s$ strictly increases with $\length(s)$.)
Let $N$ be larger than $M+1$ and $N_0$.
So assume that $s\ensuremath{\perp} t\in \Tmax$,
$\length(s)>N\geq N_0$, $A\subseteq \SUCC(t)$,
$\mu(A)>N\geq M+1$ (in particular $m_t>M$),
and $f: A\fnto \Tmax^{[s]}$.
Set $X\ensuremath{\coloneqq}\{s'\succeq s:\, s'\lhd t,\, \length(s')\geq N\}$.
Enumerate $X$ as $\{s_0,\dots,s_{l-1}\}$ (for some $l\geq 0$).
Set $A_0\ensuremath{\coloneqq} A$. Assume that $A_n$ is already defined, and
define
\[S_n\ensuremath{\coloneqq} \{s'\in \Tmax:\, (\exists t'\in A_n)\, s'\preceq f(t')\}.\]
If $n>0$ assume that $\card{\SUCC_{S_n}(s_{n-1})}\leq 1$ and that
$\card{A_n}>\card{A_{n-1}}/(2 m_t)$.
Then we define $A_{n+1}$ as follows:
Since $s_n\in X$, $\card{\SUCC(s_n)}<m_t$. By a simple pigeon-hole
argument, there is an
$A_{n+1}\subseteq A_n$ such that $\card{A_{n+1}}>\card{A_n}/(2 m_t)$
and $\card{\SUCC_{S_{n+1}}(s_{n})}\leq 1$. So in the
end we get a $B\ensuremath{\coloneqq} A_l$ with cardinality at least
\mbox{$\card{A}/(2 m_t)^{m_t}=\card{A}/M_t$}, i.e.\
\mbox{$\mu(B)>\mu(A)-1\geq M$}.
Also, $\card{\SUCC_{S_l}(s')}\leq 1$ for every $s'\in X$,
so $\mu_{S_{l}}(s')\leq 1/M$ (since $\length(s')$
was sufficiently large).
We claim that $B$ is as required.
We have to show that $S_l$ is $(N,1/M)$-meager.
Pick an $s'\in S_l$ of length $\geq N$.
We already dealt with the case $s'\in X$.
Otherwise $s'\rhd t $ (note that $s'\neq t$ since $s\ensuremath{\perp} t$).
In this case
$\card{\SUCC_{S_l}(s')}\leq \card{\SUCC_{\Tmax}(t)}\leq m_{s'}$.
So $\mu(\SUCC_{S_l}(s'))\leq 1/M$, since $\length(s')>N_0$.
\end{proof}
\begin{Lem}
If $Q$ is strongly non-homogeneous, then $P$ forces the following:\\
If $r\in\lim(\Tmax)\setminus \{\n\eta_i:\, i\in I\}$
then there is a $T\in V$ such that $r\in\lim(T)$ and
$T$ is $(1,1)$-meager.
\end{Lem}
If additionally the assumptions of Lemma~\ref{lem:card} hold, then
there are only $\al1$ many $T\in V$, and
$\al1<{2^\al0}^{V_\om2}$. This implies
that the set $\{\n\eta_i:\, i\in I\}$
is of measure 1:\\
If $r\in\lim(\Tmax)\setminus \{\eta_i:\, i\in I\}$,
then \mbox{$r\in \bigcup_{T\in V\text{ meager}}\lim(T)\in \myI$}.
\begin{proof}
Fix a $P$-name $\n r$ for a real and a $p\in P$
such that $p\ensuremath{\Vdash} \n r\notin \{\n\eta_i:\, i\in I\}$.
We will show that there is a
$p_\omega\leq p$ and a $(1,1)$-meager tree $T$ such
that $p_\omega\ensuremath{\Vdash} \n r\in\lim(T)$.
We will by induction construct $p_n\in P$,
approximations $\mathfrak{g}_n$, $k_n\in \omega$ and $i_n\in u_n=\dom(\mathfrak{g}_n)$ such that
\begin{enumerate}
\item\label{item:jkhwe_1} $p_0=p$, $p_{n+1}\leq_{\mathfrak{g}_n} p_n$,
$\mathfrak{g}_{n+1}$ is purely stronger than $\mathfrak{g}_n$.
\item\label{item:jkhwe_2} $\mathfrak{g}_n$ is $n$-dense at $i_n$.
\item\label{item:jkhwe_6} the sequence $(i_n)_{n\in\omega}$
covers $\bigcup \dom(p_n)$ infinitely often.
\item $k_n \rightarrow \max(n+1,\card{\text{Pos}(\mathfrak{g}_n)})$.
\item For each $\bar a\in \text{Pos}(\mathfrak{g}_n)$, $p_n^{[\bar a]}$
forces a value to $r\restriction k_n$.
We call this value $r^{\bar a}_n$.
\item\label{item:meager} The tree $\{r^{\bar a}_n:\, \bar a\in\text{Pos}(\mathfrak{g}_n)\}\subseteq \Tmax\restriction k_n$ is $(k_{n-1},1)$-meager.
\end{enumerate}
(1)--(3)
allow us to fuse the $(p_n)_{n\in\omega}$ into a $p_\omega\leq p$
(cf.\
\ref{lem:fusion}), and
(5),(6) imply that the tree of all
initial segments of $r$ compatible with $p_\omega$ is meager.
So assume we have found $p_n$, $\mathfrak{g}_n$ and $k_n$.
\begin{itemize}
\item
Set $p\ensuremath{\coloneqq} p_n$, $\mathfrak{g}\ensuremath{\coloneqq} \mathfrak{g}_n$, $M\ensuremath{\coloneqq} \card{\text{Pos}(\mathfrak{g}_n)}$
and $N\ensuremath{\coloneqq} k_n$.
\item
Choose the position $i_{n+1}\in\dom(p_n)$
according to some simple bookkeeping.
This takes care of (\ref{item:jkhwe_6}).
Set $j\ensuremath{\coloneqq} i_{n+1}$.
\item
Find a $q_1\leq_{\mathfrak{g}} p_n$, $m>N$ such
that $q_1\ensuremath{\Vdash}(\n\eta_j\restriction m\neq \n r\restriction m)$
and for all $\bar a\in \text{Pos}(\mathfrak{g})$ $q_1^{[\bar a]}$ determines
$\n\eta_j\restriction m$ and $\n r\restriction m$:
First we apply pure decision
\ref{cor:restrtoa}(\ref{item:puredecision}) to
get a $q'\leq_{\mathfrak{g}} p$ such that for all
$\bar a\in\text{Pos}(\mathfrak{g})$ there is an $m^{\bar a}>N$
and $\eta^*\neq r^*$
such that \mbox{$q'^{[\bar a]}\ensuremath{\Vdash} (r^*=\n r\restriction m^{\bar a},
\eta^*=\n\eta\restriction m^{\bar a})$}.
Then we apply pure decision again to get $q_1\leq_{\mathfrak{g}} q'$
determining $\n r$ and $\n \eta_j$ up to
\mbox{$\max\{m^{\bar a}:\, \bar a\in\text{Pos}(\mathfrak{g})\}$}.
\item
Pick a $q_1$-approximation $\mathfrak{h}_1$ which is $N$-dense at $j$
and (purely) stronger than $\mathfrak{g}$.
\item Pick a $k_{n+1}$ such that $k_{n+1}\rightarrow \max(n+2,\card{\text{Pos}(\mathfrak{h}_1)})$.
\item Pick a $q_2\leq_{\mathfrak{h}_1} q_1$ such that
$q_2^{[\bar a]}$ determines $\n\eta_j$ and $\n r$ up to $k_{n+1}$
for all $\bar a\in\text{Pos}(\mathfrak{h}_1)$.
\end{itemize}
So $q_2\leq_{\mathfrak{g}}p$, $\mathfrak{h}_1$ approximates $q_2$ and
witnesses $N$-density (at $j$).
However, the tree of possible values for $\n r$
could be very thick in the levels between $k_n$ and $k_{n+1}$.
We will
thin out the approximation $\mathfrak{h}_1$ so that
we still have $(n+1)$-density, and the tree of
possible values for $\n r$ gets thin.
(Recall that $N$ is much larger than $n+1$, in particular
$N\rightarrow n+1$.)
We do this in two steps:
\begin{figure}[tb]
\begin{center}
\scalebox{0.44}{\input{ramsey.pstex_t}}
\end{center}
\caption{\label{fig:ramsey}$\mathfrak{h}_2$ (bold) is a subapprox. of $\mathfrak{h}_1$ and still purely stronger than $\mathfrak{g}$.}
\end{figure}
\begin{itemize}
\item
Find a sub-approximation $\mathfrak{h}_2$ of $\mathfrak{h}_1$
that is still purely stronger than $\mathfrak{g}$
and has only as many splittings as $\mathfrak{g}$,
apart from one additional split (for each possibility)
that witnesses
$N$-density at $j$ (see Figure~\ref{fig:ramsey}).
To be more exact, we construct $\mathfrak{h}_2$ the
following way: Given $\bar a \in \text{Pos}_{<i}(\mathfrak{h}_2)$.
We have to define $\mathfrak{h}_2(i)(\bar a)$.
If $i\neq j$, pick for each $t\in \mathfrak{g}(i)(\bar a\restriction \mathfrak{g})$
exactly on successor $s\in \mathfrak{h}_1(i)(\bar a)$. So
$\mathfrak{h}_2$ makes the branches of $\mathfrak{g}$ longer,
but does not add any splittings.
At $j$, we have the front $F\ensuremath{\coloneqq}\mathfrak{g}(i)(\bar a\restriction \mathfrak{g})$
and the purely stronger $N$-dense front $F'\ensuremath{\coloneqq} \mathfrak{h}_1(i)(\bar a)$.
Recall that $T\ensuremath{\coloneqq} T_\text{cldn}^{F'}=\{s:\, s\preceq F'\}$
is the finite tree corresponding to the front $F'$.
We continue each $t\in F$ in $T$
uniquely (without splits) until we reach a node with many
(i.e.\ $N$-dense) splittings. We call this node ``splitting node''.
We take all the immediate successors of the splitting node
and continue them
uniquely in $T$ until we reach a leaf of $T$, i.e.\
an element of $F'$. This process leads to a subset
$F''$ of $F'$. Set
$\mathfrak{h}_2(j)(\bar a)\ensuremath{\coloneqq} F''$.
So we get:
There are $\card{\text{Pos}_{\leq j}(\mathfrak{g})}\leq M $ many
pairs $(\bar a,t)$, where
$\bar a\in \text{Pos}_{< j}(\mathfrak{h}_2)$ and $t$ is a splitting node.
Also, for $\bar b\in \text{Pos}_{\leq j}(\mathfrak{h}_2)$,
there are at most $ M $ continuations
of $\bar b$ to some $\bar b'\in \text{Pos}(\mathfrak{h}_2)$.
Such a $\bar b\in \text{Pos}_{\leq j}(\mathfrak{h}_2)$
corresponds to a pair $(\bar a, t)$ as
above together with a choice of an (immediate) successor of $t$.
\item
Now we are ready to apply the Ramsey property.
First fix a $\bar a\in \text{Pos}_{<j}(\mathfrak{h}_2)$ and a
splitting node $t$. (There are at most $M$ many such pairs.)
This pair corresponds to a unique $\bar b\in\text{Pos}_{\leq j}(\mathfrak{g})$.
There are at most $M$ many continuations of $\bar b$ to some
$\bar c\in\text{Pos}(g)$. Fix an enumeration $\bar c_1\dots \bar c_M$
of these possible continuations. Each $\bar c_l$ forces a value
to $\n r\restriction k_n$, call this value $r^l$.
Back to $\mathfrak{h}_2$. Set $A\ensuremath{\coloneqq} \SUCC(t)$ in the tree
$T_\text{cldn}^{\mathfrak{h}_2(j)(\bar a)}$ (or equivalently
$T_\text{cldn}^{\mathfrak{h}_1(j)(\bar a)}$). So $\mu(A)\geq N$.
For every $s\in A$ there is a unique $s'\succeq s$ such that
$\bar a \cup \{(j,s')\}\in \text{Pos}_{\leq j}(\mathfrak{h}_2)$,
and for every
$s\in A$, $l\in M$ there is a unique
$\bar b\in\text{Pos}(\mathfrak{h}_2)$ continuing $\bar c_l\in \text{Pos}(\mathfrak{g})$
and $\bar a\cup\{(j,s')\}$.
Each such $\bar b$ decides $\n r$ up to $k_{n+1}$. We
call this value $r^{s,l}$.
So $r^{s,l}\restriction k_n=r^l$.
Also $r^l\ensuremath{\perp} t$ since $t$ is a possible initial segment of $\n
\eta_i$.
For every $l\in M$
we define a function $f_l:A\fnto \Tmax^{[r^l]}\restriction k_{n+1}$
by mapping $s$ to $r^{s,l}$.
So we can apply the Ramsey property and get a $B\subseteq A$
such that $\mu(B)>M\geq n+1$, and the tree of possibilities
for $\n r$ induced by $\bar a,B$ is $1/M$-meager. We repeat that
for all pairs $(\bar a,t)$ where
$\bar a\in\text{Pos}_{<j}(\mathfrak{h}_2)$ and $t$ is a splitting node,
and get a subapproximation $\mathfrak{g}_{n+1}$ of $\mathfrak{h}_2$ such that
the tree of possibilities for $\n r$ induced by $\mathfrak{g}_{n+1}$
is $(k_n,1)$-meager (here we again use the
sub-additivity of $\mu$).
\end{itemize}
This results in a sub-approximation $\mathfrak{g}_{n+1}$ of
$\mathfrak{h}_2$ (and therefore $\mathfrak{h}_1$) which is still purely
stronger than $\mathfrak{g}=\mathfrak{g}_n$. Since $\mathfrak{g}_{n+1}$ is
a sub-approximation of $\mathfrak{h}_1$, \mbox{$\card{\text{Pos}(\mathfrak{g}_{n+1})}\leq
\card{\text{Pos}(\mathfrak{h}_1)}$}, and therefore $k_{n+1}$, $\mathfrak{g}_{n+1}$ satisfy
(4).
\end{proof}
Note that we did not use the $j_\alpha$ or automorphisms of $I$,
the proof works for all $I_0$.
$I=\{i\}$ in particular implies: If $Q\ensuremath{\Vdash} \n r\neq \n \eta_i$,
then for all $T\in Q$ there is a $S\leq_Q T$ such that $r\notin \lim(S)$, i.e.\
\begin{Cor}\label{cor:jesh}
If $Q$ is strongly non-homogeneous then $Q$
forces that $\n \eta$ is the only $Q$-generic real over $V$ in $V[G_Q]$.
\end{Cor}
\begin{uRem} A similar forcing $Q^\text{JeSh}$
(finitely splitting, rapidly increasing number of successors) was used in
\cite{JeSh:566} to construct a
complete Boolean algebra without proper atomless complete subalgebra.
$Q^\text{JeSh}$ can also be written as lim-sup forcing.
However, the difference is that the norm in
$Q^\text{JeSh}$ is ``discrete'' (as e.g. Sacks): either $s$
has a minimum number of successors, then the norm is large,
or the norm is $<1$. Such a norm cannot satisfy
a Ramsey property as the one above. For
$Q^\text{JeSh}$ we can only prove Corollary~\ref{cor:jesh}, but not
Lemma~\ref{lem:oneetalalleta}.
\end{uRem}
We have already mentioned another corollary:
\begin{Cor}
If $Q$ is strongly non-homogeneous,
then $P$ forces that
$\{\n \eta_i:\, i\in I\}$ is of measure 1.
\end{Cor}
This, together with \ref{lem:onlyetaalphamatter} and \ref{lem:oneetalalleta}
proves Theorem~\ref{thm:defaremeas}.
\begin{uRem}
There are various ways to extend the constructions in this paper. As already
mentioned, we could use non-total orders $I$ or allow $Q_i$ to be a
$P_{<i}$-name. A more difficult change would be to use lim-inf trees instead of
lim-sup trees. In this case we need additional assumptions such ad bigness and
halving. This could allow us to apply saccharinity to a ccc ideal
$\mathbb{I}$, i.e. to force (without inaccessible or amalgamation)
measurability of all definable sets.
\end{uRem}
\bibliographystyle{plain}
|
1,314,259,993,620 | arxiv | \section{Introduction} \label{sec:intro}
The stellar initial mass function (IMF), which describes the mass distribution of stars in a population at birth, plays a vital role in many fields of astrophysics. The literature generally agrees that the IMF has a power-law form $\dd N/\dd \log M \propto M^{\Gamma}$ in the high-mass end, with $\Gamma \approx -1.35$ \citep{Salpeter1955TheEvolution}, while there is an ongoing debate on the possible variations in observational estimates of the slope $\Gamma$ for extragalactic populations \citep{Bastian2010AVariations, Offner2014TheFunction, Krumholz2014TheFunction, Hopkins2018TheFunction}.
One popular candidate for determining the physics of the IMF is turbulence -- this is because the spectra of the molecular ISM, where stars are born, provide clear evidence for supersonic turbulent motions \citep{Larson1981TurbulenceClouds, Ossenkopf2002TurbulentClouds, Elmegreen2004InterstellarProcesses, Heyer2004TheClouds, Roman-Duval2011TheCalibration}. Thus, many theoretical models of the stellar IMF are based on the statistics of supersonic turbulence. \citet*{Padoan1997TheFunction} and \citet[hereafter PN02]{Padoan2002TheFragmentation} proposed that supersonic shocks create dense cores by sweeping through the ISM and compressing the gas. They then estimated the likelihood of the cores to be Jeans unstable and hence the mass distribution of collapsing cores, which may be closely linked to the IMF \citep{Andre2010FromSurvey, Offner2014TheFunction, Guszejnov2015MappingFunction}. \citet[hereafter HC08]{Hennebelle2008AnalyticalCores} and \citet[hereafter H12]{Hopkins2012TheDistribution} proposed derivations of the IMF using the \citet{Press1974FormationCondensation} and excursion set \citep{Bond1991ExcursionFluctuations} formalisms, respectively. In these models, one estimates the density variance as a function of size scale, and then determines the IMF by measuring the mass distribution of regions where the density is high enough for gravity to overcome various supporting mechanisms (such as thermal motions, turbulence, magnetic fields, and/or disc shear). The turbulence-regulated theories of the IMF by PN02, HC08, and H12 yield estimates for $\Gamma$ that are generally in good agreement with observed IMFs \citep{Miller1979TheNeighborhood, Kroupa2001OnFunction, Chabrier2003GalacticFunction, Chabrier2005The2005, Kroupa2013ThePopulations, Offner2014TheFunction}, if the parameters are chosen carefully.
In these analytic models, the power-law index $n$ of the turbulent velocity power spectrum,\footnote{We define $E(k)$ to be the one-dimensional power spectrum, so that Kolmogorov turbulence corresponds to $n=5/3$.} $E_v(k) \propto k^{-n}$, appears as a critical factor that determines the high-mass power-law slope $\Gamma$. The narrow range of $n$ in nature ($5/3 \le n < 2$) \citep{Federrath2013OnTurbulence} can be used to argue for the relatively universal high-mass slope of the IMF produced by these models and seen in observations. However, the near universality of $n$ also makes it difficult to test any particular model's prediction for the relationship between $n$ and the IMF. While the underlying functional relationship between $n$ and the IMF shape is fundamentally different in the different models, the small range of variation in $n$ yields a similarly small range in predicted IMFs.
Nonetheless, a handful of simulations have explored this question. \citet{Bate2009TheStructure} studied the effect of $n$ on the star formation within a collapsing molecular cloud by carrying out simulations with initial turbulent velocity fields characterised by $n=2$ and $n=4$, and concluded that the resultant IMFs show little dependence on $n$ overall. \citet*{Delgado-Donate2004TheFormation} conducted a set of similar simulations but in the context of low-mass ($5\,\mathrm{M_\odot}$) core fragmentation, and also found that the initial choice of $n$ does not significantly affect the stellar IMF. \citet*{Goodwin2006StarSpectrum}, on the contrary, found that a shallower velocity power spectrum ($n$ closer to zero) leads to more fragmentation in their simulations of low-mass ($\sim 5\,\mathrm{M_\odot}$) cores, although the statistical argument is weak due to the low number of sink particles used for the analysis ($N_\text{sink}<100$).
In the studies mentioned above, the authors varied only the initial velocity field, while the star formation commenced roughly after one free-fall time. The problem with this approach is that without continuous driving, most of the turbulent energy would dissipate away within a free-fall time \citep{Stone1998DissipationTurbulence, MacLow1998TheClouds, Elmegreen2004InterstellarProcesses, McKee2007TheoryFormation}, and $n$ would relax to the natural range of $5/3-2$. Therefore, while the choice of $n$ could affect the initial structure of the collapsing cloud, it would have little effect during the process of star formation. We conclude that the studies are insufficient for a direct comparison with the turbulent fragmentation theories.
The aim of this work is to test how well the turbulence-regulated IMF theories (PN02, HC08, and H12) predict the high-mass power-law slope of the IMF, by simulating star formation under hydrodynamic turbulence (i.e., without magnetic fields) with velocity power spectral index $n$ much different from what is observed in nature ($5/3-2$). We develop a turbulence driving module that is capable of driving and maintaining supersonic turbulence with arbitrary $n<2$, and create an artificial molecular cloud with $n=1$ in the computational domain. We measure the mass function of the stars, represented by sink particles, born under the $n=1$ turbulence, and compare it with the IMF from the typical $n\approx2$ supersonic turbulence. We assure the statistical significance of the study by collecting around 1000 stars represented by `sink particles' per setup through repeated simulations with different randomisation of the turbulence driving.
We note that the interaction between magnetic fields and the IMFs represents another point of difference that can be used to test the models. Magnetohydrodynamic (MHD) simulations show that magnetic fields have a variety of effects, including reducing the star formation rate and changing how gas fragments \citep{Padoan2014Infall-drivenProblem, Federrath2015InefficientFeedback, Haugbolle2018TheTurbulence,Krumholz2019TheFunction}. However, they are incorporated into IMF theories in differing ways. In the PN02 model, the presence of magnetic fields changes to which extent supersonic shocks compress the medium, which changes the mass spectrum of the density structures that may go on to collapse and form stars, whereas in the HC08 and H12 models the primary role of magnetic fields is to provide an additional form of pressure that makes it more difficult for structures to collapse. Although we present only hydrodynamic simulations here, in a forthcoming paper we explore the effects of magnetic fields as a complementary way of testing IMF theories.
The rest of the paper is organised as follows. We describe the simulation setup and the initial conditions in \S\ref{sec:methods}, and present the results in \S\ref{sec:result}. In \S\ref{sec:models} we compare our mass functions with the three turbulence-based IMF theories. We summarise our findings in \S\ref{sec:sum}.
\section{Numerical Methods} \label{sec:methods}
We simulate star formation within a turbulent, dense molecular cloud with the \textsc{flash4} adaptive mesh refinement (AMR) code \citep{Fryxell2000FLASHFlashes}. Here we use the HLL5R approximate Riemann solver \citep{Bouchut2010AWaves, Waagan2011ATests} and the multigrid Poisson gravity solver \citep{Ricker2008AMeshes} on a block-based PARAMESH AMR grid. We explain the turbulence driving method in \S\ref{sec:driving} and the sink particles in \S\ref{sec:sink}, then we outline the initial conditions and simulation procedure in \S\ref{sec:setup}.
\subsection{Turbulence driving} \label{sec:driving}
In order to drive turbulence with a prescribed velocity power spectrum of slope $n$, we add a time-varying acceleration field $\mathbf{F}_\text{stir}(\mathbf{x}, t)$ as a source term in the momentum equation \citep{Federrath2010ComparingForcing}. We utilise an Ornstein-Uhlenbeck process \citep{Eswaran1988AnTurbulence} to construct the driving field $\mathbf{F}_\text{stir}$ with an auto-correlation time matching the turbulent crossing time $T=L/2\sigma_v$, where $\sigma_v$ is the rms velocity dispersion. Inspired by observations \citep[e.g.][]{Ossenkopf2002TurbulentClouds,Elmegreen2004InterstellarProcesses,Brunt2009TurbulentClouds}, the usual procedure is to construct $\mathbf{F}_\text{stir}$ with only large-scale modes (i.e., to drive at wavenumbers\footnote{In this paper, $k$ is measured in units of the inverse box size, so $k=1$ corresponds to a mode with wavelength equal to the box scale $L$.} $k = \left|\mathbf{k}\right| \sim 2$) and let small-scale turbulence emerge naturally. The energy cascade in (supersonic) turbulence will distribute energy to smaller scales in such a way as to produce $n\approx 2$ \citep{Federrath2013OnTurbulence}.
Here, however, we want to construct velocity power spectra with $n$ significantly smaller than 2, in order to test theoretical predictions for the dependence of the IMF on $n$. Thus, we must inject energy on every resolvable scale, or in other words, the driving field needs to contain modes up to $k_N=L/(2\Delta x)$, where $\Delta x$ is the minimum computational cell size of the simulation. However, including all wavevectors within $2 \le k \le k_N$ is expensive since \textsc{flash} evaluates the acceleration field at each cell from the set of driving modes, and the number of modes in a wavenumber bin $[k,k+\dd k]$ is proportional to $k^2 \,\dd k$. To reduce the computational load, we take a heuristic approach by generating a stirring field that contains only a fraction of randomly-selected wavevectors, such that the number of modes between $k$ and $k+\dd k$ scales as $k^{0.5}\,\dd k$. This practice yields a significant gain in speed (by a factor of $\sim10^3$) while preserving the isotropy of $\mathbf{F}_\text{stir}$, and therefore the isotropy of the turbulence. The resultant driving field is constructed to have a natural mixture of solenoidal and compressive modes, which corresponds to the driving parameter $b\sim 0.4$ \citep{Federrath2010ComparingForcing}.
In order to run a set of simulations in which the power spectrum of the turbulent velocity field follows a power law with index $n=1$ or $n=2$, we construct the acceleration field $\mathbf{F}_\text{stir}$ with $\num{2.3e4}$ modes, randomly selected within $2\le k \le 256$. We show below that when the amplitude of each mode $A(\mathbf{k})$ is proportional to $k^{-0.9}$, the resulting turbulence power spectrum reaches a slope close to $n=1$. For the $n=2$ case, we use the same method, but with $A(\mathbf{k}) \propto k^{-2}$ to match the shape of the power spectrum of $\mathbf{F}_\text{stir}$ to that of the turbulent velocity typically observed in molecular clouds and simulations of supersonic turbulence \citep{Elmegreen2004InterstellarProcesses, McKee2007TheoryFormation, Federrath2013OnTurbulence}. Below we refer to simulations run with a driving field $A(\mathbf{k})\propto k^{-0.9}$ as N1 simulations, and those run with $A(\mathbf{k})\propto k^{-2}$ as N2 simulations. We show in Appendix~\ref{app:driverange} that the results we obtain for the N2 simulations using this driving procedure are nearly identical to those produced via the more common procedure of driving only at low $k$ \citep{Federrath2010ComparingForcing}, and allowing modes at higher $k$ to be produced by the turbulent cascade.
\subsection{Sink particles and AMR} \label{sec:sink}
In order to follow local collapse and accretion of gas, we use the sink particle method developed in \citet{Krumholz2004EmbeddingGrids} and extended by \citet{Federrath2010ModelingSPH}. \citet{Truelove1997TheHydrodynamics} showed that the local Jeans length $\lambda_J=(\pi c_s^2/G\rho)^{1/2}$, where $c_s$ is the sound speed, must be resolved with at least four grid cells to prevent artificial fragmentation of the collapsing gas. The sink particle technique ensures that the Jeans length is always sufficiently resolved on the highest level of AMR, and that only bound and collapsing gas is turned into sink particles. Gas above the sink particle density threshold
\begin{equation}
\rho_\mathrm{sink} = \frac{\pi c_s^2}{G\lambda_J^2} = \frac{\pi c_s^2}{G r_\mathrm{sink}^2},
\end{equation}
with the sink particle radius $r_\mathrm{sink} = 2.5\Delta x_\text{min}$, is accreted, if the gas is bound and collapsing. Since not all overdense regions that satisfy the above density condition will collapse, we adopt an additional set of sink creation criteria from \citet{Federrath2010ModelingSPH} to avoid artificial sink particle formation.
For dense regions that are not yet on the highest level of AMR, we refine based on the local Jeans density, to better resolve the gravitational collapse. In our simulations, $\lambda_J$ is resolved with at least 16 cells in all dimensions, in order to capture some solenoidal motions of the turbulence inside the Jeans scale \citep{Federrath2011ATurbulence}.
\subsection{Simulation setup} \label{sec:setup}
We simulate a small section of a molecular cloud within a three-dimensional periodic computational domain of length $L=\SI{2}{\parsec}$, mean gas density $\rho_0=\SI{1.31e-20}{\g\per\cubic\cm}$, and thus the cloud mass $M_\text{cloud}=\rho_0L^3=1550\,\mathrm{M_\odot}$. The base-grid resolution is $N_\text{base}=512^3$ grid cells, with two additional levels of AMR, which leads to a maximum effective resolution of $2048^3$ cells, i.e., a minimum cell size of $\Delta x\approx 200$~AU. At this resolution we cannot capture detailed small-scale structures and physics such as protostellar discs and radiative feedback. While radiative feedback may be crucial for setting the characteristic mass of the IMF \citep[but see \citealt{Haugbolle2018TheTurbulence}]{Bate2009TheStructure, Offner2009TheFormation, Krumholz2011OnMasses,Krumholz2016WhatSimulations, Federrath2017ConvergingStars}, at least in the theoretical models that we aim to test it has little effect on the high-mass slope of the IMF. We therefore focus solely on determining the role of the turbulence power spectrum for the high-mass tail of the IMF, and compare to predictions from IMF theories. We assume isothermal gas with constant global sound speed $c_s=\SI{0.2}{\km\per\second}$, and drive the turbulence to an rms Mach number $\mach=\sigma_v/c_s=5$ for all simulations. This ensures that all simulations have identical total kinetic energy, and thus the same global virial parameter \citep{Bertoldi1992Pressure-confinedClouds}, $\alpha_\text{vir} = 5 \sigma_v^2 L / (6GM)=0.25$, and free-fall time $t_\text{ff} = \sqrt{3\pi/(2G\rho_0)} = \SI{0.58}{\mega\year} = 0.594\,T$. While our choice of mean density is a factor of 2--3 higher than the \citet{Larson1981TurbulenceClouds} relation\footnote{According to the Larson relation, a cloud with $L=\SI{2}{\parsec}$ has $n(\text{H}_2)=\SI{1600}{\per\cm\cubed}$, or $\rho_0=\SI{5.4e-21}{\g\per\cm\cubed}$; however, there is substantial scatter around this relation \citep{Larson1981TurbulenceClouds, Falgarone1992TheClouds}.}, the choice of scaling cannot affect the shape of the IMF, which is the quantity of interest for us. We also emphasise that this commonly used approximation of $\alpha_\text{vir}$ is based on the uniform spherical approximation, and the geometry of our simulations is much different from a sphere of gas. The calculated value of $\alpha_\text{vir}$ based on its definition, $2E_\text{kin}/|E_\text{grav}|$, is more than an order of magnitude higher than the approximated value of 0.25, and is dependent on turbulence parameters such as $b$ and $n$ \citep{Federrath2012TheObservations}. This discrepancy is particularly strong for the simulations with $n=1$, which, as we show below, develop significantly less large-scale density structure than the $n=2$ case, and thus have weaker self-gravity than one might otherwise expect.
All simulations begin with uniform density distribution $\rho(\mathbf{x})=\rho_0$ and zero velocity $\mathbf{v}(\mathbf{x})=0$. We let the supersonic turbulence grow by running the models without self-gravity for two turbulent crossing times $2\,T$ \citep{Federrath2012TheObservations}, after which, gravity is turned on and sink particles are allowed to form in bound, collapsing regions of the cloud. We aim to collect around 1000 sink particles for each case to obtain tight statistical constraints on the slopes of the mass functions of the sink particles. For this reason, we run fourteen simulations where we drive with a field $A(\mathbf{k})\propto k^{-0.9}$ in order to produce $n\approx 1$ (N1A--N1N) and eight simulations where we drive with $A(\mathbf{k})\propto k^{-2}$ and thus produce $n\approx 2$ (N2A--N2H). Table~\ref{tab:params} summarises the key input parameters and derived quantities.
\begin{table*}
\centering
\caption{Key simulation parameters and measured quantities.}
\begin{tabular}{rrrrrrrrr}
\toprule
ID & $n$ & $\mach$ & $N_\text{sink}$ & $\text{SFR}_\text{ff}$ & $m_{50}$ & $m_{84}$ & $m_{98}$ \\
(1) & (2) & (3) & (4) & (5) & (6) & (7) & (8) \\
\midrule
\multicolumn{1}{l}{\textbf{N1}} & & & & & \\
\phantom{\textbf{N1}}A & \num[separate-uncertainty = true]{0.93(3)} & 5.0 & 102 & 0.14 & \num{4.7e-04} & \num{1.7e-03} & \num{5.6e-03} \\
B & \num[separate-uncertainty = true]{0.91(3)} & 4.8 & 99 & 0.53 & \num{5.3e-04} & \num{1.5e-03} & \num{6.5e-03} \\
C & \num[separate-uncertainty = true]{0.92(2)} & 4.9 & 67 & 0.21 & \num{9.1e-04} & \num{2.4e-03} & \num{7.1e-03} \\
D & \num[separate-uncertainty = true]{0.96(2)} & 4.9 & 57 & 0.14 & \num{4.3e-04} & \num{5.0e-03} & \num{1.0e-02} \\
E & \num[separate-uncertainty = true]{0.95(2)} & 5.0 & 74 & 0.17 & \num{3.5e-04} & \num{1.8e-03} & \num{1.1e-02} \\
F & \num[separate-uncertainty = true]{0.97(2)} & 5.0 & 97 & 0.22 & \num{3.9e-04} & \num{2.0e-03} & \num{7.1e-03} \\
G & \num[separate-uncertainty = true]{0.97(3)} & 5.0 & 56 & 0.14 & \num{4.6e-04} & \num{2.6e-03} & \num{1.5e-02} \\
H & \num[separate-uncertainty = true]{0.92(3)} & 4.9 & 64 & 0.13 & \num{6.3e-04} & \num{3.7e-03} & \num{6.2e-03} \\
I & \num[separate-uncertainty = true]{0.98(2)} & 4.9 & 55 & 0.04 & \num{4.5e-04} & \num{2.5e-03} & \num{1.4e-02} \\
J & \num[separate-uncertainty = true]{0.98(3)} & 5.0 & 56 & 0.31 & \num{9.9e-04} & \num{4.1e-03} & \num{6.8e-03} \\
K & \num[separate-uncertainty = true]{0.98(3)} & 5.0 & 66 & 0.13 & \num{4.6e-04} & \num{3.1e-03} & \num{9.1e-03} \\
L & \num[separate-uncertainty = true]{0.86(3)} & 4.8 & 76 & 0.17 & \num{4.7e-04} & \num{2.1e-03} & \num{1.1e-02} \\
M & \num[separate-uncertainty = true]{0.92(3)} & 5.0 & 54 & 0.19 & \num{4.5e-04} & \num{3.1e-03} & \num{1.6e-02} \\
N & \num[separate-uncertainty = true]{0.95(2)} & 5.0 & 64 & 0.20 & \num{4.3e-04} & \num{3.8e-03} & \num{7.4e-03} \\
\midrule
\textbf{total} & \textbf{\num[separate-uncertainty = true]{0.95(1)}} & \textbf{\num[separate-uncertainty = true]{4.9(1)}} & \textbf{987} & & \textbf{\num{4.9e-4}} & \textbf{\num{2.5e-3}} & \textbf{\num{1.0e-2}} \\
\midrule
\multicolumn{1}{l}{\textbf{N2}} \\
\phantom{\textbf{N2}}A & \num[separate-uncertainty = true]{1.80(1)} & 5.2 & 114 & 0.27 & \num{5.0e-04} & \num{1.9e-03} & \num{4.3e-03} \\
B & \num[separate-uncertainty = true]{1.87(1)} & 4.5 & 110 & 0.30 & \num{6.3e-04} & \num{1.8e-03} & \num{3.3e-03} \\
C & \num[separate-uncertainty = true]{1.89(1)} & 4.9 & 137 & 0.40 & \num{4.7e-04} & \num{1.3e-03} & \num{3.4e-03} \\
D & \num[separate-uncertainty = true]{1.89(1)} & 4.7 & 112 & 0.30 & \num{6.3e-04} & \num{1.6e-03} & \num{3.1e-03} \\
E & \num[separate-uncertainty = true]{1.91(1)} & 4.7 & 126 & 0.34 & \num{4.7e-04} & \num{1.4e-03} & \num{3.6e-03} \\
F & \num[separate-uncertainty = true]{1.84(1)} & 5.0 & 113 & 0.36 & \num{3.8e-04} & \num{1.6e-03} & \num{5.7e-03} \\
G & \num[separate-uncertainty = true]{1.87(1)} & 4.8 & 109 & 0.39 & \num{5.0e-04} & \num{1.9e-03} & \num{3.1e-03} \\
H & \num[separate-uncertainty = true]{1.85(1)} & 4.7 & 105 & 0.32 & \num{5.7e-04} & \num{1.7e-03} & \num{4.4e-03} \\
\midrule
\textbf{total} & \textbf{\num[separate-uncertainty = true]{1.86(1)}} & \textbf{\num[separate-uncertainty = true]{4.8(2)}} & \textbf{926} & & \textbf{\num{5.0e-4}} & \textbf{\num{1.7e-3}} & \textbf{\num{4.0e-3}} \\
\bottomrule
\end{tabular} \vspace{0.1cm}\\
{\raggedright \emph{Notes.} (1) simulation name; (2--3) power-law index $n$ and rms Mach number $\mach$ measured after two turbulent crossing times; (4--5) the number of sink particles and star formation rate (SFR) per free-fall time recorded at the star formation efficiency (SFE) of 10 per cent; (6--8) 50th, 84th, and 98th percentiles of the SMF, where masses are measured as $m = M_{\rm sink}/M_{\rm cloud}$.
\par}
\label{tab:params}
\end{table*}
\section{Results} \label{sec:result}
In this section we analyse the results of the simulations summarised in Table~\ref{tab:params}. First we examine the statistics of the velocity and density fields in \S\ref{sec:stats}, and verify that our turbulence driving method produces a range of power-law slopes as desired. We then study how the modified turbulence affects molecular cloud morphology in \S\ref{sec:cloud}. We discuss the star formation rate and temporal evolution of the simulations in \S\ref{sec:sfr}, and finally, we construct the sink mass function (SMF) and calculate its power-law slope $\Gamma$ in \S\ref{sec:imf}. Although we carry out simulations in physical units, as described in \S\ref{sec:setup}, we note that, since they are isothermal, the simulations themselves are dimensionless and can be re-scaled to arbitrary length and mass scales. For this reason, in this section we will report all results in dimensionless units, i.e., we will report all masses as fractions of $M_{\rm cloud}$, all lengths as fractions of $L$, and so forth, since these ratios are independent of the choice of dimensional scaling.
\subsection{Velocity and density statistics} \label{sec:stats}
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{Fig/ps_vel.pdf}
\caption{Turbulent velocity power spectra $E_v(k)$ (top) and the compensated power spectra $E_v(k)/k^{-n}$ (bottom), for N1 ($n=1$; blue solid line) and N2 ($n=1.9$; black dashed line). The vertical lines indicate the $1\sigma$ range of variation within the simulations, and the thick transparent lines in the top panel are power-law fits over the range $5\le k\le 30$. The $y$-axes in both panels have arbitrary units, and the compensated power spectra are normalised so that their means within the fitting range are both equal to 1.}
\label{fig:ps}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{Fig/ps_dens.pdf}
\caption{Power spectra of the density $\rho$ (top) and logarithmic density $s=\ln(\rho/\rho_0)$ (bottom) for N1 and N2 runs. Symbols and fitting methods are identical to those used in Fig.~\ref{fig:ps}. The $y$-axes have arbitrary units.}
\label{fig:psdens}
\end{figure}
To confirm that the simulations reach the intended values of the velocity power spectral index $n$ we measure the velocity power spectra $E_v(k)$ of the simulations at $t=2\,T$, i.e., when the turbulence would be fully developed and gravitational collapse begins. We interpolate the AMR grid to a $512^3$ uniform grid (i.e., at the base-grid resolution) when calculating the power spectra. Fig.~\ref{fig:ps} shows the resulting power spectra, averaged over each set of runs, i.e., the line labelled N1 in the plot is the average power spectrum of runs N1A--N1N, and similarly for N2. For both sets of simulations, the power spectra show a power-law dependence on $k$ over a broad range of length scales until $k \sim 30$, beyond which numerical dissipation begins to take effect. We therefore estimate the slope of the power-law by fitting the velocity power spectrum $E_v(k)$ over the range $5\le k \le 30$. We find best-fit values $E_v(k)\propto k^{\num[separate-uncertainty = true]{-0.95(1)}}$ for N1 and $k^{\num[separate-uncertainty = true]{-1.86(1)}}$ for N2, as shown in the top panel of Fig.~\ref{fig:ps}. The value of $n$ for N1 is in good agreement with our target, while the one for N2 is slightly shallower, because of the low target Mach number \citep[see e.g.][for comparison]{Kritsuk2007TheTurbulence,Federrath2010ComparingForcing}. Nonetheless, it is clearly steeper than the result for N1. We also present the compensated power spectra, in the bottom panel of Fig.~\ref{fig:ps}, to better visualise the deviations from the power-law scaling. In both simulations, $E_v(k)$ follows the scaling law very well within the fitting range. We conclude that we successfully drive and maintain turbulence such that its velocity power spectrum is a power-law with an index of $-1$ or $\approx -2$ for a broad range of length scales, as required for the experiment we wish to perform.
In the top panel of Fig.~\ref{fig:psdens} we plot the density power spectra, $E_\rho(k)$, which we measure and fit exactly as we do the velocity field, for N1 and N2 runs. We find turbulence with $n=1$ has considerably less power on large spatial scales (small $k$) than with $n=1.9$, due to the weaker large-scale turbulence. More interestingly, the total variance of the density fluctuations,
\begin{align}
\left<\rho^2 \right> = \int E_\rho(k) \, \dd k,
\end{align}
for N1 simulations is about 20 per cent lower than for the N2 counterpart, despite the fact that the total velocity fluctuation $\sigma_v^2 = (\mach c_s)^2$ is equal in both cases.
The bottom panel of Fig.~\ref{fig:psdens} shows the power spectra of the logarithmic density $s=\ln(\rho/\rho_0)$, $E_s(k)$, for N1 and N2 runs. We find the spectral index of $E_s(k)$, which we denote as $-n'$, to be $n'=\num[separate-uncertainty = true]{0.65(1)}$ for N1 and $n'=\num[separate-uncertainty = true]{1.48(1)}$ for N2. Although the exact scaling exponent of the density power spectrum remains in debate (our result for $E_\rho(k)$ is similar to that of \citet{Kim2005DensityFlows} and slightly shallower than found by \citet{Konstandin2016MachField}), it is important to note that $n'$ does not equal $n$ for both simulations. This contradicts a core assumption in the HC08 model and we discuss the impact this has on the shape of the HC08 IMF in detail in \S\ref{sec:hc08}.
\subsection{Cloud structure} \label{sec:cloud}
\begin{figure*}
\centering
\includegraphics[width=\linewidth]{Fig/proj_mw.pdf}
\caption{Column density maps extracted from N1A (top) and N2A (bottom), at the point when we turn on self-gravity (left) and at the times when the simulations reach a SFE of 1\% (middle) and 10\% (right). The colour scale is logarithmic and ranges from $\Sigma=0.3\Sigma_0$ (black) to $125\Sigma_0$ (white), where $\Sigma_0 = \rho_0 L$. We plot sink particles as cyan circles on top of the density projections, with sizes proportional to the logarithm of their mass $m=M_\text{sink}/M_\text{cloud}$ as indicated in the legend. Panels are annotated with the number $N_{\rm sink}$ of sink particles present in the frame and time $t$ of the simulation, where $t=0$ corresponds to the time at which we turn on self-gravity.}
\label{fig:proj}
\end{figure*}
Fig.~\ref{fig:proj} compares the column density distributions of run N1A (top) with N2A (bottom). The left panels show the structure at time $2\,T$, immediately before we turn on self-gravity. This figure confirms our speculations based on Fig.~\ref{fig:psdens}: there exist large ($k\sim 5$) density structures in the cloud with $n\approx 2$, but such structures are much less prominent in the $n=1$ model. Instead, small-scale velocity perturbations dominate the cloud, which prevent large-scale density structures from forming. As a result the overall level of density perturbation in N1A is smaller than in N2A, which explains why the integral of $E_\rho(k)$ is lower for $n=1$.
The dominance of small-scale turbulence in N1A continues after the self-gravity is switched on, as shown in the middle and right panels of Fig.~\ref{fig:proj}. While the standard supersonic turbulence ($n\approx 2$) allows gas to collapse into dense filaments, inside which dense protostellar cores emerge, gas in the $n=1$ turbulence collapses in a fairly different manner. We no longer observe gas filaments, but dense, quasi-spherical patches of gas, and fragmentation happens inside these patches. There are two explanations for the lack of gas filaments: run N1A lacks low-$k$ supersonic shocks that compresses gas in one dimension over large spatial scales, and the excessive amount of turbulent energy in high-$k$ modes would quickly destroy the filaments.
\subsection{Star formation rate} \label{sec:sfr}
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{Fig/sfr.pdf}
\caption{Star formation efficiency (top) and star formation rate measured per free-fall time (bottom) plotted as a function of time since the formation of the first sink particle, which we denote $t_{\rm sink}$. Solid lines are simulations with $n=1.9$ and dashed lines are for $n=1$. In general, N1 simulations evolve more slowly than N2 simulations and have lower star formation rates.}
\label{fig:sfr}
\end{figure}
We note in Fig.~\ref{fig:proj} that star formation is much slower in turbulence with $n=1$. N2A arrives at a star formation efficiency ($\text{SFE}=M_\text{sink}/M_\text{cloud}$) of 10 per cent after $0.89\,t_\text{ff}$, whereas it takes $2.31\,t_\text{ff}$ for N1A to convert the same amount of mass into sinks. In order to show that this is a general result and not just the case for N1A versus N2A, we plot the temporal evolution of the SFE and the star formation rate (SFR) measured per free-fall time $\text{SFR}_\text{ff} = \dd\,\text{SFE}/\dd(t/t_\text{ff})$ for all our simulations in Fig.~\ref{fig:sfr}. We observe that it takes an average of approximately 0.5 free-fall times for the N2 simulations to go from the formation of their first sink particle to the time when the SFE reaches 10\% and we stop the simulation, whereas this number grows to $\sim 1.7\,t_\text{ff}$ for N1 simulations. Similarly, we see that turbulence with $n=1$ keeps $\text{SFR}_\text{ff}\lesssim 0.2$ throughout most of the simulations, while for the N2 simulations with $n=1.9$ we have $\text{SFR}_\text{ff}\sim 0.3$.
One distinct and noteworthy feature is that some N1 simulations show a longer period of near-quiescence, even after the first sink particle appears, before the onset of vigorous star formation. Simulation N1B (light blue solid line in Fig.~\ref{fig:sfr}) is the most extreme example of this: even after the first sink forms, this run remains at $\text{SFE}\approx 0.5\%$ for almost 2 free-fall times, but then the $\text{SFR}_\text{ff}$ peaks at 0.54 near the end of the run. On the contrary, all N2 simulations show a much more regular pattern where star formation begins slowly, but then $\text{SFR}_{\text{ff}}$ rapidly increases over $\lesssim 1$ free-fall time.
\subsection{Mass function of the sink particles} \label{sec:imf}
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{Fig/imf_full.pdf}
\caption{Logarithmic mass function $\dd N/ \dd\log{m}$ ($m=M_\text{sink}/M_\text{cloud}$ is the sink mass relative to the cloud mass) of the sink particles from the N1 (blue shaded histogram) and N2 (black hatched histogram) simulations, at $\text{SFE}=10\%$. The error bars on the histograms indicate the 68\% confidence interval for each bin. The solid lines show the median values of the posterior PDF obtained from the MCMC fitting, with the surrounding shaded regions representing the 68\% (thick shades) and 95\% (light shades) confidence intervals determined from the MCMC fit. We also report the median values for the high-mass power-law slope $\Gamma$, with the 2nd to 98th percentile ranges in the legend. The red dotted line corresponds to the \citet{Salpeter1955TheEvolution} slope ($\Gamma=-1.35$). We find that the power-law slope of the SMFs generated from the simulations are shallower (for N1) and significantly steeper (for N2) than the Salpeter slope. Thus, the turbulence power spectrum plays a key role in controlling the high-mass slope of the IMF.}
\label{fig:imf}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{Fig/cdf.pdf}
\caption{Cumulative distribution function (CDF) of the sink masses for N1 (blue) and N2 (black) simulations. We observe that the CDFs with $n=1$ and 1.9 disagree only within the high-mass (beyond the median mass) region. N1 simulations produce a significantly more top-heavy CDF.}
\label{fig:cdf}
\end{figure}
We collect sink particles from the simulations when they reach $\text{SFE}=10\%$ and construct the Sink Mass Functions (SMFs) $\dd N/\dd \log m$ for each value of $n$, where $m$ is the relative mass $m=M_\text{sink}/M_\text{cloud}$ of the sinks. Fig.~\ref{fig:imf} shows the resultant SMFs, which span three orders of magnitude in mass and thus provide a sufficient dynamic range to identify differences between the N1 and N2 cases at high confidence. Quantitatively, we form sinks as small as $m= \num{5e-7}$ ($M_\mathrm{sink}= \num{8e-4}\,\mathrm{M_\odot}$), and as large as $m= \num{2.5e-2}$ ($M_\mathrm{sink}= 40\,\mathrm{M_\odot}$); the lower cutoff is imposed by the resolution of the simulation, while the upper one is due to the finite amount of mass contained in the periodic box. We observe that the N1 simulations generate significantly more sinks with $m\gtrsim\num{5e-3}$ than the N2 simulations. This makes the high-mass fall-off in N1 slightly shallower than that of the \citet{Salpeter1955TheEvolution} IMF, while the N2 SMF shows high-mass scaling visibly steeper than the Salpeter slope. The characteristic mass where the IMF peaks ($m\approx 10^{-3}$), on the other hand, appears to be fairly insensitive to the velocity power spectral index.
We compare the cumulative mass functions for the N1 and N2 runs in Fig.~\ref{fig:cdf}. The figure clearly shows that the mass distributions are statistically indistinguishable below the median mass, but that the cumulative SMF for N1 is skewed significantly towards higher mass compared to that for N2. To demonstrate this quantitatively, we report the values of the 50th, 84th, and 98th percentile of the SMF in Table~\ref{tab:params}. While we find that the median masses are almost identical for N1 and N2 ($m_{50}=\num{4.9e-4}$ for N1 and $\num{5.0e-4}$ for N2), the 86th and 98th percentile masses widely differ, as one can find from Table~\ref{tab:params}. We also conduct a Kolmogorov-Smirnov (KS) test comparing the SMFs. If we compare only the parts of the distribution below the median mass, the test returns a $p$-value $p=0.59$, consistent with the hypothesis that the N1 and N2 data are drawn from the same parent distribution. However, if we instead compare the full SMFs, we obtain $p\sim 10^{-8}$. These statistics provide additional evidence for our speculation that altering the turbulence spectral index primarily affects the high-mass tail of the IMF.
Finally, in order to quantitatively measure the difference in the slope of the SMFs ($\Gamma$), we use the Markov Chain Monte-Carlo (MCMC) sampler \texttt{emcee} \citep{Foreman-Mackey2013EmceeHammer} to fit the SMFs to a \citet{Chabrier2005The2005}-like functional form for the IMF,
\begin{align}
\frac{\dd N}{\dd \log m} = A_1
\begin{cases}
\dfrac{1}{\sqrt{2\pi \sigma^2}} \exp\left[-\dfrac{\left(\log m - \log m_0\right)^2}{2\sigma^2} \right], &\quad m<m_T, \\
A_2 m^{\Gamma}, &\quad m\ge m_T,
\end{cases} \label{eq:fit}
\end{align}
with four free parameters $\boldsymbol\theta = (m_0, \sigma, m_T, \Gamma)$, where $m_0$ and $\sigma$ are respectively the peak and standard deviation of the log-normal part, $m_T$ is the transition point between the log-normal and power-law part, and $\Gamma$ is the power-law slope. $A_1$ is a normalisation constant, set by the total mass in stars, and $A_2$ is set so as to ensure continuity at $m_T$.\footnote{We note that the derivative of Eq.~(\ref{eq:fit}) is not necessarily continuous at $m=m_T$. We allow this possibility to ensure that the slope we find for the power-law portion of the IMF at high masses is not forced to some particular value by a requirement that it match the slope favoured by the sub-peak sink population, which dominates the total number of sink particles, and thus the likelihood function.} The posterior probability distribution for $\boldsymbol\theta$ is given by Bayes' Theorem,
\begin{align}
P(\boldsymbol\theta|\{m_\text{sink}\}) = \frac{P(\boldsymbol\theta)P(\{m_\text{sink}\}|\boldsymbol\theta)}{\int P(\boldsymbol\theta')P(\{m_\text{sink}\}|\boldsymbol\theta') \,\dd \boldsymbol\theta'},
\end{align}
where the likelihood function for a given set of parameters $\boldsymbol\theta$ and sink masses $\{m_\text{sink}\}$ is
\begin{align}
P(\{m_\text{sink}\}|\boldsymbol\theta) = \prod_{m_i\in\{m_\text{sink}\}} \frac{\dd N}{\dd m}(m_i;\boldsymbol\theta).
\end{align}
In other words, $P(\{m_\text{sink}\}|\boldsymbol\theta)$ is the probability density for the particular set of sink particle masses $\{m_\text{sink}\}$ produced in our simulations, given a proposed set of parameters $\boldsymbol{\theta}$ describing the IMF. The advantage of this approach, compared to fitting a model to the histograms, is that fitting to histograms often produces results that are sensitive to the choice of bins, particularly in sparsely-populated ranges of mass; our Bayesian approach removes the need for binning.
Fitting requires some care with respect to the choice of priors. We adopt flat, uninformative priors for $m_0$, $\sigma$, and $\tan^{-1}\Gamma$, with the latter being equivalent to assuming that all angles of the power-law slope (straight line in log-log space) are equally likely \citep{Jeffreys46a}. These choices have little impact on the results of the fit parameters. For the N2 SMF, we also adopt a flat prior for $m_T$, and we obtain a good fit by doing so; we show the results of our MCMC fit in comparison to the data in Fig.~\ref{fig:imf}, indicating that the fit describes the data well. We find the high-mass slope $\Gamma(n=1.9)=-3.07\substack{+0.67 \\ -0.77}$ for N2, where the central estimate is the median of the posterior PDF, and the error bars indicate the 2nd to 98th percentile confidence interval. If we adopt a similar flat, unconstrained prior for $m_T$ for N1, we find a higher value for $m_T$ than for N2. In order to enable a meaningful comparison of the slopes between N1 and N2, we therefore adopt an informative prior on $m_T$ when fitting the N1 SMF, by setting it equal to a Gaussian approximation of the posterior distribution of $m_T$ in N2.\footnote{To be precise, the prior distribution we adopt for $m_T$ is $p_{\rm prior} \propto \exp[-(m_T-m_{T,{\rm N2,med}})^2/2\sigma_{\rm N2}^2]$, where $m_{T,{\rm N2,med}}$ is the median posterior value of $m_T$ for our fit to N2, and $\sigma_{\rm N2}$ is half the 16th to 84th percentile range for the posterior.} Intuitively, this amounts to saying that, in order to perform a meaningful comparison of slopes between N1 and N2, we demand that the turnover point $m_T$ between the lognormal and power-law portions of the SMF be at similar masses. With this prior, we find $\Gamma(n=1) = -1.20\substack{+0.23 \\ -0.27}$ for N1. We show this fit in Fig.~\ref{fig:imf}, and find that the resulting functional form is a good fit to the simulated mass distribution.
In summary, we find that the turbulence power spectrum is a key ingredient for controlling the high-mass region of the IMF, with N1 producing more massive stars than N2. The high-mass slope ($\Gamma$) of the IMF is significantly shallower for N1 compared with N2, with the Salpeter slope in between N1 and N2. We discuss possible reasons for this when we now compare the simulation results with the predictions of the IMF theories.
\section{Comparison with theoretical models of the IMF} \label{sec:models}
In this Section we compare the simulation results with the three turbulence-regulated IMF models: PN02 \citep{Padoan2002TheFragmentation}, HC08 \citep{Hennebelle2008AnalyticalCores}, and H12 \citep{Hopkins2012TheDistribution}. We summarise the comparison in Figure~\ref{fig:theory}, as well as in Table~\ref{tab:gamma}, which lists the high-mass IMF slopes estimated from the three theoretical models and calculated from our simulations for velocity power spectral indices of $n=1$ and $1.9$. We emphasise that we only compare the high-mass region of the IMF, and other features of the IMF such as the IMF peak and the sub-stellar mass function are out of the scope of this study, since we do not include the relevant physics in our simulations (\S\ref{sec:imf}).
\begin{table}
\caption{Predictions of the slope $\Gamma$ of the high-mass tail of the IMF from turbulence-regulated IMF theories.}
\def1.0{1.0}
\setlength{\tabcolsep}{19.5pt}
\begin{tabular}{rrr}
\toprule
model & \multicolumn{2}{r}{velocity spectral index} \\
& $n=1$ & $n=1.9$ \\
\midrule
& \multicolumn{2}{l}{$\Gamma=$} \\
PN02 & $-1.0$ & $-1.4$ \\
PN02~(HD) & $-1.0$ & $-2.5$ \\
HC08 & $-2.0$ & $-1.3$ \\
HC08~(exact) & $+1.3$ & $-1.1$ \\
H12 ($k=1-3$) & $-16$ & $-2.1$ \\
H12 (rms) & $-0.3$ & $-2.0$ \\
\midrule
this study & $-1.20\substack{+0.23 \\ -0.27}$ & $-3.07\substack{+0.67 \\ -0.77}$\\
\bottomrule
\end{tabular} \vspace{0.1cm} \\
\emph{Notes.} PN02: \citet{Padoan2002TheFragmentation}. PN02~(HD): PN02 with hydrodynamic shock jump conditions ($\rho'/\rho = \mach^2$). HC08: \citet{Hennebelle2008AnalyticalCores}. HC08~(exact): HC08 with the correction term discussed in \citet{Hennebelle2009AnalyticalFlow}. H12~($k=1-3$): \citet{Hopkins2012TheDistribution}, with $\mach_h$ derived by integrating the power spectrum from $k=1-3$, and slope derived by averaging between $m=\num{3e-3}$ and $10^{-2}$. H12~(rms): same as H12~($k=1-3$), but using the full rms Mach number for $\mach_h$.
\label{tab:gamma}
\end{table}
\begin{figure*}
\centering
\includegraphics[width=\linewidth]{Fig/imf_comp.pdf}
\caption{Comparison between the mass distributions obtained in the simulations (Fig.~\ref{fig:imf}) and the high-mass IMF slopes estimated by the three IMF theories by PN02 (left), HC08 (middle), and H12 (right). The blue histograms and lines correspond to the N1 ($n=1$) simulations and the black ones correspond to the N2 ($n=1.9$) simulations.
Left panel: the PN02 model, using the MHD (dashed lines) or HD (dash-dotted lines) shock jump conditions. Note that the IMF slope is identical for $n=1$, regardless of the choice of the jump condition.
Middle panel: the IMF slopes originally presented in the HC08 paper (dashed lines) and the slopes including the correction term as given in \citet[dash-dotted lines]{Hennebelle2009AnalyticalFlow}. For the PN02 and HC08 models, we anchor the power-law functions at $m=m_T=\num{3e-3}$, as obtained from the MCMC fitting of the simulation data.
Right panel: the IMFs predicted by the H12 model, with the characteristic Mach number ($\mach_h$) calculated from the velocity dispersion on the largest scales in our simulations, $1<k<3$, where $k=1$ corresponds to the box scale $L$ (dashed lines), or set to the rms Mach number of the simulations (dash-dotted lines). We arbitrarily shift both dashed lines to lower masses by a factor of 2 and both dash-dotted lines to lower masses by a factor of 10, compared with the direct prediction of the H12 model, as an attempt to match the high-mass end of the SMFs with that of the corresponding IMFs.}
\label{fig:theory}
\end{figure*}
\subsection{PN02 model} \label{sec:pn02}
In the PN02 theory, cores emerge from turbulent shocks sweeping through the molecular cloud medium, and hence the resultant IMF is dependent on the extent to which shocks compress the gas. PN02 predict that the resulting IMF will be a power law with slope
\begin{align}
\Gamma = -3/(4-n),
\end{align}
assuming a linear shock jump condition, i.e., shocks increase the density of the gas linearly with the Mach number of the shock (hereafter ``MHD condition``). On the other hand, \citet{Padoan2007TwoFormation} suggested that if there are no magnetic fields present, it is more appropriate to consider the post-shock gas density to be proportional to $\mach^2$ (``HD condition``), which leads to
\begin{align}
\Gamma = -3/(5-2n).
\end{align}
In either the HD or MHD cases, PN02 predict that a shallower velocity power spectrum produces a shallower high-mass IMF: $\Gamma(n=1)=-1$ (for both the MHD and HD condition) and $\Gamma(n=1.9)=-1.4$ (MHD condition) or $-2.5$ (HD condition). We show these theoretical predictions for $\Gamma$ together with the simulation SMFs in the left-hand panel of Fig.~\ref{fig:theory}. Overall, the PN02 prediction with the HD shock jump condition (i.e. in the absence of magnetic fields) is quantitatively consistent with both N1 and N2 simulations within the 95\% interval range.
The $n$-dependence on the high-mass slope of the PN02 model comes from the linewidth-size relation. Shocks larger in size (i.e., also with higher Mach number) can sweep up more gas and thereby produce more massive cores. However, this effect is countered by the fact that shocks with higher $\mach$ produce thinner compressed post-shock layers, which reduces the mass of the resultant dense core, because the core size is set equal to the post-shock length scale in the PN02 model. Because the velocity power spectrum controls how the velocity dispersion scales with size, namely $\mach(\ell) \propto \ell^{(n-1)/2}$, altering $n$ changes the mass of cores produced by a shock with fixed length, and hence changes the IMF shape. In addition, since more massive stars take longer to form because they require a larger core with a longer dynamical time, a shallower IMF is predicted for $n=1$ in the PN02 model, which is also consistent with our finding of a lower star formation rate for $n=1$.
\subsection{HC08 model} \label{sec:hc08}
In the HC08 model, turbulence has two roles during the star formation process: it creates dense patches of gas that may become self-gravitating, but also provides additional turbulent energy that counteracts collapse. According to the model, decreasing $n$ (i.e., making the power spectrum flatter) and hence enhancing turbulence on smaller scales both narrows the density PDF (i.e., creating dense regions less frequently) and increases the critical density for collapse. This prediction suggests that the SFR would be much lower for $n=1$, consistent with our results (see \S\ref{sec:stats}, \ref{sec:sfr}). HC08 also predict\footnote{Here we note that our $n$ is the index of the one-dimensional power spectrum, whereas HC08 work in terms of the three-dimensional spectrum, which has index $n-2$. Care should therefore be taken in comparing the expressions we give here to those given in HC08, since our $n$ does not refer to the same quantity as the $n$ that appears in their equations.}
\begin{align}
\Gamma \approx -(n+3)/(2n), \label{eq:hc08}
\end{align}
that is, turbulence with a shallower velocity power spectrum produces a steeper IMF, which is opposite to what is observed in our simulations (middle panel of Fig.~\ref{fig:theory}).
However, \citet{Hennebelle2009AnalyticalFlow} suggested a correction term for Eq.~(\ref{eq:hc08}):
\begin{align}
\Gamma = -\frac{n+3}{2n} + \frac{3(3-n)}{n}\frac{\ln{\mach_*}}{\sigma_s^2}, \label{eq:hc09}
\end{align}
where $\mach_*$ is the (one-dimensional) Mach number on the Jeans scale ($\lambda_J$) and $\sigma_s^2$ is the global variance in the logarithmic density $s$. Under usual circumstances, where $n\approx 2$ and $\mach_*\lesssim 10$, the second term is close to zero and has only minimal effect on the overall shape of the IMF. However, for $n=1$, the correction term becomes much more significant. We calculate the exact value of the high-mass slope predicted by HC08 with the correction term to be $\Gamma=+1.3$ for $n=1$, given in our $n=1$ simulations $\sigma_s^2 = 1.94$ at the beginning of gravitational collapse ($t=2\,T$) and $\mach_* = 4.9 / 3^{1/2} = 2.8$ (converting the 3D Mach number of $\sim4.9$ in the simulations, to the 1D Mach number used in the HC model). While the correction is in the right direction, it is far larger than the difference between the measured value from our simulations and the HC08 prediction, and appears implausible, since for $\Gamma=1.3$ the total mass in the high-mass tail of the IMF would diverge.
\subsection{H12 model} \label{sec:h12}
The role of the velocity power spectrum in the H12 theory is similar to that in the HC08 theory. The primary difference between the theories lies in how one estimates the density PDF and counts the number of bound regions as a function of length scale. The difference is nonetheless significant; for example, H12 speculates that the density variance is greater on small length scales and smaller on large length scales for $n\approx 1$, qualitatively similar to our results (Fig.~\ref{fig:psdens}), while in HC08 the density variance is smaller across all scales. Since the H12 IMF model generally does not have a closed form, one needs to follow the excursion-set formalism and directly rebuild the mass functions in order to study the effect of $n$ in the H12 model. We therefore developed our own Python code that reproduces the last-crossing IMF, and compared the results with our simulation.\footnote{We make one modification in our code relative to the original H12 model. In the H12 model, the barrier function includes a term representing rotational support, parameterised by the epicyclic frequency $\kappa$. Since our simulation has no systematic rotation, we take the limit $\kappa\to 0$ when evaluating the barrier function.}
In the H12 theory, the power spectral index $n$ and the characteristic Mach number $\mach_h$ are the two important parameters that determine the shape of the IMF. The parameter $n$ is straightforward to define and measure in our simulations, but there is some ambiguity in how to define $\mach_h$ for our simulation. In the context of the H12 model, $\mach_h$ is the Mach number of the velocity field measured on sizes comparable to the galactic scale height, $h$, which is identified with the outer scale of the turbulent cascade. Our simulation does not possess a scale height, since it takes place in a periodic box, and there is some ambiguity in how to define the outer scale of the turbulence, particularly for the $n=1$ case where turbulent power is not sharply peaked on large scales. We therefore consider two possibilities, which roughly bracket the range of reasonable choices. The first is simply to set $\mach_h = \mach = 5$, i.e., to set the Mach number at the outer scale of the turbulence equal to the Mach number of the simulation box as a whole. This choice is most consistent with the implicit assumption in the H12 model that the turbulent power is mostly on large scales, so as one considers larger and larger size scale, the Mach number monotonically increases, approaching the total Mach number as the size scale under consideration approaches $h$. Our second method for estimating $\mach_h$ is to integrate the velocity power spectra in the region $1<k<3$, which is roughly the outer scale of the turbulence in our periodic box. Doing so, we find $\mach_h=1.3$ for the N1 simulations and $2.9$ for the N2 simulations.
We compare the predictions of the H12 model with the aforementioned parameters to our simulations in the right-hand panel of Fig.~\ref{fig:theory} (dashed and dot-dashed lines). We first focus on the case where we measure $\mach_h$ by integrating over $k=1-3$, and observe that, while the IMF predicted for $n=1.9$ coincides fairly well with the N2 simulations for $m\gtrsim 10^{-3}$, the $n=1$ prediction is significantly steeper than that for $n=1.9$, which is the opposite of what we observe from our simulations. By contrast, if we accept a mass shifting factor\footnote{A possible justification for this shift is that in our simulations there are no density fluctuations at the box scale, whereas in the H12 model fluctuations at the galactic scale height $h$ are non-zero, and only damp to zero on scales $\ll h$ \citep[e.g.][Fig.~2]{Hopkins2013AFragmentation}.} of $10$, the predicted IMF shapes beyond the peaks are significantly closer to what we measure for both the N1 and N2 simulations in the case where we take $\mach_h=\mach=5$ (dash-dotted lines), except near $m\approx 10^{-2}$. The predicted qualitative effect of varying $n$ is also consistent with our simulation results, and with \citet{Hopkins2013AFragmentation}. According to the H12 model, the cutoff in the N1 SMF beyond $m>10^{-2}$, which is most likely a result of the finite mass in the simulation box, is explained by the suppression of density fluctuations due to mass conservation. However, we caution that, because of the ambiguity in the definition of $\mach_h$ inherent in the H12 models, as well as the necessity of an arbitrary horizontal shift, we can only tentatively identify this as a successful prediction. Finally, we note that while the H12 model in principle allows for the inclusion of magnetic fields, the dependence of the IMF on the magnetic field has not been studied in detail in \citet{Hopkins2013AFragmentation}. We aim to quantify the effects of the magnetic field on the IMF in a follow-up study.
\section{Conclusions} \label{sec:sum}
Using hydrodynamical simulations that include gravity and sink particles, we investigate the effect of the shape of the power spectrum of supersonic turbulence ($E_v(k)\propto k^{-n}$) on the stellar IMF. With the help of adaptive mesh refinement and repeated simulations with different random seeds for the turbulence, we construct statistically significant sink mass distributions with 900--1000 sink particles formed for each $n$, and a dynamic range spanning three orders of magnitude, from a low-mass cutoff imposed by the grid resolution to a high-mass cutoff imposed by the finite size of the simulation domain. From the sink particle populations, we find that turbulence with $n=1$ significantly flattens the high-mass end of the IMF compared to $n\approx 2$ (i.e., $n=1$ turbulence generates more massive stars), but has little effect on the distribution of low-mass stars and sub-stellar objects. This result is consistent with our current understanding of molecular cloud dynamics and star formation: turbulence governs the large-scale fragmentation of molecular clouds, while other mechanisms such as radiative heating play more important roles below a certain length (or mass) scale. We also find that compared to natural supersonic turbulence with $n\approx 2$, turbulence with a scaling index of $n=1$ creates less density dispersion, does not promote the formation of large-scale gas structures such as large-scale filaments, and slows down the star formation rate.
We compare our simulation results with three turbulence-regulated theoretical models of the IMF: \citet[PN02]{Padoan2002TheFragmentation}, \citet[HC08]{Hennebelle2008AnalyticalCores}, and \citet[H12]{Hopkins2012TheDistribution}. We find that the qualitative predictions of the three models vary significantly (e.g., the dependence of the high-mass slope of the IMF on $n$). Out of the three IMF models, we find that the PN02 theory is consistent with our measurement of the $n$-dependence of the high-mass IMF slope ($\Gamma$). The density statistics predicted by the HC08 model agree qualitatively with our observations, but their predicted high-mass slope diverges for $n\to 1$. We find that the H12 model can be made similar to our simulated IMFs in the high-mass range. However, the model is quite sensitive to the choice of the definition of a key parameter ($\mach_h$), which is defined somewhat ambiguously in the model, and if we adopt an alternative definition, the H12 theory predicts qualitatively different results that disagree with our simulations.
There remains one important question that is not yet answered: why did turbulence with $n\approx 2$ shape a high-mass IMF much steeper than the Salpeter IMF in our simulations? As mentioned in \S\ref{sec:intro} and \S\ref{sec:pn02}, the answer may be the absence of magnetic fields, since only the PN02 theory successfully predicts the high-mass slope for the $n\approx 2$ hydrodynamical turbulence (apart from the modified H12 theory with $\mach_h=5$), and it is the only model that explicitly encodes the role of magnetic fields in shaping the high-mass IMF. We suggest a follow-up study that includes varying levels of magnetic fields, in order to quantify the role of the magnetic field on the shape of the IMF.
\section*{Acknowledgements}
We thank {\AA}ke Nordlund for providing a detailed and constructive referee report. We also thank Patrick Hennebelle and Paolo Padoan for their interest, comments and suggestions on the manuscript. We further thank Phil Hopkins and D{\'a}vid Guszejnov for their help with reproducing the H12 IMF model. C.~F.~acknowledges funding provided by the Australian Research Council (Discovery Project DP170100603 and Future Fellowship FT180100495), and the Australia-Germany Joint Research Cooperation Scheme (UA-DAAD). M.~R.~K.~acknowledges funding from the Australian Research Council (Discovery Project DP190101258 and Future Fellowship FT180100375), and the Australia-Germany Joint Research Cooperation Scheme (UA-DAAD). We further acknowledge high-performance computing resources provided by the Leibniz Rechenzentrum and the Gauss Centre for Supercomputing (grants~pr32lo, pr48pi and GCS Large-scale project~10391), the Australian National Computational Infrastructure (grants~ek9 and jh2) in the framework of the National Computational Merit Allocation Scheme and the ANU Merit Allocation Scheme. The simulation software FLASH was in part developed by the DOE-supported Flash Center for Computational Science at the University of Chicago.
\section*{Data Availability}
The simulation data underlying this article will be shared on reasonable request to Donghee Nam at \href{mailto:u6836819@anu.edu.au}{u6836819@anu.edu.au}. Our Python code that reproduces the H12 last-crossing IMF is publicly available at \href{https://github.com/dongheenam/hopkins-imf}{https://github.com/dongheenam/hopkins-imf}.
\bibliographystyle{mnras}
|
1,314,259,993,621 | arxiv | \section{Introduction}
\label{s:intoduction}
Almost all the extensions of the Standard Model~(SM) directed towards an explanation for the neutrino masses brings in the possibility of
lepton number violation~(LNV) as an outcome. It is well known that neutrinoless double beta
decay~($0\nu\beta\beta~{\rm decay}$) which is a convincing signature for LNV, will be an inevitable consequence if
the neutrino has Majorana mass. If the main contribution to $0\nu\beta\beta~{\rm decay}$ proceeds through the Majorana neutrino propagator,
depending on the spectrum of the neutrino masses,
the expected rate for $0\nu\beta\beta~{\rm decay}$ might be too small to be observed in the experiments.
But there exist scenarios where the dominant mechanism for $0\nu\beta\beta~{\rm decay}$ is not controlled
by the Majorana neutrino propagator. In such cases we can have the possibility of large
$0\nu\beta\beta~{\rm decay}$ even when the neutrino Majorana masses are small. Many studies have been performed
in this direction in the past (see Refs.~\cite{Schechter:1981bd,Mohapatra:1998rq,Vergados:2012xy} for a general overview,
Refs.~\cite{Pati:1974yy,Mohapatra:1974gc,Senjanovic:1975rk,Hirsch:1996qw,Atre:2009rg,Tello:2010am,Blennow:2010th,Ibarra:2010xw,Mitra:2011qr,Dev:2013vxa,Dev:2014xea,Choubey:2012ux,Babu:2010vp,Gustafsson:2012vj,Liu:2016mpf,Jin:2015cla,Okada:2015hia,Ahriche:2014cda,Hatanaka:2014tba,Chen:2014ska,Geng:2015sza,Nishiwaki:2015iqa,Ahriche:2015wha} for specific models\footnote{See also \cite{Helo:2015fba} for a recent review of neutrino mass models in connection to $0\nu\beta\beta~{\rm decay}$.} and Refs.~\cite{Choi:2002bb,Engel:2003yr,deGouvea:2007qla,delAguila:2012nu} for effective field theory~(EFT) approaches). In Ref.~\cite{delAguila:2012nu}, the authors
performed an EFT analysis of the different ways of generating $0\nu\beta\beta~{\rm decay}$ and light neutrino
masses by including operators involving only leptons, Higgs and gauge bosons. This led
to a class of interesting models where $0\nu\beta\beta~{\rm decay}$ was generated at tree level whereas
neutrino masses would appear only at two-loops (see Refs.~\cite{delAguila:2011gr}
for example models in this category).
The model in Ref.~\cite{delAguila:2011gr} contains an $SU(2)_L$ singlet doubly charged
scalar like in the Zee-Babu model\cite{Zee:1985id,Babu:1988ki,Babu:2002uu}, an $SU(2)_L$
triplet scalar with hypercharge $+1$ and a real singlet scalar. A $Z_2$ symmetry, which is
later broken spontaneously, is required to prevent tree-level neutrino masses. The model is
economical in the sense that it contains no new fermions and by design, it gives new
contributions to $0\nu\beta\beta~{\rm decay}$, which, in principle, can be large. Additionally, it has a rich
phenomenology which can be probed through the searches for the lepton flavor
violating~(LFV) signals and/or the direct searches for the new scalars in the collider
experiments.
In this article we will present a simple variation of the model in Ref.~\cite{delAguila:2011gr}.
Our new model will have the same field content as in Ref.~\cite{delAguila:2011gr}, except that
the $Z_2$ symmetry will not be broken spontaneously. Consequently, $0\nu\beta\beta~{\rm decay}$ will now occur
at one-loop whereas neutrino masses will appear at three-loop order. The fact the $Z_2$ is exact makes the model simpler and allows for a viable Dark Matter~(DM) candidate: the lightest of the electrically neutral $Z_2$-odd particles.
On the other hand, the model keeps all the virtues of the previous model: very predictive neutrino mass matrix, large $0\nu\beta\beta~{\rm decay}$ decay, rich lepton flavour violation phenomenology and new scalars which are in the sub-TeV region and therefore, are within the reach of the collider experiments in the near future.
Our paper will be organized as follows. In Sec.~\ref{s:model} we lay out the scalar field content and the
physical spectrum of our model. In Sec.~\ref{s:nbb} we discuss the $0\nu\beta\beta~{\rm decay}$ and the bounds that follow from
it. Neutrino masses and constraints from LFV decays are discussed in Sec.~\ref{sec:nu-mass} and
Sec.~\ref{s:LFV} respectively. We analyze the feasibility of DM in Sec.~\ref{s:DM}. Finally, we summarize
our findings in Sec.~\ref{s:results}.
\section{The model}
\label{s:model}
The scalar sector of our model contains the following fields:
\begin{eqnarray}
\Phi = \left\{2,\frac{1}{2}\right\} \,; ~~ \chi = \{3,1\} \,; ~~ \kappa^{++} = \{1,2\} \,; ~~ \sigma = {\rm real ~ singlet} \,,
\end{eqnarray}
where, the numbers inside the curly brackets associated with the fields represent their
transformations properties under $SU(2)_L$ and $U(1)_Y$ respectively.
The normalization for the hypercharge is such that the electric charges of the component
fields are given by, $Q=T_3+Y$. The fields, $\chi$ and $\sigma$ are odd under an additional $Z_2$ symmetry which has been introduced to prevent the occurrence of tree-level neutrino masses as well as to ensure the stability of the DM particle. The most general scalar potential involving these fields is given below:
\begin{eqnarray}
\label{e:potential}
V &=& -m_{\Phi}^2\left(\Phi^\dagger\Phi\right) +m_{\chi}^2\Tr\left(\chi^\dagger\chi\right) +m_{\kappa}^2|\kappa|^2 + \frac{m_{\sigma}^2}{2}\sigma^2 +\lambda_{\Phi}\left(\Phi^\dagger\Phi\right)^2 +\lambda_{\chi}\left\{\Tr\left(\chi^\dagger\chi\right)\right\}^2
\nonumber \\
&&+\lambda'_{\chi}\Tr\left[\left(\chi^\dagger\chi\right)^2\right] +\lambda_{\kappa}|\kappa|^4 +\lambda_{\sigma}|\sigma|^4 +\lambda_{\Phi\chi}\left(\Phi^\dagger\Phi\right)\Tr\left(\chi^\dagger\chi\right) +\lambda'_{\Phi\chi}\left(\Phi^\dagger\chi \chi^\dagger\Phi \right) \nonumber \\
&&+\lambda_{\Phi\kappa}\left(\Phi^\dagger\Phi\right)|\kappa|^2 +\lambda_{\Phi\sigma}\left(\Phi^\dagger\Phi\right)\sigma^2 +\lambda_{\kappa\chi}|\kappa|^2\Tr\left(\chi^\dagger\chi\right) +\lambda_{\sigma\chi}\sigma^2\Tr\left(\chi^\dagger\chi\right) \nonumber\\
&&+\lambda_{\sigma\kappa}|\kappa|^2\sigma^2 +\left\{\mu_\kappa \kappa^{++} \Tr\left(\chi^\dagger\chi^\dagger\right) +\lambda_6\sigma\Phi^\dagger\chi\widetilde{\Phi} + {\rm h.c.} \right\} \,,
\end{eqnarray}
where `$\Tr$' represents the trace over $2\times2$ matrices and $\widetilde{\Phi}=i\sigma_2\Phi^*$, with $\sigma_2$ being the second Pauli matrix. We can take all the
parameters in the potential to be real without any loss of generality.
For the leptonic Yukawa sector, we have the following Lagrangian:
\begin{eqnarray}
\label{e:yukawa}
{\mathscr L}_Y = - (\overline{L_L})_a (Y_e)_{ab} (\ell_R)_b \Phi + f_{ab} \ell_a^T C^{-1}(\ell_R)_b \kappa^{++} + {\rm h.c.}\,,
\end{eqnarray}
where, $L_L = (\nu_\ell,~ \ell)_L^T$ denotes the left-handed lepton doublet and $\ell_R$
represents the right-handed charged lepton singlet. $C$ is the charge conjugation
operator. We choose to work in the mass basis
of the charged leptons which means, $Y_e$ is a diagonal matrix with positive entries and
$f$ is a complex symmetric matrix with three unphysical phases.
\subsection{The scalar spectrum}
\begin{table}[htbp!]
\begin{center}
{\tabulinesep=1.2mm
\begin{tabu}{|c|c|}
\hline
$Z_2$-even particles & $Z_2$-odd particles \\
\hline\hline
SM fermions and gauge bosons, $h$ and $\kappa^{\pm\pm}$ & $S$, $A$, $H$, $\chi^\pm$, $\chi^{\pm\pm}$ \\
\hline
\end{tabu}
}
\end{center}
\caption{$Z_2$ parity assignments to the physical particles in our model.}
\label{t:z2}
\end{table}
We do not want to break the $Z_2$ symmetry spontaneously. Denoting by $v$ the vacuum expectation values (vev) of the doublet the minimization conditions read
\begin{eqnarray}
m_{\Phi}^2&=& \lambda_{\Phi}v^2 \,.
\end{eqnarray}
After spontaneous symmetry breaking~(SSB) we represent the doublet and the triplet as follows:
\begin{eqnarray}
\Phi = \frac{1}{\sqrt{2}} \begin{pmatrix} \sqrt{2} \omega^+ \\ v+h+i\zeta \end{pmatrix} \,, &&
\chi = \frac{1}{\sqrt{2}} \begin{pmatrix} \chi^+ & \sqrt{2}\chi^{++} \\ h_t+iA & -\chi^+ \end{pmatrix} \,,
\end{eqnarray}
where, $\omega$ and $\zeta$ represent the Goldstones associated with the $W$ and
$Z$ bosons respectively.
Because of the unbroken $Z_2$ symmetry, only $h_t$ and $\sigma$ can have nontrivial
mixing. This leads to a very simple scalar spectrum as described below.
The masses for the doubly charged particles are given by,
\begin{eqnarray}
m_{\kappa^{++}}^2 = m_\kappa^2 +\frac{1}{2}\lambda_{\Phi\kappa} v^2 \,, ~~
m_{\chi^{++}}^2 = m_\chi^2+\frac{1}{2}\lambda_{\Phi\chi}v^2 \,.
\label{m:doubly}
\end{eqnarray}
The mass of the singly charged scalar is given by,
\begin{eqnarray}
m_{\chi^{+}}^2 = m_\chi^2+\frac{1}{4}(2\lambda_{\Phi\chi}+\lambda'_{\Phi\chi})v^2 \,.
\label{m:singly}
\end{eqnarray}
The pseudoscalar mass is given by,
\begin{eqnarray}
m_A^2 = m_\chi^2+\frac{1}{2}(\lambda_{\Phi\chi}+\lambda'_{\Phi\chi})v^2 \,.
\label{m:pseudo}
\end{eqnarray}
From Eqs.~(\ref{m:doubly}), (\ref{m:singly}) and (\ref{m:pseudo}) it is easy to see that the following correlation holds:
\begin{eqnarray}
\label{e:corr}
m_{\chi+}^2-m_{\chi++}^2 = m_{A}^2-m_{\chi+}^2 = \frac{1}{4}\lambda'_{\Phi\chi}v^2 \,.
\end{eqnarray}
In the CP even sector, the SM-like Higgs arises purely from the doublet, $\Phi$, with mass
$m_h^2 = 2\lambda_\Phi v^2$. For the other two $Z_2$-odd scalars, we obtain the following
mass matrix:
\begin{eqnarray}
&& V_{\rm mass}^{\rm S} = \frac{1}{2} \begin{pmatrix} \sigma & h_t \end{pmatrix} \begin{pmatrix}
A & -B \\ -B & C \end{pmatrix} \begin{pmatrix} \sigma \\ h_t \end{pmatrix}~~ {\rm with,} \\
&& A= m_\sigma^2+\lambda_{\Phi\sigma}v^2 \,, ~~
B= - \frac{1}{\sqrt{2}}\lambda_6v^2 \,, ~~
C=m_\chi^2+\frac{1}{2}(\lambda_{\Phi\chi}+\lambda'_{\Phi\chi})v^2 \,.
\label{e:ABC}
\end{eqnarray}
This mass matrix can be diagonalized by the following orthogonal rotation:
\begin{subequations}
\begin{eqnarray}
\begin{pmatrix} S \\ H \end{pmatrix} &=& \begin{pmatrix} \cos\alpha & -\sin\alpha \\ \sin\alpha & \cos\alpha \end{pmatrix} \begin{pmatrix} \sigma \\ h_t \end{pmatrix} \,, \\
{\rm with,}~~ m^2_{H,S} &=& \frac{1}{2}\left\{(A+C)\pm\sqrt{(A-C)^2+4B^2}\right\} \,, \\
{\rm and,}~~ \tan2\alpha &=& \frac{2B}{A-C} \,,
\end{eqnarray}
\end{subequations}
where we have implicitly assumed that `$S$' is the lighter mass eigenstate. One can easily find the following relations:
\begin{subequations}
\label{e:abc}
\begin{eqnarray}
A&=& m_H^2\sin^2\alpha+m_S^2\cos^2\alpha \,, \\
C &=& m_H^2\cos^2\alpha +m_S^2\sin^2\alpha = m_A^2 \,, \label{e:mA}\\
B &=& - \sin\alpha\cos\alpha (m_H^2-m_S^2) \,, \label{e:B}
\end{eqnarray}
\end{subequations}
which imply,
\begin{eqnarray}
\label{e:corr2}
m_S < m_A < m_H \,.
\end{eqnarray}
Combining \Eqs{e:ABC}{e:B} we can express $\lambda_6$ in terms of the physical parameter as follows:
\begin{eqnarray}
\lambda_6 = \frac{\sqrt{2}\sin\alpha\cos\alpha}{v^2}\left(m_H^2-m_S^2\right) \,.
\end{eqnarray}
The splittings between different scalar masses can be constrained further from the electroweak $T$-parameter.
The expression for the new physics contribution to the $T$-parameter is given by
\begin{eqnarray}
\label{e:T}
\Delta T &=& \frac{1}{4\pi\sin^{2}\theta_{W}M_{W}^{2}} \left[F(m_{\chi^{++}}^{2},m_{\chi^{+}}^{2})+\frac{1}{2} F(m_{\chi^{+}}^{2},m_{A}^{2}) \right. \nonumber \\
&& \left. + \frac{1}{2} \cos^{2}\alpha\left\{F(m_{\chi^{+}}^{2},m_{H}^{2})-2F(m_{A}^{2},m_{H}^{2})\right\} + \frac{1}{2}\sin^{2}\alpha\left\{F(m_{\chi^{+}}^{2},m_{S}^{2})-2F(m_{A}^{2},m_{S}^{2}) \right\} \right]\,,
\end{eqnarray}
where, $\theta_{W}$ and $M_{W}$ are the weak mixing angle and the
$W$-boson mass respectively. The function, $F(m_1^2,m_2^2)$, is given by,
\begin{eqnarray}
\label{e:fxy}
F(m_{1}^{2},m_{2}^{2})\equiv \frac{1}{2} 16\pi^{2}\int\frac{\dd[4]{k}}{(2\pi)^{4}}k^{2} \left(\frac{1}{k^{2}+m_{1}^{2}}-\frac{1}{k^{2}+m_{2}^{2}}\right)^{2}=\frac{m_{1}^{2}+m_{2}^{2}}{2}-\frac{m_{1}^{2}m_{2}^{2}}{m_{1}^{2}-m_{2}^{2}}\log\left(\frac{m_{1}^{2}}{m_{2}^{2}}\right)\,.
\end{eqnarray}
Taking the new physics contribution to the $T$-parameter as\cite{Baak:2013ppa}
\begin{eqnarray}
\Delta T = 0.05 \pm 0.12 \,,
\end{eqnarray}
we will require our model value of the $T$-parameter to be within the $2\sigma$ uncertainty range. For
small $\sin\alpha$, this leads to $|m_H-m_{\chi^{++}}|\lesssim 100$~GeV.
In passing, combining \Eqs{e:corr}{e:corr2}, we note that two types of scalar mass hierarchies are possible
depending on the sign of $\lambda'_{\Phi\chi}$,
\begin{subequations}
\label{e:hierarchies}
\begin{eqnarray}
&& m_H > m_A > m_{\chi^{+}} > m_{\chi^{++}} > m_S \,, \\
{\rm or,} && m_{\chi^{++}} > m_{\chi^{+}} > m_A > m_S ~~ {\rm and} ~~ m_H > m_A \,.
\end{eqnarray}
\end{subequations}
In both cases, $m_{\kappa^{++}}$ can be arbitrary in principle.
\section{Estimation of \texorpdfstring{$0\nu\beta\beta$}{TEXT} decay}
\label{s:nbb}
\begin{figure
\begin{centering}
\includegraphics[scale=0.55]{Figs/0nbb-loop}
\par
\end{centering}
\caption{One-loop diagram, in the mass insertion approach, contributing to
neutrinoless double beta decay.\label{fig:0nu2beta}}
\end{figure}
For new scalar masses of $\order(1~{\rm TeV})$, the Majorana mass matrix element,
$M_{ee}$, will be very small (see Sec.~\ref{sec:nu-mass} for details).
As a result, the usual neutrino exchange diagram will contribute negligibly
to $0\nu\beta\beta~{\rm decay}$. The main contribution to the $0\nu\beta\beta~{\rm decay}$ amplitude has been displayed in Fig.~\ref{fig:0nu2beta}.
From the diagram in Fig.~\ref{fig:0nu2beta} we can easily estimate
the effective $\bar{e}e^{c}(\bar{u}d)^{2}$ interaction giving rise
to $0\nu\beta\beta~{\rm decay}$
\begin{equation}
\mathcal{L}_{0\nu\beta\beta}=2\frac{f_{ee}^{*}}{16\pi^{2}}\frac{\mu_{\kappa}\lambda_{6}^{2}}{m_{\kappa^{++}}^{2}m_{A}^{4}}I_{\beta}\left(\overline{u_{L}}\gamma^{\mu}d_{L}\right)\left(\overline{u_{L}}\gamma_{\mu}d_{L}\right)\overline{e_{R}}e_{R}^{c} \,,
\label{eq:Lagrangian0n2bModel}
\end{equation}
where $I_{\beta}$ is a dimensionless function of the scalar masses
running in the loop which is expected to be $\order(1)$. For illustration, we have
chosen the common scale of the loop to be the mass of the pseudoscalar
part from the scalar triplet, $m_{A}$. Of
course the diagram in Fig.~\ref{fig:0nu2beta} is only one of the
contributions in the mass insertion approach which allows us to give
an estimate. A complete calculation of the function $I_\beta$ in the
physical basis has been presented in Appendix~\ref{ap:Neutrinoless} yielding values for $I_\beta$ which are slightly smaller than one in the range of masses of interest, $I_\beta\sim 0.1$. We will use these values for our estimates.
The interaction of \Eqn{eq:Lagrangian0n2bModel} has been considered in the literature~\cite{Pas:2000vn,Deppisch:2012nb},
where it was parametrized as follows:
\begin{equation}
\mathcal{L}_{0\nu\beta\beta}=\frac{G_{F}^{2}}{2m_{p}}\epsilon_{3}\left(\bar{u}\gamma^{\mu}(1-\gamma_{5})d\right)\left(\bar{u}\gamma_{\mu}(1-\gamma_{5})d\right)\bar{e}(1-\gamma_{5})e^{c}\,.
\label{eq:Lagrangian-0nu2beta}
\end{equation}
Comparing \Eqs{eq:Lagrangian0n2bModel}{eq:Lagrangian-0nu2beta} we obtain,
\begin{equation}
\epsilon_{3}=\frac{m_{p}}{2G_{F}^{2}}\frac{f_{ee}^{*}}{16\pi^{2}}\frac{\mu_{\kappa}\lambda_{6}^{2}}{m_{\kappa^{++}}^{2}m_{A}^{4}}I_{\beta}\,.
\label{eq:epsilon3}
\end{equation}
In Ref.~\cite{Deppisch:2012nb}, to set bounds on $\epsilon_3$, the authors used the limits on the
half-life for the $0\nu\beta\beta~{\rm decay}$ from the most sensitive experiments of that time, namely,
$T_{1/2}^{0\nu\beta\beta}(^{76}\mathrm{Ge})>1.9\times10^{25}$~yrs
(HM \cite{KlapdorKleingrothaus:2000sn}) and $T_{1/2}^{0\nu\beta\beta}(^{136}\mathrm{Xe})>1.6\times10^{25}$~yrs
(EXO-200 \cite{Auger:2012ar}). However KamLAND-Zen has recently obtained a stronger limit on
the lifetime from $^{136}\mathrm{Xe}$, $T_{1/2}^{0\nu\beta\beta}(^{136}\mathrm{Xe})>1.07\times10^{26}\,\mathrm{yr}$
\cite{KamLAND-Zen:2016pfg}, which, using the matrix elements from
\cite{Deppisch:2012nb}, translates to $\epsilon_{3}<4\times10^{-9}$ at 90\%~C.L.
On the other hand, upcoming experiments are expected to be sensitive
to lifetimes of order $10^{27}$--$10^{28}$~yrs\cite{Barabash:2011fg},
\emph{i.e.} a reduction factor on the coupling of about one order
of magnitude. Thus, for $0\nu\beta\beta~{\rm decay}$ mediated by
heavy particles to be observable in the next round of experiments we
should have $\epsilon_{3}\gtrsim4\times10^{-10}$. Therefore in order to escape the current
experimental bounds but at the same time to entertain the possibility of observing $0\nu\beta\beta~{\rm decay}$
in the near future, we require $\epsilon_3$ to be within the following range:
\begin{eqnarray}
\label{e:epsrange}
4\times 10^{-10} < \epsilon_3 < 4\times10^{-9} \,.
\end{eqnarray}
With $f_{ee},~\lambda_{6}\approx\text{1}$, $\mu_{\kappa}\approx m_{A}\approx m_{\kappa^{++}}\approx1\,\mathrm{TeV}$
and $I_{\beta}\sim 0.1$ we obtain, from \Eqn{eq:epsilon3}, $\epsilon_{3}\sim10^{-9}$ which falls
naturally within the range given in \Eqn{e:epsrange}.
\section{Estimation of the neutrino masses}
\label{sec:nu-mass}
\begin{figure}[htbp!]
\centering
\includegraphics[scale=0.55]{Figs/numass-3loops}
\caption{Sample three loop diagram, in the mass insertion approach, contributing to the neutrino masses.}
\label{fig:numass}
\end{figure}
From \Eqs{e:potential}{e:yukawa} it is obvious that simultaneous nonzero values for
$Y_e$, $f_{ab}$, $\mu_\kappa$ and $\lambda_6$ will prevent us from assigning consistent lepton numbers to all the scalar
and lepton fields. Therefore, lepton number is broken explicitly and Majorana neutrino masses will be unavoidable. The sample diagram of Fig.~\ref{fig:numass}, in the mass insertion approach, clearly
depicts the involvement of all these couplings in a multiplicative manner. Thus, we can parametrize
the neutrino mass matrix as follows:
\begin{eqnarray}
\label{e:nuelements}
M_{ab}=\frac{8\mu_{\kappa}\lambda_{6}^{2}}{(4\pi)^{6}m_{\kappa^{++}}^{2}}I_{\nu}\, m_{a}f_{ab}m_{b} \,,
\end{eqnarray}
where $m_{a}$ denotes the mass of the charged lepton, $\ell_a$, and $I_{\nu}$
represents the loop function expected to be of $\order(1)$. Detailed expression of $I_\nu$
in terms of the scalar masses has been presented in Appendix~\ref{ap:Neutrino-masses}. Eq.~(\ref{e:nuelements}) has a very particular and predictive structure, specific for this class of models, which can be constrasted with the observed spectrum of neutrino masses and mixings (see for instance Refs. \cite{delAguila:2011gr,delAguila:2012nu,Gustafsson:2014vpa}).
As before, taking $f_{\tau\tau},~\lambda_{6}\approx 1$ and $\mu_{\kappa}\approx m_{\kappa^{++}}\approx1$~TeV
and $I_{\nu}\sim 1$ we obtain the following values for the different elements
\begin{eqnarray}
M_{ee}\sim10^{-7}\,\mathrm{eV}\,, ~~ M_{e\mu}\sim10^{-4}\,\mathrm{eV}\,, ~~
M_{e\tau}\sim10^{-3}\,\mathrm{eV}\,,~~ M_{\mu\mu}\sim10^{-2}\,\mathrm{eV}\,, ~~ M_{\mu\tau}\sim10^{-1}\,\mathrm{eV}\,,~~
M_{\tau\tau}\sim 10\,\mathrm{eV}\,.
\end{eqnarray}
But of course, some of the $f_{ab}$s can be much smaller than~1. However, not all of the elements of the $f$ matrix are
arbitrary as some of them will be constrained from LFV processes. We will discuss these constraints
in Sec.~\ref{s:LFV}. But for now we wish to emphasize that the product $|f_{ee}^*f_{e\mu}|$ will receive strong
bounds from $\mu\to 3e$ as the latter can proceed at the tree-level mediated by $\kappa^{++}$. Then, one should
naturally expect the following hierarchy among the mass matrix elements:
\begin{eqnarray}
\label{e:hi1}
M_{ee},M_{e\mu}\ll M_{e\tau},M_{\mu\mu},M_{\mu\tau},M_{\tau\tau} \,,
\end{eqnarray}
which, obviously, can only accommodate a normal hierarchy among the neutrino masses. In Ref.~\cite{delAguila:2011gr} it has been shown that the above hierarchy with
\begin{eqnarray}
\label{e:hi2}
3 M_{e\tau} \sim M_{\mu\mu} \sim M_{\mu\tau} \sim M_{\tau\tau} \sim 0.02~{\rm eV}
\end{eqnarray}
can successfully reproduce the observed masses and mixings in the neutrino sector with
a prediction of $\sin^2\theta_{13} > 0.008$. \Eqn{e:hi2} will imply the following
hierarchy among the Yukawa elements:
\begin{eqnarray}
\label{e:hi3}
3 f_{e\tau} \sim \frac{m_\tau}{m_e}f_{\tau\tau} > f_{\mu\mu} \sim \frac{m_\tau^2}{m_\mu^2}f_{\tau\tau}
> f_{\mu\tau} \sim \frac{m_\tau}{m_\mu}f_{\tau\tau} > f_{\tau\tau} \,.
\end{eqnarray}
We shall also assume $f_{ee}\gg f_{e\mu}$ in such a way that $f_{ee}^*f_{e\mu}$ is still sufficiently small
to keep $\mu\to 3e$ decay under control but at the same time allowing for the possibility of large $0\nu\beta\beta~{\rm decay}$.
From \Eqs{eq:epsilon3}{e:nuelements} we see that the dimensionless factor,
\begin{eqnarray}
\label{e:gamma}
\gamma = \frac{\mu_\kappa \lambda_6^2}{m_{\kappa^{++}}} = \frac{2\sin^2\alpha\cos^2\alpha(m_H^2-m_S^2)^2}{v^4} \frac{\mu_\kappa}{m_{\kappa^{++}}} \,,
\end{eqnarray}
is common to both. In terms of $\gamma$, the explicit expression for $M_{\tau\tau}$ in \Eqn{e:hi2} reads:
\begin{eqnarray}
\label{e:mtt}
M_{\tau\tau} = \frac{8}{(4\pi)^6} \gamma I_\nu \frac{m_\tau^2f_{\tau\tau}}{m_{\kappa^{++}}} \approx 0.02 ~{\rm eV} \,.
\end{eqnarray}
As we will see in Sec.~\ref{s:LFV}, the ratio $f_{\tau\tau}/m_{\kappa^{++}}$ is bounded from LFV processes as
$f_{\tau\tau}/m_{\kappa^{++}}\lesssim 1.4\times 10^{-4}~{\rm TeV}^{-1}$. Plugging this into \Eqn{e:mtt} we obtain the following
bound for $\gamma$:
\begin{eqnarray}
\label{e:gambound}
\gamma \gtrsim \frac{22}{I_\nu} \,.
\end{eqnarray}
Having an explicit expression for the neutrino masses we can compare the light neutrino exchange contributions to $0\nu\beta\beta~{\rm decay}$ with the ones discussed in Sec.~\ref{s:nbb}.
In fact, from \Eqs{e:nuelements}{eq:epsilon3} we can express the neutrino mass matrix element $M_{ee}$, which controls the $\nu$ contributions to $0\nu\beta\beta~{\rm decay}$, in terms $\epsilon_3$, which parametrizes the new contributions
\begin{equation}\label{e:Mee-epsilon3}
M_{ee}=\frac{16m_{e}^{2}G_{F}^{2}m_{A}^{4}}{m_{p}(4\pi)^{4}}\frac{I_{\nu}}{I_{\beta}}\epsilon_{3}~.
\end{equation}
Then, it is clear that for small enough $m_A$ the new contributions will dominate over the neutrino contributions. How small? Since the nuclear matrix elements are different in the two cases we cannot make a direct comparison. However, we can use that the experimental limit
$T_{1/2}^{0\nu\beta\beta}(^{136}\mathrm{Xe})>1.07\times10^{26}\,\mathrm{yrs}$\cite{KamLAND-Zen:2016pfg}
translates into two equivalent bounds on $\epsilon_3$ and $M_{ee}$ when $0\nu\beta\beta~{\rm decay}$ is dominated by the new contributions or by neutrino masses respectively:
\begin{eqnarray}
\label{e:f1}
\epsilon_3 < 4\times 10^{-9} \,, ~~ M_{ee} < 0.1~{\rm eV} \, ,
\end{eqnarray}
which already include the appropriate nuclear matrix elements. Using these results and taking $I_\beta \sim 0.1 I_\nu$ we obtain that the new contributions will dominate for $m_A \lesssim 15$~TeV. Therefore, scalar masses must be relatively light, and this could make the model testable at the LHC and/or in LFV processes.
\begin{table}[htbp!]
\begin{center}
{\tabulinesep=1.2mm
\begin{tabu}{|c|c|c|}
\hline
Experimental Data (90\% CL) & Bounds (90\% CL) & Bounds assuming \Eqn{e:hi3} \\
\hline\hline
$\mathrm{BR}(\mu^{-}\rightarrow e^{+}e^{-}e^{-})<1.0\times10^{-12}$ & $|f_{e\mu}f_{ee}^{*}|<2.3\times10^{-5}\,\left(\frac{m_{\kappa^{++}}}{\mathrm{TeV}}\right)^{2}$ & \\
\hline
$\mathrm{BR}(\tau^{-}\rightarrow e^{+}e^{-}e^{-})<2.7\times10^{-8}$ & $|f_{e\tau}f_{ee}^{*}|<0.009\,\left(\frac{m_{\kappa^{++}}}{\mathrm{TeV}}\right)^{2}$ & $|f_{ee}^*f_{\tau\tau}|\lesssim 7.8\times10^{-6}\,\left(\frac{m_{\kappa^{++}}}{\mathrm{TeV}}\right)^{2}$ \\
\hline
$\mathrm{BR}(\tau^{-}\rightarrow e^{+}e^{-}\mu^{-})<1.8\times10^{-8}$ & $|f_{e\tau}f_{e\mu}^{*}|<0.005\,\left(\frac{m_{\kappa^{++}}}{\mathrm{TeV}}\right)^{2}$ & $|f_{e\mu}^*f_{\tau\tau}|\lesssim 4.3\times10^{-6}\,\left(\frac{m_{\kappa^{++}}}{\mathrm{TeV}}\right)^{2}$ \\
\hline
$\mathrm{BR}(\tau^{-}\rightarrow e^{+}\mu^{-}\mu^{-})<1.7\times10^{-8}$ & $|f_{e\tau}f_{\mu\mu}^{*}|<0.007\,\left(\frac{m_{\kappa^{++}}}{\mathrm{TeV}}\right)^{2}$ & $|f_{\tau\tau}|\lesssim 1.4\times10^{-4}\,\left(\frac{m_{\kappa^{++}}}{\mathrm{TeV}}\right)$ \\
\hline
$\mathrm{BR}(\mu\rightarrow e\gamma)<5.7\times10^{-13}$ & \specialcell{$|f_{ee}^{*}f_{e\mu}+f_{e\mu}^{*}f_{\mu\mu}+f_{e\tau}^{*}f_{\mu\tau}|^{2}$ \\ $<1 \times10^{-7}\,(\frac{m_{\kappa^{++}}}{\mathrm{TeV}})^{4}$} & $|f_{\tau\tau}|\lesssim 1.2\times10^{-4}\,\left(\frac{m_{\kappa^{++}}}{\mathrm{TeV}}\right)$ \\
\hline
\end{tabu}
}
\end{center}
\caption{Relevant constraints for our model from LFV
decays~\cite{Olive:2016xmw,Adam:2013mnn}. Limits on the Yukawa couplings of the doubly charged
singlet scalars have been taken from Ref.~\cite{Herrero-Garcia:2014hfa}. The constraints in the third
column are obtained from those in the second column assuming \Eqn{e:hi3} holds. The bound in
the third column corresponding to $\mu\to e\gamma$ has an additional assumption, $f_{e\mu}\approx 0$.}
\label{tab:LFV}
\end{table}
\section{Constraints from LFV processes}
\label{s:LFV}
Constraints from LFV processes come mainly from decays of the type $\ell_a^\mp \to \ell_b^\pm\ell_c^\mp\ell_d^\mp$ and
$\ell_a^\mp \to \ell_b^\mp \gamma$. In our case $\ell_a^\mp \to \ell_b^\pm\ell_c^\mp\ell_d^\mp$ will be more important
because these decays can occur at the tree-level through the exchange of the doubly charged scalar singlet, $\kappa^{\pm\pm}$.
These processes along with the kinds of constraints they imply have been reviewed in Ref.~\cite{Herrero-Garcia:2014hfa} in the context of the Zee-Babu model (see also Refs. \cite{Babu:2002uu,Nebot:2007bc}).
The experimental data has not changed much since then. In the first two columns of Table~\ref{tab:LFV} we have summarized
the experimental data and the corresponding constraints on the Yukawa couplings. In the third column of Table~\ref{tab:LFV}
we recast the constraints of the second column assuming the validity of \Eqn{e:hi3}. This allows us to express the constraints
in more specific forms. For example, using $m_ef_{e\tau} \sim m_\tau f_{\tau\tau}$ and $m_\mu^2f_{\mu \mu} \sim
m_\tau^2f_{\tau\tau}$, the constraint from $\tau\to e\mu\mu$ leads to a direct bound on $f_{\tau\tau}$ as follows:
\begin{eqnarray}
\label{e:LFV1}
|f_{\tau\tau}|\lesssim 1.4\times 10^{-4}\,\left(\frac{m_{\kappa^{++}}}{\mathrm{TeV}}\right) \,.
\end{eqnarray}
It is also worth mentioning that, using \Eqn{e:hi3}, the limit from $\tau \to 3e$ translates into
\begin{eqnarray}
\label{e:LFV2}
|f_{ee}^*f_{\tau\tau}|\lesssim 7.8 \times 10^{-6}\,\left(\frac{m_{\kappa^{++}}}{\mathrm{TeV}}\right)^{2} \,.
\end{eqnarray}
As mentioned earlier, we want to have $f_{ee}$ relatively large to have appreciable $0\nu\beta\beta~{\rm decay}$ rate in the future experiments.
Then we will need $f_{e\mu}$ to be vanishingly small to keep the constraints from $\mu\to 3e$ under control. Note
that, for $f_{ee}\sim \order(1)$ and sub TeV $\kappa^{++}$, \Eqn{e:LFV2} will imply a stronger bound on $f_{\tau\tau}$
than \Eqn{e:LFV1}.
\section{Dark Matter }
\label{s:DM}
\begin{figure}[htbp!]
\centering
\includegraphics[scale=0.35]{Figs/DM_800}
\caption{Regions corresponding to the observed relic abundance\cite{Ade:2013zuv} in the $m_S$-$\lambda_S$ plane for different values of $\sin\alpha$. We have chosen $m_H=m_{\chi^{++}}=m_{\kappa^{++}}=800$~GeV as a benchmark for this plot.
Current\cite{Tan:2016zwf,Akerib:2016vxi} and future\cite{Aprile:2015uzo} bounds
from direct detection experiments are also marked appropriately.}
\label{f:DM}
\end{figure}
Our model has a $Z_2$ symmetry which remains unbroken after the SSB. Consequently, the particle spectrum can be
divided into $Z_2$-even and odd sectors as shown in Table~\ref{t:z2}. Among the $Z_2$ odd neutral scalars, $S$, being
the lightest, is a promising candidate for DM. Notice that $S$ is and admixture of the real singlet and the triplet, and therefore, it will feel both, Higgs and gauge interactions\footnote{For recent studies of a DM candidate which is an admixture of a scalar singlet and a Y=0 triplet see for instance \cite{Fischer:2013hwa,Cheung:2013dua}.}. In spite of that, one can parametrize its couplings with the SM-like Higgs boson as follows:
\begin{eqnarray}
\label{e:lslag}
&& \mathscr L \supset -\frac{1}{2} \lambda_{S}S^{2}\abs{\Phi^{0}}^{2} \supset
-\frac{1}{2}\lambda_{S}S^{2}\left(vh+\frac{1}{2} h^{2}\right)\,, \\
\label{e:ls}
{\rm with,} && \lambda_{S} = \frac{1}{2}\left[2\lambda_{\Phi\sigma} \cos^{2}\alpha -2\sqrt{2}\lambda_{6}\sin\alpha\cos\alpha +(\lambda_{\Phi\chi}+\lambda_{\Phi\chi}^{\prime}) \sin^{2}\alpha\right]\,.
\end{eqnarray}
In Fig.~\ref{f:DM} we have displayed regions in the $m_S$-$\lambda_S$ plane, which can reproduce the observed DM
relic density\cite{Ade:2013zuv}. For this plot, we have assumed $m_H=m_{\chi^{++}}=m_{\kappa^{++}}=800$~GeV and used the MicrOMEGAs
package\cite{Belanger:2013oya} to compute the DM abundance. Note that, the region labeled as $\sin\alpha=0$
corresponds to the pure Higgs portal scenario. Barring the small window near the Higgs-pole ($m_S\approx m_h/2$, not shown explicitly in the plot), in this case, we need $m_S\gtrsim 350$~GeV\cite{Han:2015hda,Escudero:2016gzx} to evade the direct search bound.
It is worth mentioning that in the case of pure Higgs portal, for our choice of benchmark, the DM annihilates through
$f\bar{f}$, $WW$, $ZZ$ and $hh$ mainly. All these annihilation channels except $hh$ can only proceed through
s-channel $h$ exchange. But as $\sin\alpha$ is turned on, we allow for a direct $SSVV$ ($V=W,Z$) with strength
proportional to $g^2\sin^2\alpha$. For our choice of positive values for $\lambda_S$, the new contact diagram will
interfere constructively with the $h$ mediated s-channel diagram.\footnote{
A nonzero value of $\sin\alpha$ will also induce t-channel diagrams for $SS\to VV,hh$ mediated by $\chi^\pm$, $A$ or $H$.
But these amplitudes will be suppressed as long as $m_{\chi^{+}},m_A, m_H\gg m_S$. Also note that, in this limit, the gauge couplings of $S$ do not contribute to the direct detection cross section\cite{TuckerSmith:2004jv,Cirelli:2005uq}.}
This will enhance the annihilation rate for $SS\to VV$ once the corresponding threshold is reached. Therefore, we would require lower values of $\lambda_S$, compared to the pure Higgs portal case, to reproduce the relic abundance. These features have been depicted in Fig.~\ref{f:DM} where we can see that a small value of $\sin\alpha$ is sufficient to accommodate DM with mass as low as $200$~GeV, which can either be discovered or ruled out in the next run of direct detection experiments.
\section{Results and conclusions}
\label{s:results}
\begin{table
\begin{center}
{\tabulinesep=1.2mm
\begin{tabu}{|c|c|c|c|c|c|c|c|c|}
\hline
$m_{\chi^{++}}$ (GeV) & $m_{\kappa^{++}}$ (GeV) & $\sin\alpha$ & $m_H$ (GeV) & $m_S$ (GeV) & $\mu_\kappa$ (TeV) & $\abs{f_{ee}}$ & $\abs{f_{\tau\tau}}$ & $\abs{f_{e\mu}}$ \\
\hline\hline
800 & 800 & 0.08 & 800 & 200 & 20 & 0.01 & $10^{-4}$ & 0 \\
\hline
\end{tabu}
\vspace*{2mm}
\begin{tabu}{|c|c|c|c|c|c|c|c|}
\hline
$m_{\chi^{+}}$ (GeV) & $m_A$ (GeV) & $I_\beta$ & $I_\nu$ & $\epsilon_3$ & $\abs{f_{e\tau}}$ & $\abs{f_{\mu\mu}}$ & $\abs{f_{\mu\tau}}$ \\
\hline\hline
799 & 798 & 0.165 & 0.84 & $3.5\times 10^{-9}$ & $0.12$ & $0.03$ & $1.7\times 10^{-3}$ \\
\hline
\end{tabu}
}
\end{center}
\caption{Benchmark values for the input parameters (first row) and other relevant quantities derived from these inputs (second row).}
\label{tab:inputoutput}
\end{table}
Since $\kappa^{\pm\pm}$ couples directly to the charged leptons, it will be strongly constrained from the same
sign dilepton searches at the LHC. Depending on the preferred decay channel of $\kappa^{\pm\pm}$, the bound
can be as strong as $m_{\kappa^{++}}\gtrsim 500$~GeV\cite{CMS:2016cpz,ATLAS:2016pbt}. On the other hand, to keep
the $T$-parameter under control, for small $\sin\alpha$, we will need $|m_H-m_{\chi^{++}}|\lesssim 100$~GeV (see \Eqn{e:T}).
All these considerations together justify our choice of benchmark for Fig.~\ref{f:DM}. Now, to satisfy \Eqn{e:gambound}
we need to have a large splitting between $m_H$ and $m_S$. Keeping these things in mind, we have chosen the first row in
Table~\ref{tab:inputoutput} as a benchmark for the input parameters. Some relevant output quantities that follow from
these inputs have also been displayed in the second row of the same table. From the numbers of Table~\ref{tab:inputoutput}
one can easily check that the constraints of \Eqs{e:epsrange}{e:gambound} and all the
bounds in Table~\ref{tab:LFV} are satisfied. Moreover, using \Eqn{e:hi3} suitable values for $f_{e\tau}$,
$f_{\mu\mu}$ and $f_{\mu\tau}$ can be found so that the hierarchy of \Eqn{e:hi2} is satisfied.
The model has many phenomenological implications that make it special and distinguishable from similar models.
To exemplify one such feature, we note that the requirement,
$M_{ee},M_{e\mu}\ll M_{e\tau},M_{\mu\mu},M_{\mu\tau},M_{\tau\tau}$, and consequently NH among the neutrino masses, results in a strong correlation between $\delta$, the CP violating phase of the Pontecorvo–Maki–Nakagawa–Sakata mixing matrix, and the other mixing parameters. For instance, in Fig. \ref{f:deltas23} we have displayed the allowed region in the plane $s^2_{23}$--$\delta$ obtained by the NuFIT collaboration (version 3.2 of 2018)\cite{Esteban:2016qun,nufitweb18} (the different coloured contours are 68.27\%, 90\%, 95.45\%, 99\% and 99.73\% C.L. regions respectively). On top of it we superimpose the correlation obtained from the requirement $M_{ee}=M_{e\mu}=0$ for the central values of the rest of the mixing parameters (brown dashed line) and the band obtained when they are varied in 1$\sigma$.
As we can see, the prediction of the model agrees well with the fit, although with some trend to lower values of $s^2_{23}$ and $\delta$.
Moreover the model also predicts the smallest neutrino mass to be around $m_1 \sim 5\times 10^{-3}$~eV and the two Majorana phases $\alpha_1\sim 360^\circ-\delta\sim 130^\circ$ and $\alpha_2\sim \alpha_1+180^\circ \sim 310^\circ$
\footnote{Here we use the same conventions for the neutrino mixing phases used in Ref.~\cite{delAguila:2011gr} except that now we take them in the range $[0^\circ,360^\circ]$ in order to compare with NuFIT results.}
\begin{figure}[htbp!]
\centering
\includegraphics[scale=0.70]{Figs/plotdeltas23-18.png}
\caption{The NuFIT results \cite{Esteban:2016qun,nufitweb18} for the global fit to neutrino data (coloured contours correspond to 68.27\% 90\% 95.45\% 99\% 99.73\% C.L. regions in the $s^2_{23}$--$\delta$ plane) against the prediction of the model for central values of the rest of the mixing parameters (brown dashed line) and the band obtained when they are varied in 1$\sigma$.}
\label{f:deltas23}
\end{figure}
\Eqn{e:nuelements} allows us to write the couplings $f_{ab}$ in terms of the neutrino masses and mixings up to a global factor. Since these couplings control all the LFV decays mediated by the double charged scalars, all the LFV processes are, in principle, predicted in terms of neutrino masses and mixing parameters which are fixed in our model.
As can be seen from the value of $\epsilon_3$ in Table~\ref{tab:inputoutput}, our model opens up the interesting possibility of detecting $0\nu\beta\beta~{\rm decay}$ in the next
generation of experiments even if $M_{ee}\sim 0$, but, in addition, is important to remark that the process is quite different from the standard one in which two left-handed electrons are produced. If $0\nu\beta\beta~{\rm decay}$ is found and proceeds as in the mechanism suggested in this paper, the produced electrons will be right-handed and, therefore, it will be possible, in principle, to distinguish this mechanism by measuring the polarization of the emitted electrons.
We have also found a DM candidate which can reproduce the observed relic abundance
yet can survive the current constraints from the direct detection experiments.
Furthermore, our model provides the prospect of detecting new scalars with masses below $\order({\rm TeV})$ in
collider experiments (for LHC studies on lepton number violating singly and doubly charged scalars see for instance \cite{delAguila:2013yaa,delAguila:2013mia}). Among these new particles, $\chi^\pm$ and $\chi^{\pm\pm}$ being $Z_2$-odd,
cannot decay directly into the SM particles. A search strategy for these kinds of exotic charged scalars
can be interesting for the collider studies. Moreover, the decay branching ratios of the singlet doubly charged scalar $\kappa^{++}$ are controlled by the $f_{ab}$ couplings which are fixed in terms of the neutrino mass parameters, therefore, if $\kappa^{++}$ is found at the LHC it will be possible to distinguish this model from other models by comparing the $\kappa^{++}$ leptonic decay branching ratios to neutrino oscillation data and to LFV processes, which also depend on the same couplings.
\section*{Acknowledgements}
A.S. would like to acknowledge J.M. No for discussions. All Feynman diagrams have been drawn using JaxoDraw \cite{Binosi:2003yf,Binosi:2008ig}. This work has been partially supported by the Spanish MINECO under grants FPA2011-23897, FPA2014-54459-P, by the ``Centro de Excelencia Severo Ochoa'' Programme under grant SEV-2014-0398 and by the ``Generalitat Valenciana'' grant GVPROMETEOII2014-087.
\begin{appendices}
\numberwithin{equation}{section}
\section{Computation of the loop induced $\kappa WW$ vertex}
\label{ap:Neutrinoless}
\begin{figure}[htbp!]
\centering
\includegraphics[scale=0.55]{Figs/kWW1} ~~ \includegraphics[scale=0.55]{Figs/kWW3}
~~ \includegraphics[scale=0.55]{Figs/kWW2}
\caption{One loop diagrams contributing to the $\kappa WW$ vertex in the unitary
gauge.
\label{f:kww}}
\end{figure}
Here we compute the effective $\kappa^{--}W_{\mu}^{+}W^{\mu+}$ vertex
at one loop for vanishing external momenta.
Our assumption is justified in view of the fact that the momentum transfers to $\kappa$ and $W$-bosons
in Fig.~\ref{fig:0nu2beta} are much smaller than the corresponding masses. We write the effective vertex
as\begin{eqnarray}
\label{e:kwwlag}
\mathscr L_{\kappa WW} &=& C_{\kappa WW}\kappa^{--}W_{\mu}^{+}W^{\mu+} + {\rm h.c.} \,,
\end{eqnarray}
which, after spontaneous symmetry breaking, emerges from the following gauge invariant operator:
\begin{eqnarray}
\mathscr L_{\kappa\mathrm{eff}}=C_{\kappa\mathrm{eff}}\kappa^{++}\left(\Phi^{\dagger}D^{\mu}\tilde{\Phi}\right) \left(\Phi^{\dagger}D_{\mu}\tilde{\Phi}\right)+\mathrm{h.c.}
\label{eq:effective-kappa}
\end{eqnarray}
After integrating out $\kappa^{++}$, \Eqn{eq:effective-kappa} leads to the following LFV gauge invariant operator\cite{delAguila:2012nu,delAguila:2011gr}:
\begin{eqnarray}
\mathscr L_{eeWW}=C_{eeWW}\left(\overline{e_{R}}\ f_{ee}^{*}\ e_{R}^{c}\right)\left(\Phi^{\dagger}D^{\mu}\tilde{\Phi}\right)\left(\Phi^{\dagger}D_{\mu}\tilde{\Phi}\right) \,.
\end{eqnarray}
We depict in Fig.~\ref{f:kww} the three diagrams that contribute to the vertex.
Each of these diagrams seem to diverge logaritmically. But one should keep in mind that
the neutral scalar exchange must violate lepton number conservation. Thus a large cancellation among the contributions
from the three neutral scalars, $A$, $H$ and $S$, is expected. After adding all the contributions
we obtain an effective neutral scalar propagator of the following form~(for
Minkowsky momenta)
\begin{equation}
\frac{1}{2}\frac{\sin^{2}\alpha\cos^{2}\alpha(m_{H}^{2}-m_S^{2})^{2}}{(p^{2}-m_{H}^{2})(p^{2}-m_S^{2})(p^{2}-m_{A}^{2})}=\frac{\lambda_{6}^{2}\langle\Phi\rangle^{4}}{(p^{2}-m_{H}^{2})(p^{2}-m_S^{2})(p^{2}-m_{A}^{2})}\,,
\label{eq:LNVCancellation}
\end{equation}
where, $\langle \Phi \rangle = v/\sqrt{2}$. Evidently, after adding contributions from $A$, $H$ and $S$,
every diagram in Fig.~\ref{f:kww} becomes finite individually. Now we can write the expression of
$C_{\kappa WW}$ (defined in \Eqn{e:kwwlag}) as follows:
\begin{eqnarray}
\label{e:ckww}
C_{\kappa WW} &=& \mu_{\kappa}g^{2}\lambda_{6}^{2}\langle\Phi\rangle^{4}\frac{1}{16\pi^{2}m_{A}^{4}}I_{\beta} \,,
\end{eqnarray}
with $I_{\beta}$ a function of the masses of the particles running
in the loop which contains three contributions corresponding to the three diagrams in Fig.~\ref{f:kww}.
Thus, we express $I_{\beta}$ as follows:
\begin{eqnarray}
I_{\beta} &=& I_{\beta}^1 +I_{\beta}^2 +I_{\beta}^3 \,, ~~{\rm with,} \\
I_{\beta}^1 &=& m_{A}^{4}\int_{0}^{\infty}\dd{q} q^{3}\frac{q^{2}}{(q^{2}+m_{\chi^{+}}^{2})^{2}(q^{2}+m_{A}^{2})(q^{2}+m_{H}^{2})(q^{2}+m_S^{2})} \,, \\
I_{\beta}^2 &=& -2m_{A}^{4}\int_{0}^{\infty}\dd{q} q^{3}\frac{1}{(q^{2}+m_{\chi^{++}}^{2})(q^{2}+m_{A}^{2})(q^{2}+m_{H}^{2})(q^{2}+m_S^{2})} \,, \\
I_{\beta}^3 &=& 2m_{A}^{4}\int_{0}^{\infty} \dd{q} q^{3}\frac{q^{2}}{(q^{2}+m_{\chi^{++}}^{2})(q^{2}+m_{\chi^{+}}^{2})(q^{2}+m_{A}^{2})(q^{2}+m_{H}^{2})(q^{2}+m_S^{2})} \,,
\end{eqnarray}
where we have passed to Euclidean momenta and integrated over the angular variables. Adding the three contributions we simplify the expression for $I_{\beta}$ as follows:
\begin{eqnarray}
I_{\beta} = m_{A}^{4}\int_{0}^{\infty}\dd{q} q^{3}\frac{q^{4}+q^{2}(m_{\chi^{++}}^{2}-2m_{\chi^{+}}^{2})-2m_{\chi^{+}}^{4}}{(q^{2}+m_{\chi^{++}}^{2})(q^{2}+m_{\chi^{+}}^{2})^{2}(q^{2}+m_{A}^{2})(q^{2}+m_{H}^{2})(q^{2}+m_S^{2})} \,.
\end{eqnarray}
We have checked that we obtain the same result
by using the equivalence theorem where the external $W$-bosons are replaced by
the corresponding Goldstone bosons.
In the limit $m_{H}=m_{A}=m_{\chi^{++}}=m_{\chi^{+}}$ and
$m_S\ll m_A$ we obtain $I_{\beta}\sim1/4$ while if all masses are equal we get $I_{\beta}=1/24$. If we fix $\sin(\alpha)$ $m_A$ can be obtained from $m_H$ and $m_S$ using Eq.~(\ref{e:mA}) while $m_{\chi^{+}}$ can be written in terms
of $m_{\chi^{++}}$ and $m_A$ using Eq.~(\ref{e:corr}). Thus, $I_\beta$ can be written as a function of $\sin(\alpha)$, $m_{\chi^{++}}$, $m_H$ and $m_S$ only.
In Fig.~\ref{f:Ibeta} we present results for some representative values of the masses (we fix $\sin(\alpha)=0.08$ and give $I_\beta$ as a function of $m_S$ for different values of $m_H=m_{\chi^{++}}$).
\begin{figure}[htbp!]
\centering
\includegraphics[scale=0.50]{Figs/plotIbeta.png}
\caption{The $0\nu\beta\beta$ integral, $I_\beta$, as a function of $m_S$ for some representative values of the other parameters. We fix $\sin(\alpha)=0.08$, use Eq.~(\ref{e:mA}) and Eq.~(\ref{e:corr}) and take $m_H=m_{\chi^{++}}$.}
\label{f:Ibeta}
\end{figure}
\section{Details of the calculation of the neutrino masses}
\label{ap:Neutrino-masses}
We define the Majorana mass matrix for the neutrinos as follows:
\begin{eqnarray}
\mathscr L_{\rm majorana} =-\frac{1}{2}\overline{\nu_{L}^{c}}\cdot M\cdot \nu_{L}+\mathrm{h.c.}
\end{eqnarray}
Our parametrization for the elements of the neutrino mass matrix have been displayed in \Eqn{e:nuelements}
which, in terms of the physical parameters, can be rewritten as
\begin{eqnarray}
M_{ab}=\frac{8\mu_{\kappa}\sin^{2}2\alpha G_{F}^{2}(m_{H}^{2}-m_S^{2})^{2}}{(4\pi)^{6}m_{\kappa^{++}}^{2}}I_\nu m_{a}f_{ab}m_{b} \,.
\end{eqnarray}
\begin{figure}
\begin{centering}
\includegraphics[scale=0.55]{Figs/numassU1} ~~~~ \includegraphics[scale=0.55]{Figs/numassU2}
\par\end{centering}
\begin{centering}
\includegraphics[scale=0.55]{Figs/numassU3} ~~~~ \includegraphics[scale=0.55]{Figs/numassU4}
\par\end{centering}
\caption{Three loop diagrams contributing to neutrino masses in the unitary
gauge.
\label{fig:numass-unitary}}
\end{figure}
In the unitary gauge there are four diagrams contributing to the neutrino masses as displayed in Fig.~\ref{fig:numass-unitary}.
As explained in Appendix~\ref{ap:Neutrinoless}, each diagram will be finite when we add together the contributions
from $H$, $S$ and $A$. Note that the two diagrams in the last row of Fig.~\ref{fig:numass-unitary}, after some relabeling
of momenta, will give identical contributions. Taking this into account, we decompose $I_\nu$ into three pieces as follows:
\begin{eqnarray}
\label{e:inu}
I_\nu = I_\nu^1 + I_\nu^2 + I_\nu^{34} \,.
\end{eqnarray}
Explicit expressions for the individual pieces in \Eqn{e:inu} are given below (all the momenta are Euclidean):
\begin{subequations}
\label{e:pieces}
\begin{eqnarray}
I_\nu^{1} &=& (4\pi)^{6}m_{\kappa^{++}}^{2}\int_{q}P_{c}\frac{V_{1}\cdot V_{2}}{\left\{(q_{1}+q_{3})^{2}+m_{\chi^{+}}^{2}\right\}\left\{ (q_{3}-q_{2})^{2}+m_{\chi^{+}}^{2}\right\} } \,, \\
I_\nu^{2} &=& -2(4\pi)^{6}m_{\kappa^{++}}^{2}\int_{q}P_{c}\frac{4M_{W}^{4} +M_{W}^{2}(q_{1}^{2}+q_{2}^{2})+(q_{1}q_{2})^{2}}{\left\{(q_{3}+q_{1}+q_{2})^{2}+m_{\chi^{++}}^{2}\right\}} \,, \\
I_\nu^{34} &=& 2(4\pi)^{6}m_{\kappa^{++}}^{2}\int_{q}P_{c}\frac{V_{1}\cdot V_{3}}{\left\{(q_{3}+q_{1}+q_{2})^{2}+m_{\chi^{++}}^{2} \right\}\left\{(q_{3}+q_{1})^{2}+m_{\chi^{+}}^{2}\right\} } \,,
\end{eqnarray}
\end{subequations}
\begin{subequations}
\label{e:defn}
\begin{eqnarray}
\hspace*{-10mm}{\rm with,} ~~~ P_{c}&=& \frac{1}{q_{1}^{2}(q_{1}^{2}+M_{W}^{2})q_{2}^{2}(q_{2}^{2}+M_{W}^{2}) \left\{(q_{1}+q_{2})^{2}+m_{\kappa^{++}}^{2}\right\} (q_{3}^{2}+m_{H}^{2})(q_{3}^{2}+m_S^{2})(q_{3}^{2}+m_{A}^{2})}\,, \\
V_{1}^{\mu}&=& M_{W}^{2}(2q_{3}+q_{1})^{\mu}+\left\{(2q_{3}+q_{1})\cdot q_{1}\right\} q_{1}^{\mu} \,, \\
V_{2}^{\mu}&=& M_{W}^{2}\left(2q_{3}-q_{2}\right)^{\mu}+\left\{(2q_{3}-q_{2})\cdot q_{2}\right\} q_{2}^{\mu} \,, \\
V_{3}^\mu &=& M_{W}^{2}(2q_{3}+2q_{1}+q_{2})^{\mu} +\left\{(2q_{3}+2q_{1}+q_{2})\cdot q_{2}\right\}q_{2}^{\mu} \,.
\end{eqnarray}
\end{subequations}
To evaluate the integrals in \Eqn{e:pieces} we express the Euclidean four-momenta in the four dimensional
spherical polar coordinates as follows:
\begin{eqnarray}
\label{e:4vector}
q_{i}=q_{i}(\cos\psi_{i},~\sin\psi_{i}\text{\ensuremath{\cos\theta_{i}},~\ensuremath{\sin\psi_{i}\text{\ensuremath{\sin\theta_{i}\cos\phi_{i}},}~\sin\psi_{i}\text{\ensuremath{\sin\theta_{i}\sin\phi_{i}}}})} \,,
\end{eqnarray}
where, for brevity, we have used $q_i$ to denote both the four Euclidean vector and its modulus. With this, the differential
under the integral can be expressed as:
\begin{eqnarray}
\int_{q} \equiv \int\prod_{i=1}^{3}\frac{\dd{q_{i}} q_{i}^{3}}{(2\pi)^{4}} \dd{\phi_{i}} \dd{\theta_{i}} \sin\theta_{i} \dd{\psi_{i}} \sin^{2}\psi_{i}\:,\qquad\phi_{i}\in[0,2\pi]\,,\;\theta_{i}\in[0,\pi]\,,\;\psi_{i}\in[0,\pi] \,,\; q_{i}\in[0,\infty] \,.
\end{eqnarray}
Without any loss of generality we can orient our 1-axis in the direction of $q_3$ and express the momenta as follows:
\begin{eqnarray}
q_{3}=q_{3}(1,0,0,0)\,,~~ q_{2}=q_{2}(\cos\psi_{2},\, \sin\psi_{2},0,0)\,,~~ q_{1}=q_{1}(\cos\psi_{1},\, \sin\psi_{1}\text{\ensuremath{\cos\theta_{1}},~\ensuremath{\sin\psi_{1}\text{\ensuremath{\sin\theta_{1}},}}~0)} \,.
\end{eqnarray}
In this way, the integrands in \Eqn{e:pieces} will not depend on the angles $\phi_{1},\phi_{2},\theta_{2},\phi_{3},\theta_{3},\psi_{3}$
and they can be integrated out very easily. After this, the remaining six parameter integrals can be computed numerically (we have used
Mathematica along with the Cuba package for this purpose). We have also checked numerically that, in the limit $g\to 0$ and small mixing, our unitary gauge calculation agrees with the calculation discussed in Sec. \ref{sec:nu-mass}, which includes only diagrams with scalar exchanges.
In Fig.~\ref{f:Inu} we give $I_\nu$ as a function of $m_\kappa$ for different values of the other parameters. As in Sec.~\ref{ap:Neutrinoless} we use Eq.~(\ref{e:mA}) and Eq.~(\ref{e:corr}), fix $\sin(\alpha)=0.08$ and take $m_H=m_{\chi^{++}}$).
\begin{figure}[htbp!]
\centering
\includegraphics[scale=0.50]{Figs/plotInu.png}
\caption{The neutrino mass integral, $I_\nu$, as a function of $m_{\kappa^{++}}$ for some representative values of the other parameters. We fix $\sin(\alpha)=0.08$, use Eq.~(\ref{e:mA}) and Eq.~(\ref{e:corr}) and take $m_H=m_{\chi^{++}}$.}
\label{f:Inu}
\end{figure}
\end{appendices}
\bibliographystyle{JHEP}
|
1,314,259,993,622 | arxiv | \section{Introduction}
Among inverse spectral problems, let us mention the following question (see for instance \cite{ColVerd})\\
\textit{"What kind of increasing sequences of non negative numbers can be the spectrum of the Laplacian of a compact Riemannian manifold (respectively of the Dirichlet Laplacian on a domain of a fixed Euclidean space) ?"}\\
This question can be asked in other more general contexts (for other operators and for Dirichlet or Neumann boundary conditions if the manifold has boundary, and for domains of a general Riemannian manifold instead of an Euclidean space).
Such sequences, which we will call spectral, admit some restrictions given by the asymptotics of Weyl and those of Minakshisundaram-Pleijel. Thereby, concerning those sequences, a natural question, less difficult than the first one, arises \\
\textit{"Is there any restrictions on these spectral sequences, which are independent of the manifold (respectively the domain) ? "}\\
Such restrictions will be called "universal".
The first result in this direction is the universal inequality of Payne, Polya and Weinberger \cite{PPW} obtained in 1955. In fact, they proved that the eigenvalues $\{\lambda_i\}_{i=1}^{\infty}$ of the Dirichlet boundary problem for the Laplacian on a bounded domain $\Omega \subset \R^n$, must satisfy for each $k$,
\begin{equation} \label{ppw}
\displaystyle{ \lambda_{k+1}-\lambda_{k} \le \frac{4}{nk} \sum_{i=1}^{k} \lambda_{i}}
\end{equation}
(which we call henceforth the PPW inequality).\\
This result was improved in 1980 by Hile and Protter \cite{HileProt} (henceforth HP) who showed that, for $k=1,2,\ldots$
\begin{equation}
\displaystyle \frac{nk}{4} \leq \sum_{i=1}^k
\frac{\lambda_i}{\lambda_{k+1}-\lambda_i}.
\end{equation}
In 1991, H.C.Yang (see \cite{Yang.HC} and more recently
\cite{ChengYang1}) proved
\begin{equation}\label{1}
\displaystyle \sum_{i=1}^k (\lambda_{k+1}-\lambda_i)^2 \leq
\frac{4}{n} \sum_{i=1}^k \lambda_i(\lambda_{k+1}-\lambda_i),
\end{equation}
which is, until now, the best improvement of the PPW inequality (see for instance \cite{Ashb1} for a comparison of all these three inequalities).\\
Apart from this class of inequalities (PPW, HP and Yang) which was intensively studied, there exists another class, much less known, discovered by Levitin and Parnovski (see Example 4.2 and identity (4.14) of \cite{LevPar}). Indeed, they proved for the eigenvalues of the Dirichlet Laplacian of any bounded domain of $\R^{n}$ and for any $k$,
\begin{equation}\label{Levi-Parn}
\sum_{i=1}^{n} \lambda_{k+i} \le (4+n) \lambda_{k}
\end{equation}
(these inequalities, indexed by $k$, will be referred to henceforth as Levitin and Parnovski inequalities).\\
These inequalities generalize a previous inequality obtained for $k=1$ by PPW
\cite{PPW} in dimension $n=2$ and by Ashbaugh (cf section 3.2 of \cite{Ashb2}) in all dimensions.\\
\indent All these universal inequalities show that one can not prescribe arbitrarily a finite part of the spectrum of the Dirichlet Laplacian on a bounded domain of an Euclidean space. This contrasts completely with the situation of the Laplace operator on a compact manifold (or the Neumann Laplacian on a bounded Euclidean domain), for which Colin de Verdi\`ere \cite{ColVerd} showed that it is possible to prescribe any finite part of the spectrum. More precisely, Colin de Verdi\`ere proved that, if $s_{N}=\{ \lambda_{1}=0 <\lambda_{2}\le \dots \le \lambda_{N}\}$ is a finite set of real numbers and if $M$ is a compact manifold without boundary of dimension $\ge 3$, then there exists a Riemannian metric on $M$ having $s_{N}$ as the beginning of the spectrum of its Laplacian. This was generalized by Guerini \cite {Guerini} to the Hodge de Rham Laplacian acting on differential forms for compact manifolds without boundary and for bounded Euclidean domains with the relative or absolute boundary conditions. As a consequence of these prescription results of a part of the spectrum, contrary to the situation of the Dirichlet Laplacian acting on functions on Euclidean bounded domains, one cannot expect a universal inequality for the Laplacian and more generally for the Hodge de Rham Laplacian acting on forms, on a compact Riemannian manifold.
However, a generalization of the PPW universal inequality holds for some special manifolds. In fact, in 1975, Cheng \cite{Cheng} showed that the PPW inequality (\ref{ppw}) holds for domains of minimal hypersurfaces of $\R^{n+1}$ (note that his proof works also for codimension $\ge 1$). In the same spirit, Yang and Yau \cite{YangYau} obtained a generalization of the PPW inequality for the eigenvalues of the Laplacian of any compact minimal Submanifold of a Sphere. Note that these two results indicate that, a role must probably be played by the extrinsic geometry of the Submanifolds in an eventual generalization of the PPW inequality. Other generalizations were obtained (see for instance \cite{Anghel}, \cite{Ashb1}, \cite{AshbHer1}, \cite{AshbHer3}, \cite{Cheng}, \cite{Col}, \cite{Harl1}, \cite{Harl2}, \cite{HarlMichel2}, \cite{HarlMichel1}, \cite{harlStub2}, \cite{HarlStub}, \cite{HileProt},\cite{IlMa}, \cite{Lee}, \cite{LeungPF2}, \cite{LiP}, \cite{Soufi.Harl.Ilias} and \cite{Yang.HC}), among them we mention the results of Lee \cite{Lee} and Anghel \cite{Anghel} which constitute a first tentative to a generalization of the PPW inequality to the eigenvalues of the Hodge de Rham Laplacian on an Euclidean compact Submanifold. Unfortunately, these generalized inequalities depend on the intrinsic geometry of the Submanifold. Nevertheless, the results of Colin de Verdi\`ere, Guerini, Cheng and Yang and Yau, suggest in the case of Euclidean Submanifolds the following question \\
\textit{"Can one find Universal inequalities of PPW, HP, Yang or Levitin and Parnovski type for the eigenvalues of the Dirichlet Laplacian on a bounded domain of an Euclidean Submanifold or for the eigenvalues of the Hodge de Rham Laplacian on an Euclidean compact Submanifold, which depends only on the extrinsic geometry of the Submanifold (i.e its second fundamental form or its mean curvature) ? "}\\
\indent Using an algebraic commutation inequality of Harrell and Stubbe \cite{HarlStub}, we gave in \cite{IlMa}(see also the references therein for partial results) a complete answer to the first part of the question, concerning PPW, HP and Yang type inequalities. In the present article we will focus on the second part of the question. Using an algebraic identity obtained by Levitin and Parnovski and by a method completely different to that we used in \cite{IlMa}, we will give a positive answer to the second part of the question which extends the Levitin and Parnovski inequalities (\ref{Levi-Parn}) to the eigenvalues of the Hodge de Rham Laplacian of a compact Euclidean Submanifold . We observe that our proof works also for the eigenvalues of the Dirichlet Laplacian on bounded domains of Euclidean Submanifolds.\\
We must note that some partial generalizations of the Levitin and Parnovski inequality was obtained recently by Chen and Cheng \cite{ChengChen} and by Sun, Cheng and Yang \cite{ChengYang3}. But, it turns out that all these generalizations are particular cases of our results. Indeed, on one hand, a direct consequence of our work (apply Corollary \ref{cor theorem 1} with $q=0$) is that, for any bounded domain $\Omega$ of an $m$-dimensional isometrically immersed Riemannian manifold $M$ in an Euclidean space and for any $k$, we have
\begin{equation}\label{LP1}
\sum_{i=1}^{m} \lambda_{k+i} \le (4+m) \lambda_{k}+\left\|H\right\|^{2}_{\infty,\Omega}
\end{equation}
where $\left\{\lambda_{j}\right\}_{j=1}^{\infty}$ are the eigenvalues of the Dirichlet Laplacian of $\Omega$, $H$ is the mean curvature vector of the immersion of $M$ (i.e the trace of its second fundamental form) and $\left\|H\right\|^{2}_{\infty,\Omega}=\displaystyle{\sup_{\Omega}|H|^{2}}$.
When we take $k=1$ in this inequality (\ref{LP1}), we obtain as a direct consequence the generalization obtained by Chen and Cheng (see Theorem 1.1 of \cite{ChengChen}). On the other hand, if we combine inequality (\ref{LP1}) with the standard embeddings of the compact rank one symmetric spaces in an Euclidean space, we easily derive a similar inequalities for Submanifolds of a Sphere or a Projective space. Let us denote by $\overline{M}$ the Sphere $\mathbb{S}^{n}$, the real projective space $\mathbb{R}P^{n}$, the complex projective space $\mathbb{C}P^{n}$ or the quaternionic projective space $\mathbb{Q}P^{n}$ endowed with their respective standard metrics and let $M$ be an $m$-dimensional Riemannian manifold isometrically immersed in $\overline{M}$. We prove (see Corollary \ref{corRSS}) that for any bounded bounded domain of $M$ and for any $k \ge 1$,
\begin{equation}\label{LP2}
\displaystyle \sum_{i=1}^m \lambda_{k+i} \le (4+m) \lambda_{k}+\big(\left\| H \right\|^{2}_{\infty,\Omega}+d(m)\big)
\end{equation}
where, $H$ is the mean curvature of $M$ in $\overline{M}$ and $d(m)$ is the constant given by
\begin{equation*}
d(m)=
\begin{cases}
m^{2}, &\text{if $\overline{M}=\mathbb{S}^{n}$}\\
2m(m+1), &\text{if $\overline{M}= \mathbb{R}P^{n}$}\\
2m(m+2), &\text{if $\overline{M}= \mathbb{C}P^{n}$}\\
2m(m+4), &\text{if $\overline{M}= \mathbb{Q}P^{n}$}.\\
\end{cases}
\end{equation*}
If we apply inequality (\ref{LP2}) with $k=1$ to domains of $\mathbb{S}^{n}$ or $\mathbb{C}P^{n}$ (respectively for complex Submanifolds of $\mathbb{C}P^{n}$ which are in particular minimal), then we obtain Theorem 1.1 and Theorem 1.3 (respectively Theorem 1.2) of Sun, Cheng and Yang \cite{ChengYang3}.
\indent Another consequence of our work is an extension to the eigenvalues of higher order of the Reilly inequality (respectively the Asada inequality) concerning the first eigenvalue of the Laplacian (respectively the Hodge de Rham Laplacian) of a compact Riemannian manifold isometrically immersed in an Euclidean space. Indeed, for any $m$-dimensional compact Riemannian manifold immersed in an Euclidean space, Reilly \cite{Reilly} proved the following inequality between the first positive eigenvalue of its Laplacian and the mean curvature of its immersion
\begin{equation}\label{rel}
\displaystyle \lambda_{2}\leq \frac{1}{m\,{\rm Vol}(M)}\int_{M} |H|^2 dV_M
\end{equation}
where $dV_M$ and ${\rm Vol}(M)$ are respectively the Riemannian volume element and the volume of $M$.
Asada \cite{Asada} obtains an extension of this inequality to the first positive eigenvalue of the Hodge de Rham Laplacian acting on $p$-forms
\begin{equation}\label{as}
\lambda_1^{(p)}(M) \leq \frac{p}{m(m-1) {\rm Vol}(M)} \int _M
\Big[(m-p)|H|^2+(p-1)|h|^2\Big] dV_M.
\end{equation}
where $h$ denotes the second fundamental form of the immersion of $M$. To be more precise, we note that, Asada proves more. In fact, he proves this inequality for the first positive eigenvalue of the Hodge de Rham Laplacian restricted to the closed $p$-forms.\\
Using our generalizations of the Levitin and Parnovski universal inequalities, one can easily extends the Reilly and the Asada inequalities to all the eigenvalues of the Laplacian and the Hodge de Rham Laplacian of Euclidean Submanifolds. We derive (see Corollary \ref{cor Reilly general}) in particular the surprising generalization of the Reilly inequality
\begin{equation*}
\displaystyle \sum_{k=1}^m \lambda_{k+1}\leq \frac{1}{Vol(M)}\int_{M} |H|^2 dV_M.
\end{equation*}
\indent We limit ourselves to the case of the Hodge de Rham Laplacian, but all our arguments work with minor modifications in the setting of general Laplace operators on Riemannian fiber bundles. \\
\indent Another different situation which is not Riemannian involving an operator which is not elliptic, is that of the Kohn Laplacian on the Heisenberg group. In the second section, we derive a Levitin and Parnovski inequality in this case.
\section{Generalization of the Levitin-Parnovski inequality to the Hodge de Rham Laplacian}
Let $(M,g)$ be an $m$-dimensional compact Riemannian manifold. We denote by $\bigwedge^{p}(M)$ for $p \in \left\{0, \dots ,m \right\}$, and by $\Gamma(TM)$ respectively, the space of smooth differential $p$-forms and the space of smooth vector-fields of $M$.\\
For any two $p$-forms $\alpha$ and $\beta$, we let $\alpha_{i_1i_2,\ldots,i_p}=\alpha(e_{i_1},e_{i_2},\ldots,e_{i_p})$ and $\beta_{i_1i_2,\ldots,i_p}=\beta(e_{i_1},e_{i_2},\ldots,e_{i_p})$ denote the components of $\alpha$ and $\beta$, with respect to a local orthonormal frame $(e_i)_{i \leq m}$. Their pointwise inner product with respect to $g$ is given by
\begin{equation*}
\langle \alpha,\beta \rangle = \frac{1}{p!} \sum_{1 \le i_{1},\dots,i_{p}\le m}\alpha_{{i_1},\dots,{i_p}}\; \beta_{{i_1},\dots,{i_p}}.
\end{equation*}
We denote by $\Delta_{p}$ the Hodge de Rham Laplacian acting on $p$-forms
\begin{equation*}
\Delta_{p} :=(d \, \delta + \delta d),
\end{equation*}
where $d$ is the exterior derivative acting on $p$-forms and $\delta$ is the adjoint of $d$ with respect to the $L_{2}(g)$ global inner product.\\
The spectrum of $\Delta_{p}$ consists of a nondecreasing, unbounded sequence of eigenvalues with finite multiplicities
$${\rm Spec}(\Delta_{p})=\{0 \le \lambda_{1}^{(p)} \le \lambda_{2}^{(p)} \le \lambda_{3}^{(p)} \le \cdots \le \lambda_{i}^{(p)} \le \cdots \}.$$
If we denote by $\nabla$ the extension to $p-$forms of the Levi-Civita connexion of $(M,g)$ and by $\nabla^{\ast}$ its formal adjoint with respect to the metric $g$, then the Bochner-Weitzenb\"{o}ck formula gives for any $\alpha \in\bigwedge^{p}(M)$
\vspace{0.3cm}
\begin{center}
$\Delta_{p} \, \alpha = \nabla^{\ast} \nabla \alpha +\mathcal{R}_{p}(\alpha)$
\end{center}
where $\mathcal{R}_{p}$ is the curvature term which is a selfadjoint endomorphism of $\bigwedge^{p}(M)$ defined for any $X_{1},\dots,X_{p} \in \Gamma(TM)$ by
\begin{align*}
\mathcal{R}_{p}(\alpha)(X_{1},\dots,X_{p}) &
=\sum_{i,j}(-1)^{i} i_{e_{j}}(R(e_{j},X_{i})\alpha)(X_{1},\dots,\hat{X_{i}},\dots,X_{p}),\\
\end{align*}
here $(e_{i})_{i \le m}$ is a local orthonormal frame as before and $R$ is the extension of the curvature tensor to forms which is given for $X,\,Y \in \Gamma(TM)$, by
\begin{equation*} R(X,Y)\alpha= \nabla_{\left[X,Y\right]}\alpha - \left[\nabla_{X},\nabla_{Y}\right]\alpha,
\end{equation*}
An immediate consequence of the Bochner-Weitzenb\"{o}ck formula is
the following
\begin{equation}\label{weitzenbock}
\langle \Delta_{p}\alpha,\alpha \rangle= |\nabla \alpha|^{2}+\frac{1}{2} \Delta |\alpha|^{2}+\langle \mathcal{R}_{p}(\alpha),\alpha \rangle.
\end{equation}
\\
\indent In this section, the main objective is to extend the universal inequality of Levitin and Parnovski (see inequality (4.14) in \cite{LevPar}) concerning the eigenvalues of the
Dirichlet Laplacian on bounded Euclidean domains to the eigenvalues of
the Hodge-de Rham Laplacian on closed Euclidean Submanifolds.\\
\begin{theorem} \label{theorem 1} Let $X:(M^m,g)\longrightarrow (\R^n,{\rm can})$ be an isometric
immersion and $H$ be its mean curvature vector field (i.e. the trace of its second fundamental form $h$). We have, for any
$p \in \left\{1,\dots,m\right\}$ and $j \in \N^{\ast}$,
\begin{equation}\label{ineq1}
\displaystyle{\sum_{l=1}^m \lambda_{j+l}^{(p)}}\leq
\displaystyle{4\bigg[\Big(1+\frac{m}{4}\Big)\lambda_j^{(p)}
-\int_{M}\langle
\mathcal{R}_p(\omega_j),\omega_j\rangle+\frac{1}{4}\int_{M}|H|^2|\omega_j|^2\bigg]},
\end{equation}
where $\Big\{\lambda_j^{(p)}\Big\}_{j=1}^\infty$ are the eigenvalues of $\Delta_p$ and $\{\omega_j\}_{j=1}^\infty$ is a corresponding orthonormal basis of $p-$eigenforms.
\end{theorem}
\begin{proof}[Proof of Theorem \ref{theorem 1}] To prove this inequality, we need the
following algebraic identity obtained by Levitin and Parnovski (see identity 2.2 of Theorem 2.2 in \cite {LevPar}).
\begin{lemma}\label{levitin}
Let $L$ and $G$ be two self-adjoint operators with domains $D_{L}$ and $D_{G}$ contained in a same Hilbert space and such that
$G(D_L)\subseteq D_L \subseteq D_G$. Let $\lambda_j$ and $u_j$
be the eigenvalues and orthonormal eigenvectors of $L$. Then, for each $j$,
\begin{equation*} \displaystyle{\sum_{k}
\frac{|\langle[L,G]u_j,u_k\rangle|\;^2}{\lambda_k-\lambda_j}}=\displaystyle{-\frac{1}{2}\langle[[L,G],G]u_j,u_j\rangle}
\end{equation*}
(The summation is over all $k$ and is correctly defined even when
$\lambda_{k}=\lambda_{j}$ because in this case $\langle[L,G]u_j,u_k\rangle=0$
(see Lemma 2.1 in \cite{LevPar})).
\end{lemma}
Now (\ref{ineq1}) will follow by applying this Lemma \ref{levitin} with
$L=\Delta_p$ and $G=X_l$, where $X_l$ is one of the components
$(X_1,...,X_n)$ of the isometric immersion $X$. First, we have
\begin{align*}
{}\displaystyle{[[\Delta_p,X_l],X_l]\omega_j} & = [\Delta_p,X_l](X_{l}\omega_{j})-X_{l}([\Delta_p,X_{l}]\omega_{j}) \\
{} & \displaystyle = (\Delta X_{l})(X_{l}\omega_{j})-2\nabla_{\nabla X_{l}}(X_{l}\omega_{j})\\
{} & \quad \displaystyle -X_{l}\Big((\Delta X_{l})\omega_{j}-2\nabla_{\nabla X_{l}}\omega_{j}\Big)\\
{} & \displaystyle = -2 |\nabla X_{l}|^{2} \omega_{j},
\end{align*}
hence\begin{equation*}
\displaystyle{-\frac{1}{2}\langle[[\Delta_p,X_l],X_l]\omega_j,\omega_j\rangle_{L^{2}}=\int_{M}|\nabla
X_l|^2|\omega_j|^2}. \end{equation*} Thus, Lemma \ref{levitin} gives
\begin{equation}\label{i}
\displaystyle{\sum_{k }\frac{\Big(\displaystyle\int_{M}\langle[\Delta_p,X_l]\omega_j,\omega_k\rangle\Big)^2}{\lambda_k^{(p)}-\lambda_j^{(p)}}=\int_{M}|\nabla
X_l|^2|\omega_j|^2}.
\end{equation}
Now for a fixed $j$, let $A$ be the matrix
\begin{equation*}
\Big(\omega_{k,\;l}=\int_{M}\langle[\Delta_p, X_l]\omega_j,\omega_{j+k}\rangle\Big)_{1\leq k,\;l \leq n}.
\end{equation*}
Applying Gram-Schmidt orthogonalization, we can find an orthogonal
coordinate system such that $A$ has the following triangular form,
$$\left(\begin{array}{ccccc}
\omega_{1,1} \\
\omega_{2,1} &\omega_{2,2}&\text{{\huge{0}}}\\
\vdots & & \ddots \\
\omega_{n,1}&\cdots &\cdots& \omega_{n,n}
\end{array}\right)$$
where $\omega_{k,\;l}=0\;\;if\;\;k<l$.\\
Equation (\ref{i}) can be written as follows
\begin{align}\label{j}
{} \int_{M}|\nabla X_l|^2|\omega_j|^2 & = \sum_{k=1}^{j-1}\frac{\Big(\displaystyle \int_{M}\langle[\Delta_p,X_l]\omega_j,\omega_k\rangle\Big)^2}{\lambda_k^{(p)}-\lambda_j^{(p)}} \nonumber\\
{} & +\sum_{k=j+1}^{j+l-1}\frac{\Big(\displaystyle\int_{M}\langle[\Delta_p,X_l]\omega_j,\omega_k\rangle\Big)^2}{\lambda_k^{(p)}-\lambda_j^{(p)}} \nonumber\\
{} &
+\sum_{k=j+l}^{\infty}\frac{\Big(\displaystyle\int_{M}\langle[\Delta_p,X_l]\omega_j,\omega_k\rangle\Big)^2}{\lambda_k^{(p)}-\lambda_j^{(p)}}.
\end{align}
The first term of the right-hand side of equality (\ref{j}) is
nonpositive because $k <j$. The second term is equal to
\begin{equation*}\displaystyle{\sum_{k=1}^{l-1}\frac{\Big(\displaystyle\int_{M}\langle[\Delta_p,X_l]\omega_j,\omega_{j+k}\rangle\Big)^2}{\lambda_{j+k}^{(p)}-\lambda_j^{(p)}}}
\end{equation*}
which is equal to zero because $\displaystyle
\int_{M}\langle[\Delta_p,X_l]\omega_j,\omega_{j+k}\rangle=\omega_{k,\;l}=0$
if $k<l$, \\
therefore
\begin{align}\label{k}
{} \displaystyle{\int_{M}|\nabla X_l|^2|\omega_j|^2} & \leq \displaystyle{\sum_{k=j+l}^{\infty}\frac{\Big(\displaystyle\int_{M}\langle[\Delta_p,X_l]\omega_j,\omega_k\rangle\Big)^2}{\lambda_k^{(p)}-\lambda_j^{(p)}}} \nonumber\\
{} & \leq
\displaystyle{\frac{1}{\lambda_{j+l}^{(p)}-\lambda_j^{(p)}}\sum_{k=1}^\infty\Big(\int_{M}\langle[\Delta_p,X_l]\omega_j,\omega_k\rangle\Big)^2}.
\end{align}
Parceval's identity implies that
\begin{equation} \label{Parc}
\displaystyle \sum_{k=1}^\infty\Big(\int_{M}\langle[\Delta_p,X_l]\omega_j,\omega_k\rangle\Big)^2=\|[\Delta_p,X_l]\omega_j\|^2_{L^2}.
\end{equation}
Hence using (\ref{k}), (\ref{Parc}) and summing on $l$, we obtain
\begin{align}\label{l}
{} \sum_{l=1}^{n}
\Big(\lambda_{j+l}^{(p)}-\lambda_j^{(p)}\Big)\bigg(\int_{M}|\nabla
X_l|^2|\omega_j|^2\bigg) & \leq \sum_{l=1}^n \|[\Delta_p,X_l]\omega_j\|^2_{L^2}.
\end{align}
We need now to calculate $\sum_{l=1}^n \|[\Delta_p,X_l]\omega_j\|^2_{L^2}$. First, we have
\begin{equation*}
\left[\Delta_{p},X_l \right]=\left[\nabla^{\ast}\nabla,X_l \right],
\end{equation*}
because the curvature term in the Bochner-Weitzenb\"{o}ck formula is $\mathcal{C}^{\infty}(M)$-linear. Then, at a point $x \in M$, we take a local orthonormal frame $(e_{i})_{i \le m}$ of $M$ which
is normal at $x$. We have at $x$
\begin{align*}
\left[\Delta_{p}, X_l\right]\omega_j & = \nabla^{\ast}\nabla
(X_l \omega_j)-X_l \nabla^{\ast}\nabla \omega_j \\
& = - \sum_{i \le m} \nabla_{e_{i}}\nabla_{e_{i}}(X_l \omega_j)-X_l \nabla^{\ast}\nabla \omega_j\\
& = -\sum_{i\le
m}e_{i}(e_{i}(X_l))\omega_j-2\nabla_{\nabla
X_l}\omega_j+X_l\nabla^{\ast}\nabla \omega_j-X_l \nabla^{\ast}\nabla \omega_j\\
& =(\Delta X_l)\omega_j-2\nabla_{\nabla X_l}\omega_j.
\end{align*}
Hence we obtain
\begin{equation*}
\left\|\left[\Delta_p,X_l\right]\omega_j\right\|^{2}_{L^{2}}= \int_{M}
(\Delta X_l)^{2}|\omega_j|^{2} + 4 \int_{M} |\nabla_{\nabla X_l}\omega_j|^{2}-4 \int_{M}
\langle (\Delta X_l) \omega_j,\nabla_{\nabla X_l}\omega_j \rangle.
\end{equation*}
Since $X$ is an isometric immersion, we have $\sum_{l \le n} |\nabla_{\nabla
X_{l}}\omega_{j}|^{2}=|\nabla \omega_{j}|^{2}$, ${(\Delta X_{1},\dots,\Delta
X_{n})=H}$ and $\sum_{l \le n}\langle (\Delta X_{l}) \omega_{j},\nabla_{\nabla X_{l}}\omega_{j}\rangle=\frac{1}{2} \langle H,\nabla |\omega_{j}|^{2}\rangle=0.$\\
Thus it follows that
\begin{align}\label{c}
{} \sum_{l \le n}\left\|\left[\Delta_p,X_{l}\right]\omega_{j}\right\|^{2}_{L^{2}}=&\int_{M} \sum_{l \le n}(\Delta X_{l})^{2}|\omega_{j}|^{2} + 4 \int_{M} \sum_{l \le n}|\nabla_{\nabla X_{l}}\omega_{j}|^{2} \nonumber\\
{} & \quad -4\int_{M} \sum_{l \le n}\langle(\Delta X_{l})\omega_{j},\nabla_{\nabla X_{l}}\omega_{j}\rangle \nonumber \\
{}=& \int_{M} |H|^{2} |\omega_{j}|^{2}+4 \int_{M} |\nabla \omega_{j}|^{2}\nonumber\\
{}=& \int_{M} |H|^{2} |\omega_{j}|^{2}+4\lambda_{j}^{(p)}-4\int_{M} \langle \mathcal{R}_p(\omega_{j}),\omega_{j}\rangle.
\end{align}
The last equality follows from the Bochner-Weitzenb\"{o}ck formula, in fact we have
\begin{equation*}
\int_{M} |\nabla \omega_{j}|^{2}= \int_{M} \langle \Delta_p
\omega_{j},\omega_{j} \rangle-\langle
\mathcal{R}_p(\omega_{j}),\omega_{j}\rangle =\lambda_{j}^{(p)}-\int
_{M}\langle \mathcal{R}_p(\omega_{j}),\omega_{j}\rangle.
\end{equation*}
On the other hand, since the immersion $X$ is isometric we have $\displaystyle{\sum_{l=1}^{n}|\nabla X_l|^2=m}$ and therefore
\begin{align}\label{m}
{}\sum_{l=1}^n \Big(\lambda_{j+l}^{(p)}-\lambda_j^{(p)}\Big)\bigg(\int_{M}|\nabla
X_l|^2|\omega_j|^2\bigg)&=\sum_{l=1}^n\bigg(\int_{M}|\nabla X_l|^2|\omega_j|^2\bigg)\lambda_{j+l}^{(p)} \nonumber\\
{} & -m\lambda_j^{(p)},
\end{align}
then we obtain, from (\ref{l}), (\ref{c}) and (\ref{m})
\begin{align}\label{n}
{} \sum_{l=1}^{n}\bigg(\int_{M}|\nabla X_l|^2|\omega_j|^2\bigg)\lambda_{j+l}^{(p)} & \leq (4+m)\lambda_j^{(p)}-4\int_{M}\langle \mathcal{R}_p(\omega_j),\omega_j\rangle \nonumber\\
{} & +\int_{M}|H|^2|\omega_j|^2.
\end{align}
To finish the proof, we will show that
\begin{equation}\label{o}
\sum_{l=1}^n |\nabla X_l|^2 \lambda_{j+l}^{(p)} \ge \sum_{l=1}^m \lambda_{j+l}^{(p)}.
\end{equation}
In fact, let us prove inequality (\ref{o}) at an arbitrary $x\in M$. Denote by $(\epsilon_{i})_{i\le n}$ the standard Euclidean basis of $\R^{n}$. Since the immersion $X$ is isometric, we deduce that: there exist $\,l_{1},\dots, l_{m} \in \left\{1,\dots,n\right\}$ and $ i_{1},\dots, i_{m} \in \left\{1,\dots,n\right\}$ such that
\begin{itemize}
\item For any $ k \in \left\{1,\dots,m\right\}:\quad l_{k} \ge k.$
\item For any $k \in \left\{1,\dots,m\right\}: \quad \nabla X_{l_{k}}(x)=\epsilon_{i_{k}}.$
\item For any $i \notin \left\{l_{1},\dots,l_{m}\right\}:\quad \nabla X_{i}(x)=0.$
\end{itemize}
Therefore, we have at $x$
\begin{equation*}
\sum_{l=1}^n |\nabla X_l|^2 \lambda_{j+l}^{(p)}= \sum_{k=1}^{m}\lambda_{j+l_{k}}^{(p)} \ge \sum_{l=1}^m \lambda_{j+l}^{(p)}
\end{equation*}
which proves inequality (\ref{o}).
\\
Finally, we deduce, from (\ref{n}) and (\ref{o}), that
\begin{equation*}
\displaystyle{\sum_{l=1}^m \lambda_{j+l}^{(p)}\leq
(4+m)\lambda_j^{(p)}-4\int_{M}\langle
\mathcal{R}_p(\omega_j),\omega_j\rangle+\int_{M}|H|^2|\omega_j|^2}.
\end{equation*}
\end{proof}
\begin{remark}
\begin{enumerate}
\item Note that our result does not depend on the dimension of the ambient space $\R^{n}$.
\item We observe that, the same ideas work for general operators of Laplace type acting on the sections of a Riemannian vector bundle on $M$ endowed with a Riemannian connexion.
\end{enumerate}
\end{remark}
\begin{corollary} \label{cor1}
Under the conditions of Theorem \ref{theorem 1} , we have for any $j \ge 1$,
\begin{equation*}
\sum_{l=1}^m \lambda_{j+l}^{(p)}\leq 4\bigg((1+\frac{m}{4})\lambda_j^{(p)}-\delta_1+\frac{1}{4}\delta_2\bigg),
\end{equation*}
where $\displaystyle \delta_1=\inf_{x\in M}\tilde{\mathcal{R}}_{p}(x)$, $\tilde{\mathcal{R}}_{p}(x)$ being the smallest eigenvalue, at $x \in M$, of the endomorphism $(\mathcal{R}_p)_x$ of
$\bigwedge^p (T_x M)$, and $\delta_2=\sup |H|^{2}$.
\end{corollary}
The inequalities of Theorem \ref{theorem 1} and Corollary \ref{cor1} depends on the intrinsic geometry of the Submanifold $M$, because it involves the curvature term $\mathcal{R}_p$. Using as in Theorem 3.2 of \cite{IlMa}, the extrinsic estimate we derived for $\mathcal{R}_p$ in terms of the second fundamental form $h$ and the mean curvature $H$ of the immersion $X$ of $M$, we obtain
\begin{theorem} \label{PPW} Under the conditions of Theorem \ref{theorem 1}, we have for any $j \ge 1$,
\begin{equation}\label{q}
{} \displaystyle \sum_{l=1}^m \lambda_{j+l}^{(p)} \leq
4\bigg\{\Big(1+\frac{m}{4}\Big)\lambda_j^{(p)}+\int_{M}\phi(h,H)|\omega_j|^2\bigg\},
\end{equation}
where \begin{align*}
\phi(h,H)=&p^2\bigg[\Big(\frac{m-5}{4}\Big)|H|^2+|h|^2-\frac{1}{4m^2}\Big(\sqrt{m-1}(m-2)|H|\\
{}&-2\sqrt{m|h|^2-|H|^2}\,\Big)^2
\bigg]+\frac{1}{2}\sqrt{p}(p-1)\Big(|H|^2+|h|^2\Big)+\frac{1}{4}|H|^2.
\end{align*}
\end{theorem}
\begin{proof}[Proof of Theorem \ref{PPW}]
This theorem follows immediately from the estimate of $\mathcal{R}_p$ obtained in Theorem 3.2 of \cite{IlMa}
\begin{align*}
\langle\mathcal{R}_p(\omega_j),\omega_j\rangle \ge& \bigg\{-p^2\bigg[\Big(\frac{m-5}{4}\Big)|H|^2+|h|^2-\frac{1}{4m^2}\Big(\sqrt{m-1}(m-2)|H|\\
{}&-2\sqrt{m|h|^2-|H|^2}\,\Big)^2\bigg]-\frac{1}{2}\sqrt{p}(p-1)\Big(|H|^2+|h|^2\Big)\bigg\}|\omega_j|^2.
\end{align*}
\end{proof}
One can obviously eliminate the dependence on $\omega_j$ by taking
the supremum of $\phi(h,H)$, and obtain the following extension of the Levitin and Parnovski inequality which depends only on extrinsic invariants of the Submanifold $M$,
\begin{corollary} \label {corLP}
Under the conditions of Theorem \ref{theorem 1}, $\forall j \ge 1$ we have
\begin{equation}\label{r}
\displaystyle \sum_{l=1}^m \lambda_{j+l}^{(p)} \leq 4 \bigg\{ \Big(1+\frac{m}{4}\Big)\lambda_j^{(p)}+ \|\phi(h,H)\|_{\infty} \bigg\},
\end{equation}
where $\phi(h,H)$ is as in Theorem \ref{PPW}.
\end{corollary}
In the particular case where $j=1$, we obtain
\begin{corollary} Under the
conditions of Theorem \ref{theorem 1}, we have
\begin{equation*}
\displaystyle \sum_{l=1}^m \lambda_{l+1}^{(p)} \leq
\Phi(h,H),
\end{equation*}
where
\begin{align*}
{}\Phi(h,H)=4\bigg\{\frac{p}{m(m-1)Vol(M)}& \Big(1+\frac{m}{4}\Big)\int_M
\Big[(m-p) |H|^2+(p-1)|h|^2 \Big]dV_M\\
{} + \|\phi(h,H)\|_{\infty}\bigg\}.
\end{align*}
\end{corollary}
\begin{proof}
Inequality (\ref{r}) gives, for $j=1$
\begin{equation*}
\sum_{l=1}^m \lambda_{l+1}^{(p)} \leq 4\bigg\{
\bigg(1+\frac{m}{4}\bigg)\lambda_1^{(p)}+ \|\phi(h,H)\|_{\infty}
\bigg\}.
\end{equation*}
We finish the proof by using the Asada inequality \cite{Asada},
\begin{equation}\label{asada's ineq}
\lambda_1^{(p)}(M) \leq \frac{p}{m(m-1) Vol(M)} \int _M
\Big[(m-p)|H|^2+(p-1)|h|^2\Big] dV_M.
\end{equation}
\end{proof}
For any $j \ge 1$, an immediate consequence of the precedent
corollary is the following upper bound for $\lambda_{j+m}^{(p)}$ in
terms of the second fundamental form and the mean curvature,
\begin{corollary}
Under the conditions of Theorem \ref{theorem 1}, we have for any $j \ge 1$,
\begin{align}\label{t}
{}\lambda_{j+m}^{(p)} \leq
4\bigg\{d_1(m,j)\frac{p}{Vol(M)}\int_M\Big[(m-p)|H|^2+(p-1)|h|^2
\Big]dV_M \nonumber\\
{} +d_2(m,j) \|\phi(h,H)\|_{\infty}\bigg\},
\end{align}
where
$d_1(m,j)=\frac{1}{m(m-1)}\Big(1+\frac{m}{4}\Big)\Big(1+\frac{4}{m}\Big)^{j-1}$
and\\
$d_2(m,j)=\Big(1+\frac{m}{4}\Big)\Big(1+\frac{4}{m}\Big)^{j-1}-\frac{m}{4}$
\end{corollary}
\begin{proof}
We infer from (\ref{q})
\begin{equation*}
\lambda_{j+m}^{(p)} \leq 4\Big[ \Big(1+\frac{m}{4} \Big)
\lambda_j^{(p)} + \|\phi(h,H)\|_{\infty}\Big],
\end{equation*}
to obtain inequality (\ref{t}), we combine this last inequality with the following inequality obtained by us in \cite{IlMa} (see Corollary 3.6),
\begin{align}\label{s}
{}\lambda_j^{(p)} \leq \frac{1}{m(m-1)} \Big(1+\frac{4}{m}\Big)^{j-1} \frac{p}{Vol(M)} & \int_M \Big[ (m-p)
|H|^2+(p-1)|h|^2 \Big] dV_M
\nonumber\\
{} &+ \Big(\Big(1+\frac{4}{m} \Big)^{j-1}-1 \Big) \|\phi(h,H)\|_{\infty}.
\end{align}
\end{proof}
\begin{remark}
\begin{enumerate}
\item Note that inequality (\ref{t}) is sharper than inequality (\ref{s}) for
$j+m$.
\item We can obtain similar results for closed Submanifolds of compact rank one symmetric spaces but the expressions of $\phi(h,H)$ and $\Phi(h,H)$ in this case are complicated.
\end{enumerate}
\end{remark}
In the particular case where $p=0$ (i.e. for functions), all the arguments used
in the proof of Theorem \ref{theorem 1} work under the Dirichlet boundary condition, when $M$ has
boundary . The reason why all these arguments work in this case is that the product of a function $G$ by a function vanishing on $\partial M$ also vanishes on $\partial M$ (this is neither the case for the absolute boundary condition nor the relative one for $p-$forms when $p\ge1$). Then, we easily obtain
\begin{corollary} \label{cor theorem 1} Let $(M,g)$ be a compact $m$-dimensional Riemannian manifold eventually with boundary and let $X:(M,g)\longrightarrow (\mathbb{R}^{n},{\rm can})$ be an isometric immersion. For any bounded potential $q$ on $M$, the spectrum of $L=\Delta+q$ (with Dirichlet boundary condition if $\partial M \ne \emptyset$) must satisfy, for $j \ge 1$,
\begin{align*}
{} \displaystyle \sum_{l=1}^m \lambda_{j+l} & \leq 4 \Big[
(1+\frac{m}{4}) \lambda_{j}+\int_{M}\Big(\frac{1}{4}\vert H
\vert^2-q\Big)u_j^2\Big]\\
& \leq (4+m) \lambda_{j}+\left\|\vert H\vert^2-4q\right\|_{\infty},
\end{align*}
where $u_{j}$ are $L^{2}-$normalized eigenfunctions.
\end{corollary}
This corollary extends, the universal inequality of Levitin and
Parnovski to compact Submanifolds of $\R^n$ and gives for $k=1$ the main result of \cite{ChengChen}.\\
When $M$ is without boundary and $q=0$, we have $\lambda_{1}=0$ and the associated normalized eigenfunction is $u_{1}=\displaystyle{\frac{1}{\sqrt{Vol(M)}}}$; in this case, the Corollary \ref{cor theorem 1} gives the following generalization of Reilly's inequality for the first nonzero eigenvalue of the Laplacian operator on Euclidean closed Submanifolds \cite{Reilly},
\begin{corollary} \label{cor Reilly general} Let $(M,g)$ be a compact $m$-dimensional Riemannian manifold and let $X:(M,g)\longrightarrow (\mathbb{R}^{n},{\rm can})$ be an isometric immersion of mean curvature $H$. Then the spectrum of $\Delta$ must satisfy
\begin{equation*}
\displaystyle \sum_{k=1}^m \lambda_{k+1}\leq \frac{1}{Vol(M)}\int_{M} |H|^2.
\end{equation*}
\end{corollary}
As in \cite{Soufi.Harl.Ilias} (Lemma 3.1) or \cite{IlMa}, using the standard
embeddings of rank one compact symmetric spaces, we deduce easily from Corollary \ref{cor theorem 1} the following
\begin{corollary} \label{corRSS} Let $\overline{M}$ be the Sphere $\mathbb{S}^{n}$, the real projective space $\mathbb{R}P^{n}$, the complex projective space $\mathbb{C}P^{n}$ or the quaternionic projective space $\mathbb{Q}P^{n}$ endowed with their respective standard metrics. Let $(M,g)$ be a compact $m$-dimensional Riemannian manifold eventually with boundary and let $X:M\longrightarrow \overline{M}$ be an isometric immersion. For any bounded potential $q$ on $M$, the spectrum of $L=\Delta+q$ (with Dirichlet boundary condition if $\partial M \ne \emptyset$) must satisfy, for $j \ge 1$,
\begin{align*}
{} \displaystyle \sum_{l=1}^m \lambda_{j+l} & \leq 4 \Big[ \Big(1+\frac{m}{4}\Big) \lambda_{j}+\int_{M}\Big(\frac{1}{4}\big(\vert H \vert^2+d(m)\big)-q\Big)u_j^2\Big]\\
{} & \le (4+m) \lambda_{j}+\left\|\vert H \vert^2+d(m)-4q\right\|_{\infty},
\end{align*}
where $u_{j}$ are $L^{2}-$ normalized eigenfunctions and where
\begin{equation*}
d(m)=
\begin{cases}
m^{2}, &\text{if $\overline{M}=\mathbb{S}^{n}$}\\
2m(m+1), &\text{if $\overline{M}= \mathbb{R}P^{n}$}\\
2m(m+2), &\text{if $\overline{M}= \mathbb{C}P^{n}$}\\
2m(m+4), &\text{if $\overline{M}= \mathbb{Q}P^{n}$}.\\
\end{cases}
\end{equation*}
\end{corollary}
\begin{remark} \label{remRSS}
\begin{enumerate}
\item If we apply this Corollary to a bounded domain of $\mathbb{S}^{n}$ or $\mathbb{C}P^{n}$ and to complex Submanifolds of $\mathbb{C}P^{n}$, then we obtain Theorem 1.1, Theorem 1.2 and Theorem 1.3 of Sun, Cheng and Yang \cite{ChengYang3}.
\item When $M$ is without boundary and $q=0$, this gives as in corollary \ref{cor Reilly general}, the following generalized Reilly's inequality for compact Submanifolds of compact rank one symmetric spaces (with the exception of the Cayley projective space),
\begin{equation*}
\displaystyle \sum_{k=1}^m \lambda_{k+1}\leq \frac{1}{Vol(M)}\int_{M}\left(|H|^2+d(m)\right).
\end{equation*}
\end{enumerate}
\end{remark}
\section{Generalization of the Levitin-Parnovski inequality to the Kohn Laplacian on the Heisenberg group}
We first recall that the $2n+1$-dimensional Heisenberg group $\mathbb{H}^{n}$ is
the space $\mathbb{R}^{2n+1}$ equipped with the non-commutative group law
$$ (x,y,t)(x',y',t')=\left(x+x',y+y',t+t'+\frac{1}{2}\right)
(\left\langle x',y\right\rangle_{\mathbb{R}^{n}}-\left\langle x,y'\right\rangle_{\mathbb{R}^{n}}),$$
where $x,x',y,y'\in \mathbb{R}^{n},\; t \;\rm{and}\; t' \in \mathbb{R}$.
The following vector fields
$$\left\{ T=\frac{\partial}{\partial t},\ X_{i}=\frac{\partial}{\partial x_{i}}+\frac{y_{i}}{2}\frac{\partial}{\partial t},\ Y_{i}=\frac{\partial}{\partial y_{i}}-\frac{x_{i}}{2}\frac{\partial}{\partial t}\ ; \ {i \leq n}\right\}$$
form a basis of the Lie algebra of $\mathbb{H}^{n}$, denoted by $\mathcal{H}^{n}$. We notice that the only non--trivial commutators are $\left[X_{i},Y_{j}\right]= - T \delta_{ij},\; i,j=1, \cdots ,n$.
Let $\Delta_{\mathbb{H}^{n}}$ denote the real Kohn Laplacian (or the sublaplacian associated with the basis
$\left\{X_{1},\cdots,X_{n},Y_{1},\cdots,Y_{n}\right\}$):
\begin{align*}
\Delta_{\mathbb{H}^{n}}&= \sum_{i=1}^{n} X_{i}^{2}+Y_{i}^{2}\\
&= \Delta^{\mathbb{R}^{2n}}_{xy}+\frac{1}{4}(|x|^{2}+|y|^{2})\frac{\partial^{2}}{{\partial t}^{2}}+ \frac{\partial}{\partial t}\sum_{i=1}^{n}\left(y_{i}\frac{\partial}{\partial x_{i}}-x_{i}\frac{\partial}{\partial y_{i}}\right).\end{align*}
We consider the following eigenvalue problem :
\[\begin{cases}
-\Delta_{\mathbb{H}^{n}} {u} = \lambda {u}\,\,\,\, \hbox{in}\ \Omega \\
{u}=0 \,\,\,\ \ \hbox{on} \ \partial\Omega,
\end{cases}\]
\par\noindent
where $\Omega$ is a bounded domain of the Heisenberg group $\mathbb{H}^{n}$, with smooth boundary.
It is known that the Dirichlet problem (5.1) has a discrete spectrum.
In what follows, we let
$$ 0 < \lambda_{1} \leq \lambda_{2} \leq \cdots \leq \lambda_{k} \cdots \rightarrow +\infty, $$
denote its eigenvalues and orthonormalize its eigenfunctions $u_{1},\, u_{2},\,\cdots \, \in S^{1,2}_{0}(\Omega)$
so that, $\forall i,j\ge 1$,
$$\left\langle u_{i},u_{j}\right\rangle_{L^{2}}= \int_{\Omega} u_{i} u_{j} dx \, dy \, dt= \delta_{ij}. $$
Here, $S^{1,2}(\Omega)$ denotes the Hilbert space of the functions $u \in L^{2}(\Omega)$ such that $X_{i}(u),\, Y_{i}(u) \in L^{2}(\Omega)$, and $S^{1,2}_{0}$
denotes the closure of $\mathcal{C}^{\infty}_{0}(\Omega)$ with respect to the Sobolev norm
$$ \|u\|^{2}_{S^{1,2}}= \int_{\Omega} \Big(|\nabla_{\mathbb{H}^{n}}u|^{2}+ |u|^{2}\Big) dx\,dy\,dt,$$
with $\nabla_{\mathbb{H}^{n}}u= (X_{1}(u),\cdots, X_{n}(u),Y_{1}(u),\cdots,Y_{n}(u)).$\\
The main result of this paragraph is the following
\begin{theorem}
For any $j \ge 1$,
\begin{equation}\label{Lev-Parn}
\sum_{l=1}^n \lambda_{j+l} \leq (n+2) \lambda_j.
\end{equation}
\end{theorem}
\begin{proof}
Inequality (\ref{Lev-Parn}) follows by applying Lemma \ref{levitin}, with $L=-\Delta_{\mathbb{H}^n}$ and $G=x_l$ or $G=y_l$, $l=1,\ldots,n$.\\
We obtain, as in the proof of Theorem \ref{theorem 1},
\begin{equation}\label{u}
-\frac{1}{2} \langle [[L,x_l],x_l]u_j,u_j \rangle_{L^2} \leq \frac{1}{\lambda_{j+l}-\lambda_j}\|[L,x_l]u_j\|_{L^2}^2
\end{equation}
and \begin{equation}\label{v}
-\frac{1}{2} \langle [[L,y_l],y_l]u_j,u_j \rangle_{L^2} \leq \frac{1}{\lambda_{j+l}-\lambda_j}\|[L,y_l]u_j\|_{L^2}^2.
\end{equation}
Taking the sum of (\ref{u}) and (\ref{v}) and summing on $l$ from $1$ to $n$ gives
\begin{align}
{} -\frac{1}{2} \sum_{l=1}^n (\lambda_{j+l}-\lambda_j) & \langle [[L,x_l],x_l]u_j,u_j \rangle_{L^2} -\frac{1}{2}\sum_{l=1}^n (\lambda_{j+l}-\lambda_j)\langle[[L,y_l],y_l]u_j,u_j \rangle_{L^2} \nonumber \\
{} & \leq \sum_{l=1}^n \|[L,x_l]u_j\|_{L^2}^2+\sum_{l=1}^n \|[L,y_l]u_j\|_{L^2}^2. \label{y}
\end{align}
By a straightforward calculation, we obtain
$$[L,x_l]u_j=-2 X_l(u_j) \quad {\rm and}\quad [L,y_l]u_j=-2 Y_l(u_j).$$
Hence
\begin{equation}\label{z}
\sum_{l=1}^n \|[L,x_l]u_j\|_{L^2}^2+\sum_{l=1}^n \|[L,y_l]u_j\|_{L^2}^2=4 \int_{\Omega}|\nabla_{\mathbb{H}^n}u_j|^2=4\lambda_j.
\end{equation}
Now
\begin{equation}\label{a'}
[[L,x_l],x_l]u_j=-2[X_l,x_l]u_j=-2u_j
\end{equation}
and
\begin{equation}
[[L,y_l],y_l]u_j=-2[Y_l,y_l]u_j=-2u_j.
\end{equation}
Thus
\begin{equation}\label{b'}
\langle [[L,x_l],x_l]u_j,u_j \rangle_{L^2}=-2 \int_{\Omega}u_j^2=-2
\end{equation}
and
\begin{equation}\label{c'}
\langle [[L,y_l],y_l]u_j,u_j \rangle_{L^2}=-2 \int_{\Omega}u_j^2=-2.
\end{equation}
Finally, putting identities (\ref{z}), (\ref{b'}) and (\ref{c'}) in (\ref{y}), we obtain inequality (\ref{Lev-Parn}).
\end{proof}
\subsection*{Acknowledgments}
This work was partially supported by the ANR(Agence Nationale de la Recherche) through FOG project(ANR-07-BLAN-0251-01). We also wish to
thank the referee for his valuable suggestions which helped us in improving the first presentation of this article.
|
1,314,259,993,623 | arxiv | \section{Introduction}
\label{sec.Intro}
This paper is concerned with the space-time fractional stochastic partial differential equation (SPDE, for short) driven by the fractionally integrated additive noise of the form
\begin{equation} \label{eq.model}
\left\{
\begin{aligned}
& \partial_t^{\alpha} X(t) + (-\Delta)^{\beta} X(t) = F(X(t)) + \mathcal I {_t^{\gamma}} \dot{W}(t), \qquad \forall\, t \in (0,T], \\
& X(0) = X_0,
\end{aligned}
\right.
\end{equation}
which is also known as the stochastic fractional subdiffusion problem; see e.g., \cite{Kang2021IMA, Jin2019ESAIM}. Here,
\begin{itemize}
\item $\partial_t^{\alpha}$ denotes the Caputo fractional derivative of order $\alpha \in (0,1]$, and $\mathcal I {_t^{\gamma}}$ denotes the Riemann--Liouville fractional integral of order $\gamma \in [0,1]$. Both are with respect to time $t$; see e.g., \cite{Kilbas2006Book, Podlubny1999} for the specific definitions.
\item $-\Delta =: A$ is the negative Laplacian with a zero Dirichlet boundary condition in a convex polygonal domain $\mathbb D \subset \mathbb R^{d}$ ($d=1,2,3$), with its domain $\mathcal{D}(A) = H^2(\mathbb D) \cap H_0^1(\mathbb D)$, and $(-\Delta)^{\beta}$ is the fractional Laplacian of order $\beta \in (0,1]$; $\{W(t)\}_{t\in[0,T]}$ is a Wiener process with a covariance operator $Q$ on some complete filtered probability space $(\Omega, \mathscr{F}, \{\mathscr{F}_t\}_{t\in[0,T]}, \mathbb P)$; see the end of this section for more details.
\item The initial datum $X_0$ is an $\mathscr{F}_0$-measurable random variable that takes values in $H := L^2(\mathbb D)$ or its subspace, and the deterministic mapping $F:\ H \rightarrow H$ is Borel measurable.
\end{itemize}
The deterministic case of such problems has been extensively studied due to its influential applications in anomalous diffusion models; see e.g., \cite{Meerschaert2002PRE, MetzlerKlafter2000}. In order to describe the random effects on transport of particles in medium with memory or particles subject to sticking and trapping, the noise force $\mathcal I {_t^{\gamma}} \dot{W}(t)$ is usually added and used to reflect the phenomenon that the internal energy depends also on the past random effects; see e.g., \cite{AsogwaNane2017, Chen2015SPA, MijenaNane2015}. With the aid of the Laplace transform, the mild solution of the model \eqref{eq.model} can be formally formulated as
\begin{align} \label{eq.mildSol}
X(t) = \mathcal S_{1-\alpha}(t) X_0 + \int_0^t \mathcal S_{0}(t-s) F(X(s)) \mathrm d s + \int_0^t \mathcal S_{\gamma}(t-s) \mathrm d W(s), \qquad t \in [0,T].
\end{align}
Here, the solution operator $\mathcal S_{\eta}(u):\ H \rightarrow H$ with $\eta \in [0,1]$ and $u > 0$ is defined by $\mathcal S_{\eta}(u) = u^{\alpha+\eta-1} E_{\alpha,\alpha+\eta} (-A^{\beta} u^{\alpha})$, where $E_{\alpha,\alpha+\eta} (\cdot)$ denotes the Mittag--Leffler function; see Section \ref{sec.Prelim} for more details. When $\alpha \in (\frac{1}{2},1)$, $\beta \in (\frac{1}{2},1]$ and $\gamma \in [0,1]$, the existence and uniqueness of the mild solution \eqref{eq.mildSol} has been given in \cite{Kang2021IMA}. For the generalizability of the results obtained in this paper, the well-posedness of the model \eqref{eq.model} is investigated again in Section \ref{sec.wellpos} under the condition
\begin{align} \label{set.argu}
\alpha \in (0,1], \quad \beta \in (0,1] \quad \mbox{and} \quad \gamma \in [0,1] \quad \mbox{with} \quad \alpha + \gamma > \frac{1}{2}.
\end{align}
Although high-accuracy numerical methods have been used to solve the deterministic counterpart of the model \eqref{eq.model} (see e.g., \cite{Garrappa2013, LiMa2022SIAM, Nochetto2016SIAM}), the accuracy of these numerical methods will be destroyed when encountering the interaction between the fractional derivative and stochastic noise, which motivates the studies on numerical methods for the model \eqref{eq.model} or its special cases. When $\alpha = 1$, the model \eqref{eq.model} degenerates into a space-fractional SPDE, and the semi-implicit Euler method is modified in \cite{Liu2022} for the case $\gamma = 0$. When $\beta = 1$, the model \eqref{eq.model} becomes a time-fractional SPDE, and the Gr\"unwald--Letnikov method and the L1 method as two time-stepping methods are well studied in \cite{Jin2019ESAIM, HuLi2022ANM, WuYan2020ANM} for the linear equation with $\gamma \in [0,1]$. When $\alpha \in (\frac{1}{2},1)$, $\beta \in (\frac{1}{2},1]$ and $\gamma \in [0,1]$, the model \eqref{eq.model} is solved by the Galerkin finite element method combined with the Wong--Zakai approximation in \cite{Kang2021IMA}, which mainly focuses on the spatial discretization. Without being too exhaustive, we also refer to \cite{Gunzburger2019NM, Gunzburger2019MC, Kovacs2020SIAM, Kovacs2014MC, LiWangDeng2017} for other numerical methods of related models. To our best knowledge, however, there seems no work on the time-stepping method and the corresponding numerical analysis for the model \eqref{eq.model} with general $\alpha \in (0,1]$, $\beta \in (0,1]$ and $\gamma \in [0,1]$.
To fill this gap, inspired by the excellent performance of the exponential integrator in SPDEs, the present work focuses on a generalized exponential integrator named the Mittag--Leffler Euler integrator for the model \eqref{eq.model}, and is devoted to the corresponding strong convergence analysis under the condition \eqref{set.argu}. In this analysis, the main difficulty lies in the low temporal regularity of the mild solution \eqref{eq.mildSol}, particularly when $\alpha$ is small. For instance, when dealing with the integral term of the form $\Upsilon(t) := \int_0^t \mathcal S_0(t-u) G(u) \mathrm d u$ appeared in the error estimate, the common way is to decompose $\Upsilon(t)$ into the sum of two integrals on subintervals $[0,t_m]$ and $[t_m,t]$ with the same integrand. In this way, it turns out that the strong convergence rate of the Mittag--Leffler Euler integrator will not exceed order $\alpha$, which is unsatisfactory especially for the case of $\alpha$ being close to zero (see Remark \ref{rem.driftpr} with $G(\cdot) = F(X(\cdot))$ for more details). In terms of time-fractional deterministic partial differential equations, which relate to the case of $\beta = 1$ and $W \equiv 0$, the Mittag--Leffler Euler integrator has been proved in \cite{Garrappa2013} to be first-order convergent based on the asymptotic expansion of the true solution given by \cite{Lubich1983MC}. However, with the appearance of the stochastic noise, the asymptotic expansion of this type seems unavailable for the mild solution \eqref{eq.mildSol}. To get rid of this situation, we propose a new decomposition way for the integral $\Upsilon(t)$ in Section \ref{sec.Prelim}, which will be called the integral decomposition technique for the better presentation. Unlike the previous way, the newly proposed way is decomposing $\Upsilon(t)$ into the sum of two integrals with regard to distinct integrands on the same interval. The advantage of this integral decomposition technique will be reflected in the analysis of the temporal regularity of the mild solution \eqref{eq.mildSol}, as shown in Theorem \ref{th.Holder}. Furthermore, in order to facilitate the strong error analysis of the Mittag--Leffler Euler integrator for the model \eqref{eq.model}, some fine estimates associated with the solution operator are established in Section \ref{sec.Prelim}, which would be also potentially used in the analysis of some related models.
For the nonlinear case of the model \eqref{eq.model}, we prove that the Mittag--Leffler Euler integrator is convergent with order
\begin{align*}
\begin{cases}
\min\{ \frac{1}{2} + \frac{\alpha}{2\beta}(r+\lambda-\kappa), 1 \}, & \mbox{if } \alpha + \gamma = 1, \\
\min\{\frac{\alpha}{2\beta}\min\{\kappa,r+\lambda\} + (\gamma-\frac{1}{2})^{+}, 1-\varepsilon\}, & \mbox{if } \alpha + \gamma \neq 1
\end{cases}
\end{align*}
in the sense of $L^2(\Omega,H)$-norm. For the linear case of the model \eqref{eq.model}, the corresponding convergence order can attain $\min\{ \alpha + \frac{\alpha r}{2\beta} + (\gamma-\frac{1}{2})^{+}-\varepsilon, \alpha+\gamma -\varepsilon, 1 \}$; see Section \ref{sec.MitLefEuler} for more details. For the obtained results, we mention that the convergence order of the Mittag--Leffler Euler integrator could exceed $\alpha$, which also reveals that the proposed integral decomposition technique is powerful for this error analysis. When $\alpha = \beta = 1$ and $\gamma = 0$, the Mittag--Leffler Euler integrator degenerates into the accelerated exponential Euler integrator that has been well studied for the corresponding SPDE; see e.g., \cite{Jentzen2011AAP, Jentzen2009ProcA, WangQi2015}. It is worth emphasizing that the obtained results in this paper are consistent with those in the references for the case of $\alpha = \beta = 1$ and $\gamma = 0$.
This paper is organized as follows. Section \ref{sec.Prelim} provides some properties of the Mittag--Leffler function, some estimates associated with the solution operator, and the integral decomposition technique, which are instrumental for the theoretical and numerical analyses of the model \eqref{eq.model}. Section \ref{sec.wellpos} gives the existence, uniqueness and temporal regularity of the mild solution \eqref{eq.mildSol} under the condition \eqref{set.argu}. In Section \ref{sec.MitLefEuler}, the strong error analysis of the Mittag--Leffler Euler integrator for the model \eqref{eq.model} is reported for the nonlinear and linear cases separately.
\vskip 1em
\textbf{Notations}. For readability, we now list some notations frequently used in this paper.
\begin{itemize}
\item Let $\{(\lambda_k, \phi_k)\}_{k=1}^{\infty}$ be a sequence of eigenpairs of the negative Laplacian $A$ with $0<\lambda_1\leq \lambda_2\leq \cdots\le \lambda_k\leq \cdots$. In fact, $\{\phi_k\}_{k=1}^{\infty}$ forms an orthonormal basis of the separable Hilbert space $(H, \<\cdot, \cdot\>, \| \cdot \|)$.
\item Let $\mathcal L(H)$ be the space of bounded linear operators from $H$ to $H$ equipped with the usual operator norm $\| \cdot \|_{\mathcal L(H)}$. Denote by $\mathcal L_2(H)$ the space of Hilbert--Schmidt operators $\mathcal T:\ H \rightarrow H$ endowed with the Hilbert--Schmidt norm $\| \mathcal T \|_{\mathcal L_2(H)} := \left( \sum_{k=1}^{\infty} \| \mathcal T \phi_k \|^2 \right)^{\frac{1}{2}}$.
\item $\{W(t)\}_{t\in[0,T]}$ is given by $W(t) = \sum_{k=1}^{\infty} \sqrt{q_k} \xi_k(t) \phi_k$, where $\{(q_k, \phi_k)\}_{k=1}^{\infty}$ is a sequence of eigenpairs of the covariance operator $Q$, and $\{\xi_k(t)\}_{k=1}^{\infty}$ is a sequence of mutually independent real-valued Brownian motions.
\item Set $\dot{H}^{\varpi}(\mathbb D)$ or simply $\dot{H}^{\varpi}$ for any ${\varpi} \in \mathbb R$ as a Hilbert space induced by the norm $\| \cdot \|_{\varpi} := \left( \sum_{k=1}^{\infty} \lambda_k^{\varpi} \< \cdot, \phi_k \>^2 \right)^{\frac{1}{2}}$.
\item The fractional Laplacian $A^{\beta} := (-\Delta)^{\beta}$ of order $\beta \in (0,1]$ is defined by $A^{\beta} \varphi = \sum_{k=1}^{\infty} \lambda_k^{\beta} \<\varphi, \phi_k\>\phi_k$ for $\varphi \in \dot{H}^{2\beta}$.
\item Using $\Gamma(z) := \int_{0}^{\infty} u^{z-1} e^{-u} \mathrm d u$ with $z > 0$ denotes the Gamma function, and $B(a,b) := \int_{0}^{1} u^{a-1} (1-u)^{b-1} \mathrm d u$ with $a, \, b > 0$ denotes the Beta function.
\item For any $\varrho \in \mathbb R$, denote $\varrho^{+} = \max\{\varrho, 0\}$. Use $\varepsilon > 0$ to represent a sufficiently small constant. Denote by $C$ a generic constant and use $C(\cdot)$ if necessary to mention the parameters it depends on, whose values may vary from one place to another but are always independent of the discretization parameters $N$ and $M$ in \eqref{eq.NumSol}.
\end{itemize}
\section{Preliminaries}
\label{sec.Prelim}
\subsection{The Mittag--Leffler function}
Firstly, we introduce the definition and some useful properties of the Mittag--Leffler function, which are taken from \cite{Kilbas2006Book, Podlubny1999}. To this end, put the constants $a \in (0,1]$ and $b \in \mathbb R$.
The Mittag--Leffler function is defined by the series
\begin{align*}
E_{a, b} (z) = \sum_{k=0}^{\infty} \frac{z^k}{\Gamma(a k + b)}, \qquad \mbox{for } z \in \mathbb C,
\end{align*}
which can be seen as a generalization of the exponential function since $E_{1,1} (z) = e^{z}$ for all $z \in \mathbb C$. For any $\eta \in [0,1]$ and $\lambda > 0$, the following derivative formula holds
\begin{align} \label{eq.MLder}
\frac{\mathrm d}{\mathrm d t} [ t^{a+\eta-1} E_{a,a+\eta}(-\lambda t^{a}) ]
=
\begin{cases}
-\lambda t^{a-1} E_{a,a}(-\lambda t^{a}), & \mbox{if } a+\eta = 1, \\
t^{a+\eta-2} E_{a,a+\eta-1}(-\lambda t^{a}), & \mbox{if } a+\eta \neq 1.
\end{cases}
\end{align}
For any real number $c \in (\frac{a\pi}{2}, a\pi)$, there exists some constant $C = C(a,b,c) > 0$ such that
\begin{align} \label{upper.ML1}
| E_{a, b} (z) | &\leq C (1 + |z|)^{-1}, \qquad c \leq |\arg(z)| \leq \pi.
\end{align}
In particular, when $b = a$, this upper bound can be refined as
\begin{align} \label{upper.ML2}
| E_{a, a} (z) | &\leq C (1 + |z|)^{-2}, \qquad c \leq |\arg(z)| \leq \pi.
\end{align}
\subsection{Estimates of the solution operator}
Recall that the solution operator is defined by $\mathcal S_{\eta}(u) = u^{\alpha+\eta-1} E_{\alpha,\alpha+\eta} (-A^{\beta} u^{\alpha})$ for $\eta \in [0,1]$, $u > 0$, or equivalently,
\begin{align} \label{eq.St}
\mathcal S_{\eta}(u) \psi = u^{\alpha+\eta-1} \sum_{k=1}^{\infty} E_{\alpha,\alpha+\eta} (-\lambda_k^{\beta} u^{\alpha}) \<\psi, \phi_k\> \phi_k, \qquad \mbox{for } \psi \in H.
\end{align}
To facilitate the proof of the main results of this paper, let us prepare some useful estimates associated with the solution operator.
\begin{Lem} \label{lem.Opera}
Let $\alpha \in (0,1]$, $\beta \in (0,1]$, $\eta \in [0,1]$ and $0 < s < t$. Then there exists some positive constant $C$ independent of $t$ and $s$ such that $\mathcal S_{\eta}(\cdot)$ defined by \eqref{eq.St} has the following estimates:\
\begin{enumerate}
\item[(1)] For any $\varrho \leq 2\beta$,
\begin{align*}
\| A^{\frac{\varrho}{2}} \mathcal S_{\eta}(t) \|_{\mathcal L(H)} \leq C t^{ \alpha + \eta - 1 - \frac{\alpha}{2\beta}\varrho^{+} }.
\end{align*}
\item[(2)] When $\alpha + \eta = 1$, for any $\varrho \in [-2\beta, 2\beta]$,
\begin{align*}
\| A^{\frac{\varrho}{2}} ( \mathcal S_{1-\alpha}(t) - \mathcal S_{1-\alpha}(s) ) \|_{\mathcal L(H)} \leq C \int_s^t u^{-1-\frac{\alpha\varrho}{2\beta}} \mathrm d u.
\end{align*}
\item[(3)] When $\alpha + \eta \neq 1$, for any $\varrho \leq 2\beta$,
\begin{align*}
\| A^{\frac{\varrho}{2}} ( \mathcal S_{\eta}(t) - \mathcal S_{\eta}(s) ) \|_{\mathcal L(H)}
\leq C \int_s^t u^{\alpha+\eta-2-\frac{\alpha}{2\beta}\varrho^{+}} \mathrm d u.
\end{align*}
\end{enumerate}
\end{Lem}
\begin{proof}
(1) It can be obtained by \cite[Lemma 3.6]{Kang2021IMA} with some trivial extensions, so the details are omitted.
(2) By \eqref{eq.MLder} and \eqref{upper.ML2}, for any $\varrho \in [-2\beta, 2\beta]$, one can derive
\begin{align*}
\| A^{\frac{\varrho}{2}} ( \mathcal S_{1-\alpha}(t) - \mathcal S_{1-\alpha}(s) ) \|_{\mathcal L(H)}
&= \sup_{k \geq 1} \Big\{ \lambda_k^{\frac{\varrho}{2}} | E_{\alpha,1} (-\lambda_k^{\beta} t^{\alpha}) - E_{\alpha,1} (-\lambda_k^{\beta} s^{\alpha}) | \Big\} \\
&= \sup_{k \geq 1} \Big\{ \lambda_k^{\frac{\varrho}{2}} \Big| \int_s^t -\lambda_k^{\beta} u^{\alpha-1} E_{\alpha,\alpha} (-\lambda_k^{\beta} u^{\alpha}) \mathrm d u \Big| \Big\} \\
&\leq C \sup_{k \geq 1} \Big\{ \int_s^t u^{-1-\frac{\alpha\varrho}{2\beta}} \frac{(\lambda_k^{\beta}u^{\alpha})^{1+\frac{\varrho}{2\beta}}}{(1 + \lambda_k^{\beta}u^{\alpha})^2} \mathrm d u \Big\}
\leq C \int_s^t u^{-1-\frac{\alpha\varrho}{2\beta}} \mathrm d u.
\end{align*}
(3) When $\varrho \in [0, 2\beta]$, it follows from \eqref{eq.MLder} and \eqref{upper.ML1} that
\begin{align*}
\| A^{\frac{\varrho}{2}} ( \mathcal S_{\eta}(t) - \mathcal S_{\eta}(s) ) \|_{\mathcal L(H)}
&= \sup_{k \geq 1} \Big\{ \lambda_k^{\frac{\varrho}{2}} | t^{\alpha+\eta-1} E_{\alpha,\alpha+\eta} (-\lambda_k^{\beta} t^{\alpha}) - s^{\alpha+\eta-1} E_{\alpha,\alpha+\eta} (-\lambda_k^{\beta} s^{\alpha}) | \Big\} \\
&= \sup_{k \geq 1} \Big\{ \lambda_k^{\frac{\varrho}{2}} \Big| \int_s^t u^{\alpha+\eta-2} E_{\alpha,\alpha+\eta-1} (-\lambda_k^{\beta} u^{\alpha}) \mathrm d u \Big| \Big\} \\
&\leq C \sup_{k \geq 1} \Big\{ \int_s^t u^{\alpha+\eta-2-\frac{\alpha\varrho}{2\beta}} \frac{ (\lambda_k^{\beta} u^{\alpha})^{\frac{\varrho}{2\beta}} }{1 + \lambda_k^{\beta} u^{\alpha}} \mathrm d u \Big\}
\leq C \int_s^t u^{\alpha+\eta-2-\frac{\alpha\varrho}{2\beta}} \mathrm d u,
\end{align*}
which implies that for any $\varrho < 0$,
\begin{align*}
\| A^{\frac{\varrho}{2}} ( \mathcal S_{\eta}(t) - \mathcal S_{\eta}(s) ) \|_{\mathcal L(H)} \leq C \| \mathcal S_{\eta}(t) - \mathcal S_{\eta}(s) \|_{\mathcal L(H)} \leq C \int_s^t u^{\alpha+\eta-2} \mathrm d u.
\end{align*}
Hence, the proof is completed.
\end{proof}
\begin{Lem} \label{le.DiffOpera}
Let $\alpha \in (0,1]$, $\beta \in (0,1]$, $\gamma \in [0,1]$, $\alpha + \gamma > \frac{1}{2}$, $\kappa = \min\{ (\alpha+\gamma-\frac{1}{2})\frac{2\beta}{\alpha}, 2\beta \}$, $\varrho \in (-\kappa, \kappa)$ and $0 < s < t \leq T$. Then there exists some positive constant $C$ independent of $t$ and $s$ such that $\mathcal S_{\gamma}(\cdot)$ defined by \eqref{eq.St} has the following estimates:\
\begin{enumerate}
\item[(1)] When $\alpha + \gamma = 1$,
\begin{align*}
\int_0^s \| A^{\frac{\varrho}{2}} ( \mathcal S_{1-\alpha}(t-u) - \mathcal S_{1-\alpha}(s-u) ) \|_{\mathcal L(H)}^2 \mathrm d u
\leq C (t-s)^{ 1 - \frac{\alpha\varrho}{\beta} }.
\end{align*}
\item[(2)] When $\alpha + \gamma \neq 1$ and $\varrho = (\alpha+\gamma-1)\frac{2\beta}{\alpha}$,
\begin{align*}
\int_0^s \| A^{\frac{\varrho}{2}} ( \mathcal S_{\gamma}(t-u) - \mathcal S_{\gamma}(s-u) ) \|_{\mathcal L(H)}^2 \mathrm d u
\leq C (t-s)^{\min\{ 2(\alpha+\gamma-\frac{1}{2}), 1 \}}.
\end{align*}
\item[(3)] When $\alpha + \gamma \neq 1$ and $\varrho \neq (\alpha+\gamma-1)\frac{2\beta}{\alpha}$,
\begin{align*}
&\quad\ \int_0^s \| A^{\frac{\varrho}{2}} ( \mathcal S_{\gamma}(t-u) - \mathcal S_{\gamma}(s-u) ) \|_{\mathcal L(H)}^2 \mathrm d u \\
&\leq
\begin{cases}
C (t-s)^{2-\varepsilon}, & \mbox{if } \varrho \geq 0 \mbox{ and } \varrho = (\alpha+\gamma-\frac{3}{2})\frac{2\beta}{\alpha}, \\
C (t-s)^{2\min\{\alpha+\gamma-\frac{1}{2}-\frac{\alpha\varrho}{2\beta}, 1\}}, & \mbox{if } \varrho \geq 0 \mbox{ and } \varrho \neq (\alpha+\gamma-\frac{3}{2})\frac{2\beta}{\alpha}, \\
C (t-s)^{2-\varepsilon}, & \mbox{if } \varrho < 0 \mbox{ and } \alpha+\gamma = \frac{3}{2}, \\
C (t-s)^{2\min\{\alpha+\gamma-\frac{1}{2}, 1\}}, & \mbox{if } \varrho < 0 \mbox{ and } \alpha+\gamma \neq \frac{3}{2}.
\end{cases}
\end{align*}
\end{enumerate}
\end{Lem}
\begin{proof}
To facilitate the proof, we firstly give the estimate
\begin{align} \label{inequ.minus1squre}
\int_0^s \Big| \int_{s}^{t} (v-u)^{-1} \mathrm d v \Big|^2 \mathrm d u
&\leq \int_0^s \int_{s}^{t} (v-u)^{-\frac{1}{2}} \mathrm d v \int_{s}^{t} (v-u)^{-\frac{3}{2}} \mathrm d v \mathrm d u \nonumber\\
&\leq C(t-s)^{\frac{1}{2}} \int_0^s | (t-u)^{-\frac{1}{2}} - (s-u)^{-\frac{1}{2}} | \mathrm d u
\leq C(t-s).
\end{align}
(1) Applying Lemma \ref{lem.Opera}(2) and the change of variables shows
\begin{align*}
&\quad\ \int_0^s \| A^{\frac{\varrho}{2}} ( \mathcal S_{1-\alpha}(t-u) - \mathcal S_{1-\alpha}(s-u) ) \|_{\mathcal L(H)}^2 \mathrm d u \\
&\leq C \int_0^s \Big| \int_{s-u}^{t-u} v^{-1-\frac{\alpha\varrho}{2\beta}} \mathrm d v \Big|^2 \mathrm d u
= C \int_0^s \Big| \int_{s}^{t} (v-u)^{-1-\frac{\alpha\varrho}{2\beta}} \mathrm d v \Big|^2 \mathrm d u.
\end{align*}
Thus, when $\varrho = 0$, \eqref{inequ.minus1squre} implies the desired result; while $\varrho \in (-\kappa,0) \cup (0,\kappa)$, it follows from \cite[Lemma 3.2]{DaiXiaoBu2021} that
\begin{align*}
\int_0^s \| A^{\frac{\varrho}{2}} ( \mathcal S_{1-\alpha}(t-u) - \mathcal S_{1-\alpha}(s-u) ) \|_{\mathcal L(H)}^2 \mathrm d u
\leq C \int_0^s \Big( (t-u)^{-\frac{\alpha\varrho}{2\beta}} - (s-u)^{-\frac{\alpha\varrho}{2\beta}} \Big)^2 \mathrm d u
\leq C (t-s)^{1 - \frac{\alpha\varrho}{\beta}},
\end{align*}
where the last step also used the fact $-\frac{\alpha\varrho}{2\beta} \in (-\frac{1}{2},\frac{1}{2})$. Indeed, it follows from $\alpha + \gamma = 1$ and $\kappa = \min\{ (\alpha+\gamma-\frac{1}{2})\frac{2\beta}{\alpha}, 2\beta \}$ that $-\frac{\alpha\varrho}{2\beta} \in (-\frac{1}{2},\frac{1}{2})$, since
\begin{itemize}
\item if $\alpha \in (0, \frac{1}{2}]$, then $\kappa = 2\beta$ and $\varrho \in (-2\beta, 2\beta)$;
\item if $\alpha \in (\frac{1}{2}, 1]$, then $\kappa = \frac{\beta}{\alpha}$ and $\varrho \in (-\frac{\beta}{\alpha}, \frac{\beta}{\alpha})$.
\end{itemize}
(2) When $\alpha + \gamma \in (\frac{1}{2}, 1)$, using Lemma \ref{lem.Opera}(3) and \cite[Lemma 3.2]{DaiXiaoBu2021} reads
\begin{align*}
\int_0^s \| A^{\frac{\varrho}{2}} ( \mathcal S_{\gamma}(t-u) - \mathcal S_{\gamma}(s-u) ) \|_{\mathcal L(H)}^2 \mathrm d u
&\leq C\int_0^s \big( (t-u)^{\alpha+\gamma-1} - (s-u)^{\alpha+\gamma-1} \big)^2 \mathrm d u
\leq C (t-s)^{2(\alpha+\gamma-\frac{1}{2})}.
\end{align*}
While $\alpha + \gamma \in (1, 2)$, using Lemma \ref{lem.Opera}(3) with $\varrho = (\alpha+\gamma-1)\frac{2\beta}{\alpha} > 0$ as well as \eqref{inequ.minus1squre} reveals
\begin{align*}
\int_0^s \| A^{\frac{\varrho}{2}} ( \mathcal S_{\gamma}(t-u) - \mathcal S_{\gamma}(s-u) ) \|_{\mathcal L(H)}^2 \mathrm d u
\leq C \int_0^s \Big| \int_{s}^{t} (v-u)^{-1} \mathrm d v \ \Big|^2 \mathrm d u
\leq C(t-s).
\end{align*}
(3) It follows from Lemma \ref{lem.Opera}(3) and \cite[Lemma 3.2]{DaiXiaoBu2021} that
\begin{align*}
&\quad\ \int_0^s \| A^{\frac{\varrho}{2}} ( \mathcal S_{\gamma}(t-u) - \mathcal S_{\gamma}(s-u) ) \|_{\mathcal L(H)}^2 \mathrm d u \\
&\leq C \int_0^s \left( (t-u)^{\alpha+\gamma-1-\frac{\alpha}{2\beta}\varrho^{+}} - (s-u)^{\alpha+\gamma-1-\frac{\alpha}{2\beta}\varrho^{+}} \right)^2 \mathrm d u \\
&\leq
\begin{cases}
C (t-s)^{2-\varepsilon}, & \mbox{if } \varrho \geq 0 \mbox{ and } \varrho = (\alpha+\gamma-\frac{3}{2})\frac{2\beta}{\alpha}, \\
C (t-s)^{2\min\{\alpha+\gamma-\frac{1}{2}-\frac{\alpha\varrho}{2\beta}, 1\}}, & \mbox{if } \varrho \geq 0 \mbox{ and } \varrho \neq (\alpha+\gamma-\frac{3}{2})\frac{2\beta}{\alpha}, \\
C (t-s)^{2-\varepsilon}, & \mbox{if } \varrho < 0 \mbox{ and } \alpha+\gamma = \frac{3}{2}, \\
C (t-s)^{2\min\{\alpha+\gamma-\frac{1}{2}, 1\}}, & \mbox{if } \varrho < 0 \mbox{ and } \alpha+\gamma \neq \frac{3}{2}.
\end{cases}
\end{align*}
The proof is completed.
\end{proof}
In order to guarantee the existence and uniqueness of the mild solution \eqref{eq.mildSol}, we make the following assumption for the fractionally integrated noise of the model \eqref{eq.model}, which is similar to that of \cite{Jin2019ESAIM, Kang2021IMA}.
\begin{Ass} \label{ass.Noise}
Let $\alpha \in (0,1]$, $\beta \in (0,1]$, $\gamma \in [0,1]$, $\alpha + \gamma > \frac{1}{2}$, and $\kappa = \min\{ (\alpha+\gamma-\frac{1}{2})\frac{2\beta}{\alpha}, 2\beta \}$. Assume that the negative Laplacian $A$ and the covariance operator $Q$ satisfy
\begin{align*}
\| A^{\frac{ r-\kappa}{2}} Q^{\frac{1}{2}} \|_{\mathcal L_2(H)} < \infty \qquad \mbox{for some }\ r \in (0,\kappa].
\end{align*}
\end{Ass}
In virtue of Lemmas \ref{lem.Opera} and \ref{le.DiffOpera}, we can further obtain some priori estimates of the diffusion term of the mild solution \eqref{eq.mildSol}.
\begin{Lem} \label{lem.DiffOpera}
Suppose that Assumption \ref{ass.Noise} holds, and put $\varrho \in (r-2\kappa, r)$, $p \geq 2$, $0 < s < t \leq T$. Then there exists some positive constant $C$ independent of $t$ and $s$ such that
\begin{align*}
\Big\| \int_{s}^{t} \mathcal S_{\gamma}(t-u) \mathrm d W(u) \Big\|_{L^p(\Omega,\dot{H}^{\varrho})}
&\leq C (t-s)^{ \alpha + \gamma - \frac{1}{2} - \frac{\alpha}{2\beta} (\kappa- r+\varrho)^{+} }, \\
\Big\| \int_{0}^{s} \mathcal S_{\gamma}(t-u) - \mathcal S_{\gamma}(s-u) \mathrm d W(u) \Big\|_{L^p(\Omega,\dot{H}^{\varrho})}
& \leq
\begin{cases}
C (t-s)^{\frac{1}{2} - \frac{\alpha}{2\beta}(\kappa- r + \varrho) }, & \mbox{if } \alpha + \gamma = 1, \\
C (t-s)^{\min\{\alpha+\gamma-\frac{1}{2}-\frac{\alpha}{2\beta} (\kappa- r+\varrho)^{+}, 1-\varepsilon\}}, & \mbox{if } \alpha + \gamma \neq 1.
\end{cases}
\end{align*}
\end{Lem}
\begin{proof}
It follows from $\varrho \in (r-2\kappa, r)$ that $\kappa- r + \varrho \in (-\kappa,\kappa)$. Then, using the Burkholder--Davis--Gundy inequality, Assumption \ref{ass.Noise} and Lemma \ref{lem.Opera}(1) indicates
\begin{align*}
&\quad\ \Big\| \int_{s}^{t} \mathcal S_{\gamma}(t-u) \mathrm d W(u) \Big\|_{L^p(\Omega,\dot{H}^{\varrho})}
\leq C \Big( \int_s^t \| A^{\frac{\varrho}{2}} \mathcal S_{\gamma}(t-u) Q^{\frac{1}{2}} \|_{\mathcal L_2(H)}^2 \mathrm d u \Big)^{\frac{1}{2}} \nonumber\\
&\leq C \| A^{\frac{ r-\kappa}{2}} Q^{\frac{1}{2}} \|_{\mathcal L_2(H)} \Big( \int_s^t \| A^{\frac{\kappa- r + \varrho}{2}} \mathcal S_{\gamma}(t-u) \|_{\mathcal L(H)}^2 \mathrm d u \Big)^{\frac{1}{2}}
\leq C (t-s)^{ \alpha + \gamma - \frac{1}{2} - \frac{\alpha}{2\beta} (\kappa- r+\varrho)^{+} }.
\end{align*}
Utilizing the Burkholder--Davis--Gundy inequality, Assumption \ref{ass.Noise} and Lemma \ref{le.DiffOpera} reveals
\begin{align*}
&\quad\ \Big\| \int_{0}^{s} \mathcal S_{\gamma}(t-u) - \mathcal S_{\gamma}(s-u) \mathrm d W(u) \Big\|_{L^p(\Omega,\dot{H}^{\varrho})} \\
&\leq C \Big( \int_0^s \| A^{\frac{\varrho}{2}} ( \mathcal S_{\gamma}(t-u) - \mathcal S_{\gamma}(s-u) ) Q^{\frac{1}{2}} \|_{\mathcal L_2(H)}^2 \mathrm d u \Big)^{\frac{1}{2}} \nonumber\\
&\leq C \| A^{\frac{ r-\kappa}{2}} Q^{\frac{1}{2}} \|_{\mathcal L_2(H)} \Big( \int_0^s \| A^{\frac{\kappa- r + \varrho}{2}} ( \mathcal S_{\gamma}(t-u) - \mathcal S_{\gamma}(s-u) ) \|_{\mathcal L(H)}^2 \mathrm d u \Big)^{\frac{1}{2}} \nonumber\\
&\leq
\begin{cases}
C (t-s)^{\frac{1}{2} - \frac{\alpha}{2\beta}(\kappa- r + \varrho) }, & \mbox{if } \alpha + \gamma = 1, \\
C (t-s)^{\min\{\alpha+\gamma-\frac{1}{2}-\frac{\alpha}{2\beta} (\kappa- r+\varrho)^{+}, 1-\varepsilon\}}, & \mbox{if } \alpha + \gamma \neq 1.
\end{cases}
\end{align*}
The proof is completed.
\end{proof}
\subsection{The integral decomposition technique}
In this part, we are devoted to developing the integral decomposition technique, which is of vital importance to the temporal regularity analysis of the mild solution \eqref{eq.mildSol} as well as the error analysis of the Mittag--Leffler Euler integrator \eqref{eq.NumSol}; see Theorems \ref{th.Holder} and \ref{th.MLEulerErrorNonlinear} for more details.
Let $t \in (0,T]$ and the operator $\mathcal S_0(t)$ be defined by \eqref{eq.St}. Fix $p \geq 2$ and $\varpi \in \mathbb R$. For a bounded measurable mapping $G:\ [0,T] \rightarrow L^p(\Omega,\dot{H}^{\varpi})$, we are interested in the following integral
\begin{align} \label{eq.S0t}
\Upsilon(t) := \int_0^t \mathcal S_0(t-u) G(u) \mathrm d u, \qquad t \in (0,T].
\end{align}
Note that the integral $\Upsilon(\cdot)$ corresponds to the drift term of the mild solution \eqref{eq.mildSol} when taking $G(\cdot) = F(X(\cdot))$.
\begin{Theo} \label{th.integraldecompos}
Let the integral $\Upsilon(\cdot)$ be defined by \eqref{eq.S0t}. Then one has the decomposition
\begin{align*}
\Upsilon(\cdot) = \Phi(\cdot) + \Psi(\cdot)
\end{align*}
with $\Phi(\cdot)$, $\Psi(\cdot)$ and $\Upsilon(\cdot)$ satisfying the following statements$:$
\begin{enumerate}
\item[(1)] For all $0 < s < t \leq T$,
\begin{align*}
\| \Phi(t) - \Phi(s) \|_{L^p(\Omega,\dot{H}^{\varpi})} \leq C ( t^{\alpha} - s^{\alpha} ).
\end{align*}
\item[(2)] If there exist constants $K > 0$, $\mu < 1$ and $\nu \in (-\alpha,1)$ such that for all $0 < u_1 < u_2 \leq T$,
\begin{align*}
\|G(u_2) - G(u_1)\|_{L^p(\Omega,\dot{H}^{\varpi})} \leq K u_1^{-\mu} ( u_2-u_1 )^{\nu},
\end{align*}
then for all $0 < s < t \leq T$,
\begin{align*}
\| \Psi(t) - \Psi(s) \|_{L^p(\Omega,\dot{H}^{\varpi})}
&\leq C ( t^{\alpha} - s^{\alpha} ) + C s^{-\mu} (t-s)^{ \min\{\alpha+\nu-\varepsilon, 1\} }, \\
\| \Upsilon(t) - \Upsilon(s) \|_{L^p(\Omega,\dot{H}^{\varpi})}
&\leq C ( t^{\alpha} - s^{\alpha} ) + C s^{-\mu} (t-s)^{ \min\{\alpha+\nu-\varepsilon, 1\} }.
\end{align*}
\end{enumerate}
Here, $\alpha \in (0,1]$ is associated with the operator $\mathcal S_0(\cdot)$ $($i.e., \eqref{eq.St} with $\eta = 0$$)$, and all the constants $C>0$ are independent of $t$ and $s$.
\end{Theo}
\begin{proof}
First of all, one can make the following decomposition
\begin{align*}
\Upsilon(t) &= \underbrace{\int_0^t \mathcal S_0(t-u) G(0) \mathrm d u}_{=:\, \Phi(t)} + \underbrace{\int_0^t \mathcal S_0(t-u) \big( G(u) - G(0) \big) \mathrm d u}_{=:\, \Psi(t)}, \qquad \forall \, t \in (0,T].
\end{align*}
Next, we prove the claims (1) and $(2)$ separately.
(1) It follows from \eqref{eq.MLder} that
\begin{align*}
\Phi(t) - \Phi(s)
= \int_0^t \mathcal S_0(t-u) G(0) \mathrm d u - \int_0^s \mathcal S_0(s-u) G(0) \mathrm d u
= \big( \mathcal S_1(t) - \mathcal S_1(s) \big) G(0),
\end{align*}
which together with Lemma \ref{lem.Opera}(3) indicates
\begin{align*}
\| \Phi(t) - \Phi(s) \|_{L^p(\Omega,\dot{H}^{\varpi})}
\leq \| \mathcal S_1(t) - \mathcal S_1(s) \|_{\mathcal L(H)} \| G(0) \|_{L^p(\Omega,\dot{H}^{\varpi})}
\leq C (t^{\alpha} - s^{\alpha}).
\end{align*}
(2) For convenience, denote $\tau := t - s > 0$. Then, by the change of variables,
\begin{align*}
\Psi(t) - \Psi(s)
&= \int_{-\tau}^{s} \mathcal S_0(u+\tau) \big( G(s-u) - G(0) \big) \mathrm d u - \int_{0}^{s} \mathcal S_0(u) \big( G(s-u) - G(0) \big) \mathrm d u \\
&= \underbrace{\int_{-\tau}^{s} \mathcal S_0(u+\tau) \big( G(s) - G(0) \big) \mathrm d u - \int_{0}^{s} \mathcal S_0(u) \big( G(s) - G(0) \big) \mathrm d u}_{=:\, \Psi_1} \\
&\quad + \underbrace{\int_{-\tau}^{0} \mathcal S_0(u+\tau) \big( G(s-u) - G(s) \big) \mathrm d u}_{=:\, \Psi_2} + \underbrace{\int_{0}^{s} \big( \mathcal S_0(u+\tau) - \mathcal S_0(u) \big) \big( G(s-u) - G(s) \big) \mathrm d u}_{=:\, \Psi_3},
\end{align*}
which implies
\begin{align*}
\| \Psi(t) - \Psi(s) \|_{L^p(\Omega,\dot{H}^{\varpi})} \leq \| \Psi_1 \|_{L^p(\Omega,\dot{H}^{\varpi})} + \| \Psi_2 \|_{L^p(\Omega,\dot{H}^{\varpi})} + \| \Psi_3 \|_{L^p(\Omega,\dot{H}^{\varpi})}.
\end{align*}
It follows from the fact $\Psi_1 = \big( \mathcal S_1(t) - \mathcal S_1(s) \big) \big( G(s) - G(0) \big)$ and Lemma \ref{lem.Opera}(3) that
\begin{align*}
\| \Psi_1 \|_{L^p(\Omega,\dot{H}^{\varpi})}
\leq \| \mathcal S_1(t) - \mathcal S_1(s) \|_{\mathcal L(H)} \| G(s) - G(0) \|_{L^p(\Omega,\dot{H}^{\varpi})} \leq C (t^{\alpha} - s^{\alpha}).
\end{align*}
Using Lemma \ref{lem.Opera}(3) and the assumption $\|G(u_1) - G(u_2)\|_{L^p(\Omega,\dot{H}^{\varpi})} \leq K u_1^{-\mu}(u_2-u_1 )^{\nu}$ with $0<u_1<u_2 \leq T$ as well as the Beta function shows
\begin{align*}
\| \Psi_2 \|_{L^p(\Omega,\dot{H}^{\varpi})}
&\leq \int_{-\tau}^{0} \| \mathcal S_0(u+\tau) \|_{\mathcal L(H)} \| G(s-u) - G(s) \|_{L^p(\Omega,\dot{H}^{\varpi})} \mathrm d u \\
&\leq C \int_{-\tau}^{0} (u+\tau)^{\alpha-1} s^{-\mu} (-u)^{\nu} \mathrm d u
\leq C s^{-\mu} \tau^{\alpha+\nu}.
\end{align*}
And, by the estimate $\int_{u}^{u+\tau} \omega^{\alpha-2} \mathrm d \omega \leq C u^{\varepsilon-1-\nu} \tau^{ \min\{\alpha+\nu-\varepsilon, 1\} }$ with $u \in (0,T]$ and $\nu \in (-\alpha,1)$,
\begin{align*}
\| \Psi_3 \|_{L^p(\Omega,\dot{H}^{\varpi})}
&\leq \int_{0}^{s} \| \mathcal S_0(u+\tau) - \mathcal S_0(u) \|_{\mathcal L(H)} \| G(s-u) - G(s) \|_{L^p(\Omega,\dot{H}^{\varpi})} \mathrm d u \\
&\leq C \int_{0}^{s} \int_{u}^{u+\tau} \omega^{\alpha-2} \mathrm d \omega \, (s-u)^{-\mu} u^{\nu} \mathrm d u
\leq C s^{\varepsilon-\mu} \tau^{ \min\{\alpha+\nu-\varepsilon, 1\} }.
\end{align*}
Therefore, by collecting these estimates at hand,
\begin{align*}
\| \Psi(t) - \Psi(s) \|_{L^p(\Omega,\dot{H}^{\varpi})}
\leq C ( t^{\alpha} - s^{\alpha} ) + C s^{-\mu} (t-s)^{ \min\{\alpha+\nu-\varepsilon, 1\} }.
\end{align*}
Finally, recalling the claim (1) and the triangle inequality completes the proof.
\end{proof}
\section{Well-posedness and regularity}
\label{sec.wellpos}
\subsection{Existence and uniqueness of the mild solution}
The following assumption is common in the theoretical analysis of semilinear fractional SPDEs; see e.g., \cite{Kang2021IMA, Kovacs2020SIAM}.
\begin{Ass} \label{ass.Lip1}
There exists some positive constant $L$ such that $F:\ H \rightarrow H$ satisfies
\begin{align*}
\| F(\phi) \| \leq L( 1 + \| \phi \|) \quad \mbox{and} \quad \| F'(\phi)\psi \| \leq L\|\psi\|, \qquad \forall\, \phi, \psi \in H.
\end{align*}
\end{Ass}
\begin{Theo} \label{th.Wellpos}
Let $p \geq 2$ and $X_0 \in L^p(\Omega,H)$. Under Assumptions \ref{ass.Noise} and \ref{ass.Lip1}, there exists a unique mild solution $X$ given by \eqref{eq.mildSol} to the model \eqref{eq.model}. Moreover, for any $q \in [0, r)$, if $X_0 \in L^p(\Omega,\dot{H}^{q})$, then $X \in \mathcal C([0,T], L^p(\Omega,\dot{H}^{q}))$.
\end{Theo}
\begin{proof}
For any $X \in \mathcal C([0,T], L^p(\Omega,H))$, define the operator $\mathcal T$ by
\begin{align*}
\mathcal T X(t) := \mathcal S_{1-\alpha}(t) X(0) + \int_0^t \mathcal S_{0}(t-s) F(X(s)) \mathrm d s + \int_0^t \mathcal S_{\gamma}(t-s) \mathrm d W(s).
\end{align*}
Similar to the first estimate of Lemma \ref{lem.DiffOpera}, one can derive
\begin{align} \label{eq.diffUpper}
\Big\| \int_{0}^{t} \mathcal S_{\gamma}(t-s) \mathrm d W(s) \Big\|_{L^p(\Omega,\dot{H}^{q})}
\leq C t^{ \alpha + \gamma - \frac{1}{2} - \frac{\alpha}{2\beta} (\kappa- r+q)^{+} }
\leq C.
\end{align}
\textbf{Claim 1:} $\mathcal T X \in \mathcal C([0,T], L^p(\Omega,H))$ for any $X \in \mathcal C([0,T], L^p(\Omega,H))$.
Let $X \in \mathcal C([0,T], L^p(\Omega,H))$ and $t \in [0,T]$ be arbitrary. It follows from the triangle inequality, \eqref{eq.diffUpper} with $q=0$, Lemma \ref{lem.Opera}(1) and Assumption \ref{ass.Lip1} that
\begin{align*}
&\quad\ \| \mathcal T X(t) \|_{L^p(\Omega,H)} \\
&\leq \| \mathcal S_{1-\alpha}(t) X(0) \|_{L^p(\Omega,H)} + \int_0^t \| \mathcal S_{0}(t-s) F(X(s)) \|_{L^p(\Omega,H)} \mathrm d s + \Big\| \int_{0}^{t} \mathcal S_{\gamma}(t-s) \mathrm d W(s) \Big\|_{L^p(\Omega,H)} \\
&\leq \| \mathcal S_{1-\alpha}(t) \|_{\mathcal L(H)} \| X(0) \|_{L^p(\Omega,H)} + \int_0^t \| \mathcal S_{0}(t-s) \|_{\mathcal L(H)} \| F(X(s)) \|_{L^p(\Omega,H)} \mathrm d s + C \\
&\leq C + C \int_0^t (t-s)^{\alpha-1} \mathrm d s \, \Big( 1 + \| X \|_{\mathcal C([0,T], L^p(\Omega,H))} \Big)
\leq C,
\end{align*}
which implies $\mathcal T X \in \mathcal C([0,T], L^p(\Omega,H))$.
Let $\varsigma$ be a positive constant, whose value will be determined later. Set
\begin{align*}
\mathcal C([0,T], L^p(\Omega,H))_{\varsigma} := \left\{ \phi \in \mathcal C([0,T], L^p(\Omega,H)) \Big|\ \| \phi \|_{\varsigma,p} := \sup_{t\in[0,T]} \left( \frac{\mathbb E \| \phi (t) \|^p}{E_{\alpha,1}(\varsigma t^{\alpha})} \right)^{1/p} < \infty \right\}.
\end{align*}
Note that it is a Banach space, since the weighted norm $\| \cdot \|_{\varsigma,p}$ is equivalent to the standard norm of $\mathcal C([0,T], L^p(\Omega,H))$ for the fixed $\varsigma > 0$ and $p \geq 2$.
\textbf{Claim 2:} There exists a positive constant $\varsigma$ such that $\| \mathcal T X_1 - \mathcal T X_2 \|_{\varsigma,p} \leq \varpi \| X_1 - X_2 \|_{\varsigma,p}$ with $\varpi \in (0,1)$, for any $X_1, X_2 \in \mathcal C([0,T], L^p(\Omega,H))_{\varsigma}$.
Let $X_1, X_2 \in \mathcal C([0,T], L^p(\Omega,H))_{\varsigma}$ be arbitrary. According to Lemma \ref{lem.Opera}(1), Assumption \ref{ass.Lip1} and H\"older's inequality, one can read
\begin{align*}
\mathbb E \| \mathcal T X_1(t) - \mathcal T X_2(t) \|^p
&\leq \Big( \int_0^t \| \mathcal S_{0}(t-s) \|_{\mathcal L(H)} \| F(X_1(s)) - F(X_2(s)) \|_{L^p(\Omega,H)} \mathrm d s \Big)^p \\
&\leq C \Big( \int_0^t (t-s)^{\alpha-1} \| X_1(s) - X_2(s) \|_{L^p(\Omega,H)} \mathrm d s \Big)^p \\
&\leq C_0 \int_0^t (t-s)^{\alpha-1} \| X_1(s) - X_2(s) \|_{L^p(\Omega,H)}^p \mathrm d s.
\end{align*}
Then, it follows from \cite[Lemma 5]{SonHuong2018} that
\begin{align*}
\frac{\mathbb E \| \mathcal T X_1(t) - \mathcal T X_2(t) \|^p}{E_{\alpha,1}(\varsigma t^{\alpha})}
\leq \frac{C_0}{E_{\alpha,1}(\varsigma t^{\alpha})} \int_0^t (t-s)^{\alpha-1} E_{\alpha,1}(\varsigma s^{\alpha}) \mathrm d s \| X_1 - X_2 \|_{\varsigma,p}^p
\leq \frac{C_0 \Gamma(\alpha)}{\varsigma} \| X_1 - X_2 \|_{\varsigma,p}^p.
\end{align*}
Since the positive constant $C_0$ is independent of $\varsigma$, one can take sufficiently large $\varsigma$ such that $\varpi := C_0 \Gamma(\alpha) \varsigma^{-1} < 1$, which implies Claim 2. With Claims $1$ and $2$ at hand, the Banach contraction mapping theorem indicates that there exists a unique mild solution given by \eqref{eq.mildSol} to the model \eqref{eq.model}.
The remaining proof is to show $X \in \mathcal C([0,T], L^p(\Omega,\dot{H}^{q}))$ under the assumption $X_0 \in L^p(\Omega,\dot{H}^{q})$ for any $q \in [0, r)$. Using \eqref{eq.mildSol}, the triangle inequality, \eqref{eq.diffUpper}, Lemma \ref{lem.Opera}(1) and Assumption \ref{ass.Lip1} shows
\begin{align*}
&\quad\ \| X(t) \|_{L^p(\Omega,\dot{H}^{q})} \\
&\leq \| \mathcal S_{1-\alpha}(t) X_0 \|_{L^p(\Omega,\dot{H}^{q})} + \int_0^t \| \mathcal S_{0}(t-s) F(X(s)) \|_{L^p(\Omega,\dot{H}^{q})} \mathrm d s + \Big\| \int_{0}^{t} \mathcal S_{\gamma}(t-s) \mathrm d W(s) \Big\|_{L^p(\Omega,\dot{H}^{q})} \\
&\leq \| \mathcal S_{1-\alpha}(t) \|_{\mathcal L(H)} \| X_0 \|_{L^p(\Omega,\dot{H}^{q})} + \int_0^t \| A^{\frac{q}{2}} \mathcal S_{0}(t-s) \|_{\mathcal L(H)} \| F(X(s)) \|_{L^p(\Omega,H)} \mathrm d s + C \\
&\leq C + C \int_0^t (t-s)^{\alpha-1-\frac{\alpha q}{2\beta}} (1 + \|X(s)\|_{L^p(\Omega,H)}) \mathrm d s,
\end{align*}
in which the norm $\| \cdot \|_{L^p(\Omega,H)} $ can be further bounded by $C \| \cdot \|_{L^p(\Omega,\dot{H}^{q})}$. Thus, the singular version of Gr\"onwall's inequality (see e.g., \cite[Lemma A.2]{Kruse2014Book}) and the arbitrariness of $t \in [0,T]$ complete the proof.
\end{proof}
\subsection{Regularity of the mild solution}
In this part, we will provide the temporal regularity analysis of the mild solution \eqref{eq.mildSol} by making use of the integral decomposition technique; see Theorem \ref{th.integraldecompos}.
\begin{Theo} \label{th.Holder}
Suppose that Assumptions \ref{ass.Noise} and \ref{ass.Lip1} hold and let $p \geq 2$ and $X_0 \in L^p(\Omega,\dot{H}^{2\beta})$. Then for all $0\leq s < t \leq T$, there exists some positive constant $C$ independent of $t$ and $s$ such that
\begin{align} \label{Holder-1}
\| X(t) - X(s) \|_{L^p(\Omega,H)}
\leq C (t-s)^{ \min\{ \alpha, \textup{StoC} \} },
\end{align}
where
\begin{align*}
\textup{StoC} := \min\Big\{ \alpha+\gamma-\frac{1}{2}-\frac{\alpha}{2\beta} (\kappa- r), 1-\varepsilon \Big\}
= \min\Big\{ \frac{\alpha r}{2\beta} + (\gamma-\frac{1}{2})^{+}, 1-\varepsilon \Big\}.
\end{align*}
Moreover, there exists a sufficiently small constant $\varepsilon \in (0,\frac{1}{2})$ such that
\begin{align} \label{Holder-2}
\| X(t) - X(s) \|_{L^p(\Omega,H)}
\leq C ( t^{\alpha} - s^{\alpha} ) + C s^{\varepsilon-\frac{1}{2}} (t-s)^{\textup{StoC}}.
\end{align}
\end{Theo}
\begin{proof}
Note that for any $\alpha \in (0,1]$ and $0 < s < t\leq T$, there exists a positive constant $\varrho \in [\alpha,1]$ satisfying
\begin{align} \label{esti.alpha-1}
t^{\alpha} - s^{\alpha}
= t^{\alpha-\varrho} t^{\varrho} - s^{\alpha-\varrho} s^{\varrho}
\leq s^{\alpha-\varrho} t^{\varrho} - s^{\alpha-\varrho} s^{\varrho}
\leq s^{\alpha-\varrho} (t-s)^{\varrho},
\end{align}
which implies that there exists a sufficiently small constant $\varepsilon \in (0,\frac{1}{2})$ such that
\begin{align} \label{esti.alpha-2}
t^{\alpha} - s^{\alpha} \leq C s^{\varepsilon-\frac{1}{2}} (t-s)^{\min\{ \alpha+\frac{1}{2}-\varepsilon, 1 \}}.
\end{align}
It follows from \eqref{eq.mildSol} and the triangle inequality that
\begin{align*}
\| X(t) - X(s) \|_{L^p(\Omega,H)}
&\leq \underbrace{\| ( \mathcal S_{1-\alpha}(t) - \mathcal S_{1-\alpha}(s) ) X_0 \|_{L^p(\Omega,H)}}_{=:\, I_1} \\
&\quad + \underbrace{\Big\| \int_{0}^{t} \mathcal S_{0}(t-u) F(X(u)) \mathrm d u - \int_{0}^{s} \mathcal S_{0}(s-u) F(X(u)) \mathrm d u \Big\|_{L^p(\Omega,H)}}_{=:\, I_2} \\
&\quad + \underbrace{\Big\| \int_0^t \mathcal S_{\gamma}(t-u) \mathrm d W(u) - \int_0^s \mathcal S_{\gamma}(s-u) \mathrm d W(u) \Big\|_{L^p(\Omega,H)}}_{=:\, I_3}.
\end{align*}
\underline{1. Proof of \eqref{Holder-1}}. By Lemma \ref{lem.Opera}(2) and the assumption $\| X_0 \|_{L^p(\Omega,\dot{H}^{2\beta})} < \infty$,
\begin{align*}
I_1 \leq \| A^{-\beta} ( \mathcal S_{1-\alpha}(t) - \mathcal S_{1-\alpha}(s) ) \|_{\mathcal L(H)} \| X_0 \|_{L^p(\Omega,\dot{H}^{2\beta})}
\leq C \big( t^{\alpha} - s^{\alpha} \big).
\end{align*}
Using Lemma \ref{lem.Opera}, Assumption \ref{ass.Lip1}, \cite[Lemma 3.1]{DaiXiaoBu2021} and Theorem \ref{th.Wellpos} shows
\begin{align*}
I_2
&\leq \int_0^s \| ( \mathcal S_{0}(t-u) - \mathcal S_{0}(s-u) ) F(X(u)) \|_{L^p(\Omega,H)} \mathrm d u + \int_s^t \| \mathcal S_{0}(t-u) F(X(u)) \|_{L^p(\Omega,H)} \mathrm d u \nonumber\\
&\leq \int_0^s \| \mathcal S_{0}(t-u) - \mathcal S_{0}(s-u) \|_{\mathcal L(H)} \| F(X(u)) \|_{L^p(\Omega,H)} \mathrm d u
+ \int_s^t \| \mathcal S_{0}(t-u) \|_{\mathcal L(H)} \| F(X(u)) \|_{L^p(\Omega,H)} \mathrm d u \nonumber\\
&\leq C \Big( \int_0^s \int_{s-u}^{t-u} v^{\alpha-2} \mathrm d v \mathrm d u + \int_s^t (t-u)^{\alpha-1} \mathrm d u \Big) \cdot \Big( 1 + \| X \|_{\mathcal C([0,T], L^p(\Omega,H))}\Big) \\
&\leq
\begin{cases}
C (t-s)^{\alpha}, & \mbox{if } \alpha \in (0,1), \\
C (t-s)^{1-\varepsilon}, & \mbox{if } \alpha = 1.
\end{cases}
\end{align*}
In virtue of Lemma \ref{lem.DiffOpera},
\begin{align*}
I_3 \leq \Big\| \int_0^s \mathcal S_{\gamma}(t-u) - \mathcal S_{\gamma}(s-u) \mathrm d W(u) \Big\|_{L^p(\Omega,H)} + \Big\| \int_s^t \mathcal S_{\gamma}(t-u) \mathrm d W(u) \Big\|_{L^p(\Omega,H)}
\leq C (t-s)^{ \textup{StoC} },
\end{align*}
where
\begin{align} \label{esti.StoC}
\textup{StoC} := \min\{\alpha+\gamma-\frac{1}{2}-\frac{\alpha}{2\beta} (\kappa- r), 1-\varepsilon\}
\leq \min\Big\{ \alpha + \frac{1}{2}, 1 - \varepsilon \Big\}.
\end{align}
Thus, collecting these estimates at hand as well as using \eqref{esti.alpha-1} with $\varrho = \alpha$ proves \eqref{Holder-1}.
\underline{2. Proof of \eqref{Holder-2}}. From the proof of \eqref{Holder-1}, one can also read
\begin{align} \label{Holder.conclu}
\| X(t) - X(s) \|_{L^p(\Omega,H)}
\leq C(t^{\alpha} - s^{\alpha}) + I_2 + C (t-s)^{ \textup{StoC} }.
\end{align}
Based on \eqref{Holder-1} and Assumption \ref{ass.Lip1}, applying Theorem \ref{th.integraldecompos} with $\mu = 0$ and $\nu = \min\{ \alpha, \textup{StoC} \}$ to the term $I_2$ yields
\begin{align} \label{Holder.I2-2}
I_2 \leq C ( t^{\alpha} - s^{\alpha} ) + C (t-s)^{\min\{2\alpha-\varepsilon, \alpha+\textup{StoC}-\varepsilon, 1 \}}.
\end{align}
Then, it follows from \eqref{Holder.conclu}, \eqref{Holder.I2-2}, \eqref{esti.StoC} and \eqref{esti.alpha-2} that
\begin{align} \label{esti.Holder2}
\| X(t) - X(s) \|_{L^p(\Omega,H)}
\leq C ( t^{\alpha} - s^{\alpha} ) + C (t-s)^{\min\{2\alpha-\varepsilon, \textup{StoC}\}}
\leq C s^{\varepsilon-\frac{1}{2}} (t-s)^{\min\{ \alpha+\frac{1}{2}-\varepsilon, 2\alpha-\varepsilon, \textup{StoC}\}}.
\end{align}
Based on \eqref{esti.Holder2} and Assumption \ref{ass.Lip1}, applying Theorem \ref{th.integraldecompos} with $\mu = \frac{1}{2}-\varepsilon$ and $\nu = \min\{\alpha+\frac{1}{2}-\varepsilon, 2\alpha -\varepsilon, \textup{StoC}\}$ to the term $I_2$ yields
\begin{align} \label{Holder.I2-3}
I_2 \leq C ( t^{\alpha} - s^{\alpha} ) + C s^{\varepsilon-\frac{1}{2}} (t-s)^{\min\{ 2\alpha+\frac{1}{2}-2\varepsilon, 3\alpha -2\varepsilon, \alpha + \textup{StoC} -\varepsilon, 1 \}}.
\end{align}
Then, it follows from \eqref{Holder.conclu}, \eqref{Holder.I2-3}, \eqref{esti.StoC} and \eqref{esti.alpha-2} that
\begin{align} \label{esti.Holder3}
\| X(t) - X(s) \|_{L^p(\Omega,H)}
&\leq C ( t^{\alpha} - s^{\alpha} ) + C s^{\varepsilon-\frac{1}{2}} (t-s)^{\min\{2\alpha+\frac{1}{2}-2\varepsilon, 3\alpha -2\varepsilon, \textup{StoC}\}} \nonumber\\
&\leq C s^{\varepsilon-\frac{1}{2}} (t-s)^{\min\{ \alpha+\frac{1}{2}-\varepsilon, 3\alpha-2\varepsilon, \textup{StoC}\}}.
\end{align}
Repeating these derivation steps of \eqref{esti.Holder2} and \eqref{esti.Holder3}, one can arrive at
\begin{align} \label{esti.Holder-end-1}
\| X(t) - X(s) \|_{L^p(\Omega,H)}
\leq C s^{\varepsilon-\frac{1}{2}} (t-s)^{ \min\{\alpha+\frac{1}{2}-\varepsilon, \textup{StoC}\} }.
\end{align}
With the estimate \eqref{esti.Holder-end-1} at hand, applying Assumption \ref{ass.Lip1} and Theorem \ref{th.integraldecompos} with $\mu = \frac{1}{2}-\varepsilon$ and $\nu = \min\{\alpha+\frac{1}{2}-\varepsilon, \textup{StoC}\}$ to the term $I_2$ reads
\begin{align} \label{Holder.I2-end}
I_2
&\leq C ( t^{\alpha} - s^{\alpha} ) + C s^{\varepsilon-\frac{1}{2}} (t-s)^{\min\{ 2\alpha+\frac{1}{2}-2\varepsilon, \alpha + \textup{StoC} -\varepsilon, 1 \}} \nonumber\\
&\leq C ( t^{\alpha} - s^{\alpha} ) + C s^{\varepsilon-\frac{1}{2}} (t-s)^{\min\{ \alpha + \textup{StoC} -2\varepsilon, 1 \}},
\end{align}
where \eqref{esti.StoC} is also used. Finally, combining \eqref{Holder.conclu} and \eqref{Holder.I2-end} completes the proof.
\end{proof}
\section{The Mittag--Leffler Euler integrator}
\label{sec.MitLefEuler}
For a fixed integer $N \geq 2$, define a finite dimensional subspace $H_N = \text{span}\{\phi_1, \phi_2, \cdots, \phi_N\} \subset H$ and let the projection operator $P_N: H \rightarrow H_N$ be defined by $v \mapsto \sum_{k=1}^{N} \<v, \phi_k\> \phi_k$. For a fixed integer $M \geq 2$, let $\{t_m = m h:\ m = 0,1,\cdots,M \}$ be a uniform partition of $[0,T]$ with the time stepsize $h = \frac{T}{M}$.
Applying the spectral Galerkin method for the spatial discretization and the Mittag--Leffler Euler integrator for the temporal discretization (see e.g., \cite{Kovacs2020SIAM}), we obtain the fully discrete method for \eqref{eq.model} as follows:\
\begin{align} \label{eq.NumSol}
Y_m^N = \mathcal S^N_{1-\alpha}(t_m) Y_0^N + \sum_{j=0}^{m-1} \int_{t_j}^{t_{j+1}} \mathcal S^N_{0}(t_m-s) P_N F(Y_j^N) \mathrm d s + \int_0^{t_m} \mathcal S^N_{\gamma}(t_m-s) P_N \mathrm d W(s)
\end{align}
with $Y_0^N = P_N X_0$ and $\mathcal S^N_{\eta} (\cdot) = \mathcal S_{\eta} (\cdot) P_N$ for $\eta \in [0,1]$.
\subsection{The nonlinear case}
The following assumption is common in the error analysis of the exponential-type integrators for semilinear (fractional) SPDEs; see e.g., \cite{Kovacs2020SIAM, WangQi2015} for some concrete examples.
\begin{Ass} \label{ass.Lip2}
Let $\beta \in (\frac{1}{2},1]$ and $F:\ H \rightarrow H$ be twice differentiable. Assume that there exist $\delta \in [1,2\beta)$, $\lambda \in [0, r)$, $\zeta \in [1, 2\beta)$ and $L > 0$ such that
\begin{align*}
& \| F'(\varphi_1)\varphi_2 \|_{-\delta}
\leq L ( 1 + \| \varphi_1 \|_{\lambda} ) \| \varphi_2 \|_{-\lambda},
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
&& \forall\, \varphi_1 \in \dot{H}^{\lambda}, \, \varphi_2 \in \dot{H}^{-\lambda}, \\
& \| F''(u) \<\phi, \psi\> \|_{-\zeta}
\leq L \| \phi \| \| \psi \|,
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
&& \forall\, \phi, \, \psi \in H.
\end{align*}
\end{Ass}
\begin{Theo} \label{th.MLEulerErrorNonlinear}
Let $X_0 \in L^4(\Omega,\dot{H}^{2\beta})$. Under the Assumptions \ref{ass.Noise}, \ref{ass.Lip1} and \ref{ass.Lip2}, there exists some positive constant $C = C(\alpha,\beta,\gamma,r,\delta,L,T,\varepsilon)$ such that for all $2 \leq m \leq M$,
\begin{align*}
\| X(t_m) - Y_m^N \|_{L^2(\Omega,H)}
\leq
\begin{cases}
C\lambda_{N+1}^{-\frac{r-\varepsilon}{2}} + C t_m^{\alpha-1-\frac{\alpha\zeta}{2\beta}} h^{ \min\{ \frac{1}{2} + \frac{\alpha}{2\beta}(r+\lambda-\kappa), 1 \} }, & \mbox{if } \alpha + \gamma = 1, \\
C\lambda_{N+1}^{-\frac{r-\varepsilon}{2}} + C t_m^{\alpha-1-\frac{\alpha\zeta}{2\beta}} h^{\min\{\frac{\alpha}{2\beta}\min\{\kappa,r+\lambda\} + (\gamma-\frac{1}{2})^{+}, 1-\varepsilon\}}, & \mbox{if } \alpha + \gamma \neq 1.
\end{cases}
\end{align*}
\end{Theo}
\begin{proof}
In view of \eqref{eq.mildSol} and \eqref{eq.NumSol}, one can read
\begin{align} \label{err.split}
&\quad\ \| X(t_m) - Y_m^N \|_{L^2(\Omega,H)} \nonumber\\
&\leq \| \mathcal S_{1-\alpha}(t_m) ( I - P_N ) X_0 \|_{L^2(\Omega,H)} + \sum_{j=0}^{m-1} \int_{t_j}^{t_{j+1}} \| \mathcal S_{0}(t_m-s) ( I - P_N ) F(X(s)) \|_{L^2(\Omega,H)} \mathrm d s \nonumber\\
&\quad + \Big\| \sum_{j=0}^{m-1} \int_{t_j}^{t_{j+1}} \mathcal S_{0}(t_m-s) P_N ( F(X(s)) - F(Y_j^N) ) \mathrm d s \Big\|_{L^2(\Omega,H)} \nonumber\\
&\quad + \Big\| \int_0^{t_m} \mathcal S_{\gamma}(t_m-s) ( I - P_N ) \mathrm d W(s) \Big\|_{L^2(\Omega,H)} \nonumber\\
&=: J_1 + J_2 + J_3 + J_4.
\end{align}
In fact, one can find that $J_1$, $J_2$ and $J_4$ correspond to the spatial error, while $J_3$ corresponds to the temporal error.
\underline{1. Spatial error}. For this part, it will be frequently used that for any $\varrho \geq 0$,
\begin{align} \label{eq.ProjecErr}
\| A^{-\varrho} (I - P_N) \|_{\mathcal L(H)} \leq C \lambda_{N+1}^{-\varrho}.
\end{align}
Firstly, it follows from Lemma \ref{lem.Opera}(1) and \eqref{eq.ProjecErr} that
\begin{align*}
J_1
\leq \| \mathcal S_{1-\alpha}(t_m) \|_{\mathcal L(H)} \| A^{-\beta} ( I - P_N ) \|_{\mathcal L(H)} \| A^{\beta} X_0 \|_{L^2(\Omega,H)}
\leq C \lambda_{N+1}^{-\beta} \| X_0 \|_{L^2(\Omega,\dot{H}^{2\beta})}
\leq C \lambda_{N+1}^{-\frac{r-\varepsilon}{2}}.
\end{align*}
Secondly, using Lemma \ref{lem.Opera}(1), \eqref{eq.ProjecErr}, Assumption \ref{ass.Lip1} and Theorem \ref{th.Wellpos} reads
\begin{align*}
J_2
&\leq \sum_{j=0}^{m-1} \int_{t_j}^{t_{j+1}} \| A^{\frac{r-\varepsilon}{2}} \mathcal S_{0}(t_m-s) \|_{\mathcal L(H)} \| A^{-\frac{r-\varepsilon}{2}} ( I - P_N ) \|_{\mathcal L(H)} \| F(X(s)) \|_{L^2(\Omega,H)} \mathrm d s \\
&\leq C \lambda_{N+1}^{-\frac{r-\varepsilon}{2}} \int_{0}^{t_m} (t_m -s)^{\alpha-1-\frac{\alpha}{2\beta}(r-\varepsilon)} \mathrm d s \, \Big( 1 + \| X \|_{\mathcal C([0,T], L^2(\Omega,H))} \Big)
\leq C \lambda_{N+1}^{-\frac{r-\varepsilon}{2}}.
\end{align*}
Thirdly, applying It\^o's isometry, \eqref{eq.ProjecErr}, Assumption \ref{ass.Noise} and Lemma \ref{lem.Opera}(1) reveals
\begin{align*}
J_4
&= \Big( \int_0^{t_m} \| \mathcal S_{\gamma}(t_m-s) ( I - P_N ) Q^{\frac{1}{2}} \|_{\mathcal L_2(H)}^2 \mathrm d s \Big)^{\frac{1}{2}} \\
& \leq \| A^{-\frac{r-\varepsilon}{2}} ( I - P_N ) \|_{\mathcal L(H)} \| A^{\frac{ r-\kappa}{2}} Q^{\frac{1}{2}} \|_{\mathcal L_2(H)} \Big( \int_0^{t_m} \| A^{\frac{\kappa-\varepsilon}{2}} \mathcal S_{\gamma}(t_m-s) \|_{\mathcal L(H)}^2 \mathrm d s \Big)^{\frac{1}{2}} \\
&\leq C \lambda_{N+1}^{-\frac{r-\varepsilon}{2}} \Big( \int_0^{t_m} (t_m - s)^{2(\alpha+\gamma-1-\frac{\alpha}{2\beta}(\kappa-\varepsilon))} \mathrm d s \Big)^{\frac{1}{2}}
\leq C \lambda_{N+1}^{-\frac{r-\varepsilon}{2}}.
\end{align*}
Hence, one can conclude that the spatial error is bounded by
\begin{align}\label{err.spatial}
J_1 + J_2 + J_4 \leq C \lambda_{N+1}^{-\frac{r-\varepsilon}{2}}.
\end{align}
\underline{2. Temporal error}. For this part, by recalling the Taylor expansion
\begin{align*}
F(X(s)) = F(X(t_j)) + F'(X(t_j)) (X(s) - X(t_j)) + R_{F,j}(s)
\end{align*}
with $R_{F,j}(s) := \int_{0}^{1} F''(X(t_j) + \theta (X(s) - X(t_j))) \<X(s) - X(t_j), X(s) - X(t_j)\> (1-\theta) \mathrm d \theta$, one can bound the temporal error term $J_3$ as follows:\
\begin{align*}
J_3
&\leq \sum_{j=0}^{m-1} \int_{t_j}^{t_{j+1}} \| \mathcal S_{0}(t_m-s) P_N ( F(X(t_j)) - F(Y_j^N) ) \|_{L^2(\Omega,H)} \mathrm d s \\
&\quad + \Big\| \sum_{j=0}^{m-1} \int_{t_j}^{t_{j+1}} \mathcal S_{0}(t_m-s) P_N F'(X(t_j)) ( X(s) - X(t_j) ) \mathrm d s \Big\|_{L^2(\Omega,H)} \\
&\quad + \sum_{j=0}^{m-1} \int_{t_j}^{t_{j+1}} \| \mathcal S_{0}(t_m-s) P_N R_{F,j}(s) \|_{L^2(\Omega,H)} \mathrm d s.
\end{align*}
Then, taking \eqref{eq.mildSol} into account for the second term of the right hand side, one can arrive at
\begin{align} \label{eq.J3Split}
J_3 \leq J_{3,1} + J_{3,2} + J_{3,3} + J_{3,4} + J_{3,5} + J_{3,6} + J_{3,7}
\end{align}
with
\begin{align*}
J_{3,1} &:= \sum_{j=0}^{m-1} \int_{t_j}^{t_{j+1}} \| \mathcal S_{0}(t_m-s) P_N ( F(X(t_j)) - F(Y_j^N) ) \|_{L^2(\Omega,H)} \mathrm d s, \\
J_{3,2} &:= \sum_{j=0}^{m-1} \int_{t_j}^{t_{j+1}} \| \mathcal S_{0}(t_m-s) P_N F'(X(t_j)) ( \mathcal S_{1-\alpha}(s) - \mathcal S_{1-\alpha}(t_j) ) X_0 \|_{L^2(\Omega,H)} \mathrm d s, \\
J_{3,3} &:= \int_{0}^{t_{1}} \Big\| \mathcal S_{0}(t_m-s) P_N F'(X(t_0)) \int_{0}^{s} \mathcal S_{0}(s-u) F(X(u)) \mathrm d u \Big\|_{L^2(\Omega,H)} \mathrm d s, \\
J_{3,4} &:= \sum_{j=1}^{m-1} \int_{t_j}^{t_{j+1}} \Big\| \mathcal S_{0}(t_m-s) P_N F'(X(t_j)) \\
&\qquad\qquad\qquad\ \Big( \int_{0}^{s} \mathcal S_{0}(s-u) F(X(u)) \mathrm d u - \int_{0}^{t_j} \mathcal S_{0}(t_j-u) F(X(u)) \mathrm d u \Big) \Big\|_{L^2(\Omega,H)} \mathrm d s, \\
J_{3,5} &:= \Big\| \sum_{j=0}^{m-1} \int_{t_j}^{t_{j+1}} \mathcal S_{0}(t_m-s) P_N F'(X(t_j)) \int_0^{t_j} \mathcal S_{\gamma}(s-u) - \mathcal S_{\gamma}(t_j-u) \mathrm d W(u) \mathrm d s \Big\|_{L^2(\Omega,H)}, \\
J_{3,6} &:= \Big\| \sum_{j=0}^{m-1} \int_{t_j}^{t_{j+1}} \mathcal S_{0}(t_m-s) P_N F'(X(t_j)) \int_{t_j}^{s} \mathcal S_{\gamma}(s-u) \mathrm d W(u) \mathrm d s \Big\|_{L^2(\Omega,H)}, \\
J_{3,7} &:= \sum_{j=0}^{m-1} \int_{t_j}^{t_{j+1}} \| \mathcal S_{0}(t_m-s) P_N R_{F,j}(s) \|_{L^2(\Omega,H)} \mathrm d s.
\end{align*}
By Lemma \ref{lem.Opera}(1) and Assumption \ref{ass.Lip1},
\begin{align} \label{eq.J31}
J_{3,1}
&\leq \sum_{j=0}^{m-1} \int_{t_j}^{t_{j+1}} \| \mathcal S_{0}(t_m-s) \|_{\mathcal L(H)} \| F(X(t_j)) - F(Y_j^N) \|_{L^2(\Omega,H)} \mathrm d s \nonumber\\
&\leq C \sum_{j=0}^{m-1} \int_{t_j}^{t_{j+1}} (t_m-s)^{\alpha-1} \| X(t_j) - Y_j^N \|_{L^2(\Omega,H)} \mathrm d s.
\end{align}
Note that for any $\varrho \in [0,1]$, $s \in [t_j, t_{j+1}]$ with $j = 1,\cdots,m-1$,
\begin{align} \label{esti.tj->s}
t_j^{-\varrho} \leq 2^{\varrho} t_{j+1}^{-\varrho} \leq 2^{\varrho} s^{-\varrho},
\end{align}
which together with \eqref{esti.alpha-1} and the Beta function implies
\begin{align} \label{eq.usetoJ32}
\sum_{j=0}^{m-1} \int_{t_j}^{t_{j+1}} (t_m-s)^{\alpha-1} \big( s^{\alpha} - t_j^{\alpha} \big) \mathrm d s
\leq t_{m-1}^{2\alpha-1} h + C \int_{t_1}^{t_m} (t_m-s)^{\alpha-1} s^{\alpha-1} (s-t_j) \mathrm d s
\leq C t_{m}^{2\alpha-1} h.
\end{align}
Then, it follows from Assumption \ref{ass.Lip1}, Lemma \ref{lem.Opera}(1)--(2), $\| X(0) \|_{L^2(\Omega,\dot{H}^{2\beta})} < \infty$ and \eqref{eq.usetoJ32} that
\begin{align} \label{eq.J32}
J_{3,2}
&\leq C \sum_{j=0}^{m-1} \int_{t_j}^{t_{j+1}} \| \mathcal S_{0}(t_m-s) \|_{\mathcal L(H)} \| A^{-\beta} ( \mathcal S_{1-\alpha}(s) - \mathcal S_{1-\alpha}(t_j) ) \|_{\mathcal L(H)} \mathrm d s \cdot \| X_0 \|_{L^2(\Omega,\dot{H}^{2\beta})} \nonumber\\
&\leq C \sum_{j=0}^{m-1} \int_{t_j}^{t_{j+1}} (t_m-s)^{\alpha-1} \big( s^{\alpha} - t_j^{\alpha} \big) \mathrm d s
\leq C t_{m}^{2\alpha-1} h.
\end{align}
By Assumption \ref{ass.Lip1}, Lemma \ref{lem.Opera}(1) and Theorem \ref{th.Wellpos},
\begin{align} \label{eq.J33}
J_{3,3}
&\leq C \int_{0}^{t_{1}} \| \mathcal S_{0}(t_m-s) \|_{\mathcal L(H)} \int_{0}^{s} \| \mathcal S_{0}(s-u) \|_{\mathcal L(H)} \| F(X(u)) \|_{L^2(\Omega,H)} \mathrm d u \mathrm d s \nonumber\\
&\leq C \int_{0}^{t_{1}} (t_m-s)^{\alpha-1} \int_{0}^{s} (s-u)^{\alpha-1} \mathrm d u \mathrm d s
\leq C t_m^{2\alpha-1} h.
\end{align}
By Assumption \ref{ass.Lip1}, Lemma \ref{lem.Opera}(1) and a similar estimate to \eqref{Holder.I2-end} as well as \eqref{eq.usetoJ32}, \eqref{esti.tj->s} and the Beta function,
\begin{align} \label{eq.J34}
J_{3,4} &\leq \sum_{j=1}^{m-1} \int_{t_j}^{t_{j+1}} \| \mathcal S_{0}(t_m-s) \|_{\mathcal L(H)} \Big\| \int_{0}^{s} \mathcal S_{0}(s-u) F(X(u)) \mathrm d u - \int_{0}^{t_j} \mathcal S_{0}(t_j-u) F(X(u)) \mathrm d u \Big\|_{L^2(\Omega,H)} \mathrm d s \nonumber\\
&\leq C \sum_{j=1}^{m-1} \int_{t_j}^{t_{j+1}} (t_m-s)^{\alpha-1} ( s^{\alpha} - t_j^{\alpha} ) \mathrm d s + C \sum_{j=1}^{m-1} \int_{t_j}^{t_{j+1}} (t_m-s)^{\alpha-1} t_j^{\frac{\varepsilon-1}{2}} (s-t_j)^{\min\{\alpha + \textup{StoC} - \varepsilon,1\}} \mathrm d s \nonumber\\
&\leq C t_m^{2\alpha-1} h + C h^{\min\{\alpha+\textup{StoC}-\varepsilon,1 \}} \int_{t_1}^{t_m} (t_m-s)^{\alpha-1} s^{\frac{\varepsilon-1}{2}} \mathrm d s
\leq C t_m^{2\alpha-1} h^{ \min\{ 1, \alpha + \frac{\alpha r}{2\beta} + (\gamma-\frac{1}{2})^{+} - \varepsilon \} }.
\end{align}
According to Lemma \ref{lem.Opera}(1), Assumption \ref{ass.Lip2}, H\"older's inequality, Theorem \ref{th.Wellpos} and Lemma \ref{lem.DiffOpera}, one can derive
\begin{align} \label{eq.J35}
J_{3,5}
&\leq \sum_{j=0}^{m-1} \int_{t_j}^{t_{j+1}} \| A^{\frac{\delta}{2}} \mathcal S_{0}(t_m-s) \|_{\mathcal L(H)} \Big\| A^{-\frac{\delta}{2}} F'(X(t_j)) \int_0^{t_j} \mathcal S_{\gamma}(s-u) - \mathcal S_{\gamma}(t_j-u) \mathrm d W(u) \Big\|_{L^2(\Omega,H)} \mathrm d s \nonumber\\
&\leq C \sum_{j=0}^{m-1} \int_{t_j}^{t_{j+1}} (t_m-s)^{\alpha-1-\frac{\alpha\delta}{2\beta}} ( 1 + \| X(t_j) \|_{L^4(\Omega,\dot{H}^{\lambda})} ) \Big\| \int_0^{t_j} \mathcal S_{\gamma}(s-u) - \mathcal S_{\gamma}(t_j-u) \mathrm d W(u) \Big\|_{L^4(\Omega,\dot{H}^{-\lambda})} \mathrm d s \nonumber\\
&\leq
\begin{cases}
C h^{\frac{1}{2} + \frac{\alpha}{2\beta}(r+\lambda-\kappa)}, & \mbox{if } \alpha + \gamma = 1, \\
C h^{\min\{\frac{\alpha}{2\beta}\min\{\kappa,r+\lambda\} + (\gamma-\frac{1}{2})^{+}, 1-\varepsilon\}}, & \mbox{if } \alpha + \gamma \neq 1.
\end{cases}
\end{align}
Using the independence of the noise increments, Assumption \ref{ass.Lip1}, Lemmas \ref{lem.Opera}(1) and \ref{lem.DiffOpera}, the identity $\alpha + \gamma - \frac{1}{2} - \frac{\alpha\kappa}{2\beta} = (\gamma-\frac{1}{2})^{+}$ and the assumption $\lambda < r \leq \kappa$ reveals
\begin{align} \label{eq.J36}
|J_{3,6}|^2
&= \sum_{j=0}^{m-1} \Big\| \int_{t_j}^{t_{j+1}} \mathcal S_{0}(t_m-s) P_N F'(X(t_j)) \int_{t_j}^{s} \mathcal S_{\gamma}(s-u) \mathrm d W(u) \mathrm d s \Big\|_{L^2(\Omega,H)}^2 \nonumber\\
&\leq C \sum_{j=0}^{m-1} \Big( \int_{t_j}^{t_{j+1}} \| \mathcal S_{0}(t_m-s) \|_{\mathcal L(H)} \Big\| \int_{t_j}^{s} \mathcal S_{\gamma}(s-u) \mathrm d W(u) \Big\|_{L^2(\Omega,H)} \mathrm d s \Big)^2 \nonumber\\
&\leq C \sum_{j=0}^{m-1} \Big( \int_{t_j}^{t_{j+1}} (t_m-s)^{\alpha-1} (s-t_j)^{\alpha + \gamma - \frac{1}{2} - \frac{\alpha}{2\beta} (\kappa- r) } \mathrm d s \Big)^2 \nonumber\\
&\leq C h^{2( \alpha+\gamma-\frac{1}{2}-\frac{\alpha}{2\beta} (\kappa- r) )} \sum_{j=0}^{m-1} \int_{t_j}^{t_{j+1}} (t_m-s)^{2\varepsilon-1} \mathrm d s \times \int_{t_j}^{t_{j+1}} (t_m-s)^{2\alpha-1-2\varepsilon} \mathrm d s \nonumber\\
&\leq C h^{ 2\min\{ 2\alpha+\gamma-\frac{1}{2}-\frac{\alpha}{2\beta} (\kappa- r) - \varepsilon, \alpha+\gamma-\frac{\alpha}{2\beta} (\kappa- r) \} }
\leq C h^{ 2( \min\{ \alpha-\varepsilon, \frac{1}{2}\} + \frac{\alpha r}{2\beta} + (\gamma-\frac{1}{2})^{+} ) } \nonumber\\
&\leq
\begin{cases}
C h^{ 2( \frac{1}{2} + \frac{\alpha}{2\beta}(r+\lambda-\kappa) ) }, & \mbox{if } \alpha + \gamma = 1, \\
C h^{ 2( \frac{\alpha}{2\beta}\min\{\kappa,r+\lambda\} + (\gamma-\frac{1}{2})^{+} ) }, & \mbox{if } \alpha + \gamma \neq 1.
\end{cases}
\end{align}
Here, the last step is easy for the case $\alpha + \gamma = 1$, so the details are omitted. In the case $\alpha + \gamma \neq 1$, the last step also used the inequality
\begin{align*}
\frac{\alpha}{2\beta}\min\{\kappa-r, \lambda\} \leq \frac{\alpha}{2\beta}\min\{\kappa-r, r\}
\leq \frac{1}{2} \frac{\alpha\kappa}{2\beta}
= \frac{1}{2} \min\{\alpha+\gamma-\frac{1}{2}, \alpha \} \leq \frac{\alpha}{2} \leq \min\{\alpha-\varepsilon,\frac{1}{2}\}.
\end{align*}
It follows from Lemma \ref{lem.Opera}(1), Assumption \ref{ass.Lip2}, Theorem \ref{th.Holder}, \eqref{esti.alpha-1}, \eqref{esti.tj->s}, the Beta function and the assumption $\lambda < r \leq \kappa$ that
\begin{align} \label{eq.J37}
J_{3,7}
&\leq \sum_{j=0}^{m-1} \int_{t_j}^{t_{j+1}} \| A^{\frac{\zeta}{2}} \mathcal S_{0}(t_m-s) \|_{\mathcal L(H)} \| A^{-\frac{\zeta}{2}} R_{F,j}(s) \|_{L^2(\Omega,H)} \mathrm d s \nonumber\\
&\leq C \sum_{j=0}^{m-1} \int_{t_j}^{t_{j+1}} (t_m-s)^{\alpha-1-\frac{\alpha\zeta}{2\beta}} \| X(s) - X(t_j) \|_{L^4(\Omega,H)}^2 \mathrm d s \nonumber\\
&\leq C \int_{0}^{t_{1}} (t_m-s)^{\alpha-1-\frac{\alpha\zeta}{2\beta}} s^{2\min\{ \alpha, \frac{\alpha r}{2\beta} + (\gamma-\frac{1}{2})^{+}, 1-\varepsilon \}} \mathrm d s
+ C \sum_{j=1}^{m-1} \int_{t_j}^{t_{j+1}} (t_m-s)^{\alpha-1-\frac{\alpha\zeta}{2\beta}} ( s^{\alpha} - t_j^{\alpha} )^2 \mathrm d s \nonumber\\
&\quad + C \sum_{j=1}^{m-1} \int_{t_j}^{t_{j+1}} (t_m-s)^{\alpha-1-\frac{\alpha\zeta}{2\beta}} s^{2\varepsilon-1} (s-t_j)^{ 2\min\{ \frac{\alpha r}{2\beta} + (\gamma-\frac{1}{2})^{+}, 1-\varepsilon\} } \mathrm d s \nonumber\\
&\leq
\begin{cases}
C t_m^{\alpha-1-\frac{\alpha\zeta}{2\beta}} h^{ 2\min\{ \frac{\alpha r}{2\beta} + (\gamma-\frac{1}{2})^{+}, 1-\varepsilon \} }, & \mbox{if } \gamma < 1, \\
C t_m^{\alpha-1-\frac{\alpha\zeta}{2\beta}} h^{ 2\min\{ \frac{\alpha r}{2\beta} + (\gamma-\frac{1}{2})^{+} - \varepsilon, 1-\varepsilon \} }, & \mbox{if } \gamma = 1
\end{cases} \nonumber\\
&\leq
\begin{cases}
C t_m^{\alpha-1-\frac{\alpha\zeta}{2\beta}} h^{\frac{1}{2} + \frac{\alpha}{2\beta}(r+\lambda-\kappa)}, & \mbox{if } \alpha + \gamma = 1, \\
C t_m^{\alpha-1-\frac{\alpha\zeta}{2\beta}} h^{ \frac{\alpha}{2\beta}\min\{\kappa,r+\lambda\} + (\gamma-\frac{1}{2})^{+} }, & \mbox{if } \alpha + \gamma \neq 1.
\end{cases}
\end{align}
Here, the last step is easy for the case $\alpha + \gamma = 1$, so the details are omitted. In the case $\alpha + \gamma \neq 1$, the last step also used the inequality
\begin{align*}
\frac{\alpha}{2\beta}\min\{\kappa-r, \lambda\} \leq \frac{\alpha}{2\beta}\min\{\kappa-r, r\}
\leq \frac{\alpha r}{2\beta} \leq \frac{\alpha r}{2\beta} + (\gamma-\frac{1}{2})^{+}.
\end{align*}
Now, by collecting the estimates \eqref{eq.J3Split}, \eqref{eq.J31}, \eqref{eq.J32}--\eqref{eq.J37} and applying the singular version of Gr\"onwall's inequality (see e.g., \cite[Lemma A.2]{Kruse2014Book}), one can obtain that the temporal error term $J_3$ can be bounded by
\begin{align*}
J_3
\leq
\begin{cases}
C t_m^{\alpha-1-\frac{\alpha\zeta}{2\beta}} h^{ \min\{ \frac{1}{2} + \frac{\alpha}{2\beta}(r+\lambda-\kappa), 1 \} }, & \mbox{if } \alpha + \gamma = 1, \\
C t_m^{\alpha-1-\frac{\alpha\zeta}{2\beta}} h^{\min\{\frac{\alpha}{2\beta}\min\{\kappa,r+\lambda\} + (\gamma-\frac{1}{2})^{+}, 1-\varepsilon\}}, & \mbox{if } \alpha + \gamma \neq 1,
\end{cases}
\end{align*}
which together with \eqref{err.split} and \eqref{err.spatial} completes the proof.
\end{proof}
\begin{Rem} \label{rem.driftpr}
In the literature, the term $J_{3,4}$ is usually estimated in the following way.
\underline{Step 1.} Decompose the integral interval of the inner integral of $J_{3,4}$ into two parts to obtain
\begin{align*}
J_{3,4}
&\leq \sum_{j=1}^{m-1} \int_{t_j}^{t_{j+1}} \Big\| \mathcal S_{0}(t_m-s) P_N F'(X(t_j)) \int_{t_j}^{s} \mathcal S_{0}(s-u) F(X(u)) \mathrm d u \Big\|_{L^2(\Omega,H)} \mathrm d s \\
&\quad + \sum_{j=1}^{m-1} \int_{t_j}^{t_{j+1}} \Big\| \mathcal S_{0}(t_m-s) P_N F'(X(t_j)) \int_0^{t_j} ( \mathcal S_{0}(s-u) - \mathcal S_{0}(t_j-u) ) F(X(u)) \mathrm d u \Big\|_{L^2(\Omega,H)} \mathrm d s \\
&=: J_{3,4}^{\prime} + J_{3,4}^{\prime\prime}.
\end{align*}
\underline{Step 2.} Estimate $J_{3,4}^{\prime}$ and $J_{3,4}^{\prime\prime}$ separately. Use Assumption \ref{ass.Lip1}, Lemma \ref{lem.Opera}(1) and Theorem \ref{th.Wellpos} to get
\begin{align*}
J_{3,4}^{\prime}
&\leq C \sum_{j=1}^{m-1} \int_{t_j}^{t_{j+1}} \| \mathcal S_{0}(t_m-s) \|_{\mathcal L(H)} \int_{t_j}^{s} \| \mathcal S_{0}(s-u) \|_{\mathcal L(H)} \| F(X(u)) \|_{L^2(\Omega,H)} \mathrm d u \mathrm d s \\
&\leq C \sum_{j=1}^{m-1} \int_{t_j}^{t_{j+1}} (t_m-s)^{\alpha-1} \int_{t_j}^{s} (s-u)^{\alpha-1}\mathrm d u \mathrm d s
\leq C h^{\alpha},
\end{align*}
which implies $J_{3,4} \leq C h^{\alpha} + J_{3,4}^{\prime\prime}$. In this way, one can only obtain a convergence rate of not exceeding order $\alpha$, which is unsatisfactory especially for the case of $\alpha$ being close to zero. To this end, the integral decomposition technique is developed in Section \ref{sec.Prelim}, which can improve the convergence rate of $J_{3,4}$ from not exceeding order $\alpha$ to order $\min\{ 1, \alpha + \frac{\alpha r}{2\beta} + (\gamma-\frac{1}{2})^{+} - \varepsilon \}$. In addition, the integral decomposition technique is also potentially used in the estimate of $J_{3,7}$ since the estimate depends on the temporal regularity of the mild solution \eqref{eq.mildSol}.
\end{Rem}
\subsection{The linear case}
In this part, we improve the convergence order in Theorem \ref{th.MLEulerErrorNonlinear} for the linear case.
\begin{Theo} \label{th.MLEulerErrorLinear}
Suppose that Assumption \ref{ass.Noise} holds and the deterministic mapping $F$ is linear. Then there exists some positive constant $C = C(\alpha,\beta,\gamma,r,T,\varepsilon)$ such that for all $2 \leq m \leq M$,
\begin{align*}
\| X(t_m) - Y_m^N \|_{L^2(\Omega,H)}
\leq C \lambda_{N+1}^{-\frac{r-\varepsilon}{2}} + C t_m^{2\alpha-1} h^{ \min\{ \alpha + \frac{\alpha r}{2\beta} + (\gamma-\frac{1}{2})^{+}-\varepsilon, \alpha+\gamma -\varepsilon, 1 \} }.
\end{align*}
\end{Theo}
To facilitate the proof of the above theorem, we firstly prepare two useful lemmas.
\begin{Lem} \label{lem.usetoJ35sum}
Let $a \in \mathbb R$, $b < 1$, $c < 1$, $T > 0$ and the integer $M \geq 2$. For $j \in \{0,1,\cdots, M\}$, put $t_j = j\frac{T}{M}$. Then, there exists some positive constant $C=C(a,b,c,T)$ such that
\begin{align*}
\sup_{2 \leq m \leq M} \sum_{0 \leq i < j \leq m-1} \int_{t_i}^{t_{i+1}} \int_{t_j}^{t_{j+1}} (t_m-s)^{-a} (t_m-\tau)^{-b} (\tau-s)^{-c} \mathrm d \tau \mathrm d s
\leq
\begin{cases}
C, & \mbox{if } a+b+c < 2, \\
C \ln M, & \mbox{if } a+b+c = 2, \\
C M^{a+b+c-2} , & \mbox{if } a+b+c > 2.
\end{cases}
\end{align*}
\end{Lem}
\begin{proof}
Using the change of variables and the Beta function shows
\begin{align*}
\sum_{0 \leq i < j \leq m-1} & \int_{t_i}^{t_{i+1}} \int_{t_j}^{t_{j+1}} (t_m-s)^{-a} (t_m-\tau)^{-b} (\tau-s)^{-c} \mathrm d \tau \mathrm d s \\
&\leq \int_{0}^{t_{m-1}} \int_{s}^{t_m} (t_m-s)^{-a} (t_m-\tau)^{-b} (\tau-s)^{-c} \mathrm d \tau \mathrm d s \\
&= \int_{t_1}^{t_m} \int_{0}^{u} u^{-a} v^{-b} (u-v)^{-c} \mathrm d v \mathrm d u \\
&= B(1-b,1-c) \int_{t_1}^{t_m} u^{1-(a+b+c)} \mathrm d u.
\end{align*}
Then, the proof can be completed by some direct computations.
\end{proof}
\begin{Lem} \label{lem.J35linear}
Under the assumptions of Theorem \ref{th.MLEulerErrorLinear}, there exists some positive constant $C$ such that
\begin{align*}
J_{3,5}
\leq C h^{ \min\{ 1, \alpha+\gamma -\varepsilon, \alpha + \frac{\alpha r}{2\beta} + (\gamma-\frac{1}{2})^{+} - \varepsilon \} },
\end{align*}
where $J_{3,5}$ is the same as in \eqref{eq.J3Split}.
\end{Lem}
\begin{proof}
By introducing
\begin{align*}
J_{3,5}^{\star} &:= \sum_{j=0}^{m-1} \Big\| \int_{t_j}^{t_{j+1}} \mathcal S_{0}(t_m-s) \int_0^{t_j} \mathcal S_{\gamma}(s-u) - \mathcal S_{\gamma}(t_j-u) \mathrm d W(u) \mathrm d s \Big\|_{L^2(\Omega,H)}^2, \\
J_{3,5}^{\star\star} &:= \sum_{0 \leq i < j \leq m-1} \mathbb E \Big\< \int_{t_i}^{t_{i+1}} \mathcal S_{0}(t_m-s) \int_0^{t_i} \mathcal S_{\gamma}(s-u) - \mathcal S_{\gamma}(t_i-u) \mathrm d W(u) \mathrm d s, \\
&\qquad\qquad\qquad\quad\, \ \int_{t_j}^{t_{j+1}} \mathcal S_{0}(t_m-\tau) \int_0^{t_j} \mathcal S_{\gamma}(\tau-v) - \mathcal S_{\gamma}(t_j-v) \mathrm d W(v) \mathrm d \tau \Big\>,
\end{align*}
$|J_{3,5}|^2$ can be bounded by
\begin{align} \label{esti.J35tosub}
|J_{3,5}|^2
\leq C \big( J_{3,5}^{\star} + 2 J_{3,5}^{\star\star}\big),
\end{align}
where the term $J_{3,5}^{\star}$ can be estimated as
\begin{align} \label{esti.J35star}
J_{3,5}^{\star}
&\leq \sum_{j=0}^{m-1} \Big( \int_{t_j}^{t_{j+1}} \| A^{\frac{\kappa- r}{2}} \mathcal S_{0}(t_m-s) \|_{\mathcal L(H)} \Big\| \int_0^{t_j} \mathcal S_{\gamma}(s-u) - \mathcal S_{\gamma}(t_j-u) \mathrm d W(u) \Big\|_{L^2(\Omega,\dot{H}^{ r-\kappa})} \mathrm d s \Big)^2 \nonumber\\
&\leq C \sum_{j=0}^{m-1} \Big( \int_{t_j}^{t_{j+1}} (t_m-s)^{ \alpha - 1 -\frac{\alpha}{2\beta}(\kappa- r) } (s-t_j)^{ \min\{ \alpha + \gamma - \frac{1}{2}, 1-\varepsilon \} } \mathrm d s \Big)^2 \nonumber\\
&\leq C h^{ 2\min\{ \alpha + \gamma - \frac{1}{2}, 1-\varepsilon \} } \sum_{j=0}^{m-1} \int_{t_j}^{t_{j+1}} (t_m-s)^{2\varepsilon-1} \mathrm d s \int_{t_j}^{t_{j+1}} (t_m-s)^{ 2\alpha - 1 -\frac{\alpha}{\beta}(\kappa- r) - 2\varepsilon } \mathrm d s \nonumber\\
&\leq C h^{ 2\min\{2\alpha+\gamma-\frac{1}{2}-\frac{\alpha}{2\beta}(\kappa- r)-\varepsilon, 1+\alpha -\frac{\alpha}{2\beta}(\kappa- r) - 2\varepsilon, \alpha+\gamma, \frac{3}{2}-\varepsilon \} }
\leq C h^{ 2\min\{\alpha + \frac{\alpha r}{2\beta} + (\gamma-\frac{1}{2})^{+} - \varepsilon, \alpha+\gamma, 1 \} }.
\end{align}
To facilitate the estimate of the term $J_{3,5}^{\star\star}$, for $s \in (t_i,t_{i+1}]$, $\tau \in (t_j,t_{j+1}]$ with $0 \leq i < j \leq m-1$, introduce
\begin{align*}
I_{i,j}(s,\tau)
&:= \mathbb E \Big\< \mathcal S_{0}(t_m-s) \int_0^{t_i} \mathcal S_{\gamma}(s-u) - \mathcal S_{\gamma}(t_i-u) \mathrm d W(u),
\mathcal S_{0}(t_m-\tau) \int_0^{t_j} \mathcal S_{\gamma}(\tau-v) - \mathcal S_{\gamma}(t_j-v) \mathrm d W(v) \Big\>, \\
I_{i,j}^{\star}(s,\tau)
&:= \int_{0}^{t_i} \| \mathcal S_{\gamma}(s-u) - \mathcal S_{\gamma}(t_i-u) \|_{\mathcal L(H)} \| \mathcal S_{\gamma}(\tau-u) - \mathcal S_{\gamma}(t_j-u) \|_{\mathcal L(H)} \mathrm d u.
\end{align*}
Then, one can derive from \cite[Corollary 4.29]{DaPrato2014Book} that
\begin{align*}
&\quad\ I_{i,j}(s,\tau) \\
&= \int_{0}^{t_i} \textup{Tr}\Big( \mathcal S_{0}(t_m-\tau) \big( \mathcal S_{\gamma}(\tau-u) - \mathcal S_{\gamma}(t_j-u) \big) Q \big( \mathcal S_{\gamma}(s-u) - \mathcal S_{\gamma}(t_i-u) \big) \mathcal S_{0}(t_m-s) \Big) \mathrm d u \\
&\leq \int_{0}^{t_i} \| Q^{\frac{1}{2}} \big( \mathcal S_{\gamma}(s-u) - \mathcal S_{\gamma}(t_i-u) \big) \mathcal S_{0}(t_m-s) \|_{\mathcal L_2(H)} \| Q^{\frac{1}{2}} \big( \mathcal S_{\gamma}(\tau-u) - \mathcal S_{\gamma}(t_j-u) \big) \mathcal S_{0}(t_m-\tau) \|_{\mathcal L_2(H)} \mathrm d u \\
&\leq \| A^{\frac{ r-\kappa}{2}} Q^{\frac{1}{2}} \|_{\mathcal L_2(H)}^2 \| A^{\frac{\kappa- r}{2}} \mathcal S_{0}(t_m-s) \|_{\mathcal L(H)} \| A^{\frac{\kappa- r}{2}} \mathcal S_{0}(t_m-\tau) \|_{\mathcal L(H)} \times I_{i,j}^{\star}(s,\tau),
\end{align*}
which together with Assumption \ref{ass.Noise} and Lemma \ref{lem.Opera}(1) indicates
\begin{align} \label{esti.preJ35star}
J_{3,5}^{\star\star}
&= \sum_{0 \leq i < j \leq m-1} \int_{t_i}^{t_{i+1}} \int_{t_j}^{t_{j+1}} I_{i,j}(s,\tau) \mathrm d \tau \mathrm d s \nonumber\\
&\leq C\sum_{0 \leq i < j \leq m-1} \int_{t_i}^{t_{i+1}} \int_{t_j}^{t_{j+1}} (t_m-s)^{\alpha-1-\frac{\alpha}{2\beta}(\kappa-r)} (t_m-\tau)^{\alpha-1-\frac{\alpha}{2\beta}(\kappa- r)} I_{i,j}^{\star}(s,\tau) \mathrm d \tau \mathrm d s.
\end{align}
Here, it follows from Lemma \ref{lem.Opera}(2)--(3) that for $s \in (t_i,t_{i+1}]$, $\tau \in (t_j,t_{j+1}]$ with $0 \leq i < j \leq m-1$,
\begin{align*}
I_{i,j}^{\star}(s,\tau)
\leq
\begin{cases}
C \int_{0}^{t_i} \big( (t_i-u)^{\alpha+\gamma-1} - (s-u)^{\alpha+\gamma-1} \big) \big( (t_j-u)^{\alpha+\gamma-1} - (\tau-u)^{\alpha+\gamma-1} \big) \mathrm d u, & \mbox{if } \alpha + \gamma \in (\frac{1}{2},1), \\
C \int_{0}^{t_i} \int_{t_i-u}^{s-u} v^{-1} \mathrm d v \int_{t_j-u}^{\tau-u} \omega^{-1} \mathrm d \omega \mathrm d u, & \mbox{if } \alpha + \gamma =1, \\
C \int_{0}^{t_i} \big( (s-u)^{\alpha+\gamma-1} - (t_i-u)^{\alpha+\gamma-1} \big) \big( (\tau-u)^{\alpha+\gamma-1} - (t_j-u)^{\alpha+\gamma-1} \big) \mathrm d u, & \mbox{if } \alpha + \gamma \in (1,2].
\end{cases}
\end{align*}
Next, we put $s \in (t_i,t_{i+1}]$, $\tau \in (t_j,t_{j+1}]$ with $0 \leq i < j \leq m-1$ and proceed to estimate $J_{3,5}^{\star\star}$ in these three cases.
\underline{Case 1:\ $\alpha + \gamma \in (\frac{1}{2},1)$}. It follows from
\begin{align*}
&\quad\ \int_{0}^{t_i} (t_i-u)^{\alpha+\gamma-1} \big( (t_j-u)^{\alpha+\gamma-1} - (\tau-u)^{\alpha+\gamma-1} \big) \mathrm d u \\
&= t_i^{\alpha+\gamma} \int_{0}^{1} (1-v)^{\alpha+\gamma-1} \big( (t_j-t_i v)^{\alpha+\gamma-1} - (\tau-t_i v)^{\alpha+\gamma-1} \big) \mathrm d v \\
&\leq s^{\alpha+\gamma} \int_{0}^{1} (1-v)^{\alpha+\gamma-1} \big( (t_j-s v)^{\alpha+\gamma-1} - (\tau-s v)^{\alpha+\gamma-1} \big) \mathrm d v \\
&= \int_{0}^{s} (s-u)^{\alpha+\gamma-1} \big( (t_j-u)^{\alpha+\gamma-1} - (\tau-u)^{\alpha+\gamma-1} \big) \mathrm d u
\end{align*}
that
\begin{align*}
I_{i,j}^{\star}(s,\tau)
&\leq C \bigg( \int_{0}^{t_i} (t_i-u)^{\alpha+\gamma-1} \big( (t_j-u)^{\alpha+\gamma-1} - (\tau-u)^{\alpha+\gamma-1} \big) \mathrm d u \\
&\qquad\quad - \int_{0}^{s} (s-u)^{\alpha+\gamma-1} \big( (t_j-u)^{\alpha+\gamma-1} - (\tau-u)^{\alpha+\gamma-1} \big) \mathrm d u \\
&\qquad\quad + \int_{t_i}^{s} (s-u)^{\alpha+\gamma-1} \big( (t_j-u)^{\alpha+\gamma-1} - (\tau-u)^{\alpha+\gamma-1} \big) \mathrm d u \bigg) \\
&\leq C \int_{t_i}^{s} (s-u)^{\alpha+\gamma-1} \big( (t_j-u)^{\alpha+\gamma-1} - (\tau-u)^{\alpha+\gamma-1} \big) \mathrm d u.
\end{align*}
When $j > i+1$, using the mean value theorem and $\tau-s \leq 2(t_j - u)$ with $u \in [t_i,s]$ shows
\begin{align*}
I_{i,j}^{\star}(s,\tau)
&\leq C \int_{t_i}^{s} (s-u)^{\alpha+\gamma-1} (\tau-t_j) (t_j - u)^{\alpha+\gamma-2} (t_j - u)^{1-\varepsilon} \mathrm d u \, (\tau-s)^{\varepsilon-1} \\
&\leq C h (\tau-s)^{\varepsilon-1} \int_{t_i}^{s} (s-u)^{2(\alpha+\gamma-1)-\varepsilon} \mathrm d u
\leq C h^{2(\alpha+\gamma)-\varepsilon} (\tau-s)^{\varepsilon-1}.
\end{align*}
When $j = i+1$, one has $\tau-s \leq 2 h$, which indicates
\begin{align*}
I_{i,j}^{\star}(s,\tau)
&\leq C \int_{t_i}^{s} (s-u)^{\alpha+\gamma-1} (t_j-u)^{\alpha+\gamma-1} \mathrm d u \, h^{1-\varepsilon} (\tau-s)^{\varepsilon-1} \\
&\leq C h^{1-\varepsilon} (\tau-s)^{\varepsilon-1} \int_{t_i}^{s} (s-u)^{2(\alpha+\gamma-1)} \mathrm d u
\leq C h^{2(\alpha+\gamma)-\varepsilon} (\tau-s)^{\varepsilon-1}.
\end{align*}
Thus, $I_{i,j}^{\star}(s,\tau) \leq C h^{2(\alpha+\gamma)-\varepsilon} (\tau-s)^{\varepsilon-1}$. Then, by recalling \eqref{esti.preJ35star} and using Lemma \ref{lem.usetoJ35sum},
\begin{align*}
J_{3,5}^{\star\star}
\leq C h^{2\min\{\alpha+\gamma, \alpha + \frac{\alpha r}{2\beta} + (\gamma-\frac{1}{2})^{+}\} - \varepsilon}.
\end{align*}
\underline{Case 2:\ $\alpha + \gamma = 1$}. By the change of variables,
\begin{align*}
I_{i,j}^{\star}(s,\tau)
&\leq C \int_{0}^{t_i} \int_{t_i}^{s} (v-u)^{\varepsilon-1} (v-u)^{-\varepsilon} \mathrm d v \int_{t_j}^{\tau} (\omega-u)^{\varepsilon-1} (\omega-u)^{-\varepsilon} \mathrm d \omega \mathrm d u \\
&\leq C \int_{0}^{t_i} (t_i-u)^{\varepsilon-1} (t_j-u)^{\varepsilon-1} \int_{t_i}^{s} (v-u)^{-\varepsilon} \mathrm d v \int_{t_j}^{\tau} (\omega-u)^{-\varepsilon} \mathrm d \omega \mathrm d u \\
&\leq C h^{2-2\varepsilon} \int_{0}^{t_i} (t_i-u)^{\varepsilon-1} (t_j-u)^{\varepsilon-1} \mathrm d u.
\end{align*}
When $j > i+1$, it follows from $\tau-s \leq 2 (t_j - u)$ with $u\in[0,t_i]$ that
\begin{align*}
I_{i,j}^{\star}(s,\tau)
\leq C h^{2-2\varepsilon} \int_{0}^{t_i} (t_i-u)^{\varepsilon-1} (t_j-u)^{\varepsilon-1} (t_j-u)^{1-\varepsilon} \mathrm d u \, (\tau-s)^{\varepsilon-1}
\leq C h^{2-2\varepsilon} (\tau-s)^{\varepsilon-1}.
\end{align*}
When $j = i+1$, it follows from $t_j-u \geq h$ with $u \in [0,t_i]$ and $\tau-s \leq 2 h$ that
\begin{align*}
I_{i,j}^{\star}(s,\tau)
\leq C h^{1-\varepsilon} \int_{0}^{t_i} (t_i-u)^{\varepsilon-1} \mathrm d u \, (\tau-s)^{1-\varepsilon} (\tau-s)^{\varepsilon-1}
\leq C h^{2-2\varepsilon} (\tau-s)^{\varepsilon-1}.
\end{align*}
Thus, $I_{i,j}^{\star}(s,\tau) \leq C h^{2-2\varepsilon} (\tau-s)^{\varepsilon-1}$. Then, by recalling \eqref{esti.preJ35star} and using Lemma \ref{lem.usetoJ35sum},
\begin{align*}
J_{3,5}^{\star\star}
\leq C h^{ 2\min\{1, \alpha + \frac{\alpha r}{2\beta} + (\gamma-\frac{1}{2})^{+} \} - 2\varepsilon }.
\end{align*}
\underline{Case 3:\ $\alpha + \gamma \in (1,2]$}. Using the mean value theorem shows
\begin{align*}
I_{i,j}^{\star}(s,\tau)
&\leq C \int_{0}^{t_i} (t_i-u)^{\alpha+\gamma-2} (s-t_i) (t_j-u)^{\alpha+\gamma-2} (\tau-t_j) \mathrm d u \\
&\leq C h^2 \int_{0}^{t_i} (t_i-u)^{\alpha+\gamma-2} (t_j-u)^{\alpha+\gamma-2} \mathrm d u.
\end{align*}
When $\alpha + \gamma \in (\frac{3}{2},2]$, one gets $I_{i,j}^{\star}(s,\tau) \leq C h^2 \int_{0}^{t_i} (t_i-u)^{2(\alpha+\gamma-2)} \mathrm d u \leq C h^2$. When $\alpha + \gamma \in (1,\frac{3}{2}]$ and $j > i+1$, it follows from $\tau-s \leq 2 (t_j - u)$ with $u \in [0,t_i]$ that
\begin{align*}
I_{i,j}^{\star}(s,\tau)
&\leq C h^2 \int_{0}^{t_i} (t_i-u)^{\alpha+\gamma-2} (t_j-u)^{\alpha+\gamma-2} (t_j-u)^{-2(\alpha+\gamma-\frac{3}{2})+\varepsilon} \mathrm d u \, (\tau-s)^{2(\alpha+\gamma-\frac{3}{2})-\varepsilon} \\
&\leq C h^2 (\tau-s)^{2(\alpha+\gamma-\frac{3}{2})-\varepsilon} \int_{0}^{t_i} (t_i-u)^{\varepsilon-1} \mathrm d u
\leq C h^2 (\tau-s)^{2(\alpha+\gamma-\frac{3}{2})-\varepsilon}.
\end{align*}
When $\alpha + \gamma \in (1,\frac{3}{2}]$ and $j = i+1$, it follows from $t_j-u \geq h$ with $u \in [0,t_i]$ and $\tau-s \leq 2 h$ that
\begin{align*}
I_{i,j}^{\star}(s,\tau)
&\leq C h^2 \int_{0}^{t_i} (t_i-u)^{\alpha+\gamma-2} (t_j-u)^{2(\alpha+\gamma)-3-\varepsilon} (t_j-u)^{1-(\alpha+\gamma)+\varepsilon} \mathrm d u \\
&\leq C h^{2(\alpha+\gamma)-1-\varepsilon} \int_{0}^{t_i} (t_i-u)^{\varepsilon-1} \mathrm d u \, (\tau-s)^{-2(\alpha+\gamma-\frac{3}{2})+\varepsilon} (\tau-s)^{2(\alpha+\gamma-\frac{3}{2})-\varepsilon} \\
&\leq C h^2 (\tau-s)^{2(\alpha+\gamma-\frac{3}{2})-\varepsilon}.
\end{align*}
Thus, $I_{i,j}^{\star}(s,\tau) \leq C h^2 (\tau-s)^{ \min\{0, 2(\alpha+\gamma-\frac{3}{2})-\varepsilon\} }$. Then, by recalling \eqref{esti.preJ35star} and using Lemma \ref{lem.usetoJ35sum},
\begin{align*}
J_{3,5}^{\star\star}
\leq C h^{ 2\min\{1, \alpha + \frac{\alpha r}{2\beta} + (\gamma-\frac{1}{2})^{+} - \varepsilon\} }.
\end{align*}
Therefore, one can conclude
\begin{align*}
J_{3,5}^{\star\star}
\leq C h^{ 2\min\{ 1, \alpha+\gamma -\varepsilon, \alpha + \frac{\alpha r}{2\beta} + (\gamma-\frac{1}{2})^{+} - \varepsilon \} },
\end{align*}
which together with \eqref{esti.J35tosub} and \eqref{esti.J35star} completes the proof.
\end{proof}
\emph{Proof of Theorem \ref{th.MLEulerErrorLinear}}. When the deterministic mapping $F$ is linear, the estimates of $J_{3,5}$, $J_{3,6}$ and $J_{3,7}$ in the proof of Theorem \ref{th.MLEulerErrorNonlinear} can be refined. In fact, under this assumption, $J_{3,5}$ has been estimated again in Lemma \ref{lem.J35linear}, and $J_{3,7} = 0$. For $J_{3,6}$, it follows from the independence of the noise increments, Lemma \ref{lem.Opera}(1), Lemma \ref{lem.DiffOpera} and Cauchy--Schwarz's inequality that
\begin{align} \label{eq.J36linear}
|J_{3,6}|^2
&= C \sum_{j=0}^{m-1} \Big\| \int_{t_j}^{t_{j+1}} \mathcal S_{0}(t_m-s) \int_{t_j}^{s} \mathcal S_{\gamma}(s-u) \mathrm d W(u) \mathrm d s \Big\|_{L^2(\Omega,H)}^2 \nonumber\\
&\leq C \sum_{j=0}^{m-1} \Big( \int_{t_j}^{t_{j+1}} \| A^{\frac{\kappa- r}{2}} \mathcal S_{0}(t_m-s) \|_{\mathcal L(H)} \Big\| \int_{t_j}^{s} \mathcal S_{\gamma}(s-u) \mathrm d W(u) \Big\|_{L^2(\Omega,\dot{H}^{ r-\kappa})} \mathrm d s \Big)^2 \nonumber\\
&\leq C \sum_{j=0}^{m-1} \Big( \int_{t_j}^{t_{j+1}} (t_m-s)^{ \alpha - 1 -\frac{\alpha}{2\beta}(\kappa- r) } (s-t_j)^{ \alpha + \gamma - \frac{1}{2} } \mathrm d s \Big)^2 \nonumber\\
&\leq C h^{2(\alpha+\gamma-\frac{1}{2})} \sum_{j=0}^{m-1} \int_{t_j}^{t_{j+1}} (t_m-s)^{2\varepsilon-1} \mathrm d s \int_{t_j}^{t_{j+1}} (t_m-s)^{ 2\alpha - 1 -\frac{\alpha}{\beta}(\kappa- r) - 2\varepsilon } \mathrm d s \nonumber\\
&\leq C h^{ 2\min\{\alpha + \frac{\alpha r}{2\beta} + (\gamma-\frac{1}{2})^{+} - \varepsilon, \alpha+\gamma\} }.
\end{align}
Now, by combining \eqref{eq.J3Split}, \eqref{eq.J31}, \eqref{eq.J32}--\eqref{eq.J34}, Lemma \ref{lem.J35linear}, \eqref{eq.J36linear} and the fact $J_{3,7} = 0$ as well as applying the singular version of Gr\"onwall's inequality (see e.g., \cite[Lemma A.2]{Kruse2014Book}), one can conclude that the temporal error term $J_3$ can be bounded by
\begin{align*}
J_3 \leq C t_m^{2\alpha-1} h^{ \min\{ \alpha + \frac{\alpha r}{2\beta} + (\gamma-\frac{1}{2})^{+}-\varepsilon, \alpha+\gamma -\varepsilon, 1 \} },
\end{align*}
which together with \eqref{err.split} and \eqref{err.spatial} completes the proof.
\hfill$\Box$
|
1,314,259,993,624 | arxiv | \subsection[\hspace{2cm}Skeleton Scheme for A=201]{ }
\begin{figure}[h]
\begin{center}
\includegraphics[angle=90]{201skeleton-0.ps}\\
\end{center}
\end{figure}
\clearpage
\begin{figure}[h]
\begin{center}
\includegraphics[angle=90]{201skeleton-1.ps}\\
\end{center}
\end{figure}
\clearpage
\pagestyle{bob}
\begin{center}
\section[\ensuremath{^{201}_{\ 76}}Os\ensuremath{_{125}^{~}}]{ }
\vspace{-30pt}
\setcounter{chappage}{1}
\subsection[\hspace{-0.2cm}Adopted Levels]{ }
\vspace{-20pt}
\vspace{0.3cm}
\hypertarget{OS0}{{\bf \small \underline{Adopted \hyperlink{201OS_LEVEL}{Levels}}}}\\
\vspace{4pt}
\vspace{8pt}
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}Q(\ensuremath{\beta^-})=5000 {\it SY}; S(n)=4360 {\it SY}; Q(\ensuremath{\alpha})=$-$2390 {\it SY}\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2021Wa16,B}{2021Wa16}}\\
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}Estimated \textit{SY} uncertainties: 360 keV for Q(\ensuremath{\beta}\ensuremath{^{-}}), 420 keV for S(n) and 500 keV for Q(\ensuremath{\alpha}) (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2021Wa16,B}{2021Wa16}).}\\
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2009St16,B}{2009St16}: \ensuremath{^{\textnormal{201}}}Os produced and identified in \ensuremath{^{\textnormal{9}}}Be(\ensuremath{^{\textnormal{208}}}Pb,x), E=1 GeV/nucleon from the UNILAC and SIS-18 accelerator complex at}\\
\parbox[b][0.3cm]{17.7cm}{GSI. Target=2.5 g/cm\ensuremath{^{\textnormal{2}}} \ensuremath{^{\textnormal{9}}}Be.}\\
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2012Ku26,B}{2012Ku26}: \ensuremath{^{\textnormal{201}}}Os produced and identified in \ensuremath{^{\textnormal{9}}}Be(\ensuremath{^{\textnormal{238}}}U,x), E=1 GeV/nucleon from the UNILAC and SIS-18 accelerator complex at}\\
\parbox[b][0.3cm]{17.7cm}{GSI. Target=1.6 g/cm\ensuremath{^{\textnormal{2}}} \ensuremath{^{\textnormal{9}}}Be.}\\
\vspace{12pt}
\hypertarget{201OS_LEVEL}{\underline{$^{201}$Os Levels}}\\
\begin{longtable}{cccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E(level)$^{}$}&J$^{\pi}$$^{}$&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{l}{(1/2\ensuremath{^{-}})}&\parbox[t][0.3cm]{15.235281cm}{\raggedright \%\ensuremath{\beta}\ensuremath{^{-}}=100\vspace{0.1cm}}&\\
&&&\parbox[t][0.3cm]{15.235281cm}{\raggedright J\ensuremath{^{\pi}}: assuming spherical shape and systematics of N=125 isotones in the region; shell model predictions.\vspace{0.1cm}}&\\
&&&\parbox[t][0.3cm]{15.235281cm}{\raggedright T\ensuremath{_{1/2}}: \ensuremath{>}160 ns from from the time-of-flight in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2012Ku26,B}{2012Ku26}. \ensuremath{\approx} 3 s estimated in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2021Ko07,B}{2021Ko07}. Predicted T\ensuremath{_{\textnormal{1/2}}}(\ensuremath{\beta}): 56 s\vspace{0.1cm}}&\\
&&&\parbox[t][0.3cm]{15.235281cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2019Mo01,B}{2019Mo01}, FRDM16), 2.1 s (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2016Ma12,B}{2016Ma12}, CDFT) and 87 s (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2020Ne08,B}{2020Ne08}, QRPA with Skyrme EDF).\vspace{0.1cm}}&\\
&&&\parbox[t][0.3cm]{15.235281cm}{\raggedright configuration: \ensuremath{\nu} p\ensuremath{_{\textnormal{1/2}}^{\textnormal{$-$1}}} from systematics of N=125 isotones in the region; shell model predictions.\vspace{0.1cm}}&\\
\end{longtable}
\clearpage
\section[\ensuremath{^{201}_{\ 77}}Ir\ensuremath{_{124}^{~}}]{ }
\vspace{-30pt}
\setcounter{chappage}{1}
\subsection[\hspace{-0.2cm}Adopted Levels]{ }
\vspace{-20pt}
\vspace{0.3cm}
\hypertarget{IR1}{{\bf \small \underline{Adopted \hyperlink{201IR_LEVEL}{Levels}}}}\\
\vspace{4pt}
\vspace{8pt}
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}Q(\ensuremath{\beta^-})=3900 {\it SY}; S(n)=6340 {\it SY}; S(p)=8580 {\it SY}; Q(\ensuremath{\alpha})=$-$1910 {\it SY}\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2021Wa16,B}{2021Wa16}}\\
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}Estimated \textit{SY} uncertainties: 210 keV for Q(\ensuremath{\beta}\ensuremath{^{-}}), 280 keV for S(n), 360 keV for S(p) and 360 keV for Q(\ensuremath{\alpha}) (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2021WA16,B}{2021WA16}).}\\
\vspace{12pt}
\hypertarget{201IR_LEVEL}{\underline{$^{201}$Ir Levels}}\\
\begin{longtable}[c]{ll}
\multicolumn{2}{c}{\underline{Cross Reference (XREF) Flags}}\\
\\
\hyperlink{IR2}{\texttt{A }}& \ensuremath{^{\textnormal{9}}}Be(\ensuremath{^{\textnormal{208}}}Pb,X\ensuremath{\gamma})\\
\end{longtable}
\vspace{-0.5cm}
\begin{longtable}{ccccccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E(level)$^{}$}&J$^{\pi}$$^{}$&\multicolumn{2}{c}{T$_{1/2}$$^{}$}&XREF&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{l}{(3/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{21}&\multicolumn{1}{@{ }l}{s {\it 5}}&\multicolumn{1}{l}{\texttt{\hyperlink{IR2}{A}} }&\parbox[t][0.3cm]{11.952881cm}{\raggedright \%\ensuremath{\beta}\ensuremath{^{-}}=100\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.952881cm}{\raggedright J\ensuremath{^{\pi}}: assuming spherical shape and systematics of neighboring Z=77 nuclei; shell model\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.952881cm}{\raggedright {\ }{\ }{\ }predictions.\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.952881cm}{\raggedright T\ensuremath{_{1/2}}: from \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2014Mo15,B}{2014Mo15}, using Monte Carlo analysis of ion-\ensuremath{\beta}\ensuremath{\gamma}(time) data.\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.952881cm}{\raggedright configuration: \ensuremath{\pi} d\ensuremath{_{\textnormal{3/2}}^{\textnormal{$-$1}}} from systematics of known Z=77 isotopes; shell model\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.952881cm}{\raggedright {\ }{\ }{\ }predictions.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{+x}&&\multicolumn{1}{r@{}}{10}&\multicolumn{1}{@{.}l}{5 ns {\it 17}}&\multicolumn{1}{l}{\texttt{\hyperlink{IR2}{A}} }&\parbox[t][0.3cm]{11.952881cm}{\raggedright E(level): 439.6 keV, 452.0 keV and 680.9 keV \ensuremath{\gamma} rays observed below the isomer\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.952881cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2011St21,B}{2011St21}), but the ordering is unknown. Given the reported \ensuremath{\gamma}-ray intensities in\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.952881cm}{\raggedright {\ }{\ }{\ }\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2011St21,B}{2011St21}, 439.6\ensuremath{\gamma} and 452.0\ensuremath{\gamma} are most-likely in paralel in the decay scheme. Thus,\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.952881cm}{\raggedright {\ }{\ }{\ }E(level)=1132.9 keV can be expected.\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.952881cm}{\raggedright T\ensuremath{_{1/2}}: from sum of 439.6,452.0,680.9\ensuremath{\gamma}(t) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2011St21,B}{2011St21}.\vspace{0.1cm}}&\\
\end{longtable}
\clearpage
\subsection[\hspace{-0.2cm}\ensuremath{^{\textnormal{9}}}Be(\ensuremath{^{\textnormal{208}}}Pb,X\ensuremath{\gamma})]{ }
\vspace{-27pt}
\vspace{0.3cm}
\hypertarget{IR2}{{\bf \small \underline{\ensuremath{^{\textnormal{9}}}Be(\ensuremath{^{\textnormal{208}}}Pb,X\ensuremath{\gamma})\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2011St21,B}{2011St21}}}}\\
\vspace{4pt}
\vspace{8pt}
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2011St21,B}{2011St21}: \ensuremath{^{\textnormal{201}}}Ir nuclide produced by in-flight fragmentation of 1 GeV/A \ensuremath{^{\textnormal{208}}}Pb beam at the GSI UNILAC and SIS-18 accelerator}\\
\parbox[b][0.3cm]{17.7cm}{complex. Target thickness=2.526 g/cm\ensuremath{^{\textnormal{2}}}, backed by a 0.223 g/cm\ensuremath{^{\textnormal{2}}} thick \ensuremath{^{\textnormal{93}}}Nb foil. Fragments identified by the Fragment Separator}\\
\parbox[b][0.3cm]{17.7cm}{(FRS), based on time of flight, B\ensuremath{\rho} and energy loss. The ions were slowed down in Al degraders and stopped in a plastic catcher.}\\
\parbox[b][0.3cm]{17.7cm}{The stopper was surrounded by the RISING \ensuremath{\gamma}-ray spectrometer. Measured: E\ensuremath{\gamma}, I\ensuremath{\gamma}, delayed \ensuremath{\gamma} rays, isomer lifetime. Others (same}\\
\parbox[b][0.3cm]{17.7cm}{authors): \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2009St16,B}{2009St16}, \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2008StZY,B}{2008StZY}.}\\
\vspace{12pt}
\underline{$^{201}$Ir Levels}\\
\begin{longtable}{cccccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E(level)$^{}$}&J$^{\pi}$$^{{\hyperlink{IR2LEVEL0}{\dagger}}}$&\multicolumn{2}{c}{T$_{1/2}$$^{{\hyperlink{IR2LEVEL0}{\dagger}}}$}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{l}{(3/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{21}&\multicolumn{1}{@{ }l}{s {\it 5}}&&\\
\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{+x}&&\multicolumn{1}{r@{}}{10}&\multicolumn{1}{@{.}l}{5 ns {\it 17}}&\parbox[t][0.3cm]{13.172501cm}{\raggedright T\ensuremath{_{1/2}}: from sum of 440,452,681\ensuremath{\gamma}(t) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2011St21,B}{2011St21}.\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{13.172501cm}{\raggedright Experimental isomeric state population ratio \ensuremath{\geq}3\% \textit{2} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2011St21,B}{2011St21}).\vspace{0.1cm}}&\\
\end{longtable}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{IR2LEVEL0}{\dagger}}}} From Adopted Levels, unless otherwise stated.}\\
\vspace{0.5cm}
\underline{$\gamma$($^{201}$Ir)}\\
\begin{longtable}{cccccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{IR2GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{I\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{IR2GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{E\ensuremath{_{i}}(level)}&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&
\endfirsthead
\multicolumn{1}{r@{}}{$^{x}$439}&\multicolumn{1}{@{.}l}{6\ensuremath{^{\hyperlink{IR2GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{39}&\multicolumn{1}{@{ }l}{{\it 9}}&&&\\
\multicolumn{1}{r@{}}{$^{x}$452}&\multicolumn{1}{@{.}l}{0\ensuremath{^{\hyperlink{IR2GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{51}&\multicolumn{1}{@{ }l}{{\it 9}}&&&\\
\multicolumn{1}{r@{}}{$^{x}$680}&\multicolumn{1}{@{.}l}{9\ensuremath{^{\hyperlink{IR2GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{ }l}{{\it 13}}&&&\\
\end{longtable}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{IR2GAMMA0}{\dagger}}}} From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2011St21,B}{2011St21}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{IR2GAMMA1}{\ddagger}}}} observed below the 10.5-ns isomer, but the ordering is unknown.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{IR2GAMMA2}{x}}}} \ensuremath{\gamma} ray not placed in level scheme.}\\
\vspace{0.5cm}
\clearpage
\section[\ensuremath{^{201}_{\ 78}}Pt\ensuremath{_{123}^{~}}]{ }
\vspace{-30pt}
\setcounter{chappage}{1}
\subsection[\hspace{-0.2cm}Adopted Levels, Gammas]{ }
\vspace{-20pt}
\vspace{0.3cm}
\hypertarget{PT3}{{\bf \small \underline{Adopted \hyperlink{201PT_LEVEL}{Levels}, \hyperlink{201PT_GAMMA}{Gammas}}}}\\
\vspace{4pt}
\vspace{8pt}
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}Q(\ensuremath{\beta^-})=2660 {\it 50}; S(n)=5210 {\it 50}; S(p)=9460 {\it SY}; Q(\ensuremath{\alpha})=$-$1090 {\it SY}\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2021Wa16,B}{2021Wa16}}\\
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}Estimated \textit{SY} uncertainties: 200 keV for S(p) and 210 keV for Q(\ensuremath{\alpha}) (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2021WA16,B}{2021WA16}).}\\
\vspace{12pt}
\hypertarget{201PT_LEVEL}{\underline{$^{201}$Pt Levels}}\\
\begin{longtable}[c]{ll}
\multicolumn{2}{c}{\underline{Cross Reference (XREF) Flags}}\\
\\
\hyperlink{IR4}{\texttt{A }}& \ensuremath{^{\textnormal{201}}}Ir \ensuremath{\beta}\ensuremath{^{-}} decay\\
\hyperlink{PT5}{\texttt{B }}& Be(\ensuremath{^{\textnormal{208}}}Pb,X\ensuremath{\gamma})\\
\end{longtable}
\vspace{-0.5cm}
\begin{longtable}{ccccccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E(level)$^{{\hyperlink{PT3LEVEL0}{\dagger}}}$}&J$^{\pi}$$^{{\hyperlink{PT3LEVEL3}{@}}}$&\multicolumn{2}{c}{T$_{1/2}$$^{}$}&XREF&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{l}{(5/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{46 min {\it 9}}&\multicolumn{1}{l}{\texttt{\hyperlink{IR4}{A}\hyperlink{PT5}{B}} }&\parbox[t][0.3cm]{11.26808cm}{\raggedright \%\ensuremath{\beta}\ensuremath{^{-}}=100\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.26808cm}{\raggedright J\ensuremath{^{\pi}}: From systematics in neighboring N=123 nuclides and shell model predictions.\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.26808cm}{\raggedright T\ensuremath{_{1/2}}: Weighted average of 2.5 m \textit{1} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1963Go06,B}{1963Go06}) and 2.3 m \textit{2} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1962Fa06,B}{1962Fa06}).\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.26808cm}{\raggedright configuration: \ensuremath{\nu} f\ensuremath{_{\textnormal{5/2}}^{\textnormal{$-$1}}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{332}&\multicolumn{1}{@{.}l}{6\ensuremath{^{{\hyperlink{PT3LEVEL2}{\#}}}} {\it 17}}&&&&\multicolumn{1}{l}{\texttt{\hyperlink{IR4}{A}\ } }&&\\
\multicolumn{1}{r@{}}{373}&\multicolumn{1}{@{.}l}{9\ensuremath{^{{\hyperlink{PT3LEVEL1}{\ddagger}}}} {\it 5}}&\multicolumn{1}{l}{(9/2\ensuremath{^{-}})}&&&\multicolumn{1}{l}{\texttt{\ \hyperlink{PT5}{B}} }&\parbox[t][0.3cm]{11.26808cm}{\raggedright configuration: \ensuremath{\nu} (f\ensuremath{_{\textnormal{5/2}}^{\textnormal{$-$1}}})\ensuremath{\otimes}2\ensuremath{^{\textnormal{+}}}. The assignment is tentative.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{389}&\multicolumn{1}{@{.}l}{4\ensuremath{^{{\hyperlink{PT3LEVEL2}{\#}}}} {\it 13}}&&&&\multicolumn{1}{l}{\texttt{\hyperlink{IR4}{A}\ } }&&\\
\multicolumn{1}{r@{}}{655}&\multicolumn{1}{@{.}l}{0\ensuremath{^{{\hyperlink{PT3LEVEL2}{\#}}}} {\it 20}}&&&&\multicolumn{1}{l}{\texttt{\hyperlink{IR4}{A}\ } }&&\\
\multicolumn{1}{r@{}}{710}&\multicolumn{1}{@{.}l}{0\ensuremath{^{{\hyperlink{PT3LEVEL2}{\#}}}} {\it 13}}&&&&\multicolumn{1}{l}{\texttt{\hyperlink{IR4}{A}\ } }&&\\
\multicolumn{1}{r@{}}{741}&\multicolumn{1}{@{.}l}{0\ensuremath{^{{\hyperlink{PT3LEVEL2}{\#}}}} {\it 20}}&&&&\multicolumn{1}{l}{\texttt{\hyperlink{IR4}{A}\ } }&&\\
\multicolumn{1}{r@{}}{1101}&\multicolumn{1}{@{.}l}{4\ensuremath{^{{\hyperlink{PT3LEVEL1}{\ddagger}}}}}&\multicolumn{1}{l}{(13/2\ensuremath{^{-}})}&&&\multicolumn{1}{l}{\texttt{\ \hyperlink{PT5}{B}} }&\parbox[t][0.3cm]{11.26808cm}{\raggedright configuration: \ensuremath{\nu} (f\ensuremath{_{\textnormal{5/2}}^{\textnormal{$-$1}}})\ensuremath{\otimes}4\ensuremath{^{\textnormal{+}}}. The assignment is tentative.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1442}&\multicolumn{1}{@{.}l}{0\ensuremath{^{{\hyperlink{PT3LEVEL2}{\#}}}} {\it 24}}&&&&\multicolumn{1}{l}{\texttt{\hyperlink{IR4}{A}\ } }&&\\
\multicolumn{1}{r@{}}{1455}&\multicolumn{1}{@{.}l}{5\ensuremath{^{{\hyperlink{PT3LEVEL1}{\ddagger}}}}}&\multicolumn{1}{l}{(15/2\ensuremath{^{+}})}&&&\multicolumn{1}{l}{\texttt{\ \hyperlink{PT5}{B}} }&&\\
\multicolumn{1}{r@{}}{1455}&\multicolumn{1}{@{.}l}{5+x\ensuremath{^{{\hyperlink{PT3LEVEL1}{\ddagger}}}}}&\multicolumn{1}{l}{(19/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{18}&\multicolumn{1}{@{.}l}{4 ns {\it 13}}&\multicolumn{1}{l}{\texttt{\ \hyperlink{PT5}{B}} }&\parbox[t][0.3cm]{11.26808cm}{\raggedright E(level): x\ensuremath{<}90 keV in both \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2005Ca02,B}{2005Ca02} and \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2011St21,B}{2011St21}. Direct \ensuremath{\gamma}-ray decay to the\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.26808cm}{\raggedright {\ }{\ }{\ }1455.5 keV level was not observed.\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.26808cm}{\raggedright T\ensuremath{_{1/2}}: from \ensuremath{\gamma}(t) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2011St21,B}{2011St21}. Other: 21 ns \textit{3} from \ensuremath{\gamma}(t) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2005Ca02,B}{2005Ca02}.\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.26808cm}{\raggedright configuration: \ensuremath{\nu} (f\ensuremath{_{\textnormal{5/2}}^{\textnormal{$-$1}}}) \ensuremath{\pi} (d\ensuremath{_{\textnormal{3/2}}^{\textnormal{$-$1}}},h\ensuremath{_{\textnormal{11/2}}^{\textnormal{$-$1}}}). The assignment is tentative.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1706}&\multicolumn{1}{@{.}l}{3\ensuremath{^{{\hyperlink{PT3LEVEL2}{\#}}}} {\it 13}}&&&&\multicolumn{1}{l}{\texttt{\hyperlink{IR4}{A}\ } }&&\\
\end{longtable}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PT3LEVEL0}{\dagger}}}} From a least-squares fit to E\ensuremath{\gamma}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PT3LEVEL1}{\ddagger}}}} Level populated only in Be(\ensuremath{^{\textnormal{208}}}Pb,x\ensuremath{\gamma}).}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PT3LEVEL2}{\#}}}} Level populated only in \ensuremath{^{\textnormal{201}}}Ir \ensuremath{\beta}\ensuremath{^{-}} decay (\ensuremath{J^{\pi}}=(3/2\ensuremath{^{+}})).}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PT3LEVEL3}{@}}}} From Be(\ensuremath{^{\textnormal{208}}}Pb,x\ensuremath{\gamma}), unless otherwise stated.}\\
\vspace{0.5cm}
\hypertarget{201PT_GAMMA}{\underline{$\gamma$($^{201}$Pt)}}\\
\begin{longtable}{ccccccccc@{}c@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E\ensuremath{_{i}}(level)}&J\ensuremath{^{\pi}_{i}}&\multicolumn{2}{c}{E\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{PT3GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{I\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{PT3GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{E\ensuremath{_{f}}}&J\ensuremath{^{\pi}_{f}}&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{332}&\multicolumn{1}{@{.}l}{6}&&\multicolumn{1}{r@{}}{332}&\multicolumn{1}{@{.}l}{6\ensuremath{^{\hyperlink{PT3GAMMA1}{\ddagger}}} {\it 17}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PT3GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{@{}l}{(5/2\ensuremath{^{-}})}&\\
\multicolumn{1}{r@{}}{373}&\multicolumn{1}{@{.}l}{9}&\multicolumn{1}{l}{(9/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{373}&\multicolumn{1}{@{.}l}{9 {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{@{}l}{(5/2\ensuremath{^{-}})}&\\
\multicolumn{1}{r@{}}{389}&\multicolumn{1}{@{.}l}{4}&&\multicolumn{1}{r@{}}{389}&\multicolumn{1}{@{.}l}{4\ensuremath{^{\hyperlink{PT3GAMMA1}{\ddagger}}} {\it 15}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PT3GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{@{}l}{(5/2\ensuremath{^{-}})}&\\
\multicolumn{1}{r@{}}{655}&\multicolumn{1}{@{.}l}{0}&&\multicolumn{1}{r@{}}{655}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PT3GAMMA1}{\ddagger}}} {\it 2}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PT3GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{@{}l}{(5/2\ensuremath{^{-}})}&\\
\multicolumn{1}{r@{}}{710}&\multicolumn{1}{@{.}l}{0}&&\multicolumn{1}{r@{}}{710}&\multicolumn{1}{@{.}l}{1\ensuremath{^{\hyperlink{PT3GAMMA1}{\ddagger}}} {\it 15}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PT3GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{@{}l}{(5/2\ensuremath{^{-}})}&\\
\multicolumn{1}{r@{}}{741}&\multicolumn{1}{@{.}l}{0}&&\multicolumn{1}{r@{}}{741}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PT3GAMMA1}{\ddagger}}} {\it 2}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PT3GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{@{}l}{(5/2\ensuremath{^{-}})}&\\
\multicolumn{1}{r@{}}{1101}&\multicolumn{1}{@{.}l}{4}&\multicolumn{1}{l}{(13/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{726}&\multicolumn{1}{@{.}l}{9 {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{373}&\multicolumn{1}{@{.}l}{9 }&\multicolumn{1}{@{}l}{(9/2\ensuremath{^{-}})}&\\
\multicolumn{1}{r@{}}{1442}&\multicolumn{1}{@{.}l}{0}&&\multicolumn{1}{r@{}}{732}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PT3GAMMA1}{\ddagger}}} {\it 2}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PT3GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{710}&\multicolumn{1}{@{.}l}{0 }&&\\
\multicolumn{1}{r@{}}{1455}&\multicolumn{1}{@{.}l}{5}&\multicolumn{1}{l}{(15/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{353}&\multicolumn{1}{@{.}l}{6 {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{1101}&\multicolumn{1}{@{.}l}{4}&\multicolumn{1}{@{}l}{(13/2\ensuremath{^{-}})}&\\
\end{longtable}
\begin{textblock}{29}(0,27.3)
Continued on next page (footnotes at end of table)
\end{textblock}
\clearpage
\begin{longtable}{cccccccc@{}c@{\extracolsep{\fill}}c}
\\[-.4cm]
\multicolumn{10}{c}{{\bf \small \underline{Adopted \hyperlink{201PT_LEVEL}{Levels}, \hyperlink{201PT_GAMMA}{Gammas} (continued)}}}\\
\multicolumn{10}{c}{~}\\
\multicolumn{10}{c}{\underline{$\gamma$($^{201}$Pt) (continued)}}\\
\multicolumn{10}{c}{~~~}\\
\multicolumn{2}{c}{E\ensuremath{_{i}}(level)}&\multicolumn{2}{c}{E\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{PT3GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{I\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{PT3GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{E\ensuremath{_{f}}}&J\ensuremath{^{\pi}_{f}}&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endhead
\multicolumn{1}{r@{}}{1706}&\multicolumn{1}{@{.}l}{3}&\multicolumn{1}{r@{}}{996}&\multicolumn{1}{@{.}l}{5\ensuremath{^{\hyperlink{PT3GAMMA1}{\ddagger}}} {\it 16}}&\multicolumn{1}{r@{}}{42}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PT3GAMMA1}{\ddagger}}} {\it 10}}&\multicolumn{1}{r@{}}{710}&\multicolumn{1}{@{.}l}{0 }&&\\
&&\multicolumn{1}{r@{}}{1317}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PT3GAMMA1}{\ddagger}}} {\it 2}}&\multicolumn{1}{r@{}}{92}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PT3GAMMA1}{\ddagger}}} {\it 17}}&\multicolumn{1}{r@{}}{389}&\multicolumn{1}{@{.}l}{4 }&&\\
&&\multicolumn{1}{r@{}}{1706}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PT3GAMMA1}{\ddagger}}} {\it 2}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PT3GAMMA1}{\ddagger}}} {\it 17}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{@{}l}{(5/2\ensuremath{^{-}})}&\\
\end{longtable}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PT3GAMMA0}{\dagger}}}} From Be(\ensuremath{^{\textnormal{208}}}Pb,x\ensuremath{\gamma}), unless otherwise stated.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PT3GAMMA1}{\ddagger}}}} From \ensuremath{^{\textnormal{201}}}Ir \ensuremath{\beta}\ensuremath{^{-}} decay.}\\
\vspace{0.5cm}
\clearpage
\begin{figure}[h]
\begin{center}
\includegraphics{201PT3-0.ps}\\
\end{center}
\end{figure}
\clearpage
\subsection[\hspace{-0.2cm}\ensuremath{^{\textnormal{201}}}Ir \ensuremath{\beta}\ensuremath{^{-}} decay]{ }
\vspace{-27pt}
\vspace{0.3cm}
\hypertarget{IR4}{{\bf \small \underline{\ensuremath{^{\textnormal{201}}}Ir \ensuremath{\beta}\ensuremath{^{-}} decay\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2013Mo20,B}{2013Mo20}}}}\\
\vspace{4pt}
\vspace{8pt}
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}Parent: $^{201}$Ir: E=0.0; J$^{\pi}$=(3/2\ensuremath{^{+}}); T$_{1/2}$=21 s {\it 5}; Q(\ensuremath{\beta}\ensuremath{^{-}})=3900 {\it SY}; \%\ensuremath{\beta}\ensuremath{^{-}} decay=100.0
}\\
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2013Mo20,B}{2013Mo20}: \ensuremath{^{\textnormal{201}}}Ir produced in cold fragmentation reactions with E=1 GeV/A \ensuremath{^{\textnormal{208}}}Pb beam impinging a 2.5 g/cm\ensuremath{^{\textnormal{2}}} thick Be target.}\\
\parbox[b][0.3cm]{17.7cm}{The beam was provided by SIS-18 synchrotron at GSI facility. Residues were separated using Fragment Separator. Measured E\ensuremath{\gamma},}\\
\parbox[b][0.3cm]{17.7cm}{I\ensuremath{\gamma}, \ensuremath{\gamma}\ensuremath{\gamma}-coin, \ensuremath{\beta}\ensuremath{\gamma}-coin, fragment-\ensuremath{\gamma} correlated event using RISING array of 15 cluster detectors. Others (same authors): \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2011MoZP,B}{2011MoZP}.}\\
\vspace{12pt}
\underline{$^{201}$Pt Levels}\\
\begin{longtable}{cccccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E(level)$^{{\hyperlink{PT4LEVEL0}{\dagger}}}$}&J$^{\pi}$$^{}$&\multicolumn{2}{c}{T$_{1/2}$$^{}$}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{l}{(5/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{46 min {\it 9}}&\parbox[t][0.3cm]{13.226cm}{\raggedright \%\ensuremath{\beta}\ensuremath{^{-}}=100\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{13.226cm}{\raggedright J\ensuremath{^{\pi}},T\ensuremath{_{1/2}}: From Adopted Levels.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{332}&\multicolumn{1}{@{.}l}{6 {\it 17}}&&&&&\\
\multicolumn{1}{r@{}}{389}&\multicolumn{1}{@{.}l}{4 {\it 13}}&&&&&\\
\multicolumn{1}{r@{}}{655}&\multicolumn{1}{@{.}l}{0 {\it 20}}&&&&&\\
\multicolumn{1}{r@{}}{710}&\multicolumn{1}{@{.}l}{0 {\it 13}}&&&&&\\
\multicolumn{1}{r@{}}{741}&\multicolumn{1}{@{.}l}{0 {\it 20}}&&&&&\\
\multicolumn{1}{r@{}}{1442}&\multicolumn{1}{@{.}l}{0 {\it 24}}&&&&&\\
\multicolumn{1}{r@{}}{1706}&\multicolumn{1}{@{.}l}{3 {\it 13}}&&&&&\\
\end{longtable}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PT4LEVEL0}{\dagger}}}} From least-squares fit to E\ensuremath{\gamma}.}\\
\vspace{0.5cm}
\underline{$\gamma$($^{201}$Pt)}\\
\vspace{0.34cm}
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.254cm}I\ensuremath{\gamma} normalization: Since the ground state to ground state \ensuremath{\beta}-decay feeding is not known, the decay scheme is uncertain and no \ensuremath{\beta}\ensuremath{^{-}}}\\
\parbox[b][0.3cm]{17.7cm}{intensities and log ft values are provided.}\\
\vspace{0.34cm}
\begin{longtable}{cccccccc@{}c@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{IR4GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{I\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{IR4GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{E\ensuremath{_{i}}(level)}&\multicolumn{2}{c}{E\ensuremath{_{f}}}&J\ensuremath{^{\pi}_{f}}&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{332}&\multicolumn{1}{@{.}l}{6 {\it 17}}&\multicolumn{1}{r@{}}{31}&\multicolumn{1}{@{ }l}{{\it 3}}&\multicolumn{1}{r@{}}{332}&\multicolumn{1}{@{.}l}{6}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{@{}l}{(5/2\ensuremath{^{-}})}&\\
\multicolumn{1}{r@{}}{389}&\multicolumn{1}{@{.}l}{4 {\it 15}}&\multicolumn{1}{r@{}}{26}&\multicolumn{1}{@{ }l}{{\it 3}}&\multicolumn{1}{r@{}}{389}&\multicolumn{1}{@{.}l}{4}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{@{}l}{(5/2\ensuremath{^{-}})}&\\
\multicolumn{1}{r@{}}{655}&\multicolumn{1}{@{ }l}{{\it 2}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{ }l}{{\it 8}}&\multicolumn{1}{r@{}}{655}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{@{}l}{(5/2\ensuremath{^{-}})}&\\
\multicolumn{1}{r@{}}{710}&\multicolumn{1}{@{.}l}{1 {\it 15}}&\multicolumn{1}{r@{}}{46}&\multicolumn{1}{@{ }l}{{\it 4}}&\multicolumn{1}{r@{}}{710}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{@{}l}{(5/2\ensuremath{^{-}})}&\\
\multicolumn{1}{r@{}}{732}&\multicolumn{1}{@{ }l}{{\it 2}}&\multicolumn{1}{r@{}}{28}&\multicolumn{1}{@{ }l}{{\it 3}}&\multicolumn{1}{r@{}}{1442}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{r@{}}{710}&\multicolumn{1}{@{.}l}{0 }&&\\
\multicolumn{1}{r@{}}{741}&\multicolumn{1}{@{ }l}{{\it 2}}&\multicolumn{1}{r@{}}{22}&\multicolumn{1}{@{ }l}{{\it 3}}&\multicolumn{1}{r@{}}{741}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{@{}l}{(5/2\ensuremath{^{-}})}&\\
\multicolumn{1}{r@{}}{996}&\multicolumn{1}{@{.}l}{5 {\it 16}}&\multicolumn{1}{r@{}}{5}&\multicolumn{1}{@{.}l}{0 {\it 12}}&\multicolumn{1}{r@{}}{1706}&\multicolumn{1}{@{.}l}{3}&\multicolumn{1}{r@{}}{710}&\multicolumn{1}{@{.}l}{0 }&&\\
\multicolumn{1}{r@{}}{1317}&\multicolumn{1}{@{ }l}{{\it 2}}&\multicolumn{1}{r@{}}{11}&\multicolumn{1}{@{ }l}{{\it 2}}&\multicolumn{1}{r@{}}{1706}&\multicolumn{1}{@{.}l}{3}&\multicolumn{1}{r@{}}{389}&\multicolumn{1}{@{.}l}{4 }&&\\
\multicolumn{1}{r@{}}{1706}&\multicolumn{1}{@{ }l}{{\it 2}}&\multicolumn{1}{r@{}}{12}&\multicolumn{1}{@{ }l}{{\it 2}}&\multicolumn{1}{r@{}}{1706}&\multicolumn{1}{@{.}l}{3}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{@{}l}{(5/2\ensuremath{^{-}})}&\\
\end{longtable}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{IR4GAMMA0}{\dagger}}}} From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2013Mo20,B}{2013Mo20}.}\\
\vspace{0.5cm}
\clearpage
\begin{figure}[h]
\begin{center}
\includegraphics{201PT4-0.ps}\\
\end{center}
\end{figure}
\clearpage
\subsection[\hspace{-0.2cm}Be(\ensuremath{^{\textnormal{208}}}Pb,X\ensuremath{\gamma})]{ }
\vspace{-27pt}
\vspace{0.3cm}
\hypertarget{PT5}{{\bf \small \underline{Be(\ensuremath{^{\textnormal{208}}}Pb,X\ensuremath{\gamma})\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2005Ca02,B}{2005Ca02},\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2011St21,B}{2011St21}}}}\\
\vspace{4pt}
\vspace{8pt}
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2005Ca02,B}{2005Ca02}: Projectile fragmentation of \ensuremath{^{\textnormal{208}}}Pb beam at 1 GeV/A on a 1.6 g/cm\ensuremath{^{\textnormal{2}}} Be target. Fragment Recoil Separator at GSI.}\\
\parbox[b][0.3cm]{17.7cm}{Measured E\ensuremath{\gamma}, I\ensuremath{\gamma}, \ensuremath{\gamma}\ensuremath{\gamma}, \ensuremath{\gamma}\ensuremath{\gamma}(t) using four ``Clover'' type Ge detectors (providing 16 independent Ge crystals). Others (same}\\
\parbox[b][0.3cm]{17.7cm}{collaboration): \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2001Ca13,B}{2001Ca13}, \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2002Po15,B}{2002Po15}, \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2003Po14,B}{2003Po14}, \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2001MaZV,B}{2001MaZV}, \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2000PoZY,B}{2000PoZY}.}\\
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2011St21,B}{2011St21}: in-flight fragmentation of \ensuremath{^{\textnormal{208}}}Pb beam at 1 GeV/A on a 2.526 g/cm\ensuremath{^{\textnormal{2}}} Be target, backed by 0.223 g/cm\ensuremath{^{\textnormal{2}}}-thick \ensuremath{^{\textnormal{93}}}Nb foil.}\\
\parbox[b][0.3cm]{17.7cm}{Fragment Recoil Separator at GSI. Measured E\ensuremath{\gamma}, I\ensuremath{\gamma}, \ensuremath{\gamma}\ensuremath{\gamma}, \ensuremath{\gamma}\ensuremath{\gamma}(t) using the RISING \ensuremath{\gamma}-ray spectrometer. Other: \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2008StZY,B}{2008StZY}.}\\
\vspace{12pt}
\underline{$^{201}$Pt Levels}\\
\begin{longtable}{cccccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E(level)$^{{\hyperlink{PT5LEVEL0}{\dagger}}}$}&J$^{\pi}$$^{{\hyperlink{PT5LEVEL1}{\ddagger}}}$&\multicolumn{2}{c}{T$_{1/2}$$^{}$}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{l}{(5/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{46 min {\it 9}}&\parbox[t][0.3cm]{12.718821cm}{\raggedright J\ensuremath{^{\pi}},T\ensuremath{_{1/2}}: From Adopted Levels.\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{12.718821cm}{\raggedright configuration: \ensuremath{\nu} f\ensuremath{_{\textnormal{5/2}}^{\textnormal{$-$1}}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{374}&\multicolumn{1}{@{.}l}{2 {\it 10}}&\multicolumn{1}{l}{(9/2\ensuremath{^{-}})}&&&\parbox[t][0.3cm]{12.718821cm}{\raggedright configuration: \ensuremath{\nu} (f\ensuremath{_{\textnormal{5/2}}^{\textnormal{$-$1}}})\ensuremath{\otimes}2\ensuremath{^{\textnormal{+}}}. The assignment is tentative.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1101}&\multicolumn{1}{@{.}l}{4 {\it 15}}&\multicolumn{1}{l}{(13/2\ensuremath{^{-}})}&&&\parbox[t][0.3cm]{12.718821cm}{\raggedright configuration: \ensuremath{\nu} (f\ensuremath{_{\textnormal{5/2}}^{\textnormal{$-$1}}})\ensuremath{\otimes}4\ensuremath{^{\textnormal{+}}}. The assignment is tentative.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1455}&\multicolumn{1}{@{.}l}{5 {\it 18}}&\multicolumn{1}{l}{(15/2\ensuremath{^{+}})}&&&&\\
\multicolumn{1}{r@{}}{1455}&\multicolumn{1}{@{.}l}{5+x}&\multicolumn{1}{l}{(19/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{18}&\multicolumn{1}{@{.}l}{4 ns {\it 13}}&\parbox[t][0.3cm]{12.718821cm}{\raggedright E(level): x\ensuremath{<}90 keV in both \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2005Ca02,B}{2005Ca02} and \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2011St21,B}{2011St21}. Direct \ensuremath{\gamma}-ray decay to the 1455.5 keV\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{12.718821cm}{\raggedright {\ }{\ }{\ }level was not observed.\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{12.718821cm}{\raggedright T\ensuremath{_{1/2}}: from \ensuremath{\gamma}(t) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2011St21,B}{2011St21}. Other: 21 ns \textit{3} from \ensuremath{\gamma}(t) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2005Ca02,B}{2005Ca02}.\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{12.718821cm}{\raggedright configuration: \ensuremath{\nu} (f\ensuremath{_{\textnormal{5/2}}^{\textnormal{$-$1}}}) \ensuremath{\pi} (d\ensuremath{_{\textnormal{3/2}}^{\textnormal{$-$1}}},h\ensuremath{_{\textnormal{11/2}}^{\textnormal{$-$1}}}). The assignment is tentative.\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{12.718821cm}{\raggedright Experimental isomeric state population ratio \ensuremath{\geq}32\% (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2005Ca02,B}{2005Ca02}) and \ensuremath{\geq}4\% \textit{2} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2011St21,B}{2011St21}).\vspace{0.1cm}}&\\
\end{longtable}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PT5LEVEL0}{\dagger}}}} From a least-squares fit to E\ensuremath{\gamma} in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2011St21,B}{2011St21}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PT5LEVEL1}{\ddagger}}}} From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2005Ca02,B}{2005Ca02}, based on systematics and shell model predictions. Different \ensuremath{J^{\pi}} values are proposed in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2011St21,B}{2011St21}, where the}\\
\parbox[b][0.3cm]{17.7cm}{{\ }{\ }observed \ensuremath{\gamma}-ray cascade is placed above an expected, but not yet observed, \ensuremath{J^{\pi}}=13/2\ensuremath{^{+}} state. This alternative was also discussed in}\\
\parbox[b][0.3cm]{17.7cm}{{\ }{\ }\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2005Ca02,B}{2005Ca02}, but was not adopted due to the resulting large measured isomeric ratio, which would exceed the sharp-cutoff model}\\
\parbox[b][0.3cm]{17.7cm}{{\ }{\ }value.}\\
\vspace{0.5cm}
\underline{$\gamma$($^{201}$Pt)}\\
\begin{longtable}{ccccccccc@{}cc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{PT5GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{I\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{PT5GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{E\ensuremath{_{i}}(level)}&J\ensuremath{^{\pi}_{i}}&\multicolumn{2}{c}{E\ensuremath{_{f}}}&J\ensuremath{^{\pi}_{f}}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{353}&\multicolumn{1}{@{.}l}{6 {\it 5}}&\multicolumn{1}{r@{}}{76}&\multicolumn{1}{@{ }l}{{\it 6}}&\multicolumn{1}{r@{}}{1455}&\multicolumn{1}{@{.}l}{5}&\multicolumn{1}{l}{(15/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{1101}&\multicolumn{1}{@{.}l}{4 }&\multicolumn{1}{@{}l}{(13/2\ensuremath{^{-}})}&\parbox[t][0.3cm]{9.80624cm}{\raggedright E\ensuremath{_{\gamma}},I\ensuremath{_{\gamma}}: E\ensuremath{\gamma}=354.1 keV \textit{2}, I\ensuremath{\gamma}=95 \textit{6} in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2005Ca02,B}{2005Ca02}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{373}&\multicolumn{1}{@{.}l}{9 {\it 5}}&\multicolumn{1}{r@{}}{80}&\multicolumn{1}{@{ }l}{{\it 5}}&\multicolumn{1}{r@{}}{374}&\multicolumn{1}{@{.}l}{2}&\multicolumn{1}{l}{(9/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{@{}l}{(5/2\ensuremath{^{-}})}&\parbox[t][0.3cm]{9.80624cm}{\raggedright E\ensuremath{_{\gamma}},I\ensuremath{_{\gamma}}: E\ensuremath{\gamma}=374.4 keV \textit{2}, I\ensuremath{\gamma}=100 \textit{6} in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2005Ca02,B}{2005Ca02}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{726}&\multicolumn{1}{@{.}l}{9 {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{ }l}{{\it 6}}&\multicolumn{1}{r@{}}{1101}&\multicolumn{1}{@{.}l}{4}&\multicolumn{1}{l}{(13/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{374}&\multicolumn{1}{@{.}l}{2 }&\multicolumn{1}{@{}l}{(9/2\ensuremath{^{-}})}&\parbox[t][0.3cm]{9.80624cm}{\raggedright E\ensuremath{_{\gamma}},I\ensuremath{_{\gamma}}: E\ensuremath{\gamma}=727.2 keV \textit{2}, I\ensuremath{\gamma}=90 \textit{7} in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2005Ca02,B}{2005Ca02}.\vspace{0.1cm}}&\\
\end{longtable}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PT5GAMMA0}{\dagger}}}} From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2011St21,B}{2011St21}. \ensuremath{\Delta}E\ensuremath{\gamma} were estimated by the evaluator.}\\
\vspace{0.5cm}
\clearpage
\begin{figure}[h]
\begin{center}
\includegraphics{201PT5-0.ps}\\
\end{center}
\end{figure}
\clearpage
\section[\ensuremath{^{201}_{\ 79}}Au\ensuremath{_{122}^{~}}]{ }
\vspace{-30pt}
\setcounter{chappage}{1}
\subsection[\hspace{-0.2cm}Adopted Levels, Gammas]{ }
\vspace{-20pt}
\vspace{0.3cm}
\hypertarget{AU6}{{\bf \small \underline{Adopted \hyperlink{201AU_LEVEL}{Levels}, \hyperlink{201AU_GAMMA}{Gammas}}}}\\
\vspace{4pt}
\vspace{8pt}
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}Q(\ensuremath{\beta^-})=1262 {\it 3}; S(n)=7232 {\it 27}; S(p)=7090 {\it 20}; Q(\ensuremath{\alpha})=$-$561 {\it 20}\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2021Wa16,B}{2021Wa16}}\\
\vspace{12pt}
\hypertarget{201AU_LEVEL}{\underline{$^{201}$Au Levels}}\\
\begin{longtable}[c]{ll}
\multicolumn{2}{c}{\underline{Cross Reference (XREF) Flags}}\\
\\
\hyperlink{PT7}{\texttt{A }}& \ensuremath{^{\textnormal{201}}}Pt \ensuremath{\beta}\ensuremath{^{-}} decay\\
\hyperlink{AU8}{\texttt{B }}& \ensuremath{^{\textnormal{202}}}Hg(t,\ensuremath{\alpha})\\
\hyperlink{AU9}{\texttt{C }}& \ensuremath{^{\textnormal{9}}}Be(\ensuremath{^{\textnormal{208}}}Pb,X\ensuremath{\gamma})\\
\end{longtable}
\vspace{-0.5cm}
\begin{longtable}{ccccccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E(level)$^{{\hyperlink{AU6LEVEL0}{\dagger}}}$}&J$^{\pi}$$^{{\hyperlink{AU6LEVEL1}{\ddagger}}}$&\multicolumn{2}{c}{T$_{1/2}$$^{}$}&XREF&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{AU6LEVEL2}{\#}}}}}&\multicolumn{1}{l}{3/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{26}&\multicolumn{1}{@{.}l}{0 min {\it 8}}&\multicolumn{1}{l}{\texttt{\hyperlink{PT7}{A}\hyperlink{AU8}{B}\hyperlink{AU9}{C}} }&\parbox[t][0.3cm]{10.673161cm}{\raggedright \%\ensuremath{\beta}\ensuremath{^{-}}=100\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.673161cm}{\raggedright T\ensuremath{_{1/2}}: Weighted average of 26 min \textit{1} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1957Er24,B}{1957Er24}), 27 min \textit{2} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1962Fa06,B}{1962Fa06}), 22\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.673161cm}{\raggedright {\ }{\ }{\ }min \textit{3} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1962Eu01,B}{1962Eu01}) and 26.4 min \textit{16} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1972Pa24,B}{1972Pa24}). Other: 26 min (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1952Bu80,B}{1952Bu80}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{101}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{AU6LEVEL3}{@}}}} {\it 5}}&\multicolumn{1}{l}{1/2\ensuremath{^{+}}}&&&\multicolumn{1}{l}{\texttt{\hyperlink{PT7}{A}\hyperlink{AU8}{B}\ } }&\parbox[t][0.3cm]{10.673161cm}{\raggedright XREF: A(70?).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{359}&\multicolumn{1}{@{ }l}{{\it 5}}&\multicolumn{1}{l}{3/2\ensuremath{^{+}}}&&&\multicolumn{1}{l}{\texttt{\hyperlink{PT7}{A}\hyperlink{AU8}{B}\ } }&\parbox[t][0.3cm]{10.673161cm}{\raggedright XREF: A(230?).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{553}&\multicolumn{1}{@{.}l}{0 {\it 10}}&\multicolumn{1}{l}{(7/2\ensuremath{^{+}})}&&&\multicolumn{1}{l}{\texttt{\ \hyperlink{AU8}{B}\hyperlink{AU9}{C}} }&\parbox[t][0.3cm]{10.673161cm}{\raggedright XREF: B(549?).\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.673161cm}{\raggedright J\ensuremath{^{\pi}}: Similarity with \ensuremath{^{\textnormal{203}}}Au (E(7/2\ensuremath{^{+}})=563 keV); shell model predictions.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{594}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{AU6LEVEL4}{\&}}}} {\it 5}}&\multicolumn{1}{l}{11/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{34 ms {\it +90\textminus29}}&\multicolumn{1}{l}{\texttt{\ \hyperlink{AU8}{B}\hyperlink{AU9}{C}} }&\parbox[t][0.3cm]{10.673161cm}{\raggedright T\ensuremath{_{1/2}}: From 553\ensuremath{\gamma}(t) in \ensuremath{^{\textnormal{9}}}Be(\ensuremath{^{\textnormal{208}}}Pb,X\ensuremath{\gamma}) (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2011St21,B}{2011St21}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{653}&\multicolumn{1}{@{ }l}{{\it 5}}&\multicolumn{1}{l}{5/2\ensuremath{^{+}}}&&&\multicolumn{1}{l}{\texttt{\ \hyperlink{AU8}{B}\ } }&&\\
\multicolumn{1}{r@{}}{810}&\multicolumn{1}{@{ }l}{{\it 5}}&\multicolumn{1}{l}{5/2\ensuremath{^{+}}}&&&\multicolumn{1}{l}{\texttt{\ \hyperlink{AU8}{B}\ } }&&\\
\multicolumn{1}{r@{}}{897}&\multicolumn{1}{@{ }l}{{\it 5}}&\multicolumn{1}{l}{1/2\ensuremath{^{+}}}&&&\multicolumn{1}{l}{\texttt{\ \hyperlink{AU8}{B}\ } }&&\\
\multicolumn{1}{r@{}}{1055}&\multicolumn{1}{@{ }l}{{\it 5}}&\multicolumn{1}{l}{3/2\ensuremath{^{+}}}&&&\multicolumn{1}{l}{\texttt{\ \hyperlink{AU8}{B}\ } }&&\\
\multicolumn{1}{r@{}}{1216}&\multicolumn{1}{@{ }l}{{\it 5}}&\multicolumn{1}{l}{5/2\ensuremath{^{+}}}&&&\multicolumn{1}{l}{\texttt{\ \hyperlink{AU8}{B}\ } }&&\\
\multicolumn{1}{r@{}}{1232}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{AU6LEVEL5}{a}}}} {\it 6}}&\multicolumn{1}{l}{(15/2\ensuremath{^{-}})}&&&\multicolumn{1}{l}{\texttt{\ \ \hyperlink{AU9}{C}} }&\parbox[t][0.3cm]{10.673161cm}{\raggedright E(level): Introduced by the evaluator by assuming that 638\ensuremath{\gamma} directly feeds the\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.673161cm}{\raggedright {\ }{\ }{\ }\ensuremath{J^{\pi}}=11/2\ensuremath{^{-}} isomer. The existence is tentative.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1242}&\multicolumn{1}{@{ }l}{{\it 5}}&\multicolumn{1}{l}{(5/2\ensuremath{^{+}})}&&&\multicolumn{1}{l}{\texttt{\ \hyperlink{AU8}{B}\ } }&&\\
\multicolumn{1}{r@{}}{1465}&\multicolumn{1}{@{ }l}{{\it 5}}&\multicolumn{1}{l}{5/2\ensuremath{^{+}}}&&&\multicolumn{1}{l}{\texttt{\ \hyperlink{AU8}{B}\ } }&&\\
\multicolumn{1}{r@{}}{1506}&\multicolumn{1}{@{ }l}{{\it 5}}&\multicolumn{1}{l}{11/2\ensuremath{^{-}}}&&&\multicolumn{1}{l}{\texttt{\ \hyperlink{AU8}{B}\ } }&&\\
\multicolumn{1}{r@{}}{1548}&\multicolumn{1}{@{ }l}{{\it 5}}&&&&\multicolumn{1}{l}{\texttt{\ \hyperlink{AU8}{B}\ } }&&\\
\multicolumn{1}{r@{}}{1610}&\multicolumn{1}{@{ }l}{{\it 6}}&\multicolumn{1}{l}{(19/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{5}&\multicolumn{1}{@{.}l}{6 \ensuremath{\mu}s {\it 24}}&\multicolumn{1}{l}{\texttt{\ \ \hyperlink{AU9}{C}} }&\parbox[t][0.3cm]{10.673161cm}{\raggedright E(level): Introduced by the evaluator by assuming that 378.2\ensuremath{\gamma} precedes\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.673161cm}{\raggedright {\ }{\ }{\ }638.0\ensuremath{\gamma}. The assignment is tentative.\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.673161cm}{\raggedright J\ensuremath{^{\pi}}: From 378.2\ensuremath{\gamma} (M2) to (15/2\ensuremath{^{-}}). The assignment is tentative.\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.673161cm}{\raggedright T\ensuremath{_{1/2}}: From 378.2,638.0\ensuremath{\gamma}(t) in \ensuremath{^{\textnormal{9}}}Be(\ensuremath{^{\textnormal{208}}}Pb,X\ensuremath{\gamma}) (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2011St21,B}{2011St21}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1664}&\multicolumn{1}{@{ }l}{{\it 5}}&&&&\multicolumn{1}{l}{\texttt{\ \hyperlink{AU8}{B}\ } }&&\\
\multicolumn{1}{r@{}}{1698}&\multicolumn{1}{@{ }l}{{\it 5}}&&&&\multicolumn{1}{l}{\texttt{\ \hyperlink{AU8}{B}\ } }&&\\
\multicolumn{1}{r@{}}{1760?}&\multicolumn{1}{@{ }l}{{\it 20}}&&&&\multicolumn{1}{l}{\texttt{\hyperlink{PT7}{A}\ \ } }&\parbox[t][0.3cm]{10.673161cm}{\raggedright E(level): From the observed E\ensuremath{\gamma} following \ensuremath{\beta}\ensuremath{\gamma}-coin in \ensuremath{^{\textnormal{201}}}Pt \ensuremath{\beta}\ensuremath{^{-}} decay\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.673161cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1963Go06,B}{1963Go06}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1900}&\multicolumn{1}{@{ }l}{{\it 5}}&&&&\multicolumn{1}{l}{\texttt{\ \hyperlink{AU8}{B}\ } }&&\\
\multicolumn{1}{r@{}}{1941}&\multicolumn{1}{@{ }l}{{\it 5}}&&&&\multicolumn{1}{l}{\texttt{\ \hyperlink{AU8}{B}\ } }&&\\
\multicolumn{1}{r@{}}{1981}&\multicolumn{1}{@{ }l}{{\it 5}}&&&&\multicolumn{1}{l}{\texttt{\ \hyperlink{AU8}{B}\ } }&&\\
\multicolumn{1}{r@{}}{2055}&\multicolumn{1}{@{ }l}{{\it 5}}&&&&\multicolumn{1}{l}{\texttt{\ \hyperlink{AU8}{B}\ } }&&\\
\end{longtable}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{AU6LEVEL0}{\dagger}}}} From \ensuremath{^{\textnormal{202}}}Hg(t,\ensuremath{\alpha}) (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1981Fl05,B}{1981Fl05}) and from a least-squares fir to E\ensuremath{\gamma} when \ensuremath{\gamma}-ray data are available.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{AU6LEVEL1}{\ddagger}}}} From \ensuremath{\sigma}(\ensuremath{\theta},pol) and analyzing powers in \ensuremath{^{\textnormal{202}}}Hg(t,\ensuremath{\alpha}) (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1981Fl05,B}{1981Fl05}), unless otherwise stated.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{AU6LEVEL2}{\#}}}} Main configuration=\ensuremath{\pi} d\ensuremath{_{\textnormal{3/2}}^{\textnormal{$-$1}}}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{AU6LEVEL3}{@}}}} Main configuration=\ensuremath{\pi} s\ensuremath{_{\textnormal{1/2}}^{\textnormal{$-$1}}}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{AU6LEVEL4}{\&}}}} Main configuration=\ensuremath{\pi} h\ensuremath{_{\textnormal{11/2}}^{\textnormal{$-$1}}}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{AU6LEVEL5}{a}}}} Probable configuration=\ensuremath{\pi} (h\ensuremath{_{\textnormal{11/2}}^{\textnormal{$-$1}}})\ensuremath{\otimes}2\ensuremath{^{\textnormal{+}}}.}\\
\vspace{0.5cm}
\clearpage
\vspace{0.3cm}
\vspace*{-0.5cm}
{\bf \small \underline{Adopted \hyperlink{201AU_LEVEL}{Levels}, \hyperlink{201AU_GAMMA}{Gammas} (continued)}}\\
\vspace{0.3cm}
\hypertarget{201AU_GAMMA}{\underline{$\gamma$($^{201}$Au)}}\\
\begin{longtable}{ccccccccc@{}ccccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E\ensuremath{_{i}}(level)}&J\ensuremath{^{\pi}_{i}}&\multicolumn{2}{c}{E\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{AU6GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{I\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{AU6GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{E\ensuremath{_{f}}}&J\ensuremath{^{\pi}_{f}}&Mult.&\multicolumn{2}{c}{\ensuremath{\alpha}\ensuremath{^{\hyperlink{AU6GAMMA3}{@}}}}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{553}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{l}{(7/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{553}&\multicolumn{1}{@{ }l}{{\it 1}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}}}&&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
\multicolumn{1}{r@{}}{594}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{l}{11/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{(41}&\multicolumn{1}{@{ }l}{{\it 5})}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{553}&\multicolumn{1}{@{.}l}{0 }&\multicolumn{1}{@{}l}{(7/2\ensuremath{^{+}})}&\multicolumn{1}{l}{[M2]}&\multicolumn{1}{r@{}}{1.1\ensuremath{\times10^{3}}}&\multicolumn{1}{@{ }l}{{\it 9}}&\parbox[t][0.3cm]{6.192141cm}{\raggedright B(M2)(W.u.)=0.021 \textit{+79{\textminus}17}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{6.192141cm}{\raggedright E\ensuremath{_{\gamma}}: From level energy differences; not\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{6.192141cm}{\raggedright {\ }{\ }{\ }observed in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2011St21,B}{2011St21}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1232}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{l}{(15/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{638}&\multicolumn{1}{@{.}l}{0\ensuremath{^{\hyperlink{AU6GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{594}&\multicolumn{1}{@{ }l}{}&\multicolumn{1}{@{}l}{11/2\ensuremath{^{-}}}&&&&&\\
\multicolumn{1}{r@{}}{1610}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{l}{(19/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{378}&\multicolumn{1}{@{.}l}{2\ensuremath{^{\hyperlink{AU6GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{1232}&\multicolumn{1}{@{ }l}{}&\multicolumn{1}{@{}l}{(15/2\ensuremath{^{-}})}&\multicolumn{1}{l}{(M2)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{601 {\it 9}}&\parbox[t][0.3cm]{6.192141cm}{\raggedright B(M2)(W.u.)=0.013 \textit{+9{\textminus}4}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{6.192141cm}{\raggedright Mult.: From \ensuremath{\alpha}(tot,378\ensuremath{\gamma}) ~~ 0.7 from total\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{6.192141cm}{\raggedright {\ }{\ }{\ }intensity balance between 638-keV and\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{6.192141cm}{\raggedright {\ }{\ }{\ }378-keV \ensuremath{\gamma} rays in \ensuremath{^{\textnormal{9}}}Be(\ensuremath{^{\textnormal{208}}}Pb,X\ensuremath{\gamma})\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{6.192141cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2011St21,B}{2011St21}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1760?}&\multicolumn{1}{@{}l}{}&&\multicolumn{1}{r@{}}{1760}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{AU6GAMMA2}{\#}}} {\it 20}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{AU6GAMMA2}{\#}}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}}}&&&&&\\
\end{longtable}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{AU6GAMMA0}{\dagger}}}} From \ensuremath{^{\textnormal{9}}}Be(\ensuremath{^{\textnormal{208}}}Pb,X\ensuremath{\gamma}) (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2011St21,B}{2011St21}), unless otherwise stated.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{AU6GAMMA1}{\ddagger}}}} This \ensuremath{\gamma} ray shows the 5.0{\textminus}\ensuremath{\mu}s isomer half-life, but the ordering and placement in the decay scheme are not experimentally known.}\\
\parbox[b][0.3cm]{17.7cm}{{\ }{\ }The interpretation is made by the evaluator.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{AU6GAMMA2}{\#}}}} From \ensuremath{^{\textnormal{201}}}Pt \ensuremath{\beta}\ensuremath{^{-}} decay (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1963Go06,B}{1963Go06}).}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{AU6GAMMA3}{@}}}} Total theoretical internal conversion coefficients, calculated using the BrIcc code (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2008Ki07,B}{2008Ki07}) with Frozen orbital approximation}\\
\parbox[b][0.3cm]{17.7cm}{{\ }{\ }based on \ensuremath{\gamma}-ray energies, assigned multipolarities, and mixing ratios, unless otherwise specified.}\\
\vspace{0.5cm}
\begin{figure}[h]
\begin{center}
\includegraphics{201AU6-0.ps}\\
\end{center}
\end{figure}
\clearpage
\subsection[\hspace{-0.2cm}\ensuremath{^{\textnormal{201}}}Pt \ensuremath{\beta}\ensuremath{^{-}} decay]{ }
\vspace{-27pt}
\vspace{0.3cm}
\hypertarget{PT7}{{\bf \small \underline{\ensuremath{^{\textnormal{201}}}Pt \ensuremath{\beta}\ensuremath{^{-}} decay\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1963Go06,B}{1963Go06}}}}\\
\vspace{4pt}
\vspace{8pt}
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}Parent: $^{201}$Pt: E=0; J$^{\pi}$=(5/2\ensuremath{^{-}}); T$_{1/2}$=2.46 min {\it 9}; Q(\ensuremath{\beta}\ensuremath{^{-}})=2660 {\it 50}; \%\ensuremath{\beta}\ensuremath{^{-}} decay=100.0
}\\
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}\ensuremath{^{201}}Pt-Source produced using the \ensuremath{^{\textnormal{204}}}Hg(n,\ensuremath{\alpha}) reaction (\ensuremath{\sigma}=2.5 mb relative to that for \ensuremath{^{\textnormal{58}}}Ni(n,p) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1963Go06,B}{1963Go06}), following radiochemical}\\
\parbox[b][0.3cm]{17.7cm}{separation. Detectors: NaI(Tl) for gammas and scintillation spectrometer for \ensuremath{\beta}. Measured: \ensuremath{\beta}, \ensuremath{\gamma}, \ensuremath{\beta}\ensuremath{\gamma} coin.}\\
\vspace{12pt}
\underline{$^{201}$Au Levels}\\
\begin{longtable}{ccccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E(level)$^{{\hyperlink{AU7LEVEL0}{\dagger}}}$}&J$^{\pi}$$^{{\hyperlink{AU7LEVEL1}{\ddagger}}}$&\multicolumn{2}{c}{T$_{1/2}$$^{{\hyperlink{AU7LEVEL1}{\ddagger}}}$}&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&
\endfirsthead
\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{l}{3/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{26}&\multicolumn{1}{@{.}l}{0 min {\it 8}}&\\
\multicolumn{1}{r@{}}{70?}&\multicolumn{1}{@{ }l}{{\it 5}}&&&&\\
\multicolumn{1}{r@{}}{230?}&\multicolumn{1}{@{ }l}{{\it 20}}&&&&\\
\multicolumn{1}{r@{}}{1760?}&\multicolumn{1}{@{ }l}{{\it 20}}&&&&\\
\end{longtable}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{AU7LEVEL0}{\dagger}}}} From the observed E\ensuremath{\gamma} in \ensuremath{\beta}\ensuremath{\gamma}-coin data.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{AU7LEVEL1}{\ddagger}}}} From Adopted Levels.}\\
\vspace{0.5cm}
\underline{\ensuremath{\beta^-} radiations}\\
\begin{longtable}{ccccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E(decay)$$}&\multicolumn{2}{c}{E(level)}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{(9.0\ensuremath{\times10^{2}}}&\multicolumn{1}{@{ }l}{{\it 5})}&\multicolumn{1}{r@{}}{1760?}&\multicolumn{1}{@{}l}{}&&\\
\multicolumn{1}{r@{}}{(2.43\ensuremath{\times10^{3}}}&\multicolumn{1}{@{ }l}{{\it 5})}&\multicolumn{1}{r@{}}{230?}&\multicolumn{1}{@{}l}{}&&\\
\multicolumn{1}{r@{}}{2660}&\multicolumn{1}{@{ }l}{{\it 50}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{14.21664cm}{\raggedright E(decay): From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1963Go06,B}{1963Go06}.\vspace{0.1cm}}&\\
\end{longtable}
\underline{$\gamma$($^{201}$Au)}\\
\begin{longtable}{cccccc@{}cc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{PT7GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{E\ensuremath{_{i}}(level)}&\multicolumn{2}{c}{E\ensuremath{_{f}}}&J\ensuremath{^{\pi}_{f}}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{70}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PT7GAMMA1}{\ddagger}\hyperlink{PT7GAMMA2}{\#}}} {\it 5}}&\multicolumn{1}{r@{}}{70?}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}}}&\parbox[t][0.3cm]{12.88748cm}{\raggedright E\ensuremath{_{\gamma}}: Energy close to Au K\ensuremath{\alpha}{} x ray.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{150}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PT7GAMMA1}{\ddagger}}} {\it 10}}&\multicolumn{1}{r@{}}{230?}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{70?}&\multicolumn{1}{@{ }l}{}&&&\\
\multicolumn{1}{r@{}}{230}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PT7GAMMA1}{\ddagger}}} {\it 20}}&\multicolumn{1}{r@{}}{230?}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}}}&&\\
\multicolumn{1}{r@{}}{1760}&\multicolumn{1}{@{ }l}{{\it 20}}&\multicolumn{1}{r@{}}{1760?}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}}}&\parbox[t][0.3cm]{12.88748cm}{\raggedright E\ensuremath{_{\gamma}}: No coin with other gammas were observed. It shows the parent (\ensuremath{^{\textnormal{201}}}Pt) half-life of 2.5 m\vspace{0.1cm}}&\\
&&&&&&&\parbox[t][0.3cm]{12.88748cm}{\raggedright {\ }{\ }{\ }\textit{1}.\vspace{0.1cm}}&\\
\end{longtable}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PT7GAMMA0}{\dagger}}}} From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1963Go06,B}{1963Go06}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PT7GAMMA1}{\ddagger}}}} Observed only in the \ensuremath{\beta}\ensuremath{\gamma}-coin data when E(\ensuremath{\beta})\ensuremath{>}1600 keV.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PT7GAMMA2}{\#}}}} Placement of transition in the level scheme is uncertain.}\\
\vspace{0.5cm}
\clearpage
\begin{figure}[h]
\begin{center}
\includegraphics{201AU7-0.ps}\\
\end{center}
\end{figure}
\clearpage
\subsection[\hspace{-0.2cm}\ensuremath{^{\textnormal{202}}}Hg(t,\ensuremath{\alpha})]{ }
\vspace{-27pt}
\vspace{0.3cm}
\hypertarget{AU8}{{\bf \small \underline{\ensuremath{^{\textnormal{202}}}Hg(t,\ensuremath{\alpha})\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1981Fl05,B}{1981Fl05}}}}\\
\vspace{4pt}
\vspace{8pt}
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1981Fl05,B}{1981Fl05}: 17-MeV polarized triton$'$s beam; Detectors: magnetic spectrograph with FWHM=15-18 keV; Measured: \ensuremath{\sigma}(\ensuremath{\theta},pol); DWBA}\\
\parbox[b][0.3cm]{17.7cm}{analysis relative to \ensuremath{^{\textnormal{208}}}Pb(t,\ensuremath{\alpha}).}\\
\vspace{12pt}
\underline{$^{201}$Au Levels}\\
\begin{longtable}{ccccc|ccccc|ccccc|cc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E(level)$^{{\hyperlink{AU8LEVEL0}{\dagger}}}$}&J$^{\pi}$$^{{\hyperlink{AU8LEVEL1}{\ddagger}}}$&\multicolumn{2}{c|}{S$^{{\hyperlink{AU8LEVEL2}{\#}}}$}&\multicolumn{2}{c}{E(level)$^{{\hyperlink{AU8LEVEL0}{\dagger}}}$}&J$^{\pi}$$^{{\hyperlink{AU8LEVEL1}{\ddagger}}}$&\multicolumn{2}{c|}{S$^{{\hyperlink{AU8LEVEL2}{\#}}}$}&\multicolumn{2}{c}{E(level)$^{{\hyperlink{AU8LEVEL0}{\dagger}}}$}&J$^{\pi}$$^{{\hyperlink{AU8LEVEL1}{\ddagger}}}$&\multicolumn{2}{c|}{S$^{{\hyperlink{AU8LEVEL2}{\#}}}$}&\multicolumn{2}{c}{E(level)$^{{\hyperlink{AU8LEVEL0}{\dagger}}}$}&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c|}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c|}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c|}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&
\endfirsthead
\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{AU8LEVEL3}{@}}}}}&\multicolumn{1}{l}{3/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l|}{44}&\multicolumn{1}{r@{}}{653}&\multicolumn{1}{@{ }l}{{\it 5}}&\multicolumn{1}{l}{5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l|}{27}&\multicolumn{1}{r@{}}{1242}&\multicolumn{1}{@{ }l}{{\it 5}}&\multicolumn{1}{l}{(5/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l|}{07}&\multicolumn{1}{r@{}}{1698}&\multicolumn{1}{@{ }l}{{\it 5}}&\\
\multicolumn{1}{r@{}}{101}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{AU8LEVEL4}{\&}}}} {\it 5}}&\multicolumn{1}{l}{1/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l|}{23}&\multicolumn{1}{r@{}}{810}&\multicolumn{1}{@{ }l}{{\it 5}}&\multicolumn{1}{l}{5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l|}{03}&\multicolumn{1}{r@{}}{1465}&\multicolumn{1}{@{ }l}{{\it 5}}&\multicolumn{1}{l}{5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l|}{20}&\multicolumn{1}{r@{}}{1900}&\multicolumn{1}{@{ }l}{{\it 5}}&\\
\multicolumn{1}{r@{}}{359}&\multicolumn{1}{@{ }l}{{\it 5}}&\multicolumn{1}{l}{3/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l|}{05}&\multicolumn{1}{r@{}}{897}&\multicolumn{1}{@{ }l}{{\it 5}}&\multicolumn{1}{l}{1/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l|}{09}&\multicolumn{1}{r@{}}{1506}&\multicolumn{1}{@{ }l}{{\it 5}}&\multicolumn{1}{l}{11/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l|}{29}&\multicolumn{1}{r@{}}{1941}&\multicolumn{1}{@{ }l}{{\it 5}}&\\
\multicolumn{1}{r@{}}{549?}&\multicolumn{1}{@{ }l}{{\it 5}}&&&&\multicolumn{1}{r@{}}{1055}&\multicolumn{1}{@{ }l}{{\it 5}}&\multicolumn{1}{l}{3/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l|}{07}&\multicolumn{1}{r@{}}{1548}&\multicolumn{1}{@{ }l}{{\it 5}}&&&&\multicolumn{1}{r@{}}{1981}&\multicolumn{1}{@{ }l}{{\it 5}}&\\
\multicolumn{1}{r@{}}{594}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{AU8LEVEL5}{a}}}} {\it 5}}&\multicolumn{1}{l}{11/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l|}{44}&\multicolumn{1}{r@{}}{1216}&\multicolumn{1}{@{ }l}{{\it 5}}&\multicolumn{1}{l}{5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l|}{12}&\multicolumn{1}{r@{}}{1664}&\multicolumn{1}{@{ }l}{{\it 5}}&&&&\multicolumn{1}{r@{}}{2055}&\multicolumn{1}{@{ }l}{{\it 5}}&\\
\end{longtable}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{AU8LEVEL0}{\dagger}}}} From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1981Fl05,B}{1981Fl05}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{AU8LEVEL1}{\ddagger}}}} Based on angular distributions and analyzing powers in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1981Fl05,B}{1981Fl05}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{AU8LEVEL2}{\#}}}} Spectroscopic factors relative to \ensuremath{^{\textnormal{208}}}Pb(t,\ensuremath{\alpha})\ensuremath{^{\textnormal{207}}}Tl. Values indicate larger fragmentation of the proton-hole strength compared to}\\
\parbox[b][0.3cm]{17.7cm}{{\ }{\ }\ensuremath{^{\textnormal{203}}}Au, which is interpreted as a result of the larger collectivity (deformation) of the \ensuremath{^{\textnormal{202}}}Hg core relative to \ensuremath{^{\textnormal{204}}}Hg one}\\
\parbox[b][0.3cm]{17.7cm}{{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1981Fl05,B}{1981Fl05}).}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{AU8LEVEL3}{@}}}} Main configuration=\ensuremath{\pi} d\ensuremath{_{\textnormal{3/2}}^{\textnormal{$-$1}}}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{AU8LEVEL4}{\&}}}} Main configuration=\ensuremath{\pi} s\ensuremath{_{\textnormal{1/2}}^{\textnormal{$-$1}}}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{AU8LEVEL5}{a}}}} Main configuration=\ensuremath{\pi} h\ensuremath{_{\textnormal{11/2}}^{\textnormal{$-$1}}}.}\\
\vspace{0.5cm}
\clearpage
\subsection[\hspace{-0.2cm}\ensuremath{^{\textnormal{9}}}Be(\ensuremath{^{\textnormal{208}}}Pb,X\ensuremath{\gamma})]{ }
\vspace{-27pt}
\vspace{0.3cm}
\hypertarget{AU9}{{\bf \small \underline{\ensuremath{^{\textnormal{9}}}Be(\ensuremath{^{\textnormal{208}}}Pb,X\ensuremath{\gamma})\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2011St21,B}{2011St21}}}}\\
\vspace{4pt}
\vspace{8pt}
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2011St21,B}{2011St21}: in-flight fragmentation of \ensuremath{^{\textnormal{208}}}Pb beam at 1 GeV/A on a 2.526 g/cm\ensuremath{^{\textnormal{2}}} Be target, backed by 0.223 g/cm\ensuremath{^{\textnormal{2}}}-thick \ensuremath{^{\textnormal{93}}}Nb foil.}\\
\parbox[b][0.3cm]{17.7cm}{Fragment Recoil Separator at GSI. Measured E\ensuremath{\gamma}, I\ensuremath{\gamma}, \ensuremath{\gamma}\ensuremath{\gamma}, \ensuremath{\gamma}\ensuremath{\gamma}(t) using the RISING \ensuremath{\gamma}-ray spectrometer.}\\
\vspace{12pt}
\underline{$^{201}$Au Levels}\\
\begin{longtable}{cccccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E(level)$^{{\hyperlink{AU9LEVEL0}{\dagger}}}$}&J$^{\pi}$$^{{\hyperlink{AU9LEVEL1}{\ddagger}}}$&\multicolumn{2}{c}{T$_{1/2}$$^{{\hyperlink{AU9LEVEL1}{\ddagger}}}$}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{l}{3/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{26}&\multicolumn{1}{@{.}l}{0 min {\it 8}}&&\\
\multicolumn{1}{r@{}}{553}&\multicolumn{1}{@{.}l}{0 {\it 10}}&\multicolumn{1}{l}{(7/2\ensuremath{^{+}})}&&&&\\
\multicolumn{1}{r@{}}{594}&\multicolumn{1}{@{ }l}{{\it 5}}&\multicolumn{1}{l}{11/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{34 ms {\it +90\textminus29}}&\parbox[t][0.3cm]{11.89278cm}{\raggedright E(level): From Adopted Levels.\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{11.89278cm}{\raggedright T\ensuremath{_{1/2}}: From 553\ensuremath{\gamma}(t) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2011St21,B}{2011St21}.\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{11.89278cm}{\raggedright Experimental isomeric state population ratio=13\% \textit{+36{\textminus}11}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1232}&\multicolumn{1}{@{ }l}{{\it 6}}&\multicolumn{1}{l}{(15/2\ensuremath{^{-}})}&&&\parbox[t][0.3cm]{11.89278cm}{\raggedright E(level): Introduced by the evaluator by assuming that 638\ensuremath{\gamma} directly feeds the\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{11.89278cm}{\raggedright {\ }{\ }{\ }\ensuremath{J^{\pi}}=11/2\ensuremath{^{-}} isomer. The existence is tentative.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1610}&\multicolumn{1}{@{ }l}{{\it 6}}&\multicolumn{1}{l}{(19/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{5}&\multicolumn{1}{@{.}l}{6 \ensuremath{\mu}s {\it 24}}&\parbox[t][0.3cm]{11.89278cm}{\raggedright E(level): Introduced by the evaluator by assuming that 378.2\ensuremath{\gamma} precedes the 638.0\ensuremath{\gamma}.\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{11.89278cm}{\raggedright {\ }{\ }{\ }The assignment is tentative.\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{11.89278cm}{\raggedright T\ensuremath{_{1/2}}: From 378.2,638.0\ensuremath{\gamma}(t) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2011St21,B}{2011St21}.\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{11.89278cm}{\raggedright Experimental isomeric state population ratio=5\% \textit{3}.\vspace{0.1cm}}&\\
\end{longtable}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{AU9LEVEL0}{\dagger}}}} From a least-squares fit to E\ensuremath{\gamma}, unless otherwise stated.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{AU9LEVEL1}{\ddagger}}}} From Adopted Levels, unless otherwise stated.}\\
\vspace{0.5cm}
\underline{$\gamma$($^{201}$Au)}\\
\begin{longtable}{ccccccccc@{}ccccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{AU9GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{I\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{AU9GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{E\ensuremath{_{i}}(level)}&J\ensuremath{^{\pi}_{i}}&\multicolumn{2}{c}{E\ensuremath{_{f}}}&J\ensuremath{^{\pi}_{f}}&Mult.&\multicolumn{2}{c}{\ensuremath{\alpha}\ensuremath{^{\hyperlink{AU9GAMMA2}{\#}}}}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{(41}&\multicolumn{1}{@{ }l}{{\it 5})}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{594}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{l}{11/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{553}&\multicolumn{1}{@{.}l}{0 }&\multicolumn{1}{@{}l}{(7/2\ensuremath{^{+}})}&&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{6.8833604cm}{\raggedright E\ensuremath{_{\gamma}}: From level energy differences; not observed\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{6.8833604cm}{\raggedright {\ }{\ }{\ }in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2011St21,B}{2011St21}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{378}&\multicolumn{1}{@{.}l}{2\ensuremath{^{\hyperlink{AU9GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{58}&\multicolumn{1}{@{ }l}{{\it 17}}&\multicolumn{1}{r@{}}{1610}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{l}{(19/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{1232}&\multicolumn{1}{@{ }l}{}&\multicolumn{1}{@{}l}{(15/2\ensuremath{^{-}})}&\multicolumn{1}{l}{(M2)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{601 {\it 9}}&\parbox[t][0.3cm]{6.8833604cm}{\raggedright Mult.: From \ensuremath{\alpha}(tot,378\ensuremath{\gamma}) \ensuremath{\approx} 0.7 from total\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{6.8833604cm}{\raggedright {\ }{\ }{\ }intensity balance between 638-keV and 378-keV\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{6.8833604cm}{\raggedright {\ }{\ }{\ }\ensuremath{\gamma} rays.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{553}&\multicolumn{1}{@{ }l}{{\it 1}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{ }l}{{\it 51}}&\multicolumn{1}{r@{}}{553}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{l}{(7/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}}}&&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
\multicolumn{1}{r@{}}{638}&\multicolumn{1}{@{.}l}{0\ensuremath{^{\hyperlink{AU9GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{ }l}{{\it 34}}&\multicolumn{1}{r@{}}{1232}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{l}{(15/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{594}&\multicolumn{1}{@{ }l}{}&\multicolumn{1}{@{}l}{11/2\ensuremath{^{-}}}&&&&&\\
\end{longtable}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{AU9GAMMA0}{\dagger}}}} From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2011St21,B}{2011St21}. \ensuremath{\Delta}E\ensuremath{\gamma} were estimated by the evaluator.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{AU9GAMMA1}{\ddagger}}}} This \ensuremath{\gamma} ray shows the 5.0{\textminus}\ensuremath{\mu}s isomer half-life, but the ordering and placement in the decay scheme are not experimentally known.}\\
\parbox[b][0.3cm]{17.7cm}{{\ }{\ }The interpretation is made by the evaluator.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{AU9GAMMA2}{\#}}}} Total theoretical internal conversion coefficients, calculated using the BrIcc code (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2008Ki07,B}{2008Ki07}) with Frozen orbital approximation}\\
\parbox[b][0.3cm]{17.7cm}{{\ }{\ }based on \ensuremath{\gamma}-ray energies, assigned multipolarities, and mixing ratios, unless otherwise specified.}\\
\vspace{0.5cm}
\clearpage
\begin{figure}[h]
\begin{center}
\includegraphics{201AU9-0.ps}\\
\end{center}
\end{figure}
\clearpage
\section[\ensuremath{^{201}_{\ 80}}Hg\ensuremath{_{121}^{~}}]{ }
\vspace{-30pt}
\setcounter{chappage}{1}
\subsection[\hspace{-0.2cm}Adopted Levels, Gammas]{ }
\vspace{-20pt}
\vspace{0.3cm}
\hypertarget{HG10}{{\bf \small \underline{Adopted \hyperlink{201HG_LEVEL}{Levels}, \hyperlink{201HG_GAMMA}{Gammas}}}}\\
\vspace{4pt}
\vspace{8pt}
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}Q(\ensuremath{\beta^-})=$-$482 {\it 14}; S(n)=6230.6 {\it 6}; S(p)=7711 {\it 27}; Q(\ensuremath{\alpha})=332.3 {\it 8}\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2021Wa16,B}{2021Wa16}}\\
\vspace{12pt}
\hypertarget{201HG_LEVEL}{\underline{$^{201}$Hg Levels}}\\
\begin{longtable}[c]{llll}
\multicolumn{4}{c}{\underline{Cross Reference (XREF) Flags}}\\
\\
\hyperlink{AU11}{\texttt{A }}& \ensuremath{^{\textnormal{201}}}Au \ensuremath{\beta}\ensuremath{^{-}} decay & \hyperlink{HG16}{\texttt{F }}& \ensuremath{^{\textnormal{201}}}Hg(d,d\ensuremath{'}),\ensuremath{^{\textnormal{201}}}Hg(p,p\ensuremath{'})\\
\hyperlink{HG12}{\texttt{B }}& \ensuremath{^{\textnormal{201}}}Hg IT decay & \hyperlink{HG17}{\texttt{G }}& Coulomb excitation\\
\hyperlink{TL13}{\texttt{C }}& \ensuremath{^{\textnormal{201}}}Tl \ensuremath{\varepsilon} decay & \hyperlink{HG18}{\texttt{H }}& \ensuremath{^{\textnormal{202}}}Hg(d,t)\\
\hyperlink{HG14}{\texttt{D }}& \ensuremath{^{\textnormal{200}}}Hg(d,p) & \hyperlink{HG19}{\texttt{I }}& \ensuremath{^{\textnormal{203}}}Tl(\ensuremath{\mu},X\ensuremath{\gamma})\\
\hyperlink{HG15}{\texttt{E }}& \ensuremath{^{\textnormal{201}}}Hg(\ensuremath{\gamma},\ensuremath{\gamma}\ensuremath{'}) & \\
\end{longtable}
\vspace{-0.5cm}
\begin{longtable}{ccccccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E(level)$^{{\hyperlink{HG10LEVEL0}{\dagger}}}$}&J$^{\pi}$$^{}$&\multicolumn{2}{c}{T$_{1/2}$$^{}$}&XREF&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{HG10LEVEL4}{\&}}}}}&\multicolumn{1}{l}{3/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{sta}&\multicolumn{1}{@{}l}{ble}&\multicolumn{1}{l}{\texttt{\hyperlink{AU11}{A}\hyperlink{HG12}{B}\hyperlink{TL13}{C}\hyperlink{HG14}{D}\hyperlink{HG15}{E}\hyperlink{HG16}{F}\hyperlink{HG17}{G}\hyperlink{HG18}{H}\hyperlink{HG19}{I}} }&\parbox[t][0.3cm]{10.67033cm}{\raggedright \ensuremath{\mu}={\textminus}0.5602257 \textit{14} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1973Re04,B}{1973Re04},\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2014StZZ,B}{2014StZZ})\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.67033cm}{\raggedright Q=+0.387 \textit{6} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2005Bi03,B}{2005Bi03},\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2016St14,B}{2016St14})\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.67033cm}{\raggedright J\ensuremath{^{\pi}}: Optical spectroscopy (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1931Sc03,B}{1931Sc03},\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1935Sc04,B}{1935Sc04}); \ensuremath{\pi} from \ensuremath{\mu} and L(d,t)=1.\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.67033cm}{\raggedright \ensuremath{\mu}: Measured using the nuclear magnetic resonance using optically pumped\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.67033cm}{\raggedright {\ }{\ }{\ }ions technique; Other: 0.560226 \textit{3} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1961Ca21,B}{1961Ca21}).\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.67033cm}{\raggedright Q: Measured using the muonic x-ray hyperfine structure technique; Others:\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.67033cm}{\raggedright {\ }{\ }{\ }0.38 \textit{4} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986Ul02,B}{1986Ul02}), 0.39 \textit{5} or 0.27 \textit{4} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1979Ha08,B}{1979Ha08}), 0.41 \textit{4} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1965Mu15,B}{1965Mu15}), 0.46\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.67033cm}{\raggedright {\ }{\ }{\ }\textit{4} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1960Mc11,B}{1960Mc11}) and 0.53 \textit{4} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1975Ed01,B}{1975Ed01}).\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.67033cm}{\raggedright Magnetic octupole moment={\textminus}0.15 (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1976Fu06,B}{1976Fu06}).\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.67033cm}{\raggedright Isotope shift studied by \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1973BoVN,B}{1973BoVN}, \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1975Ge04,B}{1975Ge04}. Hyperfine anomalies studied\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.67033cm}{\raggedright {\ }{\ }{\ }by \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1973Re04,B}{1973Re04}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{5648\ensuremath{^{{\hyperlink{HG10LEVEL5}{a}}}} {\it 10}}&\multicolumn{1}{l}{1/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{81}&\multicolumn{1}{@{ }l}{ns {\it 5}}&\multicolumn{1}{l}{\texttt{\hyperlink{AU11}{A}\ \hyperlink{TL13}{C}\ \hyperlink{HG15}{E}\ \hyperlink{HG17}{G}\ \hyperlink{HG19}{I}} }&\parbox[t][0.3cm]{10.67033cm}{\raggedright J\ensuremath{^{\pi}}: 1.5648\ensuremath{\gamma} M1+E2 to 3/2\ensuremath{^{-}}; \ensuremath{\gamma}\ensuremath{\gamma}(\ensuremath{\theta}) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1983Sc38,B}{1983Sc38} excludes \ensuremath{J^{\pi}}=3/2\ensuremath{^{-}} and 5/2\ensuremath{^{-}}.\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.67033cm}{\raggedright T\ensuremath{_{1/2}}: From \ensuremath{\gamma}-ce(\ensuremath{\Delta}t) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2007Me12,B}{2007Me12}. Other: 56 ns \textit{19} from B(E2)\ensuremath{\uparrow}=0.127 \textit{17},\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.67033cm}{\raggedright {\ }{\ }{\ }weighted average of 0.104 \textit{25} and 0.145 \textit{22} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1979Ha08,B}{1979Ha08}). Since the\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.67033cm}{\raggedright {\ }{\ }{\ }uncertainties quoted in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1979Ha08,B}{1979Ha08} are only statistical and the Coulomb\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.67033cm}{\raggedright {\ }{\ }{\ }excitation contributions to the 26 keV level (\ensuremath{J^{\pi}}=5/2\ensuremath{^{-}}) were not taken into\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.67033cm}{\raggedright {\ }{\ }{\ }account, the value should be considered as approximate.\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.67033cm}{\raggedright B(E2)\ensuremath{\uparrow}=0.104 \textit{25} and 0.145 \textit{22} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1979Ha08,B}{1979Ha08}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{26}&\multicolumn{1}{@{.}l}{2738\ensuremath{^{{\hyperlink{HG10LEVEL6}{b}}}} {\it 3}}&\multicolumn{1}{l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{629}&\multicolumn{1}{@{ }l}{ps {\it 18}}&\multicolumn{1}{l}{\texttt{\hyperlink{AU11}{A}\hyperlink{HG12}{B}\hyperlink{TL13}{C}\hyperlink{HG14}{D}\hyperlink{HG15}{E}\ \hyperlink{HG17}{G}\hyperlink{HG18}{H}\hyperlink{HG19}{I}} }&\parbox[t][0.3cm]{10.67033cm}{\raggedright XREF: D(28)H(27).\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.67033cm}{\raggedright E(level): From nuclear resonant scattering in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2005Is19,B}{2005Is19}.\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.67033cm}{\raggedright J\ensuremath{^{\pi}}: 26.34\ensuremath{\gamma} M1+E2 to 3/2\ensuremath{^{-}}; L(d,t)=1+3; L(d,p)=1+3; \ensuremath{J^{\pi}}=5/2\ensuremath{^{-}} from \ensuremath{\gamma}(\ensuremath{\theta}) in\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.67033cm}{\raggedright {\ }{\ }{\ }Coulomb excitation (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1980Bo05,B}{1980Bo05}).\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.67033cm}{\raggedright T\ensuremath{_{1/2}}: From the time difference between the incident X-ray and the\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.67033cm}{\raggedright {\ }{\ }{\ }fluorescence signal from the \ensuremath{^{\textnormal{201}}}Hg atom in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2018Yo02,B}{2018Yo02}. Other: 630 ps \textit{50}\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.67033cm}{\raggedright {\ }{\ }{\ }from ce-\ensuremath{\gamma}(\ensuremath{\Delta}t) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1983Sc38,B}{1983Sc38}.\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.67033cm}{\raggedright B(E2)\ensuremath{\uparrow}\ensuremath{\leq}0.07 (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1983Sc38,B}{1983Sc38}), 0.00 \textit{15} and 0.00 \textit{18} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1979Ha08,B}{1979Ha08}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{32}&\multicolumn{1}{@{.}l}{155 {\it 13}}&\multicolumn{1}{l}{3/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{55}&\multicolumn{1}{@{ }l}{ps {\it 24}}&\multicolumn{1}{l}{\texttt{\hyperlink{AU11}{A}\ \hyperlink{TL13}{C}\hyperlink{HG14}{D}\hyperlink{HG15}{E}\hyperlink{HG16}{F}\hyperlink{HG17}{G}\hyperlink{HG18}{H}\hyperlink{HG19}{I}} }&\parbox[t][0.3cm]{10.67033cm}{\raggedright J\ensuremath{^{\pi}}: 30.6\ensuremath{\gamma} M1+E2 to 1/2\ensuremath{^{-}}; 32.19\ensuremath{\gamma} M1+E2 to 3/2\ensuremath{^{-}}; \ensuremath{\gamma}\ensuremath{\gamma}(\ensuremath{\theta}) in \ensuremath{^{\textnormal{201}}}Tl \ensuremath{\varepsilon} decay\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.67033cm}{\raggedright {\ }{\ }{\ }excludes \ensuremath{J^{\pi}}=1/2\ensuremath{^{-}}.\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.67033cm}{\raggedright T\ensuremath{_{1/2}}: From B(E2)\ensuremath{\uparrow}=0.14 \textit{5} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1983Sc38,B}{1983Sc38}); others: 0.2 ns (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1961Re12,B}{1961Re12}), \ensuremath{\leq}2 ns\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.67033cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1961Be29,B}{1961Be29}) and \ensuremath{>}0.1 ns (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1971Wa17,B}{1971Wa17}).\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.67033cm}{\raggedright B(E2)\ensuremath{\uparrow}=0.14 \textit{5} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1983Sc38,B}{1983Sc38}) in Coulomb excitation; Others: 0.13 \textit{15} and 0.10\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.67033cm}{\raggedright {\ }{\ }{\ }\textit{4} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1979Ha08,B}{1979Ha08}).\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.67033cm}{\raggedright Q: 0.3 \textit{15} or 0.09 \textit{20} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1979Ha08,B}{1979Ha08}) using muonic x-ray hyperfine structure\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.67033cm}{\raggedright {\ }{\ }{\ }technique.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{167}&\multicolumn{1}{@{.}l}{48 {\it 4}}&\multicolumn{1}{l}{1/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{$<$44}&\multicolumn{1}{@{ }l}{ps}&\multicolumn{1}{l}{\texttt{\hyperlink{AU11}{A}\ \hyperlink{TL13}{C}\hyperlink{HG14}{D}\ \hyperlink{HG16}{F}\hyperlink{HG17}{G}\hyperlink{HG18}{H}\ } }&\parbox[t][0.3cm]{10.67033cm}{\raggedright XREF: D(169)F(163)H(168).\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.67033cm}{\raggedright J\ensuremath{^{\pi}}: 165.88\ensuremath{\gamma} M1 to 1/2\ensuremath{^{-}}, 167.43\ensuremath{\gamma} M1+E2 to 3/2\ensuremath{^{-}}; L(d,t)=1; \ensuremath{\gamma}\ensuremath{\gamma}(\ensuremath{\theta}) in\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.67033cm}{\raggedright {\ }{\ }{\ }\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1975Ho08,B}{1975Ho08} and \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1983Sc38,B}{1983Sc38} excludes \ensuremath{J^{\pi}}=3/2\ensuremath{^{-}}.\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.67033cm}{\raggedright T\ensuremath{_{1/2}}: From B(E2)\ensuremath{\uparrow}=0.016 \textit{3}, weighted average of 0.017 \textit{4} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1983Sc38,B}{1983Sc38}) and\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.67033cm}{\raggedright {\ }{\ }{\ }0.014 \textit{4} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1980Bo05,B}{1980Bo05}). Other: \ensuremath{<}2 ns in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1961Be29,B}{1961Be29}.\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.67033cm}{\raggedright B(E2)\ensuremath{\uparrow}=0.017 \textit{4} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1983Sc38,B}{1983Sc38}) and 0.014 \textit{4} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1980Bo05,B}{1980Bo05}) in Coulomb excitation,\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.67033cm}{\raggedright {\ }{\ }{\ }and 0.017 \textit{12} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1979Ha08,B}{1979Ha08}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{384}&\multicolumn{1}{@{.}l}{604 {\it 17}}&\multicolumn{1}{l}{(5/2)\ensuremath{^{-}}}&&&\multicolumn{1}{l}{\texttt{\hyperlink{AU11}{A}\ \ \ \ \hyperlink{HG16}{F}\hyperlink{HG17}{G}\ \ } }&\parbox[t][0.3cm]{10.67033cm}{\raggedright XREF: F(382).\vspace{0.1cm}}&\\
\end{longtable}
\begin{textblock}{29}(0,27.3)
Continued on next page (footnotes at end of table)
\end{textblock}
\clearpage
\begin{longtable}{ccccccc@{\extracolsep{\fill}}c}
\\[-.4cm]
\multicolumn{8}{c}{{\bf \small \underline{Adopted \hyperlink{201HG_LEVEL}{Levels}, \hyperlink{201HG_GAMMA}{Gammas} (continued)}}}\\
\multicolumn{8}{c}{~}\\
\multicolumn{8}{c}{\underline{\ensuremath{^{201}}Hg Levels (continued)}}\\
\multicolumn{8}{c}{~}\\
\multicolumn{2}{c}{E(level)$^{{\hyperlink{HG10LEVEL0}{\dagger}}}$}&J$^{\pi}$$^{}$&\multicolumn{2}{c}{T$_{1/2}$$^{}$}&XREF&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&
\endhead
&&&&&&\parbox[t][0.3cm]{9.58535cm}{\raggedright J\ensuremath{^{\pi}}: 352.42\ensuremath{\gamma} M1(+E2) to 3/2\ensuremath{^{-}}; 358.36\ensuremath{\gamma} M1+E2 to 5/2\ensuremath{^{-}}; population in\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{9.58535cm}{\raggedright {\ }{\ }{\ }Coulomb excitation.\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{9.58535cm}{\raggedright B(E2)\ensuremath{\uparrow}=0.085 \textit{5} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1983Sc38,B}{1983Sc38}) in Coulomb excitation.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{414}&\multicolumn{1}{@{.}l}{543 {\it 18}}&\multicolumn{1}{l}{7/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{21}&\multicolumn{1}{@{.}l}{0 ps {\it 9}}&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{HG14}{D}\ \hyperlink{HG16}{F}\hyperlink{HG17}{G}\ \hyperlink{HG19}{I}} }&\parbox[t][0.3cm]{9.58535cm}{\raggedright XREF: D(417)F(412).\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{9.58535cm}{\raggedright J\ensuremath{^{\pi}}: 414.49\ensuremath{\gamma} E2 to 3/2\ensuremath{^{-}}, 388.26\ensuremath{\gamma} M1+E2 to 5/2\ensuremath{^{-}}.\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{9.58535cm}{\raggedright T\ensuremath{_{1/2}}: From B(E2)\ensuremath{\uparrow}=0.152 \textit{6} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1980Bo05,B}{1980Bo05}) in Coulomb excitation.\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{9.58535cm}{\raggedright B(E2)\ensuremath{\uparrow}=0.152 \textit{6} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1980Bo05,B}{1980Bo05}) in Coulomb excitation.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{464}&\multicolumn{1}{@{.}l}{41 {\it 4}}&\multicolumn{1}{l}{5/2\ensuremath{^{-}}}&&&\multicolumn{1}{l}{\texttt{\hyperlink{AU11}{A}\ \ \hyperlink{HG14}{D}\ \hyperlink{HG16}{F}\hyperlink{HG17}{G}\hyperlink{HG18}{H}\ } }&\parbox[t][0.3cm]{9.58535cm}{\raggedright XREF: D(466)F(465).\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{9.58535cm}{\raggedright J\ensuremath{^{\pi}}: L(d,t)=3; 464.39\ensuremath{\gamma} M1+E2 to 3/2\ensuremath{^{-}}.\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{9.58535cm}{\raggedright B(E2)\ensuremath{\uparrow}=0.209 \textit{5} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1980Bo05,B}{1980Bo05}) in Coulomb excitation.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{542}&\multicolumn{1}{@{.}l}{81 {\it 17}}&\multicolumn{1}{l}{1/2\ensuremath{^{-}},3/2,5/2}&&&\multicolumn{1}{l}{\texttt{\hyperlink{AU11}{A}\ \ \ \ \ \ \ \hyperlink{HG19}{I}} }&\parbox[t][0.3cm]{9.58535cm}{\raggedright J\ensuremath{^{\pi}}: 517.0\ensuremath{\gamma} to 5/2\ensuremath{^{-}}; 542.6\ensuremath{\gamma} to 3/2\ensuremath{^{-}}; population in \ensuremath{^{\textnormal{201}}}Au \ensuremath{\beta}\ensuremath{^{-}} decay\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{9.58535cm}{\raggedright {\ }{\ }{\ }(\ensuremath{J^{\pi}}=3/2\ensuremath{^{+}}) argues against \ensuremath{J^{\pi}}=7/2\ensuremath{^{-}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{547}&\multicolumn{1}{@{.}l}{32 {\it 10}}&\multicolumn{1}{l}{9/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{$<$20}&\multicolumn{1}{@{ }l}{ns}&\multicolumn{1}{l}{\texttt{\ \hyperlink{HG12}{B}\ \ \ \ \ \ \ } }&\parbox[t][0.3cm]{9.58535cm}{\raggedright J\ensuremath{^{\pi}}: 521.0\ensuremath{\gamma} E2 to 5/2\ensuremath{^{-}}; not population in \ensuremath{^{\textnormal{201}}}Au \ensuremath{\beta}\ensuremath{^{-}} decay (\ensuremath{J^{\pi}}=3/2\ensuremath{^{+}})\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{9.58535cm}{\raggedright {\ }{\ }{\ }argues against \ensuremath{J^{\pi}}=1/2\ensuremath{^{-}}.\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{9.58535cm}{\raggedright T\ensuremath{_{1/2}}: Upper limit from \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1964Br27,B}{1964Br27}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{553}&\multicolumn{1}{@{.}l}{01 {\it 6}}&\multicolumn{1}{l}{1/2\ensuremath{^{-}},3/2\ensuremath{^{-}},5/2\ensuremath{^{-}}}&&&\multicolumn{1}{l}{\texttt{\hyperlink{AU11}{A}\ \ \hyperlink{HG14}{D}\ \ \hyperlink{HG17}{G}\ \ } }&\parbox[t][0.3cm]{9.58535cm}{\raggedright XREF: D(550).\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{9.58535cm}{\raggedright J\ensuremath{^{\pi}}: 385.1\ensuremath{\gamma} to 1/2\ensuremath{^{-}}, 526.8\ensuremath{\gamma} to 5/2\ensuremath{^{-}}; population in Coulomb excitation\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{9.58535cm}{\raggedright {\ }{\ }{\ }would argue against \ensuremath{J^{\pi}}=3/2\ensuremath{^{+}}.\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{9.58535cm}{\raggedright B(E2)\ensuremath{\uparrow}=0.035 \textit{2} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1980Bo05,B}{1980Bo05}) in Coulomb excitation.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{645}&\multicolumn{1}{@{.}l}{4 {\it 3}}&\multicolumn{1}{l}{(5/2)\ensuremath{^{-}}}&&&\multicolumn{1}{l}{\texttt{\hyperlink{AU11}{A}\ \ \ \ \ \ \hyperlink{HG18}{H}\ } }&\parbox[t][0.3cm]{9.58535cm}{\raggedright J\ensuremath{^{\pi}}: L(d,t)=3. The feeding of this level in \ensuremath{^{\textnormal{201}}}Au \ensuremath{\beta}\ensuremath{^{-}} decay (\ensuremath{J^{\pi}}=3/2\ensuremath{^{+}})\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{9.58535cm}{\raggedright {\ }{\ }{\ }would argue against \ensuremath{J^{\pi}}=7/2\ensuremath{^{-}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{732}&\multicolumn{1}{@{.}l}{3 {\it 4}}&\multicolumn{1}{l}{1/2\ensuremath{^{-}},3/2\ensuremath{^{-}}}&&&\multicolumn{1}{l}{\texttt{\hyperlink{AU11}{A}\ \ \hyperlink{HG14}{D}\ \ \ \hyperlink{HG18}{H}\ } }&\parbox[t][0.3cm]{9.58535cm}{\raggedright XREF: D(735).\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{9.58535cm}{\raggedright J\ensuremath{^{\pi}}: L(d,t)=1.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{766}&\multicolumn{1}{@{.}l}{22\ensuremath{^{{\hyperlink{HG10LEVEL7}{c}}}} {\it 15}}&\multicolumn{1}{l}{13/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{94}&\multicolumn{1}{@{ }l}{\ensuremath{\mu}s {\it 2}}&\multicolumn{1}{l}{\texttt{\ \hyperlink{HG12}{B}\ \ \ \ \ \hyperlink{HG18}{H}\ } }&\parbox[t][0.3cm]{9.58535cm}{\raggedright J\ensuremath{^{\pi}}: 218.9\ensuremath{\gamma} M2+E3 to 9/2\ensuremath{^{-}}; L(d,t)\ensuremath{>}4.\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{9.58535cm}{\raggedright T\ensuremath{_{1/2}}: Weighted average of 92 \ensuremath{\mu}s \textit{3} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1961Kr01,B}{1961Kr01}), 100 \ensuremath{\mu}s \textit{6}\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{9.58535cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1962Eu01,B}{1962Eu01}) and 94 \ensuremath{\mu}s \textit{3} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1976Uy01,B}{1976Uy01}) in \ensuremath{^{\textnormal{201}}}Hg IT decay.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{953}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{HG10LEVEL1}{\ddagger}}}} {\it 4}}&&&&\multicolumn{1}{l}{\texttt{\ \ \ \ \ \ \ \hyperlink{HG18}{H}\ } }&&\\
\multicolumn{1}{r@{}}{1035}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{HG10LEVEL1}{\ddagger}}}} {\it 4}}&\multicolumn{1}{l}{1/2\ensuremath{^{-}},3/2\ensuremath{^{-}}}&&&\multicolumn{1}{l}{\texttt{\ \ \ \ \ \ \ \hyperlink{HG18}{H}\ } }&\parbox[t][0.3cm]{9.58535cm}{\raggedright J\ensuremath{^{\pi}}: L(d,t)=1.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1075}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{HG10LEVEL1}{\ddagger}}}} {\it 4}}&&&&\multicolumn{1}{l}{\texttt{\ \ \ \ \ \ \ \hyperlink{HG18}{H}\ } }&&\\
\multicolumn{1}{r@{}}{1187}&\multicolumn{1}{@{.}l}{8 {\it 5}}&&&&\multicolumn{1}{l}{\texttt{\hyperlink{AU11}{A}\ \ \ \ \ \ \ \ } }&&\\
\multicolumn{1}{r@{}}{1287}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{HG10LEVEL1}{\ddagger}}}} {\it 5}}&\multicolumn{1}{l}{5/2\ensuremath{^{-}},7/2\ensuremath{^{-}}}&&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{HG14}{D}\ \ \ \hyperlink{HG18}{H}\ } }&\parbox[t][0.3cm]{9.58535cm}{\raggedright XREF: D(1280).\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{9.58535cm}{\raggedright J\ensuremath{^{\pi}}: L(d,t)=3.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1336}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{HG10LEVEL2}{\#}}}} {\it 5}}&&&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{HG14}{D}\ \hyperlink{HG16}{F}\ \ \ } }&\parbox[t][0.3cm]{9.58535cm}{\raggedright XREF: F(1325).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1360}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{HG10LEVEL1}{\ddagger}}}} {\it 5}}&&&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{HG14}{D}\ \ \ \hyperlink{HG18}{H}\ } }&\parbox[t][0.3cm]{9.58535cm}{\raggedright XREF: D(1367).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1505}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{HG10LEVEL3}{@}}}} {\it 6}}&&&&\multicolumn{1}{l}{\texttt{\ \ \ \ \ \hyperlink{HG16}{F}\ \ \ } }&&\\
\multicolumn{1}{r@{}}{1583}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{HG10LEVEL1}{\ddagger}}}} {\it 6}}&&&&\multicolumn{1}{l}{\texttt{\ \ \ \ \ \ \ \hyperlink{HG18}{H}\ } }&&\\
\multicolumn{1}{r@{}}{1591}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{HG10LEVEL1}{\ddagger}}}} {\it 6}}&&&&\multicolumn{1}{l}{\texttt{\ \ \ \ \ \ \ \hyperlink{HG18}{H}\ } }&&\\
\multicolumn{1}{r@{}}{1693}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{HG10LEVEL1}{\ddagger}}}} {\it 7}}&\multicolumn{1}{l}{5/2\ensuremath{^{-}},7/2\ensuremath{^{-}}}&&&\multicolumn{1}{l}{\texttt{\ \ \ \ \ \ \ \hyperlink{HG18}{H}\ } }&\parbox[t][0.3cm]{9.58535cm}{\raggedright J\ensuremath{^{\pi}}: L(d,t)=3.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1710}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{HG10LEVEL1}{\ddagger}}}} {\it 7}}&\multicolumn{1}{l}{(5/2\ensuremath{^{-}},7/2\ensuremath{^{-}})}&&&\multicolumn{1}{l}{\texttt{\ \ \ \ \ \hyperlink{HG16}{F}\ \hyperlink{HG18}{H}\ } }&\parbox[t][0.3cm]{9.58535cm}{\raggedright XREF: F(1707).\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{9.58535cm}{\raggedright J\ensuremath{^{\pi}}: L(d,t)=(3).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1737}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{HG10LEVEL1}{\ddagger}}}} {\it 7}}&\multicolumn{1}{l}{1/2\ensuremath{^{-}},3/2\ensuremath{^{-}}}&&&\multicolumn{1}{l}{\texttt{\ \ \ \ \ \ \ \hyperlink{HG18}{H}\ } }&\parbox[t][0.3cm]{9.58535cm}{\raggedright J\ensuremath{^{\pi}}: L(d,t)=1.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1946}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{HG10LEVEL2}{\#}}}} {\it 8}}&\multicolumn{1}{l}{(5/2\ensuremath{^{-}},7/2,9/2\ensuremath{^{+}})}&&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{HG14}{D}\ \ \ \ \ } }&\parbox[t][0.3cm]{9.58535cm}{\raggedright J\ensuremath{^{\pi}}: L(d,p)=(4,3).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1971}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{HG10LEVEL1}{\ddagger}}}} {\it 8}}&\multicolumn{1}{l}{(1/2\ensuremath{^{-}},3/2\ensuremath{^{-}})}&&&\multicolumn{1}{l}{\texttt{\ \ \ \ \ \ \ \hyperlink{HG18}{H}\ } }&\parbox[t][0.3cm]{9.58535cm}{\raggedright J\ensuremath{^{\pi}}: L(d,t)=(1).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2037}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{HG10LEVEL1}{\ddagger}}}} {\it 8}}&\multicolumn{1}{l}{(\ensuremath{^{-}})}&&&\multicolumn{1}{l}{\texttt{\ \ \ \ \ \ \ \hyperlink{HG18}{H}\ } }&\parbox[t][0.3cm]{9.58535cm}{\raggedright J\ensuremath{^{\pi}}: L(d,t)=(3,5); \ensuremath{J^{\pi}}=5/2\ensuremath{^{-}},7/2\ensuremath{^{-}},9/2\ensuremath{^{-}} or 11/2\ensuremath{^{-}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2081}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{HG10LEVEL2}{\#}}}} {\it 8}}&\multicolumn{1}{l}{(5/2\ensuremath{^{-}},7/2,9/2\ensuremath{^{+}})}&&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{HG14}{D}\ \ \ \ \ } }&\parbox[t][0.3cm]{9.58535cm}{\raggedright J\ensuremath{^{\pi}}: L(d,p)=(4,3).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2096}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{HG10LEVEL1}{\ddagger}}}} {\it 8}}&\multicolumn{1}{l}{(\ensuremath{^{-}})}&&&\multicolumn{1}{l}{\texttt{\ \ \ \ \ \ \ \hyperlink{HG18}{H}\ } }&\parbox[t][0.3cm]{9.58535cm}{\raggedright J\ensuremath{^{\pi}}: L(d,t)=(3,5); \ensuremath{J^{\pi}}=5/2\ensuremath{^{-}},7/2\ensuremath{^{-}},9/2\ensuremath{^{-}} or 11/2\ensuremath{^{-}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2103}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{HG10LEVEL2}{\#}}}} {\it 8}}&\multicolumn{1}{l}{7/2\ensuremath{^{+}},9/2\ensuremath{^{+}}}&&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{HG14}{D}\ \ \ \ \ } }&\parbox[t][0.3cm]{9.58535cm}{\raggedright J\ensuremath{^{\pi}}: L(d,p)=4.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2478}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{HG10LEVEL1}{\ddagger}}}} {\it 10}}&&&&\multicolumn{1}{l}{\texttt{\ \ \ \ \ \ \ \hyperlink{HG18}{H}\ } }&&\\
\multicolumn{1}{r@{}}{2526}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{HG10LEVEL3}{@}}}} {\it 10}}&&&&\multicolumn{1}{l}{\texttt{\ \ \ \ \ \hyperlink{HG16}{F}\ \ \ } }&&\\
\multicolumn{1}{r@{}}{2628}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{HG10LEVEL2}{\#}}}} {\it 11}}&\multicolumn{1}{l}{7/2\ensuremath{^{+}},9/2\ensuremath{^{+}}}&&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{HG14}{D}\ \hyperlink{HG16}{F}\ \ \ } }&\parbox[t][0.3cm]{9.58535cm}{\raggedright J\ensuremath{^{\pi}}: L(d,p)=4.\vspace{0.1cm}}&\\
\end{longtable}
\begin{textblock}{29}(0,27.3)
Continued on next page (footnotes at end of table)
\end{textblock}
\clearpage
\begin{longtable}{ccccc@{\extracolsep{\fill}}c}
\\[-.4cm]
\multicolumn{6}{c}{{\bf \small \underline{Adopted \hyperlink{201HG_LEVEL}{Levels}, \hyperlink{201HG_GAMMA}{Gammas} (continued)}}}\\
\multicolumn{6}{c}{~}\\
\multicolumn{6}{c}{\underline{\ensuremath{^{201}}Hg Levels (continued)}}\\
\multicolumn{6}{c}{~}\\
\multicolumn{2}{c}{E(level)$^{{\hyperlink{HG10LEVEL0}{\dagger}}}$}&J$^{\pi}$$^{}$&XREF&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&\hrulefill&
\endhead
\multicolumn{1}{r@{}}{2663?}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{HG10LEVEL1}{\ddagger}}}}}&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{HG14}{D}\ \ \ \hyperlink{HG18}{H}\ } }&\parbox[t][0.3cm]{13.45306cm}{\raggedright XREF: D(2660).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2681}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{HG10LEVEL3}{@}}}} {\it 11}}&&\multicolumn{1}{l}{\texttt{\ \ \ \ \ \hyperlink{HG16}{F}\ \ \ } }&&\\
\multicolumn{1}{r@{}}{2795}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{HG10LEVEL2}{\#}}}} {\it 11}}&\multicolumn{1}{l}{(\ensuremath{^{+}})}&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{HG14}{D}\ \ \ \ \ } }&\parbox[t][0.3cm]{13.45306cm}{\raggedright J\ensuremath{^{\pi}}: L(d,p)=(4,2). \ensuremath{J^{\pi}}=7/2\ensuremath{^{+}}, 9/2\ensuremath{^{+}} or 3/2\ensuremath{^{+}}, 5/2\ensuremath{^{+}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2863}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{HG10LEVEL2}{\#}}}} {\it 12}}&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{HG14}{D}\ \ \ \ \ } }&&\\
\multicolumn{1}{r@{}}{2890}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{HG10LEVEL2}{\#}}}} {\it 12}}&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{HG14}{D}\ \hyperlink{HG16}{F}\ \ \ } }&\parbox[t][0.3cm]{13.45306cm}{\raggedright XREF: F(2891).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2911}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{HG10LEVEL2}{\#}}}} {\it 12}}&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{HG14}{D}\ \ \ \ \ } }&&\\
\multicolumn{1}{r@{}}{2938}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{HG10LEVEL2}{\#}}}} {\it 12}}&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{HG14}{D}\ \ \ \ \ } }&&\\
\multicolumn{1}{r@{}}{2976}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{HG10LEVEL2}{\#}}}} {\it 12}}&\multicolumn{1}{l}{(\ensuremath{^{+}})}&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{HG14}{D}\ \ \ \ \ } }&\parbox[t][0.3cm]{13.45306cm}{\raggedright J\ensuremath{^{\pi}}: L(d,p)=(4,2). \ensuremath{J^{\pi}}=7/2\ensuremath{^{+}}, 9/2\ensuremath{^{+}} or 3/2\ensuremath{^{+}}, 5/2\ensuremath{^{+}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2995}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{HG10LEVEL2}{\#}}}} {\it 12}}&\multicolumn{1}{l}{(\ensuremath{^{+}})}&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{HG14}{D}\ \ \ \ \ } }&\parbox[t][0.3cm]{13.45306cm}{\raggedright J\ensuremath{^{\pi}}: L(d,p)=(4,2). \ensuremath{J^{\pi}}=7/2\ensuremath{^{+}}, 9/2\ensuremath{^{+}} or 3/2\ensuremath{^{+}}, 5/2\ensuremath{^{+}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3115}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{HG10LEVEL2}{\#}}}} {\it 13}}&\multicolumn{1}{l}{(\ensuremath{^{+}})}&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{HG14}{D}\ \ \ \ \ } }&\parbox[t][0.3cm]{13.45306cm}{\raggedright J\ensuremath{^{\pi}}: L(d,p)=(2,4). \ensuremath{J^{\pi}}=3/2\ensuremath{^{+}}, 5/2\ensuremath{^{+}} or 7/2\ensuremath{^{+}}, 9/2\ensuremath{^{+}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3172}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{HG10LEVEL2}{\#}}}} {\it 13}}&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{HG14}{D}\ \ \ \ \ } }&&\\
\multicolumn{1}{r@{}}{3196}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{HG10LEVEL2}{\#}}}} {\it 13}}&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{HG14}{D}\ \ \ \ \ } }&&\\
\multicolumn{1}{r@{}}{3233}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{HG10LEVEL2}{\#}}}} {\it 13}}&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{HG14}{D}\ \ \ \ \ } }&&\\
\multicolumn{1}{r@{}}{3252}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{HG10LEVEL2}{\#}}}} {\it 13}}&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{HG14}{D}\ \ \ \ \ } }&&\\
\multicolumn{1}{r@{}}{3270}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{HG10LEVEL2}{\#}}}} {\it 13}}&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{HG14}{D}\ \ \ \ \ } }&&\\
\multicolumn{1}{r@{}}{3294}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{HG10LEVEL2}{\#}}}} {\it 13}}&\multicolumn{1}{l}{(\ensuremath{^{+}})}&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{HG14}{D}\ \ \ \ \ } }&\parbox[t][0.3cm]{13.45306cm}{\raggedright J\ensuremath{^{\pi}}: L(d,p)=(2,4). \ensuremath{J^{\pi}}=3/2\ensuremath{^{+}}, 5/2\ensuremath{^{+}} or 7/2\ensuremath{^{+}}, 9/2\ensuremath{^{+}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3539}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{HG10LEVEL2}{\#}}}} {\it 14}}&\multicolumn{1}{l}{(\ensuremath{^{+}})}&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{HG14}{D}\ \ \ \ \ } }&\parbox[t][0.3cm]{13.45306cm}{\raggedright J\ensuremath{^{\pi}}: L(d,p)=(2,4). \ensuremath{J^{\pi}}=3/2\ensuremath{^{+}}, 5/2\ensuremath{^{+}} or 7/2\ensuremath{^{+}}, 9/2\ensuremath{^{+}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3579}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{HG10LEVEL2}{\#}}}} {\it 14}}&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{HG14}{D}\ \ \ \ \ } }&&\\
\multicolumn{1}{r@{}}{3712}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{HG10LEVEL2}{\#}}}} {\it 15}}&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{HG14}{D}\ \ \ \ \ } }&&\\
\multicolumn{1}{r@{}}{3735}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{HG10LEVEL3}{@}}}} {\it 15}}&&\multicolumn{1}{l}{\texttt{\ \ \ \ \ \hyperlink{HG16}{F}\ \ \ } }&&\\
\multicolumn{1}{r@{}}{3768}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{HG10LEVEL2}{\#}}}} {\it 15}}&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{HG14}{D}\ \ \ \ \ } }&&\\
\multicolumn{1}{r@{}}{3814}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{HG10LEVEL2}{\#}}}} {\it 15}}&\multicolumn{1}{l}{(\ensuremath{^{+}})}&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{HG14}{D}\ \ \ \ \ } }&\parbox[t][0.3cm]{13.45306cm}{\raggedright J\ensuremath{^{\pi}}: L(d,p)=(4,2). \ensuremath{J^{\pi}}=7/2\ensuremath{^{+}}, 9/2\ensuremath{^{+}} or 3/2\ensuremath{^{+}}, 5/2\ensuremath{^{+}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3837}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{HG10LEVEL2}{\#}}}} {\it 15}}&\multicolumn{1}{l}{(\ensuremath{^{+}})}&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{HG14}{D}\ \ \ \ \ } }&\parbox[t][0.3cm]{13.45306cm}{\raggedright J\ensuremath{^{\pi}}: L(d,p)=(2,6). \ensuremath{J^{\pi}}=3/2\ensuremath{^{+}}, 5/2\ensuremath{^{+}} or 11/2\ensuremath{^{+}}, 13/2\ensuremath{^{+}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3870}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{HG10LEVEL2}{\#}}}} {\it 15}}&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{HG14}{D}\ \ \ \ \ } }&&\\
\multicolumn{1}{r@{}}{3884}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{HG10LEVEL2}{\#}}}} {\it 16}}&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{HG14}{D}\ \ \ \ \ } }&&\\
\multicolumn{1}{r@{}}{3900}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{HG10LEVEL2}{\#}}}} {\it 16}}&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{HG14}{D}\ \ \ \ \ } }&&\\
\multicolumn{1}{r@{}}{3921}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{HG10LEVEL2}{\#}}}} {\it 16}}&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{HG14}{D}\ \ \ \ \ } }&&\\
\multicolumn{1}{r@{}}{3965}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{HG10LEVEL3}{@}}}} {\it 16}}&&\multicolumn{1}{l}{\texttt{\ \ \ \ \ \hyperlink{HG16}{F}\ \ \ } }&&\\
\multicolumn{1}{r@{}}{4007}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{HG10LEVEL2}{\#}}}} {\it 16}}&\multicolumn{1}{l}{(\ensuremath{^{+}})}&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{HG14}{D}\ \ \ \ \ } }&\parbox[t][0.3cm]{13.45306cm}{\raggedright J\ensuremath{^{\pi}}: L(d,p)=(2,6). \ensuremath{J^{\pi}}=3/2\ensuremath{^{+}}, 5/2\ensuremath{^{+}} or 11/2\ensuremath{^{+}}, 13/2\ensuremath{^{+}}.\vspace{0.1cm}}&\\
&&&&\parbox[t][0.3cm]{13.45306cm}{\raggedright E(level): Complex peak.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{4070}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{HG10LEVEL2}{\#}}}} {\it 16}}&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{HG14}{D}\ \ \ \ \ } }&&\\
\multicolumn{1}{r@{}}{4095}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{HG10LEVEL2}{\#}}}} {\it 16}}&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{HG14}{D}\ \ \ \ \ } }&&\\
\multicolumn{1}{r@{}}{4123}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{HG10LEVEL2}{\#}}}} {\it 16}}&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{HG14}{D}\ \ \ \ \ } }&&\\
\multicolumn{1}{r@{}}{4233}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{HG10LEVEL2}{\#}}}} {\it 17}}&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{HG14}{D}\ \ \ \ \ } }&&\\
\multicolumn{1}{r@{}}{4284}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{HG10LEVEL2}{\#}}}} {\it 17}}&\multicolumn{1}{l}{(\ensuremath{^{+}})}&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{HG14}{D}\ \ \ \ \ } }&\parbox[t][0.3cm]{13.45306cm}{\raggedright J\ensuremath{^{\pi}}: L(d,p)=(2,4). \ensuremath{J^{\pi}}=3/2\ensuremath{^{+}}, 5/2\ensuremath{^{+}} or 7/2\ensuremath{^{+}}, 9/2\ensuremath{^{+}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{4313}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{HG10LEVEL2}{\#}}}} {\it 17}}&\multicolumn{1}{l}{(\ensuremath{^{+}})}&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{HG14}{D}\ \ \ \ \ } }&\parbox[t][0.3cm]{13.45306cm}{\raggedright J\ensuremath{^{\pi}}: L(d,p)=(2,4). \ensuremath{J^{\pi}}=3/2\ensuremath{^{+}}, 5/2\ensuremath{^{+}} or 7/2\ensuremath{^{+}}, 9/2\ensuremath{^{+}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{4362}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{HG10LEVEL2}{\#}}}} {\it 17}}&\multicolumn{1}{l}{(\ensuremath{^{+}})}&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{HG14}{D}\ \ \ \ \ } }&\parbox[t][0.3cm]{13.45306cm}{\raggedright J\ensuremath{^{\pi}}: L(d,p)=(4,0). \ensuremath{J^{\pi}}=7/2\ensuremath{^{+}}, 9/2\ensuremath{^{+}} or 1/2\ensuremath{^{+}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{4381}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{HG10LEVEL2}{\#}}}} {\it 18}}&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{HG14}{D}\ \ \ \ \ } }&&\\
\multicolumn{1}{r@{}}{4405}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{HG10LEVEL2}{\#}}}} {\it 18}}&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{HG14}{D}\ \ \ \ \ } }&&\\
\multicolumn{1}{r@{}}{4418?}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{HG10LEVEL2}{\#}}}}}&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{HG14}{D}\ \ \ \ \ } }&&\\
\multicolumn{1}{r@{}}{4467?}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{HG10LEVEL2}{\#}}}}}&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{HG14}{D}\ \ \ \ \ } }&&\\
\multicolumn{1}{r@{}}{4484}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{HG10LEVEL2}{\#}}}} {\it 18}}&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{HG14}{D}\ \ \ \ \ } }&&\\
\multicolumn{1}{r@{}}{4579}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{HG10LEVEL2}{\#}}}} {\it 18}}&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{HG14}{D}\ \ \ \ \ } }&&\\
\multicolumn{1}{r@{}}{4591}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{HG10LEVEL2}{\#}}}} {\it 18}}&\multicolumn{1}{l}{(\ensuremath{^{+}})}&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{HG14}{D}\ \ \ \ \ } }&\parbox[t][0.3cm]{13.45306cm}{\raggedright J\ensuremath{^{\pi}}: L(d,p)=(4,2). \ensuremath{J^{\pi}}=7/2\ensuremath{^{+}}, 9/2\ensuremath{^{+}} or 3/2\ensuremath{^{+}}, 5/2\ensuremath{^{+}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{4649}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{HG10LEVEL2}{\#}}}} {\it 19}}&\multicolumn{1}{l}{(\ensuremath{^{+}})}&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{HG14}{D}\ \ \ \ \ } }&\parbox[t][0.3cm]{13.45306cm}{\raggedright J\ensuremath{^{\pi}}: L(d,p)=(4,2). \ensuremath{J^{\pi}}=7/2\ensuremath{^{+}}, 9/2\ensuremath{^{+}} or 3/2\ensuremath{^{+}}, 5/2\ensuremath{^{+}}.\vspace{0.1cm}}&\\
\end{longtable}
\begin{textblock}{29}(0,27.3)
Continued on next page (footnotes at end of table)
\end{textblock}
\clearpage
\vspace*{-0.5cm}
{\bf \small \underline{Adopted \hyperlink{201HG_LEVEL}{Levels}, \hyperlink{201HG_GAMMA}{Gammas} (continued)}}\\
\vspace{0.3cm}
\underline{$^{201}$Hg Levels (continued)}\\
\vspace{0.3cm}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{HG10LEVEL0}{\dagger}}}} From a least-squares fit to E\ensuremath{\gamma}, unless otherwise stated.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{HG10LEVEL1}{\ddagger}}}} From \ensuremath{^{\textnormal{202}}}Hg(d,t).}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{HG10LEVEL2}{\#}}}} From \ensuremath{^{\textnormal{200}}}Hg(d,p).}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{HG10LEVEL3}{@}}}} From \ensuremath{^{\textnormal{201}}}Hg(d,d\ensuremath{'}), \ensuremath{^{\textnormal{201}}}Hg(p,p\ensuremath{'}).}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{HG10LEVEL4}{\&}}}} Configuration=\ensuremath{\nu} p\ensuremath{_{\textnormal{3/2}}^{\textnormal{$-$1}}}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{HG10LEVEL5}{a}}}} Configuration=\ensuremath{\nu} p\ensuremath{_{\textnormal{1/2}}^{\textnormal{$-$1}}}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{HG10LEVEL6}{b}}}} Dominant configuration=\ensuremath{\nu} f\ensuremath{_{\textnormal{5/2}}^{\textnormal{$-$1}}}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{HG10LEVEL7}{c}}}} Configuration=\ensuremath{\nu} i\ensuremath{_{\textnormal{13/2}}^{\textnormal{$-$1}}}.}\\
\vspace{0.5cm}
\clearpage
\vspace{0.3cm}
\begin{landscape}
\vspace*{-0.5cm}
{\bf \small \underline{Adopted \hyperlink{201HG_LEVEL}{Levels}, \hyperlink{201HG_GAMMA}{Gammas} (continued)}}\\
\vspace{0.3cm}
\hypertarget{201HG_GAMMA}{\underline{$\gamma$($^{201}$Hg)}}\\
\begin{longtable}{ccccccccc@{}ccccccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E\ensuremath{_{i}}(level)}&J\ensuremath{^{\pi}_{i}}&\multicolumn{2}{c}{E\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{HG10GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{I\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{HG10GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{E\ensuremath{_{f}}}&J\ensuremath{^{\pi}_{f}}&Mult.\ensuremath{^{\hyperlink{HG10GAMMA0}{\dagger}}}&\multicolumn{2}{c}{\ensuremath{\delta}\ensuremath{^{\hyperlink{HG10GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{\ensuremath{\alpha}\ensuremath{^{\hyperlink{HG10GAMMA4}{\&}}}}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{5648}&\multicolumn{1}{l}{1/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{5648 {\it 10}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{3/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0105 {\it 14}}&\multicolumn{1}{r@{}}{4.7\ensuremath{\times10^{4}}}&\multicolumn{1}{@{ }l}{{\it 7}}&\parbox[t][0.3cm]{8.618859cm}{\raggedright B(M1)(W.u.)=0.00151 \textit{+29{\textminus}21}; B(E2)(W.u.)=25 \textit{+9{\textminus}7}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.618859cm}{\raggedright E\ensuremath{_{\gamma}}: From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1997Ge09,B}{1997Ge09}. Other: 1.565 keV \textit{6} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1987Dr06,B}{1987Dr06}).\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.618859cm}{\raggedright Mult.: From N1/N2=0.94 \textit{31}, N1/N3=0.60 \textit{20}, N2/N3=0.64 \textit{5},\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.618859cm}{\raggedright {\ }{\ }{\ }N4/N3=0.042 \textit{18}, N5/N3=0.043 \textit{23}, N4/N5=0.98 \textit{32},\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.618859cm}{\raggedright {\ }{\ }{\ }O1/O2=0.81 \textit{16}, O2/N3=0.158 \textit{30}, O3/N3=0.20 \textit{4},\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.618859cm}{\raggedright {\ }{\ }{\ }O1/O3=0.64 \textit{12}, O1/N3=0.128 \textit{30} and O2/O3=0.79 \textit{19}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.618859cm}{\raggedright {\ }{\ }{\ }subshell ratios in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1997Ge09,B}{1997Ge09} and N1/N2=1.2 \textit{2}, N1/N3=1.1 \textit{2},\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.618859cm}{\raggedright {\ }{\ }{\ }N2/N3=0.92 \textit{15}, N4/N3=0.03 \textit{2} and N5/N3=0.04 \textit{2} subshell\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.618859cm}{\raggedright {\ }{\ }{\ }ratios in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1987Dr06,B}{1987Dr06}.\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.618859cm}{\raggedright \ensuremath{\delta}: From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1987Dr06,B}{1987Dr06}; Other: 0.0145 \textit{+19{\textminus}14} in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1997Ge09,B}{1997Ge09}.\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.618859cm}{\raggedright \ensuremath{\alpha}: 4.7E+4 7 from \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1987Dr06,B}{1987Dr06}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{26}&\multicolumn{1}{@{.}l}{2738}&\multicolumn{1}{l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{26}&\multicolumn{1}{@{.}l}{2738 {\it 3}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{3/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{012 {\it 8}}&\multicolumn{1}{r@{}}{72}&\multicolumn{1}{@{.}l}{9 {\it 13}}&\parbox[t][0.3cm]{8.618859cm}{\raggedright B(M1)(W.u.)=0.0261 \textit{9}; B(E2)(W.u.)=2.0 \textit{+37{\textminus}16}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.618859cm}{\raggedright E\ensuremath{_{\gamma}}: From nuclear resonant scattering in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2005Is19,B}{2005Is19}. Other: 26.34\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.618859cm}{\raggedright {\ }{\ }{\ }keV \textit{7} in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1983Sc38,B}{1983Sc38}.\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.618859cm}{\raggedright Mult.,\ensuremath{\delta}: From M1:M2:M3=100:12.0 \textit{20}:1.5 \textit{7} in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1983Sc38,B}{1983Sc38} and\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.618859cm}{\raggedright {\ }{\ }{\ }the briccmixing program.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{32}&\multicolumn{1}{@{.}l}{155}&\multicolumn{1}{l}{3/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{(5}&\multicolumn{1}{@{.}l}{881 {\it 13})}&\multicolumn{1}{r@{}}{$\approx$0}&\multicolumn{1}{@{.}l}{03}&\multicolumn{1}{r@{}}{26}&\multicolumn{1}{@{.}l}{2738 }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{l}{[M1]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{1451}&\multicolumn{1}{@{ }l}{{\it 22}}&\parbox[t][0.3cm]{8.618859cm}{\raggedright E\ensuremath{_{\gamma}}: Not observed directly, but required from the ce-\ensuremath{\gamma}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.618859cm}{\raggedright {\ }{\ }{\ }coincidence data in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1983Sc38,B}{1983Sc38}. E\ensuremath{\gamma} from level energy\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.618859cm}{\raggedright {\ }{\ }{\ }differences.\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{30}&\multicolumn{1}{@{.}l}{60 {\it 3}}&\multicolumn{1}{r@{}}{98}&\multicolumn{1}{@{.}l}{1 {\it 19}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{5648 }&\multicolumn{1}{@{}l}{1/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{013 {\it 5}}&\multicolumn{1}{r@{}}{46}&\multicolumn{1}{@{.}l}{4 {\it 7}}&\parbox[t][0.3cm]{8.618859cm}{\raggedright Mult.: From L3:L1=0.0136 \textit{21}, L2:L1=0.105 \textit{11} and\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.618859cm}{\raggedright {\ }{\ }{\ }L3:L2=0.130 \textit{24} in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1983Sc38,B}{1983Sc38}; L1:L2:L3:M1:M2:N:O1=50.9\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.618859cm}{\raggedright {\ }{\ }{\ }\textit{40}:5.0 \textit{6}:0.56 \textit{8}:14.2 \textit{15}:1.5 \textit{5}:4.0 \textit{5}:0.70 \textit{15} in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1960He05,B}{1960He05};\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.618859cm}{\raggedright {\ }{\ }{\ }A\ensuremath{_{\textnormal{2}}}(135.5\ensuremath{\gamma}-30.6\ensuremath{\gamma}(\ensuremath{\theta}))=0.159 \textit{26} in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1975Ho08,B}{1975Ho08}.\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.618859cm}{\raggedright \ensuremath{\delta}: From L3/L1, L2/L1 and L3/L2 in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1983Sc38,B}{1983Sc38} and the\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.618859cm}{\raggedright {\ }{\ }{\ }briccmixing program. Others: \ensuremath{\leq}0.03 from \ensuremath{\gamma}\ensuremath{\gamma}(\ensuremath{\theta}) in\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.618859cm}{\raggedright {\ }{\ }{\ }\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1975Ho08,B}{1975Ho08}; {\textminus}0.0634\ensuremath{\leq}\ensuremath{\delta}\ensuremath{\leq}+0.0515 from \ensuremath{\gamma}\ensuremath{\gamma}(\ensuremath{\theta}) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1978No06,B}{1978No06};\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.618859cm}{\raggedright {\ }{\ }{\ }0.006 \textit{16} from from L1:L2:L3 in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1960He05,B}{1960He05} and the\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.618859cm}{\raggedright {\ }{\ }{\ }briccmixing program.\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{32}&\multicolumn{1}{@{.}l}{19 {\it 3}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{.}l}{0 {\it 19}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{3/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0204 {\it 25}}&\multicolumn{1}{r@{}}{40}&\multicolumn{1}{@{.}l}{2 {\it 6}}&\parbox[t][0.3cm]{8.618859cm}{\raggedright Mult.: From L1:L2:L3=100:11.3 \textit{5}:1.75 \textit{15} in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1983Sc38,B}{1983Sc38};\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.618859cm}{\raggedright {\ }{\ }{\ }L3:L1=0.0130 \textit{20}, L2:L1=0.094 \textit{8}, L3:L2=0.138 \textit{24}, L:M=3.9\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.618859cm}{\raggedright {\ }{\ }{\ }\textit{4}, M1:M2=8.4 \textit{3} M:N=4.6 \textit{6} in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1960He05,B}{1960He05};\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.618859cm}{\raggedright {\ }{\ }{\ }A\ensuremath{_{\textnormal{2}}}(135.5\ensuremath{\gamma}-32.2\ensuremath{\gamma}(\ensuremath{\theta}))={\textminus}0.193 \textit{28} in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1975Ho08,B}{1975Ho08}.\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.618859cm}{\raggedright \ensuremath{\delta}: From L3/L1, L2/L1 and L3/L2 in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1983Sc38,B}{1983Sc38} and the\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.618859cm}{\raggedright {\ }{\ }{\ }briccmixing program. Others: \ensuremath{\leq}0.03 from \ensuremath{\gamma}\ensuremath{\gamma}(\ensuremath{\theta}) in\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.618859cm}{\raggedright {\ }{\ }{\ }\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1975Ho08,B}{1975Ho08}; {\textminus}0.0361\ensuremath{\leq}\ensuremath{\delta}\ensuremath{\leq}+0.0506 from \ensuremath{\gamma}\ensuremath{\gamma}(\ensuremath{\theta}) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1978No06,B}{1978No06};\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.618859cm}{\raggedright {\ }{\ }{\ }0.013 \textit{11} from from L1:L2:L3 in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1960He05,B}{1960He05} and the\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.618859cm}{\raggedright {\ }{\ }{\ }briccmixing program.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{167}&\multicolumn{1}{@{.}l}{48}&\multicolumn{1}{l}{1/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{135}&\multicolumn{1}{@{.}l}{34 {\it 4}}&\multicolumn{1}{r@{}}{26}&\multicolumn{1}{@{.}l}{05 {\it 18}}&\multicolumn{1}{r@{}}{32}&\multicolumn{1}{@{.}l}{155 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{$-$0}&\multicolumn{1}{@{.}l}{07 {\it 4}}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{32 {\it 5}}&\parbox[t][0.3cm]{8.618859cm}{\raggedright Mult.: From K:L1:L2:L3:M1:N1=56.0 \textit{4}:7.9 \textit{7}:0.77 \textit{15}:0.07 \textit{3}:\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.618859cm}{\raggedright {\ }{\ }{\ }2.2 \textit{3}:0.60 \textit{9}; \ensuremath{\gamma}\ensuremath{\gamma}(\ensuremath{\theta}) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1975Ho08,B}{1975Ho08} and \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1983Sc38,B}{1983Sc38}.\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.618859cm}{\raggedright \ensuremath{\delta}: From \ensuremath{\gamma}\ensuremath{\gamma}(\ensuremath{\theta}) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1975Ho08,B}{1975Ho08}; other: 0.000 \textit{4} from\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.618859cm}{\raggedright {\ }{\ }{\ }K:L1:L2:L3:M1:N1 in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1960He05,B}{1960He05} and the briccmixing\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.618859cm}{\raggedright {\ }{\ }{\ }program.\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{141}&\multicolumn{1}{@{.}l}{1 {\it 2}}&\multicolumn{1}{r@{}}{$\approx$0}&\multicolumn{1}{@{.}l}{03}&\multicolumn{1}{r@{}}{26}&\multicolumn{1}{@{.}l}{2738 }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{l}{[E2]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{389 {\it 21}}&\parbox[t][0.3cm]{8.618859cm}{\raggedright E\ensuremath{_{\gamma}}: From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1983Sc38,B}{1983Sc38}.\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{165}&\multicolumn{1}{@{.}l}{88 {\it 7}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{47 {\it 2}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{5648 }&\multicolumn{1}{@{}l}{1/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{869 {\it 26}}&\parbox[t][0.3cm]{8.618859cm}{\raggedright Mult.: From K:L1=1.65 \textit{20}:0.25 \textit{5} in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1960He05,B}{1960He05}.\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{167}&\multicolumn{1}{@{.}l}{43 {\it 7}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{3/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{07 {\it 6}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{815 {\it 29}}&\parbox[t][0.3cm]{8.618859cm}{\raggedright Mult.: From K:L1:L2:L3:M1:N1:O1=100:14.6 \textit{12}:1.6 \textit{2}:0.18\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.618859cm}{\raggedright {\ }{\ }{\ }\textit{4}:4.0 \textit{4}: 1.10 \textit{15}:0.27 \textit{6} in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1960He05,B}{1960He05}.\vspace{0.1cm}}&\\
\end{longtable}
\clearpage
\begin{longtable}{ccccccccc@{}ccccccc@{\extracolsep{\fill}}c}
\\[-.4cm]
\multicolumn{17}{c}{{\bf \small \underline{Adopted \hyperlink{201HG_LEVEL}{Levels}, \hyperlink{201HG_GAMMA}{Gammas} (continued)}}}\\
\multicolumn{17}{c}{~}\\
\multicolumn{17}{c}{\underline{$\gamma$($^{201}$Hg) (continued)}}\\
\multicolumn{17}{c}{~~~}\\
\multicolumn{2}{c}{E\ensuremath{_{i}}(level)}&J\ensuremath{^{\pi}_{i}}&\multicolumn{2}{c}{E\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{HG10GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{I\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{HG10GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{E\ensuremath{_{f}}}&J\ensuremath{^{\pi}_{f}}&Mult.\ensuremath{^{\hyperlink{HG10GAMMA0}{\dagger}}}&\multicolumn{2}{c}{\ensuremath{\delta}\ensuremath{^{\hyperlink{HG10GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{\ensuremath{\alpha}\ensuremath{^{\hyperlink{HG10GAMMA4}{\&}}}}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endhead
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.1658cm}{\raggedright \ensuremath{\delta}: From K:L1:L2:L3 in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1960He05,B}{1960He05} and the\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.1658cm}{\raggedright {\ }{\ }{\ }briccmixing program.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{384}&\multicolumn{1}{@{.}l}{604}&\multicolumn{1}{l}{(5/2)\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{352}&\multicolumn{1}{@{.}l}{42\ensuremath{^{\hyperlink{HG10GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{.}l}{0\ensuremath{^{\hyperlink{HG10GAMMA1}{\ddagger}}} {\it 16}}&\multicolumn{1}{r@{}}{32}&\multicolumn{1}{@{.}l}{155 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1(+E2)}&\multicolumn{1}{r@{}}{+0}&\multicolumn{1}{@{.}l}{07 {\it 7}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{232 {\it 4}}&\parbox[t][0.3cm]{7.1658cm}{\raggedright Mult.,\ensuremath{\delta}: From \ensuremath{\gamma}(\ensuremath{\theta}) (A\ensuremath{_{\textnormal{2}}}={\textminus}0.047 \textit{24}) in Coulomb\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.1658cm}{\raggedright {\ }{\ }{\ }excitation (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1980Bo05,B}{1980Bo05}). other (alternative): \ensuremath{\delta}={\textminus}4.5\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.1658cm}{\raggedright {\ }{\ }{\ }\textit{+1.2{\textminus}1.7} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1980Bo05,B}{1980Bo05}).\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{358}&\multicolumn{1}{@{.}l}{36\ensuremath{^{\hyperlink{HG10GAMMA1}{\ddagger}}} {\it 4}}&\multicolumn{1}{r@{}}{54}&\multicolumn{1}{@{.}l}{4\ensuremath{^{\hyperlink{HG10GAMMA1}{\ddagger}}} {\it 9}}&\multicolumn{1}{r@{}}{26}&\multicolumn{1}{@{.}l}{2738 }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1+E2}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{14 {\it 8}}&\parbox[t][0.3cm]{7.1658cm}{\raggedright Mult.,\ensuremath{\delta}: From \ensuremath{\gamma}(\ensuremath{\theta}) (A\ensuremath{_{\textnormal{2}}}=+0.08 \textit{6}) in Coulomb\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.1658cm}{\raggedright {\ }{\ }{\ }excitation (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1980Bo05,B}{1980Bo05}); {\textminus}0.3\ensuremath{\leq}\ensuremath{\delta}\ensuremath{\leq}3.9 (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1980Bo05,B}{1980Bo05}).\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{384}&\multicolumn{1}{@{.}l}{60\ensuremath{^{\hyperlink{HG10GAMMA1}{\ddagger}}} {\it 2}}&\multicolumn{1}{r@{}}{79}&\multicolumn{1}{@{.}l}{2\ensuremath{^{\hyperlink{HG10GAMMA1}{\ddagger}}} {\it 14}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{3/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{$-$0}&\multicolumn{1}{@{.}l}{23 {\it 9}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{178 {\it 6}}&\parbox[t][0.3cm]{7.1658cm}{\raggedright Mult.,\ensuremath{\delta}: From \ensuremath{\gamma}(\ensuremath{\theta}) (A\ensuremath{_{\textnormal{2}}}={\textminus}0.144 \textit{22}) in Coulomb\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.1658cm}{\raggedright {\ }{\ }{\ }excitation (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1980Bo05,B}{1980Bo05}). other (alternative): \ensuremath{\delta}={\textminus}1.8\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.1658cm}{\raggedright {\ }{\ }{\ }\textit{3} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1980Bo05,B}{1980Bo05}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{414}&\multicolumn{1}{@{.}l}{543}&\multicolumn{1}{l}{7/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{382}&\multicolumn{1}{@{.}l}{45\ensuremath{^{\hyperlink{HG10GAMMA1}{\ddagger}}} {\it 3}}&\multicolumn{1}{r@{}}{82}&\multicolumn{1}{@{.}l}{8\ensuremath{^{\hyperlink{HG10GAMMA1}{\ddagger}}} {\it 14}}&\multicolumn{1}{r@{}}{32}&\multicolumn{1}{@{.}l}{155 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{-}}}&\multicolumn{1}{l}{E2}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0535 {\it 7}}&\parbox[t][0.3cm]{7.1658cm}{\raggedright B(E2)(W.u.)=13.5 \textit{6}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.1658cm}{\raggedright Mult.: From \ensuremath{\gamma}(\ensuremath{\theta}) (A\ensuremath{_{\textnormal{2}}}=+0.202 \textit{26}) in Coulomb\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.1658cm}{\raggedright {\ }{\ }{\ }excitation (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1980Bo05,B}{1980Bo05}).\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{388}&\multicolumn{1}{@{.}l}{26\ensuremath{^{\hyperlink{HG10GAMMA1}{\ddagger}}} {\it 3}}&\multicolumn{1}{r@{}}{89}&\multicolumn{1}{@{.}l}{6\ensuremath{^{\hyperlink{HG10GAMMA1}{\ddagger}}} {\it 14}}&\multicolumn{1}{r@{}}{26}&\multicolumn{1}{@{.}l}{2738 }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{+1}&\multicolumn{1}{@{.}l}{5 {\it +5\textminus7}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{09 {\it 4}}&\parbox[t][0.3cm]{7.1658cm}{\raggedright B(M1)(W.u.)=0.0017 \textit{+14{\textminus}6}; B(E2)(W.u.)=9.4\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.1658cm}{\raggedright {\ }{\ }{\ }\textit{+15{\textminus}34}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.1658cm}{\raggedright Mult.,\ensuremath{\delta}: From \ensuremath{\gamma}(\ensuremath{\theta}) (A\ensuremath{_{\textnormal{2}}}=+0.38 \textit{3}) in Coulomb\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.1658cm}{\raggedright {\ }{\ }{\ }excitation (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1980Bo05,B}{1980Bo05}).\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{414}&\multicolumn{1}{@{.}l}{49\ensuremath{^{\hyperlink{HG10GAMMA1}{\ddagger}}} {\it 3}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{.}l}{0\ensuremath{^{\hyperlink{HG10GAMMA1}{\ddagger}}} {\it 19}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{3/2\ensuremath{^{-}}}&\multicolumn{1}{l}{E2}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0432 {\it 6}}&\parbox[t][0.3cm]{7.1658cm}{\raggedright B(E2)(W.u.)=10.9 \textit{5}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.1658cm}{\raggedright Mult.: From \ensuremath{\gamma}(\ensuremath{\theta}) (A\ensuremath{_{\textnormal{2}}}=+0.187 \textit{24}) in Coulomb\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.1658cm}{\raggedright {\ }{\ }{\ }excitation (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1980Bo05,B}{1980Bo05}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{464}&\multicolumn{1}{@{.}l}{41}&\multicolumn{1}{l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{432}&\multicolumn{1}{@{.}l}{32\ensuremath{^{\hyperlink{HG10GAMMA1}{\ddagger}}} {\it 7}}&\multicolumn{1}{r@{}}{5}&\multicolumn{1}{@{.}l}{8\ensuremath{^{\hyperlink{HG10GAMMA1}{\ddagger}}} {\it 3}}&\multicolumn{1}{r@{}}{32}&\multicolumn{1}{@{.}l}{155 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{+1}&\multicolumn{1}{@{.}l}{4 {\it 6}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{071 {\it 26}}&\parbox[t][0.3cm]{7.1658cm}{\raggedright Mult.,\ensuremath{\delta}: From \ensuremath{\gamma}(\ensuremath{\theta}) (A\ensuremath{_{\textnormal{2}}}=+0.183 \textit{26}) in Coulomb\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.1658cm}{\raggedright {\ }{\ }{\ }excitation (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1980Bo05,B}{1980Bo05}).\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{438}&\multicolumn{1}{@{.}l}{11\ensuremath{^{\hyperlink{HG10GAMMA1}{\ddagger}}} {\it 6}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{HG10GAMMA1}{\ddagger}}} {\it 3}}&\multicolumn{1}{r@{}}{26}&\multicolumn{1}{@{.}l}{2738 }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1(+E2)}&\multicolumn{1}{r@{}}{$\leq$0}&\multicolumn{1}{@{.}l}{1}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{1297 {\it 19}}&\parbox[t][0.3cm]{7.1658cm}{\raggedright Mult.,\ensuremath{\delta}: From \ensuremath{\gamma}(\ensuremath{\theta}) (A\ensuremath{_{\textnormal{2}}}=+0.077 \textit{21}) in Coulomb\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.1658cm}{\raggedright {\ }{\ }{\ }excitation (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1980Bo05,B}{1980Bo05}).\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{464}&\multicolumn{1}{@{.}l}{39\ensuremath{^{\hyperlink{HG10GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{31}&\multicolumn{1}{@{.}l}{6\ensuremath{^{\hyperlink{HG10GAMMA1}{\ddagger}}} {\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{3/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{+1}&\multicolumn{1}{@{.}l}{4 {\it +13\textminus6}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{059 {\it 22}}&\parbox[t][0.3cm]{7.1658cm}{\raggedright Mult.,\ensuremath{\delta}: From \ensuremath{\gamma}(\ensuremath{\theta}) (A\ensuremath{_{\textnormal{2}}}=+0.20 \textit{6}) in Coulomb\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.1658cm}{\raggedright {\ }{\ }{\ }excitation (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1980Bo05,B}{1980Bo05}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{542}&\multicolumn{1}{@{.}l}{81}&\multicolumn{1}{l}{1/2\ensuremath{^{-}},3/2,5/2}&\multicolumn{1}{r@{}}{517}&\multicolumn{1}{@{.}l}{0\ensuremath{^{\hyperlink{HG10GAMMA2}{\#}}} {\it 3}}&\multicolumn{1}{r@{}}{69}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{HG10GAMMA2}{\#}}} {\it 7}}&\multicolumn{1}{r@{}}{26}&\multicolumn{1}{@{.}l}{2738 }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{-}}}&&&&&&&\\
&&&\multicolumn{1}{r@{}}{542}&\multicolumn{1}{@{.}l}{6\ensuremath{^{\hyperlink{HG10GAMMA2}{\#}}} {\it 2}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{HG10GAMMA2}{\#}}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{3/2\ensuremath{^{-}}}&&&&&&&\\
\multicolumn{1}{r@{}}{547}&\multicolumn{1}{@{.}l}{32}&\multicolumn{1}{l}{9/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{521}&\multicolumn{1}{@{.}l}{05\ensuremath{^{\hyperlink{HG10GAMMA3}{@}}} {\it 10}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{HG10GAMMA3}{@}}}}&\multicolumn{1}{r@{}}{26}&\multicolumn{1}{@{.}l}{2738 }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{l}{E2\ensuremath{^{\hyperlink{HG10GAMMA3}{@}}}}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{02440 {\it 34}}&\parbox[t][0.3cm]{7.1658cm}{\raggedright B(E2)(W.u.)\ensuremath{>}0.010\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.1658cm}{\raggedright E\ensuremath{_{\gamma}}: From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1976Uy01,B}{1976Uy01}.\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.1658cm}{\raggedright Mult.: \ensuremath{\alpha}(K)exp=0.027 \textit{13}, K/L=3.4 \textit{9} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1990Lo17,B}{1990Lo17}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{553}&\multicolumn{1}{@{.}l}{01}&\multicolumn{1}{l}{1/2\ensuremath{^{-}},3/2\ensuremath{^{-}},5/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{385}&\multicolumn{1}{@{.}l}{1\ensuremath{^{\hyperlink{HG10GAMMA2}{\#}}} {\it 2}}&\multicolumn{1}{r@{}}{37}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{HG10GAMMA2}{\#}}} {\it 10}}&\multicolumn{1}{r@{}}{167}&\multicolumn{1}{@{.}l}{48 }&\multicolumn{1}{@{}l}{1/2\ensuremath{^{-}}}&&&&&&&\\
&&&\multicolumn{1}{r@{}}{520}&\multicolumn{1}{@{.}l}{9\ensuremath{^{\hyperlink{HG10GAMMA1}{\ddagger}}} {\it 1}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{HG10GAMMA1}{\ddagger}}} {\it 11}}&\multicolumn{1}{r@{}}{32}&\multicolumn{1}{@{.}l}{155 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{-}}}&&&&&&&\\
&&&\multicolumn{1}{r@{}}{526}&\multicolumn{1}{@{.}l}{8\ensuremath{^{\hyperlink{HG10GAMMA1}{\ddagger}}} {\it 1}}&\multicolumn{1}{r@{}}{84}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{HG10GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{26}&\multicolumn{1}{@{.}l}{2738 }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{-}}}&&&&&&&\\
&&&\multicolumn{1}{r@{}}{553}&\multicolumn{1}{@{.}l}{0\ensuremath{^{\hyperlink{HG10GAMMA1}{\ddagger}}} {\it 1}}&\multicolumn{1}{r@{}}{87}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{HG10GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{3/2\ensuremath{^{-}}}&&&&&&&\\
\multicolumn{1}{r@{}}{645}&\multicolumn{1}{@{.}l}{4}&\multicolumn{1}{l}{(5/2)\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{613}&\multicolumn{1}{@{.}l}{2\ensuremath{^{\hyperlink{HG10GAMMA2}{\#}}} {\it 3}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{HG10GAMMA2}{\#}}} {\it 9}}&\multicolumn{1}{r@{}}{32}&\multicolumn{1}{@{.}l}{155 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{-}}}&&&&&&&\\
&&&\multicolumn{1}{r@{}}{645}&\multicolumn{1}{@{.}l}{0\ensuremath{^{\hyperlink{HG10GAMMA2}{\#}\hyperlink{HG10GAMMA5}{a}}} {\it 4}}&\multicolumn{1}{r@{}}{55}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{HG10GAMMA2}{\#}}} {\it 6}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{3/2\ensuremath{^{-}}}&&&&&&&\\
\multicolumn{1}{r@{}}{732}&\multicolumn{1}{@{.}l}{3}&\multicolumn{1}{l}{1/2\ensuremath{^{-}},3/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{732}&\multicolumn{1}{@{.}l}{3\ensuremath{^{\hyperlink{HG10GAMMA2}{\#}}} {\it 4}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{HG10GAMMA2}{\#}}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{3/2\ensuremath{^{-}}}&&&&&&&\\
\multicolumn{1}{r@{}}{766}&\multicolumn{1}{@{.}l}{22}&\multicolumn{1}{l}{13/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{218}&\multicolumn{1}{@{.}l}{9\ensuremath{^{\hyperlink{HG10GAMMA3}{@}}} {\it 1}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{HG10GAMMA3}{@}}}}&\multicolumn{1}{r@{}}{547}&\multicolumn{1}{@{.}l}{32 }&\multicolumn{1}{@{}l}{9/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M2+E3\ensuremath{^{\hyperlink{HG10GAMMA3}{@}}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{33 {\it 20}}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{81 {\it 22}}&\parbox[t][0.3cm]{7.1658cm}{\raggedright B(M2)(W.u.)=0.00359 \textit{+31{\textminus}53}; B(E3)(W.u.)=5 \textit{+6{\textminus}4}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.1658cm}{\raggedright Mult.: \ensuremath{\alpha}(K)exp=4 \textit{2}, K/L=3.2 \textit{6}, L/M+=3.4 \textit{9}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.1658cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1990Lo17,B}{1990Lo17}).\vspace{0.1cm}}&\\
\end{longtable}
\clearpage
\begin{longtable}{cccccccc@{}cc@{\extracolsep{\fill}}c}
\\[-.4cm]
\multicolumn{11}{c}{{\bf \small \underline{Adopted \hyperlink{201HG_LEVEL}{Levels}, \hyperlink{201HG_GAMMA}{Gammas} (continued)}}}\\
\multicolumn{11}{c}{~}\\
\multicolumn{11}{c}{\underline{$\gamma$($^{201}$Hg) (continued)}}\\
\multicolumn{11}{c}{~~~}\\
\multicolumn{2}{c}{E\ensuremath{_{i}}(level)}&\multicolumn{2}{c}{E\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{HG10GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{I\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{HG10GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{E\ensuremath{_{f}}}&J\ensuremath{^{\pi}_{f}}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&
\endhead
&&&&&&&&&\parbox[t][0.3cm]{15.10876cm}{\raggedright \ensuremath{\delta}: From K/L=3.2 \textit{6}, L/M+=3.4 \textit{9} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1990Lo17,B}{1990Lo17}) and the briccmixing program.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1187}&\multicolumn{1}{@{.}l}{8}&\multicolumn{1}{r@{}}{645}&\multicolumn{1}{@{.}l}{0\ensuremath{^{\hyperlink{HG10GAMMA2}{\#}}} {\it 4}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{HG10GAMMA2}{\#}}}}&\multicolumn{1}{r@{}}{542}&\multicolumn{1}{@{.}l}{81 }&\multicolumn{1}{@{}l}{1/2\ensuremath{^{-}},3/2,5/2}&&\\
\end{longtable}
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{HG10GAMMA0}{\dagger}}}} From \ensuremath{^{\textnormal{201}}}Tl \ensuremath{\varepsilon} decay, unless otherwise stated.}\\
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{HG10GAMMA1}{\ddagger}}}} From Coulomb excitation.}\\
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{HG10GAMMA2}{\#}}}} From \ensuremath{^{\textnormal{201}}}Au \ensuremath{\beta}\ensuremath{^{-}} decay.}\\
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{HG10GAMMA3}{@}}}} From \ensuremath{^{\textnormal{201}}}Hg IT decay.}\\
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{HG10GAMMA4}{\&}}}} Total theoretical internal conversion coefficients, calculated using the BrIcc code (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2008Ki07,B}{2008Ki07}) with Frozen orbital approximation based on \ensuremath{\gamma}-ray energies,}\\
\parbox[b][0.3cm]{21.881866cm}{{\ }{\ }assigned multipolarities, and mixing ratios, unless otherwise specified.}\\
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{HG10GAMMA5}{a}}}} Placement of transition in the level scheme is uncertain.}\\
\vspace{0.5cm}
\end{landscape}\clearpage
\clearpage
\begin{figure}[h]
\begin{center}
\includegraphics{201HG10-0.ps}\\
\end{center}
\end{figure}
\clearpage
\subsection[\hspace{-0.2cm}\ensuremath{^{\textnormal{201}}}Au \ensuremath{\beta}\ensuremath{^{-}} decay]{ }
\vspace{-27pt}
\vspace{0.3cm}
\hypertarget{AU11}{{\bf \small \underline{\ensuremath{^{\textnormal{201}}}Au \ensuremath{\beta}\ensuremath{^{-}} decay\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1972Pa24,B}{1972Pa24}}}}\\
\vspace{4pt}
\vspace{8pt}
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}Parent: $^{201}$Au: E=0; J$^{\pi}$=3/2\ensuremath{^{+}}; T$_{1/2}$=26.0 min {\it 8}; Q(\ensuremath{\beta}\ensuremath{^{-}})=1262 {\it 3}; \%\ensuremath{\beta}\ensuremath{^{-}} decay=100.0
}\\
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1972Pa24,B}{1972Pa24}: Source produced by irradiating natural mercury targets of 10-100 g with 14.5 MeV neutrons; Detectors: Ge(Li), Si(Li)}\\
\parbox[b][0.3cm]{17.7cm}{and NaI(Tl); Measured: \ensuremath{\gamma} singles, \ensuremath{\gamma}\ensuremath{\gamma} coin, \ensuremath{\beta}\ensuremath{^{-}}, E\ensuremath{\gamma}, I\ensuremath{\gamma}.}\\
\vspace{12pt}
\underline{$^{201}$Hg Levels}\\
\begin{longtable}{ccc|ccc|ccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E(level)$^{{\hyperlink{HG11LEVEL0}{\dagger}}}$}&J$^{\pi}$$^{{\hyperlink{HG11LEVEL1}{\ddagger}}}$&\multicolumn{2}{c}{E(level)$^{{\hyperlink{HG11LEVEL0}{\dagger}}}$}&J$^{\pi}$$^{{\hyperlink{HG11LEVEL1}{\ddagger}}}$&\multicolumn{2}{c}{E(level)$^{{\hyperlink{HG11LEVEL0}{\dagger}}}$}&J$^{\pi}$$^{{\hyperlink{HG11LEVEL1}{\ddagger}}}$&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{l|}{3/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{167}&\multicolumn{1}{@{.}l}{48 {\it 4}}&\multicolumn{1}{l|}{1/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{552}&\multicolumn{1}{@{.}l}{89 {\it 13}}&\multicolumn{1}{l}{1/2\ensuremath{^{-}},3/2\ensuremath{^{-}},5/2\ensuremath{^{-}}}&\\
\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{5648 {\it 10}}&\multicolumn{1}{l|}{1/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{384}&\multicolumn{1}{@{.}l}{605 {\it 17}}&\multicolumn{1}{l|}{(5/2)\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{645}&\multicolumn{1}{@{.}l}{4 {\it 3}}&\multicolumn{1}{l}{(5/2)\ensuremath{^{-}}}&\\
\multicolumn{1}{r@{}}{26}&\multicolumn{1}{@{.}l}{2738 {\it 3}}&\multicolumn{1}{l|}{5/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{464}&\multicolumn{1}{@{.}l}{41 {\it 4}}&\multicolumn{1}{l|}{5/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{732}&\multicolumn{1}{@{.}l}{3 {\it 4}}&\multicolumn{1}{l}{1/2\ensuremath{^{-}},3/2\ensuremath{^{-}}}&\\
\multicolumn{1}{r@{}}{32}&\multicolumn{1}{@{.}l}{168 {\it 14}}&\multicolumn{1}{l|}{3/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{542}&\multicolumn{1}{@{.}l}{81 {\it 17}}&\multicolumn{1}{l|}{1/2\ensuremath{^{-}},3/2,5/2}&\multicolumn{1}{r@{}}{1187}&\multicolumn{1}{@{.}l}{8 {\it 5}}&&\\
\end{longtable}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{HG11LEVEL0}{\dagger}}}} From a least-squares fit to E\ensuremath{\gamma}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{HG11LEVEL1}{\ddagger}}}} From Adopted Levels.}\\
\vspace{0.5cm}
\underline{\ensuremath{\beta^-} radiations}\\
\begin{longtable}{ccccccccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E(decay)$$}&\multicolumn{2}{c}{E(level)}&\multicolumn{2}{c}{I$\beta^{- {\hyperlink{HG11DECAY0}{\dagger}}{\hyperlink{HG11DECAY1}{\ddagger}}}$}&\multicolumn{2}{c}{Log {\it ft}$^{}$}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{(530}&\multicolumn{1}{@{ }l}{{\it 3})}&\multicolumn{1}{r@{}}{732}&\multicolumn{1}{@{.}l}{3}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{43 {\it 9}}&\multicolumn{1}{r@{}}{6}&\multicolumn{1}{@{.}l}{67 {\it 10}}&&\\
\multicolumn{1}{r@{}}{(617}&\multicolumn{1}{@{ }l}{{\it 3})}&\multicolumn{1}{r@{}}{645}&\multicolumn{1}{@{.}l}{4}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{9 {\it 3}}&\multicolumn{1}{r@{}}{6}&\multicolumn{1}{@{.}l}{24 {\it 7}}&&\\
\multicolumn{1}{r@{}}{(709}&\multicolumn{1}{@{ }l}{{\it 3})}&\multicolumn{1}{r@{}}{552}&\multicolumn{1}{@{.}l}{89}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{0 {\it 4}}&\multicolumn{1}{r@{}}{6}&\multicolumn{1}{@{.}l}{25 {\it 6}}&&\\
\multicolumn{1}{r@{}}{(719}&\multicolumn{1}{@{ }l}{{\it 3})}&\multicolumn{1}{r@{}}{542}&\multicolumn{1}{@{.}l}{81}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{6 {\it 4}}&\multicolumn{1}{r@{}}{6}&\multicolumn{1}{@{.}l}{34 {\it 7}}&&\\
\multicolumn{1}{r@{}}{(798}&\multicolumn{1}{@{ }l}{{\it 3})}&\multicolumn{1}{r@{}}{464}&\multicolumn{1}{@{.}l}{41}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{47 {\it 10}}&\multicolumn{1}{r@{}}{7}&\multicolumn{1}{@{.}l}{23 {\it 10}}&&\\
\multicolumn{1}{r@{}}{(877}&\multicolumn{1}{@{ }l}{{\it 3})}&\multicolumn{1}{r@{}}{384}&\multicolumn{1}{@{.}l}{605}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{96 {\it 17}}&\multicolumn{1}{r@{}}{7}&\multicolumn{1}{@{.}l}{07 {\it 8}}&&\\
\multicolumn{1}{r@{}}{(1095}&\multicolumn{1}{@{ }l}{{\it 3})}&\multicolumn{1}{r@{}}{167}&\multicolumn{1}{@{.}l}{48}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{5 {\it 4}}&\multicolumn{1}{r@{}}{6}&\multicolumn{1}{@{.}l}{85 {\it 6}}&&\\
\multicolumn{1}{r@{}}{(1230}&\multicolumn{1}{@{ }l}{{\it 3})}&\multicolumn{1}{r@{}}{32}&\multicolumn{1}{@{.}l}{168}&\multicolumn{1}{r@{}}{$\approx$8}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{$\approx$6}&\multicolumn{1}{@{.}l}{7}&&\\
\multicolumn{1}{r@{}}{(1262}&\multicolumn{1}{@{ }l}{{\it 3})}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{r@{}}{79}&\multicolumn{1}{@{ }l}{{\it 3}}&&&\parbox[t][0.3cm]{11.672541cm}{\raggedright I$\beta^-$: An upper limit which includes contribution to the 0.0-keV, 1.5648-keV and\vspace{0.1cm}}&\\
&&&&&&&&\parbox[t][0.3cm]{11.672541cm}{\raggedright {\ }{\ }{\ }26.2738-keV levels.\vspace{0.1cm}}&\\
\end{longtable}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{HG11DECAY0}{\dagger}}}} From intensity balances, as explained in the text.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{HG11DECAY1}{\ddagger}}}} Absolute intensity per 100 decays.}\\
\vspace{0.5cm}
\clearpage
\vspace{0.3cm}
\begin{landscape}
\vspace*{-0.5cm}
{\bf \small \underline{\ensuremath{^{\textnormal{201}}}Au \ensuremath{\beta}\ensuremath{^{-}} decay\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1972Pa24,B}{1972Pa24} (continued)}}\\
\vspace{0.3cm}
\underline{$\gamma$($^{201}$Hg)}\\
\vspace{0.34cm}
\parbox[b][0.3cm]{21.881866cm}{\addtolength{\parindent}{-0.254cm}I\ensuremath{\gamma} normalization: From I\ensuremath{\beta}(167.47 keV level)=3.5\% (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1972Pa24,B}{1972Pa24}) and I(\ensuremath{\gamma}+ce) deduced from the decay scheme. Evaluator assigns 10\% uncertainty to the}\\
\parbox[b][0.3cm]{21.881866cm}{I\ensuremath{\beta}(167.47-keV level) value.}\\
\vspace{0.34cm}
\begin{longtable}{ccccccccc@{}ccccccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{AU11GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{I\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{AU11GAMMA0}{\dagger}\hyperlink{AU11GAMMA2}{\#}}}}&\multicolumn{2}{c}{E\ensuremath{_{i}}(level)}&J\ensuremath{^{\pi}_{i}}&\multicolumn{2}{c}{E\ensuremath{_{f}}}&J\ensuremath{^{\pi}_{f}}&Mult.\ensuremath{^{\hyperlink{AU11GAMMA1}{\ddagger}}}&\multicolumn{2}{c}{\ensuremath{\delta}\ensuremath{^{\hyperlink{AU11GAMMA1}{\ddagger}}}}&\multicolumn{2}{c}{\ensuremath{\alpha}\ensuremath{^{\hyperlink{AU11GAMMA3}{@}}}}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{5648\ensuremath{^{\hyperlink{AU11GAMMA1}{\ddagger}}} {\it 10}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{5648}&\multicolumn{1}{l}{1/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{@{}l}{3/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0105 {\it 14}}&\multicolumn{1}{r@{}}{4.7\ensuremath{\times10^{4}}}&\multicolumn{1}{@{ }l}{{\it 7}}&\parbox[t][0.3cm]{6.5558996cm}{\raggedright \ensuremath{\alpha}: From adopted gammas.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{(5}&\multicolumn{1}{@{.}l}{895 {\it 20})}&\multicolumn{1}{r@{}}{$\approx$0}&\multicolumn{1}{@{.}l}{002}&\multicolumn{1}{r@{}}{32}&\multicolumn{1}{@{.}l}{168}&\multicolumn{1}{l}{3/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{26}&\multicolumn{1}{@{.}l}{2738 }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{l}{[M1]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{1441}&\multicolumn{1}{@{ }l}{{\it 25}}&\parbox[t][0.3cm]{6.5558996cm}{\raggedright \%I\ensuremath{\gamma}\ensuremath{\approx}3.6\ensuremath{\times}10\ensuremath{^{\textnormal{$-$5}}}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.5558996cm}{\raggedright I\ensuremath{_{\gamma}}: From I\ensuremath{\gamma}(5.895\ensuremath{\gamma})/I\ensuremath{\gamma}(30.60\ensuremath{\gamma}) from adopted\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.5558996cm}{\raggedright {\ }{\ }{\ }gammas and I\ensuremath{\gamma}(30.60\ensuremath{\gamma})=6.9.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{26}&\multicolumn{1}{@{.}l}{2738\ensuremath{^{\hyperlink{AU11GAMMA1}{\ddagger}}} {\it 3}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{26}&\multicolumn{1}{@{.}l}{2738}&\multicolumn{1}{l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{@{}l}{3/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{012 {\it 8}}&\multicolumn{1}{r@{}}{72}&\multicolumn{1}{@{.}l}{9 {\it 13}}&&\\
\multicolumn{1}{r@{}}{30}&\multicolumn{1}{@{.}l}{60\ensuremath{^{\hyperlink{AU11GAMMA1}{\ddagger}}} {\it 3}}&\multicolumn{1}{r@{}}{$\approx$6}&\multicolumn{1}{@{.}l}{9}&\multicolumn{1}{r@{}}{32}&\multicolumn{1}{@{.}l}{168}&\multicolumn{1}{l}{3/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{5648 }&\multicolumn{1}{@{}l}{1/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{013 {\it 5}}&\multicolumn{1}{r@{}}{46}&\multicolumn{1}{@{.}l}{4 {\it 7}}&\parbox[t][0.3cm]{6.5558996cm}{\raggedright \%I\ensuremath{\gamma}\ensuremath{\approx}0.126\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.5558996cm}{\raggedright I\ensuremath{_{\gamma}}: From I(\ensuremath{\gamma}+ce) and \ensuremath{\alpha}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{32}&\multicolumn{1}{@{.}l}{19\ensuremath{^{\hyperlink{AU11GAMMA1}{\ddagger}}} {\it 3}}&\multicolumn{1}{r@{}}{$\approx$7}&\multicolumn{1}{@{.}l}{4}&\multicolumn{1}{r@{}}{32}&\multicolumn{1}{@{.}l}{168}&\multicolumn{1}{l}{3/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{@{}l}{3/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0204 {\it 25}}&\multicolumn{1}{r@{}}{40}&\multicolumn{1}{@{.}l}{2 {\it 6}}&\parbox[t][0.3cm]{6.5558996cm}{\raggedright \%I\ensuremath{\gamma}\ensuremath{\approx}0.135\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.5558996cm}{\raggedright I\ensuremath{_{\gamma}}: From I(\ensuremath{\gamma}+ce) and \ensuremath{\alpha}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{135}&\multicolumn{1}{@{.}l}{34\ensuremath{^{\hyperlink{AU11GAMMA1}{\ddagger}}} {\it 4}}&\multicolumn{1}{r@{}}{13}&\multicolumn{1}{@{.}l}{8 {\it 13}}&\multicolumn{1}{r@{}}{167}&\multicolumn{1}{@{.}l}{48}&\multicolumn{1}{l}{1/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{32}&\multicolumn{1}{@{.}l}{168 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{$-$0}&\multicolumn{1}{@{.}l}{07 {\it 4}}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{32 {\it 5}}&\parbox[t][0.3cm]{6.5558996cm}{\raggedright \%I\ensuremath{\gamma}=0.25 \textit{4}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.5558996cm}{\raggedright I\ensuremath{_{\gamma}}: From I\ensuremath{\gamma}(135.34\ensuremath{\gamma})/I\ensuremath{\gamma}(167.43\ensuremath{\gamma}) in adopted\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.5558996cm}{\raggedright {\ }{\ }{\ }gammas and I\ensuremath{\gamma}(167.43\ensuremath{\gamma})=53 \textit{5}; Other: 18 \textit{5}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.5558996cm}{\raggedright {\ }{\ }{\ }in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1972Pa24,B}{1972Pa24}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{141}&\multicolumn{1}{@{.}l}{1\ensuremath{^{\hyperlink{AU11GAMMA1}{\ddagger}}} {\it 2}}&\multicolumn{1}{r@{}}{$\approx$0}&\multicolumn{1}{@{.}l}{014}&\multicolumn{1}{r@{}}{167}&\multicolumn{1}{@{.}l}{48}&\multicolumn{1}{l}{1/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{26}&\multicolumn{1}{@{.}l}{2738 }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{l}{[E2]}&&&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{389 {\it 21}}&\parbox[t][0.3cm]{6.5558996cm}{\raggedright \%I\ensuremath{\gamma}\ensuremath{\approx}0.000255\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.5558996cm}{\raggedright I\ensuremath{_{\gamma}}: From I\ensuremath{\gamma}(141.1\ensuremath{\gamma})/I\ensuremath{\gamma}(167.43\ensuremath{\gamma}) in adopted\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.5558996cm}{\raggedright {\ }{\ }{\ }gammas and I\ensuremath{\gamma}(167.43\ensuremath{\gamma})=53 \textit{5}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{165}&\multicolumn{1}{@{.}l}{88\ensuremath{^{\hyperlink{AU11GAMMA1}{\ddagger}}} {\it 7}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{78 {\it 7}}&\multicolumn{1}{r@{}}{167}&\multicolumn{1}{@{.}l}{48}&\multicolumn{1}{l}{1/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{5648 }&\multicolumn{1}{@{}l}{1/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1}&&&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{869 {\it 26}}&\parbox[t][0.3cm]{6.5558996cm}{\raggedright \%I\ensuremath{\gamma}=0.0142 \textit{21}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.5558996cm}{\raggedright I\ensuremath{_{\gamma}}: From I\ensuremath{\gamma}(165.88\ensuremath{\gamma})/I\ensuremath{\gamma}(167.43\ensuremath{\gamma}) in adopted\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.5558996cm}{\raggedright {\ }{\ }{\ }gammas and I\ensuremath{\gamma}(167.43\ensuremath{\gamma})=53 \textit{5}; Other: 0.7\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.5558996cm}{\raggedright {\ }{\ }{\ }from I(\ensuremath{\gamma}+ce) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1972Pa24,B}{1972Pa24} and \ensuremath{\alpha}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{167}&\multicolumn{1}{@{.}l}{43\ensuremath{^{\hyperlink{AU11GAMMA1}{\ddagger}}} {\it 7}}&\multicolumn{1}{r@{}}{53}&\multicolumn{1}{@{ }l}{{\it 5}}&\multicolumn{1}{r@{}}{167}&\multicolumn{1}{@{.}l}{48}&\multicolumn{1}{l}{1/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{@{}l}{3/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{07 {\it 6}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{815 {\it 29}}&\parbox[t][0.3cm]{6.5558996cm}{\raggedright \%I\ensuremath{\gamma}=0.96 \textit{15}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{352}&\multicolumn{1}{@{.}l}{42\ensuremath{^{\hyperlink{AU11GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{19}&\multicolumn{1}{@{ }l}{{\it 4}}&\multicolumn{1}{r@{}}{384}&\multicolumn{1}{@{.}l}{605}&\multicolumn{1}{l}{(5/2)\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{32}&\multicolumn{1}{@{.}l}{168 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1(+E2)}&\multicolumn{1}{r@{}}{+0}&\multicolumn{1}{@{.}l}{07 {\it 7}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{232 {\it 4}}&\parbox[t][0.3cm]{6.5558996cm}{\raggedright \%I\ensuremath{\gamma}=0.35 \textit{8}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{358}&\multicolumn{1}{@{.}l}{36\ensuremath{^{\hyperlink{AU11GAMMA1}{\ddagger}}} {\it 4}}&\multicolumn{1}{r@{}}{10}&\multicolumn{1}{@{.}l}{3 {\it 22}}&\multicolumn{1}{r@{}}{384}&\multicolumn{1}{@{.}l}{605}&\multicolumn{1}{l}{(5/2)\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{26}&\multicolumn{1}{@{.}l}{2738 }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1+E2}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{14 {\it 8}}&\parbox[t][0.3cm]{6.5558996cm}{\raggedright \%I\ensuremath{\gamma}=0.19 \textit{5}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.5558996cm}{\raggedright I\ensuremath{_{\gamma}}: I\ensuremath{\gamma}(358.36\ensuremath{\gamma})/I\ensuremath{\gamma}(352.42\ensuremath{\gamma}) in adopted\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.5558996cm}{\raggedright {\ }{\ }{\ }gammas and I\ensuremath{\gamma}(352.42\ensuremath{\gamma})=19 \textit{4}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{384}&\multicolumn{1}{@{.}l}{60\ensuremath{^{\hyperlink{AU11GAMMA1}{\ddagger}}} {\it 2}}&\multicolumn{1}{r@{}}{15}&\multicolumn{1}{@{ }l}{{\it 3}}&\multicolumn{1}{r@{}}{384}&\multicolumn{1}{@{.}l}{605}&\multicolumn{1}{l}{(5/2)\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{@{}l}{3/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{$-$0}&\multicolumn{1}{@{.}l}{23 {\it 9}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{178 {\it 6}}&\parbox[t][0.3cm]{6.5558996cm}{\raggedright \%I\ensuremath{\gamma}=0.27 \textit{6}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.5558996cm}{\raggedright I\ensuremath{_{\gamma}}: From I\ensuremath{\gamma}(384.60\ensuremath{\gamma})/I\ensuremath{\gamma}(352.42\ensuremath{\gamma}) in adopted\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.5558996cm}{\raggedright {\ }{\ }{\ }gammas\hphantom{a}and I\ensuremath{\gamma}(352.42\ensuremath{\gamma})=19 \textit{4}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{385}&\multicolumn{1}{@{.}l}{1 {\it 2}}&\multicolumn{1}{r@{}}{19}&\multicolumn{1}{@{ }l}{{\it 5}}&\multicolumn{1}{r@{}}{552}&\multicolumn{1}{@{.}l}{89}&\multicolumn{1}{l}{1/2\ensuremath{^{-}},3/2\ensuremath{^{-}},5/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{167}&\multicolumn{1}{@{.}l}{48 }&\multicolumn{1}{@{}l}{1/2\ensuremath{^{-}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{6.5558996cm}{\raggedright \%I\ensuremath{\gamma}=0.35 \textit{10}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.5558996cm}{\raggedright I\ensuremath{_{\gamma}}: From I\ensuremath{\gamma}(385.1\ensuremath{\gamma})/I\ensuremath{\gamma}(552.8\ensuremath{\gamma}) in adopted\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.5558996cm}{\raggedright {\ }{\ }{\ }gammas\hphantom{a}and I\ensuremath{\gamma}(552.8\ensuremath{\gamma})=44 \textit{5}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{432}&\multicolumn{1}{@{.}l}{32 {\it 7}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{99 {\it 6}}&\multicolumn{1}{r@{}}{464}&\multicolumn{1}{@{.}l}{41}&\multicolumn{1}{l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{32}&\multicolumn{1}{@{.}l}{168 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{+1}&\multicolumn{1}{@{.}l}{4 {\it 6}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{071 {\it 26}}&\parbox[t][0.3cm]{6.5558996cm}{\raggedright \%I\ensuremath{\gamma}=0.0180 \textit{24}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.5558996cm}{\raggedright E\ensuremath{_{\gamma}},I\ensuremath{_{\gamma}},Mult.,\ensuremath{\delta}: From adopted gammas.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{438}&\multicolumn{1}{@{.}l}{11\ensuremath{^{\hyperlink{AU11GAMMA1}{\ddagger}}} {\it 6}}&\multicolumn{1}{r@{}}{17}&\multicolumn{1}{@{ }l}{{\it 4}}&\multicolumn{1}{r@{}}{464}&\multicolumn{1}{@{.}l}{41}&\multicolumn{1}{l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{26}&\multicolumn{1}{@{.}l}{2738 }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1(+E2)}&\multicolumn{1}{r@{}}{$\leq$0}&\multicolumn{1}{@{.}l}{1}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{1297 {\it 19}}&\parbox[t][0.3cm]{6.5558996cm}{\raggedright \%I\ensuremath{\gamma}=0.31 \textit{8}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{464}&\multicolumn{1}{@{.}l}{39\ensuremath{^{\hyperlink{AU11GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{5}&\multicolumn{1}{@{.}l}{4 {\it 13}}&\multicolumn{1}{r@{}}{464}&\multicolumn{1}{@{.}l}{41}&\multicolumn{1}{l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{@{}l}{3/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{+1}&\multicolumn{1}{@{.}l}{4 {\it +13\textminus6}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{059 {\it 22}}&\parbox[t][0.3cm]{6.5558996cm}{\raggedright \%I\ensuremath{\gamma}=0.098 \textit{26}\vspace{0.1cm}}&\\
\end{longtable}
\clearpage
\begin{longtable}{ccccccccc@{}ccccc@{\extracolsep{\fill}}c}
\\[-.4cm]
\multicolumn{15}{c}{{\bf \small \underline{\ensuremath{^{\textnormal{201}}}Au \ensuremath{\beta}\ensuremath{^{-}} decay\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1972Pa24,B}{1972Pa24} (continued)}}}\\
\multicolumn{15}{c}{~}\\
\multicolumn{15}{c}{\underline{$\gamma$($^{201}$Hg) (continued)}}\\
\multicolumn{15}{c}{~~~}\\
\multicolumn{2}{c}{E\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{AU11GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{I\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{AU11GAMMA0}{\dagger}\hyperlink{AU11GAMMA2}{\#}}}}&\multicolumn{2}{c}{E\ensuremath{_{i}}(level)}&J\ensuremath{^{\pi}_{i}}&\multicolumn{2}{c}{E\ensuremath{_{f}}}&J\ensuremath{^{\pi}_{f}}&Mult.\ensuremath{^{\hyperlink{AU11GAMMA1}{\ddagger}}}&\multicolumn{2}{c}{\ensuremath{\alpha}\ensuremath{^{\hyperlink{AU11GAMMA3}{@}}}}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endhead
\multicolumn{1}{r@{}}{517}&\multicolumn{1}{@{.}l}{0 {\it 3}}&\multicolumn{1}{r@{}}{69}&\multicolumn{1}{@{ }l}{{\it 7}}&\multicolumn{1}{r@{}}{542}&\multicolumn{1}{@{.}l}{81}&\multicolumn{1}{l}{1/2\ensuremath{^{-}},3/2,5/2}&\multicolumn{1}{r@{}}{26}&\multicolumn{1}{@{.}l}{2738 }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{l}{[M1,E2]}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{054 {\it 30}}&\parbox[t][0.3cm]{9.30348cm}{\raggedright \%I\ensuremath{\gamma}=1.26 \textit{20}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{521}&\multicolumn{1}{@{.}l}{0 {\it 4}}&\multicolumn{1}{r@{}}{51}&\multicolumn{1}{@{ }l}{{\it 5}}&\multicolumn{1}{r@{}}{552}&\multicolumn{1}{@{.}l}{89}&\multicolumn{1}{l}{1/2\ensuremath{^{-}},3/2\ensuremath{^{-}},5/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{32}&\multicolumn{1}{@{.}l}{168 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{-}}}&\multicolumn{1}{l}{[M1]}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0823 {\it 12}}&\parbox[t][0.3cm]{9.30348cm}{\raggedright \%I\ensuremath{\gamma}=0.93 \textit{14}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{9.30348cm}{\raggedright I\ensuremath{_{\gamma}}: From I\ensuremath{\gamma}(521.0\ensuremath{\gamma})/I\ensuremath{\gamma}(552.8\ensuremath{\gamma}) in adopted gammas\hphantom{a}and\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{9.30348cm}{\raggedright {\ }{\ }{\ }I\ensuremath{\gamma}(552.8\ensuremath{\gamma})=44 \textit{5}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{526}&\multicolumn{1}{@{.}l}{9 {\it 2}}&\multicolumn{1}{r@{}}{43}&\multicolumn{1}{@{ }l}{{\it 5}}&\multicolumn{1}{r@{}}{552}&\multicolumn{1}{@{.}l}{89}&\multicolumn{1}{l}{1/2\ensuremath{^{-}},3/2\ensuremath{^{-}},5/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{26}&\multicolumn{1}{@{.}l}{2738 }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{l}{[M1,E2]}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{052 {\it 28}}&\parbox[t][0.3cm]{9.30348cm}{\raggedright \%I\ensuremath{\gamma}=0.78 \textit{13}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{9.30348cm}{\raggedright I\ensuremath{_{\gamma}}: From I\ensuremath{\gamma}(526.9\ensuremath{\gamma})/I\ensuremath{\gamma}(552.8\ensuremath{\gamma}) in adopted gammas\hphantom{a}and\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{9.30348cm}{\raggedright {\ }{\ }{\ }I\ensuremath{\gamma}(552.8\ensuremath{\gamma})=44 \textit{5}.\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{9.30348cm}{\raggedright I\ensuremath{_{\gamma}}: From adopted gammas, normalized to I\ensuremath{\gamma}(553\ensuremath{\gamma}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{542}&\multicolumn{1}{@{.}l}{6 {\it 2}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{542}&\multicolumn{1}{@{.}l}{81}&\multicolumn{1}{l}{1/2\ensuremath{^{-}},3/2,5/2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{@{}l}{3/2\ensuremath{^{-}}}&\multicolumn{1}{l}{[M1,E1]}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0739 {\it 10}}&\parbox[t][0.3cm]{9.30348cm}{\raggedright \%I\ensuremath{\gamma}=1.82 \textit{22}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{9.30348cm}{\raggedright E\ensuremath{_{\gamma}},I\ensuremath{_{\gamma}}: Probably includes components to g.s. and 1.5648 keV levels.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{552}&\multicolumn{1}{@{.}l}{8 {\it 3}}&\multicolumn{1}{r@{}}{44}&\multicolumn{1}{@{ }l}{{\it 5}}&\multicolumn{1}{r@{}}{552}&\multicolumn{1}{@{.}l}{89}&\multicolumn{1}{l}{1/2\ensuremath{^{-}},3/2\ensuremath{^{-}},5/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{@{}l}{3/2\ensuremath{^{-}}}&\multicolumn{1}{l}{[M1]}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0704 {\it 10}}&\parbox[t][0.3cm]{9.30348cm}{\raggedright \%I\ensuremath{\gamma}=0.80 \textit{13}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{613}&\multicolumn{1}{@{.}l}{2 {\it 3}}&\multicolumn{1}{r@{}}{64}&\multicolumn{1}{@{ }l}{{\it 6}}&\multicolumn{1}{r@{}}{645}&\multicolumn{1}{@{.}l}{4}&\multicolumn{1}{l}{(5/2)\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{32}&\multicolumn{1}{@{.}l}{168 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{-}}}&\multicolumn{1}{l}{[M1]}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0537 {\it 8}}&\parbox[t][0.3cm]{9.30348cm}{\raggedright \%I\ensuremath{\gamma}=1.16 \textit{18}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{645}&\multicolumn{1}{@{.}l}{0\ensuremath{^{\hyperlink{AU11GAMMA4}{\&}}} {\it 4}}&\multicolumn{1}{r@{}}{35}&\multicolumn{1}{@{ }l}{{\it 4}}&\multicolumn{1}{r@{}}{645}&\multicolumn{1}{@{.}l}{4}&\multicolumn{1}{l}{(5/2)\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{@{}l}{3/2\ensuremath{^{-}}}&\multicolumn{1}{l}{[M1]}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0471 {\it 7}}&\parbox[t][0.3cm]{9.30348cm}{\raggedright \%I\ensuremath{\gamma}=0.64 \textit{11}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{645}&\multicolumn{1}{@{.}l}{0 {\it 4}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{1187}&\multicolumn{1}{@{.}l}{8}&&\multicolumn{1}{r@{}}{542}&\multicolumn{1}{@{.}l}{81 }&\multicolumn{1}{@{}l}{1/2\ensuremath{^{-}},3/2,5/2}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{9.30348cm}{\raggedright I\ensuremath{_{\gamma}}: a value of 35 \textit{4} in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1972Pa24,B}{1972Pa24} would lead to I\ensuremath{\beta}=0.64\% \textit{11} and\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{9.30348cm}{\raggedright {\ }{\ }{\ }log \textit{ft}=3.80 which seem too low.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{732}&\multicolumn{1}{@{.}l}{3 {\it 4}}&\multicolumn{1}{r@{}}{23}&\multicolumn{1}{@{ }l}{{\it 4}}&\multicolumn{1}{r@{}}{732}&\multicolumn{1}{@{.}l}{3}&\multicolumn{1}{l}{1/2\ensuremath{^{-}},3/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{@{}l}{3/2\ensuremath{^{-}}}&\multicolumn{1}{l}{[M1]}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0339 {\it 5}}&\parbox[t][0.3cm]{9.30348cm}{\raggedright \%I\ensuremath{\gamma}=0.42 \textit{9}\vspace{0.1cm}}&\\
\end{longtable}
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{AU11GAMMA0}{\dagger}}}} From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1972Pa24,B}{1972Pa24}, unless otherwise stated. K\ensuremath{\alpha}{} x ray \ensuremath{\approx}90, L\ensuremath{_{\ensuremath{\alpha}}} x ray \ensuremath{\approx}85 and L\ensuremath{_{\ensuremath{\beta}}} x ray \ensuremath{\approx}55 (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1972Pa24,B}{1972Pa24}).}\\
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{AU11GAMMA1}{\ddagger}}}} From adopted gammas.}\\
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{AU11GAMMA2}{\#}}}} For absolute intensity per 100 decays, multiply by 0.0182 \textit{22}.}\\
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{AU11GAMMA3}{@}}}} Total theoretical internal conversion coefficients, calculated using the BrIcc code (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2008Ki07,B}{2008Ki07}) with Frozen orbital approximation based on \ensuremath{\gamma}-ray energies,}\\
\parbox[b][0.3cm]{21.881866cm}{{\ }{\ }assigned multipolarities, and mixing ratios, unless otherwise specified.}\\
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{AU11GAMMA4}{\&}}}} Placement of transition in the level scheme is uncertain.}\\
\vspace{0.5cm}
\end{landscape}\clearpage
\clearpage
\begin{figure}[h]
\begin{center}
\includegraphics{201HG11-0.ps}\\
\end{center}
\end{figure}
\clearpage
\subsection[\hspace{-0.2cm}\ensuremath{^{\textnormal{201}}}Hg IT decay]{ }
\vspace{-27pt}
\vspace{0.3cm}
\hypertarget{HG12}{{\bf \small \underline{\ensuremath{^{\textnormal{201}}}Hg IT decay\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1990Lo17,B}{1990Lo17}}}}\\
\vspace{4pt}
\vspace{8pt}
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}Parent: $^{201}$Hg: E=766.22 {\it 15}; J$^{\pi}$=13/2\ensuremath{^{+}}; T$_{1/2}$=94 \ensuremath{\mu}s {\it 2}; \%IT decay=100.0
}\\
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1990Lo17,B}{1990Lo17}: \ensuremath{^{\textnormal{198}}}Pt(\ensuremath{\alpha},n\ensuremath{\gamma}); E(\ensuremath{\alpha})=18.1 MeV; Target: 1.55 mg/cm\ensuremath{^{\textnormal{2}}} thick enriched to 95.8\% in \ensuremath{^{\textnormal{198}}}Pt; Detector: HPGE, electron}\\
\parbox[b][0.3cm]{17.7cm}{spectrometer; Measured: \ensuremath{\gamma} singles, \ensuremath{\gamma}\ensuremath{\gamma} coin, \ensuremath{\gamma}(t), E\ensuremath{\gamma}, I\ensuremath{\gamma}, ce; Deduced: \ensuremath{\alpha}(K)exp, level scheme.}\\
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}Others: \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1976Uy01,B}{1976Uy01}, \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1964Br27,B}{1964Br27}, \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1962Eu01,B}{1962Eu01}, \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1961Kr01,B}{1961Kr01}.}\\
\vspace{12pt}
\underline{$^{201}$Hg Levels}\\
\begin{longtable}{cccccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E(level)$^{{\hyperlink{HG12LEVEL0}{\dagger}}}$}&J$^{\pi}$$^{{\hyperlink{HG12LEVEL1}{\ddagger}}}$&\multicolumn{2}{c}{T$_{1/2}$$^{}$}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{l}{3/2\ensuremath{^{-}}\ensuremath{^{{\hyperlink{HG12LEVEL2}{\#}}}}}&&&&\\
\multicolumn{1}{r@{}}{26}&\multicolumn{1}{@{.}l}{2738 {\it 3}}&\multicolumn{1}{l}{5/2\ensuremath{^{-}}\ensuremath{^{{\hyperlink{HG12LEVEL2}{\#}}}}}&&&&\\
\multicolumn{1}{r@{}}{547}&\multicolumn{1}{@{.}l}{32 {\it 10}}&\multicolumn{1}{l}{9/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{$<$20}&\multicolumn{1}{@{ }l}{ns}&\parbox[t][0.3cm]{13.00895cm}{\raggedright T\ensuremath{_{1/2}}: Upper limit from \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1964Br27,B}{1964Br27}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{766}&\multicolumn{1}{@{.}l}{22 {\it 15}}&\multicolumn{1}{l}{13/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{94}&\multicolumn{1}{@{ }l}{\ensuremath{\mu}s {\it 2}}&\parbox[t][0.3cm]{13.00895cm}{\raggedright T\ensuremath{_{1/2}}: Weighted average of 92 \ensuremath{\mu}s \textit{3} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1961Kr01,B}{1961Kr01}), 100 \ensuremath{\mu}s \textit{6} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1962Eu01,B}{1962Eu01}) and 94 \ensuremath{\mu}s \textit{3}\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{13.00895cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1976Uy01,B}{1976Uy01}).\vspace{0.1cm}}&\\
\end{longtable}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{HG12LEVEL0}{\dagger}}}} From a least-squares fit to E\ensuremath{\gamma}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{HG12LEVEL1}{\ddagger}}}} From deduced transition multipolarities, unless otherwise stated.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{HG12LEVEL2}{\#}}}} From Adopted Levels.}\\
\vspace{0.5cm}
\underline{$\gamma$($^{201}$Hg)}\\
\begin{longtable}{ccccccccc@{}ccccccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{HG12GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{I\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{HG12GAMMA1}{\ddagger}\hyperlink{HG12GAMMA2}{\#}}}}&\multicolumn{2}{c}{E\ensuremath{_{i}}(level)}&J\ensuremath{^{\pi}_{i}}&\multicolumn{2}{c}{E\ensuremath{_{f}}}&J\ensuremath{^{\pi}_{f}}&Mult.&\multicolumn{2}{c}{\ensuremath{\delta}}&\multicolumn{2}{c}{\ensuremath{\alpha}\ensuremath{^{\hyperlink{HG12GAMMA3}{@}}}}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{26}&\multicolumn{1}{@{.}l}{2738 {\it 3}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{353 {\it 24}}&\multicolumn{1}{r@{}}{26}&\multicolumn{1}{@{.}l}{2738}&\multicolumn{1}{l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{3/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{012 {\it 8}}&\multicolumn{1}{r@{}}{72}&\multicolumn{1}{@{.}l}{9 {\it 13}}&\parbox[t][0.3cm]{4.1657405cm}{\raggedright E\ensuremath{_{\gamma}},Mult.,\ensuremath{\delta}: From adopted\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{4.1657405cm}{\raggedright {\ }{\ }{\ }gammas.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{218}&\multicolumn{1}{@{.}l}{9 {\it 1}}&\multicolumn{1}{r@{}}{20}&\multicolumn{1}{@{.}l}{8 {\it 10}}&\multicolumn{1}{r@{}}{766}&\multicolumn{1}{@{.}l}{22}&\multicolumn{1}{l}{13/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{547}&\multicolumn{1}{@{.}l}{32 }&\multicolumn{1}{@{}l}{9/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M2+E3}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{33 {\it 20}}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{81 {\it 22}}&\parbox[t][0.3cm]{4.1657405cm}{\raggedright Mult.: \ensuremath{\alpha}(K)exp=4 \textit{2}, K/L=3.2\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{4.1657405cm}{\raggedright {\ }{\ }{\ }\textit{6}, L/M+=3.4 \textit{9} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1990Lo17,B}{1990Lo17}).\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{4.1657405cm}{\raggedright \ensuremath{\delta}: From K/L=3.2 \textit{6},\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{4.1657405cm}{\raggedright {\ }{\ }{\ }L/M+=3.4 \textit{9} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1990Lo17,B}{1990Lo17})\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{4.1657405cm}{\raggedright {\ }{\ }{\ }and the briccmixing\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{4.1657405cm}{\raggedright {\ }{\ }{\ }program.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{521}&\multicolumn{1}{@{.}l}{05 {\it 10}}&\multicolumn{1}{r@{}}{97}&\multicolumn{1}{@{.}l}{62 {\it 3}}&\multicolumn{1}{r@{}}{547}&\multicolumn{1}{@{.}l}{32}&\multicolumn{1}{l}{9/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{26}&\multicolumn{1}{@{.}l}{2738 }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{l}{E2}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{02440 {\it 34}}&\parbox[t][0.3cm]{4.1657405cm}{\raggedright Mult.: \ensuremath{\alpha}(K)exp=0.027 \textit{13},\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{4.1657405cm}{\raggedright {\ }{\ }{\ }K/L=3.4 \textit{9} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1990Lo17,B}{1990Lo17}).\vspace{0.1cm}}&\\
\end{longtable}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{HG12GAMMA0}{\dagger}}}} From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1976Uy01,B}{1976Uy01}, unless otherwise stated.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{HG12GAMMA1}{\ddagger}}}} From \ensuremath{\alpha} and by assuming I(\ensuremath{\gamma}+ce)=100 for each \ensuremath{\gamma}. I(K{} x ray):I(219\ensuremath{\gamma}):I(521\ensuremath{\gamma})=100:26:134 (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1964Br27,B}{1964Br27}).}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{HG12GAMMA2}{\#}}}} Absolute intensity per 100 decays.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{HG12GAMMA3}{@}}}} Total theoretical internal conversion coefficients, calculated using the BrIcc code (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2008Ki07,B}{2008Ki07}) with Frozen orbital approximation}\\
\parbox[b][0.3cm]{17.7cm}{{\ }{\ }based on \ensuremath{\gamma}-ray energies, assigned multipolarities, and mixing ratios, unless otherwise specified.}\\
\vspace{0.5cm}
\clearpage
\begin{figure}[h]
\begin{center}
\includegraphics{201HG12-0.ps}\\
\end{center}
\end{figure}
\clearpage
\subsection[\hspace{-0.2cm}\ensuremath{^{\textnormal{201}}}Tl \ensuremath{\varepsilon} decay]{ }
\vspace{-27pt}
\vspace{0.3cm}
\hypertarget{TL13}{{\bf \small \underline{\ensuremath{^{\textnormal{201}}}Tl \ensuremath{\varepsilon} decay\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1990Co07,B}{1990Co07},\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1990Ka08,B}{1990Ka08},\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1979De42,B}{1979De42}}}}\\
\vspace{4pt}
\vspace{8pt}
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}Parent: $^{201}$Tl: E=0.0; J$^{\pi}$=1/2\ensuremath{^{+}}; T$_{1/2}$=3.0421 d {\it 8}; Q(\ensuremath{\varepsilon})=482 {\it 14}; \%\ensuremath{\varepsilon} decay=100.0
}\\
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1990Co07,B}{1990Co07}: inter-comparison data performed at NIST, NPL and PTB metrology labs using samples produced by the same solution}\\
\parbox[b][0.3cm]{17.7cm}{of \ensuremath{^{\textnormal{201}}}Tl and the 4\ensuremath{\pi}-\ensuremath{\gamma} coincidence systems. In each case, corrections were applied for the presence of \ensuremath{^{\textnormal{200}}}Tl and \ensuremath{^{\textnormal{202}}}Tl}\\
\parbox[b][0.3cm]{17.7cm}{contaminants.}\\
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}Others: \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2007Me12,B}{2007Me12}, \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2004De02,B}{2004De02}, \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1989Pl04,B}{1989Pl04}, \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1991Dr09,B}{1991Dr09}, \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1987Dr06,B}{1987Dr06}, \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1987Fu08,B}{1987Fu08}, \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1983Fu22,B}{1983Fu22}, \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1983SC38,B}{1983SC38}, \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1978No06,B}{1978No06}, \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1977Na31,B}{1977Na31}, \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1975Ho08,B}{1975Ho08},}\\
\parbox[b][0.3cm]{17.7cm}{\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1976HiZN,B}{1976HiZN}, \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1960Gu05,B}{1960Gu05}, \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1960He05,B}{1960He05}.}\\
\vspace{12pt}
\underline{$^{201}$Hg Levels}\\
\begin{longtable}{cccccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E(level)$^{{\hyperlink{HG13LEVEL0}{\dagger}}}$}&J$^{\pi}$$^{{\hyperlink{HG13LEVEL1}{\ddagger}}}$&\multicolumn{2}{c}{T$_{1/2}$$^{}$}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{l}{3/2\ensuremath{^{-}}}&&&&\\
\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{5648 {\it 10}}&\multicolumn{1}{l}{1/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{81}&\multicolumn{1}{@{ }l}{ns {\it 5}}&\parbox[t][0.3cm]{12.994881cm}{\raggedright T\ensuremath{_{1/2}}: From \ensuremath{\gamma}-ce(\ensuremath{\Delta}t) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2007Me12,B}{2007Me12}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{26}&\multicolumn{1}{@{.}l}{2738 {\it 3}}&\multicolumn{1}{l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{629}&\multicolumn{1}{@{ }l}{ps {\it 18}}&\parbox[t][0.3cm]{12.994881cm}{\raggedright T\ensuremath{_{1/2}}: Other: 630 ps \textit{50} from ce-\ensuremath{\gamma}(\ensuremath{\Delta}t)) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1983Sc38,B}{1983Sc38}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{32}&\multicolumn{1}{@{.}l}{169 {\it 20}}&\multicolumn{1}{l}{3/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{$<$0}&\multicolumn{1}{@{.}l}{2 ns}&\parbox[t][0.3cm]{12.994881cm}{\raggedright T\ensuremath{_{1/2}}: From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1961Re12,B}{1961Re12}. Other: \ensuremath{\leq}2 ns in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1961Be29,B}{1961Be29}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{167}&\multicolumn{1}{@{.}l}{48 {\it 3}}&\multicolumn{1}{l}{1/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{$<$2}&\multicolumn{1}{@{ }l}{ns}&\parbox[t][0.3cm]{12.994881cm}{\raggedright T\ensuremath{_{1/2}}: From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1961Be29,B}{1961Be29}.\vspace{0.1cm}}&\\
\end{longtable}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{HG13LEVEL0}{\dagger}}}} From a least-squares fit to E\ensuremath{\gamma}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{HG13LEVEL1}{\ddagger}}}} From Adopted Levels.}\\
\vspace{0.5cm}
\underline{\ensuremath{\varepsilon} radiations}\\
\begin{longtable}{ccccccccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E(decay)$$}&\multicolumn{2}{c}{E(level)}&\multicolumn{2}{c}{I$\varepsilon^{{\hyperlink{HG13DECAY0}{\dagger}}{\hyperlink{HG13DECAY1}{\ddagger}}}$}&\multicolumn{2}{c}{Log {\it ft}$^{}$}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{(315}&\multicolumn{1}{@{ }l}{{\it 14})}&\multicolumn{1}{r@{}}{167}&\multicolumn{1}{@{.}l}{48}&\multicolumn{1}{r@{}}{39}&\multicolumn{1}{@{.}l}{8 {\it 4}}&\multicolumn{1}{r@{}}{6}&\multicolumn{1}{@{.}l}{09 {\it 5}}&&\\
\multicolumn{1}{r@{}}{(450}&\multicolumn{1}{@{ }l}{{\it 14})}&\multicolumn{1}{r@{}}{32}&\multicolumn{1}{@{.}l}{169}&\multicolumn{1}{r@{}}{11}&\multicolumn{1}{@{.}l}{9 {\it 4}}&\multicolumn{1}{r@{}}{6}&\multicolumn{1}{@{.}l}{99 {\it 4}}&&\\
\multicolumn{1}{r@{}}{(456}&\multicolumn{1}{@{ }l}{{\it 14})}&\multicolumn{1}{r@{}}{26}&\multicolumn{1}{@{.}l}{2738}&\multicolumn{1}{r@{}}{$\leq$0}&\multicolumn{1}{@{.}l}{5}&\multicolumn{1}{r@{}}{$\geq$8}&\multicolumn{1}{@{.}l}{2\ensuremath{^{1u}}}&\parbox[t][0.3cm]{11.8342705cm}{\raggedright I$\varepsilon$: From systematics (by the evaluator).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{(480}&\multicolumn{1}{@{ }l}{{\it 14})}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{5648}&\multicolumn{1}{r@{}}{$\approx$37}&\multicolumn{1}{@{.}l}{8}&\multicolumn{1}{r@{}}{$\approx$6}&\multicolumn{1}{@{.}l}{6}&\parbox[t][0.3cm]{11.8342705cm}{\raggedright I$\varepsilon$: From the log \textit{ft} value for a similar transition in \ensuremath{^{\textnormal{199}}}Tl \ensuremath{\varepsilon} decay (by the evaluator).\vspace{0.1cm}}&\\
&&&&&&&&\parbox[t][0.3cm]{11.8342705cm}{\raggedright {\ }{\ }{\ }I\ensuremath{\varepsilon}(3/2\ensuremath{^{-}},gs)+I\ensuremath{\varepsilon}(1/2\ensuremath{^{-}},1.56 keV)=47.8\% \textit{6} from the decay scheme. Other: 47\% \textit{23} in\vspace{0.1cm}}&\\
&&&&&&&&\parbox[t][0.3cm]{11.8342705cm}{\raggedright {\ }{\ }{\ }\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2002Kh12,B}{2002Kh12}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{(482}&\multicolumn{1}{@{ }l}{{\it 14})}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{r@{}}{$\approx$10}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{r@{}}{$\approx$7}&\multicolumn{1}{@{.}l}{1}&\parbox[t][0.3cm]{11.8342705cm}{\raggedright I$\varepsilon$: From the log \textit{ft} value for a similar transition in \ensuremath{^{\textnormal{199}}}Tl \ensuremath{\varepsilon} decay (by the evaluator).\vspace{0.1cm}}&\\
&&&&&&&&\parbox[t][0.3cm]{11.8342705cm}{\raggedright {\ }{\ }{\ }I\ensuremath{\varepsilon}(3/2\ensuremath{^{-}},gs)+I\ensuremath{\varepsilon}(1/2\ensuremath{^{-}},1.56 keV)=47.8\% \textit{6} from the decay scheme. Other: \ensuremath{<}20.9\% in\vspace{0.1cm}}&\\
&&&&&&&&\parbox[t][0.3cm]{11.8342705cm}{\raggedright {\ }{\ }{\ }\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2002Kh12,B}{2002Kh12}.\vspace{0.1cm}}&\\
\end{longtable}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{HG13DECAY0}{\dagger}}}} Estimated by the evaluator from intensity balances and the adopted decay scheme, unless otherwise stated.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{HG13DECAY1}{\ddagger}}}} Absolute intensity per 100 decays.}\\
\vspace{0.5cm}
\clearpage
\vspace{0.3cm}
\begin{landscape}
\vspace*{-0.5cm}
{\bf \small \underline{\ensuremath{^{\textnormal{201}}}Tl \ensuremath{\varepsilon} decay\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1990Co07,B}{1990Co07},\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1990Ka08,B}{1990Ka08},\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1979De42,B}{1979De42} (continued)}}\\
\vspace{0.3cm}
\underline{$\gamma$($^{201}$Hg)}\\
\vspace{0.34cm}
\parbox[b][0.3cm]{21.881866cm}{\addtolength{\parindent}{-0.254cm}I\ensuremath{\gamma} normalization: From I\ensuremath{\gamma}(167\ensuremath{\gamma})=10.00\% \textit{6} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1990Co07,B}{1990Co07}), weighted average of 9.88\% \textit{8} (NIST), 10.05\% \textit{17} (NPL) and 10.18\% \textit{10} (PTB). Others: I\ensuremath{\gamma}(167\ensuremath{\gamma}):}\\
\parbox[b][0.3cm]{21.881866cm}{9.81\% \textit{12} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1990Ka08,B}{1990Ka08}), 10.60\% \textit{15} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1989Pl04,B}{1989Pl04}), 10.25\% \textit{10} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1983Fu22,B}{1983Fu22}), 10.60\% \textit{12} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1979De42,B}{1979De42}), 10.00\% \textit{17} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1976HiZN,B}{1976HiZN}), 10.00\% (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1975Ho08,B}{1975Ho08}) and 8.4\% \textit{4}}\\
\parbox[b][0.3cm]{21.881866cm}{(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1960He05,B}{1960He05}). The total energy realized in \ensuremath{^{\textnormal{201}}}Tl \ensuremath{\varepsilon} decay is calculated using RADLST as 471 keV \textit{14}. It is in a good agreement with Q(g.s.)=482 keV \textit{14}.}\\
\vspace{0.34cm}
\raggedright\texttt{}\\
\raggedright\texttt{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x-ray\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ E\ensuremath{\gamma}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ I\ensuremath{\gamma}}\\
\raggedright\texttt{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ keV\ \ \ \ \ \ \ \ \ \ \ \ \ per\ 100\ \hspace{-0.04cm}\ensuremath{\varepsilon}\ \ decays}\\
\raggedright\texttt{\ \ \ \ \ \ \ \ \ \ \ --------------\ \ \ \ \ \ \ \ \ \ -------\ \ \ \ \ \ \ \ \ \ \ -----------------}\\
\raggedright\texttt{\ \ \ \ \ \ \ \ \ \ \ K\ensuremath{\alpha}\ensuremath{_{\textnormal{1}}}{}\ \ \ \ \ \ x\ ray\ \ \ \ \ \ \ \ \ \ \ 70.8\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 44.6\ \ \ \textit{5}}\\
\raggedright\texttt{\ \ \ \ \ \ \ \ \ \ \ K\ensuremath{\alpha}\ensuremath{_{\textnormal{2}}}{}\ \ \ \ \ \ x\ ray\ \ \ \ \ \ \ \ \ \ \ 68.9\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 26.3\ \ \ \textit{3}}\\
\raggedright\texttt{\ \ \ \ \ \ \ \ \ \ \ K\ensuremath{\alpha}{}\ \ \ \ \ \ \ x\ ray\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 71.1\ \ \ \textit{5}}\\
\raggedright\texttt{\ \ \ \ \ \ \ \ \ \ \ K\ensuremath{\beta}\ensuremath{_{\textnormal{1}}}$'${}\ \ \ \ \ \ x\ ray\ \ \ \ \ \ \ \ \ \ \ 80.2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 15.3\ \ \ \textit{4}}\\
\raggedright\texttt{\ \ \ \ \ \ \ \ \ \ \ K\ensuremath{\beta}\ensuremath{_{\textnormal{2}}}$'${}\ \ \ \ \ \ x\ ray\ \ \ \ \ \ \ \ \ \ \ 80.5\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 4.59\ \ \textit{15}}\\
\raggedright\texttt{\ \ \ \ \ \ \ \ \ \ \ K\ensuremath{\beta}{}\ \ \ \ \ \ \ x\ ray\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 20.0\ \ \ \textit{3}}\\
\raggedright\texttt{}\\
\raggedright\texttt{\ I\ensuremath{\gamma}\ \ -\ Weighted\ average\ of\ values\ given\ in\ \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1976HiZN,B}{1976HiZN},\ \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1979De42,B}{1979De42},\ \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1983Fu22,B}{1983Fu22}}\\
\raggedright\texttt{\ \ \ \ \ \ and\ \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1990Ka08,B}{1990Ka08}.}\\
\raggedright\texttt{}\\
\centering \texttt{}\\
\vspace{-0.5cm}
\begin{longtable}{ccccccccc@{}ccccccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{TL13GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{I\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{TL13GAMMA1}{\ddagger}}}}&\multicolumn{2}{c}{E\ensuremath{_{i}}(level)}&J\ensuremath{^{\pi}_{i}}&\multicolumn{2}{c}{E\ensuremath{_{f}}}&J\ensuremath{^{\pi}_{f}}&Mult.&\multicolumn{2}{c}{\ensuremath{\delta}}&\multicolumn{2}{c}{\ensuremath{\alpha}\ensuremath{^{\hyperlink{TL13GAMMA2}{\#}}}}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{5648 {\it 10}}&\multicolumn{1}{r@{}}{$\approx$0}&\multicolumn{1}{@{.}l}{011}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{5648}&\multicolumn{1}{l}{1/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{@{}l}{3/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0105 {\it 14}}&\multicolumn{1}{r@{}}{4.7\ensuremath{\times10^{4}}}&\multicolumn{1}{@{ }l}{{\it 7}}&\parbox[t][0.3cm]{8.982659cm}{\raggedright \%I\ensuremath{\gamma}\ensuremath{\approx}0.001100\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.982659cm}{\raggedright E\ensuremath{_{\gamma}}: From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1997Ge09,B}{1997Ge09}. Other: 1.565 keV \textit{6} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1987Dr06,B}{1987Dr06}).\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.982659cm}{\raggedright I\ensuremath{_{\gamma}}: Estimated by the evaluator from intensity balance and the\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.982659cm}{\raggedright {\ }{\ }{\ }adopted decay scheme.\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.982659cm}{\raggedright Mult.: From N1/N2=0.94 \textit{31}, N1/N3=0.60 \textit{20}, N2/N3=0.64 \textit{5},\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.982659cm}{\raggedright {\ }{\ }{\ }N4/N3=0.042 \textit{18}, N5/N3=0.043 \textit{23}, N4/N5=0.98 \textit{32},\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.982659cm}{\raggedright {\ }{\ }{\ }O1/O2=0.81 \textit{16}, O2/N3=0.158 \textit{30}, O3/N3=0.20 \textit{4}, O1/O3=0.64\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.982659cm}{\raggedright {\ }{\ }{\ }\textit{12}, O1/N3=0.128 \textit{30} and O2/O3=0.79 \textit{19} subshell ratios in\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.982659cm}{\raggedright {\ }{\ }{\ }\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1997Ge09,B}{1997Ge09} and N1/N2=1.2 \textit{2}, N1/N3=1.1 \textit{2}, N2/N3=0.92 \textit{15},\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.982659cm}{\raggedright {\ }{\ }{\ }N4/N3=0.03 \textit{2} and N5/N3=0.04 \textit{2} subshell ratios in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1987Dr06,B}{1987Dr06}.\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.982659cm}{\raggedright \ensuremath{\delta}: From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1987Dr06,B}{1987Dr06}; Other: 0.0145 \textit{+19{\textminus}14} in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1997Ge09,B}{1997Ge09}.\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.982659cm}{\raggedright \ensuremath{\alpha}: 4.7E+4 7 from \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1987Dr06,B}{1987Dr06}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{(5}&\multicolumn{1}{@{.}l}{895 {\it 20})}&\multicolumn{1}{r@{}}{$\approx$0}&\multicolumn{1}{@{.}l}{0007}&\multicolumn{1}{r@{}}{32}&\multicolumn{1}{@{.}l}{169}&\multicolumn{1}{l}{3/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{26}&\multicolumn{1}{@{.}l}{2738 }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{l}{[M1]}&&&\multicolumn{1}{r@{}}{1441}&\multicolumn{1}{@{ }l}{{\it 23}}&\parbox[t][0.3cm]{8.982659cm}{\raggedright \%I\ensuremath{\gamma}\ensuremath{\approx}7.00\ensuremath{\times}10\ensuremath{^{\textnormal{$-$5}}}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.982659cm}{\raggedright E\ensuremath{_{\gamma}}: Not observed directly, but required from the ce-\ensuremath{\gamma}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.982659cm}{\raggedright {\ }{\ }{\ }coincidence data in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1983Sc38,B}{1983Sc38}. E\ensuremath{\gamma} from level energy\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.982659cm}{\raggedright {\ }{\ }{\ }differences.\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.982659cm}{\raggedright I\ensuremath{_{\gamma}}: Estimated by the evaluator from intensity balance and the\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.982659cm}{\raggedright {\ }{\ }{\ }adopted decay scheme.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{26}&\multicolumn{1}{@{.}l}{2738 {\it 3}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{082 {\it 10}}&\multicolumn{1}{r@{}}{26}&\multicolumn{1}{@{.}l}{2738}&\multicolumn{1}{l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{@{}l}{3/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{012 {\it 8}}&\multicolumn{1}{r@{}}{72}&\multicolumn{1}{@{.}l}{9 {\it 13}}&\parbox[t][0.3cm]{8.982659cm}{\raggedright \%I\ensuremath{\gamma}=0.0082 \textit{10}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.982659cm}{\raggedright E\ensuremath{_{\gamma}}: From adopted gammas. Others: 26.34 keV \textit{7} in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1983Sc38,B}{1983Sc38}.\vspace{0.1cm}}&\\
\end{longtable}
\clearpage
\begin{longtable}{ccccccccc@{}ccccccc@{\extracolsep{\fill}}c}
\\[-.4cm]
\multicolumn{17}{c}{{\bf \small \underline{\ensuremath{^{\textnormal{201}}}Tl \ensuremath{\varepsilon} decay\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1990Co07,B}{1990Co07},\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1990Ka08,B}{1990Ka08},\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1979De42,B}{1979De42} (continued)}}}\\
\multicolumn{17}{c}{~}\\
\multicolumn{17}{c}{\underline{$\gamma$($^{201}$Hg) (continued)}}\\
\multicolumn{17}{c}{~~~}\\
\multicolumn{2}{c}{E\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{TL13GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{I\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{TL13GAMMA1}{\ddagger}}}}&\multicolumn{2}{c}{E\ensuremath{_{i}}(level)}&J\ensuremath{^{\pi}_{i}}&\multicolumn{2}{c}{E\ensuremath{_{f}}}&J\ensuremath{^{\pi}_{f}}&Mult.&\multicolumn{2}{c}{\ensuremath{\delta}}&\multicolumn{2}{c}{\ensuremath{\alpha}\ensuremath{^{\hyperlink{TL13GAMMA2}{\#}}}}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endhead
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{9.60112cm}{\raggedright I\ensuremath{_{\gamma}}: 0.082 \textit{10} from I(ce(M1))(26.27\ensuremath{\gamma})/I(ce(L1))(32.19\ensuremath{\gamma})=0.0131 \textit{15} in\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{9.60112cm}{\raggedright {\ }{\ }{\ }\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1983Sc38,B}{1983Sc38},\ensuremath{\alpha}(M1)(26.27\ensuremath{\gamma})=11.52 \textit{17} and \ensuremath{\alpha}(L1)(32.19\ensuremath{\gamma})=27.3 \textit{4} and\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{9.60112cm}{\raggedright {\ }{\ }{\ }I\ensuremath{\gamma}(32.19\ensuremath{\gamma})=2.63 \textit{5} from the present evaluation.\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{9.60112cm}{\raggedright Mult.,\ensuremath{\delta}: From M1:M2:M3=100:12.0 \textit{20}:1.5 \textit{7} in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1983Sc38,B}{1983Sc38} and using\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{9.60112cm}{\raggedright {\ }{\ }{\ }the briccmixing program.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{30}&\multicolumn{1}{@{.}l}{60 {\it 3}}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{58 {\it 5}}&\multicolumn{1}{r@{}}{32}&\multicolumn{1}{@{.}l}{169}&\multicolumn{1}{l}{3/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{5648 }&\multicolumn{1}{@{}l}{1/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{013 {\it 5}}&\multicolumn{1}{r@{}}{46}&\multicolumn{1}{@{.}l}{4 {\it 7}}&\parbox[t][0.3cm]{9.60112cm}{\raggedright \%I\ensuremath{\gamma}=0.258 \textit{5}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{9.60112cm}{\raggedright I\ensuremath{_{\gamma}}: Weighted average 2.2 \textit{2} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1975Ho08,B}{1975Ho08}), 3.10 \textit{13} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1976HiZN,B}{1976HiZN}), 2.57 \textit{6}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{9.60112cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1979De42,B}{1979De42}), 2.60 \textit{8} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1983Fu22,B}{1983Fu22}), 2.60 \textit{8} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1990Ka08,B}{1990Ka08}) and 2.53 \textit{5}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{9.60112cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1990Co07,B}{1990Co07}).\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{9.60112cm}{\raggedright Mult.: From L3:L1=0.0136 \textit{21}, L2:L1=0.105 \textit{11} and L3:L2=0.130 \textit{24}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{9.60112cm}{\raggedright {\ }{\ }{\ }in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1983Sc38,B}{1983Sc38}; L1:L2:L3:M1:M2:N:O1=50.9 \textit{40}:5.0 \textit{6}:0.56 \textit{8}:14.2\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{9.60112cm}{\raggedright {\ }{\ }{\ }\textit{15}:1.5 \textit{5}:4.0 \textit{5}:0.70 \textit{15} in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1960He05,B}{1960He05}; A\ensuremath{_{\textnormal{2}}}(135.5\ensuremath{\gamma}-30.6\ensuremath{\gamma}(\ensuremath{\theta}))=0.159\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{9.60112cm}{\raggedright {\ }{\ }{\ }\textit{26} in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1975Ho08,B}{1975Ho08}.\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{9.60112cm}{\raggedright \ensuremath{\delta}: From L3/L1, L2/L1 and L3/L2 in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1983Sc38,B}{1983Sc38} and the briccmixing\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{9.60112cm}{\raggedright {\ }{\ }{\ }program. Others: \ensuremath{\leq}0.03 from \ensuremath{\gamma}\ensuremath{\gamma}(\ensuremath{\theta}) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1975Ho08,B}{1975Ho08};\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{9.60112cm}{\raggedright {\ }{\ }{\ }{\textminus}0.0634\ensuremath{\leq}\ensuremath{\delta}\ensuremath{\leq}+0.0515 from \ensuremath{\gamma}\ensuremath{\gamma}(\ensuremath{\theta}) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1978No06,B}{1978No06}; 0.006 \textit{16} from from\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{9.60112cm}{\raggedright {\ }{\ }{\ }L1:L2:L3 in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1960He05,B}{1960He05} and the briccmixing program.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{32}&\multicolumn{1}{@{.}l}{19 {\it 3}}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{63 {\it 5}}&\multicolumn{1}{r@{}}{32}&\multicolumn{1}{@{.}l}{169}&\multicolumn{1}{l}{3/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{@{}l}{3/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0204 {\it 25}}&\multicolumn{1}{r@{}}{40}&\multicolumn{1}{@{.}l}{2 {\it 6}}&\parbox[t][0.3cm]{9.60112cm}{\raggedright \%I\ensuremath{\gamma}=0.263 \textit{5}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{9.60112cm}{\raggedright I\ensuremath{_{\gamma}}: Weighted average 2.2 \textit{2} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1975Ho08,B}{1975Ho08}), 2.85 \textit{12} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1976HiZN,B}{1976HiZN}), 2.60 \textit{9}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{9.60112cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1979De42,B}{1979De42}), 2.60 \textit{7} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1983Fu22,B}{1983Fu22}), 2.72 \textit{6} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1990Ka08,B}{1990Ka08})\hphantom{a}and 2.58 \textit{5}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{9.60112cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1990Co07,B}{1990Co07}).\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{9.60112cm}{\raggedright Mult.: From L1:L2:L3=100:11.3 \textit{5}:1.75 \textit{15} in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1983Sc38,B}{1983Sc38};\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{9.60112cm}{\raggedright {\ }{\ }{\ }L3:L1=0.0130 \textit{20}, L2:L1=0.094 \textit{8}, L3:L2=0.138 \textit{24}, L:M=3.9 \textit{4},\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{9.60112cm}{\raggedright {\ }{\ }{\ }M1:M2=8.4 \textit{3} M:N=4.6 \textit{6} in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1960He05,B}{1960He05};\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{9.60112cm}{\raggedright {\ }{\ }{\ }A\ensuremath{_{\textnormal{2}}}(135.5\ensuremath{\gamma}-32.2\ensuremath{\gamma}(\ensuremath{\theta}))={\textminus}0.193 \textit{28} in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1975Ho08,B}{1975Ho08}.\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{9.60112cm}{\raggedright \ensuremath{\delta}: From L3/L1, L2/L1 and L3/L2 in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1983Sc38,B}{1983Sc38} and the briccmixing\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{9.60112cm}{\raggedright {\ }{\ }{\ }program. Others: \ensuremath{\leq}0.03 from \ensuremath{\gamma}\ensuremath{\gamma}(\ensuremath{\theta}) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1975Ho08,B}{1975Ho08};\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{9.60112cm}{\raggedright {\ }{\ }{\ }{\textminus}0.0361\ensuremath{\leq}\ensuremath{\delta}\ensuremath{\leq}+0.0506 from \ensuremath{\gamma}\ensuremath{\gamma}(\ensuremath{\theta}) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1978No06,B}{1978No06}; 0.013 \textit{11} from from\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{9.60112cm}{\raggedright {\ }{\ }{\ }L1:L2:L3 in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1960He05,B}{1960He05} and the briccmixing program.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{135}&\multicolumn{1}{@{.}l}{34 {\it 4}}&\multicolumn{1}{r@{}}{26}&\multicolumn{1}{@{.}l}{05 {\it 18}}&\multicolumn{1}{r@{}}{167}&\multicolumn{1}{@{.}l}{48}&\multicolumn{1}{l}{1/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{32}&\multicolumn{1}{@{.}l}{169 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{$-$0}&\multicolumn{1}{@{.}l}{07 {\it 4}}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{32 {\it 5}}&\parbox[t][0.3cm]{9.60112cm}{\raggedright \%I\ensuremath{\gamma}=2.605 \textit{24}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{9.60112cm}{\raggedright I\ensuremath{_{\gamma}}: Weighted average 26.5 \textit{13} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1975Ho08,B}{1975Ho08}), 26.5 \textit{10} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1976HiZN,B}{1976HiZN}), 26.4\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{9.60112cm}{\raggedright {\ }{\ }{\ }\textit{3} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1979De42,B}{1979De42}), 26.5 \textit{4} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1983Fu22,B}{1983Fu22}), 27.2 \textit{5} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1990Ka08,B}{1990Ka08}) and 25.65 \textit{18}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{9.60112cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1990Co07,B}{1990Co07}). Other 27.3 \textit{2} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2004De02,B}{2004De02}).\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{9.60112cm}{\raggedright Mult.: From K:L1:L2:L3:M1:N1=56.0 \textit{4}:7.9 \textit{7}:0.77 \textit{15}:0.07 \textit{3}: 2.2\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{9.60112cm}{\raggedright {\ }{\ }{\ }\textit{3}:0.60 \textit{9}; \ensuremath{\gamma}\ensuremath{\gamma}(\ensuremath{\theta}) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1975Ho08,B}{1975Ho08}.\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{9.60112cm}{\raggedright \ensuremath{\delta}: From \ensuremath{\gamma}\ensuremath{\gamma}(\ensuremath{\theta}) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1975Ho08,B}{1975Ho08}; other: 0.000 \textit{4} from\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{9.60112cm}{\raggedright {\ }{\ }{\ }K:L1:L2:L3:M1:N1 in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1960He05,B}{1960He05} and the briccmixing program.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{141}&\multicolumn{1}{@{.}l}{1 {\it 2}}&\multicolumn{1}{r@{}}{$\approx$0}&\multicolumn{1}{@{.}l}{026}&\multicolumn{1}{r@{}}{167}&\multicolumn{1}{@{.}l}{48}&\multicolumn{1}{l}{1/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{26}&\multicolumn{1}{@{.}l}{2738 }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{l}{[E2]}&&&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{389 {\it 21}}&\parbox[t][0.3cm]{9.60112cm}{\raggedright \%I\ensuremath{\gamma}\ensuremath{\approx}0.002600\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{9.60112cm}{\raggedright E\ensuremath{_{\gamma}}: From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1983Sc38,B}{1983Sc38}.\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{9.60112cm}{\raggedright I\ensuremath{_{\gamma}}: From the decay scheme and I\ensuremath{\gamma}(141.1\ensuremath{\gamma})/I\ensuremath{\gamma}(135.34\ensuremath{\gamma})=0.11 \textit{2},\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{9.60112cm}{\raggedright {\ }{\ }{\ }using the 26\ensuremath{\gamma} as a gate in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1983Sc38,B}{1983Sc38}, and by assuming\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{9.60112cm}{\raggedright {\ }{\ }{\ }I\ensuremath{\beta}(5/2\ensuremath{^{-}},26.27 keV)=0.5\%.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{165}&\multicolumn{1}{@{.}l}{88 {\it 7}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{47 {\it 2}}&\multicolumn{1}{r@{}}{167}&\multicolumn{1}{@{.}l}{48}&\multicolumn{1}{l}{1/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{5648 }&\multicolumn{1}{@{}l}{1/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1}&&&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{869 {\it 26}}&\parbox[t][0.3cm]{9.60112cm}{\raggedright \%I\ensuremath{\gamma}=0.1470 \textit{22}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{9.60112cm}{\raggedright I\ensuremath{_{\gamma}}: Weighted average of 1.6 \textit{1} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1975Ho08,B}{1975Ho08}), 1.80 \textit{20} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1976HiZN,B}{1976HiZN}), 1.5\vspace{0.1cm}}&\\
\end{longtable}
\clearpage
\begin{longtable}{ccccccccc@{}ccccccc@{\extracolsep{\fill}}c}
\\[-.4cm]
\multicolumn{17}{c}{{\bf \small \underline{\ensuremath{^{\textnormal{201}}}Tl \ensuremath{\varepsilon} decay\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1990Co07,B}{1990Co07},\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1990Ka08,B}{1990Ka08},\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1979De42,B}{1979De42} (continued)}}}\\
\multicolumn{17}{c}{~}\\
\multicolumn{17}{c}{\underline{$\gamma$($^{201}$Hg) (continued)}}\\
\multicolumn{17}{c}{~~~}\\
\multicolumn{2}{c}{E\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{TL13GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{I\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{TL13GAMMA1}{\ddagger}}}}&\multicolumn{2}{c}{E\ensuremath{_{i}}(level)}&J\ensuremath{^{\pi}_{i}}&\multicolumn{2}{c}{E\ensuremath{_{f}}}&J\ensuremath{^{\pi}_{f}}&Mult.&\multicolumn{2}{c}{\ensuremath{\delta}}&\multicolumn{2}{c}{\ensuremath{\alpha}\ensuremath{^{\hyperlink{TL13GAMMA2}{\#}}}}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endhead
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{11.460779cm}{\raggedright {\ }{\ }{\ }\textit{2} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1979De42,B}{1979De42}), 1.46 \textit{20} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1983Fu22,B}{1983Fu22}), 1.45 \textit{2} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1990Ka08,B}{1990Ka08})\hphantom{a}and 1.55 \textit{5} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1990Co07,B}{1990Co07}).\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{11.460779cm}{\raggedright Mult.: From K:L1=1.65 \textit{20}:0.25 \textit{5} in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1960He05,B}{1960He05}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{167}&\multicolumn{1}{@{.}l}{43 {\it 7}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{r@{}}{167}&\multicolumn{1}{@{.}l}{48}&\multicolumn{1}{l}{1/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{@{}l}{3/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{07 {\it 6}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{815 {\it 29}}&\parbox[t][0.3cm]{11.460779cm}{\raggedright \%I\ensuremath{\gamma}=10.00 \textit{6}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{11.460779cm}{\raggedright I\ensuremath{_{\gamma}}: From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1990Co07,B}{1990Co07}; others: 100 (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1975Ho08,B}{1975Ho08}), 100.0 \textit{17} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1976HiZN,B}{1976HiZN}), 100.0 \textit{11}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{11.460779cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1979De42,B}{1979De42}), 100.0 \textit{10} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1983Fu22,B}{1983Fu22}) and 100.0 \textit{12} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1990Ka08,B}{1990Ka08}).\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{11.460779cm}{\raggedright Mult.: From K:L1:L2:L3:M1:N1:O1=100:14.6 \textit{12}:1.6 \textit{2}:0.18 \textit{4}:4.0 \textit{4}: 1.10 \textit{15}:0.27 \textit{6}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{11.460779cm}{\raggedright {\ }{\ }{\ }in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1960He05,B}{1960He05}.\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{11.460779cm}{\raggedright \ensuremath{\delta}: From K:L1:L2:L3 in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1960He05,B}{1960He05} and the briccmixing program.\vspace{0.1cm}}&\\
\end{longtable}
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{TL13GAMMA0}{\dagger}}}} From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1960He05,B}{1960He05}, unless otherwise stated.}\\
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{TL13GAMMA1}{\ddagger}}}} For absolute intensity per 100 decays, multiply by 0.1000 \textit{6}.}\\
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{TL13GAMMA2}{\#}}}} Total theoretical internal conversion coefficients, calculated using the BrIcc code (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2008Ki07,B}{2008Ki07}) with Frozen orbital approximation based on \ensuremath{\gamma}-ray energies,}\\
\parbox[b][0.3cm]{21.881866cm}{{\ }{\ }assigned multipolarities, and mixing ratios, unless otherwise specified.}\\
\vspace{0.5cm}
\end{landscape}\clearpage
\clearpage
\begin{figure}[h]
\begin{center}
\includegraphics{201HG13-0.ps}\\
\end{center}
\end{figure}
\clearpage
\subsection[\hspace{-0.2cm}\ensuremath{^{\textnormal{200}}}Hg(d,p)]{ }
\vspace{-27pt}
\vspace{0.3cm}
\hypertarget{HG14}{{\bf \small \underline{\ensuremath{^{\textnormal{200}}}Hg(d,p)\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1972Mo12,B}{1972Mo12}}}}\\
\vspace{4pt}
\vspace{8pt}
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}Beam: E(d)=17 MeV; Target: \ensuremath{^{\textnormal{202}}}Hg, but isotopic purity is unknown; Detectors: photographic emulsions, split-pole spectrograph,}\\
\parbox[b][0.3cm]{17.7cm}{FWHM=10-14 keV.}\\
\vspace{12pt}
\underline{$^{201}$Hg Levels}\\
\begin{longtable}{ccccccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E(level)$^{{\hyperlink{HG14LEVEL0}{\dagger}}}$}&J$^{\pi}$$^{{\hyperlink{HG14LEVEL1}{\ddagger}}}$&L$^{{\hyperlink{HG14LEVEL0}{\dagger}}}$&\multicolumn{2}{c}{S$^{{\hyperlink{HG14LEVEL2}{\#}}}$}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{HG14LEVEL3}{@}}}}}&\multicolumn{1}{l}{(3/2\ensuremath{^{-}})}&\multicolumn{1}{l}{(1)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{37}&\parbox[t][0.3cm]{12.119801cm}{\raggedright S: Probably includes strength to the 1.57 keV (\ensuremath{J^{\pi}}=1/2\ensuremath{^{-}}) level, that is unresolved in\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{12.119801cm}{\raggedright {\ }{\ }{\ }\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1972Mo12,B}{1972Mo12}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{28}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{HG14LEVEL4}{\&}}}}}&\multicolumn{1}{l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{l}{1+3}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{32}&\parbox[t][0.3cm]{12.119801cm}{\raggedright J\ensuremath{^{\pi}}: From Adopted Levels. S for L=3.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{32}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{l}{3/2\ensuremath{^{-}}}&\multicolumn{1}{l}{1+3}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{18}&\parbox[t][0.3cm]{12.119801cm}{\raggedright J\ensuremath{^{\pi}}: From Adopted Levels. S for L=1.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{169}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{l}{(1/2\ensuremath{^{-}})}&\multicolumn{1}{l}{(1)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{38}&&\\
\multicolumn{1}{r@{}}{417}&\multicolumn{1}{@{}l}{}&&&&&&\\
\multicolumn{1}{r@{}}{466}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{l}{(5/2\ensuremath{^{-}})}&\multicolumn{1}{l}{(3)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{10}&&\\
\multicolumn{1}{r@{}}{550}&\multicolumn{1}{@{}l}{}&&&&&&\\
\multicolumn{1}{r@{}}{735}&\multicolumn{1}{@{}l}{}&&&&&&\\
\multicolumn{1}{r@{}}{1280}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{l}{(7/2\ensuremath{^{-}},3/2\ensuremath{^{-}})}&\multicolumn{1}{l}{(3,1)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{11}&\parbox[t][0.3cm]{12.119801cm}{\raggedright S: Value quoted for L=3. S=0.06 for L=1 (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1972Mo12,B}{1972Mo12}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1336}&\multicolumn{1}{@{}l}{}&&&&&&\\
\multicolumn{1}{r@{}}{1367}&\multicolumn{1}{@{}l}{}&&&&&&\\
\multicolumn{1}{r@{}}{1946}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{l}{(9/2\ensuremath{^{+}},7/2\ensuremath{^{-}})}&\multicolumn{1}{l}{(4,3)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{05}&\parbox[t][0.3cm]{12.119801cm}{\raggedright S: Value quoted for L=4. S=0.07 for L=3 (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1972Mo12,B}{1972Mo12}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2081}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{l}{(9/2\ensuremath{^{+}},7/2\ensuremath{^{-}})}&\multicolumn{1}{l}{(4,3)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{08}&\parbox[t][0.3cm]{12.119801cm}{\raggedright S: Value quoted for L=4. S=0.11 for L=3 (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1972Mo12,B}{1972Mo12}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2103}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{l}{9/2\ensuremath{^{+}}}&\multicolumn{1}{l}{4}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{25}&&\\
\multicolumn{1}{r@{}}{2628}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{l}{9/2\ensuremath{^{+}}}&\multicolumn{1}{l}{4}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{06}&&\\
\multicolumn{1}{r@{}}{2660?}&\multicolumn{1}{@{}l}{}&&&&&&\\
\multicolumn{1}{r@{}}{2795}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{l}{(9/2\ensuremath{^{+}},5/2\ensuremath{^{+}})}&\multicolumn{1}{l}{(4,2)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{10}&\parbox[t][0.3cm]{12.119801cm}{\raggedright S: Value quoted for L=4. S=0.05 for L=2 (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1972Mo12,B}{1972Mo12}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2863}&\multicolumn{1}{@{}l}{}&&&&&&\\
\multicolumn{1}{r@{}}{2890}&\multicolumn{1}{@{}l}{}&&&&&&\\
\multicolumn{1}{r@{}}{2911}&\multicolumn{1}{@{}l}{}&&&&&&\\
\multicolumn{1}{r@{}}{2938}&\multicolumn{1}{@{}l}{}&&&&&&\\
\multicolumn{1}{r@{}}{2976}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{l}{(9/2\ensuremath{^{+}},5/2\ensuremath{^{+}})}&\multicolumn{1}{l}{(4,2)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{08}&\parbox[t][0.3cm]{12.119801cm}{\raggedright S: Value quoted for L=4. S=0.04 for L=2 (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1972Mo12,B}{1972Mo12}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2995}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{l}{(9/2\ensuremath{^{+}},5/2\ensuremath{^{+}})}&\multicolumn{1}{l}{(4,2)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{06}&\parbox[t][0.3cm]{12.119801cm}{\raggedright S: Value quoted for L=4. S=0.03 for L=2 (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1972Mo12,B}{1972Mo12}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3115}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{l}{(9/2\ensuremath{^{+}},5/2\ensuremath{^{+}})}&\multicolumn{1}{l}{(2,4)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{03}&\parbox[t][0.3cm]{12.119801cm}{\raggedright S: Value quoted for L=2. S=0.06 for L=4 (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1972Mo12,B}{1972Mo12}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3172}&\multicolumn{1}{@{}l}{}&&&&&&\\
\multicolumn{1}{r@{}}{3196}&\multicolumn{1}{@{}l}{}&&&&&&\\
\multicolumn{1}{r@{}}{3233}&\multicolumn{1}{@{}l}{}&&&&&&\\
\multicolumn{1}{r@{}}{3252}&\multicolumn{1}{@{}l}{}&&&&&&\\
\multicolumn{1}{r@{}}{3270}&\multicolumn{1}{@{}l}{}&&&&&&\\
\multicolumn{1}{r@{}}{3294}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{l}{(5/2\ensuremath{^{+}},9/2\ensuremath{^{+}})}&\multicolumn{1}{l}{(2,4)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{07}&\parbox[t][0.3cm]{12.119801cm}{\raggedright S: Value quoted for L=2. S=0.14 for L=4 (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1972Mo12,B}{1972Mo12}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3539}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{l}{(5/2\ensuremath{^{+}},9/2\ensuremath{^{+}})}&\multicolumn{1}{l}{(2,4)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{05}&\parbox[t][0.3cm]{12.119801cm}{\raggedright S: Value quoted for L=2. S=0.10 for L=4 (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1972Mo12,B}{1972Mo12}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3579}&\multicolumn{1}{@{}l}{}&&&&&&\\
\multicolumn{1}{r@{}}{3712}&\multicolumn{1}{@{}l}{}&&&&&&\\
\multicolumn{1}{r@{}}{3768}&\multicolumn{1}{@{}l}{}&&&&&&\\
\multicolumn{1}{r@{}}{3814}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{l}{(9/2\ensuremath{^{+}},5/2\ensuremath{^{+}})}&\multicolumn{1}{l}{(4,2)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{05}&\parbox[t][0.3cm]{12.119801cm}{\raggedright S: Value quoted for L=4. S=0.03 for L=2 (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1972Mo12,B}{1972Mo12}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3837}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{l}{(5/2\ensuremath{^{+}},13/2\ensuremath{^{+}})}&\multicolumn{1}{l}{(2,6)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{02}&\parbox[t][0.3cm]{12.119801cm}{\raggedright S: Value quoted for L=2. S=0.40 for L=6 (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1972Mo12,B}{1972Mo12}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3870}&\multicolumn{1}{@{}l}{}&&&&&&\\
\multicolumn{1}{r@{}}{3884}&\multicolumn{1}{@{}l}{}&&&&&&\\
\multicolumn{1}{r@{}}{3900}&\multicolumn{1}{@{}l}{}&&&&&&\\
\multicolumn{1}{r@{}}{3921}&\multicolumn{1}{@{}l}{}&&&&&&\\
\multicolumn{1}{r@{}}{4007}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{l}{(5/2\ensuremath{^{+}},13/2\ensuremath{^{+}})}&\multicolumn{1}{l}{(2,6)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{02}&\parbox[t][0.3cm]{12.119801cm}{\raggedright E(level): Possibly a doublet.\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{12.119801cm}{\raggedright S: Value quoted for L=2. S=0.52 for L=6 (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1972Mo12,B}{1972Mo12}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{4070}&\multicolumn{1}{@{}l}{}&&&&&&\\
\multicolumn{1}{r@{}}{4095}&\multicolumn{1}{@{}l}{}&&&&&&\\
\multicolumn{1}{r@{}}{4123}&\multicolumn{1}{@{}l}{}&&&&&&\\
\multicolumn{1}{r@{}}{4233}&\multicolumn{1}{@{}l}{}&&&&&&\\
\multicolumn{1}{r@{}}{4284}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{l}{(5/2\ensuremath{^{+}},9/2\ensuremath{^{+}})}&\multicolumn{1}{l}{(2,4)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{13}&\parbox[t][0.3cm]{12.119801cm}{\raggedright S: Value quoted for L=2. S=0.23 for L=4 (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1972Mo12,B}{1972Mo12}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{4313}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{l}{(5/2\ensuremath{^{+}},9/2\ensuremath{^{+}})}&\multicolumn{1}{l}{(2,4)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{21}&\parbox[t][0.3cm]{12.119801cm}{\raggedright S: Value quoted for L=2. S=0.37 for L=4 (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1972Mo12,B}{1972Mo12}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{4362}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{l}{(9/2\ensuremath{^{+}},1/2\ensuremath{^{+}})}&\multicolumn{1}{l}{(4,0)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{08}&\parbox[t][0.3cm]{12.119801cm}{\raggedright S: Value quoted for L=4. S=0.08 for L=0 (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1972Mo12,B}{1972Mo12}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{4381}&\multicolumn{1}{@{}l}{}&&&&&&\\
\end{longtable}
\begin{textblock}{29}(0,27.3)
Continued on next page (footnotes at end of table)
\end{textblock}
\clearpage
\begin{longtable}{ccccccc@{\extracolsep{\fill}}c}
\\[-.4cm]
\multicolumn{8}{c}{{\bf \small \underline{\ensuremath{^{\textnormal{200}}}Hg(d,p)\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1972Mo12,B}{1972Mo12} (continued)}}}\\
\multicolumn{8}{c}{~}\\
\multicolumn{8}{c}{\underline{\ensuremath{^{201}}Hg Levels (continued)}}\\
\multicolumn{8}{c}{~}\\
\multicolumn{2}{c}{E(level)$^{{\hyperlink{HG14LEVEL0}{\dagger}}}$}&J$^{\pi}$$^{{\hyperlink{HG14LEVEL1}{\ddagger}}}$&L$^{{\hyperlink{HG14LEVEL0}{\dagger}}}$&\multicolumn{2}{c}{S$^{{\hyperlink{HG14LEVEL2}{\#}}}$}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endhead
\multicolumn{1}{r@{}}{4405}&\multicolumn{1}{@{}l}{}&&&&&&\\
\multicolumn{1}{r@{}}{4418?}&\multicolumn{1}{@{}l}{}&&&&&&\\
\multicolumn{1}{r@{}}{4467?}&\multicolumn{1}{@{}l}{}&&&&&&\\
\multicolumn{1}{r@{}}{4484}&\multicolumn{1}{@{}l}{}&&&&&&\\
\multicolumn{1}{r@{}}{4579}&\multicolumn{1}{@{}l}{}&&&&&&\\
\multicolumn{1}{r@{}}{4591}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{l}{(9/2\ensuremath{^{+}},5/2\ensuremath{^{+}})}&\multicolumn{1}{l}{(4,2)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{03}&\parbox[t][0.3cm]{12.27816cm}{\raggedright S: Value quoted for L=4. S=0.02 for L=2 (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1972Mo12,B}{1972Mo12}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{4649}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{l}{(9/2\ensuremath{^{+}},5/2\ensuremath{^{+}})}&\multicolumn{1}{l}{(4,2)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{04}&\parbox[t][0.3cm]{12.27816cm}{\raggedright S: Value quoted for L=4. S=0.02 for L=2 (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1972Mo12,B}{1972Mo12}).\vspace{0.1cm}}&\\
\end{longtable}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{HG14LEVEL0}{\dagger}}}} From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1972Mo12,B}{1972Mo12}. \ensuremath{\Delta}E=0.4\% for well-resolved peaks.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{HG14LEVEL1}{\ddagger}}}} From the deduced L values and spectroscopic factors (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1972Mo12,B}{1972Mo12}), unless otherwise stated.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{HG14LEVEL2}{\#}}}} \ensuremath{\Delta}S\ensuremath{\approx}\ensuremath{\pm}50\%. S=N*(d\ensuremath{\sigma}/d\ensuremath{\Omega})\ensuremath{_{\textnormal{expt}}}/(d\ensuremath{\sigma}/d\ensuremath{\Omega})\ensuremath{_{\textnormal{DWBA}}}. N=(1/1.5)/(2J+1).}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{HG14LEVEL3}{@}}}} Dominant configuration=\ensuremath{\nu} p\ensuremath{_{\textnormal{3/2}}^{\textnormal{$-$1}}}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{HG14LEVEL4}{\&}}}} Dominant configuration=\ensuremath{\nu} f\ensuremath{_{\textnormal{5/2}}^{\textnormal{$-$1}}}.}\\
\vspace{0.5cm}
\clearpage
\subsection[\hspace{-0.2cm}\ensuremath{^{\textnormal{201}}}Hg(\ensuremath{\gamma},\ensuremath{\gamma}\ensuremath{'})]{ }
\vspace{-27pt}
\vspace{0.3cm}
\hypertarget{HG15}{{\bf \small \underline{\ensuremath{^{\textnormal{201}}}Hg(\ensuremath{\gamma},\ensuremath{\gamma}\ensuremath{'})\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1971Wa17,B}{1971Wa17},\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2005Is19,B}{2005Is19},\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2018Yo02,B}{2018Yo02}}}}\\
\vspace{4pt}
\vspace{8pt}
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1971Wa17,B}{1971Wa17}: Mossbauer transmission measurement with 32.2-keV \ensuremath{\gamma} from \ensuremath{^{\textnormal{201}}}Tl \ensuremath{\varepsilon} decay source.}\\
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2005Is19,B}{2005Is19}: synchrotron-based nuclear resonant scattering experiment at the Spring-8 facility.}\\
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2018Yo02,B}{2018Yo02}: synchrotron-based nuclear resonant scattering experiment at the Spring-8 facility.}\\
\vspace{12pt}
\underline{$^{201}$Hg Levels}\\
\begin{longtable}{cccccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E(level)$^{{\hyperlink{HG15LEVEL0}{\dagger}}}$}&J$^{\pi}$$^{{\hyperlink{HG15LEVEL0}{\dagger}}}$&\multicolumn{2}{c}{T$_{1/2}$$^{}$}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{l}{3/2\ensuremath{^{-}}}&&&&\\
\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{5648 {\it 10}}&\multicolumn{1}{l}{1/2\ensuremath{^{-}}}&&&&\\
\multicolumn{1}{r@{}}{26}&\multicolumn{1}{@{.}l}{2738 {\it 3}}&\multicolumn{1}{l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{629}&\multicolumn{1}{@{ }l}{ps {\it 18}}&\parbox[t][0.3cm]{13.15324cm}{\raggedright E(level): From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2005Is19,B}{2005Is19}.\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{13.15324cm}{\raggedright T\ensuremath{_{1/2}}: From the time difference between the incident X-ray and the fluorescence signal from the\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{13.15324cm}{\raggedright {\ }{\ }{\ }\ensuremath{^{\textnormal{201}}}Hg atom in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2018Yo02,B}{2018Yo02}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{32}&\multicolumn{1}{@{.}l}{155 {\it 13}}&\multicolumn{1}{l}{3/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{$>$0}&\multicolumn{1}{@{.}l}{1 ns}&\parbox[t][0.3cm]{13.15324cm}{\raggedright T\ensuremath{_{1/2}}: From the line width in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1971Wa17,B}{1971Wa17}.\vspace{0.1cm}}&\\
\end{longtable}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{HG15LEVEL0}{\dagger}}}} From Adopted Levels, unless otherwise stated.}\\
\vspace{0.5cm}
\underline{$\gamma$($^{201}$Hg)}\\
\begin{longtable}{ccccccc@{}c@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{HG15GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{E\ensuremath{_{i}}(level)}&J\ensuremath{^{\pi}_{i}}&\multicolumn{2}{c}{E\ensuremath{_{f}}}&J\ensuremath{^{\pi}_{f}}&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{30}&\multicolumn{1}{@{.}l}{60 {\it 3}}&\multicolumn{1}{r@{}}{32}&\multicolumn{1}{@{.}l}{155}&\multicolumn{1}{l}{3/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{5648 }&\multicolumn{1}{@{}l}{1/2\ensuremath{^{-}}}&\\
\multicolumn{1}{r@{}}{32}&\multicolumn{1}{@{.}l}{19 {\it 3}}&\multicolumn{1}{r@{}}{32}&\multicolumn{1}{@{.}l}{155}&\multicolumn{1}{l}{3/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{3/2\ensuremath{^{-}}}&\\
\end{longtable}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{HG15GAMMA0}{\dagger}}}} From adopted gammas.}\\
\vspace{0.5cm}
\begin{figure}[h]
\begin{center}
\includegraphics{201HG15-0.ps}\\
\end{center}
\end{figure}
\clearpage
\subsection[\hspace{-0.2cm}\ensuremath{^{\textnormal{201}}}Hg(d,d\ensuremath{'}),\ensuremath{^{\textnormal{201}}}Hg(p,p\ensuremath{'})]{ }
\vspace{-27pt}
\vspace{0.3cm}
\hypertarget{HG16}{{\bf \small \underline{\ensuremath{^{\textnormal{201}}}Hg(d,d\ensuremath{'}),\ensuremath{^{\textnormal{201}}}Hg(p,p\ensuremath{'})\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1972Mo12,B}{1972Mo12}}}}\\
\vspace{4pt}
\vspace{8pt}
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}Beam: E(d)=17 MeV, \ensuremath{\theta}=90\ensuremath{^\circ}; \ensuremath{^{\textnormal{201}}}Hg(p,p\ensuremath{'}), \ensuremath{\theta}=75\ensuremath{^\circ},90\ensuremath{^\circ} Target: enriched \ensuremath{^{\textnormal{202}}}Hg, but isotopic purity is unknown; Detectors:}\\
\parbox[b][0.3cm]{17.7cm}{photographic emulsions, split-pole spectrograph, FWHM=8-9 keV.}\\
\vspace{12pt}
\underline{$^{201}$Hg Levels}\\
\begin{longtable}{ccc|ccc|ccc|cc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E(level)$^{{\hyperlink{HG16LEVEL0}{\dagger}}}$}&J$^{\pi}$$^{{\hyperlink{HG16LEVEL1}{\ddagger}}}$&\multicolumn{2}{c}{E(level)$^{{\hyperlink{HG16LEVEL0}{\dagger}}}$}&J$^{\pi}$$^{{\hyperlink{HG16LEVEL1}{\ddagger}}}$&\multicolumn{2}{c}{E(level)$^{{\hyperlink{HG16LEVEL0}{\dagger}}}$}&J$^{\pi}$$^{{\hyperlink{HG16LEVEL1}{\ddagger}}}$&\multicolumn{2}{c}{E(level)$^{{\hyperlink{HG16LEVEL0}{\dagger}}}$}&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&
\endfirsthead
\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{l|}{3/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{412}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{l|}{7/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{1707}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{l|}{(5/2\ensuremath{^{-}},7/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{2891?}&\multicolumn{1}{@{}l}{}&\\
\multicolumn{1}{r@{}}{$\approx$32}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{l|}{3/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{465}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{l|}{5/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{2526}&\multicolumn{1}{@{}l}{}&&\multicolumn{1}{r@{}}{3735}&\multicolumn{1}{@{}l}{}&\\
\multicolumn{1}{r@{}}{163}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{l|}{1/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{1325}&\multicolumn{1}{@{}l}{}&&\multicolumn{1}{r@{}}{2629}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{l|}{7/2\ensuremath{^{+}},9/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{3965}&\multicolumn{1}{@{}l}{}&\\
\multicolumn{1}{r@{}}{382}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{l|}{(5/2)\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{1505}&\multicolumn{1}{@{}l}{}&&\multicolumn{1}{r@{}}{2681}&\multicolumn{1}{@{}l}{}&&&&\\
\end{longtable}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{HG16LEVEL0}{\dagger}}}} From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1972Mo12,B}{1972Mo12}. \ensuremath{\Delta}E=0.4\% for well-resolved peaks.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{HG16LEVEL1}{\ddagger}}}} From Adopted Levels.}\\
\vspace{0.5cm}
\clearpage
\subsection[\hspace{-0.2cm}Coulomb excitation]{ }
\vspace{-27pt}
\vspace{0.3cm}
\hypertarget{HG17}{{\bf \small \underline{Coulomb excitation\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1980Bo05,B}{1980Bo05},\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1983Sc38,B}{1983Sc38}}}}\\
\vspace{4pt}
\vspace{8pt}
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1980Bo05,B}{1980Bo05}: (\ensuremath{^{\textnormal{16}}}O,\ensuremath{^{\textnormal{16}}}O\ensuremath{'}), E(\ensuremath{^{\textnormal{16}}}O)=35 to 64 MeV; (\ensuremath{\alpha},\ensuremath{\alpha}\ensuremath{'}), E(\ensuremath{^{\textnormal{4}}}He)=15 MeV. 81\% \ensuremath{^{\textnormal{201}}}Hg target; Detectors: Ge(Li); Measured: \ensuremath{\gamma}-ray}\\
\parbox[b][0.3cm]{17.7cm}{yield, \ensuremath{\gamma}(\ensuremath{\theta}); Deduced: B(E2), \ensuremath{\delta}.}\\
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1983Sc38,B}{1983Sc38}: (\ensuremath{\alpha},\ensuremath{\alpha}\ensuremath{'}), E(\ensuremath{^{\textnormal{4}}}He)=16 MeV magnetic spectrograph, FWHM=14 keV. B(E2) values for levels up to 167 keV measured;}\\
\parbox[b][0.3cm]{17.7cm}{normalization based on B(E2) values from \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1980Bo05,B}{1980Bo05} for the 414 and 464 levels.}\\
\vspace{12pt}
\underline{$^{201}$Hg Levels}\\
\begin{longtable}{cccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E(level)$^{{\hyperlink{HG17LEVEL0}{\dagger}}}$}&J$^{\pi}$$^{{\hyperlink{HG17LEVEL1}{\ddagger}}}$&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{l}{3/2\ensuremath{^{-}}\ensuremath{^{{\hyperlink{HG17LEVEL2}{\#}}}}}&&\\
\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{5648 {\it 10}}&\multicolumn{1}{l}{1/2\ensuremath{^{-}}\ensuremath{^{{\hyperlink{HG17LEVEL2}{\#}}}}}&&\\
\multicolumn{1}{r@{}}{26}&\multicolumn{1}{@{.}l}{2738 {\it 3}}&\multicolumn{1}{l}{5/2\ensuremath{^{-}}}&\parbox[t][0.3cm]{14.62752cm}{\raggedright B(E2)\ensuremath{\uparrow}\ensuremath{\leq}0.07 (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1983Sc38,B}{1983Sc38})\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{32}&\multicolumn{1}{@{.}l}{149 {\it 17}}&\multicolumn{1}{l}{3/2\ensuremath{^{-}}\ensuremath{^{{\hyperlink{HG17LEVEL2}{\#}}}}}&\parbox[t][0.3cm]{14.62752cm}{\raggedright B(E2)\ensuremath{\uparrow}=0.14 \textit{5} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1983Sc38,B}{1983Sc38})\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{167}&\multicolumn{1}{@{.}l}{43 {\it 5}}&\multicolumn{1}{l}{1/2\ensuremath{^{-}}\ensuremath{^{{\hyperlink{HG17LEVEL2}{\#}}}}}&\parbox[t][0.3cm]{14.62752cm}{\raggedright B(E2)\ensuremath{\uparrow}=0.014 \textit{4} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1980Bo05,B}{1980Bo05}); B(E2)\ensuremath{\uparrow}=0.017 \textit{4} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1983Sc38,B}{1983Sc38})\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{384}&\multicolumn{1}{@{.}l}{603 {\it 17}}&\multicolumn{1}{l}{5/2\ensuremath{^{-}}}&\parbox[t][0.3cm]{14.62752cm}{\raggedright B(E2)\ensuremath{\uparrow}=0.085 \textit{5} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1980Bo05,B}{1980Bo05})\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{414}&\multicolumn{1}{@{.}l}{541 {\it 19}}&\multicolumn{1}{l}{7/2\ensuremath{^{-}}}&\parbox[t][0.3cm]{14.62752cm}{\raggedright B(E2)\ensuremath{\uparrow}=0.152 \textit{6} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1980Bo05,B}{1980Bo05})\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{464}&\multicolumn{1}{@{.}l}{41 {\it 4}}&\multicolumn{1}{l}{5/2\ensuremath{^{-}}}&\parbox[t][0.3cm]{14.62752cm}{\raggedright B(E2)\ensuremath{\uparrow}=0.209 \textit{5} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1980Bo05,B}{1980Bo05})\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{553}&\multicolumn{1}{@{.}l}{04 {\it 6}}&&\parbox[t][0.3cm]{14.62752cm}{\raggedright B(E2)\ensuremath{\uparrow}=0.035 \textit{2} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1980Bo05,B}{1980Bo05})\vspace{0.1cm}}&\\
\end{longtable}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{HG17LEVEL0}{\dagger}}}} From a least-squares fit to E\ensuremath{\gamma}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{HG17LEVEL1}{\ddagger}}}} From \ensuremath{\gamma}(\ensuremath{\theta}) and direct excitation in Coulomb excitation (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1980Bo05,B}{1980Bo05}), unless otherwise stated.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{HG17LEVEL2}{\#}}}} From Adopted Levels.}\\
\vspace{0.5cm}
\underline{$\gamma$($^{201}$Hg)}\\
\begin{longtable}{ccccccccc@{}ccccccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E\ensuremath{_{i}}(level)}&J\ensuremath{^{\pi}_{i}}&\multicolumn{2}{c}{E\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{HG17GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{I\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{HG17GAMMA2}{\#}}}}&\multicolumn{2}{c}{E\ensuremath{_{f}}}&J\ensuremath{^{\pi}_{f}}&Mult.&\multicolumn{2}{c}{\ensuremath{\delta}\ensuremath{^{\hyperlink{HG17GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{\ensuremath{\alpha}\ensuremath{^{\hyperlink{HG17GAMMA3}{@}}}}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{5648}&\multicolumn{1}{l}{1/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{5648\ensuremath{^{\hyperlink{HG17GAMMA1}{\ddagger}}} {\it 10}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{3/2\ensuremath{^{-}}}&&&&&&&\\
\multicolumn{1}{r@{}}{26}&\multicolumn{1}{@{.}l}{2738}&\multicolumn{1}{l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{26}&\multicolumn{1}{@{.}l}{2738\ensuremath{^{\hyperlink{HG17GAMMA1}{\ddagger}}} {\it 3}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{3/2\ensuremath{^{-}}}&&&&&&&\\
\multicolumn{1}{r@{}}{32}&\multicolumn{1}{@{.}l}{149}&\multicolumn{1}{l}{3/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{30}&\multicolumn{1}{@{.}l}{60\ensuremath{^{\hyperlink{HG17GAMMA1}{\ddagger}}} {\it 3}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{5648 }&\multicolumn{1}{@{}l}{1/2\ensuremath{^{-}}}&&&&&&&\\
&&&\multicolumn{1}{r@{}}{32}&\multicolumn{1}{@{.}l}{19\ensuremath{^{\hyperlink{HG17GAMMA1}{\ddagger}}} {\it 3}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{3/2\ensuremath{^{-}}}&&&&&&&\\
\multicolumn{1}{r@{}}{167}&\multicolumn{1}{@{.}l}{43}&\multicolumn{1}{l}{1/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{167}&\multicolumn{1}{@{.}l}{43 {\it 5}}&\multicolumn{1}{r@{}}{$\approx$100}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{3/2\ensuremath{^{-}}}&&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
\multicolumn{1}{r@{}}{384}&\multicolumn{1}{@{.}l}{603}&\multicolumn{1}{l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{352}&\multicolumn{1}{@{.}l}{42 {\it 5}}&\multicolumn{1}{r@{}}{42}&\multicolumn{1}{@{.}l}{8 {\it 7}}&\multicolumn{1}{r@{}}{32}&\multicolumn{1}{@{.}l}{149 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1(+E2)}&\multicolumn{1}{r@{}}{+0}&\multicolumn{1}{@{.}l}{07 {\it 7}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{4.0cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}={\textminus}0.047 \textit{24}.\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{4.0cm}{\raggedright \ensuremath{\delta}: other (alternative):\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{4.0cm}{\raggedright {\ }{\ }{\ }\ensuremath{\delta}={\textminus}4.5 \textit{+1.2{\textminus}1.7}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{4.0cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1980Bo05,B}{1980Bo05}).\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{358}&\multicolumn{1}{@{.}l}{36 {\it 4}}&\multicolumn{1}{r@{}}{23}&\multicolumn{1}{@{.}l}{3 {\it 4}}&\multicolumn{1}{r@{}}{26}&\multicolumn{1}{@{.}l}{2738 }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{15 {\it 9}}&\parbox[t][0.3cm]{4.0cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}=+0.08 \textit{6}.\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{4.0cm}{\raggedright \ensuremath{\delta}: {\textminus}0.3\ensuremath{\leq}\ensuremath{\delta}\ensuremath{\leq}3.9 (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1980Bo05,B}{1980Bo05}).\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{384}&\multicolumn{1}{@{.}l}{60 {\it 2}}&\multicolumn{1}{r@{}}{33}&\multicolumn{1}{@{.}l}{9 {\it 6}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{3/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{$-$0}&\multicolumn{1}{@{.}l}{23 {\it 9}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{4.0cm}{\raggedright I\ensuremath{_{\gamma}}: Possibly a doublet. See\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{4.0cm}{\raggedright {\ }{\ }{\ }Adopted Levels for details.\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{4.0cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}={\textminus}0.144 \textit{22}.\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{4.0cm}{\raggedright \ensuremath{\delta}: other: alternative \ensuremath{\delta}={\textminus}1.8\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{4.0cm}{\raggedright {\ }{\ }{\ }\textit{3} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1980Bo05,B}{1980Bo05}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{414}&\multicolumn{1}{@{.}l}{541}&\multicolumn{1}{l}{7/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{382}&\multicolumn{1}{@{.}l}{45 {\it 3}}&\multicolumn{1}{r@{}}{30}&\multicolumn{1}{@{.}l}{4 {\it 5}}&\multicolumn{1}{r@{}}{32}&\multicolumn{1}{@{.}l}{149 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{-}}}&\multicolumn{1}{l}{E2}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0540}&\parbox[t][0.3cm]{4.0cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}=+0.202 \textit{26}.\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{388}&\multicolumn{1}{@{.}l}{26 {\it 3}}&\multicolumn{1}{r@{}}{32}&\multicolumn{1}{@{.}l}{9 {\it 5}}&\multicolumn{1}{r@{}}{26}&\multicolumn{1}{@{.}l}{2738 }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{+1}&\multicolumn{1}{@{.}l}{5 {\it +5\textminus7}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{09 {\it 4}}&\parbox[t][0.3cm]{4.0cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}=+0.38 \textit{3}.\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{414}&\multicolumn{1}{@{.}l}{49 {\it 3}}&\multicolumn{1}{r@{}}{36}&\multicolumn{1}{@{.}l}{7 {\it 7}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{3/2\ensuremath{^{-}}}&\multicolumn{1}{l}{E2}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0436}&\parbox[t][0.3cm]{4.0cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}=+0.187 \textit{24}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{464}&\multicolumn{1}{@{.}l}{41}&\multicolumn{1}{l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{432}&\multicolumn{1}{@{.}l}{32 {\it 7}}&\multicolumn{1}{r@{}}{4}&\multicolumn{1}{@{.}l}{2 {\it 2}}&\multicolumn{1}{r@{}}{32}&\multicolumn{1}{@{.}l}{149 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{+1}&\multicolumn{1}{@{.}l}{4 {\it 6}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{07 {\it 3}}&\parbox[t][0.3cm]{4.0cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}=+0.183 \textit{26}.\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{438}&\multicolumn{1}{@{.}l}{11 {\it 6}}&\multicolumn{1}{r@{}}{72}&\multicolumn{1}{@{.}l}{8 {\it 23}}&\multicolumn{1}{r@{}}{26}&\multicolumn{1}{@{.}l}{2738 }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1(+E2)}&\multicolumn{1}{r@{}}{$\leq$0}&\multicolumn{1}{@{.}l}{1}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{09 {\it 5}}&\parbox[t][0.3cm]{4.0cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}=+0.077 \textit{21}.\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{4.0cm}{\raggedright \ensuremath{\delta}: other: 1.8 \textit{5}, if \ensuremath{J^{\pi}}=7/2\ensuremath{^{-}}.\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{464}&\multicolumn{1}{@{.}l}{39 {\it 5}}&\multicolumn{1}{r@{}}{23}&\multicolumn{1}{@{.}l}{0 {\it 7}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{3/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{+1}&\multicolumn{1}{@{.}l}{4 {\it +13\textminus6}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{061 {\it 23}}&\parbox[t][0.3cm]{4.0cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}=+0.20 \textit{6}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{553}&\multicolumn{1}{@{.}l}{04}&&\multicolumn{1}{r@{}}{520}&\multicolumn{1}{@{.}l}{9 {\it 1}}&\multicolumn{1}{r@{}}{37}&\multicolumn{1}{@{ }l}{{\it 4}}&\multicolumn{1}{r@{}}{32}&\multicolumn{1}{@{.}l}{149 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{-}}}&&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
&&&\multicolumn{1}{r@{}}{526}&\multicolumn{1}{@{.}l}{8 {\it 1}}&\multicolumn{1}{r@{}}{31}&\multicolumn{1}{@{ }l}{{\it 2}}&\multicolumn{1}{r@{}}{26}&\multicolumn{1}{@{.}l}{2738 }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{-}}}&&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
&&&\multicolumn{1}{r@{}}{553}&\multicolumn{1}{@{.}l}{0 {\it 1}}&\multicolumn{1}{r@{}}{32}&\multicolumn{1}{@{ }l}{{\it 2}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{3/2\ensuremath{^{-}}}&&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
\end{longtable}
\begin{textblock}{29}(0,27.3)
Continued on next page (footnotes at end of table)
\end{textblock}
\clearpage
\vspace*{-0.5cm}
{\bf \small \underline{Coulomb excitation\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1980Bo05,B}{1980Bo05},\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1983Sc38,B}{1983Sc38} (continued)}}\\
\vspace{0.3cm}
\underline{$\gamma$($^{201}$Hg) (continued)}\\
\vspace{0.3cm}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{HG17GAMMA0}{\dagger}}}} From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1980Bo05,B}{1980Bo05}, unless otherwise stated.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{HG17GAMMA1}{\ddagger}}}} From adopted gammas.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{HG17GAMMA2}{\#}}}} Branching intensity from each level in \% from \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1980Bo05,B}{1980Bo05}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{HG17GAMMA3}{@}}}} Total theoretical internal conversion coefficients, calculated using the BrIcc code (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2008Ki07,B}{2008Ki07}) with Frozen orbital approximation}\\
\parbox[b][0.3cm]{17.7cm}{{\ }{\ }based on \ensuremath{\gamma}-ray energies, assigned multipolarities, and mixing ratios, unless otherwise specified.}\\
\vspace{0.5cm}
\clearpage
\begin{figure}[h]
\begin{center}
\includegraphics{201HG17-0.ps}\\
\end{center}
\end{figure}
\clearpage
\subsection[\hspace{-0.2cm}\ensuremath{^{\textnormal{202}}}Hg(d,t)]{ }
\vspace{-27pt}
\vspace{0.3cm}
\hypertarget{HG18}{{\bf \small \underline{\ensuremath{^{\textnormal{202}}}Hg(d,t)\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1972Mo12,B}{1972Mo12}}}}\\
\vspace{4pt}
\vspace{8pt}
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}Beam: E(d)=17 MeV; Target: enriched \ensuremath{^{\textnormal{202}}}Hg, but isotopic purity is unknown; Detectors: photographic emulsions, split-pole}\\
\parbox[b][0.3cm]{17.7cm}{spectrograph, FWHM=8-12 keV.}\\
\vspace{12pt}
\underline{$^{201}$Hg Levels}\\
\begin{longtable}{ccccccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E(level)$^{{\hyperlink{HG18LEVEL0}{\dagger}}}$}&J$^{\pi}$$^{{\hyperlink{HG18LEVEL1}{\ddagger}}}$&L$^{{\hyperlink{HG18LEVEL0}{\dagger}}}$&\multicolumn{2}{c}{S$^{{\hyperlink{HG18LEVEL2}{\#}}}$}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{HG18LEVEL3}{@}}}}}&\multicolumn{1}{l}{3/2\ensuremath{^{-}}}&\multicolumn{1}{l}{1}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{1}&\parbox[t][0.3cm]{12.2482cm}{\raggedright S: Probably includes strength to the 1.57 keV (\ensuremath{J^{\pi}}=1/2\ensuremath{^{-}}) level, that is unresolved in\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{12.2482cm}{\raggedright {\ }{\ }{\ }\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1972Mo12,B}{1972Mo12}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{27}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{HG18LEVEL4}{\&}}}}}&\multicolumn{1}{l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{l}{1+3}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{7}&\parbox[t][0.3cm]{12.2482cm}{\raggedright J\ensuremath{^{\pi}}: From Adopted Levels. S for L=3.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{32}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{l}{3/2\ensuremath{^{-}}}&\multicolumn{1}{l}{1+3}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{85}&\parbox[t][0.3cm]{12.2482cm}{\raggedright J\ensuremath{^{\pi}}: From Adopted Levels. S for L=1; S=0.87 in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1972Mo12,B}{1972Mo12}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{168}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{l}{1/2\ensuremath{^{-}}}&\multicolumn{1}{l}{1}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{6}&&\\
\multicolumn{1}{r@{}}{464}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{l}{3}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{2}&&\\
\multicolumn{1}{r@{}}{645}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{l}{3}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{4}&&\\
\multicolumn{1}{r@{}}{732}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{l}{3/2\ensuremath{^{-}}}&\multicolumn{1}{l}{1}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{92}&&\\
\multicolumn{1}{r@{}}{766}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{HG18LEVEL5}{a}}}}}&\multicolumn{1}{l}{13/2\ensuremath{^{+}}}&\multicolumn{1}{l}{\ensuremath{>}4}&&&\parbox[t][0.3cm]{12.2482cm}{\raggedright J\ensuremath{^{\pi}}: From Adopted Levels. S=10.0 is expected for L=6 (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1972Mo12,B}{1972Mo12}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{953}&\multicolumn{1}{@{}l}{}&&&&&&\\
\multicolumn{1}{r@{}}{1035}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{l}{3/2\ensuremath{^{-}}}&\multicolumn{1}{l}{1}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{25}&&\\
\multicolumn{1}{r@{}}{1075}&\multicolumn{1}{@{}l}{}&&&&&&\\
\multicolumn{1}{r@{}}{1287}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{l}{7/2\ensuremath{^{-}}}&\multicolumn{1}{l}{3}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{2}&&\\
\multicolumn{1}{r@{}}{1360}&\multicolumn{1}{@{}l}{}&&&&&&\\
\multicolumn{1}{r@{}}{1583}&\multicolumn{1}{@{}l}{}&&&&&&\\
\multicolumn{1}{r@{}}{1591}&\multicolumn{1}{@{}l}{}&&&&&&\\
\multicolumn{1}{r@{}}{1693}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{l}{7/2\ensuremath{^{-}}}&\multicolumn{1}{l}{3}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{1}&&\\
\multicolumn{1}{r@{}}{1710}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{l}{(7/2\ensuremath{^{-}})}&\multicolumn{1}{l}{(3)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{96}&&\\
\multicolumn{1}{r@{}}{1737}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{l}{3/2\ensuremath{^{-}}}&\multicolumn{1}{l}{1}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{23}&&\\
\multicolumn{1}{r@{}}{1971}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{l}{(3/2\ensuremath{^{-}})}&\multicolumn{1}{l}{(1)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{11}&&\\
\multicolumn{1}{r@{}}{2037}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{l}{(7/2\ensuremath{^{-}},11/2\ensuremath{^{-}})}&\multicolumn{1}{l}{(3,5)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{28}&\parbox[t][0.3cm]{12.2482cm}{\raggedright S: Value quoted for L=3. S=2.8 for L=5 (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1972Mo12,B}{1972Mo12}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2096}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{l}{(7/2\ensuremath{^{-}},11/2\ensuremath{^{-}})}&\multicolumn{1}{l}{(3,5)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{35}&\parbox[t][0.3cm]{12.2482cm}{\raggedright S: Value quoted for L=3. S=4.1 for L=5 (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1972Mo12,B}{1972Mo12}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2478}&\multicolumn{1}{@{}l}{}&&&&&&\\
\multicolumn{1}{r@{}}{2663?}&\multicolumn{1}{@{}l}{}&&&&&&\\
\end{longtable}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{HG18LEVEL0}{\dagger}}}} From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1972Mo12,B}{1972Mo12}. \ensuremath{\Delta}E=0.4\% for well-resolved peaks.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{HG18LEVEL1}{\ddagger}}}} From the deduced L values and spectroscopic factors (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1972Mo12,B}{1972Mo12}), unless otherwise specified.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{HG18LEVEL2}{\#}}}} \ensuremath{\Delta}S\ensuremath{\approx}\ensuremath{\pm}50\%. S=N*(d\ensuremath{\sigma}/d\ensuremath{\Omega})\ensuremath{_{\textnormal{expt}}}/(d\ensuremath{\sigma}/d\ensuremath{\Omega})\ensuremath{_{\textnormal{DWBA}}}. N=1/3.33.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{HG18LEVEL3}{@}}}} Dominant configuration=\ensuremath{\nu} p\ensuremath{_{\textnormal{3/2}}^{\textnormal{$-$1}}}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{HG18LEVEL4}{\&}}}} Dominant configuration=\ensuremath{\nu} f\ensuremath{_{\textnormal{5/2}}^{\textnormal{$-$1}}}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{HG18LEVEL5}{a}}}} Configuration=\ensuremath{\nu}i\ensuremath{_{\textnormal{13/2}}^{\textnormal{$-$1}}}.}\\
\vspace{0.5cm}
\clearpage
\subsection[\hspace{-0.2cm}\ensuremath{^{\textnormal{203}}}Tl(\ensuremath{\mu},X\ensuremath{\gamma})]{ }
\vspace{-27pt}
\vspace{0.3cm}
\hypertarget{HG19}{{\bf \small \underline{\ensuremath{^{\textnormal{203}}}Tl(\ensuremath{\mu},X\ensuremath{\gamma})\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1972Ba53,B}{1972Ba53}}}}\\
\vspace{4pt}
\vspace{8pt}
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}Target: natural Tl; Detectors: Ge(Li); Measured: delayed \ensuremath{\gamma}$'$s in muonic Tl; E\ensuremath{\gamma}, I\ensuremath{\gamma}.}\\
\vspace{12pt}
\underline{$^{201}$Hg Levels}\\
\begin{longtable}{ccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E(level)$^{{\hyperlink{HG19LEVEL0}{\dagger}}}$}&J$^{\pi}$$^{{\hyperlink{HG19LEVEL1}{\ddagger}}}$&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{l}{3/2\ensuremath{^{-}}}&\\
\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{5648\ensuremath{^{{\hyperlink{HG19LEVEL1}{\ddagger}}}} {\it 10}}&\multicolumn{1}{l}{1/2\ensuremath{^{-}}}&\\
\multicolumn{1}{r@{}}{26}&\multicolumn{1}{@{.}l}{2738\ensuremath{^{{\hyperlink{HG19LEVEL1}{\ddagger}}}} {\it 3}}&\multicolumn{1}{l}{5/2\ensuremath{^{-}}}&\\
\multicolumn{1}{r@{}}{32}&\multicolumn{1}{@{.}l}{155\ensuremath{^{{\hyperlink{HG19LEVEL1}{\ddagger}}}} {\it 13}}&\multicolumn{1}{l}{3/2\ensuremath{^{-}}}&\\
\multicolumn{1}{r@{}}{414}&\multicolumn{1}{@{.}l}{9 {\it 4}}&\multicolumn{1}{l}{7/2\ensuremath{^{-}}}&\\
\multicolumn{1}{r@{}}{542}&\multicolumn{1}{@{.}l}{8 {\it 3}}&\multicolumn{1}{l}{1/2\ensuremath{^{-}},3/2,5/2}&\\
\end{longtable}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{HG19LEVEL0}{\dagger}}}} From a least-squares fit to E\ensuremath{\gamma}, unless otherwise stated.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{HG19LEVEL1}{\ddagger}}}} From Adopted Levels.}\\
\vspace{0.5cm}
\underline{$\gamma$($^{201}$Hg)}\\
\begin{longtable}{ccccccccc@{}c@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{HG19GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{I\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{HG19GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{E\ensuremath{_{i}}(level)}&J\ensuremath{^{\pi}_{i}}&\multicolumn{2}{c}{E\ensuremath{_{f}}}&J\ensuremath{^{\pi}_{f}}&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{388}&\multicolumn{1}{@{.}l}{17 {\it 23}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{76 {\it 19}}&\multicolumn{1}{r@{}}{414}&\multicolumn{1}{@{.}l}{9}&\multicolumn{1}{l}{7/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{26}&\multicolumn{1}{@{.}l}{2738 }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{-}}}&\\
\multicolumn{1}{r@{}}{414}&\multicolumn{1}{@{.}l}{86 {\it 36}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{68 {\it 22}}&\multicolumn{1}{r@{}}{414}&\multicolumn{1}{@{.}l}{9}&\multicolumn{1}{l}{7/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{3/2\ensuremath{^{-}}}&\\
\multicolumn{1}{r@{}}{542}&\multicolumn{1}{@{.}l}{76 {\it 31}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{78 {\it 22}}&\multicolumn{1}{r@{}}{542}&\multicolumn{1}{@{.}l}{8}&\multicolumn{1}{l}{1/2\ensuremath{^{-}},3/2,5/2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{3/2\ensuremath{^{-}}}&\\
\end{longtable}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{HG19GAMMA0}{\dagger}}}} From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1972Ba53,B}{1972Ba53}. I\ensuremath{\gamma} is per 100 \ensuremath{\mu}\ensuremath{^{-}} stopped in natural Tl. Assignment to \ensuremath{^{\textnormal{201}}}Hg was made by the evaluator based on Adopted Levels}\\
\parbox[b][0.3cm]{17.7cm}{{\ }{\ }levels and gammas properties and on syst of structures populated via (\ensuremath{\mu},xn\ensuremath{\gamma}) reactions in neighboring nuclei.}\\
\vspace{0.5cm}
\clearpage
\begin{figure}[h]
\begin{center}
\includegraphics{201HG19-0.ps}\\
\end{center}
\end{figure}
\clearpage
\section[\ensuremath{^{201}_{\ 81}}Tl\ensuremath{_{120}^{~}}]{ }
\vspace{-30pt}
\setcounter{chappage}{1}
\subsection[\hspace{-0.2cm}Adopted Levels, Gammas]{ }
\vspace{-20pt}
\vspace{0.3cm}
\hypertarget{TL20}{{\bf \small \underline{Adopted \hyperlink{201TL_LEVEL}{Levels}, \hyperlink{201TL_GAMMA}{Gammas}}}}\\
\vspace{4pt}
\vspace{8pt}
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}Q(\ensuremath{\beta^-})=$-$1910 {\it 19}; S(n)=8205 {\it 15}; S(p)=4966 {\it 14}; Q(\ensuremath{\alpha})=1534 {\it 14}\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2021Wa16,B}{2021Wa16}}\\
\vspace{12pt}
\hypertarget{201TL_LEVEL}{\underline{$^{201}$Tl Levels}}\\
\begin{longtable}[c]{llll}
\multicolumn{4}{c}{\underline{Cross Reference (XREF) Flags}}\\
\\
\hyperlink{PB21}{\texttt{A }}& \ensuremath{^{\textnormal{201}}}Pb \ensuremath{\varepsilon} decay & \hyperlink{TL25}{\texttt{E }}& \ensuremath{^{\textnormal{9}}}Be(\ensuremath{^{\textnormal{238}}}U,X\ensuremath{\gamma})\\
\hyperlink{TL22}{\texttt{B }}& \ensuremath{^{\textnormal{201}}}Tl IT decay & \hyperlink{TL26}{\texttt{F }}& \ensuremath{^{\textnormal{203}}}Tl(p,t)\\
\hyperlink{TL23}{\texttt{C }}& \ensuremath{^{\textnormal{202}}}Hg(d,3n\ensuremath{\gamma}) & \hyperlink{TL27}{\texttt{G }}& \ensuremath{^{\textnormal{204}}}Pb(p,\ensuremath{\alpha})\\
\hyperlink{TL24}{\texttt{D }}& \ensuremath{^{\textnormal{198}}}Pt(\ensuremath{^{\textnormal{7}}}Li,4n\ensuremath{\gamma}) & \hyperlink{TL28}{\texttt{H }}& \ensuremath{^{\textnormal{207}}}Pb(\ensuremath{\mu},X\ensuremath{\gamma})\\
\end{longtable}
\vspace{-0.5cm}
\begin{longtable}{ccccccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E(level)$^{{\hyperlink{TL20LEVEL0}{\dagger}}}$}&J$^{\pi}$$^{}$&\multicolumn{2}{c}{T$_{1/2}$$^{}$}&XREF&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{TL20LEVEL3}{@}}}}}&\multicolumn{1}{l}{1/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{0421 d {\it 8}}&\multicolumn{1}{l}{\texttt{\hyperlink{PB21}{A}\hyperlink{TL22}{B}\hyperlink{TL23}{C}\hyperlink{TL24}{D}\hyperlink{TL25}{E}\hyperlink{TL26}{F}\hyperlink{TL27}{G}\hyperlink{TL28}{H}} }&\parbox[t][0.3cm]{10.161421cm}{\raggedright \%\ensuremath{\varepsilon}=100\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.161421cm}{\raggedright \ensuremath{\mu}=+1.605 \textit{2} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1987Bo44,B}{1987Bo44},\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2014StZZ,B}{2014StZZ})\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.161421cm}{\raggedright J\ensuremath{^{\pi}}: Atomic beam (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1958Li45,B}{1958Li45},\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1958Ma21,B}{1958Ma21}); L(p,t)=0; \ensuremath{\mu}.\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.161421cm}{\raggedright T\ensuremath{_{1/2}}: Weighted average of 3.0408 d \textit{13} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1982La25,B}{1982La25}, original uncertainty of\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.161421cm}{\raggedright {\ }{\ }{\ }\textit{0.0040} d is 3\ensuremath{\sigma}), 3.0400 d \textit{28} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1994Si26,B}{1994Si26}), 3.0380 d \textit{17} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2004De02,B}{2004De02}),\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.161421cm}{\raggedright {\ }{\ }{\ }3.0486 d \textit{30} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2004Sc04,B}{2004Sc04}, supersedes 3.0380 d \textit{7} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1979De42,B}{1979De42}) and 3.043 d\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.161421cm}{\raggedright {\ }{\ }{\ }\textit{3} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1989Sc17,B}{1989Sc17})) and 3.0456 d \textit{15} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2014Un01,B}{2014Un01}, corrected in 2020,\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.161421cm}{\raggedright {\ }{\ }{\ }supersedes 3.0447 d \textit{9} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1982HoZJ,B}{1982HoZJ}), 3.0456 d \textit{15}\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.161421cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1992Un01,B}{1992Un01},\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2002Un02,B}{2002Un02},\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2012Fi12,B}{2012Fi12}) and 3.046 d \textit{6} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2014Un01,B}{2014Un01})). Others:\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.161421cm}{\raggedright {\ }{\ }{\ }3.00 d \textit{13} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1950Ne77,B}{1950Ne77}) and 3.063 d \textit{33} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1960He05,B}{1960He05}). A value of 3.0422 d\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.161421cm}{\raggedright {\ }{\ }{\ }\textit{17} was evaluated in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2004Wo02,B}{2004Wo02}.\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.161421cm}{\raggedright \ensuremath{\mu}: Using collinear fast beam laser spectroscopy technique. Other: 1.60 \textit{7}\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.161421cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1984Be40,B}{1984Be40},\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2014StZZ,B}{2014StZZ}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{331}&\multicolumn{1}{@{.}l}{17\ensuremath{^{{\hyperlink{TL20LEVEL4}{\&}}}} {\it 3}}&\multicolumn{1}{l}{3/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{70}&\multicolumn{1}{@{ }l}{ps {\it 20}}&\multicolumn{1}{l}{\texttt{\hyperlink{PB21}{A}\hyperlink{TL22}{B}\hyperlink{TL23}{C}\hyperlink{TL24}{D}\hyperlink{TL25}{E}\hyperlink{TL26}{F}\hyperlink{TL27}{G}\hyperlink{TL28}{H}} }&\parbox[t][0.3cm]{10.161421cm}{\raggedright XREF: F(330)G(334).\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.161421cm}{\raggedright T\ensuremath{_{1/2}}: From 360ce-331ce(\ensuremath{\Delta}t) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1960Li08,B}{1960Li08}.\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.161421cm}{\raggedright J\ensuremath{^{\pi}}: L(p,t)=2; 331.15\ensuremath{\gamma} M1+E2 to 1/2\ensuremath{^{+}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{692}&\multicolumn{1}{@{.}l}{52\ensuremath{^{{\hyperlink{TL20LEVEL5}{a}}}} {\it 4}}&\multicolumn{1}{l}{5/2\ensuremath{^{+}}}&&&\multicolumn{1}{l}{\texttt{\hyperlink{PB21}{A}\ \hyperlink{TL23}{C}\ \ \hyperlink{TL26}{F}\hyperlink{TL27}{G}\ } }&\parbox[t][0.3cm]{10.161421cm}{\raggedright XREF: F(695)G(699).\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.161421cm}{\raggedright J\ensuremath{^{\pi}}: L(p,t)=2; 692.41\ensuremath{\gamma} E2 to 1/2\ensuremath{^{+}}; \ensuremath{\gamma}\ensuremath{\gamma}(\ensuremath{\theta}) in \ensuremath{^{\textnormal{201}}}Pb \ensuremath{\varepsilon} decay.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{919}&\multicolumn{1}{@{.}l}{47 {\it 11}}&\multicolumn{1}{l}{(9/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{11 ms {\it 11}}&\multicolumn{1}{l}{\texttt{\ \hyperlink{TL22}{B}\hyperlink{TL23}{C}\hyperlink{TL24}{D}\hyperlink{TL25}{E}\ \ \ } }&\parbox[t][0.3cm]{10.161421cm}{\raggedright \%IT=100\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.161421cm}{\raggedright J\ensuremath{^{\pi}}: 588.3\ensuremath{\gamma} to 3/2\ensuremath{^{+}}; systematics of similar isomers in \ensuremath{^{\textnormal{197,199}}}Tl.\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.161421cm}{\raggedright T\ensuremath{_{1/2}}: Unweighted average of 1.8 ms \textit{1} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1962Mo19,B}{1962Mo19}), 2.3 ms \textit{2}\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.161421cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1963De38,B}{1963De38}), 2.1 ms \textit{2} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1964Br27,B}{1964Br27}), 1.8 ms \textit{1} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1965Gr04,B}{1965Gr04}), 2.65 ms \textit{20}\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.161421cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1967Co20,B}{1967Co20}), 2.1 ms \textit{1} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1975Uy01,B}{1975Uy01}) and 2.035 ms \textit{7} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1977KoZH,B}{1977KoZH}).\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.161421cm}{\raggedright {\ }{\ }{\ }Others: \ensuremath{>}60 ns \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1977Sl01,B}{1977Sl01}.\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.161421cm}{\raggedright configuration: \ensuremath{\pi} 9/2[505] (h\ensuremath{_{\textnormal{9/2}}}) Nilsson orbital at oblate deformation.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1098}&\multicolumn{1}{@{.}l}{50 {\it 4}}&\multicolumn{1}{l}{5/2\ensuremath{^{+}}}&&&\multicolumn{1}{l}{\texttt{\hyperlink{PB21}{A}\ \ \ \ \hyperlink{TL26}{F}\ \hyperlink{TL28}{H}} }&\parbox[t][0.3cm]{10.161421cm}{\raggedright XREF: F(1106).\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.161421cm}{\raggedright J\ensuremath{^{\pi}}: L(p,t)=2; 1098.52\ensuremath{\gamma} E2 to 1/2\ensuremath{^{+}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1134}&\multicolumn{1}{@{.}l}{86 {\it 6}}&\multicolumn{1}{l}{7/2\ensuremath{^{+}}}&&&\multicolumn{1}{l}{\texttt{\hyperlink{PB21}{A}\ \hyperlink{TL23}{C}\ \ \ \hyperlink{TL27}{G}\ } }&\parbox[t][0.3cm]{10.161421cm}{\raggedright XREF: G(1131).\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.161421cm}{\raggedright J\ensuremath{^{\pi}}: 803.66\ensuremath{\gamma} E2 to 3/2\ensuremath{^{+}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1157}&\multicolumn{1}{@{.}l}{43 {\it 4}}&\multicolumn{1}{l}{3/2\ensuremath{^{+}},5/2\ensuremath{^{+}}}&&&\multicolumn{1}{l}{\texttt{\hyperlink{PB21}{A}\ \ \ \ \hyperlink{TL26}{F}\ \ } }&\parbox[t][0.3cm]{10.161421cm}{\raggedright XREF: F(1159).\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.161421cm}{\raggedright J\ensuremath{^{\pi}}: 826.26\ensuremath{\gamma} M1+E2 to 3/2\ensuremath{^{+}}; log{} \textit{ft} in \ensuremath{^{\textnormal{201}}}Pb \ensuremath{\varepsilon} decay excludes 1/2\ensuremath{^{+}}.\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.161421cm}{\raggedright {\ }{\ }{\ }Note that L(p,t)=(0) is inconsistent with the adopted here assignment.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1238}&\multicolumn{1}{@{.}l}{46\ensuremath{^{{\hyperlink{TL20LEVEL6}{b}}}} {\it 15}}&\multicolumn{1}{l}{(11/2\ensuremath{^{-}})}&&&\multicolumn{1}{l}{\texttt{\ \ \hyperlink{TL23}{C}\hyperlink{TL24}{D}\hyperlink{TL25}{E}\ \ \ } }&\parbox[t][0.3cm]{10.161421cm}{\raggedright J\ensuremath{^{\pi}}: 319.0\ensuremath{\gamma} M1+E2 to (9/2\ensuremath{^{-}}). \ensuremath{\gamma}(\ensuremath{\theta}) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1977Sl01,B}{1977Sl01} does not support \ensuremath{\Delta}J=0\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.161421cm}{\raggedright {\ }{\ }{\ }assignment, and hence, 9/2\ensuremath{^{-}} possibility can be excluded. Strong\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.161421cm}{\raggedright {\ }{\ }{\ }population of this level in \ensuremath{^{\textnormal{202}}}Hg(d,3n\ensuremath{\gamma}) would favor 11/2\ensuremath{^{-}} compared to\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.161421cm}{\raggedright {\ }{\ }{\ }7/2\ensuremath{^{-}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1238}&\multicolumn{1}{@{.}l}{83 {\it 5}}&\multicolumn{1}{l}{3/2\ensuremath{^{+}}}&&&\multicolumn{1}{l}{\texttt{\hyperlink{PB21}{A}\ \ \ \ \hyperlink{TL26}{F}\ \ } }&\parbox[t][0.3cm]{10.161421cm}{\raggedright J\ensuremath{^{\pi}}: L(p,t)=(2); 546.28\ensuremath{\gamma} M1+E2 to 5/2\ensuremath{^{+}}, 1238.76\ensuremath{\gamma} M1+E2 to 1/2\ensuremath{^{+}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1277}&\multicolumn{1}{@{.}l}{12 {\it 4}}&\multicolumn{1}{l}{3/2\ensuremath{^{+}},5/2\ensuremath{^{+}}}&&&\multicolumn{1}{l}{\texttt{\hyperlink{PB21}{A}\ \ \ \ \ \ \ } }&\parbox[t][0.3cm]{10.161421cm}{\raggedright J\ensuremath{^{\pi}}: 945.96\ensuremath{\gamma} M1+E2 to 3/2\ensuremath{^{+}}, 1277.11\ensuremath{\gamma} to 1/2\ensuremath{^{+}}; log{} \textit{ft} in \ensuremath{^{\textnormal{201}}}Pb \ensuremath{\varepsilon} decay\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.161421cm}{\raggedright {\ }{\ }{\ }excludes 1/2\ensuremath{^{+}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1290}&\multicolumn{1}{@{.}l}{13 {\it 7}}&\multicolumn{1}{l}{(9/2)\ensuremath{^{+}}}&&&\multicolumn{1}{l}{\texttt{\hyperlink{PB21}{A}\ \hyperlink{TL23}{C}\ \ \hyperlink{TL26}{F}\hyperlink{TL27}{G}\ } }&\parbox[t][0.3cm]{10.161421cm}{\raggedright XREF: F(1294)G(1286).\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.161421cm}{\raggedright J\ensuremath{^{\pi}}: L(p,t)=4; 597.60\ensuremath{\gamma} E2 to 5/2\ensuremath{^{+}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1330}&\multicolumn{1}{@{.}l}{42 {\it 5}}&\multicolumn{1}{l}{3/2\ensuremath{^{+}},5/2\ensuremath{^{+}}}&&&\multicolumn{1}{l}{\texttt{\hyperlink{PB21}{A}\ \ \ \ \hyperlink{TL26}{F}\hyperlink{TL27}{G}\ } }&\parbox[t][0.3cm]{10.161421cm}{\raggedright XREF: F(1335)G(1331).\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.161421cm}{\raggedright J\ensuremath{^{\pi}}: L(p,t)=2; log{} \textit{ft} in \ensuremath{^{\textnormal{201}}}Pb \ensuremath{\varepsilon} decay rules out 1/2\ensuremath{^{+}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1401}&\multicolumn{1}{@{.}l}{26 {\it 5}}&\multicolumn{1}{l}{3/2\ensuremath{^{+}},5/2\ensuremath{^{+}}}&&&\multicolumn{1}{l}{\texttt{\hyperlink{PB21}{A}\ \ \ \ \ \ \ } }&\parbox[t][0.3cm]{10.161421cm}{\raggedright J\ensuremath{^{\pi}}: 708.75\ensuremath{\gamma} M1 to 5/2\ensuremath{^{+}}, 1401.30\ensuremath{\gamma} to 1/2\ensuremath{^{+}}.\vspace{0.1cm}}&\\
\end{longtable}
\begin{textblock}{29}(0,27.3)
Continued on next page (footnotes at end of table)
\end{textblock}
\clearpage
\begin{longtable}{ccccccc@{\extracolsep{\fill}}c}
\\[-.4cm]
\multicolumn{8}{c}{{\bf \small \underline{Adopted \hyperlink{201TL_LEVEL}{Levels}, \hyperlink{201TL_GAMMA}{Gammas} (continued)}}}\\
\multicolumn{8}{c}{~}\\
\multicolumn{8}{c}{\underline{\ensuremath{^{201}}Tl Levels (continued)}}\\
\multicolumn{8}{c}{~}\\
\multicolumn{2}{c}{E(level)$^{{\hyperlink{TL20LEVEL0}{\dagger}}}$}&J$^{\pi}$$^{}$&\multicolumn{2}{c}{T$_{1/2}$$^{}$}&XREF&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&
\endhead
\multicolumn{1}{r@{}}{1413}&\multicolumn{1}{@{.}l}{43 {\it 12}}&\multicolumn{1}{l}{11/2}&&&\multicolumn{1}{l}{\texttt{\ \ \hyperlink{TL23}{C}\ \ \ \hyperlink{TL27}{G}\ } }&\parbox[t][0.3cm]{9.994501cm}{\raggedright XREF: G(1419).\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{9.994501cm}{\raggedright J\ensuremath{^{\pi}}: 493.9\ensuremath{\gamma} D to (9/2\ensuremath{^{-}}); \ensuremath{\gamma}(\ensuremath{\theta}) in \ensuremath{^{\textnormal{202}}}Hg(d,3n\ensuremath{\gamma}) does not support \ensuremath{\Delta}J=0\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{9.994501cm}{\raggedright {\ }{\ }{\ }assignment, and hence, J=9/2 possibility can be excluded. Strong\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{9.994501cm}{\raggedright {\ }{\ }{\ }population of this level in \ensuremath{^{\textnormal{202}}}Hg(d,3n\ensuremath{\gamma}) would favor J=11/2 over 7/2.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1420}&\multicolumn{1}{@{.}l}{04 {\it 6}}&\multicolumn{1}{l}{7/2\ensuremath{^{+}}}&&&\multicolumn{1}{l}{\texttt{\hyperlink{PB21}{A}\ \ \ \ \hyperlink{TL26}{F}\ \ } }&\parbox[t][0.3cm]{9.994501cm}{\raggedright XREF: F(1423).\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{9.994501cm}{\raggedright J\ensuremath{^{\pi}}: L(p,t)=4; 1088.85\ensuremath{\gamma} to 3/2\ensuremath{^{+}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1445}&\multicolumn{1}{@{.}l}{87 {\it 6}}&\multicolumn{1}{l}{(5/2)\ensuremath{^{+}}}&&&\multicolumn{1}{l}{\texttt{\hyperlink{PB21}{A}\ \ \ \ \hyperlink{TL26}{F}\ \ } }&\parbox[t][0.3cm]{9.994501cm}{\raggedright XREF: F(1440).\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{9.994501cm}{\raggedright J\ensuremath{^{\pi}}: L(p,t)=2; 753.35\ensuremath{\gamma} (E0+M1) to 5/2\ensuremath{^{+}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1479}&\multicolumn{1}{@{.}l}{85 {\it 4}}&\multicolumn{1}{l}{5/2\ensuremath{^{+}}}&&&\multicolumn{1}{l}{\texttt{\hyperlink{PB21}{A}\ \ \ \ \hyperlink{TL26}{F}\ \ } }&\parbox[t][0.3cm]{9.994501cm}{\raggedright XREF: F(1472).\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{9.994501cm}{\raggedright J\ensuremath{^{\pi}}: 344.95\ensuremath{\gamma} M1(+E2) to 7/2\ensuremath{^{+}}, 1148.75\ensuremath{\gamma} M1+E2 to 3/2\ensuremath{^{+}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1550}&\multicolumn{1}{@{.}l}{59? {\it 15}}&\multicolumn{1}{l}{1/2,3/2,5/2\ensuremath{^{+}}}&&&\multicolumn{1}{l}{\texttt{\hyperlink{PB21}{A}\ \ \ \ \ \ \ } }&\parbox[t][0.3cm]{9.994501cm}{\raggedright J\ensuremath{^{\pi}}: 1219.4\ensuremath{\gamma} to 3/2\ensuremath{^{+}}, 1550.5\ensuremath{\gamma} to 1/2\ensuremath{^{+}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1571}&\multicolumn{1}{@{.}l}{86 {\it 17}}&\multicolumn{1}{l}{(13/2\ensuremath{^{-}})}&&&\multicolumn{1}{l}{\texttt{\ \ \hyperlink{TL23}{C}\hyperlink{TL24}{D}\hyperlink{TL25}{E}\ \ \ } }&\parbox[t][0.3cm]{9.994501cm}{\raggedright J\ensuremath{^{\pi}}: 333.4\ensuremath{\gamma} M1+E2 to (11/2\ensuremath{^{-}}), 652.0\ensuremath{\gamma} E2 to (9/2\ensuremath{^{-}}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1575}&\multicolumn{1}{@{.}l}{1 {\it 10}}&\multicolumn{1}{l}{(7/2)\ensuremath{^{+}}}&&&\multicolumn{1}{l}{\texttt{\hyperlink{PB21}{A}\ \ \ \ \hyperlink{TL26}{F}\hyperlink{TL27}{G}\ } }&\parbox[t][0.3cm]{9.994501cm}{\raggedright XREF: F(1572)G(1579).\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{9.994501cm}{\raggedright J\ensuremath{^{\pi}}: L(p,t)=4; log{} \textit{ft} in \ensuremath{^{\textnormal{201}}}Pb \ensuremath{\varepsilon} decay excludes J=9/2.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1617}&\multicolumn{1}{@{.}l}{46 {\it 15}}&\multicolumn{1}{l}{1/2,3/2,5/2\ensuremath{^{+}}}&&&\multicolumn{1}{l}{\texttt{\hyperlink{PB21}{A}\ \ \ \ \ \ \ } }&\parbox[t][0.3cm]{9.994501cm}{\raggedright J\ensuremath{^{\pi}}: 1286.3\ensuremath{\gamma} to 3/2\ensuremath{^{+}}, 1617.45\ensuremath{\gamma} to 1/2\ensuremath{^{+}};\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1639}&\multicolumn{1}{@{.}l}{36 {\it 4}}&\multicolumn{1}{l}{3/2\ensuremath{^{+}},5/2\ensuremath{^{+}}}&&&\multicolumn{1}{l}{\texttt{\hyperlink{PB21}{A}\ \ \ \ \hyperlink{TL26}{F}\hyperlink{TL27}{g}\ } }&\parbox[t][0.3cm]{9.994501cm}{\raggedright XREF: F(1636)g(1655).\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{9.994501cm}{\raggedright J\ensuremath{^{\pi}}: 1308.32\ensuremath{\gamma} M1(+E2) to 3/2\ensuremath{^{+}}; log{} \textit{ft} in \ensuremath{^{\textnormal{201}}}Pb \ensuremath{\varepsilon} decay rules out 1/2\ensuremath{^{+}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1671}&\multicolumn{1}{@{.}l}{96 {\it 5}}&\multicolumn{1}{l}{3/2\ensuremath{^{+}},5/2\ensuremath{^{+}}}&&&\multicolumn{1}{l}{\texttt{\hyperlink{PB21}{A}\ \ \ \ \hyperlink{TL26}{f}\hyperlink{TL27}{g}\ } }&\parbox[t][0.3cm]{9.994501cm}{\raggedright XREF: f(1699)g(1655).\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{9.994501cm}{\raggedright J\ensuremath{^{\pi}}: 394.86\ensuremath{\gamma} M1(+E2) to 3/2\ensuremath{^{+}},5/2\ensuremath{^{+}}, 1672.02\ensuremath{\gamma} to 1/2\ensuremath{^{+}}; log{} \textit{ft} in \ensuremath{^{\textnormal{201}}}Pb\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{9.994501cm}{\raggedright {\ }{\ }{\ }\ensuremath{\varepsilon} decay rules out 1/2\ensuremath{^{+}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1712}&\multicolumn{1}{@{.}l}{45 {\it 22}}&\multicolumn{1}{l}{3/2,5/2,7/2\ensuremath{^{+}}}&&&\multicolumn{1}{l}{\texttt{\hyperlink{PB21}{A}\ \ \ \ \hyperlink{TL26}{f}\ \ } }&\parbox[t][0.3cm]{9.994501cm}{\raggedright XREF: f(1699).\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{9.994501cm}{\raggedright J\ensuremath{^{\pi}}: 1019.8\ensuremath{\gamma} to 5/2\ensuremath{^{+}}, 1381.4\ensuremath{\gamma} to 3/2\ensuremath{^{+}}; log{} \textit{ft} in \ensuremath{^{\textnormal{201}}}Pb \ensuremath{\varepsilon} decay rules\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{9.994501cm}{\raggedright {\ }{\ }{\ }out J=1/2.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1725}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{TL20LEVEL1}{\ddagger}}}} {\it 15}}&\multicolumn{1}{l}{(7/2,9/2)}&&&\multicolumn{1}{l}{\texttt{\ \ \ \ \ \hyperlink{TL26}{F}\hyperlink{TL27}{G}\ } }&\parbox[t][0.3cm]{9.994501cm}{\raggedright XREF: F(1729).\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{9.994501cm}{\raggedright J\ensuremath{^{\pi}}: \ensuremath{\sigma}(\ensuremath{\theta}) in \ensuremath{^{\textnormal{204}}}Pb(p,\ensuremath{\alpha}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1755}&\multicolumn{1}{@{.}l}{33 {\it 7}}&\multicolumn{1}{l}{3/2,5/2\ensuremath{^{+}}}&&&\multicolumn{1}{l}{\texttt{\hyperlink{PB21}{A}\ \ \ \ \ \ \ } }&\parbox[t][0.3cm]{9.994501cm}{\raggedright J\ensuremath{^{\pi}}: 1062.79\ensuremath{\gamma} to 5/2\ensuremath{^{+}}, 1755.32\ensuremath{\gamma} to 1/2\ensuremath{^{+}}; log{} \textit{ft} in \ensuremath{^{\textnormal{201}}}Pb \ensuremath{\varepsilon} decay rules\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{9.994501cm}{\raggedright {\ }{\ }{\ }out J=1/2.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1763}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{TL20LEVEL1}{\ddagger}}}} {\it 15}}&\multicolumn{1}{l}{(7/2,9/2)}&&&\multicolumn{1}{l}{\texttt{\ \ \ \ \ \ \hyperlink{TL27}{G}\ } }&\parbox[t][0.3cm]{9.994501cm}{\raggedright J\ensuremath{^{\pi}}: \ensuremath{\sigma}(\ensuremath{\theta}) in \ensuremath{^{\textnormal{204}}}Pb(p,\ensuremath{\alpha}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1834}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{TL20LEVEL1}{\ddagger}}}} {\it 15}}&\multicolumn{1}{l}{3/2\ensuremath{^{+}},5/2\ensuremath{^{+}}}&&&\multicolumn{1}{l}{\texttt{\ \ \ \ \ \hyperlink{TL26}{F}\hyperlink{TL27}{G}\ } }&\parbox[t][0.3cm]{9.994501cm}{\raggedright XREF: F(1829).\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{9.994501cm}{\raggedright J\ensuremath{^{\pi}}: L(p,t)=2.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1908}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{TL20LEVEL1}{\ddagger}}}} {\it 15}}&\multicolumn{1}{l}{(13/2,15/2\ensuremath{^{-}})}&&&\multicolumn{1}{l}{\texttt{\ \ \ \ \ \ \hyperlink{TL27}{G}\ } }&\parbox[t][0.3cm]{9.994501cm}{\raggedright J\ensuremath{^{\pi}}: \ensuremath{\sigma}(\ensuremath{\theta}) in \ensuremath{^{\textnormal{204}}}Pb(p,\ensuremath{\alpha}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1913}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{TL20LEVEL2}{\#}}}}}&&&&\multicolumn{1}{l}{\texttt{\ \ \ \ \ \hyperlink{TL26}{F}\ \ } }&&\\
\multicolumn{1}{r@{}}{1940}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{TL20LEVEL1}{\ddagger}}}} {\it 15}}&\multicolumn{1}{l}{(7/2,9/2)}&&&\multicolumn{1}{l}{\texttt{\ \ \ \ \ \ \hyperlink{TL27}{G}\ } }&\parbox[t][0.3cm]{9.994501cm}{\raggedright J\ensuremath{^{\pi}}: \ensuremath{\sigma}(\ensuremath{\theta}) in \ensuremath{^{\textnormal{204}}}Pb(p,\ensuremath{\alpha}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1962}&\multicolumn{1}{@{.}l}{76 {\it 17}}&\multicolumn{1}{l}{(15/2\ensuremath{^{-}})}&&&\multicolumn{1}{l}{\texttt{\ \ \hyperlink{TL23}{C}\hyperlink{TL24}{D}\hyperlink{TL25}{E}\ \ \ } }&\parbox[t][0.3cm]{9.994501cm}{\raggedright J\ensuremath{^{\pi}}: 390.9\ensuremath{\gamma} M1+E2 to (13/2\ensuremath{^{-}}), 724.3\ensuremath{\gamma} E2 to (11/2\ensuremath{^{-}}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1987}&\multicolumn{1}{@{.}l}{96 {\it 18}}&\multicolumn{1}{l}{(13/2\ensuremath{^{-}})}&&&\multicolumn{1}{l}{\texttt{\ \ \hyperlink{TL23}{C}\hyperlink{TL24}{D}\ \ \hyperlink{TL27}{g}\ } }&\parbox[t][0.3cm]{9.994501cm}{\raggedright XREF: g(1996).\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{9.994501cm}{\raggedright J\ensuremath{^{\pi}}: 749.5\ensuremath{\gamma} M1(+E2) to (11/2\ensuremath{^{-}}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2015}&\multicolumn{1}{@{.}l}{2 {\it 9}}&\multicolumn{1}{l}{(17/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{9 ns {\it +19\textminus5}}&\multicolumn{1}{l}{\texttt{\ \ \hyperlink{TL23}{C}\hyperlink{TL24}{D}\hyperlink{TL25}{E}\ \hyperlink{TL27}{g}\ } }&\parbox[t][0.3cm]{9.994501cm}{\raggedright XREF: E(2016)g(1996).\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{9.994501cm}{\raggedright J\ensuremath{^{\pi}}: 443.3\ensuremath{\gamma} E2 to (13/2\ensuremath{^{-}}).\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{9.994501cm}{\raggedright T\ensuremath{_{1/2}}: From 443.3\ensuremath{\gamma}(t) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1977Sl01,B}{1977Sl01}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2041}&\multicolumn{1}{@{.}l}{3 {\it 3}}&\multicolumn{1}{l}{(15/2\ensuremath{^{+}})}&&&\multicolumn{1}{l}{\texttt{\ \ \hyperlink{TL23}{C}\hyperlink{TL24}{D}\ \ \hyperlink{TL27}{G}\ } }&\parbox[t][0.3cm]{9.994501cm}{\raggedright XREF: G(2045).\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{9.994501cm}{\raggedright J\ensuremath{^{\pi}}: 469.4\ensuremath{\gamma} E1 to (13/2\ensuremath{^{-}}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2103}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{TL20LEVEL1}{\ddagger}}}} {\it 20}}&\multicolumn{1}{l}{(9/2)}&&&\multicolumn{1}{l}{\texttt{\ \ \ \ \ \hyperlink{TL26}{F}\hyperlink{TL27}{G}\ } }&\parbox[t][0.3cm]{9.994501cm}{\raggedright XREF: F(2098).\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{9.994501cm}{\raggedright J\ensuremath{^{\pi}}: \ensuremath{\sigma}(\ensuremath{\theta}) in \ensuremath{^{\textnormal{204}}}Pb(p,\ensuremath{\alpha}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2145}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{TL20LEVEL1}{\ddagger}}}} {\it 20}}&\multicolumn{1}{l}{(9/2)}&&&\multicolumn{1}{l}{\texttt{\ \ \ \ \ \hyperlink{TL26}{F}\hyperlink{TL27}{G}\ } }&\parbox[t][0.3cm]{9.994501cm}{\raggedright XREF: F(2147).\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{9.994501cm}{\raggedright J\ensuremath{^{\pi}}: \ensuremath{\sigma}(\ensuremath{\theta}) in \ensuremath{^{\textnormal{204}}}Pb(p,\ensuremath{\alpha}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2182}&\multicolumn{1}{@{.}l}{0 {\it 9}}&\multicolumn{1}{l}{(19/2\ensuremath{^{-}})}&&&\multicolumn{1}{l}{\texttt{\ \ \hyperlink{TL23}{C}\hyperlink{TL24}{D}\ \hyperlink{TL26}{F}\ \ } }&\parbox[t][0.3cm]{9.994501cm}{\raggedright XREF: F(2183).\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{9.994501cm}{\raggedright J\ensuremath{^{\pi}}: 166.9\ensuremath{\gamma} M1(+E2) to (17/2\ensuremath{^{-}}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2196}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{TL20LEVEL1}{\ddagger}}}} {\it 20}}&\multicolumn{1}{l}{(15/2\ensuremath{^{+}})}&&&\multicolumn{1}{l}{\texttt{\ \ \ \ \ \ \hyperlink{TL27}{G}\ } }&\parbox[t][0.3cm]{9.994501cm}{\raggedright J\ensuremath{^{\pi}}: \ensuremath{\sigma}(\ensuremath{\theta}) in \ensuremath{^{\textnormal{204}}}Pb(p,\ensuremath{\alpha}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2254}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{TL20LEVEL1}{\ddagger}}}} {\it 20}}&\multicolumn{1}{l}{(15/2\ensuremath{^{-}})}&&&\multicolumn{1}{l}{\texttt{\ \ \ \ \ \ \hyperlink{TL27}{G}\ } }&\parbox[t][0.3cm]{9.994501cm}{\raggedright J\ensuremath{^{\pi}}: \ensuremath{\sigma}(\ensuremath{\theta}) in \ensuremath{^{\textnormal{204}}}Pb(p,\ensuremath{\alpha}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2271}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{TL20LEVEL2}{\#}}}}}&&&&\multicolumn{1}{l}{\texttt{\ \ \ \ \ \hyperlink{TL26}{F}\ \ } }&&\\
\multicolumn{1}{r@{}}{2343}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{TL20LEVEL2}{\#}}}}}&&&&\multicolumn{1}{l}{\texttt{\ \ \ \ \ \hyperlink{TL26}{F}\ \ } }&&\\
\multicolumn{1}{r@{}}{2379}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{TL20LEVEL1}{\ddagger}}}} {\it 20}}&\multicolumn{1}{l}{(15/2\ensuremath{^{+}})}&&&\multicolumn{1}{l}{\texttt{\ \ \ \ \ \ \hyperlink{TL27}{G}\ } }&\parbox[t][0.3cm]{9.994501cm}{\raggedright J\ensuremath{^{\pi}}: \ensuremath{\sigma}(\ensuremath{\theta}) in \ensuremath{^{\textnormal{204}}}Pb(p,\ensuremath{\alpha}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2442}&\multicolumn{1}{@{.}l}{0 {\it 9}}&\multicolumn{1}{l}{(21/2\ensuremath{^{-}})}&&&\multicolumn{1}{l}{\texttt{\ \ \hyperlink{TL23}{C}\hyperlink{TL24}{D}\ \ \hyperlink{TL27}{G}\ } }&\parbox[t][0.3cm]{9.994501cm}{\raggedright XREF: G(2449).\vspace{0.1cm}}&\\
\end{longtable}
\begin{textblock}{29}(0,27.3)
Continued on next page (footnotes at end of table)
\end{textblock}
\clearpage
\begin{longtable}{ccccccc@{\extracolsep{\fill}}c}
\\[-.4cm]
\multicolumn{8}{c}{{\bf \small \underline{Adopted \hyperlink{201TL_LEVEL}{Levels}, \hyperlink{201TL_GAMMA}{Gammas} (continued)}}}\\
\multicolumn{8}{c}{~}\\
\multicolumn{8}{c}{\underline{\ensuremath{^{201}}Tl Levels (continued)}}\\
\multicolumn{8}{c}{~}\\
\multicolumn{2}{c}{E(level)$^{{\hyperlink{TL20LEVEL0}{\dagger}}}$}&J$^{\pi}$$^{}$&\multicolumn{2}{c}{T$_{1/2}$$^{}$}&XREF&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&
\endhead
&&&&&&\parbox[t][0.3cm]{9.77836cm}{\raggedright J\ensuremath{^{\pi}}: 260.1\ensuremath{\gamma} to (19/2\ensuremath{^{-}}), 426.8\ensuremath{\gamma} E2 to (17/2\ensuremath{^{-}}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2486}&\multicolumn{1}{@{.}l}{5 {\it 8}}&\multicolumn{1}{l}{(17/2)}&&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{TL24}{D}\ \ \ \ } }&&\\
\multicolumn{1}{r@{}}{2534}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{TL20LEVEL1}{\ddagger}}}} {\it 20}}&\multicolumn{1}{l}{(9/2)}&&&\multicolumn{1}{l}{\texttt{\ \ \ \ \ \ \hyperlink{TL27}{G}\ } }&\parbox[t][0.3cm]{9.77836cm}{\raggedright J\ensuremath{^{\pi}}: \ensuremath{\sigma}(\ensuremath{\theta}) in \ensuremath{^{\textnormal{204}}}Pb(p,\ensuremath{\alpha}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2622}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{TL20LEVEL1}{\ddagger}}}} {\it 20}}&\multicolumn{1}{l}{(15/2\ensuremath{^{+}})}&&&\multicolumn{1}{l}{\texttt{\ \ \ \ \ \ \hyperlink{TL27}{G}\ } }&\parbox[t][0.3cm]{9.77836cm}{\raggedright J\ensuremath{^{\pi}}: \ensuremath{\sigma}(\ensuremath{\theta}) in \ensuremath{^{\textnormal{204}}}Pb(p,\ensuremath{\alpha}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2669}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{TL20LEVEL1}{\ddagger}}}} {\it 20}}&\multicolumn{1}{l}{(5/2)}&&&\multicolumn{1}{l}{\texttt{\ \ \ \ \ \ \hyperlink{TL27}{G}\ } }&\parbox[t][0.3cm]{9.77836cm}{\raggedright J\ensuremath{^{\pi}}: \ensuremath{\sigma}(\ensuremath{\theta}) in \ensuremath{^{\textnormal{204}}}Pb(p,\ensuremath{\alpha}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2672}&\multicolumn{1}{@{.}l}{1 {\it 9}}&\multicolumn{1}{l}{(23/2\ensuremath{^{+}})}&&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{TL24}{D}\ \ \ \ } }&\parbox[t][0.3cm]{9.77836cm}{\raggedright J\ensuremath{^{\pi}}: 230.2\ensuremath{\gamma} E1 to (21/2\ensuremath{^{-}}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2713}&\multicolumn{1}{@{.}l}{66 {\it 19}}&\multicolumn{1}{l}{(15/2\ensuremath{^{-}})}&&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{TL24}{D}\hyperlink{TL25}{E}\ \hyperlink{TL27}{G}\ } }&\parbox[t][0.3cm]{9.77836cm}{\raggedright J\ensuremath{^{\pi}}: 1141.8\ensuremath{\gamma} M1(+E2) to (13/2\ensuremath{^{-}}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2748}&\multicolumn{1}{@{.}l}{0 {\it 5}}&\multicolumn{1}{l}{(17/2\ensuremath{^{+}})}&&&\multicolumn{1}{l}{\texttt{\ \ \hyperlink{TL23}{C}\hyperlink{TL24}{D}\hyperlink{TL25}{E}\ \ \ } }&\parbox[t][0.3cm]{9.77836cm}{\raggedright J\ensuremath{^{\pi}}: 785.2\ensuremath{\gamma} E1 to (15/2\ensuremath{^{-}}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2748}&\multicolumn{1}{@{.}l}{0+x {\it 5}}&&\multicolumn{1}{r@{}}{95}&\multicolumn{1}{@{ }l}{ns {\it +39\textminus21}}&\multicolumn{1}{l}{\texttt{\ \ \ \ \hyperlink{TL25}{E}\ \ \ } }&\parbox[t][0.3cm]{9.77836cm}{\raggedright E(level): From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2013Bo18,B}{2013Bo18}. The observed 148, 158, 314 and 853 keV\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{9.77836cm}{\raggedright {\ }{\ }{\ }delayed gammas are expected above the 2748.0-keV level, but their\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{9.77836cm}{\raggedright {\ }{\ }{\ }ordering is unknown.\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{9.77836cm}{\raggedright J\ensuremath{^{\pi}}: Tentative J=(33/2) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2013Bo18,B}{2013Bo18}.\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{9.77836cm}{\raggedright T\ensuremath{_{1/2}}: From sum of (148,158,314,319,333,444,726,785,853)\ensuremath{\gamma}(t) in\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{9.77836cm}{\raggedright {\ }{\ }{\ }\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2012BoZQ,B}{2012BoZQ}, \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2013BoZT,B}{2013BoZT}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2762}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{TL20LEVEL1}{\ddagger}}}} {\it 20}}&\multicolumn{1}{l}{(7/2,9/2)}&&&\multicolumn{1}{l}{\texttt{\ \ \ \ \ \ \hyperlink{TL27}{G}\ } }&\parbox[t][0.3cm]{9.77836cm}{\raggedright J\ensuremath{^{\pi}}: \ensuremath{\sigma}(\ensuremath{\theta}) in \ensuremath{^{\textnormal{204}}}Pb(p,\ensuremath{\alpha}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2853}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{TL20LEVEL1}{\ddagger}}}} {\it 20}}&\multicolumn{1}{l}{(11/2\ensuremath{^{-}})}&&&\multicolumn{1}{l}{\texttt{\ \ \ \ \ \ \hyperlink{TL27}{G}\ } }&\parbox[t][0.3cm]{9.77836cm}{\raggedright J\ensuremath{^{\pi}}: \ensuremath{\sigma}(\ensuremath{\theta}) in \ensuremath{^{\textnormal{204}}}Pb(p,\ensuremath{\alpha}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2883}&\multicolumn{1}{@{.}l}{7 {\it 9}}&\multicolumn{1}{l}{(23/2\ensuremath{^{-}})}&&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{TL24}{D}\ \ \ \ } }&\parbox[t][0.3cm]{9.77836cm}{\raggedright J\ensuremath{^{\pi}}: 441.1\ensuremath{\gamma} M1(+E2) to (21/2\ensuremath{^{-}}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2897}&\multicolumn{1}{@{.}l}{6 {\it 9}}&\multicolumn{1}{l}{(25/2\ensuremath{^{+}})}&&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{TL24}{D}\ \ \ \ } }&\parbox[t][0.3cm]{9.77836cm}{\raggedright J\ensuremath{^{\pi}}: 225.5\ensuremath{\gamma} M1(+E2) to (23/2\ensuremath{^{+}}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2899}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{TL20LEVEL1}{\ddagger}}}} {\it 20}}&\multicolumn{1}{l}{(15/2\ensuremath{^{+}})}&&&\multicolumn{1}{l}{\texttt{\ \ \ \ \ \ \hyperlink{TL27}{G}\ } }&\parbox[t][0.3cm]{9.77836cm}{\raggedright J\ensuremath{^{\pi}}: \ensuremath{\sigma}(\ensuremath{\theta}) in \ensuremath{^{\textnormal{204}}}Pb(p,\ensuremath{\alpha}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2976}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{TL20LEVEL1}{\ddagger}}}} {\it 20}}&\multicolumn{1}{l}{(15/2\ensuremath{^{+}})}&&&\multicolumn{1}{l}{\texttt{\ \ \ \ \ \ \hyperlink{TL27}{G}\ } }&\parbox[t][0.3cm]{9.77836cm}{\raggedright J\ensuremath{^{\pi}}: \ensuremath{\sigma}(\ensuremath{\theta}) in \ensuremath{^{\textnormal{204}}}Pb(p,\ensuremath{\alpha}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2985}&\multicolumn{1}{@{.}l}{1 {\it 9}}&\multicolumn{1}{l}{(21/2\ensuremath{^{+}})}&&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{TL24}{D}\ \ \ \ } }&\parbox[t][0.3cm]{9.77836cm}{\raggedright J\ensuremath{^{\pi}}: 803.1\ensuremath{\gamma} E1 to (19/2\ensuremath{^{-}}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3011}&\multicolumn{1}{@{.}l}{7 {\it 5}}&\multicolumn{1}{l}{(19/2\ensuremath{^{+}})}&&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{TL24}{D}\ \ \hyperlink{TL27}{G}\ } }&\parbox[t][0.3cm]{9.77836cm}{\raggedright XREF: G(3030).\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{9.77836cm}{\raggedright J\ensuremath{^{\pi}}: 263.7\ensuremath{\gamma} M1(+E2) to (17/2\ensuremath{^{+}}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3044}&\multicolumn{1}{@{.}l}{9 {\it 9}}&\multicolumn{1}{l}{(25/2\ensuremath{^{-}})}&&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{TL24}{D}\ \ \ \ } }&\parbox[t][0.3cm]{9.77836cm}{\raggedright J\ensuremath{^{\pi}}: 161.2\ensuremath{\gamma} M1(+E2) to (23/2\ensuremath{^{-}}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3056}&\multicolumn{1}{@{.}l}{6 {\it 6}}&\multicolumn{1}{l}{(19/2\ensuremath{^{+}})}&&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{TL24}{D}\ \ \ \ } }&\parbox[t][0.3cm]{9.77836cm}{\raggedright J\ensuremath{^{\pi}}: 308.8\ensuremath{\gamma} M1(+E2) to (17/2\ensuremath{^{+}}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3083}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{TL20LEVEL1}{\ddagger}}}} {\it 20}}&\multicolumn{1}{l}{(9/2,11/2)}&&&\multicolumn{1}{l}{\texttt{\ \ \ \ \ \ \hyperlink{TL27}{G}\ } }&\parbox[t][0.3cm]{9.77836cm}{\raggedright J\ensuremath{^{\pi}}: \ensuremath{\sigma}(\ensuremath{\theta}) in \ensuremath{^{\textnormal{204}}}Pb(p,\ensuremath{\alpha}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3133}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{TL20LEVEL1}{\ddagger}}}} {\it 20}}&&&&\multicolumn{1}{l}{\texttt{\ \ \ \ \ \ \hyperlink{TL27}{G}\ } }&&\\
\multicolumn{1}{r@{}}{3201}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{TL20LEVEL1}{\ddagger}}}} {\it 20}}&&&&\multicolumn{1}{l}{\texttt{\ \ \ \ \ \ \hyperlink{TL27}{G}\ } }&&\\
\multicolumn{1}{r@{}}{3303}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{TL20LEVEL1}{\ddagger}}}} {\it 20}}&\multicolumn{1}{l}{(9/2,11/2)}&&&\multicolumn{1}{l}{\texttt{\ \ \ \ \ \ \hyperlink{TL27}{G}\ } }&\parbox[t][0.3cm]{9.77836cm}{\raggedright J\ensuremath{^{\pi}}: \ensuremath{\sigma}(\ensuremath{\theta}) in \ensuremath{^{\textnormal{204}}}Pb(p,\ensuremath{\alpha}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3314}&\multicolumn{1}{@{.}l}{7 {\it 6}}&\multicolumn{1}{l}{(21/2\ensuremath{^{+}})}&&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{TL24}{D}\ \ \ \ } }&\parbox[t][0.3cm]{9.77836cm}{\raggedright J\ensuremath{^{\pi}}: 258.0\ensuremath{\gamma} M1(+E2) to (19/2\ensuremath{^{+}}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3362}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{TL20LEVEL1}{\ddagger}}}} {\it 20}}&\multicolumn{1}{l}{(15/2\ensuremath{^{+}})}&&&\multicolumn{1}{l}{\texttt{\ \ \ \ \ \ \hyperlink{TL27}{G}\ } }&\parbox[t][0.3cm]{9.77836cm}{\raggedright J\ensuremath{^{\pi}}: \ensuremath{\sigma}(\ensuremath{\theta}) in \ensuremath{^{\textnormal{204}}}Pb(p,\ensuremath{\alpha}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3371}&\multicolumn{1}{@{.}l}{8 {\it 9}}&\multicolumn{1}{l}{(23/2\ensuremath{^{+}})}&&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{TL24}{D}\ \ \ \ } }&\parbox[t][0.3cm]{9.77836cm}{\raggedright J\ensuremath{^{\pi}}: 386.7\ensuremath{\gamma} M1(+E2) to (21/2\ensuremath{^{+}}), 326.7\ensuremath{\gamma} to (25/2\ensuremath{^{-}}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3401}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{TL20LEVEL1}{\ddagger}}}} {\it 20}}&\multicolumn{1}{l}{(17/2\ensuremath{^{+}})}&&&\multicolumn{1}{l}{\texttt{\ \ \ \ \ \ \hyperlink{TL27}{G}\ } }&\parbox[t][0.3cm]{9.77836cm}{\raggedright J\ensuremath{^{\pi}}: \ensuremath{\sigma}(\ensuremath{\theta}) in \ensuremath{^{\textnormal{204}}}Pb(p,\ensuremath{\alpha}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3431}&\multicolumn{1}{@{.}l}{7 {\it 9}}&\multicolumn{1}{l}{(27/2\ensuremath{^{-}})}&&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{TL24}{D}\ \ \ \ } }&\parbox[t][0.3cm]{9.77836cm}{\raggedright J\ensuremath{^{\pi}}: 386.8\ensuremath{\gamma} to (25/2\ensuremath{^{-}}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3434}&\multicolumn{1}{@{.}l}{5 {\it 6}}&\multicolumn{1}{l}{(23/2\ensuremath{^{+}})}&&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{TL24}{D}\ \ \ \ } }&\parbox[t][0.3cm]{9.77836cm}{\raggedright J\ensuremath{^{\pi}}: 422.8\ensuremath{\gamma} E2 to (19/2\ensuremath{^{+}}), 119.8\ensuremath{\gamma} M1(+E2) to (21/2\ensuremath{^{+}}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3441}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{TL20LEVEL1}{\ddagger}}}} {\it 20}}&\multicolumn{1}{l}{(9/2)}&&&\multicolumn{1}{l}{\texttt{\ \ \ \ \ \ \hyperlink{TL27}{G}\ } }&\parbox[t][0.3cm]{9.77836cm}{\raggedright J\ensuremath{^{\pi}}: \ensuremath{\sigma}(\ensuremath{\theta}) in \ensuremath{^{\textnormal{204}}}Pb(p,\ensuremath{\alpha}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3499}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{TL20LEVEL1}{\ddagger}}}} {\it 20}}&\multicolumn{1}{l}{(15/2)}&&&\multicolumn{1}{l}{\texttt{\ \ \ \ \ \ \hyperlink{TL27}{G}\ } }&\parbox[t][0.3cm]{9.77836cm}{\raggedright J\ensuremath{^{\pi}}: \ensuremath{\sigma}(\ensuremath{\theta}) in \ensuremath{^{\textnormal{204}}}Pb(p,\ensuremath{\alpha}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3530}&\multicolumn{1}{@{.}l}{6 {\it 6}}&\multicolumn{1}{l}{(25/2)}&&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{TL24}{D}\ \ \ \ } }&\parbox[t][0.3cm]{9.77836cm}{\raggedright J\ensuremath{^{\pi}}: 96.1\ensuremath{\gamma} D to (23/2\ensuremath{^{+}}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3552}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{TL20LEVEL1}{\ddagger}}}} {\it 20}}&\multicolumn{1}{l}{(15/2)}&&&\multicolumn{1}{l}{\texttt{\ \ \ \ \ \ \hyperlink{TL27}{G}\ } }&\parbox[t][0.3cm]{9.77836cm}{\raggedright J\ensuremath{^{\pi}}: \ensuremath{\sigma}(\ensuremath{\theta}) in \ensuremath{^{\textnormal{204}}}Pb(p,\ensuremath{\alpha}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3615}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{TL20LEVEL1}{\ddagger}}}} {\it 20}}&\multicolumn{1}{l}{(17/2\ensuremath{^{-}},19/2\ensuremath{^{-}})}&&&\multicolumn{1}{l}{\texttt{\ \ \ \ \ \ \hyperlink{TL27}{G}\ } }&\parbox[t][0.3cm]{9.77836cm}{\raggedright J\ensuremath{^{\pi}}: \ensuremath{\sigma}(\ensuremath{\theta}) in \ensuremath{^{\textnormal{204}}}Pb(p,\ensuremath{\alpha}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3648}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{TL20LEVEL1}{\ddagger}}}} {\it 20}}&\multicolumn{1}{l}{(15/2\ensuremath{^{+}})}&&&\multicolumn{1}{l}{\texttt{\ \ \ \ \ \ \hyperlink{TL27}{G}\ } }&\parbox[t][0.3cm]{9.77836cm}{\raggedright J\ensuremath{^{\pi}}: \ensuremath{\sigma}(\ensuremath{\theta}) in \ensuremath{^{\textnormal{204}}}Pb(p,\ensuremath{\alpha}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3674}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{TL20LEVEL1}{\ddagger}}}} {\it 20}}&\multicolumn{1}{l}{(17/2,19/2\ensuremath{^{-}})}&&&\multicolumn{1}{l}{\texttt{\ \ \ \ \ \ \hyperlink{TL27}{G}\ } }&\parbox[t][0.3cm]{9.77836cm}{\raggedright J\ensuremath{^{\pi}}: \ensuremath{\sigma}(\ensuremath{\theta}) in \ensuremath{^{\textnormal{204}}}Pb(p,\ensuremath{\alpha}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3706}&\multicolumn{1}{@{.}l}{8 {\it 6}}&\multicolumn{1}{l}{(27/2)}&&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{TL24}{D}\ \ \ \ } }&\parbox[t][0.3cm]{9.77836cm}{\raggedright J\ensuremath{^{\pi}}: 176.2\ensuremath{\gamma} D to (25/2).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3727}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{TL20LEVEL1}{\ddagger}}}} {\it 20}}&&&&\multicolumn{1}{l}{\texttt{\ \ \ \ \ \ \hyperlink{TL27}{G}\ } }&&\\
\multicolumn{1}{r@{}}{3864}&\multicolumn{1}{@{.}l}{3 {\it 6}}&\multicolumn{1}{l}{(29/2)}&&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{TL24}{D}\ \ \ \ } }&\parbox[t][0.3cm]{9.77836cm}{\raggedright J\ensuremath{^{\pi}}: 157.5\ensuremath{\gamma} D to (27/2).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3936}&\multicolumn{1}{@{.}l}{1 {\it 9}}&\multicolumn{1}{l}{(29/2)}&&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{TL24}{D}\ \ \ \ } }&\parbox[t][0.3cm]{9.77836cm}{\raggedright J\ensuremath{^{\pi}}: 504.4\ensuremath{\gamma} D to (27/2\ensuremath{^{-}}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3950}&\multicolumn{1}{@{.}l}{9 {\it 9}}&\multicolumn{1}{l}{(25/2)}&&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{TL24}{D}\ \ \ \ } }&\parbox[t][0.3cm]{9.77836cm}{\raggedright J\ensuremath{^{\pi}}: 579.1\ensuremath{\gamma} D to (23/2\ensuremath{^{+}}).\vspace{0.1cm}}&\\
\end{longtable}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{TL20LEVEL0}{\dagger}}}} From a least-squares fit to E\ensuremath{\gamma}, unless otherwise stated.}\\
\begin{textblock}{29}(0,27.3)
Continued on next page (footnotes at end of table)
\end{textblock}
\clearpage
\vspace*{-0.5cm}
{\bf \small \underline{Adopted \hyperlink{201TL_LEVEL}{Levels}, \hyperlink{201TL_GAMMA}{Gammas} (continued)}}\\
\vspace{0.3cm}
\underline{$^{201}$Tl Levels (continued)}\\
\vspace{0.3cm}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{TL20LEVEL1}{\ddagger}}}} From \ensuremath{^{\textnormal{204}}}Pb(p,\ensuremath{\alpha}), but values were lowered by 15 keV to account for differences between excitation energies in \ensuremath{^{\textnormal{204}}}Pb(p,\ensuremath{\alpha}) and}\\
\parbox[b][0.3cm]{17.7cm}{{\ }{\ }these in the Adopted Levels for E(levels) below 1600 keV.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{TL20LEVEL2}{\#}}}} From \ensuremath{^{\textnormal{203}}}Tl(p,t).}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{TL20LEVEL3}{@}}}} Configuration=Dominant \ensuremath{\pi} s\ensuremath{_{\textnormal{1/2}}^{\textnormal{$-$1}}}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{TL20LEVEL4}{\&}}}} Configuration=Dominant \ensuremath{\pi} d\ensuremath{_{\textnormal{3/2}}^{\textnormal{$-$1}}}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{TL20LEVEL5}{a}}}} Configuration=Dominant \ensuremath{\pi} (s\ensuremath{_{\textnormal{1/2}}^{\textnormal{$-$1}}})\ensuremath{\otimes}2\ensuremath{^{\textnormal{+}}}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{TL20LEVEL6}{b}}}} Configuration=\ensuremath{\pi} h\ensuremath{_{\textnormal{11/2}}^{\textnormal{$-$1}}}.}\\
\vspace{0.5cm}
\clearpage
\vspace{0.3cm}
\begin{landscape}
\vspace*{-0.5cm}
{\bf \small \underline{Adopted \hyperlink{201TL_LEVEL}{Levels}, \hyperlink{201TL_GAMMA}{Gammas} (continued)}}\\
\vspace{0.3cm}
\hypertarget{201TL_GAMMA}{\underline{$\gamma$($^{201}$Tl)}}\\
\begin{longtable}{ccccccccc@{}ccccccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E\ensuremath{_{i}}(level)}&J\ensuremath{^{\pi}_{i}}&\multicolumn{2}{c}{E\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{TL20GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{I\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{TL20GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{E\ensuremath{_{f}}}&J\ensuremath{^{\pi}_{f}}&Mult.\ensuremath{^{\hyperlink{TL20GAMMA3}{@}}}&\multicolumn{2}{c}{\ensuremath{\delta}\ensuremath{^{\hyperlink{TL20GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{\ensuremath{\alpha}\ensuremath{^{\hyperlink{TL20GAMMA4}{\&}}}}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{331}&\multicolumn{1}{@{.}l}{17}&\multicolumn{1}{l}{3/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{331}&\multicolumn{1}{@{.}l}{15 {\it 6}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{1/2\ensuremath{^{+}}}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{+1}&\multicolumn{1}{@{.}l}{33 {\it 6}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{161 {\it 5}}&\parbox[t][0.3cm]{7.6408806cm}{\raggedright B(M1)(W.u.)=0.0027 \textit{+11{\textminus}6}; B(E2)(W.u.)=16 \textit{+6{\textminus}4}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.6408806cm}{\raggedright Mult.: \ensuremath{\alpha}(K)exp=0.1069 \textit{21}, \ensuremath{\alpha}(L3)exp=0.00365 \textit{12},\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.6408806cm}{\raggedright {\ }{\ }{\ }K/L=3.76 \textit{6} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1974Ha18,B}{1974Ha18}); K/L=3.9 \textit{3}, L12/L3=7.1 \textit{9}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.6408806cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1964Aa01,B}{1964Aa01}); \ensuremath{\alpha}(K)exp=0.113 \textit{8} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1961Pe05,B}{1961Pe05});\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.6408806cm}{\raggedright {\ }{\ }{\ }K/L12=4.2 \textit{2}, L12/L3=6.5 \textit{5} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1960Li08,B}{1960Li08}); \ensuremath{\gamma}\ensuremath{\gamma}(\ensuremath{\theta}) in\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.6408806cm}{\raggedright {\ }{\ }{\ }\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1961Pe05,B}{1961Pe05}; \ensuremath{\alpha}(K)exp=0.113 \textit{8} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1979Do09,B}{1979Do09}), normalized\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.6408806cm}{\raggedright {\ }{\ }{\ }value adopted by the authors from the data of\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.6408806cm}{\raggedright {\ }{\ }{\ }\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1961Pe05,B}{1961Pe05}; \ensuremath{\alpha}(K)exp=0.111 \textit{16} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1971Hn04,B}{1971Hn04}); A\ensuremath{_{\textnormal{2}}}=0.012\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.6408806cm}{\raggedright {\ }{\ }{\ }\textit{15}, A\ensuremath{_{\textnormal{4}}}={\textminus}0.008 \textit{18} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1977Sl01,B}{1977Sl01}).\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.6408806cm}{\raggedright \ensuremath{\delta}: From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1974Ha18,B}{1974Ha18}, by taking into account the\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.6408806cm}{\raggedright {\ }{\ }{\ }penetration effect and using \ensuremath{\lambda}=+4.0 \textit{10}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{692}&\multicolumn{1}{@{.}l}{52}&\multicolumn{1}{l}{5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{361}&\multicolumn{1}{@{.}l}{25 {\it 6}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{ }l}{{\it 5}}&\multicolumn{1}{r@{}}{331}&\multicolumn{1}{@{.}l}{17 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}}}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{14 {\it 7}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{234 {\it 5}}&\parbox[t][0.3cm]{7.6408806cm}{\raggedright Mult.: \ensuremath{\alpha}(K)exp=0.193 \textit{4}, \ensuremath{\alpha}(L3)exp=0.00029 \textit{7}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.6408806cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1974Ha18,B}{1974Ha18}); K/L=5.9 \textit{6}, L12/L3=7.1 \textit{9} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1964Aa01,B}{1964Aa01});\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.6408806cm}{\raggedright {\ }{\ }{\ }\ensuremath{\alpha}(K)exp=0.210 \textit{25} and \ensuremath{\gamma}\ensuremath{\gamma}(\ensuremath{\theta}) (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1961Pe05,B}{1961Pe05});\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.6408806cm}{\raggedright {\ }{\ }{\ }\ensuremath{\alpha}(K)exp=0.22 \textit{2} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1979Do09,B}{1979Do09}); A\ensuremath{_{\textnormal{2}}}={\textminus}0.099 \textit{26},\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.6408806cm}{\raggedright {\ }{\ }{\ }A\ensuremath{_{\textnormal{4}}}=0.026 \textit{38} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1977Sl01,B}{1977Sl01}).\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.6408806cm}{\raggedright \ensuremath{\delta}: From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1974Ha18,B}{1974Ha18}, by taking into account the\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.6408806cm}{\raggedright {\ }{\ }{\ }penetration effect and using \ensuremath{\lambda}=+0.5 \textit{5}.\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{692}&\multicolumn{1}{@{.}l}{41 {\it 8}}&\multicolumn{1}{r@{}}{45}&\multicolumn{1}{@{.}l}{4 {\it 21}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{1/2\ensuremath{^{+}}}&\multicolumn{1}{l}{E2}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{01342 {\it 19}}&\parbox[t][0.3cm]{7.6408806cm}{\raggedright Mult.: \ensuremath{\alpha}(K)exp=0.010 \textit{1} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1979Do09,B}{1979Do09}); \ensuremath{\alpha}(K)exp=0.017 \textit{5}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.6408806cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1971Hn04,B}{1971Hn04}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{919}&\multicolumn{1}{@{.}l}{47}&\multicolumn{1}{l}{(9/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{588}&\multicolumn{1}{@{.}l}{3\ensuremath{^{\hyperlink{TL20GAMMA1}{\ddagger}}} {\it 1}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{TL20GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{331}&\multicolumn{1}{@{.}l}{17 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}}}&\multicolumn{1}{l}{[E3]}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0567 {\it 8}}&\parbox[t][0.3cm]{7.6408806cm}{\raggedright B(E3)(W.u.)=0.00931 \textit{+50{\textminus}47}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1098}&\multicolumn{1}{@{.}l}{50}&\multicolumn{1}{l}{5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{405}&\multicolumn{1}{@{.}l}{96 {\it 7}}&\multicolumn{1}{r@{}}{62}&\multicolumn{1}{@{ }l}{{\it 3}}&\multicolumn{1}{r@{}}{692}&\multicolumn{1}{@{.}l}{52 }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{+}}}&\multicolumn{1}{l}{M1(+E2)}&\multicolumn{1}{r@{}}{$<$0}&\multicolumn{1}{@{.}l}{4}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{164 {\it 9}}&\parbox[t][0.3cm]{7.6408806cm}{\raggedright Mult.,\ensuremath{\delta}: \ensuremath{\alpha}(K)exp=0.14 \textit{2} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1979Do09,B}{1979Do09}); \ensuremath{\alpha}(K)exp=0.15 \textit{3}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.6408806cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1971Hn04,B}{1971Hn04}).\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{767}&\multicolumn{1}{@{.}l}{26 {\it 8}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{ }l}{{\it 5}}&\multicolumn{1}{r@{}}{331}&\multicolumn{1}{@{.}l}{17 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}}}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{33 {\it 19}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0304 {\it 25}}&\parbox[t][0.3cm]{7.6408806cm}{\raggedright Mult.,\ensuremath{\delta}: \ensuremath{\alpha}(K)exp=0.024 \textit{3} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1979Do09,B}{1979Do09}); \ensuremath{\alpha}(K)exp=0.026\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.6408806cm}{\raggedright {\ }{\ }{\ }\textit{3} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1971Hn04,B}{1971Hn04}).\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{1098}&\multicolumn{1}{@{.}l}{52 {\it 7}}&\multicolumn{1}{r@{}}{57}&\multicolumn{1}{@{ }l}{{\it 3}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{1/2\ensuremath{^{+}}}&\multicolumn{1}{l}{E2}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{00528 {\it 7}}&\parbox[t][0.3cm]{7.6408806cm}{\raggedright Mult.,\ensuremath{\delta}: \ensuremath{\alpha}(K)exp=0.0055 \textit{11} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1979Do09,B}{1979Do09});\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.6408806cm}{\raggedright {\ }{\ }{\ }\ensuremath{\alpha}(K)exp=0.0061 \textit{15} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1971Hn04,B}{1971Hn04}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1134}&\multicolumn{1}{@{.}l}{86}&\multicolumn{1}{l}{7/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{803}&\multicolumn{1}{@{.}l}{66 {\it 7}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{331}&\multicolumn{1}{@{.}l}{17 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}}}&\multicolumn{1}{l}{E2}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{00982 {\it 14}}&\parbox[t][0.3cm]{7.6408806cm}{\raggedright Mult.: \ensuremath{\alpha}(K)exp=0.0074 \textit{16} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1979Do09,B}{1979Do09});\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.6408806cm}{\raggedright {\ }{\ }{\ }\ensuremath{\alpha}(K)exp=0.0079 \textit{16} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1971Hn04,B}{1971Hn04}); A\ensuremath{_{\textnormal{2}}}=0.136 \textit{19},\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.6408806cm}{\raggedright {\ }{\ }{\ }A\ensuremath{_{\textnormal{4}}}={\textminus}0.050 \textit{44} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1977Sl01,B}{1977Sl01}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1157}&\multicolumn{1}{@{.}l}{43}&\multicolumn{1}{l}{3/2\ensuremath{^{+}},5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{58}&\multicolumn{1}{@{.}l}{92 {\it 5}}&\multicolumn{1}{r@{}}{$\approx$3}&\multicolumn{1}{@{.}l}{5}&\multicolumn{1}{r@{}}{1098}&\multicolumn{1}{@{.}l}{50 }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{+}}}&\multicolumn{1}{l}{[M1]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{7}&\multicolumn{1}{@{.}l}{37 {\it 10}}&&\\
&&&\multicolumn{1}{r@{}}{464}&\multicolumn{1}{@{.}l}{90 {\it 8}}&\multicolumn{1}{r@{}}{14}&\multicolumn{1}{@{.}l}{0 {\it 7}}&\multicolumn{1}{r@{}}{692}&\multicolumn{1}{@{.}l}{52 }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{+}}}&\multicolumn{1}{l}{[M1]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{1206 {\it 17}}&&\\
&&&\multicolumn{1}{r@{}}{826}&\multicolumn{1}{@{.}l}{26 {\it 8}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{ }l}{{\it 5}}&\multicolumn{1}{r@{}}{331}&\multicolumn{1}{@{.}l}{17 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}}}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{98 {\it +43\textminus29}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0129 {\it 10}}&\parbox[t][0.3cm]{7.6408806cm}{\raggedright Mult.,\ensuremath{\delta}: \ensuremath{\alpha}(K)exp=0.010 \textit{1} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1979Do09,B}{1979Do09});\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.6408806cm}{\raggedright {\ }{\ }{\ }\ensuremath{\alpha}(K)exp=0.0110 \textit{15} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1971Hn04,B}{1971Hn04}).\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{1157}&\multicolumn{1}{@{.}l}{45 {\it 9}}&\multicolumn{1}{r@{}}{5}&\multicolumn{1}{@{.}l}{1 {\it 3}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{1/2\ensuremath{^{+}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
\multicolumn{1}{r@{}}{1238}&\multicolumn{1}{@{.}l}{46}&\multicolumn{1}{l}{(11/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{319}&\multicolumn{1}{@{.}l}{0\ensuremath{^{\hyperlink{TL20GAMMA1}{\ddagger}}} {\it 1}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{TL20GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{919}&\multicolumn{1}{@{.}l}{47 }&\multicolumn{1}{@{}l}{(9/2\ensuremath{^{-}})}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{$-$0}&\multicolumn{1}{@{.}l}{34 {\it +11\textminus8}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{307 {\it 13}}&\parbox[t][0.3cm]{7.6408806cm}{\raggedright Mult.,\ensuremath{\delta}: From A\ensuremath{_{\textnormal{2}}}={\textminus}0.451 \textit{13}, A\ensuremath{_{\textnormal{4}}}={\textminus}0.045 \textit{18}, \ensuremath{\gamma}(\ensuremath{\theta}) in\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.6408806cm}{\raggedright {\ }{\ }{\ }\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1977Sl01,B}{1977Sl01}; DCO=1.56 \textit{4} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2013Da15,B}{2013Da15}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1238}&\multicolumn{1}{@{.}l}{83}&\multicolumn{1}{l}{3/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{546}&\multicolumn{1}{@{.}l}{28 {\it 9}}&\multicolumn{1}{r@{}}{4}&\multicolumn{1}{@{.}l}{6 {\it 3}}&\multicolumn{1}{r@{}}{692}&\multicolumn{1}{@{.}l}{52 }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{+}}}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{31 {\it 24}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{074 {\it 8}}&\parbox[t][0.3cm]{7.6408806cm}{\raggedright Mult.,\ensuremath{\delta}: \ensuremath{\alpha}(K)exp=0.059 \textit{8} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1979Do09,B}{1979Do09}); \ensuremath{\alpha}(K)exp=0.065\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.6408806cm}{\raggedright {\ }{\ }{\ }\textit{14} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1971Hn04,B}{1971Hn04}).\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{907}&\multicolumn{1}{@{.}l}{67 {\it 8}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{ }l}{{\it 6}}&\multicolumn{1}{r@{}}{331}&\multicolumn{1}{@{.}l}{17 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}}}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{43 {\it +23\textminus34}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0190 {\it 20}}&\parbox[t][0.3cm]{7.6408806cm}{\raggedright Mult.,\ensuremath{\delta}: \ensuremath{\alpha}(K)exp=0.015 \textit{2} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1979Do09,B}{1979Do09}); \ensuremath{\alpha}(K)exp=0.017\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.6408806cm}{\raggedright {\ }{\ }{\ }\textit{3} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1971Hn04,B}{1971Hn04}).\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{1238}&\multicolumn{1}{@{.}l}{82 {\it 7}}&\multicolumn{1}{r@{}}{18}&\multicolumn{1}{@{.}l}{8 {\it 11}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{1/2\ensuremath{^{+}}}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{17 {\it +57\textminus35}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0065 {\it 9}}&\parbox[t][0.3cm]{7.6408806cm}{\raggedright Mult.,\ensuremath{\delta}: \ensuremath{\alpha}(K)exp=0.0053 \textit{9} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1979Do09,B}{1979Do09});\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.6408806cm}{\raggedright {\ }{\ }{\ }\ensuremath{\alpha}(K)exp=0.0053 \textit{15} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1971Hn04,B}{1971Hn04}).\vspace{0.1cm}}&\\
\end{longtable}
\clearpage
\begin{longtable}{ccccccccc@{}ccccccc@{\extracolsep{\fill}}c}
\\[-.4cm]
\multicolumn{17}{c}{{\bf \small \underline{Adopted \hyperlink{201TL_LEVEL}{Levels}, \hyperlink{201TL_GAMMA}{Gammas} (continued)}}}\\
\multicolumn{17}{c}{~}\\
\multicolumn{17}{c}{\underline{$\gamma$($^{201}$Tl) (continued)}}\\
\multicolumn{17}{c}{~~~}\\
\multicolumn{2}{c}{E\ensuremath{_{i}}(level)}&J\ensuremath{^{\pi}_{i}}&\multicolumn{2}{c}{E\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{TL20GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{I\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{TL20GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{E\ensuremath{_{f}}}&J\ensuremath{^{\pi}_{f}}&Mult.\ensuremath{^{\hyperlink{TL20GAMMA3}{@}}}&\multicolumn{2}{c}{\ensuremath{\delta}\ensuremath{^{\hyperlink{TL20GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{\ensuremath{\alpha}\ensuremath{^{\hyperlink{TL20GAMMA4}{\&}}}}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endhead
\multicolumn{1}{r@{}}{1277}&\multicolumn{1}{@{.}l}{12}&\multicolumn{1}{l}{3/2\ensuremath{^{+}},5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{120}&\multicolumn{1}{@{.}l}{0 {\it 2}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{28 {\it 8}}&\multicolumn{1}{r@{}}{1157}&\multicolumn{1}{@{.}l}{43 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}},5/2\ensuremath{^{+}}}&\multicolumn{1}{l}{[M1]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{5}&\multicolumn{1}{@{.}l}{10 {\it 8}}&&\\
&&&\multicolumn{1}{r@{}}{584}&\multicolumn{1}{@{.}l}{60 {\it 8}}&\multicolumn{1}{r@{}}{49}&\multicolumn{1}{@{.}l}{8 {\it 24}}&\multicolumn{1}{r@{}}{692}&\multicolumn{1}{@{.}l}{52 }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{+}}}&\multicolumn{1}{l}{M1(+E2)}&\multicolumn{1}{r@{}}{$<$0}&\multicolumn{1}{@{.}l}{5}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{061 {\it 5}}&\parbox[t][0.3cm]{6.6791096cm}{\raggedright Mult.,\ensuremath{\delta}: \ensuremath{\alpha}(K)exp=0.051 \textit{6} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1979Do09,B}{1979Do09});\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.6791096cm}{\raggedright {\ }{\ }{\ }\ensuremath{\alpha}(K)exp=0.049 \textit{10} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1971Hn04,B}{1971Hn04}); \ensuremath{\alpha}(K)exp=0.06\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.6791096cm}{\raggedright {\ }{\ }{\ }\textit{1} and \ensuremath{\gamma}\ensuremath{\gamma}(\ensuremath{\theta}) (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1961Pe05,B}{1961Pe05}).\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{945}&\multicolumn{1}{@{.}l}{96 {\it 8}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{ }l}{{\it 7}}&\multicolumn{1}{r@{}}{331}&\multicolumn{1}{@{.}l}{17 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}}}&\multicolumn{1}{l}{M1(+E2)}&\multicolumn{1}{r@{}}{$<$0}&\multicolumn{1}{@{.}l}{6}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0174 {\it 16}}&\parbox[t][0.3cm]{6.6791096cm}{\raggedright Mult.,\ensuremath{\delta}: \ensuremath{\alpha}(K)exp=0.014 \textit{2} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1979Do09,B}{1979Do09});\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.6791096cm}{\raggedright {\ }{\ }{\ }\ensuremath{\alpha}(K)exp=0.0160 \textit{25} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1971Hn04,B}{1971Hn04}).\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{1277}&\multicolumn{1}{@{.}l}{11 {\it 7}}&\multicolumn{1}{r@{}}{23}&\multicolumn{1}{@{.}l}{6}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{1/2\ensuremath{^{+}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
\multicolumn{1}{r@{}}{1290}&\multicolumn{1}{@{.}l}{13}&\multicolumn{1}{l}{(9/2)\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{155}&\multicolumn{1}{@{.}l}{31 {\it 10}}&\multicolumn{1}{r@{}}{43}&\multicolumn{1}{@{ }l}{{\it 5}}&\multicolumn{1}{r@{}}{1134}&\multicolumn{1}{@{.}l}{86 }&\multicolumn{1}{@{}l}{7/2\ensuremath{^{+}}}&\multicolumn{1}{l}{[M1]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{451 {\it 35}}&&\\
&&&\multicolumn{1}{r@{}}{597}&\multicolumn{1}{@{.}l}{60 {\it 9}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{ }l}{{\it 5}}&\multicolumn{1}{r@{}}{692}&\multicolumn{1}{@{.}l}{52 }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{+}}}&\multicolumn{1}{l}{E2}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{01856 {\it 26}}&&\\
\multicolumn{1}{r@{}}{1330}&\multicolumn{1}{@{.}l}{42}&\multicolumn{1}{l}{3/2\ensuremath{^{+}},5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{231}&\multicolumn{1}{@{.}l}{87 {\it 10}}&\multicolumn{1}{r@{}}{17}&\multicolumn{1}{@{.}l}{0 {\it 21}}&\multicolumn{1}{r@{}}{1098}&\multicolumn{1}{@{.}l}{50 }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{+}}}&\multicolumn{1}{l}{[M1]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{797 {\it 11}}&&\\
&&&\multicolumn{1}{r@{}}{637}&\multicolumn{1}{@{.}l}{90 {\it 9}}&\multicolumn{1}{r@{}}{57}&\multicolumn{1}{@{ }l}{{\it 3}}&\multicolumn{1}{r@{}}{692}&\multicolumn{1}{@{.}l}{52 }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{+}}}&\multicolumn{1}{l}{[M1]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0525 {\it 7}}&&\\
&&&\multicolumn{1}{r@{}}{999}&\multicolumn{1}{@{.}l}{23 {\it 7}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{ }l}{{\it 5}}&\multicolumn{1}{r@{}}{331}&\multicolumn{1}{@{.}l}{17 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}}}&\multicolumn{1}{l}{[M1]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{01650 {\it 23}}&&\\
&&&\multicolumn{1}{r@{}}{1330}&\multicolumn{1}{@{.}l}{50 {\it 15}}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{3 {\it 4}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{1/2\ensuremath{^{+}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
\multicolumn{1}{r@{}}{1401}&\multicolumn{1}{@{.}l}{26}&\multicolumn{1}{l}{3/2\ensuremath{^{+}},5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{124}&\multicolumn{1}{@{.}l}{2 {\it 2}}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{5 {\it 7}}&\multicolumn{1}{r@{}}{1277}&\multicolumn{1}{@{.}l}{12 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}},5/2\ensuremath{^{+}}}&\multicolumn{1}{l}{[M1]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{4}&\multicolumn{1}{@{.}l}{63 {\it 7}}&&\\
&&&\multicolumn{1}{r@{}}{302}&\multicolumn{1}{@{.}l}{7 {\it 4}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{90 {\it 21}}&\multicolumn{1}{r@{}}{1098}&\multicolumn{1}{@{.}l}{50 }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{+}}}&\multicolumn{1}{l}{[M1]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{383 {\it 6}}&&\\
&&&\multicolumn{1}{r@{}}{708}&\multicolumn{1}{@{.}l}{75 {\it 9}}&\multicolumn{1}{r@{}}{64}&\multicolumn{1}{@{ }l}{{\it 3}}&\multicolumn{1}{r@{}}{692}&\multicolumn{1}{@{.}l}{52 }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{+}}}&\multicolumn{1}{l}{M1}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0399 {\it 6}}&\parbox[t][0.3cm]{6.6791096cm}{\raggedright Mult.: \ensuremath{\alpha}(K)exp=0.044 \textit{7} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1979Do09,B}{1979Do09});\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.6791096cm}{\raggedright {\ }{\ }{\ }\ensuremath{\alpha}(K)exp=0.049 \textit{10} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1971Hn04,B}{1971Hn04}).\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{1070}&\multicolumn{1}{@{.}l}{04 {\it 8}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{ }l}{{\it 6}}&\multicolumn{1}{r@{}}{331}&\multicolumn{1}{@{.}l}{17 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}}}&\multicolumn{1}{l}{E2+M1}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{8 {\it +10\textminus5}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0075 {\it 11}}&\parbox[t][0.3cm]{6.6791096cm}{\raggedright Mult.,\ensuremath{\delta}: \ensuremath{\alpha}(K)exp=0.0057 \textit{10}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.6791096cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1979Do09,B}{1979Do09});\ensuremath{\alpha}(K)exp=0.0069 \textit{15} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1971Hn04,B}{1971Hn04}).\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{1401}&\multicolumn{1}{@{.}l}{30 {\it 8}}&\multicolumn{1}{r@{}}{11}&\multicolumn{1}{@{.}l}{0 {\it 6}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{1/2\ensuremath{^{+}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
\multicolumn{1}{r@{}}{1413}&\multicolumn{1}{@{.}l}{43}&\multicolumn{1}{l}{11/2}&\multicolumn{1}{r@{}}{123}&\multicolumn{1}{@{.}l}{3\ensuremath{^{\hyperlink{TL20GAMMA1}{\ddagger}}} {\it 1}}&\multicolumn{1}{r@{}}{20}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{TL20GAMMA1}{\ddagger}}} {\it 3}}&\multicolumn{1}{r@{}}{1290}&\multicolumn{1}{@{.}l}{13 }&\multicolumn{1}{@{}l}{(9/2)\ensuremath{^{+}}}&&&&&&&\\
&&&\multicolumn{1}{r@{}}{493}&\multicolumn{1}{@{.}l}{9\ensuremath{^{\hyperlink{TL20GAMMA1}{\ddagger}\hyperlink{TL20GAMMA5}{a}}} {\it 1}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{TL20GAMMA1}{\ddagger}}} {\it 14}}&\multicolumn{1}{r@{}}{919}&\multicolumn{1}{@{.}l}{47 }&\multicolumn{1}{@{}l}{(9/2\ensuremath{^{-}})}&\multicolumn{1}{l}{D}&&&&&\parbox[t][0.3cm]{6.6791096cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}={\textminus}0.203 \textit{18}, A\ensuremath{_{\textnormal{4}}}=0.055 \textit{26}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1420}&\multicolumn{1}{@{.}l}{04}&\multicolumn{1}{l}{7/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{129}&\multicolumn{1}{@{.}l}{95 {\it 10}}&\multicolumn{1}{r@{}}{12}&\multicolumn{1}{@{.}l}{1 {\it 11}}&\multicolumn{1}{r@{}}{1290}&\multicolumn{1}{@{.}l}{13 }&\multicolumn{1}{@{}l}{(9/2)\ensuremath{^{+}}}&\multicolumn{1}{l}{[M1]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{4}&\multicolumn{1}{@{.}l}{07 {\it 6}}&&\\
&&&\multicolumn{1}{r@{}}{285}&\multicolumn{1}{@{.}l}{18 {\it 13}}&\multicolumn{1}{r@{}}{$\approx$9}&\multicolumn{1}{@{.}l}{8}&\multicolumn{1}{r@{}}{1134}&\multicolumn{1}{@{.}l}{86 }&\multicolumn{1}{@{}l}{7/2\ensuremath{^{+}}}&\multicolumn{1}{l}{[M1]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{450 {\it 6}}&&\\
&&&\multicolumn{1}{r@{}}{727}&\multicolumn{1}{@{.}l}{50 {\it 9}}&\multicolumn{1}{r@{}}{13}&\multicolumn{1}{@{.}l}{4 {\it 13}}&\multicolumn{1}{r@{}}{692}&\multicolumn{1}{@{.}l}{52 }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{+}}}&\multicolumn{1}{l}{[M1]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0373 {\it 5}}&&\\
&&&\multicolumn{1}{r@{}}{1088}&\multicolumn{1}{@{.}l}{85 {\it 9}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{ }l}{{\it 6}}&\multicolumn{1}{r@{}}{331}&\multicolumn{1}{@{.}l}{17 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}}}&\multicolumn{1}{l}{[E2]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{00537 {\it 8}}&&\\
\multicolumn{1}{r@{}}{1445}&\multicolumn{1}{@{.}l}{87}&\multicolumn{1}{l}{(5/2)\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{753}&\multicolumn{1}{@{.}l}{35 {\it 9}}&\multicolumn{1}{r@{}}{90}&\multicolumn{1}{@{ }l}{{\it 8}}&\multicolumn{1}{r@{}}{692}&\multicolumn{1}{@{.}l}{52 }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{+}}}&\multicolumn{1}{l}{(E0+M1)}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{6.6791096cm}{\raggedright Mult.: \ensuremath{\alpha}(K)exp=0.087 \textit{16} (1979Do0.\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{1114}&\multicolumn{1}{@{.}l}{73 {\it 8}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{ }l}{{\it 6}}&\multicolumn{1}{r@{}}{331}&\multicolumn{1}{@{.}l}{17 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}}}&\multicolumn{1}{l}{M1(+E2)}&\multicolumn{1}{r@{}}{$<$0}&\multicolumn{1}{@{.}l}{3}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{01218 {\it 35}}&\parbox[t][0.3cm]{6.6791096cm}{\raggedright Mult.,\ensuremath{\delta}: \ensuremath{\alpha}(K)exp=0.010 \textit{2} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1979Do09,B}{1979Do09});\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.6791096cm}{\raggedright {\ }{\ }{\ }\ensuremath{\alpha}(K)exp=0.011 \textit{3} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1971Hn04,B}{1971Hn04}).\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{1445}&\multicolumn{1}{@{.}l}{80 {\it 10}}&\multicolumn{1}{r@{}}{21}&\multicolumn{1}{@{.}l}{4 {\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{1/2\ensuremath{^{+}}}&\multicolumn{1}{l}{[E2]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{00319 {\it 4}}&&\\
\multicolumn{1}{r@{}}{1479}&\multicolumn{1}{@{.}l}{85}&\multicolumn{1}{l}{5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{202}&\multicolumn{1}{@{.}l}{79 {\it 10}}&\multicolumn{1}{r@{}}{8}&\multicolumn{1}{@{.}l}{5 {\it 11}}&\multicolumn{1}{r@{}}{1277}&\multicolumn{1}{@{.}l}{12 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}},5/2\ensuremath{^{+}}}&\multicolumn{1}{l}{[M1+E2]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{157 {\it 16}}&&\\
&&&\multicolumn{1}{r@{}}{241}&\multicolumn{1}{@{.}l}{02 {\it 8}}&\multicolumn{1}{r@{}}{21}&\multicolumn{1}{@{.}l}{1 {\it 21}}&\multicolumn{1}{r@{}}{1238}&\multicolumn{1}{@{.}l}{83 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}}}&\multicolumn{1}{l}{[M1+E2]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{716 {\it 10}}&&\\
&&&\multicolumn{1}{r@{}}{322}&\multicolumn{1}{@{.}l}{42 {\it 15}}&\multicolumn{1}{r@{}}{9}&\multicolumn{1}{@{.}l}{3 {\it 13}}&\multicolumn{1}{r@{}}{1157}&\multicolumn{1}{@{.}l}{43 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}},5/2\ensuremath{^{+}}}&\multicolumn{1}{l}{[M1+E2]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{322 {\it 5}}&&\\
&&&\multicolumn{1}{r@{}}{344}&\multicolumn{1}{@{.}l}{95 {\it 7}}&\multicolumn{1}{r@{}}{39}&\multicolumn{1}{@{ }l}{{\it 3}}&\multicolumn{1}{r@{}}{1134}&\multicolumn{1}{@{.}l}{86 }&\multicolumn{1}{@{}l}{7/2\ensuremath{^{+}}}&\multicolumn{1}{l}{M1(+E2)}&\multicolumn{1}{r@{}}{$<$0}&\multicolumn{1}{@{.}l}{6}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{243 {\it 26}}&\parbox[t][0.3cm]{6.6791096cm}{\raggedright Mult.,\ensuremath{\delta}: \ensuremath{\alpha}(K)exp=0.21 \textit{11} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1979Do09,B}{1979Do09});\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.6791096cm}{\raggedright {\ }{\ }{\ }\ensuremath{\alpha}(K)exp=0.23 \textit{6} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1971Hn04,B}{1971Hn04}).\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{381}&\multicolumn{1}{@{.}l}{29 {\it 8}}&\multicolumn{1}{r@{}}{27}&\multicolumn{1}{@{.}l}{3 {\it 15}}&\multicolumn{1}{r@{}}{1098}&\multicolumn{1}{@{.}l}{50 }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{+}}}&\multicolumn{1}{l}{M1(+E2)}&\multicolumn{1}{r@{}}{$<$0}&\multicolumn{1}{@{.}l}{5}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{190 {\it 15}}&\parbox[t][0.3cm]{6.6791096cm}{\raggedright Mult.,\ensuremath{\delta}: \ensuremath{\alpha}(K)exp=0.17 \textit{3} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1979Do09,B}{1979Do09});\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.6791096cm}{\raggedright {\ }{\ }{\ }\ensuremath{\alpha}(K)exp=0.16 \textit{4} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1971Hn04,B}{1971Hn04}).\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{787}&\multicolumn{1}{@{.}l}{29 {\it 10}}&\multicolumn{1}{r@{}}{72}&\multicolumn{1}{@{ }l}{{\it 9}}&\multicolumn{1}{r@{}}{692}&\multicolumn{1}{@{.}l}{52 }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{+}}}&\multicolumn{1}{l}{[M1+E2]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0304 {\it 4}}&&\\
&&&\multicolumn{1}{r@{}}{1148}&\multicolumn{1}{@{.}l}{75 {\it 8}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{ }l}{{\it 5}}&\multicolumn{1}{r@{}}{331}&\multicolumn{1}{@{.}l}{17 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}}}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{1 {\it +4\textminus3}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0079 {\it 11}}&\parbox[t][0.3cm]{6.6791096cm}{\raggedright Mult.,\ensuremath{\delta}: \ensuremath{\alpha}(K)exp=0.0065 \textit{10} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1979Do09,B}{1979Do09});\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.6791096cm}{\raggedright {\ }{\ }{\ }\ensuremath{\alpha}(K)exp=0.0076 \textit{20} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1971Hn04,B}{1971Hn04}).\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{1479}&\multicolumn{1}{@{.}l}{91 {\it 10}}&\multicolumn{1}{r@{}}{22}&\multicolumn{1}{@{.}l}{2 {\it 11}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{1/2\ensuremath{^{+}}}&\multicolumn{1}{l}{[E2]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{00307 {\it 4}}&&\\
\multicolumn{1}{r@{}}{1550}&\multicolumn{1}{@{.}l}{59?}&\multicolumn{1}{l}{1/2,3/2,5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1219}&\multicolumn{1}{@{.}l}{40\ensuremath{^{\hyperlink{TL20GAMMA5}{a}}} {\it 15}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{ }l}{{\it 7}}&\multicolumn{1}{r@{}}{331}&\multicolumn{1}{@{.}l}{17 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}}}&&&&&&&\\
&&&\multicolumn{1}{r@{}}{1550}&\multicolumn{1}{@{.}l}{5\ensuremath{^{\hyperlink{TL20GAMMA5}{a}}} {\it 4}}&\multicolumn{1}{r@{}}{19}&\multicolumn{1}{@{ }l}{{\it 3}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{1/2\ensuremath{^{+}}}&&&&&&&\\
\end{longtable}
\clearpage
\begin{longtable}{ccccccccc@{}ccccccc@{\extracolsep{\fill}}c}
\\[-.4cm]
\multicolumn{17}{c}{{\bf \small \underline{Adopted \hyperlink{201TL_LEVEL}{Levels}, \hyperlink{201TL_GAMMA}{Gammas} (continued)}}}\\
\multicolumn{17}{c}{~}\\
\multicolumn{17}{c}{\underline{$\gamma$($^{201}$Tl) (continued)}}\\
\multicolumn{17}{c}{~~~}\\
\multicolumn{2}{c}{E\ensuremath{_{i}}(level)}&J\ensuremath{^{\pi}_{i}}&\multicolumn{2}{c}{E\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{TL20GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{I\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{TL20GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{E\ensuremath{_{f}}}&J\ensuremath{^{\pi}_{f}}&Mult.\ensuremath{^{\hyperlink{TL20GAMMA3}{@}}}&\multicolumn{2}{c}{\ensuremath{\delta}\ensuremath{^{\hyperlink{TL20GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{\ensuremath{\alpha}\ensuremath{^{\hyperlink{TL20GAMMA4}{\&}}}}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endhead
\multicolumn{1}{r@{}}{1571}&\multicolumn{1}{@{.}l}{86}&\multicolumn{1}{l}{(13/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{333}&\multicolumn{1}{@{.}l}{4\ensuremath{^{\hyperlink{TL20GAMMA2}{\#}}} {\it 1}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{TL20GAMMA2}{\#}}} {\it 5}}&\multicolumn{1}{r@{}}{1238}&\multicolumn{1}{@{.}l}{46 }&\multicolumn{1}{@{}l}{(11/2\ensuremath{^{-}})}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{$-$0}&\multicolumn{1}{@{.}l}{21 {\it +14\textminus9}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{285 {\it 9}}&\parbox[t][0.3cm]{6.751869cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}={\textminus}0.383 \textit{29}, A\ensuremath{_{\textnormal{4}}}=0.052 \textit{46} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1977Sl01,B}{1977Sl01});\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.751869cm}{\raggedright {\ }{\ }{\ }DCO=1.13 \textit{3}, POL={\textminus}0.026 \textit{19} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2013Da15,B}{2013Da15}).\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.751869cm}{\raggedright \ensuremath{\delta}: From \ensuremath{\gamma}(\ensuremath{\theta}) in \ensuremath{^{\textnormal{202}}}Hg(d,3n\ensuremath{\gamma}) (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1977Sl01,B}{1977Sl01}).\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{652}&\multicolumn{1}{@{.}l}{2\ensuremath{^{\hyperlink{TL20GAMMA2}{\#}}} {\it 5}}&\multicolumn{1}{r@{}}{44}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{TL20GAMMA2}{\#}}} {\it 3}}&\multicolumn{1}{r@{}}{919}&\multicolumn{1}{@{.}l}{47 }&\multicolumn{1}{@{}l}{(9/2\ensuremath{^{-}})}&\multicolumn{1}{l}{E2}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{01528 {\it 22}}&\parbox[t][0.3cm]{6.751869cm}{\raggedright I\ensuremath{_{\gamma}}: Other: I\ensuremath{\gamma}=29 \textit{3} in \ensuremath{^{\textnormal{202}}}Hg(d,3n\ensuremath{\gamma}) (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1977Sl01,B}{1977Sl01}).\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.751869cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}=0.204 \textit{50}, A\ensuremath{_{\textnormal{4}}}={\textminus}0.062 \textit{77} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1977Sl01,B}{1977Sl01});\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.751869cm}{\raggedright {\ }{\ }{\ }DCO=0.53 \textit{4}, POL=+0.117 \textit{24} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2013Da15,B}{2013Da15}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1575}&\multicolumn{1}{@{.}l}{1}&\multicolumn{1}{l}{(7/2)\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{285}&\multicolumn{1}{@{.}l}{0 {\it 10}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{1290}&\multicolumn{1}{@{.}l}{13 }&\multicolumn{1}{@{}l}{(9/2)\ensuremath{^{+}}}&\multicolumn{1}{l}{(M1)}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{451 {\it 8}}&\parbox[t][0.3cm]{6.751869cm}{\raggedright Mult.: \ensuremath{\alpha}(K)exp=0.37 \textit{8} in in \ensuremath{^{\textnormal{201}}}Pb \ensuremath{\varepsilon} decay\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.751869cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1979Do09,B}{1979Do09}), but the transition is multiple\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.751869cm}{\raggedright {\ }{\ }{\ }placed.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1617}&\multicolumn{1}{@{.}l}{46}&\multicolumn{1}{l}{1/2,3/2,5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1286}&\multicolumn{1}{@{.}l}{3\ensuremath{^{\hyperlink{TL20GAMMA5}{a}}} {\it 2}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{ }l}{{\it 5}}&\multicolumn{1}{r@{}}{331}&\multicolumn{1}{@{.}l}{17 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}}}&&&&&&&\\
&&&\multicolumn{1}{r@{}}{1617}&\multicolumn{1}{@{.}l}{45 {\it 15}}&\multicolumn{1}{r@{}}{37}&\multicolumn{1}{@{ }l}{{\it 3}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{1/2\ensuremath{^{+}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
\multicolumn{1}{r@{}}{1639}&\multicolumn{1}{@{.}l}{36}&\multicolumn{1}{l}{3/2\ensuremath{^{+}},5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{308}&\multicolumn{1}{@{.}l}{93 {\it 15}}&\multicolumn{1}{r@{}}{7}&\multicolumn{1}{@{.}l}{1 {\it 9}}&\multicolumn{1}{r@{}}{1330}&\multicolumn{1}{@{.}l}{42 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}},5/2\ensuremath{^{+}}}&\multicolumn{1}{l}{(M1)}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{362 {\it 5}}&\parbox[t][0.3cm]{6.751869cm}{\raggedright Mult.: \ensuremath{\alpha}(K)exp=0.67 \textit{15} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1979Do09,B}{1979Do09}), indicating\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.751869cm}{\raggedright {\ }{\ }{\ }E0 admixtures, but \ensuremath{\alpha}(K)exp=0.37 \textit{13}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.751869cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1971Hn04,B}{1971Hn04}), consistent with M1.\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{481}&\multicolumn{1}{@{.}l}{98 {\it 9}}&\multicolumn{1}{r@{}}{9}&\multicolumn{1}{@{.}l}{8 {\it 18}}&\multicolumn{1}{r@{}}{1157}&\multicolumn{1}{@{.}l}{43 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}},5/2\ensuremath{^{+}}}&\multicolumn{1}{l}{[M1]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{1096 {\it 15}}&&\\
&&&\multicolumn{1}{r@{}}{540}&\multicolumn{1}{@{.}l}{90 {\it 9}}&\multicolumn{1}{r@{}}{50}&\multicolumn{1}{@{ }l}{{\it 3}}&\multicolumn{1}{r@{}}{1098}&\multicolumn{1}{@{.}l}{50 }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{+}}}&\multicolumn{1}{l}{[M1]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0809 {\it 11}}&&\\
&&&\multicolumn{1}{r@{}}{946}&\multicolumn{1}{@{.}l}{78 {\it 4}}&\multicolumn{1}{r@{}}{86}&\multicolumn{1}{@{ }l}{{\it 31}}&\multicolumn{1}{r@{}}{692}&\multicolumn{1}{@{.}l}{52 }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{+}}}&\multicolumn{1}{l}{[M1]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{01894 {\it 27}}&&\\
&&&\multicolumn{1}{r@{}}{1308}&\multicolumn{1}{@{.}l}{32 {\it 8}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{ }l}{{\it 5}}&\multicolumn{1}{r@{}}{331}&\multicolumn{1}{@{.}l}{17 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}}}&\multicolumn{1}{l}{M1(+E2)}&\multicolumn{1}{r@{}}{$<$0}&\multicolumn{1}{@{.}l}{6}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0077 {\it 6}}&\parbox[t][0.3cm]{6.751869cm}{\raggedright Mult.,\ensuremath{\delta}: \ensuremath{\alpha}(K)exp=0.009 \textit{4} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1979Do09,B}{1979Do09});\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.751869cm}{\raggedright {\ }{\ }{\ }\ensuremath{\alpha}(K)exp=0.010 \textit{3} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1971Hn04,B}{1971Hn04}).\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{1639}&\multicolumn{1}{@{.}l}{1 {\it 5}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{61 {\it 15}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{1/2\ensuremath{^{+}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
\multicolumn{1}{r@{}}{1671}&\multicolumn{1}{@{.}l}{96}&\multicolumn{1}{l}{3/2\ensuremath{^{+}},5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{341}&\multicolumn{1}{@{.}l}{51 {\it 8}}&\multicolumn{1}{r@{}}{25}&\multicolumn{1}{@{ }l}{{\it 3}}&\multicolumn{1}{r@{}}{1330}&\multicolumn{1}{@{.}l}{42 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}},5/2\ensuremath{^{+}}}&\multicolumn{1}{l}{[M1]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{276 {\it 4}}&&\\
&&&\multicolumn{1}{r@{}}{394}&\multicolumn{1}{@{.}l}{86 {\it 9}}&\multicolumn{1}{r@{}}{40}&\multicolumn{1}{@{ }l}{{\it 3}}&\multicolumn{1}{r@{}}{1277}&\multicolumn{1}{@{.}l}{12 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}},5/2\ensuremath{^{+}}}&\multicolumn{1}{l}{M1+(E2)}&\multicolumn{1}{r@{}}{$<$0}&\multicolumn{1}{@{.}l}{4}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{177 {\it 10}}&\parbox[t][0.3cm]{6.751869cm}{\raggedright Mult.,\ensuremath{\delta}: \ensuremath{\alpha}(K)exp=0.17 \textit{5} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1979Do09,B}{1979Do09});\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.751869cm}{\raggedright {\ }{\ }{\ }\ensuremath{\alpha}(K)exp=0.22 \textit{7} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1971Hn04,B}{1971Hn04}).\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{514}&\multicolumn{1}{@{.}l}{38 {\it 9}}&\multicolumn{1}{r@{}}{34}&\multicolumn{1}{@{ }l}{{\it 7}}&\multicolumn{1}{r@{}}{1157}&\multicolumn{1}{@{.}l}{43 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}},5/2\ensuremath{^{+}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
&&&\multicolumn{1}{r@{}}{979}&\multicolumn{1}{@{.}l}{4 {\it 3}}&\multicolumn{1}{r@{}}{4}&\multicolumn{1}{@{.}l}{1 {\it 11}}&\multicolumn{1}{r@{}}{692}&\multicolumn{1}{@{.}l}{52 }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{+}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
&&&\multicolumn{1}{r@{}}{1340}&\multicolumn{1}{@{.}l}{88 {\it 9}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{ }l}{{\it 6}}&\multicolumn{1}{r@{}}{331}&\multicolumn{1}{@{.}l}{17 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
&&&\multicolumn{1}{r@{}}{1672}&\multicolumn{1}{@{.}l}{02 {\it 10}}&\multicolumn{1}{r@{}}{5}&\multicolumn{1}{@{.}l}{4 {\it 4}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{1/2\ensuremath{^{+}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
\multicolumn{1}{r@{}}{1712}&\multicolumn{1}{@{.}l}{45}&\multicolumn{1}{l}{3/2,5/2,7/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1019}&\multicolumn{1}{@{.}l}{8 {\it 3}}&\multicolumn{1}{r@{}}{82}&\multicolumn{1}{@{ }l}{{\it 36}}&\multicolumn{1}{r@{}}{692}&\multicolumn{1}{@{.}l}{52 }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{+}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
&&&\multicolumn{1}{r@{}}{1381}&\multicolumn{1}{@{.}l}{4 {\it 3}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{ }l}{{\it 18}}&\multicolumn{1}{r@{}}{331}&\multicolumn{1}{@{.}l}{17 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
\multicolumn{1}{r@{}}{1755}&\multicolumn{1}{@{.}l}{33}&\multicolumn{1}{l}{3/2,5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1062}&\multicolumn{1}{@{.}l}{79 {\it 15}}&\multicolumn{1}{r@{}}{69}&\multicolumn{1}{@{ }l}{{\it 9}}&\multicolumn{1}{r@{}}{692}&\multicolumn{1}{@{.}l}{52 }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{+}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
&&&\multicolumn{1}{r@{}}{1424}&\multicolumn{1}{@{.}l}{16 {\it 9}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{ }l}{{\it 5}}&\multicolumn{1}{r@{}}{331}&\multicolumn{1}{@{.}l}{17 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
&&&\multicolumn{1}{r@{}}{1755}&\multicolumn{1}{@{.}l}{32 {\it 10}}&\multicolumn{1}{r@{}}{11}&\multicolumn{1}{@{.}l}{2 {\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{1/2\ensuremath{^{+}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
\multicolumn{1}{r@{}}{1962}&\multicolumn{1}{@{.}l}{76}&\multicolumn{1}{l}{(15/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{390}&\multicolumn{1}{@{.}l}{9\ensuremath{^{\hyperlink{TL20GAMMA2}{\#}}} {\it 1}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{TL20GAMMA2}{\#}}} {\it 6}}&\multicolumn{1}{r@{}}{1571}&\multicolumn{1}{@{.}l}{86 }&\multicolumn{1}{@{}l}{(13/2\ensuremath{^{-}})}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{$-$0}&\multicolumn{1}{@{.}l}{20 {\it +8\textminus7}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{186 {\it 5}}&\parbox[t][0.3cm]{6.751869cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}={\textminus}0.407 \textit{26}, A\ensuremath{_{\textnormal{4}}}=0.010 \textit{37} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1977Sl01,B}{1977Sl01});\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.751869cm}{\raggedright {\ }{\ }{\ }DCO=1.48 \textit{4}, POL={\textminus}0.016 \textit{28} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2013Da15,B}{2013Da15}).\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.751869cm}{\raggedright \ensuremath{\delta}: From \ensuremath{\gamma}(\ensuremath{\theta}) in \ensuremath{^{\textnormal{202}}}Hg(d,3n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{724}&\multicolumn{1}{@{.}l}{3\ensuremath{^{\hyperlink{TL20GAMMA2}{\#}}} {\it 1}}&\multicolumn{1}{r@{}}{88}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{TL20GAMMA2}{\#}}} {\it 7}}&\multicolumn{1}{r@{}}{1238}&\multicolumn{1}{@{.}l}{46 }&\multicolumn{1}{@{}l}{(11/2\ensuremath{^{-}})}&\multicolumn{1}{l}{E2}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{01220 {\it 17}}&\parbox[t][0.3cm]{6.751869cm}{\raggedright I\ensuremath{_{\gamma}}: Other: I\ensuremath{\gamma}=38 \textit{7} in \ensuremath{^{\textnormal{202}}}Hg(d,3n\ensuremath{\gamma}) (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1977Sl01,B}{1977Sl01}).\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.751869cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}=0.33 \textit{15} with A\ensuremath{_{\textnormal{4}}} set to zero\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.751869cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1977Sl01,B}{1977Sl01}); DCO=0.54 \textit{2}, POL=+0.062 \textit{32}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.751869cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2013Da15,B}{2013Da15}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1987}&\multicolumn{1}{@{.}l}{96}&\multicolumn{1}{l}{(13/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{749}&\multicolumn{1}{@{.}l}{5\ensuremath{^{\hyperlink{TL20GAMMA2}{\#}}} {\it 1}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{TL20GAMMA2}{\#}}}}&\multicolumn{1}{r@{}}{1238}&\multicolumn{1}{@{.}l}{46 }&\multicolumn{1}{@{}l}{(11/2\ensuremath{^{-}})}&\multicolumn{1}{l}{M1(+E2)}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0345 {\it 5}}&\parbox[t][0.3cm]{6.751869cm}{\raggedright Mult.: DCO=1.41 \textit{18}, POL={\textminus}0.07 \textit{5} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2013Da15,B}{2013Da15}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2015}&\multicolumn{1}{@{.}l}{2}&\multicolumn{1}{l}{(17/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{443}&\multicolumn{1}{@{.}l}{3\ensuremath{^{\hyperlink{TL20GAMMA2}{\#}}} {\it 8}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{TL20GAMMA2}{\#}}}}&\multicolumn{1}{r@{}}{1571}&\multicolumn{1}{@{.}l}{86 }&\multicolumn{1}{@{}l}{(13/2\ensuremath{^{-}})}&\multicolumn{1}{l}{E2}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0379 {\it 6}}&\parbox[t][0.3cm]{6.751869cm}{\raggedright B(E2)(W.u.)=0.157 \textit{+33{\textminus}49}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.751869cm}{\raggedright Mult.: DCO=1.01 \textit{3}, POL=+0.077 \textit{30}\vspace{0.1cm}}&\\
\end{longtable}
\clearpage
\begin{longtable}{ccccccccc@{}ccccc@{\extracolsep{\fill}}c}
\\[-.4cm]
\multicolumn{15}{c}{{\bf \small \underline{Adopted \hyperlink{201TL_LEVEL}{Levels}, \hyperlink{201TL_GAMMA}{Gammas} (continued)}}}\\
\multicolumn{15}{c}{~}\\
\multicolumn{15}{c}{\underline{$\gamma$($^{201}$Tl) (continued)}}\\
\multicolumn{15}{c}{~~~}\\
\multicolumn{2}{c}{E\ensuremath{_{i}}(level)}&J\ensuremath{^{\pi}_{i}}&\multicolumn{2}{c}{E\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{TL20GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{I\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{TL20GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{E\ensuremath{_{f}}}&J\ensuremath{^{\pi}_{f}}&Mult.\ensuremath{^{\hyperlink{TL20GAMMA3}{@}}}&\multicolumn{2}{c}{\ensuremath{\alpha}\ensuremath{^{\hyperlink{TL20GAMMA4}{\&}}}}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endhead
&&&&&&&&&&&&&\parbox[t][0.3cm]{9.6951cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2013Da15,B}{2013Da15}); interpreted as Mult.=(M1+E2) and \ensuremath{\delta} \ensuremath{\approx} 0.3 in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1977Sl01,B}{1977Sl01}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{9.6951cm}{\raggedright {\ }{\ }{\ }(A\ensuremath{_{\textnormal{2}}}=0.162 \textit{10}, A\ensuremath{_{\textnormal{4}}}=0.002 \textit{15}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2041}&\multicolumn{1}{@{.}l}{3}&\multicolumn{1}{l}{(15/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{469}&\multicolumn{1}{@{.}l}{4\ensuremath{^{\hyperlink{TL20GAMMA2}{\#}}} {\it 2}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{TL20GAMMA2}{\#}}}}&\multicolumn{1}{r@{}}{1571}&\multicolumn{1}{@{.}l}{86 }&\multicolumn{1}{@{}l}{(13/2\ensuremath{^{-}})}&\multicolumn{1}{l}{E1}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{01054 {\it 15}}&\parbox[t][0.3cm]{9.6951cm}{\raggedright Mult.: DCO=1.8 \textit{7}, POL=+0.13 \textit{7} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2013Da15,B}{2013Da15}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2182}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{l}{(19/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{166}&\multicolumn{1}{@{.}l}{9\ensuremath{^{\hyperlink{TL20GAMMA2}{\#}}} {\it 1}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{TL20GAMMA2}{\#}}}}&\multicolumn{1}{r@{}}{2015}&\multicolumn{1}{@{.}l}{2 }&\multicolumn{1}{@{}l}{(17/2\ensuremath{^{-}})}&\multicolumn{1}{l}{M1(+E2)}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{000 {\it 28}}&\parbox[t][0.3cm]{9.6951cm}{\raggedright Mult.: DCO=1.43 \textit{4} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2013Da15,B}{2013Da15}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2442}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{l}{(21/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{260}&\multicolumn{1}{@{.}l}{1\ensuremath{^{\hyperlink{TL20GAMMA2}{\#}}} {\it 1}}&\multicolumn{1}{r@{}}{9}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{TL20GAMMA2}{\#}}} {\it 5}}&\multicolumn{1}{r@{}}{2182}&\multicolumn{1}{@{.}l}{0 }&\multicolumn{1}{@{}l}{(19/2\ensuremath{^{-}})}&&&&&\\
&&&\multicolumn{1}{r@{}}{426}&\multicolumn{1}{@{.}l}{8\ensuremath{^{\hyperlink{TL20GAMMA2}{\#}}} {\it 1}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{TL20GAMMA2}{\#}}} {\it 6}}&\multicolumn{1}{r@{}}{2015}&\multicolumn{1}{@{.}l}{2 }&\multicolumn{1}{@{}l}{(17/2\ensuremath{^{-}})}&\multicolumn{1}{l}{E2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0417 {\it 6}}&\parbox[t][0.3cm]{9.6951cm}{\raggedright Mult.: DCO=1.06 \textit{2}, POL=+0.143 \textit{23} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2013Da15,B}{2013Da15}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2486}&\multicolumn{1}{@{.}l}{5}&\multicolumn{1}{l}{(17/2)}&\multicolumn{1}{r@{}}{445}&\multicolumn{1}{@{.}l}{2\ensuremath{^{\hyperlink{TL20GAMMA2}{\#}}} {\it 7}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{TL20GAMMA2}{\#}}}}&\multicolumn{1}{r@{}}{2041}&\multicolumn{1}{@{.}l}{3 }&\multicolumn{1}{@{}l}{(15/2\ensuremath{^{+}})}&\multicolumn{1}{l}{D}&&&\parbox[t][0.3cm]{9.6951cm}{\raggedright Mult.: DCO=0.71 \textit{5} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2013Da15,B}{2013Da15}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2672}&\multicolumn{1}{@{.}l}{1}&\multicolumn{1}{l}{(23/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{230}&\multicolumn{1}{@{.}l}{2\ensuremath{^{\hyperlink{TL20GAMMA2}{\#}}} {\it 1}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{TL20GAMMA2}{\#}}}}&\multicolumn{1}{r@{}}{2442}&\multicolumn{1}{@{.}l}{0 }&\multicolumn{1}{@{}l}{(21/2\ensuremath{^{-}})}&\multicolumn{1}{l}{E1}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0539 {\it 8}}&\parbox[t][0.3cm]{9.6951cm}{\raggedright Mult.: DCO=1.30 \textit{4}, POL=+0.07 \textit{6} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2013Da15,B}{2013Da15}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2713}&\multicolumn{1}{@{.}l}{66}&\multicolumn{1}{l}{(15/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{1141}&\multicolumn{1}{@{.}l}{8\ensuremath{^{\hyperlink{TL20GAMMA2}{\#}}} {\it 1}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{TL20GAMMA2}{\#}}}}&\multicolumn{1}{r@{}}{1571}&\multicolumn{1}{@{.}l}{86 }&\multicolumn{1}{@{}l}{(13/2\ensuremath{^{-}})}&\multicolumn{1}{l}{M1(+E2)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{01174 {\it 16}}&\parbox[t][0.3cm]{9.6951cm}{\raggedright Mult.: DCO=2.13 \textit{13}, POL={\textminus}0.032 \textit{25} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2013Da15,B}{2013Da15}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2748}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{l}{(17/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{(34}&\multicolumn{1}{@{.}l}{3\ensuremath{^{\hyperlink{TL20GAMMA2}{\#}}} {\it 7})}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{TL20GAMMA2}{\#}}}}&\multicolumn{1}{r@{}}{2713}&\multicolumn{1}{@{.}l}{66 }&\multicolumn{1}{@{}l}{(15/2\ensuremath{^{-}})}&&&&\parbox[t][0.3cm]{9.6951cm}{\raggedright E\ensuremath{_{\gamma}}: From level energy differences. Not observed directly, but required\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{9.6951cm}{\raggedright {\ }{\ }{\ }by the coincidence relationship (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2013Da15,B}{2013Da15}).\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{785}&\multicolumn{1}{@{.}l}{2\ensuremath{^{\hyperlink{TL20GAMMA2}{\#}}} {\it 6}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{TL20GAMMA2}{\#}}}}&\multicolumn{1}{r@{}}{1962}&\multicolumn{1}{@{.}l}{76 }&\multicolumn{1}{@{}l}{(15/2\ensuremath{^{-}})}&\multicolumn{1}{l}{E1}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{00375 {\it 5}}&\parbox[t][0.3cm]{9.6951cm}{\raggedright Mult.: DCO=2.13 \textit{13}, POL={\textminus}0.032 \textit{25} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2013Da15,B}{2013Da15}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2883}&\multicolumn{1}{@{.}l}{7}&\multicolumn{1}{l}{(23/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{211}&\multicolumn{1}{@{.}l}{8\ensuremath{^{\hyperlink{TL20GAMMA2}{\#}}} {\it 2}}&\multicolumn{1}{r@{}}{10}&\multicolumn{1}{@{.}l}{3\ensuremath{^{\hyperlink{TL20GAMMA2}{\#}}} {\it 22}}&\multicolumn{1}{r@{}}{2672}&\multicolumn{1}{@{.}l}{1 }&\multicolumn{1}{@{}l}{(23/2\ensuremath{^{+}})}&&&&&\\
&&&\multicolumn{1}{r@{}}{441}&\multicolumn{1}{@{.}l}{1\ensuremath{^{\hyperlink{TL20GAMMA2}{\#}}} {\it 7}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{TL20GAMMA2}{\#}}} {\it 38}}&\multicolumn{1}{r@{}}{2442}&\multicolumn{1}{@{.}l}{0 }&\multicolumn{1}{@{}l}{(21/2\ensuremath{^{-}})}&\multicolumn{1}{l}{M1(+E2)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{1387 {\it 20}}&\parbox[t][0.3cm]{9.6951cm}{\raggedright Mult.: DCO=1.92 \textit{7}, POL={\textminus}0.059 \textit{29} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2013Da15,B}{2013Da15}).\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{700}&\multicolumn{1}{@{.}l}{9\ensuremath{^{\hyperlink{TL20GAMMA2}{\#}}} {\it 3}}&\multicolumn{1}{r@{}}{27}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{TL20GAMMA2}{\#}}} {\it 9}}&\multicolumn{1}{r@{}}{2182}&\multicolumn{1}{@{.}l}{0 }&\multicolumn{1}{@{}l}{(19/2\ensuremath{^{-}})}&\multicolumn{1}{l}{E2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{01308 {\it 18}}&\parbox[t][0.3cm]{9.6951cm}{\raggedright Mult.: DCO=1.09 \textit{12}\hphantom{a}(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2013Da15,B}{2013Da15}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2897}&\multicolumn{1}{@{.}l}{6}&\multicolumn{1}{l}{(25/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{225}&\multicolumn{1}{@{.}l}{5\ensuremath{^{\hyperlink{TL20GAMMA2}{\#}}} {\it 1}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{TL20GAMMA2}{\#}}}}&\multicolumn{1}{r@{}}{2672}&\multicolumn{1}{@{.}l}{1 }&\multicolumn{1}{@{}l}{(23/2\ensuremath{^{+}})}&\multicolumn{1}{l}{M1(+E2)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{861 {\it 12}}&\parbox[t][0.3cm]{9.6951cm}{\raggedright Mult.: DCO=1.42 \textit{7}, POL={\textminus}0.05 \textit{10} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2013Da15,B}{2013Da15}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2985}&\multicolumn{1}{@{.}l}{1}&\multicolumn{1}{l}{(21/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{543}&\multicolumn{1}{@{.}l}{2\ensuremath{^{\hyperlink{TL20GAMMA2}{\#}}} {\it 4}}&\multicolumn{1}{r@{}}{11}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{TL20GAMMA2}{\#}}} {\it 3}}&\multicolumn{1}{r@{}}{2442}&\multicolumn{1}{@{.}l}{0 }&\multicolumn{1}{@{}l}{(21/2\ensuremath{^{-}})}&&&&&\\
&&&\multicolumn{1}{r@{}}{803}&\multicolumn{1}{@{.}l}{1\ensuremath{^{\hyperlink{TL20GAMMA2}{\#}}} {\it 1}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{TL20GAMMA2}{\#}}} {\it 8}}&\multicolumn{1}{r@{}}{2182}&\multicolumn{1}{@{.}l}{0 }&\multicolumn{1}{@{}l}{(19/2\ensuremath{^{-}})}&\multicolumn{1}{l}{E1}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{00360 {\it 5}}&\parbox[t][0.3cm]{9.6951cm}{\raggedright Mult.: DCO=1.90 \textit{16}, POL=+0.066 \textit{21} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2013Da15,B}{2013Da15}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3011}&\multicolumn{1}{@{.}l}{7}&\multicolumn{1}{l}{(19/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{263}&\multicolumn{1}{@{.}l}{7\ensuremath{^{\hyperlink{TL20GAMMA2}{\#}}} {\it 1}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{TL20GAMMA2}{\#}}}}&\multicolumn{1}{r@{}}{2748}&\multicolumn{1}{@{.}l}{0 }&\multicolumn{1}{@{}l}{(17/2\ensuremath{^{+}})}&\multicolumn{1}{l}{M1(+E2)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{559 {\it 8}}&\parbox[t][0.3cm]{9.6951cm}{\raggedright Mult.: DCO=1.14 \textit{5}, POL={\textminus}0.30 \textit{5} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2013Da15,B}{2013Da15}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3044}&\multicolumn{1}{@{.}l}{9}&\multicolumn{1}{l}{(25/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{161}&\multicolumn{1}{@{.}l}{2\ensuremath{^{\hyperlink{TL20GAMMA2}{\#}}} {\it 1}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{TL20GAMMA2}{\#}}}}&\multicolumn{1}{r@{}}{2883}&\multicolumn{1}{@{.}l}{7 }&\multicolumn{1}{@{}l}{(23/2\ensuremath{^{-}})}&\multicolumn{1}{l}{M1(+E2)}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{206 {\it 31}}&\parbox[t][0.3cm]{9.6951cm}{\raggedright Mult.: DCO=1.30 \textit{4}\hphantom{a}(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2013Da15,B}{2013Da15}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3056}&\multicolumn{1}{@{.}l}{6}&\multicolumn{1}{l}{(19/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{308}&\multicolumn{1}{@{.}l}{8\ensuremath{^{\hyperlink{TL20GAMMA2}{\#}}} {\it 9}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{TL20GAMMA2}{\#}}}}&\multicolumn{1}{r@{}}{2748}&\multicolumn{1}{@{.}l}{0 }&\multicolumn{1}{@{}l}{(17/2\ensuremath{^{+}})}&\multicolumn{1}{l}{M1(+E2)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{362 {\it 6}}&\parbox[t][0.3cm]{9.6951cm}{\raggedright Mult.: DCO=1.29 \textit{10}, POL={\textminus}0.24 \textit{4} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2013Da15,B}{2013Da15}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3314}&\multicolumn{1}{@{.}l}{7}&\multicolumn{1}{l}{(21/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{258}&\multicolumn{1}{@{.}l}{0\ensuremath{^{\hyperlink{TL20GAMMA2}{\#}}} {\it 1}}&\multicolumn{1}{r@{}}{57}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{TL20GAMMA2}{\#}}} {\it 29}}&\multicolumn{1}{r@{}}{3056}&\multicolumn{1}{@{.}l}{6 }&\multicolumn{1}{@{}l}{(19/2\ensuremath{^{+}})}&\multicolumn{1}{l}{M1(+E2)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{593 {\it 8}}&\parbox[t][0.3cm]{9.6951cm}{\raggedright Mult.: DCO=1.17 \textit{14}\hphantom{a}(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2013Da15,B}{2013Da15}).\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{566}&\multicolumn{1}{@{.}l}{8\ensuremath{^{\hyperlink{TL20GAMMA2}{\#}}} {\it 3}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{TL20GAMMA2}{\#}}} {\it 12}}&\multicolumn{1}{r@{}}{2748}&\multicolumn{1}{@{.}l}{0 }&\multicolumn{1}{@{}l}{(17/2\ensuremath{^{+}})}&\multicolumn{1}{l}{E2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{02093 {\it 29}}&\parbox[t][0.3cm]{9.6951cm}{\raggedright Mult.: DCO=0.43 \textit{8} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2013Da15,B}{2013Da15}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3371}&\multicolumn{1}{@{.}l}{8}&\multicolumn{1}{l}{(23/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{326}&\multicolumn{1}{@{.}l}{7\ensuremath{^{\hyperlink{TL20GAMMA2}{\#}}} {\it 3}}&\multicolumn{1}{r@{}}{44}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{TL20GAMMA2}{\#}}} {\it 9}}&\multicolumn{1}{r@{}}{3044}&\multicolumn{1}{@{.}l}{9 }&\multicolumn{1}{@{}l}{(25/2\ensuremath{^{-}})}&&&&&\\
&&&\multicolumn{1}{r@{}}{386}&\multicolumn{1}{@{.}l}{7\ensuremath{^{\hyperlink{TL20GAMMA2}{\#}}} {\it 1}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{TL20GAMMA2}{\#}}} {\it 15}}&\multicolumn{1}{r@{}}{2985}&\multicolumn{1}{@{.}l}{1 }&\multicolumn{1}{@{}l}{(21/2\ensuremath{^{+}})}&\multicolumn{1}{l}{M1(+E2)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{1972 {\it 28}}&\parbox[t][0.3cm]{9.6951cm}{\raggedright Mult.: DCO=1.03 \textit{4}, POL={\textminus}0.09 \textit{5} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2013Da15,B}{2013Da15}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3431}&\multicolumn{1}{@{.}l}{7}&\multicolumn{1}{l}{(27/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{386}&\multicolumn{1}{@{.}l}{8\ensuremath{^{\hyperlink{TL20GAMMA2}{\#}}} {\it 1}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{TL20GAMMA2}{\#}}}}&\multicolumn{1}{r@{}}{3044}&\multicolumn{1}{@{.}l}{9 }&\multicolumn{1}{@{}l}{(25/2\ensuremath{^{-}})}&&&&&\\
\multicolumn{1}{r@{}}{3434}&\multicolumn{1}{@{.}l}{5}&\multicolumn{1}{l}{(23/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{119}&\multicolumn{1}{@{.}l}{8\ensuremath{^{\hyperlink{TL20GAMMA2}{\#}}} {\it 2}}&\multicolumn{1}{r@{}}{36}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{TL20GAMMA2}{\#}}} {\it 8}}&\multicolumn{1}{r@{}}{3314}&\multicolumn{1}{@{.}l}{7 }&\multicolumn{1}{@{}l}{(21/2\ensuremath{^{+}})}&\multicolumn{1}{l}{M1(+E2)}&\multicolumn{1}{r@{}}{5}&\multicolumn{1}{@{.}l}{13 {\it 8}}&\parbox[t][0.3cm]{9.6951cm}{\raggedright Mult.: DCO=0.96 \textit{14} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2013Da15,B}{2013Da15}).\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{377}&\multicolumn{1}{@{.}l}{9\ensuremath{^{\hyperlink{TL20GAMMA2}{\#}}} {\it 1}}&\multicolumn{1}{r@{}}{93}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{TL20GAMMA2}{\#}}} {\it 10}}&\multicolumn{1}{r@{}}{3056}&\multicolumn{1}{@{.}l}{6 }&\multicolumn{1}{@{}l}{(19/2\ensuremath{^{+}})}&\multicolumn{1}{l}{E2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0575 {\it 8}}&\parbox[t][0.3cm]{9.6951cm}{\raggedright Mult.: DCO=0.46 \textit{4} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2013Da15,B}{2013Da15}).\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{422}&\multicolumn{1}{@{.}l}{8\ensuremath{^{\hyperlink{TL20GAMMA2}{\#}}} {\it 2}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{TL20GAMMA2}{\#}}} {\it 19}}&\multicolumn{1}{r@{}}{3011}&\multicolumn{1}{@{.}l}{7 }&\multicolumn{1}{@{}l}{(19/2\ensuremath{^{+}})}&\multicolumn{1}{l}{E2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0427 {\it 6}}&\parbox[t][0.3cm]{9.6951cm}{\raggedright Mult.: DCO=0.52 \textit{4}, POL=+0.13 \textit{5} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2013Da15,B}{2013Da15}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3530}&\multicolumn{1}{@{.}l}{6}&\multicolumn{1}{l}{(25/2)}&\multicolumn{1}{r@{}}{96}&\multicolumn{1}{@{.}l}{1\ensuremath{^{\hyperlink{TL20GAMMA2}{\#}}} {\it 1}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{TL20GAMMA2}{\#}}}}&\multicolumn{1}{r@{}}{3434}&\multicolumn{1}{@{.}l}{5 }&\multicolumn{1}{@{}l}{(23/2\ensuremath{^{+}})}&\multicolumn{1}{l}{D}&&&\parbox[t][0.3cm]{9.6951cm}{\raggedright Mult.: DCO=0.61 \textit{6} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2013Da15,B}{2013Da15}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3706}&\multicolumn{1}{@{.}l}{8}&\multicolumn{1}{l}{(27/2)}&\multicolumn{1}{r@{}}{176}&\multicolumn{1}{@{.}l}{2\ensuremath{^{\hyperlink{TL20GAMMA2}{\#}}} {\it 1}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{TL20GAMMA2}{\#}}}}&\multicolumn{1}{r@{}}{3530}&\multicolumn{1}{@{.}l}{6 }&\multicolumn{1}{@{}l}{(25/2)}&\multicolumn{1}{l}{D}&&&\parbox[t][0.3cm]{9.6951cm}{\raggedright Mult.: DCO=0.85 \textit{4} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2013Da15,B}{2013Da15}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3864}&\multicolumn{1}{@{.}l}{3}&\multicolumn{1}{l}{(29/2)}&\multicolumn{1}{r@{}}{157}&\multicolumn{1}{@{.}l}{5\ensuremath{^{\hyperlink{TL20GAMMA2}{\#}}} {\it 1}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{TL20GAMMA2}{\#}}}}&\multicolumn{1}{r@{}}{3706}&\multicolumn{1}{@{.}l}{8 }&\multicolumn{1}{@{}l}{(27/2)}&\multicolumn{1}{l}{D}&&&\parbox[t][0.3cm]{9.6951cm}{\raggedright Mult.: DCO=0.86 \textit{4} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2013Da15,B}{2013Da15}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3936}&\multicolumn{1}{@{.}l}{1}&\multicolumn{1}{l}{(29/2)}&\multicolumn{1}{r@{}}{504}&\multicolumn{1}{@{.}l}{4\ensuremath{^{\hyperlink{TL20GAMMA2}{\#}}} {\it 1}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{TL20GAMMA2}{\#}}}}&\multicolumn{1}{r@{}}{3431}&\multicolumn{1}{@{.}l}{7 }&\multicolumn{1}{@{}l}{(27/2\ensuremath{^{-}})}&\multicolumn{1}{l}{D}&&&\parbox[t][0.3cm]{9.6951cm}{\raggedright Mult.: DCO=2.01 \textit{1} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2013Da15,B}{2013Da15}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3950}&\multicolumn{1}{@{.}l}{9}&\multicolumn{1}{l}{(25/2)}&\multicolumn{1}{r@{}}{579}&\multicolumn{1}{@{.}l}{1\ensuremath{^{\hyperlink{TL20GAMMA2}{\#}}} {\it 1}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{TL20GAMMA2}{\#}}}}&\multicolumn{1}{r@{}}{3371}&\multicolumn{1}{@{.}l}{8 }&\multicolumn{1}{@{}l}{(23/2\ensuremath{^{+}})}&\multicolumn{1}{l}{D}&&&\parbox[t][0.3cm]{9.6951cm}{\raggedright Mult.: DCO=0.68 \textit{4} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2013Da15,B}{2013Da15}).\vspace{0.1cm}}&\\
\end{longtable}
\clearpage
\vspace*{-0.5cm}
{\bf \small \underline{Adopted \hyperlink{201TL_LEVEL}{Levels}, \hyperlink{201TL_GAMMA}{Gammas} (continued)}}\\
\vspace{0.3cm}
\underline{$\gamma$($^{201}$Tl) (continued)}\\
\vspace{0.3cm}
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{TL20GAMMA0}{\dagger}}}} From \ensuremath{^{\textnormal{201}}}Pb \ensuremath{\varepsilon} decay, unless otherwise stated.}\\
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{TL20GAMMA1}{\ddagger}}}} From \ensuremath{^{\textnormal{202}}}Hg(d,3n\ensuremath{\gamma}).}\\
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{TL20GAMMA2}{\#}}}} From \ensuremath{^{\textnormal{198}}}Pt(\ensuremath{^{\textnormal{7}}}Li,4n\ensuremath{\gamma}).}\\
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{TL20GAMMA3}{@}}}} From \ensuremath{\alpha}(K)exp, \ensuremath{\alpha}(L)exp, K/L and \ensuremath{\gamma}\ensuremath{\gamma}(\ensuremath{\theta}) in \ensuremath{^{\textnormal{201}}}Pb \ensuremath{\varepsilon} decay, \ensuremath{\gamma}(\ensuremath{\theta}) in \ensuremath{^{\textnormal{202}}}Hg(d,3n\ensuremath{\gamma}), DCO and POL in \ensuremath{^{\textnormal{198}}}Pt(\ensuremath{^{\textnormal{7}}}Li,4n\ensuremath{\gamma}), unless otherwise stated.}\\
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{TL20GAMMA4}{\&}}}} Total theoretical internal conversion coefficients, calculated using the BrIcc code (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2008Ki07,B}{2008Ki07}) with Frozen orbital approximation based on \ensuremath{\gamma}-ray energies,}\\
\parbox[b][0.3cm]{21.881866cm}{{\ }{\ }assigned multipolarities, and mixing ratios, unless otherwise specified.}\\
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{TL20GAMMA5}{a}}}} Placement of transition in the level scheme is uncertain.}\\
\vspace{0.5cm}
\end{landscape}\clearpage
\clearpage
\begin{figure}[h]
\begin{center}
\includegraphics{201TL20-0.ps}\\
\end{center}
\end{figure}
\clearpage
\begin{figure}[h]
\begin{center}
\includegraphics{201TL20-1.ps}\\
\end{center}
\end{figure}
\clearpage
\begin{figure}[h]
\begin{center}
\includegraphics{201TL20-2.ps}\\
\end{center}
\end{figure}
\clearpage
\subsection[\hspace{-0.2cm}\ensuremath{^{\textnormal{201}}}Pb \ensuremath{\varepsilon} decay]{ }
\vspace{-27pt}
\vspace{0.3cm}
\hypertarget{PB21}{{\bf \small \underline{\ensuremath{^{\textnormal{201}}}Pb \ensuremath{\varepsilon} decay\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1979Do09,B}{1979Do09}}}}\\
\vspace{4pt}
\vspace{8pt}
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}Parent: $^{201}$Pb: E=0.0; J$^{\pi}$=5/2\ensuremath{^{-}}; T$_{1/2}$=9.33 h {\it 5}; Q(\ensuremath{\varepsilon})=1910 {\it 19}; \%\ensuremath{\varepsilon}+\%\ensuremath{\beta^{+}} decay=100.0
}\\
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1979Do09,B}{1979Do09}: \ensuremath{^{\textnormal{201}}}Pb source produced using \ensuremath{^{\textnormal{203}}}Tl(p,3n) reaction; E(p)=27 MeV; Target: natural thallium; Detectors: Ge(Li) and NaI;}\\
\parbox[b][0.3cm]{17.7cm}{Compton suppressed; Measured: E\ensuremath{\gamma}, I\ensuremath{\gamma}, \ensuremath{\gamma} singles, \ensuremath{\gamma}\ensuremath{\gamma} coin; Deduced: \ensuremath{\alpha}(K)exp, \ensuremath{\alpha}(L)exp, subshell ratios, \ensuremath{J^{\pi}}, T\ensuremath{_{\textnormal{1/2}}}, level scheme.}\\
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}Others: \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1974Ha18,B}{1974Ha18}, \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1971Hn04,B}{1971Hn04}, \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1970DoZT,B}{1970DoZT}, \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1964Aa01,B}{1964Aa01}, \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1961Pe05,B}{1961Pe05}, \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1960Li08,B}{1960Li08}.}\\
\vspace{12pt}
\underline{$^{201}$Tl Levels}\\
\begin{longtable}{cccccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E(level)$^{{\hyperlink{TL21LEVEL0}{\dagger}}}$}&J$^{\pi}$$^{{\hyperlink{TL21LEVEL1}{\ddagger}}}$&\multicolumn{2}{c}{T$_{1/2}$$^{}$}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{l}{1/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{0421 d {\it 17}}&&\\
\multicolumn{1}{r@{}}{331}&\multicolumn{1}{@{.}l}{17 {\it 3}}&\multicolumn{1}{l}{3/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{70}&\multicolumn{1}{@{ }l}{ps {\it 20}}&\parbox[t][0.3cm]{11.6766405cm}{\raggedright T\ensuremath{_{1/2}}: From 360ce-331ce(\ensuremath{\Delta}t) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1960Li08,B}{1960Li08}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{692}&\multicolumn{1}{@{.}l}{52 {\it 4}}&\multicolumn{1}{l}{5/2\ensuremath{^{+}}}&&&&\\
\multicolumn{1}{r@{}}{1098}&\multicolumn{1}{@{.}l}{50 {\it 4}}&\multicolumn{1}{l}{5/2\ensuremath{^{+}}}&&&&\\
\multicolumn{1}{r@{}}{1134}&\multicolumn{1}{@{.}l}{86 {\it 6}}&\multicolumn{1}{l}{7/2\ensuremath{^{+}}}&&&&\\
\multicolumn{1}{r@{}}{1157}&\multicolumn{1}{@{.}l}{43 {\it 4}}&\multicolumn{1}{l}{3/2\ensuremath{^{+}},5/2\ensuremath{^{+}}}&&&&\\
\multicolumn{1}{r@{}}{1238}&\multicolumn{1}{@{.}l}{83 {\it 5}}&\multicolumn{1}{l}{3/2\ensuremath{^{+}}}&&&&\\
\multicolumn{1}{r@{}}{1277}&\multicolumn{1}{@{.}l}{12 {\it 4}}&\multicolumn{1}{l}{3/2\ensuremath{^{+}},5/2\ensuremath{^{+}}}&&&&\\
\multicolumn{1}{r@{}}{1290}&\multicolumn{1}{@{.}l}{12 {\it 7}}&\multicolumn{1}{l}{(9/2)\ensuremath{^{+}}}&&&&\\
\multicolumn{1}{r@{}}{1330}&\multicolumn{1}{@{.}l}{42 {\it 5}}&\multicolumn{1}{l}{3/2\ensuremath{^{+}},5/2\ensuremath{^{+}}}&&&&\\
\multicolumn{1}{r@{}}{1401}&\multicolumn{1}{@{.}l}{26 {\it 5}}&\multicolumn{1}{l}{3/2\ensuremath{^{+}},5/2\ensuremath{^{+}}}&&&&\\
\multicolumn{1}{r@{}}{1420}&\multicolumn{1}{@{.}l}{04 {\it 7}}&\multicolumn{1}{l}{7/2\ensuremath{^{+}}}&&&&\\
\multicolumn{1}{r@{}}{1445}&\multicolumn{1}{@{.}l}{87 {\it 6}}&\multicolumn{1}{l}{(5/2)\ensuremath{^{+}}}&&&&\\
\multicolumn{1}{r@{}}{1479}&\multicolumn{1}{@{.}l}{85 {\it 4}}&\multicolumn{1}{l}{5/2\ensuremath{^{+}}}&&&&\\
\multicolumn{1}{r@{}}{1550}&\multicolumn{1}{@{.}l}{58? {\it 15}}&\multicolumn{1}{l}{1/2,3/2,5/2\ensuremath{^{+}}}&&&&\\
\multicolumn{1}{r@{}}{1575}&\multicolumn{1}{@{.}l}{1 {\it 10}}&\multicolumn{1}{l}{(7/2)\ensuremath{^{+}}}&&&&\\
\multicolumn{1}{r@{}}{1617}&\multicolumn{1}{@{.}l}{46 {\it 15}}&\multicolumn{1}{l}{1/2,3/2,5/2\ensuremath{^{+}}}&&&&\\
\multicolumn{1}{r@{}}{1639}&\multicolumn{1}{@{.}l}{36 {\it 4}}&\multicolumn{1}{l}{3/2\ensuremath{^{+}},5/2\ensuremath{^{+}}}&&&&\\
\multicolumn{1}{r@{}}{1671}&\multicolumn{1}{@{.}l}{96 {\it 5}}&\multicolumn{1}{l}{3/2\ensuremath{^{+}},5/2\ensuremath{^{+}}}&&&&\\
\multicolumn{1}{r@{}}{1712}&\multicolumn{1}{@{.}l}{4? {\it 3}}&\multicolumn{1}{l}{3/2,5/2,7/2\ensuremath{^{+}}}&&&&\\
\multicolumn{1}{r@{}}{1755}&\multicolumn{1}{@{.}l}{33 {\it 7}}&\multicolumn{1}{l}{3/2,5/2\ensuremath{^{+}}}&&&&\\
\end{longtable}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{TL21LEVEL0}{\dagger}}}} From a least-squares fit to E\ensuremath{\gamma}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{TL21LEVEL1}{\ddagger}}}} From Adopted Levels, unless otherwise stated.}\\
\vspace{0.5cm}
\underline{\ensuremath{\varepsilon,\beta^+} radiations}\\
\begin{longtable}{cccccccccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E(decay)$$}&\multicolumn{2}{c}{E(level)}&\multicolumn{2}{c}{I$\varepsilon^{{\hyperlink{TL21DECAY0}{\dagger}}{\hyperlink{TL21DECAY1}{\ddagger}}}$}&\multicolumn{2}{c}{Log {\it ft}$^{}$}&\multicolumn{2}{c}{I$(\varepsilon+\beta^{+})^{{\hyperlink{TL21DECAY1}{\ddagger}}}$}&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&
\endfirsthead
\multicolumn{1}{r@{}}{(155}&\multicolumn{1}{@{ }l}{{\it 19})}&\multicolumn{1}{r@{}}{1755}&\multicolumn{1}{@{.}l}{33}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{178 {\it 13}}&\multicolumn{1}{r@{}}{6}&\multicolumn{1}{@{.}l}{68 {\it 20}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{178 {\it 13}}&\\
\multicolumn{1}{r@{}}{(198}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{TL21DECAY2}{\#}}}} {\it 19})}&\multicolumn{1}{r@{}}{1712}&\multicolumn{1}{@{.}l}{4?}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{034 {\it 8}}&\multicolumn{1}{r@{}}{7}&\multicolumn{1}{@{.}l}{73 {\it 17}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{034 {\it 8}}&\\
\multicolumn{1}{r@{}}{(238}&\multicolumn{1}{@{ }l}{{\it 19})}&\multicolumn{1}{r@{}}{1671}&\multicolumn{1}{@{.}l}{96}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{02 {\it 7}}&\multicolumn{1}{r@{}}{6}&\multicolumn{1}{@{.}l}{49 {\it 11}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{02 {\it 7}}&\\
\multicolumn{1}{r@{}}{(271}&\multicolumn{1}{@{ }l}{{\it 19})}&\multicolumn{1}{r@{}}{1639}&\multicolumn{1}{@{.}l}{36}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{45 {\it 19}}&\multicolumn{1}{r@{}}{6}&\multicolumn{1}{@{.}l}{49 {\it 11}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{45 {\it 19}}&\\
\multicolumn{1}{r@{}}{(293}&\multicolumn{1}{@{ }l}{{\it 19})}&\multicolumn{1}{r@{}}{1617}&\multicolumn{1}{@{.}l}{46}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{088 {\it 6}}&\multicolumn{1}{r@{}}{7}&\multicolumn{1}{@{.}l}{80 {\it 9}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{088 {\it 6}}&\\
\multicolumn{1}{r@{}}{(335}&\multicolumn{1}{@{ }l}{{\it 19})}&\multicolumn{1}{r@{}}{1575}&\multicolumn{1}{@{.}l}{1}&\multicolumn{1}{r@{}}{$\approx$0}&\multicolumn{1}{@{.}l}{13}&\multicolumn{1}{r@{}}{$\approx$7}&\multicolumn{1}{@{.}l}{8}&\multicolumn{1}{r@{}}{$\approx$0}&\multicolumn{1}{@{.}l}{13}&\\
\multicolumn{1}{r@{}}{(359}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{TL21DECAY2}{\#}}}} {\it 19})}&\multicolumn{1}{r@{}}{1550}&\multicolumn{1}{@{.}l}{58?}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0284 {\it 23}}&\multicolumn{1}{r@{}}{8}&\multicolumn{1}{@{.}l}{51 {\it 7}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0284 {\it 23}}&\\
\multicolumn{1}{r@{}}{(430}&\multicolumn{1}{@{ }l}{{\it 19})}&\multicolumn{1}{r@{}}{1479}&\multicolumn{1}{@{.}l}{85}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{69 {\it 17}}&\multicolumn{1}{r@{}}{6}&\multicolumn{1}{@{.}l}{73 {\it 6}}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{69 {\it 17}}&\\
\multicolumn{1}{r@{}}{(464}&\multicolumn{1}{@{ }l}{{\it 19})}&\multicolumn{1}{r@{}}{1445}&\multicolumn{1}{@{.}l}{87}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{354 {\it 24}}&\multicolumn{1}{r@{}}{7}&\multicolumn{1}{@{.}l}{69 {\it 6}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{354 {\it 24}}&\\
\multicolumn{1}{r@{}}{(490}&\multicolumn{1}{@{ }l}{{\it 19})}&\multicolumn{1}{r@{}}{1420}&\multicolumn{1}{@{.}l}{04}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{58 {\it 16}}&\multicolumn{1}{r@{}}{7}&\multicolumn{1}{@{.}l}{09 {\it 6}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{58 {\it 16}}&\\
\multicolumn{1}{r@{}}{(509}&\multicolumn{1}{@{ }l}{{\it 19})}&\multicolumn{1}{r@{}}{1401}&\multicolumn{1}{@{.}l}{26}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{39 {\it 15}}&\multicolumn{1}{r@{}}{6}&\multicolumn{1}{@{.}l}{95 {\it 5}}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{39 {\it 15}}&\\
\multicolumn{1}{r@{}}{(580}&\multicolumn{1}{@{ }l}{{\it 19})}&\multicolumn{1}{r@{}}{1330}&\multicolumn{1}{@{.}l}{42}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{03 {\it 8}}&\multicolumn{1}{r@{}}{7}&\multicolumn{1}{@{.}l}{45 {\it 5}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{03 {\it 8}}&\\
\multicolumn{1}{r@{}}{(620}&\multicolumn{1}{@{ }l}{{\it 19})}&\multicolumn{1}{r@{}}{1290}&\multicolumn{1}{@{.}l}{12}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{14 {\it 14}}&\multicolumn{1}{r@{}}{8}&\multicolumn{1}{@{.}l}{5\ensuremath{^{1u}} {\it 5}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{14 {\it 14}}&\\
\multicolumn{1}{r@{}}{(633}&\multicolumn{1}{@{ }l}{{\it 19})}&\multicolumn{1}{r@{}}{1277}&\multicolumn{1}{@{.}l}{12}&\multicolumn{1}{r@{}}{12}&\multicolumn{1}{@{.}l}{4 {\it 8}}&\multicolumn{1}{r@{}}{6}&\multicolumn{1}{@{.}l}{45 {\it 5}}&\multicolumn{1}{r@{}}{12}&\multicolumn{1}{@{.}l}{4 {\it 8}}&\\
\multicolumn{1}{r@{}}{(671}&\multicolumn{1}{@{ }l}{{\it 19})}&\multicolumn{1}{r@{}}{1238}&\multicolumn{1}{@{.}l}{83}&\multicolumn{1}{r@{}}{7}&\multicolumn{1}{@{.}l}{5 {\it 5}}&\multicolumn{1}{r@{}}{6}&\multicolumn{1}{@{.}l}{73 {\it 4}}&\multicolumn{1}{r@{}}{7}&\multicolumn{1}{@{.}l}{5 {\it 5}}&\\
\multicolumn{1}{r@{}}{(753}&\multicolumn{1}{@{ }l}{{\it 19})}&\multicolumn{1}{r@{}}{1157}&\multicolumn{1}{@{.}l}{43}&\multicolumn{1}{r@{}}{$\approx$3}&\multicolumn{1}{@{.}l}{2}&\multicolumn{1}{r@{}}{$\approx$7}&\multicolumn{1}{@{.}l}{2}&\multicolumn{1}{r@{}}{$\approx$3}&\multicolumn{1}{@{.}l}{2}&\\
\multicolumn{1}{r@{}}{(775}&\multicolumn{1}{@{ }l}{{\it 19})}&\multicolumn{1}{r@{}}{1134}&\multicolumn{1}{@{.}l}{86}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{67 {\it 17}}&\multicolumn{1}{r@{}}{7}&\multicolumn{1}{@{.}l}{92 {\it 12}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{67 {\it 17}}&\\
\multicolumn{1}{r@{}}{(812}&\multicolumn{1}{@{ }l}{{\it 19})}&\multicolumn{1}{r@{}}{1098}&\multicolumn{1}{@{.}l}{50}&\multicolumn{1}{r@{}}{$\approx$6}&\multicolumn{1}{@{.}l}{2}&\multicolumn{1}{r@{}}{$\approx$7}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{r@{}}{$\approx$6}&\multicolumn{1}{@{.}l}{2}&\\
\end{longtable}
\begin{textblock}{29}(0,27.3)
Continued on next page (footnotes at end of table)
\end{textblock}
\clearpage
\begin{longtable}{ccccccccccccc@{\extracolsep{\fill}}c}
\\[-.4cm]
\multicolumn{14}{c}{{\bf \small \underline{\ensuremath{^{\textnormal{201}}}Pb \ensuremath{\varepsilon} decay\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1979Do09,B}{1979Do09} (continued)}}}\\
\multicolumn{14}{c}{~}\\
\multicolumn{14}{c}{\underline{\ensuremath{\epsilon,\beta^+} radiations (continued)}}\\
\multicolumn{14}{c}{~}\\
\multicolumn{2}{c}{E(decay)$$}&\multicolumn{2}{c}{E(level)}&\multicolumn{2}{c}{I$\beta^{+ \ {\hyperlink{TL21DECAY1}{\ddagger}}}$}&\multicolumn{2}{c}{I$\varepsilon^{{\hyperlink{TL21DECAY0}{\dagger}}{\hyperlink{TL21DECAY1}{\ddagger}}}$}&\multicolumn{2}{c}{Log {\it ft}$^{}$}&\multicolumn{2}{c}{I$(\varepsilon+\beta^{+})^{{\hyperlink{TL21DECAY1}{\ddagger}}}$}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endhead
\multicolumn{1}{r@{}}{(1217}&\multicolumn{1}{@{ }l}{{\it 19})}&\multicolumn{1}{r@{}}{692}&\multicolumn{1}{@{.}l}{52}&&&\multicolumn{1}{r@{}}{6}&\multicolumn{1}{@{.}l}{3 {\it 8}}&\multicolumn{1}{r@{}}{7}&\multicolumn{1}{@{.}l}{37 {\it 6}}&\multicolumn{1}{r@{}}{6}&\multicolumn{1}{@{.}l}{3 {\it 8}}&&\\
\multicolumn{1}{r@{}}{(1579}&\multicolumn{1}{@{ }l}{{\it 19})}&\multicolumn{1}{r@{}}{331}&\multicolumn{1}{@{.}l}{17}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{057 {\it 11}}&\multicolumn{1}{r@{}}{52}&\multicolumn{1}{@{ }l}{{\it 6}}&\multicolumn{1}{r@{}}{6}&\multicolumn{1}{@{.}l}{69 {\it 6}}&\multicolumn{1}{r@{}}{52}&\multicolumn{1}{@{ }l}{{\it 6}}&&\\
\multicolumn{1}{r@{}}{(1910}&\multicolumn{1}{@{ }l}{{\it 19})}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0009 {\it 5}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{7 {\it 4}}&\multicolumn{1}{r@{}}{9}&\multicolumn{1}{@{.}l}{84\ensuremath{^{1u}} {\it 25}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{7 {\it 4}}&\parbox[t][0.3cm]{8.61056cm}{\raggedright I$(\varepsilon+\beta^{+})$: From I(\ensuremath{\varepsilon}+\ensuremath{\beta}\ensuremath{^{\textnormal{+}}})\ensuremath{<}1.4\% in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1979Do09,B}{1979Do09} and by assuming\vspace{0.1cm}}&\\
&&&&&&&&&&&&\parbox[t][0.3cm]{8.61056cm}{\raggedright {\ }{\ }{\ }uniform probability distribution.\vspace{0.1cm}}&\\
\end{longtable}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{TL21DECAY0}{\dagger}}}} From intensity balances and the decay scheme, unless otherwise stated.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{TL21DECAY1}{\ddagger}}}} Absolute intensity per 100 decays.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{TL21DECAY2}{\#}}}} Existence of this branch is questionable.}\\
\vspace{0.5cm}
\clearpage
\vspace{0.3cm}
\begin{landscape}
\vspace*{-0.5cm}
{\bf \small \underline{\ensuremath{^{\textnormal{201}}}Pb \ensuremath{\varepsilon} decay\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1979Do09,B}{1979Do09} (continued)}}\\
\vspace{0.3cm}
\underline{$\gamma$($^{201}$Tl)}\\
\vspace{0.34cm}
\parbox[b][0.3cm]{21.881866cm}{\addtolength{\parindent}{-0.254cm}I\ensuremath{\gamma} normalization: Deduced using \ensuremath{\Sigma}(I(\ensuremath{\gamma}+ce)[g.s. \ensuremath{^{\textnormal{201}}}Tl])=100 {\textminus} I\ensuremath{\beta}\ensuremath{_{\textnormal{0}}}, with I\ensuremath{\beta}\ensuremath{_{\textnormal{0}}}=0.7\% \textit{4}.}\\
\vspace{0.34cm}
\begin{longtable}{ccccccccc@{}ccccccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{PB21GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{I\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{PB21GAMMA1}{\ddagger}\hyperlink{PB21GAMMA6}{b}}}}&\multicolumn{2}{c}{E\ensuremath{_{i}}(level)}&J\ensuremath{^{\pi}_{i}}&\multicolumn{2}{c}{E\ensuremath{_{f}}}&J\ensuremath{^{\pi}_{f}}&Mult.\ensuremath{^{\hyperlink{PB21GAMMA2}{\#}}}&\multicolumn{2}{c}{\ensuremath{\delta}\ensuremath{^{\hyperlink{PB21GAMMA3}{@}}}}&\multicolumn{2}{c}{\ensuremath{\alpha}\ensuremath{^{\hyperlink{PB21GAMMA7}{c}}}}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{58}&\multicolumn{1}{@{.}l}{92 {\it 5}}&\multicolumn{1}{r@{}}{$\approx$4}&\multicolumn{1}{@{.}l}{9}&\multicolumn{1}{r@{}}{1157}&\multicolumn{1}{@{.}l}{43}&\multicolumn{1}{l}{3/2\ensuremath{^{+}},5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1098}&\multicolumn{1}{@{.}l}{50 }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{+}}}&\multicolumn{1}{l}{[M1]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{7}&\multicolumn{1}{@{.}l}{37 {\it 10}}&\parbox[t][0.3cm]{7.42902cm}{\raggedright \%I\ensuremath{\gamma}\ensuremath{\approx}0.083\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.42902cm}{\raggedright E\ensuremath{_{\gamma}}: From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1964Aa01,B}{1964Aa01}.\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.42902cm}{\raggedright I\ensuremath{_{\gamma}}: From I(ce(L)) of \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1964Aa01,B}{1964Aa01} and assumption that\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.42902cm}{\raggedright {\ }{\ }{\ }Mult=M1.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{120}&\multicolumn{1}{@{.}l}{0 {\it 2}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{2 {\it 3}}&\multicolumn{1}{r@{}}{1277}&\multicolumn{1}{@{.}l}{12}&\multicolumn{1}{l}{3/2\ensuremath{^{+}},5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1157}&\multicolumn{1}{@{.}l}{43 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}},5/2\ensuremath{^{+}}}&\multicolumn{1}{l}{[M1,E2]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{9 {\it 12}}&\parbox[t][0.3cm]{7.42902cm}{\raggedright \%I\ensuremath{\gamma}=0.020 \textit{5}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{124}&\multicolumn{1}{@{.}l}{2 {\it 2}}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{5 {\it 5}}&\multicolumn{1}{r@{}}{1401}&\multicolumn{1}{@{.}l}{26}&\multicolumn{1}{l}{3/2\ensuremath{^{+}},5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1277}&\multicolumn{1}{@{.}l}{12 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}},5/2\ensuremath{^{+}}}&\multicolumn{1}{l}{[M1,E2]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{5 {\it 11}}&\parbox[t][0.3cm]{7.42902cm}{\raggedright \%I\ensuremath{\gamma}=0.042 \textit{9}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{129}&\multicolumn{1}{@{.}l}{95 {\it 10}}&\multicolumn{1}{r@{}}{6}&\multicolumn{1}{@{.}l}{4 {\it 6}}&\multicolumn{1}{r@{}}{1420}&\multicolumn{1}{@{.}l}{04}&\multicolumn{1}{l}{7/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1290}&\multicolumn{1}{@{.}l}{12 }&\multicolumn{1}{@{}l}{(9/2)\ensuremath{^{+}}}&\multicolumn{1}{l}{[M1,E2]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{0 {\it 10}}&\parbox[t][0.3cm]{7.42902cm}{\raggedright \%I\ensuremath{\gamma}=0.109 \textit{12}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{155}&\multicolumn{1}{@{.}l}{31 {\it 10}}&\multicolumn{1}{r@{}}{8}&\multicolumn{1}{@{.}l}{2 {\it 10}}&\multicolumn{1}{r@{}}{1290}&\multicolumn{1}{@{.}l}{12}&\multicolumn{1}{l}{(9/2)\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1134}&\multicolumn{1}{@{.}l}{86 }&\multicolumn{1}{@{}l}{7/2\ensuremath{^{+}}}&\multicolumn{1}{l}{[M1,E2]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{7 {\it 7}}&\parbox[t][0.3cm]{7.42902cm}{\raggedright \%I\ensuremath{\gamma}=0.139 \textit{18}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{202}&\multicolumn{1}{@{.}l}{79 {\it 10}}&\multicolumn{1}{r@{}}{4}&\multicolumn{1}{@{.}l}{0 {\it 5}}&\multicolumn{1}{r@{}}{1479}&\multicolumn{1}{@{.}l}{85}&\multicolumn{1}{l}{5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1277}&\multicolumn{1}{@{.}l}{12 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}},5/2\ensuremath{^{+}}}&\multicolumn{1}{l}{[M1,E2]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{8 {\it 4}}&\parbox[t][0.3cm]{7.42902cm}{\raggedright \%I\ensuremath{\gamma}=0.068 \textit{9}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{231}&\multicolumn{1}{@{.}l}{87 {\it 10}}&\multicolumn{1}{r@{}}{6}&\multicolumn{1}{@{.}l}{5 {\it 8}}&\multicolumn{1}{r@{}}{1330}&\multicolumn{1}{@{.}l}{42}&\multicolumn{1}{l}{3/2\ensuremath{^{+}},5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1098}&\multicolumn{1}{@{.}l}{50 }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{+}}}&\multicolumn{1}{l}{[M1,E2]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{52 {\it 27}}&\parbox[t][0.3cm]{7.42902cm}{\raggedright \%I\ensuremath{\gamma}=0.110 \textit{15}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{241}&\multicolumn{1}{@{.}l}{02 {\it 8}}&\multicolumn{1}{r@{}}{10}&\multicolumn{1}{@{.}l}{0 {\it 10}}&\multicolumn{1}{r@{}}{1479}&\multicolumn{1}{@{.}l}{85}&\multicolumn{1}{l}{5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1238}&\multicolumn{1}{@{.}l}{83 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}}}&\multicolumn{1}{l}{[M1,E2]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{47 {\it 25}}&\parbox[t][0.3cm]{7.42902cm}{\raggedright \%I\ensuremath{\gamma}=0.170 \textit{19}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{285}&\multicolumn{1}{@{.}l}{0\ensuremath{^{\hyperlink{PB21GAMMA5}{a}}} {\it 10}}&\multicolumn{1}{r@{}}{$\approx$5}&\multicolumn{1}{@{.}l}{2\ensuremath{^{\hyperlink{PB21GAMMA5}{a}}}}&\multicolumn{1}{r@{}}{1575}&\multicolumn{1}{@{.}l}{1}&\multicolumn{1}{l}{(7/2)\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1290}&\multicolumn{1}{@{.}l}{12 }&\multicolumn{1}{@{}l}{(9/2)\ensuremath{^{+}}}&\multicolumn{1}{l}{(M1)\ensuremath{^{\hyperlink{PB21GAMMA5}{a}}}}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{451 {\it 8}}&\parbox[t][0.3cm]{7.42902cm}{\raggedright \%I\ensuremath{\gamma}\ensuremath{\approx}0.088\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{285}&\multicolumn{1}{@{.}l}{18\ensuremath{^{\hyperlink{PB21GAMMA5}{a}}} {\it 13}}&\multicolumn{1}{r@{}}{$\approx$5}&\multicolumn{1}{@{.}l}{2\ensuremath{^{\hyperlink{PB21GAMMA5}{a}}}}&\multicolumn{1}{r@{}}{1420}&\multicolumn{1}{@{.}l}{04}&\multicolumn{1}{l}{7/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1134}&\multicolumn{1}{@{.}l}{86 }&\multicolumn{1}{@{}l}{7/2\ensuremath{^{+}}}&\multicolumn{1}{l}{[M1,E2]\ensuremath{^{\hyperlink{PB21GAMMA5}{a}}}}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{29 {\it 16}}&\parbox[t][0.3cm]{7.42902cm}{\raggedright \%I\ensuremath{\gamma}\ensuremath{\approx}0.088\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{302}&\multicolumn{1}{@{.}l}{7\ensuremath{^{\hyperlink{PB21GAMMA4}{\&}}} {\it 4}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{65 {\it 15}}&\multicolumn{1}{r@{}}{1401}&\multicolumn{1}{@{.}l}{26}&\multicolumn{1}{l}{3/2\ensuremath{^{+}},5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1098}&\multicolumn{1}{@{.}l}{50 }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{+}}}&\multicolumn{1}{l}{[M1,E2]}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{25 {\it 14}}&\parbox[t][0.3cm]{7.42902cm}{\raggedright \%I\ensuremath{\gamma}=0.011 \textit{3}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{308}&\multicolumn{1}{@{.}l}{93 {\it 15}}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{3 {\it 3}}&\multicolumn{1}{r@{}}{1639}&\multicolumn{1}{@{.}l}{36}&\multicolumn{1}{l}{3/2\ensuremath{^{+}},5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1330}&\multicolumn{1}{@{.}l}{42 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}},5/2\ensuremath{^{+}}}&\multicolumn{1}{l}{(M1)}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{362 {\it 5}}&\parbox[t][0.3cm]{7.42902cm}{\raggedright \%I\ensuremath{\gamma}=0.039 \textit{6}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.42902cm}{\raggedright Mult.: ce(K)=3.0 \textit{5} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1964Aa01,B}{1964Aa01}); \ensuremath{\alpha}(K)exp=0.67 \textit{15}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.42902cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1979Do09,B}{1979Do09}). Note that \ensuremath{\alpha}(K)exp is larger than that\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.42902cm}{\raggedright {\ }{\ }{\ }expected from theory and one may expect E0\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.42902cm}{\raggedright {\ }{\ }{\ }admixtures; \ensuremath{\alpha}(K)exp=0.37 \textit{13} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1971Hn04,B}{1971Hn04}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{322}&\multicolumn{1}{@{.}l}{42 {\it 15}}&\multicolumn{1}{r@{}}{4}&\multicolumn{1}{@{.}l}{4 {\it 6}}&\multicolumn{1}{r@{}}{1479}&\multicolumn{1}{@{.}l}{85}&\multicolumn{1}{l}{5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1157}&\multicolumn{1}{@{.}l}{43 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}},5/2\ensuremath{^{+}}}&\multicolumn{1}{l}{[M1,E2]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{21 {\it 12}}&\parbox[t][0.3cm]{7.42902cm}{\raggedright \%I\ensuremath{\gamma}=0.075 \textit{11}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{331}&\multicolumn{1}{@{.}l}{15 {\it 6}}&\multicolumn{1}{r@{}}{455\ensuremath{\times10^{1}}}&\multicolumn{1}{@{ }l}{{\it 25}}&\multicolumn{1}{r@{}}{331}&\multicolumn{1}{@{.}l}{17}&\multicolumn{1}{l}{3/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{@{}l}{1/2\ensuremath{^{+}}}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{+1}&\multicolumn{1}{@{.}l}{33 {\it 6}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{161 {\it 5}}&\parbox[t][0.3cm]{7.42902cm}{\raggedright \%I\ensuremath{\gamma}=77.2 \textit{6}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.42902cm}{\raggedright Mult.: \ensuremath{\alpha}(K)exp=0.1069 \textit{21}, \ensuremath{\alpha}(L3)exp=0.00365 \textit{12},\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.42902cm}{\raggedright {\ }{\ }{\ }K/L=3.76 \textit{6} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1974Ha18,B}{1974Ha18}) ce(K)=1000, K/L=3.9 \textit{3},\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.42902cm}{\raggedright {\ }{\ }{\ }L12/L3=7.1 \textit{9} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1964Aa01,B}{1964Aa01}); \ensuremath{\alpha}(K)exp=0.113 \textit{8}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.42902cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1961Pe05,B}{1961Pe05}); K/L12=4.2 \textit{2}, L12/L3=6.5 \textit{5}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.42902cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1960Li08,B}{1960Li08}); \ensuremath{\gamma}\ensuremath{\gamma}(\ensuremath{\theta}) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1961Pe05,B}{1961Pe05}; \ensuremath{\alpha}(K)exp=0.113 \textit{8}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.42902cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1979Do09,B}{1979Do09}), normalized value adopted by the\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.42902cm}{\raggedright {\ }{\ }{\ }authors from the data of \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1961Pe05,B}{1961Pe05}; \ensuremath{\alpha}(K)exp=0.111\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.42902cm}{\raggedright {\ }{\ }{\ }\textit{16} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1971Hn04,B}{1971Hn04}).\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.42902cm}{\raggedright \ensuremath{\delta}: From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1974Ha18,B}{1974Ha18}, by taking into account the\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.42902cm}{\raggedright {\ }{\ }{\ }penetration effect and using \ensuremath{\lambda}=+4.0 \textit{10}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{341}&\multicolumn{1}{@{.}l}{51 {\it 8}}&\multicolumn{1}{r@{}}{6}&\multicolumn{1}{@{.}l}{8 {\it 8}}&\multicolumn{1}{r@{}}{1671}&\multicolumn{1}{@{.}l}{96}&\multicolumn{1}{l}{3/2\ensuremath{^{+}},5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1330}&\multicolumn{1}{@{.}l}{42 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}},5/2\ensuremath{^{+}}}&\multicolumn{1}{l}{[M1,E2]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{18 {\it 10}}&\parbox[t][0.3cm]{7.42902cm}{\raggedright \%I\ensuremath{\gamma}=0.115 \textit{15}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{344}&\multicolumn{1}{@{.}l}{95 {\it 7}}&\multicolumn{1}{r@{}}{18}&\multicolumn{1}{@{.}l}{3 {\it 15}}&\multicolumn{1}{r@{}}{1479}&\multicolumn{1}{@{.}l}{85}&\multicolumn{1}{l}{5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1134}&\multicolumn{1}{@{.}l}{86 }&\multicolumn{1}{@{}l}{7/2\ensuremath{^{+}}}&\multicolumn{1}{l}{M1(+E2)}&\multicolumn{1}{r@{}}{$<$0}&\multicolumn{1}{@{.}l}{6}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{243 {\it 26}}&\parbox[t][0.3cm]{7.42902cm}{\raggedright \%I\ensuremath{\gamma}=0.31 \textit{3}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.42902cm}{\raggedright Mult.: ce(K)=7.4 \textit{15} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1964Aa01,B}{1964Aa01}); \ensuremath{\alpha}(K)exp=0.21 \textit{11}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.42902cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1979Do09,B}{1979Do09}); \ensuremath{\alpha}(K)exp=0.23 \textit{6} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1971Hn04,B}{1971Hn04}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{361}&\multicolumn{1}{@{.}l}{25 {\it 6}}&\multicolumn{1}{r@{}}{560}&\multicolumn{1}{@{ }l}{{\it 30}}&\multicolumn{1}{r@{}}{692}&\multicolumn{1}{@{.}l}{52}&\multicolumn{1}{l}{5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{331}&\multicolumn{1}{@{.}l}{17 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}}}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{14 {\it 7}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{234 {\it 5}}&\parbox[t][0.3cm]{7.42902cm}{\raggedright \%I\ensuremath{\gamma}=9.5 \textit{7}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.42902cm}{\raggedright Mult.: \ensuremath{\alpha}(K)exp=0.193 \textit{4}, \ensuremath{\alpha}(L3)exp=0.00029 \textit{7}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.42902cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1974Ha18,B}{1974Ha18}); ce(K)=240 \textit{10}, K/L=5.9 \textit{6}, L12/L3=7.1\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.42902cm}{\raggedright {\ }{\ }{\ }\textit{9} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1964Aa01,B}{1964Aa01}); \ensuremath{\alpha}(K)exp=0.210 \textit{25} and \ensuremath{\gamma}\ensuremath{\gamma}(\ensuremath{\theta})\vspace{0.1cm}}&\\
\end{longtable}
\clearpage
\begin{longtable}{ccccccccc@{}ccccccc@{\extracolsep{\fill}}c}
\\[-.4cm]
\multicolumn{17}{c}{{\bf \small \underline{\ensuremath{^{\textnormal{201}}}Pb \ensuremath{\varepsilon} decay\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1979Do09,B}{1979Do09} (continued)}}}\\
\multicolumn{17}{c}{~}\\
\multicolumn{17}{c}{\underline{$\gamma$($^{201}$Tl) (continued)}}\\
\multicolumn{17}{c}{~~~}\\
\multicolumn{2}{c}{E\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{PB21GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{I\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{PB21GAMMA1}{\ddagger}\hyperlink{PB21GAMMA6}{b}}}}&\multicolumn{2}{c}{E\ensuremath{_{i}}(level)}&J\ensuremath{^{\pi}_{i}}&\multicolumn{2}{c}{E\ensuremath{_{f}}}&J\ensuremath{^{\pi}_{f}}&Mult.\ensuremath{^{\hyperlink{PB21GAMMA2}{\#}}}&\multicolumn{2}{c}{\ensuremath{\delta}\ensuremath{^{\hyperlink{PB21GAMMA3}{@}}}}&\multicolumn{2}{c}{\ensuremath{\alpha}\ensuremath{^{\hyperlink{PB21GAMMA7}{c}}}}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endhead
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.372471cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1961Pe05,B}{1961Pe05}); \ensuremath{\alpha}(K)exp=0.22 \textit{2} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1979Do09,B}{1979Do09}).\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.372471cm}{\raggedright \ensuremath{\delta}: From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1974Ha18,B}{1974Ha18}, by taking into account the\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.372471cm}{\raggedright {\ }{\ }{\ }penetration effect and using \ensuremath{\lambda}=+0.5 \textit{5}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{381}&\multicolumn{1}{@{.}l}{29 {\it 8}}&\multicolumn{1}{r@{}}{12}&\multicolumn{1}{@{.}l}{9 {\it 7}}&\multicolumn{1}{r@{}}{1479}&\multicolumn{1}{@{.}l}{85}&\multicolumn{1}{l}{5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1098}&\multicolumn{1}{@{.}l}{50 }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{+}}}&\multicolumn{1}{l}{M1(+E2)}&\multicolumn{1}{r@{}}{$<$0}&\multicolumn{1}{@{.}l}{5}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{190 {\it 15}}&\parbox[t][0.3cm]{7.372471cm}{\raggedright \%I\ensuremath{\gamma}=0.219 \textit{16}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.372471cm}{\raggedright Mult.: ce(K)=4.2 \textit{7} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1964Aa01,B}{1964Aa01}); \ensuremath{\alpha}(K)exp=0.17 \textit{3}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.372471cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1979Do09,B}{1979Do09}); \ensuremath{\alpha}(K)exp=0.16 \textit{4} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1971Hn04,B}{1971Hn04}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{394}&\multicolumn{1}{@{.}l}{86 {\it 9}}&\multicolumn{1}{r@{}}{10}&\multicolumn{1}{@{.}l}{8 {\it 7}}&\multicolumn{1}{r@{}}{1671}&\multicolumn{1}{@{.}l}{96}&\multicolumn{1}{l}{3/2\ensuremath{^{+}},5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1277}&\multicolumn{1}{@{.}l}{12 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}},5/2\ensuremath{^{+}}}&\multicolumn{1}{l}{M1(+E2)}&\multicolumn{1}{r@{}}{$<$0}&\multicolumn{1}{@{.}l}{4}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{177 {\it 10}}&\parbox[t][0.3cm]{7.372471cm}{\raggedright \%I\ensuremath{\gamma}=0.183 \textit{15}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.372471cm}{\raggedright Mult.: ce(K)=3.6 \textit{10} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1964Aa01,B}{1964Aa01}); \ensuremath{\alpha}(K)exp=0.17 \textit{5}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.372471cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1979Do09,B}{1979Do09}); \ensuremath{\alpha}(K)exp=0.22 \textit{7} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1971Hn04,B}{1971Hn04}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{405}&\multicolumn{1}{@{.}l}{96 {\it 7}}&\multicolumn{1}{r@{}}{120}&\multicolumn{1}{@{ }l}{{\it 6}}&\multicolumn{1}{r@{}}{1098}&\multicolumn{1}{@{.}l}{50}&\multicolumn{1}{l}{5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{692}&\multicolumn{1}{@{.}l}{52 }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{+}}}&\multicolumn{1}{l}{M1(+E2)}&\multicolumn{1}{r@{}}{$<$0}&\multicolumn{1}{@{.}l}{4}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{164 {\it 9}}&\parbox[t][0.3cm]{7.372471cm}{\raggedright \%I\ensuremath{\gamma}=2.04 \textit{15}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.372471cm}{\raggedright Mult.: ce(K)=33 \textit{3} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1964Aa01,B}{1964Aa01}); \ensuremath{\alpha}(K)exp=0.14 \textit{2}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.372471cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1979Do09,B}{1979Do09}); \ensuremath{\alpha}(K)exp=0.15 \textit{3} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1971Hn04,B}{1971Hn04}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{464}&\multicolumn{1}{@{.}l}{90 {\it 8}}&\multicolumn{1}{r@{}}{19}&\multicolumn{1}{@{.}l}{8 {\it 10}}&\multicolumn{1}{r@{}}{1157}&\multicolumn{1}{@{.}l}{43}&\multicolumn{1}{l}{3/2\ensuremath{^{+}},5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{692}&\multicolumn{1}{@{.}l}{52 }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{+}}}&\multicolumn{1}{l}{[M1+E2]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{08 {\it 4}}&\parbox[t][0.3cm]{7.372471cm}{\raggedright \%I\ensuremath{\gamma}=0.336 \textit{24}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{481}&\multicolumn{1}{@{.}l}{98 {\it 9}}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{2 {\it 6}}&\multicolumn{1}{r@{}}{1639}&\multicolumn{1}{@{.}l}{36}&\multicolumn{1}{l}{3/2\ensuremath{^{+}},5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1157}&\multicolumn{1}{@{.}l}{43 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}},5/2\ensuremath{^{+}}}&\multicolumn{1}{l}{[M1+E2]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{07 {\it 4}}&\parbox[t][0.3cm]{7.372471cm}{\raggedright \%I\ensuremath{\gamma}=0.054 \textit{11}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{514}&\multicolumn{1}{@{.}l}{38 {\it 9}}&\multicolumn{1}{r@{}}{9}&\multicolumn{1}{@{.}l}{1 {\it 20}}&\multicolumn{1}{r@{}}{1671}&\multicolumn{1}{@{.}l}{96}&\multicolumn{1}{l}{3/2\ensuremath{^{+}},5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1157}&\multicolumn{1}{@{.}l}{43 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}},5/2\ensuremath{^{+}}}&\multicolumn{1}{l}{[M1+E2]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{059 {\it 33}}&\parbox[t][0.3cm]{7.372471cm}{\raggedright \%I\ensuremath{\gamma}=0.15 \textit{4}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{540}&\multicolumn{1}{@{.}l}{90 {\it 9}}&\multicolumn{1}{r@{}}{16}&\multicolumn{1}{@{.}l}{2 {\it 10}}&\multicolumn{1}{r@{}}{1639}&\multicolumn{1}{@{.}l}{36}&\multicolumn{1}{l}{3/2\ensuremath{^{+}},5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1098}&\multicolumn{1}{@{.}l}{50 }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{+}}}&\multicolumn{1}{l}{[M1+E2]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{052 {\it 29}}&\parbox[t][0.3cm]{7.372471cm}{\raggedright \%I\ensuremath{\gamma}=0.275 \textit{22}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{546}&\multicolumn{1}{@{.}l}{28 {\it 9}}&\multicolumn{1}{r@{}}{16}&\multicolumn{1}{@{.}l}{5 {\it 10}}&\multicolumn{1}{r@{}}{1238}&\multicolumn{1}{@{.}l}{83}&\multicolumn{1}{l}{3/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{692}&\multicolumn{1}{@{.}l}{52 }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{+}}}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{31 {\it 24}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{074 {\it 8}}&\parbox[t][0.3cm]{7.372471cm}{\raggedright \%I\ensuremath{\gamma}=0.280 \textit{22}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.372471cm}{\raggedright Mult.: ce(K)=1.9 \textit{2} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1964Aa01,B}{1964Aa01}); \ensuremath{\alpha}(K)exp=0.059 \textit{8}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.372471cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1979Do09,B}{1979Do09}); \ensuremath{\alpha}(K)exp=0.065 \textit{14} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1971Hn04,B}{1971Hn04}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$562}&\multicolumn{1}{@{.}l}{81\ensuremath{^{\hyperlink{PB21GAMMA4}{\&}}} {\it 10}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{8 {\it 4}}&&&&&&&&&&&&\parbox[t][0.3cm]{7.372471cm}{\raggedright \%I\ensuremath{\gamma}=0.031 \textit{7}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{584}&\multicolumn{1}{@{.}l}{60 {\it 8}}&\multicolumn{1}{r@{}}{211}&\multicolumn{1}{@{ }l}{{\it 10}}&\multicolumn{1}{r@{}}{1277}&\multicolumn{1}{@{.}l}{12}&\multicolumn{1}{l}{3/2\ensuremath{^{+}},5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{692}&\multicolumn{1}{@{.}l}{52 }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{+}}}&\multicolumn{1}{l}{M1(+E2)}&\multicolumn{1}{r@{}}{$<$0}&\multicolumn{1}{@{.}l}{5}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{061 {\it 5}}&\parbox[t][0.3cm]{7.372471cm}{\raggedright \%I\ensuremath{\gamma}=3.58 \textit{25}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.372471cm}{\raggedright Mult.: ce(K)=21 \textit{2} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1964Aa01,B}{1964Aa01}); \ensuremath{\alpha}(K)exp=0.051 \textit{6}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.372471cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1979Do09,B}{1979Do09}); \ensuremath{\alpha}(K)exp=0.06 \textit{1} and \ensuremath{\gamma}\ensuremath{\gamma}(\ensuremath{\theta})\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.372471cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1961Pe05,B}{1961Pe05}); \ensuremath{\alpha}(K)exp=0.049 \textit{10} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1971Hn04,B}{1971Hn04}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{597}&\multicolumn{1}{@{.}l}{60 {\it 9}}&\multicolumn{1}{r@{}}{19}&\multicolumn{1}{@{.}l}{0 {\it 10}}&\multicolumn{1}{r@{}}{1290}&\multicolumn{1}{@{.}l}{12}&\multicolumn{1}{l}{(9/2)\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{692}&\multicolumn{1}{@{.}l}{52 }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{+}}}&\multicolumn{1}{l}{E2}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{01856 {\it 26}}&\parbox[t][0.3cm]{7.372471cm}{\raggedright \%I\ensuremath{\gamma}=0.322 \textit{24}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.372471cm}{\raggedright Mult.: ce(K)=0.6 \textit{2} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1964Aa01,B}{1964Aa01}); \ensuremath{\alpha}(K)exp=0.016 \textit{6}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.372471cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1979Do09,B}{1979Do09}); \ensuremath{\alpha}(K)exp=0.017 \textit{5} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1971Hn04,B}{1971Hn04}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{637}&\multicolumn{1}{@{.}l}{90 {\it 9}}&\multicolumn{1}{r@{}}{21}&\multicolumn{1}{@{.}l}{7 {\it 10}}&\multicolumn{1}{r@{}}{1330}&\multicolumn{1}{@{.}l}{42}&\multicolumn{1}{l}{3/2\ensuremath{^{+}},5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{692}&\multicolumn{1}{@{.}l}{52 }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{+}}}&\multicolumn{1}{l}{[M1+E2]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{034 {\it 18}}&\parbox[t][0.3cm]{7.372471cm}{\raggedright \%I\ensuremath{\gamma}=0.37 \textit{3}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{692}&\multicolumn{1}{@{.}l}{41 {\it 8}}&\multicolumn{1}{r@{}}{254}&\multicolumn{1}{@{ }l}{{\it 12}}&\multicolumn{1}{r@{}}{692}&\multicolumn{1}{@{.}l}{52}&\multicolumn{1}{l}{5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{@{}l}{1/2\ensuremath{^{+}}}&\multicolumn{1}{l}{E2}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{01342 {\it 19}}&\parbox[t][0.3cm]{7.372471cm}{\raggedright \%I\ensuremath{\gamma}=4.3 \textit{3}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.372471cm}{\raggedright Mult.: ce(K)=5.0 \textit{5} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1964Aa01,B}{1964Aa01}); \ensuremath{\alpha}(K)exp=0.010 \textit{1}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.372471cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1979Do09,B}{1979Do09}); \ensuremath{\alpha}(K)exp=0.017 \textit{5} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1971Hn04,B}{1971Hn04}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{708}&\multicolumn{1}{@{.}l}{75 {\it 9}}&\multicolumn{1}{r@{}}{46}&\multicolumn{1}{@{.}l}{2 {\it 20}}&\multicolumn{1}{r@{}}{1401}&\multicolumn{1}{@{.}l}{26}&\multicolumn{1}{l}{3/2\ensuremath{^{+}},5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{692}&\multicolumn{1}{@{.}l}{52 }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{+}}}&\multicolumn{1}{l}{M1}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0399 {\it 6}}&\parbox[t][0.3cm]{7.372471cm}{\raggedright \%I\ensuremath{\gamma}=0.78 \textit{5}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.372471cm}{\raggedright Mult.: ce(K)=4.0 \textit{5} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1964Aa01,B}{1964Aa01}); \ensuremath{\alpha}(K)exp=0.044 \textit{7}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.372471cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1979Do09,B}{1979Do09}); \ensuremath{\alpha}(K)exp=0.049 \textit{10} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1971Hn04,B}{1971Hn04}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{727}&\multicolumn{1}{@{.}l}{50 {\it 9}}&\multicolumn{1}{r@{}}{7}&\multicolumn{1}{@{.}l}{1 {\it 7}}&\multicolumn{1}{r@{}}{1420}&\multicolumn{1}{@{.}l}{04}&\multicolumn{1}{l}{7/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{692}&\multicolumn{1}{@{.}l}{52 }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{+}}}&\multicolumn{1}{l}{[M1+E2]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{025 {\it 13}}&\parbox[t][0.3cm]{7.372471cm}{\raggedright \%I\ensuremath{\gamma}=0.120 \textit{13}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{753}&\multicolumn{1}{@{.}l}{35 {\it 9}}&\multicolumn{1}{r@{}}{8}&\multicolumn{1}{@{.}l}{8 {\it 8}}&\multicolumn{1}{r@{}}{1445}&\multicolumn{1}{@{.}l}{87}&\multicolumn{1}{l}{(5/2)\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{692}&\multicolumn{1}{@{.}l}{52 }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{+}}}&\multicolumn{1}{l}{(E0+M1)}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{7.372471cm}{\raggedright \%I\ensuremath{\gamma}=0.149 \textit{16}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.372471cm}{\raggedright Mult.: ce(K)=1.5 \textit{2} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1964Aa01,B}{1964Aa01}); \ensuremath{\alpha}(K)exp=0.087 \textit{16}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.372471cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1979Do09,B}{1979Do09}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{767}&\multicolumn{1}{@{.}l}{26 {\it 8}}&\multicolumn{1}{r@{}}{194}&\multicolumn{1}{@{ }l}{{\it 10}}&\multicolumn{1}{r@{}}{1098}&\multicolumn{1}{@{.}l}{50}&\multicolumn{1}{l}{5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{331}&\multicolumn{1}{@{.}l}{17 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}}}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{33 {\it 19}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0304 {\it 25}}&\parbox[t][0.3cm]{7.372471cm}{\raggedright \%I\ensuremath{\gamma}=3.29 \textit{24}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.372471cm}{\raggedright Mult.: ce(K)=9 \textit{1} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1964Aa01,B}{1964Aa01}); \ensuremath{\alpha}(K)exp=0.024 \textit{3}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.372471cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1979Do09,B}{1979Do09}); \ensuremath{\alpha}(K)exp=0.026 \textit{3} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1971Hn04,B}{1971Hn04}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{787}&\multicolumn{1}{@{.}l}{29 {\it 10}}&\multicolumn{1}{r@{}}{34}&\multicolumn{1}{@{ }l}{{\it 4}}&\multicolumn{1}{r@{}}{1479}&\multicolumn{1}{@{.}l}{85}&\multicolumn{1}{l}{5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{692}&\multicolumn{1}{@{.}l}{52 }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{+}}}&\multicolumn{1}{l}{[M1+E2]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{020 {\it 10}}&\parbox[t][0.3cm]{7.372471cm}{\raggedright \%I\ensuremath{\gamma}=0.58 \textit{7}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{803}&\multicolumn{1}{@{.}l}{66 {\it 7}}&\multicolumn{1}{r@{}}{90}&\multicolumn{1}{@{ }l}{{\it 6}}&\multicolumn{1}{r@{}}{1134}&\multicolumn{1}{@{.}l}{86}&\multicolumn{1}{l}{7/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{331}&\multicolumn{1}{@{.}l}{17 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}}}&\multicolumn{1}{l}{E2}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{00982 {\it 14}}&\parbox[t][0.3cm]{7.372471cm}{\raggedright \%I\ensuremath{\gamma}=1.53 \textit{13}\vspace{0.1cm}}&\\
\end{longtable}
\clearpage
\begin{longtable}{ccccccccc@{}ccccccc@{\extracolsep{\fill}}c}
\\[-.4cm]
\multicolumn{17}{c}{{\bf \small \underline{\ensuremath{^{\textnormal{201}}}Pb \ensuremath{\varepsilon} decay\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1979Do09,B}{1979Do09} (continued)}}}\\
\multicolumn{17}{c}{~}\\
\multicolumn{17}{c}{\underline{$\gamma$($^{201}$Tl) (continued)}}\\
\multicolumn{17}{c}{~~~}\\
\multicolumn{2}{c}{E\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{PB21GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{I\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{PB21GAMMA1}{\ddagger}\hyperlink{PB21GAMMA6}{b}}}}&\multicolumn{2}{c}{E\ensuremath{_{i}}(level)}&J\ensuremath{^{\pi}_{i}}&\multicolumn{2}{c}{E\ensuremath{_{f}}}&J\ensuremath{^{\pi}_{f}}&Mult.\ensuremath{^{\hyperlink{PB21GAMMA2}{\#}}}&\multicolumn{2}{c}{\ensuremath{\delta}\ensuremath{^{\hyperlink{PB21GAMMA3}{@}}}}&\multicolumn{2}{c}{\ensuremath{\alpha}\ensuremath{^{\hyperlink{PB21GAMMA7}{c}}}}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endhead
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.19271cm}{\raggedright Mult.: ce(K)=1.3 \textit{2} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1964Aa01,B}{1964Aa01}); \ensuremath{\alpha}(K)exp=0.0074\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.19271cm}{\raggedright {\ }{\ }{\ }\textit{16} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1979Do09,B}{1979Do09}); \ensuremath{\alpha}(K)exp=0.0079 \textit{16} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1971Hn04,B}{1971Hn04}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{826}&\multicolumn{1}{@{.}l}{26 {\it 8}}&\multicolumn{1}{r@{}}{141}&\multicolumn{1}{@{ }l}{{\it 7}}&\multicolumn{1}{r@{}}{1157}&\multicolumn{1}{@{.}l}{43}&\multicolumn{1}{l}{3/2\ensuremath{^{+}},5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{331}&\multicolumn{1}{@{.}l}{17 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}}}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{98 {\it +43\textminus29}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0129 {\it 10}}&\parbox[t][0.3cm]{7.19271cm}{\raggedright \%I\ensuremath{\gamma}=2.39 \textit{17}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.19271cm}{\raggedright Mult.: ce(K)=2.8 \textit{3} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1964Aa01,B}{1964Aa01}); \ensuremath{\alpha}(K)exp=0.010 \textit{1}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.19271cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1979Do09,B}{1979Do09}); \ensuremath{\alpha}(K)exp=0.0110 \textit{15} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1971Hn04,B}{1971Hn04}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{907}&\multicolumn{1}{@{.}l}{67 {\it 8}}&\multicolumn{1}{r@{}}{362}&\multicolumn{1}{@{ }l}{{\it 20}}&\multicolumn{1}{r@{}}{1238}&\multicolumn{1}{@{.}l}{83}&\multicolumn{1}{l}{3/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{331}&\multicolumn{1}{@{.}l}{17 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}}}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{43 {\it +23\textminus34}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0190 {\it 20}}&\parbox[t][0.3cm]{7.19271cm}{\raggedright \%I\ensuremath{\gamma}=6.1 \textit{5}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.19271cm}{\raggedright Mult.: ce(K)=10.7 \textit{10} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1964Aa01,B}{1964Aa01}); \ensuremath{\alpha}(K)exp=0.015\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.19271cm}{\raggedright {\ }{\ }{\ }\textit{2} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1979Do09,B}{1979Do09}); \ensuremath{\alpha}(K)exp=0.017 \textit{3} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1971Hn04,B}{1971Hn04}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{945}&\multicolumn{1}{@{.}l}{96 {\it 8}}&\multicolumn{1}{r@{}}{424}&\multicolumn{1}{@{ }l}{{\it 30}}&\multicolumn{1}{r@{}}{1277}&\multicolumn{1}{@{.}l}{12}&\multicolumn{1}{l}{3/2\ensuremath{^{+}},5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{331}&\multicolumn{1}{@{.}l}{17 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}}}&\multicolumn{1}{l}{M1(+E2)}&\multicolumn{1}{r@{}}{$<$0}&\multicolumn{1}{@{.}l}{6}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0174 {\it 16}}&\parbox[t][0.3cm]{7.19271cm}{\raggedright \%I\ensuremath{\gamma}=7.2 \textit{6}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.19271cm}{\raggedright Mult.: ce(K)=12 \textit{1} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1964Aa01,B}{1964Aa01}); \ensuremath{\alpha}(K)exp=0.014 \textit{2}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.19271cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1979Do09,B}{1979Do09}); \ensuremath{\alpha}(K)exp=0.0160 \textit{25} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1971Hn04,B}{1971Hn04}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{946}&\multicolumn{1}{@{.}l}{78 {\it 4}}&\multicolumn{1}{r@{}}{28}&\multicolumn{1}{@{ }l}{{\it 10}}&\multicolumn{1}{r@{}}{1639}&\multicolumn{1}{@{.}l}{36}&\multicolumn{1}{l}{3/2\ensuremath{^{+}},5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{692}&\multicolumn{1}{@{.}l}{52 }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{+}}}&\multicolumn{1}{l}{[M1+E2]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{013 {\it 6}}&\parbox[t][0.3cm]{7.19271cm}{\raggedright \%I\ensuremath{\gamma}=0.48 \textit{17}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{979}&\multicolumn{1}{@{.}l}{4 {\it 3}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{1 {\it 3}}&\multicolumn{1}{r@{}}{1671}&\multicolumn{1}{@{.}l}{96}&\multicolumn{1}{l}{3/2\ensuremath{^{+}},5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{692}&\multicolumn{1}{@{.}l}{52 }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{+}}}&\multicolumn{1}{l}{[M1+E2]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{012 {\it 5}}&\parbox[t][0.3cm]{7.19271cm}{\raggedright \%I\ensuremath{\gamma}=0.019 \textit{5}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{999}&\multicolumn{1}{@{.}l}{23 {\it 7}}&\multicolumn{1}{r@{}}{38}&\multicolumn{1}{@{.}l}{3 {\it 20}}&\multicolumn{1}{r@{}}{1330}&\multicolumn{1}{@{.}l}{42}&\multicolumn{1}{l}{3/2\ensuremath{^{+}},5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{331}&\multicolumn{1}{@{.}l}{17 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}}}&\multicolumn{1}{l}{[M1+E2]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{011 {\it 5}}&\parbox[t][0.3cm]{7.19271cm}{\raggedright \%I\ensuremath{\gamma}=0.65 \textit{5}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$1010}&\multicolumn{1}{@{.}l}{3\ensuremath{^{\hyperlink{PB21GAMMA4}{\&}}} {\it 3}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{0 {\it 2}}&&&&&&&&&&&&\parbox[t][0.3cm]{7.19271cm}{\raggedright \%I\ensuremath{\gamma}=0.017 \textit{4}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1019}&\multicolumn{1}{@{.}l}{8\ensuremath{^{\hyperlink{PB21GAMMA4}{\&}}} {\it 3}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{9 {\it 4}}&\multicolumn{1}{r@{}}{1712}&\multicolumn{1}{@{.}l}{4?}&\multicolumn{1}{l}{3/2,5/2,7/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{692}&\multicolumn{1}{@{.}l}{52 }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{+}}}&&&&&&\parbox[t][0.3cm]{7.19271cm}{\raggedright \%I\ensuremath{\gamma}=0.015 \textit{7}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1062}&\multicolumn{1}{@{.}l}{79 {\it 15}}&\multicolumn{1}{r@{}}{4}&\multicolumn{1}{@{.}l}{0 {\it 5}}&\multicolumn{1}{r@{}}{1755}&\multicolumn{1}{@{.}l}{33}&\multicolumn{1}{l}{3/2,5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{692}&\multicolumn{1}{@{.}l}{52 }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{+}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{7.19271cm}{\raggedright \%I\ensuremath{\gamma}=0.068 \textit{9}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1070}&\multicolumn{1}{@{.}l}{04 {\it 8}}&\multicolumn{1}{r@{}}{72}&\multicolumn{1}{@{ }l}{{\it 4}}&\multicolumn{1}{r@{}}{1401}&\multicolumn{1}{@{.}l}{26}&\multicolumn{1}{l}{3/2\ensuremath{^{+}},5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{331}&\multicolumn{1}{@{.}l}{17 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}}}&\multicolumn{1}{l}{E2+M1}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{8 {\it +10\textminus5}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0075 {\it 11}}&\parbox[t][0.3cm]{7.19271cm}{\raggedright \%I\ensuremath{\gamma}=1.22 \textit{9}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.19271cm}{\raggedright Mult.: ce(K)=0.8 \textit{1} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1964Aa01,B}{1964Aa01}); \ensuremath{\alpha}(K)exp=0.0057\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.19271cm}{\raggedright {\ }{\ }{\ }\textit{10} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1979Do09,B}{1979Do09}); \ensuremath{\alpha}(K)exp=0.0069 \textit{15} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1971Hn04,B}{1971Hn04}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1088}&\multicolumn{1}{@{.}l}{85 {\it 9}}&\multicolumn{1}{r@{}}{53}&\multicolumn{1}{@{ }l}{{\it 3}}&\multicolumn{1}{r@{}}{1420}&\multicolumn{1}{@{.}l}{04}&\multicolumn{1}{l}{7/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{331}&\multicolumn{1}{@{.}l}{17 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}}}&\multicolumn{1}{l}{[E2]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{00537 {\it 8}}&\parbox[t][0.3cm]{7.19271cm}{\raggedright \%I\ensuremath{\gamma}=0.90 \textit{7}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1098}&\multicolumn{1}{@{.}l}{52 {\it 7}}&\multicolumn{1}{r@{}}{111}&\multicolumn{1}{@{ }l}{{\it 6}}&\multicolumn{1}{r@{}}{1098}&\multicolumn{1}{@{.}l}{50}&\multicolumn{1}{l}{5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{@{}l}{1/2\ensuremath{^{+}}}&\multicolumn{1}{l}{E2}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{00528 {\it 7}}&\parbox[t][0.3cm]{7.19271cm}{\raggedright \%I\ensuremath{\gamma}=1.88 \textit{14}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.19271cm}{\raggedright Mult.: ce(K)=1.2 \textit{2} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1964Aa01,B}{1964Aa01}); \ensuremath{\alpha}(K)exp=0.0055\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.19271cm}{\raggedright {\ }{\ }{\ }\textit{11} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1979Do09,B}{1979Do09}); \ensuremath{\alpha}(K)exp=0.0061 \textit{15} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1971Hn04,B}{1971Hn04}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1114}&\multicolumn{1}{@{.}l}{73 {\it 8}}&\multicolumn{1}{r@{}}{9}&\multicolumn{1}{@{.}l}{8 {\it 6}}&\multicolumn{1}{r@{}}{1445}&\multicolumn{1}{@{.}l}{87}&\multicolumn{1}{l}{(5/2)\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{331}&\multicolumn{1}{@{.}l}{17 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}}}&\multicolumn{1}{l}{M1(+E2)}&\multicolumn{1}{r@{}}{$<$0}&\multicolumn{1}{@{.}l}{3}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{01218 {\it 35}}&\parbox[t][0.3cm]{7.19271cm}{\raggedright \%I\ensuremath{\gamma}=0.166 \textit{13}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.19271cm}{\raggedright Mult.: ce(K)=0.20 \textit{3} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1964Aa01,B}{1964Aa01}); \ensuremath{\alpha}(K)exp=0.010 \textit{2}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.19271cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1979Do09,B}{1979Do09}); \ensuremath{\alpha}(K)exp=0.011 \textit{3} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1971Hn04,B}{1971Hn04}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$1124}&\multicolumn{1}{@{.}l}{9\ensuremath{^{\hyperlink{PB21GAMMA4}{\&}}} {\it 2}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{57 {\it 10}}&&&&&&&&&&&&\parbox[t][0.3cm]{7.19271cm}{\raggedright \%I\ensuremath{\gamma}=0.0097 \textit{18}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1148}&\multicolumn{1}{@{.}l}{75 {\it 8}}&\multicolumn{1}{r@{}}{47}&\multicolumn{1}{@{.}l}{3 {\it 25}}&\multicolumn{1}{r@{}}{1479}&\multicolumn{1}{@{.}l}{85}&\multicolumn{1}{l}{5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{331}&\multicolumn{1}{@{.}l}{17 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}}}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{1 {\it +4\textminus3}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0079 {\it 11}}&\parbox[t][0.3cm]{7.19271cm}{\raggedright \%I\ensuremath{\gamma}=0.80 \textit{6}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.19271cm}{\raggedright Mult.: ce(K)=0.6 \textit{1} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1964Aa01,B}{1964Aa01}); \ensuremath{\alpha}(K)exp=0.0065\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.19271cm}{\raggedright {\ }{\ }{\ }\textit{10} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1979Do09,B}{1979Do09}); \ensuremath{\alpha}(K)exp=0.0076 \textit{20} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1971Hn04,B}{1971Hn04}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1157}&\multicolumn{1}{@{.}l}{45 {\it 9}}&\multicolumn{1}{r@{}}{7}&\multicolumn{1}{@{.}l}{2 {\it 4}}&\multicolumn{1}{r@{}}{1157}&\multicolumn{1}{@{.}l}{43}&\multicolumn{1}{l}{3/2\ensuremath{^{+}},5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{@{}l}{1/2\ensuremath{^{+}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{7.19271cm}{\raggedright \%I\ensuremath{\gamma}=0.122 \textit{9}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1219}&\multicolumn{1}{@{.}l}{40\ensuremath{^{\hyperlink{PB21GAMMA8}{d}}} {\it 15}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{4 {\it 1}}&\multicolumn{1}{r@{}}{1550}&\multicolumn{1}{@{.}l}{58?}&\multicolumn{1}{l}{1/2,3/2,5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{331}&\multicolumn{1}{@{.}l}{17 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}}}&&&&&&\parbox[t][0.3cm]{7.19271cm}{\raggedright \%I\ensuremath{\gamma}=0.0238 \textit{21}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1238}&\multicolumn{1}{@{.}l}{82 {\it 7}}&\multicolumn{1}{r@{}}{68}&\multicolumn{1}{@{ }l}{{\it 4}}&\multicolumn{1}{r@{}}{1238}&\multicolumn{1}{@{.}l}{83}&\multicolumn{1}{l}{3/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{@{}l}{1/2\ensuremath{^{+}}}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{17 {\it +57\textminus35}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0065 {\it 9}}&\parbox[t][0.3cm]{7.19271cm}{\raggedright \%I\ensuremath{\gamma}=1.15 \textit{9}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.19271cm}{\raggedright Mult.: ce(K)=0.7 \textit{1} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1964Aa01,B}{1964Aa01}); \ensuremath{\alpha}(K)exp=0.0053 \textit{9}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.19271cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1979Do09,B}{1979Do09}); \ensuremath{\alpha}(K)exp=0.0053 \textit{15} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1971Hn04,B}{1971Hn04}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1277}&\multicolumn{1}{@{.}l}{11 {\it 7}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{1277}&\multicolumn{1}{@{.}l}{12}&\multicolumn{1}{l}{3/2\ensuremath{^{+}},5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{@{}l}{1/2\ensuremath{^{+}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{7.19271cm}{\raggedright \%I\ensuremath{\gamma}=1.70 \textit{9}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1286}&\multicolumn{1}{@{.}l}{3\ensuremath{^{\hyperlink{PB21GAMMA8}{d}}} {\it 2}}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{75 {\it 20}}&\multicolumn{1}{r@{}}{1617}&\multicolumn{1}{@{.}l}{46}&\multicolumn{1}{l}{1/2,3/2,5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{331}&\multicolumn{1}{@{.}l}{17 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}}}&&&&&&\parbox[t][0.3cm]{7.19271cm}{\raggedright \%I\ensuremath{\gamma}=0.064 \textit{5}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1308}&\multicolumn{1}{@{.}l}{32 {\it 8}}&\multicolumn{1}{r@{}}{32}&\multicolumn{1}{@{.}l}{6 {\it 16}}&\multicolumn{1}{r@{}}{1639}&\multicolumn{1}{@{.}l}{36}&\multicolumn{1}{l}{3/2\ensuremath{^{+}},5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{331}&\multicolumn{1}{@{.}l}{17 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}}}&\multicolumn{1}{l}{M1(+E2)}&\multicolumn{1}{r@{}}{$<$0}&\multicolumn{1}{@{.}l}{6}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0077 {\it 6}}&\parbox[t][0.3cm]{7.19271cm}{\raggedright \%I\ensuremath{\gamma}=0.55 \textit{4}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.19271cm}{\raggedright Mult.: ce(K)=0.6 \textit{2} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1964Aa01,B}{1964Aa01}); \ensuremath{\alpha}(K)exp=0.009 \textit{4}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.19271cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1979Do09,B}{1979Do09}); \ensuremath{\alpha}(K)exp=0.010 \textit{3} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1971Hn04,B}{1971Hn04}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1330}&\multicolumn{1}{@{.}l}{50 {\it 15}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{86 {\it 15}}&\multicolumn{1}{r@{}}{1330}&\multicolumn{1}{@{.}l}{42}&\multicolumn{1}{l}{3/2\ensuremath{^{+}},5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{@{}l}{1/2\ensuremath{^{+}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{7.19271cm}{\raggedright \%I\ensuremath{\gamma}=0.015 \textit{3}\vspace{0.1cm}}&\\
\end{longtable}
\clearpage
\begin{longtable}{ccccccccc@{}ccccc@{\extracolsep{\fill}}c}
\\[-.4cm]
\multicolumn{15}{c}{{\bf \small \underline{\ensuremath{^{\textnormal{201}}}Pb \ensuremath{\varepsilon} decay\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1979Do09,B}{1979Do09} (continued)}}}\\
\multicolumn{15}{c}{~}\\
\multicolumn{15}{c}{\underline{$\gamma$($^{201}$Tl) (continued)}}\\
\multicolumn{15}{c}{~~~}\\
\multicolumn{2}{c}{E\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{PB21GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{I\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{PB21GAMMA1}{\ddagger}\hyperlink{PB21GAMMA6}{b}}}}&\multicolumn{2}{c}{E\ensuremath{_{i}}(level)}&J\ensuremath{^{\pi}_{i}}&\multicolumn{2}{c}{E\ensuremath{_{f}}}&J\ensuremath{^{\pi}_{f}}&Mult.\ensuremath{^{\hyperlink{PB21GAMMA2}{\#}}}&\multicolumn{2}{c}{\ensuremath{\alpha}\ensuremath{^{\hyperlink{PB21GAMMA7}{c}}}}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endhead
\multicolumn{1}{r@{}}{1340}&\multicolumn{1}{@{.}l}{88 {\it 9}}&\multicolumn{1}{r@{}}{26}&\multicolumn{1}{@{.}l}{9 {\it 15}}&\multicolumn{1}{r@{}}{1671}&\multicolumn{1}{@{.}l}{96}&\multicolumn{1}{l}{3/2\ensuremath{^{+}},5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{331}&\multicolumn{1}{@{.}l}{17 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}}}&\multicolumn{1}{l}{[M1+E2]}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0057 {\it 21}}&\parbox[t][0.3cm]{8.86264cm}{\raggedright \%I\ensuremath{\gamma}=0.46 \textit{4}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1381}&\multicolumn{1}{@{.}l}{4 {\it 3}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{1 {\it 2}}&\multicolumn{1}{r@{}}{1712}&\multicolumn{1}{@{.}l}{4?}&\multicolumn{1}{l}{3/2,5/2,7/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{331}&\multicolumn{1}{@{.}l}{17 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{8.86264cm}{\raggedright \%I\ensuremath{\gamma}=0.019 \textit{4}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1401}&\multicolumn{1}{@{.}l}{30 {\it 8}}&\multicolumn{1}{r@{}}{7}&\multicolumn{1}{@{.}l}{9 {\it 4}}&\multicolumn{1}{r@{}}{1401}&\multicolumn{1}{@{.}l}{26}&\multicolumn{1}{l}{3/2\ensuremath{^{+}},5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{@{}l}{1/2\ensuremath{^{+}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{8.86264cm}{\raggedright \%I\ensuremath{\gamma}=0.134 \textit{10}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1424}&\multicolumn{1}{@{.}l}{16 {\it 9}}&\multicolumn{1}{r@{}}{5}&\multicolumn{1}{@{.}l}{8 {\it 3}}&\multicolumn{1}{r@{}}{1755}&\multicolumn{1}{@{.}l}{33}&\multicolumn{1}{l}{3/2,5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{331}&\multicolumn{1}{@{.}l}{17 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{8.86264cm}{\raggedright \%I\ensuremath{\gamma}=0.098 \textit{7}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1445}&\multicolumn{1}{@{.}l}{80 {\it 10}}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{10 {\it 10}}&\multicolumn{1}{r@{}}{1445}&\multicolumn{1}{@{.}l}{87}&\multicolumn{1}{l}{(5/2)\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{@{}l}{1/2\ensuremath{^{+}}}&\multicolumn{1}{l}{[E2]}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{00319 {\it 4}}&\parbox[t][0.3cm]{8.86264cm}{\raggedright \%I\ensuremath{\gamma}=0.0356 \textit{25}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1479}&\multicolumn{1}{@{.}l}{91 {\it 10}}&\multicolumn{1}{r@{}}{10}&\multicolumn{1}{@{.}l}{4 {\it 5}}&\multicolumn{1}{r@{}}{1479}&\multicolumn{1}{@{.}l}{85}&\multicolumn{1}{l}{5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{@{}l}{1/2\ensuremath{^{+}}}&\multicolumn{1}{l}{[E2]}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{00307 {\it 4}}&\parbox[t][0.3cm]{8.86264cm}{\raggedright \%I\ensuremath{\gamma}=0.176 \textit{12}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{8.86264cm}{\raggedright E\ensuremath{_{\gamma}}: In table II of \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1979Do09,B}{1979Do09} E\ensuremath{\gamma}=1470.91 keV is listed, which is\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{8.86264cm}{\raggedright {\ }{\ }{\ }a typo as evident from the spectrum shown in Figure 2 in\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{8.86264cm}{\raggedright {\ }{\ }{\ }\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1979Do09,B}{1979Do09}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$1486}&\multicolumn{1}{@{.}l}{20 {\it 12}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{1 {\it 1}}&&&&&&&&&&\parbox[t][0.3cm]{8.86264cm}{\raggedright \%I\ensuremath{\gamma}=0.0187 \textit{19}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1550}&\multicolumn{1}{@{.}l}{5\ensuremath{^{\hyperlink{PB21GAMMA8}{d}}} {\it 4}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{27 {\it 4}}&\multicolumn{1}{r@{}}{1550}&\multicolumn{1}{@{.}l}{58?}&\multicolumn{1}{l}{1/2,3/2,5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{@{}l}{1/2\ensuremath{^{+}}}&&&&\parbox[t][0.3cm]{8.86264cm}{\raggedright \%I\ensuremath{\gamma}=0.0046 \textit{7}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$1587}&\multicolumn{1}{@{.}l}{6\ensuremath{^{\hyperlink{PB21GAMMA4}{\&}}} {\it 5}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{15 {\it 5}}&&&&&&&&&&\parbox[t][0.3cm]{8.86264cm}{\raggedright \%I\ensuremath{\gamma}=0.0025 \textit{9}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1617}&\multicolumn{1}{@{.}l}{45 {\it 15}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{4 {\it 1}}&\multicolumn{1}{r@{}}{1617}&\multicolumn{1}{@{.}l}{46}&\multicolumn{1}{l}{1/2,3/2,5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{@{}l}{1/2\ensuremath{^{+}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{8.86264cm}{\raggedright \%I\ensuremath{\gamma}=0.0238 \textit{21}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$1630}&\multicolumn{1}{@{.}l}{9\ensuremath{^{\hyperlink{PB21GAMMA4}{\&}}} {\it 6}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{14 {\it 4}}&&&&&&&&&&\parbox[t][0.3cm]{8.86264cm}{\raggedright \%I\ensuremath{\gamma}=0.0024 \textit{7}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1639}&\multicolumn{1}{@{.}l}{1 {\it 5}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{20 {\it 5}}&\multicolumn{1}{r@{}}{1639}&\multicolumn{1}{@{.}l}{36}&\multicolumn{1}{l}{3/2\ensuremath{^{+}},5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{@{}l}{1/2\ensuremath{^{+}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{8.86264cm}{\raggedright \%I\ensuremath{\gamma}=0.0034 \textit{9}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1672}&\multicolumn{1}{@{.}l}{02 {\it 10}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{45 {\it 10}}&\multicolumn{1}{r@{}}{1671}&\multicolumn{1}{@{.}l}{96}&\multicolumn{1}{l}{3/2\ensuremath{^{+}},5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{@{}l}{1/2\ensuremath{^{+}}}&\multicolumn{1}{l}{[E2]}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{53\ensuremath{\times10^{-3}} {\it 4}}&\parbox[t][0.3cm]{8.86264cm}{\raggedright \%I\ensuremath{\gamma}=0.0246 \textit{21}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$1678}&\multicolumn{1}{@{.}l}{96 {\it 13}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{24 {\it 3}}&&&&&&&&&&\parbox[t][0.3cm]{8.86264cm}{\raggedright \%I\ensuremath{\gamma}=0.0041 \textit{6}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1755}&\multicolumn{1}{@{.}l}{32 {\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{65 {\it 6}}&\multicolumn{1}{r@{}}{1755}&\multicolumn{1}{@{.}l}{33}&\multicolumn{1}{l}{3/2,5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{@{}l}{1/2\ensuremath{^{+}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{8.86264cm}{\raggedright \%I\ensuremath{\gamma}=0.0110 \textit{12}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$1813}&\multicolumn{1}{@{.}l}{1\ensuremath{^{\hyperlink{PB21GAMMA4}{\&}}} {\it 3}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{26 {\it 5}}&&&&&&&&&&\parbox[t][0.3cm]{8.86264cm}{\raggedright \%I\ensuremath{\gamma}=0.0044 \textit{9}\vspace{0.1cm}}&\\
\end{longtable}
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB21GAMMA0}{\dagger}}}} From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1979Do09,B}{1979Do09}, unless otherwise stated.}\\
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB21GAMMA1}{\ddagger}}}} From singles measurements in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1979Do09,B}{1979Do09}, unless otherwise stated. I\ensuremath{\gamma}(x-ray)=4980 \textit{250} and I\ensuremath{\gamma}(\ensuremath{\gamma}\ensuremath{^{\ensuremath{\pm}}})=6 \textit{1} in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1979Do09,B}{1979Do09}.}\\
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB21GAMMA2}{\#}}}} From \ensuremath{\alpha}(K)exp, \ensuremath{\alpha}(L)exp, K/L, \ensuremath{\gamma}\ensuremath{\gamma}(\ensuremath{\theta}) and multiple decay branches in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1979Do09,B}{1979Do09}, \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1960Li08,B}{1960Li08}, \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1961Pe05,B}{1961Pe05}, \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1964Aa01,B}{1964Aa01} and \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1974Ha18,B}{1974Ha18}, unless otherwise stated.}\\
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB21GAMMA3}{@}}}} From \ensuremath{\alpha}(K)exp and sub-shell ratios in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1979Do09,B}{1979Do09}, \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1974Ha18,B}{1974Ha18}, \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1971Hn04,B}{1971Hn04}, \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1964Aa01,B}{1964Aa01}, and\hphantom{a}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1961Pe05,B}{1961Pe05} and the briccmixing program, unless otherwise stated.}\\
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB21GAMMA4}{\&}}}} Assignment to \ensuremath{^{\textnormal{201}}}Pb \ensuremath{\varepsilon} decay is uncertain (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1979Do09,B}{1979Do09}).}\\
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB21GAMMA5}{a}}}} The authors in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1979Do09,B}{1979Do09} report a transition with E\ensuremath{\gamma}=285.04 keV \textit{7} and I\ensuremath{\gamma}=10.3 \textit{10} doubly placed from the 1420 and 1575 keV levels with roughly equal}\\
\parbox[b][0.3cm]{21.881866cm}{{\ }{\ }intensities. \ensuremath{\alpha}(K)exp=0.37 \textit{8}, assuming Mult=M1 for the doublet. The transition is not included in the least-squares fit. For placement from the 1420 keV level,}\\
\parbox[b][0.3cm]{21.881866cm}{{\ }{\ }the evaluator chooses E\ensuremath{\gamma}=285.18 keV 13, as given from the levels energy difference in the least-squares fit. For placement from the 1575 keV level, where the}\\
\parbox[b][0.3cm]{21.881866cm}{{\ }{\ }285\ensuremath{\gamma} is the only deexciting transition, the evaluator adopts E\ensuremath{\gamma}=285.0 keV \textit{10}. The evaluator adopts I\ensuremath{\gamma}\ensuremath{\approx}5.2 for each placement. Both transitions involve \ensuremath{\Delta}J=0}\\
\parbox[b][0.3cm]{21.881866cm}{{\ }{\ }or 1 and \ensuremath{\Delta}\ensuremath{\pi}=no, and since I\ensuremath{\gamma}$'$s are roughly equal, Mult.=M1 can be assigned to both placements.}\\
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB21GAMMA6}{b}}}} For absolute intensity per 100 decays, multiply by 0.0170 \textit{8}.}\\
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB21GAMMA7}{c}}}} Total theoretical internal conversion coefficients, calculated using the BrIcc code (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2008Ki07,B}{2008Ki07}) with Frozen orbital approximation based on \ensuremath{\gamma}-ray energies,}\\
\parbox[b][0.3cm]{21.881866cm}{{\ }{\ }assigned multipolarities, and mixing ratios, unless otherwise specified.}\\
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB21GAMMA8}{d}}}} Placement of transition in the level scheme is uncertain.}\\
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB21GAMMA9}{x}}}} \ensuremath{\gamma} ray not placed in level scheme.}\\
\vspace{0.5cm}
\end{landscape}\clearpage
\clearpage
\begin{figure}[h]
\begin{center}
\includegraphics{201TL21-0.ps}\\
\end{center}
\end{figure}
\clearpage
\begin{figure}[h]
\begin{center}
\includegraphics{201TL21-1.ps}\\
\end{center}
\end{figure}
\clearpage
\subsection[\hspace{-0.2cm}\ensuremath{^{\textnormal{201}}}Tl IT decay]{ }
\vspace{-27pt}
\vspace{0.3cm}
\hypertarget{TL22}{{\bf \small \underline{\ensuremath{^{\textnormal{201}}}Tl IT decay\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1975Uy01,B}{1975Uy01}}}}\\
\vspace{4pt}
\vspace{8pt}
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}Parent: $^{201}$Tl: E=919.47 {\it 11}; J$^{\pi}$=(9/2\ensuremath{^{-}}); T$_{1/2}$=2.11 ms {\it 11}; \%IT decay=100.0
}\\
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1975Uy01,B}{1975Uy01}: Populated following (\ensuremath{\gamma},2n) reaction using a pulsed bremsstrahlung beam with E(\ensuremath{\gamma})=25-32 MeV; Target: natural}\\
\parbox[b][0.3cm]{17.7cm}{thallium; Detectors: Ge(Li); Measured: E\ensuremath{\gamma}, \ensuremath{\gamma}(t).}\\
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}Others: \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1962Mo19,B}{1962Mo19}, \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1963De38,B}{1963De38}, \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1964Br27,B}{1964Br27}, \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1965Gr04,B}{1965Gr04}, \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1967Co20,B}{1967Co20}, \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1977KoZH,B}{1977KoZH}, \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1977Sl01,B}{1977Sl01}.}\\
\vspace{12pt}
\underline{$^{201}$Tl Levels}\\
\begin{longtable}{cccccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E(level)$^{{\hyperlink{TL22LEVEL0}{\dagger}}}$}&J$^{\pi}$$^{{\hyperlink{TL22LEVEL1}{\ddagger}}}$&\multicolumn{2}{c}{T$_{1/2}$$^{{\hyperlink{TL22LEVEL1}{\ddagger}}}$}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{l}{1/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{0421 d {\it 8}}&&\\
\multicolumn{1}{r@{}}{331}&\multicolumn{1}{@{.}l}{10 {\it 20}}&\multicolumn{1}{l}{3/2\ensuremath{^{+}}}&&&&\\
\multicolumn{1}{r@{}}{919}&\multicolumn{1}{@{.}l}{40 {\it 23}}&\multicolumn{1}{l}{(9/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{11 ms {\it 11}}&\parbox[t][0.3cm]{12.89002cm}{\raggedright T\ensuremath{_{1/2}}: Unweighted average of 1.8 ms \textit{1} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1962Mo19,B}{1962Mo19}), 2.3 ms \textit{2} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1963De38,B}{1963De38}), 2.1 ms \textit{2}\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{12.89002cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1964Br27,B}{1964Br27}), 1.8 ms \textit{1} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1965Gr04,B}{1965Gr04}), 2.65 ms \textit{20} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1967Co20,B}{1967Co20}), 2.1 ms \textit{1} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1975Uy01,B}{1975Uy01}) and 2.035\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{12.89002cm}{\raggedright {\ }{\ }{\ }ms \textit{7} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1977KoZH,B}{1977KoZH}). Others: \ensuremath{>}60 ns \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1977Sl01,B}{1977Sl01}.\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{12.89002cm}{\raggedright configuration: \ensuremath{\pi} 9/2[505] (h\ensuremath{_{\textnormal{9/2}}}) orbital at oblate deformation.\vspace{0.1cm}}&\\
\end{longtable}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{TL22LEVEL0}{\dagger}}}} From a least-squares fit to E\ensuremath{\gamma}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{TL22LEVEL1}{\ddagger}}}} From Adopted Levels, unless otherwise stated.}\\
\vspace{0.5cm}
\underline{$\gamma$($^{201}$Tl)}\\
\vspace{0.34cm}
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.254cm}Note that \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1965Gr04,B}{1965Gr04} (scin detector system) reported 225 keV \textit{10} transition depopulating the isomer to the 5/2\ensuremath{^{+}} level at 692.5 keV.}\\
\parbox[b][0.3cm]{17.7cm}{Since, no 361.3\ensuremath{\gamma} was reported by \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1965Gr04,B}{1965Gr04} (this E\ensuremath{\gamma} should follow 225\ensuremath{\gamma} in the cascade), coupled with the fact that subsequent}\\
\parbox[b][0.3cm]{17.7cm}{measurements high-resolution Ge(Li) detectors (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1975Uy01,B}{1975Uy01} and \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1977Sl01,B}{1977Sl01}) have not confirmed the \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1965Gr04,B}{1965Gr04} observation, the level}\\
\parbox[b][0.3cm]{17.7cm}{scheme proposed in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1965Gr04,B}{1965Gr04} was rejected by the evaluator.}\\
\vspace{0.34cm}
\begin{longtable}{ccccccccc@{}ccccccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{TL22GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{I\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{TL22GAMMA2}{\#}\hyperlink{TL22GAMMA3}{@}}}}&\multicolumn{2}{c}{E\ensuremath{_{i}}(level)}&J\ensuremath{^{\pi}_{i}}&\multicolumn{2}{c}{E\ensuremath{_{f}}}&J\ensuremath{^{\pi}_{f}}&Mult.\ensuremath{^{\hyperlink{TL22GAMMA1}{\ddagger}}}&\multicolumn{2}{c}{\ensuremath{\delta}\ensuremath{^{\hyperlink{TL22GAMMA1}{\ddagger}}}}&\multicolumn{2}{c}{\ensuremath{\alpha}\ensuremath{^{\hyperlink{TL22GAMMA4}{\&}}}}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{331}&\multicolumn{1}{@{.}l}{1 {\it 2}}&\multicolumn{1}{r@{}}{86}&\multicolumn{1}{@{.}l}{1 {\it 4}}&\multicolumn{1}{r@{}}{331}&\multicolumn{1}{@{.}l}{10}&\multicolumn{1}{l}{3/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{@{}l}{1/2\ensuremath{^{+}}}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{+1}&\multicolumn{1}{@{.}l}{33 {\it 6}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{161 {\it 5}}&&\\
\multicolumn{1}{r@{}}{588}&\multicolumn{1}{@{.}l}{3 {\it 1}}&\multicolumn{1}{r@{}}{94}&\multicolumn{1}{@{.}l}{63 {\it 7}}&\multicolumn{1}{r@{}}{919}&\multicolumn{1}{@{.}l}{40}&\multicolumn{1}{l}{(9/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{331}&\multicolumn{1}{@{.}l}{10 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}}}&\multicolumn{1}{l}{[E3]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0567 {\it 8}}&\parbox[t][0.3cm]{5.6958413cm}{\raggedright E\ensuremath{_{\gamma}}: From adiopted gammas; Other:\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{5.6958413cm}{\raggedright {\ }{\ }{\ }588.0 \textit{2} (975Uy01). E\ensuremath{\gamma}=582.3 keV \textit{3}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{5.6958413cm}{\raggedright {\ }{\ }{\ }was reported in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1967Co20,B}{1967Co20}, but it was\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{5.6958413cm}{\raggedright {\ }{\ }{\ }not confirmed by others.\vspace{0.1cm}}&\\
\end{longtable}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{TL22GAMMA0}{\dagger}}}} From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1975Uy01,B}{1975Uy01}, unless otherwise stated.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{TL22GAMMA1}{\ddagger}}}} From Adopted Levels.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{TL22GAMMA2}{\#}}}} From I(\ensuremath{\gamma}+ce)=100 and \ensuremath{\alpha}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{TL22GAMMA3}{@}}}} Absolute intensity per 100 decays.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{TL22GAMMA4}{\&}}}} Total theoretical internal conversion coefficients, calculated using the BrIcc code (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2008Ki07,B}{2008Ki07}) with Frozen orbital approximation}\\
\parbox[b][0.3cm]{17.7cm}{{\ }{\ }based on \ensuremath{\gamma}-ray energies, assigned multipolarities, and mixing ratios, unless otherwise specified.}\\
\vspace{0.5cm}
\clearpage
\begin{figure}[h]
\begin{center}
\includegraphics{201TL22-0.ps}\\
\end{center}
\end{figure}
\clearpage
\subsection[\hspace{-0.2cm}\ensuremath{^{\textnormal{202}}}Hg(d,3n\ensuremath{\gamma})]{ }
\vspace{-27pt}
\vspace{0.3cm}
\hypertarget{TL23}{{\bf \small \underline{\ensuremath{^{\textnormal{202}}}Hg(d,3n\ensuremath{\gamma})\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1977Sl01,B}{1977Sl01}}}}\\
\vspace{4pt}
\vspace{8pt}
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}E(d)=18-25 MeV; Target: metal oxide powder, 76.8\% enriched in \ensuremath{^{\textnormal{202}}}Hg; Detectors: Ge(Li), liquid scintillator; Measured:}\\
\parbox[b][0.3cm]{17.7cm}{excitation functions, E\ensuremath{\gamma}, I\ensuremath{\gamma}, \ensuremath{\gamma}\ensuremath{\gamma} coin, n\ensuremath{\gamma} coin, \ensuremath{\gamma}(t), \ensuremath{\gamma}(\ensuremath{\theta}); Deduced: \ensuremath{J^{\pi}}, T\ensuremath{_{\textnormal{1/2}}}, \ensuremath{\delta}.}\\
\vspace{12pt}
\underline{$^{201}$Tl Levels}\\
\begin{longtable}{cccccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E(level)$^{{\hyperlink{TL23LEVEL0}{\dagger}}}$}&J$^{\pi}$$^{{\hyperlink{TL23LEVEL1}{\ddagger}}}$&\multicolumn{2}{c}{T$_{1/2}$$^{}$}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{l}{1/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{0421 d {\it 8}}&\parbox[t][0.3cm]{11.7815cm}{\raggedright J\ensuremath{^{\pi}},T\ensuremath{_{1/2}}: From Adopted Levels.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{331}&\multicolumn{1}{@{.}l}{2 {\it 2}}&\multicolumn{1}{l}{3/2\ensuremath{^{+}}}&&&&\\
\multicolumn{1}{r@{}}{692}&\multicolumn{1}{@{.}l}{5 {\it 2}}&\multicolumn{1}{l}{5/2\ensuremath{^{+}}}&&&&\\
\multicolumn{1}{r@{}}{919}&\multicolumn{1}{@{.}l}{5 {\it 2}}&\multicolumn{1}{l}{(9/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{11 ms {\it 11}}&\parbox[t][0.3cm]{11.7815cm}{\raggedright J\ensuremath{^{\pi}},T\ensuremath{_{1/2}}: From Adopted Levels.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1134}&\multicolumn{1}{@{.}l}{8 {\it 2}}&\multicolumn{1}{l}{7/2\ensuremath{^{+}}}&&&&\\
\multicolumn{1}{r@{}}{1238}&\multicolumn{1}{@{.}l}{5 {\it 3}}&\multicolumn{1}{l}{(11/2\ensuremath{^{-}})}&&&&\\
\multicolumn{1}{r@{}}{1290}&\multicolumn{1}{@{.}l}{2 {\it 3}}&\multicolumn{1}{l}{(9/2\ensuremath{^{+}})}&&&&\\
\multicolumn{1}{r@{}}{1413}&\multicolumn{1}{@{.}l}{5 {\it 3}}&\multicolumn{1}{l}{(11/2\ensuremath{^{+}})}&&&&\\
\multicolumn{1}{r@{}}{1571}&\multicolumn{1}{@{.}l}{7 {\it 3}}&\multicolumn{1}{l}{(13/2\ensuremath{^{-}})}&&&&\\
\multicolumn{1}{r@{}}{1962}&\multicolumn{1}{@{.}l}{2 {\it 3}}&\multicolumn{1}{l}{(15/2\ensuremath{^{-}})}&&&&\\
\multicolumn{1}{r@{}}{1987}&\multicolumn{1}{@{.}l}{8?}&&&&&\\
\multicolumn{1}{r@{}}{2014}&\multicolumn{1}{@{.}l}{9 {\it 3}}&\multicolumn{1}{l}{(13/2\ensuremath{^{-}},15/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{9 ns {\it +19\textminus5}}&\parbox[t][0.3cm]{11.7815cm}{\raggedright T\ensuremath{_{1/2}}: From 443.2\ensuremath{\gamma}(t) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1977Sl01,B}{1977Sl01}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2040}&\multicolumn{1}{@{.}l}{0?}&&&&&\\
\multicolumn{1}{r@{}}{2181}&\multicolumn{1}{@{.}l}{7?}&&&&&\\
\multicolumn{1}{r@{}}{2441}&\multicolumn{1}{@{.}l}{7?}&&&&&\\
\multicolumn{1}{r@{}}{2747}&\multicolumn{1}{@{.}l}{4?}&&&&&\\
\end{longtable}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{TL23LEVEL0}{\dagger}}}} From a least-squares fit to E\ensuremath{\gamma}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{TL23LEVEL1}{\ddagger}}}} From the deduced \ensuremath{\gamma}-ray transition multipolarities using \ensuremath{\gamma}(\ensuremath{\theta}) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1977Sl01,B}{1977Sl01} and the apparent band structures, unless otherwise}\\
\parbox[b][0.3cm]{17.7cm}{{\ }{\ }stated.}\\
\vspace{0.5cm}
\underline{$\gamma$($^{201}$Tl)}\\
\begin{longtable}{ccccccccc@{}ccccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{TL23GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{I\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{TL23GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{E\ensuremath{_{i}}(level)}&J\ensuremath{^{\pi}_{i}}&\multicolumn{2}{c}{E\ensuremath{_{f}}}&J\ensuremath{^{\pi}_{f}}&Mult.\ensuremath{^{\hyperlink{TL23GAMMA2}{\#}}}&\multicolumn{2}{c}{\ensuremath{\delta}\ensuremath{^{\hyperlink{TL23GAMMA2}{\#}}}}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{123}&\multicolumn{1}{@{.}l}{3 {\it 1}}&\multicolumn{1}{r@{}}{6}&\multicolumn{1}{@{.}l}{3 {\it 9}}&\multicolumn{1}{r@{}}{1413}&\multicolumn{1}{@{.}l}{5}&\multicolumn{1}{l}{(11/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{1290}&\multicolumn{1}{@{.}l}{2 }&\multicolumn{1}{@{}l}{(9/2\ensuremath{^{+}})}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
\multicolumn{1}{r@{}}{155}&\multicolumn{1}{@{.}l}{4 {\it 1}}&\multicolumn{1}{r@{}}{4}&\multicolumn{1}{@{.}l}{6 {\it 7}}&\multicolumn{1}{r@{}}{1290}&\multicolumn{1}{@{.}l}{2}&\multicolumn{1}{l}{(9/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{1134}&\multicolumn{1}{@{.}l}{8 }&\multicolumn{1}{@{}l}{7/2\ensuremath{^{+}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
\multicolumn{1}{r@{}}{166}&\multicolumn{1}{@{.}l}{9\ensuremath{^{\hyperlink{TL23GAMMA3}{@}}} {\it 1}}&\multicolumn{1}{r@{}}{9}&\multicolumn{1}{@{.}l}{1 {\it 14}}&\multicolumn{1}{r@{}}{2181}&\multicolumn{1}{@{.}l}{7?}&&\multicolumn{1}{r@{}}{2014}&\multicolumn{1}{@{.}l}{9 }&\multicolumn{1}{@{}l}{(13/2\ensuremath{^{-}},15/2\ensuremath{^{-}})}&&&&&\\
\multicolumn{1}{r@{}}{319}&\multicolumn{1}{@{.}l}{0 {\it 1}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{1238}&\multicolumn{1}{@{.}l}{5}&\multicolumn{1}{l}{(11/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{919}&\multicolumn{1}{@{.}l}{5 }&\multicolumn{1}{@{}l}{(9/2\ensuremath{^{-}})}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{$-$0}&\multicolumn{1}{@{.}l}{34 {\it +11\textminus8}}&\parbox[t][0.3cm]{4.0cm}{\raggedright I\ensuremath{_{\gamma}}: Delayed I\ensuremath{\gamma}=22;\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.0cm}{\raggedright {\ }{\ }{\ }T\ensuremath{_{\textnormal{1/2}}}=2.4 ns \textit{+38{\textminus}8} from\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.0cm}{\raggedright {\ }{\ }{\ }319.0\ensuremath{\gamma}(t).\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.0cm}{\raggedright Mult.,\ensuremath{\delta}: A\ensuremath{_{\textnormal{2}}}={\textminus}0.451 \textit{13},\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.0cm}{\raggedright {\ }{\ }{\ }A\ensuremath{_{\textnormal{4}}}={\textminus}0.045 \textit{18}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{331}&\multicolumn{1}{@{.}l}{2 {\it 2}}&\multicolumn{1}{r@{}}{257}&\multicolumn{1}{@{ }l}{{\it 26}}&\multicolumn{1}{r@{}}{331}&\multicolumn{1}{@{.}l}{2}&\multicolumn{1}{l}{3/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{@{}l}{1/2\ensuremath{^{+}}}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{4.0cm}{\raggedright I\ensuremath{_{\gamma}}: Delayed I\ensuremath{\gamma}=184;\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.0cm}{\raggedright {\ }{\ }{\ }T\ensuremath{_{\textnormal{1/2}}}\ensuremath{\geq}60 ns from\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.0cm}{\raggedright {\ }{\ }{\ }331.2\ensuremath{\gamma}(t).\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.0cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}=0.012 \textit{15},\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.0cm}{\raggedright {\ }{\ }{\ }A\ensuremath{_{\textnormal{4}}}={\textminus}0.008 \textit{18}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{333}&\multicolumn{1}{@{.}l}{0 {\it 2}}&\multicolumn{1}{r@{}}{77}&\multicolumn{1}{@{ }l}{{\it 8}}&\multicolumn{1}{r@{}}{1571}&\multicolumn{1}{@{.}l}{7}&\multicolumn{1}{l}{(13/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{1238}&\multicolumn{1}{@{.}l}{5 }&\multicolumn{1}{@{}l}{(11/2\ensuremath{^{-}})}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{$-$0}&\multicolumn{1}{@{.}l}{21 {\it +14\textminus9}}&\parbox[t][0.3cm]{4.0cm}{\raggedright Mult.,\ensuremath{\delta}: A\ensuremath{_{\textnormal{2}}}={\textminus}0.383 \textit{29},\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.0cm}{\raggedright {\ }{\ }{\ }A\ensuremath{_{\textnormal{4}}}=0.052 \textit{46}. Alternative\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.0cm}{\raggedright {\ }{\ }{\ }solution of \ensuremath{\delta}={\textminus}2.9 \textit{{\textminus}7+9} is\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.0cm}{\raggedright {\ }{\ }{\ }also possible.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{361}&\multicolumn{1}{@{.}l}{3 {\it 1}}&\multicolumn{1}{r@{}}{21}&\multicolumn{1}{@{.}l}{3 {\it 19}}&\multicolumn{1}{r@{}}{692}&\multicolumn{1}{@{.}l}{5}&\multicolumn{1}{l}{5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{331}&\multicolumn{1}{@{.}l}{2 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}}}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{4.0cm}{\raggedright I\ensuremath{_{\gamma}}: Delayed I\ensuremath{\gamma}=0.\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.0cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}={\textminus}0.099 \textit{26},\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.0cm}{\raggedright {\ }{\ }{\ }A\ensuremath{_{\textnormal{4}}}=0.026 \textit{38}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{390}&\multicolumn{1}{@{.}l}{5 {\it 1}}&\multicolumn{1}{r@{}}{31}&\multicolumn{1}{@{ }l}{{\it 3}}&\multicolumn{1}{r@{}}{1962}&\multicolumn{1}{@{.}l}{2}&\multicolumn{1}{l}{(15/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{1571}&\multicolumn{1}{@{.}l}{7 }&\multicolumn{1}{@{}l}{(13/2\ensuremath{^{-}})}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{$-$0}&\multicolumn{1}{@{.}l}{20 {\it +8\textminus7}}&\parbox[t][0.3cm]{4.0cm}{\raggedright I\ensuremath{_{\gamma}}: Delayed I\ensuremath{\gamma}=0.\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.0cm}{\raggedright Mult.,\ensuremath{\delta}: A\ensuremath{_{\textnormal{2}}}={\textminus}0.407 \textit{26},\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.0cm}{\raggedright {\ }{\ }{\ }A\ensuremath{_{\textnormal{4}}}=0.010 \textit{37}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{426}&\multicolumn{1}{@{.}l}{9\ensuremath{^{\hyperlink{TL23GAMMA1}{\ddagger}\hyperlink{TL23GAMMA3}{@}}} {\it 3}}&\multicolumn{1}{r@{}}{9}&\multicolumn{1}{@{.}l}{1 {\it 19}}&\multicolumn{1}{r@{}}{2441}&\multicolumn{1}{@{.}l}{7?}&&\multicolumn{1}{r@{}}{2014}&\multicolumn{1}{@{.}l}{9 }&\multicolumn{1}{@{}l}{(13/2\ensuremath{^{-}},15/2\ensuremath{^{-}})}&&&&&\\
\multicolumn{1}{r@{}}{443}&\multicolumn{1}{@{.}l}{2 {\it 1}}&\multicolumn{1}{r@{}}{27}&\multicolumn{1}{@{ }l}{{\it 3}}&\multicolumn{1}{r@{}}{2014}&\multicolumn{1}{@{.}l}{9}&\multicolumn{1}{l}{(13/2\ensuremath{^{-}},15/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{1571}&\multicolumn{1}{@{.}l}{7 }&\multicolumn{1}{@{}l}{(13/2\ensuremath{^{-}})}&\multicolumn{1}{l}{(M1+E2)}&\multicolumn{1}{r@{}}{$\approx$0}&\multicolumn{1}{@{.}l}{3}&\parbox[t][0.3cm]{4.0cm}{\raggedright I\ensuremath{_{\gamma}}: Delayed I\ensuremath{\gamma}=23.\vspace{0.1cm}}&\\
\end{longtable}
\begin{textblock}{29}(0,27.3)
Continued on next page (footnotes at end of table)
\end{textblock}
\clearpage
\begin{longtable}{ccccccccc@{}ccc@{\extracolsep{\fill}}c}
\\[-.4cm]
\multicolumn{13}{c}{{\bf \small \underline{\ensuremath{^{\textnormal{202}}}Hg(d,3n\ensuremath{\gamma})\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1977Sl01,B}{1977Sl01} (continued)}}}\\
\multicolumn{13}{c}{~}\\
\multicolumn{13}{c}{\underline{$\gamma$($^{201}$Tl) (continued)}}\\
\multicolumn{13}{c}{~~~}\\
\multicolumn{2}{c}{E\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{TL23GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{I\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{TL23GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{E\ensuremath{_{i}}(level)}&J\ensuremath{^{\pi}_{i}}&\multicolumn{2}{c}{E\ensuremath{_{f}}}&J\ensuremath{^{\pi}_{f}}&Mult.\ensuremath{^{\hyperlink{TL23GAMMA2}{\#}}}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&\hrulefill&
\endhead
&&&&&&&&&&&\parbox[t][0.3cm]{7.6129203cm}{\raggedright Mult.,\ensuremath{\delta}: A\ensuremath{_{\textnormal{2}}}=0.162 \textit{10}, A\ensuremath{_{\textnormal{4}}}=0.002 \textit{15}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{468}&\multicolumn{1}{@{.}l}{4\ensuremath{^{\hyperlink{TL23GAMMA3}{@}}} {\it 3}}&\multicolumn{1}{r@{}}{6}&\multicolumn{1}{@{.}l}{2 {\it 14}}&\multicolumn{1}{r@{}}{2040}&\multicolumn{1}{@{.}l}{0?}&&\multicolumn{1}{r@{}}{1571}&\multicolumn{1}{@{.}l}{7 }&\multicolumn{1}{@{}l}{(13/2\ensuremath{^{-}})}&&&\\
\multicolumn{1}{r@{}}{493}&\multicolumn{1}{@{.}l}{9\ensuremath{^{\hyperlink{TL23GAMMA1}{\ddagger}\hyperlink{TL23GAMMA3}{@}}} {\it 1}}&\multicolumn{1}{r@{}}{31}&\multicolumn{1}{@{ }l}{{\it 4}}&\multicolumn{1}{r@{}}{1413}&\multicolumn{1}{@{.}l}{5}&\multicolumn{1}{l}{(11/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{919}&\multicolumn{1}{@{.}l}{5 }&\multicolumn{1}{@{}l}{(9/2\ensuremath{^{-}})}&\multicolumn{1}{l}{D}&\parbox[t][0.3cm]{7.6129203cm}{\raggedright I\ensuremath{_{\gamma}}: Delayed I\ensuremath{\gamma}=0.\vspace{0.1cm}}&\\
&&&&&&&&&&&\parbox[t][0.3cm]{7.6129203cm}{\raggedright E\ensuremath{_{\gamma}}: 493.9\ensuremath{\gamma} is not found to be in \ensuremath{\gamma}\ensuremath{\gamma} coin with any\vspace{0.1cm}}&\\
&&&&&&&&&&&\parbox[t][0.3cm]{7.6129203cm}{\raggedright {\ }{\ }{\ }transition in \ensuremath{^{\textnormal{201}}}Tl.\vspace{0.1cm}}&\\
&&&&&&&&&&&\parbox[t][0.3cm]{7.6129203cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}={\textminus}0.203 \textit{18}, A\ensuremath{_{\textnormal{4}}}=0.055 \textit{26}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{588}&\multicolumn{1}{@{.}l}{3 {\it 1}}&\multicolumn{1}{r@{}}{191}&\multicolumn{1}{@{ }l}{{\it 9}}&\multicolumn{1}{r@{}}{919}&\multicolumn{1}{@{.}l}{5}&\multicolumn{1}{l}{(9/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{331}&\multicolumn{1}{@{.}l}{2 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}}}&\multicolumn{1}{l}{[E3]}&\parbox[t][0.3cm]{7.6129203cm}{\raggedright I\ensuremath{_{\gamma}}: Delayed I\ensuremath{\gamma}=191; T\ensuremath{_{\textnormal{1/2}}}\ensuremath{\geq}60 ns from 588.3 \ensuremath{\gamma}(t).\vspace{0.1cm}}&\\
&&&&&&&&&&&\parbox[t][0.3cm]{7.6129203cm}{\raggedright Mult.: From adopted gammas; A\ensuremath{_{\textnormal{2}}}=0, A\ensuremath{_{\textnormal{4}}}=0 used for\vspace{0.1cm}}&\\
&&&&&&&&&&&\parbox[t][0.3cm]{7.6129203cm}{\raggedright {\ }{\ }{\ }normalization since the \ensuremath{\gamma}(\ensuremath{\theta}) is isotropic due to the\vspace{0.1cm}}&\\
&&&&&&&&&&&\parbox[t][0.3cm]{7.6129203cm}{\raggedright {\ }{\ }{\ }large T\ensuremath{_{\textnormal{1/2}}} involved.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{598}&\multicolumn{1}{@{.}l}{0 {\it 5}}&\multicolumn{1}{r@{}}{12}&\multicolumn{1}{@{ }l}{{\it 3}}&\multicolumn{1}{r@{}}{1290}&\multicolumn{1}{@{.}l}{2}&\multicolumn{1}{l}{(9/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{692}&\multicolumn{1}{@{.}l}{5 }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{+}}}&\multicolumn{1}{l}{}&&\\
\multicolumn{1}{r@{}}{652}&\multicolumn{1}{@{.}l}{0 {\it 4}}&\multicolumn{1}{r@{}}{22}&\multicolumn{1}{@{.}l}{4 {\it 25}}&\multicolumn{1}{r@{}}{1571}&\multicolumn{1}{@{.}l}{7}&\multicolumn{1}{l}{(13/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{919}&\multicolumn{1}{@{.}l}{5 }&\multicolumn{1}{@{}l}{(9/2\ensuremath{^{-}})}&\multicolumn{1}{l}{E2}&\parbox[t][0.3cm]{7.6129203cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}=0.204 \textit{50}, A\ensuremath{_{\textnormal{4}}}={\textminus}0.062 \textit{77}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{723}&\multicolumn{1}{@{.}l}{9 {\it 2}}&\multicolumn{1}{r@{}}{11}&\multicolumn{1}{@{.}l}{7 {\it 23}}&\multicolumn{1}{r@{}}{1962}&\multicolumn{1}{@{.}l}{2}&\multicolumn{1}{l}{(15/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{1238}&\multicolumn{1}{@{.}l}{5 }&\multicolumn{1}{@{}l}{(11/2\ensuremath{^{-}})}&\multicolumn{1}{l}{E2}&\parbox[t][0.3cm]{7.6129203cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}=0.33 \textit{15} with A\ensuremath{_{\textnormal{4}}} set to zero.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{749}&\multicolumn{1}{@{.}l}{2\ensuremath{^{\hyperlink{TL23GAMMA3}{@}}} {\it 3}}&\multicolumn{1}{r@{}}{9}&\multicolumn{1}{@{.}l}{1 {\it 18}}&\multicolumn{1}{r@{}}{1987}&\multicolumn{1}{@{.}l}{8?}&&\multicolumn{1}{r@{}}{1238}&\multicolumn{1}{@{.}l}{5 }&\multicolumn{1}{@{}l}{(11/2\ensuremath{^{-}})}&&&\\
\multicolumn{1}{r@{}}{785}&\multicolumn{1}{@{.}l}{3\ensuremath{^{\hyperlink{TL23GAMMA3}{@}}} {\it 3}}&\multicolumn{1}{r@{}}{9}&\multicolumn{1}{@{.}l}{2 {\it 19}}&\multicolumn{1}{r@{}}{2747}&\multicolumn{1}{@{.}l}{4?}&&\multicolumn{1}{r@{}}{1962}&\multicolumn{1}{@{.}l}{2 }&\multicolumn{1}{@{}l}{(15/2\ensuremath{^{-}})}&&&\\
\multicolumn{1}{r@{}}{803}&\multicolumn{1}{@{.}l}{6 {\it 1}}&\multicolumn{1}{r@{}}{53}&\multicolumn{1}{@{ }l}{{\it 3}}&\multicolumn{1}{r@{}}{1134}&\multicolumn{1}{@{.}l}{8}&\multicolumn{1}{l}{7/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{331}&\multicolumn{1}{@{.}l}{2 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}}}&\multicolumn{1}{l}{(E2)}&\parbox[t][0.3cm]{7.6129203cm}{\raggedright I\ensuremath{_{\gamma}}: Delayed I\ensuremath{\gamma}=0.\vspace{0.1cm}}&\\
&&&&&&&&&&&\parbox[t][0.3cm]{7.6129203cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}=0.136 \textit{19}, A\ensuremath{_{\textnormal{4}}}={\textminus}0.050 \textit{44}.\vspace{0.1cm}}&\\
\end{longtable}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{TL23GAMMA0}{\dagger}}}} From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1977Sl01,B}{1977Sl01}. I\ensuremath{\gamma} are from the E(d)=24 MeV data and were corrected for angular distribution effect.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{TL23GAMMA1}{\ddagger}}}} Assignment to \ensuremath{^{\textnormal{201}}}Tl is tentative.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{TL23GAMMA2}{\#}}}} From \ensuremath{\gamma}(\ensuremath{\theta}) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1977Sl01,B}{1977Sl01} and the apparent band structures.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{TL23GAMMA3}{@}}}} Placement of transition in the level scheme is uncertain.}\\
\vspace{0.5cm}
\clearpage
\begin{figure}[h]
\begin{center}
\includegraphics{201TL23-0.ps}\\
\end{center}
\end{figure}
\clearpage
\subsection[\hspace{-0.2cm}\ensuremath{^{\textnormal{198}}}Pt(\ensuremath{^{\textnormal{7}}}Li,4n\ensuremath{\gamma})]{ }
\vspace{-27pt}
\vspace{0.3cm}
\hypertarget{TL24}{{\bf \small \underline{\ensuremath{^{\textnormal{198}}}Pt(\ensuremath{^{\textnormal{7}}}Li,4n\ensuremath{\gamma})\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2013Da15,B}{2013Da15}}}}\\
\vspace{4pt}
\vspace{8pt}
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2013Da15,B}{2013Da15}: reaction \ensuremath{^{\textnormal{198}}}Pt(\ensuremath{^{\textnormal{7}}}Li,4n\ensuremath{\gamma}); E(\ensuremath{^{\textnormal{7}}}Li)=45 MeV; Target: 1.3 mg/cm\ensuremath{^{\textnormal{2}}}-thick self-supporting foil, 95.7\% enriched in \ensuremath{^{\textnormal{198}}}Pt;}\\
\parbox[b][0.3cm]{17.7cm}{INGA array configured with 15 Compton suppressed clover high purity germanium (HPGe) detectors; Measured: E\ensuremath{\gamma}, I\ensuremath{\gamma}, \ensuremath{\gamma}\ensuremath{\gamma}\ensuremath{\gamma}}\\
\parbox[b][0.3cm]{17.7cm}{coin, \ensuremath{\gamma}\ensuremath{\gamma}(\ensuremath{\theta}), DCO ratios and \ensuremath{\gamma}-ray polarization; Deduced: level scheme, \ensuremath{J^{\pi}}.}\\
\vspace{12pt}
\underline{$^{201}$Tl Levels}\\
\begin{longtable}{cccccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E(level)$^{{\hyperlink{TL24LEVEL0}{\dagger}}}$}&J$^{\pi}$$^{{\hyperlink{TL24LEVEL1}{\ddagger}}}$&\multicolumn{2}{c}{T$_{1/2}$$^{{\hyperlink{TL24LEVEL2}{\#}}}$}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{l}{1/2\ensuremath{^{+}}\ensuremath{^{{\hyperlink{TL24LEVEL2}{\#}}}}}&&&&\\
\multicolumn{1}{r@{}}{331}&\multicolumn{1}{@{.}l}{15 {\it 6}}&\multicolumn{1}{l}{3/2\ensuremath{^{+}}\ensuremath{^{{\hyperlink{TL24LEVEL2}{\#}}}}}&&&&\\
\multicolumn{1}{r@{}}{919}&\multicolumn{1}{@{.}l}{45 {\it 12}}&\multicolumn{1}{l}{(9/2\ensuremath{^{-}})\ensuremath{^{{\hyperlink{TL24LEVEL2}{\#}}}}}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{11 ms {\it 11}}&\parbox[t][0.3cm]{12.2523cm}{\raggedright \%IT=100\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1238}&\multicolumn{1}{@{.}l}{44 {\it 16}}&\multicolumn{1}{l}{11/2\ensuremath{^{-}}}&&&&\\
\multicolumn{1}{r@{}}{1571}&\multicolumn{1}{@{.}l}{84 {\it 17}}&\multicolumn{1}{l}{13/2\ensuremath{^{-}}}&&&&\\
\multicolumn{1}{r@{}}{1962}&\multicolumn{1}{@{.}l}{74 {\it 18}}&\multicolumn{1}{l}{15/2\ensuremath{^{-}}}&&&&\\
\multicolumn{1}{r@{}}{1987}&\multicolumn{1}{@{.}l}{95 {\it 19}}&\multicolumn{1}{l}{13/2\ensuremath{^{-}}}&&&&\\
\multicolumn{1}{r@{}}{2015}&\multicolumn{1}{@{.}l}{1 {\it 9}}&\multicolumn{1}{l}{17/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{9 ns {\it +19\textminus5}}&&\\
\multicolumn{1}{r@{}}{2041}&\multicolumn{1}{@{.}l}{2 {\it 3}}&\multicolumn{1}{l}{15/2\ensuremath{^{+}}}&&&&\\
\multicolumn{1}{r@{}}{2182}&\multicolumn{1}{@{.}l}{0 {\it 9}}&\multicolumn{1}{l}{19/2\ensuremath{^{-}}}&&&&\\
\multicolumn{1}{r@{}}{2442}&\multicolumn{1}{@{.}l}{0 {\it 9}}&\multicolumn{1}{l}{21/2\ensuremath{^{-}}}&&&&\\
\multicolumn{1}{r@{}}{2486}&\multicolumn{1}{@{.}l}{4 {\it 8}}&\multicolumn{1}{l}{(17/2)}&&&&\\
\multicolumn{1}{r@{}}{2672}&\multicolumn{1}{@{.}l}{1 {\it 9}}&\multicolumn{1}{l}{23/2\ensuremath{^{+}}}&&&&\\
\multicolumn{1}{r@{}}{2713}&\multicolumn{1}{@{.}l}{64 {\it 20}}&\multicolumn{1}{l}{15/2\ensuremath{^{-}}}&&&&\\
\multicolumn{1}{r@{}}{2747}&\multicolumn{1}{@{.}l}{9 {\it 7}}&\multicolumn{1}{l}{17/2\ensuremath{^{+}}}&&&&\\
\multicolumn{1}{r@{}}{2883}&\multicolumn{1}{@{.}l}{6 {\it 9}}&\multicolumn{1}{l}{23/2\ensuremath{^{-}}}&&&&\\
\multicolumn{1}{r@{}}{2897}&\multicolumn{1}{@{.}l}{6 {\it 9}}&\multicolumn{1}{l}{25/2\ensuremath{^{+}}}&&&&\\
\multicolumn{1}{r@{}}{2985}&\multicolumn{1}{@{.}l}{1 {\it 9}}&\multicolumn{1}{l}{21/2\ensuremath{^{+}}}&&&&\\
\multicolumn{1}{r@{}}{3011}&\multicolumn{1}{@{.}l}{7 {\it 7}}&\multicolumn{1}{l}{19/2\ensuremath{^{+}}}&&&&\\
\multicolumn{1}{r@{}}{3044}&\multicolumn{1}{@{.}l}{9 {\it 9}}&\multicolumn{1}{l}{25/2\ensuremath{^{-}}}&&&&\\
\multicolumn{1}{r@{}}{3056}&\multicolumn{1}{@{.}l}{6 {\it 7}}&\multicolumn{1}{l}{19/2\ensuremath{^{+}}}&&&&\\
\multicolumn{1}{r@{}}{3314}&\multicolumn{1}{@{.}l}{7 {\it 7}}&\multicolumn{1}{l}{21/2\ensuremath{^{+}}}&&&&\\
\multicolumn{1}{r@{}}{3371}&\multicolumn{1}{@{.}l}{8 {\it 9}}&\multicolumn{1}{l}{23/2\ensuremath{^{+}}}&&&&\\
\multicolumn{1}{r@{}}{3431}&\multicolumn{1}{@{.}l}{7 {\it 9}}&\multicolumn{1}{l}{27/2\ensuremath{^{-}}}&&&&\\
\multicolumn{1}{r@{}}{3434}&\multicolumn{1}{@{.}l}{5 {\it 7}}&\multicolumn{1}{l}{23/2\ensuremath{^{+}}}&&&&\\
\multicolumn{1}{r@{}}{3530}&\multicolumn{1}{@{.}l}{6 {\it 7}}&\multicolumn{1}{l}{25/2}&&&&\\
\multicolumn{1}{r@{}}{3706}&\multicolumn{1}{@{.}l}{8 {\it 7}}&\multicolumn{1}{l}{27/2}&&&&\\
\multicolumn{1}{r@{}}{3864}&\multicolumn{1}{@{.}l}{3 {\it 7}}&\multicolumn{1}{l}{29/2}&&&&\\
\multicolumn{1}{r@{}}{3936}&\multicolumn{1}{@{.}l}{1 {\it 9}}&\multicolumn{1}{l}{29/2}&&&&\\
\multicolumn{1}{r@{}}{3950}&\multicolumn{1}{@{.}l}{9 {\it 9}}&\multicolumn{1}{l}{(25/2\ensuremath{^{+}})}&&&&\\
\end{longtable}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{TL24LEVEL0}{\dagger}}}} From least-squares fit to E\ensuremath{\gamma}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{TL24LEVEL1}{\ddagger}}}} From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2013Da15,B}{2013Da15}, unless otherwise stated.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{TL24LEVEL2}{\#}}}} From Adopted Levels.}\\
\vspace{0.5cm}
\underline{$\gamma$($^{201}$Tl)}\\
\begin{longtable}{ccccccccc@{}ccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{TL24GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{I\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{TL24GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{E\ensuremath{_{i}}(level)}&J\ensuremath{^{\pi}_{i}}&\multicolumn{2}{c}{E\ensuremath{_{f}}}&J\ensuremath{^{\pi}_{f}}&Mult.\ensuremath{^{\hyperlink{TL24GAMMA0}{\dagger}}}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{(34}&\multicolumn{1}{@{.}l}{3 {\it 7})}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{2747}&\multicolumn{1}{@{.}l}{9}&\multicolumn{1}{l}{17/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{2713}&\multicolumn{1}{@{.}l}{64 }&\multicolumn{1}{@{}l}{15/2\ensuremath{^{-}}}&\multicolumn{1}{l}{}&\parbox[t][0.3cm]{7.9703cm}{\raggedright E\ensuremath{_{\gamma}}: From level energy differences (adopted gammas). Not\vspace{0.1cm}}&\\
&&&&&&&&&&&\parbox[t][0.3cm]{7.9703cm}{\raggedright {\ }{\ }{\ }observed directly, but required by the coincidence\vspace{0.1cm}}&\\
&&&&&&&&&&&\parbox[t][0.3cm]{7.9703cm}{\raggedright {\ }{\ }{\ }relationship (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2013Da15,B}{2013Da15}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{96}&\multicolumn{1}{@{.}l}{1 {\it 1}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{95 {\it 3}}&\multicolumn{1}{r@{}}{3530}&\multicolumn{1}{@{.}l}{6}&\multicolumn{1}{l}{25/2}&\multicolumn{1}{r@{}}{3434}&\multicolumn{1}{@{.}l}{5 }&\multicolumn{1}{@{}l}{23/2\ensuremath{^{+}}}&\multicolumn{1}{l}{(M1+E2)\ensuremath{^{\hyperlink{TL24GAMMA2}{\#}}}}&\parbox[t][0.3cm]{7.9703cm}{\raggedright DCO=0.61 \textit{6}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{119}&\multicolumn{1}{@{.}l}{8 {\it 2}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{51 {\it 35}}&\multicolumn{1}{r@{}}{3434}&\multicolumn{1}{@{.}l}{5}&\multicolumn{1}{l}{23/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{3314}&\multicolumn{1}{@{.}l}{7 }&\multicolumn{1}{@{}l}{21/2\ensuremath{^{+}}}&\multicolumn{1}{l}{M1\ensuremath{^{\hyperlink{TL24GAMMA2}{\#}}}}&\parbox[t][0.3cm]{7.9703cm}{\raggedright DCO=0.96 \textit{14}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{157}&\multicolumn{1}{@{.}l}{5 {\it 1}}&\multicolumn{1}{r@{}}{7}&\multicolumn{1}{@{.}l}{6 {\it 7}}&\multicolumn{1}{r@{}}{3864}&\multicolumn{1}{@{.}l}{3}&\multicolumn{1}{l}{29/2}&\multicolumn{1}{r@{}}{3706}&\multicolumn{1}{@{.}l}{8 }&\multicolumn{1}{@{}l}{27/2}&\multicolumn{1}{l}{M1+E2\ensuremath{^{\hyperlink{TL24GAMMA2}{\#}}}}&\parbox[t][0.3cm]{7.9703cm}{\raggedright DCO=0.86 \textit{4}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{161}&\multicolumn{1}{@{.}l}{2 {\it 1}}&\multicolumn{1}{r@{}}{6}&\multicolumn{1}{@{.}l}{1 {\it 7}}&\multicolumn{1}{r@{}}{3044}&\multicolumn{1}{@{.}l}{9}&\multicolumn{1}{l}{25/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{2883}&\multicolumn{1}{@{.}l}{6 }&\multicolumn{1}{@{}l}{23/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1+E2\ensuremath{^{\hyperlink{TL24GAMMA3}{@}}}}&\parbox[t][0.3cm]{7.9703cm}{\raggedright DCO=1.30 \textit{4}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{166}&\multicolumn{1}{@{.}l}{9 {\it 1}}&\multicolumn{1}{r@{}}{7}&\multicolumn{1}{@{.}l}{5 {\it 7}}&\multicolumn{1}{r@{}}{2182}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{l}{19/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{2015}&\multicolumn{1}{@{.}l}{1 }&\multicolumn{1}{@{}l}{17/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1+E2\ensuremath{^{\hyperlink{TL24GAMMA3}{@}}}}&\parbox[t][0.3cm]{7.9703cm}{\raggedright DCO=1.43 \textit{4}\vspace{0.1cm}}&\\
\end{longtable}
\begin{textblock}{29}(0,27.3)
Continued on next page (footnotes at end of table)
\end{textblock}
\clearpage
\begin{longtable}{ccccccccc@{}ccc@{\extracolsep{\fill}}c}
\\[-.4cm]
\multicolumn{13}{c}{{\bf \small \underline{\ensuremath{^{\textnormal{198}}}Pt(\ensuremath{^{\textnormal{7}}}Li,4n\ensuremath{\gamma})\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2013Da15,B}{2013Da15} (continued)}}}\\
\multicolumn{13}{c}{~}\\
\multicolumn{13}{c}{\underline{$\gamma$($^{201}$Tl) (continued)}}\\
\multicolumn{13}{c}{~~~}\\
\multicolumn{2}{c}{E\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{TL24GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{I\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{TL24GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{E\ensuremath{_{i}}(level)}&J\ensuremath{^{\pi}_{i}}&\multicolumn{2}{c}{E\ensuremath{_{f}}}&J\ensuremath{^{\pi}_{f}}&Mult.\ensuremath{^{\hyperlink{TL24GAMMA0}{\dagger}}}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&\hrulefill&
\endhead
\multicolumn{1}{r@{}}{176}&\multicolumn{1}{@{.}l}{2 {\it 1}}&\multicolumn{1}{r@{}}{10}&\multicolumn{1}{@{.}l}{9 {\it 8}}&\multicolumn{1}{r@{}}{3706}&\multicolumn{1}{@{.}l}{8}&\multicolumn{1}{l}{27/2}&\multicolumn{1}{r@{}}{3530}&\multicolumn{1}{@{.}l}{6 }&\multicolumn{1}{@{}l}{25/2}&\multicolumn{1}{l}{M1+E2\ensuremath{^{\hyperlink{TL24GAMMA2}{\#}}}}&\parbox[t][0.3cm]{7.015861cm}{\raggedright DCO=0.85 \textit{4}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{211}&\multicolumn{1}{@{.}l}{8 {\it 2}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{70 {\it 15}}&\multicolumn{1}{r@{}}{2883}&\multicolumn{1}{@{.}l}{6}&\multicolumn{1}{l}{23/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{2672}&\multicolumn{1}{@{.}l}{1 }&\multicolumn{1}{@{}l}{23/2\ensuremath{^{+}}}&\multicolumn{1}{l}{(E1)}&&\\
\multicolumn{1}{r@{}}{225}&\multicolumn{1}{@{.}l}{5 {\it 1}}&\multicolumn{1}{r@{}}{4}&\multicolumn{1}{@{.}l}{0 {\it 6}}&\multicolumn{1}{r@{}}{2897}&\multicolumn{1}{@{.}l}{6}&\multicolumn{1}{l}{25/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{2672}&\multicolumn{1}{@{.}l}{1 }&\multicolumn{1}{@{}l}{23/2\ensuremath{^{+}}}&\multicolumn{1}{l}{M1+E2\ensuremath{^{\hyperlink{TL24GAMMA3}{@}}}}&\parbox[t][0.3cm]{7.015861cm}{\raggedright DCO=1.42 \textit{7}\vspace{0.1cm}}&\\
&&&&&&&&&&&\parbox[t][0.3cm]{7.015861cm}{\raggedright POL={\textminus}0.05 \textit{10}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{230}&\multicolumn{1}{@{.}l}{2 {\it 1}}&\multicolumn{1}{r@{}}{9}&\multicolumn{1}{@{.}l}{0 {\it 7}}&\multicolumn{1}{r@{}}{2672}&\multicolumn{1}{@{.}l}{1}&\multicolumn{1}{l}{23/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{2442}&\multicolumn{1}{@{.}l}{0 }&\multicolumn{1}{@{}l}{21/2\ensuremath{^{-}}}&\multicolumn{1}{l}{E1\ensuremath{^{\hyperlink{TL24GAMMA3}{@}}}}&\parbox[t][0.3cm]{7.015861cm}{\raggedright DCO=1.30 \textit{4}\vspace{0.1cm}}&\\
&&&&&&&&&&&\parbox[t][0.3cm]{7.015861cm}{\raggedright POL=+0.07 \textit{6}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{258}&\multicolumn{1}{@{.}l}{0 {\it 1}}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{4 {\it 12}}&\multicolumn{1}{r@{}}{3314}&\multicolumn{1}{@{.}l}{7}&\multicolumn{1}{l}{21/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{3056}&\multicolumn{1}{@{.}l}{6 }&\multicolumn{1}{@{}l}{19/2\ensuremath{^{+}}}&\multicolumn{1}{l}{M1+E2\ensuremath{^{\hyperlink{TL24GAMMA2}{\#}}}}&\parbox[t][0.3cm]{7.015861cm}{\raggedright DCO=1.17 \textit{14}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{260}&\multicolumn{1}{@{.}l}{1 {\it 1}}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{2 {\it 12}}&\multicolumn{1}{r@{}}{2442}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{l}{21/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{2182}&\multicolumn{1}{@{.}l}{0 }&\multicolumn{1}{@{}l}{19/2\ensuremath{^{-}}}&\multicolumn{1}{l}{(M1)}&&\\
\multicolumn{1}{r@{}}{263}&\multicolumn{1}{@{.}l}{7 {\it 1}}&\multicolumn{1}{r@{}}{7}&\multicolumn{1}{@{.}l}{2 {\it 7}}&\multicolumn{1}{r@{}}{3011}&\multicolumn{1}{@{.}l}{7}&\multicolumn{1}{l}{19/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{2747}&\multicolumn{1}{@{.}l}{9 }&\multicolumn{1}{@{}l}{17/2\ensuremath{^{+}}}&\multicolumn{1}{l}{M1\ensuremath{^{\hyperlink{TL24GAMMA2}{\#}}}}&\parbox[t][0.3cm]{7.015861cm}{\raggedright DCO=1.14 \textit{5}\vspace{0.1cm}}&\\
&&&&&&&&&&&\parbox[t][0.3cm]{7.015861cm}{\raggedright POL={\textminus}0.30 \textit{5}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{308}&\multicolumn{1}{@{.}l}{8 {\it 9}}&\multicolumn{1}{r@{}}{5}&\multicolumn{1}{@{.}l}{6 {\it 6}}&\multicolumn{1}{r@{}}{3056}&\multicolumn{1}{@{.}l}{6}&\multicolumn{1}{l}{19/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{2747}&\multicolumn{1}{@{.}l}{9 }&\multicolumn{1}{@{}l}{17/2\ensuremath{^{+}}}&\multicolumn{1}{l}{M1+E2\ensuremath{^{\hyperlink{TL24GAMMA2}{\#}}}}&\parbox[t][0.3cm]{7.015861cm}{\raggedright DCO=1.29 \textit{10}\vspace{0.1cm}}&\\
&&&&&&&&&&&\parbox[t][0.3cm]{7.015861cm}{\raggedright POL={\textminus}0.24 \textit{4}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{319}&\multicolumn{1}{@{.}l}{0 {\it 1}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{ }l}{{\it 6}}&\multicolumn{1}{r@{}}{1238}&\multicolumn{1}{@{.}l}{44}&\multicolumn{1}{l}{11/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{919}&\multicolumn{1}{@{.}l}{45 }&\multicolumn{1}{@{}l}{(9/2\ensuremath{^{-}})}&\multicolumn{1}{l}{M1+E2\ensuremath{^{\hyperlink{TL24GAMMA2}{\#}}}}&\parbox[t][0.3cm]{7.015861cm}{\raggedright DCO=1.56 \textit{4}\vspace{0.1cm}}&\\
&&&&&&&&&&&\parbox[t][0.3cm]{7.015861cm}{\raggedright POL={\textminus}0.060 \textit{25}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{326}&\multicolumn{1}{@{.}l}{7 {\it 3}}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{4 {\it 5}}&\multicolumn{1}{r@{}}{3371}&\multicolumn{1}{@{.}l}{8}&\multicolumn{1}{l}{23/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{3044}&\multicolumn{1}{@{.}l}{9 }&\multicolumn{1}{@{}l}{25/2\ensuremath{^{-}}}&\multicolumn{1}{l}{(E1)}&&\\
\multicolumn{1}{r@{}}{331}&\multicolumn{1}{@{.}l}{15\ensuremath{^{\hyperlink{TL24GAMMA1}{\ddagger}}} {\it 6}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{331}&\multicolumn{1}{@{.}l}{15}&\multicolumn{1}{l}{3/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{@{}l}{1/2\ensuremath{^{+}}}&&&\\
\multicolumn{1}{r@{}}{333}&\multicolumn{1}{@{.}l}{4 {\it 1}}&\multicolumn{1}{r@{}}{75}&\multicolumn{1}{@{ }l}{{\it 4}}&\multicolumn{1}{r@{}}{1571}&\multicolumn{1}{@{.}l}{84}&\multicolumn{1}{l}{13/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{1238}&\multicolumn{1}{@{.}l}{44 }&\multicolumn{1}{@{}l}{11/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1\ensuremath{^{\hyperlink{TL24GAMMA2}{\#}}}}&\parbox[t][0.3cm]{7.015861cm}{\raggedright DCO=1.13 \textit{3}\vspace{0.1cm}}&\\
&&&&&&&&&&&\parbox[t][0.3cm]{7.015861cm}{\raggedright POL={\textminus}0.026 \textit{19}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{377}&\multicolumn{1}{@{.}l}{9 {\it 1}}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{9 {\it 4}}&\multicolumn{1}{r@{}}{3434}&\multicolumn{1}{@{.}l}{5}&\multicolumn{1}{l}{23/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{3056}&\multicolumn{1}{@{.}l}{6 }&\multicolumn{1}{@{}l}{19/2\ensuremath{^{+}}}&\multicolumn{1}{l}{E2\ensuremath{^{\hyperlink{TL24GAMMA2}{\#}}}}&\parbox[t][0.3cm]{7.015861cm}{\raggedright DCO=0.46 \textit{4}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{386}&\multicolumn{1}{@{.}l}{7 {\it 1}}&\multicolumn{1}{r@{}}{5}&\multicolumn{1}{@{.}l}{4 {\it 8}}&\multicolumn{1}{r@{}}{3371}&\multicolumn{1}{@{.}l}{8}&\multicolumn{1}{l}{23/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{2985}&\multicolumn{1}{@{.}l}{1 }&\multicolumn{1}{@{}l}{21/2\ensuremath{^{+}}}&\multicolumn{1}{l}{M1\ensuremath{^{\hyperlink{TL24GAMMA4}{\&}}}}&\parbox[t][0.3cm]{7.015861cm}{\raggedright DCO=1.03 \textit{4}\vspace{0.1cm}}&\\
&&&&&&&&&&&\parbox[t][0.3cm]{7.015861cm}{\raggedright POL={\textminus}0.09 \textit{5}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{386}&\multicolumn{1}{@{.}l}{8 {\it 1}}&\multicolumn{1}{r@{}}{4}&\multicolumn{1}{@{.}l}{65 {\it 26}}&\multicolumn{1}{r@{}}{3431}&\multicolumn{1}{@{.}l}{7}&\multicolumn{1}{l}{27/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{3044}&\multicolumn{1}{@{.}l}{9 }&\multicolumn{1}{@{}l}{25/2\ensuremath{^{-}}}&\multicolumn{1}{l}{(M1+E2)}&&\\
\multicolumn{1}{r@{}}{390}&\multicolumn{1}{@{.}l}{9 {\it 1}}&\multicolumn{1}{r@{}}{22}&\multicolumn{1}{@{.}l}{9 {\it 14}}&\multicolumn{1}{r@{}}{1962}&\multicolumn{1}{@{.}l}{74}&\multicolumn{1}{l}{15/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{1571}&\multicolumn{1}{@{.}l}{84 }&\multicolumn{1}{@{}l}{13/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1+E2\ensuremath{^{\hyperlink{TL24GAMMA2}{\#}}}}&\parbox[t][0.3cm]{7.015861cm}{\raggedright DCO=1.48 \textit{4}\vspace{0.1cm}}&\\
&&&&&&&&&&&\parbox[t][0.3cm]{7.015861cm}{\raggedright POL={\textminus}0.016 \textit{28}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{422}&\multicolumn{1}{@{.}l}{8 {\it 2}}&\multicolumn{1}{r@{}}{4}&\multicolumn{1}{@{.}l}{2 {\it 8}}&\multicolumn{1}{r@{}}{3434}&\multicolumn{1}{@{.}l}{5}&\multicolumn{1}{l}{23/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{3011}&\multicolumn{1}{@{.}l}{7 }&\multicolumn{1}{@{}l}{19/2\ensuremath{^{+}}}&\multicolumn{1}{l}{E2\ensuremath{^{\hyperlink{TL24GAMMA2}{\#}}}}&\parbox[t][0.3cm]{7.015861cm}{\raggedright DCO=0.52 \textit{4}\vspace{0.1cm}}&\\
&&&&&&&&&&&\parbox[t][0.3cm]{7.015861cm}{\raggedright POL=+0.13 \textit{5}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{426}&\multicolumn{1}{@{.}l}{8 {\it 1}}&\multicolumn{1}{r@{}}{24}&\multicolumn{1}{@{.}l}{0 {\it 15}}&\multicolumn{1}{r@{}}{2442}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{l}{21/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{2015}&\multicolumn{1}{@{.}l}{1 }&\multicolumn{1}{@{}l}{17/2\ensuremath{^{-}}}&\multicolumn{1}{l}{E2\ensuremath{^{\hyperlink{TL24GAMMA3}{@}}}}&\parbox[t][0.3cm]{7.015861cm}{\raggedright DCO=1.06 \textit{2}\vspace{0.1cm}}&\\
&&&&&&&&&&&\parbox[t][0.3cm]{7.015861cm}{\raggedright POL=+0.143 \textit{23}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{441}&\multicolumn{1}{@{.}l}{1 {\it 7}}&\multicolumn{1}{r@{}}{6}&\multicolumn{1}{@{.}l}{8 {\it 26}}&\multicolumn{1}{r@{}}{2883}&\multicolumn{1}{@{.}l}{6}&\multicolumn{1}{l}{23/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{2442}&\multicolumn{1}{@{.}l}{0 }&\multicolumn{1}{@{}l}{21/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1\ensuremath{^{\hyperlink{TL24GAMMA3}{@}}}}&\parbox[t][0.3cm]{7.015861cm}{\raggedright DCO=1.92 \textit{7}\vspace{0.1cm}}&\\
&&&&&&&&&&&\parbox[t][0.3cm]{7.015861cm}{\raggedright POL={\textminus}0.059 \textit{29}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{443}&\multicolumn{1}{@{.}l}{3 {\it 8}}&\multicolumn{1}{r@{}}{47}&\multicolumn{1}{@{ }l}{{\it 4}}&\multicolumn{1}{r@{}}{2015}&\multicolumn{1}{@{.}l}{1}&\multicolumn{1}{l}{17/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{1571}&\multicolumn{1}{@{.}l}{84 }&\multicolumn{1}{@{}l}{13/2\ensuremath{^{-}}}&\multicolumn{1}{l}{E2\ensuremath{^{\hyperlink{TL24GAMMA5}{a}}}}&\parbox[t][0.3cm]{7.015861cm}{\raggedright DCO=1.01 \textit{3}\vspace{0.1cm}}&\\
&&&&&&&&&&&\parbox[t][0.3cm]{7.015861cm}{\raggedright POL=+0.077 \textit{30}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{445}&\multicolumn{1}{@{.}l}{2 {\it 7}}&\multicolumn{1}{r@{}}{6}&\multicolumn{1}{@{.}l}{3 {\it 7}}&\multicolumn{1}{r@{}}{2486}&\multicolumn{1}{@{.}l}{4}&\multicolumn{1}{l}{(17/2)}&\multicolumn{1}{r@{}}{2041}&\multicolumn{1}{@{.}l}{2 }&\multicolumn{1}{@{}l}{15/2\ensuremath{^{+}}}&\multicolumn{1}{l}{\ensuremath{^{\hyperlink{TL24GAMMA6}{b}}}}&\parbox[t][0.3cm]{7.015861cm}{\raggedright DCO=0.71 \textit{5}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{469}&\multicolumn{1}{@{.}l}{4 {\it 2}}&\multicolumn{1}{r@{}}{11}&\multicolumn{1}{@{.}l}{0 {\it 8}}&\multicolumn{1}{r@{}}{2041}&\multicolumn{1}{@{.}l}{2}&\multicolumn{1}{l}{15/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1571}&\multicolumn{1}{@{.}l}{84 }&\multicolumn{1}{@{}l}{13/2\ensuremath{^{-}}}&\multicolumn{1}{l}{E1\ensuremath{^{\hyperlink{TL24GAMMA5}{a}}}}&\parbox[t][0.3cm]{7.015861cm}{\raggedright DCO=1.8 \textit{7}\vspace{0.1cm}}&\\
&&&&&&&&&&&\parbox[t][0.3cm]{7.015861cm}{\raggedright POL=+0.13 \textit{7}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{504}&\multicolumn{1}{@{.}l}{4 {\it 1}}&\multicolumn{1}{r@{}}{6}&\multicolumn{1}{@{.}l}{5 {\it 10}}&\multicolumn{1}{r@{}}{3936}&\multicolumn{1}{@{.}l}{1}&\multicolumn{1}{l}{29/2}&\multicolumn{1}{r@{}}{3431}&\multicolumn{1}{@{.}l}{7 }&\multicolumn{1}{@{}l}{27/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1,E1\ensuremath{^{\hyperlink{TL24GAMMA3}{@}}}}&\parbox[t][0.3cm]{7.015861cm}{\raggedright DCO=2.01 \textit{1}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{543}&\multicolumn{1}{@{.}l}{2 {\it 4}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{4 {\it 4}}&\multicolumn{1}{r@{}}{2985}&\multicolumn{1}{@{.}l}{1}&\multicolumn{1}{l}{21/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{2442}&\multicolumn{1}{@{.}l}{0 }&\multicolumn{1}{@{}l}{21/2\ensuremath{^{-}}}&\multicolumn{1}{l}{}&&\\
\multicolumn{1}{r@{}}{566}&\multicolumn{1}{@{.}l}{8 {\it 3}}&\multicolumn{1}{r@{}}{4}&\multicolumn{1}{@{.}l}{2 {\it 5}}&\multicolumn{1}{r@{}}{3314}&\multicolumn{1}{@{.}l}{7}&\multicolumn{1}{l}{21/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{2747}&\multicolumn{1}{@{.}l}{9 }&\multicolumn{1}{@{}l}{17/2\ensuremath{^{+}}}&\multicolumn{1}{l}{E2\ensuremath{^{\hyperlink{TL24GAMMA2}{\#}}}}&\parbox[t][0.3cm]{7.015861cm}{\raggedright DCO=0.43 \textit{8}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{579}&\multicolumn{1}{@{.}l}{1 {\it 1}}&\multicolumn{1}{r@{}}{4}&\multicolumn{1}{@{.}l}{8 {\it 5}}&\multicolumn{1}{r@{}}{3950}&\multicolumn{1}{@{.}l}{9}&\multicolumn{1}{l}{(25/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{3371}&\multicolumn{1}{@{.}l}{8 }&\multicolumn{1}{@{}l}{23/2\ensuremath{^{+}}}&\multicolumn{1}{l}{(M1+E2)\ensuremath{^{\hyperlink{TL24GAMMA3}{@}}}}&\parbox[t][0.3cm]{7.015861cm}{\raggedright DCO=0.68 \textit{4}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{588}&\multicolumn{1}{@{.}l}{3\ensuremath{^{\hyperlink{TL24GAMMA1}{\ddagger}}} {\it 1}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{919}&\multicolumn{1}{@{.}l}{45}&\multicolumn{1}{l}{(9/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{331}&\multicolumn{1}{@{.}l}{15 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}}}&&&\\
\multicolumn{1}{r@{}}{652}&\multicolumn{1}{@{.}l}{2 {\it 5}}&\multicolumn{1}{r@{}}{32}&\multicolumn{1}{@{.}l}{9 {\it 24}}&\multicolumn{1}{r@{}}{1571}&\multicolumn{1}{@{.}l}{84}&\multicolumn{1}{l}{13/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{919}&\multicolumn{1}{@{.}l}{45 }&\multicolumn{1}{@{}l}{(9/2\ensuremath{^{-}})}&\multicolumn{1}{l}{E2\ensuremath{^{\hyperlink{TL24GAMMA4}{\&}}}}&\parbox[t][0.3cm]{7.015861cm}{\raggedright DCO=0.53 \textit{4}\vspace{0.1cm}}&\\
&&&&&&&&&&&\parbox[t][0.3cm]{7.015861cm}{\raggedright POL=+0.117 \textit{24}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{700}&\multicolumn{1}{@{.}l}{9 {\it 3}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{8 {\it 6}}&\multicolumn{1}{r@{}}{2883}&\multicolumn{1}{@{.}l}{6}&\multicolumn{1}{l}{23/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{2182}&\multicolumn{1}{@{.}l}{0 }&\multicolumn{1}{@{}l}{19/2\ensuremath{^{-}}}&\multicolumn{1}{l}{E2\ensuremath{^{\hyperlink{TL24GAMMA3}{@}}}}&\parbox[t][0.3cm]{7.015861cm}{\raggedright DCO=1.09 \textit{12}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{724}&\multicolumn{1}{@{.}l}{3 {\it 1}}&\multicolumn{1}{r@{}}{20}&\multicolumn{1}{@{.}l}{2 {\it 17}}&\multicolumn{1}{r@{}}{1962}&\multicolumn{1}{@{.}l}{74}&\multicolumn{1}{l}{15/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{1238}&\multicolumn{1}{@{.}l}{44 }&\multicolumn{1}{@{}l}{11/2\ensuremath{^{-}}}&\multicolumn{1}{l}{E2\ensuremath{^{\hyperlink{TL24GAMMA2}{\#}}}}&\parbox[t][0.3cm]{7.015861cm}{\raggedright DCO=0.54 \textit{2}\vspace{0.1cm}}&\\
&&&&&&&&&&&\parbox[t][0.3cm]{7.015861cm}{\raggedright POL=+0.062 \textit{32}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{749}&\multicolumn{1}{@{.}l}{5 {\it 1}}&\multicolumn{1}{r@{}}{5}&\multicolumn{1}{@{.}l}{2 {\it 5}}&\multicolumn{1}{r@{}}{1987}&\multicolumn{1}{@{.}l}{95}&\multicolumn{1}{l}{13/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{1238}&\multicolumn{1}{@{.}l}{44 }&\multicolumn{1}{@{}l}{11/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1\ensuremath{^{\hyperlink{TL24GAMMA6}{b}}}}&\parbox[t][0.3cm]{7.015861cm}{\raggedright DCO=1.41 \textit{18}\vspace{0.1cm}}&\\
&&&&&&&&&&&\parbox[t][0.3cm]{7.015861cm}{\raggedright POL={\textminus}0.07 \textit{5}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{785}&\multicolumn{1}{@{.}l}{2 {\it 6}}&\multicolumn{1}{r@{}}{12}&\multicolumn{1}{@{.}l}{0 {\it 8}}&\multicolumn{1}{r@{}}{2747}&\multicolumn{1}{@{.}l}{9}&\multicolumn{1}{l}{17/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1962}&\multicolumn{1}{@{.}l}{74 }&\multicolumn{1}{@{}l}{15/2\ensuremath{^{-}}}&\multicolumn{1}{l}{E1\ensuremath{^{\hyperlink{TL24GAMMA7}{c}}}}&\parbox[t][0.3cm]{7.015861cm}{\raggedright DCO=2.13 \textit{13}\vspace{0.1cm}}&\\
&&&&&&&&&&&\parbox[t][0.3cm]{7.015861cm}{\raggedright POL={\textminus}0.032 \textit{25}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{803}&\multicolumn{1}{@{.}l}{1 {\it 1}}&\multicolumn{1}{r@{}}{12}&\multicolumn{1}{@{.}l}{9 {\it 10}}&\multicolumn{1}{r@{}}{2985}&\multicolumn{1}{@{.}l}{1}&\multicolumn{1}{l}{21/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{2182}&\multicolumn{1}{@{.}l}{0 }&\multicolumn{1}{@{}l}{19/2\ensuremath{^{-}}}&\multicolumn{1}{l}{E1\ensuremath{^{\hyperlink{TL24GAMMA3}{@}}}}&\parbox[t][0.3cm]{7.015861cm}{\raggedright DCO=1.90 \textit{16}\vspace{0.1cm}}&\\
&&&&&&&&&&&\parbox[t][0.3cm]{7.015861cm}{\raggedright POL=+0.066 \textit{21}.\vspace{0.1cm}}&\\
\end{longtable}
\begin{textblock}{29}(0,27.3)
Continued on next page (footnotes at end of table)
\end{textblock}
\clearpage
\begin{longtable}{ccccccccc@{}ccc@{\extracolsep{\fill}}c}
\\[-.4cm]
\multicolumn{13}{c}{{\bf \small \underline{\ensuremath{^{\textnormal{198}}}Pt(\ensuremath{^{\textnormal{7}}}Li,4n\ensuremath{\gamma})\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2013Da15,B}{2013Da15} (continued)}}}\\
\multicolumn{13}{c}{~}\\
\multicolumn{13}{c}{\underline{$\gamma$($^{201}$Tl) (continued)}}\\
\multicolumn{13}{c}{~~~}\\
\multicolumn{2}{c}{E\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{TL24GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{I\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{TL24GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{E\ensuremath{_{i}}(level)}&J\ensuremath{^{\pi}_{i}}&\multicolumn{2}{c}{E\ensuremath{_{f}}}&J\ensuremath{^{\pi}_{f}}&Mult.\ensuremath{^{\hyperlink{TL24GAMMA0}{\dagger}}}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&\hrulefill&
\endhead
\multicolumn{1}{r@{}}{1141}&\multicolumn{1}{@{.}l}{8 {\it 1}}&\multicolumn{1}{r@{}}{12}&\multicolumn{1}{@{.}l}{0 {\it 8}}&\multicolumn{1}{r@{}}{2713}&\multicolumn{1}{@{.}l}{64}&\multicolumn{1}{l}{15/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{1571}&\multicolumn{1}{@{.}l}{84 }&\multicolumn{1}{@{}l}{13/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1}&\parbox[t][0.3cm]{8.610161cm}{\raggedright DCO=2.13 \textit{13}\vspace{0.1cm}}&\\
&&&&&&&&&&&\parbox[t][0.3cm]{8.610161cm}{\raggedright POL={\textminus}0.032 \textit{25}.\vspace{0.1cm}}&\\
\end{longtable}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{TL24GAMMA0}{\dagger}}}} From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2013Da15,B}{2013Da15}, unless otherwise stated.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{TL24GAMMA1}{\ddagger}}}} From adopted gammas.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{TL24GAMMA2}{\#}}}} DCO using 785.2 keV (E1) gate.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{TL24GAMMA3}{@}}}} DCO using 443.3 keV (E2) gate.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{TL24GAMMA4}{\&}}}} DCO using 803.1 keV (E1) gate.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{TL24GAMMA5}{a}}}} DCO using 652.2 keV (E2) gate.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{TL24GAMMA6}{b}}}} DCO using 319.0 keV (M1+E2) gate.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{TL24GAMMA7}{c}}}} DCO using 724.3 keV (E2) gate.}\\
\vspace{0.5cm}
\clearpage
\begin{figure}[h]
\begin{center}
\includegraphics{201TL24-0.ps}\\
\end{center}
\end{figure}
\clearpage
\subsection[\hspace{-0.2cm}\ensuremath{^{\textnormal{9}}}Be(\ensuremath{^{\textnormal{238}}}U,X\ensuremath{\gamma})]{ }
\vspace{-27pt}
\vspace{0.3cm}
\hypertarget{TL25}{{\bf \small \underline{\ensuremath{^{\textnormal{9}}}Be(\ensuremath{^{\textnormal{238}}}U,X\ensuremath{\gamma})\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2012BoZQ,B}{2012BoZQ},\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2013Bo18,B}{2013Bo18},\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2013BoZT,B}{2013BoZT}}}}\\
\vspace{4pt}
\vspace{8pt}
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2013Bo18,B}{2013Bo18},\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2012BoZQ,B}{2012BoZQ},\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2013BoZT,B}{2013BoZT}: E(\ensuremath{^{\textnormal{238}}}U)=1000 MeV/A from the SIS-18 synchrotron (GSI), pulsed beam 3-4 s beam-on with 2 s}\\
\parbox[b][0.3cm]{17.7cm}{beam-off periods; Target: \ensuremath{^{\textnormal{9}}}Be, 1.63 g/cm\ensuremath{^{\textnormal{2}}}-thick; Fragments were identified in flight by the Fragment Separator (FRS), based on}\\
\parbox[b][0.3cm]{17.7cm}{time of flight, B\ensuremath{\rho} and energy loss. Transmitted ions were slowed by a degrader and stopped in a passive ion 10-mm-thick Perspex}\\
\parbox[b][0.3cm]{17.7cm}{catcher, that was surrounded by the RISING \ensuremath{\gamma}-ray spectrometer. Measured E\ensuremath{\gamma}, I\ensuremath{\gamma}, delayed \ensuremath{\gamma} rays, isomer lifetime.}\\
\vspace{12pt}
\underline{$^{201}$Tl Levels}\\
\begin{longtable}{cccccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E(level)$^{{\hyperlink{TL25LEVEL0}{\dagger}}}$}&J$^{\pi}$$^{{\hyperlink{TL25LEVEL1}{\ddagger}}}$&\multicolumn{2}{c}{T$_{1/2}$$^{{\hyperlink{TL25LEVEL1}{\ddagger}}}$}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{l}{1/2\ensuremath{^{+}}}&&&&\\
\multicolumn{1}{r@{}}{331}&\multicolumn{1}{@{.}l}{15 {\it 6}}&\multicolumn{1}{l}{3/2\ensuremath{^{+}}}&&&&\\
\multicolumn{1}{r@{}}{919}&\multicolumn{1}{@{.}l}{45 {\it 12}}&\multicolumn{1}{l}{(9/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{11 ms {\it 11}}&\parbox[t][0.3cm]{12.07896cm}{\raggedright \%IT=100\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1238}&\multicolumn{1}{@{.}l}{6 {\it 3}}&\multicolumn{1}{l}{(11/2\ensuremath{^{-}})}&&&&\\
\multicolumn{1}{r@{}}{1572}&\multicolumn{1}{@{.}l}{0 {\it 3}}&\multicolumn{1}{l}{(13/2\ensuremath{^{-}})}&&&&\\
\multicolumn{1}{r@{}}{1962}&\multicolumn{1}{@{.}l}{9 {\it 3}}&\multicolumn{1}{l}{(15/2\ensuremath{^{-}})}&&&&\\
\multicolumn{1}{r@{}}{2016}&\multicolumn{1}{@{.}l}{1 {\it 7}}&\multicolumn{1}{l}{(17/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{9 ns {\it +19\textminus5}}&\parbox[t][0.3cm]{12.07896cm}{\raggedright J\ensuremath{^{\pi}}: (13/2,15/2) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2013BoZT,B}{2013BoZT}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2713}&\multicolumn{1}{@{.}l}{8 {\it 4}}&\multicolumn{1}{l}{(15/2\ensuremath{^{-}})}&&&&\\
\multicolumn{1}{r@{}}{2747}&\multicolumn{1}{@{.}l}{9 {\it 5}}&\multicolumn{1}{l}{(17/2\ensuremath{^{+}})}&&&&\\
\multicolumn{1}{r@{}}{2747}&\multicolumn{1}{@{.}l}{9+x {\it 5}}&&\multicolumn{1}{r@{}}{95}&\multicolumn{1}{@{ }l}{ns {\it +39\textminus21}}&\parbox[t][0.3cm]{12.07896cm}{\raggedright E(level): From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2013Bo18,B}{2013Bo18}. The observed 148, 158, 314 and 853 keV delayed gammas\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{12.07896cm}{\raggedright {\ }{\ }{\ }are expected above the 2747.9-keV level, but their ordering is unknown.\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{12.07896cm}{\raggedright J\ensuremath{^{\pi}}: Tentative J=(33/2) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2013Bo18,B}{2013Bo18}.\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{12.07896cm}{\raggedright T\ensuremath{_{1/2}}: From sum of (148,158,314,319,333,444,726,785,853)\ensuremath{\gamma}(t) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2012BoZQ,B}{2012BoZQ}, \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2013BoZT,B}{2013BoZT}.\vspace{0.1cm}}&\\
\end{longtable}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{TL25LEVEL0}{\dagger}}}} From least-squares fit to E\ensuremath{\gamma}, unless otherwise stated.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{TL25LEVEL1}{\ddagger}}}} From Adopted Levels, unless otherwise stated.}\\
\vspace{0.5cm}
\underline{$\gamma$($^{201}$Tl)}\\
\begin{longtable}{ccccccccc@{}c@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{TL25GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{I\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{TL25GAMMA2}{\#}}}}&\multicolumn{2}{c}{E\ensuremath{_{i}}(level)}&J\ensuremath{^{\pi}_{i}}&\multicolumn{2}{c}{E\ensuremath{_{f}}}&J\ensuremath{^{\pi}_{f}}&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{(34}&\multicolumn{1}{@{.}l}{3 {\it 7})}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{2747}&\multicolumn{1}{@{.}l}{9}&\multicolumn{1}{l}{(17/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{2713}&\multicolumn{1}{@{.}l}{8 }&\multicolumn{1}{@{}l}{(15/2\ensuremath{^{-}})}&\\
\multicolumn{1}{r@{}}{$^{x}$148}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{TL25GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&&&&&&\\
\multicolumn{1}{r@{}}{$^{x}$158}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{TL25GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&&&&&&\\
\multicolumn{1}{r@{}}{$^{x}$314}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{TL25GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&&&&&&\\
\multicolumn{1}{r@{}}{319}&\multicolumn{1}{@{.}l}{3\ensuremath{^{\hyperlink{TL25GAMMA1}{\ddagger}}} {\it 3}}&\multicolumn{1}{r@{}}{517}&\multicolumn{1}{@{ }l}{{\it 88}}&\multicolumn{1}{r@{}}{1238}&\multicolumn{1}{@{.}l}{6}&\multicolumn{1}{l}{(11/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{919}&\multicolumn{1}{@{.}l}{45 }&\multicolumn{1}{@{}l}{(9/2\ensuremath{^{-}})}&\\
\multicolumn{1}{r@{}}{331}&\multicolumn{1}{@{.}l}{15 {\it 6}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{331}&\multicolumn{1}{@{.}l}{15}&\multicolumn{1}{l}{3/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{@{}l}{1/2\ensuremath{^{+}}}&\\
\multicolumn{1}{r@{}}{333}&\multicolumn{1}{@{.}l}{4\ensuremath{^{\hyperlink{TL25GAMMA1}{\ddagger}}} {\it 3}}&\multicolumn{1}{r@{}}{411}&\multicolumn{1}{@{ }l}{{\it 76}}&\multicolumn{1}{r@{}}{1572}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{l}{(13/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{1238}&\multicolumn{1}{@{.}l}{6 }&\multicolumn{1}{@{}l}{(11/2\ensuremath{^{-}})}&\\
\multicolumn{1}{r@{}}{390}&\multicolumn{1}{@{.}l}{9 {\it 1}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{1962}&\multicolumn{1}{@{.}l}{9}&\multicolumn{1}{l}{(15/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{1572}&\multicolumn{1}{@{.}l}{0 }&\multicolumn{1}{@{}l}{(13/2\ensuremath{^{-}})}&\\
\multicolumn{1}{r@{}}{444}&\multicolumn{1}{@{.}l}{1\ensuremath{^{\hyperlink{TL25GAMMA1}{\ddagger}}} {\it 6}}&\multicolumn{1}{r@{}}{283}&\multicolumn{1}{@{ }l}{{\it 85}}&\multicolumn{1}{r@{}}{2016}&\multicolumn{1}{@{.}l}{1}&\multicolumn{1}{l}{(17/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{1572}&\multicolumn{1}{@{.}l}{0 }&\multicolumn{1}{@{}l}{(13/2\ensuremath{^{-}})}&\\
\multicolumn{1}{r@{}}{588}&\multicolumn{1}{@{.}l}{3 {\it 1}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{919}&\multicolumn{1}{@{.}l}{45}&\multicolumn{1}{l}{(9/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{331}&\multicolumn{1}{@{.}l}{15 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}}}&\\
\multicolumn{1}{r@{}}{652}&\multicolumn{1}{@{.}l}{2 {\it 5}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{1572}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{l}{(13/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{919}&\multicolumn{1}{@{.}l}{45 }&\multicolumn{1}{@{}l}{(9/2\ensuremath{^{-}})}&\\
\multicolumn{1}{r@{}}{724}&\multicolumn{1}{@{.}l}{3 {\it 1}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{1962}&\multicolumn{1}{@{.}l}{9}&\multicolumn{1}{l}{(15/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{1238}&\multicolumn{1}{@{.}l}{6 }&\multicolumn{1}{@{}l}{(11/2\ensuremath{^{-}})}&\\
\multicolumn{1}{r@{}}{784}&\multicolumn{1}{@{.}l}{9\ensuremath{^{\hyperlink{TL25GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{167}&\multicolumn{1}{@{ }l}{{\it 61}}&\multicolumn{1}{r@{}}{2747}&\multicolumn{1}{@{.}l}{9}&\multicolumn{1}{l}{(17/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{1962}&\multicolumn{1}{@{.}l}{9 }&\multicolumn{1}{@{}l}{(15/2\ensuremath{^{-}})}&\\
\multicolumn{1}{r@{}}{$^{x}$853}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{TL25GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&&&&&&\\
\multicolumn{1}{r@{}}{1141}&\multicolumn{1}{@{.}l}{8 {\it 1}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{2713}&\multicolumn{1}{@{.}l}{8}&\multicolumn{1}{l}{(15/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{1572}&\multicolumn{1}{@{.}l}{0 }&\multicolumn{1}{@{}l}{(13/2\ensuremath{^{-}})}&\\
\end{longtable}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{TL25GAMMA0}{\dagger}}}} From adopted gammas, unless otherwise stated.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{TL25GAMMA1}{\ddagger}}}} Delayed \ensuremath{\gamma} rays observed in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2012BoZQ,B}{2012BoZQ}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{TL25GAMMA2}{\#}}}} Delayed intensities from \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2013BoZT,B}{2013BoZT}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{TL25GAMMA3}{x}}}} \ensuremath{\gamma} ray not placed in level scheme.}\\
\vspace{0.5cm}
\clearpage
\begin{figure}[h]
\begin{center}
\includegraphics{201TL25-0.ps}\\
\end{center}
\end{figure}
\clearpage
\subsection[\hspace{-0.2cm}\ensuremath{^{\textnormal{203}}}Tl(p,t)]{ }
\vspace{-27pt}
\vspace{0.3cm}
\hypertarget{TL26}{{\bf \small \underline{\ensuremath{^{\textnormal{203}}}Tl(p,t)\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1977Se15,B}{1977Se15}}}}\\
\vspace{4pt}
\vspace{8pt}
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1977Se15,B}{1977Se15}: E(p)=19 MeV; magnetic spectrograph; \ensuremath{\sigma}(\ensuremath{\theta}); \ensuremath{\Delta}E=5-7 keV; \ensuremath{J^{\pi}}(\ensuremath{^{\textnormal{203}}}Tl)=1/2\ensuremath{^{+}}.}\\
\vspace{12pt}
\underline{$^{201}$Tl Levels}\\
\begin{longtable}{cccc|cccc|cc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E(level)$^{{\hyperlink{TL26LEVEL0}{\dagger}}}$}&J$^{\pi}$$^{{\hyperlink{TL26LEVEL1}{\ddagger}}}$&L$^{{\hyperlink{TL26LEVEL3}{@}}}$&\multicolumn{2}{c}{E(level)$^{{\hyperlink{TL26LEVEL0}{\dagger}}}$}&J$^{\pi}$$^{{\hyperlink{TL26LEVEL1}{\ddagger}}}$&L$^{{\hyperlink{TL26LEVEL3}{@}}}$&\multicolumn{2}{c}{E(level)$^{{\hyperlink{TL26LEVEL0}{\dagger}}}$}&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&\multicolumn{2}{c}{\hrulefill}&
\endfirsthead
\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{l}{1/2\ensuremath{^{+}}}&\multicolumn{1}{l|}{0}&\multicolumn{1}{r@{}}{1423}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{l}{7/2\ensuremath{^{+}},9/2\ensuremath{^{+}}}&\multicolumn{1}{l|}{4}&\multicolumn{1}{r@{}}{1913}&\multicolumn{1}{@{}l}{}&\\
\multicolumn{1}{r@{}}{330}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{l}{3/2\ensuremath{^{+}}\ensuremath{^{{\hyperlink{TL26LEVEL2}{\#}}}}}&\multicolumn{1}{l|}{2}&\multicolumn{1}{r@{}}{1440}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{l}{3/2\ensuremath{^{+}},5/2\ensuremath{^{+}}}&\multicolumn{1}{l|}{2}&\multicolumn{1}{r@{}}{2098}&\multicolumn{1}{@{}l}{}&\\
\multicolumn{1}{r@{}}{695}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{l}{5/2\ensuremath{^{+}}\ensuremath{^{{\hyperlink{TL26LEVEL2}{\#}}}}}&\multicolumn{1}{l|}{2}&\multicolumn{1}{r@{}}{1472}&\multicolumn{1}{@{}l}{}&&&\multicolumn{1}{r@{}}{2147}&\multicolumn{1}{@{}l}{}&\\
\multicolumn{1}{r@{}}{1106}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{l}{3/2\ensuremath{^{+}},5/2\ensuremath{^{+}}}&\multicolumn{1}{l|}{2}&\multicolumn{1}{r@{}}{1572}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{l}{7/2\ensuremath{^{+}},9/2\ensuremath{^{+}}}&\multicolumn{1}{l|}{4}&\multicolumn{1}{r@{}}{2183}&\multicolumn{1}{@{}l}{}&\\
\multicolumn{1}{r@{}}{1159}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{l}{(1/2\ensuremath{^{+}})}&\multicolumn{1}{l|}{(0)}&\multicolumn{1}{r@{}}{1636}&\multicolumn{1}{@{}l}{}&&&\multicolumn{1}{r@{}}{2271}&\multicolumn{1}{@{}l}{}&\\
\multicolumn{1}{r@{}}{1238}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{l}{(3/2\ensuremath{^{+}},5/2\ensuremath{^{+}})}&\multicolumn{1}{l|}{(2)}&\multicolumn{1}{r@{}}{1699}&\multicolumn{1}{@{}l}{}&&&\multicolumn{1}{r@{}}{2343}&\multicolumn{1}{@{}l}{}&\\
\multicolumn{1}{r@{}}{1294}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{l}{7/2\ensuremath{^{+}},9/2\ensuremath{^{+}}}&\multicolumn{1}{l|}{4}&\multicolumn{1}{r@{}}{1729}&\multicolumn{1}{@{}l}{}&&&&&\\
\multicolumn{1}{r@{}}{1335}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{l}{3/2\ensuremath{^{+}},5/2\ensuremath{^{+}}}&\multicolumn{1}{l|}{2}&\multicolumn{1}{r@{}}{1829}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{l}{3/2\ensuremath{^{+}},5/2\ensuremath{^{+}}}&\multicolumn{1}{l|}{2}&&&\\
\end{longtable}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{TL26LEVEL0}{\dagger}}}} From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1977Se15,B}{1977Se15}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{TL26LEVEL1}{\ddagger}}}} From L transfer value in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1977Se15,B}{1977Se15}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{TL26LEVEL2}{\#}}}} From Adopted Levels.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{TL26LEVEL3}{@}}}} From \ensuremath{\sigma}(\ensuremath{\theta}) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1977Se15,B}{1977Se15}.}\\
\vspace{0.5cm}
\clearpage
\subsection[\hspace{-0.2cm}\ensuremath{^{\textnormal{204}}}Pb(p,\ensuremath{\alpha})]{ }
\vspace{-27pt}
\vspace{0.3cm}
\hypertarget{TL27}{{\bf \small \underline{\ensuremath{^{\textnormal{204}}}Pb(p,\ensuremath{\alpha})\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1985Fi05,B}{1985Fi05}}}}\\
\vspace{4pt}
\vspace{8pt}
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}E(p)=35 MeV; Target: enriched, 0.3 mg/cm\ensuremath{^{\textnormal{2}}} thick; Measured: magnetic spectrograph, \ensuremath{\sigma}(E(\ensuremath{\alpha}),\ensuremath{\theta}), FWHM(\ensuremath{\alpha})\ensuremath{\approx}35 keV; Deduced: E,}\\
\parbox[b][0.3cm]{17.7cm}{\ensuremath{J^{\pi}}. DWBA analysis.}\\
\vspace{12pt}
\underline{$^{201}$Tl Levels}\\
\begin{longtable}{ccc|ccc|ccc|ccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E(level)$^{{\hyperlink{TL27LEVEL0}{\dagger}}}$}&J$^{\pi}$$^{{\hyperlink{TL27LEVEL1}{\ddagger}}}$&\multicolumn{2}{c}{E(level)$^{{\hyperlink{TL27LEVEL0}{\dagger}}}$}&J$^{\pi}$$^{{\hyperlink{TL27LEVEL1}{\ddagger}}}$&\multicolumn{2}{c}{E(level)$^{{\hyperlink{TL27LEVEL0}{\dagger}}}$}&J$^{\pi}$$^{{\hyperlink{TL27LEVEL1}{\ddagger}}}$&\multicolumn{2}{c}{E(level)$^{{\hyperlink{TL27LEVEL0}{\dagger}}}$}&J$^{\pi}$$^{{\hyperlink{TL27LEVEL1}{\ddagger}}}$&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{l|}{1/2\ensuremath{^{+}}\ensuremath{^{{\hyperlink{TL27LEVEL2}{\#}}}}}&\multicolumn{1}{r@{}}{1834}&\multicolumn{1}{@{ }l}{{\it 15}}&\multicolumn{1}{l|}{(5/2)}&\multicolumn{1}{r@{}}{2534}&\multicolumn{1}{@{ }l}{{\it 20}}&\multicolumn{1}{l|}{(9/2)}&\multicolumn{1}{r@{}}{3201}&\multicolumn{1}{@{ }l}{{\it 20}}&&\\
\multicolumn{1}{r@{}}{334}&\multicolumn{1}{@{ }l}{{\it 8}}&\multicolumn{1}{l|}{3/2\ensuremath{^{+}}\ensuremath{^{{\hyperlink{TL27LEVEL2}{\#}}}}}&\multicolumn{1}{r@{}}{1908}&\multicolumn{1}{@{ }l}{{\it 15}}&\multicolumn{1}{l|}{(13/2,15/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{2622}&\multicolumn{1}{@{ }l}{{\it 20}}&\multicolumn{1}{l|}{(15/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{3303}&\multicolumn{1}{@{ }l}{{\it 20}}&\multicolumn{1}{l}{(9/2,11/2)}&\\
\multicolumn{1}{r@{}}{699}&\multicolumn{1}{@{ }l}{{\it 8}}&\multicolumn{1}{l|}{5/2\ensuremath{^{+}}\ensuremath{^{{\hyperlink{TL27LEVEL2}{\#}}}}}&\multicolumn{1}{r@{}}{1940}&\multicolumn{1}{@{ }l}{{\it 15}}&\multicolumn{1}{l|}{(7/2,9/2)}&\multicolumn{1}{r@{}}{2669}&\multicolumn{1}{@{ }l}{{\it 20}}&\multicolumn{1}{l|}{(5/2)}&\multicolumn{1}{r@{}}{3362}&\multicolumn{1}{@{ }l}{{\it 20}}&\multicolumn{1}{l}{(15/2\ensuremath{^{+}})}&\\
\multicolumn{1}{r@{}}{1131}&\multicolumn{1}{@{ }l}{{\it 15}}&\multicolumn{1}{l|}{7/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1996}&\multicolumn{1}{@{ }l}{{\it 15}}&\multicolumn{1}{l|}{(15/2)}&\multicolumn{1}{r@{}}{2714}&\multicolumn{1}{@{ }l}{{\it 20}}&&\multicolumn{1}{r@{}}{3401}&\multicolumn{1}{@{ }l}{{\it 20}}&\multicolumn{1}{l}{(17/2\ensuremath{^{+}})}&\\
\multicolumn{1}{r@{}}{1286}&\multicolumn{1}{@{ }l}{{\it 15}}&\multicolumn{1}{l|}{(9/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{2045}&\multicolumn{1}{@{ }l}{{\it 20}}&\multicolumn{1}{l|}{(11/2\ensuremath{^{-}},13/2)}&\multicolumn{1}{r@{}}{2762}&\multicolumn{1}{@{ }l}{{\it 20}}&\multicolumn{1}{l|}{(7/2,9/2)}&\multicolumn{1}{r@{}}{3441}&\multicolumn{1}{@{ }l}{{\it 20}}&\multicolumn{1}{l}{(9/2)}&\\
\multicolumn{1}{r@{}}{1331}&\multicolumn{1}{@{ }l}{{\it 15}}&\multicolumn{1}{l|}{(5/2,7/2,9/2)}&\multicolumn{1}{r@{}}{2103}&\multicolumn{1}{@{ }l}{{\it 20}}&\multicolumn{1}{l|}{(9/2)}&\multicolumn{1}{r@{}}{2853}&\multicolumn{1}{@{ }l}{{\it 20}}&\multicolumn{1}{l|}{(11/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{3499}&\multicolumn{1}{@{ }l}{{\it 20}}&\multicolumn{1}{l}{(15/2)}&\\
\multicolumn{1}{r@{}}{1419}&\multicolumn{1}{@{ }l}{{\it 15}}&\multicolumn{1}{l|}{(11/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{2145}&\multicolumn{1}{@{ }l}{{\it 20}}&\multicolumn{1}{l|}{(9/2)}&\multicolumn{1}{r@{}}{2899}&\multicolumn{1}{@{ }l}{{\it 20}}&\multicolumn{1}{l|}{(15/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{3552}&\multicolumn{1}{@{ }l}{{\it 20}}&\multicolumn{1}{l}{(15/2)}&\\
\multicolumn{1}{r@{}}{1579}&\multicolumn{1}{@{ }l}{{\it 15}}&\multicolumn{1}{l|}{(9/2)}&\multicolumn{1}{r@{}}{2196}&\multicolumn{1}{@{ }l}{{\it 20}}&\multicolumn{1}{l|}{(15/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{2976}&\multicolumn{1}{@{ }l}{{\it 20}}&\multicolumn{1}{l|}{(15/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{3615}&\multicolumn{1}{@{ }l}{{\it 20}}&\multicolumn{1}{l}{(17/2\ensuremath{^{-}},19/2\ensuremath{^{-}})}&\\
\multicolumn{1}{r@{}}{1655}&\multicolumn{1}{@{ }l}{{\it 15}}&\multicolumn{1}{l|}{(5/2,7/2)}&\multicolumn{1}{r@{}}{2254}&\multicolumn{1}{@{ }l}{{\it 20}}&\multicolumn{1}{l|}{(15/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{3030}&\multicolumn{1}{@{ }l}{{\it 20}}&\multicolumn{1}{l|}{(15/2\ensuremath{^{+}},17/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{3648}&\multicolumn{1}{@{ }l}{{\it 20}}&\multicolumn{1}{l}{(15/2\ensuremath{^{+}})}&\\
\multicolumn{1}{r@{}}{1725}&\multicolumn{1}{@{ }l}{{\it 15}}&\multicolumn{1}{l|}{(7/2,9/2)}&\multicolumn{1}{r@{}}{2379}&\multicolumn{1}{@{ }l}{{\it 20}}&\multicolumn{1}{l|}{(15/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{3083}&\multicolumn{1}{@{ }l}{{\it 20}}&\multicolumn{1}{l|}{(9/2,11/2)}&\multicolumn{1}{r@{}}{3674}&\multicolumn{1}{@{ }l}{{\it 20}}&\multicolumn{1}{l}{(17/2,19/2\ensuremath{^{-}})}&\\
\multicolumn{1}{r@{}}{1763}&\multicolumn{1}{@{ }l}{{\it 15}}&\multicolumn{1}{l|}{(7/2,9/2)}&\multicolumn{1}{r@{}}{2449}&\multicolumn{1}{@{ }l}{{\it 20}}&&\multicolumn{1}{r@{}}{3133}&\multicolumn{1}{@{ }l}{{\it 20}}&&\multicolumn{1}{r@{}}{3727}&\multicolumn{1}{@{ }l}{{\it 20}}&&\\
\end{longtable}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{TL27LEVEL0}{\dagger}}}} From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1985Fi05,B}{1985Fi05}, but values were lowered by 15 keV for levels above 700 keV, since from comparison of the excitation energies}\\
\parbox[b][0.3cm]{17.7cm}{{\ }{\ }in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1985Fi05,B}{1985Fi05} (up to 1600 keV) and these from the Adopted Levels, the former values appear to be {}~ 15 keV higher.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{TL27LEVEL1}{\ddagger}}}} From comparison of measured \ensuremath{\sigma}(\ensuremath{\theta}) with cluster model DWBA calculations (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1985Fi05,B}{1985Fi05}).}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{TL27LEVEL2}{\#}}}} From Adopted Levels.}\\
\vspace{0.5cm}
\clearpage
\subsection[\hspace{-0.2cm}\ensuremath{^{\textnormal{207}}}Pb(\ensuremath{\mu},X\ensuremath{\gamma})]{ }
\vspace{-27pt}
\vspace{0.3cm}
\hypertarget{TL28}{{\bf \small \underline{\ensuremath{^{\textnormal{207}}}Pb(\ensuremath{\mu},X\ensuremath{\gamma})\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1982Bu04,B}{1982Bu04},\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1983Bu02,B}{1983Bu02}}}}\\
\vspace{4pt}
\vspace{8pt}
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}Target: \ensuremath{^{\textnormal{207}}}Pb, enriched to 92.77\%; Detectors: Ge(Li); Measured: E\ensuremath{\gamma},I\ensuremath{\gamma}.}\\
\vspace{12pt}
\underline{$^{201}$Tl Levels}\\
\begin{longtable}{ccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E(level)$^{{\hyperlink{TL28LEVEL0}{\dagger}}}$}&J$^{\pi}$$^{{\hyperlink{TL28LEVEL1}{\ddagger}}}$&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{l}{1/2\ensuremath{^{+}}}&\\
\multicolumn{1}{r@{}}{331}&\multicolumn{1}{@{.}l}{13 {\it 9}}&\multicolumn{1}{l}{3/2\ensuremath{^{+}}}&\\
\multicolumn{1}{r@{}}{1098}&\multicolumn{1}{@{.}l}{2 {\it 4}}&\multicolumn{1}{l}{5/2\ensuremath{^{+}}}&\\
\end{longtable}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{TL28LEVEL0}{\dagger}}}} From a least-squares fit to E\ensuremath{\gamma}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{TL28LEVEL1}{\ddagger}}}} From Adopted Levels.}\\
\vspace{0.5cm}
\underline{$\gamma$($^{201}$Tl)}\\
\begin{longtable}{ccccccccc@{}c@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{TL28GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{I\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{TL28GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{E\ensuremath{_{i}}(level)}&J\ensuremath{^{\pi}_{i}}&\multicolumn{2}{c}{E\ensuremath{_{f}}}&J\ensuremath{^{\pi}_{f}}&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{331}&\multicolumn{1}{@{.}l}{13 {\it 9}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{8 {\it 2}}&\multicolumn{1}{r@{}}{331}&\multicolumn{1}{@{.}l}{13}&\multicolumn{1}{l}{3/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{1/2\ensuremath{^{+}}}&\\
\multicolumn{1}{r@{}}{767}&\multicolumn{1}{@{.}l}{1 {\it 4}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{2 {\it 1}}&\multicolumn{1}{r@{}}{1098}&\multicolumn{1}{@{.}l}{2}&\multicolumn{1}{l}{5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{331}&\multicolumn{1}{@{.}l}{13 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}}}&\\
\end{longtable}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{TL28GAMMA0}{\dagger}}}} From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1983Bu02,B}{1983Bu02}.}\\
\vspace{0.5cm}
\begin{figure}[h]
\begin{center}
\includegraphics{201TL28-0.ps}\\
\end{center}
\end{figure}
\clearpage
\section[\ensuremath{^{201}_{\ 82}}Pb\ensuremath{_{119}^{~}}]{ }
\vspace{-30pt}
\setcounter{chappage}{1}
\subsection[\hspace{-0.2cm}Adopted Levels, Gammas]{ }
\vspace{-20pt}
\vspace{0.3cm}
\hypertarget{PB29}{{\bf \small \underline{Adopted \hyperlink{201PB_LEVEL}{Levels}, \hyperlink{201PB_GAMMA}{Gammas}}}}\\
\vspace{4pt}
\vspace{8pt}
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}Q(\ensuremath{\beta^-})=$-$3842 {\it 18}; S(n)=7091 {\it 17}; S(p)=5513 {\it 15}; Q(\ensuremath{\alpha})=2844 {\it 14}\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2021Wa16,B}{2021Wa16}}\\
\vspace{12pt}
\hypertarget{201PB_LEVEL}{\underline{$^{201}$Pb Levels}}\\
\begin{longtable}[c]{llll}
\multicolumn{4}{c}{\underline{Cross Reference (XREF) Flags}}\\
\\
\hyperlink{BI30}{\texttt{A }}& \ensuremath{^{\textnormal{201}}}Bi \ensuremath{\varepsilon} decay & \hyperlink{PB33}{\texttt{D }}& \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma})\\
\hyperlink{PO31}{\texttt{B }}& \ensuremath{^{\textnormal{205}}}Po \ensuremath{\alpha} decay & \hyperlink{PB34}{\texttt{E }}& \ensuremath{^{\textnormal{197}}}Au(\ensuremath{^{\textnormal{207}}}Pb,X\ensuremath{\gamma})\\
\hyperlink{PB32}{\texttt{C }}& \ensuremath{^{\textnormal{200}}}Hg(\ensuremath{\alpha},3n\ensuremath{\gamma}) & \\
\end{longtable}
\vspace{-0.5cm}
\begin{longtable}{ccccccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E(level)$^{{\hyperlink{PB29LEVEL0}{\dagger}}}$}&J$^{\pi}$$^{}$&\multicolumn{2}{c}{T$_{1/2}$$^{}$}&XREF&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{PB29LEVEL1}{\ddagger}}}}}&\multicolumn{1}{l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{9}&\multicolumn{1}{@{.}l}{33 h {\it 5}}&\multicolumn{1}{l}{\texttt{\hyperlink{BI30}{A}\hyperlink{PO31}{B}\hyperlink{PB32}{C}\hyperlink{PB33}{D}\ } }&\parbox[t][0.3cm]{11.306601cm}{\raggedright \%\ensuremath{\varepsilon}+\%\ensuremath{\beta}\ensuremath{^{\textnormal{+}}}=100\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.306601cm}{\raggedright \ensuremath{\mu}=+0.6731 \textit{6} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2019StZV,B}{2019StZV})\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.306601cm}{\raggedright Q={\textminus}0.009 \textit{43} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986An06,B}{1986An06},\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2016St14,B}{2016St14})\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.306601cm}{\raggedright J\ensuremath{^{\pi}}: Atomic beam magnetic resonance (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1977Gu18,B}{1977Gu18}); Favored \ensuremath{\alpha} decay from \ensuremath{^{\textnormal{205}}}Po\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.306601cm}{\raggedright {\ }{\ }{\ }g.s. (\ensuremath{J^{\pi}}=5/2\ensuremath{^{-}}); \ensuremath{\mu}.\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.306601cm}{\raggedright T\ensuremath{_{1/2}}: Weighted average of 9.300 \textit{75} (330.7\ensuremath{\gamma}(t)), 9.40 \textit{15} (691.9\ensuremath{\gamma}(t)) and 9.350 \textit{78}\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.306601cm}{\raggedright {\ }{\ }{\ }(945.4\ensuremath{\gamma}(t)) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1981An11,B}{1981An11}. Others: 8 h \textit{2} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1950Ne77,B}{1950Ne77}), 8.4 h \textit{10} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1954Wa12,B}{1954Wa12}), 9.4\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.306601cm}{\raggedright {\ }{\ }{\ }h 2 (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1955Be12,B}{1955Be12}) and 10.0 h \textit{8} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1960Li08,B}{1960Li08}).\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.306601cm}{\raggedright \ensuremath{\mu},Q: From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986An06,B}{1986An06}, using the atomic beam with laser fluorescence spectroscopy\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.306601cm}{\raggedright {\ }{\ }{\ }technique. \ensuremath{\mu}=+0.6753 \textit{5} in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986An06,B}{1986An06}, but diamagnetic correction applied in\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.306601cm}{\raggedright {\ }{\ }{\ }\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2019StZV,B}{2019StZV}.\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.306601cm}{\raggedright \ensuremath{\Delta}\ensuremath{<}r\ensuremath{^{\textnormal{2}}}\ensuremath{>}={\textminus}0.4093 \textit{34} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986An06,B}{1986An06}). Other: {\textminus}0.4225 \textit{5} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1982Th05,B}{1982Th05}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{88}&\multicolumn{1}{@{.}l}{5\ensuremath{^{{\hyperlink{PB29LEVEL2}{\#}}}} {\it 5}}&\multicolumn{1}{l}{3/2\ensuremath{^{-}}}&&&\multicolumn{1}{l}{\texttt{\hyperlink{BI30}{A}\ \hyperlink{PB32}{C}\ \ } }&\parbox[t][0.3cm]{11.306601cm}{\raggedright J\ensuremath{^{\pi}}: 88.6\ensuremath{\gamma} M1+E2 to 5/2\ensuremath{^{-}}; systematics in neighboring nuclei.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{169}&\multicolumn{1}{@{.}l}{9\ensuremath{^{{\hyperlink{PB29LEVEL3}{@}}}} {\it 5}}&\multicolumn{1}{l}{(1/2\ensuremath{^{-}})}&&&\multicolumn{1}{l}{\texttt{\hyperlink{BI30}{A}\ \hyperlink{PB32}{C}\ \ } }&\parbox[t][0.3cm]{11.306601cm}{\raggedright J\ensuremath{^{\pi}}: 81.4\ensuremath{\gamma} to 3/2\ensuremath{^{-}}; systematics in neighboring nuclei.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{538}&\multicolumn{1}{@{.}l}{7 {\it 7}}&\multicolumn{1}{l}{(3/2\ensuremath{^{-}})}&&&\multicolumn{1}{l}{\texttt{\hyperlink{BI30}{A}\ \hyperlink{PB32}{C}\ \ } }&\parbox[t][0.3cm]{11.306601cm}{\raggedright J\ensuremath{^{\pi}}: 368.8\ensuremath{\gamma} to (1/2\ensuremath{^{-}}), 538.7\ensuremath{\gamma} to 5/2\ensuremath{^{-}}; systematics in neighboring nuclei.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{629}&\multicolumn{1}{@{.}l}{1\ensuremath{^{{\hyperlink{PB29LEVEL4}{\&}}}} {\it 3}}&\multicolumn{1}{l}{13/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{60}&\multicolumn{1}{@{.}l}{8 s {\it 18}}&\multicolumn{1}{l}{\texttt{\hyperlink{BI30}{A}\ \hyperlink{PB32}{C}\hyperlink{PB33}{D}\hyperlink{PB34}{E}} }&\parbox[t][0.3cm]{11.306601cm}{\raggedright \%IT=100\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.306601cm}{\raggedright J\ensuremath{^{\pi}}: 629.1\ensuremath{\gamma} M4 to 5/2\ensuremath{^{-}}.\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.306601cm}{\raggedright T\ensuremath{_{1/2}}: Weighted average of 60.1 s \textit{44} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1955Fi30,B}{1955Fi30}) and 61 s \textit{2} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1956St05,B}{1956St05}). Other: 50\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.306601cm}{\raggedright {\ }{\ }{\ }s (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1952Ho41,B}{1952Ho41}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{879}&\multicolumn{1}{@{.}l}{6 {\it 10}}&\multicolumn{1}{l}{(5/2)\ensuremath{^{-}}}&&&\multicolumn{1}{l}{\texttt{\hyperlink{BI30}{A}\ \hyperlink{PB32}{C}\ \ } }&\parbox[t][0.3cm]{11.306601cm}{\raggedright J\ensuremath{^{\pi}}: 791.0\ensuremath{\gamma} M1 to 3/2\ensuremath{^{-}}, 2170\ensuremath{\gamma} from 7/2\ensuremath{^{+}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{910}&\multicolumn{1}{@{.}l}{5? {\it 12}}&\multicolumn{1}{l}{5/2\ensuremath{^{-}}}&&&\multicolumn{1}{l}{\texttt{\hyperlink{BI30}{A}\ \ \ \ } }&\parbox[t][0.3cm]{11.306601cm}{\raggedright J\ensuremath{^{\pi}}: 822.6\ensuremath{\gamma} M1(+E2) to 3/2\ensuremath{^{-}}, 740.7\ensuremath{\gamma} from 7/2\ensuremath{^{+}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{936}&\multicolumn{1}{@{.}l}{1\ensuremath{^{{\hyperlink{PB29LEVEL6}{b}}}} {\it 3}}&\multicolumn{1}{l}{7/2\ensuremath{^{-}}}&&&\multicolumn{1}{l}{\texttt{\hyperlink{BI30}{A}\ \hyperlink{PB32}{C}\ \ } }&\parbox[t][0.3cm]{11.306601cm}{\raggedright J\ensuremath{^{\pi}}: 847.7\ensuremath{\gamma} E2 to 3/2\ensuremath{^{-}}; direct population in \ensuremath{^{\textnormal{201}}}Bi \ensuremath{\varepsilon} decay (\ensuremath{J^{\pi}}=9/2\ensuremath{^{-}}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{990}&\multicolumn{1}{@{.}l}{5\ensuremath{^{{\hyperlink{PB29LEVEL7}{c}}}} {\it 3}}&\multicolumn{1}{l}{7/2\ensuremath{^{-}}}&&&\multicolumn{1}{l}{\texttt{\hyperlink{BI30}{A}\ \ \ \ } }&\parbox[t][0.3cm]{11.306601cm}{\raggedright J\ensuremath{^{\pi}}: 902.0\ensuremath{\gamma} E2 to 3/2\ensuremath{^{-}}; direct population in \ensuremath{^{\textnormal{201}}}Bi \ensuremath{\varepsilon} decay (\ensuremath{J^{\pi}}=9/2\ensuremath{^{-}}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1014}&\multicolumn{1}{@{.}l}{2\ensuremath{^{{\hyperlink{PB29LEVEL6}{b}}}} {\it 4}}&\multicolumn{1}{l}{9/2\ensuremath{^{-}}}&&&\multicolumn{1}{l}{\texttt{\hyperlink{BI30}{A}\ \hyperlink{PB32}{C}\ \ } }&\parbox[t][0.3cm]{11.306601cm}{\raggedright J\ensuremath{^{\pi}}: 1014.1\ensuremath{\gamma} E2 to 5/2\ensuremath{^{-}}; direct population in \ensuremath{^{\textnormal{201}}}Bi \ensuremath{\varepsilon} decay (\ensuremath{J^{\pi}}=9/2\ensuremath{^{-}}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1185}&\multicolumn{1}{@{.}l}{8 {\it 4}}&\multicolumn{1}{l}{(7/2)\ensuremath{^{-}}}&&&\multicolumn{1}{l}{\texttt{\hyperlink{BI30}{A}\ \hyperlink{PB32}{C}\ \ } }&\parbox[t][0.3cm]{11.306601cm}{\raggedright J\ensuremath{^{\pi}}: 171.7\ensuremath{\gamma} M1(+E2) to 9/2\ensuremath{^{-}}, 1186.5\ensuremath{\gamma} to 5/2\ensuremath{^{-}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1325}&\multicolumn{1}{@{.}l}{4\ensuremath{^{{\hyperlink{PB29LEVEL5}{a}}}} {\it 5}}&\multicolumn{1}{l}{7/2\ensuremath{^{-}}}&&&\multicolumn{1}{l}{\texttt{\hyperlink{BI30}{A}\ \ \ \ } }&\parbox[t][0.3cm]{11.306601cm}{\raggedright J\ensuremath{^{\pi}}: 1325.2\ensuremath{\gamma} M1+E2 to 5/2\ensuremath{^{-}}; direct population in \ensuremath{^{\textnormal{201}}}Bi \ensuremath{\varepsilon} decay (\ensuremath{J^{\pi}}=9/2\ensuremath{^{-}}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1415}&\multicolumn{1}{@{.}l}{4\ensuremath{^{{\hyperlink{PB29LEVEL8}{d}}}} {\it 4}}&\multicolumn{1}{l}{9/2\ensuremath{^{+}}}&&&\multicolumn{1}{l}{\texttt{\hyperlink{BI30}{A}\ \ \ \ } }&\parbox[t][0.3cm]{11.306601cm}{\raggedright J\ensuremath{^{\pi}}: 786.4\ensuremath{\gamma} E2 to 13/2\ensuremath{^{+}}, 424.5\ensuremath{\gamma} (E1) to 7/2\ensuremath{^{-}}; direct population in \ensuremath{^{\textnormal{201}}}Bi \ensuremath{\varepsilon} decay\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.306601cm}{\raggedright {\ }{\ }{\ }(\ensuremath{J^{\pi}}=9/2\ensuremath{^{-}}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1447}&\multicolumn{1}{@{.}l}{9 {\it 5}}&\multicolumn{1}{l}{(11/2)\ensuremath{^{+}}}&&&\multicolumn{1}{l}{\texttt{\hyperlink{BI30}{A}\ \hyperlink{PB32}{C}\ \ } }&\parbox[t][0.3cm]{11.306601cm}{\raggedright J\ensuremath{^{\pi}}: 818.9\ensuremath{\gamma} E2+M1 to 13/2\ensuremath{^{+}}; direct population in \ensuremath{^{\textnormal{201}}}Bi \ensuremath{\varepsilon} decay (\ensuremath{J^{\pi}}=9/2\ensuremath{^{-}}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1490}&\multicolumn{1}{@{.}l}{3 {\it 6}}&\multicolumn{1}{l}{7/2\ensuremath{^{-}},9/2\ensuremath{^{-}}}&&&\multicolumn{1}{l}{\texttt{\hyperlink{BI30}{A}\ \ \ \ } }&\parbox[t][0.3cm]{11.306601cm}{\raggedright J\ensuremath{^{\pi}}: 499.9\ensuremath{\gamma} M1(+E2) to 7/2\ensuremath{^{-}}; direct population in \ensuremath{^{\textnormal{201}}}Bi \ensuremath{\varepsilon} decay (\ensuremath{J^{\pi}}=9/2\ensuremath{^{-}}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1541}&\multicolumn{1}{@{.}l}{8\ensuremath{^{{\hyperlink{PB29LEVEL9}{e}}}} {\it 4}}&\multicolumn{1}{l}{17/2\ensuremath{^{+}}}&&&\multicolumn{1}{l}{\texttt{\ \ \hyperlink{PB32}{C}\hyperlink{PB33}{D}\hyperlink{PB34}{E}} }&\parbox[t][0.3cm]{11.306601cm}{\raggedright J\ensuremath{^{\pi}}: 912.7\ensuremath{\gamma} E2 to 13/2\ensuremath{^{+}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1545}&\multicolumn{1}{@{.}l}{8 {\it 4}}&\multicolumn{1}{l}{15/2\ensuremath{^{+}}}&&&\multicolumn{1}{l}{\texttt{\ \ \hyperlink{PB32}{C}\hyperlink{PB33}{D}\hyperlink{PB34}{E}} }&\parbox[t][0.3cm]{11.306601cm}{\raggedright J\ensuremath{^{\pi}}: 916.7\ensuremath{\gamma} M1+E2 to 13/2\ensuremath{^{+}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1651}&\multicolumn{1}{@{.}l}{0 {\it 4}}&\multicolumn{1}{l}{7/2\ensuremath{^{+}}}&&&\multicolumn{1}{l}{\texttt{\hyperlink{BI30}{A}\ \ \ \ } }&\parbox[t][0.3cm]{11.306601cm}{\raggedright J\ensuremath{^{\pi}}: 1650.9\ensuremath{\gamma} E1 to 5/2\ensuremath{^{-}}; direct population in \ensuremath{^{\textnormal{201}}}Bi \ensuremath{\varepsilon} decay (\ensuremath{J^{\pi}}=9/2\ensuremath{^{-}}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1737}&\multicolumn{1}{@{.}l}{3 {\it 4}}&\multicolumn{1}{l}{9/2\ensuremath{^{+}}}&&&\multicolumn{1}{l}{\texttt{\hyperlink{BI30}{A}\ \ \ \ } }&\parbox[t][0.3cm]{11.306601cm}{\raggedright J\ensuremath{^{\pi}}: 746.8\ensuremath{\gamma} E1 to 7/2\ensuremath{^{-}}, 1108.1\ensuremath{\gamma} E2 to 13/2\ensuremath{^{+}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1843}&\multicolumn{1}{@{.}l}{8 {\it 5}}&\multicolumn{1}{l}{11/2\ensuremath{^{+}}}&&&\multicolumn{1}{l}{\texttt{\hyperlink{BI30}{A}\ \ \ \ } }&\parbox[t][0.3cm]{11.306601cm}{\raggedright J\ensuremath{^{\pi}}: 1214.5\ensuremath{\gamma} M1(+E2) to 13/2\ensuremath{^{+}}; direct population in \ensuremath{^{\textnormal{201}}}Bi \ensuremath{\varepsilon} decay (\ensuremath{J^{\pi}}=9/2\ensuremath{^{-}}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1875}&\multicolumn{1}{@{.}l}{6 {\it 5}}&\multicolumn{1}{l}{9/2\ensuremath{^{+}}}&&&\multicolumn{1}{l}{\texttt{\hyperlink{BI30}{A}\ \ \ \ } }&\parbox[t][0.3cm]{11.306601cm}{\raggedright J\ensuremath{^{\pi}}: 224.5\ensuremath{\gamma} M1(+E2) to 7/2\ensuremath{^{+}}, 428.0\ensuremath{\gamma} M1(+E2) to 11/2\ensuremath{^{+}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1896}&\multicolumn{1}{@{.}l}{1\ensuremath{^{{\hyperlink{PB29LEVEL10}{f}}}} {\it 4}}&\multicolumn{1}{l}{19/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{2 ns {\it 6}}&\multicolumn{1}{l}{\texttt{\ \ \hyperlink{PB32}{C}\hyperlink{PB33}{D}\hyperlink{PB34}{E}} }&\parbox[t][0.3cm]{11.306601cm}{\raggedright J\ensuremath{^{\pi}}: 350.3\ensuremath{\gamma} E2 to 15/2\ensuremath{^{+}}, 354.3\ensuremath{\gamma} M1 to 17/2\ensuremath{^{+}}.\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.306601cm}{\raggedright T\ensuremath{_{1/2}}: From (350\ensuremath{\gamma},354\ensuremath{\gamma},913\ensuremath{\gamma},917\ensuremath{\gamma})(t) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1981He07,B}{1981He07} (\ensuremath{^{\textnormal{200}}}Hg(\ensuremath{\alpha},3n\ensuremath{\gamma})).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1902}&\multicolumn{1}{@{.}l}{2\ensuremath{^{{\hyperlink{PB29LEVEL11}{g}}}} {\it 5}}&\multicolumn{1}{l}{21/2\ensuremath{^{+}}}&&&\multicolumn{1}{l}{\texttt{\ \ \hyperlink{PB32}{C}\hyperlink{PB33}{D}\ } }&\parbox[t][0.3cm]{11.306601cm}{\raggedright J\ensuremath{^{\pi}}: 360.4\ensuremath{\gamma} E2 to 17/2\ensuremath{^{+}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1977}&\multicolumn{1}{@{.}l}{6 {\it 5}}&\multicolumn{1}{l}{7/2\ensuremath{^{+}},9/2\ensuremath{^{+}}}&&&\multicolumn{1}{l}{\texttt{\hyperlink{BI30}{A}\ \ \ \ } }&\parbox[t][0.3cm]{11.306601cm}{\raggedright J\ensuremath{^{\pi}}: 986.5\ensuremath{\gamma} E1 to 7/2\ensuremath{^{-}}, 529.8\ensuremath{\gamma} to 11/2\ensuremath{^{+}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2068}&\multicolumn{1}{@{.}l}{5 {\it 5}}&\multicolumn{1}{l}{(21/2\ensuremath{^{+}})}&&&\multicolumn{1}{l}{\texttt{\ \ \hyperlink{PB32}{C}\hyperlink{PB33}{D}\ } }&\parbox[t][0.3cm]{11.306601cm}{\raggedright J\ensuremath{^{\pi}}: 166.2\ensuremath{\gamma} M1,E2 \ensuremath{\Delta}J=0 to 21/2\ensuremath{^{+}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2119}&\multicolumn{1}{@{.}l}{5 {\it 5}}&\multicolumn{1}{l}{9/2\ensuremath{^{+}}}&&&\multicolumn{1}{l}{\texttt{\hyperlink{BI30}{A}\ \ \ \ } }&\parbox[t][0.3cm]{11.306601cm}{\raggedright J\ensuremath{^{\pi}}: 275.5\ensuremath{\gamma} M1(+E2) to 11/2\ensuremath{^{+}}, 1183.7\ensuremath{\gamma} to 7/2\ensuremath{^{-}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2151}&\multicolumn{1}{@{.}l}{9 {\it 6}}&\multicolumn{1}{l}{7/2\ensuremath{^{+}}}&&&\multicolumn{1}{l}{\texttt{\hyperlink{BI30}{A}\ \ \ \ } }&\parbox[t][0.3cm]{11.306601cm}{\raggedright J\ensuremath{^{\pi}}: 414.6\ensuremath{\gamma} M1(+E2) to 9/2\ensuremath{^{+}}, 1241.4\ensuremath{\gamma} (E1) to 5/2\ensuremath{^{-}}.\vspace{0.1cm}}&\\
\end{longtable}
\begin{textblock}{29}(0,27.3)
Continued on next page (footnotes at end of table)
\end{textblock}
\clearpage
\begin{longtable}{ccccccc@{\extracolsep{\fill}}c}
\\[-.4cm]
\multicolumn{8}{c}{{\bf \small \underline{Adopted \hyperlink{201PB_LEVEL}{Levels}, \hyperlink{201PB_GAMMA}{Gammas} (continued)}}}\\
\multicolumn{8}{c}{~}\\
\multicolumn{8}{c}{\underline{\ensuremath{^{201}}Pb Levels (continued)}}\\
\multicolumn{8}{c}{~}\\
\multicolumn{2}{c}{E(level)$^{{\hyperlink{PB29LEVEL0}{\dagger}}}$}&J$^{\pi}$$^{}$&\multicolumn{2}{c}{T$_{1/2}$$^{}$}&XREF&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&
\endhead
\multicolumn{1}{r@{}}{2208}&\multicolumn{1}{@{.}l}{9 {\it 6}}&\multicolumn{1}{l}{(9/2\ensuremath{^{+}})}&&&\multicolumn{1}{l}{\texttt{\hyperlink{BI30}{A}\ \ \ \ } }&\parbox[t][0.3cm]{10.09322cm}{\raggedright J\ensuremath{^{\pi}}: 557.7\ensuremath{\gamma} to 7/2\ensuremath{^{+}}, 1579.8\ensuremath{\gamma} to 13/2\ensuremath{^{+}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2279}&\multicolumn{1}{@{.}l}{8 {\it 8}}&\multicolumn{1}{l}{(9/2)\ensuremath{^{+}}}&&&\multicolumn{1}{l}{\texttt{\hyperlink{BI30}{A}\ \ \ \ } }&\parbox[t][0.3cm]{10.09322cm}{\raggedright J\ensuremath{^{\pi}}: 832.0\ensuremath{\gamma} M1(+E2) to 11/2\ensuremath{^{+}}, 1265.7\ensuremath{\gamma} to 9/2\ensuremath{^{-}}, 1400.3\ensuremath{\gamma} to (5/2)\ensuremath{^{-}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2439}&\multicolumn{1}{@{.}l}{4 {\it 7}}&\multicolumn{1}{l}{7/2\ensuremath{^{+}},9/2\ensuremath{^{+}}}&&&\multicolumn{1}{l}{\texttt{\hyperlink{BI30}{A}\ \ \ \ } }&\parbox[t][0.3cm]{10.09322cm}{\raggedright J\ensuremath{^{\pi}}: 1024.4\ensuremath{\gamma} M1(+E2) to 9/2\ensuremath{^{+}}, 1503.0\ensuremath{\gamma} to 7/2\ensuremath{^{-}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2459}&\multicolumn{1}{@{.}l}{9 {\it 6}}&\multicolumn{1}{l}{7/2\ensuremath{^{+}},9/2\ensuremath{^{+}}}&&&\multicolumn{1}{l}{\texttt{\hyperlink{BI30}{A}\ \ \ \ } }&\parbox[t][0.3cm]{10.09322cm}{\raggedright J\ensuremath{^{\pi}}: 250.9\ensuremath{\gamma} M1+E2 to (9/2\ensuremath{^{+}}), 1469.5\ensuremath{\gamma} to 7/2\ensuremath{^{-}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2474}&\multicolumn{1}{@{.}l}{6 {\it 5}}&\multicolumn{1}{l}{(7/2\ensuremath{^{+}},9/2\ensuremath{^{+}})}&&&\multicolumn{1}{l}{\texttt{\hyperlink{BI30}{A}\ \ \ \ } }&\parbox[t][0.3cm]{10.09322cm}{\raggedright J\ensuremath{^{\pi}}: 1288.9\ensuremath{\gamma} (E1) to (7/2)\ensuremath{^{-}}; direct population in \ensuremath{^{\textnormal{201}}}Bi \ensuremath{\varepsilon} decay (\ensuremath{J^{\pi}}=9/2\ensuremath{^{-}}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2496}&\multicolumn{1}{@{.}l}{3\ensuremath{^{{\hyperlink{PB29LEVEL12}{h}}}} {\it 4}}&\multicolumn{1}{l}{21/2\ensuremath{^{-}}}&&&\multicolumn{1}{l}{\texttt{\ \ \hyperlink{PB32}{C}\hyperlink{PB33}{D}\hyperlink{PB34}{E}} }&\parbox[t][0.3cm]{10.09322cm}{\raggedright J\ensuremath{^{\pi}}: 594.5\ensuremath{\gamma} (E1) to 21/2\ensuremath{^{+}}, 600.5\ensuremath{\gamma} (E1) to 19/2\ensuremath{^{+}}, 222.4\ensuremath{\gamma} E2 from 25/2\ensuremath{^{-}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2506}&\multicolumn{1}{@{.}l}{8 {\it 4}}&\multicolumn{1}{l}{9/2\ensuremath{^{+}}}&&&\multicolumn{1}{l}{\texttt{\hyperlink{BI30}{A}\ \ \ \ } }&\parbox[t][0.3cm]{10.09322cm}{\raggedright J\ensuremath{^{\pi}}: 1570.8\ensuremath{\gamma} E1 to 7/2\ensuremath{^{-}}, 1877.4\ensuremath{\gamma} to 13/2\ensuremath{^{+}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2548}&\multicolumn{1}{@{.}l}{9 {\it 6}}&\multicolumn{1}{l}{(11/2\ensuremath{^{-}})}&&&\multicolumn{1}{l}{\texttt{\hyperlink{BI30}{A}\ \ \ \ } }&\parbox[t][0.3cm]{10.09322cm}{\raggedright J\ensuremath{^{\pi}}: 1558.6\ensuremath{\gamma} (E2) to 7/2\ensuremath{^{-}}, 1919.4\ensuremath{\gamma} to 13/2\ensuremath{^{+}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2604}&\multicolumn{1}{@{.}l}{0 {\it 7}}&\multicolumn{1}{l}{21/2,23/2}&&&\multicolumn{1}{l}{\texttt{\ \ \hyperlink{PB32}{C}\ \ } }&\parbox[t][0.3cm]{10.09322cm}{\raggedright J\ensuremath{^{\pi}}: 707.9\ensuremath{\gamma} D to 19/2\ensuremath{^{+}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2718}&\multicolumn{1}{@{.}l}{5\ensuremath{^{{\hyperlink{PB29LEVEL13}{i}}}} {\it 4}}&\multicolumn{1}{l}{25/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{63}&\multicolumn{1}{@{ }l}{ns {\it 3}}&\multicolumn{1}{l}{\texttt{\ \ \hyperlink{PB32}{C}\hyperlink{PB33}{D}\hyperlink{PB34}{E}} }&\parbox[t][0.3cm]{10.09322cm}{\raggedright \ensuremath{\mu}={\textminus}0.79 \textit{4} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1988Ro08,B}{1988Ro08},\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2014StZZ,B}{2014StZZ}); Q=0.46 \textit{2} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1979MaYQ,B}{1979MaYQ},\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2016St14,B}{2016St14})\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.09322cm}{\raggedright J\ensuremath{^{\pi}}: 222.4\ensuremath{\gamma} E2 to 21/2\ensuremath{^{-}}; \ensuremath{\mu}.\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.09322cm}{\raggedright T\ensuremath{_{1/2}}: From (222.3\ensuremath{\gamma},350.3\ensuremath{\gamma},354.3\ensuremath{\gamma},600.3\ensuremath{\gamma},913.2\ensuremath{\gamma},917.1\ensuremath{\gamma})(t) in\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.09322cm}{\raggedright {\ }{\ }{\ }\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1988Ro08,B}{1988Ro08} (\ensuremath{^{\textnormal{200}}}Hg(\ensuremath{\alpha},3n\ensuremath{\gamma})). Other: \ensuremath{\approx}55 ns in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1981He07,B}{1981He07}.\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.09322cm}{\raggedright \ensuremath{\mu},Q: Using the time dependent perturbed angular distribution technique.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2718}&\multicolumn{1}{@{.}l}{5+x\ensuremath{^{{\hyperlink{PB29LEVEL14}{j}}}} {\it 4}}&\multicolumn{1}{l}{(29/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{508}&\multicolumn{1}{@{ }l}{ns {\it 5}}&\multicolumn{1}{l}{\texttt{\ \ \hyperlink{PB32}{C}\hyperlink{PB33}{D}\hyperlink{PB34}{E}} }&\parbox[t][0.3cm]{10.09322cm}{\raggedright \ensuremath{\mu}={\textminus}1.011 \textit{6} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1988Ro08,B}{1988Ro08},\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2014StZZ,B}{2014StZZ})\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.09322cm}{\raggedright E(level): X\ensuremath{<}70 keV in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1981He07,B}{1981He07}.\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.09322cm}{\raggedright J\ensuremath{^{\pi}}: From systematics. Consistent with the proposed configuration and \ensuremath{\mu}.\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.09322cm}{\raggedright T\ensuremath{_{1/2}}: From (222.3\ensuremath{\gamma},350.3\ensuremath{\gamma},354.3\ensuremath{\gamma},600.3\ensuremath{\gamma},913.2\ensuremath{\gamma},917.1\ensuremath{\gamma})(t) in\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.09322cm}{\raggedright {\ }{\ }{\ }\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1988Ro08,B}{1988Ro08} (\ensuremath{^{\textnormal{200}}}Hg(\ensuremath{\alpha},3n\ensuremath{\gamma})). Other: 540 ns \textit{40} \ensuremath{\gamma}(t) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1981He07,B}{1981He07}\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.09322cm}{\raggedright {\ }{\ }{\ }(\ensuremath{^{\textnormal{200}}}Hg(\ensuremath{\alpha},3n\ensuremath{\gamma})).\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.09322cm}{\raggedright \ensuremath{\mu}: Using the time dependent perturbed angular distribution technique.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2732}&\multicolumn{1}{@{.}l}{8 {\it 6}}&&&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{PB33}{D}\ } }&&\\
\multicolumn{1}{r@{}}{2736}&\multicolumn{1}{@{.}l}{2 {\it 6}}&&&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{PB33}{D}\ } }&&\\
\multicolumn{1}{r@{}}{2788}&\multicolumn{1}{@{.}l}{8 {\it 6}}&\multicolumn{1}{l}{11/2\ensuremath{^{-}}}&&&\multicolumn{1}{l}{\texttt{\hyperlink{BI30}{A}\ \ \ \ } }&\parbox[t][0.3cm]{10.09322cm}{\raggedright J\ensuremath{^{\pi}}: 239.7\ensuremath{\gamma} M1+E2 to 11/2\ensuremath{^{-}}, 1051.6\ensuremath{\gamma} E1 to 9/2\ensuremath{^{+}}, 2159.7 to 13/2\ensuremath{^{+}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2794}&\multicolumn{1}{@{.}l}{2 {\it 5}}&\multicolumn{1}{l}{(19/2,21/2,23/2)\ensuremath{^{-}}}&&&\multicolumn{1}{l}{\texttt{\ \ \hyperlink{PB32}{C}\ \ } }&\parbox[t][0.3cm]{10.09322cm}{\raggedright J\ensuremath{^{\pi}}: 297.9\ensuremath{\gamma} M1+E2 to 21/2\ensuremath{^{-}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2961}&\multicolumn{1}{@{.}l}{8 {\it 6}}&\multicolumn{1}{l}{(7/2,9/2\ensuremath{^{-}})}&&&\multicolumn{1}{l}{\texttt{\hyperlink{BI30}{A}\ \ \ \ } }&\parbox[t][0.3cm]{10.09322cm}{\raggedright J\ensuremath{^{\pi}}: 1971.0\ensuremath{\gamma} to 7/2\ensuremath{^{-}}, 2082.0\ensuremath{\gamma} to (5/2)\ensuremath{^{-}}; direct population in \ensuremath{^{\textnormal{201}}}Bi \ensuremath{\varepsilon}\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.09322cm}{\raggedright {\ }{\ }{\ }decay (\ensuremath{J^{\pi}}=9/2\ensuremath{^{-}}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3050}&\multicolumn{1}{@{.}l}{7 {\it 5}}&\multicolumn{1}{l}{(7/2)\ensuremath{^{+}}}&&&\multicolumn{1}{l}{\texttt{\hyperlink{BI30}{A}\ \ \ \ } }&\parbox[t][0.3cm]{10.09322cm}{\raggedright J\ensuremath{^{\pi}}: 931.6\ensuremath{\gamma} M1(+E2) to 9/2\ensuremath{^{+}}, 2170.4\ensuremath{\gamma} to (5/2)\ensuremath{^{-}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3509}&\multicolumn{1}{@{.}l}{5+x {\it 4}}&\multicolumn{1}{l}{(31/2\ensuremath{^{-}})}&&&\multicolumn{1}{l}{\texttt{\ \ \hyperlink{PB32}{C}\hyperlink{PB33}{D}\ } }&\parbox[t][0.3cm]{10.09322cm}{\raggedright J\ensuremath{^{\pi}}: 791.0\ensuremath{\gamma} M1+E2 to (29/2\ensuremath{^{-}}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3545}&\multicolumn{1}{@{.}l}{0+x {\it 4}}&\multicolumn{1}{l}{(33/2\ensuremath{^{-}})}&&&\multicolumn{1}{l}{\texttt{\ \ \hyperlink{PB32}{C}\hyperlink{PB33}{D}\hyperlink{PB34}{E}} }&\parbox[t][0.3cm]{10.09322cm}{\raggedright J\ensuremath{^{\pi}}: 826.6\ensuremath{\gamma} E2 to (29/2\ensuremath{^{-}}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3638}&\multicolumn{1}{@{.}l}{0+x {\it 4}}&\multicolumn{1}{l}{(31/2\ensuremath{^{-}})}&&&\multicolumn{1}{l}{\texttt{\ \ \hyperlink{PB32}{C}\hyperlink{PB33}{D}\ } }&\parbox[t][0.3cm]{10.09322cm}{\raggedright J\ensuremath{^{\pi}}: 919.4\ensuremath{\gamma} M1+E2 to (29/2\ensuremath{^{-}}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3832}&\multicolumn{1}{@{.}l}{3+x {\it 6}}&\multicolumn{1}{l}{(35/2\ensuremath{^{-}})}&&&\multicolumn{1}{l}{\texttt{\ \ \hyperlink{PB32}{C}\hyperlink{PB33}{D}\hyperlink{PB34}{E}} }&\parbox[t][0.3cm]{10.09322cm}{\raggedright J\ensuremath{^{\pi}}: 287.2\ensuremath{\gamma} M1+E2 to (33/2\ensuremath{^{-}}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3932}&\multicolumn{1}{@{.}l}{0+x\ensuremath{^{{\hyperlink{PB29LEVEL15}{k}}}} {\it 4}}&\multicolumn{1}{l}{(33/2\ensuremath{^{+}})}&&&\multicolumn{1}{l}{\texttt{\ \ \hyperlink{PB32}{C}\hyperlink{PB33}{D}\ } }&\parbox[t][0.3cm]{10.09322cm}{\raggedright J\ensuremath{^{\pi}}: 293.9\ensuremath{\gamma} D to (31/2\ensuremath{^{-}}),387.0\ensuremath{\gamma} (D) to (33/2\ensuremath{^{-}}), 422.5\ensuremath{\gamma} (D) to (31/2\ensuremath{^{-}});\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.09322cm}{\raggedright {\ }{\ }{\ }proposed configuration.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{4059}&\multicolumn{1}{@{.}l}{5 {\it 7}}&&&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{PB33}{D}\ } }&&\\
\multicolumn{1}{r@{}}{4059}&\multicolumn{1}{@{.}l}{5+y\ensuremath{^{{\hyperlink{PB29LEVEL18}{n}}}} {\it 7}}&&&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{PB33}{D}\ } }&&\\
\multicolumn{1}{r@{}}{4168}&\multicolumn{1}{@{.}l}{7+y\ensuremath{^{{\hyperlink{PB29LEVEL18}{n}}}} {\it 5}}&&&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{PB33}{D}\ } }&&\\
\multicolumn{1}{r@{}}{4350}&\multicolumn{1}{@{.}l}{3+y\ensuremath{^{{\hyperlink{PB29LEVEL18}{n}}}} {\it 7}}&&&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{PB33}{D}\ } }&&\\
\multicolumn{1}{r@{}}{4505}&\multicolumn{1}{@{.}l}{1+x {\it 6}}&\multicolumn{1}{l}{(35/2)}&&&\multicolumn{1}{l}{\texttt{\ \ \hyperlink{PB32}{C}\hyperlink{PB33}{D}\ } }&\parbox[t][0.3cm]{10.09322cm}{\raggedright J\ensuremath{^{\pi}}: 573.2\ensuremath{\gamma} D to (33/2\ensuremath{^{+}}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{4560}&\multicolumn{1}{@{.}l}{2+x {\it 6}}&\multicolumn{1}{l}{(37/2\ensuremath{^{+}})}&&&\multicolumn{1}{l}{\texttt{\ \ \hyperlink{PB32}{C}\hyperlink{PB33}{D}\hyperlink{PB34}{E}} }&\parbox[t][0.3cm]{10.09322cm}{\raggedright J\ensuremath{^{\pi}}: 628.4\ensuremath{\gamma} E2 to (33/2\ensuremath{^{+}}), 728.0 D to (35/2\ensuremath{^{-}}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{4614}&\multicolumn{1}{@{.}l}{1+y\ensuremath{^{{\hyperlink{PB29LEVEL18}{n}}}} {\it 9}}&&&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{PB33}{D}\ } }&&\\
\multicolumn{1}{r@{}}{4640}&\multicolumn{1}{@{.}l}{0+x\ensuremath{^{{\hyperlink{PB29LEVEL16}{l}}}} {\it 6}}&\multicolumn{1}{l}{(41/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{46}&\multicolumn{1}{@{ }l}{ns {\it 3}}&\multicolumn{1}{l}{\texttt{\ \ \hyperlink{PB32}{C}\hyperlink{PB33}{D}\hyperlink{PB34}{E}} }&\parbox[t][0.3cm]{10.09322cm}{\raggedright \ensuremath{\mu}={\textminus}3.7 \textit{8} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1988Ro08,B}{1988Ro08},\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2014StZZ,B}{2014StZZ})\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.09322cm}{\raggedright J\ensuremath{^{\pi}}: 79.5\ensuremath{\gamma} E2 to (37/2\ensuremath{^{+}}).\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.09322cm}{\raggedright T\ensuremath{_{1/2}}: Unweighted average of 52 ns \textit{2} from \ensuremath{\gamma}\ensuremath{\gamma}(\ensuremath{\Delta}t) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2019Ro12,B}{2019Ro12}, using\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.09322cm}{\raggedright {\ }{\ }{\ }902\ensuremath{\gamma}-(728\ensuremath{\gamma},827\ensuremath{\gamma}) and 447\ensuremath{\gamma}-(728\ensuremath{\gamma},827\ensuremath{\gamma}) in \ensuremath{^{\textnormal{197}}}Au(\ensuremath{^{\textnormal{207}}}Pb,X\ensuremath{\gamma}), 43 ns\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.09322cm}{\raggedright {\ }{\ }{\ }\textit{3} from 80.1\ensuremath{\gamma}(t) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1989Su12,B}{1989Su12} (\ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma})),\hphantom{a}and 43 ns \textit{3} from\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.09322cm}{\raggedright {\ }{\ }{\ }(727.7\ensuremath{\gamma},287.0\ensuremath{\gamma},825.6\ensuremath{\gamma})(t) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1988Ro08,B}{1988Ro08} (\ensuremath{^{\textnormal{200}}}Hg(\ensuremath{\alpha},3n\ensuremath{\gamma})).\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.09322cm}{\raggedright \ensuremath{\mu}: Using time dependent perturbed angular distribution technique.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{4640}&\multicolumn{1}{@{.}l}{0+u\ensuremath{^{{\hyperlink{PB29LEVEL20}{p}}}} {\it 7}}&&&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{PB33}{D}\ } }&&\\
\multicolumn{1}{r@{}}{4640}&\multicolumn{1}{@{.}l}{0+v\ensuremath{^{{\hyperlink{PB29LEVEL21}{q}}}} {\it 7}}&&&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{PB33}{D}\ } }&&\\
\multicolumn{1}{r@{}}{4640}&\multicolumn{1}{@{.}l}{0+z\ensuremath{^{{\hyperlink{PB29LEVEL19}{o}}}} {\it 7}}&&&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{PB33}{D}\ } }&&\\
\multicolumn{1}{r@{}}{4640}&\multicolumn{1}{@{.}l}{1+x {\it 5}}&\multicolumn{1}{l}{(35/2)}&&&\multicolumn{1}{l}{\texttt{\ \ \hyperlink{PB32}{C}\ \ } }&\parbox[t][0.3cm]{10.09322cm}{\raggedright J\ensuremath{^{\pi}}: 708.1\ensuremath{\gamma} to (33/2\ensuremath{^{+}}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{4647}&\multicolumn{1}{@{.}l}{6+x {\it 6}}&\multicolumn{1}{l}{(35/2)}&&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{PB33}{D}\ } }&\parbox[t][0.3cm]{10.09322cm}{\raggedright J\ensuremath{^{\pi}}: 142.5\ensuremath{\gamma} (D) to (35/2\ensuremath{^{+}}), 715.7\ensuremath{\gamma} D to (33/2\ensuremath{^{+}}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{4780}&\multicolumn{1}{@{.}l}{5+z\ensuremath{^{{\hyperlink{PB29LEVEL19}{o}}}} {\it 5}}&&&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{PB33}{D}\ } }&&\\
\end{longtable}
\begin{textblock}{29}(0,27.3)
Continued on next page (footnotes at end of table)
\end{textblock}
\clearpage
\begin{longtable}{ccccc@{\extracolsep{\fill}}c}
\\[-.4cm]
\multicolumn{6}{c}{{\bf \small \underline{Adopted \hyperlink{201PB_LEVEL}{Levels}, \hyperlink{201PB_GAMMA}{Gammas} (continued)}}}\\
\multicolumn{6}{c}{~}\\
\multicolumn{6}{c}{\underline{\ensuremath{^{201}}Pb Levels (continued)}}\\
\multicolumn{6}{c}{~}\\
\multicolumn{2}{c}{E(level)$^{{\hyperlink{PB29LEVEL0}{\dagger}}}$}&J$^{\pi}$$^{}$&XREF&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&\hrulefill&
\endhead
\multicolumn{1}{r@{}}{4793}&\multicolumn{1}{@{.}l}{8+v\ensuremath{^{{\hyperlink{PB29LEVEL21}{q}}}} {\it 5}}&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{PB33}{D}\ } }&&\\
\multicolumn{1}{r@{}}{4817}&\multicolumn{1}{@{.}l}{4+u\ensuremath{^{{\hyperlink{PB29LEVEL20}{p}}}} {\it 5}}&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{PB33}{D}\ } }&&\\
\multicolumn{1}{r@{}}{4830}&\multicolumn{1}{@{.}l}{2+x {\it 6}}&\multicolumn{1}{l}{(39/2)}&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{PB33}{D}\ } }&\parbox[t][0.3cm]{12.45988cm}{\raggedright J\ensuremath{^{\pi}}: 190.4\ensuremath{\gamma} D to (41/2\ensuremath{^{+}}), 269.9\ensuremath{\gamma} D to (37/2\ensuremath{^{+}}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{4954}&\multicolumn{1}{@{.}l}{9+y\ensuremath{^{{\hyperlink{PB29LEVEL18}{n}}}} {\it 10}}&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{PB33}{D}\ } }&&\\
\multicolumn{1}{r@{}}{4956}&\multicolumn{1}{@{.}l}{3+z\ensuremath{^{{\hyperlink{PB29LEVEL19}{o}}}} {\it 7}}&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{PB33}{D}\ } }&&\\
\multicolumn{1}{r@{}}{4992}&\multicolumn{1}{@{.}l}{4+v\ensuremath{^{{\hyperlink{PB29LEVEL21}{q}}}} {\it 7}}&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{PB33}{D}\ } }&&\\
\multicolumn{1}{r@{}}{5000}&\multicolumn{1}{@{.}l}{1+x {\it 7}}&&\multicolumn{1}{l}{\texttt{\ \ \hyperlink{PB32}{C}\ \ } }&&\\
\multicolumn{1}{r@{}}{5043}&\multicolumn{1}{@{.}l}{1+u\ensuremath{^{{\hyperlink{PB29LEVEL20}{p}}}} {\it 7}}&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{PB33}{D}\ } }&&\\
\multicolumn{1}{r@{}}{5087}&\multicolumn{1}{@{.}l}{1+x {\it 6}}&\multicolumn{1}{l}{(43/2)}&\multicolumn{1}{l}{\texttt{\ \ \hyperlink{PB32}{C}\hyperlink{PB33}{D}\hyperlink{PB34}{E}} }&\parbox[t][0.3cm]{12.45988cm}{\raggedright J\ensuremath{^{\pi}}: 447.3\ensuremath{\gamma} D to (41/2\ensuremath{^{+}}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{5172}&\multicolumn{1}{@{.}l}{4+x {\it 7}}&\multicolumn{1}{l}{(35/2,37/2)}&\multicolumn{1}{l}{\texttt{\ \ \hyperlink{PB32}{C}\ \ } }&\parbox[t][0.3cm]{12.45988cm}{\raggedright J\ensuremath{^{\pi}}: 667.3\ensuremath{\gamma} D to (35/2\ensuremath{^{+}}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{5178}&\multicolumn{1}{@{.}l}{6+z\ensuremath{^{{\hyperlink{PB29LEVEL19}{o}}}} {\it 9}}&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{PB33}{D}\ } }&&\\
\multicolumn{1}{r@{}}{5242}&\multicolumn{1}{@{.}l}{4+v\ensuremath{^{{\hyperlink{PB29LEVEL21}{q}}}} {\it 9}}&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{PB33}{D}\ } }&&\\
\multicolumn{1}{r@{}}{5321}&\multicolumn{1}{@{.}l}{3+u\ensuremath{^{{\hyperlink{PB29LEVEL20}{p}}}} {\it 9}}&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{PB33}{D}\ } }&&\\
\multicolumn{1}{r@{}}{5358}&\multicolumn{1}{@{.}l}{7+y\ensuremath{^{{\hyperlink{PB29LEVEL18}{n}}}} {\it 10}}&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{PB33}{D}\ } }&&\\
\multicolumn{1}{r@{}}{5389}&\multicolumn{1}{@{.}l}{1+x {\it 7}}&\multicolumn{1}{l}{(45/2)}&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{PB33}{D}\ } }&\parbox[t][0.3cm]{12.45988cm}{\raggedright J\ensuremath{^{\pi}}: 302.0\ensuremath{\gamma} D to (43/2).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{5425}&\multicolumn{1}{@{.}l}{5+x {\it 7}}&&\multicolumn{1}{l}{\texttt{\ \ \hyperlink{PB32}{C}\ \ } }&&\\
\multicolumn{1}{r@{}}{5455}&\multicolumn{1}{@{.}l}{0+z\ensuremath{^{{\hyperlink{PB29LEVEL19}{o}}}} {\it 10}}&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{PB33}{D}\ } }&&\\
\multicolumn{1}{r@{}}{5554}&\multicolumn{1}{@{.}l}{4+v\ensuremath{^{{\hyperlink{PB29LEVEL21}{q}}}} {\it 10}}&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{PB33}{D}\ } }&&\\
\multicolumn{1}{r@{}}{5581}&\multicolumn{1}{@{.}l}{9+x {\it 7}}&\multicolumn{1}{l}{(39/2\ensuremath{^{-}})}&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{PB33}{D}\ } }&\parbox[t][0.3cm]{12.45988cm}{\raggedright J\ensuremath{^{\pi}}: 1749.5\ensuremath{\gamma} E2 to (35/2\ensuremath{^{-}}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{5648}&\multicolumn{1}{@{.}l}{0+u\ensuremath{^{{\hyperlink{PB29LEVEL20}{p}}}} {\it 10}}&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{PB33}{D}\ } }&&\\
\multicolumn{1}{r@{}}{5787}&\multicolumn{1}{@{.}l}{3+z\ensuremath{^{{\hyperlink{PB29LEVEL19}{o}}}} {\it 12}}&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{PB33}{D}\ } }&&\\
\multicolumn{1}{r@{}}{5817}&\multicolumn{1}{@{.}l}{7+y\ensuremath{^{{\hyperlink{PB29LEVEL18}{n}}}} {\it 10}}&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{PB33}{D}\ } }&&\\
\multicolumn{1}{r@{}}{5830}&\multicolumn{1}{@{.}l}{0+x {\it 7}}&\multicolumn{1}{l}{(45/2\ensuremath{^{+}})}&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{PB33}{D}\ } }&\parbox[t][0.3cm]{12.45988cm}{\raggedright J\ensuremath{^{\pi}}: 1190.1\ensuremath{\gamma} E2 to (41/2\ensuremath{^{+}}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{5835}&\multicolumn{1}{@{.}l}{7+x {\it 6}}&\multicolumn{1}{l}{(41/2)}&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{PB33}{D}\ } }&\parbox[t][0.3cm]{12.45988cm}{\raggedright J\ensuremath{^{\pi}}: 253.7\ensuremath{\gamma} (D) to (39/2\ensuremath{^{-}}), 1005.5\ensuremath{\gamma} D to (39/2\ensuremath{^{-}}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{5891}&\multicolumn{1}{@{.}l}{3+x {\it 7}}&\multicolumn{1}{l}{(43/2)}&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{PB33}{D}\ } }&\parbox[t][0.3cm]{12.45988cm}{\raggedright J\ensuremath{^{\pi}}: 1251.3\ensuremath{\gamma} D to (41/2\ensuremath{^{+}}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{5928}&\multicolumn{1}{@{.}l}{8+v\ensuremath{^{{\hyperlink{PB29LEVEL21}{q}}}} {\it 12}}&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{PB33}{D}\ } }&&\\
\multicolumn{1}{r@{}}{5989}&\multicolumn{1}{@{.}l}{4+x {\it 6}}&\multicolumn{1}{l}{(45/2)}&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{PB33}{D}\hyperlink{PB34}{E}} }&\parbox[t][0.3cm]{12.45988cm}{\raggedright J\ensuremath{^{\pi}}: 98.2\ensuremath{\gamma} D to (43/2), 153.7\ensuremath{\gamma} E2 to (41/2), 159.4\ensuremath{\gamma} (D) to (45/2\ensuremath{^{+}}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{6028}&\multicolumn{1}{@{.}l}{4+u\ensuremath{^{{\hyperlink{PB29LEVEL20}{p}}}} {\it 12}}&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{PB33}{D}\ } }&&\\
\multicolumn{1}{r@{}}{6145}&\multicolumn{1}{@{.}l}{1+x\ensuremath{^{{\hyperlink{PB29LEVEL17}{m}}}} {\it 7}}&\multicolumn{1}{l}{(35/2)}&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{PB33}{D}\ } }&\parbox[t][0.3cm]{12.45988cm}{\raggedright J\ensuremath{^{\pi}}: 1640.0\ensuremath{\gamma} D to (35/2).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{6175}&\multicolumn{1}{@{.}l}{4+z\ensuremath{^{{\hyperlink{PB29LEVEL19}{o}}}} {\it 12}}&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{PB33}{D}\ } }&&\\
\multicolumn{1}{r@{}}{6246}&\multicolumn{1}{@{.}l}{8+x\ensuremath{^{{\hyperlink{PB29LEVEL17}{m}}}} {\it 8}}&\multicolumn{1}{l}{(37/2)}&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{PB33}{D}\ } }&\parbox[t][0.3cm]{12.45988cm}{\raggedright J\ensuremath{^{\pi}}: 101.7\ensuremath{\gamma} (M1) to (35/2); band structure.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{6323}&\multicolumn{1}{@{.}l}{4+y\ensuremath{^{{\hyperlink{PB29LEVEL18}{n}}}} {\it 11}}&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{PB33}{D}\ } }&&\\
\multicolumn{1}{r@{}}{6323}&\multicolumn{1}{@{.}l}{9+x {\it 7}}&\multicolumn{1}{l}{(45/2\ensuremath{^{+}})}&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{PB33}{D}\ } }&\parbox[t][0.3cm]{12.45988cm}{\raggedright J\ensuremath{^{\pi}}: 1683.8\ensuremath{\gamma} E2 to (41/2\ensuremath{^{+}}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{6336}&\multicolumn{1}{@{.}l}{1+y {\it 11}}&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{PB33}{D}\ } }&&\\
\multicolumn{1}{r@{}}{6364}&\multicolumn{1}{@{.}l}{8+v\ensuremath{^{{\hyperlink{PB29LEVEL21}{q}}}} {\it 13}}&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{PB33}{D}\ } }&&\\
\multicolumn{1}{r@{}}{6376}&\multicolumn{1}{@{.}l}{4+x\ensuremath{^{{\hyperlink{PB29LEVEL17}{m}}}} {\it 9}}&\multicolumn{1}{l}{(39/2)}&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{PB33}{D}\ } }&\parbox[t][0.3cm]{12.45988cm}{\raggedright J\ensuremath{^{\pi}}: 129.7\ensuremath{\gamma} (M1) to (37/2); band structure.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{6458}&\multicolumn{1}{@{.}l}{1+u\ensuremath{^{{\hyperlink{PB29LEVEL20}{p}}}} {\it 13}}&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{PB33}{D}\ } }&&\\
\multicolumn{1}{r@{}}{6460}&\multicolumn{1}{@{.}l}{1+x {\it 7}}&\multicolumn{1}{l}{(47/2)}&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{PB33}{D}\ } }&\parbox[t][0.3cm]{12.45988cm}{\raggedright J\ensuremath{^{\pi}}: 136.2\ensuremath{\gamma} D to (45/2\ensuremath{^{+}}), 470.7\ensuremath{\gamma} D to (45/2).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{6548}&\multicolumn{1}{@{.}l}{0+x\ensuremath{^{{\hyperlink{PB29LEVEL17}{m}}}} {\it 9}}&\multicolumn{1}{l}{(41/2)}&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{PB33}{D}\ } }&\parbox[t][0.3cm]{12.45988cm}{\raggedright J\ensuremath{^{\pi}}: 171.6\ensuremath{\gamma} (M1) to (39/2); band structure.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{6616}&\multicolumn{1}{@{.}l}{7+z\ensuremath{^{{\hyperlink{PB29LEVEL19}{o}}}} {\it 12}}&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{PB33}{D}\ } }&&\\
\multicolumn{1}{r@{}}{6706}&\multicolumn{1}{@{.}l}{7+x {\it 7}}&\multicolumn{1}{l}{(49/2)}&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{PB33}{D}\ } }&\parbox[t][0.3cm]{12.45988cm}{\raggedright J\ensuremath{^{\pi}}: 717.3\ensuremath{\gamma} E2 to (45/2).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{6768}&\multicolumn{1}{@{.}l}{5+x\ensuremath{^{{\hyperlink{PB29LEVEL17}{m}}}} {\it 8}}&\multicolumn{1}{l}{(43/2)}&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{PB33}{D}\ } }&\parbox[t][0.3cm]{12.45988cm}{\raggedright J\ensuremath{^{\pi}}: 220.5\ensuremath{\gamma} (M1) to (41/2); band structure.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{6858}&\multicolumn{1}{@{.}l}{2+v\ensuremath{^{{\hyperlink{PB29LEVEL21}{q}}}} {\it 14}}&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{PB33}{D}\ } }&&\\
\multicolumn{1}{r@{}}{6881}&\multicolumn{1}{@{.}l}{9+y\ensuremath{^{{\hyperlink{PB29LEVEL18}{n}}}} {\it 12}}&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{PB33}{D}\ } }&&\\
\multicolumn{1}{r@{}}{6910}&\multicolumn{1}{@{.}l}{1+x {\it 8}}&\multicolumn{1}{l}{(47/2)}&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{PB33}{D}\ } }&\parbox[t][0.3cm]{12.45988cm}{\raggedright J\ensuremath{^{\pi}}: 586.1\ensuremath{\gamma} D to (45/2\ensuremath{^{+}}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{6941}&\multicolumn{1}{@{.}l}{2+u\ensuremath{^{{\hyperlink{PB29LEVEL20}{p}}}} {\it 14}}&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{PB33}{D}\ } }&&\\
\multicolumn{1}{r@{}}{7008}&\multicolumn{1}{@{.}l}{4+x {\it 8}}&\multicolumn{1}{l}{(49/2)}&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{PB33}{D}\ } }&\parbox[t][0.3cm]{12.45988cm}{\raggedright J\ensuremath{^{\pi}}: 548.3\ensuremath{\gamma} D to (47/2).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{7044}&\multicolumn{1}{@{.}l}{3+x\ensuremath{^{{\hyperlink{PB29LEVEL17}{m}}}} {\it 7}}&\multicolumn{1}{l}{(45/2)}&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{PB33}{D}\ } }&\parbox[t][0.3cm]{12.45988cm}{\raggedright J\ensuremath{^{\pi}}: 275.9\ensuremath{\gamma} (M1) to (43/2); band structure.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{7108}&\multicolumn{1}{@{.}l}{4+z\ensuremath{^{{\hyperlink{PB29LEVEL19}{o}}}} {\it 13}}&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{PB33}{D}\ } }&&\\
\multicolumn{1}{r@{}}{7142}&\multicolumn{1}{@{.}l}{3+x {\it 7}}&\multicolumn{1}{l}{(49/2\ensuremath{^{+}})}&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{PB33}{D}\ } }&\parbox[t][0.3cm]{12.45988cm}{\raggedright J\ensuremath{^{\pi}}: 232.2\ensuremath{\gamma} D to (47/2), 682.3\ensuremath{\gamma} D to (47/2), 1312.3\ensuremath{\gamma} E2 to (45/2\ensuremath{^{+}}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{7339}&\multicolumn{1}{@{.}l}{5+x {\it 8}}&\multicolumn{1}{l}{(51/2)}&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{PB33}{D}\ } }&\parbox[t][0.3cm]{12.45988cm}{\raggedright J\ensuremath{^{\pi}}: 197.2\ensuremath{\gamma} D to (49/2\ensuremath{^{+}}), 331.1\ensuremath{\gamma} D to (49/2).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{7377}&\multicolumn{1}{@{.}l}{5+x {\it 7}}&\multicolumn{1}{l}{(47/2)}&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{PB33}{D}\ } }&\parbox[t][0.3cm]{12.45988cm}{\raggedright J\ensuremath{^{\pi}}: 333.1\ensuremath{\gamma} (M1) to (45/2,47/2), 1388.1\ensuremath{\gamma} D to (45/2).\vspace{0.1cm}}&\\
\end{longtable}
\begin{textblock}{29}(0,27.3)
Continued on next page (footnotes at end of table)
\end{textblock}
\clearpage
\begin{longtable}{ccccc@{\extracolsep{\fill}}c}
\\[-.4cm]
\multicolumn{6}{c}{{\bf \small \underline{Adopted \hyperlink{201PB_LEVEL}{Levels}, \hyperlink{201PB_GAMMA}{Gammas} (continued)}}}\\
\multicolumn{6}{c}{~}\\
\multicolumn{6}{c}{\underline{\ensuremath{^{201}}Pb Levels (continued)}}\\
\multicolumn{6}{c}{~}\\
\multicolumn{2}{c}{E(level)$^{{\hyperlink{PB29LEVEL0}{\dagger}}}$}&J$^{\pi}$$^{}$&XREF&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&\hrulefill&
\endhead
\multicolumn{1}{r@{}}{7378}&\multicolumn{1}{@{.}l}{9+x\ensuremath{^{{\hyperlink{PB29LEVEL17}{m}}}} {\it 7}}&\multicolumn{1}{l}{(47/2)}&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{PB33}{D}\ } }&\parbox[t][0.3cm]{13.1083cm}{\raggedright J\ensuremath{^{\pi}}: 334.6\ensuremath{\gamma} (M1) to (45/2,47/2), 1389.4\ensuremath{\gamma} D to (45/2).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{7471}&\multicolumn{1}{@{.}l}{4+u\ensuremath{^{{\hyperlink{PB29LEVEL20}{p}}}} {\it 15}}&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{PB33}{D}\ } }&&\\
\multicolumn{1}{r@{}}{7648}&\multicolumn{1}{@{.}l}{2+z\ensuremath{^{{\hyperlink{PB29LEVEL19}{o}}}} {\it 13}}&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{PB33}{D}\ } }&&\\
\multicolumn{1}{r@{}}{7759}&\multicolumn{1}{@{.}l}{5+x {\it 7}}&\multicolumn{1}{l}{(49/2)}&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{PB33}{D}\ } }&\parbox[t][0.3cm]{13.1083cm}{\raggedright J\ensuremath{^{\pi}}: 380.6\ensuremath{\gamma} D to (47/2), 382.0\ensuremath{\gamma} D to (47/2).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{7772}&\multicolumn{1}{@{.}l}{2+x\ensuremath{^{{\hyperlink{PB29LEVEL17}{m}}}} {\it 8}}&\multicolumn{1}{l}{(49/2)}&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{PB33}{D}\ } }&\parbox[t][0.3cm]{13.1083cm}{\raggedright J\ensuremath{^{\pi}}: 393.3\ensuremath{\gamma} (M1) to (47/2), 394.8\ensuremath{\gamma} (M1) to (47/2).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{8003}&\multicolumn{1}{@{.}l}{4+x {\it 10}}&\multicolumn{1}{l}{(53/2)}&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{PB33}{D}\ } }&\parbox[t][0.3cm]{13.1083cm}{\raggedright J\ensuremath{^{\pi}}: 663.9\ensuremath{\gamma} D to (51/2).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{8018}&\multicolumn{1}{@{.}l}{7+x {\it 8}}&\multicolumn{1}{l}{(51/2)}&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{PB33}{D}\ } }&\parbox[t][0.3cm]{13.1083cm}{\raggedright J\ensuremath{^{\pi}}: 259.2\ensuremath{\gamma} D to (49/2), 1312.0\ensuremath{\gamma} D to (49/2).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{8198}&\multicolumn{1}{@{.}l}{0+x {\it 9}}&\multicolumn{1}{l}{(53/2)}&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{PB33}{D}\ } }&\parbox[t][0.3cm]{13.1083cm}{\raggedright J\ensuremath{^{\pi}}: 179.3\ensuremath{\gamma} D to (51/2).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{8214}&\multicolumn{1}{@{.}l}{7+x {\it 9}}&\multicolumn{1}{l}{(51/2)}&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{PB33}{D}\ } }&\parbox[t][0.3cm]{13.1083cm}{\raggedright J\ensuremath{^{\pi}}: 442.5\ensuremath{\gamma} (M1) to (49/2).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{8226}&\multicolumn{1}{@{.}l}{1+x\ensuremath{^{{\hyperlink{PB29LEVEL17}{m}}}} {\it 9}}&\multicolumn{1}{l}{(51/2)}&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{PB33}{D}\ } }&\parbox[t][0.3cm]{13.1083cm}{\raggedright J\ensuremath{^{\pi}}: 453.9\ensuremath{\gamma} (M1) to (49/2).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{8653}&\multicolumn{1}{@{.}l}{8+x {\it 11}}&\multicolumn{1}{l}{(55/2)}&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{PB33}{D}\ } }&\parbox[t][0.3cm]{13.1083cm}{\raggedright J\ensuremath{^{\pi}}: 455.8\ensuremath{\gamma} (M1) to (53/2).\vspace{0.1cm}}&\\
\end{longtable}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB29LEVEL0}{\dagger}}}} From a least squares fit to E\ensuremath{\gamma}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB29LEVEL1}{\ddagger}}}} Configuration=\ensuremath{\nu} f\ensuremath{_{\textnormal{5/2}}^{\textnormal{$-$1}}}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB29LEVEL2}{\#}}}} Configuration=\ensuremath{\nu} p\ensuremath{_{\textnormal{3/2}}^{\textnormal{$-$1}}}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB29LEVEL3}{@}}}} Configuration=\ensuremath{\nu} p\ensuremath{_{\textnormal{1/2}}^{\textnormal{$-$1}}}. The assignment is tentative.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB29LEVEL4}{\&}}}} Configuration=\ensuremath{\nu} i\ensuremath{_{\textnormal{13/2}}^{\textnormal{$-$1}}}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB29LEVEL5}{a}}}} Configuration=\ensuremath{\nu} f\ensuremath{_{\textnormal{7/2}}^{\textnormal{$-$1}}}. The assignment is tentative.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB29LEVEL6}{b}}}} Configuration=\ensuremath{\nu} (f\ensuremath{_{\textnormal{5/2}}^{\textnormal{$-$1}}})\ensuremath{\otimes}2\ensuremath{^{\textnormal{+}}}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB29LEVEL7}{c}}}} Configuration=\ensuremath{\nu} (p\ensuremath{_{\textnormal{3/2}}^{\textnormal{$-$1}}})\ensuremath{\otimes}2\ensuremath{^{\textnormal{+}}}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB29LEVEL8}{d}}}} Configuration=\ensuremath{\nu} (i\ensuremath{_{\textnormal{13/2}}^{\textnormal{$-$1}}})\ensuremath{\otimes}2\ensuremath{^{\textnormal{+}}}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB29LEVEL9}{e}}}} Probably an admixture of configuration=\ensuremath{\nu} (f\ensuremath{_{\textnormal{5/2}}^{\textnormal{$-$1}}},p\ensuremath{_{\textnormal{1/2}}^{\textnormal{$-$1}}},i\ensuremath{_{\textnormal{13/2}}^{\textnormal{$-$1}}})\ensuremath{\otimes}2\ensuremath{^{\textnormal{+}}} and configuration=\ensuremath{\nu} (i\ensuremath{_{\textnormal{13/2}}^{\textnormal{$-$1}}})\ensuremath{\otimes}2\ensuremath{^{\textnormal{+}}}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB29LEVEL10}{f}}}} Probably an admixture of configuration=\ensuremath{\nu} (f\ensuremath{_{\textnormal{5/2}}^{\textnormal{$-$1}}},p\ensuremath{_{\textnormal{1/2}}^{\textnormal{$-$1}}},i\ensuremath{_{\textnormal{13/2}}^{\textnormal{$-$1}}})\ensuremath{\otimes}4\ensuremath{^{\textnormal{+}}} and configuration=\ensuremath{\nu} (i\ensuremath{_{\textnormal{13/2}}^{\textnormal{$-$1}}})\ensuremath{\otimes}4\ensuremath{^{\textnormal{+}}}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB29LEVEL11}{g}}}} Configuration=\ensuremath{\nu} (f\ensuremath{_{\textnormal{5/2}}^{\textnormal{$-$2}}},i\ensuremath{_{\textnormal{13/2}}^{\textnormal{$-$1}}}).}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB29LEVEL12}{h}}}} Configuration=\ensuremath{\nu} [p\ensuremath{_{\textnormal{3/2}}^{\textnormal{$-$1}}},(i\ensuremath{_{\textnormal{13/2}}^{\textnormal{$-$2}}})\ensuremath{_{\textnormal{12+}}}].}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB29LEVEL13}{i}}}} Probably an admixture of configuration=\ensuremath{\nu} [f\ensuremath{_{\textnormal{5/2}}^{\textnormal{$-$1}}},(i\ensuremath{_{\textnormal{13/2}}^{\textnormal{$-$2}}})\ensuremath{_{\textnormal{10+}}}], configuration=\ensuremath{\nu} [p\ensuremath{_{\textnormal{3/2}}^{\textnormal{$-$1}}},(i\ensuremath{_{\textnormal{13/2}}^{\textnormal{$-$2}}})\ensuremath{_{\textnormal{12+}}}] and configuration=\ensuremath{\nu}}\\
\parbox[b][0.3cm]{17.7cm}{{\ }{\ }[p\ensuremath{_{\textnormal{1/2}}^{\textnormal{$-$1}}},(i\ensuremath{_{\textnormal{13/2}}^{\textnormal{$-$2}}})\ensuremath{_{\textnormal{12+}}}].}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB29LEVEL14}{j}}}} Configuration=\ensuremath{\nu} [f\ensuremath{_{\textnormal{5/2}}^{\textnormal{$-$1}}},(i\ensuremath{_{\textnormal{13/2}}^{\textnormal{$-$2}}})\ensuremath{_{\textnormal{12+}}}].}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB29LEVEL15}{k}}}} Configuration=\ensuremath{\nu} (i\ensuremath{_{\textnormal{13/2}}^{\textnormal{$-$3}}}).}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB29LEVEL16}{l}}}} Configuration=\ensuremath{\nu} (p\ensuremath{_{\textnormal{3/2}}^{\textnormal{$-$1}}},f\ensuremath{_{\textnormal{5/2}}^{\textnormal{$-$1}}},i\ensuremath{_{\textnormal{13/2}}^{\textnormal{$-$3}}}).}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB29LEVEL17}{m}}}} Band(A): configuration=\ensuremath{\nu} [p\ensuremath{_{\textnormal{3/2}}^{\textnormal{$-$1}}},(i\ensuremath{_{\textnormal{13/2}}})\ensuremath{^{\textnormal{$-$2}}})\ensuremath{_{\textnormal{12+}}}]\ensuremath{\otimes} \ensuremath{\pi} (h\ensuremath{_{\textnormal{9/2}}^{\textnormal{+1}}},i\ensuremath{_{\textnormal{13/2}}^{\textnormal{+1}}})\ensuremath{_{\textnormal{11$-$}}}.\hphantom{a}Band 2 in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1995Ba70,B}{1995Ba70}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB29LEVEL18}{n}}}} Band(B): configuration=\ensuremath{\nu} (i\ensuremath{_{\textnormal{13/2}}^{\textnormal{$-$1}}}) \ensuremath{\otimes}\ensuremath{\pi} (h\ensuremath{_{\textnormal{9/2}}^{\textnormal{+1}}},i\ensuremath{_{\textnormal{13/2}}^{\textnormal{+1}}})\ensuremath{_{\textnormal{11$-$}}}. Band 1 in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1995Ba70,B}{1995Ba70}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB29LEVEL19}{o}}}} Band(C): Band 3 in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1995Ba70,B}{1995Ba70}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB29LEVEL20}{p}}}} Band(D): Band 4 in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1995Ba70,B}{1995Ba70}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB29LEVEL21}{q}}}} Band(E): Band 5 in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1995Ba70,B}{1995Ba70}.}\\
\vspace{0.5cm}
\clearpage
\vspace{0.3cm}
\begin{landscape}
\vspace*{-0.5cm}
{\bf \small \underline{Adopted \hyperlink{201PB_LEVEL}{Levels}, \hyperlink{201PB_GAMMA}{Gammas} (continued)}}\\
\vspace{0.3cm}
\hypertarget{201PB_GAMMA}{\underline{$\gamma$($^{201}$Pb)}}\\
\begin{longtable}{ccccccccc@{}ccccccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E\ensuremath{_{i}}(level)}&J\ensuremath{^{\pi}_{i}}&\multicolumn{2}{c}{E\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{PB29GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{I\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{PB29GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{E\ensuremath{_{f}}}&J\ensuremath{^{\pi}_{f}}&Mult.\ensuremath{^{\hyperlink{PB29GAMMA4}{\&}}}&\multicolumn{2}{c}{\ensuremath{\delta}\ensuremath{^{\hyperlink{PB29GAMMA5}{a}}}}&\multicolumn{2}{c}{\ensuremath{\alpha}\ensuremath{^{\hyperlink{PB29GAMMA6}{b}}}}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{88}&\multicolumn{1}{@{.}l}{5}&\multicolumn{1}{l}{3/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{88}&\multicolumn{1}{@{.}l}{6 {\it 10}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{6}&\multicolumn{1}{@{ }l}{{\it 6}}&\parbox[t][0.3cm]{8.549472cm}{\raggedright Mult.: From \ensuremath{\alpha}(exp) in \ensuremath{^{\textnormal{201}}}Bi \ensuremath{\varepsilon} decay.\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.549472cm}{\raggedright \ensuremath{\alpha}: From intensity balance in \ensuremath{^{\textnormal{201}}}Bi \ensuremath{\varepsilon} decay.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{169}&\multicolumn{1}{@{.}l}{9}&\multicolumn{1}{l}{(1/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{(81}&\multicolumn{1}{@{.}l}{40 {\it 10})}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{88}&\multicolumn{1}{@{.}l}{5 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{-}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{8.549472cm}{\raggedright E\ensuremath{_{\gamma}}: Not observed experimentally; E\ensuremath{\gamma} from E(level) difference.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{538}&\multicolumn{1}{@{.}l}{7}&\multicolumn{1}{l}{(3/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{368}&\multicolumn{1}{@{.}l}{8 {\it 10}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{ }l}{{\it 19}}&\multicolumn{1}{r@{}}{169}&\multicolumn{1}{@{.}l}{9 }&\multicolumn{1}{@{}l}{(1/2\ensuremath{^{-}})}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
&&&\multicolumn{1}{r@{}}{450}&\multicolumn{1}{@{.}l}{3 {\it 10}}&\multicolumn{1}{r@{}}{70}&\multicolumn{1}{@{ }l}{{\it 14}}&\multicolumn{1}{r@{}}{88}&\multicolumn{1}{@{.}l}{5 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{-}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
&&&\multicolumn{1}{r@{}}{538}&\multicolumn{1}{@{.}l}{7 {\it 10}}&\multicolumn{1}{r@{}}{50}&\multicolumn{1}{@{ }l}{{\it 11}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
\multicolumn{1}{r@{}}{629}&\multicolumn{1}{@{.}l}{1}&\multicolumn{1}{l}{13/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{629}&\multicolumn{1}{@{.}l}{1 {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M4}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{813 {\it 12}}&\parbox[t][0.3cm]{8.549472cm}{\raggedright B(M4)(W.u.)=3.08 \textit{+11{\textminus}10}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.549472cm}{\raggedright Mult.: From \ensuremath{\alpha}(K)exp=0.6 \textit{2}, K/L=2.3 \textit{3} and L12/L3=4 \textit{1}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.549472cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1956St05,B}{1956St05}) and \ensuremath{\alpha}(L)exp=0.21 \textit{1} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1978Ri04,B}{1978Ri04}) in \ensuremath{^{\textnormal{201}}}Bi \ensuremath{\varepsilon}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.549472cm}{\raggedright {\ }{\ }{\ }decay.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{879}&\multicolumn{1}{@{.}l}{6}&\multicolumn{1}{l}{(5/2)\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{710}&\multicolumn{1}{@{.}l}{0\ensuremath{^{\hyperlink{PB29GAMMA7}{c}}} {\it 10}}&\multicolumn{1}{r@{}}{7}&\multicolumn{1}{@{.}l}{4 {\it 14}}&\multicolumn{1}{r@{}}{169}&\multicolumn{1}{@{.}l}{9 }&\multicolumn{1}{@{}l}{(1/2\ensuremath{^{-}})}&&&&&&&\\
&&&\multicolumn{1}{r@{}}{791}&\multicolumn{1}{@{.}l}{0\ensuremath{^{\hyperlink{PB29GAMMA7}{c}}} {\it 10}}&\multicolumn{1}{r@{}}{27}&\multicolumn{1}{@{ }l}{{\it 6}}&\multicolumn{1}{r@{}}{88}&\multicolumn{1}{@{.}l}{5 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1(+E2)}&\multicolumn{1}{r@{}}{$<$1}&\multicolumn{1}{@{.}l}{1}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{027 {\it 6}}&\parbox[t][0.3cm]{8.549472cm}{\raggedright Mult.,\ensuremath{\delta}: From \ensuremath{\alpha}(K)exp=0.028 \textit{11} in \ensuremath{^{\textnormal{201}}}Bi \ensuremath{\varepsilon} decay.\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{879}&\multicolumn{1}{@{.}l}{6 {\it 10}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{ }l}{{\it 20}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{l}{E2+M1}&\multicolumn{1}{r@{}}{7}&\multicolumn{1}{@{.}l}{6 {\it 13}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{00885 {\it 18}}&\parbox[t][0.3cm]{8.549472cm}{\raggedright E\ensuremath{_{\gamma}}: Other: 879.6 keV \textit{5} in \ensuremath{^{\textnormal{200}}}Hg(\ensuremath{\alpha},3n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.549472cm}{\raggedright Mult.,\ensuremath{\delta}: From \ensuremath{\alpha}(K)exp=0.007 \textit{3} in \ensuremath{^{\textnormal{201}}}Bi \ensuremath{\varepsilon} decay.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{910}&\multicolumn{1}{@{.}l}{5?}&\multicolumn{1}{l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{372}&\multicolumn{1}{@{.}l}{3\ensuremath{^{\hyperlink{PB29GAMMA7}{c}}} {\it 10}}&\multicolumn{1}{r@{}}{5}&\multicolumn{1}{@{.}l}{4 {\it 11}}&\multicolumn{1}{r@{}}{538}&\multicolumn{1}{@{.}l}{7 }&\multicolumn{1}{@{}l}{(3/2\ensuremath{^{-}})}&&&&&&&\\
&&&\multicolumn{1}{r@{}}{822}&\multicolumn{1}{@{.}l}{6\ensuremath{^{\hyperlink{PB29GAMMA7}{c}}} {\it 10}}&\multicolumn{1}{r@{}}{43}&\multicolumn{1}{@{ }l}{{\it 9}}&\multicolumn{1}{r@{}}{88}&\multicolumn{1}{@{.}l}{5 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1(+E2)}&\multicolumn{1}{r@{}}{$<$1}&\multicolumn{1}{@{.}l}{7}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{022 {\it 7}}&\parbox[t][0.3cm]{8.549472cm}{\raggedright Mult.,\ensuremath{\delta}: From \ensuremath{\alpha}(K)exp=0.019 \textit{8} in \ensuremath{^{\textnormal{201}}}Bi \ensuremath{\varepsilon} decay.\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{911}&\multicolumn{1}{@{.}l}{0\ensuremath{^{\hyperlink{PB29GAMMA7}{c}}} {\it 10}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{ }l}{{\it 20}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1(+E2)}&\multicolumn{1}{r@{}}{$<$0}&\multicolumn{1}{@{.}l}{5}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0211 {\it 15}}&\parbox[t][0.3cm]{8.549472cm}{\raggedright Mult.,\ensuremath{\delta}: From \ensuremath{\alpha}(K)exp=0.028 \textit{7} in \ensuremath{^{\textnormal{201}}}Bi \ensuremath{\varepsilon} decay.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{936}&\multicolumn{1}{@{.}l}{1}&\multicolumn{1}{l}{7/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{847}&\multicolumn{1}{@{.}l}{7 {\it 10}}&\multicolumn{1}{r@{}}{16}&\multicolumn{1}{@{ }l}{{\it 3}}&\multicolumn{1}{r@{}}{88}&\multicolumn{1}{@{.}l}{5 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{-}}}&\multicolumn{1}{l}{E2}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{00924 {\it 13}}&\parbox[t][0.3cm]{8.549472cm}{\raggedright Mult.: From \ensuremath{\alpha}(K)exp=0.009 \textit{5} in \ensuremath{^{\textnormal{201}}}Bi \ensuremath{\varepsilon} decay.\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{936}&\multicolumn{1}{@{.}l}{2 {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{ }l}{{\it 5}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1,E2}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{014 {\it 7}}&\parbox[t][0.3cm]{8.549472cm}{\raggedright Mult.: From \ensuremath{\alpha}(K)exp=0.004 \textit{3} in \ensuremath{^{\textnormal{201}}}Bi \ensuremath{\varepsilon} decay.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{990}&\multicolumn{1}{@{.}l}{5}&\multicolumn{1}{l}{7/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{902}&\multicolumn{1}{@{.}l}{0 {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{ }l}{{\it 5}}&\multicolumn{1}{r@{}}{88}&\multicolumn{1}{@{.}l}{5 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{-}}}&\multicolumn{1}{l}{E2}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{00816 {\it 11}}&\parbox[t][0.3cm]{8.549472cm}{\raggedright Mult.: From \ensuremath{\alpha}(K)exp=0.006 \textit{2} in \ensuremath{^{\textnormal{201}}}Bi \ensuremath{\varepsilon} decay.\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{990}&\multicolumn{1}{@{.}l}{6 {\it 5}}&\multicolumn{1}{r@{}}{38}&\multicolumn{1}{@{.}l}{8 {\it 20}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{l}{E2+M1}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{2 {\it 6}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0087 {\it 13}}&\parbox[t][0.3cm]{8.549472cm}{\raggedright Mult.,\ensuremath{\delta}: From \ensuremath{\alpha}(K)exp=0.007 \textit{2} in \ensuremath{^{\textnormal{201}}}Bi \ensuremath{\varepsilon} decay.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1014}&\multicolumn{1}{@{.}l}{2}&\multicolumn{1}{l}{9/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{1014}&\multicolumn{1}{@{.}l}{1 {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{l}{E2}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{00648 {\it 9}}&\parbox[t][0.3cm]{8.549472cm}{\raggedright Mult.: From \ensuremath{\alpha}(K)exp=0.005 \textit{2} in \ensuremath{^{\textnormal{201}}}Bi \ensuremath{\varepsilon} decay.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1185}&\multicolumn{1}{@{.}l}{8}&\multicolumn{1}{l}{(7/2)\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{171}&\multicolumn{1}{@{.}l}{7 {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{ }l}{{\it 5}}&\multicolumn{1}{r@{}}{1014}&\multicolumn{1}{@{.}l}{2 }&\multicolumn{1}{@{}l}{9/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1(+E2)}&\multicolumn{1}{r@{}}{$<$0}&\multicolumn{1}{@{.}l}{5}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{88 {\it 13}}&\parbox[t][0.3cm]{8.549472cm}{\raggedright Mult.,\ensuremath{\delta}: From \ensuremath{\alpha}(L)exp=0.27 \textit{3}, L12/L3\ensuremath{>}66\hphantom{a}in \ensuremath{^{\textnormal{201}}}Bi \ensuremath{\varepsilon} decay.\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{305}&\multicolumn{1}{@{.}l}{7\ensuremath{^{\hyperlink{PB29GAMMA7}{c}}} {\it 10}}&\multicolumn{1}{r@{}}{4}&\multicolumn{1}{@{.}l}{1 {\it 9}}&\multicolumn{1}{r@{}}{879}&\multicolumn{1}{@{.}l}{6 }&\multicolumn{1}{@{}l}{(5/2)\ensuremath{^{-}}}&&&&&&&\\
&&&\multicolumn{1}{r@{}}{1186}&\multicolumn{1}{@{.}l}{5 {\it 10}}&\multicolumn{1}{r@{}}{8}&\multicolumn{1}{@{.}l}{4 {\it 16}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
\multicolumn{1}{r@{}}{1325}&\multicolumn{1}{@{.}l}{4}&\multicolumn{1}{l}{7/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{1325}&\multicolumn{1}{@{.}l}{2 {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{6 {\it 4}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0074 {\it 11}}&\parbox[t][0.3cm]{8.549472cm}{\raggedright Mult.,\ensuremath{\delta}: From \ensuremath{\alpha}(K)exp=0.006 \textit{1} in \ensuremath{^{\textnormal{201}}}Bi \ensuremath{\varepsilon} decay.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1415}&\multicolumn{1}{@{.}l}{4}&\multicolumn{1}{l}{9/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{424}&\multicolumn{1}{@{.}l}{5 {\it 10}}&\multicolumn{1}{r@{}}{9}&\multicolumn{1}{@{.}l}{3 {\it 18}}&\multicolumn{1}{r@{}}{990}&\multicolumn{1}{@{.}l}{5 }&\multicolumn{1}{@{}l}{7/2\ensuremath{^{-}}}&\multicolumn{1}{l}{(E1)}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{01354 {\it 20}}&\parbox[t][0.3cm]{8.549472cm}{\raggedright Mult.: From \ensuremath{\alpha}(K)exp=0.050 \textit{20} in \ensuremath{^{\textnormal{201}}}Bi \ensuremath{\varepsilon} decay.\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{786}&\multicolumn{1}{@{.}l}{4 {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{ }l}{{\it 5}}&\multicolumn{1}{r@{}}{629}&\multicolumn{1}{@{.}l}{1 }&\multicolumn{1}{@{}l}{13/2\ensuremath{^{+}}}&\multicolumn{1}{l}{E2}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{01077 {\it 15}}&\parbox[t][0.3cm]{8.549472cm}{\raggedright Mult.: From \ensuremath{\alpha}(K)exp=0.0095 \textit{8} in \ensuremath{^{\textnormal{201}}}Bi \ensuremath{\varepsilon} decay.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1447}&\multicolumn{1}{@{.}l}{9}&\multicolumn{1}{l}{(11/2)\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{818}&\multicolumn{1}{@{.}l}{9 {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{629}&\multicolumn{1}{@{.}l}{1 }&\multicolumn{1}{@{}l}{13/2\ensuremath{^{+}}}&\multicolumn{1}{l}{E2+M1}&\multicolumn{1}{r@{}}{7}&\multicolumn{1}{@{.}l}{8 {\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{01023 {\it 17}}&\parbox[t][0.3cm]{8.549472cm}{\raggedright Mult.,\ensuremath{\delta}: From \ensuremath{\alpha}(K)exp=0.0080 \textit{20} in \ensuremath{^{\textnormal{201}}}Bi \ensuremath{\varepsilon} decay;\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.549472cm}{\raggedright {\ }{\ }{\ }A\ensuremath{_{\textnormal{2}}}={\textminus}0.21 \textit{5}, A\ensuremath{_{\textnormal{4}}}=0.01 \textit{7} using \ensuremath{\gamma}(\ensuremath{\theta}) in \ensuremath{^{\textnormal{200}}}Hg(\ensuremath{\alpha},3n\ensuremath{\gamma}) are\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.549472cm}{\raggedright {\ }{\ }{\ }consistent with Mult=D.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1490}&\multicolumn{1}{@{.}l}{3}&\multicolumn{1}{l}{7/2\ensuremath{^{-}},9/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{499}&\multicolumn{1}{@{.}l}{9 {\it 10}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{ }l}{{\it 20}}&\multicolumn{1}{r@{}}{990}&\multicolumn{1}{@{.}l}{5 }&\multicolumn{1}{@{}l}{7/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1(+E2)}&\multicolumn{1}{r@{}}{$<$1}&\multicolumn{1}{@{.}l}{2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{085 {\it 23}}&\parbox[t][0.3cm]{8.549472cm}{\raggedright Mult.: From \ensuremath{\alpha}(K)exp=0.08 \textit{3} in \ensuremath{^{\textnormal{201}}}Bi \ensuremath{\varepsilon} decay.\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{610}&\multicolumn{1}{@{.}l}{4\ensuremath{^{\hyperlink{PB29GAMMA7}{c}}} {\it 10}}&\multicolumn{1}{r@{}}{30}&\multicolumn{1}{@{ }l}{{\it 6}}&\multicolumn{1}{r@{}}{879}&\multicolumn{1}{@{.}l}{6 }&\multicolumn{1}{@{}l}{(5/2)\ensuremath{^{-}}}&&&&&&&\\
&&&\multicolumn{1}{r@{}}{1490}&\multicolumn{1}{@{.}l}{1 {\it 10}}&\multicolumn{1}{r@{}}{39}&\multicolumn{1}{@{ }l}{{\it 7}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
\multicolumn{1}{r@{}}{1541}&\multicolumn{1}{@{.}l}{8}&\multicolumn{1}{l}{17/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{912}&\multicolumn{1}{@{.}l}{7\ensuremath{^{\hyperlink{PB29GAMMA2}{\#}}} {\it 2}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA2}{\#}}}}&\multicolumn{1}{r@{}}{629}&\multicolumn{1}{@{.}l}{1 }&\multicolumn{1}{@{}l}{13/2\ensuremath{^{+}}}&\multicolumn{1}{l}{E2}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{00797 {\it 11}}&\parbox[t][0.3cm]{8.549472cm}{\raggedright Mult.: From A\ensuremath{_{\textnormal{2}}}=0.32 \textit{5}, A\ensuremath{_{\textnormal{4}}}={\textminus}0.01 \textit{6} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1988Ro08,B}{1988Ro08}) and A\ensuremath{_{\textnormal{2}}}=0.21\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.549472cm}{\raggedright {\ }{\ }{\ }\textit{3}, A\ensuremath{_{\textnormal{4}}}={\textminus}0.05 \textit{4} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1981He07,B}{1981He07}) using \ensuremath{\gamma}(\ensuremath{\theta}) in \ensuremath{^{\textnormal{200}}}Hg(\ensuremath{\alpha},3n\ensuremath{\gamma});\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.549472cm}{\raggedright {\ }{\ }{\ }R(DCO)=1.08 \textit{8} in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1545}&\multicolumn{1}{@{.}l}{8}&\multicolumn{1}{l}{15/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{916}&\multicolumn{1}{@{.}l}{7\ensuremath{^{\hyperlink{PB29GAMMA2}{\#}}} {\it 2}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA2}{\#}}}}&\multicolumn{1}{r@{}}{629}&\multicolumn{1}{@{.}l}{1 }&\multicolumn{1}{@{}l}{13/2\ensuremath{^{+}}}&\multicolumn{1}{l}{M1+E2}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{015 {\it 7}}&\parbox[t][0.3cm]{8.549472cm}{\raggedright Mult.: From A\ensuremath{_{\textnormal{2}}}={\textminus}0.51 \textit{4}, A\ensuremath{_{\textnormal{4}}}=0.15 \textit{6} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1988Ro08,B}{1988Ro08}) and\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.549472cm}{\raggedright {\ }{\ }{\ }A\ensuremath{_{\textnormal{2}}}={\textminus}0.28 \textit{3}, A\ensuremath{_{\textnormal{4}}}=0.04 \textit{4} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1981He07,B}{1981He07}) using \ensuremath{\gamma}(\ensuremath{\theta}) in\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.549472cm}{\raggedright {\ }{\ }{\ }\ensuremath{^{\textnormal{200}}}Hg(\ensuremath{\alpha},3n\ensuremath{\gamma}); R(DCO)=0.81 \textit{11} in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1651}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{l}{7/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{325}&\multicolumn{1}{@{.}l}{7\ensuremath{^{\hyperlink{PB29GAMMA7}{c}}} {\it 10}}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{9 {\it 6}}&\multicolumn{1}{r@{}}{1325}&\multicolumn{1}{@{.}l}{4 }&\multicolumn{1}{@{}l}{7/2\ensuremath{^{-}}}&&&&&&&\\
\end{longtable}
\clearpage
\begin{longtable}{ccccccccc@{}ccccccc@{\extracolsep{\fill}}c}
\\[-.4cm]
\multicolumn{17}{c}{{\bf \small \underline{Adopted \hyperlink{201PB_LEVEL}{Levels}, \hyperlink{201PB_GAMMA}{Gammas} (continued)}}}\\
\multicolumn{17}{c}{~}\\
\multicolumn{17}{c}{\underline{$\gamma$($^{201}$Pb) (continued)}}\\
\multicolumn{17}{c}{~~~}\\
\multicolumn{2}{c}{E\ensuremath{_{i}}(level)}&J\ensuremath{^{\pi}_{i}}&\multicolumn{2}{c}{E\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{PB29GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{I\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{PB29GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{E\ensuremath{_{f}}}&J\ensuremath{^{\pi}_{f}}&Mult.\ensuremath{^{\hyperlink{PB29GAMMA4}{\&}}}&\multicolumn{2}{c}{\ensuremath{\delta}\ensuremath{^{\hyperlink{PB29GAMMA5}{a}}}}&\multicolumn{2}{c}{\ensuremath{\alpha}\ensuremath{^{\hyperlink{PB29GAMMA6}{b}}}}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endhead
\multicolumn{1}{r@{}}{1651}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{l}{7/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{740}&\multicolumn{1}{@{.}l}{7\ensuremath{^{\hyperlink{PB29GAMMA7}{c}}} {\it 10}}&\multicolumn{1}{r@{}}{9}&\multicolumn{1}{@{.}l}{1 {\it 16}}&\multicolumn{1}{r@{}}{910}&\multicolumn{1}{@{.}l}{5? }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{l}{E1}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{00436 {\it 6}}&\parbox[t][0.3cm]{7.293291cm}{\raggedright Mult.: From \ensuremath{\alpha}(K)exp=0.0030 \textit{20} in \ensuremath{^{\textnormal{201}}}Bi \ensuremath{\varepsilon} decay.\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{1650}&\multicolumn{1}{@{.}l}{9 {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{ }l}{{\it 5}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{l}{E1}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{33\ensuremath{\times10^{-3}} {\it 2}}&\parbox[t][0.3cm]{7.293291cm}{\raggedright Mult.: From \ensuremath{\alpha}(K)exp=0.0006 \textit{4} in \ensuremath{^{\textnormal{201}}}Bi \ensuremath{\varepsilon} decay.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1737}&\multicolumn{1}{@{.}l}{3}&\multicolumn{1}{l}{9/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{288}&\multicolumn{1}{@{.}l}{6 {\it 10}}&\multicolumn{1}{r@{}}{7}&\multicolumn{1}{@{.}l}{4 {\it 15}}&\multicolumn{1}{r@{}}{1447}&\multicolumn{1}{@{.}l}{9 }&\multicolumn{1}{@{}l}{(11/2)\ensuremath{^{+}}}&\multicolumn{1}{l}{M1(+E2)}&\multicolumn{1}{r@{}}{$<$0}&\multicolumn{1}{@{.}l}{24}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{465 {\it 12}}&\parbox[t][0.3cm]{7.293291cm}{\raggedright Mult.,\ensuremath{\delta}: From \ensuremath{\alpha}(K)exp=0.45 \textit{5} in \ensuremath{^{\textnormal{201}}}Bi \ensuremath{\varepsilon} decay.\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{411}&\multicolumn{1}{@{.}l}{6 {\it 10}}&\multicolumn{1}{r@{}}{13}&\multicolumn{1}{@{.}l}{4 {\it 24}}&\multicolumn{1}{r@{}}{1325}&\multicolumn{1}{@{.}l}{4 }&\multicolumn{1}{@{}l}{7/2\ensuremath{^{-}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
&&&\multicolumn{1}{r@{}}{723}&\multicolumn{1}{@{.}l}{5 {\it 10}}&\multicolumn{1}{r@{}}{6}&\multicolumn{1}{@{.}l}{3 {\it 13}}&\multicolumn{1}{r@{}}{1014}&\multicolumn{1}{@{.}l}{2 }&\multicolumn{1}{@{}l}{9/2\ensuremath{^{-}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
&&&\multicolumn{1}{r@{}}{746}&\multicolumn{1}{@{.}l}{8 {\it 10}}&\multicolumn{1}{r@{}}{29}&\multicolumn{1}{@{ }l}{{\it 6}}&\multicolumn{1}{r@{}}{990}&\multicolumn{1}{@{.}l}{5 }&\multicolumn{1}{@{}l}{7/2\ensuremath{^{-}}}&\multicolumn{1}{l}{E1}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{00430 {\it 6}}&\parbox[t][0.3cm]{7.293291cm}{\raggedright Mult.: From \ensuremath{\alpha}(K)exp=0.0050 \textit{10} in \ensuremath{^{\textnormal{201}}}Bi \ensuremath{\varepsilon} decay.\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{1108}&\multicolumn{1}{@{.}l}{1 {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{ }l}{{\it 5}}&\multicolumn{1}{r@{}}{629}&\multicolumn{1}{@{.}l}{1 }&\multicolumn{1}{@{}l}{13/2\ensuremath{^{+}}}&\multicolumn{1}{l}{E2}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{00546 {\it 8}}&\parbox[t][0.3cm]{7.293291cm}{\raggedright Mult.: From \ensuremath{\alpha}(K)exp=0.005 \textit{1} in \ensuremath{^{\textnormal{201}}}Bi \ensuremath{\varepsilon} decay.\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{1737}&\multicolumn{1}{@{.}l}{6 {\it 10}}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{3 {\it 7}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
\multicolumn{1}{r@{}}{1843}&\multicolumn{1}{@{.}l}{8}&\multicolumn{1}{l}{11/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{396}&\multicolumn{1}{@{.}l}{1 {\it 10}}&\multicolumn{1}{r@{}}{11}&\multicolumn{1}{@{.}l}{1 {\it 22}}&\multicolumn{1}{r@{}}{1447}&\multicolumn{1}{@{.}l}{9 }&\multicolumn{1}{@{}l}{(11/2)\ensuremath{^{+}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
&&&\multicolumn{1}{r@{}}{1214}&\multicolumn{1}{@{.}l}{5 {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{ }l}{{\it 5}}&\multicolumn{1}{r@{}}{629}&\multicolumn{1}{@{.}l}{1 }&\multicolumn{1}{@{}l}{13/2\ensuremath{^{+}}}&\multicolumn{1}{l}{M1(+E2)}&\multicolumn{1}{r@{}}{$<$2}&\multicolumn{1}{@{.}l}{3}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0082 {\it 26}}&\parbox[t][0.3cm]{7.293291cm}{\raggedright Mult.,\ensuremath{\delta}: From \ensuremath{\alpha}(K)exp=0.006 \textit{3} in \ensuremath{^{\textnormal{201}}}Bi \ensuremath{\varepsilon} decay.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1875}&\multicolumn{1}{@{.}l}{6}&\multicolumn{1}{l}{9/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{224}&\multicolumn{1}{@{.}l}{5 {\it 10}}&\multicolumn{1}{r@{}}{55}&\multicolumn{1}{@{ }l}{{\it 10}}&\multicolumn{1}{r@{}}{1651}&\multicolumn{1}{@{.}l}{0 }&\multicolumn{1}{@{}l}{7/2\ensuremath{^{+}}}&\multicolumn{1}{l}{M1(+E2)}&\multicolumn{1}{r@{}}{$<$0}&\multicolumn{1}{@{.}l}{44}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{90 {\it 6}}&\parbox[t][0.3cm]{7.293291cm}{\raggedright Mult.,\ensuremath{\delta}: From \ensuremath{\alpha}(K)exp=0.75 \textit{8} in \ensuremath{^{\textnormal{201}}}Bi \ensuremath{\varepsilon} decay.\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{384}&\multicolumn{1}{@{.}l}{4 {\it 10}}&\multicolumn{1}{r@{}}{18}&\multicolumn{1}{@{ }l}{{\it 4}}&\multicolumn{1}{r@{}}{1490}&\multicolumn{1}{@{.}l}{3 }&\multicolumn{1}{@{}l}{7/2\ensuremath{^{-}},9/2\ensuremath{^{-}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
&&&\multicolumn{1}{r@{}}{428}&\multicolumn{1}{@{.}l}{0 {\it 10}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{ }l}{{\it 17}}&\multicolumn{1}{r@{}}{1447}&\multicolumn{1}{@{.}l}{9 }&\multicolumn{1}{@{}l}{(11/2)\ensuremath{^{+}}}&\multicolumn{1}{l}{M1(+E2)}&\multicolumn{1}{r@{}}{$<$0}&\multicolumn{1}{@{.}l}{9}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{136 {\it 27}}&\parbox[t][0.3cm]{7.293291cm}{\raggedright Mult.,\ensuremath{\delta}: From \ensuremath{\alpha}(K)exp=0.12 \textit{3} in \ensuremath{^{\textnormal{201}}}Bi \ensuremath{\varepsilon} decay.\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{460}&\multicolumn{1}{@{.}l}{1 {\it 10}}&\multicolumn{1}{r@{}}{55}&\multicolumn{1}{@{ }l}{{\it 10}}&\multicolumn{1}{r@{}}{1415}&\multicolumn{1}{@{.}l}{4 }&\multicolumn{1}{@{}l}{9/2\ensuremath{^{+}}}&\multicolumn{1}{l}{M1(+E2)}&\multicolumn{1}{r@{}}{$<$0}&\multicolumn{1}{@{.}l}{7}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{118 {\it 16}}&\parbox[t][0.3cm]{7.293291cm}{\raggedright Mult.,\ensuremath{\delta}: From \ensuremath{\alpha}(K)exp=0.16 \textit{6} in \ensuremath{^{\textnormal{201}}}Bi \ensuremath{\varepsilon} decay.\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{885}&\multicolumn{1}{@{.}l}{0 {\it 10}}&\multicolumn{1}{r@{}}{48}&\multicolumn{1}{@{ }l}{{\it 10}}&\multicolumn{1}{r@{}}{990}&\multicolumn{1}{@{.}l}{5 }&\multicolumn{1}{@{}l}{7/2\ensuremath{^{-}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
\multicolumn{1}{r@{}}{1896}&\multicolumn{1}{@{.}l}{1}&\multicolumn{1}{l}{19/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{350}&\multicolumn{1}{@{.}l}{3\ensuremath{^{\hyperlink{PB29GAMMA2}{\#}}} {\it 2}}&\multicolumn{1}{r@{}}{54}&\multicolumn{1}{@{.}l}{4\ensuremath{^{\hyperlink{PB29GAMMA2}{\#}}} {\it 22}}&\multicolumn{1}{r@{}}{1545}&\multicolumn{1}{@{.}l}{8 }&\multicolumn{1}{@{}l}{15/2\ensuremath{^{+}}}&\multicolumn{1}{l}{E2}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0739 {\it 10}}&\parbox[t][0.3cm]{7.293291cm}{\raggedright B(E2)(W.u.)=0.141 \textit{+33{\textminus}23}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.293291cm}{\raggedright I\ensuremath{_{\gamma}}: From \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}); I\ensuremath{\gamma}=68 \textit{6} in\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.293291cm}{\raggedright {\ }{\ }{\ }\ensuremath{^{\textnormal{200}}}Hg(\ensuremath{\alpha},3n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.293291cm}{\raggedright Mult.: From A\ensuremath{_{\textnormal{2}}}=0.25 \textit{5}, A\ensuremath{_{\textnormal{4}}}={\textminus}0.05 \textit{6} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1988Ro08,B}{1988Ro08})\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.293291cm}{\raggedright {\ }{\ }{\ }using \ensuremath{\gamma}(\ensuremath{\theta}) in \ensuremath{^{\textnormal{200}}}Hg(\ensuremath{\alpha},3n\ensuremath{\gamma}). Note, that A\ensuremath{_{\textnormal{2}}}=0.09\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.293291cm}{\raggedright {\ }{\ }{\ }\textit{2}, A\ensuremath{_{\textnormal{4}}}={\textminus}0.07 \textit{4} in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1981He07,B}{1981He07} (\ensuremath{^{\textnormal{200}}}Hg(\ensuremath{\alpha},3n\ensuremath{\gamma}))\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.293291cm}{\raggedright {\ }{\ }{\ }would imply Mult=M1+E2; R(DCO)=0.95 \textit{13} in\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.293291cm}{\raggedright {\ }{\ }{\ }\ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{354}&\multicolumn{1}{@{.}l}{3\ensuremath{^{\hyperlink{PB29GAMMA2}{\#}}} {\it 2}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA2}{\#}}} {\it 3}}&\multicolumn{1}{r@{}}{1541}&\multicolumn{1}{@{.}l}{8 }&\multicolumn{1}{@{}l}{17/2\ensuremath{^{+}}}&\multicolumn{1}{l}{M1}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{271 {\it 4}}&\parbox[t][0.3cm]{7.293291cm}{\raggedright B(M1)(W.u.)=8.3\ensuremath{\times}10\ensuremath{^{\textnormal{$-$5}}} \textit{+19{\textminus}13}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.293291cm}{\raggedright I\ensuremath{_{\gamma}}: From \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.293291cm}{\raggedright Mult.: From A\ensuremath{_{\textnormal{2}}}=0.70 \textit{5}, A\ensuremath{_{\textnormal{4}}}=0.04 \textit{6} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1988Ro08,B}{1988Ro08}) and\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.293291cm}{\raggedright {\ }{\ }{\ }A\ensuremath{_{\textnormal{2}}}=0.19 \textit{3}, A\ensuremath{_{\textnormal{4}}}=0.01 \textit{4} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1981He07,B}{1981He07}) using \ensuremath{\gamma}(\ensuremath{\theta}) in\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.293291cm}{\raggedright {\ }{\ }{\ }\ensuremath{^{\textnormal{200}}}Hg(\ensuremath{\alpha},3n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1902}&\multicolumn{1}{@{.}l}{2}&\multicolumn{1}{l}{21/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{360}&\multicolumn{1}{@{.}l}{4\ensuremath{^{\hyperlink{PB29GAMMA2}{\#}}} {\it 3}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA2}{\#}}}}&\multicolumn{1}{r@{}}{1541}&\multicolumn{1}{@{.}l}{8 }&\multicolumn{1}{@{}l}{17/2\ensuremath{^{+}}}&\multicolumn{1}{l}{E2}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0683 {\it 10}}&\parbox[t][0.3cm]{7.293291cm}{\raggedright Mult.: From A\ensuremath{_{\textnormal{2}}}=0.25 \textit{3}, A\ensuremath{_{\textnormal{4}}}={\textminus}0.08 \textit{4} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1981He07,B}{1981He07})\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.293291cm}{\raggedright {\ }{\ }{\ }using \ensuremath{\gamma}(\ensuremath{\theta}) in \ensuremath{^{\textnormal{200}}}Hg(\ensuremath{\alpha},3n\ensuremath{\gamma}); R(DCO)=0.96 \textit{9} in\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.293291cm}{\raggedright {\ }{\ }{\ }\ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1977}&\multicolumn{1}{@{.}l}{6}&\multicolumn{1}{l}{7/2\ensuremath{^{+}},9/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{529}&\multicolumn{1}{@{.}l}{8 {\it 10}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{ }l}{{\it 20}}&\multicolumn{1}{r@{}}{1447}&\multicolumn{1}{@{.}l}{9 }&\multicolumn{1}{@{}l}{(11/2)\ensuremath{^{+}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
&&&\multicolumn{1}{r@{}}{562}&\multicolumn{1}{@{.}l}{5 {\it 10}}&\multicolumn{1}{r@{}}{43}&\multicolumn{1}{@{ }l}{{\it 9}}&\multicolumn{1}{r@{}}{1415}&\multicolumn{1}{@{.}l}{4 }&\multicolumn{1}{@{}l}{9/2\ensuremath{^{+}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
&&&\multicolumn{1}{r@{}}{651}&\multicolumn{1}{@{.}l}{8 {\it 10}}&\multicolumn{1}{r@{}}{20}&\multicolumn{1}{@{ }l}{{\it 4}}&\multicolumn{1}{r@{}}{1325}&\multicolumn{1}{@{.}l}{4 }&\multicolumn{1}{@{}l}{7/2\ensuremath{^{-}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
&&&\multicolumn{1}{r@{}}{986}&\multicolumn{1}{@{.}l}{5 {\it 10}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{ }l}{{\it 20}}&\multicolumn{1}{r@{}}{990}&\multicolumn{1}{@{.}l}{5 }&\multicolumn{1}{@{}l}{7/2\ensuremath{^{-}}}&\multicolumn{1}{l}{E1}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{00256 {\it 4}}&\parbox[t][0.3cm]{7.293291cm}{\raggedright Mult.: From \ensuremath{\alpha}(K)exp=0.005 \textit{3} in \ensuremath{^{\textnormal{201}}}Bi \ensuremath{\varepsilon} decay.\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{1042}&\multicolumn{1}{@{.}l}{8 {\it 10}}&\multicolumn{1}{r@{}}{27}&\multicolumn{1}{@{ }l}{{\it 5}}&\multicolumn{1}{r@{}}{936}&\multicolumn{1}{@{.}l}{1 }&\multicolumn{1}{@{}l}{7/2\ensuremath{^{-}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
\multicolumn{1}{r@{}}{2068}&\multicolumn{1}{@{.}l}{5}&\multicolumn{1}{l}{(21/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{166}&\multicolumn{1}{@{.}l}{2\ensuremath{^{\hyperlink{PB29GAMMA2}{\#}}} {\it 2}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA2}{\#}}}}&\multicolumn{1}{r@{}}{1902}&\multicolumn{1}{@{.}l}{2 }&\multicolumn{1}{@{}l}{21/2\ensuremath{^{+}}}&\multicolumn{1}{l}{M1,E2}&&&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{5 {\it 7}}&\parbox[t][0.3cm]{7.293291cm}{\raggedright Mult.: From A\ensuremath{_{\textnormal{2}}}=0.36 \textit{5}, A\ensuremath{_{\textnormal{4}}}=0.01 \textit{5} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1988Ro08,B}{1988Ro08})\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.293291cm}{\raggedright {\ }{\ }{\ }using \ensuremath{\gamma}(\ensuremath{\theta}) in \ensuremath{^{\textnormal{200}}}Hg(\ensuremath{\alpha},3n\ensuremath{\gamma}); R(DCO)=1.08 \textit{9} in\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.293291cm}{\raggedright {\ }{\ }{\ }\ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}); consistent with \ensuremath{\Delta}J=0 transition.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2119}&\multicolumn{1}{@{.}l}{5}&\multicolumn{1}{l}{9/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{142}&\multicolumn{1}{@{.}l}{6 {\it 10}}&\multicolumn{1}{r@{}}{5}&\multicolumn{1}{@{.}l}{2 {\it 12}}&\multicolumn{1}{r@{}}{1977}&\multicolumn{1}{@{.}l}{6 }&\multicolumn{1}{@{}l}{7/2\ensuremath{^{+}},9/2\ensuremath{^{+}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
&&&\multicolumn{1}{r@{}}{243}&\multicolumn{1}{@{.}l}{1 {\it 10}}&\multicolumn{1}{r@{}}{10}&\multicolumn{1}{@{.}l}{3 {\it 21}}&\multicolumn{1}{r@{}}{1875}&\multicolumn{1}{@{.}l}{6 }&\multicolumn{1}{@{}l}{9/2\ensuremath{^{+}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
&&&\multicolumn{1}{r@{}}{275}&\multicolumn{1}{@{.}l}{5 {\it 10}}&\multicolumn{1}{r@{}}{67}&\multicolumn{1}{@{ }l}{{\it 15}}&\multicolumn{1}{r@{}}{1843}&\multicolumn{1}{@{.}l}{8 }&\multicolumn{1}{@{}l}{11/2\ensuremath{^{+}}}&\multicolumn{1}{l}{M1(+E2)}&\multicolumn{1}{r@{}}{$<$0}&\multicolumn{1}{@{.}l}{25}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{527 {\it 15}}&\parbox[t][0.3cm]{7.293291cm}{\raggedright Mult.,\ensuremath{\delta}: From \ensuremath{\alpha}(K)exp=0.52 \textit{9} in \ensuremath{^{\textnormal{201}}}Bi \ensuremath{\varepsilon} decay.\vspace{0.1cm}}&\\
\end{longtable}
\clearpage
\begin{longtable}{ccccccccc@{}ccccccc@{\extracolsep{\fill}}c}
\\[-.4cm]
\multicolumn{17}{c}{{\bf \small \underline{Adopted \hyperlink{201PB_LEVEL}{Levels}, \hyperlink{201PB_GAMMA}{Gammas} (continued)}}}\\
\multicolumn{17}{c}{~}\\
\multicolumn{17}{c}{\underline{$\gamma$($^{201}$Pb) (continued)}}\\
\multicolumn{17}{c}{~~~}\\
\multicolumn{2}{c}{E\ensuremath{_{i}}(level)}&J\ensuremath{^{\pi}_{i}}&\multicolumn{2}{c}{E\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{PB29GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{I\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{PB29GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{E\ensuremath{_{f}}}&J\ensuremath{^{\pi}_{f}}&Mult.\ensuremath{^{\hyperlink{PB29GAMMA4}{\&}}}&\multicolumn{2}{c}{\ensuremath{\delta}\ensuremath{^{\hyperlink{PB29GAMMA5}{a}}}}&\multicolumn{2}{c}{\ensuremath{\alpha}\ensuremath{^{\hyperlink{PB29GAMMA6}{b}}}}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endhead
\multicolumn{1}{r@{}}{2119}&\multicolumn{1}{@{.}l}{5}&\multicolumn{1}{l}{9/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{671}&\multicolumn{1}{@{.}l}{9 {\it 10}}&\multicolumn{1}{r@{}}{13}&\multicolumn{1}{@{ }l}{{\it 3}}&\multicolumn{1}{r@{}}{1447}&\multicolumn{1}{@{.}l}{9 }&\multicolumn{1}{@{}l}{(11/2)\ensuremath{^{+}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
&&&\multicolumn{1}{r@{}}{703}&\multicolumn{1}{@{.}l}{9 {\it 10}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{ }l}{{\it 21}}&\multicolumn{1}{r@{}}{1415}&\multicolumn{1}{@{.}l}{4 }&\multicolumn{1}{@{}l}{9/2\ensuremath{^{+}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
&&&\multicolumn{1}{r@{}}{1183}&\multicolumn{1}{@{.}l}{7 {\it 10}}&\multicolumn{1}{r@{}}{31}&\multicolumn{1}{@{ }l}{{\it 6}}&\multicolumn{1}{r@{}}{936}&\multicolumn{1}{@{.}l}{1 }&\multicolumn{1}{@{}l}{7/2\ensuremath{^{-}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
\multicolumn{1}{r@{}}{2151}&\multicolumn{1}{@{.}l}{9}&\multicolumn{1}{l}{7/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{414}&\multicolumn{1}{@{.}l}{6 {\it 10}}&\multicolumn{1}{r@{}}{92}&\multicolumn{1}{@{ }l}{{\it 18}}&\multicolumn{1}{r@{}}{1737}&\multicolumn{1}{@{.}l}{3 }&\multicolumn{1}{@{}l}{9/2\ensuremath{^{+}}}&\multicolumn{1}{l}{M1(+E2)}&\multicolumn{1}{r@{}}{$<$0}&\multicolumn{1}{@{.}l}{6}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{160 {\it 17}}&\parbox[t][0.3cm]{6.8695707cm}{\raggedright Mult.,\ensuremath{\delta}: From \ensuremath{\alpha}(K)exp=0.14 \textit{3} in \ensuremath{^{\textnormal{201}}}Bi \ensuremath{\varepsilon} decay.\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{661}&\multicolumn{1}{@{.}l}{5\ensuremath{^{\hyperlink{PB29GAMMA7}{c}}} {\it 10}}&\multicolumn{1}{r@{}}{63}&\multicolumn{1}{@{ }l}{{\it 13}}&\multicolumn{1}{r@{}}{1490}&\multicolumn{1}{@{.}l}{3 }&\multicolumn{1}{@{}l}{7/2\ensuremath{^{-}},9/2\ensuremath{^{-}}}&&&&&&&\\
&&&\multicolumn{1}{r@{}}{736}&\multicolumn{1}{@{.}l}{4 {\it 10}}&\multicolumn{1}{r@{}}{88}&\multicolumn{1}{@{ }l}{{\it 18}}&\multicolumn{1}{r@{}}{1415}&\multicolumn{1}{@{.}l}{4 }&\multicolumn{1}{@{}l}{9/2\ensuremath{^{+}}}&\multicolumn{1}{l}{M1}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0391 {\it 6}}&\parbox[t][0.3cm]{6.8695707cm}{\raggedright Mult.: From \ensuremath{\alpha}(K)exp=0.048 \textit{10} in \ensuremath{^{\textnormal{201}}}Bi \ensuremath{\varepsilon} decay.\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{1137}&\multicolumn{1}{@{.}l}{7\ensuremath{^{\hyperlink{PB29GAMMA7}{c}}} {\it 10}}&\multicolumn{1}{r@{}}{11}&\multicolumn{1}{@{.}l}{2 {\it 22}}&\multicolumn{1}{r@{}}{1014}&\multicolumn{1}{@{.}l}{2 }&\multicolumn{1}{@{}l}{9/2\ensuremath{^{-}}}&&&&&&&\\
&&&\multicolumn{1}{r@{}}{1161}&\multicolumn{1}{@{.}l}{2 {\it 10}}&\multicolumn{1}{r@{}}{22}&\multicolumn{1}{@{ }l}{{\it 4}}&\multicolumn{1}{r@{}}{990}&\multicolumn{1}{@{.}l}{5 }&\multicolumn{1}{@{}l}{7/2\ensuremath{^{-}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
&&&\multicolumn{1}{r@{}}{1241}&\multicolumn{1}{@{.}l}{4\ensuremath{^{\hyperlink{PB29GAMMA7}{c}}} {\it 10}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{ }l}{{\it 20}}&\multicolumn{1}{r@{}}{910}&\multicolumn{1}{@{.}l}{5? }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{l}{(E1)}&&&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{74\ensuremath{\times10^{-3}} {\it 2}}&\parbox[t][0.3cm]{6.8695707cm}{\raggedright Mult.: From \ensuremath{\alpha}(K)exp=0.004 \textit{3} in \ensuremath{^{\textnormal{201}}}Bi \ensuremath{\varepsilon} decay.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2208}&\multicolumn{1}{@{.}l}{9}&\multicolumn{1}{l}{(9/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{557}&\multicolumn{1}{@{.}l}{7 {\it 10}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{ }l}{{\it 22}}&\multicolumn{1}{r@{}}{1651}&\multicolumn{1}{@{.}l}{0 }&\multicolumn{1}{@{}l}{7/2\ensuremath{^{+}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
&&&\multicolumn{1}{r@{}}{1298}&\multicolumn{1}{@{.}l}{4 {\it 10}}&\multicolumn{1}{r@{}}{78}&\multicolumn{1}{@{ }l}{{\it 17}}&\multicolumn{1}{r@{}}{910}&\multicolumn{1}{@{.}l}{5? }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
&&&\multicolumn{1}{r@{}}{1579}&\multicolumn{1}{@{.}l}{8 {\it 10}}&\multicolumn{1}{r@{}}{19}&\multicolumn{1}{@{ }l}{{\it 4}}&\multicolumn{1}{r@{}}{629}&\multicolumn{1}{@{.}l}{1 }&\multicolumn{1}{@{}l}{13/2\ensuremath{^{+}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
\multicolumn{1}{r@{}}{2279}&\multicolumn{1}{@{.}l}{8}&\multicolumn{1}{l}{(9/2)\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{832}&\multicolumn{1}{@{.}l}{0 {\it 10}}&\multicolumn{1}{r@{}}{82}&\multicolumn{1}{@{ }l}{{\it 18}}&\multicolumn{1}{r@{}}{1447}&\multicolumn{1}{@{.}l}{9 }&\multicolumn{1}{@{}l}{(11/2)\ensuremath{^{+}}}&\multicolumn{1}{l}{M1(+E2)}&\multicolumn{1}{r@{}}{$<$1}&\multicolumn{1}{@{.}l}{3}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{023 {\it 6}}&\parbox[t][0.3cm]{6.8695707cm}{\raggedright Mult.: From \ensuremath{\alpha}(K)exp=0.028 \textit{14} in \ensuremath{^{\textnormal{201}}}Bi \ensuremath{\varepsilon} decay.\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{1265}&\multicolumn{1}{@{.}l}{6 {\it 10}}&\multicolumn{1}{r@{}}{47}&\multicolumn{1}{@{ }l}{{\it 10}}&\multicolumn{1}{r@{}}{1014}&\multicolumn{1}{@{.}l}{2 }&\multicolumn{1}{@{}l}{9/2\ensuremath{^{-}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
&&&\multicolumn{1}{r@{}}{1400}&\multicolumn{1}{@{.}l}{3\ensuremath{^{\hyperlink{PB29GAMMA7}{c}}} {\it 10}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{ }l}{{\it 18}}&\multicolumn{1}{r@{}}{879}&\multicolumn{1}{@{.}l}{6 }&\multicolumn{1}{@{}l}{(5/2)\ensuremath{^{-}}}&&&&&&&\\
\multicolumn{1}{r@{}}{2439}&\multicolumn{1}{@{.}l}{4}&\multicolumn{1}{l}{7/2\ensuremath{^{+}},9/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1024}&\multicolumn{1}{@{.}l}{0 {\it 10}}&\multicolumn{1}{r@{}}{42}&\multicolumn{1}{@{ }l}{{\it 9}}&\multicolumn{1}{r@{}}{1415}&\multicolumn{1}{@{.}l}{4 }&\multicolumn{1}{@{}l}{9/2\ensuremath{^{+}}}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{9 {\it 11}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0075 {\it 13}}&\parbox[t][0.3cm]{6.8695707cm}{\raggedright Mult.,\ensuremath{\delta}: From \ensuremath{\alpha}(K)exp=0.006 \textit{3} in \ensuremath{^{\textnormal{201}}}Bi \ensuremath{\varepsilon}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.8695707cm}{\raggedright {\ }{\ }{\ }decay.\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{1253}&\multicolumn{1}{@{.}l}{8 {\it 10}}&\multicolumn{1}{r@{}}{72}&\multicolumn{1}{@{ }l}{{\it 14}}&\multicolumn{1}{r@{}}{1185}&\multicolumn{1}{@{.}l}{8 }&\multicolumn{1}{@{}l}{(7/2)\ensuremath{^{-}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
&&&\multicolumn{1}{r@{}}{1503}&\multicolumn{1}{@{.}l}{0 {\it 10}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{ }l}{{\it 21}}&\multicolumn{1}{r@{}}{936}&\multicolumn{1}{@{.}l}{1 }&\multicolumn{1}{@{}l}{7/2\ensuremath{^{-}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
\multicolumn{1}{r@{}}{2459}&\multicolumn{1}{@{.}l}{9}&\multicolumn{1}{l}{7/2\ensuremath{^{+}},9/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{250}&\multicolumn{1}{@{.}l}{9 {\it 10}}&\multicolumn{1}{r@{}}{13}&\multicolumn{1}{@{ }l}{{\it 3}}&\multicolumn{1}{r@{}}{2208}&\multicolumn{1}{@{.}l}{9 }&\multicolumn{1}{@{}l}{(9/2\ensuremath{^{+}})}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{6 {\it 4}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{57 {\it 12}}&\parbox[t][0.3cm]{6.8695707cm}{\raggedright Mult.: From \ensuremath{\alpha}(K)exp=0.46 \textit{12} in \ensuremath{^{\textnormal{201}}}Bi \ensuremath{\varepsilon} decay.\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{584}&\multicolumn{1}{@{.}l}{3 {\it 10}}&\multicolumn{1}{r@{}}{10}&\multicolumn{1}{@{.}l}{9 {\it 21}}&\multicolumn{1}{r@{}}{1875}&\multicolumn{1}{@{.}l}{6 }&\multicolumn{1}{@{}l}{9/2\ensuremath{^{+}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
&&&\multicolumn{1}{r@{}}{969}&\multicolumn{1}{@{.}l}{9 {\it 10}}&\multicolumn{1}{r@{}}{19}&\multicolumn{1}{@{ }l}{{\it 4}}&\multicolumn{1}{r@{}}{1490}&\multicolumn{1}{@{.}l}{3 }&\multicolumn{1}{@{}l}{7/2\ensuremath{^{-}},9/2\ensuremath{^{-}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
&&&\multicolumn{1}{r@{}}{1469}&\multicolumn{1}{@{.}l}{5 {\it 10}}&\multicolumn{1}{r@{}}{26}&\multicolumn{1}{@{ }l}{{\it 6}}&\multicolumn{1}{r@{}}{990}&\multicolumn{1}{@{.}l}{5 }&\multicolumn{1}{@{}l}{7/2\ensuremath{^{-}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
&&&\multicolumn{1}{r@{}}{1523}&\multicolumn{1}{@{.}l}{3 {\it 10}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{ }l}{{\it 20}}&\multicolumn{1}{r@{}}{936}&\multicolumn{1}{@{.}l}{1 }&\multicolumn{1}{@{}l}{7/2\ensuremath{^{-}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
&&&\multicolumn{1}{r@{}}{1547}&\multicolumn{1}{@{.}l}{6\ensuremath{^{\hyperlink{PB29GAMMA7}{c}}} {\it 10}}&\multicolumn{1}{r@{}}{22}&\multicolumn{1}{@{ }l}{{\it 4}}&\multicolumn{1}{r@{}}{910}&\multicolumn{1}{@{.}l}{5? }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{-}}}&&&&&&&\\
\multicolumn{1}{r@{}}{2474}&\multicolumn{1}{@{.}l}{6}&\multicolumn{1}{l}{(7/2\ensuremath{^{+}},9/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{322}&\multicolumn{1}{@{.}l}{3 {\it 10}}&\multicolumn{1}{r@{}}{6}&\multicolumn{1}{@{.}l}{8 {\it 14}}&\multicolumn{1}{r@{}}{2151}&\multicolumn{1}{@{.}l}{9 }&\multicolumn{1}{@{}l}{7/2\ensuremath{^{+}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
&&&\multicolumn{1}{r@{}}{1059}&\multicolumn{1}{@{.}l}{7 {\it 10}}&\multicolumn{1}{r@{}}{18}&\multicolumn{1}{@{ }l}{{\it 4}}&\multicolumn{1}{r@{}}{1415}&\multicolumn{1}{@{.}l}{4 }&\multicolumn{1}{@{}l}{9/2\ensuremath{^{+}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
&&&\multicolumn{1}{r@{}}{1288}&\multicolumn{1}{@{.}l}{9 {\it 10}}&\multicolumn{1}{r@{}}{69}&\multicolumn{1}{@{ }l}{{\it 14}}&\multicolumn{1}{r@{}}{1185}&\multicolumn{1}{@{.}l}{8 }&\multicolumn{1}{@{}l}{(7/2)\ensuremath{^{-}}}&\multicolumn{1}{l}{(E1)}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{65\ensuremath{\times10^{-3}} {\it 2}}&\parbox[t][0.3cm]{6.8695707cm}{\raggedright Mult.: From \ensuremath{\alpha}(K)exp=0.003 \textit{1} in \ensuremath{^{\textnormal{201}}}Bi \ensuremath{\varepsilon} decay.\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{1538}&\multicolumn{1}{@{.}l}{4 {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{ }l}{{\it 5}}&\multicolumn{1}{r@{}}{936}&\multicolumn{1}{@{.}l}{1 }&\multicolumn{1}{@{}l}{7/2\ensuremath{^{-}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
\multicolumn{1}{r@{}}{2496}&\multicolumn{1}{@{.}l}{3}&\multicolumn{1}{l}{21/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{594}&\multicolumn{1}{@{.}l}{1\ensuremath{^{\hyperlink{PB29GAMMA2}{\#}}} {\it 3}}&\multicolumn{1}{r@{}}{4}&\multicolumn{1}{@{.}l}{2\ensuremath{^{\hyperlink{PB29GAMMA2}{\#}}} {\it 4}}&\multicolumn{1}{r@{}}{1902}&\multicolumn{1}{@{.}l}{2 }&\multicolumn{1}{@{}l}{21/2\ensuremath{^{+}}}&\multicolumn{1}{l}{(E1)}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{00672 {\it 9}}&\parbox[t][0.3cm]{6.8695707cm}{\raggedright I\ensuremath{_{\gamma}}: From \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.8695707cm}{\raggedright Mult.: From A\ensuremath{_{\textnormal{2}}}={\textminus}0.06 \textit{5}, A\ensuremath{_{\textnormal{4}}}=0.08 \textit{6} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1988Ro08,B}{1988Ro08})\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.8695707cm}{\raggedright {\ }{\ }{\ }using \ensuremath{\gamma}(\ensuremath{\theta}) in \ensuremath{^{\textnormal{200}}}Hg(\ensuremath{\alpha},3n\ensuremath{\gamma}); R(DCO)=0.73 \textit{21}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.8695707cm}{\raggedright {\ }{\ }{\ }in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{600}&\multicolumn{1}{@{.}l}{2\ensuremath{^{\hyperlink{PB29GAMMA2}{\#}}} {\it 1}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{.}l}{0\ensuremath{^{\hyperlink{PB29GAMMA2}{\#}}} {\it 20}}&\multicolumn{1}{r@{}}{1896}&\multicolumn{1}{@{.}l}{1 }&\multicolumn{1}{@{}l}{19/2\ensuremath{^{+}}}&\multicolumn{1}{l}{(E1)}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{00658 {\it 9}}&\parbox[t][0.3cm]{6.8695707cm}{\raggedright I\ensuremath{_{\gamma}}: From \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.8695707cm}{\raggedright Mult.: From A\ensuremath{_{\textnormal{2}}}={\textminus}0.21 \textit{4}, A\ensuremath{_{\textnormal{4}}}={\textminus}0.02 \textit{6} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1988Ro08,B}{1988Ro08})\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.8695707cm}{\raggedright {\ }{\ }{\ }and A\ensuremath{_{\textnormal{2}}}={\textminus}0.10 \textit{3}, A\ensuremath{_{\textnormal{4}}}={\textminus}0.03 \textit{4} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1981He07,B}{1981He07}) using\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.8695707cm}{\raggedright {\ }{\ }{\ }\ensuremath{\gamma}(\ensuremath{\theta}) in \ensuremath{^{\textnormal{200}}}Hg(\ensuremath{\alpha},3n\ensuremath{\gamma}); R(DCO)=0.80 \textit{8} in\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.8695707cm}{\raggedright {\ }{\ }{\ }\ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2506}&\multicolumn{1}{@{.}l}{8}&\multicolumn{1}{l}{9/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{387}&\multicolumn{1}{@{.}l}{3 {\it 10}}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{4 {\it 7}}&\multicolumn{1}{r@{}}{2119}&\multicolumn{1}{@{.}l}{5 }&\multicolumn{1}{@{}l}{9/2\ensuremath{^{+}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
&&&\multicolumn{1}{r@{}}{855}&\multicolumn{1}{@{.}l}{8 {\it 10}}&\multicolumn{1}{r@{}}{57}&\multicolumn{1}{@{ }l}{{\it 12}}&\multicolumn{1}{r@{}}{1651}&\multicolumn{1}{@{.}l}{0 }&\multicolumn{1}{@{}l}{7/2\ensuremath{^{+}}}&\multicolumn{1}{l}{M1(+E2)}&\multicolumn{1}{r@{}}{$<$2}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{020 {\it 7}}&\parbox[t][0.3cm]{6.8695707cm}{\raggedright Mult.: From \ensuremath{\alpha}(K)exp=0.020 \textit{10} in \ensuremath{^{\textnormal{201}}}Bi \ensuremath{\varepsilon} decay.\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{1091}&\multicolumn{1}{@{.}l}{5 {\it 10}}&\multicolumn{1}{r@{}}{26}&\multicolumn{1}{@{ }l}{{\it 6}}&\multicolumn{1}{r@{}}{1415}&\multicolumn{1}{@{.}l}{4 }&\multicolumn{1}{@{}l}{9/2\ensuremath{^{+}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
&&&\multicolumn{1}{r@{}}{1320}&\multicolumn{1}{@{.}l}{9 {\it 10}}&\multicolumn{1}{r@{}}{31}&\multicolumn{1}{@{ }l}{{\it 6}}&\multicolumn{1}{r@{}}{1185}&\multicolumn{1}{@{.}l}{8 }&\multicolumn{1}{@{}l}{(7/2)\ensuremath{^{-}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
\end{longtable}
\clearpage
\begin{longtable}{ccccccccc@{}ccccccc@{\extracolsep{\fill}}c}
\\[-.4cm]
\multicolumn{17}{c}{{\bf \small \underline{Adopted \hyperlink{201PB_LEVEL}{Levels}, \hyperlink{201PB_GAMMA}{Gammas} (continued)}}}\\
\multicolumn{17}{c}{~}\\
\multicolumn{17}{c}{\underline{$\gamma$($^{201}$Pb) (continued)}}\\
\multicolumn{17}{c}{~~~}\\
\multicolumn{2}{c}{E\ensuremath{_{i}}(level)}&J\ensuremath{^{\pi}_{i}}&\multicolumn{2}{c}{E\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{PB29GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{I\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{PB29GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{E\ensuremath{_{f}}}&J\ensuremath{^{\pi}_{f}}&Mult.\ensuremath{^{\hyperlink{PB29GAMMA4}{\&}}}&\multicolumn{2}{c}{\ensuremath{\delta}\ensuremath{^{\hyperlink{PB29GAMMA5}{a}}}}&\multicolumn{2}{c}{\ensuremath{\alpha}\ensuremath{^{\hyperlink{PB29GAMMA6}{b}}}}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endhead
\multicolumn{1}{r@{}}{2506}&\multicolumn{1}{@{.}l}{8}&\multicolumn{1}{l}{9/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1516}&\multicolumn{1}{@{.}l}{4 {\it 10}}&\multicolumn{1}{r@{}}{38}&\multicolumn{1}{@{ }l}{{\it 8}}&\multicolumn{1}{r@{}}{990}&\multicolumn{1}{@{.}l}{5 }&\multicolumn{1}{@{}l}{7/2\ensuremath{^{-}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
&&&\multicolumn{1}{r@{}}{1570}&\multicolumn{1}{@{.}l}{8 {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{ }l}{{\it 5}}&\multicolumn{1}{r@{}}{936}&\multicolumn{1}{@{.}l}{1 }&\multicolumn{1}{@{}l}{7/2\ensuremath{^{-}}}&\multicolumn{1}{l}{E1}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{36\ensuremath{\times10^{-3}} {\it 2}}&\parbox[t][0.3cm]{6.381651cm}{\raggedright Mult.: From \ensuremath{\alpha}(K)exp=0.0009 \textit{5} in \ensuremath{^{\textnormal{201}}}Bi \ensuremath{\varepsilon}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.381651cm}{\raggedright {\ }{\ }{\ }decay.\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{1877}&\multicolumn{1}{@{.}l}{4 {\it 10}}&\multicolumn{1}{r@{}}{6}&\multicolumn{1}{@{.}l}{0 {\it 12}}&\multicolumn{1}{r@{}}{629}&\multicolumn{1}{@{.}l}{1 }&\multicolumn{1}{@{}l}{13/2\ensuremath{^{+}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
\multicolumn{1}{r@{}}{2548}&\multicolumn{1}{@{.}l}{9}&\multicolumn{1}{l}{(11/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{339}&\multicolumn{1}{@{.}l}{7 {\it 10}}&\multicolumn{1}{r@{}}{10}&\multicolumn{1}{@{.}l}{7 {\it 21}}&\multicolumn{1}{r@{}}{2208}&\multicolumn{1}{@{.}l}{9 }&\multicolumn{1}{@{}l}{(9/2\ensuremath{^{+}})}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
&&&\multicolumn{1}{r@{}}{1558}&\multicolumn{1}{@{.}l}{6 {\it 10}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{ }l}{{\it 20}}&\multicolumn{1}{r@{}}{990}&\multicolumn{1}{@{.}l}{5 }&\multicolumn{1}{@{}l}{7/2\ensuremath{^{-}}}&\multicolumn{1}{l}{(E2)}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{00296 {\it 4}}&\parbox[t][0.3cm]{6.381651cm}{\raggedright Mult.: From \ensuremath{\alpha}(K)exp=0.0020 \textit{10} in \ensuremath{^{\textnormal{201}}}Bi \ensuremath{\varepsilon}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.381651cm}{\raggedright {\ }{\ }{\ }decay.\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{1638}&\multicolumn{1}{@{.}l}{7\ensuremath{^{\hyperlink{PB29GAMMA7}{c}}} {\it 10}}&\multicolumn{1}{r@{}}{13}&\multicolumn{1}{@{.}l}{4 {\it 21}}&\multicolumn{1}{r@{}}{910}&\multicolumn{1}{@{.}l}{5? }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{-}}}&&&&&&&\\
&&&\multicolumn{1}{r@{}}{1919}&\multicolumn{1}{@{.}l}{4 {\it 10}}&\multicolumn{1}{r@{}}{12}&\multicolumn{1}{@{.}l}{5 {\it 25}}&\multicolumn{1}{r@{}}{629}&\multicolumn{1}{@{.}l}{1 }&\multicolumn{1}{@{}l}{13/2\ensuremath{^{+}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
\multicolumn{1}{r@{}}{2604}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{l}{21/2,23/2}&\multicolumn{1}{r@{}}{707}&\multicolumn{1}{@{.}l}{9\ensuremath{^{\hyperlink{PB29GAMMA2}{\#}}} {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA2}{\#}}}}&\multicolumn{1}{r@{}}{1896}&\multicolumn{1}{@{.}l}{1 }&\multicolumn{1}{@{}l}{19/2\ensuremath{^{+}}}&\multicolumn{1}{l}{D}&&&&&\parbox[t][0.3cm]{6.381651cm}{\raggedright Mult.: From A\ensuremath{_{\textnormal{2}}}={\textminus}0.10 \textit{4}, A\ensuremath{_{\textnormal{4}}}={\textminus}0.02 \textit{6}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.381651cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1988Ro08,B}{1988Ro08}) using \ensuremath{\gamma}(\ensuremath{\theta}) in \ensuremath{^{\textnormal{200}}}Hg(\ensuremath{\alpha},3n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2718}&\multicolumn{1}{@{.}l}{5}&\multicolumn{1}{l}{25/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{222}&\multicolumn{1}{@{.}l}{2\ensuremath{^{\hyperlink{PB29GAMMA2}{\#}}} {\it 1}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA2}{\#}}}}&\multicolumn{1}{r@{}}{2496}&\multicolumn{1}{@{.}l}{3 }&\multicolumn{1}{@{}l}{21/2\ensuremath{^{-}}}&\multicolumn{1}{l}{E2}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{299 {\it 4}}&\parbox[t][0.3cm]{6.381651cm}{\raggedright B(E2)(W.u.)=0.181 \textit{9}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.381651cm}{\raggedright Mult.: From A\ensuremath{_{\textnormal{2}}}=0.22 \textit{5}, A\ensuremath{_{\textnormal{4}}}=0.02 \textit{6}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.381651cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1988Ro08,B}{1988Ro08}) using \ensuremath{\gamma}(\ensuremath{\theta}) and \ensuremath{\alpha}(exp)=0.34 \textit{3}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.381651cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1981He07,B}{1981He07}) in \ensuremath{^{\textnormal{200}}}Hg(\ensuremath{\alpha},3n\ensuremath{\gamma});\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.381651cm}{\raggedright {\ }{\ }{\ }R(DCO)=0.93 \textit{10} in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2732}&\multicolumn{1}{@{.}l}{8}&&\multicolumn{1}{r@{}}{664}&\multicolumn{1}{@{.}l}{1\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{48}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 4}}&\multicolumn{1}{r@{}}{2068}&\multicolumn{1}{@{.}l}{5 }&\multicolumn{1}{@{}l}{(21/2\ensuremath{^{+}})}&\multicolumn{1}{l}{D,E2}&&&&&\parbox[t][0.3cm]{6.381651cm}{\raggedright Mult.: From R(DCO)=1.08 \textit{25} in\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.381651cm}{\raggedright {\ }{\ }{\ }\ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{830}&\multicolumn{1}{@{.}l}{7\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 11}}&\multicolumn{1}{r@{}}{1902}&\multicolumn{1}{@{.}l}{2 }&\multicolumn{1}{@{}l}{21/2\ensuremath{^{+}}}&\multicolumn{1}{l}{D,E2}&&&&&\parbox[t][0.3cm]{6.381651cm}{\raggedright Mult.: From R(DCO)=1.13 \textit{16} in\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.381651cm}{\raggedright {\ }{\ }{\ }\ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2736}&\multicolumn{1}{@{.}l}{2}&&\multicolumn{1}{r@{}}{667}&\multicolumn{1}{@{.}l}{6\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{74}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{2068}&\multicolumn{1}{@{.}l}{5 }&\multicolumn{1}{@{}l}{(21/2\ensuremath{^{+}})}&\multicolumn{1}{l}{D,E2}&&&&&\parbox[t][0.3cm]{6.381651cm}{\raggedright Mult.: From R(DCO)= 1.03 \textit{16} in\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.381651cm}{\raggedright {\ }{\ }{\ }\ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{834}&\multicolumn{1}{@{.}l}{0\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 8}}&\multicolumn{1}{r@{}}{1902}&\multicolumn{1}{@{.}l}{2 }&\multicolumn{1}{@{}l}{21/2\ensuremath{^{+}}}&\multicolumn{1}{l}{D,E2}&&&&&\parbox[t][0.3cm]{6.381651cm}{\raggedright Mult.: From R(DCO)= 0.94 \textit{11} in\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.381651cm}{\raggedright {\ }{\ }{\ }\ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2788}&\multicolumn{1}{@{.}l}{8}&\multicolumn{1}{l}{11/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{239}&\multicolumn{1}{@{.}l}{7 {\it 10}}&\multicolumn{1}{r@{}}{14}&\multicolumn{1}{@{ }l}{{\it 3}}&\multicolumn{1}{r@{}}{2548}&\multicolumn{1}{@{.}l}{9 }&\multicolumn{1}{@{}l}{(11/2\ensuremath{^{-}})}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{47 {\it 7}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{690 {\it 28}}&\parbox[t][0.3cm]{6.381651cm}{\raggedright Mult.,\ensuremath{\delta}: From \ensuremath{\alpha}(K)exp=0.55 \textit{2} in \ensuremath{^{\textnormal{201}}}Bi \ensuremath{\varepsilon}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.381651cm}{\raggedright {\ }{\ }{\ }decay.\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{1051}&\multicolumn{1}{@{.}l}{6 {\it 10}}&\multicolumn{1}{r@{}}{47}&\multicolumn{1}{@{ }l}{{\it 10}}&\multicolumn{1}{r@{}}{1737}&\multicolumn{1}{@{.}l}{3 }&\multicolumn{1}{@{}l}{9/2\ensuremath{^{+}}}&\multicolumn{1}{l}{E1}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{29\ensuremath{\times10^{-3}} {\it 3}}&\parbox[t][0.3cm]{6.381651cm}{\raggedright Mult.: From \ensuremath{\alpha}(K)exp=0.0011 \textit{10} in \ensuremath{^{\textnormal{201}}}Bi \ensuremath{\varepsilon}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.381651cm}{\raggedright {\ }{\ }{\ }decay.\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{1603}&\multicolumn{1}{@{.}l}{8 {\it 10}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{ }l}{{\it 18}}&\multicolumn{1}{r@{}}{1185}&\multicolumn{1}{@{.}l}{8 }&\multicolumn{1}{@{}l}{(7/2)\ensuremath{^{-}}}&\multicolumn{1}{l}{(E2)}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{00283 {\it 4}}&\parbox[t][0.3cm]{6.381651cm}{\raggedright Mult.: From \ensuremath{\alpha}(K)exp=0.0040 \textit{10} in \ensuremath{^{\textnormal{201}}}Bi \ensuremath{\varepsilon}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.381651cm}{\raggedright {\ }{\ }{\ }decay.\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{1851}&\multicolumn{1}{@{.}l}{9 {\it 10}}&\multicolumn{1}{r@{}}{5}&\multicolumn{1}{@{.}l}{3 {\it 10}}&\multicolumn{1}{r@{}}{936}&\multicolumn{1}{@{.}l}{1 }&\multicolumn{1}{@{}l}{7/2\ensuremath{^{-}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
&&&\multicolumn{1}{r@{}}{2159}&\multicolumn{1}{@{.}l}{7 {\it 10}}&\multicolumn{1}{r@{}}{67}&\multicolumn{1}{@{ }l}{{\it 14}}&\multicolumn{1}{r@{}}{629}&\multicolumn{1}{@{.}l}{1 }&\multicolumn{1}{@{}l}{13/2\ensuremath{^{+}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
\multicolumn{1}{r@{}}{2794}&\multicolumn{1}{@{.}l}{2}&\multicolumn{1}{l}{(19/2,21/2,23/2)\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{297}&\multicolumn{1}{@{.}l}{9\ensuremath{^{\hyperlink{PB29GAMMA2}{\#}}} {\it 3}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA2}{\#}}}}&\multicolumn{1}{r@{}}{2496}&\multicolumn{1}{@{.}l}{3 }&\multicolumn{1}{@{}l}{21/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1+E2}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{28 {\it 16}}&\parbox[t][0.3cm]{6.381651cm}{\raggedright Mult.: From A\ensuremath{_{\textnormal{2}}}={\textminus}0.99 \textit{13}, A\ensuremath{_{\textnormal{4}}}=0.13 \textit{5}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.381651cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1981He07,B}{1981He07}) using \ensuremath{\gamma}(\ensuremath{\theta}) in \ensuremath{^{\textnormal{200}}}Hg(\ensuremath{\alpha},3n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2961}&\multicolumn{1}{@{.}l}{8}&\multicolumn{1}{l}{(7/2,9/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{1472}&\multicolumn{1}{@{.}l}{1 {\it 10}}&\multicolumn{1}{r@{}}{71}&\multicolumn{1}{@{ }l}{{\it 14}}&\multicolumn{1}{r@{}}{1490}&\multicolumn{1}{@{.}l}{3 }&\multicolumn{1}{@{}l}{7/2\ensuremath{^{-}},9/2\ensuremath{^{-}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
&&&\multicolumn{1}{r@{}}{1775}&\multicolumn{1}{@{.}l}{7 {\it 10}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{ }l}{{\it 19}}&\multicolumn{1}{r@{}}{1185}&\multicolumn{1}{@{.}l}{8 }&\multicolumn{1}{@{}l}{(7/2)\ensuremath{^{-}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
&&&\multicolumn{1}{r@{}}{1971}&\multicolumn{1}{@{.}l}{0 {\it 10}}&\multicolumn{1}{r@{}}{31}&\multicolumn{1}{@{ }l}{{\it 6}}&\multicolumn{1}{r@{}}{990}&\multicolumn{1}{@{.}l}{5 }&\multicolumn{1}{@{}l}{7/2\ensuremath{^{-}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
&&&\multicolumn{1}{r@{}}{2025}&\multicolumn{1}{@{.}l}{6 {\it 10}}&\multicolumn{1}{r@{}}{50}&\multicolumn{1}{@{ }l}{{\it 10}}&\multicolumn{1}{r@{}}{936}&\multicolumn{1}{@{.}l}{1 }&\multicolumn{1}{@{}l}{7/2\ensuremath{^{-}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
&&&\multicolumn{1}{r@{}}{2082}&\multicolumn{1}{@{.}l}{0\ensuremath{^{\hyperlink{PB29GAMMA7}{c}}} {\it 10}}&\multicolumn{1}{r@{}}{14}&\multicolumn{1}{@{ }l}{{\it 3}}&\multicolumn{1}{r@{}}{879}&\multicolumn{1}{@{.}l}{6 }&\multicolumn{1}{@{}l}{(5/2)\ensuremath{^{-}}}&&&&&&&\\
\multicolumn{1}{r@{}}{3050}&\multicolumn{1}{@{.}l}{7}&\multicolumn{1}{l}{(7/2)\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{931}&\multicolumn{1}{@{.}l}{6 {\it 10}}&\multicolumn{1}{r@{}}{44}&\multicolumn{1}{@{ }l}{{\it 9}}&\multicolumn{1}{r@{}}{2119}&\multicolumn{1}{@{.}l}{5 }&\multicolumn{1}{@{}l}{9/2\ensuremath{^{+}}}&\multicolumn{1}{l}{M1(+E2)}&\multicolumn{1}{r@{}}{$<$0}&\multicolumn{1}{@{.}l}{6}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0195 {\it 18}}&\parbox[t][0.3cm]{6.381651cm}{\raggedright Mult.,\ensuremath{\delta}: From \ensuremath{\alpha}(K)exp=0.020 \textit{5} in \ensuremath{^{\textnormal{201}}}Bi \ensuremath{\varepsilon}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.381651cm}{\raggedright {\ }{\ }{\ }decay.\vspace{0.1cm}}&\\
\end{longtable}
\clearpage
\begin{longtable}{ccccccccc@{}ccccccc@{\extracolsep{\fill}}c}
\\[-.4cm]
\multicolumn{17}{c}{{\bf \small \underline{Adopted \hyperlink{201PB_LEVEL}{Levels}, \hyperlink{201PB_GAMMA}{Gammas} (continued)}}}\\
\multicolumn{17}{c}{~}\\
\multicolumn{17}{c}{\underline{$\gamma$($^{201}$Pb) (continued)}}\\
\multicolumn{17}{c}{~~~}\\
\multicolumn{2}{c}{E\ensuremath{_{i}}(level)}&J\ensuremath{^{\pi}_{i}}&\multicolumn{2}{c}{E\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{PB29GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{I\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{PB29GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{E\ensuremath{_{f}}}&J\ensuremath{^{\pi}_{f}}&Mult.\ensuremath{^{\hyperlink{PB29GAMMA4}{\&}}}&\multicolumn{2}{c}{\ensuremath{\delta}\ensuremath{^{\hyperlink{PB29GAMMA5}{a}}}}&\multicolumn{2}{c}{\ensuremath{\alpha}\ensuremath{^{\hyperlink{PB29GAMMA6}{b}}}}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endhead
\multicolumn{1}{r@{}}{3050}&\multicolumn{1}{@{.}l}{7}&\multicolumn{1}{l}{(7/2)\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1313}&\multicolumn{1}{@{.}l}{2 {\it 10}}&\multicolumn{1}{r@{}}{28}&\multicolumn{1}{@{ }l}{{\it 5}}&\multicolumn{1}{r@{}}{1737}&\multicolumn{1}{@{.}l}{3 }&\multicolumn{1}{@{}l}{9/2\ensuremath{^{+}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
&&&\multicolumn{1}{r@{}}{1634}&\multicolumn{1}{@{.}l}{9 {\it 10}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{ }l}{{\it 20}}&\multicolumn{1}{r@{}}{1415}&\multicolumn{1}{@{.}l}{4 }&\multicolumn{1}{@{}l}{9/2\ensuremath{^{+}}}&\multicolumn{1}{l}{M1(+E2)}&\multicolumn{1}{r@{}}{$<$1}&\multicolumn{1}{@{.}l}{2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0045 {\it 7}}&\parbox[t][0.3cm]{8.15143cm}{\raggedright Mult.,\ensuremath{\delta}: From \ensuremath{\alpha}(K)exp=0.0040 \textit{10} in \ensuremath{^{\textnormal{201}}}Bi \ensuremath{\varepsilon} decay.\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{2035}&\multicolumn{1}{@{.}l}{8 {\it 10}}&\multicolumn{1}{r@{}}{4}&\multicolumn{1}{@{.}l}{3 {\it 9}}&\multicolumn{1}{r@{}}{1014}&\multicolumn{1}{@{.}l}{2 }&\multicolumn{1}{@{}l}{9/2\ensuremath{^{-}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
&&&\multicolumn{1}{r@{}}{2060}&\multicolumn{1}{@{.}l}{9 {\it 10}}&\multicolumn{1}{r@{}}{9}&\multicolumn{1}{@{.}l}{4 {\it 19}}&\multicolumn{1}{r@{}}{990}&\multicolumn{1}{@{.}l}{5 }&\multicolumn{1}{@{}l}{7/2\ensuremath{^{-}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
&&&\multicolumn{1}{r@{}}{2114}&\multicolumn{1}{@{.}l}{7 {\it 10}}&\multicolumn{1}{r@{}}{5}&\multicolumn{1}{@{.}l}{6 {\it 11}}&\multicolumn{1}{r@{}}{936}&\multicolumn{1}{@{.}l}{1 }&\multicolumn{1}{@{}l}{7/2\ensuremath{^{-}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
&&&\multicolumn{1}{r@{}}{2170}&\multicolumn{1}{@{.}l}{4\ensuremath{^{\hyperlink{PB29GAMMA7}{c}}} {\it 10}}&\multicolumn{1}{r@{}}{8}&\multicolumn{1}{@{.}l}{9 {\it 18}}&\multicolumn{1}{r@{}}{879}&\multicolumn{1}{@{.}l}{6 }&\multicolumn{1}{@{}l}{(5/2)\ensuremath{^{-}}}&&&&&&&\\
\multicolumn{1}{r@{}}{3509}&\multicolumn{1}{@{.}l}{5+x}&\multicolumn{1}{l}{(31/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{791}&\multicolumn{1}{@{.}l}{0\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{2718}&\multicolumn{1}{@{.}l}{5+x }&\multicolumn{1}{@{}l}{(29/2\ensuremath{^{-}})}&\multicolumn{1}{l}{M1+E2}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{022 {\it 11}}&\parbox[t][0.3cm]{8.15143cm}{\raggedright Mult.: From A\ensuremath{_{\textnormal{2}}}={\textminus}1.06 \textit{3}, A\ensuremath{_{\textnormal{4}}}={\textminus}0.01 \textit{5} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1988Ro08,B}{1988Ro08}) using\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.15143cm}{\raggedright {\ }{\ }{\ }\ensuremath{\gamma}(\ensuremath{\theta}) in \ensuremath{^{\textnormal{200}}}Hg(\ensuremath{\alpha},3n\ensuremath{\gamma}) and R(DCO)=0.61 \textit{4} in\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.15143cm}{\raggedright {\ }{\ }{\ }\ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3545}&\multicolumn{1}{@{.}l}{0+x}&\multicolumn{1}{l}{(33/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{826}&\multicolumn{1}{@{.}l}{6\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{2718}&\multicolumn{1}{@{.}l}{5+x }&\multicolumn{1}{@{}l}{(29/2\ensuremath{^{-}})}&\multicolumn{1}{l}{E2}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{00973 {\it 14}}&\parbox[t][0.3cm]{8.15143cm}{\raggedright Mult.: From A\ensuremath{_{\textnormal{2}}}=0.28 \textit{5}, A\ensuremath{_{\textnormal{4}}}={\textminus}0.03 \textit{6} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1988Ro08,B}{1988Ro08}) using\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.15143cm}{\raggedright {\ }{\ }{\ }\ensuremath{\gamma}(\ensuremath{\theta}) in \ensuremath{^{\textnormal{200}}}Hg(\ensuremath{\alpha},3n\ensuremath{\gamma}) and R(DCO)=1.00 \textit{14} in\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.15143cm}{\raggedright {\ }{\ }{\ }\ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3638}&\multicolumn{1}{@{.}l}{0+x}&\multicolumn{1}{l}{(31/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{919}&\multicolumn{1}{@{.}l}{4\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{2718}&\multicolumn{1}{@{.}l}{5+x }&\multicolumn{1}{@{}l}{(29/2\ensuremath{^{-}})}&\multicolumn{1}{l}{M1+E2}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{015 {\it 7}}&\parbox[t][0.3cm]{8.15143cm}{\raggedright Mult.: From A\ensuremath{_{\textnormal{2}}}={\textminus}0.48 \textit{4}, A\ensuremath{_{\textnormal{4}}}=0.31 \textit{7} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1988Ro08,B}{1988Ro08}) using\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.15143cm}{\raggedright {\ }{\ }{\ }\ensuremath{\gamma}(\ensuremath{\theta}) in \ensuremath{^{\textnormal{200}}}Hg(\ensuremath{\alpha},3n\ensuremath{\gamma}) and R(DCO)=0.87 \textit{12} in\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.15143cm}{\raggedright {\ }{\ }{\ }\ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3832}&\multicolumn{1}{@{.}l}{3+x}&\multicolumn{1}{l}{(35/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{287}&\multicolumn{1}{@{.}l}{2\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{3545}&\multicolumn{1}{@{.}l}{0+x }&\multicolumn{1}{@{}l}{(33/2\ensuremath{^{-}})}&\multicolumn{1}{l}{M1+E2}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{31 {\it 17}}&\parbox[t][0.3cm]{8.15143cm}{\raggedright Mult.: From A\ensuremath{_{\textnormal{2}}}={\textminus}0.51 \textit{5}, A\ensuremath{_{\textnormal{4}}}=0.02 \textit{8} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1988Ro08,B}{1988Ro08}) using\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.15143cm}{\raggedright {\ }{\ }{\ }\ensuremath{\gamma}(\ensuremath{\theta}) in \ensuremath{^{\textnormal{200}}}Hg(\ensuremath{\alpha},3n\ensuremath{\gamma}); R(DCO)=0.76 \textit{5} and K/L\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.15143cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1989Su12,B}{1989Su12}) in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3932}&\multicolumn{1}{@{.}l}{0+x}&\multicolumn{1}{l}{(33/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{293}&\multicolumn{1}{@{.}l}{9\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{39}&\multicolumn{1}{@{.}l}{4\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 25}}&\multicolumn{1}{r@{}}{3638}&\multicolumn{1}{@{.}l}{0+x }&\multicolumn{1}{@{}l}{(31/2\ensuremath{^{-}})}&\multicolumn{1}{l}{D}&&&&&\parbox[t][0.3cm]{8.15143cm}{\raggedright I\ensuremath{_{\gamma}}: Note, that I\ensuremath{\gamma}=76 \textit{8} in \ensuremath{^{\textnormal{200}}}Hg(\ensuremath{\alpha},3n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.15143cm}{\raggedright Mult.: From A\ensuremath{_{\textnormal{2}}}={\textminus}0.28 \textit{4}, A\ensuremath{_{\textnormal{4}}}=0.07 \textit{6} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1988Ro08,B}{1988Ro08}) using\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.15143cm}{\raggedright {\ }{\ }{\ }\ensuremath{\gamma}(\ensuremath{\theta}) in \ensuremath{^{\textnormal{200}}}Hg(\ensuremath{\alpha},3n\ensuremath{\gamma}) and R(DCO)=0.78 \textit{7} in\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.15143cm}{\raggedright {\ }{\ }{\ }\ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{387}&\multicolumn{1}{@{.}l}{0\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{10}&\multicolumn{1}{@{.}l}{2\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 3}}&\multicolumn{1}{r@{}}{3545}&\multicolumn{1}{@{.}l}{0+x }&\multicolumn{1}{@{}l}{(33/2\ensuremath{^{-}})}&\multicolumn{1}{l}{(D)}&&&&&\parbox[t][0.3cm]{8.15143cm}{\raggedright Mult.: From R(DCO)=0.90 \textit{15} in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{422}&\multicolumn{1}{@{.}l}{5\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{3509}&\multicolumn{1}{@{.}l}{5+x }&\multicolumn{1}{@{}l}{(31/2\ensuremath{^{-}})}&\multicolumn{1}{l}{(D)}&&&&&\parbox[t][0.3cm]{8.15143cm}{\raggedright Mult.: From R(DCO)=0.83 \textit{14} in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{4059}&\multicolumn{1}{@{.}l}{5}&&\multicolumn{1}{r@{}}{1341}&\multicolumn{1}{@{.}l}{0\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{2718}&\multicolumn{1}{@{.}l}{5 }&\multicolumn{1}{@{}l}{25/2\ensuremath{^{-}}}&&&&&&\parbox[t][0.3cm]{8.15143cm}{\raggedright Mult.: R(DCO)=0.89 \textit{16}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{4168}&\multicolumn{1}{@{.}l}{7+y}&&\multicolumn{1}{r@{}}{109}&\multicolumn{1}{@{.}l}{2\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{4059}&\multicolumn{1}{@{.}l}{5+y }&&\multicolumn{1}{l}{(M1)}&&&\multicolumn{1}{r@{}}{7}&\multicolumn{1}{@{.}l}{27 {\it 14}}&\parbox[t][0.3cm]{8.15143cm}{\raggedright Mult.: From R(DCO)=0.71 \textit{15} in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{4350}&\multicolumn{1}{@{.}l}{3+y}&&\multicolumn{1}{r@{}}{181}&\multicolumn{1}{@{.}l}{6\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{4168}&\multicolumn{1}{@{.}l}{7+y }&&\multicolumn{1}{l}{(M1)}&&&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{716 {\it 27}}&\parbox[t][0.3cm]{8.15143cm}{\raggedright Mult.: From R(DCO)=0.66 \textit{6} in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{4505}&\multicolumn{1}{@{.}l}{1+x}&\multicolumn{1}{l}{(35/2)}&\multicolumn{1}{r@{}}{573}&\multicolumn{1}{@{.}l}{2\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{3932}&\multicolumn{1}{@{.}l}{0+x }&\multicolumn{1}{@{}l}{(33/2\ensuremath{^{+}})}&\multicolumn{1}{l}{D}&&&&&\parbox[t][0.3cm]{8.15143cm}{\raggedright Mult.: From A\ensuremath{_{\textnormal{2}}}={\textminus}0.87 \textit{5}, A\ensuremath{_{\textnormal{4}}}=0.14 \textit{8} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1988Ro08,B}{1988Ro08}) using\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.15143cm}{\raggedright {\ }{\ }{\ }\ensuremath{\gamma}(\ensuremath{\theta}) in \ensuremath{^{\textnormal{200}}}Hg(\ensuremath{\alpha},3n\ensuremath{\gamma}) and R(DCO)=0.63 \textit{11} in\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.15143cm}{\raggedright {\ }{\ }{\ }\ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{4560}&\multicolumn{1}{@{.}l}{2+x}&\multicolumn{1}{l}{(37/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{628}&\multicolumn{1}{@{.}l}{4\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{43}&\multicolumn{1}{@{.}l}{9\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 23}}&\multicolumn{1}{r@{}}{3932}&\multicolumn{1}{@{.}l}{0+x }&\multicolumn{1}{@{}l}{(33/2\ensuremath{^{+}})}&\multicolumn{1}{l}{E2}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{01736 {\it 24}}&\parbox[t][0.3cm]{8.15143cm}{\raggedright I\ensuremath{_{\gamma}}: Note, that I\ensuremath{\gamma}\ensuremath{\approx}10 in \ensuremath{^{\textnormal{200}}}Hg(\ensuremath{\alpha},3n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.15143cm}{\raggedright Mult.: From R(DCO)=1.00 \textit{9} in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{728}&\multicolumn{1}{@{.}l}{0\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 3}}&\multicolumn{1}{r@{}}{3832}&\multicolumn{1}{@{.}l}{3+x }&\multicolumn{1}{@{}l}{(35/2\ensuremath{^{-}})}&\multicolumn{1}{l}{D}&&&&&\parbox[t][0.3cm]{8.15143cm}{\raggedright Mult.: From A\ensuremath{_{\textnormal{2}}}={\textminus}0.13 \textit{5}, A\ensuremath{_{\textnormal{4}}}={\textminus}0.02 \textit{7} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1988Ro08,B}{1988Ro08}) using\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.15143cm}{\raggedright {\ }{\ }{\ }\ensuremath{\gamma}(\ensuremath{\theta}) in \ensuremath{^{\textnormal{200}}}Hg(\ensuremath{\alpha},3n\ensuremath{\gamma}) and R(DCO)=0.89 \textit{4} in\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.15143cm}{\raggedright {\ }{\ }{\ }\ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{4614}&\multicolumn{1}{@{.}l}{1+y}&&\multicolumn{1}{r@{}}{263}&\multicolumn{1}{@{.}l}{8\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{4350}&\multicolumn{1}{@{.}l}{3+y }&&\multicolumn{1}{l}{(M1)}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{607 {\it 9}}&\parbox[t][0.3cm]{8.15143cm}{\raggedright Mult.: From R(DCO)=0.71 \textit{6} in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{4640}&\multicolumn{1}{@{.}l}{0+x}&\multicolumn{1}{l}{(41/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{80}&\multicolumn{1}{@{.}l}{1\ensuremath{^{\hyperlink{PB29GAMMA3}{@}}} {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA3}{@}}}}&\multicolumn{1}{r@{}}{4560}&\multicolumn{1}{@{.}l}{2+x }&\multicolumn{1}{@{}l}{(37/2\ensuremath{^{+}})}&\multicolumn{1}{l}{E2}&&&\multicolumn{1}{r@{}}{16}&\multicolumn{1}{@{.}l}{5 {\it 6}}&\parbox[t][0.3cm]{8.15143cm}{\raggedright B(E2)(W.u.)=3.05 \textit{+26{\textminus}23}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.15143cm}{\raggedright Mult.: From R(DCO)=1.1 \textit{4} and K/L in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
\end{longtable}
\clearpage
\begin{longtable}{ccccccccc@{}ccccc@{\extracolsep{\fill}}c}
\\[-.4cm]
\multicolumn{15}{c}{{\bf \small \underline{Adopted \hyperlink{201PB_LEVEL}{Levels}, \hyperlink{201PB_GAMMA}{Gammas} (continued)}}}\\
\multicolumn{15}{c}{~}\\
\multicolumn{15}{c}{\underline{$\gamma$($^{201}$Pb) (continued)}}\\
\multicolumn{15}{c}{~~~}\\
\multicolumn{2}{c}{E\ensuremath{_{i}}(level)}&J\ensuremath{^{\pi}_{i}}&\multicolumn{2}{c}{E\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{PB29GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{I\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{PB29GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{E\ensuremath{_{f}}}&J\ensuremath{^{\pi}_{f}}&Mult.\ensuremath{^{\hyperlink{PB29GAMMA4}{\&}}}&\multicolumn{2}{c}{\ensuremath{\alpha}\ensuremath{^{\hyperlink{PB29GAMMA6}{b}}}}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endhead
\multicolumn{1}{r@{}}{4640}&\multicolumn{1}{@{.}l}{0+u}&&\multicolumn{1}{r@{}}{u}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{4640}&\multicolumn{1}{@{.}l}{0+x }&\multicolumn{1}{@{}l}{(41/2\ensuremath{^{+}})}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
\multicolumn{1}{r@{}}{4640}&\multicolumn{1}{@{.}l}{0+v}&&\multicolumn{1}{r@{}}{v}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{4640}&\multicolumn{1}{@{.}l}{0+x }&\multicolumn{1}{@{}l}{(41/2\ensuremath{^{+}})}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
\multicolumn{1}{r@{}}{4640}&\multicolumn{1}{@{.}l}{0+z}&&\multicolumn{1}{r@{}}{z}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{4640}&\multicolumn{1}{@{.}l}{0+x }&\multicolumn{1}{@{}l}{(41/2\ensuremath{^{+}})}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
\multicolumn{1}{r@{}}{4640}&\multicolumn{1}{@{.}l}{1+x}&\multicolumn{1}{l}{(35/2)}&\multicolumn{1}{r@{}}{708}&\multicolumn{1}{@{.}l}{1\ensuremath{^{\hyperlink{PB29GAMMA2}{\#}}} {\it 2}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA2}{\#}}}}&\multicolumn{1}{r@{}}{3932}&\multicolumn{1}{@{.}l}{0+x }&\multicolumn{1}{@{}l}{(33/2\ensuremath{^{+}})}&&&&&\\
\multicolumn{1}{r@{}}{4647}&\multicolumn{1}{@{.}l}{6+x}&\multicolumn{1}{l}{(35/2)}&\multicolumn{1}{r@{}}{142}&\multicolumn{1}{@{.}l}{5\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{20}&\multicolumn{1}{@{.}l}{0\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 18}}&\multicolumn{1}{r@{}}{4505}&\multicolumn{1}{@{.}l}{1+x }&\multicolumn{1}{@{}l}{(35/2)}&\multicolumn{1}{l}{(D)}&&&\parbox[t][0.3cm]{8.367391cm}{\raggedright Mult.: From R(DCO)=1.00 \textit{35} in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}),\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{8.367391cm}{\raggedright {\ }{\ }{\ }consistent with \ensuremath{\Delta}J=0 transition.\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{715}&\multicolumn{1}{@{.}l}{7\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 15}}&\multicolumn{1}{r@{}}{3932}&\multicolumn{1}{@{.}l}{0+x }&\multicolumn{1}{@{}l}{(33/2\ensuremath{^{+}})}&\multicolumn{1}{l}{D}&&&\parbox[t][0.3cm]{8.367391cm}{\raggedright Mult.: From R(DCO)=0.66 \textit{12} in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{4780}&\multicolumn{1}{@{.}l}{5+z}&&\multicolumn{1}{r@{}}{139}&\multicolumn{1}{@{.}l}{6\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{4640}&\multicolumn{1}{@{.}l}{0+z }&&\multicolumn{1}{l}{(M1)}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{61 {\it 6}}&\parbox[t][0.3cm]{8.367391cm}{\raggedright Mult.: From R(DCO)=0.69 \textit{17} in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{4793}&\multicolumn{1}{@{.}l}{8+v}&&\multicolumn{1}{r@{}}{152}&\multicolumn{1}{@{.}l}{9\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{4640}&\multicolumn{1}{@{.}l}{0+v }&&\multicolumn{1}{l}{(M1)}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{79 {\it 5}}&\parbox[t][0.3cm]{8.367391cm}{\raggedright Mult.: From R(DCO)=0.55 \textit{23} in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{4817}&\multicolumn{1}{@{.}l}{4+u}&&\multicolumn{1}{r@{}}{176}&\multicolumn{1}{@{.}l}{5\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{4640}&\multicolumn{1}{@{.}l}{0+u }&&\multicolumn{1}{l}{(M1)}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{860 {\it 30}}&\parbox[t][0.3cm]{8.367391cm}{\raggedright Mult.: From R(DCO)=0.83 \textit{19} in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{4830}&\multicolumn{1}{@{.}l}{2+x}&\multicolumn{1}{l}{(39/2)}&\multicolumn{1}{r@{}}{190}&\multicolumn{1}{@{.}l}{4\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{$<$25}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{4640}&\multicolumn{1}{@{.}l}{0+x }&\multicolumn{1}{@{}l}{(41/2\ensuremath{^{+}})}&\multicolumn{1}{l}{D}&&&\parbox[t][0.3cm]{8.367391cm}{\raggedright Mult.: From R(DCO)=0.81 \textit{18} in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{269}&\multicolumn{1}{@{.}l}{9\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 18}}&\multicolumn{1}{r@{}}{4560}&\multicolumn{1}{@{.}l}{2+x }&\multicolumn{1}{@{}l}{(37/2\ensuremath{^{+}})}&\multicolumn{1}{l}{D}&&&\parbox[t][0.3cm]{8.367391cm}{\raggedright Mult.: From R(DCO)=0.64 \textit{12} in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{4954}&\multicolumn{1}{@{.}l}{9+y}&&\multicolumn{1}{r@{}}{340}&\multicolumn{1}{@{.}l}{8\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{4614}&\multicolumn{1}{@{.}l}{1+y }&&\multicolumn{1}{l}{(M1)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{301 {\it 4}}&\parbox[t][0.3cm]{8.367391cm}{\raggedright Mult.: From R(DCO)=0.69 \textit{5} in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{4956}&\multicolumn{1}{@{.}l}{3+z}&&\multicolumn{1}{r@{}}{175}&\multicolumn{1}{@{.}l}{8\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{4780}&\multicolumn{1}{@{.}l}{5+z }&&\multicolumn{1}{l}{(M1)}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{881 {\it 30}}&\parbox[t][0.3cm]{8.367391cm}{\raggedright Mult.: From R(DCO)=0.68 \textit{19} in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{4992}&\multicolumn{1}{@{.}l}{4+v}&&\multicolumn{1}{r@{}}{198}&\multicolumn{1}{@{.}l}{6\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{4793}&\multicolumn{1}{@{.}l}{8+v }&&\multicolumn{1}{l}{(M1)}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{335 {\it 21}}&\parbox[t][0.3cm]{8.367391cm}{\raggedright Mult.: From R(DCO)=0.63 \textit{11} in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{5000}&\multicolumn{1}{@{.}l}{1+x}&&\multicolumn{1}{r@{}}{360}&\multicolumn{1}{@{.}l}{0\ensuremath{^{\hyperlink{PB29GAMMA2}{\#}}} {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA2}{\#}}}}&\multicolumn{1}{r@{}}{4640}&\multicolumn{1}{@{.}l}{1+x }&\multicolumn{1}{@{}l}{(35/2)}&&&&&\\
\multicolumn{1}{r@{}}{5043}&\multicolumn{1}{@{.}l}{1+u}&&\multicolumn{1}{r@{}}{225}&\multicolumn{1}{@{.}l}{7\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{4817}&\multicolumn{1}{@{.}l}{4+u }&&\multicolumn{1}{l}{(M1)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{935 {\it 14}}&\parbox[t][0.3cm]{8.367391cm}{\raggedright Mult.: From R(DCO)=0.88 \textit{17} in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{5087}&\multicolumn{1}{@{.}l}{1+x}&\multicolumn{1}{l}{(43/2)}&\multicolumn{1}{r@{}}{446}&\multicolumn{1}{@{.}l}{9\ensuremath{^{\hyperlink{PB29GAMMA2}{\#}}} {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA2}{\#}}}}&\multicolumn{1}{r@{}}{4640}&\multicolumn{1}{@{.}l}{0+x }&\multicolumn{1}{@{}l}{(41/2\ensuremath{^{+}})}&\multicolumn{1}{l}{D}&&&\parbox[t][0.3cm]{8.367391cm}{\raggedright Mult.: From R(DCO)=0.76 \textit{5} in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{5172}&\multicolumn{1}{@{.}l}{4+x}&\multicolumn{1}{l}{(35/2,37/2)}&\multicolumn{1}{r@{}}{667}&\multicolumn{1}{@{.}l}{3\ensuremath{^{\hyperlink{PB29GAMMA2}{\#}}} {\it 3}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA2}{\#}}}}&\multicolumn{1}{r@{}}{4505}&\multicolumn{1}{@{.}l}{1+x }&\multicolumn{1}{@{}l}{(35/2)}&\multicolumn{1}{l}{D}&&&\parbox[t][0.3cm]{8.367391cm}{\raggedright Mult.: From A\ensuremath{_{\textnormal{2}}}={\textminus}0.38 \textit{6}, A\ensuremath{_{\textnormal{4}}}={\textminus}0.16 \textit{9} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1981He07,B}{1981He07}) using\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{8.367391cm}{\raggedright {\ }{\ }{\ }\ensuremath{\gamma}(\ensuremath{\theta}) in \ensuremath{^{\textnormal{200}}}Hg(\ensuremath{\alpha},3n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{5178}&\multicolumn{1}{@{.}l}{6+z}&&\multicolumn{1}{r@{}}{222}&\multicolumn{1}{@{.}l}{3\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{4956}&\multicolumn{1}{@{.}l}{3+z }&&\multicolumn{1}{l}{(M1)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{975 {\it 15}}&\parbox[t][0.3cm]{8.367391cm}{\raggedright Mult.: From DCO in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{5242}&\multicolumn{1}{@{.}l}{4+v}&&\multicolumn{1}{r@{}}{250}&\multicolumn{1}{@{.}l}{0\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{4992}&\multicolumn{1}{@{.}l}{4+v }&&\multicolumn{1}{l}{(M1)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{704 {\it 11}}&\parbox[t][0.3cm]{8.367391cm}{\raggedright Mult.: From DCO in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{5321}&\multicolumn{1}{@{.}l}{3+u}&&\multicolumn{1}{r@{}}{278}&\multicolumn{1}{@{.}l}{2\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{5043}&\multicolumn{1}{@{.}l}{1+u }&&\multicolumn{1}{l}{(M1)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{524 {\it 8}}&\parbox[t][0.3cm]{8.367391cm}{\raggedright Mult.: From DCO in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{5358}&\multicolumn{1}{@{.}l}{7+y}&&\multicolumn{1}{r@{}}{404}&\multicolumn{1}{@{.}l}{0\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 10}}&\multicolumn{1}{r@{}}{4954}&\multicolumn{1}{@{.}l}{9+y }&&\multicolumn{1}{l}{(M1)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{1905 {\it 27}}&\parbox[t][0.3cm]{8.367391cm}{\raggedright Mult.: From DCO in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{744}&\multicolumn{1}{@{.}l}{6\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{18}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 6}}&\multicolumn{1}{r@{}}{4614}&\multicolumn{1}{@{.}l}{1+y }&&\multicolumn{1}{l}{E2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{01206 {\it 17}}&\parbox[t][0.3cm]{8.367391cm}{\raggedright Mult.: From DCO in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{5389}&\multicolumn{1}{@{.}l}{1+x}&\multicolumn{1}{l}{(45/2)}&\multicolumn{1}{r@{}}{302}&\multicolumn{1}{@{.}l}{0\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{5087}&\multicolumn{1}{@{.}l}{1+x }&\multicolumn{1}{@{}l}{(43/2)}&\multicolumn{1}{l}{D}&&&\parbox[t][0.3cm]{8.367391cm}{\raggedright Mult.: From DCO in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{5425}&\multicolumn{1}{@{.}l}{5+x}&&\multicolumn{1}{r@{}}{785}&\multicolumn{1}{@{.}l}{4\ensuremath{^{\hyperlink{PB29GAMMA2}{\#}}} {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA2}{\#}}}}&\multicolumn{1}{r@{}}{4640}&\multicolumn{1}{@{.}l}{1+x }&\multicolumn{1}{@{}l}{(35/2)}&&&&&\\
\multicolumn{1}{r@{}}{5455}&\multicolumn{1}{@{.}l}{0+z}&&\multicolumn{1}{r@{}}{276}&\multicolumn{1}{@{.}l}{4\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{5178}&\multicolumn{1}{@{.}l}{6+z }&&\multicolumn{1}{l}{(M1)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{534 {\it 8}}&\parbox[t][0.3cm]{8.367391cm}{\raggedright Mult.: From R(DCO)=0.66 \textit{9} in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{5554}&\multicolumn{1}{@{.}l}{4+v}&&\multicolumn{1}{r@{}}{312}&\multicolumn{1}{@{.}l}{0\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{5242}&\multicolumn{1}{@{.}l}{4+v }&&\multicolumn{1}{l}{(M1)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{383 {\it 6}}&\parbox[t][0.3cm]{8.367391cm}{\raggedright Mult.: From R(DCO)=0.58 \textit{11} in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{5581}&\multicolumn{1}{@{.}l}{9+x}&\multicolumn{1}{l}{(39/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{1749}&\multicolumn{1}{@{.}l}{5\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{3832}&\multicolumn{1}{@{.}l}{3+x }&\multicolumn{1}{@{}l}{(35/2\ensuremath{^{-}})}&\multicolumn{1}{l}{E2}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{48\ensuremath{\times10^{-3}} {\it 4}}&\parbox[t][0.3cm]{8.367391cm}{\raggedright Mult.: From R(DCO)=0.93 \textit{19} in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{5648}&\multicolumn{1}{@{.}l}{0+u}&&\multicolumn{1}{r@{}}{326}&\multicolumn{1}{@{.}l}{7\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{5321}&\multicolumn{1}{@{.}l}{3+u }&&\multicolumn{1}{l}{(M1)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{338 {\it 5}}&\parbox[t][0.3cm]{8.367391cm}{\raggedright Mult.: From R(DCO)=0.79 \textit{16} in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{5787}&\multicolumn{1}{@{.}l}{3+z}&&\multicolumn{1}{r@{}}{332}&\multicolumn{1}{@{.}l}{3\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{5455}&\multicolumn{1}{@{.}l}{0+z }&&\multicolumn{1}{l}{(M1)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{323 {\it 5}}&\parbox[t][0.3cm]{8.367391cm}{\raggedright Mult.: From R(DCO)=0.69 \textit{13} in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{5817}&\multicolumn{1}{@{.}l}{7+y}&&\multicolumn{1}{r@{}}{459}&\multicolumn{1}{@{.}l}{0\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 11}}&\multicolumn{1}{r@{}}{5358}&\multicolumn{1}{@{.}l}{7+y }&&\multicolumn{1}{l}{(M1)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{1355 {\it 19}}&\parbox[t][0.3cm]{8.367391cm}{\raggedright Mult.: From R(DCO)=0.70 \textit{6} in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{862}&\multicolumn{1}{@{.}l}{8\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{22}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 9}}&\multicolumn{1}{r@{}}{4954}&\multicolumn{1}{@{.}l}{9+y }&&\multicolumn{1}{l}{E2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{00892 {\it 13}}&\parbox[t][0.3cm]{8.367391cm}{\raggedright Mult.: From R(DCO)=0.92 \textit{34} in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{5830}&\multicolumn{1}{@{.}l}{0+x}&\multicolumn{1}{l}{(45/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{1190}&\multicolumn{1}{@{.}l}{1\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{4640}&\multicolumn{1}{@{.}l}{0+x }&\multicolumn{1}{@{}l}{(41/2\ensuremath{^{+}})}&\multicolumn{1}{l}{E2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{00477 {\it 7}}&\parbox[t][0.3cm]{8.367391cm}{\raggedright Mult.: From R(DCO)=1.07 \textit{13} in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{5835}&\multicolumn{1}{@{.}l}{7+x}&\multicolumn{1}{l}{(41/2)}&\multicolumn{1}{r@{}}{253}&\multicolumn{1}{@{.}l}{7\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{47}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{5581}&\multicolumn{1}{@{.}l}{9+x }&\multicolumn{1}{@{}l}{(39/2\ensuremath{^{-}})}&\multicolumn{1}{l}{(D)}&&&\parbox[t][0.3cm]{8.367391cm}{\raggedright Mult.: From R(DCO)=0.78 \textit{28} in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
\end{longtable}
\clearpage
\begin{longtable}{ccccccccc@{}ccccc@{\extracolsep{\fill}}c}
\\[-.4cm]
\multicolumn{15}{c}{{\bf \small \underline{Adopted \hyperlink{201PB_LEVEL}{Levels}, \hyperlink{201PB_GAMMA}{Gammas} (continued)}}}\\
\multicolumn{15}{c}{~}\\
\multicolumn{15}{c}{\underline{$\gamma$($^{201}$Pb) (continued)}}\\
\multicolumn{15}{c}{~~~}\\
\multicolumn{2}{c}{E\ensuremath{_{i}}(level)}&J\ensuremath{^{\pi}_{i}}&\multicolumn{2}{c}{E\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{PB29GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{I\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{PB29GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{E\ensuremath{_{f}}}&J\ensuremath{^{\pi}_{f}}&Mult.\ensuremath{^{\hyperlink{PB29GAMMA4}{\&}}}&\multicolumn{2}{c}{\ensuremath{\alpha}\ensuremath{^{\hyperlink{PB29GAMMA6}{b}}}}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endhead
\multicolumn{1}{r@{}}{5835}&\multicolumn{1}{@{.}l}{7+x}&\multicolumn{1}{l}{(41/2)}&\multicolumn{1}{r@{}}{1005}&\multicolumn{1}{@{.}l}{5\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{4830}&\multicolumn{1}{@{.}l}{2+x }&\multicolumn{1}{@{}l}{(39/2)}&\multicolumn{1}{l}{(D)}&&&\parbox[t][0.3cm]{9.722919cm}{\raggedright Mult.: From R(DCO)=0.76 \textit{20} in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{5891}&\multicolumn{1}{@{.}l}{3+x}&\multicolumn{1}{l}{(43/2)}&\multicolumn{1}{r@{}}{1251}&\multicolumn{1}{@{.}l}{3\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{4640}&\multicolumn{1}{@{.}l}{0+x }&\multicolumn{1}{@{}l}{(41/2\ensuremath{^{+}})}&\multicolumn{1}{l}{D}&&&\parbox[t][0.3cm]{9.722919cm}{\raggedright Mult.: From R(DCO)=0.74 \textit{17} in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{5928}&\multicolumn{1}{@{.}l}{8+v}&&\multicolumn{1}{r@{}}{374}&\multicolumn{1}{@{.}l}{4\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{5554}&\multicolumn{1}{@{.}l}{4+v }&&\multicolumn{1}{l}{(M1)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{2338 {\it 34}}&\parbox[t][0.3cm]{9.722919cm}{\raggedright Mult.: From R(DCO)=0.68 \textit{14} in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{5989}&\multicolumn{1}{@{.}l}{4+x}&\multicolumn{1}{l}{(45/2)}&\multicolumn{1}{r@{}}{98}&\multicolumn{1}{@{.}l}{2\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{4\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 4}}&\multicolumn{1}{r@{}}{5891}&\multicolumn{1}{@{.}l}{3+x }&\multicolumn{1}{@{}l}{(43/2)}&\multicolumn{1}{l}{D}&&&\parbox[t][0.3cm]{9.722919cm}{\raggedright Mult.: From R(DCO)=0.74 \textit{21} in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{153}&\multicolumn{1}{@{.}l}{7\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{8}&\multicolumn{1}{@{.}l}{4\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 4}}&\multicolumn{1}{r@{}}{5835}&\multicolumn{1}{@{.}l}{7+x }&\multicolumn{1}{@{}l}{(41/2)}&\multicolumn{1}{l}{E2}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{114 {\it 21}}&\parbox[t][0.3cm]{9.722919cm}{\raggedright Mult.: From R(DCO)=1.02 \textit{15}\hphantom{a}in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{159}&\multicolumn{1}{@{.}l}{4\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{9}&\multicolumn{1}{@{.}l}{6\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 4}}&\multicolumn{1}{r@{}}{5830}&\multicolumn{1}{@{.}l}{0+x }&\multicolumn{1}{@{}l}{(45/2\ensuremath{^{+}})}&\multicolumn{1}{l}{(D)}&&&\parbox[t][0.3cm]{9.722919cm}{\raggedright Mult.: From R(DCO)=1.06 \textit{13} in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{600}&\multicolumn{1}{@{.}l}{2\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{12}&\multicolumn{1}{@{.}l}{1\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 17}}&\multicolumn{1}{r@{}}{5389}&\multicolumn{1}{@{.}l}{1+x }&\multicolumn{1}{@{}l}{(45/2)}&\multicolumn{1}{l}{(D)}&&&\parbox[t][0.3cm]{9.722919cm}{\raggedright Mult.: From R(DCO)=0.98 \textit{16} in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{902}&\multicolumn{1}{@{.}l}{2\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{.}l}{0\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 8}}&\multicolumn{1}{r@{}}{5087}&\multicolumn{1}{@{.}l}{1+x }&\multicolumn{1}{@{}l}{(43/2)}&\multicolumn{1}{l}{D}&&&\parbox[t][0.3cm]{9.722919cm}{\raggedright Mult.: From R(DCO)=0.55 \textit{5} in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{6028}&\multicolumn{1}{@{.}l}{4+u}&&\multicolumn{1}{r@{}}{380}&\multicolumn{1}{@{.}l}{4\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{5648}&\multicolumn{1}{@{.}l}{0+u }&&\multicolumn{1}{l}{(M1)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{2240 {\it 32}}&\parbox[t][0.3cm]{9.722919cm}{\raggedright Mult.: From R(DCO)=0.89 \textit{11} in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{6145}&\multicolumn{1}{@{.}l}{1+x}&\multicolumn{1}{l}{(35/2)}&\multicolumn{1}{r@{}}{1640}&\multicolumn{1}{@{.}l}{0\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{4505}&\multicolumn{1}{@{.}l}{1+x }&\multicolumn{1}{@{}l}{(35/2)}&\multicolumn{1}{l}{D}&&&\parbox[t][0.3cm]{9.722919cm}{\raggedright Mult.: From R(DCO)=1.13 \textit{26} in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{6175}&\multicolumn{1}{@{.}l}{4+z}&&\multicolumn{1}{r@{}}{388}&\multicolumn{1}{@{.}l}{1\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{5787}&\multicolumn{1}{@{.}l}{3+z }&&\multicolumn{1}{l}{(M1)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{2122 {\it 31}}&\parbox[t][0.3cm]{9.722919cm}{\raggedright Mult.: From R(DCO)=0.59 \textit{14} in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{6246}&\multicolumn{1}{@{.}l}{8+x}&\multicolumn{1}{l}{(37/2)}&\multicolumn{1}{r@{}}{101}&\multicolumn{1}{@{.}l}{7\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{6145}&\multicolumn{1}{@{.}l}{1+x }&\multicolumn{1}{@{}l}{(35/2)}&\multicolumn{1}{l}{(M1)}&\multicolumn{1}{r@{}}{8}&\multicolumn{1}{@{.}l}{91 {\it 18}}&\parbox[t][0.3cm]{9.722919cm}{\raggedright Mult.: From R(DCO)=0.55 \textit{14} in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{6323}&\multicolumn{1}{@{.}l}{4+y}&&\multicolumn{1}{r@{}}{505}&\multicolumn{1}{@{.}l}{7\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 33}}&\multicolumn{1}{r@{}}{5817}&\multicolumn{1}{@{.}l}{7+y }&&\multicolumn{1}{l}{(M1)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{1048 {\it 15}}&\parbox[t][0.3cm]{9.722919cm}{\raggedright Mult.: From R(DCO)=0.67 \textit{11} in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{964}&\multicolumn{1}{@{.}l}{7\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{67}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 20}}&\multicolumn{1}{r@{}}{5358}&\multicolumn{1}{@{.}l}{7+y }&&\multicolumn{1}{l}{(E2)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{00714 {\it 10}}&\parbox[t][0.3cm]{9.722919cm}{\raggedright Mult.: From R(DCO)=0.77 \textit{18} in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{6323}&\multicolumn{1}{@{.}l}{9+x}&\multicolumn{1}{l}{(45/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{1683}&\multicolumn{1}{@{.}l}{8\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{4640}&\multicolumn{1}{@{.}l}{0+x }&\multicolumn{1}{@{}l}{(41/2\ensuremath{^{+}})}&\multicolumn{1}{l}{E2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{00263 {\it 4}}&\parbox[t][0.3cm]{9.722919cm}{\raggedright Mult.: From R(DCO)=1.01 \textit{17} in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{6336}&\multicolumn{1}{@{.}l}{1+y}&&\multicolumn{1}{r@{}}{518}&\multicolumn{1}{@{.}l}{4\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{5817}&\multicolumn{1}{@{.}l}{7+y }&&\multicolumn{1}{l}{D}&&&\parbox[t][0.3cm]{9.722919cm}{\raggedright Mult.: R(DCO)=0.69 \textit{10} in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{6364}&\multicolumn{1}{@{.}l}{8+v}&&\multicolumn{1}{r@{}}{436}&\multicolumn{1}{@{.}l}{0\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{5928}&\multicolumn{1}{@{.}l}{8+v }&&\multicolumn{1}{l}{(M1)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{1554 {\it 22}}&\parbox[t][0.3cm]{9.722919cm}{\raggedright Mult.: From R(DCO)=0.77 \textit{21} in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{6376}&\multicolumn{1}{@{.}l}{4+x}&\multicolumn{1}{l}{(39/2)}&\multicolumn{1}{r@{}}{129}&\multicolumn{1}{@{.}l}{7\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{6246}&\multicolumn{1}{@{.}l}{8+x }&\multicolumn{1}{@{}l}{(37/2)}&\multicolumn{1}{l}{(M1)}&\multicolumn{1}{r@{}}{4}&\multicolumn{1}{@{.}l}{45 {\it 8}}&\parbox[t][0.3cm]{9.722919cm}{\raggedright Mult.: From R(DCO)=0.57 \textit{8} in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{6458}&\multicolumn{1}{@{.}l}{1+u}&&\multicolumn{1}{r@{}}{429}&\multicolumn{1}{@{.}l}{7\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{6028}&\multicolumn{1}{@{.}l}{4+u }&&\multicolumn{1}{l}{(M1)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{1615 {\it 23}}&\parbox[t][0.3cm]{9.722919cm}{\raggedright Mult.: From R(DCO)=0.79 \textit{30} in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{6460}&\multicolumn{1}{@{.}l}{1+x}&\multicolumn{1}{l}{(47/2)}&\multicolumn{1}{r@{}}{136}&\multicolumn{1}{@{.}l}{2\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{8}&\multicolumn{1}{@{.}l}{5\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 14}}&\multicolumn{1}{r@{}}{6323}&\multicolumn{1}{@{.}l}{9+x }&\multicolumn{1}{@{}l}{(45/2\ensuremath{^{+}})}&\multicolumn{1}{l}{D}&&&\parbox[t][0.3cm]{9.722919cm}{\raggedright Mult.: From R(DCO)=0.89 \textit{28} in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{470}&\multicolumn{1}{@{.}l}{7\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 6}}&\multicolumn{1}{r@{}}{5989}&\multicolumn{1}{@{.}l}{4+x }&\multicolumn{1}{@{}l}{(45/2)}&\multicolumn{1}{l}{D}&&&\parbox[t][0.3cm]{9.722919cm}{\raggedright Mult.: From R(DCO)=0.94 \textit{12} in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{6548}&\multicolumn{1}{@{.}l}{0+x}&\multicolumn{1}{l}{(41/2)}&\multicolumn{1}{r@{}}{171}&\multicolumn{1}{@{.}l}{6\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{6376}&\multicolumn{1}{@{.}l}{4+x }&\multicolumn{1}{@{}l}{(39/2)}&\multicolumn{1}{l}{(M1)}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{013 {\it 33}}&\parbox[t][0.3cm]{9.722919cm}{\raggedright Mult.: From R(DCO)=0.58 \textit{6} in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{6616}&\multicolumn{1}{@{.}l}{7+z}&&\multicolumn{1}{r@{}}{441}&\multicolumn{1}{@{.}l}{3\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 27}}&\multicolumn{1}{r@{}}{6175}&\multicolumn{1}{@{.}l}{4+z }&&\multicolumn{1}{l}{(M1)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{1505 {\it 22}}&\parbox[t][0.3cm]{9.722919cm}{\raggedright Mult.: From R(DCO)=0.64 \textit{16} in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{829}&\multicolumn{1}{@{.}l}{4\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{35}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 15}}&\multicolumn{1}{r@{}}{5787}&\multicolumn{1}{@{.}l}{3+z }&&\multicolumn{1}{l}{E2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{00966 {\it 14}}&\parbox[t][0.3cm]{9.722919cm}{\raggedright Mult.: From R(DCO)=1.2 \textit{7} in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{6706}&\multicolumn{1}{@{.}l}{7+x}&\multicolumn{1}{l}{(49/2)}&\multicolumn{1}{r@{}}{717}&\multicolumn{1}{@{.}l}{3\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{5989}&\multicolumn{1}{@{.}l}{4+x }&\multicolumn{1}{@{}l}{(45/2)}&\multicolumn{1}{l}{E2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{01305 {\it 18}}&\parbox[t][0.3cm]{9.722919cm}{\raggedright Mult.: From R(DCO)=1.05 \textit{6} in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{6768}&\multicolumn{1}{@{.}l}{5+x}&\multicolumn{1}{l}{(43/2)}&\multicolumn{1}{r@{}}{220}&\multicolumn{1}{@{.}l}{5\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{6548}&\multicolumn{1}{@{.}l}{0+x }&\multicolumn{1}{@{}l}{(41/2)}&\multicolumn{1}{l}{(M1)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{997 {\it 15}}&\parbox[t][0.3cm]{9.722919cm}{\raggedright Mult.: From R(DCO)=0.58 \textit{5} in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{6858}&\multicolumn{1}{@{.}l}{2+v}&&\multicolumn{1}{r@{}}{493}&\multicolumn{1}{@{.}l}{4\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{6364}&\multicolumn{1}{@{.}l}{8+v }&&\multicolumn{1}{l}{(M1)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{1118 {\it 16}}&\parbox[t][0.3cm]{9.722919cm}{\raggedright Mult.: From R(DCO)=0.76 \textit{20} in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{6881}&\multicolumn{1}{@{.}l}{9+y}&&\multicolumn{1}{r@{}}{558}&\multicolumn{1}{@{.}l}{5\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{6323}&\multicolumn{1}{@{.}l}{4+y }&&\multicolumn{1}{l}{(M1)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0806 {\it 11}}&\parbox[t][0.3cm]{9.722919cm}{\raggedright Mult.: From R(DCO)=0.65 \textit{18} in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{6910}&\multicolumn{1}{@{.}l}{1+x}&\multicolumn{1}{l}{(47/2)}&\multicolumn{1}{r@{}}{586}&\multicolumn{1}{@{.}l}{1\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{6323}&\multicolumn{1}{@{.}l}{9+x }&\multicolumn{1}{@{}l}{(45/2\ensuremath{^{+}})}&\multicolumn{1}{l}{D}&&&\parbox[t][0.3cm]{9.722919cm}{\raggedright Mult.: From R(DCO)=0.76 \textit{13} in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{6941}&\multicolumn{1}{@{.}l}{2+u}&&\multicolumn{1}{r@{}}{483}&\multicolumn{1}{@{.}l}{1\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{6458}&\multicolumn{1}{@{.}l}{1+u }&&\multicolumn{1}{l}{(M1)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{1183 {\it 17}}&\parbox[t][0.3cm]{9.722919cm}{\raggedright Mult.: From R(DCO)=0.73 \textit{26} in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{7008}&\multicolumn{1}{@{.}l}{4+x}&\multicolumn{1}{l}{(49/2)}&\multicolumn{1}{r@{}}{548}&\multicolumn{1}{@{.}l}{3\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{6460}&\multicolumn{1}{@{.}l}{1+x }&\multicolumn{1}{@{}l}{(47/2)}&\multicolumn{1}{l}{D}&&&\parbox[t][0.3cm]{9.722919cm}{\raggedright Mult.: From R(DCO)=0.67 \textit{9} in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{7044}&\multicolumn{1}{@{.}l}{3+x}&\multicolumn{1}{l}{(45/2)}&\multicolumn{1}{r@{}}{275}&\multicolumn{1}{@{.}l}{9\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{6768}&\multicolumn{1}{@{.}l}{5+x }&\multicolumn{1}{@{}l}{(43/2)}&\multicolumn{1}{l}{(M1)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{536 {\it 8}}&\parbox[t][0.3cm]{9.722919cm}{\raggedright Mult.: From R(DCO)=0.56 \textit{3} in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{7108}&\multicolumn{1}{@{.}l}{4+z}&&\multicolumn{1}{r@{}}{491}&\multicolumn{1}{@{.}l}{7\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 22}}&\multicolumn{1}{r@{}}{6616}&\multicolumn{1}{@{.}l}{7+z }&&\multicolumn{1}{l}{(M1)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{1129 {\it 16}}&\parbox[t][0.3cm]{9.722919cm}{\raggedright Mult.: From R(DCO)=0.64 \textit{17} in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{933}&\multicolumn{1}{@{.}l}{1\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{44}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 17}}&\multicolumn{1}{r@{}}{6175}&\multicolumn{1}{@{.}l}{4+z }&&\multicolumn{1}{l}{E2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{00763 {\it 11}}&\parbox[t][0.3cm]{9.722919cm}{\raggedright Mult.: From R(DCO)=0.8 \textit{3} in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{7142}&\multicolumn{1}{@{.}l}{3+x}&\multicolumn{1}{l}{(49/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{232}&\multicolumn{1}{@{.}l}{2\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{10}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{6910}&\multicolumn{1}{@{.}l}{1+x }&\multicolumn{1}{@{}l}{(47/2)}&\multicolumn{1}{l}{D}&&&\parbox[t][0.3cm]{9.722919cm}{\raggedright Mult.: From R(DCO)=0.81 \textit{25} in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
\end{longtable}
\clearpage
\begin{longtable}{ccccccccc@{}ccccc@{\extracolsep{\fill}}c}
\\[-.4cm]
\multicolumn{15}{c}{{\bf \small \underline{Adopted \hyperlink{201PB_LEVEL}{Levels}, \hyperlink{201PB_GAMMA}{Gammas} (continued)}}}\\
\multicolumn{15}{c}{~}\\
\multicolumn{15}{c}{\underline{$\gamma$($^{201}$Pb) (continued)}}\\
\multicolumn{15}{c}{~~~}\\
\multicolumn{2}{c}{E\ensuremath{_{i}}(level)}&J\ensuremath{^{\pi}_{i}}&\multicolumn{2}{c}{E\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{PB29GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{I\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{PB29GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{E\ensuremath{_{f}}}&J\ensuremath{^{\pi}_{f}}&Mult.\ensuremath{^{\hyperlink{PB29GAMMA4}{\&}}}&\multicolumn{2}{c}{\ensuremath{\alpha}\ensuremath{^{\hyperlink{PB29GAMMA6}{b}}}}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endhead
\multicolumn{1}{r@{}}{7142}&\multicolumn{1}{@{.}l}{3+x}&\multicolumn{1}{l}{(49/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{682}&\multicolumn{1}{@{.}l}{3\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{17}&\multicolumn{1}{@{.}l}{2\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 17}}&\multicolumn{1}{r@{}}{6460}&\multicolumn{1}{@{.}l}{1+x }&\multicolumn{1}{@{}l}{(47/2)}&\multicolumn{1}{l}{D}&&&\parbox[t][0.3cm]{9.881279cm}{\raggedright Mult.: From R(DCO)=0.73 \textit{18} in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{1312}&\multicolumn{1}{@{.}l}{3\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 7}}&\multicolumn{1}{r@{}}{5830}&\multicolumn{1}{@{.}l}{0+x }&\multicolumn{1}{@{}l}{(45/2\ensuremath{^{+}})}&\multicolumn{1}{l}{E2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{00398 {\it 6}}&\parbox[t][0.3cm]{9.881279cm}{\raggedright Mult.: From R(DCO)=0.97 \textit{11} in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{7339}&\multicolumn{1}{@{.}l}{5+x}&\multicolumn{1}{l}{(51/2)}&\multicolumn{1}{r@{}}{197}&\multicolumn{1}{@{.}l}{2\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{.}l}{0\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 24}}&\multicolumn{1}{r@{}}{7142}&\multicolumn{1}{@{.}l}{3+x }&\multicolumn{1}{@{}l}{(49/2\ensuremath{^{+}})}&\multicolumn{1}{l}{D}&&&\parbox[t][0.3cm]{9.881279cm}{\raggedright Mult.: From R(DCO)=0.70 \textit{11} in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{331}&\multicolumn{1}{@{.}l}{1\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{42}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{7008}&\multicolumn{1}{@{.}l}{4+x }&\multicolumn{1}{@{}l}{(49/2)}&\multicolumn{1}{l}{D}&&&\parbox[t][0.3cm]{9.881279cm}{\raggedright Mult.: From R(DCO)=0.77 \textit{18} in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{7377}&\multicolumn{1}{@{.}l}{5+x}&\multicolumn{1}{l}{(47/2)}&\multicolumn{1}{r@{}}{333}&\multicolumn{1}{@{.}l}{1\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 20}}&\multicolumn{1}{r@{}}{7044}&\multicolumn{1}{@{.}l}{3+x }&\multicolumn{1}{@{}l}{(45/2)}&\multicolumn{1}{l}{(M1)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{321 {\it 5}}&\parbox[t][0.3cm]{9.881279cm}{\raggedright Mult.: From R(DCO)=0.57 \textit{5} in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{1388}&\multicolumn{1}{@{.}l}{1\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{5}&\multicolumn{1}{@{.}l}{6\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 3}}&\multicolumn{1}{r@{}}{5989}&\multicolumn{1}{@{.}l}{4+x }&\multicolumn{1}{@{}l}{(45/2)}&\multicolumn{1}{l}{D}&&&\parbox[t][0.3cm]{9.881279cm}{\raggedright Mult.: From R(DCO)=0.72 \textit{29} in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{7378}&\multicolumn{1}{@{.}l}{9+x}&\multicolumn{1}{l}{(47/2)}&\multicolumn{1}{r@{}}{334}&\multicolumn{1}{@{.}l}{6\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 24}}&\multicolumn{1}{r@{}}{7044}&\multicolumn{1}{@{.}l}{3+x }&\multicolumn{1}{@{}l}{(45/2)}&\multicolumn{1}{l}{(M1)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{317 {\it 5}}&\parbox[t][0.3cm]{9.881279cm}{\raggedright Mult.: From R(DCO)=0.60 \textit{5} in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{1389}&\multicolumn{1}{@{.}l}{4\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{4}&\multicolumn{1}{@{.}l}{4\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 3}}&\multicolumn{1}{r@{}}{5989}&\multicolumn{1}{@{.}l}{4+x }&\multicolumn{1}{@{}l}{(45/2)}&\multicolumn{1}{l}{D}&&&\parbox[t][0.3cm]{9.881279cm}{\raggedright Mult.: From R(DCO)=0.64 \textit{37} in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{7471}&\multicolumn{1}{@{.}l}{4+u}&&\multicolumn{1}{r@{}}{530}&\multicolumn{1}{@{.}l}{2\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{6941}&\multicolumn{1}{@{.}l}{2+u }&&\multicolumn{1}{l}{(M1)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0925 {\it 13}}&\parbox[t][0.3cm]{9.881279cm}{\raggedright Mult.: From R(DCO)=0.85 \textit{20} in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{7648}&\multicolumn{1}{@{.}l}{2+z}&&\multicolumn{1}{r@{}}{539}&\multicolumn{1}{@{.}l}{8\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{92}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 25}}&\multicolumn{1}{r@{}}{7108}&\multicolumn{1}{@{.}l}{4+z }&&\multicolumn{1}{l}{(M1)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0882 {\it 13}}&\parbox[t][0.3cm]{9.881279cm}{\raggedright Mult.: From R(DCO)=0.66 \textit{26} in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{1031}&\multicolumn{1}{@{.}l}{4\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 33}}&\multicolumn{1}{r@{}}{6616}&\multicolumn{1}{@{.}l}{7+z }&&\multicolumn{1}{l}{E2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{00627 {\it 9}}&\parbox[t][0.3cm]{9.881279cm}{\raggedright Mult.: From R(DCO)=1.2 \textit{5} in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{7759}&\multicolumn{1}{@{.}l}{5+x}&\multicolumn{1}{l}{(49/2)}&\multicolumn{1}{r@{}}{380}&\multicolumn{1}{@{.}l}{6\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{84}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 8}}&\multicolumn{1}{r@{}}{7378}&\multicolumn{1}{@{.}l}{9+x }&\multicolumn{1}{@{}l}{(47/2)}&\multicolumn{1}{l}{D}&&&\parbox[t][0.3cm]{9.881279cm}{\raggedright Mult.: From R(DCO)=0.84 \textit{29} in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{382}&\multicolumn{1}{@{.}l}{0\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 3}}&\multicolumn{1}{r@{}}{7377}&\multicolumn{1}{@{.}l}{5+x }&\multicolumn{1}{@{}l}{(47/2)}&\multicolumn{1}{l}{D}&&&\parbox[t][0.3cm]{9.881279cm}{\raggedright Mult.: From R(DCO)=0.66 \textit{15} in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{7772}&\multicolumn{1}{@{.}l}{2+x}&\multicolumn{1}{l}{(49/2)}&\multicolumn{1}{r@{}}{393}&\multicolumn{1}{@{.}l}{3\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{89}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 12}}&\multicolumn{1}{r@{}}{7378}&\multicolumn{1}{@{.}l}{9+x }&\multicolumn{1}{@{}l}{(47/2)}&\multicolumn{1}{l}{(M1)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{2048 {\it 30}}&\parbox[t][0.3cm]{9.881279cm}{\raggedright Mult.: From R(DCO)=0.64 \textit{10} in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{394}&\multicolumn{1}{@{.}l}{8\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 23}}&\multicolumn{1}{r@{}}{7377}&\multicolumn{1}{@{.}l}{5+x }&\multicolumn{1}{@{}l}{(47/2)}&\multicolumn{1}{l}{(M1)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{2027 {\it 29}}&\parbox[t][0.3cm]{9.881279cm}{\raggedright Mult.: From R(DCO)=0.60 \textit{13} in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{8003}&\multicolumn{1}{@{.}l}{4+x}&\multicolumn{1}{l}{(53/2)}&\multicolumn{1}{r@{}}{663}&\multicolumn{1}{@{.}l}{9\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{7339}&\multicolumn{1}{@{.}l}{5+x }&\multicolumn{1}{@{}l}{(51/2)}&\multicolumn{1}{l}{D}&&&\parbox[t][0.3cm]{9.881279cm}{\raggedright Mult.: From R(DCO)=0.80 \textit{12} in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}). Note, that the\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{9.881279cm}{\raggedright {\ }{\ }{\ }authors give R(DCO)=0.8 \textit{12}, which is probably a typo.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{8018}&\multicolumn{1}{@{.}l}{7+x}&\multicolumn{1}{l}{(51/2)}&\multicolumn{1}{r@{}}{259}&\multicolumn{1}{@{.}l}{2\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{9}&\multicolumn{1}{@{.}l}{2\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{7759}&\multicolumn{1}{@{.}l}{5+x }&\multicolumn{1}{@{}l}{(49/2)}&\multicolumn{1}{l}{D}&&&\parbox[t][0.3cm]{9.881279cm}{\raggedright Mult.: From R(DCO)=0.63 \textit{13} in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{1312}&\multicolumn{1}{@{.}l}{0\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 6}}&\multicolumn{1}{r@{}}{6706}&\multicolumn{1}{@{.}l}{7+x }&\multicolumn{1}{@{}l}{(49/2)}&\multicolumn{1}{l}{D}&&&\parbox[t][0.3cm]{9.881279cm}{\raggedright Mult.: From R(DCO)=0.73 \textit{13} in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{8198}&\multicolumn{1}{@{.}l}{0+x}&\multicolumn{1}{l}{(53/2)}&\multicolumn{1}{r@{}}{179}&\multicolumn{1}{@{.}l}{3\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{8018}&\multicolumn{1}{@{.}l}{7+x }&\multicolumn{1}{@{}l}{(51/2)}&\multicolumn{1}{l}{D}&&&\parbox[t][0.3cm]{9.881279cm}{\raggedright Mult.: From R(DCO)=0.70 \textit{6} in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{8214}&\multicolumn{1}{@{.}l}{7+x}&\multicolumn{1}{l}{(51/2)}&\multicolumn{1}{r@{}}{442}&\multicolumn{1}{@{.}l}{5\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{7772}&\multicolumn{1}{@{.}l}{2+x }&\multicolumn{1}{@{}l}{(49/2)}&\multicolumn{1}{l}{(M1)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{1494 {\it 21}}&\parbox[t][0.3cm]{9.881279cm}{\raggedright Mult.: From R(DCO)=0.68 \textit{16} in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{8226}&\multicolumn{1}{@{.}l}{1+x}&\multicolumn{1}{l}{(51/2)}&\multicolumn{1}{r@{}}{453}&\multicolumn{1}{@{.}l}{9\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{7772}&\multicolumn{1}{@{.}l}{2+x }&\multicolumn{1}{@{}l}{(49/2)}&\multicolumn{1}{l}{(M1)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{1396 {\it 20}}&\parbox[t][0.3cm]{9.881279cm}{\raggedright Mult.: From R(DCO)=0.62 \textit{14} in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{8653}&\multicolumn{1}{@{.}l}{8+x}&\multicolumn{1}{l}{(55/2)}&\multicolumn{1}{r@{}}{455}&\multicolumn{1}{@{.}l}{8\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PB29GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{8198}&\multicolumn{1}{@{.}l}{0+x }&\multicolumn{1}{@{}l}{(53/2)}&\multicolumn{1}{l}{(M1)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{1380 {\it 20}}&\parbox[t][0.3cm]{9.881279cm}{\raggedright Mult.: From R(DCO)=0.71 \textit{7} in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
\end{longtable}
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB29GAMMA0}{\dagger}}}} From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1978Ri04,B}{1978Ri04} in \ensuremath{^{\textnormal{201}}}Bi \ensuremath{\varepsilon} decay, unless otherwise specified.}\\
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB29GAMMA1}{\ddagger}}}} From \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}).}\\
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB29GAMMA2}{\#}}}} From \ensuremath{^{\textnormal{200}}}Hg(\ensuremath{\alpha},3n\ensuremath{\gamma}).}\\
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB29GAMMA3}{@}}}} From \ensuremath{^{\textnormal{197}}}Au(\ensuremath{^{\textnormal{207}}}Pb,X\ensuremath{\gamma}).}\\
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB29GAMMA4}{\&}}}} From \ensuremath{\alpha}(K)exp, \ensuremath{\alpha}(L)exp and subshell ratios in \ensuremath{^{\textnormal{201}}}Bi \ensuremath{\varepsilon} decay (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1978Ri04,B}{1978Ri04}), \ensuremath{\gamma}(\ensuremath{\theta}) in \ensuremath{^{\textnormal{200}}}Hg(\ensuremath{\alpha},3n\ensuremath{\gamma}) and DCO in \ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma}), coupled together with the}\\
\parbox[b][0.3cm]{21.881866cm}{{\ }{\ }observed multiple decay branches and band structures. For rotational band transitions whose multipolarity is determined from \ensuremath{\gamma}(\ensuremath{\theta}) or DCO, Mult.=(M1), instead}\\
\parbox[b][0.3cm]{21.881866cm}{{\ }{\ }of D, is assigned in this evaluation.}\\
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB29GAMMA5}{a}}}} From \ensuremath{\alpha}(K)exp, \ensuremath{\alpha}(L)exp and subshell ratios in \ensuremath{^{\textnormal{201}}}Bi \ensuremath{\varepsilon} decay (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1978Ri04,B}{1978Ri04}) and the briccmixing program, unless otherwise stated.}\\
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB29GAMMA6}{b}}}} Total theoretical internal conversion coefficients, calculated using the BrIcc code (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2008Ki07,B}{2008Ki07}) with Frozen orbital approximation based on \ensuremath{\gamma}-ray energies,}\\
\parbox[b][0.3cm]{21.881866cm}{{\ }{\ }assigned multipolarities, and mixing ratios, unless otherwise specified.}\\
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB29GAMMA7}{c}}}} Placement of transition in the level scheme is uncertain.}\\
\vspace{0.5cm}
\end{landscape}\clearpage
\clearpage
\begin{figure}[h]
\begin{center}
\includegraphics{201PB29-0.ps}\\
\end{center}
\end{figure}
\clearpage
\begin{figure}[h]
\begin{center}
\includegraphics{201PB29-1.ps}\\
\end{center}
\end{figure}
\clearpage
\begin{figure}[h]
\begin{center}
\includegraphics{201PB29-2.ps}\\
\end{center}
\end{figure}
\clearpage
\begin{figure}[h]
\begin{center}
\includegraphics{201PB29-3.ps}\\
\end{center}
\end{figure}
\clearpage
\begin{figure}[h]
\begin{center}
\includegraphics{201PB29-4.ps}\\
\end{center}
\end{figure}
\clearpage
\begin{figure}[h]
\begin{center}
\includegraphics{201PB29-5.ps}\\
\end{center}
\end{figure}
\clearpage
\clearpage
\begin{figure}[h]
\begin{center}
\includegraphics{201PB29B-0.ps}\\
\end{center}
\end{figure}
\clearpage
\subsection[\hspace{-0.2cm}\ensuremath{^{\textnormal{201}}}Bi \ensuremath{\varepsilon} decay]{ }
\vspace{-27pt}
\vspace{0.3cm}
\hypertarget{BI30}{{\bf \small \underline{\ensuremath{^{\textnormal{201}}}Bi \ensuremath{\varepsilon} decay\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1978Ri04,B}{1978Ri04}}}}\\
\vspace{4pt}
\vspace{8pt}
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}Parent: $^{201}$Bi: E=0; J$^{\pi}$=9/2\ensuremath{^{-}}; T$_{1/2}$=103.2 min {\it 17}; Q(\ensuremath{\varepsilon})=3842 {\it 18}; \%\ensuremath{\varepsilon}+\%\ensuremath{\beta^{+}} decay=100.0
}\\
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1978Ri04,B}{1978Ri04}: mass-separated source following (p,xn) reaction of 73-MeV protons on natural lead; Detectors: Ge(Li) and Si(Li);}\\
\parbox[b][0.3cm]{17.7cm}{Radiochemical separation of bismuth from its lead and thallium daughters; Measured: E\ensuremath{\gamma}, I\ensuremath{\gamma}, \ensuremath{\gamma}\ensuremath{\gamma} coin, \ensuremath{\alpha}(K)exp, \ensuremath{\alpha}(L)exp;}\\
\parbox[b][0.3cm]{17.7cm}{Deduced: \ensuremath{J^{\pi}}, level scheme.}\\
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}Others: \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1956St05,B}{1956St05}, \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1970DaZM,B}{1970DaZM}, \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1970Jo26,B}{1970Jo26}.}\\
\vspace{12pt}
\underline{$^{201}$Pb Levels}\\
\begin{longtable}{ccccc|ccc|ccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E(level)$^{{\hyperlink{PB30LEVEL0}{\dagger}}}$}&J$^{\pi}$$^{{\hyperlink{PB30LEVEL9}{e}}}$&\multicolumn{2}{c|}{T$_{1/2}$$^{{\hyperlink{PB30LEVEL9}{e}}}$}&\multicolumn{2}{c}{E(level)$^{{\hyperlink{PB30LEVEL0}{\dagger}}}$}&J$^{\pi}$$^{{\hyperlink{PB30LEVEL9}{e}}}$&\multicolumn{2}{c}{E(level)$^{{\hyperlink{PB30LEVEL0}{\dagger}}}$}&J$^{\pi}$$^{{\hyperlink{PB30LEVEL9}{e}}}$&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c|}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{PB30LEVEL1}{\ddagger}}}}}&\multicolumn{1}{l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{9}&\multicolumn{1}{@{.}l|}{33 h {\it 5}}&\multicolumn{1}{r@{}}{1325}&\multicolumn{1}{@{.}l}{4\ensuremath{^{{\hyperlink{PB30LEVEL5}{a}}}} {\it 5}}&\multicolumn{1}{l|}{7/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{2208}&\multicolumn{1}{@{.}l}{9 {\it 6}}&\multicolumn{1}{l}{(9/2\ensuremath{^{+}})}&\\
\multicolumn{1}{r@{}}{88}&\multicolumn{1}{@{.}l}{5\ensuremath{^{{\hyperlink{PB30LEVEL2}{\#}}}} {\it 5}}&\multicolumn{1}{l}{3/2\ensuremath{^{-}}}&&&\multicolumn{1}{r@{}}{1415}&\multicolumn{1}{@{.}l}{4\ensuremath{^{{\hyperlink{PB30LEVEL8}{d}}}} {\it 4}}&\multicolumn{1}{l|}{9/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{2279}&\multicolumn{1}{@{.}l}{8 {\it 8}}&\multicolumn{1}{l}{(9/2)\ensuremath{^{+}}}&\\
\multicolumn{1}{r@{}}{169}&\multicolumn{1}{@{.}l}{9\ensuremath{^{{\hyperlink{PB30LEVEL3}{@}}}} {\it 9}}&\multicolumn{1}{l}{(1/2\ensuremath{^{-}})}&&&\multicolumn{1}{r@{}}{1447}&\multicolumn{1}{@{.}l}{9 {\it 5}}&\multicolumn{1}{l|}{(11/2)\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{2439}&\multicolumn{1}{@{.}l}{5 {\it 7}}&\multicolumn{1}{l}{7/2\ensuremath{^{+}},9/2\ensuremath{^{+}}}&\\
\multicolumn{1}{r@{}}{538}&\multicolumn{1}{@{.}l}{8 {\it 7}}&\multicolumn{1}{l}{(3/2\ensuremath{^{-}})}&&&\multicolumn{1}{r@{}}{1490}&\multicolumn{1}{@{.}l}{3 {\it 6}}&\multicolumn{1}{l|}{7/2\ensuremath{^{-}},9/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{2459}&\multicolumn{1}{@{.}l}{9 {\it 6}}&\multicolumn{1}{l}{7/2\ensuremath{^{+}},9/2\ensuremath{^{+}}}&\\
\multicolumn{1}{r@{}}{629}&\multicolumn{1}{@{.}l}{1\ensuremath{^{{\hyperlink{PB30LEVEL4}{\&}}}} {\it 3}}&\multicolumn{1}{l}{13/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{60}&\multicolumn{1}{@{.}l|}{8 s {\it 18}}&\multicolumn{1}{r@{}}{1651}&\multicolumn{1}{@{.}l}{0 {\it 4}}&\multicolumn{1}{l|}{7/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{2474}&\multicolumn{1}{@{.}l}{6 {\it 5}}&\multicolumn{1}{l}{(7/2\ensuremath{^{+}},9/2\ensuremath{^{+}})}&\\
\multicolumn{1}{r@{}}{879}&\multicolumn{1}{@{.}l}{9? {\it 6}}&\multicolumn{1}{l}{(5/2)\ensuremath{^{-}}}&&&\multicolumn{1}{r@{}}{1737}&\multicolumn{1}{@{.}l}{3 {\it 4}}&\multicolumn{1}{l|}{9/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{2506}&\multicolumn{1}{@{.}l}{8 {\it 4}}&\multicolumn{1}{l}{9/2\ensuremath{^{+}}}&\\
\multicolumn{1}{r@{}}{910}&\multicolumn{1}{@{.}l}{5? {\it 12}}&\multicolumn{1}{l}{5/2\ensuremath{^{-}}}&&&\multicolumn{1}{r@{}}{1843}&\multicolumn{1}{@{.}l}{8 {\it 5}}&\multicolumn{1}{l|}{11/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{2548}&\multicolumn{1}{@{.}l}{9 {\it 6}}&\multicolumn{1}{l}{(11/2\ensuremath{^{-}})}&\\
\multicolumn{1}{r@{}}{936}&\multicolumn{1}{@{.}l}{1\ensuremath{^{{\hyperlink{PB30LEVEL6}{b}}}} {\it 3}}&\multicolumn{1}{l}{7/2\ensuremath{^{-}}}&&&\multicolumn{1}{r@{}}{1875}&\multicolumn{1}{@{.}l}{6 {\it 5}}&\multicolumn{1}{l|}{9/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{2788}&\multicolumn{1}{@{.}l}{8 {\it 6}}&\multicolumn{1}{l}{11/2\ensuremath{^{-}}}&\\
\multicolumn{1}{r@{}}{990}&\multicolumn{1}{@{.}l}{5\ensuremath{^{{\hyperlink{PB30LEVEL7}{c}}}} {\it 3}}&\multicolumn{1}{l}{7/2\ensuremath{^{-}}}&&&\multicolumn{1}{r@{}}{1977}&\multicolumn{1}{@{.}l}{6 {\it 5}}&\multicolumn{1}{l|}{7/2\ensuremath{^{+}},9/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{2961}&\multicolumn{1}{@{.}l}{8 {\it 6}}&\multicolumn{1}{l}{(7/2,9/2\ensuremath{^{-}})}&\\
\multicolumn{1}{r@{}}{1014}&\multicolumn{1}{@{.}l}{2\ensuremath{^{{\hyperlink{PB30LEVEL6}{b}}}} {\it 4}}&\multicolumn{1}{l}{9/2\ensuremath{^{-}}}&&&\multicolumn{1}{r@{}}{2119}&\multicolumn{1}{@{.}l}{5 {\it 5}}&\multicolumn{1}{l|}{9/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{3050}&\multicolumn{1}{@{.}l}{7 {\it 5}}&\multicolumn{1}{l}{(7/2)\ensuremath{^{+}}}&\\
\multicolumn{1}{r@{}}{1185}&\multicolumn{1}{@{.}l}{8 {\it 4}}&\multicolumn{1}{l}{(7/2)\ensuremath{^{-}}}&&&\multicolumn{1}{r@{}}{2151}&\multicolumn{1}{@{.}l}{9 {\it 6}}&\multicolumn{1}{l|}{7/2\ensuremath{^{+}}}&&&&\\
\end{longtable}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB30LEVEL0}{\dagger}}}} From a least squares fit to E\ensuremath{\gamma}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB30LEVEL1}{\ddagger}}}} Configuration=\ensuremath{\nu} f\ensuremath{_{\textnormal{5/2}}^{\textnormal{$-$1}}}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB30LEVEL2}{\#}}}} Configuration=\ensuremath{\nu} p\ensuremath{_{\textnormal{3/2}}^{\textnormal{$-$1}}}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB30LEVEL3}{@}}}} Configuration=\ensuremath{\nu} p\ensuremath{_{\textnormal{1/2}}^{\textnormal{$-$1}}}. The assignment is tentative.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB30LEVEL4}{\&}}}} Configuration=\ensuremath{\nu} i\ensuremath{_{\textnormal{13/2}}^{\textnormal{$-$1}}}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB30LEVEL5}{a}}}} Configuration=\ensuremath{\nu} f\ensuremath{_{\textnormal{7/2}}^{\textnormal{$-$1}}}. The assignment is tentative.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB30LEVEL6}{b}}}} Configuration=\ensuremath{\nu} (f\ensuremath{_{\textnormal{5/2}}^{\textnormal{$-$1}}})\ensuremath{\otimes}2\ensuremath{^{\textnormal{+}}}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB30LEVEL7}{c}}}} Configuration=\ensuremath{\nu} (p\ensuremath{_{\textnormal{3/2}}^{\textnormal{$-$1}}})\ensuremath{\otimes}2\ensuremath{^{\textnormal{+}}}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB30LEVEL8}{d}}}} Configuration=\ensuremath{\nu} (i\ensuremath{_{\textnormal{13/2}}^{\textnormal{$-$1}}})\ensuremath{\otimes}2\ensuremath{^{\textnormal{+}}}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB30LEVEL9}{e}}}} From Adopted Levels.}\\
\vspace{0.5cm}
\underline{\ensuremath{\varepsilon,\beta^+} radiations}\\
\begin{longtable}{cccccccccccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E(decay)$$}&\multicolumn{2}{c}{E(level)}&\multicolumn{2}{c}{I$\beta^{+ \ {\hyperlink{PB30DECAY1}{\ddagger}}}$}&\multicolumn{2}{c}{I$\varepsilon^{{\hyperlink{PB30DECAY1}{\ddagger}}}$}&\multicolumn{2}{c}{Log {\it ft}$^{}$}&\multicolumn{2}{c}{I$(\varepsilon+\beta^{+})^{{\hyperlink{PB30DECAY0}{\dagger}}{\hyperlink{PB30DECAY1}{\ddagger}}}$}&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&
\endfirsthead
\multicolumn{1}{r@{}}{(791}&\multicolumn{1}{@{ }l}{{\it 18})}&\multicolumn{1}{r@{}}{3050}&\multicolumn{1}{@{.}l}{7}&&&\multicolumn{1}{r@{}}{4}&\multicolumn{1}{@{.}l}{0 {\it 5}}&\multicolumn{1}{r@{}}{6}&\multicolumn{1}{@{.}l}{46 {\it 6}}&\multicolumn{1}{r@{}}{4}&\multicolumn{1}{@{.}l}{0 {\it 5}}&\\
\multicolumn{1}{r@{}}{(880}&\multicolumn{1}{@{ }l}{{\it 18})}&\multicolumn{1}{r@{}}{2961}&\multicolumn{1}{@{.}l}{8}&&&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{38 {\it 15}}&\multicolumn{1}{r@{}}{7}&\multicolumn{1}{@{.}l}{02 {\it 6}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{38 {\it 15}}&\\
\multicolumn{1}{r@{}}{(1053}&\multicolumn{1}{@{ }l}{{\it 18})}&\multicolumn{1}{r@{}}{2788}&\multicolumn{1}{@{.}l}{8}&&&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{0 {\it 4}}&\multicolumn{1}{r@{}}{6}&\multicolumn{1}{@{.}l}{85 {\it 6}}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{0 {\it 4}}&\\
\multicolumn{1}{r@{}}{(1293}&\multicolumn{1}{@{ }l}{{\it 18})}&\multicolumn{1}{r@{}}{2548}&\multicolumn{1}{@{.}l}{9}&&&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{6 {\it 3}}&\multicolumn{1}{r@{}}{7}&\multicolumn{1}{@{.}l}{31 {\it 9}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{6 {\it 3}}&\\
\multicolumn{1}{r@{}}{(1335}&\multicolumn{1}{@{ }l}{{\it 18})}&\multicolumn{1}{r@{}}{2506}&\multicolumn{1}{@{.}l}{8}&&&\multicolumn{1}{r@{}}{6}&\multicolumn{1}{@{.}l}{5 {\it 5}}&\multicolumn{1}{r@{}}{6}&\multicolumn{1}{@{.}l}{73 {\it 4}}&\multicolumn{1}{r@{}}{6}&\multicolumn{1}{@{.}l}{5 {\it 5}}&\\
\multicolumn{1}{r@{}}{(1367}&\multicolumn{1}{@{ }l}{{\it 18})}&\multicolumn{1}{r@{}}{2474}&\multicolumn{1}{@{.}l}{6}&&&\multicolumn{1}{r@{}}{5}&\multicolumn{1}{@{.}l}{6 {\it 5}}&\multicolumn{1}{r@{}}{6}&\multicolumn{1}{@{.}l}{82 {\it 5}}&\multicolumn{1}{r@{}}{5}&\multicolumn{1}{@{.}l}{6 {\it 5}}&\\
\multicolumn{1}{r@{}}{(1382}&\multicolumn{1}{@{ }l}{{\it 18})}&\multicolumn{1}{r@{}}{2459}&\multicolumn{1}{@{.}l}{9}&&&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{7 {\it 3}}&\multicolumn{1}{r@{}}{7}&\multicolumn{1}{@{.}l}{15 {\it 5}}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{7 {\it 3}}&\\
\multicolumn{1}{r@{}}{(1403}&\multicolumn{1}{@{ }l}{{\it 18})}&\multicolumn{1}{r@{}}{2439}&\multicolumn{1}{@{.}l}{5}&&&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{3 {\it 3}}&\multicolumn{1}{r@{}}{7}&\multicolumn{1}{@{.}l}{23 {\it 6}}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{3 {\it 3}}&\\
\multicolumn{1}{r@{}}{(1562}&\multicolumn{1}{@{ }l}{{\it 18})}&\multicolumn{1}{r@{}}{2279}&\multicolumn{1}{@{.}l}{8}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{00113 {\it 22}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{26 {\it 16}}&\multicolumn{1}{r@{}}{7}&\multicolumn{1}{@{.}l}{59 {\it 6}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{26 {\it 16}}&\\
\multicolumn{1}{r@{}}{(1633}&\multicolumn{1}{@{ }l}{{\it 18})}&\multicolumn{1}{r@{}}{2208}&\multicolumn{1}{@{.}l}{9}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{00074 {\it 24}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{50 {\it 15}}&\multicolumn{1}{r@{}}{8}&\multicolumn{1}{@{.}l}{03 {\it 13}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{50 {\it 15}}&\\
\multicolumn{1}{r@{}}{(1690}&\multicolumn{1}{@{ }l}{{\it 18})}&\multicolumn{1}{r@{}}{2151}&\multicolumn{1}{@{.}l}{9}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{012 {\it 2}}&\multicolumn{1}{r@{}}{5}&\multicolumn{1}{@{.}l}{6 {\it 6}}&\multicolumn{1}{r@{}}{7}&\multicolumn{1}{@{.}l}{02 {\it 5}}&\multicolumn{1}{r@{}}{5}&\multicolumn{1}{@{.}l}{6 {\it 6}}&\\
\multicolumn{1}{r@{}}{(1723}&\multicolumn{1}{@{ }l}{{\it 18})}&\multicolumn{1}{r@{}}{2119}&\multicolumn{1}{@{.}l}{5}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0035 {\it 8}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{4 {\it 3}}&\multicolumn{1}{r@{}}{7}&\multicolumn{1}{@{.}l}{63 {\it 10}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{4 {\it 3}}&\\
\multicolumn{1}{r@{}}{(1864}&\multicolumn{1}{@{ }l}{{\it 18})}&\multicolumn{1}{r@{}}{1977}&\multicolumn{1}{@{.}l}{6}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{012 {\it 2}}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{4 {\it 3}}&\multicolumn{1}{r@{}}{7}&\multicolumn{1}{@{.}l}{47 {\it 6}}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{4 {\it 3}}&\\
\multicolumn{1}{r@{}}{(1966}&\multicolumn{1}{@{ }l}{{\it 18})}&\multicolumn{1}{r@{}}{1875}&\multicolumn{1}{@{.}l}{6}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{017 {\it 3}}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{2 {\it 3}}&\multicolumn{1}{r@{}}{7}&\multicolumn{1}{@{.}l}{56 {\it 6}}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{2 {\it 3}}&\\
\multicolumn{1}{r@{}}{(1998}&\multicolumn{1}{@{ }l}{{\it 18})}&\multicolumn{1}{r@{}}{1843}&\multicolumn{1}{@{.}l}{8}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{017 {\it 3}}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{0 {\it 3}}&\multicolumn{1}{r@{}}{7}&\multicolumn{1}{@{.}l}{62 {\it 7}}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{0 {\it 3}}&\\
\multicolumn{1}{r@{}}{(2105}&\multicolumn{1}{@{ }l}{{\it 18})}&\multicolumn{1}{r@{}}{1737}&\multicolumn{1}{@{.}l}{3}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{048 {\it 7}}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{9 {\it 5}}&\multicolumn{1}{r@{}}{7}&\multicolumn{1}{@{.}l}{38 {\it 6}}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{9 {\it 5}}&\\
\end{longtable}
\begin{textblock}{29}(0,27.3)
Continued on next page (footnotes at end of table)
\end{textblock}
\clearpage
\begin{longtable}{cccccccccccc@{\extracolsep{\fill}}c}
\\[-.4cm]
\multicolumn{13}{c}{{\bf \small \underline{\ensuremath{^{\textnormal{201}}}Bi \ensuremath{\varepsilon} decay\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1978Ri04,B}{1978Ri04} (continued)}}}\\
\multicolumn{13}{c}{~}\\
\multicolumn{13}{c}{\underline{\ensuremath{\epsilon,\beta^+} radiations (continued)}}\\
\multicolumn{13}{c}{~}\\
\multicolumn{2}{c}{E(decay)$$}&\multicolumn{2}{c}{E(level)}&\multicolumn{2}{c}{I$\beta^{+ \ {\hyperlink{PB30DECAY1}{\ddagger}}}$}&\multicolumn{2}{c}{I$\varepsilon^{{\hyperlink{PB30DECAY1}{\ddagger}}}$}&\multicolumn{2}{c}{Log {\it ft}$^{}$}&\multicolumn{2}{c}{I$(\varepsilon+\beta^{+})^{{\hyperlink{PB30DECAY0}{\dagger}}{\hyperlink{PB30DECAY1}{\ddagger}}}$}&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&
\endhead
\multicolumn{1}{r@{}}{(2191}&\multicolumn{1}{@{ }l}{{\it 18})}&\multicolumn{1}{r@{}}{1651}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{064 {\it 9}}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{9 {\it 5}}&\multicolumn{1}{r@{}}{7}&\multicolumn{1}{@{.}l}{40 {\it 6}}&\multicolumn{1}{r@{}}{4}&\multicolumn{1}{@{.}l}{0 {\it 5}}&\\
\multicolumn{1}{r@{}}{(2352}&\multicolumn{1}{@{ }l}{{\it 18})}&\multicolumn{1}{r@{}}{1490}&\multicolumn{1}{@{.}l}{3}&\multicolumn{1}{r@{}}{$<$0}&\multicolumn{1}{@{.}l}{010}&\multicolumn{1}{r@{}}{$<$0}&\multicolumn{1}{@{.}l}{4}&\multicolumn{1}{r@{}}{$>$8}&\multicolumn{1}{@{.}l}{5}&\multicolumn{1}{r@{}}{$<$0}&\multicolumn{1}{@{.}l}{4}&\\
\multicolumn{1}{r@{}}{(2394}&\multicolumn{1}{@{ }l}{{\it 18})}&\multicolumn{1}{r@{}}{1447}&\multicolumn{1}{@{.}l}{9}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{12 {\it 1}}&\multicolumn{1}{r@{}}{4}&\multicolumn{1}{@{.}l}{5 {\it 5}}&\multicolumn{1}{r@{}}{7}&\multicolumn{1}{@{.}l}{43 {\it 5}}&\multicolumn{1}{r@{}}{4}&\multicolumn{1}{@{.}l}{6 {\it 5}}&\\
\multicolumn{1}{r@{}}{(2427}&\multicolumn{1}{@{ }l}{{\it 18})}&\multicolumn{1}{r@{}}{1415}&\multicolumn{1}{@{.}l}{4}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{12 {\it 2}}&\multicolumn{1}{r@{}}{4}&\multicolumn{1}{@{.}l}{1 {\it 8}}&\multicolumn{1}{r@{}}{7}&\multicolumn{1}{@{.}l}{48 {\it 9}}&\multicolumn{1}{r@{}}{4}&\multicolumn{1}{@{.}l}{2 {\it 8}}&\\
\multicolumn{1}{r@{}}{(2517}&\multicolumn{1}{@{ }l}{{\it 18})}&\multicolumn{1}{r@{}}{1325}&\multicolumn{1}{@{.}l}{4}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{19 {\it 2}}&\multicolumn{1}{r@{}}{5}&\multicolumn{1}{@{.}l}{2 {\it 4}}&\multicolumn{1}{r@{}}{7}&\multicolumn{1}{@{.}l}{40 {\it 4}}&\multicolumn{1}{r@{}}{5}&\multicolumn{1}{@{.}l}{4 {\it 4}}&\\
\multicolumn{1}{r@{}}{(2656}&\multicolumn{1}{@{ }l}{{\it 18})}&\multicolumn{1}{r@{}}{1185}&\multicolumn{1}{@{.}l}{8}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{12 {\it 3}}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{4 {\it 7}}&\multicolumn{1}{r@{}}{7}&\multicolumn{1}{@{.}l}{79 {\it 13}}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{5 {\it 7}}&\\
\multicolumn{1}{r@{}}{(2828}&\multicolumn{1}{@{ }l}{{\it 18})}&\multicolumn{1}{r@{}}{1014}&\multicolumn{1}{@{.}l}{2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{18 {\it 5}}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{7 {\it 7}}&\multicolumn{1}{r@{}}{7}&\multicolumn{1}{@{.}l}{79 {\it 12}}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{9 {\it 8}}&\\
\multicolumn{1}{r@{}}{(2852}&\multicolumn{1}{@{ }l}{{\it 18})}&\multicolumn{1}{r@{}}{990}&\multicolumn{1}{@{.}l}{5}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{28 {\it 5}}&\multicolumn{1}{r@{}}{4}&\multicolumn{1}{@{.}l}{0 {\it 7}}&\multicolumn{1}{r@{}}{7}&\multicolumn{1}{@{.}l}{63 {\it 8}}&\multicolumn{1}{r@{}}{4}&\multicolumn{1}{@{.}l}{3 {\it 7}}&\\
\multicolumn{1}{r@{}}{(2906}&\multicolumn{1}{@{ }l}{{\it 18})}&\multicolumn{1}{r@{}}{936}&\multicolumn{1}{@{.}l}{1}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{36 {\it 7}}&\multicolumn{1}{r@{}}{4}&\multicolumn{1}{@{.}l}{6 {\it 8}}&\multicolumn{1}{r@{}}{7}&\multicolumn{1}{@{.}l}{58 {\it 8}}&\multicolumn{1}{r@{}}{5}&\multicolumn{1}{@{.}l}{0 {\it 9}}&\\
\multicolumn{1}{r@{}}{(3213}&\multicolumn{1}{@{ }l}{{\it 18})}&\multicolumn{1}{r@{}}{629}&\multicolumn{1}{@{.}l}{1}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{71 {\it 11}}&\multicolumn{1}{r@{}}{18}&\multicolumn{1}{@{ }l}{{\it 3}}&\multicolumn{1}{r@{}}{8}&\multicolumn{1}{@{.}l}{65\ensuremath{^{1u}} {\it 7}}&\multicolumn{1}{r@{}}{19}&\multicolumn{1}{@{ }l}{{\it 3}}&\\
\end{longtable}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB30DECAY0}{\dagger}}}} Deduced from the decay scheme using intensity balances considerations and by assuming no direct feeding to the g.s.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB30DECAY1}{\ddagger}}}} Absolute intensity per 100 decays.}\\
\vspace{0.5cm}
\clearpage
\vspace{0.3cm}
\begin{landscape}
\vspace*{-0.5cm}
{\bf \small \underline{\ensuremath{^{\textnormal{201}}}Bi \ensuremath{\varepsilon} decay\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1978Ri04,B}{1978Ri04} (continued)}}\\
\vspace{0.3cm}
\underline{$\gamma$($^{201}$Pb)}\\
\vspace{0.34cm}
\parbox[b][0.3cm]{21.881866cm}{\addtolength{\parindent}{-0.254cm}I\ensuremath{\gamma} normalization: Deduced using \ensuremath{\Sigma}(I(\ensuremath{\gamma}+ce)[g.s. \ensuremath{^{\textnormal{201}}}Pb])=100\% and by assuming that there is no direct feeding to the \ensuremath{^{\textnormal{201}}}Pb g.s. (\ensuremath{J^{\pi}}=5/2\ensuremath{^{-}}).}\\
\vspace{0.34cm}
\begin{longtable}{ccccccccc@{}ccccccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{BI30GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{I\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{BI30GAMMA1}{\ddagger}\hyperlink{BI30GAMMA4}{\&}}}}&\multicolumn{2}{c}{E\ensuremath{_{i}}(level)}&J\ensuremath{^{\pi}_{i}}&\multicolumn{2}{c}{E\ensuremath{_{f}}}&J\ensuremath{^{\pi}_{f}}&Mult.\ensuremath{^{\hyperlink{BI30GAMMA2}{\#}}}&\multicolumn{2}{c}{\ensuremath{\delta}\ensuremath{^{\hyperlink{BI30GAMMA3}{@}}}}&\multicolumn{2}{c}{\ensuremath{\alpha}\ensuremath{^{\hyperlink{BI30GAMMA5}{a}}}}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{(81}&\multicolumn{1}{@{.}l}{4 {\it 10})}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{39 {\it 5}}&\multicolumn{1}{r@{}}{169}&\multicolumn{1}{@{.}l}{9}&\multicolumn{1}{l}{(1/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{88}&\multicolumn{1}{@{.}l}{5 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{-}}}&\multicolumn{1}{l}{[M1]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{16 {\it 12}}&\parbox[t][0.3cm]{7.8432703cm}{\raggedright \%I\ensuremath{\gamma}=0.096 \textit{13}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.8432703cm}{\raggedright E\ensuremath{_{\gamma}}: Not observed experimentally; E\ensuremath{\gamma} from E(level)\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.8432703cm}{\raggedright {\ }{\ }{\ }difference.\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.8432703cm}{\raggedright I\ensuremath{_{\gamma}}: From intensity balance at the 169.9-keV level.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{88}&\multicolumn{1}{@{.}l}{6 {\it 10}}&\multicolumn{1}{r@{}}{4}&\multicolumn{1}{@{.}l}{4 {\it 9}}&\multicolumn{1}{r@{}}{88}&\multicolumn{1}{@{.}l}{5}&\multicolumn{1}{l}{3/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{6}&\multicolumn{1}{@{ }l}{{\it 6}}&\parbox[t][0.3cm]{7.8432703cm}{\raggedright \%I\ensuremath{\gamma}=1.08 \textit{20}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.8432703cm}{\raggedright Mult.: \ensuremath{\alpha}(exp)=10.4 \textit{24} from the intensity balance at the\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.8432703cm}{\raggedright {\ }{\ }{\ }88.5 level.\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.8432703cm}{\raggedright \ensuremath{\alpha}: From intensity balance at the 88.5-keV level.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$138}&\multicolumn{1}{@{.}l}{8 {\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{51 {\it 10}}&&&&&&&\multicolumn{1}{l}{M1}&&&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{67 {\it 9}}&\parbox[t][0.3cm]{7.8432703cm}{\raggedright \%I\ensuremath{\gamma}=0.126 \textit{25}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.8432703cm}{\raggedright Mult.: \ensuremath{\alpha}(K)exp=3.7 \textit{5}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{142}&\multicolumn{1}{@{.}l}{6 {\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{17 {\it 4}}&\multicolumn{1}{r@{}}{2119}&\multicolumn{1}{@{.}l}{5}&\multicolumn{1}{l}{9/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1977}&\multicolumn{1}{@{.}l}{6 }&\multicolumn{1}{@{}l}{7/2\ensuremath{^{+}},9/2\ensuremath{^{+}}}&\multicolumn{1}{l}{[M1]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{40 {\it 8}}&\parbox[t][0.3cm]{7.8432703cm}{\raggedright \%I\ensuremath{\gamma}=0.042 \textit{10}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{171}&\multicolumn{1}{@{.}l}{7 {\it 5}}&\multicolumn{1}{r@{}}{10}&\multicolumn{1}{@{.}l}{4 {\it 5}}&\multicolumn{1}{r@{}}{1185}&\multicolumn{1}{@{.}l}{8}&\multicolumn{1}{l}{(7/2)\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{1014}&\multicolumn{1}{@{.}l}{2 }&\multicolumn{1}{@{}l}{9/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1(+E2)}&\multicolumn{1}{r@{}}{$<$0}&\multicolumn{1}{@{.}l}{5}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{88 {\it 13}}&\parbox[t][0.3cm]{7.8432703cm}{\raggedright \%I\ensuremath{\gamma}=2.56 \textit{17}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.8432703cm}{\raggedright Mult.: From \ensuremath{\alpha}(L)exp=0.27 \textit{3}, L12/L3\ensuremath{>}66.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$181}&\multicolumn{1}{@{.}l}{2 {\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{26 {\it 5}}&&&&&&&&&&&&\parbox[t][0.3cm]{7.8432703cm}{\raggedright \%I\ensuremath{\gamma}=0.064 \textit{13}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$185}&\multicolumn{1}{@{.}l}{5 {\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{34 {\it 7}}&&&&&&&&&&&&\parbox[t][0.3cm]{7.8432703cm}{\raggedright \%I\ensuremath{\gamma}=0.084 \textit{18}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$186}&\multicolumn{1}{@{.}l}{9 {\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{34 {\it 7}}&&&&&&&&&&&&\parbox[t][0.3cm]{7.8432703cm}{\raggedright \%I\ensuremath{\gamma}=0.084 \textit{18}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{224}&\multicolumn{1}{@{.}l}{5 {\it 10}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{6 {\it 3}}&\multicolumn{1}{r@{}}{1875}&\multicolumn{1}{@{.}l}{6}&\multicolumn{1}{l}{9/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1651}&\multicolumn{1}{@{.}l}{0 }&\multicolumn{1}{@{}l}{7/2\ensuremath{^{+}}}&\multicolumn{1}{l}{M1(+E2)}&\multicolumn{1}{r@{}}{$<$0}&\multicolumn{1}{@{.}l}{44}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{90 {\it 6}}&\parbox[t][0.3cm]{7.8432703cm}{\raggedright \%I\ensuremath{\gamma}=0.39 \textit{8}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.8432703cm}{\raggedright Mult.: From \ensuremath{\alpha}(K)exp=0.75 \textit{8}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{239}&\multicolumn{1}{@{.}l}{7 {\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{69 {\it 14}}&\multicolumn{1}{r@{}}{2788}&\multicolumn{1}{@{.}l}{8}&\multicolumn{1}{l}{11/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{2548}&\multicolumn{1}{@{.}l}{9 }&\multicolumn{1}{@{}l}{(11/2\ensuremath{^{-}})}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{47 {\it 7}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{690 {\it 28}}&\parbox[t][0.3cm]{7.8432703cm}{\raggedright \%I\ensuremath{\gamma}=0.170 \textit{35}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.8432703cm}{\raggedright Mult.: From \ensuremath{\alpha}(K)exp=0.55 \textit{2}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{243}&\multicolumn{1}{@{.}l}{1 {\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{34 {\it 7}}&\multicolumn{1}{r@{}}{2119}&\multicolumn{1}{@{.}l}{5}&\multicolumn{1}{l}{9/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1875}&\multicolumn{1}{@{.}l}{6 }&\multicolumn{1}{@{}l}{9/2\ensuremath{^{+}}}&\multicolumn{1}{l}{[M1,E2]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{49 {\it 27}}&\parbox[t][0.3cm]{7.8432703cm}{\raggedright \%I\ensuremath{\gamma}=0.084 \textit{18}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{250}&\multicolumn{1}{@{.}l}{9 {\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{69 {\it 14}}&\multicolumn{1}{r@{}}{2459}&\multicolumn{1}{@{.}l}{9}&\multicolumn{1}{l}{7/2\ensuremath{^{+}},9/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{2208}&\multicolumn{1}{@{.}l}{9 }&\multicolumn{1}{@{}l}{(9/2\ensuremath{^{+}})}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{6 {\it 4}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{57 {\it 12}}&\parbox[t][0.3cm]{7.8432703cm}{\raggedright \%I\ensuremath{\gamma}=0.170 \textit{35}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.8432703cm}{\raggedright Mult.: From \ensuremath{\alpha}(K)exp=0.46 \textit{12}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$273}&\multicolumn{1}{@{.}l}{2 {\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{09 {\it 2}}&&&&&&&&&&&&\parbox[t][0.3cm]{7.8432703cm}{\raggedright \%I\ensuremath{\gamma}=0.022 \textit{5}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{275}&\multicolumn{1}{@{.}l}{5 {\it 10}}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{2 {\it 5}}&\multicolumn{1}{r@{}}{2119}&\multicolumn{1}{@{.}l}{5}&\multicolumn{1}{l}{9/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1843}&\multicolumn{1}{@{.}l}{8 }&\multicolumn{1}{@{}l}{11/2\ensuremath{^{+}}}&\multicolumn{1}{l}{M1(+E2)}&\multicolumn{1}{r@{}}{$<$0}&\multicolumn{1}{@{.}l}{25}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{527 {\it 15}}&\parbox[t][0.3cm]{7.8432703cm}{\raggedright \%I\ensuremath{\gamma}=0.54 \textit{13}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.8432703cm}{\raggedright Mult.: From \ensuremath{\alpha}(K)exp=0.52 \textit{9}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{288}&\multicolumn{1}{@{.}l}{6 {\it 10}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{21 {\it 24}}&\multicolumn{1}{r@{}}{1737}&\multicolumn{1}{@{.}l}{3}&\multicolumn{1}{l}{9/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1447}&\multicolumn{1}{@{.}l}{9 }&\multicolumn{1}{@{}l}{(11/2)\ensuremath{^{+}}}&\multicolumn{1}{l}{M1(+E2)}&\multicolumn{1}{r@{}}{$<$0}&\multicolumn{1}{@{.}l}{24}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{465 {\it 12}}&\parbox[t][0.3cm]{7.8432703cm}{\raggedright \%I\ensuremath{\gamma}=0.30 \textit{6}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.8432703cm}{\raggedright Mult.: From \ensuremath{\alpha}(K)exp=0.45 \textit{5}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$295}&\multicolumn{1}{@{.}l}{7 {\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{43 {\it 9}}&&&&&&&&&&&&\parbox[t][0.3cm]{7.8432703cm}{\raggedright \%I\ensuremath{\gamma}=0.106 \textit{23}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{305}&\multicolumn{1}{@{.}l}{7\ensuremath{^{\hyperlink{BI30GAMMA6}{b}}} {\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{43 {\it 9}}&\multicolumn{1}{r@{}}{1185}&\multicolumn{1}{@{.}l}{8}&\multicolumn{1}{l}{(7/2)\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{879}&\multicolumn{1}{@{.}l}{9? }&\multicolumn{1}{@{}l}{(5/2)\ensuremath{^{-}}}&\multicolumn{1}{l}{[M1]}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{405 {\it 7}}&\parbox[t][0.3cm]{7.8432703cm}{\raggedright \%I\ensuremath{\gamma}=0.106 \textit{23}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{322}&\multicolumn{1}{@{.}l}{3 {\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{78 {\it 16}}&\multicolumn{1}{r@{}}{2474}&\multicolumn{1}{@{.}l}{6}&\multicolumn{1}{l}{(7/2\ensuremath{^{+}},9/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{2151}&\multicolumn{1}{@{.}l}{9 }&\multicolumn{1}{@{}l}{7/2\ensuremath{^{+}}}&\multicolumn{1}{l}{[M1]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{351 {\it 6}}&\parbox[t][0.3cm]{7.8432703cm}{\raggedright \%I\ensuremath{\gamma}=0.19 \textit{4}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{325}&\multicolumn{1}{@{.}l}{7\ensuremath{^{\hyperlink{BI30GAMMA6}{b}}} {\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{69 {\it 14}}&\multicolumn{1}{r@{}}{1651}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{l}{7/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1325}&\multicolumn{1}{@{.}l}{4 }&\multicolumn{1}{@{}l}{7/2\ensuremath{^{-}}}&\multicolumn{1}{l}{[E1]}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0244 {\it 4}}&\parbox[t][0.3cm]{7.8432703cm}{\raggedright \%I\ensuremath{\gamma}=0.170 \textit{35}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{339}&\multicolumn{1}{@{.}l}{7 {\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{60 {\it 12}}&\multicolumn{1}{r@{}}{2548}&\multicolumn{1}{@{.}l}{9}&\multicolumn{1}{l}{(11/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{2208}&\multicolumn{1}{@{.}l}{9 }&\multicolumn{1}{@{}l}{(9/2\ensuremath{^{+}})}&\multicolumn{1}{l}{[E1]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{02220 {\it 34}}&\parbox[t][0.3cm]{7.8432703cm}{\raggedright \%I\ensuremath{\gamma}=0.148 \textit{30}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{368}&\multicolumn{1}{@{.}l}{8 {\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{86 {\it 16}}&\multicolumn{1}{r@{}}{538}&\multicolumn{1}{@{.}l}{8}&\multicolumn{1}{l}{(3/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{169}&\multicolumn{1}{@{.}l}{9 }&\multicolumn{1}{@{}l}{(1/2\ensuremath{^{-}})}&\multicolumn{1}{l}{[M1]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{243 {\it 4}}&\parbox[t][0.3cm]{7.8432703cm}{\raggedright \%I\ensuremath{\gamma}=0.21 \textit{4}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{372}&\multicolumn{1}{@{.}l}{3\ensuremath{^{\hyperlink{BI30GAMMA6}{b}}} {\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{43 {\it 9}}&\multicolumn{1}{r@{}}{910}&\multicolumn{1}{@{.}l}{5?}&\multicolumn{1}{l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{538}&\multicolumn{1}{@{.}l}{8 }&\multicolumn{1}{@{}l}{(3/2\ensuremath{^{-}})}&\multicolumn{1}{l}{[M1]}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{237 {\it 4}}&\parbox[t][0.3cm]{7.8432703cm}{\raggedright \%I\ensuremath{\gamma}=0.106 \textit{23}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{384}&\multicolumn{1}{@{.}l}{4 {\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{52 {\it 10}}&\multicolumn{1}{r@{}}{1875}&\multicolumn{1}{@{.}l}{6}&\multicolumn{1}{l}{9/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1490}&\multicolumn{1}{@{.}l}{3 }&\multicolumn{1}{@{}l}{7/2\ensuremath{^{-}},9/2\ensuremath{^{-}}}&\multicolumn{1}{l}{[E1]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{01683 {\it 25}}&\parbox[t][0.3cm]{7.8432703cm}{\raggedright \%I\ensuremath{\gamma}=0.128 \textit{25}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{387}&\multicolumn{1}{@{.}l}{3 {\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{34 {\it 7}}&\multicolumn{1}{r@{}}{2506}&\multicolumn{1}{@{.}l}{8}&\multicolumn{1}{l}{9/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{2119}&\multicolumn{1}{@{.}l}{5 }&\multicolumn{1}{@{}l}{9/2\ensuremath{^{+}}}&\multicolumn{1}{l}{[M1]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{2134 {\it 33}}&\parbox[t][0.3cm]{7.8432703cm}{\raggedright \%I\ensuremath{\gamma}=0.084 \textit{18}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$393}&\multicolumn{1}{@{.}l}{4 {\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{60 {\it 12}}&&&&&&&&&&&&\parbox[t][0.3cm]{7.8432703cm}{\raggedright \%I\ensuremath{\gamma}=0.148 \textit{30}\vspace{0.1cm}}&\\
\end{longtable}
\clearpage
\begin{longtable}{ccccccccc@{}ccccccc@{\extracolsep{\fill}}c}
\\[-.4cm]
\multicolumn{17}{c}{{\bf \small \underline{\ensuremath{^{\textnormal{201}}}Bi \ensuremath{\varepsilon} decay\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1978Ri04,B}{1978Ri04} (continued)}}}\\
\multicolumn{17}{c}{~}\\
\multicolumn{17}{c}{\underline{$\gamma$($^{201}$Pb) (continued)}}\\
\multicolumn{17}{c}{~~~}\\
\multicolumn{2}{c}{E\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{BI30GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{I\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{BI30GAMMA1}{\ddagger}\hyperlink{BI30GAMMA4}{\&}}}}&\multicolumn{2}{c}{E\ensuremath{_{i}}(level)}&J\ensuremath{^{\pi}_{i}}&\multicolumn{2}{c}{E\ensuremath{_{f}}}&J\ensuremath{^{\pi}_{f}}&Mult.\ensuremath{^{\hyperlink{BI30GAMMA2}{\#}}}&\multicolumn{2}{c}{\ensuremath{\delta}\ensuremath{^{\hyperlink{BI30GAMMA3}{@}}}}&\multicolumn{2}{c}{\ensuremath{\alpha}\ensuremath{^{\hyperlink{BI30GAMMA5}{a}}}}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endhead
\multicolumn{1}{r@{}}{396}&\multicolumn{1}{@{.}l}{1 {\it 10}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{12 {\it 22}}&\multicolumn{1}{r@{}}{1843}&\multicolumn{1}{@{.}l}{8}&\multicolumn{1}{l}{11/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1447}&\multicolumn{1}{@{.}l}{9 }&\multicolumn{1}{@{}l}{(11/2)\ensuremath{^{+}}}&\multicolumn{1}{l}{[M1]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{2009 {\it 31}}&\parbox[t][0.3cm]{8.292671cm}{\raggedright \%I\ensuremath{\gamma}=0.28 \textit{6}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{411}&\multicolumn{1}{@{.}l}{6 {\it 10}}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{2 {\it 4}}&\multicolumn{1}{r@{}}{1737}&\multicolumn{1}{@{.}l}{3}&\multicolumn{1}{l}{9/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1325}&\multicolumn{1}{@{.}l}{4 }&\multicolumn{1}{@{}l}{7/2\ensuremath{^{-}}}&\multicolumn{1}{l}{[E1]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{01448 {\it 22}}&\parbox[t][0.3cm]{8.292671cm}{\raggedright \%I\ensuremath{\gamma}=0.54 \textit{10}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{414}&\multicolumn{1}{@{.}l}{6 {\it 10}}&\multicolumn{1}{r@{}}{5}&\multicolumn{1}{@{.}l}{5 {\it 11}}&\multicolumn{1}{r@{}}{2151}&\multicolumn{1}{@{.}l}{9}&\multicolumn{1}{l}{7/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1737}&\multicolumn{1}{@{.}l}{3 }&\multicolumn{1}{@{}l}{9/2\ensuremath{^{+}}}&\multicolumn{1}{l}{M1(+E2)}&\multicolumn{1}{r@{}}{$<$0}&\multicolumn{1}{@{.}l}{6}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{160 {\it 17}}&\parbox[t][0.3cm]{8.292671cm}{\raggedright \%I\ensuremath{\gamma}=1.36 \textit{28}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.292671cm}{\raggedright Mult.: From \ensuremath{\alpha}(K)exp=0.14 \textit{3}, \ensuremath{\alpha}(L)exp=0.040 \textit{10}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{424}&\multicolumn{1}{@{.}l}{5 {\it 10}}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{7 {\it 7}}&\multicolumn{1}{r@{}}{1415}&\multicolumn{1}{@{.}l}{4}&\multicolumn{1}{l}{9/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{990}&\multicolumn{1}{@{.}l}{5 }&\multicolumn{1}{@{}l}{7/2\ensuremath{^{-}}}&\multicolumn{1}{l}{(E1)}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{01354 {\it 20}}&\parbox[t][0.3cm]{8.292671cm}{\raggedright \%I\ensuremath{\gamma}=0.91 \textit{18}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.292671cm}{\raggedright Mult.: From \ensuremath{\alpha}(K)exp=0.050 \textit{20}, implies E1+M2 with \ensuremath{\delta}=2.1\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.292671cm}{\raggedright {\ }{\ }{\ }\textit{5}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{428}&\multicolumn{1}{@{.}l}{0 {\it 10}}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{9 {\it 5}}&\multicolumn{1}{r@{}}{1875}&\multicolumn{1}{@{.}l}{6}&\multicolumn{1}{l}{9/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1447}&\multicolumn{1}{@{.}l}{9 }&\multicolumn{1}{@{}l}{(11/2)\ensuremath{^{+}}}&\multicolumn{1}{l}{M1(+E2)}&\multicolumn{1}{r@{}}{$<$0}&\multicolumn{1}{@{.}l}{9}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{136 {\it 27}}&\parbox[t][0.3cm]{8.292671cm}{\raggedright \%I\ensuremath{\gamma}=0.71 \textit{13}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.292671cm}{\raggedright Mult.: From \ensuremath{\alpha}(K)exp=0.12 \textit{3}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{450}&\multicolumn{1}{@{.}l}{3 {\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{60 {\it 12}}&\multicolumn{1}{r@{}}{538}&\multicolumn{1}{@{.}l}{8}&\multicolumn{1}{l}{(3/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{88}&\multicolumn{1}{@{.}l}{5 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{-}}}&\multicolumn{1}{l}{[M1]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{1426 {\it 22}}&\parbox[t][0.3cm]{8.292671cm}{\raggedright \%I\ensuremath{\gamma}=0.148 \textit{30}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{460}&\multicolumn{1}{@{.}l}{1 {\it 10}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{6 {\it 3}}&\multicolumn{1}{r@{}}{1875}&\multicolumn{1}{@{.}l}{6}&\multicolumn{1}{l}{9/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1415}&\multicolumn{1}{@{.}l}{4 }&\multicolumn{1}{@{}l}{9/2\ensuremath{^{+}}}&\multicolumn{1}{l}{M1(+E2)}&\multicolumn{1}{r@{}}{$<$0}&\multicolumn{1}{@{.}l}{7}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{118 {\it 16}}&\parbox[t][0.3cm]{8.292671cm}{\raggedright \%I\ensuremath{\gamma}=0.39 \textit{8}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.292671cm}{\raggedright Mult.: From \ensuremath{\alpha}(K)exp=0.16 \textit{6}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$490}&\multicolumn{1}{@{.}l}{8 {\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{78 {\it 16}}&&&&&&&\multicolumn{1}{l}{M1,E2}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{07 {\it 4}}&\parbox[t][0.3cm]{8.292671cm}{\raggedright \%I\ensuremath{\gamma}=0.19 \textit{4}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.292671cm}{\raggedright Mult.: \ensuremath{\alpha}(K)exp=0.06 \textit{4}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$495}&\multicolumn{1}{@{.}l}{2 {\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{43 {\it 9}}&&&&&&&&&&&&\parbox[t][0.3cm]{8.292671cm}{\raggedright \%I\ensuremath{\gamma}=0.106 \textit{23}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{499}&\multicolumn{1}{@{.}l}{9 {\it 10}}&\multicolumn{1}{r@{}}{4}&\multicolumn{1}{@{.}l}{1 {\it 8}}&\multicolumn{1}{r@{}}{1490}&\multicolumn{1}{@{.}l}{3}&\multicolumn{1}{l}{7/2\ensuremath{^{-}},9/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{990}&\multicolumn{1}{@{.}l}{5 }&\multicolumn{1}{@{}l}{7/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1(+E2)}&\multicolumn{1}{r@{}}{$<$1}&\multicolumn{1}{@{.}l}{2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{085 {\it 23}}&\parbox[t][0.3cm]{8.292671cm}{\raggedright \%I\ensuremath{\gamma}=1.01 \textit{20}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.292671cm}{\raggedright Mult.: From \ensuremath{\alpha}(K)exp=0.08 \textit{3}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$511}&\multicolumn{1}{@{.}l}{1\ensuremath{^{\hyperlink{BI30GAMMA6}{b}}} {\it 10}}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{3 {\it 5}}&&&&&&&&&&&&\parbox[t][0.3cm]{8.292671cm}{\raggedright \%I\ensuremath{\gamma}=0.57 \textit{13}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.292671cm}{\raggedright E\ensuremath{_{\gamma}}: Possibly \ensuremath{\gamma}\ensuremath{^{\ensuremath{\pm}}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{529}&\multicolumn{1}{@{.}l}{8 {\it 10}}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{5 {\it 7}}&\multicolumn{1}{r@{}}{1977}&\multicolumn{1}{@{.}l}{6}&\multicolumn{1}{l}{7/2\ensuremath{^{+}},9/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1447}&\multicolumn{1}{@{.}l}{9 }&\multicolumn{1}{@{}l}{(11/2)\ensuremath{^{+}}}&\multicolumn{1}{l}{[M1,E2]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{059 {\it 34}}&\parbox[t][0.3cm]{8.292671cm}{\raggedright \%I\ensuremath{\gamma}=0.86 \textit{18}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{538}&\multicolumn{1}{@{.}l}{7 {\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{43 {\it 9}}&\multicolumn{1}{r@{}}{538}&\multicolumn{1}{@{.}l}{8}&\multicolumn{1}{l}{(3/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{l}{[M1]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0887 {\it 13}}&\parbox[t][0.3cm]{8.292671cm}{\raggedright \%I\ensuremath{\gamma}=0.106 \textit{23}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$548}&\multicolumn{1}{@{.}l}{1 {\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{60 {\it 12}}&&&&&&&&&&&&\parbox[t][0.3cm]{8.292671cm}{\raggedright \%I\ensuremath{\gamma}=0.148 \textit{30}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$552}&\multicolumn{1}{@{.}l}{7 {\it 10}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{21 {\it 24}}&&&&&&&&&&&&\parbox[t][0.3cm]{8.292671cm}{\raggedright \%I\ensuremath{\gamma}=0.30 \textit{6}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{557}&\multicolumn{1}{@{.}l}{7 {\it 10}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{8 {\it 4}}&\multicolumn{1}{r@{}}{2208}&\multicolumn{1}{@{.}l}{9}&\multicolumn{1}{l}{(9/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{1651}&\multicolumn{1}{@{.}l}{0 }&\multicolumn{1}{@{}l}{7/2\ensuremath{^{+}}}&\multicolumn{1}{l}{[M1]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0809 {\it 12}}&\parbox[t][0.3cm]{8.292671cm}{\raggedright \%I\ensuremath{\gamma}=0.44 \textit{10}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{562}&\multicolumn{1}{@{.}l}{5 {\it 10}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{5 {\it 3}}&\multicolumn{1}{r@{}}{1977}&\multicolumn{1}{@{.}l}{6}&\multicolumn{1}{l}{7/2\ensuremath{^{+}},9/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1415}&\multicolumn{1}{@{.}l}{4 }&\multicolumn{1}{@{}l}{9/2\ensuremath{^{+}}}&\multicolumn{1}{l}{[M1,E2]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{051 {\it 28}}&\parbox[t][0.3cm]{8.292671cm}{\raggedright \%I\ensuremath{\gamma}=0.37 \textit{8}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$564}&\multicolumn{1}{@{.}l}{7 {\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{86 {\it 17}}&&&&&&&&&&&&\parbox[t][0.3cm]{8.292671cm}{\raggedright \%I\ensuremath{\gamma}=0.21 \textit{4}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{584}&\multicolumn{1}{@{.}l}{3 {\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{60 {\it 12}}&\multicolumn{1}{r@{}}{2459}&\multicolumn{1}{@{.}l}{9}&\multicolumn{1}{l}{7/2\ensuremath{^{+}},9/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1875}&\multicolumn{1}{@{.}l}{6 }&\multicolumn{1}{@{}l}{9/2\ensuremath{^{+}}}&\multicolumn{1}{l}{[M1]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0716 {\it 11}}&\parbox[t][0.3cm]{8.292671cm}{\raggedright \%I\ensuremath{\gamma}=0.148 \textit{30}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{610}&\multicolumn{1}{@{.}l}{4\ensuremath{^{\hyperlink{BI30GAMMA6}{b}}} {\it 10}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{21 {\it 24}}&\multicolumn{1}{r@{}}{1490}&\multicolumn{1}{@{.}l}{3}&\multicolumn{1}{l}{7/2\ensuremath{^{-}},9/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{879}&\multicolumn{1}{@{.}l}{9? }&\multicolumn{1}{@{}l}{(5/2)\ensuremath{^{-}}}&\multicolumn{1}{l}{[M1,E2]}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{041 {\it 23}}&\parbox[t][0.3cm]{8.292671cm}{\raggedright \%I\ensuremath{\gamma}=0.30 \textit{6}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$614}&\multicolumn{1}{@{.}l}{4 {\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{78 {\it 16}}&&&&&&&&&&&&\parbox[t][0.3cm]{8.292671cm}{\raggedright \%I\ensuremath{\gamma}=0.19 \textit{4}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$618}&\multicolumn{1}{@{.}l}{9 {\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{34 {\it 7}}&&&&&&&&&&&&\parbox[t][0.3cm]{8.292671cm}{\raggedright \%I\ensuremath{\gamma}=0.084 \textit{18}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{629}&\multicolumn{1}{@{.}l}{1 {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{ }l}{{\it 5}}&\multicolumn{1}{r@{}}{629}&\multicolumn{1}{@{.}l}{1}&\multicolumn{1}{l}{13/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M4}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{813 {\it 12}}&\parbox[t][0.3cm]{8.292671cm}{\raggedright \%I\ensuremath{\gamma}=24.7 \textit{12}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.292671cm}{\raggedright Mult.: From \ensuremath{\alpha}(K)exp=0.6 \textit{2}, K/L=2.3 \textit{3} and L12/L3=4 \textit{1} in\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.292671cm}{\raggedright {\ }{\ }{\ }\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1956St05,B}{1956St05}; \ensuremath{\alpha}(L)exp=0.21 \textit{1} in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1978Ri04,B}{1978Ri04}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$642}&\multicolumn{1}{@{.}l}{0 {\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{86 {\it 17}}&&&&&&&&&&&&\parbox[t][0.3cm]{8.292671cm}{\raggedright \%I\ensuremath{\gamma}=0.21 \textit{4}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{651}&\multicolumn{1}{@{.}l}{8 {\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{69 {\it 14}}&\multicolumn{1}{r@{}}{1977}&\multicolumn{1}{@{.}l}{6}&\multicolumn{1}{l}{7/2\ensuremath{^{+}},9/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1325}&\multicolumn{1}{@{.}l}{4 }&\multicolumn{1}{@{}l}{7/2\ensuremath{^{-}}}&\multicolumn{1}{l}{[E1]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{00559 {\it 8}}&\parbox[t][0.3cm]{8.292671cm}{\raggedright \%I\ensuremath{\gamma}=0.170 \textit{35}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{661}&\multicolumn{1}{@{.}l}{5\ensuremath{^{\hyperlink{BI30GAMMA6}{b}}} {\it 10}}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{8 {\it 8}}&\multicolumn{1}{r@{}}{2151}&\multicolumn{1}{@{.}l}{9}&\multicolumn{1}{l}{7/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1490}&\multicolumn{1}{@{.}l}{3 }&\multicolumn{1}{@{}l}{7/2\ensuremath{^{-}},9/2\ensuremath{^{-}}}&\multicolumn{1}{l}{[E1]}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{00543 {\it 8}}&\parbox[t][0.3cm]{8.292671cm}{\raggedright \%I\ensuremath{\gamma}=0.94 \textit{20}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{671}&\multicolumn{1}{@{.}l}{9 {\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{43 {\it 9}}&\multicolumn{1}{r@{}}{2119}&\multicolumn{1}{@{.}l}{5}&\multicolumn{1}{l}{9/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1447}&\multicolumn{1}{@{.}l}{9 }&\multicolumn{1}{@{}l}{(11/2)\ensuremath{^{+}}}&\multicolumn{1}{l}{[M1]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0497 {\it 7}}&\parbox[t][0.3cm]{8.292671cm}{\raggedright \%I\ensuremath{\gamma}=0.106 \textit{23}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$675}&\multicolumn{1}{@{.}l}{4 {\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{34 {\it 7}}&&&&&&&&&&&&\parbox[t][0.3cm]{8.292671cm}{\raggedright \%I\ensuremath{\gamma}=0.084 \textit{18}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$698}&\multicolumn{1}{@{.}l}{8 {\it 10}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{03 {\it 21}}&&&&&&&&&&&&\parbox[t][0.3cm]{8.292671cm}{\raggedright \%I\ensuremath{\gamma}=0.25 \textit{5}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{703}&\multicolumn{1}{@{.}l}{9 {\it 10}}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{3 {\it 7}}&\multicolumn{1}{r@{}}{2119}&\multicolumn{1}{@{.}l}{5}&\multicolumn{1}{l}{9/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1415}&\multicolumn{1}{@{.}l}{4 }&\multicolumn{1}{@{}l}{9/2\ensuremath{^{+}}}&\multicolumn{1}{l}{[M1,E2]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{029 {\it 15}}&\parbox[t][0.3cm]{8.292671cm}{\raggedright \%I\ensuremath{\gamma}=0.81 \textit{18}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{710}&\multicolumn{1}{@{.}l}{0\ensuremath{^{\hyperlink{BI30GAMMA6}{b}}} {\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{52 {\it 10}}&\multicolumn{1}{r@{}}{879}&\multicolumn{1}{@{.}l}{9?}&\multicolumn{1}{l}{(5/2)\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{169}&\multicolumn{1}{@{.}l}{9 }&\multicolumn{1}{@{}l}{(1/2\ensuremath{^{-}})}&\multicolumn{1}{l}{[E2]}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{01333 {\it 19}}&\parbox[t][0.3cm]{8.292671cm}{\raggedright \%I\ensuremath{\gamma}=0.128 \textit{25}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$716}&\multicolumn{1}{@{.}l}{3 {\it 10}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{12 {\it 22}}&&&&&&&&&&&&\parbox[t][0.3cm]{8.292671cm}{\raggedright \%I\ensuremath{\gamma}=0.28 \textit{6}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{723}&\multicolumn{1}{@{.}l}{5 {\it 10}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{03 {\it 21}}&\multicolumn{1}{r@{}}{1737}&\multicolumn{1}{@{.}l}{3}&\multicolumn{1}{l}{9/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1014}&\multicolumn{1}{@{.}l}{2 }&\multicolumn{1}{@{}l}{9/2\ensuremath{^{-}}}&\multicolumn{1}{l}{[E1]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{00457 {\it 7}}&\parbox[t][0.3cm]{8.292671cm}{\raggedright \%I\ensuremath{\gamma}=0.25 \textit{5}\vspace{0.1cm}}&\\
\end{longtable}
\clearpage
\begin{longtable}{ccccccccc@{}ccccccc@{\extracolsep{\fill}}c}
\\[-.4cm]
\multicolumn{17}{c}{{\bf \small \underline{\ensuremath{^{\textnormal{201}}}Bi \ensuremath{\varepsilon} decay\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1978Ri04,B}{1978Ri04} (continued)}}}\\
\multicolumn{17}{c}{~}\\
\multicolumn{17}{c}{\underline{$\gamma$($^{201}$Pb) (continued)}}\\
\multicolumn{17}{c}{~~~}\\
\multicolumn{2}{c}{E\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{BI30GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{I\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{BI30GAMMA1}{\ddagger}\hyperlink{BI30GAMMA4}{\&}}}}&\multicolumn{2}{c}{E\ensuremath{_{i}}(level)}&J\ensuremath{^{\pi}_{i}}&\multicolumn{2}{c}{E\ensuremath{_{f}}}&J\ensuremath{^{\pi}_{f}}&Mult.\ensuremath{^{\hyperlink{BI30GAMMA2}{\#}}}&\multicolumn{2}{c}{\ensuremath{\delta}\ensuremath{^{\hyperlink{BI30GAMMA3}{@}}}}&\multicolumn{2}{c}{\ensuremath{\alpha}\ensuremath{^{\hyperlink{BI30GAMMA5}{a}}}}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endhead
\multicolumn{1}{r@{}}{736}&\multicolumn{1}{@{.}l}{4 {\it 10}}&\multicolumn{1}{r@{}}{5}&\multicolumn{1}{@{.}l}{3 {\it 11}}&\multicolumn{1}{r@{}}{2151}&\multicolumn{1}{@{.}l}{9}&\multicolumn{1}{l}{7/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1415}&\multicolumn{1}{@{.}l}{4 }&\multicolumn{1}{@{}l}{9/2\ensuremath{^{+}}}&\multicolumn{1}{l}{M1}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0391 {\it 6}}&\parbox[t][0.3cm]{8.207071cm}{\raggedright \%I\ensuremath{\gamma}=1.31 \textit{28}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.207071cm}{\raggedright Mult.: From \ensuremath{\alpha}(K)exp=0.048 \textit{10}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{740}&\multicolumn{1}{@{.}l}{7\ensuremath{^{\hyperlink{BI30GAMMA6}{b}}} {\it 10}}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{2 {\it 4}}&\multicolumn{1}{r@{}}{1651}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{l}{7/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{910}&\multicolumn{1}{@{.}l}{5? }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{l}{E1}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{00436 {\it 6}}&\parbox[t][0.3cm]{8.207071cm}{\raggedright \%I\ensuremath{\gamma}=0.54 \textit{10}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.207071cm}{\raggedright Mult.: From \ensuremath{\alpha}(K)exp=0.0030 \textit{20}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{746}&\multicolumn{1}{@{.}l}{8 {\it 10}}&\multicolumn{1}{r@{}}{4}&\multicolumn{1}{@{.}l}{7 {\it 9}}&\multicolumn{1}{r@{}}{1737}&\multicolumn{1}{@{.}l}{3}&\multicolumn{1}{l}{9/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{990}&\multicolumn{1}{@{.}l}{5 }&\multicolumn{1}{@{}l}{7/2\ensuremath{^{-}}}&\multicolumn{1}{l}{E1}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{00430 {\it 6}}&\parbox[t][0.3cm]{8.207071cm}{\raggedright \%I\ensuremath{\gamma}=1.16 \textit{23}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.207071cm}{\raggedright Mult.: From \ensuremath{\alpha}(K)exp=0.0050 \textit{10}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$768}&\multicolumn{1}{@{.}l}{2 {\it 10}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{4 {\it 3}}&&&&&&&&&&&&\parbox[t][0.3cm]{8.207071cm}{\raggedright \%I\ensuremath{\gamma}=0.35 \textit{8}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$772}&\multicolumn{1}{@{.}l}{3 {\it 10}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{6 {\it 3}}&&&&&&&&&&&&\parbox[t][0.3cm]{8.207071cm}{\raggedright \%I\ensuremath{\gamma}=0.39 \textit{8}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{786}&\multicolumn{1}{@{.}l}{4 {\it 5}}&\multicolumn{1}{r@{}}{39}&\multicolumn{1}{@{.}l}{7 {\it 20}}&\multicolumn{1}{r@{}}{1415}&\multicolumn{1}{@{.}l}{4}&\multicolumn{1}{l}{9/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{629}&\multicolumn{1}{@{.}l}{1 }&\multicolumn{1}{@{}l}{13/2\ensuremath{^{+}}}&\multicolumn{1}{l}{E2}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{01077 {\it 15}}&\parbox[t][0.3cm]{8.207071cm}{\raggedright \%I\ensuremath{\gamma}=9.8 \textit{7}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.207071cm}{\raggedright Mult.: From \ensuremath{\alpha}(K)exp=0.0095 \textit{8}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{791}&\multicolumn{1}{@{.}l}{0\ensuremath{^{\hyperlink{BI30GAMMA6}{b}}} {\it 10}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{9 {\it 4}}&\multicolumn{1}{r@{}}{879}&\multicolumn{1}{@{.}l}{9?}&\multicolumn{1}{l}{(5/2)\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{88}&\multicolumn{1}{@{.}l}{5 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1(+E2)}&\multicolumn{1}{r@{}}{$<$1}&\multicolumn{1}{@{.}l}{1}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{027 {\it 6}}&\parbox[t][0.3cm]{8.207071cm}{\raggedright \%I\ensuremath{\gamma}=0.47 \textit{10}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.207071cm}{\raggedright Mult.: From \ensuremath{\alpha}(K)exp=0.028 \textit{11}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{818}&\multicolumn{1}{@{.}l}{9 {\it 5}}&\multicolumn{1}{r@{}}{30}&\multicolumn{1}{@{.}l}{6 {\it 15}}&\multicolumn{1}{r@{}}{1447}&\multicolumn{1}{@{.}l}{9}&\multicolumn{1}{l}{(11/2)\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{629}&\multicolumn{1}{@{.}l}{1 }&\multicolumn{1}{@{}l}{13/2\ensuremath{^{+}}}&\multicolumn{1}{l}{E2+M1}&\multicolumn{1}{r@{}}{7}&\multicolumn{1}{@{.}l}{8 {\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{01023 {\it 17}}&\parbox[t][0.3cm]{8.207071cm}{\raggedright \%I\ensuremath{\gamma}=7.5 \textit{5}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.207071cm}{\raggedright Mult.: From \ensuremath{\alpha}(K)exp=0.0080 \textit{20}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{822}&\multicolumn{1}{@{.}l}{6\ensuremath{^{\hyperlink{BI30GAMMA6}{b}}} {\it 10}}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{4 {\it 7}}&\multicolumn{1}{r@{}}{910}&\multicolumn{1}{@{.}l}{5?}&\multicolumn{1}{l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{88}&\multicolumn{1}{@{.}l}{5 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1(+E2)}&\multicolumn{1}{r@{}}{$<$1}&\multicolumn{1}{@{.}l}{7}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{022 {\it 7}}&\parbox[t][0.3cm]{8.207071cm}{\raggedright \%I\ensuremath{\gamma}=0.84 \textit{18}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.207071cm}{\raggedright Mult.: From \ensuremath{\alpha}(K)exp=0.019 \textit{8}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{832}&\multicolumn{1}{@{.}l}{0 {\it 10}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{8 {\it 4}}&\multicolumn{1}{r@{}}{2279}&\multicolumn{1}{@{.}l}{8}&\multicolumn{1}{l}{(9/2)\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1447}&\multicolumn{1}{@{.}l}{9 }&\multicolumn{1}{@{}l}{(11/2)\ensuremath{^{+}}}&\multicolumn{1}{l}{M1(+E2)}&\multicolumn{1}{r@{}}{$<$1}&\multicolumn{1}{@{.}l}{3}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{023 {\it 6}}&\parbox[t][0.3cm]{8.207071cm}{\raggedright \%I\ensuremath{\gamma}=0.44 \textit{10}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.207071cm}{\raggedright Mult.: From \ensuremath{\alpha}(K)exp=0.028 \textit{14}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$839}&\multicolumn{1}{@{.}l}{7 {\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{34 {\it 7}}&&&&&&&&&&&&\parbox[t][0.3cm]{8.207071cm}{\raggedright \%I\ensuremath{\gamma}=0.084 \textit{18}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{847}&\multicolumn{1}{@{.}l}{7 {\it 10}}&\multicolumn{1}{r@{}}{7}&\multicolumn{1}{@{.}l}{4 {\it 15}}&\multicolumn{1}{r@{}}{936}&\multicolumn{1}{@{.}l}{1}&\multicolumn{1}{l}{7/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{88}&\multicolumn{1}{@{.}l}{5 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{-}}}&\multicolumn{1}{l}{E2}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{00924 {\it 13}}&\parbox[t][0.3cm]{8.207071cm}{\raggedright \%I\ensuremath{\gamma}=1.8 \textit{4}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.207071cm}{\raggedright Mult.: From \ensuremath{\alpha}(K)exp=0.009 \textit{5}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{855}&\multicolumn{1}{@{.}l}{8 {\it 10}}&\multicolumn{1}{r@{}}{5}&\multicolumn{1}{@{.}l}{7 {\it 12}}&\multicolumn{1}{r@{}}{2506}&\multicolumn{1}{@{.}l}{8}&\multicolumn{1}{l}{9/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1651}&\multicolumn{1}{@{.}l}{0 }&\multicolumn{1}{@{}l}{7/2\ensuremath{^{+}}}&\multicolumn{1}{l}{M1(+E2)}&\multicolumn{1}{r@{}}{$<$2}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{020 {\it 7}}&\parbox[t][0.3cm]{8.207071cm}{\raggedright \%I\ensuremath{\gamma}=1.41 \textit{30}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.207071cm}{\raggedright Mult.: From \ensuremath{\alpha}(K)exp=0.020 \textit{10}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$867}&\multicolumn{1}{@{.}l}{0 {\it 10}}&\multicolumn{1}{r@{}}{8}&\multicolumn{1}{@{.}l}{4 {\it 17}}&&&&&&&\multicolumn{1}{l}{E2}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{00883 {\it 13}}&\parbox[t][0.3cm]{8.207071cm}{\raggedright \%I\ensuremath{\gamma}=2.1 \textit{4}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.207071cm}{\raggedright Mult.: \ensuremath{\alpha}(K)exp=0.007 \textit{3}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{879}&\multicolumn{1}{@{.}l}{6\ensuremath{^{\hyperlink{BI30GAMMA6}{b}}} {\it 10}}&\multicolumn{1}{r@{}}{7}&\multicolumn{1}{@{.}l}{0 {\it 14}}&\multicolumn{1}{r@{}}{879}&\multicolumn{1}{@{.}l}{9?}&\multicolumn{1}{l}{(5/2)\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{l}{E2+M1}&\multicolumn{1}{r@{}}{7}&\multicolumn{1}{@{.}l}{3 {\it 16}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{00888 {\it 22}}&\parbox[t][0.3cm]{8.207071cm}{\raggedright \%I\ensuremath{\gamma}=1.73 \textit{35}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.207071cm}{\raggedright Mult.: From \ensuremath{\alpha}(K)exp=0.007 \textit{3}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{885}&\multicolumn{1}{@{.}l}{0 {\it 10}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{4 {\it 3}}&\multicolumn{1}{r@{}}{1875}&\multicolumn{1}{@{.}l}{6}&\multicolumn{1}{l}{9/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{990}&\multicolumn{1}{@{.}l}{5 }&\multicolumn{1}{@{}l}{7/2\ensuremath{^{-}}}&\multicolumn{1}{l}{[E1]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{00313 {\it 4}}&\parbox[t][0.3cm]{8.207071cm}{\raggedright \%I\ensuremath{\gamma}=0.35 \textit{8}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{902}&\multicolumn{1}{@{.}l}{0 {\it 5}}&\multicolumn{1}{r@{}}{34}&\multicolumn{1}{@{.}l}{8 {\it 17}}&\multicolumn{1}{r@{}}{990}&\multicolumn{1}{@{.}l}{5}&\multicolumn{1}{l}{7/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{88}&\multicolumn{1}{@{.}l}{5 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{-}}}&\multicolumn{1}{l}{E2}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{00816 {\it 11}}&\parbox[t][0.3cm]{8.207071cm}{\raggedright \%I\ensuremath{\gamma}=8.6 \textit{6}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.207071cm}{\raggedright Mult.: From \ensuremath{\alpha}(K)exp=0.006 \textit{2}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{911}&\multicolumn{1}{@{.}l}{0\ensuremath{^{\hyperlink{BI30GAMMA6}{b}}} {\it 10}}&\multicolumn{1}{r@{}}{7}&\multicolumn{1}{@{.}l}{9 {\it 16}}&\multicolumn{1}{r@{}}{910}&\multicolumn{1}{@{.}l}{5?}&\multicolumn{1}{l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1(+E2)}&\multicolumn{1}{r@{}}{$<$0}&\multicolumn{1}{@{.}l}{5}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0211 {\it 15}}&\parbox[t][0.3cm]{8.207071cm}{\raggedright \%I\ensuremath{\gamma}=1.9 \textit{4}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.207071cm}{\raggedright Mult.: From \ensuremath{\alpha}(K)exp=0.028 \textit{7}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$916}&\multicolumn{1}{@{.}l}{9 {\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{95 {\it 19}}&&&&&&&&&&&&\parbox[t][0.3cm]{8.207071cm}{\raggedright \%I\ensuremath{\gamma}=0.23 \textit{5}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$924}&\multicolumn{1}{@{.}l}{2 {\it 10}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{12 {\it 22}}&&&&&&&&&&&&\parbox[t][0.3cm]{8.207071cm}{\raggedright \%I\ensuremath{\gamma}=0.28 \textit{6}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{931}&\multicolumn{1}{@{.}l}{6 {\it 10}}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{5 {\it 7}}&\multicolumn{1}{r@{}}{3050}&\multicolumn{1}{@{.}l}{7}&\multicolumn{1}{l}{(7/2)\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{2119}&\multicolumn{1}{@{.}l}{5 }&\multicolumn{1}{@{}l}{9/2\ensuremath{^{+}}}&\multicolumn{1}{l}{M1(+E2)}&\multicolumn{1}{r@{}}{$<$0}&\multicolumn{1}{@{.}l}{6}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0195 {\it 18}}&\parbox[t][0.3cm]{8.207071cm}{\raggedright \%I\ensuremath{\gamma}=0.86 \textit{18}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.207071cm}{\raggedright Mult.: From \ensuremath{\alpha}(K)exp=0.020 \textit{5}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{936}&\multicolumn{1}{@{.}l}{2 {\it 5}}&\multicolumn{1}{r@{}}{47}&\multicolumn{1}{@{.}l}{1 {\it 24}}&\multicolumn{1}{r@{}}{936}&\multicolumn{1}{@{.}l}{1}&\multicolumn{1}{l}{7/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1,E2}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{014 {\it 7}}&\parbox[t][0.3cm]{8.207071cm}{\raggedright \%I\ensuremath{\gamma}=11.6 \textit{7}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.207071cm}{\raggedright Mult.: From \ensuremath{\alpha}(K)exp=0.004 \textit{3}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$957}&\multicolumn{1}{@{.}l}{0 {\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{43 {\it 9}}&&&&&&&&&&&&\parbox[t][0.3cm]{8.207071cm}{\raggedright \%I\ensuremath{\gamma}=0.106 \textit{23}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$960}&\multicolumn{1}{@{.}l}{0 {\it 10}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{21 {\it 24}}&&&&&&&&&&&&\parbox[t][0.3cm]{8.207071cm}{\raggedright \%I\ensuremath{\gamma}=0.30 \textit{6}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{970}&\multicolumn{1}{@{.}l}{0 {\it 10}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{03 {\it 21}}&\multicolumn{1}{r@{}}{2459}&\multicolumn{1}{@{.}l}{9}&\multicolumn{1}{l}{7/2\ensuremath{^{+}},9/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1490}&\multicolumn{1}{@{.}l}{3 }&\multicolumn{1}{@{}l}{7/2\ensuremath{^{-}},9/2\ensuremath{^{-}}}&\multicolumn{1}{l}{[E1]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{00264 {\it 4}}&\parbox[t][0.3cm]{8.207071cm}{\raggedright \%I\ensuremath{\gamma}=0.25 \textit{5}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$978}&\multicolumn{1}{@{.}l}{3 {\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{60 {\it 12}}&&&&&&&&&&&&\parbox[t][0.3cm]{8.207071cm}{\raggedright \%I\ensuremath{\gamma}=0.148 \textit{30}\vspace{0.1cm}}&\\
\end{longtable}
\clearpage
\begin{longtable}{ccccccccc@{}ccccccc@{\extracolsep{\fill}}c}
\\[-.4cm]
\multicolumn{17}{c}{{\bf \small \underline{\ensuremath{^{\textnormal{201}}}Bi \ensuremath{\varepsilon} decay\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1978Ri04,B}{1978Ri04} (continued)}}}\\
\multicolumn{17}{c}{~}\\
\multicolumn{17}{c}{\underline{$\gamma$($^{201}$Pb) (continued)}}\\
\multicolumn{17}{c}{~~~}\\
\multicolumn{2}{c}{E\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{BI30GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{I\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{BI30GAMMA1}{\ddagger}\hyperlink{BI30GAMMA4}{\&}}}}&\multicolumn{2}{c}{E\ensuremath{_{i}}(level)}&J\ensuremath{^{\pi}_{i}}&\multicolumn{2}{c}{E\ensuremath{_{f}}}&J\ensuremath{^{\pi}_{f}}&Mult.\ensuremath{^{\hyperlink{BI30GAMMA2}{\#}}}&\multicolumn{2}{c}{\ensuremath{\delta}\ensuremath{^{\hyperlink{BI30GAMMA3}{@}}}}&\multicolumn{2}{c}{\ensuremath{\alpha}\ensuremath{^{\hyperlink{BI30GAMMA5}{a}}}}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endhead
\multicolumn{1}{r@{}}{986}&\multicolumn{1}{@{.}l}{5 {\it 10}}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{5 {\it 7}}&\multicolumn{1}{r@{}}{1977}&\multicolumn{1}{@{.}l}{6}&\multicolumn{1}{l}{7/2\ensuremath{^{+}},9/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{990}&\multicolumn{1}{@{.}l}{5 }&\multicolumn{1}{@{}l}{7/2\ensuremath{^{-}}}&\multicolumn{1}{l}{E1}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{00256 {\it 4}}&\parbox[t][0.3cm]{7.3146906cm}{\raggedright \%I\ensuremath{\gamma}=0.86 \textit{18}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.3146906cm}{\raggedright Mult.: From \ensuremath{\alpha}(K)exp=0.005 \textit{3}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{990}&\multicolumn{1}{@{.}l}{6 {\it 5}}&\multicolumn{1}{r@{}}{13}&\multicolumn{1}{@{.}l}{5 {\it 7}}&\multicolumn{1}{r@{}}{990}&\multicolumn{1}{@{.}l}{5}&\multicolumn{1}{l}{7/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{l}{E2+M1}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{2 {\it 6}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0087 {\it 13}}&\parbox[t][0.3cm]{7.3146906cm}{\raggedright \%I\ensuremath{\gamma}=3.33 \textit{22}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.3146906cm}{\raggedright Mult.: From \ensuremath{\alpha}(K)exp=0.007 \textit{2}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$998}&\multicolumn{1}{@{.}l}{6 {\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{52 {\it 10}}&&&&&&&&&&&&\parbox[t][0.3cm]{7.3146906cm}{\raggedright \%I\ensuremath{\gamma}=0.128 \textit{25}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$1005}&\multicolumn{1}{@{.}l}{7 {\it 10}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{29 {\it 25}}&&&&&&&&&&&&\parbox[t][0.3cm]{7.3146906cm}{\raggedright \%I\ensuremath{\gamma}=0.32 \textit{6}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1014}&\multicolumn{1}{@{.}l}{1 {\it 5}}&\multicolumn{1}{r@{}}{44}&\multicolumn{1}{@{.}l}{6 {\it 22}}&\multicolumn{1}{r@{}}{1014}&\multicolumn{1}{@{.}l}{2}&\multicolumn{1}{l}{9/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{l}{E2}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{00648 {\it 9}}&\parbox[t][0.3cm]{7.3146906cm}{\raggedright \%I\ensuremath{\gamma}=11.0 \textit{7}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.3146906cm}{\raggedright Mult.: From \ensuremath{\alpha}(K)exp=0.005 \textit{2}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$1019}&\multicolumn{1}{@{.}l}{7 {\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{78 {\it 16}}&&&&&&&&&&&&\parbox[t][0.3cm]{7.3146906cm}{\raggedright \%I\ensuremath{\gamma}=0.19 \textit{4}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1024}&\multicolumn{1}{@{.}l}{2 {\it 10}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{8 {\it 4}}&\multicolumn{1}{r@{}}{2439}&\multicolumn{1}{@{.}l}{5}&\multicolumn{1}{l}{7/2\ensuremath{^{+}},9/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1415}&\multicolumn{1}{@{.}l}{4 }&\multicolumn{1}{@{}l}{9/2\ensuremath{^{+}}}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{9 {\it 11}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0075 {\it 13}}&\parbox[t][0.3cm]{7.3146906cm}{\raggedright \%I\ensuremath{\gamma}=0.44 \textit{10}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.3146906cm}{\raggedright Mult.: From \ensuremath{\alpha}(K)exp=0.006 \textit{3}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$1033}&\multicolumn{1}{@{.}l}{0 {\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{69 {\it 14}}&&&&&&&&&&&&\parbox[t][0.3cm]{7.3146906cm}{\raggedright \%I\ensuremath{\gamma}=0.170 \textit{35}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1042}&\multicolumn{1}{@{.}l}{8 {\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{95 {\it 19}}&\multicolumn{1}{r@{}}{1977}&\multicolumn{1}{@{.}l}{6}&\multicolumn{1}{l}{7/2\ensuremath{^{+}},9/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{936}&\multicolumn{1}{@{.}l}{1 }&\multicolumn{1}{@{}l}{7/2\ensuremath{^{-}}}&\multicolumn{1}{l}{[E1]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{32\ensuremath{\times10^{-3}} {\it 3}}&\parbox[t][0.3cm]{7.3146906cm}{\raggedright \%I\ensuremath{\gamma}=0.23 \textit{5}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1051}&\multicolumn{1}{@{.}l}{6 {\it 10}}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{3 {\it 5}}&\multicolumn{1}{r@{}}{2788}&\multicolumn{1}{@{.}l}{8}&\multicolumn{1}{l}{11/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{1737}&\multicolumn{1}{@{.}l}{3 }&\multicolumn{1}{@{}l}{9/2\ensuremath{^{+}}}&\multicolumn{1}{l}{E1}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{29\ensuremath{\times10^{-3}} {\it 3}}&\parbox[t][0.3cm]{7.3146906cm}{\raggedright \%I\ensuremath{\gamma}=0.57 \textit{13}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.3146906cm}{\raggedright Mult.: From \ensuremath{\alpha}(K)exp=0.0011 \textit{10}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1059}&\multicolumn{1}{@{.}l}{7 {\it 10}}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{1 {\it 4}}&\multicolumn{1}{r@{}}{2474}&\multicolumn{1}{@{.}l}{6}&\multicolumn{1}{l}{(7/2\ensuremath{^{+}},9/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{1415}&\multicolumn{1}{@{.}l}{4 }&\multicolumn{1}{@{}l}{9/2\ensuremath{^{+}}}&\multicolumn{1}{l}{[M1]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{01533 {\it 22}}&\parbox[t][0.3cm]{7.3146906cm}{\raggedright \%I\ensuremath{\gamma}=0.52 \textit{10}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1091}&\multicolumn{1}{@{.}l}{5 {\it 10}}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{6 {\it 6}}&\multicolumn{1}{r@{}}{2506}&\multicolumn{1}{@{.}l}{8}&\multicolumn{1}{l}{9/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1415}&\multicolumn{1}{@{.}l}{4 }&\multicolumn{1}{@{}l}{9/2\ensuremath{^{+}}}&\multicolumn{1}{l}{[M1]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{01421 {\it 20}}&\parbox[t][0.3cm]{7.3146906cm}{\raggedright \%I\ensuremath{\gamma}=0.64 \textit{15}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1108}&\multicolumn{1}{@{.}l}{1 {\it 5}}&\multicolumn{1}{r@{}}{16}&\multicolumn{1}{@{.}l}{4 {\it 8}}&\multicolumn{1}{r@{}}{1737}&\multicolumn{1}{@{.}l}{3}&\multicolumn{1}{l}{9/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{629}&\multicolumn{1}{@{.}l}{1 }&\multicolumn{1}{@{}l}{13/2\ensuremath{^{+}}}&\multicolumn{1}{l}{E2}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{00546 {\it 8}}&\parbox[t][0.3cm]{7.3146906cm}{\raggedright \%I\ensuremath{\gamma}=4.04 \textit{27}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.3146906cm}{\raggedright Mult.: From \ensuremath{\alpha}(K)exp=0.005 \textit{1}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1137}&\multicolumn{1}{@{.}l}{7\ensuremath{^{\hyperlink{BI30GAMMA6}{b}}} {\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{67 {\it 13}}&\multicolumn{1}{r@{}}{2151}&\multicolumn{1}{@{.}l}{9}&\multicolumn{1}{l}{7/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1014}&\multicolumn{1}{@{.}l}{2 }&\multicolumn{1}{@{}l}{9/2\ensuremath{^{-}}}&\multicolumn{1}{l}{[E1]}&&&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{99\ensuremath{\times10^{-3}} {\it 3}}&\parbox[t][0.3cm]{7.3146906cm}{\raggedright \%I\ensuremath{\gamma}=0.165 \textit{33}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$1147}&\multicolumn{1}{@{.}l}{1 {\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{86 {\it 17}}&&&&&&&&&&&&\parbox[t][0.3cm]{7.3146906cm}{\raggedright \%I\ensuremath{\gamma}=0.21 \textit{4}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$1151}&\multicolumn{1}{@{.}l}{5 {\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{34 {\it 7}}&&&&&&&&&&&&\parbox[t][0.3cm]{7.3146906cm}{\raggedright \%I\ensuremath{\gamma}=0.084 \textit{18}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1161}&\multicolumn{1}{@{.}l}{2 {\it 10}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{29 {\it 25}}&\multicolumn{1}{r@{}}{2151}&\multicolumn{1}{@{.}l}{9}&\multicolumn{1}{l}{7/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{990}&\multicolumn{1}{@{.}l}{5 }&\multicolumn{1}{@{}l}{7/2\ensuremath{^{-}}}&\multicolumn{1}{l}{[E1]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{92\ensuremath{\times10^{-3}} {\it 3}}&\parbox[t][0.3cm]{7.3146906cm}{\raggedright \%I\ensuremath{\gamma}=0.32 \textit{6}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$1175}&\multicolumn{1}{@{.}l}{0 {\it 10}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{5 {\it 3}}&&&&&&&&&&&&\parbox[t][0.3cm]{7.3146906cm}{\raggedright \%I\ensuremath{\gamma}=0.37 \textit{8}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1183}&\multicolumn{1}{@{.}l}{7 {\it 10}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{03 {\it 21}}&\multicolumn{1}{r@{}}{2119}&\multicolumn{1}{@{.}l}{5}&\multicolumn{1}{l}{9/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{936}&\multicolumn{1}{@{.}l}{1 }&\multicolumn{1}{@{}l}{7/2\ensuremath{^{-}}}&\multicolumn{1}{l}{[E1]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{87\ensuremath{\times10^{-3}} {\it 3}}&\parbox[t][0.3cm]{7.3146906cm}{\raggedright \%I\ensuremath{\gamma}=0.25 \textit{5}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1186}&\multicolumn{1}{@{.}l}{5 {\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{87 {\it 17}}&\multicolumn{1}{r@{}}{1185}&\multicolumn{1}{@{.}l}{8}&\multicolumn{1}{l}{(7/2)\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{l}{[M1]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{01149 {\it 16}}&\parbox[t][0.3cm]{7.3146906cm}{\raggedright \%I\ensuremath{\gamma}=0.21 \textit{4}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$1193}&\multicolumn{1}{@{.}l}{1 {\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{69 {\it 14}}&&&&&&&&&&&&\parbox[t][0.3cm]{7.3146906cm}{\raggedright \%I\ensuremath{\gamma}=0.170 \textit{35}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$1196}&\multicolumn{1}{@{.}l}{2 {\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{60 {\it 12}}&&&&&&&&&&&&\parbox[t][0.3cm]{7.3146906cm}{\raggedright \%I\ensuremath{\gamma}=0.148 \textit{30}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$1203}&\multicolumn{1}{@{.}l}{2 {\it 10}}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{4 {\it 5}}&&&&&&&\multicolumn{1}{l}{M1+E2}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0079 {\it 32}}&\parbox[t][0.3cm]{7.3146906cm}{\raggedright \%I\ensuremath{\gamma}=0.59 \textit{13}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.3146906cm}{\raggedright Mult.: \ensuremath{\alpha}(K)exp=0.009 \textit{5}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1214}&\multicolumn{1}{@{.}l}{5 {\it 5}}&\multicolumn{1}{r@{}}{10}&\multicolumn{1}{@{.}l}{1 {\it 5}}&\multicolumn{1}{r@{}}{1843}&\multicolumn{1}{@{.}l}{8}&\multicolumn{1}{l}{11/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{629}&\multicolumn{1}{@{.}l}{1 }&\multicolumn{1}{@{}l}{13/2\ensuremath{^{+}}}&\multicolumn{1}{l}{M1(+E2)}&\multicolumn{1}{r@{}}{$<$2}&\multicolumn{1}{@{.}l}{3}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0082 {\it 26}}&\parbox[t][0.3cm]{7.3146906cm}{\raggedright \%I\ensuremath{\gamma}=2.49 \textit{17}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.3146906cm}{\raggedright Mult.: From \ensuremath{\alpha}(K)exp=0.006 \textit{3}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$1225}&\multicolumn{1}{@{.}l}{6 {\it 10}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{4 {\it 3}}&&&&&&&&&&&&\parbox[t][0.3cm]{7.3146906cm}{\raggedright \%I\ensuremath{\gamma}=0.35 \textit{8}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$1234}&\multicolumn{1}{@{.}l}{7 {\it 10}}&\multicolumn{1}{r@{}}{7}&\multicolumn{1}{@{.}l}{3 {\it 15}}&&&&&&&&&&&&\parbox[t][0.3cm]{7.3146906cm}{\raggedright \%I\ensuremath{\gamma}=1.8 \textit{4}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1241}&\multicolumn{1}{@{.}l}{4\ensuremath{^{\hyperlink{BI30GAMMA6}{b}}} {\it 10}}&\multicolumn{1}{r@{}}{6}&\multicolumn{1}{@{.}l}{0 {\it 12}}&\multicolumn{1}{r@{}}{2151}&\multicolumn{1}{@{.}l}{9}&\multicolumn{1}{l}{7/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{910}&\multicolumn{1}{@{.}l}{5? }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{l}{(E1)}&&&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{74\ensuremath{\times10^{-3}} {\it 2}}&\parbox[t][0.3cm]{7.3146906cm}{\raggedright \%I\ensuremath{\gamma}=1.48 \textit{30}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.3146906cm}{\raggedright Mult.: From \ensuremath{\alpha}(K)exp=0.004 \textit{3}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$1244}&\multicolumn{1}{@{.}l}{8 {\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{77 {\it 15}}&&&&&&&&&&&&\parbox[t][0.3cm]{7.3146906cm}{\raggedright \%I\ensuremath{\gamma}=0.19 \textit{4}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1253}&\multicolumn{1}{@{.}l}{8 {\it 10}}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{1 {\it 6}}&\multicolumn{1}{r@{}}{2439}&\multicolumn{1}{@{.}l}{5}&\multicolumn{1}{l}{7/2\ensuremath{^{+}},9/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1185}&\multicolumn{1}{@{.}l}{8 }&\multicolumn{1}{@{}l}{(7/2)\ensuremath{^{-}}}&\multicolumn{1}{l}{[E1]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{71\ensuremath{\times10^{-3}} {\it 2}}&\parbox[t][0.3cm]{7.3146906cm}{\raggedright \%I\ensuremath{\gamma}=0.76 \textit{15}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1265}&\multicolumn{1}{@{.}l}{6 {\it 10}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{03 {\it 21}}&\multicolumn{1}{r@{}}{2279}&\multicolumn{1}{@{.}l}{8}&\multicolumn{1}{l}{(9/2)\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1014}&\multicolumn{1}{@{.}l}{2 }&\multicolumn{1}{@{}l}{9/2\ensuremath{^{-}}}&\multicolumn{1}{l}{[E1]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{69\ensuremath{\times10^{-3}} {\it 2}}&\parbox[t][0.3cm]{7.3146906cm}{\raggedright \%I\ensuremath{\gamma}=0.25 \textit{5}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$1269}&\multicolumn{1}{@{.}l}{0 {\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{76 {\it 15}}&&&&&&&&&&&&\parbox[t][0.3cm]{7.3146906cm}{\raggedright \%I\ensuremath{\gamma}=0.19 \textit{4}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$1275}&\multicolumn{1}{@{.}l}{1 {\it 10}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{01 {\it 20}}&&&&&&&&&&&&\parbox[t][0.3cm]{7.3146906cm}{\raggedright \%I\ensuremath{\gamma}=0.25 \textit{5}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$1278}&\multicolumn{1}{@{.}l}{1 {\it 10}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{29 {\it 25}}&&&&&&&&&&&&\parbox[t][0.3cm]{7.3146906cm}{\raggedright \%I\ensuremath{\gamma}=0.32 \textit{6}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1288}&\multicolumn{1}{@{.}l}{9 {\it 10}}&\multicolumn{1}{r@{}}{7}&\multicolumn{1}{@{.}l}{9 {\it 16}}&\multicolumn{1}{r@{}}{2474}&\multicolumn{1}{@{.}l}{6}&\multicolumn{1}{l}{(7/2\ensuremath{^{+}},9/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{1185}&\multicolumn{1}{@{.}l}{8 }&\multicolumn{1}{@{}l}{(7/2)\ensuremath{^{-}}}&\multicolumn{1}{l}{(E1)}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{65\ensuremath{\times10^{-3}} {\it 2}}&\parbox[t][0.3cm]{7.3146906cm}{\raggedright \%I\ensuremath{\gamma}=1.9 \textit{4}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.3146906cm}{\raggedright Mult.: From \ensuremath{\alpha}(K)exp=0.003 \textit{1}.\vspace{0.1cm}}&\\
\end{longtable}
\clearpage
\begin{longtable}{ccccccccc@{}ccccccc@{\extracolsep{\fill}}c}
\\[-.4cm]
\multicolumn{17}{c}{{\bf \small \underline{\ensuremath{^{\textnormal{201}}}Bi \ensuremath{\varepsilon} decay\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1978Ri04,B}{1978Ri04} (continued)}}}\\
\multicolumn{17}{c}{~}\\
\multicolumn{17}{c}{\underline{$\gamma$($^{201}$Pb) (continued)}}\\
\multicolumn{17}{c}{~~~}\\
\multicolumn{2}{c}{E\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{BI30GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{I\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{BI30GAMMA1}{\ddagger}\hyperlink{BI30GAMMA4}{\&}}}}&\multicolumn{2}{c}{E\ensuremath{_{i}}(level)}&J\ensuremath{^{\pi}_{i}}&\multicolumn{2}{c}{E\ensuremath{_{f}}}&J\ensuremath{^{\pi}_{f}}&Mult.\ensuremath{^{\hyperlink{BI30GAMMA2}{\#}}}&\multicolumn{2}{c}{\ensuremath{\delta}\ensuremath{^{\hyperlink{BI30GAMMA3}{@}}}}&\multicolumn{2}{c}{\ensuremath{\alpha}\ensuremath{^{\hyperlink{BI30GAMMA5}{a}}}}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endhead
\multicolumn{1}{r@{}}{1298}&\multicolumn{1}{@{.}l}{4 {\it 10}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{4 {\it 3}}&\multicolumn{1}{r@{}}{2208}&\multicolumn{1}{@{.}l}{9}&\multicolumn{1}{l}{(9/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{910}&\multicolumn{1}{@{.}l}{5? }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{l}{[M2]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{02165 {\it 31}}&\parbox[t][0.3cm]{7.100691cm}{\raggedright \%I\ensuremath{\gamma}=0.35 \textit{8}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1313}&\multicolumn{1}{@{.}l}{2 {\it 10}}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{2 {\it 4}}&\multicolumn{1}{r@{}}{3050}&\multicolumn{1}{@{.}l}{7}&\multicolumn{1}{l}{(7/2)\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1737}&\multicolumn{1}{@{.}l}{3 }&\multicolumn{1}{@{}l}{9/2\ensuremath{^{+}}}&\multicolumn{1}{l}{[M1]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{00889 {\it 13}}&\parbox[t][0.3cm]{7.100691cm}{\raggedright \%I\ensuremath{\gamma}=0.54 \textit{10}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1320}&\multicolumn{1}{@{.}l}{9 {\it 10}}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{1 {\it 6}}&\multicolumn{1}{r@{}}{2506}&\multicolumn{1}{@{.}l}{8}&\multicolumn{1}{l}{9/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1185}&\multicolumn{1}{@{.}l}{8 }&\multicolumn{1}{@{}l}{(7/2)\ensuremath{^{-}}}&\multicolumn{1}{l}{[E1]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{59\ensuremath{\times10^{-3}} {\it 2}}&\parbox[t][0.3cm]{7.100691cm}{\raggedright \%I\ensuremath{\gamma}=0.76 \textit{15}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1325}&\multicolumn{1}{@{.}l}{2 {\it 5}}&\multicolumn{1}{r@{}}{25}&\multicolumn{1}{@{.}l}{4 {\it 13}}&\multicolumn{1}{r@{}}{1325}&\multicolumn{1}{@{.}l}{4}&\multicolumn{1}{l}{7/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{6 {\it 4}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0074 {\it 11}}&\parbox[t][0.3cm]{7.100691cm}{\raggedright \%I\ensuremath{\gamma}=6.3 \textit{4}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.100691cm}{\raggedright Mult.: From \ensuremath{\alpha}(K)exp=0.006 \textit{1}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$1358}&\multicolumn{1}{@{.}l}{1 {\it 10}}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{4 {\it 7}}&&&&&&&&&&&&\parbox[t][0.3cm]{7.100691cm}{\raggedright \%I\ensuremath{\gamma}=0.84 \textit{18}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$1380}&\multicolumn{1}{@{.}l}{4 {\it 10}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{4 {\it 3}}&&&&&&&&&&&&\parbox[t][0.3cm]{7.100691cm}{\raggedright \%I\ensuremath{\gamma}=0.35 \textit{8}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$1389}&\multicolumn{1}{@{.}l}{1 {\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{75 {\it 15}}&&&&&&&&&&&&\parbox[t][0.3cm]{7.100691cm}{\raggedright \%I\ensuremath{\gamma}=0.18 \textit{4}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$1394}&\multicolumn{1}{@{.}l}{2 {\it 10}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{5 {\it 3}}&&&&&&&&&&&&\parbox[t][0.3cm]{7.100691cm}{\raggedright \%I\ensuremath{\gamma}=0.37 \textit{8}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1400}&\multicolumn{1}{@{.}l}{3\ensuremath{^{\hyperlink{BI30GAMMA6}{b}}} {\it 10}}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{2 {\it 4}}&\multicolumn{1}{r@{}}{2279}&\multicolumn{1}{@{.}l}{8}&\multicolumn{1}{l}{(9/2)\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{879}&\multicolumn{1}{@{.}l}{9? }&\multicolumn{1}{@{}l}{(5/2)\ensuremath{^{-}}}&\multicolumn{1}{l}{[M2]}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{01781 {\it 25}}&\parbox[t][0.3cm]{7.100691cm}{\raggedright \%I\ensuremath{\gamma}=0.54 \textit{10}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$1411}&\multicolumn{1}{@{.}l}{0 {\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{95 {\it 19}}&&&&&&&&&&&&\parbox[t][0.3cm]{7.100691cm}{\raggedright \%I\ensuremath{\gamma}=0.23 \textit{5}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$1417}&\multicolumn{1}{@{.}l}{9 {\it 10}}&\multicolumn{1}{r@{}}{4}&\multicolumn{1}{@{.}l}{8 {\it 10}}&&&&&&&\multicolumn{1}{l}{(E2)}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{00347 {\it 5}}&\parbox[t][0.3cm]{7.100691cm}{\raggedright \%I\ensuremath{\gamma}=1.18 \textit{25}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.100691cm}{\raggedright Mult.: \ensuremath{\alpha}(K)exp=0.003 \textit{2}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$1420}&\multicolumn{1}{@{.}l}{9 {\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{52 {\it 10}}&&&&&&&&&&&&\parbox[t][0.3cm]{7.100691cm}{\raggedright \%I\ensuremath{\gamma}=0.128 \textit{25}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1469}&\multicolumn{1}{@{.}l}{5 {\it 10}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{4 {\it 3}}&\multicolumn{1}{r@{}}{2459}&\multicolumn{1}{@{.}l}{9}&\multicolumn{1}{l}{7/2\ensuremath{^{+}},9/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{990}&\multicolumn{1}{@{.}l}{5 }&\multicolumn{1}{@{}l}{7/2\ensuremath{^{-}}}&\multicolumn{1}{l}{[E1]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{43\ensuremath{\times10^{-3}} {\it 2}}&\parbox[t][0.3cm]{7.100691cm}{\raggedright \%I\ensuremath{\gamma}=0.35 \textit{8}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1472}&\multicolumn{1}{@{.}l}{1 {\it 10}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{5 {\it 3}}&\multicolumn{1}{r@{}}{2961}&\multicolumn{1}{@{.}l}{8}&\multicolumn{1}{l}{(7/2,9/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{1490}&\multicolumn{1}{@{.}l}{3 }&\multicolumn{1}{@{}l}{7/2\ensuremath{^{-}},9/2\ensuremath{^{-}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{7.100691cm}{\raggedright \%I\ensuremath{\gamma}=0.37 \textit{8}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1490}&\multicolumn{1}{@{.}l}{1 {\it 10}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{6 {\it 3}}&\multicolumn{1}{r@{}}{1490}&\multicolumn{1}{@{.}l}{3}&\multicolumn{1}{l}{7/2\ensuremath{^{-}},9/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{l}{[M1,E2]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0049 {\it 17}}&\parbox[t][0.3cm]{7.100691cm}{\raggedright \%I\ensuremath{\gamma}=0.39 \textit{8}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1503}&\multicolumn{1}{@{.}l}{0 {\it 10}}&\multicolumn{1}{r@{}}{4}&\multicolumn{1}{@{.}l}{3 {\it 9}}&\multicolumn{1}{r@{}}{2439}&\multicolumn{1}{@{.}l}{5}&\multicolumn{1}{l}{7/2\ensuremath{^{+}},9/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{936}&\multicolumn{1}{@{.}l}{1 }&\multicolumn{1}{@{}l}{7/2\ensuremath{^{-}}}&\multicolumn{1}{l}{[E1]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{40\ensuremath{\times10^{-3}} {\it 2}}&\parbox[t][0.3cm]{7.100691cm}{\raggedright \%I\ensuremath{\gamma}=1.06 \textit{23}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$1505}&\multicolumn{1}{@{.}l}{7 {\it 10}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{9 {\it 4}}&&&&&&&&&&&&\parbox[t][0.3cm]{7.100691cm}{\raggedright \%I\ensuremath{\gamma}=0.47 \textit{10}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1516}&\multicolumn{1}{@{.}l}{4 {\it 10}}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{8 {\it 8}}&\multicolumn{1}{r@{}}{2506}&\multicolumn{1}{@{.}l}{8}&\multicolumn{1}{l}{9/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{990}&\multicolumn{1}{@{.}l}{5 }&\multicolumn{1}{@{}l}{7/2\ensuremath{^{-}}}&\multicolumn{1}{l}{[E1]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{39\ensuremath{\times10^{-3}} {\it 2}}&\parbox[t][0.3cm]{7.100691cm}{\raggedright \%I\ensuremath{\gamma}=0.94 \textit{20}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1523}&\multicolumn{1}{@{.}l}{3 {\it 10}}&\multicolumn{1}{r@{}}{5}&\multicolumn{1}{@{.}l}{5 {\it 11}}&\multicolumn{1}{r@{}}{2459}&\multicolumn{1}{@{.}l}{9}&\multicolumn{1}{l}{7/2\ensuremath{^{+}},9/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{936}&\multicolumn{1}{@{.}l}{1 }&\multicolumn{1}{@{}l}{7/2\ensuremath{^{-}}}&\multicolumn{1}{l}{[E1]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{39\ensuremath{\times10^{-3}} {\it 2}}&\parbox[t][0.3cm]{7.100691cm}{\raggedright \%I\ensuremath{\gamma}=1.36 \textit{28}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1538}&\multicolumn{1}{@{.}l}{4 {\it 5}}&\multicolumn{1}{r@{}}{11}&\multicolumn{1}{@{.}l}{4 {\it 6}}&\multicolumn{1}{r@{}}{2474}&\multicolumn{1}{@{.}l}{6}&\multicolumn{1}{l}{(7/2\ensuremath{^{+}},9/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{936}&\multicolumn{1}{@{.}l}{1 }&\multicolumn{1}{@{}l}{7/2\ensuremath{^{-}}}&\multicolumn{1}{l}{[E1]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{38\ensuremath{\times10^{-3}} {\it 2}}&\parbox[t][0.3cm]{7.100691cm}{\raggedright \%I\ensuremath{\gamma}=2.81 \textit{19}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1547}&\multicolumn{1}{@{.}l}{6\ensuremath{^{\hyperlink{BI30GAMMA6}{b}}} {\it 10}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{21 {\it 24}}&\multicolumn{1}{r@{}}{2459}&\multicolumn{1}{@{.}l}{9}&\multicolumn{1}{l}{7/2\ensuremath{^{+}},9/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{910}&\multicolumn{1}{@{.}l}{5? }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{l}{[E1]}&&&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{37\ensuremath{\times10^{-3}} {\it 2}}&\parbox[t][0.3cm]{7.100691cm}{\raggedright \%I\ensuremath{\gamma}=0.30 \textit{6}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$1553}&\multicolumn{1}{@{.}l}{0 {\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{90 {\it 18}}&&&&&&&&&&&&\parbox[t][0.3cm]{7.100691cm}{\raggedright \%I\ensuremath{\gamma}=0.22 \textit{5}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1558}&\multicolumn{1}{@{.}l}{6 {\it 10}}&\multicolumn{1}{r@{}}{5}&\multicolumn{1}{@{.}l}{6 {\it 11}}&\multicolumn{1}{r@{}}{2548}&\multicolumn{1}{@{.}l}{9}&\multicolumn{1}{l}{(11/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{990}&\multicolumn{1}{@{.}l}{5 }&\multicolumn{1}{@{}l}{7/2\ensuremath{^{-}}}&\multicolumn{1}{l}{(E2)}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{00296 {\it 4}}&\parbox[t][0.3cm]{7.100691cm}{\raggedright \%I\ensuremath{\gamma}=1.38 \textit{28}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.100691cm}{\raggedright Mult.: From \ensuremath{\alpha}(K)exp=0.0020 \textit{10}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1570}&\multicolumn{1}{@{.}l}{8 {\it 5}}&\multicolumn{1}{r@{}}{10}&\multicolumn{1}{@{.}l}{0 {\it 5}}&\multicolumn{1}{r@{}}{2506}&\multicolumn{1}{@{.}l}{8}&\multicolumn{1}{l}{9/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{936}&\multicolumn{1}{@{.}l}{1 }&\multicolumn{1}{@{}l}{7/2\ensuremath{^{-}}}&\multicolumn{1}{l}{E1}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{36\ensuremath{\times10^{-3}} {\it 2}}&\parbox[t][0.3cm]{7.100691cm}{\raggedright \%I\ensuremath{\gamma}=2.47 \textit{16}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.100691cm}{\raggedright Mult.: From \ensuremath{\alpha}(K)exp=0.0009 \textit{5}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1579}&\multicolumn{1}{@{.}l}{8 {\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{34 {\it 7}}&\multicolumn{1}{r@{}}{2208}&\multicolumn{1}{@{.}l}{9}&\multicolumn{1}{l}{(9/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{629}&\multicolumn{1}{@{.}l}{1 }&\multicolumn{1}{@{}l}{13/2\ensuremath{^{+}}}&\multicolumn{1}{l}{[E2]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{00290 {\it 4}}&\parbox[t][0.3cm]{7.100691cm}{\raggedright \%I\ensuremath{\gamma}=0.084 \textit{18}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1603}&\multicolumn{1}{@{.}l}{8 {\it 10}}&\multicolumn{1}{r@{}}{4}&\multicolumn{1}{@{.}l}{9 {\it 9}}&\multicolumn{1}{r@{}}{2788}&\multicolumn{1}{@{.}l}{8}&\multicolumn{1}{l}{11/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{1185}&\multicolumn{1}{@{.}l}{8 }&\multicolumn{1}{@{}l}{(7/2)\ensuremath{^{-}}}&\multicolumn{1}{l}{(E2)}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{00283 {\it 4}}&\parbox[t][0.3cm]{7.100691cm}{\raggedright \%I\ensuremath{\gamma}=1.21 \textit{23}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.100691cm}{\raggedright Mult.: From \ensuremath{\alpha}(K)exp=0.0040 \textit{10}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$1626}&\multicolumn{1}{@{.}l}{0 {\it 10}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{4 {\it 3}}&&&&&&&&&&&&\parbox[t][0.3cm]{7.100691cm}{\raggedright \%I\ensuremath{\gamma}=0.35 \textit{8}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1634}&\multicolumn{1}{@{.}l}{9 {\it 10}}&\multicolumn{1}{r@{}}{8}&\multicolumn{1}{@{.}l}{0 {\it 16}}&\multicolumn{1}{r@{}}{3050}&\multicolumn{1}{@{.}l}{7}&\multicolumn{1}{l}{(7/2)\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1415}&\multicolumn{1}{@{.}l}{4 }&\multicolumn{1}{@{}l}{9/2\ensuremath{^{+}}}&\multicolumn{1}{l}{M1(+E2)}&\multicolumn{1}{r@{}}{$<$1}&\multicolumn{1}{@{.}l}{2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0045 {\it 7}}&\parbox[t][0.3cm]{7.100691cm}{\raggedright \%I\ensuremath{\gamma}=2.0 \textit{4}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.100691cm}{\raggedright Mult.: From \ensuremath{\alpha}(K)exp=0.0040 \textit{10}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1638}&\multicolumn{1}{@{.}l}{7\ensuremath{^{\hyperlink{BI30GAMMA6}{b}}} {\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{75 {\it 15}}&\multicolumn{1}{r@{}}{2548}&\multicolumn{1}{@{.}l}{9}&\multicolumn{1}{l}{(11/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{910}&\multicolumn{1}{@{.}l}{5? }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{l}{[M3]}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{02050 {\it 29}}&\parbox[t][0.3cm]{7.100691cm}{\raggedright \%I\ensuremath{\gamma}=0.18 \textit{4}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1650}&\multicolumn{1}{@{.}l}{9 {\it 5}}&\multicolumn{1}{r@{}}{24}&\multicolumn{1}{@{.}l}{2 {\it 12}}&\multicolumn{1}{r@{}}{1651}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{l}{7/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{l}{E1}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{33\ensuremath{\times10^{-3}} {\it 2}}&\parbox[t][0.3cm]{7.100691cm}{\raggedright \%I\ensuremath{\gamma}=6.0 \textit{4}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.100691cm}{\raggedright Mult.: From \ensuremath{\alpha}(K)exp=0.0006 \textit{4}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$1664}&\multicolumn{1}{@{.}l}{0 {\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{60 {\it 12}}&&&&&&&&&&&&\parbox[t][0.3cm]{7.100691cm}{\raggedright \%I\ensuremath{\gamma}=0.148 \textit{30}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$1703}&\multicolumn{1}{@{.}l}{1 {\it 10}}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{2 {\it 4}}&&&&&&&&&&&&\parbox[t][0.3cm]{7.100691cm}{\raggedright \%I\ensuremath{\gamma}=0.54 \textit{10}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$1709}&\multicolumn{1}{@{.}l}{4 {\it 10}}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{3 {\it 5}}&&&&&&&&&&&&\parbox[t][0.3cm]{7.100691cm}{\raggedright \%I\ensuremath{\gamma}=0.57 \textit{13}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$1718}&\multicolumn{1}{@{.}l}{9 {\it 10}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{12 {\it 22}}&&&&&&&&&&&&\parbox[t][0.3cm]{7.100691cm}{\raggedright \%I\ensuremath{\gamma}=0.28 \textit{6}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$1730}&\multicolumn{1}{@{.}l}{8 {\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{80 {\it 16}}&&&&&&&&&&&&\parbox[t][0.3cm]{7.100691cm}{\raggedright \%I\ensuremath{\gamma}=0.20 \textit{4}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1737}&\multicolumn{1}{@{.}l}{6 {\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{54 {\it 11}}&\multicolumn{1}{r@{}}{1737}&\multicolumn{1}{@{.}l}{3}&\multicolumn{1}{l}{9/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{l}{[M2]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{01038 {\it 15}}&\parbox[t][0.3cm]{7.100691cm}{\raggedright \%I\ensuremath{\gamma}=0.133 \textit{28}\vspace{0.1cm}}&\\
\end{longtable}
\clearpage
\begin{longtable}{ccccccccc@{}ccccc@{\extracolsep{\fill}}c}
\\[-.4cm]
\multicolumn{15}{c}{{\bf \small \underline{\ensuremath{^{\textnormal{201}}}Bi \ensuremath{\varepsilon} decay\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1978Ri04,B}{1978Ri04} (continued)}}}\\
\multicolumn{15}{c}{~}\\
\multicolumn{15}{c}{\underline{$\gamma$($^{201}$Pb) (continued)}}\\
\multicolumn{15}{c}{~~~}\\
\multicolumn{2}{c}{E\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{BI30GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{I\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{BI30GAMMA1}{\ddagger}\hyperlink{BI30GAMMA4}{\&}}}}&\multicolumn{2}{c}{E\ensuremath{_{i}}(level)}&J\ensuremath{^{\pi}_{i}}&\multicolumn{2}{c}{E\ensuremath{_{f}}}&J\ensuremath{^{\pi}_{f}}&Mult.\ensuremath{^{\hyperlink{BI30GAMMA2}{\#}}}&\multicolumn{2}{c}{\ensuremath{\alpha}\ensuremath{^{\hyperlink{BI30GAMMA5}{a}}}}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endhead
\multicolumn{1}{r@{}}{$^{x}$1758}&\multicolumn{1}{@{.}l}{5 {\it 10}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{03 {\it 21}}&&&&&&&&&&\parbox[t][0.3cm]{10.32212cm}{\raggedright \%I\ensuremath{\gamma}=0.25 \textit{5}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$1767}&\multicolumn{1}{@{.}l}{7 {\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{45 {\it 9}}&&&&&&&&&&\parbox[t][0.3cm]{10.32212cm}{\raggedright \%I\ensuremath{\gamma}=0.111 \textit{23}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1775}&\multicolumn{1}{@{.}l}{7 {\it 10}}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{1 {\it 4}}&\multicolumn{1}{r@{}}{2961}&\multicolumn{1}{@{.}l}{8}&\multicolumn{1}{l}{(7/2,9/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{1185}&\multicolumn{1}{@{.}l}{8 }&\multicolumn{1}{@{}l}{(7/2)\ensuremath{^{-}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{10.32212cm}{\raggedright \%I\ensuremath{\gamma}=0.52 \textit{10}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$1788}&\multicolumn{1}{@{.}l}{3 {\it 10}}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{2 {\it 4}}&&&&&&&&&&\parbox[t][0.3cm]{10.32212cm}{\raggedright \%I\ensuremath{\gamma}=0.54 \textit{10}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$1790}&\multicolumn{1}{@{.}l}{9 {\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{34 {\it 7}}&&&&&&&&&&\parbox[t][0.3cm]{10.32212cm}{\raggedright \%I\ensuremath{\gamma}=0.084 \textit{18}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$1798}&\multicolumn{1}{@{.}l}{8 {\it 10}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{03 {\it 21}}&&&&&&&&&&\parbox[t][0.3cm]{10.32212cm}{\raggedright \%I\ensuremath{\gamma}=0.25 \textit{5}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1851}&\multicolumn{1}{@{.}l}{9 {\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{26 {\it 5}}&\multicolumn{1}{r@{}}{2788}&\multicolumn{1}{@{.}l}{8}&\multicolumn{1}{l}{11/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{936}&\multicolumn{1}{@{.}l}{1 }&\multicolumn{1}{@{}l}{7/2\ensuremath{^{-}}}&\multicolumn{1}{l}{[E2]}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{30\ensuremath{\times10^{-3}} {\it 3}}&\parbox[t][0.3cm]{10.32212cm}{\raggedright \%I\ensuremath{\gamma}=0.064 \textit{13}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$1855}&\multicolumn{1}{@{.}l}{3 {\it 10}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{29 {\it 25}}&&&&&&&&&&\parbox[t][0.3cm]{10.32212cm}{\raggedright \%I\ensuremath{\gamma}=0.32 \textit{6}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$1866}&\multicolumn{1}{@{.}l}{2 {\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{80 {\it 16}}&&&&&&&&&&\parbox[t][0.3cm]{10.32212cm}{\raggedright \%I\ensuremath{\gamma}=0.20 \textit{4}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1877}&\multicolumn{1}{@{.}l}{4 {\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{60 {\it 12}}&\multicolumn{1}{r@{}}{2506}&\multicolumn{1}{@{.}l}{8}&\multicolumn{1}{l}{9/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{629}&\multicolumn{1}{@{.}l}{1 }&\multicolumn{1}{@{}l}{13/2\ensuremath{^{+}}}&\multicolumn{1}{l}{[E2]}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{26\ensuremath{\times10^{-3}} {\it 3}}&\parbox[t][0.3cm]{10.32212cm}{\raggedright \%I\ensuremath{\gamma}=0.148 \textit{30}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$1889}&\multicolumn{1}{@{.}l}{7 {\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{75 {\it 15}}&&&&&&&&&&\parbox[t][0.3cm]{10.32212cm}{\raggedright \%I\ensuremath{\gamma}=0.18 \textit{4}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$1897}&\multicolumn{1}{@{.}l}{9 {\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{33 {\it 7}}&&&&&&&&&&\parbox[t][0.3cm]{10.32212cm}{\raggedright \%I\ensuremath{\gamma}=0.081 \textit{18}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$1911}&\multicolumn{1}{@{.}l}{6 {\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{55 {\it 11}}&&&&&&&&&&\parbox[t][0.3cm]{10.32212cm}{\raggedright \%I\ensuremath{\gamma}=0.136 \textit{28}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1919}&\multicolumn{1}{@{.}l}{4 {\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{70 {\it 14}}&\multicolumn{1}{r@{}}{2548}&\multicolumn{1}{@{.}l}{9}&\multicolumn{1}{l}{(11/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{629}&\multicolumn{1}{@{.}l}{1 }&\multicolumn{1}{@{}l}{13/2\ensuremath{^{+}}}&\multicolumn{1}{l}{[E1]}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{29\ensuremath{\times10^{-3}} {\it 2}}&\parbox[t][0.3cm]{10.32212cm}{\raggedright \%I\ensuremath{\gamma}=0.173 \textit{35}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$1928}&\multicolumn{1}{@{.}l}{7 {\it 10}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{6 {\it 3}}&&&&&&&&&&\parbox[t][0.3cm]{10.32212cm}{\raggedright \%I\ensuremath{\gamma}=0.39 \textit{8}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$1949}&\multicolumn{1}{@{.}l}{7 {\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{45 {\it 9}}&&&&&&&&&&\parbox[t][0.3cm]{10.32212cm}{\raggedright \%I\ensuremath{\gamma}=0.111 \textit{23}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1971}&\multicolumn{1}{@{.}l}{0 {\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{65 {\it 13}}&\multicolumn{1}{r@{}}{2961}&\multicolumn{1}{@{.}l}{8}&\multicolumn{1}{l}{(7/2,9/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{990}&\multicolumn{1}{@{.}l}{5 }&\multicolumn{1}{@{}l}{7/2\ensuremath{^{-}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{10.32212cm}{\raggedright \%I\ensuremath{\gamma}=0.160 \textit{33}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$1980}&\multicolumn{1}{@{.}l}{4 {\it 10}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{4 {\it 3}}&&&&&&&&&&\parbox[t][0.3cm]{10.32212cm}{\raggedright \%I\ensuremath{\gamma}=0.35 \textit{8}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$2021}&\multicolumn{1}{@{.}l}{3 {\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{43 {\it 9}}&&&&&&&&&&\parbox[t][0.3cm]{10.32212cm}{\raggedright \%I\ensuremath{\gamma}=0.106 \textit{23}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2025}&\multicolumn{1}{@{.}l}{6 {\it 10}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{05 {\it 21}}&\multicolumn{1}{r@{}}{2961}&\multicolumn{1}{@{.}l}{8}&\multicolumn{1}{l}{(7/2,9/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{936}&\multicolumn{1}{@{.}l}{1 }&\multicolumn{1}{@{}l}{7/2\ensuremath{^{-}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{10.32212cm}{\raggedright \%I\ensuremath{\gamma}=0.26 \textit{5}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2035}&\multicolumn{1}{@{.}l}{8 {\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{34 {\it 7}}&\multicolumn{1}{r@{}}{3050}&\multicolumn{1}{@{.}l}{7}&\multicolumn{1}{l}{(7/2)\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1014}&\multicolumn{1}{@{.}l}{2 }&\multicolumn{1}{@{}l}{9/2\ensuremath{^{-}}}&\multicolumn{1}{l}{[E1]}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{30\ensuremath{\times10^{-3}} {\it 2}}&\parbox[t][0.3cm]{10.32212cm}{\raggedright \%I\ensuremath{\gamma}=0.084 \textit{18}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2060}&\multicolumn{1}{@{.}l}{9 {\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{75 {\it 15}}&\multicolumn{1}{r@{}}{3050}&\multicolumn{1}{@{.}l}{7}&\multicolumn{1}{l}{(7/2)\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{990}&\multicolumn{1}{@{.}l}{5 }&\multicolumn{1}{@{}l}{7/2\ensuremath{^{-}}}&\multicolumn{1}{l}{[E1]}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{30\ensuremath{\times10^{-3}} {\it 2}}&\parbox[t][0.3cm]{10.32212cm}{\raggedright \%I\ensuremath{\gamma}=0.18 \textit{4}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$2064}&\multicolumn{1}{@{.}l}{8 {\it 10}}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{8 {\it 6}}&&&&&&&&&&\parbox[t][0.3cm]{10.32212cm}{\raggedright \%I\ensuremath{\gamma}=0.69 \textit{15}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2082}&\multicolumn{1}{@{.}l}{0\ensuremath{^{\hyperlink{BI30GAMMA6}{b}}} {\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{30 {\it 6}}&\multicolumn{1}{r@{}}{2961}&\multicolumn{1}{@{.}l}{8}&\multicolumn{1}{l}{(7/2,9/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{879}&\multicolumn{1}{@{.}l}{9? }&\multicolumn{1}{@{}l}{(5/2)\ensuremath{^{-}}}&&&&\parbox[t][0.3cm]{10.32212cm}{\raggedright \%I\ensuremath{\gamma}=0.074 \textit{15}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$2091}&\multicolumn{1}{@{.}l}{8 {\it 10}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{50 {\it 3}}&&&&&&&&&&\parbox[t][0.3cm]{10.32212cm}{\raggedright \%I\ensuremath{\gamma}=0.370 \textit{18}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$2105}&\multicolumn{1}{@{.}l}{3 {\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{53 {\it 11}}&&&&&&&&&&\parbox[t][0.3cm]{10.32212cm}{\raggedright \%I\ensuremath{\gamma}=0.131 \textit{28}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2114}&\multicolumn{1}{@{.}l}{7 {\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{45 {\it 9}}&\multicolumn{1}{r@{}}{3050}&\multicolumn{1}{@{.}l}{7}&\multicolumn{1}{l}{(7/2)\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{936}&\multicolumn{1}{@{.}l}{1 }&\multicolumn{1}{@{}l}{7/2\ensuremath{^{-}}}&\multicolumn{1}{l}{[E1]}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{31\ensuremath{\times10^{-3}} {\it 2}}&\parbox[t][0.3cm]{10.32212cm}{\raggedright \%I\ensuremath{\gamma}=0.111 \textit{23}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$2124}&\multicolumn{1}{@{.}l}{7 {\it 10}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{03 {\it 21}}&&&&&&&&&&\parbox[t][0.3cm]{10.32212cm}{\raggedright \%I\ensuremath{\gamma}=0.25 \textit{5}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$2129}&\multicolumn{1}{@{.}l}{1 {\it 10}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{5 {\it 3}}&&&&&&&&&&\parbox[t][0.3cm]{10.32212cm}{\raggedright \%I\ensuremath{\gamma}=0.37 \textit{8}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$2145}&\multicolumn{1}{@{.}l}{0 {\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{43 {\it 9}}&&&&&&&&&&\parbox[t][0.3cm]{10.32212cm}{\raggedright \%I\ensuremath{\gamma}=0.106 \textit{23}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2159}&\multicolumn{1}{@{.}l}{7 {\it 10}}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{3 {\it 7}}&\multicolumn{1}{r@{}}{2788}&\multicolumn{1}{@{.}l}{8}&\multicolumn{1}{l}{11/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{629}&\multicolumn{1}{@{.}l}{1 }&\multicolumn{1}{@{}l}{13/2\ensuremath{^{+}}}&\multicolumn{1}{l}{[E1]}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{32\ensuremath{\times10^{-3}} {\it 2}}&\parbox[t][0.3cm]{10.32212cm}{\raggedright \%I\ensuremath{\gamma}=0.81 \textit{18}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2170}&\multicolumn{1}{@{.}l}{4\ensuremath{^{\hyperlink{BI30GAMMA6}{b}}} {\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{71 {\it 14}}&\multicolumn{1}{r@{}}{3050}&\multicolumn{1}{@{.}l}{7}&\multicolumn{1}{l}{(7/2)\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{879}&\multicolumn{1}{@{.}l}{9? }&\multicolumn{1}{@{}l}{(5/2)\ensuremath{^{-}}}&\multicolumn{1}{l}{[E1]}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{32\ensuremath{\times10^{-3}} {\it 2}}&\parbox[t][0.3cm]{10.32212cm}{\raggedright \%I\ensuremath{\gamma}=0.175 \textit{35}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$2189}&\multicolumn{1}{@{.}l}{3 {\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{35 {\it 7}}&&&&&&&&&&\parbox[t][0.3cm]{10.32212cm}{\raggedright \%I\ensuremath{\gamma}=0.086 \textit{18}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$2196}&\multicolumn{1}{@{.}l}{1 {\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{62 {\it 12}}&&&&&&&&&&\parbox[t][0.3cm]{10.32212cm}{\raggedright \%I\ensuremath{\gamma}=0.153 \textit{30}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$2201}&\multicolumn{1}{@{.}l}{0 {\it 10}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{6 {\it 3}}&&&&&&&&&&\parbox[t][0.3cm]{10.32212cm}{\raggedright \%I\ensuremath{\gamma}=0.39 \textit{8}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$2210}&\multicolumn{1}{@{.}l}{5 {\it 10}}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{3 {\it 5}}&&&&&&&&&&\parbox[t][0.3cm]{10.32212cm}{\raggedright \%I\ensuremath{\gamma}=0.57 \textit{13}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$2219}&\multicolumn{1}{@{.}l}{6 {\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{11 {\it 2}}&&&&&&&&&&\parbox[t][0.3cm]{10.32212cm}{\raggedright \%I\ensuremath{\gamma}=0.027 \textit{5}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$2238}&\multicolumn{1}{@{.}l}{3 {\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{57 {\it 12}}&&&&&&&&&&\parbox[t][0.3cm]{10.32212cm}{\raggedright \%I\ensuremath{\gamma}=0.141 \textit{30}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$2242}&\multicolumn{1}{@{.}l}{4 {\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{45 {\it 9}}&&&&&&&&&&\parbox[t][0.3cm]{10.32212cm}{\raggedright \%I\ensuremath{\gamma}=0.111 \textit{23}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$2261}&\multicolumn{1}{@{.}l}{0 {\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{72 {\it 15}}&&&&&&&&&&\parbox[t][0.3cm]{10.32212cm}{\raggedright \%I\ensuremath{\gamma}=0.18 \textit{4}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$2313}&\multicolumn{1}{@{.}l}{6 {\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{69 {\it 14}}&&&&&&&&&&\parbox[t][0.3cm]{10.32212cm}{\raggedright \%I\ensuremath{\gamma}=0.170 \textit{35}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$2321}&\multicolumn{1}{@{.}l}{6 {\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{78 {\it 16}}&&&&&&&&&&\parbox[t][0.3cm]{10.32212cm}{\raggedright \%I\ensuremath{\gamma}=0.19 \textit{4}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$2403}&\multicolumn{1}{@{.}l}{2 {\it 10}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{6 {\it 3}}&&&&&&&&&&\parbox[t][0.3cm]{10.32212cm}{\raggedright \%I\ensuremath{\gamma}=0.39 \textit{8}\vspace{0.1cm}}&\\
\end{longtable}
\clearpage
\vspace*{-0.5cm}
{\bf \small \underline{\ensuremath{^{\textnormal{201}}}Bi \ensuremath{\varepsilon} decay\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1978Ri04,B}{1978Ri04} (continued)}}\\
\vspace{0.3cm}
\underline{$\gamma$($^{201}$Pb) (continued)}\\
\vspace{0.3cm}
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{BI30GAMMA0}{\dagger}}}} From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1978Ri04,B}{1978Ri04} where \ensuremath{\Delta}E\ensuremath{\gamma}\ensuremath{\leq}0.5 keV for I\ensuremath{\gamma}\ensuremath{\geq}10 and \ensuremath{\Delta}E\ensuremath{\gamma}\ensuremath{\leq}1.0 keV for I\ensuremath{\gamma}\ensuremath{\leq}1 were reported. The evaluator assigns \ensuremath{\Delta}E\ensuremath{\gamma}=1 keV for I\ensuremath{\gamma}\ensuremath{<}10 and 0.5 keV for}\\
\parbox[b][0.3cm]{21.881866cm}{{\ }{\ }I\ensuremath{\gamma}\ensuremath{\geq}10.}\\
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{BI30GAMMA1}{\ddagger}}}} From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1978Ri04,B}{1978Ri04} where \ensuremath{\Delta}I\ensuremath{\gamma}\ensuremath{\leq}5\% for I\ensuremath{\gamma}\ensuremath{\geq}10 and \ensuremath{\leq}20\% for I\ensuremath{\gamma}\ensuremath{\leq}1 were reported. The evaluator assigns \ensuremath{\Delta}I\ensuremath{\gamma}=5\% for I\ensuremath{\gamma}\ensuremath{\geq}10 and 20\% for I\ensuremath{\gamma}\ensuremath{<}10.}\\
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{BI30GAMMA2}{\#}}}} Based on \ensuremath{\alpha}(K)exp, \ensuremath{\alpha}(L)exp and subshell ratios in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1978Ri04,B}{1978Ri04}, unless otherwise stated; \ensuremath{\alpha}(K)exp and \ensuremath{\alpha}(L)exp values were normalized using M4 mult. for 629.5\ensuremath{\gamma},}\\
\parbox[b][0.3cm]{21.881866cm}{{\ }{\ }as determined in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1956St05,B}{1956St05}.}\\
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{BI30GAMMA3}{@}}}} From \ensuremath{\alpha}(K)exp, \ensuremath{\alpha}(L)exp and subshell ratios in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1978Ri04,B}{1978Ri04} and the briccmixing program, unless otherwise stated.}\\
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{BI30GAMMA4}{\&}}}} For absolute intensity per 100 decays, multiply by 0.247 \textit{11}.}\\
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{BI30GAMMA5}{a}}}} Total theoretical internal conversion coefficients, calculated using the BrIcc code (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2008Ki07,B}{2008Ki07}) with Frozen orbital approximation based on \ensuremath{\gamma}-ray energies,}\\
\parbox[b][0.3cm]{21.881866cm}{{\ }{\ }assigned multipolarities, and mixing ratios, unless otherwise specified.}\\
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{BI30GAMMA6}{b}}}} Placement of transition in the level scheme is uncertain.}\\
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{BI30GAMMA7}{x}}}} \ensuremath{\gamma} ray not placed in level scheme.}\\
\vspace{0.5cm}
\end{landscape}\clearpage
\clearpage
\begin{figure}[h]
\begin{center}
\includegraphics{201PB30-0.ps}\\
\end{center}
\end{figure}
\clearpage
\begin{figure}[h]
\begin{center}
\includegraphics{201PB30-1.ps}\\
\end{center}
\end{figure}
\clearpage
\begin{figure}[h]
\begin{center}
\includegraphics{201PB30-2.ps}\\
\end{center}
\end{figure}
\clearpage
\subsection[\hspace{-0.2cm}\ensuremath{^{\textnormal{205}}}Po \ensuremath{\alpha} decay]{ }
\vspace{-27pt}
\vspace{0.3cm}
\hypertarget{PO31}{{\bf \small \underline{\ensuremath{^{\textnormal{205}}}Po \ensuremath{\alpha} decay\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1967Ti04,B}{1967Ti04},\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1970Jo26,B}{1970Jo26}}}}\\
\vspace{4pt}
\vspace{8pt}
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}Parent: $^{205}$Po: E=0; J$^{\pi}$=5/2\ensuremath{^{-}}; T$_{1/2}$=1.74 h {\it 8}; Q(\ensuremath{\alpha})=5325 {\it 10}; \%\ensuremath{\alpha} decay=0.040 {\it 12}
}\\
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}\ensuremath{^{205}}Po-J$^{\pi}$,T$_{1/2}$: From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2020Ko17,B}{2020Ko17}.}\\
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}\ensuremath{^{205}}Po-Q(\ensuremath{\alpha}): From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2021Wa16,B}{2021Wa16}.}\\
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}\ensuremath{^{205}}Po-\%\ensuremath{\alpha} decay: From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2020Ko17,B}{2020Ko17}.}\\
\vspace{12pt}
\underline{$^{201}$Pb Levels}\\
\begin{longtable}{ccccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E(level)$^{}$}&J$^{\pi}$$^{{\hyperlink{PB31LEVEL0}{\dagger}}}$&\multicolumn{2}{c}{T$_{1/2}$$^{{\hyperlink{PB31LEVEL0}{\dagger}}}$}&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&
\endfirsthead
\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{9}&\multicolumn{1}{@{.}l}{33 h {\it 5}}&\\
\end{longtable}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB31LEVEL0}{\dagger}}}} From Adopted Levels.}\\
\vspace{0.5cm}
\underline{\ensuremath{\alpha} radiations}\\
\begin{longtable}{ccccccccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E$\alpha^{{}}$}&\multicolumn{2}{c}{E(level)}&\multicolumn{2}{c}{I$\alpha^{{\hyperlink{PB31DECAY1}{\ddagger}}}$}&\multicolumn{2}{c}{HF$^{{\hyperlink{PB31DECAY0}{\dagger}}}$}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{5220}&\multicolumn{1}{@{ }l}{{\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{1 {\it 4}}&\parbox[t][0.3cm]{12.8046cm}{\raggedright E$\alpha$: From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1967Ti04,B}{1967Ti04}; Other: 5224 \textit{10} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1970Jo26,B}{1970Jo26}).\vspace{0.1cm}}&\\
\end{longtable}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB31DECAY0}{\dagger}}}} Using r\ensuremath{_{\textnormal{0}}}(\ensuremath{^{\textnormal{201}}}Pb)=1.4586 \textit{16} from \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2020Si16,B}{2020Si16}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB31DECAY1}{\ddagger}}}} For absolute intensity per 100 decays, multiply by 0.00040 \textit{12}.}\\
\vspace{0.5cm}
\clearpage
\subsection[\hspace{-0.2cm}\ensuremath{^{\textnormal{200}}}Hg(\ensuremath{\alpha},3n\ensuremath{\gamma})]{ }
\vspace{-27pt}
\vspace{0.3cm}
\hypertarget{PB32}{{\bf \small \underline{\ensuremath{^{\textnormal{200}}}Hg(\ensuremath{\alpha},3n\ensuremath{\gamma})\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1988Ro08,B}{1988Ro08},\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1981He07,B}{1981He07}}}}\\
\vspace{4pt}
\vspace{8pt}
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1988Ro08,B}{1988Ro08}: E(\ensuremath{\alpha})=53 MeV; Target: enriched liquid \ensuremath{^{\textnormal{200}}}Hg; Detectors: intrinsic germanium and Ge(Li); Measured: \ensuremath{\gamma}, \ensuremath{\gamma}(\ensuremath{\theta}),}\\
\parbox[b][0.3cm]{17.7cm}{\ensuremath{\gamma}(\ensuremath{\theta},\ensuremath{\beta},t) and \ensuremath{\gamma}\ensuremath{\gamma}(t).}\\
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1981He07,B}{1981He07}: E(\ensuremath{\alpha})=40 MeV; Target: enriched to 95.7\% \ensuremath{^{\textnormal{200}}}Hg oxide; Detectors: Ge(Li); Measured: \ensuremath{\gamma}, \ensuremath{\gamma}\ensuremath{\gamma}, \ensuremath{\gamma}(\ensuremath{\theta}), \ensuremath{\gamma}(t) and \ensuremath{\gamma}\ensuremath{\gamma}(t).}\\
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}Other: \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1977He06,B}{1977He06}.}\\
\vspace{12pt}
\underline{$^{201}$Pb Levels}\\
\begin{longtable}{cccccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E(level)$^{{\hyperlink{PB32LEVEL0}{\dagger}}}$}&J$^{\pi}$$^{{\hyperlink{PB32LEVEL1}{\ddagger}}}$&\multicolumn{2}{c}{T$_{1/2}$$^{{\hyperlink{PB32LEVEL2}{\#}}}$}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{PB32LEVEL3}{@}}}}}&\multicolumn{1}{l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{9}&\multicolumn{1}{@{.}l}{33 h {\it 5}}&&\\
\multicolumn{1}{r@{}}{88}&\multicolumn{1}{@{.}l}{6\ensuremath{^{{\hyperlink{PB32LEVEL4}{\&}}}} {\it 10}}&\multicolumn{1}{l}{3/2\ensuremath{^{-}}}&&&&\\
\multicolumn{1}{r@{}}{170}&\multicolumn{1}{@{.}l}{0\ensuremath{^{{\hyperlink{PB32LEVEL5}{a}}}} {\it 15}}&\multicolumn{1}{l}{(1/2\ensuremath{^{-}})}&&&&\\
\multicolumn{1}{r@{}}{538}&\multicolumn{1}{@{.}l}{7 {\it 6}}&\multicolumn{1}{l}{(3/2\ensuremath{^{-}})}&&&&\\
\multicolumn{1}{r@{}}{628}&\multicolumn{1}{@{.}l}{8\ensuremath{^{{\hyperlink{PB32LEVEL6}{b}}}} {\it 4}}&\multicolumn{1}{l}{13/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{60}&\multicolumn{1}{@{.}l}{8 s {\it 18}}&&\\
\multicolumn{1}{r@{}}{879}&\multicolumn{1}{@{.}l}{6 {\it 5}}&\multicolumn{1}{l}{(5/2\ensuremath{^{-}})}&&&&\\
\multicolumn{1}{r@{}}{936}&\multicolumn{1}{@{.}l}{1 {\it 3}}&\multicolumn{1}{l}{7/2\ensuremath{^{-}}}&&&&\\
\multicolumn{1}{r@{}}{1014}&\multicolumn{1}{@{.}l}{4 {\it 5}}&\multicolumn{1}{l}{9/2\ensuremath{^{-}}}&&&&\\
\multicolumn{1}{r@{}}{1186}&\multicolumn{1}{@{.}l}{3 {\it 7}}&\multicolumn{1}{l}{11/2\ensuremath{^{-}}}&&&&\\
\multicolumn{1}{r@{}}{1447}&\multicolumn{1}{@{.}l}{3 {\it 5}}&\multicolumn{1}{l}{11/2\ensuremath{^{+}}}&&&&\\
\multicolumn{1}{r@{}}{1541}&\multicolumn{1}{@{.}l}{5\ensuremath{^{{\hyperlink{PB32LEVEL7}{c}}}} {\it 5}}&\multicolumn{1}{l}{17/2\ensuremath{^{+}}}&&&&\\
\multicolumn{1}{r@{}}{1545}&\multicolumn{1}{@{.}l}{5 {\it 5}}&\multicolumn{1}{l}{15/2\ensuremath{^{+}}}&&&&\\
\multicolumn{1}{r@{}}{1895}&\multicolumn{1}{@{.}l}{8\ensuremath{^{{\hyperlink{PB32LEVEL8}{d}}}} {\it 5}}&\multicolumn{1}{l}{19/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{2 ns {\it 6}}&\parbox[t][0.3cm]{12.07468cm}{\raggedright T\ensuremath{_{1/2}}: From (350\ensuremath{\gamma},354\ensuremath{\gamma},913\ensuremath{\gamma},917\ensuremath{\gamma})(t) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1981He07,B}{1981He07}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1901}&\multicolumn{1}{@{.}l}{9\ensuremath{^{{\hyperlink{PB32LEVEL9}{e}}}} {\it 5}}&\multicolumn{1}{l}{21/2\ensuremath{^{+}}}&&&&\\
\multicolumn{1}{r@{}}{2068}&\multicolumn{1}{@{.}l}{1 {\it 6}}&\multicolumn{1}{l}{21/2\ensuremath{^{+}}}&&&&\\
\multicolumn{1}{r@{}}{2496}&\multicolumn{1}{@{.}l}{0\ensuremath{^{{\hyperlink{PB32LEVEL10}{f}}}} {\it 5}}&\multicolumn{1}{l}{21/2\ensuremath{^{-}}}&&&&\\
\multicolumn{1}{r@{}}{2603}&\multicolumn{1}{@{.}l}{7 {\it 7}}&\multicolumn{1}{l}{21/2,23/2}&&&&\\
\multicolumn{1}{r@{}}{2718}&\multicolumn{1}{@{.}l}{2\ensuremath{^{{\hyperlink{PB32LEVEL11}{g}}}} {\it 5}}&\multicolumn{1}{l}{25/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{63}&\multicolumn{1}{@{ }l}{ns {\it 3}}&\parbox[t][0.3cm]{12.07468cm}{\raggedright T\ensuremath{_{1/2}}: From (222.3\ensuremath{\gamma},350.3\ensuremath{\gamma},354.3\ensuremath{\gamma},600.3\ensuremath{\gamma},913.2\ensuremath{\gamma},917.1\ensuremath{\gamma})(t) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1988Ro08,B}{1988Ro08}. Other: 55 ns\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{12.07468cm}{\raggedright {\ }{\ }{\ }in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1981He07,B}{1981He07}.\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{12.07468cm}{\raggedright g-factor={\textminus}0.063 \textit{3} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1988Ro08,B}{1988Ro08}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2718}&\multicolumn{1}{@{.}l}{2+x\ensuremath{^{{\hyperlink{PB32LEVEL12}{h}}}}}&\multicolumn{1}{l}{29/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{508}&\multicolumn{1}{@{ }l}{ns {\it 5}}&\parbox[t][0.3cm]{12.07468cm}{\raggedright E(level): \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1981He07,B}{1981He07} stated that X\ensuremath{<}70 keV.\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{12.07468cm}{\raggedright T\ensuremath{_{1/2}}: From (222.3\ensuremath{\gamma},350.3\ensuremath{\gamma},354.3\ensuremath{\gamma},600.3\ensuremath{\gamma},913.2\ensuremath{\gamma},917.1\ensuremath{\gamma})(t) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1988Ro08,B}{1988Ro08}. Other: 540\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{12.07468cm}{\raggedright {\ }{\ }{\ }ns \textit{40} \ensuremath{\gamma}(t) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1981He07,B}{1981He07};\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{12.07468cm}{\raggedright g-factor={\textminus}0.0697 \textit{4} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1988Ro08,B}{1988Ro08}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2793}&\multicolumn{1}{@{.}l}{9 {\it 6}}&&&&&\\
\multicolumn{1}{r@{}}{3508}&\multicolumn{1}{@{.}l}{8+x {\it 4}}&\multicolumn{1}{l}{31/2\ensuremath{^{-}}}&&&&\\
\multicolumn{1}{r@{}}{3543}&\multicolumn{1}{@{.}l}{8+x {\it 4}}&\multicolumn{1}{l}{33/2\ensuremath{^{-}}}&&&&\\
\multicolumn{1}{r@{}}{3637}&\multicolumn{1}{@{.}l}{5+x {\it 4}}&\multicolumn{1}{l}{31/2}&&&&\\
\multicolumn{1}{r@{}}{3830}&\multicolumn{1}{@{.}l}{8+x {\it 6}}&\multicolumn{1}{l}{35/2\ensuremath{^{-}}}&&&&\\
\multicolumn{1}{r@{}}{3931}&\multicolumn{1}{@{.}l}{3+x\ensuremath{^{{\hyperlink{PB32LEVEL13}{i}}}} {\it 4}}&\multicolumn{1}{l}{33/2\ensuremath{^{+}}}&&&&\\
\multicolumn{1}{r@{}}{4504}&\multicolumn{1}{@{.}l}{4+x {\it 7}}&\multicolumn{1}{l}{35/2}&&&&\\
\multicolumn{1}{r@{}}{4558}&\multicolumn{1}{@{.}l}{5+x {\it 6}}&\multicolumn{1}{l}{37/2\ensuremath{^{+}}}&&&&\\
\multicolumn{1}{r@{}}{4638}&\multicolumn{1}{@{.}l}{0+x\ensuremath{^{{\hyperlink{PB32LEVEL14}{j}}}} {\it 8}}&\multicolumn{1}{l}{41/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{43}&\multicolumn{1}{@{ }l}{ns {\it 3}}&\parbox[t][0.3cm]{12.07468cm}{\raggedright T\ensuremath{_{1/2}}: From (727.7\ensuremath{\gamma},287.0\ensuremath{\gamma},825.6\ensuremath{\gamma})(t) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1988Ro08,B}{1988Ro08}.\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{12.07468cm}{\raggedright g-factor={\textminus}0.18 \textit{4} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1988Ro08,B}{1988Ro08}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{4639}&\multicolumn{1}{@{.}l}{4+x {\it 5}}&\multicolumn{1}{l}{(35/2)}&&&&\\
\multicolumn{1}{r@{}}{4999}&\multicolumn{1}{@{.}l}{4+x {\it 7}}&&&&&\\
\multicolumn{1}{r@{}}{5084}&\multicolumn{1}{@{.}l}{9+x {\it 9}}&&&&&\\
\multicolumn{1}{r@{}}{5171}&\multicolumn{1}{@{.}l}{7+x {\it 7}}&&&&&\\
\multicolumn{1}{r@{}}{5423}&\multicolumn{1}{@{.}l}{4+x {\it 9}}&&&&&\\
\end{longtable}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB32LEVEL0}{\dagger}}}} From a least-squares fit to E\ensuremath{\gamma}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB32LEVEL1}{\ddagger}}}} From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1988Ro08,B}{1988Ro08}, unless otherwise stated.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB32LEVEL2}{\#}}}} From Adopted Levels, unless otherwise stated.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB32LEVEL3}{@}}}} Configuration=\ensuremath{\nu} f\ensuremath{_{\textnormal{5/2}}^{\textnormal{$-$1}}}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB32LEVEL4}{\&}}}} Configuration=\ensuremath{\nu} p\ensuremath{_{\textnormal{3/2}}^{\textnormal{$-$1}}}.}\\
\begin{textblock}{29}(0,27.3)
Continued on next page (footnotes at end of table)
\end{textblock}
\clearpage
\vspace*{-0.5cm}
{\bf \small \underline{\ensuremath{^{\textnormal{200}}}Hg(\ensuremath{\alpha},3n\ensuremath{\gamma})\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1988Ro08,B}{1988Ro08},\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1981He07,B}{1981He07} (continued)}}\\
\vspace{0.3cm}
\underline{$^{201}$Pb Levels (continued)}\\
\vspace{0.3cm}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB32LEVEL5}{a}}}} Configuration=\ensuremath{\nu} p\ensuremath{_{\textnormal{1/2}}^{\textnormal{$-$1}}}. The assignment is tentative.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB32LEVEL6}{b}}}} Configuration=\ensuremath{\nu} i\ensuremath{_{\textnormal{13/2}}^{\textnormal{$-$1}}}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB32LEVEL7}{c}}}} Probably an admixture of configuration=\ensuremath{\nu} (f\ensuremath{_{\textnormal{5/2}}^{\textnormal{$-$1}}},p\ensuremath{_{\textnormal{1/2}}^{\textnormal{$-$1}}},i\ensuremath{_{\textnormal{13/2}}^{\textnormal{$-$1}}})\ensuremath{\otimes}2\ensuremath{^{\textnormal{+}}} and configuration=\ensuremath{\nu} (i\ensuremath{_{\textnormal{13/2}}^{\textnormal{$-$1}}})\ensuremath{\otimes}2\ensuremath{^{\textnormal{+}}}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB32LEVEL8}{d}}}} Probably an admixture of configuration=\ensuremath{\nu} (f\ensuremath{_{\textnormal{5/2}}^{\textnormal{$-$1}}},p\ensuremath{_{\textnormal{1/2}}^{\textnormal{$-$1}}},i\ensuremath{_{\textnormal{13/2}}^{\textnormal{$-$1}}})\ensuremath{\otimes}4\ensuremath{^{\textnormal{+}}} and configuration=\ensuremath{\nu} (i\ensuremath{_{\textnormal{13/2}}^{\textnormal{$-$1}}})\ensuremath{\otimes}4\ensuremath{^{\textnormal{+}}}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB32LEVEL9}{e}}}} Configuration=\ensuremath{\nu} (f\ensuremath{_{\textnormal{5/2}}^{\textnormal{$-$2}}},i\ensuremath{_{\textnormal{13/2}}^{\textnormal{$-$1}}}).}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB32LEVEL10}{f}}}} Configuration=\ensuremath{\nu} [p\ensuremath{_{\textnormal{3/2}}^{\textnormal{$-$1}}},(i\ensuremath{_{\textnormal{13/2}}^{\textnormal{$-$2}}})\ensuremath{_{\textnormal{12+}}}].}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB32LEVEL11}{g}}}} Probably an admixture of configuration=\ensuremath{\nu} [f\ensuremath{_{\textnormal{5/2}}^{\textnormal{$-$1}}},(i\ensuremath{_{\textnormal{13/2}}^{\textnormal{$-$2}}})\ensuremath{_{\textnormal{10+}}}], configuration=\ensuremath{\nu} [p\ensuremath{_{\textnormal{3/2}}^{\textnormal{$-$1}}},(i\ensuremath{_{\textnormal{13/2}}^{\textnormal{$-$2}}})\ensuremath{_{\textnormal{12+}}}] and configuration=\ensuremath{\nu}}\\
\parbox[b][0.3cm]{17.7cm}{{\ }{\ }[p\ensuremath{_{\textnormal{1/2}}^{\textnormal{$-$1}}},(i\ensuremath{_{\textnormal{13/2}}^{\textnormal{$-$2}}})\ensuremath{_{\textnormal{12+}}}].}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB32LEVEL12}{h}}}} Configuration=\ensuremath{\nu} [f\ensuremath{_{\textnormal{5/2}}^{\textnormal{$-$1}}},(i\ensuremath{_{\textnormal{13/2}}^{\textnormal{$-$2}}})\ensuremath{_{\textnormal{12+}}}].}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB32LEVEL13}{i}}}} Configuration=\ensuremath{\nu} (i\ensuremath{_{\textnormal{13/2}}^{\textnormal{$-$3}}}).}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB32LEVEL14}{j}}}} Configuration=\ensuremath{\nu} (p\ensuremath{_{\textnormal{3/2}}^{\textnormal{$-$1}}},f\ensuremath{_{\textnormal{5/2}}^{\textnormal{$-$1}}},i\ensuremath{_{\textnormal{13/2}}^{\textnormal{$-$3}}}).}\\
\vspace{0.5cm}
\underline{$\gamma$($^{201}$Pb)}\\
\begin{longtable}{ccccccccc@{}ccccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{PB32GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{I\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{PB32GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{E\ensuremath{_{i}}(level)}&J\ensuremath{^{\pi}_{i}}&\multicolumn{2}{c}{E\ensuremath{_{f}}}&J\ensuremath{^{\pi}_{f}}&Mult.\ensuremath{^{\hyperlink{PB32GAMMA3}{@}}}&\multicolumn{2}{c}{\ensuremath{\alpha}\ensuremath{^{\hyperlink{PB32GAMMA4}{\&}}}}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{79}&\multicolumn{1}{@{.}l}{5 {\it 5}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{4638}&\multicolumn{1}{@{.}l}{0+x}&\multicolumn{1}{l}{41/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{4558}&\multicolumn{1}{@{.}l}{5+x }&\multicolumn{1}{@{}l}{37/2\ensuremath{^{+}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
\multicolumn{1}{r@{}}{(81}&\multicolumn{1}{@{.}l}{40\ensuremath{^{\hyperlink{PB32GAMMA2}{\#}}})}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{170}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{l}{(1/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{88}&\multicolumn{1}{@{.}l}{6 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{-}}}&&&&&\\
\multicolumn{1}{r@{}}{88}&\multicolumn{1}{@{.}l}{6\ensuremath{^{\hyperlink{PB32GAMMA2}{\#}}} {\it 10}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{88}&\multicolumn{1}{@{.}l}{6}&\multicolumn{1}{l}{3/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{5/2\ensuremath{^{-}}}&&&&&\\
\multicolumn{1}{r@{}}{166}&\multicolumn{1}{@{.}l}{2\ensuremath{^{\hyperlink{PB32GAMMA1}{\ddagger}}} {\it 2}}&\multicolumn{1}{r@{}}{4}&\multicolumn{1}{@{.}l}{3 {\it 4}}&\multicolumn{1}{r@{}}{2068}&\multicolumn{1}{@{.}l}{1}&\multicolumn{1}{l}{21/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1901}&\multicolumn{1}{@{.}l}{9 }&\multicolumn{1}{@{}l}{21/2\ensuremath{^{+}}}&\multicolumn{1}{l}{E2+M1}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{5 {\it 7}}&\parbox[t][0.3cm]{5.7491612cm}{\raggedright I\ensuremath{_{\gamma}}: 4.1 \textit{4} in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1981He07,B}{1981He07}.\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{5.7491612cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}=0.36 \textit{5}, A\ensuremath{_{\textnormal{4}}}=0.01 \textit{5} in\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{5.7491612cm}{\raggedright {\ }{\ }{\ }\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1988Ro08,B}{1988Ro08}, consistent with \ensuremath{\Delta}J=0\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{5.7491612cm}{\raggedright {\ }{\ }{\ }assignment.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{171}&\multicolumn{1}{@{.}l}{9 {\it 5}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{9 {\it 1}}&\multicolumn{1}{r@{}}{1186}&\multicolumn{1}{@{.}l}{3}&\multicolumn{1}{l}{11/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{1014}&\multicolumn{1}{@{.}l}{4 }&\multicolumn{1}{@{}l}{9/2\ensuremath{^{-}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
\multicolumn{1}{r@{}}{222}&\multicolumn{1}{@{.}l}{2\ensuremath{^{\hyperlink{PB32GAMMA1}{\ddagger}}} {\it 1}}&\multicolumn{1}{r@{}}{44}&\multicolumn{1}{@{ }l}{{\it 4}}&\multicolumn{1}{r@{}}{2718}&\multicolumn{1}{@{.}l}{2}&\multicolumn{1}{l}{25/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{2496}&\multicolumn{1}{@{.}l}{0 }&\multicolumn{1}{@{}l}{21/2\ensuremath{^{-}}}&\multicolumn{1}{l}{E2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{299 {\it 4}}&\parbox[t][0.3cm]{5.7491612cm}{\raggedright I\ensuremath{_{\gamma}}: 32 \textit{3} in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1981He07,B}{1981He07}.\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{5.7491612cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}=0.22 \textit{5}, A\ensuremath{_{\textnormal{4}}}=0.02 \textit{6} in\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{5.7491612cm}{\raggedright {\ }{\ }{\ }\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1988Ro08,B}{1988Ro08}; A\ensuremath{_{\textnormal{2}}}=0.00 \textit{3}, A\ensuremath{_{\textnormal{4}}}={\textminus}0.06 \textit{4} in\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{5.7491612cm}{\raggedright {\ }{\ }{\ }\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1981He07,B}{1981He07}; \ensuremath{\alpha}(exp)=0.34 \textit{3} in\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{5.7491612cm}{\raggedright {\ }{\ }{\ }\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1981He07,B}{1981He07}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{287}&\multicolumn{1}{@{.}l}{0 {\it 5}}&\multicolumn{1}{r@{}}{7}&\multicolumn{1}{@{.}l}{7 {\it 8}}&\multicolumn{1}{r@{}}{3830}&\multicolumn{1}{@{.}l}{8+x}&\multicolumn{1}{l}{35/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{3543}&\multicolumn{1}{@{.}l}{8+x }&\multicolumn{1}{@{}l}{33/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{31 {\it 17}}&\parbox[t][0.3cm]{5.7491612cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}={\textminus}0.51 \textit{5}, A\ensuremath{_{\textnormal{4}}}=0.02 \textit{8} in\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{5.7491612cm}{\raggedright {\ }{\ }{\ }\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1988Ro08,B}{1988Ro08}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{293}&\multicolumn{1}{@{.}l}{8 {\it 5}}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{8 {\it 4}}&\multicolumn{1}{r@{}}{3931}&\multicolumn{1}{@{.}l}{3+x}&\multicolumn{1}{l}{33/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{3637}&\multicolumn{1}{@{.}l}{5+x }&\multicolumn{1}{@{}l}{31/2}&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{5.7491612cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}={\textminus}0.28 \textit{4}, A\ensuremath{_{\textnormal{4}}}=0.07 \textit{6} in\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{5.7491612cm}{\raggedright {\ }{\ }{\ }\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1988Ro08,B}{1988Ro08}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{297}&\multicolumn{1}{@{.}l}{9\ensuremath{^{\hyperlink{PB32GAMMA1}{\ddagger}}} {\it 3}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{2 {\it 12}}&\multicolumn{1}{r@{}}{2793}&\multicolumn{1}{@{.}l}{9}&&\multicolumn{1}{r@{}}{2496}&\multicolumn{1}{@{.}l}{0 }&\multicolumn{1}{@{}l}{21/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{28 {\it 16}}&\parbox[t][0.3cm]{5.7491612cm}{\raggedright I\ensuremath{_{\gamma}}: 0.7 \textit{1} in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1981He07,B}{1981He07}.\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{5.7491612cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}={\textminus}0.99 \textit{13}, A\ensuremath{_{\textnormal{4}}}=0.13 \textit{5} in\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{5.7491612cm}{\raggedright {\ }{\ }{\ }\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1981He07,B}{1981He07}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{350}&\multicolumn{1}{@{.}l}{3\ensuremath{^{\hyperlink{PB32GAMMA1}{\ddagger}}} {\it 2}}&\multicolumn{1}{r@{}}{32}&\multicolumn{1}{@{ }l}{{\it 3}}&\multicolumn{1}{r@{}}{1895}&\multicolumn{1}{@{.}l}{8}&\multicolumn{1}{l}{19/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1545}&\multicolumn{1}{@{.}l}{5 }&\multicolumn{1}{@{}l}{15/2\ensuremath{^{+}}}&\multicolumn{1}{l}{E2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0739 {\it 10}}&\parbox[t][0.3cm]{5.7491612cm}{\raggedright I\ensuremath{_{\gamma}}: 26.0 \textit{25} in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1981He07,B}{1981He07}.\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{5.7491612cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}=0.25 \textit{5}, A\ensuremath{_{\textnormal{4}}}={\textminus}0.05 \textit{6} in\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{5.7491612cm}{\raggedright {\ }{\ }{\ }\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1988Ro08,B}{1988Ro08}; Note, that A\ensuremath{_{\textnormal{2}}}=0.09 \textit{2},\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{5.7491612cm}{\raggedright {\ }{\ }{\ }A\ensuremath{_{\textnormal{4}}}={\textminus}0.07 \textit{4} in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1981He07,B}{1981He07} would imply\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{5.7491612cm}{\raggedright {\ }{\ }{\ }an M1+E2 assignment.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{354}&\multicolumn{1}{@{.}l}{3\ensuremath{^{\hyperlink{PB32GAMMA1}{\ddagger}}} {\it 2}}&\multicolumn{1}{r@{}}{47}&\multicolumn{1}{@{ }l}{{\it 5}}&\multicolumn{1}{r@{}}{1895}&\multicolumn{1}{@{.}l}{8}&\multicolumn{1}{l}{19/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1541}&\multicolumn{1}{@{.}l}{5 }&\multicolumn{1}{@{}l}{17/2\ensuremath{^{+}}}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{171 {\it 99}}&\parbox[t][0.3cm]{5.7491612cm}{\raggedright I\ensuremath{_{\gamma}}: 44 \textit{4} in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1981He07,B}{1981He07}.\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{5.7491612cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}=0.70 \textit{5}, A\ensuremath{_{\textnormal{4}}}=0.04 \textit{6} in\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{5.7491612cm}{\raggedright {\ }{\ }{\ }\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1988Ro08,B}{1988Ro08}; A\ensuremath{_{\textnormal{2}}}=0.19 \textit{3}, A\ensuremath{_{\textnormal{4}}}=0.01 \textit{4} in\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{5.7491612cm}{\raggedright {\ }{\ }{\ }\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1981He07,B}{1981He07}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{360}&\multicolumn{1}{@{.}l}{0 {\it 5}}&\multicolumn{1}{r@{}}{$\approx$1}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{4999}&\multicolumn{1}{@{.}l}{4+x}&&\multicolumn{1}{r@{}}{4639}&\multicolumn{1}{@{.}l}{4+x }&\multicolumn{1}{@{}l}{(35/2)}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
\multicolumn{1}{r@{}}{360}&\multicolumn{1}{@{.}l}{4\ensuremath{^{\hyperlink{PB32GAMMA1}{\ddagger}}} {\it 3}}&\multicolumn{1}{r@{}}{17}&\multicolumn{1}{@{ }l}{{\it 2}}&\multicolumn{1}{r@{}}{1901}&\multicolumn{1}{@{.}l}{9}&\multicolumn{1}{l}{21/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1541}&\multicolumn{1}{@{.}l}{5 }&\multicolumn{1}{@{}l}{17/2\ensuremath{^{+}}}&\multicolumn{1}{l}{E2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0683 {\it 10}}&\parbox[t][0.3cm]{5.7491612cm}{\raggedright I\ensuremath{_{\gamma}}: 25.3 \textit{25} in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1981He07,B}{1981He07}.\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{5.7491612cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}=+ve in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1988Ro08,B}{1988Ro08}; A\ensuremath{_{\textnormal{2}}}=0.25 \textit{3},\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{5.7491612cm}{\raggedright {\ }{\ }{\ }A\ensuremath{_{\textnormal{4}}}={\textminus}0.08 \textit{4} in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1981He07,B}{1981He07}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{368}&\multicolumn{1}{@{.}l}{8\ensuremath{^{\hyperlink{PB32GAMMA2}{\#}\hyperlink{PB32GAMMA5}{a}}} {\it 10}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{538}&\multicolumn{1}{@{.}l}{7}&\multicolumn{1}{l}{(3/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{170}&\multicolumn{1}{@{.}l}{0 }&\multicolumn{1}{@{}l}{(1/2\ensuremath{^{-}})}&&&&&\\
\multicolumn{1}{r@{}}{387}&\multicolumn{1}{@{.}l}{5 {\it 5}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{0 {\it 1}}&\multicolumn{1}{r@{}}{3931}&\multicolumn{1}{@{.}l}{3+x}&\multicolumn{1}{l}{33/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{3543}&\multicolumn{1}{@{.}l}{8+x }&\multicolumn{1}{@{}l}{33/2\ensuremath{^{-}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
\multicolumn{1}{r@{}}{422}&\multicolumn{1}{@{.}l}{5 {\it 5}}&\multicolumn{1}{r@{}}{5}&\multicolumn{1}{@{.}l}{0 {\it 5}}&\multicolumn{1}{r@{}}{3931}&\multicolumn{1}{@{.}l}{3+x}&\multicolumn{1}{l}{33/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{3508}&\multicolumn{1}{@{.}l}{8+x }&\multicolumn{1}{@{}l}{31/2\ensuremath{^{-}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
\multicolumn{1}{r@{}}{446}&\multicolumn{1}{@{.}l}{9 {\it 5}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{70 {\it 7}}&\multicolumn{1}{r@{}}{5084}&\multicolumn{1}{@{.}l}{9+x}&&\multicolumn{1}{r@{}}{4638}&\multicolumn{1}{@{.}l}{0+x }&\multicolumn{1}{@{}l}{41/2\ensuremath{^{+}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
\end{longtable}
\begin{textblock}{29}(0,27.3)
Continued on next page (footnotes at end of table)
\end{textblock}
\clearpage
\begin{longtable}{ccccccccc@{}ccccc@{\extracolsep{\fill}}c}
\\[-.4cm]
\multicolumn{15}{c}{{\bf \small \underline{\ensuremath{^{\textnormal{200}}}Hg(\ensuremath{\alpha},3n\ensuremath{\gamma})\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1988Ro08,B}{1988Ro08},\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1981He07,B}{1981He07} (continued)}}}\\
\multicolumn{15}{c}{~}\\
\multicolumn{15}{c}{\underline{$\gamma$($^{201}$Pb) (continued)}}\\
\multicolumn{15}{c}{~~~}\\
\multicolumn{2}{c}{E\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{PB32GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{I\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{PB32GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{E\ensuremath{_{i}}(level)}&J\ensuremath{^{\pi}_{i}}&\multicolumn{2}{c}{E\ensuremath{_{f}}}&J\ensuremath{^{\pi}_{f}}&Mult.\ensuremath{^{\hyperlink{PB32GAMMA3}{@}}}&\multicolumn{2}{c}{\ensuremath{\alpha}\ensuremath{^{\hyperlink{PB32GAMMA4}{\&}}}}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endhead
\multicolumn{1}{r@{}}{573}&\multicolumn{1}{@{.}l}{1 {\it 5}}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{1 {\it 21}}&\multicolumn{1}{r@{}}{4504}&\multicolumn{1}{@{.}l}{4+x}&\multicolumn{1}{l}{35/2}&\multicolumn{1}{r@{}}{3931}&\multicolumn{1}{@{.}l}{3+x }&\multicolumn{1}{@{}l}{33/2\ensuremath{^{+}}}&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{4.8739cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}={\textminus}0.87 \textit{5}, A\ensuremath{_{\textnormal{4}}}=0.14 \textit{8} in\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.8739cm}{\raggedright {\ }{\ }{\ }\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1988Ro08,B}{1988Ro08}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{594}&\multicolumn{1}{@{.}l}{1\ensuremath{^{\hyperlink{PB32GAMMA1}{\ddagger}}} {\it 3}}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{4 {\it 4}}&\multicolumn{1}{r@{}}{2496}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{l}{21/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{1901}&\multicolumn{1}{@{.}l}{9 }&\multicolumn{1}{@{}l}{21/2\ensuremath{^{+}}}&\multicolumn{1}{l}{D}&&&\parbox[t][0.3cm]{4.8739cm}{\raggedright I\ensuremath{_{\gamma}}: 1.1 \textit{2} in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1981He07,B}{1981He07}.\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.8739cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}={\textminus}0.06 \textit{5}, A\ensuremath{_{\textnormal{4}}}=0.08 \textit{6} in\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.8739cm}{\raggedright {\ }{\ }{\ }\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1988Ro08,B}{1988Ro08}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{600}&\multicolumn{1}{@{.}l}{2\ensuremath{^{\hyperlink{PB32GAMMA1}{\ddagger}}} {\it 1}}&\multicolumn{1}{r@{}}{74}&\multicolumn{1}{@{ }l}{{\it 8}}&\multicolumn{1}{r@{}}{2496}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{l}{21/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{1895}&\multicolumn{1}{@{.}l}{8 }&\multicolumn{1}{@{}l}{19/2\ensuremath{^{+}}}&\multicolumn{1}{l}{D}&&&\parbox[t][0.3cm]{4.8739cm}{\raggedright I\ensuremath{_{\gamma}}: 46 \textit{5} in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1981He07,B}{1981He07}.\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.8739cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}={\textminus}0.21 \textit{4}, A\ensuremath{_{\textnormal{4}}}={\textminus}0.02 \textit{6} in\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.8739cm}{\raggedright {\ }{\ }{\ }\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1988Ro08,B}{1988Ro08}; A\ensuremath{_{\textnormal{2}}}={\textminus}0.10 \textit{3},\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.8739cm}{\raggedright {\ }{\ }{\ }A\ensuremath{_{\textnormal{4}}}={\textminus}0.03 \textit{4} in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1981He07,B}{1981He07}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{627}&\multicolumn{1}{@{.}l}{2 {\it 5}}&\multicolumn{1}{r@{}}{$\approx$1}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{4558}&\multicolumn{1}{@{.}l}{5+x}&\multicolumn{1}{l}{37/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{3931}&\multicolumn{1}{@{.}l}{3+x }&\multicolumn{1}{@{}l}{33/2\ensuremath{^{+}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
\multicolumn{1}{r@{}}{628}&\multicolumn{1}{@{.}l}{8\ensuremath{^{\hyperlink{PB32GAMMA1}{\ddagger}}} {\it 4}}&\multicolumn{1}{r@{}}{194}&\multicolumn{1}{@{ }l}{{\it 20}}&\multicolumn{1}{r@{}}{628}&\multicolumn{1}{@{.}l}{8}&\multicolumn{1}{l}{13/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M4}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{815 {\it 12}}&\parbox[t][0.3cm]{4.8739cm}{\raggedright I\ensuremath{_{\gamma}}: 98 \textit{9} in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1981He07,B}{1981He07}.\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.8739cm}{\raggedright Mult.: From adopted gammas.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{667}&\multicolumn{1}{@{.}l}{3\ensuremath{^{\hyperlink{PB32GAMMA1}{\ddagger}}} {\it 3}}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{1 {\it 2}}&\multicolumn{1}{r@{}}{5171}&\multicolumn{1}{@{.}l}{7+x}&&\multicolumn{1}{r@{}}{4504}&\multicolumn{1}{@{.}l}{4+x }&\multicolumn{1}{@{}l}{35/2}&\multicolumn{1}{l}{D}&&&\parbox[t][0.3cm]{4.8739cm}{\raggedright I\ensuremath{_{\gamma}}: 2.1 \textit{2} in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1981He07,B}{1981He07}.\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.8739cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}={\textminus}0.38 \textit{6}, A\ensuremath{_{\textnormal{4}}}={\textminus}0.16 \textit{9} in\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.8739cm}{\raggedright {\ }{\ }{\ }\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1981He07,B}{1981He07}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{707}&\multicolumn{1}{@{.}l}{9 {\it 5}}&\multicolumn{1}{r@{}}{10}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{2603}&\multicolumn{1}{@{.}l}{7}&\multicolumn{1}{l}{21/2,23/2}&\multicolumn{1}{r@{}}{1895}&\multicolumn{1}{@{.}l}{8 }&\multicolumn{1}{@{}l}{19/2\ensuremath{^{+}}}&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{4.8739cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}={\textminus}0.10 \textit{4}, A\ensuremath{_{\textnormal{4}}}={\textminus}0.02 \textit{6} in\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.8739cm}{\raggedright {\ }{\ }{\ }\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1988Ro08,B}{1988Ro08}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{708}&\multicolumn{1}{@{.}l}{1\ensuremath{^{\hyperlink{PB32GAMMA1}{\ddagger}}} {\it 2}}&\multicolumn{1}{r@{}}{$\approx$1}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{4639}&\multicolumn{1}{@{.}l}{4+x}&\multicolumn{1}{l}{(35/2)}&\multicolumn{1}{r@{}}{3931}&\multicolumn{1}{@{.}l}{3+x }&\multicolumn{1}{@{}l}{33/2\ensuremath{^{+}}}&&&&\parbox[t][0.3cm]{4.8739cm}{\raggedright Placement of this gamma is from\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.8739cm}{\raggedright {\ }{\ }{\ }\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1988Ro08,B}{1988Ro08}.\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.8739cm}{\raggedright I\ensuremath{_{\gamma}}: 5.6 \textit{5} in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1981He07,B}{1981He07}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{727}&\multicolumn{1}{@{.}l}{7 {\it 5}}&\multicolumn{1}{r@{}}{10}&\multicolumn{1}{@{ }l}{{\it 1}}&\multicolumn{1}{r@{}}{4558}&\multicolumn{1}{@{.}l}{5+x}&\multicolumn{1}{l}{37/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{3830}&\multicolumn{1}{@{.}l}{8+x }&\multicolumn{1}{@{}l}{35/2\ensuremath{^{-}}}&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{4.8739cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}={\textminus}0.13 \textit{5}, A\ensuremath{_{\textnormal{4}}}={\textminus}0.02 \textit{7} in\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.8739cm}{\raggedright {\ }{\ }{\ }\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1988Ro08,B}{1988Ro08}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{785}&\multicolumn{1}{@{.}l}{4 {\it 5}}&\multicolumn{1}{r@{}}{5}&\multicolumn{1}{@{.}l}{9 {\it 6}}&\multicolumn{1}{r@{}}{5423}&\multicolumn{1}{@{.}l}{4+x}&&\multicolumn{1}{r@{}}{4638}&\multicolumn{1}{@{.}l}{0+x }&\multicolumn{1}{@{}l}{41/2\ensuremath{^{+}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
\multicolumn{1}{r@{}}{790}&\multicolumn{1}{@{.}l}{6 {\it 5}}&\multicolumn{1}{r@{}}{5}&\multicolumn{1}{@{.}l}{7 {\it 6}}&\multicolumn{1}{r@{}}{3508}&\multicolumn{1}{@{.}l}{8+x}&\multicolumn{1}{l}{31/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{2718}&\multicolumn{1}{@{.}l}{2+x}&\multicolumn{1}{@{}l}{29/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{022 {\it 11}}&\parbox[t][0.3cm]{4.8739cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}={\textminus}1.06 \textit{3}, A\ensuremath{_{\textnormal{4}}}={\textminus}0.01 \textit{5} in\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.8739cm}{\raggedright {\ }{\ }{\ }\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1988Ro08,B}{1988Ro08}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{818}&\multicolumn{1}{@{.}l}{5\ensuremath{^{\hyperlink{PB32GAMMA1}{\ddagger}}} {\it 2}}&\multicolumn{1}{r@{}}{4}&\multicolumn{1}{@{.}l}{5 {\it 5}}&\multicolumn{1}{r@{}}{1447}&\multicolumn{1}{@{.}l}{3}&\multicolumn{1}{l}{11/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{628}&\multicolumn{1}{@{.}l}{8 }&\multicolumn{1}{@{}l}{13/2\ensuremath{^{+}}}&\multicolumn{1}{l}{D}&&&\parbox[t][0.3cm]{4.8739cm}{\raggedright I\ensuremath{_{\gamma}}: 4.7 \textit{5} in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1981He07,B}{1981He07}.\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.8739cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}={\textminus}0.21 \textit{5}, A\ensuremath{_{\textnormal{4}}}=0.01 \textit{7} in\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.8739cm}{\raggedright {\ }{\ }{\ }\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1988Ro08,B}{1988Ro08}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{825}&\multicolumn{1}{@{.}l}{6 {\it 5}}&\multicolumn{1}{r@{}}{17}&\multicolumn{1}{@{ }l}{{\it 2}}&\multicolumn{1}{r@{}}{3543}&\multicolumn{1}{@{.}l}{8+x}&\multicolumn{1}{l}{33/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{2718}&\multicolumn{1}{@{.}l}{2+x}&\multicolumn{1}{@{}l}{29/2\ensuremath{^{-}}}&\multicolumn{1}{l}{E2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{00975 {\it 14}}&\parbox[t][0.3cm]{4.8739cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}=0.28 \textit{5}, A\ensuremath{_{\textnormal{4}}}={\textminus}0.03 \textit{6} in\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.8739cm}{\raggedright {\ }{\ }{\ }\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1988Ro08,B}{1988Ro08}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{847}&\multicolumn{1}{@{.}l}{7\ensuremath{^{\hyperlink{PB32GAMMA2}{\#}\hyperlink{PB32GAMMA5}{a}}} {\it 10}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{936}&\multicolumn{1}{@{.}l}{1}&\multicolumn{1}{l}{7/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{88}&\multicolumn{1}{@{.}l}{6 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{-}}}&&&&&\\
\multicolumn{1}{r@{}}{879}&\multicolumn{1}{@{.}l}{6 {\it 5}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{8 {\it 1}}&\multicolumn{1}{r@{}}{879}&\multicolumn{1}{@{.}l}{6}&\multicolumn{1}{l}{(5/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
\multicolumn{1}{r@{}}{912}&\multicolumn{1}{@{.}l}{7\ensuremath{^{\hyperlink{PB32GAMMA1}{\ddagger}}} {\it 2}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{ }l}{{\it 10}}&\multicolumn{1}{r@{}}{1541}&\multicolumn{1}{@{.}l}{5}&\multicolumn{1}{l}{17/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{628}&\multicolumn{1}{@{.}l}{8 }&\multicolumn{1}{@{}l}{13/2\ensuremath{^{+}}}&\multicolumn{1}{l}{E2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{00797 {\it 11}}&\parbox[t][0.3cm]{4.8739cm}{\raggedright I\ensuremath{_{\gamma}}: 100 in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1981He07,B}{1981He07}.\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.8739cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}=0.32 \textit{5}, A\ensuremath{_{\textnormal{4}}}={\textminus}0.01 \textit{6} in\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.8739cm}{\raggedright {\ }{\ }{\ }\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1988Ro08,B}{1988Ro08}; A\ensuremath{_{\textnormal{2}}}=0.21 \textit{3}, A\ensuremath{_{\textnormal{4}}}={\textminus}0.05\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.8739cm}{\raggedright {\ }{\ }{\ }\textit{4} in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1981He07,B}{1981He07}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{916}&\multicolumn{1}{@{.}l}{7\ensuremath{^{\hyperlink{PB32GAMMA1}{\ddagger}}} {\it 2}}&\multicolumn{1}{r@{}}{41}&\multicolumn{1}{@{ }l}{{\it 4}}&\multicolumn{1}{r@{}}{1545}&\multicolumn{1}{@{.}l}{5}&\multicolumn{1}{l}{15/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{628}&\multicolumn{1}{@{.}l}{8 }&\multicolumn{1}{@{}l}{13/2\ensuremath{^{+}}}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{015 {\it 7}}&\parbox[t][0.3cm]{4.8739cm}{\raggedright I\ensuremath{_{\gamma}}: 35 \textit{3} in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1981He07,B}{1981He07}.\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.8739cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}={\textminus}0.51 \textit{4}, A\ensuremath{_{\textnormal{4}}}=0.15 \textit{6} in\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.8739cm}{\raggedright {\ }{\ }{\ }\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1988Ro08,B}{1988Ro08}; A\ensuremath{_{\textnormal{2}}}={\textminus}0.28 \textit{3}, A\ensuremath{_{\textnormal{4}}}=0.04\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.8739cm}{\raggedright {\ }{\ }{\ }\textit{4} in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1981He07,B}{1981He07}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{919}&\multicolumn{1}{@{.}l}{3 {\it 5}}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{0 {\it 3}}&\multicolumn{1}{r@{}}{3637}&\multicolumn{1}{@{.}l}{5+x}&\multicolumn{1}{l}{31/2}&\multicolumn{1}{r@{}}{2718}&\multicolumn{1}{@{.}l}{2+x}&\multicolumn{1}{@{}l}{29/2\ensuremath{^{-}}}&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{4.8739cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}={\textminus}0.48 \textit{4}, A\ensuremath{_{\textnormal{4}}}=0.31 \textit{7} in\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.8739cm}{\raggedright {\ }{\ }{\ }\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1988Ro08,B}{1988Ro08}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{936}&\multicolumn{1}{@{.}l}{2\ensuremath{^{\hyperlink{PB32GAMMA2}{\#}\hyperlink{PB32GAMMA5}{a}}} {\it 5}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{936}&\multicolumn{1}{@{.}l}{1}&\multicolumn{1}{l}{7/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{5/2\ensuremath{^{-}}}&&&&&\\
\multicolumn{1}{r@{}}{1014}&\multicolumn{1}{@{.}l}{4 {\it 5}}&\multicolumn{1}{r@{}}{8}&\multicolumn{1}{@{.}l}{8 {\it 9}}&\multicolumn{1}{r@{}}{1014}&\multicolumn{1}{@{.}l}{4}&\multicolumn{1}{l}{9/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{l}{E2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{00648 {\it 9}}&\parbox[t][0.3cm]{4.8739cm}{\raggedright Mult.: From adopted gammas.\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.8739cm}{\raggedright {\ }{\ }{\ }Note, that A\ensuremath{_{\textnormal{2}}}=0.07 \textit{4}, A\ensuremath{_{\textnormal{4}}}=0.01 \textit{6}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.8739cm}{\raggedright {\ }{\ }{\ }in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1988Ro08,B}{1988Ro08} are inconsistent\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.8739cm}{\raggedright {\ }{\ }{\ }with the adopted multipolarity.\vspace{0.1cm}}&\\
\end{longtable}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB32GAMMA0}{\dagger}}}} From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1988Ro08,B}{1988Ro08}, unless otherwise specified. Evaluator assigns a 0.5 keV uncertainty for E\ensuremath{\gamma} and a 10\% uncertainty for I\ensuremath{\gamma}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB32GAMMA1}{\ddagger}}}} From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1981He07,B}{1981He07}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB32GAMMA2}{\#}}}} From adopted gammas.}\\
\begin{textblock}{29}(0,27.3)
Continued on next page (footnotes at end of table)
\end{textblock}
\clearpage
\vspace*{-0.5cm}
{\bf \small \underline{\ensuremath{^{\textnormal{200}}}Hg(\ensuremath{\alpha},3n\ensuremath{\gamma})\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1988Ro08,B}{1988Ro08},\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1981He07,B}{1981He07} (continued)}}\\
\vspace{0.3cm}
\underline{$\gamma$($^{201}$Pb) (continued)}\\
\vspace{0.3cm}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB32GAMMA3}{@}}}} From \ensuremath{\gamma}(\ensuremath{\theta}) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1981He07,B}{1981He07} and \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1988Ro08,B}{1988Ro08}, unless otherwise stated.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB32GAMMA4}{\&}}}} Total theoretical internal conversion coefficients, calculated using the BrIcc code (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2008Ki07,B}{2008Ki07}) with Frozen orbital approximation}\\
\parbox[b][0.3cm]{17.7cm}{{\ }{\ }based on \ensuremath{\gamma}-ray energies, assigned multipolarities, and mixing ratios, unless otherwise specified.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB32GAMMA5}{a}}}} Placement of transition in the level scheme is uncertain.}\\
\vspace{0.5cm}
\clearpage
\begin{figure}[h]
\begin{center}
\includegraphics{201PB32-0.ps}\\
\end{center}
\end{figure}
\clearpage
\subsection[\hspace{-0.2cm}\ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma})]{ }
\vspace{-27pt}
\vspace{0.3cm}
\hypertarget{PB33}{{\bf \small \underline{\ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma})\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1995Ba70,B}{1995Ba70}}}}\\
\vspace{4pt}
\vspace{8pt}
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}E(\ensuremath{^{\textnormal{14}}}C)=76 MeV; Target: \ensuremath{^{\textnormal{192}}}Os, enriched to 99\% with average thickness of 100 mg/cm\ensuremath{^{\textnormal{2}}}; Detectors: 12 Compton suppressed Ge}\\
\parbox[b][0.3cm]{17.7cm}{detectors and 48 BGO scintillation counters; Measured: E\ensuremath{\gamma}, I\ensuremath{\gamma}, \ensuremath{\gamma}\ensuremath{\gamma} coin, \ensuremath{\gamma}\ensuremath{\gamma}(\ensuremath{\theta})(DCO); Deduced: level scheme, \ensuremath{J^{\pi}}.}\\
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}Other: \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1989Su12,B}{1989Su12}, \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1992Ba39,B}{1992Ba39} (superseded by \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1995Ba70,B}{1995Ba70}).}\\
\vspace{12pt}
\underline{$^{201}$Pb Levels}\\
\begin{longtable}{cccccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E(level)$^{{\hyperlink{PB33LEVEL0}{\dagger}}}$}&J$^{\pi}$$^{{\hyperlink{PB33LEVEL1}{\ddagger}}}$&\multicolumn{2}{c}{T$_{1/2}$$^{}$}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{9}&\multicolumn{1}{@{.}l}{33 h {\it 5}}&\parbox[t][0.3cm]{12.1667cm}{\raggedright J\ensuremath{^{\pi}},T\ensuremath{_{1/2}}: From Adopted Levels.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{629}&\multicolumn{1}{@{.}l}{1\ensuremath{^{{\hyperlink{PB33LEVEL2}{\#}}}} {\it 3}}&\multicolumn{1}{l}{13/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{60}&\multicolumn{1}{@{.}l}{8 s {\it 18}}&\parbox[t][0.3cm]{12.1667cm}{\raggedright E(level),J\ensuremath{^{\pi}},T\ensuremath{_{1/2}}: From Adopted Levels.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1542}&\multicolumn{1}{@{.}l}{1\ensuremath{^{{\hyperlink{PB33LEVEL3}{@}}}} {\it 5}}&\multicolumn{1}{l}{17/2\ensuremath{^{+}}}&&&&\\
\multicolumn{1}{r@{}}{1546}&\multicolumn{1}{@{.}l}{1 {\it 5}}&\multicolumn{1}{l}{15/2\ensuremath{^{+}}}&&&&\\
\multicolumn{1}{r@{}}{1896}&\multicolumn{1}{@{.}l}{6\ensuremath{^{{\hyperlink{PB33LEVEL4}{\&}}}} {\it 5}}&\multicolumn{1}{l}{19/2\ensuremath{^{+}}}&&&&\\
\multicolumn{1}{r@{}}{1902}&\multicolumn{1}{@{.}l}{7\ensuremath{^{{\hyperlink{PB33LEVEL5}{a}}}} {\it 6}}&\multicolumn{1}{l}{21/2\ensuremath{^{+}}}&&&&\\
\multicolumn{1}{r@{}}{2069}&\multicolumn{1}{@{.}l}{1 {\it 7}}&&&&&\\
\multicolumn{1}{r@{}}{2497}&\multicolumn{1}{@{.}l}{2\ensuremath{^{{\hyperlink{PB33LEVEL6}{b}}}} {\it 6}}&\multicolumn{1}{l}{21/2\ensuremath{^{-}}}&&&&\\
\multicolumn{1}{r@{}}{2719}&\multicolumn{1}{@{.}l}{6\ensuremath{^{{\hyperlink{PB33LEVEL7}{c}}}} {\it 8}}&\multicolumn{1}{l}{25/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{63}&\multicolumn{1}{@{ }l}{ns {\it 3}}&\parbox[t][0.3cm]{12.1667cm}{\raggedright J\ensuremath{^{\pi}},T\ensuremath{_{1/2}}: From Adopted Levels.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2719}&\multicolumn{1}{@{.}l}{6+x\ensuremath{^{{\hyperlink{PB33LEVEL8}{d}}}}}&\multicolumn{1}{l}{(29/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{508}&\multicolumn{1}{@{ }l}{ns {\it 5}}&\parbox[t][0.3cm]{12.1667cm}{\raggedright J\ensuremath{^{\pi}},T\ensuremath{_{1/2}}: From Adopted Levels.\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{12.1667cm}{\raggedright E(level): Deexcites to the 2719.5 level via a low-energy (E\ensuremath{\gamma}\ensuremath{<}80 keV) \ensuremath{\gamma} ray.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2733}&\multicolumn{1}{@{.}l}{3 {\it 7}}&&&&&\\
\multicolumn{1}{r@{}}{2736}&\multicolumn{1}{@{.}l}{7 {\it 7}}&&&&&\\
\multicolumn{1}{r@{}}{3510}&\multicolumn{1}{@{.}l}{6+x {\it 4}}&\multicolumn{1}{l}{(31/2\ensuremath{^{-}})}&&&&\\
\multicolumn{1}{r@{}}{3546}&\multicolumn{1}{@{.}l}{2+x {\it 4}}&\multicolumn{1}{l}{(33/2\ensuremath{^{-}})}&&&&\\
\multicolumn{1}{r@{}}{3639}&\multicolumn{1}{@{.}l}{1+x {\it 4}}&\multicolumn{1}{l}{(31/2)}&&&&\\
\multicolumn{1}{r@{}}{3833}&\multicolumn{1}{@{.}l}{4+x {\it 6}}&\multicolumn{1}{l}{(35/2\ensuremath{^{-}})}&&&&\\
\multicolumn{1}{r@{}}{3933}&\multicolumn{1}{@{.}l}{0+x\ensuremath{^{{\hyperlink{PB33LEVEL9}{e}}}} {\it 4}}&\multicolumn{1}{l}{(33/2\ensuremath{^{+}})}&&&&\\
\multicolumn{1}{r@{}}{4060}&\multicolumn{1}{@{.}l}{6 {\it 10}}&&&&&\\
\multicolumn{1}{r@{}}{4060}&\multicolumn{1}{@{.}l}{6+y\ensuremath{^{{\hyperlink{PB33LEVEL12}{h}}}}}&&&&&\\
\multicolumn{1}{r@{}}{4169}&\multicolumn{1}{@{.}l}{8+y\ensuremath{^{{\hyperlink{PB33LEVEL12}{h}}}} {\it 5}}&&&&&\\
\multicolumn{1}{r@{}}{4351}&\multicolumn{1}{@{.}l}{4+y\ensuremath{^{{\hyperlink{PB33LEVEL12}{h}}}} {\it 7}}&&&&&\\
\multicolumn{1}{r@{}}{4506}&\multicolumn{1}{@{.}l}{2+x {\it 6}}&\multicolumn{1}{l}{(35/2)}&&&&\\
\multicolumn{1}{r@{}}{4561}&\multicolumn{1}{@{.}l}{4+x {\it 6}}&\multicolumn{1}{l}{(37/2\ensuremath{^{+}})}&&&&\\
\multicolumn{1}{r@{}}{4615}&\multicolumn{1}{@{.}l}{2+y\ensuremath{^{{\hyperlink{PB33LEVEL12}{h}}}} {\it 9}}&&&&&\\
\multicolumn{1}{r@{}}{4640}&\multicolumn{1}{@{.}l}{9+x\ensuremath{^{{\hyperlink{PB33LEVEL10}{f}}}} {\it 7}}&\multicolumn{1}{l}{(41/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{43}&\multicolumn{1}{@{ }l}{ns {\it 3}}&\parbox[t][0.3cm]{12.1667cm}{\raggedright T\ensuremath{_{1/2}}: From 80.1\ensuremath{\gamma}(t) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1989Su12,B}{1989Su12}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{4640}&\multicolumn{1}{@{.}l}{9+z\ensuremath{^{{\hyperlink{PB33LEVEL13}{i}}}}}&&&&\parbox[t][0.3cm]{12.1667cm}{\raggedright E(level): Decays to levels between 2719+x to 4506+x.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{4640}&\multicolumn{1}{@{.}l}{9+u\ensuremath{^{{\hyperlink{PB33LEVEL14}{j}}}}}&&&&\parbox[t][0.3cm]{12.1667cm}{\raggedright E(level): Decays to levels between 2719+x to 4641+x.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{4640}&\multicolumn{1}{@{.}l}{9+v\ensuremath{^{{\hyperlink{PB33LEVEL15}{k}}}}}&&&&\parbox[t][0.3cm]{12.1667cm}{\raggedright E(level): Decays to levels above 2719+x.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{4648}&\multicolumn{1}{@{.}l}{7+x {\it 6}}&\multicolumn{1}{l}{(35/2)}&&&&\\
\multicolumn{1}{r@{}}{4780}&\multicolumn{1}{@{.}l}{5+z\ensuremath{^{{\hyperlink{PB33LEVEL13}{i}}}} {\it 5}}&&&&&\\
\multicolumn{1}{r@{}}{4793}&\multicolumn{1}{@{.}l}{8+v\ensuremath{^{{\hyperlink{PB33LEVEL15}{k}}}} {\it 5}}&&&&&\\
\multicolumn{1}{r@{}}{4817}&\multicolumn{1}{@{.}l}{4+u\ensuremath{^{{\hyperlink{PB33LEVEL14}{j}}}} {\it 5}}&&&&&\\
\multicolumn{1}{r@{}}{4831}&\multicolumn{1}{@{.}l}{3+x {\it 7}}&\multicolumn{1}{l}{(39/2)}&&&&\\
\multicolumn{1}{r@{}}{4956}&\multicolumn{1}{@{.}l}{0+y\ensuremath{^{{\hyperlink{PB33LEVEL12}{h}}}} {\it 10}}&&&&&\\
\multicolumn{1}{r@{}}{4956}&\multicolumn{1}{@{.}l}{3+z\ensuremath{^{{\hyperlink{PB33LEVEL13}{i}}}} {\it 7}}&&&&&\\
\multicolumn{1}{r@{}}{4992}&\multicolumn{1}{@{.}l}{4+v\ensuremath{^{{\hyperlink{PB33LEVEL15}{k}}}} {\it 7}}&&&&&\\
\multicolumn{1}{r@{}}{5043}&\multicolumn{1}{@{.}l}{1+u\ensuremath{^{{\hyperlink{PB33LEVEL14}{j}}}} {\it 7}}&&&&&\\
\multicolumn{1}{r@{}}{5088}&\multicolumn{1}{@{.}l}{2+x {\it 7}}&\multicolumn{1}{l}{(43/2)}&&&&\\
\multicolumn{1}{r@{}}{5178}&\multicolumn{1}{@{.}l}{6+z\ensuremath{^{{\hyperlink{PB33LEVEL13}{i}}}} {\it 9}}&&&&&\\
\multicolumn{1}{r@{}}{5242}&\multicolumn{1}{@{.}l}{4+v\ensuremath{^{{\hyperlink{PB33LEVEL15}{k}}}} {\it 9}}&&&&&\\
\multicolumn{1}{r@{}}{5321}&\multicolumn{1}{@{.}l}{3+u\ensuremath{^{{\hyperlink{PB33LEVEL14}{j}}}} {\it 9}}&&&&&\\
\multicolumn{1}{r@{}}{5359}&\multicolumn{1}{@{.}l}{8+y\ensuremath{^{{\hyperlink{PB33LEVEL12}{h}}}} {\it 10}}&&&&&\\
\multicolumn{1}{r@{}}{5390}&\multicolumn{1}{@{.}l}{2+x {\it 8}}&\multicolumn{1}{l}{(45/2)}&&&&\\
\multicolumn{1}{r@{}}{5455}&\multicolumn{1}{@{.}l}{0+z\ensuremath{^{{\hyperlink{PB33LEVEL13}{i}}}} {\it 10}}&&&&&\\
\end{longtable}
\begin{textblock}{29}(0,27.3)
Continued on next page (footnotes at end of table)
\end{textblock}
\clearpage
\begin{longtable}{ccc|ccc|ccc|ccc@{\extracolsep{\fill}}c}
\\[-.4cm]
\multicolumn{13}{c}{{\bf \small \underline{\ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma})\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1995Ba70,B}{1995Ba70} (continued)}}}\\
\multicolumn{13}{c}{~}\\
\multicolumn{13}{c}{\underline{\ensuremath{^{201}}Pb Levels (continued)}}\\
\multicolumn{3}{c}{~}&\multicolumn{3}{c}{~}&\multicolumn{3}{c}{~}&\multicolumn{3}{c}{~}&\\
\multicolumn{2}{c}{E(level)$^{{\hyperlink{PB33LEVEL0}{\dagger}}}$}&J$^{\pi}$$^{{\hyperlink{PB33LEVEL1}{\ddagger}}}$&\multicolumn{2}{c}{E(level)$^{{\hyperlink{PB33LEVEL0}{\dagger}}}$}&J$^{\pi}$$^{{\hyperlink{PB33LEVEL1}{\ddagger}}}$&\multicolumn{2}{c}{E(level)$^{{\hyperlink{PB33LEVEL0}{\dagger}}}$}&J$^{\pi}$$^{{\hyperlink{PB33LEVEL1}{\ddagger}}}$&\multicolumn{2}{c}{E(level)$^{{\hyperlink{PB33LEVEL0}{\dagger}}}$}&J$^{\pi}$$^{{\hyperlink{PB33LEVEL1}{\ddagger}}}$&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endhead
\multicolumn{1}{r@{}}{5554}&\multicolumn{1}{@{.}l}{4+v\ensuremath{^{{\hyperlink{PB33LEVEL15}{k}}}} {\it 10}}&&\multicolumn{1}{r@{}}{6175}&\multicolumn{1}{@{.}l}{4+z\ensuremath{^{{\hyperlink{PB33LEVEL13}{i}}}} {\it 12}}&&\multicolumn{1}{r@{}}{6769}&\multicolumn{1}{@{.}l}{5+x\ensuremath{^{{\hyperlink{PB33LEVEL11}{g}}}} {\it 8}}&\multicolumn{1}{l|}{(43/2)}&\multicolumn{1}{r@{}}{7471}&\multicolumn{1}{@{.}l}{4+u\ensuremath{^{{\hyperlink{PB33LEVEL14}{j}}}} {\it 15}}&&\\
\multicolumn{1}{r@{}}{5583}&\multicolumn{1}{@{.}l}{0+x {\it 7}}&\multicolumn{1}{l|}{(39/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{6247}&\multicolumn{1}{@{.}l}{8+x\ensuremath{^{{\hyperlink{PB33LEVEL11}{g}}}} {\it 8}}&\multicolumn{1}{l|}{(37/2)}&\multicolumn{1}{r@{}}{6858}&\multicolumn{1}{@{.}l}{2+v\ensuremath{^{{\hyperlink{PB33LEVEL15}{k}}}} {\it 14}}&&\multicolumn{1}{r@{}}{7648}&\multicolumn{1}{@{.}l}{2+z\ensuremath{^{{\hyperlink{PB33LEVEL13}{i}}}} {\it 13}}&&\\
\multicolumn{1}{r@{}}{5648}&\multicolumn{1}{@{.}l}{0+u\ensuremath{^{{\hyperlink{PB33LEVEL14}{j}}}} {\it 10}}&&\multicolumn{1}{r@{}}{6324}&\multicolumn{1}{@{.}l}{5+y\ensuremath{^{{\hyperlink{PB33LEVEL12}{h}}}} {\it 11}}&&\multicolumn{1}{r@{}}{6883}&\multicolumn{1}{@{.}l}{0+y\ensuremath{^{{\hyperlink{PB33LEVEL12}{h}}}} {\it 12}}&&\multicolumn{1}{r@{}}{7760}&\multicolumn{1}{@{.}l}{5+x {\it 8}}&\multicolumn{1}{l}{(49/2)}&\\
\multicolumn{1}{r@{}}{5787}&\multicolumn{1}{@{.}l}{3+z\ensuremath{^{{\hyperlink{PB33LEVEL13}{i}}}} {\it 12}}&&\multicolumn{1}{r@{}}{6324}&\multicolumn{1}{@{.}l}{8+x {\it 8}}&\multicolumn{1}{l|}{(45/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{6911}&\multicolumn{1}{@{.}l}{0+x {\it 8}}&\multicolumn{1}{l|}{(47/2)}&\multicolumn{1}{r@{}}{7773}&\multicolumn{1}{@{.}l}{3+x\ensuremath{^{{\hyperlink{PB33LEVEL11}{g}}}} {\it 8}}&\multicolumn{1}{l}{(49/2)}&\\
\multicolumn{1}{r@{}}{5818}&\multicolumn{1}{@{.}l}{8+y\ensuremath{^{{\hyperlink{PB33LEVEL12}{h}}}} {\it 10}}&&\multicolumn{1}{r@{}}{6337}&\multicolumn{1}{@{.}l}{2+y {\it 11}}&&\multicolumn{1}{r@{}}{6941}&\multicolumn{1}{@{.}l}{2+u\ensuremath{^{{\hyperlink{PB33LEVEL14}{j}}}} {\it 14}}&&\multicolumn{1}{r@{}}{8004}&\multicolumn{1}{@{.}l}{4+x {\it 10}}&\multicolumn{1}{l}{(53/2)}&\\
\multicolumn{1}{r@{}}{5831}&\multicolumn{1}{@{.}l}{0+x {\it 7}}&\multicolumn{1}{l|}{(45/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{6364}&\multicolumn{1}{@{.}l}{8+v\ensuremath{^{{\hyperlink{PB33LEVEL15}{k}}}} {\it 13}}&&\multicolumn{1}{r@{}}{7009}&\multicolumn{1}{@{.}l}{4+x {\it 8}}&\multicolumn{1}{l|}{(49/2)}&\multicolumn{1}{r@{}}{8019}&\multicolumn{1}{@{.}l}{7+x {\it 8}}&\multicolumn{1}{l}{(51/2)}&\\
\multicolumn{1}{r@{}}{5836}&\multicolumn{1}{@{.}l}{7+x {\it 7}}&\multicolumn{1}{l|}{(41/2)}&\multicolumn{1}{r@{}}{6377}&\multicolumn{1}{@{.}l}{5+x\ensuremath{^{{\hyperlink{PB33LEVEL11}{g}}}} {\it 9}}&\multicolumn{1}{l|}{(39/2)}&\multicolumn{1}{r@{}}{7045}&\multicolumn{1}{@{.}l}{4+x\ensuremath{^{{\hyperlink{PB33LEVEL11}{g}}}} {\it 8}}&\multicolumn{1}{l|}{(45/2)}&\multicolumn{1}{r@{}}{8199}&\multicolumn{1}{@{.}l}{0+x {\it 10}}&\multicolumn{1}{l}{(53/2)}&\\
\multicolumn{1}{r@{}}{5892}&\multicolumn{1}{@{.}l}{2+x {\it 7}}&\multicolumn{1}{l|}{(43/2)}&\multicolumn{1}{r@{}}{6458}&\multicolumn{1}{@{.}l}{1+u\ensuremath{^{{\hyperlink{PB33LEVEL14}{j}}}} {\it 13}}&&\multicolumn{1}{r@{}}{7108}&\multicolumn{1}{@{.}l}{4+z\ensuremath{^{{\hyperlink{PB33LEVEL13}{i}}}} {\it 13}}&&\multicolumn{1}{r@{}}{8215}&\multicolumn{1}{@{.}l}{8+x {\it 10}}&\multicolumn{1}{l}{(51/2)}&\\
\multicolumn{1}{r@{}}{5928}&\multicolumn{1}{@{.}l}{8+v\ensuremath{^{{\hyperlink{PB33LEVEL15}{k}}}} {\it 12}}&&\multicolumn{1}{r@{}}{6461}&\multicolumn{1}{@{.}l}{1+x {\it 8}}&\multicolumn{1}{l|}{(47/2)}&\multicolumn{1}{r@{}}{7143}&\multicolumn{1}{@{.}l}{3+x {\it 8}}&\multicolumn{1}{l|}{(49/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{8227}&\multicolumn{1}{@{.}l}{2+x\ensuremath{^{{\hyperlink{PB33LEVEL11}{g}}}} {\it 10}}&\multicolumn{1}{l}{(51/2)}&\\
\multicolumn{1}{r@{}}{5990}&\multicolumn{1}{@{.}l}{4+x {\it 7}}&\multicolumn{1}{l|}{(45/2)}&\multicolumn{1}{r@{}}{6549}&\multicolumn{1}{@{.}l}{1+x\ensuremath{^{{\hyperlink{PB33LEVEL11}{g}}}} {\it 9}}&\multicolumn{1}{l|}{(41/2)}&\multicolumn{1}{r@{}}{7340}&\multicolumn{1}{@{.}l}{5+x {\it 9}}&\multicolumn{1}{l|}{(51/2)}&\multicolumn{1}{r@{}}{8654}&\multicolumn{1}{@{.}l}{8+x {\it 11}}&\multicolumn{1}{l}{(55/2)}&\\
\multicolumn{1}{r@{}}{6028}&\multicolumn{1}{@{.}l}{4+u\ensuremath{^{{\hyperlink{PB33LEVEL14}{j}}}} {\it 12}}&&\multicolumn{1}{r@{}}{6616}&\multicolumn{1}{@{.}l}{7+z\ensuremath{^{{\hyperlink{PB33LEVEL13}{i}}}} {\it 12}}&&\multicolumn{1}{r@{}}{7378}&\multicolumn{1}{@{.}l}{5+x {\it 8}}&\multicolumn{1}{l|}{(47/2)}&&&&\\
\multicolumn{1}{r@{}}{6146}&\multicolumn{1}{@{.}l}{2+x\ensuremath{^{{\hyperlink{PB33LEVEL11}{g}}}} {\it 7}}&\multicolumn{1}{l|}{(35/2)}&\multicolumn{1}{r@{}}{6707}&\multicolumn{1}{@{.}l}{7+x {\it 8}}&\multicolumn{1}{l|}{(49/2)}&\multicolumn{1}{r@{}}{7379}&\multicolumn{1}{@{.}l}{9+x\ensuremath{^{{\hyperlink{PB33LEVEL11}{g}}}} {\it 8}}&\multicolumn{1}{l|}{(47/2)}&&&&\\
\end{longtable}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB33LEVEL0}{\dagger}}}} From a least-squares fit to E\ensuremath{\gamma}. Energies are relative to E(13/2\ensuremath{^{+}})=629.11 \textit{18} keV. For levels labeled with +X, +Y, +Z, +U and}\\
\parbox[b][0.3cm]{17.7cm}{{\ }{\ }+V the excitation energies are relative to the 2719.6+Y, 4060.6+y, 0+Z, 0+U and 0+V states, respectively.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB33LEVEL1}{\ddagger}}}} From deduced transition multipolarities and multiple decay branches in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1995Ba70,B}{1995Ba70}, unless otherwise stated.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB33LEVEL2}{\#}}}} Configuration=\ensuremath{\nu} i\ensuremath{_{\textnormal{13/2}}^{\textnormal{$-$1}}}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB33LEVEL3}{@}}}} Probably an admixture of configuration=\ensuremath{\nu} (f\ensuremath{_{\textnormal{5/2}}^{\textnormal{$-$1}}},p\ensuremath{_{\textnormal{1/2}}^{\textnormal{$-$1}}},i\ensuremath{_{\textnormal{13/2}}^{\textnormal{$-$1}}})\ensuremath{\otimes}2\ensuremath{^{\textnormal{+}}} and configuration=\ensuremath{\nu} (i\ensuremath{_{\textnormal{13/2}}^{\textnormal{$-$1}}})\ensuremath{\otimes}2\ensuremath{^{\textnormal{+}}}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB33LEVEL4}{\&}}}} Probably an admixture of configuration=\ensuremath{\nu} (f\ensuremath{_{\textnormal{5/2}}^{\textnormal{$-$1}}},p\ensuremath{_{\textnormal{1/2}}^{\textnormal{$-$1}}},i\ensuremath{_{\textnormal{13/2}}^{\textnormal{$-$1}}})\ensuremath{\otimes}4\ensuremath{^{\textnormal{+}}} and configuration=\ensuremath{\nu} (i\ensuremath{_{\textnormal{13/2}}^{\textnormal{$-$1}}})\ensuremath{\otimes}4\ensuremath{^{\textnormal{+}}}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB33LEVEL5}{a}}}} Configuration=\ensuremath{\nu} (f\ensuremath{_{\textnormal{5/2}}^{\textnormal{$-$2}}},i\ensuremath{_{\textnormal{13/2}}^{\textnormal{$-$1}}}).}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB33LEVEL6}{b}}}} Configuration=\ensuremath{\nu} [p\ensuremath{_{\textnormal{3/2}}^{\textnormal{$-$1}}},(i\ensuremath{_{\textnormal{13/2}}^{\textnormal{$-$2}}})\ensuremath{_{\textnormal{12+}}}].}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB33LEVEL7}{c}}}} Probably an admixture of configuration=\ensuremath{\nu} [f\ensuremath{_{\textnormal{5/2}}^{\textnormal{$-$1}}},(i\ensuremath{_{\textnormal{13/2}}^{\textnormal{$-$2}}})\ensuremath{_{\textnormal{10+}}}], configuration=\ensuremath{\nu} [p\ensuremath{_{\textnormal{3/2}}^{\textnormal{$-$1}}},(i\ensuremath{_{\textnormal{13/2}}^{\textnormal{$-$2}}})\ensuremath{_{\textnormal{12+}}}] and configuration=\ensuremath{\nu}}\\
\parbox[b][0.3cm]{17.7cm}{{\ }{\ }[p\ensuremath{_{\textnormal{1/2}}^{\textnormal{$-$1}}},(i\ensuremath{_{\textnormal{13/2}}^{\textnormal{$-$2}}})\ensuremath{_{\textnormal{12+}}}].}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB33LEVEL8}{d}}}} Configuration=\ensuremath{\nu} [f\ensuremath{_{\textnormal{5/2}}^{\textnormal{$-$1}}},(i\ensuremath{_{\textnormal{13/2}}^{\textnormal{$-$2}}})\ensuremath{_{\textnormal{12+}}}].}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB33LEVEL9}{e}}}} Configuration=\ensuremath{\nu} (i\ensuremath{_{\textnormal{13/2}}^{\textnormal{$-$3}}}).}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB33LEVEL10}{f}}}} Configuration=\ensuremath{\nu} (p\ensuremath{_{\textnormal{3/2}}^{\textnormal{$-$1}}},f\ensuremath{_{\textnormal{5/2}}^{\textnormal{$-$1}}},i\ensuremath{_{\textnormal{13/2}}^{\textnormal{$-$3}}}).}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB33LEVEL11}{g}}}} Band(A): configuration=\ensuremath{\nu} [p\ensuremath{_{\textnormal{3/2}}^{\textnormal{$-$1}}},(i\ensuremath{_{\textnormal{13/2}}})\ensuremath{^{\textnormal{$-$2}}})\ensuremath{_{\textnormal{12+}}}]\ensuremath{\otimes} \ensuremath{\pi} (h\ensuremath{_{\textnormal{9/2}}^{\textnormal{+1}}},i\ensuremath{_{\textnormal{13/2}}^{\textnormal{+1}}})\ensuremath{_{\textnormal{11$-$}}}.\hphantom{a}Band 2 in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1995Ba70,B}{1995Ba70}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB33LEVEL12}{h}}}} Band(B): configuration=\ensuremath{\nu} (i\ensuremath{_{\textnormal{13/2}}^{\textnormal{$-$1}}}) \ensuremath{\otimes}\ensuremath{\pi} (h\ensuremath{_{\textnormal{9/2}}^{\textnormal{+1}}},i\ensuremath{_{\textnormal{13/2}}^{\textnormal{+1}}})\ensuremath{_{\textnormal{11$-$}}}. Band 1 in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1995Ba70,B}{1995Ba70}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB33LEVEL13}{i}}}} Band(C): Band 3 in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1995Ba70,B}{1995Ba70}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB33LEVEL14}{j}}}} Band(D): Band 4 in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1995Ba70,B}{1995Ba70}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB33LEVEL15}{k}}}} Band(E): Band 5 in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1995Ba70,B}{1995Ba70}.}\\
\vspace{0.5cm}
\clearpage
\vspace{0.3cm}
\begin{landscape}
\vspace*{-0.5cm}
{\bf \small \underline{\ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma})\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1995Ba70,B}{1995Ba70} (continued)}}\\
\vspace{0.3cm}
\underline{$\gamma$($^{201}$Pb)}\\
\begin{longtable}{ccccccccc@{}ccccccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{PB33GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{I\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{PB33GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{E\ensuremath{_{i}}(level)}&J\ensuremath{^{\pi}_{i}}&\multicolumn{2}{c}{E\ensuremath{_{f}}}&J\ensuremath{^{\pi}_{f}}&Mult.\ensuremath{^{\hyperlink{PB33GAMMA1}{\ddagger}}}&\multicolumn{2}{c}{\ensuremath{\alpha}\ensuremath{^{\hyperlink{PB33GAMMA3}{@}}}}&\multicolumn{2}{c}{I\ensuremath{_{(\gamma+ce)}}\ensuremath{^{\hyperlink{PB33GAMMA2}{\#}}}}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{79}&\multicolumn{1}{@{.}l}{5 {\it 5}}&\multicolumn{1}{r@{}}{10}&\multicolumn{1}{@{.}l}{2 {\it 19}}&\multicolumn{1}{r@{}}{4640}&\multicolumn{1}{@{.}l}{9+x}&\multicolumn{1}{l}{(41/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{4561}&\multicolumn{1}{@{.}l}{4+x }&\multicolumn{1}{@{}l}{(37/2\ensuremath{^{+}})}&\multicolumn{1}{l}{E2}&\multicolumn{1}{r@{}}{17}&\multicolumn{1}{@{.}l}{1 {\it 6}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{9.04595cm}{\raggedright E\ensuremath{_{\gamma}}: Other: 80.1 keV \textit{5} in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1989Su12,B}{1989Su12}.\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{9.04595cm}{\raggedright Mult.: R(DCO)=1.1 \textit{4}; L/M in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1989Su12,B}{1989Su12}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{98}&\multicolumn{1}{@{.}l}{2 {\it 5}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{8 {\it 1}}&\multicolumn{1}{r@{}}{5990}&\multicolumn{1}{@{.}l}{4+x}&\multicolumn{1}{l}{(45/2)}&\multicolumn{1}{r@{}}{5892}&\multicolumn{1}{@{.}l}{2+x }&\multicolumn{1}{@{}l}{(43/2)}&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{9.04595cm}{\raggedright Mult.: R(DCO)=0.74 \textit{21}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{101}&\multicolumn{1}{@{.}l}{7 {\it 5}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{6247}&\multicolumn{1}{@{.}l}{8+x}&\multicolumn{1}{l}{(37/2)}&\multicolumn{1}{r@{}}{6146}&\multicolumn{1}{@{.}l}{2+x }&\multicolumn{1}{@{}l}{(35/2)}&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{72}&\multicolumn{1}{@{ }l}{{\it 23}}&\parbox[t][0.3cm]{9.04595cm}{\raggedright Mult.: R(DCO)=0.55 \textit{14}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{109}&\multicolumn{1}{@{.}l}{2 {\it 5}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{4169}&\multicolumn{1}{@{.}l}{8+y}&&\multicolumn{1}{r@{}}{4060}&\multicolumn{1}{@{.}l}{6+y}&&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{89}&\multicolumn{1}{@{ }l}{{\it 28}}&\parbox[t][0.3cm]{9.04595cm}{\raggedright Mult.: R(DCO)=0.71 \textit{15}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{129}&\multicolumn{1}{@{.}l}{7 {\it 5}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{6377}&\multicolumn{1}{@{.}l}{5+x}&\multicolumn{1}{l}{(39/2)}&\multicolumn{1}{r@{}}{6247}&\multicolumn{1}{@{.}l}{8+x }&\multicolumn{1}{@{}l}{(37/2)}&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{77}&\multicolumn{1}{@{ }l}{{\it 15}}&\parbox[t][0.3cm]{9.04595cm}{\raggedright Mult.: R(DCO)=0.57 \textit{8}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{136}&\multicolumn{1}{@{.}l}{2 {\it 5}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{6 {\it 1}}&\multicolumn{1}{r@{}}{6461}&\multicolumn{1}{@{.}l}{1+x}&\multicolumn{1}{l}{(47/2)}&\multicolumn{1}{r@{}}{6324}&\multicolumn{1}{@{.}l}{8+x }&\multicolumn{1}{@{}l}{(45/2\ensuremath{^{+}})}&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{9.04595cm}{\raggedright Mult.: R(DCO)=0.89 \textit{28}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{139}&\multicolumn{1}{@{.}l}{6 {\it 5}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{4780}&\multicolumn{1}{@{.}l}{5+z}&&\multicolumn{1}{r@{}}{4640}&\multicolumn{1}{@{.}l}{9+z}&&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{ }l}{{\it 18}}&\parbox[t][0.3cm]{9.04595cm}{\raggedright Mult.: R(DCO)=0.69 \textit{17}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{142}&\multicolumn{1}{@{.}l}{5 {\it 5}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{1 {\it 1}}&\multicolumn{1}{r@{}}{4648}&\multicolumn{1}{@{.}l}{7+x}&\multicolumn{1}{l}{(35/2)}&\multicolumn{1}{r@{}}{4506}&\multicolumn{1}{@{.}l}{2+x }&\multicolumn{1}{@{}l}{(35/2)}&\multicolumn{1}{l}{(D)}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{9.04595cm}{\raggedright Mult.: R(DCO)=1.00 \textit{35}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{152}&\multicolumn{1}{@{.}l}{9 {\it 5}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{4793}&\multicolumn{1}{@{.}l}{8+v}&&\multicolumn{1}{r@{}}{4640}&\multicolumn{1}{@{.}l}{9+v}&&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{94}&\multicolumn{1}{@{ }l}{{\it 21}}&\parbox[t][0.3cm]{9.04595cm}{\raggedright Mult.: R(DCO)=0.55 \textit{23}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{153}&\multicolumn{1}{@{.}l}{7 {\it 5}}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{0 {\it 1}}&\multicolumn{1}{r@{}}{5990}&\multicolumn{1}{@{.}l}{4+x}&\multicolumn{1}{l}{(45/2)}&\multicolumn{1}{r@{}}{5836}&\multicolumn{1}{@{.}l}{7+x }&\multicolumn{1}{@{}l}{(41/2)}&\multicolumn{1}{l}{E2}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{114 {\it 21}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{9.04595cm}{\raggedright Mult.: R(DCO)=1.02 \textit{15}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{159}&\multicolumn{1}{@{.}l}{4 {\it 5}}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{3 {\it 1}}&\multicolumn{1}{r@{}}{5990}&\multicolumn{1}{@{.}l}{4+x}&\multicolumn{1}{l}{(45/2)}&\multicolumn{1}{r@{}}{5831}&\multicolumn{1}{@{.}l}{0+x }&\multicolumn{1}{@{}l}{(45/2\ensuremath{^{+}})}&\multicolumn{1}{l}{D,E2}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{9.04595cm}{\raggedright Mult.: R(DCO)=1.06 \textit{13}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{166}&\multicolumn{1}{@{.}l}{4 {\it 5}}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{3 {\it 2}}&\multicolumn{1}{r@{}}{2069}&\multicolumn{1}{@{.}l}{1}&&\multicolumn{1}{r@{}}{1902}&\multicolumn{1}{@{.}l}{7 }&\multicolumn{1}{@{}l}{21/2\ensuremath{^{+}}}&\multicolumn{1}{l}{D,E2}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{9.04595cm}{\raggedright Mult.: R(DCO)=1.08 \textit{9}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{171}&\multicolumn{1}{@{.}l}{6 {\it 5}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{6549}&\multicolumn{1}{@{.}l}{1+x}&\multicolumn{1}{l}{(41/2)}&\multicolumn{1}{r@{}}{6377}&\multicolumn{1}{@{.}l}{5+x }&\multicolumn{1}{@{}l}{(39/2)}&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{86}&\multicolumn{1}{@{ }l}{{\it 12}}&\parbox[t][0.3cm]{9.04595cm}{\raggedright Mult.: R(DCO)=0.58 \textit{6}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{175}&\multicolumn{1}{@{.}l}{8 {\it 5}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{4956}&\multicolumn{1}{@{.}l}{3+z}&&\multicolumn{1}{r@{}}{4780}&\multicolumn{1}{@{.}l}{5+z }&&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{99}&\multicolumn{1}{@{ }l}{{\it 28}}&\parbox[t][0.3cm]{9.04595cm}{\raggedright Mult.: R(DCO)=0.68 \textit{19}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{176}&\multicolumn{1}{@{.}l}{5 {\it 5}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{4817}&\multicolumn{1}{@{.}l}{4+u}&&\multicolumn{1}{r@{}}{4640}&\multicolumn{1}{@{.}l}{9+u}&&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{93}&\multicolumn{1}{@{ }l}{{\it 19}}&\parbox[t][0.3cm]{9.04595cm}{\raggedright Mult.: R(DCO)=0.83 \textit{19}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{179}&\multicolumn{1}{@{.}l}{3 {\it 5}}&\multicolumn{1}{r@{}}{6}&\multicolumn{1}{@{.}l}{8 {\it 7}}&\multicolumn{1}{r@{}}{8199}&\multicolumn{1}{@{.}l}{0+x}&\multicolumn{1}{l}{(53/2)}&\multicolumn{1}{r@{}}{8019}&\multicolumn{1}{@{.}l}{7+x }&\multicolumn{1}{@{}l}{(51/2)}&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{9.04595cm}{\raggedright Mult.: R(DCO)=0.70 \textit{6}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{181}&\multicolumn{1}{@{.}l}{6 {\it 5}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{4351}&\multicolumn{1}{@{.}l}{4+y}&&\multicolumn{1}{r@{}}{4169}&\multicolumn{1}{@{.}l}{8+y }&&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{ }l}{{\it 21}}&\parbox[t][0.3cm]{9.04595cm}{\raggedright Mult.: R(DCO)=0.66 \textit{6}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{190}&\multicolumn{1}{@{.}l}{4 {\it 5}}&\multicolumn{1}{r@{}}{$<$1}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{r@{}}{4831}&\multicolumn{1}{@{.}l}{3+x}&\multicolumn{1}{l}{(39/2)}&\multicolumn{1}{r@{}}{4640}&\multicolumn{1}{@{.}l}{9+x }&\multicolumn{1}{@{}l}{(41/2\ensuremath{^{+}})}&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{9.04595cm}{\raggedright Mult.: R(DCO)=0.81 \textit{18}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{197}&\multicolumn{1}{@{.}l}{2 {\it 5}}&\multicolumn{1}{r@{}}{4}&\multicolumn{1}{@{.}l}{1 {\it 1}}&\multicolumn{1}{r@{}}{7340}&\multicolumn{1}{@{.}l}{5+x}&\multicolumn{1}{l}{(51/2)}&\multicolumn{1}{r@{}}{7143}&\multicolumn{1}{@{.}l}{3+x }&\multicolumn{1}{@{}l}{(49/2\ensuremath{^{+}})}&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{9.04595cm}{\raggedright Mult.: R(DCO)=0.70 \textit{11}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{198}&\multicolumn{1}{@{.}l}{6 {\it 5}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{4992}&\multicolumn{1}{@{.}l}{4+v}&&\multicolumn{1}{r@{}}{4793}&\multicolumn{1}{@{.}l}{8+v }&&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{96}&\multicolumn{1}{@{ }l}{{\it 9}}&\parbox[t][0.3cm]{9.04595cm}{\raggedright Mult.: R(DCO)=0.63 \textit{11}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{220}&\multicolumn{1}{@{.}l}{5 {\it 5}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{6769}&\multicolumn{1}{@{.}l}{5+x}&\multicolumn{1}{l}{(43/2)}&\multicolumn{1}{r@{}}{6549}&\multicolumn{1}{@{.}l}{1+x }&\multicolumn{1}{@{}l}{(41/2)}&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{ }l}{{\it 15}}&\parbox[t][0.3cm]{9.04595cm}{\raggedright Mult.: R(DCO)=0.58 \textit{5}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{222}&\multicolumn{1}{@{.}l}{3 {\it 5}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{5178}&\multicolumn{1}{@{.}l}{6+z}&&\multicolumn{1}{r@{}}{4956}&\multicolumn{1}{@{.}l}{3+z }&&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{ }l}{{\it 15}}&\parbox[t][0.3cm]{9.04595cm}{\raggedright Mult.: R(DCO)=0.64 \textit{10}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{222}&\multicolumn{1}{@{.}l}{4 {\it 5}}&\multicolumn{1}{r@{}}{58}&\multicolumn{1}{@{.}l}{0 {\it 12}}&\multicolumn{1}{r@{}}{2719}&\multicolumn{1}{@{.}l}{6}&\multicolumn{1}{l}{25/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{2497}&\multicolumn{1}{@{.}l}{2 }&\multicolumn{1}{@{}l}{21/2\ensuremath{^{-}}}&\multicolumn{1}{l}{(E2)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{298 {\it 5}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{9.04595cm}{\raggedright Mult.: R(DCO)=0.93 \textit{10}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{225}&\multicolumn{1}{@{.}l}{7 {\it 5}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{5043}&\multicolumn{1}{@{.}l}{1+u}&&\multicolumn{1}{r@{}}{4817}&\multicolumn{1}{@{.}l}{4+u }&&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{ }l}{{\it 17}}&\parbox[t][0.3cm]{9.04595cm}{\raggedright Mult.: R(DCO)=0.88 \textit{17}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{232}&\multicolumn{1}{@{.}l}{2 {\it 5}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{6 {\it 3}}&\multicolumn{1}{r@{}}{7143}&\multicolumn{1}{@{.}l}{3+x}&\multicolumn{1}{l}{(49/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{6911}&\multicolumn{1}{@{.}l}{0+x }&\multicolumn{1}{@{}l}{(47/2)}&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{9.04595cm}{\raggedright Mult.: R(DCO)=0.81 \textit{25}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{250}&\multicolumn{1}{@{.}l}{0 {\it 5}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{5242}&\multicolumn{1}{@{.}l}{4+v}&&\multicolumn{1}{r@{}}{4992}&\multicolumn{1}{@{.}l}{4+v }&&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{ }l}{{\it 11}}&\parbox[t][0.3cm]{9.04595cm}{\raggedright Mult.: R(DCO)=0.68 \textit{11}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{253}&\multicolumn{1}{@{.}l}{7 {\it 5}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{9 {\it 1}}&\multicolumn{1}{r@{}}{5836}&\multicolumn{1}{@{.}l}{7+x}&\multicolumn{1}{l}{(41/2)}&\multicolumn{1}{r@{}}{5583}&\multicolumn{1}{@{.}l}{0+x }&\multicolumn{1}{@{}l}{(39/2\ensuremath{^{-}})}&\multicolumn{1}{l}{(D)}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{9.04595cm}{\raggedright Mult.: R(DCO)=0.78 \textit{28}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{259}&\multicolumn{1}{@{.}l}{2 {\it 5}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{8 {\it 1}}&\multicolumn{1}{r@{}}{8019}&\multicolumn{1}{@{.}l}{7+x}&\multicolumn{1}{l}{(51/2)}&\multicolumn{1}{r@{}}{7760}&\multicolumn{1}{@{.}l}{5+x }&\multicolumn{1}{@{}l}{(49/2)}&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{9.04595cm}{\raggedright Mult.: R(DCO)=0.63 \textit{13}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{263}&\multicolumn{1}{@{.}l}{8 {\it 5}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{4615}&\multicolumn{1}{@{.}l}{2+y}&&\multicolumn{1}{r@{}}{4351}&\multicolumn{1}{@{.}l}{4+y }&&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{99}&\multicolumn{1}{@{ }l}{{\it 10}}&\parbox[t][0.3cm]{9.04595cm}{\raggedright Mult.: R(DCO)=0.71 \textit{6}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{269}&\multicolumn{1}{@{.}l}{9 {\it 5}}&\multicolumn{1}{r@{}}{4}&\multicolumn{1}{@{.}l}{0 {\it 7}}&\multicolumn{1}{r@{}}{4831}&\multicolumn{1}{@{.}l}{3+x}&\multicolumn{1}{l}{(39/2)}&\multicolumn{1}{r@{}}{4561}&\multicolumn{1}{@{.}l}{4+x }&\multicolumn{1}{@{}l}{(37/2\ensuremath{^{+}})}&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{9.04595cm}{\raggedright Mult.: R(DCO)=0.64 \textit{12}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{275}&\multicolumn{1}{@{.}l}{9 {\it 5}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{7045}&\multicolumn{1}{@{.}l}{4+x}&\multicolumn{1}{l}{(45/2)}&\multicolumn{1}{r@{}}{6769}&\multicolumn{1}{@{.}l}{5+x }&\multicolumn{1}{@{}l}{(43/2)}&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{92}&\multicolumn{1}{@{ }l}{{\it 10}}&\parbox[t][0.3cm]{9.04595cm}{\raggedright Mult.: R(DCO)=0.56 \textit{3}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{276}&\multicolumn{1}{@{.}l}{4 {\it 5}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{5455}&\multicolumn{1}{@{.}l}{0+z}&&\multicolumn{1}{r@{}}{5178}&\multicolumn{1}{@{.}l}{6+z }&&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{94}&\multicolumn{1}{@{ }l}{{\it 15}}&\parbox[t][0.3cm]{9.04595cm}{\raggedright Mult.: R(DCO)=0.66 \textit{9}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{278}&\multicolumn{1}{@{.}l}{2 {\it 5}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{5321}&\multicolumn{1}{@{.}l}{3+u}&&\multicolumn{1}{r@{}}{5043}&\multicolumn{1}{@{.}l}{1+u }&&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{91}&\multicolumn{1}{@{ }l}{{\it 10}}&\parbox[t][0.3cm]{9.04595cm}{\raggedright Mult.: R(DCO)=0.85 \textit{14}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{287}&\multicolumn{1}{@{.}l}{2 {\it 5}}&\multicolumn{1}{r@{}}{42}&\multicolumn{1}{@{.}l}{1 {\it 8}}&\multicolumn{1}{r@{}}{3833}&\multicolumn{1}{@{.}l}{4+x}&\multicolumn{1}{l}{(35/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{3546}&\multicolumn{1}{@{.}l}{2+x }&\multicolumn{1}{@{}l}{(33/2\ensuremath{^{-}})}&\multicolumn{1}{l}{M1}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{481 {\it 7}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{9.04595cm}{\raggedright Mult.: R(DCO)=0.76 \textit{5}; K/L in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1989Su12,B}{1989Su12}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{293}&\multicolumn{1}{@{.}l}{9 {\it 5}}&\multicolumn{1}{r@{}}{12}&\multicolumn{1}{@{.}l}{8 {\it 8}}&\multicolumn{1}{r@{}}{3933}&\multicolumn{1}{@{.}l}{0+x}&\multicolumn{1}{l}{(33/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{3639}&\multicolumn{1}{@{.}l}{1+x }&\multicolumn{1}{@{}l}{(31/2)}&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{9.04595cm}{\raggedright Mult.: R(DCO)=0.78 \textit{7}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{302}&\multicolumn{1}{@{.}l}{0 {\it 5}}&\multicolumn{1}{r@{}}{4}&\multicolumn{1}{@{.}l}{5 {\it 1}}&\multicolumn{1}{r@{}}{5390}&\multicolumn{1}{@{.}l}{2+x}&\multicolumn{1}{l}{(45/2)}&\multicolumn{1}{r@{}}{5088}&\multicolumn{1}{@{.}l}{2+x }&\multicolumn{1}{@{}l}{(43/2)}&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{9.04595cm}{\raggedright Mult.: R(DCO)=0.79 \textit{13}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{312}&\multicolumn{1}{@{.}l}{0 {\it 5}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{5554}&\multicolumn{1}{@{.}l}{4+v}&&\multicolumn{1}{r@{}}{5242}&\multicolumn{1}{@{.}l}{4+v }&&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{75}&\multicolumn{1}{@{ }l}{{\it 10}}&\parbox[t][0.3cm]{9.04595cm}{\raggedright Mult.: R(DCO)=0.58 \textit{11}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{326}&\multicolumn{1}{@{.}l}{7 {\it 5}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{5648}&\multicolumn{1}{@{.}l}{0+u}&&\multicolumn{1}{r@{}}{5321}&\multicolumn{1}{@{.}l}{3+u }&&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{64}&\multicolumn{1}{@{ }l}{{\it 12}}&\parbox[t][0.3cm]{9.04595cm}{\raggedright Mult.: R(DCO)=0.79 \textit{16}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{331}&\multicolumn{1}{@{.}l}{1 {\it 5}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{7 {\it 2}}&\multicolumn{1}{r@{}}{7340}&\multicolumn{1}{@{.}l}{5+x}&\multicolumn{1}{l}{(51/2)}&\multicolumn{1}{r@{}}{7009}&\multicolumn{1}{@{.}l}{4+x }&\multicolumn{1}{@{}l}{(49/2)}&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{9.04595cm}{\raggedright Mult.: R(DCO)=0.77 \textit{18}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{332}&\multicolumn{1}{@{.}l}{3 {\it 5}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{5787}&\multicolumn{1}{@{.}l}{3+z}&&\multicolumn{1}{r@{}}{5455}&\multicolumn{1}{@{.}l}{0+z }&&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{69}&\multicolumn{1}{@{ }l}{{\it 11}}&\parbox[t][0.3cm]{9.04595cm}{\raggedright Mult.: R(DCO)=0.69 \textit{13}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{333}&\multicolumn{1}{@{.}l}{1 {\it 5}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{7378}&\multicolumn{1}{@{.}l}{5+x}&\multicolumn{1}{l}{(47/2)}&\multicolumn{1}{r@{}}{7045}&\multicolumn{1}{@{.}l}{4+x }&\multicolumn{1}{@{}l}{(45/2)}&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{46}&\multicolumn{1}{@{ }l}{{\it 9}}&\parbox[t][0.3cm]{9.04595cm}{\raggedright Mult.: R(DCO)=0.57 \textit{5}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{334}&\multicolumn{1}{@{.}l}{6 {\it 5}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{7379}&\multicolumn{1}{@{.}l}{9+x}&\multicolumn{1}{l}{(47/2)}&\multicolumn{1}{r@{}}{7045}&\multicolumn{1}{@{.}l}{4+x }&\multicolumn{1}{@{}l}{(45/2)}&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{43}&\multicolumn{1}{@{ }l}{{\it 11}}&\parbox[t][0.3cm]{9.04595cm}{\raggedright Mult.: R(DCO)=0.60 \textit{5}.\vspace{0.1cm}}&\\
\end{longtable}
\clearpage
\begin{longtable}{ccccccccc@{}ccccccc@{\extracolsep{\fill}}c}
\\[-.4cm]
\multicolumn{17}{c}{{\bf \small \underline{\ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma})\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1995Ba70,B}{1995Ba70} (continued)}}}\\
\multicolumn{17}{c}{~}\\
\multicolumn{17}{c}{\underline{$\gamma$($^{201}$Pb) (continued)}}\\
\multicolumn{17}{c}{~~~}\\
\multicolumn{2}{c}{E\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{PB33GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{I\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{PB33GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{E\ensuremath{_{i}}(level)}&J\ensuremath{^{\pi}_{i}}&\multicolumn{2}{c}{E\ensuremath{_{f}}}&J\ensuremath{^{\pi}_{f}}&Mult.\ensuremath{^{\hyperlink{PB33GAMMA1}{\ddagger}}}&\multicolumn{2}{c}{\ensuremath{\alpha}\ensuremath{^{\hyperlink{PB33GAMMA3}{@}}}}&\multicolumn{2}{c}{I\ensuremath{_{(\gamma+ce)}}\ensuremath{^{\hyperlink{PB33GAMMA2}{\#}}}}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endhead
\multicolumn{1}{r@{}}{340}&\multicolumn{1}{@{.}l}{8 {\it 5}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{4956}&\multicolumn{1}{@{.}l}{0+y}&&\multicolumn{1}{r@{}}{4615}&\multicolumn{1}{@{.}l}{2+y }&&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{87}&\multicolumn{1}{@{ }l}{{\it 8}}&\parbox[t][0.3cm]{8.905619cm}{\raggedright Mult.: R(DCO)=0.69 \textit{5}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{350}&\multicolumn{1}{@{.}l}{5 {\it 5}}&\multicolumn{1}{r@{}}{34}&\multicolumn{1}{@{.}l}{5 {\it 14}}&\multicolumn{1}{r@{}}{1896}&\multicolumn{1}{@{.}l}{6}&\multicolumn{1}{l}{19/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1546}&\multicolumn{1}{@{.}l}{1 }&\multicolumn{1}{@{}l}{15/2\ensuremath{^{+}}}&\multicolumn{1}{l}{E2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0738 {\it 11}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{8.905619cm}{\raggedright Mult.: R(DCO)=0.95 \textit{13}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{354}&\multicolumn{1}{@{.}l}{5 {\it 5}}&\multicolumn{1}{r@{}}{63}&\multicolumn{1}{@{.}l}{4 {\it 19}}&\multicolumn{1}{r@{}}{1896}&\multicolumn{1}{@{.}l}{6}&\multicolumn{1}{l}{19/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1542}&\multicolumn{1}{@{.}l}{1 }&\multicolumn{1}{@{}l}{17/2\ensuremath{^{+}}}&\multicolumn{1}{l}{M1}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{8.905619cm}{\raggedright Mult.: From adopted gammas. R(DCO)=1.10 \textit{11}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{360}&\multicolumn{1}{@{.}l}{6 {\it 5}}&\multicolumn{1}{r@{}}{26}&\multicolumn{1}{@{.}l}{3 {\it 11}}&\multicolumn{1}{r@{}}{1902}&\multicolumn{1}{@{.}l}{7}&\multicolumn{1}{l}{21/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1542}&\multicolumn{1}{@{.}l}{1 }&\multicolumn{1}{@{}l}{17/2\ensuremath{^{+}}}&\multicolumn{1}{l}{E2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0682 {\it 10}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{8.905619cm}{\raggedright Mult.: R(DCO)=0.96 \textit{9}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{374}&\multicolumn{1}{@{.}l}{4 {\it 5}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{5928}&\multicolumn{1}{@{.}l}{8+v}&&\multicolumn{1}{r@{}}{5554}&\multicolumn{1}{@{.}l}{4+v }&&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{57}&\multicolumn{1}{@{ }l}{{\it 7}}&\parbox[t][0.3cm]{8.905619cm}{\raggedright Mult.: R(DCO)=0.68 \textit{14}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{380}&\multicolumn{1}{@{.}l}{4 {\it 5}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{6028}&\multicolumn{1}{@{.}l}{4+u}&&\multicolumn{1}{r@{}}{5648}&\multicolumn{1}{@{.}l}{0+u }&&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{36}&\multicolumn{1}{@{ }l}{{\it 7}}&\parbox[t][0.3cm]{8.905619cm}{\raggedright Mult.: R(DCO)=0.89 \textit{11}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{380}&\multicolumn{1}{@{.}l}{6 {\it 5}}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{2 {\it 3}}&\multicolumn{1}{r@{}}{7760}&\multicolumn{1}{@{.}l}{5+x}&\multicolumn{1}{l}{(49/2)}&\multicolumn{1}{r@{}}{7379}&\multicolumn{1}{@{.}l}{9+x }&\multicolumn{1}{@{}l}{(47/2)}&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{8.905619cm}{\raggedright Mult.: R(DCO)=0.84 \textit{29}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{382}&\multicolumn{1}{@{.}l}{0 {\it 5}}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{8 {\it 1}}&\multicolumn{1}{r@{}}{7760}&\multicolumn{1}{@{.}l}{5+x}&\multicolumn{1}{l}{(49/2)}&\multicolumn{1}{r@{}}{7378}&\multicolumn{1}{@{.}l}{5+x }&\multicolumn{1}{@{}l}{(47/2)}&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{8.905619cm}{\raggedright Mult.: R(DCO)=0.66 \textit{15}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{387}&\multicolumn{1}{@{.}l}{0 {\it 5}}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{3 {\it 1}}&\multicolumn{1}{r@{}}{3933}&\multicolumn{1}{@{.}l}{0+x}&\multicolumn{1}{l}{(33/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{3546}&\multicolumn{1}{@{.}l}{2+x }&\multicolumn{1}{@{}l}{(33/2\ensuremath{^{-}})}&\multicolumn{1}{l}{(D)}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{8.905619cm}{\raggedright Mult.: R(DCO)=0.90 \textit{15}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{388}&\multicolumn{1}{@{.}l}{1 {\it 5}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{6175}&\multicolumn{1}{@{.}l}{4+z}&&\multicolumn{1}{r@{}}{5787}&\multicolumn{1}{@{.}l}{3+z }&&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{51}&\multicolumn{1}{@{ }l}{{\it 7}}&\parbox[t][0.3cm]{8.905619cm}{\raggedright Mult.: R(DCO)=0.59 \textit{14}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{393}&\multicolumn{1}{@{.}l}{3 {\it 5}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{7773}&\multicolumn{1}{@{.}l}{3+x}&\multicolumn{1}{l}{(49/2)}&\multicolumn{1}{r@{}}{7379}&\multicolumn{1}{@{.}l}{9+x }&\multicolumn{1}{@{}l}{(47/2)}&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{28}&\multicolumn{1}{@{ }l}{{\it 4}}&\parbox[t][0.3cm]{8.905619cm}{\raggedright Mult.: R(DCO)=0.64 \textit{10}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{394}&\multicolumn{1}{@{.}l}{8 {\it 5}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{7773}&\multicolumn{1}{@{.}l}{3+x}&\multicolumn{1}{l}{(49/2)}&\multicolumn{1}{r@{}}{7378}&\multicolumn{1}{@{.}l}{5+x }&\multicolumn{1}{@{}l}{(47/2)}&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{31}&\multicolumn{1}{@{ }l}{{\it 7}}&\parbox[t][0.3cm]{8.905619cm}{\raggedright Mult.: R(DCO)=0.60 \textit{13}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{404}&\multicolumn{1}{@{.}l}{0 {\it 5}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{5359}&\multicolumn{1}{@{.}l}{8+y}&&\multicolumn{1}{r@{}}{4956}&\multicolumn{1}{@{.}l}{0+y }&&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{75}&\multicolumn{1}{@{ }l}{{\it 7}}&\parbox[t][0.3cm]{8.905619cm}{\raggedright Mult.: R(DCO)=0.70 \textit{6}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{422}&\multicolumn{1}{@{.}l}{5 {\it 5}}&\multicolumn{1}{r@{}}{32}&\multicolumn{1}{@{.}l}{5 {\it 16}}&\multicolumn{1}{r@{}}{3933}&\multicolumn{1}{@{.}l}{0+x}&\multicolumn{1}{l}{(33/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{3510}&\multicolumn{1}{@{.}l}{6+x }&\multicolumn{1}{@{}l}{(31/2\ensuremath{^{-}})}&\multicolumn{1}{l}{(D)}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{8.905619cm}{\raggedright Mult.: R(DCO)=0.83 \textit{14}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{429}&\multicolumn{1}{@{.}l}{7 {\it 5}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{6458}&\multicolumn{1}{@{.}l}{1+u}&&\multicolumn{1}{r@{}}{6028}&\multicolumn{1}{@{.}l}{4+u }&&\multicolumn{1}{l}{(D)}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{16}&\multicolumn{1}{@{ }l}{{\it 7}}&\parbox[t][0.3cm]{8.905619cm}{\raggedright Mult.: R(DCO)=0.79 \textit{30}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{436}&\multicolumn{1}{@{.}l}{0 {\it 5}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{6364}&\multicolumn{1}{@{.}l}{8+v}&&\multicolumn{1}{r@{}}{5928}&\multicolumn{1}{@{.}l}{8+v }&&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{39}&\multicolumn{1}{@{ }l}{{\it 10}}&\parbox[t][0.3cm]{8.905619cm}{\raggedright Mult.: R(DCO)=0.77 \textit{21}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{441}&\multicolumn{1}{@{.}l}{3 {\it 5}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{6616}&\multicolumn{1}{@{.}l}{7+z}&&\multicolumn{1}{r@{}}{6175}&\multicolumn{1}{@{.}l}{4+z }&&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{39}&\multicolumn{1}{@{ }l}{{\it 10}}&\parbox[t][0.3cm]{8.905619cm}{\raggedright Mult.: R(DCO)=0.64 \textit{16}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{442}&\multicolumn{1}{@{.}l}{5 {\it 5}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{8215}&\multicolumn{1}{@{.}l}{8+x}&\multicolumn{1}{l}{(51/2)}&\multicolumn{1}{r@{}}{7773}&\multicolumn{1}{@{.}l}{3+x }&\multicolumn{1}{@{}l}{(49/2)}&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{9}&\multicolumn{1}{@{ }l}{{\it 5}}&\parbox[t][0.3cm]{8.905619cm}{\raggedright Mult.: R(DCO)=0.68 \textit{16}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{447}&\multicolumn{1}{@{.}l}{3 {\it 5}}&\multicolumn{1}{r@{}}{31}&\multicolumn{1}{@{.}l}{5 {\it 6}}&\multicolumn{1}{r@{}}{5088}&\multicolumn{1}{@{.}l}{2+x}&\multicolumn{1}{l}{(43/2)}&\multicolumn{1}{r@{}}{4640}&\multicolumn{1}{@{.}l}{9+x }&\multicolumn{1}{@{}l}{(41/2\ensuremath{^{+}})}&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{8.905619cm}{\raggedright Mult.: R(DCO)=0.76 \textit{5}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{453}&\multicolumn{1}{@{.}l}{9 {\it 5}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{8227}&\multicolumn{1}{@{.}l}{2+x}&\multicolumn{1}{l}{(51/2)}&\multicolumn{1}{r@{}}{7773}&\multicolumn{1}{@{.}l}{3+x }&\multicolumn{1}{@{}l}{(49/2)}&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{19}&\multicolumn{1}{@{ }l}{{\it 6}}&\parbox[t][0.3cm]{8.905619cm}{\raggedright Mult.: R(DCO)=0.62 \textit{14}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{455}&\multicolumn{1}{@{.}l}{8 {\it 5}}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{2 {\it 2}}&\multicolumn{1}{r@{}}{8654}&\multicolumn{1}{@{.}l}{8+x}&\multicolumn{1}{l}{(55/2)}&\multicolumn{1}{r@{}}{8199}&\multicolumn{1}{@{.}l}{0+x }&\multicolumn{1}{@{}l}{(53/2)}&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{8.905619cm}{\raggedright Mult.: R(DCO)=0.71 \textit{7}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{459}&\multicolumn{1}{@{.}l}{0 {\it 5}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{5818}&\multicolumn{1}{@{.}l}{8+y}&&\multicolumn{1}{r@{}}{5359}&\multicolumn{1}{@{.}l}{8+y }&&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{52}&\multicolumn{1}{@{ }l}{{\it 6}}&\parbox[t][0.3cm]{8.905619cm}{\raggedright Mult.: R(DCO)=0.70 \textit{6}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{470}&\multicolumn{1}{@{.}l}{7 {\it 5}}&\multicolumn{1}{r@{}}{7}&\multicolumn{1}{@{.}l}{1 {\it 4}}&\multicolumn{1}{r@{}}{6461}&\multicolumn{1}{@{.}l}{1+x}&\multicolumn{1}{l}{(47/2)}&\multicolumn{1}{r@{}}{5990}&\multicolumn{1}{@{.}l}{4+x }&\multicolumn{1}{@{}l}{(45/2)}&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{8.905619cm}{\raggedright Mult.: R(DCO)=0.94 \textit{12}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{483}&\multicolumn{1}{@{.}l}{1 {\it 5}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{6941}&\multicolumn{1}{@{.}l}{2+u}&&\multicolumn{1}{r@{}}{6458}&\multicolumn{1}{@{.}l}{1+u }&&\multicolumn{1}{l}{(D)}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{19}&\multicolumn{1}{@{ }l}{{\it 7}}&\parbox[t][0.3cm]{8.905619cm}{\raggedright Mult.: R(DCO)=0.73 \textit{26}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{491}&\multicolumn{1}{@{.}l}{7 {\it 5}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{7108}&\multicolumn{1}{@{.}l}{4+z}&&\multicolumn{1}{r@{}}{6616}&\multicolumn{1}{@{.}l}{7+z }&&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{26}&\multicolumn{1}{@{ }l}{{\it 5}}&\parbox[t][0.3cm]{8.905619cm}{\raggedright Mult.: R(DCO)=0.64 \textit{17}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{493}&\multicolumn{1}{@{.}l}{4 {\it 5}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{6858}&\multicolumn{1}{@{.}l}{2+v}&&\multicolumn{1}{r@{}}{6364}&\multicolumn{1}{@{.}l}{8+v }&&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{37}&\multicolumn{1}{@{ }l}{{\it 7}}&\parbox[t][0.3cm]{8.905619cm}{\raggedright Mult.: R(DCO)=0.76 \textit{20}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{505}&\multicolumn{1}{@{.}l}{7 {\it 5}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{6324}&\multicolumn{1}{@{.}l}{5+y}&&\multicolumn{1}{r@{}}{5818}&\multicolumn{1}{@{.}l}{8+y }&&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{17}&\multicolumn{1}{@{ }l}{{\it 6}}&\parbox[t][0.3cm]{8.905619cm}{\raggedright Mult.: R(DCO)=0.67 \textit{11}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{518}&\multicolumn{1}{@{.}l}{4 {\it 5}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{6337}&\multicolumn{1}{@{.}l}{2+y}&&\multicolumn{1}{r@{}}{5818}&\multicolumn{1}{@{.}l}{8+y }&&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{10}&\multicolumn{1}{@{ }l}{{\it 2}}&\parbox[t][0.3cm]{8.905619cm}{\raggedright Mult.: R(DCO)=0.69 \textit{10}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{530}&\multicolumn{1}{@{.}l}{2 {\it 5}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{7471}&\multicolumn{1}{@{.}l}{4+u}&&\multicolumn{1}{r@{}}{6941}&\multicolumn{1}{@{.}l}{2+u }&&\multicolumn{1}{l}{(D)}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{15}&\multicolumn{1}{@{ }l}{{\it 3}}&\parbox[t][0.3cm]{8.905619cm}{\raggedright Mult.: R(DCO)=0.85 \textit{20}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{539}&\multicolumn{1}{@{.}l}{8 {\it 5}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{7648}&\multicolumn{1}{@{.}l}{2+z}&&\multicolumn{1}{r@{}}{7108}&\multicolumn{1}{@{.}l}{4+z }&&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{12}&\multicolumn{1}{@{ }l}{{\it 3}}&\parbox[t][0.3cm]{8.905619cm}{\raggedright Mult.: R(DCO)=0.66 \textit{26}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{548}&\multicolumn{1}{@{.}l}{3 {\it 5}}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{7 {\it 1}}&\multicolumn{1}{r@{}}{7009}&\multicolumn{1}{@{.}l}{4+x}&\multicolumn{1}{l}{(49/2)}&\multicolumn{1}{r@{}}{6461}&\multicolumn{1}{@{.}l}{1+x }&\multicolumn{1}{@{}l}{(47/2)}&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{8.905619cm}{\raggedright Mult.: R(DCO)=0.67 \textit{9}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{558}&\multicolumn{1}{@{.}l}{5 {\it 5}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{6883}&\multicolumn{1}{@{.}l}{0+y}&&\multicolumn{1}{r@{}}{6324}&\multicolumn{1}{@{.}l}{5+y }&&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{7}&\multicolumn{1}{@{ }l}{{\it 1}}&\parbox[t][0.3cm]{8.905619cm}{\raggedright Mult.: R(DCO)=0.65 \textit{18}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{573}&\multicolumn{1}{@{.}l}{2 {\it 5}}&\multicolumn{1}{r@{}}{9}&\multicolumn{1}{@{.}l}{0 {\it 8}}&\multicolumn{1}{r@{}}{4506}&\multicolumn{1}{@{.}l}{2+x}&\multicolumn{1}{l}{(35/2)}&\multicolumn{1}{r@{}}{3933}&\multicolumn{1}{@{.}l}{0+x }&\multicolumn{1}{@{}l}{(33/2\ensuremath{^{+}})}&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{8.905619cm}{\raggedright Mult.: R(DCO)=0.63 \textit{11}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{586}&\multicolumn{1}{@{.}l}{1 {\it 5}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{8 {\it 3}}&\multicolumn{1}{r@{}}{6911}&\multicolumn{1}{@{.}l}{0+x}&\multicolumn{1}{l}{(47/2)}&\multicolumn{1}{r@{}}{6324}&\multicolumn{1}{@{.}l}{8+x }&\multicolumn{1}{@{}l}{(45/2\ensuremath{^{+}})}&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{8.905619cm}{\raggedright Mult.: R(DCO)=0.76 \textit{13}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{594}&\multicolumn{1}{@{.}l}{5 {\it 5}}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{0 {\it 3}}&\multicolumn{1}{r@{}}{2497}&\multicolumn{1}{@{.}l}{2}&\multicolumn{1}{l}{21/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{1902}&\multicolumn{1}{@{.}l}{7 }&\multicolumn{1}{@{}l}{21/2\ensuremath{^{+}}}&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{8.905619cm}{\raggedright Mult.: R(DCO)=0.73 \textit{21}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{600}&\multicolumn{1}{@{.}l}{2 {\it 5}}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{9 {\it 4}}&\multicolumn{1}{r@{}}{5990}&\multicolumn{1}{@{.}l}{4+x}&\multicolumn{1}{l}{(45/2)}&\multicolumn{1}{r@{}}{5390}&\multicolumn{1}{@{.}l}{2+x }&\multicolumn{1}{@{}l}{(45/2)}&\multicolumn{1}{l}{D,E2}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{8.905619cm}{\raggedright Mult.: R(DCO)=0.98 \textit{16}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{600}&\multicolumn{1}{@{.}l}{5 {\it 5}}&\multicolumn{1}{r@{}}{70}&\multicolumn{1}{@{.}l}{9 {\it 14}}&\multicolumn{1}{r@{}}{2497}&\multicolumn{1}{@{.}l}{2}&\multicolumn{1}{l}{21/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{1896}&\multicolumn{1}{@{.}l}{6 }&\multicolumn{1}{@{}l}{19/2\ensuremath{^{+}}}&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{8.905619cm}{\raggedright Mult.: R(DCO)=0.80 \textit{8}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$610}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{36}&\multicolumn{1}{@{ }l}{{\it 4}}&&&&&&&&&&&&\parbox[t][0.3cm]{8.905619cm}{\raggedright E\ensuremath{_{\gamma}}: Depopulates band 1 in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1995Ba70,B}{1995Ba70}, but the exact placement is\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.905619cm}{\raggedright {\ }{\ }{\ }not known.\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.905619cm}{\raggedright I\ensuremath{_{\gamma}}: \% branching of the total population of band 1 in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1995Ba70,B}{1995Ba70}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{628}&\multicolumn{1}{@{.}l}{4 {\it 5}}&\multicolumn{1}{r@{}}{27}&\multicolumn{1}{@{.}l}{0 {\it 14}}&\multicolumn{1}{r@{}}{4561}&\multicolumn{1}{@{.}l}{4+x}&\multicolumn{1}{l}{(37/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{3933}&\multicolumn{1}{@{.}l}{0+x }&\multicolumn{1}{@{}l}{(33/2\ensuremath{^{+}})}&\multicolumn{1}{l}{E2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{01736 {\it 24}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{8.905619cm}{\raggedright Mult.: R(DCO)=1.00 \textit{9}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{663}&\multicolumn{1}{@{.}l}{9 {\it 5}}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{1 {\it 2}}&\multicolumn{1}{r@{}}{8004}&\multicolumn{1}{@{.}l}{4+x}&\multicolumn{1}{l}{(53/2)}&\multicolumn{1}{r@{}}{7340}&\multicolumn{1}{@{.}l}{5+x }&\multicolumn{1}{@{}l}{(51/2)}&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{8.905619cm}{\raggedright Mult.: R(DCO)=0.80 \textit{12}. Note, that the authors give\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.905619cm}{\raggedright {\ }{\ }{\ }R(DCO)=0.8 \textit{12} that is probably a typo.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{664}&\multicolumn{1}{@{.}l}{1 {\it 5}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{3 {\it 1}}&\multicolumn{1}{r@{}}{2733}&\multicolumn{1}{@{.}l}{3}&&\multicolumn{1}{r@{}}{2069}&\multicolumn{1}{@{.}l}{1 }&&\multicolumn{1}{l}{D,E2}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{8.905619cm}{\raggedright Mult.: R(DCO)=1.08 \textit{25}.\vspace{0.1cm}}&\\
\end{longtable}
\clearpage
\begin{longtable}{ccccccccc@{}ccccccc@{\extracolsep{\fill}}c}
\\[-.4cm]
\multicolumn{17}{c}{{\bf \small \underline{\ensuremath{^{\textnormal{192}}}Os(\ensuremath{^{\textnormal{14}}}C,5n\ensuremath{\gamma})\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1995Ba70,B}{1995Ba70} (continued)}}}\\
\multicolumn{17}{c}{~}\\
\multicolumn{17}{c}{\underline{$\gamma$($^{201}$Pb) (continued)}}\\
\multicolumn{17}{c}{~~~}\\
\multicolumn{2}{c}{E\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{PB33GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{I\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{PB33GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{E\ensuremath{_{i}}(level)}&J\ensuremath{^{\pi}_{i}}&\multicolumn{2}{c}{E\ensuremath{_{f}}}&J\ensuremath{^{\pi}_{f}}&Mult.\ensuremath{^{\hyperlink{PB33GAMMA1}{\ddagger}}}&\multicolumn{2}{c}{\ensuremath{\alpha}\ensuremath{^{\hyperlink{PB33GAMMA3}{@}}}}&\multicolumn{2}{c}{I\ensuremath{_{(\gamma+ce)}}\ensuremath{^{\hyperlink{PB33GAMMA2}{\#}}}}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endhead
\multicolumn{1}{r@{}}{667}&\multicolumn{1}{@{.}l}{6 {\it 5}}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{8 {\it 2}}&\multicolumn{1}{r@{}}{2736}&\multicolumn{1}{@{.}l}{7}&&\multicolumn{1}{r@{}}{2069}&\multicolumn{1}{@{.}l}{1 }&&\multicolumn{1}{l}{D,E2}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{7.7179203cm}{\raggedright Mult.: R(DCO)= 1.03 \textit{16}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{682}&\multicolumn{1}{@{.}l}{3 {\it 5}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{0 {\it 1}}&\multicolumn{1}{r@{}}{7143}&\multicolumn{1}{@{.}l}{3+x}&\multicolumn{1}{l}{(49/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{6461}&\multicolumn{1}{@{.}l}{1+x }&\multicolumn{1}{@{}l}{(47/2)}&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{7.7179203cm}{\raggedright Mult.: R(DCO)=0.73 \textit{18}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{715}&\multicolumn{1}{@{.}l}{7 {\it 5}}&\multicolumn{1}{r@{}}{5}&\multicolumn{1}{@{.}l}{5 {\it 8}}&\multicolumn{1}{r@{}}{4648}&\multicolumn{1}{@{.}l}{7+x}&\multicolumn{1}{l}{(35/2)}&\multicolumn{1}{r@{}}{3933}&\multicolumn{1}{@{.}l}{0+x }&\multicolumn{1}{@{}l}{(33/2\ensuremath{^{+}})}&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{7.7179203cm}{\raggedright Mult.: R(DCO)=0.66 \textit{12}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{717}&\multicolumn{1}{@{.}l}{3 {\it 5}}&\multicolumn{1}{r@{}}{27}&\multicolumn{1}{@{.}l}{7 {\it 6}}&\multicolumn{1}{r@{}}{6707}&\multicolumn{1}{@{.}l}{7+x}&\multicolumn{1}{l}{(49/2)}&\multicolumn{1}{r@{}}{5990}&\multicolumn{1}{@{.}l}{4+x }&\multicolumn{1}{@{}l}{(45/2)}&\multicolumn{1}{l}{E2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{01305 {\it 18}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{7.7179203cm}{\raggedright Mult.: R(DCO)=1.05 \textit{6}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{728}&\multicolumn{1}{@{.}l}{0 {\it 5}}&\multicolumn{1}{r@{}}{61}&\multicolumn{1}{@{.}l}{5 {\it 18}}&\multicolumn{1}{r@{}}{4561}&\multicolumn{1}{@{.}l}{4+x}&\multicolumn{1}{l}{(37/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{3833}&\multicolumn{1}{@{.}l}{4+x }&\multicolumn{1}{@{}l}{(35/2\ensuremath{^{-}})}&\multicolumn{1}{l}{(D)}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{7.7179203cm}{\raggedright Mult.: R(DCO)=0.89 \textit{4}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{744}&\multicolumn{1}{@{.}l}{6 {\it 5}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{5359}&\multicolumn{1}{@{.}l}{8+y}&&\multicolumn{1}{r@{}}{4615}&\multicolumn{1}{@{.}l}{2+y }&&\multicolumn{1}{l}{E2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{01206 {\it 17}}&\multicolumn{1}{r@{}}{11}&\multicolumn{1}{@{ }l}{{\it 4}}&\parbox[t][0.3cm]{7.7179203cm}{\raggedright Mult.: R(DCO)=1.1 \textit{3}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{791}&\multicolumn{1}{@{.}l}{0 {\it 5}}&\multicolumn{1}{r@{}}{22}&\multicolumn{1}{@{.}l}{8 {\it 11}}&\multicolumn{1}{r@{}}{3510}&\multicolumn{1}{@{.}l}{6+x}&\multicolumn{1}{l}{(31/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{2719}&\multicolumn{1}{@{.}l}{6+x}&\multicolumn{1}{@{}l}{(29/2\ensuremath{^{-}})}&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{7.7179203cm}{\raggedright Mult.: R(DCO)=0.61 \textit{4}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{826}&\multicolumn{1}{@{.}l}{6 {\it 5}}&\multicolumn{1}{r@{}}{89}&\multicolumn{1}{@{.}l}{4 {\it 18}}&\multicolumn{1}{r@{}}{3546}&\multicolumn{1}{@{.}l}{2+x}&\multicolumn{1}{l}{(33/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{2719}&\multicolumn{1}{@{.}l}{6+x}&\multicolumn{1}{@{}l}{(29/2\ensuremath{^{-}})}&\multicolumn{1}{l}{E2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{00973 {\it 14}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{7.7179203cm}{\raggedright Mult.: R(DCO)=1.00 \textit{14}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{829}&\multicolumn{1}{@{.}l}{4 {\it 5}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{6616}&\multicolumn{1}{@{.}l}{7+z}&&\multicolumn{1}{r@{}}{5787}&\multicolumn{1}{@{.}l}{3+z }&&\multicolumn{1}{l}{E2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{00966 {\it 14}}&\multicolumn{1}{r@{}}{12}&\multicolumn{1}{@{ }l}{{\it 5}}&\parbox[t][0.3cm]{7.7179203cm}{\raggedright Mult.: R(DCO)=1.2 \textit{7}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{830}&\multicolumn{1}{@{.}l}{7 {\it 5}}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{7 {\it 3}}&\multicolumn{1}{r@{}}{2733}&\multicolumn{1}{@{.}l}{3}&&\multicolumn{1}{r@{}}{1902}&\multicolumn{1}{@{.}l}{7 }&\multicolumn{1}{@{}l}{21/2\ensuremath{^{+}}}&\multicolumn{1}{l}{D,E2}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{7.7179203cm}{\raggedright Mult.: R(DCO)=1.13 \textit{16}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{834}&\multicolumn{1}{@{.}l}{0 {\it 5}}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{8 {\it 3}}&\multicolumn{1}{r@{}}{2736}&\multicolumn{1}{@{.}l}{7}&&\multicolumn{1}{r@{}}{1902}&\multicolumn{1}{@{.}l}{7 }&\multicolumn{1}{@{}l}{21/2\ensuremath{^{+}}}&\multicolumn{1}{l}{D,E2}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{7.7179203cm}{\raggedright Mult.: R(DCO)= 0.94 \textit{11}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{862}&\multicolumn{1}{@{.}l}{8 {\it 5}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{5818}&\multicolumn{1}{@{.}l}{8+y}&&\multicolumn{1}{r@{}}{4956}&\multicolumn{1}{@{.}l}{0+y }&&\multicolumn{1}{l}{E2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{00892 {\it 13}}&\multicolumn{1}{r@{}}{10}&\multicolumn{1}{@{ }l}{{\it 4}}&\parbox[t][0.3cm]{7.7179203cm}{\raggedright Mult.: R(DCO)=0.92 \textit{34}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{902}&\multicolumn{1}{@{.}l}{2 {\it 5}}&\multicolumn{1}{r@{}}{23}&\multicolumn{1}{@{.}l}{9 {\it 2}}&\multicolumn{1}{r@{}}{5990}&\multicolumn{1}{@{.}l}{4+x}&\multicolumn{1}{l}{(45/2)}&\multicolumn{1}{r@{}}{5088}&\multicolumn{1}{@{.}l}{2+x }&\multicolumn{1}{@{}l}{(43/2)}&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{7.7179203cm}{\raggedright Mult.: R(DCO)=0.55 \textit{5}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{913}&\multicolumn{1}{@{.}l}{0 {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{.}l}{0 {\it 20}}&\multicolumn{1}{r@{}}{1542}&\multicolumn{1}{@{.}l}{1}&\multicolumn{1}{l}{17/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{629}&\multicolumn{1}{@{.}l}{1 }&\multicolumn{1}{@{}l}{13/2\ensuremath{^{+}}}&\multicolumn{1}{l}{E2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{00797 {\it 11}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{7.7179203cm}{\raggedright Mult.: R(DCO)=1.08 \textit{8}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{917}&\multicolumn{1}{@{.}l}{0 {\it 5}}&\multicolumn{1}{r@{}}{37}&\multicolumn{1}{@{.}l}{0 {\it 15}}&\multicolumn{1}{r@{}}{1546}&\multicolumn{1}{@{.}l}{1}&\multicolumn{1}{l}{15/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{629}&\multicolumn{1}{@{.}l}{1 }&\multicolumn{1}{@{}l}{13/2\ensuremath{^{+}}}&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{7.7179203cm}{\raggedright Mult.: R(DCO)=0.81 \textit{11}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{919}&\multicolumn{1}{@{.}l}{4 {\it 5}}&\multicolumn{1}{r@{}}{8}&\multicolumn{1}{@{.}l}{8 {\it 25}}&\multicolumn{1}{r@{}}{3639}&\multicolumn{1}{@{.}l}{1+x}&\multicolumn{1}{l}{(31/2)}&\multicolumn{1}{r@{}}{2719}&\multicolumn{1}{@{.}l}{6+x}&\multicolumn{1}{@{}l}{(29/2\ensuremath{^{-}})}&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{7.7179203cm}{\raggedright Mult.: R(DCO)=0.87 \textit{12}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{933}&\multicolumn{1}{@{.}l}{1 {\it 5}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{7108}&\multicolumn{1}{@{.}l}{4+z}&&\multicolumn{1}{r@{}}{6175}&\multicolumn{1}{@{.}l}{4+z }&&\multicolumn{1}{l}{E2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{00763 {\it 11}}&\multicolumn{1}{r@{}}{10}&\multicolumn{1}{@{ }l}{{\it 4}}&\parbox[t][0.3cm]{7.7179203cm}{\raggedright Mult.: R(DCO)=0.8 \textit{3}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{964}&\multicolumn{1}{@{.}l}{7 {\it 5}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{6324}&\multicolumn{1}{@{.}l}{5+y}&&\multicolumn{1}{r@{}}{5359}&\multicolumn{1}{@{.}l}{8+y }&&\multicolumn{1}{l}{(E2)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{00714 {\it 10}}&\multicolumn{1}{r@{}}{10}&\multicolumn{1}{@{ }l}{{\it 3}}&\parbox[t][0.3cm]{7.7179203cm}{\raggedright Mult.: R(DCO)=0.77 \textit{18}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1005}&\multicolumn{1}{@{.}l}{5 {\it 5}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{9 {\it 1}}&\multicolumn{1}{r@{}}{5836}&\multicolumn{1}{@{.}l}{7+x}&\multicolumn{1}{l}{(41/2)}&\multicolumn{1}{r@{}}{4831}&\multicolumn{1}{@{.}l}{3+x }&\multicolumn{1}{@{}l}{(39/2)}&\multicolumn{1}{l}{(D)}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{7.7179203cm}{\raggedright Mult.: R(DCO)=0.76 \textit{20}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1031}&\multicolumn{1}{@{.}l}{4 {\it 5}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{7648}&\multicolumn{1}{@{.}l}{2+z}&&\multicolumn{1}{r@{}}{6616}&\multicolumn{1}{@{.}l}{7+z }&&\multicolumn{1}{l}{E2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{00627 {\it 9}}&\multicolumn{1}{r@{}}{12}&\multicolumn{1}{@{ }l}{{\it 4}}&\parbox[t][0.3cm]{7.7179203cm}{\raggedright Mult.: R(DCO)=1.2 \textit{5}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1190}&\multicolumn{1}{@{.}l}{1 {\it 5}}&\multicolumn{1}{r@{}}{13}&\multicolumn{1}{@{.}l}{6 {\it 4}}&\multicolumn{1}{r@{}}{5831}&\multicolumn{1}{@{.}l}{0+x}&\multicolumn{1}{l}{(45/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{4640}&\multicolumn{1}{@{.}l}{9+x }&\multicolumn{1}{@{}l}{(41/2\ensuremath{^{+}})}&\multicolumn{1}{l}{E2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{00477 {\it 7}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{7.7179203cm}{\raggedright Mult.: R(DCO)=1.07 \textit{13}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1251}&\multicolumn{1}{@{.}l}{3 {\it 5}}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{5 {\it 1}}&\multicolumn{1}{r@{}}{5892}&\multicolumn{1}{@{.}l}{2+x}&\multicolumn{1}{l}{(43/2)}&\multicolumn{1}{r@{}}{4640}&\multicolumn{1}{@{.}l}{9+x }&\multicolumn{1}{@{}l}{(41/2\ensuremath{^{+}})}&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{7.7179203cm}{\raggedright Mult.: R(DCO)=0.74 \textit{17}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1312}&\multicolumn{1}{@{.}l}{0 {\it 5}}&\multicolumn{1}{r@{}}{19}&\multicolumn{1}{@{.}l}{6 {\it 12}}&\multicolumn{1}{r@{}}{8019}&\multicolumn{1}{@{.}l}{7+x}&\multicolumn{1}{l}{(51/2)}&\multicolumn{1}{r@{}}{6707}&\multicolumn{1}{@{.}l}{7+x }&\multicolumn{1}{@{}l}{(49/2)}&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{7.7179203cm}{\raggedright Mult.: R(DCO)=0.73 \textit{13}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1312}&\multicolumn{1}{@{.}l}{3 {\it 5}}&\multicolumn{1}{r@{}}{5}&\multicolumn{1}{@{.}l}{8 {\it 4}}&\multicolumn{1}{r@{}}{7143}&\multicolumn{1}{@{.}l}{3+x}&\multicolumn{1}{l}{(49/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{5831}&\multicolumn{1}{@{.}l}{0+x }&\multicolumn{1}{@{}l}{(45/2\ensuremath{^{+}})}&\multicolumn{1}{l}{E2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{00398 {\it 6}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{7.7179203cm}{\raggedright Mult.: R(DCO)=0.97 \textit{11}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1341}&\multicolumn{1}{@{.}l}{0 {\it 5}}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{4 {\it 1}}&\multicolumn{1}{r@{}}{4060}&\multicolumn{1}{@{.}l}{6}&&\multicolumn{1}{r@{}}{2719}&\multicolumn{1}{@{.}l}{6 }&\multicolumn{1}{@{}l}{25/2\ensuremath{^{-}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{7.7179203cm}{\raggedright Mult.: R(DCO)=0.89 \textit{16}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1388}&\multicolumn{1}{@{.}l}{1 {\it 5}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{9 {\it 1}}&\multicolumn{1}{r@{}}{7378}&\multicolumn{1}{@{.}l}{5+x}&\multicolumn{1}{l}{(47/2)}&\multicolumn{1}{r@{}}{5990}&\multicolumn{1}{@{.}l}{4+x }&\multicolumn{1}{@{}l}{(45/2)}&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{7.7179203cm}{\raggedright Mult.: R(DCO)=0.72 \textit{29}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1389}&\multicolumn{1}{@{.}l}{4 {\it 5}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{4 {\it 1}}&\multicolumn{1}{r@{}}{7379}&\multicolumn{1}{@{.}l}{9+x}&\multicolumn{1}{l}{(47/2)}&\multicolumn{1}{r@{}}{5990}&\multicolumn{1}{@{.}l}{4+x }&\multicolumn{1}{@{}l}{(45/2)}&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{7.7179203cm}{\raggedright Mult.: R(DCO)=0.64 \textit{37}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1640}&\multicolumn{1}{@{.}l}{0 {\it 5}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{7 {\it 1}}&\multicolumn{1}{r@{}}{6146}&\multicolumn{1}{@{.}l}{2+x}&\multicolumn{1}{l}{(35/2)}&\multicolumn{1}{r@{}}{4506}&\multicolumn{1}{@{.}l}{2+x }&\multicolumn{1}{@{}l}{(35/2)}&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{7.7179203cm}{\raggedright Mult.: R(DCO)=1.13 \textit{26}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1683}&\multicolumn{1}{@{.}l}{8 {\it 5}}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{8 {\it 2}}&\multicolumn{1}{r@{}}{6324}&\multicolumn{1}{@{.}l}{8+x}&\multicolumn{1}{l}{(45/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{4640}&\multicolumn{1}{@{.}l}{9+x }&\multicolumn{1}{@{}l}{(41/2\ensuremath{^{+}})}&\multicolumn{1}{l}{E2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{00263 {\it 4}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{7.7179203cm}{\raggedright Mult.: R(DCO)=1.01 \textit{17}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1749}&\multicolumn{1}{@{.}l}{5 {\it 5}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{9 {\it 1}}&\multicolumn{1}{r@{}}{5583}&\multicolumn{1}{@{.}l}{0+x}&\multicolumn{1}{l}{(39/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{3833}&\multicolumn{1}{@{.}l}{4+x }&\multicolumn{1}{@{}l}{(35/2\ensuremath{^{-}})}&\multicolumn{1}{l}{E2}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{48\ensuremath{\times10^{-3}} {\it 4}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{7.7179203cm}{\raggedright Mult.: R(DCO)=0.93 \textit{19}.\vspace{0.1cm}}&\\
\end{longtable}
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB33GAMMA0}{\dagger}}}} From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1995Ba70,B}{1995Ba70}, unless otherwise stated. \ensuremath{\Delta}E\ensuremath{\gamma}=0.5 keV estimated by the evaluator.}\\
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB33GAMMA1}{\ddagger}}}} From DCO ratios and multiple decay branches in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1995Ba70,B}{1995Ba70}, unless otherwise stated.}\\
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB33GAMMA2}{\#}}}} Relative total intensity within each band from \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1995Ba70,B}{1995Ba70}. Values were corrected in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1995Ba70,B}{1995Ba70} for internal electron conversion by assuming pure M1 and E2}\\
\parbox[b][0.3cm]{21.881866cm}{{\ }{\ }characters for the \ensuremath{\Delta}J=1 and \ensuremath{\Delta}J=2 transitions, respectively.}\\
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB33GAMMA3}{@}}}} Total theoretical internal conversion coefficients, calculated using the BrIcc code (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2008Ki07,B}{2008Ki07}) with Frozen orbital approximation based on \ensuremath{\gamma}-ray energies,}\\
\parbox[b][0.3cm]{21.881866cm}{{\ }{\ }assigned multipolarities, and mixing ratios, unless otherwise specified.}\\
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB33GAMMA4}{x}}}} \ensuremath{\gamma} ray not placed in level scheme.}\\
\vspace{0.5cm}
\end{landscape}\clearpage
\clearpage
\begin{figure}[h]
\begin{center}
\includegraphics{201PB33-0.ps}\\
\end{center}
\end{figure}
\clearpage
\begin{figure}[h]
\begin{center}
\includegraphics{201PB33-1.ps}\\
\end{center}
\end{figure}
\clearpage
\begin{figure}[h]
\begin{center}
\includegraphics{201PB33-2.ps}\\
\end{center}
\end{figure}
\clearpage
\begin{figure}[h]
\begin{center}
\includegraphics{201PB33-3.ps}\\
\end{center}
\end{figure}
\clearpage
\clearpage
\begin{figure}[h]
\begin{center}
\includegraphics{201PB33B-0.ps}\\
\end{center}
\end{figure}
\clearpage
\subsection[\hspace{-0.2cm}\ensuremath{^{\textnormal{197}}}Au(\ensuremath{^{\textnormal{207}}}Pb,X\ensuremath{\gamma})]{ }
\vspace{-27pt}
\vspace{0.3cm}
\hypertarget{PB34}{{\bf \small \underline{\ensuremath{^{\textnormal{197}}}Au(\ensuremath{^{\textnormal{207}}}Pb,X\ensuremath{\gamma})\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2019Ro12,B}{2019Ro12}}}}\\
\vspace{4pt}
\vspace{8pt}
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2019Ro12,B}{2019Ro12}: E=1430{\textminus}MeV \ensuremath{^{\textnormal{207}}}Pb beam was produced from the ATLAS accelerator at ANL. Target was \ensuremath{\approx}50 mg/cm\ensuremath{^{\textnormal{2}}} \ensuremath{^{\textnormal{197}}}Au. \ensuremath{\gamma} rays}\\
\parbox[b][0.3cm]{17.7cm}{were detected with the Gammasphere array comprising of 100 HPGe detectors. Measured: E\ensuremath{\gamma}, I\ensuremath{\gamma}, \ensuremath{\gamma}\ensuremath{\gamma}(t). Deduced: level scheme,}\\
\parbox[b][0.3cm]{17.7cm}{T\ensuremath{_{\textnormal{1/2}}}.}\\
\vspace{12pt}
\underline{$^{201}$Pb Levels}\\
\begin{longtable}{cccccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E(level)$^{{\hyperlink{PB34LEVEL0}{\dagger}}}$}&J$^{\pi}$$^{{\hyperlink{PB34LEVEL1}{\ddagger}}}$&\multicolumn{2}{c}{T$_{1/2}$$^{}$}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{629}&\multicolumn{1}{@{.}l}{1 {\it 3}}&\multicolumn{1}{l}{13/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{60}&\multicolumn{1}{@{.}l}{8 s {\it 18}}&\parbox[t][0.3cm]{12.68672cm}{\raggedright E(level),J\ensuremath{^{\pi}},T\ensuremath{_{1/2}}: From Adopted Levels.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1542}&\multicolumn{1}{@{.}l}{1 {\it 9}}&\multicolumn{1}{l}{17/2\ensuremath{^{+}}}&&&&\\
\multicolumn{1}{r@{}}{1546}&\multicolumn{1}{@{.}l}{1 {\it 9}}&\multicolumn{1}{l}{15/2\ensuremath{^{+}}}&&&&\\
\multicolumn{1}{r@{}}{1896}&\multicolumn{1}{@{.}l}{1 {\it 10}}&\multicolumn{1}{l}{19/2\ensuremath{^{+}}}&&&&\\
\multicolumn{1}{r@{}}{2497}&\multicolumn{1}{@{.}l}{1 {\it 15}}&\multicolumn{1}{l}{21/2\ensuremath{^{-}}}&&&&\\
\multicolumn{1}{r@{}}{2719}&\multicolumn{1}{@{.}l}{1 {\it 18}}&\multicolumn{1}{l}{25/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{63}&\multicolumn{1}{@{ }l}{ns {\it 3}}&\parbox[t][0.3cm]{12.68672cm}{\raggedright T\ensuremath{_{1/2}}: From Adopted Levels.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2719}&\multicolumn{1}{@{.}l}{1+x}&\multicolumn{1}{l}{29/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{508}&\multicolumn{1}{@{ }l}{ns {\it 5}}&\parbox[t][0.3cm]{12.68672cm}{\raggedright T\ensuremath{_{1/2}}: From Adopted Levels.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3546}&\multicolumn{1}{@{.}l}{0+x {\it 10}}&\multicolumn{1}{l}{33/2\ensuremath{^{-}}}&&&&\\
\multicolumn{1}{r@{}}{3833}&\multicolumn{1}{@{.}l}{0+x {\it 15}}&\multicolumn{1}{l}{35/2\ensuremath{^{-}}}&&&&\\
\multicolumn{1}{r@{}}{4561}&\multicolumn{1}{@{.}l}{0+x {\it 18}}&\multicolumn{1}{l}{37/2\ensuremath{^{+}}}&&&&\\
\multicolumn{1}{r@{}}{4641}&\multicolumn{1}{@{.}l}{0+x {\it 20}}&\multicolumn{1}{l}{41/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{52}&\multicolumn{1}{@{ }l}{ns {\it 2}}&\parbox[t][0.3cm]{12.68672cm}{\raggedright T\ensuremath{_{1/2}}: From \ensuremath{\gamma}\ensuremath{\gamma}(\ensuremath{\Delta}t) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2019Ro12,B}{2019Ro12}, using 902\ensuremath{\gamma}-(728\ensuremath{\gamma},827\ensuremath{\gamma}) and 447\ensuremath{\gamma}-(728\ensuremath{\gamma},827\ensuremath{\gamma}).\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{12.68672cm}{\raggedright configuration: Dominant \ensuremath{\nu} (p\ensuremath{_{\textnormal{3/2}}^{\textnormal{$-$1}}},f\ensuremath{_{\textnormal{5/2}}^{\textnormal{$-$1}}},i\ensuremath{_{\textnormal{13/2}}^{\textnormal{$-$3}}}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{5088}&\multicolumn{1}{@{.}l}{0+x {\it 23}}&\multicolumn{1}{l}{43/2}&&&&\\
\multicolumn{1}{r@{}}{5990}&\multicolumn{1}{@{.}l}{0+x {\it 25}}&\multicolumn{1}{l}{45/2}&&&&\\
\end{longtable}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB34LEVEL0}{\dagger}}}} From a least-squares fit to E\ensuremath{\gamma}, unless otherwise stated. \ensuremath{\Delta}E\ensuremath{\gamma}=1 keV were estimated by the evaluator.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB34LEVEL1}{\ddagger}}}} From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2019Ro12,B}{2019Ro12}, based on previous studies (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1981He07,B}{1981He07},\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1988Ro08,B}{1988Ro08}, \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1995Ba70,B}{1995Ba70}), shell-model and systematics.}\\
\vspace{0.5cm}
\underline{$\gamma$($^{201}$Pb)}\\
\begin{longtable}{ccccccc@{}c|ccccccc@{}c@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{PB34GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{E\ensuremath{_{i}}(level)}&J\ensuremath{^{\pi}_{i}}&\multicolumn{2}{c}{E\ensuremath{_{f}}}&J\ensuremath{^{\pi}_{f}}&\multicolumn{2}{c}{E\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{PB34GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{E\ensuremath{_{i}}(level)}&J\ensuremath{^{\pi}_{i}}&\multicolumn{2}{c}{E\ensuremath{_{f}}}&J\ensuremath{^{\pi}_{f}}&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{80}&\multicolumn{1}{@{ }l}{{\it 1}}&\multicolumn{1}{r@{}}{4641}&\multicolumn{1}{@{.}l}{0+x}&\multicolumn{1}{l}{41/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{4561}&\multicolumn{1}{@{.}l}{0+x }&\multicolumn{1}{@{}l|}{37/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{601}&\multicolumn{1}{@{ }l}{{\it 1}}&\multicolumn{1}{r@{}}{2497}&\multicolumn{1}{@{.}l}{1}&\multicolumn{1}{l}{21/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{1896}&\multicolumn{1}{@{.}l}{1 }&\multicolumn{1}{@{}l}{19/2\ensuremath{^{+}}}&\\
\multicolumn{1}{r@{}}{222}&\multicolumn{1}{@{ }l}{{\it 1}}&\multicolumn{1}{r@{}}{2719}&\multicolumn{1}{@{.}l}{1}&\multicolumn{1}{l}{25/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{2497}&\multicolumn{1}{@{.}l}{1 }&\multicolumn{1}{@{}l|}{21/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{728}&\multicolumn{1}{@{ }l}{{\it 1}}&\multicolumn{1}{r@{}}{4561}&\multicolumn{1}{@{.}l}{0+x}&\multicolumn{1}{l}{37/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{3833}&\multicolumn{1}{@{.}l}{0+x }&\multicolumn{1}{@{}l}{35/2\ensuremath{^{-}}}&\\
\multicolumn{1}{r@{}}{287}&\multicolumn{1}{@{ }l}{{\it 1}}&\multicolumn{1}{r@{}}{3833}&\multicolumn{1}{@{.}l}{0+x}&\multicolumn{1}{l}{35/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{3546}&\multicolumn{1}{@{.}l}{0+x }&\multicolumn{1}{@{}l|}{33/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{827}&\multicolumn{1}{@{ }l}{{\it 1}}&\multicolumn{1}{r@{}}{3546}&\multicolumn{1}{@{.}l}{0+x}&\multicolumn{1}{l}{33/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{2719}&\multicolumn{1}{@{.}l}{1+x}&\multicolumn{1}{@{}l}{29/2\ensuremath{^{-}}}&\\
\multicolumn{1}{r@{}}{350}&\multicolumn{1}{@{ }l}{{\it 1}}&\multicolumn{1}{r@{}}{1896}&\multicolumn{1}{@{.}l}{1}&\multicolumn{1}{l}{19/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1546}&\multicolumn{1}{@{.}l}{1 }&\multicolumn{1}{@{}l|}{15/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{902}&\multicolumn{1}{@{ }l}{{\it 1}}&\multicolumn{1}{r@{}}{5990}&\multicolumn{1}{@{.}l}{0+x}&\multicolumn{1}{l}{45/2}&\multicolumn{1}{r@{}}{5088}&\multicolumn{1}{@{.}l}{0+x }&\multicolumn{1}{@{}l}{43/2}&\\
\multicolumn{1}{r@{}}{354}&\multicolumn{1}{@{ }l}{{\it 1}}&\multicolumn{1}{r@{}}{1896}&\multicolumn{1}{@{.}l}{1}&\multicolumn{1}{l}{19/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1542}&\multicolumn{1}{@{.}l}{1 }&\multicolumn{1}{@{}l|}{17/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{913}&\multicolumn{1}{@{ }l}{{\it 1}}&\multicolumn{1}{r@{}}{1542}&\multicolumn{1}{@{.}l}{1}&\multicolumn{1}{l}{17/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{629}&\multicolumn{1}{@{.}l}{1 }&\multicolumn{1}{@{}l}{13/2\ensuremath{^{+}}}&\\
\multicolumn{1}{r@{}}{447}&\multicolumn{1}{@{ }l}{{\it 1}}&\multicolumn{1}{r@{}}{5088}&\multicolumn{1}{@{.}l}{0+x}&\multicolumn{1}{l}{43/2}&\multicolumn{1}{r@{}}{4641}&\multicolumn{1}{@{.}l}{0+x }&\multicolumn{1}{@{}l|}{41/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{917}&\multicolumn{1}{@{ }l}{{\it 1}}&\multicolumn{1}{r@{}}{1546}&\multicolumn{1}{@{.}l}{1}&\multicolumn{1}{l}{15/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{629}&\multicolumn{1}{@{.}l}{1 }&\multicolumn{1}{@{}l}{13/2\ensuremath{^{+}}}&\\
\end{longtable}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PB34GAMMA0}{\dagger}}}} From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2019Ro12,B}{2019Ro12}.}\\
\vspace{0.5cm}
\clearpage
\begin{figure}[h]
\begin{center}
\includegraphics{201PB34-0.ps}\\
\end{center}
\end{figure}
\clearpage
\section[\ensuremath{^{201}_{\ 83}}Bi\ensuremath{_{118}^{~}}]{ }
\vspace{-30pt}
\setcounter{chappage}{1}
\subsection[\hspace{-0.2cm}Adopted Levels, Gammas]{ }
\vspace{-20pt}
\vspace{0.3cm}
\hypertarget{BI35}{{\bf \small \underline{Adopted \hyperlink{201BI_LEVEL}{Levels}, \hyperlink{201BI_GAMMA}{Gammas}}}}\\
\vspace{4pt}
\vspace{8pt}
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}Q(\ensuremath{\beta^-})=$-$4908 {\it 13}; S(n)=9130 {\it 26}; S(p)=2467 {\it 16}; Q(\ensuremath{\alpha})=4500 {\it 6}\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2021Wa16,B}{2021Wa16}}\\
\vspace{12pt}
\hypertarget{201BI_LEVEL}{\underline{$^{201}$Bi Levels}}\\
\begin{longtable}[c]{llll}
\multicolumn{4}{c}{\underline{Cross Reference (XREF) Flags}}\\
\\
\hyperlink{PO36}{\texttt{A }}& \ensuremath{^{\textnormal{201}}}Po \ensuremath{\varepsilon} decay (15.50 min) & \hyperlink{BI39}{\texttt{D }}& \ensuremath{^{\textnormal{203}}}Tl(\ensuremath{\alpha},6n\ensuremath{\gamma})\\
\hyperlink{PO37}{\texttt{B }}& \ensuremath{^{\textnormal{201}}}Po \ensuremath{\varepsilon} decay (8.96 min) & \hyperlink{BI40}{\texttt{E }}& \ensuremath{^{\textnormal{196}}}Pt(\ensuremath{^{\textnormal{10}}}B,5n\ensuremath{\gamma})\\
\hyperlink{AT38}{\texttt{C }}& \ensuremath{^{\textnormal{205}}}At \ensuremath{\alpha} decay & \\
\end{longtable}
\vspace{-0.5cm}
\begin{longtable}{ccccccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E(level)$^{{\hyperlink{BI35LEVEL0}{\dagger}}}$}&J$^{\pi}$$^{{\hyperlink{BI35LEVEL1}{\ddagger}}}$&\multicolumn{2}{c}{T$_{1/2}$$^{}$}&XREF&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{BI35LEVEL2}{\#}}}}}&\multicolumn{1}{l}{9/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{103}&\multicolumn{1}{@{.}l}{2 min {\it 17}}&\multicolumn{1}{l}{\texttt{\hyperlink{PO36}{A}\hyperlink{PO37}{B}\hyperlink{AT38}{C}\hyperlink{BI39}{D}\hyperlink{BI40}{E}} }&\parbox[t][0.3cm]{10.0012cm}{\raggedright \%\ensuremath{\varepsilon}+\%\ensuremath{\beta}\ensuremath{^{\textnormal{+}}}=100\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.0012cm}{\raggedright \ensuremath{\mu}=4.8 \textit{3} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1988Wo12,B}{1988Wo12},\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2019StZV,B}{2019StZV})\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.0012cm}{\raggedright J\ensuremath{^{\pi}}: Atomic beam (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1960Li03,B}{1960Li03}); favored \ensuremath{\alpha}-decay from \ensuremath{^{\textnormal{205}}}At\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.0012cm}{\raggedright {\ }{\ }{\ }(\ensuremath{J^{\pi}}=9/2\ensuremath{^{-}},\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2018Cu02,B}{2018Cu02}); \ensuremath{\mu}.\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.0012cm}{\raggedright T\ensuremath{_{1/2}}: Weighted average of 110 min \textit{10} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1950Ne77,B}{1950Ne77}), 111 min \textit{4}\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.0012cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1956St05,B}{1956St05}), 94 min \textit{3} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1966KaZY,B}{1966KaZY}), 96 min \textit{12} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1970Jo26,B}{1970Jo26}) and 106.2\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.0012cm}{\raggedright {\ }{\ }{\ }min \textit{24} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1970DaZM,B}{1970DaZM}).\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.0012cm}{\raggedright \ensuremath{\mu}: Using static nuclear orientation with \ensuremath{\gamma}-ray\hphantom{a}detection technique.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{846}&\multicolumn{1}{@{.}l}{35\ensuremath{^{{\hyperlink{BI35LEVEL3}{@}}}} {\it 18}}&\multicolumn{1}{l}{1/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{58}&\multicolumn{1}{@{.}l}{5 min {\it 6}}&\multicolumn{1}{l}{\texttt{\hyperlink{PO36}{A}\ \ \ \ } }&\parbox[t][0.3cm]{10.0012cm}{\raggedright \%\ensuremath{\varepsilon}+\%\ensuremath{\beta}\ensuremath{^{\textnormal{+}}}=88.7 \textit{14}; \%IT=11.0 \textit{14}; \%\ensuremath{\alpha}\ensuremath{\approx}0.3\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.0012cm}{\raggedright \%IT from \ensuremath{^{\textnormal{201}}}Po \ensuremath{\varepsilon} decay (15.6 min), \%\ensuremath{\alpha} from \ensuremath{\alpha} HF syst.\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.0012cm}{\raggedright J\ensuremath{^{\pi}}: 846.3\ensuremath{\gamma} M4 to 9/2\ensuremath{^{-}}.\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.0012cm}{\raggedright T\ensuremath{_{1/2}}: Weighted average of 62 min \textit{8} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1950Ne77,B}{1950Ne77}), 52 min \textit{2} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1964Si11,B}{1964Si11}),\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.0012cm}{\raggedright {\ }{\ }{\ }59.1 min \textit{6} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1966Ma51,B}{1966Ma51}) and 57 min \textit{5} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1970DaZM,B}{1970DaZM}).\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.0012cm}{\raggedright E\ensuremath{_{\ensuremath{\alpha}}}=5240 keV \textit{6} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1966Ma51,B}{1966Ma51}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{890}&\multicolumn{1}{@{.}l}{24\ensuremath{^{{\hyperlink{BI35LEVEL4}{\&}}}} {\it 13}}&\multicolumn{1}{l}{5/2\ensuremath{^{-}}}&&&\multicolumn{1}{l}{\texttt{\hyperlink{PO36}{A}\ \ \ \ } }&\parbox[t][0.3cm]{10.0012cm}{\raggedright J\ensuremath{^{\pi}}: 890.1\ensuremath{\gamma} E2 to 9/2\ensuremath{^{-}}; direct \ensuremath{\varepsilon} feeding of this level in \ensuremath{^{\textnormal{201}}}Po \ensuremath{\varepsilon} decay\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.0012cm}{\raggedright {\ }{\ }{\ }(15.6 min, \ensuremath{J^{\pi}}=3/2\ensuremath{^{-}}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{904}&\multicolumn{1}{@{.}l}{23\ensuremath{^{{\hyperlink{BI35LEVEL4}{\&}}}} {\it 12}}&\multicolumn{1}{l}{(7/2)\ensuremath{^{-}}}&&&\multicolumn{1}{l}{\texttt{\hyperlink{PO36}{A}\ \ \ \ } }&\parbox[t][0.3cm]{10.0012cm}{\raggedright J\ensuremath{^{\pi}}: 904.2\ensuremath{\gamma} M1+E2 to 9/2\ensuremath{^{-}}; observation in \ensuremath{^{\textnormal{201}}}Po \ensuremath{\varepsilon} decay (15.6 min,\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.0012cm}{\raggedright {\ }{\ }{\ }\ensuremath{J^{\pi}}=3/2\ensuremath{^{-}}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{964}&\multicolumn{1}{@{.}l}{40\ensuremath{^{{\hyperlink{BI35LEVEL4}{\&}}}} {\it 15}}&\multicolumn{1}{l}{11/2\ensuremath{^{-}}}&&&\multicolumn{1}{l}{\texttt{\ \hyperlink{PO37}{B}\ \ \hyperlink{BI40}{E}} }&\parbox[t][0.3cm]{10.0012cm}{\raggedright J\ensuremath{^{\pi}}: 964.3\ensuremath{\gamma} M1(+E2) to 9/2\ensuremath{^{-}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{967}&\multicolumn{1}{@{.}l}{49\ensuremath{^{{\hyperlink{BI35LEVEL4}{\&}}}} {\it 17}}&\multicolumn{1}{l}{13/2\ensuremath{^{-}}}&&&\multicolumn{1}{l}{\texttt{\ \hyperlink{PO37}{B}\ \hyperlink{BI39}{D}\hyperlink{BI40}{E}} }&\parbox[t][0.3cm]{10.0012cm}{\raggedright J\ensuremath{^{\pi}}: 967.4\ensuremath{\gamma} E2 to 9/2\ensuremath{^{-}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1086}&\multicolumn{1}{@{.}l}{21\ensuremath{^{{\hyperlink{BI35LEVEL5}{a}}}} {\it 18}}&\multicolumn{1}{l}{3/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{260}&\multicolumn{1}{@{ }l}{ps {\it 30}}&\multicolumn{1}{l}{\texttt{\hyperlink{PO36}{A}\ \ \ \ } }&\parbox[t][0.3cm]{10.0012cm}{\raggedright J\ensuremath{^{\pi}}: 195.9\ensuremath{\gamma} E1 to 5/2\ensuremath{^{-}}; 240.1\ensuremath{\gamma} M1+E2 to 1/2\ensuremath{^{+}}.\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.0012cm}{\raggedright T\ensuremath{_{1/2}}: From 188.5ce(K){\textminus}239.8ce(K)(\ensuremath{\Delta}t) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986Be07,B}{1986Be07} (\ensuremath{^{\textnormal{201}}}Po \ensuremath{\varepsilon} decay\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.0012cm}{\raggedright {\ }{\ }{\ }(15.6 min)).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1186}&\multicolumn{1}{@{.}l}{59\ensuremath{^{{\hyperlink{BI35LEVEL7}{c}}}} {\it 17}}&\multicolumn{1}{l}{(7/2)\ensuremath{^{-}}}&&&\multicolumn{1}{l}{\texttt{\hyperlink{PO36}{A}\ \ \ \ } }&\parbox[t][0.3cm]{10.0012cm}{\raggedright J\ensuremath{^{\pi}}: 296.1\ensuremath{\gamma} to 5/2\ensuremath{^{-}}; 1186.7\ensuremath{\gamma} M1(+E2) to 9/2\ensuremath{^{-}}; observation in \ensuremath{^{\textnormal{201}}}Po \ensuremath{\varepsilon}\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.0012cm}{\raggedright {\ }{\ }{\ }decay (15.6 min, \ensuremath{J^{\pi}}=3/2\ensuremath{^{-}}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1274}&\multicolumn{1}{@{.}l}{45 {\it 19}}&\multicolumn{1}{l}{(5/2)\ensuremath{^{+}}}&&&\multicolumn{1}{l}{\texttt{\hyperlink{PO36}{A}\ \ \ \ } }&\parbox[t][0.3cm]{10.0012cm}{\raggedright J\ensuremath{^{\pi}}: 188.6\ensuremath{\gamma} M1+E2 to 3/2\ensuremath{^{+}}; 428.2\ensuremath{\gamma} E2 to 1/2\ensuremath{^{+}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1379}&\multicolumn{1}{@{.}l}{34\ensuremath{^{{\hyperlink{BI35LEVEL6}{b}}}} {\it 24}}&\multicolumn{1}{l}{15/2\ensuremath{^{-}}}&&&\multicolumn{1}{l}{\texttt{\ \hyperlink{PO37}{B}\ \hyperlink{BI39}{D}\hyperlink{BI40}{E}} }&\parbox[t][0.3cm]{10.0012cm}{\raggedright J\ensuremath{^{\pi}}: 411.86\ensuremath{\gamma} M1+E2 to 13/2\ensuremath{^{-}}; 414.9\ensuremath{\gamma} to 11/2\ensuremath{^{-}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1441}&\multicolumn{1}{@{.}l}{71 {\it 18}}&\multicolumn{1}{l}{7/2\ensuremath{^{-}}}&&&\multicolumn{1}{l}{\texttt{\hyperlink{PO36}{A}\ \ \ \ } }&\parbox[t][0.3cm]{10.0012cm}{\raggedright J\ensuremath{^{\pi}}: 535.5\ensuremath{\gamma} M1 to 7/2\ensuremath{^{-}}; 551.9\ensuremath{\gamma} M1(+E2) to 5/2\ensuremath{^{-}}; 1442.2\ensuremath{\gamma} M1(+E2) to\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.0012cm}{\raggedright {\ }{\ }{\ }9/2\ensuremath{^{-}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1470}&\multicolumn{1}{@{.}l}{87 {\it 22}}&\multicolumn{1}{l}{(5/2,7/2)\ensuremath{^{-}}}&&&\multicolumn{1}{l}{\texttt{\hyperlink{PO36}{A}\ \ \ \ } }&\parbox[t][0.3cm]{10.0012cm}{\raggedright J\ensuremath{^{\pi}}: 566.6\ensuremath{\gamma} M1(+E2) to 7/2\ensuremath{^{-}}, 1470.9\ensuremath{\gamma} to 9/2\ensuremath{^{-}}. Direct \ensuremath{\varepsilon} feeding of this\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.0012cm}{\raggedright {\ }{\ }{\ }level in \ensuremath{^{\textnormal{201}}}Po \ensuremath{\varepsilon} decay (15.6 min, \ensuremath{J^{\pi}}=3/2\ensuremath{^{-}}) excludes 9/2\ensuremath{^{-}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1474}&\multicolumn{1}{@{.}l}{6\ensuremath{^{{\hyperlink{BI35LEVEL6}{b}}}} {\it 3}}&\multicolumn{1}{l}{17/2\ensuremath{^{-}}}&&&\multicolumn{1}{l}{\texttt{\ \hyperlink{PO37}{B}\ \hyperlink{BI39}{D}\hyperlink{BI40}{E}} }&\parbox[t][0.3cm]{10.0012cm}{\raggedright J\ensuremath{^{\pi}}: 96.25\ensuremath{\gamma} M1(+E2) to 15/2\ensuremath{^{-}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1483}&\multicolumn{1}{@{.}l}{54 {\it 24}}&\multicolumn{1}{l}{(3/2\ensuremath{^{-}})}&&&\multicolumn{1}{l}{\texttt{\hyperlink{PO36}{A}\ \ \ \ } }&\parbox[t][0.3cm]{10.0012cm}{\raggedright J\ensuremath{^{\pi}}: 593.3\ensuremath{\gamma} (M1) to 5/2\ensuremath{^{-}}, 636.5\ensuremath{\gamma} to 1/2\ensuremath{^{+}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1501}&\multicolumn{1}{@{.}l}{89 {\it 17}}&\multicolumn{1}{l}{(13/2\ensuremath{^{+}})}&&&\multicolumn{1}{l}{\texttt{\ \hyperlink{PO37}{B}\ \ \ } }&\parbox[t][0.3cm]{10.0012cm}{\raggedright J\ensuremath{^{\pi}}: 534.2\ensuremath{\gamma} to 13/2\ensuremath{^{-}}, 537.5\ensuremath{\gamma} to 11/2\ensuremath{^{-}}, 1502.4\ensuremath{\gamma} to 9/2\ensuremath{^{-}}. Strong\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.0012cm}{\raggedright {\ }{\ }{\ }population of this level in \ensuremath{^{\textnormal{201}}}Po \ensuremath{\varepsilon} decay (8.96 min, \ensuremath{J^{\pi}}=13/2\ensuremath{^{+}}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1504}&\multicolumn{1}{@{.}l}{40 {\it 23}}&\multicolumn{1}{l}{(9/2,11/2)\ensuremath{^{+}}}&&&\multicolumn{1}{l}{\texttt{\ \hyperlink{PO37}{B}\ \ \ } }&\parbox[t][0.3cm]{10.0012cm}{\raggedright J\ensuremath{^{\pi}}: 540.1\ensuremath{\gamma} to 11/2\ensuremath{^{-}}, 1504.3\ensuremath{\gamma} to 9/2\ensuremath{^{-}}; 217.6\ensuremath{\gamma} M1(+E2) from\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.0012cm}{\raggedright {\ }{\ }{\ }(11/2,13/2)\ensuremath{^{+}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1616}&\multicolumn{1}{@{.}l}{3 {\it 4}}&\multicolumn{1}{l}{1/2\ensuremath{^{+}},3/2\ensuremath{^{+}},5/2\ensuremath{^{+}}}&&&\multicolumn{1}{l}{\texttt{\hyperlink{PO36}{A}\ \ \ \ } }&\parbox[t][0.3cm]{10.0012cm}{\raggedright J\ensuremath{^{\pi}}: 530.1\ensuremath{\gamma} M1+E2 to 3/2\ensuremath{^{+}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1665}&\multicolumn{1}{@{.}l}{1 {\it 3}}&&&&\multicolumn{1}{l}{\texttt{\ \hyperlink{PO37}{B}\ \ \ } }&&\\
\multicolumn{1}{r@{}}{1719}&\multicolumn{1}{@{.}l}{15 {\it 22}}&\multicolumn{1}{l}{(11/2,13/2)\ensuremath{^{+}}}&&&\multicolumn{1}{l}{\texttt{\ \hyperlink{PO37}{B}\ \ \hyperlink{BI40}{E}} }&\parbox[t][0.3cm]{10.0012cm}{\raggedright J\ensuremath{^{\pi}}: 754.6\ensuremath{\gamma} E1 to 11/2\ensuremath{^{-}}, 217.6\ensuremath{\gamma} M1+E2 to (13/2\ensuremath{^{+}}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1746}&\multicolumn{1}{@{.}l}{5\ensuremath{^{{\hyperlink{BI35LEVEL8}{d}}}} {\it 4}}&\multicolumn{1}{l}{17/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{5}&\multicolumn{1}{@{.}l}{1 ns {\it 13}}&\multicolumn{1}{l}{\texttt{\ \hyperlink{PO37}{B}\ \hyperlink{BI39}{D}\hyperlink{BI40}{E}} }&\parbox[t][0.3cm]{10.0012cm}{\raggedright J\ensuremath{^{\pi}}: 271.91\ensuremath{\gamma} E1 to 17/2\ensuremath{^{-}}. \ensuremath{\gamma}(\ensuremath{\theta})$'$s are consistent with \ensuremath{\Delta}J=0, dipole\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.0012cm}{\raggedright {\ }{\ }{\ }transition.\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.0012cm}{\raggedright T\ensuremath{_{1/2}}: From \ensuremath{\gamma}\ensuremath{\gamma}(t) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1985Pi05,B}{1985Pi05} (\ensuremath{^{\textnormal{196}}}Pt(\ensuremath{^{\textnormal{10}}}B,5n\ensuremath{\gamma})) using the centroid shift\vspace{0.1cm}}&\\
\end{longtable}
\begin{textblock}{29}(0,27.3)
Continued on next page (footnotes at end of table)
\end{textblock}
\clearpage
\begin{longtable}{ccccccc@{\extracolsep{\fill}}c}
\\[-.4cm]
\multicolumn{8}{c}{{\bf \small \underline{Adopted \hyperlink{201BI_LEVEL}{Levels}, \hyperlink{201BI_GAMMA}{Gammas} (continued)}}}\\
\multicolumn{8}{c}{~}\\
\multicolumn{8}{c}{\underline{\ensuremath{^{201}}Bi Levels (continued)}}\\
\multicolumn{8}{c}{~}\\
\multicolumn{2}{c}{E(level)$^{{\hyperlink{BI35LEVEL0}{\dagger}}}$}&J$^{\pi}$$^{{\hyperlink{BI35LEVEL1}{\ddagger}}}$&\multicolumn{2}{c}{T$_{1/2}$$^{}$}&XREF&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&
\endhead
&&&&&&\parbox[t][0.3cm]{10.06449cm}{\raggedright {\ }{\ }{\ }analysis. Other: 9.6 ns \textit{6} from \ensuremath{\gamma}\ensuremath{\gamma}(t) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1982Br21,B}{1982Br21} (\ensuremath{^{\textnormal{203}}}Tl(\ensuremath{\alpha},6n\ensuremath{\gamma})) using\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.06449cm}{\raggedright {\ }{\ }{\ }gates on 272\ensuremath{\gamma}, 412\ensuremath{\gamma} and 967\ensuremath{\gamma} below the isomer (stop) and 186\ensuremath{\gamma}\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.06449cm}{\raggedright {\ }{\ }{\ }above the isomer (start). The prompt response function is obtained\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.06449cm}{\raggedright {\ }{\ }{\ }using \ensuremath{\gamma}\ensuremath{\gamma}(t) spectrum with gates on 412\ensuremath{\gamma} and 967\ensuremath{\gamma} (start) and 272\ensuremath{\gamma}\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.06449cm}{\raggedright {\ }{\ }{\ }(stop). One should note, however, that there is a difference between the\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.06449cm}{\raggedright {\ }{\ }{\ }time walk for the 186\ensuremath{\gamma} and 272\ensuremath{\gamma}, and hence, this value may be not so\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.06449cm}{\raggedright {\ }{\ }{\ }accurate.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1762}&\multicolumn{1}{@{.}l}{9 {\it 3}}&\multicolumn{1}{l}{(11/2,13/2\ensuremath{^{-}})}&&&\multicolumn{1}{l}{\texttt{\ \hyperlink{PO37}{B}\ \ \ } }&\parbox[t][0.3cm]{10.06449cm}{\raggedright J\ensuremath{^{\pi}}: 798.5\ensuremath{\gamma} to 11/2\ensuremath{^{-}}, 1762.9 to 9/2\ensuremath{^{-}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1778}&\multicolumn{1}{@{.}l}{92 {\it 22}}&\multicolumn{1}{l}{(3/2\ensuremath{^{-}},5/2)}&&&\multicolumn{1}{l}{\texttt{\hyperlink{PO36}{A}\ \ \ \ } }&\parbox[t][0.3cm]{10.06449cm}{\raggedright J\ensuremath{^{\pi}}: 874.6\ensuremath{\gamma} to 7/2\ensuremath{^{-}}, 889.2\ensuremath{\gamma} to 5/2\ensuremath{^{-}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1817}&\multicolumn{1}{@{.}l}{86 {\it 22}}&\multicolumn{1}{l}{1/2\ensuremath{^{+}},3/2,5/2\ensuremath{^{+}}}&&&\multicolumn{1}{l}{\texttt{\hyperlink{PO36}{A}\ \ \ \ } }&\parbox[t][0.3cm]{10.06449cm}{\raggedright J\ensuremath{^{\pi}}: 543.4\ensuremath{\gamma} to 5/2\ensuremath{^{+}}, 731.7\ensuremath{\gamma} to 3/2\ensuremath{^{+}}, 971.4\ensuremath{\gamma} to 1/2\ensuremath{^{+}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1848}&\multicolumn{1}{@{.}l}{16 {\it 25}}&\multicolumn{1}{l}{(5/2\ensuremath{^{-}})}&&&\multicolumn{1}{l}{\texttt{\hyperlink{PO36}{A}\ \ \ \ } }&\parbox[t][0.3cm]{10.06449cm}{\raggedright J\ensuremath{^{\pi}}: 944.2\ensuremath{\gamma} to 7/2\ensuremath{^{-}}, 1848.0\ensuremath{\gamma} to 9/2\ensuremath{^{-}}; direct feeding in \ensuremath{^{\textnormal{201}}}Po \ensuremath{\varepsilon} decay\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.06449cm}{\raggedright {\ }{\ }{\ }(\ensuremath{J^{\pi}}=3/2\ensuremath{^{-}}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1858}&\multicolumn{1}{@{.}l}{02 {\it 24}}&\multicolumn{1}{l}{3/2\ensuremath{^{+}}}&&&\multicolumn{1}{l}{\texttt{\hyperlink{PO36}{A}\ \ \ \ } }&\parbox[t][0.3cm]{10.06449cm}{\raggedright J\ensuremath{^{\pi}}: 771.8\ensuremath{\gamma} E0+M1 to 3/2\ensuremath{^{+}}, 583.6\ensuremath{\gamma} M1+E2 to 5/2\ensuremath{^{+}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1927}&\multicolumn{1}{@{.}l}{32 {\it 19}}&\multicolumn{1}{l}{(5/2\ensuremath{^{-}})}&&&\multicolumn{1}{l}{\texttt{\hyperlink{PO36}{A}\ \ \ \ } }&\parbox[t][0.3cm]{10.06449cm}{\raggedright J\ensuremath{^{\pi}}: 1023.0\ensuremath{\gamma} to 7/2\ensuremath{^{-}}, 1927.5\ensuremath{\gamma} to 9/2\ensuremath{^{-}}; direct feeding in \ensuremath{^{\textnormal{201}}}Po \ensuremath{\varepsilon} decay\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.06449cm}{\raggedright {\ }{\ }{\ }(\ensuremath{J^{\pi}}=3/2\ensuremath{^{-}}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1932}&\multicolumn{1}{@{.}l}{3\ensuremath{^{{\hyperlink{BI35LEVEL9}{e}}}} {\it 4}}&\multicolumn{1}{l}{21/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{$<$40}&\multicolumn{1}{@{ }l}{ns}&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{BI39}{D}\hyperlink{BI40}{E}} }&\parbox[t][0.3cm]{10.06449cm}{\raggedright J\ensuremath{^{\pi}}: 185.77\ensuremath{\gamma} E2 to 17/2\ensuremath{^{+}}.\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.06449cm}{\raggedright T\ensuremath{_{1/2}}: Estimated from the 185.8\ensuremath{\gamma}(t) data in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1985Pi05,B}{1985Pi05} (\ensuremath{^{\textnormal{196}}}Pt(\ensuremath{^{\textnormal{10}}}B,5n\ensuremath{\gamma})).\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.06449cm}{\raggedright {\ }{\ }{\ }The absence of prompt component in the time spectrum produced by\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.06449cm}{\raggedright {\ }{\ }{\ }gating on the 185.8\ensuremath{\gamma}, shown in figure 4(c) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1985Pi05,B}{1985Pi05}, suggests that\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.06449cm}{\raggedright {\ }{\ }{\ }185.8\ensuremath{\gamma} depopulates an isomeric state.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1932}&\multicolumn{1}{@{.}l}{3+x\ensuremath{^{{\hyperlink{BI35LEVEL10}{f}}}} {\it 4}}&\multicolumn{1}{l}{(25/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{118}&\multicolumn{1}{@{ }l}{ns {\it 28}}&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{BI39}{D}\hyperlink{BI40}{E}} }&\parbox[t][0.3cm]{10.06449cm}{\raggedright J\ensuremath{^{\pi}}: From syst (analogy to \ensuremath{^{\textnormal{203}}}Bi and \ensuremath{^{\textnormal{205}}}Bi) and shell-model calculations\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.06449cm}{\raggedright {\ }{\ }{\ }in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1985Pi05,B}{1985Pi05}.\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.06449cm}{\raggedright T\ensuremath{_{1/2}}: From \ensuremath{\gamma}\ensuremath{\gamma}(t) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1985Pi05,B}{1985Pi05} (\ensuremath{^{\textnormal{196}}}Pt(\ensuremath{^{\textnormal{10}}}B,5n\ensuremath{\gamma})) using time-difference\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.06449cm}{\raggedright {\ }{\ }{\ }spectra between 617.3\ensuremath{\gamma} with 967.4\ensuremath{\gamma}, 411.9\ensuremath{\gamma}, 271.9\ensuremath{\gamma} and 185.8\ensuremath{\gamma}.\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.06449cm}{\raggedright {\ }{\ }{\ }Other: 210 ns \textit{20} from \ensuremath{\gamma}\ensuremath{\gamma}(t) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1982Br21,B}{1982Br21} (\ensuremath{^{\textnormal{203}}}Tl(\ensuremath{\alpha},6n\ensuremath{\gamma})) using gates\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.06449cm}{\raggedright {\ }{\ }{\ }on 617\ensuremath{\gamma} above the isomer (start) and 186\ensuremath{\gamma}, 272\ensuremath{\gamma}, 412\ensuremath{\gamma} and 967\ensuremath{\gamma}\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.06449cm}{\raggedright {\ }{\ }{\ }below the isomer (stop).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1944}&\multicolumn{1}{@{.}l}{24 {\it 17}}&\multicolumn{1}{l}{(5/2\ensuremath{^{-}})}&&&\multicolumn{1}{l}{\texttt{\hyperlink{PO36}{A}\ \ \ \ } }&\parbox[t][0.3cm]{10.06449cm}{\raggedright J\ensuremath{^{\pi}}: 1054.7\ensuremath{\gamma} to 5/2\ensuremath{^{-}} and 1944.2\ensuremath{\gamma} to 9/2\ensuremath{^{-}}; direct feeding in \ensuremath{^{\textnormal{201}}}Po \ensuremath{\varepsilon}\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.06449cm}{\raggedright {\ }{\ }{\ }decay (\ensuremath{J^{\pi}}=3/2\ensuremath{^{-}}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1971}&\multicolumn{1}{@{.}l}{3+x\ensuremath{^{{\hyperlink{BI35LEVEL10}{f}}}} {\it 3}}&\multicolumn{1}{l}{(27/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{105}&\multicolumn{1}{@{ }l}{ns {\it 75}}&\multicolumn{1}{l}{\texttt{\ \ \ \ \hyperlink{BI40}{E}} }&\parbox[t][0.3cm]{10.06449cm}{\raggedright J\ensuremath{^{\pi}}: From systematics and shell-model calculations in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1985Pi05,B}{1985Pi05}.\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.06449cm}{\raggedright T\ensuremath{_{1/2}}: From \ensuremath{\gamma}\ensuremath{\gamma}(t) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1985Pi05,B}{1985Pi05} (\ensuremath{^{\textnormal{196}}}Pt(\ensuremath{^{\textnormal{10}}}B,5n\ensuremath{\gamma})) using time-difference\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.06449cm}{\raggedright {\ }{\ }{\ }spectra between 679.8\ensuremath{\gamma} with 967.4\ensuremath{\gamma}, 411.9\ensuremath{\gamma}, 271.9\ensuremath{\gamma} and 185.8\ensuremath{\gamma}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2034}&\multicolumn{1}{@{.}l}{3 {\it 7}}&&&&\multicolumn{1}{l}{\texttt{\ \hyperlink{PO37}{B}\ \ \ } }&&\\
\multicolumn{1}{r@{}}{2053}&\multicolumn{1}{@{.}l}{59 {\it 21}}&\multicolumn{1}{l}{(5/2)\ensuremath{^{+}}}&&&\multicolumn{1}{l}{\texttt{\hyperlink{PO36}{A}\ \ \ \ } }&\parbox[t][0.3cm]{10.06449cm}{\raggedright J\ensuremath{^{\pi}}: 779.4\ensuremath{\gamma} (M1) to 5/2\ensuremath{^{+}}; 1207.1\ensuremath{\gamma} E2(+M1) to 1/2\ensuremath{^{+}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2065}&\multicolumn{1}{@{.}l}{82 {\it 17}}&\multicolumn{1}{l}{5/2\ensuremath{^{+}}}&&&\multicolumn{1}{l}{\texttt{\hyperlink{PO36}{A}\ \ \ \ } }&\parbox[t][0.3cm]{10.06449cm}{\raggedright J\ensuremath{^{\pi}}: 624.7\ensuremath{\gamma} to 7/2\ensuremath{^{-}}, 1175.3\ensuremath{\gamma} (E1) to 5/2\ensuremath{^{-}}, 1219.3\ensuremath{\gamma} to 1/2\ensuremath{^{+}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2299}&\multicolumn{1}{@{.}l}{1+x {\it 3}}&\multicolumn{1}{l}{(27/2\ensuremath{^{+}})}&&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{BI39}{D}\hyperlink{BI40}{E}} }&\parbox[t][0.3cm]{10.06449cm}{\raggedright J\ensuremath{^{\pi}}: 366.6\ensuremath{\gamma} (M1) to (25/2\ensuremath{^{+}}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2386}&\multicolumn{1}{@{.}l}{7 {\it 5}}&&&&\multicolumn{1}{l}{\texttt{\hyperlink{PO36}{A}\ \ \ \ } }&&\\
\multicolumn{1}{r@{}}{2422}&\multicolumn{1}{@{.}l}{1 {\it 3}}&\multicolumn{1}{l}{(3/2\ensuremath{^{-}},5/2)}&&&\multicolumn{1}{l}{\texttt{\hyperlink{PO36}{A}\ \ \ \ } }&\parbox[t][0.3cm]{10.06449cm}{\raggedright J\ensuremath{^{\pi}}: 1518.2\ensuremath{\gamma} to 7/2\ensuremath{^{-}}, 1531.7\ensuremath{\gamma} to 5/2\ensuremath{^{-}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2434}&\multicolumn{1}{@{.}l}{9 {\it 3}}&\multicolumn{1}{l}{1/2\ensuremath{^{+}},3/2,5/2\ensuremath{^{+}}}&&&\multicolumn{1}{l}{\texttt{\hyperlink{PO36}{A}\ \ \ \ } }&\parbox[t][0.3cm]{10.06449cm}{\raggedright J\ensuremath{^{\pi}}: 1160.6\ensuremath{\gamma} to 5/2\ensuremath{^{+}}, 1588.5\ensuremath{\gamma} to 1/2\ensuremath{^{+}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2455}&\multicolumn{1}{@{.}l}{5 {\it 3}}&\multicolumn{1}{l}{1/2\ensuremath{^{+}},3/2,5/2\ensuremath{^{+}}}&&&\multicolumn{1}{l}{\texttt{\hyperlink{PO36}{A}\ \ \ \ } }&\parbox[t][0.3cm]{10.06449cm}{\raggedright J\ensuremath{^{\pi}}: 1181.3\ensuremath{\gamma} to 5/2\ensuremath{^{+}}, 1609.0\ensuremath{\gamma} to 1/2\ensuremath{^{+}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2484}&\multicolumn{1}{@{.}l}{3 {\it 3}}&\multicolumn{1}{l}{1/2,3/2,5/2\ensuremath{^{+}}}&&&\multicolumn{1}{l}{\texttt{\hyperlink{PO36}{A}\ \ \ \ } }&\parbox[t][0.3cm]{10.06449cm}{\raggedright J\ensuremath{^{\pi}}: 1398.0\ensuremath{\gamma} to 3/2\ensuremath{^{+}}, 1638.1\ensuremath{\gamma} to 1/2\ensuremath{^{+}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2549}&\multicolumn{1}{@{.}l}{50+x {\it 19}}&\multicolumn{1}{l}{(27/2\ensuremath{^{+}})}&&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{BI39}{D}\hyperlink{BI40}{E}} }&\parbox[t][0.3cm]{10.06449cm}{\raggedright J\ensuremath{^{\pi}}: 250.2\ensuremath{\gamma} M1+E2 to (27/2\ensuremath{^{+}}), 617.27\ensuremath{\gamma} M1+E2 to (25/2\ensuremath{^{+}}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2589}&\multicolumn{1}{@{.}l}{7+x {\it 3}}&&&&\multicolumn{1}{l}{\texttt{\ \ \ \ \hyperlink{BI40}{E}} }&&\\
\multicolumn{1}{r@{}}{2592}&\multicolumn{1}{@{.}l}{88 {\it 20}}&\multicolumn{1}{l}{(3/2\ensuremath{^{-}},5/2\ensuremath{^{+}})}&&&\multicolumn{1}{l}{\texttt{\hyperlink{PO36}{A}\ \ \ \ } }&\parbox[t][0.3cm]{10.06449cm}{\raggedright J\ensuremath{^{\pi}}: 1689.3\ensuremath{\gamma} to 7/2\ensuremath{^{-}}; 1746.8\ensuremath{\gamma} to 1/2\ensuremath{^{+}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2612}&\multicolumn{1}{@{.}l}{00+x {\it 10}}&\multicolumn{1}{l}{(27/2)}&&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{BI39}{D}\ } }&\parbox[t][0.3cm]{10.06449cm}{\raggedright J\ensuremath{^{\pi}}: 679.7\ensuremath{\gamma} D to (25/2\ensuremath{^{+}}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2651}&\multicolumn{1}{@{.}l}{13+x {\it 24}}&\multicolumn{1}{l}{(29/2\ensuremath{^{+}})}&&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{BI39}{D}\hyperlink{BI40}{E}} }&\parbox[t][0.3cm]{10.06449cm}{\raggedright J\ensuremath{^{\pi}}: 679.8\ensuremath{\gamma} M1+E2 to (27/2\ensuremath{^{+}}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2668}&\multicolumn{1}{@{.}l}{31+x {\it 21}}&\multicolumn{1}{l}{(29/2\ensuremath{^{+}})}&&&\multicolumn{1}{l}{\texttt{\ \ \ \ \hyperlink{BI40}{E}} }&\parbox[t][0.3cm]{10.06449cm}{\raggedright J\ensuremath{^{\pi}}: 118.81\ensuremath{\gamma} M1+E2 to (27/2\ensuremath{^{+}}), 736.0\ensuremath{\gamma} E2 to (25/2\ensuremath{^{+}}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2740}&\multicolumn{1}{@{.}l}{01+x\ensuremath{^{{\hyperlink{BI35LEVEL11}{g}}}} {\it 22}}&\multicolumn{1}{l}{(29/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{124}&\multicolumn{1}{@{ }l}{ns {\it 4}}&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{BI39}{D}\hyperlink{BI40}{E}} }&\parbox[t][0.3cm]{10.06449cm}{\raggedright J\ensuremath{^{\pi}}: 88.88\ensuremath{\gamma} to (29/2\ensuremath{^{+}}), 150.5\ensuremath{\gamma} (E1) to (25/2\ensuremath{^{-}},27/2), 190.49\ensuremath{\gamma} to (27/2\ensuremath{^{+}}),\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.06449cm}{\raggedright {\ }{\ }{\ }441.0\ensuremath{\gamma} to (27/2\ensuremath{^{+}}).\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.06449cm}{\raggedright T\ensuremath{_{1/2}}: From 617.3\ensuremath{\gamma}(t) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1985Pi05,B}{1985Pi05} (\ensuremath{^{\textnormal{196}}}Pt(\ensuremath{^{\textnormal{10}}}B,5n\ensuremath{\gamma})). Other: 160 ns 30\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.06449cm}{\raggedright {\ }{\ }{\ }from \ensuremath{\gamma}\ensuremath{\gamma}(t) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1982Br21,B}{1982Br21} (\ensuremath{^{\textnormal{203}}}Tl(\ensuremath{\alpha},6n\ensuremath{\gamma})) using gates on 617\ensuremath{\gamma} above\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{10.06449cm}{\raggedright {\ }{\ }{\ }the isomer and 186\ensuremath{\gamma}, 272\ensuremath{\gamma}, 412\ensuremath{\gamma} and 967\ensuremath{\gamma} below the isomer.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2902}&\multicolumn{1}{@{.}l}{09 {\it 25}}&\multicolumn{1}{l}{1/2\ensuremath{^{+}},3/2,5/2\ensuremath{^{+}}}&&&\multicolumn{1}{l}{\texttt{\hyperlink{PO36}{A}\ \ \ \ } }&\parbox[t][0.3cm]{10.06449cm}{\raggedright J\ensuremath{^{\pi}}: 2055.7\ensuremath{\gamma} to 1/2\ensuremath{^{+}}, 1627.7\ensuremath{\gamma} to (5/2)\ensuremath{^{+}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2905}&\multicolumn{1}{@{.}l}{9 {\it 3}}&\multicolumn{1}{l}{1/2,3/2,5/2\ensuremath{^{+}}}&&&\multicolumn{1}{l}{\texttt{\hyperlink{PO36}{A}\ \ \ \ } }&\parbox[t][0.3cm]{10.06449cm}{\raggedright J\ensuremath{^{\pi}}: 2059.4\ensuremath{\gamma} to 1/2\ensuremath{^{+}}, 1819.8\ensuremath{\gamma} to 3/2\ensuremath{^{+}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2994}&\multicolumn{1}{@{.}l}{7+x {\it 8}}&&&&\multicolumn{1}{l}{\texttt{\ \ \ \ \hyperlink{BI40}{E}} }&&\\
\end{longtable}
\begin{textblock}{29}(0,27.3)
Continued on next page (footnotes at end of table)
\end{textblock}
\clearpage
\begin{longtable}{ccccc@{\extracolsep{\fill}}c}
\\[-.4cm]
\multicolumn{6}{c}{{\bf \small \underline{Adopted \hyperlink{201BI_LEVEL}{Levels}, \hyperlink{201BI_GAMMA}{Gammas} (continued)}}}\\
\multicolumn{6}{c}{~}\\
\multicolumn{6}{c}{\underline{\ensuremath{^{201}}Bi Levels (continued)}}\\
\multicolumn{6}{c}{~}\\
\multicolumn{2}{c}{E(level)$^{{\hyperlink{BI35LEVEL0}{\dagger}}}$}&J$^{\pi}$$^{{\hyperlink{BI35LEVEL1}{\ddagger}}}$&XREF&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&\hrulefill&
\endhead
\multicolumn{1}{r@{}}{3011}&\multicolumn{1}{@{.}l}{5+x {\it 5}}&&\multicolumn{1}{l}{\texttt{\ \ \ \ \hyperlink{BI40}{E}} }&&\\
\multicolumn{1}{r@{}}{3238}&\multicolumn{1}{@{.}l}{9+x {\it 3}}&\multicolumn{1}{l}{(31/2\ensuremath{^{-}})}&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{BI39}{D}\hyperlink{BI40}{E}} }&\parbox[t][0.3cm]{13.06336cm}{\raggedright J\ensuremath{^{\pi}}: 498.95\ensuremath{\gamma} M1+E2 to (29/2\ensuremath{^{-}}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3422}&\multicolumn{1}{@{.}l}{9+x {\it 9}}&&\multicolumn{1}{l}{\texttt{\ \ \ \ \hyperlink{BI40}{E}} }&&\\
\multicolumn{1}{r@{}}{3526}&\multicolumn{1}{@{.}l}{5+x {\it 4}}&\multicolumn{1}{l}{(33/2\ensuremath{^{-}})}&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{BI39}{D}\hyperlink{BI40}{E}} }&\parbox[t][0.3cm]{13.06336cm}{\raggedright J\ensuremath{^{\pi}}: 287.3\ensuremath{\gamma} to (31/2\ensuremath{^{-}}), 786.3\ensuremath{\gamma} (E2) to (29/2\ensuremath{^{-}}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3592}&\multicolumn{1}{@{.}l}{3+x? {\it 10}}&&\multicolumn{1}{l}{\texttt{\ \ \ \ \hyperlink{BI40}{E}} }&&\\
\multicolumn{1}{r@{}}{3638}&\multicolumn{1}{@{.}l}{4+x {\it 5}}&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{BI39}{D}\hyperlink{BI40}{E}} }&&\\
\multicolumn{1}{r@{}}{3706}&\multicolumn{1}{@{.}l}{7+x {\it 5}}&&\multicolumn{1}{l}{\texttt{\ \ \ \ \hyperlink{BI40}{E}} }&&\\
\multicolumn{1}{r@{}}{3727}&\multicolumn{1}{@{.}l}{61+x {\it 24}}&\multicolumn{1}{l}{(33/2\ensuremath{^{-}})}&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{BI39}{D}\ } }&\parbox[t][0.3cm]{13.06336cm}{\raggedright J\ensuremath{^{\pi}}: 987.6\ensuremath{\gamma} E2 to (29/2\ensuremath{^{-}}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3810}&\multicolumn{1}{@{.}l}{8+x {\it 4}}&\multicolumn{1}{l}{(33/2\ensuremath{^{-}})}&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{BI39}{D}\hyperlink{BI40}{E}} }&\parbox[t][0.3cm]{13.06336cm}{\raggedright J\ensuremath{^{\pi}}: 284.19\ensuremath{\gamma} M1(+E2) to (33/2\ensuremath{^{-}}), 572.4\ensuremath{\gamma} M1+E2 to (31/2\ensuremath{^{-}}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3922}&\multicolumn{1}{@{.}l}{9+x {\it 6}}&\multicolumn{1}{l}{(35/2\ensuremath{^{-}})}&\multicolumn{1}{l}{\texttt{\ \ \ \ \hyperlink{BI40}{E}} }&\parbox[t][0.3cm]{13.06336cm}{\raggedright J\ensuremath{^{\pi}}: 396.4\ensuremath{\gamma} M1+E2 to (33/2\ensuremath{^{-}}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{4075}&\multicolumn{1}{@{.}l}{3+x? {\it 7}}&&\multicolumn{1}{l}{\texttt{\ \ \ \ \hyperlink{BI40}{E}} }&&\\
\multicolumn{1}{r@{}}{4484}&\multicolumn{1}{@{.}l}{5+x {\it 9}}&&\multicolumn{1}{l}{\texttt{\ \ \ \ \hyperlink{BI40}{E}} }&&\\
\multicolumn{1}{r@{}}{5282}&\multicolumn{1}{@{.}l}{3+x {\it 10}}&&\multicolumn{1}{l}{\texttt{\ \ \ \ \hyperlink{BI40}{E}} }&&\\
\end{longtable}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{BI35LEVEL0}{\dagger}}}} From a least squares fit to E\ensuremath{\gamma}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{BI35LEVEL1}{\ddagger}}}} From deduced \ensuremath{\gamma}-ray transition multipolarities using \ensuremath{\gamma}(\ensuremath{\theta}) and DCO in \ensuremath{^{\textnormal{196}}}Pt(\ensuremath{^{\textnormal{10}}}B,5n\ensuremath{\gamma}), \ensuremath{\gamma}(\ensuremath{\theta}) in\hphantom{a}\ensuremath{^{\textnormal{203}}}Tl(\ensuremath{\alpha},6n\ensuremath{\gamma}) and \ensuremath{\alpha}(K)exp,}\\
\parbox[b][0.3cm]{17.7cm}{{\ }{\ }\ensuremath{\alpha}(L)exp in \ensuremath{^{\textnormal{201}}}Po \ensuremath{\varepsilon} decay (15.6\hphantom{a}min) and \ensuremath{^{\textnormal{201}}}Po \ensuremath{\varepsilon} decay (8.96 min), unless otherwise stated.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{BI35LEVEL2}{\#}}}} Configuration=\ensuremath{\pi} h\ensuremath{_{\textnormal{9/2}}^{\textnormal{+1}}}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{BI35LEVEL3}{@}}}} Configuration=\ensuremath{\pi} s\ensuremath{_{\textnormal{1/2}}^{\textnormal{$-$1}}}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{BI35LEVEL4}{\&}}}} Configuration=\ensuremath{\pi} (h\ensuremath{_{\textnormal{9/2}}^{\textnormal{+1}}})\ensuremath{\otimes}2\ensuremath{^{\textnormal{+}}}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{BI35LEVEL5}{a}}}} Configuration=\ensuremath{\pi}\hphantom{a}d\ensuremath{_{\textnormal{3/2}}^{\textnormal{$-$1}}}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{BI35LEVEL6}{b}}}} Configuration=\ensuremath{\pi} (h\ensuremath{_{\textnormal{9/2}}^{\textnormal{+1}}})\ensuremath{\otimes}4\ensuremath{^{\textnormal{+}}}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{BI35LEVEL7}{c}}}} Configuration=\ensuremath{\pi} f\ensuremath{_{\textnormal{7/2}}^{\textnormal{+1}}}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{BI35LEVEL8}{d}}}} Admixture of configuration= \ensuremath{\pi} (h\ensuremath{_{\textnormal{9/2}}^{\textnormal{+1}}}) \ensuremath{\nu} (f\ensuremath{_{\textnormal{5/2}}^{\textnormal{$-$1}}},i\ensuremath{_{\textnormal{13/2}}^{\textnormal{$-$1}}})\ensuremath{_{\textnormal{5$-$}}} and configuration=\ensuremath{\pi} (h\ensuremath{_{\textnormal{9/2}}^{\textnormal{+1}}}) \ensuremath{\nu} (p\ensuremath{_{\textnormal{3/2}}^{\textnormal{$-$1}}},i\ensuremath{_{\textnormal{13/2}}^{\textnormal{$-$1}}})\ensuremath{_{\textnormal{5$-$}}}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{BI35LEVEL9}{e}}}} Admixture of configuration= \ensuremath{\pi} (h\ensuremath{_{\textnormal{9/2}}^{\textnormal{+1}}}) \ensuremath{\nu} (f\ensuremath{_{\textnormal{5/2}}^{\textnormal{$-$1}}},i\ensuremath{_{\textnormal{13/2}}^{\textnormal{$-$1}}})\ensuremath{_{\textnormal{7$-$}}} and configuration=\ensuremath{\pi} (h\ensuremath{_{\textnormal{9/2}}^{\textnormal{+1}}}) \ensuremath{\nu} (p\ensuremath{_{\textnormal{3/2}}^{\textnormal{$-$1}}},i\ensuremath{_{\textnormal{13/2}}^{\textnormal{$-$1}}})\ensuremath{_{\textnormal{7$-$}}}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{BI35LEVEL10}{f}}}} Configuration=\ensuremath{\pi} (h\ensuremath{_{\textnormal{9/2}}^{\textnormal{+1}}}) \ensuremath{\nu} (f\ensuremath{_{\textnormal{5/2}}^{\textnormal{$-$1}}},i\ensuremath{_{\textnormal{13/2}}^{\textnormal{$-$1}}})\ensuremath{_{\textnormal{9$-$}}}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{BI35LEVEL11}{g}}}} Configuration=\ensuremath{\pi} (h\ensuremath{_{\textnormal{9/2}}^{\textnormal{+1}}}) \ensuremath{\nu} (i\ensuremath{_{\textnormal{13/2}}^{\textnormal{$-$2}}})\ensuremath{_{\textnormal{12+}}}.}\\
\vspace{0.5cm}
\clearpage
\vspace{0.3cm}
\begin{landscape}
\vspace*{-0.5cm}
{\bf \small \underline{Adopted \hyperlink{201BI_LEVEL}{Levels}, \hyperlink{201BI_GAMMA}{Gammas} (continued)}}\\
\vspace{0.3cm}
\hypertarget{201BI_GAMMA}{\underline{$\gamma$($^{201}$Bi)}}\\
\begin{longtable}{ccccccccc@{}ccccccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E\ensuremath{_{i}}(level)}&J\ensuremath{^{\pi}_{i}}&\multicolumn{2}{c}{E\ensuremath{_{\gamma}}}&\multicolumn{2}{c}{I\ensuremath{_{\gamma}}}&\multicolumn{2}{c}{E\ensuremath{_{f}}}&J\ensuremath{^{\pi}_{f}}&Mult.&\multicolumn{2}{c}{\ensuremath{\delta}}&\multicolumn{2}{c}{\ensuremath{\alpha}\ensuremath{^{\hyperlink{BI35GAMMA4}{\&}}}}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{846}&\multicolumn{1}{@{.}l}{35}&\multicolumn{1}{l}{1/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{846}&\multicolumn{1}{@{.}l}{3\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 3}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{9/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M4}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{292 {\it 4}}&\parbox[t][0.3cm]{7.834711cm}{\raggedright B(M4)(W.u.)=5.7\ensuremath{\times}10\ensuremath{^{\textnormal{$-$4}}} \textit{7}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.834711cm}{\raggedright Mult.: From \ensuremath{\alpha}(K)exp and K/Lexp in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986Br28,B}{1986Br28} (\ensuremath{^{\textnormal{201}}}Po\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.834711cm}{\raggedright {\ }{\ }{\ }\ensuremath{\varepsilon} decay (15.6 min)).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{890}&\multicolumn{1}{@{.}l}{24}&\multicolumn{1}{l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{890}&\multicolumn{1}{@{.}l}{1\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 2}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{9/2\ensuremath{^{-}}}&\multicolumn{1}{l}{E2}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{00880 {\it 12}}&\parbox[t][0.3cm]{7.834711cm}{\raggedright Mult.: From \ensuremath{\alpha}(K)exp in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986Br28,B}{1986Br28} (\ensuremath{^{\textnormal{201}}}Po \ensuremath{\varepsilon} decay (15.6\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.834711cm}{\raggedright {\ }{\ }{\ }min)).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{904}&\multicolumn{1}{@{.}l}{23}&\multicolumn{1}{l}{(7/2)\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{904}&\multicolumn{1}{@{.}l}{2\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 2}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{9/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{5 {\it 3}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0216 {\it 31}}&\parbox[t][0.3cm]{7.834711cm}{\raggedright Mult.,\ensuremath{\delta}: From \ensuremath{\alpha}(K)exp in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986Br28,B}{1986Br28} (\ensuremath{^{\textnormal{201}}}Po \ensuremath{\varepsilon} decay\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.834711cm}{\raggedright {\ }{\ }{\ }(15.6 min)).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{964}&\multicolumn{1}{@{.}l}{40}&\multicolumn{1}{l}{11/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{964}&\multicolumn{1}{@{.}l}{3\ensuremath{^{\hyperlink{BI35GAMMA2}{\#}}} {\it 2}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{BI35GAMMA2}{\#}}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{9/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1(+E2)}&\multicolumn{1}{r@{}}{$-$0}&\multicolumn{1}{@{.}l}{04 {\it 7}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{02109 {\it 33}}&\parbox[t][0.3cm]{7.834711cm}{\raggedright Mult.: From \ensuremath{\alpha}(K)exp in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986Br28,B}{1986Br28} (\ensuremath{^{\textnormal{201}}}Po \ensuremath{\varepsilon} decay (15.6\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.834711cm}{\raggedright {\ }{\ }{\ }min)); \ensuremath{\gamma}(\ensuremath{\theta}) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1985Pi05,B}{1985Pi05} (\ensuremath{^{\textnormal{196}}}Pt(\ensuremath{^{\textnormal{10}}}B,5n\ensuremath{\gamma})).\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.834711cm}{\raggedright \ensuremath{\delta}: From \ensuremath{\gamma}(\ensuremath{\theta}) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1985Pi05,B}{1985Pi05} (\ensuremath{^{\textnormal{196}}}Pt(\ensuremath{^{\textnormal{10}}}B,5n\ensuremath{\gamma})).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{967}&\multicolumn{1}{@{.}l}{49}&\multicolumn{1}{l}{13/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{967}&\multicolumn{1}{@{.}l}{4\ensuremath{^{\hyperlink{BI35GAMMA2}{\#}}} {\it 2}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{BI35GAMMA2}{\#}}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{9/2\ensuremath{^{-}}}&\multicolumn{1}{l}{E2}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{00746 {\it 10}}&\parbox[t][0.3cm]{7.834711cm}{\raggedright Mult.: From \ensuremath{\alpha}(K)exp in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986Br28,B}{1986Br28} (\ensuremath{^{\textnormal{201}}}Po \ensuremath{\varepsilon} decay (8.96\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.834711cm}{\raggedright {\ }{\ }{\ }min)); \ensuremath{\gamma}(\ensuremath{\theta}) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1982Br21,B}{1982Br21} (\ensuremath{^{\textnormal{203}}}Tl(\ensuremath{\alpha},6n\ensuremath{\gamma})) and\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.834711cm}{\raggedright {\ }{\ }{\ }\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1985Pi05,B}{1985Pi05} (\ensuremath{^{\textnormal{196}}}Pt(\ensuremath{^{\textnormal{10}}}B,5n\ensuremath{\gamma})).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1086}&\multicolumn{1}{@{.}l}{21}&\multicolumn{1}{l}{3/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{195}&\multicolumn{1}{@{.}l}{9\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 3}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{60\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 9}}&\multicolumn{1}{r@{}}{890}&\multicolumn{1}{@{.}l}{24 }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{l}{E1}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0841 {\it 12}}&\parbox[t][0.3cm]{7.834711cm}{\raggedright B(E1)(W.u.)=4.6\ensuremath{\times}10\ensuremath{^{\textnormal{$-$7}}} \textit{+10{\textminus}8}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.834711cm}{\raggedright Mult.: From \ensuremath{\alpha}(K)exp in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986Br28,B}{1986Br28} (\ensuremath{^{\textnormal{201}}}Po \ensuremath{\varepsilon} decay (15.6\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.834711cm}{\raggedright {\ }{\ }{\ }min)).\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{240}&\multicolumn{1}{@{.}l}{1\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 2}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 4}}&\multicolumn{1}{r@{}}{846}&\multicolumn{1}{@{.}l}{35 }&\multicolumn{1}{@{}l}{1/2\ensuremath{^{+}}}&\multicolumn{1}{l}{E2+M1}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{0 {\it +4\textminus3}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{303 {\it 13}}&\parbox[t][0.3cm]{7.834711cm}{\raggedright B(M1)(W.u.)=0.00047 \textit{+12{\textminus}10}; B(E2)(W.u.)=26.8 \textit{+36{\textminus}29}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.834711cm}{\raggedright Mult.,\ensuremath{\delta}: From K/L in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986Br28,B}{1986Br28} (\ensuremath{^{\textnormal{201}}}Po \ensuremath{\varepsilon} decay (15.6\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.834711cm}{\raggedright {\ }{\ }{\ }min)).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1186}&\multicolumn{1}{@{.}l}{59}&\multicolumn{1}{l}{(7/2)\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{296}&\multicolumn{1}{@{.}l}{1\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 3}}&\multicolumn{1}{r@{}}{9}&\multicolumn{1}{@{.}l}{3\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 8}}&\multicolumn{1}{r@{}}{890}&\multicolumn{1}{@{.}l}{24 }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{l}{[M1]}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{481 {\it 7}}&&\\
&&&\multicolumn{1}{r@{}}{1186}&\multicolumn{1}{@{.}l}{7\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 2}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 4}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{9/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1(+E2)}&\multicolumn{1}{r@{}}{$<$0}&\multicolumn{1}{@{.}l}{9}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0107 {\it 17}}&\parbox[t][0.3cm]{7.834711cm}{\raggedright Mult.,\ensuremath{\delta}: From \ensuremath{\alpha}(K)exp in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986Br28,B}{1986Br28} (\ensuremath{^{\textnormal{201}}}Po \ensuremath{\varepsilon} decay\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.834711cm}{\raggedright {\ }{\ }{\ }(15.6 min)).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1274}&\multicolumn{1}{@{.}l}{45}&\multicolumn{1}{l}{(5/2)\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{188}&\multicolumn{1}{@{.}l}{6\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 3}}&\multicolumn{1}{r@{}}{55}&\multicolumn{1}{@{.}l}{9\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 24}}&\multicolumn{1}{r@{}}{1086}&\multicolumn{1}{@{.}l}{21 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}}}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{53 {\it 9}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{43 {\it 7}}&\parbox[t][0.3cm]{7.834711cm}{\raggedright Mult.,\ensuremath{\delta}: From \ensuremath{\alpha}(K)exp and K/L in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986Br28,B}{1986Br28} (\ensuremath{^{\textnormal{201}}}Po \ensuremath{\varepsilon}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.834711cm}{\raggedright {\ }{\ }{\ }decay (15.6 min)).\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{428}&\multicolumn{1}{@{.}l}{2\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 3}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 4}}&\multicolumn{1}{r@{}}{846}&\multicolumn{1}{@{.}l}{35 }&\multicolumn{1}{@{}l}{1/2\ensuremath{^{+}}}&\multicolumn{1}{l}{(E2)}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0451 {\it 6}}&\parbox[t][0.3cm]{7.834711cm}{\raggedright Mult.: From \ensuremath{\alpha}(K)exp in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986Br28,B}{1986Br28} (\ensuremath{^{\textnormal{201}}}Po \ensuremath{\varepsilon} decay (15.6\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.834711cm}{\raggedright {\ }{\ }{\ }min)).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1379}&\multicolumn{1}{@{.}l}{34}&\multicolumn{1}{l}{15/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{411}&\multicolumn{1}{@{.}l}{86\ensuremath{^{\hyperlink{BI35GAMMA0}{\dagger}}} {\it 20}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{BI35GAMMA0}{\dagger}}} {\it 3}}&\multicolumn{1}{r@{}}{967}&\multicolumn{1}{@{.}l}{49 }&\multicolumn{1}{@{}l}{13/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{$-$0}&\multicolumn{1}{@{.}l}{023 {\it 17}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{1966 {\it 28}}&\parbox[t][0.3cm]{7.834711cm}{\raggedright Mult.: From \ensuremath{\gamma}(\ensuremath{\theta}) and \ensuremath{\alpha}(exp) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1982Br21,B}{1982Br21}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.834711cm}{\raggedright {\ }{\ }{\ }(\ensuremath{^{\textnormal{203}}}Tl(\ensuremath{\alpha},6n\ensuremath{\gamma})), \ensuremath{\gamma}(\ensuremath{\theta}) and DCO in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1985Pi05,B}{1985Pi05}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.834711cm}{\raggedright {\ }{\ }{\ }(\ensuremath{^{\textnormal{196}}}Pt(\ensuremath{^{\textnormal{10}}}B,5n\ensuremath{\gamma})).\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.834711cm}{\raggedright \ensuremath{\delta}: From \ensuremath{\gamma}(\ensuremath{\theta}) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1985Pi05,B}{1985Pi05} (\ensuremath{^{\textnormal{196}}}Pt(\ensuremath{^{\textnormal{10}}}B,5n\ensuremath{\gamma})). Other:\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.834711cm}{\raggedright {\ }{\ }{\ }\ensuremath{\approx}0.1 from \ensuremath{\gamma}(\ensuremath{\theta}) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1982Br21,B}{1982Br21} (\ensuremath{^{\textnormal{203}}}Tl(\ensuremath{\alpha},6n\ensuremath{\gamma})).\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{414}&\multicolumn{1}{@{.}l}{9\ensuremath{^{\hyperlink{BI35GAMMA0}{\dagger}}} {\it 5}}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{57\ensuremath{^{\hyperlink{BI35GAMMA0}{\dagger}}} {\it 17}}&\multicolumn{1}{r@{}}{964}&\multicolumn{1}{@{.}l}{40 }&\multicolumn{1}{@{}l}{11/2\ensuremath{^{-}}}&\multicolumn{1}{l}{[E2]}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0489 {\it 7}}&&\\
\multicolumn{1}{r@{}}{1441}&\multicolumn{1}{@{.}l}{71}&\multicolumn{1}{l}{7/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{537}&\multicolumn{1}{@{.}l}{5\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 2}}&\multicolumn{1}{r@{}}{58}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 7}}&\multicolumn{1}{r@{}}{904}&\multicolumn{1}{@{.}l}{23 }&\multicolumn{1}{@{}l}{(7/2)\ensuremath{^{-}}}&\multicolumn{1}{l}{M1}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0968 {\it 14}}&\parbox[t][0.3cm]{7.834711cm}{\raggedright Mult.: From \ensuremath{\alpha}(K)exp in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986Br28,B}{1986Br28} (\ensuremath{^{\textnormal{201}}}Po \ensuremath{\varepsilon} decay (15.6\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.834711cm}{\raggedright {\ }{\ }{\ }min)).\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{551}&\multicolumn{1}{@{.}l}{9\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 3}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 4}}&\multicolumn{1}{r@{}}{890}&\multicolumn{1}{@{.}l}{24 }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1(+E2)}&\multicolumn{1}{r@{}}{$<$0}&\multicolumn{1}{@{.}l}{7}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{079 {\it 11}}&\parbox[t][0.3cm]{7.834711cm}{\raggedright Mult.,\ensuremath{\delta}: From \ensuremath{\alpha}(K)exp in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986Br28,B}{1986Br28} (\ensuremath{^{\textnormal{201}}}Po \ensuremath{\varepsilon} decay\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.834711cm}{\raggedright {\ }{\ }{\ }(15.6 min)).\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{1442}&\multicolumn{1}{@{.}l}{2\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 6}}&\multicolumn{1}{r@{}}{42}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 4}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{9/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1(+E2)}&\multicolumn{1}{r@{}}{$<$1}&\multicolumn{1}{@{.}l}{4}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0063 {\it 13}}&\parbox[t][0.3cm]{7.834711cm}{\raggedright Mult.,\ensuremath{\delta}: From \ensuremath{\alpha}(K)exp in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986Br28,B}{1986Br28} (\ensuremath{^{\textnormal{201}}}Po \ensuremath{\varepsilon} decay\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.834711cm}{\raggedright {\ }{\ }{\ }(15.6 min)).\vspace{0.1cm}}&\\
\end{longtable}
\clearpage
\begin{longtable}{ccccccccc@{}ccccccc@{\extracolsep{\fill}}c}
\\[-.4cm]
\multicolumn{17}{c}{{\bf \small \underline{Adopted \hyperlink{201BI_LEVEL}{Levels}, \hyperlink{201BI_GAMMA}{Gammas} (continued)}}}\\
\multicolumn{17}{c}{~}\\
\multicolumn{17}{c}{\underline{$\gamma$($^{201}$Bi) (continued)}}\\
\multicolumn{17}{c}{~~~}\\
\multicolumn{2}{c}{E\ensuremath{_{i}}(level)}&J\ensuremath{^{\pi}_{i}}&\multicolumn{2}{c}{E\ensuremath{_{\gamma}}}&\multicolumn{2}{c}{I\ensuremath{_{\gamma}}}&\multicolumn{2}{c}{E\ensuremath{_{f}}}&J\ensuremath{^{\pi}_{f}}&Mult.&\multicolumn{2}{c}{\ensuremath{\delta}}&\multicolumn{2}{c}{\ensuremath{\alpha}\ensuremath{^{\hyperlink{BI35GAMMA4}{\&}}}}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endhead
\multicolumn{1}{r@{}}{1470}&\multicolumn{1}{@{.}l}{87}&\multicolumn{1}{l}{(5/2,7/2)\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{566}&\multicolumn{1}{@{.}l}{6\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 3}}&\multicolumn{1}{r@{}}{32}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 4}}&\multicolumn{1}{r@{}}{904}&\multicolumn{1}{@{.}l}{23 }&\multicolumn{1}{@{}l}{(7/2)\ensuremath{^{-}}}&\multicolumn{1}{l}{M1(+E2)}&\multicolumn{1}{r@{}}{$<$0}&\multicolumn{1}{@{.}l}{9}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{071 {\it 14}}&\parbox[t][0.3cm]{6.520749cm}{\raggedright Mult.,\ensuremath{\delta}: From \ensuremath{\alpha}(K)exp in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986Br28,B}{1986Br28} (\ensuremath{^{\textnormal{201}}}Po \ensuremath{\varepsilon}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.520749cm}{\raggedright {\ }{\ }{\ }decay (15.6 min)).\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{1470}&\multicolumn{1}{@{.}l}{9\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 3}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 7}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{9/2\ensuremath{^{-}}}&&&&&&&\\
\multicolumn{1}{r@{}}{1474}&\multicolumn{1}{@{.}l}{6}&\multicolumn{1}{l}{17/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{95}&\multicolumn{1}{@{.}l}{26\ensuremath{^{\hyperlink{BI35GAMMA0}{\dagger}}} {\it 15}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{BI35GAMMA0}{\dagger}}}}&\multicolumn{1}{r@{}}{1379}&\multicolumn{1}{@{.}l}{34 }&\multicolumn{1}{@{}l}{15/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1(+E2)}&&&\multicolumn{1}{r@{}}{10}&\multicolumn{1}{@{.}l}{0 {\it 17}}&\parbox[t][0.3cm]{6.520749cm}{\raggedright Mult.: From \ensuremath{\gamma}(\ensuremath{\theta}) and \ensuremath{\alpha}(exp) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1982Br21,B}{1982Br21}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.520749cm}{\raggedright {\ }{\ }{\ }(\ensuremath{^{\textnormal{203}}}Tl(\ensuremath{\alpha},6n\ensuremath{\gamma})), \ensuremath{\gamma}(\ensuremath{\theta}) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1985Pi05,B}{1985Pi05}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.520749cm}{\raggedright {\ }{\ }{\ }(\ensuremath{^{\textnormal{196}}}Pt(\ensuremath{^{\textnormal{10}}}B,5n\ensuremath{\gamma})).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1483}&\multicolumn{1}{@{.}l}{54}&\multicolumn{1}{l}{(3/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{593}&\multicolumn{1}{@{.}l}{3\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 2}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 4}}&\multicolumn{1}{r@{}}{890}&\multicolumn{1}{@{.}l}{24 }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1(+E2)}&\multicolumn{1}{r@{}}{$<$0}&\multicolumn{1}{@{.}l}{33}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0720 {\it 28}}&\parbox[t][0.3cm]{6.520749cm}{\raggedright Mult.,\ensuremath{\delta}: From \ensuremath{\alpha}(K)exp in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986Br28,B}{1986Br28} (\ensuremath{^{\textnormal{201}}}Po \ensuremath{\varepsilon}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.520749cm}{\raggedright {\ }{\ }{\ }decay (15.6 min)).\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{636}&\multicolumn{1}{@{.}l}{5\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}\hyperlink{BI35GAMMA5}{a}}} {\it 2}}&\multicolumn{1}{r@{}}{12}&\multicolumn{1}{@{.}l}{1\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 15}}&\multicolumn{1}{r@{}}{846}&\multicolumn{1}{@{.}l}{35 }&\multicolumn{1}{@{}l}{1/2\ensuremath{^{+}}}&\multicolumn{1}{l}{[E1]}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{00609 {\it 9}}&&\\
\multicolumn{1}{r@{}}{1501}&\multicolumn{1}{@{.}l}{89}&\multicolumn{1}{l}{(13/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{534}&\multicolumn{1}{@{.}l}{2\ensuremath{^{\hyperlink{BI35GAMMA2}{\#}}} {\it 3}}&\multicolumn{1}{r@{}}{42}&\multicolumn{1}{@{.}l}{5\ensuremath{^{\hyperlink{BI35GAMMA2}{\#}}} {\it 17}}&\multicolumn{1}{r@{}}{967}&\multicolumn{1}{@{.}l}{49 }&\multicolumn{1}{@{}l}{13/2\ensuremath{^{-}}}&\multicolumn{1}{l}{[E1]}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{00865 {\it 12}}&&\\
&&&\multicolumn{1}{r@{}}{537}&\multicolumn{1}{@{.}l}{5\ensuremath{^{\hyperlink{BI35GAMMA2}{\#}}} {\it 2}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{BI35GAMMA2}{\#}}} {\it 6}}&\multicolumn{1}{r@{}}{964}&\multicolumn{1}{@{.}l}{40 }&\multicolumn{1}{@{}l}{11/2\ensuremath{^{-}}}&\multicolumn{1}{l}{[E1]}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{00854 {\it 12}}&&\\
&&&\multicolumn{1}{r@{}}{1502}&\multicolumn{1}{@{.}l}{4\ensuremath{^{\hyperlink{BI35GAMMA2}{\#}}} {\it 3}}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{1\ensuremath{^{\hyperlink{BI35GAMMA2}{\#}}} {\it 5}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{9/2\ensuremath{^{-}}}&\multicolumn{1}{l}{(E3)}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{00677 {\it 9}}&\parbox[t][0.3cm]{6.520749cm}{\raggedright Mult.: From \ensuremath{\alpha}(K)exp in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986Br28,B}{1986Br28} (\ensuremath{^{\textnormal{201}}}Po \ensuremath{\varepsilon}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.520749cm}{\raggedright {\ }{\ }{\ }decay (8.96 min)).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1504}&\multicolumn{1}{@{.}l}{40}&\multicolumn{1}{l}{(9/2,11/2)\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{540}&\multicolumn{1}{@{.}l}{1\ensuremath{^{\hyperlink{BI35GAMMA2}{\#}}} {\it 3}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{BI35GAMMA2}{\#}}} {\it 5}}&\multicolumn{1}{r@{}}{964}&\multicolumn{1}{@{.}l}{40 }&\multicolumn{1}{@{}l}{11/2\ensuremath{^{-}}}&&&&&&&\\
&&&\multicolumn{1}{r@{}}{1504}&\multicolumn{1}{@{.}l}{3\ensuremath{^{\hyperlink{BI35GAMMA2}{\#}}} {\it 3}}&\multicolumn{1}{r@{}}{9}&\multicolumn{1}{@{.}l}{3\ensuremath{^{\hyperlink{BI35GAMMA2}{\#}}} {\it 21}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{9/2\ensuremath{^{-}}}&&&&&&&\\
\multicolumn{1}{r@{}}{1616}&\multicolumn{1}{@{.}l}{3}&\multicolumn{1}{l}{1/2\ensuremath{^{+}},3/2\ensuremath{^{+}},5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{530}&\multicolumn{1}{@{.}l}{1\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 3}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{1086}&\multicolumn{1}{@{.}l}{21 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}}}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{4 {\it +5\textminus3}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{052 {\it 9}}&\parbox[t][0.3cm]{6.520749cm}{\raggedright Mult.,\ensuremath{\delta}: From \ensuremath{\alpha}(K)exp in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986Br28,B}{1986Br28} (\ensuremath{^{\textnormal{201}}}Po \ensuremath{\varepsilon}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.520749cm}{\raggedright {\ }{\ }{\ }decay (15.6 min)).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1665}&\multicolumn{1}{@{.}l}{1}&&\multicolumn{1}{r@{}}{697}&\multicolumn{1}{@{.}l}{6\ensuremath{^{\hyperlink{BI35GAMMA2}{\#}}} {\it 2}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{BI35GAMMA2}{\#}}}}&\multicolumn{1}{r@{}}{967}&\multicolumn{1}{@{.}l}{49 }&\multicolumn{1}{@{}l}{13/2\ensuremath{^{-}}}&&&&&&&\\
\multicolumn{1}{r@{}}{1719}&\multicolumn{1}{@{.}l}{15}&\multicolumn{1}{l}{(11/2,13/2)\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{217}&\multicolumn{1}{@{.}l}{6\ensuremath{^{\hyperlink{BI35GAMMA2}{\#}}} {\it 3}}&\multicolumn{1}{r@{}}{8}&\multicolumn{1}{@{.}l}{7\ensuremath{^{\hyperlink{BI35GAMMA2}{\#}}} {\it 14}}&\multicolumn{1}{r@{}}{1501}&\multicolumn{1}{@{.}l}{89 }&\multicolumn{1}{@{}l}{(13/2\ensuremath{^{+}})}&\multicolumn{1}{l}{M1(+E2)}&\multicolumn{1}{r@{}}{$<$1}&\multicolumn{1}{@{.}l}{7}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{83 {\it 29}}&\parbox[t][0.3cm]{6.520749cm}{\raggedright Mult.,\ensuremath{\delta}: From \ensuremath{\alpha}(K)exp in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986Br28,B}{1986Br28} (\ensuremath{^{\textnormal{201}}}Po \ensuremath{\varepsilon}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.520749cm}{\raggedright {\ }{\ }{\ }decay (8.96 min)).\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{754}&\multicolumn{1}{@{.}l}{6\ensuremath{^{\hyperlink{BI35GAMMA2}{\#}}} {\it 2}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{BI35GAMMA2}{\#}}} {\it 6}}&\multicolumn{1}{r@{}}{964}&\multicolumn{1}{@{.}l}{40 }&\multicolumn{1}{@{}l}{11/2\ensuremath{^{-}}}&\multicolumn{1}{l}{E1}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{00438 {\it 6}}&\parbox[t][0.3cm]{6.520749cm}{\raggedright Mult.: From \ensuremath{\alpha}(K)exp in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986Br28,B}{1986Br28} (\ensuremath{^{\textnormal{201}}}Po \ensuremath{\varepsilon}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.520749cm}{\raggedright {\ }{\ }{\ }decay (8.96 min)). Note, that \ensuremath{\gamma}(\ensuremath{\theta}) in\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.520749cm}{\raggedright {\ }{\ }{\ }\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1985Pi05,B}{1985Pi05} (\ensuremath{^{\textnormal{196}}}Pt(\ensuremath{^{\textnormal{10}}}B,5n\ensuremath{\gamma})) implies M1+E2\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.520749cm}{\raggedright {\ }{\ }{\ }multipolarity.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1746}&\multicolumn{1}{@{.}l}{5}&\multicolumn{1}{l}{17/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{271}&\multicolumn{1}{@{.}l}{91\ensuremath{^{\hyperlink{BI35GAMMA0}{\dagger}}} {\it 20}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{BI35GAMMA0}{\dagger}}}}&\multicolumn{1}{r@{}}{1474}&\multicolumn{1}{@{.}l}{6 }&\multicolumn{1}{@{}l}{17/2\ensuremath{^{-}}}&\multicolumn{1}{l}{E1}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0383 {\it 5}}&\parbox[t][0.3cm]{6.520749cm}{\raggedright B(E1)(W.u.)=1.9\ensuremath{\times}10\ensuremath{^{\textnormal{$-$6}}} \textit{+6{\textminus}4}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.520749cm}{\raggedright Mult.: From \ensuremath{\alpha}(K)exp in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986Br28,B}{1986Br28} (\ensuremath{^{\textnormal{201}}}Po \ensuremath{\varepsilon}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.520749cm}{\raggedright {\ }{\ }{\ }decay (8.96 min)), \ensuremath{\gamma}(\ensuremath{\theta}) and \ensuremath{\alpha}(exp) in\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.520749cm}{\raggedright {\ }{\ }{\ }\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1982Br21,B}{1982Br21} (\ensuremath{^{\textnormal{203}}}Tl(\ensuremath{\alpha},6n\ensuremath{\gamma})), \ensuremath{\gamma}(\ensuremath{\theta}) and DCO in\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.520749cm}{\raggedright {\ }{\ }{\ }\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1985Pi05,B}{1985Pi05} (\ensuremath{^{\textnormal{196}}}Pt(\ensuremath{^{\textnormal{10}}}B,5n\ensuremath{\gamma})).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1762}&\multicolumn{1}{@{.}l}{9}&\multicolumn{1}{l}{(11/2,13/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{798}&\multicolumn{1}{@{.}l}{5\ensuremath{^{\hyperlink{BI35GAMMA2}{\#}}} {\it 3}}&\multicolumn{1}{r@{}}{55}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{BI35GAMMA2}{\#}}} {\it 5}}&\multicolumn{1}{r@{}}{964}&\multicolumn{1}{@{.}l}{40 }&\multicolumn{1}{@{}l}{11/2\ensuremath{^{-}}}&&&&&&&\\
&&&\multicolumn{1}{r@{}}{1762}&\multicolumn{1}{@{.}l}{9\ensuremath{^{\hyperlink{BI35GAMMA2}{\#}}} {\it 6}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{BI35GAMMA2}{\#}}} {\it 8}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{9/2\ensuremath{^{-}}}&&&&&&&\\
\multicolumn{1}{r@{}}{1778}&\multicolumn{1}{@{.}l}{92}&\multicolumn{1}{l}{(3/2\ensuremath{^{-}},5/2)}&\multicolumn{1}{r@{}}{874}&\multicolumn{1}{@{.}l}{6\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 2}}&\multicolumn{1}{r@{}}{87}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 6}}&\multicolumn{1}{r@{}}{904}&\multicolumn{1}{@{.}l}{23 }&\multicolumn{1}{@{}l}{(7/2)\ensuremath{^{-}}}&&&&&&&\\
&&&\multicolumn{1}{r@{}}{889}&\multicolumn{1}{@{.}l}{2\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 11}}&\multicolumn{1}{r@{}}{890}&\multicolumn{1}{@{.}l}{24 }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{-}}}&&&&&&&\\
\multicolumn{1}{r@{}}{1817}&\multicolumn{1}{@{.}l}{86}&\multicolumn{1}{l}{1/2\ensuremath{^{+}},3/2,5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{543}&\multicolumn{1}{@{.}l}{4\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 3}}&\multicolumn{1}{r@{}}{63}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 4}}&\multicolumn{1}{r@{}}{1274}&\multicolumn{1}{@{.}l}{45 }&\multicolumn{1}{@{}l}{(5/2)\ensuremath{^{+}}}&&&&&&&\\
&&&\multicolumn{1}{r@{}}{731}&\multicolumn{1}{@{.}l}{7\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 2}}&\multicolumn{1}{r@{}}{65}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 6}}&\multicolumn{1}{r@{}}{1086}&\multicolumn{1}{@{.}l}{21 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}}}&&&&&&&\\
&&&\multicolumn{1}{r@{}}{971}&\multicolumn{1}{@{.}l}{4\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 3}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 6}}&\multicolumn{1}{r@{}}{846}&\multicolumn{1}{@{.}l}{35 }&\multicolumn{1}{@{}l}{1/2\ensuremath{^{+}}}&&&&&&&\\
\multicolumn{1}{r@{}}{1848}&\multicolumn{1}{@{.}l}{16}&\multicolumn{1}{l}{(5/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{944}&\multicolumn{1}{@{.}l}{2\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 4}}&\multicolumn{1}{r@{}}{54}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 8}}&\multicolumn{1}{r@{}}{904}&\multicolumn{1}{@{.}l}{23 }&\multicolumn{1}{@{}l}{(7/2)\ensuremath{^{-}}}&&&&&&&\\
\end{longtable}
\clearpage
\begin{longtable}{ccccccccc@{}ccccccc@{\extracolsep{\fill}}c}
\\[-.4cm]
\multicolumn{17}{c}{{\bf \small \underline{Adopted \hyperlink{201BI_LEVEL}{Levels}, \hyperlink{201BI_GAMMA}{Gammas} (continued)}}}\\
\multicolumn{17}{c}{~}\\
\multicolumn{17}{c}{\underline{$\gamma$($^{201}$Bi) (continued)}}\\
\multicolumn{17}{c}{~~~}\\
\multicolumn{2}{c}{E\ensuremath{_{i}}(level)}&J\ensuremath{^{\pi}_{i}}&\multicolumn{2}{c}{E\ensuremath{_{\gamma}}}&\multicolumn{2}{c}{I\ensuremath{_{\gamma}}}&\multicolumn{2}{c}{E\ensuremath{_{f}}}&J\ensuremath{^{\pi}_{f}}&Mult.&\multicolumn{2}{c}{\ensuremath{\delta}}&\multicolumn{2}{c}{\ensuremath{\alpha}\ensuremath{^{\hyperlink{BI35GAMMA4}{\&}}}}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endhead
\multicolumn{1}{r@{}}{1848}&\multicolumn{1}{@{.}l}{16}&\multicolumn{1}{l}{(5/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{1848}&\multicolumn{1}{@{.}l}{0\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 3}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{9/2\ensuremath{^{-}}}&&&&&&&\\
\multicolumn{1}{r@{}}{1858}&\multicolumn{1}{@{.}l}{02}&\multicolumn{1}{l}{3/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{583}&\multicolumn{1}{@{.}l}{6\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 3}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 6}}&\multicolumn{1}{r@{}}{1274}&\multicolumn{1}{@{.}l}{45 }&\multicolumn{1}{@{}l}{(5/2)\ensuremath{^{+}}}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{0 {\it +10\textminus5}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{050 {\it 17}}&\parbox[t][0.3cm]{6.34741cm}{\raggedright Mult.,\ensuremath{\delta}: From \ensuremath{\alpha}(K)exp in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986Br28,B}{1986Br28} (\ensuremath{^{\textnormal{201}}}Po\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.34741cm}{\raggedright {\ }{\ }{\ }\ensuremath{\varepsilon} decay (15.6 min)).\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{771}&\multicolumn{1}{@{.}l}{8\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 2}}&\multicolumn{1}{r@{}}{85}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 6}}&\multicolumn{1}{r@{}}{1086}&\multicolumn{1}{@{.}l}{21 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}}}&\multicolumn{1}{l}{E0+M1}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{10 {\it 3}}&\parbox[t][0.3cm]{6.34741cm}{\raggedright Mult.,\ensuremath{\alpha}: From \ensuremath{^{\textnormal{201}}}Po \ensuremath{\varepsilon} decay (15.6 min).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1927}&\multicolumn{1}{@{.}l}{32}&\multicolumn{1}{l}{(5/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{1023}&\multicolumn{1}{@{.}l}{0\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 2}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 9}}&\multicolumn{1}{r@{}}{904}&\multicolumn{1}{@{.}l}{23 }&\multicolumn{1}{@{}l}{(7/2)\ensuremath{^{-}}}&&&&&&&\\
&&&\multicolumn{1}{r@{}}{1927}&\multicolumn{1}{@{.}l}{5\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 3}}&\multicolumn{1}{r@{}}{97}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 11}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{9/2\ensuremath{^{-}}}&&&&&&&\\
\multicolumn{1}{r@{}}{1932}&\multicolumn{1}{@{.}l}{3}&\multicolumn{1}{l}{21/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{185}&\multicolumn{1}{@{.}l}{77\ensuremath{^{\hyperlink{BI35GAMMA0}{\dagger}}} {\it 20}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{BI35GAMMA0}{\dagger}}}}&\multicolumn{1}{r@{}}{1746}&\multicolumn{1}{@{.}l}{5 }&\multicolumn{1}{@{}l}{17/2\ensuremath{^{+}}}&\multicolumn{1}{l}{E2}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{583 {\it 8}}&\parbox[t][0.3cm]{6.34741cm}{\raggedright B(E2)(W.u.)\ensuremath{>}0.57\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.34741cm}{\raggedright Mult.: From \ensuremath{\gamma}(\ensuremath{\theta}) and \ensuremath{\alpha}(exp) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1982Br21,B}{1982Br21}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.34741cm}{\raggedright {\ }{\ }{\ }(\ensuremath{^{\textnormal{203}}}Tl(\ensuremath{\alpha},6n\ensuremath{\gamma})), \ensuremath{\gamma}(\ensuremath{\theta}) and DCO in\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.34741cm}{\raggedright {\ }{\ }{\ }\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1985Pi05,B}{1985Pi05} (\ensuremath{^{\textnormal{196}}}Pt(\ensuremath{^{\textnormal{10}}}B,5n\ensuremath{\gamma})).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1932}&\multicolumn{1}{@{.}l}{3+x}&\multicolumn{1}{l}{(25/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{($<$80)}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{1932}&\multicolumn{1}{@{.}l}{3 }&\multicolumn{1}{@{}l}{21/2\ensuremath{^{+}}}&&&&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{6.34741cm}{\raggedright E\ensuremath{_{\gamma}}: Anticipated in the decay of this level by\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.34741cm}{\raggedright {\ }{\ }{\ }analogy to \ensuremath{^{\textnormal{201}}}Bi and \ensuremath{^{\textnormal{205}}}Bi, and the\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.34741cm}{\raggedright {\ }{\ }{\ }non-observation of low-energy E\ensuremath{\gamma} in the\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.34741cm}{\raggedright {\ }{\ }{\ }corresponding \ensuremath{\gamma}-ray spectra.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1944}&\multicolumn{1}{@{.}l}{24}&\multicolumn{1}{l}{(5/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{1039}&\multicolumn{1}{@{.}l}{7\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 2}}&\multicolumn{1}{r@{}}{75}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 9}}&\multicolumn{1}{r@{}}{904}&\multicolumn{1}{@{.}l}{23 }&\multicolumn{1}{@{}l}{(7/2)\ensuremath{^{-}}}&&&&&&&\\
&&&\multicolumn{1}{r@{}}{1054}&\multicolumn{1}{@{.}l}{7\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 3}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 9}}&\multicolumn{1}{r@{}}{890}&\multicolumn{1}{@{.}l}{24 }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{-}}}&&&&&&&\\
&&&\multicolumn{1}{r@{}}{1944}&\multicolumn{1}{@{.}l}{2\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 3}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 10}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{9/2\ensuremath{^{-}}}&&&&&&&\\
\multicolumn{1}{r@{}}{1971}&\multicolumn{1}{@{.}l}{3+x}&\multicolumn{1}{l}{(27/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{(39}&\multicolumn{1}{@{.}l}{0 {\it 5})}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{1932}&\multicolumn{1}{@{.}l}{3+x }&\multicolumn{1}{@{}l}{(25/2\ensuremath{^{+}})}&\multicolumn{1}{l}{[M1]}&&&\multicolumn{1}{r@{}}{30}&\multicolumn{1}{@{.}l}{1 {\it 12}}&\parbox[t][0.3cm]{6.34741cm}{\raggedright B(M1)(W.u.)=0.00011 \textit{+14{\textminus}5}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.34741cm}{\raggedright E\ensuremath{_{\gamma}}: From level energy difference. The\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.34741cm}{\raggedright {\ }{\ }{\ }transition was not observed directly and the\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.34741cm}{\raggedright {\ }{\ }{\ }existence is based on the \ensuremath{\gamma}\ensuremath{\gamma}-coincidence\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.34741cm}{\raggedright {\ }{\ }{\ }relationships in \ensuremath{^{\textnormal{196}}}Pt(\ensuremath{^{\textnormal{10}}}B,5n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2034}&\multicolumn{1}{@{.}l}{3}&&\multicolumn{1}{r@{}}{532}&\multicolumn{1}{@{.}l}{4\ensuremath{^{\hyperlink{BI35GAMMA2}{\#}}} {\it 6}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{BI35GAMMA2}{\#}}}}&\multicolumn{1}{r@{}}{1501}&\multicolumn{1}{@{.}l}{89 }&\multicolumn{1}{@{}l}{(13/2\ensuremath{^{+}})}&&&&&&&\\
\multicolumn{1}{r@{}}{2053}&\multicolumn{1}{@{.}l}{59}&\multicolumn{1}{l}{(5/2)\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{779}&\multicolumn{1}{@{.}l}{4\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 3}}&\multicolumn{1}{r@{}}{35}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 3}}&\multicolumn{1}{r@{}}{1274}&\multicolumn{1}{@{.}l}{45 }&\multicolumn{1}{@{}l}{(5/2)\ensuremath{^{+}}}&\multicolumn{1}{l}{(M1)}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0366 {\it 5}}&\parbox[t][0.3cm]{6.34741cm}{\raggedright Mult.: From \ensuremath{^{\textnormal{201}}}Po \ensuremath{\varepsilon} decay (15.6 min).\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{967}&\multicolumn{1}{@{.}l}{4\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 2}}&\multicolumn{1}{r@{}}{28}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 4}}&\multicolumn{1}{r@{}}{1086}&\multicolumn{1}{@{.}l}{21 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}}}&\multicolumn{1}{l}{(E2)}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{00746 {\it 10}}&\parbox[t][0.3cm]{6.34741cm}{\raggedright Mult.: From \ensuremath{\alpha}(K)exp in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986Br28,B}{1986Br28} (\ensuremath{^{\textnormal{201}}}Po \ensuremath{\varepsilon}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.34741cm}{\raggedright {\ }{\ }{\ }decay (15.6 min)).\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{1207}&\multicolumn{1}{@{.}l}{1\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 2}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{846}&\multicolumn{1}{@{.}l}{35 }&\multicolumn{1}{@{}l}{1/2\ensuremath{^{+}}}&\multicolumn{1}{l}{E2}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{00488 {\it 7}}&\parbox[t][0.3cm]{6.34741cm}{\raggedright Mult.: From \ensuremath{\alpha}(K)exp in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986Br28,B}{1986Br28} (\ensuremath{^{\textnormal{201}}}Po \ensuremath{\varepsilon}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.34741cm}{\raggedright {\ }{\ }{\ }decay (15.6 min)).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2065}&\multicolumn{1}{@{.}l}{82}&\multicolumn{1}{l}{5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{624}&\multicolumn{1}{@{.}l}{7\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 3}}&\multicolumn{1}{r@{}}{17}&\multicolumn{1}{@{.}l}{3\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 12}}&\multicolumn{1}{r@{}}{1441}&\multicolumn{1}{@{.}l}{71 }&\multicolumn{1}{@{}l}{7/2\ensuremath{^{-}}}&\multicolumn{1}{l}{[E1]}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{00631 {\it 9}}&&\\
&&&\multicolumn{1}{r@{}}{791}&\multicolumn{1}{@{.}l}{4\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 2}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 4}}&\multicolumn{1}{r@{}}{1274}&\multicolumn{1}{@{.}l}{45 }&\multicolumn{1}{@{}l}{(5/2)\ensuremath{^{+}}}&\multicolumn{1}{l}{M1(+E2)}&\multicolumn{1}{r@{}}{$\leq$0}&\multicolumn{1}{@{.}l}{4}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0335 {\it 17}}&\parbox[t][0.3cm]{6.34741cm}{\raggedright Mult.,\ensuremath{\delta}: From \ensuremath{\alpha}(K)exp in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986Br28,B}{1986Br28} (\ensuremath{^{\textnormal{201}}}Po\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.34741cm}{\raggedright {\ }{\ }{\ }\ensuremath{\varepsilon} decay (15.6 min)).\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{979}&\multicolumn{1}{@{.}l}{7\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 3}}&\multicolumn{1}{r@{}}{31}&\multicolumn{1}{@{.}l}{9\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 21}}&\multicolumn{1}{r@{}}{1086}&\multicolumn{1}{@{.}l}{21 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}}}&\multicolumn{1}{l}{[M1]}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{02026 {\it 28}}&&\\
&&&\multicolumn{1}{r@{}}{1175}&\multicolumn{1}{@{.}l}{3\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 2}}&\multicolumn{1}{r@{}}{76}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 4}}&\multicolumn{1}{r@{}}{890}&\multicolumn{1}{@{.}l}{24 }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{l}{(E1)}&&&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{97\ensuremath{\times10^{-3}} {\it 3}}&\parbox[t][0.3cm]{6.34741cm}{\raggedright Mult.: From \ensuremath{\alpha}(K)exp in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986Br28,B}{1986Br28} (\ensuremath{^{\textnormal{201}}}Po \ensuremath{\varepsilon}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.34741cm}{\raggedright {\ }{\ }{\ }decay (15.6 min)).\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{1219}&\multicolumn{1}{@{.}l}{3\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 3}}&\multicolumn{1}{r@{}}{22}&\multicolumn{1}{@{.}l}{7\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 21}}&\multicolumn{1}{r@{}}{846}&\multicolumn{1}{@{.}l}{35 }&\multicolumn{1}{@{}l}{1/2\ensuremath{^{+}}}&\multicolumn{1}{l}{[E2]}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{00479 {\it 7}}&&\\
\multicolumn{1}{r@{}}{2299}&\multicolumn{1}{@{.}l}{1+x}&\multicolumn{1}{l}{(27/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{366}&\multicolumn{1}{@{.}l}{6\ensuremath{^{\hyperlink{BI35GAMMA0}{\dagger}}} {\it 4}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{BI35GAMMA0}{\dagger}}}}&\multicolumn{1}{r@{}}{1932}&\multicolumn{1}{@{.}l}{3+x }&\multicolumn{1}{@{}l}{(25/2\ensuremath{^{+}})}&\multicolumn{1}{l}{(M1)}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{269 {\it 4}}&\parbox[t][0.3cm]{6.34741cm}{\raggedright Mult.: From DCO in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1985Pi05,B}{1985Pi05}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.34741cm}{\raggedright {\ }{\ }{\ }(\ensuremath{^{\textnormal{196}}}Pt(\ensuremath{^{\textnormal{10}}}B,5n\ensuremath{\gamma})) and \ensuremath{\gamma}(\ensuremath{\theta}) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1982Br21,B}{1982Br21}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.34741cm}{\raggedright {\ }{\ }{\ }(\ensuremath{^{\textnormal{203}}}Tl(\ensuremath{\alpha},6n\ensuremath{\gamma})).\vspace{0.1cm}}&\\
\end{longtable}
\clearpage
\begin{longtable}{ccccccccc@{}ccccccc@{\extracolsep{\fill}}c}
\\[-.4cm]
\multicolumn{17}{c}{{\bf \small \underline{Adopted \hyperlink{201BI_LEVEL}{Levels}, \hyperlink{201BI_GAMMA}{Gammas} (continued)}}}\\
\multicolumn{17}{c}{~}\\
\multicolumn{17}{c}{\underline{$\gamma$($^{201}$Bi) (continued)}}\\
\multicolumn{17}{c}{~~~}\\
\multicolumn{2}{c}{E\ensuremath{_{i}}(level)}&J\ensuremath{^{\pi}_{i}}&\multicolumn{2}{c}{E\ensuremath{_{\gamma}}}&\multicolumn{2}{c}{I\ensuremath{_{\gamma}}}&\multicolumn{2}{c}{E\ensuremath{_{f}}}&J\ensuremath{^{\pi}_{f}}&Mult.&\multicolumn{2}{c}{\ensuremath{\delta}}&\multicolumn{2}{c}{\ensuremath{\alpha}\ensuremath{^{\hyperlink{BI35GAMMA4}{\&}}}}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endhead
\multicolumn{1}{r@{}}{2386}&\multicolumn{1}{@{.}l}{7}&&\multicolumn{1}{r@{}}{1300}&\multicolumn{1}{@{.}l}{5\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 4}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{1086}&\multicolumn{1}{@{.}l}{21 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}}}&&&&&&&\\
\multicolumn{1}{r@{}}{2422}&\multicolumn{1}{@{.}l}{1}&\multicolumn{1}{l}{(3/2\ensuremath{^{-}},5/2)}&\multicolumn{1}{r@{}}{1518}&\multicolumn{1}{@{.}l}{2\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 4}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 6}}&\multicolumn{1}{r@{}}{904}&\multicolumn{1}{@{.}l}{23 }&\multicolumn{1}{@{}l}{(7/2)\ensuremath{^{-}}}&&&&&&&\\
&&&\multicolumn{1}{r@{}}{1531}&\multicolumn{1}{@{.}l}{7\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 3}}&\multicolumn{1}{r@{}}{26}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{890}&\multicolumn{1}{@{.}l}{24 }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{-}}}&&&&&&&\\
\multicolumn{1}{r@{}}{2434}&\multicolumn{1}{@{.}l}{9}&\multicolumn{1}{l}{1/2\ensuremath{^{+}},3/2,5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1160}&\multicolumn{1}{@{.}l}{6\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{64}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 8}}&\multicolumn{1}{r@{}}{1274}&\multicolumn{1}{@{.}l}{45 }&\multicolumn{1}{@{}l}{(5/2)\ensuremath{^{+}}}&&&&&&&\\
&&&\multicolumn{1}{r@{}}{1348}&\multicolumn{1}{@{.}l}{4\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 6}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 10}}&\multicolumn{1}{r@{}}{1086}&\multicolumn{1}{@{.}l}{21 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}}}&&&&&&&\\
&&&\multicolumn{1}{r@{}}{1588}&\multicolumn{1}{@{.}l}{5\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 3}}&\multicolumn{1}{r@{}}{69}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 10}}&\multicolumn{1}{r@{}}{846}&\multicolumn{1}{@{.}l}{35 }&\multicolumn{1}{@{}l}{1/2\ensuremath{^{+}}}&&&&&&&\\
\multicolumn{1}{r@{}}{2455}&\multicolumn{1}{@{.}l}{5}&\multicolumn{1}{l}{1/2\ensuremath{^{+}},3/2,5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1181}&\multicolumn{1}{@{.}l}{3\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{27}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{1274}&\multicolumn{1}{@{.}l}{45 }&\multicolumn{1}{@{}l}{(5/2)\ensuremath{^{+}}}&&&&&&&\\
&&&\multicolumn{1}{r@{}}{1369}&\multicolumn{1}{@{.}l}{3\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 4}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 8}}&\multicolumn{1}{r@{}}{1086}&\multicolumn{1}{@{.}l}{21 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}}}&&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{005 {\it 4}}&\parbox[t][0.3cm]{6.1107807cm}{\raggedright Mult.: From \ensuremath{\alpha}(K)exp in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986Br28,B}{1986Br28} (\ensuremath{^{\textnormal{201}}}Po\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.1107807cm}{\raggedright {\ }{\ }{\ }\ensuremath{\varepsilon} decay (15.6 min)).\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{1609}&\multicolumn{1}{@{.}l}{0\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 3}}&\multicolumn{1}{r@{}}{98}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 8}}&\multicolumn{1}{r@{}}{846}&\multicolumn{1}{@{.}l}{35 }&\multicolumn{1}{@{}l}{1/2\ensuremath{^{+}}}&&&&&&&\\
\multicolumn{1}{r@{}}{2484}&\multicolumn{1}{@{.}l}{3}&\multicolumn{1}{l}{1/2,3/2,5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1398}&\multicolumn{1}{@{.}l}{0\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 3}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 6}}&\multicolumn{1}{r@{}}{1086}&\multicolumn{1}{@{.}l}{21 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}}}&&&&&&&\\
&&&\multicolumn{1}{r@{}}{1638}&\multicolumn{1}{@{.}l}{1\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{29}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 4}}&\multicolumn{1}{r@{}}{846}&\multicolumn{1}{@{.}l}{35 }&\multicolumn{1}{@{}l}{1/2\ensuremath{^{+}}}&&&&&&&\\
\multicolumn{1}{r@{}}{2549}&\multicolumn{1}{@{.}l}{50+x}&\multicolumn{1}{l}{(27/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{250}&\multicolumn{1}{@{.}l}{2\ensuremath{^{\hyperlink{BI35GAMMA0}{\dagger}}} {\it 4}}&\multicolumn{1}{r@{}}{4}&\multicolumn{1}{@{.}l}{37\ensuremath{^{\hyperlink{BI35GAMMA0}{\dagger}}} {\it 25}}&\multicolumn{1}{r@{}}{2299}&\multicolumn{1}{@{.}l}{1+x }&\multicolumn{1}{@{}l}{(27/2\ensuremath{^{+}})}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{$>$1}&\multicolumn{1}{@{.}l}{2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{32 {\it 11}}&\parbox[t][0.3cm]{6.1107807cm}{\raggedright Mult.,\ensuremath{\delta}: From \ensuremath{\gamma}(\ensuremath{\theta}) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1985Pi05,B}{1985Pi05}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.1107807cm}{\raggedright {\ }{\ }{\ }(\ensuremath{^{\textnormal{196}}}Pt(\ensuremath{^{\textnormal{10}}}B,5n\ensuremath{\gamma})).\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{617}&\multicolumn{1}{@{.}l}{27\ensuremath{^{\hyperlink{BI35GAMMA0}{\dagger}}} {\it 25}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{.}l}{0\ensuremath{^{\hyperlink{BI35GAMMA0}{\dagger}}} {\it 9}}&\multicolumn{1}{r@{}}{1932}&\multicolumn{1}{@{.}l}{3+x }&\multicolumn{1}{@{}l}{(25/2\ensuremath{^{+}})}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{+0}&\multicolumn{1}{@{.}l}{046 {\it 28}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0671 {\it 10}}&\parbox[t][0.3cm]{6.1107807cm}{\raggedright Mult.,\ensuremath{\delta}: From \ensuremath{\gamma}(\ensuremath{\theta}) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1985Pi05,B}{1985Pi05}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.1107807cm}{\raggedright {\ }{\ }{\ }(\ensuremath{^{\textnormal{196}}}Pt(\ensuremath{^{\textnormal{10}}}B,5n\ensuremath{\gamma})).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2589}&\multicolumn{1}{@{.}l}{7+x}&&\multicolumn{1}{r@{}}{657}&\multicolumn{1}{@{.}l}{4\ensuremath{^{\hyperlink{BI35GAMMA0}{\dagger}}} {\it 3}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{BI35GAMMA0}{\dagger}}}}&\multicolumn{1}{r@{}}{1932}&\multicolumn{1}{@{.}l}{3+x }&\multicolumn{1}{@{}l}{(25/2\ensuremath{^{+}})}&&&&&&&\\
\multicolumn{1}{r@{}}{2592}&\multicolumn{1}{@{.}l}{88}&\multicolumn{1}{l}{(3/2\ensuremath{^{-}},5/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{1318}&\multicolumn{1}{@{.}l}{0\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{37}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{1274}&\multicolumn{1}{@{.}l}{45 }&\multicolumn{1}{@{}l}{(5/2)\ensuremath{^{+}}}&&&&&&&\\
&&&\multicolumn{1}{r@{}}{1506}&\multicolumn{1}{@{.}l}{6\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 3}}&\multicolumn{1}{r@{}}{62}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 6}}&\multicolumn{1}{r@{}}{1086}&\multicolumn{1}{@{.}l}{21 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}}}&&&&&&&\\
&&&\multicolumn{1}{r@{}}{1689}&\multicolumn{1}{@{.}l}{3\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 3}}&\multicolumn{1}{r@{}}{54}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 6}}&\multicolumn{1}{r@{}}{904}&\multicolumn{1}{@{.}l}{23 }&\multicolumn{1}{@{}l}{(7/2)\ensuremath{^{-}}}&&&&&&&\\
&&&\multicolumn{1}{r@{}}{1702}&\multicolumn{1}{@{.}l}{1\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 3}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 6}}&\multicolumn{1}{r@{}}{890}&\multicolumn{1}{@{.}l}{24 }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{-}}}&&&&&&&\\
&&&\multicolumn{1}{r@{}}{1746}&\multicolumn{1}{@{.}l}{8\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{44}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 6}}&\multicolumn{1}{r@{}}{846}&\multicolumn{1}{@{.}l}{35 }&\multicolumn{1}{@{}l}{1/2\ensuremath{^{+}}}&&&&&&&\\
\multicolumn{1}{r@{}}{2612}&\multicolumn{1}{@{.}l}{00+x}&\multicolumn{1}{l}{(27/2)}&\multicolumn{1}{r@{}}{679}&\multicolumn{1}{@{.}l}{7 {\it 1}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{1932}&\multicolumn{1}{@{.}l}{3+x }&\multicolumn{1}{@{}l}{(25/2\ensuremath{^{+}})}&\multicolumn{1}{l}{D}&&&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{6.1107807cm}{\raggedright E\ensuremath{_{\gamma}},I\ensuremath{_{\gamma}}: From \ensuremath{^{\textnormal{203}}}Tl(\ensuremath{\alpha},6n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.1107807cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}={\textminus}0.47 \textit{3}, A\ensuremath{_{\textnormal{4}}}={\textminus}0.06 \textit{4} in\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.1107807cm}{\raggedright {\ }{\ }{\ }\ensuremath{^{\textnormal{203}}}Tl(\ensuremath{\alpha},6n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2651}&\multicolumn{1}{@{.}l}{13+x}&\multicolumn{1}{l}{(29/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{679}&\multicolumn{1}{@{.}l}{8\ensuremath{^{\hyperlink{BI35GAMMA0}{\dagger}}} {\it 3}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{BI35GAMMA0}{\dagger}}} {\it 1}}&\multicolumn{1}{r@{}}{1971}&\multicolumn{1}{@{.}l}{3+x }&\multicolumn{1}{@{}l}{(27/2\ensuremath{^{+}})}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{$-$0}&\multicolumn{1}{@{.}l}{15 {\it 12}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0514 {\it 18}}&\parbox[t][0.3cm]{6.1107807cm}{\raggedright Mult.,\ensuremath{\delta}: From \ensuremath{\gamma}(\ensuremath{\theta}) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1985Pi05,B}{1985Pi05}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.1107807cm}{\raggedright {\ }{\ }{\ }(\ensuremath{^{\textnormal{196}}}Pt(\ensuremath{^{\textnormal{10}}}B,5n\ensuremath{\gamma})).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2668}&\multicolumn{1}{@{.}l}{31+x}&\multicolumn{1}{l}{(29/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{118}&\multicolumn{1}{@{.}l}{81\ensuremath{^{\hyperlink{BI35GAMMA0}{\dagger}}} {\it 15}}&\multicolumn{1}{r@{}}{52}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{BI35GAMMA0}{\dagger}}} {\it 3}}&\multicolumn{1}{r@{}}{2549}&\multicolumn{1}{@{.}l}{50+x }&\multicolumn{1}{@{}l}{(27/2\ensuremath{^{+}})}&\multicolumn{1}{l}{M1(+E2)}&&&\multicolumn{1}{r@{}}{4}&\multicolumn{1}{@{.}l}{8 {\it 15}}&\parbox[t][0.3cm]{6.1107807cm}{\raggedright Mult.: From \ensuremath{\gamma}(\ensuremath{\theta}) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1985Pi05,B}{1985Pi05}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.1107807cm}{\raggedright {\ }{\ }{\ }(\ensuremath{^{\textnormal{196}}}Pt(\ensuremath{^{\textnormal{10}}}B,5n\ensuremath{\gamma})).\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{736}&\multicolumn{1}{@{.}l}{0\ensuremath{^{\hyperlink{BI35GAMMA0}{\dagger}}} {\it 4}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{BI35GAMMA0}{\dagger}}} {\it 5}}&\multicolumn{1}{r@{}}{1932}&\multicolumn{1}{@{.}l}{3+x }&\multicolumn{1}{@{}l}{(25/2\ensuremath{^{+}})}&\multicolumn{1}{l}{E2}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{01296 {\it 18}}&\parbox[t][0.3cm]{6.1107807cm}{\raggedright Mult.: From \ensuremath{\gamma}(\ensuremath{\theta}) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1985Pi05,B}{1985Pi05}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.1107807cm}{\raggedright {\ }{\ }{\ }(\ensuremath{^{\textnormal{196}}}Pt(\ensuremath{^{\textnormal{10}}}B,5n\ensuremath{\gamma})).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2740}&\multicolumn{1}{@{.}l}{01+x}&\multicolumn{1}{l}{(29/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{(71}&\multicolumn{1}{@{.}l}{7 {\it 3})}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{2668}&\multicolumn{1}{@{.}l}{31+x }&\multicolumn{1}{@{}l}{(29/2\ensuremath{^{+}})}&&&&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{6.1107807cm}{\raggedright E\ensuremath{_{\gamma}}: From E(level) difference.\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{88}&\multicolumn{1}{@{.}l}{88\ensuremath{^{\hyperlink{BI35GAMMA0}{\dagger}}} {\it 12}}&\multicolumn{1}{r@{}}{93}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{BI35GAMMA0}{\dagger}}} {\it 27}}&\multicolumn{1}{r@{}}{2651}&\multicolumn{1}{@{.}l}{13+x }&\multicolumn{1}{@{}l}{(29/2\ensuremath{^{+}})}&\multicolumn{1}{l}{[E1]}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{1292 {\it 19}}&\parbox[t][0.3cm]{6.1107807cm}{\raggedright B(E1)(W.u.)=7.3\ensuremath{\times}10\ensuremath{^{\textnormal{$-$7}}} \textit{+13{\textminus}16}\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{128}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{BI35GAMMA5}{a}}} {\it 1}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{2612}&\multicolumn{1}{@{.}l}{00+x }&\multicolumn{1}{@{}l}{(27/2)}&&&&&&\parbox[t][0.3cm]{6.1107807cm}{\raggedright E\ensuremath{_{\gamma}}: From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1982Br21,B}{1982Br21} in \ensuremath{^{\textnormal{203}}}Tl(\ensuremath{\alpha},6n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{150}&\multicolumn{1}{@{.}l}{5\ensuremath{^{\hyperlink{BI35GAMMA0}{\dagger}}} {\it 6}}&\multicolumn{1}{r@{}}{10}&\multicolumn{1}{@{.}l}{9\ensuremath{^{\hyperlink{BI35GAMMA0}{\dagger}}} {\it 7}}&\multicolumn{1}{r@{}}{2589}&\multicolumn{1}{@{.}l}{7+x }&&\multicolumn{1}{l}{D}&&&&&\parbox[t][0.3cm]{6.1107807cm}{\raggedright Mult.: From \ensuremath{\gamma}(\ensuremath{\theta}) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1985Pi05,B}{1985Pi05}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.1107807cm}{\raggedright {\ }{\ }{\ }(\ensuremath{^{\textnormal{196}}}Pt(\ensuremath{^{\textnormal{10}}}B,5n\ensuremath{\gamma})).\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{190}&\multicolumn{1}{@{.}l}{49\ensuremath{^{\hyperlink{BI35GAMMA0}{\dagger}}} {\it 25}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{.}l}{0\ensuremath{^{\hyperlink{BI35GAMMA0}{\dagger}}} {\it 12}}&\multicolumn{1}{r@{}}{2549}&\multicolumn{1}{@{.}l}{50+x }&\multicolumn{1}{@{}l}{(27/2\ensuremath{^{+}})}&\multicolumn{1}{l}{(E1)}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0901 {\it 13}}&\parbox[t][0.3cm]{6.1107807cm}{\raggedright B(E1)(W.u.)=8.0\ensuremath{\times}10\ensuremath{^{\textnormal{$-$8}}} \textit{+10{\textminus}8}\vspace{0.1cm}}&\\
\end{longtable}
\clearpage
\begin{longtable}{ccccccccc@{}ccccccc@{\extracolsep{\fill}}c}
\\[-.4cm]
\multicolumn{17}{c}{{\bf \small \underline{Adopted \hyperlink{201BI_LEVEL}{Levels}, \hyperlink{201BI_GAMMA}{Gammas} (continued)}}}\\
\multicolumn{17}{c}{~}\\
\multicolumn{17}{c}{\underline{$\gamma$($^{201}$Bi) (continued)}}\\
\multicolumn{17}{c}{~~~}\\
\multicolumn{2}{c}{E\ensuremath{_{i}}(level)}&J\ensuremath{^{\pi}_{i}}&\multicolumn{2}{c}{E\ensuremath{_{\gamma}}}&\multicolumn{2}{c}{I\ensuremath{_{\gamma}}}&\multicolumn{2}{c}{E\ensuremath{_{f}}}&J\ensuremath{^{\pi}_{f}}&Mult.&\multicolumn{2}{c}{\ensuremath{\delta}}&\multicolumn{2}{c}{\ensuremath{\alpha}\ensuremath{^{\hyperlink{BI35GAMMA4}{\&}}}}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endhead
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.201889cm}{\raggedright Mult.: From \ensuremath{\gamma}(\ensuremath{\theta}) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1985Pi05,B}{1985Pi05}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.201889cm}{\raggedright {\ }{\ }{\ }(\ensuremath{^{\textnormal{196}}}Pt(\ensuremath{^{\textnormal{10}}}B,5n\ensuremath{\gamma})) and \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1982Br21,B}{1982Br21}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.201889cm}{\raggedright {\ }{\ }{\ }(\ensuremath{^{\textnormal{203}}}Tl(\ensuremath{\alpha},6n\ensuremath{\gamma})).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2740}&\multicolumn{1}{@{.}l}{01+x}&\multicolumn{1}{l}{(29/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{440}&\multicolumn{1}{@{.}l}{9\ensuremath{^{\hyperlink{BI35GAMMA0}{\dagger}}} {\it 4}}&\multicolumn{1}{r@{}}{60}&\multicolumn{1}{@{.}l}{8\ensuremath{^{\hyperlink{BI35GAMMA0}{\dagger}}} {\it 14}}&\multicolumn{1}{r@{}}{2299}&\multicolumn{1}{@{.}l}{1+x }&\multicolumn{1}{@{}l}{(27/2\ensuremath{^{+}})}&\multicolumn{1}{l}{(E1)}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{01292 {\it 18}}&\parbox[t][0.3cm]{6.201889cm}{\raggedright B(E1)(W.u.)=3.91\ensuremath{\times}10\ensuremath{^{\textnormal{$-$9}}} \textit{+49{\textminus}41}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.201889cm}{\raggedright Mult.: From \ensuremath{\gamma}(\ensuremath{\theta}) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1985Pi05,B}{1985Pi05}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.201889cm}{\raggedright {\ }{\ }{\ }(\ensuremath{^{\textnormal{196}}}Pt(\ensuremath{^{\textnormal{10}}}B,5n\ensuremath{\gamma})).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2902}&\multicolumn{1}{@{.}l}{09}&\multicolumn{1}{l}{1/2\ensuremath{^{+}},3/2,5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1627}&\multicolumn{1}{@{.}l}{7\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 3}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 11}}&\multicolumn{1}{r@{}}{1274}&\multicolumn{1}{@{.}l}{45 }&\multicolumn{1}{@{}l}{(5/2)\ensuremath{^{+}}}&&&&&&&\\
&&&\multicolumn{1}{r@{}}{1815}&\multicolumn{1}{@{.}l}{8\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 4}}&\multicolumn{1}{r@{}}{50}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 9}}&\multicolumn{1}{r@{}}{1086}&\multicolumn{1}{@{.}l}{21 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}}}&&&&&&&\\
&&&\multicolumn{1}{r@{}}{2055}&\multicolumn{1}{@{.}l}{7\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 3}}&\multicolumn{1}{r@{}}{20}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 6}}&\multicolumn{1}{r@{}}{846}&\multicolumn{1}{@{.}l}{35 }&\multicolumn{1}{@{}l}{1/2\ensuremath{^{+}}}&&&&&&&\\
\multicolumn{1}{r@{}}{2905}&\multicolumn{1}{@{.}l}{9}&\multicolumn{1}{l}{1/2,3/2,5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1819}&\multicolumn{1}{@{.}l}{8\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 3}}&\multicolumn{1}{r@{}}{83}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 20}}&\multicolumn{1}{r@{}}{1086}&\multicolumn{1}{@{.}l}{21 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}}}&&&&&&&\\
&&&\multicolumn{1}{r@{}}{2059}&\multicolumn{1}{@{.}l}{4\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 3}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{BI35GAMMA1}{\ddagger}}} {\it 16}}&\multicolumn{1}{r@{}}{846}&\multicolumn{1}{@{.}l}{35 }&\multicolumn{1}{@{}l}{1/2\ensuremath{^{+}}}&&&&&&&\\
\multicolumn{1}{r@{}}{2994}&\multicolumn{1}{@{.}l}{7+x}&&\multicolumn{1}{r@{}}{1062}&\multicolumn{1}{@{.}l}{4\ensuremath{^{\hyperlink{BI35GAMMA0}{\dagger}}} {\it 8}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{BI35GAMMA0}{\dagger}}}}&\multicolumn{1}{r@{}}{1932}&\multicolumn{1}{@{.}l}{3+x }&\multicolumn{1}{@{}l}{(25/2\ensuremath{^{+}})}&&&&&&&\\
\multicolumn{1}{r@{}}{3011}&\multicolumn{1}{@{.}l}{5+x}&&\multicolumn{1}{r@{}}{421}&\multicolumn{1}{@{.}l}{8\ensuremath{^{\hyperlink{BI35GAMMA0}{\dagger}}} {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{BI35GAMMA0}{\dagger}}} {\it 16}}&\multicolumn{1}{r@{}}{2589}&\multicolumn{1}{@{.}l}{7+x }&&&&&&&&\\
&&&\multicolumn{1}{r@{}}{462}&\multicolumn{1}{@{.}l}{2\ensuremath{^{\hyperlink{BI35GAMMA0}{\dagger}}} {\it 8}}&\multicolumn{1}{r@{}}{60}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{BI35GAMMA0}{\dagger}}} {\it 16}}&\multicolumn{1}{r@{}}{2549}&\multicolumn{1}{@{.}l}{50+x }&\multicolumn{1}{@{}l}{(27/2\ensuremath{^{+}})}&&&&&&&\\
\multicolumn{1}{r@{}}{3238}&\multicolumn{1}{@{.}l}{9+x}&\multicolumn{1}{l}{(31/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{498}&\multicolumn{1}{@{.}l}{95\ensuremath{^{\hyperlink{BI35GAMMA0}{\dagger}\hyperlink{BI35GAMMA3}{@}}} {\it 25}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{BI35GAMMA0}{\dagger}}}}&\multicolumn{1}{r@{}}{2740}&\multicolumn{1}{@{.}l}{01+x }&\multicolumn{1}{@{}l}{(29/2\ensuremath{^{-}})}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{$-$0}&\multicolumn{1}{@{.}l}{33 {\it 11}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{109 {\it 6}}&\parbox[t][0.3cm]{6.201889cm}{\raggedright Mult.: From \ensuremath{\gamma}(\ensuremath{\theta}) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1985Pi05,B}{1985Pi05}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.201889cm}{\raggedright {\ }{\ }{\ }(\ensuremath{^{\textnormal{196}}}Pt(\ensuremath{^{\textnormal{10}}}B,5n\ensuremath{\gamma})) and \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1982Br21,B}{1982Br21}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.201889cm}{\raggedright {\ }{\ }{\ }(\ensuremath{^{\textnormal{203}}}Tl(\ensuremath{\alpha},6n\ensuremath{\gamma})).\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.201889cm}{\raggedright \ensuremath{\delta}: From \ensuremath{\gamma}(\ensuremath{\theta}) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1985Pi05,B}{1985Pi05}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.201889cm}{\raggedright {\ }{\ }{\ }(\ensuremath{^{\textnormal{196}}}Pt(\ensuremath{^{\textnormal{10}}}B,5n\ensuremath{\gamma})).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3422}&\multicolumn{1}{@{.}l}{9+x}&&\multicolumn{1}{r@{}}{428}&\multicolumn{1}{@{.}l}{2\ensuremath{^{\hyperlink{BI35GAMMA0}{\dagger}}} {\it 4}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{BI35GAMMA0}{\dagger}}}}&\multicolumn{1}{r@{}}{2994}&\multicolumn{1}{@{.}l}{7+x }&&&&&&&&\\
\multicolumn{1}{r@{}}{3526}&\multicolumn{1}{@{.}l}{5+x}&\multicolumn{1}{l}{(33/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{287}&\multicolumn{1}{@{.}l}{3\ensuremath{^{\hyperlink{BI35GAMMA0}{\dagger}}} {\it 4}}&\multicolumn{1}{r@{}}{26}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{BI35GAMMA0}{\dagger}}} {\it 4}}&\multicolumn{1}{r@{}}{3238}&\multicolumn{1}{@{.}l}{9+x }&\multicolumn{1}{@{}l}{(31/2\ensuremath{^{-}})}&\multicolumn{1}{l}{[M1]}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{522 {\it 8}}&&\\
&&&\multicolumn{1}{r@{}}{786}&\multicolumn{1}{@{.}l}{3\ensuremath{^{\hyperlink{BI35GAMMA0}{\dagger}\hyperlink{BI35GAMMA3}{@}}} {\it 3}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{.}l}{0\ensuremath{^{\hyperlink{BI35GAMMA0}{\dagger}}} {\it 14}}&\multicolumn{1}{r@{}}{2740}&\multicolumn{1}{@{.}l}{01+x }&\multicolumn{1}{@{}l}{(29/2\ensuremath{^{-}})}&\multicolumn{1}{l}{(E2)}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{01130 {\it 16}}&\parbox[t][0.3cm]{6.201889cm}{\raggedright Mult.: From \ensuremath{\gamma}(\ensuremath{\theta}) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1985Pi05,B}{1985Pi05}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.201889cm}{\raggedright {\ }{\ }{\ }(\ensuremath{^{\textnormal{196}}}Pt(\ensuremath{^{\textnormal{10}}}B,5n\ensuremath{\gamma})).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3592}&\multicolumn{1}{@{.}l}{3+x?}&&\multicolumn{1}{r@{}}{169}&\multicolumn{1}{@{.}l}{4\ensuremath{^{\hyperlink{BI35GAMMA0}{\dagger}}} {\it 4}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{BI35GAMMA0}{\dagger}}}}&\multicolumn{1}{r@{}}{3422}&\multicolumn{1}{@{.}l}{9+x }&&\multicolumn{1}{l}{M1(+E2)}&&&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{5 {\it 7}}&\parbox[t][0.3cm]{6.201889cm}{\raggedright Mult.: From \ensuremath{\gamma}(\ensuremath{\theta}) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1985Pi05,B}{1985Pi05}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.201889cm}{\raggedright {\ }{\ }{\ }(\ensuremath{^{\textnormal{196}}}Pt(\ensuremath{^{\textnormal{10}}}B,5n\ensuremath{\gamma})).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3638}&\multicolumn{1}{@{.}l}{4+x}&&\multicolumn{1}{r@{}}{987}&\multicolumn{1}{@{.}l}{3\ensuremath{^{\hyperlink{BI35GAMMA0}{\dagger}}} {\it 4}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{BI35GAMMA0}{\dagger}}}}&\multicolumn{1}{r@{}}{2651}&\multicolumn{1}{@{.}l}{13+x }&\multicolumn{1}{@{}l}{(29/2\ensuremath{^{+}})}&&&&&&&\\
\multicolumn{1}{r@{}}{3706}&\multicolumn{1}{@{.}l}{7+x}&&\multicolumn{1}{r@{}}{180}&\multicolumn{1}{@{.}l}{0\ensuremath{^{\hyperlink{BI35GAMMA0}{\dagger}}} {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{BI35GAMMA0}{\dagger}}} {\it 6}}&\multicolumn{1}{r@{}}{3526}&\multicolumn{1}{@{.}l}{5+x }&\multicolumn{1}{@{}l}{(33/2\ensuremath{^{-}})}&\multicolumn{1}{l}{D}&&&&&\parbox[t][0.3cm]{6.201889cm}{\raggedright Mult.: From \ensuremath{\gamma}(\ensuremath{\theta}) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1985Pi05,B}{1985Pi05}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.201889cm}{\raggedright {\ }{\ }{\ }(\ensuremath{^{\textnormal{196}}}Pt(\ensuremath{^{\textnormal{10}}}B,5n\ensuremath{\gamma})).\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{468}&\multicolumn{1}{@{.}l}{0\ensuremath{^{\hyperlink{BI35GAMMA0}{\dagger}}} {\it 4}}&\multicolumn{1}{r@{}}{98}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{BI35GAMMA0}{\dagger}}} {\it 49}}&\multicolumn{1}{r@{}}{3238}&\multicolumn{1}{@{.}l}{9+x }&\multicolumn{1}{@{}l}{(31/2\ensuremath{^{-}})}&&&&&&&\\
\multicolumn{1}{r@{}}{3727}&\multicolumn{1}{@{.}l}{61+x}&\multicolumn{1}{l}{(33/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{987}&\multicolumn{1}{@{.}l}{6 {\it 1}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{2740}&\multicolumn{1}{@{.}l}{01+x }&\multicolumn{1}{@{}l}{(29/2\ensuremath{^{-}})}&\multicolumn{1}{l}{E2}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{00717 {\it 10}}&\parbox[t][0.3cm]{6.201889cm}{\raggedright E\ensuremath{_{\gamma}},I\ensuremath{_{\gamma}}: From \ensuremath{^{\textnormal{203}}}Tl(\ensuremath{\alpha},6n\ensuremath{\gamma}).\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.201889cm}{\raggedright Mult.: From A\ensuremath{_{\textnormal{2}}}=+0.31 \textit{5}, A\ensuremath{_{\textnormal{4}}}={\textminus}0.01 \textit{5}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.201889cm}{\raggedright {\ }{\ }{\ }(\ensuremath{^{\textnormal{203}}}Tl(\ensuremath{\alpha},6n\ensuremath{\gamma})).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3810}&\multicolumn{1}{@{.}l}{8+x}&\multicolumn{1}{l}{(33/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{284}&\multicolumn{1}{@{.}l}{19\ensuremath{^{\hyperlink{BI35GAMMA0}{\dagger}}} {\it 25}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{.}l}{0\ensuremath{^{\hyperlink{BI35GAMMA0}{\dagger}}} {\it 22}}&\multicolumn{1}{r@{}}{3526}&\multicolumn{1}{@{.}l}{5+x }&\multicolumn{1}{@{}l}{(33/2\ensuremath{^{-}})}&\multicolumn{1}{l}{M1(+E2)}&\multicolumn{1}{r@{}}{$\leq$0}&\multicolumn{1}{@{.}l}{58}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{49 {\it 5}}&\parbox[t][0.3cm]{6.201889cm}{\raggedright Mult.,\ensuremath{\delta}: From \ensuremath{\gamma}(\ensuremath{\theta}) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1985Pi05,B}{1985Pi05}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.201889cm}{\raggedright {\ }{\ }{\ }(\ensuremath{^{\textnormal{196}}}Pt(\ensuremath{^{\textnormal{10}}}B,5n\ensuremath{\gamma})).\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{572}&\multicolumn{1}{@{.}l}{4\ensuremath{^{\hyperlink{BI35GAMMA0}{\dagger}}} {\it 4}}&\multicolumn{1}{r@{}}{19}&\multicolumn{1}{@{.}l}{6\ensuremath{^{\hyperlink{BI35GAMMA0}{\dagger}}} {\it 17}}&\multicolumn{1}{r@{}}{3238}&\multicolumn{1}{@{.}l}{9+x }&\multicolumn{1}{@{}l}{(31/2\ensuremath{^{-}})}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{$-$0}&\multicolumn{1}{@{.}l}{41 {\it 11}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{073 {\it 4}}&\parbox[t][0.3cm]{6.201889cm}{\raggedright Mult.,\ensuremath{\delta}: From \ensuremath{\gamma}(\ensuremath{\theta}) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1985Pi05,B}{1985Pi05}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.201889cm}{\raggedright {\ }{\ }{\ }(\ensuremath{^{\textnormal{196}}}Pt(\ensuremath{^{\textnormal{10}}}B,5n\ensuremath{\gamma})).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3922}&\multicolumn{1}{@{.}l}{9+x}&\multicolumn{1}{l}{(35/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{396}&\multicolumn{1}{@{.}l}{4\ensuremath{^{\hyperlink{BI35GAMMA0}{\dagger}}} {\it 4}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{BI35GAMMA0}{\dagger}}}}&\multicolumn{1}{r@{}}{3526}&\multicolumn{1}{@{.}l}{5+x }&\multicolumn{1}{@{}l}{(33/2\ensuremath{^{-}})}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{$-$0}&\multicolumn{1}{@{.}l}{24 {\it 11}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{209 {\it 9}}&\parbox[t][0.3cm]{6.201889cm}{\raggedright Mult.,\ensuremath{\delta}: From \ensuremath{\gamma}(\ensuremath{\theta}) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1985Pi05,B}{1985Pi05}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.201889cm}{\raggedright {\ }{\ }{\ }(\ensuremath{^{\textnormal{196}}}Pt(\ensuremath{^{\textnormal{10}}}B,5n\ensuremath{\gamma})).\vspace{0.1cm}}&\\
\end{longtable}
\clearpage
\begin{longtable}{ccccccccc@{}ccc@{\extracolsep{\fill}}c}
\\[-.4cm]
\multicolumn{13}{c}{{\bf \small \underline{Adopted \hyperlink{201BI_LEVEL}{Levels}, \hyperlink{201BI_GAMMA}{Gammas} (continued)}}}\\
\multicolumn{13}{c}{~}\\
\multicolumn{13}{c}{\underline{$\gamma$($^{201}$Bi) (continued)}}\\
\multicolumn{13}{c}{~~~}\\
\multicolumn{2}{c}{E\ensuremath{_{i}}(level)}&J\ensuremath{^{\pi}_{i}}&\multicolumn{2}{c}{E\ensuremath{_{\gamma}}}&\multicolumn{2}{c}{I\ensuremath{_{\gamma}}}&\multicolumn{2}{c}{E\ensuremath{_{f}}}&J\ensuremath{^{\pi}_{f}}&Mult.&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&\hrulefill&
\endhead
\multicolumn{1}{r@{}}{4075}&\multicolumn{1}{@{.}l}{3+x?}&&\multicolumn{1}{r@{}}{152}&\multicolumn{1}{@{.}l}{4\ensuremath{^{\hyperlink{BI35GAMMA0}{\dagger}}} {\it 4}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{BI35GAMMA0}{\dagger}}}}&\multicolumn{1}{r@{}}{3922}&\multicolumn{1}{@{.}l}{9+x }&\multicolumn{1}{@{}l}{(35/2\ensuremath{^{-}})}&\multicolumn{1}{l}{D}&\parbox[t][0.3cm]{14.18232cm}{\raggedright Mult.: From \ensuremath{\gamma}(\ensuremath{\theta}) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1985Pi05,B}{1985Pi05} (\ensuremath{^{\textnormal{196}}}Pt(\ensuremath{^{\textnormal{10}}}B,5n\ensuremath{\gamma})).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{4484}&\multicolumn{1}{@{.}l}{5+x}&&\multicolumn{1}{r@{}}{846}&\multicolumn{1}{@{.}l}{1\ensuremath{^{\hyperlink{BI35GAMMA0}{\dagger}}} {\it 7}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{BI35GAMMA0}{\dagger}}}}&\multicolumn{1}{r@{}}{3638}&\multicolumn{1}{@{.}l}{4+x }&&&&\\
\multicolumn{1}{r@{}}{5282}&\multicolumn{1}{@{.}l}{3+x}&&\multicolumn{1}{r@{}}{797}&\multicolumn{1}{@{.}l}{8\ensuremath{^{\hyperlink{BI35GAMMA0}{\dagger}}} {\it 4}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{BI35GAMMA0}{\dagger}}}}&\multicolumn{1}{r@{}}{4484}&\multicolumn{1}{@{.}l}{5+x }&&&&\\
\end{longtable}
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{BI35GAMMA0}{\dagger}}}} From \ensuremath{^{\textnormal{196}}}Pt(\ensuremath{^{\textnormal{10}}}B,5n\ensuremath{\gamma}).}\\
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{BI35GAMMA1}{\ddagger}}}} From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986Br28,B}{1986Br28} in \ensuremath{^{\textnormal{201}}}Po \ensuremath{\varepsilon} decay (15.6 min).}\\
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{BI35GAMMA2}{\#}}}} From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986Br28,B}{1986Br28} in \ensuremath{^{\textnormal{201}}}Po \ensuremath{\varepsilon} decay (8.96 min).}\\
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{BI35GAMMA3}{@}}}} Reported by \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1985Pi05,B}{1985Pi05} to have a delayed component with T\ensuremath{_{\textnormal{1/2}}}=14 ns \textit{3}. Similar delayed component (T\ensuremath{_{\textnormal{1/2}}}\ensuremath{>}10 ns) is reported in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1982Br21,B}{1982Br21} above the 3812+X}\\
\parbox[b][0.3cm]{21.881866cm}{{\ }{\ }keV level.}\\
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{BI35GAMMA4}{\&}}}} Total theoretical internal conversion coefficients, calculated using the BrIcc code (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2008Ki07,B}{2008Ki07}) with Frozen orbital approximation based on \ensuremath{\gamma}-ray energies,}\\
\parbox[b][0.3cm]{21.881866cm}{{\ }{\ }assigned multipolarities, and mixing ratios, unless otherwise specified.}\\
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{BI35GAMMA5}{a}}}} Placement of transition in the level scheme is uncertain.}\\
\vspace{0.5cm}
\end{landscape}\clearpage
\clearpage
\begin{figure}[h]
\begin{center}
\includegraphics{201BI35-0.ps}\\
\end{center}
\end{figure}
\clearpage
\begin{figure}[h]
\begin{center}
\includegraphics{201BI35-1.ps}\\
\end{center}
\end{figure}
\clearpage
\begin{figure}[h]
\begin{center}
\includegraphics{201BI35-2.ps}\\
\end{center}
\end{figure}
\clearpage
\subsection[\hspace{-0.2cm}\ensuremath{^{\textnormal{201}}}Po \ensuremath{\varepsilon} decay (15.50 min)]{ }
\vspace{-27pt}
\vspace{0.3cm}
\hypertarget{PO36}{{\bf \small \underline{\ensuremath{^{\textnormal{201}}}Po \ensuremath{\varepsilon} decay (15.50 min)\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986Br28,B}{1986Br28}}}}\\
\vspace{4pt}
\vspace{8pt}
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}Parent: $^{201}$Po: E=0; J$^{\pi}$=3/2\ensuremath{^{-}}; T$_{1/2}$=15.50 min {\it 14}; Q(\ensuremath{\varepsilon})=4908 {\it 13}; \%\ensuremath{\varepsilon}+\%\ensuremath{\beta^{+}} decay=100.0
}\\
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986Br28,B}{1986Br28}:\ensuremath{^{\textnormal{193}}}Ir(\ensuremath{^{\textnormal{14}}}N,6n), E=116 MeV, mass separated source; Detectors: Ge(Li) and cooled Si(Li); Measured: \ensuremath{\gamma}, \ensuremath{\gamma}\ensuremath{\gamma} coin, \ensuremath{\gamma}\ensuremath{\gamma}(t),}\\
\parbox[b][0.3cm]{17.7cm}{\ensuremath{\gamma}(x-ray)(t), ce, and T\ensuremath{_{\textnormal{1/2}}}.}\\
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}Others: \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986Be07,B}{1986Be07}, \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1980Br23,B}{1980Br23}, \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1976Ko13,B}{1976Ko13}, \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1970DaZM,B}{1970DaZM}, \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1970Jo26,B}{1970Jo26}, \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1969Al10,B}{1969Al10}.}\\
\vspace{12pt}
\underline{$^{201}$Bi Levels}\\
\begin{longtable}{cccccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E(level)$^{{\hyperlink{BI36LEVEL0}{\dagger}}}$}&J$^{\pi}$$^{{\hyperlink{BI36LEVEL1}{\ddagger}}}$&\multicolumn{2}{c}{T$_{1/2}$$^{{\hyperlink{BI36LEVEL1}{\ddagger}}}$}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{BI36LEVEL2}{\#}}}}}&\multicolumn{1}{l}{9/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{103}&\multicolumn{1}{@{.}l}{2 min {\it 17}}&&\\
\multicolumn{1}{r@{}}{846}&\multicolumn{1}{@{.}l}{35\ensuremath{^{{\hyperlink{BI36LEVEL3}{@}}}} {\it 18}}&\multicolumn{1}{l}{1/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{58}&\multicolumn{1}{@{.}l}{5 min {\it 6}}&\parbox[t][0.3cm]{11.34708cm}{\raggedright \%\ensuremath{\varepsilon}+\%\ensuremath{\beta}\ensuremath{^{\textnormal{+}}}=88.7 \textit{14}; \%IT=11.0 \textit{14}; \%\ensuremath{\alpha}\ensuremath{\approx}0.3\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{11.34708cm}{\raggedright \%IT determined by the evaluator from intensity balance considerations at the\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{11.34708cm}{\raggedright {\ }{\ }{\ }846.35-keV level, using I(\ensuremath{\gamma}+ce)(1/2\ensuremath{^{+}})=130 \textit{4} and I(\ensuremath{\varepsilon}+\ensuremath{\beta}\ensuremath{^{\textnormal{+}}})(1/2\ensuremath{^{+}})=20 \textit{5},\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{11.34708cm}{\raggedright {\ }{\ }{\ }determined from I(\ensuremath{\varepsilon}+\ensuremath{\beta}\ensuremath{^{\textnormal{+}}})(3/2\ensuremath{^{+}},1086-keV)=18 \textit{4} and by assuming log\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{11.34708cm}{\raggedright {\ }{\ }{\ }\textit{ft}(1/2\ensuremath{^{+}})=log \textit{ft}(3/2\ensuremath{^{+}},1086-keV). Other: \%IT=6.8\% in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1980Br23,B}{1980Br23} (same\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{11.34708cm}{\raggedright {\ }{\ }{\ }collaboration as \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986Br28,B}{1986Br28}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{890}&\multicolumn{1}{@{.}l}{24\ensuremath{^{{\hyperlink{BI36LEVEL4}{\&}}}} {\it 13}}&\multicolumn{1}{l}{5/2\ensuremath{^{-}}}&&&&\\
\multicolumn{1}{r@{}}{904}&\multicolumn{1}{@{.}l}{23\ensuremath{^{{\hyperlink{BI36LEVEL4}{\&}}}} {\it 12}}&\multicolumn{1}{l}{(7/2)\ensuremath{^{-}}}&&&&\\
\multicolumn{1}{r@{}}{1086}&\multicolumn{1}{@{.}l}{21\ensuremath{^{{\hyperlink{BI36LEVEL5}{a}}}} {\it 18}}&\multicolumn{1}{l}{3/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{260}&\multicolumn{1}{@{ }l}{ps {\it 30}}&\parbox[t][0.3cm]{11.34708cm}{\raggedright T\ensuremath{_{1/2}}: From 188.5ce-239.8ce(\ensuremath{\Delta}t) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986Be07,B}{1986Be07}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1186}&\multicolumn{1}{@{.}l}{59\ensuremath{^{{\hyperlink{BI36LEVEL6}{b}}}} {\it 17}}&\multicolumn{1}{l}{(7/2)\ensuremath{^{-}}}&&&&\\
\multicolumn{1}{r@{}}{1274}&\multicolumn{1}{@{.}l}{45 {\it 19}}&\multicolumn{1}{l}{(5/2)\ensuremath{^{+}}}&&&&\\
\multicolumn{1}{r@{}}{1441}&\multicolumn{1}{@{.}l}{71 {\it 18}}&\multicolumn{1}{l}{7/2\ensuremath{^{-}}}&&&&\\
\multicolumn{1}{r@{}}{1470}&\multicolumn{1}{@{.}l}{87 {\it 22}}&\multicolumn{1}{l}{(5/2,7/2)\ensuremath{^{-}}}&&&&\\
\multicolumn{1}{r@{}}{1483}&\multicolumn{1}{@{.}l}{54 {\it 24}}&\multicolumn{1}{l}{(3/2\ensuremath{^{-}})}&&&&\\
\multicolumn{1}{r@{}}{1616}&\multicolumn{1}{@{.}l}{3 {\it 4}}&\multicolumn{1}{l}{1/2\ensuremath{^{+}},3/2\ensuremath{^{+}},5/2\ensuremath{^{+}}}&&&&\\
\multicolumn{1}{r@{}}{1778}&\multicolumn{1}{@{.}l}{92 {\it 22}}&\multicolumn{1}{l}{(3/2\ensuremath{^{-}},5/2)}&&&&\\
\multicolumn{1}{r@{}}{1817}&\multicolumn{1}{@{.}l}{86 {\it 22}}&\multicolumn{1}{l}{1/2\ensuremath{^{+}},3/2,5/2\ensuremath{^{+}}}&&&&\\
\multicolumn{1}{r@{}}{1848}&\multicolumn{1}{@{.}l}{16 {\it 25}}&\multicolumn{1}{l}{(5/2\ensuremath{^{-}})}&&&&\\
\multicolumn{1}{r@{}}{1858}&\multicolumn{1}{@{.}l}{02 {\it 24}}&\multicolumn{1}{l}{3/2\ensuremath{^{+}}}&&&&\\
\multicolumn{1}{r@{}}{1927}&\multicolumn{1}{@{.}l}{32 {\it 19}}&\multicolumn{1}{l}{(5/2\ensuremath{^{-}})}&&&&\\
\multicolumn{1}{r@{}}{1944}&\multicolumn{1}{@{.}l}{24 {\it 17}}&\multicolumn{1}{l}{(5/2\ensuremath{^{-}},7/2)}&&&&\\
\multicolumn{1}{r@{}}{2053}&\multicolumn{1}{@{.}l}{59 {\it 21}}&\multicolumn{1}{l}{(5/2)\ensuremath{^{+}}}&&&&\\
\multicolumn{1}{r@{}}{2065}&\multicolumn{1}{@{.}l}{82 {\it 17}}&\multicolumn{1}{l}{5/2\ensuremath{^{+}}}&&&&\\
\multicolumn{1}{r@{}}{2386}&\multicolumn{1}{@{.}l}{7 {\it 5}}&&&&&\\
\multicolumn{1}{r@{}}{2422}&\multicolumn{1}{@{.}l}{1 {\it 3}}&\multicolumn{1}{l}{(3/2\ensuremath{^{-}},5/2)}&&&&\\
\multicolumn{1}{r@{}}{2434}&\multicolumn{1}{@{.}l}{9 {\it 3}}&\multicolumn{1}{l}{1/2\ensuremath{^{+}},3/2,5/2\ensuremath{^{+}}}&&&&\\
\multicolumn{1}{r@{}}{2455}&\multicolumn{1}{@{.}l}{5 {\it 3}}&\multicolumn{1}{l}{1/2\ensuremath{^{+}},3/2,5/2\ensuremath{^{+}}}&&&&\\
\multicolumn{1}{r@{}}{2484}&\multicolumn{1}{@{.}l}{3 {\it 3}}&\multicolumn{1}{l}{1/2,3/2,5/2\ensuremath{^{+}}}&&&&\\
\multicolumn{1}{r@{}}{2592}&\multicolumn{1}{@{.}l}{88 {\it 20}}&\multicolumn{1}{l}{(3/2\ensuremath{^{-}},5/2\ensuremath{^{+}})}&&&&\\
\multicolumn{1}{r@{}}{2902}&\multicolumn{1}{@{.}l}{09 {\it 25}}&\multicolumn{1}{l}{1/2\ensuremath{^{+}},3/2,5/2\ensuremath{^{+}}}&&&&\\
\multicolumn{1}{r@{}}{2905}&\multicolumn{1}{@{.}l}{9 {\it 3}}&\multicolumn{1}{l}{1/2,3/2,5/2\ensuremath{^{+}}}&&&&\\
\end{longtable}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{BI36LEVEL0}{\dagger}}}} From a least squares fit to E\ensuremath{\gamma}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{BI36LEVEL1}{\ddagger}}}} From Adopted Levels, unless otherwise stated.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{BI36LEVEL2}{\#}}}} Configuration=\ensuremath{\pi} h\ensuremath{_{\textnormal{9/2}}^{\textnormal{+1}}}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{BI36LEVEL3}{@}}}} Configuration=\ensuremath{\pi} s\ensuremath{_{\textnormal{1/2}}^{\textnormal{$-$1}}}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{BI36LEVEL4}{\&}}}} Configuration=\ensuremath{\pi} (h\ensuremath{_{\textnormal{9/2}}^{\textnormal{+1}}})\ensuremath{\otimes}2\ensuremath{^{\textnormal{+}}}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{BI36LEVEL5}{a}}}} Configuration=\ensuremath{\pi} d\ensuremath{_{\textnormal{3/2}}^{\textnormal{$-$1}}}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{BI36LEVEL6}{b}}}} Configuration=\ensuremath{\pi} f\ensuremath{_{\textnormal{7/2}}^{\textnormal{+1}}}.}\\
\vspace{0.5cm}
\clearpage
\vspace{0.3cm}
\vspace*{-0.5cm}
{\bf \small \underline{\ensuremath{^{\textnormal{201}}}Po \ensuremath{\varepsilon} decay (15.50 min)\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986Br28,B}{1986Br28} (continued)}}\\
\vspace{0.3cm}
\underline{\ensuremath{\varepsilon,\beta^+} radiations}\\
\vspace{0.34cm}
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.254cm}The I(\ensuremath{\beta}\ensuremath{^{\textnormal{+}}}+\ensuremath{\varepsilon}), I\ensuremath{\beta}, I\ensuremath{\varepsilon} and log{} \textit{ft} values are approximate, given the uncertain \%IT value for the 846 keV, \ensuremath{J^{\pi}}=1/2\ensuremath{^{+}} state.}\\
\vspace{0.34cm}
\begin{longtable}{ccccccccccccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E(decay)$$}&\multicolumn{2}{c}{E(level)}&\multicolumn{2}{c}{I$\beta^{+ \ {\hyperlink{BI36DECAY1}{\ddagger}}}$}&\multicolumn{2}{c}{I$\varepsilon^{{\hyperlink{BI36DECAY1}{\ddagger}}}$}&\multicolumn{2}{c}{Log {\it ft}$^{}$}&\multicolumn{2}{c}{I$(\varepsilon+\beta^{+})^{{\hyperlink{BI36DECAY0}{\dagger}}{\hyperlink{BI36DECAY1}{\ddagger}}}$}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{(2002}&\multicolumn{1}{@{ }l}{{\it 13})}&\multicolumn{1}{r@{}}{2905}&\multicolumn{1}{@{.}l}{9}&\multicolumn{1}{r@{}}{$\approx$0}&\multicolumn{1}{@{.}l}{0052}&\multicolumn{1}{r@{}}{$\approx$0}&\multicolumn{1}{@{.}l}{63}&\multicolumn{1}{r@{}}{$\approx$7}&\multicolumn{1}{@{.}l}{3}&\multicolumn{1}{r@{}}{$\approx$0}&\multicolumn{1}{@{.}l}{64}&&\\
\multicolumn{1}{r@{}}{(2006}&\multicolumn{1}{@{ }l}{{\it 13})}&\multicolumn{1}{r@{}}{2902}&\multicolumn{1}{@{.}l}{09}&\multicolumn{1}{r@{}}{$\approx$0}&\multicolumn{1}{@{.}l}{0114}&\multicolumn{1}{r@{}}{$\approx$1}&\multicolumn{1}{@{.}l}{39}&\multicolumn{1}{r@{}}{$\approx$7}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{r@{}}{$\approx$1}&\multicolumn{1}{@{.}l}{40}&&\\
\multicolumn{1}{r@{}}{(2315}&\multicolumn{1}{@{ }l}{{\it 13})}&\multicolumn{1}{r@{}}{2592}&\multicolumn{1}{@{.}l}{88}&\multicolumn{1}{r@{}}{$\approx$0}&\multicolumn{1}{@{.}l}{098}&\multicolumn{1}{r@{}}{$\approx$4}&\multicolumn{1}{@{.}l}{59}&\multicolumn{1}{r@{}}{$\approx$6}&\multicolumn{1}{@{.}l}{6}&\multicolumn{1}{r@{}}{$\approx$4}&\multicolumn{1}{@{.}l}{69}&&\\
\multicolumn{1}{r@{}}{(2424}&\multicolumn{1}{@{ }l}{{\it 13})}&\multicolumn{1}{r@{}}{2484}&\multicolumn{1}{@{.}l}{3}&\multicolumn{1}{r@{}}{$\approx$0}&\multicolumn{1}{@{.}l}{0680}&\multicolumn{1}{r@{}}{$\approx$2}&\multicolumn{1}{@{.}l}{44}&\multicolumn{1}{r@{}}{$\approx$6}&\multicolumn{1}{@{.}l}{9}&\multicolumn{1}{r@{}}{$\approx$2}&\multicolumn{1}{@{.}l}{51}&&\\
\multicolumn{1}{r@{}}{(2453}&\multicolumn{1}{@{ }l}{{\it 13})}&\multicolumn{1}{r@{}}{2455}&\multicolumn{1}{@{.}l}{5}&\multicolumn{1}{r@{}}{$\approx$0}&\multicolumn{1}{@{.}l}{0790}&\multicolumn{1}{r@{}}{$\approx$2}&\multicolumn{1}{@{.}l}{65}&\multicolumn{1}{r@{}}{$\approx$6}&\multicolumn{1}{@{.}l}{9}&\multicolumn{1}{r@{}}{$\approx$2}&\multicolumn{1}{@{.}l}{73}&&\\
\multicolumn{1}{r@{}}{(2473}&\multicolumn{1}{@{ }l}{{\it 13})}&\multicolumn{1}{r@{}}{2434}&\multicolumn{1}{@{.}l}{9}&\multicolumn{1}{r@{}}{$\approx$0}&\multicolumn{1}{@{.}l}{0624}&\multicolumn{1}{r@{}}{$\approx$2}&\multicolumn{1}{@{.}l}{00}&\multicolumn{1}{r@{}}{$\approx$7}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{r@{}}{$\approx$2}&\multicolumn{1}{@{.}l}{06}&&\\
\multicolumn{1}{r@{}}{(2486}&\multicolumn{1}{@{ }l}{{\it 13})}&\multicolumn{1}{r@{}}{2422}&\multicolumn{1}{@{.}l}{1}&\multicolumn{1}{r@{}}{$\approx$0}&\multicolumn{1}{@{.}l}{0620}&\multicolumn{1}{r@{}}{$\approx$1}&\multicolumn{1}{@{.}l}{93}&\multicolumn{1}{r@{}}{$\approx$7}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{r@{}}{$\approx$1}&\multicolumn{1}{@{.}l}{99}&&\\
\multicolumn{1}{r@{}}{(2521}&\multicolumn{1}{@{ }l}{{\it 13})}&\multicolumn{1}{r@{}}{2386}&\multicolumn{1}{@{.}l}{7}&\multicolumn{1}{r@{}}{$\approx$0}&\multicolumn{1}{@{.}l}{0410}&\multicolumn{1}{r@{}}{$\approx$1}&\multicolumn{1}{@{.}l}{18}&\multicolumn{1}{r@{}}{$\approx$7}&\multicolumn{1}{@{.}l}{3}&\multicolumn{1}{r@{}}{$\approx$1}&\multicolumn{1}{@{.}l}{22}&&\\
\multicolumn{1}{r@{}}{(2842}&\multicolumn{1}{@{ }l}{{\it 13})}&\multicolumn{1}{r@{}}{2065}&\multicolumn{1}{@{.}l}{82}&\multicolumn{1}{r@{}}{$\approx$0}&\multicolumn{1}{@{.}l}{658}&\multicolumn{1}{r@{}}{$\approx$10}&\multicolumn{1}{@{.}l}{1}&\multicolumn{1}{r@{}}{$\approx$6}&\multicolumn{1}{@{.}l}{4}&\multicolumn{1}{r@{}}{$\approx$10}&\multicolumn{1}{@{.}l}{8}&&\\
\multicolumn{1}{r@{}}{(2854}&\multicolumn{1}{@{ }l}{{\it 13})}&\multicolumn{1}{r@{}}{2053}&\multicolumn{1}{@{.}l}{59}&\multicolumn{1}{r@{}}{$\approx$0}&\multicolumn{1}{@{.}l}{34}&\multicolumn{1}{r@{}}{$\approx$5}&\multicolumn{1}{@{.}l}{2}&\multicolumn{1}{r@{}}{$\approx$6}&\multicolumn{1}{@{.}l}{7}&\multicolumn{1}{r@{}}{$\approx$5}&\multicolumn{1}{@{.}l}{5}&&\\
\multicolumn{1}{r@{}}{(2964}&\multicolumn{1}{@{ }l}{{\it 13})}&\multicolumn{1}{r@{}}{1944}&\multicolumn{1}{@{.}l}{24}&\multicolumn{1}{r@{}}{$\approx$0}&\multicolumn{1}{@{.}l}{179}&\multicolumn{1}{r@{}}{$\approx$2}&\multicolumn{1}{@{.}l}{25}&\multicolumn{1}{r@{}}{$\approx$7}&\multicolumn{1}{@{.}l}{1}&\multicolumn{1}{r@{}}{$\approx$2}&\multicolumn{1}{@{.}l}{43}&&\\
\multicolumn{1}{r@{}}{(2981}&\multicolumn{1}{@{ }l}{{\it 13})}&\multicolumn{1}{r@{}}{1927}&\multicolumn{1}{@{.}l}{32}&\multicolumn{1}{r@{}}{$\approx$0}&\multicolumn{1}{@{.}l}{0996}&\multicolumn{1}{r@{}}{$\approx$1}&\multicolumn{1}{@{.}l}{22}&\multicolumn{1}{r@{}}{$\approx$7}&\multicolumn{1}{@{.}l}{4}&\multicolumn{1}{r@{}}{$\approx$1}&\multicolumn{1}{@{.}l}{32}&&\\
\multicolumn{1}{r@{}}{(3050}&\multicolumn{1}{@{ }l}{{\it 13})}&\multicolumn{1}{r@{}}{1858}&\multicolumn{1}{@{.}l}{02}&\multicolumn{1}{r@{}}{$\approx$0}&\multicolumn{1}{@{.}l}{267}&\multicolumn{1}{r@{}}{$\approx$2}&\multicolumn{1}{@{.}l}{93}&\multicolumn{1}{r@{}}{$\approx$7}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{r@{}}{$\approx$3}&\multicolumn{1}{@{.}l}{20}&&\\
\multicolumn{1}{r@{}}{(3060}&\multicolumn{1}{@{ }l}{{\it 13})}&\multicolumn{1}{r@{}}{1848}&\multicolumn{1}{@{.}l}{16}&\multicolumn{1}{r@{}}{$\approx$0}&\multicolumn{1}{@{.}l}{115}&\multicolumn{1}{r@{}}{$\approx$1}&\multicolumn{1}{@{.}l}{25}&\multicolumn{1}{r@{}}{$\approx$7}&\multicolumn{1}{@{.}l}{4}&\multicolumn{1}{r@{}}{$\approx$1}&\multicolumn{1}{@{.}l}{36}&&\\
\multicolumn{1}{r@{}}{(3090}&\multicolumn{1}{@{ }l}{{\it 13})}&\multicolumn{1}{r@{}}{1817}&\multicolumn{1}{@{.}l}{86}&\multicolumn{1}{r@{}}{$\approx$0}&\multicolumn{1}{@{.}l}{337}&\multicolumn{1}{r@{}}{$\approx$3}&\multicolumn{1}{@{.}l}{48}&\multicolumn{1}{r@{}}{$\approx$7}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{r@{}}{$\approx$3}&\multicolumn{1}{@{.}l}{82}&&\\
\multicolumn{1}{r@{}}{(3129}&\multicolumn{1}{@{ }l}{{\it 13})}&\multicolumn{1}{r@{}}{1778}&\multicolumn{1}{@{.}l}{92}&\multicolumn{1}{r@{}}{$\approx$0}&\multicolumn{1}{@{.}l}{291}&\multicolumn{1}{r@{}}{$\approx$2}&\multicolumn{1}{@{.}l}{84}&\multicolumn{1}{r@{}}{$\approx$7}&\multicolumn{1}{@{.}l}{1}&\multicolumn{1}{r@{}}{$\approx$3}&\multicolumn{1}{@{.}l}{13}&&\\
\multicolumn{1}{r@{}}{(3292}&\multicolumn{1}{@{ }l}{{\it 13})}&\multicolumn{1}{r@{}}{1616}&\multicolumn{1}{@{.}l}{3}&\multicolumn{1}{r@{}}{$\approx$0}&\multicolumn{1}{@{.}l}{371}&\multicolumn{1}{r@{}}{$\approx$2}&\multicolumn{1}{@{.}l}{89}&\multicolumn{1}{r@{}}{$\approx$7}&\multicolumn{1}{@{.}l}{1}&\multicolumn{1}{r@{}}{$\approx$3}&\multicolumn{1}{@{.}l}{26}&&\\
\multicolumn{1}{r@{}}{(3424}&\multicolumn{1}{@{ }l}{{\it 13})}&\multicolumn{1}{r@{}}{1483}&\multicolumn{1}{@{.}l}{54}&\multicolumn{1}{r@{}}{$\approx$0}&\multicolumn{1}{@{.}l}{627}&\multicolumn{1}{r@{}}{$\approx$4}&\multicolumn{1}{@{.}l}{10}&\multicolumn{1}{r@{}}{$\approx$7}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{r@{}}{$\approx$4}&\multicolumn{1}{@{.}l}{73}&&\\
\multicolumn{1}{r@{}}{(3437}&\multicolumn{1}{@{ }l}{{\it 13})}&\multicolumn{1}{r@{}}{1470}&\multicolumn{1}{@{.}l}{87}&\multicolumn{1}{r@{}}{$\approx$0}&\multicolumn{1}{@{.}l}{242}&\multicolumn{1}{r@{}}{$\approx$1}&\multicolumn{1}{@{.}l}{56}&\multicolumn{1}{r@{}}{$\approx$7}&\multicolumn{1}{@{.}l}{4}&\multicolumn{1}{r@{}}{$\approx$1}&\multicolumn{1}{@{.}l}{80}&&\\
\multicolumn{1}{r@{}}{(3466}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{BI36DECAY2}{\#}}}} {\it 13})}&\multicolumn{1}{r@{}}{1441}&\multicolumn{1}{@{.}l}{71}&\multicolumn{1}{r@{}}{$\approx$0}&\multicolumn{1}{@{.}l}{563}&\multicolumn{1}{r@{}}{$\approx$3}&\multicolumn{1}{@{.}l}{50}&\multicolumn{1}{r@{}}{$\approx$7}&\multicolumn{1}{@{.}l}{1}&\multicolumn{1}{r@{}}{$\approx$4}&\multicolumn{1}{@{.}l}{06}&\parbox[t][0.3cm]{8.59986cm}{\raggedright I$(\varepsilon+\beta^{+})$: The existence of this decay branch is unlikely.\vspace{0.1cm}}&\\
&&&&&&&&&&&&\parbox[t][0.3cm]{8.59986cm}{\raggedright {\ }{\ }{\ }Imbalance is probably due to a missing de-exciting \ensuremath{\gamma}-ray\vspace{0.1cm}}&\\
&&&&&&&&&&&&\parbox[t][0.3cm]{8.59986cm}{\raggedright {\ }{\ }{\ }transition to the (7/2)\ensuremath{^{-}} level at 91186.59 keV.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{(3721}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{BI36DECAY2}{\#}}}} {\it 13})}&\multicolumn{1}{r@{}}{1186}&\multicolumn{1}{@{.}l}{59}&\multicolumn{1}{r@{}}{$\approx$1}&\multicolumn{1}{@{.}l}{2}&\multicolumn{1}{r@{}}{$\approx$5}&\multicolumn{1}{@{.}l}{5}&\multicolumn{1}{r@{}}{$\approx$6}&\multicolumn{1}{@{.}l}{9}&\multicolumn{1}{r@{}}{$\approx$6}&\multicolumn{1}{@{.}l}{7}&\parbox[t][0.3cm]{8.59986cm}{\raggedright I$(\varepsilon+\beta^{+})$: The existence of this decay branch is unlikely.\vspace{0.1cm}}&\\
&&&&&&&&&&&&\parbox[t][0.3cm]{8.59986cm}{\raggedright {\ }{\ }{\ }Imbalance is probably due to a missing de-exciting \ensuremath{\gamma}-ray\vspace{0.1cm}}&\\
&&&&&&&&&&&&\parbox[t][0.3cm]{8.59986cm}{\raggedright {\ }{\ }{\ }transition to the (7/2)\ensuremath{^{-}} level at 904.23 keV.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{(3822}&\multicolumn{1}{@{ }l}{{\it 13})}&\multicolumn{1}{r@{}}{1086}&\multicolumn{1}{@{.}l}{21}&\multicolumn{1}{r@{}}{$\approx$1}&\multicolumn{1}{@{.}l}{1}&\multicolumn{1}{r@{}}{$\approx$4}&\multicolumn{1}{@{.}l}{5}&\multicolumn{1}{r@{}}{$\approx$7}&\multicolumn{1}{@{.}l}{1}&\multicolumn{1}{r@{}}{$\approx$5}&\multicolumn{1}{@{.}l}{6}&&\\
\multicolumn{1}{r@{}}{(4004}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{BI36DECAY2}{\#}}}} {\it 13})}&\multicolumn{1}{r@{}}{904}&\multicolumn{1}{@{.}l}{23}&\multicolumn{1}{r@{}}{$\approx$1}&\multicolumn{1}{@{.}l}{8}&\multicolumn{1}{r@{}}{$\approx$6}&\multicolumn{1}{@{.}l}{3}&\multicolumn{1}{r@{}}{$\approx$6}&\multicolumn{1}{@{.}l}{9}&\multicolumn{1}{r@{}}{$\approx$8}&\multicolumn{1}{@{.}l}{1}&\parbox[t][0.3cm]{8.59986cm}{\raggedright I$(\varepsilon+\beta^{+})$: The existence of this decay branch is unlikely.\vspace{0.1cm}}&\\
&&&&&&&&&&&&\parbox[t][0.3cm]{8.59986cm}{\raggedright {\ }{\ }{\ }Imbalance is probably due to a missing de-exciting \ensuremath{\gamma}-ray\vspace{0.1cm}}&\\
&&&&&&&&&&&&\parbox[t][0.3cm]{8.59986cm}{\raggedright {\ }{\ }{\ }transition to the 5/2\ensuremath{^{-}} level at 890.24 keV.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{(4018}&\multicolumn{1}{@{ }l}{{\it 13})}&\multicolumn{1}{r@{}}{890}&\multicolumn{1}{@{.}l}{24}&\multicolumn{1}{r@{}}{$\approx$2}&\multicolumn{1}{@{.}l}{80}&\multicolumn{1}{r@{}}{$\approx$9}&\multicolumn{1}{@{.}l}{50}&\multicolumn{1}{r@{}}{$\approx$6}&\multicolumn{1}{@{.}l}{8}&\multicolumn{1}{r@{}}{$\approx$12}&\multicolumn{1}{@{.}l}{3}&&\\
\multicolumn{1}{r@{}}{(4062}&\multicolumn{1}{@{ }l}{{\it 13})}&\multicolumn{1}{r@{}}{846}&\multicolumn{1}{@{.}l}{35}&\multicolumn{1}{r@{}}{$\approx$1}&\multicolumn{1}{@{.}l}{4}&\multicolumn{1}{r@{}}{$\approx$4}&\multicolumn{1}{@{.}l}{7}&\multicolumn{1}{r@{}}{$\approx$7}&\multicolumn{1}{@{.}l}{1}&\multicolumn{1}{r@{}}{$\approx$6}&\multicolumn{1}{@{.}l}{1}&&\\
\end{longtable}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{BI36DECAY0}{\dagger}}}} Deduced from the decay scheme using intensity balances and by assuming no direct feeding to the g.s. (\ensuremath{J^{\pi}}=9/2\ensuremath{^{-}}).}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{BI36DECAY1}{\ddagger}}}} Absolute intensity per 100 decays.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{BI36DECAY2}{\#}}}} Existence of this branch is questionable.}\\
\vspace{0.5cm}
\clearpage
\vspace{0.3cm}
\begin{landscape}
\vspace*{-0.5cm}
{\bf \small \underline{\ensuremath{^{\textnormal{201}}}Po \ensuremath{\varepsilon} decay (15.50 min)\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986Br28,B}{1986Br28} (continued)}}\\
\vspace{0.3cm}
\underline{$\gamma$($^{201}$Bi)}\\
\vspace{0.34cm}
\parbox[b][0.3cm]{21.881866cm}{\addtolength{\parindent}{-0.254cm}I\ensuremath{\gamma} normalization: Using \ensuremath{\Sigma}I(\ensuremath{\gamma}+ce)(to g.s.)=100\% and by assuming that there is no direct feeding to the g.s. (\ensuremath{J^{\pi}}=9/2\ensuremath{^{-}}).}\\
\vspace{0.34cm}
\begin{longtable}{ccccccccc@{}ccccccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{PO36GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{I\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{PO36GAMMA0}{\dagger}\hyperlink{PO36GAMMA4}{\&}}}}&\multicolumn{2}{c}{E\ensuremath{_{i}}(level)}&J\ensuremath{^{\pi}_{i}}&\multicolumn{2}{c}{E\ensuremath{_{f}}}&J\ensuremath{^{\pi}_{f}}&Mult.\ensuremath{^{\hyperlink{PO36GAMMA1}{\ddagger}}}&\multicolumn{2}{c}{\ensuremath{\delta}\ensuremath{^{\hyperlink{PO36GAMMA2}{\#}}}}&\multicolumn{2}{c}{\ensuremath{\alpha}\ensuremath{^{\hyperlink{PO36GAMMA5}{a}}}}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{188}&\multicolumn{1}{@{.}l}{6 {\it 3}}&\multicolumn{1}{r@{}}{7}&\multicolumn{1}{@{.}l}{1 {\it 3}}&\multicolumn{1}{r@{}}{1274}&\multicolumn{1}{@{.}l}{45}&\multicolumn{1}{l}{(5/2)\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1086}&\multicolumn{1}{@{.}l}{21 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}}}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{53 {\it 8}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{43 {\it 6}}&\parbox[t][0.3cm]{7.5436707cm}{\raggedright \%I\ensuremath{\gamma}=2.16 \textit{10}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.5436707cm}{\raggedright Mult.,\ensuremath{\delta}: \ensuremath{\alpha}(K)exp=1.13 \textit{15} and K/L=4.5 \textit{3} in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986Br28,B}{1986Br28}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{195}&\multicolumn{1}{@{.}l}{9 {\it 3}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{39 {\it 6}}&\multicolumn{1}{r@{}}{1086}&\multicolumn{1}{@{.}l}{21}&\multicolumn{1}{l}{3/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{890}&\multicolumn{1}{@{.}l}{24 }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{l}{E1}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0841 {\it 12}}&\parbox[t][0.3cm]{7.5436707cm}{\raggedright \%I\ensuremath{\gamma}=0.119 \textit{18}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.5436707cm}{\raggedright Mult.: \ensuremath{\alpha}(K)exp\ensuremath{<}0.01 in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986Br28,B}{1986Br28}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{240}&\multicolumn{1}{@{.}l}{1 {\it 2}}&\multicolumn{1}{r@{}}{65}&\multicolumn{1}{@{.}l}{3 {\it 23}}&\multicolumn{1}{r@{}}{1086}&\multicolumn{1}{@{.}l}{21}&\multicolumn{1}{l}{3/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{846}&\multicolumn{1}{@{.}l}{35 }&\multicolumn{1}{@{}l}{1/2\ensuremath{^{+}}}&\multicolumn{1}{l}{E2+M1}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{0 {\it +4\textminus3}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{303 {\it 13}}&\parbox[t][0.3cm]{7.5436707cm}{\raggedright \%I\ensuremath{\gamma}=19.8 \textit{8}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.5436707cm}{\raggedright Mult.,\ensuremath{\delta}: From K/L=1.75 \textit{15} in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986Be07,B}{1986Be07};\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.5436707cm}{\raggedright {\ }{\ }{\ }\ensuremath{\alpha}(K)exp=0.152 \textit{24}, K/L=1.68 \textit{6} and L/M=4.1 \textit{4} in\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.5436707cm}{\raggedright {\ }{\ }{\ }\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986Br28,B}{1986Br28}; Other: \ensuremath{\alpha}(K)exp=0.55 \textit{25} in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1976Ko13,B}{1976Ko13}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{296}&\multicolumn{1}{@{.}l}{1 {\it 3}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{78 {\it 15}}&\multicolumn{1}{r@{}}{1186}&\multicolumn{1}{@{.}l}{59}&\multicolumn{1}{l}{(7/2)\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{890}&\multicolumn{1}{@{.}l}{24 }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{l}{[M1]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{481 {\it 7}}&\parbox[t][0.3cm]{7.5436707cm}{\raggedright \%I\ensuremath{\gamma}=0.54 \textit{5}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{428}&\multicolumn{1}{@{.}l}{2 {\it 3}}&\multicolumn{1}{r@{}}{12}&\multicolumn{1}{@{.}l}{7 {\it 5}}&\multicolumn{1}{r@{}}{1274}&\multicolumn{1}{@{.}l}{45}&\multicolumn{1}{l}{(5/2)\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{846}&\multicolumn{1}{@{.}l}{35 }&\multicolumn{1}{@{}l}{1/2\ensuremath{^{+}}}&\multicolumn{1}{l}{(E2)}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0451 {\it 6}}&\parbox[t][0.3cm]{7.5436707cm}{\raggedright \%I\ensuremath{\gamma}=3.86 \textit{18}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.5436707cm}{\raggedright Mult.: \ensuremath{\alpha}(K)exp=0.038 \textit{7} in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986Br28,B}{1986Br28}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$506}&\multicolumn{1}{@{.}l}{8 {\it 4}}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{96 {\it 22}}&&&&&&&&&&&&\parbox[t][0.3cm]{7.5436707cm}{\raggedright \%I\ensuremath{\gamma}=1.20 \textit{7}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$516}&\multicolumn{1}{@{.}l}{7 {\it 5}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{97 {\it 17}}&&&&&&&&&&&&\parbox[t][0.3cm]{7.5436707cm}{\raggedright \%I\ensuremath{\gamma}=0.60 \textit{5}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{530}&\multicolumn{1}{@{.}l}{1 {\it 3}}&\multicolumn{1}{r@{}}{10}&\multicolumn{1}{@{.}l}{2 {\it 4}}&\multicolumn{1}{r@{}}{1616}&\multicolumn{1}{@{.}l}{3}&\multicolumn{1}{l}{1/2\ensuremath{^{+}},3/2\ensuremath{^{+}},5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1086}&\multicolumn{1}{@{.}l}{21 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}}}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{4 {\it +5\textminus3}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{052 {\it 9}}&\parbox[t][0.3cm]{7.5436707cm}{\raggedright \%I\ensuremath{\gamma}=3.10 \textit{14}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.5436707cm}{\raggedright Mult.,\ensuremath{\delta}: \ensuremath{\alpha}(K)exp=0.040 \textit{8} in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986Br28,B}{1986Br28}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{537}&\multicolumn{1}{@{.}l}{5 {\it 2}}&\multicolumn{1}{r@{}}{4}&\multicolumn{1}{@{.}l}{3\ensuremath{^{\hyperlink{PO36GAMMA3}{@}}} {\it 5}}&\multicolumn{1}{r@{}}{1441}&\multicolumn{1}{@{.}l}{71}&\multicolumn{1}{l}{7/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{904}&\multicolumn{1}{@{.}l}{23 }&\multicolumn{1}{@{}l}{(7/2)\ensuremath{^{-}}}&\multicolumn{1}{l}{M1}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0968 {\it 14}}&\parbox[t][0.3cm]{7.5436707cm}{\raggedright \%I\ensuremath{\gamma}=1.31 \textit{15}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.5436707cm}{\raggedright Mult.: \ensuremath{\alpha}(K)exp=0.05 \textit{3} in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1976Ko13,B}{1976Ko13}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{543}&\multicolumn{1}{@{.}l}{4 {\it 3}}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{48 {\it 21}}&\multicolumn{1}{r@{}}{1817}&\multicolumn{1}{@{.}l}{86}&\multicolumn{1}{l}{1/2\ensuremath{^{+}},3/2,5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1274}&\multicolumn{1}{@{.}l}{45 }&\multicolumn{1}{@{}l}{(5/2)\ensuremath{^{+}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{7.5436707cm}{\raggedright \%I\ensuremath{\gamma}=1.06 \textit{7}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{551}&\multicolumn{1}{@{.}l}{9 {\it 3}}&\multicolumn{1}{r@{}}{7}&\multicolumn{1}{@{.}l}{4 {\it 3}}&\multicolumn{1}{r@{}}{1441}&\multicolumn{1}{@{.}l}{71}&\multicolumn{1}{l}{7/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{890}&\multicolumn{1}{@{.}l}{24 }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1(+E2)}&\multicolumn{1}{r@{}}{$<$0}&\multicolumn{1}{@{.}l}{7}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{079 {\it 11}}&\parbox[t][0.3cm]{7.5436707cm}{\raggedright \%I\ensuremath{\gamma}=2.25 \textit{10}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.5436707cm}{\raggedright Mult.,\ensuremath{\delta}: \ensuremath{\alpha}(K)exp=0.074 \textit{17} in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986Br28,B}{1986Br28}; Other:\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.5436707cm}{\raggedright {\ }{\ }{\ }\ensuremath{\alpha}(K)exp=0.07 \textit{3} in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1976Ko13,B}{1976Ko13}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{566}&\multicolumn{1}{@{.}l}{6 {\it 3}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{41 {\it 18}}&\multicolumn{1}{r@{}}{1470}&\multicolumn{1}{@{.}l}{87}&\multicolumn{1}{l}{(5/2,7/2)\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{904}&\multicolumn{1}{@{.}l}{23 }&\multicolumn{1}{@{}l}{(7/2)\ensuremath{^{-}}}&\multicolumn{1}{l}{M1(+E2)}&\multicolumn{1}{r@{}}{$<$0}&\multicolumn{1}{@{.}l}{9}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{071 {\it 14}}&\parbox[t][0.3cm]{7.5436707cm}{\raggedright \%I\ensuremath{\gamma}=0.43 \textit{6}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.5436707cm}{\raggedright Mult.,\ensuremath{\delta}: \ensuremath{\alpha}(K)exp=0.079 \textit{32} in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986Br28,B}{1986Br28}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{583}&\multicolumn{1}{@{.}l}{6 {\it 3}}&\multicolumn{1}{r@{}}{5}&\multicolumn{1}{@{.}l}{3 {\it 3}}&\multicolumn{1}{r@{}}{1858}&\multicolumn{1}{@{.}l}{02}&\multicolumn{1}{l}{3/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1274}&\multicolumn{1}{@{.}l}{45 }&\multicolumn{1}{@{}l}{(5/2)\ensuremath{^{+}}}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{0 {\it +10\textminus5}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{050 {\it 17}}&\parbox[t][0.3cm]{7.5436707cm}{\raggedright \%I\ensuremath{\gamma}=1.61 \textit{10}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.5436707cm}{\raggedright Mult.,\ensuremath{\delta}: \ensuremath{\alpha}(K)exp=0.040 \textit{15} in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986Br28,B}{1986Br28}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{593}&\multicolumn{1}{@{.}l}{3 {\it 2}}&\multicolumn{1}{r@{}}{14}&\multicolumn{1}{@{.}l}{5 {\it 6}}&\multicolumn{1}{r@{}}{1483}&\multicolumn{1}{@{.}l}{54}&\multicolumn{1}{l}{(3/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{890}&\multicolumn{1}{@{.}l}{24 }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1(+E2)}&\multicolumn{1}{r@{}}{$<$0}&\multicolumn{1}{@{.}l}{33}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0720 {\it 28}}&\parbox[t][0.3cm]{7.5436707cm}{\raggedright \%I\ensuremath{\gamma}=4.41 \textit{21}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.5436707cm}{\raggedright Mult.,\ensuremath{\delta}: \ensuremath{\alpha}(K)exp=0.072 \textit{11} in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986Br28,B}{1986Br28}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{624}&\multicolumn{1}{@{.}l}{7 {\it 3}}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{44 {\it 17}}&\multicolumn{1}{r@{}}{2065}&\multicolumn{1}{@{.}l}{82}&\multicolumn{1}{l}{5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1441}&\multicolumn{1}{@{.}l}{71 }&\multicolumn{1}{@{}l}{7/2\ensuremath{^{-}}}&\multicolumn{1}{l}{[E1]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{00631 {\it 9}}&\parbox[t][0.3cm]{7.5436707cm}{\raggedright \%I\ensuremath{\gamma}=0.74 \textit{5}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{636}&\multicolumn{1}{@{.}l}{5\ensuremath{^{\hyperlink{PO36GAMMA6}{b}}} {\it 2}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{75 {\it 21}}&\multicolumn{1}{r@{}}{1483}&\multicolumn{1}{@{.}l}{54}&\multicolumn{1}{l}{(3/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{846}&\multicolumn{1}{@{.}l}{35 }&\multicolumn{1}{@{}l}{1/2\ensuremath{^{+}}}&\multicolumn{1}{l}{[E1]}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{00609 {\it 9}}&\parbox[t][0.3cm]{7.5436707cm}{\raggedright \%I\ensuremath{\gamma}=0.53 \textit{6}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$650}&\multicolumn{1}{@{.}l}{9 {\it 3}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{41 {\it 18}}&&&&&&&&&&&&\parbox[t][0.3cm]{7.5436707cm}{\raggedright \%I\ensuremath{\gamma}=0.43 \textit{6}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{731}&\multicolumn{1}{@{.}l}{7 {\it 2}}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{6 {\it 3}}&\multicolumn{1}{r@{}}{1817}&\multicolumn{1}{@{.}l}{86}&\multicolumn{1}{l}{1/2\ensuremath{^{+}},3/2,5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1086}&\multicolumn{1}{@{.}l}{21 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{7.5436707cm}{\raggedright \%I\ensuremath{\gamma}=1.09 \textit{9}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{771}&\multicolumn{1}{@{.}l}{8 {\it 2}}&\multicolumn{1}{r@{}}{4}&\multicolumn{1}{@{.}l}{5 {\it 3}}&\multicolumn{1}{r@{}}{1858}&\multicolumn{1}{@{.}l}{02}&\multicolumn{1}{l}{3/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1086}&\multicolumn{1}{@{.}l}{21 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}}}&\multicolumn{1}{l}{E0+M1}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{10 {\it 3}}&\parbox[t][0.3cm]{7.5436707cm}{\raggedright \%I\ensuremath{\gamma}=1.37 \textit{10}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.5436707cm}{\raggedright Mult.: \ensuremath{\alpha}(K)exp=0.079 \textit{27} in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986Br28,B}{1986Br28}.\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.5436707cm}{\raggedright \ensuremath{\alpha}: From \ensuremath{\alpha}(K)exp in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986Br28,B}{1986Br28} and by assuming 10\%\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.5436707cm}{\raggedright {\ }{\ }{\ }contribution from \ensuremath{\alpha}(L).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{779}&\multicolumn{1}{@{.}l}{4 {\it 3}}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{8 {\it 3}}&\multicolumn{1}{r@{}}{2053}&\multicolumn{1}{@{.}l}{59}&\multicolumn{1}{l}{(5/2)\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1274}&\multicolumn{1}{@{.}l}{45 }&\multicolumn{1}{@{}l}{(5/2)\ensuremath{^{+}}}&\multicolumn{1}{l}{(M1)}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0366 {\it 6}}&\parbox[t][0.3cm]{7.5436707cm}{\raggedright \%I\ensuremath{\gamma}=1.15 \textit{9}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.5436707cm}{\raggedright Mult.: \ensuremath{\alpha}(K)exp=0.11 \textit{5} in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986Br28,B}{1986Br28}; E0 contribution\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.5436707cm}{\raggedright {\ }{\ }{\ }is possible.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{791}&\multicolumn{1}{@{.}l}{4 {\it 2}}&\multicolumn{1}{r@{}}{14}&\multicolumn{1}{@{.}l}{1 {\it 6}}&\multicolumn{1}{r@{}}{2065}&\multicolumn{1}{@{.}l}{82}&\multicolumn{1}{l}{5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1274}&\multicolumn{1}{@{.}l}{45 }&\multicolumn{1}{@{}l}{(5/2)\ensuremath{^{+}}}&\multicolumn{1}{l}{M1(+E2)}&\multicolumn{1}{r@{}}{$\leq$0}&\multicolumn{1}{@{.}l}{4}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0335 {\it 17}}&\parbox[t][0.3cm]{7.5436707cm}{\raggedright \%I\ensuremath{\gamma}=4.28 \textit{21}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.5436707cm}{\raggedright Mult.,\ensuremath{\delta}: \ensuremath{\alpha}(K)exp=0.033 \textit{6} in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986Br28,B}{1986Br28}.\vspace{0.1cm}}&\\
\end{longtable}
\clearpage
\begin{longtable}{ccccccccc@{}ccccccccc@{\extracolsep{\fill}}c}
\\[-.4cm]
\multicolumn{19}{c}{{\bf \small \underline{\ensuremath{^{\textnormal{201}}}Po \ensuremath{\varepsilon} decay (15.50 min)\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986Br28,B}{1986Br28} (continued)}}}\\
\multicolumn{19}{c}{~}\\
\multicolumn{19}{c}{\underline{$\gamma$($^{201}$Bi) (continued)}}\\
\multicolumn{19}{c}{~~~}\\
\multicolumn{2}{c}{E\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{PO36GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{I\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{PO36GAMMA0}{\dagger}\hyperlink{PO36GAMMA4}{\&}}}}&\multicolumn{2}{c}{E\ensuremath{_{i}}(level)}&J\ensuremath{^{\pi}_{i}}&\multicolumn{2}{c}{E\ensuremath{_{f}}}&J\ensuremath{^{\pi}_{f}}&Mult.\ensuremath{^{\hyperlink{PO36GAMMA1}{\ddagger}}}&\multicolumn{2}{c}{\ensuremath{\delta}\ensuremath{^{\hyperlink{PO36GAMMA2}{\#}}}}&\multicolumn{2}{c}{\ensuremath{\alpha}\ensuremath{^{\hyperlink{PO36GAMMA5}{a}}}}&\multicolumn{2}{c}{I\ensuremath{_{(\gamma+ce)}}\ensuremath{^{\hyperlink{PO36GAMMA4}{\&}}}}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endhead
\multicolumn{1}{r@{}}{$^{x}$809}&\multicolumn{1}{@{.}l}{9 {\it 4}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{63 {\it 21}}&&&&&&&&&&&&&&\parbox[t][0.3cm]{5.55793cm}{\raggedright \%I\ensuremath{\gamma}=0.50 \textit{6}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{846}&\multicolumn{1}{@{.}l}{3 {\it 3}}&\multicolumn{1}{r@{}}{12}&\multicolumn{1}{@{.}l}{8 {\it 15}}&\multicolumn{1}{r@{}}{846}&\multicolumn{1}{@{.}l}{35}&\multicolumn{1}{l}{1/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{9/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M4}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{292 {\it 4}}&\multicolumn{1}{r@{}}{150}&\multicolumn{1}{@{ }l}{{\it 6}}&\parbox[t][0.3cm]{5.55793cm}{\raggedright \%I\ensuremath{\gamma}=35.3 \textit{9}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&&&\parbox[t][0.3cm]{5.55793cm}{\raggedright I\ensuremath{_{\gamma}}: Corrected for equilibrium in\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&&&\parbox[t][0.3cm]{5.55793cm}{\raggedright {\ }{\ }{\ }\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986Br28,B}{1986Br28}.\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&&&\parbox[t][0.3cm]{5.55793cm}{\raggedright Mult.: \ensuremath{\alpha}(K)exp=0.20 \textit{3} and K/L=3.1 \textit{4}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&&&\parbox[t][0.3cm]{5.55793cm}{\raggedright {\ }{\ }{\ }in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1980Br23,B}{1980Br23}, where the 856\ensuremath{\gamma} was\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&&&\parbox[t][0.3cm]{5.55793cm}{\raggedright {\ }{\ }{\ }found to be doublet; Other:\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&&&\parbox[t][0.3cm]{5.55793cm}{\raggedright {\ }{\ }{\ }\ensuremath{\alpha}(K)exp=0.13 \textit{4} and K/L=2.3 \textit{4} in\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&&&\parbox[t][0.3cm]{5.55793cm}{\raggedright {\ }{\ }{\ }\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1969Al10,B}{1969Al10}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{874}&\multicolumn{1}{@{.}l}{6 {\it 2}}&\multicolumn{1}{r@{}}{4}&\multicolumn{1}{@{.}l}{8 {\it 3}}&\multicolumn{1}{r@{}}{1778}&\multicolumn{1}{@{.}l}{92}&\multicolumn{1}{l}{(3/2\ensuremath{^{-}},5/2)}&\multicolumn{1}{r@{}}{904}&\multicolumn{1}{@{.}l}{23 }&\multicolumn{1}{@{}l}{(7/2)\ensuremath{^{-}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{5.55793cm}{\raggedright \%I\ensuremath{\gamma}=1.46 \textit{10}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{889}&\multicolumn{1}{@{.}l}{2 {\it 5}}&\multicolumn{1}{r@{}}{5}&\multicolumn{1}{@{.}l}{5\ensuremath{^{\hyperlink{PO36GAMMA3}{@}}} {\it 6}}&\multicolumn{1}{r@{}}{1778}&\multicolumn{1}{@{.}l}{92}&\multicolumn{1}{l}{(3/2\ensuremath{^{-}},5/2)}&\multicolumn{1}{r@{}}{890}&\multicolumn{1}{@{.}l}{24 }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{-}}}&&&&&&&&\parbox[t][0.3cm]{5.55793cm}{\raggedright \%I\ensuremath{\gamma}=1.67 \textit{19}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{890}&\multicolumn{1}{@{.}l}{1 {\it 2}}&\multicolumn{1}{r@{}}{92}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PO36GAMMA3}{@}}} {\it 4}}&\multicolumn{1}{r@{}}{890}&\multicolumn{1}{@{.}l}{24}&\multicolumn{1}{l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{9/2\ensuremath{^{-}}}&\multicolumn{1}{l}{E2}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{00880 {\it 12}}&&&\parbox[t][0.3cm]{5.55793cm}{\raggedright \%I\ensuremath{\gamma}=28.0 \textit{10}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&&&\parbox[t][0.3cm]{5.55793cm}{\raggedright Mult.: \ensuremath{\alpha}(K)exp=0.0056 \textit{8} in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986Br28,B}{1986Br28};\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&&&\parbox[t][0.3cm]{5.55793cm}{\raggedright {\ }{\ }{\ }Other: \ensuremath{\alpha}(K)exp=0.030 \textit{15} in\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&&&\parbox[t][0.3cm]{5.55793cm}{\raggedright {\ }{\ }{\ }\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1976Ko13,B}{1976Ko13}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{904}&\multicolumn{1}{@{.}l}{2 {\it 2}}&\multicolumn{1}{r@{}}{50}&\multicolumn{1}{@{.}l}{4 {\it 20}}&\multicolumn{1}{r@{}}{904}&\multicolumn{1}{@{.}l}{23}&\multicolumn{1}{l}{(7/2)\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{9/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{5 {\it 4}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{022 {\it 4}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{5.55793cm}{\raggedright \%I\ensuremath{\gamma}=15.3 \textit{6}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&&&\parbox[t][0.3cm]{5.55793cm}{\raggedright Mult.,\ensuremath{\delta}: \ensuremath{\alpha}(K)exp=0.018 \textit{3} in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986Br28,B}{1986Br28};\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&&&\parbox[t][0.3cm]{5.55793cm}{\raggedright {\ }{\ }{\ }Other: \ensuremath{\alpha}(K)exp=0.036 \textit{15} in\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&&&\parbox[t][0.3cm]{5.55793cm}{\raggedright {\ }{\ }{\ }\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1976Ko13,B}{1976Ko13}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$918}&\multicolumn{1}{@{.}l}{7 {\it 3}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{46 {\it 21}}&&&&&&&&&&&&&&\parbox[t][0.3cm]{5.55793cm}{\raggedright \%I\ensuremath{\gamma}=0.44 \textit{6}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$926}&\multicolumn{1}{@{.}l}{5 {\it 3}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{74 {\it 18}}&&&&&&&&&&&&&&\parbox[t][0.3cm]{5.55793cm}{\raggedright \%I\ensuremath{\gamma}=0.53 \textit{6}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{944}&\multicolumn{1}{@{.}l}{2 {\it 4}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{57 {\it 22}}&\multicolumn{1}{r@{}}{1848}&\multicolumn{1}{@{.}l}{16}&\multicolumn{1}{l}{(5/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{904}&\multicolumn{1}{@{.}l}{23 }&\multicolumn{1}{@{}l}{(7/2)\ensuremath{^{-}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{5.55793cm}{\raggedright \%I\ensuremath{\gamma}=0.48 \textit{7}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{967}&\multicolumn{1}{@{.}l}{4 {\it 2}}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{1\ensuremath{^{\hyperlink{PO36GAMMA3}{@}}} {\it 4}}&\multicolumn{1}{r@{}}{2053}&\multicolumn{1}{@{.}l}{59}&\multicolumn{1}{l}{(5/2)\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1086}&\multicolumn{1}{@{.}l}{21 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}}}&\multicolumn{1}{l}{(E2)}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{00746 {\it 10}}&&&\parbox[t][0.3cm]{5.55793cm}{\raggedright \%I\ensuremath{\gamma}=0.94 \textit{12}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&&&\parbox[t][0.3cm]{5.55793cm}{\raggedright Mult.: \ensuremath{\alpha}(K)exp=0.0058 \textit{11} in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986Br28,B}{1986Br28}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{971}&\multicolumn{1}{@{.}l}{4 {\it 3}}&\multicolumn{1}{r@{}}{5}&\multicolumn{1}{@{.}l}{5 {\it 3}}&\multicolumn{1}{r@{}}{1817}&\multicolumn{1}{@{.}l}{86}&\multicolumn{1}{l}{1/2\ensuremath{^{+}},3/2,5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{846}&\multicolumn{1}{@{.}l}{35 }&\multicolumn{1}{@{}l}{1/2\ensuremath{^{+}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{5.55793cm}{\raggedright \%I\ensuremath{\gamma}=1.67 \textit{10}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{979}&\multicolumn{1}{@{.}l}{7 {\it 3}}&\multicolumn{1}{r@{}}{4}&\multicolumn{1}{@{.}l}{5 {\it 3}}&\multicolumn{1}{r@{}}{2065}&\multicolumn{1}{@{.}l}{82}&\multicolumn{1}{l}{5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1086}&\multicolumn{1}{@{.}l}{21 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}}}&\multicolumn{1}{l}{[M1]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{02026 {\it 28}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{5.55793cm}{\raggedright \%I\ensuremath{\gamma}=1.37 \textit{10}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1023}&\multicolumn{1}{@{.}l}{0 {\it 2}}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{21 {\it 23}}&\multicolumn{1}{r@{}}{1927}&\multicolumn{1}{@{.}l}{32}&\multicolumn{1}{l}{(5/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{904}&\multicolumn{1}{@{.}l}{23 }&\multicolumn{1}{@{}l}{(7/2)\ensuremath{^{-}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{5.55793cm}{\raggedright \%I\ensuremath{\gamma}=0.67 \textit{7}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$1031}&\multicolumn{1}{@{.}l}{2 {\it 2}}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{01 {\it 23}}&&&&&&&&&&&&&&\parbox[t][0.3cm]{5.55793cm}{\raggedright \%I\ensuremath{\gamma}=0.61 \textit{7}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1039}&\multicolumn{1}{@{.}l}{7 {\it 2}}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{18 {\it 25}}&\multicolumn{1}{r@{}}{1944}&\multicolumn{1}{@{.}l}{24}&\multicolumn{1}{l}{(5/2\ensuremath{^{-}},7/2)}&\multicolumn{1}{r@{}}{904}&\multicolumn{1}{@{.}l}{23 }&\multicolumn{1}{@{}l}{(7/2)\ensuremath{^{-}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{5.55793cm}{\raggedright \%I\ensuremath{\gamma}=0.66 \textit{8}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1054}&\multicolumn{1}{@{.}l}{7 {\it 3}}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{90 {\it 25}}&\multicolumn{1}{r@{}}{1944}&\multicolumn{1}{@{.}l}{24}&\multicolumn{1}{l}{(5/2\ensuremath{^{-}},7/2)}&\multicolumn{1}{r@{}}{890}&\multicolumn{1}{@{.}l}{24 }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{5.55793cm}{\raggedright \%I\ensuremath{\gamma}=0.88 \textit{8}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1160}&\multicolumn{1}{@{.}l}{6 {\it 5}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{86 {\it 23}}&\multicolumn{1}{r@{}}{2434}&\multicolumn{1}{@{.}l}{9}&\multicolumn{1}{l}{1/2\ensuremath{^{+}},3/2,5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1274}&\multicolumn{1}{@{.}l}{45 }&\multicolumn{1}{@{}l}{(5/2)\ensuremath{^{+}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{5.55793cm}{\raggedright \%I\ensuremath{\gamma}=0.57 \textit{7}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1175}&\multicolumn{1}{@{.}l}{3 {\it 2}}&\multicolumn{1}{r@{}}{10}&\multicolumn{1}{@{.}l}{7 {\it 5}}&\multicolumn{1}{r@{}}{2065}&\multicolumn{1}{@{.}l}{82}&\multicolumn{1}{l}{5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{890}&\multicolumn{1}{@{.}l}{24 }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{l}{(E1)}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{97\ensuremath{\times10^{-3}} {\it 3}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{5.55793cm}{\raggedright \%I\ensuremath{\gamma}=3.25 \textit{17}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&&&\parbox[t][0.3cm]{5.55793cm}{\raggedright Mult.: \ensuremath{\alpha}(K)exp\ensuremath{<}0.005 in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986Br28,B}{1986Br28}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&&&\parbox[t][0.3cm]{5.55793cm}{\raggedright {\ }{\ }{\ }allows mult=E1 or E2. The placement\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&&&\parbox[t][0.3cm]{5.55793cm}{\raggedright {\ }{\ }{\ }in the decay scheme requires\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&&&\parbox[t][0.3cm]{5.55793cm}{\raggedright {\ }{\ }{\ }\ensuremath{\Delta}\ensuremath{\pi}=yes.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1181}&\multicolumn{1}{@{.}l}{3 {\it 5}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{08 {\it 21}}&\multicolumn{1}{r@{}}{2455}&\multicolumn{1}{@{.}l}{5}&\multicolumn{1}{l}{1/2\ensuremath{^{+}},3/2,5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1274}&\multicolumn{1}{@{.}l}{45 }&\multicolumn{1}{@{}l}{(5/2)\ensuremath{^{+}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{5.55793cm}{\raggedright \%I\ensuremath{\gamma}=0.33 \textit{6}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1186}&\multicolumn{1}{@{.}l}{7 {\it 2}}&\multicolumn{1}{r@{}}{19}&\multicolumn{1}{@{.}l}{1 {\it 8}}&\multicolumn{1}{r@{}}{1186}&\multicolumn{1}{@{.}l}{59}&\multicolumn{1}{l}{(7/2)\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{9/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1(+E2)}&\multicolumn{1}{r@{}}{$<$0}&\multicolumn{1}{@{.}l}{9}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0107 {\it 17}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{5.55793cm}{\raggedright \%I\ensuremath{\gamma}=5.80 \textit{26}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&&&\parbox[t][0.3cm]{5.55793cm}{\raggedright Mult.,\ensuremath{\delta}: \ensuremath{\alpha}(K)exp=0.012 \textit{4} in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986Br28,B}{1986Br28}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$1196}&\multicolumn{1}{@{.}l}{3 {\it 3}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{16 {\it 22}}&&&&&&&&&&&&&&\parbox[t][0.3cm]{5.55793cm}{\raggedright \%I\ensuremath{\gamma}=0.35 \textit{7}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1207}&\multicolumn{1}{@{.}l}{1 {\it 2}}&\multicolumn{1}{r@{}}{11}&\multicolumn{1}{@{.}l}{0 {\it 5}}&\multicolumn{1}{r@{}}{2053}&\multicolumn{1}{@{.}l}{59}&\multicolumn{1}{l}{(5/2)\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{846}&\multicolumn{1}{@{.}l}{35 }&\multicolumn{1}{@{}l}{1/2\ensuremath{^{+}}}&\multicolumn{1}{l}{E2}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0084 {\it 35}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{5.55793cm}{\raggedright \%I\ensuremath{\gamma}=3.34 \textit{17}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&&&\parbox[t][0.3cm]{5.55793cm}{\raggedright Mult.: \ensuremath{\alpha}(K)exp\ensuremath{<}0.007 in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986Br28,B}{1986Br28}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&&&\parbox[t][0.3cm]{5.55793cm}{\raggedright {\ }{\ }{\ }allows mult=E1 or E2. The placement\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&&&\parbox[t][0.3cm]{5.55793cm}{\raggedright {\ }{\ }{\ }in the decay scheme requires \ensuremath{\Delta}\ensuremath{\pi}=no.\vspace{0.1cm}}&\\
\end{longtable}
\clearpage
\begin{longtable}{ccccccccc@{}ccccccc@{\extracolsep{\fill}}c}
\\[-.4cm]
\multicolumn{17}{c}{{\bf \small \underline{\ensuremath{^{\textnormal{201}}}Po \ensuremath{\varepsilon} decay (15.50 min)\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986Br28,B}{1986Br28} (continued)}}}\\
\multicolumn{17}{c}{~}\\
\multicolumn{17}{c}{\underline{$\gamma$($^{201}$Bi) (continued)}}\\
\multicolumn{17}{c}{~~~}\\
\multicolumn{2}{c}{E\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{PO36GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{I\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{PO36GAMMA0}{\dagger}\hyperlink{PO36GAMMA4}{\&}}}}&\multicolumn{2}{c}{E\ensuremath{_{i}}(level)}&J\ensuremath{^{\pi}_{i}}&\multicolumn{2}{c}{E\ensuremath{_{f}}}&J\ensuremath{^{\pi}_{f}}&Mult.\ensuremath{^{\hyperlink{PO36GAMMA1}{\ddagger}}}&\multicolumn{2}{c}{\ensuremath{\delta}\ensuremath{^{\hyperlink{PO36GAMMA2}{\#}}}}&\multicolumn{2}{c}{\ensuremath{\alpha}\ensuremath{^{\hyperlink{PO36GAMMA5}{a}}}}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endhead
\multicolumn{1}{r@{}}{1219}&\multicolumn{1}{@{.}l}{3 {\it 3}}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{2 {\it 3}}&\multicolumn{1}{r@{}}{2065}&\multicolumn{1}{@{.}l}{82}&\multicolumn{1}{l}{5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{846}&\multicolumn{1}{@{.}l}{35 }&\multicolumn{1}{@{}l}{1/2\ensuremath{^{+}}}&\multicolumn{1}{l}{[E2]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{00479 {\it 7}}&\parbox[t][0.3cm]{8.65861cm}{\raggedright \%I\ensuremath{\gamma}=0.97 \textit{9}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1300}&\multicolumn{1}{@{.}l}{5 {\it 4}}&\multicolumn{1}{r@{}}{4}&\multicolumn{1}{@{.}l}{0 {\it 3}}&\multicolumn{1}{r@{}}{2386}&\multicolumn{1}{@{.}l}{7}&&\multicolumn{1}{r@{}}{1086}&\multicolumn{1}{@{.}l}{21 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{8.65861cm}{\raggedright \%I\ensuremath{\gamma}=1.22 \textit{10}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$1306}&\multicolumn{1}{@{.}l}{9 {\it 3}}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{5 {\it 3}}&&&&&&&&&&&&\parbox[t][0.3cm]{8.65861cm}{\raggedright \%I\ensuremath{\gamma}=1.06 \textit{9}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1318}&\multicolumn{1}{@{.}l}{0 {\it 5}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{93 {\it 25}}&\multicolumn{1}{r@{}}{2592}&\multicolumn{1}{@{.}l}{88}&\multicolumn{1}{l}{(3/2\ensuremath{^{-}},5/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{1274}&\multicolumn{1}{@{.}l}{45 }&\multicolumn{1}{@{}l}{(5/2)\ensuremath{^{+}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{8.65861cm}{\raggedright \%I\ensuremath{\gamma}=0.59 \textit{8}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$1346}&\multicolumn{1}{@{.}l}{2 {\it 8}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{8 {\it 3}}&&&&&&&&&&&&\parbox[t][0.3cm]{8.65861cm}{\raggedright \%I\ensuremath{\gamma}=0.55 \textit{9}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1348}&\multicolumn{1}{@{.}l}{4 {\it 6}}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{9 {\it 3}}&\multicolumn{1}{r@{}}{2434}&\multicolumn{1}{@{.}l}{9}&\multicolumn{1}{l}{1/2\ensuremath{^{+}},3/2,5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1086}&\multicolumn{1}{@{.}l}{21 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{8.65861cm}{\raggedright \%I\ensuremath{\gamma}=0.88 \textit{9}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1369}&\multicolumn{1}{@{.}l}{3 {\it 4}}&\multicolumn{1}{r@{}}{4}&\multicolumn{1}{@{.}l}{0 {\it 3}}&\multicolumn{1}{r@{}}{2455}&\multicolumn{1}{@{.}l}{5}&\multicolumn{1}{l}{1/2\ensuremath{^{+}},3/2,5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1086}&\multicolumn{1}{@{.}l}{21 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{8.65861cm}{\raggedright \%I\ensuremath{\gamma}=1.22 \textit{10}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.65861cm}{\raggedright Mult.: \ensuremath{\alpha}(K)exp\ensuremath{<}0.009 in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986Br28,B}{1986Br28}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1398}&\multicolumn{1}{@{.}l}{0 {\it 3}}&\multicolumn{1}{r@{}}{6}&\multicolumn{1}{@{.}l}{4 {\it 4}}&\multicolumn{1}{r@{}}{2484}&\multicolumn{1}{@{.}l}{3}&\multicolumn{1}{l}{1/2,3/2,5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1086}&\multicolumn{1}{@{.}l}{21 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{8.65861cm}{\raggedright \%I\ensuremath{\gamma}=1.94 \textit{13}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1442}&\multicolumn{1}{@{.}l}{2 {\it 6}}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{1 {\it 3}}&\multicolumn{1}{r@{}}{1441}&\multicolumn{1}{@{.}l}{71}&\multicolumn{1}{l}{7/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{9/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1(+E2)}&\multicolumn{1}{r@{}}{$<$1}&\multicolumn{1}{@{.}l}{4}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0063 {\it 13}}&\parbox[t][0.3cm]{8.65861cm}{\raggedright \%I\ensuremath{\gamma}=0.94 \textit{9}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.65861cm}{\raggedright Mult.,\ensuremath{\delta}: \ensuremath{\alpha}(K)exp=0.008 \textit{4} in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986Br28,B}{1986Br28}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1470}&\multicolumn{1}{@{.}l}{9 {\it 3}}&\multicolumn{1}{r@{}}{4}&\multicolumn{1}{@{.}l}{4 {\it 3}}&\multicolumn{1}{r@{}}{1470}&\multicolumn{1}{@{.}l}{87}&\multicolumn{1}{l}{(5/2,7/2)\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{9/2\ensuremath{^{-}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{8.65861cm}{\raggedright \%I\ensuremath{\gamma}=1.34 \textit{10}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1506}&\multicolumn{1}{@{.}l}{6 {\it 3}}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{2 {\it 3}}&\multicolumn{1}{r@{}}{2592}&\multicolumn{1}{@{.}l}{88}&\multicolumn{1}{l}{(3/2\ensuremath{^{-}},5/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{1086}&\multicolumn{1}{@{.}l}{21 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{8.65861cm}{\raggedright \%I\ensuremath{\gamma}=0.97 \textit{9}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1518}&\multicolumn{1}{@{.}l}{2 {\it 4}}&\multicolumn{1}{r@{}}{5}&\multicolumn{1}{@{.}l}{2 {\it 3}}&\multicolumn{1}{r@{}}{2422}&\multicolumn{1}{@{.}l}{1}&\multicolumn{1}{l}{(3/2\ensuremath{^{-}},5/2)}&\multicolumn{1}{r@{}}{904}&\multicolumn{1}{@{.}l}{23 }&\multicolumn{1}{@{}l}{(7/2)\ensuremath{^{-}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{8.65861cm}{\raggedright \%I\ensuremath{\gamma}=1.58 \textit{10}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$1521}&\multicolumn{1}{@{.}l}{5 {\it 4}}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{32 {\it 23}}&&&&&&&&&&&&\parbox[t][0.3cm]{8.65861cm}{\raggedright \%I\ensuremath{\gamma}=0.70 \textit{7}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1531}&\multicolumn{1}{@{.}l}{7 {\it 3}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{36 {\it 24}}&\multicolumn{1}{r@{}}{2422}&\multicolumn{1}{@{.}l}{1}&\multicolumn{1}{l}{(3/2\ensuremath{^{-}},5/2)}&\multicolumn{1}{r@{}}{890}&\multicolumn{1}{@{.}l}{24 }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{8.65861cm}{\raggedright \%I\ensuremath{\gamma}=0.41 \textit{7}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1588}&\multicolumn{1}{@{.}l}{5 {\it 3}}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{0 {\it 3}}&\multicolumn{1}{r@{}}{2434}&\multicolumn{1}{@{.}l}{9}&\multicolumn{1}{l}{1/2\ensuremath{^{+}},3/2,5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{846}&\multicolumn{1}{@{.}l}{35 }&\multicolumn{1}{@{}l}{1/2\ensuremath{^{+}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{8.65861cm}{\raggedright \%I\ensuremath{\gamma}=0.61 \textit{9}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$1593}&\multicolumn{1}{@{.}l}{9 {\it 3}}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{3 {\it 3}}&&&&&&&&&&&&\parbox[t][0.3cm]{8.65861cm}{\raggedright \%I\ensuremath{\gamma}=1.00 \textit{9}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1609}&\multicolumn{1}{@{.}l}{0 {\it 3}}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{9 {\it 3}}&\multicolumn{1}{r@{}}{2455}&\multicolumn{1}{@{.}l}{5}&\multicolumn{1}{l}{1/2\ensuremath{^{+}},3/2,5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{846}&\multicolumn{1}{@{.}l}{35 }&\multicolumn{1}{@{}l}{1/2\ensuremath{^{+}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{8.65861cm}{\raggedright \%I\ensuremath{\gamma}=1.19 \textit{10}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1627}&\multicolumn{1}{@{.}l}{7 {\it 3}}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{7 {\it 3}}&\multicolumn{1}{r@{}}{2902}&\multicolumn{1}{@{.}l}{09}&\multicolumn{1}{l}{1/2\ensuremath{^{+}},3/2,5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1274}&\multicolumn{1}{@{.}l}{45 }&\multicolumn{1}{@{}l}{(5/2)\ensuremath{^{+}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{8.65861cm}{\raggedright \%I\ensuremath{\gamma}=0.82 \textit{9}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1638}&\multicolumn{1}{@{.}l}{1 {\it 5}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{87 {\it 25}}&\multicolumn{1}{r@{}}{2484}&\multicolumn{1}{@{.}l}{3}&\multicolumn{1}{l}{1/2,3/2,5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{846}&\multicolumn{1}{@{.}l}{35 }&\multicolumn{1}{@{}l}{1/2\ensuremath{^{+}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{8.65861cm}{\raggedright \%I\ensuremath{\gamma}=0.57 \textit{8}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$1676}&\multicolumn{1}{@{.}l}{3 {\it 4}}&\multicolumn{1}{r@{}}{4}&\multicolumn{1}{@{.}l}{5 {\it 3}}&&&&&&&&&&&&\parbox[t][0.3cm]{8.65861cm}{\raggedright \%I\ensuremath{\gamma}=1.37 \textit{10}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1689}&\multicolumn{1}{@{.}l}{3 {\it 3}}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{8 {\it 3}}&\multicolumn{1}{r@{}}{2592}&\multicolumn{1}{@{.}l}{88}&\multicolumn{1}{l}{(3/2\ensuremath{^{-}},5/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{904}&\multicolumn{1}{@{.}l}{23 }&\multicolumn{1}{@{}l}{(7/2)\ensuremath{^{-}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{8.65861cm}{\raggedright \%I\ensuremath{\gamma}=0.85 \textit{9}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1702}&\multicolumn{1}{@{.}l}{1 {\it 3}}&\multicolumn{1}{r@{}}{5}&\multicolumn{1}{@{.}l}{2 {\it 3}}&\multicolumn{1}{r@{}}{2592}&\multicolumn{1}{@{.}l}{88}&\multicolumn{1}{l}{(3/2\ensuremath{^{-}},5/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{890}&\multicolumn{1}{@{.}l}{24 }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{8.65861cm}{\raggedright \%I\ensuremath{\gamma}=1.58 \textit{10}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1746}&\multicolumn{1}{@{.}l}{8 {\it 5}}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{3 {\it 3}}&\multicolumn{1}{r@{}}{2592}&\multicolumn{1}{@{.}l}{88}&\multicolumn{1}{l}{(3/2\ensuremath{^{-}},5/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{846}&\multicolumn{1}{@{.}l}{35 }&\multicolumn{1}{@{}l}{1/2\ensuremath{^{+}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{8.65861cm}{\raggedright \%I\ensuremath{\gamma}=0.70 \textit{9}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1815}&\multicolumn{1}{@{.}l}{8 {\it 4}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{36 {\it 24}}&\multicolumn{1}{r@{}}{2902}&\multicolumn{1}{@{.}l}{09}&\multicolumn{1}{l}{1/2\ensuremath{^{+}},3/2,5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1086}&\multicolumn{1}{@{.}l}{21 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{8.65861cm}{\raggedright \%I\ensuremath{\gamma}=0.41 \textit{7}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1819}&\multicolumn{1}{@{.}l}{8 {\it 3}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{95 {\it 23}}&\multicolumn{1}{r@{}}{2905}&\multicolumn{1}{@{.}l}{9}&\multicolumn{1}{l}{1/2,3/2,5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1086}&\multicolumn{1}{@{.}l}{21 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{8.65861cm}{\raggedright \%I\ensuremath{\gamma}=0.29 \textit{7}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$1833}&\multicolumn{1}{@{.}l}{1 {\it 3}}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{2 {\it 3}}&&&&&&&&&&&&\parbox[t][0.3cm]{8.65861cm}{\raggedright \%I\ensuremath{\gamma}=0.97 \textit{9}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$1841}&\multicolumn{1}{@{.}l}{6 {\it 4}}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{0 {\it 3}}&&&&&&&&&&&&\parbox[t][0.3cm]{8.65861cm}{\raggedright \%I\ensuremath{\gamma}=0.61 \textit{9}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1848}&\multicolumn{1}{@{.}l}{0 {\it 3}}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{9 {\it 3}}&\multicolumn{1}{r@{}}{1848}&\multicolumn{1}{@{.}l}{16}&\multicolumn{1}{l}{(5/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{9/2\ensuremath{^{-}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{8.65861cm}{\raggedright \%I\ensuremath{\gamma}=0.88 \textit{9}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1927}&\multicolumn{1}{@{.}l}{5 {\it 3}}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{14 {\it 25}}&\multicolumn{1}{r@{}}{1927}&\multicolumn{1}{@{.}l}{32}&\multicolumn{1}{l}{(5/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{9/2\ensuremath{^{-}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{8.65861cm}{\raggedright \%I\ensuremath{\gamma}=0.65 \textit{8}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$1930}&\multicolumn{1}{@{.}l}{2 {\it 6}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{94 {\it 24}}&&&&&&&&&&&&\parbox[t][0.3cm]{8.65861cm}{\raggedright \%I\ensuremath{\gamma}=0.29 \textit{7}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1944}&\multicolumn{1}{@{.}l}{2 {\it 3}}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{9 {\it 3}}&\multicolumn{1}{r@{}}{1944}&\multicolumn{1}{@{.}l}{24}&\multicolumn{1}{l}{(5/2\ensuremath{^{-}},7/2)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{9/2\ensuremath{^{-}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{8.65861cm}{\raggedright \%I\ensuremath{\gamma}=0.88 \textit{9}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$2028}&\multicolumn{1}{@{.}l}{9 {\it 5}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{8 {\it 3}}&&&&&&&&&&&&\parbox[t][0.3cm]{8.65861cm}{\raggedright \%I\ensuremath{\gamma}=0.55 \textit{9}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2055}&\multicolumn{1}{@{.}l}{7 {\it 3}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{53 {\it 17}}&\multicolumn{1}{r@{}}{2902}&\multicolumn{1}{@{.}l}{09}&\multicolumn{1}{l}{1/2\ensuremath{^{+}},3/2,5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{846}&\multicolumn{1}{@{.}l}{35 }&\multicolumn{1}{@{}l}{1/2\ensuremath{^{+}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{8.65861cm}{\raggedright \%I\ensuremath{\gamma}=0.16 \textit{5}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2059}&\multicolumn{1}{@{.}l}{4 {\it 3}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{14 {\it 18}}&\multicolumn{1}{r@{}}{2905}&\multicolumn{1}{@{.}l}{9}&\multicolumn{1}{l}{1/2,3/2,5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{846}&\multicolumn{1}{@{.}l}{35 }&\multicolumn{1}{@{}l}{1/2\ensuremath{^{+}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{8.65861cm}{\raggedright \%I\ensuremath{\gamma}=0.35 \textit{6}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$2065}&\multicolumn{1}{@{.}l}{3 {\it 4}}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{5 {\it 3}}&&&&&&&&&&&&\parbox[t][0.3cm]{8.65861cm}{\raggedright \%I\ensuremath{\gamma}=0.76 \textit{9}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$2073}&\multicolumn{1}{@{.}l}{2 {\it 3}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{48 {\it 21}}&&&&&&&&&&&&\parbox[t][0.3cm]{8.65861cm}{\raggedright \%I\ensuremath{\gamma}=0.15 \textit{6}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$2128}&\multicolumn{1}{@{.}l}{0 {\it 4}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{52 {\it 18}}&&&&&&&&&&&&\parbox[t][0.3cm]{8.65861cm}{\raggedright \%I\ensuremath{\gamma}=0.16 \textit{5}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$2191}&\multicolumn{1}{@{.}l}{5 {\it 3}}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{0 {\it 3}}&&&&&&&&&&&&\parbox[t][0.3cm]{8.65861cm}{\raggedright \%I\ensuremath{\gamma}=0.91 \textit{9}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$2321}&\multicolumn{1}{@{.}l}{7 {\it 3}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{53 {\it 23}}&&&&&&&&&&&&\parbox[t][0.3cm]{8.65861cm}{\raggedright \%I\ensuremath{\gamma}=0.46 \textit{7}\vspace{0.1cm}}&\\
\end{longtable}
\clearpage
\vspace*{-0.5cm}
{\bf \small \underline{\ensuremath{^{\textnormal{201}}}Po \ensuremath{\varepsilon} decay (15.50 min)\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986Br28,B}{1986Br28} (continued)}}\\
\vspace{0.3cm}
\underline{$\gamma$($^{201}$Bi) (continued)}\\
\vspace{0.3cm}
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PO36GAMMA0}{\dagger}}}} From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986Br28,B}{1986Br28}, unless otherwise stated.}\\
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PO36GAMMA1}{\ddagger}}}} From the ce measurements in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986Br28,B}{1986Br28} and \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1969Al10,B}{1969Al10}, unless otherwise stated.}\\
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PO36GAMMA2}{\#}}}} From \ensuremath{\alpha}(K)exp and subshell ratios in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986Br28,B}{1986Br28} and the briccmixing program, unless otherwise stated.}\\
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PO36GAMMA3}{@}}}} Estimated from coincidence intensity.}\\
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PO36GAMMA4}{\&}}}} For absolute intensity per 100 decays, multiply by 0.304 \textit{7}.}\\
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PO36GAMMA5}{a}}}} Total theoretical internal conversion coefficients, calculated using the BrIcc code (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2008Ki07,B}{2008Ki07}) with Frozen orbital approximation based on \ensuremath{\gamma}-ray energies,}\\
\parbox[b][0.3cm]{21.881866cm}{{\ }{\ }assigned multipolarities, and mixing ratios, unless otherwise specified.}\\
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PO36GAMMA6}{b}}}} Placement of transition in the level scheme is uncertain.}\\
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PO36GAMMA7}{x}}}} \ensuremath{\gamma} ray not placed in level scheme.}\\
\vspace{0.5cm}
\end{landscape}\clearpage
\clearpage
\begin{figure}[h]
\begin{center}
\includegraphics[angle=90]{201BI36-0.ps}\\
\end{center}
\end{figure}
\clearpage
\subsection[\hspace{-0.2cm}\ensuremath{^{\textnormal{201}}}Po \ensuremath{\varepsilon} decay (8.96 min)]{ }
\vspace{-27pt}
\vspace{0.3cm}
\hypertarget{PO37}{{\bf \small \underline{\ensuremath{^{\textnormal{201}}}Po \ensuremath{\varepsilon} decay (8.96 min)\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986Br28,B}{1986Br28},\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1976Ko13,B}{1976Ko13}}}}\\
\vspace{4pt}
\vspace{8pt}
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}Parent: $^{201}$Po: E=423.41 {\it 22}; J$^{\pi}$=13/2\ensuremath{^{+}}; T$_{1/2}$=8.96 min {\it 12}; Q(\ensuremath{\varepsilon})=4908 {\it 13}; \%\ensuremath{\varepsilon}+\%\ensuremath{\beta^{+}} decay$\approx$55.0
}\\
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986Br28,B}{1986Br28}:\ensuremath{^{\textnormal{193}}}Ir(\ensuremath{^{\textnormal{14}}}N,6n), E=116 MeV; Detectors: Ge(Li) and cooled Si(Li); Measured: \ensuremath{\gamma}, \ensuremath{\gamma}\ensuremath{\gamma} coin, \ensuremath{\gamma}\ensuremath{\gamma}(t), \ensuremath{\gamma}(x-ray)(t), ce, and}\\
\parbox[b][0.3cm]{17.7cm}{T\ensuremath{_{\textnormal{1/2}}}.}\\
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1976Ko13,B}{1976Ko13}: \ensuremath{^{\textnormal{197}}}Au(\ensuremath{^{\textnormal{10}}}B,6n), E(\ensuremath{^{\textnormal{10}}}B)\ensuremath{\approx}90 MeV; Ge(Li) and Si(Li); Measured \ensuremath{\gamma}, \ensuremath{\gamma}\ensuremath{\gamma}, I(ce), T\ensuremath{_{\textnormal{1/2}}}.}\\
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}Other: \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1971Jo19,B}{1971Jo19}.}\\
\vspace{12pt}
\underline{$^{201}$Bi Levels}\\
\begin{longtable}{ccccc|ccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E(level)$^{{\hyperlink{BI37LEVEL0}{\dagger}}}$}&J$^{\pi}$$^{{\hyperlink{BI37LEVEL1}{\ddagger}}}$&\multicolumn{2}{c|}{T$_{1/2}$$^{{\hyperlink{BI37LEVEL1}{\ddagger}}}$}&\multicolumn{2}{c}{E(level)$^{{\hyperlink{BI37LEVEL0}{\dagger}}}$}&J$^{\pi}$$^{{\hyperlink{BI37LEVEL1}{\ddagger}}}$&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c|}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{l}{9/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{103}&\multicolumn{1}{@{.}l|}{2 min {\it 17}}&\multicolumn{1}{r@{}}{1504}&\multicolumn{1}{@{.}l}{40 {\it 23}}&\multicolumn{1}{l}{(9/2,11/2)\ensuremath{^{+}}}&\\
\multicolumn{1}{r@{}}{964}&\multicolumn{1}{@{.}l}{40 {\it 15}}&\multicolumn{1}{l}{11/2\ensuremath{^{-}}}&&&\multicolumn{1}{r@{}}{1665}&\multicolumn{1}{@{.}l}{1 {\it 3}}&&\\
\multicolumn{1}{r@{}}{967}&\multicolumn{1}{@{.}l}{49 {\it 18}}&\multicolumn{1}{l}{13/2\ensuremath{^{-}}}&&&\multicolumn{1}{r@{}}{1719}&\multicolumn{1}{@{.}l}{15 {\it 22}}&\multicolumn{1}{l}{(11/2,13/2)\ensuremath{^{+}}}&\\
\multicolumn{1}{r@{}}{1379}&\multicolumn{1}{@{.}l}{4 {\it 3}}&\multicolumn{1}{l}{15/2\ensuremath{^{-}}}&&&\multicolumn{1}{r@{}}{1746}&\multicolumn{1}{@{.}l}{5 {\it 9}}&\multicolumn{1}{l}{17/2\ensuremath{^{+}}}&\\
\multicolumn{1}{r@{}}{1474}&\multicolumn{1}{@{.}l}{6 {\it 9}}&\multicolumn{1}{l}{17/2\ensuremath{^{-}}}&&&\multicolumn{1}{r@{}}{1762}&\multicolumn{1}{@{.}l}{9 {\it 3}}&\multicolumn{1}{l}{(11/2,13/2\ensuremath{^{-}})}&\\
\multicolumn{1}{r@{}}{1501}&\multicolumn{1}{@{.}l}{89 {\it 17}}&\multicolumn{1}{l}{(13/2\ensuremath{^{+}})}&&&\multicolumn{1}{r@{}}{2034}&\multicolumn{1}{@{.}l}{3 {\it 7}}&&\\
\end{longtable}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{BI37LEVEL0}{\dagger}}}} From a least-squares fit to E\ensuremath{\gamma}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{BI37LEVEL1}{\ddagger}}}} From Adopted Levels.}\\
\vspace{0.5cm}
\underline{\ensuremath{\varepsilon,\beta^+} radiations}\\
\begin{longtable}{ccccccccccccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E(decay)$$}&\multicolumn{2}{c}{E(level)}&\multicolumn{2}{c}{I$\beta^{+ \ {\hyperlink{BI37DECAY0}{\dagger}}}$}&\multicolumn{2}{c}{I$\varepsilon^{{\hyperlink{BI37DECAY0}{\dagger}}}$}&\multicolumn{2}{c}{Log {\it ft}$^{}$}&\multicolumn{2}{c}{I$(\varepsilon+\beta^{+})^{{\hyperlink{BI37DECAY0}{\dagger}}}$}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{(3297}&\multicolumn{1}{@{ }l}{{\it 13})}&\multicolumn{1}{r@{}}{2034}&\multicolumn{1}{@{.}l}{3}&\multicolumn{1}{r@{}}{$\approx$0}&\multicolumn{1}{@{.}l}{1}&\multicolumn{1}{r@{}}{$\approx$0}&\multicolumn{1}{@{.}l}{8}&\multicolumn{1}{r@{}}{$\approx$7}&\multicolumn{1}{@{.}l}{4}&\multicolumn{1}{r@{}}{$\approx$0}&\multicolumn{1}{@{.}l}{9}&&\\
\multicolumn{1}{r@{}}{(3569}&\multicolumn{1}{@{ }l}{{\it 13})}&\multicolumn{1}{r@{}}{1762}&\multicolumn{1}{@{.}l}{9}&\multicolumn{1}{r@{}}{$\approx$0}&\multicolumn{1}{@{.}l}{26}&\multicolumn{1}{r@{}}{$\approx$1}&\multicolumn{1}{@{.}l}{4}&\multicolumn{1}{r@{}}{$\approx$7}&\multicolumn{1}{@{.}l}{2}&\multicolumn{1}{r@{}}{$\approx$1}&\multicolumn{1}{@{.}l}{7}&&\\
\multicolumn{1}{r@{}}{(3585}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{BI37DECAY1}{\ddagger}}}} {\it 13})}&\multicolumn{1}{r@{}}{1746}&\multicolumn{1}{@{.}l}{5}&\multicolumn{1}{r@{}}{$\approx$0}&\multicolumn{1}{@{.}l}{31}&\multicolumn{1}{r@{}}{$\approx$1}&\multicolumn{1}{@{.}l}{7}&\multicolumn{1}{r@{}}{$\approx$7}&\multicolumn{1}{@{.}l}{2}&\multicolumn{1}{r@{}}{$\approx$2}&\multicolumn{1}{@{.}l}{0}&&\\
\multicolumn{1}{r@{}}{(3612}&\multicolumn{1}{@{ }l}{{\it 13})}&\multicolumn{1}{r@{}}{1719}&\multicolumn{1}{@{.}l}{15}&\multicolumn{1}{r@{}}{$\approx$0}&\multicolumn{1}{@{.}l}{39}&\multicolumn{1}{r@{}}{$\approx$2}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{r@{}}{$\approx$7}&\multicolumn{1}{@{.}l}{1}&\multicolumn{1}{r@{}}{$\approx$2}&\multicolumn{1}{@{.}l}{4}&&\\
\multicolumn{1}{r@{}}{(3666}&\multicolumn{1}{@{ }l}{{\it 13})}&\multicolumn{1}{r@{}}{1665}&\multicolumn{1}{@{.}l}{1}&\multicolumn{1}{r@{}}{$\approx$0}&\multicolumn{1}{@{.}l}{22}&\multicolumn{1}{r@{}}{$\approx$1}&\multicolumn{1}{@{.}l}{1}&\multicolumn{1}{r@{}}{$\approx$7}&\multicolumn{1}{@{.}l}{4}&\multicolumn{1}{r@{}}{$\approx$1}&\multicolumn{1}{@{.}l}{3}&&\\
\multicolumn{1}{r@{}}{(3827}&\multicolumn{1}{@{ }l}{{\it 13})}&\multicolumn{1}{r@{}}{1504}&\multicolumn{1}{@{.}l}{40}&\multicolumn{1}{r@{}}{$\approx$0}&\multicolumn{1}{@{.}l}{53}&\multicolumn{1}{r@{}}{$\approx$2}&\multicolumn{1}{@{.}l}{2}&\multicolumn{1}{r@{}}{$\approx$7}&\multicolumn{1}{@{.}l}{1}&\multicolumn{1}{r@{}}{$\approx$2}&\multicolumn{1}{@{.}l}{7}&&\\
\multicolumn{1}{r@{}}{(3830}&\multicolumn{1}{@{ }l}{{\it 13})}&\multicolumn{1}{r@{}}{1501}&\multicolumn{1}{@{.}l}{89}&\multicolumn{1}{r@{}}{$\approx$1}&\multicolumn{1}{@{.}l}{8}&\multicolumn{1}{r@{}}{$\approx$7}&\multicolumn{1}{@{.}l}{3}&\multicolumn{1}{r@{}}{$\approx$6}&\multicolumn{1}{@{.}l}{6}&\multicolumn{1}{r@{}}{$\approx$9}&\multicolumn{1}{@{.}l}{1}&&\\
\multicolumn{1}{r@{}}{(3857}&\multicolumn{1}{@{ }l}{{\it 13})}&\multicolumn{1}{r@{}}{1474}&\multicolumn{1}{@{.}l}{6}&\multicolumn{1}{r@{}}{$<$0}&\multicolumn{1}{@{.}l}{07}&\multicolumn{1}{r@{}}{$<$0}&\multicolumn{1}{@{.}l}{8}&\multicolumn{1}{r@{}}{$>$9}&\multicolumn{1}{@{.}l}{3\ensuremath{^{1u}}}&\multicolumn{1}{r@{}}{$<$0}&\multicolumn{1}{@{.}l}{9}&&\\
\multicolumn{1}{r@{}}{(3952}&\multicolumn{1}{@{ }l}{{\it 13})}&\multicolumn{1}{r@{}}{1379}&\multicolumn{1}{@{.}l}{4}&\multicolumn{1}{r@{}}{$\approx$2}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{r@{}}{$\approx$7}&\multicolumn{1}{@{.}l}{3}&\multicolumn{1}{r@{}}{$\approx$6}&\multicolumn{1}{@{.}l}{6}&\multicolumn{1}{r@{}}{$\approx$9}&\multicolumn{1}{@{.}l}{3}&&\\
\multicolumn{1}{r@{}}{(4364}&\multicolumn{1}{@{ }l}{{\it 13})}&\multicolumn{1}{r@{}}{967}&\multicolumn{1}{@{.}l}{49}&\multicolumn{1}{r@{}}{$\approx$3}&\multicolumn{1}{@{.}l}{78}&\multicolumn{1}{r@{}}{$\approx$9}&\multicolumn{1}{@{.}l}{32}&\multicolumn{1}{r@{}}{$\approx$6}&\multicolumn{1}{@{.}l}{6}&\multicolumn{1}{r@{}}{$\approx$13}&\multicolumn{1}{@{.}l}{1}&&\\
\multicolumn{1}{r@{}}{(4367}&\multicolumn{1}{@{ }l}{{\it 13})}&\multicolumn{1}{r@{}}{964}&\multicolumn{1}{@{.}l}{40}&\multicolumn{1}{r@{}}{$\approx$3}&\multicolumn{1}{@{.}l}{58}&\multicolumn{1}{r@{}}{$\approx$8}&\multicolumn{1}{@{.}l}{82}&\multicolumn{1}{r@{}}{$\approx$6}&\multicolumn{1}{@{.}l}{6}&\multicolumn{1}{r@{}}{$\approx$12}&\multicolumn{1}{@{.}l}{4}&&\\
\multicolumn{1}{r@{}}{(5331}&\multicolumn{1}{@{ }l}{{\it 13})}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{$\approx$11}&\multicolumn{1}{@{.}l}{2}&\multicolumn{1}{r@{}}{$\approx$33}&\multicolumn{1}{@{.}l}{8}&\multicolumn{1}{r@{}}{$\approx$8}&\multicolumn{1}{@{.}l}{2\ensuremath{^{1u}}}&\multicolumn{1}{r@{}}{$\approx$45}&\multicolumn{1}{@{.}l}{0}&\parbox[t][0.3cm]{8.98934cm}{\raggedright I$(\varepsilon+\beta^{+})$: From log \textit{ft}\ensuremath{^{\textnormal{I1u}}}=8.2 in \ensuremath{^{\textnormal{197}}}Pb \ensuremath{\varepsilon} decay (\ensuremath{J^{\pi}}=13/2\ensuremath{^{+}}) to\vspace{0.1cm}}&\\
&&&&&&&&&&&&\parbox[t][0.3cm]{8.98934cm}{\raggedright {\ }{\ }{\ }the \ensuremath{J^{\pi}}=9/2\ensuremath{^{-}} level in \ensuremath{^{\textnormal{193}}}Tl.\vspace{0.1cm}}&\\
\end{longtable}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{BI37DECAY0}{\dagger}}}} For absolute intensity per 100 decays, multiply by{ }\ensuremath{\approx}0.55.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{BI37DECAY1}{\ddagger}}}} Existence of this branch is questionable.}\\
\vspace{0.5cm}
\clearpage
\vspace{0.3cm}
\begin{landscape}
\vspace*{-0.5cm}
{\bf \small \underline{\ensuremath{^{\textnormal{201}}}Po \ensuremath{\varepsilon} decay (8.96 min)\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986Br28,B}{1986Br28},\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1976Ko13,B}{1976Ko13} (continued)}}\\
\vspace{0.3cm}
\underline{$\gamma$($^{201}$Bi)}\\
\vspace{0.34cm}
\parbox[b][0.3cm]{21.881866cm}{\addtolength{\parindent}{-0.254cm}I\ensuremath{\gamma} normalization: Using \ensuremath{\Sigma}I(\ensuremath{\gamma}+ce)(to g.s.)=100 {\textminus} I\ensuremath{\beta}(g.s.) where I\ensuremath{\beta}(g.s.) \ensuremath{\approx} 45\% was deduced by the evaluator from log \textit{ft}\ensuremath{^{\textnormal{I1u}}}=8.2 in \ensuremath{^{\textnormal{197}}}Pb \ensuremath{\varepsilon} decay (\ensuremath{J^{\pi}}=13/2\ensuremath{^{+}})}\\
\parbox[b][0.3cm]{21.881866cm}{to the \ensuremath{J^{\pi}}=9/2\ensuremath{^{-}} level in \ensuremath{^{\textnormal{193}}}Tl.}\\
\vspace{0.34cm}
\begin{longtable}{ccccccccc@{}ccccccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{PO37GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{I\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{PO37GAMMA1}{\ddagger}\hyperlink{PO37GAMMA3}{@}}}}&\multicolumn{2}{c}{E\ensuremath{_{i}}(level)}&J\ensuremath{^{\pi}_{i}}&\multicolumn{2}{c}{E\ensuremath{_{f}}}&J\ensuremath{^{\pi}_{f}}&Mult.\ensuremath{^{\hyperlink{PO37GAMMA0}{\dagger}}}&\multicolumn{2}{c}{\ensuremath{\delta}\ensuremath{^{\hyperlink{PO37GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{\ensuremath{\alpha}\ensuremath{^{\hyperlink{PO37GAMMA4}{\&}}}}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{95}&\multicolumn{1}{@{.}l}{26 {\it 15}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{6 {\it 3}}&\multicolumn{1}{r@{}}{1474}&\multicolumn{1}{@{.}l}{6}&\multicolumn{1}{l}{17/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{1379}&\multicolumn{1}{@{.}l}{4 }&\multicolumn{1}{@{}l}{15/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1(+E2)}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{10}&\multicolumn{1}{@{.}l}{0 {\it 17}}&\parbox[t][0.3cm]{7.194851cm}{\raggedright \%I\ensuremath{\gamma}\ensuremath{\approx}0.10\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.194851cm}{\raggedright I\ensuremath{_{\gamma}}: Estimated from cascade intensity (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986Br28,B}{1986Br28}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{217}&\multicolumn{1}{@{.}l}{6 {\it 3}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{61 {\it 10}}&\multicolumn{1}{r@{}}{1719}&\multicolumn{1}{@{.}l}{15}&\multicolumn{1}{l}{(11/2,13/2)\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1501}&\multicolumn{1}{@{.}l}{89 }&\multicolumn{1}{@{}l}{(13/2\ensuremath{^{+}})}&\multicolumn{1}{l}{M1(+E2)}&\multicolumn{1}{r@{}}{$<$1}&\multicolumn{1}{@{.}l}{7}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{83 {\it 29}}&\parbox[t][0.3cm]{7.194851cm}{\raggedright \%I\ensuremath{\gamma}\ensuremath{\approx}0.099\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.194851cm}{\raggedright Mult.,\ensuremath{\delta}: From \ensuremath{\alpha}(K)exp=0.75 \textit{41} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986Br28,B}{1986Br28}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{271}&\multicolumn{1}{@{.}l}{91 {\it 20}}&\multicolumn{1}{r@{}}{6}&\multicolumn{1}{@{.}l}{9 {\it 3}}&\multicolumn{1}{r@{}}{1746}&\multicolumn{1}{@{.}l}{5}&\multicolumn{1}{l}{17/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1474}&\multicolumn{1}{@{.}l}{6 }&\multicolumn{1}{@{}l}{17/2\ensuremath{^{-}}}&\multicolumn{1}{l}{E1}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0383 {\it 5}}&\parbox[t][0.3cm]{7.194851cm}{\raggedright \%I\ensuremath{\gamma}\ensuremath{\approx}1.12\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.194851cm}{\raggedright Mult.: From \ensuremath{\alpha}(K)exp\ensuremath{<}0.066 (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986Br28,B}{1986Br28}). Note, that\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.194851cm}{\raggedright {\ }{\ }{\ }\ensuremath{\alpha}(K)exp=0.42 \textit{30} and K/L=5.7 in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1976Ko13,B}{1976Ko13} are\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.194851cm}{\raggedright {\ }{\ }{\ }consistent with Mult=M1.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{411}&\multicolumn{1}{@{.}l}{86 {\it 20}}&\multicolumn{1}{r@{}}{32}&\multicolumn{1}{@{.}l}{0 {\it 11}}&\multicolumn{1}{r@{}}{1379}&\multicolumn{1}{@{.}l}{4}&\multicolumn{1}{l}{15/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{967}&\multicolumn{1}{@{.}l}{49 }&\multicolumn{1}{@{}l}{13/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{$-$0}&\multicolumn{1}{@{.}l}{023 {\it 17}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{1966 {\it 28}}&\parbox[t][0.3cm]{7.194851cm}{\raggedright \%I\ensuremath{\gamma}\ensuremath{\approx}5.19\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.194851cm}{\raggedright Mult.: Other: \ensuremath{\alpha}(K)exp=0.22 \textit{10} and K/L=5.3 in\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.194851cm}{\raggedright {\ }{\ }{\ }\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1976Ko13,B}{1976Ko13}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{532}&\multicolumn{1}{@{.}l}{4 {\it 6}}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{12 {\it 18}}&\multicolumn{1}{r@{}}{2034}&\multicolumn{1}{@{.}l}{3}&&\multicolumn{1}{r@{}}{1501}&\multicolumn{1}{@{.}l}{89 }&\multicolumn{1}{@{}l}{(13/2\ensuremath{^{+}})}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{7.194851cm}{\raggedright \%I\ensuremath{\gamma}\ensuremath{\approx}0.506\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{534}&\multicolumn{1}{@{.}l}{2 {\it 3}}&\multicolumn{1}{r@{}}{10}&\multicolumn{1}{@{.}l}{2 {\it 4}}&\multicolumn{1}{r@{}}{1501}&\multicolumn{1}{@{.}l}{89}&\multicolumn{1}{l}{(13/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{967}&\multicolumn{1}{@{.}l}{49 }&\multicolumn{1}{@{}l}{13/2\ensuremath{^{-}}}&\multicolumn{1}{l}{[E1]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{00865 {\it 12}}&\parbox[t][0.3cm]{7.194851cm}{\raggedright \%I\ensuremath{\gamma}\ensuremath{\approx}1.65\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{537}&\multicolumn{1}{@{.}l}{5 {\it 2}}&\multicolumn{1}{r@{}}{24}&\multicolumn{1}{@{.}l}{0\ensuremath{^{\hyperlink{PO37GAMMA2}{\#}}} {\it 15}}&\multicolumn{1}{r@{}}{1501}&\multicolumn{1}{@{.}l}{89}&\multicolumn{1}{l}{(13/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{964}&\multicolumn{1}{@{.}l}{40 }&\multicolumn{1}{@{}l}{11/2\ensuremath{^{-}}}&\multicolumn{1}{l}{[E1]}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{00854 {\it 12}}&\parbox[t][0.3cm]{7.194851cm}{\raggedright \%I\ensuremath{\gamma}\ensuremath{\approx}3.89\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{540}&\multicolumn{1}{@{.}l}{1 {\it 3}}&\multicolumn{1}{r@{}}{8}&\multicolumn{1}{@{.}l}{4 {\it 4}}&\multicolumn{1}{r@{}}{1504}&\multicolumn{1}{@{.}l}{40}&\multicolumn{1}{l}{(9/2,11/2)\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{964}&\multicolumn{1}{@{.}l}{40 }&\multicolumn{1}{@{}l}{11/2\ensuremath{^{-}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{7.194851cm}{\raggedright \%I\ensuremath{\gamma}\ensuremath{\approx}1.36\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{697}&\multicolumn{1}{@{.}l}{6 {\it 2}}&\multicolumn{1}{r@{}}{4}&\multicolumn{1}{@{.}l}{5 {\it 3}}&\multicolumn{1}{r@{}}{1665}&\multicolumn{1}{@{.}l}{1}&&\multicolumn{1}{r@{}}{967}&\multicolumn{1}{@{.}l}{49 }&\multicolumn{1}{@{}l}{13/2\ensuremath{^{-}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{7.194851cm}{\raggedright \%I\ensuremath{\gamma}\ensuremath{\approx}0.73\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{754}&\multicolumn{1}{@{.}l}{6 {\it 2}}&\multicolumn{1}{r@{}}{7}&\multicolumn{1}{@{.}l}{0 {\it 4}}&\multicolumn{1}{r@{}}{1719}&\multicolumn{1}{@{.}l}{15}&\multicolumn{1}{l}{(11/2,13/2)\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{964}&\multicolumn{1}{@{.}l}{40 }&\multicolumn{1}{@{}l}{11/2\ensuremath{^{-}}}&\multicolumn{1}{l}{E1}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{00438 {\it 6}}&\parbox[t][0.3cm]{7.194851cm}{\raggedright \%I\ensuremath{\gamma}\ensuremath{\approx}1.13\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.194851cm}{\raggedright Mult.: \ensuremath{\alpha}(K)exp\ensuremath{<}0.007 (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986Br28,B}{1986Br28}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{798}&\multicolumn{1}{@{.}l}{5 {\it 3}}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{02 {\it 20}}&\multicolumn{1}{r@{}}{1762}&\multicolumn{1}{@{.}l}{9}&\multicolumn{1}{l}{(11/2,13/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{964}&\multicolumn{1}{@{.}l}{40 }&\multicolumn{1}{@{}l}{11/2\ensuremath{^{-}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{7.194851cm}{\raggedright \%I\ensuremath{\gamma}\ensuremath{\approx}0.328\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{964}&\multicolumn{1}{@{.}l}{3 {\it 2}}&\multicolumn{1}{r@{}}{82}&\multicolumn{1}{@{ }l}{{\it 3}}&\multicolumn{1}{r@{}}{964}&\multicolumn{1}{@{.}l}{40}&\multicolumn{1}{l}{11/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{9/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1(+E2)}&\multicolumn{1}{r@{}}{$-$0}&\multicolumn{1}{@{.}l}{04 {\it 7}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{02109 {\it 33}}&\parbox[t][0.3cm]{7.194851cm}{\raggedright \%I\ensuremath{\gamma}\ensuremath{\approx}13.29\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.194851cm}{\raggedright Mult.: Other: \ensuremath{\alpha}(K)exp=0.0231 \textit{33} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986Br28,B}{1986Br28}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{967}&\multicolumn{1}{@{.}l}{4 {\it 2}}&\multicolumn{1}{r@{}}{96}&\multicolumn{1}{@{.}l}{9 {\it 15}}&\multicolumn{1}{r@{}}{967}&\multicolumn{1}{@{.}l}{49}&\multicolumn{1}{l}{13/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{9/2\ensuremath{^{-}}}&\multicolumn{1}{l}{E2}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{00746 {\it 10}}&\parbox[t][0.3cm]{7.194851cm}{\raggedright \%I\ensuremath{\gamma}\ensuremath{\approx}15.71\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.194851cm}{\raggedright Mult.: Other: \ensuremath{\alpha}(K)exp=0.0058 \textit{11} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986Br28,B}{1986Br28}). Note\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.194851cm}{\raggedright {\ }{\ }{\ }that \ensuremath{\alpha}(K)exp=0.011 \textit{5} in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1976Ko13,B}{1976Ko13} is consistent\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.194851cm}{\raggedright {\ }{\ }{\ }with M.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1502}&\multicolumn{1}{@{.}l}{4 {\it 3}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{74 {\it 11}}&\multicolumn{1}{r@{}}{1501}&\multicolumn{1}{@{.}l}{89}&\multicolumn{1}{l}{(13/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{9/2\ensuremath{^{-}}}&\multicolumn{1}{l}{(E3)}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{00677 {\it 9}}&\parbox[t][0.3cm]{7.194851cm}{\raggedright \%I\ensuremath{\gamma}\ensuremath{\approx}0.120\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.194851cm}{\raggedright Mult.: \ensuremath{\alpha}(K)exp=0.005 \textit{2} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986Br28,B}{1986Br28}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1504}&\multicolumn{1}{@{.}l}{3 {\it 3}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{78 {\it 18}}&\multicolumn{1}{r@{}}{1504}&\multicolumn{1}{@{.}l}{40}&\multicolumn{1}{l}{(9/2,11/2)\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{9/2\ensuremath{^{-}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{7.194851cm}{\raggedright \%I\ensuremath{\gamma}\ensuremath{\approx}0.126\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1762}&\multicolumn{1}{@{.}l}{9 {\it 6}}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{7 {\it 3}}&\multicolumn{1}{r@{}}{1762}&\multicolumn{1}{@{.}l}{9}&\multicolumn{1}{l}{(11/2,13/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{9/2\ensuremath{^{-}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{7.194851cm}{\raggedright \%I\ensuremath{\gamma}\ensuremath{\approx}0.60\vspace{0.1cm}}&\\
\end{longtable}
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PO37GAMMA0}{\dagger}}}} From adopted gammas, unless otherwise stated.}\\
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PO37GAMMA1}{\ddagger}}}} From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986Br28,B}{1986Br28}.}\\
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PO37GAMMA2}{\#}}}} Estimated from coincidence intensities in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986Br28,B}{1986Br28}.}\\
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PO37GAMMA3}{@}}}} For absolute intensity per 100 decays, multiply by{ }\ensuremath{\approx}0.162.}\\
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PO37GAMMA4}{\&}}}} Total theoretical internal conversion coefficients, calculated using the BrIcc code (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2008Ki07,B}{2008Ki07}) with Frozen orbital approximation based on \ensuremath{\gamma}-ray energies,}\\
\parbox[b][0.3cm]{21.881866cm}{{\ }{\ }assigned multipolarities, and mixing ratios, unless otherwise specified.}\\
\vspace{0.5cm}
\end{landscape}\clearpage
\clearpage
\begin{figure}[h]
\begin{center}
\includegraphics{201BI37-0.ps}\\
\end{center}
\end{figure}
\clearpage
\subsection[\hspace{-0.2cm}\ensuremath{^{\textnormal{205}}}At \ensuremath{\alpha} decay]{ }
\vspace{-27pt}
\vspace{0.3cm}
\hypertarget{AT38}{{\bf \small \underline{\ensuremath{^{\textnormal{205}}}At \ensuremath{\alpha} decay}}}\\
\vspace{4pt}
\vspace{8pt}
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}Parent: $^{205}$At: E=0.0; J$^{\pi}$=9/2\ensuremath{^{-}}; T$_{1/2}$=26.9 min {\it 8}; Q(\ensuremath{\alpha})=6019.6 {\it 17}; \%\ensuremath{\alpha} decay=10 {\it 2}
}\\
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}\ensuremath{^{205}}At-\ensuremath{J^{\pi}}, T\ensuremath{_{\textnormal{1/2}}} and \%\ensuremath{\alpha} decay from \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2020Ko17,B}{2020Ko17}; Q(\ensuremath{\alpha}) from \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2021Wa16,B}{2021Wa16}.}\\
\vspace{12pt}
\underline{$^{201}$Bi Levels}\\
\begin{longtable}{ccccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E(level)$^{}$}&J$^{\pi}$$^{{\hyperlink{BI38LEVEL0}{\dagger}}}$&\multicolumn{2}{c}{T$_{1/2}$$^{{\hyperlink{BI38LEVEL0}{\dagger}}}$}&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&
\endfirsthead
\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{l}{9/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{103}&\multicolumn{1}{@{.}l}{2 min {\it 17}}&\\
\end{longtable}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{BI38LEVEL0}{\dagger}}}} From Adopted Levels.}\\
\vspace{0.5cm}
\underline{\ensuremath{\alpha} radiations}\\
\begin{longtable}{ccccccccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E$\alpha^{{}}$}&\multicolumn{2}{c}{E(level)}&\multicolumn{2}{c}{I$\alpha^{{\hyperlink{BI38DECAY1}{\ddagger}}}$}&\multicolumn{2}{c}{HF$^{{\hyperlink{BI38DECAY0}{\dagger}}}$}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{5902}&\multicolumn{1}{@{.}l}{1 {\it 7}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{5 {\it 4}}&\parbox[t][0.3cm]{12.72328cm}{\raggedright E$\alpha$: From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1991Ry01,B}{1991Ry01}, based on E\ensuremath{\alpha}=5901 keV \textit{5} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1974Ho27,B}{1974Ho27}), 5910 keV \textit{10} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1970DaZM,B}{1970DaZM}),\vspace{0.1cm}}&\\
&&&&&&&&\parbox[t][0.3cm]{12.72328cm}{\raggedright {\ }{\ }{\ }5903 keV \textit{2} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1968Go12,B}{1968Go12}) and 5896 keV \textit{4} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1963Ho18,B}{1963Ho18}); Other: 5900 keV \textit{40} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1954Bu67,B}{1954Bu67}).\vspace{0.1cm}}&\\
\end{longtable}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{BI38DECAY0}{\dagger}}}} Using r\ensuremath{_{\textnormal{0}}}(\ensuremath{^{\textnormal{201}}}Bi)=1.4771 \textit{24} from \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2020Si16,B}{2020Si16}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{BI38DECAY1}{\ddagger}}}} For absolute intensity per 100 decays, multiply by 0.10 \textit{2}.}\\
\vspace{0.5cm}
\clearpage
\subsection[\hspace{-0.2cm}\ensuremath{^{\textnormal{203}}}Tl(\ensuremath{\alpha},6n\ensuremath{\gamma})]{ }
\vspace{-27pt}
\vspace{0.3cm}
\hypertarget{BI39}{{\bf \small \underline{\ensuremath{^{\textnormal{203}}}Tl(\ensuremath{\alpha},6n\ensuremath{\gamma})\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1982Br21,B}{1982Br21}}}}\\
\vspace{4pt}
\vspace{8pt}
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1982Br21,B}{1982Br21}: E(\ensuremath{\alpha})=66, 70 and 77 MeV; Detectors: Ge(Li); Measured: \ensuremath{\gamma}\ensuremath{\gamma} coin, \ensuremath{\gamma}(t), \ensuremath{\gamma}(\ensuremath{\theta}); Deduced: level scheme, \ensuremath{J^{\pi}}, T\ensuremath{_{\textnormal{1/2}}}.}\\
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}Other: \ensuremath{^{\textnormal{204}}}Pb(p,4n\ensuremath{\gamma}),\ensuremath{^{\textnormal{206}}}Pb(p,6n\ensuremath{\gamma}) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1975OHZZ,B}{1975OHZZ} (E(p)=33-52 MeV; Detectors: Ge(Li)), where no delayed \ensuremath{\gamma} rays with T\ensuremath{_{\textnormal{1/2}}}\ensuremath{\geq}1}\\
\parbox[b][0.3cm]{17.7cm}{\ensuremath{\mu}s were observed.}\\
\vspace{12pt}
\underline{$^{201}$Bi Levels}\\
\begin{longtable}{cccccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E(level)$^{{\hyperlink{BI39LEVEL0}{\dagger}}}$}&J$^{\pi}$$^{{\hyperlink{BI39LEVEL1}{\ddagger}}}$&\multicolumn{2}{c}{T$_{1/2}$$^{{\hyperlink{BI39LEVEL2}{\#}}}$}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{l}{9/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{103}&\multicolumn{1}{@{.}l}{2 min {\it 17}}&\parbox[t][0.3cm]{12.188101cm}{\raggedright J\ensuremath{^{\pi}},T\ensuremath{_{1/2}}: From Adopted Levels.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{967}&\multicolumn{1}{@{.}l}{50 {\it 10}}&\multicolumn{1}{l}{13/2\ensuremath{^{-}}}&&&&\\
\multicolumn{1}{r@{}}{1379}&\multicolumn{1}{@{.}l}{40 {\it 15}}&\multicolumn{1}{l}{15/2\ensuremath{^{-}}}&&&&\\
\multicolumn{1}{r@{}}{1474}&\multicolumn{1}{@{.}l}{60 {\it 18}}&\multicolumn{1}{l}{17/2\ensuremath{^{-}}}&&&&\\
\multicolumn{1}{r@{}}{1746}&\multicolumn{1}{@{.}l}{50 {\it 20}}&\multicolumn{1}{l}{17/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{9}&\multicolumn{1}{@{.}l}{6 ns {\it 6}}&\parbox[t][0.3cm]{12.188101cm}{\raggedright T\ensuremath{_{1/2}}: From \ensuremath{\gamma}\ensuremath{\gamma}(t) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1982Br21,B}{1982Br21} using gates on 272\ensuremath{\gamma}, 412\ensuremath{\gamma} and 967\ensuremath{\gamma} below the isomer\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{12.188101cm}{\raggedright {\ }{\ }{\ }(stop) and 186\ensuremath{\gamma} above the isomer (start). The result is compared to the \ensuremath{\gamma}\ensuremath{\gamma}(t) value\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{12.188101cm}{\raggedright {\ }{\ }{\ }deduced using 412\ensuremath{\gamma} and 967\ensuremath{\gamma} (start) and 272\ensuremath{\gamma} (stop), but one should note that there is\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{12.188101cm}{\raggedright {\ }{\ }{\ }a difference between the time walk for the 186\ensuremath{\gamma} and 272\ensuremath{\gamma}, and hence, the value quoted\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{12.188101cm}{\raggedright {\ }{\ }{\ }for the half-life may be somewhat higher than it should be.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1932}&\multicolumn{1}{@{.}l}{20 {\it 23}}&\multicolumn{1}{l}{21/2\ensuremath{^{+}}}&&&&\\
\multicolumn{1}{r@{}}{1932}&\multicolumn{1}{@{.}l}{20+x {\it 23}}&\multicolumn{1}{l}{(25/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{210}&\multicolumn{1}{@{ }l}{ns {\it 20}}&\parbox[t][0.3cm]{12.188101cm}{\raggedright T\ensuremath{_{1/2}}: From \ensuremath{\gamma}\ensuremath{\gamma}(t) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1982Br21,B}{1982Br21} using gates on 617\ensuremath{\gamma} above the isomer (start) and 186\ensuremath{\gamma},\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{12.188101cm}{\raggedright {\ }{\ }{\ }272\ensuremath{\gamma}, 412\ensuremath{\gamma} and 967\ensuremath{\gamma} below the isomer (stop).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2298}&\multicolumn{1}{@{.}l}{98+x {\it 9}}&\multicolumn{1}{l}{(27/2)}&&&&\\
\multicolumn{1}{r@{}}{2549}&\multicolumn{1}{@{.}l}{43+x {\it 9}}&\multicolumn{1}{l}{(27/2)}&&&&\\
\multicolumn{1}{r@{}}{2611}&\multicolumn{1}{@{.}l}{90+x {\it 10}}&\multicolumn{1}{l}{(27/2)}&&&&\\
\multicolumn{1}{r@{}}{2739}&\multicolumn{1}{@{.}l}{95+x {\it 10}}&\multicolumn{1}{l}{(29/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{160}&\multicolumn{1}{@{ }l}{ns {\it 30}}&\parbox[t][0.3cm]{12.188101cm}{\raggedright T\ensuremath{_{1/2}}: From \ensuremath{\gamma}\ensuremath{\gamma}(t) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1982Br21,B}{1982Br21} using gates on 617\ensuremath{\gamma} above the isomer (start) and 186\ensuremath{\gamma},\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{12.188101cm}{\raggedright {\ }{\ }{\ }272\ensuremath{\gamma}, 412\ensuremath{\gamma} and 967\ensuremath{\gamma} below the isomer (stop).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3238}&\multicolumn{1}{@{.}l}{89+x {\it 13}}&\multicolumn{1}{l}{(31/2\ensuremath{^{-}})}&&&&\\
\multicolumn{1}{r@{}}{3526}&\multicolumn{1}{@{.}l}{22+x {\it 13}}&&&&&\\
\multicolumn{1}{r@{}}{3727}&\multicolumn{1}{@{.}l}{55+x {\it 15}}&\multicolumn{1}{l}{(33/2\ensuremath{^{-}})}&&&&\\
\multicolumn{1}{r@{}}{3810}&\multicolumn{1}{@{.}l}{42+x {\it 17}}&&&&&\\
\end{longtable}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{BI39LEVEL0}{\dagger}}}} From a least-squares fit to E\ensuremath{\gamma}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{BI39LEVEL1}{\ddagger}}}} Based on \ensuremath{\gamma}(\ensuremath{\theta}) and \ensuremath{\alpha}(exp) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1982Br21,B}{1982Br21}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{BI39LEVEL2}{\#}}}} T\ensuremath{_{\textnormal{1/2}}}\ensuremath{>}10 ns is reported in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1982Br21,B}{1982Br21} for a level above 3810+X keV.}\\
\vspace{0.5cm}
\underline{$\gamma$($^{201}$Bi)}\\
\begin{longtable}{ccccccccc@{}ccccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{BI39GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{I\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{BI39GAMMA1}{\ddagger}}}}&\multicolumn{2}{c}{E\ensuremath{_{i}}(level)}&J\ensuremath{^{\pi}_{i}}&\multicolumn{2}{c}{E\ensuremath{_{f}}}&J\ensuremath{^{\pi}_{f}}&Mult.\ensuremath{^{\hyperlink{BI39GAMMA2}{\#}}}&\multicolumn{2}{c}{I\ensuremath{\gamma} (delayed)\ensuremath{^{\hyperlink{BI39GAMMA3}{@}}}}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{95}&\multicolumn{1}{@{.}l}{2 {\it 1}}&\multicolumn{1}{r@{}}{6}&\multicolumn{1}{@{.}l}{4 {\it 6}}&\multicolumn{1}{r@{}}{1474}&\multicolumn{1}{@{.}l}{60}&\multicolumn{1}{l}{17/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{1379}&\multicolumn{1}{@{.}l}{40 }&\multicolumn{1}{@{}l}{15/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1(+E2)}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{4.514381cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}={\textminus}0.11 \textit{5}, A\ensuremath{_{\textnormal{4}}}={\textminus}0.07 \textit{6};\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.514381cm}{\raggedright {\ }{\ }{\ }\ensuremath{\alpha}(exp)=11 \textit{2} from intensity\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.514381cm}{\raggedright {\ }{\ }{\ }balance considerations in\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.514381cm}{\raggedright {\ }{\ }{\ }\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1982Br21,B}{1982Br21}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{128}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{BI39GAMMA4}{\&}}} {\it 1}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{2739}&\multicolumn{1}{@{.}l}{95+x}&\multicolumn{1}{l}{(29/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{2611}&\multicolumn{1}{@{.}l}{90+x }&\multicolumn{1}{@{}l}{(27/2)}&&&&&\\
\multicolumn{1}{r@{}}{185}&\multicolumn{1}{@{.}l}{7 {\it 1}}&\multicolumn{1}{r@{}}{58}&\multicolumn{1}{@{ }l}{{\it 6}}&\multicolumn{1}{r@{}}{1932}&\multicolumn{1}{@{.}l}{20}&\multicolumn{1}{l}{21/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1746}&\multicolumn{1}{@{.}l}{50 }&\multicolumn{1}{@{}l}{17/2\ensuremath{^{+}}}&\multicolumn{1}{l}{E2}&\multicolumn{1}{r@{}}{65}&\multicolumn{1}{@{ }l}{{\it 7}}&\parbox[t][0.3cm]{4.514381cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}=+0.31 \textit{1}, A\ensuremath{_{\textnormal{4}}}={\textminus}0.09 \textit{1};\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.514381cm}{\raggedright {\ }{\ }{\ }\ensuremath{\alpha}(exp)=0.58 \textit{6} deduced from\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.514381cm}{\raggedright {\ }{\ }{\ }the out-of-beam intensity\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.514381cm}{\raggedright {\ }{\ }{\ }balance in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1982Br21,B}{1982Br21}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{190}&\multicolumn{1}{@{.}l}{5 {\it 1}}&\multicolumn{1}{r@{}}{7}&\multicolumn{1}{@{.}l}{0 {\it 7}}&\multicolumn{1}{r@{}}{2739}&\multicolumn{1}{@{.}l}{95+x}&\multicolumn{1}{l}{(29/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{2549}&\multicolumn{1}{@{.}l}{43+x }&\multicolumn{1}{@{}l}{(27/2)}&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{5}&\multicolumn{1}{@{.}l}{8 {\it 6}}&\parbox[t][0.3cm]{4.514381cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}={\textminus}0.27 \textit{4}, A\ensuremath{_{\textnormal{4}}}={\textminus}0.09 \textit{5}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$250}&\multicolumn{1}{@{.}l}{0 {\it 1}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{0 {\it 1}}&&&&&&&&&&\parbox[t][0.3cm]{4.514381cm}{\raggedright E\ensuremath{_{\gamma}}: Suggested to populates the\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.514381cm}{\raggedright {\ }{\ }{\ }17/2\ensuremath{^{+}} leve in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1982Br21,B}{1982Br21}, but\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.514381cm}{\raggedright {\ }{\ }{\ }not placed in the level scheme\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.514381cm}{\raggedright {\ }{\ }{\ }by the authors.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{271}&\multicolumn{1}{@{.}l}{9 {\it 1}}&\multicolumn{1}{r@{}}{85}&\multicolumn{1}{@{ }l}{{\it 9}}&\multicolumn{1}{r@{}}{1746}&\multicolumn{1}{@{.}l}{50}&\multicolumn{1}{l}{17/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1474}&\multicolumn{1}{@{.}l}{60 }&\multicolumn{1}{@{}l}{17/2\ensuremath{^{-}}}&\multicolumn{1}{l}{E1}&\multicolumn{1}{r@{}}{97}&\multicolumn{1}{@{ }l}{{\it 10}}&\parbox[t][0.3cm]{4.514381cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}=+0.37 \textit{1}, A\ensuremath{_{\textnormal{4}}}={\textminus}0.01 \textit{1};\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.514381cm}{\raggedright {\ }{\ }{\ }\ensuremath{\alpha}(exp)=0.03 \textit{4} deduced from\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.514381cm}{\raggedright {\ }{\ }{\ }the out-of-beam intensity\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.514381cm}{\raggedright {\ }{\ }{\ }balance in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1982Br21,B}{1982Br21}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$275}&\multicolumn{1}{@{.}l}{3 {\it 1}}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{0 {\it 2}}&&&&&&&\multicolumn{1}{l}{D}&&&\parbox[t][0.3cm]{4.514381cm}{\raggedright E\ensuremath{_{\gamma}}: Suggested to populates the\vspace{0.1cm}}&\\
\end{longtable}
\begin{textblock}{29}(0,27.3)
Continued on next page (footnotes at end of table)
\end{textblock}
\clearpage
\begin{longtable}{ccccccccc@{}ccccc@{\extracolsep{\fill}}c}
\\[-.4cm]
\multicolumn{15}{c}{{\bf \small \underline{\ensuremath{^{\textnormal{203}}}Tl(\ensuremath{\alpha},6n\ensuremath{\gamma})\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1982Br21,B}{1982Br21} (continued)}}}\\
\multicolumn{15}{c}{~}\\
\multicolumn{15}{c}{\underline{$\gamma$($^{201}$Bi) (continued)}}\\
\multicolumn{15}{c}{~~~}\\
\multicolumn{2}{c}{E\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{BI39GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{I\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{BI39GAMMA1}{\ddagger}}}}&\multicolumn{2}{c}{E\ensuremath{_{i}}(level)}&J\ensuremath{^{\pi}_{i}}&\multicolumn{2}{c}{E\ensuremath{_{f}}}&J\ensuremath{^{\pi}_{f}}&Mult.\ensuremath{^{\hyperlink{BI39GAMMA2}{\#}}}&\multicolumn{2}{c}{I\ensuremath{\gamma} (delayed)\ensuremath{^{\hyperlink{BI39GAMMA3}{@}}}}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endhead
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.4822803cm}{\raggedright {\ }{\ }{\ }17/2\ensuremath{^{+}} leve in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1982Br21,B}{1982Br21}, but\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.4822803cm}{\raggedright {\ }{\ }{\ }not placed in the level scheme\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.4822803cm}{\raggedright {\ }{\ }{\ }by the authors.\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.4822803cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}={\textminus}0.21 \textit{12}, A\ensuremath{_{\textnormal{4}}}=+0.12\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.4822803cm}{\raggedright {\ }{\ }{\ }\textit{15}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{284}&\multicolumn{1}{@{.}l}{2 {\it 1}}&\multicolumn{1}{r@{}}{9}&\multicolumn{1}{@{.}l}{1 {\it 9}}&\multicolumn{1}{r@{}}{3810}&\multicolumn{1}{@{.}l}{42+x}&&\multicolumn{1}{r@{}}{3526}&\multicolumn{1}{@{.}l}{22+x }&&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{1 {\it 1}}&\parbox[t][0.3cm]{4.4822803cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}=+0.25 \textit{3}, A\ensuremath{_{\textnormal{4}}}={\textminus}0.07 \textit{4}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{287}&\multicolumn{1}{@{.}l}{3 {\it 1}}&\multicolumn{1}{r@{}}{8}&\multicolumn{1}{@{.}l}{1 {\it 8}}&\multicolumn{1}{r@{}}{3526}&\multicolumn{1}{@{.}l}{22+x}&&\multicolumn{1}{r@{}}{3238}&\multicolumn{1}{@{.}l}{89+x }&\multicolumn{1}{@{}l}{(31/2\ensuremath{^{-}})}&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{2 {\it 1}}&\parbox[t][0.3cm]{4.4822803cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}={\textminus}0.28 \textit{3}, A\ensuremath{_{\textnormal{4}}}=+0.01 \textit{4}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{366}&\multicolumn{1}{@{.}l}{8 {\it 1}}&\multicolumn{1}{r@{}}{12}&\multicolumn{1}{@{ }l}{{\it 1}}&\multicolumn{1}{r@{}}{2298}&\multicolumn{1}{@{.}l}{98+x}&\multicolumn{1}{l}{(27/2)}&\multicolumn{1}{r@{}}{1932}&\multicolumn{1}{@{.}l}{20+x }&\multicolumn{1}{@{}l}{(25/2\ensuremath{^{+}})}&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{12}&\multicolumn{1}{@{ }l}{{\it 1}}&\parbox[t][0.3cm]{4.4822803cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}={\textminus}0.14 \textit{3}, A\ensuremath{_{\textnormal{4}}}={\textminus}0.01 \textit{4}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{411}&\multicolumn{1}{@{.}l}{9 {\it 1}}&\multicolumn{1}{r@{}}{83}&\multicolumn{1}{@{ }l}{{\it 8}}&\multicolumn{1}{r@{}}{1379}&\multicolumn{1}{@{.}l}{40}&\multicolumn{1}{l}{15/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{967}&\multicolumn{1}{@{.}l}{50 }&\multicolumn{1}{@{}l}{13/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{88}&\multicolumn{1}{@{ }l}{{\it 9}}&\parbox[t][0.3cm]{4.4822803cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}={\textminus}0.34 \textit{1}, A\ensuremath{_{\textnormal{4}}}={\textminus}0.01 \textit{1};\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.4822803cm}{\raggedright {\ }{\ }{\ }\ensuremath{\alpha}(exp)=0.15 \textit{8} from the\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.4822803cm}{\raggedright {\ }{\ }{\ }out-of-beam intensity balance\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.4822803cm}{\raggedright {\ }{\ }{\ }in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1982Br21,B}{1982Br21}.\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.4822803cm}{\raggedright \ensuremath{\delta}: \ensuremath{\approx}0.03 or \ensuremath{\approx}4.4 from\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.4822803cm}{\raggedright {\ }{\ }{\ }\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1982Br21,B}{1982Br21}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{441}&\multicolumn{1}{@{.}l}{0 {\it 1}}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{5 {\it 3}}&\multicolumn{1}{r@{}}{2739}&\multicolumn{1}{@{.}l}{95+x}&\multicolumn{1}{l}{(29/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{2298}&\multicolumn{1}{@{.}l}{98+x }&\multicolumn{1}{@{}l}{(27/2)}&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{5 {\it 3}}&\parbox[t][0.3cm]{4.4822803cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}={\textminus}0.64 \textit{12}, A\ensuremath{_{\textnormal{4}}}={\textminus}0.19\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.4822803cm}{\raggedright {\ }{\ }{\ }\textit{16}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{498}&\multicolumn{1}{@{.}l}{9 {\it 1}}&\multicolumn{1}{r@{}}{22}&\multicolumn{1}{@{ }l}{{\it 2}}&\multicolumn{1}{r@{}}{3238}&\multicolumn{1}{@{.}l}{89+x}&\multicolumn{1}{l}{(31/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{2739}&\multicolumn{1}{@{.}l}{95+x }&\multicolumn{1}{@{}l}{(29/2\ensuremath{^{-}})}&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{4 {\it 2}}&\parbox[t][0.3cm]{4.4822803cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}={\textminus}0.39 \textit{2}, A\ensuremath{_{\textnormal{4}}}={\textminus}0.05 \textit{3}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{617}&\multicolumn{1}{@{.}l}{2 {\it 1}}&\multicolumn{1}{r@{}}{34}&\multicolumn{1}{@{ }l}{{\it 3}}&\multicolumn{1}{r@{}}{2549}&\multicolumn{1}{@{.}l}{43+x}&\multicolumn{1}{l}{(27/2)}&\multicolumn{1}{r@{}}{1932}&\multicolumn{1}{@{.}l}{20+x }&\multicolumn{1}{@{}l}{(25/2\ensuremath{^{+}})}&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{32}&\multicolumn{1}{@{ }l}{{\it 3}}&\parbox[t][0.3cm]{4.4822803cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}={\textminus}0.19 \textit{2}, A\ensuremath{_{\textnormal{4}}}={\textminus}0.02 \textit{3}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{679}&\multicolumn{1}{@{.}l}{7 {\it 1}}&\multicolumn{1}{r@{}}{16}&\multicolumn{1}{@{.}l}{0 {\it 16}}&\multicolumn{1}{r@{}}{2611}&\multicolumn{1}{@{.}l}{90+x}&\multicolumn{1}{l}{(27/2)}&\multicolumn{1}{r@{}}{1932}&\multicolumn{1}{@{.}l}{20+x }&\multicolumn{1}{@{}l}{(25/2\ensuremath{^{+}})}&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{11}&\multicolumn{1}{@{ }l}{{\it 1}}&\parbox[t][0.3cm]{4.4822803cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}={\textminus}0.47 \textit{3}, A\ensuremath{_{\textnormal{4}}}={\textminus}0.06 \textit{4}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{786}&\multicolumn{1}{@{.}l}{3 {\it 1}}&\multicolumn{1}{r@{}}{14}&\multicolumn{1}{@{.}l}{0 {\it 14}}&\multicolumn{1}{r@{}}{3526}&\multicolumn{1}{@{.}l}{22+x}&&\multicolumn{1}{r@{}}{2739}&\multicolumn{1}{@{.}l}{95+x }&\multicolumn{1}{@{}l}{(29/2\ensuremath{^{-}})}&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{4.4822803cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}={\textminus}0.22 \textit{4}, A\ensuremath{_{\textnormal{4}}}={\textminus}0.08 \textit{5}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{967}&\multicolumn{1}{@{.}l}{5 {\it 1}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{ }l}{{\it 10}}&\multicolumn{1}{r@{}}{967}&\multicolumn{1}{@{.}l}{50}&\multicolumn{1}{l}{13/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{9/2\ensuremath{^{-}}}&\multicolumn{1}{l}{E2}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{ }l}{{\it 10}}&\parbox[t][0.3cm]{4.4822803cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}=+0.30 \textit{1}, A\ensuremath{_{\textnormal{4}}}={\textminus}0.05 \textit{1}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{987}&\multicolumn{1}{@{.}l}{6 {\it 1}}&\multicolumn{1}{r@{}}{12}&\multicolumn{1}{@{ }l}{{\it 1}}&\multicolumn{1}{r@{}}{3727}&\multicolumn{1}{@{.}l}{55+x}&\multicolumn{1}{l}{(33/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{2739}&\multicolumn{1}{@{.}l}{95+x }&\multicolumn{1}{@{}l}{(29/2\ensuremath{^{-}})}&\multicolumn{1}{l}{E2}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{8 {\it 4}}&\parbox[t][0.3cm]{4.4822803cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}=+0.31 \textit{5}, A\ensuremath{_{\textnormal{4}}}={\textminus}0.01 \textit{5}.\vspace{0.1cm}}&\\
\end{longtable}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{BI39GAMMA0}{\dagger}}}} From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1982Br21,B}{1982Br21}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{BI39GAMMA1}{\ddagger}}}} From E(\ensuremath{\alpha})=77 MeV in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1982Br21,B}{1982Br21}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{BI39GAMMA2}{\#}}}} Based on \ensuremath{\gamma}(\ensuremath{\theta}) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1982Br21,B}{1982Br21}, unless otherwise stated.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{BI39GAMMA3}{@}}}} From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1982Br21,B}{1982Br21}. Measured in the out-of-beam time region of 50 to 130 ns.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{BI39GAMMA4}{\&}}}} Placement of transition in the level scheme is uncertain.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{BI39GAMMA5}{x}}}} \ensuremath{\gamma} ray not placed in level scheme.}\\
\vspace{0.5cm}
\clearpage
\begin{figure}[h]
\begin{center}
\includegraphics{201BI39-0.ps}\\
\end{center}
\end{figure}
\clearpage
\subsection[\hspace{-0.2cm}\ensuremath{^{\textnormal{196}}}Pt(\ensuremath{^{\textnormal{10}}}B,5n\ensuremath{\gamma})]{ }
\vspace{-27pt}
\vspace{0.3cm}
\hypertarget{BI40}{{\bf \small \underline{\ensuremath{^{\textnormal{196}}}Pt(\ensuremath{^{\textnormal{10}}}B,5n\ensuremath{\gamma})\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1985Pi05,B}{1985Pi05}}}}\\
\vspace{4pt}
\vspace{8pt}
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}E(\ensuremath{^{\textnormal{10}}}B)=57-72 MeV; Target: \ensuremath{^{\textnormal{196}}}Pt, enriched \ensuremath{>}95\% and 3.6 mg/cm\ensuremath{^{\textnormal{2}}} thick; Detectors: two Ge(Li) and one planar Ge(intrinsic);}\\
\parbox[b][0.3cm]{17.7cm}{Measured: excitation functions, \ensuremath{\gamma}(\ensuremath{\theta}),DCO, \ensuremath{\gamma}\ensuremath{\gamma}, \ensuremath{\gamma}(t) {\textminus} pulsed beam with 10 ns on and 2 \ensuremath{\mu}s off periods. Deduced: level scheme,}\\
\parbox[b][0.3cm]{17.7cm}{\ensuremath{J^{\pi}}, T\ensuremath{_{\textnormal{1/2}}}.}\\
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}Others: \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1974GiZX,B}{1974GiZX}, \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1973GiZW,B}{1973GiZW}.}\\
\vspace{12pt}
\underline{$^{201}$Bi Levels}\\
\begin{longtable}{cccccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E(level)$^{{\hyperlink{BI40LEVEL0}{\dagger}}}$}&J$^{\pi}$$^{{\hyperlink{BI40LEVEL1}{\ddagger}}}$&\multicolumn{2}{c}{T$_{1/2}$$^{{\hyperlink{BI40LEVEL2}{\#}}}$}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{BI40LEVEL3}{@}}}}}&\multicolumn{1}{l}{9/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{103}&\multicolumn{1}{@{.}l}{2 min {\it 17}}&\parbox[t][0.3cm]{11.812691cm}{\raggedright J\ensuremath{^{\pi}},T\ensuremath{_{1/2}}: From Adopted Levels.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{964}&\multicolumn{1}{@{.}l}{2\ensuremath{^{{\hyperlink{BI40LEVEL4}{\&}}}} {\it 5}}&\multicolumn{1}{l}{11/2\ensuremath{^{-}}}&&&&\\
\multicolumn{1}{r@{}}{967}&\multicolumn{1}{@{.}l}{39\ensuremath{^{{\hyperlink{BI40LEVEL4}{\&}}}} {\it 25}}&\multicolumn{1}{l}{13/2\ensuremath{^{-}}}&&&&\\
\multicolumn{1}{r@{}}{1379}&\multicolumn{1}{@{.}l}{2\ensuremath{^{{\hyperlink{BI40LEVEL5}{a}}}} {\it 3}}&\multicolumn{1}{l}{15/2\ensuremath{^{-}}}&&&&\\
\multicolumn{1}{r@{}}{1474}&\multicolumn{1}{@{.}l}{5\ensuremath{^{{\hyperlink{BI40LEVEL5}{a}}}} {\it 4}}&\multicolumn{1}{l}{17/2\ensuremath{^{-}}}&&&&\\
\multicolumn{1}{r@{}}{1719}&\multicolumn{1}{@{.}l}{1 {\it 7}}&\multicolumn{1}{l}{(13/2\ensuremath{^{-}})}&&&&\\
\multicolumn{1}{r@{}}{1746}&\multicolumn{1}{@{.}l}{4\ensuremath{^{{\hyperlink{BI40LEVEL6}{b}}}} {\it 4}}&\multicolumn{1}{l}{17/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{5}&\multicolumn{1}{@{.}l}{1 ns {\it 13}}&\parbox[t][0.3cm]{11.812691cm}{\raggedright T\ensuremath{_{1/2}}: From 185.8\ensuremath{\gamma}-271.91\ensuremath{\gamma}(\ensuremath{\Delta}t) and the centroid shift analysis in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1985Pi05,B}{1985Pi05}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1932}&\multicolumn{1}{@{.}l}{2\ensuremath{^{{\hyperlink{BI40LEVEL7}{c}}}} {\it 5}}&\multicolumn{1}{l}{21/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{$<$40}&\multicolumn{1}{@{ }l}{ns}&\parbox[t][0.3cm]{11.812691cm}{\raggedright E(level): The absence of prompt component in the time spectrum produced by gating\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{11.812691cm}{\raggedright {\ }{\ }{\ }on the 185.8\ensuremath{\gamma}, shown in figure 4(c) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1985Pi05,B}{1985Pi05}, suggests that 185.8\ensuremath{\gamma} directly\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{11.812691cm}{\raggedright {\ }{\ }{\ }depopulates an isomeric state.\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{11.812691cm}{\raggedright T\ensuremath{_{1/2}}: Estimated value from 185.8\ensuremath{\gamma}(t) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1985Pi05,B}{1985Pi05}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1932}&\multicolumn{1}{@{.}l}{2+x\ensuremath{^{{\hyperlink{BI40LEVEL8}{d}}}}}&\multicolumn{1}{l}{(25/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{118}&\multicolumn{1}{@{ }l}{ns {\it 28}}&\parbox[t][0.3cm]{11.812691cm}{\raggedright E(level): X is expected to be less than 80 keV (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1985Pi05,B}{1985Pi05}), otherwise a \ensuremath{\gamma}-ray\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{11.812691cm}{\raggedright {\ }{\ }{\ }transition would be observed. The assignment is consistent with the energy difference\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{11.812691cm}{\raggedright {\ }{\ }{\ }between the \ensuremath{J^{\pi}}=7\ensuremath{^{-}} and 9\ensuremath{^{-}} states in \ensuremath{^{\textnormal{200}}}Pb, as well as with the systematics in\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{11.812691cm}{\raggedright {\ }{\ }{\ }neighboring \ensuremath{^{\textnormal{203}}}Bi and \ensuremath{^{\textnormal{205}}}Bi isotopes.\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{11.812691cm}{\raggedright T\ensuremath{_{1/2}}: From \ensuremath{\gamma}\ensuremath{\gamma}(\ensuremath{\Delta}t) using time-difference spectra between 617.3\ensuremath{\gamma} with 967.4\ensuremath{\gamma},\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{11.812691cm}{\raggedright {\ }{\ }{\ }411.9\ensuremath{\gamma}, 271.9\ensuremath{\gamma} and 185.8\ensuremath{\gamma}\hphantom{a}(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1985Pi05,B}{1985Pi05}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1971}&\multicolumn{1}{@{.}l}{2+x\ensuremath{^{{\hyperlink{BI40LEVEL8}{d}}}} {\it 4}}&\multicolumn{1}{l}{27/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{105}&\multicolumn{1}{@{ }l}{ns {\it 75}}&\parbox[t][0.3cm]{11.812691cm}{\raggedright T\ensuremath{_{1/2}}: From \ensuremath{\gamma}\ensuremath{\gamma}(\ensuremath{\Delta}t) using time-difference spectra between 679.8\ensuremath{\gamma} with 967.4\ensuremath{\gamma},\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{11.812691cm}{\raggedright {\ }{\ }{\ }411.9\ensuremath{\gamma}, 271.9\ensuremath{\gamma} and 185.8\ensuremath{\gamma}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2299}&\multicolumn{1}{@{.}l}{0+x {\it 3}}&\multicolumn{1}{l}{27/2\ensuremath{^{+}}}&&&&\\
\multicolumn{1}{r@{}}{2549}&\multicolumn{1}{@{.}l}{40+x {\it 19}}&\multicolumn{1}{l}{27/2\ensuremath{^{+}}}&&&&\\
\multicolumn{1}{r@{}}{2589}&\multicolumn{1}{@{.}l}{6+x {\it 3}}&&&&&\\
\multicolumn{1}{r@{}}{2651}&\multicolumn{1}{@{.}l}{0+x {\it 3}}&\multicolumn{1}{l}{29/2\ensuremath{^{+}}}&&&&\\
\multicolumn{1}{r@{}}{2668}&\multicolumn{1}{@{.}l}{21+x {\it 22}}&\multicolumn{1}{l}{(29/2\ensuremath{^{+}})}&&&&\\
\multicolumn{1}{r@{}}{2739}&\multicolumn{1}{@{.}l}{91+x\ensuremath{^{{\hyperlink{BI40LEVEL9}{e}}}} {\it 25}}&\multicolumn{1}{l}{29/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{124}&\multicolumn{1}{@{ }l}{ns {\it 4}}&\parbox[t][0.3cm]{11.812691cm}{\raggedright T\ensuremath{_{1/2}}: From 617.3\ensuremath{\gamma}(t) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1985Pi05,B}{1985Pi05}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2994}&\multicolumn{1}{@{.}l}{6+x {\it 8}}&&&&&\\
\multicolumn{1}{r@{}}{3011}&\multicolumn{1}{@{.}l}{4+x {\it 5}}&&&&&\\
\multicolumn{1}{r@{}}{3238}&\multicolumn{1}{@{.}l}{8+x {\it 4}}&\multicolumn{1}{l}{31/2\ensuremath{^{-}}}&&&&\\
\multicolumn{1}{r@{}}{3422}&\multicolumn{1}{@{.}l}{8+x {\it 9}}&&&&&\\
\multicolumn{1}{r@{}}{3526}&\multicolumn{1}{@{.}l}{4+x {\it 4}}&\multicolumn{1}{l}{33/2\ensuremath{^{-}}}&&&&\\
\multicolumn{1}{r@{}}{3592}&\multicolumn{1}{@{.}l}{2+x? {\it 10}}&&&&&\\
\multicolumn{1}{r@{}}{3638}&\multicolumn{1}{@{.}l}{3+x {\it 5}}&\multicolumn{1}{l}{(33/2\ensuremath{^{-}})}&&&&\\
\multicolumn{1}{r@{}}{3706}&\multicolumn{1}{@{.}l}{6+x {\it 5}}&&&&&\\
\multicolumn{1}{r@{}}{3810}&\multicolumn{1}{@{.}l}{7+x {\it 4}}&\multicolumn{1}{l}{33/2\ensuremath{^{-}}}&&&&\\
\multicolumn{1}{r@{}}{3922}&\multicolumn{1}{@{.}l}{8+x {\it 6}}&\multicolumn{1}{l}{35/2\ensuremath{^{-}}}&&&&\\
\multicolumn{1}{r@{}}{4075}&\multicolumn{1}{@{.}l}{2+x? {\it 7}}&&&&&\\
\multicolumn{1}{r@{}}{4484}&\multicolumn{1}{@{.}l}{4+x {\it 9}}&\multicolumn{1}{l}{(35/2\ensuremath{^{-}})}&&&&\\
\multicolumn{1}{r@{}}{5282}&\multicolumn{1}{@{.}l}{2+x {\it 10}}&\multicolumn{1}{l}{(37/2\ensuremath{^{-}})}&&&&\\
\end{longtable}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{BI40LEVEL0}{\dagger}}}} From a least-squares fit to E\ensuremath{\gamma}. X is expected to be less than 80 keV, otherwise a \ensuremath{\gamma}-ray transition would be observed. The}\\
\parbox[b][0.3cm]{17.7cm}{{\ }{\ }assignment is based on similarities with the \ensuremath{J^{\pi}}=7\ensuremath{^{-}} and 9\ensuremath{^{-}} states in \ensuremath{^{\textnormal{200}}}Pb, as well as with the systematics in neighboring \ensuremath{^{\textnormal{203}}}Bi}\\
\parbox[b][0.3cm]{17.7cm}{{\ }{\ }and \ensuremath{^{\textnormal{205}}}Bi isotopes.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{BI40LEVEL1}{\ddagger}}}} From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1985Pi05,B}{1985Pi05}, unless otherwise stated.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{BI40LEVEL2}{\#}}}} An isomer with T\ensuremath{_{\textnormal{1/2}}}=14 ns \textit{3} was found above the 3526.4+X keV level in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1985Pi05,B}{1985Pi05}. Note, that an isomer with T\ensuremath{_{\textnormal{1/2}}}\ensuremath{\approx}10 ns was}\\
\parbox[b][0.3cm]{17.7cm}{{\ }{\ }also reported in \ensuremath{^{\textnormal{203}}}Tl(\ensuremath{\alpha},6n\ensuremath{\gamma}) (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1982Br21,B}{1982Br21}) at or above the 3810+X level.}\\
\begin{textblock}{29}(0,27.3)
Continued on next page (footnotes at end of table)
\end{textblock}
\clearpage
\vspace*{-0.5cm}
{\bf \small \underline{\ensuremath{^{\textnormal{196}}}Pt(\ensuremath{^{\textnormal{10}}}B,5n\ensuremath{\gamma})\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1985Pi05,B}{1985Pi05} (continued)}}\\
\vspace{0.3cm}
\underline{$^{201}$Bi Levels (continued)}\\
\vspace{0.3cm}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{BI40LEVEL3}{@}}}} Configuration=\ensuremath{\pi} h\ensuremath{_{\textnormal{9/2}}^{\textnormal{+1}}}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{BI40LEVEL4}{\&}}}} Configuration=\ensuremath{\pi} (h\ensuremath{_{\textnormal{9/2}}^{\textnormal{+1}}})\ensuremath{\otimes}2\ensuremath{^{\textnormal{+}}}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{BI40LEVEL5}{a}}}} Configuration=\ensuremath{\pi} (h\ensuremath{_{\textnormal{9/2}}^{\textnormal{+1}}})\ensuremath{\otimes}4\ensuremath{^{\textnormal{+}}}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{BI40LEVEL6}{b}}}} Admixture of configuration= \ensuremath{\pi} (h\ensuremath{_{\textnormal{9/2}}^{\textnormal{+1}}}) \ensuremath{\nu} (f\ensuremath{_{\textnormal{5/2}}^{\textnormal{$-$1}}},i\ensuremath{_{\textnormal{13/2}}^{\textnormal{$-$1}}})\ensuremath{_{\textnormal{5$-$}}} and configuration=\ensuremath{\pi} (h\ensuremath{_{\textnormal{9/2}}^{\textnormal{+1}}}) \ensuremath{\nu} (p\ensuremath{_{\textnormal{3/2}}^{\textnormal{$-$1}}},i\ensuremath{_{\textnormal{13/2}}^{\textnormal{$-$1}}})\ensuremath{_{\textnormal{5$-$}}}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{BI40LEVEL7}{c}}}} Admixture of configuration= \ensuremath{\pi} (h\ensuremath{_{\textnormal{9/2}}^{\textnormal{+1}}}) \ensuremath{\nu} (f\ensuremath{_{\textnormal{5/2}}^{\textnormal{$-$1}}},i\ensuremath{_{\textnormal{13/2}}^{\textnormal{$-$1}}})\ensuremath{_{\textnormal{7$-$}}} and configuration=\ensuremath{\pi} (h\ensuremath{_{\textnormal{9/2}}^{\textnormal{+1}}}) \ensuremath{\nu} (p\ensuremath{_{\textnormal{3/2}}^{\textnormal{$-$1}}},i\ensuremath{_{\textnormal{13/2}}^{\textnormal{$-$1}}})\ensuremath{_{\textnormal{7$-$}}}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{BI40LEVEL8}{d}}}} Configuration=\ensuremath{\pi} (h\ensuremath{_{\textnormal{9/2}}^{\textnormal{+1}}}) \ensuremath{\nu} (f\ensuremath{_{\textnormal{5/2}}^{\textnormal{$-$1}}},i\ensuremath{_{\textnormal{13/2}}^{\textnormal{$-$1}}})\ensuremath{_{\textnormal{9$-$}}}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{BI40LEVEL9}{e}}}} Configuration=\ensuremath{\pi} (h\ensuremath{_{\textnormal{9/2}}^{\textnormal{+1}}}) \ensuremath{\nu} (i\ensuremath{_{\textnormal{13/2}}^{\textnormal{$-$2}}})\ensuremath{_{\textnormal{12+}}}.}\\
\vspace{0.5cm}
\underline{$\gamma$($^{201}$Bi)}\\
\begin{longtable}{ccccccccc@{}ccccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{BI40GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{I\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{BI40GAMMA1}{\ddagger}}}}&\multicolumn{2}{c}{E\ensuremath{_{i}}(level)}&J\ensuremath{^{\pi}_{i}}&\multicolumn{2}{c}{E\ensuremath{_{f}}}&J\ensuremath{^{\pi}_{f}}&Mult.\ensuremath{^{\hyperlink{BI40GAMMA2}{\#}}}&\multicolumn{2}{c}{\ensuremath{\delta}\ensuremath{^{\hyperlink{BI40GAMMA0}{\dagger}}}}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{(39}&\multicolumn{1}{@{.}l}{0 {\it 4})}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{1971}&\multicolumn{1}{@{.}l}{2+x}&\multicolumn{1}{l}{27/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1932}&\multicolumn{1}{@{.}l}{2+x}&\multicolumn{1}{@{}l}{(25/2\ensuremath{^{+}})}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{4.90204cm}{\raggedright E\ensuremath{_{\gamma}}: From level energy difference.\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.90204cm}{\raggedright {\ }{\ }{\ }The transition was not observed\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.90204cm}{\raggedright {\ }{\ }{\ }directly and the existence is\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.90204cm}{\raggedright {\ }{\ }{\ }based on the \ensuremath{\gamma}\ensuremath{\gamma}-coincidence\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.90204cm}{\raggedright {\ }{\ }{\ }relationships.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{(71}&\multicolumn{1}{@{.}l}{7 {\it 3})}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{2739}&\multicolumn{1}{@{.}l}{91+x}&\multicolumn{1}{l}{29/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{2668}&\multicolumn{1}{@{.}l}{21+x }&\multicolumn{1}{@{}l}{(29/2\ensuremath{^{+}})}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{4.90204cm}{\raggedright E\ensuremath{_{\gamma}}: From level energy difference.\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.90204cm}{\raggedright {\ }{\ }{\ }The transition was not observed\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.90204cm}{\raggedright {\ }{\ }{\ }directly and the existence is\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.90204cm}{\raggedright {\ }{\ }{\ }based on the observed delayed\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.90204cm}{\raggedright {\ }{\ }{\ }component for 118.8\ensuremath{\gamma} and\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.90204cm}{\raggedright {\ }{\ }{\ }736.0\ensuremath{\gamma}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{($<$80)}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{1932}&\multicolumn{1}{@{.}l}{2+x}&\multicolumn{1}{l}{(25/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{1932}&\multicolumn{1}{@{.}l}{2 }&\multicolumn{1}{@{}l}{21/2\ensuremath{^{+}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
\multicolumn{1}{r@{}}{88}&\multicolumn{1}{@{.}l}{88 {\it 12}}&\multicolumn{1}{r@{}}{7}&\multicolumn{1}{@{.}l}{8 {\it 23}}&\multicolumn{1}{r@{}}{2739}&\multicolumn{1}{@{.}l}{91+x}&\multicolumn{1}{l}{29/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{2651}&\multicolumn{1}{@{.}l}{0+x }&\multicolumn{1}{@{}l}{29/2\ensuremath{^{+}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
\multicolumn{1}{r@{}}{95}&\multicolumn{1}{@{.}l}{26 {\it 15}}&\multicolumn{1}{r@{}}{4}&\multicolumn{1}{@{.}l}{7 {\it 12}}&\multicolumn{1}{r@{}}{1474}&\multicolumn{1}{@{.}l}{5}&\multicolumn{1}{l}{17/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{1379}&\multicolumn{1}{@{.}l}{2 }&\multicolumn{1}{@{}l}{15/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1(+E2)}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{4.90204cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}={\textminus}0.04 \textit{18}, A\ensuremath{_{\textnormal{4}}}={\textminus}0.09 \textit{28};\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.90204cm}{\raggedright {\ }{\ }{\ }DCO=0.65 \textit{36}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{118}&\multicolumn{1}{@{.}l}{81 {\it 15}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{61 {\it 9}}&\multicolumn{1}{r@{}}{2668}&\multicolumn{1}{@{.}l}{21+x}&\multicolumn{1}{l}{(29/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{2549}&\multicolumn{1}{@{.}l}{40+x }&\multicolumn{1}{@{}l}{27/2\ensuremath{^{+}}}&\multicolumn{1}{l}{M1(+E2)}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{4.90204cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}={\textminus}0.08 \textit{12}, A\ensuremath{_{\textnormal{4}}}=0.05 \textit{19}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{150}&\multicolumn{1}{@{.}l}{5 {\it 6}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{92 {\it 6}}&\multicolumn{1}{r@{}}{2739}&\multicolumn{1}{@{.}l}{91+x}&\multicolumn{1}{l}{29/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{2589}&\multicolumn{1}{@{.}l}{6+x }&&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{4.90204cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}={\textminus}0.18 \textit{14}, A\ensuremath{_{\textnormal{4}}}={\textminus}0.18 \textit{22}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{152}&\multicolumn{1}{@{.}l}{4 {\it 4}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{46 {\it 15}}&\multicolumn{1}{r@{}}{4075}&\multicolumn{1}{@{.}l}{2+x?}&&\multicolumn{1}{r@{}}{3922}&\multicolumn{1}{@{.}l}{8+x }&\multicolumn{1}{@{}l}{35/2\ensuremath{^{-}}}&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{4.90204cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}={\textminus}0.23 \textit{9}, A\ensuremath{_{\textnormal{4}}}={\textminus}0.07 \textit{15}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{169}&\multicolumn{1}{@{.}l}{4 {\it 4}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{61 {\it 7}}&\multicolumn{1}{r@{}}{3592}&\multicolumn{1}{@{.}l}{2+x?}&&\multicolumn{1}{r@{}}{3422}&\multicolumn{1}{@{.}l}{8+x }&&\multicolumn{1}{l}{M1(+E2)}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{4.90204cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}={\textminus}0.68 \textit{21}, A\ensuremath{_{\textnormal{4}}}={\textminus}0.02 \textit{33}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{180}&\multicolumn{1}{@{.}l}{0 {\it 5}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{43 {\it 8}}&\multicolumn{1}{r@{}}{3706}&\multicolumn{1}{@{.}l}{6+x}&&\multicolumn{1}{r@{}}{3526}&\multicolumn{1}{@{.}l}{4+x }&\multicolumn{1}{@{}l}{33/2\ensuremath{^{-}}}&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{4.90204cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}={\textminus}0.13 \textit{11}, A\ensuremath{_{\textnormal{4}}}=0.35 \textit{18}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{185}&\multicolumn{1}{@{.}l}{77 {\it 20}}&\multicolumn{1}{r@{}}{51}&\multicolumn{1}{@{.}l}{3 {\it 5}}&\multicolumn{1}{r@{}}{1932}&\multicolumn{1}{@{.}l}{2}&\multicolumn{1}{l}{21/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1746}&\multicolumn{1}{@{.}l}{4 }&\multicolumn{1}{@{}l}{17/2\ensuremath{^{+}}}&\multicolumn{1}{l}{(E2)}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{4.90204cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}=+0.157 \textit{10}, A\ensuremath{_{\textnormal{4}}}={\textminus}0.017\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.90204cm}{\raggedright {\ }{\ }{\ }\textit{17}; DCO=0.95 \textit{6}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{190}&\multicolumn{1}{@{.}l}{49 {\it 25}}&\multicolumn{1}{r@{}}{8}&\multicolumn{1}{@{.}l}{41 {\it 10}}&\multicolumn{1}{r@{}}{2739}&\multicolumn{1}{@{.}l}{91+x}&\multicolumn{1}{l}{29/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{2549}&\multicolumn{1}{@{.}l}{40+x }&\multicolumn{1}{@{}l}{27/2\ensuremath{^{+}}}&\multicolumn{1}{l}{(E1)}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{4.90204cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}={\textminus}0.128 \textit{24}, A\ensuremath{_{\textnormal{4}}}={\textminus}0.029\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.90204cm}{\raggedright {\ }{\ }{\ }\textit{40}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$192}&\multicolumn{1}{@{.}l}{5 {\it 7}}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{7 {\it 7}}&&&&&&&&&&&\\
\multicolumn{1}{r@{}}{$^{x}$197}&\multicolumn{1}{@{.}l}{2 {\it 4}}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{77 {\it 10}}&&&&&&&&&&\parbox[t][0.3cm]{4.90204cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}={\textminus}0.05 \textit{8}, A\ensuremath{_{\textnormal{4}}}={\textminus}0.05 \textit{13}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{250}&\multicolumn{1}{@{.}l}{2 {\it 4}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{41 {\it 8}}&\multicolumn{1}{r@{}}{2549}&\multicolumn{1}{@{.}l}{40+x}&\multicolumn{1}{l}{27/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{2299}&\multicolumn{1}{@{.}l}{0+x }&\multicolumn{1}{@{}l}{27/2\ensuremath{^{+}}}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{$>$1}&\multicolumn{1}{@{.}l}{2}&\parbox[t][0.3cm]{4.90204cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}={\textminus}0.24 \textit{11}, A\ensuremath{_{\textnormal{4}}}=0.03 \textit{18}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$258}&\multicolumn{1}{@{.}l}{4 {\it 4}}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{33 {\it 8}}&&&&&&&&&&\parbox[t][0.3cm]{4.90204cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}={\textminus}0.36 \textit{7}, A\ensuremath{_{\textnormal{4}}}=0.04 \textit{11}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{271}&\multicolumn{1}{@{.}l}{91 {\it 20}}&\multicolumn{1}{r@{}}{90}&\multicolumn{1}{@{.}l}{0 {\it 9}}&\multicolumn{1}{r@{}}{1746}&\multicolumn{1}{@{.}l}{4}&\multicolumn{1}{l}{17/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1474}&\multicolumn{1}{@{.}l}{5 }&\multicolumn{1}{@{}l}{17/2\ensuremath{^{-}}}&\multicolumn{1}{l}{(E1)}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{4.90204cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}=+0.189 \textit{6}, A\ensuremath{_{\textnormal{4}}}={\textminus}0.012 \textit{9};\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.90204cm}{\raggedright {\ }{\ }{\ }DCO=0.922 \textit{44}; consistent with\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.90204cm}{\raggedright {\ }{\ }{\ }\ensuremath{\Delta}J=0 transition.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{284}&\multicolumn{1}{@{.}l}{19 {\it 25}}&\multicolumn{1}{r@{}}{10}&\multicolumn{1}{@{.}l}{01 {\it 22}}&\multicolumn{1}{r@{}}{3810}&\multicolumn{1}{@{.}l}{7+x}&\multicolumn{1}{l}{33/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{3526}&\multicolumn{1}{@{.}l}{4+x }&\multicolumn{1}{@{}l}{33/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1(+E2)}&\multicolumn{1}{r@{}}{$\leq$0}&\multicolumn{1}{@{.}l}{58}&\parbox[t][0.3cm]{4.90204cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}=+0.238 \textit{42}, A\ensuremath{_{\textnormal{4}}}={\textminus}0.098\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.90204cm}{\raggedright {\ }{\ }{\ }\textit{70}; DCO=0.67 \textit{27}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{287}&\multicolumn{1}{@{.}l}{3 {\it 4}}&\multicolumn{1}{r@{}}{6}&\multicolumn{1}{@{.}l}{4 {\it 9}}&\multicolumn{1}{r@{}}{3526}&\multicolumn{1}{@{.}l}{4+x}&\multicolumn{1}{l}{33/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{3238}&\multicolumn{1}{@{.}l}{8+x }&\multicolumn{1}{@{}l}{31/2\ensuremath{^{-}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
\multicolumn{1}{r@{}}{366}&\multicolumn{1}{@{.}l}{6 {\it 4}}&\multicolumn{1}{r@{}}{4}&\multicolumn{1}{@{.}l}{6 {\it 8}}&\multicolumn{1}{r@{}}{2299}&\multicolumn{1}{@{.}l}{0+x}&\multicolumn{1}{l}{27/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1932}&\multicolumn{1}{@{.}l}{2+x}&\multicolumn{1}{@{}l}{(25/2\ensuremath{^{+}})}&\multicolumn{1}{l}{(M1)}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{4.90204cm}{\raggedright Mult.: DCO=0.58 \textit{21}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$382}&\multicolumn{1}{@{.}l}{2 {\it 4}}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{6 {\it 23}}&&&&&&&&&&&\\
\multicolumn{1}{r@{}}{396}&\multicolumn{1}{@{.}l}{4 {\it 4}}&\multicolumn{1}{r@{}}{5}&\multicolumn{1}{@{.}l}{18 {\it 8}}&\multicolumn{1}{r@{}}{3922}&\multicolumn{1}{@{.}l}{8+x}&\multicolumn{1}{l}{35/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{3526}&\multicolumn{1}{@{.}l}{4+x }&\multicolumn{1}{@{}l}{33/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{$-$0}&\multicolumn{1}{@{.}l}{24 {\it 11}}&\parbox[t][0.3cm]{4.90204cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}={\textminus}0.325 \textit{32}, A\ensuremath{_{\textnormal{4}}}={\textminus}0.010\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.90204cm}{\raggedright {\ }{\ }{\ }\textit{53}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{411}&\multicolumn{1}{@{.}l}{86 {\it 20}}&\multicolumn{1}{r@{}}{80}&\multicolumn{1}{@{.}l}{7 {\it 22}}&\multicolumn{1}{r@{}}{1379}&\multicolumn{1}{@{.}l}{2}&\multicolumn{1}{l}{15/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{967}&\multicolumn{1}{@{.}l}{39 }&\multicolumn{1}{@{}l}{13/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{$-$0}&\multicolumn{1}{@{.}l}{023 {\it 17}}&\parbox[t][0.3cm]{4.90204cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}={\textminus}0.128 \textit{4}, A\ensuremath{_{\textnormal{4}}}={\textminus}0.010 \textit{6};\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.90204cm}{\raggedright {\ }{\ }{\ }DCO=1.41 \textit{8}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{414}&\multicolumn{1}{@{.}l}{9 {\it 5}}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{88 {\it 14}}&\multicolumn{1}{r@{}}{1379}&\multicolumn{1}{@{.}l}{2}&\multicolumn{1}{l}{15/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{964}&\multicolumn{1}{@{.}l}{2 }&\multicolumn{1}{@{}l}{11/2\ensuremath{^{-}}}&\multicolumn{1}{l}{[E2]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{4.90204cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}=+0.07 \textit{6}, A\ensuremath{_{\textnormal{4}}}=0.00 \textit{9},\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.90204cm}{\raggedright {\ }{\ }{\ }but values are inconsistent with\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.90204cm}{\raggedright {\ }{\ }{\ }the expected Milt=E2\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.90204cm}{\raggedright {\ }{\ }{\ }assignment.\vspace{0.1cm}}&\\
\end{longtable}
\begin{textblock}{29}(0,27.3)
Continued on next page (footnotes at end of table)
\end{textblock}
\clearpage
\begin{longtable}{ccccccccc@{}ccccc@{\extracolsep{\fill}}c}
\\[-.4cm]
\multicolumn{15}{c}{{\bf \small \underline{\ensuremath{^{\textnormal{196}}}Pt(\ensuremath{^{\textnormal{10}}}B,5n\ensuremath{\gamma})\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1985Pi05,B}{1985Pi05} (continued)}}}\\
\multicolumn{15}{c}{~}\\
\multicolumn{15}{c}{\underline{$\gamma$($^{201}$Bi) (continued)}}\\
\multicolumn{15}{c}{~~~}\\
\multicolumn{2}{c}{E\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{BI40GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{I\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{BI40GAMMA1}{\ddagger}}}}&\multicolumn{2}{c}{E\ensuremath{_{i}}(level)}&J\ensuremath{^{\pi}_{i}}&\multicolumn{2}{c}{E\ensuremath{_{f}}}&J\ensuremath{^{\pi}_{f}}&Mult.\ensuremath{^{\hyperlink{BI40GAMMA2}{\#}}}&\multicolumn{2}{c}{\ensuremath{\delta}\ensuremath{^{\hyperlink{BI40GAMMA0}{\dagger}}}}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endhead
\multicolumn{1}{r@{}}{421}&\multicolumn{1}{@{.}l}{8 {\it 5}}&\multicolumn{1}{r@{}}{4}&\multicolumn{1}{@{.}l}{3 {\it 7}}&\multicolumn{1}{r@{}}{3011}&\multicolumn{1}{@{.}l}{4+x}&&\multicolumn{1}{r@{}}{2589}&\multicolumn{1}{@{.}l}{6+x }&&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
\multicolumn{1}{r@{}}{428}&\multicolumn{1}{@{.}l}{2 {\it 4}}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{1 {\it 7}}&\multicolumn{1}{r@{}}{3422}&\multicolumn{1}{@{.}l}{8+x}&&\multicolumn{1}{r@{}}{2994}&\multicolumn{1}{@{.}l}{6+x }&&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
\multicolumn{1}{r@{}}{440}&\multicolumn{1}{@{.}l}{9 {\it 4}}&\multicolumn{1}{r@{}}{5}&\multicolumn{1}{@{.}l}{11 {\it 12}}&\multicolumn{1}{r@{}}{2739}&\multicolumn{1}{@{.}l}{91+x}&\multicolumn{1}{l}{29/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{2299}&\multicolumn{1}{@{.}l}{0+x }&\multicolumn{1}{@{}l}{27/2\ensuremath{^{+}}}&\multicolumn{1}{l}{(E1)}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{4.59784cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}={\textminus}0.17 \textit{5}, A\ensuremath{_{\textnormal{4}}}=0.12 \textit{8}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{462}&\multicolumn{1}{@{.}l}{2 {\it 8}}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{6 {\it 7}}&\multicolumn{1}{r@{}}{3011}&\multicolumn{1}{@{.}l}{4+x}&&\multicolumn{1}{r@{}}{2549}&\multicolumn{1}{@{.}l}{40+x }&\multicolumn{1}{@{}l}{27/2\ensuremath{^{+}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
\multicolumn{1}{r@{}}{468}&\multicolumn{1}{@{.}l}{0 {\it 4}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{4 {\it 7}}&\multicolumn{1}{r@{}}{3706}&\multicolumn{1}{@{.}l}{6+x}&&\multicolumn{1}{r@{}}{3238}&\multicolumn{1}{@{.}l}{8+x }&\multicolumn{1}{@{}l}{31/2\ensuremath{^{-}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
\multicolumn{1}{r@{}}{498}&\multicolumn{1}{@{.}l}{95 {\it 25}}&\multicolumn{1}{r@{}}{23}&\multicolumn{1}{@{.}l}{66 {\it 24}}&\multicolumn{1}{r@{}}{3238}&\multicolumn{1}{@{.}l}{8+x}&\multicolumn{1}{l}{31/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{2739}&\multicolumn{1}{@{.}l}{91+x }&\multicolumn{1}{@{}l}{29/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{$-$0}&\multicolumn{1}{@{.}l}{33 {\it 11}}&\parbox[t][0.3cm]{4.59784cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}={\textminus}0.390 \textit{13}, A\ensuremath{_{\textnormal{4}}}=0.032\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.59784cm}{\raggedright {\ }{\ }{\ }\textit{22}; DCO=1.14 \textit{22}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$552}&\multicolumn{1}{@{.}l}{5 {\it 4}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{16 {\it 12}}&&&&&&&&&&\parbox[t][0.3cm]{4.59784cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}={\textminus}0.70 \textit{22}, A\ensuremath{_{\textnormal{4}}}={\textminus}0.40\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.59784cm}{\raggedright {\ }{\ }{\ }\textit{34}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{572}&\multicolumn{1}{@{.}l}{4 {\it 4}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{96 {\it 17}}&\multicolumn{1}{r@{}}{3810}&\multicolumn{1}{@{.}l}{7+x}&\multicolumn{1}{l}{33/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{3238}&\multicolumn{1}{@{.}l}{8+x }&\multicolumn{1}{@{}l}{31/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{$-$0}&\multicolumn{1}{@{.}l}{41 {\it 11}}&\parbox[t][0.3cm]{4.59784cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}={\textminus}0.67 \textit{17}, A\ensuremath{_{\textnormal{4}}}={\textminus}0.23\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.59784cm}{\raggedright {\ }{\ }{\ }\textit{27}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{617}&\multicolumn{1}{@{.}l}{27 {\it 25}}&\multicolumn{1}{r@{}}{32}&\multicolumn{1}{@{.}l}{3 {\it 3}}&\multicolumn{1}{r@{}}{2549}&\multicolumn{1}{@{.}l}{40+x}&\multicolumn{1}{l}{27/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1932}&\multicolumn{1}{@{.}l}{2+x}&\multicolumn{1}{@{}l}{(25/2\ensuremath{^{+}})}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{+0}&\multicolumn{1}{@{.}l}{046 {\it 28}}&\parbox[t][0.3cm]{4.59784cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}={\textminus}0.087 \textit{15}, A\ensuremath{_{\textnormal{4}}}={\textminus}0.015\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.59784cm}{\raggedright {\ }{\ }{\ }\textit{23}; DCO=1.02 \textit{10}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{657}&\multicolumn{1}{@{.}l}{4 {\it 3}}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{9 {\it 7}}&\multicolumn{1}{r@{}}{2589}&\multicolumn{1}{@{.}l}{6+x}&&\multicolumn{1}{r@{}}{1932}&\multicolumn{1}{@{.}l}{2+x}&\multicolumn{1}{@{}l}{(25/2\ensuremath{^{+}})}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
\multicolumn{1}{r@{}}{679}&\multicolumn{1}{@{.}l}{8 {\it 3}}&\multicolumn{1}{r@{}}{14}&\multicolumn{1}{@{.}l}{41 {\it 15}}&\multicolumn{1}{r@{}}{2651}&\multicolumn{1}{@{.}l}{0+x}&\multicolumn{1}{l}{29/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1971}&\multicolumn{1}{@{.}l}{2+x }&\multicolumn{1}{@{}l}{27/2\ensuremath{^{+}}}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{$-$0}&\multicolumn{1}{@{.}l}{15 {\it 12}}&\parbox[t][0.3cm]{4.59784cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}={\textminus}0.312 \textit{15}, A\ensuremath{_{\textnormal{4}}}=0.020\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.59784cm}{\raggedright {\ }{\ }{\ }\textit{25}; DCO=1.42 \textit{31}. Value may\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.59784cm}{\raggedright {\ }{\ }{\ }be obscured by unresolved\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.59784cm}{\raggedright {\ }{\ }{\ }transition.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{736}&\multicolumn{1}{@{.}l}{0 {\it 4}}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{10 {\it 16}}&\multicolumn{1}{r@{}}{2668}&\multicolumn{1}{@{.}l}{21+x}&\multicolumn{1}{l}{(29/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{1932}&\multicolumn{1}{@{.}l}{2+x}&\multicolumn{1}{@{}l}{(25/2\ensuremath{^{+}})}&\multicolumn{1}{l}{E2}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{4.59784cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}=+0.20 \textit{11}, A\ensuremath{_{\textnormal{4}}}={\textminus}0.06\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.59784cm}{\raggedright {\ }{\ }{\ }\textit{17}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{754}&\multicolumn{1}{@{.}l}{9 {\it 4}}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{60 {\it 12}}&\multicolumn{1}{r@{}}{1719}&\multicolumn{1}{@{.}l}{1}&\multicolumn{1}{l}{(13/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{964}&\multicolumn{1}{@{.}l}{2 }&\multicolumn{1}{@{}l}{11/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1(+E2)}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{4.59784cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}={\textminus}0.02 \textit{9}, A\ensuremath{_{\textnormal{4}}}=0.60 \textit{15}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{786}&\multicolumn{1}{@{.}l}{3 {\it 3}}&\multicolumn{1}{r@{}}{25}&\multicolumn{1}{@{.}l}{04 {\it 35}}&\multicolumn{1}{r@{}}{3526}&\multicolumn{1}{@{.}l}{4+x}&\multicolumn{1}{l}{33/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{2739}&\multicolumn{1}{@{.}l}{91+x }&\multicolumn{1}{@{}l}{29/2\ensuremath{^{-}}}&\multicolumn{1}{l}{(E2)}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{4.59784cm}{\raggedright E\ensuremath{_{\gamma}}: This \ensuremath{\gamma} ray shows delayed\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.59784cm}{\raggedright {\ }{\ }{\ }component with T\ensuremath{_{\textnormal{1/2}}}=14 ns \textit{3}.\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.59784cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}=+0.113 \textit{14}, A\ensuremath{_{\textnormal{4}}}=0.031\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.59784cm}{\raggedright {\ }{\ }{\ }\textit{23}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{797}&\multicolumn{1}{@{.}l}{8 {\it 4}}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{26 {\it 11}}&\multicolumn{1}{r@{}}{5282}&\multicolumn{1}{@{.}l}{2+x}&\multicolumn{1}{l}{(37/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{4484}&\multicolumn{1}{@{.}l}{4+x }&\multicolumn{1}{@{}l}{(35/2\ensuremath{^{-}})}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{4.59784cm}{\raggedright A\ensuremath{_{\textnormal{2}}}=+0.63 \textit{10}, A\ensuremath{_{\textnormal{4}}}={\textminus}0.07 \textit{16}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$844}&\multicolumn{1}{@{.}l}{1 {\it 8}}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{59 {\it 10}}&&&&&&&&&&\parbox[t][0.3cm]{4.59784cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}=+0.05 \textit{6}, A\ensuremath{_{\textnormal{4}}}={\textminus}0.03 \textit{10}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{846}&\multicolumn{1}{@{.}l}{1 {\it 7}}&\multicolumn{1}{r@{}}{4}&\multicolumn{1}{@{.}l}{4 {\it 10}}&\multicolumn{1}{r@{}}{4484}&\multicolumn{1}{@{.}l}{4+x}&\multicolumn{1}{l}{(35/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{3638}&\multicolumn{1}{@{.}l}{3+x }&\multicolumn{1}{@{}l}{(33/2\ensuremath{^{-}})}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{4.59784cm}{\raggedright A\ensuremath{_{\textnormal{2}}}=+0.134 \textit{53}, A\ensuremath{_{\textnormal{4}}}=0.086 \textit{90}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.59784cm}{\raggedright {\ }{\ }{\ }Value may be obscured by\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.59784cm}{\raggedright {\ }{\ }{\ }unresolved transition.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{963}&\multicolumn{1}{@{.}l}{9 {\it 8}}&\multicolumn{1}{r@{}}{12}&\multicolumn{1}{@{.}l}{52 {\it 22}}&\multicolumn{1}{r@{}}{964}&\multicolumn{1}{@{.}l}{2}&\multicolumn{1}{l}{11/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{9/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1(+E2)}&\multicolumn{1}{r@{}}{$-$0}&\multicolumn{1}{@{.}l}{04 {\it 7}}&\parbox[t][0.3cm]{4.59784cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}={\textminus}0.152 \textit{35}, A\ensuremath{_{\textnormal{4}}}={\textminus}0.023\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.59784cm}{\raggedright {\ }{\ }{\ }\textit{57}. Value may be obscured by\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.59784cm}{\raggedright {\ }{\ }{\ }unresolved transition.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{967}&\multicolumn{1}{@{.}l}{42 {\it 25}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{.}l}{0 {\it 5}}&\multicolumn{1}{r@{}}{967}&\multicolumn{1}{@{.}l}{39}&\multicolumn{1}{l}{13/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{9/2\ensuremath{^{-}}}&\multicolumn{1}{l}{E2}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{4.59784cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}=+0.152 \textit{11},\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.59784cm}{\raggedright {\ }{\ }{\ }A\ensuremath{_{\textnormal{4}}}={\textminus}0.025 \textit{18}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{987}&\multicolumn{1}{@{.}l}{3 {\it 4}}&\multicolumn{1}{r@{}}{9}&\multicolumn{1}{@{.}l}{97 {\it 14}}&\multicolumn{1}{r@{}}{3638}&\multicolumn{1}{@{.}l}{3+x}&\multicolumn{1}{l}{(33/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{2651}&\multicolumn{1}{@{.}l}{0+x }&\multicolumn{1}{@{}l}{29/2\ensuremath{^{+}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{4.59784cm}{\raggedright A\ensuremath{_{\textnormal{2}}}=+0.231 \textit{27}, A\ensuremath{_{\textnormal{4}}}={\textminus}0.056 \textit{46};\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{4.59784cm}{\raggedright {\ }{\ }{\ }DCO=0.75 \textit{40}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1062}&\multicolumn{1}{@{.}l}{4 {\it 8}}&\multicolumn{1}{r@{}}{4}&\multicolumn{1}{@{.}l}{34 {\it 15}}&\multicolumn{1}{r@{}}{2994}&\multicolumn{1}{@{.}l}{6+x}&&\multicolumn{1}{r@{}}{1932}&\multicolumn{1}{@{.}l}{2+x}&\multicolumn{1}{@{}l}{(25/2\ensuremath{^{+}})}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{4.59784cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}=+0.16 \textit{8}, A\ensuremath{_{\textnormal{4}}}={\textminus}0.01 \textit{11}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$1358}&\multicolumn{1}{@{.}l}{7 {\it 5}}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{26 {\it 11}}&&&&&&&&&&\parbox[t][0.3cm]{4.59784cm}{\raggedright A\ensuremath{_{\textnormal{2}}}=+0.23 \textit{7}, A\ensuremath{_{\textnormal{4}}}={\textminus}0.18 \textit{12}.\vspace{0.1cm}}&\\
\end{longtable}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{BI40GAMMA0}{\dagger}}}} From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1985Pi05,B}{1985Pi05}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{BI40GAMMA1}{\ddagger}}}} From E(\ensuremath{^{\textnormal{10}}}B)=67 MeV in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1985Pi05,B}{1985Pi05}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{BI40GAMMA2}{\#}}}} Based on \ensuremath{\gamma}(\ensuremath{\theta}) and DCO, unless otherwise stated. DCO values were obtained by gating on stretched E2 transitions.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{BI40GAMMA3}{x}}}} \ensuremath{\gamma} ray not placed in level scheme.}\\
\vspace{0.5cm}
\clearpage
\begin{figure}[h]
\begin{center}
\includegraphics{201BI40-0.ps}\\
\end{center}
\end{figure}
\clearpage
\section[\ensuremath{^{201}_{\ 84}}Po\ensuremath{_{117}^{~}}]{ }
\vspace{-30pt}
\setcounter{chappage}{1}
\subsection[\hspace{-0.2cm}Adopted Levels, Gammas]{ }
\vspace{-20pt}
\vspace{0.3cm}
\hypertarget{PO41}{{\bf \small \underline{Adopted \hyperlink{201PO_LEVEL}{Levels}, \hyperlink{201PO_GAMMA}{Gammas}}}}\\
\vspace{4pt}
\vspace{8pt}
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}Q(\ensuremath{\beta^-})=$-$5732 {\it 10}; S(n)=7651 {\it 9}; S(p)=3440 {\it 23}; Q(\ensuremath{\alpha})=5799.3 {\it 17}\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2021Wa16,B}{2021Wa16}}\\
\vspace{12pt}
\hypertarget{201PO_LEVEL}{\underline{$^{201}$Po Levels}}\\
\begin{longtable}[c]{ll}
\multicolumn{2}{c}{\underline{Cross Reference (XREF) Flags}}\\
\\
\hyperlink{PO42}{\texttt{A }}& \ensuremath{^{\textnormal{201}}}Po IT decay (8.96 min)\\
\hyperlink{AT43}{\texttt{B }}& \ensuremath{^{\textnormal{201}}}At \ensuremath{\varepsilon} decay\\
\hyperlink{RN44}{\texttt{C }}& \ensuremath{^{\textnormal{205}}}Rn \ensuremath{\alpha} decay\\
\hyperlink{PO45}{\texttt{D }}& \ensuremath{^{\textnormal{194}}}Pt(\ensuremath{^{\textnormal{12}}}C,5n\ensuremath{\gamma})\\
\end{longtable}
\vspace{-0.5cm}
\begin{longtable}{ccccccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E(level)$^{{\hyperlink{PO41LEVEL0}{\dagger}}}$}&J$^{\pi}$$^{}$&\multicolumn{2}{c}{T$_{1/2}$$^{}$}&XREF&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{PO41LEVEL3}{@}}}}}&\multicolumn{1}{l}{3/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{15}&\multicolumn{1}{@{.}l}{50 min {\it 14}}&\multicolumn{1}{l}{\texttt{\hyperlink{PO42}{A}\hyperlink{AT43}{B}\hyperlink{RN44}{C}\hyperlink{PO45}{D}} }&\parbox[t][0.3cm]{11.45854cm}{\raggedright \%\ensuremath{\alpha}=1.13 \textit{3}; \%\ensuremath{\varepsilon}+\%\ensuremath{\beta}\ensuremath{^{\textnormal{+}}}=98.87 \textit{3}\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.45854cm}{\raggedright \ensuremath{\mu}={\textminus}0.98 \textit{7} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2014Se07,B}{2014Se07},\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2019StZV,B}{2019StZV})\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.45854cm}{\raggedright Q=+0.10 \textit{10} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2014Se07,B}{2014Se07})\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.45854cm}{\raggedright \ensuremath{\delta}\ensuremath{\nu}(\ensuremath{^{\textnormal{201}}}Po,\ensuremath{^{\textnormal{196}}}Po)={\textminus}1.51 GHz \textit{15}; \ensuremath{\delta}\ensuremath{\langle}r\ensuremath{^{\textnormal{2}}}\ensuremath{\rangle}(\ensuremath{^{\textnormal{201}}}Po,\ensuremath{^{\textnormal{210}}}Po)={\textminus}0.510 fm\ensuremath{^{\textnormal{2}}} \textit{13} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2013Se03,B}{2013Se03}).\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.45854cm}{\raggedright \ensuremath{\langle}\ensuremath{\beta}\ensuremath{_{\textnormal{2}}^{\textnormal{2}}}\ensuremath{\rangle}\ensuremath{^{\textnormal{1/2}}}=0.10 (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2013Se03,B}{2013Se03},\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2014Se07,B}{2014Se07}).\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.45854cm}{\raggedright \%\ensuremath{\alpha} is unweighted average of \%\ensuremath{\alpha}=1.15\% \textit{1} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1967Le21,B}{1967Le21}) and 1.10\% \textit{4} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1993Wa04,B}{1993Wa04}).\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.45854cm}{\raggedright {\ }{\ }{\ }Other: 1.6\% \textit{3} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1971Ho01,B}{1971Ho01}). \%\ensuremath{\varepsilon}+\%\ensuremath{\beta}\ensuremath{^{\textnormal{+}}} has not been directly measured.\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.45854cm}{\raggedright J\ensuremath{^{\pi}}: atomic beam (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1962Ax02,B}{1962Ax02}) and \ensuremath{\mu}.\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.45854cm}{\raggedright T\ensuremath{_{1/2}}: Weighted average of 15.3 min \textit{8} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1963Ho18,B}{1963Ho18}), 15.8 min \textit{3} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1967Le08,B}{1967Le08}), 15.1\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.45854cm}{\raggedright {\ }{\ }{\ }min \textit{3} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1967Ti04,B}{1967Ti04}), 16.0 min \textit{15} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1968Go12,B}{1968Go12}), 15 min \textit{3} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1970Jo26,B}{1970Jo26}), 15.2 min \textit{3}\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.45854cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1970Ra14,B}{1970Ra14}), 15.5 min \textit{6} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1970DaZM,B}{1970DaZM}), 15.1 min \textit{4} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1971Ho01,B}{1971Ho01}), 17.5 min \textit{5}\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.45854cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1964Br23,B}{1964Br23}) and 14.5 min \textit{10} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1976Ko13,B}{1976Ko13}).\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.45854cm}{\raggedright E\ensuremath{\alpha}=5683.3 keV \textit{16}, recommended by \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1991Ry01,B}{1991Ry01}. Values from individual\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.45854cm}{\raggedright {\ }{\ }{\ }measurements are 5674 keV 9 (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1963Ho18,B}{1963Ho18}), 5670 keV 10 (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1967Le08,B}{1967Le08}), 5677 keV \textit{5}\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.45854cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1967Tr06,B}{1967Tr06}), 5684 keV \textit{6} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1967Ti04,B}{1967Ti04}), 5684 keV 2 (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1968Go12,B}{1968Go12}), 5689 keV \textit{10}\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.45854cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1970Jo26,B}{1970Jo26}), 5680 keV \textit{10} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1970DaZM,B}{1970DaZM}) and 5685 keV \textit{4} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1970Ra14,B}{1970Ra14}).\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.45854cm}{\raggedright \ensuremath{\mu},Q: hyperfine structure studies using in-source resonance ionization spectroscopy at\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.45854cm}{\raggedright {\ }{\ }{\ }CERN-ISOLDE facility (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2014Se07,B}{2014Se07}). Total (statistical uncertainties=0.010 for \ensuremath{\mu}\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.45854cm}{\raggedright {\ }{\ }{\ }and 0.08 for Q, and systematic) uncertainties are given. Others: \ensuremath{\mu}=0.94 \textit{8}\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.45854cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1991Wo04,B}{1991Wo04}, using the static nuclear orientation technique), 0.74 \textit{11} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1988Wo12,B}{1988Wo12}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{5}&\multicolumn{1}{@{.}l}{61\ensuremath{^{{\hyperlink{PO41LEVEL1}{\ddagger}}}} {\it 13}}&\multicolumn{1}{l}{5/2\ensuremath{^{-}}}&&&\multicolumn{1}{l}{\texttt{\hyperlink{PO42}{A}\hyperlink{AT43}{B}\hyperlink{RN44}{C}\hyperlink{PO45}{D}} }&\parbox[t][0.3cm]{11.45854cm}{\raggedright J\ensuremath{^{\pi}}: Favored \ensuremath{\alpha}-decay from the \ensuremath{^{\textnormal{205}}}Rn g.s. (\ensuremath{J^{\pi}}=5/2\ensuremath{^{-}}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{142}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{PO41LEVEL2}{\#}}}} {\it 3}}&\multicolumn{1}{l}{(1/2\ensuremath{^{-}})}&&&\multicolumn{1}{l}{\texttt{\ \ \hyperlink{RN44}{C}\ } }&\parbox[t][0.3cm]{11.45854cm}{\raggedright E(level): From \ensuremath{^{\textnormal{205}}}Rn \ensuremath{\alpha} decay.\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.45854cm}{\raggedright J\ensuremath{^{\pi}}: Unfavored \ensuremath{\alpha}-decay from \ensuremath{^{\textnormal{205}}}Rn g.s. (\ensuremath{J^{\pi}}=5/2\ensuremath{^{-}}). Systematics of structures in\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.45854cm}{\raggedright {\ }{\ }{\ }neighboring nuclei.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{423}&\multicolumn{1}{@{.}l}{41\ensuremath{^{{\hyperlink{PO41LEVEL4}{\&}}}} {\it 22}}&\multicolumn{1}{l}{13/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{8}&\multicolumn{1}{@{.}l}{96 min {\it 12}}&\multicolumn{1}{l}{\texttt{\hyperlink{PO42}{A}\hyperlink{AT43}{B}\ \hyperlink{PO45}{D}} }&\parbox[t][0.3cm]{11.45854cm}{\raggedright \%IT\ensuremath{\approx}42.6; \%\ensuremath{\alpha}=2.4 \textit{5}; \%\ensuremath{\varepsilon}+\%\ensuremath{\beta}\ensuremath{^{\textnormal{+}}}\ensuremath{\approx}55\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.45854cm}{\raggedright \ensuremath{\mu}={\textminus}1.00 \textit{7} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2014Se07,B}{2014Se07},\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2019StZV,B}{2019StZV})\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.45854cm}{\raggedright Q=+1.26 \textit{20} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2014Se07,B}{2014Se07})\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.45854cm}{\raggedright \ensuremath{\delta}\ensuremath{\nu}(\ensuremath{^{\textnormal{201}}}Po,\ensuremath{^{\textnormal{196}}}Po)={\textminus}2.20 GHz \textit{15}; \ensuremath{\delta}\ensuremath{\langle}r\ensuremath{^{\textnormal{2}}}\ensuremath{\rangle}(\ensuremath{^{\textnormal{201}}}Po,\ensuremath{^{\textnormal{210}}}Po)={\textminus}0.452 fm\ensuremath{^{\textnormal{2}}} \textit{13} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2013Se03,B}{2013Se03}).\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.45854cm}{\raggedright \ensuremath{\langle}\ensuremath{\beta}\ensuremath{_{\textnormal{2}}^{\textnormal{2}}}\ensuremath{\rangle}\ensuremath{^{\textnormal{1/2}}}=0.12 (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2013Se03,B}{2013Se03},\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2014Se07,B}{2014Se07}).\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.45854cm}{\raggedright \%IT, \%\ensuremath{\alpha}, and \%\ensuremath{\varepsilon}+\%\ensuremath{\beta}\ensuremath{^{\textnormal{+}}} from \%IT + \%\ensuremath{\alpha} + \%(\ensuremath{\varepsilon}+\ensuremath{\beta}\ensuremath{^{\textnormal{+}}})=\%100 and \%IT/(\%\ensuremath{\varepsilon}+\%\ensuremath{\beta}\ensuremath{^{\textnormal{+}}})\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.45854cm}{\raggedright {\ }{\ }{\ }\ensuremath{\approx} 0.76, deduced by the evaluator from the decay scheme and \ensuremath{\gamma}-ray intensities of\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.45854cm}{\raggedright {\ }{\ }{\ }\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986Br28,B}{1986Br28}. \%\ensuremath{\alpha} is unweighted average of 2.9\% \textit{2} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1967Le21,B}{1967Le21}) and 1.9\% \textit{4}\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.45854cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1970Jo26,B}{1970Jo26}). Others: \%IT=28 \textit{+12 {\textminus}7} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1971Jo19,B}{1971Jo19}).\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.45854cm}{\raggedright E(level): Other: 423.8 keV \textit{24} in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2021Ko07,B}{2021Ko07} from E\ensuremath{_{\ensuremath{\alpha}}}.\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.45854cm}{\raggedright J\ensuremath{^{\pi}}: 417.8\ensuremath{\gamma} M4 to 5/2\ensuremath{^{-}}; favored \ensuremath{\alpha} decay to the 319.31 keV level (\ensuremath{J^{\pi}}=13/2\ensuremath{^{+}}) in\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.45854cm}{\raggedright {\ }{\ }{\ }\ensuremath{^{\textnormal{197}}}Pb.\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.45854cm}{\raggedright T\ensuremath{_{1/2}}: Weighted average of 8.7 min \textit{6} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1963Ho18,B}{1963Ho18}), 9.0 min \textit{2} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1967Le08,B}{1967Le08}), 8.9 min \textit{4}\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.45854cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1967Ti04,B}{1967Ti04}), 10.0 min \textit{15} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1968Go12,B}{1968Go12}), 9 min \textit{3} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1970Jo26,B}{1970Jo26}), 8.8 min \textit{4} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1970Ra14,B}{1970Ra14}),\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.45854cm}{\raggedright {\ }{\ }{\ }8.9 min \textit{8} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1970DaZM,B}{1970DaZM}), 9.0 min \textit{3} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1976Ko13,B}{1976Ko13}) and 9.0 min \textit{3} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986Br28,B}{1986Br28}).\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.45854cm}{\raggedright E\ensuremath{\alpha}=5786.0 keV \textit{16} recommended by \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1991Ry01,B}{1991Ry01}. Values from individual\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.45854cm}{\raggedright {\ }{\ }{\ }measurements are 5780 keV 7 (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1963Ho18,B}{1963Ho18}), 5770 keV 10 (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1967Le08,B}{1967Le08}), 5780 keV \textit{5}\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.45854cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1967Tr06,B}{1967Tr06}), 5788 keV \textit{7} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1967Ti04,B}{1967Ti04}), 5787 keV 2 (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1968Go12,B}{1968Go12}), 5778 keV \textit{10}\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.45854cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1970Jo26,B}{1970Jo26}), 5780 keV \textit{10} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1970DaZM,B}{1970DaZM}) and 5786 keV \textit{4} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1970Ra14,B}{1970Ra14}).\vspace{0.1cm}}&\\
\end{longtable}
\begin{textblock}{29}(0,27.3)
Continued on next page (footnotes at end of table)
\end{textblock}
\clearpage
\begin{longtable}{ccccc@{\extracolsep{\fill}}c}
\\[-.4cm]
\multicolumn{6}{c}{{\bf \small \underline{Adopted \hyperlink{201PO_LEVEL}{Levels}, \hyperlink{201PO_GAMMA}{Gammas} (continued)}}}\\
\multicolumn{6}{c}{~}\\
\multicolumn{6}{c}{\underline{\ensuremath{^{201}}Po Levels (continued)}}\\
\multicolumn{6}{c}{~}\\
\multicolumn{2}{c}{E(level)$^{{\hyperlink{PO41LEVEL0}{\dagger}}}$}&J$^{\pi}$$^{}$&XREF&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&\hrulefill&
\endhead
&&&&\parbox[t][0.3cm]{12.43634cm}{\raggedright \ensuremath{\mu},Q: hyperfine structure studies using in-source resonance ionization spectroscopy at\vspace{0.1cm}}&\\
&&&&\parbox[t][0.3cm]{12.43634cm}{\raggedright {\ }{\ }{\ }CERN-ISOLDE facility (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2014Se07,B}{2014Se07}). Total (statistical uncertainties=0.055 for \ensuremath{\mu} and 0.20\vspace{0.1cm}}&\\
&&&&\parbox[t][0.3cm]{12.43634cm}{\raggedright {\ }{\ }{\ }for Q, and systematic) uncertainties are given. Others: \ensuremath{\mu}=1.00 \textit{8} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1991Wo04,B}{1991Wo04}, using the\vspace{0.1cm}}&\\
&&&&\parbox[t][0.3cm]{12.43634cm}{\raggedright {\ }{\ }{\ }static nuclear orientation technique), 0.99 \textit{11} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1988Wo12,B}{1988Wo12}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{621}&\multicolumn{1}{@{.}l}{66 {\it 16}}&\multicolumn{1}{l}{(7/2)\ensuremath{^{-}}}&\multicolumn{1}{l}{\texttt{\ \hyperlink{AT43}{B}\ \ } }&\parbox[t][0.3cm]{12.43634cm}{\raggedright J\ensuremath{^{\pi}}: 616.1\ensuremath{\gamma} M1+E2 to 5/2\ensuremath{^{-}}, 621.6\ensuremath{\gamma} (E2) to 3/2\ensuremath{^{-}}; direct feeding in \ensuremath{^{\textnormal{201}}}Ae \ensuremath{\varepsilon}\hphantom{a}decay\vspace{0.1cm}}&\\
&&&&\parbox[t][0.3cm]{12.43634cm}{\raggedright {\ }{\ }{\ }(\ensuremath{J^{\pi}}=(9/2\ensuremath{^{-}})).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{623}&\multicolumn{1}{@{.}l}{3? {\it 3}}&\multicolumn{1}{l}{(5/2\ensuremath{^{-}})}&\multicolumn{1}{l}{\texttt{\ \hyperlink{AT43}{B}\ \ } }&\parbox[t][0.3cm]{12.43634cm}{\raggedright J\ensuremath{^{\pi}}: 617.7\ensuremath{\gamma} to 5/2\ensuremath{^{-}}, 623.3\ensuremath{\gamma} to 3/2\ensuremath{^{-}}; no direct feeding in \ensuremath{^{\textnormal{201}}}Ae \ensuremath{\varepsilon}\hphantom{a}decay (\ensuremath{J^{\pi}}=(9/2\ensuremath{^{-}})).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{722}&\multicolumn{1}{@{.}l}{44 {\it 18}}&\multicolumn{1}{l}{7/2\ensuremath{^{-}}}&\multicolumn{1}{l}{\texttt{\ \hyperlink{AT43}{B}\ \ } }&\parbox[t][0.3cm]{12.43634cm}{\raggedright J\ensuremath{^{\pi}}: 722.5\ensuremath{\gamma} E2 to 3/2\ensuremath{^{-}}; direct feeding in \ensuremath{^{\textnormal{201}}}Ae \ensuremath{\varepsilon}\hphantom{a}decay (\ensuremath{J^{\pi}}=(9/2\ensuremath{^{-}})).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{758}&\multicolumn{1}{@{.}l}{30? {\it 20}}&\multicolumn{1}{l}{(7/2)\ensuremath{^{-}}}&\multicolumn{1}{l}{\texttt{\ \hyperlink{AT43}{B}\ \ } }&\parbox[t][0.3cm]{12.43634cm}{\raggedright J\ensuremath{^{\pi}}: 758.3\ensuremath{\gamma} E2 to 3/2\ensuremath{^{-}}; direct feeding in \ensuremath{^{\textnormal{201}}}Ae \ensuremath{\varepsilon}\hphantom{a}decay (\ensuremath{J^{\pi}}=(9/2\ensuremath{^{-}})).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{766}&\multicolumn{1}{@{.}l}{31? {\it 24}}&\multicolumn{1}{l}{(9/2)\ensuremath{^{-}}}&\multicolumn{1}{l}{\texttt{\ \hyperlink{AT43}{B}\ \ } }&\parbox[t][0.3cm]{12.43634cm}{\raggedright J\ensuremath{^{\pi}}: 760.7\ensuremath{\gamma} E2 to 5/2\ensuremath{^{-}}; direct feeding in \ensuremath{^{\textnormal{201}}}Ae \ensuremath{\varepsilon}\hphantom{a}decay (\ensuremath{J^{\pi}}=(9/2\ensuremath{^{-}})).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1006}&\multicolumn{1}{@{.}l}{7? {\it 3}}&\multicolumn{1}{l}{(11/2)\ensuremath{^{+}}}&\multicolumn{1}{l}{\texttt{\ \hyperlink{AT43}{B}\ \ } }&\parbox[t][0.3cm]{12.43634cm}{\raggedright J\ensuremath{^{\pi}}: 583.3\ensuremath{\gamma} M1+E2 to 13/2\ensuremath{^{+}}; direct feeding in \ensuremath{^{\textnormal{201}}}Ae \ensuremath{\varepsilon}\hphantom{a}decay (\ensuremath{J^{\pi}}=(9/2\ensuremath{^{-}})).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1015}&\multicolumn{1}{@{.}l}{2 {\it 3}}&\multicolumn{1}{l}{(11/2)\ensuremath{^{+}}}&\multicolumn{1}{l}{\texttt{\ \hyperlink{AT43}{B}\ \ } }&\parbox[t][0.3cm]{12.43634cm}{\raggedright J\ensuremath{^{\pi}}: 591.8\ensuremath{\gamma} M1+E2 to 13/2\ensuremath{^{+}}; direct feeding in \ensuremath{^{\textnormal{201}}}Ae \ensuremath{\varepsilon}\hphantom{a}decay (\ensuremath{J^{\pi}}=(9/2\ensuremath{^{-}})).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1037}&\multicolumn{1}{@{.}l}{0\ensuremath{^{{\hyperlink{PO41LEVEL5}{a}}}} {\it 11}}&\multicolumn{1}{l}{17/2\ensuremath{^{+}}}&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{PO45}{D}} }&\parbox[t][0.3cm]{12.43634cm}{\raggedright J\ensuremath{^{\pi}}: 613.6\ensuremath{\gamma} (E2) to 13/2\ensuremath{^{+}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1059}&\multicolumn{1}{@{.}l}{5 {\it 3}}&\multicolumn{1}{l}{(7/2)\ensuremath{^{-}}}&\multicolumn{1}{l}{\texttt{\ \hyperlink{AT43}{B}\ \ } }&\parbox[t][0.3cm]{12.43634cm}{\raggedright J\ensuremath{^{\pi}}: 436.2\ensuremath{\gamma} M1+E2 to (5/2\ensuremath{^{-}}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1124}&\multicolumn{1}{@{.}l}{8? {\it 5}}&\multicolumn{1}{l}{(7/2,9/2,11/2)}&\multicolumn{1}{l}{\texttt{\ \hyperlink{AT43}{B}\ \ } }&\parbox[t][0.3cm]{12.43634cm}{\raggedright J\ensuremath{^{\pi}}: 358.5\ensuremath{\gamma} to (5/2)\ensuremath{^{-}};\hphantom{a}direct feeding in \ensuremath{^{\textnormal{201}}}Ae \ensuremath{\varepsilon}\hphantom{a}decay (\ensuremath{J^{\pi}}=(9/2\ensuremath{^{-}})).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1242}&\multicolumn{1}{@{.}l}{9? {\it 5}}&\multicolumn{1}{l}{(7/2,9/2,11/2)}&\multicolumn{1}{l}{\texttt{\ \hyperlink{AT43}{B}\ \ } }&\parbox[t][0.3cm]{12.43634cm}{\raggedright J\ensuremath{^{\pi}}: 476.6\ensuremath{\gamma} to (5/2)\ensuremath{^{-}};\hphantom{a}direct feeding in \ensuremath{^{\textnormal{201}}}Ae \ensuremath{\varepsilon}\hphantom{a}decay (\ensuremath{J^{\pi}}=(9/2\ensuremath{^{-}})).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1552}&\multicolumn{1}{@{.}l}{2 {\it 3}}&\multicolumn{1}{l}{(9/2)\ensuremath{^{+}}}&\multicolumn{1}{l}{\texttt{\ \hyperlink{AT43}{B}\ \ } }&\parbox[t][0.3cm]{12.43634cm}{\raggedright J\ensuremath{^{\pi}}: 492.7\ensuremath{\gamma} E1 to (7/2)\ensuremath{^{-}}, 537\ensuremath{\gamma} M1+E2 to (11/2)\ensuremath{^{+}};\hphantom{a}direct feeding in \ensuremath{^{\textnormal{201}}}Ae \ensuremath{\varepsilon}\hphantom{a}decay\vspace{0.1cm}}&\\
&&&&\parbox[t][0.3cm]{12.43634cm}{\raggedright {\ }{\ }{\ }(\ensuremath{J^{\pi}}=(9/2\ensuremath{^{-}})).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1574}&\multicolumn{1}{@{.}l}{3 {\it 4}}&\multicolumn{1}{l}{(9/2,11/2)\ensuremath{^{+}}}&\multicolumn{1}{l}{\texttt{\ \hyperlink{AT43}{B}\ \ } }&\parbox[t][0.3cm]{12.43634cm}{\raggedright J\ensuremath{^{\pi}}: 559.1\ensuremath{\gamma} M1+E2 to (11/2)\ensuremath{^{+}};\hphantom{a}direct feeding in \ensuremath{^{\textnormal{201}}}Ae \ensuremath{\varepsilon}\hphantom{a}decay (\ensuremath{J^{\pi}}=(9/2\ensuremath{^{-}})).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1593}&\multicolumn{1}{@{.}l}{6\ensuremath{^{{\hyperlink{PO41LEVEL6}{b}}}} {\it 15}}&\multicolumn{1}{l}{21/2\ensuremath{^{+}}}&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{PO45}{D}} }&\parbox[t][0.3cm]{12.43634cm}{\raggedright J\ensuremath{^{\pi}}: 556.6\ensuremath{\gamma} E2 to 17/2\ensuremath{^{+}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1912}&\multicolumn{1}{@{.}l}{3\ensuremath{^{{\hyperlink{PO41LEVEL7}{c}}}} {\it 18}}&\multicolumn{1}{l}{25/2\ensuremath{^{+}}}&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{PO45}{D}} }&\parbox[t][0.3cm]{12.43634cm}{\raggedright J\ensuremath{^{\pi}}: 318.7\ensuremath{\gamma} E2 to 21/2\ensuremath{^{+}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2044}&\multicolumn{1}{@{.}l}{0 {\it 4}}&\multicolumn{1}{l}{(9/2)\ensuremath{^{+}}}&\multicolumn{1}{l}{\texttt{\ \hyperlink{AT43}{B}\ \ } }&\parbox[t][0.3cm]{12.43634cm}{\raggedright J\ensuremath{^{\pi}}: 491.8\ensuremath{\gamma} E2 to (9/2)\ensuremath{^{+}};\hphantom{a}direct feeding in \ensuremath{^{\textnormal{201}}}Ae \ensuremath{\varepsilon}\hphantom{a}decay (\ensuremath{J^{\pi}}=(9/2\ensuremath{^{-}})).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2101}&\multicolumn{1}{@{.}l}{6 {\it 21}}&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{PO45}{D}} }&&\\
\multicolumn{1}{r@{}}{2133}&\multicolumn{1}{@{.}l}{8 {\it 21}}&\multicolumn{1}{l}{25/2\ensuremath{^{+}},(29/2\ensuremath{^{+}})}&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{PO45}{D}} }&\parbox[t][0.3cm]{12.43634cm}{\raggedright J\ensuremath{^{\pi}}: 221.5\ensuremath{\gamma} D,Q to 25/2\ensuremath{^{+}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2202}&\multicolumn{1}{@{.}l}{9 {\it 4}}&\multicolumn{1}{l}{(9/2,11/2)\ensuremath{^{+}}}&\multicolumn{1}{l}{\texttt{\ \hyperlink{AT43}{B}\ \ } }&\parbox[t][0.3cm]{12.43634cm}{\raggedright J\ensuremath{^{\pi}}: 628.6\ensuremath{\gamma} E2 to (9/2,11/2)\ensuremath{^{+}};\hphantom{a}direct feeding in \ensuremath{^{\textnormal{201}}}Ae \ensuremath{\varepsilon}\hphantom{a}decay (\ensuremath{J^{\pi}}=(9/2\ensuremath{^{-}})).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2239}&\multicolumn{1}{@{.}l}{6 {\it 23}}&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{PO45}{D}} }&&\\
\multicolumn{1}{r@{}}{2332}&\multicolumn{1}{@{.}l}{2 {\it 21}}&\multicolumn{1}{l}{(27/2)}&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{PO45}{D}} }&\parbox[t][0.3cm]{12.43634cm}{\raggedright J\ensuremath{^{\pi}}: 419.9\ensuremath{\gamma} D to 25/2\ensuremath{^{+}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2347}&\multicolumn{1}{@{.}l}{6 {\it 18}}&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{PO45}{D}} }&&\\
\multicolumn{1}{r@{}}{2354}&\multicolumn{1}{@{.}l}{7 {\it 21}}&\multicolumn{1}{l}{(27/2\ensuremath{^{+}})}&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{PO45}{D}} }&\parbox[t][0.3cm]{12.43634cm}{\raggedright J\ensuremath{^{\pi}}: 442.4\ensuremath{\gamma} (M1) to 25/2\ensuremath{^{+}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2463}&\multicolumn{1}{@{.}l}{9 {\it 21}}&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{PO45}{D}} }&&\\
\multicolumn{1}{r@{}}{2570}&\multicolumn{1}{@{.}l}{2 {\it 21}}&\multicolumn{1}{l}{(27/2\ensuremath{^{+}})}&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{PO45}{D}} }&\parbox[t][0.3cm]{12.43634cm}{\raggedright J\ensuremath{^{\pi}}: 657.9\ensuremath{\gamma} (M1) to 25/2\ensuremath{^{+}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2627}&\multicolumn{1}{@{.}l}{5 {\it 23}}&\multicolumn{1}{l}{(29/2\ensuremath{^{+}})}&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{PO45}{D}} }&\parbox[t][0.3cm]{12.43634cm}{\raggedright J\ensuremath{^{\pi}}: 272.8\ensuremath{\gamma} (M1) to (27/2\ensuremath{^{+}}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2770}&\multicolumn{1}{@{.}l}{1 {\it 21}}&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{PO45}{D}} }&&\\
\multicolumn{1}{r@{}}{2979}&\multicolumn{1}{@{.}l}{0 {\it 23}}&\multicolumn{1}{l}{27/2\ensuremath{^{+}},31/2\ensuremath{^{+}}}&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{PO45}{D}} }&\parbox[t][0.3cm]{12.43634cm}{\raggedright J\ensuremath{^{\pi}}: 408.8\ensuremath{\gamma} M1,E2 to (27/2\ensuremath{^{+}}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3039}&\multicolumn{1}{@{.}l}{6 {\it 23}}&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{PO45}{D}} }&&\\
\multicolumn{1}{r@{}}{3196}&\multicolumn{1}{@{.}l}{5 {\it 23}}&\multicolumn{1}{l}{(29/2)}&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{PO45}{D}} }&\parbox[t][0.3cm]{12.43634cm}{\raggedright J\ensuremath{^{\pi}}: 626.3\ensuremath{\gamma} D to (27/2\ensuremath{^{+}}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3210}&\multicolumn{1}{@{.}l}{3 {\it 23}}&\multicolumn{1}{l}{(31/2\ensuremath{^{+}})}&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{PO45}{D}} }&\parbox[t][0.3cm]{12.43634cm}{\raggedright J\ensuremath{^{\pi}}: 640.1\ensuremath{\gamma} E2 to (27/2\ensuremath{^{+}}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3333}&\multicolumn{1}{@{.}l}{1 {\it 25}}&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{PO45}{D}} }&&\\
\multicolumn{1}{r@{}}{3710}&\multicolumn{1}{@{.}l}{1 {\it 25}}&\multicolumn{1}{l}{(35/2\ensuremath{^{+}})}&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{PO45}{D}} }&\parbox[t][0.3cm]{12.43634cm}{\raggedright J\ensuremath{^{\pi}}: 499.8\ensuremath{\gamma} to (31/2\ensuremath{^{+}}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{4153?}&\multicolumn{1}{@{ }l}{{\it 3}}&&\multicolumn{1}{l}{\texttt{\ \ \ \hyperlink{PO45}{D}} }&&\\
\end{longtable}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PO41LEVEL0}{\dagger}}}} From a least-squares fit to E\ensuremath{\gamma}. \ensuremath{\Delta}E\ensuremath{\gamma}=0.5 keV is assumed for E\ensuremath{\gamma}$'$s without uncertainties.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PO41LEVEL1}{\ddagger}}}} Configuration=\ensuremath{\nu} f\ensuremath{_{\textnormal{5/2}}^{\textnormal{$-$1}}}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PO41LEVEL2}{\#}}}} Configuration=\ensuremath{\nu} p\ensuremath{_{\textnormal{1/2}}^{\textnormal{$-$1}}}. The assignment is tentative.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PO41LEVEL3}{@}}}} Configuration=\ensuremath{\nu} p\ensuremath{_{\textnormal{3/2}}^{\textnormal{$-$1}}}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PO41LEVEL4}{\&}}}} Configuration=\ensuremath{\nu} i\ensuremath{_{\textnormal{13/2}}^{\textnormal{$-$1}}}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PO41LEVEL5}{a}}}} Configuration=\ensuremath{\nu} (i\ensuremath{_{\textnormal{13/2}}^{\textnormal{$-$1}}})\ensuremath{\otimes}2\ensuremath{^{\textnormal{+}}}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PO41LEVEL6}{b}}}} Configuration=\ensuremath{\nu} (i\ensuremath{_{\textnormal{13/2}}^{\textnormal{$-$1}}})\ensuremath{\otimes}4\ensuremath{^{\textnormal{+}}}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PO41LEVEL7}{c}}}} Possibly a mixture between configuration=\ensuremath{\nu} (i\ensuremath{_{\textnormal{13/2}}^{\textnormal{$-$1}}})\ensuremath{\otimes}6\ensuremath{^{\textnormal{+}}} and configuration=\ensuremath{\nu} (i\ensuremath{_{\textnormal{13/2}}^{\textnormal{$-$1}}}) \ensuremath{\pi} (h\ensuremath{_{\textnormal{9/2}}^{\textnormal{+2}}})\ensuremath{_{\textnormal{8+}}}.}\\
\vspace{0.5cm}
\clearpage
\vspace{0.3cm}
\begin{landscape}
\vspace*{-0.5cm}
{\bf \small \underline{Adopted \hyperlink{201PO_LEVEL}{Levels}, \hyperlink{201PO_GAMMA}{Gammas} (continued)}}\\
\vspace{0.3cm}
\hypertarget{201PO_GAMMA}{\underline{$\gamma$($^{201}$Po)}}\\
\begin{longtable}{ccccccccc@{}ccccccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E\ensuremath{_{i}}(level)}&J\ensuremath{^{\pi}_{i}}&\multicolumn{2}{c}{E\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{PO41GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{I\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{PO41GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{E\ensuremath{_{f}}}&J\ensuremath{^{\pi}_{f}}&Mult.\ensuremath{^{\hyperlink{PO41GAMMA2}{\#}}}&\multicolumn{2}{c}{\ensuremath{\delta}}&\multicolumn{2}{c}{\ensuremath{\alpha}\ensuremath{^{\hyperlink{PO41GAMMA3}{@}}}}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{5}&\multicolumn{1}{@{.}l}{61}&\multicolumn{1}{l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{(5}&\multicolumn{1}{@{.}l}{61 {\it 13})}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{3/2\ensuremath{^{-}}}&\multicolumn{1}{l}{}&&&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{7.8698597cm}{\raggedright E\ensuremath{_{\gamma}}: From level energy difference.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{423}&\multicolumn{1}{@{.}l}{41}&\multicolumn{1}{l}{13/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{417}&\multicolumn{1}{@{.}l}{8 {\it 2}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{5}&\multicolumn{1}{@{.}l}{61 }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M4}&&&\multicolumn{1}{r@{}}{4}&\multicolumn{1}{@{.}l}{84 {\it 7}}&\parbox[t][0.3cm]{7.8698597cm}{\raggedright B(M4)(W.u.)\ensuremath{\approx}1.8\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.8698597cm}{\raggedright E\ensuremath{_{\gamma}}: Weighted average of 418.5 keV \textit{6} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1976Ko13,B}{1976Ko13}), 417.6\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.8698597cm}{\raggedright {\ }{\ }{\ }keV \textit{3} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1971Jo19,B}{1971Jo19}) an 417.8 keV \textit{6} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986Br28,B}{1986Br28}).\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.8698597cm}{\raggedright Mult.: \ensuremath{\alpha}(K)exp=2.49 \textit{32}, K/L=1.8 \textit{2}, L/M=2.5 \textit{3}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.8698597cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986Br28,B}{1986Br28}); K/L=2.0 \textit{3} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1976Ko13,B}{1976Ko13}), 1.6 \textit{2} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1971Jo19,B}{1971Jo19}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{621}&\multicolumn{1}{@{.}l}{66}&\multicolumn{1}{l}{(7/2)\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{616}&\multicolumn{1}{@{.}l}{1 {\it 2}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{ }l}{{\it 9}}&\multicolumn{1}{r@{}}{5}&\multicolumn{1}{@{.}l}{61 }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{72 {\it 20}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0334 {\it 27}}&\parbox[t][0.3cm]{7.8698597cm}{\raggedright Mult.,\ensuremath{\delta}: \ensuremath{\alpha}(K)exp=0.026 \textit{4} in \ensuremath{^{\textnormal{201}}}Ae \ensuremath{\varepsilon}\hphantom{a}decay\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.8698597cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2010De04,B}{2010De04}).\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{621}&\multicolumn{1}{@{.}l}{6 {\it 2}}&\multicolumn{1}{r@{}}{57}&\multicolumn{1}{@{ }l}{{\it 5}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{3/2\ensuremath{^{-}}}&\multicolumn{1}{l}{(E2)}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{01950 {\it 27}}&\parbox[t][0.3cm]{7.8698597cm}{\raggedright Mult.: \ensuremath{\alpha}(K)exp=0.019 \textit{3} in \ensuremath{^{\textnormal{201}}}Ae \ensuremath{\varepsilon}\hphantom{a}decay (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2010De04,B}{2010De04}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{623}&\multicolumn{1}{@{.}l}{3?}&\multicolumn{1}{l}{(5/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{(617}&\multicolumn{1}{@{.}l}{9 {\it 3})}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{5}&\multicolumn{1}{@{.}l}{61 }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{l}{}&&&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{7.8698597cm}{\raggedright E\ensuremath{_{\gamma}}: From level energy difference.\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{(623}&\multicolumn{1}{@{.}l}{3 {\it 3})}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{3/2\ensuremath{^{-}}}&\multicolumn{1}{l}{}&&&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{7.8698597cm}{\raggedright E\ensuremath{_{\gamma}}: From level energy difference.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{722}&\multicolumn{1}{@{.}l}{44}&\multicolumn{1}{l}{7/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{716}&\multicolumn{1}{@{.}l}{6 {\it 4}}&\multicolumn{1}{r@{}}{20}&\multicolumn{1}{@{.}l}{9 {\it 18}}&\multicolumn{1}{r@{}}{5}&\multicolumn{1}{@{.}l}{61 }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{l}{[M1,E2]}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{032 {\it 18}}&&\\
&&&\multicolumn{1}{r@{}}{722}&\multicolumn{1}{@{.}l}{5 {\it 2}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{ }l}{{\it 9}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{3/2\ensuremath{^{-}}}&\multicolumn{1}{l}{E2}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{01413 {\it 20}}&\parbox[t][0.3cm]{7.8698597cm}{\raggedright Mult.: \ensuremath{\alpha}(K)exp=0.008 \textit{1} in \ensuremath{^{\textnormal{201}}}Ae \ensuremath{\varepsilon}\hphantom{a}decay (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2010De04,B}{2010De04}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{758}&\multicolumn{1}{@{.}l}{30?}&\multicolumn{1}{l}{(7/2)\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{758}&\multicolumn{1}{@{.}l}{3 {\it 2}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{3/2\ensuremath{^{-}}}&\multicolumn{1}{l}{E2}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{01278 {\it 18}}&\parbox[t][0.3cm]{7.8698597cm}{\raggedright Mult.: \ensuremath{\alpha}(K)exp=0.007 \textit{1} in \ensuremath{^{\textnormal{201}}}Ae \ensuremath{\varepsilon}\hphantom{a}decay (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2010De04,B}{2010De04}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{766}&\multicolumn{1}{@{.}l}{31?}&\multicolumn{1}{l}{(9/2)\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{760}&\multicolumn{1}{@{.}l}{7 {\it 2}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{5}&\multicolumn{1}{@{.}l}{61 }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{l}{E2}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{01269 {\it 18}}&\parbox[t][0.3cm]{7.8698597cm}{\raggedright Mult.: \ensuremath{\alpha}(K)exp=0.009 \textit{1} in \ensuremath{^{\textnormal{201}}}Ae \ensuremath{\varepsilon}\hphantom{a}decay (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2010De04,B}{2010De04}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1006}&\multicolumn{1}{@{.}l}{7?}&\multicolumn{1}{l}{(11/2)\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{583}&\multicolumn{1}{@{.}l}{3 {\it 2}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{423}&\multicolumn{1}{@{.}l}{41 }&\multicolumn{1}{@{}l}{13/2\ensuremath{^{+}}}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{61 {\it 17}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0304 {\it 11}}&\parbox[t][0.3cm]{7.8698597cm}{\raggedright Mult.,\ensuremath{\delta}: \ensuremath{\alpha}(K)exp=0.023 \textit{3} in \ensuremath{^{\textnormal{201}}}Ae \ensuremath{\varepsilon}\hphantom{a}decay\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.8698597cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2010De04,B}{2010De04}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1015}&\multicolumn{1}{@{.}l}{2}&\multicolumn{1}{l}{(11/2)\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{591}&\multicolumn{1}{@{.}l}{8 {\it 2}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{423}&\multicolumn{1}{@{.}l}{41 }&\multicolumn{1}{@{}l}{13/2\ensuremath{^{+}}}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{67 {\it 18}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0291 {\it 10}}&\parbox[t][0.3cm]{7.8698597cm}{\raggedright Mult.,\ensuremath{\delta}: \ensuremath{\alpha}(K)exp=0.022 \textit{3} in \ensuremath{^{\textnormal{201}}}Ae \ensuremath{\varepsilon}\hphantom{a}decay\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.8698597cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2010De04,B}{2010De04}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1037}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{l}{17/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{613}&\multicolumn{1}{@{.}l}{6\ensuremath{^{\hyperlink{PO41GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PO41GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{423}&\multicolumn{1}{@{.}l}{41 }&\multicolumn{1}{@{}l}{13/2\ensuremath{^{+}}}&\multicolumn{1}{l}{(E2)}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{02007 {\it 28}}&\parbox[t][0.3cm]{7.8698597cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}=0.5 \textit{2}, A\ensuremath{_{\textnormal{4}}}=0.1 \textit{3}, but values are distorted since\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.8698597cm}{\raggedright {\ }{\ }{\ }the 613.6\ensuremath{\gamma} is situated on the slope of both the neutron\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.8698597cm}{\raggedright {\ }{\ }{\ }bump and the stronger 611.2\ensuremath{\gamma} in \ensuremath{^{\textnormal{194}}}Pt(\ensuremath{^{\textnormal{12}}}C,5n\ensuremath{\gamma})\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.8698597cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1985We05,B}{1985We05}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1059}&\multicolumn{1}{@{.}l}{5}&\multicolumn{1}{l}{(7/2)\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{436}&\multicolumn{1}{@{.}l}{2 {\it 2}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{623}&\multicolumn{1}{@{.}l}{3? }&\multicolumn{1}{@{}l}{(5/2\ensuremath{^{-}})}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{93 {\it 23}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{119 {\it 19}}&\parbox[t][0.3cm]{7.8698597cm}{\raggedright Mult.,\ensuremath{\delta}: \ensuremath{\alpha}(K)exp=0.094 \textit{14} in \ensuremath{^{\textnormal{201}}}Ae \ensuremath{\varepsilon}\hphantom{a}decay\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.8698597cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2010De04,B}{2010De04}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1124}&\multicolumn{1}{@{.}l}{8?}&\multicolumn{1}{l}{(7/2,9/2,11/2)}&\multicolumn{1}{r@{}}{358}&\multicolumn{1}{@{.}l}{5 {\it 4}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{766}&\multicolumn{1}{@{.}l}{31? }&\multicolumn{1}{@{}l}{(9/2)\ensuremath{^{-}}}&\multicolumn{1}{l}{}&&&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
\multicolumn{1}{r@{}}{1242}&\multicolumn{1}{@{.}l}{9?}&\multicolumn{1}{l}{(7/2,9/2,11/2)}&\multicolumn{1}{r@{}}{476}&\multicolumn{1}{@{.}l}{6 {\it 4}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{766}&\multicolumn{1}{@{.}l}{31? }&\multicolumn{1}{@{}l}{(9/2)\ensuremath{^{-}}}&\multicolumn{1}{l}{}&&&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
\multicolumn{1}{r@{}}{1552}&\multicolumn{1}{@{.}l}{2}&\multicolumn{1}{l}{(9/2)\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{492}&\multicolumn{1}{@{.}l}{7 {\it 2}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{ }l}{{\it 15}}&\multicolumn{1}{r@{}}{1059}&\multicolumn{1}{@{.}l}{5 }&\multicolumn{1}{@{}l}{(7/2)\ensuremath{^{-}}}&\multicolumn{1}{l}{E1}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{01058 {\it 15}}&\parbox[t][0.3cm]{7.8698597cm}{\raggedright Mult.: \ensuremath{\alpha}(K)exp=0.010 \textit{4} in \ensuremath{^{\textnormal{201}}}Ae \ensuremath{\varepsilon}\hphantom{a}decay (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2010De04,B}{2010De04}).\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{537}&\multicolumn{1}{@{.}l}{0 {\it 2}}&\multicolumn{1}{r@{}}{95}&\multicolumn{1}{@{ }l}{{\it 8}}&\multicolumn{1}{r@{}}{1015}&\multicolumn{1}{@{.}l}{2 }&\multicolumn{1}{@{}l}{(11/2)\ensuremath{^{+}}}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{58 {\it 17}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0328 {\it 7}}&\parbox[t][0.3cm]{7.8698597cm}{\raggedright Mult.,\ensuremath{\delta}: \ensuremath{\alpha}(K)exp=0.024 \textit{3} in \ensuremath{^{\textnormal{201}}}Ae \ensuremath{\varepsilon}\hphantom{a}decay\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.8698597cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2010De04,B}{2010De04}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1574}&\multicolumn{1}{@{.}l}{3}&\multicolumn{1}{l}{(9/2,11/2)\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{559}&\multicolumn{1}{@{.}l}{1 {\it 2}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{1015}&\multicolumn{1}{@{.}l}{2 }&\multicolumn{1}{@{}l}{(11/2)\ensuremath{^{+}}}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{78 {\it 20}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0415 {\it 33}}&\parbox[t][0.3cm]{7.8698597cm}{\raggedright Mult.,\ensuremath{\delta}: \ensuremath{\alpha}(K)exp=0.032 \textit{5} in \ensuremath{^{\textnormal{201}}}Ae \ensuremath{\varepsilon}\hphantom{a}decay\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.8698597cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2010De04,B}{2010De04}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1593}&\multicolumn{1}{@{.}l}{6}&\multicolumn{1}{l}{21/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{556}&\multicolumn{1}{@{.}l}{6\ensuremath{^{\hyperlink{PO41GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PO41GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{1037}&\multicolumn{1}{@{.}l}{0 }&\multicolumn{1}{@{}l}{17/2\ensuremath{^{+}}}&\multicolumn{1}{l}{E2}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{02499 {\it 35}}&\parbox[t][0.3cm]{7.8698597cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}=0.24 \textit{4}, A\ensuremath{_{\textnormal{4}}}={\textminus}0.01 \textit{5} in \ensuremath{^{\textnormal{194}}}Pt(\ensuremath{^{\textnormal{12}}}C,5n\ensuremath{\gamma})\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.8698597cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1985We05,B}{1985We05}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1912}&\multicolumn{1}{@{.}l}{3}&\multicolumn{1}{l}{25/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{318}&\multicolumn{1}{@{.}l}{7\ensuremath{^{\hyperlink{PO41GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PO41GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{1593}&\multicolumn{1}{@{.}l}{6 }&\multicolumn{1}{@{}l}{21/2\ensuremath{^{+}}}&\multicolumn{1}{l}{E2}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{1054 {\it 15}}&\parbox[t][0.3cm]{7.8698597cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}=0.22 \textit{1}, A\ensuremath{_{\textnormal{4}}}={\textminus}0.05 \textit{1} in \ensuremath{^{\textnormal{194}}}Pt(\ensuremath{^{\textnormal{12}}}C,5n\ensuremath{\gamma})\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.8698597cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1985We05,B}{1985We05}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2044}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{l}{(9/2)\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{491}&\multicolumn{1}{@{.}l}{8 {\it 2}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{1552}&\multicolumn{1}{@{.}l}{2 }&\multicolumn{1}{@{}l}{(9/2)\ensuremath{^{+}}}&\multicolumn{1}{l}{E2}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0334 {\it 5}}&\parbox[t][0.3cm]{7.8698597cm}{\raggedright Mult.: \ensuremath{\alpha}(K)exp=0.023 \textit{7} in \ensuremath{^{\textnormal{201}}}Ae \ensuremath{\varepsilon}\hphantom{a}decay (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2010De04,B}{2010De04}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2101}&\multicolumn{1}{@{.}l}{6}&&\multicolumn{1}{r@{}}{189}&\multicolumn{1}{@{.}l}{3\ensuremath{^{\hyperlink{PO41GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PO41GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{1912}&\multicolumn{1}{@{.}l}{3 }&\multicolumn{1}{@{}l}{25/2\ensuremath{^{+}}}&\multicolumn{1}{l}{D,Q}&&&&&\parbox[t][0.3cm]{7.8698597cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}=0.1 \textit{4}, A\ensuremath{_{\textnormal{4}}}=0.2 \textit{3} in \ensuremath{^{\textnormal{194}}}Pt(\ensuremath{^{\textnormal{12}}}C,5n\ensuremath{\gamma})\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{7.8698597cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1985We05,B}{1985We05}).\vspace{0.1cm}}&\\
\end{longtable}
\clearpage
\begin{longtable}{ccccccccc@{}ccccc@{\extracolsep{\fill}}c}
\\[-.4cm]
\multicolumn{15}{c}{{\bf \small \underline{Adopted \hyperlink{201PO_LEVEL}{Levels}, \hyperlink{201PO_GAMMA}{Gammas} (continued)}}}\\
\multicolumn{15}{c}{~}\\
\multicolumn{15}{c}{\underline{$\gamma$($^{201}$Po) (continued)}}\\
\multicolumn{15}{c}{~~~}\\
\multicolumn{2}{c}{E\ensuremath{_{i}}(level)}&J\ensuremath{^{\pi}_{i}}&\multicolumn{2}{c}{E\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{PO41GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{I\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{PO41GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{E\ensuremath{_{f}}}&J\ensuremath{^{\pi}_{f}}&Mult.\ensuremath{^{\hyperlink{PO41GAMMA2}{\#}}}&\multicolumn{2}{c}{\ensuremath{\alpha}\ensuremath{^{\hyperlink{PO41GAMMA3}{@}}}}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endhead
\multicolumn{1}{r@{}}{2133}&\multicolumn{1}{@{.}l}{8}&\multicolumn{1}{l}{25/2\ensuremath{^{+}},(29/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{221}&\multicolumn{1}{@{.}l}{5\ensuremath{^{\hyperlink{PO41GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PO41GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{1912}&\multicolumn{1}{@{.}l}{3 }&\multicolumn{1}{@{}l}{25/2\ensuremath{^{+}}}&\multicolumn{1}{l}{D,Q}&&&\parbox[t][0.3cm]{9.5774cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}=0.10 \textit{6}, A\ensuremath{_{\textnormal{4}}}={\textminus}0.11 \textit{9} in \ensuremath{^{\textnormal{194}}}Pt(\ensuremath{^{\textnormal{12}}}C,5n\ensuremath{\gamma}) (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1985We05,B}{1985We05}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2202}&\multicolumn{1}{@{.}l}{9}&\multicolumn{1}{l}{(9/2,11/2)\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{628}&\multicolumn{1}{@{.}l}{6 {\it 2}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{1574}&\multicolumn{1}{@{.}l}{3 }&\multicolumn{1}{@{}l}{(9/2,11/2)\ensuremath{^{+}}}&\multicolumn{1}{l}{E2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{01903 {\it 27}}&\parbox[t][0.3cm]{9.5774cm}{\raggedright Mult.: \ensuremath{\alpha}(K)exp=0.014 \textit{3} in \ensuremath{^{\textnormal{201}}}Ae \ensuremath{\varepsilon}\hphantom{a}decay (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2010De04,B}{2010De04}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2239}&\multicolumn{1}{@{.}l}{6}&&\multicolumn{1}{r@{}}{138}&\multicolumn{1}{@{.}l}{0\ensuremath{^{\hyperlink{PO41GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PO41GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{2101}&\multicolumn{1}{@{.}l}{6 }&&&&&&\\
\multicolumn{1}{r@{}}{2332}&\multicolumn{1}{@{.}l}{2}&\multicolumn{1}{l}{(27/2)}&\multicolumn{1}{r@{}}{419}&\multicolumn{1}{@{.}l}{9\ensuremath{^{\hyperlink{PO41GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PO41GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{1912}&\multicolumn{1}{@{.}l}{3 }&\multicolumn{1}{@{}l}{25/2\ensuremath{^{+}}}&\multicolumn{1}{l}{D}&&&\parbox[t][0.3cm]{9.5774cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}={\textminus}0.2 \textit{3}, A\ensuremath{_{\textnormal{4}}}={\textminus}0.1 \textit{4} in \ensuremath{^{\textnormal{194}}}Pt(\ensuremath{^{\textnormal{12}}}C,5n\ensuremath{\gamma}) (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1985We05,B}{1985We05}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2347}&\multicolumn{1}{@{.}l}{6}&&\multicolumn{1}{r@{}}{754}&\multicolumn{1}{@{.}l}{0\ensuremath{^{\hyperlink{PO41GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PO41GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{1593}&\multicolumn{1}{@{.}l}{6 }&\multicolumn{1}{@{}l}{21/2\ensuremath{^{+}}}&&&&&\\
\multicolumn{1}{r@{}}{2354}&\multicolumn{1}{@{.}l}{7}&\multicolumn{1}{l}{(27/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{442}&\multicolumn{1}{@{.}l}{4\ensuremath{^{\hyperlink{PO41GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PO41GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{1912}&\multicolumn{1}{@{.}l}{3 }&\multicolumn{1}{@{}l}{25/2\ensuremath{^{+}}}&\multicolumn{1}{l}{(M1)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{1766 {\it 25}}&\parbox[t][0.3cm]{9.5774cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}={\textminus}0.05 \textit{7}, A\ensuremath{_{\textnormal{4}}}=0.27 \textit{9} in \ensuremath{^{\textnormal{194}}}Pt(\ensuremath{^{\textnormal{12}}}C,5n\ensuremath{\gamma}) (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1985We05,B}{1985We05}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2463}&\multicolumn{1}{@{.}l}{9}&&\multicolumn{1}{r@{}}{551}&\multicolumn{1}{@{.}l}{6\ensuremath{^{\hyperlink{PO41GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PO41GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{1912}&\multicolumn{1}{@{.}l}{3 }&\multicolumn{1}{@{}l}{25/2\ensuremath{^{+}}}&&&&&\\
\multicolumn{1}{r@{}}{2570}&\multicolumn{1}{@{.}l}{2}&\multicolumn{1}{l}{(27/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{657}&\multicolumn{1}{@{.}l}{9\ensuremath{^{\hyperlink{PO41GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PO41GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{1912}&\multicolumn{1}{@{.}l}{3 }&\multicolumn{1}{@{}l}{25/2\ensuremath{^{+}}}&\multicolumn{1}{l}{(M1)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0618 {\it 9}}&\parbox[t][0.3cm]{9.5774cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}={\textminus}0.30 \textit{18}, A\ensuremath{_{\textnormal{4}}}={\textminus}0.2 \textit{2} in \ensuremath{^{\textnormal{194}}}Pt(\ensuremath{^{\textnormal{12}}}C,5n\ensuremath{\gamma}) (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1985We05,B}{1985We05}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2627}&\multicolumn{1}{@{.}l}{5}&\multicolumn{1}{l}{(29/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{272}&\multicolumn{1}{@{.}l}{8\ensuremath{^{\hyperlink{PO41GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PO41GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{2354}&\multicolumn{1}{@{.}l}{7 }&\multicolumn{1}{@{}l}{(27/2\ensuremath{^{+}})}&\multicolumn{1}{l}{(M1)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{656 {\it 9}}&\parbox[t][0.3cm]{9.5774cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}=0.3 \textit{3}, A\ensuremath{_{\textnormal{4}}}=0.4 \textit{3} in \ensuremath{^{\textnormal{194}}}Pt(\ensuremath{^{\textnormal{12}}}C,5n\ensuremath{\gamma}) (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1985We05,B}{1985We05}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2770}&\multicolumn{1}{@{.}l}{1}&&\multicolumn{1}{r@{}}{857}&\multicolumn{1}{@{.}l}{8\ensuremath{^{\hyperlink{PO41GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PO41GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{1912}&\multicolumn{1}{@{.}l}{3 }&\multicolumn{1}{@{}l}{25/2\ensuremath{^{+}}}&&&&&\\
\multicolumn{1}{r@{}}{2979}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{l}{27/2\ensuremath{^{+}},31/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{408}&\multicolumn{1}{@{.}l}{8\ensuremath{^{\hyperlink{PO41GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PO41GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{2570}&\multicolumn{1}{@{.}l}{2 }&\multicolumn{1}{@{}l}{(27/2\ensuremath{^{+}})}&\multicolumn{1}{l}{M1,E2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{14 {\it 8}}&\parbox[t][0.3cm]{9.5774cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}=0.5 \textit{2}, A\ensuremath{_{\textnormal{4}}}=0.1 \textit{3} in \ensuremath{^{\textnormal{194}}}Pt(\ensuremath{^{\textnormal{12}}}C,5n\ensuremath{\gamma}) (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1985We05,B}{1985We05}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3039}&\multicolumn{1}{@{.}l}{6}&&\multicolumn{1}{r@{}}{905}&\multicolumn{1}{@{.}l}{8\ensuremath{^{\hyperlink{PO41GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PO41GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{2133}&\multicolumn{1}{@{.}l}{8 }&\multicolumn{1}{@{}l}{25/2\ensuremath{^{+}},(29/2\ensuremath{^{+}})}&\multicolumn{1}{l}{D}&&&\parbox[t][0.3cm]{9.5774cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}={\textminus}0.1 \textit{3}, A\ensuremath{_{\textnormal{4}}}={\textminus}0.1 \textit{5} in \ensuremath{^{\textnormal{194}}}Pt(\ensuremath{^{\textnormal{12}}}C,5n\ensuremath{\gamma}) (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1985We05,B}{1985We05}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3196}&\multicolumn{1}{@{.}l}{5}&\multicolumn{1}{l}{(29/2)}&\multicolumn{1}{r@{}}{626}&\multicolumn{1}{@{.}l}{3\ensuremath{^{\hyperlink{PO41GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PO41GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{2570}&\multicolumn{1}{@{.}l}{2 }&\multicolumn{1}{@{}l}{(27/2\ensuremath{^{+}})}&\multicolumn{1}{l}{D}&&&\parbox[t][0.3cm]{9.5774cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}={\textminus}0.36 \textit{7}, A\ensuremath{_{\textnormal{4}}}={\textminus}0.48 \textit{10} in \ensuremath{^{\textnormal{194}}}Pt(\ensuremath{^{\textnormal{12}}}C,5n\ensuremath{\gamma}) (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1985We05,B}{1985We05}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3210}&\multicolumn{1}{@{.}l}{3}&\multicolumn{1}{l}{(31/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{640}&\multicolumn{1}{@{.}l}{1\ensuremath{^{\hyperlink{PO41GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PO41GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{2570}&\multicolumn{1}{@{.}l}{2 }&\multicolumn{1}{@{}l}{(27/2\ensuremath{^{+}})}&\multicolumn{1}{l}{(E2)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{01829 {\it 26}}&\parbox[t][0.3cm]{9.5774cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}=0.44 \textit{19}, A\ensuremath{_{\textnormal{4}}}={\textminus}0.2 \textit{3} in \ensuremath{^{\textnormal{194}}}Pt(\ensuremath{^{\textnormal{12}}}C,5n\ensuremath{\gamma}) (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1985We05,B}{1985We05}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3333}&\multicolumn{1}{@{.}l}{1}&&\multicolumn{1}{r@{}}{354}&\multicolumn{1}{@{.}l}{1\ensuremath{^{\hyperlink{PO41GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PO41GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{2979}&\multicolumn{1}{@{.}l}{0 }&\multicolumn{1}{@{}l}{27/2\ensuremath{^{+}},31/2\ensuremath{^{+}}}&\multicolumn{1}{l}{D}&&&\parbox[t][0.3cm]{9.5774cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}=0.02 \textit{16}, A\ensuremath{_{\textnormal{4}}}=0.1 \textit{2} in \ensuremath{^{\textnormal{194}}}Pt(\ensuremath{^{\textnormal{12}}}C,5n\ensuremath{\gamma}) (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1985We05,B}{1985We05}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3710}&\multicolumn{1}{@{.}l}{1}&\multicolumn{1}{l}{(35/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{499}&\multicolumn{1}{@{.}l}{8\ensuremath{^{\hyperlink{PO41GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PO41GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{3210}&\multicolumn{1}{@{.}l}{3 }&\multicolumn{1}{@{}l}{(31/2\ensuremath{^{+}})}&&&&&\\
\multicolumn{1}{r@{}}{4153?}&\multicolumn{1}{@{}l}{}&&\multicolumn{1}{r@{}}{442}&\multicolumn{1}{@{.}l}{4\ensuremath{^{\hyperlink{PO41GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PO41GAMMA1}{\ddagger}}}}&\multicolumn{1}{r@{}}{3710}&\multicolumn{1}{@{.}l}{1 }&\multicolumn{1}{@{}l}{(35/2\ensuremath{^{+}})}&&&&&\\
\end{longtable}
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PO41GAMMA0}{\dagger}}}} From \ensuremath{^{\textnormal{201}}}At \ensuremath{\varepsilon} decay, unless otherwise stated.}\\
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PO41GAMMA1}{\ddagger}}}} From \ensuremath{^{\textnormal{194}}}Pt(\ensuremath{^{\textnormal{12}}}C,5n\ensuremath{\gamma}).}\\
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PO41GAMMA2}{\#}}}} From \ensuremath{\alpha}(K)exp in \ensuremath{^{\textnormal{201}}}At \ensuremath{\varepsilon} decay and \ensuremath{\gamma}(\ensuremath{\theta}) in \ensuremath{^{\textnormal{194}}}Pt(\ensuremath{^{\textnormal{12}}}C,5n\ensuremath{\gamma}), unless otherwise stated.}\\
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PO41GAMMA3}{@}}}} Total theoretical internal conversion coefficients, calculated using the BrIcc code (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2008Ki07,B}{2008Ki07}) with Frozen orbital approximation based on \ensuremath{\gamma}-ray energies,}\\
\parbox[b][0.3cm]{21.881866cm}{{\ }{\ }assigned multipolarities, and mixing ratios, unless otherwise specified.}\\
\vspace{0.5cm}
\end{landscape}\clearpage
\clearpage
\begin{figure}[h]
\begin{center}
\includegraphics{201PO41-0.ps}\\
\end{center}
\end{figure}
\clearpage
\begin{figure}[h]
\begin{center}
\includegraphics{201PO41-1.ps}\\
\end{center}
\end{figure}
\clearpage
\subsection[\hspace{-0.2cm}\ensuremath{^{\textnormal{201}}}Po IT decay (8.96 min)]{ }
\vspace{-27pt}
\vspace{0.3cm}
\hypertarget{PO42}{{\bf \small \underline{\ensuremath{^{\textnormal{201}}}Po IT decay (8.96 min)}}}\\
\vspace{4pt}
\vspace{8pt}
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}Parent: $^{201}$Po: E=423.41 {\it 22}; J$^{\pi}$=13/2\ensuremath{^{+}}; T$_{1/2}$=8.96 min {\it 12}; \%IT decay$\approx$42.6
}\\
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}\ensuremath{^{201}}Po-\%IT decay: From Adopted Levels.}\\
\vspace{12pt}
\underline{$^{201}$Po Levels}\\
\begin{longtable}{cccccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E(level)$^{{\hyperlink{PO42LEVEL0}{\dagger}}}$}&J$^{\pi}$$^{{\hyperlink{PO42LEVEL0}{\dagger}}}$&\multicolumn{2}{c}{T$_{1/2}$$^{{\hyperlink{PO42LEVEL0}{\dagger}}}$}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{PO42LEVEL1}{\ddagger}}}}}&\multicolumn{1}{l}{3/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{15}&\multicolumn{1}{@{.}l}{50 min {\it 14}}&&\\
\multicolumn{1}{r@{}}{5}&\multicolumn{1}{@{.}l}{61\ensuremath{^{{\hyperlink{PO42LEVEL2}{\#}}}} {\it 13}}&\multicolumn{1}{l}{5/2\ensuremath{^{-}}}&&&&\\
\multicolumn{1}{r@{}}{423}&\multicolumn{1}{@{.}l}{41\ensuremath{^{{\hyperlink{PO42LEVEL3}{@}}}} {\it 22}}&\multicolumn{1}{l}{13/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{8}&\multicolumn{1}{@{.}l}{96 min {\it 12}}&\parbox[t][0.3cm]{12.691cm}{\raggedright \%IT\ensuremath{\approx}42.6; \%\ensuremath{\alpha}=2.4 \textit{5}; \%\ensuremath{\varepsilon}+\%\ensuremath{\beta}\ensuremath{^{\textnormal{+}}}\ensuremath{\approx}55\vspace{0.1cm}}&\\
\end{longtable}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PO42LEVEL0}{\dagger}}}} From Adopted Levels.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PO42LEVEL1}{\ddagger}}}} Configuration=\ensuremath{\nu} p\ensuremath{_{\textnormal{3/2}}^{\textnormal{$-$1}}}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PO42LEVEL2}{\#}}}} Configuration=\ensuremath{\nu} f\ensuremath{_{\textnormal{5/2}}^{\textnormal{$-$1}}}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PO42LEVEL3}{@}}}} Configuration=\ensuremath{\nu} i\ensuremath{_{\textnormal{13/2}}^{\textnormal{$-$1}}}.}\\
\vspace{0.5cm}
\underline{$\gamma$($^{201}$Po)}\\
\begin{longtable}{ccccccccc@{}ccccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E\ensuremath{_{\gamma}}}&\multicolumn{2}{c}{I\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{PO42GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{E\ensuremath{_{i}}(level)}&J\ensuremath{^{\pi}_{i}}&\multicolumn{2}{c}{E\ensuremath{_{f}}}&J\ensuremath{^{\pi}_{f}}&Mult.&\multicolumn{2}{c}{\ensuremath{\alpha}\ensuremath{^{\hyperlink{PO42GAMMA1}{\ddagger}}}}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{(5}&\multicolumn{1}{@{.}l}{61 {\it 13})}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{5}&\multicolumn{1}{@{.}l}{61}&\multicolumn{1}{l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{3/2\ensuremath{^{-}}}&&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
\multicolumn{1}{r@{}}{417}&\multicolumn{1}{@{.}l}{8 {\it 2}}&\multicolumn{1}{r@{}}{17}&\multicolumn{1}{@{.}l}{1 {\it 2}}&\multicolumn{1}{r@{}}{423}&\multicolumn{1}{@{.}l}{41}&\multicolumn{1}{l}{13/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{5}&\multicolumn{1}{@{.}l}{61 }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M4}&\multicolumn{1}{r@{}}{4}&\multicolumn{1}{@{.}l}{84 {\it 7}}&\parbox[t][0.3cm]{7.9790406cm}{\raggedright E\ensuremath{_{\gamma}}: Weighted average of 418.5 keV \textit{6} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1976Ko13,B}{1976Ko13}), 417.6\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{7.9790406cm}{\raggedright {\ }{\ }{\ }keV \textit{3} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1971Jo19,B}{1971Jo19}) and 417.8 keV \textit{6} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986Br28,B}{1986Br28}).\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{7.9790406cm}{\raggedright Mult.: \ensuremath{\alpha}(K)exp=2.49 \textit{32}, K/L=1.8 \textit{2}, L/M=2.5 \textit{3}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{7.9790406cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986Br28,B}{1986Br28}); K/L=2.0 \textit{3} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1976Ko13,B}{1976Ko13}) and 1.6 \textit{2}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{7.9790406cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1971Jo19,B}{1971Jo19}).\vspace{0.1cm}}&\\
\end{longtable}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PO42GAMMA0}{\dagger}}}} For absolute intensity per 100 decays, multiply by{ }\ensuremath{\approx}0.426.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PO42GAMMA1}{\ddagger}}}} Total theoretical internal conversion coefficients, calculated using the BrIcc code (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2008Ki07,B}{2008Ki07}) with Frozen orbital approximation}\\
\parbox[b][0.3cm]{17.7cm}{{\ }{\ }based on \ensuremath{\gamma}-ray energies, assigned multipolarities, and mixing ratios, unless otherwise specified.}\\
\vspace{0.5cm}
\clearpage
\begin{figure}[h]
\begin{center}
\includegraphics{201PO42-0.ps}\\
\end{center}
\end{figure}
\clearpage
\subsection[\hspace{-0.2cm}\ensuremath{^{\textnormal{201}}}At \ensuremath{\varepsilon} decay]{ }
\vspace{-27pt}
\vspace{0.3cm}
\hypertarget{AT43}{{\bf \small \underline{\ensuremath{^{\textnormal{201}}}At \ensuremath{\varepsilon} decay\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2010De04,B}{2010De04}}}}\\
\vspace{4pt}
\vspace{8pt}
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}Parent: $^{201}$At: E=0; J$^{\pi}$=9/2\ensuremath{^{-}}; T$_{1/2}$=87.6 s {\it 13}; Q(\ensuremath{\varepsilon})=5732 {\it 10}; \%\ensuremath{\varepsilon}+\%\ensuremath{\beta^{+}} decay=29 {\it 7}
}\\
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}\ensuremath{^{201}}At-Q(\ensuremath{\varepsilon}): From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2021Wa16,B}{2021Wa16}.}\\
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2010De04,B}{2010De04}: 1.4 GeV proton beam induced spallation on a 49 mg/cm\ensuremath{^{\textnormal{2}}} UC\ensuremath{_{\textnormal{2}}}-C target at ISOLDE-CERN facility. Francium was}\\
\parbox[b][0.3cm]{17.7cm}{surface ionized, accelerated to 30 keV an a mass separated by the ISOLDE General Purpose Separator (GPS). Using tape systems,}\\
\parbox[b][0.3cm]{17.7cm}{measured E\ensuremath{\gamma}, I\ensuremath{\gamma}, \ensuremath{\gamma}\ensuremath{\gamma}, ce, \ensuremath{\gamma}(ce) coin; Detectors: two HPGe detectors located at 90\ensuremath{^\circ} and 180\ensuremath{^\circ} around Si(Li) detector placed in a}\\
\parbox[b][0.3cm]{17.7cm}{MINI-ORANGE spectrometer. \ensuremath{^{\textnormal{201}}}At source is produced from \ensuremath{\alpha} decay of \ensuremath{^{\textnormal{205}}}Fr.}\\
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}Other: \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1970DaZM,B}{1970DaZM}.}\\
\vspace{12pt}
\underline{$^{201}$Po Levels}\\
\begin{longtable}{cccccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E(level)$^{{\hyperlink{PO43LEVEL0}{\dagger}}}$}&J$^{\pi}$$^{{\hyperlink{PO43LEVEL1}{\ddagger}}}$&\multicolumn{2}{c}{T$_{1/2}$$^{{\hyperlink{PO43LEVEL1}{\ddagger}}}$}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{l}{3/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{15}&\multicolumn{1}{@{.}l}{50 min {\it 14}}&&\\
\multicolumn{1}{r@{}}{5}&\multicolumn{1}{@{.}l}{61 {\it 13}}&\multicolumn{1}{l}{5/2\ensuremath{^{-}}}&&&&\\
\multicolumn{1}{r@{}}{423}&\multicolumn{1}{@{.}l}{4 {\it 3}}&\multicolumn{1}{l}{13/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{8}&\multicolumn{1}{@{.}l}{96 min {\it 12}}&\parbox[t][0.3cm]{11.730141cm}{\raggedright \%IT\ensuremath{\approx}42.6; \%\ensuremath{\alpha}=2.4 \textit{5}; \%\ensuremath{\varepsilon}+\%\ensuremath{\beta}\ensuremath{^{\textnormal{+}}}\ensuremath{\approx}55\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{11.730141cm}{\raggedright \%IT, \%A and \%EC+\%B+ are from Adopted Levels.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{621}&\multicolumn{1}{@{.}l}{66 {\it 17}}&\multicolumn{1}{l}{(7/2)\ensuremath{^{-}}}&&&&\\
\multicolumn{1}{r@{}}{623}&\multicolumn{1}{@{.}l}{3? {\it 5}}&\multicolumn{1}{l}{(5/2\ensuremath{^{-}})}&&&\parbox[t][0.3cm]{11.730141cm}{\raggedright E(level): No \ensuremath{\gamma}-rays reported in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2010De04,B}{2010De04} to depopulate this level, presumably due to\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{11.730141cm}{\raggedright {\ }{\ }{\ }small branchings to low-lying states. Evidence for the existence of this level is from\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{11.730141cm}{\raggedright {\ }{\ }{\ }\ensuremath{\gamma}\ensuremath{\gamma} coin data.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{722}&\multicolumn{1}{@{.}l}{44 {\it 19}}&\multicolumn{1}{l}{7/2\ensuremath{^{-}}}&&&&\\
\multicolumn{1}{r@{}}{758}&\multicolumn{1}{@{.}l}{30? {\it 20}}&\multicolumn{1}{l}{(7/2)\ensuremath{^{-}}}&&&&\\
\multicolumn{1}{r@{}}{766}&\multicolumn{1}{@{.}l}{3? {\it 3}}&\multicolumn{1}{l}{(9/2)\ensuremath{^{-}}}&&&&\\
\multicolumn{1}{r@{}}{1006}&\multicolumn{1}{@{.}l}{7? {\it 4}}&\multicolumn{1}{l}{(11/2)\ensuremath{^{+}}}&&&&\\
\multicolumn{1}{r@{}}{1015}&\multicolumn{1}{@{.}l}{2 {\it 4}}&\multicolumn{1}{l}{(11/2)\ensuremath{^{+}}}&&&&\\
\multicolumn{1}{r@{}}{1059}&\multicolumn{1}{@{.}l}{5 {\it 4}}&\multicolumn{1}{l}{(7/2)\ensuremath{^{-}}}&&&&\\
\multicolumn{1}{r@{}}{1124}&\multicolumn{1}{@{.}l}{8? {\it 5}}&\multicolumn{1}{l}{(7/2,9/2,11/2)}&&&&\\
\multicolumn{1}{r@{}}{1242}&\multicolumn{1}{@{.}l}{9? {\it 5}}&\multicolumn{1}{l}{(7/2,9/2,11/2)}&&&&\\
\multicolumn{1}{r@{}}{1552}&\multicolumn{1}{@{.}l}{2 {\it 4}}&\multicolumn{1}{l}{(9/2)\ensuremath{^{+}}}&&&&\\
\multicolumn{1}{r@{}}{1574}&\multicolumn{1}{@{.}l}{3 {\it 4}}&\multicolumn{1}{l}{(9/2,11/2)\ensuremath{^{+}}}&&&&\\
\multicolumn{1}{r@{}}{2044}&\multicolumn{1}{@{.}l}{0 {\it 5}}&\multicolumn{1}{l}{(9/2)\ensuremath{^{+}}}&&&&\\
\multicolumn{1}{r@{}}{2202}&\multicolumn{1}{@{.}l}{9 {\it 5}}&\multicolumn{1}{l}{(9/2,11/2)\ensuremath{^{+}}}&&&&\\
\end{longtable}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PO43LEVEL0}{\dagger}}}} From a least-squares fit to E\ensuremath{\gamma}$'$s, unless otherwise stated.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PO43LEVEL1}{\ddagger}}}} From Adopted Levels.}\\
\vspace{0.5cm}
\underline{\ensuremath{\varepsilon,\beta^+} radiations}\\
\begin{longtable}{cccccccccccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E(decay)$$}&\multicolumn{2}{c}{E(level)}&\multicolumn{2}{c}{I$\beta^{+ \ {\hyperlink{PO43DECAY1}{\ddagger}}}$}&\multicolumn{2}{c}{I$\varepsilon^{{\hyperlink{PO43DECAY1}{\ddagger}}}$}&\multicolumn{2}{c}{Log {\it ft}$^{}$}&\multicolumn{2}{c}{I$(\varepsilon+\beta^{+})^{{\hyperlink{PO43DECAY0}{\dagger}}{\hyperlink{PO43DECAY1}{\ddagger}}}$}&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&
\endfirsthead
\multicolumn{1}{r@{}}{(3529}&\multicolumn{1}{@{ }l}{{\it 10})}&\multicolumn{1}{r@{}}{2202}&\multicolumn{1}{@{.}l}{9}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{17 {\it 6}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{0 {\it 3}}&\multicolumn{1}{r@{}}{6}&\multicolumn{1}{@{.}l}{62 {\it 18}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{2 {\it 4}}&\\
\multicolumn{1}{r@{}}{(3688}&\multicolumn{1}{@{ }l}{{\it 10})}&\multicolumn{1}{r@{}}{2044}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{47 {\it 15}}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{4 {\it 8}}&\multicolumn{1}{r@{}}{6}&\multicolumn{1}{@{.}l}{29 {\it 17}}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{9 {\it 9}}&\\
\multicolumn{1}{r@{}}{(4158}&\multicolumn{1}{@{ }l}{{\it 10})}&\multicolumn{1}{r@{}}{1574}&\multicolumn{1}{@{.}l}{3}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{09 {\it 6}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{27 {\it 19}}&\multicolumn{1}{r@{}}{7}&\multicolumn{1}{@{.}l}{3 {\it 4}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{36 {\it 25}}&\\
\multicolumn{1}{r@{}}{(4180}&\multicolumn{1}{@{ }l}{{\it 10})}&\multicolumn{1}{r@{}}{1552}&\multicolumn{1}{@{.}l}{2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{9 {\it 3}}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{8 {\it 9}}&\multicolumn{1}{r@{}}{6}&\multicolumn{1}{@{.}l}{34 {\it 18}}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{7 {\it 12}}&\\
\multicolumn{1}{r@{}}{(4489}&\multicolumn{1}{@{ }l}{{\it 10})}&\multicolumn{1}{r@{}}{1242}&\multicolumn{1}{@{.}l}{9?}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{12 {\it 3}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{28 {\it 8}}&\multicolumn{1}{r@{}}{7}&\multicolumn{1}{@{.}l}{40 {\it 16}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{40 {\it 11}}&\\
\multicolumn{1}{r@{}}{(4607}&\multicolumn{1}{@{ }l}{{\it 10})}&\multicolumn{1}{r@{}}{1124}&\multicolumn{1}{@{.}l}{8?}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{11 {\it 3}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{23 {\it 6}}&\multicolumn{1}{r@{}}{7}&\multicolumn{1}{@{.}l}{51 {\it 16}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{34 {\it 9}}&\\
\multicolumn{1}{r@{}}{(4717}&\multicolumn{1}{@{ }l}{{\it 10})}&\multicolumn{1}{r@{}}{1015}&\multicolumn{1}{@{.}l}{2}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{3 {\it 3}}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{6 {\it 7}}&\multicolumn{1}{r@{}}{6}&\multicolumn{1}{@{.}l}{48 {\it 16}}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{9 {\it 10}}&\\
\multicolumn{1}{r@{}}{(4725}&\multicolumn{1}{@{ }l}{{\it 10})}&\multicolumn{1}{r@{}}{1006}&\multicolumn{1}{@{.}l}{7?}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{6 {\it 4}}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{3 {\it 9}}&\multicolumn{1}{r@{}}{6}&\multicolumn{1}{@{.}l}{38 {\it 16}}&\multicolumn{1}{r@{}}{4}&\multicolumn{1}{@{.}l}{9 {\it 13}}&\\
\multicolumn{1}{r@{}}{(4966}&\multicolumn{1}{@{ }l}{{\it 10})}&\multicolumn{1}{r@{}}{766}&\multicolumn{1}{@{.}l}{3?}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{0 {\it 5}}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{2 {\it 9}}&\multicolumn{1}{r@{}}{6}&\multicolumn{1}{@{.}l}{43 {\it 16}}&\multicolumn{1}{r@{}}{5}&\multicolumn{1}{@{.}l}{2 {\it 14}}&\\
\multicolumn{1}{r@{}}{(4974}&\multicolumn{1}{@{ }l}{{\it 10})}&\multicolumn{1}{r@{}}{758}&\multicolumn{1}{@{.}l}{30?}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{76 {\it 23}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{2 {\it 4}}&\multicolumn{1}{r@{}}{6}&\multicolumn{1}{@{.}l}{85 {\it 17}}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{0 {\it 6}}&\\
\multicolumn{1}{r@{}}{(5010}&\multicolumn{1}{@{ }l}{{\it 10})}&\multicolumn{1}{r@{}}{722}&\multicolumn{1}{@{.}l}{44}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{4 {\it 3}}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{2 {\it 6}}&\multicolumn{1}{r@{}}{6}&\multicolumn{1}{@{.}l}{60 {\it 16}}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{6 {\it 9}}&\\
\multicolumn{1}{r@{}}{(5110}&\multicolumn{1}{@{ }l}{{\it 10})}&\multicolumn{1}{r@{}}{621}&\multicolumn{1}{@{.}l}{66}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{0 {\it 3}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{6 {\it 4}}&\multicolumn{1}{r@{}}{6}&\multicolumn{1}{@{.}l}{77 {\it 16}}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{6 {\it 7}}&\\
\end{longtable}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PO43DECAY0}{\dagger}}}} From the decay scheme and the intensity balances.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PO43DECAY1}{\ddagger}}}} Absolute intensity per 100 decays.}\\
\vspace{0.5cm}
\clearpage
\vspace{0.3cm}
\begin{landscape}
\vspace*{-0.5cm}
{\bf \small \underline{\ensuremath{^{\textnormal{201}}}At \ensuremath{\varepsilon} decay\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2010De04,B}{2010De04} (continued)}}\\
\vspace{0.3cm}
\underline{$\gamma$($^{201}$Po)}\\
\vspace{0.34cm}
\parbox[b][0.3cm]{21.881866cm}{\addtolength{\parindent}{-0.254cm}I\ensuremath{\gamma} normalization: \ensuremath{\Sigma}I(\ensuremath{\gamma}+ce)(to g.s.)=100\% and by assuming that there is no direct feeding to the g.s. (\ensuremath{J^{\pi}}=3/2\ensuremath{^{-}}), 5.61-keV level (\ensuremath{J^{\pi}}=5/2\ensuremath{^{-}}), 423.4-keV level}\\
\parbox[b][0.3cm]{21.881866cm}{(Jp=13/2\ensuremath{^{+}}) and the 623.3-keV level (\ensuremath{J^{\pi}}=(5/2)\ensuremath{^{-}}).}\\
\vspace{0.34cm}
\begin{longtable}{ccccccccc@{}ccccccccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{AT43GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{I\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{AT43GAMMA0}{\dagger}\hyperlink{AT43GAMMA4}{\&}}}}&\multicolumn{2}{c}{E\ensuremath{_{i}}(level)}&J\ensuremath{^{\pi}_{i}}&\multicolumn{2}{c}{E\ensuremath{_{f}}}&J\ensuremath{^{\pi}_{f}}&Mult.\ensuremath{^{\hyperlink{AT43GAMMA2}{\#}}}&\multicolumn{2}{c}{\ensuremath{\delta}\ensuremath{^{\hyperlink{AT43GAMMA3}{@}}}}&\multicolumn{2}{c}{\ensuremath{\alpha}\ensuremath{^{\hyperlink{AT43GAMMA5}{a}}}}&\multicolumn{2}{c}{I\ensuremath{_{(\gamma+ce)}}\ensuremath{^{\hyperlink{AT43GAMMA4}{\&}}}}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{(5}&\multicolumn{1}{@{.}l}{61 {\it 13})}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{5}&\multicolumn{1}{@{.}l}{61}&\multicolumn{1}{l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{@{}l}{3/2\ensuremath{^{-}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{264}&\multicolumn{1}{@{ }l}{{\it 9}}&&\\
\multicolumn{1}{r@{}}{358}&\multicolumn{1}{@{.}l}{5 {\it 4}}&\multicolumn{1}{r@{}}{4}&\multicolumn{1}{@{.}l}{0 {\it 4}}&\multicolumn{1}{r@{}}{1124}&\multicolumn{1}{@{.}l}{8?}&\multicolumn{1}{l}{(7/2,9/2,11/2)}&\multicolumn{1}{r@{}}{766}&\multicolumn{1}{@{.}l}{3? }&\multicolumn{1}{@{}l}{(9/2)\ensuremath{^{-}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{5.6315994cm}{\raggedright \%I\ensuremath{\gamma}=0.34 \textit{9}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$392}&\multicolumn{1}{@{.}l}{2\ensuremath{^{\hyperlink{AT43GAMMA6}{b}}}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&&&&&&&&&&&&&\parbox[t][0.3cm]{5.6315994cm}{\raggedright E\ensuremath{_{\gamma}}: Weak \ensuremath{\gamma} ray reported in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2010De04,B}{2010De04}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&&&\parbox[t][0.3cm]{5.6315994cm}{\raggedright {\ }{\ }{\ }to depopulate \ensuremath{J^{\pi}}=(11/2\ensuremath{^{+}}) level, but\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&&&\parbox[t][0.3cm]{5.6315994cm}{\raggedright {\ }{\ }{\ }the placement is unlikely given the\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&&&\parbox[t][0.3cm]{5.6315994cm}{\raggedright {\ }{\ }{\ }expected Mult=[E3].\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{417}&\multicolumn{1}{@{.}l}{8 {\it 2}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{423}&\multicolumn{1}{@{.}l}{4}&\multicolumn{1}{l}{13/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{5}&\multicolumn{1}{@{.}l}{61 }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M4}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{4}&\multicolumn{1}{@{.}l}{84 {\it 7}}&\multicolumn{1}{r@{}}{161}&\multicolumn{1}{@{ }l}{{\it 6}}&\parbox[t][0.3cm]{5.6315994cm}{\raggedright \%I\ensuremath{\gamma}=2.4 \textit{6}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&&&\parbox[t][0.3cm]{5.6315994cm}{\raggedright E\ensuremath{_{\gamma}},Mult.: From adopted gammas.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{436}&\multicolumn{1}{@{.}l}{2 {\it 2}}&\multicolumn{1}{r@{}}{11}&\multicolumn{1}{@{.}l}{5 {\it 10}}&\multicolumn{1}{r@{}}{1059}&\multicolumn{1}{@{.}l}{5}&\multicolumn{1}{l}{(7/2)\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{623}&\multicolumn{1}{@{.}l}{3? }&\multicolumn{1}{@{}l}{(5/2\ensuremath{^{-}})}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{93 {\it 23}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{119 {\it 19}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{5.6315994cm}{\raggedright \%I\ensuremath{\gamma}=0.98 \textit{25}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&&&\parbox[t][0.3cm]{5.6315994cm}{\raggedright \ensuremath{\alpha}(K)exp=0.094 \textit{14}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{476}&\multicolumn{1}{@{.}l}{6 {\it 4}}&\multicolumn{1}{r@{}}{4}&\multicolumn{1}{@{.}l}{7 {\it 5}}&\multicolumn{1}{r@{}}{1242}&\multicolumn{1}{@{.}l}{9?}&\multicolumn{1}{l}{(7/2,9/2,11/2)}&\multicolumn{1}{r@{}}{766}&\multicolumn{1}{@{.}l}{3? }&\multicolumn{1}{@{}l}{(9/2)\ensuremath{^{-}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{5.6315994cm}{\raggedright \%I\ensuremath{\gamma}=0.40 \textit{11}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{491}&\multicolumn{1}{@{.}l}{8 {\it 2}}&\multicolumn{1}{r@{}}{33}&\multicolumn{1}{@{ }l}{{\it 5}}&\multicolumn{1}{r@{}}{2044}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{l}{(9/2)\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1552}&\multicolumn{1}{@{.}l}{2 }&\multicolumn{1}{@{}l}{(9/2)\ensuremath{^{+}}}&\multicolumn{1}{l}{E2}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0334 {\it 5}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{5.6315994cm}{\raggedright \%I\ensuremath{\gamma}=2.8 \textit{8}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&&&\parbox[t][0.3cm]{5.6315994cm}{\raggedright \ensuremath{\alpha}(K)exp=0.023 \textit{7}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{492}&\multicolumn{1}{@{.}l}{7 {\it 2}}&\multicolumn{1}{r@{}}{39}&\multicolumn{1}{@{ }l}{{\it 6}}&\multicolumn{1}{r@{}}{1552}&\multicolumn{1}{@{.}l}{2}&\multicolumn{1}{l}{(9/2)\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1059}&\multicolumn{1}{@{.}l}{5 }&\multicolumn{1}{@{}l}{(7/2)\ensuremath{^{-}}}&\multicolumn{1}{l}{E1}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{01058 {\it 15}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{5.6315994cm}{\raggedright \%I\ensuremath{\gamma}=3.3 \textit{10}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&&&\parbox[t][0.3cm]{5.6315994cm}{\raggedright \ensuremath{\alpha}(K)exp=0.010 \textit{4}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{537}&\multicolumn{1}{@{.}l}{0 {\it 2}}&\multicolumn{1}{r@{}}{37}&\multicolumn{1}{@{ }l}{{\it 3}}&\multicolumn{1}{r@{}}{1552}&\multicolumn{1}{@{.}l}{2}&\multicolumn{1}{l}{(9/2)\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1015}&\multicolumn{1}{@{.}l}{2 }&\multicolumn{1}{@{}l}{(11/2)\ensuremath{^{+}}}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{58 {\it 17}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0328 {\it 7}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{5.6315994cm}{\raggedright \%I\ensuremath{\gamma}=3.2 \textit{8}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&&&\parbox[t][0.3cm]{5.6315994cm}{\raggedright \ensuremath{\alpha}(K)exp=0.024 \textit{3}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{559}&\multicolumn{1}{@{.}l}{1 {\it 2}}&\multicolumn{1}{r@{}}{17}&\multicolumn{1}{@{.}l}{8 {\it 16}}&\multicolumn{1}{r@{}}{1574}&\multicolumn{1}{@{.}l}{3}&\multicolumn{1}{l}{(9/2,11/2)\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1015}&\multicolumn{1}{@{.}l}{2 }&\multicolumn{1}{@{}l}{(11/2)\ensuremath{^{+}}}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{78 {\it 20}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0415 {\it 33}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{5.6315994cm}{\raggedright \%I\ensuremath{\gamma}=1.5 \textit{4}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&&&\parbox[t][0.3cm]{5.6315994cm}{\raggedright \ensuremath{\alpha}(K)exp=0.032 \textit{5}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{583}&\multicolumn{1}{@{.}l}{3 {\it 2}}&\multicolumn{1}{r@{}}{56}&\multicolumn{1}{@{ }l}{{\it 5}}&\multicolumn{1}{r@{}}{1006}&\multicolumn{1}{@{.}l}{7?}&\multicolumn{1}{l}{(11/2)\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{423}&\multicolumn{1}{@{.}l}{4 }&\multicolumn{1}{@{}l}{13/2\ensuremath{^{+}}}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{61 {\it 17}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0304 {\it 11}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{5.6315994cm}{\raggedright \%I\ensuremath{\gamma}=4.8 \textit{12}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&&&\parbox[t][0.3cm]{5.6315994cm}{\raggedright \ensuremath{\alpha}(K)exp=0.023 \textit{3}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{591}&\multicolumn{1}{@{.}l}{8 {\it 2}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{1015}&\multicolumn{1}{@{.}l}{2}&\multicolumn{1}{l}{(11/2)\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{423}&\multicolumn{1}{@{.}l}{4 }&\multicolumn{1}{@{}l}{13/2\ensuremath{^{+}}}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{67 {\it 18}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0291 {\it 10}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{5.6315994cm}{\raggedright \%I\ensuremath{\gamma}=8.5 \textit{21}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&&&\parbox[t][0.3cm]{5.6315994cm}{\raggedright \ensuremath{\alpha}(K)exp=0.022 \textit{3}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{616}&\multicolumn{1}{@{.}l}{1 {\it 2}}&\multicolumn{1}{r@{}}{18}&\multicolumn{1}{@{.}l}{6 {\it 16}}&\multicolumn{1}{r@{}}{621}&\multicolumn{1}{@{.}l}{66}&\multicolumn{1}{l}{(7/2)\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{5}&\multicolumn{1}{@{.}l}{61 }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{72 {\it 20}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0334 {\it 27}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{5.6315994cm}{\raggedright \%I\ensuremath{\gamma}=1.6 \textit{4}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&&&\parbox[t][0.3cm]{5.6315994cm}{\raggedright \ensuremath{\alpha}(K)exp=0.026 \textit{4}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{(617}&\multicolumn{1}{@{.}l}{7 {\it 5})}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{623}&\multicolumn{1}{@{.}l}{3?}&\multicolumn{1}{l}{(5/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{5}&\multicolumn{1}{@{.}l}{61 }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{6}&\multicolumn{1}{@{.}l}{45 {\it 60}}&\parbox[t][0.3cm]{5.6315994cm}{\raggedright E\ensuremath{_{\gamma}}: From level energy difference.\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&&&\parbox[t][0.3cm]{5.6315994cm}{\raggedright I\ensuremath{_{(\gamma+ce)}}: Taken as half of\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&&&\parbox[t][0.3cm]{5.6315994cm}{\raggedright {\ }{\ }{\ }I(\ensuremath{\gamma}+ce)(623.3-keV level)=12.9 \textit{12},\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&&&\parbox[t][0.3cm]{5.6315994cm}{\raggedright {\ }{\ }{\ }determined from intensity balance.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{621}&\multicolumn{1}{@{.}l}{6 {\it 2}}&\multicolumn{1}{r@{}}{10}&\multicolumn{1}{@{.}l}{5 {\it 9}}&\multicolumn{1}{r@{}}{621}&\multicolumn{1}{@{.}l}{66}&\multicolumn{1}{l}{(7/2)\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{@{}l}{3/2\ensuremath{^{-}}}&\multicolumn{1}{l}{(E2)}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{01950 {\it 27}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{5.6315994cm}{\raggedright \%I\ensuremath{\gamma}=0.90 \textit{23}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&&&\parbox[t][0.3cm]{5.6315994cm}{\raggedright \ensuremath{\alpha}(K)exp=0.019 \textit{3}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{(623}&\multicolumn{1}{@{.}l}{3 {\it 5})}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{623}&\multicolumn{1}{@{.}l}{3?}&\multicolumn{1}{l}{(5/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{@{}l}{3/2\ensuremath{^{-}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{6}&\multicolumn{1}{@{.}l}{45 {\it 60}}&\parbox[t][0.3cm]{5.6315994cm}{\raggedright E\ensuremath{_{\gamma}}: From level energy difference.\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&&&\parbox[t][0.3cm]{5.6315994cm}{\raggedright I\ensuremath{_{(\gamma+ce)}}: Taken as half of\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&&&\parbox[t][0.3cm]{5.6315994cm}{\raggedright {\ }{\ }{\ }I(\ensuremath{\gamma}+ce)(623.3-keV level)=12.9 \textit{12},\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&&&\parbox[t][0.3cm]{5.6315994cm}{\raggedright {\ }{\ }{\ }determined from intensity balance.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{628}&\multicolumn{1}{@{.}l}{6 {\it 2}}&\multicolumn{1}{r@{}}{14}&\multicolumn{1}{@{.}l}{1 {\it 20}}&\multicolumn{1}{r@{}}{2202}&\multicolumn{1}{@{.}l}{9}&\multicolumn{1}{l}{(9/2,11/2)\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1574}&\multicolumn{1}{@{.}l}{3 }&\multicolumn{1}{@{}l}{(9/2,11/2)\ensuremath{^{+}}}&\multicolumn{1}{l}{E2}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{01903 {\it 27}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{5.6315994cm}{\raggedright \%I\ensuremath{\gamma}=1.21 \textit{34}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&&&\parbox[t][0.3cm]{5.6315994cm}{\raggedright \ensuremath{\alpha}(K)exp=0.014 \textit{3}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{716}&\multicolumn{1}{@{.}l}{6 {\it 4}}&\multicolumn{1}{r@{}}{7}&\multicolumn{1}{@{.}l}{1 {\it 6}}&\multicolumn{1}{r@{}}{722}&\multicolumn{1}{@{.}l}{44}&\multicolumn{1}{l}{7/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{5}&\multicolumn{1}{@{.}l}{61 }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{l}{[M1,E2]}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{032 {\it 18}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{5.6315994cm}{\raggedright \%I\ensuremath{\gamma}=0.61 \textit{16}\vspace{0.1cm}}&\\
\end{longtable}
\clearpage
\begin{longtable}{ccccccccc@{}ccccc@{\extracolsep{\fill}}c}
\\[-.4cm]
\multicolumn{15}{c}{{\bf \small \underline{\ensuremath{^{\textnormal{201}}}At \ensuremath{\varepsilon} decay\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2010De04,B}{2010De04} (continued)}}}\\
\multicolumn{15}{c}{~}\\
\multicolumn{15}{c}{\underline{$\gamma$($^{201}$Po) (continued)}}\\
\multicolumn{15}{c}{~~~}\\
\multicolumn{2}{c}{E\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{AT43GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{I\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{AT43GAMMA0}{\dagger}\hyperlink{AT43GAMMA4}{\&}}}}&\multicolumn{2}{c}{E\ensuremath{_{i}}(level)}&J\ensuremath{^{\pi}_{i}}&\multicolumn{2}{c}{E\ensuremath{_{f}}}&J\ensuremath{^{\pi}_{f}}&Mult.\ensuremath{^{\hyperlink{AT43GAMMA2}{\#}}}&\multicolumn{2}{c}{\ensuremath{\alpha}\ensuremath{^{\hyperlink{AT43GAMMA5}{a}}}}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endhead
\multicolumn{1}{r@{}}{722}&\multicolumn{1}{@{.}l}{5 {\it 2}}&\multicolumn{1}{r@{}}{34}&\multicolumn{1}{@{ }l}{{\it 3}}&\multicolumn{1}{r@{}}{722}&\multicolumn{1}{@{.}l}{44}&\multicolumn{1}{l}{7/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{@{}l}{3/2\ensuremath{^{-}}}&\multicolumn{1}{l}{E2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{01413 {\it 20}}&\parbox[t][0.3cm]{11.678881cm}{\raggedright \%I\ensuremath{\gamma}=2.9 \textit{7}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{11.678881cm}{\raggedright \ensuremath{\alpha}(K)exp=0.008 \textit{1}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{758}&\multicolumn{1}{@{.}l}{3\ensuremath{^{\hyperlink{AT43GAMMA1}{\ddagger}}} {\it 2}}&\multicolumn{1}{r@{}}{23}&\multicolumn{1}{@{.}l}{4 {\it 20}}&\multicolumn{1}{r@{}}{758}&\multicolumn{1}{@{.}l}{30?}&\multicolumn{1}{l}{(7/2)\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{@{}l}{3/2\ensuremath{^{-}}}&\multicolumn{1}{l}{E2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{01278 {\it 18}}&\parbox[t][0.3cm]{11.678881cm}{\raggedright \%I\ensuremath{\gamma}=2.0 \textit{5}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{11.678881cm}{\raggedright \ensuremath{\alpha}(K)exp=0.007 \textit{1}\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{760}&\multicolumn{1}{@{.}l}{7\ensuremath{^{\hyperlink{AT43GAMMA1}{\ddagger}}} {\it 2}}&\multicolumn{1}{r@{}}{69}&\multicolumn{1}{@{ }l}{{\it 6}}&\multicolumn{1}{r@{}}{766}&\multicolumn{1}{@{.}l}{3?}&\multicolumn{1}{l}{(9/2)\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{5}&\multicolumn{1}{@{.}l}{61 }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{l}{E2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{01269 {\it 18}}&\parbox[t][0.3cm]{11.678881cm}{\raggedright \%I\ensuremath{\gamma}=5.9 \textit{15}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{11.678881cm}{\raggedright \ensuremath{\alpha}(K)exp=0.009 \textit{1}\vspace{0.1cm}}&\\
\end{longtable}
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{AT43GAMMA0}{\dagger}}}} From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2010De04,B}{2010De04}, unless otherwise stated.}\\
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{AT43GAMMA1}{\ddagger}}}} Placement in the decay scheme is not unambiguous.}\\
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{AT43GAMMA2}{\#}}}} From multiple decay branches and the comparison of \ensuremath{\alpha}(K)exp (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2010De04,B}{2010De04}) with theoretical values.}\\
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{AT43GAMMA3}{@}}}} From \ensuremath{\alpha}(K)exp and the briccmixing program.}\\
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{AT43GAMMA4}{\&}}}} For absolute intensity per 100 decays, multiply by 0.086 \textit{21}.}\\
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{AT43GAMMA5}{a}}}} Total theoretical internal conversion coefficients, calculated using the BrIcc code (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2008Ki07,B}{2008Ki07}) with Frozen orbital approximation based on \ensuremath{\gamma}-ray energies,}\\
\parbox[b][0.3cm]{21.881866cm}{{\ }{\ }assigned multipolarities, and mixing ratios, unless otherwise specified.}\\
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{AT43GAMMA6}{b}}}} Placement of transition in the level scheme is uncertain.}\\
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{AT43GAMMA7}{x}}}} \ensuremath{\gamma} ray not placed in level scheme.}\\
\vspace{0.5cm}
\end{landscape}\clearpage
\clearpage
\begin{figure}[h]
\begin{center}
\includegraphics{201PO43-0.ps}\\
\end{center}
\end{figure}
\clearpage
\subsection[\hspace{-0.2cm}\ensuremath{^{\textnormal{205}}}Rn \ensuremath{\alpha} decay]{ }
\vspace{-27pt}
\vspace{0.3cm}
\hypertarget{RN44}{{\bf \small \underline{\ensuremath{^{\textnormal{205}}}Rn \ensuremath{\alpha} decay}}}\\
\vspace{4pt}
\vspace{8pt}
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}Parent: $^{205}$Rn: E=0; J$^{\pi}$=5/2\ensuremath{^{-}}; T$_{1/2}$=170 s {\it 4}; Q(\ensuremath{\alpha})=6386.5 {\it 18}; \%\ensuremath{\alpha} decay=24.6 {\it 9}
}\\
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}\ensuremath{^{205}}Rn-\ensuremath{J^{\pi}}, T\ensuremath{_{\textnormal{1/2}}} and \%\ensuremath{\alpha} decay from \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2020Ko17,B}{2020Ko17}; Q(\ensuremath{\alpha}) from \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2021Wa16,B}{2021Wa16}.}\\
\vspace{12pt}
\underline{$^{201}$Po Levels}\\
\begin{longtable}{cccccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E(level)$^{{\hyperlink{PO44LEVEL0}{\dagger}}}$}&J$^{\pi}$$^{{\hyperlink{PO44LEVEL0}{\dagger}}}$&\multicolumn{2}{c}{T$_{1/2}$$^{{\hyperlink{PO44LEVEL0}{\dagger}}}$}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{PO44LEVEL3}{@}}}}}&\multicolumn{1}{l}{3/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{15}&\multicolumn{1}{@{.}l}{50 min {\it 14}}&&\\
\multicolumn{1}{r@{}}{5}&\multicolumn{1}{@{.}l}{61\ensuremath{^{{\hyperlink{PO44LEVEL1}{\ddagger}}}} {\it 15}}&\multicolumn{1}{l}{5/2\ensuremath{^{-}}}&&&\parbox[t][0.3cm]{12.67816cm}{\raggedright E(level): Other: 0.3 keV \textit{25} using Q(\ensuremath{\alpha})=6386.5 \textit{18} and E(\ensuremath{\alpha}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{142}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{PO44LEVEL2}{\#}}}} {\it 3}}&\multicolumn{1}{l}{(1/2\ensuremath{^{-}})}&&&\parbox[t][0.3cm]{12.67816cm}{\raggedright E(level): From \ensuremath{\alpha}-\ensuremath{\gamma} coincidences in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1971Jo19,B}{1971Jo19}.\vspace{0.1cm}}&\\
\end{longtable}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PO44LEVEL0}{\dagger}}}} From Adopted Levels, unless otherwise stated.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PO44LEVEL1}{\ddagger}}}} Configuration=\ensuremath{\nu} f\ensuremath{_{\textnormal{5/2}}^{\textnormal{$-$1}}}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PO44LEVEL2}{\#}}}} Configuration=\ensuremath{\nu} p\ensuremath{_{\textnormal{1/2}}^{\textnormal{$-$1}}}. The assignment is tentative.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PO44LEVEL3}{@}}}} Configuration=\ensuremath{\nu} p\ensuremath{_{\textnormal{3/2}}^{\textnormal{$-$1}}}.}\\
\vspace{0.5cm}
\underline{\ensuremath{\alpha} radiations}\\
\begin{longtable}{ccccccccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E$\alpha^{{}}$}&\multicolumn{2}{c}{E(level)}&\multicolumn{2}{c}{I$\alpha^{{\hyperlink{PO44DECAY1}{\ddagger}}}$}&\multicolumn{2}{c}{HF$^{{\hyperlink{PO44DECAY0}{\dagger}}}$}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{6125}&\multicolumn{1}{@{ }l}{{\it 4}}&\multicolumn{1}{r@{}}{142}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{$\leq$0}&\multicolumn{1}{@{.}l}{1}&\multicolumn{1}{r@{}}{$\geq$327}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{12.077cm}{\raggedright E$\alpha$: From Q(\ensuremath{\alpha})=6386.5 \textit{18} and E(level).\vspace{0.1cm}}&\\
&&&&&&&&\parbox[t][0.3cm]{12.077cm}{\raggedright I$\alpha$: Estimated by evaluator from Fig. 6 in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1971Ho01,B}{1971Ho01}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{6261}&\multicolumn{1}{@{.}l}{4 {\it 18}}&\multicolumn{1}{r@{}}{5}&\multicolumn{1}{@{.}l}{61}&\multicolumn{1}{r@{}}{$\approx$98}&\multicolumn{1}{@{.}l}{5}&\multicolumn{1}{r@{}}{$\approx$1}&\multicolumn{1}{@{.}l}{3}&\parbox[t][0.3cm]{12.077cm}{\raggedright E$\alpha$: Weighted average of E\ensuremath{\alpha}=6262 keV \textit{3} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1967Va17,B}{1967Va17}), 6262 keV \textit{6} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1971Ho01,B}{1971Ho01}) and\vspace{0.1cm}}&\\
&&&&&&&&\parbox[t][0.3cm]{12.077cm}{\raggedright {\ }{\ }{\ }6260.9 keV \textit{25} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1993Wa04,B}{1993Wa04}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{(6268}&\multicolumn{1}{@{ }l}{{\it 4})}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{4 {\it SY}}&\multicolumn{1}{r@{}}{99}&\multicolumn{1}{@{ }l}{{\it SY}}&\parbox[t][0.3cm]{12.077cm}{\raggedright I$\alpha$: Based on \ensuremath{\alpha} decay HF values for configuration=\ensuremath{\nu} f\ensuremath{_{\textnormal{5/2}}^{\textnormal{$-$1}}} to configuration=\ensuremath{\nu} p\ensuremath{_{\textnormal{3/2}}^{\textnormal{$-$1}}}\vspace{0.1cm}}&\\
&&&&&&&&\parbox[t][0.3cm]{12.077cm}{\raggedright {\ }{\ }{\ }decay in \ensuremath{^{\textnormal{203}}}Po (HF=95 \textit{15}) and \ensuremath{^{\textnormal{205}}}Po (HF=105 \textit{17}).\vspace{0.1cm}}&\\
\end{longtable}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PO44DECAY0}{\dagger}}}} Using r\ensuremath{_{\textnormal{0}}}(\ensuremath{^{\textnormal{201}}}Po)=1.497 \textit{2} from \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2020Si16,B}{2020Si16}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PO44DECAY1}{\ddagger}}}} For absolute intensity per 100 decays, multiply by 0.246 \textit{9}.}\\
\vspace{0.5cm}
\clearpage
\subsection[\hspace{-0.2cm}\ensuremath{^{\textnormal{194}}}Pt(\ensuremath{^{\textnormal{12}}}C,5n\ensuremath{\gamma})]{ }
\vspace{-27pt}
\vspace{0.3cm}
\hypertarget{PO45}{{\bf \small \underline{\ensuremath{^{\textnormal{194}}}Pt(\ensuremath{^{\textnormal{12}}}C,5n\ensuremath{\gamma})\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1985We05,B}{1985We05}}}}\\
\vspace{4pt}
\vspace{8pt}
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1985We05,B}{1985We05}: Produced in \ensuremath{^{\textnormal{194}}}Pt(\ensuremath{^{\textnormal{12}}}C,5n) (E(\ensuremath{^{\textnormal{12}}}C)=102 and 106 MeV) and \ensuremath{^{\textnormal{195}}}Pt(\ensuremath{^{\textnormal{12}}}C,6n) (E(\ensuremath{^{\textnormal{12}}}C)=100 MeV) reactions; Detectors:}\\
\parbox[b][0.3cm]{17.7cm}{two n-type HPGE and two liquid scin neutron detectors; Measured: E\ensuremath{\gamma}, I\ensuremath{\gamma}, \ensuremath{\gamma} singles, \ensuremath{\gamma}-\ensuremath{\gamma} coin, n-\ensuremath{\gamma} and n-\ensuremath{\gamma}-\ensuremath{\gamma} coin, \ensuremath{\gamma}(\ensuremath{\theta});}\\
\parbox[b][0.3cm]{17.7cm}{Deduced: level scheme, \ensuremath{J^{\pi}}.}\\
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}Others: \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2004Ro09,B}{2004Ro09}, but no \ensuremath{^{\textnormal{201}}}Po levels and transitions were reported.}\\
\vspace{12pt}
\underline{$^{201}$Po Levels}\\
\begin{longtable}{ccccc|ccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E(level)$^{{\hyperlink{PO45LEVEL0}{\dagger}}}$}&J$^{\pi}$$^{{\hyperlink{PO45LEVEL2}{\#}}}$&\multicolumn{2}{c|}{T$_{1/2}$$^{{\hyperlink{PO45LEVEL3}{@}}}$}&\multicolumn{2}{c}{E(level)$^{{\hyperlink{PO45LEVEL0}{\dagger}}}$}&J$^{\pi}$$^{{\hyperlink{PO45LEVEL2}{\#}}}$&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c|}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{\ensuremath{^{{\hyperlink{PO45LEVEL1}{\ddagger}}{\hyperlink{PO45LEVEL5}{a}}}}}&\multicolumn{1}{l}{3/2\ensuremath{^{-}}\ensuremath{^{{\hyperlink{PO45LEVEL1}{\ddagger}}}}}&\multicolumn{1}{r@{}}{15}&\multicolumn{1}{@{.}l|}{50 min {\it 14}}&\multicolumn{1}{r@{}}{2463}&\multicolumn{1}{@{.}l}{9 {\it 21}}&&\\
\multicolumn{1}{r@{}}{5}&\multicolumn{1}{@{.}l}{61\ensuremath{^{{\hyperlink{PO45LEVEL1}{\ddagger}}{\hyperlink{PO45LEVEL4}{\&}}}} {\it 13}}&\multicolumn{1}{l}{5/2\ensuremath{^{-}}\ensuremath{^{{\hyperlink{PO45LEVEL1}{\ddagger}}}}}&&&\multicolumn{1}{r@{}}{2570}&\multicolumn{1}{@{.}l}{2 {\it 21}}&\multicolumn{1}{l}{27/2\ensuremath{^{+}}}&\\
\multicolumn{1}{r@{}}{423}&\multicolumn{1}{@{.}l}{41\ensuremath{^{{\hyperlink{PO45LEVEL1}{\ddagger}}{\hyperlink{PO45LEVEL6}{b}}}} {\it 22}}&\multicolumn{1}{l}{13/2\ensuremath{^{+}}\ensuremath{^{{\hyperlink{PO45LEVEL1}{\ddagger}}}}}&\multicolumn{1}{r@{}}{8}&\multicolumn{1}{@{.}l|}{96 min {\it 12}}&\multicolumn{1}{r@{}}{2627}&\multicolumn{1}{@{.}l}{5 {\it 23}}&\multicolumn{1}{l}{(29/2\ensuremath{^{+}})}&\\
\multicolumn{1}{r@{}}{1037}&\multicolumn{1}{@{.}l}{0\ensuremath{^{{\hyperlink{PO45LEVEL7}{c}}{\hyperlink{PO45LEVEL10}{f}}}} {\it 11}}&\multicolumn{1}{l}{17/2\ensuremath{^{+}}}&&&\multicolumn{1}{r@{}}{2770}&\multicolumn{1}{@{.}l}{1 {\it 21}}&&\\
\multicolumn{1}{r@{}}{1593}&\multicolumn{1}{@{.}l}{6\ensuremath{^{{\hyperlink{PO45LEVEL8}{d}}{\hyperlink{PO45LEVEL10}{f}}}} {\it 15}}&\multicolumn{1}{l}{21/2\ensuremath{^{+}}}&&&\multicolumn{1}{r@{}}{2979}&\multicolumn{1}{@{.}l}{0 {\it 23}}&\multicolumn{1}{l}{27/2\ensuremath{^{+}},31/2\ensuremath{^{+}}}&\\
\multicolumn{1}{r@{}}{1912}&\multicolumn{1}{@{.}l}{3\ensuremath{^{{\hyperlink{PO45LEVEL9}{e}}{\hyperlink{PO45LEVEL10}{f}}}} {\it 18}}&\multicolumn{1}{l}{25/2\ensuremath{^{+}}}&&&\multicolumn{1}{r@{}}{3039}&\multicolumn{1}{@{.}l}{6 {\it 23}}&&\\
\multicolumn{1}{r@{}}{2101}&\multicolumn{1}{@{.}l}{6 {\it 21}}&&&&\multicolumn{1}{r@{}}{3196}&\multicolumn{1}{@{.}l}{5 {\it 23}}&\multicolumn{1}{l}{(29/2\ensuremath{^{+}})}&\\
\multicolumn{1}{r@{}}{2133}&\multicolumn{1}{@{.}l}{8 {\it 21}}&\multicolumn{1}{l}{25/2\ensuremath{^{+}},(29/2\ensuremath{^{+}})}&&&\multicolumn{1}{r@{}}{3210}&\multicolumn{1}{@{.}l}{3 {\it 23}}&\multicolumn{1}{l}{31/2\ensuremath{^{+}}}&\\
\multicolumn{1}{r@{}}{2239}&\multicolumn{1}{@{.}l}{6 {\it 23}}&&&&\multicolumn{1}{r@{}}{3333}&\multicolumn{1}{@{.}l}{1 {\it 25}}&&\\
\multicolumn{1}{r@{}}{2332}&\multicolumn{1}{@{.}l}{2 {\it 21}}&\multicolumn{1}{l}{(27/2)}&&&\multicolumn{1}{r@{}}{3710}&\multicolumn{1}{@{.}l}{1 {\it 25}}&\multicolumn{1}{l}{(35/2\ensuremath{^{+}})}&\\
\multicolumn{1}{r@{}}{2347}&\multicolumn{1}{@{.}l}{6 {\it 18}}&&&&\multicolumn{1}{r@{}}{4153?}&\multicolumn{1}{@{ }l}{{\it 3}}&&\\
\multicolumn{1}{r@{}}{2354}&\multicolumn{1}{@{.}l}{7 {\it 21}}&\multicolumn{1}{l}{27/2\ensuremath{^{+}}}&&&&&&\\
\end{longtable}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PO45LEVEL0}{\dagger}}}} From a least-squares fit to E\ensuremath{\gamma} and by assuming \ensuremath{\Delta}E\ensuremath{\gamma}=0.5 keV.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PO45LEVEL1}{\ddagger}}}} From Adopted Levels.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PO45LEVEL2}{\#}}}} From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1985We05,B}{1985We05}, unless otherwise stated.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PO45LEVEL3}{@}}}} From Adopted Levels.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PO45LEVEL4}{\&}}}} Configuration=\ensuremath{\nu} f\ensuremath{_{\textnormal{5/2}}^{\textnormal{$-$1}}}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PO45LEVEL5}{a}}}} Configuration=\ensuremath{\nu} p\ensuremath{_{\textnormal{3/2}}^{\textnormal{$-$1}}}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PO45LEVEL6}{b}}}} Configuration=\ensuremath{\nu} i\ensuremath{_{\textnormal{13/2}}^{\textnormal{$-$1}}}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PO45LEVEL7}{c}}}} Configuration=\ensuremath{\nu} (i\ensuremath{_{\textnormal{13/2}}^{\textnormal{$-$1}}})\ensuremath{\otimes}2\ensuremath{^{\textnormal{+}}}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PO45LEVEL8}{d}}}} Configuration=\ensuremath{\nu} (i\ensuremath{_{\textnormal{13/2}}^{\textnormal{$-$1}}})\ensuremath{\otimes}4\ensuremath{^{\textnormal{+}}}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PO45LEVEL9}{e}}}} Possibly a mixture between configuration=\ensuremath{\nu} (i\ensuremath{_{\textnormal{13/2}}^{\textnormal{$-$1}}})\ensuremath{\otimes}6\ensuremath{^{\textnormal{+}}} and configuration=\ensuremath{\nu} (i\ensuremath{_{\textnormal{13/2}}^{\textnormal{$-$1}}}) \ensuremath{\pi} (h\ensuremath{_{\textnormal{9/2}}^{\textnormal{+2}}})\ensuremath{_{\textnormal{8+}}}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PO45LEVEL10}{f}}}} The \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1985We05,B}{1985We05} authors stated that the ordering of the 318.7\ensuremath{\gamma}, 556.6\ensuremath{\gamma} and 613.6\ensuremath{\gamma}, and hence, the placement of corresponding}\\
\parbox[b][0.3cm]{17.7cm}{{\ }{\ }level energies, is based on systematics and their relative population.}\\
\vspace{0.5cm}
\underline{$\gamma$($^{201}$Po)}\\
\begin{longtable}{ccccccccc@{}ccccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{PO45GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{I\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{PO45GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{E\ensuremath{_{i}}(level)}&J\ensuremath{^{\pi}_{i}}&\multicolumn{2}{c}{E\ensuremath{_{f}}}&J\ensuremath{^{\pi}_{f}}&Mult.\ensuremath{^{\hyperlink{PO45GAMMA2}{\#}}}&\multicolumn{2}{c}{\ensuremath{\alpha}\ensuremath{^{\hyperlink{PO45GAMMA3}{@}}}}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{(5}&\multicolumn{1}{@{.}l}{61\ensuremath{^{\hyperlink{PO45GAMMA1}{\ddagger}}} {\it 13})}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{5}&\multicolumn{1}{@{.}l}{61}&\multicolumn{1}{l}{5/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{3/2\ensuremath{^{-}}}&&&&&\\
\multicolumn{1}{r@{}}{138}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{r@{}}{$\approx$2}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{2239}&\multicolumn{1}{@{.}l}{6}&&\multicolumn{1}{r@{}}{2101}&\multicolumn{1}{@{.}l}{6 }&&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
\multicolumn{1}{r@{}}{189}&\multicolumn{1}{@{.}l}{3}&\multicolumn{1}{r@{}}{6}&\multicolumn{1}{@{ }l}{{\it 2}}&\multicolumn{1}{r@{}}{2101}&\multicolumn{1}{@{.}l}{6}&&\multicolumn{1}{r@{}}{1912}&\multicolumn{1}{@{.}l}{3 }&\multicolumn{1}{@{}l}{25/2\ensuremath{^{+}}}&\multicolumn{1}{l}{D,Q}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{5.0900416cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}=0.1 \textit{4}, A\ensuremath{_{\textnormal{4}}}=0.2 \textit{3}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{221}&\multicolumn{1}{@{.}l}{5}&\multicolumn{1}{r@{}}{14}&\multicolumn{1}{@{ }l}{{\it 1}}&\multicolumn{1}{r@{}}{2133}&\multicolumn{1}{@{.}l}{8}&\multicolumn{1}{l}{25/2\ensuremath{^{+}},(29/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{1912}&\multicolumn{1}{@{.}l}{3 }&\multicolumn{1}{@{}l}{25/2\ensuremath{^{+}}}&\multicolumn{1}{l}{D,Q}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{5.0900416cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}=0.10 \textit{6}, A\ensuremath{_{\textnormal{4}}}={\textminus}0.11 \textit{9}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{272}&\multicolumn{1}{@{.}l}{8}&\multicolumn{1}{r@{}}{7}&\multicolumn{1}{@{ }l}{{\it 2}}&\multicolumn{1}{r@{}}{2627}&\multicolumn{1}{@{.}l}{5}&\multicolumn{1}{l}{(29/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{2354}&\multicolumn{1}{@{.}l}{7 }&\multicolumn{1}{@{}l}{27/2\ensuremath{^{+}}}&\multicolumn{1}{l}{D,Q}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{5.0900416cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}=0.3 \textit{3}, A\ensuremath{_{\textnormal{4}}}=0.4 \textit{3}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{318}&\multicolumn{1}{@{.}l}{7}&\multicolumn{1}{r@{}}{94}&\multicolumn{1}{@{ }l}{{\it 1}}&\multicolumn{1}{r@{}}{1912}&\multicolumn{1}{@{.}l}{3}&\multicolumn{1}{l}{25/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1593}&\multicolumn{1}{@{.}l}{6 }&\multicolumn{1}{@{}l}{21/2\ensuremath{^{+}}}&\multicolumn{1}{l}{E2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{107}&\parbox[t][0.3cm]{5.0900416cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}=0.22 \textit{1}, A\ensuremath{_{\textnormal{4}}}={\textminus}0.05 \textit{1}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{354}&\multicolumn{1}{@{.}l}{1}&\multicolumn{1}{r@{}}{4}&\multicolumn{1}{@{ }l}{{\it 1}}&\multicolumn{1}{r@{}}{3333}&\multicolumn{1}{@{.}l}{1}&&\multicolumn{1}{r@{}}{2979}&\multicolumn{1}{@{.}l}{0 }&\multicolumn{1}{@{}l}{27/2\ensuremath{^{+}},31/2\ensuremath{^{+}}}&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{5.0900416cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}=0.02 \textit{16}, A\ensuremath{_{\textnormal{4}}}=0.1 \textit{2}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{408}&\multicolumn{1}{@{.}l}{8}&\multicolumn{1}{r@{}}{12}&\multicolumn{1}{@{ }l}{{\it 1}}&\multicolumn{1}{r@{}}{2979}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{l}{27/2\ensuremath{^{+}},31/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{2570}&\multicolumn{1}{@{.}l}{2 }&\multicolumn{1}{@{}l}{27/2\ensuremath{^{+}}}&\multicolumn{1}{l}{Q}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{5.0900416cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}=0.5 \textit{2}, A\ensuremath{_{\textnormal{4}}}=0.1 \textit{3}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{417}&\multicolumn{1}{@{.}l}{8\ensuremath{^{\hyperlink{PO45GAMMA1}{\ddagger}}} {\it 2}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{423}&\multicolumn{1}{@{.}l}{41}&\multicolumn{1}{l}{13/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{5}&\multicolumn{1}{@{.}l}{61 }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{-}}}&&&&&\\
\multicolumn{1}{r@{}}{419}&\multicolumn{1}{@{.}l}{9}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{2332}&\multicolumn{1}{@{.}l}{2}&\multicolumn{1}{l}{(27/2)}&\multicolumn{1}{r@{}}{1912}&\multicolumn{1}{@{.}l}{3 }&\multicolumn{1}{@{}l}{25/2\ensuremath{^{+}}}&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{5.0900416cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}={\textminus}0.2 \textit{3}, A\ensuremath{_{\textnormal{4}}}={\textminus}0.1 \textit{4}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{442}&\multicolumn{1}{@{.}l}{4\ensuremath{^{\hyperlink{PO45GAMMA4}{\&}}}}&\multicolumn{1}{r@{}}{19}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PO45GAMMA4}{\&}}} {\it 1}}&\multicolumn{1}{r@{}}{2354}&\multicolumn{1}{@{.}l}{7}&\multicolumn{1}{l}{27/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1912}&\multicolumn{1}{@{.}l}{3 }&\multicolumn{1}{@{}l}{25/2\ensuremath{^{+}}}&\multicolumn{1}{l}{D}&&&\parbox[t][0.3cm]{5.0900416cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}={\textminus}0.05 \textit{7}, A\ensuremath{_{\textnormal{4}}}=0.27 \textit{9}.\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{5.0900416cm}{\raggedright {\ }{\ }{\ }Doublet.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{442}&\multicolumn{1}{@{.}l}{4\ensuremath{^{\hyperlink{PO45GAMMA4}{\&}}}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{PO45GAMMA4}{\&}}}}&\multicolumn{1}{r@{}}{4153?}&\multicolumn{1}{@{}l}{}&&\multicolumn{1}{r@{}}{3710}&\multicolumn{1}{@{.}l}{1 }&\multicolumn{1}{@{}l}{(35/2\ensuremath{^{+}})}&&&&&\\
\end{longtable}
\begin{textblock}{29}(0,27.3)
Continued on next page (footnotes at end of table)
\end{textblock}
\clearpage
\begin{longtable}{ccccccccc@{}ccccc@{\extracolsep{\fill}}c}
\\[-.4cm]
\multicolumn{15}{c}{{\bf \small \underline{\ensuremath{^{\textnormal{194}}}Pt(\ensuremath{^{\textnormal{12}}}C,5n\ensuremath{\gamma})\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1985We05,B}{1985We05} (continued)}}}\\
\multicolumn{15}{c}{~}\\
\multicolumn{15}{c}{\underline{$\gamma$($^{201}$Po) (continued)}}\\
\multicolumn{15}{c}{~~~}\\
\multicolumn{2}{c}{E\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{PO45GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{I\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{PO45GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{E\ensuremath{_{i}}(level)}&J\ensuremath{^{\pi}_{i}}&\multicolumn{2}{c}{E\ensuremath{_{f}}}&J\ensuremath{^{\pi}_{f}}&Mult.\ensuremath{^{\hyperlink{PO45GAMMA2}{\#}}}&\multicolumn{2}{c}{\ensuremath{\alpha}\ensuremath{^{\hyperlink{PO45GAMMA3}{@}}}}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endhead
\multicolumn{1}{r@{}}{499}&\multicolumn{1}{@{.}l}{8}&\multicolumn{1}{r@{}}{$\approx$9}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{3710}&\multicolumn{1}{@{.}l}{1}&\multicolumn{1}{l}{(35/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{3210}&\multicolumn{1}{@{.}l}{3 }&\multicolumn{1}{@{}l}{31/2\ensuremath{^{+}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{6.3462214cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}=0.07 \textit{3}, A\ensuremath{_{\textnormal{4}}}={\textminus}0.16 \textit{4} consistent with\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{6.3462214cm}{\raggedright {\ }{\ }{\ }M1(+E2), but the \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1985We05,B}{1985We05} level scheme\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{6.3462214cm}{\raggedright {\ }{\ }{\ }requires E2.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{551}&\multicolumn{1}{@{.}l}{6}&\multicolumn{1}{r@{}}{8}&\multicolumn{1}{@{ }l}{{\it 1}}&\multicolumn{1}{r@{}}{2463}&\multicolumn{1}{@{.}l}{9}&&\multicolumn{1}{r@{}}{1912}&\multicolumn{1}{@{.}l}{3 }&\multicolumn{1}{@{}l}{25/2\ensuremath{^{+}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{6.3462214cm}{\raggedright A\ensuremath{_{\textnormal{2}}}\ensuremath{>}0.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{556}&\multicolumn{1}{@{.}l}{6}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{1593}&\multicolumn{1}{@{.}l}{6}&\multicolumn{1}{l}{21/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1037}&\multicolumn{1}{@{.}l}{0 }&\multicolumn{1}{@{}l}{17/2\ensuremath{^{+}}}&\multicolumn{1}{l}{E2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0253}&\parbox[t][0.3cm]{6.3462214cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}=0.24 \textit{4}, A\ensuremath{_{\textnormal{4}}}={\textminus}0.01 \textit{5}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{613}&\multicolumn{1}{@{.}l}{6}&\multicolumn{1}{r@{}}{$\approx$100}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{1037}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{l}{17/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{423}&\multicolumn{1}{@{.}l}{41 }&\multicolumn{1}{@{}l}{13/2\ensuremath{^{+}}}&\multicolumn{1}{l}{(E2)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0203}&\parbox[t][0.3cm]{6.3462214cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}=0.5 \textit{2}, A\ensuremath{_{\textnormal{4}}}=0.1 \textit{3}, but values are\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{6.3462214cm}{\raggedright {\ }{\ }{\ }distorted since the 613.6\ensuremath{\gamma} is situated on the\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{6.3462214cm}{\raggedright {\ }{\ }{\ }slope of both the neutron bump and the\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{6.3462214cm}{\raggedright {\ }{\ }{\ }stronger 611.2\ensuremath{\gamma}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{626}&\multicolumn{1}{@{.}l}{3}&\multicolumn{1}{r@{}}{8}&\multicolumn{1}{@{ }l}{{\it 2}}&\multicolumn{1}{r@{}}{3196}&\multicolumn{1}{@{.}l}{5}&\multicolumn{1}{l}{(29/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{2570}&\multicolumn{1}{@{.}l}{2 }&\multicolumn{1}{@{}l}{27/2\ensuremath{^{+}}}&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{6.3462214cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}={\textminus}0.36 \textit{7}, A\ensuremath{_{\textnormal{4}}}={\textminus}0.48 \textit{10}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{640}&\multicolumn{1}{@{.}l}{1}&\multicolumn{1}{r@{}}{12}&\multicolumn{1}{@{ }l}{{\it 1}}&\multicolumn{1}{r@{}}{3210}&\multicolumn{1}{@{.}l}{3}&\multicolumn{1}{l}{31/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{2570}&\multicolumn{1}{@{.}l}{2 }&\multicolumn{1}{@{}l}{27/2\ensuremath{^{+}}}&\multicolumn{1}{l}{Q}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{6.3462214cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}=0.44 \textit{19}, A\ensuremath{_{\textnormal{4}}}={\textminus}0.2 \textit{3}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{657}&\multicolumn{1}{@{.}l}{9}&\multicolumn{1}{r@{}}{34}&\multicolumn{1}{@{ }l}{{\it 5}}&\multicolumn{1}{r@{}}{2570}&\multicolumn{1}{@{.}l}{2}&\multicolumn{1}{l}{27/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1912}&\multicolumn{1}{@{.}l}{3 }&\multicolumn{1}{@{}l}{25/2\ensuremath{^{+}}}&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{6.3462214cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}={\textminus}0.30 \textit{18}, A\ensuremath{_{\textnormal{4}}}={\textminus}0.2 \textit{2}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{754}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{r@{}}{$\approx$2}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{2347}&\multicolumn{1}{@{.}l}{6}&&\multicolumn{1}{r@{}}{1593}&\multicolumn{1}{@{.}l}{6 }&\multicolumn{1}{@{}l}{21/2\ensuremath{^{+}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
\multicolumn{1}{r@{}}{857}&\multicolumn{1}{@{.}l}{8}&\multicolumn{1}{r@{}}{10}&\multicolumn{1}{@{ }l}{{\it 2}}&\multicolumn{1}{r@{}}{2770}&\multicolumn{1}{@{.}l}{1}&&\multicolumn{1}{r@{}}{1912}&\multicolumn{1}{@{.}l}{3 }&\multicolumn{1}{@{}l}{25/2\ensuremath{^{+}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{6.3462214cm}{\raggedright A\ensuremath{_{\textnormal{2}}}=0.1 \textit{5}, A\ensuremath{_{\textnormal{4}}}=0.3 \textit{7}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{905}&\multicolumn{1}{@{.}l}{8}&\multicolumn{1}{r@{}}{11}&\multicolumn{1}{@{ }l}{{\it 1}}&\multicolumn{1}{r@{}}{3039}&\multicolumn{1}{@{.}l}{6}&&\multicolumn{1}{r@{}}{2133}&\multicolumn{1}{@{.}l}{8 }&\multicolumn{1}{@{}l}{25/2\ensuremath{^{+}},(29/2\ensuremath{^{+}})}&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{6.3462214cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}={\textminus}0.1 \textit{3}, A\ensuremath{_{\textnormal{4}}}={\textminus}0.1 \textit{5}.\vspace{0.1cm}}&\\
\end{longtable}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PO45GAMMA0}{\dagger}}}} From \ensuremath{^{\textnormal{194}}}Pt(\ensuremath{^{\textnormal{12}}}C,5n) reaction at 106 MeV in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1985We05,B}{1985We05}, unless otherwise stated.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PO45GAMMA1}{\ddagger}}}} From adopted gammas.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PO45GAMMA2}{\#}}}} From \ensuremath{\gamma}(\ensuremath{\theta}) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1985We05,B}{1985We05}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PO45GAMMA3}{@}}}} Total theoretical internal conversion coefficients, calculated using the BrIcc code (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2008Ki07,B}{2008Ki07}) with Frozen orbital approximation}\\
\parbox[b][0.3cm]{17.7cm}{{\ }{\ }based on \ensuremath{\gamma}-ray energies, assigned multipolarities, and mixing ratios, unless otherwise specified.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{PO45GAMMA4}{\&}}}} Multiply placed with undivided intensity.}\\
\vspace{0.5cm}
\clearpage
\begin{figure}[h]
\begin{center}
\includegraphics{201PO45-0.ps}\\
\end{center}
\end{figure}
\clearpage
\section[\ensuremath{^{201}_{\ 85}}At\ensuremath{_{116}^{~}}]{ }
\vspace{-30pt}
\setcounter{chappage}{1}
\subsection[\hspace{-0.2cm}Adopted Levels, Gammas]{ }
\vspace{-20pt}
\vspace{0.3cm}
\hypertarget{AT46}{{\bf \small \underline{Adopted \hyperlink{201AT_LEVEL}{Levels}, \hyperlink{201AT_GAMMA}{Gammas}}}}\\
\vspace{4pt}
\vspace{8pt}
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}Q(\ensuremath{\beta^-})=$-$6682 {\it 13}; S(n)=9873 {\it 26}; S(p)=1137 {\it 11}; Q(\ensuremath{\alpha})=6472.8 {\it 16}\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2021Wa16,B}{2021Wa16}}\\
\vspace{12pt}
\hypertarget{201AT_LEVEL}{\underline{$^{201}$At Levels}}\\
\begin{longtable}[c]{ll}
\multicolumn{2}{c}{\underline{Cross Reference (XREF) Flags}}\\
\\
\hyperlink{FR47}{\texttt{A }}& \ensuremath{^{\textnormal{205}}}Fr \ensuremath{\alpha} decay\\
\hyperlink{AT48}{\texttt{B }}& \ensuremath{^{\textnormal{192}}}Pt(\ensuremath{^{\textnormal{14}}}N,5n\ensuremath{\gamma})\\
\hyperlink{AT49}{\texttt{C }}& \ensuremath{^{\textnormal{165}}}Ho(\ensuremath{^{\textnormal{40}}}Ar,4n\ensuremath{\gamma})\\
\end{longtable}
\vspace{-0.5cm}
\begin{longtable}{ccccccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E(level)$^{{\hyperlink{AT46LEVEL0}{\dagger}}}$}&J$^{\pi}$$^{{\hyperlink{AT46LEVEL1}{\ddagger}}}$&\multicolumn{2}{c}{T$_{1/2}$$^{}$}&XREF&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0\ensuremath{^{{\hyperlink{AT46LEVEL4}{\&}}}}}&\multicolumn{1}{l}{9/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{87}&\multicolumn{1}{@{.}l}{6 s {\it 13}}&\multicolumn{1}{l}{\texttt{\hyperlink{FR47}{A}\hyperlink{AT48}{B}\hyperlink{AT49}{C}} }&\parbox[t][0.3cm]{11.405041cm}{\raggedright \%\ensuremath{\alpha}=71 \textit{7}; \%\ensuremath{\varepsilon}+\%\ensuremath{\beta}\ensuremath{^{\textnormal{+}}}=29 \textit{7} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1974Ho27,B}{1974Ho27})\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.405041cm}{\raggedright \ensuremath{\mu}=4.03 \textit{7}\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.405041cm}{\raggedright Q={\textminus}0.96 \textit{52}\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.405041cm}{\raggedright J\ensuremath{^{\pi}}: Favored \ensuremath{\alpha}-decay to the \ensuremath{^{\textnormal{197}}}Bi g.s.(\ensuremath{J^{\pi}}=9/2\ensuremath{^{-}},\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2016Ba42,B}{2016Ba42}); J=(9/2) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2018Cu02,B}{2018Cu02};\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.405041cm}{\raggedright {\ }{\ }{\ }\ensuremath{\mu}.\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.405041cm}{\raggedright T\ensuremath{_{1/2}}: Unweighted average of 90 s \textit{6} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1963Ho18,B}{1963Ho18}), 90 s \textit{4} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1967Tr06,B}{1967Tr06}), 87.0 s \textit{6}\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.405041cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1970DaZM,B}{1970DaZM}), 88 s \textit{5} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1974Ho27,B}{1974Ho27}) and 83 s \textit{2} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1996Ta18,B}{1996Ta18}).\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.405041cm}{\raggedright \ensuremath{\mu},Q: from the measured hyperfine-structure constants and isotope shifts using the\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.405041cm}{\raggedright {\ }{\ }{\ }in-source resonance-ionization spectroscopy method (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2018Cu02,B}{2018Cu02}). \ensuremath{\mu} from 4.025\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.405041cm}{\raggedright {\ }{\ }{\ }\textit{45}(stat)\textit{57}(syst), deduced using a reference value of \ensuremath{\mu}(\ensuremath{^{\textnormal{211}}}At)=4.139 \textit{37}, with\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.405041cm}{\raggedright {\ }{\ }{\ }statistical and systematic uncertainties added in quadrature (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2018Cu02,B}{2018Cu02}). Q from\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.405041cm}{\raggedright {\ }{\ }{\ }{\textminus}0.96 \textit{15}(stat)\textit{50}(syst) with statistical and systematic uncertainties added in\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.405041cm}{\raggedright {\ }{\ }{\ }quadrature (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2018Cu02,B}{2018Cu02}).\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.405041cm}{\raggedright \ensuremath{\delta}\ensuremath{<}r\ensuremath{^{\textnormal{2}}}\ensuremath{>}(\ensuremath{^{\textnormal{201}}}At,\ensuremath{^{\textnormal{205}}}At)={\textminus}0.197 fm\ensuremath{^{\textnormal{2}}}\textit{7}(stat) \textit{10}(syst).\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.405041cm}{\raggedright \ensuremath{\delta}\ensuremath{\nu}(\ensuremath{^{\textnormal{201}}}At,\ensuremath{^{\textnormal{205}}}At)=2299 MHz \textit{75}(stat), using 795-nm transition.\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.405041cm}{\raggedright E\ensuremath{\alpha}=6342 keV \textit{6} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1963Ho18,B}{1963Ho18}), 6342 keV \textit{3} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1967Tr06,B}{1967Tr06}), 6340 keV \textit{10} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1970DaZM,B}{1970DaZM}),\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.405041cm}{\raggedright {\ }{\ }{\ }6347 keV \textit{5} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1974Ho27,B}{1974Ho27}), 6345 keV \textit{2} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1975BaYJ,B}{1975BaYJ}), 6344 keV (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986Wo03,B}{1986Wo03}), 6344\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.405041cm}{\raggedright {\ }{\ }{\ }keV \textit{3} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1996Ta18,B}{1996Ta18}) and 6343 keV \textit{4} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2005De01,B}{2005De01}).\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.405041cm}{\raggedright configuration: \ensuremath{\pi} h\ensuremath{_{\textnormal{9/2}}^{\textnormal{+1}}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{190}&\multicolumn{1}{@{.}l}{10 {\it 10}}&\multicolumn{1}{l}{7/2\ensuremath{^{-}}}&&&\multicolumn{1}{l}{\texttt{\ \ \hyperlink{AT49}{C}} }&\parbox[t][0.3cm]{11.405041cm}{\raggedright J\ensuremath{^{\pi}}: 190.1\ensuremath{\gamma} M1+E2 to 9/2\ensuremath{^{-}}.\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.405041cm}{\raggedright configuration: \ensuremath{\pi} f\ensuremath{_{\textnormal{7/2}}^{\textnormal{+1}}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{459}&\multicolumn{1}{@{.}l}{20 {\it 14}}&\multicolumn{1}{l}{1/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{45}&\multicolumn{1}{@{ }l}{ms {\it 3}}&\multicolumn{1}{l}{\texttt{\ \ \hyperlink{AT49}{C}} }&\parbox[t][0.3cm]{11.405041cm}{\raggedright \%IT=100\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.405041cm}{\raggedright J\ensuremath{^{\pi}}: 269.1\ensuremath{\gamma} E3 to 7/2\ensuremath{^{-}}.\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.405041cm}{\raggedright T\ensuremath{_{1/2}}: From recoil-ce(\ensuremath{\Delta}t) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2014Au03,B}{2014Au03}, where recoils were correlated with the\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.405041cm}{\raggedright {\ }{\ }{\ }173\ensuremath{\gamma} and 433\ensuremath{\gamma} above the isomer, and ce were in coincidence with 190\ensuremath{\gamma} and\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.405041cm}{\raggedright {\ }{\ }{\ }269\ensuremath{\gamma}, below the isomer.\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.405041cm}{\raggedright configuration: \ensuremath{\pi} s\ensuremath{_{\textnormal{1/2}}^{\textnormal{$-$1}}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{631}&\multicolumn{1}{@{.}l}{8\ensuremath{^{{\hyperlink{AT46LEVEL2}{\#}}}} {\it 4}}&\multicolumn{1}{l}{3/2\ensuremath{^{+}}}&&&\multicolumn{1}{l}{\texttt{\ \ \hyperlink{AT49}{C}} }&\parbox[t][0.3cm]{11.405041cm}{\raggedright J\ensuremath{^{\pi}}: 172.6\ensuremath{\gamma} (M1+E2) to 1/2\ensuremath{^{+}}.\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.405041cm}{\raggedright configuration: \ensuremath{\pi} d\ensuremath{_{\textnormal{3/2}}^{\textnormal{$-$1}}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{635}&\multicolumn{1}{@{.}l}{17\ensuremath{^{{\hyperlink{AT46LEVEL4}{\&}}}} {\it 16}}&\multicolumn{1}{l}{13/2\ensuremath{^{-}}}&&&\multicolumn{1}{l}{\texttt{\ \hyperlink{AT48}{B}\hyperlink{AT49}{C}} }&\parbox[t][0.3cm]{11.405041cm}{\raggedright J\ensuremath{^{\pi}}: 635.1\ensuremath{\gamma} (E2) to 9/2\ensuremath{^{-}}.\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.405041cm}{\raggedright configuration: dominant \ensuremath{\pi} (h\ensuremath{_{\textnormal{9/2}}^{\textnormal{+1}}})\ensuremath{\otimes}2\ensuremath{^{\textnormal{+}}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{690}&\multicolumn{1}{@{.}l}{98 {\it 16}}&\multicolumn{1}{l}{(11/2\ensuremath{^{-}})}&&&\multicolumn{1}{l}{\texttt{\ \ \hyperlink{AT49}{C}} }&\parbox[t][0.3cm]{11.405041cm}{\raggedright J\ensuremath{^{\pi}}: 691.1\ensuremath{\gamma} to 9/2\ensuremath{^{-}}.\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.405041cm}{\raggedright configuration: dominant \ensuremath{\pi} (h\ensuremath{_{\textnormal{9/2}}^{\textnormal{+1}}})\ensuremath{\otimes}2\ensuremath{^{\textnormal{+}}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{749}&\multicolumn{1}{@{.}l}{36 {\it 14}}&\multicolumn{1}{l}{13/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{15}&\multicolumn{1}{@{.}l}{9 ns {\it 14}}&\multicolumn{1}{l}{\texttt{\ \hyperlink{AT48}{B}\hyperlink{AT49}{C}} }&\parbox[t][0.3cm]{11.405041cm}{\raggedright J\ensuremath{^{\pi}}: 114.1\ensuremath{\gamma} to 13/2\ensuremath{^{-}}, 749.3\ensuremath{\gamma} (M2) to 9/2\ensuremath{^{-}}; systematics and shell-model predictions.\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.405041cm}{\raggedright T\ensuremath{_{1/2}}: From 749.0\ensuremath{\gamma}(t) in \ensuremath{^{\textnormal{192}}}Pt(\ensuremath{^{\textnormal{14}}}N,5n\ensuremath{\gamma}) (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1983Dy02,B}{1983Dy02}). Other \ensuremath{\approx} 20 ns in\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.405041cm}{\raggedright {\ }{\ }{\ }\ensuremath{^{\textnormal{165}}}Ho(\ensuremath{^{\textnormal{40}}}Ar,4n\ensuremath{\gamma}) (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2015Au01,B}{2015Au01}).\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.405041cm}{\raggedright configuration: \ensuremath{\pi} i\ensuremath{_{\textnormal{13/2}}^{\textnormal{+1}}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{804}&\multicolumn{1}{@{.}l}{2\ensuremath{^{{\hyperlink{AT46LEVEL3}{@}}}} {\it 6}}&\multicolumn{1}{l}{(5/2\ensuremath{^{+}})}&&&\multicolumn{1}{l}{\texttt{\ \ \hyperlink{AT49}{C}} }&\parbox[t][0.3cm]{11.405041cm}{\raggedright J\ensuremath{^{\pi}}: 172.5\ensuremath{\gamma} to 3/2\ensuremath{^{+}}.\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.405041cm}{\raggedright configuration: Probable \ensuremath{\pi} d\ensuremath{_{\textnormal{5/2}}^{\textnormal{$-$1}}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1065}&\multicolumn{1}{@{.}l}{2\ensuremath{^{{\hyperlink{AT46LEVEL2}{\#}}}} {\it 6}}&\multicolumn{1}{l}{(7/2\ensuremath{^{+}})}&&&\multicolumn{1}{l}{\texttt{\ \ \hyperlink{AT49}{C}} }&\parbox[t][0.3cm]{11.405041cm}{\raggedright J\ensuremath{^{\pi}}: 433.3\ensuremath{\gamma} (E2) to 3/2\ensuremath{^{+}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1228}&\multicolumn{1}{@{.}l}{96\ensuremath{^{{\hyperlink{AT46LEVEL4}{\&}}}} {\it 22}}&\multicolumn{1}{l}{17/2\ensuremath{^{-}}}&&&\multicolumn{1}{l}{\texttt{\ \hyperlink{AT48}{B}\hyperlink{AT49}{C}} }&\parbox[t][0.3cm]{11.405041cm}{\raggedright J\ensuremath{^{\pi}}: 583.8\ensuremath{\gamma} (E2) to 13/2\ensuremath{^{-}}.\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.405041cm}{\raggedright configuration: \ensuremath{\pi} (h\ensuremath{_{\textnormal{9/2}}^{\textnormal{+1}}})\ensuremath{\otimes}4\ensuremath{^{\textnormal{+}}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1261}&\multicolumn{1}{@{.}l}{29 {\it 20}}&\multicolumn{1}{l}{15/2\ensuremath{^{+}}}&&&\multicolumn{1}{l}{\texttt{\ \ \hyperlink{AT49}{C}} }&\parbox[t][0.3cm]{11.405041cm}{\raggedright J\ensuremath{^{\pi}}: 511.8\ensuremath{\gamma} D to 13/2\ensuremath{^{+}}; 364.1\ensuremath{\gamma} (M1) from 17/2\ensuremath{^{+}}.\vspace{0.1cm}}&\\
\end{longtable}
\begin{textblock}{29}(0,27.3)
Continued on next page (footnotes at end of table)
\end{textblock}
\clearpage
\begin{longtable}{ccccccc@{\extracolsep{\fill}}c}
\\[-.4cm]
\multicolumn{8}{c}{{\bf \small \underline{Adopted \hyperlink{201AT_LEVEL}{Levels}, \hyperlink{201AT_GAMMA}{Gammas} (continued)}}}\\
\multicolumn{8}{c}{~}\\
\multicolumn{8}{c}{\underline{\ensuremath{^{201}}At Levels (continued)}}\\
\multicolumn{8}{c}{~}\\
\multicolumn{2}{c}{E(level)$^{{\hyperlink{AT46LEVEL0}{\dagger}}}$}&J$^{\pi}$$^{{\hyperlink{AT46LEVEL1}{\ddagger}}}$&\multicolumn{2}{c}{T$_{1/2}$$^{}$}&XREF&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&
\endhead
\multicolumn{1}{r@{}}{1288}&\multicolumn{1}{@{.}l}{6\ensuremath{^{{\hyperlink{AT46LEVEL3}{@}}}} {\it 6}}&\multicolumn{1}{l}{(9/2\ensuremath{^{+}})}&&&\multicolumn{1}{l}{\texttt{\ \ \hyperlink{AT49}{C}} }&\parbox[t][0.3cm]{11.16445cm}{\raggedright J\ensuremath{^{\pi}}: 223.3\ensuremath{\gamma} (M1) to (7/2\ensuremath{^{+}}), 484.5\ensuremath{\gamma} (E2) to (5/2\ensuremath{^{+}}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1494}&\multicolumn{1}{@{.}l}{85 {\it 19}}&\multicolumn{1}{l}{17/2\ensuremath{^{+}}}&&&\multicolumn{1}{l}{\texttt{\ \hyperlink{AT48}{B}\hyperlink{AT49}{C}} }&\parbox[t][0.3cm]{11.16445cm}{\raggedright J\ensuremath{^{\pi}}: 233.4\ensuremath{\gamma} (M1) to 15/2\ensuremath{^{+}}, 745.5\ensuremath{\gamma} (E2) to 13/2\ensuremath{^{+}}.\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.16445cm}{\raggedright configuration: Dominant \ensuremath{\pi} (i\ensuremath{_{\textnormal{13/2}}^{\textnormal{+1}}})\ensuremath{\otimes}2\ensuremath{^{\textnormal{+}}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1613}&\multicolumn{1}{@{.}l}{3\ensuremath{^{{\hyperlink{AT46LEVEL2}{\#}}}} {\it 7}}&\multicolumn{1}{l}{(11/2\ensuremath{^{+}})}&&&\multicolumn{1}{l}{\texttt{\ \ \hyperlink{AT49}{C}} }&\parbox[t][0.3cm]{11.16445cm}{\raggedright J\ensuremath{^{\pi}}: 548.1\ensuremath{\gamma} (E2) to (7/2\ensuremath{^{+}}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1625}&\multicolumn{1}{@{.}l}{34 {\it 19}}&\multicolumn{1}{l}{17/2\ensuremath{^{+}}}&&&\multicolumn{1}{l}{\texttt{\ \ \hyperlink{AT49}{C}} }&\parbox[t][0.3cm]{11.16445cm}{\raggedright J\ensuremath{^{\pi}}: 130.3\ensuremath{\gamma} (M1) to 17/2\ensuremath{^{+}}, 876.1\ensuremath{\gamma} (E2) to 13/2\ensuremath{^{+}}.\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.16445cm}{\raggedright configuration: Dominant \ensuremath{\pi} (h\ensuremath{_{\textnormal{9/2}}^{\textnormal{+1}}}) \ensuremath{\nu} (f\ensuremath{_{\textnormal{5/2}}^{\textnormal{$-$1}}},i\ensuremath{_{\textnormal{13/2}}^{\textnormal{$-$1}}})\ensuremath{_{\textnormal{5$-$}}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1705}&\multicolumn{1}{@{.}l}{03\ensuremath{^{{\hyperlink{AT46LEVEL4}{\&}}}} {\it 24}}&\multicolumn{1}{l}{21/2\ensuremath{^{-}}}&&&\multicolumn{1}{l}{\texttt{\ \hyperlink{AT48}{B}\hyperlink{AT49}{C}} }&\parbox[t][0.3cm]{11.16445cm}{\raggedright J\ensuremath{^{\pi}}: 476.2\ensuremath{\gamma} (E2) to 17/2\ensuremath{^{-}}.\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.16445cm}{\raggedright configuration: a mixture between \ensuremath{\pi} (h\ensuremath{_{\textnormal{9/2}}^{\textnormal{+1}}})\ensuremath{\otimes}6\ensuremath{^{\textnormal{+}}} and \ensuremath{\pi} [f\ensuremath{_{\textnormal{7/2}}^{\textnormal{+1}}},(h\ensuremath{_{\textnormal{9/2}}^{\textnormal{+2}}})\ensuremath{_{\textnormal{8+}}}].\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1790}&\multicolumn{1}{@{.}l}{3 {\it 4}}&&&&\multicolumn{1}{l}{\texttt{\ \hyperlink{AT48}{B}\ } }&&\\
\multicolumn{1}{r@{}}{1856}&\multicolumn{1}{@{.}l}{0\ensuremath{^{{\hyperlink{AT46LEVEL3}{@}}}} {\it 7}}&\multicolumn{1}{l}{(13/2\ensuremath{^{+}})}&&&\multicolumn{1}{l}{\texttt{\ \ \hyperlink{AT49}{C}} }&\parbox[t][0.3cm]{11.16445cm}{\raggedright J\ensuremath{^{\pi}}: 242.8\ensuremath{\gamma} to (11/2\ensuremath{^{+}}), 567.3\ensuremath{\gamma} (E2) to (9/2\ensuremath{^{+}}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1921}&\multicolumn{1}{@{.}l}{30 {\it 21}}&\multicolumn{1}{l}{21/2\ensuremath{^{+}}}&&&\multicolumn{1}{l}{\texttt{\ \hyperlink{AT48}{B}\hyperlink{AT49}{C}} }&\parbox[t][0.3cm]{11.16445cm}{\raggedright J\ensuremath{^{\pi}}: 216.3\ensuremath{\gamma} to 21/2\ensuremath{^{-}}, 295.9\ensuremath{\gamma} (E2) to 17/2\ensuremath{^{+}}.\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.16445cm}{\raggedright configuration: Dominant \ensuremath{\pi} (h\ensuremath{_{\textnormal{9/2}}^{\textnormal{+1}}}) \ensuremath{\nu} (f\ensuremath{_{\textnormal{5/2}}^{\textnormal{$-$1}}},i\ensuremath{_{\textnormal{13/2}}^{\textnormal{$-$1}}})\ensuremath{_{\textnormal{7$-$}}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1980}&\multicolumn{1}{@{.}l}{6 {\it 3}}&\multicolumn{1}{l}{23/2\ensuremath{^{-}}}&&&\multicolumn{1}{l}{\texttt{\ \ \hyperlink{AT49}{C}} }&\parbox[t][0.3cm]{11.16445cm}{\raggedright J\ensuremath{^{\pi}}: 275.5\ensuremath{\gamma} (M1) to 21/2\ensuremath{^{-}}.\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.16445cm}{\raggedright configuration: Dominant \ensuremath{\pi} [f\ensuremath{_{\textnormal{7/2}}^{\textnormal{+1}}},(h\ensuremath{_{\textnormal{9/2}}^{\textnormal{+2}}})\ensuremath{_{\textnormal{8+}}}].\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2004}&\multicolumn{1}{@{.}l}{3 {\it 3}}&\multicolumn{1}{l}{23/2\ensuremath{^{+}}}&&&\multicolumn{1}{l}{\texttt{\ \ \hyperlink{AT49}{C}} }&\parbox[t][0.3cm]{11.16445cm}{\raggedright J\ensuremath{^{\pi}}: 83.0\ensuremath{\gamma} to 21/2\ensuremath{^{+}}, 299.3\ensuremath{\gamma} to 21/2\ensuremath{^{-}}.\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.16445cm}{\raggedright configuration: Dominant \ensuremath{\pi} (h\ensuremath{_{\textnormal{9/2}}^{\textnormal{+1}}}) \ensuremath{\nu} (f\ensuremath{_{\textnormal{5/2}}^{\textnormal{$-$1}}},i\ensuremath{_{\textnormal{13/2}}^{\textnormal{$-$1}}})\ensuremath{_{\textnormal{8$-$}}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2050}&\multicolumn{1}{@{.}l}{8 {\it 3}}&\multicolumn{1}{l}{25/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{$<$20}&\multicolumn{1}{@{ }l}{ns}&\multicolumn{1}{l}{\texttt{\ \ \hyperlink{AT49}{C}} }&\parbox[t][0.3cm]{11.16445cm}{\raggedright J\ensuremath{^{\pi}}: 46.5\ensuremath{\gamma} to 23/2\ensuremath{^{+}}.\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.16445cm}{\raggedright T\ensuremath{_{1/2}}: An estimate from \ensuremath{\gamma}(t) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2015Au01,B}{2015Au01}.\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.16445cm}{\raggedright configuration: Dominant \ensuremath{\pi} (h\ensuremath{_{\textnormal{9/2}}^{\textnormal{+1}}}) \ensuremath{\nu} (f\ensuremath{_{\textnormal{5/2}}^{\textnormal{$-$1}}},i\ensuremath{_{\textnormal{13/2}}^{\textnormal{$-$1}}})\ensuremath{_{\textnormal{9$-$}}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2076}&\multicolumn{1}{@{.}l}{9 {\it 5}}&\multicolumn{1}{l}{(23/2\ensuremath{^{-}})}&&&\multicolumn{1}{l}{\texttt{\ \hyperlink{AT48}{B}\ } }&\parbox[t][0.3cm]{11.16445cm}{\raggedright J\ensuremath{^{\pi}}: 371.9\ensuremath{\gamma} D to 21/2\ensuremath{^{-}}; proposed configuration.\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.16445cm}{\raggedright configuration: Probable \ensuremath{\pi} [f\ensuremath{_{\textnormal{7/2}}^{\textnormal{+1}}},(h\ensuremath{_{\textnormal{9/2}}^{\textnormal{+2}}})\ensuremath{_{\textnormal{8+}}}].\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2147}&\multicolumn{1}{@{.}l}{2\ensuremath{^{{\hyperlink{AT46LEVEL4}{\&}}}} {\it 4}}&\multicolumn{1}{l}{21/2\ensuremath{^{-}}}&&&\multicolumn{1}{l}{\texttt{\ \ \hyperlink{AT49}{C}} }&\parbox[t][0.3cm]{11.16445cm}{\raggedright J\ensuremath{^{\pi}}: 917.8\ensuremath{\gamma} (E2) to 17/2\ensuremath{^{-}}.\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.16445cm}{\raggedright configuration: a mixture between \ensuremath{\pi} (h\ensuremath{_{\textnormal{9/2}}^{\textnormal{+1}}})\ensuremath{\otimes}6\ensuremath{^{\textnormal{+}}} and \ensuremath{\pi} (f\ensuremath{_{\textnormal{7/2}}^{\textnormal{+1}}},(h\ensuremath{_{\textnormal{9/2}}^{\textnormal{+2}}})\ensuremath{_{\textnormal{8+}}}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2232}&\multicolumn{1}{@{.}l}{1\ensuremath{^{{\hyperlink{AT46LEVEL2}{\#}}}} {\it 9}}&\multicolumn{1}{l}{(15/2\ensuremath{^{+}})}&&&\multicolumn{1}{l}{\texttt{\ \ \hyperlink{AT49}{C}} }&\parbox[t][0.3cm]{11.16445cm}{\raggedright J\ensuremath{^{\pi}}: 618.8\ensuremath{\gamma} (E2) to (11/2\ensuremath{^{+}}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2319}&\multicolumn{1}{@{.}l}{8 {\it 3}}&\multicolumn{1}{l}{29/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{39 \ensuremath{\mu}s {\it 9}}&\multicolumn{1}{l}{\texttt{\ \ \hyperlink{AT49}{C}} }&\parbox[t][0.3cm]{11.16445cm}{\raggedright \%IT=100\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.16445cm}{\raggedright J\ensuremath{^{\pi}}: 269.0\ensuremath{\gamma} E2 to 25/2\ensuremath{^{+}}, 339.2\ensuremath{\gamma} E3 to 23/2\ensuremath{^{-}}.\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.16445cm}{\raggedright T\ensuremath{_{1/2}}: From recoil-269\ensuremath{\gamma}(t) using the planar detector data and the logarithmic time\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.16445cm}{\raggedright {\ }{\ }{\ }scale method (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2015Au01,B}{2015Au01}). A prompt coincidence with a 296\ensuremath{\gamma}, 427\ensuremath{\gamma}, 594\ensuremath{\gamma},\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.16445cm}{\raggedright {\ }{\ }{\ }635\ensuremath{\gamma},\hphantom{a}746\ensuremath{\gamma} or 749\ensuremath{\gamma} in any of the focal plane clover detectors was also\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.16445cm}{\raggedright {\ }{\ }{\ }required.\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.16445cm}{\raggedright configuration: \ensuremath{\pi} [i\ensuremath{_{\textnormal{13/2}}^{\textnormal{+1}}},(h\ensuremath{_{\textnormal{9/2}}^{\textnormal{+2}}})\ensuremath{_{\textnormal{8+}}}].\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2518}&\multicolumn{1}{@{.}l}{9\ensuremath{^{{\hyperlink{AT46LEVEL4}{\&}}}} {\it 5}}&\multicolumn{1}{l}{25/2\ensuremath{^{-}}}&&&\multicolumn{1}{l}{\texttt{\ \ \hyperlink{AT49}{C}} }&\parbox[t][0.3cm]{11.16445cm}{\raggedright J\ensuremath{^{\pi}}: 371.7\ensuremath{\gamma} (E2) to 21/2\ensuremath{^{-}}.\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.16445cm}{\raggedright configuration: Dominant \ensuremath{\pi} (h\ensuremath{_{\textnormal{9/2}}^{\textnormal{+1}}})\ensuremath{\otimes}8\ensuremath{^{\textnormal{+}}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2637}&\multicolumn{1}{@{.}l}{6 {\it 5}}&\multicolumn{1}{l}{(25/2\ensuremath{^{+}})}&&&\multicolumn{1}{l}{\texttt{\ \ \hyperlink{AT49}{C}} }&\parbox[t][0.3cm]{11.16445cm}{\raggedright J\ensuremath{^{\pi}}: 716.3\ensuremath{\gamma} (E2) to 21/2\ensuremath{^{+}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2990}&\multicolumn{1}{@{.}l}{2\ensuremath{^{{\hyperlink{AT46LEVEL5}{a}}}} {\it 5}}&\multicolumn{1}{l}{(23/2\ensuremath{^{-}})}&&&\multicolumn{1}{l}{\texttt{\ \ \hyperlink{AT49}{C}} }&\parbox[t][0.3cm]{11.16445cm}{\raggedright J\ensuremath{^{\pi}}: 1068.9\ensuremath{\gamma} D to 21/2\ensuremath{^{+}}; proposed configuration.\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.16445cm}{\raggedright E(level): The total transition intensity of 145\ensuremath{\gamma} that feeds this level is much higher\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.16445cm}{\raggedright {\ }{\ }{\ }than the 1069\ensuremath{\gamma} one that depopulates it, thus suggesting the existence of at least\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.16445cm}{\raggedright {\ }{\ }{\ }one additional, unobserved decay branch.\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.16445cm}{\raggedright configuration: \ensuremath{\pi} (i\ensuremath{_{\textnormal{13/2}}^{\textnormal{+1}}})\ensuremath{\otimes}\ensuremath{\nu} (f\ensuremath{_{\textnormal{5/2}}^{\textnormal{$-$1}}},i\ensuremath{_{\textnormal{13/2}}^{\textnormal{$-$1}}})\ensuremath{_{\textnormal{5$-$}}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3135}&\multicolumn{1}{@{.}l}{2\ensuremath{^{{\hyperlink{AT46LEVEL5}{a}}}} {\it 6}}&\multicolumn{1}{l}{(25/2\ensuremath{^{-}})}&&&\multicolumn{1}{l}{\texttt{\ \ \hyperlink{AT49}{C}} }&\parbox[t][0.3cm]{11.16445cm}{\raggedright J\ensuremath{^{\pi}}: 145.0\ensuremath{\gamma} (M1) to (23/2\ensuremath{^{-}}); band assignment.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3219}&\multicolumn{1}{@{.}l}{2 {\it 6}}&&&&\multicolumn{1}{l}{\texttt{\ \ \hyperlink{AT49}{C}} }&&\\
\multicolumn{1}{r@{}}{3240}&\multicolumn{1}{@{.}l}{9 {\it 5}}&\multicolumn{1}{l}{(33/2\ensuremath{^{+}})}&&&\multicolumn{1}{l}{\texttt{\ \ \hyperlink{AT49}{C}} }&\parbox[t][0.3cm]{11.16445cm}{\raggedright J\ensuremath{^{\pi}}: 921.1\ensuremath{\gamma} (E2) to (29/2\ensuremath{^{+}}).\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.16445cm}{\raggedright configuration: Possible \ensuremath{\pi} [i\ensuremath{_{\textnormal{13/2}}^{\textnormal{+1}}},(h\ensuremath{_{\textnormal{9/2}}^{\textnormal{+2}}})\ensuremath{_{\textnormal{8+}}}]\ensuremath{\otimes}2\ensuremath{^{\textnormal{+}}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3245}&\multicolumn{1}{@{.}l}{4 {\it 6}}&\multicolumn{1}{l}{(29/2\ensuremath{^{+}})}&&&\multicolumn{1}{l}{\texttt{\ \ \hyperlink{AT49}{C}} }&\parbox[t][0.3cm]{11.16445cm}{\raggedright J\ensuremath{^{\pi}}: 607.8\ensuremath{\gamma} (E2) to (25/2\ensuremath{^{+}}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3369}&\multicolumn{1}{@{.}l}{6 {\it 5}}&&&&\multicolumn{1}{l}{\texttt{\ \ \hyperlink{AT49}{C}} }&&\\
\multicolumn{1}{r@{}}{3379}&\multicolumn{1}{@{.}l}{6\ensuremath{^{{\hyperlink{AT46LEVEL5}{a}}}} {\it 7}}&\multicolumn{1}{l}{(27/2\ensuremath{^{-}})}&&&\multicolumn{1}{l}{\texttt{\ \ \hyperlink{AT49}{C}} }&\parbox[t][0.3cm]{11.16445cm}{\raggedright J\ensuremath{^{\pi}}: 244.4\ensuremath{\gamma} (M1) to (25/2\ensuremath{^{-}}); band assignment.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3504}&\multicolumn{1}{@{.}l}{4 {\it 4}}&&&&\multicolumn{1}{l}{\texttt{\ \ \hyperlink{AT49}{C}} }&&\\
\multicolumn{1}{r@{}}{3621}&\multicolumn{1}{@{.}l}{8 {\it 7}}&&&&\multicolumn{1}{l}{\texttt{\ \ \hyperlink{AT49}{C}} }&&\\
\multicolumn{1}{r@{}}{3666}&\multicolumn{1}{@{.}l}{5\ensuremath{^{{\hyperlink{AT46LEVEL5}{a}}}} {\it 8}}&\multicolumn{1}{l}{(29/2\ensuremath{^{-}})}&&&\multicolumn{1}{l}{\texttt{\ \ \hyperlink{AT49}{C}} }&\parbox[t][0.3cm]{11.16445cm}{\raggedright J\ensuremath{^{\pi}}: 286.9\ensuremath{\gamma} (M1) to (27/2\ensuremath{^{-}}); band assignment.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3693}&\multicolumn{1}{@{.}l}{9 {\it 7}}&&&&\multicolumn{1}{l}{\texttt{\ \ \hyperlink{AT49}{C}} }&&\\
\multicolumn{1}{r@{}}{3699}&\multicolumn{1}{@{.}l}{7 {\it 6}}&&&&\multicolumn{1}{l}{\texttt{\ \ \hyperlink{AT49}{C}} }&&\\
\multicolumn{1}{r@{}}{3779}&\multicolumn{1}{@{.}l}{1 {\it 6}}&&&&\multicolumn{1}{l}{\texttt{\ \ \hyperlink{AT49}{C}} }&&\\
\multicolumn{1}{r@{}}{3785}&\multicolumn{1}{@{.}l}{9 {\it 8}}&&&&\multicolumn{1}{l}{\texttt{\ \ \hyperlink{AT49}{C}} }&&\\
\end{longtable}
\begin{textblock}{29}(0,27.3)
Continued on next page (footnotes at end of table)
\end{textblock}
\clearpage
\begin{longtable}{ccccc@{\extracolsep{\fill}}c}
\\[-.4cm]
\multicolumn{6}{c}{{\bf \small \underline{Adopted \hyperlink{201AT_LEVEL}{Levels}, \hyperlink{201AT_GAMMA}{Gammas} (continued)}}}\\
\multicolumn{6}{c}{~}\\
\multicolumn{6}{c}{\underline{\ensuremath{^{201}}At Levels (continued)}}\\
\multicolumn{6}{c}{~}\\
\multicolumn{2}{c}{E(level)$^{{\hyperlink{AT46LEVEL0}{\dagger}}}$}&J$^{\pi}$$^{{\hyperlink{AT46LEVEL1}{\ddagger}}}$&XREF&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&\hrulefill&
\endhead
\multicolumn{1}{r@{}}{3853}&\multicolumn{1}{@{.}l}{3 {\it 7}}&&\multicolumn{1}{l}{\texttt{\ \ \hyperlink{AT49}{C}} }&&\\
\multicolumn{1}{r@{}}{3952}&\multicolumn{1}{@{.}l}{8 {\it 6}}&&\multicolumn{1}{l}{\texttt{\ \ \hyperlink{AT49}{C}} }&&\\
\multicolumn{1}{r@{}}{3983}&\multicolumn{1}{@{.}l}{8\ensuremath{^{{\hyperlink{AT46LEVEL5}{a}}}} {\it 9}}&\multicolumn{1}{l}{(31/2\ensuremath{^{-}})}&\multicolumn{1}{l}{\texttt{\ \ \hyperlink{AT49}{C}} }&\parbox[t][0.3cm]{13.33086cm}{\raggedright J\ensuremath{^{\pi}}: 317.3\ensuremath{\gamma} (M1) to (29/2\ensuremath{^{-}}); band assignment.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{4111}&\multicolumn{1}{@{.}l}{4 {\it 6}}&&\multicolumn{1}{l}{\texttt{\ \ \hyperlink{AT49}{C}} }&&\\
\multicolumn{1}{r@{}}{4159}&\multicolumn{1}{@{.}l}{0 {\it 7}}&&\multicolumn{1}{l}{\texttt{\ \ \hyperlink{AT49}{C}} }&&\\
\multicolumn{1}{r@{}}{4256}&\multicolumn{1}{@{.}l}{1\ensuremath{^{{\hyperlink{AT46LEVEL5}{a}}}} {\it 10}}&\multicolumn{1}{l}{(33/2\ensuremath{^{-}})}&\multicolumn{1}{l}{\texttt{\ \ \hyperlink{AT49}{C}} }&\parbox[t][0.3cm]{13.33086cm}{\raggedright J\ensuremath{^{\pi}}: 272.3\ensuremath{\gamma} (M1) to (31/2\ensuremath{^{-}}); band assignment.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{4454}&\multicolumn{1}{@{.}l}{0\ensuremath{^{{\hyperlink{AT46LEVEL5}{a}}}} {\it 11}}&\multicolumn{1}{l}{(35/2\ensuremath{^{-}})}&\multicolumn{1}{l}{\texttt{\ \ \hyperlink{AT49}{C}} }&\parbox[t][0.3cm]{13.33086cm}{\raggedright J\ensuremath{^{\pi}}: 197.9\ensuremath{\gamma} (M1) to (33/2\ensuremath{^{-}}); band assignment.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{4789}&\multicolumn{1}{@{.}l}{0\ensuremath{^{{\hyperlink{AT46LEVEL5}{a}}}} {\it 12}}&\multicolumn{1}{l}{(37/2\ensuremath{^{-}})}&\multicolumn{1}{l}{\texttt{\ \ \hyperlink{AT49}{C}} }&\parbox[t][0.3cm]{13.33086cm}{\raggedright J\ensuremath{^{\pi}}: 335.0\ensuremath{\gamma} (M1) to (35/2\ensuremath{^{-}}); band assignment.\vspace{0.1cm}}&\\
\end{longtable}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{AT46LEVEL0}{\dagger}}}} From a least-squares fit to E\ensuremath{\gamma}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{AT46LEVEL1}{\ddagger}}}} From deduced transition multipolarities and the observed decay pattern in \ensuremath{^{\textnormal{165}}}Ho(\ensuremath{^{\textnormal{40}}}Ar,4n\ensuremath{\gamma}) and \ensuremath{^{\textnormal{192}}}Pt(\ensuremath{^{\textnormal{14}}}N,5n\ensuremath{\gamma}), systematics in}\\
\parbox[b][0.3cm]{17.7cm}{{\ }{\ }the region and shell-model predictions, unless otherwise stated.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{AT46LEVEL2}{\#}}}} Seq.(B): Based on \ensuremath{\pi} (d\ensuremath{_{\textnormal{3/2}}^{\textnormal{$-$1}}})\ensuremath{\otimes}\ensuremath{^{\textnormal{202}}}Rn core states (\ensuremath{J^{\pi}}=2\ensuremath{^{+}},4\ensuremath{^{+}},6\ensuremath{^{+}}).}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{AT46LEVEL3}{@}}}} Seq.(C): Based on \ensuremath{\pi} (d\ensuremath{_{\textnormal{5/2}}^{\textnormal{$-$1}}})\ensuremath{\otimes}\ensuremath{^{\textnormal{202}}}Rn core states (\ensuremath{J^{\pi}}=2\ensuremath{^{+}},4\ensuremath{^{+}}).}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{AT46LEVEL4}{\&}}}} Seq.(D): Based on \ensuremath{\pi} (h\ensuremath{_{\textnormal{9/2}}^{\textnormal{+1}}})\ensuremath{\otimes}\ensuremath{^{\textnormal{202}}}Rn core states (\ensuremath{J^{\pi}}=2\ensuremath{^{+}},4\ensuremath{^{+}},6\ensuremath{^{+}}, 8\ensuremath{^{+}}).}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{AT46LEVEL5}{a}}}} Band(A): Magnetic-dipole, shears band. Configuration=\ensuremath{\pi} (i\ensuremath{_{\textnormal{13/2}}^{\textnormal{+1}}})\ensuremath{\otimes}\ensuremath{\nu} (f\ensuremath{_{\textnormal{5/2}}^{\textnormal{$-$1}}},i\ensuremath{_{\textnormal{13/2}}^{\textnormal{$-$1}}})\ensuremath{_{\textnormal{9$-$}}} for the lower cascade and}\\
\parbox[b][0.3cm]{17.7cm}{{\ }{\ }Configuration= \ensuremath{\pi} (h\ensuremath{_{\textnormal{9/2}}^{\textnormal{+2}}},i\ensuremath{_{\textnormal{13/2}}^{\textnormal{+1}}})\ensuremath{\otimes}\ensuremath{\nu} (f\ensuremath{_{\textnormal{5/2}}^{\textnormal{$-$1}}},i\ensuremath{_{\textnormal{13/2}}^{\textnormal{$-$1}}})\ensuremath{_{\textnormal{5$-$}}} above the band crossing.}\\
\vspace{0.5cm}
\hypertarget{201AT_GAMMA}{\underline{$\gamma$($^{201}$At)}}\\
\begin{longtable}{ccccccccc@{}ccccccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E\ensuremath{_{i}}(level)}&J\ensuremath{^{\pi}_{i}}&\multicolumn{2}{c}{E\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{AT46GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{I\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{AT46GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{E\ensuremath{_{f}}}&J\ensuremath{^{\pi}_{f}}&Mult.\ensuremath{^{\hyperlink{AT46GAMMA0}{\dagger}}}&\multicolumn{2}{c}{\ensuremath{\delta}\ensuremath{^{\hyperlink{AT46GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{\ensuremath{\alpha}\ensuremath{^{\hyperlink{AT46GAMMA2}{\#}}}}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{190}&\multicolumn{1}{@{.}l}{10}&\multicolumn{1}{l}{7/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{190}&\multicolumn{1}{@{.}l}{1 {\it 1}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{@{}l}{9/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{65 {\it 8}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{55 {\it 7}}&\parbox[t][0.3cm]{4.4482207cm}{\raggedright Mult.,\ensuremath{\delta}: From\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{4.4482207cm}{\raggedright {\ }{\ }{\ }K/(L+M+...)exp=3.1 \textit{2}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{4.4482207cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2014Au03,B}{2014Au03}). \ensuremath{\delta} was\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{4.4482207cm}{\raggedright {\ }{\ }{\ }determined by the evaluator\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{4.4482207cm}{\raggedright {\ }{\ }{\ }using the briccmixing\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{4.4482207cm}{\raggedright {\ }{\ }{\ }program.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{459}&\multicolumn{1}{@{.}l}{20}&\multicolumn{1}{l}{1/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{269}&\multicolumn{1}{@{.}l}{1 {\it 1}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{190}&\multicolumn{1}{@{.}l}{10 }&\multicolumn{1}{@{}l}{7/2\ensuremath{^{-}}}&\multicolumn{1}{l}{E3}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{231 {\it 17}}&\parbox[t][0.3cm]{4.4482207cm}{\raggedright B(E3)(W.u.)=0.0493 \textit{+35{\textminus}31}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{4.4482207cm}{\raggedright Mult.: From\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{4.4482207cm}{\raggedright {\ }{\ }{\ }K/(L+M+...)exp=0.24 \textit{1}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{4.4482207cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2014Au03,B}{2014Au03}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{631}&\multicolumn{1}{@{.}l}{8}&\multicolumn{1}{l}{3/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{172}&\multicolumn{1}{@{.}l}{6 {\it 4}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{459}&\multicolumn{1}{@{.}l}{20 }&\multicolumn{1}{@{}l}{1/2\ensuremath{^{+}}}&\multicolumn{1}{l}{(M1+E2)}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{7 {\it 9}}&\parbox[t][0.3cm]{4.4482207cm}{\raggedright Mult.: R=0.83 \textit{5} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2014Au03,B}{2014Au03}).\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{4.4482207cm}{\raggedright {\ }{\ }{\ }The x-ray intensity in a\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{4.4482207cm}{\raggedright {\ }{\ }{\ }spectrum produced by gating\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{4.4482207cm}{\raggedright {\ }{\ }{\ }on 533\ensuremath{\gamma} is consistent with\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{4.4482207cm}{\raggedright {\ }{\ }{\ }Mult=M1+E2, but not with a\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{4.4482207cm}{\raggedright {\ }{\ }{\ }pure Mult=E2.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{635}&\multicolumn{1}{@{.}l}{17}&\multicolumn{1}{l}{13/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{635}&\multicolumn{1}{@{.}l}{1 {\it 2}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{@{}l}{9/2\ensuremath{^{-}}}&\multicolumn{1}{l}{(E2)}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{01951 {\it 27}}&\parbox[t][0.3cm]{4.4482207cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}=0.11 \textit{3} in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1983Dy02,B}{1983Dy02}.\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{4.4482207cm}{\raggedright {\ }{\ }{\ }Other: A2=+0.09 \textit{4}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{4.4482207cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2015Au01,B}{2015Au01}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{690}&\multicolumn{1}{@{.}l}{98}&\multicolumn{1}{l}{(11/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{691}&\multicolumn{1}{@{.}l}{1 {\it 2}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{@{}l}{9/2\ensuremath{^{-}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
\multicolumn{1}{r@{}}{749}&\multicolumn{1}{@{.}l}{36}&\multicolumn{1}{l}{13/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{58}&\multicolumn{1}{@{.}l}{5 {\it 2}}&\multicolumn{1}{r@{}}{4}&\multicolumn{1}{@{.}l}{5\ensuremath{^{\hyperlink{AT46GAMMA1}{\ddagger}}} {\it 15}}&\multicolumn{1}{r@{}}{690}&\multicolumn{1}{@{.}l}{98 }&\multicolumn{1}{@{}l}{(11/2\ensuremath{^{-}})}&\multicolumn{1}{l}{[E1]}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{424 {\it 7}}&\parbox[t][0.3cm]{4.4482207cm}{\raggedright B(E1)(W.u.)=2.1\ensuremath{\times}10\ensuremath{^{\textnormal{$-$6}}} \textit{+8{\textminus}7}\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{114}&\multicolumn{1}{@{.}l}{1 {\it 2}}&\multicolumn{1}{r@{}}{9}&\multicolumn{1}{@{.}l}{6\ensuremath{^{\hyperlink{AT46GAMMA1}{\ddagger}}} {\it 9}}&\multicolumn{1}{r@{}}{635}&\multicolumn{1}{@{.}l}{17 }&\multicolumn{1}{@{}l}{13/2\ensuremath{^{-}}}&\multicolumn{1}{l}{[E1]}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{332 {\it 5}}&\parbox[t][0.3cm]{4.4482207cm}{\raggedright B(E1)(W.u.)=6.1\ensuremath{\times}10\ensuremath{^{\textnormal{$-$7}}} \textit{+11{\textminus}9}\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{749}&\multicolumn{1}{@{.}l}{3 {\it 2}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{AT46GAMMA1}{\ddagger}}} {\it 12}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{@{}l}{9/2\ensuremath{^{-}}}&\multicolumn{1}{l}{(M2)}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{1204 {\it 17}}&\parbox[t][0.3cm]{4.4482207cm}{\raggedright B(M2)(W.u.)=0.183 \textit{+18{\textminus}16}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{4.4482207cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}=0.05 \textit{3} in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1983Dy02,B}{1983Dy02}.\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{4.4482207cm}{\raggedright {\ }{\ }{\ }E3 admixtures are possible.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{804}&\multicolumn{1}{@{.}l}{2}&\multicolumn{1}{l}{(5/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{172}&\multicolumn{1}{@{.}l}{5 {\it 4}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{631}&\multicolumn{1}{@{.}l}{8 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
\multicolumn{1}{r@{}}{1065}&\multicolumn{1}{@{.}l}{2}&\multicolumn{1}{l}{(7/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{433}&\multicolumn{1}{@{.}l}{3 {\it 4}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{631}&\multicolumn{1}{@{.}l}{8 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}}}&\multicolumn{1}{l}{(E2)}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0478 {\it 7}}&\parbox[t][0.3cm]{4.4482207cm}{\raggedright Mult.: R=1.21 \textit{5} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2014Au03,B}{2014Au03}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1228}&\multicolumn{1}{@{.}l}{96}&\multicolumn{1}{l}{17/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{593}&\multicolumn{1}{@{.}l}{8 {\it 2}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{635}&\multicolumn{1}{@{.}l}{17 }&\multicolumn{1}{@{}l}{13/2\ensuremath{^{-}}}&\multicolumn{1}{l}{(E2)}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{02262 {\it 32}}&\parbox[t][0.3cm]{4.4482207cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}=0.16 \textit{5} in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1983Dy02,B}{1983Dy02}.\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{4.4482207cm}{\raggedright {\ }{\ }{\ }Other: A2=+0.08 \textit{2}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{4.4482207cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2015Au01,B}{2015Au01}).\vspace{0.1cm}}&\\
\end{longtable}
\begin{textblock}{29}(0,27.3)
Continued on next page (footnotes at end of table)
\end{textblock}
\clearpage
\begin{longtable}{ccccccccc@{}ccccc@{\extracolsep{\fill}}c}
\\[-.4cm]
\multicolumn{15}{c}{{\bf \small \underline{Adopted \hyperlink{201AT_LEVEL}{Levels}, \hyperlink{201AT_GAMMA}{Gammas} (continued)}}}\\
\multicolumn{15}{c}{~}\\
\multicolumn{15}{c}{\underline{$\gamma$($^{201}$At) (continued)}}\\
\multicolumn{15}{c}{~~~}\\
\multicolumn{2}{c}{E\ensuremath{_{i}}(level)}&J\ensuremath{^{\pi}_{i}}&\multicolumn{2}{c}{E\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{AT46GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{I\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{AT46GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{E\ensuremath{_{f}}}&J\ensuremath{^{\pi}_{f}}&Mult.\ensuremath{^{\hyperlink{AT46GAMMA0}{\dagger}}}&\multicolumn{2}{c}{\ensuremath{\alpha}\ensuremath{^{\hyperlink{AT46GAMMA2}{\#}}}}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endhead
\multicolumn{1}{r@{}}{1261}&\multicolumn{1}{@{.}l}{29}&\multicolumn{1}{l}{15/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{511}&\multicolumn{1}{@{.}l}{8 {\it 2}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{749}&\multicolumn{1}{@{.}l}{36 }&\multicolumn{1}{@{}l}{13/2\ensuremath{^{+}}}&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{5.4838004cm}{\raggedright Mult.: A2={\textminus}0.14 \textit{3} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2015Au01,B}{2015Au01}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1288}&\multicolumn{1}{@{.}l}{6}&\multicolumn{1}{l}{(9/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{223}&\multicolumn{1}{@{.}l}{3 {\it 4}}&\multicolumn{1}{r@{}}{41}&\multicolumn{1}{@{.}l}{5 {\it 24}}&\multicolumn{1}{r@{}}{1065}&\multicolumn{1}{@{.}l}{2 }&\multicolumn{1}{@{}l}{(7/2\ensuremath{^{+}})}&\multicolumn{1}{l}{(M1)}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{245 {\it 19}}&\parbox[t][0.3cm]{5.4838004cm}{\raggedright Mult.: A2={\textminus}0.29 \textit{7} and R=0.74 \textit{12}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{5.4838004cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2014Au03,B}{2014Au03}).\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{484}&\multicolumn{1}{@{.}l}{5 {\it 4}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{ }l}{{\it 7}}&\multicolumn{1}{r@{}}{804}&\multicolumn{1}{@{.}l}{2 }&\multicolumn{1}{@{}l}{(5/2\ensuremath{^{+}})}&\multicolumn{1}{l}{(E2)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0362 {\it 5}}&\parbox[t][0.3cm]{5.4838004cm}{\raggedright Mult.: A2=+0.70 \textit{6} and R=1.4 \textit{2}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{5.4838004cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2014Au03,B}{2014Au03}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1494}&\multicolumn{1}{@{.}l}{85}&\multicolumn{1}{l}{17/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{233}&\multicolumn{1}{@{.}l}{4 {\it 2}}&\multicolumn{1}{r@{}}{19}&\multicolumn{1}{@{.}l}{6 {\it 5}}&\multicolumn{1}{r@{}}{1261}&\multicolumn{1}{@{.}l}{29 }&\multicolumn{1}{@{}l}{15/2\ensuremath{^{+}}}&\multicolumn{1}{l}{(M1)}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{101 {\it 16}}&\parbox[t][0.3cm]{5.4838004cm}{\raggedright Mult.: A2={\textminus}0.05 \textit{2} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2015Au01,B}{2015Au01}).\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{745}&\multicolumn{1}{@{.}l}{5 {\it 2}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{ }l}{{\it 3}}&\multicolumn{1}{r@{}}{749}&\multicolumn{1}{@{.}l}{36 }&\multicolumn{1}{@{}l}{13/2\ensuremath{^{+}}}&\multicolumn{1}{l}{(E2)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{01389 {\it 19}}&\parbox[t][0.3cm]{5.4838004cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}=0.24 \textit{9} in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1983Dy02,B}{1983Dy02}. Other:\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{5.4838004cm}{\raggedright {\ }{\ }{\ }A2=+0.4 \textit{2} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2015Au01,B}{2015Au01}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1613}&\multicolumn{1}{@{.}l}{3}&\multicolumn{1}{l}{(11/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{548}&\multicolumn{1}{@{.}l}{1 {\it 4}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{1065}&\multicolumn{1}{@{.}l}{2 }&\multicolumn{1}{@{}l}{(7/2\ensuremath{^{+}})}&\multicolumn{1}{l}{(E2)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0271 {\it 4}}&\parbox[t][0.3cm]{5.4838004cm}{\raggedright Mult.: A2=+0.5 \textit{2} and R=1.26 \textit{12}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{5.4838004cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2014Au03,B}{2014Au03}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1625}&\multicolumn{1}{@{.}l}{34}&\multicolumn{1}{l}{17/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{130}&\multicolumn{1}{@{.}l}{3 {\it 2}}&\multicolumn{1}{r@{}}{20}&\multicolumn{1}{@{.}l}{1 {\it 21}}&\multicolumn{1}{r@{}}{1494}&\multicolumn{1}{@{.}l}{85 }&\multicolumn{1}{@{}l}{17/2\ensuremath{^{+}}}&\multicolumn{1}{l}{(M1)}&\multicolumn{1}{r@{}}{5}&\multicolumn{1}{@{.}l}{69 {\it 8}}&\parbox[t][0.3cm]{5.4838004cm}{\raggedright Mult.: A2={\textminus}0.36 \textit{7} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2015Au01,B}{2015Au01}).\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{364}&\multicolumn{1}{@{.}l}{1 {\it 3}}&\multicolumn{1}{r@{}}{33}&\multicolumn{1}{@{.}l}{5 {\it 16}}&\multicolumn{1}{r@{}}{1261}&\multicolumn{1}{@{.}l}{29 }&\multicolumn{1}{@{}l}{15/2\ensuremath{^{+}}}&\multicolumn{1}{l}{(M1)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{325 {\it 5}}&\parbox[t][0.3cm]{5.4838004cm}{\raggedright Mult.: A2={\textminus}0.11 \textit{3} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2015Au01,B}{2015Au01}).\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{876}&\multicolumn{1}{@{.}l}{1 {\it 2}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{ }l}{{\it 3}}&\multicolumn{1}{r@{}}{749}&\multicolumn{1}{@{.}l}{36 }&\multicolumn{1}{@{}l}{13/2\ensuremath{^{+}}}&\multicolumn{1}{l}{(E2)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{01002 {\it 14}}&\parbox[t][0.3cm]{5.4838004cm}{\raggedright Mult.: A2=+0.16 \textit{3} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2015Au01,B}{2015Au01}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1705}&\multicolumn{1}{@{.}l}{03}&\multicolumn{1}{l}{21/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{476}&\multicolumn{1}{@{.}l}{2 {\it 2}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{1228}&\multicolumn{1}{@{.}l}{96 }&\multicolumn{1}{@{}l}{17/2\ensuremath{^{-}}}&\multicolumn{1}{l}{(E2)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0378 {\it 5}}&\parbox[t][0.3cm]{5.4838004cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}=0.22 \textit{7} in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1983Dy02,B}{1983Dy02}. Other:\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{5.4838004cm}{\raggedright {\ }{\ }{\ }A2=+0.13 \textit{4} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2015Au01,B}{2015Au01}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1790}&\multicolumn{1}{@{.}l}{3}&&\multicolumn{1}{r@{}}{295}&\multicolumn{1}{@{.}l}{5 {\it 3}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{1494}&\multicolumn{1}{@{.}l}{85 }&\multicolumn{1}{@{}l}{17/2\ensuremath{^{+}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
\multicolumn{1}{r@{}}{1856}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{l}{(13/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{242}&\multicolumn{1}{@{.}l}{8 {\it 7}}&\multicolumn{1}{r@{}}{15}&\multicolumn{1}{@{.}l}{4 {\it 14}}&\multicolumn{1}{r@{}}{1613}&\multicolumn{1}{@{.}l}{3 }&\multicolumn{1}{@{}l}{(11/2\ensuremath{^{+}})}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
&&&\multicolumn{1}{r@{}}{567}&\multicolumn{1}{@{.}l}{3 {\it 4}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{ }l}{{\it 5}}&\multicolumn{1}{r@{}}{1288}&\multicolumn{1}{@{.}l}{6 }&\multicolumn{1}{@{}l}{(9/2\ensuremath{^{+}})}&\multicolumn{1}{l}{(E2)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{02506 {\it 35}}&\parbox[t][0.3cm]{5.4838004cm}{\raggedright Mult.: A2=+0.5 \textit{3} and R=1.2 \textit{3}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{5.4838004cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2014Au03,B}{2014Au03}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1921}&\multicolumn{1}{@{.}l}{30}&\multicolumn{1}{l}{21/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{216}&\multicolumn{1}{@{.}l}{3 {\it 3}}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{3\ensuremath{^{\hyperlink{AT46GAMMA1}{\ddagger}}} {\it 6}}&\multicolumn{1}{r@{}}{1705}&\multicolumn{1}{@{.}l}{03 }&\multicolumn{1}{@{}l}{21/2\ensuremath{^{-}}}&\multicolumn{1}{l}{[E1]}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0696 {\it 10}}&&\\
&&&\multicolumn{1}{r@{}}{295}&\multicolumn{1}{@{.}l}{9 {\it 2}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{ }l}{{\it 5}}&\multicolumn{1}{r@{}}{1625}&\multicolumn{1}{@{.}l}{34 }&\multicolumn{1}{@{}l}{17/2\ensuremath{^{+}}}&\multicolumn{1}{l}{(E2)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{1374 {\it 19}}&\parbox[t][0.3cm]{5.4838004cm}{\raggedright Mult.: A2=+0.11 \textit{4} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2015Au01,B}{2015Au01}).\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{426}&\multicolumn{1}{@{.}l}{5 {\it 2}}&\multicolumn{1}{r@{}}{98}&\multicolumn{1}{@{ }l}{{\it 5}}&\multicolumn{1}{r@{}}{1494}&\multicolumn{1}{@{.}l}{85 }&\multicolumn{1}{@{}l}{17/2\ensuremath{^{+}}}&\multicolumn{1}{l}{(E2)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0497 {\it 7}}&\parbox[t][0.3cm]{5.4838004cm}{\raggedright Mult.: A2=+0.14 \textit{7} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2015Au01,B}{2015Au01}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1980}&\multicolumn{1}{@{.}l}{6}&\multicolumn{1}{l}{23/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{275}&\multicolumn{1}{@{.}l}{5 {\it 2}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{1705}&\multicolumn{1}{@{.}l}{03 }&\multicolumn{1}{@{}l}{21/2\ensuremath{^{-}}}&\multicolumn{1}{l}{(M1)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{695 {\it 10}}&\parbox[t][0.3cm]{5.4838004cm}{\raggedright Mult.: A2={\textminus}0.47 \textit{2} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2015Au01,B}{2015Au01}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2004}&\multicolumn{1}{@{.}l}{3}&\multicolumn{1}{l}{23/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{83}&\multicolumn{1}{@{.}l}{0 {\it 4}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{AT46GAMMA1}{\ddagger}}} {\it 33}}&\multicolumn{1}{r@{}}{1921}&\multicolumn{1}{@{.}l}{30 }&\multicolumn{1}{@{}l}{21/2\ensuremath{^{+}}}&\multicolumn{1}{l}{[M1]}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{99 {\it 8}}&&\\
&&&\multicolumn{1}{r@{}}{299}&\multicolumn{1}{@{.}l}{3 {\it 2}}&\multicolumn{1}{r@{}}{20}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{AT46GAMMA1}{\ddagger}}} {\it 3}}&\multicolumn{1}{r@{}}{1705}&\multicolumn{1}{@{.}l}{03 }&\multicolumn{1}{@{}l}{21/2\ensuremath{^{-}}}&\multicolumn{1}{l}{[E1]}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0325 {\it 5}}&&\\
\multicolumn{1}{r@{}}{2050}&\multicolumn{1}{@{.}l}{8}&\multicolumn{1}{l}{25/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{46}&\multicolumn{1}{@{.}l}{5 {\it 2}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{2004}&\multicolumn{1}{@{.}l}{3 }&\multicolumn{1}{@{}l}{23/2\ensuremath{^{+}}}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
\multicolumn{1}{r@{}}{2076}&\multicolumn{1}{@{.}l}{9}&\multicolumn{1}{l}{(23/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{371}&\multicolumn{1}{@{.}l}{9 {\it 4}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{1705}&\multicolumn{1}{@{.}l}{03 }&\multicolumn{1}{@{}l}{21/2\ensuremath{^{-}}}&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{5.4838004cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}={\textminus}0.1 \textit{1} in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1983Dy02,B}{1983Dy02}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2147}&\multicolumn{1}{@{.}l}{2}&\multicolumn{1}{l}{21/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{442}&\multicolumn{1}{@{.}l}{6 {\it 4}}&\multicolumn{1}{r@{}}{86}&\multicolumn{1}{@{ }l}{{\it 3}}&\multicolumn{1}{r@{}}{1705}&\multicolumn{1}{@{.}l}{03 }&\multicolumn{1}{@{}l}{21/2\ensuremath{^{-}}}&\multicolumn{1}{l}{(E2)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0453 {\it 6}}&\parbox[t][0.3cm]{5.4838004cm}{\raggedright Mult.: A2=+0.35 \textit{8} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2015Au01,B}{2015Au01});\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{5.4838004cm}{\raggedright {\ }{\ }{\ }consistent with \ensuremath{\Delta}J=0 transition.\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{917}&\multicolumn{1}{@{.}l}{8 {\it 4}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{ }l}{{\it 6}}&\multicolumn{1}{r@{}}{1228}&\multicolumn{1}{@{.}l}{96 }&\multicolumn{1}{@{}l}{17/2\ensuremath{^{-}}}&\multicolumn{1}{l}{(E2)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{00914 {\it 13}}&\parbox[t][0.3cm]{5.4838004cm}{\raggedright Mult.: A2=+0.37 \textit{6} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2015Au01,B}{2015Au01}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2232}&\multicolumn{1}{@{.}l}{1}&\multicolumn{1}{l}{(15/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{618}&\multicolumn{1}{@{.}l}{8 {\it 6}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{1613}&\multicolumn{1}{@{.}l}{3 }&\multicolumn{1}{@{}l}{(11/2\ensuremath{^{+}})}&\multicolumn{1}{l}{(E2)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{02065 {\it 29}}&\parbox[t][0.3cm]{5.4838004cm}{\raggedright Mult.: A2=+0.6 \textit{4} and R=1.1 \textit{4}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{5.4838004cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2014Au03,B}{2014Au03}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2319}&\multicolumn{1}{@{.}l}{8}&\multicolumn{1}{l}{29/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{269}&\multicolumn{1}{@{.}l}{0 {\it 2}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{AT46GAMMA1}{\ddagger}}} {\it 9}}&\multicolumn{1}{r@{}}{2050}&\multicolumn{1}{@{.}l}{8 }&\multicolumn{1}{@{}l}{25/2\ensuremath{^{+}}}&\multicolumn{1}{l}{E2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{1842 {\it 26}}&\parbox[t][0.3cm]{5.4838004cm}{\raggedright B(E2)(W.u.)=1.27\ensuremath{\times}10\ensuremath{^{\textnormal{$-$3}}} \textit{6}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{5.4838004cm}{\raggedright Mult.: From K/(L+M+...)exp=0.93 \textit{5}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{5.4838004cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2015Au01,B}{2015Au01}).\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{339}&\multicolumn{1}{@{.}l}{2 {\it 2}}&\multicolumn{1}{r@{}}{10}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{AT46GAMMA1}{\ddagger}}} {\it 3}}&\multicolumn{1}{r@{}}{1980}&\multicolumn{1}{@{.}l}{6 }&\multicolumn{1}{@{}l}{23/2\ensuremath{^{-}}}&\multicolumn{1}{l}{E3}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{464 {\it 7}}&\parbox[t][0.3cm]{5.4838004cm}{\raggedright B(E3)(W.u.)=22 \textit{6}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{5.4838004cm}{\raggedright Mult.: From K/(L+M+...)exp=0.45 \textit{4}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{5.4838004cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2015Au01,B}{2015Au01}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2518}&\multicolumn{1}{@{.}l}{9}&\multicolumn{1}{l}{25/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{371}&\multicolumn{1}{@{.}l}{7 {\it 4}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{2147}&\multicolumn{1}{@{.}l}{2 }&\multicolumn{1}{@{}l}{21/2\ensuremath{^{-}}}&\multicolumn{1}{l}{(E2)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0715 {\it 10}}&\parbox[t][0.3cm]{5.4838004cm}{\raggedright Mult.: A2=+0.16 \textit{6} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2015Au01,B}{2015Au01}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2637}&\multicolumn{1}{@{.}l}{6}&\multicolumn{1}{l}{(25/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{716}&\multicolumn{1}{@{.}l}{3 {\it 4}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{1921}&\multicolumn{1}{@{.}l}{30 }&\multicolumn{1}{@{}l}{21/2\ensuremath{^{+}}}&\multicolumn{1}{l}{(E2)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{01509 {\it 21}}&\parbox[t][0.3cm]{5.4838004cm}{\raggedright Mult.: A2=+0.46 \textit{4}\hphantom{a}(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2015Au01,B}{2015Au01}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2990}&\multicolumn{1}{@{.}l}{2}&\multicolumn{1}{l}{(23/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{1068}&\multicolumn{1}{@{.}l}{9\ensuremath{^{\hyperlink{AT46GAMMA3}{@}}} {\it 4}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{1921}&\multicolumn{1}{@{.}l}{30 }&\multicolumn{1}{@{}l}{21/2\ensuremath{^{+}}}&\multicolumn{1}{l}{D}&&&\parbox[t][0.3cm]{5.4838004cm}{\raggedright A2={\textminus}0.47 \textit{5} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2015Au01,B}{2015Au01}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3135}&\multicolumn{1}{@{.}l}{2}&\multicolumn{1}{l}{(25/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{145}&\multicolumn{1}{@{.}l}{0 {\it 4}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{2990}&\multicolumn{1}{@{.}l}{2 }&\multicolumn{1}{@{}l}{(23/2\ensuremath{^{-}})}&\multicolumn{1}{l}{(M1)}&\multicolumn{1}{r@{}}{4}&\multicolumn{1}{@{.}l}{20 {\it 7}}&\parbox[t][0.3cm]{5.4838004cm}{\raggedright Mult.: A2={\textminus}0.5 \textit{2} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2015Au01,B}{2015Au01}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3219}&\multicolumn{1}{@{.}l}{2}&&\multicolumn{1}{r@{}}{581}&\multicolumn{1}{@{.}l}{6 {\it 4}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{2637}&\multicolumn{1}{@{.}l}{6 }&\multicolumn{1}{@{}l}{(25/2\ensuremath{^{+}})}&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{5.4838004cm}{\raggedright Mult.: A2={\textminus}0.44 \textit{7} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2015Au01,B}{2015Au01}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3240}&\multicolumn{1}{@{.}l}{9}&\multicolumn{1}{l}{(33/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{921}&\multicolumn{1}{@{.}l}{1 {\it 4}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{2319}&\multicolumn{1}{@{.}l}{8 }&\multicolumn{1}{@{}l}{29/2\ensuremath{^{+}}}&\multicolumn{1}{l}{(E2)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{00908 {\it 13}}&\parbox[t][0.3cm]{5.4838004cm}{\raggedright Mult.: A2=+0.20 \textit{3} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2015Au01,B}{2015Au01}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3245}&\multicolumn{1}{@{.}l}{4}&\multicolumn{1}{l}{(29/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{607}&\multicolumn{1}{@{.}l}{8 {\it 4}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{2637}&\multicolumn{1}{@{.}l}{6 }&\multicolumn{1}{@{}l}{(25/2\ensuremath{^{+}})}&\multicolumn{1}{l}{(E2)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{02148 {\it 30}}&\parbox[t][0.3cm]{5.4838004cm}{\raggedright Mult.: A2=+0.35 \textit{8} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2015Au01,B}{2015Au01}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3369}&\multicolumn{1}{@{.}l}{6}&&\multicolumn{1}{r@{}}{1049}&\multicolumn{1}{@{.}l}{9\ensuremath{^{\hyperlink{AT46GAMMA3}{@}}} {\it 4}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{2319}&\multicolumn{1}{@{.}l}{8 }&\multicolumn{1}{@{}l}{29/2\ensuremath{^{+}}}&\multicolumn{1}{l}{D}&&&\parbox[t][0.3cm]{5.4838004cm}{\raggedright Mult.: A2={\textminus}0.7 \textit{3} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2015Au01,B}{2015Au01}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3379}&\multicolumn{1}{@{.}l}{6}&\multicolumn{1}{l}{(27/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{244}&\multicolumn{1}{@{.}l}{4 {\it 4}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{3135}&\multicolumn{1}{@{.}l}{2 }&\multicolumn{1}{@{}l}{(25/2\ensuremath{^{-}})}&\multicolumn{1}{l}{(M1)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{968 {\it 14}}&\parbox[t][0.3cm]{5.4838004cm}{\raggedright Mult.: A2={\textminus}0.59 \textit{9} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2015Au01,B}{2015Au01}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3504}&\multicolumn{1}{@{.}l}{4}&&\multicolumn{1}{r@{}}{135}&\multicolumn{1}{@{.}l}{0\ensuremath{^{\hyperlink{AT46GAMMA3}{@}}} {\it 4}}&\multicolumn{1}{r@{}}{29}&\multicolumn{1}{@{.}l}{4 {\it 24}}&\multicolumn{1}{r@{}}{3369}&\multicolumn{1}{@{.}l}{6 }&&&&&&\\
&&&\multicolumn{1}{r@{}}{263}&\multicolumn{1}{@{.}l}{6 {\it 4}}&\multicolumn{1}{r@{}}{71}&\multicolumn{1}{@{ }l}{{\it 12}}&\multicolumn{1}{r@{}}{3240}&\multicolumn{1}{@{.}l}{9 }&\multicolumn{1}{@{}l}{(33/2\ensuremath{^{+}})}&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{5.4838004cm}{\raggedright Mult.: A2={\textminus}0.26 \textit{9} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2015Au01,B}{2015Au01}).\vspace{0.1cm}}&\\
&&&\multicolumn{1}{r@{}}{1184}&\multicolumn{1}{@{.}l}{5 {\it 4}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{ }l}{{\it 12}}&\multicolumn{1}{r@{}}{2319}&\multicolumn{1}{@{.}l}{8 }&\multicolumn{1}{@{}l}{29/2\ensuremath{^{+}}}&\multicolumn{1}{l}{D,Q}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{5.4838004cm}{\raggedright Mult.: A2=+0.40 \textit{8} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2015Au01,B}{2015Au01}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3621}&\multicolumn{1}{@{.}l}{8}&&\multicolumn{1}{r@{}}{402}&\multicolumn{1}{@{.}l}{6 {\it 4}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{3219}&\multicolumn{1}{@{.}l}{2 }&&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{5.4838004cm}{\raggedright Mult.: A2={\textminus}0.7 \textit{2} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2015Au01,B}{2015Au01}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3666}&\multicolumn{1}{@{.}l}{5}&\multicolumn{1}{l}{(29/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{286}&\multicolumn{1}{@{.}l}{9 {\it 4}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{3379}&\multicolumn{1}{@{.}l}{6 }&\multicolumn{1}{@{}l}{(27/2\ensuremath{^{-}})}&\multicolumn{1}{l}{(M1)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{622 {\it 9}}&\parbox[t][0.3cm]{5.4838004cm}{\raggedright Mult.: A2={\textminus}0.47 \textit{3}\hphantom{a}(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2015Au01,B}{2015Au01}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3693}&\multicolumn{1}{@{.}l}{9}&&\multicolumn{1}{r@{}}{448}&\multicolumn{1}{@{.}l}{5 {\it 4}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{3245}&\multicolumn{1}{@{.}l}{4 }&\multicolumn{1}{@{}l}{(29/2\ensuremath{^{+}})}&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{5.4838004cm}{\raggedright Mult.: A2={\textminus}0.4 \textit{2} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2015Au01,B}{2015Au01}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3699}&\multicolumn{1}{@{.}l}{7}&&\multicolumn{1}{r@{}}{1379}&\multicolumn{1}{@{.}l}{9\ensuremath{^{\hyperlink{AT46GAMMA3}{@}}} {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{2319}&\multicolumn{1}{@{.}l}{8 }&\multicolumn{1}{@{}l}{29/2\ensuremath{^{+}}}&&&&&\\
\end{longtable}
\begin{textblock}{29}(0,27.3)
Continued on next page (footnotes at end of table)
\end{textblock}
\clearpage
\begin{longtable}{ccccccccc@{}ccccc@{\extracolsep{\fill}}c}
\\[-.4cm]
\multicolumn{15}{c}{{\bf \small \underline{Adopted \hyperlink{201AT_LEVEL}{Levels}, \hyperlink{201AT_GAMMA}{Gammas} (continued)}}}\\
\multicolumn{15}{c}{~}\\
\multicolumn{15}{c}{\underline{$\gamma$($^{201}$At) (continued)}}\\
\multicolumn{15}{c}{~~~}\\
\multicolumn{2}{c}{E\ensuremath{_{i}}(level)}&J\ensuremath{^{\pi}_{i}}&\multicolumn{2}{c}{E\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{AT46GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{I\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{AT46GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{E\ensuremath{_{f}}}&J\ensuremath{^{\pi}_{f}}&Mult.\ensuremath{^{\hyperlink{AT46GAMMA0}{\dagger}}}&\multicolumn{2}{c}{\ensuremath{\alpha}\ensuremath{^{\hyperlink{AT46GAMMA2}{\#}}}}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endhead
\multicolumn{1}{r@{}}{3779}&\multicolumn{1}{@{.}l}{1}&&\multicolumn{1}{r@{}}{538}&\multicolumn{1}{@{.}l}{2 {\it 4}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{3240}&\multicolumn{1}{@{.}l}{9 }&\multicolumn{1}{@{}l}{(33/2\ensuremath{^{+}})}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
\multicolumn{1}{r@{}}{3785}&\multicolumn{1}{@{.}l}{9}&&\multicolumn{1}{r@{}}{540}&\multicolumn{1}{@{.}l}{5 {\it 5}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{3245}&\multicolumn{1}{@{.}l}{4 }&\multicolumn{1}{@{}l}{(29/2\ensuremath{^{+}})}&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
\multicolumn{1}{r@{}}{3853}&\multicolumn{1}{@{.}l}{3}&&\multicolumn{1}{r@{}}{153}&\multicolumn{1}{@{.}l}{6\ensuremath{^{\hyperlink{AT46GAMMA3}{@}}} {\it 4}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{3699}&\multicolumn{1}{@{.}l}{7 }&&&&&&\\
\multicolumn{1}{r@{}}{3952}&\multicolumn{1}{@{.}l}{8}&&\multicolumn{1}{r@{}}{448}&\multicolumn{1}{@{.}l}{4 {\it 4}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{3504}&\multicolumn{1}{@{.}l}{4 }&&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{6.947561cm}{\raggedright Mult.: A2={\textminus}0.3 \textit{2} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2015Au01,B}{2015Au01}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3983}&\multicolumn{1}{@{.}l}{8}&\multicolumn{1}{l}{(31/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{317}&\multicolumn{1}{@{.}l}{3 {\it 4}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{3666}&\multicolumn{1}{@{.}l}{5 }&\multicolumn{1}{@{}l}{(29/2\ensuremath{^{-}})}&\multicolumn{1}{l}{(M1)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{472 {\it 7}}&\parbox[t][0.3cm]{6.947561cm}{\raggedright Mult.: A2={\textminus}0.81 \textit{9} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2015Au01,B}{2015Au01}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{4111}&\multicolumn{1}{@{.}l}{4}&&\multicolumn{1}{r@{}}{870}&\multicolumn{1}{@{.}l}{5\ensuremath{^{\hyperlink{AT46GAMMA3}{@}}} {\it 4}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{3240}&\multicolumn{1}{@{.}l}{9 }&\multicolumn{1}{@{}l}{(33/2\ensuremath{^{+}})}&&&&&\\
\multicolumn{1}{r@{}}{4159}&\multicolumn{1}{@{.}l}{0}&&\multicolumn{1}{r@{}}{206}&\multicolumn{1}{@{.}l}{2 {\it 4}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{3952}&\multicolumn{1}{@{.}l}{8 }&&\multicolumn{1}{l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&&\\
\multicolumn{1}{r@{}}{4256}&\multicolumn{1}{@{.}l}{1}&\multicolumn{1}{l}{(33/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{272}&\multicolumn{1}{@{.}l}{3 {\it 4}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{3983}&\multicolumn{1}{@{.}l}{8 }&\multicolumn{1}{@{}l}{(31/2\ensuremath{^{-}})}&\multicolumn{1}{l}{(M1)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{718 {\it 10}}&\parbox[t][0.3cm]{6.947561cm}{\raggedright Mult.: A2={\textminus}0.45 \textit{11} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2015Au01,B}{2015Au01}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{4454}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{l}{(35/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{197}&\multicolumn{1}{@{.}l}{9 {\it 4}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{4256}&\multicolumn{1}{@{.}l}{1 }&\multicolumn{1}{@{}l}{(33/2\ensuremath{^{-}})}&\multicolumn{1}{l}{(M1)}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{745 {\it 26}}&\parbox[t][0.3cm]{6.947561cm}{\raggedright Mult.: A2={\textminus}0.80 \textit{12} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2015Au01,B}{2015Au01}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{4789}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{l}{(37/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{335}&\multicolumn{1}{@{.}l}{0 {\it 4}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{4454}&\multicolumn{1}{@{.}l}{0 }&\multicolumn{1}{@{}l}{(35/2\ensuremath{^{-}})}&\multicolumn{1}{l}{(M1)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{407 {\it 6}}&\parbox[t][0.3cm]{6.947561cm}{\raggedright Mult.: A2={\textminus}0.66 \textit{4} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2015Au01,B}{2015Au01}).\vspace{0.1cm}}&\\
\end{longtable}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{AT46GAMMA0}{\dagger}}}} From \ensuremath{^{\textnormal{165}}}Ho(\ensuremath{^{\textnormal{40}}}Ar,4n\ensuremath{\gamma}), unless otherwise stated.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{AT46GAMMA1}{\ddagger}}}} Determined by the evaluator from I(\ensuremath{\gamma}+ce) in \ensuremath{^{\textnormal{165}}}Ho(\ensuremath{^{\textnormal{40}}}Ar,4n\ensuremath{\gamma}) and \ensuremath{\alpha}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{AT46GAMMA2}{\#}}}} Total theoretical internal conversion coefficients, calculated using the BrIcc code (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2008Ki07,B}{2008Ki07}) with Frozen orbital approximation}\\
\parbox[b][0.3cm]{17.7cm}{{\ }{\ }based on \ensuremath{\gamma}-ray energies, assigned multipolarities, and mixing ratios, unless otherwise specified.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{AT46GAMMA3}{@}}}} Placement of transition in the level scheme is uncertain.}\\
\vspace{0.5cm}
\clearpage
\begin{figure}[h]
\begin{center}
\includegraphics{201AT46-0.ps}\\
\end{center}
\end{figure}
\clearpage
\begin{figure}[h]
\begin{center}
\includegraphics{201AT46-1.ps}\\
\end{center}
\end{figure}
\clearpage
\begin{figure}[h]
\begin{center}
\includegraphics{201AT46-2.ps}\\
\end{center}
\end{figure}
\clearpage
\clearpage
\begin{figure}[h]
\begin{center}
\includegraphics{201AT46B-0.ps}\\
\end{center}
\end{figure}
\clearpage
\subsection[\hspace{-0.2cm}\ensuremath{^{\textnormal{205}}}Fr \ensuremath{\alpha} decay]{ }
\vspace{-27pt}
\vspace{0.3cm}
\hypertarget{FR47}{{\bf \small \underline{\ensuremath{^{\textnormal{205}}}Fr \ensuremath{\alpha} decay\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1974Ho27,B}{1974Ho27},\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1967Va20,B}{1967Va20}}}}\\
\vspace{4pt}
\vspace{8pt}
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}Parent: $^{205}$Fr: E=0; J$^{\pi}$=9/2\ensuremath{^{-}}; T$_{1/2}$=3.90 s {\it 7}; Q(\ensuremath{\alpha})=7054.7 {\it 24}; \%\ensuremath{\alpha} decay=98.5 {\it 4}
}\\
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}\ensuremath{^{205}}Fr-J$^{\pi}$,T$_{1/2}$: From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2020Ko17,B}{2020Ko17}.}\\
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}\ensuremath{^{205}}Fr-Q(\ensuremath{\alpha}): From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2021Wa16,B}{2021Wa16}.}\\
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}\ensuremath{^{205}}Fr-\%\ensuremath{\alpha} decay: From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2020Ko17,B}{2020Ko17}.}\\
\vspace{12pt}
\underline{$^{201}$At Levels}\\
\begin{longtable}{ccccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E(level)$^{}$}&J$^{\pi}$$^{{\hyperlink{AT47LEVEL0}{\dagger}}}$&\multicolumn{2}{c}{T$_{1/2}$$^{{\hyperlink{AT47LEVEL0}{\dagger}}}$}&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&
\endfirsthead
\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{l}{9/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{87}&\multicolumn{1}{@{.}l}{6 s {\it 13}}&\\
\end{longtable}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{AT47LEVEL0}{\dagger}}}} From Adopted Levels.}\\
\vspace{0.5cm}
\underline{\ensuremath{\alpha} radiations}\\
\begin{longtable}{ccccccccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E$\alpha^{{}}$}&\multicolumn{2}{c}{E(level)}&\multicolumn{2}{c}{I$\alpha^{{\hyperlink{AT47DECAY1}{\ddagger}}}$}&\multicolumn{2}{c}{HF$^{{\hyperlink{AT47DECAY0}{\dagger}}}$}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{6915}&\multicolumn{1}{@{.}l}{4 {\it 24}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{70 {\it 4}}&\parbox[t][0.3cm]{12.406561cm}{\raggedright E$\alpha$: Weighted average E\ensuremath{\alpha}=6910 keV \textit{20} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1964Gr04,B}{1964Gr04}), 6917 keV \textit{5} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1967Va20,B}{1967Va20}), 6912 keV \textit{5}\vspace{0.1cm}}&\\
&&&&&&&&\parbox[t][0.3cm]{12.406561cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1974Ho27,B}{1974Ho27}), 6917 keV \textit{5} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1981Ri04,B}{1981Ri04}), 6915 \textit{7} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1995Le04,B}{1995Le04}) and 6916 keV \textit{5} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2005De01,B}{2005De01}).\vspace{0.1cm}}&\\
\end{longtable}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{AT47DECAY0}{\dagger}}}} Using r\ensuremath{_{\textnormal{0}}}(\ensuremath{^{\textnormal{201}}}At)=1.516 \textit{3} from \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2020Si16,B}{2020Si16}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{AT47DECAY1}{\ddagger}}}} For absolute intensity per 100 decays, multiply by 0.985 \textit{4}.}\\
\vspace{0.5cm}
\clearpage
\subsection[\hspace{-0.2cm}\ensuremath{^{\textnormal{192}}}Pt(\ensuremath{^{\textnormal{14}}}N,5n\ensuremath{\gamma})]{ }
\vspace{-27pt}
\vspace{0.3cm}
\hypertarget{AT48}{{\bf \small \underline{\ensuremath{^{\textnormal{192}}}Pt(\ensuremath{^{\textnormal{14}}}N,5n\ensuremath{\gamma})\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1983Dy02,B}{1983Dy02}}}}\\
\vspace{4pt}
\vspace{8pt}
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}\ensuremath{^{\textnormal{192}}}Pt(\ensuremath{^{\textnormal{14}}}N,5n\ensuremath{\gamma}), E(\ensuremath{^{\textnormal{14}}}N)=85-100 MeV; Target: 3 mg/cm\ensuremath{^{\textnormal{2}}} thick, enriched to 57 \% in \ensuremath{^{\textnormal{192}}}Pt; Detectors: Ge(Li) with a typical}\\
\parbox[b][0.3cm]{17.7cm}{energy resolution (FWHM) of 2 keV at 1.33 MeV; Measured: excitation functions, \ensuremath{\gamma}(t), \ensuremath{\gamma}(\ensuremath{\theta}), \ensuremath{\gamma}\ensuremath{\gamma} coin (two Ge(Li) detectors);}\\
\parbox[b][0.3cm]{17.7cm}{Deduced: level scheme, \ensuremath{J^{\pi}}, T\ensuremath{_{\textnormal{1/2}}}.}\\
\vspace{12pt}
\underline{$^{201}$At Levels}\\
\begin{longtable}{cccccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E(level)$^{{\hyperlink{AT48LEVEL0}{\dagger}}}$}&J$^{\pi}$$^{{\hyperlink{AT48LEVEL1}{\ddagger}}}$&\multicolumn{2}{c}{T$_{1/2}$$^{}$}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0\ensuremath{^{{\hyperlink{AT48LEVEL2}{\#}}}}}&\multicolumn{1}{l}{9/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{87}&\multicolumn{1}{@{.}l}{6 s {\it 13}}&\parbox[t][0.3cm]{12.32934cm}{\raggedright J\ensuremath{^{\pi}},T\ensuremath{_{1/2}}: From Adopted Levels.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{634}&\multicolumn{1}{@{.}l}{90\ensuremath{^{{\hyperlink{AT48LEVEL3}{@}}}} {\it 20}}&\multicolumn{1}{l}{(13/2\ensuremath{^{-}})}&&&&\\
\multicolumn{1}{r@{}}{749}&\multicolumn{1}{@{.}l}{0\ensuremath{^{{\hyperlink{AT48LEVEL4}{\&}}}} {\it 3}}&\multicolumn{1}{l}{(13/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{15}&\multicolumn{1}{@{.}l}{9 ns {\it 14}}&\parbox[t][0.3cm]{12.32934cm}{\raggedright T\ensuremath{_{1/2}}: From 749.0\ensuremath{\gamma}(t) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1983Dy02,B}{1983Dy02}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1228}&\multicolumn{1}{@{.}l}{9\ensuremath{^{{\hyperlink{AT48LEVEL5}{a}}}} {\it 4}}&\multicolumn{1}{l}{(17/2\ensuremath{^{-}})}&&&&\\
\multicolumn{1}{r@{}}{1494}&\multicolumn{1}{@{.}l}{6\ensuremath{^{{\hyperlink{AT48LEVEL6}{b}}}} {\it 5}}&\multicolumn{1}{l}{(17/2\ensuremath{^{+}})}&&&&\\
\multicolumn{1}{r@{}}{1705}&\multicolumn{1}{@{.}l}{5\ensuremath{^{{\hyperlink{AT48LEVEL7}{c}}}} {\it 5}}&\multicolumn{1}{l}{(21/2\ensuremath{^{-}})}&&&&\\
\multicolumn{1}{r@{}}{1790}&\multicolumn{1}{@{.}l}{1 {\it 6}}&&&&&\\
\multicolumn{1}{r@{}}{1921}&\multicolumn{1}{@{.}l}{3 {\it 6}}&&&&&\\
\multicolumn{1}{r@{}}{2077}&\multicolumn{1}{@{.}l}{4\ensuremath{^{{\hyperlink{AT48LEVEL8}{d}}}} {\it 7}}&\multicolumn{1}{l}{(23/2\ensuremath{^{-}})}&&&\parbox[t][0.3cm]{12.32934cm}{\raggedright J\ensuremath{^{\pi}}: Systematics in neighboring nuclei suggests negative parity.\vspace{0.1cm}}&\\
\end{longtable}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{AT48LEVEL0}{\dagger}}}} From a least-squares fit to E\ensuremath{\gamma}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{AT48LEVEL1}{\ddagger}}}} From deduced transition multipolarities, unless otherwise stated.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{AT48LEVEL2}{\#}}}} Configuration=\ensuremath{\pi} h\ensuremath{_{\textnormal{9/2}}^{\textnormal{+1}}}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{AT48LEVEL3}{@}}}} Configuration=\ensuremath{\pi} (h\ensuremath{_{\textnormal{9/2}}^{\textnormal{+1}}})\ensuremath{\otimes}2\ensuremath{^{\textnormal{+}}}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{AT48LEVEL4}{\&}}}} Configuration=\ensuremath{\pi} i\ensuremath{_{\textnormal{13/2}}^{\textnormal{+1}}}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{AT48LEVEL5}{a}}}} Configuration=\ensuremath{\pi} (h\ensuremath{_{\textnormal{9/2}}^{\textnormal{+1}}})\ensuremath{\otimes}4\ensuremath{^{\textnormal{+}}}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{AT48LEVEL6}{b}}}} Configuration=\ensuremath{\pi} (i\ensuremath{_{\textnormal{13/2}}^{\textnormal{+1}}})\ensuremath{\otimes}2\ensuremath{^{\textnormal{+}}}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{AT48LEVEL7}{c}}}} Configuration=\ensuremath{\pi} (h\ensuremath{_{\textnormal{9/2}}^{\textnormal{+3}}})\ensuremath{_{\textnormal{21/2$-$}}}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{AT48LEVEL8}{d}}}} Configuration=\ensuremath{\pi} (h\ensuremath{_{\textnormal{9/2}}^{\textnormal{+2}}})\ensuremath{_{\textnormal{8$-$}}},f\ensuremath{_{\textnormal{7/2}}^{\textnormal{+1}}}).}\\
\vspace{0.5cm}
\underline{$\gamma$($^{201}$At)}\\
\begin{longtable}{ccccccccc@{}ccccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{AT48GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{I\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{AT48GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{E\ensuremath{_{i}}(level)}&J\ensuremath{^{\pi}_{i}}&\multicolumn{2}{c}{E\ensuremath{_{f}}}&J\ensuremath{^{\pi}_{f}}&Mult.\ensuremath{^{\hyperlink{AT48GAMMA0}{\dagger}}}&\multicolumn{2}{c}{\ensuremath{\alpha}\ensuremath{^{\hyperlink{AT48GAMMA2}{\#}}}}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{(114}&\multicolumn{1}{@{.}l}{1)}&\multicolumn{1}{r@{}}{$\approx$15}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{749}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{l}{(13/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{634}&\multicolumn{1}{@{.}l}{90 }&\multicolumn{1}{@{}l}{(13/2\ensuremath{^{-}})}&\multicolumn{1}{l}{[E1]}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{332 {\it 5}}&\parbox[t][0.3cm]{5.954601cm}{\raggedright E\ensuremath{_{\gamma}}: Not observed directly, but required by\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{5.954601cm}{\raggedright {\ }{\ }{\ }the out-of-beam coincidence relationship.\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{5.954601cm}{\raggedright {\ }{\ }{\ }E\ensuremath{\gamma} is from level energy differences.\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{5.954601cm}{\raggedright I\ensuremath{_{\gamma}}: Estimated from the reported 20\%\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{5.954601cm}{\raggedright {\ }{\ }{\ }out-of-beam intensity for the 634.9\ensuremath{\gamma}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{295}&\multicolumn{1}{@{.}l}{5 {\it 3}}&\multicolumn{1}{r@{}}{15}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{AT48GAMMA1}{\ddagger}}} {\it 15}}&\multicolumn{1}{r@{}}{1790}&\multicolumn{1}{@{.}l}{1}&&\multicolumn{1}{r@{}}{1494}&\multicolumn{1}{@{.}l}{6 }&\multicolumn{1}{@{}l}{(17/2\ensuremath{^{+}})}&&&&&\\
\multicolumn{1}{r@{}}{371}&\multicolumn{1}{@{.}l}{9 {\it 4}}&\multicolumn{1}{r@{}}{25}&\multicolumn{1}{@{ }l}{{\it 8}}&\multicolumn{1}{r@{}}{2077}&\multicolumn{1}{@{.}l}{4}&\multicolumn{1}{l}{(23/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{1705}&\multicolumn{1}{@{.}l}{5 }&\multicolumn{1}{@{}l}{(21/2\ensuremath{^{-}})}&\multicolumn{1}{l}{D}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{5.954601cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}={\textminus}0.1 \textit{1}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{426}&\multicolumn{1}{@{.}l}{7 {\it 3}}&\multicolumn{1}{r@{}}{26}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{AT48GAMMA1}{\ddagger}}} {\it 8}}&\multicolumn{1}{r@{}}{1921}&\multicolumn{1}{@{.}l}{3}&&\multicolumn{1}{r@{}}{1494}&\multicolumn{1}{@{.}l}{6 }&\multicolumn{1}{@{}l}{(17/2\ensuremath{^{+}})}&&&&&\\
\multicolumn{1}{r@{}}{476}&\multicolumn{1}{@{.}l}{6 {\it 3}}&\multicolumn{1}{r@{}}{38}&\multicolumn{1}{@{ }l}{{\it 5}}&\multicolumn{1}{r@{}}{1705}&\multicolumn{1}{@{.}l}{5}&\multicolumn{1}{l}{(21/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{1228}&\multicolumn{1}{@{.}l}{9 }&\multicolumn{1}{@{}l}{(17/2\ensuremath{^{-}})}&\multicolumn{1}{l}{(E2)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0377 {\it 5}}&\parbox[t][0.3cm]{5.954601cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}=0.22 \textit{7}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{594}&\multicolumn{1}{@{.}l}{0 {\it 3}}&\multicolumn{1}{r@{}}{59}&\multicolumn{1}{@{ }l}{{\it 2}}&\multicolumn{1}{r@{}}{1228}&\multicolumn{1}{@{.}l}{9}&\multicolumn{1}{l}{(17/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{634}&\multicolumn{1}{@{.}l}{90 }&\multicolumn{1}{@{}l}{(13/2\ensuremath{^{-}})}&\multicolumn{1}{l}{(E2)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{02260 {\it 32}}&\parbox[t][0.3cm]{5.954601cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}=0.16 \textit{5}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{634}&\multicolumn{1}{@{.}l}{9 {\it 2}}&\multicolumn{1}{r@{}}{85}&\multicolumn{1}{@{ }l}{{\it 10}}&\multicolumn{1}{r@{}}{634}&\multicolumn{1}{@{.}l}{90}&\multicolumn{1}{l}{(13/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{@{}l}{9/2\ensuremath{^{-}}}&\multicolumn{1}{l}{(E2)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{01952 {\it 27}}&\parbox[t][0.3cm]{5.954601cm}{\raggedright I\ensuremath{_{\gamma}}: Estimated from the pulsed beam data.\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{5.954601cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}=0.11 \textit{3}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{745}&\multicolumn{1}{@{.}l}{6 {\it 3}}&\multicolumn{1}{r@{}}{57}&\multicolumn{1}{@{ }l}{{\it 4}}&\multicolumn{1}{r@{}}{1494}&\multicolumn{1}{@{.}l}{6}&\multicolumn{1}{l}{(17/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{749}&\multicolumn{1}{@{.}l}{0 }&\multicolumn{1}{@{}l}{(13/2\ensuremath{^{+}})}&\multicolumn{1}{l}{(E2)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{01389 {\it 19}}&\parbox[t][0.3cm]{5.954601cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}=0.24 \textit{9}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{749}&\multicolumn{1}{@{.}l}{0 {\it 3}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{ }l}{{\it 2}}&\multicolumn{1}{r@{}}{749}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{l}{(13/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{@{}l}{9/2\ensuremath{^{-}}}&\multicolumn{1}{l}{(M2)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{1206 {\it 17}}&\parbox[t][0.3cm]{5.954601cm}{\raggedright Mult.: A\ensuremath{_{\textnormal{2}}}=0.05 \textit{3}. E3 admixtures are\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{5.954601cm}{\raggedright {\ }{\ }{\ }possible.\vspace{0.1cm}}&\\
\end{longtable}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{AT48GAMMA0}{\dagger}}}} From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1983Dy02,B}{1983Dy02}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{AT48GAMMA1}{\ddagger}}}} Estimated by the authors from the \ensuremath{\gamma}\ensuremath{\gamma} coin data.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{AT48GAMMA2}{\#}}}} Total theoretical internal conversion coefficients, calculated using the BrIcc code (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2008Ki07,B}{2008Ki07}) with Frozen orbital approximation}\\
\parbox[b][0.3cm]{17.7cm}{{\ }{\ }based on \ensuremath{\gamma}-ray energies, assigned multipolarities, and mixing ratios, unless otherwise specified.}\\
\vspace{0.5cm}
\clearpage
\begin{figure}[h]
\begin{center}
\includegraphics{201AT48-0.ps}\\
\end{center}
\end{figure}
\clearpage
\subsection[\hspace{-0.2cm}\ensuremath{^{\textnormal{165}}}Ho(\ensuremath{^{\textnormal{40}}}Ar,4n\ensuremath{\gamma})]{ }
\vspace{-27pt}
\vspace{0.3cm}
\hypertarget{AT49}{{\bf \small \underline{\ensuremath{^{\textnormal{165}}}Ho(\ensuremath{^{\textnormal{40}}}Ar,4n\ensuremath{\gamma})\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2014Au03,B}{2014Au03},\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2015Au01,B}{2015Au01}}}}\\
\vspace{4pt}
\vspace{8pt}
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2014Au03,B}{2014Au03},\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2015Au01,B}{2015Au01}: \ensuremath{^{\textnormal{40}}}Ar\ensuremath{^{\textnormal{8+}}}, E=205{\textminus}MeV, I=11 pnA beam from the K-130 cyclotron at the University of Jyvaskyla Accelerator}\\
\parbox[b][0.3cm]{17.7cm}{Lab. Target: self-supporting 350{\textminus}\ensuremath{\mu}g/cm\ensuremath{^{\textnormal{2}}}\hphantom{a}thick \ensuremath{^{\textnormal{165}}}Ho. Detectors: The JUROGAM2 array consisting of Compton-suppressed 24}\\
\parbox[b][0.3cm]{17.7cm}{Clover and 15 Phase-1 and GASP type HPGe detectors. The fusion-evaporation residues (recoils) were separated from the primary}\\
\parbox[b][0.3cm]{17.7cm}{beam and other unwanted particles using the gas-filled recoil separator RITU and studied at the focal plane using the GREAT}\\
\parbox[b][0.3cm]{17.7cm}{spectrometer. The recoils were implanted onto 300 \ensuremath{\mu}m DSSD surrounded by 28 Si PIN diodes to measure the \ensuremath{\alpha} particle and ce}\\
\parbox[b][0.3cm]{17.7cm}{energies. \ensuremath{\gamma} rays at the focal plane were measured using three Clover and one planar type HPGe detector. Measured: E\ensuremath{\gamma}, I\ensuremath{\gamma}, ce,}\\
\parbox[b][0.3cm]{17.7cm}{\ensuremath{\gamma}\ensuremath{\gamma} coin, \ensuremath{\gamma}-recoil e\ensuremath{^{-}}coin, recoil e\ensuremath{^{-}} tagged prompt \ensuremath{\gamma}, recoil e\ensuremath{^{-}} tagged prompt \ensuremath{\gamma}-delayed \ensuremath{\gamma} coin, recoil e\ensuremath{^{-}} tagged prompt \ensuremath{\gamma}-prompt}\\
\parbox[b][0.3cm]{17.7cm}{\ensuremath{\gamma} coin, \ensuremath{\gamma}(\ensuremath{\theta}), and isomer half-life. Deduced levels, isomer, J, \ensuremath{\pi}, multipolarity, configuration, bands.}\\
\vspace{12pt}
\underline{$^{201}$At Levels}\\
\begin{longtable}{cccccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E(level)$^{{\hyperlink{AT49LEVEL0}{\dagger}}}$}&J$^{\pi}$$^{{\hyperlink{AT49LEVEL1}{\ddagger}}}$&\multicolumn{2}{c}{T$_{1/2}$$^{}$}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{l}{9/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{87}&\multicolumn{1}{@{.}l}{6 s {\it 13}}&\parbox[t][0.3cm]{12.38407cm}{\raggedright J\ensuremath{^{\pi}},T\ensuremath{_{1/2}}: From Adopted Levels.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{190}&\multicolumn{1}{@{.}l}{10 {\it 10}}&\multicolumn{1}{l}{7/2\ensuremath{^{-}}}&&&\parbox[t][0.3cm]{12.38407cm}{\raggedright Possible configuration=\ensuremath{\pi} f\ensuremath{_{\textnormal{7/2}}^{\textnormal{+1}}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{459}&\multicolumn{1}{@{.}l}{20 {\it 14}}&\multicolumn{1}{l}{1/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{45}&\multicolumn{1}{@{ }l}{ms {\it 3}}&\parbox[t][0.3cm]{12.38407cm}{\raggedright \%IT=100\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{12.38407cm}{\raggedright Proposed configuration=\ensuremath{\pi} s\ensuremath{_{\textnormal{1/2}}^{\textnormal{$-$1}}}.\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{12.38407cm}{\raggedright T\ensuremath{_{1/2}}: From recoil-ce(\ensuremath{\Delta}t) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2014Au03,B}{2014Au03}, where recoils were correlated with the 173\ensuremath{\gamma} and\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{12.38407cm}{\raggedright {\ }{\ }{\ }433\ensuremath{\gamma} above the isomer, and ce were in coincidence with 190\ensuremath{\gamma} and 269\ensuremath{\gamma} below the\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{12.38407cm}{\raggedright {\ }{\ }{\ }isomer. The value was deduced using the the logarithmic time-scale method.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{631}&\multicolumn{1}{@{.}l}{8\ensuremath{^{{\hyperlink{AT49LEVEL2}{\#}}}} {\it 4}}&\multicolumn{1}{l}{3/2\ensuremath{^{+}}}&&&\parbox[t][0.3cm]{12.38407cm}{\raggedright Proposed configuration=\ensuremath{\pi} d\ensuremath{_{\textnormal{3/2}}^{\textnormal{$-$1}}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{635}&\multicolumn{1}{@{.}l}{17 {\it 16}}&\multicolumn{1}{l}{13/2\ensuremath{^{-}}}&&&\parbox[t][0.3cm]{12.38407cm}{\raggedright Configuration=\ensuremath{\pi} (h\ensuremath{_{\textnormal{9/2}}^{\textnormal{+1}}})\ensuremath{\otimes}2\ensuremath{^{\textnormal{+}}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{690}&\multicolumn{1}{@{.}l}{98 {\it 16}}&\multicolumn{1}{l}{11/2\ensuremath{^{-}}}&&&\parbox[t][0.3cm]{12.38407cm}{\raggedright Configuration=\ensuremath{\pi} (h\ensuremath{_{\textnormal{9/2}}^{\textnormal{+1}}})\ensuremath{\otimes}2\ensuremath{^{\textnormal{+}}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{749}&\multicolumn{1}{@{.}l}{36 {\it 14}}&\multicolumn{1}{l}{13/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{$\approx$20}&\multicolumn{1}{@{ }l}{ns}&\parbox[t][0.3cm]{12.38407cm}{\raggedright T\ensuremath{_{1/2}}: from \ensuremath{\gamma}(t) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2015Au01,B}{2015Au01}.\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{12.38407cm}{\raggedright Configuration=\ensuremath{\pi} i\ensuremath{_{\textnormal{13/2}}^{\textnormal{+1}}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{804}&\multicolumn{1}{@{.}l}{2\ensuremath{^{{\hyperlink{AT49LEVEL3}{@}}}} {\it 6}}&\multicolumn{1}{l}{5/2\ensuremath{^{+}}}&&&\parbox[t][0.3cm]{12.38407cm}{\raggedright Proposed configuration=\ensuremath{\pi} d\ensuremath{_{\textnormal{5/2}}^{\textnormal{$-$1}}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1065}&\multicolumn{1}{@{.}l}{2\ensuremath{^{{\hyperlink{AT49LEVEL2}{\#}}}} {\it 6}}&\multicolumn{1}{l}{7/2\ensuremath{^{+}}}&&&&\\
\multicolumn{1}{r@{}}{1228}&\multicolumn{1}{@{.}l}{96 {\it 22}}&\multicolumn{1}{l}{17/2\ensuremath{^{-}}}&&&\parbox[t][0.3cm]{12.38407cm}{\raggedright Configuration=\ensuremath{\pi} (h\ensuremath{_{\textnormal{9/2}}^{\textnormal{+1}}})\ensuremath{\otimes}4\ensuremath{^{\textnormal{+}}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1261}&\multicolumn{1}{@{.}l}{29 {\it 20}}&\multicolumn{1}{l}{15/2\ensuremath{^{+}}}&&&&\\
\multicolumn{1}{r@{}}{1288}&\multicolumn{1}{@{.}l}{6\ensuremath{^{{\hyperlink{AT49LEVEL3}{@}}}} {\it 6}}&\multicolumn{1}{l}{9/2\ensuremath{^{+}}}&&&&\\
\multicolumn{1}{r@{}}{1494}&\multicolumn{1}{@{.}l}{85 {\it 19}}&\multicolumn{1}{l}{17/2\ensuremath{^{+}}}&&&\parbox[t][0.3cm]{12.38407cm}{\raggedright Dominant configuration=\ensuremath{\pi} (i\ensuremath{_{\textnormal{13/2}}^{\textnormal{+1}}})\ensuremath{\otimes}2\ensuremath{^{\textnormal{+}}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1613}&\multicolumn{1}{@{.}l}{3\ensuremath{^{{\hyperlink{AT49LEVEL2}{\#}}}} {\it 7}}&\multicolumn{1}{l}{11/2\ensuremath{^{+}}}&&&&\\
\multicolumn{1}{r@{}}{1625}&\multicolumn{1}{@{.}l}{34 {\it 19}}&\multicolumn{1}{l}{17/2\ensuremath{^{+}}}&&&\parbox[t][0.3cm]{12.38407cm}{\raggedright Dominant configuration=\ensuremath{\pi} (h\ensuremath{_{\textnormal{9/2}}^{\textnormal{+1}}}) \ensuremath{\nu} (f\ensuremath{_{\textnormal{5/2}}^{\textnormal{$-$1}}},i\ensuremath{_{\textnormal{13/2}}^{\textnormal{$-$1}}})\ensuremath{_{\textnormal{5$-$}}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1705}&\multicolumn{1}{@{.}l}{03 {\it 24}}&\multicolumn{1}{l}{21/2\ensuremath{^{-}}}&&&\parbox[t][0.3cm]{12.38407cm}{\raggedright Dominant configuration=\ensuremath{\pi} (h\ensuremath{_{\textnormal{9/2}}^{\textnormal{+1}}})\ensuremath{\otimes}6\ensuremath{^{\textnormal{+}}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1856}&\multicolumn{1}{@{.}l}{0\ensuremath{^{{\hyperlink{AT49LEVEL3}{@}}}} {\it 7}}&\multicolumn{1}{l}{13/2\ensuremath{^{+}}}&&&&\\
\multicolumn{1}{r@{}}{1921}&\multicolumn{1}{@{.}l}{30 {\it 21}}&\multicolumn{1}{l}{21/2\ensuremath{^{+}}}&&&\parbox[t][0.3cm]{12.38407cm}{\raggedright Dominant configuration=\ensuremath{\pi} (h\ensuremath{_{\textnormal{9/2}}^{\textnormal{+1}}}) \ensuremath{\nu} (f\ensuremath{_{\textnormal{5/2}}^{\textnormal{$-$1}}},i\ensuremath{_{\textnormal{13/2}}^{\textnormal{$-$1}}})\ensuremath{_{\textnormal{7$-$}}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1980}&\multicolumn{1}{@{.}l}{6 {\it 3}}&\multicolumn{1}{l}{23/2\ensuremath{^{-}}}&&&\parbox[t][0.3cm]{12.38407cm}{\raggedright Dominant configuration=\ensuremath{\pi} [f\ensuremath{_{\textnormal{7/2}}^{\textnormal{+1}}},(h\ensuremath{_{\textnormal{9/2}}^{\textnormal{+2}}})\ensuremath{_{\textnormal{8+}}}].\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2004}&\multicolumn{1}{@{.}l}{3 {\it 3}}&\multicolumn{1}{l}{23/2\ensuremath{^{+}}}&&&\parbox[t][0.3cm]{12.38407cm}{\raggedright Dominant configuration=\ensuremath{\pi} (h\ensuremath{_{\textnormal{9/2}}^{\textnormal{+1}}}) \ensuremath{\nu} (f\ensuremath{_{\textnormal{5/2}}^{\textnormal{$-$1}}},i\ensuremath{_{\textnormal{13/2}}^{\textnormal{$-$1}}})\ensuremath{_{\textnormal{8$-$}}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2050}&\multicolumn{1}{@{.}l}{8 {\it 3}}&\multicolumn{1}{l}{25/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{$<$20}&\multicolumn{1}{@{ }l}{ns}&\parbox[t][0.3cm]{12.38407cm}{\raggedright T\ensuremath{_{1/2}}: prompt-like time distribution suggests that the half-life is much shorter than 20 ns.\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{12.38407cm}{\raggedright Dominant configuration=\ensuremath{\pi} (h\ensuremath{_{\textnormal{9/2}}^{\textnormal{+1}}}) \ensuremath{\nu} (f\ensuremath{_{\textnormal{5/2}}^{\textnormal{$-$1}}},i\ensuremath{_{\textnormal{13/2}}^{\textnormal{$-$1}}})\ensuremath{_{\textnormal{9$-$}}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2147}&\multicolumn{1}{@{.}l}{2 {\it 4}}&\multicolumn{1}{l}{21/2\ensuremath{^{-}}}&&&\parbox[t][0.3cm]{12.38407cm}{\raggedright Dominant configuration=\ensuremath{\pi} (f\ensuremath{_{\textnormal{7/2}}^{\textnormal{+1}}})\ensuremath{\otimes}\ensuremath{\nu} (i\ensuremath{_{\textnormal{13/2}}^{\textnormal{$-$2}}})\ensuremath{_{\textnormal{8+}}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2232}&\multicolumn{1}{@{.}l}{1\ensuremath{^{{\hyperlink{AT49LEVEL2}{\#}}}} {\it 9}}&\multicolumn{1}{l}{15/2\ensuremath{^{+}}}&&&&\\
\multicolumn{1}{r@{}}{2319}&\multicolumn{1}{@{.}l}{8 {\it 3}}&\multicolumn{1}{l}{29/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{39 \ensuremath{\mu}s {\it 9}}&\parbox[t][0.3cm]{12.38407cm}{\raggedright \%IT=100\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{12.38407cm}{\raggedright T\ensuremath{_{1/2}}: From recoil-269\ensuremath{\gamma}(t) using the planar detector data and the logarithmic time scale\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{12.38407cm}{\raggedright {\ }{\ }{\ }method (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2015Au01,B}{2015Au01}). A prompt coincidence with a 296\ensuremath{\gamma}, 427\ensuremath{\gamma}, 594\ensuremath{\gamma}, 635\ensuremath{\gamma},\hphantom{a}746\ensuremath{\gamma} or\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{12.38407cm}{\raggedright {\ }{\ }{\ }749\ensuremath{\gamma} in any of the focal plane clover detectors was also required.\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{12.38407cm}{\raggedright Dominant configuration=\ensuremath{\pi} [i\ensuremath{_{\textnormal{13/2}}^{\textnormal{+1}}},(h\ensuremath{_{\textnormal{9/2}}^{\textnormal{+2}}})\ensuremath{_{\textnormal{8+}}}].\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2518}&\multicolumn{1}{@{.}l}{9 {\it 5}}&\multicolumn{1}{l}{25/2\ensuremath{^{-}}}&&&\parbox[t][0.3cm]{12.38407cm}{\raggedright Configuration=\ensuremath{\pi} (h\ensuremath{_{\textnormal{9/2}}^{\textnormal{+1}}})\ensuremath{\otimes}8\ensuremath{^{\textnormal{+}}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{2637}&\multicolumn{1}{@{.}l}{6 {\it 5}}&\multicolumn{1}{l}{(25/2\ensuremath{^{+}})}&&&&\\
\multicolumn{1}{r@{}}{2990}&\multicolumn{1}{@{.}l}{2\ensuremath{^{{\hyperlink{AT49LEVEL4}{\&}}}} {\it 5}}&\multicolumn{1}{l}{(23/2\ensuremath{^{-}})}&&&\parbox[t][0.3cm]{12.38407cm}{\raggedright E(level): The total transition intensity of 145\ensuremath{\gamma} that feeds this level is much higher than the\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{12.38407cm}{\raggedright {\ }{\ }{\ }1069\ensuremath{\gamma} one that depopulates it, thus suggesting the existence of at least one additional,\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{12.38407cm}{\raggedright {\ }{\ }{\ }unobserved decay branch.\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{12.38407cm}{\raggedright Configuration=\ensuremath{\pi} (i\ensuremath{_{\textnormal{13/2}}^{\textnormal{+1}}})\ensuremath{\otimes}\ensuremath{\nu} (f\ensuremath{_{\textnormal{5/2}}^{\textnormal{$-$1}}},i\ensuremath{_{\textnormal{13/2}}^{\textnormal{$-$1}}})\ensuremath{_{\textnormal{5$-$}}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3135}&\multicolumn{1}{@{.}l}{2\ensuremath{^{{\hyperlink{AT49LEVEL4}{\&}}}} {\it 6}}&\multicolumn{1}{l}{(25/2\ensuremath{^{-}})}&&&&\\
\multicolumn{1}{r@{}}{3219}&\multicolumn{1}{@{.}l}{2 {\it 6}}&&&&&\\
\end{longtable}
\begin{textblock}{29}(0,27.3)
Continued on next page (footnotes at end of table)
\end{textblock}
\clearpage
\begin{longtable}{cccc@{\extracolsep{\fill}}c}
\\[-.4cm]
\multicolumn{5}{c}{{\bf \small \underline{\ensuremath{^{\textnormal{165}}}Ho(\ensuremath{^{\textnormal{40}}}Ar,4n\ensuremath{\gamma})\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2014Au03,B}{2014Au03},\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2015Au01,B}{2015Au01} (continued)}}}\\
\multicolumn{5}{c}{~}\\
\multicolumn{5}{c}{\underline{\ensuremath{^{201}}At Levels (continued)}}\\
\multicolumn{5}{c}{~}\\
\multicolumn{2}{c}{E(level)$^{{\hyperlink{AT49LEVEL0}{\dagger}}}$}&J$^{\pi}$$^{{\hyperlink{AT49LEVEL1}{\ddagger}}}$&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&
\endhead
\multicolumn{1}{r@{}}{3240}&\multicolumn{1}{@{.}l}{9 {\it 5}}&\multicolumn{1}{l}{(33/2\ensuremath{^{+}})}&\parbox[t][0.3cm]{14.48628cm}{\raggedright Possible configuration=\ensuremath{\pi} [i\ensuremath{_{\textnormal{13/2}}^{\textnormal{+1}}},(h\ensuremath{_{\textnormal{9/2}}^{\textnormal{+2}}})\ensuremath{_{\textnormal{8+}}}]\ensuremath{\otimes}2\ensuremath{^{\textnormal{+}}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{3245}&\multicolumn{1}{@{.}l}{4 {\it 6}}&\multicolumn{1}{l}{(29/2\ensuremath{^{+}})}&&\\
\multicolumn{1}{r@{}}{3369}&\multicolumn{1}{@{.}l}{6 {\it 5}}&&&\\
\multicolumn{1}{r@{}}{3379}&\multicolumn{1}{@{.}l}{6\ensuremath{^{{\hyperlink{AT49LEVEL4}{\&}}}} {\it 7}}&\multicolumn{1}{l}{(27/2\ensuremath{^{-}})}&&\\
\multicolumn{1}{r@{}}{3504}&\multicolumn{1}{@{.}l}{4 {\it 4}}&&&\\
\multicolumn{1}{r@{}}{3621}&\multicolumn{1}{@{.}l}{8 {\it 7}}&&&\\
\multicolumn{1}{r@{}}{3666}&\multicolumn{1}{@{.}l}{5?\ensuremath{^{{\hyperlink{AT49LEVEL4}{\&}}}} {\it 8}}&\multicolumn{1}{l}{(29/2\ensuremath{^{-}})}&&\\
\multicolumn{1}{r@{}}{3693}&\multicolumn{1}{@{.}l}{9 {\it 7}}&&&\\
\multicolumn{1}{r@{}}{3699}&\multicolumn{1}{@{.}l}{7 {\it 6}}&&&\\
\multicolumn{1}{r@{}}{3779}&\multicolumn{1}{@{.}l}{1 {\it 6}}&&&\\
\multicolumn{1}{r@{}}{3785}&\multicolumn{1}{@{.}l}{9 {\it 8}}&&&\\
\multicolumn{1}{r@{}}{3853}&\multicolumn{1}{@{.}l}{3 {\it 7}}&&&\\
\multicolumn{1}{r@{}}{3952}&\multicolumn{1}{@{.}l}{8 {\it 6}}&&&\\
\multicolumn{1}{r@{}}{3983}&\multicolumn{1}{@{.}l}{8\ensuremath{^{{\hyperlink{AT49LEVEL4}{\&}}}} {\it 9}}&\multicolumn{1}{l}{(31/2\ensuremath{^{-}})}&&\\
\multicolumn{1}{r@{}}{4111}&\multicolumn{1}{@{.}l}{4 {\it 6}}&&&\\
\multicolumn{1}{r@{}}{4159}&\multicolumn{1}{@{.}l}{0 {\it 7}}&&&\\
\multicolumn{1}{r@{}}{4256}&\multicolumn{1}{@{.}l}{1\ensuremath{^{{\hyperlink{AT49LEVEL4}{\&}}}} {\it 10}}&\multicolumn{1}{l}{(33/2\ensuremath{^{-}})}&&\\
\multicolumn{1}{r@{}}{4454}&\multicolumn{1}{@{.}l}{0\ensuremath{^{{\hyperlink{AT49LEVEL4}{\&}}}} {\it 11}}&\multicolumn{1}{l}{(35/2\ensuremath{^{-}})}&&\\
\multicolumn{1}{r@{}}{4789}&\multicolumn{1}{@{.}l}{0\ensuremath{^{{\hyperlink{AT49LEVEL4}{\&}}}} {\it 12}}&\multicolumn{1}{l}{(37/2\ensuremath{^{-}})}&&\\
\end{longtable}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{AT49LEVEL0}{\dagger}}}} From a least-squares fit to E\ensuremath{\gamma}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{AT49LEVEL1}{\ddagger}}}} From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2014Au03,B}{2014Au03} and \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2015Au01,B}{2015Au01}, based on the deduced \ensuremath{\gamma}-ray transition multipolarities using the \ensuremath{\gamma}(\ensuremath{\theta}) analysis, observed decay}\\
\parbox[b][0.3cm]{17.7cm}{{\ }{\ }pattern, systematics in the region and shell-model assignments.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{AT49LEVEL2}{\#}}}} Seq.(B): Based on \ensuremath{\pi} (d\ensuremath{_{\textnormal{3/2}}^{\textnormal{$-$1}}})\ensuremath{\otimes}\ensuremath{^{\textnormal{202}}}Rn core states (\ensuremath{J^{\pi}}=2\ensuremath{^{+}},4\ensuremath{^{+}},6\ensuremath{^{+}}).}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{AT49LEVEL3}{@}}}} Seq.(C): Based on \ensuremath{\pi} (d\ensuremath{_{\textnormal{5/2}}^{\textnormal{$-$1}}})\ensuremath{\otimes}\ensuremath{^{\textnormal{202}}}Rn core states (\ensuremath{J^{\pi}}=2\ensuremath{^{+}},4\ensuremath{^{+}}).}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{AT49LEVEL4}{\&}}}} Band(A): Magnetic-dipole, shears band. Configuration=\ensuremath{\pi} (i\ensuremath{_{\textnormal{13/2}}^{\textnormal{+1}}})\ensuremath{\otimes}\ensuremath{\nu} (f\ensuremath{_{\textnormal{5/2}}^{\textnormal{$-$1}}},i\ensuremath{_{\textnormal{13/2}}^{\textnormal{$-$1}}})\ensuremath{_{\textnormal{9$-$}}} for the lower cascade and}\\
\parbox[b][0.3cm]{17.7cm}{{\ }{\ }Configuration= \ensuremath{\pi} (h\ensuremath{_{\textnormal{9/2}}^{\textnormal{+2}}},i\ensuremath{_{\textnormal{13/2}}^{\textnormal{+1}}})\ensuremath{\otimes}\ensuremath{\nu} (f\ensuremath{_{\textnormal{5/2}}^{\textnormal{$-$1}}},i\ensuremath{_{\textnormal{13/2}}^{\textnormal{$-$1}}})\ensuremath{_{\textnormal{5$-$}}} above the band crossing.}\\
\vspace{0.5cm}
\clearpage
\vspace{0.3cm}
\begin{landscape}
\vspace*{-0.5cm}
{\bf \small \underline{\ensuremath{^{\textnormal{165}}}Ho(\ensuremath{^{\textnormal{40}}}Ar,4n\ensuremath{\gamma})\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2014Au03,B}{2014Au03},\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2015Au01,B}{2015Au01} (continued)}}\\
\vspace{0.3cm}
\underline{$\gamma$($^{201}$At)}\\
\begin{longtable}{ccccccccc@{}ccccccccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{AT49GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{I\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{AT49GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{E\ensuremath{_{i}}(level)}&J\ensuremath{^{\pi}_{i}}&\multicolumn{2}{c}{E\ensuremath{_{f}}}&J\ensuremath{^{\pi}_{f}}&Mult.\ensuremath{^{\hyperlink{AT49GAMMA10}{f}}}&\multicolumn{2}{c}{\ensuremath{\delta}}&\multicolumn{2}{c}{\ensuremath{\alpha}\ensuremath{^{\hyperlink{AT49GAMMA11}{g}}}}&\multicolumn{2}{c}{I\ensuremath{_{(\gamma+ce)}}\ensuremath{^{\hyperlink{AT49GAMMA6}{b}}}}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{46}&\multicolumn{1}{@{.}l}{5 {\it 2}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{2050}&\multicolumn{1}{@{.}l}{8}&\multicolumn{1}{l}{25/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{2004}&\multicolumn{1}{@{.}l}{3 }&\multicolumn{1}{@{}l}{23/2\ensuremath{^{+}}}&\multicolumn{1}{l}{}&&&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{90}&\multicolumn{1}{@{ }l}{{\it 30}}&\parbox[t][0.3cm]{6.9027605cm}{\raggedright E\ensuremath{_{\gamma}},I\ensuremath{_{(\gamma+ce)}}: from recoil-gated planar singles\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.9027605cm}{\raggedright {\ }{\ }{\ }spectrum.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{58}&\multicolumn{1}{@{.}l}{5 {\it 2}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{749}&\multicolumn{1}{@{.}l}{36}&\multicolumn{1}{l}{13/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{690}&\multicolumn{1}{@{.}l}{98 }&\multicolumn{1}{@{}l}{11/2\ensuremath{^{-}}}&\multicolumn{1}{l}{[E1]}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{424 {\it 7}}&\multicolumn{1}{r@{}}{6}&\multicolumn{1}{@{ }l}{{\it 2}}&\parbox[t][0.3cm]{6.9027605cm}{\raggedright E\ensuremath{_{\gamma}}: from recoil-gated planar singles spectrum.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{83}&\multicolumn{1}{@{.}l}{0 {\it 4}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{2004}&\multicolumn{1}{@{.}l}{3}&\multicolumn{1}{l}{23/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1921}&\multicolumn{1}{@{.}l}{30 }&\multicolumn{1}{@{}l}{21/2\ensuremath{^{+}}}&\multicolumn{1}{l}{[M1]}&&&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{99 {\it 8}}&\multicolumn{1}{r@{}}{120}&\multicolumn{1}{@{ }l}{{\it 40}}&\parbox[t][0.3cm]{6.9027605cm}{\raggedright E\ensuremath{_{\gamma}}: calculated value using the assumption that in\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.9027605cm}{\raggedright {\ }{\ }{\ }a closed loop of transitions, the energy shift is\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.9027605cm}{\raggedright {\ }{\ }{\ }zero. Two loops were used, weighted average of\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.9027605cm}{\raggedright {\ }{\ }{\ }those results given here; partially overlaps with\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.9027605cm}{\raggedright {\ }{\ }{\ }the K\ensuremath{\alpha} x-ray peak.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{114}&\multicolumn{1}{@{.}l}{1\ensuremath{^{\hyperlink{AT49GAMMA5}{a}}} {\it 2}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{749}&\multicolumn{1}{@{.}l}{36}&\multicolumn{1}{l}{13/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{635}&\multicolumn{1}{@{.}l}{17 }&\multicolumn{1}{@{}l}{13/2\ensuremath{^{-}}}&\multicolumn{1}{l}{[E1]}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{332 {\it 5}}&\multicolumn{1}{r@{}}{12}&\multicolumn{1}{@{ }l}{{\it 1}}&&\\
\multicolumn{1}{r@{}}{130}&\multicolumn{1}{@{.}l}{3\ensuremath{^{\hyperlink{AT49GAMMA5}{a}}} {\it 2}}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{9 {\it 4}}&\multicolumn{1}{r@{}}{1625}&\multicolumn{1}{@{.}l}{34}&\multicolumn{1}{l}{17/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1494}&\multicolumn{1}{@{.}l}{85 }&\multicolumn{1}{@{}l}{17/2\ensuremath{^{+}}}&\multicolumn{1}{l}{(M1)}&&&\multicolumn{1}{r@{}}{5}&\multicolumn{1}{@{.}l}{69 {\it 8}}&\multicolumn{1}{r@{}}{20}&\multicolumn{1}{@{ }l}{{\it 3}}&\parbox[t][0.3cm]{6.9027605cm}{\raggedright Mult.: A2={\textminus}0.36 \textit{7} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2015Au01,B}{2015Au01}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{135}&\multicolumn{1}{@{.}l}{0\ensuremath{^{\hyperlink{AT49GAMMA12}{h}}} {\it 4}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{50\ensuremath{^{\hyperlink{AT49GAMMA3}{@}}} {\it 4}}&\multicolumn{1}{r@{}}{3504}&\multicolumn{1}{@{.}l}{4}&&\multicolumn{1}{r@{}}{3369}&\multicolumn{1}{@{.}l}{6 }&&&&&&&&&&\\
\multicolumn{1}{r@{}}{145}&\multicolumn{1}{@{.}l}{0 {\it 4}}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{7 {\it 3}}&\multicolumn{1}{r@{}}{3135}&\multicolumn{1}{@{.}l}{2}&\multicolumn{1}{l}{(25/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{2990}&\multicolumn{1}{@{.}l}{2 }&\multicolumn{1}{@{}l}{(23/2\ensuremath{^{-}})}&\multicolumn{1}{l}{(M1)\ensuremath{^{\hyperlink{AT49GAMMA4}{\&}}}}&&&\multicolumn{1}{r@{}}{4}&\multicolumn{1}{@{.}l}{20 {\it 7}}&&&\parbox[t][0.3cm]{6.9027605cm}{\raggedright Mult.: A2={\textminus}0.5 \textit{2} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2015Au01,B}{2015Au01}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{153}&\multicolumn{1}{@{.}l}{6\ensuremath{^{\hyperlink{AT49GAMMA12}{h}}} {\it 4}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{47\ensuremath{^{\hyperlink{AT49GAMMA3}{@}}} {\it 4}}&\multicolumn{1}{r@{}}{3853}&\multicolumn{1}{@{.}l}{3}&&\multicolumn{1}{r@{}}{3699}&\multicolumn{1}{@{.}l}{7 }&&&&&&&&&&\\
\multicolumn{1}{r@{}}{172}&\multicolumn{1}{@{.}l}{5\ensuremath{^{\hyperlink{AT49GAMMA8}{d}\hyperlink{AT49GAMMA7}{c}}} {\it 4}}&\multicolumn{1}{r@{}}{18}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{AT49GAMMA8}{d}\hyperlink{AT49GAMMA7}{c}}} {\it 1}}&\multicolumn{1}{r@{}}{804}&\multicolumn{1}{@{.}l}{2}&\multicolumn{1}{l}{5/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{631}&\multicolumn{1}{@{.}l}{8 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}}}&&&&&&&&&\\
\multicolumn{1}{r@{}}{172}&\multicolumn{1}{@{.}l}{6\ensuremath{^{\hyperlink{AT49GAMMA8}{d}\hyperlink{AT49GAMMA7}{c}}} {\it 4}}&\multicolumn{1}{r@{}}{113}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{AT49GAMMA8}{d}\hyperlink{AT49GAMMA7}{c}}} {\it 5}}&\multicolumn{1}{r@{}}{631}&\multicolumn{1}{@{.}l}{8}&\multicolumn{1}{l}{3/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{459}&\multicolumn{1}{@{.}l}{20 }&\multicolumn{1}{@{}l}{1/2\ensuremath{^{+}}}&\multicolumn{1}{l}{(M1+E2)}&&&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{7 {\it 9}}&&&\parbox[t][0.3cm]{6.9027605cm}{\raggedright Mult.: R=0.83 \textit{5} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2014Au03,B}{2014Au03}). The x-ray intensity\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.9027605cm}{\raggedright {\ }{\ }{\ }in a spectrum produced by gating on 533\ensuremath{\gamma} is\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.9027605cm}{\raggedright {\ }{\ }{\ }consistent with Mult=M1+E2, but not with a\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.9027605cm}{\raggedright {\ }{\ }{\ }pure Mult=E2.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{190}&\multicolumn{1}{@{.}l}{1 {\it 1}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{190}&\multicolumn{1}{@{.}l}{10}&\multicolumn{1}{l}{7/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{@{}l}{9/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M1+E2}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{65 {\it 8}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{55 {\it 7}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{6.9027605cm}{\raggedright E\ensuremath{_{\gamma}}: From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2014Au03,B}{2014Au03}.\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.9027605cm}{\raggedright Mult.,\ensuremath{\delta}: From K/(L+M+...)exp=3.1 \textit{2} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2014Au03,B}{2014Au03}).\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.9027605cm}{\raggedright {\ }{\ }{\ }\ensuremath{\delta} was determined by the evaluator using the\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.9027605cm}{\raggedright {\ }{\ }{\ }briccmixing program.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{197}&\multicolumn{1}{@{.}l}{9 {\it 4}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{6 {\it 2}}&\multicolumn{1}{r@{}}{4454}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{l}{(35/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{4256}&\multicolumn{1}{@{.}l}{1 }&\multicolumn{1}{@{}l}{(33/2\ensuremath{^{-}})}&\multicolumn{1}{l}{(M1)\ensuremath{^{\hyperlink{AT49GAMMA4}{\&}}}}&&&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{745 {\it 26}}&&&\parbox[t][0.3cm]{6.9027605cm}{\raggedright Mult.: A2={\textminus}0.80 \textit{12} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2015Au01,B}{2015Au01}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{206}&\multicolumn{1}{@{.}l}{2 {\it 4}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{31\ensuremath{^{\hyperlink{AT49GAMMA3}{@}}} {\it 5}}&\multicolumn{1}{r@{}}{4159}&\multicolumn{1}{@{.}l}{0}&&\multicolumn{1}{r@{}}{3952}&\multicolumn{1}{@{.}l}{8 }&&&&&&&&&&\\
\multicolumn{1}{r@{}}{216}&\multicolumn{1}{@{.}l}{3\ensuremath{^{\hyperlink{AT49GAMMA5}{a}}} {\it 3}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{1921}&\multicolumn{1}{@{.}l}{30}&\multicolumn{1}{l}{21/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1705}&\multicolumn{1}{@{.}l}{03 }&\multicolumn{1}{@{}l}{21/2\ensuremath{^{-}}}&\multicolumn{1}{l}{[E1]}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0696 {\it 10}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{3 {\it 3}}&&\\
\multicolumn{1}{r@{}}{223}&\multicolumn{1}{@{.}l}{3\ensuremath{^{\hyperlink{AT49GAMMA7}{c}}} {\it 4}}&\multicolumn{1}{r@{}}{17}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{AT49GAMMA7}{c}}} {\it 1}}&\multicolumn{1}{r@{}}{1288}&\multicolumn{1}{@{.}l}{6}&\multicolumn{1}{l}{9/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1065}&\multicolumn{1}{@{.}l}{2 }&\multicolumn{1}{@{}l}{7/2\ensuremath{^{+}}}&\multicolumn{1}{l}{(M1)}&&&&&&&\parbox[t][0.3cm]{6.9027605cm}{\raggedright Mult.: A2={\textminus}0.29 \textit{7} and R=0.74 \textit{12} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2014Au03,B}{2014Au03}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{233}&\multicolumn{1}{@{.}l}{4\ensuremath{^{\hyperlink{AT49GAMMA5}{a}}} {\it 2}}&\multicolumn{1}{r@{}}{14}&\multicolumn{1}{@{.}l}{5 {\it 4}}&\multicolumn{1}{r@{}}{1494}&\multicolumn{1}{@{.}l}{85}&\multicolumn{1}{l}{17/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1261}&\multicolumn{1}{@{.}l}{29 }&\multicolumn{1}{@{}l}{15/2\ensuremath{^{+}}}&\multicolumn{1}{l}{(M1)}&&&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{101 {\it 16}}&\multicolumn{1}{r@{}}{24}&\multicolumn{1}{@{ }l}{{\it 3}}&\parbox[t][0.3cm]{6.9027605cm}{\raggedright Mult.: A2={\textminus}0.05 \textit{2} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2015Au01,B}{2015Au01}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{242}&\multicolumn{1}{@{.}l}{8\ensuremath{^{\hyperlink{AT49GAMMA7}{c}}} {\it 7}}&\multicolumn{1}{r@{}}{5}&\multicolumn{1}{@{.}l}{7\ensuremath{^{\hyperlink{AT49GAMMA9}{e}\hyperlink{AT49GAMMA7}{c}}} {\it 5}}&\multicolumn{1}{r@{}}{1856}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{l}{13/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1613}&\multicolumn{1}{@{.}l}{3 }&\multicolumn{1}{@{}l}{11/2\ensuremath{^{+}}}&&&&&&&&&\\
\multicolumn{1}{r@{}}{244}&\multicolumn{1}{@{.}l}{4 {\it 4}}&\multicolumn{1}{r@{}}{8}&\multicolumn{1}{@{.}l}{3 {\it 5}}&\multicolumn{1}{r@{}}{3379}&\multicolumn{1}{@{.}l}{6}&\multicolumn{1}{l}{(27/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{3135}&\multicolumn{1}{@{.}l}{2 }&\multicolumn{1}{@{}l}{(25/2\ensuremath{^{-}})}&\multicolumn{1}{l}{(M1)\ensuremath{^{\hyperlink{AT49GAMMA4}{\&}}}}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{968 {\it 14}}&&&\parbox[t][0.3cm]{6.9027605cm}{\raggedright Mult.: A2={\textminus}0.59 \textit{9} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2015Au01,B}{2015Au01}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{263}&\multicolumn{1}{@{.}l}{6 {\it 4}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{2\ensuremath{^{\hyperlink{AT49GAMMA3}{@}}} {\it 2}}&\multicolumn{1}{r@{}}{3504}&\multicolumn{1}{@{.}l}{4}&&\multicolumn{1}{r@{}}{3240}&\multicolumn{1}{@{.}l}{9 }&\multicolumn{1}{@{}l}{(33/2\ensuremath{^{+}})}&\multicolumn{1}{l}{D}&&&&&&&\parbox[t][0.3cm]{6.9027605cm}{\raggedright Mult.: A2={\textminus}0.26 \textit{9} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2015Au01,B}{2015Au01}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{269}&\multicolumn{1}{@{.}l}{0 {\it 2}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{2319}&\multicolumn{1}{@{.}l}{8}&\multicolumn{1}{l}{29/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{2050}&\multicolumn{1}{@{.}l}{8 }&\multicolumn{1}{@{}l}{25/2\ensuremath{^{+}}}&\multicolumn{1}{l}{E2}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{1842 {\it 26}}&\multicolumn{1}{r@{}}{118}&\multicolumn{1}{@{ }l}{{\it 11}}&\parbox[t][0.3cm]{6.9027605cm}{\raggedright Mult.: From K/(L+M+...)exp=0.93 \textit{5} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2015Au01,B}{2015Au01}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{269}&\multicolumn{1}{@{.}l}{1 {\it 1}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{459}&\multicolumn{1}{@{.}l}{20}&\multicolumn{1}{l}{1/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{190}&\multicolumn{1}{@{.}l}{10 }&\multicolumn{1}{@{}l}{7/2\ensuremath{^{-}}}&\multicolumn{1}{l}{E3}&&&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{231 {\it 17}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{6.9027605cm}{\raggedright E\ensuremath{_{\gamma}}: From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2014Au03,B}{2014Au03}.\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&&&\parbox[t][0.3cm]{6.9027605cm}{\raggedright Mult.: From K/(L+M+...)exp=0.24 \textit{1} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2014Au03,B}{2014Au03}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{272}&\multicolumn{1}{@{.}l}{3 {\it 4}}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{5 {\it 3}}&\multicolumn{1}{r@{}}{4256}&\multicolumn{1}{@{.}l}{1}&\multicolumn{1}{l}{(33/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{3983}&\multicolumn{1}{@{.}l}{8 }&\multicolumn{1}{@{}l}{(31/2\ensuremath{^{-}})}&\multicolumn{1}{l}{(M1)\ensuremath{^{\hyperlink{AT49GAMMA4}{\&}}}}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{718 {\it 10}}&&&\parbox[t][0.3cm]{6.9027605cm}{\raggedright Mult.: A2={\textminus}0.45 \textit{11} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2015Au01,B}{2015Au01}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{275}&\multicolumn{1}{@{.}l}{5\ensuremath{^{\hyperlink{AT49GAMMA5}{a}}} {\it 2}}&\multicolumn{1}{r@{}}{6}&\multicolumn{1}{@{.}l}{6 {\it 2}}&\multicolumn{1}{r@{}}{1980}&\multicolumn{1}{@{.}l}{6}&\multicolumn{1}{l}{23/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{1705}&\multicolumn{1}{@{.}l}{03 }&\multicolumn{1}{@{}l}{21/2\ensuremath{^{-}}}&\multicolumn{1}{l}{(M1)}&&&&&\multicolumn{1}{r@{}}{12}&\multicolumn{1}{@{ }l}{{\it 2}}&\parbox[t][0.3cm]{6.9027605cm}{\raggedright Mult.: A2={\textminus}0.47 \textit{2} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2015Au01,B}{2015Au01}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{286}&\multicolumn{1}{@{.}l}{9 {\it 4}}&\multicolumn{1}{r@{}}{6}&\multicolumn{1}{@{.}l}{6 {\it 4}}&\multicolumn{1}{r@{}}{3666}&\multicolumn{1}{@{.}l}{5?}&\multicolumn{1}{l}{(29/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{3379}&\multicolumn{1}{@{.}l}{6 }&\multicolumn{1}{@{}l}{(27/2\ensuremath{^{-}})}&\multicolumn{1}{l}{(M1)\ensuremath{^{\hyperlink{AT49GAMMA4}{\&}}}}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{622 {\it 9}}&&&\parbox[t][0.3cm]{6.9027605cm}{\raggedright Mult.: A2={\textminus}0.47 \textit{3}\hphantom{a}(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2015Au01,B}{2015Au01}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{295}&\multicolumn{1}{@{.}l}{9\ensuremath{^{\hyperlink{AT49GAMMA5}{a}}} {\it 2}}&\multicolumn{1}{r@{}}{40}&\multicolumn{1}{@{ }l}{{\it 2}}&\multicolumn{1}{r@{}}{1921}&\multicolumn{1}{@{.}l}{30}&\multicolumn{1}{l}{21/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1625}&\multicolumn{1}{@{.}l}{34 }&\multicolumn{1}{@{}l}{17/2\ensuremath{^{+}}}&\multicolumn{1}{l}{(E2)}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{1374 {\it 19}}&\multicolumn{1}{r@{}}{60}&\multicolumn{1}{@{ }l}{{\it 6}}&\parbox[t][0.3cm]{6.9027605cm}{\raggedright Mult.: A2=+0.11 \textit{4} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2015Au01,B}{2015Au01}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{$^{x}$297}&\multicolumn{1}{@{.}l}{5\ensuremath{^{\hyperlink{AT49GAMMA1}{\ddagger}}} {\it 4}}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{0\ensuremath{^{\hyperlink{AT49GAMMA3}{@}}} {\it 2}}&&&&&&&&&&&&&&&\\
\multicolumn{1}{r@{}}{299}&\multicolumn{1}{@{.}l}{3\ensuremath{^{\hyperlink{AT49GAMMA5}{a}}} {\it 2}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{2004}&\multicolumn{1}{@{.}l}{3}&\multicolumn{1}{l}{23/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1705}&\multicolumn{1}{@{.}l}{03 }&\multicolumn{1}{@{}l}{21/2\ensuremath{^{-}}}&\multicolumn{1}{l}{[E1]}&&&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0325 {\it 5}}&\multicolumn{1}{r@{}}{5}&\multicolumn{1}{@{.}l}{0 {\it 7}}&&\\
\end{longtable}
\clearpage
\begin{longtable}{ccccccccc@{}ccccccc@{\extracolsep{\fill}}c}
\\[-.4cm]
\multicolumn{17}{c}{{\bf \small \underline{\ensuremath{^{\textnormal{165}}}Ho(\ensuremath{^{\textnormal{40}}}Ar,4n\ensuremath{\gamma})\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2014Au03,B}{2014Au03},\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2015Au01,B}{2015Au01} (continued)}}}\\
\multicolumn{17}{c}{~}\\
\multicolumn{17}{c}{\underline{$\gamma$($^{201}$At) (continued)}}\\
\multicolumn{17}{c}{~~~}\\
\multicolumn{2}{c}{E\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{AT49GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{I\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{AT49GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{E\ensuremath{_{i}}(level)}&J\ensuremath{^{\pi}_{i}}&\multicolumn{2}{c}{E\ensuremath{_{f}}}&J\ensuremath{^{\pi}_{f}}&Mult.\ensuremath{^{\hyperlink{AT49GAMMA10}{f}}}&\multicolumn{2}{c}{\ensuremath{\alpha}\ensuremath{^{\hyperlink{AT49GAMMA11}{g}}}}&\multicolumn{2}{c}{I\ensuremath{_{(\gamma+ce)}}\ensuremath{^{\hyperlink{AT49GAMMA6}{b}}}}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endhead
\multicolumn{1}{r@{}}{317}&\multicolumn{1}{@{.}l}{3 {\it 4}}&\multicolumn{1}{r@{}}{6}&\multicolumn{1}{@{.}l}{7 {\it 4}}&\multicolumn{1}{r@{}}{3983}&\multicolumn{1}{@{.}l}{8}&\multicolumn{1}{l}{(31/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{3666}&\multicolumn{1}{@{.}l}{5? }&\multicolumn{1}{@{}l}{(29/2\ensuremath{^{-}})}&\multicolumn{1}{l}{(M1)\ensuremath{^{\hyperlink{AT49GAMMA4}{\&}}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{472 {\it 7}}&&&\parbox[t][0.3cm]{8.306419cm}{\raggedright Mult.: A2={\textminus}0.81 \textit{9} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2015Au01,B}{2015Au01}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{335}&\multicolumn{1}{@{.}l}{0 {\it 4}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{5 {\it 2}}&\multicolumn{1}{r@{}}{4789}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{l}{(37/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{4454}&\multicolumn{1}{@{.}l}{0 }&\multicolumn{1}{@{}l}{(35/2\ensuremath{^{-}})}&\multicolumn{1}{l}{(M1)\ensuremath{^{\hyperlink{AT49GAMMA4}{\&}}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{407 {\it 6}}&&&\parbox[t][0.3cm]{8.306419cm}{\raggedright Mult.: A2={\textminus}0.66 \textit{4} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2015Au01,B}{2015Au01}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{339}&\multicolumn{1}{@{.}l}{2 {\it 2}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{2319}&\multicolumn{1}{@{.}l}{8}&\multicolumn{1}{l}{29/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1980}&\multicolumn{1}{@{.}l}{6 }&\multicolumn{1}{@{}l}{23/2\ensuremath{^{-}}}&\multicolumn{1}{l}{E3}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{464 {\it 7}}&\multicolumn{1}{r@{}}{14}&\multicolumn{1}{@{ }l}{{\it 5}}&\parbox[t][0.3cm]{8.306419cm}{\raggedright Mult.: From K/(L+M+...)exp=0.45 \textit{4} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2015Au01,B}{2015Au01}). The 339\ensuremath{\gamma}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.306419cm}{\raggedright {\ }{\ }{\ }K-conversion peak overlaps with the L+M+... conversion\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.306419cm}{\raggedright {\ }{\ }{\ }peaks from the 269- and 276-keV transitions.\hphantom{a}The number\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.306419cm}{\raggedright {\ }{\ }{\ }of 269- and 276-keV L+M+... conversion events were\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.306419cm}{\raggedright {\ }{\ }{\ }estimated using I\ensuremath{\gamma}(269\ensuremath{\gamma})/I\ensuremath{\gamma}(276\ensuremath{\gamma}), the number of\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.306419cm}{\raggedright {\ }{\ }{\ }observed 269- and 276-keV K conversion events, and the\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.306419cm}{\raggedright {\ }{\ }{\ }theoretical K/(L+M+...) ratios for the 269-keV E2 and\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.306419cm}{\raggedright {\ }{\ }{\ }276-keV M1 transitions.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{364}&\multicolumn{1}{@{.}l}{1\ensuremath{^{\hyperlink{AT49GAMMA5}{a}}} {\it 3}}&\multicolumn{1}{r@{}}{6}&\multicolumn{1}{@{.}l}{5 {\it 3}}&\multicolumn{1}{r@{}}{1625}&\multicolumn{1}{@{.}l}{34}&\multicolumn{1}{l}{17/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1261}&\multicolumn{1}{@{.}l}{29 }&\multicolumn{1}{@{}l}{15/2\ensuremath{^{+}}}&\multicolumn{1}{l}{(M1)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{325 {\it 5}}&\multicolumn{1}{r@{}}{8}&\multicolumn{1}{@{ }l}{{\it 2}}&\parbox[t][0.3cm]{8.306419cm}{\raggedright Mult.: A2={\textminus}0.11 \textit{3} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2015Au01,B}{2015Au01}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{371}&\multicolumn{1}{@{.}l}{7 {\it 4}}&\multicolumn{1}{r@{}}{10}&\multicolumn{1}{@{.}l}{9 {\it 4}}&\multicolumn{1}{r@{}}{2518}&\multicolumn{1}{@{.}l}{9}&\multicolumn{1}{l}{25/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{2147}&\multicolumn{1}{@{.}l}{2 }&\multicolumn{1}{@{}l}{21/2\ensuremath{^{-}}}&\multicolumn{1}{l}{(E2)}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{8.306419cm}{\raggedright Mult.: A2=+0.16 \textit{6} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2015Au01,B}{2015Au01}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{402}&\multicolumn{1}{@{.}l}{6 {\it 4}}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{6\ensuremath{^{\hyperlink{AT49GAMMA2}{\#}}} {\it 3}}&\multicolumn{1}{r@{}}{3621}&\multicolumn{1}{@{.}l}{8}&&\multicolumn{1}{r@{}}{3219}&\multicolumn{1}{@{.}l}{2 }&&\multicolumn{1}{l}{D}&&&&&\parbox[t][0.3cm]{8.306419cm}{\raggedright Mult.: A2={\textminus}0.7 \textit{2} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2015Au01,B}{2015Au01}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{426}&\multicolumn{1}{@{.}l}{5\ensuremath{^{\hyperlink{AT49GAMMA5}{a}}} {\it 2}}&\multicolumn{1}{r@{}}{39}&\multicolumn{1}{@{ }l}{{\it 2}}&\multicolumn{1}{r@{}}{1921}&\multicolumn{1}{@{.}l}{30}&\multicolumn{1}{l}{21/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1494}&\multicolumn{1}{@{.}l}{85 }&\multicolumn{1}{@{}l}{17/2\ensuremath{^{+}}}&\multicolumn{1}{l}{(E2)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0497 {\it 7}}&\multicolumn{1}{r@{}}{69}&\multicolumn{1}{@{ }l}{{\it 9}}&\parbox[t][0.3cm]{8.306419cm}{\raggedright Mult.: A2=+0.14 \textit{7} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2015Au01,B}{2015Au01}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{433}&\multicolumn{1}{@{.}l}{3\ensuremath{^{\hyperlink{AT49GAMMA7}{c}}} {\it 4}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{AT49GAMMA7}{c}}} {\it 4}}&\multicolumn{1}{r@{}}{1065}&\multicolumn{1}{@{.}l}{2}&\multicolumn{1}{l}{7/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{631}&\multicolumn{1}{@{.}l}{8 }&\multicolumn{1}{@{}l}{3/2\ensuremath{^{+}}}&\multicolumn{1}{l}{(E2)}&&&&&\parbox[t][0.3cm]{8.306419cm}{\raggedright Mult.: R=1.21 \textit{5} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2014Au03,B}{2014Au03}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{442}&\multicolumn{1}{@{.}l}{6 {\it 4}}&\multicolumn{1}{r@{}}{6}&\multicolumn{1}{@{.}l}{0 {\it 2}}&\multicolumn{1}{r@{}}{2147}&\multicolumn{1}{@{.}l}{2}&\multicolumn{1}{l}{21/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{1705}&\multicolumn{1}{@{.}l}{03 }&\multicolumn{1}{@{}l}{21/2\ensuremath{^{-}}}&\multicolumn{1}{l}{(E2)}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{8.306419cm}{\raggedright Mult.: A2=+0.35 \textit{8} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2015Au01,B}{2015Au01}); consistent with \ensuremath{\Delta}J=0\vspace{0.1cm}}&\\
&&&&&&&&&&&&&&&\parbox[t][0.3cm]{8.306419cm}{\raggedright {\ }{\ }{\ }transition.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{448}&\multicolumn{1}{@{.}l}{4 {\it 4}}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{9\ensuremath{^{\hyperlink{AT49GAMMA3}{@}}} {\it 2}}&\multicolumn{1}{r@{}}{3952}&\multicolumn{1}{@{.}l}{8}&&\multicolumn{1}{r@{}}{3504}&\multicolumn{1}{@{.}l}{4 }&&\multicolumn{1}{l}{D}&&&&&\parbox[t][0.3cm]{8.306419cm}{\raggedright Mult.: A2={\textminus}0.3 \textit{2} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2015Au01,B}{2015Au01}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{448}&\multicolumn{1}{@{.}l}{5 {\it 4}}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{8\ensuremath{^{\hyperlink{AT49GAMMA2}{\#}}} {\it 2}}&\multicolumn{1}{r@{}}{3693}&\multicolumn{1}{@{.}l}{9}&&\multicolumn{1}{r@{}}{3245}&\multicolumn{1}{@{.}l}{4 }&\multicolumn{1}{@{}l}{(29/2\ensuremath{^{+}})}&\multicolumn{1}{l}{D}&&&&&\parbox[t][0.3cm]{8.306419cm}{\raggedright Mult.: A2={\textminus}0.4 \textit{2} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2015Au01,B}{2015Au01}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{476}&\multicolumn{1}{@{.}l}{2\ensuremath{^{\hyperlink{AT49GAMMA5}{a}}} {\it 2}}&\multicolumn{1}{r@{}}{33}&\multicolumn{1}{@{ }l}{{\it 1}}&\multicolumn{1}{r@{}}{1705}&\multicolumn{1}{@{.}l}{03}&\multicolumn{1}{l}{21/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{1228}&\multicolumn{1}{@{.}l}{96 }&\multicolumn{1}{@{}l}{17/2\ensuremath{^{-}}}&\multicolumn{1}{l}{(E2)}&&&\multicolumn{1}{r@{}}{21}&\multicolumn{1}{@{ }l}{{\it 4}}&\parbox[t][0.3cm]{8.306419cm}{\raggedright Mult.: A2=+0.13 \textit{4} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2015Au01,B}{2015Au01}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{484}&\multicolumn{1}{@{.}l}{5\ensuremath{^{\hyperlink{AT49GAMMA7}{c}}} {\it 4}}&\multicolumn{1}{r@{}}{41}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{AT49GAMMA7}{c}}} {\it 3}}&\multicolumn{1}{r@{}}{1288}&\multicolumn{1}{@{.}l}{6}&\multicolumn{1}{l}{9/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{804}&\multicolumn{1}{@{.}l}{2 }&\multicolumn{1}{@{}l}{5/2\ensuremath{^{+}}}&\multicolumn{1}{l}{(E2)}&&&&&\parbox[t][0.3cm]{8.306419cm}{\raggedright Mult.: A2=+0.70 \textit{6} and R=1.4 \textit{2} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2014Au03,B}{2014Au03}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{511}&\multicolumn{1}{@{.}l}{8\ensuremath{^{\hyperlink{AT49GAMMA5}{a}}} {\it 2}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{1261}&\multicolumn{1}{@{.}l}{29}&\multicolumn{1}{l}{15/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{749}&\multicolumn{1}{@{.}l}{36 }&\multicolumn{1}{@{}l}{13/2\ensuremath{^{+}}}&\multicolumn{1}{l}{D}&&&\multicolumn{1}{r@{}}{38}&\multicolumn{1}{@{ }l}{{\it 7}}&\parbox[t][0.3cm]{8.306419cm}{\raggedright Mult.: A2={\textminus}0.14 \textit{3} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2015Au01,B}{2015Au01}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{538}&\multicolumn{1}{@{.}l}{2 {\it 4}}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{9\ensuremath{^{\hyperlink{AT49GAMMA3}{@}}} {\it 2}}&\multicolumn{1}{r@{}}{3779}&\multicolumn{1}{@{.}l}{1}&&\multicolumn{1}{r@{}}{3240}&\multicolumn{1}{@{.}l}{9 }&\multicolumn{1}{@{}l}{(33/2\ensuremath{^{+}})}&&&&&&&\\
\multicolumn{1}{r@{}}{540}&\multicolumn{1}{@{.}l}{5 {\it 5}}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{1\ensuremath{^{\hyperlink{AT49GAMMA2}{\#}}} {\it 3}}&\multicolumn{1}{r@{}}{3785}&\multicolumn{1}{@{.}l}{9}&&\multicolumn{1}{r@{}}{3245}&\multicolumn{1}{@{.}l}{4 }&\multicolumn{1}{@{}l}{(29/2\ensuremath{^{+}})}&&&&&&&\\
\multicolumn{1}{r@{}}{548}&\multicolumn{1}{@{.}l}{1\ensuremath{^{\hyperlink{AT49GAMMA7}{c}}} {\it 4}}&\multicolumn{1}{r@{}}{48}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{AT49GAMMA7}{c}}} {\it 2}}&\multicolumn{1}{r@{}}{1613}&\multicolumn{1}{@{.}l}{3}&\multicolumn{1}{l}{11/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1065}&\multicolumn{1}{@{.}l}{2 }&\multicolumn{1}{@{}l}{7/2\ensuremath{^{+}}}&\multicolumn{1}{l}{(E2)}&&&&&\parbox[t][0.3cm]{8.306419cm}{\raggedright Mult.: A2=+0.5 \textit{2} and R=1.26 \textit{12} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2014Au03,B}{2014Au03}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{567}&\multicolumn{1}{@{.}l}{3\ensuremath{^{\hyperlink{AT49GAMMA7}{c}}} {\it 4}}&\multicolumn{1}{r@{}}{37}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{AT49GAMMA7}{c}}} {\it 2}}&\multicolumn{1}{r@{}}{1856}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{l}{13/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1288}&\multicolumn{1}{@{.}l}{6 }&\multicolumn{1}{@{}l}{9/2\ensuremath{^{+}}}&\multicolumn{1}{l}{(E2)}&&&&&\parbox[t][0.3cm]{8.306419cm}{\raggedright Mult.: A2=+0.5 \textit{3} and R=1.2 \textit{3} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2014Au03,B}{2014Au03}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{581}&\multicolumn{1}{@{.}l}{6 {\it 4}}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{1\ensuremath{^{\hyperlink{AT49GAMMA2}{\#}}} {\it 2}}&\multicolumn{1}{r@{}}{3219}&\multicolumn{1}{@{.}l}{2}&&\multicolumn{1}{r@{}}{2637}&\multicolumn{1}{@{.}l}{6 }&\multicolumn{1}{@{}l}{(25/2\ensuremath{^{+}})}&\multicolumn{1}{l}{D}&&&&&\parbox[t][0.3cm]{8.306419cm}{\raggedright Mult.: A2={\textminus}0.44 \textit{7} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2015Au01,B}{2015Au01}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{593}&\multicolumn{1}{@{.}l}{8\ensuremath{^{\hyperlink{AT49GAMMA5}{a}}} {\it 2}}&\multicolumn{1}{r@{}}{77}&\multicolumn{1}{@{ }l}{{\it 3}}&\multicolumn{1}{r@{}}{1228}&\multicolumn{1}{@{.}l}{96}&\multicolumn{1}{l}{17/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{635}&\multicolumn{1}{@{.}l}{17 }&\multicolumn{1}{@{}l}{13/2\ensuremath{^{-}}}&&&&\multicolumn{1}{r@{}}{25}&\multicolumn{1}{@{ }l}{{\it 4}}&\parbox[t][0.3cm]{8.306419cm}{\raggedright A2=+0.08 \textit{2} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2015Au01,B}{2015Au01}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{607}&\multicolumn{1}{@{.}l}{8 {\it 4}}&\multicolumn{1}{r@{}}{8}&\multicolumn{1}{@{.}l}{4\ensuremath{^{\hyperlink{AT49GAMMA2}{\#}}} {\it 8}}&\multicolumn{1}{r@{}}{3245}&\multicolumn{1}{@{.}l}{4}&\multicolumn{1}{l}{(29/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{2637}&\multicolumn{1}{@{.}l}{6 }&\multicolumn{1}{@{}l}{(25/2\ensuremath{^{+}})}&\multicolumn{1}{l}{(E2)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{02148 {\it 30}}&&&\parbox[t][0.3cm]{8.306419cm}{\raggedright Mult.: A2=+0.35 \textit{8} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2015Au01,B}{2015Au01}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{618}&\multicolumn{1}{@{.}l}{8\ensuremath{^{\hyperlink{AT49GAMMA7}{c}}} {\it 6}}&\multicolumn{1}{r@{}}{7}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{AT49GAMMA9}{e}\hyperlink{AT49GAMMA7}{c}}} {\it 3}}&\multicolumn{1}{r@{}}{2232}&\multicolumn{1}{@{.}l}{1}&\multicolumn{1}{l}{15/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1613}&\multicolumn{1}{@{.}l}{3 }&\multicolumn{1}{@{}l}{11/2\ensuremath{^{+}}}&\multicolumn{1}{l}{(E2)}&&&&&\parbox[t][0.3cm]{8.306419cm}{\raggedright Mult.: A2=+0.6 \textit{4} and R=1.1 \textit{4} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2014Au03,B}{2014Au03}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{635}&\multicolumn{1}{@{.}l}{1\ensuremath{^{\hyperlink{AT49GAMMA5}{a}}} {\it 2}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{ }l}{{\it 3}}&\multicolumn{1}{r@{}}{635}&\multicolumn{1}{@{.}l}{17}&\multicolumn{1}{l}{13/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{@{}l}{9/2\ensuremath{^{-}}}&&&&\multicolumn{1}{r@{}}{42}&\multicolumn{1}{@{ }l}{{\it 6}}&\parbox[t][0.3cm]{8.306419cm}{\raggedright A2=+0.09 \textit{4} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2015Au01,B}{2015Au01}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{691}&\multicolumn{1}{@{.}l}{1\ensuremath{^{\hyperlink{AT49GAMMA5}{a}}} {\it 2}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{690}&\multicolumn{1}{@{.}l}{98}&\multicolumn{1}{l}{11/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{@{}l}{9/2\ensuremath{^{-}}}&&&&\multicolumn{1}{r@{}}{11}&\multicolumn{1}{@{ }l}{{\it 2}}&&\\
\multicolumn{1}{r@{}}{716}&\multicolumn{1}{@{.}l}{3 {\it 4}}&\multicolumn{1}{r@{}}{11}&\multicolumn{1}{@{.}l}{9\ensuremath{^{\hyperlink{AT49GAMMA2}{\#}}} {\it 7}}&\multicolumn{1}{r@{}}{2637}&\multicolumn{1}{@{.}l}{6}&\multicolumn{1}{l}{(25/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{1921}&\multicolumn{1}{@{.}l}{30 }&\multicolumn{1}{@{}l}{21/2\ensuremath{^{+}}}&\multicolumn{1}{l}{(E2)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{01509 {\it 21}}&&&\parbox[t][0.3cm]{8.306419cm}{\raggedright Mult.: A2=+0.46 \textit{4}\hphantom{a}(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2015Au01,B}{2015Au01}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{745}&\multicolumn{1}{@{.}l}{5\ensuremath{^{\hyperlink{AT49GAMMA5}{a}}} {\it 2}}&\multicolumn{1}{r@{}}{74}&\multicolumn{1}{@{ }l}{{\it 2}}&\multicolumn{1}{r@{}}{1494}&\multicolumn{1}{@{.}l}{85}&\multicolumn{1}{l}{17/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{749}&\multicolumn{1}{@{.}l}{36 }&\multicolumn{1}{@{}l}{13/2\ensuremath{^{+}}}&\multicolumn{1}{l}{(E2)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{01389 {\it 19}}&\multicolumn{1}{r@{}}{72}&\multicolumn{1}{@{ }l}{{\it 8}}&\parbox[t][0.3cm]{8.306419cm}{\raggedright Mult.: A2=+0.4 \textit{2} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2015Au01,B}{2015Au01}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{749}&\multicolumn{1}{@{.}l}{3\ensuremath{^{\hyperlink{AT49GAMMA5}{a}}} {\it 2}}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{749}&\multicolumn{1}{@{.}l}{36}&\multicolumn{1}{l}{13/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{@{}l}{9/2\ensuremath{^{-}}}&\multicolumn{1}{l}{[M2]}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{1204 {\it 17}}&\multicolumn{1}{r@{}}{105}&\multicolumn{1}{@{ }l}{{\it 12}}&&\\
\multicolumn{1}{r@{}}{$^{x}$774}&\multicolumn{1}{@{.}l}{9\ensuremath{^{\hyperlink{AT49GAMMA1}{\ddagger}}} {\it 4}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{7\ensuremath{^{\hyperlink{AT49GAMMA3}{@}}} {\it 2}}&&&&&&&&&&&&\parbox[t][0.3cm]{8.306419cm}{\raggedright A2=+0.29 4 (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2015Au01,B}{2015Au01}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{870}&\multicolumn{1}{@{.}l}{5\ensuremath{^{\hyperlink{AT49GAMMA12}{h}}} {\it 4}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{93\ensuremath{^{\hyperlink{AT49GAMMA3}{@}}} {\it 8}}&\multicolumn{1}{r@{}}{4111}&\multicolumn{1}{@{.}l}{4}&&\multicolumn{1}{r@{}}{3240}&\multicolumn{1}{@{.}l}{9 }&\multicolumn{1}{@{}l}{(33/2\ensuremath{^{+}})}&&&&&&&\\
\multicolumn{1}{r@{}}{876}&\multicolumn{1}{@{.}l}{1\ensuremath{^{\hyperlink{AT49GAMMA5}{a}}} {\it 2}}&\multicolumn{1}{r@{}}{19}&\multicolumn{1}{@{.}l}{4 {\it 6}}&\multicolumn{1}{r@{}}{1625}&\multicolumn{1}{@{.}l}{34}&\multicolumn{1}{l}{17/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{749}&\multicolumn{1}{@{.}l}{36 }&\multicolumn{1}{@{}l}{13/2\ensuremath{^{+}}}&\multicolumn{1}{l}{(E2)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{01002 {\it 14}}&\multicolumn{1}{r@{}}{24}&\multicolumn{1}{@{ }l}{{\it 3}}&\parbox[t][0.3cm]{8.306419cm}{\raggedright Mult.: A2=+0.16 \textit{3} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2015Au01,B}{2015Au01}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{917}&\multicolumn{1}{@{.}l}{8 {\it 4}}&\multicolumn{1}{r@{}}{7}&\multicolumn{1}{@{.}l}{0 {\it 4}}&\multicolumn{1}{r@{}}{2147}&\multicolumn{1}{@{.}l}{2}&\multicolumn{1}{l}{21/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{1228}&\multicolumn{1}{@{.}l}{96 }&\multicolumn{1}{@{}l}{17/2\ensuremath{^{-}}}&\multicolumn{1}{l}{(E2)}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{}&\multicolumn{1}{@{}l}{}&\parbox[t][0.3cm]{8.306419cm}{\raggedright Mult.: A2=+0.37 \textit{6} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2015Au01,B}{2015Au01}).\vspace{0.1cm}}&\\
\end{longtable}
\clearpage
\begin{longtable}{ccccccccc@{}ccccc@{\extracolsep{\fill}}c}
\\[-.4cm]
\multicolumn{15}{c}{{\bf \small \underline{\ensuremath{^{\textnormal{165}}}Ho(\ensuremath{^{\textnormal{40}}}Ar,4n\ensuremath{\gamma})\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2014Au03,B}{2014Au03},\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2015Au01,B}{2015Au01} (continued)}}}\\
\multicolumn{15}{c}{~}\\
\multicolumn{15}{c}{\underline{$\gamma$($^{201}$At) (continued)}}\\
\multicolumn{15}{c}{~~~}\\
\multicolumn{2}{c}{E\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{AT49GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{I\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{AT49GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{E\ensuremath{_{i}}(level)}&J\ensuremath{^{\pi}_{i}}&\multicolumn{2}{c}{E\ensuremath{_{f}}}&J\ensuremath{^{\pi}_{f}}&Mult.\ensuremath{^{\hyperlink{AT49GAMMA10}{f}}}&\multicolumn{2}{c}{\ensuremath{\alpha}\ensuremath{^{\hyperlink{AT49GAMMA11}{g}}}}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endhead
\multicolumn{1}{r@{}}{921}&\multicolumn{1}{@{.}l}{1 {\it 4}}&\multicolumn{1}{r@{}}{10}&\multicolumn{1}{@{.}l}{1\ensuremath{^{\hyperlink{AT49GAMMA3}{@}}} {\it 6}}&\multicolumn{1}{r@{}}{3240}&\multicolumn{1}{@{.}l}{9}&\multicolumn{1}{l}{(33/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{2319}&\multicolumn{1}{@{.}l}{8 }&\multicolumn{1}{@{}l}{29/2\ensuremath{^{+}}}&\multicolumn{1}{l}{(E2)}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{00908 {\it 13}}&\parbox[t][0.3cm]{10.52756cm}{\raggedright Mult.: A2=+0.20 \textit{3} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2015Au01,B}{2015Au01}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1049}&\multicolumn{1}{@{.}l}{9\ensuremath{^{\hyperlink{AT49GAMMA12}{h}}} {\it 4}}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{2\ensuremath{^{\hyperlink{AT49GAMMA3}{@}}} {\it 2}}&\multicolumn{1}{r@{}}{3369}&\multicolumn{1}{@{.}l}{6}&&\multicolumn{1}{r@{}}{2319}&\multicolumn{1}{@{.}l}{8 }&\multicolumn{1}{@{}l}{29/2\ensuremath{^{+}}}&\multicolumn{1}{l}{D}&&&\parbox[t][0.3cm]{10.52756cm}{\raggedright Mult.: A2={\textminus}0.7 \textit{3} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2015Au01,B}{2015Au01}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1068}&\multicolumn{1}{@{.}l}{9\ensuremath{^{\hyperlink{AT49GAMMA12}{h}}} {\it 4}}&\multicolumn{1}{r@{}}{4}&\multicolumn{1}{@{.}l}{5 {\it 3}}&\multicolumn{1}{r@{}}{2990}&\multicolumn{1}{@{.}l}{2}&\multicolumn{1}{l}{(23/2\ensuremath{^{-}})}&\multicolumn{1}{r@{}}{1921}&\multicolumn{1}{@{.}l}{30 }&\multicolumn{1}{@{}l}{21/2\ensuremath{^{+}}}&&&&\parbox[t][0.3cm]{10.52756cm}{\raggedright A2={\textminus}0.47 \textit{5} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2015Au01,B}{2015Au01}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1184}&\multicolumn{1}{@{.}l}{5 {\it 4}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{7\ensuremath{^{\hyperlink{AT49GAMMA3}{@}}} {\it 2}}&\multicolumn{1}{r@{}}{3504}&\multicolumn{1}{@{.}l}{4}&&\multicolumn{1}{r@{}}{2319}&\multicolumn{1}{@{.}l}{8 }&\multicolumn{1}{@{}l}{29/2\ensuremath{^{+}}}&\multicolumn{1}{l}{D,Q}&&&\parbox[t][0.3cm]{10.52756cm}{\raggedright Mult.: A2=+0.40 \textit{8} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2015Au01,B}{2015Au01}).\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1379}&\multicolumn{1}{@{.}l}{9\ensuremath{^{\hyperlink{AT49GAMMA12}{h}}} {\it 5}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{4\ensuremath{^{\hyperlink{AT49GAMMA3}{@}}} {\it 2}}&\multicolumn{1}{r@{}}{3699}&\multicolumn{1}{@{.}l}{7}&&\multicolumn{1}{r@{}}{2319}&\multicolumn{1}{@{.}l}{8 }&\multicolumn{1}{@{}l}{29/2\ensuremath{^{+}}}&&&&&\\
\end{longtable}
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{AT49GAMMA0}{\dagger}}}} From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2015Au01,B}{2015Au01} using the JUROGAM2 data, unless otherwise stated. I\ensuremath{\gamma} normalized to I\ensuremath{\gamma}(635\ensuremath{\gamma})=100.}\\
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{AT49GAMMA1}{\ddagger}}}} Transition probably feeds the 2319-keV, 29/2\ensuremath{^{+}} level (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2015Au01,B}{2015Au01}).}\\
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{AT49GAMMA2}{\#}}}} From summed spectra gated on 296\ensuremath{\gamma} and 427\ensuremath{\gamma} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2015Au01,B}{2015Au01}).}\\
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{AT49GAMMA3}{@}}}} From 269\ensuremath{\gamma}, 427\ensuremath{\gamma}, 635\ensuremath{\gamma}, 746\ensuremath{\gamma}, or 749\ensuremath{\gamma} delayed \ensuremath{\gamma}-ray tagged singles spectrum (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2015Au01,B}{2015Au01}).}\\
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{AT49GAMMA4}{\&}}}} In addition to stretched dipole angular distributions, M1 character supported by high x-ray yield in coincidence with transition (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2015Au01,B}{2015Au01}).}\\
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{AT49GAMMA5}{a}}}} From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2015Au01,B}{2015Au01} using the focal-plane Clover data.}\\
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{AT49GAMMA6}{b}}}} From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2015Au01,B}{2015Au01}. The focal plane values deduced from the focal-plane Clover data, unless otherwise stated. Normalized to I\ensuremath{\gamma}(269\ensuremath{\gamma})=100. Internal conversion}\\
\parbox[b][0.3cm]{21.881866cm}{{\ }{\ }coefficients that were used to calculate I(g\ensuremath{\pm}ce) were taken from \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2008Ki07,B}{2008Ki07}.}\\
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{AT49GAMMA7}{c}}}} From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2014Au03,B}{2014Au03}. I\ensuremath{\gamma} above the \ensuremath{J^{\pi}}=1/2\ensuremath{^{+}} isomer are from recoil-ce-tagged singles \ensuremath{\gamma}-ray data.}\\
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{AT49GAMMA8}{d}}}} Doublet in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2014Au03,B}{2014Au03}. Intensities of the two components from 173-gated, recoil-corrected \ensuremath{\gamma}\ensuremath{\gamma}-coin data.}\\
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{AT49GAMMA9}{e}}}} Weak transition in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2014Au03,B}{2014Au03}, intensity from 173-keV gated, recoil-correlated \ensuremath{\gamma}\ensuremath{\gamma}-coin data.}\\
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{AT49GAMMA10}{f}}}} From \ensuremath{\gamma}(\ensuremath{\theta}) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2014Au03,B}{2014Au03} and \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2015Au01,B}{2015Au01}, unless otherwise stated. The reported in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2014Au03,B}{2014Au03} correlation ratios, R,\hphantom{a}are defined as}\\
\parbox[b][0.3cm]{21.881866cm}{{\ }{\ }R=[I\ensuremath{\gamma}(133.6\ensuremath{^\circ})+I\ensuremath{\gamma}(157.6\ensuremath{^\circ})]/I\ensuremath{\gamma}(104.5\ensuremath{^\circ} or 75.5\ensuremath{^\circ}). Expected values are 1.30 \textit{7} for \ensuremath{\Delta}J=2, quadrupole, and 0.70 \textit{6} for \ensuremath{\Delta}J=1, dipole transitions.}\\
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{AT49GAMMA11}{g}}}} Total theoretical internal conversion coefficients, calculated using the BrIcc code (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2008Ki07,B}{2008Ki07}) with Frozen orbital approximation based on \ensuremath{\gamma}-ray energies,}\\
\parbox[b][0.3cm]{21.881866cm}{{\ }{\ }assigned multipolarities, and mixing ratios, unless otherwise specified.}\\
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{AT49GAMMA12}{h}}}} Placement of transition in the level scheme is uncertain.}\\
\parbox[b][0.3cm]{21.881866cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{AT49GAMMA13}{x}}}} \ensuremath{\gamma} ray not placed in level scheme.}\\
\vspace{0.5cm}
\end{landscape}\clearpage
\clearpage
\begin{figure}[h]
\begin{center}
\includegraphics{201AT49-0.ps}\\
\end{center}
\end{figure}
\clearpage
\begin{figure}[h]
\begin{center}
\includegraphics{201AT49-1.ps}\\
\end{center}
\end{figure}
\clearpage
\begin{figure}[h]
\begin{center}
\includegraphics{201AT49-2.ps}\\
\end{center}
\end{figure}
\clearpage
\clearpage
\begin{figure}[h]
\begin{center}
\includegraphics{201AT49B-0.ps}\\
\end{center}
\end{figure}
\clearpage
\section[\ensuremath{^{201}_{\ 86}}Rn\ensuremath{_{115}^{~}}]{ }
\vspace{-30pt}
\setcounter{chappage}{1}
\subsection[\hspace{-0.2cm}Adopted Levels, Gammas]{ }
\vspace{-20pt}
\vspace{0.3cm}
\hypertarget{RN50}{{\bf \small \underline{Adopted \hyperlink{201RN_LEVEL}{Levels}, \hyperlink{201RN_GAMMA}{Gammas}}}}\\
\vspace{4pt}
\vspace{8pt}
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}Q(\ensuremath{\beta^-})=$-$7696 {\it 14}; S(n)=8178 {\it 12}; S(p)=2408 {\it 26}; Q(\ensuremath{\alpha})=6860.7 {\it 23}\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2021Wa16,B}{2021Wa16}}\\
\vspace{12pt}
\hypertarget{201RN_LEVEL}{\underline{$^{201}$Rn Levels}}\\
\begin{longtable}[c]{ll}
\multicolumn{2}{c}{\underline{Cross Reference (XREF) Flags}}\\
\\
\hyperlink{RA51}{\texttt{A }}& \ensuremath{^{\textnormal{205}}}Ra \ensuremath{\alpha} decay (210 ms)\\
\hyperlink{RA52}{\texttt{B }}& \ensuremath{^{\textnormal{205}}}Ra \ensuremath{\alpha} decay (170 ms)\\
\hyperlink{RN53}{\texttt{C }}& \ensuremath{^{\textnormal{122}}}Sn(\ensuremath{^{\textnormal{82}}}Kr,3n\ensuremath{\gamma})\\
\end{longtable}
\vspace{-0.5cm}
\begin{longtable}{ccccccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E(level)$^{{\hyperlink{RN50LEVEL0}{\dagger}}}$}&J$^{\pi}$$^{{\hyperlink{RN50LEVEL1}{\ddagger}}}$&\multicolumn{2}{c}{T$_{1/2}$$^{}$}&XREF&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{l}{3/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{7}&\multicolumn{1}{@{.}l}{0 s {\it 4}}&\multicolumn{1}{l}{\texttt{\hyperlink{RA51}{A}\ \hyperlink{RN53}{C}} }&\parbox[t][0.3cm]{11.88868cm}{\raggedright \%\ensuremath{\alpha}=?; \%\ensuremath{\varepsilon}+\%\ensuremath{\beta}\ensuremath{^{\textnormal{+}}}=?\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.88868cm}{\raggedright Using \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2019Mo01,B}{2019Mo01} predictions of T\ensuremath{_{\textnormal{1/2}}}(\ensuremath{\beta}\ensuremath{^{\textnormal{+}}}+\ensuremath{\varepsilon})=18.05 s\hphantom{a}and T\ensuremath{_{\textnormal{1/2}}}(\ensuremath{\alpha})=4.68 s, one gets\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.88868cm}{\raggedright {\ }{\ }{\ }\%\ensuremath{\alpha}=55 and \%\ensuremath{\varepsilon}+\%\ensuremath{\beta}\ensuremath{^{\textnormal{+}}}=41 using T\ensuremath{_{\textnormal{1/2}}}(exp)=7.0 s.\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.88868cm}{\raggedright J\ensuremath{^{\pi}}: Favored \ensuremath{\alpha}-decay to \ensuremath{^{\textnormal{197}}}Po g.s. (\ensuremath{J^{\pi}}=3/2\ensuremath{^{-}}, \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2021Ko07,B}{2021Ko07}); systematics of levels in\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.88868cm}{\raggedright {\ }{\ }{\ }neighboring nuclei.\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.88868cm}{\raggedright T\ensuremath{_{1/2}}: Weighted average of 7.0 s \textit{4} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1971Ho01,B}{1971Ho01}) and 7.1 s \textit{8} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1996Ta18,B}{1996Ta18}). Other: 6.7 s\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.88868cm}{\raggedright {\ }{\ }{\ }\textit{+51{\textminus}20} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1987He10,B}{1987He10}).\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.88868cm}{\raggedright E\ensuremath{\alpha}1=6725 keV \textit{2}, correlated with E\ensuremath{\alpha}2(\ensuremath{^{\textnormal{197}}}Po)=6281 keV \textit{2} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1996Ta18,B}{1996Ta18}). The quoted\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.88868cm}{\raggedright {\ }{\ }{\ }uncertainty is statistical only. Others: 6727 keV \textit{7} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1995Le04,B}{1995Le04}), 6723.7 keV \textit{25}\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.88868cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1993Wa04,B}{1993Wa04}), 6730 keV \textit{20} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1987He10,B}{1987He10}) and 6721 keV \textit{8} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1971Ho01,B}{1971Ho01}).\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.88868cm}{\raggedright configuration: \ensuremath{\nu} p\ensuremath{_{\textnormal{3/2}}^{\textnormal{$-$1}}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{245}&\multicolumn{1}{@{ }l}{{\it 12}}&\multicolumn{1}{l}{13/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{8 s {\it 1}}&\multicolumn{1}{l}{\texttt{\ \hyperlink{RA52}{B}\hyperlink{RN53}{C}} }&\parbox[t][0.3cm]{11.88868cm}{\raggedright \%\ensuremath{\alpha}=?; \%\ensuremath{\varepsilon}+\%\ensuremath{\beta}\ensuremath{^{\textnormal{+}}}=?\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.88868cm}{\raggedright E(level): From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2021Ko07,B}{2021Ko07}, based on the least-square adjustment of the atomic masses\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.88868cm}{\raggedright {\ }{\ }{\ }and the \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2017Al34,B}{2017Al34} data.\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.88868cm}{\raggedright J\ensuremath{^{\pi}}: Favored \ensuremath{\alpha}-decay to \ensuremath{^{\textnormal{197}}}Po isomeric state (\ensuremath{J^{\pi}}=13/2\ensuremath{^{+}}, \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2021Ko07,B}{2021Ko07}); systematics of\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.88868cm}{\raggedright {\ }{\ }{\ }levels in neighboring nuclei.\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.88868cm}{\raggedright T\ensuremath{_{1/2}}: Weighted average of 3.8 s \textit{1} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1996Ta18,B}{1996Ta18}) and 3.8 s \textit{4} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1971Ho01,B}{1971Ho01}). Others: 3.0 s\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.88868cm}{\raggedright {\ }{\ }{\ }\textit{15} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1967Va17,B}{1967Va17}) and 2.7 s \textit{+14{\textminus}7} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1987He10,B}{1987He10}).\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.88868cm}{\raggedright E\ensuremath{\alpha}1=6773 keV \textit{2}, correlated with E\ensuremath{\alpha}2(\ensuremath{^{\textnormal{197}}}Po)=6380 keV \textit{2} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1996Ta18,B}{1996Ta18}). The quoted\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.88868cm}{\raggedright {\ }{\ }{\ }uncertainty is statistical only. Others: 6787 keV \textit{30} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2010He25,B}{2010He25}), 6778 keV \textit{7}\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.88868cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1995Le04,B}{1995Le04}), 6772.1 keV \textit{25} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1993Wa04,B}{1993Wa04}), 6770 keV \textit{20} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1987He10,B}{1987He10}), 6770 keV \textit{8}\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.88868cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1971Ho01,B}{1971Ho01}) and 6768 keV \textit{5} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1967Va17,B}{1967Va17}).\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.88868cm}{\raggedright configuration: \ensuremath{\nu} i\ensuremath{_{\textnormal{13/2}}^{\textnormal{$-$1}}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{718}&\multicolumn{1}{@{.}l}{30 {\it 20}}&\multicolumn{1}{l}{(17/2\ensuremath{^{+}})}&&&\multicolumn{1}{l}{\texttt{\ \ \hyperlink{RN53}{C}} }&\parbox[t][0.3cm]{11.88868cm}{\raggedright J\ensuremath{^{\pi}}: 473.3\ensuremath{\gamma} to 13/2\ensuremath{^{+}}; systematics of similar structures in neighboring nuclei.\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.88868cm}{\raggedright configuration: \ensuremath{\nu} (i\ensuremath{_{\textnormal{13/2}}^{\textnormal{$-$1}}})\ensuremath{\otimes}2\ensuremath{^{\textnormal{+}}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1266}&\multicolumn{1}{@{.}l}{2 {\it 3}}&\multicolumn{1}{l}{(21/2\ensuremath{^{+}})}&&&\multicolumn{1}{l}{\texttt{\ \ \hyperlink{RN53}{C}} }&\parbox[t][0.3cm]{11.88868cm}{\raggedright J\ensuremath{^{\pi}}: 548.1\ensuremath{\gamma} to (17/2\ensuremath{^{+}}); systematics of similar structures in neighboring nuclei.\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.88868cm}{\raggedright configuration: \ensuremath{\nu} (i\ensuremath{_{\textnormal{13/2}}^{\textnormal{$-$1}}})\ensuremath{\otimes}4\ensuremath{^{\textnormal{+}}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1364}&\multicolumn{1}{@{.}l}{9? {\it 4}}&\multicolumn{1}{l}{(21/2\ensuremath{^{+}})}&&&\multicolumn{1}{l}{\texttt{\ \ \hyperlink{RN53}{C}} }&\parbox[t][0.3cm]{11.88868cm}{\raggedright J\ensuremath{^{\pi}}: 646.2g to (17/2\ensuremath{^{+}}); systematics of similar structures in neighboring nuclei.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1819}&\multicolumn{1}{@{.}l}{3? {\it 4}}&\multicolumn{1}{l}{(25/2\ensuremath{^{+}})}&&&\multicolumn{1}{l}{\texttt{\ \ \hyperlink{RN53}{C}} }&\parbox[t][0.3cm]{11.88868cm}{\raggedright J\ensuremath{^{\pi}}: 553.5\ensuremath{\gamma} to (21/2\ensuremath{^{+}}); systematics of similar structures in neighboring nuclei.\vspace{0.1cm}}&\\
\end{longtable}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{RN50LEVEL0}{\dagger}}}} From a least-squares fit to E\ensuremath{\gamma}, unless otherwise stated.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{RN50LEVEL1}{\ddagger}}}} From \ensuremath{^{\textnormal{122}}}Sn(\ensuremath{^{\textnormal{82}}}Kr,3n\ensuremath{\gamma}) (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2008An05,B}{2008An05}), unless otherwise stated.}\\
\vspace{0.5cm}
\hypertarget{201RN_GAMMA}{\underline{$\gamma$($^{201}$Rn)}}\\
\begin{longtable}{ccccccccc@{}c@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E\ensuremath{_{i}}(level)}&J\ensuremath{^{\pi}_{i}}&\multicolumn{2}{c}{E\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{RN50GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{I\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{RN50GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{E\ensuremath{_{f}}}&J\ensuremath{^{\pi}_{f}}&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{718}&\multicolumn{1}{@{.}l}{30}&\multicolumn{1}{l}{(17/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{473}&\multicolumn{1}{@{.}l}{3 {\it 2}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{245}&\multicolumn{1}{@{ }l}{}&\multicolumn{1}{@{}l}{13/2\ensuremath{^{+}}}&\\
\multicolumn{1}{r@{}}{1266}&\multicolumn{1}{@{.}l}{2}&\multicolumn{1}{l}{(21/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{548}&\multicolumn{1}{@{.}l}{1 {\it 2}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{718}&\multicolumn{1}{@{.}l}{30 }&\multicolumn{1}{@{}l}{(17/2\ensuremath{^{+}})}&\\
\multicolumn{1}{r@{}}{1364}&\multicolumn{1}{@{.}l}{9?}&\multicolumn{1}{l}{(21/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{646}&\multicolumn{1}{@{.}l}{2\ensuremath{^{\hyperlink{RN50GAMMA1}{\ddagger}}} {\it 3}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{718}&\multicolumn{1}{@{.}l}{30 }&\multicolumn{1}{@{}l}{(17/2\ensuremath{^{+}})}&\\
\end{longtable}
\begin{textblock}{29}(0,27.3)
Continued on next page (footnotes at end of table)
\end{textblock}
\clearpage
\begin{longtable}{ccccccccc@{}c@{\extracolsep{\fill}}c}
\\[-.4cm]
\multicolumn{11}{c}{{\bf \small \underline{Adopted \hyperlink{201RN_LEVEL}{Levels}, \hyperlink{201RN_GAMMA}{Gammas} (continued)}}}\\
\multicolumn{11}{c}{~}\\
\multicolumn{11}{c}{\underline{$\gamma$($^{201}$Rn) (continued)}}\\
\multicolumn{11}{c}{~~~}\\
\multicolumn{2}{c}{E\ensuremath{_{i}}(level)}&J\ensuremath{^{\pi}_{i}}&\multicolumn{2}{c}{E\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{RN50GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{I\ensuremath{_{\gamma}}\ensuremath{^{\hyperlink{RN50GAMMA0}{\dagger}}}}&\multicolumn{2}{c}{E\ensuremath{_{f}}}&J\ensuremath{^{\pi}_{f}}&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endhead
\multicolumn{1}{r@{}}{1819}&\multicolumn{1}{@{.}l}{3?}&\multicolumn{1}{l}{(25/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{454}&\multicolumn{1}{@{.}l}{0\ensuremath{^{\hyperlink{RN50GAMMA1}{\ddagger}}} {\it 3}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{ }l}{{\it 17}}&\multicolumn{1}{r@{}}{1364}&\multicolumn{1}{@{.}l}{9? }&\multicolumn{1}{@{}l}{(21/2\ensuremath{^{+}})}&\\
&&&\multicolumn{1}{r@{}}{553}&\multicolumn{1}{@{.}l}{5\ensuremath{^{\hyperlink{RN50GAMMA1}{\ddagger}}} {\it 3}}&\multicolumn{1}{r@{}}{83}&\multicolumn{1}{@{ }l}{{\it 14}}&\multicolumn{1}{r@{}}{1266}&\multicolumn{1}{@{.}l}{2 }&\multicolumn{1}{@{}l}{(21/2\ensuremath{^{+}})}&\\
\end{longtable}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{RN50GAMMA0}{\dagger}}}} From \ensuremath{^{\textnormal{122}}}Sn(\ensuremath{^{\textnormal{82}}}Kr,3n\ensuremath{\gamma}) (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2008An05,B}{2008An05}).}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{RN50GAMMA1}{\ddagger}}}} Placement of transition in the level scheme is uncertain.}\\
\vspace{0.5cm}
\begin{figure}[h]
\begin{center}
\includegraphics{201RN50-0.ps}\\
\end{center}
\end{figure}
\clearpage
\subsection[\hspace{-0.2cm}\ensuremath{^{\textnormal{205}}}Ra \ensuremath{\alpha} decay (210 ms)]{ }
\vspace{-27pt}
\vspace{0.3cm}
\hypertarget{RA51}{{\bf \small \underline{\ensuremath{^{\textnormal{205}}}Ra \ensuremath{\alpha} decay (210 ms)\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1996Le09,B}{1996Le09},\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1987He10,B}{1987He10}}}}\\
\vspace{4pt}
\vspace{8pt}
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}Parent: $^{205}$Ra: E=0.0; J$^{\pi}$=(3/2\ensuremath{^{-}}); T$_{1/2}$=210 ms {\it +60\textminus40}; Q(\ensuremath{\alpha})=7486 {\it 20}; \%\ensuremath{\alpha} decay$\approx$100.0
}\\
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}\ensuremath{^{205}}Ra-J$^{\pi}$,T$_{1/2}$: From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2020Ko17,B}{2020Ko17}.}\\
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}\ensuremath{^{205}}Ra-Q(\ensuremath{\alpha}): From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2021Wa16,B}{2021Wa16}.}\\
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}\ensuremath{^{205}}Ra-\%\ensuremath{\alpha} decay: From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2020Ko17,B}{2020Ko17}.}\\
\vspace{12pt}
\underline{$^{201}$Rn Levels}\\
\begin{longtable}{ccccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E(level)$^{}$}&J$^{\pi}$$^{}$&\multicolumn{2}{c}{T$_{1/2}$$^{}$}&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&
\endfirsthead
\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{l}{3/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{7}&\multicolumn{1}{@{.}l}{0 s {\it 4}}&\\
\end{longtable}
\underline{\ensuremath{\alpha} radiations}\\
\begin{longtable}{ccccccccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E$\alpha^{{}}$}&\multicolumn{2}{c}{E(level)}&\multicolumn{2}{c}{I$\alpha^{{\hyperlink{RN51DECAY1}{\ddagger}}}$}&\multicolumn{2}{c}{HF$^{{\hyperlink{RN51DECAY0}{\dagger}}}$}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{7340}&\multicolumn{1}{@{ }l}{{\it 20}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{r@{}}{$\approx$100}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{$\approx$1}&\multicolumn{1}{@{.}l}{4}&\parbox[t][0.3cm]{12.855961cm}{\raggedright E$\alpha$: From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1996Le09,B}{1996Le09}. Others:\hphantom{a}7350 keV \textit{25} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1995Le15,B}{1995Le15}), 7355 keV \textit{10} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1995Le04,B}{1995Le04}) and 7360\vspace{0.1cm}}&\\
&&&&&&&&\parbox[t][0.3cm]{12.855961cm}{\raggedright {\ }{\ }{\ }keV \textit{20} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1987He10,B}{1987He10}).\vspace{0.1cm}}&\\
\end{longtable}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{RN51DECAY0}{\dagger}}}} Using r\ensuremath{_{\textnormal{0}}}(\ensuremath{^{\textnormal{201}}}Rn)=1.527 \textit{9} from \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2020Si16,B}{2020Si16}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{RN51DECAY1}{\ddagger}}}} For absolute intensity per 100 decays, multiply by{ }\ensuremath{\approx}1.0.}\\
\vspace{0.5cm}
\clearpage
\subsection[\hspace{-0.2cm}\ensuremath{^{\textnormal{205}}}Ra \ensuremath{\alpha} decay (170 ms)]{ }
\vspace{-27pt}
\vspace{0.3cm}
\hypertarget{RA52}{{\bf \small \underline{\ensuremath{^{\textnormal{205}}}Ra \ensuremath{\alpha} decay (170 ms)\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1996Le09,B}{1996Le09},\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1987He10,B}{1987He10}}}}\\
\vspace{4pt}
\vspace{8pt}
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}Parent: $^{205}$Ra: E=263 {\it 25}; J$^{\pi}$=13/2\ensuremath{^{+}}; T$_{1/2}$=170 ms {\it +60\textminus40}; Q(\ensuremath{\alpha})=7486 {\it 20}; \%\ensuremath{\alpha} decay$\approx$100.0
}\\
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}\ensuremath{^{205}}Ra-T$_{1/2}$: From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2020Ko17,B}{2020Ko17}.}\\
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}\ensuremath{^{205}}Ra-E,J$^{\pi}$: From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2021Ko07,B}{2021Ko07}.}\\
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}\ensuremath{^{205}}Ra-Q(\ensuremath{\alpha}): From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2021Wa16,B}{2021Wa16}.}\\
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}\ensuremath{^{205}}Ra-\%\ensuremath{\alpha} decay: From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2020Ko17,B}{2020Ko17}.}\\
\vspace{12pt}
\underline{$^{201}$Rn Levels}\\
\begin{longtable}{ccccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E(level)$^{}$}&J$^{\pi}$$^{}$&\multicolumn{2}{c}{T$_{1/2}$$^{}$}&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&
\endfirsthead
\multicolumn{1}{r@{}}{245}&\multicolumn{1}{@{ }l}{{\it 12}}&\multicolumn{1}{l}{13/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{8 s {\it 1}}&\\
\end{longtable}
\underline{\ensuremath{\alpha} radiations}\\
\begin{longtable}{ccccccccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E$\alpha^{{}}$}&\multicolumn{2}{c}{E(level)}&\multicolumn{2}{c}{I$\alpha^{{\hyperlink{RN52DECAY1}{\ddagger}}}$}&\multicolumn{2}{c}{HF$^{{\hyperlink{RN52DECAY0}{\dagger}}}$}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{7359}&\multicolumn{1}{@{ }l}{{\it 9}}&\multicolumn{1}{r@{}}{245}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{$\approx$100}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{$\approx$1}&\multicolumn{1}{@{.}l}{3}&\parbox[t][0.3cm]{13.01432cm}{\raggedright E$\alpha$: From Q(\ensuremath{\alpha})=7505 keV \textit{9} in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2021Hu06,B}{2021Hu06} (a least-squares adjustment of the atomic masses).\vspace{0.1cm}}&\\
&&&&&&&&\parbox[t][0.3cm]{13.01432cm}{\raggedright {\ }{\ }{\ }Individual E\ensuremath{\alpha} values are 7370 keV \textit{20} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1996Le09,B}{1996Le09}), 7355 keV \textit{10} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1995Le04,B}{1995Le04}), 7375 keV \textit{25}\vspace{0.1cm}}&\\
&&&&&&&&\parbox[t][0.3cm]{13.01432cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1995Le15,B}{1995Le15}) and 7379 keV \textit{30} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2010He25,B}{2010He25}).\vspace{0.1cm}}&\\
\end{longtable}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{RN52DECAY0}{\dagger}}}} Using r\ensuremath{_{\textnormal{0}}}(\ensuremath{^{\textnormal{201}}}Rn)=1.527 \textit{9} from \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2020Si16,B}{2020Si16}.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{RN52DECAY1}{\ddagger}}}} For absolute intensity per 100 decays, multiply by{ }\ensuremath{\approx}1.0.}\\
\vspace{0.5cm}
\clearpage
\subsection[\hspace{-0.2cm}\ensuremath{^{\textnormal{122}}}Sn(\ensuremath{^{\textnormal{82}}}Kr,3n\ensuremath{\gamma})]{ }
\vspace{-27pt}
\vspace{0.3cm}
\hypertarget{RN53}{{\bf \small \underline{\ensuremath{^{\textnormal{122}}}Sn(\ensuremath{^{\textnormal{82}}}Kr,3n\ensuremath{\gamma})\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2008An05,B}{2008An05}}}}\\
\vspace{4pt}
\vspace{8pt}
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}\ensuremath{^{\textnormal{122}}}Sn(\ensuremath{^{\textnormal{82}}}Kr,3n\ensuremath{\gamma}), E=355 MeV beam delivered at JYFL, Finland. Measured E\ensuremath{\gamma}, I\ensuremath{\gamma}, \ensuremath{\gamma}\ensuremath{\gamma}, ce, \ensuremath{\gamma}\ensuremath{\alpha} coin using recoil-decay tagging}\\
\parbox[b][0.3cm]{17.7cm}{method with the JUROGAM array of 43 EUROGAM type escape-suppressed HPGe detectors at angles of 72\ensuremath{^\circ}, 86\ensuremath{^\circ}, 94\ensuremath{^\circ}, 108\ensuremath{^\circ}, 134\ensuremath{^\circ}}\\
\parbox[b][0.3cm]{17.7cm}{and 158\ensuremath{^\circ}. Reaction products were separated with the RITU recoil separator and implanted in the double-sided silicon strip detectors}\\
\parbox[b][0.3cm]{17.7cm}{of the GREAT spectrometer. Reaction \ensuremath{^{\textnormal{152}}}Sm(\ensuremath{^{\textnormal{52}}}Cr,3n\ensuremath{\gamma}), E=231 MeV was also used.}\\
\vspace{12pt}
\underline{$^{201}$Rn Levels}\\
\begin{longtable}{cccccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E(level)$^{{\hyperlink{RN53LEVEL0}{\dagger}}}$}&J$^{\pi}$$^{{\hyperlink{RN53LEVEL1}{\ddagger}}}$&\multicolumn{2}{c}{T$_{1/2}$$^{}$}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{.}l}{0}&\multicolumn{1}{l}{3/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{7}&\multicolumn{1}{@{.}l}{0 s {\it 4}}&\parbox[t][0.3cm]{13.1083cm}{\raggedright \%\ensuremath{\alpha}=?; \%\ensuremath{\varepsilon}+\%\ensuremath{\beta}\ensuremath{^{\textnormal{+}}}=?\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{13.1083cm}{\raggedright J\ensuremath{^{\pi}},T\ensuremath{_{1/2}}: From Adopted Levels.\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{13.1083cm}{\raggedright configuration: \ensuremath{\nu} p\ensuremath{_{\textnormal{3/2}}^{\textnormal{$-$1}}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{245}&\multicolumn{1}{@{ }l}{{\it 12}}&\multicolumn{1}{l}{13/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{3}&\multicolumn{1}{@{.}l}{8 s {\it 1}}&\parbox[t][0.3cm]{13.1083cm}{\raggedright \%\ensuremath{\alpha}=?; \%\ensuremath{\varepsilon}+\%\ensuremath{\beta}\ensuremath{^{\textnormal{+}}}=?\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{13.1083cm}{\raggedright J\ensuremath{^{\pi}},T\ensuremath{_{1/2}},E(level): From Adopted Levels.\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{13.1083cm}{\raggedright configuration: \ensuremath{\nu} i\ensuremath{_{\textnormal{13/2}}^{\textnormal{$-$1}}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{718}&\multicolumn{1}{@{.}l}{30 {\it 20}}&\multicolumn{1}{l}{(17/2\ensuremath{^{+}})}&&&\parbox[t][0.3cm]{13.1083cm}{\raggedright configuration: \ensuremath{\nu} (i\ensuremath{_{\textnormal{13/2}}^{\textnormal{$-$1}}})\ensuremath{\otimes}2\ensuremath{^{\textnormal{+}}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1266}&\multicolumn{1}{@{.}l}{2 {\it 3}}&\multicolumn{1}{l}{(21/2\ensuremath{^{+}})}&&&\parbox[t][0.3cm]{13.1083cm}{\raggedright configuration: \ensuremath{\nu} (i\ensuremath{_{\textnormal{13/2}}^{\textnormal{$-$1}}})\ensuremath{\otimes}4\ensuremath{^{\textnormal{+}}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{1364}&\multicolumn{1}{@{.}l}{9? {\it 4}}&\multicolumn{1}{l}{(21/2\ensuremath{^{+}})}&&&&\\
\multicolumn{1}{r@{}}{1819}&\multicolumn{1}{@{.}l}{3? {\it 4}}&\multicolumn{1}{l}{(25/2\ensuremath{^{+}})}&&&&\\
\end{longtable}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{RN53LEVEL0}{\dagger}}}} From a least-squares fit to E\ensuremath{\gamma}, unless otherwise stated.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{RN53LEVEL1}{\ddagger}}}} From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2008An05,B}{2008An05}, unless otherwise stated.}\\
\vspace{0.5cm}
\underline{$\gamma$($^{201}$Rn)}\\
\begin{longtable}{ccccccccc@{}c@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E\ensuremath{_{\gamma}}}&\multicolumn{2}{c}{I\ensuremath{_{\gamma}}}&\multicolumn{2}{c}{E\ensuremath{_{i}}(level)}&J\ensuremath{^{\pi}_{i}}&\multicolumn{2}{c}{E\ensuremath{_{f}}}&J\ensuremath{^{\pi}_{f}}&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{$^{x}$323}&\multicolumn{1}{@{.}l}{4 {\it 5}}&\multicolumn{1}{r@{}}{8}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{RN53GAMMA1}{\ddagger}}} {\it 3}}&&&&&&&\\
\multicolumn{1}{r@{}}{$^{x}$337}&\multicolumn{1}{@{.}l}{3 {\it 4}}&\multicolumn{1}{r@{}}{12}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{RN53GAMMA1}{\ddagger}}} {\it 3}}&&&&&&&\\
\multicolumn{1}{r@{}}{$^{x}$346}&\multicolumn{1}{@{.}l}{3 {\it 3}}&\multicolumn{1}{r@{}}{22}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{RN53GAMMA1}{\ddagger}}} {\it 4}}&&&&&&&\\
\multicolumn{1}{r@{}}{$^{x}$370}&\multicolumn{1}{@{.}l}{7 {\it 5}}&\multicolumn{1}{r@{}}{6}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{RN53GAMMA1}{\ddagger}}} {\it 3}}&&&&&&&\\
\multicolumn{1}{r@{}}{$^{x}$397}&\multicolumn{1}{@{.}l}{8 {\it 3}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{RN53GAMMA0}{\dagger}}} {\it 32}}&&&&&&&\\
\multicolumn{1}{r@{}}{454}&\multicolumn{1}{@{.}l}{0\ensuremath{^{\hyperlink{RN53GAMMA3}{@}}} {\it 3}}&\multicolumn{1}{r@{}}{29}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{RN53GAMMA1}{\ddagger}}} {\it 5}}&\multicolumn{1}{r@{}}{1819}&\multicolumn{1}{@{.}l}{3?}&\multicolumn{1}{l}{(25/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{1364}&\multicolumn{1}{@{.}l}{9? }&\multicolumn{1}{@{}l}{(21/2\ensuremath{^{+}})}&\\
\multicolumn{1}{r@{}}{$^{x}$464}&\multicolumn{1}{@{.}l}{7 {\it 6}}&\multicolumn{1}{r@{}}{9}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{RN53GAMMA1}{\ddagger}}} {\it 3}}&&&&&&&\\
\multicolumn{1}{r@{}}{473}&\multicolumn{1}{@{.}l}{3 {\it 2}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{RN53GAMMA1}{\ddagger}}} {\it 7}}&\multicolumn{1}{r@{}}{718}&\multicolumn{1}{@{.}l}{30}&\multicolumn{1}{l}{(17/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{245}&\multicolumn{1}{@{ }l}{}&\multicolumn{1}{@{}l}{13/2\ensuremath{^{+}}}&\\
\multicolumn{1}{r@{}}{$^{x}$536}&\multicolumn{1}{@{.}l}{8 {\it 5}}&\multicolumn{1}{r@{}}{24}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{RN53GAMMA0}{\dagger}}} {\it 17}}&&&&&&&\\
\multicolumn{1}{r@{}}{548}&\multicolumn{1}{@{.}l}{1 {\it 2}}&\multicolumn{1}{r@{}}{68}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{RN53GAMMA1}{\ddagger}}} {\it 7}}&\multicolumn{1}{r@{}}{1266}&\multicolumn{1}{@{.}l}{2}&\multicolumn{1}{l}{(21/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{718}&\multicolumn{1}{@{.}l}{30 }&\multicolumn{1}{@{}l}{(17/2\ensuremath{^{+}})}&\\
\multicolumn{1}{r@{}}{553}&\multicolumn{1}{@{.}l}{5\ensuremath{^{\hyperlink{RN53GAMMA3}{@}}} {\it 3}}&\multicolumn{1}{r@{}}{24}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{RN53GAMMA1}{\ddagger}}} {\it 4}}&\multicolumn{1}{r@{}}{1819}&\multicolumn{1}{@{.}l}{3?}&\multicolumn{1}{l}{(25/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{1266}&\multicolumn{1}{@{.}l}{2 }&\multicolumn{1}{@{}l}{(21/2\ensuremath{^{+}})}&\\
\multicolumn{1}{r@{}}{$^{x}$564}&\multicolumn{1}{@{.}l}{3 {\it 2}}&\multicolumn{1}{r@{}}{43}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{RN53GAMMA1}{\ddagger}}} {\it 5}}&&&&&&&\\
\multicolumn{1}{r@{}}{$^{x}$583}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{RN53GAMMA2}{\#}}} {\it 2}}&\multicolumn{1}{r@{}}{55}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{RN53GAMMA0}{\dagger}}} {\it 40}}&&&&&&&\\
\multicolumn{1}{r@{}}{$^{x}$587}&\multicolumn{1}{@{.}l}{1 {\it 5}}&\multicolumn{1}{r@{}}{13}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{RN53GAMMA1}{\ddagger}}} {\it 3}}&&&&&&&\\
\multicolumn{1}{r@{}}{$^{x}$590}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{RN53GAMMA2}{\#}}} {\it 2}}&\multicolumn{1}{r@{}}{42}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{RN53GAMMA0}{\dagger}}} {\it 32}}&&&&&&&\\
\multicolumn{1}{r@{}}{$^{x}$619}&\multicolumn{1}{@{.}l}{6 {\it 7}}&\multicolumn{1}{r@{}}{12}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{RN53GAMMA1}{\ddagger}}} {\it 4}}&&&&&&&\\
\multicolumn{1}{r@{}}{$^{x}$623}&\multicolumn{1}{@{.}l}{6 {\it 5}}&\multicolumn{1}{r@{}}{19}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{RN53GAMMA1}{\ddagger}}} {\it 4}}&&&&&&&\\
\multicolumn{1}{r@{}}{646}&\multicolumn{1}{@{.}l}{2\ensuremath{^{\hyperlink{RN53GAMMA3}{@}}} {\it 3}}&\multicolumn{1}{r@{}}{33}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{RN53GAMMA1}{\ddagger}}} {\it 4}}&\multicolumn{1}{r@{}}{1364}&\multicolumn{1}{@{.}l}{9?}&\multicolumn{1}{l}{(21/2\ensuremath{^{+}})}&\multicolumn{1}{r@{}}{718}&\multicolumn{1}{@{.}l}{30 }&\multicolumn{1}{@{}l}{(17/2\ensuremath{^{+}})}&\\
\multicolumn{1}{r@{}}{$^{x}$700}&\multicolumn{1}{@{ }l}{{\it 1}}&\multicolumn{1}{r@{}}{39}&\multicolumn{1}{@{}l}{\ensuremath{^{\hyperlink{RN53GAMMA0}{\dagger}}} {\it 23}}&&&&&&&\\
\end{longtable}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{RN53GAMMA0}{\dagger}}}} \ensuremath{\gamma} ray associated with the \ensuremath{J^{\pi}}=3/2\ensuremath{^{-}} ground state. Intensities are normalized to 100 for the 397.8 \ensuremath{\gamma} ray.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{RN53GAMMA1}{\ddagger}}}} \ensuremath{\gamma} ray associated with the \ensuremath{J^{\pi}}=13/2\ensuremath{^{+}} isomer. Intensities are normalized to 100 for the 473.3 \ensuremath{\gamma} ray.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{RN53GAMMA2}{\#}}}} Doublet.}\\
\begin{textblock}{29}(0,27.3)
Continued on next page (footnotes at end of table)
\end{textblock}
\clearpage
\vspace*{-0.5cm}
{\bf \small \underline{\ensuremath{^{\textnormal{122}}}Sn(\ensuremath{^{\textnormal{82}}}Kr,3n\ensuremath{\gamma})\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2008An05,B}{2008An05} (continued)}}\\
\vspace{0.3cm}
\underline{$\gamma$($^{201}$Rn) (continued)}\\
\vspace{0.3cm}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{RN53GAMMA3}{@}}}} Placement of transition in the level scheme is uncertain.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{RN53GAMMA4}{x}}}} \ensuremath{\gamma} ray not placed in level scheme.}\\
\vspace{0.5cm}
\begin{figure}[h]
\begin{center}
\includegraphics{201RN53-0.ps}\\
\end{center}
\end{figure}
\clearpage
\section[\ensuremath{^{201}_{\ 87}}Fr\ensuremath{_{114}^{~}}]{ }
\vspace{-30pt}
\setcounter{chappage}{1}
\subsection[\hspace{-0.2cm}Adopted Levels, Gammas]{ }
\vspace{-20pt}
\vspace{0.3cm}
\hypertarget{FR54}{{\bf \small \underline{Adopted \hyperlink{201FR_LEVEL}{Levels}, \hyperlink{201FR_GAMMA}{Gammas}}}}\\
\vspace{4pt}
\vspace{8pt}
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}Q(\ensuremath{\beta^-})=$-$8348 {\it 22}; S(n)=10620 {\it 30}; S(p)=$-$300 {\it 11}; Q(\ensuremath{\alpha})=7519 {\it 4}\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2021Wa16,B}{2021Wa16}}\\
\vspace{12pt}
\hypertarget{201FR_LEVEL}{\underline{$^{201}$Fr Levels}}\\
\begin{longtable}[c]{ll}
\multicolumn{2}{c}{\underline{Cross Reference (XREF) Flags}}\\
\\
\hyperlink{AC55}{\texttt{A }}& \ensuremath{^{\textnormal{205}}}Ac \ensuremath{\alpha} decay\\
\hyperlink{FR56}{\texttt{B }}& (HI,xn\ensuremath{\gamma})\\
\end{longtable}
\vspace{-0.5cm}
\begin{longtable}{ccccccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E(level)$^{}$}&J$^{\pi}$$^{}$&\multicolumn{2}{c}{T$_{1/2}$$^{}$}&XREF&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{l}{9/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{62}&\multicolumn{1}{@{.}l}{8 ms {\it 19}}&\multicolumn{1}{l}{\texttt{\hyperlink{AC55}{A}\hyperlink{FR56}{B}} }&\parbox[t][0.3cm]{11.60192cm}{\raggedright \%\ensuremath{\alpha}\ensuremath{\approx}100\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.60192cm}{\raggedright J\ensuremath{^{\pi}}: Favorite \ensuremath{\alpha} decay to \ensuremath{^{\textnormal{197}}}At (\ensuremath{J^{\pi}}=(9/2\ensuremath{^{-}}),\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2018Cu02,B}{2018Cu02}) and subsequent favorite \ensuremath{\alpha}\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.60192cm}{\raggedright {\ }{\ }{\ }decay to \ensuremath{^{\textnormal{193}}}Bi (\ensuremath{J^{\pi}}=9/2\ensuremath{^{-}},\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2016Ba42,B}{2016Ba42}).\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.60192cm}{\raggedright T\ensuremath{_{1/2}}: Weighted average of 64 \textit{3} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2014Ka23,B}{2014Ka23}), 53 ms \textit{4} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2005Uu02,B}{2005Uu02}),\hphantom{a}and 67 ms \textit{3}\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.60192cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2005De01,B}{2005De01}). Others: 48 ms \textit{15} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1980Ew03,B}{1980Ew03}) and 69 ms \textit{+16{\textminus}11} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1996En01,B}{1996En01}).\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.60192cm}{\raggedright E\ensuremath{\alpha}1=7369 keV \textit{5} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2014Ka23,B}{2014Ka23}; E\ensuremath{\alpha}1=7369 keV \textit{8}, correlated with E\ensuremath{\alpha}2(\ensuremath{^{\textnormal{197}}}At)=6959\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.60192cm}{\raggedright {\ }{\ }{\ }keV \textit{6} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2005Uu02,B}{2005Uu02}); E\ensuremath{\alpha}1=7379 keV \textit{7} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2005De01,B}{2005De01}); E\ensuremath{\alpha}1=7361 keV \textit{7}, correlated\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.60192cm}{\raggedright {\ }{\ }{\ }with E\ensuremath{\alpha}2(\ensuremath{^{\textnormal{197}}}At)=6956 keV \textit{5} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1996En01,B}{1996En01}); E\ensuremath{\alpha}1=7388 keV \textit{15} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1980Ew03,B}{1980Ew03}).\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.60192cm}{\raggedright configuration: \ensuremath{\pi} h\ensuremath{_{\textnormal{9/2}}^{\textnormal{+1}}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{129}&\multicolumn{1}{@{ }l}{{\it 10}}&\multicolumn{1}{l}{1/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{37}&\multicolumn{1}{@{ }l}{ms {\it +14\textminus8}}&\multicolumn{1}{l}{\texttt{\ \hyperlink{FR56}{B}} }&\parbox[t][0.3cm]{11.60192cm}{\raggedright \%\ensuremath{\alpha}=100\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.60192cm}{\raggedright E(level): From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2021Ko07,B}{2021Ko07}.\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.60192cm}{\raggedright J\ensuremath{^{\pi}}: Favorite \ensuremath{\alpha} decay to \ensuremath{^{\textnormal{197m}}}At (\ensuremath{J^{\pi}}=1/2\ensuremath{^{+}},\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2018Cu02,B}{2018Cu02}) and subsequent favorite \ensuremath{\alpha}\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.60192cm}{\raggedright {\ }{\ }{\ }decay to \ensuremath{^{\textnormal{193m}}}Bi (\ensuremath{J^{\pi}}=1/2\ensuremath{^{+}},\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2016Ba42,B}{2016Ba42}).\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.60192cm}{\raggedright T\ensuremath{_{1/2}}: From \ensuremath{\alpha}(t) in\hphantom{a}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2020Au01,B}{2020Au01} (14 \ensuremath{\alpha}1{\textminus}\ensuremath{\alpha}2{\textminus}\ensuremath{\alpha}3 correlated events). Others: 8 ms \textit{+12{\textminus}3}\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.60192cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2014Ka23,B}{2014Ka23}) and 19 ms \textit{+19{\textminus}6} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2005Uu02,B}{2005Uu02}).\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.60192cm}{\raggedright E\ensuremath{\alpha}1=7457 keV \textit{9} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2020Au01,B}{2020Au01}); E\ensuremath{\alpha}1=7445 keV \textit{8} correlated with E\ensuremath{\alpha}2(\ensuremath{^{\textnormal{197}}}At)=6698\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.60192cm}{\raggedright {\ }{\ }{\ }keV \textit{16} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2014Ka23,B}{2014Ka23}); E\ensuremath{\alpha}1=7454 keV \textit{8} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2005Uu02,B}{2005Uu02}).\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.60192cm}{\raggedright configuration: \ensuremath{\pi} s\ensuremath{_{\textnormal{1/2}}^{\textnormal{$-$1}}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{289}&\multicolumn{1}{@{.}l}{5 {\it 4}}&\multicolumn{1}{l}{13/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{720}&\multicolumn{1}{@{ }l}{ns {\it 40}}&\multicolumn{1}{l}{\texttt{\ \hyperlink{FR56}{B}} }&\parbox[t][0.3cm]{11.60192cm}{\raggedright \%IT=100\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.60192cm}{\raggedright E(level): From E\ensuremath{\gamma} in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2020Au01,B}{2020Au01}.\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.60192cm}{\raggedright J\ensuremath{^{\pi}}: 289.5\ensuremath{\gamma} M2 to 9/2\ensuremath{^{-}}; systematics in the region.\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.60192cm}{\raggedright T\ensuremath{_{1/2}}: From ce(t) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2020Au01,B}{2020Au01}. Others: 0.7 \ensuremath{\mu}s \textit{+5{\textminus}2} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2014Ka23,B}{2014Ka23}).\vspace{0.1cm}}&\\
&&&&&&\parbox[t][0.3cm]{11.60192cm}{\raggedright configuration: \ensuremath{\pi} i\ensuremath{_{\textnormal{13/2}}^{\textnormal{+1}}}.\vspace{0.1cm}}&\\
\end{longtable}
\hypertarget{201FR_GAMMA}{\underline{$\gamma$($^{201}$Fr)}}\\
\begin{longtable}{ccccccccc@{}ccccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E\ensuremath{_{i}}(level)}&J\ensuremath{^{\pi}_{i}}&\multicolumn{2}{c}{E\ensuremath{_{\gamma}}}&\multicolumn{2}{c}{I\ensuremath{_{\gamma}}}&\multicolumn{2}{c}{E\ensuremath{_{f}}}&J\ensuremath{^{\pi}_{f}}&Mult.&\multicolumn{2}{c}{\ensuremath{\alpha}\ensuremath{^{\hyperlink{FR54GAMMA0}{\dagger}}}}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{289}&\multicolumn{1}{@{.}l}{5}&\multicolumn{1}{l}{13/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{289}&\multicolumn{1}{@{.}l}{5 {\it 4}}&\multicolumn{1}{r@{}}{100}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{9/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M2}&\multicolumn{1}{r@{}}{2}&\multicolumn{1}{@{.}l}{60 {\it 4}}&\parbox[t][0.3cm]{8.954881cm}{\raggedright B(M2)(W.u.)=0.172 \textit{+11{\textminus}9}\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{8.954881cm}{\raggedright E\ensuremath{_{\gamma}},I\ensuremath{_{\gamma}}: From\hphantom{a}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2020Au01,B}{2020Au01}.\vspace{0.1cm}}&\\
&&&&&&&&&&&&&\parbox[t][0.3cm]{8.954881cm}{\raggedright Mult.: From K/LMN+=3.0 \textit{9} in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2020Au01,B}{2020Au01}.\vspace{0.1cm}}&\\
\end{longtable}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{FR54GAMMA0}{\dagger}}}} Total theoretical internal conversion coefficients, calculated using the BrIcc code (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2008Ki07,B}{2008Ki07}) with Frozen orbital approximation}\\
\parbox[b][0.3cm]{17.7cm}{{\ }{\ }based on \ensuremath{\gamma}-ray energies, assigned multipolarities, and mixing ratios, unless otherwise specified.}\\
\vspace{0.5cm}
\clearpage
\begin{figure}[h]
\begin{center}
\includegraphics{201FR54-0.ps}\\
\end{center}
\end{figure}
\clearpage
\subsection[\hspace{-0.2cm}\ensuremath{^{\textnormal{205}}}Ac \ensuremath{\alpha} decay]{ }
\vspace{-27pt}
\vspace{0.3cm}
\hypertarget{AC55}{{\bf \small \underline{\ensuremath{^{\textnormal{205}}}Ac \ensuremath{\alpha} decay\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2014Zh03,B}{2014Zh03}}}}\\
\vspace{4pt}
\vspace{8pt}
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}Parent: $^{205}$Ac: E=0; J$^{\pi}$=9/2\ensuremath{^{-}}; T$_{1/2}$=20 ms {\it +97\textminus9}; Q(\ensuremath{\alpha})=8090 {\it 60}; \%\ensuremath{\alpha} decay$\approx$100.0
}\\
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}\ensuremath{^{205}}Ac-E,J$^{\pi}$: From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2021Ko07,B}{2021Ko07}.}\\
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}\ensuremath{^{205}}Ac-T$_{1/2}$: From 7935\ensuremath{\alpha}(t) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2014Zh03,B}{2014Zh03}.}\\
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}\ensuremath{^{205}}Ac-Q(\ensuremath{\alpha}): From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2021Wa16,B}{2021Wa16}.}\\
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}\ensuremath{^{205}}Ac-\%\ensuremath{\alpha} decay: From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2021Ko07,B}{2021Ko07}.}\\
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2014Zh03,B}{2014Zh03}: \ensuremath{^{\textnormal{205}}}Ac produced in the \ensuremath{^{\textnormal{169}}}Tm(\ensuremath{^{\textnormal{40}}}Ca,4n) reaction, E(\ensuremath{^{\textnormal{40}}}Ca=196 MeV at the HIRFL facility, Lanzhou. Target: 400}\\
\parbox[b][0.3cm]{17.7cm}{\ensuremath{\mu}g/cm\ensuremath{^{\textnormal{2}}} thick covered with a 10 \ensuremath{\mu}g/cm\ensuremath{^{\textnormal{2}}}-thick carbon layer. Evaporation residues were separated in flight using SHANS recoil}\\
\parbox[b][0.3cm]{17.7cm}{separator, and implanted into position sensitive DSSD (48 vertical strips of 3 mm width). Eight non-position sensitive Si detectors}\\
\parbox[b][0.3cm]{17.7cm}{were used to detect escaping \ensuremath{\alpha} particles. Measured: recoil-\ensuremath{\alpha}\ensuremath{_{\textnormal{1}}}(t)-\ensuremath{\alpha}\ensuremath{_{\textnormal{2}}}(t)-\ensuremath{\alpha}\ensuremath{_{\textnormal{3}}}(t) correlated events. Deduced: E\ensuremath{\alpha} and half-life of \ensuremath{^{\textnormal{205}}}Ac.}\\
\vspace{12pt}
\underline{$^{201}$Fr Levels}\\
\begin{longtable}{ccccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E(level)$^{}$}&J$^{\pi}$$^{}$&\multicolumn{2}{c}{T$_{1/2}$$^{}$}&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&
\endfirsthead
\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{l}{9/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{62}&\multicolumn{1}{@{.}l}{8 ms {\it 19}}&\\
\end{longtable}
\underline{\ensuremath{\alpha} radiations}\\
\begin{longtable}{ccccccccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E$\alpha^{{}}$}&\multicolumn{2}{c}{E(level)}&\multicolumn{2}{c}{I$\alpha^{{\hyperlink{FR55DECAY1}{\ddagger}}}$}&\multicolumn{2}{c}{HF$^{{\hyperlink{FR55DECAY0}{\dagger}}}$}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{7935}&\multicolumn{1}{@{ }l}{{\it 30}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{$\approx$100}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{r@{}}{$\approx$2}&\multicolumn{1}{@{.}l}{3}&\parbox[t][0.3cm]{12.855961cm}{\raggedright E$\alpha$,I$\alpha$: From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2014Zh03,B}{2014Zh03}.\hphantom{a}E\ensuremath{\alpha}1=7935 keV \textit{30} correlated with E\ensuremath{\alpha}2=7406 keV \textit{30} (\ensuremath{^{\textnormal{201}}}Fr) and\vspace{0.1cm}}&\\
&&&&&&&&\parbox[t][0.3cm]{12.855961cm}{\raggedright {\ }{\ }{\ }E\ensuremath{\alpha}3=6997 keV \textit{30} (\ensuremath{^{\textnormal{197}}}At).\vspace{0.1cm}}&\\
\end{longtable}
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{FR55DECAY0}{\dagger}}}} Using r\ensuremath{_{\textnormal{0}}}=1.498 \textit{2}, unweighted average of r\ensuremath{_{\textnormal{0}}}=1.4803 \textit{26} for \ensuremath{^{\textnormal{200}}}Po and 1.516 \textit{7} for \ensuremath{^{\textnormal{202}}}Rn.}\\
\parbox[b][0.3cm]{17.7cm}{\makebox[1ex]{\ensuremath{^{\hypertarget{FR55DECAY1}{\ddagger}}}} For absolute intensity per 100 decays, multiply by{ }\ensuremath{\approx}1.}\\
\vspace{0.5cm}
\clearpage
\subsection[\hspace{-0.2cm}(HI,xn\ensuremath{\gamma})]{ }
\vspace{-27pt}
\vspace{0.3cm}
\hypertarget{FR56}{{\bf \small \underline{(HI,xn\ensuremath{\gamma})\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2020Au01,B}{2020Au01},\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2014Ka23,B}{2014Ka23},\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2005Uu02,B}{2005Uu02}}}}\\
\vspace{4pt}
\vspace{8pt}
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2020Au01,B}{2020Au01}: \ensuremath{^{\textnormal{169}}}Tm(\ensuremath{^{\textnormal{36}}}Ar,4n\ensuremath{\gamma}) at E(\ensuremath{^{\textnormal{36}}}Ar)=178, 184, and 187 MeV. Evaporation residues separated with RITU separator and}\\
\parbox[b][0.3cm]{17.7cm}{implanted into a DSSD. Measured E\ensuremath{\gamma}, I\ensuremath{\gamma}, E\ensuremath{\alpha}, \ensuremath{\alpha}(t), ce(t) using an array of silicon PIN diodes and three Clover-type HPGe}\\
\parbox[b][0.3cm]{17.7cm}{detectors.}\\
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2014Ka23,B}{2014Ka23}: \ensuremath{^{\textnormal{149}}}Sm(\ensuremath{^{\textnormal{56}}}Fe,p3n) at E(\ensuremath{^{\textnormal{56}}}Fe)=275 MeV produced by the GSI accelerator facility. Target=370 \ensuremath{\mu}g/cm\ensuremath{^{\textnormal{2}}} thick enriched to}\\
\parbox[b][0.3cm]{17.7cm}{96.4\% in \ensuremath{^{\textnormal{147}}}Sm, with 40 \ensuremath{\mu}g/cm\ensuremath{^{\textnormal{2}}} thick carbon backing and covered with a 10 \ensuremath{\mu}g/cm\ensuremath{^{\textnormal{2}}} layer of carbon, and mounted on a rotating}\\
\parbox[b][0.3cm]{17.7cm}{wheel. Detectors: SHIP recoil separator, 16-strip position sensitive Si detectors (PSSD), six Si strip detectors to detect escaping \ensuremath{\alpha}}\\
\parbox[b][0.3cm]{17.7cm}{particles and one HPGe clover detector behind the PSDD. Measured: recoil-\ensuremath{\alpha}-\ensuremath{\gamma}(t) and recoil-\ensuremath{\alpha}-\ensuremath{\alpha}(t). Deduced: E\ensuremath{\alpha} and T\ensuremath{_{\textnormal{1/2}}}.}\\
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2005Uu02,B}{2005Uu02}: produced using \ensuremath{^{\textnormal{170}}}Yb(\ensuremath{^{\textnormal{36}}}Ar,p4n),E(\ensuremath{^{\textnormal{36}}}Ar)=180 and 185 MeV. Target: 70 \% enriched in \ensuremath{^{\textnormal{170}}}Yb. Detectors: gas filled}\\
\parbox[b][0.3cm]{17.7cm}{mass separator, position sensitive silicon detectors with a typical resolution (FWHM) of 30 keV, multi-wire proportional gas}\\
\parbox[b][0.3cm]{17.7cm}{counter. Measured: E\ensuremath{\alpha}, T\ensuremath{_{\textnormal{1/2}}}.}\\
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2005De01,B}{2005De01}: produced in a bombardment with a 1.4 GeV pulsed proton beam on 51 g/cm\ensuremath{^{\textnormal{2}}} thorium/graphite target. Detectors: on-line}\\
\parbox[b][0.3cm]{17.7cm}{mass separator, recoils were implanted on a carbon foil for 100 ms and subsequent \ensuremath{\alpha}-decay counted using a 400 mm\ensuremath{^{\textnormal{2}}}, 1 mm thick}\\
\parbox[b][0.3cm]{17.7cm}{silicon detector for 1100 ms; Measured: E\ensuremath{\alpha}, T\ensuremath{_{\textnormal{1/2}}}.}\\
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}Others: \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1996En01,B}{1996En01}: produced using \ensuremath{^{\textnormal{170}}}Yb(\ensuremath{^{\textnormal{35}}}Cl,4n), E(\ensuremath{^{\textnormal{35}}}Cl)=205 and 213 MeV; Target: 72 \% enriched in \ensuremath{^{\textnormal{170}}}Yb; Detectors: gas}\\
\parbox[b][0.3cm]{17.7cm}{filled mass separator, position sensitive silicon detectors with a typical resolution (FWHM) of 35 keV; Measured: E\ensuremath{\alpha}, T\ensuremath{_{\textnormal{1/2}}}.}\\
\parbox[b][0.3cm]{17.7cm}{Assignment to \ensuremath{^{\textnormal{201}}}Fr is based on the observed E\ensuremath{\alpha}1-E\ensuremath{\alpha}2 correlation with the characteristic daughter \ensuremath{\alpha}-decay;\hphantom{a}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1980Ew03,B}{1980Ew03}: produced}\\
\parbox[b][0.3cm]{17.7cm}{using \ensuremath{^{\textnormal{238}}}U(p,spallation); E(p)=600 MeV; Detectors: on-line mass separator, silicon charged particle detector; Measured: E\ensuremath{\alpha}, T\ensuremath{_{\textnormal{1/2}}}.}\\
\vspace{12pt}
\underline{$^{201}$Fr Levels}\\
\begin{longtable}{cccccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E(level)$^{}$}&J$^{\pi}$$^{}$&\multicolumn{2}{c}{T$_{1/2}$$^{}$}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{l}{9/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{62}&\multicolumn{1}{@{.}l}{8 ms {\it 19}}&\parbox[t][0.3cm]{12.82154cm}{\raggedright \%\ensuremath{\alpha}\ensuremath{\approx}100\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{12.82154cm}{\raggedright T\ensuremath{_{1/2}}: From Adopted Levels. Individual measurements: 64 \textit{3} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2014Ka23,B}{2014Ka23}), 53 ms \textit{4} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2005Uu02,B}{2005Uu02}),\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{12.82154cm}{\raggedright {\ }{\ }{\ }67 ms \textit{3} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2005De01,B}{2005De01}), 48 ms \textit{15} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1980Ew03,B}{1980Ew03}) and 69 ms \textit{+16{\textminus}11} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1996En01,B}{1996En01}).\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{12.82154cm}{\raggedright E\ensuremath{\alpha}1=7369 keV \textit{5} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2014Ka23,B}{2014Ka23}; E\ensuremath{\alpha}1=7369 keV \textit{8}, correlated with E\ensuremath{\alpha}2(\ensuremath{^{\textnormal{197}}}At)=6959 keV \textit{6}\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{12.82154cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2005Uu02,B}{2005Uu02}); E\ensuremath{\alpha}1=7379 keV \textit{7} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2005De01,B}{2005De01}); E\ensuremath{\alpha}1=7361 keV \textit{7}, correlated with\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{12.82154cm}{\raggedright {\ }{\ }{\ }E\ensuremath{\alpha}2(\ensuremath{^{\textnormal{197}}}At)=6956 keV \textit{5} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1996En01,B}{1996En01}); E\ensuremath{\alpha}1=7388 keV \textit{15} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1980Ew03,B}{1980Ew03}).\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{12.82154cm}{\raggedright \ensuremath{\sigma}(\ensuremath{^{\textnormal{201}}}Fr)=4.0 nb \textit{4} at 275 MeV (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2014Ka23,B}{2014Ka23}).\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{12.82154cm}{\raggedright configuration: \ensuremath{\pi} h\ensuremath{_{\textnormal{9/2}}^{\textnormal{+1}}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{129}&\multicolumn{1}{@{ }l}{{\it 10}}&\multicolumn{1}{l}{1/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{37}&\multicolumn{1}{@{ }l}{ms {\it +14\textminus8}}&\parbox[t][0.3cm]{12.82154cm}{\raggedright \%\ensuremath{\alpha}=100\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{12.82154cm}{\raggedright T\ensuremath{_{1/2}}: From \ensuremath{\alpha}(t) in\hphantom{a}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2020Au01,B}{2020Au01} (14 \ensuremath{\alpha}1{\textminus}\ensuremath{\alpha}2{\textminus}\ensuremath{\alpha}3 correlated events). Others: 8 ms \textit{+12{\textminus}3}\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{12.82154cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2014Ka23,B}{2014Ka23}) and 19 ms \textit{+19{\textminus}6} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2005Uu02,B}{2005Uu02}).\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{12.82154cm}{\raggedright E\ensuremath{\alpha}=7457 keV \textit{9} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2020Au01,B}{2020Au01}), E\ensuremath{\alpha}=7445 keV \textit{8} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2014Ka23,B}{2014Ka23}) and E\ensuremath{\alpha}=7454 keV \textit{8} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2005Uu02,B}{2005Uu02}).\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{12.82154cm}{\raggedright Isomeric Ratio: I(1/2\ensuremath{^{+}} isomer)/I(9/2\ensuremath{^{-}} g.s.)=0.02 \textit{1} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2014Ka23,B}{2014Ka23}).\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{12.82154cm}{\raggedright configuration: \ensuremath{\pi} s\ensuremath{_{\textnormal{1/2}}^{\textnormal{$-$1}}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{289}&\multicolumn{1}{@{.}l}{5 {\it 4}}&\multicolumn{1}{l}{13/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{720}&\multicolumn{1}{@{ }l}{ns {\it 40}}&\parbox[t][0.3cm]{12.82154cm}{\raggedright \%IT=100\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{12.82154cm}{\raggedright T\ensuremath{_{1/2}}: from ce(t) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2020Au01,B}{2020Au01}. Others: 0.7 \ensuremath{\mu}s \textit{+5{\textminus}2} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2014Ka23,B}{2014Ka23}).\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{12.82154cm}{\raggedright configuration: \ensuremath{\pi} i\ensuremath{_{\textnormal{13/2}}^{\textnormal{+1}}}.\vspace{0.1cm}}&\\
\end{longtable}
\underline{$\gamma$($^{201}$Fr)}\\
\begin{longtable}{ccccccc@{}ccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E\ensuremath{_{\gamma}}}&\multicolumn{2}{c}{E\ensuremath{_{i}}(level)}&J\ensuremath{^{\pi}_{i}}&\multicolumn{2}{c}{E\ensuremath{_{f}}}&J\ensuremath{^{\pi}_{f}}&Mult.&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&\hrulefill&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{289}&\multicolumn{1}{@{.}l}{5 {\it 4}}&\multicolumn{1}{r@{}}{289}&\multicolumn{1}{@{.}l}{5}&\multicolumn{1}{l}{13/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{@{}l}{9/2\ensuremath{^{-}}}&\multicolumn{1}{l}{M2}&\parbox[t][0.3cm]{11.103081cm}{\raggedright E\ensuremath{_{\gamma}}: From\hphantom{a}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2020Au01,B}{2020Au01}.\vspace{0.1cm}}&\\
&&&&&&&&&\parbox[t][0.3cm]{11.103081cm}{\raggedright Mult.: From K/LMN+=3.0 \textit{9} in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2020Au01,B}{2020Au01}.\vspace{0.1cm}}&\\
\end{longtable}
\clearpage
\begin{figure}[h]
\begin{center}
\includegraphics{201FR56-0.ps}\\
\end{center}
\end{figure}
\clearpage
\section[\ensuremath{^{201}_{\ 88}}Ra\ensuremath{_{113}^{~}}]{ }
\vspace{-30pt}
\setcounter{chappage}{1}
\subsection[\hspace{-0.2cm}Adopted Levels]{ }
\vspace{-20pt}
\vspace{0.3cm}
\hypertarget{RA57}{{\bf \small \underline{Adopted \hyperlink{201RA_LEVEL}{Levels}}}}\\
\vspace{4pt}
\vspace{8pt}
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}S(p)=1490 {\it 40}; Q(\ensuremath{\alpha})=8002 {\it 12}\hspace{0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2021Wa16,B}{2021Wa16}}\\
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2005Uu02,B}{2005Uu02}: \ensuremath{^{\textnormal{201}}}Ra produced using the \ensuremath{^{\textnormal{141}}}Pr(\ensuremath{^{\textnormal{63}}}Cu,3n) reaction, E(\ensuremath{^{\textnormal{63}}}Cu)=278 and 288 MeV. Detectors: gas filled mass separator,}\\
\parbox[b][0.3cm]{17.7cm}{position sensitive silicon detectors with a typical energy resolution (FWHM) of 30 keV; multi-wire proportional gas counters;}\\
\parbox[b][0.3cm]{17.7cm}{Measured: E\ensuremath{\alpha}, \ensuremath{\alpha}-\ensuremath{\alpha} correlations, T\ensuremath{_{\textnormal{1/2}}}.}\\
\parbox[b][0.3cm]{17.7cm}{\addtolength{\parindent}{-0.2in}\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2014Ka23,B}{2014Ka23}: \ensuremath{^{\textnormal{201}}}Ra produced using the \ensuremath{^{\textnormal{147}}}Sm(\ensuremath{^{\textnormal{56}}}Fe,2n) reactions, E(\ensuremath{^{\textnormal{56}}}Fe)=249 MeV. Target=370 \ensuremath{\mu}g/cm\ensuremath{^{\textnormal{2}}} thick enriched to 96.4\%}\\
\parbox[b][0.3cm]{17.7cm}{in \ensuremath{^{\textnormal{147}}}Sm, with 40 \ensuremath{\mu}g/cm\ensuremath{^{\textnormal{2}}} thick carbon backing and covered with a 10 \ensuremath{\mu}g/cm\ensuremath{^{\textnormal{2}}} layer of carbon, and mounted on a rotating}\\
\parbox[b][0.3cm]{17.7cm}{wheel. Detectors: SHIP recoil separator, 16-strip position sensitive Si detectors (PSSD), six Si strip detectors to detect escaping \ensuremath{\alpha}}\\
\parbox[b][0.3cm]{17.7cm}{particles and one HPGe clover detector behind the PSDD. Measured: recoil-\ensuremath{\alpha}-\ensuremath{\gamma}(t) and recoil-\ensuremath{\alpha}-\ensuremath{\alpha}(t). Deduced: E\ensuremath{\alpha} and T\ensuremath{_{\textnormal{1/2}}}.}\\
\vspace{12pt}
\hypertarget{201RA_LEVEL}{\underline{$^{201}$Ra Levels}}\\
\begin{longtable}{cccccc@{\extracolsep{\fill}}c}
\multicolumn{2}{c}{E(level)$^{}$}&J$^{\pi}$$^{}$&\multicolumn{2}{c}{T$_{1/2}$$^{}$}&Comments&\\[-.2cm]
\multicolumn{2}{c}{\hrulefill}&\hrulefill&\multicolumn{2}{c}{\hrulefill}&\hrulefill&
\endfirsthead
\multicolumn{1}{r@{}}{0}&\multicolumn{1}{@{}l}{}&\multicolumn{1}{l}{3/2\ensuremath{^{-}}}&\multicolumn{1}{r@{}}{8}&\multicolumn{1}{@{ }l}{ms {\it +40\textminus4}}&\parbox[t][0.3cm]{12.898581cm}{\raggedright \%\ensuremath{\alpha}\ensuremath{\approx}100\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{12.898581cm}{\raggedright J\ensuremath{^{\pi}}: Favorite \ensuremath{\alpha} decay to \ensuremath{^{\textnormal{197}}}Rn (\ensuremath{J^{\pi}}=3/2\ensuremath{^{-}}), subsequent favorite \ensuremath{\alpha} decay to \ensuremath{^{\textnormal{193}}}Po\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{12.898581cm}{\raggedright {\ }{\ }{\ }(\ensuremath{J^{\pi}}=3/2\ensuremath{^{-}},\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2014Se07,B}{2014Se07}) and subsequent favorite \ensuremath{\alpha} decay to \ensuremath{^{\textnormal{189}}}Pb (\ensuremath{J^{\pi}}=3/2\ensuremath{^{-}},\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2009Se13,B}{2009Se13}).\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{12.898581cm}{\raggedright T\ensuremath{_{1/2}}: From \ensuremath{\alpha}(t) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2014Ka23,B}{2014Ka23}.\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{12.898581cm}{\raggedright E\ensuremath{\alpha}1=7842 \textit{12}, correlated with E\ensuremath{\alpha}2(\ensuremath{^{\textnormal{197}}}Rn) (escape) and E\ensuremath{\alpha}3(\ensuremath{^{\textnormal{193}}}Po)=6949 keV \textit{12}\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{12.898581cm}{\raggedright {\ }{\ }{\ }(\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2014Ka23,B}{2014Ka23}).\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{12.898581cm}{\raggedright \ensuremath{\sigma}(\ensuremath{^{\textnormal{201}}}Ra)=4 pb \textit{+80{\textminus}30} at 249 MeV (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2014Ka23,B}{2014Ka23}).\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{12.898581cm}{\raggedright configuration: \ensuremath{\nu} p\ensuremath{_{\textnormal{3/2}}^{\textnormal{$-$1}}}.\vspace{0.1cm}}&\\
\multicolumn{1}{r@{}}{263}&\multicolumn{1}{@{ }l}{{\it 26}}&\multicolumn{1}{l}{13/2\ensuremath{^{+}}}&\multicolumn{1}{r@{}}{1}&\multicolumn{1}{@{.}l}{6 ms {\it +77\textminus7}}&\parbox[t][0.3cm]{12.898581cm}{\raggedright \%\ensuremath{\alpha}\ensuremath{\approx}100\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{12.898581cm}{\raggedright E(level): From \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2021Ko07,B}{2021Ko07}.\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{12.898581cm}{\raggedright J\ensuremath{^{\pi}}: Favorite \ensuremath{\alpha} decay to \ensuremath{^{\textnormal{197m}}}Rn (\ensuremath{J^{\pi}}=13/2\ensuremath{^{+}}), subsequent favorite \ensuremath{\alpha} decay to \ensuremath{^{\textnormal{193m}}}Po\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{12.898581cm}{\raggedright {\ }{\ }{\ }(\ensuremath{J^{\pi}}=13/2\ensuremath{^{+}},\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2014Se07,B}{2014Se07}) and subsequent favorite \ensuremath{\alpha} decay to \ensuremath{^{\textnormal{189m}}}Pb (\ensuremath{J^{\pi}}=13/2\ensuremath{^{+}},\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2009Se13,B}{2009Se13}).\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{12.898581cm}{\raggedright T\ensuremath{_{1/2}}: From \ensuremath{\alpha}(t) in \href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2005Uu02,B}{2005Uu02}.\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{12.898581cm}{\raggedright E\ensuremath{\alpha}1=7905 keV \textit{20}, correlated with E\ensuremath{\alpha}2(\ensuremath{^{\textnormal{197m}}}Rn)=7358 keV \textit{14} (\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2005Uu02,B}{2005Uu02}).\vspace{0.1cm}}&\\
&&&&&\parbox[t][0.3cm]{12.898581cm}{\raggedright configuration: \ensuremath{\nu} i\ensuremath{_{\textnormal{13/2}}^{\textnormal{$-$1}}}.\vspace{0.1cm}}&\\
\end{longtable}
\end{center}
\clearpage
\newpage
\pagestyle{plain}
\section[References]{ }
\vspace{-30pt}
\begin{longtable}{l@{\hskip 0.9cm}l}
\multicolumn{2}{c}{REFERENCES FOR A=201}\\
&\endfirsthead
\multicolumn{2}{c}{REFERENCES FOR A=201(CONTINUED)}\\
&\endhead
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1931Sc03,B}{1931Sc03}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}H.Schuler, J.E.Keyston - Z.Physik 72, 423 (1931).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Hyperfeinstrukturen und Kernmomente des Quecksilbers.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1935Sc04,B}{1935Sc04}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}H.Schuler, T.Schmidt - Z.Physik 98, 430 (1935); See Also 38Sc10 (Bi\ensuremath{^{\textnormal{209}}}).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Bemerkungen zu den elektrischen Quadrupolmomenten einiger Atomkerne und dem magnetischen Moment des protons.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1950Ne77,B}{1950Ne77}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}H.M.Neumann, I.Perlman - Phys.Rev. 78, 191 (1950).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Isotopic Assignments of Bismuth Isotopes Produced with High Energy Particles.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1952Bu80,B}{1952Bu80}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}F.D.S.Butement, R.Shillito - Proc.Phys.Soc.(London) 65A, 945 (1952).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Radioactive Gold Isotopes.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1952Ho41,B}{1952Ho41}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}N.J.Hopkins - Phys.Rev. 88, 680 (1952).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Isomeric Levels in Pb\ensuremath{^{\textnormal{201}}} and Pb\ensuremath{^{\textnormal{202}}}.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1954Bu67,B}{1954Bu67}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}W.E.Burcham - Proc.Phys.Soc.(London) 67A, 555 (1954).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{The \ensuremath{\alpha}-Activity Induced in Gold by Bombardment with Nitrogen Ions.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1954Wa12,B}{1954Wa12}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}A.H.Wapstra, D.Maeder, G.J.Nijgh, L.T.M.Ornstein - Physica 20, 169 (1954).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{The Decay of \ensuremath{^{\textnormal{203}}}Hg, \ensuremath{^{\textnormal{203}}}Pb and \ensuremath{^{\textnormal{201}}}Pb.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1955Be12,B}{1955Be12}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}K.E.Bergkvist, I.Bergstrom, C.J.Herrlander, S.Hultberg, H.Slatis, E.Sokolowski, A.H.Wapstra, T.Wiedling - Phil.Mag. 46, 65 (1955).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Neutron Deficient Isotopes of Pb and Tl {\textminus} II. Mass Numbers 204, 202, 201 and 200.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1955Fi30,B}{1955Fi30}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}V.K.Fischer - Phys.Rev. 99, 764 (1955).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Metastable States of Re\ensuremath{^{\textnormal{180}}}, Ir\ensuremath{^{\textnormal{191}}}, Au\ensuremath{^{\textnormal{193}}}, Pb\ensuremath{^{\textnormal{201}}} and Pb\ensuremath{^{\textnormal{203}}}.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1956St05,B}{1956St05}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}R.Stockendal, J.A.McDonnell, M.Schmorak, I.Bergstrom - Arkiv Fysik 11, 165 (1956).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Nuclear Isomerism in Odd Lead Isotopes.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1957Er24,B}{1957Er24}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}P.Erdos, P.Scherrer, P.Stoll - Helv.Phys.Acta 30, 639 (1957).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Die kunstliche Alphaaktivitat gammaangeregter Atomkerne.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1958Li45,B}{1958Li45}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}I.Lindgren, C.M.Johansson, S.Axensten - Phys.Rev.Letters 1, 473 (1958).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Nuclear Ground-State Spins of Thallium-198 and {\textminus}201.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1958Ma21,B}{1958Ma21}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}L.L.Marino, G.O.Brink, W.B.Ewbank, H.A.Shugart, H.B.Silsbee - Bull.Am.Phys.Soc. 3, No.3, 186, J6 (1958).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Nuclear Spins of Tl\ensuremath{^{\textnormal{200}}} and Tl\ensuremath{^{\textnormal{201}}}.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1960Gu05,B}{1960Gu05}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}R.K.Gupta - Arkiv Fysik 17, 337 (1960).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{On the Determination of the Electron-Capture Decay Energies of Tl\ensuremath{^{\textnormal{201}}} and Tl\ensuremath{^{\textnormal{202}}}.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1960He05,B}{1960He05}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}C.J.Herrlander, R.Stockendal, R.K.Gupta - Arkiv Fysik 17, 315 (1960).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{The Decay of Tl\ensuremath{^{\textnormal{201}}}.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1960Li03,B}{1960Li03}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}I.Lindgren, C.M.Johansson-Olsmats, S.Axensten - Arkiv Fysik 16, 506A (1960).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Atomic Beam Investigation of Radioactive Isotopes in the Lead Region.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1960Li08,B}{1960Li08}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}J.Lindskog, E.Bashandy, T.R.Gerholm - Nuclear Phys. 16, 175 (1960).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Determination of a 7 x 10\ensuremath{^{\textnormal{$-$11}}} sec Halflife for the First Excited State in Thallium 201.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1960Mc11,B}{1960Mc11}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}M.N.McDermott, W.L.Lichten - Phys.Rev. 119, 134 (1960).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Hyperfine Structure of the 6\ensuremath{^{\textnormal{3}}}P\ensuremath{_{\textnormal{2}}} State of Hg\ensuremath{^{\textnormal{199}}} and Hg\ensuremath{^{\textnormal{201}}}.Properties of Metastable States of Mercury.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1961Be29,B}{1961Be29}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}I.Bergstrom, C.J.Herrlander, P.Thieberger, J.Uhler - Arkiv Fysik 20, 93 (1961).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{On Nuclear f\ensuremath{_{\textnormal{5}}}/\ensuremath{_{\textnormal{2}}}, p\ensuremath{_{\textnormal{3}}}/\ensuremath{_{\textnormal{2}}}, p\ensuremath{_{\textnormal{1}}}/\ensuremath{_{\textnormal{2}}} Level Spacings and Transition Probabilities in the Lead Region.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1961Ca21,B}{1961Ca21}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}B.Cagnac - Ann.Phys.(Paris) 6, 467 (1961).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Orientation Nucleaire par Pumpage Optique des Isotopes Impaires du Mercure.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1961Kr01,B}{1961Kr01}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}H.Krehbiel, U.Meyer-Berkhout - Z.Physik 165, 99 (1961).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Erzeugung 21 neuer kurzlebiger Isomere durch Kernphotoreaktionen.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1961Pe05,B}{1961Pe05}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}B.G.Pettersson, T.R.Gerholm, Z.Grabowski, B.Van Nooijen - Nuclear Phys. 24, 196 (1961).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Penetration Matrix Elements and Nuclear Structure Effects in Tl\ensuremath{^{\textnormal{201}}}.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1961Re12,B}{1961Re12}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}P.Reyes-Suter, T.Suter - Arkiv Fysik 20, 415 (1961).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Electromagnetic Transition in Odd-A Mercury Isotopes.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1962Ax02,B}{1962Ax02}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}S.Axensten, G.Liljegren, I.Lindgren, C.M.Olsmats - Arkiv Fysik 22, 392 (1962).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Nuclear Spin Measurements on Tl, Po, and Te Isotopes.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1962Eu01,B}{1962Eu01}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}P.Euthymiou, P.Axel - Phys.Rev. 128, 274 (1962).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Photoproduction of a 100{\textminus}\ensuremath{\mu}sec Isomer and its Tentative Assignment toHg-201.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1962Fa06,B}{1962Fa06}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}J.Facetti, E.Trabal, R.McClin, S.Torres - Phys.Rev. 127, 1690 (1962).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{A New Isotope, Pt\ensuremath{^{\textnormal{201}}}.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1962Mo19,B}{1962Mo19}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}A.M.Morozov, V.V.Remaev - Zhur.Eksptl.i Teoret.Fiz. 43, 438 (1962); Soviet Phys.JETP 16, 314 (1963).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Investigation of the Millisecond Isomers Detected in Nuclear Reactions Involving Fast Protons.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1963De38,B}{1963De38}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}A.G.Demin, Y.P.Kushakevich, E.A.Makoveev, I.M.Rozman, A.F.Chachakov - Zh.Eksperim.i Teor.Fiz. 45, 1344 (1963); Soviet Phys.JETP 18, 925 (1964).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Millisecond Thallium Isomers.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1963Go06,B}{1963Go06}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}K.P.Gopinathan, M.C.Joshi, M.Radhamenon - Nuovo Cimento 30, 14 (1963).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Decay of the New 2.5 min \ensuremath{^{\textnormal{201}}}Pt.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1963Ho18,B}{1963Ho18}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}R.W.Hoff, F.Asaro, I.Perlman - J.Inorg.Nucl.Chem. 25, 1303 (1963).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Alpha Decay of Neutron-Deficient Astatine Isotopes.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1964Aa01,B}{1964Aa01}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}E.Aasa, T.Sundstrom, O.Bergman, J.Lindskog, K.Sevier - Arkiv Fysik 27, 133 (1964).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{A Conversion Electron Study of the Decay of Pb\ensuremath{^{\textnormal{201}}}.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1964Br23,B}{1964Br23}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}C.Brun, M.Lefort - J. Inorg. Nucl. Chem. 26, 1633 (1964).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Production des Isotopes 196 a 202 du Polonium par Reactions Nucleaires (p,xn) sur le Bismuth.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1964Br27,B}{1964Br27}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}K.Brandi, R.Engelmann, V.Hepp, E.Kluge, H.Krehbiel, U.Meyer-Berkhout - Nucl. Phys. 59, 33 (1964).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Investigation of Short-Lived Isomeric States with Half-Lives in the \ensuremath{\mu}s and ms Region.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1964Gr04,B}{1964Gr04}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}R.D.Griffioen, R.D.Macfarlane - Phys.Rev. 133, B1373 (1964).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Alpha-Decay Properties of Some Francium Isotopes Near the 126-NeutronClosed Shell.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1964Si11,B}{1964Si11}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}A.Siivola, P.Kauranen, B.Jung, J.Svedberg - Nucl.Phys. 52, 449 (1964).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Alpha Active Isomers in Bi\ensuremath{^{\textnormal{201}}} and Bi\ensuremath{^{\textnormal{199}}}.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1965Gr04,B}{1965Gr04}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}V.T.Gritsyna, H.H.Forster - Nucl.Phys. 61, 129 (1965).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{New Short-Lived Isomeric Levels in Tl\ensuremath{^{\textnormal{201m}}} and Tb\ensuremath{^{\textnormal{153m}}}.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1965Mu15,B}{1965Mu15}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}K.Murakawa - J.Phys.Soc.Japan 20, 1094 (1965).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Quadrupole Coupling in the Hyperfine Structure of Hg I.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1966KaZY,B}{1966KaZY}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}P.Karol, C.Stearns - CU-1019-51, p.8 (1966).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Decay Schemes of Bi\ensuremath{^{\textnormal{201}}}\ensuremath{\rightarrow}\ensuremath{^{\textnormal{204}}}.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1966Ma51,B}{1966Ma51}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}I.Mahunka, L.Tron, T.Fenyes, V.A.Khalkin - Izv.Akad.Nauk SSSR, Ser.Fiz. 30, 1375 (1966); Bull.Acad.Sci.USSR, Phys.Ser. 30, 1436 (1967).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Investigation of the Alpha Spectrum of Bismuth Isotopes.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1967Co20,B}{1967Co20}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}T.W.Conlon - Nucl.Phys. A100, 545 (1967).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{High-Resolution Studies of the Gamma Rays from Isomeric States with Half-Lives of 10\ensuremath{\mu}s-30ms in Nuclei with Z = 63-83.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1967Le08,B}{1967Le08}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}Y.Le Beyec, M.Lefort - Nucl.Phys. A99, 131 (1967).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Reactions entre Protons de 30 a 155 MeV et Noyaux Lourds Complexes.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1967Le21,B}{1967Le21}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}Y.Le Beyec, M.Lefort - Arkiv Fysik 36, 183 (1967).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Alpha Emission from Light Polonium Isotopes.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1967Ti04,B}{1967Ti04}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}E.Tielsch-Cassel - Nucl.Phys. A100, 425 (1967).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Alpha-Decay Studies of Polonium Isotopes in the Mass Range 199 to 208.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1967Tr06,B}{1967Tr06}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}W.Treytl, K.Valli - Nucl.Phys. A97, 405 (1967).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Alpha Decay of Neutron Deficient Astatine Isotopes.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1967Va17,B}{1967Va17}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}K.Valli, M.J.Nurmia, E.K.Hyde - Phys.Rev. 159, 1013 (1967).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Alpha-Decay Properties of Neutron-Deficient Isotopes of Emanation.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1967Va20,B}{1967Va20}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}K.Valli, E.K.Hyde, W.Treytl - J.Inorg.Nucl.Chem. 29, 2503 (1967).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Alpha Decay of Neutron-Deficient Francium Isotopes.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1968Go12,B}{1968Go12}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}N.A.Golovkov, R.B.Ivanov, Y.V.Norseev, So Ki Kvan, V.A.Khalkin, V.G.Chumin - Contrib.Intern.Conf.Nucl.Struct., Dubna, p.54 (1968).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Determination of Energies of Principle \ensuremath{\alpha}-Groups of Short-Lived Po and At Isotopes.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1969Al10,B}{1969Al10}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}M.Alpsten, G.Astner - Nucl.Phys. A134, 407 (1969).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Positive-Parity States in Some Odd-Mass Bismuth Isotopes.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1970DaZM,B}{1970DaZM}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}J.M.Dairiki - Thesis, Univ. California (1970); UCRL-20412 (1970).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Decay Studies of Neutron-Deficient Isotopes of Astatine, Polonium, and Bismuth.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1970DoZT,B}{1970DoZT}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}R.E.Doebler - Thesis, Michigan State Univ. (1970).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Gamma Ray Spectroscopic Studies of States in Neutron-Deficient Pb andTi Isotopes.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1970Jo26,B}{1970Jo26}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}A.G.Jones, A.H.W.Aten, Jr. - Radiochim.Acta 13, 176 (1970).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Neutron-Deficient Isotopes of Polonium.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1970Ra14,B}{1970Ra14}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}K.Raichev, L.Tron - Acta Phys. 28, 263 (1970).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Investigation of the Alpha Spectra of Po Isotopes.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1971Hn04,B}{1971Hn04}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}V.Hnatowicz, J.Kristak, R.D.Connor - Nucl.Phys. A175, 539 (1971).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{The Decay of \ensuremath{^{\textnormal{201}}}Pb.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1971Ho01,B}{1971Ho01}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}P.Hornshoj, K.Wilsky, P.G.Hansen, A.Lindahl, O.B.Nielsen - Nucl.Phys. A163, 277 (1971).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Alpha Decay of Neutron-Deficient Radon and Polonium Isotopes.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1971Jo19,B}{1971Jo19}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}B.Jonson, M.Alpsten, A.Appelqvist, G.Astner - Nucl.Phys. A174, 225 (1971).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Properties of Neutron-Deficient Odd-Mass Polonium Isotopes.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1971Wa17,B}{1971Wa17}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}D.Walcher - Z.Phys. 246, 123 (1971).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Mossbaueruntersuchungen an \ensuremath{^{\textnormal{195}}}Pt und \ensuremath{^{\textnormal{201}}}Hg.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1972Ba53,B}{1972Ba53}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}H.Backe, R.Engfer, U.Jahnke, E.Kankeleit, R.M.Pearce, C.Petitjean, L.Schellenberg, H.Schneuwly, W.U.Schroder, H.K.Walter, A.Zehnder - Nucl.Phys. A189, 472 (1972).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Study of X-Rays and Nuclear \ensuremath{\gamma}-Rays in Muonic Thallium.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1972Mo12,B}{1972Mo12}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}R.A.Moyer - Phys.Rev. C5, 1678 (1972).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Deuteron-Induced Reactions on the Even-Even Isotopes of Mercury.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1972Pa24,B}{1972Pa24}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}A.Pakkanen, H.Helppi, T.Komppa, P.Puumalainen - Z.Phys. 254, 98 (1972).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Low-Lying Levels of \ensuremath{^{\textnormal{201}}}Hg from the Decay of \ensuremath{^{\textnormal{201}}}Au.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1973BoVN,B}{1973BoVN}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}J.Bonn, G.Huber, H.-J.Kluge, U.Kopf, L.Kugler, E.-W.Otten, J.Rodriguez - Proc.Int.Conf.Nucl.Moments and Nucl.Struct., Osaka, Japan (1972), H.Horie, K.Sugimoto, Eds., p.317 (1973); J.Phys.Soc.Jap. 34 Suppl. (1973).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Determination of Nuclear Spins Moments and Charge Volumes Far from Stability by Optical Pumping.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1973GiZW,B}{1973GiZW}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}P.Gippner, K.-H.Kaun, W.Neubert, F.Stary, W.Schulze - Proc.Int.Conf.Nucl.Phys., Munich, J.de Boer, H.J.Mang, Eds., North-Holland Publ.Co., Amsterdam, Vol.1, p.235 (1973).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Isomeric States in 199-Bi and 201-Bi.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1973Re04,B}{1973Re04}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}R.J.Reimann, M.N.McDermott - Phys.Rev. C7, 2065 (1973).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Precision Magnetic Moment Determinations for 43-min \ensuremath{^{\textnormal{190m}}}Hg and Other Isomers of Mercury.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1974GiZX,B}{1974GiZX}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}P.Gippner, K.-H.Kaun, W.Neubert, F.Stary, W.Schulze - JINR-E6-7392 (1974).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Highly Excited Isomeric States in \ensuremath{^{\textnormal{199}}}Bi and \ensuremath{^{\textnormal{201}}}Bi.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1974Ha18,B}{1974Ha18}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}A.Hanser - Z.Phys. 267, 135 (1974).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Bestimmung des Kernstrukturparameters der inneren Konversion und der E2-Beimischung fur M1-Ubergange in \ensuremath{^{\textnormal{201}}}Tl und \ensuremath{^{\textnormal{203}}}Tl.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1974Ho27,B}{1974Ho27}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}P.Hornshoj, P.G.Hansen, B.Jonson - Nucl.Phys. A230, 380 (1974).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Alpha-Decay Widths of Neutron-Deficient Francium and Astatine Isotopes.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1975BaYJ,B}{1975BaYJ}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}G.Bastin, C.F.Liang - CSNSM-1973-1975 Prog.Rept., p.35 (1975).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Spectres \ensuremath{\alpha} d$'$Isotopes d$'$Astate Deficients en Neutrons Produits et Separes en Ligne.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1975Ed01,B}{1975Ed01}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}W.A.Edelstein, R.V.Pound - Phys.Rev. B11, 985 (1975).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Time-Dependent Directional Correlations of \ensuremath{^{\textnormal{199}}}Hg-m in Liquid Hg, Solid Hg, and HgCl\ensuremath{_{\textnormal{2}}}.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1975Ge04,B}{1975Ge04}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}S.Gerstenkorn, J.Verges - J.Phys.(Paris) 36, 481 (1975).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Interpretation des Deplacements Isotopiques Pairs-Impairs Anormaux dans le Spectre d$'$Arc du Mercure.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1975Ho08,B}{1975Ho08}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}S.Hofmann, D.Walcher - Z.Phys. A272, 351 (1975).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Spin Values of the Low Lying Nuclear Levels of \ensuremath{^{\textnormal{201}}}Hg.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1975OhZZ,B}{1975OhZZ}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}S.Ohya, Y.Shida, N.Yoshikawa - Inst.Nucl.Study, Univ.Tokyo, Ann.Rept. 1974, p.54 (1975).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Search for Isomers in Bi Isotopes.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1975Uy01,B}{1975Uy01}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}J.Uyttenhove, K.Heyde, H.Vincx, M.Waroquier - Nucl.Phys. A241, 135 (1975).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Possible Evidence for Microsecond Shape Isomerism in Tl Isotopes.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1976Fu06,B}{1976Fu06}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}G.H.Fuller - J.Phys.Chem.Ref.Data 5, 835 (1976).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Nuclear Spins and Moments.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1976HiZN,B}{1976HiZN}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}A.T.Hirshfeld, D.D.Hoppes, F.J.Schima - Priv.Comm. (October 1976).}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1976Ko13,B}{1976Ko13}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}A.Korman, D.Chlebowska, T.Kempisty, S.Chojnacki - Acta Phys.Pol. B7, 141 (1976).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Excited States in \ensuremath{^{\textnormal{199}}}Bi, \ensuremath{^{\textnormal{201}}}Bi, \ensuremath{^{\textnormal{203}}}Bi Isotopes.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1976Uy01,B}{1976Uy01}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}J.Uyttenhove, J.Demuynck - Nucl.Instrum.Methods 136, 529 (1976).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{A Ge(Li) Spectrometer with Improved Overload Recovery Performance.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1977Gu18,B}{1977Gu18}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}M.Gustafsson, I.Lindgren, J.Lindgren, A.Rosen, H.Rubinsztein - Phys.Lett. 72B, 166 (1977).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Hyperfine Structure Measurements in Metastable Atomic States {\textminus} Nuclear Spins of \ensuremath{^{\textnormal{201}}}Pb and \ensuremath{^{\textnormal{203}}}Pb.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1977He06,B}{1977He06}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}H.Helppi, S.K.Saha, P.J.Daly, S.R.Faber, T.L.Khoo, F.M.Bernthal - Phys.Lett. 67B, 279 (1977).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{High-Spin Neutron Hole Excitations in Light Odd-A Pb Nuclei.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1977KoZH,B}{1977KoZH}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}T.Kobayashi, T.Numao, H.Nakayama, J.Imazato - UTPN-100, p.40 (1977).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Lifetime Measurements in \ensuremath{\mu}sec-nsec Region.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1977Na31,B}{1977Na31}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}H.W.Nass - J.Nucl.Med. 18, 1047 (1977).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{New Tl-201 Nuclear Decay Data.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1977Se15,B}{1977Se15}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}R.Seltz - Nukleonika 22, 31 (1977).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Is the (p,t) Reaction a Useful Tool for the Investigation of Shape Tra.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1977Sl01,B}{1977Sl01}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}M.G.Slocombe, J.O.Newton, G.D.Dracoulis - Nucl.Phys. A275, 166 (1977).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{A Study of States in \ensuremath{^{\textnormal{201}}},\ensuremath{^{\textnormal{203}}}Tl Using the (d,3n\ensuremath{\gamma}) Reaction: A New 9/2\ensuremath{^{-}} Band.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1978No06,B}{1978No06}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}E.F.Novak, A.P.Arya, R.L.Martz - Phys.Rev. C18, 1045 (1978).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Gamma-Gamma Directional Correlations in \ensuremath{^{\textnormal{201}}}Hg.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1978Ri04,B}{1978Ri04}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}H.Richel, G.Albouy, G.Auger, F.Hanappe, J.M.Lagrange, M.Pautrat, C.Roulet, H.Sergolle, J.Vanhorenbeeck - Nucl.Phys. A303, 483 (1978).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{On the Decay of \ensuremath{^{\textnormal{201}}}Bi and \ensuremath{^{\textnormal{199}}}Bi.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1979De42,B}{1979De42}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}K.Debertin, W.Pessara, U.Schotzig, K.F.Walz - Int.J.Appl.Radiat.Isotop. 30, 551 (1979).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Decay Data of \ensuremath{^{\textnormal{67}}}Ga and \ensuremath{^{\textnormal{201}}}Tl.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1979Do09,B}{1979Do09}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}R.E.Doebler, W.C.McHarris, W.H.Kelly - Phys.Rev. C20, 1112 (1979).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Decay of \ensuremath{^{\textnormal{201}}}Pb.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1979Ha08,B}{1979Ha08}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}A.A.Hahn, J.P.Miller, R.J.Powers, A.Zehnder, A.M.Rushton, R.E.Welsh, A.R.Kunselman, P.Roberson, H.K.Walter - Nucl.Phys. A314, 361 (1979).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{An Experimental Study of Muonic X-Ray Transitions in Mercury Isotopes.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1979MaYQ,B}{1979MaYQ}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}H.-E.Mahnke, O.Hausser, T.K.Alexander, H.R.Andrews, J.F.Sharpey-Schafer, P.Taras, D.Ward, I.S.Towner - AECL-6680, p.27 (1979).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Quadrupole moments of isomeric states in \ensuremath{^{\textnormal{200$-$206}}}Pb.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1980Bo05,B}{1980Bo05}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}A.Bockisch, M.Miller, K.P.Lieb, K.Heyde - Phys.Lett. 90B, 61 (1980).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Identification of the Missing f\ensuremath{_{\textnormal{5}}}/\ensuremath{_{\textnormal{2}}} State in \ensuremath{^{\textnormal{201}}}Hg by CoulombExcitation.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1980Br23,B}{1980Br23}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}R.A.Braga, W.R.Western, J.L.Wood, R.W.Fink, R.Stone, C.R.Bingham, L.L.Riedinger - Nucl.Phys. A349, 61 (1980).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Very Slow M4 Transitions and Shell-Model Intruder States in \ensuremath{^{\textnormal{199}}},\ensuremath{^{\textnormal{201}}}Bi.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1980Ew03,B}{1980Ew03}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}G.T.Ewan, E.Hagberg, B.Jonson, S.Mattsson, P.Tidemand-Petersson - Z.Phys. A296, 223 (1980).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Alpha-Decay Studies of New Neutron-Deficient Francium Isotopes and Their Daughters.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1981An11,B}{1981An11}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}O.Ando, K.Miyano - Int.J.Appl.Radiat.Isotop. 32, 381 (1981).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Half-Life Determination with High Counting Rate and Half-Lives of \ensuremath{^{\textnormal{202m}}}Pb and \ensuremath{^{\textnormal{201}}}Pb.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1981Fl05,B}{1981Fl05}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}E.R.Flynn, R.E.Brown, J.W.Sunier, J.M.Gursky, J.A.Cizewski, D.G.Burke - Phys.Lett. 105B, 125 (1981).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Proton-Hole Excitations in \ensuremath{^{\textnormal{201}}},\ensuremath{^{\textnormal{203}}}Au from \ensuremath{^{\textnormal{202}}},\ensuremath{^{\textnormal{204}}}Hg(t(pol),\ensuremath{\alpha}) Reactions.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1981He07,B}{1981He07}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}H.Helppi, S.K.Saha, P.J.Daly, S.R.Faber, T.L.Khoo, F.M.Bernthal - Phys.Rev. C23, 1446 (1981).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{High-Spin Level Spectra of the Nuclei \ensuremath{^{\textnormal{195}}}Pb,\ensuremath{^{\textnormal{197}}}Pb,\ensuremath{^{\textnormal{199}}}Pb,and \ensuremath{^{\textnormal{201}}}Pb.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1981Ri04,B}{1981Ri04}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}B.G.Ritchie, K.S.Toth, H.K.Carter, R.L.Mlekodaj, E.H.Spejewski - Phys.Rev. C23, 2342 (1981).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Alpha-Decay Properties of \ensuremath{^{\textnormal{205}}},\ensuremath{^{\textnormal{206}}},\ensuremath{^{\textnormal{207}}},\ensuremath{^{\textnormal{208}}}Fr: Identification of \ensuremath{^{\textnormal{206m}}}Fr.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1982Br21,B}{1982Br21}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}R.Broda, C.Gunther, B.V.Thirumala Rao - Nucl.Phys. A389, 366 (1982).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{High-Spin States in \ensuremath{^{\textnormal{201}}}Bi Populated in the (\ensuremath{\alpha},6n) Reaction.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1982Bu04,B}{1982Bu04}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}B.Budick, R.Anigstein, J.W.Kast - Phys.Lett. 110B, 375 (1982).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Nuclear Excitation in the Reaction \ensuremath{^{\textnormal{207}}}Pb(\ensuremath{\mu}\ensuremath{^{-}},\ensuremath{\nu}6n)\ensuremath{^{\textnormal{201}}}Tl.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1982HoZJ,B}{1982HoZJ}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}D.D.Hoppes, J.M.R.Hutchinson, F.J.Schima, M.P.Unterweger - NBS-SP-626, p.85 (1982).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Nuclear Data for X- or Gamma-Ray Spectrometer Efficiency Calibrations.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1982La25,B}{1982La25}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}F.Lagoutine, J.Legrand - Int.J.Appl.Radiat.Isotop. 33, 711 (1982).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Periodes de Neuf Radionucleides.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1982Th05,B}{1982Th05}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}R.C.Thompson, A.Hanser, K.Bekk, G.Meisel, D.Frolich - Z.Phys. A305, 89 (1982).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{High Resolution Measurements of Isotope Shifts in Lead.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1983Bu02,B}{1983Bu02}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}B.Budick, R.Anigstein, J.W.Kast - Nucl.Phys. A393, 469 (1983).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Delayed Gamma Rays from Muon Capture on \ensuremath{^{\textnormal{207}}}Pb.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1983Dy02,B}{1983Dy02}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}K.Dybdal, T.Chapuran, D.B.Fossan, W.F.Piel,Jr., D.Horn, E.K.Warburton - Phys.Rev. C28, 1171 (1983).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{High-Spin States in \ensuremath{^{\textnormal{201}}},\ensuremath{^{\textnormal{203}}}At and the Systematic Behavior of Z = 85 Isotopes.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1983Fu22,B}{1983Fu22}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}E.Funck, K.Debertin, K.F.Walz - Int.J.Nucl.Med.Biol. 10, 137 (1983).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Standardization and Decay Data of \ensuremath{^{\textnormal{201}}}Tl.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1983Sc38,B}{1983Sc38}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}P.Schuler, K.Hardt, C.Gunther, K.Freitag, P.Herzog, H.Niederwestberg,H.Reif, E.B.Shera, M.V.Hoehn - Z.Phys. A313, 305 (1983).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Experimental Investigation of Low Lying Excited States in \ensuremath{^{\textnormal{201}}}Hg.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1984Be40,B}{1984Be40}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}C.Bengtsson, C.Ekstrom, L.Robertsson, J.Heinemeier, and the ISOLDE Collaboration - Phys.Scr. 30, 164 (1984).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Nuclear Spins of Some Indium and Thallium Isotopes.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1985Fi05,B}{1985Fi05}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}J.E.Finck, G.M.Crawley, D.Weber, P.A.Smith - Nucl.Phys. A441, 57 (1985).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{The \ensuremath{^{\textnormal{206}}},\ensuremath{^{\textnormal{204}}}Pb(p,\ensuremath{\alpha})\ensuremath{^{\textnormal{203}}},\ensuremath{^{\textnormal{201}}}Tl Reactions at 35 MeV.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1985Pi05,B}{1985Pi05}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}W.F.Piel,Jr., T.Chapuran, K.Dybdal, D.B.Fossan, T.Lonnroth, D.Horn, E.K.Warburton - Phys.Rev. C31, 2087 (1985).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{High-Spin \ensuremath{\gamma}-Ray Spectroscopy in Z = 83 Isotopes: \ensuremath{^{\textnormal{199}}},\ensuremath{^{\textnormal{201}}}Bi.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1985We05,B}{1985We05}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}T.Weckstrom, B.Fant, T.Lonnroth, V.Rahkonen, A.Kallberg, C.-J.Herrlander - Z.Phys. A321, 231 (1985).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{In-Beam Study of High-Spin States in \ensuremath{^{\textnormal{199}}},\ensuremath{^{\textnormal{200}}},\ensuremath{^{\textnormal{201}}}Po and Systematical Features of Z = 84 Polonium Isotopes.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986An06,B}{1986An06}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}M.Anselment, W.Faubel, S.Goring, A.Hanser, G.Meisel, H.Rebel, G.Schatz - Nucl.Phys. A451, 471 (1986).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{The Odd-Even Staggering of the Nuclear Charge Radii of Pb Isotopes.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986Be07,B}{1986Be07}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}V.Berg, J.Oms, K.Fransson, Z.Hu, and the ISOCELE Collaboration - Nucl.Phys. A453, 93 (1986).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Half-Life and Electromagentic Deexcitation Rate Measurements for (1/2)\ensuremath{^{\textnormal{+}}} and (3/2)\ensuremath{^{\textnormal{+}}} States in Odd-Proton Nuclei around Z = 82.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986Br28,B}{1986Br28}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}R.A.Braga, P.B.Semmes, W.R.Western, R.W.Fink - Nucl.Phys. A459, 359 (1986).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Decay of Mass-Separated \ensuremath{^{\textnormal{201m}}}Po and \ensuremath{^{\textnormal{201g}}}Po.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986Ul02,B}{1986Ul02}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}G.Ulm, S.K.Bhattacherjee, P.Dabkiewicz, G.Huber, H.-J.Kluge, T.Kuhl, H.Lochmann, E.-W.Otten, K.Wendt, S.A.Ahmad, W.Klempt, R.Neugart, and the ISOLDE Collaboration - Z.Phys. A325, 247 (1986).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Isotope Shift of \ensuremath{^{\textnormal{182}}}Hg and an Update of Nuclear Moments and ChargeRadii in the Isotope Range \ensuremath{^{\textnormal{181}}}Hg {\textminus} \ensuremath{^{\textnormal{206}}}Hg.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1986Wo03,B}{1986Wo03}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}J.Wouters, D.Vandeplassche, E.van Walle, N.Severijns, L.Vanneste - Phys.Rev.Lett. 56, 1901 (1986).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Anisotropic Alpha Emission from On-Line-Separated Isotopes.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1987Bo44,B}{1987Bo44}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}J.A.Bounds, C.R.Bingham, H.K.Carter, G.A.Leander, R.L.Mlekodaj, E.H.Spejewski, W.M.Fairbank, Jr. - Phys.Rev. C36, 2560 (1987).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Nuclear Structure of Light Thallium Isotopes as Deduced from Laser Spectroscopy on a Fast Atom Beam.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1987Dr06,B}{1987Dr06}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}O.Dragoun, V.Brabec, M.Rysavy, A.Spalek, K.Freitag - Z.Phys. A326, 279 (1987).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Some Nuclear and Atomic Properties of \ensuremath{^{\textnormal{201}}}Hg Determined by Conversion Electron Spectroscopy.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1987Fu08,B}{1987Fu08}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}E.Funck - Appl.Radiat.Isot. 38, 771 (1987).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{A Two-Dimensional Extrapolation for the Standardization of \ensuremath{^{\textnormal{201}}}Tl by the 4\ensuremath{\pi}\ensuremath{\beta}-\ensuremath{\gamma}- Coincidence Method.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1987He10,B}{1987He10}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}F.P.Hessberger, S.Hofmann, G.Munzenberg, A.B.Quint, K.Summerer, P.Armbruster - Europhys.Lett. 3, 895 (1987).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Observation of Two New Alpha Emitters with Z = 88.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1988Ro08,B}{1988Ro08}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}U.Rosengard, P.Carle, A.Kallberg, L.O.Norlin, K.-G.Rensfelt, H.C.Jain, B.Fant, T.Weckstrom - Nucl.Phys. A482, 573 (1988).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Yrast Spectroscopy and g-Factor Measurements in \ensuremath{^{\textnormal{199}}}Pb,\ensuremath{^{\textnormal{201}}}Pb and\ensuremath{^{\textnormal{203}}}Pb.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1988Wo12,B}{1988Wo12}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}J.Wouters, N.Severijns, J.Vanhaverbeke, W.Vanderpoorten, L.Vanneste - Hyperfine Interactions 43, 401 (1988).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Magnetic Moments Close to \ensuremath{^{\textnormal{208}}}Pb from On-Line NO.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1989Pl04,B}{1989Pl04}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}J.Plch, P.Kovar, P.Dryak - Appl.Radiat.Isot. 40, 513 (1989).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Absolute Measurement of Activity and Total Photon Yield in the Decay of \ensuremath{^{\textnormal{201}}}Tl.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1989Sc17,B}{1989Sc17}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}H.Schrader - Appl.Radiat.Isot. 40, 381 (1989).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Measurement of the Half-Lives of \ensuremath{^{\textnormal{18}}}F,\ensuremath{^{\textnormal{56}}}Co,\ensuremath{^{\textnormal{125}}}I,\ensuremath{^{\textnormal{195}}}Au and \ensuremath{^{\textnormal{201}}}Tl.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1989Su12,B}{1989Su12}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}X.Sun, U.Rosengard, H.Grawe, H.Haas, H.Kluge, A.Kuhnert, K.H.Maier - Z.Phys. A333, 281 (1989).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Conversion Electron Measurement of Isomeric Primary Transitions in \ensuremath{^{\textnormal{196}}},\ensuremath{^{\textnormal{198}}},\ensuremath{^{\textnormal{199}}},\ensuremath{^{\textnormal{200}}},\ensuremath{^{\textnormal{201}}}Pb.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1990Co07,B}{1990Co07}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}B.M.Coursey, D.D.Hoppes, A.T.Hirshfeld, S.M.Judge, D.H.Woods, M.J.Woods, E.Funck, H.Schrader, A.G.Tuck - Appl.Radiat.Isot. 41, 289 (1990).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Standardization and Decay Scheme of \ensuremath{^{\textnormal{201}}}Tl.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1990Ka08,B}{1990Ka08}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}Y.Kawada, Y.Hino, W.Gatot - Nucl.Instrum.Methods Phys.Res. A286, 539 (1990).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Photon Emission Probabilities of \ensuremath{^{\textnormal{201}}}Tl.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1990Lo17,B}{1990Lo17}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}T.Lonnroth, P.Ahonen, R.Julin, S.Juutinen, A.Lampinen, A.Pakkanen, S.Tormanen, W.Trzaska, A.Virtanen - Z.Phys. A337, 11 (1990).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Properties of the (13/2)\ensuremath{^{\textnormal{+}}} Isomeric Decay in \ensuremath{^{\textnormal{201}}}Hg.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1991Dr09,B}{1991Dr09}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}O.Dragoun - J.Phys.(London) G17, S91 (1991).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Rare Effects in Gamma-Ray Internal Conversion.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1991Ry01,B}{1991Ry01}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}A.Rytz - At.Data Nucl.Data Tables 47, 205 (1991).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Recommended Energy and Intensity Values of Alpha Particles from Radioactive Decay.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1991Wo04,B}{1991Wo04}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}J.Wouters, N.Severijns, J.Vanhaverbeke, L.Vanneste - J.Phys.(London) G17, 1673 (1991).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Magnetic Moments of Po Isotopes and the Quenching of Nuclear Magnetism in the \ensuremath{^{\textnormal{208}}}Pb Region.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1992Ba39,B}{1992Ba39}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}G.Baldsiefen, H.Hubel, F.Azaiez, C.Bourgeois, D.Hojman, A.Korichi, N.Perrin, H.Sergolle - Z.Phys. A343, 245 (1992).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Rotational Bands in \ensuremath{^{\textnormal{201}}}Pb.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1992Un01,B}{1992Un01}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}M.P.Unterweger, D.D.Hoppes, F.J.Schima - Nucl.Instrum.Methods Phys.Res. A312, 349 (1992).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{New and Revised Half-Life Measurements Results.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1993Wa04,B}{1993Wa04}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}J.Wauters, P.Dendooven, M.Huyse, G.Reusen, P.Van Duppen, P.Lievens, and the ISOLDE Collaboration - Phys.Rev. C47, 1447 (1993).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{\ensuremath{\alpha}-Decay Properties of Neutron-Deficient Polonium and Radon Nuclei.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1994Si26,B}{1994Si26}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}B.R.S.Simpson, B.R.Meyer - Appl.Radiat.Isot. 45, 669 (1994).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Standardization and Half-Life of \ensuremath{^{\textnormal{201}}}Tl by the 4\ensuremath{\pi}(x,e)-\ensuremath{\gamma} Coincidence Method with Liquid Scintillation Counting in the 4\ensuremath{\pi}-Channel.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1995Ba70,B}{1995Ba70}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}G.Baldsiefen, P.Maagh, H.Hubel, W.Korten, S.Chmel, M.Neffgen, W.Pohler, H.Grawe, K.H.Maier, K.Spohr, R.Schubart, S.Frauendorf, H.J.Maier - Nucl.Phys. A592, 365 (1995).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Shears Bands in \ensuremath{^{\textnormal{201}}}Pb and \ensuremath{^{\textnormal{202}}}Pb.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1995Le04,B}{1995Le04}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}M.J.Leddy, S.J.Freeman, J.L.Durell, A.G.Smith, S.J.Warburton, D.J.Blumenthal, C.N.Davids, C.J.Lister, H.T.Penttila - Phys.Rev. C51, R1047 (1995).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{\ensuremath{\alpha} Decay of a New Isotope, \ensuremath{^{\textnormal{204}}}Ra.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1995Le15,B}{1995Le15}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}M.Leino, J.Aysto, T.Enqvist, A.Jokinen, M.Nurmia, A.Ostrowski, W.H.Trzaska, J.Uusitalo, K.Eskola - Acta Phys.Pol. B26, 309 (1995).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Research on Heavy Elements Using the JYFL Gas-Filled Recoil SeparatorRITU.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1996En01,B}{1996En01}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}T.Enqvist, K.Eskola, A.Jokinen, M.Leino, W.H.Trzaska, J.Uusitalo, V.Ninov, P.Armbruster - Z.Phys. A354, 1 (1996).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Alpha Decay Properties of \ensuremath{^{\textnormal{200}}~^{-}~^{\textnormal{202}}}Fr.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1996Le09,B}{1996Le09}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}M.Leino, J.Uusitalo, R.G.Allatt, P.Armbruster, T.Enqvist, K.Eskola, S.Hofmann, S.Hurskanen, A.Jokinen, V.Ninov, R.D.Page, W.H.Trzaska - Z.Phys. A355, 157 (1996).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Alpha Decay Studies of Neutron-Deficient Radium Isotopes.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1996Ta18,B}{1996Ta18}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}R.B.E.Taylor, S.J.Freeman, J.L.Durell, M.J.Leddy, A.G.Smith, D.J.Blumenthal, M.P.Carpenter, C.N.Davids, C.J.Lister, R.V.F.Janssens, D.Seweryniak - Phys.Rev. C54, 2926 (1996).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{\ensuremath{\gamma} Decay from States at Low Excitation Energy in the Neutron-Deficient Isotope,\ensuremath{^{\textnormal{200}}}Rn,Identified by Correlated Radioactive Decay.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?1997Ge09,B}{1997Ge09}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}V.N.Gerasimov, D.V.Grebennikov, V.M.Kulakov, S.K.Lisin, V.V.Kharitonov - Yad.Fiz. 60, No 11, 1948 (1997); Phys.Atomic Nuclei 60, 1780 (1997).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Spectrum of Conversion Electrons from the 1.56-keV (M1 + E2) Transition in \ensuremath{^{\textnormal{201}}}Hg.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2000PoZY,B}{2000PoZY}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}Zs.Podolyak, P.H.Regan, M.Pfutzner, J.Gerl, M.Hellstrom, M.Caamano, P.Mayet, M.Mineva, M.Sawicka, Ch.Schlegel, for the GSI Isomer Collaboration - Proc.2nd Intern.Conf Fission and Properties of Neutron-Rich Nuclei, St Andrews, Scotland, June 28-July 3, 1999, J.H.Hamilton, W.R.Phillips, H.K.Carter, Eds., World Scientific, Singapore, p.156 (2000).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{High Spin Fragmentation Spectroscopy.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2001Ca13,B}{2001Ca13}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}M.Caamano, P.H.Regan, Zs.Podolyak, C.J.Pearson, P.Mayet, J.Gerl, Ch.Schlegel, M.Pfutzner, M.Hellstrom, M.Mineva, and the GSI ISOMER Collaboration - Nucl.Phys. A682, 223c (2001).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Nuclear Structure of Heavy Neutron Rich Systems: Fragmentation spectroscopy with a 1 GeV per nucleon \ensuremath{^{\textnormal{208}}}Pb beam.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2001MaZV,B}{2001MaZV}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}P.Mayet, J.Gerl, Ch.Schlegel, Zs.Podolyak, P.H.Regan, M.Caamano, M.Pfutzner, M.Hellstrom, M.Mineva, for the GSI ISOMER Collaboration - GSI 2001-1, p.13 (2001).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{High Spin States Populated via Projectile Fragmentation in Very Neutron-Rich Nuclei Around Mass 180.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2002Kh12,B}{2002Kh12}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}V.V.Kharitonov, V.N.Gerasimov - Yad.Fiz. 65, 1411 (2002); Phys.Atomic Nuclei 65, 1377 (2002).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Study of \ensuremath{^{\textnormal{201}}}Tl \ensuremath{\rightarrow} \ensuremath{^{\textnormal{201}}}Hg Decay Properties.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2002Po15,B}{2002Po15}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}Zs.Podolyak, M.Caamano, P.H.Regan, P.M.Walker, P.Mayet, J.Gerl, Ch.Schlegel, M.Hellstrom, M.Mineva, M.Pfutzner, and the GSI ISOMER Collaboration - Prog.Theor.Phys.(Kyoto), Suppl. 146, 467 (2002).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Isomeric Decays in Neutron-Rich W, Os and Pt Nuclei.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2002Un02,B}{2002Un02}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}M.P.Unterweger - Appl.Radiat.Isot. 56, 125 (2002).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Half-Life Measurements at the National Institute of Standards and Technology.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2003Po14,B}{2003Po14}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}Zs.Podolyak, P.H.Regan, P.M.Walker, M.Caamano, K.Gladnishki, J.Gerl, M.Hellstrom, P.Mayet, M.Pfutzner, M.Mineva, for the GSI ISOMER Collaboration - Nucl.Phys. A722, 273c (2003).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Heavy nuclei studied in projectile fragmentation.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2004De02,B}{2004De02}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}K.C.de Souza, M.L.da Silva, J.U.Delgado, R.Poledna, R.T.Lopes, C.J.daSilva - Appl.Radiat.Isot. 60, 307 (2004).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Measurements of nuclear data parameters of \ensuremath{^{\textnormal{201}}}Tl by gamma-ray spectrometry.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2004Ro09,B}{2004Ro09}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}B.Rosse, O.Stezowski, M.Meyer, A.Prevost, N.Redon, J.Styczen, M.Brekiesz, J.Grebosz, A.Maj, W.Meczynski, T.Pawlat, M.Zieblinski, K.Zuber, P.Bednarczyk, D.Curien, O.Dorvaux, B.Gall, P.Papka, J.Robin, J.P.Vivien, A.Astier, I.Deloncle, M.-G.Porquet, K.Keyes, A.Papenberg, K.M.Spohr, H.Hubel, A.Bracco, F.Camera, J.Bastin, M.Gorska - Int.J.Mod.Phys. E13, 47 (2004).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Structure of polonium isotopes at high spin with RFD + Euroball IV.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2004Sc04,B}{2004Sc04}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}H.Schrader - Appl.Radiat.Isot. 60, 317 (2004).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Half-life measurements with ionization chambers {\textminus} A study of systematic effects and results.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2004Wo02,B}{2004Wo02}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}M.J.Woods, S.M.Collins - Appl.Radiat.Isot. 60, 257 (2004).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Half-life data {\textminus} a critical review of TECDOC-619 update.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2005Bi03,B}{2005Bi03}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}J.Bieron, P.Pyykko, P.Jonsson - Phys.Rev. A 71, 012502 (2005).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Nuclear quadrupole moment of \ensuremath{^{\textnormal{201}}}Hg.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2005Ca02,B}{2005Ca02}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}M.Caamano, P.M.Walker, P.H.Regan, M.Pfutzner, Zs.Podolyak, J.Gerl, M.Hellstrom, P.Mayet, M.N.Mineva, A.Aprahamian, J.Benlliure, A.M.Bruce, P.A.Butler, D.Cortina Gil, D.M.Cullen, J.Doring, T.Enqvist, C.Fox, J.Garces Narro, H.Geissel, W.Gelletly, J.Giovinazzo, M.Gorska, H.Grawe, R.Grzywacz, A.Kleinbohl, W.Korten, M.Lewitowicz, R.Lucas, H.Mach, C.D.O$'$Leary, F.De Oliveira, C.J.Pearson, F.Rejmund, M.Rejmund, M.Sawicka, H.Schaffner, C.Schlegel, K.Schmidt, K.-H.Schmidt, P.D.Stevenson, Ch.Theisen, F.Vives, D.D.Warner, C.Wheldon, H.J.Wollersheim, S.Wooding, F.Xu, O.Yordanov - Eur.Phys.J. A 23, 201 (2005).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Isomers in neutron-rich A \ensuremath{\approx} 190 nuclides from \ensuremath{^{\textnormal{208}}}Pb fragmentation.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2005De01,B}{2005De01}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}H.De Witte, A.N.Andreyev, S.Dean, S.Franchoo, M.Huyse, O.Ivanov, U.Koster, W.Kurcewicz, J.Kurpeta, A.Plochocki, K.Van de Vel, J.Van de Walle, P.Van Duppen - Eur.Phys.J. A 23, 243 (2005).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Alpha-decay of neutron-deficient \ensuremath{^{\textnormal{200}}}Fr and heavier neighbours.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2005Is19,B}{2005Is19}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}D.Ishikawa, A.Q.R.Baron, T.Ishikawa - Phys.Rev. B 72, 140301 (2005).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Nuclear resonant scattering from the subnanosecond lifetime excited state of \ensuremath{^{\textnormal{201}}}Hg.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2005Uu02,B}{2005Uu02}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}J.Uusitalo, M.Leino, T.Enqvist, K.Eskola, T.Grahn, P.T.Greenlees, P.Jones, R.Julin, S.Juutinen, A.Keenan, H.Kettunen, H.Koivisto, P.Kuusiniemi, A.-P.Leppanen, P.Nieminen, J.Pakarinen, P.Rahkila, C.Scholey - Phys.Rev. C 71, 024306 (2005).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{\ensuremath{\alpha} decay studies of very neutron-deficient francium and radium isotopes.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2007Ko06,B}{2007Ko06}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}F.G.Kondev - Nucl.Data Sheets 108, 365 (2007).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Nuclear Data Sheets for A = 201.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2007Me12,B}{2007Me12}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}V.Meot, J.Aupiais, P.Morel, G.Gosselin, F.Gobet, J.N.Scheurer, M.Tarisien - Phys.Rev. C 75, 064306 (2007).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Half-life of the first excited state of \ensuremath{^{\textnormal{201}}}Hg.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2008An05,B}{2008An05}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}K.Andgren, B.Cederwall, J.Uusitalo, A.N.Andreyev, S.J.Freeman, P.T.Greenlees, B.Hadinia, U.Jakobsson, A.Johnson, P.M.Jones, D.T.Joss, S.Juutinen, R.Julin, S.Ketelhut, A.Khaplanov, M.Leino, M.Nyman, R.D.Page, P.Rahkila, M.Sandzelius, P.Sapple, J.Saren, C.Scholey, J.Simpson, J.Sorri, J.Thomson, R.Wyss - Phys.Rev. C 77, 054303 (2008).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Excited states in the neutron-deficient nuclei \ensuremath{^{\textnormal{197,199,201}}}Rn.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2008Ki07,B}{2008Ki07}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}T.Kibedi, T.W.Burrows, M.B.Trzhaskovskaya, P.M.Davidson, C.W.Nestor,Jr. - Nucl.Instrum.Methods Phys.Res. A589, 202 (2008).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Evaluation of theoretical conversion coefficients using BrIcc.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2008StZY,B}{2008StZY}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}S.J.Steer - University of Surrey (2008).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Isomer Decay Spectroscopy of N \ensuremath{\leq} 126 Neutron-Rich Nuclei.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2009Se13,B}{2009Se13}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}M.D.Seliverstov, A.N.Andreyev, N.Barre, A.E.Barzakh, S.Dean, H.De Witte, D.V.Fedorov, V.N.Fedoseyev, L.M.Fraile, S.Franchoo, J.Genevey, G.Huber, M.Huyse, U.Koster, P.Kunz, S.R.Lesher, B.A.Marsh, I.Mukha, B.Roussiere, J.Sauvage, I.Stefanescu, K.Van de Vel, P.Van Duppen, Yu.M.Volkov - Eur.Phys.J. A 41, 315 (2009).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Charge radii and magnetic moments of odd- A \ensuremath{^{\textnormal{183$-$189}}}Pb isotopes.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2009St16,B}{2009St16}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}S.J.Steer, Zs.Podolyak, S.Pietri, M.Gorska, G.F.Farrelly, P.H.Regan, D.Rudolph, A.B.Garnsworthy, R.Hoischen, J.Gerl, H.J.Wollersheim, H.Grawe, K.H.Maier, F.Becker, P.Bednarczyk, L.Caceres, P.Doornenbal, H.Geissel, J.Grebosz, A.Kelic, I.Kojouharov, N.Kurz, F.Montes, W.Prokopowicz, T.Saito, H.Schaffner, S.Tashenov, A.Heinz, T.Kurtukian-nieto, G.Benzoni, M.Pfutzner, A.Jungclaus, D.L.Balabanski, C.Brandau, A.Brown, A.M.Bruce, W.N.Catford, I.J.Cullen, Zs.Dombradi, M.E.Estevez, W.Gelletly, G.Ilie, J.Jolie, G.A.Jones, M.Kmiecik, F.G.Kondev, R.Krucken, S.Lalkovski, Z.Liu, A.Maj, S.Myalski, S.Schwertel, T.Shizuma, P.M.Walker, E.Werner-Malento, O.Wieland - Int.J.Mod.Phys. E18, 1002 (2009).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Isomeric decay studies in neutron-rich N\ensuremath{\approx}126 nuclei.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2010De04,B}{2010De04}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}A.Y.Deo, Zs.Podolyak, P.M.Walker, A.Algora, B.Rubio, J.Agramunt, L.M.Fraile, N.Al-Dahan, N.Alkhomashi, J.A.Briz, E.Estevez, G.Farrelly, W.Gelletly, A.Herlert, U.Koster, A.Maira, S.Singla - Phys.Rev. C 81, 024322 (2010).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Structures of \ensuremath{^{\textnormal{201}}}Po and \ensuremath{^{\textnormal{205}}}Rn from EC/\ensuremath{\beta}\ensuremath{^{\textnormal{+}}}-decay studies.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2010He25,B}{2010He25}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}J.A.Heredia, A.N.Andreyev, S.Antalic, S.Hofmann, D.Ackermann, V.F.Comas, S.Heinz, F.P.Hessberger, B.Kindler, J.Khuyagbaatar, B.Lommel, R.Mann - Eur.Phys.J. A 46, 337 (2010).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{The new isotope \ensuremath{^{\textnormal{208}}}Th.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2011MoZP,B}{2011MoZP}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}A.I.Morales Lopez - Thesis, Universidad de Santiago de Compostela (2011).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{\ensuremath{\beta}-delayed \ensuremath{\gamma}-ray spectroscopy of heavy neutron-rich nuclei produced by cold-fragmentation of \ensuremath{^{\textnormal{208}}}Pb.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2011St21,B}{2011St21}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}S.J.Steer, Zs.Podolyak, S.Pietri, M.Gorska, H.Grawe, K.H.Maier, P.H.Regan, D.Rudolph, A.B.Garnsworthy, R.Hoischen, J.Gerl, H.J.Wollersheim, F.Becker, P.Bednarczyk, L.Caceres, P.Doornenbal, H.Geissel, J.Grebosz, A.Kelic, I.Kojouharov, N.Kurz, F.Montes, W.Prokopwicz, T.Saito, H.Schaffner, S.Tashenov, A.Heinz, M.Pfutzner, T.Kurtukian-Nieto, G.Benzoni, A.Jungclaus, D.L.Balabanski, M.Bowry, C.Brandau, A.Brown, A.M.Bruce, W.N.Catford, I.J.Cullen, Zs.Dombradi, M.E.Estevez, W.Gelletly, G.Ilie, J.Jolie, G.A.Jones, M.Kmiecik, F.G.Kondev, R.Krucken, S.Lalkovski, Z.Liu, A.Maj, S.Myalski, S.Schwertel, T.Shizuma, P.M.Walker, E.Werner-Malento, O.Wieland - Phys.Rev. C 84, 044313 (2011).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Isomeric states observed in heavy neutron-rich nuclei populated in the fragmentation of a \ensuremath{^{\textnormal{208}}}Pb beam.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2012BoZQ,B}{2012BoZQ}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}M.Bowry, Zs.Podolyak, J.Kurcewicz, S.Pietri, M.Bunce, P.H.Regan, F.Farinon, H.Geissel, C.Nociforo, A.Prochazka, H.Weick, P.Allegro, J.Benlliure, G.Benzoni, P.Boutachkov, J.Gerl, M.Gorska, A.Gottardo, N.Gregor, R.Janik, R.Knobel, I.Kojouharov, T.Kubo, Y.A.Litvinov, E.Merchan, I.Mukha, F.Naqvi, B.Pfeiffer, M.Pfutzner, W.Plass, M.Pomorski, B.Riese, M.V.Ricciardi, K.-H.Schmidt, H.Schaffner, N.Kurz, A.M.D.Bacelar, A.M.Bruce, G.F.Farrelly, N.Alkhomashi, N.Al-Dahan, C.Scheidenberger, B.Sitar, P.Spiller, J.Stadlmann, P.Strmen, B.Sun, H.Takeda, I.Tanihata, S.Terashima, J.J.Valiente Dobon, J.S.Winfield, H.-J.Wollersheim, P.J.Woods - Proc.Intern.Conf.on Nuclear Structure and Dynamics,12, Opatija, Croatia, 9-13 July, 2012, T.Niksic, M.Milin, D.Vretenar, S.Szilner, Eds., p.317 (2012); AIP Conf.Proc.1491 (2012).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Half-life measurements of isomeric states populated in projectile fragmentation.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2012Fi12,B}{2012Fi12}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}R.Fitzgerald - J.Res.Natl.Inst.Stand.Technol. 117, 80 (2012).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{NIST Ionization Chamber ``A'' Sample-Height Corrections.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2012Ku26,B}{2012Ku26}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}J.Kurcewicz, F.Farinon, H.Geissel, S.Pietri, C.Nociforo, A.Prochazka,H.Weick, J.S.Winfield, A.Estrade, P.R.P.Allegro, A.Bail, G.Belier, J.Benlliure, G.Benzoni, M.Bunce, M.Bowry, R.Caballero-Folch, I.Dillmann, A.Evdokimov, J.Gerl, A.Gottardo,\hphantom{a}E.Gregor, R.Janik, A.Kelic-Heil, R.Knobel, T.Kubo, Yu.A.Litvinov, E.Merchan, I.Mukha, F.Naqvi, M.Pfutzner, M.Pomorski, Zs.Podolyak, P.H.Regan, B.Riese, M.V.Ricciardi, C.Scheidenberger, B.Sitar, P.Spiller, J.Stadlmann, P.Strmen, B.Sun, I.Szarka, J.Taieb, S.Terashima, J.J.Valiente-Dobon, M.Winkler, Ph.Woods - Phys.Lett. B 717, 371 (2012).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Discovery and cross-section measurement of neutron-rich isotopes in the element range from neodymium to platinum with the FRS.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2013Bo18,B}{2013Bo18}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}M.Bowry, Zs.Podolyak, S.Pietri, J.Kurcewicz, M.Bunce, P.H.Regan, F.Farinon, H.Geissel, C.Nociforo, A.Prochazka, H.Weick, N.Al-Dahan, N.Alkhomashi, P.R.P.Allegro, J.Benlliure, G.Benzoni, P.Boutachkov, A.M.Bruce, A.M.D.Bacelar, G.F.Farrelly, J.Gerl, M.Gorska, A.Gottardo, J.Grebosz, N.Gregor, R.Janik, R.Knobel, I.Kojouharov, T.Kubo, N.Kurz, Yu.A.Litvinov, E.Merchan, I.Mukha, F.Naqvi, B.Pfeiffer, M.Pfutzner, W.Plass, M.Pomorski, B.Riese, M.V.Ricciardi, K.-H.Schmidt, H.Schaffner, C.Scheidenberger, E.C.Simpson, B.Sitar, P.Spiller, J.Stadlmann, P.Strmen, B.Sun, I.Tanihata, S.Terashima, J.J.Valiente Dobon, J.S.Winfield, H.-J.Wollersheim, P.J.Woods - Phys.Rev. C 88, 024611 (2013).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Population of high-spin isomeric states following fragmentation of \ensuremath{^{\textnormal{238}}}U.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2013BoZT,B}{2013BoZT}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}M.D.Bowry - Thesis, Univ. of Surrey (2013).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Population of high-spin states following projectile fragmentation of \ensuremath{^{\textnormal{238}}}U.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2013Da15,B}{2013Da15}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}S.Das Gupta, S.Bhattacharyya, H.Pai, G.Mukherjee, S.Bhattacharya, R.Palit, A.Shrivastava, A.Chatterjee, S.Chanda, V.Nanal, S.K.Pandit, S.Saha, J.Sethi, S.Thakur - Phys.Rev. C 88, 044328 (2013).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{High spin spectroscopy of \ensuremath{^{\textnormal{201}}}Tl.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2013Mo20,B}{2013Mo20}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}A.I.Morales, J.Benlliure, M.Gorska, H.Grawe, S.Verma, P.H.Regan, Zs.Podolyak, S.Pietri, R.Kumar, E.Casarejos, A.Algora, N.Alkhomashi, H.Alvarez-Pol, G.Benzoni, A.Blazhev, P.Boutachkov, A.M.Bruce, L.S.Caceres, I.J.Cullen, A.M.D.Bacelar, P.Doornenbal, M.E.Estevez Aguado, G.Farrelly, Y.Fujita, A.B.Garnsworthy, W.Gelletly, J.Gerl, J.Grebosz, R.Hoischen, I.Kojouharov, N.Kurz, S.Lalkovski, Z.Liu, C.Mihai, F.Molina, D.Mucher, W.Prokopowicz, B.Rubio, H.Schaffner, S.J.Steer, A.Tamii, S.Tashenov, J.J.Valiente-Dobon, P.M.Walker, H.J.Wollersheim, P.J.Woods - Phys.Rev. C 88, 014319 (2013).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{\ensuremath{\beta}-delayed \ensuremath{\gamma}-ray spectroscopy of \ensuremath{^{\textnormal{203,204}}}Au and \ensuremath{^{\textnormal{200$-$202}}}Pt.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2013Se03,B}{2013Se03}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}M.D.Seliverstov, T.E.Cocolios, W.Dexters, A.N.Andreyev, S.Antalic, A.E.Barzakh, B.Bastin, J.Buscher, I.G.Darby, D.V.Fedorov, V.N.Fedoseyev, K.T.Flanagan, S.Franchoo, S.Fritzsche, G.Huber, M.Huyse, M.Keupers, U.Koster, Yu.Kudryavtsev, B.A.Marsh, P.L.Molkanov, R.D.Page, A.M.Sjodin, I.Stefan, J.Van de Walle, P.Van Duppen, M.Venhart, S.G.Zemlyanoy - Phys.Lett. B 719, 362 (2013).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Charge radii of odd-A \ensuremath{^{\textnormal{191$-$211}}}Po isotopes.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2014Au03,B}{2014Au03}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}K.Auranen, J.Uusitalo, S.Juutinen, U.Jakobsson, T.Grahn, P.T.Greenlees, K.Hauschild, A.Herzan, R.Julin, J.Konki, M.Leino, J.Pakarinen, J.Partanen, P.Peura, P.Rahkila, P.Ruotsalainen, M.Sandzelius, J.Saren, C.Scholey, J.Sorri, S.Stolze - Phys.Rev. C 90, 024310 (2014).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Experimental study of 1/2\ensuremath{^{\textnormal{+}}} isomers in \ensuremath{^{\textnormal{199,201}}}At.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2014Ka23,B}{2014Ka23}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}Z.Kalaninova, S.Antalic, A.N.Andreyev, F.P.Hessberger, D.Ackermann, B.Andel, L.Bianco, S.Hofmann, M.Huyse, B.Kindler, B.Lommel, R.Mann, R.D.Page, P.J.Sapple, J.Thomson, P.Van Duppen, M.Venhart - Phys.Rev. C 89, 054312 (2014).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Decay of \ensuremath{^{\textnormal{201$-$203}}}Ra and \ensuremath{^{\textnormal{200$-$202}}}Fr.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2014Mo15,B}{2014Mo15}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}A.I.Morales, J.Benlliure, T.Kurtukian-Nieto, K.-H.Schmidt, S.Verma, P.H.Regan, Z.Podolyak, M.Gorska, S.Pietri, R.Kumar, E.Casarejos, N.Al-Dahan, A.Algora, N.Alkhomashi, H.Alvarez-Pol, G.Benzoni, A.Blazhev, P.Boutachkov, A.M.Bruce, L.S.Caceres, I.J.Cullen, A.M.D.Bacelar, P.Doornenbal, M.E.Estevez Aguado, G.Farrelly, Y.Fujita, A.B.Garnsworthy, W.Gelletly, J.Gerl, J.Grebosz, R.Hoischen, I.Kojouharov, N.Kurz, S.Lalkovski, Z.Liu, C.Mihai, F.Molina, D.Mucher, B.Rubio, H.Shaffner, S.J.Steer, A.Tamii, S.Tashenov, J.J.Valiente-Dobon, P.M.Walker, H.J.Wollersheim, P.J.Woods - Phys.Rev.Lett. 113, 022702 (2014).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Half-Life Systematics across the N=126 Shell Closure: Role of First-Forbidden Transitions in the \ensuremath{\beta} Decay of Heavy Neutron-Rich Nuclei.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2014Pr09,B}{2014Pr09}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}B.Pritychenko, E.Betak, B.Singh, J.Totans - Nucl.Data Sheets 120, 291 (2014).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Nuclear Science References Database.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2014Se07,B}{2014Se07}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}M.D.Seliverstov, T.E.Cocolios, W.Dexters, A.N.Andreyev, S.Antalic, A.E.Barzakh, B.Bastin, J.Buscher, I.G.Darby, D.V.Fedorov, V.N.Fedosseev, K.T.Flanagan, S.Franchoo, G.Huber, M.Huyse, M.Keupers, U.Koster, Yu.Kudryavtsev, B.A.Marsh, P.L.Molkanov, R.D.Page, A.M.Sjodin, I.Stefan, P.Van Duppen, M.Venhart, S.G.Zemlyanoy - Phys.Rev. C 89, 034323 (2014).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Electromagnetic moments of odd-A \ensuremath{^{\textnormal{193$-$203,211}}}Po isotopes.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2014StZZ,B}{2014StZZ}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}N.J.Stone - REPT INDC(NDS){\textminus}0658 (2014).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Table of nuclear magnetic dipole and electric quadrupole moments.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2014Un01,B}{2014Un01}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}M.P.Unterweger, R.Fitzgerald - Appl.Radiat.Isot. 87, 92 (2014); Erratum Appl.Radiat.Isot. 159, 108976 (2020).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Update of NIST half-life results corrected for ionization chamber source-holder instability.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2014Zh03,B}{2014Zh03}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}Z.Y.Zhang, Z.G.Gan, L.Ma, L.Yu, H.B.Yang, T.H.Huang, G.S.Li, Y.L.Tian, Y.S.Wang, X.X.Xu, X.L.Wu, M.H.Huang, C.Luo, Z.Z.Ren, S.G.Zhou, X.H.Zhou, H.S.Xu, G.Q.Xiao - Phys.Rev. C 89, 014308 (2014).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{\ensuremath{\alpha} decay of the new neutron-deficient isotope \ensuremath{^{\textnormal{205}}}Ac.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2015Au01,B}{2015Au01}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}K.Auranen, J.Uusitalo, S.Juutinen, U.Jakobsson, T.Grahn, P.T.Greenlees, K.Hauschild, A.Herzan, R.Julin, J.Konki, M.Leino, J.Pakarinen, J.Partanen, P.Peura, P.Rahkila, P.Ruotsalainen, M.Sandzelius, J.Saren, C.Scholey, J.Sorri, S.Stolze - Phys.Rev. C 91, 024324 (2015); Erratum Phys.Rev. C 92, 039901 (2015).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Spectroscopy of \ensuremath{^{\textnormal{201}}}At including the observation of a shears band and the 29/2\ensuremath{^{\textnormal{+}}} isomeric state.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2016Ba42,B}{2016Ba42}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}A.E.Barzakh, D.V.Fedorov, V.S.Ivanov, P.L.Molkanov, F.V.Moroz, S.Yu.Orlov, V.N.Panteleev, M.D.Seliverstov, Yu.M.Volkov - Phys.Rev. C 94, 024334 (2016).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Laser spectroscopy studies of intruder states in \ensuremath{^{\textnormal{193,195,197}}}Bi.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2016Ma12,B}{2016Ma12}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}T.Marketin, L.Huther, G.Martinez-Pinedo - Phys.Rev. C 93, 025805 (2016).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Large-scale evaluation of \ensuremath{\beta}-decay rates of r-process nuclei with theinclusion of first-forbidden transitions.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2016St14,B}{2016St14}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}N.J.Stone - At.Data Nucl.Data Tables 111-112, 1 (2016).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Table of nuclear electric quadrupole moments.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2017Al34,B}{2017Al34}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}N.A.Althubiti, D.Atanasov, K.Blaum, T.E.Cocolios, T.Day Goodacre, G.J.Farooq-Smith, D.V.Fedorov, V.N.Fedosseev, S.George, F.Herfurth, K.Heyde, S.Kreim, D.Lunney, K.M.Lynch, V.Manea, B.A.Marsh, D.Neidherr, M.Rosenbusch, R.E.Rossel, S.Rothe, L.Schweikhard, M.D.Seliverstov, A.Welker, F.Wienholtz, R.N.Wolf, K.Zuber, for the ISOLTRAP Collaboration - Phys.Rev. C 96, 044325 (2017).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Spectroscopy of the long-lived excited state in the neutron-deficientnuclides \ensuremath{^{\textnormal{195,197,199}}}Po by precision mass measurements.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2018Cu02,B}{2018Cu02}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}J.G.Cubiss, A.E.Barzakh, M.D.Seliverstov, A.N.Andreyev, B.Andel, S.Antalic, P.Ascher, D.Atanasov, D.Beck, J.Bieron, K.Blaum, Ch.Borgmann, M.Breitenfeldt, L.Capponi, T.E.Cocolios, T.Day Goodacre, X.Derkx, H.De Witte, J.Elseviers, D.V.Fedorov, V.N.Fedosseev, S.Fritzsche, L.P.Gaffney, S.George, L.Ghys, F.P.Hessberger, M.Huyse, N.Imai, Z.Kalaninova, D.Kisler, U.Koster, M.Kowalska, S.Kreim, J.F.W.Lane, V.Liberati, D.Lunney, K.M.Lynch, V.Manea, B.A.Marsh, S.Mitsuoka, P.L.Molkanov, Y.Nagame, D.Neidherr, K.Nishio, S.Ota, D.Pauwels, L.Popescu, D.Radulov, E.Rapisarda, J.P.Revill, M.Rosenbusch, R.E.Rossel, S.Rothe, K.Sandhu, L.Schweikhard, S.Sels, V.L.Truesdale, C.Van Beveren, P.Van den Bergh, Y.Wakabayashi, P.Van Duppen, K.D.A.Wendt, F.Wienholtz, B.W.Whitmore, G.L.Wilson, R.N.Wolf, K.Zuber - Phys.Rev. C 97, 054327 (2018).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Charge radii and electromagnetic moments of \ensuremath{^{\textnormal{195$-$211}}}At.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2018Yo02,B}{2018Yo02}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}A.Yoshimi, H.Hara, T.Hiraki, Y.Kasamatsu, S.Kitao, Y.Kobayashi, K.Konashi, R.Masuda, T.Masuda, Y.Miyamoto, K.Okai, S.Okubo, R.Ozaki, N.Sasao, O.Sato, M.Seto, T.Schumm, Y.Shigekawa, S.Stellmer, K.Suzuki, S.Uetake, M.Watanabe, A.Yamaguchi, Y.Yasuda, Y.Yoda, K.Yoshimura, M.Yoshimura - Phys.Rev. C 97, 024607 (2018).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Nuclear resonant scattering experiment with fast time response: Photonuclear excitation of \ensuremath{^{\textnormal{201}}}Hg.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2019Mo01,B}{2019Mo01}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}P.Moller, M.R.Mumpower, T.Kawano, W.D.Myers - At.Data Nucl.Data Tables 125, 1 (2019).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Nuclear properties for astrophysical and radioactive-ion-beam applications (II).}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2019Ro12,B}{2019Ro12}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}P.Roy, S.K.Tandel, S.Suman, P.Chowdhury, R.V.F.Janssens, M.P.Carpenter, T.L.Khoo, F.G.Kondev, T.Lauritsen, C.J.Lister, D.Seweryniak, S.Zhu - Phys.Rev. C 100, 024320 (2019).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Isomers from intrinsic excitations in \ensuremath{^{\textnormal{200}}}Tl and \ensuremath{^{\textnormal{201,202}}}Pb.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2019StZV,B}{2019StZV}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}N.J.Stone - INDC(NDS){\textminus}0794 (2019).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Table of Recommended Nuclear Magnetic Dipole Moments: Part I {\textminus} Long-lived States.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2020Au01,B}{2020Au01}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}K.Auranen, U.Jakobsson, H.Badran, T.Grahn, P.T.Greenlees, A.Herzan, R.Julin, S.Juutinen, J.Konki, M.Leino, A.-P.Leppanen, G.O$'$Neill, J.Pakarinen, P.Papadakis, J.Partanen, P.Rahkila, P.Ruotsalainen, M.Sandzelius, J.Saren, C.Scholey, L.Sinclair, J.Sorri, S.Stolze, J.Uusitalo, A.Voss - Phys.Rev. C 101, 024306 (2020).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Isomeric 13/2\ensuremath{^{\textnormal{+}}} state in \ensuremath{^{\textnormal{201}}}Fr.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2020Ko17,B}{2020Ko17}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}F.G.Kondev - Nucl.Data Sheets 166, 1 (2020).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Nuclear Data Sheets for A = 205.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2020Ne08,B}{2020Ne08}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}E.M.Ney, J.Engel, T.Li, N.Schunck - Phys.Rev. C 102, 034326 (2020).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Global description of \ensuremath{\beta}\ensuremath{^{-}} decay with the axially deformed Skyrme finite-amplitude method: Extension to odd-mass and odd-odd nuclei.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2020Si16,B}{2020Si16}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}S.Singh, S.Kumar, B.Singh, A.K.Jain - Nucl.Data Sheets 167, 1 (2020).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{Nuclear radius parameters (r\ensuremath{_{\textnormal{0}}}) for even-even nuclei from alpha decay.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2021Hu06,B}{2021Hu06}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}W.J.Huang, M.Wang, F.G.Kondev, G.Audi, S.Naimi - Chin.Phys.C 45, 030002 (2021).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{The AME 2020 atomic mass evaluation (I). Evaluation of input data, and adjustment procedures.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2021Ko07,B}{2021Ko07}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}F.G.Kondev, M.Wang, W.J.Huang, S.Naimi, G.Audi - Chin.Phys.C 45, 030001 (2021).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{The NUBASE2020 evaluation of nuclear physics properties.}}\\
\href{https://www.nndc.bnl.gov/nsr/nsrlink.jsp?2021Wa16,B}{2021Wa16}&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm}M.Wang, W.J.Huang, F.G.Kondev, G.Audi, S.Naimi - Chin.Phys.C 45, 030003 (2021).}\\
&\parbox[t]{6in}{\addtolength{\parindent}{-0.25cm} \textit{The AME 2020 atomic mass evaluation (II). Tables, graphs and references.}}\\
\end{longtable}
\end{document}
|
1,314,259,993,625 | arxiv | \section{Introduction}
\label{Sec:Introduction}
Gianfausto Dell'Antonio has been always transmitting to younger collaborators the attitude
to understand -- and explain -- a mathematical phenomenon in the simplest possible model which
still captures its essential features. Remembering his recommendation, we devote this contribution
to explain a recent, model-independent result -- namely the \emph{Localization Dicothomy} for
gapped periodic quantum systems, proved in \cite{MonacoPanatiPisanteTeufel2018} --
by illustrating its essential features in a simple, but yet physically relevant, discrete model.
We consider the model proposed by Haldane in \cite{Haldane88}, which has become one of the paradigmatic models to describe Chern insulators, a subclass of topological insulators \cite{Ando, HasanKane, FruchartCarpentier2013}.
Haldane argued that the essential ingredient in the Quantum Hall Effect (QHE) is the breaking of time-reversal symmetry, an effect that can be obtained either by an external magnetic field (as in a QHE setup) or, alternatively, by some mechanism internal to the sample,
as {\sl e.\,g.\ } the presence of strong magnetic dipole moments of the ionic cores.
In Haldane's words \cite{Haldane88}:
\begin{quote}
``{\it
While the particular model presented here is unlikely to be directly physically realizable, it indicates that,
at least in principle, {the QHE can be placed in the wider context of phenomena associated with
broken time-reversal invariance, and does not necessarily require external magnetic fields}, but could occur as a consequence
of magnetic ordering in a quasi-two-dimensional system.}''
\end{quote}
\noindent Remarkably, the first sentence turned out to be too pessimistic: after three decades, Chern insulators predicted in \cite{Haldane88} have been experimentally synthesized as crystalline solids \cite{Experiment,Bestwick et al 2015, Chang et al 2015} and the Haldane model can also be physically simulated by Bose-Einstein condensates in suitably arranged optical lattices.
In this paper, we first provide a pedagogical introduction to the Haldane model, which is here presented in the first-quantization formalism, as opposed to most of the physics literature, which uses instead a second-quantization language.
In Section~\ref{Sec:Singularities}, we recall the definition of {\it Bloch functions} and of the {\it Chern number} associated to an isolated Bloch band, and
we exhibit, in the Haldane model, a Bloch function producing a non-zero Chern number and having a singular derivative: more precisely, its $H^1$-norm diverges. This quantitative relation between non-trivial topology and the allowed singularities of Bloch functions in the Haldane model was investigated numerically in \cite{ThonhauserVanderbilt}. This situation exemplifies a recent model-independent mathematical result \cite{MonacoPanatiPisanteTeufel2018}, which shows that a non-zero Chern number indeed forces a divergence of the $H^1$-norm of the corresponding Bloch functions in any Bloch gauge.
We explain in Section~\ref{Sec:Dichotomy} how the latter divergence reflects into the delocalization of the corresponding {\it Wannier functions}, and we illustrate to the reader the more general {\it Localization Dichotomy} mentioned above.
A natural question is whether the previous result -- whose formulation heavily relies on periodicity -- can be recast in the broader context of non-periodic models. Some preliminary results in this direction, still unpublished \cite{MarcelliMoscolariPanati}, are announced in Section~\ref{Sec:GeneralizedDichotomy}.
We hope that the introductory style of this contribution will be useful to fill the linguistic gap between mathematics and physics, as they represent a unity in the scientific vision of the person to whom the paper is dedicated.
\bigskip
\noindent \textbf{Dedication.} The senior author of this paper moved his first steps into the scientific world
under the precious guidance of Gianfausto Dell'Antonio. From his example,
as a scientist and as a human being, he learned not only how to do mathematics, but how to be a Mathematical Physicist.
We all -- authors of different generations -- consider Gianfausto as our {\it Maestro}, and we gratefully acknowledge the unvaluable contribution he gave to the development of Quantum Mathematics in Italy over more than half a century.
\bigskip
\noindent \textbf{Acknowledgements.}
We are grateful to Cl\'ement Tauber for many useful discussions on the related Kane-Mele model,
and for his precious help with some Figures.
\section{The Haldane model and its symmetries}
\label{Sec:Haldane}
The tight-binding model proposed by Haldane \cite{Haldane88} has become a paradigm in solid-state physics, as it is
presumably the simplest physically-reasonable model which is invariant by lattice-translations (a unitary ${\mathbb{Z}^2}$-symmetry) and simultaneously breaks, for some values of the parameters $(\phi, M)$ labeling the model, time-reversal symmetry (an antiunitary $\mathbb{Z}_2$-symmetry). In view of that, it has become one of the most popular models to study materials in the Altland-Zirnbauer symmetry class A, which includes Quantum Hall systems and Chern insulators \cite{Ando, HasanKane, FruchartCarpentier2013}.
The Haldane model is usually presented by using a second-quantization formalism \cite{FruchartCarpentier2013, Santoro_LN,GiulianiMastropietroPorta2017}, which makes it difficult to readers unfamiliar with the latter to appreciate the simplicity and elegance
of the essential ideas. Since second quantization is not needed at all to describe non-interacting electrons,
we review in this Section the essential features of the Haldane model, in a pedagogical style, by using the usual language of discrete Schr\"odinger operators ({\sl i.\,e.\ } a first-quantization formalism).
\medskip
\subsection{The honeycomb structure}
The Haldane model describes independent electrons on a honeycomb structur
\footnote{The physics literature usually refers to the latter as a ``honeycomb lattice''.
We prefer to avoid here this ambiguous use of the word ``lattice'', since this word has a
precise meaning in mathematics: a lattice is a discrete subgroup of $(\mathbb{R}^d, +)$ with maximal rank.
The ambiguity does not arise when speaking about the \emph{Bravais lattice}, which is a lattice for both
physicists and mathematicians.}\
$\mathcal{C}\subset \mathbb{R}^2$, illustrated in Figure \ref{fig:honeycomb}. The structure is characterized by the \emph{displacement vectors}
$$
\V d_1 = d\begin{pmatrix} \frac{1}{2} & -\frac{\sqrt{3}}{2} \end{pmatrix}, \qquad \V d_2 = d\begin{pmatrix} \frac{1}{2} &\frac{\sqrt{3}}{2} \end{pmatrix}, \qquad \V d_3 = d\begin{pmatrix} -1 & 0 \end{pmatrix} = -\V d_1-\V d_2,
$$
where $d$ is the {smallest} distance between two points of $\mathcal{C}$.
The periodicity of the structure is expressed by the \emph{periodicity vectors}
\begin{equation}
\V a_1 = \V d_2 - \V d_3, \qquad \V a_2 = \V d_3-\V d_1, \qquad \V a_3 = \V d_1-\V d_2 = -\V a_1 - \V a_2.
\end{equation}
\begin{figure}[htb]
\centering
\includegraphics{Honeycomb.pdf}
\caption{\footnotesize The honeycomb structure, with the displacement vectors $\set{\ve d_1, \ve d_2,\ve d_3}$ and the periodicity vectors $\set{\ve a_1, \ve a_2,\ve a_3}$ (color online).
\label{fig:honeycomb}}
\end{figure}
\begin{figure}[htb]
\centering
\includegraphics[width=\textwidth]{UnitCells1.pdf}
\caption{\footnotesize Three possible dimerizations of the honeycomb structure, corresponding to three different periodicity cells
(color online). \label{fig:unitcells}}
\end{figure}
The vectors $\V a_i$ generate a Bravais lattice $\Gamma := \mathrm{Span}_\mathbb{Z}\{ \V a_1, \V a_2, \V a_3\} \cong \mathbb Z^2$ where one $\V a_i$ is redundant as it is an integer linear combination of the other two.
Any point $\mathbf{x} \in \cry$ can be written by using a Bravais lattice vector and one of the $\V d_i$ vectors. It is then sufficient to pick two $\V a_i$-vectors and one $\V d_i$-vector to generate the whole crystal. This choice, which is often called a \emph{dimerization} of $\mathcal C$, is not unique, as illustrated in Figure \ref{fig:unitcells}. The above procedure is equivalent to the choice of a periodicity cell that contains two non-equivalent sites $A$ and $B$ (black and white dots in Figure \ref{fig:honeycomb}, respectively), and is a fundamental cell {w.r.t.\ the action of $\Gamma$}. Hence, each choice of a periodicity cell provides an identification
$\cry \iso \Gamma \times \set{\ve 0, \ve \nu}$, where $\nu$ is one of the displacement vectors,
yielding an isomorphis
\footnote{From an abstract viewpoint, we are just using the fact that the $L^2$-functor, from measure spaces to Hilbert spaces, preserves the product structure, mapping the cartesian product into the tensor product.
}
$\ell^2(\mathcal C) \cong \ell^2(\Gamma) \otimes \mathbb{C}^2 \iso \ell^2(\Gamma, \mathbb{C}^2)$. We will often use this ``dimerization
isomorphism'' and the following typographic convention:
\begin{itemize}
\item a small letter for a function $\psi \in \ell^2(\cry)$, with complex values $\psi_\mathbf{x}$ for $\mathbf{x} \in \cry$;
\item capital letter for a function $\Psi \in \ell^2(\Gamma, \mathbb{C}^2)$; we make use of a pseudo-spin notation, namely
$$
\Psi_{\gamma} =
\left(
\begin{array}{c}
\psi_{\gamma, A} \\
\psi_{\gamma, B}
\end{array}
\right)
\qquad \text{ for } \gamma \in \Gamma,
$$
where the labels $A$ and $B$ refer respectively to {the sublattices $\Gamma_A$ and $\Gamma_B$},
so that $\cry = \Gamma_A \cup \Gamma_B$.
For example, in the three dimerizations appearing in Figure~2, one has {$\Gamma_A = \Gamma$ and $\Gamma_B =
\Gamma + \ve \nu$}, where $\ve \nu \in \set{\ve d_1, \ve d_2,\ve d_3}$ depends on the chosen dimerization.
\end{itemize}
{Finally, notice that the honeycomb structure has an interesting {\it \textbf{inversion symmetry}}, namely a reflection w.r.t.\ a specific line, which exchanges the role of the sublattices $\Gamma_A$ and $\Gamma_B$. Thus, it yields a $\mathbb{Z}_2$-symmetry which can be easily broken by adding an on-site $\Gamma$-periodic potential which distinguishes between $\Gamma_A$ and $\Gamma_B$.} The latter procedure corresponds to a variation of the parameter $M$ in the Haldane Hamiltonian, to be introduced shortly, and to the transition from graphene to boron-nitride sheets in physical reality.
\goodbreak
\subsection{The Hamiltonian}
The Haldane model is defined, in a first quantization formalism, through a Hamiltonian operator acting on
$\ell^2(\mathcal C) \iso \ell^2(\Gamma, \mathbb{C}^2)$, and depending on two real parameters $(\phi, M)$, with
${\phi \in (-\pi, \pi]}$ representing a magnetic flux and $M \in \mathbb{R}$ corresponding to an on-site energy which distinguishes among the two sublattices {$\Gamma_A$ and $\Gamma_B$}.
\noindent The translation operator $T_{\V u}$, corresponding to a translation by $\ve u \in \mathbb{R}^2$, is defined by
\begin{equation}\label{Translations}
(T_{\V u} \psi)_{\V x} = \begin{cases}
\psi_{\V x -\V u} & \text{ if } {\V x-\V u} \in \mathcal{C}\\
0 & \text{ otherwise}
\end{cases} \qquad \text{for all } \psi \in \ell^2(\mathcal{C}).
\end{equation}
Moreover, we denote by $\chi_A$ (resp.\ $\chi_B$) the charachteristic function of the sublattice {$\Gamma_A$ (resp.\ $\Gamma_B$)}.
Equipped with this notation, one defines the Haldane operator $H \equiv H_{(\phi, M)}$ acting in
$\ell^2(\cry)$
({\sl i.\,e.\ } without reference to a specific dimerization) as a sum of three terms
\begin{equation} \label{Haldane}
{H = H\sub{NN} + H\sub{NNN} + V .}
\end{equation}
{The nearest neighbor (NN) term is defined -- by using the displacement vectors -- by}
\begin{equation} \label{H_NN}
H\sub{NN} = t_1 \sum_{j = 1}^3 (T_{\ve{d}_j} + T_{- \ve{d}_j})\quad \text{with $t_1\in \mathbb{R}$}.
\end{equation}
{The next nearest neighbor (NNN) term uses instead the periodicity vectors and reads}
\begin{equation} \label{H_NNN}
H\sub{NNN} =
t_2 (\cos \phi) \sum_{j=1}^3 (T_{\ve{a}_j} + T_{- \ve{a}_j}) \,+\,
t_2 (\mathrm{i} \sin \phi) (\chi_A - \chi_B) \sum_{j=1}^3 (T_{\ve{a}_j} - T_{- \ve{a}_j})
\end{equation}
{with $t_2\in \mathbb{R}$}. The last term is a potential that distinguishes sites in sublattices {$\Gamma_A$ and $\Gamma_B$}, namely
\begin{equation} \label{H_site}
V_\mathbf{x} = M (\chi_{A} - \chi_{B})_\mathbf{x} =
\begin{cases}
+ M & \text{ if } \mathbf{x} \in {\Gamma_A} \\
- M & \text{ if } \mathbf{x} \in {\Gamma_B}.
\end{cases}
\end{equation}
\begin{remark}[{Comparison with the honeycomb Hofstadter model}]
\label{Rem:simpler}
By analogy with the Hofstadter model \cite{Hofstadter76}, one might be tempted to replace
the NNN term by the more symmetric expression
\begin{align} \label{tilde_H_NNN}
\widetilde H\sub{NNN}
&= {t_2} \,\, \sum_{j=1}^3 (\mathrm{e}^{\mathrm{i} \phi} T_{\ve{a}_j} + \mathrm{e}^{-\mathrm{i} \phi} T_{- \ve{a}_j}) \\
&= t_2 (\cos \phi) \sum_{j=1}^3 (T_{\ve{a}_j} + T_{- \ve{a}_j})
+ t_2 (\mathrm{i} \sin \phi) \sum_{j=1}^3 (T_{\ve{a}_j} - T_{- \ve{a}_j}).
\nonumber
\end{align}
Notice, however, that the latter operator does not distinguish between the sublattices {$\Gamma_A$ and $\Gamma_B$}, yielding
an operator which acts diagonally on the $\mathbb{C}^2$-factor in $\ell^2(\Gamma) \otimes \mathbb{C}^2$.
The operator \eqref{H_NNN} acts instead in a non-diagonal way, and offers the opportunity to model subtler
physical effects. \hfill $\diamond$
\end{remark}
\noindent One can easily check that the Haldane model enjoys some relevant symmetries:
\renewcommand{\labelenumi}{{\rm(\roman{enumi})}}
\begin{enumerate}
\item {$\Gamma$-periodicity}: indeed, one checks that $[T_\gamma, H] =0$ for every $\gamma \in \Gamma$;
\item {$\frac{2\pi}{3}$-rotation symmetry}: indeed $[U_R, H] =0$ where $U_R$ is defined as usual by
$(U_R\psi)_{\mathbf{x}} = \psi_{R^{-1}\mathbf{x}}$, with $R \in \mathrm{SO}(2)$ a rotation by a $\frac{2\pi}{3}$ angle in the plane;
\item {broken time-reversal symmetry (TRS)}: for $\phi \in {\{0,\pi\}}$ the Hamiltonian commutes with the time-reversal operator,
given by complex conjugation in $\ell^2(\cry)$. As far as $\sin \phi \neq 0$, TRS is broken, as it clearly appears from
\eqref{H_NNN}.
\end{enumerate}
\subsection{The Fourier decomposition}
\label{Sec:Fourier}
The $\Gamma$-periodicity of the model allows to use Fourier transform or, more intrinsically, the Bloch-Floquet decomposition.
Since the Fourier transform unitarily maps $\ell^2(\mathbb{Z}^d)$ into $L^2(\mathbb{T}^d)$, after a choice of dimerization one
obtains an isomorphism $\ell^2(\cry) \iso \ell^2(\Gamma, \mathbb{C}^2) \iso L^2(\mathbb{T}^2_*, \mathbb{C}^2)$ where the torus $\mathbb{T}^2_* = \mathbb{R}^2/\Gamma^*$, called \emph{Brillouin torus} by physicists, is defined as a quotient by the reciprocal or dual lattice
\begin{equation} \label{Def:Reciprocal}
\Gamma^* = \set{ \ve k \in {\mathbb{R}^2} : \ve k \cdot \ve \gamma \in 2\pi\mathbb{Z} \text{ for all } \gamma \in \Gamma}.
\end{equation}
We choose any dimerization such that the sublattices are identified with $\Gamma$ and
$\Gamma + \ve \nu$, respectively, for a suitable $\ve \nu \in \set{\ve d_1, \ve d_2, \ve d_3}$ (compare Figure \ref{fig:unitcells}).
With this convention, an isomorphism is exhibited by
\begin{equation} \label{Def:Fourier}
(\mathcal{F}_\nu \psi)(\ve{k}) = \sum_{\ve \gamma \in \Gamma} \mathrm{e}^{- \mathrm{i} \ve{k} \cdot \ve\gamma} \Psi_{- \ve \gamma}
= \sum_{\ve \gamma \in \Gamma} \mathrm{e}^{- \mathrm{i} \ve{k} \cdot \ve\gamma}
\left(
\begin{array}{c}
\psi_{- \ve \gamma} \\
\psi_{- \ve \gamma + \ve \nu}
\end{array}
\right).
\end{equation}
The operator $\mathcal{F}_\nu$ establishes a unitary transformation
\begin{equation}
\label{eqn:defn Fnu}
\mathcal{F}_\nu\colon\ell^2(\cry)\to\mathcal{H}:
L^2(\mathbb{T}^2_*,\mathbb{C}^2),
\end{equation}
where $\mathcal{H}$ is equipped with the inner product (notice the normalization)
$$
\inner{\varphi_1}{\varphi_2}_{\mathcal{H}}:=\frac{1}{|\mathbb{T}^2_*|}\int_{\mathbb{T}^2_*}\mathrm{d} \ve{k}\,
\inner{\varphi_1(\ve{k})}{\varphi_2(\ve{k})}_{\mathbb{C}^2}.
$$
Every operator $A$ acting in $\ell^2(\cry)$ which is $\Gamma$-periodic, in the sense that
\begin{equation} \label{Periodicity}
[A, T_{\gamma}] =0 \text{ for every } \gamma \in \Gamma,
\end{equation}
is conjugated to an operator
$\mathcal{F}_{\nu} \, A \, \mathcal{F}_{\nu}^{-1} =: A_{\nu}$ acting in $\mathcal{H}$. Notice that $A_{\nu}$ is decomposabl
\footnote{{For the sake of brevity we omit the dependence of the operator $A_{\nu}$ on the dimerization procedure, {\sl i.\,e.\ } we remove the subscript $\nu$.}
}
in the sense that
$$
(A_\nu \varphi) (\ve k) = A(\ve k) \varphi(\ve k) \qquad \text{ for all } \ve k \in \mathbb{T}^2_* \, ,
$$
where $\mathbb{T}^2_*\ni \ve k \mapsto A(\ve k)\in \mathcal{B}(\mathbb{C}^2)$ due to \eqref{eqn:defn Fnu}.
Moreover, the $\Gamma$-periodicity of $A$ reflects in the following property:
\begin{equation} \label{A periodic}
A(\ve{k} + \la) = A(\ve{k}) \qquad \text{ for all } \la \in \Gamma^*, \ve{k} \in \mathbb{T}^2_{*}.
\end{equation}
\noindent The latter is understood as an equality of matrices. The matrix
$A(\ve k)$ is called the \emph{fiber of the operator $A$} at the point $\ve k$, and we use
the notation $A \longleftrightarrow A(\ve k)$ to indicate the correspondence between the $\Gamma$-periodic
operator $A$ and the operator $\mathcal{F}_{\nu} \, A \, \mathcal{F}_{\nu}^{-1}$, acting in $L^2(\mathbb{T}^2_*, \mathbb{C}^2)$,
given {fiberwise} by the multiplication operator times the (matrix-valued) function $A(\ve{k})$.
Notice that everything above depends -- in general -- on the choice of a dimerization, as the subscript in $\mathcal{F}_{\nu}$ suggests.
\bigskip
The Haldane Hamiltonian \eqref{Haldane} is $\Gamma$-periodic, and its fibers $H(\ve{k})$ over the Brillouin torus
can be conveniently decomposed on the Pauli basis $\set{\sigma_0 = \id, \sigma_1, \sigma_2, \sigma_3}$
as
$$
H(\ve{k}) = \sum_{j=0}^3 R_j(\ve{k}) \, \sigma_j.
$$
It is easy to show that
\begin{eqnarray} \label{R0}
R_0(\ve{k}) &=& 2 t_2 (\cos {\phi}) \sum_{j=1}^3 \cos(\ve{k} \cdot \ve{a}_j), \\
R_3(\ve{k}) &=& M - 2 t_2 (\sin {\phi}) \sum_{j=1}^3 \sin(\ve{k} \cdot \ve{a}_j).
\label{R3}
\end{eqnarray}
Indeed, one exploits the fact that the Fourier transform intertwines the translation operator
$T_{\gamma}$, for $\gamma \in \Gamma$, with the multiplication times $\mathrm{e}^{\mathrm{i} \ve{k} \cdot \gamma} \, \id$.
Since $T_{\ve{a}_j} \longleftrightarrow \mathrm{e}^{\mathrm{i} \ve{k} \cdot \ve{a}_j} \id$, one concludes that
$$
T_{\ve{a}_j} + T_{-\ve{a}_j} \longleftrightarrow 2 \cos(\ve{k} \cdot \ve{a}_j) \, \id
$$
which immediately gives \eqref{R0}. Analogously, since
$
(\chi_A - \chi_B) T_{\ve{a}_j} \longleftrightarrow \mathrm{e}^{\mathrm{i} \ve{k} \cdot \ve{a}_j } \, \sigma_3
$,
one concludes that
$$
(\chi_A - \chi_B) \left(T_{\ve{a}_j} - T_{- \ve{a}_j} \right) \longleftrightarrow 2\mathrm{i} \sin(\ve k \cdot \ve a_j) \, \sigma_3
$$
which gives \eqref{R3}. Notice that the previous terms do not depend on a specific choice of the dimerization, provided one
of the sublattices agrees with $\Gamma$.
As for the off-diagonal terms, one has however to be more careful, since the computation \crucial{does depend on the choice of ``the'' periodicity cell}, as pointed out for example in \cite{BenaMontambaux,Fruchart2014}.
We make here the specific choice
\begin{equation} \label{cellY}
Y = \set{\ve{x} \in \mathbb{R}^2: \ve{x} = \alpha_1 \ve a_1 + \alpha_2 \ve a_2 \text{ with } \alpha_j \in [-\half,+\half] }
\end{equation}
so that $Y \cap \, \cry = \set{\ve 0, \ve d_3}$, as illustrated in the first panel in Figure 2.
One has that $\cry \iso \Gamma \times \set{0, \ve{d}_3}$ as a measure space, and the dimerization isomorphism is exhibited by
\begin{equation} \label{Specific_dimer}
\Psi_{\gamma} =
\left(
\begin{array}{c}
\psi_{\gamma, A} \\
\psi_{\gamma, B}
\end{array}
\right)
=
\left(
\begin{array}{c}
\psi_{\gamma + 0} \\
\psi_{\gamma + \ve{d}_3}
\end{array}
\right) \, .
\end{equation}
\noindent With this specific choice, the remaining terms are
\begin{eqnarray} \label{R1}
R_1(\ve{k}) &=& t_1 \( 1 + \cos(\ve{k} \cdot \ve{a}_1) + \cos(\ve{k} \cdot \ve{a}_2) \), \\
R_2(\ve{k}) &=& t_1 \( \sin(\ve{k} \cdot \ve{a}_1) - \sin(\ve{k} \cdot \ve{a}_2) \).
\label{R2}
\end{eqnarray}
\noindent These expressions are easily derived.
By using \eqref{Translations}, one computes
\begin{eqnarray*}
\( T_{ +\ve{d}_3} \psi \)_{\gamma} =
\left(
\begin{array}{c}
\psi_{\gamma - \ve{d}_3} \\
\psi_{(\gamma + \ve{d}_3) - \ve{d}_3}
\end{array}
\right)
=
\left(
\begin{array}{c}
0 \\
\psi_{\gamma, A}
\end{array}
\right) , \\
\( T_{- \ve{d}_3} \Psi \)_{\gamma} =
\left(
\begin{array}{c}
\psi_{\gamma + \ve{d}_3} \\
\psi_{(\gamma + \ve{d}_3) + \ve{d}_3}
\end{array}
\right)
=
\left(
\begin{array}{c}
\psi_{\gamma, B} \\
0
\end{array}
\right). \\
\end{eqnarray*}
Thus $T_{\ve{d}_3} + T_{- \ve{d}_3} = \Id \otimes \sigma_1$, so that the Fourier transform $\mathcal{F}_{\ve d_3}$ yields
\begin{equation} \label{}
T_{\ve{d}_3} + T_{- \ve{d}_3} \longleftrightarrow \Id \otimes \sigma_1 .
\end{equation}
The coordinate $j=3$ is privileged in view of our choice of the periodicity cell.
As for the next term, one uses that $\ve{a}_1 = \ve{d}_2 - \ve{d}_3$ so that
\begin{eqnarray*}
\( T_{ +\ve{d}_2} \psi \)_{\gamma} =
\left(
\begin{array}{c}
\psi_{\gamma - \ve{d}_3 - \ve a_1} \\
\psi_{(\gamma + \ve{d}_3 ) - \ve{d}_3 - \ve{a}_1}
\end{array}
\right)
=
\left(
\begin{array}{c}
0 \\
\psi_{\gamma - \ve{a}_1, A}
\end{array}
\right) , \\
\( T_{- \ve{d}_2} \psi \)_{\gamma} =
\left(
\begin{array}{c}
\psi_{\gamma + \ve{d}_3 + \ve{a}_1} \\
\psi_{(\gamma + \ve{d}_3) + \ve{d}_3 + \ve a_1}
\end{array}
\right)
=
\left(
\begin{array}{c}
\psi_{\gamma + \ve{a}_1, B} \\
0
\end{array}
\right). \\
\end{eqnarray*}
After Fourier transform one obtains
\begin{eqnarray*}
\(\mathcal{F}_{\ve d_3} (T_{\ve{d}_2} + T_{- \ve{d}_2})\psi \) (\ve{k}) &=&
\left(
\begin{array}{cc}
0 & \mathrm{e}^{{-} \mathrm{i} \ve{k} \cdot \ve{a}_1} \\
\mathrm{e}^{{+}\mathrm{i} \ve{k} \cdot \ve{a}_1} & 0
\end{array}
\right)
\(\mathcal{F}_{\ve d_3} \psi \) (\ve{k}).
\end{eqnarray*}
\noindent Analogously, in view of $\ve{a}_2 = \ve{d}_3 - \ve{d}_1$ one has
\begin{eqnarray*}
\( T_{ +\ve{d}_1} \psi \)_{\gamma} =
\left(
\begin{array}{c}
\psi_{\gamma - \ve{d}_3 + \ve a_2} \\
\psi_{(\gamma + \ve{d}_3) - \ve{d}_3 + \ve a_2}
\end{array}
\right)
=
\left(
\begin{array}{c}
0 \\
\psi_{\gamma + \ve{a}_2, A}
\end{array}
\right) , \\
\( T_{- \ve{d}_1} \psi \)_{\gamma} =
\left(
\begin{array}{c}
\psi_{\gamma + \ve{d}_3 -\ve a_2} \\
\psi_{(\gamma + \ve{d}_3) +\ve{d}_3 -\ve a_2}
\end{array}
\right)
=
\left(
\begin{array}{c}
\psi_{\gamma - \ve{a}_2, B} \\
0
\end{array}
\right) , \\
\end{eqnarray*}
which gives
\begin{eqnarray*}
\(\mathcal{F}_{\ve d_3} (T_{\ve{d}_1} + T_{- \ve{d}_1})\psi \) (\ve{k}) &=&
\left(
\begin{array}{cc}
0 & \mathrm{e}^{+ \mathrm{i} \ve{k} \cdot \ve{a}_2}\\
\mathrm{e}^{- \mathrm{i} \ve{k} \cdot \ve{a}_2} & 0
\end{array}
\right)
\(\mathcal{F}_{\ve d_3} \psi \) (\ve{k}).
\end{eqnarray*}
Summarizing the information above, one concludes that
\begin{equation} \label{dm1}
\sum_{j = 1}^3 (T_{\ve{d}_j} + T_{- \ve{d}_j})
\quad \longleftrightarrow \quad
\left(
\begin{array}{cc}
0 & 1 + \mathrm{e}^{ {-} \mathrm{i} \ve{k} \cdot \ve{a}_1} + \mathrm{e}^{+ \mathrm{i} \ve{k} \cdot \ve{a}_2}\\
1 + \mathrm{e}^{{+}\mathrm{i} \ve{k} \cdot \ve{a}_1} + \mathrm{e}^{- \mathrm{i} \ve{k} \cdot \ve{a}_2} & 0
\end{array}
\right)
\end{equation}
which immediately gives \eqref{R1} and \eqref{R2}.
\begin{remark} \label{Rmk:comparison}
Our definition of the Haldane model agrees with the ones in the cited references \cite{FruchartCarpentier2013,
GiulianiMastropietroPorta2017, Santoro_LN}, up to the translation to first-quantization formalism and some trivial relabelling.
\hfill $\diamond$
\end{remark}
\goodbreak
\section{Bloch functions and their singularities}
\label{Sec:Singularities}
In this Section, we will be interested in studying the spectral properties of the Haldane Hamiltonian, which we rewrite as
\[
H(\ve{k}) = \sum_{j=0}^{3} R_j(\ve{k}) \, \sigma_j =
\begin{pmatrix}
R_0(\ve{k}) + R_3(\ve{k}) & \overline{R(\ve{k})} \\[3pt]
R(\ve{k}) & R_0(\ve{k}) - R_3(\ve{k})
\end{pmatrix},
\]
where we have abbreviated
\[ R(\ve{k}) := R_1(\ve{k}) + \mathrm{i}\, R_2(\ve{k}) = t_1 \big(1 + \mathrm{e}^{\mathrm{i} \ve{k} \cdot \ve{a}_1} + \mathrm{e}^{-\mathrm{i} \ve{k} \cdot \ve{a}_2}\big)\]
(compare \eqref{dm1}). It is then immediate to see that the eigenvalues of $H(\ve{k})$ are given by
\[ E_{\pm}(\ve{k}) := R_0(\ve{k}) \pm \sqrt{\sum_{j=1}^{3} R_j(\ve{k})^2} = R_0(\ve{k}) \pm \sqrt{R_3(\ve{k})^2 + \left| R(\ve{k}) \right|^2}. \]
These two energy bands will not overlap (that is, $E_-(\ve{k}) \le E_+(\ve{k})$ for all $\ve{k} \in \mathbb{R}^2$) provided that $t_1 \neq 0$. For simplicity, in the following we will assume $t_1,t_2 >0$. The bands can still touch at the points in the Brillouin torus which are determined by the equation
\[ \sum_{j=1}^{3} R_j(\ve{k})^2 = 0 \quad \Longleftrightarrow \quad R(\ve{k}) = 0 \text{ and } R_3(\ve{k}) = 0. \]
We see then that there are (at most) two such points in the Brillouin torus, usually labeled $\ve{K}$ and $\ve{K}'$, determined by the zeroes of $R$: these are obtained from the conditions
\[ \mathrm{e}^{\mathrm{i} \ve{K}' \cdot \ve{a}_1} = \mathrm{e}^{\mathrm{i} 2 \pi /3} \text{ and } \mathrm{e}^{-\mathrm{i} \ve{K}' \cdot \ve{a}_2} = \mathrm{e}^{-\mathrm{i} 2 \pi /3}, \quad \mathrm{e}^{\mathrm{i} \ve{K} \cdot \ve{a}_1} = \mathrm{e}^{-\mathrm{i} 2 \pi /3} \text{ and } \mathrm{e}^{-\mathrm{i} \ve{K} \cdot \ve{a}_2} = \mathrm{e}^{\mathrm{i} 2 \pi /3}, \]
which in particular imply $\ve{K}'=-\ve{K} \bmod \Gamma^*$.
Since locally around these points the dispersion of the energy bands is linear when they produce band intersections, {\sl i.\,e.\ } $E_{\pm}(\ve{k})=E_{\pm}(\ve{K}^{\sharp})\pm v\sub{F} | \ve{k}-\ve{K}^{\sharp}| + O\big(| \ve{k}-\ve{K}^{\sharp}|^2\big)$ for $\ve{K}^{\sharp} \in \set{\ve{K},\ve{K}'}$, the points $\ve{K}$ and $\ve{K}'$ are usually called \emph{Dirac points}.
The equation $R_3(\ve{k})=0$ then determines the locus in the space of parameters $(\phi,M)$ where either $\ve{K}$ or $\ve{K}'$ (or both) are points of degeneracy for the eigenvalues of the Haldane Hamiltonian, namely%
\footnote{
Notice that when $R(\ve{k})=0$, that is, when $R_1(\ve{k})=t_1 \big( 1+\cos(\ve{k} \cdot \ve{a}_1) + \cos(\ve{k} \cdot \ve{a}_2) \big) = 0$ and $R_2(\ve{k})= t_1 \big(\sin(\ve{k} \cdot \ve{a}_1) - \sin(\ve{k} \cdot \ve{a}_2)\big) =0$, then using $\ve{a}_3 = - \ve{a}_1 - \ve{a}_2$
\begin{align*}
\sin(\ve{k} \cdot \ve{a}_3) &= - \sin(\ve{k} \cdot \ve{a}_1 + \ve{k} \cdot \ve{a}_2) = - \big( \sin(\ve{k} \cdot \ve{a}_1) \, \cos(\ve{k} \cdot \ve{a}_2) + \sin(\ve{k} \cdot \ve{a}_2) \, \cos(\ve{k} \cdot \ve{a}_1) \big) \\
& = - \big( \sin(\ve{k} \cdot \ve{a}_1) \,(-1 -\cos(\ve{k} \cdot \ve{a}_1) )+ \sin(\ve{k} \cdot \ve{a}_1) \, \cos(\ve{k} \cdot \ve{a}_1) \big) \\
& = \sin(\ve{k} \cdot \ve{a}_1),
\end{align*}
so that $\sum_{j=1}^{3} \sin(\ve{k} \cdot \ve{a}_j) = 3 \, \sin(\ve{k}\cdot\ve{a}_1)$. Now $\sin(\ve{K}\cdot\ve{a}_1) = \sin(-2\pi/3) = - \sqrt{3}/2$, while $\sin(\ve{K}'\cdot\ve{a}_1) = \sin(2\pi/3) = \sqrt{3}/2$.
}
\[ R_3(\ve{K}) = M + 3 \sqrt{3}\, t_2 \, \sin \phi, \quad R_3(\ve{K}') = M - 3 \sqrt{3} \, t_2 \, \sin \phi. \]
\begin{figure}[htb]
\centering
\includegraphics[width=.8\textwidth]{Phase_Diagram.pdf}
\caption{\footnotesize The topological phase diagram of the Haldane model. In cyan, the region $\set{R_3(\ve{K}) > 0, R_3(\ve{K}') < 0}$, characterized by a Chern number $c_1=-1$; in orange, the region $\set{R_3(\ve{K}) < 0, R_3(\ve{K}') > 0}$, characterized by a Chern number $c_1=+1$ (color online). In the rest of the phase diagram, $c_1=0$.}
\label{PhaseDiagram}
\end{figure}
We see that the parameter space $(\phi,M)$ gets divided into four regions where the Hamiltonian is gapped (see Figure~\ref{PhaseDiagram}), characterized by the signs of $R_3(\ve{K})$ and $R_3(\ve{K}')$. We will show now how it is possible to assign a \emph{topological label} to each of the four gapped phases, determining also the ``quantum anomalous Hall conductivity'' of the Haldane model for all parameters in the region. To this end, it is convenient to introduce the eigenvector $u_-(\ve{k})$, that is, the \emph{Bloch function}, associated to the lower band $E_-(\ve{k})$ of the Haldane Hamiltonian. This reads
\[ u_-(\ve{k}) = N(\ve{k})^{-1} \,
\begin{pmatrix}
\sqrt{R_3(\ve{k})^2 + |R(\ve{k})|^2} - R_3(\ve{k}) \\
-R(\ve{k})
\end{pmatrix}, \]
where $N(\ve{k}) := \left[ 2 \sqrt{R_3(\ve{k})^2 + |R(\ve{k})|^2} \left( \sqrt{R_3(\ve{k})^2 + |R(\ve{k})|^2} - R_3(\ve{k}) \right) \right]^{1/2}$ is a normalizing factor ensuring $\norm{u_-(\ve{k})}_{\mathbb{C}^2} = 1$ for all $\ve{k} \in \mathbb{R}^2$ (compare \cite[Appendix B]{GiulianiMastropietroPorta2017}). The \emph{Bloch gauge} (that is, the phase within the complex one-dimensional eigenspace associated to the lower energy band) is chosen so that the first component $u_{-,1}(\ve{k})$ is real.
If $\ve{K}^\sharp$ denotes either of the Dirac points, then $R(\ve{K}^\sharp)$ vanishes, as $\ve{K}$ and $\ve{K}'$ are precisely the zeroes of $R$, while $R_3(\ve{K}^\sharp) \ne 0$ due to the gap condition. Consequently,
\[ \sqrt{R_3(\ve{k})^2 + |R(\ve{k})|^2} - R_3(\ve{k}) \Big|_{\ve{k}=\ve{K}^\sharp} = \big|R_3(\ve{K}^\sharp)\big|-R_3(\ve{K}^\sharp) , \]
and similarly
\[ N(\ve{K}^\sharp) = \left[ 2 \big|R_3(\ve{K}^\sharp)\big| \left( \big|R_3(\ve{K}^\sharp)\big| - R_3(\ve{K}^\sharp) \right) \right]^{1/2}. \]
We see that $u_-(\ve{k})$ may have singularities at the Dirac points, depending on the signs of $R_3(\ve{K})$ and $R_3(\ve{K}')$. In particular, it holds that $u_-(\ve{k})$ is \emph{singular at $\ve{K}$} in the region $\set{R_3(\ve{K})>0, R_3(\ve{K}')<0}$ of the parameter space $(\phi,M)$ (depicted in cyan in Figure \ref{PhaseDiagram}), while it is \emph{analytic on the whole Brillouin torus} in the region $\set{R_3(\ve{K})<0, R_3(\ve{K}')<0}$ (the lower white region in Figure \ref{PhaseDiagram}). The qualitative features of this singularity (or lack thereof) are illustrated in Figures \ref{fig:phi!=0} and \ref{fig:phi=0}.
\begin{figure}[ht]
\centering
\begin{subfigure}{.3\textwidth}
\centering
\includegraphics[width=\textwidth]{phin=0_u1_Density_BZ.pdf}
\caption{$u_{-,1}(\ve{k})$}
\label{subfig:A}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\centering
\includegraphics[width=\textwidth]{phin=0_Reu2_Density_BZ.pdf}
\caption{$\operatorname{Re} u_{-,2}(\ve{k})$}
\label{subfig:B}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\centering
\includegraphics[width=\textwidth]{phin=0_Imu2_Density_BZ.pdf}
\caption{$\operatorname{Im} u_{-,2}(\ve{k})$}
\label{subfig:C}
\end{subfigure}
\begin{minipage}[c]{.3\textwidth}
\centering
\vspace{0pt}
\begin{subfigure}{\textwidth}
\centering
\includegraphics[width=\textwidth]{phin=0_u2_Density_BZ.pdf}
\caption{$|u_{-,2}(\ve{k})|$}
\label{subfig:D}
\end{subfigure}
\end{minipage}
\begin{minipage}[c]{.6\textwidth}
\centering
\vspace{0pt}
\captionsetup{width=.9\textwidth}
\caption{Density plots for the components of $u_-(\ve{k})$ (color online). The parameters chosen to produce these plots are as follows: $d=1$ for the lattice constant, $t_1=1$, $t_2=1/4$, $M=0$, $\phi = \pi/2$. The rhomboidal region is the Brillouin zone $\set{k_1 \, \ve{b}_1 + k_2 \, \ve{b}_2: k_1,k_2 \in [0,1]}$, where the vectors $\ve{b}_1,\ve{b}_2$ spanning the dual lattice $\Gamma^*$ are determined by the conditions $\ve{a}_i \cdot \ve{b}_j = 2 \pi \delta_{ij}$. The circle points to the position of the Dirac point $\ve{K}$. The rapid change of both $\operatorname{Re} u_{-,2}(\ve{k})$ and $\operatorname{Im} u_{-,2}(\ve{k})$ around $\ve{k}=\ve{K}$ are evident from \eqref{subfig:B} and \eqref{subfig:C}, respectively, signalling a discontinuity. Instead, $u_{-,1}(\ve{k})$ is seen to be regular (and actually vanishing) at $\ve{k}=\ve{K}$ from \eqref{subfig:A}. In the last plot \eqref{subfig:D}, contour lines for the absolute value $|u_{-,2}(\ve{k})|$ are plotted, while the color code indicates the value of the argument of the phase of $u_{-,2}(\ve{k})$ as in the legend. In agreement with the previous comments, it is possible to see a phase singularity of $u_{-,2}(\ve{k})$ around $\ve{k}=\ve{K}$, while the absolute value $|u_{-,2}(\ve{k})| = \sqrt{1-u_{-,1}(\ve{k})^2}$ remains smooth.}
\label{fig:phi!=0}
\end{minipage}
\end{figure}
\begin{figure}[ht]
\centering
\begin{subfigure}{.3\textwidth}
\centering
\includegraphics[width=\textwidth]{phi=0_u1_Density_BZ.pdf}
\caption{$u_{-,1}(\ve{k})$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\centering
\includegraphics[width=\textwidth]{phi=0_Reu2_Density_BZ.pdf}
\caption{$\operatorname{Re} u_{-,2}(\ve{k})$}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\centering
\includegraphics[width=\textwidth]{phi=0_Imu2_Density_BZ.pdf}
\caption{$\operatorname{Im} u_{-,2}(\ve{k})$}
\end{subfigure}
\begin{minipage}[c]{.3\textwidth}
\centering
\vspace{0pt}
\begin{subfigure}{\textwidth}
\centering
\includegraphics[width=\textwidth]{phi=0_u2_Density_BZ.pdf}
\caption{$|u_{-,2}(\ve{k})|$}
\end{subfigure}
\end{minipage}
\begin{minipage}[c]{.6\textwidth}
\centering
\vspace{0pt}
\captionsetup{width=.9\textwidth}
\caption{Similar plots to those of Figure \ref{fig:phi!=0} (color online), this time corresponding to the parameters $M=-3\sqrt{3}$ and $\phi = 0$. All other parameters where left as specified in Figure \ref{fig:phi!=0}. In this case, $u_-(\ve{k})$ is analytic over the whole Brillouin zone.}
\label{fig:phi=0}
\end{minipage}
\end{figure}
To investigate further the singularity of $u_-(\ve{k})$, we restrict our attention to parameters $(\phi,M)$ so that $R_3(\ve{K})>0$ and $R_3(\ve{K}')<0$. As discussed, in this region $\ve{K}$ is the only singular point of $u_-(\ve{k})$. By rewriting, after a few simple algebraic manipulations,
\[ u_-(\ve{k}) = \frac{1}{\sqrt{2}}
\begin{pmatrix}
\dfrac{|R(\ve{k})|}{\left(R_3(\ve{k})^2 + |R(\ve{k})|^2\right)^{1/4} \left( \sqrt{R_3(\ve{k})^2 + |R(\ve{k})|^2} + R_3(\ve{k}) \right)^{1/2}} \\[3pt]
-\dfrac{R(\ve{k})}{|R(\ve{k})|} \, \left( 1 + \dfrac{R_3(\ve{k})}{\sqrt{R_3(\ve{k})^2 + |R(\ve{k})|^2}} \right)^{1/2}
\end{pmatrix} ,
\]
we see that in this region the first component of $u_-(\ve{k})$ is smooth, while it is the second component that has a singularity, due to the explicit dependence on the phase of $R(\ve{k})$. This implies in particular that, locally around $\ve{k}=\ve{K}$, $u_{-,2}(\ve{k})$ is homogeneous of degree zero in the radial coordinate $r = |\ve{k}-\ve{K}|$, so that the derivatives of $u_{-,2}(\ve{k})$ have a $(1/r)$-singularity, making the $H^1$-norm of $u_-$
\begin{align*}
\norm{u_-}_{H^1} & := \left( \norm{u_-}_{L^2}^2 + \norm{\partial_{k_1} u_-}_{L^2}^2 + \norm{\partial_{k_2} u_-}_{L^2}^2 \right)^{1/2} \\
& = \left[ \int_{\mathbb{T}^2_*} \left( \norm{u_-(\ve{k})}_{\mathbb{C}^2}^2 + \norm{\partial_{k_1} u_-(\ve{k})}_{\mathbb{C}^2}^2 + \norm{\partial_{k_2} u_-(\ve{k})}_{\mathbb{C}^2}^2 \right) \mathrm{d} \ve{k} \right]^{1/2}
\end{align*}
divergent. From the same type of homogeneity argument, one can also deduce that all the fractional Sobolev norms $\norm{u_-}_{H^s}$ for $s\in [0,1)$ are instead finite.
The singularity of $u_-(\ve{k})$ at $\ve{k} = \ve{K}$ carries also a topological information. This can be accessed by means of the \emph{Berry connection}, defined as the differential $1$-form
\[ \mathcal{A} := \operatorname{Im} \, \inner{u_-(\ve{k})}{\mathrm{d} u_-(\ve{k})}_{\mathbb{C}^2} = \sum_{j=1}^{2} \operatorname{Im} \, \inner{u_-(\ve{k})}{\partial_{k_j} u_-(\ve{k})}_{\mathbb{C}^2} \mathrm{d} k_j. \]
We argue as above: the $(1/r)$-singularity of the derivatives of $u_{-,2}(\ve{k})$ around $\ve{k}=\ve{K}$ is integrable (even though not square-integrable), so that the integral of $\mathcal{A}$ around a small loop $\ell_{\varepsilon}$ (say, of diameter $\varepsilon \ll 1$) encircling the singularity of $u_-(\ve{k})$ stays bounded even in the limit $\varepsilon \to 0$. Denoting by $D_{\varepsilon}$ the region bounded by the loop $\ell_{\varepsilon}$ (which then bounds also $\mathbb{T}^2_* \setminus D_{\varepsilon}$, as the Brillouin torus is closed), in the limit of a very small loop one obtains from Stokes' theorem
\begin{equation} \label{ChernSingularity}
\lim_{\varepsilon \to 0} \oint_{\ell_{\varepsilon}} \mathcal{A} = - \lim_{\varepsilon \to 0} \oint_{\partial(\mathbb{T}^2_* \setminus D_{\varepsilon})} \mathcal{A} = - \lim_{\varepsilon \to 0} \int_{\mathbb{T}^2_* \setminus D_{\varepsilon}} \mathrm{d} \mathcal{A} = - \int_{\mathbb{T}^2_*} \mathcal{F}.
\end{equation}
In the last step, we introduced the \emph{Berry curvature} $2$-form
\begin{equation} \label{Berry}
\begin{aligned}
\mathcal{F} & := \mathrm{d} \mathcal{A} = 2 \operatorname{Im} \, \inner{\partial_{k_1} u_-(\ve{k})}{\partial_{k_2} u_-(\ve{k})} \mathrm{d} k_1 \wedge \mathrm{d} k_2 \\
& = - \mathrm{i} \, \Tr_{\mathbb{C}^2} \big( P_-(\ve{k}) \left[ \partial_{k_1} P_-(\ve{k}), \partial_{k_2} P_-(\ve{k}) \right] \big) \, \mathrm{d} \ve{k}.
\end{aligned}
\end{equation}
The last equality (see {\sl e.\,g.\ } \cite[Lemma 7.2]{MonacoPanatiPisanteTeufel2018}) shows that $\mathcal{F}$ can be expressed directly in terms of the family of projections
\begin{align*}
& P_-(\ve{k}) := \ket{u_-(\ve{k})} \bra{u_-(\ve{k})}= \\
& \frac{1}{2 \sqrt{R_3(\ve{k})^2 + |R(\ve{k})|^2}}
\begin{pmatrix}
\sqrt{R_3(\ve{k})^2 + |R(\ve{k})|^2} - R_3(\ve{k}) & - R(\ve{k}) \\[3pt]
- \overline{R(\ve{k})} & \sqrt{R_3(\ve{k})^2 + |R(\ve{k})|^2} + R_3(\ve{k})
\end{pmatrix}
\end{align*}
on the eigenspace corresponding to the lower energy band. Contrary to the Bloch function, these projections depend analytically on $\ve{k}$ over the whole Brillouin torus, making it possible to compute the last limit in \eqref{ChernSingularity}.
The Berry curvature is a geometric object. In fact, its integral over the Brillouin torus is an integer multiple of $2\pi$:
\begin{equation} \label{Chern}
c_1 := \frac{1}{2\pi} \int_{\mathbb{T}^2_*} \mathcal{F} \quad \in \mathbb{Z}.
\end{equation}
This integer, called the \emph{Chern number}, is the topological invariant which underlies the quantization of the (anomalous) Hall conductivity in Chern insulators \cite{Haldane88,Experiment,Bestwick et al 2015,Chang et al 2015} and quantum Hall insulators \cite{TKNN,Graf review}. In the specific case under investigation of the Haldane Hamiltonian, the four regions of parameters $(\phi,M)$ in which $H(\ve{k})$ is gapped can be labelled by the Chern number \cite{Haldane88}: with reference to the colors of Figure \ref{PhaseDiagram}, the Chern number can be computed, {\sl e.\,g.\ } starting from \eqref{ChernSingularity}, to be $c_1=-1$ for the cyan region, $c_1=+1$ for the orange region, and $c_1=0$ for the two white regions. In analogy with the thermodynamical phases of statistical mechanics, one then speaks of \emph{topological phases of matter} distinguished by different topological invariants, and refers to Figure \ref{PhaseDiagram} as the \emph{topological phase diagram} for the Haldane Hamiltonian.
\begin{remark}
It is interesting to notice that the topological content associated to singularities of the Bloch function at the Dirac points persists, in an appropriate sense, also in the \emph{gapless} regime. If for example the parameters $(\phi,M)$ are threaded from the cyan region to the lower white region of Figure \ref{PhaseDiagram}, passing through a point in parameter space $(\phi_*,M_*)$ where $R_3(\ve{K})=0$, then at $(\phi_*,M_*)$ not only the Bloch function $u_-$ but also the projection $P_-$ becomes singular at $\ve{K}$. Nonetheless, the topological charge exchanged through the gapless phase can be quantified by means of a local topological invariant, the \emph{eigenspace vorticity}, associated to family of projections $P_-$ around the singular point $\ve{K}$ \cite{MonacoPanati2014}. In the situation described above, this eigenspace vorticity equals $\Delta c_1 = 0 - (-1) = 1$.
\hfill $\diamond$
\end{remark}
\goodbreak
\section{The localization dichotomy for periodic insulators}
\label{Sec:Dichotomy}
It is astounding to discover the predictive power of the Haldane model. In fact, it turns out that the features discussed in the previous Section are completely generic in two and three dimensions: indeed, the close connection between the structure of the singularities of the Bloch functions and the topology of the associated eigenspaces persists in a much wider context and for more general models.
This was recently proved and quantified in a precise way in \cite{MonacoPanatiPisanteTeufel2018}. To formulate the main result, we need to set up the more general framework. Let $d\le 3$. The configuration space of a crystalline system is modeled by the space $X$, which can be either $\mathbb{R}^d$ or a $d$-dimensional crystalline structure ({\sl e.\,g.\ } the honeycomb structure presented in Section \ref{Sec:Haldane}): $X$ carries an action of the lattice $\Gamma \simeq \mathbb{Z}^d$ by translations, which is assumed to lift to translation operators $T_\gamma \in \mathrm{U}(L^2(X))$, $\gamma \in \Gamma$.
Associated to these translation operators, there is a \emph{Bloch--Floquet--Zak transform} $\mathcal{U} \colon L^2(X) \to L^2_\tau(\mathbb{B};L^2\sub{per}(Y)) \simeq \int_{\mathbb{B}}^{\oplus} L^2\sub{per}(Y) \mathrm{d} \ve{k}$, defined by
\begin{equation} \label{BF}
\left(\mathcal{U} \psi\right)(\ve{k},\ve{y}) := \sum_{\gamma \in \Gamma} \mathrm{e}^{-\mathrm{i} \ve{k} \cdot (\ve{y}-\gamma)} \, (T_\gamma \psi)(\ve{y}), \quad \ve{k} \in \mathbb{B}, \, \ve{y} \in Y,
\end{equation}
on suitable $\psi \in L^2(X)$. Here $\mathbb{B}$ stands for the fundamental cell of the dual lattice $\Gamma^*$ (the \emph{Brillouin zone} in the physics literature), $Y$ stands for the fundamental cell of the lattice $\Gamma$ (compare \eqref{cellY}), and $L^2_\tau(\mathbb{B};L^2\sub{per}(Y))$ is the Hilbert space
\[
L^2_\tau(\mathbb{B};L^2\sub{per}(Y)) := \left\{
\begin{gathered}
u \in L^2\sub{loc}(\mathbb{R}^d;L^2\sub{loc}(\mathbb{R}^d)) : \\
u(\ve{k}+\lambda,\ve{y}) = (\tau_\lambda u)(\ve{k},\ve{y}) := \mathrm{e}^{-\mathrm{i} \lambda \cdot \ve{y}} \, u(\ve{k},\ve{y}) \\
\text{and } T_\gamma u(\ve{k},\cdot) = u(\ve{k},\cdot) \\
\forall \ve{k} \in \mathbb{R}^d, \, \ve{y} \in \mathbb{R}^d, \, \lambda \in \Gamma^*, \, \gamma \in \Gamma
\end{gathered} \right\}
\]
of functions of the \emph{Bloch momentum} $\ve{k}$ and of the degrees of freedom in the unit cell $\ve{y}$ which are quasi-periodic (\emph{$\tau$-covariant}) in $\ve{k}$ and periodic in $\ve{y}$ (see \cite{MonacoPanati2015} for details). For crystalline structures of the type described in Section \ref{Sec:Haldane}, $\mathcal{U}$ coincides with the Fourier transform \eqref{Def:Fourier} up to the extra phase factor $\mathrm{e}^{-\mathrm{i} \ve{k} \cdot \ve{y}}$ in \eqref{BF}, which turns periodic functions of $\ve{k}$ into quasi-periodic, but makes the boundary conditions on the unit cell $Y$ in direct space $\ve{k}$-independent (namely, exactly periodic). A Bloch--Floquet--Zak transform is defined by \eqref{BF} also in the continuum case $X=\mathbb{R}^d$, where $T_\gamma$ can be the standard translation $(T_\gamma \psi)(\ve{y}) := \psi(\ve{y}-\gamma)$ or, more interestingly, a \emph{magnetic translation} generated by a uniform magnetic field with flux per unit cell which is commensurate to the flux quantum (equal to $2\pi$ in Hartree units), see \cite{Zak1964} and the discussion in \cite[Sec.~3]{MonacoPanatiPisanteTeufel2018}. Also in this case we will denote by $A \longleftrightarrow A(\ve{k})$ the correspondence between a periodic operator $A$ on $L^2(X)$ such that $[A,T_\gamma]=0$ for all $\gamma \in\ \Gamma$ and its decomposition into fibers in the Bloch--Floquet--Zak representation: $\mathcal{U} \, A \, \mathcal{U}^{-1} = \int_{\mathbb{B}}^{\oplus} A(\ve{k}) \, \mathrm{d} \ve{k}$.
Now that the framework of crystalline systems is clear, we can formulate the main hypothesis of the central result from \cite{MonacoPanatiPisanteTeufel2018}, which abstracts the predominant features of the Haldane Hamiltonian described in the previous Sections.
\medskip
\noindent \textbf{Assumption.} Let $H$ be a periodic self-adjoint operator on $L^2(X)$ with $H \longleftrightarrow H(\ve{k})$ where $H(\ve{k})$ defines a family of operators on $L^2\sub{per}(Y)$ such that
\begin{enumerate}
\item $\set{H(\boldsymbol{\kappa})}_{\boldsymbol{\kappa} \in \mathbb{C}^d}$ defines an \emph{entire analytic family in the sense of Kato} with compact resolvent \cite{RS4};
\item the family is \emph{$\tau$-covariant}, that is, $H(\ve{k}+\lambda) = \tau_{\lambda} \, H(\ve{k}) \, \tau_{\lambda}^{-1}$ for all $\ve{k} \in \mathbb{R}^d$ and $\lambda \in \Gamma^*$;
\item the family is \emph{gapped}, namely there exists a set $\mathcal{I} \subset \mathbb{N}$ with $|\mathcal{I}| = m < \infty$ such that
\[ \inf_{\ve{k} \in \mathbb{R}^d} \inf_{\substack{n \in \mathcal{I} \\ m \in \mathbb{N} \setminus \mathcal{I}}} \big| E_n(\ve{k}) - E_m(\ve{k}) \big| \ge g > 0 \]
where $\sigma(H(\ve{k}))=\set{E_n(\ve{k})}_{n \in \mathbb{N}}$ denotes the spectrum of $H(\ve{k})$ (consisting of discrete eigenvalues, the \emph{Bloch bands}, by the compact resolvent assumption).
\end{enumerate}
\medskip
In the discrete case ({\sl e.\,g.\ } for the Haldane Hamiltonian), the regularity assumption is easy to verify, as it is equivalent in position space to having sufficiently fast decaying hoppings between different sites of the crystal (say, exponential in the distance between the sites), and thus is in particular satisfied whenever the hopping Hamiltonian has finite range, as often happens in applications. For (magnetic, periodic) Schr\"{o}dinger operators, there are standard $L^p$-regularity assumptions on the electro-magnetic potentials that guarantee analyticity of the corresponding fiber Hamiltonians \cite{RS4}.
Notice moreover that the gap assumption allows to define the family of spectral projections $P(\ve{k})$ onto the spectral island $\sigma_0(\ve{k}) := \set{E_n(\ve{k}) : n \in \mathcal{I}}$, for example through the \emph{Riesz formula}
\[ P(\ve{k}) = \frac{\mathrm{i}}{2 \pi} \oint_{C(\ve{k})} (H(\ve{k}) - z)^{-1} \mathrm{d} z, \]
where $C(\ve{k})$ is a positively oriented contour in the complex energy plane, locally constant in $\ve{k}$, which lies in the resolvent set of $H(\ve{k})$ and encircles only the eigenvalues in $\sigma_0(\ve{k})$. This family of projections is then $\tau$-covariant and depends analytically on $\boldsymbol{\kappa} \in \Omega_\alpha$, where $\Omega_\alpha \subset \mathbb{C}^d$ is a complex strip of half-width $\alpha>0$ around the ``real axis'' $\mathbb{R}^d \subset \mathbb{C}^d$ \cite[Prop.~2.1]{PanatiPisante}.
As in \eqref{Chern}, we can define the \emph{Chern numbers} associated to $\set{P(\ve{k})}_{\ve{k} \in \mathbb{R}^d}$ as
\begin{equation} \label{Chern_ij}
c_1(P)_{ij} := \frac{1}{2\pi} \int_{\mathbb{B}_{ij}} \Tr_{L^2\sub{per}(Y)} \left( P(\ve{k}) \left[ \partial_{k_i} P(\ve{k}), \partial_{k_j} P(\ve{k}) \right] \right) \mathrm{d} k_i \wedge \mathrm{d} k_j, \quad 1 \le i < j \le d,
\end{equation}
where $\mathbb{B}_{ij} \subset \mathbb{B}$ is the $2$-dimensional sub-torus of $\mathbb{B}$ where the coordinate different from the $i$-th and $j$-th is fixed ({\sl e.\,g.\ } to zero).
We are finally able to state the main result from \cite{MonacoPanatiPisanteTeufel2018}, generalizing the analysis on the Haldane Hamiltonian from the previous Section.
\begin{theorem}[{\cite{MonacoPanatiPisanteTeufel2018}}] \label{LocDic_Bloch}
Let $H \longleftrightarrow H(\ve{k})$ be as in the above Assumption, and $P(\ve{k})$ be the spectral projection onto the gapped spectral island of $H(\ve{k})$. Then for all $s \in [0,1)$ there exists a \emph{Bloch frame} $\set{u_1, \ldots, u_m} \subset H^s_\tau(\mathbb{B};L^2\sub{per}(Y))$ for $\set{P(\ve{k})}_{\ve{k} \in \mathbb{R}^d}$, namely a set of functions $u_a \in H^s\sub{loc}(\mathbb{R}^d;L^2\sub{per}(Y))$ such that
\[ u_a(\ve{k}+\lambda) = \tau_\lambda u_a(\ve{k}), \quad \inner{u_a}{u_b}_{L^2} = \delta_{ab}, \quad \text{and} \quad P(\ve{k}) = \sum_{a=1}^{m} \ket{u_a(\ve{k})} \bra{u_a(\ve{k})}. \]
Moreover, the following statements are equivalent:
\begin{enumerate}
\item there exists a Bloch frame in $H^1_\tau(\mathbb{B};L^2\sub{per}(Y))$;
\item there exists a Bloch frame in $C^\omega_\tau(\Omega_\alpha; L^2\sub{per}(Y))$, the space of $\tau$-covariant analytic functions on $\Omega_\alpha$ with values in $L^2\sub{per}(Y)$;
\item the Chern numbers $c_1(P)_{ij}$, $1 \le i < j \le d$, defined in \eqref{Chern_ij}, vanish.
\end{enumerate}
\end{theorem}
The above result can be interpreted as a \emph{Localization--Topology Correspondence}, having implications also for the transport properties of the model under scrutiny for a crystalline insulator. To better clarify this point, we need to introduce one further notion. Given a periodic Hamiltonian $H \longleftrightarrow H(\ve{k})$ as in the Assumption above, denote by $P = \mathcal{U}^{-1} \left( \int_{\mathbb{B}}^{\oplus} P(\ve{k}) \mathrm{d} \ve{k} \right) \mathcal{U}$ the periodic projection on $L^2(X)$ onto the subspace corresponding to the isolated spectral island in momentum space. The Hamiltonian $H$ has generically absolutely continuous spectrum (which is given by $\sigma(H) = \left\{\lambda \in \mathbb{R} : \lambda = E_n(\ve{k}) \text{ for some } n \in \mathbb{N}, \, \ve{k} \in \mathbb{R}^d\right\}$), so it is not possible in general to find a basis of the range of $P$ given by eigenstates of the Hamiltonian. Nonetheless, if $\set{u_a(\ve{k})}_{1 \le a \le m}$ is an orthonormal basis for $\Ran P(\ve{k})$ --- a Bloch frame, in the terminology introduced above --- then it is possible to define (\emph{composite}) \emph{Wannier functions} \cite{MarzariEtAl12} by
\[ w_a(\ve{y}-\gamma) := (\mathcal{U}^{-1} u_a)(\ve{y}-\gamma) = \frac{1}{|\mathbb{B}|} \int_{\mathbb{B}} \mathrm{e}^{\mathrm{i} \ve{k} \cdot (\ve{y}-\gamma)} u_a(\ve{k},\ve{y}) \, \mathrm{d} \ve{k} , \quad 1 \le a \le m, \]
where $\ve{y} \in Y$ and $ \gamma \in \Gamma$.
The functions $w_a$ will automatically be in $\Ran P \subset L^2(X)$, and so will the translates $T_\gamma w_a$ by periodicity of $P$. One can then check \cite{Kuchment16} that $\set{T_\gamma w_a}_{\gamma \in \Gamma, \, 1 \le a \le m}$ constitutes an orthonormal basis for $\Ran P$ if the Bloch frame is $\tau$-covariant. \emph{Localized} Wannier functions are found to describe accurately the orbitals of the crystalline insulator \cite{MarzariEtAl12}, and it is hence important to understand their decay properties at infinity. Since the Bloch--Floquet--Zak transform shares with the standard Fourier transform the property of intertwining the multiplication operator by $\ve{x}$ on $L^2(X)$ and the gradient $\nabla_{\ve{k}}$ with respect to the crystal momentum, one can read off these decay properties of Wannier functions by looking at the smoothness with respect to $\ve{k}$ of the corresponding Bloch frame. More precisely, it holds that
\begin{align*}
&\langle \ve{x} \rangle^s w_a \in L^2(X) \quad &&\Longleftrightarrow \quad u_a \in H^s(\mathbb{B};L^2\sub{per}(Y)), \quad s \ge 0, \\
&\mathrm{e}^{\beta |\ve{x}|} w_a \in L^2(X) \; , \forall \beta \in [0,\alpha) &&\Longleftrightarrow \quad u_a \in C^\omega(\Omega_\alpha;L^2\sub{per}(Y)),
\end{align*}
where we have denoted $\langle \ve{x} \rangle := (1 + |\ve{x}|^2)^{1/2}$.
The existence of a basis of well-localized (say, exponentially) Wannier functions signals the absence of charge transport in the crystal; on the contrary, a power-law decay of the Wannier functions is an indication of topological transport. If the Hall conductivity is non-zero, one then expects Wannier functions to be poorly localized. This is exactly the content of the above Theorem, which can be recast in terms of Wannier functions as a \emph{Localization Dichotomy}: \textsl{either} Wannier functions are exponentially localized (and this happens exactly when the Hall conductivity vanishes), \textsl{or} they are delocalized in the sense that they yield an infinite expectation of the squared position operator $|\ve{x}|^2$; no intermediate regimes of decay are allowed. The precise result is as follows.
\begin{theorem}[Localization Dichotomy {\cite{MonacoPanatiPisanteTeufel2018}}] \label{LocDic_WF}
Let $H$ be as in the above Assumption, and $P$ be the spectral projection onto the gapped spectral island. Then for all $s \in [0,1)$ there exists a \emph{Wannier basis} for $\Ran P$, that is, an orthonormal basis $\set{T_\gamma w_a}_{\gamma \in \Gamma, 1 \le a \le m}$ of $\Ran P$, such that
\[ \sup_{\gamma \in \Gamma} \int_{X} \langle \ve{x} - \gamma \rangle^{2s} \left|(T_\gamma w_a)(\ve{x}) \right|^2 \mathrm{d} \ve{x} \le C_s < \infty \quad \text{for all } s \in [0,1).
\]
\smallskip
\noindent Moreover, the following statements are equivalent:
\begin{enumerate}
\item there exists a Wannier basis such that
\[ \sup_{\gamma \in \Gamma} \int_{X} \langle \ve{x} - \gamma \rangle^2 \left|(T_\gamma w_a)(\ve{x}) \right|^2 \mathrm{d} \ve{x} \le C_1 < \infty ; \]
\item there exists a Wannier basis such that
\[ \sup_{\gamma \in \Gamma} \int_{X} \mathrm{e}^{2 \beta |\ve{x} - \gamma|} \left|(T_\gamma w_a)(\ve{x}) \right|^2 \mathrm{d} \ve{x} \le C_\omega < \infty \quad \text{for all } \beta \in [0,\alpha); \]
\item the Chern numbers $c_1(P)_{ij}$, $1 \le i < j \le d$, defined in \eqref{Chern_ij}, vanish.
\end{enumerate}
\end{theorem}
We sketch here the main ideas from the proof of Theorem \ref{LocDic_Bloch}: for the detailed argument, the reader is referred to \cite{MonacoPanatiPisanteTeufel2018}.
The first part consists in exhibiting a Bloch frame which is in $H^s_\tau$ for all $s \in [0,1)$. In $2d$, this is obtained via \emph{parallel transport}, a procedure which allows to construct a smooth ($C^\infty$) and $\tau$-covariant Bloch frame on the $1$-dimensional boundary of the Brillouin zone $\mathbb{B}$, and to extend this to the interior. The end result is a Bloch frame which is $\tau$-covariant and smooth except at one point in the Brillouin zone. The technique of parallel transport gives a precise control also on the type of singularity of the constructed Bloch frame, which is seen to be consistent with the claimed $H^s$-regularity (the derivatives of the Bloch functions have a $(1/r)$-divergence at the singular point). This situation should be compared with the Bloch function for the Haldane Hamiltonian exhibited in the previous Section. In $3d$, one needs to further extend an already singular datum at the $2$-dimensional boundary of the Brillouin zone to the $3$-dimensional ``bulk'': this can be done again by parallel transport, and produces this time lines of singularities, which dictate in turn the $H^s$-regularity in the statement of the Theorem.
The next part of the proof requires to show that if a Bloch frame in $H^1_\tau$ exists, then the Chern numbers of the family of projections vanish. The proof relies on a very subtle approximation of $H^1_\tau$ frames by $C^\infty_\tau$ frames. The subtlety lies in the fact that the space of frames in an Hilbert space is a non-linear manifold; the approximation of Sobolev maps with values in a manifold by regular maps becomes more involved, and requires in general certain topological conditions to be satisfied (see \cite{HangLin} and \cite[App.~B]{MonacoPanatiPisanteTeufel2018}). Nonetheless, in our setting $H^1$ maps can indeed be approximated by $C^\infty$ ones; when calculating an ``approximate'' Berry curvature \eqref{Berry} with the regular frames, its integrals over the tori $\mathbb{B}_{ij}$ are zero, so that in the limit the Chern numbers for the family of projections $P(\ve{k})$ must also vanish. It is then well-known \cite{Panati,PanatiPisante} how to modify the $H^1$-regular Bloch frame to an analytic one, provided the Chern numbers vanish.
As a side remark, note how in $2d$ the ``threshold'' Sobolev regularity $H^1$ coincides also with the ``threshold'' of the Sobolev embedding $H^s \hookrightarrow C^0$, which holds for $s>1$. Geometric arguments, based on the theory of vector bundles, yield that a non-zero Chern number forbids the existence of $\tau$-covariant \emph{continuous} Bloch frames \cite{Panati}: Theorem \ref{LocDic_Bloch} improves this result, claiming that also Bloch frames in $H^1_\tau$ cannot exist when the Chern numbers are non-vanishing. In $3d$, the result is even more stringent, as the threshold for the Sobolev embedding of $H^s$ into continuous functions is at $s=3/2$.
\section{The localization dichotomy for non-periodic insulators}
\label{Sec:GeneralizedDichotomy}
The results presented so far on the interplay between the localization properties and the topological features of crystalline systems exhibit a clear ``logical order'': even though the Wannier functions are position-space objects, they are defined in terms of the Bloch functions, which are intrinsically $\mathbf{k}$-space objects. In the same way, the topological marker, namely the Chern number(s), is defined in terms of the $\mathbf{k}$-space fibering of the projection on the gapped part of the spectrum by means of the Bloch--Floquet--Zak transform. However, there is no physical reasons for the dichotomic behaviour illustrated by Theorem \ref{LocDic_WF} to hold only in systems for which a $\mathbf{k}$-space description is possible. Indeed, perfect crystalline systems do not exist in nature and an extension of the localization dichotomy to non-periodic gapped quantum systems is very tempting and desired.
As a starting point for this ambitious goal, it is necessary to extend the notions of Wannier functions and Chern number to non-periodic systems. Distilling the true essence of Wannier functions, we get that they are a set of localized functions that, together with their copies obtained by lattice translation, form an orthonormal basis of a given spectral subspace, and encode the topological and transport information about the latter. Following this direction, we can extend the concept of Wannier basis to non-periodic systems.
In order to proceed further, we need the notion of \emph{localization function}. For the sake of the presentation, we restrict to $2$-dimensional continuum systems, namely we consider the Hilbert space $L^2(\mathbb{R}^2)$.
\begin{definition}[Localization function]
\label{LocalizationFunction}
We say that a continuous function $G: [0,+\infty) \to (0,+\infty)$ is a \emph{localization function} if $\lim_{|\mathbf{x}| \to + \infty} G (|\mathbf{x}|) \; = \; + \infty$ and there exists a constant $C_{G}>0$ such that
\begin{equation*}
\label{GTriang}
G(|\mathbf{x}-\mathbf{y}|) \leq C_{G} \; G(|\mathbf{x}-\mathbf{z}|) \; G(|\mathbf{z}-\mathbf{y}|) \, , \qquad \forall \, \mathbf{x},\mathbf{y},\mathbf{z} \in \mathbb{R}^2 \, .
\end{equation*}
\end{definition}
\noindent Following the seminal ideas in \cite{NenciuNenciu1993,NenciuNenciu1998}, we give a definition of generalized Wannier basis \cite{MarcelliMoscolariPanati,Moscolari}.
\begin{definition}[Generalized Wannier Basis]
\label{GWB}
Let $P \in \mathcal{B}(L^2(\mathbb{R}^2))$ be an orthogonal projection. Assume that there exist:
\begin{enumerate}
\item a Delone set $\mathfrak{D} \subseteq \mathbb{R}^2$, {\sl i.\,e.\ } a discrete set such that for some $0<r<R<\infty$
it holds true that:
\begin{enumerate}
\item[(a)] for all $\mathbf{x} \in \mathbb{R}^2$ there is at most one element of $\mathfrak{D}$ in the ball of radius $r$ centered at $\mathbf{x}$
(in particular, the set has no accumulation points);
\item[(b)] for all $\mathbf{x} \in \mathbb{R}^2$ there is at least one element of $\mathfrak{D}$ in the ball of radius $R$ centered at $\mathbf{x}$ (the set is ``not sparse'');
\end{enumerate}
\item \label{L2norm} a localization function $G$, constants $M >0$ and $m_* \in \mathbb{N}$ independent of $\gamma \in \mathfrak{D}$, and an orthonormal basis of $\Ran P$, denoted by $\{\psi_{\gamma,a}\}_{\gamma \in \mathfrak{D}, 1 \leq a \leq m(\gamma)}$ with $m(\gamma) \leq m_* \; \forall \gamma \in \mathfrak{D}$, satisfying
\begin{equation*}
\int_{\mathbb{R}^2} |\psi_{\gamma,a}(\mathbf{x})|^2 \, G(|\mathbf{x}-\gamma|) \, \mathrm{d} \mathbf{x} \leq M
\end{equation*}
for all $\gamma \in \mathfrak{D}, a \in \set{1,\ldots, m(\gamma)}$.
\end{enumerate}
Then we call $\psi_{\gamma,a}$ a \emph{generalized Wannier function} (GWF) with \emph{centre} $\gamma$, and we say that $P$ admits a \emph{generalized Wannier basis} (GWB) $\{\psi_{\gamma,a}\}_{\gamma \in \mathfrak{D}, 1 \leq a \leq m(\gamma)}$.
\end{definition}
When the localization function $G$ is an exponential function, that is $G(|\mathbf{x}|)=e^{2 \beta |\mathbf{x}|}$ for some $\beta > 0$,
we say that the GWB is \emph{exponentially localized}. If the localization function $G$ is of polynomial type, that is, $G(|\mathbf{x}|)= \langle \ve{x}\rangle^{2s} = \big( 1+ |\mathbf{x}|^2 \big)^{s}$ for some $s > 0$, we say that the GWB is \emph{$s$-localized}.
Notice that an orthonormal basis made of composite Wannier functions, as defined in Section \ref{Sec:Dichotomy},
is an example of GWB. Moreover, from the results in \cite{CorneanNenciuNenciu2008}, one concludes that for generic gapped $1$-dimensional systems there always exists an orthonormal basis for the range of the gapped spectral projection satisfying the requirements of Definition \ref{GWB} with the exception of some properties of the set $\mathfrak{D}$ (in particular $\mathfrak{D}$ is only proven to be a discrete set).
As already mentioned, the Chern number has been historically related to the quantized conductivity in the QHE \cite{Graf review}. Since the experiments revealing the QHE have been realized with real materials, a generalization of the Chern number in presence of impurities and disorder has been considered long ago. Indeed, Bellissard, van Elst and Schulz-Baldes (see \cite{BellissardVanElstSchulzBaldes1994} and references therein), inspired by ideas in Non Commutative Geometry, extended the concept of Chern number to ergodic systems and connected it to the transverse conductivity in the QHE. Moreover, more recently, in the physics community there has been a growing interest in the analysis of topological materials by means of topological marker defined directly in position space \cite{BiancoResta2011, Caio et al 2019, Irsigler et al 2019}. Inspired by these ideas, we give the following definition of \emph{Chern marker} (which is called Chern character in \cite{CorneanMonacoMoscolari2018}, where the setting is slightly different).
\begin{definition}[Chern marker]
\label{def:Cherncharacter}
Let $P$ be a projection on $L^2(\mathbb{R}^2)$ and $\chi_{L}$ be the indicator function of the set $(-L,L]^2$. The \emph{Chern marker} of $P$ is defined by
\begin{equation*}
C(P):=\lim\limits_{L \to \infty} \frac{2\pi}{4L^2} \Tr \Big(\mathrm{i}\chi_{L} P \Big[\left[X_1, P \right],\left[X_2,P \right]\Big]P \chi_L\Big)
\end{equation*}
\noindent whenever the limit on the right hand side exists.
\end{definition}
Notice that, in case $P$ is an integral operator then the integral kernel of the operator $P [\left[X_1, P \right],\left[X_2,P \right]]P$ coincides with the definition of local Chern number given in \cite{BiancoResta2011} and with the definition of local Chern marker given in \cite{Caio et al 2019}.
Whenever the projection is periodic and hence can be fibered by the Bloch--Floquet--Zak transform, the Chern marker coincides with the Chern number, appearing in \eqref{Chern}, and hence it is an integer. Furthermore, one can show that the Chern marker is stable against regular perturbations and against the addition of a constant magnetic field, provided that these perturbations do not close the gap \cite{CorneanMonacoMoscolari2018}.
Hence, guided by the results in the periodic case, in \cite{MarcelliMoscolariPanati} the authors conjecture the existence of a relation between the localization properties of a GWB for a projection and the Chern marker of the projection itself. To formulate this conjecture properly, we focus on physical systems that can be described by a Hamiltonian operator $H$, acting in the Hilbert space $L^2(\mathbb{R}^2)$, and of the form
\begin{equation}
\label{Hamiltonian}
H=-\frac{1}{2}\Delta_{\bf A} + V \, ,
\end{equation}
where $V$ is a scalar potential such that $V$ is in $L^2\sub{u-loc}(\mathbb{R}^2)$, and $-\Delta_{\bf A}:=\left(-\mathrm{i} \nabla-\mathbf{A}\right)^2$ is the magnetic Laplacian. The magnetic potential $\mathbf{A}$ is such that $\mathbf{A} \in L^4\sub{loc}(\mathbb{R}^2,\mathbb{R}^2)$ and the distributional derivative $\nabla \cdot \mathbf{A}$ is in $L^2\sub{loc}(\mathbb{R}^2)$. The assumptions on the potentials are the usual assumptions which allow to apply the diamagnetic inequality. Under these hypotheses, the Hamiltonian is essentially selfadjoint on the dense core $C^\infty_0(\mathbb{R}^2)$. Moreover, we assume that the spectrum of the Hamiltonian has a spectral island $\sigma_0(H)$ isolated from the rest of the spectrum of $H$, that is ,
$$
\dist(\sigma_0(H),\sigma(H)\setminus \sigma_0(H))= g > 0 \, .
$$
\noindent We can now formulate the
\medskip
\noindent \textbf{Localization Dichotomy Conjecture.}
{\it
Under the above assumptions, let $P$ be the spectral projection onto the spectral island $\sigma_0(H)$ of a Hamiltonian operator of the form \eqref{Hamiltonian}. Then the following statements are equivalent:
\begin{enumerate}[label=(\alph*),ref=(\alph*)]
\item \label{exp-loc} $P$ admits a generalized Wannier basis that is exponentially localized.
\item \label{s-loc} $P$ admits a generalized Wannier basis that is $s_*$-localized for $s_*=1$.
\item \label{c-zero} $P$ is topologically trivial in the sense that its Chern marker $C(P)$ exists and is equal to zero.
\end{enumerate}
}
\medskip
Notice that \ref{exp-loc} easily implies \ref{s-loc} by a simple inequality, while the opposite implication is not trivial. The Localization Dichotomy Conjecture generalizes the results proved in the periodic setting in Theorem \ref{LocDic_WF} \cite{MonacoPanatiPisanteTeufel2018}.
A new result, still unpublished, covering the non-periodic setting is the following \cite{Moscolari,MarcelliMoscolariPanati}.
While we expect the Conjecture to be true for $s_*=1$, the theorem is restricted to $s_*>5$ for technical reasons.
\begin{theorem}[Localization implies topological triviality]
\label{LocDicTheorem}
Under the above assumptions, let $P$ be the spectral projection onto the spectral island $\sigma_0(H)$ of a Hamiltonian operator of the form \eqref{Hamiltonian}. Suppose that $P$ admits a $s_*$-localized generalized Wannier basis $\{\psi_{\gamma,a}\}_{\gamma \in \mathfrak{D}, 1 \leq a \leq m(\gamma)}$
for $s_*>5$, that is: there exists $M > 0$ and $m_* \in \mathbb{N}$ such that
\begin{equation*}
\int_{\mathbb{R}^2} |\psi_{\gamma,a}(\mathbf{x})|^2 (1+\|\mathbf{x}-\gamma\|^2)^{s_*} \, d\mathbf{x} \leq M\, , \qquad \forall\; \gamma \in \mathfrak{D} \, , \forall a \in \left\{1, \dots, m(\gamma)\right\}
\end{equation*}
with $m(\gamma)\leq m_*$ for all $\gamma \in \mathfrak{D}$.
Then the Chern marker of $P$ is zero, namely the following limit exists and
\begin{equation*} \label{ZeroCN}
\lim\limits_{L \to \infty} \frac{2\pi}{4L^2} \Tr \left(\mathrm{i}\chi_{L} P \left[\left[X_1, P \right],\left[X_2,P \right]\right]P \chi_L\right) \;=\; 0.
\end{equation*}
\end{theorem}
The proof of Theorem \ref{LocDicTheorem} is obtained in two steps: first one proves exponential localization estimates for the integral kernel of the projection and for some auxiliary operators by using Combes-Thomas-type estimates. Then, by using those estimates, one can gain an explicit control on the asymptotic behaviour of the trace of $ \mathrm{i}\chi_{L} P \left[\left[X_1, P \right],\left[X_2,P \right]\right]P \chi_L$ as $L \to \infty$ and therefore can prove the desired limit. Notice that the threshold $s_*>5$ is only due to technical reasons and, as we mentioned before, a full generalization of the localization dichotomy proved in \cite{MonacoPanatiPisanteTeufel2018} would require $s_*=1$, as well as a proof of the opposite implication. Further investigations are planned for the future in order to move the threshold to $s_*=1$.
\begin{remark}
As a by-product of Theorem \ref{LocDicTheorem}, it follows that the dichotomic behaviour of the Wannier basis in Theorem \ref{LocDic_WF} is ``stable'' with respect to regular perturbations. Indeed, consider a periodic system such that its Chern number is different from zero and suppose that we perturb the system with a small non-periodic term, for example by adding some impurities modelled by Coulomb potentials. By contradiction, suppose that the perturbed system has an exponentially localized GWB in the sense of Definition \ref{GWB}. By a result proved by A. Nenciu and G. Nenciu \cite{NenciuNenciu1993}, it is possible to unitarily transport the GWB back to the original system. Then, Theorem \ref{LocDicTheorem} implies that the Chern marker is zero. As we have mentioned before, for periodic systems the Chern marker equals the Chern number. Therefore, this implies that the original periodic system has a vanishing Chern number and yields a contradiction.
\hfill $\diamond$
\end{remark}
Despite a proof of the implication \ref{c-zero} $\Rightarrow$ \ref{exp-loc} is still missing in the non-periodic setting, Theorem \ref{LocDicTheorem} provides a clear relation between the GWB and the Chern marker. Whenever a sufficiently localized GWB for a given gapped quantum system exists, one can be sure that such physical system does not exhibit Hall transport. This relation is completely independent of the periodicity of the system.
|
1,314,259,993,626 | arxiv | \section{Introduction}
\label{I}
The main mission of statistical physics is to bridge the gap between different levels of description of physical systems, from a microscopic level where we know the laws governing each of the myriad of components of the system and their interactions, to a macroscopic level where the whole system is described through emergent laws relating only a handful of global observables. The law of ideal gases is an extreme example of that, where three quantities are enough to describe the typical state of a somewhat caricatural system in a rather specific set-up, but it is essentially the same question that is asked when one want to describe, for instance, a model of particles moving and interacting on a lattice, by a Langevin equation on just the local average density of particles with a conserved Gaussian noise. Obtaining such a hydrodynamic description, and determining whether the Gaussian noise does reproduce faithfully the fluctuations of the lattice model at that scale, is in general an extremely challenging problem.
For systems that are close to equilibrium, where detailed balance is broken only infinitesimally in the large size limit, such as boundary-driven or weakly bulk-driven models, the aforementioned macroscopic hydrodynamic description is known to hold quite generally, under the names of macroscopic fluctuation theory (MFT, \cite{Bertini2007}) and additivity principle \cite{PhysRevLett.92.180601}. The question becomes more interesting when one considers systems that are far from equilibrium, where no such general result exists. There, from an analytical perspective, one is mostly restricted to study specific models or classes of models which are amenable to calculations, which often means exactly solvable, such as zero-range processes \cite{Harris2005,Chleboun2016} or exclusion processes \cite{Derrida199865,0034-4885-74-11-116601}. Moreover, since the question involves not only the typical behaviour of the systems at a macroscopic scale, but their fluctuations as well, the relevant framework is that of large deviation functions \cite{Touchette20091}, which are the dynamical equivalent of equilibrium free energies.
The most interesting feature of those large deviation functions are so-called \textit{dynamical phase transitions}. Just like their equilibrium equivalents, they appear as non-analyticities which correspond to boundaries between qualitatively different behaviours of the system when changing its parameters. Unlike for equilibrium phase transitions, however, dynamical control parameters are usually abstract quantities which do not correspond to actual tunable parameters of the physical model \cite{Jack2015a}, and those different qualitative behaviours correspond to the best way in which the system can fluctuate to maintain an atypical value of some observable for a long time, rather than stable state changes that could be observed experimentally by tuning the environment. This is not to say that dynamical phase transitions have no experimentally measurable consequences, as for instance models which are critical in that respect (i.e. where the stationary state sits precisely at a dynamical phase transition) will show non-Gaussian fluctuations, anomalous diffusive behaviours and special dynamical exponents, a famous example being models in the KPZ universality class \cite{Spohn2016,Corwin2016}. Identifying those dynamical phase transitions is quite crucial when attempting to describe the macroscopic behaviour of stochastic systems: they define the domains of applicability of any proposed description.
~~
In this paper, we focus on one-dimensional bulk-driven open simple exclusion processes (SEP), where particles jump from site to neighbouring site on a one dimensional lattice, with a bias towards one side (e.g. the right), and can enter or exit the system only at the boundaries. The \textit{simple exclusion} constraint means that the particles interact through hard-core repulsion: at most one particle can be on a given site at a given time. These models are among the most studied in non-equilibrium statistical physics, both analytically for the exactly solvable versions \cite{derrida1993exact,1751-8121-40-46-R01,Lazarescu2015} (which can also have periodic boundary conditions \cite{derrida1998exact,Prolhac2009} and symmetric or weakly asymmetric jumps \cite{Enaud2004,Prolhac2009a}), and numerically in the case of non-solvable generalisations \cite{1742-5468-2008-06-P06009,Greulich2008,Ciandrini2010}.
For such transport models, the most important observables are usually the macroscopic currents of particles (or whatever charges are being transported), which are a direct consequence of the system being driven out of equilibrium, and which can be expected to play an important role in its macroscopic behaviour. In particular, any dynamical phase transition that the system might undergo will most likely be visible in the large deviation function of those currents, which is a contraction of the full joint large deviation function of currents and densities. Indeed, such transitions have been found, for instance in the weakly asymmetric version of the model, both for periodic boundary conditions \cite{PhysRevE.72.066110,Bodineau2008,appert2008universal,Simon2011,Espigares2013} and open ones \cite{Lecomte2010,Baek2016a}.
A dynamical phase transition has also been found in the large deviations of the current of the \textit{asymmetric} simple exclusion process, both for total asymmetry (TASEP, where particles jump only to the right) and partial asymmetry (PASEP or simply ASEP, where particles can jump backwards as well), and both in the periodic and the open geometry. Those models are integrable (in the sense of quantum integrability \cite{Faddeev1996,Lazarescu2014}, through its connexion with the XXZ spin chain \cite{sandow1994partially}), meaning that one can in principle obtain an exact expression for the large deviation function, and describe the transition analytically. This was first done in a periodic and totally asymmetric setting in \cite{derrida1998exact} and \cite{derrida1999universal}, which was then extended to the partially asymmetric case \cite{Prolhac2009,prolhac2010tree}. In the open setting, part of the large deviation function was correctly conjectured in \cite{Bodineau2006} and later confirmed in \cite{DeGier2011a}, and the full large deviation function for the TASEP was obtained soon afterwards \cite{Lazarescu2011} and later extended to the ASEP \cite{gorissen2012exact,Lazarescu2014}. In all of these works, the results were obtained using integrability methods such as the Bethe Ansatz, and are in principle restricted to small fluctuations of the current because of approximations made towards the large size limit, except for \cite{Popkov2010} which deals with the limit of very high currents in the periodic ASEP.
Finally, the complete dynamical phase diagram of the current for the open ASEP was obtained in \cite{Lazarescu2013,Lazarescu2015}, by combining the aforementioned results for small fluctuations with exact diagonalisation methods for extreme values of the current. It was found that in the very high current limit, the system effectively behaves like a discrete Coulomb gas, as it does in the periodic case \cite{Popkov2010}, with a large deviation function proportional to the size $L$ of the system ; in the very low current limit, the system has an effective dynamics involving only anti-shock states, and the large deviation function does not depend on $L$. The whole phase diagram is then obtained by interpolating between those tree regimes, and by conjecturing that the MFT can be used to obtain a large part of it (which is compatible with all the exact results). A dynamical phase transition is identified, which includes the stationary maximal current phase, and separates the two different scalings in $L$.
~
As we mentioned, a large part of those results are obtained by integrability methods, which are unfortunately entirely specific to that very special model. Luckily, the aforementioned exact diagonalisation methods for extreme currents do not require the model to be integrable, which means that they can be applied in more general settings. Indeed, in this paper, we consider a much broader class of models, by adding two features to the ASEP: arbitrary spatial inhomogeneities in the jump rates, and an arbitrary short range interaction between the particles. We will see that the same methods apply (albeit in a much more involved way) and yield essentially the same results: the large deviation function of the current is proportional to $L$ for high currents, and independent of $L$ for low currents. This points to the existence of a dynamical phase transition separating those two regimes, although this time no description of the transition itself is available, and in particular no information on the critical exponents associated to it, which leads us to qualify the transition as \textit{generic} rather than \textit{universal}.
The structure of this paper is as follows. In section \ref{II}, we first introduce the reader to the models and formalism we will be using throughout the paper, along with a few useful standard results. In section \ref{III}, we focus on the high current limit, where we first recall the results pertaining to the regular ASEP, before extending them rather straightforwardly to our generalised versions. In section \ref{IV}, which contains the main technical result of this paper, we do the same for the low current limit, by finding bounds on the behaviour of both the large deviation function of the current, and the shape of the typical states associated to those fluctuations. We then provide numerical illustrations for those results in section \ref{V}, as well as an interpretation of the origin of the dynamical phase transition in terms of maximal hydrodynamic currents, and discuss its relation to the KPZ universality class \cite{BenArous2011} and third order phase transitions \cite{LeDoussal2016}. We finally conclude and discuss a few extensions that our result could receive in the future.
\newpage
\section{Models and formalism}
\label{II}
In this first section, we define the models that we intend to study, and present the formalism in which they will be treated, as well as the mathematical tools we will use in order to access the fluctuations of the current of particles flowing through them.
\subsection{Markov matrix and master equation}
\label{IIa}
The basic model upon which we aim to build is the asymmetric simple exclusion process (ASEP). It is a Markov process in continuous time defined on a finite one-dimensional lattice of size $L$, be it periodic or open and connected to reservoirs at both ends. Each site can be either empty, or holding one particle, and particles hop from site to site with a rate $p$ to the right and $q<p$ to the left (as if driven by a field $\log(p/q)$). In the open case, particles can enter the system on the first site with rate $p_0$ and on the last with rate $q_L$, and exit from the first site with rate $q_0$ and from the last with rate $p_L$. In all these cases, if the target site is already occupied, the exclusion rule means the particle cannot jump. The rates for the open case are summarised on fig.\ref{fig1}.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.8\textwidth]{PASEP.pdf}
\caption{Dynamical rules for the ASEP with open boundaries. The jumps shown in green are allowed by the exclusion constraint. Those shown in red and crossed out are forbidden.}
\label{fig1}
\end{center}
\end{figure}
We will also be considering a simpler version called the totally asymmetric simple exclusion process (TASEP), where the particles can only jump to the right, which is to say that $q_0=q=q_L=0$. Note that we will be focusing on the open case, but that everything we will say is easily transposed to the periodic case, as will be regularly pointed out.
~
To put that into equations: states of the system are written as configurations ${\cal C}=\{\tau_i\}$, where $\tau_i=0$ if site $i$ is empty and $\tau_i=1$ if it is occupied. The probability vector $|P_{t}\rangle$, of which an entry $P_{t}({\cal C})$ gives the probability to be in configuration ${\cal C}$ at time $t$, obeys the master equation
\begin{equation}\label{MP}
\frac{d}{dt}|P_{t}\rangle=M|P_{t}\rangle
\end{equation}
where $M$ is the Markov matrix of the open ASEP:
\begin{equation}\label{M}
M=m_0+\sum\limits_{i=1}^{L-1}M_i +m_L
\end{equation}
with
\begin{equation}\label{M2}
m_0=\begin{bmatrix} -p_0 & q_0 \\ p_0 & -q_0 \end{bmatrix}~,~ M_{i}=\begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & -q & p & 0 \\ 0 & q & -p & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix}~,~m_L=\begin{bmatrix} -q_L & p_L \\ q_L & -q_L \end{bmatrix}.
\end{equation}
It is implied that $m_0$ acts as written on site $1$ (and is represented in basis $\{0,1\}$ for the occupancy of the first site), and as the identity on all the other sites. Likewise, $m_L$ acts as written on site $L$, and $M_i$ on sites $i$ and $i+1$ (and is represented in basis $\{00,01,10,11\}$ for the occupancy of those two sites). Each of the non-diagonal entries represents a transition between two configurations that are one particle jump away from each other. Note that we will be using the convention where $w(\cal{C},\cal{C}')$ is a transition from $\cal{C}'$ to $\cal{C}$, consistently with $M$ acting to the right on $|P_{t}\rangle$.
~
We will be generalising this model in two ways. First, by adding an interaction potential $V({\cal C})$ to configurations, which does not have to be two-body or translation-invariant (note that we will later assume $V$ to be short-range and bounded in order to prove the results in section \ref{IV}). This is done by multiplying the transition rate from ${\cal C}'$ to ${\cal C}$ by ${\rm e}^{(V({\cal C}')-V({\cal C}))/2}$. If the system were in equilibrium (if for instance $p=q$ in the periodic case), the stationary probability of ${\cal C}$ would then be ${\rm e}^{-V({\cal C})}$ up to a normalisation. Note that in order not to introduce too many irrelevant quantities, the Boltzmann inverse temperature is not written explicitly (i.e. either taken equal to $1$, or as an inverse unit of energy).
Secondly, by adding on-site inhomogeneities in the jump rates, i.e. by having them depend on space: $p$ and $q$ become $p_i$ and $q_i$, where $i$ is the label of the bond involved in the transition, starting at $1$ between the first and second site (so that a particle on site $i$ jumps to the right with rate $p_i$ and to the left with rate $q_{i-1}$), consistently with the notation used for the boundary rates. This second generalisation merely adds an index $i$ to the entries of $M_i$, but the first one modifies the structure of $M$: the entries of $M_i$ can now depend on all the details of the initial and final configurations, so that they are not effectively of dimension $4$ any more. Note that part of the inhomogeneity can be absorbed in $V$, so that these two generalisations are not orthogonal, but this will not be important for us.
~
We will be generically writing the transition rates of our process as $w(\cal{C},\cal{C}')$, and we will take ${\cal C}\sim{\cal C}'$ to mean that there is a transition from ${\cal C}'$ to ${\cal C}$. It will be useful for future calculations to decompose the Markov matrix of this generalised simple exclusion process into three pieces:
\begin{itemize}
\item $M^+$, containing the rates for jumps to the right, of the form ${\rm e}^{-V({\cal C})/2}~p_i~{\rm e}^{V({\cal C}')/2}$ ;
\item $M^-$, containing the rates for jumps to the left, of the form ${\rm e}^{-V({\cal C})/2}~q_i~{\rm e}^{V({\cal C}')/2}$ ;
\item $M^d$, containing the escape rates, i.e. the diagonal part of $M$, of the form $-\sum_{{\cal C}\sim{\cal C}'}w(\cal{C},\cal{C}')$.
\end{itemize}
Note that the addition of the potential $V$ to $M^+$ and $M^-$ is a conjugation (division to the left, multiplication to the right) of the potential-less rates with a diagonal matrix ${\rm e}^{V/2}$, but this is not the case for $M^d$.
\subsection{Deformed Markov matrix for the current}
\label{IIb}
Now that we have defined our processes, we can see how to access the statistics of the stationary current. The simplest, most tractable way to do that, for our purposes, is through the cumulants of the current.
Let us say that we want to know, for instance, the stationary statistics of the time-averaged current of particles $j_0$ across the bond between the left reservoir and the first site. We should, in principle, from a given initial condition, look at all the possible histories of the system for a certain runtime $t$, count the number of particles crossing that bond in either direction, and compute the probability that the difference of these two numbers is close to $t j_0$. A simpler way to proceed is to introduce a fugacity (or counter or Lagrange multiplier) ${\rm e}^{\mu_0}$ for that algebraic number of ingoing jumps, and multiply, in $M$, the rate which increases that count by one (i.e. in $M^+$) by ${\rm e}^{\mu_0}$ and the rate which decreases it (i.e. in $M^-$) by ${\rm e}^{-\mu_0}$. By using these deformed rates instead of the original ones in our process (which is not a proper Markov process any more, since we have not modified $M^d$), every history will be multiplied by as many fugacities as there was jumps, i.e. by a factor ${\rm e}^{t j_0 \mu_0}$. Summing these weighted probabilities and taking derivatives with respect to $\mu_0$ will then yield the moments of $t j_0$, which we can then relate to their probability distribution.
~
This can be rephrased neatly in a few equations. For the sake of compactness, we will consider the case where $V=0$, so that we can write $M$ in a simple way, but it is straightforward to include the potential as well. Let us now assume that we want to monitor all of the time-averaged currents $j_i$, across each of the $L+1$ bonds in the system, independently, in order to obtain their large deviation function $g(j)$ defined through
\begin{equation}
\mathbb{P}_{0..t}\bigl[\{[\#~i\rightarrow i+1]\}\sim \{tj_i\}\bigr]\approx \mathrm{e}^{-tg(j)}
\end{equation}
in the large $t$ limit, where $[\#~i\rightarrow i+1]$ is the algebraic number of jumps made between sites $i$ and $i+1$ from time $0$ till $t$. We associate a fugacity ${\rm e}^{\mu_i}$ to each current (fig.\ref{Currents}), and write simply $\mu$ for the vector containing the $\mu_i$'s.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.6\textwidth]{Currents.pdf}
\caption{Transitions contributing to each of the physical currents $j_i$, along with the fugacities applied to each of the (non-diagonal) transition rates.}
\label{Currents}
\end{center}
\end{figure}
Consider then the following matrix
\begin{equation}\label{Mmu}
M_{\mu}=m_0(\mu_0)+\sum_{i=1}^{L-1} M_{i}(\mu_i)+m_L(\mu_l)
\end{equation}
with
\begin{equation}\label{Mmui}
m_0(\mu_0)=\begin{bmatrix} -p_0 & q_0{\rm e}^{-\mu_0} \\ p_0{\rm e}^{\mu_0} & -q_0 \end{bmatrix}~,~ M_{i}(\mu_i)=\begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & -q_i & p_i{\rm e}^{\mu_i} & 0 \\ 0 & q_i{\rm e}^{-\mu_i}& -p_i & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix}~,~m_L(\mu_L)=\begin{bmatrix} -q_L & p_L{\rm e}^{\mu_L} \\ q_L{\rm e}^{-\mu_L} & -p_L \end{bmatrix}
\end{equation}
(where, as before, it is implied that $m_0$ acts as written on site $0$ in the basis $\{0,1\}$ and as the identity on the other sites, and the same goes for $m_L$ on site $L$; similarly, $M_i$ is expressed by its action on sites $i$ and $i\!+\!1$ in the basis $\{00,01,10,11\}$ and acts as the identity on the rest of the system).
~
As we said earlier, summing the weights of histories obtained by using this matrix as a generator for a time $t$ from an initial state $P_0$ yields the generating function of the joint moments of all the currents up to a trivial dependence on $t$. Taking the logarithm of that generating function and dividing by $t$ therefore gives the (rescaled) generating function of the cumulants of the current $E$, in the limit of large times:
\begin{equation}\label{Emui}
E(\mu)=\lim\limits_{t\rightarrow\infty}\frac{1}{t}\bigl( \langle 1|{\rm e}^{tM_{\mu}}|P_0\rangle \bigr)
\end{equation}
It is straightforward to show from that expression that $E(\mu)$ is in fact the largest eigenvalue of $M_{\mu}$: for $t$ large, we can write
\begin{equation}
{\rm e}^{tM_{\mu}}\sim {\rm e}^{t\Lambda(\mu)}|P_{\mu}\rangle \langle \tilde{P}_{\mu}|
\end{equation}
where $\Lambda(\mu)$ is the largest eigenvalue of $M_{\mu}$ and $|P_{\mu}\rangle$ and $ \langle \tilde{P}_{\mu}|$ are the corresponding eigenvectors. Injecting this in (\ref{Emui}) yields $E=\Lambda$. All of these elements hold important information regarding the stationary fluctuations of the current and the fluctuating states themselves. We can show, by making a saddle-point approximation for large $t$ in the definition of $E$, that \cite{Touchette20091,Lazarescu2015}:
\begin{itemize}
\item G\"artner-Ellis theorem: the large deviation function of the currents, $g(j)$, where $j$ is the vector holding all the individual currents $j_i$, is the Legendre transform of $E(\mu)$, i.e.
\begin{equation}\label{gEi}
g(j)=j.\mu-E(\mu)~~~~{\rm with}~~~~j_i=\frac{{\rm d}}{{\rm d}\mu_i} E(\mu).
\end{equation}
\item The right eigenvector holds probabilities of final configurations conditioned on the currents that have been observed, up to a normalisation, i.e.
\begin{equation}
P_\mu({\mathcal C})\propto{\rm P}\Bigl({\mathcal C}_t={\mathcal C}~\Big|~j_i\!=\!\frac{{\rm d}}{{\rm d}\mu_i} E(\mu)\Bigr).
\end{equation}
\item The left eigenvector holds probabilities of initial configurations conditioned on the currents that will be observed, up to a normalisation, i.e.
\begin{equation}
\tilde{P}_\mu({\mathcal C})\propto{\rm P}\Bigl({\mathcal C}_0={\mathcal C}~\Big|~j_i\!=\!\frac{{\rm d}}{{\rm d}\mu_i} E(\mu)\Bigr).
\end{equation}
\item The product of these two probabilities gives the stationary probability of configurations (far from the initial and final time) conditioned on the currents that are observed, up to a normalisation, i.e.
\begin{equation}\label{sensemble}
P_\mu({\mathcal C})\tilde{P}_\mu({\mathcal C})\propto{\rm P}\Bigl({\mathcal C}~\Big|~j_i\!=\!\frac{{\rm d}}{{\rm d}\mu_i} E(\mu)\Bigr).
\end{equation}
\end{itemize}
In all of these cases, the currents we consider are time-averaged, but for a finite-sized system, they also correspond to stationary instantaneous currents (in other words, a given average current is best realised, over a long period, by a constant instantaneous current of the same value). These properties are not specific to one-dimensional simple exclusion processes, but are in fact valid for any currents in any Markov process on a finite configuration space.
~
We can significantly simplify these expressions for our models, due to the fact that they are one-dimensional, with conservative dynamics in the bulk (no particles are created or destroyed). This means that stationary currents cannot depend on space: there can be no prolonged build-up or depletion of particles anywhere in the system. Perhaps the easiest way to see this formally is through the following procedure.
Consider the diagonal matrix ${\rm e}^{\lambda \tau_i}$ (with $1\leq i\leq L$) with an entry ${\rm e}^{\lambda}$ for all configurations for which site $i$ is occupied, and $1$ otherwise. One may easily check that the matrix similarity ${\rm e}^{-\lambda \tau_i}M_{\mu}{\rm e}^{\lambda \tau_i}$ simply replaces $M_{i-1}(\mu_{i-1})$ and $M_{i}(\mu_{i})$ by, respectively, $M_{i-1}(\mu_{i-1}-\lambda)$ and $M_{i}(\mu_{i}+\lambda)$, and leaves the rest of $M_\mu$ unchanged. That is to say that part of the deformation is transferred from $M_{i-1}(\mu_{i-1})$ to $M_{i}(\mu_{i})$. Using combinations of these transformations for any sites and parameters $\lambda$, we conclude that all the Markov matrices deformed with respect to the currents are similar, and therefore have the same eigenvalues, as long as the sum of the deformation parameters $\sum_{i=0}^{L}\mu_i$ is fixed. Note that the eigenvectors are different, but related to each other through those simple transformations, and that the product of the two eigenvectors, $P_\mu({\mathcal C})\tilde{P}_\mu({\mathcal C})$, is invariant as well.
~
In short, the two quantities that give information on the stationary fluctuations of the current in the system, $E(\mu)$ and $P_\mu({\mathcal C})\tilde{P}_\mu({\mathcal C})$, only depend on the sum of the $\mu_i$'s, which we will write as $\mu$, and all the $j_i$'s have the same value $j$. We may then rewrite eq.(\ref{gEi}) as
\begin{equation}\label{gE}\boxed{
g(j)=j\mu-E(\mu)~~~~{\rm with}~~~~j=\frac{{\rm d}}{{\rm d}\mu} E(\mu)
}\end{equation}
where all the variables are now scalars. The freedom we have in distributing the $\mu_i$'s for a given $\mu$ is quite useful in practice for calculations. In this paper, we will always choose $\mu_i=\mu/(L+1)$.
\subsection{Current and entropy production}
\label{IIc}
A physically meaningful consequence of this simplification, which also highlights the importance of the current as an observable, is that this space-independent current is exactly proportional to the entropy production in the system. Consider the particular set of weights $\{\mu_i\}$ defined by $\mu_i=\nu\log{(p_i/q_i)}$, for which $M_{\mu}$ becomes $M_\nu$:
\begin{equation}\label{MLambda}
m_0=\begin{bmatrix} -p_0 & q_0^{1+\nu}p_0^{-\nu} \\ p_0^{1+\nu}q_0^{-\nu} & -q_0 \end{bmatrix}~,~ M_{i}=\begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & -q_i & p_i^{1+\nu}q_i^{-\nu} & 0 \\ 0 & q_i^{1+\nu}p_i^{-\nu}& -p_i & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix}~,~m_L=\begin{bmatrix} -q_L & p_L^{1+\nu}q_L^{-\nu} \\ q_L^{1+\nu}p_L^{-\nu} & -p_L \end{bmatrix}
\end{equation}
which is the deformed Markov matrix measuring the entropy production. We see immediately that
\begin{equation}
M_{-1-\nu}= ~^t\!M_\nu
\end{equation}
which implies the Gallavotti-Cohen symmetry \cite{PhysRevLett.74.2694,Lebowitz99agallavotti-cohen} for the eigenvalues and between the left and right eigenvectors of $M_\nu$ with respect to the transformation $\nu\leftrightarrow(-1\!-\!\nu)$.
Considering that $\mu=\nu \log{\Bigl(\frac{\prod_i p_i}{\prod_i q_i}\Bigr)}$, we also obtain the Gallavotti-Cohen symmetry related to the current, namely
\begin{equation}\boxed{
E(\mu)=E\biggl(-\log{\Bigl(\frac{\prod_i p_i}{\prod_i q_i}\Bigr)}-\mu\biggr)
}
\end{equation}
which is also valid for the other eigenvalues of $M_\mu$, and the corresponding relations between the right and left eigenvectors, as well as a simple relation between the microscopic entropy production $s$, conjugate to $\nu$, and the macroscopic current $j$, conjugate to $\mu$:
\begin{equation}\label{sj}
s= j\log{\Bigl(\frac{\prod_i p_i}{\prod_i q_i}\Bigr)}.
\end{equation}
Two remarks are to be made here. First of all, those weights are ill-defined for the TASEP: micro-reversibility (i.e. the fact that for any allowed transition, the reverse transition is also allowed) is essential to have a fluctuation theorem. Moreover, if we take either the $q_i\rightarrow 0$ or the $L\rightarrow\infty$ limit (with all the $q_i$'s being finitely smaller than the $p_i$'s, so that the logarithm is of order $L$), the centre of the Gallavotti-Cohen symmetry $\mu^\star=-\frac{1}{2}\log{\Bigl(\frac{\prod_i p_i}{\prod_i q_i}\Bigr)}$ is rejected to $-\infty$, so that the `negative current' part of the fluctuations is lost. This will in fact be useful to us: assuming that we can exchange the two limits (the validity of this assumption will be discussed in the conclusion), we will be able to get information about the $j\rightarrow 0$ limit by taking $\mu$ to $-\infty$ in a totally asymmetric situation, which is much simpler than taking it to $-\frac{1}{2}\log{\Bigl(\frac{\prod_i p_i}{\prod_i q_i}\Bigr)}$ in a partially asymmetric one.
Secondly, we may consider the detailed balance case, where $\prod_i p_i=\prod_i q_i$. In that case, there is no entropy production whatsoever, i.e. that $s= 0$, as we see from eq.(\ref{sj}). This does not mean, however, that $j=0$: the deformations through $\mu$ and $\nu$ are in that case not equivalent. The only implication this has on $E(\mu)$ is that it is an even function: $E(\mu)=E(-\mu)$, all the odd cumulants are zero, and positive and negative currents of the same amplitude are equiprobable.
Note that, apart from the precise expressions of $M_\mu$ and $M_\nu$, all of this holds equally well for the case with an extra interaction potential $V$.
\subsection{A note on asymptotics and Legendre transforms}
In the rest of this paper, we will see how analysing the asymptotic behaviour of $M_\mu$ for $\mu\rightarrow\pm\infty$ can give us information on the nature of extreme fluctuations of the current, and indicate the existence of a generic dynamical phase transition. This will be done by obtaining asymptotic expressions or bounds on the generating function of cumulants $E(\mu)$ from diagonalising $M_\mu$ for $\mu\rightarrow\pm\infty$, and taking their Legendre transforms. In order to do that, one has to be careful to check that the Legendre transform of the asymptotic equivalent of $E(\mu)$ is indeed an asymptotic equivalent of the Legendre transform of the real $E(\mu)$. This will be done in appendix \ref{A1}.
~~
Moreover, all calculations will be done at a size $L$ large but finite, so that the limits $\mu\rightarrow\pm\infty$ are taken before $L\rightarrow\infty$. These two limits do not commute in principle, meaning that our results will not be a proof of the existence of a dynamical phase transition but rather a strong indication of it. In particular, in the case of a very inhomogeneous system (possibly with quenched disorder), or a very unphysical potential, the behaviour of $E(\mu)$ and the presence of a sharp dynamical transition will depend on how the $L\rightarrow\infty$ limit is taken on $p_i$ and $V$. On the contrary, for a well-behaved and local $V$, and slow-varying disorder $p_i$, the phase transition is likely to be similar to that which can be observed in the simple ASEP \cite{derrida1999universal,Lazarescu2015}, although it is unclear which of its features are universal. This will be illustrated with a few numerical plots in section \ref{V}.
\newpage
\section{High current limit}
\label{III}
We will first consider the limit where $j\rightarrow\infty$, which corresponds to $\mu\rightarrow\infty$.
As before, we can write
\begin{equation}
M_\mu=M^+_\mu+M^-_\mu+M^d.
\end{equation}
Note that the diagonal part of $M_\mu$ is not deformed. The generic form of the entries of $M^+_\mu$ and $M^-_\mu$ for a given transition and its reverse is
\begin{equation}
{\rm e}^{-V({\cal C})/2}~p_i~{\rm e}^{\mu_i}~{\rm e}^{V({\cal C}')/2}~~~~{\rm and}~~~~{\rm e}^{-V({\cal C}')/2}~q_i~{\rm e}^{-\mu_i}~{\rm e}^{V({\cal C})/2},
\end{equation}
From what we saw in section \ref{IIb}, we can find a function $\phi({\cal C})$ such that the matrices ${\rm e}^{\phi}M^+_\mu{\rm e}^{-\phi}$ and ${\rm e}^{\phi}M^-_\mu{\rm e}^{-\phi}$ have rates
\begin{equation}
{\rm e}^{\frac{\mu}{L+1}}~{\rm e}^{-V({\cal C})/2}~p_i~{\rm e}^{V({\cal C}')/2}~~~~{\rm and}~~~~{\rm e}^{-\frac{\mu}{L+1}}~{\rm e}^{-V({\cal C}')/2}~q_i~{\rm e}^{V({\cal C})/2},
\end{equation}
so that in the $\mu\rightarrow\infty$ limit, $M^d$ and $M^-_\mu$ become negligible, and the problem reduces to the analysis of $M^+_\mu$.
\subsection{Mapping to an XX spin chain}
\label{IIIa}
We are now considering only the right-moving part of $M_\mu$, with entries
\begin{equation}
{\rm e}^{-V({\cal C})/2}~p_i~{\rm e}^{\mu_i}~{\rm e}^{V({\cal C}')/2}
\end{equation}
between configurations which differ by one jump to the right. Using matrix similarities, we can significantly simplify the problem. First of all, it is clear that we can get rid of $V$ through the similarity ${\rm e}^{-V/2}M^+_\mu{\rm e}^{V/2}$, so that $V$ has no influence on large positive fluctuations of the current (as long as $V$ takes finite values). We are left with entries of the form $p_i~{\rm e}^{\mu_i}={\rm e}^{\mu_i+\log(p_i)}$, which is to say that we can treat the inhomogeneity in the rates $p_i$ as an inhomogeneity in the deformations $\mu_i$. Since only their sum matters, only $\prod_i p_i$ will appear in the fluctuations of the current, so that the inhomogeneity is also virtually irrelevant.
~
Putting all this together, we are left with a much simpler matrix to analyse : the right-moving part of a standard ASEP, with all $p_i=1$, and with a pre-factor $({\rm e}^{\mu}\prod_i p_i )^{\frac{1}{L-1}}$. It turns out that it is in fact more convenient to keep a factor $\frac{1}{\sqrt{2}}$ in the boundary matrices, so that we are left to study the matrix
\begin{equation}
M_\mu\sim\Bigl(2{\rm e}^{\mu}\prod_i p_i \Bigr)^{\frac{1}{L-1}}\Biggl(\frac{1}{\sqrt{2}} S_1^+ +\sum\limits_{n=1}^{L-1}S_n^- S_{n+1}^+ +\frac{1}{\sqrt{2}}S_L^-\Biggr)
\end{equation}
where $S_i^+$ and $S_i^-$ are matrices which respectively add and remove a particle at site $i$.
~
We may recognise this to be the upper half of the Hamiltonian of an open XX spin chain \cite{Bilstein1999}. Moreover, it happens to commute with its transpose, thanks to the factors $\frac{1}{\sqrt{2}}$ on each side. We know, from the Perron-Frobenius theorem, that the highest eigenvalue of that matrix is real and non-degenerate. It is therefore also the highest eigenvalue of its transpose, with the same eigenvectors (because they commute). This allows us to define their average $H$, which has the same highest eigenvalue and the same eigenvectors as $M^+$. Forgetting about the constant pre-factor $(2{\rm e}^{\mu}\prod_i p_i )^{\frac{1}{L-1}}$ for the time being, $H$ is given by:
\begin{equation}\boxed{
H=\frac{1}{\sqrt{8}} S_1^x +\frac{1}{2}\sum\limits_{n=1}^{L-1}(S_n^- S_{n+1}^++S_n^+ S_{n+1}^-) +\frac{1}{\sqrt{8}}S_L^x
}\end{equation}
which is the Hamiltonian for the open XX chain with spin $1/2$ and extra boundary terms $S_1^x$ and $S_L^x$ (with $S^x=S^++S^-$). Luckily, we can diagonalise it exactly.
\subsection{Largest eigenvalue of H}
\label{IIIb}
We will not give here the full derivation of the diagonalisation of $H$, which involves standard free fermion techniques, but only the results relevant to this paper. For more details, the reader can refer to section V B of \cite{Lazarescu2015}, or to \cite{Bilstein1999}, where this spin chain is studied with more general boundary conditions.
This system has $2L+2$ independent excitations, with energies
\begin{equation}
{\cal E}_k=\sin\biggl(\frac{(2k-1)\pi}{2L+2}\biggr)~~~~~~~~{\rm for}~~ k\in\llbracket1,2L+2\rrbracket
\end{equation}
and a vacuum energy of
\begin{equation}
{\cal E}_0=-\frac{1}{2}\sum\limits_{k=1}^{L+1}{\cal E}_k.
\end{equation}
The highest eigenvalue is therefore obtained by adding all the positive excitation energies, which yields
\begin{equation}
{\cal E}_0+\sum\limits_{k=1}^{L+1}{\cal E}_k=\frac{1}{2}\sin\Bigl(\frac{\pi}{2L+2}\Bigr)^{-1}\sim \frac{L}{\pi}.
\end{equation}
Remembering the global pre-factor $(2{\rm e}^{\mu}\prod_i p_i )^{\frac{1}{L-1}}$ which we removed earlier, and after simplification, we finally get:
\begin{equation}
E(\mu)\sim(\prod_i p_i )^{\frac{1}{L}}~\frac{L}{\pi}~{\rm e}^{\mu/L}
\end{equation}
Depending on how the $p_i$'s are chosen, the first factor in this expression might be negligible or not. However, we can simply decide to always impose that $\prod_i p_i =1$ or is of order $1$, which means choosing the average time it takes for one free particle to go through the system as a natural time scale. Without a great loss of generality, we therefore have
\begin{equation}\boxed{
E(\mu)\sim\frac{L}{\pi}{\rm e}^{\mu/L}
}\end{equation}
and
\begin{equation}\label{IV-3-gMMC}\boxed{\boxed{
g(j)\sim L\bigl( j\log(j)-j(1-\log(\pi))\bigl)
}}\end{equation}
which is proportional to $L$, and {\it depends neither on $V$ nor on the $p_i$'s} (apart from a trivial re-scaling).
This expression, and its scaling with respect to $L$, are what we were mainly after, but it will also be informative to look at the corresponding stationary state.
\subsection{Stationary state conditioned on a high current}
\label{IIIc}
As stated before, the state that we want to examine is the highest energy state of a free fermions system, so that it will not come as a big surprise that the probability of each configuration can be expressed as a Vandermonde determinant. Once more, we will only state the relevant results, and let the curious readers refer to section V B of \cite{Lazarescu2015} for the details of the calculations.
The un-normalised stationary probability of a configuration ${\mathcal C}$ can be expressed, as we saw, as a product of the right and left eigenvectors corresponding to the eigenvalue $E(\mu)$. In the $\mu\rightarrow\infty$ limit, the eigenvectors of $M_\mu$ do not depend on $\mu$ except up to a matrix similarity which leaves this product invariant, so that we will simply write it as ${\rm P}_{\mu\rightarrow\infty}({\mathcal C})$. We have
\begin{equation}
{\rm P}_{\mu\rightarrow\infty}({\mathcal C})={\rm P}\Bigl({\mathcal C}~\Big|~j\!=\!\frac{{\rm d}}{{\rm d}\mu} E(\mu)\Bigr)=P_\mu({\mathcal C})\tilde{P}_\mu({\mathcal C}),
\end{equation}
conditioned on a current $j\sim\frac{1}{\pi}{\rm e}^{\mu/L}$. We find that this probability can be expressed as:
\begin{equation}\boxed{
{\rm P}_{\mu\rightarrow\infty}({\mathcal C})=\prod\limits_{\tau_i=\tau_j}[\sin(r_j - r_i)]\prod\limits_{\tau_i\neq \tau_j}[\sin(r_j + r_i)]
}\end{equation}
where $\tau_i=0$ or $1$ is the occupancy of site $i$, and $r_i=i \pi/(2L+2)$. Note that all these probabilities are still un-normalised.
This distribution is that of a Dyson-Gaudin gas \cite{Gaudin1973}, which is a discrete version of the Coulomb gas, on a periodic lattice of size $2L+2$, with two defect sites (at $0$ and $L+1$) that have no occupancy, and a reflection anti-symmetry between one side of the system and the other (fig.-\ref{fig-DGgas}). The first (upper) part of the gas is given by the configuration we are considering, and the second (lower) is deduced by anti-symmetry. The interaction potential between two particles at angles $r_i$ and $r_j$ is then given by:
\begin{equation}
V(r_i,r_j)=-\log\bigl(\sin(r_j - r_i)\bigr).
\end{equation}
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.5\textwidth]{DGgas.pdf}
\caption{Dyson-Gaudin gas equivalent for the configuration $(110101000110111)$ for the open ASEP conditioned on a large current. The lower part of the system is deduced from the upper part by an axial anti-symmetry.}
\label{fig-DGgas}
\end{center}
\end{figure}
~
An important feature of that state is the centred density correlation $C_{ij}$ between two sites $i$ and $j$, i.e.
\begin{equation}
C_{ij}=\langle\tau_i\tau_j\rangle-\langle\tau_i\rangle\langle\tau_j\rangle.
\end{equation}
Using the probability distribution we just obtained, we can compute it to be given by
\begin{equation}
C_{ij}=\frac{1}{4(L+1)^2\sin^2\Bigl(\frac{\pi(i+j)}{(2L+2)}\Bigr)}-\frac{1}{4(L+1)^2\sin^2\Bigl(\frac{\pi(i-j)}{(2L+2)}\Bigr)}.
\end{equation}
The correlations are therefore exactly $0$ for sites which are an even number of bonds apart (as is the case for a half-filled periodic chain \cite{Popkov2010}), and behave as
\begin{equation}\boxed{
C_{ij}\sim -\frac{1}{\pi^2 (i-j)^2}
}\end{equation}
otherwise, if the two sites are far away enough from the boundaries. Note that those correlations do not vanish with the size of the system, in contrast with the steady state of the ASEP at $\mu=0$, where they behave as $L^{-1}$ in the maximal current phase and vanish exponentially in the high and low density phases \cite{Derrida1993a}.
~
In the periodic case, it was shown in \cite{Popkov2010} that the large current limit of the steady state of the ASEP of size $L$ converges to a simple periodic Dyson-Gaudin gas (without defects or symmetry). The inhomogeneous interacting version of that model can be treated in the exact same way as we did here, yielding the same result.
We should note that the trick consisting in taking the sum of $M^+_\mu$ and its transpose to reconstruct an XX spin chain is not in fact necessary. All the results can be obtained, in a slightly different way, on $M^+_\mu$ directly, which has the added advantage that the imaginary part of the other eigenvalues is not lost \cite{Karevski2016}.
\newpage
\section{Low current limit}
\label{IV}
We now consider the $j\rightarrow0$ limit.
This case is somewhat more subtle than the previous one. As we saw in section \ref{IIc}, the centre of the Gallavotti-Cohen symmetry is at $\mu^\star=-\frac{1}{2}\log{\Bigl(\frac{\prod_i p_i}{\prod_i q_i}\Bigr)}$, and is also the point where $j=\frac{{\rm d}}{{\rm d}\mu} E(\mu)=0$, by symmetry. Trying to analyse the behaviour of $E(\mu)$ around that point is, in general, not much simpler that the complete problem. However, as we remarked, if we consider the TASEP instead, where all the backward rates are set to $0$, this point becomes $\mu=-\infty$, which greatly simplifies the problem, as we will shortly demonstrate.
The question then remains of the relevance of this special case. Notice that even without taking $q_i=0$, the value of $\mu^\star$ will go to $-\infty$ in the large size limit, as long as the amplitude of the inhomogeneities is bounded (i.e. does not get arbitrarily small or large when $L$ goes to $\infty$), so that $\frac{\prod_i p_i}{\prod_i q_i}$ grows exponentially with $L$. This suggests that the two limits might be consistent (first $L\rightarrow\infty$, then $q_i\rightarrow 0$, or the reverse), but in no way proves it. We will come back on this assumption in the conclusion of this paper, but in all rigour the computations in this section apply only to totally asymmetric models.
~
We will first present the problem, and the tools appropriate for solving it. We will then look at the simple case of the TASEP (which can also be found in \cite{Lazarescu2015} but is reproduced here, in less detail, for pedagogical purposes), where everything can be done fairly explicitly. We will generalise the proof to interacting inhomogeneous systems, where the calculations cannot be done in detail, but where we can still obtain bounds on $E(\mu)$ that will be sufficient for the main result to hold ; that subsection is particularly technical, and the reader interested in the results rather than the methods and proofs may jump directly to section \ref{IVc4}. Finally, we will give a few illustrative examples of cases where the result holds (\ref{IVd1}) or doesn't (\ref{IVd2}). The results and techniques presented in this section constitute the core of this paper, and our main new contribution to the study of dynamical phase transitions.
\subsection{Matrix perturbation and resolvant method}
\label{IVa}
Our starting point here is the deformed Markov matrix for a totally asymmetric model, with all the deformations set to $\varepsilon={\rm e}^{\frac{\mu}{L+1}}$ for simplicity, as in the previous section. In the limit $\mu\rightarrow-\infty$, we have $\varepsilon\rightarrow 0$. We can then write the deformed Markov matrix as
\begin{equation}
M_\mu=M^d+\varepsilon M^+.
\end{equation}
We need to extract the largest eigenvalue of this matrix. Unlike for $\mu\rightarrow\infty$, we cannot simply keep only the leading term, which is this time $M^d$, because it is constant, and therefore not sufficient to give us the behaviour of $g(j)$ through a Legendre transform. We will need to treat the non-diagonal part perturbatively in order to obtain the first correction to that constant term as well.
We will go into the details of how this can be obtained, but the reader familiar with perturbation theory may jump directly to the last paragraph of this sub-section, where these preliminary calculations are summarised.
~
The first step is to find the eigenspace of $M^d$ with the largest eigenvalue. Since $M^d$ is a diagonal matrix containing the opposite of the escape rate from every configuration (i.e. minus the sum of all out-going rates), that eigenspace will be the space of all configurations having the smallest possible escape rate. In other terms, the best states to be stuck into to produce an extremely small current are the ones that have the longest life-time.
Let us write that space as ${\cal S}$, the configurations that live in it as $\{{\cal C}_i^\star\}$, and the corresponding eigenvalue of $M^d$ as $-z_0$. Once we have that space, we need to find the largest eigenvalue of the perturbation $\varepsilon M^+$ restricted to it.
While that perturbative calculation can be performed without too much effort in the simplest cases (where the dominant eigenvalue of $M^d$ is not degenerate), we will need to cover all possible complications. This can be done in a very compact way using the so-called \textit{resolvent formalism} \cite{Fredholm1903}. Not only is it a clean and systematic way to deal with perturbative expansions in degenerate spaces, but it also provides with a rigorous definition of an effective dynamics within that space.
This formalism can be stated as follows: for a diagonalisable matrix $A$ with eigenvalues $E_i$ and eigenvectors $ |P_i\rangle$ and $\langle P_i|$, we may write
\begin{equation}\label{IV-3-Eproj}
\oint_{C}\frac{dz}{{\rm i} 2\pi}\frac{z}{z-A}=\sum\limits_{E_i\in C} E_i|P_i\rangle\langle P_i|
\end{equation}
where the sum is over the eigenvalues of $A$ which lie inside of the contour $C$.
Since we are only interested in the eigenvalues of $M_\mu$ which are close to the dominant eigenvalue $-z_0$ of $M^d$, we can apply this formula to $M_\mu$ with a small contour around $-z_0$ to get our effective biased Markov matrix, which we will write as $-z_0+M_{eff}$ (keeping the scalar term out for convenience). Shifting the origin of the complex plane to $-z_0$ for simplicity, we get
\begin{equation}\label{IV-3-Eproj2}
M_{eff}=\oint_{C}\frac{dz}{{\rm i} 2\pi}\frac{z}{z-z_0-M^d-\varepsilon M^+}
\end{equation}
where $C$ is a small circle centred at $0$. We can now expand this expression in powers of $\varepsilon$:
\begin{equation}\label{MeffPath}
M_{eff}=\oint_{C}\frac{dz}{{\rm i} 2\pi} \sum\limits_{k=0}^{\infty}\frac{z}{z-(M^d+z_0)}\Bigl(M^+ \frac{1}{z-(M^d+z_0)} \Bigr)^k \varepsilon ^k
\end{equation}
which is a sum over paths of length $k$, with transitions given by $M^+$ and a configuration weight given by $(z-M^d-z_0)^{-1}$. We see that, if $C$ is small enough to contain only the poles of $(z-M^d-z_0)^{-1}$ which are at $0$, the only terms which contribute to the integral (i.e. that give first order poles which yield non-zero residues) are those for which $M^d$ is taken at $-z_0$ at least twice. In other terms, they are the paths which go through at least two of the ${\cal C}_i^\star$'s.
Moreover, since we are only interested in the leading term in the largest eigenvalue of $M_{eff}$, the corrections to the eigenvectors will not be relevant, so that $M_{eff}$ is equivalent to its projection onto ${\cal S}$. This means that we can restrict the expression (\ref{MeffPath}) to paths that start and end in ${\cal S}$, from some ${\cal C}_i^\star$ to ${\cal C}_j^\star$, accounting for the previous requirement.
Finally, notice that the weight of each of those paths is proportional to $\varepsilon$ to the power of the number of jumps performed, with a pre-factor which is the ratio of all the jump rates used on that path over all the escape rates of the intermediate configurations minus $z_0$. Since we are only interested in the leading order in $\varepsilon$, we only need to consider the paths with the least number of jumps between the initial and final configuration. The corresponding entry in $M_{eff}$ will have a pre-factor, being the sum of the ones for each of the paths that have that minimal length, and which we can calculate in the simplest cases, but which will ultimately be inessential to our result.
~
Let us summarise this first section before getting into specific calculations. If $M^d$ has a highest eigenvalue $-z_0$ with eigenspace ${\cal S}$ generated by configurations $\{{\cal C}_i^\star\}$, then the highest eigenvalue of $M_\mu$ is given by $-z_0$ plus a first correction which is the largest eigenvalue of a matrix $M_{eff}$ with entries $A_{ij}\varepsilon^{d_{ij}}$, where $d_{ij}$ is the smallest number of jumps connecting ${\cal C}_j^\star$ to ${\cal C}_i^\star$, and $A_{ij}$ is a numerical pre-factor which can be obtained from expression (\ref{MeffPath}) although it will not be necessary.
\subsection{TASEP conditioned on low current}
\label{IVb}
We first look at the simple case of the regular TASEP, with all bulk rates equal to $p$, and boundary rates $p_0$ and $p_L$. It is customary to consider $p_0\leq p$ and $p_L\leq p$, which does not restrict the behaviour of the system by much. We will focus on those case and merely comment on the remaining ones, which are covered in all generality by section \ref{IVc}.
~
For this model, we can easily narrow down the set of candidates for ${\cal S}$: since all the transition rates are independent (i.e. the rate of a jump does not depend on which other jumps are possible), the escape rate of a state with several possible jumps is the sum of escape rates of states which have only one possible jump, and is therefore larger than each one of those. We only need to consider states with a single allowed jump, which is to say states that are completely full up to a given site and then completely empty.
We then see that there are three qualitatively different situations (fig.-\ref{fig-DiagLC}):
\begin{itemize}
\item if $p_0<p_L$ and $p_0<p$, then the best configuration is empty ($\tau_i=0$ for all $i$'s), with $z_0=p_0$. If $p_L<p_0$ and $p_L<p$, we have the same in reverse: the best configuration is full ($\tau_i=1$ for all $i$'s) and $z_0=p_L$ (those two first cases are symmetric to one another, and we will only be considering the first one) ;
\item if $p_0=p_L<p$, then $z_0=p_0$, and we have two competing configurations: empty or full ;
\item if $p_0\geq p$ and $p_L\geq p$, any configuration with a block of $1$'s followed by a block of $0$'s has an eigenvalue of $-z_0=-p$, which is the highest, except possibly the completely full and empty configurations depending on whether those inequalities are strict or saturated ; all those situations being essentially identical in the large size limit, we will focus on $p_0=p_L=p$, in which case there are $L+1$ states in ${\cal S}$.
\end{itemize}
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.4\textwidth]{PhaseDiagLC.pdf}
\caption{Phase diagram of the open ASEP for very low current. The mean density profiles are represented in red the insets. The profiles in orange are the individual configurations which compose the steady state.}
\label{fig-DiagLC}
\end{center}
\end{figure}
We will now construct $M_{eff}$ in each of these cases, and analyse its largest eigenvalue.
\subsubsection{Empty/full phases}
\label{IV-1-a}
This first case, where $z_0=p_0 <p_L$, is fairly straightforward: ${\cal S}$ contains only the empty configuration, and $M_{eff}$ has only one entry, equal to the weight of a path going from the empty configuration to itself. That is achieved in the least number of steps, which is $L+1$, by having one single particle go through the whole system from left to right. The contribution of $\varepsilon$ is therefore $\varepsilon^{L+1}={\rm e}^{\mu}$, and the numerical pre-factor is simply given by $\frac{p_0}{p-p_0}$.
This gives us
\begin{equation}\label{IV-3-EmuLD}
E(\mu)\sim -p_0+{\rm e}^{\mu}\frac{p_0}{p-p_0}
\end{equation}
and
\begin{equation}\label{IV-3-gjLD}\boxed{
g(j)=p_0+j\log(j)-j\bigl(\log(p_0/(p-p_0))+1\bigr).
}\end{equation}
Finally, note that in this case, the second largest eigenvalue is $-p_L$ for $\varepsilon\rightarrow 0$ (and corresponds to a completely full system), so that the gap between the first two eigenvalues of $M_\mu$ is finite and equal, at leading order, to $\Delta E=(p_L-p_0)$.
The corresponding results for $p_L<p_0$ (the `full' phase) can be obtained by exchanging $p_0$ with $p_L$.
\subsubsection{Empty-full line}
\label{IV-1-b}
We now consider the slightly more complex case where $p_0=p_L<p$. This time, there are two states with equal eigenvalues for $\mu=-\infty$:
\begin{equation}\label{IV-3-P0}
|0\rangle=|00\dots00\rangle~~~~{\rm and}~~~~|1\rangle=|11\dots11\rangle.
\end{equation}
As in the previous case, the diagonal part of $M_{eff}$ is given by the shortest paths going from these configurations to themselves, with the same weight as before (them being equal to each other because of the particle-hole symmetry). For the off-diagonal elements, we have to consider the shortest way to go from $|0\rangle$ to $|1\rangle$, or the opposite. This means completely filling or emptying the system, which can be done in $L(L+1)/2$ steps, but in this case the pre-factor contains contributions from many paths and is not straightforward to calculate. We therefore have something of the form
\begin{equation}
M_{eff}=\begin{bmatrix} {\rm e}^{\mu}\frac{p_0}{p-p_0} &X {\rm e}^{\frac{L}{2}\mu}\\ X {\rm e}^{\frac{L}{2}\mu} &{\rm e}^{\mu}\frac{p_0}{p-p_0} \end{bmatrix}
\end{equation}
where $X$ is said pre-factor.
From this, we see that the difference between the two highest eigenvalues is of order $\varepsilon^{L(L+1)/2}={\rm e}^{\frac{L}{2}\mu}$. For symmetry reasons, the main eigenvector is then $\frac{1}{2}(|0\rangle+|1\rangle)$, and the second one is $\frac{1}{2}(|0\rangle-|1\rangle)$. The leading terms of the largest eigenvalue are the same as before:
\begin{equation}\label{IV-3-EmuLDHD}
E(\mu)\sim -p_0+{\rm e}^{\mu}\frac{p_0}{p-p_0}
\end{equation}
and
\begin{equation}\label{IV-3-gjLDHD}\boxed{
g(j)=p_0+j\log(j)-j\bigl(\log(p_0/(p-p_0))+1\bigr)
}\end{equation}
but this time, the gap behaves as $\Delta E\sim{\rm e}^{\frac{L}{2}\mu}$.
\subsubsection{Anti-shock zone}
\label{IV-1-c}
For the last case, where all the jumping rates are equal ($p_0=p_L=p$), we find $L+1$ states with an eigenvalue equal to $-p$ for $\mu=-\infty$. Those states are given by $|k\rangle=|\{1\}_{(k)} \{0\}_{(L-k)}\rangle$, i.e. configurations made of a block of $1$'s followed by a block of $0$'s. Those are called \textit{anti-shocks}, being symmetric to the usual shocks observed in the steady state of the TASEP for $p_0=p_L\leq p/2$, which have a low density region followed by a high density one.
Using the resolvent formalism, we find:
\begin{align}
\langle k| M_{eff} |k\rangle&\sim\varepsilon^{L+1},\\
\langle k+1| M_{eff} |k\rangle&\sim\varepsilon^{k+1},\\
\langle k-1| M_{eff} |k\rangle&\sim\varepsilon^{L-k+1},
\end{align}
as well as terms of the type
\begin{align}
\langle k+2| M_{eff} |k\rangle&\sim X\varepsilon^{2k+3},\\
\langle k-2| M_{eff} |k\rangle&\sim Y\varepsilon^{2L-2k+3},\\
\langle k+3| M_{eff} |k\rangle&\sim Z\varepsilon^{3k+6},
\end{align}
and so on. We can check those last terms to be of sub-leading order in $E(\mu)$, and we will neglect them right away, which will allow us to complete our calculation. A more rigorous approach will be taken in section \ref{IVc}.
We are left with
\begin{equation}
M_{eff}=p\varepsilon^{L+1}+p\sum\limits_{k=1}^{L}\Bigl(\varepsilon^{k}|k\rangle\langle k-1|+\varepsilon^{L-k+1}|k-1\rangle\langle k|\Bigr).
\end{equation}
This matrix is similar to
\begin{equation}
M_{eff}=p\varepsilon^{L+1}+p\varepsilon^{(L+1)/2}\sum\limits_{k=1}^{L}\Bigl(|k\rangle\langle k-1|+|k-1\rangle\langle k|\Bigr)
\end{equation}
and can be diagonalised exactly (see \cite{Lazarescu2015} for more details). This time the dominant contribution to the largest eigenvalue turns out to come from the non-diagonal part, and be equal to $-2p\varepsilon^{(L+1)/2}\cos(\pi/(L+2))$, which yields
\begin{equation}\label{IV-3-EER-}
E(\mu)\sim-p+2p~{\rm e}^{\mu/2}
\end{equation}
and
\begin{equation}\label{IV-3-gjLDc}\boxed{
g(j)=p+2j\log(j)-2j.
}\end{equation}
In this case, the gap is equal to
\begin{equation}
\Delta E=2p\varepsilon^{(L+1)/2}\Bigl(\cos\bigl(\pi/(L+2)\bigr)-\cos\bigl(2 \pi/(L+2)\bigr)\Bigr)\sim \frac{3p\pi^2}{L^2}{\rm e}^{\mu/2}.
\end{equation}
The cases where $p_0>p$ and/or $p_L>p$ are almost identical, with the first state $|0\rangle$ and/or the last state $|L\rangle$ removed, and show the same large-size behaviour.
\subsection{Interacting inhomogeneous TASEP}
\label{IVc}
In this section, which contains the main new result of the present paper, we generalise the ones obtained for the standard TASEP to the inhomogeneous and interacting processes defined in section \ref{II}. That result can be stated as follows: in the limit of small currents ($\mu\rightarrow-\infty$), the first correction to the generating function of the cumulants of the current scales exponentially with $\mu$, with a rate that does not vanish in the limit of large sizes $L\rightarrow\infty$ (as it does in the high current limit):
\begin{equation}
\exists \{A,B\}>0: ~~~~ E(\mu)+z_0\sim{\rm e}^{B\mu}\ll {\rm e}^{A\mu}~~~~{\rm for}~~~~\mu\rightarrow -\infty,~~L\rightarrow\infty
\end{equation}
where $B$ can depend on $L$ but not $A$.
The fact that it scales exponentially with $\mu$ simply comes from it being an eigenvalue of a finite matrix $M_{eff}$ whose elements are powers of $\varepsilon$ (with unimportant numerical pre-factors). The proof of the bound can then be achieved in three steps:
\begin{itemize}
\item we will first show that, under certain assumptions, the states in $\mathcal{S}$, which have the longest lifetime, are similar to anti-shocks, with a region transiting from completely full to completely empty which can have any occupancy but whose size cannot grow with $L$ ;
\item we will then show that a simple cycle of such states takes a number of steps that is at least one order of $L$ larger than the number of states in the cycle ;
\item finally, we will use that bound to estimate the principal minors of $M_{eff}$, in order to show that the leading power of $\varepsilon$ in the largest eigenvalue of $M_{eff}$ is linear in $L$, which will complete the proof.
\end{itemize}
Note that all of the estimates we will be making are broad enough to account for the worst cases, but can certainly be made more precise for specific models.
\subsubsection{Longest-lived states}
\label{IVc1}
We will first show that all the states in $\mathcal{S}$ are of the form $|\{1\}_{(k^-)} \{\tau_i\}_{(k^+-k^-)}\{0\}_{(L-k^+)}\rangle$ with $k^+-k^-\leq K$ where $K$ is independent of $L$, i.e. anti-shocks with a maximal width $K$. In this expression, $k^-$ is the site of the first particle that can jump, and $k^+$ that of the last one (it is assumed that site $k^-+1$ is empty and that site $k^+$ is occupied, unless $k^-=k^+$).
In order to prove that first step, we need to make two assumptions:
\begin{itemize}
\item first, that the values of the inhomogeneous rates do not scale with the size of the system, i.e. that the values of $p_i$ are bounded on both sides for any value of $L$:
\begin{equation}\label{pbound}
\exists \{p_{min},p_{max}\},\forall L,\forall i: ~~~~0< p_{min}<p_i<p_{max}
\end{equation}
\item secondly, that the potential $V$ is such that any difference of $V$ over a transition (i.e. $V(\mathcal{C}')-V(\mathcal{C})$ if $\mathcal{C}\sim \mathcal{C}'$) is bounded on both sides:
\begin{equation}\label{vbound}
\exists \{\delta V_{min},\delta V_{max}\},\forall L,\forall \mathcal{C}\sim \mathcal{C}':~~~~2\delta V_{min}< V(\mathcal{C}')-V(\mathcal{C})< 2\delta V_{max},
\end{equation}
and that it is local, in the sense that the potential difference for a jump between sites $i$ and $i+1$ doesn't depend strongly on the part of the configuration that is far enough from the jump:
\begin{align}\label{vlocal}
\forall \alpha> 0, &~\exists~l_{\alpha}, \forall L, \forall i,\forall \mathcal{C}_1\sim \mathcal{C}_1',\forall \mathcal{C}_2\sim \mathcal{C}_2':\\
&\mathcal{C}_1|_{[i-l_\alpha,i+1+l_\alpha]}=\mathcal{C}_2|_{[i-l_\alpha,i+1+l_\alpha]}~{\rm and}~\mathcal{C}'_1|_{[i-l_\alpha,i+1+l_\alpha]}=\mathcal{C}'_2|_{[i-l_\alpha,i+1+l_\alpha]}\nonumber\\
&~~~~~~\Rightarrow~~|V(\mathcal{C}_1')-V(\mathcal{C}_1)-V(\mathcal{C}_2')+V(\mathcal{C}_2)|<2\log(1+\alpha)\nonumber
\end{align}
where $\mathcal{C}|_{[i-l_\alpha,i+1+l_\alpha]}$ is the portion of $\mathcal{C}$ that is between sites $i-l_\alpha$ and $i+1+l_\alpha$, and the $2\log$ is there for later convenience ; these two conditions are for instance verified if $V$ is a short-range two-body potential, and are not verified for typical long-range potentials, but are less restrictive than that.
\end{itemize}
From assumptions (\ref{pbound}) and (\ref{vbound}), we get that the transition rates $w(\mathcal{C},\mathcal{C}')$ are bounded independently of $L$:
\begin{equation}
w_{min}=p_{min}~{\rm e}^{\delta V_{min}}<w(\mathcal{C},\mathcal{C}')<p_{max}~{\rm e}^{\delta V_{max}}=w_{max}.
\end{equation}
~~
We can then prove the following crucial statement: any state with the minimal escape rate $z_0$ has at most $\bigl\lfloor w_{max}/w_{min}\bigr\rfloor$ possible transitions. The proof is straightforward: if a longest-lived state $\mathcal{C}$ has $n_t$ possible transitions, then, from the left side of the previous inequality, its escape rate is larger than $n_t~w_{min}$ ; from the right side of the inequality, it also has to be smaller than $w_{max}$, because there are, for instance, states with only one transition whose escape rate is one single jump rate and which is smaller than that bound ; therefore
\begin{equation}
n_t< \frac{w_{max}}{w_{min}}.
\end{equation}
~~
Moreover, those $n_t$ transitions cannot be too far from each other, because of the locality assumption (\ref{vlocal}) which ensures that particles cannot stabilise each-other from afar. Consider for instance a configuration $\mathcal{C}'_1$ with two successive possible jumps involving particles at sites $i$ and $j$ such that $j-i>2l$. This means that the first of the two particles has more than $l$ holes in front of it, or that the second has more than $l$ particles behind it. Let us assume it is the former (but the latter can be treated in the exact same way). Consider also the configurations $\mathcal{C}_1^{(k)}$ resulting from one of the jumps, from site $k$ to $k+1$, the configuration $\mathcal{C}'_2$ which is the same as $\mathcal{C}'_1$ up to site $i$ and empty afterwards, and the configurations $\mathcal{C}_2^{(k)}$ resulting from the jump from site $k$ to $k+1$ in $\mathcal{C}'_2$ (cf. fig.\ref{fig-Distance}). $\mathcal{C}'_2$ has at most $n_t-1$ possible jumps.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.5\textwidth]{Distance.pdf}
\caption{Example of configurations $\mathcal{C}'_1$ and $\mathcal{C}'_2$ with $l=3$, $i=6$ and $j=13$. The area in the dashed red box represents the part of those configurations which could have a strong influence on the value of $w(\mathcal{C}_1^{(6)},\mathcal{C}'_1)$ or $w(\mathcal{C}_2^{(6)},\mathcal{C}'_2)$, and which is identical in both configurations.}
\label{fig-Distance}
\end{center}
\end{figure}
Choosing now $l=l_\alpha$ for $\alpha=(w_{min}/w_{max})^2$, we have, according to (\ref{vlocal}):
\begin{equation}
w(\mathcal{C}_2^{(k)},\mathcal{C}'_2)<(1+\alpha)w(\mathcal{C}_1^{(k)},\mathcal{C}'_1)~~~~{\rm for}~~~~k\leq i
\end{equation}
because $\mathcal{C}'_1$ and $\mathcal{C}'_2$ are identical up to a distance $l_\alpha$ from any transitions that they have in common. We can now compare the escape rates from $\mathcal{C}'_1$ and $\mathcal{C}'_2$:
\begin{equation}
\sum\limits_k w(\mathcal{C}_2^{(k)},\mathcal{C}'_2)<(1+\alpha)\sum\limits_{k\leq i} w(\mathcal{C}_1^{(k)},\mathcal{C}'_1)<\sum\limits_{k\leq i} w(\mathcal{C}_1^{(k)},\mathcal{C}'_1)+(n_t-1)~\alpha~ w_{max}<\sum\limits_{k\leq i} w(\mathcal{C}_1^{(k)},\mathcal{C}'_1)+w_{min}.
\end{equation}
The right-hand side is smaller than the escape rate from $\mathcal{C}'_1$, since there is at least one transition to the right of $i$. Therefore, it cannot be one of the longest-lived states, because $\mathcal{C}'_2$ has a strictly longer lifetime.
From this, we conclude that longest-lived states cannot have transitions that are further apart than $2 l_\alpha$ with $\alpha=(w_{min}/w_{max})^2$.
~~
Combining this with the previous result, we find that all the transitions from a longest-lived state have to be within a region of size $K=2 l_\alpha \bigl(\bigl\lfloor w_{max}/w_{min}\bigr\rfloor-1\bigr)$, which is independent of $L$. All the sites to the left of that region have to be full, and all those to the right have to be empty. This concludes this first part of the proof.
\subsubsection{Length of simple cycles in $\mathcal{S}$}
\label{IVc2}
For this second step, we will be estimating the minimal number of jumps performed along a simple cycle of $n$ states from $\mathcal{S}$: $\mathcal{C}_1\rightarrow\mathcal{C}_2\rightarrow\dots\rightarrow\mathcal{C}_n\rightarrow\mathcal{C}_1$, where no state is visited more than once. Let us write $d=\sum\limits_{i\rightarrow j} d_{ji}$ the total minimal number of steps along that cycle. As we saw in the previous section, the states in $\mathcal{S}$ can be indexed by the position $k^-$ of the first jump, the position $k^+$ of the last jump (with $k^+-k^-\leq K$), and the configuration $\{\tau_i\}$ of the sites in between. We will also write the number of particles in a state as $N_{\mathcal{C}}=k^-+\sum\tau_i$.
We want to show that there is a constant $A$ independent of $L$ such that $d>(L+1)~n~A$.
~~
Consider first a cycle of any length, however small. Since we are coming back to the initial state at the end, the total number of jumps has to be a multiple of $L+1$ (the jumps can be reordered so that every particle does an integral number of loops around the system before coming back to its initial position). Since at least one step is taken, we have then $d>L$. This is true in particular for cycles of length $1$, so that $d_{ii}>L$ for any $i$.
To obtain a bound on the number of steps of a larger cycle, we can first simplify the problem by noticing that any state $\mathcal{C}$ is a finite number of steps away from the totally ordered state with as many particles, which we will call $\mathcal{C}^{(o)}$, with $N_{\mathcal{C}}$ particless followed by $L-N_{\mathcal{C}}$ holes (i.e. the state that we called $|N_{\mathcal{C}}\rangle$ for the simple TASEP). In the worst case, it takes $K^2/4$ steps to go from $\mathcal{C}^{(o)}$ to $\mathcal{C}$, which happens if $K$ is even and if $\mathcal{C}$ is of the form $|\{1\}_{(k^-)} \{0\}_{(K/2)}\{1\}_{(K/2)}\{0\}_{(L-k^--K)}\rangle$. Considering now the number of steps from $\mathcal{C}_1^{(o)}$ to $\mathcal{C}_2^{(o)}$, we know that it is larger than that from $\mathcal{C}_1^{(o)}$ to $\mathcal{C}_1$ plus that from $\mathcal{C}_1$ to $\mathcal{C}_2$ plus that from $\mathcal{C}_2$ to $\mathcal{C}_2^{(o)}$. Writing $d_{ij}^{(o)}$ as the minimal number of steps from $\mathcal{C}_j^{(o)}$ to $\mathcal{C}_i^{(o)}$, we have therefore that
\begin{equation}
d_{ij}>d_{ij}^{(o)}-\frac{K^2}{2}.
\end{equation}
Moreover, depending on $N_{\mathcal{C}_1}$ and $N_{\mathcal{C}_2}$, it is straightforward to obtain $d_{21}^{(o)}$:
\begin{itemize}
\item if $N_{\mathcal{C}_1}>N_{\mathcal{C}_2}$, then the last $N_{\mathcal{C}_1}-N_{\mathcal{C}_2}$ particles have to leave the system through the right boundary, which can be done with
\begin{equation}
d_{21}^{(o)}=\frac{1}{2}(2L+1-N_{\mathcal{C}_1}-N_{\mathcal{C}_2})(N_{\mathcal{C}_1}-N_{\mathcal{C}_2}).
\end{equation}
\item if $N_{\mathcal{C}_1}<N_{\mathcal{C}_2}$, then the last $N_{\mathcal{C}_2}-N_{\mathcal{C}_1}$ particles have to enter the system from the left boundary, which can be done with
\begin{equation}
d_{21}^{(o)}=\frac{1}{2}(1+N_{\mathcal{C}_1}+N_{\mathcal{C}_2})(N_{\mathcal{C}_2}-N_{\mathcal{C}_1}).
\end{equation}
\end{itemize}
Note that in all cases, $d_{21}^{(o)}+d_{12}^{(o)}=(L+1)|N_{\mathcal{C}_2}-N_{\mathcal{C}_1}|$, so that
\begin{equation}
d_{21}+d_{12}>(L+1)|N_{\mathcal{C}_2}-N_{\mathcal{C}_1}|-K^2.
\end{equation}
It follows that, for a cycle with $N^-=\min[\{N_{\mathcal{C}_i}\}]$ and $N^+=\max[\{N_{\mathcal{C}_i}\}]$, the total number of steps has to be larger than the direct path back and forth between two states that realise those extrema, so that
\begin{equation}\label{dNN}\boxed{
d>(L+1)(N^+-N^-)-K^2.
}\end{equation}
One final remark to be made is that a certain value of $N_{\mathcal{C}}$ can correspond to at most $2^{K-1}$ different states from $\mathcal{S}$. This can be seen by considering a state with $k^-=N_{\mathcal{C}}-n$, with the other $n$ particles being confined to the $K-1$ sites following the first hole at $k^-+1$. There are at most ${K-1 \choose n}$ such states, for $n$ from $1$ to $K-1$, plus the state with $n=0$, which all sum up to $2^{K-1}$.
~~
Consider now a cycle of length $n>2^{K+1}$. From what we just saw, $N_{\mathcal{C}}$ takes at least $n~2^{-K+1}$ different values along the cycle, so that $N^+-N^->n~2^{-K+1}-1=4n~2^{-K-1}-1>n~2^{-K-1}+2$. Combining this with the previous inequality (\ref{dNN}) gives
\begin{equation}
d>(L+1)(n~2^{-K-1}+2)-K^2,
\end{equation}
which, for $L+1>K^2/2$, finally gives
\begin{equation}\boxed{
d>(L+1) ~n~A
}\end{equation}
with $A=2^{-K-1}$.
\subsubsection{Equivalent and bound for $E(\mu)$}
\label{IVc3}
We will now find a bound on the eigenvalues of $M_{eff}$ using the bounds we have on the weight of its cycles. This can be done by looking at the characteristic polynomial of $M_{eff}$:
\begin{equation}
P_\varepsilon(x)={\rm det}\bigl[x~\delta_{ij} -A_{ij}\varepsilon^{d_{ij}} \bigr]=\sum_{k=0}^{N} a_k~x^{N-k}
\end{equation}
where $N=|\mathcal{S}|$ and $a_0=1$. It is well known that $a_k$ can be expressed as a sum of the principal minors of size $k$ of $M_{eff}$, which is to say a sum of weights of all composite cycles from $M_{eff}$ of total length $k$. Each $a_k$ is therefore a polynomial in $\varepsilon$, the valuation (smallest exponent) of which we will write as $m_k$. Let us also define
\begin{equation}\label{mindn}
C= \min_{k:1..N}\biggl[\frac{m_{k}}{k}\biggr]=\min_{\rm cycles}\biggl[\frac{d}{n}\biggr],
\end{equation}
where, as in the previous section, $d$ is the number of steps in a cycle of length $n$. We have shown that $C>(L+1)A$.
Consider now the rescaled polynomial
\begin{equation}
Q_\varepsilon(x)=\varepsilon^{-N C} P_\varepsilon(\varepsilon^{C}x)=\sum_{k=0}^{N} a_k\varepsilon^{-kC}~x^{N-k}
\end{equation}
which has at least one finite coefficient $a_k\varepsilon^{-kC}$ ($k\neq 0$), all the others being infinitesimal in $\varepsilon$. The roots of this polynomial are therefore finite in the limit $\varepsilon\rightarrow 0$, and at least one of them is non-vanishing. It is not obvious that the highest root, which is the one we are interested in, is among those roots, as they could in principle be all negative. We will however see that it is the case here.
Consider the matrix $\tilde{M}_{eff}$ where only the entries that contribute to the cycles that realise the minimum (\ref{mindn}) are kept:
\begin{equation}
\tilde{M}_{eff}=\sum\limits_{i\sim j}A_{ij}\varepsilon^{d_{ij}},
\end{equation}
where $i\sim j$ means that the transition $j\rightarrow i$ is on at least one cycle such that $d=nC$. In particular, for transitions which can be done in the same number of steps through different paths, only the paths with the most intermediate states (the highest $k$) will be kept, which explains why some terms were sub-dominant in $M_{eff}$ in section \ref{IV-1-c}. By construction, the characteristic polynomial of $\tilde{M}_{eff}$ is $\varepsilon^{N C}Q_0(\varepsilon^{-C}x)$: we can obtain it by rescaling $P_\varepsilon$, taking $\varepsilon$ to $0$ so that only the finite terms survive, and finally taking it back to the original scaling. Consider also $\hat{M}_{eff}=\tilde{M}_{eff}\big|_{\varepsilon=1}$
\begin{equation}
\hat{M}_{eff}=\sum\limits_{i\sim j}A_{ij},
\end{equation}
whose characteristic polynomial is therefore $Q_0(x)$. Both $\hat{M}_{eff}$ and $\varepsilon^{-C}\tilde{M}_{eff}$ are diagonalisable, because all of their entries are on at least one cycle, and they have the same characteristic polynomial, from which we conclude that they are similar. In particular, the largest eigenvalue of $\varepsilon^{-C}\tilde{M}_{eff}$ is the same as that of $\hat{M}_{eff}$, which is a positive matrix. It is therefore strictly positive.
Moreover, the matrix $\hat{M}_{eff}$ contains all the information relative to the effective process at low current. We will describe that process in two simple cases in section \ref{IVd1} (fig.\ref{fig-Effective}).
~~
Going back to $M_{eff}$, we conclude that its largest eigenvalue scales as $\varepsilon^{C}\sim {\rm e}^{B\mu}$ with
\begin{equation}\boxed{
B=\frac{C}{L+1}\ll A
}\end{equation}
for $\mu\rightarrow-\infty$. This concludes our proof.
\subsubsection{Conclusion: asymptotics of the large deviation function of the currents}
\label{IVc4}
We conclude by re-stating the assumptions and the result from this section, and examining its consequence on the large deviation function of the current.
We have seen that, for a generalised TASEP with bounded inhomogeneities and a short-range potential, we can find a quantity $A$ independent of the size of the system such that
\begin{equation}
|E(\mu)+z_0|\ll {\rm e}^{A\mu}
\end{equation}
for $L\rightarrow\infty$ and $\mu\rightarrow-\infty$. Moreover, $E(\mu)+z_0$ is equivalent to the largest eigenvalue of a matrix $M_{eff}$ who behaves as
\begin{equation}\boxed{
E(\mu)+z_0\sim {\rm e}^{B\mu}
}\end{equation}
for $\mu\rightarrow-\infty$ with $B$ bounded as a function of $L$. The states in $\mathcal{S}$, which are the ones involved in the dynamics in that limit, are anti-shocks, of the form $|\{1\}_{(k^-)} \{\tau_i\}_{(k^+-k^-)}\{0\}_{(L-k^+)}\rangle$ with $k^+-k^-\leq K$ where the maximal width $K$ of the anti-shock is independent of $L$
The statements and proofs did not require any specific form for the inhomogeneous rates and the potential, other than conditions (\ref{pbound}-\ref{vlocal}), which means that there is no reason for $B$ to have a limit for $L\rightarrow\infty$. However, if $V$ is local and well-behaved and $p_i$ is slowly varying in space, we can expect that limit to exist.
Having then an equivalent of that form, we can deduce that
\begin{equation}\label{LowCurrtEquiv}\boxed{\boxed{
g(j)\sim z_0+B^{-1}j\log(j)
}}\end{equation}
which does not scale with the size of the system. This and the localised nature of the states in $\mathcal{S}$, which are the typical states occupied by the system at low current, are compatible with a hydrodynamic description of the fluctuating current \cite{Bodineau2006,Lazarescu2015}, where the cost of maintaining such a fluctuation in the system comes from one localised defect and hence is independent of $L$.
\subsection{Illustrative examples}
\label{IVd}
In this section, we give a few simple examples to illustrate our result beyond the excessively simple case of the TASEP. We start with a family of models with finite-range interaction for which the width $K$ of the anti-shock region can be tuned to any value, and the number of relevant states can be adjusted as well. We also explicitly build the effective dynamics $M_{eff}$ in the simplest non-trivial case. Finally, to illustrate the necessity of having local interactions, we give an example of a system with long-range interactions where our result does not hold.
\subsubsection{Anti-shock regime for finite-range interactions}
\label{IVd1}
We will see here how we can easily build models with a prescribed maximal width $K$ of the anti-shock and various numbers of longest-lived states in $\mathcal{S}$.
We consider a homogeneous system, with $p=1$ except for $p_0$ and $p_L$ at the boundaries, and a symmetric two-bodies interaction $V$ which we will write as
\begin{equation}
V(\{\tau_i\})=-2\sum\limits_{i>j}\log(a_{i-j})\tau_i \tau_j.
\end{equation}
Note that we count every pair of sites $(i,j)$ only once. To make sure that $V$ is short-range, we will take $a_k=1$ for $k>K$. With that notation, the transition rates from anti-shock states in the bulk of the system take a simple form, as seen on fig.\ref{fig-LongStates}: for a configuration with domain walls $10$ at positions $i_k$ and $01$ at positions $j_l$, the jump rate of the particle at position $j$ is given by
\begin{equation}
\frac{\prod\limits_{k}~~a_{i_k-j}}{a_1\prod\limits_{l, j_l\neq j}a_{j_l-j}}.
\end{equation}
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.72\textwidth]{LongStates.pdf}
\caption{Jump rates from the few simplest anti-shock states.}
\label{fig-LongStates}
\end{center}
\end{figure}
~
By choosing $a_k=k$, we get that every escape rate from states with anti-shocks of width less than $K$ is exactly $1$, and all other escape rates are larger than $1$ (the boundary rates also need to be tuned to insure that, which is straightforward). This surprising identity can be easily checked on the examples shown in fig.\ref{fig-LongStates}, and a formal proof can be found in appendix \ref{A2}. This choice of rates yields a number of longest-lived states of order $LK^2/2$. Considering, for instance, the cycle of states shown on fig.\ref{fig-Cycle}, we see that $C<(L+1)/2K$ (as defined in eq.(\ref{mindn})). The full effective process in this case is still quite complicated, so we will not go into more detail.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.72\textwidth]{Cycle.pdf}
\caption{Example of a cycle with $K=4$, $8$ states, and $L+1$ steps. The red dot indicates an arbitrary reference site $k$.}
\label{fig-Cycle}
\end{center}
\end{figure}
~
A simpler example is obtained by choosing $a_K=2$ and $a_k=1$ otherwise. In this case, only the anti-shocks of width $0$ or of width $K$ with two possible jumps have an escape rate equal to $1$, all the others being larger. We then have a number of longest-lived states of order $L(K-1)$, and an effective process with a structure given in fig.\ref{fig-Effective}. Note that, except for $K=2$, only three types of anti-shocks end up contributing to the effective dynamics, as all the others only contribute to sub-dominant terms in $\tilde{M}_{eff}$. In all cases, we find that $C=(L+1)/4$, so that $E(\mu)+1\sim {\rm e}^{\mu/4}$ and $g(j)\sim1+4j\log(j)$.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.72\textwidth]{EffectiveProcesses.pdf}
\caption{Structure of the effective process for $K=2$ (above) and $K=4$ (below). The red dot indicates an arbitrary reference site $k$. The full arrows indicate transitions which contribute to the effective process at leading order, while the dotted arrows indicate those of lower order, which are not taken into account to estimate the equivalent of $E(\mu)$.}
\label{fig-Effective}
\end{center}
\end{figure}
\subsubsection{Non-hydrodynamic behaviour for long-range interactions}
\label{IVd2}
We now exhibit an system with long-range interactions where our result is not valid, to illustrate the importance of that condition. We will construct that example step by step starting from the TASEP with $p_0=p_L=1$, changing one element at a time in order to obtain a model where there is a cycle $\mathcal{O}(L)$ of longest-lived states which makes only $\mathcal{O}(L)$ steps in total.
The simplest such cycle that one could think of is that of one-particle states $|k\rangle=|\{\delta_{i,k}\}\rangle$ plus the empty state $|0\rangle$, where one particle enter from the left, jumps through the whole system, and exits from the right. The escape rates for those states in the case of the TASEP are $1$ from $|0\rangle$ and $|1\rangle$, but $2$ from the other states. Moreover, all perfect (i.e. of width $0$) anti-shock states have an escape-rate of $1$ as well, which we don't want (except for $|0\rangle$ and $|1\rangle$). We need to correct those two issues.
First, we increase the escape-rate of unwanted anti-shocks by adding a repulsive nearest-neighbour interaction
\begin{equation}
V_A(\{\tau_i\})=2\log(A)\sum\limits_{i}\tau_i \tau_{i+1}
\end{equation}
with $A>2$, so that particles leaving a neighbour behind do it with a rate $A>2$, thus disqualifying those states. This has the adverse consequence of facilitating the first jumps from states $|010000...\rangle$ and $|1010000...\rangle$, giving them smaller escape-rates. We correct this by setting $p_1>A$ and $p_2>2-1/A$, so that those states and $|1\rangle$ now have an escape-rate larger than $2$. At this stage, all states $|k>2\rangle$ have an escape-rate of $2$, except for $|0\rangle$ which has an escape-rate of $1$, and all other states have an escape-rate of $2$ or more.
It only remains for us to set the transition rate for $|0\rangle\rightarrow|1\rangle$ to $2$ without modifying the other rates. That is equivalent to adding an interaction potential
\begin{equation}
V_0(\{\tau_i\})=-2\log(2)\prod\limits_{i=1}^{L}(1-\tau_i)
\end{equation}
which is non-zero only for the empty state $|0\rangle$. It is clear that this term does not satisfy the locality condition (\ref{vlocal}), which allows the escape rate from $|L\rangle$ to be the same as that from $|0\rangle$, even though the two possible jumps out of $|L\rangle$ are as far away from each-other as possible and one of them is the same as the jump out of $|0\rangle$.
It follows from that cycle of states that $C=(L+1)/(L-1)$, as defined in eq.(\ref{mindn}), and that $E(\mu)+z_0\sim {\rm e}^{\mu/(L-1)}$, so that
\begin{equation}\boxed{
g(j)\sim 2+Lj\log(j).
}\end{equation}
The factor $L$ means that the probability of observing a small current scales with the size of the system, as expected. This can be seen clearly on numerical evaluations of $\log(E(\mu)+z_0)$ for small system sizes. On fig.\ref{fig-LogE}, we plot that quantity for systems of various sizes and with or without the non-local interaction $V_0$. As we can see, it is independent of $L$ even for small negative values of $\mu$ when $V_0$ is not introduced, and depends strongly on $L$ when it is.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.5\textwidth]{TASEPLogE.pdf}
\caption{Numerical evaluations of $\log(E(\mu)+z_0)$ with $V_0$ (top curves) and without $V_0$ (bottom curves), for various system sizes: from bottom to top, $L=4$ (blue), $L=6$ (purple), $L=8$ (orange), $L=10$ (green).}
\label{fig-LogE}
\end{center}
\end{figure}
\newpage
\section{Illustration and discussion}
\label{V}
In this final section, we illustrate our results with a variety of numerical plots, and discuss a possible physical interpretation of the dynamical transition in terms of maximum hydrodynamic current as well as a possible connexion to the KPZ universality class.
Considering the broad class of models that we have been looking at, constrained only by (\ref{pbound}), (\ref{vbound}) and (\ref{vlocal}), we need to somewhat restrict ourselves in order to obtain something meaningful in the large size limit (i.e. an identifiable phase transition). Above all, that limit itself has to make sense, which restricts the sequence of models of increasing size (or possibly the sequence of ensembles of models of increasing size) that we may consider. We first have to distinguish between disordered models, for which $p_i$ and/or $V$ might be drawn from a distribution, and models with fixed parameters. We then have to define $p_i$ and $V$ so that a large size limit can be taken, which might involve taking $p_i$ to be a discretisation of a fixed smooth function $p(x)$, and $V$ to be a combination of simple short- or finite-range $n-$body interactions, with perhaps a slow space-dependence.
In this section, we will only be considering (and conjecturing about) homogeneous systems with simple short-range potentials, of which the ASEP is the simplest example (and we will use it as a guide throughout, with all the related results taken from \cite{Lazarescu2015}).
~
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.6\textwidth]{TASEPE2.pdf}
\caption{Rescaled real part of the spectrum of $M_{\mu}$ for a model of size $L=8$, with homogeneous jumps $p_i=1$ and next-to-nearest-neighbour interactions $a_2=2$. The colours are only there to help differentiate overlapping curves.}
\label{fig-E3}
\end{center}
\end{figure}
We start by simply looking at the qualitative behaviour of the spectrum $\{E_i\}$ of $M_\mu$ as a function of $\mu$. For this, we choose a simple case where all $p_i=1$, with next-to-nearest-neighbour interactions $a_2=2$ in the notation of section \ref{IVd1}, and a system size $L=8$. We plot, on fig.\ref{fig-E3}, the real part of the eigenvalues of $M_\mu/(1+{\rm e}^{\mu/(L+1)})$, where the rescaling is introduced so that those eigenvalues converge to a constant for $\mu\rightarrow\infty$ rather than diverge, which makes the plot clearer. We also plot on fig.\ref{fig-EEEE} the rescaled complex spectrum of the same model with $L=12$ for a few values of $\mu$.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.8\textwidth]{TASEPEEEE.pdf}
\caption{Rescaled complex spectra of $M_{\mu}$ for a model of size $L=12$, with homogeneous jumps $p_i=1$ and nearest-neighbour interactions $a_2=2$, at various values of $\mu$: $50$ (red, top left), $0$ (purple, top right), $-10$ (blue, bottom left) and $-20$ (blue, bottom right). Note that each plot has a different scale.}
\label{fig-EEEE}
\end{center}
\end{figure}
As we can clearly see, the aspect of the spectrum is quite different between negative and positive values of $\mu$. For $\mu\rightarrow\infty$, we have a Fermionic spectrum, where the eigenvalues are distributed according to the semi-circle law (as is clear on fig.\ref{fig-EEEE} ; the highest and lowest eigenvalues on fig.\ref{fig-E3} seem to be separated from the rest of the spectrum by a rather large gap, but this is due to the small system size and would not be the case in the large size limit). In the $\mu\rightarrow-\infty$ limit, on the other hand, we have a quasi-discrete real part of the spectrum where eigenvalues accumulate around specific values with a high degeneracy (this is helped by the fact that we chose an homogeneous system with a simple next-to-nearest-neighbour potential, in order to have a high degeneracy of escape rates even for a small size~; in the large size limit, we would observe such a high degeneracy even for more complex potentials and slowly-varying jump rates). The transition between these two regimes is clearly visible on fig.\ref{fig-E3} as an area dense with bifurcations and crossings, extending between $\mu\sim-10$ and $\mu\sim8$, and is an indication of the potential existence of a phase transition in the large size limit, even though for such a small size no sign of a non-analiticity can be yet observed for the highest eigenvalue $E(\mu)$. It is unclear how that area itself behaves in the large size limit, but it would be reasonable to expect that it converge to a single non-analiticity at $\mu=0$ under the proper rescaling.
~
That behaviour is similar to that of the open ASEP, as analysed in \cite{Bodineau2006,Lazarescu2015}. In the low current regime, we have a \textit{hydrodynamic} phase where the large deviation function of the current is consistent with applying the macroscopic fluctuation theory (MFT, \cite{Bertini2007}) or the additivity principle \cite{PhysRevLett.92.180601} to the continuous limit of the model, with a diffusion constant of order $L^{-1}$, which is to say that the long time large deviation function $g(j,\rho)$ of the current $j$ and mean local density $\rho(x)=\langle\tau_{\lfloor xL\rfloor}\rangle$ has the form
\begin{equation}\label{gjr}
g(j,\rho)=L\int_0^1\frac{\bigl[j-J^\star(\rho)+\frac{D(\rho)}{L}\nabla\rho\bigr]^2}{2\sigma(\rho)}dx
\end{equation}
with boundary conditions $\rho(0)=\rho_a$ and $\rho(1)=\rho_b$ and can be minimised over $\rho$ in order to obtain $g(j)$. In the case of the ASEP, we have that $\sigma(\rho)=\rho(1-\rho)$, $D=\frac{p}{2}$ and $J^\star(\rho)=p~\sigma(\rho)$ (this proportionality being a far-from-equilibrium version of Einstein's relation, which might be specific to the ASEP and a few other models and is not to be expected in general). Moreover, the minimisation produces not only the most probable density $\rho^\star$, which is associated to $E(\mu)$ through the Legendre transform of $g(j,\rho^\star)$, but also a family of metastable states $\rho_i$, which turn out to be related to the other eigenvalues $E_i(\mu)$ in the same way (c.f. \cite{Lazarescu} for more details). The structure of $\{E_i(\mu)\}$ thus obtained is similar to what is observed on fig.\ref{fig-E3} for $\mu$ low enough. The first group of eigenvalues then account for the effective process discussed in section \ref{IVa} at lowest order, and subsequent groups account for higher orders. Moreover, those eigenvalues undergo many first-order phase transitions as the boundary parameters $\rho_a$ and $\rho_b$ are varied.
In the high current regime, as we have seen, the behaviour of the ASEP is exactly the same as that of the inhomogeneous interacting versions that we have considered: we find a \textit{correlated} free Fermion phase characterised by an average density of $\frac{1}{2}$ with long-range correlations, which is sometimes called a hyperuniform phase \cite{Jack2015}, and which can be described by a conformal field theory \cite{Karevski2016}.
Between those two regimes is a dynamical phase transition, which occurs when the fluctuating current $j$ goes through the critical value $j=\frac{p}{4}$. That transition is of second order, and can be observed directly on the minimisation of eq.(\ref{gjr}), since a value $j>\frac{p}{4}$ yields a minimum of order $L$ (the integrand is finite for every $x$) whereas for $j<\frac{p}{4}$ it is of order $1$ (the integrand is non-zero only on a fraction of order $L^{-1}$ of $[0,1]$). However, the behaviour of $g(j)$ for $j\rightarrow \frac{p}{4}^+$ is wrongly predicted by the MFT as $(j-\frac{p}{4})^2$, whereas exact calculations using integrability methods \cite{Lazarescu2014,Lazarescu2015} yield instead a true scaling as $(j-\frac{p}{4})^{\frac{5}{2}}$.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.8\textwidth]{PhaseSigma.pdf}
\caption{a) Dynamical phase diagram of the current for the open ASEP, with respect to the boundary densities $\rho_a$ and $\rho_b$, and the biasing parameter $\mu$. The green dashed square corresponds to the static phase diagram of the current in the stationary state $\mu=0$. The hydrodynamic regime (top) is highlighted in blue and the correlated one (bottom) in red, with the second-order dynamical phase transition in between in purple. \newline b) Density/current diagram for a hypothetical model with finite-range interactions. The black curve $J^\star(\rho)$ corresponds to the hydrodynamic (mean-field) current associated to a uniform density $\rho$. An example of a trajectory $(j(\mu),\rho(\mu))$ for fixed boundary densities is represented with colours corresponding to those on the phase diagram.}
\label{fig-PhaseSigma}
\end{center}
\end{figure}
The dynamical phase diagram of $E(\mu)$ is shown on fig.\ref{fig-PhaseSigma}.a, as a function of the boundary parameters $\rho_a$ and $\rho_b$. The purple surface corresponds to the hydrodynamic/correlated phase transition (the distinction between the light and dark purple zones will be made later). The region above, marked in blue, is the hydrodynamic phase corresponding to $j<\frac{p}{4}$ and contains a variety of first-order phase transitions between different typical states $\rho^\star$. The region below, marked in red, is the correlated phase, and does not contain other phase transitions as far as we know, although a precise description of the bulk of the phase is yet to be obtained.
~
In light of these similarities, we are led to give the same interpretation to the physical origin of the dynamical phase transition as in \cite{Lazarescu2015} for the ASEP: for low enough currents, the system behaves, in the large size limit, in accordance with a Langevin equation with conserved noise
\begin{equation}\label{LangevinGas}
{\rm d}_t \rho=-\nabla j~~~~{\rm with}~~~~j=J^\star(\rho)-\frac{D(\rho)}{L}\nabla\rho+\sqrt{\sigma(\rho)}\xi,
\end{equation}
where $\xi$ is a Gaussian white noise. The so-called transport coefficients $J^\star(\rho)$, $D(\rho)$ and $\sigma(\rho)$ are in general very difficult to obtain from the microscopic process \cite{Arita2014,Arita2016}, but one important property of $J^\star(\rho)$ can be deduced from the exclusion property alone: it vanishes at $\rho=0$ and at $\rho=1$, since those densities cannot sustain any current, which means that $J^\star(\rho)$ is bounded from above, with a maximum value $J^\star_{\mathrm{max}}$.
For a fluctuating current smaller than that maximum (as well as for the steady state, which lies somewhere along $j=J^\star(\rho)$), there are hydrodynamic states which produce it through localised defects at a cost $g(j)$ which doesn't grow with $L$. In the limit of very low currents, for instance, the typical densities are $0$ and $1$, and the localised defects in question are the anti-shocks that bridge those two densities.
For a current larger than that maximum, the best hydrodynamic states have a cost proportional to $L$, but it becomes more efficient to introduce correlations everywhere in the system, producing a hyperuniform state which is optimised to produce a large current by forcing the particles and holes to alternate more than at random. The dynamical phase transition therefore corresponds to the appearance of correlations in the system when the current is pushed beyond its hydrodynamic regime.
This is illustrated schematically on fig.\ref{fig-PhaseSigma}.b, where a hypothetical $J^\star$ is represented (as would appear for instance with nearest-neighbour interactions, in a KLS-type model \cite{PhysRevB.28.1655,Popkov1999}), along with a trajectory $(j(\mu),\rho(\mu))$ obtained by varying $\mu$ from $-\infty$ to $\infty$ for certain fixed boundary densities $\rho_a$ and $\rho_b$ (which is to say a vertical line from the diagram on fig.\ref{fig-PhaseSigma}.a): the blue curve represents the fluctuating hydrodynamic regime, from a completely empty state $\rho=0$ to a state (purple dot) sustaining the maximal hydrodynamic current (purple dashed line), passing through the typical stationary state (blue dot) at $\mu=0$ ; the red dotted curve corresponds to the correlated regime for currents larger than that maximum.
A more recent study \cite{Baek2016} makes a very similar conjecture for boundary-driven systems, consistent with ours: for systems obeying Einstein's relation $J^\star\propto D\sigma$, one can observe a hydrodynamic behaviour for currents even far from equilibrium, as long as they are lower than the maximum of $J^\star$ if that maximum exists. When $J^\star$ is unbounded, as for some zero-range processes, no such dynamical phase transition can be found.
~
Finally, we should note that the hydrodynamic/correlated dynamical phase transition for the ASEP is closely related to the appearance of Tracy-Widom distributions in the statistics of the relaxation of $j$ towards stationarity, as seen in models from the KPZ universality class \cite{Spohn2016,Corwin2016}. The context there is quite different: the models are of infinite size and observed at long times, whereas we put ourselves at infinite time and then increase the size. However, it would make sense that both approaches be at least somewhat connected, and a first confirmation of this is the fact that the phase diagrams of small deviations around the typical currents are identical (although the current in question and the two parameters are not exactly the same in both cases).
More precisely, we compare:
\begin{itemize}
\item 1) the vicinity of the central slice (marked by a green dashed line) of fig.\ref{fig-PhaseSigma}.a, which corresponds to small fluctuations of the time-averaged stationary current of the open ASEP with boundary densities $\rho_a$ and $\rho_b$, in the large size limit (we refer to section VI.B.5 of \cite{Lazarescu2015} for the names and full description of the phases) ;
\item 2) fig.2 from \cite{BenArous2011}, which corresponds to the fluctuations of the time-integrated current $tj$ across the middle bond of an ASEP on an infinite line with a product state initial condition with densities $\rho_a$ on the left ($i<0$) and $\rho_b$ on the right ($i>0$), in the large time $t$ limit, on a certain time-scale (which is $t^{\frac{1}{2}}$ outside of the maximal current phase $(\rho_a>\frac{1}{2},\rho_b<\frac{1}{2})$, and $t^{\frac{1}{3}}$ inside of it, including the boundaries).
\end{itemize}
The comparison is quite straightforward:
\begin{itemize}
\item the parts of 1) which do not sit at a phase transition, i.e. the LD phase $(\rho_a<\frac{1}{2},1-\rho_b>\rho_a)$ and HD phase $(1-\rho_b<\frac{1}{2},\rho_a>1-\rho_b$), correspond to Gaussian fluctuations on a diffusive time-scale $t^{\frac{1}{2}}$ in 2) ;
\item the line which corresponds to a first order phase transition between two states in 1), i.e. the S line $(\rho_a<\frac{1}{2},1-\rho_b=\rho_a)$, corresponds to the maximum of two Gaussian distributions on a diffusive time-scale $t^{\frac{1}{2}}$ in 2) ;
\item most importantly, the parts of 1) which sit at the hydrodynamic/correlated dynamical phase transition, i.e. the MC phase $(\rho_a>\frac{1}{2},1-\rho_b<\frac{1}{2})$ and its boundaries, correspond to three different Tracy-Widom distributions on a subdiffusive time-scale $t^{\frac{1}{3}}$ in 2).
\end{itemize}
On this last point, in both cases, the special statistics of the current arise from the fact that it is much more difficult for the system to accommodate for currents higher than average than for lower ones. It is then natural to wonder which features of these statistics are universal, and which are model-dependent, or even parameter-dependent within the same model. Considering even the standard ASEP, the transition surface naturally splits in four areas, corresponding to the four sectors $\rho_{\{a,b\}}\lessgtr\frac{1}{2}$, of which we have mentioned only one so far, namely $(\rho_a>\frac{1}{2},\rho_b<\frac{1}{2})$. It turns out that, because of a very special symmetry of the model (proven in appendix \ref{A3}, extending on a result from \cite{Torkaman2015}), the sector $(\rho_a<\frac{1}{2},\rho_b>\frac{1}{2})$ has the same properties. On the other hand, the line $(\rho_a=\frac{1}{2},\rho_b>\frac{1}{2})$ (which is, mysteriously, equivalent to a half-filled periodic system \cite{derrida1999universal}), on the boundary of the MC phase, and the point $(\rho_a=\frac{1}{2},\rho_b=\frac{1}{2})$, have slightly different properties (same exponents but different pre-factors and distributions), and we expect the sector $(\rho_a>\frac{1}{2},\rho_b>\frac{1}{2})$ and its symmetric to make for yet another universality subclass.
All that being said, we expect models with more complex hydrodynamic currents $J^\star$ to all be in the same universality class. However, situations where several densities produce the same maximal current (for instance a version of fig.\ref{fig-PhaseSigma}.b with a symmetric $J^\star$) would most probably show different behaviours in the appropriate regimes, and there is undoubtedly much more to be understood about the hydrodynamic/correlated dynamical phase transition. In particular, we should be able to find some correspondence between the exponents and pre-factors found around the transition in the stationary case and those obtained in the infinite volume case from the perspective of so-called third order phase transitions \cite{Majumdar2014,LeDoussal2016}, which we believe to be the long-time relaxation equivalent to our stationary dynamical phase transition.
\newpage
\section{Conclusion}
In this paper, we have analysed the large deviations of the current $g(j)$ in extreme limits for a very general class of models based on the TASEP, with inhomogeneities and short-range interactions. After defining the models and formalism relevant to our endeavour, we reduced the problem to that of obtaining the approximate behaviour of the largest eigenvalue $E(\mu)$ of the Markov matrix deformed by a counting parameter $\mu$, in the limits of $\mu\rightarrow\pm\infty$.
In the $\mu\rightarrow\infty$ limit, corresponding to a high current, the deformed Markov matrix is equivalent to a free Fermions Hamiltonian and depends only trivially on the disorder and interactions. We found that $g(j)$ is proportional to the size of the system $L$ and that the typical states are Coulomb gases, with an average density equal to $\frac{1}{2}$ and strong nearest-neighbour anti-correlations, resulting in a \textit{correlated} phase. In the $\mu\rightarrow -\infty$ limit, corresponding to a low current, the deformed Markov matrix is a high-order perturbation of a diagonal matrix. We were able to show that the typical states are of the anti-shock type, with a block of particles followed by a block of holes, separated by an area no larger than a certain constant, and in particular not growing with the system size $L$. Moreover, $g(j)$ was also shown not to scale with $L$, consistently with being in a \textit{hydrodynamic} phase. Those two very different limits, and in particular the different scaling of $g(j)$ with respect to $L$, indicating the possible existence of a dynamical phase transition in between.
We then looked at a specific model for illustration, with homogeneous rates and next-to-nearest-neighbour interactions. We saw how the transition between a hydrodynamic regime and a correlated one manifested itself on the whole spectrum of the deformed Markov matrix, and interpreted that transition in terms of $j$ pushing beyond the maximal hydrodynamic current $J^\star_{\mathrm{max}}$ at the price of introducing correlations in the system. We also showed a connexion between that dynamical phase transition and the appearance of Tracy-Widom distributions, as for all models in the KPZ universality class, in the infinite volume case.
~~
These results are a quite encouraging step towards understanding the large size limit of interacting particle models far from equilibrium, especially because of the non-solvable nature of the models we have considered: the behaviour that we have described comes only from the geometric structure of the models (i.e. a lattice gas, with physical inhomogeneities and interactions), and not from a very special algebraic structure of the Markov matrix. This makes it likely that our methods could be applied for many other models with different components or geometries (it would for instance be quite straightforward to apply them to the totally asymmetric partial exclusion process, where the number of particle per site is not limited to one but to some integer \cite{Arita2014}, or to a multispecies TASEP \cite{Crampe2016,Crampe}). Of course, we were only able to perform calculations in extreme limits where the problem is greatly simplified, and it is more than likely that we will have to rely at least partly on numerics if we want to go further, for instance in describing the dynamical transition itself rather than the phases on each side of it. Luckily, a lot of progress is being made on the variety and effectiveness of the numerical methods available for that purpose \cite{Gorissen2009,giardina2011simulating,Espigares2013,Nemoto2016}.
There is of course a lot more to be done on the subject. One of the outstanding problems in this context, for instance, is to obtain the hydrodynamic transport coefficients, as seen in eq.(\ref{LangevinGas}), from the microscopic dynamics of the systems, even in an equilibrium setting \cite{Arita2014,Arita2016}. Once those coefficients are known, and the MFT is assumed to be valid, one can analyse the full spectrum of the model in the bulk-driven case, and identify the effective dynamics to any order (this will be the subject of a future work \cite{Lazarescu}), as well as relate them to the relaxation paths and the pseudopotential which have already been studied \cite{Bahadoran2010}. On the other side of the transition, a good description of the correlated phase remains to be found except in the infinite current limit \cite{Karevski2016}, and in particular the appropriate order parameters have not been clearly identified, although it seems likely that they would consist of correlation functions (as the local density becomes irrelevant in that phase, and the correlations grow from $0$ to a finite value). As for the transition itself, we have already mentioned that different situations will give rise to different pre-factors to the scaling of $g(j)$, related to different sub-classes within KPZ universality, and a precise classification of those does not exist yet as far as we know. Finally, we have focused on models with well-behaved jump rates and potentials, but our general result holds for disordered systems as well, although the consequences are more mysterious in that case and remain to be analysed. It is for instance unclear under which conditions we can expect a hydrodynamic phase to survive, as it does for dilute disorder \cite{Bahadoran2016}.
~
We conclude by examining in more detail a few of the natural extensions of our results as well as how they relate to other existing works.
\begin{itemize}
\item \textit{Partially asymmetric models}: In order to take the $j\rightarrow0$ limit through $\mu\rightarrow-\infty$, we had to restrict ourselves to totally asymmetric models. It is quite likely that the phenomenology of partially asymmetric models would be exactly the same, as was shown for the ASEP in \cite{Lazarescu2015}, where the only difference is a simple factor $(p-q)$ which rescales the current (meaning that, by some miracle, Einstein's relation $J^\star\propto D\sigma$ is valid even far from equilibrium), although in general we would expect $J^\star$ to depend on the backwards rates in a more complicated way. However, extending our method to that case does not seem easy: first of all, the expansion around $j=0$ now has to be done at a finite value of $\mu$, which means that the deformed Markov matrix is not a perturbation of a diagonal matrix any more ; and secondly, the possibility of backward jumps makes eq.(\ref{MeffPath}) much more complex, since every term now contains an infinity of paths that needs to be re-summed.
\item \textit{Close-to-equilibrium systems}: The case of boundary-driven or weakly bulk-driven systems is more significantly different. Unlike the asymmetric case, the MFT is rigorously proven for low currents (note that in that case, the stationary current is itself ``low", as it is of order $L^{-1}$ if we measure it on one bond only, and it is the space-integrated current which converges to a finite value), but the typical states are not discontinuous, as there is no difference in scaling between the drift and diffusion terms in the MFT equation equivalent to eq.(\ref{LangevinGas}). On the other hand, the large current limit is exactly the same as here since the $\mu\rightarrow\infty$ limit is identical. We therefore expect to find a dynamical phase transition, and we already mentioned consistent recent results for large fluctuating currents in close-to-equilibrium models \cite{Baek2016}. However, the nature of the transition might well be different : in periodic systems, a dynamical phase transition has been identified \cite{PhysRevE.72.066110,Bodineau2008,appert2008universal,Simon2011}, in which travelling waves states seem to play an important role \cite{PhysRevE.72.066110,Espigares2013}, although that might be an effect of the total density constraint in periodic systems, as no such states are found in open systems \cite{Lecomte2010,Baek2016a}. Moreover, the appropriate scaling for the current is not the same as in the bulk-driven case (i.e. the space-integrated current is finite, not the one-bond current), which means that the high current part of the large deviation function loses its scaling with respect to $L$.
\item \textit{Large deviations of the activity}: Another quite natural extension would be to consider observables other than the current, such as for instance the dynamical activity, defined as a symmetric average of the number of jumps in both directions, rather than an antisymmetric one. Note that, unlike the current, the activity is not conserved throughout the system, and so every different weighting of the single bond activities is a different observable, the standard one being the uniform average (or unweighted sum). For totally asymmetric models, the current and average activity are one and the same, but they play different roles close to equilibrium (as the current appears explicitly in the MFT action, and the activity does not). The activity is known to undergo dynamical phase transitions for exclusion processes \cite{Bodineau2008,Lecomte2012,Jack2015} as well as for kinetically constrained models \cite{Garrahan2009,Nemoto2016,Nemoto2014}. For the former, the transition must be somewhat linked to that of the current, since they are identical in the totally asymmetric limit, although one should note that the choice of scaling of the observable (total activity or average activity, the latter being divided by the system size $L$) will have an effect on the aspect of the phase transition, which will appear to be of first order for the total activity, as it would for the total current \cite{Jack2015,1751-8121-44-11-115005}. For more general models, or more general definitions of the activity, we expect our methods to be applicable, although perhaps not as straightforwardly as for the current. In the high activity limit, the deformed Markov matrix would be equivalent to an inhomogeneous XX spin chain, solvable in principle, and the same scaling would be found for the large deviation function. In the low activity limit, we would still have a perturbation around the diagonal, but with backward jumps allowed, which seems to result in $\varepsilon^2$ terms being present in $E(\mu)$, meaning that the large deviation function of the activity would scale linearly in $L$ (excluding the constant part) as it does in the high activity regime. That would invalidate our scaling argument for the existence of a dynamical phase transition. A more careful analysis is in order.
\item \textit{Higher dimensions}: Finally, we might wonder if our methods can be extended to other geometries. The first step would be to consider the model on a tree, where it is known that the stationary density to current relation is consistent with a hydrodynamic behaviour at least in some cases \cite{Mottishaw2013}. We expect to be able to extend the low current method without major issues. In the high current limit, although the standard free Fermion techniques we used here are expected to fail, one might still be able to perform calculations using auxiliary spins at every fork, as done in \cite{Crampe2013} for a star graph. In the case of higher-dimensional regular lattices, things might be less straightforward, as there are more ways to be far for equilibrium than in one dimension: the system can be driven along loops rather than from one side to the opposite one. Moreover, the stationary current itself will be more complex than in one dimension, as the zero divergence condition allows for vortices in addition to a constant overall flux. In the relatively simple case of a system driven along one of the lattice directions, between reservoirs, with periodic or closed boundary conditions in the other directions, we expect our methods to be applicable but to produce highly degenerate dominant states at leading order, which then have to be separated by a perturbation to a higher order (especially in the high current limit, where the system essentially splits into disconnected one-dimensional chains at leading order). The low current limit can then be compared to the MFT approach, where a dynamical phase transition has already been found close to equilibrium \cite{Hurtado2011}.
\end{itemize}
~~
~~
\textit{Acknowledgements}: I would like to thank M. Esposito and his group, as well as C. Maes, G. Schutz and D. Karevski, for interesting and useful discussions. I am grateful to R. Jack for helping me correct a mistake in eq.(\ref{LowCurrtEquiv}). This work was supported by the Interuniversity Attraction Pole - Phase VII/18 (Dynamics, Geometry
and Statistical Physics) at KU Leuven and the AFR PDR 2014-2 Grant No. 9202381 at the University of Luxembourg.
\newpage
|
1,314,259,993,627 | arxiv |
\section{Method details}
In this section, we explain detailed settings of the model designs and experimental analyzes conducted in main text.
\smallskip
\noindent\textbf{Texture control analysis.}
Here, we describe detailed setups of the texture control analysis (Section 3.1, main text).
In Figure 4a (main text), each stroke thickness case was generated by the separately trained \textsc{Cartooner}.
We used three models and these are trained with the cartoon images resolutions of $\{256^2, 416^2, 800^2\}$.
We also set the texture controller to only have a single branch.
With these setups, the models trained with $\{256^2, 416^2, 800^2\}$ resolutions generated thin, moderately thick, and thick strokes respectively.
To conduct abstraction change experiment, as shown in Figure 4b (main text), we trained multiple \textsc{Cartooner} similar to the stroke change scenario except for the receptive field (RF) of the generator.
We differentiated the RF of the network by changing the kernel size of conv layers in the texture controller by $\{3, 11, 19\}$ each, which corresponds to the low, moderate, and high abstraction scenes.
\smallskip
\noindent\textbf{Network architecture.} \Tref{table:network_architecture} presents the network architecture of \textsc{Cartooner}.
We applied LeakyReLU for all conv layers and did not use any normalization layer.
In our experiments, we observed that the existences of normalization layer (batchnorm~\cite{ioffe2015batch} and instance norm~\cite{ulyanov2016instance}) drops the cartoonization quality.
The cardinality of conv layers in the ResNeXt blocks were set to 32.
We used bilinear interpolation method for \textit{Upsample} layer.
In \textit{col2} and \textit{col3} layers in the color decoder, the additional 3-channel of each first conv layer is for the color cue injection.
\smallskip
\noindent\textbf{Abstraction control unit.}
We designed this unit to be a shared multi-branch system to increase the quality robustness and to reduce the model size.
Specifically, the multi-branch module was composed of conv layers with varying kernel size, $K_1 < ... < K_N$, where $K$ denotes kernel size and $N$ is the number of branches.
Instead of using $N$ conv kernels, we only initialized a single conv layer with $K_N$ kernel size.
All other conv kernels were set to be a subset of $K_N$ kernel as illustrated in \Fref{fig:shared_unit}.
By doing so, the abstraction control unit can produce robust outcomes for the different abstraction level.
We will demonstrate how crucial this design is in \Sref{sec:suppl_model_analysis}.
Such design also successfully reduces the model parameters; without a shared scheme, the model parameters of \textsc{Cartooner} becomes 26.5M, while the shared kernel version (ours) is 5.9M.
\tableNetworkArchitecture
\smallskip
\noindent\textbf{Model training.}
We trained \textsc{Cartooner} using Adam~\cite{kingma2014adam} for 100K steps with batchsize of 32 and learning rate of $2\times10^{-4}$.
Unlike previous deep methods~\cite{chen2018cartoongan,chen2019animegan,wang2020learning}, we did not perform network pre-training~\cite{chen2018cartoongan}.
For all results shown in this paper, we used the same hyper-parameters: $\lambda^1_{texture}=$ 1.0, $\lambda^2_{texture}=$ 0.0025, $\lambda^3_{texture}=$ 0.0045, $\lambda^4_{texture}=$ 0.0015.
We utilized official Caffe-version of VGG19~\cite{simonyan2014very}, and when creating a Gram matrix, we divided it by the product of \# of the channels, width, and height.
When generating an initial color map, $C^{RGB}_{src}$ from an input photo $I^{RGB}_{src}$, we adopted Zhu et al.~\cite{zhu2021learning} for both training and inference.
We selected this algorithm since it is GPU-friendly, nevertheless, any off-the-shelf super-pixel algorithm can be adopted.
We tested other algorithms, such as SLIC~\cite{achanta2012slic}, and observed no performance drop even when we use a different approach at train and inference.
\figSharedUnit
\section{Experimental settings}
\noindent\textbf{Dataset.}
In our study, we mainly focused on outdoor scenes and landscapes, to better target the domain of cartoonizing background scenes.
We collected 8,227 real-world outdoor images from the \textit{monet2photo} dataset~\cite{zhu2017unpaired} for the source photo domain.
Then, this was split into 6,227 and 2,000 images for the train and test set.
For the target cartoon domain, we collected cartoon datasets from Japanese animations and Webtoons.
Specifically, we acquired animation images from `The Garden of Words' and `Your Name' by Shinkai Makato, and `Spirited Away' by Miyazaki Hayao.
For the Webtoon dataset, we collected comics of titles `FreeDraw'\footnote{\url{https://comic.naver.com/webtoon/list?titleId=597447}} and `Barkhan'\footnote{\url{https://comic.naver.com/webtoon/list?titleId=650305}} from the NAVER Webtoon platform.
We resized source domain images as 256$\times$256 resolution.
For cartoon datasets, we cropped images to be a resolution of 512$\times$512, and applied $\times$2 super-resolution~\cite{wang2018esrgan} beforehand when the raw image is $<$1024$\times$1024 resolution.
In total, we gathered images of 5,480 Hayao, 5,647 Shinkai, 5,308 FreeDraw, and 6,186 Barkhan datasets.
In our experiment, we treated the above four style datasets as independent since each artwork has a unique style.
\smallskip
\noindent\textbf{User study.}
We asked 26 participants to pick the best results for how well the outputs follow both the cartoon styles and source photos.
Each of them was asked to vote on 16 questions, thus we collected 416 samples in total.
In every question, we presented source photography, exemplar cartoon image and the results of the previous and our methods.
For better visibility, we also showed cropped patches for all the results.
We computed the \textit{quality preference} score by the ratio of the voted (as the best) cases.
\figAblPretrain
\figResultPreAnalysisSuppl
\smallskip
\noindent\textbf{Interactive UI.}
\Fref{fig:ui} shows an interactive UI of \textsc{Cartooner}.
In the left panel, selection tools (\textit{e.g.,} selection, quick selection, and eraser) offer mask-based region selection so the user can easily manipulate local region.
In the right panel, style change tools offer texture and color control over the selected region.
Here, the creators can change the \textbf{1)} target cartoon style, \textbf{2)} texture (\textit{i.e.,} stroke thickness and abstraction), and \textbf{3)} color of the cartoonized outcomes.
For the color control, we provide both a color picker and an HSV control slider UIs since we observed that the latter is straightforward to utilize for unskilled users.
Throughout this, the creators easily render given natural photos into the cartoon styles as well as manipulate the results with their own desired texture and color.
\figAblAbsShared
\smallskip
\noindent\textbf{Reference image-based color control.}
It can be achieved with a simple pipeline.
With a given reference image, we extract the color palette through the K-means clustering.
In our demonstration (Figure 13, main text), we used eight palettes (\textit{i.e.,} clusters).
Note that when the user selects the specific region (via selection tools) to transfer the color, we generate a color palette from that region instead.
Then, the user chooses the region to be changed in a source photo and the framework calculates the average color of the region, denoted as $c$.
In the meantime, the user also picks the color $c'$ from the palette.
Using these colors, $c$ and $c'$, the framework performs a palette-based color transfer algorithm~\cite{chang2015palette} to the initial color map $C_{src}$ and generates $\bar{C}_{src}$.
However, we observed that a simple color transfer in RGB space (\Eref{eq:color_transfer}) also produces robust results.
With the manipulated color map, \textsc{Cartooner} now generates appropriate cartoonized outputs that fulfill the users' color intention.
\begin{equation}
\label{eq:color_transfer}
\bar{C}_{src}^{RGB} = C_{src}^{RGB} + (c' - c)
\end{equation}
\section{Model analysis}
\label{sec:suppl_model_analysis}
\noindent\textbf{Model training.}
We analyzed the pre-training that has been a prevalent strategy on deep cartoonization~\cite{chen2018cartoongan,chen2019animegan,wang2020learning}.
Previous studies reported that the warm-up process, which optimizes the network through the content loss only in advance, guarantees better convergence and cartoonization quality.
However, we found that the network pre-training does not require to \textsc{Cartooner} (\Fref{fig:abl_pretrain}).
We claim that a separate design of texture and color enables robust training since the texture and color decoders can solely concentrate on synthesizing texture and color alone, respectively.
\figAblAbsKsize
\figUI
\smallskip
\noindent\textbf{What affects the abstraction?}
In \Fref{fig:result_preanalysis_suppl}, we present results of the abstraction analysis where we only change receptive field (RF) of the generator (\Fref{fig:result_preanalysis_suppl}a), or change both RF of the generator and the image resolution of cartoon domain dataset (\Fref{fig:result_preanalysis_suppl}b).
Note that the latter result was shown in the main text.
When we alter RF of the generator alone, the abstraction seems not much affected since the low-complexity cartoon scene does not guide the generator, hence the generator would not have any incentive to increase the abstraction.
On the other hand, as we discussed in Figure 5 (main text), increasing both RF and the resolution effectively affects the abstraction due to the scene complexity guidance from the cartoon images.
\smallskip
\noindent\textbf{Abstraction control unit.}
We designed this unit to have a shared conv kernel scheme in a multi-branch system.
When the conv kernels are not shared (\Fref{fig:abl_abs_shared}a), it is not guaranteed consistent and smooth abstraction transitions.
For example, the detailed textural gradations near window edges intensify even when we increase the abstraction (2nd column).
This is because each branch learns separate representations without communication, thus, they are not tuned to each other to generate smooth abstraction change.
In contrast, the shared conv kernel approach (\Fref{fig:abl_abs_shared}b) adequately produces continuous abstraction modification.
We claim that robustness can be achieved since RF of the current abstraction is gradually evolved based on the previous RF (\Fref{fig:shared_unit}).
By doing so, all the branches share the viewpoint near the center point (of RF), and the kernels with larger RF look wider region while maintaining the perspective of the previous ones.
As a result, they produce consistent and gradually transition on the abstraction.
We demonstrate the results of the different kernel size cases.
In \Fref{fig:abl_abs_ksize}a, we excessively expand the kernel sizes and the network generates blurred outputs since the model is guided from too many flat regions.
In our experiment, we found that kernel size in \Fref{fig:abl_abs_ksize}b shows the best abstraction change in terms of the perceptual quality.
However, the other settings (such as \{3, 5, 9, 11, 13\}) also make plausible outcomes as long as kernel sizes are in increasing order.
\section{Additional results}
\Fref{fig:interactivity_suppl} displays interactive cartoonization scenarios.
We demonstrate visual comparison to the state-of-the-art deep cartoonization approaches in \Fref{fig:comp_suppl_one}, \ref{fig:comp_suppl_two}, \ref{fig:comp_suppl_three}, \ref{fig:comp_suppl_four}.
\figInteractivitySuppl
\figCompSupplOne
\figCompSupplTwo
\figCompSupplThree
\figCompSupplFour
\section{Experiment}
\smallskip
\noindent\textbf{Baselines.}
We compare \textsc{Cartooner} with the state-of-the-art deep learning-based cartoonization methods, CartoonGAN~\cite{chen2018cartoongan}, AnimeGANv2~\cite{chen2019animegan}, and WhiteboxGAN~\cite{wang2020learning}.
Since they have trained their cartoonization network on different datasets and setups from each other, we retrained using our cartoon datasets using official codes.
\smallskip
\noindent\textbf{Datasets.}
We built datasets focused on landscape, to better target the domain of cartoonizing background scenes.
We used \textit{monet2photo}~\cite{zhu2017unpaired} as the photo domain.
We collected cartoon datasets from Japanese animations and Webtoons.
Specifically, we acquired artworks by Miyazaki Hayao and Shinkai Makoto, and comics of titles ``FreeDraw" and ``Barkhan" from the NAVER Webtoon platform.
Detailed dataset generation protocols are described in Suppl.
\smallskip
\noindent\textbf{Metrics.}
We evaluated the cartoonization with Fre\'chet Inception Distance (FID)~\cite{heusel2017gans} and $\text{FID}_{\text{CLIP}}$~\cite{kynkaanniemi2022role}.
We additionally conducted a user study to measure perceptual quality.
We asked 26 users to select the best results for how well the outputs follow both the cartoon styles and source photos.
\figInteractivity
\subsection{Comparison with state-of-the-art method}
When comparing \textsc{Cartooner} with others, we generate images to reflect the target cartoon since FIDs can be influenced by color information.
In addition, we set the texture levels $(\alpha_s, \alpha_a)$ as zero, which is identical stylization setting to others.
\Tref{table:comp_quant} shows the quantitative comparison.
\textsc{Cartooner} achieves exceeding performance on both FID and $\text{FID}_{\text{CLIP}}$ with significant margins for all the cases.
We also present the visual comparison in \Fref{fig:comp_qual}.
Separation of the texture and color decoders helps prevent image artifacts, for instance, \textsc{Cartooner} produces fewer color bleeding (\Fref{fig:comp_qual}, 2nd row).
The visual quality is also profoundly enhanced and \textsc{Cartooner} can capture adequate stroke and color nuance of the target cartoon.
A user study shows the superiority of \textsc{Cartooner} as well (\Tref{table:user_study}).
\subsection{Interactivity}
\label{sec:interactivity}
As shown in \Fref{fig:interactivity}, \textsc{Cartooner} creates diverse results according to user interaction.
When the artist manipulates the colors to their tastes (with any color adjustment UI), \textsc{Cartooner} automatically reflects the intention.
They can also edit textural details by simply controlling the stroke or abstraction factors to match the output in various cartoon situations.
These editings can be performed locally or globally through a simple mask-based region control UI (shown in Suppl.).
Our cartoonization workflow is more compact than the traditional editing tools, while still maintaining an adequate level of user intervention.
Although \textsc{Cartooner} may not achieve the degree of meticulous editing workflows (which requires the effort of skilled artists), it can provide a broader range of user experiences with suitable quality.
We would like to emphasize that none of the deep cartoonization methods can provide controllability nor produce diverse results of a given source photography.
\figAblColor
\figFeatmap
\subsection{Model analysis}
\noindent\textbf{Color module.}
In \Fref{fig:abl_color}, we present the result where the color change is performed before or after cartoonization, unlike ours that jointly models the color and texture.
The pre-execution of color change (\Fref{fig:abl_color}b) cannot adequately handle the delicate color alters and produces uneven texture level since the model has not observed the re-colorized input image at training, which becomes out-of-distribution.
The pipeline of color change after cartoonization (\Fref{fig:abl_color}c) cannot generate cartoon-style colors at all.
\smallskip
\noindent\textbf{What does texture controller learn?}
We visualize the output feature map of the texture controller in \Fref{fig:featmap}.
The stroke control unit produces features that more concentrate on the high-frequency edge regions, which empirically demonstrates why this can control the stroke thickness.
On the other hand, the abstraction control unit focuses on a wide range region including flat texture and some mid-frequency details.
As a result, this unit can deliver helpful clues of the abstraction change to the decoder.
\smallskip
\noindent\textbf{Stroke control unit.}
We decrease the number of stroke level at training and examine how the models react at inference.
\Fref{fig:abl_stroke} shows that the model trained with 2-levels cannot capture high stroke thickness; we observed that it produces saturated thickness only.
In contrast, the model with increased stroke levels adequately expresses a wide dynamic range of stroke thickness.
In Suppl., we demonstrate a similar analysis regards on the abstraction control unit.
\figAblStroke
\section{Introduction}
Cartoons gain steep popularity in a recent, and the number of cartoon creators have also increased.
The universal workflow of cartoon painters is as follows:
character drawing, which is then composed into a background scene. Post-processing such as shading is added afterward.
Professional tools~\cite{clipstudio,adobeae} provide helpful plugins to assist the artist. Despite this, cartoon creation still remains an arduous task even for the more skilled creators.
We follow the observation that cartoon-styled scene generation has received notable attention.
Many artists convert real-world photographs into cartoon styles to utilize as a background scene, dubbed as \textit{image cartoonization}.
This allows creators to more focus on effective decisions in making cartoons, such as character generation.
It is shown that deep learning-based cartoonization approach is able to produce cartoon-stylized output with a prominent quality, that is possible to be utilized in real service production~\cite{chen2018cartoongan,chen2019animegan,wang2020learning}.
\figBackgroundProcess
However, the previous deep methods skip the intermediate procedures of cartoon-making processes, thus disabling the creators from controlling outputs.
The artists follow a series of structured steps when creating a cartoon background from a photo (\Fref{fig:background_process}).
\textbf{1)} Color stylization, where the author changes the color both locally and globally.
Sky synthesis is performed along with this procedure.
\textbf{2)} Texture stylization, where additional sketch lines are drawn, and fine details are selectively removed to achieve the different \textit{levels of abstraction}. \textbf{3)} Post-processing, which includes lighting and image filters.
Unfortunately, due to the end-to-end inference nature of the previous deep cartoonization methods, the artist has no control over the generation process.
The creators may only intervene with a source photo (\Fref{fig:background_process}a) or the final output (\Fref{fig:background_process}e), which harms the usability of the cartoonization methods in artists' workflow.
In this study, we present an effective approach to embedding interactivity in cartoonization.
The proposed solution focuses on building a pipeline for more controllable texture and color.
We define texture control as the manipulation of stroke thickness and abstractions.
This concept can be utilized in many scenarios; the artist can abstract the details of the far-distance scene to depict the natural perspective or emphasize the details of the character.
The creators can also choose to change the delicacy of the brushstroke to match the texture of the foreground objects when composing the scene.
As for color control, we aim to build a control system in which the creator freely manipulates arbitrary regions with the desired color.
This is designed to assist the artist in the color stylization procedure (\Fref{fig:background_process}c).
To obtain user controllability in cartoonization, we separately build texture and color decoders to minimize interference across the features (\Fref{fig:model_overview}).
We also found that the decomposed architecture provides a robust and superb quality of texture stylization.
For texture control, we investigated the role of the receptive field and the target image resolution in the level of stroke thickness and abstraction.
Based on these observations, we present a \textit{texture controller}, which adjusts the receptive field of the network through a dynamic replacement of the intermediate features.
For color control, we jointly train the color decoder in a supervised manner with the paired dataset that is built based on the proposed HSV augmentation.
Throughout this training strategy, the color module gains the ability to produce diverse colors.
With the combination of the decoupled texture and color modules, we achieve a two-dimension of control space that can create a variety of cartoonized results upon user communication.
Such a design also provides robust and perceptually high-quality cartoonized outcomes.
To the best of our knowledge, our framework is the first approach that presents interactivity to deep learning-based cartoonization.
Based on the proposed solution, we demonstrate application scenarios that permit user intentions to create cartoonized images with diverse settings.
Extensive experiments demonstrate that the proposed solution outperforms the previous cartoonization methods in terms of perceptual quality, while also being able to generate multiple images based on the user's choices of texture and color.
\figModelOverview
\section{Application}
\smallskip
\noindent\textbf{Reference image-based color control.}
In \Sref{sec:interactivity}, we demonstrated a simple interactive cartoonization workflow with \textsc{Cartooner}.
However, unskilled users might struggle to choose appropriate color tones if they have little experience in coloring.
To increase usability for inexpert users, we present reference image-based color control (\Fref{fig:ref_based}).
Instead of direct color manipulation, a user prepares a color guidance image and chooses which regions to be referred via region masking UI.
After this, \textsc{Cartooner} transfers the color information of the selected area to the cartoonized outputs.
To implement this, we first extract a color palette from a reference image and then manipulate color map, $C_{src}$ using palette-based color transfer algorithm~\cite{chang2015palette}.
\smallskip
\noindent\textbf{Semi-automatic cartoon making.}
As discussed, making background scenes is repetitive and time-consuming.
\textsc{Cartooner} can help to reduce the burden of background creation with interactive texture-color editing so the artists can focus more on other creative tasks.
\Fref{fig:cartoon_making} shows an example where one can use \textsc{Cartooner} to effectively create a cartoon cut, consisting of a background scene blended with character(s) and/or speech balloons, which would have been a strenuous task for previous pipelines.
\figRefBased
\figCartoonMaking
\section{Conclusion}
We proposed an interactive cartoonization model, \textsc{Cartooner}.
The proposed method accepts user-guided texture control in the form of abstraction and stroke strength levels, which are passed to a \textit{texture controller} to dynamically control the overall texture of the generated image.
The user can also manipulate the color scheme through a color module, which is reinforced by the HSV augmentation.
Experimental results demonstrate \textsc{Cartooner}'s superiority in both quality and usability as applications for cartoon creators.
Although we provided effective control space, there exist more controlling factors, especially for texture control (\Fref{fig:background_process}d).
In the future, it is worth exploring the other aspects of texture editing such as brush stroke's style~\cite{kotovenko2021rethinking}.
\section{Related work}
\noindent\textbf{Non-photorealistic rendering.}
Non-photorealistic rendering (NPR) is a computer vision/graphics task that focuses on representing diverse expressive styles for digital art.
Among a variety of subtasks of NPR, we briefly describe the methods that stylize natural domain image to the specific artistic style.
Because of its usefulness in digital art creation, NPR has been expanded into various applicable scenarios such as line drawing~\cite{winnemoller2012xdog}, image abstraction~\cite{xu2011image}, and cartoonization~\cite{wang2004video}.
Style transfer methods~\cite{gatys2016image,johnson2016perceptual} are notable approach in NPR.
By jointly optimizing the content and style losses, they can generate decent-quality of stylization.
\smallskip
\noindent\textbf{Cartoonization.}
Deep learning-based methods show profound improvement over conventional NPR algorithms on this task.
CartoonGAN~\cite{chen2018cartoongan}, a pioneering study on deep cartoonization, adopts adversarial training~\cite{goodfellow2014generative} along with an edge-promoting loss to improve cartoon style.
AnimeGAN~\cite{chen2019animegan} enhances CartoonGAN with advanced losses suitable for cartoon style such as Gram-based loss~\cite{gatys2016image}.
With a careful inspection of cartoon drawing process, WhiteboxGAN~\cite{wang2020learning} decomposed cartoon images as surface, structure, and texture representations to tackle each factor with tailored losses. This approach achieves superior cartoonization quality compared to previous methods.
Despite the imposing cartoonization results, none of the deep cartoonization methods support interaction, failing to create diverse conditional outputs. Our method aims to enable user control while maintaining perceptually appealing cartoonization, moving closer to the actual service level.
\section{Method}
We describe interactive cartoonization method dubbed as \textsc{Cartooner}.
It uses separate decoders for texture and color (\Fref{fig:model_overview}), contrary to a single decoder architecture of the previous methods.
The decision was made from observing professional artists' workflow, where they separate color modification from texture editing.
We further inspect that isolated modeling of texture and color produces reliable and high-quality cartoonized results.
The controllable features are defined as texture level vector $\pmb{\alpha}$ and users' color modification $\pmb{c}$.
Given a photo $I_{src}$, the goal is to generate a cartoonized image $\hat{I}_{tgt}$ that follows the user intention $\pmb{\alpha}$ and $\pmb{c}$.
To achieve this, \textsc{Cartooner} encodes an image to the latent feature through $E_{shared}$, then delivers it to the separate decoders, $S_{texture}$ and $S_{color}$.
Note that we use Lab color space instead of RGB, hence the texture module produces an L-channel texture map, while the color module generates an ab-channel color map.
These outputs are finalized by converting back to RGB space.
\subsection{Texture module}
\label{subsec:texture_module}
\noindent\textbf{Analysis of texture level.}
In this module, the primary goal is to provide a fine control mechanism and we define texture control as altering \textit{stroke thickness} and image \textit{abstraction}.
To do that, we first analyze which components influence stroke thickness and abstraction change.
In our preliminary experiments, we observed that increasing the resolution of target cartoon images affects stroke thickness, and expanding the receptive field (RF) of the generator along with the increased resolution inflates the abstraction level (\Fref{fig:result_preanalysis}).
\figPreAnalysis
For stroke thickness change, we argue that the \textit{loss network with a fixed RF} (\textit{e.g.,} VGG or discriminator) is involved as shown in \Fref{fig:preanalysis}a.
Given the fixed loss network, when we increase the resolution of cartoon images (\textcolor{ForestGreen}{green box}), the strokes are enlarged within an RF window, thus inevitably, the generator learns to produce thick strokes at training.
When we decrease the resolution (\red{red box}), the opposite behavior occurs.
This impacts cartoonization more since cartoon images mostly have flat texture regions.
For abstraction change, we argue that \textit{scene complexity} affects to this as shown in \Fref{fig:preanalysis}b.
When the RF of the generator is expanded (\textcolor{blue}{blue box}), the network can perceive wider region of a content image, which results in high scene complexity.
In contrast, when the resolution of a cartoon image grows, it's scene complexity becomes lower since the loss network can only see relatively tiny region (\textcolor{ForestGreen}{green box}).
With these, if we utilize high-resolution cartoon images $I^{HR}_{tgt}$ to train the generator with a large RF (which are \textcolor{ForestGreen}{green} and \textcolor{blue}{blue box}, respectively), the generator is guided to reduce the complexity of the high-complexity scene.
This arise from the loss calculation with the low-complexity (of cartoon) scene extracted from the loss network.
As a result, the generator with a large RF gains the ability to ``abstract" the complex details.
For the lower RF (\textcolor{magenta}{pink window}), the contrary behavior happens.
However, the abstraction change is not as dramatic as the high one since in general, the scene complexity of cartoon is lower than the content images.
We also analyzed the scenario where \textit{only RF of the generator is expanded}, however, the results are not drastic as \Fref{fig:result_preanalysis}b since the generator is not guided by the different-complexity cartoon scene.
Note that Jing et al.~\cite{jing2018stroke} inspected the role of the resolution and RF in style transfer literature, however, our in-depth analysis reveals that their behavior patterns are disparate in cartoonization.
\figTextureController
\smallskip
\noindent\textbf{Texture controller.}
The above analysis requires multiple networks to handle diverse levels and it cannot produce consistent styles for each other.
Also, it only supports discrete control levels, making it challenging to be used as a real-world solution.
Hence, based on the analysis, we introduce a simple but effective texture control module, dubbed as \textit{texture controller} (\Fref{fig:texture_controller}).
It consists of the stroke and abstraction control units and we design both units as to be a multi-branch architecture.
In the stroke unit, each branch is composed of two consecutive 3$\times$3 conv layers, and these are fused by the gating module.
The abstraction unit is identical structure to the stroke unit except it uses conv layers with large kernel size, $K_1 < K_2 < ... < K_N$.
The texture controller is influenced by texture level $\pmb{\alpha}=\{\alpha_s, \alpha_a\}$, specifically, the stroke and abstraction units are guided by stroke thickness $\alpha_s$ and abstraction $\alpha_a$ levels, respectively.
With the feature $f$ from the encoder, the stroke unit generates a feature set $\pmb{g}_{s} = \{g_s^1,...,g_s^N\}$ through conv branches, and the abstraction unit produces $\pmb{g}_{a}$ as the same way.
Then, according to texture levels $\alpha_{\{s,a\}}$, which are a positive rational number, the two features of $\pmb{g}_{\{s, a\}}$ with indices closest to a texture level are chosen.
The chosen features are then interpolated based on the respective distance between $\alpha_{\{s,a\}}$ and indices.
Finally, these are combined by an element-wise addition operation.
We design the stroke control unit to have all 3$\times$3 conv layers since the texture level analysis showed that RF of the generator does not affect the stroke thickness.
Instead, each branch is trained by different resolutions of target cartoon images.
At inference, the feature interpolation via $\alpha_s$ enables continuous control over stroke thickness.
For the abstraction unit, we also construct a single module based on the analysis.
However, unlike the stroke unit, each branch includes conv layers with different kernel sizes (with increasing order) because changing both RF of the generator and the resolution of target images alters the abstraction.
The output features are interpolate through $\alpha_a$, as identical to the stroke unit.
As we design the decoupled structure of stroke and abstraction in parallel, each unit can concentrate on a different aspect and it provides the ability that can control the texture as a two-dimensional space.
In addition, to incorporate adversarial learning~\cite{goodfellow2014generative} into texture control, we utilize a multi-texture discriminator.
It is based on the multi-task discriminator~\cite{mescheder2018training,choi2020stargan}, which consists of multiple output branches.
Each branch corresponds to a different texture level and learns to distinguish whether a given image is from a real cartoon domain or generated.
\figPreprocessTrain
\figCompQual
\smallskip
\noindent\textbf{Loss function.}
We use adversarial loss $L^{adv}_{texture}$ to guide the model mimicking the texture of the target cartoon.
\begin{equation}
L^{adv}_{texture} = logD_{\pmb{\alpha}}(I^{L, \pmb{\alpha}}_{tgt}) + log(1-D_{\pmb{\alpha}}(G(I^{Lab}_{src}, \pmb{\alpha})))
\end{equation}
where $G$ is the generator and $D_{\pmb{\alpha}}$ denotes the multi-texture discriminator with a given texture factors $\pmb{\alpha}$. $I^{L, \pmb{\alpha}}_{tgt}$ is a resized target cartoon image to fit a respective texture level $\pmb{\alpha}$.
To ensure a cartoonized output well preserves the semantic information of a source photo, we employ content loss.
\begin{equation}
L^{vgg}_{content} = ||VGG(I^{L}_{src}) - VGG(G(I^{Lab}_{src}, \pmb{\alpha}))||_1.
\end{equation}
We use a \textit{conv4\_4} layer of the pre-trained VGG19~\cite{simonyan2014very}.
In addition, we enforce the generator to learn high-level texture representation via Gram-based loss as:
\begin{equation}
L^{vgg}_{texture} = ||Gram(I^{L, \pmb{\alpha}}_{tgt}) - Gram(G(I^{Lab}_{src}, \pmb{\alpha}))||_1.
\end{equation}
$Gram$ indicates the Gram calculation with VGG feature extraction (of \textit{conv4\_4}).
We also use total variation loss~\cite{aly2005image} to impose spatial smoothness on the output.
\begin{equation}
L^{tv}_{texture} = || \nabla x (G(I^{Lab}_{src}, \pmb{\alpha})) + \nabla y (G(I^{Lab}_{src}, \pmb{\alpha})) ||_1
\end{equation}
With balancing parameters $\lambda_{texture}^{1,...,4}$, the final loss of the texture decoder (and the shared encoder) is defined as:
\begin{align}
L_{texture} &= \lambda_{texture}^1 * L^{adv}_{texture} + \lambda_{texture}^2 * L^{vgg}_{content} \nonumber\\
&+ \lambda_{texture}^3 * L^{vgg}_{texture} + \lambda_{texture}^4 * L^{tv}_{texture}.
\end{align}
\subsection{Color module}
\label{subsec:color_module}
The goal is to transfer the color of a given source photo to a provided color intention, while reflecting the color \textit{nuance} of target cartoon.
\textsc{Cartooner} takes an input photo $I^{Lab}_{src}$ as well as an input color map $\bar{C}^{Lab}_{src}$ and generates an ab-channel image $\hat{I}^{ab}_{src}$, which is later concatenated with the texture map $\hat{I}^{L}_{tgt}$ generated from the texture decoder.
To simulate control manipulation, we synthetically generate color map, $\bar{C}^{Lab}_{src}$ (\Fref{fig:preprocess_train}).
Given an input photo $I^{RGB}_{src}$, we create an initial color map $C^{RGB}_{src}$ by applying a superpixel algorithm.
Without this, fine details of an input image become too noisy and thus not adequate to be utilized as a color cue; a superpixel is used as a noise reduction procedure.
Then, the HSV augmentation changes the color of $I^{RGB}_{src}$ and $C^{RGB}_{src}$, creating color manipulated images $\bar{I}^{RGB}_{src}$ and $\bar{C}^{RGB}_{src}$.
These are converted to Lab space.
Note that we observed that color transfer to either input or output image, unlike ours, cannot achieve faithful visual quality.
HSV augmentation is a simple but effective method that can reflect diverse color control intentions from a user.
It randomly alters all color channels of HSV; hue, saturation, and value (brightness).
To prevent the color shifting from generating perceptually implausible outputs, we further apply the L caching trick~\cite{cho2017palettenet} prior to the color augmentation, which caches luminance (L) of image and reverts the luminance of augmented image to a cached one.
We cache L instead of V-channel since V indirectly interferes with L, which is important in regard to diverse cartoonization.
\tableCompQuant
\smallskip
\noindent\textbf{Loss function.}
We use a simple mean squared error-based reconstruction loss as shown in below.
\begin{equation}
L_{color} = ||\bar{I}^{ab}_{src} - G(I^{Lab}_{src}, \; \bar{C}^{Lab}_{src})||_2
\end{equation}
In our experiment, additional adversarial loss or regularization shows marginal improvement in color quality.
We suspect that the decomposed color modeling as well as the color cue ($\bar{C}^{Lab}_{src}$) provision ease the training difficulty.
\smallskip
\noindent\textbf{Reflecting target cartoon's color.}
We designed to preserve the color information of an input image to increase the controllability, however, one might want to generate an image that has a similar color distribution to the target cartoon.
To handle this scenario, we additionally fine-tune the color decoder with an assist of adversarial loss as in below.
\begin{align}
L^{tgt}_{color} &= \lambda^1_{color} * L_{color} + \lambda^2_{color} * L^{adv}_{color} \;\;\;\; \text{where,}\nonumber\\
L^{adv}_{color} &= logD(I^{ab}_{tgt}) + log(1-D(G(I^{Lab}_{src}, \; C^{Lab}_{src})))
\end{align}
Note that we use color cue that is generated from an original image ($C^{Lab}_{src}$), instead of $\bar{C}^{Lab}_{src}$.
With the aforementioned color decoder parts, a user can choose which ``color mode" to use interchangeably depending on the situation.
\subsection{Model training}
\label{subsec:training}
Unlike previous deep cartoonization methods, we do not perform network warm-up~\cite{chen2018cartoongan}.
We train the entire framework with a loss of $L = L_{texture} + L_{color}$, except for the abstraction control unit.
Then, the abstraction unit is trained (via $L_{texture}$) while other components are all frozen.
To provide various resolution images to the generator, we resize $I^{RGB}_{tgt}$ according to texture level $\pmb{\alpha}$ (\Fref{fig:preprocess_train}).
We set kernel sizes of the abstraction unit, $\{K_1, K_2,...,K_N\}$, as $\{3, 7, 11, 15, 19\}$ each.
When training \textsc{Cartooner}, we randomly choose $\alpha_{\{s, a\}} \in \{1,...,5\}$, which respectively resize $I^{RGB}_{tgt}$ to be $\{256^2, 320^2, 416^2, 544^2, 800^2\}$ resolutions, but $\alpha_{\{s, a\}}$ can be expanded to arbitrary numbers at inference.
More detailed setups are described in Suppl. |
1,314,259,993,628 | arxiv | \section{Introduction}
In recent years, research in deep neural networks (DNNs) has been fueled by new available computational resources, which have brought a wide variety of new techniques for visual object recognition, object detection and speech recognition among many others~\cite{lecun2015deep}.
The rise of DNNs in many applications, such as medicine \cite{zeleznik2021deep,de2018clinically}, climate \cite{waldmann2019mapping}, wildlife ecology \cite{norouzzadeh2018automatically}, physics \cite{udrescu2020ai,kwon2020magnetic} or sustainability \cite{vinuesa_et_al_2020}, has not been overlooked in fluid-mechanics research \cite{kutz2017deep,brunton2020machine}.
Some of the outstanding applications of DNNs in fluid mechanics are the improvement of Reynolds-averaged Navier--Stokes simulations \cite{ling2016reynolds}, the extraction of turbulence theory for two-dimensional flow \cite{jimenez2018machine}, prediction of temporal dynamics~\cite{srinivasan2019predictions,eivazi2021recurrent} or the embedding of physical laws in DNN predictions \cite{raissi_et_al}.
Generative adversarial networks (GANs), firstly introduced in Ref.~\onlinecite{goodfellow2014generative}, are one of the latest advances in DNN research.
Based on game theory, GANs are composed of two competing networks: a generator that tries to produce an artificial output which mimics reality; and a discriminator, which is in charge of distinguishing between reality and artificial outputs.
During training, the generator network makes its output more realistic by improving the features that the discriminator identified as artificial.
Among the different areas in which GANs have been applied successfully, their use to enhance image resolution stands out \cite{ledig2017photo,wang2018esrgan}.
In fluid-mechanics research, they have been successfully applied to recover high-resolution fields in different types of flow, such as the wake behind one or two side-by-side cylinders \cite{deng2019super} or volumetric smoke data \cite{werhahn2019multi}.
While in these works the training has been carried out with a supervised approach i.e., with paired high- and low-resolution flow fields, GANs have been recently applied with an unsupervised approach to enhance the resolution of homogeneous turbulence and channel flows \cite{kim2021unsupervised}.
GANs are now challenging other resolution-enhancement strategies based on convolutional neural networks (CNNs), which showed to be successful for the cases of the flow around a cylinder, two-dimensional decaying isotropic turbulence \cite{fukami2019super} and channel flows \cite{liu2020deep}.
More recently, Ref.~\onlinecite{fukami2021machine} has proposed a methodology based on CNNs to recover high-resolution sequences of flow fields in homogeneous isotropic and wall turbulence from the low-resolution fields at the beginning and end of the sequence.
CNNs have also been used successfully to estimate flow fields using field measurements of wall shear and/or pressure.
Several methods have been proposed, such as the direct reconstruction of the flow field from the wall quantities using fully-convolutional networks (FCNs) ~\cite{guastoni,guastoni2020convolutional}, or the use of proper orthogonal decomposition\cite{lumley1967structure} (POD) in combination with CNNs \cite{guemes2019sensing} and FCNs \cite{guastoni2020convolutional}.
Moreover, Ref.~\onlinecite{guemes2019sensing} studied the effect of the wall-resolution measurements on the predictions accuracy, showing that their architecture was able to continue providing predictions of similar accuracy for downsampling factors up to 4.
When a limited number of sensors is available, shallow neural networks (SNNs) offer another option for this task.
Ref.~\onlinecite{erichson2020shallow} compared SNNs with POD for the reconstruction of a circular cylinder wake, sea surface temperature, and decaying homogeneous isotropic turbulence, showing that the new data-driven approach outperforms the traditional one.
In the first part of the present work, a GAN-based methodology is proposed to recover high-resolution fields of wall measurements.
Because the results are very positive when performing this task, and it has already been shown that GANs can be used successfully in enhancing turbulent-flow resolution\cite{kim2021unsupervised}, the second part of this work extends their use to reconstruct high-resolution wall-parallel flow fields from coarse wall measurements.
This method is compared with the FCN-POD architecture proposed in Ref.~\onlinecite{guastoni2020convolutional}.
The choice is based on the proven capability of this network to deal with low-resolution input information \cite{guemes2019sensing}.
The paper is organized as follows: \S\ref{sec:metho} outlines the details of the numerical database used for this study and presents the different DNNs employed for that purpose; the main results for wall-resolution enhancement are provided in \S\ref{sec:wall}, while the flow-reconstruction results are reported in \S\ref{sec:flow}.
To close the paper, \S\ref{sec:concl} presents the main conclusions of the work.
\section{Methodology}\label{sec:metho}
This section presents the details of the numerical database employed for this study, as well as the DNN architectures and the training methodology with which they have been optimized.
Throughtout the paper $x$, $y$, and $z$ denote the streamwise, wall-normal, and spanwise directions respectively, with $u$, $v$, and $w$ referring to their corresponding instantaneous velocity fluctuations.
Streamwise and spanwise wall-shear-stress fluctuations are referred to as $\tau_{w_x}$ and $\tau_{w_z}$ respectively, with $p_w$ denoting the pressure fluctuations at the wall.
\subsection{Dataset description}
The methodology proposed in this work has been tested with a direct numerical simulation (DNS) of a turbulent open-channel flow generated with the pseudo-spectral code SIMSON \citep{chevalier}.
The simulation domain extends $4\pi h \times h \times 2\pi h$ (where $h$ is the channel height) in the streamwise, wall-normal and spanwise directions respectively, with the flow represented by 65 Chebyshev modes in the wall-normal direction and with 192 Fourier modes in the streamwise and spanwise directions.
The simulation is characterized by a friction Reynolds number $Re_{\tau}=180$, which is based on $h$ and the friction velocity $u_{\tau}=\sqrt{\tau_w/\rho}$ (where $\tau_w$ is the magnitude of the wall-shear stress and $\rho$ is the fluid density).
The superscript '+' denotes inner-scaled quantities, using $u_{\tau}$ for the velocities and the viscous length $\ell^*=\nu/u_{\tau}$ (where $\nu$ is the fluid kinematic viscosity) for the distances.
DNNs have been trained with 50,400 samples separated by $\Delta t^+=5.08$, while 3,125 samples with time separation $\Delta t^+=1.69$ have been used for testing them.
For further simulation details, see Ref.~\onlinecite{guastoni2020convolutional}.
Wall information, used as input to reconstruct wall-parallel fluctuating velocity fields, is composed of streamwise and spanwise shear stress, as well as pressure fluctuations.
To assess the capability of our methodology to reconstruct turbulent wall and flow fields from coarse wall measurements, three different sets of downsampled wall fields have been generated, with downsampling factors $f_d=[4,8,16]$.
Note that $f_d$ is defined as the resolution reduction in each direction, thus a downsampling factor of $f_d$ yields a downsampled field with a number of points equal to $f_d^{-2}$ times the original.
It has to be noted that $f_d$ values of 2 and 4 were considered in Ref.~\onlinecite{guemes2019sensing}, although for a test case with larger $Re_{\tau}$.
The reconstruction of the fluctuating velocity fields is evaluated at four different inner-scaled wall-normal distances: $y^+=[15,30,50,100]$.
\subsection{Super-resolution generative adversarial networks}
Super-resolution GAN (SRGAN) is proposed as a method to reconstruct turbulent wall-measurement fields.
Additionally, SRGANs are explored also for direct estimation of velocity fields in wall-parallel planes.
A typical SRGAN architecture consists of two networks: a generator ($G$) and a discriminator ($D$); $G$ is in charge of generating a high-resolution artificial image $\widetilde{H}_R$ from its low-resolution counterpart $L_R$, whereas $D$ is in charge of distinguishing between high-resolution real images $H_R$ and artificial ones.
Note that the purpose of this work is not to generate a custom architecture to tackle fluid-mechanics cases, since these types of DNNs are already available in the literature~\cite{deng2019super,werhahn2019multi,kim2021unsupervised}.
Therefore, the architecture presented in Ref.~\onlinecite{ledig2017photo} was used in this study.
It uses a CNN as generator, where the main core is composed of 16 residual blocks \cite{he2016deep}, and the resolution increase is carried out at the end of the network by means of $\log_2(f_d)$ sub-pixel convolution layers \citep{shi2016real}.
In the case of flow-field reconstruction with full-resolution wall data as input, the sub-pixel convolution layers are removed.
For the discriminator, convolution layers are also used before adding two fully-connected layers, using a sigmoid activation in the last one to obtain a probability to discern whether the high-resolution input is real or not.
A schematic view of the generator network is shown in Figure~\ref{fig:01}a) and the rest of details can be found in Ref.~\onlinecite{ledig2017photo}.
The discriminator loss is defined as:
\begin{equation}
\mathcal{L}_D=-\mathbb{E}[\log D(H_R)] - \mathbb{E}[\log(1-D(G(L_R)))].
\end{equation}
For the generator loss, we have used the perceptual loss \cite{ledig2017photo}, where the content loss is evaluated with the pixel-based mean-squared error between $H_R$ and $\widetilde{H}_{R}$, leading to:
\begin{equation}
\mathcal{L}_G=\frac{1}{N_xN_z}\sum^{Nx}_{i=1}\sum^{Nz}_{j=1}|G(L_R)_{i,j} - H_R{_{i,j}}|^2 - \lambda \mathcal{L}_D,
\end{equation}
\noindent where $N_x$ and $N_z$ are the number of grid points in the streamwise and spanwise directions for the high-resolution images (192 for both of them in our case) and $\lambda$ is a scalar to weight the value of the adversarial loss, set to $10^{-3}$.
The weights of the model for each downsampling case have been optimized for 30 epochs using the Adam algorithm~\cite{kingmaba} with learning rate $10^{-4}$.
\begin{figure*}
\centerline{
\includegraphics[width=\linewidth]{figs/figure01}}
\caption{Schematic view of the DNN architectures for a) generator network in SRGAN, and b) FCN-POD. The colour coding for each layer is: 2D-convolution \sy{conv}{b}, parametric-ReLU-activation \sy{prel}{b}, batch-normalization \sy{batc}{b}, sub-pix-convolution \sy{subp}{b}, ReLU-activation \sy{relu}{b}, and max-pooling \sy{pool}{b} layers. The kernel size and the number of filters are shown at the bottom of the convolution layers.}
\label{fig:01}
\end{figure*}
\subsection{POD-based fully-convolutional networks}
The baseline method for assessing the quality of flow reconstructions achieved by SRGAN is the FCN-POD approach \cite{guemes2019sensing,guastoni2020convolutional}.
This method divides the turbulent flow fields into $N_s$ two-dimensional subdomains of $N_p\times N_p$ grid points, and POD is performed on each of these subdomains.
The number of subdomains is chosen based on $Re_{\tau}$, with the purpose of ensuring that ~90\% of the flow kinetic energy is contained within $\mathcal{O}(10^2)$ POD modes which can be represented by convolutional filters.
For the $Re_{\tau}=180$ case, each field is divided into $12\times12$ subdomains, each of them with $16\times16$ grid points.
The proposed architecture will reconstruct this three-dimensional tensor of POD coefficients from the wall quantities; this tensor is later converted into the flow field by projecting the POD coefficients of each subdomain into its corresponding basis.
Note that this method does not ensure continuity between subdomains; nevertheless, the convolutional layers have been shown to provide reasonably smooth flow fields\cite{guastoni2020convolutional}.
For each wall-normal distance a different model has been used, the weights of which have been optimized for 30 epochs using the Adam optimizer \cite{kingmaba} with $\epsilon=0.1$, learning rate $10^{-3}$ and an exponential decay starting from epoch 10.
A schematic representation of the architecture is shown in Figure \ref{fig:01}b), and the rest of the implementation details can be found in Ref.~\onlinecite{guastoni2020convolutional}.
For the case of coarse input data, a modified version of the FCN-POD model has been used.
To deal with the different sizes of the input and output tensors, $\log_2(f_d)$ pooling layers have been removed from the original model.
\section{Assessment of resolution enhancement for wall measurements}\label{sec:wall}
The quality of the resolution enhancement of the wall fields is evaluated first.
Figure~\ref{fig:02} shows an instantaneous field of the streamwise and spanwise wall-shear-stress and pressure fluctuations for the DNS reference and the SRGAN predictions.
While the reconstructions from fields with $f_d=4$ and $f_d=8$ recover almost all the flow features present in the DNS references, the instantaneous field for $f_d=16$ exhibits loss of small-scale details.
Moreover, it appears that the high-intensity regions are attenuated for the latter case.
Note however, that the locations and sizes of the largest flow structures are very well represented even for $f_d=16$.
\begin{figure*}
\begin{center}
\includegraphics[width=\linewidth]{figs/pof_fig02.pdf}
\end{center}
\caption{Comparison of the wall-quantity fluctuating fields at $Re_{\tau} = 180$, scaled with their corresponding standard deviation. Reference DNS is reported at left panel, while the six-row panels report the different $f_d$ cases, covering $f_d=4$ (left), $f_d=8$ (center), and $f_d=16$ (right). Odd rows refer to low-resolution inputs, and even ones to the SRGAN predictions. Top two-row panels report streamwise wall-shear stress, middle ones report spanwise wall-shear stress and bottom ones refer to pressure fluctuations.}
\label{fig:02}
\end{figure*}
The first observations on the resolution-enhancement performance with respect to $f_d$ obtained from the inspection of instantaneous fields are confirmed when analyzing the mean-squared-error of those fields.
The errors, normalized with the standard deviation of each quantity, are reported in Table~\ref{tab:01}. SRGANs show excellent results for $f_d=4$ in the three wall quantities, and confirm the performance decay between $f_d=8$ and $f_d=16$.
When assessing the performance differences among wall quantities, it is clear that with larger downsampling factors the errors in the streamwise wall-shear-stress fields are lower than for the other two wall quantities.
This behaviour can be ascribed to the spatial organization of streawise wall-shear-stress fluctuations, which exhibit a characteristic alignment in the streamwise direction.
\begin{table}
\caption{\label{tab:01}Mean-squared-error in the instantaneous wall fields scaled with their corresponding standard deviations.}
\begin{ruledtabular}
\begin{tabular}{lccc}
$f_d$ & $\tau_{w_x}$ & $\tau_{w_z}$ & $p_w$\\
\hline
4 & 0.0187 & 0.0244 & 0.0153 \\
8 & 0.2240 & 0.3041 & 0.2741 \\
16 & 0.6531 & 0.7732 & 0.7461 \\
\end{tabular}
\end{ruledtabular}
\end{table}
The pre-multiplied two-dimensional inner-scaled spectra for the three wall quantities are reported in Figure~\ref{fig:03}.
The high-energy peak containing 90\% or more of $\tau_{w_x}$ is well captured by the predictions with $f_d=4$ and $f_d=8$, while for $f_d=16$ this is not recovered, even showing a significant attenuation of 50\% of the total energy content.
The energy attenuation is even stronger for $\tau_{w_z}$ and $p_w$, where the predictions of $f_d=4$ are the only ones capturing the energy distribution for both quantities.
In the case of $f_d=16$, the attenuation is so significant that even the 50\% energy-content level is not recovered.
The distribution of scales over a larger range also explains why the $\tau_{w_x}$ error is smaller than that of the other two wall quantities, since for the first one the SRGAN architecture deals with a lower parametric space.
\begin{figure}
\centerline{\includegraphics[width=\columnwidth]{figs/pof_fig03.pdf}}
\caption{Pre-multiplied two-dimensional power-spectral densities for a) fluctuating streamwise wall-shear-stress, b) fluctuating spanwise wall-shear-stress, and c) wall-pressure fluctuations. The contour levels contain 10\%, 50\% and 90\% of the maximum DNS power-spectral density. Shaded contours refer to the reference DNS data, while coloured lines denote $f_d=4$ \lcap{-}{fd04}, $f_d=8$ \lcap{-}{fd08}, and $f_d=16$ \lcap{-}{fd16}.}
\label{fig:03}
\end{figure}
Although the scope of this work is not to develop a customized SRGAN architecture for wall turbulence, here we briefly compare our results with those of other studies in the literature.
For example, Ref.~\onlinecite{kim2021unsupervised} used an unsupervised GAN to reconstruct wall-parallel velocity fields at $y^+=15$ and $y^+=100$ in a turbulent channel flow at $Re_{\tau}=1000$ with $f_d=8$.
They report good resolution-enhancing results in terms of instantaneous fields, turbulence statistics and one-dimensional spectra, similar to ours for the same $f_d$.
With respect to the spectra, their work and ours coincide in identifying the small-scale structures as those most difficult to recover.
Because of the different $Re_{\tau}$ in both studies, it is important to highlight that $f_d$ is a pixel ratio between the high- and low-resolution fields, and it does not take into account how many viscous lengths are contained in a single pixel.
For a fair comparison in turbulent flows, we propose the following normalized downsampling factor:
\begin{equation}
\tilde{f}_{d}=f_d\sqrt{\Delta x^{{+}^2} + \Delta z^{{+}^2}},
\label{eq:03}
\end{equation}
\noindent where $\Delta x^{+}$ and $\Delta z^{+}$ are the inner-scaled grid spacing in physical space for $x$ and $z$ respectively.
Using equation~(\ref{eq:03}) yields a normalized downsampling factor $\tilde{f}_{d}\approx 105$, while the work of Ref.~\onlinecite{kim2021unsupervised} tackles a problem with $\tilde{f}_{d}\approx 109$, therefore showing that the comparison is fair.
\section{Prediction of turbulent flow fields \\from coarse wall measurements}\label{sec:flow}
This section presents the reconstruction performance of wall-parallel velocity fields from wall measurements.
Reconstructions are performed at four different wall-normal distances: $y^+=[15,30,50,100]$, considering 4 downsampling cases for the input wall measurements: $f_d=[1,4,8,16]$.
Note that $f_d=1$ means that no information is lost at the wall with respect to the DNS reference.
Figure~\ref{fig:04} shows instantaneous fields of the streamwise velocity fluctuations at the four wall-normal distances of interest in this study.
Predictions generated with SRGAN and FCN-POD networks are compared to the DNS reference.
Note that the FCN-POD predictions are only reported for $f_d=1$ case, and they are analogous to the results presented in Ref.~\onlinecite{guastoni2020convolutional}.
Inspecting the fields, it can be seen that the best results are obtained closer to the wall, with the lowest downsamplings.
When moving away from the wall or reducing the information provided by the wall, the small-scale fluctuations in the fields start to disappear, and one of the networks recover the high-intensity fluctuating regions of the flow.
However, there is a clear performance difference between FCN-POD and SRGAN predictions.
While the loss of small-scale fluctuations is clearly observable at $y^+=30$ for the FCN-POD predictions, SRGAN is able to capture most of them at $y^+=50$.
It is not until $y^+=100$ that it is clearly observed that small-scale fluctuations are not recovered by the SRGAN architecture.
Nonetheless, the results of $f_d=8$ and $f_d=16$ at $y^+=15$ are successful in capturing most of the flow features present in the DNS reference, and the same can be said for $f_d=8$ at $y^+=30$.
Since most of the flow-control techniques actuate over this region \cite{choi1994active,lee1997application,bai2014active}, these results indicate that equally-distributed probes would be sufficient to feed flow information to these control techniques, instead of using image-based acquisition systems, which are more expensive and difficult to implement.
\begin{figure*}
\centerline{\includegraphics[width=\textwidth]{figs/pof_fig04.pdf}}
\caption{Countour map for the streamwise velocity fluctuation fields scaled with the corresponding standard deviation. From top to bottom, rows denote FCN-POD, reference DNS, and SRGAN predictions with $f_d=[1,4,8,16]$ for the wall information. From left to right, columns indicate $y^+=15$, $y^+=30$, $y^+=50$ and $y^+=100$.}
\label{fig:04}
\end{figure*}
A global view of the flow-reconstruction performance is provided in terms of mean-squared-error.
Figure~\ref{fig:05} reports the evolution of the error with respect to the wall-normal distance for the three flow quantities, the four $f_d$ values and the two reconstruction techniques.
There are two aspects to analyze: the performance difference between the two networks, and the evolution of the error with respect to $f_d$.
When comparing the error evolution for both networks, it can be seen that SRGAN outperforms FCN-POD predictions for all $f_d=4$ and $f_d=8$ cases, where the errors for the predictions generated with SRGAN are better than when using FCN-POD approach.
However, for the $f_d=16$ case both errors collapse, and therefore the benefit of using SRGAN disappears.
This deterioration of the flow reconstruction can be ascribed to the low amount of information contained by the coarse wall measurements.
In the case of $f_d=1$, the wall data contains information of the fluctuations with characteristic lengths as small as $\sim10\ell^*$ in the streamwise direction, while for $f_d=16$ this increases up to $\sim160\ell^*$.
While $f_d=4$ and $f_d=8$ recover the small scales present in the DNS reference, $f_d=16$ does not succeed in this task.
In any case, it is important to remark the significant accuracy improvement of SRGAN for $f_d=1$ with respect to the FCN-POD method, especially in the wall-normal and spanwise components.
Although not presented here, this improvement is noticeable also if compared with the FCN method used in Ref.~\onlinecite{guastoni2020convolutional}.
\begin{figure}
\centerline{\includegraphics[width=\columnwidth]{figs/pof_fig05.pdf}}
\caption{Mean-squared-error in the instantaneous fields of a) streamwise, b) wall-normal, and c) spanwise velocity fluctuations scaled with their corresponding standard deviations. Line styles refer to \lcap{-}{fd01} SRGAN, and \lcap{--}{fd01} FCN-POD predictions, respectively. Colours and symbols denote $f_d=1$ \sy{fd01}{o*}, $f_d=4$ \sy{fd04}{t*}, $f_d=8$ \sy{fd08}{dt*}, and $f_d=16$ \sy{fd16}{s*}.}
\label{fig:05}
\end{figure}
The second factor to analyze is the performance decay of the predictions when increasing $f_d$.
In a previous study\cite{guemes2019sensing}, the effect of $f_d$ when reconstructing the large-scale structures present in wall-parallel flow fields from wall measurements on a turbulent channel flow of $Re_{\tau}=1000$ was analyzed.
The analyzed effect of $f_d=[1,2,4]$ reported only a weak deterioration effect due to the increase of $f_d$.
However, the results presented in Figure~\ref{fig:06} show a clear dependency between $f_d$ and the mean-squared-error. Once again, the question arises whether $f_d$ is adequate to characterize the downsampling effect in wall turbulence.
If we used the normalized downsampling factor proposed in equation~(\ref{eq:03}), $f_d=4$ becomes $\tilde{f}_{d}\approx44$ for Ref.~\onlinecite{guemes2019sensing}, while in our case it is $\tilde{f}_{d}\approx52$, increasing to 105 and 210 for $f_d$ values equal to 8 and 16 respectively.
Therefore, it can be argued that in this work we are facing a more challenging wall-information loss.
Furthermore, it must be recalled that the flow scales to be predicted also affect the performance of the method.
Ref.~\onlinecite{guemes2019sensing} only targeted the flow scales in the first 10 POD modes, while this work targets the entire energy spectra.
The first 10 POD modes of Ref.~\onlinecite{guemes2019sensing} refer to the most energetic structures present in the flow.
Large coherent structures are more persistent over time, with lives proportional to their scale\cite{lozano2014time}.
These characteristic length and time scales make them less sensitive to the changes in the resolution of the wall data.
However, small-scale structures are affected, both because they are smaller than the scales contained in the coarse wall data and because the modulation effect of large scales~\cite{hutchins2007large,dogan_modulation} is also hidden by the low-resolution data.
The pre-multiplied two-dimensional energy spectra of the flow quantities at the four wall-normal locations discussed above are shown in Figure~\ref{fig:07}.
As reported in Ref.~\onlinecite{guastoni2020convolutional}, the amount of energy captured by the predictions decreases moving farther from the wall.
Moreover, it is important to note that the FCN-POD method is able to recover scales larger than the subdomain size, although a discontinuity in the spectra can be observed at that wavelength, especially in the wall-normal and spanwise components.
With respect to the effect of using SRGAN as a reconstruction method, the findings presented above are corroborated by the spectra.
The predictions generated with SRGAN recover a wider range of energetic scales in both the streamwise and spanwise wavelengths for the three velocity fluctuations, even up to the case $f_d=8$, while for $f_d=16$ both methods have been shown to provide the same mean-squared error.
Nonetheless, it is also important to mention that for $f_d=16$ at $y^+=100$ no energetic scales above the 10\% of the DNS reference has been recovered in the wall-normal and spanwise spectra.
This also occurs in the spanwise fluctuation spectra at $y^+=50$, but only for the predictions generated without SRGAN.
\begin{figure*}
\centerline{\includegraphics[width=\textwidth]{figs/pof_fig06.pdf}}
\caption{Pre-multiplied two-dimensional power-spectral densities for streamwise (first row), wall-normal (second row), and spanwise (third row) velocity fluctuations. From left to right, columns refer to inner-scaled wall distance $y^+$ equal to 15, 30, 50, and 100. The contour levels contain 10\%, 50\% and 90\% of the maximum DNS power-spectral density. Shaded contours refer to the reference DNS data, while contour lines refer to \lcap{-}{fd01} SRGAN-FCN-POD, and \lcap{--}{fd01} FCN-POD predictions, respectively. Colours denote $f_d=1$ \sy{fd01}{s*}, $f_d=4$ \sy{fd04}{s*}, $f_d=8$ \sy{fd08}{s*}, and $f_d=16$ \sy{fd16}{s*}.}
\label{fig:06}
\end{figure*}
While Figure~\ref{fig:04} shows a large error for the extreme cases i.e., those reconstructions farther from the wall or with high $f_d$ values, a visual inspection of Figure~\ref{fig:05} reveals that even in these case the SRGAN predictions are able to capture the large-scale organization of the instantaneous flow field.
Therefore, the large observed errors can be ascribed to the attenuation of the velocity fluctuations, which makes it necessary to define a metric that evaluates the error based on the scale wavelengths.
Following Ref.~\onlinecite{encinar2019logarithmic}, a spectral fractional error can be defined as:
\begin{equation}
R_{ab}(k_x,y,k_z) = \frac{\mathcal{R}e\langle(a-a^{\dagger})(b-b^{\dagger})^*\rangle(k_x,y,k_z)}{\mathcal{R}e\langle ab^*\rangle(k_x,y,k_z)},
\label{eq:04}
\end{equation}
\noindent where $k_x$ and $k_z$ are the wave numbers in the streamwise and spanwise directions respectively, superscripts `*' and `$\dagger$' refer to complex conjugate and estimated quantities respectively, $\mathcal{R}e$ denotes real part, while $a$ and $b$ stand for either $u$, $v$, or $w$.
Note that equation~(\ref{eq:04}) is related to the linear cohrence spectrum \cite{baars2016spectral,encinar2019logarithmic,tanarro2020effect}.
Figure~\ref{fig:07} shows the iso-contours of $R_{ab}=0.5$ for each case.
The reconstruction performance of $f_d=1$ and $f_d=4$ in the viscous region, which covers entirely the wavelengths above 10\% energy content of the streamwise fluctations, is particularly remarkable.
It can also be observed that the farther from the wall the fewer small scales are recovered in the reconstruction.
A similar behaviour is observed when increasing the downsampling factor $f_d$.
\begin{figure*}[!t]
\centerline{\includegraphics[width=\textwidth]{figs/pof_fig07.pdf}}
\caption{Fractional spectral error for streamwise (first row), wall-normal (second row), and spanwise (third row) velocity fluctuations. From left to right, columns refer to inner-scaled wall-normal locations $y^+$ equal to 15, 30, 50, and 100. The contour level corresponds to $R_{ab}=0.5$. Contour lines refer to \lcap{-}{fd01} SRGAN, and \lcap{--}{fd01} FCN-POD predictions, respectively. Colours denote $f_d=1$ \sy{fd01}{s*}, $f_d=4$ \sy{fd04}{s*}, $f_d=8$ \sy{fd08}{s*}, and $f_d=16$ \sy{fd16}{s*}. Shaded contours refer to pre-multiplied two-dimensional power-spectral densities for the reference DNS data.}
\label{fig:07}
\end{figure*}
Since the SRGAN predictions in the most challenging configurations exhibit some similarities with the filtered fields of the DNS reference, it is of interest to conduct the comparison.
Low-pass filtering has been applied to the DNS reference, where the cut-off lengths are adjusted to retain those scale with $R_{ab}<0.5$.
Figure~\ref{fig:08} shows this comparison for the case at $y^+=50$ with $f_d=8$ at the wall input data.
While the SRGAN prediction does not yield small-scale details, it exhibits a remarkable resemblance in terms of the streak patterns when compared with the filtered DNS.
Note that for the case of Figure~\ref{fig:08}, the cut-off wavelengths are set to $\lambda_x^+\approx500$ and $\lambda_z^+\approx100$.
If the mean-squared error displayed in Figure~\ref{fig:05} is computed with this filtered reference, the error reduces from 0.603 to 0.317 for the case of Figure~\ref{fig:08}.
\begin{figure}
\centerline{\includegraphics[width=\columnwidth]{figs/pof_fig08.pdf}}
\caption{Countour map of a sample streamwise velocity fluctuation field scaled with the corresponding standard deviation at $y^+=50$. Top and bottom panels denote reference filtered DNS and SRGAN prediction with $f_d=8$, respectively.}
\label{fig:08}
\end{figure}
\section{Summary and conclusions}\label{sec:concl}
The reconstruction of wall-parallel velocity fields from coarse measurements at the wall in a wall-bounded turbulent flow has been evaluated in this work, together with the resolution enhancement of the wall measurements.
For that purpose, SRGAN has been proposed for both tasks.
In the case of flow reconstruction from wall measurements, this architecture has been compared with the FCN-POD method proposed by Ref.~\onlinecite{guastoni2020convolutional}.
The resolution enhancement of wall fields from their coarse counterparts has been carried out with donwsampling factors $f_d=[4,8,16]$.
In the case of flow reconstruction, the methods have been evaluated at the following wall-normal locations: $y^+=[15,30,50,100]$ with wall downsampling factors $f_d=[1,4,8,16]$.
SRGAN is shown to provide accurate reconstructions for the case of resolution enhancement of wall fields at $f_d=[4,8]$.
In the most challenging case $(f_d=16)$, it can be observed that small-scale contributions are not recovered in the reconstruction, but the large-scale footprint of the flow at the wall is very well represented.
With respect to the flow reconstruction with full resolution at the wall, SRGAN is shown to provide a significant improvement with respect to the baseline FCN-POD method\cite{guastoni2020convolutional}.
The effect of increasing $f_d$ is also evaluated, showing a clear performance decrease unlike in the work of Ref.~\onlinecite{guemes2019sensing}, where only a weak effect is reported.
This difference is ascribed to two reasons: First, the range of scales targeted in Ref.~\onlinecite{guemes2019sensing} only cover the large wavelengths, while this study does it for the entire spectrum.
Small-scale structures have characteristic time and length scales smaller than that of the filtering bandwidth from the coarse measurements, thus losing relevant information for the reconstruction.
Second, $f_d$ is not an adequate parameter to compare different databases of wall-bounded turbulent flows.
To overcome this issue, we propose to use $\tilde{f}_{d}$, which takes into account the fraction of viscous length covered by a pixel.
With this parameter, the effect of the downsampling is homogenized among the various works, showing a clear trend between the results of Ref.~\onlinecite{guemes2019sensing} and the ones presented here.
To the authors' knowledge this is the first study where DNNs are used to reconstruct flow fields from coarse wall measurements in a turbulent flow, and this approach has great potential in the context of closed-loop control.
In any case, it is observed that the accuracy improvements of SRGAN start to decrease when increasing $f_d$.
The capability of SRGAN to recover the large-scale structures present in the flow is also evaluated by means of the fractional spectral error\cite{encinar2019logarithmic} and assessment of filtered instantaneous fields.
It is shown that the SRGAN predictions are in good agreement with large-scale patterns obtained from the filtered DNS reference.
The present study has used high-resolution DNS data to train the proposed GAN network.
However, the computational cost and requirements for producing this data increase with $Re_{\tau}$, being impossible to obtain for Reynolds numbers from real-life applications.
Consequently, it is advisable to look for alternatives that allow the use of the proposed network in real-life scenarios.
For instance, transfer learning could be explored as in Guastoni et al. (2021), to confirm that GANs are able to generalize from one $Re_{\tau}$ to another.
Another option could be to rely on experimental data obtained from those real-life scenarios.
However, experimental data might be contaminated by noise and have lower spatial and temporal resolution than well-resolved DNS data.
Therefore, future investigations should focus on how to combine different neural networks with incomplete or noisy turbulent data.
\section*{Acknowledgements}
RV acknowledges the support by the G\"oran Gustafsson Foundation.
SD and AI acknowledge the support by the European Research Council, under the COTURB grant ERC-2014.AdG-669505.
HA acknowledges the support by Wallenberg AI, Autonomous Systems, and Software Program (WASP-AI).
We would also like to acknowledge Hampus Tober for useful discussions throughout the present study.
\section*{Data Availability Statement}
The data that support the findings of this study are available from the corresponding author upon reasonable request.
The codes that support the findings of this study are openly available in GitHub at https://doi.org/10.5281/zenodo.5067426.
|
1,314,259,993,629 | arxiv | \section{Introduction}
Automatic generation and tuning of convolutional neural network (CNN) architectures is a growing research topic. The majority of approaches in the literature (for a deep overview, see Section~\ref{sec:related_works}) are rooted into the fundamental idea of large-scale explorations; more precisely, they can be based either on evolution and mutations~\cite{real2017large,DBLP:journals/corr/XieY17,DBLP:journals/corr/MiikkulainenLMR17}, or on reinforcement learning~\cite{DBLP:journals/corr/abs-1708-05552,DBLP:journals/corr/ZophL16,zoph2017learning,cai2018efficient,baker}. All these algorithms require a large amount of training experiments which quickly leads to massive resource and time to solution requirements.
Recently, the concept of performance prediction for architecture search has emerged. The fundamental idea is to drastically reduce exploration cost, by forecasting accuracy of networks without (or with very limited) training. Prediction is obtained either from partial learning curves~\cite{lce,bnn,svr,freeze}, or from a database of trained experiments~\cite{peephole}. The former approach requires partial training of each specific network. The latter one, implies training hundreds of networks on the given input dataset, to build a reliable ground-truth. Thus, none of them can be used out-of-the-box for near real-time architecture search.
In this work, we introduce a train-less accuracy predictor for architecture search~(\TAPAS), that provides reliable architecture peak accuracy predictions when used with unseen (i.e., not previously seen by the predictor) datasets. This is achieved by adapting the prediction to the \emph{difficulty} of the dataset, that is automatically determined by the framework. In addition, we reuse experience accumulated from previous experiments. The main features of our framework are summarized as follows: (i)~it is not bounded to any specific dataset, (ii)~it learns from previous experiments, whatever dataset they involve, improving prediction over usage and (iii)~it allows to run large-scale architecture search on a single GPU device within a few minutes.
In summary, our main contributions are the following:
\begin{itemize}
\item a fast, scalable, and reliable framework for CNN architecture performance prediction;
\item a flexible prediction algorithm, that dynamically adapts to the \emph{difficulty} of the input;
\item an extensive comparison with preexisting methods/results, clearly illustrating the advantages of our approach.
\end{itemize}
The outline of the paper is the following: in Section~\ref{sec:related_works} we briefly review literature approaches and analyze pros and cons of each of them. In Section~\ref{sec:methodology} we present the design of our prediction framework, with a deep dive into its three main components. Then, in Section~\ref{sec:experiments} we compare experimental results with current state-of-the-art. Finally, conclusions are summarized in Section~\ref{sec:conclusion}.
\section{Related work}\label{sec:related_works}
This paper follows a similar design idea as Peephole~\cite{peephole}, which predicts a network accuracy by only analyzing the network structure.
Similar to our approach, a long short-term memory (LSTM) based network receives a layer-by-layer encoding. In contrast to our approach, they encode an epoch number and predict the accuracy at the given epoch. Peephole delivers good performance on MNIST and CIFAR-10, however it has not been designed to transfer knowledge from familiar datasets to unseen ones. Given a new dataset, hundreds of networks need to be trained before Peephole makes a prediction. In contrast, our framework is designed to operate on unseen datasets, without the need of expensive training.
Accuracy predictors such as learning curves extrapolation (LCE)~\cite{lce}, BNN~\cite{bnn}, $\nu$-SVR~\cite{svr} forecast network performance based on partial learning curves. These algorithms are designed in the context of hyperparameter optimization or meta-learning. Both cases require extensive use of training and thus result in high computational costs. Moreover, they are all dataset and network specific, i.e., the prediction cannot be transferred to another network or dataset, without re-training. In particular, LCE employs a weighted probabilistic model to predict network performance. BNN uses Bayesian Neural Networks to fit completely new learning curves and extrapolate partially observed ones. This approach yields superior performance compared to~LCE, particularly at stages where the initial observed learning curve is not sufficient for the parametric algorithm to converge. Nevertheless, both methods rely on expensive Markov~Chain Monte~Carlo sampling procedures.
$\nu$-SVR~\cite{svr} complements the information on the learning curve with network architecture details and a list of predefined hyperparameters. These are used to train a sequence of regression models, that outperform LCE and~BNN. Although these methods exhibit good performance, they require a considerable part of the initial learning curve to provide reliable performance.
Large-scale exploration algorithms~\cite{real2017large,DBLP:journals/corr/XieY17,DBLP:journals/corr/MiikkulainenLMR17,DBLP:journals/corr/abs-1708-05552,DBLP:journals/corr/ZophL16,zoph2017learning,cai2018efficient,baker,pham2018efficient} employ genetic mutations or reinforcement learning to explore a large space of architecture configurations.
Regardless of the approach, all these methods train a large number of networks, some of them employing hundreds of GPUs for more than ten days~\cite{real2017large}. ENAS~\cite{pham2018efficient} uses a controller to discover CNN architectures, by searching for an optimal subgraph within a large computational graph. With this approach it discovers a $97.11\%$ accurate network for CIFAR-10, on a single GPU in 10~hours. While ENAS reduces drastically the time-to-solution compared to previous results, the model is applied to only one dataset and not generalized to the case of multiple datasets. Indeed, sharing parameters among child models for different datasets is not straightforward.
\section{Methodology} \label{sec:methodology}
In this section, we provide a detailed overview of the main building blocks of the \TAPAS framework. \TAPAS aims to reliably estimate peak accuracy at low cost for a variety of CNN architectures. This is achieved by leveraging a compact characterization of the user-provided input dataset, as well as a dynamically growing database of trained neural networks and associated performance.
The \TAPAS framework, depicted in Figure~\ref{fig:tapas_workflow}, is built on three main components:
\begin{enumerate}
\item \textbf{Dataset Characterization (\DCC):}
Receives an unseen dataset and computes a scalar score, namely the Dataset Characterization Number (DCN\xspace)~\cite{dataset_char}, which is used to rank datasets;
\item \textbf{Lifelong Database of Experiments (\LCDB):}
Ingests training experiments of NNs on a variety of image classification datasets executed inside the \TAPAS framework;
\item \textbf{Train-less Accuracy Predictor (\AP):} Given an NN architecture and a DCN\xspace, it predicts the potentially reachable peak accuracy without training the network.
\end{enumerate}
\begin{figure}[!t]
\includegraphics[width=\textwidth]{./fig/tapas_framework_2.pdf}
\caption{Schematic \TAPAS workflow. First row: the Dataset Characterization (\DCC) takes a new, unseen dataset and characterizes its difficulty by computing the Dataset Characterization Number (DCN\xspace). This number is then used to select a subset of experiments executed on similarly difficult datasets from the Lifelong Database of Experiments (\LCDB). Subsequently, the filtered experiments are used to train the Train-less Accuracy Predictor (\AP), an operation that takes up to a few minutes. Second row: the trained \AP takes the network architecture structure and the dataset DCN\xspace and predict the peak accuracy reachable after training. This phase scales very efficiently in a few seconds over a large number of networks.}
\vspace*{-1em}
\label{fig:tapas_workflow}
\end{figure}
\subsection{Dataset characterization (\DCC)} \label{sec:dataset_characterization}
The same CNN can yield different results if trained on an easy dataset (e.g., MNIST~\cite{data_mnist}) or on a more challenging one (e.g., CIFAR-100~\cite{data_cifar10_100}), although the two datasets might share features such as number of classes, number of images, and resolution. Therefore, in order to reliably estimate a CNN performance on a dataset we argue that we must first analyze the dataset difficulty. We compute the DCN\xspace by training a \emph{probe net} to obtain a dataset difficulty estimation~\cite{dataset_char}. We use the DCN\xspace for filtering datasets from the \LCDB and directly as input score in the \AP training and prediction phases as described in Section~\ref{sec:ap}.
\subsubsection{DCN\xspace computation}
\emph{Prob nets} are modest-sized neural networks designed to characterize the difficulty of an image classification dataset~\cite{dataset_char}. We compute the DCN\xspace as peak accuracy, ranged in $[0, 1]$, obtained by training the \textit{Deep normalized ProbeNet} on a specific dataset for ten epochs. The DCN\xspace calculation cost is low due the following reasons: (i)~\textit{Deep norm ProbeNet} is a modest-size network, (ii)~the characterization step is performed only once at the entry of the dataset in the framework (the \LCDB stores the DCN\xspace afterwards), (iii)~the DCN\xspace does not require an extremely accurate training, thus reducing the cost to a few epochs, and (iv)~large datasets can be subsampled both in terms of number of images and of pixels.
The DCN\xspace is a rough estimation of the dataset difficulty, and is thus tolerant to approximations. In Section~\ref{sec:experiments} we provide evidence of the effect of the DCN\xspace on the \AP.
\subsection{Lifelong database of experiments (\LCDB)}\label{sec:LDE}
\LCDB is a continuously growing DB, which ingests every new experiment effectuated inside the framework. An experiment includes the CNN architecture description, the training hyper-parameters, the employed dataset (with its DCN\xspace), as well as the achieved accuracy.
\subsubsection{\LCDB initialization}\label{sec:lde_initialization}
At the very beginning, the \LCDB is empty. Thus we perform a massive initialization procedure to populate it with experiments.
For each available dataset in Figure~\ref{fig:datasets} we sample 800~networks from a slight variation of the space of MetaQNN~\cite{baker}. For convolution layers we use strides with values in $\{1, 2\}$, receptive fields with values in $\{3, 4,.. 256\}$, padding in $\{same, valid\}$ and whether is batch normalized or not. We also add two more layer types to the search space: residual blocks and skip connections. The hyperparameters of the residual blocks are the receptive field, stride and the repeat factor. The receptive field and the stride have the same bounds as in the convolution layer, while the repeat factor varies between 1 and 6 inclusively. The skip connection has only one hyperparameter, namely the previous layer to be connected to.
To speed up the process, we train the networks one layer at a time using the incremental method described in~\cite{incremental}. In this way we obtain the accuracies of all intermediary sub-networks at the same cost of the entire one. To facilitate the \AP, we train all networks with the same hyper-parameters, i.e., same optimizer, learning rate, batch size, and weights initiallizer. Although the fixed hyper-parameter setting seems a strong limitation and might limit peak accuracy by a few percent, it is enough to trim poorly performing networks and, in the case of an architecture search, to fairly rank competitive networks, the performance of which can later be optimized further, as discussed in Section~\ref{sec:experiments}. As data augmentation we use standard horizontal flips, when possible, and left/right shifts with four pixels. For all datasets we perform feature-wise standardization.
\subsubsection{\LCDB selection}\label{sec:lde_prefiltering}
Let us consider an \LCDB populated with experiments from $N_d$~different datasets $D_j$, with $j=1,\dots,N_d$. Given a new input dataset~$\hat{D}$ and its corresponding characterization $\textrm{DCN\xspace}(\hat{D})$, the \LCDB block returns all experiments performed with datasets that satisfy the following relation
\begin{equation}
\| \textrm{DCN\xspace}(\hat{D}) - \textrm{DCN\xspace}(D_j) \| \leq \tau \qquad j \in [1,N_d],
\label{eq:florian_DCN}
\end{equation}
where $\tau$ is a predefined threshold that, in our experiments, is set to 0.05.
\subsection{Train-less accuracy predictor (\AP)}\label{sec:ap}
\AP is designed to perform fast and reliable CNN accuracy predictions. Compared to Peephole~\cite{peephole}, \AP leverages knowledge accumulated through experiments of datasets of similar difficulty filtered from the \LCDB based on the DCN\xspace. Additionally, \AP does not first analyze the entire NN structure and then makes a prediction, but instead performs an iterative prediction as depicted in Figure~\ref{fig:encoding}. In other words, it aims to predict the accuracy of a sub-network $l_{1:i+1}$, assuming the accuracy of the sub-network $l_{1:i}$ is known. The main building elements of the predictor are: (i) a compact encoding vector that represents the main network characteristics, (ii) a quickly-trainable network of LSTMs, and (iii) a layer-by-layer prediction mechanism.
\subsubsection{Neural network architecture encoding} \label{sec:network_encoding}
Similar to Peephole, \AP employs a layer-by-layer encoding vector as described in Figure~\ref{fig:encoding}. Unlike Peephole, we encode more complex information of the network architecture for a better prediction.
\begin{figure}[!t]
\includegraphics[width=\textwidth]{./fig/encoding_and_prediction.pdf}
\caption{Encoding vector structure and its usage in the iterative prediction. a) The encoding vector contains two blocks: $i$-layer information and from input to $i$-layer sub-network information. b) The encoding vector is used by the \AP following an iterative scheme. Starting from Layer~1 (input) we encode and concatenate two layers at a time and feed them to the \AP. In the concatenated vector, the \emph{Accuracy} field $\textrm{A}_i$ of $l_i$ is set to the predicted accuracy obtained from the previous \AP evaluation, whereas the one of $A_{i+1}$ corresponding to $l_{i+1}$ is always set to zero. For the input layer, we set $\textrm{A}_0$ to $1/N_c$, where $N_c$ is the number of classes, assuming a random distribution. The final predicted accuracy $A_{N_l}$ is the accuracy of the complete network.}
\label{fig:encoding}
\vspace*{-1.0em}
\end{figure}
Let us consider a network with $N_l$~layers, $l_i$ being the $i$-th layer counting from the input, with $i=1,\dots,N_l$. We define a CNN sub-network as $l_{a:b}$ with $1 \le a < b \le N_l$. Our encoding vector contains two types of information as depicted in Figure~\ref{fig:encoding}~a): (i)~$i$-th layer information and (ii)~$l_{1:i}$~sub-network information. For the current $i$-th layer we make the following selection of parameters: \emph{Layer type} is a one-hot encoding that identifies either convolution, pooling, batch normalization, dropout, residual block, skip connection, or fully connected. Note that for the shortcut connection of the residual block we use both the identity and the projection shortcuts~\cite{resnet}. The projection is employed only when the residual block decreases the number of filters as compared to the previous layer. Moreover, as compared to~\cite{peephole}, our networks do not follow a fixed skeleton in the convolutional pipeline, allowing for more generality. We only force a fixed block at the end, by using a global pooling and a fully connected layer to prevent networks from overfitting~\cite{nin}.
The \emph{ratio between the output height and input height} of each layer accounts for different strides or paddings, whereas the \emph{ratio between the output depth and input depth} accounts for modifications of the number of kernels. The \textit{number of weights} specifies the total of learnable parameters in $l_i$. This value helps the \AP differentiate between layers that increase the learning power of the network (e.g., convolution, fully connected layers) and layers that reduce the dimensionality or avoid overfitting (e.g., pooling, dropout). In the second part of the encoding vector, we include: \emph{Total number of layers}, counting from input to $l_i$, \emph{Inference FLOPs} and \emph{Inference memory} that are an accurate estimate of the computational cost and memory requirements of the sub-network, and finally \emph{Accuracy}, which is set either to 1/$N_c$, for the first layer, where $N_c$ is the number of classes to predict, zero for prediction purposes, or a specific value $\mathrm{A}_i \in [0, 1]$ that is obtained from the previous layer prediction. Before training, we perform a feature-wise standardization of the data, meaning that for each feature of the encoding vector, we subtract the mean and divide by the standard deviation.
\subsubsection{\AP architecture}
\AP is a neural network consisting of two stacked LSTMs of 50 and 100 hidden units, respectively, followed by a single-output fully connected layer with sigmoid activation. The \AP network has two inputs. The first input is a concatenation of two encoding vectors corresponding to layer $l_i$ and $l_{i+1}$, respectively. This input is fed into the first LSTM. The second input is the DCN\xspace and is concatenated with the output of the second LSTM and then fed into the fully connected layer.
\subsubsection{\AP training}\label{sec:training}
\AP requires a significant amount of training data to make reliable predictions. The \LCDB provides this data as described in Section~\ref{sec:LDE}. As mentioned above, all our generated networks are trained in an incremental fashion, as presented in \cite{incremental}, meaning that for each network of length $N_l$ we train all intermediary sub-networks $l_{1:k}$ with $1 < k \leq N_l$ and save their performance $A_{k}$. We encode each set of two consecutive layers $l_i$ and $l_{i+1}$ following the schema detailed in \ref{sec:network_encoding}, setting the accuracy field in the encoding vector of $l_i$ to $A_i$, which was obtained through training, and aiming to predict $A_{i+1}$.
\AP is trained with RMSprop~\cite{rmsprop}, using a learning rate of $10^{-3}$, a HeNormal weight initialization~\cite{heinit}, and a batch size of 512. As the architecture of the \AP is very small, the training process is of the order of a few minutes on a single GPU device. Moreover, the trained \AP can be stored and reapplied to other datasets with similar DCN numbers without the need for retraining.
\subsubsection{\AP prediction}\label{sec:tap_prediction}
\AP employs a layer-by-layer prediction mechanism. The accuracy~$\textrm{A}_{i}$ of the sub-network~$l_{1:i}$ predicted by the previous \AP evaluation is subsequently fed as input into the next \AP evaluation, which returns the predicted accuracy~$\textrm{A}_{i+1}$ of the sub-network~$l_{1:i+1}$. This mechanism is described more in detail in Figure~\ref{fig:encoding} b).
\section{Experiments} \label{sec:experiments}
In this section, we demonstrate \TAPAS performance over a wide range of experiments. Results are compared with reference works from the literature.
All runs involve single-precision arithmetic and are performed on
IBM\footnote{\fontsize{8}{6}\selectfont IBM, the IBM logo, ibm.com, OpenPOWER are trademarks or registered trademarks of International Business Machines Corporation in the United States, other countries, or both. Other product and service names might be trademarks of IBM or other companies.} POWER8 compute nodes, equipped with four NVIDIA~P100 GPUs.
\subsection{Dataset selection for \LCDB initialization}
All the experiments are based on a \LCDB populated with nineteen datasets, ranked by difficulty in Figure~\ref{fig:datasets}. Eleven of them are publicly available. The other eight are generated by sub-sampling the ImageNet dataset~\cite{imagenet_cvpr09} varying the number of classes and the number of images per class. The result is a finer distribution of datasets per DCN\xspace value, that improves the predictions by biasing \AP closer to the relevant data, particularly for the leave-one-out cross-validation experiment presented later. Additional details are provided in the Appendix.
We resize all dataset images to $32 \times 32$ pixels. On the one hand, this reduces the cost of \LCDB initialization, on the other hand, it allows us to potentially test networks and datasets in an \emph{All2All} fashion. We remark that this choice does not lead to a loss of generality, as images of different sizes can be employed in the same pipeline.
For every dataset, we generate 800~networks based on the procedure described in Section~\ref{sec:lde_initialization}. All networks are trained under the same settings: RMSprop optimizer with a learning rate of $10^{-3}$, weight decay $10^{-4}$, batch size 64, and HeNormal weight initialization.
\begin{figure}[!t]
\centering
\includegraphics[width=0.7\textwidth]{./fig/dataset_ranking_2.pdf}
\caption{List of image classification datasets used for characterization. The datasets are sorted by the DCN\xspace value from the easiest (left) to the hardest (right).
}
\label{fig:datasets}
\vspace*{-1em}
\end{figure}
\subsection{\TAPAS performance evaluation} \label{sec:experiments_predictor}
In this section, we define three different scenarios to compare \TAPAS with LCE~\cite{lce}, BNN~\cite{bnn}, $\nu$-SVR~\cite{svr} and Peephole~\cite{peephole}. We employ three evaluation metrics: (i)~the \emph{mean squared error (MSE)}, which measures the difference between the estimator and what is estimated, (ii)~\emph{Kendall's Tau (Tau)}, which measures the similarities of the ordering between the predictions and the real values, and (iii)~the \emph{coefficient of determination} ($R^2$), which measures the proportion of the variance in the dependent variable that is predictable from the independent variable. In the first metric, lower is better (zero is best); in the others, higher is better (one is best).
\subsubsection{Scenario A: Prediction based on experiments on a single dataset}
\begin{figure}[!t]
\centering
\includegraphics[width=\textwidth]{./fig/scenario_a_and_b_2.pdf}
\caption{Superior predictive performance of \AP compared with state-of-the-art methods, both when trained on only one dataset (Scenario A) or on multiple datasets (Scenario B).}
\label{fig:scenarios_a_and_b}
\vspace*{-1em}
\end{figure}
We train the \AP on a filtered list of experiments from the \LCDB based on the CIFAR-10 dataset. We recognize that this scenario is very favorable for prediction, however it is used in reference publications, and therefore allows for a fair comparison.
We perform ten-fold cross validation and present the results for Peephole, LCE, and \TAPAS in the first row of Figure~\ref{fig:scenarios_a_and_b}. For BNN and $\nu$-SVR we rely on published numbers. When presented with 20\% of the initial learning curve, BNN states an MSE of 0.007, while $\nu$-SVR states an $R^2$ of 0.9. \AP outperforms all methods, in terms of all the considered metrics. Moreover, if we modify \AP to not use the DCN\xspace, we still get better predictions than with all the other methods. The \AP prediction performance is not strongly affected because the training and prediction involve only one dataset.
We argue that the lower results of the Peephole method, as compared to the original paper, are due to the more complicated structure of the network we used in our benchmark. Specifically, the Peephole-encoding tuple (layer type, kernel height, kernel width, channels ratio) is not sufficient to predict complicated structures like ResNets.
\subsubsection{Scenario B: Prediction based on experiments on all datasets}
This scenario is similar to Scenario~A, but we do not filter experiments by dataset. The second row of Figure~\ref{fig:scenarios_a_and_b} shows results when \AP is trained on all datasets, regardless of their DCN\xspace. Also in this scenario, \AP outperforms all methods in all of the considered metrics. We recognize that Peephole is designed to be dataset-specific. However, compared to \AP without DCN\xspace the comparison is fair, as neither of these algorithms contain information about the dataset difficulty.
\subsubsection{Scenario C: Prediction based on experiments on unseen datasets}
This scenario aims (i)~to demonstrate \TAPAS performance when targeting completely unseen datasets and (ii)~to highlight importance of dataset-difficulty characterization and \LCDB pre-filtering. To do that, we consider the list of datasets in Figure~\ref{fig:datasets} and perform eleven leave-one-out cross-validation benchmarks, considering only the real datasets.
The result of this experiment is presented in Figure~\ref{fig:scenario_c}. From left to right, we observe the cumulative impact of the DCN\xspace awareness in the \AP training, as well as of the pre-filtering of the experiments in the \LCDB according to~\eqref{eq:florian_DCN}. Moreover, by comparing the rightmost plot and metrics with previous results in Figure~\ref{fig:scenarios_a_and_b}, we observe that \TAPAS performance does not diminish significantly when applied to an unknown dataset.
\begin{figure}
\centering
\includegraphics[width=0.9\textwidth]{./fig/scenario_c.pdf}
\caption{Predicted vs real performance (i.e., after training) for Scenario~C. Left plot: \AP trained without DCN\xspace or \LCDB pre-filtering. Middle plot: \AP trained with DCN\xspace, but \LCDB is not pre-filtered. Right plot: \AP trained only on \LCDB experiments with similar dataset difficulty, according to \eqref{eq:florian_DCN}.}
\label{fig:scenario_c}
\vspace*{-0.7em}
\end{figure}
\subsection{Simulated large-scale evolution of image classifiers} \label{sec:search}
The \AP can be plugged into any large-scale evolution algorithm to perform train-less architecture search. In this work, we use the genetic algorithm introduced in~\cite{real2017large}. As described in the original paper, the evolution algorithm begins with a small population, consisting of one~thousand single-layered networks. After training, two candidates are randomly chosen from the population: the less accurate one is removed, whereas the other one undergoes a mutation. The mutated network is evaluated in roughly 30 epochs and then put back in the population. The operation repeats until convergence is achieved.
The above algorithm is very expensive: 250~parallel workers are used for training the population and the entire process takes 256~hours~\cite[Figure~1]{real2017large}. The \AP can simulate the large-scale evolution search in only 400~seconds on a single GPU~device performing 20k~mutations. We employ the same mutations as in~\cite{real2017large}, apart from those that do not make sense in a simulation, such as altering the learning rate and resetting the weights. No network is trained during the entire process.
Figure~\ref{fig:large_scale_evolution} presents results of the simulated evolution for both CIFAR-10 and CIFAR-100 datasets. To verify that the \AP discovers good networks, we select the top three networks (according to accuracy prediction) and train them a-posteriori. For CIFAR-10 our best network reaches 93.67\%, whereas for CIFAR-100 we achieve 81.01\%, an improvement of 4\% w.r.t.\ the reference work~\cite{real2017large}. Moreover, we observe that all the top three networks perform well, and prediction values are reasonably close to those after training.
\begin{figure}
\centering
\begin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=\linewidth]{./fig/cifar10_evolution.pdf}
\label{fig:sub1}
\end{subfigure}%
\begin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=\linewidth]{./fig/cifar100_evolution.pdf}
\label{fig:sub2}
\end{subfigure}
\vspace*{-0.7em}
\caption{Simulation of large-scale evolution, with 20k~mutations. The table compares top three networks (predicted and trained accuracy) with reference work~\cite{real2017large}. The simulations require only $2\times10^{11}$ FLOPs per dataset, while training the top-three networks for 100 epochs is an additional $3\times10^{15}$ FLOPs, causing a 6 hour runtime on a single GPU. The reference work employs $9\times10^{19}$~(CIFAR-10) and $2\times10^{20}$ FLOPs (CIFAR-100) causing a runtime of 256 hours on 250 GPUs.}
\label{fig:large_scale_evolution}
\vspace*{-1em}
\end{figure}
\section{Conclusion}\label{sec:conclusion}
In this paper we propose \TAPAS, a novel prediction framework that given a CNN architecture, accurately forecasts its performance at convergence (i.e., peak validation accuracy) for any given input dataset. \TAPAS know-how originates from a lifelong database of experiments, based on a wide variety of datasets. Reliance on dataset-\emph{difficulty} characterization, is our key differentiation to outperform state-of-the-art methods by a large margin. Indeed, we demonstrated that \TAPAS outperforms preexisting methods, both in the favourable case when the methods are tuned for a specific dataset, as well as when they are applied on a wide range of datasets, without any bias.
\TAPAS does not require new training experiments, even in the case scenario when it is applied to a completely new dataset.
This facilitates large-scale network architecture searches, that do not require executions of training jobs. Indeed, \TAPAS enabled us to identify very accurate CNN architectures, in a few minutes, using only a single GPU. This is a performance that is several orders of magnitude faster than any training-based approach.
\small
|
1,314,259,993,630 | arxiv | \section{Introduction}
Surfers on the Internet frequently use search engines to find pages satisfying their query.
However, there are typically hundreds or thousands of relevant pages available on the
Web. Thus, listing them in a proper order is a crucial and non-trivial task. One can
use several criteria to sort relevant answers. It turns out that the link-based criteria
provide rankings that appear to be
very satisfactory to Internet users. The examples of link-based criteria are PageRank~\cite{PB98}
used by search engine Google, HITS \cite{K99} used by search engines Teoma and Ask, and
SALSA \cite{LM00}.
In the present work we restrict ourselves to the analysis of the PageRank
criterion and use the following definition of PageRank from~\cite{LM06}. Denote by $n$ the total
number of pages on the Web and define the $n\times n$ hyperlink matrix $P$ as follows:
\begin{equation}
\label{P}
p_{ij} =
\left\{ \begin{array}{ll}
1/d_i, & \mbox{if page $i$ links to $j$},\\
1/n, & \mbox{if page $i$ is dangling},\\
0, & \mbox{otherwise},
\end{array} \right.
\end{equation}
for $i,j=1,...,n$, where $d_i$ is the number of outgoing links from page $i$. We recall that the page is
called dangling if it does not have outgoing links.
In order to make the hyperlink graph connected, it is assumed that at each step, with some probability, a random surfer goes to an arbitrary Web page sampled from the uniform distribution. Thus, the PageRank is defined as a stationary
distribution of a Markov chain whose state space is the set of all Web pages, and the
transition matrix is
\begin{equation}
\label{GoogleMatrix}
G = cP + (1-c)(1/n)E,
\end{equation}
where $E$ is a matrix whose all entries are equal to one, and $c \in (0,1)$ is a probability of following a hyperlink. The constant $c$ is often
referred to as a damping factor. The Google matrix $G$ is stochastic, aperiodic, and irreducible,
so there exists a unique row vector $\pi$ such that
\begin{equation}
\label{BalanceEq}
\pi G = \pi, \quad \pi {\bf 1} =1,
\end{equation}
where ${\bf 1}$ is a column vector of ones. The row vector $\pi$ satisfying (\ref{BalanceEq})
is called a PageRank vector, or simply PageRank. If we consider a surfer that follows a hyperlink with probability $c$ and jumps to a random page with probability $1-c$, then $\pi_i$ can be interpreted as
a stationary probability that the surfer is at page $i$.
The damping factor $c$ is a crucial parameter in the PageRank definition. It regulates
the level of the uniform noise introduced to the system. Based on the publicly available
information Google originally used $c=0.85$. There is the following empirical explanation of this
choice, see e.g. \cite{LM06}: it seems that the closer the value of the damping factor to
one, the better the graph structure of the Web is represented in the PageRank vector.
However, when the value of $c$ approaches one, the rate of power iteration method slows
down significantly. The choice $c=0.85$ appears to be a reasonable compromise between
the two antagonistic objectives. However, in \cite{Boldi1} the authors argue that choosing
the value of $c$ too close to one is not necessarily a good thing to do. Not only the
power iteration method becomes very slowly convergent but also the ranking of the important
pages becomes distorted. Independently of \cite{Boldi1}, this phenomenon was also mentioned in~\cite{AL06}. We also remark that another incentive to reduce $c$ is that it will increase the robustness of the PageRank towards small changes in the link structure. That is, with smaller $c$, one can bound the influence of outgoing links of a page (or a small group of pages) on the PageRank of other groups~\cite{Bianchini} and on its own PageRank~\cite{AL06}.
In the present work, we go further than \cite{Boldi1, AL06} and suggest that even the value
$c=0.85$ is by far too large. Our argument is that one has to make a choice of $c$ to reflect
the natural intensity of the probability flow in the absorbing Markov chain associated with the
Web Graph. Our argument is based on the singular perturbation theory \cite{A99,KT93,PG88,YZ05}.
It turns out that
the value of $c$ that adequately reflects the flow of probability is very close to $1/2$. We note that the value $c=1/2$ was used in \cite{PRcitations} to find gems in scientific citations, where the authors justified this choice by intuitive argument discussed in more detail in Section~\ref{sec:conclusions}. In this work, we present a mathematical evidence for setting $c=1/2$ in the PageRank formula.
Of course, a drastic reduction of $c$ considerably accelerates the computation
of PageRank by numerical methods \cite{ALNO06,B05,LM06}. We would like to mention
that choosing smaller value for the damping factor could have similar effect on numerical
methods as choosing fast decreasing damping function \cite{Boldi2}.
As a by-product of the application of the singular perturbation approach we obtain
a refinement of the graph structure of the Web. We demonstrate that the dead-end strongly
connected components have unjustifiably large PageRank with damping factor $c=0.85$ and
by taking $c=0.5$ one can mitigate this problem. The results presented in this work are confirmed by experimental data that we obtained from two large samples of the Web Graph, described in Section~\ref{sec:datasets}.
The main contributions of this paper are as follows. First, in Section~\ref{sec:ergodic}, we describe the ergodic structure of the Web Graph and show how this structure changes under assumption that the dangling pages have a link to all pages in the Web, as in (\ref{P}). In particular, we discover an Extended Strongly Connected Component (ESCC) that contains a majority of the Web pages. Using the theory of singular perturbations, we find an exact formula for the limiting PageRank distribution when $c\to 1$. This result immediately implies that the limiting PageRank mass of ESCC equals zero. Next, in Section~\ref{sec:bounds}, we analytically characterize the PageRank mass of ESCC as a function of $c$, and we obtain simple bounds for this function. Further, in Section~\ref{sec:c=1/2}, we argue that $c=1/2$ ensures that ESCC receives a fair share of total PageRank mass. We conclude with a short discussion of the present results and future research directions in Section~\ref{sec:conclusions}.
\section{Datasets}
\label{sec:datasets}
For our numerical experiments, we have collected two Web Graphs, which we denote by INRIA and FrMathInfo.
The Web Graph INRIA was taken from the site of INRIA, the French Research Institute of Informatics and
Automatics. The seed for the INRIA collection was Web page {\tt www.inria.fr}. It is a typical large Web
site with around 300.000 pages and 2 millions hyperlinks. We have crawled the INRIA site until we have
collected all pages belonging to INRIA.
The Web Graph FrMathInfo was crawled with
the initial seeds of 50 mathematics and informatics laboratories of France, taken from Google Directory. The crawl was executed by breadth first search and
the depth of this crawl was 6. The FrMathInfo Web Graph contains around 700.000 pages and
8 millions hyperlinks. We expect our datasets to be enough representative. This is justified by
the fractal structure of the Web \cite{self-similar}.
The link structure of these two Web Graphs is stored in Oracle database. Due to sparsity
of the Web Graph and reasonable sizes of our datasets, we can store the adjacency lists in RAM to speed up
the computation of PageRank and other quantities of interest. This enables us to make
more iterations, which is extremely important in the case when the damping factor $c$ is close to one.
Our PageRank computation program consumes about one hour to make 500
iterations for the FrMathInfo dataset and about haft an hour for the INRIA dataset for the same
number of iterations. Our algorithms for discovering the ergodic structures of the Web Graph are based on
Breadth First Search and Depth First Search methods, which are
linear in the sum of number of nodes and links.
\section{Ergodic structure of the Web Graph}
\label{sec:ergodic}
In \cite{WebGraph1,WebGraph2} the authors have studied the graph structure of the Web.
In particular, in \cite{WebGraph1,WebGraph2}
it was shown that the Web Graph can be divided into three principle components: the Giant Strongly
Connected Component, to which we simply refer as SCC component, the IN component and the OUT component.
The SCC component is the largest strongly connected component in the Web Graph. In fact, it is
larger than the second largest strongly connected component by several orders of magnitude.
Following hyperlinks one can come from the IN component to the SCC component but it is not possible to return
back. Then, from the SCC component one can come to the OUT component and it is not possible to
return to SCC from the OUT component.
With this structure in mind, we would like to analyze ergodic properties of the random walk on the Web Graph. If a node has outgoing links, then such random walk follows one of these links with uniform distribution. However, as in the definition of PageRank, we have to define how the process evolves when it reaches one of dangling nodes. This choice has a crucial influence on the ergodic structure of the associated Markov chain. There are three natural possibilities: 1)~the process is absorbed in the
dangling node; 2)~the process moves to the predecessor node, or 3)~the process moves to an
arbitrary node. In this paper we focus on the latter option, which is used in the original PageRank model \cite{PB98}. The first two options are definitely worthy to be considered as well, and it is a nice topic for future research.
Thus, throughout the paper we consider a random walk with transition matrix $P$ given by (\ref{P}). As we shall see below, the analysis of the ergodic structure of $P$ leads to a more detailed description of the OUT component, and it allows us to evaluate the effect of damping factor
on PageRank. Obviously, the graph induced by $P$ has a much higher connectivity than the original Web Graph. In particular, if the random walk
can move from a dangling node to an arbitrary node with the uniform distribution, then the
Giant SCC component increases further in size. We refer to this new strongly
connected component as the Extended Strongly Connected Component (ESCC). First, we note that due to the artificial links from the dangling nodes,
the SCC component and IN component are now inter-connected and are parts of the Extended
SCC. Then, if there are dangling nodes emanating from some nodes in the OUT component,
these nodes together with all their predecessors become a part of the Extended SCC.
Let us consider an example of the graph presented in Figure~\ref{fig:smallgraph}.
Node~0 represents the IN component, nodes from 1 to 3 form the SCC component, and
the rest of the nodes, nodes from 4 to 11, are in the OUT component. Node~5 is a dangling node, thus,
artificial links go from the dangling node~5 to all other nodes. After addition of the
artificial links, all nodes from 0 to 5 form a new strongly connected component, which is the ESCC
in this example.
\begin{figure}[hbt]
\centering {\epsfxsize=2.8in \epsfbox{GraphStructure.eps}}
\caption{\small Example of a graph}
\label{fig:smallgraph}
\end{figure}
By renumbering the nodes, the transition matrix $P$ can be then transformed to the
following form
\begin{equation}
\label{ESCC}
P=\left[ \begin{array}{cc}
Q & 0 \\
R & T \end{array} \right],
\end{equation}
where the block $T$ corresponds to the Extended SCC, the block $Q$ corresponds to
the part of the OUT component without dangling nodes and their predecessors, and the block $R$ corresponds to the transitions from ESCC to the nodes in block $Q$. We refer to the set of nodes in the block $Q$ as Pure OUT component.
In the example of graph on Figure~\ref{fig:smallgraph} the Pure OUT component consists of
nodes from 6 to 11. Typically, the Pure OUT component is much smaller than the Extended SCC
component. The sizes of all components for our two datasets are displayed in Table~1.
We would like to note that the zero size of the IN components should not come as a surprise.
To crawl the Web Graph we have used the Breadth First Search method and have started from
important pages. Therefore, it is natural that the seed pages belong to the Giant SCC and
there is no IN component. For the purposes of the present research the absence of the IN
component is not a problem as the dangling nodes unite the IN and the Giant SCC into the
Extended SCC.
As was observed in \cite{moler}, the PageRank vector can be expressed by the following
formula
\begin{equation}
\label{PRformula}
\pi=\frac{1-c}{n} {\bf 1}^T [I-cP]^{-1}.
\end{equation}
If we substitute the expression (\ref{ESCC}) for the transition matrix $P$
into (\ref{PRformula}), we obtain the following formula for the part of the
PageRank vector corresponding to the nodes in ESCC:
$$
\pi_{T}=\frac{1-c}{n} {\bf 1}^T [I-cT]^{-1},
$$
or, equivalently,
\begin{equation}
\label{PRESCC}
\pi_{T} = \alpha (1-c) u_{T} [I-cT]^{-1},
\end{equation}
where $\alpha=n_T/n$ and $n_T$ is the number of nodes in ESCC,
and where $u_T$ is the uniform distribution over all ESCC nodes.
We shall also use $n_Q=n-n_T$, which is the number of nodes in the
Pure OUT component.
First, we note that since matrix $T$ is substochastic, the inverse
$[I-T]^{-1}$ exists and consequently $\pi_T \to 0$ as $c \to 1$.
Clearly, as was also observed in \cite{Boldi1}, it is not good to take
the value of $c$ too close to one. Next, we argue that even the value
of $0.85$ is too large.
Let us analyze the structure of the Pure OUT component in more detail.
It turns out that there are many disjoint strongly connected components
inside the Pure OUT component. One can see the histograms of the SCCs'
sizes of the Pure OUT for two our datasets INRIA and FrMathInfo in
Figures~\ref{fig:histINRIA}~and~\ref{fig:histFrMathInfo}. In particular,
there are many SCCs of size 2 and 3 in the Pure OUT component.
\begin{figure}[hbt]
\centering {\epsfxsize=3.0in \epsfbox{HistINRIA.eps}}
\caption{\small Histogram of SCCs' sizes of Pure OUT, INRIA dataset}
\label{fig:histINRIA}
\end{figure}
\begin{figure}[hbt]
\centering {\epsfxsize=3.0in \epsfbox{HistFrMathInfo.eps}}
\caption{\small Histogram of SCCs' sizes of Pure OUT, FrMathInfo dataset}
\label{fig:histFrMathInfo}
\end{figure}
By appropriate renumbering of the states, we can refine (\ref{ESCC}) as
follows:
\begin{equation}
\label{ESCCdetail0}
P=\left[ \begin{array}{ccccc}
Q_1 & & & & 0 \\
& \ddots & & & \\
0 & & Q_m & & \\
S_1 & \cdots & S_m & S_0 & 0 \\
R_1 & \cdots & R_m & R_0 & T
\end{array} \right],
\end{equation}
For instance, in example of the graph from Figure~\ref{fig:smallgraph}, the nodes
8 and 9 correspond to block $Q_1$, nodes 10 and 11 correspond to block $Q_2$, and
nodes 6 and 7 correspond to blocks $S$.
Since the random walk will be eventually absorbed in one of the $Q$ blocks,
we can simplify notations for our further analysis. Namely, define the submatrices
$$
\tilde{R}_i=\left[\begin{array}{c}S_i\{\mathbb R}_i\end{array}\right],\;i=1,\ldots,m; \quad \tilde{T}=
\left[\begin{array}{cc}S_0&0\{\mathbb R}_0&T\end{array}\right].
$$
Then the structure (\ref{ESCCdetail0}) becomes
\begin{equation}
\label{ESCCdetail}
P=\left[ \begin{array}{cccc}
Q_1 & & 0 & 0 \\
& \ddots & & \\
0 & & Q_m & 0 \\
\tilde{R}_1 & \cdots & \tilde{R}_m & \tilde{T}
\end{array} \right].
\end{equation}
Next, we note
that if $c<1$, then the Markov chain induced by matrix $G$ is ergodic.
However, if $c=1$, the Markov chain becomes non-ergodic. In particular,
if the process moves to one of the $Q_i$ blocks,
it will
never leave this
block. Hence, the random walk governed by the Google transition
matrix (\ref{GoogleMatrix}) is in fact a singularly perturbed Markov
chain.
According to the singular perturbation theory (see e.g., \cite{A99,KT93,PG88,YZ05}),
the PageRank vector goes to
some limit as the damping factor goes to one. Using the results
of the singular perturbation theory we can characterize explicitly
this limit.
\begin{proposition}
Let $\mu_i$ be a limiting stationary distribution of the Markov process
governed by
$P$ when the process settles in $Q_i$, the $i$-th SCC of
the Pure OUT component. Namely, vector $\mu_i$ is a unique solution
of the equations
$$
\mu_i Q_i = \mu_i, \quad \mu_i {\bf 1} = 1.
$$
Then, we have
$$
\lim_{c \to 1} \pi(c)=
\left[\bar{\pi}_1 \ \cdots \ \bar{\pi}_m \ 0 \right],
$$
where
\begin{equation}
\label{barpi}
\bar{\pi}_i=
\left(\frac{n_i}{n}+\frac{n_{\tilde{T}}}{n}u_{\tilde{T}}[I-\tilde{T}]^{-1}\tilde{R}_i{\bf 1}\right)\mu_i,
\end{equation}
for $i=1,...,m$ and the zeros at the end of the limiting vector correspond to all nodes,
which are not in $Q_i$, $i=1,\ldots,n$, that is, not in any SCC of the Pure OUT component.
\end{proposition}
{\bf Proof:}
First, we note that if we make a change of variables $\eps = 1-c$ the Google matrix
becomes a transition matrix of a singularly perturbed Markov chain as in Lemma~\ref{lm:SPMC}
with $C=\frac{1}{n}{\bf 1} {\bf 1}^T -P$. Let us calculate the aggregated generator matrix $D$.
$$
D=MCQ=\frac{1}{n}{\bf 1}\one^TQ-MPQ
$$
Using $MP=M$, $MQ=I$, and $M{\bf 1}={\bf 1}$ where vectors ${\bf 1}$ are of appropriate dimensions,
we obtain
$$
D=\frac{1}{n}{\bf 1}\one^TQ-I=
$$
$$
\frac{1}{n}{\bf 1}[n_1+n_{\tilde{T}}u_{\tilde{T}}[I-\tilde{T}]^{-1}\tilde{R}_1{\bf 1},\cdots,
n_m+n_{\tilde{T}}u_{\tilde{T}}[I-\tilde{T}]^{-1}\tilde{R}_m{\bf 1}]-I
$$
Since the aggregated transition matrix $D+I$ has identical rows, its stationary
distribution $\nu$ is just equal to these rows. Thus, invoking Lemma~\ref{lm:SPMC}
we obtain (\ref{barpi}).
\qed
The second term inside the brackets in formula (\ref{barpi}) corresponds to the PageRank
mass that an SCC component in Pure OUT receives from the Extended SCC. If $c$ is close to
one, then this contribution can outweight by far the fair share of the PageRank
which is given by $\frac{n_i}{n}$. For instance, in our numerical experiments with $c=0.85$, the PageRank mass of the Pure OUT component in the INRIA dataset equals $1.95n_Q/n$, whereas a `fair share' is $n_Q/n$. In the other dataset, FrMathInfo, the unfairness is even more pronounced: the PageRank mass of the Pure OUT component is $3.44n_Q/n$. This gives users an incentive to create `dead-ends': groups of pages that link only to each other. In the next sections we quantify the influence of parameter $c$ and show that in order to obtain balanced probability flow, $c$ should be taken around 1/2.
\begin{table}[htb]
\label{tab:table1}
\centerline{\begin{tabular}{|r|r|r|}
\hline
$ $&$INRIA$&$FrMathInfo$\\
\hline \hline
Total size & 318585 & 764119\\
Number of nodes in SCC & 154142 & 333175\\
Number of nodes in IN & 0 & 0 \\
Number of nodes in OUT & 164443 & 430944 \\
Number of nodes in ESCC & 300682 & 760016\\
Number of nodes in Pure OUT & 17903 & 4103\\
Number of SCCs in OUT & 1148 & 1382\\
Number of SCCs in Pure Out & 631 & 379\\
\hline
\end{tabular}}
\caption{Component sizes in INRIA and FrMathInfo datasets}
\end{table}
\section{PageRank mass of ESCC}
\label{sec:bounds}
Let us consider the PageRank mass of the Extended SCC component (ESCC) described in the previous section. Thus, we continue to analyze the transition matrix in the form presented in (\ref{ESCC}).
Our goal now is to characterize the behavior of the total PageRank mass of the ESCC component as a function of $c \in [0,1]$. From (\ref{PRESCC}) we have
\begin{align}
\nonumber ||\pi_T(c)||_1&=
\pi_{T}(c){\bf 1}=(1-c)\alpha{u_T}[I-cT]^{-1}{\bf 1}\\
\label{piscc_sum}
&=(1-c)\alpha{u_T}\sum_{k=0}^\infty c^kT^k{\bf 1}.\end{align}
Clearly, since $T$ is substochastic, we have $||\pi_{T}(0)||_1=\alpha$ and $||\pi_{T}(1)||_1=0$. Also, it is easy to show that
\[\frac{d}{dc}||\pi_{T}(c)||_1=-\alpha{u_T}[I-cT]^{-2}[I-T]{\bf 1}<0\]
and
\[\frac{d^2}{dc^2}||\pi_{T}(c)||_1=-2\alpha{u_T}[I-cT]^{-3}T[I-T]{\bf 1}<0.\]
Hence, $||\pi_{T}(c)||_1$ is a concave decreasing function.
In order to get a better idea about the behavior of this function, we derive a series of bounds. If we define\[\underline{p}=\inf_{k\ge 1}[uT^k{\bf 1}]^{1/k},\quad
\overline{p}=\sup_{k\ge 1}[uT^k{\bf 1}]^{1/k},\]
then it follows immediately from (\ref{piscc_sum}) that
\begin{equation}
\label{b1}
\frac{\alpha(1-c)}{1-c\underline{p}}\le ||\pi_T(c)||_1\le\frac{\alpha(1-c)}{1-c\overline{p}}.
\end{equation}
Now, let $\lambda_1$ be the Perron-Frobenius eigenvalue of $T$, and let $\tau$ be a random time when a random walk induced by $T$ leaves ESCC given that the initial distribution is uniform on ESCC. It is well known that
\[\lambda_1=\lim_{k\to\infty}{\mathbb P}[\tau>k|\tau>k-1]=\lim_{k\to\infty}\frac{uT^k{\bf 1}}{uT^{k-1}{\bf 1}}\]
Thus, we evaluate $\lambda_1$ iteratively by computing
\begin{equation}
\label{lambdak}
\lambda_1^{(k)}=\frac{uT^k{\bf 1}}{uT^{k-1}{\bf 1}},\quad k\ge 1,\end{equation}
where the numerator and denominator are simply results of the power iterations of $T$.
From the definition of $\lambda_1^{(k)}$ it is easy to see that if the sequence $\lambda_1^{(k)}$, $k\ge 1$, is increasing then
the sequence $(uT^k{\bf 1})^{1/k}$, $k\ge 1$, is also increasing, and thus in this case $\overline{p}=\lambda_1$ and $\underline{p}=p_1$, where $p_1={u_T} T{\bf 1}={\mathbb P}(\tau>1)$. Then equation (\ref{b1}) becomes
\begin{equation}
\label{b}
\frac{\alpha(1-c)}{1-cp_1}\le ||\pi_T(c)||_1\le\frac{\alpha(1-c)}{1-c\lambda_1}.\end{equation}
Although in our experiments we indeed observed that the sequence $\lambda_1^{(k)}$, $k\ge 1$, is increasing for both INRIA and FrMathInfo datasets, this condition is still too strong and we presume that it may fail in some cases. In the next proposition we provide much milder and more intuitive conditions under which (\ref{b}) still holds.
\begin{proposition}
\label{prop1}
Let $\lambda_1$ be the Perron-Frobenius eigenvalue of $T$, and define
$p_1={u_T} T{\bf 1}$.
\begin{itemize}
\item[(i)]
If $p_1<\lambda_1$ then
\begin{equation}
\label{ub}
||\pi_{T}(c)||_1<\frac{\alpha(1-c)}{1-c\lambda_1},\quad c\in(0,1).\end{equation}
\item[(ii)] If $1/(1-p_1)<{u_T}[I-T]^{-1}{\bf 1}$ then
\begin{equation}
\label{lb}
||\pi_{T}(c)||_1>\frac{\alpha(1-c)}{1-cp_1},\quad c\in(0,1).\end{equation}
\end{itemize}
\end{proposition}
{\bf Proof.} (i) The function $f(c)=\alpha(1-c)/(1-\lambda_1 c)$ is decreasing and concave, and so is $||\pi_{T}(c)||_1$. Also, $||\pi_{T}(0)||_1=f(0)=\alpha$, and $||\pi_{T}(1)||_1=f(1)=0$. Thus, for $c\in (0,1)$, the plot of $||\pi_{T}(c)||_1$ is either entirely above or entirely below $f(c)$. In particular, if the first derivatives satisfy $||\pi'_{T}(0)||_1<f'(0)$, then $||\pi_{T}(c)||_1<f(c)$ for any $c\in(0,1)$. Since $f'(0)=\alpha(\lambda_1-1)$ and $||\pi'_{T}(0)||_1=\alpha(p_1-1)$, we see that
$p_1<\lambda_1$ implies (\ref{ub}).
The proof of (ii) is similar. We consider a concave decreasing function $g(c)=\alpha(1-c)/(1-p_1 c)$ and note that $g(0)=\alpha$, $g(1)=0$. Now, if the condition in (ii) holds then $g'(1)>||\pi'_{T}(1)||_1$, which implies (\ref{lb}). \qed
Note that the conditions of Proposition~\ref{prop1} are satisfied when the sequence $\lambda_1^{(k)}$, $k\ge 1$, is increasing in $k$.
The condition $p_1<\lambda_1$, which gives the upper bound, has a clear intuitive interpretation. Let $\tilde{\pi}_T$ be a quasi-stationary distribution of $T$. By definition, $\tilde{\pi}_T$ is the probability-normed left Perron-Frobenius eigenvector of $T$, and it is well-known that $\tilde{\pi}_T$ is a limiting probability distribution obtained under condition that the random walk does not leave the ESCC component (see e.g. \cite{seneta}). Hence, $\tilde{\pi}_TT=\lambda_1\tilde{\pi}_T$, and the condition $p_1<\lambda_1$ means that the chance to stay in ESCC for one step in the quasi-stationary regime is higher than starting from the uniform distribution ${u_T}$. This inequality looks quite natural, since the quasi-stationary distribution should somehow favor states, from which the chance to leave ESCC is lower. Therefore, although $p_1<\lambda_1$ does not hold in general, one may expect that it should hold for transition matrices describing large entangled graphs.
With the help of the derived bounds we can conclude that the function $||\pi_{T}(c)||_1$ decreases very slowly for small and moderate values of $c$, and it decreases extremely fast when $c$ becomes close to 1. This typical behavior is clearly seen in Figures~\ref{fig:INRIA_escc},~\ref{fig:FrMathInfo_escc}, where $||\pi_{T}(c)||_1$ is plotted with a solid line.
\begin{figure}[hbt]
\centering {\epsfxsize=2.8in \epsfbox{PRbounds_inrian.eps}}
\caption{\small PageRank mass of ESCC and bounds, INRIA dataset}
\label{fig:INRIA_escc}
\end{figure}
\begin{figure}[hbt]
\centering {\epsfxsize=2.8in \epsfbox{PRbounds_FrMathInfo.eps}}
\caption{\small PageRank mass of ESCC and bounds, FrMathInfo dataset}
\label{fig:FrMathInfo_escc}
\end{figure}
In order to evaluate $||\pi_{T}(c)||_1$ we did not compute it separately for different values of $c$ but rather presented it as a function of $c$ so that {\it any} value of $c$ could be substituted. For that, we stored the values $\lambda_1^{(k)}$, $k\ge 1$, and then used (\ref{piscc_sum}) and (\ref{lambdak}) to obtain
\[||\pi_{T}(c)||_1=\alpha\sum_{k=0}^{\infty}c^k\prod_{l=1}^k\lambda_1^{(k)}, \quad c\in[0,1].\]
This is a more direct approach compared to \cite{Boldi1}, where the authors used derivatives of the PageRank to present $\pi$ as a function of $c$.
As for the bounds, the values $\lambda_1$ and $p_1$ can be directly substituted in (\ref{ub}) and (\ref{lb}), respectively. For the INRIA dataset we have $p_1=\lambda_1^{(1)}=0.97557$, $\lambda_1=0.99954$, and for the FrMathInfo dataset we have $p_1=0.99659$, $\lambda_1=0.99937$.
In the next section we use the above results on $||\pi_{T}(c)||_1$ and its bounds to determine the values of $c$ that reflect natural probability flows through the ESCC component.
\section{Why the damping factor should be 1/2}
\label{sec:c=1/2}
Since ESCC is by far more important and interesting part of the Web than the Pure~OUT component, it would be reasonable to ensure that the PageRank mass of ESCC is at least the fraction of nodes in this component (we denoted this fraction by $\alpha$). However, because $||\pi_{T}(c)||_1$ is decreasing, and $||\pi_{T}(0)||_1=\alpha$,
it follows that the total PageRank mass of ESCC is smaller than $\alpha$ for any value $c>0$.
Now let us discuss an `optimal' choice of $c$. First of all, $c$ can not be too close to one because in this case the PageRank mass of the giant ESCC component will be close to 0. This was observed independently in \cite{AL06,Boldi1}. Specifically, from the analysis above it follows that the value of $c$ should not be chosen in the critical region where the PageRank mass of the ESCC component is rapidly decreasing. Luckily, the shape of the function $||\pi_{T}(c)||_1$ is such that it decreases drastically only when $c$ is really close to one, which leaves a lot of freedom for choosing $c$. In particular, the famous Google constant $c=0.85$ is small enough to ensure a reasonably large PageRank mass of ESCC.
However, as we have observed in Section~\ref{sec:ergodic}, even moderately large values of $c$ result in an unfairly large PageRank mass of the Pure OUT component. Now, our goal is to find the values of $c$ that lead to a `fair' distribution of the PageRank mass between the Pure OUT and the ESCC components.
Formally, we would like to define a number $\gamma\in(0,1)$ such that a desirable PageRank mass of ESCC could be written as $\gamma\alpha$, and then find the value $c^*$ that satisfies
\[||\pi_{T}(c^*)||_1=\gamma\alpha.\]
Then $c\le c^*$ will ensure that $||\pi_{T}(c)||_1\ge\gamma\alpha$. Naturally, $\gamma$ should somehow reflect the properties of the substochastic block $T$. For instance, as $T$ becomes closer to stochastic matrix, $\gamma$ should also increase. One possibility to do it is to define
\[\gamma={{v}}T{\bf 1},\]
where ${v}$ is a row vector representing some probability distribution on ESCC. Then the damping factor $c$ should satisfy
\[c\le c^*,\]
where $c^*$ is given by
\begin{equation}
\label{c*}
||\pi_{T}(c^*)||_1=\alpha{{v}}T{\bf 1}.
\end{equation}
In this setting, $\gamma$ is a probability to stay in ESCC for one step if initial distribution is ${v}$. For given ${v}$, this number increases as $T$ becomes closer to stochastic matrix. Now, the problem of choosing $\gamma$ comes down to the problem of choosing ${v}$. The advantage of this approach is twofold. First, we still have all the flexibility because, depending on ${v}$, the value of $\gamma$ may be literally anything between zero and one. Second, we can use a probabilistic interpretation of ${v}$ to make a reasonable choice. In this paper we consider three appealing choices of ${v}$:
\begin{enumerate}
\item $\tilde{\pi}_T$, the quasi-stationary distribution of $T$,
\item the uniform vector ${u_T}$, and
\item the normalized PageRank vector $\pi_T(c)/||\pi_T(c)||_1$. Note that in this case both ${v}$ and $\gamma$ depend on $c$.
\end{enumerate}
First, let us take ${v}=\tilde{\pi}_T$, the quasi-stationary distribution of $T$. The motivation for taking ${v}=\tilde{\pi}_T$ is that $\tilde{\pi}_T$ weights the states according to their quasi-stationary probabilities, which captures the structure of $T$.
With $v=\tilde{\pi}_T$, equation (\ref{c*}) becomes
\[\pi_{T}(c^*){\bf 1}= \alpha\tilde{\pi}_T T{\bf 1}=\alpha\lambda_1.
\]
In this case, $\gamma=\lambda_1$ is the probability that the random walk stays in ESCC given that it did not leave this block for infinitely long time. Hence, $\lambda_1$ is a natural measure of proximity of $T$ to stochastic matrix.
If conditions of Proposition~\ref{prop1} are satisfied, then (\ref{ub}) and (\ref{lb}) hold, and thus the value of $c^*$ satisfying (\ref{c*}) must be in the interval $(c_1,c_2)$, where
\[(1-c_1)/(1-p_1 c_1)=\lambda_1,\quad (1-c_2)/(1-\lambda_1 c_2)=\lambda_1.\]
It is easy to check that $c_1=(1-\lambda_1)/(1-\lambda_1p_1)$ and $c_2=1/(\lambda_1+1)$. Since $\lambda_1$ is very close to 1, it follows that $c$ is bounded from above by the number $c^*\le c_2$, where $c_2$ is only slightly larger than $1/2$! Numerical results for our two datasets are presented in Table~\ref{tab:c}.
\begin{table}[htb]
\centerline{
\begin{tabular}{|c|c|r|r|}
\hline
$v$&$c$&INRIA&FrMathInfo\\
\hline \hline
$\tilde{\pi}_T$&$c_1$&0.0184&0.1571\\
&$c_2$&0.5001&0.5002\\
&$c^*$&.02&.16\\
\hline
${u_T}$&$c_3$&0.5062&0.5009\\
&$c_4$&0.9820&0.8051\\
&$c^*$&.604&.535\\
\hline
$\pi_T/||\pi_T||_1$&$1/(1+\lambda_1)$&0.5001&0.5002\\
&$1/(1+p_1)$&0.5062&0.5009\\
\hline
\end{tabular}}
\caption{Values of $c^*$ with bounds.}
\label{tab:c}
\end{table}
From the numerical results we can see that in case when $v=\tilde{\pi}_T$, we obtain $c^*$ close to zero. This however leads to ranking that takes into account only local information about the Web Graph. Specifically, the number of incoming links will play a dominant role in the PageRank value (see e.g. \cite{Fortunato2}). Furthermore, the interpretation of ${v}=\tilde{\pi}_T$ also suggests that it is not the best choice because the `easily bored surfer' random walk that is used in PageRank computations never follows a quasi-stationary distribution. Indeed, with probability $(1-c)$, this random walk restarts itself from the uniform probability vector. Clearly, the intervals between subsequent restarting points are too short to reach a quasi-stationary regime.
Our second choice is the uniform vector ${v}={u_T}$. In this case, (\ref{c*}) becomes
\[||\pi_{T}(c^*)||_1= \alpha{u_T} T{\bf 1}=\alpha p_1.
\]
If the conditions of Proposition~\ref{prop1} hold then we again can use (\ref{ub}) and (\ref{lb}) to establish that $c^*\in (c_3,c_4)$, where
\[(1-c_3)/(1-p_1c_3)=p_1, \quad (1-c_4)/(1-\lambda_1c_4)=p_1.\]
The values of $c_3=1/(1+p_1)$, $c_4=(1-p_1)/(1-\lambda_1p_1)$, and $c^*$ for our datasets are given in Table~1.
As we see, in this case, we have obtained a higher upper bound. However, the values of $c^*$ are still much smaller than $0.85$.
Note that ${v}=\tilde{\pi}_T$ implies $\gamma=\lambda_1$, which is a probability to stay in ESCC for one step after infinitely long time, and ${v}={u_T}$ leads to $\gamma=p_1$, which is the probability to stay in ESCC for one step after starting afresh. Our third choice, the normalized PageRank vector ${v}=\pi_T/||\pi_T||_1$, is a symbiosis of the previous two cases. With this choice of ${v}$, according to (\ref{c*}), the value $c=c^*$ solves the equation
\begin{align*}
||\pi_{T}(c)||_1&= \frac{\alpha}{||\pi_{T}(c)||_1}\pi_{T}(c)T{\bf 1}\\
&= \frac{\alpha^2(1-c)}{||\pi_{T}(c)||_1}{{u_T}}[I-cT]^{-1}T{\bf 1},
\end{align*}
where the last equality follows from (\ref{PRESCC}).
Multiplying by $||\pi_{T}(c)||_1$, we obtain
\begin{align*}
||\pi_{T}(c)||_1^2&=\alpha^2(1-c){u_T}\frac{1}{c}\,cT[I-cT]^{-1}{\bf 1}\\
&=\alpha^2(1-c){u_T}\frac{1}{c}\,cT[I-cT]^{-1}{\bf 1}\\
&=\alpha^2(1-c){u_T}\frac{1}{c}\,\left([I-cT]^{-1}-I\right){\bf 1}\\
&=
\frac{\alpha}{c}\,||\pi_{T}(c)||_1-\frac{(1-c)\alpha^2}{c}.
\end{align*}
Solving the quadratic equation for $||\pi_{T}(c)||_1$, we get
\[||\pi_{T}(c)||_1=r(c)=\left\{\begin{array}{ll}\alpha&\mbox{if }c\le 1/2,\\
\frac{\alpha(1-c)}{c}&\mbox{if }c> 1/2.\end{array}\right.\]
Hence, the value $c^*$ solving (\ref{c*}) corresponds to the point where the graphs of $||\pi_{T}(c)||_1$ and $r(c)$ cross each other. First, note that there is only one such point on (0,1). Furthermore, since $||\pi_{T}(c)||_1$ decreases very slowly unless $c$ is close to one, and $r(c)$ starts decreasing relatively fast for $c>1/2$, one can expect that $c^*$ is only slightly larger than $1/2$. This is illustrated in Figure~\ref{fig:plots_for_c}, where we depict $||\pi_{T}(c)||_1$ and $r(c)$ for INRIA and FrMathInfo datasets.
\begin{figure}[hbt]
\centering {\epsfxsize=2.8in \epsfbox{plots_for_c.eps}}
\caption{\small The value $c^*$ with $v=\pi_{T}/||\pi_{T}||_1$ is the crossing point of $r(c)$ and $||\pi_{T}(c)||_1$.}
\label{fig:plots_for_c}
\end{figure}
Under conditions of Proposition~\ref{prop1}, we may use (\ref{ub}) and (\ref{lb}) to deduce that $r(c)$ first crosses the line $\alpha(1-c)/(1-\lambda_1c)$, then $||\pi_{T}(c)||_1$, and then $\alpha(1-c)/(1-p_1c)$. Thus, we yield
\[\frac{1}{1+\lambda_1}<c^*< \frac{1}{1+p_1}.\]
Since both $\lambda_1$ and $p_1$ are close to 1, this clearly indicates that $c$ should be chosen close to 1/2. The values for lower and upper bounds for $c^*$ are given in Table~1. Since these bounds are tight we did not compute $c^*$ explicitly.
To summarize, our results indicate that with $c=0.85$, the ESCC component does not receive a fair share of the PageRank mass. Remarkably, in order to satisfy any of the three intuitive criteria of fairness presented above, the value of $c$ should be drastically reduced. In particular, the value $c=1/2$ looks like a well justified choice.
In future, it would be interesting to design and analyze other criteria for choosing the most `fair' value of $c$. However, given the outcome of our studies, we foresee that any criterion based on the PageRank mass of ESCC will lead to similar results.
\section{Conclusions}
\label{sec:conclusions}
The choice of the PageRank damping factor is not evident. The old motivation for the value $c=0.85$ was a compromise between the true reflection of the Web structure and numerical efficiency. However, the Markov random walk on the Web Graph does not reflect the importance of the pages because it absorbs in dead ends. Thus, the damping factor is needed not only for speeding up the computations but also for establishing a fair ranking of pages.
In this paper, we proposed new criteria for choosing the damping factor, based on the ergodic structure and probability flows. Our approach leads to the conclusion that the value $c=0.85$ is too high, and in fact the damping factor should be chosen close to 1/2.
As we already mentioned before, the value $c=1/2$ was used in \cite{PRcitations} to find gems in scientific citations. This choice was justified intuitively by stating that researchers may check references in cited papers but on average they hardly go deeper than two levels, which results in probability 1/2 of `giving up'. Nowadays, when search engines work really fast, this argument also applies to Web search. Indeed, it is easier for the user to refine a query and receive a proper page in fraction of seconds than to look for this page by clicking on hyperlinks. Therefore, we may assume that a surfer searching for a page, on average, does not go deeper than two clicks.
Even if our statement that $c$ should be 1/2, might be received with a healthy skepticism, we hope to
have convinced the reader that the study of ergodic structure of the Web helps in choosing the value
of the damping factor, and in improving link-based ranking criteria in general. We believe that future research in this
direction will yield new reasoning for a well grounded choice of the ranking criteria and help
to discover new fascinating properties of the Web Graph.
\section*{Acknowledgments}
This work is supported by EGIDE ECO-NET grant no. 10191XC and by NWO Meervoud grant no.~632.002.401.
We also would like to thank Danil Nemirovsky for the collection of the Web Graph data.
\section*{Appendix: A Singular Perturbation Lemma}
\begin{lemma}
\label{lm:SPMC}
Let $G(\eps)=P+\eps C$ be a transition matrix of perturbed Markov chain.
The perturbed Markov chain is assumed to be ergodic for sufficiently small
$\eps$ different from zero. And let the unperturbed Markov chain $(\eps =0)$
have $m$ ergodic classes. Namely, the transition matrix $P$ can be written
in the form
$$
P=\left[ \begin{array}{cccc}
Q_1 & & 0 & 0 \\
& \ddots & & \\
0 & & Q_m & 0 \\
R_1 & \cdots & R_m & T
\end{array} \right] \in R^{n\times n}.
$$
Then, the stationary distribution of the perturbed Markov chain has a limit
$$
\lim_{\eps \to 0} \pi(\eps) =[\nu_1\mu_1 \ \cdots \ \nu_m\mu_m \ 0],
$$
where zeros correspond to the set of transient states in the unperturbed
Markov chain, $\mu_i$ is a stationary distribution of the unperturbed Markov
chain corresponding to the $i$-th ergodic set, and $\nu_i$ is the $i$-th
element of the aggregated stationary distribution vector that can be found
by solution
$$
\nu D = \nu, \quad \nu {\bf 1} =1,
$$
where $D=MCQ$ is the generator of the aggregated Markov chain and
$$
M=\left[ \begin{array}{cccc}
\mu_1 & & 0 & 0 \\
& \ddots & & \\
0 & & \mu_m & 0
\end{array} \right] \in R^{m\times n}.
$$
$$
Q=
\left[ \begin{array}{cccc}
{\bf 1} & & 0 \\
& \ddots & \\
0 & & {\bf 1} \\
\phi_1 & \cdots & \phi_m
\end{array} \right] \in R^{n\times m}.
$$
with $\phi_i=[I-T]^{-1}R_i{\bf 1}$.
\end{lemma}
The proof of this lemma can be found in \cite{A99,KT93,YZ05}.
|
1,314,259,993,631 | arxiv |
\section{Introduction}
The theory of polymer adsorption at an impenetrable surface is a well established subject.
Useful reviews can be found in \cite{DeBell} and \cite{Rensburg2000}. One of
the standard models is self-avoiding walks,
or SAWs, confined to a half-space and interacting
with the confining line or plane. For this problem we have some rigorous
results \cite{HTW1982,Rensburg1998} that establish the existence of a phase transition and provide
useful information about the behaviour of the free energy as the temperature
is varied. More detailed information comes from a variety of numerical investigations
that, among other things, give quite precise information about the location of
the phase transition \cite{Beaton2012,Guim1989,Guttmann2014,Hegger1994,Rensburg2004}
and strongly suggest that the transition is second order
\cite{Guim1989,Hegger1994,Rensburg2004}.
With the advent of atomic force microscopy \cite{Haupt1999,Zhang2003} it has become
possible to pull an adsorbed polymer off a surface at which it is adsorbed. In principle it is possible
to measure the temperature dependence of the critical force for desorption and the stress-strain
curves. It is only quite recently that the effects of a force have been investigated
for the self-avoiding walk model \cite{Beaton2015,Guttmann2014,Rensburg2009,JvRW2013,Krawczyk2005,Mishra2005,Binder2012}.
Consider the $d$-dimensional hypercubic lattice, ${\mathbb Z}^d$, and attach a coordinate system
$(x_1,x_2, \dots x_d)$ so that each vertex of the lattice has integer coordinates. Suppose
that $c_n^+$ is the number of $n$-edge self-avoiding walks that start at $(0,0, \ldots 0)$ and have
all vertices in the half-space $x_d \ge 0$.
It is known \cite{Whittington1975} that
\begin{equation}
\lim_{n\to\infty} n^{-1} \log c_n^+ = \kappa_d
\end{equation}
where $\kappa_d$ is the \emph{connective constant} of the lattice \cite{HM54}.
Each vertex of the walk in the hyperplane
$x_d=0$ is called a \emph{visit}. Let $c_n^+(v,h)$ be the number of these walks
with $v+1$ visits and having the $x_d$-coordinate of their last vertex equal to
$h$, which we call the \emph{height} of the last vertex. Define the
partition function
\begin{equation}
C_n^+(a,y) = \sum_{v,h} c_n^+(v,h) a^v y^h.
\label{eqn:sawpf}
\end{equation}
It is known \cite{JvRW2013} that the limit
\begin{equation}
\lim_{n\to\infty} n^{-1} \log C_n^+(a,y) = \psi(a,y)
\label{eqn:sawfreeenergy}
\end{equation}
exists. $\psi (a,y)$ is the (reduced, limiting) free energy.
We can interpret the two fugacities $a$ and $y$ as
\begin{equation}
a = \exp (-\epsilon /k_BT) \quad \mbox{and} \quad y = \exp (f/k_BT)
\label{eqn:fugacities}
\end{equation}
where $k_B$ is Boltzmann's constant, $T$ is the absolute temperature, $\epsilon$
is the energy associated with a vertex in the surface and $f$ is the force applied at
the last vertex, normal to the surface. For adsorption to occur $\epsilon < 0$ so that
the interaction with the surface is attractive. For the walk to be desorbed
by the action of the force, $f > 0$.
If we set $f=0$, so that $y=1$, we have the pure adsorption problem. Write
$\psi(a,1) = \kappa (a)$, the free energy for pure adsorption.
We know \cite{HTW1982,Rensburg1998}
that there is a critical value of $a$, $a_c > 1$, such that $\kappa (a) = \kappa (1)=\kappa_d$
for $a \le a_c$ and $\kappa (a) > \kappa (1)$ for $a > a_c$ so that the free energy is
singular at $a=a_c$, corresponding to the adsorption transition. Similarly if we
set $\epsilon = 0$ so that $a=1$ we have no (attractive) interaction with the surface
and there is no adsorbed phase. The free energy is $\lambda (y) = \psi (1,y)$
and we know that $\lambda (y)$ is singular at $y=y_c =1$ \cite{Beaton2015}.
There is a transition from a \emph{free phase} to a \emph{ballistic phase} at $y=1$
\cite{Beaton2015}, see also \cite{Ioffe_Velenik_2008,Ioffe_Velenik_2010}.
Returning to the full problem, we know \cite{JvRW2013} that
for $a \ge a_c$ and $y \ge 1$
\begin{equation}
\psi(a,y) = \max [\kappa (a), \lambda (y)].
\label{eqn:sawferelation}
\end{equation}
This gives a complete characterization of the phase boundary between the
adsorbed and ballistic phases in terms of the behaviour when $\epsilon = 0$ and
when $f=0$. This was used in \cite{Guttmann2014} to give very precise numerical estimates
of the location of the phase boundary when $d=2$. Numerical estimates
using a different approach are given in \cite{Mishra2005} for both
$d=2$ and $d=3$. In addition we know
\cite{Guttmann2014} that the phase transition from the adsorbed to the ballistic
phase is first order.
These results raise a variety of new questions. For the self-avoiding walk model,
what happens if the force is applied somewhere other than at the last vertex? What
happens if the force is applied at an angle to the surface? Not all polymers are
linear and there are interesting questions about the behaviour of ring polymers or
branched polymers when they are pulled off a surface at which they are adsorbed.
That is, how does the architecture of the polymer affect its behaviour?
Does it matter where the
force is applied? If the force is applied at the \emph{top} vertex, \emph{i.e.} at the
vertex furthest from the surface, this is equivalent to confining the walk (or polygon, etc.)
between two parallel lines or planes and requiring at least one vertex in each plane,
then applying a force to move the confining plane. For the walk problem
we know that the limiting free energy is the same when the force is applied in this way
or at the terminal vertex \cite{RensburgWhittington2016a,RensburgWhittington2016b}.
If the force is applied at an interior vertex, the free energy depends on which
vertex the force is applied to, and in some circumstances, an additional phase can be present
\cite{RensburgWhittington2017}. Some results for staircase polygons \cite{Beaton2017}
suggest a phase diagram with a mixed adsorbed and ballistic phase.
In this paper we begin to investigate the issue of polymer
architecture. We consider a ring polymer
adsorbed at a surface, being pulled off the surface by a force applied
in a particular way.
\begin{figure}[t]
\beginpicture
\setcoordinatesystem units <1.5pt,1.5pt>
\setplotarea x from -140 to 110, y from -10 to 100
\setplotarea x from -80 to 80, y from 0 to 90
\color{Tan}
\grid 16 9
\color{Tan}
\setplotsymbol ({$\bullet$})
\plot -80 93 80 93 /
\plot -80 -3 80 -3 /
\color{RoyalBlue}
\plot -60 0 -50 0 -40 0 -40 10 -40 20 -50 20 -50 30 -50 40 -40 40 -30 40 -30 30
-20 30 -20 40 -20 50 -20 60 -20 70 -10 70 0 70 10 70 10 60 10 50 10 40 0 40 0 30
0 20 -10 20 -10 10 -10 0 -10 0 0 0 10 0 20 0 20 10 30 10 30 20 40 20 40 30 50 30 50 20 50 10
50 0 60 0 60 10 60 20 60 30 60 40 60 50 50 50 50 60 50 70 40 70 30 70 30 80 20 80
10 80 0 80 -10 80 -10 90 -20 90 -20 80 -30 80 -40 80 -40 70 -50 70 -50 60 -50 50 -60 50
-60 40 -60 30 -60 20 -60 10 -60 0 /
\color{Blue}
\multiput {\LARGE$\bullet$} at
-60 0 -50 0 -40 0 -40 10 -40 20 -50 20 -50 30 -50 40 -40 40 -30 40 -30 30
-20 30 -20 40 -20 50 -20 60 -20 70 -10 70 0 70 10 70 10 60 10 50 10 40 0 40 0 30
0 20 -10 20 -10 10 -10 0 -10 0 0 0 10 0 20 0 20 10 30 10 30 20 40 20 40 30 50 30 50 20 50 10
50 0 60 0 60 10 60 20 60 30 60 40 60 50 50 50 50 60 50 70 40 70 30 70 30 80 20 80
10 80 0 80 -10 80 -10 90 -20 90 -20 80 -30 80 -40 80 -40 70 -50 70 -50 60 -50 50 -60 50
-60 40 -60 30 -60 20 -60 10 -60 0 /
\color{red}
\multiput {\Large$\blacksquare$} at -60 1 /
\color{black} \normalcolor
\setplotsymbol ({.})
\arrow <5pt> [.2,.67] from 85 0 to 85 90
\put {\Large$w$} at 90 45
\endpicture
\label{fig1}
\caption{A polygon in a slab of width $w$. If $w = \infty$ the polygon is in a half-space. }
\end{figure}
\begin{figure}[h!]
\beginpicture
\setcoordinatesystem units <2.1pt,2pt>
\setplotarea x from -40 to 100, y from -10 to 100
\color{black}
\setplotsymbol ({$\cdot$})
\plot -2 40 0 40 / \plot 36 -2 36 0 /
\setsolid
\setplotsymbol ({\tiny$\bullet$})
\plot 0 100 0 0 100 0 /
\color{black}
\put {\Large$y$} at -8 90
\put {\Large$a$} at 90 -7
\put {$1$} at -5 40
\put {$1$} at 36 -5
\put {\Large$y_c(a)$} at 80 82
\put {\large$(a_c^0,1)$} at 50 40
\put {\Large\hbox{$\psi_0=\log \mu$}} at 20 20
\put {\Large\hbox{$\psi_0=\lambda_0(y)$}} at 40 80
\put {\Large\hbox{$\psi_0=\kappa_0(a)$}} at 70 20
\color{blue}
\plot 40 0 40 40 0 40 /
\setplotsymbol ({\LARGE$\cdot$})
\color{red}
\setquadratic
\plot 40 40 59 69 82 100 /
\setlinear
\color{Maroon}
\put {\huge$\bullet$} at 40 40
\color{black}
\normalcolor
\endpicture
\caption{The phase diagram of pulled adsorbing polygons in $d\geq 3$ dimensions.
For $y\leq 1$ and $a\leq a_c^0$ the free energy is $\psi_0 = \log \mu$.
This corresponds to a free phase with phase boundaries separating it
from the ballistic phase at $y=1$ and from the adsorbed phase.
In the ballistic phase the free energy is $\psi_0 = \lambda_0(y) =
\lambda(\sqrt{y})$. In the adsorbed phase the free energy is
$\psi_0 = \kappa_0(a) = \kappa(a)$. The solution of
$\lambda_0(y) = \kappa_0(a)$ is a critical curve $y_c(a)$ separating the ballistic
and adsorbed phases. This phase boundary is first order.}
\label{figure22}
\end{figure}
\section{Definition of the model and statement of results}
\label{sec:definition}
A standard model of ring polymers is \emph{self-avoiding lattice polygons} or \emph{polygons},
or SAPs, for short.
These are embeddings of the circle graph in a lattice. Each vertex has degree 2.
Let $p_n$ be the number of (undirected, unrooted) $n$-edge polygons in ${\mathbb Z}^d$,
counted modulo translation. In two dimensions, $p_4 = 1$, $p_6 = 2$, $p_8=7$, etc. Clearly
$p_{2m+1} = 0$. Let $p_n(v,s)$ be the number of $n$-edge polygons with all
vertices having $x_d \ge 0$, $v+2$ visits ($v \ge 0$)
and span in the $x_d$-direction equal to $s$, counted modulo translation parallel to the surface.
Define the partition function
\begin{equation}
P_n(a,y) = \sum_{v \ge 0,s \ge 0} p_n(v,s) a^vy^s.
\label{eqn:polygonpf}
\end{equation}
The fugacities are again interpreted as in equation (\ref{eqn:fugacities}) and now
the force is applied at the highest vertex.
In this paper we show that in $d\geq 3$ dimensions this model has
a thermodynamic limit with free energy defined by
\begin{equation}
\psi_0(a,y) = \lim_{n\to\infty} \sfrac{1}{n} \log P_n(a,y) .
\end{equation}
In addition, we show that, in this case, for $a\geq a_c^0$ (where $a_c^0$
is the critical adsorption fugacity for adsorbing polygons) and $y\geq 1$,
\begin{equation}
\psi_0(a,y) = \max [ \kappa_0(a),\lambda_0(y) ]
\end{equation}
where $\kappa_0(a)$ is the free energy of adsorbing polygons in the absence of a
pulling force, and $\lambda_0(y)$ is the free energy of a pulled polygon in the
absence of an adsorption fugacity. The resulting phase diagram of this model is
sketched in Figure \ref{figure22}. We know that the phase boundary between the
free phase and the adsorbed phase is a vertical line, and the phase boundary between the
free phase and the ballistic phase is a horizontal line.
In $d=2$ dimensions, our results are less complete.
\section{Polygons adsorbing at a surface}
\label{sec:pureadsorb}
If we consider polygons adsorbing at a surface with no applied force ($y=1$) we have the
following theorem:
\begin{theo}[Soteros]
The thermodynamic limit
\begin{equation}
\lim_{n\to\infty} n^{-1} \log P_n(a,1) \equiv \kappa_0(a) < \infty
\end{equation}
exists. Moreover, $\kappa_0(a)$ is a convex function of $\log a$ and hence
is continuous and differentiable almost everywhere.
\label{theo:Soteros1}
\end{theo}
The proof of this theorem follows from results in \cite{Soteros}, in Sections 3 and 4.
When $d \ge 3$ it is relatively straightforward to show that $\kappa_0(a) = \kappa (a)$ so the location
of the adsorption transition is the same as that of walks. See
Section 3 of \cite{Soteros} for a proof when $d=3$ that can be extended to $d > 3$
without much difficulty.
When $d=2$ the situation is slightly different because a polygon cannot lie
entirely in the confining line. We have the following theorem.
\begin{theo}[Soteros]
Suppose that $d=2$. When $a \le 1$ then $\kappa_0(a) = \kappa_0(1)
=\kappa_2$. When $a > 1$
\begin{equation}
\max[\kappa_2, (1/2) \log a] \le \kappa_0(a) \le \kappa_2 + (1/2) \log a.
\end{equation}
\label{theo:Soteros2}
\end{theo}
A proof is given in Section 4 of \cite{Soteros}.
This theorem implies that there is an adsorption transition at $a=a_c^0$ where
\begin{equation}
1 \le a_c^0 \le \exp[2 \kappa_2]
\label{eqn:inequalities}
\end{equation}
and, with a little more effort, the upper bound
can be made strict. By deleting a suitable edge each polygon can be converted
into a terminally attached self-avoiding walk so $\kappa_0(a) \le \kappa (a)$
and this implies that $a_c^0 \ge a_c$. Since we know
\cite{HTW1982} that $a_c > 1$ this implies that $a_c^0 > 1$ so both
inequalities in (\ref{eqn:inequalities}) are strict. It is an open question as to whether the two
critical points are identical or not.
\section{Polygons pulled from a non-interacting surface}
\label{sec:purepull}
In this section we consider a polygon attached to an impenetrable surface and pulled
away from the surface. There is no attractive interaction so $a=1$. The force is conjugate
to the span ($s$) of the polygon in the $x_d$-direction and the partition function is
\begin{equation}
P_n(1,y) = \sum_{v\ge 0,s \ge 0} p_n(v,s) y^s.
\end{equation}
The existence of the limit
\begin{equation}
\lim_{n\to\infty} n^{-1} \log P_n(1,y) = \lambda_0(y)
\end{equation}
is established in \cite{Rensburg2008}. In addition we know \cite{Rensburg2008}
that $\lambda_0(y)$ is
a convex function of $\log y$ (and hence continuous) and, for $y \ge 1$,
$\lambda_0(y)$ satisfies the bounds
\begin{equation}
\max [\lambda_0(1), (1/2) \log y ] \le \lambda_0(y) \le \lambda_0 (1) + (1/2) \log y.
\label{eqn:lambdabound}
\end{equation}
Note that $\lambda_0(1) = \kappa_d$.
\subsection{Polygons in a slit or slab}
\label{ssectionA}
It will be useful to consider polygons with at least one vertex in the bottom boundary
of a slit or slab of width $w$ in the hypercubic lattice. Let $\pi_n(w)$ be the number of
polygons of length $n$ with highest vertices at height $w$, so
that they fit in a slab or slit of width $w$ (see Figure \ref{fig1}), counted modulo translation parallel to the
confining boundaries.
The generating function of this model is
\begin{equation}
\Pi(t,y) = \sum_{n=0}^\infty \sum_{w=0}^{n/2} \pi_n(w)\,y^wt^n ,
\end{equation}
where $\pi_n(w) = \sum_v p_n(v,w)$.
If $0 \leq y \leq 1$, by monotonicity,
\begin{equation}
G(t) = \Pi(t,1) \geq \Pi(t,y) ,
\end{equation}
where $G(t)$ is the generating function of polygons in a half-lattice.
Then it is known that $G(t)$ is singular at $t=t_c = \frac{1}{\mu} = e^{-\kappa_d}$.
Suppose that $d \ge 3$. Then it is known
\begin{equation}
\lim_{n\to \infty} n^{-1} \log \pi_n(w) = \log \mu_w
\end{equation}
exists where $\mu_w$ is the growth constant for self-avoiding walks in a
slab of width $w$ \cite{HamWhitt1985}. It is proved there that
$\mu_w < \mu_{w+1}$ and $\lim_{w \to \infty} \mu_w = \mu$.
$\Pi(t,y)$ is bounded from below by any term in its defining series:
\begin{equation}
\Pi (t,y) \geq y^w\sum_{n=2w}^\infty \pi_n(w)\,t^n = y^w\, G_w(t).
\end{equation}
$G_w(t)$ is singular at $t_w = \frac{1}{\mu_w}$.
The above gives
\begin{equation}
G(t) \geq \Pi(t,y) \geq y^w\, G_w(t),\quad \hbox{if $y \leq 1$, for every $w> 0$}.
\end{equation}
The generating functions $G(t)$ and $G_w(t)$ have radius of convergence
$\frac{1}{\mu}$ and $\frac{1}{\mu_w}$ respectively, where $\mu_w < \mu$.
This shows, that for any fixed $0<y\leq1$, the radius of convergence of
$\Pi (t,y)$ is $\frac{1}{\mu_0(y)}$, where
\begin{equation}
\mu \geq \mu_0(y) \geq \mu_w .
\end{equation}
Since $\mu_w \nearrow \mu$ as $w\to\infty$, this shows that
\begin{equation}
\mu_0(y) = \mu , \quad\hbox{for all $0<y\leq 1$}
\end{equation}
for all $d \ge 3$.
It remains to consider the case of $d=2$. There we know that the growth constant
$\mu_w^0$ for polygons in a slab of width $w$ is strictly less than that of walks in a
slab of width $w$, \emph{i.e.} $\mu_w^0 < \mu_w$
for all $w < \infty$ \cite{Soteros1988}. The same argument shows
that
\begin{equation}
\mu \ge \mu_0(y) \ge \mu_w^0
\label{eqn:2dbounds}
\end{equation}
and we give some properties of $\mu_w^0$ in the next theorem.
\begin{theo}
For polygons in a two dimensional slit $\mu_w^0$ is an increasing function of
$w$ and $\sup_w \mu_w^0 = \mu$.
\end{theo}
{\it Proof: }
Every polygon with $n$ edges and span $w$ can be converted to a polygon with $n+2$
edges and span $w+1$ so $\pi_{n+2}(w+1) \ge \pi_n(w)$, from which we have
$\mu_{w+1}^0 \ge \mu_w^0$. The existence of the limit $\lim_{w \to \infty} \mu_w^0$
and the fact that it is equal to $\sup_w \mu_w^0$ and that this in turn is equal to $\mu$
is a consequence of the arguments in Sections 4 and 6 of \cite{HamWhitt1985}.
$\Box$
Since $\sup_w \mu_w^0 = \mu$ (by the above Theorem) it follows from
(\ref{eqn:2dbounds}) that $\mu_0(y) = \mu$ for all $y \le 1$ in two dimensions, and
therefore for all $d \ge 2$.
\vspace{0.1in}
\subsection{Polygons pulled at a middle vertex}
Consider $n$-edge polygons ($n$ is automatically even)
with at least one vertex in the adsorbing plane and with all vertices in or on one
side of this plane. Translate the polygon so that the lexicographically first vertex
in the surface is at the origin. The \emph{middle vertex}
is the vertex joined to the origin by two sub-walks each of length $n/2$.
Let $c_n^+(h)$ be the number of positive walks from the origin with endpoint at height $h$ above the adsorbing
plane, and $p_n^+(h)$ be the number of polygons with middle vertex
at height $h$. Note that $\sum_h c_n^+(h) = c_n^+$.
Let $C_n^+(y)$ be the partition function of $c_n^+(h)$ and $P_n^+(y)$ be the partition function of $p_n^+(h)$,
with growth constants $\mu_c^+(y)$ and $\mu_p^+(y)$. The free energies are $\lambda^+(y) = \log \mu_c^+(y)$ and $\lambda_p^+(y)
= \log \mu_p^+(y)$. The partition function $P_n^+(y)$ is the partition function of
polygons pulled at their middle vertex and $C_n^+(y)$ is the corresponding partition function
for walks pulled at their last vertex.
We define the generating functions
\begin{equation}
W^+(t,y) = \sum_n C_n^+(y) t^n \quad \quad P^+(t,y) = \sum_n P_n^+(y) t^n
\end{equation}
and we write their radii of convergence as $t_c^+(y)$ and $t_p^+(y)$.
We now define similar quantities for bridges. A bridge is a positive walk that takes its first step
away from the adsorbing plane, never returns to the adsorbing plane and whose last vertex is
in the top plane of the walk. Let $b_n(h)$ be the number of $n$-edge bridges of height $h$. The partition
function and generating function are defined as
\begin{equation}
B_n(y) = \sum_h b_n(h) y^h , \quad \quad B(t,y) = \sum_n B_n(y) t^n.
\end{equation}
A bridge can be doubly unfolded in the first coordinate direction so that
the origin is left-most and the first coordinate of the last vertex is
right-most \cite{HammersleyWelsh}. See Figure \ref{figure2}.
Let $b_n^\dagger(h)$ be the number of doubly unfolded bridges with $n$ edges and height $h$.
Then
\[ b_n^\dagger(h) \leq b_n (h) \leq e^{o(n)} b_n^\dagger(h). \]
Their partition functions are related by the inequalities,
\begin{equation}
B_n^\dagger(y) \leq B_n(y) \leq e^{o(n)} B_n^\dagger(y),
\end{equation}
and so
$\lim_{n\to\infty} (B_n^\dagger(y))^{1/n} = \mu_c^B(y)$ where $\mu_c^B(y) = 1/t_c^B(y)=1/t_c^+(y)$. This last
equality follows from Lemma 1 and Theorem 3 in \cite{RensburgWhittington2016a}.
\subsection{A connection between polygon and walk partition functions}
Clearly
$t_c^+(y) \leq t_p^+(y)$ for all $y>0$ and $t_c^+(1) = t_p^+(1) = \mu^{-1}$.
If $y>1$ then $P_n^+ (y) \leq p_n y^{n/2}$ and $C_n^+(y) \geq y^n$. Taking powers $1/n$ and then letting $n\to\infty$
gives $\mu_p^+(y) \leq \mu\,\sqrt{y} < y \leq \mu_c^+(y)$ for $y > \mu^2$, or
\begin{equation}
t_p^+(y) > t_c^+(y), \quad\hbox{for $y > \mu^2$}.
\label{eqn:polywalk1inequality}
\end{equation}
We next show how this inequality can be strengthened.
\begin{theo}
$$t_p^+(y^2) \ge t_c^+(y), \: \mbox{for all } y \ge 1.$$
\label{theo:polywalk}
\end{theo}
{\it Proof: }
By cutting the polygon at its middle vertex (where the force is applied) into two walks, $p_n^+ (h) \leq (c_{n/2}^+(h))^2$, so that
\begin{equation}
\fl \quad
P_n^+(y) = \sum_h p_n^+(h) y^h \leq \sum_h \left(} \def\R{\right) c_{n/2}^+(h) y^{h/2} \R^2
\leq \left(} \def\R{\right) \sum_h c_{n/2}^+(h) y^{h/2} \R^2 = (C_{n/2}^+(\sqrt{y}))^2 .
\end{equation}
Take the power $1/n$ and let $n\to\infty$ to obtain $\mu_p^+(y) \leq \mu_c^+(\sqrt{y})$. That is
\[ t_p^+(y^2) \geq t_c^+(y), \quad\hbox{for $y \geq 1$}. \]
$\Box$
A corollary of this Theorem is as follows:
The free energy of pulled walks is $-\log t_c^+(y)$, and this is strictly increasing with $y>1$, since the model is ballistic.
That is, $t_c^+(y) > t_c^+(y^2)$ if $y>1$. This shows with the above that
$t_p^+(y^2) \geq t_c^+(y) > t_c^+(y^2)$ for $y>1$, or $t_p^+(y) > t_c^+(y)$ whenever $y>1$. This strengthens (\ref{eqn:polywalk1inequality}).
\def\Vec#1{\overset{\to}{#1}}
We next prove the corresponding inequality in the other direction. We obtain a lower bound by constructing polygons
by concatenating four doubly unfolded bridges.
The unfolded bridge has a width $\Vec{w}$ and a height $h$, where $\Vec{w}$ is the vector of
widths in all except the vertical direction, and may look as illustrated in Figure
\ref{figure2}.
\begin{figure}[t]
\beginpicture
\setcoordinatesystem units <1.5pt,1.5pt>
\setplotarea x from -130 to 90, y from -10 to 80
\setplotarea x from -50 to 50, y from 0 to 80
\arrow <5pt> [.2,.67] from 60 0 to 60 80
\put {\large $h$} at 65 40
\arrow <5pt> [.2,.67] from -50 -10 to 50 -10
\put {\large $\overset{\longrightarrow}{w}$} at 0 -15
\color{Tan}
\grid 10 8
\color{Tan}
\setplotsymbol ({$\bullet$})
\plot -50 -3 50 -3 /
\color{Blue}
\plot -50 0 -50 10 -40 10 -40 20 -50 20 -50 30 -40 30 -30 30 -30 20 -30 10 -20 10 -20 20
-20 30 -20 40 -20 50 -20 60 -30 60 -30 70 -30 80 -20 80 -20 70 -10 70 0 70 10 70 10 60 10 50
10 40 0 40 0 30 10 30 20 30 20 40 20 50 30 50 40 50 50 50 50 60 40 60 40 70 50 70
50 80 /
\color{NavyBlue}
\multiput {\LARGE$\bullet$} at
-50 0 -50 10 -40 10 -40 20 -50 20 -50 30 -40 30 -30 30 -30 20 -30 10 -20 10 -20 20
-20 30 -20 40 -20 50 -20 60 -30 60 -30 70 -30 80 -20 80 -20 70 -10 70 0 70 10 70 10 60 10 50
10 40 0 40 0 30 10 30 20 30 20 40 20 50 30 50 40 50 50 50 50 60 40 60 40 70 50 70
50 80 /
\color{red}
\multiput {\huge$\bullet$} at -50 0 50 80 /
\color{black} \normalcolor
\endpicture
\caption{A doubly unfolded bridge. The vertex in the surface is in the left-most plane and the last vertex
is in the right-most plane. The bridge steps away from the surface at its first step and
never returns.}
\label{figure2}
\end{figure}
Next, we have to choose values for $h$ and $\Vec{w}$. Let $b_n^{\dagger}(h,\Vec{w})$ be the number of $n$-edge
doubly unfolded bridges with width $\Vec{w}$ and height $h$.
In the partition function for doubly unfolded bridges, $B_n^\dagger (y) = \sum_{h,\Vec{w}} b_n^\dagger (h,\Vec{w}) y^h$
there are \textit{most popular values} of $h$ and $\Vec{w}$ (these are functions of $y$); denote them by $h^*$ and $\Vec{w}^{\,*}$.
Then $b_n^\dagger(h^*,\Vec{w}^{\,*}) y^{h^*}$ is a largest term in $B_n^\dagger(y) = \sum_{h,\Vec{w}} b_n^\dagger(h,\Vec{w}) y^h$, so that
\begin{equation}
b_n^\dagger(h^*,\Vec{w}^{\,*}) y^{h^*} \leq B_n^\dagger(y) \leq (n+1)^d\, b_n^\dagger(h^*,\Vec{w}^{\,*}) y^{h^*}.
\label{eqn:unfoldedbridge}
\end{equation}
Taking powers $1/n$, letting $n\to\infty$, and noting that $(B_n^\dagger(y))^{1/n} \to \mu_c^B(y)$, it follows that
\begin{equation} \lim_{n\to\infty} (b_n^\dagger(h^*,\Vec{w}^{\,*})y^{h^*})^{1/n} = \mu^B_c(y) = 1/t_c^B(y)= 1/t_c^+(y).
\label{eqn:bridgemu}
\end{equation}
\begin{theo}
$$t_p^+(y^2) = t_c^+(y) \quad \hbox{for all $y \ge 1$}.$$
\end{theo}
{\it Proof: }
Because of Theorem \ref{theo:polywalk} above we only need to prove an inequality in one direction.
By reflecting and rotating unfolded bridges of most popular widths and heights
they can be concatenated to form a polygon as in Figure \ref{figure3}.
\begin{figure}[t]
\beginpicture
\setcoordinatesystem units <1.0pt,1.0pt>
\setplotarea x from -130 to 170, y from -10 to 160
\setplotarea x from -50 to 50, y from 0 to 160
\arrow <5pt> [.2,.67] from -50 -10 to 50 -10
\put {\large $\overset{\longrightarrow}{w}^*$} at 0 -20
\arrow <5pt> [.2,.67] from 60 -10 to 160 -10
\put {\large $\overset{\longrightarrow}{w}^*$} at 110 -20
\arrow <5pt> [.2,.67] from 170 0 to 170 80
\put {\large $h^*$} at 180 40
\arrow <5pt> [.2,.67] from 170 80 to 170 160
\put {\large $h^*$} at 180 120
\color{Tan}
\grid 10 16
\color{Tan}
\setplotsymbol ({\scriptsize$\bullet$})
\plot -50 80 160 80 /
\setplotsymbol ({$\bullet$})
\plot -50 -3 160 -3 /
\color{Blue}
\plot -50 80 -40 80 -30 80 -30 70 -30 60 -40 60 -50 60 -50 50 -40 50 -30 50 -20 50
-20 60 -10 60 -10 50 -10 40 -10 30 -10 20 -10 10 0 10 0 20 0 30 0 40 10 40 10 50
20 50 20 60 10 60 10 70 10 80 20 80 30 80 30 70 30 60 30 50 30 40 20 40 20 30
30 30 40 30 40 20 50 20 50 10 50 0 /
\color{NavyBlue}
\multiput {\LARGE$\bullet$} at
-50 80 -40 80 -30 80 -30 70 -30 60 -40 60 -50 60 -50 50 -40 50 -30 50 -20 50
-20 60 -10 60 -10 50 -10 40 -10 30 -10 20 -10 10 0 10 0 20 0 30 0 40 10 40 10 50
20 50 20 60 10 60 10 70 10 80 20 80 30 80 30 70 30 60 30 50 30 40 20 40 20 30
30 30 40 30 40 20 50 20 50 10 50 0 /
\color{red}
\multiput {\huge$\bullet$} at -50 80 50 0 /
\color{black} \normalcolor
\setcoordinatesystem units <1.0pt,1.0pt> point at -110 0
\setplotarea x from -50 to 50, y from 0 to 160
\color{Tan}
\grid 10 16
\color{Blue}
\plot -50 0 -50 10 -40 10 -40 20 -50 20 -50 30 -40 30 -30 30 -30 20 -30 10 -20 10
-20 20 -20 30 -20 40 -20 50 -20 60 -30 60 -30 70 -20 70 -10 70 0 70 10 70 10
60 10 50 10 40 0 40 0 30 10 30 20 30 20 40 20 50 30 50 40 50 40 40 40
30 50 30 50 40 50 50 50 60 40 60 40 70 50 70 50 80 /
\color{NavyBlue}
\multiput {\LARGE$\bullet$} at
-50 0 -50 10 -40 10 -40 20 -50 20 -50 30 -40 30 -30 30 -30 20 -30 10 -20 10
-20 20 -20 30 -20 40 -20 50 -20 60 -30 60 -30 70 -20 70 -10 70 0 70 10 70 10
60 10 50 10 40 0 40 0 30 10 30 20 30 20 40 20 50 30 50 40 50 40 40 40
30 50 30 50 40 50 50 50 60 40 60 40 70 50 70 50 80 /
\color{red}
\multiput {\huge$\bullet$} at -50 0 50 80 /
\color{black} \normalcolor
\setcoordinatesystem units <1.0pt,1.0pt> point at 0 -80
\setplotarea x from -50 to 50, y from 0 to 80
\color{Blue}
\plot -50 0 -50 10 -40 10 -40 20 -50 20 -50 30 -40 30 -30 30 -30 20 -30 10 -20 10 -20 20
-20 30 -20 40 -20 50 -20 60 -30 60 -30 70 -20 70 -10 70 0 70 10 70 10 60 10 50
10 40 0 40 0 30 0 20 0 10 10 10 10 20 10 30 20 30 20 40 20 50 30 50 40 50 50 50
50 60 40 60 40 70 50 70 50 80 /
\color{NavyBlue}
\multiput {\LARGE$\bullet$} at
-50 0 -50 10 -40 10 -40 20 -50 20 -50 30 -40 30 -30 30 -30 20 -30 10 -20 10 -20 20
-20 30 -20 40 -20 50 -20 60 -30 60 -30 70 -20 70 -10 70 0 70 10 70 10 60 10 50
10 40 0 40 0 30 0 20 0 10 10 10 10 20 10 30 20 30 20 40 20 50 30 50 40 50 50 50
50 60 40 60 40 70 50 70 50 80 /
\color{red}
\multiput {\huge$\bullet$} at -50 0 50 80 /
\color{black} \normalcolor
\setcoordinatesystem units <1.0pt,1.0pt> point at -110 -80
\setplotarea x from -50 to 50, y from 0 to 80
\color{Blue}
\plot -50 80 -50 70 -50 60 -50 50 -50 40 -50 30 -40 30 -40 20 -40 20 -40 10
-30 10 -20 10 -10 10 -10 20 -20 20 -30 20 -30 30 -30 40 -20 40 -20 50 -10 50
0 50 0 60 0 70 0 80 10 80 20 80 20 70 20 60 20 60 10 60 10 50 20 50
30 50 30 40 40 40 40 30 30 30 30 20 30 10 40 10 50 10 50 0 /
\color{NavyBlue}
\multiput {\LARGE$\bullet$} at
-50 80 -50 70 -50 60 -50 50 -50 40 -50 30 -40 30 -40 20 -40 20 -40 10
-30 10 -20 10 -10 10 -10 20 -20 20 -30 20 -30 30 -30 40 -20 40 -20 50 -10 50
0 50 0 60 0 70 0 80 10 80 20 80 20 70 20 60 20 60 10 60 10 50 20 50
30 50 30 40 40 40 40 30 30 30 30 20 30 10 40 10 50 10 50 0 /
\color{red}
\multiput {\huge$\bullet$} at -50 80 50 0 /
\color{black} \normalcolor
\endpicture
\caption{Four doubly unfolded bridges can be concatenated to form a polygon. The bridges are selected
so that their heights and widths have their most popular values.}
\label{figure3}
\end{figure}
This arrangement gives a lower bound on polygons of height $2h^*$ and of length $4n+2$ which gives a lower bound
on the polygon partition function:
\begin{equation}
\left(} \def\R{\right) b_n^\dagger(h^*,\Vec{w}^{\,*}) y^{h^*} \R^{\!4} \leq p^+_{4n+2}(2h^*) y^{4h^*} \leq P^+_{4n+2}(y^2) .
\label{eqn:bridgepol}
\end{equation}
Take the power $1/4n$ and then let $n\to\infty$. The left hand side goes to $1/t_c^+(y)$, and the right hand side goes to
$\mu_p^+(y^2) = 1/t_p^+(y^2)$. This shows that
\[ t_p^+(y^2) \leq t_c^+(y). \]
Since we already know that $t_p^+(y^2) \geq t_c^+(y)$ the result is that
\begin{equation}
t_p^+(y^2) = t_c^+(y) .
\label{eqn1}
\end{equation}
This shows that polygons also become ballistic at $y_c=1$.
$\Box$
We turn our attention now to the case where $y \le 1$. The construction in Figure \ref{figure3},
together with equations (\ref{eqn:unfoldedbridge}) and (\ref{eqn:bridgepol}), shows that
\begin{eqnarray}
\frac{1}{(n+1)^{4d}}(B_n^{\dagger}(y))^4 & \le & (b_n^{\dagger}(h^*,\Vec{w}^*)y^{h^*})^4 \nonumber \\
& \le & p_{4n+2}^+(2h^*)y^{4h^*} \le P_{4n+2}^+(y^2).
\end{eqnarray}
Recall that $h^*$ and $\Vec{w}^*$ are the most popular values of the height and width.
We know that, for $y \le 1$, $t_c^+(y) = t_c^B(y)= 1/\mu$ \cite{RensburgWhittington2016a}.
Hence $1/\mu \ge t_p^+(y^2) \ge t_c^+(y^2) = 1/\mu$ for all $y \le 1$. This proves the following
theorem:
\begin{theo}
When $y \le 1$, $ t_p^+(y) = \frac{1}{\mu}$.
\label{theo:QQ}
\end{theo}
\section{Polygons pulled from their top plane}
\label{section5}
Define $p_n(\ell)$ to be the number of polygons, with at least one vertex in $x_d=0$
and with \textit{highest vertices} at height $\ell$. A model of pulled polygons,
where the highest vertices are pulled vertically by a force $f$, is defined by
the partition function
\begin{equation}
P_n (y) = \sum_{\ell\geq 0} p_n(\ell)\, y^\ell.
\end{equation}
Here, the activity $y=e^{f /k_BT}$ is introduced and is equal to the exponential of the
reduced pulling force. The generating function of this model is given by
$P(t,y)$ and its radius of convergence is denoted $t_p(y)$.
It remains to relate the limiting free energy of this model to that of polygons
pulled in the middle, considered above. Note that the midpoint
of a polygon of height $\ell$ is itself at most at height $\ell$. Thus, assuming
that $y\geq 1$, $P_n^+(y) \leq P_n(y)$. This shows that
\begin{equation}
\lim_{n\to\infty} \frac{1}{n} \log P_n^+(y) \leq \liminf_{n\to\infty}
\frac{1}{n} \log P_n(y) .
\label{eqn2}
\end{equation}
This, in particular, shows that $t_c^+(\sqrt{y}) = t_p^+(y) \geq t_p(y)$.
Existence of the free energy will now be shown by bounding the limiting supremum.
Cut the polygon at its lexicographically first vertex in the surface, and unfold it into a
bridge in the $x_1$-direction (by
adding a single edge in the horizontal direction). This is
schematically illustrated in Figure \ref{figure4}.
\begin{figure}[t]
\beginpicture
\setcoordinatesystem units <1.5pt,1.5pt>
\setplotarea x from -150 to 130, y from -10 to 50
\setplotarea x from -100 to 130, y from 0 to 40
\color{Tan}
\grid 23 4
\color{Tan}
\setplotsymbol ({$\bullet$})
\plot -100 -3 130 -3 /
\plot -100 43 130 43 /
\color{Blue}
\plot -50 0 -50 10 -60 10 -70 10 -80 10 -80 20 -70 20 -70 30 -70 40
-60 40 -50 40 -50 30 -50 20 -40 20 -40 30 -30 30 -30 20 -30 10 -40 10 -40 0 -50 0 /
\plot 0 0 0 10 10 10 20 10 30 10 30 20 40 20 40 30 40 40 50 40 60 40
60 30 60 20 70 20 70 30 80 30 80 20 80 10 90 10 90 0 100 0 110 0 /
\color{NavyBlue}
\multiput {\LARGE$\bullet$} at
-50 0 -50 10 -60 10 -70 10 -80 10 -80 20 -70 20 -70 30 -70 40
-60 40 -50 40 -50 30 -50 20 -40 20 -40 30 -30 30 -30 20 -30 10 -40 10 -40 0 -50 0
0 0 0 10 10 10 20 10 30 10 30 20 40 20 40 30 40 40 50 40 60 40
60 30 60 20 70 20 70 30 80 30 80 20 80 10 90 10 90 0 100 0 110 0 /
\color{red}
\multiput {\huge$\bullet$} at -50 0 0 0 100 0 110 0 /
\color{black} \normalcolor
\setplotsymbol ({.})
\arrow <5pt> [.2,.67] from -25 20 to -5 20
\arrow <5pt> [.2,.67] from 135 0 to 135 40
\put {\Large$\ell$} at 140 20
\endpicture
\caption{A polygon in a slit or slab unfolded to form a loop.}
\label{figure4}
\end{figure}
The polygon is unfolded into a loop of height $\ell$.
Denote the partition function of these (unfolded) loops by $L_n^\ddagger(y)$,
and it follows that $P_n(y) \leq e^{o(n)} L_n^\ddagger (y)$. Let the
number of unfolded loops of
length $n$ and height $\ell$ be denoted by $l_n^\ddagger(\ell)$. Then
\begin{equation}
L_n^\ddagger(y) = \sum_{\ell=0}^{n/2} l_n^\ddagger(\ell)\,y^\ell.
\end{equation}
For each value of $y>0$ there is a most popular value of $\ell$ in this summation,
and this is denoted by $\ell^*$ (this is dependent on $y$ and on $n$). In particular,
\begin{equation}
l_n^\ddagger(\ell^*)\,y^{\ell^*} \leq L_n^\ddagger(y) \leq \sfrac{1}{2}n\, l_n^\ddagger(\ell^*)\,y^{\ell^*}.
\end{equation}
The loops in this most popular class are schematically illustrated
in Figure \ref{figure5}.
\begin{figure}[t]
\beginpicture
\setcoordinatesystem units <1.5pt,1.5pt>
\setplotarea x from -50 to 120, y from -10 to 50
\setplotarea x from -10 to 120, y from 0 to 40
\color{Tan}
\grid 13 4
\color{Tan}
\setplotsymbol ({$\bullet$})
\plot -10 -3 120 -3 /
\plot -10 43 120 43 /
\color{Blue}
\plot 0 0 0 10 10 10 10 0 20 0 20 10 30 10 30 20 40 20 40 30 40 40 50 40 60 40
60 30 60 20 70 20 70 30 80 30 80 20 80 10 90 10 90 0 100 0 110 0 /
\color{NavyBlue}
\multiput {\LARGE$\bullet$} at
0 0 0 10 10 10 10 0 20 0 20 10 30 10 30 20 40 20 40 30 40 40 50 40 60 40
60 30 60 20 70 20 70 30 80 30 80 20 80 10 90 10 90 0 100 0 110 0 /
\color{red}
\multiput {\huge$\bullet$} at 0 0 100 0 110 0 /
\color{black} \normalcolor
\setplotsymbol ({.})
\arrow <5pt> [.2,.67] from -15 0 to -15 40
\put {\LARGE$\ell^*$} at -20 20
\put {\LARGE$=$} at 130 20
\put {
\beginpicture
\setplotarea x from -10 to 75, y from -5 to 45
\color{black}
\arrow <5pt> [.2,.67] from 70 0 to 70 40
\put {\LARGE$\ell^*$} at 75 20
\put {\LARGE$n$} at 30 32
\color{Tan}
\setplotsymbol ({$\bullet$})
\plot 0 -4 70 -4 / \plot 0 44 70 44 /
\color{Blue}
\plot 60 0 70 0 /
\setquadratic
\plot 0 0 30 40 60 0 /
\setlinear
\color{red}
\multiput {\huge$\bullet$} at 0 0 60 0 70 0 /
\color{black}
\endpicture } at 180 20
\color{black}
\normalcolor
\endpicture
\caption{An unfolded loop with its last step in the surface and with height equal to the most
popular height $\ell^*$. On the right is a schematic diagram
of this class of loops.}
\label{figure5}
\end{figure}
\begin{figure}[t]
\beginpicture
\setplotarea x from -100 to 205, y from -5 to 45
\setplotarea x from -10 to 205, y from -5 to 45
\arrow <8pt> [.2,.67] from 120 20 to 140 20
\multiput {
\beginpicture
\setplotarea x from -10 to 75, y from -5 to 45
\color{black}
\arrow <5pt> [.2,.67] from 70 0 to 70 40
\put {\large$\ell^*$} at 65 20
\put {\large$n$} at 30 30
\color{Tan}
\setplotsymbol ({$\bullet$})
\plot 0 -4 70 -4 / \plot 0 44 70 44 /
\color{Blue}
\plot 60 0 70 0 /
\setquadratic
\plot 0 0 30 40 60 0 /
\setlinear
\color{red}
\multiput {\huge$\bullet$} at 0 0 70 0 /
\color{black}
\endpicture } at 0 20 70 20 /
\setplotarea x from 150 to 250, y from -5 to 85
\color{black}
\arrow <8pt> [.2,.67] from 290 0 to 290 80
\put {\large$2\ell^*$} at 299 40
\setsolid
\multiput {\large$n$} at 180 30 250 30 /
\color{Tan}
\setplotsymbol ({$\bullet$})
\plot 150 -4 290 -4 / \plot 150 84 290 84 /
\color{blue}
\plot 210 80 220 80 /
\plot 280 0 290 0 /
\setquadratic
\plot 150 0 165 30 180 40 195 50 210 80 /
\plot 220 80 235 50 250 40 265 30 280 0 /
\setlinear
\color{red}
\multiput {\LARGE$\bullet$} at 150 0 220 80 290 0 /
\color{black}
\normalcolor
\endpicture
\caption{Two unfolded loops are concatenated in a slab of height $\ell^*$ and
then reflected through the top boundary of the slab to obtain a loop in a slab of
height $2\ell^*$.}
\label{figure6}
\end{figure}
Two loops in this most popular class can be concatenated as illustrated
schematically in Figure \ref{figure6}.
If the middle part of the concatenated loops is reflected through
the top plane as shown, then a loop of height $2\ell^*$ is obtained
with the property that its middle vertex is also in the top plane. This
loop consists of two self-avoiding walks of length $n$ and height $2\ell^*$.
Thus
\begin{eqnarray}
\frac{4}{n^2} L_n^{\ddagger 2}(y) & \leq
\left(} \def\R{\right) l_n^\ddagger(\ell^*)\, y^{\ell^*} \R^2 \leq
\left(} \def\R{\right) c_n^+(2\ell^*) \, y^{\ell^*} \R^2 \nonumber \\
& \leq \sum_\ell \left(} \def\R{\right) c_n^+(\ell) \, y^{\ell/2} \R^2 \leq
\left(} \def\R{\right) \sum_\ell c_n^+(\ell)\, y^{\ell/2} \R^2 = (C_n^+ (\sqrt{y}))^2 . \label{eqn5.5}
\end{eqnarray}
That is, since $P_n (y) \leq e^{o(n)} L_n^\ddagger(y)$,
\begin{equation}
\limsup_{n\to\infty} \frac{1}{n} \log P_n(y) \leq \lim_{n\to\infty} \frac{1}{n} \log C_n^+(\sqrt{y})
\end{equation}
with the result that by equation \Ref{eqn2},
\begin{eqnarray}
- \log t_p^+(y) &=
\lim_{n\to\infty} \frac{1}{n} \log P_n^+(y) \leq \liminf_{n\to\infty}
\frac{1}{n} \log P_n(y) \nonumber \\
& \leq \limsup_{n\to\infty} \frac{1}{n} \log P_n(y) \leq \lim_{n\to\infty} \frac{1}{n} \log C_n^+(\sqrt{y})
\nonumber \\
&= - \log t_c^+(\sqrt{y})= - \log t_p^+(y).
\end{eqnarray}
Here, recall that $P_n^+(y)$ is the partition function of polygons pulled in
the middle, $P_n(y)$ is the partition function of polygons pulled in
their highest plane, and $C_n^+(\sqrt{y})$ is the partition function of
walks pulled at their endpoint, and $y\geq 1$.
By equation \Ref{eqn1} this shows that for polygons pulled in their top plane,
\begin{equation}
\lim_{n\to\infty} \frac{1}{n} \log P_n (y) = - \log t_c^+(\sqrt{y}) = - \log t_p^+(y).
\end{equation}
That is, the free energy of polygons pulled at their middle point is equal to the
free energy of polygons pulled in their top plane if $y\geq 1$. The case of $y \le 1$
follows from Figure \ref{figure3} and the proof of Theorem \ref{theo:QQ}. This gives the
following theorem:
\begin{theo}
$$\lambda_0(y) = \lim_{n \to \infty} \frac{1}{n} \log P_n(y) = - \log t_c^+(\sqrt{y}) = -\log t_p^+(y)$$
for all $ y > 0$. When $y \le 1$ this is equal to $\log \mu$.
\end{theo}
For polygons, as well as for walks, there is a phase transition to a ballistic
phase at $y=1$ but the response of polygons and walks is different in
this ballistic phase.
\section{Polygons pulled from an interacting surface}
\label{sec:ayplane}
In this section we consider the full problem where the polygon interacts with
the surface ($a \ne 1$) and the applied force is pulling the polymer
off the surface in its top plane ($y > 1$). We derive some results about the
$a$-dependence of the free energy at fixed $y > 1$ and show that there is a phase transition from
an adsorbed phase to a ballistic phase at $a=a_c^0(y)$.
First consider the situation when $y > 0$ and $a \le 1$.
\begin{theo}
For $y > 0$ and $a \le 1$ the thermodynamic limit
\begin{equation}
\lim_{n\to\infty} n^{-1} \log P_n (a,y) \equiv \psi_0(a,y)
\end{equation}
exists so we have a well defined limiting free energy. Moreover, in this region
of the $(a,y)$-plane the free energy is independent of $a$ so that $\psi_0(a,y) = \psi_0(1,y) = \lambda_0(y)$.
\end{theo}
{\it Proof: } For fixed $y > 0$ and for all $a \le 1$, by monotonicity
\begin{equation}
P_n (0,y) \le P_n (a,y) \le P_n (1,y).
\label{eqn:Pbounds}
\end{equation}
Consider an $n$-edge polygon $\omega$ with span $s$.
Suppose that $V(\omega)$ is the vertex of
$\omega$ in the surface (\emph{i.e.} that is a visit) with lexicographically
first coordinates in the surface. When $d=2$ there is exactly one edge of $\omega$
in the surface that is incident
on $V$. For $d>2$, if there are two such edges, choose the one with
lexicographically first mid-point.
Delete this edge in the surface, incident on $V$, translate $\omega$ by unit distance away from the surface,
add two edges to connect the resulting walk to the surface and add an edge in the
surface to obtain a polygon $\omega'$ with $n+2$ edges and span $s+1$ with exactly two
visits.
\begin{equation}
\sum_v p_n(v,s) \leq p_{n+2}(0,s+1)
.
\end{equation}
This implies that
\begin{equation}
P_n(1,y) \leq \sfrac{1}{y}\, P_{n+2}(0,y)
\end{equation}
and hence, from equation (\ref{eqn:Pbounds}),
\begin{equation}
y P_{n-2}(1,y) \le P_n (0,y) \le P_n (a,y) \le P_n (1,y).
\end{equation}
Take logarithms, divide by $n$, let $n\to\infty$,
and the result follows.
$\Box$
When $y \ge 1$ and $a \ge 1$ we have two useful lower bounds on the partition function
that follow from monotonicity, namely
\begin{equation}
P_n(a,y) \ge P_n(a,1) \quad \mbox{and} \quad P_n(a,y) \ge P_n(1,y).
\end{equation}
These bounds imply that
\begin{equation}
\liminf_{n\to\infty} n^{-1} \log P_n(a,y) \ge \max[\kappa_0(a), \lambda_0(y)].
\end{equation}
We can also construct an upper bound similar to the one derived
for the self-avoiding walk model \cite{JvRW2013}. Let $l_n(h)$ be the number of
loops (or arches) with $n$ edges and span in the $x_d$-direction equal to $h$.
Write the partition function as
\begin{equation}
L_n(y) = \sum_h l_n(h) y^h.
\end{equation}
\begin{theo}
For all $d \ge 2$
\begin{equation}
P_n(a,y) \le y^{-1} \sum_m C_m(a,1)\, L_{n-m+2}(y)
\label{eqn:upperbound}
\end{equation}
\end{theo}
{\it Proof: }
Every $n$-edge polygon with span $h$ must have at least one edge with $x_d=h$.
Either the polygon has every edge in $x_d=0$ so that $h=0$, or $h \ge 1$.
If $h=0$ choose the edge that is lexicographically first
and add two edges to the polygon just before and just after the
distinguished edge in $x_d=0$ to produce a unique edge in $x_d=1$
in a single loop of 3 edges. If $h \ge 1$
choose the edge in $x_d=h$ that is lexicographically first. This edge must be
in a loop with, say, $n-m$ edges. The remainder of the polygon is a
walk with $m$ edges. Add two edges to the loop just before and just after the
distinguished edge in $x_d=h$ to produce a unique edge in $x_d=h+1$. This
unique edge distinguishes the loop of length $n-m+2$
from the rest of the polygon. By noting that the
rest of the polygon is a walk of length $m$, the following inequality is obtained:
\begin{equation}
p_n (v,h) \le \sum_m c_m(v) \, l_{n-m+2}(h+1)
\end{equation}
where $c_m(v) = \sum_h c_m^+(v,h)$. Now multiply both sides by
$y^h a^v$ and sum over $h$ and $v$ giving (\ref{eqn:upperbound}) which proves
the theorem.
$\Box$
Define the generating functions
\begin{equation}
\widehat{L}(t,y) = \sum_n L_n(y)t^n \quad \mbox{and} \quad \widehat{C}(t,a) = \sum_n C_n^+(a,1)t^n.
\end{equation}
By the convolution theorem
\begin{equation}
\sum_n \sum_m C_m(a,1) L_{n-m+2}(y) t^n \le t^{-2} \widehat{C}(a,t) \widehat{L}(y,t) = t^{-2}
\widehat{B}(a,y,t).
\end{equation}
The radius of convergence of $\widehat{C}(t,a)$ is $t_1(a)= \exp[-\kappa (a)]$ and the
radius of convergence of $\widehat{L}(t,y)$ is $t_2(y)$ so the radius of
convergence of $\widehat{B}(a,y,z)$ is $\min[t_1(a),t_2(y)]$. This implies that
\begin{equation}
\limsup_{n\to\infty} n^{-1} \log P_n(a,y)
\le \max[\kappa (a), \liminf_{n\to\infty} n^{-1} \log L_n(y)]
\label{eqn6.13}
\end{equation}
or, roughly, the free energy of pulled polygons interacting with the
surface is bounded above by the maximum of the free energy of
walks interacting with the surface and pulled loops.
By the results in Section \ref{section5} (see equation \Ref{eqn5.5}) it follows
\begin{equation}
e^{o(n)}P_n(y) \leq L_n^\ddagger (y) \leq e^{o(n)}C_n^+(\sqrt{y}) = e^{o(n)} P_n (y) .
\end{equation}
Taking logarithms, dividing by $n$, and letting $n\to\infty$, it follows that
\[ \lim_{n\to\infty} \sfrac{1}{n} \log L_n^\ddagger (y) = -\log t_p(y) . \]
But, by unfolding loops, it follows that $e^{o(n)} L_n(y) \leq L_n^\ddagger(y) \leq
L_n(y)$. Thus
\begin{equation}
\lim_{n\to\infty} \sfrac{1}{n} \log L_n(y) = -\log t_p(y) .
\end{equation}
In equation \Ref{eqn6.13} this gives
\begin{equation}
\limsup_{n\to\infty} n^{-1} \log P_n(a,y)
\le \max[\kappa (a), \lambda_0(y) ]
\label{eqn6.16}
\end{equation}
When $d > 2$ we know that $\kappa_0(a) = \kappa (a)$
\cite{Soteros} so the above result can be replaced with
\begin{equation}
\lim_{n\to\infty} n^{-1} \log P_n(a,y) = \max[\kappa_0 (a), \lambda_0(y)], \quad d \ge 3.
\label{eqn:phaseboundarycondition}
\end{equation}
This gives a complete characterization of the phase boundary when $d\geq 3$ and we state this as a theorem.
\begin{theo}
When $d \ge 3$ the phase boundary between the ballistic and adsorbed phases
for $y \ge 1$ is determined by the solution of the equation $\kappa_0(a) = \lambda_0(y)$.
\end{theo}
{\it Proof: }
This follows immediately from (\ref{eqn:phaseboundarycondition}).
$\Box$
Since $\kappa_0(a)=\kappa (a)$ \cite{Soteros} and $\lambda_0(y) = \lambda(\sqrt{y})$ the
phase boundary is determined by the properties of the self-avoiding
walk problem. We know the asymptotics of both $\kappa (a)$ \cite{Rychlewski}
and $\lambda(y)$ \cite{JvRW2013} so we know the behaviour of the phase boundary for polygons at large
values of $a$. We can switch into the force-temperature plane and this corresponds to the
behaviour at small values of the temperature. In particular, the critical force - temperature curve
is reentrant for all $d > 2$. The phase transition between the ballistic and adsorbed phases is
first order, except perhaps at $(a_c^0,1)$. This follows \emph{mutatis mutandis} from the
arguments in \cite{Guttmann2014} for the self-avoiding walk model.
When $d=2$ we know that
\begin{eqnarray}
\max[\kappa_0 (a), \lambda_0(y)]
&\leq \liminf_{n\to\infty} n^{-1} \log P_n(a,y) \nonumber \\
&\leq \limsup_{n\to\infty} n^{-1} \log P_n(a,y)
\leq \max[\kappa (a), \lambda_0(y)] .
\end{eqnarray}
Unlike the self-avoiding walk problem \cite{JvRW2013} we do not have a
precise condition for locating the phase boundary when $d=2$, since we only
have lower and upper bounds on the free energy when $a>1$ and $y>1$.
\section{Pushing a polygon towards an interacting surface}
\label{sec:pushing}
We now consider the situation when $d\geq 3$ and when $0\leq a\leq a_c^0$ and $0<y \leq 1$. First consider
$0 \leq a \leq 1$ and $0 < y \leq 1$. By theorem 8 we know that
\begin{equation}
\psi_0(a,y) = \psi_0(1,y) .
\end{equation}
By theorem 7 it follows that
$\lambda_0(y) = \psi_0(1,y) = \psi(1,\sqrt{y}) = \lambda(\sqrt{y})$ (see the discussion of the walk
problem in the introduction). But $\lambda(\sqrt{y}) = \log \mu$ for $0 < y \leq 1$
\cite{RensburgWhittington2016a}. Thus $\psi_0(a,y) = \log \mu$ for $0\leq a\leq 1$ and $0<y \leq 1$.
Since $\psi_0(a,y)$ is convex in each of its variables, the critical curve $y_c(a)$ in Figure
\ref{figure22} is a non-decreasing function of $a$. We now show why $y_c(a)$ has a jump
discontinuity at $a=a_c^0$ so that the phase boundary between the free and adsorbed phases
is a vertical line segment in Figure \ref{figure22}. Take $a=a_c^0-\epsilon$ for any
$\epsilon>0$. At $y=1$, $\psi_0(a,1) = \log \mu$. But $\psi_0(a,y)$ is monotone
non-decreasing in $y$ so it cannot be greater than $\log \mu$ for $y<1$, so
that $\psi_0(a,y) = \log \mu$ since it cannot be smaller than $\log \mu$.
We now give an alternative proof that the phase boundary between the free and adsorbed phases
is a vertical line segment in Figure \ref{figure22}. In fact this argument proves considerably
more and essentially completes our knowledge of the phase diagram
when $d \ge 3$. We shall need some preliminary lemmas. Let $\pi_n^w(a) = \sum_v p_n(v,w)a^v$.
By a concatenation argument we can show that the limit
\begin{equation}
\kappa_0^w(a) = \lim_{n\to\infty} n^{-1} \log \pi_n^w(a)
\end{equation}
exists. For instance, one can use a modification of the concatenation construction used in Section
4 of \cite{HamWhitt1985} coupled with a generalized supermultiplicative inequality. Let
$\widehat{\pi}_n^w(a) = \sum_{u \le w}\pi_n^u(a)$. This sum includes all polygons with span at most $w$.
\begin{lemm}
When $d \ge 2$,
$\kappa_0^w(a) \le \kappa_0^{w+1}(a)$.
\end{lemm}
{\it Proof: }
Each polygon contributing to the sum $\pi_n^w(a) = \sum_v p_n(v,w)a^v$ has at least one
edge in the hyperplane $x_d=w$. If there is more than one such edge, take the one with
lexicographically first midpoint. Translate this edge unit distance into the hyperplane
$x_d=w+1$ and add two edges to reconnect the polygon. Then
\begin{equation}
\pi_n^w(a) \le \pi_{n+2}^{w+1}(a)
\end{equation}
and taking logarithms, dividing by $n$ and letting $n \to \infty$ completes the proof.
$\Box$
\begin{lemm}
The exponential growth rate of $\widehat{\pi}_n^w(a)$ is identical to that
of $\pi_n^w(a)$ for all $w < \infty$.
\end{lemm}
{\it Proof: }
The result follows from the following inequalities:
\begin{equation}
\pi_n^w(a) \le \widehat{\pi}_n^w(a) \le (w+1) \pi_n^w(a) = \exp[\kappa_0^w(a) n + o(n)].
\end{equation}
$\Box$
\begin{lemm}
When $d \ge 2$,
$\sup_w \kappa_0^w(a) = \kappa_0(a)$.
\end{lemm}
{\it Proof: }
Clearly $\kappa_0^w(a) \le \kappa_0(a)$ for all $w$.
Write $n=Nr + q$, $0 \le q < N$. Concatenate $r$ polygons each with $N$ edges,
and a final polygon with $q$ edges,
using the concatenation construction detailed in Section 4 of \cite{HamWhitt1985}.
Each polygon with $N$ edges has span no larger than $w=N/2$ so the resulting
polygon (with $Nr+q$ edges) has span no larger than $w=N/2$. By the
argument leading to (4.14) in \cite{HamWhitt1985} this gives
\begin{equation}
\fl \quad \quad
n^{-1} \log \widehat{\pi}_n^w(a) \ge N^{-1} (1-q/n) \log P_N(a,1) -2 N^{-1} (1-q/N) \log[(d-1) N^{d-1}].
\end{equation}
Hence
\begin{equation}
\kappa_0^w(a) \ge N^{-1} \log P_N(a,1),
\end{equation}
where we recall that $w = N/2$. As $N \to \infty$ the right hand side goes to $\kappa_0(a)$
so $\sup_w \kappa_0^w(a) \ge \kappa_0(a)$ which completes the proof.
$\Box$
This allows us to prove the following:
\begin{theo}
When $d \ge 2$
$$\psi_0(a,y) = \psi_0(a,1) = \kappa_0(a)$$
for all $0 < y \le 1$.
\end{theo}
{\it Proof: }
Fix $0 < y \le 1$.
By monotonicity
$P_n(a,y) \le P_n(a,1)$
so
\begin{equation}
\limsup_{n\to\infty} n^{-1} \log P_n(a,y) \le \kappa_0(a).
\label{eqn:limsupslab}
\end{equation}
By considering one term in the partition function
\begin{equation}
P_n(a,y) \ge y^w \sum_v p_n(v,w)a^v = \pi_n^w(a),
\end{equation}
and
\begin{equation}
\liminf_{n\to\infty} n^{-1} \log P_n(a,y) \ge \lim_{n\to\infty} n^{-1} \log \pi_n^w(a) = \kappa_0^w(a)
\end{equation}
for all $w > 0$. Hence
\begin{equation}
\liminf_{n\to\infty} n^{-1} \log P_n(a,y) \ge \sup_w \kappa_0^w(a) = \kappa_0(a)
\label{eqn:liminfslab}
\end{equation}
Then (\ref{eqn:limsupslab}) and (\ref{eqn:liminfslab}) prove the theorem.
$\Box$
In particular, this proves that the phase boundary between the free and adsorbed phases
is a vertical line in the $(a,y)$-phase diagram, at $a=a_c^0$.
\section{The phase diagram}
\label{sec:phasediagram}
In this section we give a brief summary of what is known rigorously about the
form of the phase diagram in the $(a,y)$-plane.
There is a free phase when $a < a_c^0$ and $y < 1$. When $a > a_c^0$ and $y < 1$ the system
is in an adsorbed phase and $\psi^0(a,y) = \kappa^0(a)$, independent of $y$.
Suppose that $y_I(a)$ is the solution of the equation $\lambda_0(y)=\kappa(a)$ and
suppose that $y_{II}(a)$ is the solution of the equation $\lambda_0(y) = \kappa_0(a)$.
When $d \ge 3$ $\kappa_0(a) = \kappa(a)$ so $y_I(a) =y_{II}(a)$.
For this case ($d \ge 3$) there is a ballistic phase (where $\psi_0(a,y) = \lambda_0(y)$)
when $y > \max[1,y_I(a)]$ and an adsorbed
phase (where $\psi_0(a,y) = \kappa_0(a)$) when $a > a_c^0$ and $y < y_I(a)$.
When $d=2$ we know less. The system is in a ballistic phase when $y > \max[1,y_I(a)]$ but we
do not know whether $y = y_I(a)$ is a boundary of this phase. When $y < y_{II}(a)$ the system is
no longer ballistic. There are two possibilities:
\begin{enumerate}
\item
The phase boundary of the ballistic phase, $y_B(a)$, is equal to $y_{II}(a)$. In this case there are three
phases: a free phase, a ballistic phase and an adsorbed phase (when $a>a_c^0$ and $y < y_{II}(a)$).
\item
The phase boundary of the ballistic phase satisfies $y_{II}(a) < y_B(a) \le y_{I}(a)$. Then there at
least four phases: a free phase, a ballistic phase, an adsorbed phase (including the region defined
by $a > a_c^0$ and $y<1$) and at least one additional phase where the free energy depends on both
$a$ and $y$, in the region defined by $a > a_c^0$ and $1 < y < y_B(a)$. We do not know if $y=1$ is a
phase boundary.
\end{enumerate}
\begin{figure}[t]
\input figureBFACF.tex
\caption{BFACF moves in the square lattice \cite{BF81}. A positive move (or a
\textit{positive BFACF move}) increases the length of a polygon by
2 steps by replacing an edge with three edges. The reverse of this move
reduces the length of the polygon by 2 steps and is a
\textit{negative BFACF move}. \textit{Neutral BFACF moves}
change the polygon locally as shown without changing its length.}
\label{figureBFACF}
\end{figure}
\section{Numerical results from Monte Carlo data}
\label{sec:numericalMC}
In order to investigate the phase behaviour in the half square lattice, we collected
approximate enumeration data for polygons as a function of $a$ and $y$
using the GARM algorithm \cite{RJvR08}. This algorithm was implemented
using BFACF-style elementary moves \cite{BF81} (see figure \ref{figureBFACF})
on polygons in the half-lattice.
Polygons in the upper half square lattice and constrained to have at least one edge in
the boundary of the half-lattice (the $x$-axis) can be sampled by executing BFACF
moves. This is done as follows using a GARM implementation of BFACF moves.
A positive BFACF move can be done along the polygon
by replacing one edge by three
\beginpicture
\setcoordinatesystem units <0.75pt,0.75pt>
\plot 0 0 10 0 /
\arrow <5pt> [.2,.67] from 15 5 to 30 5
\plot 40 0 40 10 50 10 50 0 /
\multiput {\Large.} at 0 0 10 0 40 0 40 10 50 10 50 0 /
\put {$ $} at 58 0
\endpicture
while maintaining the constraints that the polygon has to step at least
once in the $x$-axis and must stay in the upper half square lattice. The collection
of all possible positive BFACF moves is the \textit{positive atmosphere} of the polygon.
The number of possible BFACF moves in the positive atmosphere of a polygon $\omega$
is denoted $a_+(\omega)$.
A negative BFACF move is the reversal of a positive BFACF move, and is
implemented by replacing three edges by one:
\beginpicture
\setcoordinatesystem units <0.75pt,0.75pt>
\plot 0 0 0 10 10 10 10 0 /
\arrow <5pt> [.2,.67] from 15 5 to 30 5
\plot 40 0 50 0 /
\multiput {\Large.} at 0 0 0 10 10 10 10 0 40 0 50 0 /
\put {$ $} at 58 0
\endpicture,
while maintaining the constraints that the polygon has to contain an edge in the
$x$-axis and stay in the upper half square lattice. The collection of all possible
negative BFACF moves is the \textit{negative atmosphere} of the polygon.
The number of possible BFACF moves in the negative atmosphere of a
polygon $\omega$ is denoted $a_-(\omega)$.
Neutral BFACF moves are implemented by local changes involving
two edges of the polygon:
\beginpicture
\setcoordinatesystem units <0.75pt,0.75pt>
\plot 0 0 0 10 10 10 /
\arrow <5pt> [.2,.67] from 15 5 to 40 5
\arrow <5pt> [.2,.67] from 40 5 to 15 5
\plot 50 0 60 0 60 10 /
\multiput {\Large.} at 0 0 0 10 10 10 50 0 60 0 60 10 /
\put {$ $} at 68 0
\endpicture
while maintaining the constraints that the polygon has to contain an edge in the
$x$-axis and stay in the upper half square lattice. The collection of all possible
neutral BFACF moves is the \textit{neutral atmosphere} of the polygon.
The number of possible BFACF moves in the neutral atmosphere of a
polygon $\omega$ is denoted $a_0 (\omega)$.
A sequence $\phi = \langle \phi_0, \phi_1 , \phi_2 , \ldots, \phi_n \rangle$
of polygons in the half square lattice can be sampled by executing a move
uniformly selected from the positive and neutral atmospheres of
$\phi_j$ to find $\phi_{j+1}$. The sequence is started in the polygon $\phi_0$ of
length $4$ with one edge in the $x$-axis. Notice that any polygon
$\phi_j$ has at least one atmospheric move which can be executed on it,
and that every polygon $\omega$ can be obtained in the upper half
square lattice from $\phi_0$ by executing positive and neutral BFACF
moves. (To see this, reverse the steps by starting at $\omega = \phi_n$,
and show that it can be made shorter by executing a negative atmospheric
move, sometimes after a neutral move was done a number of times).
Let $\phi_n$ be a state with $v$ visits and top plane of height $h$. The
probability of generating a sequence $\phi$, starting at $\phi_0$ and
ending in $\phi_n$, is given by
\begin{equation}
P(\phi) = \prod_{j=0}^{n-1} \frac{1}{a_0(\phi_j) + a_+(\phi_j)} .
\end{equation}
Assign a weight
\begin{equation}
W(\phi) = \prod_{j=0}^{n-1} \frac{a_0(\phi_j) + a_+(\phi_j)}{a_0(\phi_{j+1}) + a_-(\phi_{j+1})}
\end{equation}
to the squence $\phi$.
The \textit{average weight} of sequences ending in the state $\phi_n$ is given by
\begin{equation}
\langle W(\phi_n) \rangle = \sum_{\phi \to \phi_n} P(\phi) \, W(\phi)
= \sum_{\phi \to \phi_n} \prod_{j=1}^{n} \frac{1}{a_0(\phi_j) + a_-(\phi_j)} .
\end{equation}
This, however, is the probability of the \textit{reverse sequence} starting in the state
$\phi_n$ and ending in the state $\phi_0$ if only negative and neutral moves are
done. This probability is equal to $1$,
since these sequences end up in state $\phi_0$ with probability $1$. In other words,
\begin{equation}
\langle W(\phi_n) \rangle
= \sum_{\phi \to \phi_n} \prod_{j=1}^{n} \frac{1}{a_0(\phi_j) + a_-(\phi_j)} = 1 .
\end{equation}
This is the GARM counting theorem \cite{RJvR08}.
If $S_n(v,w)$ is the set of all polygons of length $n$ in the half-lattice with $v$ visits
and height $w$, then the average weight of sequences ending up in states
in $S_n(v,w)$ is given by
\begin{eqnarray}
W_{v,w} &=& \sum_{\phi_n\in S(v,w)} \langle W(\phi_n) \rangle \cr
&=& \sum_{\phi_n\in S(v,w)}\sum_{\phi \to \phi_n} \prod_{j=1}^{n} \frac{1}{a_0(\phi_j) + a_-(\phi_j)} =
\sum_{\phi_n\in S(v,h)}1 = p_n(v,w).
\end{eqnarray}
In other words, by computing the average weight $W_{v,w}$ of polygons of length $n$
with $v$ visits and height $w$, estimates of the microcanonical partition function
$p_n(v,w)$ are obtained. This is an example of approximate enumeration
\cite{Rensburg2010} and these data can be used to determine average number of visits or height
for polygons of fixed length.
The algorithm is implemented with pruning and enrichment in exactly the same
way the PERM or flatPERM algorithms are implemented. For details, see
references \cite{Grassberger1997} and \cite{Prellberg2004}. The resulting implementation
of \textit{flatGARM} is a flat histogram sampling method which continually
prunes states of low weight that do not contribute much to the partition
function, and otherwise enriches states of high weight in the sampling. The algorithm was run to complete
about $11,000$ GARM sequences and we collected data for polygons with up to
$200$ edges and computed the free energy, the mean number of visits and
the mean height of the polygons, as well as the variances of these quantities.
\begin{figure}[th!]
\input figureEa-N.tex
\caption{Energies of pulled adsorbing polygons.
Left panels: The average density of visits $V_n$ as a function of $\log y$ for $a=3$
(top left panel), $a=4$ (middle left panel) and $a=5$ (bottom left panel).
Right panels: The average height $H_n$ as a function of $\log y$ for $a=3$
(top right panel), $a=4$ (middle right panel) and $a=5$ (bottom right panel).
In all these graphs the value of $n$ increased from $40$ to $200$
in steps of $10$, with curves progressively darker as $n$ increases.}
\label{figureEa}
\end{figure}
\begin{figure}[t]
\input figureCa-N.tex
\caption{Variances of pulled adsorbing polygons.
Left panels: The variance $\C{V}_n$ of the density of visits as a function of $\log y$ for $a=3$
(top left panel), $a=4$ (middle left panel) and $a=5$ (bottom left panel).
Right panels: The variance $\C{H}_n$ of the scaled height as a function of $\log y$ for $a=3$
(top right panel), $a=4$ (middle right panel) and $a=5$ (bottom right panel).
In all these graphs the value of $n$ increased from $40$ to $200$
in steps of $10$, with curves progressively darker as $n$ increases.}
\label{figureCa}
\end{figure}
In Figure \ref{figureEa} we show the average value of the density of visits
$V_n$ and the average value of the scaled height $H_n$ plotted against
$\log y$ for $a=3$, $a=4$ and $a=5$. There is clear
evidence for two phase transitions in these figures. We know that for $\log y < 0$
the density of visits and scaled height are independent of $y$. In all four
panels the curves seem to be approaching horizontal lines as $n$ increases in this
regime. There is a transition to a phase with a reduced number of visits and a larger
value of the height around $y=1$, and a second, more marked, transition at a location
whose value depends on $a$ from this phase to a ballistic phase where the density of
visits approaches zero. In the intermediate regime between the two transitions the
density of visits and the average scaled height depend on both $a$ and $y$. This regime
is referred to as a \textit{mixed phase}.
For instance, when $a = 5$, the curves for $V_n$ are approaching a horizontal line
when $\log y \leq 0$, as we know must happen from our rigorous arguments. For
$\log y > 0$ there is a regime in which $V_n$ is a decreasing function of $\log y$ and
$H_n$ is an increasing function of $\log y$, so the free energy depends on $y$.
It also depends on $a$, as can be seen by comparing with the results at
$a=3$ and $a=4$.
When $\log y$ is somewhat larger than $1$ there is a rapid decrease in $V_n$ and a rapid increase
in $H_n$ and then at larger values of $y$ both quantities become less dependent on $y$.
All of this suggests two transitions, one from an adsorbed phase where the free
energy only depends on $a$ to a mixed phase where the free energy depends on both
$a$ and $y$, and a second transition at larger $y$ to a ballistic phase. In this third
(ballistic) phase the free energy depends on $y$ but is essentially independent of $a$.
We know rigorously that in the infinite $n$ limit the free energy is independent of $a$
in the ballistic phase.
In Figure \ref{figureCa} we show the corresponding fluctuation quantities $\C{V}_n$ and
$\C{H}_n$ of the data in
Figure \ref{figureEa}. In all four panels there are two peaks consistent with two phase
transitions. If we look first at the transitions at larger values of $y$ then the peaks
are growing and moving to the left with increasing $n$. This is clear evidence of
a phase transition although it is difficult to determine a precise location for the
transition. The peaks at smaller values of $y$ are more difficult to interpret. At
$a=3$ the peaks largely occur for $y<1$ but the positions may be moving towards
$y=1$ with increasing $n$. However, we know rigorously that there is no transition
for $y<1$. When $a=5$ the picture is clearer. The peaks occur for values of
$y>1$ though their positions do not move smoothly with increasing $n$.
We interpret these results as showing the existence of a mixed phase in
this model. There is a transition from the mixed phase to the ballistic phase occurring
at a critical value of $y$ that is a function of $a$. There is also a second transition
from the adsorbed phase to the mixed phase which must occur at a value of
$y\geq 1$, but we cannot be sure exactly where it occurs, or whether its location is
a function of $a$.
\section{Series Analysis}
\label{sec:numerical}
\subsection{Exact enumerations \label{sec:enum}}
Our algorithm for the enumeration of self-avoiding polygons (SAP) on the square lattice
is based on the work of Enting \cite{Enting80} who pioneered the use of the finite lattice method.
The first terms in the generating function for SAP are calculated using transfer matrix (TM)
techniques to count the number of polygons in rectangles $W$ unit cells wide and $L$ cells long.
Any polygon spanning such a rectangle has a size of at least $2(W+L)$ edges. Adding contributions
from all rectangles of width $W \leq W_{\rm max}$ and length $W \leq L \leq 2W_{\rm max}-W+1$
the number of polygons per vertex of an infinite lattice is obtained correctly up to length
$N=4W_{\rm max}+2$. Normally one can use the symmetry of the square lattice to restrict the TM
calculations to rectangles with $W\leq W_{\rm max}/2$ and $L\geq W$ by counting contributions for rectangles
with $L>W$ twice. The interactions with the surface breaks the symmetry and therefore we have to consider all
rectangles with $W\leq W_{\rm max}$. The size of the transfer matrix grows exponentially with $W$ and to
partially overcome this hurdle we break the TM calculation on the set of rectangles
into two sub-sets with $L\geq W$ and $L<W$, respectively. In the calculations for the
sub-set with $L\geq W$ the surface is placed on the bottom of the rectangle and
for the sub-set with $L< W$ the surface is placed on the left side of the rectangle.
The height (or span) $h$ of the SAP is simply $W$ for the first sub-set and $L$
for the second sub-set.
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.45\textwidth]{sapex-eps-converted-to.pdf}
\hspace{1cm}
\includegraphics[width=0.45\textwidth]{saptm-eps-converted-to.pdf}
\end{center}
\caption{
The first panel shows an example of a self-avoiding polygon on a $8\times 10$ rectangle with a surface on
the bottom having 8 vertices in the surface and span 8. Alternatively we can view it as a SAP on a $10\times 8$ rectangle
with the surface on the left having 7 vertices in the surface and span 10.
The cut-line (dashed-line) splits the SAP into a set of arcs to the left (right) of the cut-line.
The second panel illustrates how the cut-line is moved in order to build the rectangle vertex by vertex.}
\label{fig:sapex}
\end{figure}
The basic idea of the algorithm is illustrated by the example in Figure~\ref{fig:sapex}.
Clearly any SAP is topologically equivalent to a circle and when cut by
a vertical line (the dashed line in Figure~\ref{fig:sapex}) it is broken into
several arcs on {\em either} side of the cut-line connecting two occupied edges.
As the cut-line is moved from left to right we keep track of the ever changing connections (arcs)
between occupied edges on the cut-line. Each end of an arc is assigned one of two labels
depending on whether it is the lower or upper edge. Any configuration along the
cut-line can thus be represented by a set of edge states $\{\sigma_i\}$, where
\begin{equation}\label{eq:states}
\sigma_i = \left\{ \begin{array}{rl}
0 &\;\;\; \mbox{empty edge}, \\
1 &\;\;\; \mbox{lower edge}, \\
2 &\;\;\; \mbox{upper edge}. \\
\end{array} \right.
\end{equation}
\noindent
Since crossings are not permitted this encoding uniquely describes
how the occupied edges are connected. Reading from bottom to top the configuration or signature $S$
along the cut-line of the SAP in Figure~\ref{fig:sapex} is $S=\{111020022 \}$ encoding the arcs to the left of the cut-line.
The most efficient implementation of the TM algorithm involves moving
the cut-line in such a way as to build up the lattice vertex by vertex (see the second panel of Figure~\ref{fig:sapex}).
The sum over all contributing SAP is calculated as the cut-line is moved through the lattice.
For each signature we maintain a generating function $G_S$ for partially completed polygons. Here $G_S$ is a truncated
polynomial $G_S(x,a)$ where $x$ is conjugate to the number of edges in the partially completed polygon and $a$
to the number of visited vertices in the surface. In a TM update each source signature $S$ (before the boundary is moved) gives rise
to only a few new target signatures $S'$ (after the move of the boundary line).
In a specific update $k=0, 1$ or 2 new edges are occupied and $m=0$ or 1 surface vertices are added
(on the bottom or left of the rectangle depending on the sub-set we are dealing with) leading to the
update $G_{S'}(x,a)=G_{S'}(x,a)+x^ka^mG_S(x,a)$. In the case illustrated in Figure~\ref{fig:sapex} the two `new' edges
intersected by the dotted lines are either empty ($k=0$) or occupied ($k=2$) if a new arc is inserted.
We calculated the number of SAP up to length $N=100$. The calculation was
performed in parallel using up to 32 processors, a maximum of some 70GB of memory
and using a total of just under 1000 CPU hours. Details of the implementation
and parallelization of our algorithm can be found in \cite{Jensen99,Jensen03,GJ09}.
\subsection{Results}
\label{sec:results}
For SAPs in the bulk, on a bi-partite lattice such as the square, simple-cubic, or indeed hyper-cubic lattice, it is universally believed (though not proved) that
\begin{equation}
p_{2n} \sim const \cdot \mu^{2n} \cdot n^{\alpha_b-3},
\end{equation}
where, for the square lattice, $\alpha_b = 1/2,$ while for the simple-cubic lattice the best estimate \cite{C10} is $\alpha_b \approx 0.237209.$
However if the polygon sits at a surface and a compressive force (i.e. $y < 1$) is applied to the top of the polygon, then it has recently been shown by Beaton {\em et al.} \cite{BGJL15} from probability arguments and particularly assuming SLE predictions that the
expected asymptotics now includes a stretched-exponential term. More precisely, for the square lattice,
\begin{equation}
p_{2n} \sim const \cdot \mu^{2n}\cdot \mu_1^{n^{3/7}} \cdot n^{-11/7},
\end{equation}
where both the constant and $\mu_1$ are $y$-dependent, and the $y$ dependence of $\mu_1$ is also predicted.
In this Section we describe the results from series analysis, chiefly using the method of
differential approximants (DAs) \cite{GJ09}. Unfortunately, as discussed in \cite{G15}, this method has some problems when applied to generating functions whose coefficients have stretched-exponential terms. In particular the estimate of the dominant growth constant $\mu$ produced is much less precise than is usually the case, while the estimates of the critical exponent vary wildly from approximant to approximant. In practical terms, we expect accuracy of 2-4 significant digits in the critical point estimate, while the critical exponent estimate is unobtainable by this method.
We first discuss the $y$-dependence of the free-energy
$\lambda_0 (y)$ when there is no surface interaction (i.e. $a=1$), then the $a$-dependence of the
free-energy $\kappa_0 (a)$ when there is no applied force (i.e., $y=1$) and finally the two variable
free-energy $\psi_0 (a,y)$ when there is both a surface interaction and an applied force.
\subsubsection{No surface interaction. $a=1.$}\label{sec:lambda}
\begin{figure}
\centering
\includegraphics[scale =0.5] {a1.jpg}
\caption{The $y$-dependence of the free-energy $\lambda_0 (y)$.
}
\label{fig:lambda}
\end{figure}
If we write
\begin{equation}
H(x,y) = \sum_n P_{2n}(1,y) x^n=\sum_n e^{2\lambda_0(y) n + o(n)} x^n
\end{equation}
where $x$ is the generating variable conjugate to the half-length of the polygon,
then $H(x,y)$ will be singular at $x=x_c(y) = \exp[-2\lambda_0 (y)]$
and, close to this singularity, $H(x,y)$ is expected to behave as
\begin{equation}
H(x,y) \sim {A}\, {[x_c(y)-x]^{\alpha (y)}}
\end{equation}
where $\alpha(y)$ is a critical exponent whose value depends on $y$.
In the last three columns of Table~\ref{tab:y1} below we give the results of an analysis of the series $H(x,y)$ for various values of $y$.
The resulting estimates of the free-energy $\lambda_0 (y) = -{\frac{1}{2}\log x_c}$ are plotted in Figure~\ref{fig:lambda}.
The series were analysed using second and third order differential approximants \cite{GJ09}. At $y=1$ the series is well behaved and has critical point $1/\mu^2$ with exponent $\alpha=3/2,$ the exponent for self-avoiding polygons, which is unchanged if we consider SAPs attached to a surface (unlike the SAW case).
For $y$ just below 1 the series are quite difficult to analyse, due to the presence of the stretched exponential term. Estimates of $x_c$ are moderately close to the known value $ 1/\mu^2$ in magnitude, but just below $y=1$ they have a small imaginary part. As we lower $y$ further below 1, we get approximants moderately close to $ 1/\mu^2$ with very large exponent values, and poor convergence. This is exactly the behaviour discussed in \cite{G15} when using the method of differential approximants to analyse series with a stretched-exponential term. The data are consistent with $\mu$ fixed at the bulk (no force) value, and indeed we have proved that in the free region the free-energy stays at the bulk value.
For $y \ge 1.5$ the series are beautifully behaved, the singularity is clearly seen to be a square root, and we can provide 10 digit (or more) accuracy in estimates of the critical point.
For $1 < y < 1.5$ we get the sort of behaviour we expect with a discontinuous change in exponent as we transition from an exponent $3/2$ to a square root.
So, in summary, it appears that for $y < 1$ we have $x_c = 1/\mu^2$ and stretched exponential behaviour; for $y=1$ we have $x_c=1/\mu^2$ and exponent $\alpha = 3/2$ and for $y > 1$ we have $x_c$ monotonically decreasing as $y$ increases, and with a square root singularity.
\subsubsection{No applied force. $y=1.$}\label{sec:kappa}
Define the generating function
\begin{equation}
K(x,a) = \sum_n P_{2n}(a,1) x^n=\sum_n e^{2\kappa_0(a) n + o(n)} x^n.
\end{equation}
$K(x,a)$ will be singular at $x=x_c(a) = \exp[-2\kappa_0 (a)]$
and, close to this singularity, $K(x,a)$ should behave as
\begin{equation}
K(x,a) \sim {B}\,{[x_c(a)-x]^{\alpha (a)}}
\end{equation}
where $\alpha (a)$ is a critical exponent whose value depends on $a$.
\begin{figure}
\centering
\includegraphics[scale =0.5] {y1.jpg}
\caption{The $a$-dependence of the free-energy $\kappa_0 (a)$.
}
\label{fig:kappa}
\end{figure}
We have analysed the series $K(x,a),$ corresponding to the ``no force'' situation, so that $y=1.$ Here the differential approximants work well, as there are no stretched-exponential terms and the critical point and exponent are well estimated. The results are shown in the first three columns of Table~\ref{tab:y1}. If we denote the transition
from the free phase to the adsorbed phase by $a_c^o=\exp(-\epsilon/k_B T_c^o)$, and denote the corresponding quantity for adsorbed self-avoiding walks
by $a_c=\exp(-\epsilon/k_B T_c),$ we have proved that $a_c \le a_c^o.$ The numerical evidence is extremely strong that equality holds.
The best estimate \cite{Guttmann2014} for the SAW case is $a_c = 1.775615 \pm 0.000005.$ From Table \ref{tab:y1}, we see that at this value of $a$ the exponent is estimated to be $0.754$ which is reasonably close to the conjectured exact value \cite{Duplantier1990} $\alpha^{sp}=3/4,$ where the superscript refers to the ``special'' transition that takes place right at the adsorption temperature.
Note that for $a < a_c$ the exponent is 3/2. At $a_c$ it has changed (presumably discontinuously) to $3/4.$ We looked at nearby values, and found that at $a=1.774$ the exponent appeared to be $0.763$ reflecting a cross-over from 1.5 to 0.75, while at $a=1.776$ the free-energy has started to change, as the estimate of $x_c$ was 0.1436799, while the exponent was around $0.751.$ Thus from the exponent value at $a=1.774$ and the free-energy value at $1.776$ we conclude that $a_c$ lies between these two values. It therefore seems very likely that $a_c^o=a_c,$ and if not, they differ by less than 1 part in a thousand, which seems very unlikely.
So in summary it seems that for $a=a_c$ the singularity is characterised by an exponent $3/4,$ and that this changes discontinuously to a square root for $a > a_c.$ For $a < a_c$ the exponent is, as we would expect, given by $\alpha=3/2.$
In Figure~\ref{fig:kappa} we give our estimates of the free-energy $\kappa_0 (a) = -{\frac{1}{2}}\log x_c$ as a function of $\log a$.
\begin{table}
\centering
\begin{tabular}{|l|l|l||l|l|l|}
\hline
$a \,\,(y=1)$ & $x_c$ & Exponent& $y \,\,(a=1)$& $x_c$ & Exponent\\
\hline
0.5 & 0.143680629 & 1.5000 & 0.4 & 0.147 & 20 \\
1 & 0.143680629 &1.5000 & 0.7 & 0.1456 &11 \\
1.775385 & 0.143680629 & 0.754 &0.9 & 0.1432 $\pm$ 0.0004$i$ & complex \\
2.1 & 0.1406445 & 0.5000 & 1 & 0.143680629 &1.5000 \\
2.5 & 0.1332540 & 0.5000 & 1.2 & 0.1377 & 0.6 \\
2.75 & 0.1282078 & 0.5000 & 1.5 & 0.12702 & 0.495\\
3.3 & 0.1175624 & 0.5000 & 2 & 0.1118410 & 0.4998 \\
4 & 0.1058177 & 0.5000 & 2.5 & 0.1000544& 0.5000\\
5 & 0.09243473 & 0.5000 & 3 & 0.09075811& 0.5000\\
7 & 0.07390853 & 0.5000 & 4 & 0.07703333& 0.5000\\
9 & 0.06179279 & 0.5000 & 5 & 0.06733037& 0.5000\\
11 & 0.05324182 & 0.5000 & 7 & 0.05436885& 0.5000\\
13 & 0.04686712 & 0.5000 & 9 & 0.04598874& 0.5000\\
16 & 0.03983702 & 0.5000 & 12 & 0.03768982& 0.5000\\
20 & 0.03330407 & 0.5000 & 15 & 0.03213498& 0.5000\\
25 & 0.02772693 & 0.5000 & 19 & 0.02701707& 0.5000\\
32 & 0.02253994 & 0.5000 & 25 & 0.02196582& 0.5000\\
40 & 0.01862508 & 0.5000 & 32 & 0.01814603& 0.5000\\
50 & 0.01534307 & 0.5000& 40 & 0.01521102& 0.5000\\
65 & 0.01217273 & 0.5000& 50 & 0.01270767& 0.5000\\
\hline
\end{tabular}
\caption{SAPs at a surface. Estimates of $x_c$ and exponents for $y=1$ and various $a$ values and estimates of $x_c$ and exponents for $a=1$ and various $y$ values.}
\label{tab:y1}
\end{table}
\subsubsection{The region $y > 1,$ $a < a_c.$}
In this region (and indeed for larger values of $a,$ the precise limit depending on the value of $y$), we are in the ballistic regime. For fixed $y,$ we expect the free-energy to be independent of $a,$ until we cross a phase boundary. That this is the case is shown in the first three columns of Table ~\ref{tab:y2}, where we show the results for $y=5.$ As the value of $a$ increases, the free-energy remains constant until, for $a$ sufficiently large it starts to change with $a.$ This constancy is the expected behaviour in the ballistic regime,
and it is clear that already at $a=4$ we have transitioned to another regime. We give a second example in Table~\ref{tab:y3} where we show data for $y=2.$ Here we see a transition occuring around $a=2.5.$ We examine the nature of this regime below.
\subsubsection{The region $y < 1$ and $a > a_c.$}
We have seen that for $y < 1$ and $a < a_c$ we are in the so-called ``free'' region, the free-energy is constant, but one has stretched-exponential behaviour. For SAWs, when $y < 1$ and $a> a_c$ one is in the adsorbed regime, and that
is the case also for polygons. However in this regime we still observe stretched exponential behaviour, and a free-energy that depends only on the value of $a,$ and agrees with the value given in Table~\ref{tab:y1} for $y=1,$ though our estimates of the free-energy in this regime are less precise than elsewhere because of the stretched exponential behaviour. For this reason, we are also unable to estimate the associated critical exponent in this region. At $y=1$ for $a=a_c$ the generating function has a square-root singularity. So this is an adsorbed regime, but with a phase boundary at $y=1,$ the nature of which we now examine.
\subsubsection{The region $y > 1$ and $a > a_c.$}
If one chooses a value of $a > a_c,$ then as $y$ increases above 1, we find the free-energy changes monotonically with $y.$ An example of this is shown in Table~\ref{tab:y2}, where in the last three columns we show the results of our analysis with $a=3.45,$ which is about double the value of $a_c.$ As $ y$ increases the estimates of $x_c$ decrease, and though they are not as stable as we might like, the exponent values are initially around $-1,$ suggesting a simple pole. As $y$ gets sufficiently large, the exponents switch to a square-root. What has happened is that we have gone from a mixed regime where the free-energy depends on both $y$ and $a$, to the ballistic regime.
\begin{table}
\centering
\begin{tabular}{|l|l|l||l|l|l|}
\hline
$a \,\,(y=5)$ & $x_c$ & Exponent& $y \,\,(a=3.45)$& $x_c$ & Exponent\\
\hline
1 & 0.067330372 & 0.5000 & 1.0 & 0.11148643447 & 0.500000 \\
2.5 & 0.06733 & 0.4999 & 1.25 & 0.11 &-1.3 \\
2.75 & 0.06733 & 0.498 & 1.5 & 0.11078& -0.83 \\
3.0 & 0.067332 & 0.477 & 1.75 & 0.1073 & -0.8 \\
3.25 & 0.06734 & 0.3 & 3.5 & 0.0824875 & -1.1\\
3.35 & 0.06734 & 0.1 & 4 & 0.076809 & -1.06\\
3.4 & 0.067343 & -0.1 & 4.5 & 0.07177& -0.73\\
3.5 & 0.06733 & -0.452 & 5 & 0.06734 &-0.27 \\
4.0 & 0.066687 & -1.08 & 5.5 & 0.063464 & 0.13 \\
5.0 & 0.0632077 & -1.0000 & 6 & 0.060063 & 0.34\\
8.0 &0.05137318 & -1.00000 & 7 & 0.054369 & 0.48 \\
\hline
\end{tabular}
\caption{SAPs at a surface. Estimates of $x_c$ and exponents for $y=5$ and various $a$ values and estimates of $x_c$ and exponents for $a=3.45$ and various $y$ values.}
\label{tab:y2}
\end{table}
\subsubsection{Phase diagram calculation}
For SAWs, we were able to locate the phase boundary between the ballistic and the adsorbed region by solving $\kappa(a)=\lambda(y)$ \cite{Guttmann2014}. But for SAPs in the mixed region the free energy, $\psi_0$, depends on both $a$ and $y,$ so we cannot locate the phase boundary in this way. We know that the phase boundary is on or between the solutions of $\kappa(a)=\lambda(\sqrt{y})$ and $\kappa_0(a)=\lambda(\sqrt{y}).$ As there is a mixed phase, it follows that the boundary of the ballistic phase cannot coincide with the latter solution, and must be at strictly smaller values of $a$ for each $y > 1.$
In fact it seems from our numerical data that the phase boundary does indeed lie on the solution of $\kappa(a)=\lambda(\sqrt{y}).$ Consider the data in Table \ref{tab:y2}. In the first three columns, the data for $y=5$ are given. As $a$ increases, both the free-energy and exponent initially remain essentially constant, as we expect in the ballistic regime. Then between $a=3.25$ and $a=3.5$ the exponent has changed dramatically, from $0.3$ to $-0.45,$ reflecting, we suggest, the transition from the square-root singularity characteristic of the ballistic regime to the simple-pole behaviour characteristic of the mixed regime. So the phase boundary should lie between these values of $a.$ The mid-point is $a=3.375.$ From the phase boundary for SAWs, given in \cite{Guttmann2014}, we find at $y=\sqrt{5}$ that the point on the phase boundary is at $a=3.379,$ remarkably close to our crude estimate.
\begin{table}
\centering
\begin{tabular}{|l|l|l||}
\hline
$a \,\,(y=2)$ & $x_c$ & Exponent\\
\hline
1.0 & 0.111841 & 0.4998 \\
1.5 & 0.111841 &0.4996 \\
2.0 & 0.111841 & 0.49 \\
2.5 & 0.111807 & -0.56 \\
2.6 & 0.11157 & -0.94\\
3.0& 0.108584 & -1.01 \\
\hline
\end{tabular}
\caption{SAPs at a surface. Estimates of $x_c$ and exponents for $y=2$ and various $a$ values, showing exponent change as one crosses the phase boundary.}
\label{tab:y3}
\end{table}
Now consider the data in the last three columns of Table \ref{tab:y2}. Here $a=3.45,$ and as the value of $y$ increases it is clear that there is a transition from the simple pole behavior for $y \le 4.5,$ toward the square-root behaviour of the ballistic regime when $y > 5.5.$ So we expect the phase boundary to be at around $y=5.$ Again from the phase boundary for SAWs, given in \cite{Guttmann2014}, we find at $a=3.45$ that the point on the phase boundary is at $y=2.2593.$ Squaring this, we expect the corresponding point on the SAP phase boundary to be at $y \approx 5.10,$ again close to our observed transition point.
Our third such calculation involves the data in Table \ref{tab:y3}. Here $y=2,$ and it is clear that the transition from the ballistic to the mixed regime takes place at a value of $a$ around 2.5. Turning to the SAW phase boundary, we find that at $y=\sqrt{2},$ the point on the phase boundary is at $a=2.498.$
So while we cannot identify the phase boundary with the precision that was achieved in the SAW case, all the evidence is consistent with the hypothesis that the phase boundary between the ballistic and mixed phases is given by the solution of $\kappa(a)=\lambda(\sqrt{y}).$ Taking this as our working assumption, we show in Figure \ref{fig:phaseboundary2} the phase boundary
between the mixed and ballistic phases for SAPs (upper point-plot) and the phase boundary between the ballistic and adsorbed phases for SAWs in the lower point-plot, in the $(\log{a},\log{y})$-plane.
\begin{figure}
\centering
\includegraphics[scale =0.7] {pb2.jpg}
\caption{The conjectured phase boundary between the mixed and ballistic phases in the
$(\log{a},\log{y})$-plane for SAPs (upper point-plot) and between the adsorbed and ballistic phases for SAWs (lower point-plot). }
\label{fig:phaseboundary2}
\end{figure}
We can switch to physical variables (force and temperature)
using equation (\ref{eqn:fugacities}). Without much loss of generality
we can set $\epsilon = -1$ and work in units where $k_B=1$. The corresponding phase
boundary in the force-temperature plane is given in Figure~\ref{fig:ft1}. Notice that
the force at zero $T$ is $2$ and the limiting slope at $T=0$ is zero. For the
self-avoiding walk the zero derivative at $T=0$ was predicted in reference
\cite{JvRW2013}. The curve is monotone decreasing as $T$ increases, with no re-entrance.
\section{Conclusions}
\label{sec:discussion}
When a polymer is adsorbed at a surface it can be desorbed by applying a
force normal to the surface to pull the polymer away from the surface. We
already have a number of rigorous results available for the self-avoiding walk
model of a linear
polymer \cite{JvRW2013,RensburgWhittington2016b,RensburgWhittington2017}.
The behaviour might depend on polymer
architecture and we begin to investigate this issue in this paper by considering
a self-avoiding polygon model of a ring polymer.
When the dimension is $d \ge 3$ we show that the critical force-temperature curve
(\emph{i.e.} the temperature dependence of the force required to desorb the
polygon) can be characterized in terms of the free energy of the adsorbed
polygon without a force and the free energy of the polygon subject to a force but not
interacting with the surface. Similar results are known for
the self-avoiding walk model \cite{JvRW2013}, though we also show that the critical
force-temperature curve is different for the polygon case. We are
able to determine the phase boundaries in the phase diagram in terms
of these free energies. When $d=2$ the situation is more complicated
because at most half of the vertices of the polygon can be in contact
with the surface. For $d=2$ we have bounds on the free energy but
our results are less complete.
Our Monte Carlo and exact enumeration results suggest the existence
of a \emph{mixed phase} in two dimensions where the free energy
depends on both $a$ and $y$.
The results on adsorbing and pulled staircase polygons in reference \cite{Beaton2017}
similarly show a mixed phase which is adsorbed-ballistic in the phase
diagram. Mixed phases were also seen in a directed model of copolymer
adsorption in reference \cite{IJvR12}.
The model of adsorbing and pulled staircase polygons is a directed
version of our model of two dimensional adsorbing
and pulled polygons.
\vspace{5mm}
\begin{figure}[htbp]
\centering
\includegraphics[scale=0.5]{f-t.jpg}
\caption{The phase boundary given as a force-temperature diagram.
The horizontal axis is the temperature $T=\frac{1}{\log(a)}$, the vertical axis is
the force, given by $f=\frac{\log(y)}{\log(a)}.$}
\label{fig:ft1}
\end{figure}
\section*{Acknowledgements}
EJJvR and SGW acknowledge support in the form of Discovery Grants from NSERC (Canada).
SGW was partially supported by the Leverhulme Trust Research Programme Grant
No. RP2013-K-009, SPOCK: Scientific Properties of Complex Knots. He would like to
acknowledge the hospitality of University of Bristol where some of this research was carried out.
AJG and IJ acknowledge support in the form of a Discovery Grant DP140101110 from the
ARC (Australia). The computational work of IJ was
undertaken with the assistance of resources and services from the National
Computational Infrastructure (NCI), which is supported by the Australian Government.
\vspace{5mm}
\section*{References}
|
1,314,259,993,632 | arxiv | \section{Introduction}
\vspace{-5pt}
Entity Resolution (ER) is the task of identifying different entity profiles that describe the same real-world object \cite{DBLP:journals/tkde/ElmagarmidIV07,DBLP:series/synthesis/2015Christophides}. It is a core task for Data Integration, applying to any kind of data, from the structured entities of relational databases \cite{DBLP:books/daglib/0030287} to the semi-structured entities of the Linked Open Data Cloud (\url{https://lod-cloud.net}) \cite{DBLP:series/synthesis/2015Dong,DBLP:series/synthesis/2015Christophides} and the unstructured entities that are automatically extracted from free text \cite{DBLP:journals/tkde/ShenWH15}.
ER consists of two parts: (i) the \textit{candidate selection step}, which determines the entities worth comparing,
and (ii) the \textit{candidate matching step}, or simply \textit{Matching}, which compares the selected entities to determine whether they represent the same real-world object. The latter step
involves \textit{pairwise comparisons}, i.e., time-consuming operations that typically apply string similarity measures to pairs of entities, dominating the overall cost of ER \cite{DBLP:books/daglib/0030287,DBLP:series/synthesis/2015Christophides,DBLP:series/synthesis/2015Dong}.
In this survey, we focus on the
candidate selection step, which is the crucial part of ER with respect to time efficiency and scalability. Without it, ER suffers from a quadratic time complexity, $O(n^2)$, as every entity profile has to be compared with all others. Reducing this computational cost is the goal of numerous techniques from
two dominant frameworks: Blocking and Filtering. The former
attempts to identify
entity pairs that are likely to match,
restricting comparisons only between them, while
the latter
attempts to quickly discard pairs that are guaranteed to not match,
executing comparisons only between the rest.
The former operates without knowledge of the
Matching step, while the latter is based on
it,
assuming that two entities match if their similarity exceeds a specified threshold.
Hence, Blocking and Filtering
share the same goal, but are complementary, as they operate under different settings and assumptions.
So far, though,
they have been developed independently of one another: their combination and, more generally, their relation
have been overlooked in the literature, with the exception of very few works (e.g., \cite{DBLP:journals/pvldb/KopckeTR10}).
Moreover, the rise of Big Data poses new challenges for both Blocking and Filtering approaches \cite{DBLP:series/synthesis/2015Dong,DBLP:series/synthesis/2015Christophides}: \textit{Volume} requires techniques to scale to millions of entities, while \textit{Variety} calls for techniques that can cope with an unprecedented schema heterogeneity. Both Blocking and Filtering address Volume primarily through paralellization. Existing techniques were adapted to split their workload into smaller chunks that are distributed across different processing units so that they are executed in parallel. This can be done on a cluster (distributed methods), or through the modern multi-core and multi-socket hardware architectures. Variety, though, is addressed differently in each field. For Blocking, the schema-aware methods are replaced by schema-agnostic techniques, which disregard any schema information, creating blocks of very high recall but low precision. Additionally, a whole new category of methods, called \textit{Block Processing}, intervenes between Blocking and Matching
to refine the original blocks,
significantly increasing precision at a negligible (if any) cost in recall. For Filtering, techniques that employ more relaxed matching criteria (e.g., fuzzy set matching or local string similarity join) are proposed, while the case of low similarity thresholds~is~also~considered.
To the best of our knowledge, this is the first survey to comprehensively cover the aforementioned aspects and to jointly review the two frameworks for efficient ER. We formally define Blocking, Block Processing and Filtering, introducing a common terminology that facilitates their understanding. For each field, we propose a new taxonomy with categories that highlight the distinguishing characteristics of the corresponding methods. Based on these taxonomies, we provide a broad overview of every field, elucidating the functionality of the main techniques as well as the relations among them. As a result, established techniques are now seen in a different light - Canopy Clustering~\cite{DBLP:conf/kdd/McCallumNU00}, for instance, may now be viewed as a Block Processing method.
We also elaborate on the parallelization methods for each field.
Most importantly, this survey attempts to
place Blocking and Filtering under a common context,
taking special care to stress hybrid methods that combine features from both Blocking and Filtering, to analyze works that experimentally compare the two frameworks
(e.g., \cite{DBLP:conf/semweb/SongH11})
and to qualitatively outline their commonalities and differences.
We also investigate the ER tools that incorporate established efficiency techniques and propose a series of open challenges that constitute promising directions for future research.
Parts of the material included in this survey have been presented in tutorials at WWW 2014~\cite{DBLP:conf/www/StefanidisEHC14}, ICDE 2016~\cite{7498364}, ICDE 2017~\cite{DBLP:conf/icde/StefanidisCE17}, and WWW 2018~\cite{PapadakisTutorialWww18}.
A past survey \cite{DBLP:journals/tkde/Christen12} also covers efficiency ER techniques, but is restricted to the schema-aware Blocking methods.
Other surveys \cite{DBLP:journals/tkde/ElmagarmidIV07} and textbooks \cite{DBLP:books/daglib/0030287,DBLP:series/synthesis/2015Dong,DBLP:series/synthesis/2015Christophides} provide a holistic overview of ER, merely examining the main Blocking and Block Processing techniques. Closer to our work is a recent survey on Blocking \cite{o2019review}, which however offers a more limited coverage and refers neither to parallelization nor to Filtering works.
Recent surveys on string and set similarity joins also exist, but
they focus exclusively on
centralized \cite{DBLP:journals/pvldb/JiangLFL14,DBLP:journals/fcsc/YuLDF16,DBLP:journals/pvldb/MannAB16} or distributed approaches \cite{DBLP:journals/pvldb/FierABLF18}, with the purpose of experimental comparison, and without covering approximate techniques
that allow for more relaxed matching criteria. Most importantly, none of these surveys considers similarity joins in the broader context of ER.
The rest of the paper is structured as follows: Section \ref{sec:er} provides background knowledge on ER and its efficiency techniques, while Sections \ref{sec:blocking} and \ref{sec:blockProcessing} delve into Blocking and Block Processing, respectively. Section \ref{sec:filtering} is devoted to Filtering, whereas Section \ref{sec:hybrid} elaborates on works that combine Blocking with Filtering.
Section \ref{sec:tools} enumerates the main ER tools that incorporate efficiency methods, Section \ref{sec:discussion} provides a high-level discussion of the relation between Blocking and Filtering, Section \ref{sec:futureDirections} provides the main directions for future work, and Section \ref{sec:conclusions} concludes the paper.
\section{Preliminaries}
\label{sec:er}
At the core of ER lies the notion of \textit{entity profile}, which constitutes a uniquely identified description of a real-world object in the form of name-value pairs. Assuming infinite sets of attribute names $\mathcal{N}$, attribute values $\mathcal{V}$, and unique identifiers $\mathcal{I}$, an entity profile is formally defined~as~follows~\cite{DBLP:series/synthesis/2015Christophides,DBLP:journals/tkde/PapadakisIPNN13}:
\begin{definition}[Entity Profile]
An \emph{entity profile} $\mathbf{e_{id}}$ is a tuple $\langle id, A_{id} \rangle$, where $id \in \mathcal{I}$ is a unique identifier, and $A_{id}$ is a set of name-value pairs $\langle n, v \rangle$, with $n \in \mathcal{N}$ and $v \in (\mathcal{V} \cup \mathcal{I})$. A set of entity profiles $\mathbf{\mathcal{E}}$ is called \emph{entity collection}.
\end{definition}
This definition is simple, but flexible enough to accommodate a wide variety of (semi-)structured
representations.
E.g., nested attributes can be transformed into a flat set of name-value pairs, while links
may be represented by assigning the id of one entity as the attribute value of the other.
\begin{definition}[Entity Resolution]
Two entity profiles $e_i$ and $e_j$ \emph{match}, $\mathbf{e_i\equiv e_j}$, if they refer to the same real-world entity.
Matching entities are also
called \emph{duplicates}. The task of Entity Resolution (ER) is to find all matching entities within an entity collection or across two or more entity collections.
\end{definition}
In particular, we distinguish between the following two cases \cite{DBLP:journals/tkde/Christen12,DBLP:books/daglib/0030287}:
\begin{enumerate}
\item \textit{Deduplication} receives as input an entity collection $\mathcal{E}$ and produces as output the set of all pairs of matching entity profiles within $\mathcal{E}$, i.e., $\mathcal{D}(\mathcal{E}) = \{ (e_i, e_j) : e_i \in \mathcal{E}, \, e_j \in \mathcal{E}, \, e_i\equiv e_j \}$.
\item \textit{Record Linkage} receives
two duplicate-free entity collections, $\mathcal{E}_1$ and $\mathcal{E}_2$,
and
returns the pairs of matching entity profiles between them, i.e., $\mathcal{D}(\mathcal{E}_1$, $\mathcal{E}_2)$=$\{ (e_i, e_j) : e_i \in \mathcal{E}_1, \, e_j \in \mathcal{E}_2, \, e_i\equiv e_j \}$.
\end{enumerate}
\textit{Multi-source Entity Resolution} involves three or more entity collections and can be performed by applying Deduplication to the union of all collections, or by executing a sequence of pairwise Record Linkage tasks, provided that every input collection is duplicate-free.
ER performance
is characterized by
its \textit{effectiveness} and its \textit{efficiency}. The former refers to how many of the actual duplicates are detected, while the latter expresses the computational cost for detecting them -- usually
in terms of the number of performed comparisons,
which is referred to as \textit{cardinality} and
denoted by $||\mathcal{E}||$. The naive, brute-force approach performs all pairwise comparisons between the input entity profiles, having a quadratic complexity that does not scale to large datasets;
for Record Linkage, $||\mathcal{E}|| = |\mathcal{E}_1| \times |\mathcal{E}_2|$, while for Deduplication $||\mathcal{E}|| = |\mathcal{E}| \cdot (|\mathcal{E}| -1)/2$.
\vspace{2pt}
\textbf{Blocking.} To tackle ER's inherently quadratic complexity, Blocking
trades slightly lower effectiveness for significantly higher efficiency. Its goal is to reduce the number of performed comparisons, while missing as few matches as possible. Ideally, one would compare only the pairs of duplicates, whose number grows \textit{linearly} with the number of the input entity profiles~\cite{DBLP:journals/pvldb/GetoorM12,DBLP:conf/icde/StefanidisCE17}. To this end, Blocking clusters potentially matching entities in common blocks
and exclusively compares entity profiles that co-occur in at least one block.
Internally, a blocking method employs a \textit{blocking scheme}, which applies to one or more entity collections to yield a set of blocks $\mathcal{B}$,
called
\textit{block collection}. Cardinality $||\mathcal{B}||$
denotes the number of comparisons in $\mathcal{B}$,
given that only entity pairs within the same block are compared, i.e., $||\mathcal{B}||$=$\sum_{b_i \in \mathcal{B}} ||b_i||$, where $||b_i||$ stands for the number of comparisons contained in an individual block $b_i$. We denote the set of \textit{detectable duplicates} in $\mathcal{B}$ as $\mathcal{D}(\mathcal{B})$, while $\mathcal{D}(\mathcal{E})$ stands for all existing duplicates. Since $\mathcal{B}$ reduces the number of performed comparisons,~$\mathcal{D}(\mathcal{B})$$\subseteq$$\mathcal{D}(\mathcal{E})$.
A common assumption in the literature is the \textit{oracle}, i.e., a perfect matching function that, for each pair of entity profiles,
decides correctly whether they match or not \cite{DBLP:conf/icde/StefanidisCE17,DBLP:journals/tkde/Christen12,DBLP:series/synthesis/2015Dong,DBLP:journals/tkde/PapadakisIPNN13,DBLP:journals/tkde/PapadakisKPN14}.
Using an oracle,
a pair of duplicates is detected as long as they share at least one block. This allows for reasoning about the performance of blocking methods independently of matching methods: there is a clear trade-off between the effectiveness and the efficiency of a blocking scheme \cite{DBLP:conf/icde/StefanidisCE17,DBLP:journals/tkde/Christen12,DBLP:series/synthesis/2015Dong}: the more comparisons are contained in the resulting block
collection $\mathcal{B}$ (i.e., higher $||\mathcal{B}||$), the more duplicates
will be detected (i.e., higher $|\mathcal{D}(\mathcal{B})|$), raising
effectiveness
at the cost of lower efficiency.
Thus, a blocking scheme
should achieve a good balance between these two competing objectives as expressed through
the following
measures~\cite{DBLP:conf/icdm/BilenkoKM06,DBLP:conf/cikm/VriesKCC09,DBLP:conf/aaai/MichelsonK06,DBLP:conf/wsdm/PapadakisINF11}:
\begin{enumerate}
\item \textit{Pair Completeness ($PC$)} corresponds to \textit{recall}, estimating the portion of the detectable duplicates in $\mathcal{B}$ with respect to those in $\mathcal{E}$:
$PC(\mathcal{B}) = |\mathcal{D}(\mathcal{B})| / |\mathcal{D}(\mathcal{E})| \in [0,1]$.
\item \textit{Pairs Quality ($PQ$)} corresponds to \textit{precision}, estimating the portion of
comparisons in $\mathcal{B}$ that correspond to real duplicates:
$PQ(\mathcal{B})= |\mathcal{D}(\mathcal{B})|/||\mathcal{B}|| \in [0,1]$.
\item \textit{Reduction Ratio ($RR$)} measures the reduction in the number of pairwise comparisons
in $\mathcal{B}$ with respect to the brute-force approach:
$RR(\mathcal{B},\mathcal{E}) = 1 - ||\mathcal{B}||/||\mathcal{E}|| \in [0, 1]$..
\end{enumerate}
Higher values for $PC$ indicate higher \textit{effectiveness} of the blocking scheme, while higher values for $PQ$ and $RR$ indicate
higher \textit{efficiency}.
Note that $PC$ provides an optimistic estimation of recall, presuming
the existence of an oracle, while $PQ$ provides a pessimistic estimation
of precision, treating as false positives the repeated comparisons between duplicates (i.e., only the non-repeated duplicate pairs are considered as true positives).
In this context,
we can define Blocking as follows:
\begin{definition}[Blocking]
Given an entity collection $\mathcal{E}$, Blocking clusters similar entities into a block collection $\mathcal{B}$ such that $PC(\mathcal{B})$, $PQ(\mathcal{B})$ and $RR(\mathcal{B}, \mathcal{E})$ are simultaneously maximized.
\label{def:blocking}
\end{definition}
This definition refers to Deduplication, but can be easily extended to Record Linkage. Simultaneously maximizing $PC$, $PQ$ and $RR$ necessitates that the enhancements in efficiency do not affect the effectiveness of Blocking,
carefully removing comparisons between non-matching entities.
Conceptually, Blocking can be viewed as an optimization task, but
this implies that the real duplicate collection $\mathcal{D}(\mathcal{E})$ is known, which is actually what ER tries to compute. Hence,
Blocking is typically treated as an engineering task that
provides an approximate solution for the data at hand.
\begin{figure}[t]\centering
\includegraphics[width=0.75\linewidth]{preliminariesFigure.png}
\vspace{-9pt}
\caption{{\small (a) The internal functionality of Blocking modeled as a deterministic finite automaton with three states: Block Building (\textsf{BlBu}), Block Cleaning (\textsf{BlCl}) and Comparison Cleaning (\textsf{CoCl}). (b) The end-to-end workflow for non-learning Entity Resolution \cite{DBLP:journals/pvldb/KopckeTR10}. (c) The relative computational cost for the brute-force approach, Blocking, Filtering and the ideal solution (Duplicate Pairs) over Deduplication.}
}
\label{fig:computationalCostPlusWorkflow}
\vspace{-10pt}
\end{figure}
A blocking-based ER workflow may comprise several stages.
First, \textit{Block Building} (BlBu)
applies a blocking scheme to produce a block collection $\mathcal{B}$ from the input entity collection(s).
This step may be repeated several times on the same input,
applying multiple blocking schemes, in order to achieve a more robust performance in the context of highly noisy data. Often, there is a second, optional stage, called \textit{Block Processing}, which
refines $\mathcal{B}$ through additional optimizations that further reduce the number of performed comparisons. This may involve discarding \textit{entire blocks} that primarily contain unnecessary comparisons,
called \textit{Block Cleaning} (BlCl), and/or discarding \textit{individual comparisons} within certain blocks,
called \textit{Comparison Cleaning} (CoCl).
The former may be applied repeatedly, each time enforcing a different, complementary method to discard blocks, but the latter
can be performed only once;
CoCl comprises competitive methods that
serve exactly the same purpose and, once applied to a block collection, they alter it in such a way that turns all other methods inapplicable.
Figure \ref{fig:computationalCostPlusWorkflow}(a) models this workflow as a deterministic finite automaton with three states, where each state corresponds to one of the blocking sub-tasks.
\vspace{2pt}
\textbf{Filtering.}
Given two entity collections $\mathcal{E}_1$ and $\mathcal{E}_2$, a similarity function $f_S : \mathcal{E}_1 \times \mathcal{E}_2 \rightarrow {\rm I\!R}$, and a similarity threshold $\theta$, a \textit{similarity join} identifies all pairs of entity profiles in $\mathcal{E}_1$ and $\mathcal{E}_2$ that have similarity at least $\theta$, i.e., $\mathcal{E}_1 \Join_{\theta} \mathcal{E}_2 = \{ (e_i, e_j) \in \mathcal{E}_1 \times \mathcal{E}_2 : f_S(e_i, e_j) \geq \theta \}$.
\begin{figure}[t]\centering
\includegraphics[width=0.69\linewidth]{measures.png}
\vspace{-10pt}
\caption{{\small Definition of the main similarity measures used by string similarity join algorithms, and how the input threshold $\theta$ for each measure can be transformed into an equivalent Overlap threshold $\tau$.}
}
\label{fig:measures}
\vspace{-14pt}
\end{figure}
Similarity joins can be used for defining ER under the intuitive assumption that matching entity profiles are highly similar. In fact, the above formulation corresponds to Record Linkage, while Deduplication can be defined analogously as a self-join operation, where $\mathcal{E}_1 \equiv \mathcal{E}_2$.
To avoid exhaustive pairwise comparisons,
similarity joins typically follow the \textit{filter-verification} framework, which involves two~stages~\cite{DBLP:series/synthesis/2013Augsten,DBLP:journals/pvldb/JiangLFL14}:
\begin{enumerate}
\item \textit{Filtering} computes a set of \textit{candidates} for each entity $e_i$, excluding all those that cannot match with $e_i$. In other words, it prunes all true negatives, but allows some false positives.
\item \textit{Verification} computes the actual similarity between candidates (or a sufficient upper bound) to remove the false positives.
\end{enumerate}
Due to the relatively straightforward implementation of Verification, in the following we exclusively focus on Filtering. The relevant techniques are defined with respect to three parameters: (i) the representation for each entity, (ii) the similarity function between entity pairs under this representation, and (iii) the similarity threshold above which two entities are considered to match.
The representation typically relies on
\textit{signatures} extracted from each entity such that two entities match only if their signatures overlap.
Given that we address ER
over entities described by one or more textual attributes, we focus on string similarity joins, which
can be \textit{character-} or \textit{token-based}. The former compare two strings by representing them as sequences of characters and by considering the character transformations required to transform one string into the other. The latter are also called \textit{set similarity joins}, since they transform the strings into sets, typically via tokenization or $q$-gram extraction, and then compare strings using a set-based similarity measure.
Regarding the similarity function, the most common one for character-based similarity joins is Edit Distance,
which measures the minimum number of edit operations (i.e., insertions, deletions and substitutions) that are required to transform one string to the other \cite{DBLP:series/synthesis/2013Augsten}. For token-based similarity joins, the most commonly used similarity measures include Overlap, Jaccard, Cosine or Dice. The last three are normalized variants of the Overlap \cite{DBLP:series/synthesis/2013Augsten,DBLP:journals/pvldb/MannAB16,DBLP:journals/pvldb/JiangLFL14}.
Finally, the similarity threshold depends on the data at hand. Note, though, that the join algorithms do not operate directly with thresholds on Jaccard, Cosine or Dice similarity, but
first translate the given threshold $\theta$ into an equivalent set overlap threshold $\tau$ that depends
on the size of the sets, as shown in Figure~\ref{fig:measures}.
A similar transformation is also possible for Edit Distance, which means
that
set similarity joins
may be applied to this measure as well \cite{DBLP:series/synthesis/2013Augsten}.
\vspace{2pt}
\textbf{Blocking vs Filtering.} The relation between the two frameworks is illustrated in Figure \ref{fig:computationalCostPlusWorkflow}(b). Blocking, in the sense of the entire process in Figure \ref{fig:computationalCostPlusWorkflow}(a), is applied first, reducing the pairwise comparisons that are considered by Matching. These comparisons are further cut down by Filtering, which is subsequently applied, as the initial part of Matching, given that it requires specifying both a similarity measure and a similarity threshold. Next, Verification is applied to estimate the actual similarity between the compared attribute values. The Entity Resolution process concludes with \textit{Match Decision}, which synthesizes the estimated similarity between multiple attribute values to determine whether the compared entity profiles are indeed duplicates.
Both Blocking and Filtering are optional steps, but at least one of them should be applied in order to tame the otherwise quadratic computational cost of ER. As shown in Figure~\ref{fig:computationalCostPlusWorkflow}(c), Blocking yields a \textit{super-linear}, but \textit{sub-quadratic} time complexity, lying between the two extremes: the brute-force solution and the ideal one (i.e., Duplicate Pairs). The same applies to the computational cost of Filtering, except that it typically constitutes an \textit{exact} procedure that produces no false negatives, i.e., missed duplicates. It exclusively allows false positives, which are later removed by Verification \cite{DBLP:series/synthesis/2013Augsten}. For this reason, Filtering corresponds to a superset of Duplicate Pairs in Figure~\ref{fig:computationalCostPlusWorkflow}(c). In contrast, Blocking constitutes an inherently \textit{approximate} solution that increases ER efficiency
at the cost of allowing both false positives and false negatives
\cite{DBLP:series/synthesis/2015Christophides}.
Thus, it intersects Duplicate Pairs, such that the area of their intersection is inversely proportional to the duplicates that are missed by Blocking, while the relative complement of the Duplicate Pairs in Blocking is analogous to the executed comparisons between non-matching entities.
Note that Figure~\ref{fig:computationalCostPlusWorkflow}(c) corresponds to Deduplication, but can be easily generalized to Record Linkage, as well. Moreover, the relative performance of Blocking and Filtering, i.e., the relative position of their circles, depends on the methods and the data at hand. In most cases, though, the best solution is to use both frameworks, yielding the computational cost that corresponds to their intersection. However, this approach is rarely used in the literature (e.g., \cite{DBLP:journals/pvldb/KopckeTR10}). Most works on Blocking typically omit Filtering (e.g., \cite{DBLP:journals/pvldb/0001APK15,DBLP:journals/pvldb/0001SGP16,DBLP:journals/tkde/Christen12}), whereas most works on Filtering disregard Blocking, applying directly to the input entity collections (e.g., \cite{DBLP:journals/pvldb/MannAB16,DBLP:journals/pvldb/JiangLFL14}). The goal of the present survey is to cover this gap, elucidating the complementarity of the two frameworks.
\section{Block Building}
\label{sec:blocking}
Block Building receives as input one or more entity collections and produces as output a block collection $\mathcal{B}$. The process is guided by a \textit{blocking scheme}, which determines how entity profiles are assigned to blocks. This scheme typically comprises two parts. First, every entity is processed to extract \textit{signatures} (e.g., tokens),
such that the similarity of signatures reflects the similarity of the corresponding
profiles. Second, every entity is
mapped to one or more blocks based on these signatures. Let $\mathcal{P(S)}$ denote the power set of a set $S$ and $\mathcal{K}$ denote the universe of signatures appearing in entity profiles. We formally define a blocking scheme as follows:
\begin{definition}[Blocking Scheme]
Given an entity collection $\mathcal{E}$, a \emph{blocking scheme} is a function $f_B : \mathcal{E} \rightarrow \mathcal{P}(\mathcal{B})$ that maps entity profiles to blocks. It is composed of two functions: (a) a \emph{transformation} function $f_{T} : \mathcal{E} \rightarrow \mathcal{P}(\mathcal{K})$ that maps an entity profile to a set of \emph{signatures} (also called \emph{blocking keys}), and (b) an \emph{assignment} function $f_{A} : \mathcal{K} \rightarrow \mathcal{P}(\mathcal{B})$ that maps each signature to one or more blocks.
\end{definition}
This definition applies to Deduplication, but can be easily extended to Record Linkage.
\begin{table*}[h]
\centering
\caption{Taxonomy of the Block Building methods discussed in Sections \ref{sec:schemaBasedBB} and \ref{sec:schemaAgnosticBB}.}
\label{tb:bbTaxonomy}
\vspace{-5pt}
{\scriptsize
\begin{tabular}{| l || c | c | c | c | }
\hline
\multicolumn{1}{|c||}{\textbf{Method}} & \textbf{Key} & \textbf{Redundancy} & \textbf{Constraint} & \textbf{Matching} \\
& \textbf{type} & \textbf{awareness} & \textbf{awareness} & \textbf{awareness} \\
\hline
\hline
Standard Blocking (\textsf{SB}) \cite{fellegi1969theory} & hash-based & redundancy-free & lazy & static \\
Suffix Arrays Blocking (\textsf{SA}) \cite{DBLP:conf/wiri/AizawaO05} & hash-based & redundancy-positive & proactive & static \\
Extended Suffix Arrays Blocking \cite{DBLP:journals/tkde/Christen12,DBLP:journals/pvldb/0001APK15} & hash-based & redundancy-positive & proactive & static \\
Improved Suffix Arrays Blocking \cite{DBLP:conf/cikm/VriesKCC09} & hash-based & redundancy-positive & proactive & static \\
Q-Grams Blocking \cite{DBLP:journals/tkde/Christen12,DBLP:journals/pvldb/0001APK15} & hash-based & redundancy-positive & lazy & static \\
Extended Q-Grams Blocking \cite{baxter2003comparison,DBLP:journals/tkde/Christen12,DBLP:journals/pvldb/0001APK15} & hash-based & redundancy-positive & lazy & static \\
MFIBlocks \cite{DBLP:journals/is/KenigG13} & hash-based & redundancy-positive & proactive & static \\
\hline
Sorted Neighborhood (\textsf{SN}) \cite{DBLP:conf/sigmod/HernandezS95, DBLP:journals/datamine/HernandezS98,DBLP:conf/edbt/PuhlmannWN06} & sort-based & redundancy-neutral & proactive & static \\
Extended Sorted Neighborhood \cite{DBLP:journals/tkde/Christen12} & sort-based & redundancy-neutral & lazy & static \\
Incrementally Adaptive SN \cite{DBLP:conf/jcdl/YanLKG07} & sort-based & redundancy-neutral & proactive & static \\
Accumulative Adaptive SN \cite{DBLP:conf/jcdl/YanLKG07} & sort-based & redundancy-neutral & proactive & static \\
Duplicate Count Strategy (\textsf{DCS}) \cite{DBLP:conf/icde/DraisbachNSW12} & sort-based & redundancy-neutral & proactive & dynamic \\
\textsf{DCS++} \cite{DBLP:conf/icde/DraisbachNSW12} & sort-based & redundancy-neutral & proactive & dynamic \\
\hline
Sorted Blocks \cite{DBLP:conf/nss/DraisbachN11} & hybrid & redundancy-neutral & lazy & static \\
Sorted Blocks New Partition \cite{DBLP:conf/nss/DraisbachN11} & hybrid & redundancy-neutral & proactive & static \\
Sorted Blocks Sliding Window \cite{DBLP:conf/nss/DraisbachN11} & hybrid & redundancy-neutral & proactive & static \\
\hline
\multicolumn{5}{c}{\textbf{(a) Non-learning, schema-aware methods.}}\\
\hline
ApproxRBSetCover \cite{DBLP:conf/icdm/BilenkoKM06} & hash-based & redundancy-positive & lazy & static \\
ApproxDNF \cite{DBLP:conf/icdm/BilenkoKM06} & hash-based & redundancy-positive & lazy & static \\
Blocking Scheme Learner (\textsf{BSL}) \cite{DBLP:conf/aaai/MichelsonK06} & hash-based & redundancy-positive & lazy & static \\
Conjunction Learner \cite{DBLP:conf/ijcai/CaoCZYLY11} (semi-supervised) & hash-based & redundancy-positive & lazy & static \\
\textsf{BGP} \cite{DBLP:journals/jidm/EvangelistaCSM10} & hash-based & redundancy-positive & lazy & static \\
CBlock \cite{DBLP:conf/cikm/SarmaJMB12} & hash-based & redundancy-positive & proactive & static \\
DNF Learner \cite{giang2015machine} & hash-based & redundancy-positive & lazy & dynamic \\
\hline
FisherDisjunctive \cite{DBLP:conf/icdm/KejriwalM13} (unsupervised) & hash-based & redundancy-positive & lazy & static \\
\hline
\multicolumn{5}{c}{\textbf{(b) Learning-based (supervised), schema-aware methods.}}\\
\hline
Token Blocking (\textsf{TB}) \cite{DBLP:conf/wsdm/PapadakisINF11} & hash-based & redundancy-positive & lazy & static \\
Attribute Clustering Blocking \cite{DBLP:journals/tkde/PapadakisIPNN13} & hash-based & redundancy-positive & lazy & static \\
RDFKeyLearner \cite{DBLP:conf/semweb/SongH11} & hash-based & redundancy-positive & lazy & static \\
Prefix-Infix(-Suffix) Blocking \cite{DBLP:conf/wsdm/PapadakisINPN12} & hash-based & redundancy-positive & lazy & static \\
TYPiMatch \cite{DBLP:conf/wsdm/MaT13} & hash-based & redundancy-positive & lazy & static \\
Semantic Graph Blocking \cite{DBLP:conf/ideas/NinMML07} & - & redundancy-neutral & proactive & static \\
\hline
\multicolumn{5}{c}{\textbf{(c) Non-learning, schema-agnostic methods.}}\\
\hline
Hetero \cite{DBLP:conf/semweb/KejriwalM14a} & hash-based & redundancy-positive & lazy & static \\
Extended DNF BSL \cite{DBLP:journals/corr/KejriwalM15} & hash-based & redundancy-positive & lazy & static \\
\hline
\multicolumn{5}{c}{\textbf{(d) Learning-based (unsupervised), schema-agnostic methods.}}
\end{tabular}
}
\vspace{-12pt}
\end{table*}
The set of comparisons in the resulting block collection $\mathcal{B}$ is called \textit{comparison collection} and is denoted by $\mathcal{C}(\mathcal{B})$. Every comparison $c_{i,j} \in \mathcal{C}(\mathcal{B})$ belongs to one of the following types~\cite{DBLP:journals/tkde/PapadakisKPN14,DBLP:journals/tkde/PapadakisIPNN13}:
\begin{itemize}
\item \textit{Matching comparison}, if $e_i$ and $e_j$ match.
\item \textit{Superfluous comparison}, if $e_i$ and $e_j$ do not match.
\item \textit{Redundant comparison}, if $e_i$ and $e_j$ have already been compared in a previous block.
\end{itemize}
We collectively call the last two types \textit{unnecessary comparisons}, as their execution brings no gain.
Note that the resulting block collection $\mathcal{B}$ can be modelled as an inverted index that points from block ids to entity ids.
For this reason, Block Building is also called \textit{Indexing}
\cite{DBLP:journals/tkde/Christen12,DBLP:books/daglib/0030287}.
\subsection{Taxonomy}
\label{sec:taxonomy}
To facilitate the understanding of the main methods for Block Building, we organize them into a novel taxonomy
that consists of the following dimensions:
\begin{itemize}
\item \textit{Key selection} distinguishes between \textit{non-learning} and \textit{learning-based} methods. The former
rely on rules derived from expert knowledge or mere heuristics, while the latter
require a training set to learn the best blocking keys using Machine Learning techniques.
\item \textit{Schema-awareness} distinguishes between \textit{schema-aware} and \textit{schema-agnostic} methods. The former
extract blocking keys from specific attributes that are considered to be more appropriate for matching (e.g., more distinctive or less noisy), while the latter
disregard schema knowledge, extracting blocking keys from all attributes.
\item \textit{Key type} distinguishes between \textit{hash-} or \textit{equality-based} methods, which map a pair of entities to the same block if they have a common key, and \textit{sort-} or \textit{similarity-based} methods, which map a pair of entities to the same block if they have a similar key. There exist also \textit{hybrid} methods, which combine hash- with sort-based functionality.
\item \textit{Redundancy-awareness} classifies methods into three categories based on the relation between their blocks.
\textit{Redundancy-free} methods assign every entity to a single block, thus creating disjoint blocks. \textit{Redundancy-positive} methods place every entity into multiple blocks, yielding overlapping blocks. The more blocks two entities share, the more similar their profiles are. The number of blocks shared by a pair of entities is thus proportional to their matching likelihood. \textit{Redundancy-neutral} methods create overlapping blocks, where most entity pairs share the same number of blocks, or the degree of redundancy is arbitrary, having no implications.
\item \textit{Constraint-awareness} distinguishes blocking methods into \textit{lazy}, which impose no constraints on the blocks they create, and \textit{proactive}, which enforce
constraints on their blocks~(e.g., maximum block size), or
refine their comparisons by discarding
unnecessary ones.
\item \textit{Matching-awareness} distinguishes between \textit{static} methods, which are independent of the subsequent matching process, producing an immutable block collection, and \textit{dynamic} methods, which intertwine Block Building with Matching, updating or processing their blocks dynamically, as more duplicates are detected.
\end{itemize}
Table \ref{tb:bbTaxonomy} maps all methods discussed in Sections \ref{sec:schemaBasedBB} and \ref{sec:schemaAgnosticBB} to our taxonomy.
\subsection{Schema-aware Block Building}
\label{sec:schemaBasedBB}
Methods of this type assume that the input entity profiles adhere to a known schema and, based on this schema and respective domain knowledge, one can select the attributes that are most suitable for Blocking.
We distinguish between non-learning methods, reviewed in Section~\ref{sec:nonlearningBlBu}, and learning-based methods, reviewed in Section~\ref{sec:learningBlBu}.
\subsubsection{Non-learning Methods}
\label{sec:nonlearningBlBu}
The family tree of the methods in this category is shown in Figure \ref{fig:schemaBasedBlocking}(a); a parent-child edge implies that the latter method improves upon the former one. Below, we elaborate on these methods based on their key type.
\textbf{Hash-based Methods.}
\textit{Standard Blocking} (\textsf{SB}) \cite{fellegi1969theory}
involves the simplest
functionality:
an expert selects the most suitable attributes, and a transformation function concatenates (parts of) their values to form blocking keys. For every distinct key, a block is created containing all corresponding entities. In short, \textsf{SB} operates as a hash function, conveying two main advantages: (i) it yields redundancy-free blocks,
and (ii) it has a linear time complexity, $O(|E|)$.
On the flip side, its effectiveness is very sensitive to noise,
as the slightest difference in the blocking keys of duplicates places them in different blocks.
\textsf{SB} is also a lazy method that imposes no limit on block sizes.
To address these issues, \textit{Suffix Arrays Blocking} (\textsf{SA})~\cite{DBLP:conf/wiri/AizawaO05} converts each blocking key of \textsf{SB} into the list of its suffixes that are longer than a predetermined minimum length $l_{min}$. Then, it defines a block for every suffix that does not exceed a predetermined frequency threshold $b_{max}$, which essentially specifies the maximum block size. This proactive functionality is necessary, as very frequent suffixes (e.g., ``ing") result in large blocks that are dominated by unnecessary comparisons.
\begin{figure}[t]\centering
\includegraphics[width=0.86\linewidth]{genealogy.png}
\vspace{-8pt}
\caption{The genealogy trees of non-learning (a) schema-aware and (b) schema-agnostic Block Building techniques. Hybrid, hash- and sort-based methods are marked in {\color{blue}blue}, black and {\color{red}red}, respectively.
}
\label{fig:schemaBasedBlocking}
\vspace{-10pt}
\end{figure}
\textsf{SA}
has two major advantages \cite{DBLP:conf/cikm/VriesKCC09}: (i) it has low time complexity, $O(|E|$$\cdot$$log|E|)$~\cite{DBLP:journals/dke/AllamSK18}, and is very efficient, as it results in a small but relevant set of candidate matches; (ii) it is very effective, due to the robustness to the noise at the beginning of blocking keys and the high levels of redundancy (i.e., it places every entity into multiple blocks). On the downside, \textsf{SA} does not handle noise at the end of \textsf{SB} keys. E.g., two matches with \textsf{SB} keys ``JohnSnith" and ``JohnSmith" have no common suffix if $l_{min}$=4, while for $l_{min}$=3, they co-occur in a block only if the frequency of ``ith" is lower than $b_{max}$.
This problem is addressed by \textit{Extended Suffix Arrays Blocking} \cite{DBLP:journals/tkde/Christen12,DBLP:journals/pvldb/0001APK15}, which uses as keys not just the suffixes
of \textsf{SB} keys,
but all their substrings with more than $l_{min}$ characters. E.g., for $l_{min}$=4, \textsf{SA} extracts from ``JohnSnith" the keys ``JohnSnith", ``ohnSnith", ``hnSnith", ``nSnith", ``Snith" and ``nith", while \textit{Extended SA} additionally extracts the keys ``John", ``ohnS", ``hnSn", ``nSni", ``Snit" as well as all substrings of ``JohnSnith"
ranging from 5 to 8 characters.
Another extension of \textsf{SB}
is \textit{Q-grams Blocking} \cite{DBLP:journals/tkde/Christen12,DBLP:journals/pvldb/0001APK15}. Its transformation function converts the blocking keys of \textsf{SB} into sub-sequences of $q$ characters (\textit{$q$-grams}) and defines a block for every distinct $q$-gram. For example, for $q$=3, the key \textit{france} is transformed into the trigrams \textit{fra}, \textit{ran}, \textit{anc}, \textit{nce}. This approach
differs from \textsf{Extended SA} in that it does not restrict block sizes (lazy method). Also, it is more resilient to noise than \textsf{SB},
but results in more and larger blocks.
To improve it,
\textit{Extended Q-Grams Blocking} \cite{baxter2003comparison,DBLP:journals/tkde/Christen12,DBLP:journals/pvldb/0001APK15} uses combinations of $q$-grams, instead of individual $q$-grams. Its transformation function concatenates at least $l$ $q$-grams, where $l = max(1,\lfloor k \cdot t \rfloor)$, with $k$ denoting the number of $q$-grams and $t \in [0, 1)$ standing for a user-defined threshold. The larger $t$ is, the larger $l$ gets, yielding less keys from the $k$ $q$-grams. For $T=0.9$ and $q$=3, the key \textit{france} is transformed into the following four signatures ($k$=4 and $l$=3): [\textit{fra}, \textit{ran}, \textit{anc}, \textit{nce}], [\textit{fra}, \textit{ran}, \textit{anc}], [\textit{fra}, \textit{anc}, \textit{nce}], [\textit{ran}, \textit{anc}, \textit{nce}]. In this way, $q$-gram-based blocking keys become more distinctive, decreasing the number and cardinality of blocks.
A more advanced $q$-gram-based approach is \textit{MFIBlocks} \cite{DBLP:journals/is/KenigG13}. Its transformation function concatenates keys of Q-Grams Blocking into itemsets and uses a maximal frequent itemset algorithm to define as new blocking keys those
exceeding a predetermined support threshold.
\textbf{Sort-based Methods.}
\textit{Sorted Neighborhood} (\textsf{SN}) \cite{DBLP:conf/sigmod/HernandezS95}
sorts all blocking keys in alphabetical order and arranges the associated entities accordingly. Subsequently, a window of fixed size $w$ slides over the sorted list of entities and compares the entity at the last position with all other entities placed within the same window. The underlying assumption is that the closer the blocking keys of two entities are in the lexicographical order, the more likely they are to be matching. Originally crafted for relational data, \textsf{SN} is extended to hierarchical/XML data based on user-defined keys in \cite{DBLP:conf/edbt/PuhlmannWN06}.
\textsf{SN} has three major advantages \cite{DBLP:journals/tkde/Christen12}: (i) it has low time complexity, $O(|E|\cdot log |E|)$, (ii) it results in linear ER complexity, $O(w \cdot |E|)$, and (iii) it is robust to noise, supporting errors at the end of blocking keys.
However, it may place two entities in the same block even if their keys are dissimilar (e.g.,
"alphabet" and "apple",
if no other key intervenes between them).
Its performance also depends heavily on the window size $w$, which is difficult to configure,
especially in Deduplication, where the matching entities form clusters of varying size \cite{DBLP:conf/nss/DraisbachN11,DBLP:journals/tkde/Christen12}.
To ameliorate the effect of $w$,
a common solution is the \textit{Multi-pass SN}
\cite{DBLP:journals/datamine/HernandezS98}, which
applies the core algorithm multiple times, using a different transformation function in each iteration. In this way, more matches can be identified, even if the window is set to low size. Another solution is the \textit{Extended Sorted Neighborhood} \cite{DBLP:journals/tkde/Christen12,DBLP:journals/pvldb/0001APK15}, which slides a window of fixed size over the sorted list of blocking keys rather than the list of entities; this means that each block merges $w$ \textsf{SB} blocks.
More advanced strategies adapt the window size dynamically
to optimize the balance between effectiveness and efficiency.
They are grouped into
three
categories, depending on the criterion for moving the
boundaries of the window \cite{DBLP:journals/cj/MaDY15}:
1) \textit{Key similarity strategy.} The window size increases if the similarity of the blocking keys exceeds a predetermined threshold, which indicates that more similar entities should be expected \cite{DBLP:journals/cj/MaDY15}.
2) \textit{Entity similarity strategy.} The window size relies on the similarity of the entities within the current window. \textit{Incrementally Adaptive SN} \cite{DBLP:conf/jcdl/YanLKG07} increases the window size if
the distance of the first and the last element in the window is smaller than a predetermined threshold. The actual increase
depends on the current window size and the selected threshold. \textit{Accumulative Adaptive SN} \cite{DBLP:conf/jcdl/YanLKG07} creates windows with a single overlapping entity and exploits transitivity to group multiple adjacent windows into the same block, as long as the last entity of one window is a potential duplicate of the last entity in the next
window. After expanding the window, both algorithms apply a retrenchment phase that decreases the window size until all entities
are potential duplicates.
3) \textit{Dynamic strategy.}
The core assumption is that the more duplicates are found within a window, the more are expected to be found by increasing its size.
\textit{Duplicate Count Strategy} (\textsf{DCS}) \cite{DBLP:conf/icde/DraisbachNSW12} defines a window
$w$ for every entity in \textsf{SN}'s sorted list and executes all its comparisons
to compute the ratio $d/c$,
where $d$ denotes the newly detected duplicates and $c$ the executed comparisons. The window size is then incremented by one position at a time as long as $d/c \geq \phi$, where $\phi \in (0,1)$ is a threshold that expresses the average number of duplicates per comparison. \textsf{DCS++} \cite{DBLP:conf/icde/DraisbachNSW12} improves \textsf{DCS} by increasing the window size with the next $w-1$ entities, even if the new ratio becomes lower than $\phi$.
Using transitive closure,
it skips some windows, saving part of the comparisons.
\textbf{Hybrid methods.} \textit{Sorted Blocks} \cite{DBLP:conf/nss/DraisbachN11} combines the benefits of \textsf{SB}
and \textsf{SN}.
First, it sorts all blocking keys and the corresponding entities in lexicographical order, like \textsf{SN}. Then, it partitions the sorted entities
into disjoint blocks, like \textsf{SB}, using a prefix of the blocking keys. Next, all pairwise comparisons are executed within each block. To avoid missing any matches, an overlap parameter $o$
defines a
window of fixed size that includes
the $o$ last entities in the current block together with the first entity of the next block. The window slides by one position at a time until reaching the $o^{th}$ entity of the next block, executing
all pairwise comparisons between entities from different blocks.
Sorted Blocks is a lazy approach that does not restrict block sizes.
Thus, it may result in large blocks that dominate its processing time. To address this, two proactive variants set a limit on the maximum block size. \textit{Sorted Blocks New Partition} \cite{DBLP:conf/nss/DraisbachN11}
creates a new block when the maximum
size is reached for a
(prefix of) blocking key; the overlap between the blocks ensures that every entity is compared with its predecessors and successors in the sorting order. \textit{Sorted Blocks Sliding Window} \cite{DBLP:conf/nss/DraisbachN11} avoids executing all comparisons within a block that is
larger than the upper limit; instead, it slides a window
equal to the maximum block size over the entities of the current block.
Finally, \textit{Improved Suffix Arrays Blocking} \cite{DBLP:conf/cikm/VriesKCC09}
employs the same blocking keys as \textsf{SA}, but sorts them in alphabetical order, like \textsf{SN}. Then, it compares the consecutive keys with a string similarity measure. If the similarity of two suffixes exceeds a predetermined threshold, the corresponding blocks are merged
in an effort to detect duplicates even when there is noise at the end of \textsf{SB} keys, or their sole common key is too frequent. For example, \textit{Improved SA} detects the high string similarity of the keys ``JohnSnith" and ``JohnSmith", placing the corresponding entities into the same block.
\subsubsection{Learning-based Methods}
\label{sec:learningBlBu}
We distinguish these methods into supervised and unsupervised ones.
Both rely on a labelled dataset that includes pairs of matching and non-matching entities, called \textit{positive} and \textit{negative instances}, respectively. This dataset is used to learn \textit{blocking predicates}, i.e., combinations of an attribute name and a transformation function (e.g., $\{title, First3Characters\}$). Entities sharing the same output for a particular blocking predicate are considered candidate matches (i.e., hash-based functionality). Disjunctions of conjunctions of predicates, i.e., composite blocking schemes, are learned by optimizing an objective function.
\textbf{Supervised Methods.}
\textit{ApproxRBSetCover} \cite{DBLP:conf/icdm/BilenkoKM06}
learns disjunctive
blocking schemes by solving a standard weighted set cover problem. The cover is iteratively constructed by adding in each turn the blocking predicate
that maximizes the ratio of the previously uncovered positive pairs over the covered negative pairs. This is a "soft cover", since some positive instances may remain uncovered.
\textit{ApproxDNF} \cite{DBLP:conf/icdm/BilenkoKM06} alters ApproxRBSetCover so that it learns
blocking schemes in Disjunctive Normal Form (DNF). Instead of individual predicates, each turn greedily learns a conjunction of up to $k$ predicates that maximizes the ratio of positive and negative covered instances.
A similar approach is \textit{Blocking Scheme Learner} (\textsf{BSL}) \cite{DBLP:conf/aaai/MichelsonK06}. Based on an adaptation of the
Sequential Covering Algorithm, it learns
blocking schemes that maximize $RR$, while maintaining $PC$ above a predetermined threshold. Its output is a disjunction of conjunctions of blocking predicates.
\textsf{BSL} is improved by \textit{Conjunction Learner} \cite{DBLP:conf/ijcai/CaoCZYLY11}, which minimizes the candidate matches not only in the labelled, but also in the \textit{unlabelled} data, while maintaining high $PC$. The effect of the unlabelled data is determined through a weight $w \in [0,1]$; $w=0$ disregards unlabelled data completely, falling back to \textsf{BSL}, while
$w=1$ indicates that they are equally important as the labelled ones.
On another line of research, \textit{Blocking based on Genetic Programming} (\textsf{BGP}) \cite{DBLP:journals/jidm/EvangelistaCSM10} employs a tree representation of supervised blocking schemes, where every leaf node corresponds to a blocking predicate. In every turn, a set of genetic programming operators, such as copy, mutation and crossover, are applied to the initial, random set of
blocking schemes. Then, a fitness function
infers the performance of the new schemes from the harmonic mean of $PC$ and $RR$, and the best ones
are returned as output.
Yet, \textsf{BGP} involves numerous internal parameters
that are hard to fine-tune.
Another tree-based approach
is \textit{CBLOCK} \cite{DBLP:conf/cikm/SarmaJMB12}. In this case, every edge is annotated with a hash (i.e., transformation) function and every node $n_i$ comprises the set of entities that result after applying all hash functions from the root to $n_i$. \textit{CBLOCK} is the only proactive learning-based method, restricting the maximum size of its blocks. Every node that exceeds this limit is split into smaller, disjoint blocks through a greedy algorithm that picks the best hash function based on the resulting $PC$. To minimize the human effort, a drill down approach is proposed for bootstrapping.
\textbf{Unsupervised Methods.}
\textit{FisherDisjunctive} \cite{DBLP:conf/icdm/KejriwalM13}
uses a weak training set generated by
the TF-IDF similarity between pairs of entities. Pairs with very low (high) values are considered as negative (positive) instances. A boolean feature vector is then associated with every labelled instance. The discovery of DNF
blocking schemes is finally cast as a Fisher feature selection problem.
Similarly, \textit{DNF Learner}~\cite{giang2015machine}
applies a matching algorithm
to a sample of entity pairs to automatically create a labelled dataset. Then, the learning of
blocking schemes is cast as a DNF learning problem. To scale it to the exponential search space of possible schemes, their complexity is restricted to manageable levels (e.g., they comprise at most $k$ predicates).
\vspace{-8pt}
\subsection{Schema-agnostic Block Building}
\label{sec:schemaAgnosticBB}
Methods of this type make no assumptions about schema knowledge, disregarding completely attribute names; they extract blocks from all attribute values.
Thus, they inherently support noise in both attribute names and values and are suitable for
highly heterogeneous, loosely structured entity profiles, such as those stemming from the Web of Data~\cite{DBLP:conf/wsdm/PapadakisINF11,DBLP:conf/wsdm/PapadakisINPN12,DBLP:journals/tkde/PapadakisIPNN13}.
\textbf{Non-learning Methods.}
The family tree of this category appears in Figure \ref{fig:schemaBasedBlocking}(b). The cornerstone approach is \textit{Token Blocking} (\textsf{TB}) \cite{DBLP:conf/wsdm/PapadakisINF11}. Assuming that duplicates share at least one common token, its transformation function extracts all tokens from all attribute values of every entity. A block $b_t$ is then defined for every distinct token $t$.
Hence, two entities co-occur in block $b_t$ if they share token $t$ in their values, regardless of the associated attribute names.
To improve \textsf{TB}, \textit{Attribute Clustering Blocking} \cite{DBLP:journals/tkde/PapadakisIPNN13} requires the common tokens of two entities to appear in \textit{syntactically similar attributes}. These are attribute names that correspond to similar values, but are not necessarily semantically matching (unlike Schema Matching). First, it clusters attributes based on the similarities of their aggregate values.
Each attribute is connected to its most similar one and the transitive closure of the connected attributes forms disjoint clusters. A block $b_{k,t}$ is then defined for every token $t$ in the values of the attributes belonging to cluster $k$.
\textit{RDFKeyLearner} \cite{DBLP:conf/semweb/SongH11} applies \textsf{TB} independently to the values of specific attributes, which are selected through the following process:
each attribute is
associated with a \textit{discriminability} score, which amounts to the portion of
distinct values over all values in the given dataset. If this is lower than a predetermined threshold, the attribute is ignored due to limited diversity, i.e., too many entities have the same value(s). For each attribute with high discriminability, its \textit{coverage} is estimated, i.e., the portion of entities that contain it. The harmonic mean of discriminability and coverage is then computed for all valid attributes and the one with the maximum score is selected for defining blocking keys as long as its score exceeds another predetermined threshold. If not, the selected attribute
is combined with all other attributes and the process is repeated.
\textit{Prefix-Infix(-Suffix) Blocking} \cite{DBLP:conf/wsdm/PapadakisINPN12}
exploits the naming pattern in entity URIs. The \textit{prefix} describes the domain of the URI, the \textit{infix} is a local identifier, and the optional \textit{suffix} contains details about the format, or a named anchor \cite{DBLP:conf/iiwas/PapadakisDFK10}. E.g., in the URI {\small\texttt{https://en.wikipedia.org/wiki/France\#History}}, the prefix is {\small\texttt{https://en.wikipedia.org/wiki}}, the infix is {\small\texttt{France}} and the suffix is {\small\texttt{History}}. In this context, this method uses as
keys all (URI) infixes along with all tokens in the literal values.
\textit{TYPiMatch} \cite{DBLP:conf/wsdm/MaT13} improves \textsf{TB}
by automatically detecting the entity types in the input data.
It creates a co-occurrence graph, where every node corresponds to a token in any attribute value and every edge connects two tokens if both conditional probabilities of co-occurrence exceed a predetermined threshold. The maximal cliques are extracted
and merged if their overlap exceeds another threshold. The resulting clusters correspond to the entity types,
with every entity participating in all types to which its tokens belong.
\textsf{TB} is then applied independently to the profiles of each type.
Finally, \textit{Semantic Graph Blocking} \cite{DBLP:conf/ideas/NinMML07} is based exclusively on the relations between entities, be it foreign keys in a database or links in RDF data. It completely disregards attribute values, building a collaborative graph, where every node corresponds to an entity and every edge connects two associated entities. For instance, the collaborative graph for a bibliographic data collection can be formed by mapping every author to a node and adding edges between co-authors.
Then a new block $b_i$ is formed for each node $n_i$, containing all nodes connected with $n_i$ through a path, provided that the path length or the block size do not exceed predetermined limits (proactive functionality).
\textbf{Learning-based Methods.}
\textit{Hetero} \cite{DBLP:conf/semweb/KejriwalM14a}
converts the input data into heterogeneous structured datasets using property tables. Then, it maps every entity to a normalized TF vector, and applies an adapted Hungarian algorithm with linear scalability to produce positive and negative feature vectors. Finally, it applies \textit{FisherDisjunctive} \cite{DBLP:conf/icdm/KejriwalM13} with bagging to achieve robust performance.
Similarly, \textit{Extended DNF BSL}
\cite{DBLP:journals/corr/KejriwalM15} combines an established instance-based schema matcher with weighted set covering to learn DNF
blocking schemes with at most $k$ predicates.
\subsection{Parallelization Approaches}
\label{sec:parallelizationBlBu}
To scale Blocking methods to massive entity collections without altering their functionality,
the \textit{MapReduce framework} \cite{DeanG04} is typically used,
as it offers fault-tolerant, optimized execution for applications distributed across a set of independent nodes.
\textbf{Schema-aware methods.}
The hash-based, non-learning methods are adapted to MapReduce in a straightforward way. The \texttt{map}
phase implements the transformation function(s), emitting {\small \texttt{(key, entity\_id)}} pairs for each entity. Every reducer acts as an assignment function, placing all entities with blocking key $t$ in block $b_t$.
Dedoop~\cite{DBLP:journals/pvldb/KolbTR12} provides such implementations for various methods.
For sort-based methods,
the adaptation of \textsf{SN}
to MapReduce in \cite{DBLP:journals/ife/KolbTR12} can be used as a template.
The \texttt{map} function extracts the blocking key(s) from each input entity, while the ensuing \textit{partitioning} phase
sorts all entities in alphabetical order of their keys.
The \texttt{reduce} function slides a window of fixed size within every reduce partition. Inevitably, entities close to the partition boundaries need to be compared across different reduce tasks. Thus, the \texttt{map} function is extended to replicate those entities, forwarding them to the respective reduce task and its successor.
DCS and DCS++ are adapted to the MapReduce framework in \cite{DBLP:conf/sac/MestrePN15}, using three jobs. The first one sorts the originally unordered entities of the data partition assigned to each mapper according to the selected blocking keys. It also selects the boundary pairs of the sorted partitions. The second job generates the Partition Allocation Matrix, which specifies the sorted partitions to be replicated, while the third job performs DCS(++) locally, to the data assigned to every reducer.
\textbf{Schema-agnostic methods.}
A single MapReduce job is required for parallelizing \textsf{TB} \cite{DBLP:series/synthesis/2015Christophides,DBLP:conf/bigdataconf/EfthymiouSC15}. For every input entity $e_i$, the \texttt{map} function emits a ($t$, $e_i$) pair for every token $t$ in the values of $e_i$. Then, all entities sharing a particular token are directed to the same \texttt{reducer} to form a new block.
For Attribute Clustering Blocking, four MapReduce jobs are required \cite{DBLP:series/synthesis/2015Christophides,DBLP:conf/bigdataconf/EfthymiouSC15}. The first assembles all values
per attribute.
The second computes the pairwise similarities between all attributes,
even if they are placed in different data partitions. The third connects every attribute
to its most similar one. The fourth associates every attribute name with a cluster id and adapts
\textsf{TB}'s \texttt{map} function to emit pairs of the form ($k$.$t$, $i$), where $k$ is the cluster id of $e_i$'s attribute name that contains token $t$.
Finally, the parallelization of Prefix-Infix(-Suffix) Blocking involves three MapReduce jobs \cite{DBLP:series/synthesis/2015Christophides,DBLP:conf/bigdataconf/EfthymiouSC15}. The first parallelizes the algorithm that extracts the prefixes from a set of URIs \cite{DBLP:conf/iiwas/PapadakisDFK10}. The second
extracts the URI suffixes.
The third
applies \textsf{TB}'s
mapper
to the literal values simultaneously with an infix mapper that emits a pair ($j$, $e_i$) for every infix $j$ that is extracted from $e_i$'s profile.
The final \texttt{reduce} phase ensures that all entities having a common token or infix
are placed in the same block.
\textbf{Load Balancing.}
For MapReduce, it is crucial to distribute evenly the overall workload among the available nodes, avoiding potential bottlenecks.
The following methods distribute the execution of comparisons
in a block collection -
not the cost of building the blocks.
\textit{BlockSplit} \cite{DBLP:conf/icde/KolbTR12} partitions large blocks into smaller sub-blocks and processes them in parallel.
Every entity is compared to all entities in its sub-block as well as to all entities of its super-block, even if their sub-block is initially assigned to a different node. This yields an additional network and I/O overhead
and may still lead to unbalanced workload, due to sub-blocks of different size.
To overcome this, \textit{PairRange} \cite{DBLP:conf/icde/KolbTR12} splits evenly the comparisons in a set of blocks into a predefined number of partitions. It involves a single MapReduce job with a mapper that associates every entity $e_i$ in block $b_k$ with the output key $p.k.i$, where $p$ denotes the partition id.
The reducer assembles
all entities that have the same $p$ and block id, reproducing the comparisons of each partition.
The space requirements of these two algorithms are improved in \cite{DBLP:conf/ipccc/YanXM13}, which minimizes their memory consumption by adapting them so that they work with sketches.
Finally, \textit{Dis-Dedup} \cite{DBLP:journals/pvldb/ChuIK16} is the only method that takes into account both the computational and the communication cost (e.g., network transfer time, local disk I/O time). Dis-Dedup considers all possible cases, from disjoint blocks produced by a single blocking technique to overlapping blocks derived from multiple techniques. It also provides strong theoretical guarantees that the overall maximum cost per reducer is within a small constant factor from the lower bounds.
\subsection{Discussion \& Experimental Results}
The performance of the above techniques is examined both qualitatively and quantitatively in a series of individual of papers (e.g., \cite{DBLP:journals/tkdd/VriesKCC11,o2018new,DBLP:journals/tkde/PapadakisIPNN13,DBLP:conf/wsdm/PapadakisINF11}) and experimental analyses (e.g., \cite{DBLP:journals/tkde/Christen12,DBLP:journals/pvldb/0001APK15,DBLP:journals/pvldb/0001SGP16}). Below, we summarize the main findings in order to facilitate the use of Block Building techniques.
Starting with \textit{Standard Blocking} (\textsf{SB}),
its performance depends heavily on the frequency distribution of attribute values and, thus, of blocking keys. The best case corresponds to a uniform distribution, where $||B||=||E||/|B|$ \cite{DBLP:journals/tkde/Christen12}. Due to its lazy functionality, though, all other key distributions yield a portion of large blocks with many superfluous comparisons, i.e., low $PQ$~and~$RR$.
\textit{Suffix Arrays Blocking} (\textsf{SA}) improves \textsf{SB}'s $PC$, by supporting errors at the beginning of blocking keys. The higher $l_{min}$ is and the lower $b_{max}$ is, the lower $||B||$ and $PC$ get. For the same settings, \textit{Extended SA} raises $PC$ at the cost of higher $||B||$, which inevitably lowers both $PQ$ and $RR$. \textit{Improved SA} is theoretically proven in \cite{DBLP:journals/tkdd/VriesKCC11} to result in a $PC$ greater or equal to that of \textsf{SA}, though at the cost of a higher computational cost and more comparisons, which lower $PQ$ and $RR$.
\textit{Q-grams Blocking} yields higher $PC$ than \textsf{SB}, but decreases both $PQ$ and $RR$. \textit{Extended Q-grams Blocking} raises $PQ$ and $RR$ at a limited, if any, cost in $PC$. \textsf{MFIBlocks} reduces significantly the number of blocks and matching candidates (i.e., very high $PQ$ and $RR$) \cite{DBLP:journals/is/KenigG13}, but it may come at the cost of missed matches (insufficient $PC$) in case the resulting blocking keys are very restrictive for matches with noisy descriptions \cite{DBLP:journals/pvldb/0001SGP16}.
For \textit{Sorted Neighborhood} (\textsf{SN}), a small $w$ leads to high $PQ$ and $RR$ but low $PC$ and vice versa for a large $w$. For \textit{Extended SN}, variations in the window size have a large impact on efficiency ($PQ$ and $RR$), affecting the portion of unnecessary comparisons, but $PC$ is more stable. Among the other \textsf{SN} variants, \textsf{DCS++} is theoretically proven to miss no matches with an appropriate value for $\phi$, while being at least as~efficient~as~\textsf{SN}. \textit{Sorted Blocks New Partition} outperforms most SN-based algorithms, but includes more parameters than \textsf{SN}, involving a more complex configuration.
Most importantly, all these non-learning schema-aware methods
are quite parameter-sensitive: even small parameter value modifications may yield significantly different performance \cite{DBLP:journals/tkde/Christen12,DBLP:journals/pvldb/0001APK15,DBLP:conf/cikm/VriesKCC09,o2018new}. Their most important parameter is the definition of the blocking keys, which requires fine-tuning by an expert. Otherwise, their $PC$ remains insufficient, placing most duplicates in no common block \cite{DBLP:journals/tkde/Christen12,DBLP:journals/pvldb/0001APK15}. This applies even to methods that employ redundancy for higher recall.
This shortcoming is ameliorated by schema-agnostic methods,
which consistently achieve much higher $PC$ than their schema-aware counterparts \cite{DBLP:journals/pvldb/0001APK15}. They also simplify the configuration of Block Building, reducing its sensitivity through the automatic definition of blocking keys \cite{DBLP:journals/pvldb/0001APK15,DBLP:journals/pvldb/0001SGP16}. Rather than human intervention or expert knowledge, their robustness emanates from the high levels of redundancy they employ, placing every entity in a multitude of blocks. On the downside, they yield a considerably higher number of comparisons, resulting in very low $PQ$ and $RR$. Both, however, can be significantly improved by Block Processing \cite{DBLP:conf/icde/PapadakisN11,DBLP:journals/pvldb/0001SGP16}.
Regarding the relative performance of schema-agnostic methods, \textsf{TB} yields very high $PC$, at the cost of very low $PQ$ and $RR$. It constitutes a very efficient approach, iterating only once over the input entities, and it is the sole parameter-free Block Building technique in the literature as well as the most generic one, applying to any entity collection with textual values. Its performance is improved by
\textit{Attribute Clustering} and \textit{Prefix-Infix(-Suffix) Blocking} for specific type of datasets: highly heterogeneous ones, with a large variety of attribute names \cite{DBLP:journals/tkde/PapadakisIPNN13,DBLP:journals/pvldb/0001SGP16}, and semi-structured (RDF) ones \cite{DBLP:conf/wsdm/PapadakisINF11}, respectively. In these cases, both methods yield a much larger number of smaller blocks, significantly raising $PQ$ at a minor cost in $PC$. Both methods, though, involve a much higher computational cost than \textsf{TB}. The same applies to
\textit{TYPiMatch}, where the detection of entity types is a rather time-consuming process. Yet, its $PC$ is consistently insufficient, because it falsely divides duplicate entities different entity types, due to the sensitivity to its parameter~configuration~\cite{DBLP:journals/pvldb/0001SGP16}.
Finally, the learning-based Block Building techniques
typically suffer from the scarcity of labelled datasets; even if a training set is available for a particular dataset, it cannot be directly used for learning supervised blocking schemes for another dataset. Instead, a complex transfer learning procedure is typically required \cite{DBLP:conf/cikm/NegahbanRG12,DBLP:journals/corr/abs-1809-11084}. Regarding their efficiency, \textsf{BSL} is typically faster than \textit{ApproxRBSetCover} and \textit{ApproxDNF}, as it exclusively considers positive instances, thus requiring a smaller training set. \textit{Conjunction Learner}
requires every supervised blocking scheme to be applied to the large set of unlabelled data, which is impractical. To accelerate it, a random sample of the unlabelled data is used in practice. \textit{CBLOCK} is also the only learning-based method that is suitable for the MapReduce framework: every entity runs through the learned tree and is directed to the machine corresponding to its leaf node. In terms of effectiveness, there is no clear winner. \textsf{BSL} and \textit{FisherDisjunctive} achieve the top performance in \cite{o2018new}. The latter addresses the scarcity of labelled data, but is not scalable to large datasets.
\section{Block Processing}
\label{sec:blockProcessing}
Block Processing receives as input an existing block collection $\mathcal{B}$ and produces as output a new block collection $\mathcal{B'}$ that improves the balance between effectiveness and efficiency, i.e., $PQ(B) \ll PQ(B')$, $RR(B',B)\gg 0$, while $PC(B) \sim PC(B')$. We distinguish Block Processing methods into \textit{Block Cleaning} ones, which decide whether
entire blocks
should be retained or modified, and \textit{Comparison Cleaning} ones, which
decide whether
individual comparisons
are unnecessary.
\vspace{-10pt}
\subsection{Block Cleaning}
\label{sec:blcl}
We classify Block Cleaning methods into two categories: (i) \textit{static}, which are independent of matching results, and (ii) \textit{dynamic}, which are interwoven with the matching process.
\textbf{Static Methods.} A core idea is the assumption that the larger a block is, the less likely it is to contain unique duplicates, i.e., matches that share no other block. Such large blocks are typically produced by lazy schema-agnostic techniques and correspond to stop words.
In this context, \textit{Block Purging}
discards
blocks that exceed an upper limit on block cardinality \cite{DBLP:journals/tkde/PapadakisIPNN13} or size \cite{DBLP:conf/wsdm/PapadakisINPN12}.
\textit{Block Filtering} \cite{DBLP:conf/edbt/0001PPK16} applies this assumption to individual entities, removing every entity from the largest blocks that contain it. In other words, it
retains every entity in $r\%$ of its smallest blocks.
On a different line of research, \textit{Size-based Block Clustering} \cite{DBLP:conf/kdd/FisherCWR15} applies hierarchical clustering to transform a set of blocks into a new one where all block sizes lie within a specified size range. It merges recursively small blocks that correspond to similar blocking keys, while splitting large blocks into smaller ones. A penalty function controls the trade-off between block quality and block size.
A similar approach is the MapReduce-based dynamic blocking algorithm
in~\cite{mcneill2012dynamic}, which splits large blocks into sub-blocks.
\textit{MaxIntersectionMerge} \cite{nascimento2019exploiting} ensures that all blocks involve at least $|b|_{min}$ entities.
To this end, it merges each block smaller than $|b|_{min}$ entities with the block that has the most entities in common and is larger than $|b|_{min}$.
Similarly,
\textit{Rollup Canopies} \cite{DBLP:conf/cikm/SarmaJMB12} receives as input a training set with positive examples, a limit on the maximum block size and a set of disjoint blocks; using a learning-based greedy algorithm, it merges pairs of small blocks to increase $PC$.
Finally, \cite{DBLP:conf/icdm/RanbadugeVC16} generalizes Meta-blocking (see Section \ref{sec:cocl}) to Multi-source ER: it constructs a graph, where the nodes correspond to blocks and the edges connect blocks whose blocking keys are more similar than a predetermined threshold. The edges are weighted using various functions and all pairs of blocks are then processed in descending edge weights in an effort to maximize the redundant and superfluous comparisons that are skipped.
\textbf{Dynamic Methods.}
\textit{Iterative Blocking} \cite{DBLP:conf/sigmod/WhangMKTG09} merges
any new pair of detected duplicates, $e_i$ and $e_j$, into a new entity, $e_{i,j}$,
and replaces both $e_i$ and $e_j$ with $e_{i,j}$ in all blocks that contain them, even if they have already been processed.
The new entity $e_{i,j}$ is compared with all co-occurring entities, as
the new content in $e_{i,j}$ might
identify previously missed matches.
The ER process terminates when all blocks have been processed without finding new duplicates.
Iterative Blocking applies exclusively to Deduplication. In Record Linkage, there is no need for merging two matching entities,
due to the 1-1 restriction. Still,
the detected duplicates should be propagated in order to save the superfluous comparisons with
their co-occurring entities
in the subsequently processed blocks.
The earlier the matches are detected, the more superfluous comparisons are saved.
To this end, \textit{Block Scheduling}
optimizes the processing order
of blocks in a non-iterative way, sorting them in decreasing order of
the probability $p_i(d)$ that a block $b_i$ contains a pair of duplicates. This is set inversely proportional to block cardinality, i.e., $p_i(d)=1/||b_i||$ \cite{simonini2018schema}, or to the minimum size of the inner block, i.e., $p_i(d)=1/min{|b_{i,1}|,|b_{i,2}|}$, where $|b_{i,k}| \subset \mathcal{E}_k$ \cite{DBLP:conf/wsdm/PapadakisINF11}.
The former definition also applies to Iterative Blocking, which does not specify the exact block processing order, even though this affects significantly the resulting performance \cite{DBLP:journals/pvldb/0001SGP16}.
\textit{Block Pruning} \cite{DBLP:conf/wsdm/PapadakisINF11} extends Block Scheduling by exploiting the decreasing density of detected matches in its processing order (i.e., the later a block is processed, the less
unique duplicates it contains).
After processing the latest block, it estimates the average number of executed comparisons per new duplicate.
If this ratio falls below a specific threshold, it terminates the ER process.
\subsection{Comparison Cleaning}
\label{sec:cocl}
\begin{figure}[t]\centering
\includegraphics[width=0.59\linewidth]{ccGenealogy.png}
\vspace{-8pt}
\caption{The genealogy tree of non-learning Comparison Cleaning methods. Methods in black conform to the Meta-blocking framework in Figure \ref{fig:coclExample}, methods in {\color{blue}blue} are Meta-blocking techniques following a (partially) different approach and methods in {\color{red}red} are not part of the Meta-blocking framework.
}
\label{fig:taxonomyCC}
\vspace{-14pt}
\end{figure}
\textbf{Non-learning Methods.}
Figure \ref{fig:taxonomyCC} illustrates the family tree of the methods belonging to this category.
The cornerstone method is \textit{Comparison Propagation} \cite{DBLP:conf/jcdl/PapadakisINPN11}, which propagates all executed comparisons to the subsequently processed blocks. In this manner, it eliminates all redundant comparisons in a given block collection without losing any pair of duplicates, thus raising $PQ$ and $RR$ at no cost in $PC$.
It builds an inverted index that points from entity ids to block ids, called \textit{Entity Index}, and with its help, it compares two entities $e_i$ and $e_j$ in block $b_k$ only if $k$ is their least common block id.
For example, consider the blocks in Figure \ref{fig:coclExample}(a) and their Entity Index in Figure \ref{fig:coclExample}(b). The least common block id of $e_1$ and $e_3$ is 2. Thus, they are compared in $b_2$, but neither in $b_4$~nor~in~$b_5$.
Given a redundancy-positive block collection, the Entity Index allows for identifying the blocks shared by a pair of co-occurring entities. This allows for weighting all pairwise comparisons in proportion to the matching likelihood of the corresponding entities, based on the principle that the more blocks two entities share, the more likely they are to be matching. This gives rises to a family of \textit{Meta-blocking} techniques \cite{DBLP:journals/tkde/PapadakisKPN14,DBLP:conf/edbt/0001PPK16,DBLP:journals/pvldb/SimoniniBJ16} that
go beyond Comparison Propagation by discarding not only all redundant comparisons, but also the vast majority of the superfluous ones.
The first relevant method is \textit{Comparison Pruning} \cite{DBLP:conf/sigmod/PapadakisINPN11}, which
computes the Jaccard co-efficient of the block lists of two entities. If it does not exceed a conservative threshold that depends on the average number of blocks per entity, the
comparison is pruned, as it designates an unlikely match.
Meta-blocking was formalized into a more principled approach in \cite{DBLP:journals/tkde/PapadakisKPN14}. The given redundancy-positive block collection $\mathcal{B}$ is converted into a blocking graph $G_B$, where the nodes correspond to entities and the edges connect every pair of co-occurring entities - see Figure \ref{fig:coclExample}(c). Given that no parallel edges are allowed, all redundant comparisons are discarded by definition. The edges are then weighted proportionately to the likelihood that the adjacent entities are matching. In Figure \ref{fig:coclExample}(d), the edge weights indicate the number of common blocks. Edges with low weights are pruned, because they correspond to superfluous comparisons. In Figure \ref{fig:coclExample}(e), all edges with a weight lower than the average one are discarded. The resulting pruned blocking graph $G_{B'}$ is transformed into a restructured block collection $\mathcal{B}'$ by forming one block for every retained edge - see Figure \ref{fig:coclExample}(f). As a result, $\mathcal{B}'$ exhibits a much higher efficiency, $PQ(B')$$\gg$$PQ(B)$ and $RR(B',B)$$\gg$$0$, for similar effectiveness, $PC(B')$$\sim$$PC(B)$; in our example, the 12 comparisons in the input blocks of Figure \ref{fig:coclExample}(a) are reduced to 2 matching comparisons in the output blocks in Figure \ref{fig:coclExample}(f).
Four main pruning algorithms exist: (i) \textit{Weighted Edge Pruning} (\textsf{WEP}) removes all edges that do not exceed a specific threshold, e.g., the average edge weight \cite{DBLP:journals/tkde/PapadakisKPN14}; (ii) \textit{Cardinality Edge Pruning} (\textsf{CEP}) retains the globally $K$ top weighted edges, where $K$ is static \cite{DBLP:journals/tkde/PapadakisKPN14} or dynamic \cite{zhang2017pruning};
(iii)~\textit{Weighted Node Pruning} (\textsf{WNP}) retains in each node neighborhood the entities that exceed a local threshold, which may be the average edge weight of each neighborhood \cite{DBLP:journals/tkde/PapadakisKPN14}, or the average of the maximum weights in the two adjacent node neighborhoods, as in \textit{BLAST}
\cite{DBLP:journals/pvldb/SimoniniBJ16};
(iv) \textit{Cardinality Node Pruning} (\textsf{CNP}) retains the top-$k$ weighted edges in each node neighborhood \cite{DBLP:journals/tkde/PapadakisKPN14}.
\textit{Reciprocal WNP} and \textit{CNP} \cite{DBLP:conf/edbt/0001PPK16} apply an aggressive pruning that retains edges satisfying the pruning criteria
in both adjacent node neighborhoods. \textsf{WNP} and \textsf{WEP} are combined through the weighted sum of their thresholds in \cite{DBLP:conf/iscc/AraujoPN17}.
Another family of pruning algorithms is presented in \cite{nascimento2019exploiting}, focusing on the
edge weights between the entities in each block. \textit{Low Entity Co-occurrence Pruning} (\textsf{LECP}) cleans every block from a specific portion of the entities with the lowest average edge weights. \textit{Large Block Size Pruning} (\textsf{LBSP}) applies \textsf{LECP} only to the blocks whose size exceeds the average block size in the input block collection. \textit{Low Block Co-occurrence Pruning} (\textsf{LBCP}) removes every entity from the blocks, where it is connected with the lowest weights, on average, with the rest of the entities. \textit{CooSlicer} enforces a maximum block size constraint, $|b|_{max}$, to all input blocks. In blocks larger than $|b|_{max}$ all entities are sorted in decreasing order of average edge weight, and the $|b|_{max}$ top-ranked entities are iteratively placed into a new block. \textit{Low Block Co-occurrence Excluder} (\textsf{LBCE}) discards a specific portion of the blocks with the lowest average edge weight among their entities.
All these pruning algorithms can be coupled with any
\textit{edge weighting scheme} \cite{DBLP:journals/tkde/PapadakisKPN14}. \textsf{ARCS} sums the inverse cardinalities of the common blocks, giving higher weights to
entity pairs that co-occur in smaller blocks. \textsf{CBS} counts the number of blocks shared by two entities, as in Figure \ref{fig:coclExample}(c), with \textsf{ECBS} extending it to discount the contribution from entities placed in many blocks.
\textsf{JS} corresponds to the Jaccard coefficient of two block lists, while
\textsf{EJS} extends it to discount the contribution from entities appearing in many non-redundant comparisons.
Finally, Pearson's $\chi^2$
test assesses whether two adjacent entities appear independently in blocks and can be combined with
the aggregate attribute entropy associated with the tokens forming their common blocks~\cite{DBLP:journals/pvldb/SimoniniBJ16}.
Note that
Meta-blocking covers established methods that are
considered as Block Building methods in the literature:
given that Block Building is equivalent to indexing \cite{DBLP:journals/tkde/Christen12}, any method based on indexes is in fact a Meta-blocking technique.
For example, \textit{Transitive LSH} \cite{DBLP:conf/psd/SteortsVSF14}
converts the blocks extracted from \textsf{LSH} into an unweighted blocking graph and applies a community detection algorithm (e.g., \cite{clauset2004finding})
to partition the graph nodes into disjoint clusters, which will become the new blocks. The process finishes when the size of the largest cluster is lower than a predetermined threshold. This approach can be applied on top of any Block Building method, not just \textsf{LSH}.
The generalization principle also applies to \textit{Canopy Clustering} \cite{DBLP:conf/kdd/McCallumNU00}, which
places all entities in a pool and, in every iteration, it removes a random entity $e_i$ from the pool to create a new block. Using a cheap similarity measure, all entities still in the pool are compared with $e_i$. Those exceeding a threshold $t_{ex}$ are removed from the pool and placed into the new block. Entities exceeding another threshold $t_{in}$ ($< t_{ex}$) are also placed in the new block, without being removed from the pool.
As the cheap similarity measure, we can use any of the above weighting schemes on top of any Block Building method,
thus turning Canopy Clustering into a pruning algorithm for Meta-blocking.
\begin{figure}[t!]\centering
\includegraphics[width=0.99\linewidth]{comparisonCleaningExample.png}
\vspace{-8pt}
\caption{(a) A block collection $B$ with $e_1$$\equiv$$e_3$ and $e_2$$\equiv$$e_4$, (b) the corresponding Entity Index, (c) the corresponding blocking graph $G_B$, (d) the weighted $G_B$, (e) the pruned $G_B$, and (f) the new block collection $B'$.}
\vspace{-19pt}
\label{fig:coclExample}
\end{figure}
The generalization applies to \textit{Extended Canopy Clustering} \cite{DBLP:journals/tkde/Christen12,DBLP:journals/pvldb/0001APK15}, too,
which replaces the sensitive
weight thresholds
with cardinality ones:
for each randomly selected entity, the $n_1$ nearest entities are placed in its block, while the $n_2 (\leq n_1)$ nearest entities are removed from the pool.
On another line of research, \textit{SPAN} \cite{DBLP:conf/icde/ShuCXM11} converts a block collection into a matrix $M$, where the rows correspond to entities and the columns to the tf-idf of blocking keys (tokens or $q$-grams). Then, the entity-entity matrix is defined as $A=MM^T$. A spectral clustering algorithm converts $A$ into a binary tree, where the root node contains all entities and every leaf node is a disjoint subset of entities. The Newman-Girvan modularity is used as the stopping criterion for the bipartition of the tree. Blocks are then derived from a search procedure that carries out pairwise comparisons based on the blocking keys, inside the leaf nodes and across the neighboring ones.
Finally, the sole dynamic non-learning method
is \textit{Comparison Scheduling} \cite{DBLP:journals/tkde/PapadakisIPNN13}. Its goal is
to detect most matches upfront so as to maximize the superfluous comparisons that are skipped, due to the 1-1 restriction. It orders all comparisons in decreasing matching likelihood (edge weight) and executes a comparison only if none of the involved entities has already been matched.
\textbf{Learning-based Approaches.}
\textit{Supervised Meta-blocking} \cite{DBLP:journals/pvldb/0001PK14}
treats edge pruning
as a binary classification problem, where every edge is labelled "\texttt{likely match}" or "\texttt{unlikely match}".
Every edge is represented by a feature vector that comprises five features: \textsf{ARCS}, \textsf{ECBS}, \textsf{JS} and the Node Degrees of the adjacent entities. Undersampling is employed to tackle the class imbalance problem:
the training set comprises just 5\% of the minority class ("\texttt{likely match}") and an equal number of majority class instances. Several established classification algorithms are used for \textsf{WEP}, \textsf{CEP} and \textsf{CNP}, with all of them exhibiting robust performance with respect to their internal configuration.
\textit{BLOSS} \cite{DBLP:journals/is/BiancoGD18}
restricts the labelling cost of Supervised Meta-blocking by
carefully selecting a training set that is up to 40 times smaller, but retains the original performance. Using $ECBS$ weights, it partitions the unlabelled instances into similarity levels and applies rule-based active sampling inside every level. Then, it cleans the sample from non-matching outliers with high $JS$ weights.
\textbf{Parallelization Approaches.}
Meta-blocking is adapted to the MapReduce framework in three ways \cite{DBLP:journals/is/Efthymiou0PSP17}:
(i) The \textit{edge-based strategy} stores the blocking graph
on the disk, bearing a significant I/O cost.
(ii) The \textit{comparison-based strategy} builds the blocking graph \textit{implicitly}. A pre-processing job enriches every block with the list of block ids associated with every entity. The Map phase of the second job computes the edge weights and discards all redundant comparisons, while the ensuing Reduce phase prunes superfluous comparisons.
This strategy maximizes the efficiency of \textsf{WEP} and \textsf{CEP}
and is adapted to Apache Spark
in \cite{DBLP:conf/iscc/AraujoPN17}. (iii) The \textit{entity-based strategy}
aggregates for every entity the bag of all entities that co-occur with it in at least one block. Then, it estimates the edge weight that corresponds to each neighbor based on its frequency in the co-occurrence bag. This approach offers the best implementation for \textsf{WNP} and \textsf{CNP} and their variations (e.g., \textsf{BLAST}). It is adapted to Apache Spark in \cite{DBLP:journals/is/SimoniniGBJ19}, leveraging the broadcast join for higher efficiency.
To avoid the underutilization of the available resources, these strategies employ \textit{MaxBlock} \cite{DBLP:journals/is/Efthymiou0PSP17} for load balancing. Based on the highly skewed distribution of block sizes in redundancy-positive block collections, it splits the input blocks into partitions of equivalent computational cost, which is equal to the total number of comparisons in the largest input block.
The \textit{multi-core parallelization} of Meta-blocking is examined in \cite{DBLP:conf/i-semantics/0001BPK17}.
The input is transformed into
an array of chunks, with an index indicating the next chunk to be processed. Following the established fork-join model, every thread retrieves the current value of the index and is assigned to process the corresponding chunk. Depending on the definition of chunks, three alternative strategies are proposed: (i) \textit{Naive Parallelization} treats every entity as a separate chunk, ordering all entities in decreasing computational cost (i.e., the aggregate number of comparisons in the associated blocks). (ii) \textit{Partition Parallelization} uses MaxBlock to group the input entities into an arbitrary number of disjoint clusters with identical computational cost. (iii) \textit{Segment Parallelization}
sets the number of clusters equal to the number of available cores.
\subsection{Discussion \& Experimental Results}
\begin{figure}[t]\centering
\includegraphics[width=0.40\linewidth]{ccRelativePerformance.png}
\vspace{-8pt}
\caption{The relative performance of the main Comparison Cleaning methods.
}
\label{fig:relativePerformance}
\vspace{-14pt}
\end{figure}
The core characteristic of Block Processing methods is their
schema-agnostic functionality, which typically
relies
on block features, such as size, cardinality and overlap. This is no surprise, as they are primarily crafted for boosting the performance of schema-agnostic Block Building methods. In fact, extensive experiments demonstrate that Block Processing is indispensable for these methods, raising precision
by whole orders of magnitude, at a minor cost in
recall
\cite{DBLP:journals/pvldb/0001SGP16,DBLP:journals/bdr/PapadakisPPK16,DBLP:journals/tkde/PapadakisIPNN13}.
Regarding their relative performance, there is no clear winner among the Block Cleaning methods. For example,
both Block Filtering and Block Purging boost $PQ$ and $RR$ by orders of magnitude, while exhibiting a low computational cost and a negligible impact on $PC$ \cite{DBLP:journals/tkde/PapadakisIPNN13,DBLP:conf/edbt/0001PPK16}. However, the top performer among them depends not only on their parameter configuration, but also on the data at hand \cite{DBLP:journals/pvldb/0001SGP16}.
Most importantly, though, Block Cleaning techniques are usually complementary in the sense that multiple ones can be applied consecutively in a single blocking workflow, as depicted in Figure \ref{fig:computationalCostPlusWorkflow}(b). For example, Block Filtering is typically applied after Block Purging by lowering $r$ to $50\%$ instead of $80\%$, which is the best configuration when applied independently \cite{DBLP:journals/bdr/PapadakisPPK16,DBLP:conf/edbt/0001PPK16,DBLP:journals/pvldb/0001SGP16}.
In contrast, Comparison Cleaning methods are incompatible with each other in the sense that at most one of them can be part of a blocking workflow. The reason is that applying any Comparison Cleaning technique to a redundancy-positive block collection deprives it from its co-occurrence patterns and renders all other techniques inapplicable.
These techniques also involve a much higher computational cost than Block Cleaning methods, due to their finer level of granularity. Their relative performance is summarized in Figure \ref{fig:relativePerformance}, based on empirical evidence from experimental studies \cite{DBLP:journals/pvldb/0001SGP16} and individual publications \cite{DBLP:journals/is/BiancoGD18,DBLP:journals/pvldb/0001PK14,DBLP:journals/pvldb/SimoniniBJ16,DBLP:journals/bdr/PapadakisPPK16,DBLP:conf/edbt/0001PPK16,DBLP:journals/is/SimoniniGBJ19}. Note that we exclude methods not compared to other Comparison Cleaning techniques (e.g., the techniques presented in \cite{nascimento2019exploiting}).
In more detail, Figure \ref{fig:relativePerformance} maps the performance of the main Comparison Cleaning methods to a two dimensional space defined by $\Delta PC$=$PC(\mathcal{B}')-PC(\mathcal{B})$ on the vertical axis and $\Delta PQ$=$PQ(\mathcal{B}')-PQ(\mathcal{B})$ on the horizontal axis, where $B$ and $B'$ stand for the input and the output block collections, respectively. Given that Comparison Cleaning techniques trade lower recall ($PC$) for higher precision ($PQ$), $\Delta PC$ and $\Delta PQ$ take exclusively negative and positive values, respectively. Therefore, the higher a method is placed, the better recall it achieves, whereas the further to the right it lies, the better is its precision. This means that the ideal overall performance corresponds to the upper~right~corner.
We observe that $\Delta PC$ is delimited by two extremes: Comparison Cleaning on the top left corner and \textsf{CEP} on the bottom right corner. The former has no impact on recall, as it
increases precision only by removing redundant comparisons. All other Comparison Cleaning techniques discard superfluous comparisons, too, thus achieving larger $\Delta PQ$ at the cost of a negative $\Delta PC$. On the other extreme, \textsf{CEP} prunes a large portion of superfluous comparisons, yielding very high precision, but the lowest recall. \textsf{WEP} replaces \textsf{CEP}'s cardinality constraint with a weight threshold, dropping precision to a large extent for a significantly higher recall. Still, \textsf{WEP}'s performance is a major improvement over the input block collection, while being rather robust across numerous datasets. \textsf{WNP} moves further towards this direction, shrinking the decrease in recall and the increase in precision. This is further improved by \textit{Reciprocal WNP}, which significantly raises \textsf{WNP}'s precision for slightly lower recall. Thus, it dominates \textsf{WEP}, albeit being sensitive to the characteristics of the data at hand. Compared to \textsf{CEP}, \textsf{CNP} confines its pruning inside individual node neighborhoods. In this way, it achieves a much higher recall for a limited decrease in precision. This is further improved by \textit{Reciprocal CNP}, which reduces \textsf{CNP}'s recall slightly for much higher precision and, thus, it often dominates \textsf{CEP}.
\textsf{WNP}, \textsf{CNP} and their variants are improved by \textit{Supervised Meta-blocking} and \textsf{BLAST}, which achieve comparable recall for significantly higher precision. \textsf{BLAST} takes a lead in precision, partially because it employs the most effective weighting scheme, namely Pearson's $\chi^2$ test. Another advantage is that \textsf{BLAST} requires no labeling effort, due to its unsupervised functionality. \textsf{BLOSS}, however, achieves almost perfect recall ($\Delta PC \approx 0$) for the highest precision among all Comparison Cleaning techniques, while requiring merely $\sim$50 labeled instances. Note that exceptions to these general patterns of performance are possible for a particular dataset.
\section{Filtering}
\label{sec:filtering}
\begin{table*}[t]
\centering
\setlength{\tabcolsep}{3.5pt}
\caption{Overview of string and set similarity join methods.}
\label{tab:filtering_table}
\vspace{-5pt}
{\scriptsize
\begin{tabular}{| l || l | l | l | l |}
\hline
\textbf{Method} & \textbf{Operation} & \textbf{Similarity} & \textbf{Filters} & \textbf{Index} \\
\hline
\hline
GramCount~\cite{DBLP:conf/vldb/GravanoIJKMS01} & string join & Edit Distance & length, count, position & $q$-grams table \\
MergeOpt~\cite{DBLP:conf/sigmod/Sarawagi04} & set join & Overlap & count & inverted index \\
FastSS~\cite{BoHuSt07} & string join & Edit Distance & deletion neighborhood & dictionary \\
\hline
SSJoin~\cite{DBLP:conf/icde/ChaudhuriGK06} & set join & Overlap & prefix & DBMS \\
All-Pairs~\cite{DBLP:conf/www/BayardoMS07} & vector join & Cosine & prefix & inverted index \\
DivideSkip~\cite{DBLP:conf/icde/LiLL08} & string search & Edit Distance, Overlap & length, position, prefix & inverted index \\
Ed-Join~\cite{DBLP:journals/pvldb/XiaoWL08} & string join & Edit Distance & prefix+mismatching $q$-grams & inverted index \\
QChunk~\cite{DBLP:conf/sigmod/QinWLXL11} & string join & Edit Distance & prefix+$q$-chunks & inverted index \\
VChunkJoin~\cite{DBLP:journals/tkde/WangQXLS13} & string join & Edit Distance & prefix+chunks & inverted index \\
PPJoin~\cite{DBLP:conf/www/XiaoWLY08,DBLP:journals/tods/XiaoWLYW11} & set join & Overlap & prefix, positional & inverted index \\
PPJoin+~\cite{DBLP:conf/www/XiaoWLY08,DBLP:journals/tods/XiaoWLYW11} & set join & Overlap & prefix, positional, suffix & inverted index \\
MPJoin~\cite{DBLP:journals/is/RibeiroH11} & set join & Overlap & min-prefix & inverted index \\
GroupJoin~\cite{DBLP:journals/pvldb/BourosGM12} & set join & Overlap & prefix+grouping & inverted index \\
AdaptJoin~\cite{DBLP:conf/sigmod/WangLF12} & set join & Overlap & adaptive prefix & inverted index \\
SKJ~\cite{DBLP:journals/pvldb/WangQLZC17} & set join & Overlap & prefix-based+set relations & inverted index \\
TopkJoin~\cite{DBLP:conf/icde/XiaoWLS09} & top-$k$ set join & Overlap & prefix-based & inverted index \\
JOSIE~\cite{DBLP:conf/sigmod/ZhuDNM19} & top-$k$ set search & Overlap & prefix, position & inverted index \\
\hline
PartEnum~\cite{DBLP:conf/vldb/ArasuGK06} & set join & Hamming, Jaccard & partition-based & clustered index \\
PassJoin~\cite{DBLP:journals/pvldb/LiDWF11} & string join & Edit Distance & partition-based & inverted index \\
PTJ~\cite{DBLP:journals/pvldb/DengLWF15} & set join & Overlap & partition-based & inverted index \\
\hline
B$^{ed}-$Tree~\cite{DBLP:conf/sigmod/ZhangHOS10} & string search/join & Edit Distance & string orders & B$^+$-tree \\
PBI~\cite{DBLP:journals/tkde/LuDHO14} & string search & Edit Distance & reference strings & B$^+$-tree \\
MultiTree~\cite{DBLP:conf/icde/ZhangLWZXY17} & set search & Jaccard & tree traversal & B$^+$-tree \\
Trie-Join~\cite{DBLP:journals/pvldb/WangLF10} & string join & Edit Distance & subtrie pruning & trie \\
HSTree~\cite{DBLP:journals/vldb/YuWLZDF17} & string search & Edit Distance & partition-based & segment tree \\
Trans~\cite{zhang2018transformation} & top-$k$ set search & Jaccard & transformation distance & R-tree \\
\hline
\multicolumn{5}{c}{\textbf{(a) Exact, centralized, single predicate algorithms}}\\
\hline
FuzzyJoin~\cite{DBLP:conf/icde/AfratiSMPU12} & set/string join & Hamming, ED, Jaccard & ball-hashing, splitting, anchor points & lookup tables \\
VernicaJoin~\cite{DBLP:conf/sigmod/VernicaCL10} & set join & Overlap & prefix, positional, suffix & inverted index \\
MGJoin~\cite{DBLP:journals/tkde/RongLWDCT13} & set join & Overlap & multiple prefix & inverted index \\
MRGroupJoin~\cite{DBLP:journals/pvldb/DengLWF15} & set join & Overlap & partition-based & inverted index \\
FS-Join~\cite{DBLP:conf/icde/RongLSWLD17} & set join & Overlap & segment-based & inverted index \\
Dima~\cite{DBLP:journals/pvldb/SunSLDB17,DBLP:journals/pvldb/SunS0BD19} & search, join, top-$k$ & Jaccard, ED & segment-based & global \& local \\
\hline
\multicolumn{5}{c}{\textbf{(b) Parallel \& distributed algorithms}}\\
\hline
ATLAS~\cite{DBLP:conf/sigmod/ZhaiLG11} & vector join & Jaccard, Cosine & random permutations & inverted index \\
BayesLSH~\cite{DBLP:journals/pvldb/SatuluriP12} & set join & Jaccard, Cosine & All-Pairs / LSH & All-Pairs / LSH \\
CPSJoin~\cite{DBLP:conf/icde/ChristianiPS18} & set join & Jaccard & LSH-based & sketches \\
\hline
\multicolumn{5}{c}{\textbf{(c) Approximate algorithms}}\\
\hline
LS-Join~\cite{DBLP:journals/tkde/WangYWL17} & local string join & Edit Distance & length, count & inverted index \\
pkwise~\cite{DBLP:conf/sigmod/WangXQWZI16} & local set join & Overlap & $k$-wise signatures & inverted index \\
pkduck~\cite{DBLP:journals/pvldb/TaoDS17} & abbreviation matching & Custom & extension of prefix filter & trie \\
\hline
Fast-Join~\cite{DBLP:journals/tods/WangLF14} & fuzzy set join & Bipart. graph matching & token sensitive signatures & inverted index \\
SilkMoth~\cite{DBLP:journals/pvldb/DengKMS17} & fuzzy set join & Bipart. graph matching & weighted token signatures & inverted index \\
MF-Join~\cite{DBLP:conf/icde/WangLZ19} & fuzzy set join & Bipart. graph matching & partion-based & inverted index \\
\hline
MultiAttr~\cite{DBLP:conf/sigmod/LiHDL15} & set search/join & Overlap & tree traversal & prefix tree \\
Smurf~\cite{DBLP:journals/pvldb/CADA18} & string matching & Jaccard, Edit Distance & random forest & inverted indexes \\
AU-Join~\cite{DBLP:journals/pvldb/XuL19} & string join & Syntactic, Synonym, Taxonomy & pebbles & inverted indexes \\
\hline
\multicolumn{5}{c}{\textbf{(d) Algorithms for complex matching}}\\
\end{tabular}
\vspace{-15pt}
}
\end{table*}
Given specific similarity predicates, comprising a similarity measure and a corresponding threshold, Filtering techniques receive as input an entity or a block collection and produce as output pairs of entities satisfying these predicates.
Next, we present the main filtering methods in the literature, organized in four groups: basic filters proposed by earlier works; prefix filtering and its extensions; partition-based filtering; and methods using tree indexes.
An overview of the discussed methods is presented in Table~\ref{tab:filtering_table}, characterized by the type of operation they perform (e.g., search or join), the similarity measure they assume (e.g., token- or character-based), the type of filters they use (e.g., prefix- or partition-based) and the index structure they employ (e.g., inverted index or tree).
\textbf{Basic filtering.}
\texttt{GramCount}~\cite{DBLP:conf/vldb/GravanoIJKMS01} focuses on incorporating string similarity joins inside a
DBMS based on $q$-grams and edit distance. It is the first work to propose
the following techniques:
\textit{Length filtering} states that if two strings $r$ and $s$ are within edit distance $\theta$, their lengths cannot differ by more than $\theta$. In the case of set similarity joins, the length filter has been adapted to deal with set sizes~\cite{DBLP:conf/vldb/ArasuGK06}; e.g., for Jaccard similarity threshold $\theta$, the condition becomes $\theta \cdot |s| \leq |r| \leq |s| / \theta$. Length filtering is a simple but effective criterion that is employed by many other works alongside more advanced filters.
A \textit{position-enhanced} length filter
offers a tighter upper bound
\cite{DBLP:conf/gvd/MannA14}.
\textit{Count filtering} states that if two strings $r$ and $s$ are within edit distance $\theta$, they must have at least $max(|r|, |s|) - 1 - (\theta - 1) \cdot q$ common $q$-grams. This filter has also been adapted to sets, in particular in \texttt{MergeOpt}~\cite{DBLP:conf/sigmod/Sarawagi04}, which proposed various optimizations for applying count filtering with both character-based and token-based similarity measures, and in \texttt{DivideSkip}~\cite{DBLP:conf/icde/LiLL08}, which proposed efficient techniques for merging the inverted lists of signatures.
\textit{Position filtering} also considers the positions of $q$-grams in the strings. It states that if two strings $r$ and $s$ are within edit distance $\theta$, a positional $q$-gram in one cannot correspond to a positional $q$-gram in the other that differs from it by more than $\theta$ positions.
On another line of research, \texttt{FastSS}~\cite{BoHuSt07}
introduces the concept of \textit{deletion neighborhood}, a filtering criterion specifically tailored to edit distance. For a string $s$, its deletion neighborhood contains substrings of $s$ derived by deleting a certain number of characters. These are then used as signatures for filtering. However, this method is practical only for very short strings.
\textbf{Prefix-based filtering.}
\textit{Prefix filtering} has been proposed by \texttt{SSJoin}~\cite{DBLP:conf/icde/ChaudhuriGK06}, which focuses on
similarity joins inside a DBMS,
and \texttt{All-Pairs}~\cite{DBLP:conf/www/BayardoMS07},
which is a main memory algorithm. Prefix filter applies to sets and can also be used for strings represented as sets of $q$-grams. The elements of each set are first sorted in a global order, typically in increasing order of frequency. Then, the $\pi$-prefix of each set is formed by selecting its $\pi$ first elements in that order. Prefix filter states that for two sets to be similar, their prefixes must contain at least one common element. The prefix size $\pi$ of a set $r$ is determined based on the similarity measure and threshold being used; e.g., for edit distance threshold $\theta$, $\pi = q \cdot \theta + 1$, while for Jaccard similarity threshold $\theta$, $\pi = \lfloor (1 - \theta) \cdot |r| \rfloor + 1$. As described next, numerous subsequent algorithms have adopted prefix filtering and proposed various optimizations and extensions over it, both for edit distance and set-based similarity joins.
For edit distance, \texttt{DivideSkip}~\cite{DBLP:conf/icde/LiLL08} uses prefix filtering in combination with length and position filtering,
taking special care to efficiently merge the inverted lists of signatures. \texttt{Ed-Join}~\cite{DBLP:journals/pvldb/XiaoWL08} proposes two optimizations
based on analyzing the locations and contents of mismatching $q$-grams to further reduce the prefix length by removing unnecessary elements. \texttt{QChunk}~\cite{DBLP:conf/sigmod/QinWLXL11} introduces the concept of \textit{$q$-chunks}, which are substrings of length $q$ that start at 1+$i$$\cdot$$q$ positions in the string, for $i \in [0, (|r|-1)/q]$. Given two strings $r$ and $s$, QChunk extracts $q$-grams from the one and $q$-chunks from the other;
if $r$ and $s$ are within edit distance $\theta$, the size of the intersection between the $q$-grams of $r$ and the $q$-chunks of $s$ should be at least $\lceil|s|/q\rceil$-$\theta$.
\texttt{VChunkJoin}~\cite{DBLP:journals/tkde/WangQXLS13} uses non-overlapping substrings called \textit{chunks}, ensuring that each edit operation destroys at most two chunks. This yields a tight lower bound on the number of common chunks that two strings must share if they match.
For set similarity joins,
\texttt{PPJoin}~\cite{DBLP:conf/www/XiaoWLY08,DBLP:journals/tods/XiaoWLYW11} extends prefix filtering with \textit{positional filtering}. This takes also into consideration the positions where the common tokens in the prefix occur, thus deriving a tighter upper bound for the overlap between the two sets. In addition, \texttt{PPJoin}+~\cite{DBLP:conf/www/XiaoWLY08,DBLP:journals/tods/XiaoWLYW11} further uses \textit{suffix filtering}. Following a divide-and-conquer strategy, this partitions the suffix of the one set into two subsets of similar sizes. The token separating the two partitions is called \textit{pivot} and is used to split the suffix of the other set. This allows to calculate the maximum number of tokens in each pair of corresponding partitions between the two sets that can match.
\texttt{MPJoin}~\cite{DBLP:journals/is/RibeiroH11} adds a further optimization over \texttt{PPJoin} that allows for dynamically pruning the length of the inverted lists. This reduces the computational cost of candidate generation, rather than the number of candidates. \texttt{GroupJoin}~\cite{DBLP:journals/pvldb/BourosGM12} extended \texttt{PPJoin} with \textit{group filtering}, whose candidate generation
treats all sets with identical prefixes as a single set.
Multiple candidates may thus be pruned in batches. \texttt{AdaptJoin}~\cite{DBLP:conf/sigmod/WangLF12} proposed \textit{adaptive prefix filtering}, which generalizes prefix filtering by adaptively selecting an appropriate prefix length for each set. It supports longer prefixes dynamically, extending their length by $n-1$, and then prunes a pair of sets if they contain less than $n$ common tokens in their extended prefixes. Prefix filtering is a special case where $n=1$.
A different perspective for speeding up set similarity joins is proposed by \texttt{SKJ}~\cite{DBLP:journals/pvldb/WangQLZC17}. The idea is based on the following observation: existing approaches examine each set individually when computing the join; however, it is possible to improve efficiency through computational cost sharing between \textit{related sets}. To this end, the \texttt{SKJ} algorithm introduces \textit{index-level skipping}, which groups related sets in the index into blocks, and \textit{answer-level skipping}, which incrementally generates the answer of one set from an already computed answer of another related set.
Finally, there are Filtering techniques
for computing top-$k$ results progressively, instead of requiring the user to select a similarity threshold.
\texttt{TopkJoin}~\cite{DBLP:conf/icde/XiaoWLS09} retrieves the top-$k$ pairs of sets ranked by their similarity score,
based on prefix filtering and
on the monotonicity of maximum possible scores of unseen pairs.
\texttt{JOSIE}~\cite{DBLP:conf/sigmod/ZhuDNM19}
presents a method for top-$k$ set similarity search. It exploits prefix and position filtering but,
instead of dealing with sets of relatively small size (e.g., $\sim$100 tokens), it
is crafted for
finding joinable tables in data lakes, where sets represent the distinct values of a table column,
comprising millions of tokens.
This introduces new challenges, which are tackled by proposing an algorithm that minimizes the cost of set reads and inverted index probes.
\textbf{Partition-based filtering.} The algorithms in this category partition each string or set into multiple disjoint segments in such a way that matching pairs have at least one common segment.
\texttt{PartEnum}~\cite{DBLP:conf/vldb/ArasuGK06} generates a signature scheme based on the principles of \textit{partitioning} and \textit{enumeration}. The former states that if two vectors with Hamming distance not higher than $k$ are partitioned into $k$ + 1 equi-sized partitions, then they must have at least one common partition. The latter states that if these vectors are partitioned instead into $n > k$ equi-sized partitions, then they must have in common at least $n - k$ partitions. \texttt{PassJoin}~\cite{DBLP:journals/pvldb/LiDWF11} partitions a string into a set of segments and creates inverted indices for the segments. Then, for each string, it selects some of its substrings and uses them to retrieve candidates from the index. A method is proposed to minimize the number of segments required to find the candidates pairs. \texttt{PTJ}~\cite{DBLP:journals/pvldb/DengLWF15} proposes an approach to increase the pruning power of partition-based filtering by using a mixture of the subsets and their 1-deletion neighborhoods, which are subsets derived from a set after eliminating one element.
Essentially, these methods are based on the \textit{pigeonhole principle}, which states that if $n$ items are contained in $m$ boxes,
at least one box has no more than $\lfloor n / m \rfloor$ items. This is extended by
the \textit{pigeonring principle}
\cite{DBLP:journals/pvldb/QinX18}, which organizes the boxes in a ring and constrains the number of items in multiple boxes rather than a single one, thus offering tighter bounds.
Applying it to various similarity search problems
shows that pigeonring always produces less or equal number of candidates than the pigeonhole principle does and that pigeonring-based algorithms can be implemented on top of existing pigeonhole-based ones with minor modifications \cite{DBLP:journals/pvldb/QinX18}.
\textbf{Tree-based filtering.} Most methods presented so far build inverted indexes on the signatures extracted from the strings or sets. Next, we present algorithms employing tree-based indexes.
Most approaches are based on the B$^+$-tree. \texttt{B}$^{ed}$-\texttt{Tree}~\cite{DBLP:conf/sigmod/ZhangHOS10} proposes a B$^+$-tree based index for range and top-$k$ similarity queries as well as similarity joins, using edit distance. It is based on a mapping from the string space to the integer space to support efficient searching and pruning. \texttt{PBI}~\cite{DBLP:journals/tkde/LuDHO14} uses a B$^+$-tree index and exploits the fact that edit distance is a metric. The string collection is partitioned according to a set of selected \textit{reference strings}. Then, the strings in each partition are indexed based on their distances to their corresponding reference strings. The proposed approach supports both range and $k$-NN queries and can be integrated inside a DBMS. In \texttt{MultiTree}~\cite{DBLP:conf/icde/ZhangLWZXY17}, each element in a set is represented as a vector and is mapped to an integer according to a defined global ordering, which is then used to insert the element in the B$^+$-tree index. Searching for similar elements is then done via a range query on the index.
On another line of research, \texttt{Trie-Join}~\cite{DBLP:journals/pvldb/WangLF10}
proposes a trie-based technique for string similarity joins with edit distance. Each trie node represents a character in the string. Thus, strings with a common prefix share the same ancestors.
A trie node is called an \textit{active node} of a string $s$ if their edit distance is not larger than the given threshold. This leads to a technique called \textit{subtrie pruning}:
given a trie $T$ and a string $s$, if node $n$ is not an active node for every prefix of $s$, then $n$'s descendants cannot be similar to $s$. \texttt{HSTree}~\cite{DBLP:journals/vldb/YuWLZDF17} recursively partitions strings into disjoint segments and builds a hierarchical segment tree index. This is then used to support both threshold-based and top-$k$ string similarity
search based on edit distance. Finally, a transformation-based framework for top-$k$ set similarity search is presented in~\cite{zhang2018transformation}. It transforms sets of various lengths into fixed-length vectors in such a way that similar sets are mapped closer to each other. An R-tree is then used to index these records and prune the space during search.
\subsection{Parallel \& Distributed Algorithms}
\label{subsec:filtering_distributed}
MapReduce-based approaches have been proposed to tackle scalability issues when dealing with very large collections of sets or strings. A theoretical analysis of different methods for performing similarity joins on MapReduce is presented in~\cite{DBLP:conf/icde/AfratiSMPU12}.
It considers algorithms that
operate in a single MapReduce job, avoiding the overhead associated with initiating multiple ones.
It shows that different algorithms provide different tradeoffs with respect to map, reduce and
communication~cost.
\texttt{VernicaJoin}~\cite{DBLP:conf/sigmod/VernicaCL10} is based on prefix filtering. It computes prefix tokens and builds an inverted index on them. Then, it generates candidate pairs from the inverted lists, using additionally the length, positional and suffix filters to prune candidates. A deduplication step is finally employed to remove duplicate result pairs generated from different reducers. \texttt{MGJoin}~\cite{DBLP:journals/tkde/RongLWDCT13} follows a similar approach to \texttt{VernicaJoin}, but introduces multiple prefix orders and a load balancing technique that partitions sets based on their length. \texttt{MRGroupJoin}~\cite{DBLP:journals/pvldb/DengLWF15} is a MapReduce extension of \texttt{PTJ}~\cite{DBLP:journals/pvldb/DengLWF15}. It applies a partition-based technique, where records are grouped by length and are partitioned in subrecords, such that matching records share at least one subrecord. The process is performed in a single MapReduce job. \texttt{FS-Join}~\cite{DBLP:conf/icde/RongLSWLD17} sorts the tokens in each set in increasing order of frequency, and then splits each set into disjoint subsets using appropriate pivot tokens. These subsets are then grouped together so that subsets from different groups are non-overlapping.
Finally, \texttt{Dima}~\cite{DBLP:journals/pvldb/SunSLDB17,DBLP:journals/pvldb/SunS0BD19} is a distributed in-memory system built on top of Spark that supports threshold and top-$k$ similarity search and join with both token-based and character-based similarities. It relies on signature-based global and local indexes for efficiency. The proposed signatures are adaptively selectable based on the workload, which allows to balance the workload among partitions. \texttt{Dima} extends the Catalyst optimizer of Spark SQL to introduce cost-based optimizations.
\subsection{Approximate Algorithms}
\label{subsec:filtering_approx}
Approximate algorithms for similarity search and join
increase the efficiency of Filtering step at the cost of allowing both false positives and false negatives, thus missing some matches
\cite{DBLP:journals/corr/WangSSJ14,DBLP:series/synthesis/2013Augsten}. They typically rely on \textit{locality sensitive hashing} (\textsf{LSH})~\cite{DBLP:conf/vldb/GionisIM99}, which transforms an item to a low-dimensional representation such that similar items have much higher probability to be mapped to the same hash code than dissimilar ones. This property allows \textsf{LSH} to be exploited in the filtering phase to generate candidates~\cite{DBLP:conf/vldb/LvJWCL07,DBLP:conf/sigmod/TaoYSK09,DBLP:conf/compgeom/DatarIIM04}. The basic idea is that each object is hashed several times using randomly chosen hash functions. Then, candidates are those pairs of objects that have been hashed to the same code by at least one hash function.
\texttt{ATLAS}~\cite{DBLP:conf/sigmod/ZhaiLG11} is a probabilistic algorithm that is based on random permutations both to generate candidates and to estimate the similarity between candidate pairs. It also proposes a method to efficiently detect cluster structures within the data, which are then exploited to search for similar pairs only within each cluster.
\texttt{BayesLSH}~\cite{DBLP:journals/pvldb/SatuluriP12} combines Bayesian inference with \textsf{LSH} to estimate similarities to a user-specified level of accuracy. It uses \textsf{LSH}
for both Filtering
and Verification,
providing probabilistic guarantees on the resulting
accuracy and recall. \texttt{CPSJoin}~\cite{DBLP:conf/icde/ChristianiPS18} is a randomized algorithm for set similarity joins. It uses a recursive filtering technique, building upon a previously proposed index for set similarity search~\cite{DBLP:conf/stoc/ChristianiP17}, as well as sketches for estimating set similarity. The algorithm has 100\% precision and provides a probabilistic guarantee on recall.
\subsection{Algorithms for Complex Matching}
\label{subsec:filtering_advanced}
The works discussed so far assume a single similarity predicate, i.e, they apply to the values of a specific attribute. Moreover, when comparing sets, they assume binary matching between their
elements,
while in the case of strings,
they compare strings in their entirety. In the following, we present methods that employ \textit{multiple} similarity predicates or more complex ones.
\textbf{Local matching.}
A local string similarity join
finds pairs of strings that contain similar \textit{substrings}. Under edit distance constraints, it can be defined as matching any $l$-length substring with up to $k$ errors. \texttt{LS-Join}~\cite{DBLP:journals/tkde/WangYWL17} is based on the observation that if two strings are locally similar, they must share at least one common $q$-gram, for a suitably calculated gram length $q$. An inverted index is constructed incrementally during the search. For every examined string, its $q$-grams are generated and the candidates are retrieved from the index by finding those strings that have matching $q$-grams.
\texttt{pkwise}~\cite{DBLP:conf/sigmod/WangXQWZI16} detects pieces of text in a given collection that share similar \emph{sliding windows}, i.e., multisets containing $w$ consecutive tokens of a given document. The similarity of two sliding windows is defined as the overlap of those sets. Prefix filtering is used but instead of relying on single tokens to build the signatures, it proposes \emph{$k$-wise signatures}, which comprise combinations of $k$ tokens. Larger values of $k$ increase the signatures' selectivity but also the cost of signature generation.
An additional optimization is to share common signatures across adjacent windows.
Finally, \texttt{pkduck}~\cite{DBLP:journals/pvldb/TaoDS17}
matches strings with \textit{abbreviations},
based on a new similarity measure
that accounts for
abbreviations.
It also proposes an appropriate signature scheme that extends prefix filtering and generates signatures without iterating over all strings derived from an
abbreviation.
\textbf{Fuzzy matching.} Rather than assuming a binary match, in this setting, the similarity between the elements of two sets may take any value between 0 and 1. In fact, it is defined
as the maximum matching score in the bipartite graph representing the matches between their elements.
In \texttt{Fast}-\texttt{Join}~\cite{DBLP:journals/tods/WangLF14}, edge weights in this bipartite graph denote the edit similarities between matching elements. The proposed method follows the filter-verification framework, creating a signature for each set such that matching sets have overlapping signatures. The signature of a set comprises an appropriately selected subset of its tokens. \texttt{SilkMoth}~\cite{DBLP:journals/pvldb/DengKMS17} generalizes and improves upon this work, providing a formal characterization of the space of valid signatures.
Given
that finding the optimal signature is NP-complete, it proposes heuristics to select signatures. To further reduce candidates, a refinement step is added: it compares each set with its candidates and rejects those for which certain bounds do not hold.
Both edit distance and Jaccard coefficient are supported
for measuring the similarity between elements. \texttt{MF-Join}~\cite{DBLP:conf/icde/WangLZ19} performs element-~and~record-level filtering. The former utilizes a partition-based signature scheme with a frequency-aware partition strategy, while the latter exploits count filtering and an upper bound on record-level similarity.
\textbf{Multiple predicates.} A method for similarity search and join on \textit{multi-attribute} data is presented in~\cite{DBLP:conf/sigmod/LiHDL15}. For instance, given an entity collection
where each entity is described
by its name and address, this work identifies pairs of entities having \textit{both} similar names and similar addresses. To enable simultaneous filtering on multiple attributes, a combined prefix tree index is built on these attributes. The construction of the index is guided by a cost model and a greedy algorithm. In another direction, \texttt{Smurf}~\cite{DBLP:journals/pvldb/CADA18} performs string matching between two collections of strings based on multiple-predicate matching conditions in the form of a \textit{random forest} classifier that is learned via active learning. Filtering techniques for string similarity joins are exploited to speed up the execution of the random forest. The focus and novelty of this work is on how to reuse computations across the trees in the forest to further increase efficiency. Finally, \texttt{AU}-\texttt{Join}~\cite{DBLP:journals/pvldb/XuL19} presents a new framework for string similarity joins that supports not only syntactic similarity measures, such as Jaccard similarity on $q$-grams, but also \textit{semantic} similarities, including \textit{synonym-based} and \textit{taxonomy-based} matching. It
partitions strings into segments and applies
different types of similarity measures on different pairs of segments. A new signature scheme, called \textit{pebble},
handles multiple similarity measures:
pebbles are $q$-grams for gram-based similarity,
the left-hand side of a synonym rule for synonym similarity, and
ancestor nodes in the taxonomy for taxonomy similarity.
\vspace{-10pt}
\subsection{Discussion \& Experimental Results}
\label{subsec:filtering_discussion}
Filtering techniques for string and set similarity joins have attracted a lot of research interest over the past two decades. Early works
view this operation as an extension of the standard join operator in relational databases, where the join condition is based on similarity rather than equality~\cite{DBLP:conf/vldb/GravanoIJKMS01,DBLP:conf/icde/ChaudhuriGK06}. The same perspective is shared by more recent works, like those
proposing B$^+$-tree based indexes, which
can be easily integrated into an existing DBMS
\cite{DBLP:conf/sigmod/ZhangHOS10,DBLP:journals/tkde/LuDHO14,DBLP:conf/icde/ZhangLWZXY17}. Another characteristic example is Dima~\cite{DBLP:journals/pvldb/SunSLDB17,DBLP:journals/pvldb/SunS0BD19}, which extends the Catalyst optimizer of Spark SQL to
support
similarity-based queries. In this sense, similarity joins are sometimes referred to as \textit{approximate} or \textit{fuzzy} joins, although this should not be confused with the approximate algorithms in Sec.~\ref{subsec:filtering_approx}, or the fuzzy set joins in Sec.~\ref{subsec:filtering_advanced}. Numerous Filtering techniques have been proposed by more recent works, which focus on main memory execution.
\textit{Prefix-based} filtering is the most popular
approach~\cite{DBLP:conf/icde/ChaudhuriGK06,DBLP:conf/www/BayardoMS07,DBLP:conf/icde/LiLL08,DBLP:journals/pvldb/XiaoWL08,DBLP:conf/sigmod/QinWLXL11,DBLP:journals/tkde/WangQXLS13,DBLP:conf/www/XiaoWLY08,DBLP:journals/tods/XiaoWLYW11,DBLP:conf/www/XiaoWLY08,DBLP:journals/tods/XiaoWLYW11,DBLP:journals/is/RibeiroH11,DBLP:journals/pvldb/BourosGM12,DBLP:conf/sigmod/WangLF12},
followed by
\textit{partition-based} filtering
\cite{DBLP:conf/vldb/ArasuGK06,DBLP:journals/pvldb/LiDWF11,DBLP:journals/pvldb/DengLWF15}. Furthermore, to scale similarity joins to large collections, distributed~\cite{DBLP:conf/icde/AfratiSMPU12,DBLP:conf/sigmod/VernicaCL10,DBLP:journals/tkde/RongLWDCT13,DBLP:journals/pvldb/DengLWF15,DBLP:conf/icde/RongLSWLD17,DBLP:journals/pvldb/SunSLDB17,DBLP:journals/pvldb/SunS0BD19} and approximate~\cite{DBLP:conf/sigmod/ZhaiLG11,DBLP:journals/pvldb/SatuluriP12,DBLP:conf/icde/ChristianiPS18} algorithms have been proposed.
More recently, there has been an increasing focus and interest on works that deal with more complex similarity predicates. These include the matching of strings based on \textit{substrings} or \textit{abbreviations}~\cite{DBLP:journals/tkde/WangYWL17,DBLP:conf/sigmod/WangXQWZI16,DBLP:journals/pvldb/TaoDS17}, matching of sets based on \textit{fuzzy matching} of their elements~\cite{DBLP:journals/tods/WangLF14,DBLP:journals/pvldb/DengKMS17,DBLP:conf/icde/WangLZ19}, and the combination of \textit{multiple} similarity predicates~\cite{DBLP:conf/sigmod/LiHDL15,DBLP:journals/pvldb/CADA18}. These works can be considered as more closely relevant to matching entity profiles in Entity Resolution.
Regarding performance,
a series of experimental analyses provides interesting insights \cite{DBLP:journals/pvldb/JiangLFL14,DBLP:journals/pvldb/MannAB16,DBLP:journals/pvldb/FierABLF18}.
However, each study focuses on a certain subset of the aforementioned methods.
Below, we
briefly summarize their findings,
including
additional results from individual papers to fill the gaps.
\textit{Similarity joins using Edit Distance}. A comparison between \texttt{FastSS}, \texttt{All-Pairs}, \texttt{DivideSkip}, \texttt{Ed-Join}, \texttt{QChunk}, \texttt{VChunkJoin}, \texttt{PPJoin}, \texttt{PPJoin}+, \texttt{AdaptJoin}, \texttt{PartEnum}, \texttt{PassJoin} and \texttt{Trie}-\texttt{Join} is conducted in~\cite{DBLP:journals/pvldb/JiangLFL14}. The results demonstrate that \texttt{PassJoin} is the most efficient
algorithm, with \texttt{FastSS} providing a reliable alternative
in the case of very short strings.
\textit{Similarity joins using set-based measures}. \texttt{AdaptJoin} and \texttt{PPJoin}+ are reported as the best algorithms in the aforementioned study~\cite{DBLP:journals/pvldb/JiangLFL14}. Different results, though, are reported in
a subsequent
study
that compares \texttt{All-Pairs}, \texttt{PPJoin}, \texttt{PPJoin}+, \texttt{MPJoin}, \texttt{AdaptJoin} and \texttt{GroupJoin}.
It indicates that the plain prefix filtering, i.e., \texttt{All-Pairs}, is still quite competitive,
winning in the majority of cases.
\texttt{PPJoin} and \texttt{GroupJoin}
exhibit
the best median and average performance, respectively, while
more sophisticated filters are found to provide only moderate improvements in some cases or even to negatively affect performance.
The difference with the
results
in \cite{DBLP:journals/pvldb/JiangLFL14} is attributed to the
more efficient
verification step;
reducing the cost of Verification means that
complex and, thus, time-consuming filters
often do not pay off, despite
reducing the number of candidate pairs.
\textit{Prefix vs. partition filtering}. \texttt{PTJ} is compared against \texttt{PPJoin+} and \texttt{AdaptJoin} in \cite{DBLP:journals/pvldb/DengLWF15}, showing that it outperforms both methods.
The same comparison is performed in \cite{DBLP:journals/pvldb/WangQLZC17}, showing that \texttt{PTJ} does not outperform those methods in most cases. As noted in~\cite{DBLP:journals/pvldb/WangQLZC17}, this discrepancy seems to be
caused by
differences in implementation; specifically, the comparison in~\cite{DBLP:journals/pvldb/DengLWF15}
uses the original implementations of \texttt{PPJoin+} and \texttt{AdaptJoin}, while the one in~\cite{DBLP:journals/pvldb/WangQLZC17} uses the optimized implementations provided by~\cite{DBLP:journals/pvldb/MannAB16}. Overall, \texttt{PTJ} may generate fewer candidates, but uses complex index structures, thus spending much more time on the filtering phase compared to prefix-based
algorithms. Another factor that affects the performance of prefix filtering is the frequency distribution of the tokens in the dataset. The core idea of prefix filtering is to select rare tokens as signatures
so as to reduce the number of candidates. In~\cite{DBLP:journals/pvldb/JiangLFL14}, an experiment
involving different dataset distributions shows that
\texttt{PPJoin(+)} and \texttt{AdaptJoin} perform better in datasets with Zipfian distribution than uniform one.
\textit{Set relations}. Another interesting finding is that set relations can be effectively exploited to speed up the computation of similarity joins~\cite{DBLP:journals/pvldb/WangQLZC17}. In the presented experiments, the proposed algorithm, \texttt{SKJ},
consistently outperforms \texttt{PPJoin}, \texttt{PPJoin+}, \texttt{AdaptJoin} and \texttt{PTJ} across all datasets.
\textit{Tree-based algorithms}. These algorithms typically focus on similarity search rather than join. \texttt{HSTree} and \texttt{PBI} are compared against \texttt{B}$^{ed}$-\texttt{Tree} in~\cite{DBLP:journals/vldb/YuWLZDF17} and~\cite{DBLP:journals/tkde/LuDHO14}, respectively, reporting better performance. Also, \texttt{Trans} shows better performance than \texttt{MultiTree} in~\cite{zhang2018transformation}. \texttt{BiTrieJoin}, an improved variant of \texttt{TrieJoin},
is reported in~\cite{DBLP:journals/pvldb/JiangLFL14} to have comparable performance to \texttt{PassJoin} for short strings, but it underperforms
for medium and long strings.
\textit{Distributed algorithms}. \texttt{VernicaJoin}, \texttt{MGJoin}, \texttt{MRGroupJoin} and \texttt{FS}-\texttt{Join} are experimentally compared in~\cite{DBLP:journals/pvldb/FierABLF18}.
\texttt{VernicaJoin} exhibits the best performance
in most cases, but
all algorithms are often outperformed by non-distributed ones. This should be attributed
to the overhead introduced by the MapReduce framework
as well as
to high or skewed data replication between map and reduce tasks. The latter constitutes
an inherent limitation of the distributed algorithms that cannot be overcome by simply increasing the number of nodes in the cluster. In~\cite{DBLP:journals/pvldb/SunS0BD19}, \texttt{Dima} is shown to outperform
the adaptation of \texttt{VernicaJoin}
to Apache Spark.
\textit{Approximate algorithms}. The experimental survey in~\cite{DBLP:journals/pvldb/JiangLFL14} included a comparison between \texttt{BayesLSH}-\texttt{lite} and exact algorithms. Moreover, \texttt{ATLAS}, \texttt{BayesLSH} and \texttt{CPSJoin} have been compared against \texttt{All-Pairs} in~\cite{DBLP:conf/sigmod/ZhaiLG11}, \cite{DBLP:journals/pvldb/SatuluriP12} and \cite{DBLP:conf/icde/ChristianiPS18}, respectively. Overall, the experiments indicate that approximate algorithms are preferable for low similarity thresholds, e.g., for Jaccard similarity below 0.5, while exact algorithms perform better for high thresholds.
\section{Join-based Blocking Methods}
\label{sec:hybrid}
\begin{figure}[t]\centering
\includegraphics[width=0.46\linewidth]{hybridTaxonomy.png}
\includegraphics[width=0.46\linewidth]{TimelineV2.png}
\vspace{-8pt}
\caption{(a) The taxonomy of the hybrid, join-based blocking methods. (b) Timeline of the landmarks in the evolution of {\color{blue}Blocking}, {\color{red}Filtering} and {\color{purple}their convergence}.
}
\label{fig:taxonomyTimeline}
\vspace{-14pt}
\end{figure}
We now elaborate on Block Building
methods that incorporate Filtering techniques,
converting Blocking into a nearest neighbor search. As illustrated in Figure \ref{fig:taxonomyTimeline}(a), we categorize these hybrid techniques into three major categories according to the filtering techniques they employ: the \textit{lossless} ones rely on exact, single predicate filtering techniques (cf. Table \ref{tab:filtering_table}(a)), the \textit{lossy} ones
rely on approximate filtering (cf. Section \ref{subsec:filtering_approx}), while the \textit{spatial} ones
leverage spatial join techniques for filtering. Note that the lossy hybrid methods are further distinguished into \textit{static} and \textit{dynamic} ones, depending on whether they are independent or interwoven with Matching, respectively.
Starting with the lossless hybrid methods, the simplest approach is to combine Prefix Filtering with Token Blocking, creating one block for every token that appears in the prefix of at least two entities \cite{DBLP:series/synthesis/2015Christophides}. Another approach is \textit{Adaptive Filtering} \cite{DBLP:conf/sdm/GuB04}, which couples schema-aware, non-learning Block Building techniques with two filtering methods. First, blocks are created by extracting keys from specific attributes. In every block with a size exceeding a predetermined threshold, Length and Count Filtering are applied for Comparison Cleaning, using an edit distance threshold on an attribute that is not considered by the initial transformation function.
Another lossless hybrid method is \textit{LIMES}, which
operates only on metric spaces \cite{DBLP:conf/ijcai/NgomoA11}. Its core idea is to
leverage the triangle inequality to approximate the distance between entities based on previous comparisons. Utilizing sets of entities as reference points, called \textit{exemplars}, this method computes lower and upper bounds to filter out superfluous comparisons before their execution.
In another direction, \textit{MultiBlock} \cite{isele2011efficient}
optimizes the execution of complex matching rules that comprise special similarity functions for textual, geographic and numeric values. A block collection is created for every similarity function such that similar entities
share multiple blocks. E.g., edit distance is supported for textual values and blocks are created for character $q$-grams such that entity pairs satisfying the distance threshold co-occur in a sufficient number of blocks. Then, all block collections are aggregated into a multidimensional index
that respects the co-occurrence patterns of similar entities and guarantees no false dismissals, i.e., $PC$=1.
Regarding the lossy approaches, they are dominated by techniques based
on \textsf{LSH} \cite{DBLP:conf/vldb/GionisIM99}, which
efficiently estimates the similarity between two attribute values $v_i$ and $v_j$ by randomly sampling hash functions $f$ from a \textit{sim-sensitive} family $F$ such that the probability $Pr(f(v_i) = f(v_j))$ equals to $sim(v_i, v_j)$ for any pair of attribute values and any function $f \in F$. This means that \textsf{LSH} derives $sim(v_i, v_j)$ from the proportion of hash functions $f$ such that $f(v_i)$ = $f(v_j$). Typically, the required number of these functions is relatively small for a sufficiently small sampling error; e.g., for 500 functions, the maximum sampling error is about $\pm$4.5\% with 95\% confidence interval \cite{DBLP:conf/semweb/DuanFHKSW12}.
In the context of ER, LSH is typically combined with MinHash signatures \cite{DBLP:conf/sequences/Broder97}, which efficiently estimate the Jaccard similarity as follows \cite{DBLP:journals/corr/abs-1907-08667,DBLP:conf/psd/SteortsVSF14}. Given an entity collection $\mathcal{E}$, the values of selected attribute names are converted into a bag of $k$-\textit{shingles}, i.e., $k$ consecutive words or characters. Then, a matrix $M$ of size $K \times |\mathcal{E}|$ is formed, with the rows corresponding to the $K$ distinct shingles that appear in all attribute values and the columns to the input entities. The value of every cell $M(i,j)$ indicates whether the entity $e_j$ contains the shingle $s_j$, $M(i,j)$=1, or not, $M(i,j)$=0. Given that $M$ is a sparse matrix, $p$ random minhash functions are used to reduce its dimensionality: they are applied to each column, deriving a new matrix $M'$ of size $p \times |\mathcal{E}|$.
The $p$ rows are then partitioned into $b$ non-overlapping bands and a hash function is applied to every band of each column. The resulting buckets are treated as blocks that provide probabilistic guarantees that the pairs of similar entities co-occur in at least one block. In fact, the desired probabilistic guarantees can be used for configuring the parameters of \textsf{LSH}, i.e., the number of hash functions, rows and bands \cite{DBLP:journals/corr/abs-1907-08667}.
In this context, LSH is combined with K-Means in \textsf{KLSH} \cite{DBLP:conf/psd/SteortsVSF14}.
KMeans is applied to the low-dimensional columns of $M'$, which represent the input entities. The resulting clusters form a disjoint block collection $\mathcal{B}$, with $|\mathcal{B}|$ determined by the desired average number of entities per block.
\textsf{LSH} is applied to the distributed representations (i.e., embeddings) of the input entities in \textit{DeepER} \cite{DBLP:journals/corr/abs-1710-00597}. Every entity is transformed into a dense, real-valued vector by aggregating the embeddings of all attribute value tokens, which are pre-trained by word2vec \cite{DBLP:conf/nips/MikolovSCCD13}, Glove \cite{DBLP:conf/emnlp/PenningtonSM14} etc. This vector is then hashed into multiple buckets with \textsf{LSH}. A block is then created for every entity containing its top-$N$ most likely matches, which are detected using Multiprobe-LSH \cite{DBLP:conf/vldb/LvJWCL07}.
\textsf{LSH} is also combined with a semantic similarity in \textsf{SA-LSH} (i.e., semantic-aware LSH) \cite{DBLP:journals/tkde/WangCL16}.
A taxonomy tree is used to model the concepts that describe the input entity collection.
The semantic similarity of two entities is inversely proportional to the length of the paths that connect the corresponding concepts and their children: the longer the paths, the lower the semantic similarity.
The concepts of every entity are converted into a hash signature through a semantic hashing algorithm. The resulting low-dimensional signatures are directly combined with the
signatures that are extracted from the n-grams of selected attribute values, capturing the textual similarity of entities. However, the construction of the taxonomy tree requires heavy human intervention.
Regarding the dynamic lossy methods, \textsf{LSH} is combined with \textsf{R-Swoosh} \cite{DBLP:journals/vldb/BenjellounGMSWW09}
in \cite{DBLP:conf/edbt/MalhotraAS14} through a MapReduce parallelization. Initially, a job is used for defining blocks using \textsf{LSH}. Then, a graph-parallel Pregel-based platform
applies R-Swoosh,
iteratively executing the non-redundant comparisons in the blocks and computing the transitive closure of the detected duplicates.
LSH also lies at the core of \textsf{cBV-HB} \cite{DBLP:conf/edbt/KarapiperisVVC16,DBLP:journals/kais/KarapiperisV16}, which embeds the textual values of selected attributes into a compact binary Hamming space that is efficient, due to the limited size of its embeddings (e.g., 120 bits for 4 attributes), and preserves the original distances in the sense that certain types of errors correspond to specific distance bounds. Special care is taken to support composite matching rules that involve the main logical operators (i.e., AND, OR and NOT).
Similarly, \textit{HARRA} \cite{DBLP:conf/edbt/KimL10}
uses \textsf{LSH} to hash similar entities into the same buckets.
Inside every bucket, all pairwise comparisons are executed and
duplicates
are merged into new profiles. The new profiles are hashed into the existing hash tables and the process is repeated until no entities are merged or another stopping criterion is met (e.g., the portion of merged profiles drops below a predetermined threshold). In every iteration, special care is taken to avoid redundant and superfluous comparisons.
Finally, spatial hybrid methods combine
spatial joins with Block Building. The core approach is \textit{StringMap}~\cite{DBLP:conf/dasfaa/JinLM03}, which converts schema-aware blocking keys to a similarity-preserving Euclidean space, whose dimensionality $d$ is heuristically derived from a random sample (typically, $d \in [15, 25]$). For each dimension, a linear algorithm initially selects two pivot attribute values that are (ideally) as far apart as possible. Subsequently, the coordinates of all other attribute values are determined through a comparison with the pivot strings. Using an R-tree or a grid-based index
in combination
with two weight
thresholds, similar attribute values are clustered together into overlapping blocks.
This approach is enhanced by \textit{Extended StringMap} \cite{DBLP:journals/tkde/Christen12}, which replaces the weight thresholds with cardinality ones, and the \textit{Double embedding scheme}~\cite{DBLP:conf/dmin/Adly09}. The latter initially maps the input entities to the same $d$-dimensional Euclidean space. Next, the embedded attribute values are mapped to another Euclidean space of lower dimensionality $d' < d$. A similarity join is performed in the second Euclidean space using a $k$-d tree index. The resulting candidate matches are then clustered in the first, $d$-dimensional Euclidean space. The experimental study suggests that the $d'$-dimensional space significantly reduces the runtime of StringMap by 30\% to 60\%.
\section{Blocking vs Filtering: Commonalities and Differences}
\label{sec:discussion}
The timeline in Figure \ref{fig:taxonomyTimeline}(b)
summarizes the landmarks in the evolution of the two frameworks
showing their gradual convergence.
We observe that Blocking is the oldest discipline, with the first relevant technique, namely \textsf{SB}, presented in 1969 \cite{fellegi1969theory}. For several decades, research focused on schema-based techniques, with the most significant breakthrough taking place in 1995, with the introduction of \textsf{SN} \cite{DBLP:conf/sigmod/HernandezS95}. The first schema-agnostic Block Building technique is \textit{Semantic Graph Blocking} \cite{DBLP:conf/ideas/NinMML07}, introduced in 2007, but it considers only entity links. In 2011, it was followed by
\textsf{TB} \cite{DBLP:conf/wsdm/PapadakisINF11}, which exclusively applies to textual values. Block Processing was introduced in 2009 by Iterative Blocking \cite{DBLP:conf/sigmod/WhangMKTG09}, followed by the use of Canopy Clustering for Blocking in 2012 \cite{DBLP:journals/tkde/Christen12} and the introduction of Meta-blocking in 2014 \cite{DBLP:journals/tkde/PapadakisKPN14}.
For Filtering, the first similarity join to be used in an RDBMS can be traced back to 2001 \cite{DBLP:conf/vldb/GravanoIJKMS01}, while the techniques for in-memory execution were coined in 2007 \cite{DBLP:conf/www/BayardoMS07}. Attempts to further increase efficiency by allowing approximate results were first presented in 2011~\cite{DBLP:conf/sigmod/ZhaiLG11}. The first works on massive parallelization for Filtering appear in 2010 \cite{DBLP:conf/sigmod/VernicaCL10}, for Blocking in 2012 \cite{DBLP:journals/pvldb/KolbTR12}, and for Block Processing in 2015 \cite{DBLP:conf/bigdataconf/Efthymiou0PSP15}. The convergence of the two frameworks essentially starts in 2011 with MultiBlock \cite{isele2011efficient}, which introduces Join-based Blocking,
whereas
multiple-predicate Filtering for efficient Matching,
is introduced by \textsf{Smurf} in 2018 \cite{DBLP:journals/pvldb/CADA18}.
Regarding the qualitative comparison of the two frameworks, we observe that they
have a number of commonalities: (i) Both serve the same purpose: they increase ER efficiency by reducing the number of performed comparisons. To this end, both employ a stage producing candidate matches, which are subsequently examined analytically in order to remove false positives. (ii) Both usually operate either on two clean but overlapping data collections (Record Linkage for Blocking, Cross-table Join for Filtering) or on a single dirty data collection (Deduplication for Blocking, Self-join for Filtering). (iii) Both extract signatures such that the similarity of two entities is reflected in the similarity of their signatures. (iv) Both also apply similar implementation-level optimizations, representing signatures with integer ids, instead of strings, so as to reduce the memory footprint and facilitate in-memory execution. (v) Both include character- and token-based methods. For Blocking, the former methods mainly pertain to schema-aware techniques that apply character-level transformations to blocking keys (e.g., $q$-grams, suffixes etc), while token-based methods primarily pertain to schema-agnostic methods. For Filtering, similarity measures can also be distinguished between character-based (e.g., edit similarity) and token-based ones (e.g., Jaccard), even though many algorithms can be adapted to handle both. (vi) In both cases, textual data have been combined with other types of data, particularly with spatial or spatio-temporal data, including \cite{DBLP:conf/semweb/Ngomo13} for Blocking and \cite{DBLP:reference/db/Gao09, DBLP:journals/tods/JacoxS07, DBLP:journals/pvldb/BourosGM12, DBLP:journals/vldb/BelesiotisSEKP18} for Filtering. (vii) Both
can be used in real-time applications, where the input comprises a query entity and the goal is to identify the most similar ones in the minimum possible time. This is called Similarity Search in the case of Filtering and Real-time ER in the case of Blocking (see Section \ref{sec:futureDirections} for more details).
Due to these commonalities, several works use the two frameworks interchangeably, considering Filtering as a means for Blocking (e.g., \cite{DBLP:series/synthesis/2015Christophides}).
In reality, though, Blocking and Filtering have several distinguishing characteristics: (i)
By definition, a blocking scheme applies to a single entity, considering all its attribute values (schema-agnostic methods), or combinations of multiple values (schema-aware techniques). In contrast, Filtering usually applies to a pair of values from the same attribute of two entities. (ii) Blocking relies on positive evidence, clustering together similar entities, while Filtering relies on negative evidence, detecting dissimilar entities early on. (iii) Blocking is typically independent of Entity Matching, whereas Filtering is interwoven with it, as its goal is to optimize the execution of a matching rule. (iv) Blocking is an inherently approximate procedure that falls short of perfect recall ($PC$), even when providing probabilistic guarantees (e.g., LSH Blocking in DeepER \cite{DBLP:journals/corr/abs-1710-00597}). In contrast, most Filtering methods provide an exact solution, returning all pairs of values that exceed the predetermined threshold along with false positives. (v) Blocking trades slightly lower recall ($PC$) for much higher precision ($PQ$), while Filtering trades filtering power for filtering cost. (vi) Blocking may be modelled as a learning problem, where the goal is to define supervised blocking schemes that simultaneously optimize $PC$, $PQ$ and $RR$, but Filtering requires no labelled set for learning to mark a comparison as true negative. Instead, it relies on a theoretical analysis based on the given similarity measure and threshold. (vii) Preserving privacy is orthogonal to Filtering, with very few works examining privacy-preserving similarity joins \cite{DBLP:conf/icde/LiC08,DBLP:journals/dke/KantarciogluIJM09,DBLP:journals/tifs/YuanWWYN17}. In contrast, Blocking constitutes an integral part of privacy-preserving ER, with several relevant works (for details, refer to a recent survey~\cite{DBLP:journals/is/VatsalanCV13}). (viii) Blocking constitutes an integral part of pay-as-you-go ER applications, conveying a significant body of relevant works, as described below. This does not apply to Filtering, given that the only relevant technique is TopkJoin~\cite{DBLP:conf/icde/XiaoWLS09}.
Regarding the quantitative comparison between
Blocking and Filtering, few works have actually examined their relative performance. The two frameworks are experimentally juxtaposed in \cite{DBLP:conf/semweb/SongH11,DBLP:journals/tkde/SongLH17,DBLP:conf/semweb/Song12} in terms of effectiveness and time efficiency.
Using a series of real-world datasets,
\textsf{RDFKeyLearner} is compared against \textsf{AllPairs}, \textsf{PPJoin}(+) and \textsf{EdJoin} in \cite{DBLP:conf/semweb/SongH11,DBLP:conf/semweb/Song12}
and against \textsf{EdJoin}, \textsf{PPJoin+} and \textsf{FastJoin} in \cite{DBLP:journals/tkde/SongLH17}.
All methods are fine-tuned using a sample of each dataset.
The outcomes indicate no significant difference in effectiveness,
but regarding time efficiency, Filtering is consistently faster in generating candidate matches and consistently slower in executing the corresponding pairwise comparisons, due to their larger number. In \cite{DBLP:journals/tkde/SongLH17}, the relative scalability of \textsf{RDFKeyLearner} and \textsf{EdJoin} is examined over synthetic datasets of 10$^5$, 2$\cdot$10$^5$, ..., 10$^6$ entities. Again, \textsf{EdJoin} produces more candidate matches and, thus, is slower than \textsf{RDFKeyLearner}.
In \cite{DBLP:journals/pvldb/KopckeTR10}, an experimental analysis over 4 real-world datasets investigates the combined effect of Blocking and Filtering on ER efficiency, implementing the workflow in Figure \ref{fig:computationalCostPlusWorkflow}(b). The results suggest that together, the two frameworks reduce the overall ER running time from 33\% to 76\%, with an average of 50\%. However, only one method per framework is considered:
the manually fine-tuned \textsf{SB} and \textsf{PPJoin} in combination with Cosine and Jaccard similarity. Note that, due to its careful, manual fine-tuning, Blocking has no impact on ER effectiveness.
However, more experimental analyses are required for drawing safe conclusions about the relative performance of Blocking and Filtering. These analyses should include
a large, representative variety of techniques per framework along with several established benchmark datasets and should examine the benefits of combining the two frameworks in more depth.
\section{Blocking and Filtering in Entity Resolution Systems}
\label{sec:tools}
We now present the main systems that address ER, examining whether they incorporate any of the aforementioned methods to improve the runtime and the scalability of their workflows. We analytically examined the 18 non-commercial and 15 commercial systems listed in the extended version of \cite{konda2016magellan}\footnote{The extended version of \cite{konda2016magellan} is available here: \url{http://pages.cs.wisc.edu/~anhai/papers/magellan-tr.pdf}.} along with the 10 Link Discovery frameworks surveyed in \cite{DBLP:journals/semweb/NentwigHNR17}.
Table \ref{tab:LinkDiscoveryToolkits} summarizes the characteristics of 12 open-source ER systems that include at least one Blocking~or~Filtering~method.
Half of the tools offer a graphical user interface and are implemented in Java. Regarding the type of the input data, most systems support structured data. The only exceptions are the three Link Discovery frameworks, which are crafted for semi-structured data. JedAI is the only tool that applies uniformly to both structured and semi-structured data.
We also observe that all systems include Blocking methods, with Standard Blocking (\textsf{SB}) and Sorted Neighborhood (\textsf{SN}) being the most popular ones. The first four systems are Link Discovery frameworks that implement custom approaches: KnoFuss and SERIMI apply Token Blocking only to the literal values of RDF tiples, while Silk and LIMES implement hybrid methods, MultiBlock and LIMES, respectively (see Section \ref{sec:hybrid}).
Febrl and JedAI offer the largest variety of established techniques. The former provides their original, schema-aware implementation, while the latter provides their schema-agnostic adaptations. For this reason, JedAI is the only tool that implements Block Processing techniques, as well.
Note that Block Building is also a core part of the ER workflow in several commercial systems, such as IBM Infosphere and Informatica Data Quality \cite{konda2016magellan}. These systems are generally required to handle diverse types of data, focusing on data exploration and cleaning. They typically provide variations of \textsf{SB}, allowing users to extract blocking keys from specific attributes through a sophisticated GUI that provides statistics and data analysis. As a result, users' expertise and experience with specific domains is critical for the performance of these systems' blocking components.
Surprisingly, only two systems currently include Filtering algorithms for improving the runtime of their matching process: LIMES and Magellan. The latter actually offers the largest variety of established techniques through the \texttt{py\_stringsimjoin} package. Filtering techniques are also provided by FEVER \cite{DBLP:journals/pvldb/KopckeTR09}, which is a closed-source ER tool, as well as by JedAI's forthcoming version 3. Still, a mere minority of ER tools enables users to combine the benefits of Blocking and Filtering, despite the promising potential of their synergy (see below for more details). Most importantly, these tools exclusively consider traditional Filtering algorithms that apply to the values of individual attributes. Hence, they disregard the recent Filtering techniques for Complex Matching (cf. Section \ref{subsec:filtering_advanced}), which are more suitable for Entity Resolution. Therefore, more effort should be devoted on developing ER tools that make the most of the synergy between Blocking and Filtering.
\begin{table*}[tbp]
\centering
\caption{Blocking and Filtering methods in open-source systems for Entity Resolution.
}
\label{tab:LinkDiscoveryToolkits}
\vspace{-10pt}
\begin{scriptsize}
\begin{tabular}{|p{1.5cm}|p{4cm}|p{1.5cm}|p{0.4cm}|p{1.0cm}|p{3cm}|}
\toprule
\textbf{Tool} & \textbf{Blocking} & \textbf{Filtering} & \textbf{GUI} & \textbf{Language} & \textbf{Data Formats} \\
\midrule
KnoFuss \cite{nikolov2007knofuss} & Literal Blocking & - & No & Java & RDF, SPARQL \\ \hline
SERIMI \cite{DBLP:journals/tkde/AraujoTVS15} & Literal Blocking & - & No & Ruby & SPARQL \\ \hline
Silk \cite{volz2009silk} & Multiblock & - & Yes & Scala & RDF, SPARQL, CSV \\ \hline
LIMES \cite{DBLP:conf/ijcai/NgomoA11} & custom methods & PPJoin+, EdJoin,
custom methods, e.g., ORCHID \cite{DBLP:conf/semweb/Ngomo13}
& Yes & Java & RDF, SPARQL, CSV \\ \hline
Dedupe \cite{bilenko2003adaptive} & SB with learning-based techniques & - & No & Python & CSV, SQL \\ \hline
DuDe \cite{draisbach2010dude} & SB, \textsf{SN}, Sorted blocks & - & No & Java & CSV, JSON, XML, BibTex, Databases(Oracle, DB2, MySQL and PostgreSQL) \\ \hline
Febrl \cite{christen2008febrl} & SB, \textsf{SN}, Sorted Blocks,
Suffix Arrays, Extended Q-Grams, Canopy Clustering, StringMap& - & Yes & Python & CSV, text-based \\ \hline
FRIL \cite{jurczyk2008fine} & SB, \textsf{SN} & - & Yes & Java & CSV, Excel, COL, Database \\ \hline
OYSTER \cite{nelson2011entity} & SB & - & No & Java & text-based \\ \hline
RecordLinkage \cite{sariyar2011controlling} & SB (with SOUNDEX) & - & No & R & Database \\ \hline
Magellan \cite{konda2016magellan} & SB, \textsf{SN}, it also supports user-specified blocking methods & Overlap, Length, Prefix, Position, Suffix & Yes & Python & CSV \\ \hline
JedAI \cite{DBLP:journals/pvldb/PapadakisTTGPK18} & SB, \textsf{SN}, Extended \textsf{SN}, Suffix Arrays, Extended Suffix Arrays, LSH, Q-Grams, Extended Q-Grams + Block Processing & to be added in the forthcoming version 3 & Yes & Java & CSV, RDF, SPARQL, XML, Database \\
& & & & & \\
\bottomrule
\end{tabular}
\end{scriptsize}
\vspace{-12pt}
\end{table*}
\section{Future Directions}
\label{sec:futureDirections}
Various directions seem promising for future work, from entity evolution \cite{DBLP:conf/jcdl/PapadakisGNPN11} to deep learning \cite{DBLP:journals/corr/abs-1710-00597} and summarization algorithms \cite{DBLP:conf/edbt/KarapiperisGV18}, which minimize the memory footprint of blocks, while accelerating their processing.
The following are more mature fields, having assembled a critical mass of methods already.
\textbf{Progressive Entity Resolution.}
Due to the constant increase of data volumes, new \textit{progressive} or \textit{pay-as-you-go} ER
methods provide the best possible \textit{partial solution} within a limited budget of temporal or computational resources.
Based on Blocking, they
schedule the processing of entities, comparisons or blocks according to the likelihood that they involve duplicates.
Among the schema-based methods,
\textit{Progressive Sorted Neighborhood} (PSN) \cite{DBLP:journals/tkde/WhangMG13}
applies an incremental window size $w$ to a sorted list of entities
until reaching the available budget.
\textit{Dynamic PSN} \cite{DBLP:journals/tkde/PapenbrockHN15} adjusts \textsf{PSN}'s processing order on-the-fly, according to the results of an oracle, while
\textit{Hierarchy of Record Partitions} \cite{DBLP:journals/tkde/WhangMG13}
creates a hierarchy of blocks that is resolved
level by level, from the leaves, which contain the most likely matches, to the root.
A variation of this approach is adapted to
MapReduce
in \cite{DBLP:conf/icde/AltowimM17}, while
the \textit{Ordered List of Records} \cite{DBLP:journals/tkde/WhangMG13} converts it
into a list of entities that are sorted by their likelihood to produce matches.
A progressive solution for relational Multi-source ER over different entity types is proposed in \cite{DBLP:journals/pvldb/AltowimKM14}. \textsf{P-RDS} adapts LSH-based blocking to a progressive operation by rearranging the processing order of hash tables according to the number of matching and unnecessary comparisons in the buckets examined
so far.
Among the schema-agnostic
methods,
\textit{Local Schema-agnostic Progressive SN} \cite{simonini2018schema} slides an incremental window over the sorted list of entities that is created by schema-agnostic \textsf{SN} and, for each window size, it
orders the non-redundant comparisons
according to
the co-occurrence frequency of their entities.
\textit{Global Schema-agnostic Progressive SN} \cite{simonini2018schema} does the same for a predetermined range of windows, eliminating all redundant comparisons they contain.
\textit{Progressive Block Scheduling} \cite{simonini2018schema} orders the blocks in ascending number of comparisons and then prioritizes all comparisons
per block
in decreasing weight. \textit{Progressive Profile Scheduling} \cite{simonini2018schema} orders entities in decreasing average comparison weight
and then prioritizes
all comparisons per
entity in decreasing weight.
The schema-agnostic methods excel in
recall and precision \cite{simonini2018schema}, but
exclusively support \textit{static} prioritization, defining an immutable processing order that disregards the detection of duplicates. Hence, more research is needed for developing \textit{dynamic schema-agnostic} progressive methods.
\textbf{Real-time Entity Resolution.}
This task matches a query entity
to the available entity collections in (ideally) sub-second run-time.
An early solution
is presented in \cite{christenDI}, which
pre-calculates the similarities between the attribute values of entities co-occurring in \textsf{SB} blocks
to avoid similarity calculations at query time.
Its indexes
are dynamically adapted to query entities in
\cite{10.1007/978-3-642-40319-4_5}.
Other dynamic indexing
techniques extend \textsf{SN}.
\textit{F-DySNI} \cite{DBLP:conf/cikm/RamadanC14,Ramadan:2015:DSN:2836847.2816821} converts the sorted list of blocking keys into
a braided AVL tree \cite{rice2007braided}
that is updated
whenever a query entity arrives. The window is
fixed
or adaptive,
considering as neighbors the nodes exceeding a similarity threshold. F-DySNI is extended in \cite{DBLP:conf/pakdd/RamadanC15} with automatic
blocking keys: the weak training set of \cite{DBLP:conf/icdm/KejriwalM13} is
coupled
with a scoring function that considers both key coverage
and block size distribution.
Another group of methods relies on LSH.
MinHash LSH is combined with \textsf{SN}
in \cite{DBLP:conf/pakdd/LiangWCG14}: when searching for the nearest neighbors of a query entity, the entities in large LSH blocks are sorted via a custom scoring function and, then, a window of fixed size slides over the sorted list of entities. \textsf{CF-RDS} \cite{DBLP:journals/datamine/KarapiperisGV18} leverages Hamming LSH, ranking the most similar entities to each query without performing any profile comparison. Instead, it merely aggregates the number of occurrences of each candidate match in the buckets associated with the query entity.
On another line of research, \textit{BlockSketch} \cite{DBLP:conf/edbt/KarapiperisGV18} organizes the entities inside every block into sub-blocks according to their similarity. A representative is assigned to each sub-block based on its distance from the corresponding blocking key. In this way, every query suffices to be compared with a constant number of entities in the target block in order to detect its most similar entities. \textit{SBlockSketch} \cite{DBLP:conf/edbt/KarapiperisGV18} adapts this approach to a stream of query entities through an eviction strategy that bounds the number of blocks that need to be maintained in memory.
All these methods are crafted for structured data, assuming a fixed schema of known quality. New techniques are required, though, for the noisy, heterogeneous entities of semi-structured data.
\textbf{Parameter Configuration.}
Except \textsf{TB},
all Blocking
methods involve at least one internal parameter that affects their performance to a large extent \cite{DBLP:journals/tkde/Christen12,DBLP:journals/pvldb/0001SGP16}. This
also affects their relative performance, rendering the selection of the best performing method
into a non-trivial task.
To mitigate this issue, parameter fine-tuning is modelled as an optimization problem in \cite{DBLP:journals/tlsdkcs/MaskatPE16}. The
configuration space is searched through a genetic algorithm, whose fitness function exploits the labels
of some candidate matches. After
several generations, the configuration maximizing the fitness function is set as optimal. Yet, this approach involves a large number of parameters itself.
\textit{MatchCatcher} \cite{DBLP:conf/edbt/LiKCDSPKDR18} implements a human-in-the-loop approach combining expert knowledge with labelled instances in order to learn composite blocking schemes.
Using string similarity joins,
missed duplicates, which share no block, are efficiently detected. To capture them, the expert user adapts the transformation and assignment functions iteratively.
Finally, a method's performance over several labelled datasets is used for fine-tuning its parameters over an unlabelled dataset in \cite{o2018new}.
The best choice corresponds to the method that achieves the best combination of run-time and F-Measure across most datasets.
However, this is a rather time-consuming approach, given the large number of computations it requires.
None of the above methods satisfies the requirement for automatic, data-driven, a-priori parameter configuration of Blocking methods, which thus remains an open problem.
\textbf{Filtering for Entity Resolution.}
An interesting direction is
to investigate
to what extent similarity joins suffice for ER, i.e., representing entity profiles by strings or sets and defining a matching function based on a similarity threshold. Probably, techniques supporting relaxed matching criteria and/or lower similarity thresholds will be required to achieve high recall, but
relatively few Filtering
techniques are designed for these cases (see Section~\ref{subsec:filtering_advanced}). We also believe that
scalability remains an open challenge for string and set similarity joins \cite{DBLP:journals/pvldb/FierABLF18} and that more opportunities exist for transferring ideas and approaches between Blocking and Filtering.
Finally, there is a need for extensible, open-source ER tools that incorporate the majority of established Blocking and Filtering methods and apply seamlessly to structured, semi-structured and unstructured~data~\cite{DBLP:conf/pods/GolshanHMT17}.
\section{Conclusions}
\label{sec:conclusions}
Efficiency techniques are an integral part of Entity Resolution, since its infancy. We organize the
relevant works
into Blocking, Filtering and hybrid techniques, facilitating their understanding and use. We also provide an in-depth coverage of each category, further classifying its
works into novel sub-categories. Lately,
the rise of big semi-structured data
poses challenges
to the scalability of efficiency techniques and
to their core assumptions: the requirement of Blocking for schema knowledge and of Filtering for high similarity thresholds. The former led to the introduction of schema-agnostic Blocking and of Block Processing techniques, while the latter led to
more relaxed criteria of similarity. We cover these new fields in detail, putting in context all relevant works.
\vspace{4pt}
\noindent
\textbf{Acknowledgements.} This work was partially funded by EU H2020 projects ExtremeEarth (825258) and SmartDataLake (825041).
\vspace{-14pt}
\def\thebibliography#1{
\section*{References}
\vspace{-2pt}
\scriptsize
\list
{[\arabic{enumi}]}
{\settowidth\labelwidth{[#1]}
\leftmargin\labelwidth
\parsep 0pt
\itemsep 0pt
\advance\leftmargin\labelsep
\usecounter{enumi}
}
\def\hskip .11em plus .33em minus .07em{\hskip .11em plus .33em minus .07em}
\sloppy\clubpenalty10000\widowpenalty10000
\sfcode`\.=1000\relax
}
\balance
\bibliographystyle{abbrv}
\section{Introduction}
\vspace{-5pt}
Entity Resolution (ER) is the task of identifying different entity profiles that describe the same real-world object \cite{DBLP:journals/tkde/ElmagarmidIV07,DBLP:series/synthesis/2015Christophides}. It is a core task for Data Integration, applying to any kind of data, from the structured entities of relational databases \cite{DBLP:books/daglib/0030287} to the semi-structured entities of the Linked Open Data Cloud (\url{https://lod-cloud.net}) \cite{DBLP:series/synthesis/2015Dong,DBLP:series/synthesis/2015Christophides} and the unstructured entities that are automatically extracted from free text \cite{DBLP:journals/tkde/ShenWH15}.
ER consists of two parts: (i) the \textit{candidate selection step}, which determines the entities worth comparing,
and (ii) the \textit{candidate matching step}, or simply \textit{Matching}, which compares the selected entities to determine whether they represent the same real-world object. The latter step
involves \textit{pairwise comparisons}, i.e., time-consuming operations that typically apply string similarity measures to pairs of entities, dominating the overall cost of ER \cite{DBLP:books/daglib/0030287,DBLP:series/synthesis/2015Christophides,DBLP:series/synthesis/2015Dong}.
In this survey, we focus on the
candidate selection step, which is the crucial part of ER with respect to time efficiency and scalability. Without it, ER suffers from a quadratic time complexity, $O(n^2)$, as every entity profile has to be compared with all others. Reducing this computational cost is the goal of numerous techniques from
two dominant frameworks: Blocking and Filtering. The former
attempts to identify
entity pairs that are likely to match,
restricting comparisons only between them, while
the latter
attempts to quickly discard pairs that are guaranteed to not match,
executing comparisons only between the rest.
The former operates without knowledge of the
Matching step, while the latter is based on
it,
assuming that two entities match if their similarity exceeds a specified threshold.
Hence, Blocking and Filtering
share the same goal, but are complementary, as they operate under different settings and assumptions.
So far, though,
they have been developed independently of one another: their combination and, more generally, their relation
have been overlooked in the literature, with the exception of very few works (e.g., \cite{DBLP:journals/pvldb/KopckeTR10}).
Moreover, the rise of Big Data poses new challenges for both Blocking and Filtering approaches \cite{DBLP:series/synthesis/2015Dong,DBLP:series/synthesis/2015Christophides}: \textit{Volume} requires techniques to scale to millions of entities, while \textit{Variety} calls for techniques that can cope with an unprecedented schema heterogeneity. Both Blocking and Filtering address Volume primarily through paralellization. Existing techniques were adapted to split their workload into smaller chunks that are distributed across different processing units so that they are executed in parallel. This can be done on a cluster (distributed methods), or through the modern multi-core and multi-socket hardware architectures. Variety, though, is addressed differently in each field. For Blocking, the schema-aware methods are replaced by schema-agnostic techniques, which disregard any schema information, creating blocks of very high recall but low precision. Additionally, a whole new category of methods, called \textit{Block Processing}, intervenes between Blocking and Matching
to refine the original blocks,
significantly increasing precision at a negligible (if any) cost in recall. For Filtering, techniques that employ more relaxed matching criteria (e.g., fuzzy set matching or local string similarity join) are proposed, while the case of low similarity thresholds~is~also~considered.
To the best of our knowledge, this is the first survey to comprehensively cover the aforementioned aspects and to jointly review the two frameworks for efficient ER. We formally define Blocking, Block Processing and Filtering, introducing a common terminology that facilitates their understanding. For each field, we propose a new taxonomy with categories that highlight the distinguishing characteristics of the corresponding methods. Based on these taxonomies, we provide a broad overview of every field, elucidating the functionality of the main techniques as well as the relations among them. As a result, established techniques are now seen in a different light - Canopy Clustering~\cite{DBLP:conf/kdd/McCallumNU00}, for instance, may now be viewed as a Block Processing method.
We also elaborate on the parallelization methods for each field.
Most importantly, this survey attempts to
place Blocking and Filtering under a common context,
taking special care to stress hybrid methods that combine features from both Blocking and Filtering, to analyze works that experimentally compare the two frameworks
(e.g., \cite{DBLP:conf/semweb/SongH11})
and to qualitatively outline their commonalities and differences.
We also investigate the ER tools that incorporate established efficiency techniques and propose a series of open challenges that constitute promising directions for future research.
Parts of the material included in this survey have been presented in tutorials at WWW 2014~\cite{DBLP:conf/www/StefanidisEHC14}, ICDE 2016~\cite{7498364}, ICDE 2017~\cite{DBLP:conf/icde/StefanidisCE17}, and WWW 2018~\cite{PapadakisTutorialWww18}.
A past survey \cite{DBLP:journals/tkde/Christen12} also covers efficiency ER techniques, but is restricted to the schema-aware Blocking methods.
Other surveys \cite{DBLP:journals/tkde/ElmagarmidIV07} and textbooks \cite{DBLP:books/daglib/0030287,DBLP:series/synthesis/2015Dong,DBLP:series/synthesis/2015Christophides} provide a holistic overview of ER, merely examining the main Blocking and Block Processing techniques. Closer to our work is a recent survey on Blocking \cite{o2019review}, which however offers a more limited coverage and refers neither to parallelization nor to Filtering works.
Recent surveys on string and set similarity joins also exist, but
they focus exclusively on
centralized \cite{DBLP:journals/pvldb/JiangLFL14,DBLP:journals/fcsc/YuLDF16,DBLP:journals/pvldb/MannAB16} or distributed approaches \cite{DBLP:journals/pvldb/FierABLF18}, with the purpose of experimental comparison, and without covering approximate techniques
that allow for more relaxed matching criteria. Most importantly, none of these surveys considers similarity joins in the broader context of ER.
The rest of the paper is structured as follows: Section \ref{sec:er} provides background knowledge on ER and its efficiency techniques, while Sections \ref{sec:blocking} and \ref{sec:blockProcessing} delve into Blocking and Block Processing, respectively. Section \ref{sec:filtering} is devoted to Filtering, whereas Section \ref{sec:hybrid} elaborates on works that combine Blocking with Filtering.
Section \ref{sec:tools} enumerates the main ER tools that incorporate efficiency methods, Section \ref{sec:discussion} provides a high-level discussion of the relation between Blocking and Filtering, Section \ref{sec:futureDirections} provides the main directions for future work, and Section \ref{sec:conclusions} concludes the paper.
\section{Preliminaries}
\label{sec:er}
At the core of ER lies the notion of \textit{entity profile}, which constitutes a uniquely identified description of a real-world object in the form of name-value pairs. Assuming infinite sets of attribute names $\mathcal{N}$, attribute values $\mathcal{V}$, and unique identifiers $\mathcal{I}$, an entity profile is formally defined~as~follows~\cite{DBLP:series/synthesis/2015Christophides,DBLP:journals/tkde/PapadakisIPNN13}:
\begin{definition}[Entity Profile]
An \emph{entity profile} $\mathbf{e_{id}}$ is a tuple $\langle id, A_{id} \rangle$, where $id \in \mathcal{I}$ is a unique identifier, and $A_{id}$ is a set of name-value pairs $\langle n, v \rangle$, with $n \in \mathcal{N}$ and $v \in (\mathcal{V} \cup \mathcal{I})$. A set of entity profiles $\mathbf{\mathcal{E}}$ is called \emph{entity collection}.
\end{definition}
This definition is simple, but flexible enough to accommodate a wide variety of (semi-)structured
representations.
E.g., nested attributes can be transformed into a flat set of name-value pairs, while links
may be represented by assigning the id of one entity as the attribute value of the other.
\begin{definition}[Entity Resolution]
Two entity profiles $e_i$ and $e_j$ \emph{match}, $\mathbf{e_i\equiv e_j}$, if they refer to the same real-world entity.
Matching entities are also
called \emph{duplicates}. The task of Entity Resolution (ER) is to find all matching entities within an entity collection or across two or more entity collections.
\end{definition}
In particular, we distinguish between the following two cases \cite{DBLP:journals/tkde/Christen12,DBLP:books/daglib/0030287}:
\begin{enumerate}
\item \textit{Deduplication} receives as input an entity collection $\mathcal{E}$ and produces as output the set of all pairs of matching entity profiles within $\mathcal{E}$, i.e., $\mathcal{D}(\mathcal{E}) = \{ (e_i, e_j) : e_i \in \mathcal{E}, \, e_j \in \mathcal{E}, \, e_i\equiv e_j \}$.
\item \textit{Record Linkage} receives
two duplicate-free entity collections, $\mathcal{E}_1$ and $\mathcal{E}_2$,
and
returns the pairs of matching entity profiles between them, i.e., $\mathcal{D}(\mathcal{E}_1$, $\mathcal{E}_2)$=$\{ (e_i, e_j) : e_i \in \mathcal{E}_1, \, e_j \in \mathcal{E}_2, \, e_i\equiv e_j \}$.
\end{enumerate}
\textit{Multi-source Entity Resolution} involves three or more entity collections and can be performed by applying Deduplication to the union of all collections, or by executing a sequence of pairwise Record Linkage tasks, provided that every input collection is duplicate-free.
ER performance
is characterized by
its \textit{effectiveness} and its \textit{efficiency}. The former refers to how many of the actual duplicates are detected, while the latter expresses the computational cost for detecting them -- usually
in terms of the number of performed comparisons,
which is referred to as \textit{cardinality} and
denoted by $||\mathcal{E}||$. The naive, brute-force approach performs all pairwise comparisons between the input entity profiles, having a quadratic complexity that does not scale to large datasets;
for Record Linkage, $||\mathcal{E}|| = |\mathcal{E}_1| \times |\mathcal{E}_2|$, while for Deduplication $||\mathcal{E}|| = |\mathcal{E}| \cdot (|\mathcal{E}| -1)/2$.
\vspace{2pt}
\textbf{Blocking.} To tackle ER's inherently quadratic complexity, Blocking
trades slightly lower effectiveness for significantly higher efficiency. Its goal is to reduce the number of performed comparisons, while missing as few matches as possible. Ideally, one would compare only the pairs of duplicates, whose number grows \textit{linearly} with the number of the input entity profiles~\cite{DBLP:journals/pvldb/GetoorM12,DBLP:conf/icde/StefanidisCE17}. To this end, Blocking clusters potentially matching entities in common blocks
and exclusively compares entity profiles that co-occur in at least one block.
Internally, a blocking method employs a \textit{blocking scheme}, which applies to one or more entity collections to yield a set of blocks $\mathcal{B}$,
called
\textit{block collection}. Cardinality $||\mathcal{B}||$
denotes the number of comparisons in $\mathcal{B}$,
given that only entity pairs within the same block are compared, i.e., $||\mathcal{B}||$=$\sum_{b_i \in \mathcal{B}} ||b_i||$, where $||b_i||$ stands for the number of comparisons contained in an individual block $b_i$. We denote the set of \textit{detectable duplicates} in $\mathcal{B}$ as $\mathcal{D}(\mathcal{B})$, while $\mathcal{D}(\mathcal{E})$ stands for all existing duplicates. Since $\mathcal{B}$ reduces the number of performed comparisons,~$\mathcal{D}(\mathcal{B})$$\subseteq$$\mathcal{D}(\mathcal{E})$.
A common assumption in the literature is the \textit{oracle}, i.e., a perfect matching function that, for each pair of entity profiles,
decides correctly whether they match or not \cite{DBLP:conf/icde/StefanidisCE17,DBLP:journals/tkde/Christen12,DBLP:series/synthesis/2015Dong,DBLP:journals/tkde/PapadakisIPNN13,DBLP:journals/tkde/PapadakisKPN14}.
Using an oracle,
a pair of duplicates is detected as long as they share at least one block. This allows for reasoning about the performance of blocking methods independently of matching methods: there is a clear trade-off between the effectiveness and the efficiency of a blocking scheme \cite{DBLP:conf/icde/StefanidisCE17,DBLP:journals/tkde/Christen12,DBLP:series/synthesis/2015Dong}: the more comparisons are contained in the resulting block
collection $\mathcal{B}$ (i.e., higher $||\mathcal{B}||$), the more duplicates
will be detected (i.e., higher $|\mathcal{D}(\mathcal{B})|$), raising
effectiveness
at the cost of lower efficiency.
Thus, a blocking scheme
should achieve a good balance between these two competing objectives as expressed through
the following
measures~\cite{DBLP:conf/icdm/BilenkoKM06,DBLP:conf/cikm/VriesKCC09,DBLP:conf/aaai/MichelsonK06,DBLP:conf/wsdm/PapadakisINF11}:
\begin{enumerate}
\item \textit{Pair Completeness ($PC$)} corresponds to \textit{recall}, estimating the portion of the detectable duplicates in $\mathcal{B}$ with respect to those in $\mathcal{E}$:
$PC(\mathcal{B}) = |\mathcal{D}(\mathcal{B})| / |\mathcal{D}(\mathcal{E})| \in [0,1]$.
\item \textit{Pairs Quality ($PQ$)} corresponds to \textit{precision}, estimating the portion of
comparisons in $\mathcal{B}$ that correspond to real duplicates:
$PQ(\mathcal{B})= |\mathcal{D}(\mathcal{B})|/||\mathcal{B}|| \in [0,1]$.
\item \textit{Reduction Ratio ($RR$)} measures the reduction in the number of pairwise comparisons
in $\mathcal{B}$ with respect to the brute-force approach:
$RR(\mathcal{B},\mathcal{E}) = 1 - ||\mathcal{B}||/||\mathcal{E}|| \in [0, 1]$..
\end{enumerate}
Higher values for $PC$ indicate higher \textit{effectiveness} of the blocking scheme, while higher values for $PQ$ and $RR$ indicate
higher \textit{efficiency}.
Note that $PC$ provides an optimistic estimation of recall, presuming
the existence of an oracle, while $PQ$ provides a pessimistic estimation
of precision, treating as false positives the repeated comparisons between duplicates (i.e., only the non-repeated duplicate pairs are considered as true positives).
In this context,
we can define Blocking as follows:
\begin{definition}[Blocking]
Given an entity collection $\mathcal{E}$, Blocking clusters similar entities into a block collection $\mathcal{B}$ such that $PC(\mathcal{B})$, $PQ(\mathcal{B})$ and $RR(\mathcal{B}, \mathcal{E})$ are simultaneously maximized.
\label{def:blocking}
\end{definition}
This definition refers to Deduplication, but can be easily extended to Record Linkage. Simultaneously maximizing $PC$, $PQ$ and $RR$ necessitates that the enhancements in efficiency do not affect the effectiveness of Blocking,
carefully removing comparisons between non-matching entities.
Conceptually, Blocking can be viewed as an optimization task, but
this implies that the real duplicate collection $\mathcal{D}(\mathcal{E})$ is known, which is actually what ER tries to compute. Hence,
Blocking is typically treated as an engineering task that
provides an approximate solution for the data at hand.
\begin{figure}[t]\centering
\includegraphics[width=0.75\linewidth]{preliminariesFigure.png}
\vspace{-9pt}
\caption{{\small (a) The internal functionality of Blocking modeled as a deterministic finite automaton with three states: Block Building (\textsf{BlBu}), Block Cleaning (\textsf{BlCl}) and Comparison Cleaning (\textsf{CoCl}). (b) The end-to-end workflow for non-learning Entity Resolution \cite{DBLP:journals/pvldb/KopckeTR10}. (c) The relative computational cost for the brute-force approach, Blocking, Filtering and the ideal solution (Duplicate Pairs) over Deduplication.}
}
\label{fig:computationalCostPlusWorkflow}
\vspace{-10pt}
\end{figure}
A blocking-based ER workflow may comprise several stages.
First, \textit{Block Building} (BlBu)
applies a blocking scheme to produce a block collection $\mathcal{B}$ from the input entity collection(s).
This step may be repeated several times on the same input,
applying multiple blocking schemes, in order to achieve a more robust performance in the context of highly noisy data. Often, there is a second, optional stage, called \textit{Block Processing}, which
refines $\mathcal{B}$ through additional optimizations that further reduce the number of performed comparisons. This may involve discarding \textit{entire blocks} that primarily contain unnecessary comparisons,
called \textit{Block Cleaning} (BlCl), and/or discarding \textit{individual comparisons} within certain blocks,
called \textit{Comparison Cleaning} (CoCl).
The former may be applied repeatedly, each time enforcing a different, complementary method to discard blocks, but the latter
can be performed only once;
CoCl comprises competitive methods that
serve exactly the same purpose and, once applied to a block collection, they alter it in such a way that turns all other methods inapplicable.
Figure \ref{fig:computationalCostPlusWorkflow}(a) models this workflow as a deterministic finite automaton with three states, where each state corresponds to one of the blocking sub-tasks.
\vspace{2pt}
\textbf{Filtering.}
Given two entity collections $\mathcal{E}_1$ and $\mathcal{E}_2$, a similarity function $f_S : \mathcal{E}_1 \times \mathcal{E}_2 \rightarrow {\rm I\!R}$, and a similarity threshold $\theta$, a \textit{similarity join} identifies all pairs of entity profiles in $\mathcal{E}_1$ and $\mathcal{E}_2$ that have similarity at least $\theta$, i.e., $\mathcal{E}_1 \Join_{\theta} \mathcal{E}_2 = \{ (e_i, e_j) \in \mathcal{E}_1 \times \mathcal{E}_2 : f_S(e_i, e_j) \geq \theta \}$.
\begin{figure}[t]\centering
\includegraphics[width=0.69\linewidth]{measures.png}
\vspace{-10pt}
\caption{{\small Definition of the main similarity measures used by string similarity join algorithms, and how the input threshold $\theta$ for each measure can be transformed into an equivalent Overlap threshold $\tau$.}
}
\label{fig:measures}
\vspace{-14pt}
\end{figure}
Similarity joins can be used for defining ER under the intuitive assumption that matching entity profiles are highly similar. In fact, the above formulation corresponds to Record Linkage, while Deduplication can be defined analogously as a self-join operation, where $\mathcal{E}_1 \equiv \mathcal{E}_2$.
To avoid exhaustive pairwise comparisons,
similarity joins typically follow the \textit{filter-verification} framework, which involves two~stages~\cite{DBLP:series/synthesis/2013Augsten,DBLP:journals/pvldb/JiangLFL14}:
\begin{enumerate}
\item \textit{Filtering} computes a set of \textit{candidates} for each entity $e_i$, excluding all those that cannot match with $e_i$. In other words, it prunes all true negatives, but allows some false positives.
\item \textit{Verification} computes the actual similarity between candidates (or a sufficient upper bound) to remove the false positives.
\end{enumerate}
Due to the relatively straightforward implementation of Verification, in the following we exclusively focus on Filtering. The relevant techniques are defined with respect to three parameters: (i) the representation for each entity, (ii) the similarity function between entity pairs under this representation, and (iii) the similarity threshold above which two entities are considered to match.
The representation typically relies on
\textit{signatures} extracted from each entity such that two entities match only if their signatures overlap.
Given that we address ER
over entities described by one or more textual attributes, we focus on string similarity joins, which
can be \textit{character-} or \textit{token-based}. The former compare two strings by representing them as sequences of characters and by considering the character transformations required to transform one string into the other. The latter are also called \textit{set similarity joins}, since they transform the strings into sets, typically via tokenization or $q$-gram extraction, and then compare strings using a set-based similarity measure.
Regarding the similarity function, the most common one for character-based similarity joins is Edit Distance,
which measures the minimum number of edit operations (i.e., insertions, deletions and substitutions) that are required to transform one string to the other \cite{DBLP:series/synthesis/2013Augsten}. For token-based similarity joins, the most commonly used similarity measures include Overlap, Jaccard, Cosine or Dice. The last three are normalized variants of the Overlap \cite{DBLP:series/synthesis/2013Augsten,DBLP:journals/pvldb/MannAB16,DBLP:journals/pvldb/JiangLFL14}.
Finally, the similarity threshold depends on the data at hand. Note, though, that the join algorithms do not operate directly with thresholds on Jaccard, Cosine or Dice similarity, but
first translate the given threshold $\theta$ into an equivalent set overlap threshold $\tau$ that depends
on the size of the sets, as shown in Figure~\ref{fig:measures}.
A similar transformation is also possible for Edit Distance, which means
that
set similarity joins
may be applied to this measure as well \cite{DBLP:series/synthesis/2013Augsten}.
\vspace{2pt}
\textbf{Blocking vs Filtering.} The relation between the two frameworks is illustrated in Figure \ref{fig:computationalCostPlusWorkflow}(b). Blocking, in the sense of the entire process in Figure \ref{fig:computationalCostPlusWorkflow}(a), is applied first, reducing the pairwise comparisons that are considered by Matching. These comparisons are further cut down by Filtering, which is subsequently applied, as the initial part of Matching, given that it requires specifying both a similarity measure and a similarity threshold. Next, Verification is applied to estimate the actual similarity between the compared attribute values. The Entity Resolution process concludes with \textit{Match Decision}, which synthesizes the estimated similarity between multiple attribute values to determine whether the compared entity profiles are indeed duplicates.
Both Blocking and Filtering are optional steps, but at least one of them should be applied in order to tame the otherwise quadratic computational cost of ER. As shown in Figure~\ref{fig:computationalCostPlusWorkflow}(c), Blocking yields a \textit{super-linear}, but \textit{sub-quadratic} time complexity, lying between the two extremes: the brute-force solution and the ideal one (i.e., Duplicate Pairs). The same applies to the computational cost of Filtering, except that it typically constitutes an \textit{exact} procedure that produces no false negatives, i.e., missed duplicates. It exclusively allows false positives, which are later removed by Verification \cite{DBLP:series/synthesis/2013Augsten}. For this reason, Filtering corresponds to a superset of Duplicate Pairs in Figure~\ref{fig:computationalCostPlusWorkflow}(c). In contrast, Blocking constitutes an inherently \textit{approximate} solution that increases ER efficiency
at the cost of allowing both false positives and false negatives
\cite{DBLP:series/synthesis/2015Christophides}.
Thus, it intersects Duplicate Pairs, such that the area of their intersection is inversely proportional to the duplicates that are missed by Blocking, while the relative complement of the Duplicate Pairs in Blocking is analogous to the executed comparisons between non-matching entities.
Note that Figure~\ref{fig:computationalCostPlusWorkflow}(c) corresponds to Deduplication, but can be easily generalized to Record Linkage, as well. Moreover, the relative performance of Blocking and Filtering, i.e., the relative position of their circles, depends on the methods and the data at hand. In most cases, though, the best solution is to use both frameworks, yielding the computational cost that corresponds to their intersection. However, this approach is rarely used in the literature (e.g., \cite{DBLP:journals/pvldb/KopckeTR10}). Most works on Blocking typically omit Filtering (e.g., \cite{DBLP:journals/pvldb/0001APK15,DBLP:journals/pvldb/0001SGP16,DBLP:journals/tkde/Christen12}), whereas most works on Filtering disregard Blocking, applying directly to the input entity collections (e.g., \cite{DBLP:journals/pvldb/MannAB16,DBLP:journals/pvldb/JiangLFL14}). The goal of the present survey is to cover this gap, elucidating the complementarity of the two frameworks.
\section{Block Building}
\label{sec:blocking}
Block Building receives as input one or more entity collections and produces as output a block collection $\mathcal{B}$. The process is guided by a \textit{blocking scheme}, which determines how entity profiles are assigned to blocks. This scheme typically comprises two parts. First, every entity is processed to extract \textit{signatures} (e.g., tokens),
such that the similarity of signatures reflects the similarity of the corresponding
profiles. Second, every entity is
mapped to one or more blocks based on these signatures. Let $\mathcal{P(S)}$ denote the power set of a set $S$ and $\mathcal{K}$ denote the universe of signatures appearing in entity profiles. We formally define a blocking scheme as follows:
\begin{definition}[Blocking Scheme]
Given an entity collection $\mathcal{E}$, a \emph{blocking scheme} is a function $f_B : \mathcal{E} \rightarrow \mathcal{P}(\mathcal{B})$ that maps entity profiles to blocks. It is composed of two functions: (a) a \emph{transformation} function $f_{T} : \mathcal{E} \rightarrow \mathcal{P}(\mathcal{K})$ that maps an entity profile to a set of \emph{signatures} (also called \emph{blocking keys}), and (b) an \emph{assignment} function $f_{A} : \mathcal{K} \rightarrow \mathcal{P}(\mathcal{B})$ that maps each signature to one or more blocks.
\end{definition}
This definition applies to Deduplication, but can be easily extended to Record Linkage.
\begin{table*}[h]
\centering
\caption{Taxonomy of the Block Building methods discussed in Sections \ref{sec:schemaBasedBB} and \ref{sec:schemaAgnosticBB}.}
\label{tb:bbTaxonomy}
\vspace{-5pt}
{\scriptsize
\begin{tabular}{| l || c | c | c | c | }
\hline
\multicolumn{1}{|c||}{\textbf{Method}} & \textbf{Key} & \textbf{Redundancy} & \textbf{Constraint} & \textbf{Matching} \\
& \textbf{type} & \textbf{awareness} & \textbf{awareness} & \textbf{awareness} \\
\hline
\hline
Standard Blocking (\textsf{SB}) \cite{fellegi1969theory} & hash-based & redundancy-free & lazy & static \\
Suffix Arrays Blocking (\textsf{SA}) \cite{DBLP:conf/wiri/AizawaO05} & hash-based & redundancy-positive & proactive & static \\
Extended Suffix Arrays Blocking \cite{DBLP:journals/tkde/Christen12,DBLP:journals/pvldb/0001APK15} & hash-based & redundancy-positive & proactive & static \\
Improved Suffix Arrays Blocking \cite{DBLP:conf/cikm/VriesKCC09} & hash-based & redundancy-positive & proactive & static \\
Q-Grams Blocking \cite{DBLP:journals/tkde/Christen12,DBLP:journals/pvldb/0001APK15} & hash-based & redundancy-positive & lazy & static \\
Extended Q-Grams Blocking \cite{baxter2003comparison,DBLP:journals/tkde/Christen12,DBLP:journals/pvldb/0001APK15} & hash-based & redundancy-positive & lazy & static \\
MFIBlocks \cite{DBLP:journals/is/KenigG13} & hash-based & redundancy-positive & proactive & static \\
\hline
Sorted Neighborhood (\textsf{SN}) \cite{DBLP:conf/sigmod/HernandezS95, DBLP:journals/datamine/HernandezS98,DBLP:conf/edbt/PuhlmannWN06} & sort-based & redundancy-neutral & proactive & static \\
Extended Sorted Neighborhood \cite{DBLP:journals/tkde/Christen12} & sort-based & redundancy-neutral & lazy & static \\
Incrementally Adaptive SN \cite{DBLP:conf/jcdl/YanLKG07} & sort-based & redundancy-neutral & proactive & static \\
Accumulative Adaptive SN \cite{DBLP:conf/jcdl/YanLKG07} & sort-based & redundancy-neutral & proactive & static \\
Duplicate Count Strategy (\textsf{DCS}) \cite{DBLP:conf/icde/DraisbachNSW12} & sort-based & redundancy-neutral & proactive & dynamic \\
\textsf{DCS++} \cite{DBLP:conf/icde/DraisbachNSW12} & sort-based & redundancy-neutral & proactive & dynamic \\
\hline
Sorted Blocks \cite{DBLP:conf/nss/DraisbachN11} & hybrid & redundancy-neutral & lazy & static \\
Sorted Blocks New Partition \cite{DBLP:conf/nss/DraisbachN11} & hybrid & redundancy-neutral & proactive & static \\
Sorted Blocks Sliding Window \cite{DBLP:conf/nss/DraisbachN11} & hybrid & redundancy-neutral & proactive & static \\
\hline
\multicolumn{5}{c}{\textbf{(a) Non-learning, schema-aware methods.}}\\
\hline
ApproxRBSetCover \cite{DBLP:conf/icdm/BilenkoKM06} & hash-based & redundancy-positive & lazy & static \\
ApproxDNF \cite{DBLP:conf/icdm/BilenkoKM06} & hash-based & redundancy-positive & lazy & static \\
Blocking Scheme Learner (\textsf{BSL}) \cite{DBLP:conf/aaai/MichelsonK06} & hash-based & redundancy-positive & lazy & static \\
Conjunction Learner \cite{DBLP:conf/ijcai/CaoCZYLY11} (semi-supervised) & hash-based & redundancy-positive & lazy & static \\
\textsf{BGP} \cite{DBLP:journals/jidm/EvangelistaCSM10} & hash-based & redundancy-positive & lazy & static \\
CBlock \cite{DBLP:conf/cikm/SarmaJMB12} & hash-based & redundancy-positive & proactive & static \\
DNF Learner \cite{giang2015machine} & hash-based & redundancy-positive & lazy & dynamic \\
\hline
FisherDisjunctive \cite{DBLP:conf/icdm/KejriwalM13} (unsupervised) & hash-based & redundancy-positive & lazy & static \\
\hline
\multicolumn{5}{c}{\textbf{(b) Learning-based (supervised), schema-aware methods.}}\\
\hline
Token Blocking (\textsf{TB}) \cite{DBLP:conf/wsdm/PapadakisINF11} & hash-based & redundancy-positive & lazy & static \\
Attribute Clustering Blocking \cite{DBLP:journals/tkde/PapadakisIPNN13} & hash-based & redundancy-positive & lazy & static \\
RDFKeyLearner \cite{DBLP:conf/semweb/SongH11} & hash-based & redundancy-positive & lazy & static \\
Prefix-Infix(-Suffix) Blocking \cite{DBLP:conf/wsdm/PapadakisINPN12} & hash-based & redundancy-positive & lazy & static \\
TYPiMatch \cite{DBLP:conf/wsdm/MaT13} & hash-based & redundancy-positive & lazy & static \\
Semantic Graph Blocking \cite{DBLP:conf/ideas/NinMML07} & - & redundancy-neutral & proactive & static \\
\hline
\multicolumn{5}{c}{\textbf{(c) Non-learning, schema-agnostic methods.}}\\
\hline
Hetero \cite{DBLP:conf/semweb/KejriwalM14a} & hash-based & redundancy-positive & lazy & static \\
Extended DNF BSL \cite{DBLP:journals/corr/KejriwalM15} & hash-based & redundancy-positive & lazy & static \\
\hline
\multicolumn{5}{c}{\textbf{(d) Learning-based (unsupervised), schema-agnostic methods.}}
\end{tabular}
}
\vspace{-12pt}
\end{table*}
The set of comparisons in the resulting block collection $\mathcal{B}$ is called \textit{comparison collection} and is denoted by $\mathcal{C}(\mathcal{B})$. Every comparison $c_{i,j} \in \mathcal{C}(\mathcal{B})$ belongs to one of the following types~\cite{DBLP:journals/tkde/PapadakisKPN14,DBLP:journals/tkde/PapadakisIPNN13}:
\begin{itemize}
\item \textit{Matching comparison}, if $e_i$ and $e_j$ match.
\item \textit{Superfluous comparison}, if $e_i$ and $e_j$ do not match.
\item \textit{Redundant comparison}, if $e_i$ and $e_j$ have already been compared in a previous block.
\end{itemize}
We collectively call the last two types \textit{unnecessary comparisons}, as their execution brings no gain.
Note that the resulting block collection $\mathcal{B}$ can be modelled as an inverted index that points from block ids to entity ids.
For this reason, Block Building is also called \textit{Indexing}
\cite{DBLP:journals/tkde/Christen12,DBLP:books/daglib/0030287}.
\subsection{Taxonomy}
\label{sec:taxonomy}
To facilitate the understanding of the main methods for Block Building, we organize them into a novel taxonomy
that consists of the following dimensions:
\begin{itemize}
\item \textit{Key selection} distinguishes between \textit{non-learning} and \textit{learning-based} methods. The former
rely on rules derived from expert knowledge or mere heuristics, while the latter
require a training set to learn the best blocking keys using Machine Learning techniques.
\item \textit{Schema-awareness} distinguishes between \textit{schema-aware} and \textit{schema-agnostic} methods. The former
extract blocking keys from specific attributes that are considered to be more appropriate for matching (e.g., more distinctive or less noisy), while the latter
disregard schema knowledge, extracting blocking keys from all attributes.
\item \textit{Key type} distinguishes between \textit{hash-} or \textit{equality-based} methods, which map a pair of entities to the same block if they have a common key, and \textit{sort-} or \textit{similarity-based} methods, which map a pair of entities to the same block if they have a similar key. There exist also \textit{hybrid} methods, which combine hash- with sort-based functionality.
\item \textit{Redundancy-awareness} classifies methods into three categories based on the relation between their blocks.
\textit{Redundancy-free} methods assign every entity to a single block, thus creating disjoint blocks. \textit{Redundancy-positive} methods place every entity into multiple blocks, yielding overlapping blocks. The more blocks two entities share, the more similar their profiles are. The number of blocks shared by a pair of entities is thus proportional to their matching likelihood. \textit{Redundancy-neutral} methods create overlapping blocks, where most entity pairs share the same number of blocks, or the degree of redundancy is arbitrary, having no implications.
\item \textit{Constraint-awareness} distinguishes blocking methods into \textit{lazy}, which impose no constraints on the blocks they create, and \textit{proactive}, which enforce
constraints on their blocks~(e.g., maximum block size), or
refine their comparisons by discarding
unnecessary ones.
\item \textit{Matching-awareness} distinguishes between \textit{static} methods, which are independent of the subsequent matching process, producing an immutable block collection, and \textit{dynamic} methods, which intertwine Block Building with Matching, updating or processing their blocks dynamically, as more duplicates are detected.
\end{itemize}
Table \ref{tb:bbTaxonomy} maps all methods discussed in Sections \ref{sec:schemaBasedBB} and \ref{sec:schemaAgnosticBB} to our taxonomy.
\subsection{Schema-aware Block Building}
\label{sec:schemaBasedBB}
Methods of this type assume that the input entity profiles adhere to a known schema and, based on this schema and respective domain knowledge, one can select the attributes that are most suitable for Blocking.
We distinguish between non-learning methods, reviewed in Section~\ref{sec:nonlearningBlBu}, and learning-based methods, reviewed in Section~\ref{sec:learningBlBu}.
\subsubsection{Non-learning Methods}
\label{sec:nonlearningBlBu}
The family tree of the methods in this category is shown in Figure \ref{fig:schemaBasedBlocking}(a); a parent-child edge implies that the latter method improves upon the former one. Below, we elaborate on these methods based on their key type.
\textbf{Hash-based Methods.}
\textit{Standard Blocking} (\textsf{SB}) \cite{fellegi1969theory}
involves the simplest
functionality:
an expert selects the most suitable attributes, and a transformation function concatenates (parts of) their values to form blocking keys. For every distinct key, a block is created containing all corresponding entities. In short, \textsf{SB} operates as a hash function, conveying two main advantages: (i) it yields redundancy-free blocks,
and (ii) it has a linear time complexity, $O(|E|)$.
On the flip side, its effectiveness is very sensitive to noise,
as the slightest difference in the blocking keys of duplicates places them in different blocks.
\textsf{SB} is also a lazy method that imposes no limit on block sizes.
To address these issues, \textit{Suffix Arrays Blocking} (\textsf{SA})~\cite{DBLP:conf/wiri/AizawaO05} converts each blocking key of \textsf{SB} into the list of its suffixes that are longer than a predetermined minimum length $l_{min}$. Then, it defines a block for every suffix that does not exceed a predetermined frequency threshold $b_{max}$, which essentially specifies the maximum block size. This proactive functionality is necessary, as very frequent suffixes (e.g., ``ing") result in large blocks that are dominated by unnecessary comparisons.
\begin{figure}[t]\centering
\includegraphics[width=0.86\linewidth]{genealogy.pdf}
\vspace{-8pt}
\caption{The genealogy trees of non-learning (a) schema-aware and (b) schema-agnostic Block Building techniques. Hybrid, hash- and sort-based methods are marked in {\color{blue}blue}, black and {\color{red}red}, respectively.
}
\label{fig:schemaBasedBlocking}
\vspace{-10pt}
\end{figure}
\textsf{SA}
has two major advantages \cite{DBLP:conf/cikm/VriesKCC09}: (i) it has low time complexity, $O(|E|$$\cdot$$log|E|)$~\cite{DBLP:journals/dke/AllamSK18}, and is very efficient, as it results in a small but relevant set of candidate matches; (ii) it is very effective, due to the robustness to the noise at the beginning of blocking keys and the high levels of redundancy (i.e., it places every entity into multiple blocks). On the downside, \textsf{SA} does not handle noise at the end of \textsf{SB} keys. E.g., two matches with \textsf{SB} keys ``JohnSnith" and ``JohnSmith" have no common suffix if $l_{min}$=4, while for $l_{min}$=3, they co-occur in a block only if the frequency of ``ith" is lower than $b_{max}$.
This problem is addressed by \textit{Extended Suffix Arrays Blocking} \cite{DBLP:journals/tkde/Christen12,DBLP:journals/pvldb/0001APK15}, which uses as keys not just the suffixes
of \textsf{SB} keys,
but all their substrings with more than $l_{min}$ characters. E.g., for $l_{min}$=4, \textsf{SA} extracts from ``JohnSnith" the keys ``JohnSnith", ``ohnSnith", ``hnSnith", ``nSnith", ``Snith" and ``nith", while \textit{Extended SA} additionally extracts the keys ``John", ``ohnS", ``hnSn", ``nSni", ``Snit" as well as all substrings of ``JohnSnith"
ranging from 5 to 8 characters.
Another extension of \textsf{SB}
is \textit{Q-grams Blocking} \cite{DBLP:journals/tkde/Christen12,DBLP:journals/pvldb/0001APK15}. Its transformation function converts the blocking keys of \textsf{SB} into sub-sequences of $q$ characters (\textit{$q$-grams}) and defines a block for every distinct $q$-gram. For example, for $q$=3, the key \textit{france} is transformed into the trigrams \textit{fra}, \textit{ran}, \textit{anc}, \textit{nce}. This approach
differs from \textsf{Extended SA} in that it does not restrict block sizes (lazy method). Also, it is more resilient to noise than \textsf{SB},
but results in more and larger blocks.
To improve it,
\textit{Extended Q-Grams Blocking} \cite{baxter2003comparison,DBLP:journals/tkde/Christen12,DBLP:journals/pvldb/0001APK15} uses combinations of $q$-grams, instead of individual $q$-grams. Its transformation function concatenates at least $l$ $q$-grams, where $l = max(1,\lfloor k \cdot t \rfloor)$, with $k$ denoting the number of $q$-grams and $t \in [0, 1)$ standing for a user-defined threshold. The larger $t$ is, the larger $l$ gets, yielding less keys from the $k$ $q$-grams. For $T=0.9$ and $q$=3, the key \textit{france} is transformed into the following four signatures ($k$=4 and $l$=3): [\textit{fra}, \textit{ran}, \textit{anc}, \textit{nce}], [\textit{fra}, \textit{ran}, \textit{anc}], [\textit{fra}, \textit{anc}, \textit{nce}], [\textit{ran}, \textit{anc}, \textit{nce}]. In this way, $q$-gram-based blocking keys become more distinctive, decreasing the number and cardinality of blocks.
A more advanced $q$-gram-based approach is \textit{MFIBlocks} \cite{DBLP:journals/is/KenigG13}. Its transformation function concatenates keys of Q-Grams Blocking into itemsets and uses a maximal frequent itemset algorithm to define as new blocking keys those
exceeding a predetermined support threshold.
\textbf{Sort-based Methods.}
\textit{Sorted Neighborhood} (\textsf{SN}) \cite{DBLP:conf/sigmod/HernandezS95}
sorts all blocking keys in alphabetical order and arranges the associated entities accordingly. Subsequently, a window of fixed size $w$ slides over the sorted list of entities and compares the entity at the last position with all other entities placed within the same window. The underlying assumption is that the closer the blocking keys of two entities are in the lexicographical order, the more likely they are to be matching. Originally crafted for relational data, \textsf{SN} is extended to hierarchical/XML data based on user-defined keys in \cite{DBLP:conf/edbt/PuhlmannWN06}.
\textsf{SN} has three major advantages \cite{DBLP:journals/tkde/Christen12}: (i) it has low time complexity, $O(|E|\cdot log |E|)$, (ii) it results in linear ER complexity, $O(w \cdot |E|)$, and (iii) it is robust to noise, supporting errors at the end of blocking keys.
However, it may place two entities in the same block even if their keys are dissimilar (e.g.,
"alphabet" and "apple",
if no other key intervenes between them).
Its performance also depends heavily on the window size $w$, which is difficult to configure,
especially in Deduplication, where the matching entities form clusters of varying size \cite{DBLP:conf/nss/DraisbachN11,DBLP:journals/tkde/Christen12}.
To ameliorate the effect of $w$,
a common solution is the \textit{Multi-pass SN}
\cite{DBLP:journals/datamine/HernandezS98}, which
applies the core algorithm multiple times, using a different transformation function in each iteration. In this way, more matches can be identified, even if the window is set to low size. Another solution is the \textit{Extended Sorted Neighborhood} \cite{DBLP:journals/tkde/Christen12,DBLP:journals/pvldb/0001APK15}, which slides a window of fixed size over the sorted list of blocking keys rather than the list of entities; this means that each block merges $w$ \textsf{SB} blocks.
More advanced strategies adapt the window size dynamically
to optimize the balance between effectiveness and efficiency.
They are grouped into
three
categories, depending on the criterion for moving the
boundaries of the window \cite{DBLP:journals/cj/MaDY15}:
1) \textit{Key similarity strategy.} The window size increases if the similarity of the blocking keys exceeds a predetermined threshold, which indicates that more similar entities should be expected \cite{DBLP:journals/cj/MaDY15}.
2) \textit{Entity similarity strategy.} The window size relies on the similarity of the entities within the current window. \textit{Incrementally Adaptive SN} \cite{DBLP:conf/jcdl/YanLKG07} increases the window size if
the distance of the first and the last element in the window is smaller than a predetermined threshold. The actual increase
depends on the current window size and the selected threshold. \textit{Accumulative Adaptive SN} \cite{DBLP:conf/jcdl/YanLKG07} creates windows with a single overlapping entity and exploits transitivity to group multiple adjacent windows into the same block, as long as the last entity of one window is a potential duplicate of the last entity in the next
window. After expanding the window, both algorithms apply a retrenchment phase that decreases the window size until all entities
are potential duplicates.
3) \textit{Dynamic strategy.}
The core assumption is that the more duplicates are found within a window, the more are expected to be found by increasing its size.
\textit{Duplicate Count Strategy} (\textsf{DCS}) \cite{DBLP:conf/icde/DraisbachNSW12} defines a window
$w$ for every entity in \textsf{SN}'s sorted list and executes all its comparisons
to compute the ratio $d/c$,
where $d$ denotes the newly detected duplicates and $c$ the executed comparisons. The window size is then incremented by one position at a time as long as $d/c \geq \phi$, where $\phi \in (0,1)$ is a threshold that expresses the average number of duplicates per comparison. \textsf{DCS++} \cite{DBLP:conf/icde/DraisbachNSW12} improves \textsf{DCS} by increasing the window size with the next $w-1$ entities, even if the new ratio becomes lower than $\phi$.
Using transitive closure,
it skips some windows, saving part of the comparisons.
\textbf{Hybrid methods.} \textit{Sorted Blocks} \cite{DBLP:conf/nss/DraisbachN11} combines the benefits of \textsf{SB}
and \textsf{SN}.
First, it sorts all blocking keys and the corresponding entities in lexicographical order, like \textsf{SN}. Then, it partitions the sorted entities
into disjoint blocks, like \textsf{SB}, using a prefix of the blocking keys. Next, all pairwise comparisons are executed within each block. To avoid missing any matches, an overlap parameter $o$
defines a
window of fixed size that includes
the $o$ last entities in the current block together with the first entity of the next block. The window slides by one position at a time until reaching the $o^{th}$ entity of the next block, executing
all pairwise comparisons between entities from different blocks.
Sorted Blocks is a lazy approach that does not restrict block sizes.
Thus, it may result in large blocks that dominate its processing time. To address this, two proactive variants set a limit on the maximum block size. \textit{Sorted Blocks New Partition} \cite{DBLP:conf/nss/DraisbachN11}
creates a new block when the maximum
size is reached for a
(prefix of) blocking key; the overlap between the blocks ensures that every entity is compared with its predecessors and successors in the sorting order. \textit{Sorted Blocks Sliding Window} \cite{DBLP:conf/nss/DraisbachN11} avoids executing all comparisons within a block that is
larger than the upper limit; instead, it slides a window
equal to the maximum block size over the entities of the current block.
Finally, \textit{Improved Suffix Arrays Blocking} \cite{DBLP:conf/cikm/VriesKCC09}
employs the same blocking keys as \textsf{SA}, but sorts them in alphabetical order, like \textsf{SN}. Then, it compares the consecutive keys with a string similarity measure. If the similarity of two suffixes exceeds a predetermined threshold, the corresponding blocks are merged
in an effort to detect duplicates even when there is noise at the end of \textsf{SB} keys, or their sole common key is too frequent. For example, \textit{Improved SA} detects the high string similarity of the keys ``JohnSnith" and ``JohnSmith", placing the corresponding entities into the same block.
\subsubsection{Learning-based Methods}
\label{sec:learningBlBu}
We distinguish these methods into supervised and unsupervised ones.
Both rely on a labelled dataset that includes pairs of matching and non-matching entities, called \textit{positive} and \textit{negative instances}, respectively. This dataset is used to learn \textit{blocking predicates}, i.e., combinations of an attribute name and a transformation function (e.g., $\{title, First3Characters\}$). Entities sharing the same output for a particular blocking predicate are considered candidate matches (i.e., hash-based functionality). Disjunctions of conjunctions of predicates, i.e., composite blocking schemes, are learned by optimizing an objective function.
\textbf{Supervised Methods.}
\textit{ApproxRBSetCover} \cite{DBLP:conf/icdm/BilenkoKM06}
learns disjunctive
blocking schemes by solving a standard weighted set cover problem. The cover is iteratively constructed by adding in each turn the blocking predicate
that maximizes the ratio of the previously uncovered positive pairs over the covered negative pairs. This is a "soft cover", since some positive instances may remain uncovered.
\textit{ApproxDNF} \cite{DBLP:conf/icdm/BilenkoKM06} alters ApproxRBSetCover so that it learns
blocking schemes in Disjunctive Normal Form (DNF). Instead of individual predicates, each turn greedily learns a conjunction of up to $k$ predicates that maximizes the ratio of positive and negative covered instances.
A similar approach is \textit{Blocking Scheme Learner} (\textsf{BSL}) \cite{DBLP:conf/aaai/MichelsonK06}. Based on an adaptation of the
Sequential Covering Algorithm, it learns
blocking schemes that maximize $RR$, while maintaining $PC$ above a predetermined threshold. Its output is a disjunction of conjunctions of blocking predicates.
\textsf{BSL} is improved by \textit{Conjunction Learner} \cite{DBLP:conf/ijcai/CaoCZYLY11}, which minimizes the candidate matches not only in the labelled, but also in the \textit{unlabelled} data, while maintaining high $PC$. The effect of the unlabelled data is determined through a weight $w \in [0,1]$; $w=0$ disregards unlabelled data completely, falling back to \textsf{BSL}, while
$w=1$ indicates that they are equally important as the labelled ones.
On another line of research, \textit{Blocking based on Genetic Programming} (\textsf{BGP}) \cite{DBLP:journals/jidm/EvangelistaCSM10} employs a tree representation of supervised blocking schemes, where every leaf node corresponds to a blocking predicate. In every turn, a set of genetic programming operators, such as copy, mutation and crossover, are applied to the initial, random set of
blocking schemes. Then, a fitness function
infers the performance of the new schemes from the harmonic mean of $PC$ and $RR$, and the best ones
are returned as output.
Yet, \textsf{BGP} involves numerous internal parameters
that are hard to fine-tune.
Another tree-based approach
is \textit{CBLOCK} \cite{DBLP:conf/cikm/SarmaJMB12}. In this case, every edge is annotated with a hash (i.e., transformation) function and every node $n_i$ comprises the set of entities that result after applying all hash functions from the root to $n_i$. \textit{CBLOCK} is the only proactive learning-based method, restricting the maximum size of its blocks. Every node that exceeds this limit is split into smaller, disjoint blocks through a greedy algorithm that picks the best hash function based on the resulting $PC$. To minimize the human effort, a drill down approach is proposed for bootstrapping.
\textbf{Unsupervised Methods.}
\textit{FisherDisjunctive} \cite{DBLP:conf/icdm/KejriwalM13}
uses a weak training set generated by
the TF-IDF similarity between pairs of entities. Pairs with very low (high) values are considered as negative (positive) instances. A boolean feature vector is then associated with every labelled instance. The discovery of DNF
blocking schemes is finally cast as a Fisher feature selection problem.
Similarly, \textit{DNF Learner}~\cite{giang2015machine}
applies a matching algorithm
to a sample of entity pairs to automatically create a labelled dataset. Then, the learning of
blocking schemes is cast as a DNF learning problem. To scale it to the exponential search space of possible schemes, their complexity is restricted to manageable levels (e.g., they comprise at most $k$ predicates).
\vspace{-8pt}
\subsection{Schema-agnostic Block Building}
\label{sec:schemaAgnosticBB}
Methods of this type make no assumptions about schema knowledge, disregarding completely attribute names; they extract blocks from all attribute values.
Thus, they inherently support noise in both attribute names and values and are suitable for
highly heterogeneous, loosely structured entity profiles, such as those stemming from the Web of Data~\cite{DBLP:conf/wsdm/PapadakisINF11,DBLP:conf/wsdm/PapadakisINPN12,DBLP:journals/tkde/PapadakisIPNN13}.
\textbf{Non-learning Methods.}
The family tree of this category appears in Figure \ref{fig:schemaBasedBlocking}(b). The cornerstone approach is \textit{Token Blocking} (\textsf{TB}) \cite{DBLP:conf/wsdm/PapadakisINF11}. Assuming that duplicates share at least one common token, its transformation function extracts all tokens from all attribute values of every entity. A block $b_t$ is then defined for every distinct token $t$.
Hence, two entities co-occur in block $b_t$ if they share token $t$ in their values, regardless of the associated attribute names.
To improve \textsf{TB}, \textit{Attribute Clustering Blocking} \cite{DBLP:journals/tkde/PapadakisIPNN13} requires the common tokens of two entities to appear in \textit{syntactically similar attributes}. These are attribute names that correspond to similar values, but are not necessarily semantically matching (unlike Schema Matching). First, it clusters attributes based on the similarities of their aggregate values.
Each attribute is connected to its most similar one and the transitive closure of the connected attributes forms disjoint clusters. A block $b_{k,t}$ is then defined for every token $t$ in the values of the attributes belonging to cluster $k$.
\textit{RDFKeyLearner} \cite{DBLP:conf/semweb/SongH11} applies \textsf{TB} independently to the values of specific attributes, which are selected through the following process:
each attribute is
associated with a \textit{discriminability} score, which amounts to the portion of
distinct values over all values in the given dataset. If this is lower than a predetermined threshold, the attribute is ignored due to limited diversity, i.e., too many entities have the same value(s). For each attribute with high discriminability, its \textit{coverage} is estimated, i.e., the portion of entities that contain it. The harmonic mean of discriminability and coverage is then computed for all valid attributes and the one with the maximum score is selected for defining blocking keys as long as its score exceeds another predetermined threshold. If not, the selected attribute
is combined with all other attributes and the process is repeated.
\textit{Prefix-Infix(-Suffix) Blocking} \cite{DBLP:conf/wsdm/PapadakisINPN12}
exploits the naming pattern in entity URIs. The \textit{prefix} describes the domain of the URI, the \textit{infix} is a local identifier, and the optional \textit{suffix} contains details about the format, or a named anchor \cite{DBLP:conf/iiwas/PapadakisDFK10}. E.g., in the URI {\small\texttt{https://en.wikipedia.org/wiki/France\#History}}, the prefix is {\small\texttt{https://en.wikipedia.org/wiki}}, the infix is {\small\texttt{France}} and the suffix is {\small\texttt{History}}. In this context, this method uses as
keys all (URI) infixes along with all tokens in the literal values.
\textit{TYPiMatch} \cite{DBLP:conf/wsdm/MaT13} improves \textsf{TB}
by automatically detecting the entity types in the input data.
It creates a co-occurrence graph, where every node corresponds to a token in any attribute value and every edge connects two tokens if both conditional probabilities of co-occurrence exceed a predetermined threshold. The maximal cliques are extracted
and merged if their overlap exceeds another threshold. The resulting clusters correspond to the entity types,
with every entity participating in all types to which its tokens belong.
\textsf{TB} is then applied independently to the profiles of each type.
Finally, \textit{Semantic Graph Blocking} \cite{DBLP:conf/ideas/NinMML07} is based exclusively on the relations between entities, be it foreign keys in a database or links in RDF data. It completely disregards attribute values, building a collaborative graph, where every node corresponds to an entity and every edge connects two associated entities. For instance, the collaborative graph for a bibliographic data collection can be formed by mapping every author to a node and adding edges between co-authors.
Then a new block $b_i$ is formed for each node $n_i$, containing all nodes connected with $n_i$ through a path, provided that the path length or the block size do not exceed predetermined limits (proactive functionality).
\textbf{Learning-based Methods.}
\textit{Hetero} \cite{DBLP:conf/semweb/KejriwalM14a}
converts the input data into heterogeneous structured datasets using property tables. Then, it maps every entity to a normalized TF vector, and applies an adapted Hungarian algorithm with linear scalability to produce positive and negative feature vectors. Finally, it applies \textit{FisherDisjunctive} \cite{DBLP:conf/icdm/KejriwalM13} with bagging to achieve robust performance.
Similarly, \textit{Extended DNF BSL}
\cite{DBLP:journals/corr/KejriwalM15} combines an established instance-based schema matcher with weighted set covering to learn DNF
blocking schemes with at most $k$ predicates.
\subsection{Parallelization Approaches}
\label{sec:parallelizationBlBu}
To scale Blocking methods to massive entity collections without altering their functionality,
the \textit{MapReduce framework} \cite{DeanG04} is typically used,
as it offers fault-tolerant, optimized execution for applications distributed across a set of independent nodes.
\textbf{Schema-aware methods.}
The hash-based, non-learning methods are adapted to MapReduce in a straightforward way. The \texttt{map}
phase implements the transformation function(s), emitting {\small \texttt{(key, entity\_id)}} pairs for each entity. Every reducer acts as an assignment function, placing all entities with blocking key $t$ in block $b_t$.
Dedoop~\cite{DBLP:journals/pvldb/KolbTR12} provides such implementations for various methods.
For sort-based methods,
the adaptation of \textsf{SN}
to MapReduce in \cite{DBLP:journals/ife/KolbTR12} can be used as a template.
The \texttt{map} function extracts the blocking key(s) from each input entity, while the ensuing \textit{partitioning} phase
sorts all entities in alphabetical order of their keys.
The \texttt{reduce} function slides a window of fixed size within every reduce partition. Inevitably, entities close to the partition boundaries need to be compared across different reduce tasks. Thus, the \texttt{map} function is extended to replicate those entities, forwarding them to the respective reduce task and its successor.
DCS and DCS++ are adapted to the MapReduce framework in \cite{DBLP:conf/sac/MestrePN15}, using three jobs. The first one sorts the originally unordered entities of the data partition assigned to each mapper according to the selected blocking keys. It also selects the boundary pairs of the sorted partitions. The second job generates the Partition Allocation Matrix, which specifies the sorted partitions to be replicated, while the third job performs DCS(++) locally, to the data assigned to every reducer.
\textbf{Schema-agnostic methods.}
A single MapReduce job is required for parallelizing \textsf{TB} \cite{DBLP:series/synthesis/2015Christophides,DBLP:conf/bigdataconf/EfthymiouSC15}. For every input entity $e_i$, the \texttt{map} function emits a ($t$, $e_i$) pair for every token $t$ in the values of $e_i$. Then, all entities sharing a particular token are directed to the same \texttt{reducer} to form a new block.
For Attribute Clustering Blocking, four MapReduce jobs are required \cite{DBLP:series/synthesis/2015Christophides,DBLP:conf/bigdataconf/EfthymiouSC15}. The first assembles all values
per attribute.
The second computes the pairwise similarities between all attributes,
even if they are placed in different data partitions. The third connects every attribute
to its most similar one. The fourth associates every attribute name with a cluster id and adapts
\textsf{TB}'s \texttt{map} function to emit pairs of the form ($k$.$t$, $i$), where $k$ is the cluster id of $e_i$'s attribute name that contains token $t$.
Finally, the parallelization of Prefix-Infix(-Suffix) Blocking involves three MapReduce jobs \cite{DBLP:series/synthesis/2015Christophides,DBLP:conf/bigdataconf/EfthymiouSC15}. The first parallelizes the algorithm that extracts the prefixes from a set of URIs \cite{DBLP:conf/iiwas/PapadakisDFK10}. The second
extracts the URI suffixes.
The third
applies \textsf{TB}'s
mapper
to the literal values simultaneously with an infix mapper that emits a pair ($j$, $e_i$) for every infix $j$ that is extracted from $e_i$'s profile.
The final \texttt{reduce} phase ensures that all entities having a common token or infix
are placed in the same block.
\textbf{Load Balancing.}
For MapReduce, it is crucial to distribute evenly the overall workload among the available nodes, avoiding potential bottlenecks.
The following methods distribute the execution of comparisons
in a block collection -
not the cost of building the blocks.
\textit{BlockSplit} \cite{DBLP:conf/icde/KolbTR12} partitions large blocks into smaller sub-blocks and processes them in parallel.
Every entity is compared to all entities in its sub-block as well as to all entities of its super-block, even if their sub-block is initially assigned to a different node. This yields an additional network and I/O overhead
and may still lead to unbalanced workload, due to sub-blocks of different size.
To overcome this, \textit{PairRange} \cite{DBLP:conf/icde/KolbTR12} splits evenly the comparisons in a set of blocks into a predefined number of partitions. It involves a single MapReduce job with a mapper that associates every entity $e_i$ in block $b_k$ with the output key $p.k.i$, where $p$ denotes the partition id.
The reducer assembles
all entities that have the same $p$ and block id, reproducing the comparisons of each partition.
The space requirements of these two algorithms are improved in \cite{DBLP:conf/ipccc/YanXM13}, which minimizes their memory consumption by adapting them so that they work with sketches.
Finally, \textit{Dis-Dedup} \cite{DBLP:journals/pvldb/ChuIK16} is the only method that takes into account both the computational and the communication cost (e.g., network transfer time, local disk I/O time). Dis-Dedup considers all possible cases, from disjoint blocks produced by a single blocking technique to overlapping blocks derived from multiple techniques. It also provides strong theoretical guarantees that the overall maximum cost per reducer is within a small constant factor from the lower bounds.
\subsection{Discussion \& Experimental Results}
The performance of the above techniques is examined both qualitatively and quantitatively in a series of individual of papers (e.g., \cite{DBLP:journals/tkdd/VriesKCC11,o2018new,DBLP:journals/tkde/PapadakisIPNN13,DBLP:conf/wsdm/PapadakisINF11}) and experimental analyses (e.g., \cite{DBLP:journals/tkde/Christen12,DBLP:journals/pvldb/0001APK15,DBLP:journals/pvldb/0001SGP16}). Below, we summarize the main findings in order to facilitate the use of Block Building techniques.
Starting with \textit{Standard Blocking} (\textsf{SB}),
its performance depends heavily on the frequency distribution of attribute values and, thus, of blocking keys. The best case corresponds to a uniform distribution, where $||B||=||E||/|B|$ \cite{DBLP:journals/tkde/Christen12}. Due to its lazy functionality, though, all other key distributions yield a portion of large blocks with many superfluous comparisons, i.e., low $PQ$~and~$RR$.
\textit{Suffix Arrays Blocking} (\textsf{SA}) improves \textsf{SB}'s $PC$, by supporting errors at the beginning of blocking keys. The higher $l_{min}$ is and the lower $b_{max}$ is, the lower $||B||$ and $PC$ get. For the same settings, \textit{Extended SA} raises $PC$ at the cost of higher $||B||$, which inevitably lowers both $PQ$ and $RR$. \textit{Improved SA} is theoretically proven in \cite{DBLP:journals/tkdd/VriesKCC11} to result in a $PC$ greater or equal to that of \textsf{SA}, though at the cost of a higher computational cost and more comparisons, which lower $PQ$ and $RR$.
\textit{Q-grams Blocking} yields higher $PC$ than \textsf{SB}, but decreases both $PQ$ and $RR$. \textit{Extended Q-grams Blocking} raises $PQ$ and $RR$ at a limited, if any, cost in $PC$. \textsf{MFIBlocks} reduces significantly the number of blocks and matching candidates (i.e., very high $PQ$ and $RR$) \cite{DBLP:journals/is/KenigG13}, but it may come at the cost of missed matches (insufficient $PC$) in case the resulting blocking keys are very restrictive for matches with noisy descriptions \cite{DBLP:journals/pvldb/0001SGP16}.
For \textit{Sorted Neighborhood} (\textsf{SN}), a small $w$ leads to high $PQ$ and $RR$ but low $PC$ and vice versa for a large $w$. For \textit{Extended SN}, variations in the window size have a large impact on efficiency ($PQ$ and $RR$), affecting the portion of unnecessary comparisons, but $PC$ is more stable. Among the other \textsf{SN} variants, \textsf{DCS++} is theoretically proven to miss no matches with an appropriate value for $\phi$, while being at least as~efficient~as~\textsf{SN}. \textit{Sorted Blocks New Partition} outperforms most SN-based algorithms, but includes more parameters than \textsf{SN}, involving a more complex configuration.
Most importantly, all these non-learning schema-aware methods
are quite parameter-sensitive: even small parameter value modifications may yield significantly different performance \cite{DBLP:journals/tkde/Christen12,DBLP:journals/pvldb/0001APK15,DBLP:conf/cikm/VriesKCC09,o2018new}. Their most important parameter is the definition of the blocking keys, which requires fine-tuning by an expert. Otherwise, their $PC$ remains insufficient, placing most duplicates in no common block \cite{DBLP:journals/tkde/Christen12,DBLP:journals/pvldb/0001APK15}. This applies even to methods that employ redundancy for higher recall.
This shortcoming is ameliorated by schema-agnostic methods,
which consistently achieve much higher $PC$ than their schema-aware counterparts \cite{DBLP:journals/pvldb/0001APK15}. They also simplify the configuration of Block Building, reducing its sensitivity through the automatic definition of blocking keys \cite{DBLP:journals/pvldb/0001APK15,DBLP:journals/pvldb/0001SGP16}. Rather than human intervention or expert knowledge, their robustness emanates from the high levels of redundancy they employ, placing every entity in a multitude of blocks. On the downside, they yield a considerably higher number of comparisons, resulting in very low $PQ$ and $RR$. Both, however, can be significantly improved by Block Processing \cite{DBLP:conf/icde/PapadakisN11,DBLP:journals/pvldb/0001SGP16}.
Regarding the relative performance of schema-agnostic methods, \textsf{TB} yields very high $PC$, at the cost of very low $PQ$ and $RR$. It constitutes a very efficient approach, iterating only once over the input entities, and it is the sole parameter-free Block Building technique in the literature as well as the most generic one, applying to any entity collection with textual values. Its performance is improved by
\textit{Attribute Clustering} and \textit{Prefix-Infix(-Suffix) Blocking} for specific type of datasets: highly heterogeneous ones, with a large variety of attribute names \cite{DBLP:journals/tkde/PapadakisIPNN13,DBLP:journals/pvldb/0001SGP16}, and semi-structured (RDF) ones \cite{DBLP:conf/wsdm/PapadakisINF11}, respectively. In these cases, both methods yield a much larger number of smaller blocks, significantly raising $PQ$ at a minor cost in $PC$. Both methods, though, involve a much higher computational cost than \textsf{TB}. The same applies to
\textit{TYPiMatch}, where the detection of entity types is a rather time-consuming process. Yet, its $PC$ is consistently insufficient, because it falsely divides duplicate entities different entity types, due to the sensitivity to its parameter~configuration~\cite{DBLP:journals/pvldb/0001SGP16}.
Finally, the learning-based Block Building techniques
typically suffer from the scarcity of labelled datasets; even if a training set is available for a particular dataset, it cannot be directly used for learning supervised blocking schemes for another dataset. Instead, a complex transfer learning procedure is typically required \cite{DBLP:conf/cikm/NegahbanRG12,DBLP:journals/corr/abs-1809-11084}. Regarding their efficiency, \textsf{BSL} is typically faster than \textit{ApproxRBSetCover} and \textit{ApproxDNF}, as it exclusively considers positive instances, thus requiring a smaller training set. \textit{Conjunction Learner}
requires every supervised blocking scheme to be applied to the large set of unlabelled data, which is impractical. To accelerate it, a random sample of the unlabelled data is used in practice. \textit{CBLOCK} is also the only learning-based method that is suitable for the MapReduce framework: every entity runs through the learned tree and is directed to the machine corresponding to its leaf node. In terms of effectiveness, there is no clear winner. \textsf{BSL} and \textit{FisherDisjunctive} achieve the top performance in \cite{o2018new}. The latter addresses the scarcity of labelled data, but is not scalable to large datasets.
\section{Block Processing}
\label{sec:blockProcessing}
Block Processing receives as input an existing block collection $\mathcal{B}$ and produces as output a new block collection $\mathcal{B'}$ that improves the balance between effectiveness and efficiency, i.e., $PQ(B) \ll PQ(B')$, $RR(B',B)\gg 0$, while $PC(B) \sim PC(B')$. We distinguish Block Processing methods into \textit{Block Cleaning} ones, which decide whether
entire blocks
should be retained or modified, and \textit{Comparison Cleaning} ones, which
decide whether
individual comparisons
are unnecessary.
\vspace{-10pt}
\subsection{Block Cleaning}
\label{sec:blcl}
We classify Block Cleaning methods into two categories: (i) \textit{static}, which are independent of matching results, and (ii) \textit{dynamic}, which are interwoven with the matching process.
\textbf{Static Methods.} A core idea is the assumption that the larger a block is, the less likely it is to contain unique duplicates, i.e., matches that share no other block. Such large blocks are typically produced by lazy schema-agnostic techniques and correspond to stop words.
In this context, \textit{Block Purging}
discards
blocks that exceed an upper limit on block cardinality \cite{DBLP:journals/tkde/PapadakisIPNN13} or size \cite{DBLP:conf/wsdm/PapadakisINPN12}.
\textit{Block Filtering} \cite{DBLP:conf/edbt/0001PPK16} applies this assumption to individual entities, removing every entity from the largest blocks that contain it. In other words, it
retains every entity in $r\%$ of its smallest blocks.
On a different line of research, \textit{Size-based Block Clustering} \cite{DBLP:conf/kdd/FisherCWR15} applies hierarchical clustering to transform a set of blocks into a new one where all block sizes lie within a specified size range. It merges recursively small blocks that correspond to similar blocking keys, while splitting large blocks into smaller ones. A penalty function controls the trade-off between block quality and block size.
A similar approach is the MapReduce-based dynamic blocking algorithm
in~\cite{mcneill2012dynamic}, which splits large blocks into sub-blocks.
\textit{MaxIntersectionMerge} \cite{nascimento2019exploiting} ensures that all blocks involve at least $|b|_{min}$ entities.
To this end, it merges each block smaller than $|b|_{min}$ entities with the block that has the most entities in common and is larger than $|b|_{min}$.
Similarly,
\textit{Rollup Canopies} \cite{DBLP:conf/cikm/SarmaJMB12} receives as input a training set with positive examples, a limit on the maximum block size and a set of disjoint blocks; using a learning-based greedy algorithm, it merges pairs of small blocks to increase $PC$.
Finally, \cite{DBLP:conf/icdm/RanbadugeVC16} generalizes Meta-blocking (see Section \ref{sec:cocl}) to Multi-source ER: it constructs a graph, where the nodes correspond to blocks and the edges connect blocks whose blocking keys are more similar than a predetermined threshold. The edges are weighted using various functions and all pairs of blocks are then processed in descending edge weights in an effort to maximize the redundant and superfluous comparisons that are skipped.
\textbf{Dynamic Methods.}
\textit{Iterative Blocking} \cite{DBLP:conf/sigmod/WhangMKTG09} merges
any new pair of detected duplicates, $e_i$ and $e_j$, into a new entity, $e_{i,j}$,
and replaces both $e_i$ and $e_j$ with $e_{i,j}$ in all blocks that contain them, even if they have already been processed.
The new entity $e_{i,j}$ is compared with all co-occurring entities, as
the new content in $e_{i,j}$ might
identify previously missed matches.
The ER process terminates when all blocks have been processed without finding new duplicates.
Iterative Blocking applies exclusively to Deduplication. In Record Linkage, there is no need for merging two matching entities,
due to the 1-1 restriction. Still,
the detected duplicates should be propagated in order to save the superfluous comparisons with
their co-occurring entities
in the subsequently processed blocks.
The earlier the matches are detected, the more superfluous comparisons are saved.
To this end, \textit{Block Scheduling}
optimizes the processing order
of blocks in a non-iterative way, sorting them in decreasing order of
the probability $p_i(d)$ that a block $b_i$ contains a pair of duplicates. This is set inversely proportional to block cardinality, i.e., $p_i(d)=1/||b_i||$ \cite{simonini2018schema}, or to the minimum size of the inner block, i.e., $p_i(d)=1/min{|b_{i,1}|,|b_{i,2}|}$, where $|b_{i,k}| \subset \mathcal{E}_k$ \cite{DBLP:conf/wsdm/PapadakisINF11}.
The former definition also applies to Iterative Blocking, which does not specify the exact block processing order, even though this affects significantly the resulting performance \cite{DBLP:journals/pvldb/0001SGP16}.
\textit{Block Pruning} \cite{DBLP:conf/wsdm/PapadakisINF11} extends Block Scheduling by exploiting the decreasing density of detected matches in its processing order (i.e., the later a block is processed, the less
unique duplicates it contains).
After processing the latest block, it estimates the average number of executed comparisons per new duplicate.
If this ratio falls below a specific threshold, it terminates the ER process.
\subsection{Comparison Cleaning}
\label{sec:cocl}
\begin{figure}[t]\centering
\includegraphics[width=0.59\linewidth]{ccGenealogy.pdf}
\vspace{-8pt}
\caption{The genealogy tree of non-learning Comparison Cleaning methods. Methods in black conform to the Meta-blocking framework in Figure \ref{fig:coclExample}, methods in {\color{blue}blue} are Meta-blocking techniques following a (partially) different approach and methods in {\color{red}red} are not part of the Meta-blocking framework.
}
\label{fig:taxonomyCC}
\vspace{-14pt}
\end{figure}
\textbf{Non-learning Methods.}
Figure \ref{fig:taxonomyCC} illustrates the family tree of the methods belonging to this category.
The cornerstone method is \textit{Comparison Propagation} \cite{DBLP:conf/jcdl/PapadakisINPN11}, which propagates all executed comparisons to the subsequently processed blocks. In this manner, it eliminates all redundant comparisons in a given block collection without losing any pair of duplicates, thus raising $PQ$ and $RR$ at no cost in $PC$.
It builds an inverted index that points from entity ids to block ids, called \textit{Entity Index}, and with its help, it compares two entities $e_i$ and $e_j$ in block $b_k$ only if $k$ is their least common block id.
For example, consider the blocks in Figure \ref{fig:coclExample}(a) and their Entity Index in Figure \ref{fig:coclExample}(b). The least common block id of $e_1$ and $e_3$ is 2. Thus, they are compared in $b_2$, but neither in $b_4$~nor~in~$b_5$.
Given a redundancy-positive block collection, the Entity Index allows for identifying the blocks shared by a pair of co-occurring entities. This allows for weighting all pairwise comparisons in proportion to the matching likelihood of the corresponding entities, based on the principle that the more blocks two entities share, the more likely they are to be matching. This gives rises to a family of \textit{Meta-blocking} techniques \cite{DBLP:journals/tkde/PapadakisKPN14,DBLP:conf/edbt/0001PPK16,DBLP:journals/pvldb/SimoniniBJ16} that
go beyond Comparison Propagation by discarding not only all redundant comparisons, but also the vast majority of the superfluous ones.
The first relevant method is \textit{Comparison Pruning} \cite{DBLP:conf/sigmod/PapadakisINPN11}, which
computes the Jaccard co-efficient of the block lists of two entities. If it does not exceed a conservative threshold that depends on the average number of blocks per entity, the
comparison is pruned, as it designates an unlikely match.
Meta-blocking was formalized into a more principled approach in \cite{DBLP:journals/tkde/PapadakisKPN14}. The given redundancy-positive block collection $\mathcal{B}$ is converted into a blocking graph $G_B$, where the nodes correspond to entities and the edges connect every pair of co-occurring entities - see Figure \ref{fig:coclExample}(c). Given that no parallel edges are allowed, all redundant comparisons are discarded by definition. The edges are then weighted proportionately to the likelihood that the adjacent entities are matching. In Figure \ref{fig:coclExample}(d), the edge weights indicate the number of common blocks. Edges with low weights are pruned, because they correspond to superfluous comparisons. In Figure \ref{fig:coclExample}(e), all edges with a weight lower than the average one are discarded. The resulting pruned blocking graph $G_{B'}$ is transformed into a restructured block collection $\mathcal{B}'$ by forming one block for every retained edge - see Figure \ref{fig:coclExample}(f). As a result, $\mathcal{B}'$ exhibits a much higher efficiency, $PQ(B')$$\gg$$PQ(B)$ and $RR(B',B)$$\gg$$0$, for similar effectiveness, $PC(B')$$\sim$$PC(B)$; in our example, the 12 comparisons in the input blocks of Figure \ref{fig:coclExample}(a) are reduced to 2 matching comparisons in the output blocks in Figure \ref{fig:coclExample}(f).
Four main pruning algorithms exist: (i) \textit{Weighted Edge Pruning} (\textsf{WEP}) removes all edges that do not exceed a specific threshold, e.g., the average edge weight \cite{DBLP:journals/tkde/PapadakisKPN14}; (ii) \textit{Cardinality Edge Pruning} (\textsf{CEP}) retains the globally $K$ top weighted edges, where $K$ is static \cite{DBLP:journals/tkde/PapadakisKPN14} or dynamic \cite{zhang2017pruning};
(iii)~\textit{Weighted Node Pruning} (\textsf{WNP}) retains in each node neighborhood the entities that exceed a local threshold, which may be the average edge weight of each neighborhood \cite{DBLP:journals/tkde/PapadakisKPN14}, or the average of the maximum weights in the two adjacent node neighborhoods, as in \textit{BLAST}
\cite{DBLP:journals/pvldb/SimoniniBJ16};
(iv) \textit{Cardinality Node Pruning} (\textsf{CNP}) retains the top-$k$ weighted edges in each node neighborhood \cite{DBLP:journals/tkde/PapadakisKPN14}.
\textit{Reciprocal WNP} and \textit{CNP} \cite{DBLP:conf/edbt/0001PPK16} apply an aggressive pruning that retains edges satisfying the pruning criteria
in both adjacent node neighborhoods. \textsf{WNP} and \textsf{WEP} are combined through the weighted sum of their thresholds in \cite{DBLP:conf/iscc/AraujoPN17}.
Another family of pruning algorithms is presented in \cite{nascimento2019exploiting}, focusing on the
edge weights between the entities in each block. \textit{Low Entity Co-occurrence Pruning} (\textsf{LECP}) cleans every block from a specific portion of the entities with the lowest average edge weights. \textit{Large Block Size Pruning} (\textsf{LBSP}) applies \textsf{LECP} only to the blocks whose size exceeds the average block size in the input block collection. \textit{Low Block Co-occurrence Pruning} (\textsf{LBCP}) removes every entity from the blocks, where it is connected with the lowest weights, on average, with the rest of the entities. \textit{CooSlicer} enforces a maximum block size constraint, $|b|_{max}$, to all input blocks. In blocks larger than $|b|_{max}$ all entities are sorted in decreasing order of average edge weight, and the $|b|_{max}$ top-ranked entities are iteratively placed into a new block. \textit{Low Block Co-occurrence Excluder} (\textsf{LBCE}) discards a specific portion of the blocks with the lowest average edge weight among their entities.
All these pruning algorithms can be coupled with any
\textit{edge weighting scheme} \cite{DBLP:journals/tkde/PapadakisKPN14}. \textsf{ARCS} sums the inverse cardinalities of the common blocks, giving higher weights to
entity pairs that co-occur in smaller blocks. \textsf{CBS} counts the number of blocks shared by two entities, as in Figure \ref{fig:coclExample}(c), with \textsf{ECBS} extending it to discount the contribution from entities placed in many blocks.
\textsf{JS} corresponds to the Jaccard coefficient of two block lists, while
\textsf{EJS} extends it to discount the contribution from entities appearing in many non-redundant comparisons.
Finally, Pearson's $\chi^2$
test assesses whether two adjacent entities appear independently in blocks and can be combined with
the aggregate attribute entropy associated with the tokens forming their common blocks~\cite{DBLP:journals/pvldb/SimoniniBJ16}.
Note that
Meta-blocking covers established methods that are
considered as Block Building methods in the literature:
given that Block Building is equivalent to indexing \cite{DBLP:journals/tkde/Christen12}, any method based on indexes is in fact a Meta-blocking technique.
For example, \textit{Transitive LSH} \cite{DBLP:conf/psd/SteortsVSF14}
converts the blocks extracted from \textsf{LSH} into an unweighted blocking graph and applies a community detection algorithm (e.g., \cite{clauset2004finding})
to partition the graph nodes into disjoint clusters, which will become the new blocks. The process finishes when the size of the largest cluster is lower than a predetermined threshold. This approach can be applied on top of any Block Building method, not just \textsf{LSH}.
The generalization principle also applies to \textit{Canopy Clustering} \cite{DBLP:conf/kdd/McCallumNU00}, which
places all entities in a pool and, in every iteration, it removes a random entity $e_i$ from the pool to create a new block. Using a cheap similarity measure, all entities still in the pool are compared with $e_i$. Those exceeding a threshold $t_{ex}$ are removed from the pool and placed into the new block. Entities exceeding another threshold $t_{in}$ ($< t_{ex}$) are also placed in the new block, without being removed from the pool.
As the cheap similarity measure, we can use any of the above weighting schemes on top of any Block Building method,
thus turning Canopy Clustering into a pruning algorithm for Meta-blocking.
\begin{figure}[t!]\centering
\includegraphics[width=0.99\linewidth]{comparisonCleaningExample.png}
\vspace{-8pt}
\caption{(a) A block collection $B$ with $e_1$$\equiv$$e_3$ and $e_2$$\equiv$$e_4$, (b) the corresponding Entity Index, (c) the corresponding blocking graph $G_B$, (d) the weighted $G_B$, (e) the pruned $G_B$, and (f) the new block collection $B'$.}
\vspace{-19pt}
\label{fig:coclExample}
\end{figure}
The generalization applies to \textit{Extended Canopy Clustering} \cite{DBLP:journals/tkde/Christen12,DBLP:journals/pvldb/0001APK15}, too,
which replaces the sensitive
weight thresholds
with cardinality ones:
for each randomly selected entity, the $n_1$ nearest entities are placed in its block, while the $n_2 (\leq n_1)$ nearest entities are removed from the pool.
On another line of research, \textit{SPAN} \cite{DBLP:conf/icde/ShuCXM11} converts a block collection into a matrix $M$, where the rows correspond to entities and the columns to the tf-idf of blocking keys (tokens or $q$-grams). Then, the entity-entity matrix is defined as $A=MM^T$. A spectral clustering algorithm converts $A$ into a binary tree, where the root node contains all entities and every leaf node is a disjoint subset of entities. The Newman-Girvan modularity is used as the stopping criterion for the bipartition of the tree. Blocks are then derived from a search procedure that carries out pairwise comparisons based on the blocking keys, inside the leaf nodes and across the neighboring ones.
Finally, the sole dynamic non-learning method
is \textit{Comparison Scheduling} \cite{DBLP:journals/tkde/PapadakisIPNN13}. Its goal is
to detect most matches upfront so as to maximize the superfluous comparisons that are skipped, due to the 1-1 restriction. It orders all comparisons in decreasing matching likelihood (edge weight) and executes a comparison only if none of the involved entities has already been matched.
\textbf{Learning-based Approaches.}
\textit{Supervised Meta-blocking} \cite{DBLP:journals/pvldb/0001PK14}
treats edge pruning
as a binary classification problem, where every edge is labelled "\texttt{likely match}" or "\texttt{unlikely match}".
Every edge is represented by a feature vector that comprises five features: \textsf{ARCS}, \textsf{ECBS}, \textsf{JS} and the Node Degrees of the adjacent entities. Undersampling is employed to tackle the class imbalance problem:
the training set comprises just 5\% of the minority class ("\texttt{likely match}") and an equal number of majority class instances. Several established classification algorithms are used for \textsf{WEP}, \textsf{CEP} and \textsf{CNP}, with all of them exhibiting robust performance with respect to their internal configuration.
\textit{BLOSS} \cite{DBLP:journals/is/BiancoGD18}
restricts the labelling cost of Supervised Meta-blocking by
carefully selecting a training set that is up to 40 times smaller, but retains the original performance. Using $ECBS$ weights, it partitions the unlabelled instances into similarity levels and applies rule-based active sampling inside every level. Then, it cleans the sample from non-matching outliers with high $JS$ weights.
\textbf{Parallelization Approaches.}
Meta-blocking is adapted to the MapReduce framework in three ways \cite{DBLP:journals/is/Efthymiou0PSP17}:
(i) The \textit{edge-based strategy} stores the blocking graph
on the disk, bearing a significant I/O cost.
(ii) The \textit{comparison-based strategy} builds the blocking graph \textit{implicitly}. A pre-processing job enriches every block with the list of block ids associated with every entity. The Map phase of the second job computes the edge weights and discards all redundant comparisons, while the ensuing Reduce phase prunes superfluous comparisons.
This strategy maximizes the efficiency of \textsf{WEP} and \textsf{CEP}
and is adapted to Apache Spark
in \cite{DBLP:conf/iscc/AraujoPN17}. (iii) The \textit{entity-based strategy}
aggregates for every entity the bag of all entities that co-occur with it in at least one block. Then, it estimates the edge weight that corresponds to each neighbor based on its frequency in the co-occurrence bag. This approach offers the best implementation for \textsf{WNP} and \textsf{CNP} and their variations (e.g., \textsf{BLAST}). It is adapted to Apache Spark in \cite{DBLP:journals/is/SimoniniGBJ19}, leveraging the broadcast join for higher efficiency.
To avoid the underutilization of the available resources, these strategies employ \textit{MaxBlock} \cite{DBLP:journals/is/Efthymiou0PSP17} for load balancing. Based on the highly skewed distribution of block sizes in redundancy-positive block collections, it splits the input blocks into partitions of equivalent computational cost, which is equal to the total number of comparisons in the largest input block.
The \textit{multi-core parallelization} of Meta-blocking is examined in \cite{DBLP:conf/i-semantics/0001BPK17}.
The input is transformed into
an array of chunks, with an index indicating the next chunk to be processed. Following the established fork-join model, every thread retrieves the current value of the index and is assigned to process the corresponding chunk. Depending on the definition of chunks, three alternative strategies are proposed: (i) \textit{Naive Parallelization} treats every entity as a separate chunk, ordering all entities in decreasing computational cost (i.e., the aggregate number of comparisons in the associated blocks). (ii) \textit{Partition Parallelization} uses MaxBlock to group the input entities into an arbitrary number of disjoint clusters with identical computational cost. (iii) \textit{Segment Parallelization}
sets the number of clusters equal to the number of available cores.
\subsection{Discussion \& Experimental Results}
\begin{figure}[t]\centering
\includegraphics[width=0.40\linewidth]{ccRelativePerformance.png}
\vspace{-8pt}
\caption{The relative performance of the main Comparison Cleaning methods.
}
\label{fig:relativePerformance}
\vspace{-14pt}
\end{figure}
The core characteristic of Block Processing methods is their
schema-agnostic functionality, which typically
relies
on block features, such as size, cardinality and overlap. This is no surprise, as they are primarily crafted for boosting the performance of schema-agnostic Block Building methods. In fact, extensive experiments demonstrate that Block Processing is indispensable for these methods, raising precision
by whole orders of magnitude, at a minor cost in
recall
\cite{DBLP:journals/pvldb/0001SGP16,DBLP:journals/bdr/PapadakisPPK16,DBLP:journals/tkde/PapadakisIPNN13}.
Regarding their relative performance, there is no clear winner among the Block Cleaning methods. For example,
both Block Filtering and Block Purging boost $PQ$ and $RR$ by orders of magnitude, while exhibiting a low computational cost and a negligible impact on $PC$ \cite{DBLP:journals/tkde/PapadakisIPNN13,DBLP:conf/edbt/0001PPK16}. However, the top performer among them depends not only on their parameter configuration, but also on the data at hand \cite{DBLP:journals/pvldb/0001SGP16}.
Most importantly, though, Block Cleaning techniques are usually complementary in the sense that multiple ones can be applied consecutively in a single blocking workflow, as depicted in Figure \ref{fig:computationalCostPlusWorkflow}(b). For example, Block Filtering is typically applied after Block Purging by lowering $r$ to $50\%$ instead of $80\%$, which is the best configuration when applied independently \cite{DBLP:journals/bdr/PapadakisPPK16,DBLP:conf/edbt/0001PPK16,DBLP:journals/pvldb/0001SGP16}.
In contrast, Comparison Cleaning methods are incompatible with each other in the sense that at most one of them can be part of a blocking workflow. The reason is that applying any Comparison Cleaning technique to a redundancy-positive block collection deprives it from its co-occurrence patterns and renders all other techniques inapplicable.
These techniques also involve a much higher computational cost than Block Cleaning methods, due to their finer level of granularity. Their relative performance is summarized in Figure \ref{fig:relativePerformance}, based on empirical evidence from experimental studies \cite{DBLP:journals/pvldb/0001SGP16} and individual publications \cite{DBLP:journals/is/BiancoGD18,DBLP:journals/pvldb/0001PK14,DBLP:journals/pvldb/SimoniniBJ16,DBLP:journals/bdr/PapadakisPPK16,DBLP:conf/edbt/0001PPK16,DBLP:journals/is/SimoniniGBJ19}. Note that we exclude methods not compared to other Comparison Cleaning techniques (e.g., the techniques presented in \cite{nascimento2019exploiting}).
In more detail, Figure \ref{fig:relativePerformance} maps the performance of the main Comparison Cleaning methods to a two dimensional space defined by $\Delta PC$=$PC(\mathcal{B}')-PC(\mathcal{B})$ on the vertical axis and $\Delta PQ$=$PQ(\mathcal{B}')-PQ(\mathcal{B})$ on the horizontal axis, where $B$ and $B'$ stand for the input and the output block collections, respectively. Given that Comparison Cleaning techniques trade lower recall ($PC$) for higher precision ($PQ$), $\Delta PC$ and $\Delta PQ$ take exclusively negative and positive values, respectively. Therefore, the higher a method is placed, the better recall it achieves, whereas the further to the right it lies, the better is its precision. This means that the ideal overall performance corresponds to the upper~right~corner.
We observe that $\Delta PC$ is delimited by two extremes: Comparison Cleaning on the top left corner and \textsf{CEP} on the bottom right corner. The former has no impact on recall, as it
increases precision only by removing redundant comparisons. All other Comparison Cleaning techniques discard superfluous comparisons, too, thus achieving larger $\Delta PQ$ at the cost of a negative $\Delta PC$. On the other extreme, \textsf{CEP} prunes a large portion of superfluous comparisons, yielding very high precision, but the lowest recall. \textsf{WEP} replaces \textsf{CEP}'s cardinality constraint with a weight threshold, dropping precision to a large extent for a significantly higher recall. Still, \textsf{WEP}'s performance is a major improvement over the input block collection, while being rather robust across numerous datasets. \textsf{WNP} moves further towards this direction, shrinking the decrease in recall and the increase in precision. This is further improved by \textit{Reciprocal WNP}, which significantly raises \textsf{WNP}'s precision for slightly lower recall. Thus, it dominates \textsf{WEP}, albeit being sensitive to the characteristics of the data at hand. Compared to \textsf{CEP}, \textsf{CNP} confines its pruning inside individual node neighborhoods. In this way, it achieves a much higher recall for a limited decrease in precision. This is further improved by \textit{Reciprocal CNP}, which reduces \textsf{CNP}'s recall slightly for much higher precision and, thus, it often dominates \textsf{CEP}.
\textsf{WNP}, \textsf{CNP} and their variants are improved by \textit{Supervised Meta-blocking} and \textsf{BLAST}, which achieve comparable recall for significantly higher precision. \textsf{BLAST} takes a lead in precision, partially because it employs the most effective weighting scheme, namely Pearson's $\chi^2$ test. Another advantage is that \textsf{BLAST} requires no labeling effort, due to its unsupervised functionality. \textsf{BLOSS}, however, achieves almost perfect recall ($\Delta PC \approx 0$) for the highest precision among all Comparison Cleaning techniques, while requiring merely $\sim$50 labeled instances. Note that exceptions to these general patterns of performance are possible for a particular dataset.
\section{Filtering}
\label{sec:filtering}
\begin{table*}[t]
\centering
\setlength{\tabcolsep}{3.5pt}
\caption{Overview of string and set similarity join methods.}
\label{tab:filtering_table}
\vspace{-5pt}
{\scriptsize
\begin{tabular}{| l || l | l | l | l |}
\hline
\textbf{Method} & \textbf{Operation} & \textbf{Similarity} & \textbf{Filters} & \textbf{Index} \\
\hline
\hline
GramCount~\cite{DBLP:conf/vldb/GravanoIJKMS01} & string join & Edit Distance & length, count, position & $q$-grams table \\
MergeOpt~\cite{DBLP:conf/sigmod/Sarawagi04} & set join & Overlap & count & inverted index \\
FastSS~\cite{BoHuSt07} & string join & Edit Distance & deletion neighborhood & dictionary \\
\hline
SSJoin~\cite{DBLP:conf/icde/ChaudhuriGK06} & set join & Overlap & prefix & DBMS \\
All-Pairs~\cite{DBLP:conf/www/BayardoMS07} & vector join & Cosine & prefix & inverted index \\
DivideSkip~\cite{DBLP:conf/icde/LiLL08} & string search & Edit Distance, Overlap & length, position, prefix & inverted index \\
Ed-Join~\cite{DBLP:journals/pvldb/XiaoWL08} & string join & Edit Distance & prefix+mismatching $q$-grams & inverted index \\
QChunk~\cite{DBLP:conf/sigmod/QinWLXL11} & string join & Edit Distance & prefix+$q$-chunks & inverted index \\
VChunkJoin~\cite{DBLP:journals/tkde/WangQXLS13} & string join & Edit Distance & prefix+chunks & inverted index \\
PPJoin~\cite{DBLP:conf/www/XiaoWLY08,DBLP:journals/tods/XiaoWLYW11} & set join & Overlap & prefix, positional & inverted index \\
PPJoin+~\cite{DBLP:conf/www/XiaoWLY08,DBLP:journals/tods/XiaoWLYW11} & set join & Overlap & prefix, positional, suffix & inverted index \\
MPJoin~\cite{DBLP:journals/is/RibeiroH11} & set join & Overlap & min-prefix & inverted index \\
GroupJoin~\cite{DBLP:journals/pvldb/BourosGM12} & set join & Overlap & prefix+grouping & inverted index \\
AdaptJoin~\cite{DBLP:conf/sigmod/WangLF12} & set join & Overlap & adaptive prefix & inverted index \\
SKJ~\cite{DBLP:journals/pvldb/WangQLZC17} & set join & Overlap & prefix-based+set relations & inverted index \\
TopkJoin~\cite{DBLP:conf/icde/XiaoWLS09} & top-$k$ set join & Overlap & prefix-based & inverted index \\
JOSIE~\cite{DBLP:conf/sigmod/ZhuDNM19} & top-$k$ set search & Overlap & prefix, position & inverted index \\
\hline
PartEnum~\cite{DBLP:conf/vldb/ArasuGK06} & set join & Hamming, Jaccard & partition-based & clustered index \\
PassJoin~\cite{DBLP:journals/pvldb/LiDWF11} & string join & Edit Distance & partition-based & inverted index \\
PTJ~\cite{DBLP:journals/pvldb/DengLWF15} & set join & Overlap & partition-based & inverted index \\
\hline
B$^{ed}-$Tree~\cite{DBLP:conf/sigmod/ZhangHOS10} & string search/join & Edit Distance & string orders & B$^+$-tree \\
PBI~\cite{DBLP:journals/tkde/LuDHO14} & string search & Edit Distance & reference strings & B$^+$-tree \\
MultiTree~\cite{DBLP:conf/icde/ZhangLWZXY17} & set search & Jaccard & tree traversal & B$^+$-tree \\
Trie-Join~\cite{DBLP:journals/pvldb/WangLF10} & string join & Edit Distance & subtrie pruning & trie \\
HSTree~\cite{DBLP:journals/vldb/YuWLZDF17} & string search & Edit Distance & partition-based & segment tree \\
Trans~\cite{zhang2018transformation} & top-$k$ set search & Jaccard & transformation distance & R-tree \\
\hline
\multicolumn{5}{c}{\textbf{(a) Exact, centralized, single predicate algorithms}}\\
\hline
FuzzyJoin~\cite{DBLP:conf/icde/AfratiSMPU12} & set/string join & Hamming, ED, Jaccard & ball-hashing, splitting, anchor points & lookup tables \\
VernicaJoin~\cite{DBLP:conf/sigmod/VernicaCL10} & set join & Overlap & prefix, positional, suffix & inverted index \\
MGJoin~\cite{DBLP:journals/tkde/RongLWDCT13} & set join & Overlap & multiple prefix & inverted index \\
MRGroupJoin~\cite{DBLP:journals/pvldb/DengLWF15} & set join & Overlap & partition-based & inverted index \\
FS-Join~\cite{DBLP:conf/icde/RongLSWLD17} & set join & Overlap & segment-based & inverted index \\
Dima~\cite{DBLP:journals/pvldb/SunSLDB17,DBLP:journals/pvldb/SunS0BD19} & search, join, top-$k$ & Jaccard, ED & segment-based & global \& local \\
\hline
\multicolumn{5}{c}{\textbf{(b) Parallel \& distributed algorithms}}\\
\hline
ATLAS~\cite{DBLP:conf/sigmod/ZhaiLG11} & vector join & Jaccard, Cosine & random permutations & inverted index \\
BayesLSH~\cite{DBLP:journals/pvldb/SatuluriP12} & set join & Jaccard, Cosine & All-Pairs / LSH & All-Pairs / LSH \\
CPSJoin~\cite{DBLP:conf/icde/ChristianiPS18} & set join & Jaccard & LSH-based & sketches \\
\hline
\multicolumn{5}{c}{\textbf{(c) Approximate algorithms}}\\
\hline
LS-Join~\cite{DBLP:journals/tkde/WangYWL17} & local string join & Edit Distance & length, count & inverted index \\
pkwise~\cite{DBLP:conf/sigmod/WangXQWZI16} & local set join & Overlap & $k$-wise signatures & inverted index \\
pkduck~\cite{DBLP:journals/pvldb/TaoDS17} & abbreviation matching & Custom & extension of prefix filter & trie \\
\hline
Fast-Join~\cite{DBLP:journals/tods/WangLF14} & fuzzy set join & Bipart. graph matching & token sensitive signatures & inverted index \\
SilkMoth~\cite{DBLP:journals/pvldb/DengKMS17} & fuzzy set join & Bipart. graph matching & weighted token signatures & inverted index \\
MF-Join~\cite{DBLP:conf/icde/WangLZ19} & fuzzy set join & Bipart. graph matching & partion-based & inverted index \\
\hline
MultiAttr~\cite{DBLP:conf/sigmod/LiHDL15} & set search/join & Overlap & tree traversal & prefix tree \\
Smurf~\cite{DBLP:journals/pvldb/CADA18} & string matching & Jaccard, Edit Distance & random forest & inverted indexes \\
AU-Join~\cite{DBLP:journals/pvldb/XuL19} & string join & Syntactic, Synonym, Taxonomy & pebbles & inverted indexes \\
\hline
\multicolumn{5}{c}{\textbf{(d) Algorithms for complex matching}}\\
\end{tabular}
\vspace{-15pt}
}
\end{table*}
Given specific similarity predicates, comprising a similarity measure and a corresponding threshold, Filtering techniques receive as input an entity or a block collection and produce as output pairs of entities satisfying these predicates.
Next, we present the main filtering methods in the literature, organized in four groups: basic filters proposed by earlier works; prefix filtering and its extensions; partition-based filtering; and methods using tree indexes.
An overview of the discussed methods is presented in Table~\ref{tab:filtering_table}, characterized by the type of operation they perform (e.g., search or join), the similarity measure they assume (e.g., token- or character-based), the type of filters they use (e.g., prefix- or partition-based) and the index structure they employ (e.g., inverted index or tree).
\textbf{Basic filtering.}
\texttt{GramCount}~\cite{DBLP:conf/vldb/GravanoIJKMS01} focuses on incorporating string similarity joins inside a
DBMS based on $q$-grams and edit distance. It is the first work to propose
the following techniques:
\textit{Length filtering} states that if two strings $r$ and $s$ are within edit distance $\theta$, their lengths cannot differ by more than $\theta$. In the case of set similarity joins, the length filter has been adapted to deal with set sizes~\cite{DBLP:conf/vldb/ArasuGK06}; e.g., for Jaccard similarity threshold $\theta$, the condition becomes $\theta \cdot |s| \leq |r| \leq |s| / \theta$. Length filtering is a simple but effective criterion that is employed by many other works alongside more advanced filters.
A \textit{position-enhanced} length filter
offers a tighter upper bound
\cite{DBLP:conf/gvd/MannA14}.
\textit{Count filtering} states that if two strings $r$ and $s$ are within edit distance $\theta$, they must have at least $max(|r|, |s|) - 1 - (\theta - 1) \cdot q$ common $q$-grams. This filter has also been adapted to sets, in particular in \texttt{MergeOpt}~\cite{DBLP:conf/sigmod/Sarawagi04}, which proposed various optimizations for applying count filtering with both character-based and token-based similarity measures, and in \texttt{DivideSkip}~\cite{DBLP:conf/icde/LiLL08}, which proposed efficient techniques for merging the inverted lists of signatures.
\textit{Position filtering} also considers the positions of $q$-grams in the strings. It states that if two strings $r$ and $s$ are within edit distance $\theta$, a positional $q$-gram in one cannot correspond to a positional $q$-gram in the other that differs from it by more than $\theta$ positions.
On another line of research, \texttt{FastSS}~\cite{BoHuSt07}
introduces the concept of \textit{deletion neighborhood}, a filtering criterion specifically tailored to edit distance. For a string $s$, its deletion neighborhood contains substrings of $s$ derived by deleting a certain number of characters. These are then used as signatures for filtering. However, this method is practical only for very short strings.
\textbf{Prefix-based filtering.}
\textit{Prefix filtering} has been proposed by \texttt{SSJoin}~\cite{DBLP:conf/icde/ChaudhuriGK06}, which focuses on
similarity joins inside a DBMS,
and \texttt{All-Pairs}~\cite{DBLP:conf/www/BayardoMS07},
which is a main memory algorithm. Prefix filter applies to sets and can also be used for strings represented as sets of $q$-grams. The elements of each set are first sorted in a global order, typically in increasing order of frequency. Then, the $\pi$-prefix of each set is formed by selecting its $\pi$ first elements in that order. Prefix filter states that for two sets to be similar, their prefixes must contain at least one common element. The prefix size $\pi$ of a set $r$ is determined based on the similarity measure and threshold being used; e.g., for edit distance threshold $\theta$, $\pi = q \cdot \theta + 1$, while for Jaccard similarity threshold $\theta$, $\pi = \lfloor (1 - \theta) \cdot |r| \rfloor + 1$. As described next, numerous subsequent algorithms have adopted prefix filtering and proposed various optimizations and extensions over it, both for edit distance and set-based similarity joins.
For edit distance, \texttt{DivideSkip}~\cite{DBLP:conf/icde/LiLL08} uses prefix filtering in combination with length and position filtering,
taking special care to efficiently merge the inverted lists of signatures. \texttt{Ed-Join}~\cite{DBLP:journals/pvldb/XiaoWL08} proposes two optimizations
based on analyzing the locations and contents of mismatching $q$-grams to further reduce the prefix length by removing unnecessary elements. \texttt{QChunk}~\cite{DBLP:conf/sigmod/QinWLXL11} introduces the concept of \textit{$q$-chunks}, which are substrings of length $q$ that start at 1+$i$$\cdot$$q$ positions in the string, for $i \in [0, (|r|-1)/q]$. Given two strings $r$ and $s$, QChunk extracts $q$-grams from the one and $q$-chunks from the other;
if $r$ and $s$ are within edit distance $\theta$, the size of the intersection between the $q$-grams of $r$ and the $q$-chunks of $s$ should be at least $\lceil|s|/q\rceil$-$\theta$.
\texttt{VChunkJoin}~\cite{DBLP:journals/tkde/WangQXLS13} uses non-overlapping substrings called \textit{chunks}, ensuring that each edit operation destroys at most two chunks. This yields a tight lower bound on the number of common chunks that two strings must share if they match.
For set similarity joins,
\texttt{PPJoin}~\cite{DBLP:conf/www/XiaoWLY08,DBLP:journals/tods/XiaoWLYW11} extends prefix filtering with \textit{positional filtering}. This takes also into consideration the positions where the common tokens in the prefix occur, thus deriving a tighter upper bound for the overlap between the two sets. In addition, \texttt{PPJoin}+~\cite{DBLP:conf/www/XiaoWLY08,DBLP:journals/tods/XiaoWLYW11} further uses \textit{suffix filtering}. Following a divide-and-conquer strategy, this partitions the suffix of the one set into two subsets of similar sizes. The token separating the two partitions is called \textit{pivot} and is used to split the suffix of the other set. This allows to calculate the maximum number of tokens in each pair of corresponding partitions between the two sets that can match.
\texttt{MPJoin}~\cite{DBLP:journals/is/RibeiroH11} adds a further optimization over \texttt{PPJoin} that allows for dynamically pruning the length of the inverted lists. This reduces the computational cost of candidate generation, rather than the number of candidates. \texttt{GroupJoin}~\cite{DBLP:journals/pvldb/BourosGM12} extended \texttt{PPJoin} with \textit{group filtering}, whose candidate generation
treats all sets with identical prefixes as a single set.
Multiple candidates may thus be pruned in batches. \texttt{AdaptJoin}~\cite{DBLP:conf/sigmod/WangLF12} proposed \textit{adaptive prefix filtering}, which generalizes prefix filtering by adaptively selecting an appropriate prefix length for each set. It supports longer prefixes dynamically, extending their length by $n-1$, and then prunes a pair of sets if they contain less than $n$ common tokens in their extended prefixes. Prefix filtering is a special case where $n=1$.
A different perspective for speeding up set similarity joins is proposed by \texttt{SKJ}~\cite{DBLP:journals/pvldb/WangQLZC17}. The idea is based on the following observation: existing approaches examine each set individually when computing the join; however, it is possible to improve efficiency through computational cost sharing between \textit{related sets}. To this end, the \texttt{SKJ} algorithm introduces \textit{index-level skipping}, which groups related sets in the index into blocks, and \textit{answer-level skipping}, which incrementally generates the answer of one set from an already computed answer of another related set.
Finally, there are Filtering techniques
for computing top-$k$ results progressively, instead of requiring the user to select a similarity threshold.
\texttt{TopkJoin}~\cite{DBLP:conf/icde/XiaoWLS09} retrieves the top-$k$ pairs of sets ranked by their similarity score,
based on prefix filtering and
on the monotonicity of maximum possible scores of unseen pairs.
\texttt{JOSIE}~\cite{DBLP:conf/sigmod/ZhuDNM19}
presents a method for top-$k$ set similarity search. It exploits prefix and position filtering but,
instead of dealing with sets of relatively small size (e.g., $\sim$100 tokens), it
is crafted for
finding joinable tables in data lakes, where sets represent the distinct values of a table column,
comprising millions of tokens.
This introduces new challenges, which are tackled by proposing an algorithm that minimizes the cost of set reads and inverted index probes.
\textbf{Partition-based filtering.} The algorithms in this category partition each string or set into multiple disjoint segments in such a way that matching pairs have at least one common segment.
\texttt{PartEnum}~\cite{DBLP:conf/vldb/ArasuGK06} generates a signature scheme based on the principles of \textit{partitioning} and \textit{enumeration}. The former states that if two vectors with Hamming distance not higher than $k$ are partitioned into $k$ + 1 equi-sized partitions, then they must have at least one common partition. The latter states that if these vectors are partitioned instead into $n > k$ equi-sized partitions, then they must have in common at least $n - k$ partitions. \texttt{PassJoin}~\cite{DBLP:journals/pvldb/LiDWF11} partitions a string into a set of segments and creates inverted indices for the segments. Then, for each string, it selects some of its substrings and uses them to retrieve candidates from the index. A method is proposed to minimize the number of segments required to find the candidates pairs. \texttt{PTJ}~\cite{DBLP:journals/pvldb/DengLWF15} proposes an approach to increase the pruning power of partition-based filtering by using a mixture of the subsets and their 1-deletion neighborhoods, which are subsets derived from a set after eliminating one element.
Essentially, these methods are based on the \textit{pigeonhole principle}, which states that if $n$ items are contained in $m$ boxes,
at least one box has no more than $\lfloor n / m \rfloor$ items. This is extended by
the \textit{pigeonring principle}
\cite{DBLP:journals/pvldb/QinX18}, which organizes the boxes in a ring and constrains the number of items in multiple boxes rather than a single one, thus offering tighter bounds.
Applying it to various similarity search problems
shows that pigeonring always produces less or equal number of candidates than the pigeonhole principle does and that pigeonring-based algorithms can be implemented on top of existing pigeonhole-based ones with minor modifications \cite{DBLP:journals/pvldb/QinX18}.
\textbf{Tree-based filtering.} Most methods presented so far build inverted indexes on the signatures extracted from the strings or sets. Next, we present algorithms employing tree-based indexes.
Most approaches are based on the B$^+$-tree. \texttt{B}$^{ed}$-\texttt{Tree}~\cite{DBLP:conf/sigmod/ZhangHOS10} proposes a B$^+$-tree based index for range and top-$k$ similarity queries as well as similarity joins, using edit distance. It is based on a mapping from the string space to the integer space to support efficient searching and pruning. \texttt{PBI}~\cite{DBLP:journals/tkde/LuDHO14} uses a B$^+$-tree index and exploits the fact that edit distance is a metric. The string collection is partitioned according to a set of selected \textit{reference strings}. Then, the strings in each partition are indexed based on their distances to their corresponding reference strings. The proposed approach supports both range and $k$-NN queries and can be integrated inside a DBMS. In \texttt{MultiTree}~\cite{DBLP:conf/icde/ZhangLWZXY17}, each element in a set is represented as a vector and is mapped to an integer according to a defined global ordering, which is then used to insert the element in the B$^+$-tree index. Searching for similar elements is then done via a range query on the index.
On another line of research, \texttt{Trie-Join}~\cite{DBLP:journals/pvldb/WangLF10}
proposes a trie-based technique for string similarity joins with edit distance. Each trie node represents a character in the string. Thus, strings with a common prefix share the same ancestors.
A trie node is called an \textit{active node} of a string $s$ if their edit distance is not larger than the given threshold. This leads to a technique called \textit{subtrie pruning}:
given a trie $T$ and a string $s$, if node $n$ is not an active node for every prefix of $s$, then $n$'s descendants cannot be similar to $s$. \texttt{HSTree}~\cite{DBLP:journals/vldb/YuWLZDF17} recursively partitions strings into disjoint segments and builds a hierarchical segment tree index. This is then used to support both threshold-based and top-$k$ string similarity
search based on edit distance. Finally, a transformation-based framework for top-$k$ set similarity search is presented in~\cite{zhang2018transformation}. It transforms sets of various lengths into fixed-length vectors in such a way that similar sets are mapped closer to each other. An R-tree is then used to index these records and prune the space during search.
\subsection{Parallel \& Distributed Algorithms}
\label{subsec:filtering_distributed}
MapReduce-based approaches have been proposed to tackle scalability issues when dealing with very large collections of sets or strings. A theoretical analysis of different methods for performing similarity joins on MapReduce is presented in~\cite{DBLP:conf/icde/AfratiSMPU12}.
It considers algorithms that
operate in a single MapReduce job, avoiding the overhead associated with initiating multiple ones.
It shows that different algorithms provide different tradeoffs with respect to map, reduce and
communication~cost.
\texttt{VernicaJoin}~\cite{DBLP:conf/sigmod/VernicaCL10} is based on prefix filtering. It computes prefix tokens and builds an inverted index on them. Then, it generates candidate pairs from the inverted lists, using additionally the length, positional and suffix filters to prune candidates. A deduplication step is finally employed to remove duplicate result pairs generated from different reducers. \texttt{MGJoin}~\cite{DBLP:journals/tkde/RongLWDCT13} follows a similar approach to \texttt{VernicaJoin}, but introduces multiple prefix orders and a load balancing technique that partitions sets based on their length. \texttt{MRGroupJoin}~\cite{DBLP:journals/pvldb/DengLWF15} is a MapReduce extension of \texttt{PTJ}~\cite{DBLP:journals/pvldb/DengLWF15}. It applies a partition-based technique, where records are grouped by length and are partitioned in subrecords, such that matching records share at least one subrecord. The process is performed in a single MapReduce job. \texttt{FS-Join}~\cite{DBLP:conf/icde/RongLSWLD17} sorts the tokens in each set in increasing order of frequency, and then splits each set into disjoint subsets using appropriate pivot tokens. These subsets are then grouped together so that subsets from different groups are non-overlapping.
Finally, \texttt{Dima}~\cite{DBLP:journals/pvldb/SunSLDB17,DBLP:journals/pvldb/SunS0BD19} is a distributed in-memory system built on top of Spark that supports threshold and top-$k$ similarity search and join with both token-based and character-based similarities. It relies on signature-based global and local indexes for efficiency. The proposed signatures are adaptively selectable based on the workload, which allows to balance the workload among partitions. \texttt{Dima} extends the Catalyst optimizer of Spark SQL to introduce cost-based optimizations.
\subsection{Approximate Algorithms}
\label{subsec:filtering_approx}
Approximate algorithms for similarity search and join
increase the efficiency of Filtering step at the cost of allowing both false positives and false negatives, thus missing some matches
\cite{DBLP:journals/corr/WangSSJ14,DBLP:series/synthesis/2013Augsten}. They typically rely on \textit{locality sensitive hashing} (\textsf{LSH})~\cite{DBLP:conf/vldb/GionisIM99}, which transforms an item to a low-dimensional representation such that similar items have much higher probability to be mapped to the same hash code than dissimilar ones. This property allows \textsf{LSH} to be exploited in the filtering phase to generate candidates~\cite{DBLP:conf/vldb/LvJWCL07,DBLP:conf/sigmod/TaoYSK09,DBLP:conf/compgeom/DatarIIM04}. The basic idea is that each object is hashed several times using randomly chosen hash functions. Then, candidates are those pairs of objects that have been hashed to the same code by at least one hash function.
\texttt{ATLAS}~\cite{DBLP:conf/sigmod/ZhaiLG11} is a probabilistic algorithm that is based on random permutations both to generate candidates and to estimate the similarity between candidate pairs. It also proposes a method to efficiently detect cluster structures within the data, which are then exploited to search for similar pairs only within each cluster.
\texttt{BayesLSH}~\cite{DBLP:journals/pvldb/SatuluriP12} combines Bayesian inference with \textsf{LSH} to estimate similarities to a user-specified level of accuracy. It uses \textsf{LSH}
for both Filtering
and Verification,
providing probabilistic guarantees on the resulting
accuracy and recall. \texttt{CPSJoin}~\cite{DBLP:conf/icde/ChristianiPS18} is a randomized algorithm for set similarity joins. It uses a recursive filtering technique, building upon a previously proposed index for set similarity search~\cite{DBLP:conf/stoc/ChristianiP17}, as well as sketches for estimating set similarity. The algorithm has 100\% precision and provides a probabilistic guarantee on recall.
\subsection{Algorithms for Complex Matching}
\label{subsec:filtering_advanced}
The works discussed so far assume a single similarity predicate, i.e, they apply to the values of a specific attribute. Moreover, when comparing sets, they assume binary matching between their
elements,
while in the case of strings,
they compare strings in their entirety. In the following, we present methods that employ \textit{multiple} similarity predicates or more complex ones.
\textbf{Local matching.}
A local string similarity join
finds pairs of strings that contain similar \textit{substrings}. Under edit distance constraints, it can be defined as matching any $l$-length substring with up to $k$ errors. \texttt{LS-Join}~\cite{DBLP:journals/tkde/WangYWL17} is based on the observation that if two strings are locally similar, they must share at least one common $q$-gram, for a suitably calculated gram length $q$. An inverted index is constructed incrementally during the search. For every examined string, its $q$-grams are generated and the candidates are retrieved from the index by finding those strings that have matching $q$-grams.
\texttt{pkwise}~\cite{DBLP:conf/sigmod/WangXQWZI16} detects pieces of text in a given collection that share similar \emph{sliding windows}, i.e., multisets containing $w$ consecutive tokens of a given document. The similarity of two sliding windows is defined as the overlap of those sets. Prefix filtering is used but instead of relying on single tokens to build the signatures, it proposes \emph{$k$-wise signatures}, which comprise combinations of $k$ tokens. Larger values of $k$ increase the signatures' selectivity but also the cost of signature generation.
An additional optimization is to share common signatures across adjacent windows.
Finally, \texttt{pkduck}~\cite{DBLP:journals/pvldb/TaoDS17}
matches strings with \textit{abbreviations},
based on a new similarity measure
that accounts for
abbreviations.
It also proposes an appropriate signature scheme that extends prefix filtering and generates signatures without iterating over all strings derived from an
abbreviation.
\textbf{Fuzzy matching.} Rather than assuming a binary match, in this setting, the similarity between the elements of two sets may take any value between 0 and 1. In fact, it is defined
as the maximum matching score in the bipartite graph representing the matches between their elements.
In \texttt{Fast}-\texttt{Join}~\cite{DBLP:journals/tods/WangLF14}, edge weights in this bipartite graph denote the edit similarities between matching elements. The proposed method follows the filter-verification framework, creating a signature for each set such that matching sets have overlapping signatures. The signature of a set comprises an appropriately selected subset of its tokens. \texttt{SilkMoth}~\cite{DBLP:journals/pvldb/DengKMS17} generalizes and improves upon this work, providing a formal characterization of the space of valid signatures.
Given
that finding the optimal signature is NP-complete, it proposes heuristics to select signatures. To further reduce candidates, a refinement step is added: it compares each set with its candidates and rejects those for which certain bounds do not hold.
Both edit distance and Jaccard coefficient are supported
for measuring the similarity between elements. \texttt{MF-Join}~\cite{DBLP:conf/icde/WangLZ19} performs element-~and~record-level filtering. The former utilizes a partition-based signature scheme with a frequency-aware partition strategy, while the latter exploits count filtering and an upper bound on record-level similarity.
\textbf{Multiple predicates.} A method for similarity search and join on \textit{multi-attribute} data is presented in~\cite{DBLP:conf/sigmod/LiHDL15}. For instance, given an entity collection
where each entity is described
by its name and address, this work identifies pairs of entities having \textit{both} similar names and similar addresses. To enable simultaneous filtering on multiple attributes, a combined prefix tree index is built on these attributes. The construction of the index is guided by a cost model and a greedy algorithm. In another direction, \texttt{Smurf}~\cite{DBLP:journals/pvldb/CADA18} performs string matching between two collections of strings based on multiple-predicate matching conditions in the form of a \textit{random forest} classifier that is learned via active learning. Filtering techniques for string similarity joins are exploited to speed up the execution of the random forest. The focus and novelty of this work is on how to reuse computations across the trees in the forest to further increase efficiency. Finally, \texttt{AU}-\texttt{Join}~\cite{DBLP:journals/pvldb/XuL19} presents a new framework for string similarity joins that supports not only syntactic similarity measures, such as Jaccard similarity on $q$-grams, but also \textit{semantic} similarities, including \textit{synonym-based} and \textit{taxonomy-based} matching. It
partitions strings into segments and applies
different types of similarity measures on different pairs of segments. A new signature scheme, called \textit{pebble},
handles multiple similarity measures:
pebbles are $q$-grams for gram-based similarity,
the left-hand side of a synonym rule for synonym similarity, and
ancestor nodes in the taxonomy for taxonomy similarity.
\vspace{-10pt}
\subsection{Discussion \& Experimental Results}
\label{subsec:filtering_discussion}
Filtering techniques for string and set similarity joins have attracted a lot of research interest over the past two decades. Early works
view this operation as an extension of the standard join operator in relational databases, where the join condition is based on similarity rather than equality~\cite{DBLP:conf/vldb/GravanoIJKMS01,DBLP:conf/icde/ChaudhuriGK06}. The same perspective is shared by more recent works, like those
proposing B$^+$-tree based indexes, which
can be easily integrated into an existing DBMS
\cite{DBLP:conf/sigmod/ZhangHOS10,DBLP:journals/tkde/LuDHO14,DBLP:conf/icde/ZhangLWZXY17}. Another characteristic example is Dima~\cite{DBLP:journals/pvldb/SunSLDB17,DBLP:journals/pvldb/SunS0BD19}, which extends the Catalyst optimizer of Spark SQL to
support
similarity-based queries. In this sense, similarity joins are sometimes referred to as \textit{approximate} or \textit{fuzzy} joins, although this should not be confused with the approximate algorithms in Sec.~\ref{subsec:filtering_approx}, or the fuzzy set joins in Sec.~\ref{subsec:filtering_advanced}. Numerous Filtering techniques have been proposed by more recent works, which focus on main memory execution.
\textit{Prefix-based} filtering is the most popular
approach~\cite{DBLP:conf/icde/ChaudhuriGK06,DBLP:conf/www/BayardoMS07,DBLP:conf/icde/LiLL08,DBLP:journals/pvldb/XiaoWL08,DBLP:conf/sigmod/QinWLXL11,DBLP:journals/tkde/WangQXLS13,DBLP:conf/www/XiaoWLY08,DBLP:journals/tods/XiaoWLYW11,DBLP:conf/www/XiaoWLY08,DBLP:journals/tods/XiaoWLYW11,DBLP:journals/is/RibeiroH11,DBLP:journals/pvldb/BourosGM12,DBLP:conf/sigmod/WangLF12},
followed by
\textit{partition-based} filtering
\cite{DBLP:conf/vldb/ArasuGK06,DBLP:journals/pvldb/LiDWF11,DBLP:journals/pvldb/DengLWF15}. Furthermore, to scale similarity joins to large collections, distributed~\cite{DBLP:conf/icde/AfratiSMPU12,DBLP:conf/sigmod/VernicaCL10,DBLP:journals/tkde/RongLWDCT13,DBLP:journals/pvldb/DengLWF15,DBLP:conf/icde/RongLSWLD17,DBLP:journals/pvldb/SunSLDB17,DBLP:journals/pvldb/SunS0BD19} and approximate~\cite{DBLP:conf/sigmod/ZhaiLG11,DBLP:journals/pvldb/SatuluriP12,DBLP:conf/icde/ChristianiPS18} algorithms have been proposed.
More recently, there has been an increasing focus and interest on works that deal with more complex similarity predicates. These include the matching of strings based on \textit{substrings} or \textit{abbreviations}~\cite{DBLP:journals/tkde/WangYWL17,DBLP:conf/sigmod/WangXQWZI16,DBLP:journals/pvldb/TaoDS17}, matching of sets based on \textit{fuzzy matching} of their elements~\cite{DBLP:journals/tods/WangLF14,DBLP:journals/pvldb/DengKMS17,DBLP:conf/icde/WangLZ19}, and the combination of \textit{multiple} similarity predicates~\cite{DBLP:conf/sigmod/LiHDL15,DBLP:journals/pvldb/CADA18}. These works can be considered as more closely relevant to matching entity profiles in Entity Resolution.
Regarding performance,
a series of experimental analyses provides interesting insights \cite{DBLP:journals/pvldb/JiangLFL14,DBLP:journals/pvldb/MannAB16,DBLP:journals/pvldb/FierABLF18}.
However, each study focuses on a certain subset of the aforementioned methods.
Below, we
briefly summarize their findings,
including
additional results from individual papers to fill the gaps.
\textit{Similarity joins using Edit Distance}. A comparison between \texttt{FastSS}, \texttt{All-Pairs}, \texttt{DivideSkip}, \texttt{Ed-Join}, \texttt{QChunk}, \texttt{VChunkJoin}, \texttt{PPJoin}, \texttt{PPJoin}+, \texttt{AdaptJoin}, \texttt{PartEnum}, \texttt{PassJoin} and \texttt{Trie}-\texttt{Join} is conducted in~\cite{DBLP:journals/pvldb/JiangLFL14}. The results demonstrate that \texttt{PassJoin} is the most efficient
algorithm, with \texttt{FastSS} providing a reliable alternative
in the case of very short strings.
\textit{Similarity joins using set-based measures}. \texttt{AdaptJoin} and \texttt{PPJoin}+ are reported as the best algorithms in the aforementioned study~\cite{DBLP:journals/pvldb/JiangLFL14}. Different results, though, are reported in
a subsequent
study
that compares \texttt{All-Pairs}, \texttt{PPJoin}, \texttt{PPJoin}+, \texttt{MPJoin}, \texttt{AdaptJoin} and \texttt{GroupJoin}.
It indicates that the plain prefix filtering, i.e., \texttt{All-Pairs}, is still quite competitive,
winning in the majority of cases.
\texttt{PPJoin} and \texttt{GroupJoin}
exhibit
the best median and average performance, respectively, while
more sophisticated filters are found to provide only moderate improvements in some cases or even to negatively affect performance.
The difference with the
results
in \cite{DBLP:journals/pvldb/JiangLFL14} is attributed to the
more efficient
verification step;
reducing the cost of Verification means that
complex and, thus, time-consuming filters
often do not pay off, despite
reducing the number of candidate pairs.
\textit{Prefix vs. partition filtering}. \texttt{PTJ} is compared against \texttt{PPJoin+} and \texttt{AdaptJoin} in \cite{DBLP:journals/pvldb/DengLWF15}, showing that it outperforms both methods.
The same comparison is performed in \cite{DBLP:journals/pvldb/WangQLZC17}, showing that \texttt{PTJ} does not outperform those methods in most cases. As noted in~\cite{DBLP:journals/pvldb/WangQLZC17}, this discrepancy seems to be
caused by
differences in implementation; specifically, the comparison in~\cite{DBLP:journals/pvldb/DengLWF15}
uses the original implementations of \texttt{PPJoin+} and \texttt{AdaptJoin}, while the one in~\cite{DBLP:journals/pvldb/WangQLZC17} uses the optimized implementations provided by~\cite{DBLP:journals/pvldb/MannAB16}. Overall, \texttt{PTJ} may generate fewer candidates, but uses complex index structures, thus spending much more time on the filtering phase compared to prefix-based
algorithms. Another factor that affects the performance of prefix filtering is the frequency distribution of the tokens in the dataset. The core idea of prefix filtering is to select rare tokens as signatures
so as to reduce the number of candidates. In~\cite{DBLP:journals/pvldb/JiangLFL14}, an experiment
involving different dataset distributions shows that
\texttt{PPJoin(+)} and \texttt{AdaptJoin} perform better in datasets with Zipfian distribution than uniform one.
\textit{Set relations}. Another interesting finding is that set relations can be effectively exploited to speed up the computation of similarity joins~\cite{DBLP:journals/pvldb/WangQLZC17}. In the presented experiments, the proposed algorithm, \texttt{SKJ},
consistently outperforms \texttt{PPJoin}, \texttt{PPJoin+}, \texttt{AdaptJoin} and \texttt{PTJ} across all datasets.
\textit{Tree-based algorithms}. These algorithms typically focus on similarity search rather than join. \texttt{HSTree} and \texttt{PBI} are compared against \texttt{B}$^{ed}$-\texttt{Tree} in~\cite{DBLP:journals/vldb/YuWLZDF17} and~\cite{DBLP:journals/tkde/LuDHO14}, respectively, reporting better performance. Also, \texttt{Trans} shows better performance than \texttt{MultiTree} in~\cite{zhang2018transformation}. \texttt{BiTrieJoin}, an improved variant of \texttt{TrieJoin},
is reported in~\cite{DBLP:journals/pvldb/JiangLFL14} to have comparable performance to \texttt{PassJoin} for short strings, but it underperforms
for medium and long strings.
\textit{Distributed algorithms}. \texttt{VernicaJoin}, \texttt{MGJoin}, \texttt{MRGroupJoin} and \texttt{FS}-\texttt{Join} are experimentally compared in~\cite{DBLP:journals/pvldb/FierABLF18}.
\texttt{VernicaJoin} exhibits the best performance
in most cases, but
all algorithms are often outperformed by non-distributed ones. This should be attributed
to the overhead introduced by the MapReduce framework
as well as
to high or skewed data replication between map and reduce tasks. The latter constitutes
an inherent limitation of the distributed algorithms that cannot be overcome by simply increasing the number of nodes in the cluster. In~\cite{DBLP:journals/pvldb/SunS0BD19}, \texttt{Dima} is shown to outperform
the adaptation of \texttt{VernicaJoin}
to Apache Spark.
\textit{Approximate algorithms}. The experimental survey in~\cite{DBLP:journals/pvldb/JiangLFL14} included a comparison between \texttt{BayesLSH}-\texttt{lite} and exact algorithms. Moreover, \texttt{ATLAS}, \texttt{BayesLSH} and \texttt{CPSJoin} have been compared against \texttt{All-Pairs} in~\cite{DBLP:conf/sigmod/ZhaiLG11}, \cite{DBLP:journals/pvldb/SatuluriP12} and \cite{DBLP:conf/icde/ChristianiPS18}, respectively. Overall, the experiments indicate that approximate algorithms are preferable for low similarity thresholds, e.g., for Jaccard similarity below 0.5, while exact algorithms perform better for high thresholds.
\section{Join-based Blocking Methods}
\label{sec:hybrid}
\begin{figure}[t]\centering
\includegraphics[width=0.46\linewidth]{hybridTaxonomy.pdf}
\includegraphics[width=0.46\linewidth]{TimelineV2.png}
\vspace{-8pt}
\caption{(a) The taxonomy of the hybrid, join-based blocking methods. (b) Timeline of the landmarks in the evolution of {\color{blue}Blocking}, {\color{red}Filtering} and {\color{purple}their convergence}.
}
\label{fig:taxonomyTimeline}
\vspace{-14pt}
\end{figure}
We now elaborate on Block Building
methods that incorporate Filtering techniques,
converting Blocking into a nearest neighbor search. As illustrated in Figure \ref{fig:taxonomyTimeline}(a), we categorize these hybrid techniques into three major categories according to the filtering techniques they employ: the \textit{lossless} ones rely on exact, single predicate filtering techniques (cf. Table \ref{tab:filtering_table}(a)), the \textit{lossy} ones
rely on approximate filtering (cf. Section \ref{subsec:filtering_approx}), while the \textit{spatial} ones
leverage spatial join techniques for filtering. Note that the lossy hybrid methods are further distinguished into \textit{static} and \textit{dynamic} ones, depending on whether they are independent or interwoven with Matching, respectively.
Starting with the lossless hybrid methods, the simplest approach is to combine Prefix Filtering with Token Blocking, creating one block for every token that appears in the prefix of at least two entities \cite{DBLP:series/synthesis/2015Christophides}. Another approach is \textit{Adaptive Filtering} \cite{DBLP:conf/sdm/GuB04}, which couples schema-aware, non-learning Block Building techniques with two filtering methods. First, blocks are created by extracting keys from specific attributes. In every block with a size exceeding a predetermined threshold, Length and Count Filtering are applied for Comparison Cleaning, using an edit distance threshold on an attribute that is not considered by the initial transformation function.
Another lossless hybrid method is \textit{LIMES}, which
operates only on metric spaces \cite{DBLP:conf/ijcai/NgomoA11}. Its core idea is to
leverage the triangle inequality to approximate the distance between entities based on previous comparisons. Utilizing sets of entities as reference points, called \textit{exemplars}, this method computes lower and upper bounds to filter out superfluous comparisons before their execution.
In another direction, \textit{MultiBlock} \cite{isele2011efficient}
optimizes the execution of complex matching rules that comprise special similarity functions for textual, geographic and numeric values. A block collection is created for every similarity function such that similar entities
share multiple blocks. E.g., edit distance is supported for textual values and blocks are created for character $q$-grams such that entity pairs satisfying the distance threshold co-occur in a sufficient number of blocks. Then, all block collections are aggregated into a multidimensional index
that respects the co-occurrence patterns of similar entities and guarantees no false dismissals, i.e., $PC$=1.
Regarding the lossy approaches, they are dominated by techniques based
on \textsf{LSH} \cite{DBLP:conf/vldb/GionisIM99}, which
efficiently estimates the similarity between two attribute values $v_i$ and $v_j$ by randomly sampling hash functions $f$ from a \textit{sim-sensitive} family $F$ such that the probability $Pr(f(v_i) = f(v_j))$ equals to $sim(v_i, v_j)$ for any pair of attribute values and any function $f \in F$. This means that \textsf{LSH} derives $sim(v_i, v_j)$ from the proportion of hash functions $f$ such that $f(v_i)$ = $f(v_j$). Typically, the required number of these functions is relatively small for a sufficiently small sampling error; e.g., for 500 functions, the maximum sampling error is about $\pm$4.5\% with 95\% confidence interval \cite{DBLP:conf/semweb/DuanFHKSW12}.
In the context of ER, LSH is typically combined with MinHash signatures \cite{DBLP:conf/sequences/Broder97}, which efficiently estimate the Jaccard similarity as follows \cite{DBLP:journals/corr/abs-1907-08667,DBLP:conf/psd/SteortsVSF14}. Given an entity collection $\mathcal{E}$, the values of selected attribute names are converted into a bag of $k$-\textit{shingles}, i.e., $k$ consecutive words or characters. Then, a matrix $M$ of size $K \times |\mathcal{E}|$ is formed, with the rows corresponding to the $K$ distinct shingles that appear in all attribute values and the columns to the input entities. The value of every cell $M(i,j)$ indicates whether the entity $e_j$ contains the shingle $s_j$, $M(i,j)$=1, or not, $M(i,j)$=0. Given that $M$ is a sparse matrix, $p$ random minhash functions are used to reduce its dimensionality: they are applied to each column, deriving a new matrix $M'$ of size $p \times |\mathcal{E}|$.
The $p$ rows are then partitioned into $b$ non-overlapping bands and a hash function is applied to every band of each column. The resulting buckets are treated as blocks that provide probabilistic guarantees that the pairs of similar entities co-occur in at least one block. In fact, the desired probabilistic guarantees can be used for configuring the parameters of \textsf{LSH}, i.e., the number of hash functions, rows and bands \cite{DBLP:journals/corr/abs-1907-08667}.
In this context, LSH is combined with K-Means in \textsf{KLSH} \cite{DBLP:conf/psd/SteortsVSF14}.
KMeans is applied to the low-dimensional columns of $M'$, which represent the input entities. The resulting clusters form a disjoint block collection $\mathcal{B}$, with $|\mathcal{B}|$ determined by the desired average number of entities per block.
\textsf{LSH} is applied to the distributed representations (i.e., embeddings) of the input entities in \textit{DeepER} \cite{DBLP:journals/corr/abs-1710-00597}. Every entity is transformed into a dense, real-valued vector by aggregating the embeddings of all attribute value tokens, which are pre-trained by word2vec \cite{DBLP:conf/nips/MikolovSCCD13}, Glove \cite{DBLP:conf/emnlp/PenningtonSM14} etc. This vector is then hashed into multiple buckets with \textsf{LSH}. A block is then created for every entity containing its top-$N$ most likely matches, which are detected using Multiprobe-LSH \cite{DBLP:conf/vldb/LvJWCL07}.
\textsf{LSH} is also combined with a semantic similarity in \textsf{SA-LSH} (i.e., semantic-aware LSH) \cite{DBLP:journals/tkde/WangCL16}.
A taxonomy tree is used to model the concepts that describe the input entity collection.
The semantic similarity of two entities is inversely proportional to the length of the paths that connect the corresponding concepts and their children: the longer the paths, the lower the semantic similarity.
The concepts of every entity are converted into a hash signature through a semantic hashing algorithm. The resulting low-dimensional signatures are directly combined with the
signatures that are extracted from the n-grams of selected attribute values, capturing the textual similarity of entities. However, the construction of the taxonomy tree requires heavy human intervention.
Regarding the dynamic lossy methods, \textsf{LSH} is combined with \textsf{R-Swoosh} \cite{DBLP:journals/vldb/BenjellounGMSWW09}
in \cite{DBLP:conf/edbt/MalhotraAS14} through a MapReduce parallelization. Initially, a job is used for defining blocks using \textsf{LSH}. Then, a graph-parallel Pregel-based platform
applies R-Swoosh,
iteratively executing the non-redundant comparisons in the blocks and computing the transitive closure of the detected duplicates.
LSH also lies at the core of \textsf{cBV-HB} \cite{DBLP:conf/edbt/KarapiperisVVC16,DBLP:journals/kais/KarapiperisV16}, which embeds the textual values of selected attributes into a compact binary Hamming space that is efficient, due to the limited size of its embeddings (e.g., 120 bits for 4 attributes), and preserves the original distances in the sense that certain types of errors correspond to specific distance bounds. Special care is taken to support composite matching rules that involve the main logical operators (i.e., AND, OR and NOT).
Similarly, \textit{HARRA} \cite{DBLP:conf/edbt/KimL10}
uses \textsf{LSH} to hash similar entities into the same buckets.
Inside every bucket, all pairwise comparisons are executed and
duplicates
are merged into new profiles. The new profiles are hashed into the existing hash tables and the process is repeated until no entities are merged or another stopping criterion is met (e.g., the portion of merged profiles drops below a predetermined threshold). In every iteration, special care is taken to avoid redundant and superfluous comparisons.
Finally, spatial hybrid methods combine
spatial joins with Block Building. The core approach is \textit{StringMap}~\cite{DBLP:conf/dasfaa/JinLM03}, which converts schema-aware blocking keys to a similarity-preserving Euclidean space, whose dimensionality $d$ is heuristically derived from a random sample (typically, $d \in [15, 25]$). For each dimension, a linear algorithm initially selects two pivot attribute values that are (ideally) as far apart as possible. Subsequently, the coordinates of all other attribute values are determined through a comparison with the pivot strings. Using an R-tree or a grid-based index
in combination
with two weight
thresholds, similar attribute values are clustered together into overlapping blocks.
This approach is enhanced by \textit{Extended StringMap} \cite{DBLP:journals/tkde/Christen12}, which replaces the weight thresholds with cardinality ones, and the \textit{Double embedding scheme}~\cite{DBLP:conf/dmin/Adly09}. The latter initially maps the input entities to the same $d$-dimensional Euclidean space. Next, the embedded attribute values are mapped to another Euclidean space of lower dimensionality $d' < d$. A similarity join is performed in the second Euclidean space using a $k$-d tree index. The resulting candidate matches are then clustered in the first, $d$-dimensional Euclidean space. The experimental study suggests that the $d'$-dimensional space significantly reduces the runtime of StringMap by 30\% to 60\%.
\section{Blocking vs Filtering: Commonalities and Differences}
\label{sec:discussion}
The timeline in Figure \ref{fig:taxonomyTimeline}(b)
summarizes the landmarks in the evolution of the two frameworks
showing their gradual convergence.
We observe that Blocking is the oldest discipline, with the first relevant technique, namely \textsf{SB}, presented in 1969 \cite{fellegi1969theory}. For several decades, research focused on schema-based techniques, with the most significant breakthrough taking place in 1995, with the introduction of \textsf{SN} \cite{DBLP:conf/sigmod/HernandezS95}. The first schema-agnostic Block Building technique is \textit{Semantic Graph Blocking} \cite{DBLP:conf/ideas/NinMML07}, introduced in 2007, but it considers only entity links. In 2011, it was followed by
\textsf{TB} \cite{DBLP:conf/wsdm/PapadakisINF11}, which exclusively applies to textual values. Block Processing was introduced in 2009 by Iterative Blocking \cite{DBLP:conf/sigmod/WhangMKTG09}, followed by the use of Canopy Clustering for Blocking in 2012 \cite{DBLP:journals/tkde/Christen12} and the introduction of Meta-blocking in 2014 \cite{DBLP:journals/tkde/PapadakisKPN14}.
For Filtering, the first similarity join to be used in an RDBMS can be traced back to 2001 \cite{DBLP:conf/vldb/GravanoIJKMS01}, while the techniques for in-memory execution were coined in 2007 \cite{DBLP:conf/www/BayardoMS07}. Attempts to further increase efficiency by allowing approximate results were first presented in 2011~\cite{DBLP:conf/sigmod/ZhaiLG11}. The first works on massive parallelization for Filtering appear in 2010 \cite{DBLP:conf/sigmod/VernicaCL10}, for Blocking in 2012 \cite{DBLP:journals/pvldb/KolbTR12}, and for Block Processing in 2015 \cite{DBLP:conf/bigdataconf/Efthymiou0PSP15}. The convergence of the two frameworks essentially starts in 2011 with MultiBlock \cite{isele2011efficient}, which introduces Join-based Blocking,
whereas
multiple-predicate Filtering for efficient Matching,
is introduced by \textsf{Smurf} in 2018 \cite{DBLP:journals/pvldb/CADA18}.
Regarding the qualitative comparison of the two frameworks, we observe that they
have a number of commonalities: (i) Both serve the same purpose: they increase ER efficiency by reducing the number of performed comparisons. To this end, both employ a stage producing candidate matches, which are subsequently examined analytically in order to remove false positives. (ii) Both usually operate either on two clean but overlapping data collections (Record Linkage for Blocking, Cross-table Join for Filtering) or on a single dirty data collection (Deduplication for Blocking, Self-join for Filtering). (iii) Both extract signatures such that the similarity of two entities is reflected in the similarity of their signatures. (iv) Both also apply similar implementation-level optimizations, representing signatures with integer ids, instead of strings, so as to reduce the memory footprint and facilitate in-memory execution. (v) Both include character- and token-based methods. For Blocking, the former methods mainly pertain to schema-aware techniques that apply character-level transformations to blocking keys (e.g., $q$-grams, suffixes etc), while token-based methods primarily pertain to schema-agnostic methods. For Filtering, similarity measures can also be distinguished between character-based (e.g., edit similarity) and token-based ones (e.g., Jaccard), even though many algorithms can be adapted to handle both. (vi) In both cases, textual data have been combined with other types of data, particularly with spatial or spatio-temporal data, including \cite{DBLP:conf/semweb/Ngomo13} for Blocking and \cite{DBLP:reference/db/Gao09, DBLP:journals/tods/JacoxS07, DBLP:journals/pvldb/BourosGM12, DBLP:journals/vldb/BelesiotisSEKP18} for Filtering. (vii) Both
can be used in real-time applications, where the input comprises a query entity and the goal is to identify the most similar ones in the minimum possible time. This is called Similarity Search in the case of Filtering and Real-time ER in the case of Blocking (see Section \ref{sec:futureDirections} for more details).
Due to these commonalities, several works use the two frameworks interchangeably, considering Filtering as a means for Blocking (e.g., \cite{DBLP:series/synthesis/2015Christophides}).
In reality, though, Blocking and Filtering have several distinguishing characteristics: (i)
By definition, a blocking scheme applies to a single entity, considering all its attribute values (schema-agnostic methods), or combinations of multiple values (schema-aware techniques). In contrast, Filtering usually applies to a pair of values from the same attribute of two entities. (ii) Blocking relies on positive evidence, clustering together similar entities, while Filtering relies on negative evidence, detecting dissimilar entities early on. (iii) Blocking is typically independent of Entity Matching, whereas Filtering is interwoven with it, as its goal is to optimize the execution of a matching rule. (iv) Blocking is an inherently approximate procedure that falls short of perfect recall ($PC$), even when providing probabilistic guarantees (e.g., LSH Blocking in DeepER \cite{DBLP:journals/corr/abs-1710-00597}). In contrast, most Filtering methods provide an exact solution, returning all pairs of values that exceed the predetermined threshold along with false positives. (v) Blocking trades slightly lower recall ($PC$) for much higher precision ($PQ$), while Filtering trades filtering power for filtering cost. (vi) Blocking may be modelled as a learning problem, where the goal is to define supervised blocking schemes that simultaneously optimize $PC$, $PQ$ and $RR$, but Filtering requires no labelled set for learning to mark a comparison as true negative. Instead, it relies on a theoretical analysis based on the given similarity measure and threshold. (vii) Preserving privacy is orthogonal to Filtering, with very few works examining privacy-preserving similarity joins \cite{DBLP:conf/icde/LiC08,DBLP:journals/dke/KantarciogluIJM09,DBLP:journals/tifs/YuanWWYN17}. In contrast, Blocking constitutes an integral part of privacy-preserving ER, with several relevant works (for details, refer to a recent survey~\cite{DBLP:journals/is/VatsalanCV13}). (viii) Blocking constitutes an integral part of pay-as-you-go ER applications, conveying a significant body of relevant works, as described below. This does not apply to Filtering, given that the only relevant technique is TopkJoin~\cite{DBLP:conf/icde/XiaoWLS09}.
Regarding the quantitative comparison between
Blocking and Filtering, few works have actually examined their relative performance. The two frameworks are experimentally juxtaposed in \cite{DBLP:conf/semweb/SongH11,DBLP:journals/tkde/SongLH17,DBLP:conf/semweb/Song12} in terms of effectiveness and time efficiency.
Using a series of real-world datasets,
\textsf{RDFKeyLearner} is compared against \textsf{AllPairs}, \textsf{PPJoin}(+) and \textsf{EdJoin} in \cite{DBLP:conf/semweb/SongH11,DBLP:conf/semweb/Song12}
and against \textsf{EdJoin}, \textsf{PPJoin+} and \textsf{FastJoin} in \cite{DBLP:journals/tkde/SongLH17}.
All methods are fine-tuned using a sample of each dataset.
The outcomes indicate no significant difference in effectiveness,
but regarding time efficiency, Filtering is consistently faster in generating candidate matches and consistently slower in executing the corresponding pairwise comparisons, due to their larger number. In \cite{DBLP:journals/tkde/SongLH17}, the relative scalability of \textsf{RDFKeyLearner} and \textsf{EdJoin} is examined over synthetic datasets of 10$^5$, 2$\cdot$10$^5$, ..., 10$^6$ entities. Again, \textsf{EdJoin} produces more candidate matches and, thus, is slower than \textsf{RDFKeyLearner}.
In \cite{DBLP:journals/pvldb/KopckeTR10}, an experimental analysis over 4 real-world datasets investigates the combined effect of Blocking and Filtering on ER efficiency, implementing the workflow in Figure \ref{fig:computationalCostPlusWorkflow}(b). The results suggest that together, the two frameworks reduce the overall ER running time from 33\% to 76\%, with an average of 50\%. However, only one method per framework is considered:
the manually fine-tuned \textsf{SB} and \textsf{PPJoin} in combination with Cosine and Jaccard similarity. Note that, due to its careful, manual fine-tuning, Blocking has no impact on ER effectiveness.
However, more experimental analyses are required for drawing safe conclusions about the relative performance of Blocking and Filtering. These analyses should include
a large, representative variety of techniques per framework along with several established benchmark datasets and should examine the benefits of combining the two frameworks in more depth.
\section{Blocking and Filtering in Entity Resolution Systems}
\label{sec:tools}
We now present the main systems that address ER, examining whether they incorporate any of the aforementioned methods to improve the runtime and the scalability of their workflows. We analytically examined the 18 non-commercial and 15 commercial systems listed in the extended version of \cite{konda2016magellan}\footnote{The extended version of \cite{konda2016magellan} is available here: \url{http://pages.cs.wisc.edu/~anhai/papers/magellan-tr.pdf}.} along with the 10 Link Discovery frameworks surveyed in \cite{DBLP:journals/semweb/NentwigHNR17}.
Table \ref{tab:LinkDiscoveryToolkits} summarizes the characteristics of 12 open-source ER systems that include at least one Blocking~or~Filtering~method.
Half of the tools offer a graphical user interface and are implemented in Java. Regarding the type of the input data, most systems support structured data. The only exceptions are the three Link Discovery frameworks, which are crafted for semi-structured data. JedAI is the only tool that applies uniformly to both structured and semi-structured data.
We also observe that all systems include Blocking methods, with Standard Blocking (\textsf{SB}) and Sorted Neighborhood (\textsf{SN}) being the most popular ones. The first four systems are Link Discovery frameworks that implement custom approaches: KnoFuss and SERIMI apply Token Blocking only to the literal values of RDF tiples, while Silk and LIMES implement hybrid methods, MultiBlock and LIMES, respectively (see Section \ref{sec:hybrid}).
Febrl and JedAI offer the largest variety of established techniques. The former provides their original, schema-aware implementation, while the latter provides their schema-agnostic adaptations. For this reason, JedAI is the only tool that implements Block Processing techniques, as well.
Note that Block Building is also a core part of the ER workflow in several commercial systems, such as IBM Infosphere and Informatica Data Quality \cite{konda2016magellan}. These systems are generally required to handle diverse types of data, focusing on data exploration and cleaning. They typically provide variations of \textsf{SB}, allowing users to extract blocking keys from specific attributes through a sophisticated GUI that provides statistics and data analysis. As a result, users' expertise and experience with specific domains is critical for the performance of these systems' blocking components.
Surprisingly, only two systems currently include Filtering algorithms for improving the runtime of their matching process: LIMES and Magellan. The latter actually offers the largest variety of established techniques through the \texttt{py\_stringsimjoin} package. Filtering techniques are also provided by FEVER \cite{DBLP:journals/pvldb/KopckeTR09}, which is a closed-source ER tool, as well as by JedAI's forthcoming version 3. Still, a mere minority of ER tools enables users to combine the benefits of Blocking and Filtering, despite the promising potential of their synergy (see below for more details). Most importantly, these tools exclusively consider traditional Filtering algorithms that apply to the values of individual attributes. Hence, they disregard the recent Filtering techniques for Complex Matching (cf. Section \ref{subsec:filtering_advanced}), which are more suitable for Entity Resolution. Therefore, more effort should be devoted on developing ER tools that make the most of the synergy between Blocking and Filtering.
\begin{table*}[tbp]
\centering
\caption{Blocking and Filtering methods in open-source systems for Entity Resolution.
}
\label{tab:LinkDiscoveryToolkits}
\vspace{-10pt}
\begin{scriptsize}
\begin{tabular}{|p{1.5cm}|p{4cm}|p{1.5cm}|p{0.4cm}|p{1.0cm}|p{3cm}|}
\toprule
\textbf{Tool} & \textbf{Blocking} & \textbf{Filtering} & \textbf{GUI} & \textbf{Language} & \textbf{Data Formats} \\
\midrule
KnoFuss \cite{nikolov2007knofuss} & Literal Blocking & - & No & Java & RDF, SPARQL \\ \hline
SERIMI \cite{DBLP:journals/tkde/AraujoTVS15} & Literal Blocking & - & No & Ruby & SPARQL \\ \hline
Silk \cite{volz2009silk} & Multiblock & - & Yes & Scala & RDF, SPARQL, CSV \\ \hline
LIMES \cite{DBLP:conf/ijcai/NgomoA11} & custom methods & PPJoin+, EdJoin,
custom methods, e.g., ORCHID \cite{DBLP:conf/semweb/Ngomo13}
& Yes & Java & RDF, SPARQL, CSV \\ \hline
Dedupe \cite{bilenko2003adaptive} & SB with learning-based techniques & - & No & Python & CSV, SQL \\ \hline
DuDe \cite{draisbach2010dude} & SB, \textsf{SN}, Sorted blocks & - & No & Java & CSV, JSON, XML, BibTex, Databases(Oracle, DB2, MySQL and PostgreSQL) \\ \hline
Febrl \cite{christen2008febrl} & SB, \textsf{SN}, Sorted Blocks,
Suffix Arrays, Extended Q-Grams, Canopy Clustering, StringMap& - & Yes & Python & CSV, text-based \\ \hline
FRIL \cite{jurczyk2008fine} & SB, \textsf{SN} & - & Yes & Java & CSV, Excel, COL, Database \\ \hline
OYSTER \cite{nelson2011entity} & SB & - & No & Java & text-based \\ \hline
RecordLinkage \cite{sariyar2011controlling} & SB (with SOUNDEX) & - & No & R & Database \\ \hline
Magellan \cite{konda2016magellan} & SB, \textsf{SN}, it also supports user-specified blocking methods & Overlap, Length, Prefix, Position, Suffix & Yes & Python & CSV \\ \hline
JedAI \cite{DBLP:journals/pvldb/PapadakisTTGPK18} & SB, \textsf{SN}, Extended \textsf{SN}, Suffix Arrays, Extended Suffix Arrays, LSH, Q-Grams, Extended Q-Grams + Block Processing & to be added in the forthcoming version 3 & Yes & Java & CSV, RDF, SPARQL, XML, Database \\
& & & & & \\
\bottomrule
\end{tabular}
\end{scriptsize}
\vspace{-12pt}
\end{table*}
It is worth noting that Filtering plays an important role in modern systems.
For example, \textit{Corleone} \cite{DBLP:conf/sigmod/GokhaleDDNRSZ14} introduced a novel filtering approach that leverages machine learning: active learning is used to minimize the number of examples labeled by users, and then, random forests are trained to learn the matching rules that will be used in filtering. The resulting robust model is scaled by \textit{Falcon} \cite{DBLP:conf/sigmod/DasCDNKDARP17}, while \textit{CloudMatcher} \cite{DBLP:journals/pvldb/GovindPNCDPFCCS18} provides an end-to-end implementation based on this approach. These systems have been successfully applied to real-world domains~\cite{DBLP:conf/sigmod/GovindKCMNLSMBZ19}.
\section{Future Directions}
\label{sec:futureDirections}
Various directions seem promising for future work, from entity evolution \cite{DBLP:conf/jcdl/PapadakisGNPN11} to deep learning \cite{DBLP:journals/corr/abs-1710-00597} and summarization algorithms \cite{DBLP:conf/edbt/KarapiperisGV18}, which minimize the memory footprint of blocks, while accelerating their processing. The following are more mature fields, having assembled a critical mass of methods already.
\textbf{Progressive Entity Resolution.} Due to the constant increase of data volumes, new \textit{progressive} or \textit{pay-as-you-go} ER applications have emerged. Their goal is to provide the best possible \textit{partial solution} within a limited budget of temporal or computational resources. In such applications, Blocking lays the ground for \textit{Prioritization}, which schedules the processing of entities, comparisons or blocks according to the likelihood that they involve duplicates. We distinguish the relevant techniques into schema-aware and schema-agnostic ones.
The schema-aware progressive methods require domain knowledge \cite{DBLP:journals/tkde/PapenbrockHN15,DBLP:journals/tkde/WhangMG13}. \textit{Progressive Sorted Neighborhood} (PSN) \cite{DBLP:journals/tkde/WhangMG13} uses schema-based \textsf{SN} to create a sorted list of entities and then applies an incremental window size $w$. Starting from the top of the list, all entities in consecutive positions ($w$=1) are compared; then, all entities at distance $w$=2 are compared and so on and so forth, until reaching the user-defined budget. \textit{Dynamic PSN} \cite{DBLP:journals/tkde/PapenbrockHN15} extends this static approach by adjusting the processing order of comparisons on-the-fly, according to the results of a perfect matcher. It arranges the sorted entities in a two-dimensional array $A$, and if $A(i, j)$ corresponds to a pair of duplicates, the processing moves on to check $A(i+1, j)$ and $A(i, j+1)$, as well. \textit{Progressive Blocking} \cite{DBLP:journals/tkde/PapenbrockHN15} generalizes this principle to \textsf{SB}. \textit{Hierarchy of Record Partitions} \cite{DBLP:journals/tkde/WhangMG13} creates a static hierarchy of blocks, where the matching likelihood of two entities is proportional to the level in which they co-occur for the first time. This hierarchy is then progressively resolved, level by level, from leaves to root. A variation of this approach is adapted to MapReduce in \cite{DBLP:conf/icde/AltowimM17}, while the \textit{Ordered List of Records} \cite{DBLP:journals/tkde/WhangMG13} converts it into a list of entities that are sorted by their likelihood to produce matches. A progressive solution for relational Multi-source ER over different entity types is proposed in \cite{DBLP:journals/pvldb/AltowimKM14}. Finally, \textsf{P-RDS} adapts LSH-based blocking to a progressive functionality by rearranging the processing order of its hash tables according to the number of matching and unnecessary comparisons in their buckets that have been resolved so far.
The schema-agnostic methods, which disregard any domain knowledge, are classified into two
types \cite{simonini2018schema}: (i) The \textit{sort-based methods} order all entities alphabetically, according to their attribute value tokens, leveraging schema-agnostic SN. \textit{Local Schema-agnostic Progressive SN} \cite{simonini2018schema} slides an incremental window over the sorted list of entities and, for each window size, it orders the non-redundant comparisons according to the co-occurrence frequency of their entities and the number of blocking keys per entity. \textit{Global Schema-agnostic Progressive SN} \cite{simonini2018schema} does the same, but for a predetermined range of windows, eliminating all redundant comparisons they contain. (ii) The \textit{hash-based methods} leverage the blocking graph for Prioritization. \textit{Progressive Block Scheduling} \cite{simonini2018schema} orders the blocks in ascending number of comparisons and then prioritizes all comparisons per block in decreasing edge weight. \textit{Progressive Profile Scheduling} \cite{simonini2018schema} orders entities in decreasing average edge weight and then prioritizes all comparisons per entity in decreasing edge weight.
The schema-agnostic methods excel in recall and precision \cite{simonini2018schema}, but exclusively support \textit{static} prioritization, defining an immutable processing order that disregards the detection of duplicates. Hence, more research is needed for developing \textit{dynamic schema-agnostic} progressive methods.
\textbf{Real-time Entity Resolution.} This is the task of matching an entity that is given as query
to the available entity collections in (ideally) sub-second run-time. To meet this goal, several
specialized \textit{dynamic indexing} techniques have been proposed in the literature. An early approach is presented in \cite{christenDI}. The core idea is to pre-calculate similarities between the attribute values of entities co-occurring in the blocks of Standard Blocking, thus avoiding similarity calculations at query time. Three indexes are created for this purpose, containing all the necessary information. This approach is extended by \textit{DySimII} \cite{10.1007/978-3-642-40319-4_5} so that all three indexes are updated as query entities arrive. The experimental results demonstrate that both the average record insertion time and the average query time remain practically stable, even when the index size grows.
Another family of relevant techniques extends SN. \textit{F-DySNI} \cite{DBLP:conf/cikm/RamadanC14,Ramadan:2015:DSN:2836847.2816821} converts the sorted list of blocking keys into an index that is faster to search: it creates a braided AVL tree \cite{rice2007braided} that combines a height balanced binary tree with a double-linked list, where every node is linked to its alphabetically sorted predecessor node, to its successor node and to the ids of all entities that correspond to its blocking key. There is one tree for each blocking key definition that gets updated whenever a query entity arrives. The window is fixed or adaptive, considering as neighbors the nodes that exceed a specific similarity threshold. F-DySNI is extended in \cite{DBLP:conf/pakdd/RamadanC15} with an automatic approach for selecting blocking keys; the weak training set of \cite{DBLP:conf/icdm/KejriwalM13} is coupled with a scoring function that assesses the coverage of each key along with the distribution of its block sizes.
Another group of methods relies on LSH. MinHash LSH is combined with SN in \cite{DBLP:conf/pakdd/LiangWCG14}: when searching for the nearest neighbors of a query entity, the entities in large LSH blocks are sorted via a custom scoring function and, then, a window of fixed size slides over the sorted list of entities. \textsf{CF-RDS} \cite{DBLP:journals/datamine/KarapiperisGV18} leverages Hamming LSH, ranking the most similar entities to each query without performing any profile comparison. Instead, it merely aggregates the number of occurrences of each candidate match in the buckets associated with the query entity.
On another line of research, \textit{BlockSketch} \cite{DBLP:conf/edbt/KarapiperisGV18} organizes the entities inside every block into sub-blocks according to their similarity. A representative is assigned to each sub-block based on its distance from the corresponding blocking key. In this way, every query suffices to be compared with a constant number of entities in the target block in order to detect its most similar entities. \textit{SBlockSketch} \cite{DBLP:conf/edbt/KarapiperisGV18} adapts this approach to a stream of query entities through an eviction strategy that bounds the number of blocks that need to be maintained in memory.
All these methods are crafted for structured data, assuming a fixed schema of known quality.
New techniques are required, though, for the noisy, heterogeneous entities of semi-structured data.
\textbf{Parameter Configuration.} Except \textsf{TB}, all Blocking methods involve at least one internal parameter that affects their performance to a large extent \cite{DBLP:journals/tkde/Christen12,DBLP:journals/pvldb/0001SGP16}. This affects their relative performance, rendering the selection of the best performing method for the data at hand into a non-trivial task.
To mitigate this issue, parameter fine-tuning is modelled as an optimization problem in \cite{DBLP:journals/tlsdkcs/MaskatPE16}. The large, heterogeneous space of possible configurations is searched through a genetic algorithm, whose fitness function exploits the labels (i.e., \texttt{match} vs \texttt{non-match}) of part of the candidate matches. After applying the typical series of genetic operators, (i.e., mutation, crossover, elite capture and parental selection) is applied for a specific number of generations, the configuration maximizing the fitness function is selected as optimal. However, this approach involves a large number of parameters itself. In another direction, \textit{MatchCatcher} \cite{DBLP:conf/edbt/LiKCDSPKDR18} implements a human-in-the-loop approach combining expert knowledge with labelled instances in order to learn composite blocking schemes. Using string similarity joins, duplicates sharing no block are efficiently detected. To capture them, the expert user adapts the transformation and assignment functions iteratively. Finally, a method’s performance over several labelled datasets is used for fine-tuning its parameters over a given unlabelled dataset in \cite{o2018new}. At its core lies a two-dimensional metric space formed by the overall running time and F-Measure (horizontal and vertical axis, respectively). The closer a method is mapped to the ideal point (0,1), the better is its performance. A graph is then built such that every node corresponds to a different configuration or blocking method, while a directed edge points from node $n_i$ to $n_j$ if $n_j$ is closer to (0,1). The node with no outgoing edges or the largest difference between incoming and outgoing edges corresponds to the best choice. However, this is a rather time-consuming approach, given the large number of computations it requires.
None of the above methods satisfies the requirement for automatic, data-driven, a-priori parameter configuration of Blocking methods, which thus remains an open problem.
\textbf{Filtering for Entity Resolution.} We believe that more opportunities exist for transferring ideas and approaches between Blocking and Filtering. Another interesting direction is to investigate in practical settings to what extent similarity joins suffice for ER, i.e., representing entity profiles by strings or sets and defining a matching function based on a similarity threshold. We expect that techniques supporting relaxed matching criteria and/or lower similarity thresholds will be required to achieve high recall. Yet, as explained in Section \ref{subsec:filtering_advanced}, relatively few Filtering techniques are designed for these cases. Moreover, scalability remains an open challenge for string and set similarity joins, as shown in \cite{DBLP:journals/pvldb/FierABLF18}. Finally, there is a need for extensible, open-source ER tools that incorporate the majority of established Blocking and Filtering methods and apply seamlessly to structured, semi-structured and unstructured data \cite{DBLP:conf/pods/GolshanHMT17}.
\section{Conclusions}
\label{sec:conclusions}
Efficiency techniques are an integral part of Entity Resolution, since its infancy. We organize the
relevant works
into Blocking, Filtering and hybrid techniques, facilitating their understanding and use. We also provide an in-depth coverage of each category, further classifying its
works into novel sub-categories. Lately,
the rise of big semi-structured data
poses challenges
to the scalability of efficiency techniques and
to their core assumptions: the requirement of Blocking for schema knowledge and of Filtering for high similarity thresholds. The former led to the introduction of schema-agnostic Blocking and of Block Processing techniques, while the latter led to
more relaxed criteria of similarity. We cover these new fields in detail, putting in context all relevant works.
\vspace{4pt}
\noindent
\textbf{Acknowledgements.} This work was partially funded by EU H2020 projects ExtremeEarth (825258) and SmartDataLake (825041).
\def\thebibliography#1{
\section*{References}
\vspace{-2pt}
\scriptsize
\list
{[\arabic{enumi}]}
{\settowidth\labelwidth{[#1]}
\leftmargin\labelwidth
\parsep 0pt
\itemsep 0pt
\advance\leftmargin\labelsep
\usecounter{enumi}
}
\def\hskip .11em plus .33em minus .07em{\hskip .11em plus .33em minus .07em}
\sloppy\clubpenalty10000\widowpenalty10000
\sfcode`\.=1000\relax
}
\balance
\bibliographystyle{abbrv}
|
1,314,259,993,633 | arxiv | \section{Introduction}
Magnetite (Fe$_3$O$_4$) is the oldest known magnet and a
fascinating material both for understanding the fundamental physics that emerge from electronic correlations,
and for novel technologies related to oxide electronics \cite{Orozco-1999,Wu-2013,Liu-2013}.
At ambient conditions, it crystallizes in the inverse spinel structure with cubic $Fd\overline{3}m$ symmetry.
The O atoms form a fcc lattice, with Fe$_A$ atoms in tetrahedral sites acting with a nominal $+3$ valence, and
Fe$_B$ cations with $+2.5$ valence in octahedral positions.
The Fe$_A$ and Fe$_B$ sublattices are antiferromagnetically coupled,
and the minority spin $t_{2g}$ states of the Fe$_B$ atoms cross the Fermi level,
leading to a half-metallic ferrimagnet with high
magnetic moment, 4 $\mu_B$ per formula unit (f.u.).
At a critical temperature T$_V \sim 120$ K, magnetite undergoes the first-order Verwey transition (VT),
that manifests in a structural modification to a monoclinic symmetry
accompanied by a drop of the conductivity of 2-3 orders of magnitude \cite{Verwey}. The decrease of the conductivity
is due to a freezing of the electron hopping between different octahedral Fe sites, causing
a charge disproportionation that results in two types of Fe$_B$ atoms acting with a slightly enhanced (Fe$^{3+}$)
or reduced (Fe$^{2+}$) valence. The distribution of the different Fe$_B$ atoms at the unit cell configures
the charge order (CO), intimately linked to the orbital order \cite{Leonov-2004,Huang-2006},
and determines the full monoclinic $Cc$ symmetry \cite{Iizumi-1982,Jeng-2006,Yamauchi-2009,Garcia-2011,Attfield-nature}.
Decades of efforts have been devoted to the understanding of the VT \cite{Walz-2002,Garcia-2004},
and yet some puzzling fundamental aspects remain
unanswered, such as the structural or electronic origin of the transition, or the extent of the
short range order above T$_V$. The present consensus is that
the phase transition is governed by electron-phonon couplings in the presence of strong electronic correlations
\cite{Piekarz-2013}.
A local perturbation of the extended CO has been recently identified in the form of trimerons: linear chains of three adjacent
Fe$_B$ cations dominantly formed by a central Fe$^{2+}$ and two Fe$^{3+}$, with a significant reduction of the interatomic Fe-Fe distances
and a polaronic distribution of shared charge \cite{Attfield-nature}.
Trimerons reveal as the essential short-range unit in the electronic phase transitions of magnetite \cite{Piekarz-2014}.
Furthermore, laser pump-probe experiments have created a non-equilibrium version of the VT by introducing holes in the trimeron lattice \cite{Jong-2013}.
This invokes the possibility to obtain analogs of the VT under sizes much lower than those required by a full $Cc$ cell,
and in fact, as a first result of this work, we will demonstrate the ability of trimerons to emerge in a reduced unit cell of $P2/m$ symmetry.
One of the handicaps for the exploitation of the VT in novel technologies is the low value of T$_V$, well below
room temperature (RT).
The measurement at the Fe$_3$O$_4$(001) surface of an insulating gap
at RT \cite{Jordan-2006} and the further prediction that its existence
was accompanied by a subsurface CO similar to the bulk one \cite{Lodziana-2007}, caused thus great excitement.
Fe$_3$O$_4$(001) presents a $(\sqrt{2}\times\sqrt{2})R45^o$ reconstruction
which corresponds to a bulk truncation at an Fe$_B$-O
plane \cite{Stanka-2000,Shvets-2004,Novotny-2013}. Its origin has been assigned to a Jahn-Teller distortion causing
a wavelike displacement of the Fe$_B$ surface atoms along $<110>$ directions \cite{Pentcheva-2005,Pentcheva-2008}.
Seemingly this RT
reconstruction is not significantly altered across the VT \cite{Pentcheva-2008,JdF-roof}. However, the evolution in
depth of the subsurface CO has not been investigated, setting forth interrogants about the formation of surface trimerons
and the relation between the surface and bulk COs below T$_V$.
A recent study proves that the bulk low temperature phase (LTP) manifests at the surface in distinct structural features
than the reconstruction \cite{JdF-roof}.
Furthermore, the $(\sqrt{2}\times\sqrt{2})R45^o$ symmetry is lost in favor of a $(1 \times 1)$ structure
at a temperature T$_S \sim 720$ K through a second-order transition involving loss of long-range CO
\cite{Norm-2013}.
These results suggest the existence of fundamental differences between the surface and bulk insulating phases.
In this work we provide firm proof of this fact, calculating the evolution with temperature of the electronic structure
of the Fe$_3$O$_4$(001) surface.
Our results evidence that the surface insulating state
originates from the combination of large O electron affinity and loss of O bonds,
and remains unaltered across the bulk and surface transitions.
This has an impact for the disappearance of trimerons close to the surface,
replaced by bipolaronic structures below T$_S$. As a consequence,
a competition between the local bulk and surface COs emerges below T$_V$, that manifests in
modulations of the surface CO arising both from bulk trimerons and from the surface
reconstruction.
\section{Theoretical method}
We have performed first principles calculations of both bulk magnetite and its (001) termination,
based on the density functional theory including correlation effects.
We employ a plane wave basis set \cite{vasp1} and the projector augmented waves (PAW) method to describe the
core electrons \cite{paw}, with an energy cutoff of 400 eV and a Monhorst-Pack sampling of the Brillouin zone (BZ)
of $(7 \times 7 \times 5)$ for the bulk and up to $(6 \times 6 \times 2)$ for the surface slabs,
that guarantee convergence in the total energy better than 0.1 meV/f.u.
We use the exchange-correlation functional parametrization of Perdew-Burke-Erzenhof (PBE), adding
an effective on-site Coulomb repulsion term U=4 eV \cite{dudarev}. This choice of U is based on the recovery of
an equilibrium value of the cubic lattice parameter $a=8.4$ $\text{\AA}$ in excellent agreement with experiments, and
the adequate description of the Verwey transition in terms of charge disproportionation (0.27 $e$) and electronic
band gap (0.2 eV) when reducing the symmetry from the cubic $Fd\overline 3m$
Our description of bulk magnetite is based on a $P2/m$ unit cell formed by 28 atoms.
We have determined the equilibrium structures above and below T$_V$ starting from the ideal cubic lattice and allowing relaxation
of the lattice vectors and atomic positions, with no symmetry constraints for the low temperature phase (LTP).
Even at the high temperature phase (HTP) there exists a noticeable distortion of the O sublattice, that introduces a slight tetragonal deformation
of the unit cell with a small reduction of the total volume of 3 $\text{\AA}^3$.
At the LTP, relaxation of the lattice vectors leads to an orthorhombic symmetry, but again the distortion of the unit cell is small,
with a similar reduction of the total volume.
To model the Fe$_3$O$_4$(001) surface, we have used slabs of different thicknesses, containing from 8 to 16
atomic planes, supported on a Au substrate and including a vacuum region of at least 12 $\text{\AA}$ that avoids interaction
between opposite slab surfaces. The choice of the substrate has been performed to
minimize interface effects and to confine them to the interface layer. In all cases we have employed
$(\sqrt{2}\times\sqrt{2})R45^o$ two-dimensional unit cells, starting our calculations either from
the $(1 \times 1)$ termination or from the Jahn-Teller induced wavelike pattern,
and allowing to relax the atomic positions of the 3 outermost surface layers until the forces on all atoms are
below 0.01 eV/$\text{\AA}$. We have done this for slabs constructed both from the HTP and the LTP bulk structures.
The slabs of 12 planes provide the minimum thickness to recover the bulk structure at the inner layers below T$_V$
including the distribution of trimerons, and all the results presented here correspond to this configuration.
We have also modelled thicker unsupported symmetric slabs of 16 planes to check the independence of our
conclusions on the slab configuration, particularly concerning the penetration of surface effects.
\section{Bulk $\text{Fe}_3\text{O}_4$}
Figure \ref{fbulk} and Table \ref{tabla-dist} summarize our results
for the density of states (DOS), the Bader charges (Q$_B$) and the interatomic distances at
both the HTP and the LTP of bulk magnetite. The energy barrier between both phases is 170 meV/f.u.
The higher symmetry of the HTP reflects in
the existence of only one type of O and Fe$_B$ sites with a bond length of 2.06 $\text{\AA}$, and
in the uniform value of the Fe$_B$-Fe$_B$ interatomic distance (d$^{FF}=2.96\text{\AA}$).
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=\columnwidth,clip]{Bernal-magnetite-fig1.eps}
\caption{
(Color online)
Total DOS of bulk Fe$_3$O$_4$ at the (a) HTP and (b) LTP of bulk magnetite, showing the projections on the
Fe$_A$ (thick black) and inequivalent Fe$_B$ (red/blue) sites.
Positive (negative) DOS values correspond to majority (minority) spin projections.}
\label{fbulk}
\end{center}
\end{figure}
Below T$_V$, while Fe$_A$ remains essentially unaffected by the transition,
a charge disproportionation of 0.27 $e$ appears in the Fe$_B$ sublattice,
opening a band gap of 0.2 eV.
Within our reduced $P2/m$ cell,
the Fe$^{2+}$ and Fe$^{3+}$ ions alternate along the [001] direction, as evidenced in figure \ref{fbulk-trims}.
Different values of the d$^{FF}$ can be found depending on the Fe valence.
This is accompanied by a noticeable dispersion of the Fe$_B$-O bond lengths,
with larger average values for Fe$^{2+}$ (2.08$\text{\AA}$) than for Fe$^{3+}$ (2.03$\text{\AA}$).
The result is a non-uniform distribution of charge and magnetic moments that leads to slightly different
O atoms at the Fe$^{3+}$ (O$_1$) and Fe$^{2+}$ (O$_2$) planes,
as reflected in the dispersion of the Q$_B$ values.
However, the same net magnetization of 4 $\mu_B$/f.u. is obtained above and below T$_V$.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=.65\columnwidth,clip]{Bernal-magnetite-fig2.eps}
\caption{Distribution of trimerons at the $P2/m$ unit cell of the LTP of bulk Fe$_3$O$_4$.}
\label{fbulk-trims}
\end{center}
\end{figure}
\begin{table}[hbtp]
\begin{center}
\caption{Mean Fe-O bond-lengths (d(Fe-O)) and values of the interatomic distances between first Fe$_B$ neighbors
(d$^{FF}$) at the HTP and LTP of bulk Fe$_3$O$_4$. Units are $\text{\AA}$.
\label{tabla-dist}}
\renewcommand{\arraystretch}{1.5}
\renewcommand{\tabcolsep}{0.4pc}
\begin{tabular}{c c c c }
\hline
\hline
d(Fe-O) &Fe$^{2+}$-O &Fe$^{3+}$-O &Fe$_A$-O \\ \hline
HTP &2.06 &\textendash\ &1.89 \\
LTP &2.03 &2.08 &1.89 \\
\hline
\hline
d$^{FF}$ &Fe$^{2+}$-Fe$^{2+}$ &Fe$^{2+}$-Fe$^{3+}$ &Fe$^{3+}$-Fe$^{3+}$\\ \hline
HTP &2.96 &\textendash\ & \textendash\ \\
LTP &2.95 &2.89/3.03 &2.95 \\ \hline\hline
\end{tabular}
\end{center}
\end{table}
The inhomogeneities in the d$^{FF}$ at the LTP have important consequences for the emergence
of trimerons. Regarding figure \ref{fbulk-trims},
every Fe$^{2+}$ is surrounded by 4 Fe$^{3+}$ placed at the adjacent upper and lower (001) layers.
Two of them are at 2.89$\text{\AA}$ (solid colored lines) and the other two are at 3.03$\text{\AA}$ (dotted lines),
while the interatomic distance between coplanar Fe$_B$ atoms is 2.95$\text{\AA}$, as shown in table \ref{tabla-dist}.
This defines linear Fe$^{3+}$-Fe$^{2+}$-Fe$^{3+}$ chains of shortened lengths, with a charge accumulation over 0.027 $e/\text{\AA}^3$
at the middle of each Fe$^{3+}$-Fe$^{2+}$ segment, in analogy with the experimental features assigned to trimerons
\cite{Attfield-nature}.
The orbital character of the electronic states confirms the polaronic charge distribution, with the occupied
Fe$^{2+}$ $t_{2g}$ minority spin states lying along the central axis of the
chain and inducing a small contribution of the same orbital character at the closer Fe$^{3+}$.
Trimerons are uniquely characterized by the coexistence of all these features $-$short d$^{FF}$ ($<2.93$ $\text{\AA}$),
enhanced charge accumulation ($>0.027$ e/$\text{\AA}^3$) and orbital directionality$-$, as confirmed by exploring
alternative solutions without CO along the (001) direction where trimerons do not form.
Moreover, the existence of these solutions points to the complex link between the long- and short-range COs \cite{Piekarz-2014}.
We have observed that,
already under a cubic lattice, the reduced $P2/m$ unit cell is enough for the Verwey metal-insulator transition to emerge,
merely by relaxing the symmetry constraints of the HTP in the presence of electronic correlations
(U$> 2$ eV) \cite{Leonov-2004}.
The additional full relaxation of the lattice vectors and atomic positions introduces a slight orthorhombic distortion
at the LTP, and is accompanied by the formation of the local trimeron structures.
This links the metal-insulator transition to the extended CO, and separates it from the short-range correlations,
in good agreement with recent evidence \cite{Piekarz-2014}.
Though the intricate relation between the different COs can only be ultimately integrated under the full $Cc$ symmetry,
the results presented in this section prove that our reduced unit cell contains the main features of the charge distribution
at the LTP: a dominant CO along the $[001]$ axis \cite{Wright-2002}, and the existence of trimerons as short-range
features that are distinct to the low temperature CO but intimately connected to it.
This supports the use of the $P2/m$ cell as a basis to explore the surface properties below T$_V$.
\section{The $\text{Fe}_3\text{O}_4$(001) surface above T$_V$}
We will first focus on the unreconstructed surface of the HTP above T$_S$.
A sketch of the structure corresponding to our ground state is depicted in figure \ref{fsurf-struc1}, where
layers are numbered from the surface (L1) towards the bulk.
As each surface O atom has lost one donor neighbor,
they reduce the bond lengths to the remaining Fe$_B$ cations to $\sim 1.97$ $\text{\AA}$ in order to recover the bulk-like charge.
This leads to a significant rearrangement of the atomic positions,
where the compression of the first interlayer distance (d$_{12} =0.78$ $\text{\AA}$,
to be compared to the bulk value $1.04$ $\text{\AA}$) is followed by the expansion of
the subsequent interlayer spacings (d$_{23}=1.17$ $\text{\AA}$, d$_{34}=1.07$ $\text{\AA}$).
As indicated in figure \ref{fsurf-struc1},
at L1 there are two types of O sites, either bonded to a subsurface Fe$_B$ (O$_B$) or to Fe$_A$ (O$_A$).
In order to avoid the excessive shortening of the O-Fe$_A$ distance,
the O$_A$ atoms move outwards, inducing at L1
a large corrugation of 0.11 $\text{\AA}$
and a slight in-plane wavelike distortion of the O rows.
The asymmetry persists at L3, where the O corrugation attenuates to 0.04 $\text{\AA}$.
While Fe$_A$ remains essentially unaffected by the large distortion of the O sublattice,
the opposite occurs for the Fe$_B$ at the two outermost layers, L1 and L3. Each Fe$_B$
along the surface $[110]$ and subsurface $[1\overline{1}0]$ rows
approaches one of their adjacent Fe neighbors at the cost of farthening from the opposite.
As shown in figure \ref{fsurf-struc1}(b), the movement is more pronounced at the subsurface.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.75\columnwidth,clip]{Bernal-magnetite-fig3.eps}
\caption{
(Color online)
(a) Top and (b) side views of the Fe$_3$O$_4$(001) surface above T$_S$.
Panel (b) only shows the 3 outermost planes,
indicating the different in-plane Fe$_B$-Fe$_B$ distances in $\text{\AA}$.
Also, for clarity, the leftmost surface row of O atoms is not depicted.}
\label{fsurf-struc1}
\end{center}
\end{figure}
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.8\columnwidth,clip]{Bernal-magnetite-fig4.eps}
\caption{
(Color online)
Spin-resolved DOS of all inequivalent atoms (blue for Fe, red/green for surface O)
at the outermost Fe$_B$-O planes of figure \protect\ref{fsurf-struc1},
providing the corresponding Q$_B$.}
\label{fsurf-struc}
\end{center}
\end{figure}
The result of this restructuration in the electronic properties can be seen in figure \ref{fsurf-struc},
that provides the atomic-resolved DOS and the corresponding Q$_B$ at the outermost Fe$_B$-O planes,
where all surface effects are contained.
Although the O charges show significant dispersion, the
differences are not apparent in the DOS, and their Q$_B$ are close to bulk values throughout the structure.
The Fe$_B$ atoms at L1 behave as Fe$^{3+}$, opening an insulating gap.
However, the emergence of the gap is not accompanied by any charge disproportionation at the Fe$_B$ sublattice.
A gradual recovery of bulk-like behavior starts at L3, and is almost restored at L5.
As our slabs are not completely free from confinement effects, we cannot discard that it could be restored even at L3,
as inferred from STM observations of the structure of antiphase boundaries (APB) \cite{Norm-2013,Parkinson-2012}.
It is also important to remark that although
uncompensated and slightly enhanced magnetic moments emerge at the surface plane (4.16$\mu_B$ for Fe, 0.4$\mu_B$ for O),
the antiferromagnetic coupling between the Fe$_A$ and Fe$_B$ sublattices
remains unaltered.
This preserves the bulk-like high magnetic moment of Fe$_3$O$_4$ also at the high temperature surface,
validating it as a promising material for spintronics applications.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.75\columnwidth,clip]{Bernal-magnetite-fig5.eps}
\caption{
(Color online)
Same as figure \protect\ref{fsurf-struc1} for the HTP below T$_S$.
Values of the Fe$_B$-Fe$_B$ distances in $\text{\AA}$ are provided (a) between planes
and (b) along the surface and subsurface Fe$_B$ rows.
Arrows in (b) are a guide to indicate the wavelike displacements of surface Fe, and only
the central row of surface O atoms is depicted for clarity.}
\label{fsurf-is2-dos1}
\end{center}
\end{figure}
When the temperature is lowered below T$_S$, the $(\sqrt{2}\times\sqrt{2})R45^\circ$ reconstruction sets in.
This surface has already been studied in detail, but there are yet controversies about the origin of the reconstruction
and its dependence on electronic correlations \cite{Lodziana-2007,Pentcheva-2005}. Our results indicate that all
effects described for the unreconstructed surface are still present below T$_S$,
with only minor modificationis of d$_{23}$ and d$_{34}$ of less than 0.04 $\text{\AA}$,
and slightly more
asymmetric Fe-O coordination units.
Figure \ref{fsurf-is2-dos1} shows a sketch of the structure
and figure \ref{fsurf-is2-dos} the corresponding DOS and Q$_B$ at the 3 outermost
Fe$_B$-O planes.
The most relevant feature introduced by the reconstruction is the emergence of a charge disproportionation of
$\sim 0.10$$e$ between Fe sites at L3,
that defines a CO pattern within the (001) plane reducing the dispersion of O charges.
This subsurface CO was already proposed on the basis of purely electronic effects \cite{Lodziana-2007}.
However, we obtain that the
atomic wavelike displacement at the surface Fe rows lowers the energy by 28 meV/f.u.
with respect to the $(1 \times 1)$ surface also in the presence of electronic correlations.
Reminiscence of this CO persists at L5, though half-metallicity is recovered.
In fact, we cannot discard some penetration of the surface effects at deeper layers in real samples,
where the existence of defects or APB may contribute to alterations of
the CO, as the energy barrier between different charge distributions is of only a few meV \cite{Parkinson-2012}.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.8\columnwidth,clip]{Bernal-magnetite-fig6.eps}
\caption{
(Color online)
Same as figure \protect\ref{fsurf-struc} for the structure in figure \protect\ref{fsurf-is2-dos1}.}
\label{fsurf-is2-dos}
\end{center}
\end{figure}
From these results it is clear that the surface transition arises from the interplay between CO and electron-lattice
couplings, as already proposed on the basis of thermodynamic models \cite{Norm-2013}.
But although an insulating and charge-ordered state exists below T$_S$, the surface introduces significant differences
with the bulk LTP.
At L3, the Fe charge and DOS width are influenced by the demand of charge from surface O, and show
reduced values with respect to the bulk Fe$^{3+}$ and Fe$^{2+}$. More important, as we will prove now,
neither the surface structure nor the
orbital character of the surface $t_{2g}$ states support the definition of trimerons.
Regarding figure \ref{fsurf-is2-dos1},
the wavelike Fe$_B$ surface displacements define narrow and wide regions occupied respectively by Fe$^{2+}$ and Fe$^{3+}$.
As a result, along each subsurface $[1\overline{1}0]$ row, pairs of Fe$^{2+}$ and Fe$^{3+}$ alternate, inhibiting
the formation of linear Fe$^{3+}$-Fe$^{2+}$-Fe$^{3+}$ chains within the (001) plane.
Eventhough the longitudinal movement of the surface Fe$_B$ along (110) rows (not shown in the figure) is similar to that above T$_S$,
at the subsurface the
displacement of Fe$^{3+}$ is suppressed,
originating shortened d$^{FF}$=2.70 $\text{\AA}$ between Fe$^{2+}$-Fe$^{2+}$ and large d$^{FF}$=3.09 $\text{\AA}$
between Fe$^{3+}$-Fe$^{2+}$.
This leads to in-plane charge sharing between Fe$^{2+}$ sites, forming a kind of localized bipolarons
\cite{Lodziana-2007,Shvets-2004}
with a large charge accumulation of 0.035 $e/\text{\AA}^3$,
but opposes to the structure of bulk trimerons.
This tendency persists with respect to the adjacent planes: as shown in figure \ref{fsurf-is2-dos1}(a),
the d$^{FF}$ to the Fe$_B$ neighbors at L1
is similar for Fe$^{3+}$ and Fe$^{2+}$, and much larger than 2.70 $\text{\AA}$.
Similarly, the Fe$_B$ closer to subsurface Fe$^{3+}$ (Fe$^{2+}$) at L5 are those of Fe$^{3+}$ (Fe$^{2+}$) type, and are
also farther than 2.70 $\text{\AA}$.
In conclusion, neither the interatomic distances nor the charge distribution arising from the surface
reconstruction support the formation of bulk-like trimerons.
\section{The $\text{Fe}_3\text{O}_4$(001) surface below T$_V$}
The different nature of the low temperature surface and bulk phases discards that the
$(\sqrt{2}\times\sqrt{2})R45^o$ reconstruction acts as the first stage for the development of the VT.
However, it suggests the possibility of a competition between surface and bulk COs
below T$_V$. In order to explore this, we have modelled the Fe$_3$O$_4$(001) surface of the LTP departing from
our $P2/m$ bulk unit cell. Although this cell contains limited information of the actual long-range CO,
we will show that yet important insights about the mutual influence of the bulk and surface short-range
correlations become evident.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.75\columnwidth,clip]{Bernal-magnetite-fig7.eps}
\caption{
(Color online)
Side view of the Fe sublattice at the Fe$^{2+}$-ended Fe$_3$O$_4$(001) surface below T$_V$,
indicating the bulk-like trimerons and providing selected d$^{FF}$ values (in $\text{\AA}$).}
\label{fsurf-is01}
\end{center}
\end{figure}
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.8\columnwidth,clip]{Bernal-magnetite-fig8.eps}
\caption{
(Color online)
Layer- and spin-resolved DOS of the Fe$^{3+}$ (blue), Fe$^{2+}$ (red) and O (black) atoms of the structure
in figure \ref{fsurf-is01}.}
\label{fsurf-is0}
\end{center}
\end{figure}
The surface can be constructed exposing either Fe$^{3+}$ or Fe$^{2+}$ planes,
which has implications for the continuity of bulk trimerons close to the surface,
as shown in figures \ref{fsurf-is01} and \ref{fsurf-is0-Fe3+str}.
The most stable situation by $\sim 70$ meV/f.u. corresponds to the Fe$^{2+}$-ended case in figure \ref{fsurf-is01},
that at difference with the Fe$^{3+}$ termination, preserves the bulk CO up to the subsurface.
This energy difference is much larger than that between the
$(\sqrt{2}\times\sqrt{2})R45^o$ and $(1 \times 1)$ surfaces at high temperatures, evidencing the
high impact of the bulk CO on the surface properties.
We have estimated that the loss of the bulk CO at the subsurface lowers the work function by 0.30 eV, a variation close to
that induced by the adsorption of water \cite{Kendelewicz-2013}.
On the other hand, surface effects are similar to those at the HTP
under both terminations: an insulating Fe$^{3+}$ surface layer, shortened surface O-Fe$_B$
bonds, and a similar pattern of outermost interlayer distances and longitudinal atomic displacements within the Fe$_B$ rows.
This helps to attain bulk values of the O charge, though the surface causes an additional dispersion in Q$_B$,
as shown in figures \ref{fsurf-is0} and \ref{fsurf-is0-Fe3+} for the Fe$^{2+}$- and Fe$^{3+}$-ended cases, respectively.
Surprisingly, at the Fe$^{2+}$ termination the same electronic structure
corresponds to the $(1 \times 1)$ and $(\sqrt{2}\times\sqrt{2})R45^o$
surfaces, separated by less than 7 meV/f.u.
This is because the LTP bulk structure introduces an additional charge modulation within (001) planes,
that obscures that induced by the reconstruction:
regarding figure \ref{fsurf-is01},
half of the Fe$^{3+}$ sites at L3 would develop trimerons with the upper Fe$_B$, but these have changed their valence inhibiting
the polaronic charge distribution. As the other half participate in trimerons with the layers below, two types of Fe$_B$ sites exist
at the subsurface, with similar DOS but slightly different Q$_B$ and interatomic distances to the surface Fe$_B$.
Again this proves the influence of the bulk CO on the surface properties below T$_V$.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.75\columnwidth,clip]{Bernal-magnetite-fig9.eps}
\caption{
(Color online)
Same as figure \protect\ref{fsurf-is01} for the Fe$^{3+}$-ended surface of the LTP.}
\label{fsurf-is0-Fe3+str}
\end{center}
\end{figure}
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.8\columnwidth,clip]{Bernal-magnetite-fig10.eps}
\caption{
(Color online)
Same as figure \protect\ref{fsurf-is0} for the structure in figure \ref{fsurf-is0-Fe3+str}.}
\label{fsurf-is0-Fe3+}
\end{center}
\end{figure}
In turn, the robust insulating surface layer, which seems to be a universal feature of magnetite even under metastable
terminations \cite{Jordan-2006,Lodziana-2007}, has an important local effect on the bulk CO.
This better manifests at the Fe$^{3+}$ termination in figures \ref{fsurf-is0-Fe3+str} and \ref{fsurf-is0-Fe3+}, where the lack of continuity
of the trimerons at the subsurface allows for the emergence of localized bipolarons, indicating the possible coexistence
of local surface and bulk COs.
But figure \ref{fsurf-is01} evidences that also at the Fe$^{2+}$ termination those trimerons closer to the surface are slightly affected by it: the Fe$^{3+}$-Fe$^{2+}$
distances between L3 and L5 are moderately enlarged, which introduces an asymmetry in the
Fe chain weakening the charge sharing in its upper branch.
In summary, though preservation of the bulk CO seems to have a dominant effect on the surface stability,
it is conditioned by the insulating Fe$^{3+}$ surface layer, and there is a mutual influence of the bulk and
surface properties that extends several layers below the surface plane.
\section{Summary and conclusions}
Our results prove that the Fe$_3$O$_4$(001) surface shows a robust insulating state that persists across the surface and
bulk phase transitions.
It is originated by the large demand of charge from surface O arising from bond breaking,
and causes a significant restructuration at the outermost planes that inhibits the formation of trimerons.
Below T$_S$, a surface CO distinct from that of the bulk LTP emerges. Its distinct nature manifests
in a lower charge disproportionation as compared to the bulk LTP, and in the preferential bipolaronic CO
within (001) planes. When the temperature is lowered below T$_V$, this surface CO
competes with the dominant bulk one. This competition is conditioned by the
insulating Fe$^{3+}$ character of the surface, which weakens the trimeron structures.
Besides its intrinsic interest, the relation between CO and dimensionality has implications for the multifunctional
properties of magnetite, since
the emergence of ferroelectric polarization \cite{Yamauchi-2009}, the orientation of the magnetic easy axis \cite{JdF-roof} or the
catalytic activity \cite{Skomurski-2010,Parkinson-catal-2013}
are related to the existence of different Fe$_B$ sites and the resulting charge distribution.
Direct evidence from surface measurements is challenging, as most effects will manifest at the subsurface level.
Additional complications emerge from the existence of APB in real samples, and from differences in the relative orientation of
the monoclinic and cubic crystal axes in single crystals and thin films.
From the theoretical side, the inclusion of the full $Cc$ symmetry, with additional modulations of the CO within (001) planes,
may show an even richer scenario.
However, our results unequivocally demonstrate the existence of a mutual influence of the surface and bulk COs,
providing and partially quantifying the main features involved in it. We hope they will motivate further work in this fascinating subject.
\section*{Acknowledgments}
This work has been financed by the Spanish Ministry of Science under contracts MAT2009-14578-C03-03 and MAT2012-38045-C04-04.
I.B. acknowledges financial support from the JAE program of the CSIC.
\providecommand{\noopsort}[1]{}\providecommand{\singleletter}[1]{#1}%
|
1,314,259,993,634 | arxiv | \section{Introduction}
Several experiments have been conducted worldwide, with the goal of observing low-energy nuclear recoils induced by WIMPs scattering off target nuclei in ultra-sensitive, low-background detectors. In the last few decades noble liquid detectors designed to search for dark matter in the form of WIMPs have been extremely successful in improving their sensitivities and setting the best limits. Current dark matter detectors using noble liquids have an effective target mass ranging from 100 kg to the ton-scale (e.g. LUX~\cite{[LUX]}, Xenon-1T ~\cite{[Xenon]}, DarkSide ~\cite{[DS50],[DS50a],[DS50b]}). Hundreds of 3 inch PMTs are used in these detectors for the accurate measurement of scintillation light from liquid argon (128 nm shifted to 420 nm) and liquid xenon (170 nm).
An attractive alternative to photomultipliers is offered by silicon photomultipliers (SiPMs), a type of avalanche photodiode operated in Geiger mode, which have much lower intrinsic radioactive background~\cite{[Cebrian]} and smaller mass in addition to unrivalled performances in single photon detection. SiPMs behave linearly with a satisfactory gain of 10$^6$-10$^7$ and offer low intrinsic radioactive background, low operating voltage and power consumption, and have possibilities for inexpensive mass production.
\subsection {Experimental setup and data acquisition}
\label{sec:data_taking}
\indent
In Figure~\ref{fig:exp_sketch} a schematic of the experimental setup is shown. The SiPM array, mounted on its cryogenic front-end board, is housed in a stainless-steel dewar of 25~cm diameter and 100~cm height, with an inner volume of approximately 50~liters. The dewar is closed by a stainless-steel flange equipped with a series of smaller size feedthrough flanges for: the liquid argon (LAr) input line, the evacuation line, the cryocooler head, the readout of the SiPM signals, supplying the bias voltage to the SiPM, connecting the temperature sensors, and for the transmission of laser pulses through the optical fiber.
\begin{figure}[!ht]
\center\includegraphics[height=7cm]{Figures/Figure_1.pdf}
\caption{Sketch of the experimental setup.}
\label{fig:exp_sketch}
\end{figure}
The support structure of the SiPM array consists of a set of three copper plates (Figure~\ref{fig:copper_holders}). The upper one is screwed to the cryocooler head to allow for good thermal contact. In addition, three copper bars support to plates, placed at different heights to host the optical fiber connector (the middle plate) and the SiPM array readout board and a temperature sensor (lower plate). The plate hosting the optical fiber connector is fixed at about 10~cm above the SiPM array, so as to illuminate all the SiPMs dies. Light pulses are generated by a Hamamatsu PLP10 light pulser, equipped with a 408~nm laser head with a pulse width of 70~ps. The laser beam is passed through an optical attenuator mount in which discrete filters of various attenuation coefficients were used to select the magnitude of illumination reaching the photosensor.
Before the data taking starts, the cryostat is first pumped out until a residual pressure of 10$^{-4}$~mbar is achieved and then filled with LAr. The level of the liquid argon in the cryostat is monitored with two PT1000 resistors which are readout by a calibrated CRYOCON 32 controller. The first PT1000 (low level) is placed at the same height as the SiPM array, while the second (high level) is placed a few cm above the optical fibre output. The liquid argon filling operation is stopped 15~min after the high level is reached. The cryocooler maintains a constant LAr level, in order to ensure constant thermodynamic conditions throughout data taking.
The Bias voltage was supplied via a low noise power\footnote{TTi QL355TP} supply through a 10~k$\Omega$ resistor. The SiPM array readout board, described in section~\ref{sec:readout}, has two outputs that are conveyed outside the cryostat through a signal feedthrough flange and are fed to an external (custom made) NIM amplifier (gain x10). The output of the amplifier is fed in to a CAEN V1720E digitizer with 4~ns sampling and 12~bit resolution, connected via optical link to a PC for data handling, storage and analysis.
\begin{figure}[!ht]
\center\includegraphics[height=7cm]{Figures/Figure_2.pdf}
\caption{Sketch of the support structure of the photodetector: it consists of a set of three copper plates. The upper one is screwed to the cryocooler head for allowing a good thermal contact. The second and the third plate are placed at different heights to host the optical fiber connector (middle plate) and the SiPM readout board (lower plate). Two PT1000 temperature sensors are placed on the middle and on the lower plate, acting as a level meter for the liquid argon.}
\label{fig:copper_holders}
\end{figure}
\subsection {Cryogenic readout board}
\label{sec:readout}
\indent
Until very recently, the small size of the active area of SiPM dies were the main obstacle for them to be considered a valid alternative to PMTs in noble liquid direct dark matter search experiments. In the last few years much progress has been made to enlarge the effective area of silicon detectors by bonding SiPM dies together into arrays. These have now reached sizes as large as 5x5cm$^2$~\cite{[SensL]}. Large commercially produced arrays are usually provided with a common connector to bias all the dies and a number of output pins equivalent to the number of SiPM dies that are mounted in the array. In a large cryogenic apparatus {\em O}(10~ton) one should try to minimize the acquisition channels of the photosensors for several reasons. The number of signal output cables should be kept low because cables increase the heat load of the system and the radioactive budget. Moreover, a large number of channels increases both the complexity of the detector and the cryogenic system. It is therefore mandatory to find a solution for reducing the number of channels to be readout from a SiPM array.
Our approach to solve this issue has been the use of an active front-end board placed near the SiPM array working at cryogenic temperature. The aim of the front-end amplifying board is to sum up the output of multiple SiPM dies without distorting the pulse shape (e.g. recharge time should stay constant).
The front-end board is coupled to the SensL ArrayB-30035-16P (Figure~\ref{fig:array_pic}), which is composed of 4x4 SiPM dies of 3x3mm$^2$ each (see Table~\ref{tab:data_sheet}), and it conveys the output of 8 SiPMs dies together through the electric scheme shown in Figure~\ref{fig:electric_sketch}, resulting in two summed output channels of 8 SiPM dies each.
The board and the SiPM array showed good performance in terms of mechanical robustness, undergoing multiple cooling/warming cycles without any failure.
\begin{figure}[!ht]
\center\includegraphics[height=7cm]{Figures/Figure_3.jpg}
\caption{Picture of the SiPM array and of the cryogenic front-end board.}
\label{fig:array_pic}
\end{figure}
\begin{figure}[!ht]
\center\includegraphics[height=7cm]{Figures/Figure_4.pdf}
\caption{SiPM readout board electric sketch. The feedback resistor value is R$_\mathrm{f}$=560~$\Omega$ and C$_\mathrm{b}$=220~nF.}
\label{fig:electric_sketch}
\end{figure}
\begin{table}[h!]
\begin{center}
\begin{tabular}{c c c c c c c}
\hline \hline
Model & Size & PDE$_{max}$ & $\mu$cell size & cells & Recharge time~[ns] & C$_\mathrm{SiPM}$ [pF] \\
& mm$^2$ & \% & $\mu$m & & at 300~K & at 300~K \\
SensL MicroSB-30035 & 3x3 & 41 & 35 & 4774 & 180 & 850 \\
\hline \hline
\end{tabular}
\caption{Summary of the characteristics at room temperature (from data sheet) of the SIPM dies forming the array under test.}
\label{tab:data_sheet}
\end{center}
\end{table}
\subsection {Data Taking}
\indent
A study of the performance of the readout board coupled with the SiPM array has been performed at the liquid argon temperature. The main characteristics under study have been the breakdown voltage, the recharge time, the single photoelectron (SPE) spectrum and resolution, the dark rate, the correlated pulses and the relative PDE. A scan of the performance of the SiPM array as a function of V$_\mathrm{bias}$ has been performed. Each data taking run consisted of 100.000 triggers at a given V$_\mathrm{bias}$, with a memory buffer of 5~$\mu$s, with 2~$\mu$s pre-trigger. For the SPE runs the laser was triggered by an external pulser at a repetition rate of 10~kHz and its illumination was set, through a system of discrete attenuators in order to send a few photons for each trigger to the photosensor.
The single photoelectron spectrum, was reconstructed by integrating the waveform region around the trigger for 800~ns\footnote{The integration window was set as large as to contain at least three time the recharge time (3$\tau$).} for a given run.
\begin{figure}[!ht]
\center\includegraphics[height=7cm]{Figures/Figure_5.pdf}
\caption{Example of Gaussian fit of the first photoelectron spectrum.}
\label{fig:SPE_fit}
\end{figure}
The SPE response of the SiPM array at each V$_\mathrm{bias}$ is calculated by fitting the first photoelectron peak of the SPE spectrum with a Gaussian shape, as shown in Figure~\ref{fig:SPE_fit}.
In addition, we define the SPE resolution as
\begin{equation}
Res=\frac{\sigma_\mathrm{SPE}}{\mu_\mathrm{SPE}}
\end{equation}
where $\sigma_\mathrm{SPE}$ is the sigma of the gaussian fit of the first photoelectron peak.
\section {Results}
\label{sec:analysis}
\indent
\subsection {SPE response and resolution}
\indent
Figure~\ref{fig:workfunction} shows the SPE response of the SiPM readout board (workfunction) as a function of the V$_\mathrm{bias}$ at the liquid argon temperature. Points have been fit with line. The workfunction is found to be linear in the whole range of V$_\mathrm{bias}$ explored. The result of the fit indicates, as expected, that the $V_\mathrm{bd}^\mathrm{87K}$=20.75$\pm$0.13~V is much lower than the corresponding one at room temperature ($V_\mathrm{bd}^\mathrm{300K}\sim$25~V).
\begin{figure}[!ht]
\center\includegraphics[height=7cm]{Figures/Figure_6.pdf}
\caption{SPE response of the SiPM array as a function of V$_\mathrm{bias}$. Points are fit with a line. Results of the fit indicates that $V_\mathrm{bd}^{87K}$=20.75$\pm$0.13~V.}
\label{fig:workfunction}
\end{figure}
An important parameter that stresses the great performance of the SiPMs with respect to the PMTs is the excellent SPE resolution. SiPMs can be as good as 4\% while cryogenic PMTs usually show performance of 25-35\%. Summing up several channels of a SiPM array through a readout board could spoil the SPE resolution for two reasons: firstly, there is a spread of $V_\mathrm{bd}$ among the SiPM dies installed on an array, and hence a slightly different gain for each channel, secondly, the large input capacitance of each SiPM die (about 1~nF) might increase the electronic noise and reduce the signal-to-noise ratio.
Figure~\ref{fig:SPE-res} shows the SPE resolution as a function of the overvoltage (V$_\mathrm{OV}$=V$_\mathrm{bias}$ -V$_\mathrm{bd}$). The SPE resolution decreases with the overvoltage as expected and it goes as low as 5\% at V$_\mathrm{OV}\sim$5~V. Results of our measurements are quite promising and highlight the encouraging performance of the readout board. At this stage, the possibility of summing up even larger-size arrays with more channels with the same technique seems possible.
\begin{figure}[!ht]
\center\includegraphics[height=7cm]{Figures/Figure_7.pdf}
\caption{Single photoelectron resolution as a function of the overvoltage.}
\label{fig:SPE-res}
\end{figure}
\subsection {Pulse shape studies}
\indent
Recharge time is an important parameter for a SiPM affecting the pulse shape. It is given by $\tau_{rec}=C_d\cdot R_q$ where $C_d$ is the diode capacitance and $R_q$ is the quenching resistor. For polysilicon quenching resistors, $R_q$ increases with decreasing the temperature. Thus, the pulse duration at liquid argon temperature is larger than the corresponding one at room temperature. This feature might have an impact in liquid argon-based dark matter experiments since it might spoil the pulse shape discrimination capabilities in case the recharge time is too large with respect to the fast component of the scintillation light. A common integration window is 90~ns~\cite{[DS50b]}.
We performed a precise evaluation of the recharge time constant of the 8 channel summed output by fitting the falling edge of the average waveform of 100k events with an exponential curve. Results of the fit are shown in Figure~\ref{fig:RecTime}, where the recharge time can be estimated to be $\tau=343\pm5$~ns. This results is consistent with the value we obtained by measuring the recharge time of one of the single SiPM dies forming the array at the liquid argon temperature.
\begin{figure}[!ht]
\center\includegraphics[height=7cm]{Figures/Figure_8.pdf}
\caption{Fit of the falling edge of the average (inverted) waveform of 100000 triggers at the liquid argon temperature.}
\label{fig:RecTime}
\end{figure}
The parallel of the capacitance of the 8 summed SiPM dies reduces the bandwidth of the amplifier. In Figure~\ref{fig:Riseup}, a close-up of the rising edge of the average waveform of 100k events is shown. The resulting rising edge is slowed slightly (about $\sim$20~ns) with respect to the rise time of a single SiPM die (a few ns).
\begin{figure}[!ht]
\center\includegraphics[height=7cm]{Figures/Figure_9.pdf}
\caption{Zoom of the the rise time region of the averaged waveform at the liquid argon temperature.}
\label{fig:Riseup}
\end{figure}
\subsection {Relative PDE and correlated pulses}
\indent
Every specific application of SiPM requires optimal selection of operating voltage to balance the signal performance (gain, photon detection efficiency, timing resolution, etc.) with noise (dark count rate, cross-talk, and afterpulsing).
This section reports about the measurements performed to highlight the influence of cross-talk and afterpulsing on photodetection characteristics that are useful for such optimization by using the method described in~\cite{[Vinogradov]}.
The typical experimental result of a few photon short pulse detection is the single photoelectron spectrum (SPE). Figure~\ref{fig:SPE_fit} shows an example of a histogram of SPE where the distribution of output charge reflects the probabilities to detect photoresponse pulses equal to 0, 1, 2, and more fired pixels (referred to as photoelectrons). The histogram peaks are very narrow due to low excess noise of charge multiplication, and thus the superb photon counting capabilities of SiPMs. In the absence of correlated pulses, the probability distribution should follow a Poisson law. However, when cross-talk and afterpulsing are considerable, deviations from Poissonian behaviour can be observed.
We performed measurements of the relative PDE ($\lambda$) and of the correlated pulses as a function of $V_\mathrm{bias}$. Measurements have been performed through the following steps. The SPE spectra at each voltage and at different temperatures have been discretized, and the mean value and variance have been evaluated. Following~\cite{[Vinogradov]} the mean value ($\mu_{SPE}$) of the discrete SPE distribution can be rewritten as:
\begin{equation}
\mu_{SPE}=\frac{\lambda}{1-p}
\end{equation}
while the variance can be rewritten as:
\begin{equation}
Var=\frac{\lambda(1+p)}{(1-p)^2}
\end{equation}
The parameter $p$, the duplication probability, takes into account the deviations of the SPE spectrum from the Poisson law, while the $\lambda$ value is widely used in the evaluation of photon detection efficiency, when the mean number of photons per detected pulse $N_{ph}$ is known $\lambda=N_{ph}\cdot PDE$.
Absolute PDE evaluation was not possible in our experimental setup. However, we performed a comparative measurement of the PDE as a function of the overvoltage, as shown in Figure~\ref{fig:relPDEvsBias}. Specifically, we plot the ratio of $\lambda$ at a given overvoltage divided by the $\lambda$ at 0.8~V overvoltage (at the smallest overvoltage value acquired). This plot shows that the PDE increases with overvoltage until it plateaus. It is worthwhile note that the maximal PDE is already reached at about V$_\mathrm{OV}$=3.5~V.
\begin{figure}[!ht]
\center\includegraphics[height=7cm]{Figures/Figure_10.pdf}
\caption{Relative Photon Detection Efficency (PDE) as a function of the overvoltage.}
\label{fig:relPDEvsBias}
\end{figure}
Once the V$_\mathrm{OV}$ has been set to maximize the PDE, the other figure of merit can be estimated. Figure~\ref{fig:pvalue} shows the probability of having a correlated pulse as a function of the overvoltage. The probability of correlated pulses increases like a pol2 function with the V$_\mathrm{OV}$ and at V$_\mathrm{OV}$=3.5~V it is about 30\%.
\begin{figure}[!ht]
\center\includegraphics[height=7cm]{Figures/Figure_11.pdf}
\caption{Probability of Correlated pulses as a function of the overvoltage.}
\label{fig:pvalue}
\end{figure}
\subsection {Dark rate at liquid argon temperature}
\indent
The dark rate has been evaluated through a peak finder algorithm that searched for peaks in the acquisition window of each triggered waveform for each run. The peaks found in the pre-trigger time window (i.e. before the reference laser pulse) have been counted. The dark rate is calculated with the following formula:
\begin{equation}
DR\mathrm{[Hz/mm^2]}=\frac{N_\mathrm{pulses}}{T_\mathrm{pre}~E_\mathrm{number}^\mathrm{tot}~A~N_{dies}}
\end{equation}
where $T_\mathrm{pre}$=2$\cdot10^{-6}$s is the pre-trigger window, $E_\mathrm{number}^\mathrm{tot}$=100000 is the number of triggered events for each run, $A=9~mm^2$ is the active surface area per SiPM die and $N_{dies}=8$ is the number of SiPM dies summed up by the readout board. The measured dark rate (expressed in Hz/mm$^2$) as a function of the overvoltage V$_\mathrm{OV}$=V$_\mathrm{bias}$-V$_\mathrm{bd}$ at the liquid argon temperature is shown in Figure~\ref{fig:dr}. This figure of merit surpassed the expectations, being four orders of magnitude smaller than the corresponding dark rate at room temperature ($\sim$MHz/mm$^2$). With this value a SiPM array equivalent to a 3 inch PMT (3000~mm$^2$) would have a dark rate of about 10~kHz at the liquid argon temperature.
This result is very promising and sets a very important milestone in the possibility of replacing cryogenic PMTs with large-size SiPMs arrays.
\begin{figure}[!ht]
\center\includegraphics[height=7cm]{Figures/Figure_12.pdf}
\caption{Dark rate as a function of the overvoltage at the liquid argon temperature.}
\label{fig:dr}
\end{figure}
\section {Conclusions}
SiPMs appear to be very promising devices for next generation noble liquids direct dark matter search experiments. SiPM arrays sizes are nowadays comparable to PMTs of 3 inch size. The manufacturing progress of last years have made SiPMs arrays very appealing for substituting PMTs in cryogenic environment, as also shown in~\cite{[Whitt]}. In particular, SensL-ArrayB-30035-16P can be operated at liquid argon temperature coupled with a cryogenic readout amplifying board, to reduce the number of output channels without distorting the pulse shape. Moreover, the array performance at V$_\mathrm{OV}$=3.5~V (corresponding to a gain of about 3$\cdot10^6$) where the PDE is maximal, are very promising in terms of SPE resolution (about 8\%), dark rate (about 250~Hz for the whole array) and correlated pulses (30\%).
\section* {Acknowledgments}
The work presented in this paper was conducted thanks to grant PHY-1314507 National Science Foundation and from INFN. We warmly acknowledge all the institutions. We thank A. Razeto for the useful support he provided and D. Sablone for the assembling of the cryogenic readout board. We thank L. Tatananni, N. Canci and the LNGS mechanical workshop for the realization of the cryostat top flange. We also acknowledge A. Anastasio, A. Boiano, P. Di Meo and A. Vanzanella and the electronic workshop of INFN Napoli for the fruitful discussions and their support during the assembling of the experimental setup.
\section*{References}
|
1,314,259,993,635 | arxiv | \section{Introduction}
The unitary group $U(1)$ is compact with an invariant measure, and which may be modeled as acting by rotation on the circle $S^1\subset\C$ taken to have length 1. It is well known in this model that $U(1)$ is (monogenically) topologically generated by a rotation by any irrational angle.\footnote{For a nonabelian consideration, see e.g. Parzanchevski--Sarnak \cite{PS}.} A natural question here is which of these topological generators is the {\em best}. To answer this inquiry, we introduce the following function:
\begin{definition-non}[cf. \cite{GV}]
Let $[[m]]=\{0,\dots,m\}$.\footnote{This is in contrast to $[m]$, which denotes $\{1,\dots,m\}$.} Define $d_\gt(m)$ as
$$\sup\{\abs{I} : \text{$I\subset\R$ an interval},(I+\Z)\cap[[m]]\gt=\emptyset\}.$$
\end{definition-non}
$d_\gt(m)$ measures the largest ``gap,'' modulo 1, of $m+1$ consecutive integer multiples of the real number $\gt$. It is clear that if $\gt$ is rational with the reduced fraction representation $\gt=\frac{a}{b}$, then $d_\gt(m)=\frac{1}{b}$ for all $m\ge b-1$. Meanwhile, when $\gt$ is irrational it is a topological generator of $U(1)$, so
$$\lim\limits_{m\to\infty}d_\gt(m)=0$$
weakly monotonically. For all choices of $\gt$, $(m+1)d_\gt(m)\ge1$ since equality is attained precisely when $d_\gt(m)=\frac{1}{m+1}$, but by the pigeonhole principle, $d_\gt(m)\ge\frac{1}{m+1}$. Therefore, $d_\gt(m)$ can be thought of as the {\em discrepancy} between the first $m+1$ iterates of $\gt$ and an equidistribution, and $(m+1)d_\gt(m)$ can be thought of as measuring how quickly $d_\gt(m)$ tends to 0 for irrational $\gt$.
Graham and van Lint \cite{GV} studied asymptotic behavior of this quantity, using the language of continued fractions. We say that two continued fractions $\gt$ and $\gs$ are {\em equivalent}, written $\gt\asymp\gs$, if there are positive integers $m$ and $n$ such that $\gt$ and $\gs$ agree after removing the length-$m$ and length-$n$ prefixes, respectively. The golden ratio is $\vf=\frac{1+\sqrt{5}}{2}$, and has continued fraction consisting of all 1's.
\begin{theorem-non}[\cite{GV}, Theorem 2]
For any irrational $\gt$,
$$\limsup\limits_{m\to\infty}(m+1)d_\gt(m)\ge1+\frac{2}{\sqrt{5}}$$
with equality iff $\gt\asymp\vf$.
\end{theorem-non}
Here, we prove a stronger result about these asymptotics:
\begin{theorem}\label{under nec}
Given $\gt\in\R$, there exists $M\in\N$ for which $m\ge M$ implies $(m+1)d_\gt(m)<1+\frac{2}{\sqrt{5}}$ if and only if $\gt\asymp\vf$.
\end{theorem}
Letting $\cT$ be the set of values $\gt$ for which the condition on $d_\gt(m)$ in \thmref{under nec} holds, we will see, as is well known, that $\cT$ is the set of linear fractional transformations by $GL_2(\Z)$ of $\vf$, a dense countable subset of $\R$.
For many choices of $\gt$, $(m+1)d_\gt(m)$ rises above $1+\frac{2}{\sqrt{5}}$ before settling below, i.e. $M=1$ as in \thmref{under nec} does not suffice for us here. To study this new sought-after phenomenon---a global generalization of $\limsup\limits_{m\to\infty}(m+1)d_\gt(m)=1+\frac{2}{\sqrt{5}}$---we introduce a new measure of quality for topological generators.
\begin{definition-non}
$D(\gt)=\sup\limits_{m\in\N}(m+1)d_m(\gt)$.
\end{definition-non}
From \cite{GV}, $D(\gt)\ge1+\frac{2}{\sqrt{5}}$ with equality on some (possibly empty) subset $\cS\subset\cT$. Sarnak conjectured, and Mozzochi recently proved, the following (the ``golden mean conjecture''):
\begin{theorem-non}[\cite{Moz}]
$D(\vf)=1+\frac{2}{\sqrt{5}}$.
\end{theorem-non}
This can be expanded to a surprising result completely characterizing $\cS$.
\begin{theorem}\label{under suff}
There exist exactly 16 values $\gt$, modulo 1, for which $D(\gt)=1+\frac{2}{\sqrt{5}}$, which are specified in \figref{unders}.\footnote{$d_\gt(m)=d_{1-\gt}(m)$, so $\gt\in\cS$ if and only if $1-\gt\in\cS$, which is why only 8 values are specified in the table.}
\end{theorem}
Unsurprisingly, $\vf$ (and $\vf^2=\vf+1$) is in one of these 16 modulo-1 classes: note that $\vf+\eta_7=2$.
One way to measure the ``quality'' of a generator on $1\le m\le M$ is by the largest value of $(m+1)d_\gt(m)$ attained on that range. To put this formally, we introduce:
\begin{definition-non}
$D_M(\gt)=\max\limits_{m\in[M]}(m+1)d_\gt(m)$.
\end{definition-non}
Then, there is no single ``best'' generator, in the sense of minimizing this quantity:
\begin{theorem}\label{no best}
For each $\gt_0\in\cS$, there are infinitely many values $M\in\N$ for which $\gt_0=\argmin\limits_{\gt\in\cS}D_M(\gt)$.
\end{theorem}
\begin{figure
\centering
\begin{tabular}{|r|cccccc|c|c|c|}
& 0 & 1 & 2 & 3 & 4 & 5 & matrix & exact & num.~val.\\\hline
$\eta_{7}$ & 0 & 2 & 1 & 1 & 1 & $\dot1$ & $\begin{pmatrix}1 \\ 2 & 1\end{pmatrix}$ & $\frac{3-\sqrt{5}}{2}$ & 0.381\dots \\\hline
$\eta_{6}$ & 0 & 2 & 1 & 2 & 1 & $\dot1$ & $\begin{pmatrix}3 & 1 \\ 8 & 3\end{pmatrix}$ & $\frac{25-\sqrt{5}}{62}$ & 0.367\dots \\\hline
$\eta_{8}$ & 0 & 2 & 2 & 1 & 1 & $\dot1$ & $\begin{pmatrix}2 & 1 \\ 5 & 2\end{pmatrix}$ & $\frac{7+\sqrt{5}}{22}$ & 0.419\dots \\\hline
$\eta_{4}$ & 0 & 3 & 1 & 1 & 1 & $\dot1$ & $\begin{pmatrix}1 \\ 3 & 1\end{pmatrix}$ & $\frac{5-\sqrt{5}}{10}$ & 0.276\dots \\\hline
$\eta_{5}$ & 0 & 3 & 2 & 1 & 1 & $\dot1$ & $\begin{pmatrix}2 & 1 \\ 7 & 3\end{pmatrix}$ & $\frac{9+\sqrt{5}}{38}$ & 0.295\dots \\\hline
$\eta_{2}$ & 0 & 4 & 1 & 1 & 1 & $\dot1$ & $\begin{pmatrix}1 \\ 4 & 1\end{pmatrix}$ & $\frac{7-\sqrt{5}}{22}$ & 0.216\dots \\\hline
$\eta_{3}$ & 0 & 4 & 2 & 1 & 1 & $\dot1$ & $\begin{pmatrix}2 & 1 \\ 9 & 4\end{pmatrix}$ & $\frac{11+\sqrt{5}}{58}$ & 0.228\dots \\\hline
$\eta_{1}$ & 0 & 5 & 2 & 1 & 1 & $\dot1$ & $\begin{pmatrix}2 & 1 \\ 11 & 5\end{pmatrix}$ & $\frac{13+\sqrt{5}}{82}$ & 0.185\dots \\\hline
\end{tabular}
\caption{The values in $(\cS\mod1)\cap\lbr{0,\half}$ in lexicographic order of continued fraction. Indices reflect the canonical order with respect to embedding the $\cS\mod1\hookrightarrow[0,1]$ in the obvious way.}
\label{unders}
\end{figure}
\section{Definitions and past results}
Henceforth let $\gt$ be irrational. $d_\gt(m)$ may be evaluated exactly, using the language of continued fractions. We recall the following from \cite{GV,HW}:
\begin{definition-non}
Consider the infinite continued fraction $\gt=[a_0,a_1,\dots]$.\footnote{It is elementary that $\gt$ must have a continued fraction and that it cannot be finite.} We have the following notation, for nonnegative integers $n$:
\begin{itemize}
\item $\frac{h_n}{k_n}=\frac{a_nh_{n-1}+h_{n-2}}{a_nk_{n-1}+k_{n-2}}=[a_0,a_1,\dots,a_n]$ is the $n$th convergent.
\item $x_n=[a_{n+1},\dots,a_1]$.
\item $\gt_n=[a_n,a_{n+1},\dots]$.
\item $[a_0,\dots,a_{n-1},\dot1]=[a_0,\dots,a_{n-1},1,1,1,\dots]$.
\end{itemize}
\end{definition-non}
\begin{remark}\label{rmk}
Let $\gt=[a_0,\dots,a_N,\dot{1}]$, where for $n>N$ we have $a_n=1$. Then, for such $n=N+d$, $k_n=F_{d+1}k_N+F_dk_{N-1}$. By the recurrence $k_n=a_nk_{n-1}+k_{n-2}$ and the stipulation that $a_n\in\N$, $k_n\ge F_{n+1}$.
\end{remark}
Indeed, the $n$th convergent $g_n=[1,\dots,1]$ to $\vf=[\dot1]$ equals $\frac{F_{n+2}}{F_{n+1}}$, for $F_n$ the $n$th Fibonacci number, indexed from $F_0=0$ and $F_1=1$, and so in this way $\vf$ has the smallest convergents.
Using our new notation, we can write more concisely that if $\gt\asymp\gs$ then there exist positive integers $m$ and $n$ for which $\gt_m=\gs_n$. The relationship between equivalent continued fractions can be made even more explicit:
\begin{theorem-non}[cf. \cite{HW}, Theorems 174 and 176]
Equivalence of continued fractions is an equivalence relation, and two continued fractions $\gt$ and $\gs$ are equivalent if and only if there exists $\fM=\begin{pmatrix}a&b\\c&d\end{pmatrix}\in GL_2(\Z)$ for which $\gt=\frac{a\gs+b}{c\gs+d}$, denoted by $\fM\gs$ in this context.
\end{theorem-non}
In this terminology, the aforementioned theorem of \cite{GV} and \thmref{under nec} can be thought of as a biconditional with $\gt\asymp\vf$, and $\cT$ can be seen as $GL_2(\Z)\vf$.
The following are long-established results about continued fractions:
\begin{lemma}[cf. \cite{HW}, pp.140]
Fixing again $\frac{h_n}{k_n}$ and $\gt_n$ with respect to $\gt$:
\begin{align}
\gt-\frac{h_n}{k_n}&=\frac{(-1)^n}{k_n(k_{n-1}+k_n\gt_{n+1})}.\label{diff w convergent}\tag{$*$}
\end{align}
\end{lemma}
With these notions in hand, the following is proved by Slater \cite{Sla} and S\'os \cite{Sos} and used extensively in \cite{GV}:
\begin{lemma}
Given $\gt=[a_0,a_1,\dots]$ and nonnegative integers $\ga$ and $m$ satisfying $\ga<a_{n+2}$ and $k_n+(\ga+1)k_{n+1}-1\le m\le k_n+(\ga+2)k_{n+1}-2$, it is the case that
\begin{equation}
d_\gt(m)=\abs{(k_n\gt-h_n)-\ga(k_{n+1}\gt-h_{n+1})}.\label{gv lem 1}\tag{$**$}
\end{equation}
\end{lemma}
Combining \eqref{diff w convergent} into \eqref{gv lem 1}, with some algebraic manipulation we have:
\begin{cor}\label{cor:useful}
Given $\gt=[a_0,a_1,\dots]$ and nonnegative integers $\ga$ and $m$ satisfying $\ga<a_{n+2}$ and $k_n+(\ga+1)k_{n+1}-1\le m\le k_n+(\ga+2)k_{n+1}-2$, it is the case that
$$d_\gt(m)=\frac{\gt_{n+2}-\ga}{k_n+k_{n+1}\gt_{n+2}}.$$
\end{cor}
Henceforth, let $\rho=1+\frac{2}{\sqrt{5}}$.
\section{Proofs of \thmref{under nec} and \thmref{under suff}}\label{main}
\thmref{under nec} asserts that $\cT=GL_2(\Z)\vf$, via linear fractional transformation; that is, $\cT$ is the set of continued fractions $\gt\asymp\vf$. Towards the proof of this result, we first prove a useful lemma. Of course, this lemma can be generalized considerably, but this is not needed to prove the result in mind.
\begin{lemma}\label{monotone}
Let $f(x)=[1,\dots,1,x]$ be a function on $\R^+$, where the continued fraction is length $n+2$. Then $f$ is monotonic (either increasing or decreasing).
\end{lemma}
\begin{proof}
Fix $x$. Then
$$f(x)=\frac{h_nx+h_{n-1}}{k_nx+k_{n-1}}=\frac{F_{n+2}x+F_{n+1}}{F_{n+1}x+F_n}$$
which is clearly differentiable on $\R^+$, so taking the derivative gives
$$f'(x)=\frac{F_nF_{n+2}-F_{n+1}^2}{(F_{n+1}x+F_n)^2}=\pm\frac{1}{(F_{n+1}x+F_n)^2}$$
which has constant sign in $x$.
\end{proof}
This simple lemma equips us to characterize the set $\cT$.
\begin{proof}[Proof of \thmref{under nec}]
We know from \cite{GV} that equivalence to $\vf$ is necessary, since if $\gt\not\asymp\vf$ then $\limsup(m+1)d_\gt(m)=\ell>\rho$ so for all $M_0\in\N$, there is $m>M_0$ with $(m+1)d_\gt(m)>\half(\ell+\rho)>\rho$.
We now show that equivalence to $\vf$ is sufficient. Write $\gt=[0,a_1\dots,a_N,\dot{1}]$, where for $n>N$ we have $a_n=1$. \cite{Moz} shows that when $a_n=1$ and $k_n+k_{n+1}-1\le m\le k_n+2k_{n+1}-2$,
$$\max(m+1)d_\gt(m)=\frac{1+2x_n-\frac{1}{k_n}}{\gt_{n+1}+\frac{1}{x_{n-1}}}=\frac{1+2x_n-\frac{1}{k_n}}{\vf+x_n-1}.$$
We see that
\begin{equation}
x_n<\vf+\frac{5+2\sqrt{5}}{k_n}\label{eq:thm}\tag{$\star$}
\end{equation}
is necessary and sufficient to show $\max(m+1)d_\gt(m)<\rho$ over that range for $m$, by algebraic manipulation. $x_n=[1,\dots,1,a_N,\dots,a_1]$ with $d=n-N$ 1's. By \lemref{monotone} and since $a_N\in\N$ implies $a_N\ge1$, $x_n$ is bounded between $g_{d+1}$ and $g_d$, so
$$\abs{\vf-x_n}\le\max\lcr{\abs{\vf-g_d},\abs{\vf-g_{d+1}}}<\frac{1}{F_{d+1}^2}.$$
Since $k_n=F_{d+1}k_N+F_dk_{N-1}$, we simply require $F_{d+1}^2>\frac{F_{d+1}k_N+F_dk_{N-1}}{5+2\sqrt{5}}$. This holds if
\begin{equation}
F_{d+1}>\frac{k_N+k_{N-1}}{5+2\sqrt{5}}\label{ineq:thm}\tag{$\star\star$}
\end{equation}
which, since $d$ is variable while $N$ is fixed, is eventually true. If we let $d_0$ be the least $d$ for which \eqref{ineq:thm} holds, and let $N_0=N+d_0$, then we see that \eqref{eq:thm} holds for $n\ge N_0$ and so the theorem holds for $M_0=k_{N_0}+k_{N_0+1}-1$.
\end{proof}
We now investigate when the lower bound can be made $M_0=1$, and we let $\cS$ denote the set of such irrational numbers. Of course, by \thmref{under nec}, any such generator is equivalent to $\vf$. While $\cT$ is dense in $\R$, \thmref{under suff} asserts that $\cS$ is remarkably sparse: $\#(\cS\mod1)=16$. Towards this result, we prove two lemmas. The first establishes when the continued fractions of $\cS$'s elements must become $\dot1$. The second establishes upper bounds on the values that can appear in the prefix of those continued fractions. It is then merely a matter of verifying with the aid of a short computer program (\secref{python app}) which values suffice.
\begin{lemma}\label{when ones}
Write $\gt=[0,a_1,\dots]$. Suppose $n\ge6$. If $a_n>1$ then $\gt\not\in\cS$.
\end{lemma}
\begin{proof}
If $\gt\not\in GL_2(\z)\vf$ then we already know the result to hold, by \thmref{under nec}. So, we take $\gt\in GL_2(\z)\vf$.
Suppose towards contradiction that for some $N\ge5$, $a_n=1$ for all $n\ge N+2$, but $a_{N+1}>1$, yet $\gt\in\cS$. From \corref{cor:useful}, we have for $k_N+k_{N+1}-1\le m\le k_N+2k_{N+1}-2$:
$$d_\gt(m)=\frac{1}{\gt_{N+1}k_N+k_{N-1}}.$$
It therefore follows that for $m=k_N+2k_{N+1}-2$:
$$(m+1)d_\gt(m)=\frac{2k_{N+1}+k_N-1}{\gt_{N+1}k_N+k_{N-1}}.$$
Since $\gt\in\cS$, $(m+1)d_\gt(m)<\rho$. Rearranging the inequality, along with the substitutions
\begin{align*}
\gt_{N+1}&=a_{N+1}-1+\vf\\
k_{N+1}&=a_{N+1}k_N+k_{N-1},
\end{align*}
yields the following:
$$((2-\rho)a_{N+1}+1+\rho-\rho\vf)k_N+(2-\rho)k_{N-1}<1.$$
Using \rmkref{rmk}, the fact that $a_{N+1}\ge2$ by hypothesis, and numerical values of $\vf$ and $\rho$, we note that the left-hand side is lower-bounded by $0.04F_{N+1}+0.1F_N$, which, since $F_6=13$ and $F_5=8$, is lower-bounded by 1.3. This provides the desired contradiction and proves the result.
\end{proof}
\begin{lemma}\label{abcde bounds}
If $[0,a,b,c,d,e,\dot{1}]\in\cS$, then:
\begin{align*}
a&\le18, & b&\le18, & c&\le14, & d&\le12, & e&\le11.
\end{align*}
\end{lemma}
\begin{proof}
Consider any $\gt=[0,a_1,\dots]\in\cS$, and fix $n\in[5]$. We know that we have for $k_{n-1}+(\ga+1)k_n-1\le m\le k_{n-1}+(\ga+2)k_n-2$, $d_\gt(m)=\frac{1}{\gt_nk_{n-1}+k_{n-2}}$ and so $(m+1)d_\gt(m)$ attains its maximum on this range:
$$\frac{k_{n-1}+(\ga+2)k_n-1}{\gt_nk_{n-1}+k_{n-2}}.$$
In order for this value to be less than $\rho$ (a necessary---but far from sufficient---condition for $\gt\in\cS$), we must have, for $\ga=0$:
$$k_{n-1}+2k_n-1<\rho(\gt_nk_{n-1}+k_{n-2}).$$
Using the substitutions
\begin{align*}
\gt_n&=a_n+\frac{1}{\gt_{n+1}}\\
k_n&=a_nk_{n-1}+k_{n-2}
\end{align*}
we apply the fact that $\gt_{n+1}\ge1$ and rearrange to obtain
$$(2-\rho)k_{n-1}a_n+k_{n-1}\lpr{1-\frac{\rho}{\gt_{n+1}}}+(2-\rho)k_{n-2}<1$$
and therefore
\begin{align*}
a_n&<\frac{\rho-1}{2-\rho}+\frac{k_{n-2}}{k_{n-1}}+\frac{1}{(2-\rho)k_{n-1}}\\
&<\frac{\rho-1}{2-\rho}+1+\frac{1}{(2-\rho)F_n}.
\end{align*}
Using the numerical value of $\rho$ and letting $n$ range on [5] gives the desired bounds.
\end{proof}
\begin{proof}[Proof of \thmref{under suff}]
\lemref{when ones} and \lemref{abcde bounds} are sufficient to prove that $\#(S\mod1)<\infty$. Running the code specified in \secref{python app} reveals the values specified in \figref{unders}. All that remains to be shown is the correctness of the program; each step is evident except for why \texttt{n} only needs to be checked up to 29. This is merely a consequence of \eqref{ineq:thm} for $N=5$, specifically in the ``worst case'' (in terms of the sizes of $k_4$ and $k_5$) of $\gt=[0,18,18,14,12,11,\dot{1}]$, where $k_4=55141$ and $k_5=611119$ so $F_{d+1}>\frac{k_5+k_4}{5+2\sqrt{5}}\approx70000$, hence $d=24$. Because this justifies the code used, the Theorem is true.
\end{proof}
To demonstrate the empirical difference between $\cS$ and a worse choice of $\gt$, see \figref{eye test} for the partition of the circle for $m=75$ for each element of $\cS$ as well as $\gt=\pi$. Stylistically, these diagrams are inspired by Motta, Shipman, and Springer's Figure 1 \cite{MSS}. When there are three distinct lengths, the longest one is colored red and the shortest green; when there are two distinct lengths (\figref{eye test}(d)), the longer one is colored orange and the shorter black. The code for this figure is found in \secref{mathematica app}.
\begin{figure}
\begin{tabular}{ccc}
\includegraphics[width=0.25\textwidth]{a1-75.pdf} & \includegraphics[width=0.25\textwidth]{a2-75.pdf} & \includegraphics[width=0.25\textwidth]{a3-75.pdf} \\
(a) $\eta_1$ & (b) $\eta_2$ & (c) $\eta_3$ \bigstrut[b] \\
\includegraphics[width=0.25\textwidth]{a4-75.pdf} & \includegraphics[width=0.25\textwidth]{pi-75.pdf} & \includegraphics[width=0.25\textwidth]{a5-75.pdf} \\
(d) $\eta_4$ & (e) $\pi$ & (f) $\eta_5$ \bigstrut[b] \\
\includegraphics[width=0.25\textwidth]{a6-75.pdf} & \includegraphics[width=0.25\textwidth]{a7-75.pdf} & \includegraphics[width=0.25\textwidth]{a8-75.pdf} \\
(g) $\eta_6$ & (h) $\eta_7$ & (i) $\eta_8$
\end{tabular}
\caption{The partition of $S^1$ for nine values of $\gt$ with $m=75$. Note that for (e), the partition is far less uniform than in the other figures.}
\label{eye test}
\end{figure}
\section{Proof of \thmref{no best}}
\begin{remark-non}
Let $\bowtie$ be the equivalence relation on $\cS$ of $\gt\bowtie\gu$ iff $\gt\pm\gu\in\Z$. Clearly $\#(\cS/\bowtie)=8$, and for $\gt,\gu\in\cS$, $d_\gt(m)=d_\gu(m)$ iff $\gt\bowtie\gu$. Therefore, $f_m:(\cS/\bowtie)\to\R^+$ with $f_m(\gt)=d_\gt(m)$ is well-defined. $\cS/\bowtie$ has the convenient choice of representatives $\{\eta_i:i\in[8]\}$.
\end{remark-non}
As a consequence of this remark, we treat $\cS$ implicitly as $\cS/\bowtie$ because of our primary concern with the context of $d_\gt(m)$. We now introduce some further notation.
\begin{definition-non}
Define the functions $w:\N\to\cS$ and $W:\cS\times\N\to\R$ as
\begin{align*}
w(M)&=\argmin\limits_{\gt\in\cS} D_M(\gt)\\
W_\gt(M)&=\#\{m\in[M]:\gt=w(m)\}.
\end{align*}
We have the shorthand
\begin{align*}
LI(i)&=\liminf\limits_{M\to\infty}\frac{W_{\eta_i}(M)}{M}\\
LS(i)&=\limsup\limits_{M\to\infty}\frac{W_{\eta_i}(M)}{M}.
\end{align*}
\end{definition-non}
We now begin our approach towards \thmref{no best}. It is an immediate corollary to the following:
\begin{theorem}\label{help 3}
We have the following asymptotics, where the third and fifth column the give the percentages rounded to the nearest tenth:
$$
\begin{array}{c||c|r||c|r||}
i & \multicolumn{2}{c||}{LI(i)} & \multicolumn{2}{c||}{LS(i)} \\\hline\hline
1 & \frac{4\sqrt{5}-6}{11} & 26.8 &\frac{13+\sqrt{5}}{41} & 37.2 \bigstrut \\\hline
2 & \frac{7-2\sqrt{5}}{29} & 8.7 & \frac{2-3\sqrt{5}}{11} & 13.4 \bigstrut \\ \hline
3 & 9-4\sqrt{5} & 5.6 & \frac{7-2\sqrt{5}}{29} & 8.7 \bigstrut \\\hline
4 & \frac{11-3\sqrt{5}}{38} & 11.3 & \frac{3\sqrt{5}-5}{10} & 17.1 \bigstrut \\ \hline
5 & \frac{19-8\sqrt{5}}{41} & 2.7 & \frac{12-5\sqrt{5}}{19} & 4.3 \bigstrut \\ \hline
6 & \frac{7\sqrt{5}-15}{10} & 6.5 & \frac{13-3\sqrt{5}}{62} & 10.1 \bigstrut \\ \hline
7 & \frac{4-\sqrt{5}}{11} & 16.0 & \sqrt{5}-2 & 23.6 \bigstrut \\\hline
8 & \frac{27-11\sqrt{5}}{62} & 3.9 & \frac{17-7\sqrt{5}}{22} & 6.1 \bigstrut \\ \hline
\end{array}
$$
In particular, each of the $\liminf$s is positive.
\end{theorem}
As an illustration of the alternating nature for small $M$, see \figref{small M}, where if $\eta_i=\argmin\limits_{\gt\in\cS}D_M(\gt)$ then the $M$th data point $\lpr{M,\min\limits_{\gt\in\cS}D_M(\gt)}$ is colored with the $i$th color in the following list: red, orange, purple, green, blue, brown, black, aquamarine. The code used to generate this figure can be found in \secref{code:small M}.
\begin{figure}
\centering
\includegraphics[width=0.75\textwidth]{fig}
\caption{A plot of $\min\limits_{\gt\in\cS}D_M(\gt)$ for $M\in[200]$, colored corresponding to $\argmin\limits_{\gt\in\cS}D_M(\gt)$.}
\label{small M}
\end{figure}
The proof of this Theorem involves indirectly computing particular values of $W$ by computing the values at which each $\gt\in\cS$ is the minimizer, in terms of the convergents. It is now convenient to look at the convergents as functions $k_5,k_6:\cS\to\N$:
\begin{align*}
k_5(\eta_1) &= 43 & k_6(\eta_1) &= 70 \\
k_5(\eta_2) &= 23 & k_6(\eta_2) &= 37 \\
k_5(\eta_3) &= 35 & k_6(\eta_3) &= 57 \\
k_5(\eta_4) &= 18 & k_6(\eta_4) &= 29 \\
k_5(\eta_5) &= 27 & k_6(\eta_5) &= 44 \\
k_5(\eta_6) &= 19 & k_6(\eta_6) &= 30 \\
k_5(\eta_7) &= 13 & k_6(\eta_7) &= 21 \\
k_5(\eta_8) &= 19 & k_6(\eta_8) &= 31 \\
\end{align*}
We then define new sequences
\begin{align*}
K_n(1) &= 70F_n+43F_{n-1} \\
K_n(2) &= 71F_n+44F_{n-1} \\
K_n(3) &= 76F_n+47F_{n-1} \\
K_n(4) &= 79F_n+49F_{n-1} \\
K_n(5) &= 81F_n+50F_{n-1} \\
K_n(6) &= 89F_n+55F_{n-1} \\
K_n(7) &= 92F_n+57F_{n-1} \\
K_n(8) &= 97F_n+60F_{n-1}
\end{align*}
with the further convention that for any $n\in\N$, $m\in\Z$, and $i\in[8]$,
$$K_n(i+8m)=K_{n-m}(i).$$
So, for instance, $K_n(0)=K_{n-1}(8)$.
Note that this is merely a reindexing of each $k_n(\cdot)$ by the permutation $\pi=(2 \ 8 \ 5)(3 \ 7 \ 6 \ 4)\in S_8$ (that is, $K_n(j)$ is a shift of the convergents $k_{n}(\eta_i)$ for $j=\pi i$). Call $\copi=\pi\inv$.
\begin{lemma}\label{cycle}
For all positive integers $n$,
$$K_n(1)<K_n(2)<K_n(3)<K_n(4)<K_n(5)<K_n(6)<K_n(7)<K_n(8)<K_{n+1}(1).$$
\end{lemma}
\begin{proof}
Equivalently, $K_n(i)<K_n(j)<K_{n+1}(i)$ for all $1\le i<j\le 8$. The first inequality is obvious: if $K_n(i)=a_iF_n+b_iF_{n-1}$, then by inspection, $a_i<a_j$ whenever $i<j$. The second inequality comes from observing that $K_{n+1}(i)=a_iF_{n+1}+b_iF_n=(a_i+b_i)F_n+a_iF_{n-1}$ and since $a_i>b_j$ for all $i,j$.
\end{proof}
\begin{lemma}\label{sigma and tau formulas}
Define the sequences $\gs_n(i)$ and $\tau_n(i)$, where $\gs_n(i)<\tau_n(i)<\gs_{n+1}(i)-1$, as follows:
$$\{M\in\N : \eta_i=w(M)\}=\bigsqcup\limits_{n\in\N}[\gs_n(i),\tau_n(i)].$$
Then, we have that $j=\pi i$ and
\begin{align*}
\gs_n(i)&=\ceil{(K_{n+3}(j-2)-3)\lpr{\frac{K_{n-1}(j-1)+K_n(j-1)\vf}{K_{n-1}(j-2)+K_n(j-2)\vf}}}-1\\
\tau_n(i)&=\ceil{(K_{n+3}(j-1)-3)\lpr{\frac{K_{n-1}(j)+K_n(j)\vf}{K_{n-1}(j-1)+K_n(j-1)\vf}}}-2.
\end{align*}
\end{lemma}
\begin{proof}
We first establish that these sequences are well-defined for all $i$.
$w(M)$ is $\eta_i$ for which $D_M(\eta_i)<D_M(\eta_j)$ for all $j\neq i$. However, for all choices of $i\neq j$ and $n$, with $M_n(i)=K_n(\pi i)+2K_{n+1}(\pi i)-2=K_{n+3}(\pi i)-2$, we have
$$D_{M_n(i)}(\eta_i)\ge(M_n(i)+1)d_{\eta_i}(M_n(i))>D_{M_n(i)}(\eta_j).$$
The first inequality is trivial.
The second follows by considering $m+1$ and $d_{\eta_j}(m)$ separately: clearly on $m\in[M_n(i)]$, $m+1\le M_n(i)+1$. Then, say for fixed $j$ that $K_{n+2}(\pi j)-1\le M_n(i)\le K_{n+3}(\pi j)-2$. By \lemref{cycle}, $M_{n-1}(i)<K_{n+2}(\pi j)<M_n(i)$, from which we conclude that $K_n(\pi i)<K_n(\pi j)$. By \corref{cor:useful}, we have that
\begin{align*}
d_{\eta_i}(M_n(i))&=\frac{1}{K_{n-1}(\pi i)+K_n(\pi i)\vf} \\
d_{\eta_j}(M_n(i))&=\frac{1}{K_{n-1}(\pi j)+K_n(\pi j)\vf}
\end{align*}
Therefore $d_{\eta_i}(M_n(i))>d_{\eta_j}(M_n(i))$, concluding the second inequality. Thus, there are infinitely many values $M$ (e.g. those of the form $M_n(i)$) at which $\eta_i\neq w(M)$. Hence $\gs_n(i)$ and $\tau_n(i)$ are well-defined sequences for all $i$.
Further, it is evident from the above argument that the ``order of succession'' for $M$ sufficiently large, e.g. $M\ge K_1(1)=70$, is $\eta_{\copi i}$ for $i=1,2,\dots,8$ and repeating---that is, $w(M)=\eta_1$ for $M$ on some interval $[s_1,s_2-1]$, followed by $w(M)=\eta_2$ on $[s_2,s_3-1]$, etc., up to $w(M)=\eta_8$ on $[s_8,s_1'-1]$, and then this cycle repeats with $w(M)=\eta_1$ on $[s_1',s_2'-1]$. Therefore we just need to compare $\eta_{\copi i}$ against $\eta_{\copi(i-1)}$ and $\eta_{\copi(i+1)}$. See \figref{runs} for an illustration of the interval-based behvaior.
\begin{figure}
\centering
\includegraphics[width=0.75\textwidth]{fig2}
\caption{A plot of $\min\limits_{\gt\in\cS}D_M(\gt)$ for $M\in[1200,1400]$, colored corresponding to $\argmin\limits_{\gt\in\cS}D_M(\gt)$.}
\label{runs}
\end{figure}
It is now convenient to define ``dual'' sequences $\hgs_n(i)$ and $\htau_n(i)$ defined as $[\hgs_n(i),\htau_n(i)]\ni M$ the $n$th range on $\N\cap[70,\infty)$ for which $\eta_{\copi i}$ {\em maximizes} $D_M(\gt)$ over $\gt\in\cS$. We see that for similar reasons, this maximizer cycles through $1,2,\dots,8$. We compute $\hgs_n(i)$ by considering $\eta_{\copi(i-1)},\eta_{\copi i}$: at what value $m>M_n(\copi(i-1))$ does it first occur that
$$(m+1)d_{\eta_{\copi i}}(m)\ge(M_n(\copi(i-1))+1)d_{\eta_{\copi(i-1)}}(M_n(\copi(i-1)))?$$
Algebraic manipulation gives $m\ge(M_n(\copi(i-1))-1)\lpr{\frac{K_{n-1}(\copi i)+K_n(\copi i)\vf}{K_{n-1}(\copi(i-1))+K_n(\copi(i-1))\vf}}$, hence
$$\hgs_n(i)=\ceil{(M_n(\copi(i-1))-1)\lpr{\frac{K_{n-1}(\copi i)+K_n(\copi i)\vf}{K_{n-1}(\copi(i-1))+K_n(\copi(i-1))\vf}}-1}.$$
Then, since $\N\cap[70,\infty)\subset\bigsqcup\limits_{n\in\N}\bigsqcup\limits_{i\in[8]}[\hgs_n(i),\htau_n(i)]$, we immediately obtain the relationship
$$\htau_n(i)=\hgs_n(i+1)-1.$$
Finally, we observe that
\begin{align*}
\gs_n(i)&=\hgs_n(\pi i-1)\\
\tau_n(i)&=\htau_n(\pi i-1)=\hgs_n(\pi i)-1
\end{align*}
because by that interval, all $j\neq i$ will have already achieved a maximum surpassing $\eta_{\copi i}$'s.
\end{proof}
\begin{proof}[Proof of \thmref{help 3}]
Given $i$ and $M$, let $j=\pi i$ and let $n$ be the greatest integer such that $K_n(j)\le M$.
$W_{\eta_i}(M)\in\gT\lpr{\tau_n(j)-\gs_n(j)}$ and so we have the following asymptotic tendencies:
\begin{align*}
\liminf\limits_{M\to\infty}\frac{W_{\eta_i}(M)}{M}&=\lim\limits_{n\to\infty}\frac{\tau_n(j)-\gs_n(j)}{\gs_{n+1}(j)-\gs_n(j)} \\
\limsup\limits_{M\to\infty}\frac{W_{\eta_i}(M)}{M}&=\lim\limits_{n\to\infty}\frac{\tau_n(j)-\gs_n(j)}{\tau_n(j)-\tau_{n-1}(j)}
\end{align*}
and using the exact values computed in \lemref{sigma and tau formulas} gives the stated values.
\end{proof}
We can interpret this result as saying that as $M$ grows, each element of $\cS$ is represented as $w(M)$ infinitely many times. Further, $\eta_1=w(M)$ with marginally higher probability than the alternatives.
There is an interesting parallel to be drawn with Theorems \ref{no best} and \ref{help 3} and with work in analytic number theory on prime distributions. In 1914, Littlewood \cite{Lit} proved the unexpected fact that the difference $\pi(x)-\li(x)$ alternates infinitely often.\footnote{Here, $\pi(x,q,a)$ counts primes $p<x$ with $p\equiv a\pmod{q}$ with $\pi(x)$ implicitly having $(q,a)=(1,0)$ and $\li$ is the logarithmic integral $\int_0^x\frac{dt}{\log t}$.} Likewise, \thmref{no best} gives eightfold (rather than twofold) alternation. Earlier, in 1853, Chebyshev noticed that $\pi(x,4,3)>\pi(x,4,1)$ despite the asymptotic behavior $\frac{\pi(x,4,3)}{\pi(x,4,1)}\to1$, a result strengthened and generalized considerably by Rubinstein--Sarnak \cite{RS} and termed ``Chebbyshev's bias.'' Here we see a much stronger emergent bias in the statement of \thmref{help 3}, where there exists some $M_0\in\N$ where for all $M>M_0$, we have
$$W_{\eta_1}(M)>W_{\eta_7}(M)>W_{\eta_4}(M)>W_{\eta_2}(M)>W_{\eta_6}(M)>W_{\eta_3}(M)>W_{\eta_8}(M)>W_{\eta_5}(M).$$
In preliminary explorations that became this paper, an attempt was made at the related problem of
\begin{center}
\bf for each $M\in[49]$, minimize $D_M(\gt)$ over all $\gt\in\lbr{0,\half}$.
\end{center}
The approach was to na\"{i}vely sample from the interval a large number of times (100000) for each $M$. Except when $M$ takes the values 30 and 31---where the optimum is approximately $\frac{1}{30}$ and $\frac{1}{31}$, respectively, to within one part in $10^6$---the values agree with the problem constrained for $\gt\in\cS$ as is solved in this section of the text to within one part in at least $10^3$.
We can also compare these results with Ridley \cite{Rid}, which studies a related problem in packing efficiency of features in plants which grow at fixed divergence angles. There, the optimal angle (out of total angle 1) is determined to be $(\vf-1)^2$; note that $\eta_7=(\vf-1)^2$ (as enumerated in \figref{unders}). Therefore, we see that Ridley's notion of optimality coincides with the notion explored here using $D_M(\gt)$ when $M$ takes the values 2, 5, 7--10, 29, 45, and 47--49, where in Ridley's model, $M$ represents the number of generations, that is, the number of features (e.g. petals on a flower) that have grown using the constant divergence angle $\gt$.
\section*{Acknowledgements}
This work was completed as part of my senior thesis at Princeton University. I am grateful to my advisor Peter Sarnak for suggesting this problem and for his guidance throughout.
|
1,314,259,993,636 | arxiv | \section{Introduction}\label{sec:introduction}
Massive stars largely drive the dynamical and chemical evolution of gas in galaxies \citep[e.g.][]{Hopkins:2014}. They accomplish this via their stellar winds, eruptions, and explosive deaths, ultimately producing neutron stars and black holes \citep{Langer:2012}.
These compact remnants can merge and generate the gravitational waves observed by LIGO/Virgo \citep{2016PhRvL.116f1102A}. The evolutionary trajectory starting with a massive star burning hydrogen in its core and ending with a compact remnant is understood only qualitatively. We still do not know how to map initial properties of the star, like mass, rotation rate, and metallicity, to e.g. the final mass and spin of the compact remnant it leaves behind. The picture is further complicated by the fact that the majority of massive stars are found in multiple systems \citep{Sana:2012}, with a large fraction expected to interact with their companions \citep{deMink:2014}.
The path towards a detailed understanding of massive stars begins with a quantitative study of their internal structure on the main sequence. In recent years asteroseismology has opened a new window on these challenging astrophysical environments, with high precision photometry from space delivering many new exciting results \citep[e.g., MOST, CoRoT, BRITE, Kepler/K2 and TESS, see ][]{BowmanReview:2020}.
The latest discovery is the detection of a new ubiquitous phenomenon in massive stars: stochastic low-frequency photometric variability \citep[SLF variability;][]{Blomme:2011,2019NatAs...3..760B,Bowman:2019,Pedersen:2019,Bowman:2020,Rauw:2021}.
This joins a number of other surface and wind phenomena that are routinely observed in early-type stars, including surface velocity fluctuations \citep[Macroturbulence;][]{Simon-Diaz:2014}, line profile variability \citep{Fullerton:1996}, and discrete absorption components in UV spectra \citep{Howarth:1989,Cranmer:1996,Fullerton:1997,Kaper:1997}. Surface magnetism and bright spots are harder to observe but could still be common in these stars \citep[e.g.][]{Ramiaramanantsoa:2014}.
The origin of this SLF variability is currently debated.
It could be caused by sub-surface convection zones \citep{Cantiello:2009,Blomme:2011,Lecoanet:2020} or by internal gravity waves (IGWs) launched by turbulent core convection \citep{Edelmann:2019,Ratnasingam:2020}\footnote{Classical heat-mechanism pulsations could be responsible for
spectroscopic and photometric variability in specific parts of the HRD, but
they can hardly justify the apparent ubiquity of macroturbulence and SLF in
massive stars \citep{Godart:2017,Simon-Diaz:2017}.}. Instabilities in the stellar wind could also play a role \citep{Krticka:2021}.
Regardless of its origin, this photometric signal likely carries important information about stellar structure, complementing asteroseismic studies that use well-identified oscillation modes \citep[e.g.][]{Aerts:2019,Aerts_araa:2019,BowmanReview:2020}.
Recently the use of high resolution ground-based spectroscopy for the targets observed by K2 and TESS \citep{Bowman:2020} has allowed precise determination of stellar parameters, including spectroscopic mass, luminosity, and macroturbulence \citep{Burssens:2020}.
The latter is particularly important if, as seems likely, the mechanism exciting surface turbulent velocities is the same as that which produces the observed SLF variability \citep{Grassitelli:2016,Bowman:2020}.
Here we combine spectroscopic and photometric data to compare the observed properties of the stochastic photometric variability and macroturbulence with predictions from non-rotating 1D stellar models.
We make simple predictions for the amplitude and frequency of the variability that subsurface convection induces at the stellar surface and examine how these vary with stellar temperature and luminosity.
We find that the predicted trends coming from a suburface convection zone driven by the iron opacity peak (FeCZ) at $\approx150$kK match the observations well.
We next show that one way to differentiate between the two proposed mechanisms is by examining macroturbulence and SLF in massive stars with surface magnetic fields, since magnetic effects have a larger impact on core IGWs than on the FeCZ.
Macroturbulence is observed in stars with fairly strong magnetic fields, sufficient to suppress IGWs from the core, favoring a model based on subsurface convection.
Furthermore, the only stars with no observed macroturbulence are ones where the magnetic field is strong enough to shut off the FeCZ, and so far as we know all stars with such strong magnetic fields lack macroturbulence, consistent with a subsurface origin of surface perturbations~\citep{Jermyn:2020}.
Based on these considerations we suggest that subsurface convection represents a possible unifying mechanism causing SLF variability, surface turbulence, and magnetic spots in massive stars.
\begin{figure}[ht!]
\begin{center}
\includegraphics[width=1.0\columnwidth]{figures/kipp_radius_norm_20.pdf}
\caption{\label{fig:kipp20} Evolution of the normalized radial location of the FeCZ in a 20${\rm M}_\odot$ model during the main sequence (first 5 Myr of evolution). The HeCZ is also visible very close to the surface. Locations of P(r)/P($R_*$) = $e^n$ for $n=[1,3,5,7]$ are also shown and labelled as $n {\rm H}_{\rm P}$.}
\end{center}
\end{figure}
\section{Methods}
We calculated stellar evolution models using the Modules for Experiments in Stellar Astrophysics
\citep[MESA][]{Paxton2011, Paxton2013, Paxton2015, Paxton2018, Paxton2019} software instrument.
Details on the microphysics inputs to this software instrument are given in Appendix~\ref{sec:software}.
Our models have initial mass ranging from 5 to 120${\rm M}_\odot$ and are non-rotating. Since OB stars are known to be rapidly rotating \citep[e.g.][]{Maeder:2000,Dufton:2013,Ramirez:2013}, we discuss the potential impact of rotation on our results in Section~\ref{sec:rotation}. We also neglect the effect of wind mass-loss. While this process is important for the evolution of massive stars \citep{Smith:2014}, the conditions in subsurface convective zones and the properties of convection depend almost exclusively from the location in the Hertzsprung-Russell diagram. This is true as long as the outer layers composition is not substantially altered.
We focused on the structure and convective properties of our models in order to calculate typical frequencies and convective fluxes.
Convection is calculated in the framework of the Mixing Length Theory \citep[][MLT hereafter]{1958ZA.....46..108B}, and we adopted $\alpha_{\rm MLT} = 1.6$. While the properties of efficient convection zones (e.g. stellar cores) are insensitive to the choice of this parameter, those of inefficient convective regions close to the stellar surface \emph{do} depend on $\alpha_{\rm MLT}$ \citep{2015ApJ...813...74J,Cantiello:2019}. In Section~\ref{sec:discussion} we discuss how uncertainties in the choice of the $\alpha_{\rm MLT}$ parameter affect our results.
Since we are interested in the excitation of surface phenomena we focus on the upper part of convective zones.
Following \citet{Cantiello:2009} we define the average of a generic quantity $q$ as
\begin{equation}
\overline{q} \equiv \frac{1}{\alpha_{\rm MLT} \mathrm{H}_{\mathrm{P}}} \int_{\mathrm{R}_{\mathrm{c}}-\alpha_{\rm MLT}\mathrm{H}_{\mathrm{P}}}^{\mathrm{R}_{\mathrm{c}}} q(r)\; dr,
\label{aver}
\end{equation}
where $\mathrm{H}_{\mathrm{P}}$ is the pressure scale height calculated at the upper boundary ($\mathrm{R}_{\mathrm{c}}$) of the
convective zone of interest. We tested a variety of different average prescriptions and found that our results do not depend much on the specific choice of prescription.
Using equation~\eqref{aver} we extracted the average Mach number $\overline{\rm M}_{\rm c}$, convective velocity $\langle{\varv_{c}}\rangle$ and density $\overline{\rho}_{\rm c}$ in the convective core and in the subsurface convection zones of our models.
The most important subsurface convection zone for the massive stars we focus on is the FeCZ (see Fig.~\ref{fig:kipp20}), although in the low luminosity regime He-driven convection zones could play a role as well \citep{Cantiello:2009,Cantiello:2019}.
We compare relevant properties of our theoretical models with the observed characteristic frequency $\nu_{\rm char}$ and amplitude $\alpha_0$ of SLF variability. These quantities are derived by fitting the stochastically-excited, low-frequency power excess in a power
density spectrum using a Lorentzian function \citep[e.g.][]{Bowman:2019}:
\begin{equation}\label{eq:rednoise}
\alpha(\nu) = \frac{\alpha_0}{1+\big(\frac{\nu}{\nu_{\rm char}}\big)^{\gamma}} \, + P_{\rm W}.
\end{equation}
This shows that $\alpha_0$ represents the amplitude at zero
frequency. $\nu_{\rm char}$ is defined as in eqn.~\ref{eq:nuchar} with $\tau$ the typical timescale of the SLF variability.
Finally, $\gamma$ is the gradient of the linear part of the profile and $P_{\rm W}$ is a white noise term.
Due to its stochastic, low-frequency manifestation in the power density spectrum, in the literature the SLF variability is also referred to as ``red noise'' \citep[e.g.][]{Blomme:2011}.
\section{Results}
We want to test a possible correlation between the properties of the FeCZ and observed photospheric phenomena, in particular SLF variability and turbulent velocities at the stellar surface (macroturbulence).
We proceed by calculating quantities that measure the amplitude of perturbations in the FeCZ. We then check if some of these properties correlate with the amplitude of observed photospheric phenomena, including turbulent velocity fluctuations and SLF variability.
We define the characteristic frequency as
\begin{equation}\label{eq:nuchar}
\nu_{\rm char} \equiv \frac{1}{2\pi\tau},
\end{equation}
where $\tau$ is a characteristic timescale.
For comparing with observations we set $\tau$ to be the average convective turnover time
\begin{equation}\label{eq:turnover}
\tau_{\rm c} = \alpha_{\rm MLT} \overline{\mathrm{H}}_{\mathrm P} / \langle{\varv_{c}}\rangle,
\end{equation}
calculated either in the FeCZ or in the convective core.
We calculated the convective flux $F_{\rm c} = \overline{\rho}_{\rm c} \langle{\varv_{c}}\rangle^3$, where $\overline{\rho}_{\rm c}$ and $\langle{\varv_{c}}\rangle$ are the average density and convective velocity calculated according to eqn.~\ref{aver}.
We did this as a function of both mass and evolutionary history for stars with initial mass ranging from 5 to 120$~M_\odot$.
Here we present results for an initial metallicity of Z=0.02, but in Appendix~\ref{appen:grids} we report results for model grids with Z=0.006 and 0.002 as well.
\subsection{FeCZ and Macroturbulence}
\begin{figure*}
\centering
\subfloat{\includegraphics[width=1\columnwidth]{figures/vc_FeCZ_macroturbulence.pdf}}\hfill
\subfloat{\includegraphics[width=1.005\columnwidth]{figures/vc_FeCZ_turbulentpressure.pdf}}
\caption{\label{fig:vconv} Left: Average convective velocities in the FeCZ as a function of the location of stellar models in the spectroscopic H-R Diagram ($\mathscr{L} \equiv {\rm T}^4_{\rm eff}/g$). We also show observed stars with detected macroturbulent velocity as grey circles. The area of the circle is proportional to the observed macroturbulence \citep[data from][]{Burssens:2020,Bowman:2020}. The FeCZ is absent in models with $\log \mathscr{L}/\mathscr{L}_\odot \lesssim 2.5$. Right: Same as left, but showing predictions for the maximum of the ratio between turbulent pressure and total pressure in any subsurface convection zone. The FeCZ largely dominates, except for stars at low luminosities where turbulent pressure is provided by a helium convection zone (HeCZ).}\label{fig:macroturbulence}
\end{figure*}
Macroturbulence is a spectroscopic measure of velocity fields with a scale larger than the photons mean free path in the stellar atmosphere. The shape of spectral lines can be used not only to measure the amplitude of the velocity field, but also to infer some of its directionality.
Note that \citet{Simon-Diaz:2014} showed that the line profiles are fitted better by a radial-tangential velocity function than a Gaussian one, but the observations do not tell if the dominant velocity component is radial or horizontal\footnote{The claim is that the line profiles are fitted better by the radial-tangential velocity function, compared to an isotropic one, with either $v_{\rm r} \gg v_{\rm t}$ or $v_{\rm r} \ll v_{\rm t}$, where $v_{\rm r}$ and $v_{\rm t}$ are the radial and tangential components of the velocity, respectively (Sim\'on-D\'iaz, private comm.).
Below we compare the observed amplitude of macroturbulent velocities with predicted properties of the FeCZ.}
We show the average convective velocity in the FeCZ in Fig.~\ref{fig:vconv}, left panel. Velocities of the order $10 \dots 100\ \rm km s^{-1}$ are found across the upper spectroscopic H-R diagram \citep[$\mathscr{L} \equiv {\rm T}^4_{\rm eff}/g$;][]{2014A&A...564A..52L}, with a trend of increasing $\varv_{c}$ for higher luminosities. Note that, contrary to core convection, the FeCZ is just mildly subsonic, with Mach numbers ranging from 0.01 to 0.3, though these are uncertain by a factor of $\approx 8$ due to a dependence on the uncertain $\alpha_{\rm MLT}$ \citep{Cantiello:2019}.
In the right panel of the same figure we also show the maximum of the ratio of turbulent pressure to total pressure (P$_{\rm turb} \propto \varv_{c}^2$), which, in agreement with the results of \citet{Grassitelli:2015}, shows a strong correlation with the spectroscopically-derived macroturbulent velocities in massive stars.
The two quantities in Fig.~\ref{fig:vconv} measure the strength of the inefficient convection.
We do not yet know the exact mechanism connecting the FeCZ with the surface velocity perturbations.
If one assumes convective elements conserve their inertia as they reach layers stable against convection \citep[e.g., because they are thermally diffusive, see][]{2015ApJ...813...74J}, then the surface velocities should be proportional to the convective velocity; see the left panel of Fig.~\ref{fig:vconv}.
Alternatively, one can assume IGWs are excited with pressure perturbations $\delta p \sim P_{\rm turb}\propto \varv_{c}^2$, the turbulent pressure of the convection \citep{Press:1981}.
This stochastic excitation can lead to both running waves and standing modes. Using the polarization relations of adiabatic IGWs \citep[e.g.,][]{sutherland_2010}, a wave with pressure perturbation $\delta p$ has an associated horizontal velocity $u_h \sim (\delta p/\rho_0) k_h/\omega$, where $k_h$ is the horizontal wavenumber of the wave and $\omega$ is its frequency.
The dominant waves will have $k_h\sim1/{\mathrm{H}}_{\mathrm P}$ and $\omega\sim1/\tau_c$ \citep{Cantiello:2009}.
So running IGWs would have surface velocities which also scale like $u_h \sim \varv_{c}$ at the surface.
Concerning mode excitation, \citet{Grassitelli:2015} argue that the ratio of turbulent pressure to total pressure in the FeCZ traces the stochastic Lagrangian pressure perturbation associated with the convective motions. This is responsible for local deviation from hydrostatic equilibrium and the excitation of high-order pulsations with frequencies close to the spectrum of the fluctuations. The ratio of the turbulent to total pressure in the FeCZ is reported in the right panel of Fig.~\ref{fig:vconv}.
We confirm that convective velocities and the ratio of turbulent pressure to total pressure in the FeCZ correlate very well with the amplitude of macroturbulence.
\subsection{FeCZ and Stochastic, Low-Frequency Variability}
\begin{figure*}[htp]
\centering
\subfloat{\includegraphics[width=1.0\columnwidth]{figures/freq_FeCZ_data.pdf}}\hfill
\subfloat{\includegraphics[width=1\columnwidth]{figures/flux_FeCZ_rednoise.pdf}}
\caption{ Left panel: Characteristic frequency $\nu_{\rm char}$ in the FeCZ as a function of the location of stellar models in the spectroscopic H-R Diagram (black contour lines). Evolutionary tracks are shown as grey solid lines. We also show the observed stars with detected stochastic photometric variability as grey circles. The area of the circle is proportional to the observed $\nu_{\rm char}$, derived from eqn.~\eqref{eq:rednoise} fitting the data in the range $0.1 \le
\nu \le 360\,$d$^{-1}$ \citep{Burssens:2020,Bowman:2020}. Right panel: Ratio of FeCZ convective flux to the total stellar flux in the spectroscopic H-R Diagram ($\mathscr{L} \equiv {\rm T}^4_{\rm eff}/g$). We also show the observed stars with detected stochastic, low-frequency photometric variability as grey circles. The area of the circle is proportional to $\alpha_0$.}
\label{fig:nuchar_fecz}
\end{figure*}
If the SLF variability is caused by the FeCZ, a natural choice of proxy for the typical timescale of this variability is the convective turnover timescale. We find typical values of the convective turnover timescale to be about $\sim 0.1\dots2$ d in the FeCZ, with a tendency for shorter values in models with high effective temperature and surface luminosity \citep{Cantiello:2009}.
We computed $\nu_{\rm char}$ using eqn.~\ref{eq:nuchar} and plot this alongside observed characteristic frequencies\footnote{These were inferred via eqn.~\eqref{eq:rednoise}.} of SLF variability on the spectroscopic H-R diagram, see left panel in Fig.~\ref{fig:nuchar_fecz}.
We see good agreement, with our models reproducing both the typical values of the observed $\nu_{\rm char}$ and the trend with $\log \mathscr{L}$ and $\log {\rm T}_{\rm eff}$.
While the characteristic frequencies found by \citet{Bowman:2020} seem to be larger by a factor of $\approx$ 3, our predictions for the turnover timescale are affected by uncertainty in the convective velocities arising from the MLT treatment ($\varv_{c} \propto \alpha_{\rm MLT}^{3}$, so $\nu_{\rm char} \propto \alpha_{\rm MLT}^{2}$), as well as our limited knowledge of the frequency spectrum generated by turbulent convection. Since $\alpha_{\rm MLT}$ is uncertain by a factor of 2 or so, our estimates of the characteristic frequency in the FeCZ are uncertain by a factor of $\approx$ 4 and so are consistent with the observations.
We also compare the amplitude of SLF variability with the relative convective flux in the FeCZ (right panel in Fig.~\ref{fig:nuchar_fecz}). Values of F$_{\rm c}$/F$_{*}$ tend to increase with increasing $\log \mathscr{L}$ and decreasing $\log {\rm T}_{\rm eff}$, a trend that is also found for the amplitude of observed SLF variability.
Overall the turnover timescale and relative flux in the FeCZ correlate very well with the observed timescale and amplitude of SLF variability.
\subsection{Core Convection and Stochastic, Low-Frequency Variability}
\begin{figure*}[htp]
\centering
\subfloat{\includegraphics[width=1\columnwidth]{figures/F0_relative_flux_IGW_ahp_core_rednoise_data.pdf}}\hfill
\subfloat{\includegraphics[width=1.03\columnwidth]{figures/freq_core_data_days.pdf}}
\caption{ Left panel: Square root of the ratio between the flux of gravity waves launched by the convective core and $F_0 = 1/2 \rho(r) \,r^3 \omega^2 \sqrt{N(r)^2-\omega^2}$, evaluated at the stellar surface. In the absence of damping, this quantity is expected to be proportional to the relative radial displacement (Appendix~\ref{appen:xir}) and hence to the relative luminosity fluctuations at the surface. The IGW flux is calculated multiplying the core convective flux by the average convective Mach number in the top pressure scale height of the convective region. We also show the observed stars with detected stochastic, low-frequency photometric variability as grey circles. The area of the circle is proportional to $\alpha_0$, derived from eqn.~\eqref{eq:rednoise} fitting the data in the range $0.1 \le
\nu \le 360\,$d$^{-1}$ \citep{Burssens:2020,Bowman:2020}.
Right panel: Characteristic frequency $\nu_{\rm char}$ in the convective core as function of the location of stellar models in the spectroscopic H-R diagram. We also show the observed stars with detected photometric variability as grey circles. The area of the circle is proportional to $\nu_{\rm char}$. }
\label{fig:nuchar_core}
\end{figure*}
Early-type stars posses convective cores during their main sequence. Turbulent convection in these convective cores excites internal gravity waves \citep{1990ApJ...363..694G,2013ApJ...772...21R,2013MNRAS.430.2363L,Shiode:2013,Edelmann:2019,Horst:2020}, that can propagate through the stellar envelope and reach the stellar surface \citep[e.g.][]{Ratnasingam:2019,Lecoanet:2020,Ratnasingam:2020}.
Some groups have argued that such waves could be responsible for both the observed macroturbulence and SLF variability in early-type stars \citep{2009A&A...508..409A,2017A&A...597A..22S,Bowman:2019,2019NatAs...3..760B}.
There is substantial uncertainty in the surface brightness fluctuations from IGWs generated by core convection. \citet{Shiode:2013} predicted the typical amplitude of g-modes excited by convection to be $\approx 10^{-2}-10^2 \, \mu{\rm mag}$, seemingly at odds with the relatively large amplitudes $\approx 10-10^4 \mu{\rm mag}$ observed by e.g.~\citet{Bowman:2020}. However, \citet{Lecoanet:2021} recently found an error in the \citet{Shiode:2013} prediction. The g-mode amplitude should be larger by a factor of $\approx \sqrt{\nu/\gamma}$, where $\gamma$ is the mode's damping rate. This missing factor could increase the predicted g-mode amplitude by a factor of $10^4$ or larger for high-frequency modes near the Brunt-V\"ais\"al\"a frequency (e.g., $\sim 10\, d^{-1}$ for a $10 M_\odot$ ZAMS star), but does not change the predicted mode amplitudes for lower frequency waves (e.g., $\sim 0.3\, d^{-1}$ for a $10 M_\odot$ ZAMS star). \citet{Lecoanet:2020} argued that there should be very low wave power at low frequencies due to radiative damping, while the wave signal at frequencies above $0.5\, d^{-1}$ should be dominated by g-modes, as predicted by \citet{Shiode:2013}. These features do not seem to be present in the observed spectra.
Recent numerical simulations of wave generation by convection in a $3M_\odot$ star \citep{Edelmann:2019,Horst:2020} produce wave fluctuation spectra which are qualitatively similar to those observed. However, those simulations artificially boost the stellar luminosity by factors ranging from $10^3$ to $10^7$. Boosting the luminosity should both enhance the wave amplitude and increase the typical frequency of excited waves, making it difficult to quantitatively compare to observations.
Although the detailed physics of wave generation by convection is uncertain, we can still analyze the properties of core convective of our models.
If the surface variability is due to core convection, one would expect the characteristic frequency and amplitude of the SLF variability to correlate with the properties of the core convection.
The flux of IGWs excited by turbulent convection at the core boundary is of order F$_{\rm IGW} = {\rm F}_* \overline{\rm M}_{\rm c}$ \citep{1990ApJ...363..694G}, where we evaluate the average Mach number $\overline{\rm M}_{\rm c}$ using eqn.~\ref{aver} and the local adiabatic sound speed. We expect the luminosity fluctuations to be proportional to the relative surface radial displacement $\xi_r/R$ produced by these waves at the stellar surface \citep[e.g.][]{Dziembowski:1977,Aerts:2010}. It can be shown that $\xi_r/R \propto \sqrt{{\rm F}_{\rm IGW}/ {\rm F}_0} $ (see Appendix~\ref{appen:xir}). This quantity is presented in Fig.~\ref{fig:nuchar_core} along with the observed amplitudes of SFL variability.
One important caveat is that this estimate neglects the important role of radiative damping, which is essential in shaping both the amplitude and the shape of the spectrum of waves at the surface \citep[e.g.][]{Lecoanet:2020}. Rotation might also be key in setting the amplitude and shape of the surface fluctuations spectrum (See Section~\ref{sec:rotation}).
Despite neglecting the important effect of radiative damping, the trend for the relative radial displacement at the surface do show some interesting correlations with the observed trend in SLF variability, though they generally proceed the wrong way, with increasing amplitude towards higher luminosities where we expect IGW to show the smallest effects (left panel in Fig.~\ref{fig:nuchar_core}).
Next, we focus on the characteristic timescale of waves excited by core convection.
We expect the maximum of the IGW flux to be launched at frequencies close to $\nu_{\rm char} = (2\pi\tau_{\rm c})^{-1}$, with $\tau_{\rm c}$ defined in eqn.~\ref{eq:turnover}.
Fig.~\ref{fig:nuchar_core} shows that the predicted values of $\nu_{\rm char}$ are in the range $0.02\dots0.008\,{\rm d}^{-1}$ ($0.2\dots0.08\, \mu\textrm{Hz}$), in agreement with the results of \citet{Shiode:2013}.
These values are about 2 orders of magnitude smaller than the typical characteristic frequencies of SLF variability observed by \citep{Bowman:2019,2019NatAs...3..760B}.
As mentioned earlier, a significant caveat in correlating core and surface quantities is that wave propagation through the stellar envelope affects the spectrum, changing the frequency of maximum power of waves reaching the surface \citep{Lecoanet:2020}.
Radiative damping efficiently suppresses low-frequency IGWs (because these waves have high radial wavenumbers), and the peak of the spectrum observed at the surface is expected to move to higher frequencies. The amplitude of this effect depends on the envelope properties, which are a function of $\log {\rm T}_{\rm eff}$ and $\log \mathscr{L}$. Nevertheless we will assume that trends of $\nu_{\rm char}$, as function of $\log {\rm T}_{\rm eff}$ and $\log \mathscr{L}$, are still set by the core convective properties.
Under this assumption, Fig.~\ref{fig:nuchar_core} shows that $\nu_{\rm char}$ should increase with both $\log {\rm T}_{\rm eff}$ and $\log \mathscr{L}$, so that the characteristic frequencies of peak IGWs should increase as stars evolve on the main sequence. This is exactly the opposite of what is observed: the characteristic frequencies of SLF variability is largest for stars early on on the main sequence, and seem to decreases as stars evolve.
Therefore either radiative damping is able to revert this trend or else the observed variability is unlikely to be caused by IGWs launched by the core.
\section{Macroturbulence in Magnetic Stars}
The hypothesis that subsurface convection is responsible for surface turbulence is corroborated by the match between trends in the FeCZ properties (i.e. convective velocities and turbulent pressure) and the observed micro and macroturbulence \citep{Cantiello:2009,Grassitelli:2015}.
One important test to this hypothesis is provided by the disappearance of macroturbulence in stars with surface magnetic fields above a critical strength \citep{10.1093/mnras/stt921}, closely corresponding to
the critical field needed to stabilize the FeCZ~\citep{2019MNRAS.487.3904M,Jermyn:2020}. Note that in OB stars, the FeCZ is deeper and more vigorous than the H and He convection zones, so a magnetic field stabilizing the FeCZ will necessarily also stabilize the other subsurface convection zones.
We can analogously define a critical magnetic field strength $B_{\rm crit}$ which suffices to reflect IGWs before they reach the photosphere.
Using the dispersion relation for IGWs in a magnetic medium, the radial component of this field is~\citep{2015Sci...350..423F}
\begin{align}
B_{r, \rm crit} = \frac{\omega}{2 k_r}\sqrt{4\pi \rho},
\end{align}
where $\omega=2\pi\nu$ is the angular frequency of the waves, $\rho$ is the density, and $k_r$ is the radial wave-number.
This critical field may be thought of as the field strength at which the Alfv{\'e}n frequency computed with the length-scale $1/k_r$ is comparable to the wave frequency and is therefore analogous to the effect of rotation, which enters in when the rotation angular velocity is faster than the wave frequency.
For IGWs the radial wave-number is related to the spherical harmonic degree $\ell$ by
\begin{align}\label{eq:kr}
k_r \approx \frac{\sqrt{\ell(\ell+1)}}{r}\left(\frac{N}{\omega}\right),
\end{align}
where $N$ is the Br\"unt-V\"ais\"al\"a\ frequency.
So
\begin{align}
\label{eq:Brcrit}
B_{r, \rm crit} &\approx \frac{\omega^2 r}{N}\sqrt{\frac{4\pi \rho}{2\ell(\ell+1)}} \\
&\approx \frac{\omega^2 r}{\ell N}\sqrt{4\pi \rho}.
\end{align}
Note that because this decreases with increasing $\ell$, all waves of a given frequency are reflected if the $\ell=1$ waves are reflected.
We computed $B_{r, \rm crit}$ for several values of $\ell$ and $\nu$ for a main-sequence model of a $30 M_\odot$ star as a function of radius, shown in Fig.~\ref{fig:magnetic}.
For frequencies comparable to those of core convection the critical magnetic field strength is of order $10^{-2}\,\mathrm{G}$ to $10^{-1}\,\mathrm{G}$.
Strong macroturbulence is observed in similar-mass O-type stars with magnetic fields up to $2.5\,\mathrm{kG}$ \citep[e.g. HD~191612,][]{10.1093/mnras/stt921}, so macroturbulence in those stars is unlikely to be due to IGWs coming from their cores if those waves have similar frequencies to that of core convection.
For frequencies comparable to the observed $\nu_{\rm char}$, on the order of $3\,\mathrm{d}^{-1}$, the critical magnetic field is much larger, on the order of $300\,\mathrm{G}$ to $1\,\mathrm{kG}$.
This makes an explanation of macroturbulence based on IGWs marginally inconsistent with observations showing strong macroturbulence up to field strengths of $2.5\,\mathrm{kG}$.
However, because eqn.~\ref{eq:Brcrit} is a strong function of $\omega$, and hence of $\nu$, it is possible that these strongly magnetized stars just have larger $\nu_{\rm char}$.
The full range of observed characteristic frequencies spans $0.2-10\,\mathrm{d^{-1}}$, corresponding to critical field strengths at the surface of $3\,\mathrm{G}-10\mathrm{kG}$ for the $\ell=1$ mode.
Thus while IGWs with lower frequencies are inconsistent with observations of strongly-magnetized stars with substantial macroturbulence, those at higher frequencies likely make it to the surface and could contribute to the observed macroturbulence.
A further prediction of this calculation is that, if IGWs are the cause of macroturbulence in these stars, we should expect the strength of macroturbulence to decline with increasing magnetic field strength as more and more modes are reflected before they reach the surface.
\citet{10.1093/mnras/stt921} find no such trend, though they do that macroturbulence vanishes when the magnetic field exceeds the FeCZ shutoff strength~\citep{2019MNRAS.487.3904M}, and this is consistent with observations of other strongly-magnetized O/B stars such as HD~215441~\citep{1989ApJ...344..876L} and HD~54879~\citep{2015A&A...581A..81C}, both of which show little or no macroturbulence and magnetic fields stronger than the theoretical shutoff field strength.
This again points against an explanation of macroturbulence based on core-generated IGWs.
\begin{figure}[ht!]
\begin{center}
\includegraphics[width=1.0\columnwidth]{figures/composite.pdf}
\caption{\label{fig:magnetic} The critical radial magnetic field strength needed to reflect internal gravity waves of different $\ell$ and $\nu$ is show as a function of fractional depth for a $30 M_\odot$ stellar model at an age of $4\,\mathrm{Myr}$. The frequency $\nu_{\rm char,MLT}$ is that given by eqn.~\ref{eq:nuchar} but evaluated for the core convection zone instead of the subsurface FeCZ. The frequency $\nu_{\rm char,Obs}$ is a typical value of $3\,\mathrm{d}^{-1}$ for the observations. We further show frequencies of $0.2\,\mathrm{d^{-1}}$ and $10\,\mathrm{d^{-1}}$ as these span the full range of observed characteristic frequencies.}
\end{center}
\end{figure}
\section{Rotation}\label{sec:rotation}
\subsection{Subsurface Convection}
The sample of stars discussed here have projected rotational velocities $v_{\rm eq} \sin i$ in the range 7 to 320 $\rm km s^{-1}$, suggesting rotation could have an impact on the properties of subsurface convection.
The effect of rotation is usually measured by the convective Rossby number $R_0 = 1 / ( 2 \, \Omega \, \tau_c)$, where $\Omega$ is the stellar rotational frequency. For $R_0 > 1$ we expect rotation to have a moderate to negligible impact.
On the other hand when the rotational period becomes comparable or shorter than the convective turnover time ($R_0 \le 1$), the properties of convection can be altered substantially \citep[e.g.][]{Stevenson:1979,Augustson:2019}.
The typical rotational period of OB stars is about 3 days (assuming a typical equatorial rotational velocity $v_{\rm eq} \approx 150 \rm km s^{-1}$), while the convective turnover timescale in the FeCZ is a few hours (See e.g. Fig.~\ref{fig:nuchar_fecz}).
Therefore the $R_0$ in the observed sample is likely in the range $1\dots10$. For these values, the convective velocities as calculated in the 1D MLT approximation are affected only at the $\sim 10\%$ level \citep[see e.g. Fig.4 in][]{Cantiello:2009}.
This said, the latitudinal structure of the FeCZ zone is substantially altered at the highest rotation rates \citep{Maeder:2008}, which could have an impact on the way these regions affect the stellar surface.
\subsection{Core Convection}
In the stellar cores of intermediate and massive stars $R_0$ is very likely $<1$, so
rotation is expected to change the properties of convection \citep[e.g.][]{Stevenson:1979,Augustson:2019}.
At the same time, in the presence of rotation, gravity waves can be perturbed by the Coriolis acceleration and combine with inertial waves (gravito-inertial waves, GIWs).
The stochastic excitation of gravity and GIWs by rotating convective zones was studied by \citet{Mathis:2014} and \citet{Augustson:2020}.
The main result is that rotation can enhance the amplitude of stochastically excited waves \citep{Mathis:2014}.
The work of \citet{Neiner:2020} shows that in some rapidly rotating stars, stochastically excited GIWs from the core could explain part of the observed low-frequency variability.
However, we note that it is unlikely that core-generated GIWs are responsible for the ubiquitous SLF variability. This is because the visibility of GIWs depends on the inclination of the stellar rotation axis respect to the observer.
Internal waves cannot propagate at the poles for $\omega < 2\Omega$, where $\Omega$ is the stellar rotational frequency and $\omega$ is the wave frequency. The largest wave flux is expected at low latitudes,
with the degree of equatorial confinement proportional to $\Omega$. This means that the propagation domain of subinertial ($\omega < 2\Omega$) GIWs excludes the pole and it becomes increasingly concentrated toward the equator for faster rotation
rate \citep{Dintrans:2000,Prat:2016,Augustson:2020}.
Assuming that the stars in \citet{Bowman:2020} have spin vectors randomly oriented, some should be observed nearly pole-on.
Then if the SLF variability was due to GIWs, these objects would show very little power at frequencies less than $2\Omega$.
Such a sharp decline in variability at low
frequencies is not observed in any of the stars, suggesting GIWs are not the culprit \citep{Lecoanet:2020}.
A detailed study of the surface amplitude of waves excited by core convection in rotating early-type stars is beyond the scope of this paper.
\section{Discussion}\label{sec:discussion}
The presence of a subsurface convection zone can result in variability of photospheric properties via a number of processes, including wave excitation \citep{Cantiello:2009,Grassitelli:2015} and magnetic buoyancy \citep{Cantiello:2011}.
A linear perturbative analysis is limited, since the turbulent fluctuations in these convective regions can be large \citep{Grassitelli:2015}.
Multi dimensional simulations including radiation have been performed in a restricted range of the parameter space, and show that the full turbulent manifestation of these convective regions extends up to the stellar surface \citep{2015ApJ...813...74J,Jiang:2017,2018Natur.561..498J,Schultz:2020}.
In their calculations of OB stars envelopes including the stellar photosphere, \citet{2015ApJ...813...74J} observe turbulent velocities reaching the isothermal sound speed ($\approx 50 \rm km s^{-1}$) at the stellar surface, demonstrating that velocity fields of amplitude comparable to the observed macroturbulence are naturally explained by the presence of the FeCZ.
Therefore it could be that the observed SLF variability and macroturbulence simply represent the direct manifestation of turbulent, radiation-dominated convection at the stellar surface.
The simplified MLT treatment in our one dimensional calculations is unable to capture the complex phenomenology of these layers, but the fact that it can reproduce both the timescales and the trends in amplitude of the observed SLF variabilty is compelling. It calls for extending the radiation hydrodynamics simulations to cover the parameter space of the TESS observations, in order to unravel the precise mechanism connecting subsurface convection zones to the observed surface variability.
\subsection{Perturbation Lengthscale}
If the perturbation is due to stochastically excited modes driven by the FeCZ \citep{Grassitelli:2015}, then we expect the largest fluctuations to be produced by modes with $\ell \lesssim 20$ \citep[see e.g.][]{Godart:2017}.
The situation is different if instead the perturbation is provided by running waves or by convective motions extending to the stellar surface \citep{Cantiello:2009,2015ApJ...813...74J}.
In this case a good proxy for the typical scale of surface perturbations is provided by the size of convective cells in the FeCZ.
This in turn is quantified by the average pressure scale height in the subsurface convection zone. Note that the pressure scale height only decreases slightly moving from the FeCZ to the stellar surface.
In general one expects that velocity perturbations induced by the FeCZ should have scales that are comparable or slightly larger than the line forming region, so macroturbulence can be explained via this mechanism.
Rotation could also be responsible for organizing the convective flow on slightly larger scales (see Sec.~\ref{sec:rotation}).
Since convective turbulence also results in smaller-scale motions, the FeCZ could also be responsible for the excitation of surface microturbulence \citep{Cantiello:2009}.
We show in Fig.~\ref{fig:cells} the number of convective cells $N_{\rm CC} = (R_\star/\overline{\rm H}_{\rm P})^2$, calculated using the stellar radius and the average pressure scale height in the FeCZ.
We expect approximately $10^2...10^4$ convective cells in the FeCZ of OB stars. Therefore the order of the perturbation $\ell \approx \sqrt{N_{\rm CC}}\approx 10...100$.
While it might seem impossible for such high-degree perturbations to leave a visible signature on the stellar disc integrated properties, we point out that in this case the surface fluctuations are uncorrelated. So even high degree $\ell$ fluctuations do not undergo the dramatic cancellation effects experienced by highly-correlated stellar oscillations. Similarly to granulation, we expect the amplitude of the integrated surface perturbations to scale as $1/\sqrt{N_{\rm CC}}$.
It is important to distinguish the signal induced by (sub)surface convection zones in OBA stars with the granulation pattern expected in cool stars with convective envelopes. The driving mechanism, properties, and location of these convection zones change substantially from late- to early-type stars \citep{Cantiello:2019}. This is why is not surprising that the characteristic frequency of the SLF variability in the early-type stars observed by \citet{Bowman:2019} and \citet{Szewczuk:2021} does not follow the granulation scaling of \citet{Kjeldsen:1995}, which was derived for late-type stars. On the other hand we have shown here that the observed frequencies are consistent with the expectation of perturbations arising from subsurface convection zones (see Fig.~\ref{fig:nuchar_fecz}).
\begin{figure}[ht!]
\begin{center}
\includegraphics[width=1.0\columnwidth]{figures/ncells_fecz.pdf}
\caption{\label{fig:cells} Number of convective cells in the FeCZ. This was calculated using the average pressure scale height in the FeCZ.}
\end{center}
\end{figure}
\subsection{Metallicity}
If the FeCZ is responsible for surface macroturbulence and SLF variability, then these phenomena should be affected by the stellar metallicity. On the other hand we do not expect a significant metallicity dependence in
the context of a core-convection origin.
The results of \citep{Bowman:2019} show that SLF variability is also observed in low-metallicity, LMC stars.
The presence of the FeCZ depends on the luminosity and metallicity of the star, and in 1D stellar evolution calculations the FeCZ occurs above a luminosity of L$\approx10^{3.2} L_{\odot}$ for Z=0.02.
This corresponds to a zero age main sequence star of about 7${\rm M}_\odot$ (see e.g. Fig.~\ref{fig:macroturbulence}).
The LMC has a metallicity about half solar, and in models at Z=0.008 the FeCZ only appears at L$\approx10^{3.9} L_{\odot}$, corresponding to a zero age main sequence star of about 11${\rm M}_\odot$ \citep{Cantiello:2009}. In our models with metallicity Z=0.006 (See Appendix~\ref{appen:grids}) the FeCZ appears at $\log \mathscr{L}/\mathscr{L}_\odot \approx 3.2$.
Note that these limits could move downward in the presence of atomic diffusion and radiative acceleration \citep{Richer:2000}, or due to an upward revision of the uncertain values of Fe opacity \citep[e.g.][]{Bailey:2015}. We notice that the TESS observations of LMC stars reported by \citet{2019NatAs...3..760B} reveal a trend of lower $\nu_{\rm char}$ compared to the galactic sample (See their Fig.~4). Interestingly, this trend is reproduced by the characteristic frequency of convection in the FeCZ in our models (compare Fig.~\ref{fig:nuchar_fecz} with Fig.~\ref{fig:nuchar_fecz_LMC}). It will be interesting to see if the TESS observations can probe the transition region between stars with and without a FeCZ, and determine a possible change in surface properties.
At the same time, X-shooter within the Ulysses program could more firmly establish the metallicity-dependence of the macroturbulent
line-broadening \citep{Penny:2009}.
It is important to note that some low-luminosity, main sequence A stars in \citet{Bowman:2019} show SLF variability. For these stars models do not predict the presence of the FeCZ. However, these stars still show (sub)surface convection zones triggered by ionization of H and He \citep{Cantiello:2019}. The characteristic frequencies of convection in these regions are also in the range $\sim$tens of $\mu$Hz, although the amplitude of the velocity fluctuations they can induce is much smaller than for the FeCZ, so it is not clear if they could be linked to observed SLF variability and macroturbulence. On the other hand, since the relative perturbation induced by subsurface convection is weakest in the regime of A and late-B type \citep{Cantiello:2019,Jermyn:2020},
these stars are the best targets for detecting core-generated IGWs. Depending on the amplitude of core generated IGWs, the impact of subsurface convection could well be subdominant in these objects, allowing for a detection.
\subsection{Stochastic, Low-Frequency Variability in Evolved Stars}
SLF variability with similar properties to the massive main sequence stars discussed by \citep{Bowman:2019,Bowman:2020} was recently observed in evolved massive stars.
\citet{Naze:2021} detected SLF variability in both luminous blue variables (LBV) and Wolf-Rayet (WR) stars, and \citet{Dorn:2020} found the same photometric signature in yellow supergiants (YSG).
Compared to OB stars, LBV, WR and YSG correspond to later stages of evolution. In particular, WR stars and YSG are likely burning helium in their cores.
The internal structure of OB, LBV, WR and YSG stars changes dramatically, and this can affect substantially the generation and propagation of internal gravity waves.
The amount of radiative damping is expected to change due to the large differences in envelope temperature and densities. We recall here that the radiative damping rate $ \gamma_{\rm rad}$ for a traveling g-mode is given by
\begin{align*}\label{eqn:gammarad}
&\rm \gamma_{rad} (\omega, \ell, r) = K_{\rm rad} (r) \, k_{r}^{2},\\
&K_{\rm rad} (r) = \frac{16 \, \sigma \, T(r)^{3}}{3 \, \rho(r)^{2} \kappa(r) \, c_{p}(r)}
\end{align*}
where $k_{r}$ is the radial wavenumber (eqn.~\ref{eq:kr}) and $\kappa$ is the opacity.
Compact WR stars have surface temperatures that can exceed $\approx 10^5$K, while OB stars and LBVs have effective temperatures $\approx 10^{4\dots4.7}$K, with LBVs found at the cooler end of this range. The surface of extended, low-density, YSGs is cooler than $10^4$K. With different types of core convection (H-burning vs He-burning) and radiative damping rates, it would be surprising if core-generated IGWs in e.g. OB and WR stars showed up at the surface with similar properties. On the other hand, the FeCZ is present below/at the surface in both OB and WR stars, and with similar velocities and convective turnover times. This seems to strengthen the main thesis of this work, adding support to a (sub)surface origin of the observed SLF variability.
The discussion is more complicated for LBVs and YSGs, since at lower temperatures other convective regions can become prominent \citep[driven by H and He recombination, see e.g.][]{2018Natur.561..498J}. \citet{Dorn:2020} disfavor a near-surface origin for the observed SLF variability in YSG, on the ground that the observed timescale do not follow the predicted scaling for granulation \citep{Kallinger:2014}. However, we believe that the scaling of \citet{Kallinger:2014} is not applicable in the regime explored by \citet{Dorn:2020}. This scaling was derived for a sample of red giants and solar-like stars, which all have temperatures well-below the temperature for the recombination of hydrogen, and it is not directly applicable to earlier
spectral subtypes.
Interestingly, the characteristic timescales of 0.1-1 days found by \citet{Dorn:2020} at $\log {\rm T}_{\rm eff} \approx$4 are consistent with the convective turnover timescale in the FeCZ (e.g. Fig.~\ref{fig:nuchar_fecz}). Moreover, in their lower temperature sample ($\log {\rm T}_{\rm eff} < 3.75$) the rapid increase in amplitudes and characteristic frequencies of the variability is in agreement with the development of near-surface convection induced by the large opacities associated with the recombination of hydrogen \citep{Grassitelli:2015b}. We suggest that surface and near-surface convection could be indeed responsible for the variability observed by \citet{Dorn:2020}, and we plan to systematically study the properties of (sub)surface convection in these evolved, cool stars in future work.
\subsection{Towards a unified model for surface phenomena in massive stars}
The presence of subsurface convection can simultaneously account for a large variety of puzzling phenomena that appear ubiquitous at the surface of early-type stars.
\begin{itemize}
\item {\bf Microturbulence and Macroturbulence} can be accounted for by velocity fields excited by the underlying FeCZ subsurface convection zone \citep{Cantiello:2009,Grassitelli:2015,2015ApJ...813...74J}. The only stars with no macroturbulence appear to have magnetic fields strong enough to shut off convection in the FeCZ, while stars with slightly lower surface magnetic fields show normal values of macroturbulence \citep{10.1093/mnras/stt921}.
\item {\bf Bright spots} in early-type stars have been observed \citep[e.g.][]{Ramiaramanantsoa:2014}, and can be explained with the presence of magnetic spots rising from subsurface convective layers \citep{Cantiello:2011,Cantiello:2019}.
\item {\bf Discrete absorption components (DACs)} in UV spectra \citep[e.g.][]{Howarth:1989,Cranmer:1996,Fullerton:1997,Kaper:1997} can then be caused by the aforementioned bright spots \citep{Cantiello:2011} and associated prominences \citep{Sudnik:2016}.
\item {\bf Line profile variability} is another ubiquitous phenomena in hot stars \citep{Fullerton:1996}, and can be explained by surface velocity and density perturbations seeded by subsurface convection \citep[e.g.][]{Cantiello:2009,2015ApJ...813...74J}.
\item {\bf Wind clumping} can also be seeded by these surface density and velocity perturbations, which are amplified by the development of instabilities in the stellar wind \citep[][but see also \citet{Sundqvist:2013}]{Owocki:1988}.
\item {\bf SLF variability} can also be caused by the presence of subsurface convection zones, as discussed in this work.
\end{itemize}
An economical hypothesis emerges: the presence of subsurface convection, and in particular of the FeCZ, could be the common underlying physical cause for the appearance of turbulence, magnetic spots, SLF variability, as well spectroscopic variability in early-type stars.
\section{Conclusions}
We used one-dimensional, non-rotating stellar evolution calculations to study the predicted trends in the properties of subsurface convection in the spectroscopic H-R diagram.
We found that the trends of relative convective flux and convective turnover timescale in the FeCZ of our models match very well the trends in timescale
and amplitude of stochastic, low-frequency photometric variability in OB stars observed by TESS and K2.
Similar to previous works, we show that the observed trends in stellar macroturbulence are also well reproduced assuming the FeCZ is its driver. This connection is also supported by the observations of strongly magnetized early-type stars,
showing no macroturbulence only for magnetic fields above the critical value required to shut-off turbulent convection in the FeCZ. We find that IGWs coming from the stellar core would be reflected or damped for values of the magnetic field well below this critical value. The fact that stars with strong but subcritical magnetic fields show typical values of macroturbulence points against a convective core origin of this surface perturbation.
In the presence of rotation, GIWs are also expected to propagate and reach the stellar surface. These waves are increasingly confined to stellar equatorial regions in rapidly-rotating stars, and for stars seen close to pole-on a sharp decline in their variability is expected below twice their rotational frequency. This feature is not detected in any of the early-type stars observed, suggesting GIWs are also unlikely to explain the observed surface variability.
Overall the observations support a picture in which subsurface convection, and in particular the FeCZ, is responsible for the ubiquitous low-frequency, stochastic photometric variability and macroturbulence detected in OB stars.
These surface manifestations join a number of phenomena observed in early-type stars and attributed to the presence of subsurface convection, including the observations of (magnetic) bright spots as well as wind and spectroscopic variabilty. Radiation (magneto)hydrodynamics simulations of the outer envelope regions of early-type stars are required to understand the details of how subsurface convection zones cause the observed surface perturbations.
\acknowledgments
We thank the anonymous referee for a constructive report which helped improve the manuscript. We also thank Evan Anders for useful discussions on stellar convection.
The Center for Computational Astrophysics at the Flatiron Institute is supported by the Simons Foundation.
This research was supported in part by the National Science Foundation under Grant No. NSF PHY-1748958 and by the Gordon and Betty Moore Foundation through Grant GBMF7392. DL is supported by a Lyman Spitzer Jr.~Fellowship.
|
1,314,259,993,637 | arxiv | \section{Introduction}
\noindent
The study of traversable wormholes has received considerable
attention from researchers for the past two decades. Although
lacking observational evidence, wormholes are just as good a
prediction of the general theory of relativity as black holes.
In particular, we refer to the pioneering work of Visser
\cite{Visser1989}, who proposed a theoretical method for
constructing a new class of traversable Lorentzian wormholes
from black-hole spacetimes. This construction proceeds by
surgically grafting two Schwarzschild spacetimes
together in such a way that no event horizon is permitted to
form. The resulting structure is a wormhole spacetime in which
the throat is a three-dimensional thin shell. In recent years,
Visser's approach was adopted by various authors for constructing
thin-shell wormholes by similar methods, generally requiring
spherical symmetry
\cite{Poisson1995,Lobo2003,Lobo2004,Eiroa2004a,Eiroa2004b,Eiroa2005,
Thibeault2005,Lobo2005,Rahaman2006,
Eiroa2007,Rahaman2007a,Rahaman2007b,Rahaman2007c,
Lemos2007,Richarte2008,Rahaman2008a,
Rahaman2008b,Eiroa2008a,Eiroa2008b}. The approach is of
special interest because it minimizes the amount of exotic matter
required. All the exotic matter is confined to the shell.
More recently, Sur, Das, and SenGupta \cite{Sur2005} discovered
a new black-hole solution for Einstein-Maxwell scalar field
systems inspired by low-energy string theory. They considered
a generalized action in which two scalar fields are minimally
coupled to an Einstein-Hilbert-Maxwell field in four
dimensions,
\begin{equation}
I = \frac{1}{2 \kappa} \int d^4x \sqrt{-g}\left[ R -
\frac{1}{2}\partial_\mu\varphi \partial^\mu\varphi - W \right],
\end{equation}
where
\begin{multline*}
W=\frac{1}{2}\omega(\varphi) \partial_\mu \zeta
\partial^\mu\zeta - \alpha (\varphi,\zeta)
F_{\mu\nu}F^{\mu\nu}\\
- \beta (\varphi,\zeta)F_{\mu\nu} F^{\mu\nu\ast},
\end{multline*}
$\kappa = 8 \pi G $, $R$ is the curvature scalar, $F_{\mu\nu}$
is the Maxwell field tensor, while $\varphi$ and $\zeta$ are
two massless scalar or pseudo scalar fields, which are
coupled to the Maxwell field. This coupling is described by
the functions $\alpha $ and $\beta$. Here $\zeta $ acquires a
non-minimal kinetic term $\omega$. In the context of low-energy
string theory, fields $\phi$ and $\xi$ can be identified as
massless scalar dilaton and pseudo scalar axion fields,
respectively.
With the above action, Eq. (1), Sur, \emph{et al.}, \cite{Sur2005}
found the most general class of black-hole solutions and
obtained two types of metrics, classified as asymptotically flat
and asymptotically non-flat. Since we are interested in
obtaining a thin-shell wormhole from this new black hole, we
adopt the asymptotically flat metric given by
\begin{equation}\label{E:line1}
ds^2 = -f(r) dt^2 + f(r)^{-1}dr^2 + h(r) (d\theta^2+\sin^2\theta
d\phi^2),
\end{equation}
where
\begin{equation}
f(r) = \frac{(r-r_-)(r-r_+)}{(r-r_0)^{2-2n}(r+r_0)^{2n}},
\end{equation}
and
\begin{equation}
h(r) = \frac{(r+r_0)^{2n}}{(r-r_0)^{2n-2}},
\end{equation}
where, according to Ref. \cite{Sur2005}, the exponent $n$
is a dimensionless constant stricly greater than 0 and
stricly less than 1. In addition, various other parameters
are given by
\begin{eqnarray}
r_{\pm} &=& m_0 \pm \sqrt{m_0^2 +r_0^2 -\frac{1}{8}\left(
\frac{K_1}{n} + \frac{K_2}{1-n}\right)},\\ r_0 &=&\frac{1}{16m_0}
\left(\frac{K_1}{n} - \frac{K_2}{1-n}\right),\\ m_0 &=& m -
(2n-1)r_0,\\ K_1 &=& 4n[ 4r_0^2 + 2r_0(r_+ +r_-) + r_+r_-],\\ K_2
&=& 4(1-n)r_+r_-, \ \ 0<n<1,\\ m &=& \frac{1}{16r_0} \left(
\frac{K_1}{n} - \frac{K_2}{1-n}\right) + (2n-1)r_0,
\end{eqnarray}
where $m$ is the mass of the black hole. The parameters
$r_+$ and $r_-$ are the inner and outer event horizons,
respectively. Also, $r=r_0$ is a curvature singularity;
the parameters obey the condition $r_0<r_-<r_+$.
In this paper we present a new kind of thin-shell wormhole
by surgically grafting two charged black holes in generalized
dilaton-axion gravity. The exotic matter required for its
physical existence may possibly be collected from scalar
fields that built the black holes. Various aspects of this
thin-shell wormhole are analyzed, particularly the equation
of state relating pressure and density. Also discussed is
the attractive or repulsive nature of the wormhole, as well
as the energy conditions on the shell. Our final topic is
a stability analysis to determine the conditions under which
the wormhole is stable to linearized radial perturbations.
A comparison to the stability of other thin-shell
wormholes in the literature is also made.
\section{Thin-shell wormhole construction}
\noindent
The mathematical construction of our thin-shell wormhole begins
by taking two copies of the black hole and removing from each
the four-dimensional region
\[
\Omega^\pm = \{r\leq a \mid a>r_+\}.
\]
We now identify (in the sense of topology) the timelike
hypersurfaces
\[
\partial\Omega^\pm = \{r=a \mid a>r_+\},
\]
denoted by $\Sigma$. The resulting manifold is geodesically
complete and consists of two asymptotically flat regions
connected by a throat. The induced metric on $\Sigma$ is
given by
\begin{equation}
ds^2 = - d\tau^2 + a(\tau)^2( d\theta^2 +
\sin^2\theta d\phi^2),
\end{equation}
where $\tau$ is the proper time on the junction surface. Using
the Lanczos equations
\cite{Visser1989,Poisson1995,Lobo2003,Lobo2004,Eiroa2004a,Eiroa2004b,Eiroa2005,
Thibeault2005,Lobo2005,Rahaman2006,
Eiroa2007,Rahaman2007a,Rahaman2007b,Rahaman2007c,Lemos2007,Richarte2008,Rahaman2008a,
Rahaman2008b,Eiroa2008a,Eiroa2008b}, one can obtain the surface
stress energy tensor $ S_{\phantom{i}j}^i=\text{diag}(-\sigma,p_{\theta},
p_{\phi})$, where $\sigma$ is the surface energy density and
$p_{\theta}$ and $p_{\phi}$ are the surface pressures.
The Lanczos equations now yield \cite{Eiroa2005}
\begin{equation}\label{E:sigma1}
\sigma = - \frac{1}{4\pi }\frac{h^\prime(a)}{h(a)}\sqrt{f(a) +
\dot{a}^2}
\end{equation}
and
\begin{equation}\label{E:pressure1}
p_{\theta} = p_{\phi} = p = \frac{1}{8\pi
}\frac{h^\prime(a)}{h(a)}\sqrt{f(a) + \dot{a}^2} + \frac{1}{8\pi
}\frac{2\ddot{a} + f^\prime(a) }{\sqrt{f(a) + \dot{a}^2}}.
\end{equation}
To understand the dynamics of the wormhole, we assume the radius
of the throat to be a function of proper time, or $ a =
a(\tau)$. Also, overdot and prime denote, respectively, the
derivatives with respect to $\tau$ and $a$. For a static
configuration of radius $a$, we obtain the respective values
of the surface energy density and the surface pressures. For
a static configeration of radius $a$, we obtain (assuming
$\dot{a} = 0 $ and $\ddot{a}= 0 $) from Eqs. (\ref{E:sigma1})
and (\ref{E:pressure1}),
\begin{equation}\label{E:sigma2}
\sigma = -
\frac{4[a+(1-2n)r_0]}{D}\frac{(a-r_-)(a-r_+)}{(a-r_0)(a+r_0)}
\end{equation}
and
\begin{equation}\label{E:pressure2}
p_{\theta}= p_{\phi} =p= \frac{2a-r_- -r_+}{D}
\end{equation}
where
\begin{equation}\label{E:D}
D=8\pi (a-r_0)^{1-n}(a+r_0)^{n} \sqrt{(a-r_-)(a-r_+)}.
\end{equation}
Observe that the energy density $\sigma$ is negative. The
pressure $p$ may be positive, however. This would depend on
the position of the throat and hence on the physical parameters
$r_0$, $r_-$, and $r_+$ defining the wormhole. Similarly, $p +
\sigma$, $ \sigma + 2p $ $ $, and $\sigma+3p $, obtained by
using the above equations, may also be positive under certain
conditions, in which case the strong energy condition is
satisfied.
Keeping in mind the condition $r_+>r_->r_0$ for
different radii defining the wormhole, we plot $p$ versus $a$ in
Fig.~\ref{fig1}. We choose typical wormholes whose radii ($r_0$,
$r_-$, and $r_+$) fall within the range $2$ to $12$ kms.
Also taken into account is the sensitivity of the plots with
respect to $n$, as described in the caption of the figure.
\begin{figure}
\begin{center}
\vspace{0.5cm}
\includegraphics[width=0.5\textwidth]{fig1.eps}
\caption{Plot for $p$ versus $a$. The black, blue, and red colors
represent $n=0.98$, 0.5 and 0.02, respectively. For every color,
thin, thick, and thicker curves, respectively, represent $r_+=10$,
$8$, and $6$. For every combination of $r_+$ and $n$, we plot three
different sets, ($r_-=5$, $r_0=2$), ($r_-=5$, $r_0=3$), and
($r_-=4$, $r_0=2$), which are represented by chain and solid
curves, respectively. } \label{fig1}
\end{center}
\end{figure}
\section{The gravitational field}
\noindent
We now turn our attention to the attractive or repulsive
nature of our wormhole. To perform the analysis, we
calculate the observer's four-acceleration
$a^\mu = u^\mu_{\,\,;\nu} u^\nu$, where
$u^{\nu} = d x^{\nu}/d {\tau}
=(1/\sqrt{f(r)}, 0,0,0)$. In view of the line
element, Eq. (\ref{E:line1}), the only non-zero component
is given by
\begin{equation}\label{E:acceleration}
a^r = \Gamma^r_{tt}
\left(\frac{dt}{d\tau}\right)^2 = \frac{1}{2}
\frac{Ar^2-Br+C}{(r-r_0)^{3-2n}(r+r_0)^{2n+1}} \end{equation}
where,
\[A= r_- +r_+ + 4nr_0 -2r_0,\]
\[ B= 2r_0^2 +(r_- + r_+)(4nr_0-2r_0)+ 2r_- r_+, \]
and
\[ C= r_0^2 (r_- + r_+)+(4nr_0-2r_0) r_- r_+ .\]
A radially moving test particle initially at rest obeys
the equation of motion
\begin{equation}\label{E:motion}
\frac{d^2r}{d\tau^2}= -\Gamma^r_{tt}\left(\frac{dt}{d\tau}
\right)^2 =-a^r.
\end{equation}
If $a^r=0$, we obtain the geodesic equation. Moreover, a
wormhole is attractive if $a^r>0$ and repulsive if $a^r<0$.
These characteristics depend on the parameters $r_0$, $r_-$,
$r_+$, and $n$, the conditions on which can be conveniently
expressed in terms of the coefficients $A$, $B$, and $C$.
To avoid negative values for $r$, let us consider only the root
$r=(B+\sqrt{B^2-4AC})/(2A)$ of the quadratic equation
$Ar^2-Br+C=0$. It now follows from Eq.~(\ref{E:acceleration})
that $a^r=0$ whenever
\[ \left( r -\frac{B}{2A} \right)^2
= \frac{ B^2 -4AC}{4A^2} .\]
For the attractive case, $a^r>0$, the condition becomes
\[ \left(r -\frac{B}{2A} \right)^2
>\frac{ B^2 -4AC}{4A^2}. \]
For the repulsive case, $a^r<0$, the sense of the inequality
is reversed.
\section{The total amount of exotic matter}
\noindent
In this section we determine the total amount of exotic matter
for the thin-shell wormhole. This total can be quantified by the
integral
\cite{Eiroa2005,Thibeault2005,Lobo2005,Rahaman2006,
Eiroa2007,Rahaman2007a,Rahaman2007b}
\begin{equation}
\Omega_{\sigma}=\int [\rho+p]\sqrt{-g}d^3x.
\end{equation}
By introducing the radial coordinate $R=r-a$, w get
\[
\Omega_{\sigma}=\int^{2\pi}_0\int^{\pi}_0\int^{\infty}_{-\infty}
[\rho+p]\sqrt{-g}\,dR\,d\theta\,d\phi.
\]
Since the shell is infinitely thin, it does not exert any radial
pressure. Moreover, $\rho=\delta(R)\sigma(a)$. So
\begin{multline}\label{E:amount}
\Omega_{\sigma}=\int^{2\pi}_0\int^{\pi}_0\left.[\rho\sqrt{-g}]
\right|_{r=a}d\theta\,d\phi=4\pi h(a)\sigma(a)\\
=-\frac{16\pi[a+(1-2n)r_0]}{D}\left[\frac{(a-r_-)(a-r_+)}
{(a-r_0)^{2n-1}(a+r_0)^{1-2n}}\right].
\end{multline}
Here $D$ is given in Eq. (\ref{E:D}).
This NEC violating matter can be reduced by taking the
value of $a$ closer to $r_+$, the location of the outer
event horizon. The closer $a$ is to $r_+$, however, the
closer the wormhole is to a black hole: incoming microwave
background radiation would get blueshifted to an extremely
high temperature \cite{tR93}. On the other hand, it follows
from Eq. (\ref{E:amount}) that for $a\gg r_+$,
$\Omega_{\sigma}$ will depend linearly on $a$: \\
\begin{equation}
\Omega_{\sigma} \approx -2a. \end{equation}
\section{An equation of state}
\noindent
Taking the form of the equation of state (EoS) to be
$p=w\sigma$, we obtain from Eqs. (\ref{E:sigma2}) and
(\ref{E:pressure2}),
\begin{equation}\label{E:EoS}
\frac{p}{\sigma} = w = \frac{1}{4} \frac{(2a-r_- -
r_+)(r_0^2-a^2)}{(a-r_-)(a-r_+)[a+(1-2n)r_0]}.
\end{equation}
Observe that if the location of the wormhole throat is
very large, i.e., if $a\rightarrow +\infty$, then
$w\rightarrow -\frac{1}{2}$. On the other hand, if
$a\rightarrow r_{+}$ (from the right),
then $\omega\rightarrow -\infty$.
So the distribution of matter in the shell is of the
dark-energy type. Now, purely mathematicall speaking,
if $a \rightarrow \frac{1}{2}(r_-+ r_+)$, then
$p \rightarrow 0$. Since $\frac{1}{2}(r_-+ r_+)<r_+$,
however, such a dust shell is never found.
Our spacetime metric implies that the surface mass of this
thin shell is given by $M_{shell} =
4 \pi h(a) \sigma$. (For a static solution, we have
$\dot{a} = 0$ and $\ddot{a}= 0$.) Thus
\begin{multline}
M_{shell} = 2 \frac{[a+(1-2n)r_0]}{(r_0^2-a^2)}\times\\
(a-r_0)^{1-n}(a+r_0)^n\sqrt{(a-r_-)(a-r_+)} .
\end{multline}
Now observe that for $n=\frac{1}{2}$, the mass of the black
hole in Eq. (10) is increasing with $r_0$. At the same time,
for a fixed value of the throat radius $a$, the mass of the
thin shell is decreasing with $r_0$, as long as $r_0$ remains
much less than $a$.
\begin{figure}
\begin{center}
\vspace{0.5cm}
\includegraphics[width=0.5\textwidth]{fig2.eps}
\caption{Plot for $\sigma$ versus $a$. The description of the
curves is the same as in FIG.~\ref{fig1}. } \label{fig2}
\end{center}
\end{figure}
\section{Stability}
\noindent
Now we will focus our attention on the stability of the
configuration under small perturbations around a static solution
at $a_0$. The starting point is the definition of a potential,
extended to our metric [Eq. (\ref{E:line1})]. Rearranging Eq.
(\ref{E:sigma1}), we obtain the thin shell's equation of
motion
\begin{equation}
\dot{a}^2 + V(a)= 0.
\end{equation}
Here the potential $V(a)$ is defined as
\begin{equation}
V(a) = f(a) - \left[\frac{4\pi
h(a)\sigma(a)}{h^{\prime}(a)}\right]^2.
\end{equation}
Expanding $V(a)$ around $a_0$, we obtain
\begin{eqnarray}
V(a) &=& V(a_0) + V^\prime(a_0) ( a - a_0) +
\frac{1}{2} V^{\prime\prime}(a_0) ( a - a_0)^2 \nonumber \\
&\;& + O\left[( a - a_0)^3\right],
\end{eqnarray}
where the prime denotes the derivative with respect to $a$,
assuming a static solution situated at $a_0$. Since we are
linearizing around $ a = a_0 $, we must have $ V(a_0) = 0 $
and $V^\prime(a_0)= 0 $. The configuration will then be in
stable equilibrium if $V^{\prime\prime}(a_0)>0$.
To carry out this analysis, we start with the energy
conservation equation. Using Eqs. (\ref{E:sigma1}) and
(\ref{E:pressure1}), one can verify that
\begin{multline}\label{E:conservation}
\frac{d}{d\tau}(\sigma\mathcal{A})+p\frac{d\mathcal{A}}
{d\tau}=\\
\{[h'(a)]^2-2h(a)h''(a)\}
\frac{\dot{a}\sqrt{f(a)+\dot{a}^2}}{2h(a)},
\end{multline}
where $\mathcal{A}=4\pi h(a)$ by Eq.~(\ref{E:line1}).
The first term on the left side corresponds to a
change in the throat's internal energy, while the
second term corresponds to the work done by the
throat's internal forces. According to
Ref. \cite{Eiroa2008a}, the right side
represents a flux. From Eq. (\ref{E:conservation}), we get
\begin{multline*}
\frac{d}{da}[\sigma h(a)]+\mathcal{P}
\frac{d}{da}[h(a)]\\
=-\{[h'(a)]^2-2h(a)h''(a)\}\frac{\sigma}{2h'(a)},
\end{multline*}
and, finally,
\begin{multline}\label{E:alternate}
h(a)\sigma'+h'(a)(\sigma+p)+\{[h'(a)]^2-2h(a)
h''(a)\}\frac{\sigma}{2h'(a)}\\=0.
\end{multline}
It is also shown in Ref. \cite{Eiroa2008a} that
\begin{multline}\label{E:Vdoubleprime}
V''(a)=f''(a)+16\pi^2\times\\\left\{\left[\frac{h(a)}{h'(a)}
\sigma'(a)+\left(1-\frac{h(a)h''(a)}{[h'(a)]^2}\right)
\sigma(a)\right][\sigma(a)+2p(a)]\right.\\
\left.+\frac{h(a)}{h'(a)}\sigma(a)[\sigma'(a)+2p'(a)]\right\}.
\end{multline}
Next, we define a parameter $\beta$, which is interpreted as the
subluminal sound speed, by the relation
\begin{equation}
\beta^2(\sigma) =\left. \frac{ \partial p}{\partial
\sigma}\right\vert_\sigma.
\end{equation}
To do so, observe that
\begin{multline*}
\sigma'(a)+2p'(a)\\=\sigma'(a)[1+2p'(a)/\sigma'(a)]=
\sigma'(a)(1+\beta^2).
\end{multline*}
Using Eq.~(\ref{E:alternate}),
we can now rewrite $V''(a)$ as follows:
\begin{multline}\label{E:potential}
V''(a)=f''(a)-8\pi^2\left\{[\sigma(a)+2p(a)]^2
\phantom{\frac{h}{h'}}\right.\\
\left.+2\sigma(a)\left[\left(\frac{3}{2}-\frac{h(a)h''(a)}
{[h'(a)]^2}\right)\sigma(a)+p(a)\right](1+2\beta^2)
\right\}.
\end{multline}
At the static solution $a=a_0$, the conditions $V(a_0)=0$
and $V'(a_0)=0$ are indeed met.
Now consider the stability criterion $V''(a_0)>0$
starting with Eq. (\ref{E:potential}): first let
$V''(a_0)=0$ and solve for $\beta^2$. We then find
that the graph of
\begin{equation}\label{E:beta}
\beta^2=-\frac{1}{2}+\frac{f''/8\pi^2-(\sigma+2p)^2}
{4\sigma\left[\left(\frac{3}{2}-\frac{hh''}
{(h')^2}\right)\sigma+p\right]}
\end{equation}
has a single vertical asymptote (Fig. 3.) To the
right of the asymptote,
\begin{equation}\label{E:signchange}
4\sigma\left[\left(\frac{3}{2}-\frac{hh''}
{(h')^2}\right)\sigma+p\right]>0,
\end{equation}
also determined graphically. Returning now to the
inequality $V''(a_0)>0$, we therefore have at $a=a_0$
\begin{equation}\label{E:above}
\beta^2<-\frac{1}{2}+\frac{f''/8\pi^2-(\sigma+2p)^2}
{4\sigma\left[\left(\frac{3}{2}-\frac{hh''}
{(h')^2}\right)\sigma+p\right]}.
\end{equation}
So to the right of the asymptote, the region of
stability is below the graph of
Eq. (\ref{E:beta}), as shown in Fig. 3.
To the left of the asymptote, the sense of the
inequality in (\ref{E:signchange}) is reversed and
we obtain at $a=a_0$
\begin{equation}\label{E:below}
\beta^2>-\frac{1}{2}+\frac{f''/8\pi^2-(\sigma+2p)^2}
{4\sigma\left[\left(\frac{3}{2}-\frac{hh''}
{(h')^2}\right)\sigma+p\right]}.
\end{equation}
So to the left of the asymptote, the region of stability
is above the graph.
\begin{figure}
\begin{center}
\vspace{0.5cm}
\includegraphics[width=0.5\textwidth]{fig3.eps}
\caption{Plot for $\beta^2$ versus $a_0$.
} \label{fig3}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\vspace{0.5cm}
\includegraphics[width=0.5\textwidth]{fig4.eps}
\caption{All functions are the same. Some are cut off. See Fig.~3.
} \label{fig4}
\end{center}
\end{figure}
Fig.~3 does indeed show typical regions of stability
using somewhat arbitrary values of the various
parameters: $r_0=1$, $r_-=2$, $r_+=3$, and $n=1/2$.
As noted above, the region is below the curve on the
right and above the curve on the left. The sign change
is determined by inequality (\ref{E:signchange}).
These results, including the graphs, are similar to
those in Refs. \cite{Poisson1995} and \cite{Eiroa2008a}
(dealing with Schwarzschild and dilaton thin-shell
wormholes, respectively) in the sense that the regions
do not correspond to any value in the interval
$0<\beta^2\le 1$. Since $\beta$ is ordinarily
interpreted as the speed of sound, it is highly desirable
to obtain a region for which $\beta^2<1$. This is
indeed possible for our wormhole: if we choose $r=r_-$
close to $r=r_+$, then we typically get a region of
stability for $\beta^2<1$. For example, in Fig. 4,
$r_0=1$, $r_-=2$, $r_+=2.05$, and $n=0.8$. The closer
$r_-$ is to $r_+$, the more the region of stability
extends below $\beta^2=1$.
\\
\\
\\
\section{Conclusion}\noindent
A new black-hole solution by Sur, \emph{et al.}, for
Einstein-Maxwell scalar fields was inspired by low-energy
string theory. This paper discusses a new thin-shell
wormhole constructed by applying the cut-and-paste
technique to two copies of such black holes. We analyzed
various aspects of this wormhole, such as the amount of
exotic matter required, the attractive or repulsive
nature of the wormhole, and a possible equation of state
for the thin shell. The stability analysis concentrated
on the parameter $\beta$, normally interpreted as the
speed of sound. It was found that whenever the two
event horizons are close together, a stability
region exists for some values of $\beta^2$ less
than unity, unlike the the cases discussed in
Refs. \cite{Poisson1995} and \cite{Eiroa2008a}
for Schwarzschild and dilaton thin-shell
wormholes, respectively.
\subsection*{Acknowledgments}
AAU, FR and SR are thankful to Inter-University Centre for
Astronomy and Astrophysics, Pune, India for providing Visiting
Associateship under which a part of this work is carried out. FR
and ZH are also grateful to UGC, Govt. of India and D.S. Kothari
fellowship,
for financial
support.
|
1,314,259,993,638 | arxiv | \section{Introduction}
Symmetry plays an important role in the electromagnetic (EM) response of matter. It is
revealed in the form of susceptibilities relating electric and magnetic polarization
($\bVec{P}$ and $\bVec{M}$) with source EM field. In high symmetry case, $\bVec{P}$
and $\bVec{M}$ consist of (the superpositions of) independent groups of excitations
belonging to different irreducible representations of the symmetry group in consideration.
This allows us to treat electric and magnetic properties of matter independently.
When a medium lacks in certain mirror symmetry, i.e., the case of chiral symmetry,
however, some (or all the) components of $\bVec{P}$ and $\bVec{M}$ cannot be
distinguished, so that they can be induced by both electric and magnetic source fields.
In addition, there is also a mixing between electric dipole (E1) and electric quadrupole
(E2) transitions.
The study of chiral symmetry in the EM response of matter has a long history
(Introduction of \cite{Lakhtakia}). Chiral substances have been considered as unconventional
materials for a long time, but now it is regarded as an important source of new materials
and states, providing hot topics in the studies of metamaterials \cite{Meta-M},
multiferroics \cite{multiF}, and superconductivity \cite{super-c}.
In spite of its long history, theoretical description of chirality does not
seem to be standardized. In the
documents of IUPAP and IUPAC dealing with the standard definitions of physical
and chemical quantities \cite{IUPAP, IUPAC}, there is no mentioning about the
chiral susceptibilities. Correspondingly, there are two or more different forms
of phenomenological constitutive equations in use for macroscopic response.
Though the effect of chiral symmetry is expected also in microscopic responses,
its first-principles thoery has been made only very recently \cite{Cho}.
From the viewpoit that all the different forms of EM response theories should
belong to a single hierarchy with logical ranking, one should be able to choose
the most appropriate form of the constitutive equations for the macroscopic
chiral response on the basis of the microscopic theory.
A typical effect of chirality is the difference in the phase velocity of EM waves
with right- and left-circular polarizations, which appears in the off-resonant
region of susceptibilities. However, this is not the only aspect of our interest
in discussing chirality. In fact, the dispersion curves in the resonant region of
susceptibility show a remarkable behavior, by which we can select the correct
constitutive equations. \\
Macroscopic EM response of matter is usually calculated by the combination
of Maxwell and constitutive equations. The standard form of the latter is
\begin{equation}
\label{eqn:DEBH}
\bVec{D} = \epsilon \bVec{E} \ , \ \ \ \bVec{B} = \mu \bVec{H}
\end{equation}
with the dielectric constant (permittivity) $\epsilon$ and permeability $\mu$.
However, if the medium in consideration has chiral symmetry, these constitutive
equations need to be generalized. A popular form of such an extention is
\begin{eqnarray}
\label{eqn:DBFa}
\bVec{D} &=& \epsilon (\bVec{E} + \beta \nabla \times \bVec{E} ) \ , \\
\label{eqn:DBFb}
\bVec{B} &=& \mu (\bVec{H} + \beta \nabla \times \bVec{H} ) \ ,
\end{eqnarray}
which is called Drude-Born-Fedorov equations (DBF eqs) \cite{DBF, Band}.
The parameter (chiral admittance) $\beta$ describes the chirality of the medium.
This is a phnomenology for uniform and isotropic media.
However, this is not the only way of generalization. From
the viewpoint that the fundamental variables of EM field are $\bVec{E}$ and
$\bVec{B}$, both electric and magnetic polarizations $\bVec{P}$ and $\bVec{M}$
should consist both of the $\{\bVec{E}$ and $\bVec{B}\}$-induced components,
so that the definition $\bVec{D} = \bVec{E} + 4\pi \bVec{P}$,
$\bVec{H} = \bVec{B} - 4\pi \bVec{M}$ leads to the extension
\begin{eqnarray}
\label{eqn:ChiCa}
\bVec{D} &=& \hat{\epsilon} \bVec{E} + i\xi \bVec{B} \ , \\
\label{eqn:ChiCb}
\bVec{H} &=& (1/\hat{\mu}) \bVec{B} + i\eta \bVec{E} \ ,
\end{eqnarray}
where the terms with $\xi$ and $\eta$ take care of the chirality. For later
convenience, let us call them chiral constitutive equations (ChC eqs). Though they
are a result of phenomenological consideration on the one hand, a first-principles
calculation of macroscopic constitutive equations can be put also in this form
on the other hand \cite{Cho}.
As to the difference or similarity of DBF and ChC eqs, there is a controversy.
There have been arguments in the metamaterials community that DBF and ChC eqs
are essentially same \cite{Engheta}, and also it is argued
that the former can be derived from the latter by assuming the uniformity and
isotropy of matter \cite{Lakhtakia}(Sec.4.4). But there is also other group of people
preferring ChC to DBF eqs. The purpose of this article is to show that there
is a clear difference between the two, and that ChC eqs should be preferred.
In view of the fact that DBF eqs are frequently used in metamaterials studies
and also in recent textbook of standard electromagnetism \cite{Band},
it will be important to clarify the difference between DBF and ChC eqs. \\
We first note the relation between the parameters of DBF and ChC eqs. By means
of the relation
\begin{equation}
\label{eqn:AmpFar}
\nabla \times \bVec{E} = (i\omega/c) \bVec{B} \ , \ \ \
\nabla \times \bVec{H} = (-i\omega/c) \bVec{D} \ ,
\end{equation}
DBF eqs can be rewritten as
\begin{eqnarray}
\label{eqn:DBFc}
\bVec{D} &=& \epsilon \bVec{E} + (i\omega/c) \epsilon \beta \bVec{B} \ , \\
\label{eqn:DBFd}
\bVec{H} &=& (i\omega/c)\epsilon \beta \bVec{E}
+ (1/\mu)[1 - (\omega \beta/c)^2\epsilon \mu]\bVec{B} \ .
\end{eqnarray}
If DBF and ChC eqs are equivalent, the DBF parameters can be written
in terms of the ChC parameters by comparing (\ref{eqn:DBFc}) and
(\ref{eqn:DBFd}) with (\ref{eqn:ChiCa}) and (\ref{eqn:ChiCb}) as
\begin{equation}
\label{eqn:ChC-DBF}
\hat{\epsilon} = \epsilon \ , \ \ \ \xi = \eta = (\omega/c)\epsilon\beta\ ,
\ \ \ (1/\hat{\mu}) = (1/\mu) - (\omega\beta/c)^2 \epsilon \ .
\end{equation}
This relation will be shown later to lead to contradiction, which disproves
the equivalence of DBF and ChC eqs. \\
The first-principles derivation of micro- and macroscopic constitutive equations is
done in the following way \cite{Cho}. We assume
a general form of non-relativistic Hamiltonian (including relativistic correction
terms, such as spin-orbit interaction and spin Zeeman term, etc.) for a many particle
system in an EM field, and calculate the microscopic current density induced by
the EM field, which is in general given as a functional of the transverse (T) part
of vector potential $\bVec{A}^{(\rm T)}$ and the longitudinal (L) external electric
field $\bVec{E}_{\rm ext}^{(L)}$. The integral kernel of
the functional is the microscopic susceptibility of a separable form with respect to
position coordinates. When the relevant quantum mechanical states have spatial
extension much less than the wavelength of the EM field, we may apply long wavelength
approximation to the microscopic current density, which leads to the macroscopic
constitutive equations to be used for macroscopic Maxwell eqs. In this macroscopic scheme,
we need only a single $3\times 3$ tensor to relate induced current density and source
EM field, covering all the electric, magnetic and chiral polarizations of matter.
This macroscopic constitutive equation is given in the form \cite{Cho}
\begin{equation}
\label{eqn:const-fp}
\bVec{J}(\bVec{k}, \omega) = \chi_{\rm em}(\bVec{k}, \omega)\ \{\bVec{A}^{(\rm T)}(\bVec{k}, \omega)
- (ic/\omega) \bVec{E}_{\rm ext}^{(L)}(\bVec{k}, \omega) \} \ .
\end{equation}
The internal L field does not appear in the source field, since it is taken into account
as the Coulomb potential in the matter Hamiltonian. The susceptibility $\chi_{\rm em}$
is written in terms of the quantum mechanical transition energies and the lower moments
of the corresponding transition matrix elements of current density operator.
Using the identity $\bVec{J} = -i\omega \bVec{P} + i c \bVec{k} \times \bVec{M}$ in
Fourier representation, we can rigorously rewrite the constitutive equation into the form
\begin{equation}
\label{eqn:const-PM}
\bVec{P} = \chi_{\rm eE} \bVec{E} +\chi_{\rm eB} \bVec{B} \ , \ \ \
\bVec{M} = \chi_{\rm mE} \bVec{E} +\chi_{\rm mB} \bVec{B}
\end{equation}
The four susceptibilities $\chi_{\rm eE}$, $\chi_{\rm eB}$, $\chi_{\rm mE}$, $\chi_{\rm mB}$
are again written in terms of the quantum mechanical transition energies and lower transition
moments of electric dipole (E1), electric quadrupole (E2), and magnetic dipole (M1) characters.
Details are given in sec.3.1 of \cite{Cho}. The lowest order terms of them are
\begin{eqnarray}
\chi_{\rm eE} &=& \frac{1}{\omega^2 V} \sum_{\nu} \big[
\bar{g}_{\nu} \bar{\bVec{J}}_{0\nu} \bar{\bVec{J}}_{\nu 0}
+ \bar{h}_{\nu} \bar{\bVec{J}}_{\nu 0} \bar{\bVec{J}}_{0\nu} \big]
\ , \nonumber \\
\chi_{\rm mB} &=& \frac{1}{V} \sum_{\nu} \big[
\bar{g}_{\nu} \bar{\bVec{M}}_{0\nu} \bar{\bVec{M}}_{\nu 0}
+ \bar{h}_{\nu} \bar{\bVec{M}}_{\nu 0} \bar{\bVec{M}}_{0\nu} \big]
\ , \nonumber \\
\chi_{\rm eB} &=& \frac{i}{\omega V} \sum_{\nu} \big[
\bar{g}_{\nu} \bar{\bVec{J}}_{0\nu} \bar{\bVec{M}}_{\nu 0}
+ \bar{h}_{\nu} \bar{\bVec{J}}_{\nu 0} \bar{\bVec{M}}_{0\nu} \big]
\ , \nonumber \\
\chi_{\rm mE} &=& \frac{-i}{\omega V} \sum_{\nu} \big[
\bar{g}_{\nu} \bar{\bVec{M}}_{0\nu} \bar{\bVec{J}}_{\nu 0}
+ \bar{h}_{\nu} \bar{\bVec{M}}_{\nu 0} \bar{\bVec{J}}_{0\nu} \big]
\ , \\
\bar{g}_{\nu} &=& \frac{1}{E_{\nu 0} - \hbar \omega - i 0^+} - \frac{1}{E_{\nu 0}} \ , \ \ \
\bar{h}_{\nu} = \frac{1}{E_{\nu 0} + \hbar \omega + i 0^+} - \frac{1}{E_{\nu 0}} \ ,
\end{eqnarray}
where $V$ is the volume of a cell for periodic boundary condition to define $\bVec{k}$, and
$\bar{\bVec{J}}_{0\nu}$ and $\bar{\bVec{M}}_{0 \nu}$ are, respectively, the E1 and M1
transition moments of current density and (orbital and spin) magnetization operators between
the matter eigenstates $|0\rangle$ (ground state) and $|\nu\rangle$ with transition energy
$E_{\nu 0}$ between them. (E2 moments appear in the $\bar{\bVec{J}}_{\mu\nu}$ terms
in the next higher order.) Chiral symmetry allows the existence of the transitions
with mixed (E1 and M1) or (E1 and E2) character, leading to the $O(k^1)$ terms in
$\chi_{\rm em}$.
Though there appear four susceptibilities, the single susceptibility nature is intact,
since the rewriting of (\ref{eqn:const-fp}) into (\ref{eqn:const-PM}) is reversible.
Combining the new form of constitutive equations with the definition of $\bVec{D}$ and $\bVec{H}$,
we obtain
\begin{eqnarray}
\bVec{D} &=& \bVec{E} + 4\pi \bVec{P} = (1 + 4\pi \chi_{\rm eE}) \bVec{E}
+ 4\pi \chi_{\rm eB} \bVec{B} \ , \\
\bVec{H} &=& \bVec{B} - 4\pi \bVec{M} = (1 - 4\pi \chi_{\rm mB}) \bVec{B}
- 4\pi \chi_{\rm mE} \bVec{E} \ ,
\end{eqnarray}
which is essentially equivalent to ChC eqs. The parameters of ChC eqs are given as
\begin{equation}
\hat{\epsilon} = 1 + 4\pi \chi_{\rm eE} \ , \ \
i \xi = 4\pi \chi_{\rm eB} \ , \ \
i \eta = -4\pi \chi_{\rm mE} \ , \ \
\hat{\mu} = \frac{1}{1 - 4\pi \chi_{\rm mB}} \ .
\end{equation}
all of which are tensors, with no assumption of isotropy and homogeneity as for DBF eqs.
It should also be noted that the poles of $\chi_{\rm mB}$, i.e., the magnetic transition
energies, are, not the poles, but the zeros of $\hat{\mu}$. This is due to the definition
of $\chi_{\rm mB}$, $\bVec{M} = \chi_{\rm mB} \bVec{B}$ as required in the
first-principles approach, in contrast to the conventional one
$\bVec{M} = \chi_{\rm m} \bVec{H}$.
At this stage, the ChC eqs are not a phenomenology, but a first-principles theory.
In contrast, this kind of first-principles derivation does not exist for DBF eqs.
\section{Dispersion equation}
In order to show the difference between DBF and ChC eqs, it is
sufficient to give a single example. For this purpose,we compare the dispersion
relations obtained from DBF and ChC eqs.
\subsection{Case of DBF eqs}
If we solve DBF eqs and eq.(\ref{eqn:AmpFar}) for $\nabla \times \bVec{H}$ and
$\nabla \times \bVec{E}$, we obtain
\begin{eqnarray}
\label{eqn:DBF2a}
\nabla \times \bVec{H} &=& a \bVec{H} + b \bVec{E} \ , \\
\label{eqn:DBF2b}
\nabla \times \bVec{E} &=& d \bVec{H} + e \bVec{E} \ ,
\end{eqnarray}
where
\begin{equation}
a = e = -\epsilon \mu \beta / \Delta \ , \ \ \ b = -i c \epsilon /\omega \Delta \ , \ \ \
d = + i c \mu/ \omega \Delta \ ,
\end{equation}
and $\Delta = \epsilon \mu \beta^2 - c^2/\omega^2$. From eqs (\ref{eqn:AmpFar}),
(\ref{eqn:DBFc}), (\ref{eqn:DBFd}) and $\nabla\cdot \bVec{B} = 0$, both $\bVec{E}$ and
$\bVec{H}$ are transverse, so that
\begin{equation}
\nabla \times \nabla \times \bVec{E} = k^2 \bVec{E}, \ \ \ \ \
\nabla \times \nabla \times \bVec{H} = k^2 \bVec{H} \
\end{equation}
for Fourier components. Then, by operating $\nabla \times $ to (\ref{eqn:DBF2a}) and
(\ref{eqn:DBF2b}), we obtain a set of homogeneous linear equations of $\bVec{H}, \bVec{E}$.
The condition for the existence of non-trivial solution gives us the dispersion equation
\begin{equation}
{\rm det}| k^2 {\bf 1} - {\cal A}^2 | = 0 \ ,
\end{equation}
where ${\cal A}$ is a $2\times 2$ matrix with the components $a, b, d, e$
\begin{equation}
{\cal A} = \left[
\begin{array}{cc}
a & b \\ d & e
\end{array} \right] \ .
\end{equation}
This dispersion equation can be rewritten as
\begin{equation}
{\rm det}| k {\bf 1} + {\cal A} | = 0 \ \ \ {\rm or} \ \ \ {\rm det}| k {\bf 1} - {\cal A} | = 0 \ ,
\end{equation}
so that the solution is
\begin{equation}
k = \pm a \pm \sqrt{bd} \ ,
\end{equation}
with all the combinations of $\pm$ being allowed, which finally leads to a compact expression
\begin{equation}
\label{eqn:disp1}
\frac{ck}{\omega} = \pm \frac{\sqrt{\epsilon \mu}}{1 \pm (\omega\beta/c)\sqrt{\epsilon \mu}} \ .
\end{equation}
This gives two branches of dispersion curve. In homogeneous isotropic media,
the two modes correspond to right and left circular polarizations. It should be noted that
the condition for the existence of real solution is $\epsilon \mu \geq 0$. This means that
the left-handed medium is defined in the same way as in non-chiral medium, in contrast to
the case of ChC eqs.
\subsection{Case of ChC eqs}
A same way of solution is possible in this case, too. After eliminating $\bVec{D}, \bVec{B}$
from the ChC eqs, the solution for $\nabla \times \bVec{H}, \ \nabla \times \bVec{E}$
has a same form as the one, where $a, b, d, e$ of previous section are replaced with
the following $ f, g, h, j$, respectively
\begin{eqnarray}
f &=& -i(\omega/c) \xi\hat{\mu} \ , \\
g &=& -i(\omega/c) (\hat{\epsilon} + \hat{\mu} \xi \eta) \ , \\
h &=& -i(\omega/c) \hat{\mu} \ , \\
j &=& -i(\omega/c) \hat{\mu} \eta \ .
\end{eqnarray}
Further transformation of the equations of $\nabla \times \bVec{H}, \ \nabla \times \bVec{E}$
into a set of homogeneous equations of $\bVec{E}, \bVec{H}$ allows us to obtain the dispersion
equation, as the condition for the existence of non-trivial solution,
\begin{equation}
\label{eqn:disp2}
\frac{ck}{\omega} = \pm \frac{1}{2} \big[\hat{\mu}(\eta - \xi)
\pm \sqrt{\{\hat{\mu}(\eta - \xi)\}^2 + 4\hat{\epsilon} \hat{\mu}}\ \big] \ ,
\end{equation}
where we take all the combinations of $\pm$. The condition for real solutions is
\begin{equation}
\label{eqn:real-r}
\{\hat{\mu}(\eta - \xi)\}^2 + 4\hat{\epsilon} \hat{\mu} \geq 0 \ ,
\end{equation}
which is less restrictive than non-chiral case.
\section{Discussions}
First of all, we note that, for both DBF and ChC eqs, the well-known
result of $(ck/\omega)^2 = \epsilon \mu$ is obtained in the absence of chirality
($\beta = 0$ and $\xi = 0, \eta = 0$). Also both of the constitutive equations exhibit
the typical behavior of chiral medium, i.e., the existence of the two branches with
polarization dependent refractive indices. In the non-resonant region, both of them
could be used to fit experimental results via appropriate choice of parameter values.
\subsection{Resonant region of left-handed chiral medium}
A decisive difference appears in resonant region. An example will be the left-handed
behavior emerging in the neighborhood of a chiral resonance with E1-M1 mixed character.
Such a case has been treated by the first-principles theory of macroscopic constitutive
equation \cite{Cho} (Sec.3.8.1 and Sec.4.1.1). It shows a pair of dispersion curves
for left and right circularly polarized modes, which have a linear crossing at $k=0$.
It will be a test for the phenomenologies whether such a linear crossing can be
realized or not by choosing parameter values.
The dispersion equation in the first-principles macroscopic formalism is
\begin{equation}
0 = {\rm det} | \frac{c^2k^2}{\omega^2} {\bf 1} - \big[ {\bf 1} + \frac{4\pi c}{\omega^2}
\chi_{\rm em}^{(\rm T)}(\bVec{k}, \omega) \big] | \ ,
\end{equation}
where $\chi_{\rm em}^{(\rm T)}(\bVec{k}, \omega)$ is the T component of susceptibility
tensor (sec.2.5 of \cite{Cho}). Let us choose a chiral form of susceptibility tensor
$\chi_{\rm em}^{(\rm T)}$ as
\begin{equation}
{\bf 1} + \frac{4\pi c}{\omega^2}\chi_{\rm em}^{(\rm T)} =
(\epsilon_{\rm b} + a' + c'k^2) {\bf 1} + \big[
\begin{array}{cc}
0 & ib'k \\ -ib'k & 0
\end{array}
\big] \ ,
\end{equation}
where the terms with $a', b', c'$ represents the contribution of a pole
$\sim 1/(\omega_{0} - \omega)$ with mixed (E1, M1) character, while
$\epsilon_{\rm b}$ is the background dielectgric constant due to all
the other resonances. We assume that the resonance with mixed E1 and M1 characters
occurs in the frquency region of $\epsilon_{\rm b} < 0$, i.e., a chiral version of
left-handed medium. The dispersion equation is
\begin{equation}
\big(\frac{ck}{\omega}\big)^2 = \bar{\epsilon} \bar{\mu}
\pm \bar{\beta} \bar{\mu} \ \frac{ck}{\omega} \ ,
\end{equation}
and its solution is given as
\begin{equation}
\label{eqn:disp3}
\frac{ck}{\omega} = \pm \frac{1}{2} \big[ \pm \bar{\beta} \bar{\mu}
+ \sqrt{\bar{\beta}^2 \bar{\mu}^2 + 4 \bar{\epsilon} \bar{\mu}} \ \big] \ ,
\end{equation}
where we take all the combinations of $\pm$, and
\begin{equation}
\bar{\beta} = \omega b'/c \ , \ \ \ \bar{\epsilon} = \epsilon_{\rm b} + a' \ \ \
\bar{\mu} = 1/[1 - (\omega/c)^2 c'] \ .
\end{equation}
Noting that $b'$ is a chiral parameter corresponding to $\xi, \eta$ of the ChC eqs,
we see that this equation is the same type as eq.(\ref{eqn:disp2}), but not as
eq.(\ref{eqn:disp1}). An example of this dispersion relation is given in Fig.1.
The characteristic behavior of the dispersion curves is a linear crossing at k=0
(Fig.4.1 of \cite{Cho}).
\begin{figure}
\begin{center}
\includegraphics[width=10cm,clip]{Fig4-1revz.eps}
\end{center}
\caption{Dispersion curves of a chiral left-handed medium for the model in the text.
Both ordinate and abscissa are normalized by the frequency of the pole as
$\omega/\omega_{0}$ and $ck/\omega_{0}$. Two horizontal lines show the frequncies of
$\epsilon = 0$ and $\mu=0$.}
\label{fig:1}
\end{figure}
In order to check whether the dispersion equations (\ref{eqn:disp1}) and (\ref{eqn:disp2})
have this typical behavior of "linear crossing at $k=0$", we focus on the behavior of
the dispersion equations near $k=0$. Since both of the dispersion equations are given in
the form "$ck/\omega = F(\omega)$", we only need to examine how the function $F(\omega)$
approaches to zero in each case.
The microscopic model of left-handed chiral medium given above consists of
a matter excitation level with (E1, M1) mixed character in the frequency range of
$\epsilon_{\rm b} < 0$. This means that all of $\chi_{\rm eE},
\chi_{\rm eB},\chi_{\rm mE}, \chi_{\rm mB}$ have a common pole
(at $\omega = \omega_{0}$) and $\chi_{\rm eE}$ is largely negative
in the frequency range of interest to make $\epsilon_{\rm b} < 0$. Namely,
$\hat{\epsilon}, 1/\hat{\mu}, \xi, \eta$ of ChC eqs have a common pole at
$\omega = \omega_{0}$ and
$\hat{\epsilon} = \epsilon_{\rm b} + \bar{a}/(\omega_{0} - \omega)$.
It may appear that the r.h.s. of eq.(\ref{eqn:disp2}) becoms zero for frequency
satisfying $\hat{\mu} = 0$ or $\hat{\epsilon} = 0$. However, as mentioned before, the zero
of $\hat{\mu}$ corresponds to the pole of M1 transition, which in this case is common
to the pole of $\xi$ and $\eta$. Therefore the zero of $\hat{\mu}$ is cancelled in the
product $\hat{\mu}(\xi - \eta)$. Thus, the only remaining possibility of zero arises
from $\hat{\epsilon} = 0$. The $\omega$-dependence of the r.h.s. eq.(\ref{eqn:disp2})
near zero point can be found by rewriting it as
\begin{equation}
\pm \frac{1}{2} \frac{\hat{\epsilon} \hat{\mu}}{\hat{\mu}(\xi - \eta) \pm
\sqrt{\{\hat{\mu}(\xi - \eta)\}^2 + 4 \hat{\epsilon} \hat{\mu}}} \ .
\end{equation}
At the frequency satisfying $\hat{\epsilon} = 0$, i.e., $\epsilon_{\rm b} + a' = 0$, all of
$\hat{\mu}, \xi, \eta$ remain finite, and one of the $\pm$ combinations in the denominator
remains finite, so that the whole expression becomes zero for this combination.
This occurs for both signs of $\pm$ in front of the whole expression.
For negative $\epsilon_{\rm b}$ and positive numerator of the pole
$\sim 1/(\omega_{0} - \omega)$, $\epsilon_{\rm b} + a' = 0$ occurs at
$\omega=\omega_{\rm z} < \omega_{0}$ and
in its neighborhood $\hat{\epsilon} \sim (\omega - \omega_{\rm z})$. This shows that
the r.h.s. of eq.(\ref{eqn:disp2}) behaves like $\sim (\omega - \omega_{z})$, which means
the linear crossing of the two branches. Note also that $\omega_{\rm z}$ lies inside the
frequency range of eq.(\ref{eqn:real-r}).
Now we check whether the same behavior is obtained for DBF eqs by assuming eq.
(\ref{eqn:ChC-DBF}), from which we obtain
\begin{equation}
\frac{1}{\mu} = \frac{1}{\hat{\mu}} + \frac{\xi^2}{\hat{\epsilon}} \ ,\ \ \
\frac{\omega^2\beta^2}{c^2} \epsilon \mu = 1 - \frac{\mu}{\hat{\mu}} \ .
\end{equation}
This shows that $1/\mu$ has the same pole as $1/\hat{\mu}$ at $\omega = \omega_{0}$,
so that the factor $\mu/\hat{\mu}$ on the r.h.s. of the second equation does not
have the pole at $\omega = \omega_{0}$ via cancellation. Therefore, there is no
chance for the denominator of the r.h.s. of eq.(\ref{eqn:disp1}) to diverge. Hence
the only possibility of its becoming zero comes from the factor $\sqrt{\epsilon\mu}$
on the numerator. In view of the fact that the zeros of $\epsilon, \mu$ occur at
different $\omega$'s, e.g., at $\omega_{\rm z1}$ and $\omega_{\rm z2}$
($\omega_{\rm z1}$ $>$ $\omega_{\rm z2}$), the $\omega$-dependence of
$\sqrt{\epsilon \mu}$ should be
$\sim \sqrt{\omega_{z1} - \omega}$ or $\sim \sqrt{\omega - \omega_{z2}}$
in the neigborhood of the zeros.
Therefore, no linear crossing is possible in the DBF dispersion curves.
The two zeros are the boundaries of the region of left-handed behavior.
One might argue that other type of $\omega$-dependence than eq. (\ref{eqn:ChC-DBF})
could lead to the linear crossing behavior. But one cannot freely give the
$\omega$-dependence even as a phenomenology. Linear susceptibilities should be
a sum of single pole functions. In the absence of the first-principles theory
for DBF eqs, it would be quite difficult to give an appropriate model on a reliable
basis. \\
\subsection{Conventionality vs. Logical Consistency}
DBF eqs have been popularly used in the macroscopic argument of chiral systems,
especially in the field of metamaterials research. As long as they are used for
nonresonant phenomena as a practical tool, there is not much to say against it,
except for the difficulty in assigning microscopic meaning to the parameter
$\beta$. However, the restriction to the nonresonant phenomena does not
seem to be widely recognized, to the knowledge of the present author. In fact
there are examples of its use for resonant phenomena \cite{Luan, Tomita}. (The
constitutive equations used in \cite{Tomita} are not exactly DBF eqs, but
$\bVec{D} = \epsilon \bVec{E} - i\xi \bVec{H} \ , \ \bVec{B} = \mu \bVec{H}
+ i\xi \bVec{E}$, different also from ChC eqs.)
From the qualitative difference of the two dispersion equations (\ref{eqn:disp1})
and (\ref{eqn:disp2}) in resonant region, and from the fact that DBF eqs have no
support by microscopic theory in contrast to ChC eqs, the use of DBF eqs for
resonant phenomena is risky. As a conventional approach with a long history,
DBF eqs might be kept in use further, but the validity limit should be kept in
mind. However, if we consider that ChC eqs can be handled as easily as DBF eqs,
and that they are consistent with the microscopically derived macroscopic
constitutive equationn, it is highly recommended to use ChC eqs. For problems
requiring severe distinction, logical consistency should be preferred to
conventionality.
\section{Conclusion}
The DBF eqs, popularly used as constitutive equations of chiral media, should be regarded as
a phenomenological theory applicable only in nonresonant region. In resonant region,
it would lead to a qualitatively erroneous result. On the other hand, the ChC eqs,
consistent with the first-principles microscopic constitutive equations, can be used both for
resonant and nonresonant problems. \\
\begin{flushleft}
\underline{Acknowledgment}\\
\end{flushleft}
This work is supported by Grant-in-Aid for scientific research on Innovative Areas
Electromagnetic Metamaterials of MEXT Japan (Grant No. 22109001).
|
1,314,259,993,639 | arxiv | \section{Introduction}
\label{sec:Introduction}
A central issue of fundamental research is the unification of quantum theory (QT) and general relativity (GR) in the framework of quantum gravity (QG). A critical scale in the context of this unification is the Planck scale defined as $l_{pl}=\sqrt{\frac{\hbar G}{c^3}}=10^{-35}m$ (see Ref.\cite{Garay:1994en} for a review) which has been shown to be the minimum measurable scale if both QT and GR are applicable. Indeed it may be shown \cite{Plato:2016azz} that the high energies required to probe scales smaller than the Planck scale would lead to the formation of a black hole through the gravitational disturbances of spacetime structure which would prohibit any measurement on smaller scales. The existence of such a minimum measurable length would lead to a modification of the Heisenberg Uncertainty Principle \cite{aHeisenberg:1927zz,Robertson:1929zz}(HUP) to the so-called Generalized (Gravitational) Uncertainty Principle (GUP)(see Ref.\cite{Tawfik:2015rva} for a review)
\be
\Delta x \Delta p \geq \frac{\hbar}{2} (1+\beta \Delta p^2)
\label{gup1}
\ee
where $\beta$ is the GUP parameter defined as $\beta =\beta_0 /M_{pl} c^2=\beta_0 l_{pl}^2/\hbar^2$, $M_{pl}c^2=10^{19}GeV$, $l_{pl}$ is the 4-dimensional fundamental Planck scale and $\beta_0$ is a dimensionless parameter expected to be of order unity.
Such a GUP is closely related to the concept of noncommutative geometry \cite{Connes:1994yd} and has been extensively investigated in Refs. \cite{Mead:1964zz,Maggiore:1993kv,Maggiore:1993rv,Maggiore:1993zu,Kempf:1994su,Hinrichsen:1995mf,Kempf:1996ss,Kempf:1996nm,Snyder:1946qz,Yang:1947ud,Karolyhazy:1966zz,Ashoorioon:2004vm,Ashoorioon:2004rs,Ashoorioon:2004wd,Ashoorioon:2005ep,Faizal:2014mba,Ali:2015ola,Mohammadi:2015upa,Faizal:2016zlo,Zhao:2017xjj}. In particular interest in a minimum measurable length or equivalently in a ultraviolet cutoff has been motivated by studies of string theory \cite{Veneziano:1986zf,Gross:1987kza,Gross:1987ar,Amati:1987wq,Amati:1988tn,Konishi:1989wk,Kato:1990bd}, loop quantum gravity \cite{Rovelli:1989za,Rovelli:1994ge,Carr:2011pr,rovelli_2004,Ashtekar:2004eh,Thiemann:2006cf,Thiemann:2002nj}, quantum geometry \cite{Capozziello:1999wx}, doubly special relativity (DSR)\cite{AmelinoCamelia:2000mn,Cortes:2004qn,Magueijo:2001cr,Magueijo:2002am,AmelinoCamelia:2002wr,Magueijo:2004vv} and by black hole physics \cite{Maggiore:1993rv,Adler:2001vs,Nozari:2011gj,Alasfar:2017loh} or even Gedanken experiments \cite{Scardigli:1999jh} and thermodynamic properties of gravity \cite{Zhu:2008cg}. Several phenomenological implications of minimal length theories and quantum gravity phenomenology were investigated and a number of researchers have studied phenomenological aspects of GUP effects in several contexts (e.g. in Refs. \cite{Das:2008kaa,Das:2009hs} atomic physics experiments such as Lamb’s shift and Landau levels have been considered and constraints on the minimum length scale parameter $\beta$ have been estimated ). In Refs. \cite{Ali:2009zq,Ali:2011fa,Nozari:2012gd,Das:2010zf,Basilakos:2010vs} a model that is consistent with string theory, black hole physics and DSR is presented and discussed. This model of GUP predicts both a minimal observable length and a maximal momentum simultaneously \cite{Ali:2011fa,Das:2011tq}.
The existence of a minimum measurable length is closely related to the existence of the black hole horizon which tends to form if length scales below the Planck scale are probed. Correspondingly, there is a maximum measurable length associated with the cosmological particle horizon \cite{Faraoni:2011hf,Davis:2003ze} which provides due to causality a maximum measurable length scale in the Universe. The particle horizon corresponds to the length scale of the boundary between the observable and the unobservable regions of the universe. This scale at any time defines the size of the observable universe. The physical distance to this maximum observable scale at the cosmic time $t$ is given by (see e.g \cite{Kolb:1990vq,Hobson:2006se})
\be
l_{max}(t)=a(t)\int_0^t \frac{c\; dt}{a(t)}
\label{parthorscale}
\ee
where $a(t)$ is the cosmic scale factor.
For the best fit \lcdm cosmic background at the present time $t_0$ we have
\be
l_{max}(t_0)\simeq 14 Gpc \simeq 10^{26} m
\label{parthort0}
\ee
This existence of such a maximum measurable length would lead to modified version of the GUP of the form \footnote{A perturbative version of this GUP was introduced in \cite{Park:2007az} as $\Delta x \Delta p \geq 1+ \alpha\frac{ \Delta x^2 }{L_*^2}$ (where $\alpha$ is a constant of order unity and $L_*$ is the characteristic, large length scale) and called extended uncertainty principle (EUP) by many authors \cite{Park:2007az,Bambi:2007ty,Zhu:2008cg,Mignemi:2009ji,Ghosh:2009hp,COSTAFILHO2016367,Schurmann:2018yuz,Mureika:2018gxl,Dabrowski:2019wjk}. Here we keep the notation `GUP' instead of `EUP' for consistency with Ref. \cite{Perivolaropoulos:2017rgq}.} \cite{Perivolaropoulos:2017rgq}
\be
\Delta x \Delta p \geq \frac{\hbar}{2} \frac{1}{1- \alpha \Delta x^2}
\label{gupmaxlength}
\ee
As shown in Fig. \ref{lmax}, this GUP indicates the existence of maximum position uncertainty (see Ref.\cite{Perivolaropoulos:2017rgq})
\be
l_{max}\equiv \Delta x_{max} = \alpha^{-1/2}
\ee
due to the divergence of the RHS of eq. (\ref{gupmaxlength}). As shown in Fig. \ref{lmax} the existence of a maximum length scale is associated with the presence of a minimum momentum scale $\Delta p_{min}$.
\begin{figure}
\begin{centering}
\includegraphics[width=0.45 \textwidth]{gupmax1.pdf}
\par\end{centering}
\caption{The deformation of the HUP in accordance with eq. (\ref{gupmaxlength}) after rescaling to dimensionless form using a characteristic length scale of the quantum system (from Ref. \cite{Perivolaropoulos:2017rgq}). }
\label{lmax}
\end{figure}
The GUP (\ref{gupmaxlength}) originates from a commutation relation of the form
\be
[x,p]=i \hbar \frac{1}{1-\alpha x^2}
\label{comrelgupmaxlength}
\ee
It is straightforward to show (see in Appendix \ref{Appendix}) that this commutation relation leads to the GUP (\ref{gupmaxlength}) using the general uncertainty principle for the pair of non-commuting observables $x$, $p$
\be
\Delta x\Delta p \geq\frac{\hbar}{2}\mid\left\langle\left[\hat{x},\hat{p}\right]\right\rangle\mid
\label{genupAB}
\ee
with
\be
\Delta x\equiv \sqrt{\left\langle \left(\hat{x}-\left\langle \hat{x}\right\rangle\right)^2\right\rangle}
\ee
\be
\Delta p\equiv \sqrt{\left\langle \left(\hat{p}-\left\langle \hat{p}\right\rangle\right)^2\right\rangle}
\ee
where $\hat{x},\hat{p}$ are the operator representations of the observables $x$, $p$ .
The commutation relation (\ref{comrelgupmaxlength}) may be represented as shown in Appendix \ref{Appendix} by position and momentum operators of the form
\ba
p &=& \frac{1}{1 - \alpha x_0^2}p_0=(1+\alpha x_0^2 + \alpha^2 x_0^4 + ...)p_0
\label{reproperp} \\
x &=& x_0
\label{reproperx}
\ea
where $x_0$ and $p_0$ are the usual position and momentum operators satisfying the Heisenberg commutation relation $[x_0,p_0]=i\hbar$.
The representation (\ref{reproperp}), (\ref{reproperx}) may be used to solve the Schrodinger equation for simple quantum systems to find the dependence of the energy spectrum on the maximum measurable scale $l_{max}$. Such an analysis has indicated \cite{Perivolaropoulos:2017rgq} that the current cosmic particle horizon is too large to lead to any observable effects in present day quantum systems. This however is not necessarily the case in the Early Universe when the particle horizon scale is much smaller and could leave an observable signature in the quantum generation of the primordial fluctuations during inflation. Thus, in the present analysis we wish to address the following questions
\begin{itemize}
\item
What is the deformation of the scale invariant spectrum of perturbations produced during inflation due to the Heisenberg algebra deformation (\ref{comrelgupmaxlength}) corresponding to the existence of a maximum measurable scale?
\item
What constraints can be imposed on the fundamental parameter $\alpha=l_{max}^{-2}$ from the observed power spectrum of primordial fluctuations?
\end{itemize}
The structure of this paper is the following: In the next section \ref{QHO} we consider a simple harmonic oscillator in the presence of a large maximum measurable scale and find the variance of the position as a function of the parameter $\alpha$ and the corresponding variance in the context of the HUP ($\alpha=0$). In section \ref{PSCF} we generalize this analysis to the case of systems with infinite degrees of freedom (fields) and derive the spectrum and the spectral index of tensor and scalar perturbations generated during inflation as a function of the parameter $\alpha$ and of the corresponding spectrum obtained in the context of the HUP. In section \ref{OBCON} we use the derived theoretical expression for the (running) spectral index along with the corresponding observationally allowed range of the index as a function of the scale $k$ to derive constraints on the fundamental parameter $\alpha$ of the GUP. Finally in section \ref{Discussion} we conclude, summarize and discuss the implications and possible extensions of our analysis.\\
\section{Toy Model: The position variance of the Harmonic Oscillator under GUP}
\label{QHO}
In order to quantize the simple harmonic oscillator under the assumption of the GUP (\ref{gupmaxlength}) we need to generalize the expressions of the creation and annihilation operators $\hat{a}^{\dagger}$ and $\hat{a}$ in terms of $x,p$ so that the commutation relation \cite{Sakurai:1167961}
\be
[\hat{a},\hat{a}^{\dagger}]=1
\label{comrelaad}
\ee
is retained while at the same time the GUP commutation relation (\ref{comrelgupmaxlength}) is also respected. Thus, in order to satisfy these conditions, we generalize the analysis of Refs. \cite{Camacho:2003dm,Nozari:2005it} which applies to the GUP (\ref{gup1}) and define
\ba
\hat{a}&=&\frac{1}{\sqrt{2\hbar \omega}}\left(\omega\left[x+f(\alpha,x)\right]+ip\right)
\label{anngup} \\
\hat{a}^{\dagger}&=&\frac{1}{\sqrt{2\hbar \omega}}\left(\omega\left[x+f(\alpha,x)\right]-ip\right)
\label{cregup}
\ea
where $f(\alpha,x)$ is a function chosen so that the commutation relations (\ref{comrelaad}) and (\ref{comrelgupmaxlength}) are respected.
It is straightforward to show that the following function satisfies the aforementioned conditions simultaneously
\be
f(\alpha,x)= \sum_{n=1}^{\infty}\frac{(-\alpha)^n}{2n+1}x^{2n+1}
\label{fx}
\ee
while it reduces to 0 in the limit $\alpha\rightarrow 0$ as it should.
Thus, we can rewrite eqs.(\ref{anngup}) and (\ref{cregup}) as
\be
\hat{a}=\frac{1}{\sqrt{2\hbar \omega}}\left(\omega\frac{1}{\sqrt{\alpha}}arctan(\sqrt{\alpha}x)+ip\right)
\label{anntan}
\ee
\be
\hat{a}^{\dagger}=\frac{1}{\sqrt{2\hbar \omega}}\left(\omega\frac{1}{\sqrt{\alpha}}arctan(\sqrt{\alpha}x)-ip\right)
\label{cretan}
\ee
and the $p$ and $x$ operators are
\be
p=-i\sqrt{\frac{\hbar \omega}{2}}\left(\hat{a}-\hat{a}^{\dagger}\right)
\ee
\be
x=\frac{1}{\sqrt{\alpha}}tan \left(\sqrt{\frac{\hbar \alpha}{2\omega}}(\hat{a}+\hat{a}^{\dagger})\right)
\label{operxtan}
\ee\
Using $tanx=x+\frac{x^3}{3}+\frac{2x^5}{15}+ ...$ , we have
\be
x=x_0+\frac{\alpha x_0^3}{3}+\frac{2\alpha^2 x_0^5}{15}+ ...
\ee
where
\be
x_0=\sqrt{\frac{\hbar}{2\omega}}(\hat{a}+\hat{a}^{\dagger})
\ee
is the position operator in the case of the HUP ($\alpha=0$).
Keeping the lower order terms in $\alpha$ (assuming $\frac{\alpha \hbar}{6\omega} \ll 1$) we obtain
\be
x=x_0+\frac{\alpha x_0^3}{3}\Rightarrow x=\sqrt{\frac{\hbar}{2\omega}}(\hat{a}+\hat{a}^{\dagger})\left[1+\frac{\alpha \hbar}{6\omega}(\hat{a}+\hat{a}^{\dagger})^2 \right]
\label{operxa}
\ee
For $\alpha=0$ we have
\be
x_0=\upsilon(\omega,t)\tilde{a}+\upsilon^*(\omega,t)\tilde{a}^{\dagger}
\label{operxo}
\ee
where
\be
\upsilon(\omega,t)=\sqrt{\frac{\hbar}{2\omega}}e^{-i\omega t}
\ee
is the properly normalized solution of the classical evolution equation of the harmonic oscillator $\frac{d^2\upsilon}{dt^2}+ \omega^2 \upsilon = 0 $.
Therefore the position operator may be expressed as
\be
x=\left(\upsilon\tilde{a}+\upsilon^*\tilde{a}^{\dagger}\right)\left[1+\frac{\alpha}{3}(\upsilon\tilde{a}+\upsilon^*\tilde{a}^{\dagger})^2\right]
\ee
Thus the variance of the position in the ground state takes the form
\begin{widetext}
\be
\langle |x|^2\rangle\equiv\langle 0| x^\dagger x |0\rangle\Rightarrow \langle |x|^2\rangle=|\upsilon(\omega,t)|^2\left[1+2\alpha|\upsilon(\omega,t)|^2\right]
\label{qfoperx}
\ee
\end{widetext}
which reduces to the familiar result for $\alpha=0$ (see e.g. \cite{Dodelson:2003ft,Baumann:2009ds}).
In the next section we generalize the above analysis to the case of quantum field fluctuations involving infinite degrees of freedom aiming to derive the perturbation power spectrum generated during inflation in the context of the GUP.\\
\section{PRIMORDIAL SPECTRA OF COSMOLOGICAL FLUCTUATIONS with GUP}
\label{PSCF}
According to the decomposition theorem \cite{Lifshitz:1945du} the perturbations of each type evolve independently (at the linear level) and we can treat tensor (T) and scalar (S) perturbations of the metric separately. Therefore for spatially flat Friedmann-Robertson-Walker (FRW) background plus the perturbations we can write
\be
ds_T^2=a^2\left[-d\tau^2+(\delta_{ij}+H_{ij})dx^idx^j\right]
\ee
and in conformal Newtonian gauge \cite{Mukhanov:1990me}
\be
ds_S^2=a^2\left[-(1+2\Psi)d\tau^2+\delta_{ij}(1+2\Phi)dx^idx^j\right]
\ee
where $a$ is the scale factor, $\tau$ is the conformal time, $\Psi$ corresponds to the gravitational potential of the perturbations, $\Phi$ is the perturbation of the spatial curvature\footnote{In the absence of anisotropic stress ($\Pi=0$) we have $\Psi=-\Phi$ \cite{Bertschinger:2001is}} and $H_{ij}$ is the tensor perturbation which has the form \footnote{It has this form in a coordinate system where wavevector $\bold{k}$ points along the z-axis.}
\be
\left[H_{ij}\right]=\left[
\begin{array}{ccc}
h_+ & h_{\times} & 0 \\
h_{\times} &-h_+ & 0 \\
0 & 0 & 0 \\
\end{array}
\right]
\ee
The classical evolution equations for the tensor mode perturbations $h_T$ (where $T={+},{\times}$ for two polarization states \cite{Misner:1974qy}) of the FRW metric during inflation in conformal time are obtained from the linearized Einstein equations and may be written as \cite{Grishchuk:1974ny}
\be
h_T''+ 2 \frac{a'}{a}h_T'+k^2h_T=0
\label{tfh}
\ee
where primes denote derivatives with respect to $\tau$. This becomes a simple harmonic oscillator equation by defining
\be
\tilde{h}_T\equiv\frac{a h_T}{\sqrt{16\pi G}}
\ee
and eq. (\ref{tfh}) takes the form
\be
\tilde{h}_T''+\omega^2\tilde{h}_T=0
\label{qfha}
\ee
where
\be
\omega^2 =k^2-\frac{a''}{a}
\ee
During slow roll inflation when the Hubble rate $H$ is nearly constant \cite{Lyth:1994dc}, the conformal time is \cite{Lyth:1998xn,Baumann:2009ds}
\be
\tau\simeq\frac{-1}{aH}
\label{ct}
\ee
Thus we obtain
\be
\omega^2 =k^2-\frac{2}{\tau^2}
\ee
We now quantize the tensor field fluctuations by promoting them to operators and imposing a generalized field commutation (GFC) relation \cite{Matsuo:2005fb,Kober:2011uj} corresponding to (\ref{comrelgupmaxlength}). This GFC takes the form ($\hbar=1$)
\be
[\tilde{h}_T(\bold{k}),\pi_{\tilde{h}_T}(\bold{k'})]=i\delta(\bold{k}-\bold{k}')\frac{1}{1-\mu\tilde{h}_T^2(\bold{k})}
\label{guph}
\ee
where $\pi_{\tilde{h}_T}$ is the conjugate momentum to $\tilde{h}_T$ which is given by
\be
\pi_{\tilde{h}_T}=\tilde{h}_T'-\frac{a'}{a}\tilde{h}_T
\ee
and $\mu$ is a GFC parameter
\be
\mu \simeq \alpha^2 = l_{max}^{-4}
\label{param}
\ee
where $\alpha$ is the parameter of the GUP (\ref{gupmaxlength}).
Thus we have an infinite number of decoupled harmonic oscillators corresponding to eq. (\ref{qfha}) which may be quantized in accordance with the GFC (\ref{guph}).
Using the results of the previous section we connect the field normal modes with the creation and annihilation operators which satisfy the commutation relation $[\hat{a}_{\bold{k}},\hat{a}_{\bold{k'}}^{\dagger}]=\delta^3(\bold{k}-\bold{k'})$, as
\be
\tilde{h}_T(\bold{k})=\frac{1}{\sqrt{\mu}}tan \left(\sqrt{\frac{\mu}{2\omega}}(\hat{a}_{\bold{k}}+\hat{a}_{\bold{k}}^{\dagger})\right)
\ee
\be
\pi_{\tilde{h}_T}(\bold{k})=-i\sqrt{\frac{\omega}{2}}\left(\hat{a}_{\bold{k}}-\hat{a}_{\bold{k}}^{\dagger}\right)
\ee
and obtain the variance of the perturbations as
\begin{widetext}
\be
\langle h_T^\dagger(\bold{k},\tau)h_T(\bold{k'},\tau)\rangle = \frac{16\pi G}{a^2}|\upsilon(\bold{k},\tau)|^2\left[1+2\bar{\mu}|\upsilon(\bold{k},\tau)|^2\right](2\pi)^3 \delta^3(\bold{k}-\bold{k'})\equiv (2\pi)^3 P_h(k)\delta^3(\bold{k}-\bold{k'})
\label{psh}
\ee
\end{widetext}
where $P_h$ is the power spectrum of the primordial tensor perturbations of the metric, the Dirac delta function enforces the independence of the different modes ($ h(\bold{k},\tau)$ is uncorrelated with $h(\bold{k'},\tau)$ if $\bold{k}\neq\bold{k}'$ ) and
\be
\bar{\mu}=\mu V_*
\label{barmu}
\ee
Here the volume scale $V_*=\delta^3(0)\simeq l_{max}^3$ is an infrared regulator \cite{Oblak:2016eij} while $\upsilon$ satisfies the Mukhanov-Sasaki equation \cite{Mukhanov:1988jd,Kodama:1985bj,Stewart:1993bc}
\be
\upsilon''(k,\tau)+(k^2-\frac{a''}{a})\upsilon (k,\tau)=0
\label{ups}
\ee
During slow-roll inflation with initial condition $\upsilon (k,\tau)=\frac{1}{\sqrt{2k}}e^{-ik\tau}$ and by virtue of eq. (\ref{ct}) (as in spatially flat de Sitter background) we obtain the Bunch-Davies solution of eq. (\ref{ups}) \cite{Birrell:1982ix,Parker:2009uva,Kinney:2009vz,Baumann:2009ds}
\be
\upsilon(k,\tau)=\frac{e^{-ik\tau}}{\sqrt{2k}}\left(1-\frac{i}{k\tau}\right)
\label{mod}
\ee
Using eq. (\ref{psh}) we can write the primordial power spectrum for tensor modes as
\be
P_h(k)=P_h^{(0)}(k)\left[1+\frac{\bar{\mu}a^2}{8\pi G} P_h^{(0)}(k)\right]
\label{psh0}
\ee
where
\be
P_h^{(0)}(k)= \frac{16\pi G}{a^2}|\upsilon(k,\tau)|^2
\label{psh00}
\ee
Once $k|\tau|<1 $ , the mode leaves the horizon, after which $h$ remains constant. Thus, using eqs. (\ref{mod}) and (\ref{psh00}) we obtain
\be
P_h^{(0)}(k)= \frac{16\pi G}{a^2}\frac{1}{2k^3\tau^2}=\frac{8\pi G H^2}{k^3}
\ee
where the equality on the second line holds because we have assumed that $H$ is constant and $\tau=-\frac{1}{a H}$.\footnote{We evaluate $H$ at the time when the mode leaves the horizon.}
In a similar manner we may investigate scalar perturbations induced by quantum fluctuations of the inflaton scalar field $\phi$ \cite{Liddle:1993fq,Lidsey:1995np,Baumann:2009ds} of the form
\be
\phi(\bold{x},t)=\phi^{(0)}(t)+\delta\phi(\bold{x},t)
\ee
where $\phi^{(0)}$ is the zero-order part and $\delta\phi$ is the first-order perturbation.
The fluctuations $\delta\phi$ of the scalar field driving inflation evolve in conformal time $\tau$ according to the equation (see e.g. \cite{Kolb:1990vq})
\be
\delta\phi''+ 2 \frac{a'}{a}\delta\phi'+k^2\delta\phi=0
\label{ifphi}
\ee
Using the definition
\be
\varphi=a\delta\phi
\ee
eq. (\ref{ifphi}) becomes
\be
\varphi''+\omega^2\varphi=0
\label{qfha2}
\ee
with $\omega^2 =k^2-\frac{a''}{a}$.
In the context of the maximal measurable length GUP as applied to the case of the inflaton fluctuations, the field commutation relation gets generalized as
\be
[\varphi(\bold{k}),\pi_{\varphi}(\bold{k'})]=i\delta(\bold{k}-\bold{k'})\frac{1}{1-\mu\varphi^2(\bold{k})}
\label{gupphi}
\ee
where $\pi_{\varphi}$ is the conjugate momentum to $\varphi$ which is given by
\be
\pi_{\varphi}=\varphi'-\frac{a'}{a}\varphi
\ee
Since eq. (\ref{ifphi}) is identical to eq. (\ref{tfh}) we can use the result of eq. (\ref{psh0}) without the factor $16\pi G$ in order to turn the dimensionless $h$ into a field $\delta\phi$ with dimensions of mass
\be
P_{\delta\phi}(k)=P_{\delta\phi}^{(0)}(k)\left[1+2\bar{\mu}a^2 P_{\delta\phi}^{(0)}(k)\right]
\label{psphi0}
\ee
where
\be
P_{\delta\phi}^{(0)}(k)= \frac{H^2}{2 k^3}
\ee
In the case $\bar{\mu}=0$ eqs. (\ref{psh0}) and (\ref{psphi0}) reduce to the familiar results of HUP \cite{Mukhanov:1990me}.
The perturbation from the scalar field driving inflation $\delta\phi$ gets transferred to the gravitational potential $\Phi$. The post inflation power spectrum of $\Phi$ is related to the horizon-crossing power spectrum of $\delta\phi$ via \cite{Dodelson:2003ft}
\be
P_\Phi=\frac{16\pi G}{9\epsilon}P_{\delta\phi}
\ee
where $\epsilon$ is the Hubble slow-roll parameter, defined as
\be
\epsilon\equiv\frac{d}{dt}\left(\frac{1}{H}\right)
\label{epsilon}
\ee
We note that the Hubble slow-roll parameter $\epsilon$ is equal to the first potential slow-roll parameter $\epsilon_V$, to leading order in the slow-roll approximation \cite{Liddle:2000cg,Liddle:1994dx,Liddle:1992wi,Lyth:1998xn,Baumann:2009ds}
\be
\epsilon\simeq\epsilon_V\equiv \frac{1}{16\pi G}(\frac{V'}{V})^2
\label{epsilonV}
\ee
where $V'$ is defined as the derivative of the potential $V$ with respect to the field $\phi^{(0)}$.
In the case of single-field slow-roll models of inflation for modes which are outside the horizon ($k|\tau|\ll 1$) at the end of inflation, the primordial spectra of scalar and tensor perturbations do not depend on time\footnote{We assume that non-adiabatic pressure terms are negligible.} and it is conventional to write \cite{Lyth:1998xn}
\be
P_S(k)\equiv k^3 P_\Phi(k)\equiv A_Sk^{n_s -1}
\label{psr}
\ee
\be
P_T(k)\equiv k^3 P_{h}(k)\equiv A_Tk^{n_T}
\label{pst}
\ee
where $A_S (A_T)$ is the scalar (tensor) amplitude and $n_s (n_T)$ is the scalar (tensor) spectral index. The special case with $n_s= 1$ ($n_T=0$) results in the scale-invariant spectrum.
From eqs. (\ref{psh0}) and (\ref{pst}) we obtain
\be
P_T(k)=P_T^{(0)}(k)\left[1+\frac{\bar{\mu} a^2}{8\pi G k^3} P_T^{(0)}(k)\right]
\label{pst0}
\ee
where (for $k|\tau|\ll 1$ )
\be
P_T^{(0)}(k)= \frac{8\pi G}{a^2\tau^2}=8\pi G H^2
\label{ptkmo}
\ee
It is straightforward to show at the horizon crossing time ($k=a H$)
\be
P_T(k)=P_T^{(0)}(k)\left(1+\frac{\bar{\mu}}{k}\right)
\label{ptkm}
\ee
In eq. (\ref{pst}) the tensor spectral index is defined as
\be
n_T\equiv\frac{d \ln P_T}{d \ln k}
\label{lnnT}
\ee
Also by virtue of eq. (\ref{epsilon}) we have that the logarithmic derivative of Hubble rate at horizon crossing is
\be
\frac{d\ln H}{d\ln k}=-\epsilon
\ee
Therefore using eqs. (\ref{ptkmo}), (\ref{ptkm}) and (\ref{lnnT}) we obtain that the tensor spectral index runs as
\be
n_T=-2\epsilon-\frac{\bar{\mu}}{k}
\ee
Similarly, from eq. (\ref{psphi0}) and using $P_S=k^3\frac{16\pi G}{9\epsilon}P_{\delta\phi}$ we obtain at horizon crossing time ($k=a H$)
\be
P_S(k)=P_S^{(0)}(k)\left[1+\frac{9\bar{\mu} \epsilon}{8\pi G H^2 k} P_S^{(0)}(k)\right]
\label{pss0}
\ee
where
\be
P_S^{(0)}(k)= \frac{8\pi G H^2}{9\epsilon}
\label{pseo}
\ee
It is straightforward to show that the
\be
P_S(k)=P_S^{(0)}(k)\left(1+\frac{\bar{\mu}}{k}\right)
\label{psmo}
\ee
Notice that Eqs. (\ref{pss0}) and (\ref{pseo}) have a generic form which could have been guessed even on the basis of dimensional analysis. However, here we have demonstrated in detail that these equations are not simply well motivated parametrizations based on dimensional analysis. Instead they constitute the unique and generic prediction of the inflationary power spectrum of fluctuations generated in the context of the GUP eq. (\ref{gupphi}) as derived in the context of our analysis. Thus there is no room to modify eq. (\ref{pss0}) without violating the physical principle corresponding to the GUP (\ref{gupphi}).
In eq. (\ref{psr}) the scalar spectal index is defined as
\be
n_s-1\equiv\frac{d \ln P_{\Phi}}{d \ln k}
\label{lnnS}
\ee
Now using the eq. (\ref{epsilonV}) and the Hubble slow-roll parameter \cite{Liddle:1992wi}
\be
\delta\equiv\frac{1}{H}\frac{d^2\phi^{(0)}/dt^2}{d\phi^{(0)}/dt}
\ee
we have that the logarithmic derivative of the slow-roll parameter $\epsilon$ is
\be
\frac{d\ln \epsilon}{d\ln k}=2(\epsilon+\delta)
\ee
Therefore using eqs. (\ref{pseo}), (\ref{psmo}) and (\ref{lnnS}) we obtain that the scalar spectral index runs as
\be
n_s=1-4\epsilon-2\delta-\frac{\bar{\mu}}{k}
\ee
Alternatively using the the second potential slow-roll parameter $\eta\equiv\frac{1}{8\pi G}\frac{V''}{V}$ and the relation $\delta=\epsilon-\eta$\footnote{The second slow-roll parameter $\delta$ and the second potential slow-roll parameter $\eta$ are sometimes defined as $\eta$ and $\eta_V$ respectively, so that the relation has the form $\eta=\epsilon_V-\eta_V$} \cite{Lyth:1998xn}, we obtain
\be
n_s=1-6\epsilon+2\eta-\frac{\bar{\mu}}{k}
\label{nsehm}
\ee
In the next subsection we use observational scalar spectral index data to obtain bounds on $\bar{\mu}$.
\begin{figure*}
\begin{centering}
\includegraphics[width=0.75\textwidth]{figerbf.pdf}
\par\end{centering}
\caption{The best fit forms of the scalar spectral index eq. (\ref{nslm}) (blue curve for HUP and red curve for GFC eq. (\ref{gupphi})) on the observed data (thick dots). The green and brown continuous curves correspond to $-1\sigma$ and $+1\sigma$ deviation of the parameter $\bar{\mu}$ respectively. The light green and the orange dashed curves correspond to observationally allowed range for the spectral index $n_S$ at approximatelly $2\sigma$ level. } \label{figerbf}
\end{figure*}
\section{Observational Constraints}
\label{OBCON}
The predicted form of the running spectral index eq. (\ref{nsehm}) reduces to the standard form \cite{Lyth:1994dc,Lyth:1998xn} for the HUP ($\bar{\mu}=0$) and may be used along with observational constraints of the spectral index to impose constraints on the GFC parameter $\bar{\mu}$.
The parameters that can lead to deviations from scale invariance of the spectral index are the GFC parameter $\mu$ and the slow-roll parameter $\lambda$ defined as
\be
\lambda=6\epsilon-2\eta
\ee
Thus using eq. (\ref{nsehm}), the scalar spectral index takes the form
\be
n_s=1-\lambda-\frac{\bar{\mu}}{k}
\label{nslm}
\ee
In order to impose constraints on the parameters $\lambda, \bar{\mu}$ we use constraints on the scalar spectral index of Ref. \cite{Peiris:2009wp} which are based on the angular power spectrum data of the 5 year Wilkinson Microwave Anisotropy Probe (WMAP5) Cosmic Microwave Background (CMB) temperature and polarization, the Large Scale Structure (LSS) data of the Sloan Digital Sky Survey (SDSS) data release 7 (DR7) Luminous Red Galaxy (LRG) power spectrum, and the Lyman-alpha forest (Lya) power spectrum constraints. The allowed range on $n_s$ is shown in Fig. \ref{figerbf}.
Expressing this range as a set of $N=60$ datapoints leads to constraints on the parameters $\lambda, \bar{\mu}$ through the maximum likelihood method \cite{DBLP:books/daglib/0072312}. As a first step for the construction of $\chi^2$, we consider the vector \cite{ Verde:2009tu}
\be
V^i(k_i,\lambda,\bar{\mu} )\equiv n_{s,i}^{obs}(k_i)-n_{s,i}^{th}(k_i,\lambda,\bar{\mu})
\ee\
where $n_{s,i}^{obs}(k_i)$ and $n_{s,i}^{th}(k_i,\lambda,\bar{\mu})$ are the observational and the theoretical spectral index at wavenumber $k_i$ respectively ( $i=1,...,N$ with $N$ corresponds to the number of datapoints).
Then we obtain $\chi^2$ as
\be
\chi^2=V^i F_{ij}V^j
\label{x2}
\ee
where $F_{ij}$ is the Fisher matrix \cite{10.2307/2342435} (the inverse of the covariance matrix $C_{ij}$ of the data).
The $N\times N$ covariance matrix is assumed to be of the form
\be
\left[C_{ij}\right]=\left[
\begin{array}{cccc}
\sigma_1^2 & 0 & 0 & \cdots \\
0 & \sigma_2^2 & 0& \cdots \\
0 & 0 & \cdots & \sigma_N^2 \\
\end{array}
\right] \label{eq:totalcijother}
\ee
where $\sigma_i$ denotes the $1\sigma$ error of data point $i$.
The $68.3\%$ ($1\sigma$), $95.4\%$ ($2\sigma$) and $99.7\%$ ($3\sigma$) confidence contours in the $\lambda$ and $\bar{\mu}$ parametric space are shown in Fig. \ref{cont}. The contours correspond to confidence regions obtained from the full data set (left panel), the large scales ($k<0.015$ $h/Mpc$) data (middle panel), and the small scales ($k>0.015$ $h/Mpc$) data (right panel). The $1\sigma$-$3\sigma$ contours for parameters $\lambda$ and $\bar{\mu}$ correspond to the curves $\chi^2(\lambda,\bar{\mu})=\chi^2_{min}+2.3$, $\chi^2(\lambda,\bar{\mu})=\chi^2_{min}+6.17$ and $\chi^2(\lambda,\bar{\mu})=\chi^2_{min}+9.21$ respectively. Notice (in Fig. \ref{cont}) that the large scales are most efficient in constraining the GFC parameter $\bar{\mu}$.
The largest scales that correspond to small $k$ give the largest value for the correction ${\bar \mu} /k$ of the power spectrum and the spectral index eq. (\ref{nslm}). Thus it is these scales that are more sensitive to the correction and lead to the strongest constraints as shown in Fig. \ref{cont}.
\begin{widetext}
\begin{figure*}
\includegraphics[width=0.98\textwidth]{contourfig.pdf}
\caption{The $1\sigma-3\sigma$ contours in the ($\lambda,\bar{\mu}$) parametric space. The contours describe the corresponding confidence regions obtained from the full data set (left panel), large scales ($k<0.015$ $h/Mpc$) data (middle panel), and small scales ($k>0.015$ $h/Mpc$) data (right panel). The red and green points correspond to the HUP and GUP best fits respectively.} \label{cont}
\end{figure*}
\end{widetext}
In Table \ref{table:surveys} we show the best fit values of parameters $\lambda$ and $\bar{\mu}$ with the corresponding $1\sigma$ standard deviations. In the case of HUP ($\bar{\mu}=0$) the result agrees with the current best fit values of the scalar spectral index from Planck which indicate that $\lambda\simeq 0.04$ \cite{Aghanim:2018eyx} .
\begin{center}
\begin{table}
\centering
\begin{tabular}{c c c c c c}
\hhline{======}
\multicolumn{6}{c}{ }\\
\multicolumn{6}{c}{GFC}\\
\multicolumn{6}{c}{ }\\
\hline
&&&&& \\
Parameter &Full & &Large Scales &&Small Scales \\
& Data ($1\sigma$)&&Data ($1\sigma$) &&Data ($1\sigma$)\\
&&&&& \\
\hline
&&& \\
$\bar{\mu} $ &$0.9 \pm 7.6$& &$2.1 \pm 8.1$ & &$-149 \pm 535$ \\
&$[\times 10^{-6}h/Mpc]$&&$[\times 10^{-6}h/Mpc]$& &$[\times 10^{-6}h/Mpc]$\\
&&& \\
$\lambda $ & $0.042\pm 0.0067$&&$0.039\pm 0.0095$&& $0.048\pm 0.0146$\\
\hhline{======}
\end{tabular}
\caption{\small The best fit values of parameters $\lambda$ and $\bar{\mu}$ with the corresponding $1\sigma$ standard deviations for the fitted spectral index on the observed data \cite{Peiris:2009wp}.}
\label{table:surveys}
\end{table}
\end{center}
Using eq. (\ref{param}) and the $1\sigma$ constraint on the GFC parameter $\bar{\mu}\lesssim 10^{-5}h/Mpc$ we can obtain the single GUP free parameter as
\be
\alpha=\bar{\mu}^2\lesssim 10^{-54}m^{-2}
\ee
and the corresponding maximum measurable scale as
\be
l_{max}=\bar{\mu}^{-1}\gtrsim 10^{27}m
\ee
This result is one order of magnitude larger than the present day particle horizon ($l_{max}(t_0)\simeq 10^{26}m$) given in eq. (\ref{parthort0}).
However, at about $2\sigma$ level the physically anticipated maximum measurable scale (the particle horizon scale) is included in the observationally allowed range of the maximum measurable scales. Thus, the emergence of the parameter $\mu$ in (\ref{guph}) and (\ref{gupphi}) as a consequence of a maximum measurable length associated with the cosmological particle horizon remains an observationally viable hypothesis. The parameter $\bar{\mu}$ is a fundamental parameter connected to the GUP (\ref{gupphi}) and it is not necessarily connected with the detailed physics of inflation.
Thus our analysis can not directly impose constraints on models of inflation even though there may be an indirect connection of the present day value of $l_{max}$ with the scale of inflation. Such a connection would require a time dependent form fo $l_{max}$ and is beyond the scope of the present analysis.
\section{CONCLUSIONS-DISCUSSION}
\label{Discussion}
We have derived the generalized form of the primordial power spectrum of cosmological perturbations generated during inflation due to the quantum fluctuations of scalar and tensor degrees of freedom in the context of a generalization of quantum mechanics involving a maximum measurable length scale. The existence of such a scale is motivated by the existence of the particle horizon in cosmology and would lead to a generalization of the uncertainty principle (GUP) to the form $\Delta x \Delta p \geq \frac{\hbar}{2}\frac{1}{1-\alpha\Delta x^2} $, which implies the existence of a maximum position and a minimum momentum uncertainty (infrared cutoff)\cite{Perivolaropoulos:2017rgq}. The GUP implies a generalization of the commutation relation between conjugate operators including fields and their conjugate momenta. For example we showed that the generalized field commutation (GFC) relation between a scalar field and its conjugate momentum $[\varphi(\bold{k}),\pi_{\varphi}(\bold{k}')]=i\delta(\bold{k}-\bold{k}')\frac{1}{1-\mu\varphi^2(\bold{k})}$ which is implied by the GUP leads to a modified primordial spectrum of scalar perturbation are $P_S(k)=P_S^{(0)}(k)\left(1+\frac{\bar{\mu}}{k}\right)$ with a running spectral index of the form $n_s=1-\lambda-\frac{\bar{\mu}}{k}$ with $\lambda=6\epsilon-2\eta$.
Using cosmological constraints of the scalar perturbations spectral index as a function of the scale $k$ \cite{Verde:2009tu} we imposed constraints on the parameter of the GFC $\bar{\mu}\simeq l_{max}^{-1}$. We found that $\bar{\mu}=(0.9\pm 7.6)\cdot 10^{-6} h/Mpc$ at the $1\sigma$ level which corresponds to an upper bound scale $l_{max}$ larger than the present horizon scale. At $2\sigma$ level we find that the observationally allowed range of $l_{max}$ includes the current cosmological horizon scale $l_{max}\simeq 10^{26} m $. Thus at $2\sigma$ level, the derived observational constraints on $l_{max}$ are consistent with the physically anticipated maximum measurable scale which is the current cosmological particle horizon and are much more powerful than the corresponding constraints obtained using laboratory data measuring the energy spectrum of simple quantum systems obtained in Ref. \cite{Perivolaropoulos:2017rgq}.
An interesting extension of our analysis would be the consideration of other types of GUP (e.g. the UV cutoff GUP of eq. (\ref{gup1})) and the derivation of constraints on the corresponding fundamental parameters using cosmological data and constraints on the power spectrum index.
An alternative approach in deriving the effects of a GUP on the primordial perturbation spectrum involves the generalization of the position and momentum operators as described in the Introduction, but with an ultraviolet rather than infrared cutoff, while keeping the field theoretical commutation relations unchanged \cite{Kempf:2000ac,Palma:2008tx}. According to \cite{Kempf:2000ac,Palma:2008tx}, this approach would also lead to a modification of the evolution of the field perturbation modes eq. (\ref{ifphi}) even though this equation is derived before quantization at the classical level. This approach is questionable as it is implemented at the classical level. Nevertheless, it would be of interest to extend our analysis to include such effects of modification of the classical evolution of field perturbations due to a generalization of position and momentum operators. \\
\textbf{Supplemental Material:} The Mathematica file used for the numerical analysis and for construction of the figures can be found in \cite{suppl}.\\
\section*{ACKNOWLEDGEMENTS}
This article has benefited from COST Action CA15117 (CANTATA), supported by COST (European Cooperation in Science and Technology).\\
|
1,314,259,993,640 | arxiv |
\section{Introduction}
Since the the discovery of the $2.7$~K cosmic microwave background (CMB) by
\citet{penzias65}, rapid progress in instrumental sensitivity has permitted the
detection of progressively subtler effects. The $\sim100~\mu$K temperature
anisotropies, measured to high precision by the \textit{WMAP} and \textit{Planck} satellites
\citep{wmap9params,planckxvi} and by ground-based telescopes
\citep{sievers13,story13,das13,hou14}, are $\sim10^{-5}$ fluctuations in the
2.7~K background. The degree scale primary CMB temperature anisotropies are polarized at the
$\sim1\%$ level \citep{kovac02}, with fluctuations of the order of $1~\mu$K. This
polarization, which arises as a natural consequence of the same acoustic
oscillations that source the temperature anisotropies \citep{bond84}, is
curl-free ($E$-mode) and its angular power spectrum is uniquely predicted given the temperature ($T$)
spectrum with the addition of no additional cosmological parameters. The
agreement of the $E$-mode\ spectrum with the predictions given the best fitting $T$
spectrum is a striking, independent confirmation of $\Lambda$CDM, modern cosmology's
basic paradigm \citep{pryke09,quiet12,barkats13,crites14,actpol14}.
Fainter still is the divergence-free ($B$-mode) polarization of the CMB that would
be caused by gravitational waves present in the Universe at the time of
recombination \citep{polnarev85,kamikosostebbins97,seljak97a,seljak97b}. Because the production of a
stochastic background of gravitational waves is a generic prediction of
inflationary models
\citep{grishchuk75,starobinskii79,rubakov82,fabbri83,abbott84}, the detection of the
cosmological $B$-mode\ polarization would constitute direct evidence for an era of
cosmic inflation. The amplitude of the cosmological $B$-mode\ spectrum is
parametrized by the tensor/scalar ratio $r$. An $r=0.1$ $B$-mode\ signal has
degree scale fluctuations of the order of $100~$nK, a factor $10$ smaller than the
$E$-mode\ anisotropy, a factor $10^3$ smaller than the unpolarized anisotropy, and
a factor $10^8$ smaller than the CMB monopole.
Measuring CMB polarization anisotropy is made difficult by its weakness relative
to the unpolarized anisotropy and by the additional sources of systematic error
specific to polarization measurements. Effects that convert CMB temperature
anisotropy into a false polarization signal are of particular importance. This
is especially true for $B$-mode\ measurements because both the temperature and the
expected inflationary $B$-mode\ spectra peak at similar angular scales. Detecting
and characterizing a $B$-mode\ polarization signal of this magnitude requires controlling
systematics to a level to match the experiment's unprecedentedly low
instrumental noise.
In \citet{biceptwoI}, hereafter the \textit{Results Paper}, we present a
detection of $B$-mode\ power in $>5\sigma$ excess over
the lensed-$\Lambda$CDM\ CMB expectation.
In this paper, we present extensive studies of possible systematic
contamination in this measurement using
detailed calibration data that allow us to
directly predict or place stringent upper limits on it. We find
that systematics contribute power at a level subdominant to \bicep2's
statistical noise and negligible compared to the measured $B$-mode\ spectrum.
The structure of this paper is as follows. In Section~\ref{sec:instrument} we briefly
review the aspects of the \bicep2\ instrument that are most important for an
understanding of potential systematic contamination.
In Section~\ref{sec:noisemodel} we review the noise estimation procedure and show
that our debiased auto spectrum procedure is equivalent to a cross
spectral analysis.
In Section~\ref{sec:cancellation}
we review how \bicep2's specific observing strategy modulates the contamination
from beam systematics in the signal maps and in our internal consistency
checks.
In Section~\ref{sec:deprojection} we introduce the deprojection algorithm we use
to mitigate contamination from beam imperfections.
In Section~\ref{sec:beammeas} we review external beam shape measurements.
In Section~\ref{sec:simpipeline} we detail the simulation
pipeline used to predict the level of spurious polarization due to
imperfect beam shapes.
In Section~\ref{sec:jackknives} we review \bicep2's
``jackknife'' internal consistency null tests and discuss the classes of systematics to
which each is sensitive.
In Section~\ref{sec:deprojperformance} we check that deprojection of CMB
data does indeed recover the known beam non-idealities within
uncertainties, even in the presence of realistic template noise.
In Section~\ref{sec:syslevels} we present the constraints on
many potential sources of systematic contamination.
We conclude in Section~\ref{sec:conclusions}.
In a series of four appendices we provide the formal
definition of our elliptical Gaussian beam parametrization
(Appendix~\ref{sec:beamparam}), an expanded discussion of beam
shape mismatch (Appendix~\ref{sec:beamheuristic}), the mathematical and
practical details of deprojection (Appendix~\ref{sec:deprojmath}), and a
discussion of the uncertainties in the beam mismatch simulations
(Appendix~\ref{sec:beammapsimappendix}).
\section{Instrument design and observational strategy}
\label{sec:instrument}
The \bicep2\ instrument is discussed in depth in \citet{biceptwoII}, hereafter
the \textit{Instrument Paper}. Here we highlight the details most relevant to
systematics, and in particular those that can cause false polarization. In this section we describe how
effects can arise in the antennas (beam shape and pointing), in the bolometers
(thermal mismatch), or in the readout (crosstalk). We also describe several
aspects of the observing strategy that serve to suppress these systematics
and/or to aid in identifying them.
\subsection{Instrument Design}
\label{sec:instdesign}
Each camera ``pixel'' in \bicep2's focal plane consists of two orthogonally
polarized beam-forming antennas \citep{obrient12,biceptwoV} that couple
incoming radiation to two bolometric detectors (each antenna is coupled to its
own detector). We label the members of an antenna/detector pair (which we refer
to simply as a ``detector pair'') ``A'' and ``B.'' The A and B antennas within
a pair are spatially coincident in the focal plane so they nominally observe the
same location on the sky. The time-ordered data, or ``timestreams,'' from the A
and B detectors are summed to measure the total intensity of the incoming
radiation and differenced to measure its polarized component. Therefore, any
mechanism other than the intrinsic polarization of the sky signal that produces
a differential signal in the A and B detectors will produce spurious
polarization if not properly accounted for.
The response of an antenna to incoming radiation as a function of angle is called
its beam. One class of systematics that can cause a false polarization is a
difference in the beam shape or beam center (``centroid') of the A and B
detectors. Beam shape imperfections or centroid offsets that are common to A
and B do not cause a false polarization. We observe that \bicep2's beams
exhibit significant systematic centroid mismatch within a pair, which we call
``differential pointing,'' and which we have precisely characterized.
In the time-reversed sense, each antenna illuminates the telescope aperture with
a nearly Gaussian pattern \citep{kuo08}. The illumination pattern (i.e. the
``near-field beam'') is truncated on a $26.4$~cm cold aperture stop. The
asymmetric truncation of the near-field beams will induce an expected far-field
beam asymmetry. We observe an expected dependence of detectors' beam
ellipticity on the radial position in the focal plane. Because we treat beam shapes
and centroids fully empirically, a precise understanding of the mechanisms
governing them is not required for assessing systematic contamination. A brief
review of the parametrization and measurements of \bicep2's beams is given
in Section~\ref{sec:ellipparam} and Section~\ref{sec:beammeas}, respectively. A fully
detailed treatment is given in \citet{biceptwoIV}, hereafter the \textit{Beams
Paper}.
We have designed the telescope shielding system and our observation strategy to
mitigate contamination from the ground and the Galaxy. A co-moving forebaffle
and fixed ground shield ensure that at the lowest observing elevation rays
originating from the ground must diffract twice before entering the telescope
aperture. The brightest parts of the Galaxy are always well outside of the angle
intersected by the co-moving forebaffle. The lowest galactic latitude of the
observations is $b=-39^\circ$, and we have measured that for a typical detector
$<0.1\%$ of the total integrated power is found outside of $25^\circ$ from the
main beam with the co-moving forebaffle installed. Details are in the Beams
Paper.
\bicep2's bolometers are transition edge sensors (TESs). We measure the amount
of incident radiation by tracking, as a function of time, the
amount of electrical power (presumed to be in addition to the
radiative power) required to maintain the TES at a fixed point in the
superconducting/normal transition. Thermal drifts in the focal plane thus
produce spurious signals in the detector timestreams. A false
\textit{polarization} signal arises if the responses of the A and B bolometers to
these thermal fluctuations are different. We mitigate thermal drift using
a combination of passive thermal filters and active thermal control
\citep{kaufmanthesis}. We then continuously measure any remaining thermal
fluctuations to high precision using neutron transmutation doped (NTD) germanium
thermistors located on the focal plane, allowing us to directly constrain
spurious signal from thermal drift (see Section~\ref{sec:thermalinstability}).
The bolometers are read out using multiplexed superconducting quantum
interference devices (SQUIDs) \citep{irwin02}. The use of SQUID readouts
introduces susceptibility to pickup from magnetic fields. \bicep2\ employs a
combination of high magnetic permeability and superconducting shielding to block
external magnetic fields, and its scan strategy allows for nearly perfect
filtering (``ground subtraction'') of pickup that is constant in time and a
function of telescope pointing direction, as is expected of most magnetic
fields. The multiplexing of detector timestreams \citep{dekorte03} creates
crosstalk between channels in the cryogenic and room temperature readout
hardware. Crosstalk, which we have measured in a variety of ways, can also
produce false polarization.
Using calibration data we make detailed calculations of the impact of the above
effects in Section~\ref{sec:syslevels} below.
\subsection{Observational Strategy and Data Cuts}
\label{sec:obsstrat}
The \bicep2\ telescope was situated on an azimuth/elevation mount that performed
constant elevation scans at a fixed azimuth center. The scans spanned just over
$60^\circ$ in azimuth and were re-centered on a new azimuth at approximately one
hour intervals, during which time the sky moved in azimuth by $15^\circ$. Because the sky
changed position with respect to the scan boundaries, we can differentiate
between signals that are scan synchronous (ground-fixed signal), and signals that
rotate with the sky. By subtracting the mean of all scans from each scan we
exactly remove any contaminating signal that is a function of scan position and
is constant over hour-long timescales. We refer to this filtering as ``ground
subtraction.'' This method was used successfully by \bicep1
\citep{chiang10,barkats13} and by the QUIET experiment \citep{quiet12}.
The \bicep2\ mount also allowed for a third axis of motion, the rotation of the
entire telescope about the boresight. \bicep2\ observed at four distinct
boresight orientations, or ``deck angles'': $68^\circ$, $113^\circ$, $248^\circ$, and
$293^\circ$.\footnotemark[1]\footnotetext[1]{The Instrument Paper notes that different deck angles were
used early in the 2010 season. Given their low weights in the final data set,
however, they are largely irrelevant for the present analysis.} (At $0^\circ$,
the rows of \bicep2's focal plane were roughly perpendicular to the horizon.)
Because \bicep2's detector polarization angles were all aligned in the focal
plane, reconstructing maps of Stokes $Q$ and $U$ requires a minimum of two deck
angles, optimally separated by $45^\circ$, $135^\circ$, or $225^\circ$. A valid deck
angle pair cannot be separated by $180^\circ$.
With \bicep2's
four deck angles, a map formed from one valid deck angle pair (e.g. $68^\circ$
and $113^\circ$) is complementary to the map made from the other deck angle pair
(e.g. $248^\circ$ and $293^\circ$). The deck angle pair that is complementary to any
of the four valid pairs is rotated $180^\circ$ from it.
We guard against systematics arising from unusually functioning detectors by
removing them during map making. The map making process uses data from only a
subset of the nominally functioning (i.e. optically responsive) detectors. We
implement a series of channel cuts that exclude detector pairs having certain
properties outside a pre-defined range. The details are discussed
in Section~ 13.7 of the Instrument Paper. When we have \textit{a
priori} reason to believe that a systematic will contaminate a few detectors
much more strongly than others, we can also perform a detector pair exclusion
test in which we remake maps cutting the most contamination-prone pairs.
For the test to be considered passed, we
require that the change in the resulting maps and power spectra is consistent
with the corresponding changes in systematics-free simulations.
\subsection{Summary}
\label{sec:instrsummary}
We address systematics using a combination of five general strategies. Three
strategies reduce contamination in the final maps.
\begin{enumerate}
\item \textit{Natural mitigation}: \bicep2's maps are built up from observations
made with many detectors. A systematic that varies between detector pairs will
thus statistically average down in the final map. \bicep2's maps are also built
from observations at four deck angles. Some systematics cancel with
instrument rotation. This is discussed further in Section~\ref{sec:cancellation}.
\item \textit{Time-domain filtering}: We remove atmospheric $1/f$ noise by applying
a third-order polynomial filter to the timestreams. Atmospheric noise is not a
systematic because it averages down over time and is accounted for in the noise model,
but such a filter also removes any large angular scale contamination
that might not average down. In addition, we also exactly remove any
remaining signal that is fixed with respect to the ground or scan
(as opposed to the sky) by applying the ground subtraction filter discussed
in Section~\ref{sec:obsstrat}.
\item \textit{Deprojection}: We also filter out the map modes most contaminated by
beam imperfections.
If they are ignored, differences in beam shape between the two detectors of
a detector pair will transform bright temperature anisotropies into false
polarization anisotropies. We have developed a technique
to explicitly filter the handful of map modes contaminated by several major
types of beam mismatch, and to account for this removal in power spectrum
estimation. This technique is described in Section~\ref{sec:deprojection} and in
Appendices A-D.
\end{enumerate}
\noindent Two strategies characterize the level of contamination remaining in the maps.
\begin{enumerate}
\item \textit{Jackknife maps}: Many classes of systematics produce different
contamination in different subsets of data. As part of our internal
consistency checks, we split \bicep2's data set into two halves, form $Q$ and
$U$ maps from each of the halves, difference these maps, and test whether the
resulting residuals are consistent with the difference of systematics-free,
signal-plus-noise simulations. We refer to these null tests as ``jackknives,'' and they
are discussed in more detail in Section~\ref{sec:jackknives}. We refer to the
un-differenced maps, made with the full data set, from which the science
analysis in the Results Paper derives, as the ``signal'' maps. We refer to the
angular power spectra of those maps as the signal spectra.
\item \textit{Time-domain simulations}: Our analysis pipeline generates simulated
realizations of time-ordered data (signal and noise) for each detector, which
is then processed in exactly the same manner as our real data. We have
extended our pipeline to optionally incorporate the effects of various
instrumental systematics into these simulated data, which allows us to model
their effects on the final power spectra and $r$ estimate. This pipeline is
described in Section~\ref{sec:simpipeline}, with particular regard given to
simulating beam mismatch effects. Measurements of beam mismatch are presented
in Section~\ref{sec:beammeas}. The results of these studies are presented
in Section~\ref{sec:syslevels}.
\end{enumerate}
Generally speaking, time-domain simulations allow us to model the consequences
of known systematic effects. Jackknife maps are useful for empirically
constraining contamination from both known and unknown systematics.
\section{Noise estimation}
\label{sec:noisemodel}
The Results Paper describes the construction of ``noise pseudosimulations'' that
we use to estimate the noise bias and uncertainty of our measured auto
spectrum. We construct these pseudosimulations by differencing the two maps made
from two halves of a random permutation of $17,000$ temporal subsets of the full
data set, which are long enough (approximately 1h each) to have minimal noise correlations.
We impose a constraint that each half have the same total weight. Jackknife noise
pseudosimulations are similarly constructed by randomly permuting the subsets
within a jackknife half and differencing the two maps in each half separately.
As described in the Results Paper, this noise estimation procedure has been
checked against two alternative techniques and all are found to yield equivalent results.
More formally, the $j$th random permutation splits the full data set to
define two half maps $M_{1j},M_{2j}$, which can be
recombined by summing or differencing:
\begin{eqnarray}
M & = & {\textstyle\frac{1}{2}}(M_{1j}+M_{2j}) \nonumber \\
N_{j} & = & {\textstyle\frac{1}{2}}(M_{1j}-M_{2j}) .
\end{eqnarray}
\noindent $M$ is our standard full map and is the same for any split,
while $N_j$ is the noise realization. The auto spectra of
these two maps can be written
\begin{eqnarray}
M \!\! \times \!\! M & = & {\textstyle\frac{1}{4}}
[M_{1j} \!\! \times \!\! M_{1j} + 2(M_{1j} \!\! \times \!\! M_{2j})
+ M_{2j} \!\! \times \!\! M_{2j}] \\
N_j \!\! \times \!\! N_j & = & {\textstyle\frac{1}{4}}
[M_{1j} \!\! \times \!\! M_{1j} - 2(M_{1j} \!\! \times \!\! M_{2j})
+ M_{2j} \!\! \times \!\! M_{2j}] .
\end{eqnarray}
\noindent Subtracting these gives
\begin{equation}
M \!\! \times \!\! M - N_j \!\! \times \!\! N_j = M_{1j} \!\! \times \!\! M_{2j} .
\end{equation}
\noindent We see that subtracting the auto spectrum of a single noise pseudosimulation
$N_j$ from that of the full map is identical to taking the cross-spectrum of the
two corresponding half maps.
Our actual noise bias and uncertainty estimation uses an ensemble of $N\sim500$
noise pseudosimulations. We noise debias the auto spectrum of the full map by
subtracting the mean of the auto spectra of the noise realizations,
\begin{equation}
M \!\! \times \!\! M - \left< N_j \!\! \times \!\! N_j \right> =
\left< M \!\! \times \!\! M - N_j \!\! \times \!\! N_j \right> =
\left< M_{1j} \!\! \times \!\! M_{2j} \right> .
\end{equation}
\noindent where brackets represent mean over the $j=1...N$ realizations of the ensemble.
This shows that our debiasing procedure is equivalent to computing the
mean of cross-spectra between data subsets for a large number of splits.
Similarly,
the higher order statistics (variance, skewness, etc.) of the noise
pseudosimulations are mathematically equivalent to the higher order statistics
of the cross-spectra formed between the data subsets.
One can go on to demonstrate that our procedure is also equivalent to taking
the mean of cross-spectra between many smaller data split
chunks~\citep{fowler10,lueker10,story13}.
As in any such cross-spectrum
analysis, in the limit of uncorrelated noise
between data subsets, there can be no residual noise bias from incorrect noise
modeling, as our ``noise model'' is in fact not a model, but rather a linear
combination of the data themselves.
The main effect that could possibly correlate noise among data subsets is
anisotropic turbulent structure in the atmosphere. The spatial structure of the
turbulence above the telescope averages down over time but persists on
timescales of the order or the height of the turbulent layer divided by the wind speed
at that altitude. (The timescale only becomes shorter if the turbulent structure
is not assumed to be ``frozen in'' in the frame of the moving atmosphere but
instead also evolves in time.) For a height of $5$~km and a wind speed of
$5$~m~s$^{-1}$, the timescale is $\sim15$~minutes. The data subsets we use are
approximately $1$~hr in duration, so even in the unpolarized pair sum
timestreams, the noise properties of which are dominated by turbulent atmospheric
emission, we expect very little noise correlation between data
subsets. Furthermore, because the atmosphere is almost totally unpolarized,
pair-differencing of detector pairs almost completely eliminates the noise due
to
atmospheric turbulence, leaving only the white noise of random photon arrival
times. The cancellation of unpolarized atmospheric turbulent emission is
apparent in Figure 22 of the Instrument Paper, which shows that the
instantaneous temporal power spectrum of the unfiltered pair-difference
timestreams is dominated by white noise, with a possible contribution
from atmospheric turbulence at most a few percent at the lowest frequencies.
Lastly, any remaining polarization noise correlations are further suppressed by
the time-domain filtering described in Section~\ref{sec:instrsummary}, which downweights
the lowest frequency Fourier modes along the scan direction. These are the modes
with the highest fractional contribution of atmospheric turbulence to the total
noise.
\section{Beam systematics in maps}
\label{sec:cancellation}
\begin{deluxetable*}{lccc}
\tablecolumns{4} \tablewidth{0pc} \tablecaption{Transformation of beam mismatch
leakage under rotation\label{tab:cancellation}} \tablehead{\colhead{Rotation}
& \colhead{\shortstack{Monopole \\ (e.g.\ Diff. Gain, Beamwidth)}} &
\colhead{\shortstack{Dipole \\ (e.g.\ Centroid Offset)}} &
\colhead{\shortstack{Quadrupole \\ (e.g.\ Diff. Ellipticity)}}} \startdata $45^\circ$
& $E \rightarrow B$, $B \rightarrow E$ & $E \rightarrow (E+E')/\sqrt{2}$,
$B \rightarrow (B+B')/\sqrt{2}$ & $E \rightarrow E$, $B \rightarrow B$ \\ $90^\circ$ &
$E \rightarrow -E$, $B \rightarrow -B$ & $E \rightarrow E'$, $B \rightarrow B'$ &
$E \rightarrow E$, $B \rightarrow B$ \\ $180^\circ$ & $E \rightarrow E$, $B \rightarrow B$
& $E \rightarrow -E$, $B \rightarrow -B$ & $E \rightarrow E$, $B \rightarrow B$ \enddata
\tablecomments{In a map formed by a detector pair at one set of projected orientations
on the sky, this table summarizes how the spurious signal from beam mismatch
of the given symmetry is transformed in a second map made from the same
detector pair at a second set of orientations rotated from the first by the given
angle.}
\end{deluxetable*}
We refer to any differential response to incoming unpolarized radiation between
the A and B members of a detector pair as ``beam mismatch.'' In the presence of
beam mismatch, the pair-difference signal will, in general, be non-zero even
when observing an unpolarized source. This signal directly enters polarization
maps and so must be filtered out or otherwise accounted for.
One can think of such potential contamination as the
unpolarized temperature field ``leaking'' into the pair-difference signal of a
given detector pair. We refer to this as temperature-to-polarization (\textit{T}$\rightarrow$\textit{P})
leakage. At high galactic latitude at $150$~GHz, CMB $T$ is much brighter than
foregrounds and is the dominant unpolarized signal sourcing \textit{T}$\rightarrow$\textit{P}\ leakage.
The leaked signal, $d$, that enters the pair-difference data of a given
detector pair is the convolution of the unpolarized sky with the difference of
the pair's A and B beams,
\begin{eqnarray}\label{eq:diff}
d_{T \rightarrow P} & = & T(\mathbf{\hat{n}}) \ast \left[
B_A(\mathbf{\hat{n}}) - B_B(\mathbf{\hat{n}}) \right] \nonumber \\
& \equiv & T(\mathbf{\hat{n}}) \ast B_{\delta}(\mathbf{\hat{n}})
\end{eqnarray}
\noindent where $T$ is the unpolarized temperature field, $B$ is the response of
a detector, and $\mathbf{\hat{n}}$ is the sky coordinate. If the difference
beam, $B_{\delta}$, is non-axially symmetric, then $d_{T\rightarrow P}$ is a
function of both the pointing direction of the detector pair and the projected
orientation of $B_\delta$ on the sky.
Given measurements of $T(\mathbf{\hat{n}})$ and $B_{\delta}(\mathbf{\hat{n}})$,
Equation~\ref{eq:diff} is sufficient to predict the instantaneous \textit{T}$\rightarrow$\textit{P}\ leakage
in a detector pair's pair-difference timestream as a function of that pair's
pointing direction. Predicting how this timestream level contamination manifests
in polarization maps requires
knowledge of the observing strategy. In principle, timestream level simulations of beam
mismatch that go all the way to final maps capture the map
level contamination without the need for any heuristic understanding.
Nonetheless, to gain confidence that these simulations accurately reflect
reality, it is helpful to build intuition about the way
in which different classes of beam mismatch interact with the observing strategy
to produce the map level contamination. The remainder of this section attempts
to develop this intuition.
We treat each detector pair's difference beam as the linear combination of different
components, or modes,
\begin{equation}\label{eq:modesum}
B_{\delta}(\mathbf{\hat{n}}) = \sum_{k} a_k B_{\delta k}(\mathbf{\hat{n}})
\end{equation}
\noindent Our map making procedure is a linear process. Thus,
the contamination in the final maps is a linear combination of the contamination
produced by each of these modes individually. How each mode contaminates the
final map depends upon its {\it amplitude} $a_k$, its {\it coherence} across detector
pairs in the focal plane, and its {\it symmetry} under rotation of the
instrument with respect to the sky. Amplitude sets the magnitude of the
systematic in time-ordered data, while coherence and symmetry determine the
degree of cancellation in maps made from multiple detectors and at multiple deck
angles.
\subsection{Incoherence Across the Focal Plane}
\label{sec:incor}
When combining data from multiple detector pairs to form a map, the
\textit{T}$\rightarrow$\textit{P}\ leakage from a difference beam mode that randomly varies among detector
pairs will average down if $\left< a_k \right> = 0$,
where the expectation value of the $k$th mode is over detector pairs. Since any
map pixel is only sampled by a finite number of detectors, the averaging
down is only partial. Nonetheless, because the contamination in maps made from
different subsets of detector pairs will be different, the jackknife tests
described in Section~\ref{sec:jackknives} that check for consistency between detector
pairs will fail. In general, jackknife maps have the same noise level as the
signal maps. Because a randomly varying beam systematic will contaminate the
signal map as much as a pair selection jackknife,
we expect pair selection jackknives to fail when the contamination in the signal map
is comparable to \bicep2's statistical uncertainty.
More worrisome are beam systematics that are correlated between detector pairs,
the leakage from which does not necessarily average down and can potentially
evade jackknives. \bicep2's many pair selection jackknives test for consistency
between subsets of detectors whose beam mismatch is expected to be different for
various mechanisms, e.g.\ varying by position in the focal plane or by multiplex
column.
\subsection{Symmetry}
A difference beam mode that is common to all
detector pairs (i.e. fully coherent across the focal plane) will not produce any
contamination of pair selection jackknives and will not average down when
combining data from detector pairs. However, under an azimuthal rotation of the
beam about its center, the leakage from modes of certain symmetries will change
sign. When combining data from detectors at different projected orientations on
the sky, the leakage from even fully coherent mismatch will sometimes nearly
exactly cancel in the signal maps \citep{odea07,shimon08,quiet11}. Whether or
not this occurs depends on the azimuthal symmetry of the mode. \bicep2 heavily
exploits this cancellation effect by performing deck angle rotation. When this
cancellation occurs in the signal maps, the contaminating signals in both halves
of the corresponding deck angle jackknife map are equal to each other but
opposite in sign, so that the jackknife experiences no such
cancellation. In this case, the deck angle jackknife will fail for levels of
contamination that are negligible in the signal map. Appropriate deck angle jackknifes are
thus highly sensitive probes of \textit{T}$\rightarrow$\textit{P}\ leakage from these beam systematics.
In analogy with the azimuthal symmetry of pure monopoles, dipoles, and
quadrupoles, we classify difference beam modes as having monopolar symmetry
(i.e. invariant under rotation, i.e. azimuthally symmetric), dipolar symmetry
(reversing sign under $180^\circ$ rotation), or quadrupolar symmetry (reversing
sign under $90^\circ$ rotation); other symmetries are possible for complex beam
shapes, but are not modeled here. Table~\ref{tab:cancellation} summarizes how
$d_{T \rightarrow P}$ from these modes is reconstructed as a false polarization
signal in a polarization map depending on the mode's projected orientation on the sky.
The reconstructed leakage from a monopole symmetric mode changes sign under a
$90^\circ$ rotation. (Thus, leakage to $+E$ and $+B$ at one orientation leaks to
$-E$ and $-B$ at the second; adding these two maps results in cancellation of
the leakage, and subtracting them to form a jackknife multiplies the
contamination by two.)
The reconstructed leakage from a dipole symmetric mode changes sign under a
$180^\circ$ rotation. The reconstructed leakage from a quadrupole symmetric mode
is invariant under rotation.
In a given map pixel, the cancellation of \textit{T}$\rightarrow$\textit{P}\ leakage from a monopole or
dipole symmetric mode will occur if that pixel is sampled at appropriate
orientations by the same detector pair. If the pixel is sampled by different
detector pairs, then it is only the leakage from the common component that
cancels due to the rotation. Full cancellation of the \textit{T}$\rightarrow$\textit{P}\ leakage from the monopole or dipole
symmetric modes thus requires that one of two corresponding criteria
be met: either (1) the sky coverage of any given detector pair is the same at
all deck angles, or (2) the contribution to any final map pixel is from detector
pairs with identical $a_k$.
Boresight rotation, in addition to rotating a detector pair's beam, also changes
its pointing direction. Because the instantaneous field of view of \bicep2's
focal plane is large compared to the overall map boundaries, the area of
sky mapped by a given detector is different at different deck angles. This is
illustrated in Figure~\ref{fig:perchmap}, which shows the map regions sampled by
two detector pairs, one located near the center of the focal plane and one
located near the edge. The coverage of the detector pair located near the center
of the focal plane is largely the same at different deck angles; the coverage of
the detector pair located near the edge of the focal plane is very different at
different deck angles. As a consequence, detector pairs near the center of the
focal plane (and thus the central regions of the signal maps) satisfy criterion
(1) and experience highly efficient cancellation. Detector pairs near the edge
of the focal plane still experience cancellation, but only in so far as they
satisfy criterion (2).
The remainder of this section considers in more detail the cancellation of
leakage from difference beams of different symmetries.
\begin{figure}[t]
\begin{center}
\begin{tabular}{c}
\includegraphics[width=1\columnwidth]{outlines.pdf}
\end{tabular}
\end{center}
\caption[example] { \label{fig:perchmap} Map coverage of a single \bicep2 detector
pair located (top panel) near the edge of the focal plane and (bottom
panel) near the center of the focal plane. The coverage at different deck
angles overlaps significantly for central detectors but not at all for edge
detectors. The inset (not drawn to scale) indicates the location of the
detector pair in the focal plane.}
\end{figure}
\begin{deluxetable*}{lccc}[h]
\tablecolumns{4} \tablewidth{0pt}
\tablecaption{Summary of Beam Mismatch Leakage Effects\label{tab:cancelsummary}}
\tablehead{\colhead{Symmetry:} & \colhead{Monopole} & \colhead{Dipole} & \colhead{Quadrupole}}
\startdata
\textit{Incoherent across focal plane} & & & \\
In signal map: & Averages down & Averages down & Averages down \\
In pair selection jackknife: & Potentially contaminates & Potentially contaminates & Potentially contaminates\\
\\
\textit{Coherent across focal plane} & & & \\
In signal map: & Cancels under $90^\circ$ rot.\ & Cancels under $180^\circ$ rot.\ & Does not cancel \\
In deck angle jackknife: & Contaminates in $90^\circ$ jackknife & Contaminates in $180^\circ$ jackknife & Does not contaminate \\
\enddata
\tablecomments{In a map formed by many detector pairs at multiple projected
focal plane orientations on the sky this table summarizes the
behavior of \textit{T}$\rightarrow$\textit{P}\ beam systematics having various symmetries.}
\end{deluxetable*}
\subsubsection{Monopole Symmetric Difference Beam}
\label{sec:monopole}
Examples of monopole symmetric difference beams are the difference of two
circular Gaussians with different peak heights or widths, as illustrated in the
upper and lower left panels of Figure~\ref{fig:differencebeams}. We focus on
these particular modes because the calibration measurements presented
in Section~\ref{sec:beammeas} indicate that they describe the majority of \bicep2's
monopole symmetric beam mismatch. However, we note
that the discussion here is generally applicable to \emph{any} monopole
symmetric difference beam.
If $d_{T \rightarrow P}$ for a detector pair pointed at some location on the sky
is from a monopole symmetric difference beam, it remains constant under rotation
of the difference beam. However, because the polarization sensitivity of the pair
(i.e. the interpretation of that signal under the assumption that it is not a
systematic and ``on the sky'') rotates as well, how $d_{T \rightarrow P}$ is reconstructed in the final
map does change. If the leakage is reconstructed as a false polarization with
some magnitude and direction at one orientation, rotating the detector pair
$90^\circ$ causes it to be reconstructed as false polarization with equal
magnitude but rotated $90^\circ$ from the first. Rotating a polarization vector by
$90^\circ$ simply transforms $+Q\rightarrow-Q$ and $+U\rightarrow-U$, so combining
the measurements cancels the \textit{T}$\rightarrow$\textit{P}\ leakage.
\bicep2's scan strategy did not cancel leakage from monopole symmetric
difference beams in this way. \bicep2's observation strategy included only
$180^\circ$ deck angle pair complements and no $90^\circ$ complements. In maps made
from deck angle pairs separated by $180^\circ$, the \textit{T}$\rightarrow$\textit{P}\ leakage from monopole
symmetric difference beams is reconstructed as $Q$ and $U$ identically. This
leakage adds in the signal map and cancels in the deck jackknife, making the
deck jackknives \bicep2\ forms insensitive to this type of leakage.
\bicep2's successor experiment, the \textit{Keck Array}\ \citep{sheehy10,kernasovskiy12},
consists of five \bicep2-like receivers with common boresight pointing and
oriented at $72^\circ$ increments to one another. This leads to an effective
fivefold increase in the number of deck angles and thus a certain degree of
cancellation of monopole symmetric beam mismatch in the final coadded map.
Monopole symmetric mismatch that is common between the focal planes of the two
experiments will thus be suppressed in cross-spectra taken between them.
Beginning in 2013, the \textit{Keck Array}\ added the additional four $90^\circ$ complementary
deck angles necessary to fully cancel leakage from coherent monopole symmetric
difference beams and to form deck jackknives that can test for it. Recently,
\citet{k15} demonstrated consistency between \bicep2 and \textit{Keck Array}'s auto
and cross spectra.
The predecessor experiment to \bicep2\ was \bicep1\ \citep{yoon06}. \bicep1 also
observed at the same deck angle intervals as \bicep2, but because the
polarization angles of \bicep1's detector pairs were not uniformly oriented in
the focal plane like \bicep2\ and \textit{Keck Array}'s, a monopole symmetric difference beam
common to \bicep1\ and \bicep2\ will also be suppressed in a cross-spectrum.
In summary, even though monopole symmetric beam mismatch does not contaminate
\bicep2's deck jackknives, it will (1) contaminate the \textit{Keck Array}'s $90^\circ$ deck angle
jackknife, (2) contaminate \bicep1's pair selection jackknives, and (3) not
produce fully correlated power in cross-spectra formed between any of these
experiments. Lastly, we expect the deprojection technique described
in Section~\ref{sec:deprojection} to fully remove \textit{T}$\rightarrow$\textit{P}\ leakage from gain and
beamwidth mismatch, both of which have monopole symmetric difference beams. We
empirically test this last proposition via the beam map simulations described
in Section~\ref{sec:simpipeline}.
\subsubsection{Dipole Symmetric Difference Beam}
\label{sec:dipolecancel}
An example of a difference beam having dipolar symmetry is the difference of two
identical circular Gaussians with offset centroids, as illustrated in the top middle
and right panels of Figure~\ref{fig:differencebeams}. As discussed in
Section~\ref{sec:beammeas}, this ``differential pointing'' is also \bicep2's dominant
source of \textit{T}$\rightarrow$\textit{P}\ leakage.
Dipole symmetric difference beam $d_{T\rightarrow P}$ changes sign under
a $180^\circ$ rotation. Because the rotation of the detector polarization angles
is also $180^\circ$, the reconstructed spurious polarization is equal in magnitude
and opposite in sign. Again, averaging the maps cancels the leakage; subtracting
the maps to form \bicep2's deck jackknife boosts the contamination by a factor
of two. \bicep2's set of deck angles does include $180^\circ$ complements. The
high degree of cancellation in the signal map relative to the deck jackknife
makes the deck jackknife a powerful probe of dipole symmetric contamination. This is discussed in
more detail in Section~\ref{sec:deprojperformance}.
\subsubsection{Quadrupole Symmetric Difference Beam}
\label{sec:quadrupolecancel}
An example of a quadrupole symmetric difference beam is the difference of two
elliptical Gaussians with mismatched magnitudes and/or directions of
their elongations, and
is illustrated in the bottom middle and right panels of
Figure~\ref{fig:differencebeams}.
In this case it is the difference between the pair polarization sensitivity
angle and the orientation angle of the quadrupolar pattern
which determines the nature of the leakage --- $0^\circ$ and $90^\circ$ leak
$T\rightarrow \pm E$ while $\pm 45^\circ$ leak $T\rightarrow \pm B$ \citep{shimon08}.
A quadrupole symmetric difference beam $d_{T \rightarrow P}$ changes sign under a
$90^\circ$ rotation. This is the same periodicity as a real polarized
sky signal, so no amount of boresight rotation can distinguish it from real
polarization for a single pair.
As explained in Section~\ref{sec:incor}, leakage from incoherent
beam mismatch with any symmetry averages down over pairs in the signal map
and potentially contaminates pair selection jackknives.
Coherent quadrupolar mismatch
produces leakage that is indistinguishable from real sky polarization. No
possible jackknife can test for this. For this reason, coherent quadrupole
symmetric beam mismatch is especially pernicious and must be carefully
controlled. In Section~\ref{sec:beamsim}, we accurately simulate the real
beam mismatch and correctly predict the effects of
ellipticity mismatch in our data (this being the dominant quadrupole symmetric
component).
\subsection{Summary}
Table~\ref{tab:cancelsummary} summarizes the situation. Any component of
$B_{\delta}(\mathbf{\hat{n}})$ that varies randomly across the focal plane(s)
averages down to at least some degree --- even for quadrupolar effects so long
as the orientations are random --- and in general we expect residual
contamination to be as strong in the jackknife maps as in the signal map. For a
component of $B_{\delta}(\mathbf{\hat{n}})$ that is coherent across the focal
plane(s), whether or not there is cancellation in the signal map under instrument
rotation depends on the symmetry of the component, as does the jackknife split
required to expose the systematic. A subtlety is the issue of whether each pair
self-cancels under instrument rotation. This will be true in the limit that the
focal plane field of view is small compared to the size of the map, and becomes
less true as the field of view approaches the size of the map
(as is the case for \bicep2).
\section{Deprojection technique}
\label{sec:deprojection}
As introduced in Section~{IV.F} of the Results Paper, we have developed an analysis
technique, which we call ``deprojection,'' to filter out \textit{T}$\rightarrow$\textit{P}\ leakage from beam
mismatch (and potentially other effects). Such a filter renders our analysis
immune to contamination from leading order beam imperfections.
In this section, we describe the
technique as we have implemented it for the \bicep2 analysis. Testing of the
performance of the algorithm in our case is deferred
to Section~\ref{sec:deprojperformance}.
\subsection{Beam Parametrization}
\label{sec:ellipparam}
We model $B_{\delta}(\mathbf{\hat{n}})$ as the difference of two elliptical
Gaussians. In principle, we are free to choose any model with which to
parametrize and mitigate \textit{T}$\rightarrow$\textit{P}\ leakage, but the elliptical Gaussian
parametrization is convenient.
Six parameters define an elliptical Gaussian: one for peak height, two for the
center of the ellipse (centroid), one for width, and two specifying
ellipticity. The two parameters for ellipticity are often taken as a magnitude
and orientation. We choose an alternate but equivalent basis --- plus- and
cross-ellipticity, denoted $p$ and $c$ --- that describes an ellipse oriented
either vertically/horizontally or at $\pm45^\circ$ to the horizontal axis. The
mathematical details of the parametrization are given in
Appendix~\ref{sec:beamparam}.
We model intra-pair gain mismatch (differential gain) as a difference in
Gaussian peak height; the difference beam mode for differential gain,
$B_{\delta g}(\mathbf{\hat{n}})$, is therefore just a circular
Gaussian. We model differential pointing as a centroid offset
in an $x/y$ coordinate system fixed with respect to the focal plane and centered
on the nominal beam center; the corresponding difference beam modes,
$B_{\delta x}(\mathbf{\hat{n}})$ and $B_{\delta y}(\mathbf{\hat{n}})$, are the
differences of circular Gaussians offset in either the $x$ or $y$
direction. (\bicep2's beams are $\sim0.5^\circ$~FWHM, so making
the flat sky approximation and parametrizing the ellipse on a Cartesian
coordinate system centered on each beam center is an adequate approximation.)
Beamwidth mismatch is parametrized by a difference in Gaussian width
$\sigma$. Differential plus- and cross-ellipticity are defined as the
differences of purely plus-elliptical or purely cross-elliptical Gaussians whose
orientations are defined with respect to the same focal plane fixed coordinate
system in which differential pointing is described.
Figure~\ref{fig:differencebeams} shows the differential elliptical Gaussian
modes. We consider the total difference beam to be a linear combination of
these modes in isolation, so that the sum in Equation~\ref{eq:modesum} is over
$k=\{g,x,y,\sigma,p,c\}$.
\subsection{Algorithm}
\label{sec:deprojalgorithm}
\begin{figure}
\begin{center}
\begin{tabular}{c}
\includegraphics[width=1\columnwidth]{difference_beams.pdf}
\end{tabular}
\end{center}
\caption[example] { \label{fig:differencebeams} Differences of elliptical
Gaussian beams, which we choose for $B_{\delta k}(\mathbf{\hat{n}})$. The
total difference beam, $B_{\delta}(\mathbf{\hat{n}})$, is a linear
combination of these modes. Differential gain and beamwidth produce
monopole symmetric difference beams, differential pointing a dipole
symmetric difference beam, and differential ellipticity a quadrupole
symmetric difference beam. These difference beams couple to different
derivatives of the underlying CMB temperature field. }
\end{figure}
Because the \textit{T}$\rightarrow$\textit{P}\ leakage from beam mismatch is deterministic and beam shapes
are constant in time, we can filter some of it out by
constructing leakage templates corresponding to the
differential modes of elliptical Gaussians, fitting them to our data, and subtracting
them. Such a method prevents contamination arising from the component of \bicep2's beams
described by elliptical Gaussians from entering the maps. It
requires no \textit{a priori}
knowledge of the actual magnitude of the mismatch
\citep{aikinthesis,sheehythesis}.
To second order, the individual modes of a differential elliptical Gaussian
couple to distinct linear combinations of $T(\mathbf{\hat{n}})$ and its first
and second derivatives \citep{hu03}. Appendix~\ref{sec:beamheuristic} provides a
heuristic description of this coupling. Given maps of $T(\mathbf{\hat{n}})$ and
its spatial derivatives (which we refer to as the
``template maps'') and knowledge of the pointing of each of \bicep2's detector
pairs as a function of time (as required for map making), we sample the template
maps along each detector pair's pointing trajectory to create derivative
timestreams. We use the chain
rule for derivatives to express the derivatives with respect to the
\bicep2\ focal plane coordinate system as projected on the sky at each step in
the time series. The derivative timestreams are given by
\begin{equation}
d_{i,j}(t)=\nabla^i_j \tilde{T}(t)
\end{equation}
\noindent where the $i$th spatial derivative is defined with respect to the
focal plane coordinate $j=\{x,y\}$,
\begin{equation}
\nabla^i_j \equiv \frac{\partial^i}{\partial j^i},
\end{equation}
\noindent and the tilde denotes that the template map
has been pre-convolved by a circular Gaussian beam of nominal width, $\sigma$.
We then form the linear combinations of $ d_{i,j}(t)$ that correspond to leakage
from differential elliptical Gaussian modes. We call these linear combinations
the ``leakage templates'' and denote them $d_{\delta k}(t)$ for the $k$th mode.
The net leakage corresponding to mismatched elliptical Gaussians is then a
linear combination of the leakage templates,
\begin{equation}
d_{\delta}(t) = \sum_{k=g,x,y,\sigma,p,c} a_k d_{\delta k}(t).
\end{equation}
We fit the leakage templates to a detector pair's timestreams to obtain
$a_k$ and subtract the fitted templates to filter out the leakage. We also have the
option to directly measure differential beam parameters from external
calibration data, in which case we can fix $a_k$ at its measured value and
subtract scaled leakage templates to remove leakage.
Table~\ref{tab:deprojection} summarizes the proportionality between the fit
coefficients, $a_k$, and the differential beam parameters, $\delta k$, for the six modes of
our elliptical Gaussian beam parametrization. Table~\ref{tab:deprojection} also
summarizes the linear combinations of $d_{i,j}(t)$ that
comprise the leakage templates, $d_{\delta k}(t)$. The derivation of the leakage templates and a
discussion of the practical implementation of deprojection is given in
Appendix~\ref{sec:deprojmath}.
\begin{deluxetable}{lcccc}
\tablecolumns{5} \tablewidth{0pc} \tablecaption{Deprojection templates and fit
coefficients \label{tab:deprojection}} \tablehead{\colhead{Differential Mode}
& \colhead{Symbol} & \colhead{Definition} & \colhead{Fit Coefficient} &
\colhead{Template}} \startdata Gain & $\delta g$ & $g_A-g_B$ & $\delta g$ &
$\tilde{T}$ \\ Pointing, x & $\delta x$ & $x_A-x_B$ & $\delta x $ &
$\nabla_x\tilde{T}$ \\ Pointing, y & $\delta y$ & $y_A-y_B$ & $\delta y $ &
$\nabla_y\tilde{T}$ \\ Beamwidth & $\delta\sigma$ & $\sigma_A-\sigma_B$ &
$\sigma\delta\sigma$ & $(\nabla^2_x+\nabla^2_y)\tilde{T}$ \\ Ellipticity, +
& $\delta p$ & $p_A-p_B$ & $(\sigma^2/2)\delta p$ &
$(\nabla^2_x-\nabla^2_y)\tilde{T}$ \\ Ellipticity, $\times$ & $\delta c$ &
$c_A-c_B$ & $(\sigma^2/2)\delta c$ & $2\nabla_x\nabla_y\tilde{T}$ \enddata
\tablecomments{A qualitative description of the coupling of elliptical Gaussian
beam mismatch to the first and second spatial derivatives of the nominal beam
convolved temperature field, $\tilde{T}$, is given in
Appendix~\ref{sec:beamheuristic}. The formal derivations of the templates are
given in Appendix~\ref{sec:deprojmath}}.
\end{deluxetable}
Like any filtering, deprojection removes non-leakage signal modes from the final
map, and thus affects the inferred power spectra.
In practice, only a tiny fraction of the $Q$ and $U$ maps are removed.
However, along with timestream filtering and sky cut effects,
deprojection does cause relevant mixing of $E$ into $B$.
This can be corrected for in the mean using simulations, but instead
we remove the contaminated spatial modes from the map using the
``matrix purification'' method described in Section~{VI.B} of the Results Paper.
\section{``External'' beam measurements}
\label{sec:beammeas}
\begin{figure}[t]
\begin{center}
\includegraphics[width=1\columnwidth]{abscal1.pdf}
\end{center}
\caption
{Measured absolute gain for each detector included in \bicep2's maps.
The gains
are normalized such that the median gain is one. The distribution within
the focal plane is represented schematically. Each detector pair is
depicted as a small square. The A (B) member of a detector pair is
depicted as the lower (upper) triangle of the square.}
\label{fig:abscal}
\end{figure}
\begin{figure}[]
\begin{center}
\includegraphics[width=1\columnwidth]{abscal2.pdf}
\end{center}
\caption
{Measured fractional differential gain, $(g_A-g_B)/[g_A+g_B)/2]$, for each
detector pair included in \bicep2's maps.}
\label{fig:relgain}
\end{figure}
We emphasize that the deprojection algorithm described above does not
require any external measurements of beam imperfections --- the necessary
coefficients, $a_k$, are fit for (marginalized over) from the CMB data itself.
However, checking the operation of the technique and determining the residual
contamination remaining
after deprojection of any given set of modes requires external measurements of
the actual instrument beams.
As summarized in Section~11.2 of the Instrument Paper, we have made high
signal-to-noise beam maps of each detector by rastering the telescope over
a chopped thermal source located $195$~m from the telescope's aperture --- for
full details, see the Beams Paper. In
this paper, we use these beam maps in two ways: (1) we fit elliptical Gaussians to
them and cross check the fit parameters against those derived from the
deprojection algorithm (Section~\ref{sec:beammapconsist}), and (2) we use them as direct inputs to simulations to
predict the \textit{T}$\rightarrow$\textit{P}\ leakage in the real data signal and jackknife maps while varying the set of modes
deprojected (Section~\ref{sec:beamsim}). Both offer highly robust checks that the beam
maps correspond to reality.
During beam mapping, the instrument is put in a rather different state than that
used for routine CMB observing, and the frequency spectrum of the source is not
the same as that of the CMB. Beam shapes (especially differential beam shapes)
and centroids are relatively
insensitive to changes in the source spectrum, but differential gain --- which
typically arises from the coupling of intra-pair bandpass mismatch to
the difference between the frequency spectrum of the atmosphere and the CMB ---
is not. Therefore, the differential gain measured in beam
maps is not a reliable estimate of the CMB value. Instead we estimate it by
cross-correlating single detector $T$ maps coadded over the full data set against the
\textit{Planck} 143~GHz map in a per-detector analog of the absolute gain calibration
described in Section~13.3 of the Instrument Paper. Figure~\ref{fig:abscal} shows the
results, the measured absolute gain, $g$ for each of \bicep2's
detectors. Figure~\ref{fig:relgain} shows the measured fractional differential gain for
each of \bicep2's detector pairs, $(g_A-g_B)/[(g_A+g_B)/2]$.
Differential pointing can be measured either from the beam maps or from the
per-detector cross-correlation against the Planck 143~GHz maps described in Section~11.9 of
the Instrument Paper. The results are very similar.
Figure~\ref{fig:dipolequiver} shows \bicep2's differential pointing measured
from per-detector cross-correlation,
which shows a strong coherent component across the focal plane. The coherent
part of the pattern will cancel in the final signal map and be enhanced in a
$180^\circ$ split jackknife as described in Section~\ref{sec:dipolecancel}. The
incoherent part will average down in the signal map and also potentially cause
jackknife failure.
Figure~\ref{fig:ellipquiver} shows \bicep2's measured beam
ellipticity and differential ellipticity. The differential ellipticity shows
strong pair to pair variation in angle, so we expect some averaging down of
leakage in the signal maps as described in Section~\ref{sec:quadrupolecancel}. We also
expect that jackknife tests that split the data according to pair will be
sensitive to it.
\begin{figure}
\begin{center}
\includegraphics[width=1\columnwidth]{diffpoint_beamspaper_mod.pdf}
\end{center}
\caption{Differential pointing in the \bicep2 focal plane as projected onto the
sky at deck$=90^\circ$. As drawn, the vectors originate at the nominal beam
center and point from detector B to detector A. Their magnitudes are
drawn $\times20$ for display purposes. All functioning pairs are
plotted, but grayed out vectors indicate detector pairs that are excluded from
the final maps. }
\label{fig:dipolequiver}
\end{figure}
\begin{figure}[!t]
\begin{center}
\includegraphics[width=1\columnwidth]{ellip_bicep2.pdf}
\end{center}
\caption
{Top: per-detector beam ellipticity in the \bicep2 focal plane as
projected onto the sky at deck$=90^\circ$. Ellipticity is exaggerated for
clarity. Red and blue denote A and B members of a detector pair,
respectively. All functioning pairs are plotted, but light colors indicate
detectors that are excluded from the final map. Bottom: per-pair
differential ellipticity, defined as $\sqrt{(\delta p)^2+(\delta c)^2}$. The
orientation of the ellipse indicates the orientation of the difference beam
quadrupole. Detector polarization angles are aligned with the horizontal and
vertical axes.}
\label{fig:ellipquiver}
\end{figure}
\section{Simulation pipeline}
\label{sec:simpipeline}
\bicep2's power spectrum analysis is Monte-Carlo-based, requiring simulations of
maps ``as seen'' by the experiment \citep{hivon02}. We simulate both noiseless
(signal-only) and noise-only maps. The standard simulations introduced in
Section~{V.A} of the Results Paper include only differential pointing at the
measured values shown in Figure~\ref{fig:dipolequiver}. Here we extend the
signal-only simulations to include many different types of systematics.
\subsection{Input Maps and Interpolation}
\label{sec:standardsim}
The simulation pipeline produces signal-only timestreams by
sampling an input Healpix map along individual detectors' trajectories. The
simulated timestream data is then passed through the same map maker as the real
data to produce simulated $T$, $Q$, and $U$ maps that are filtered identically to
the data. Our pipeline extensions optionally introduce many different
systematics at the timestream generation stage, allowing us to model their
effects on the final maps.
Both the main simulations and our dedicated systematics simulations use input
maps of Nside=2048.
We perform the simulations of systematic \textit{T}$\rightarrow$\textit{P}\ using the \textit{Planck} HFI 143~GHz $T$
map --- pre-smoothed by \bicep2's nominal, circular Gaussian beam as described in
Appendix~\ref{sec:practical} --- as input. We use the downgraded resolution,
Nside=512 version of the same map as the deprojection template. To predict \textit{T}$\rightarrow$\textit{P},
we set the input $Q$ and $U$ maps to zero so that any
non-zero signal in the resulting polarization maps and spectra are due entirely to
leakage. To simulate systematics that primarily leak \textit{E}$\rightarrow$\textit{B}\ we use as input maps
\texttt{synfast} generated realizations of $\Lambda$CDM\ and do not set the $Q$ and $U$
maps to zero. We difference the spectra simulated with and without the
systematic included and average over 10 realizations to estimate the
\textit{E}$\rightarrow$\textit{B}\ leakage.
All the simulations except the beam map simulations described
in Section~\ref{sec:beamsimpipeline} interpolate the input map to
timestreams using a second order Taylor expansion around the $T$, $Q$, and $U$ pixel
centers using the derivative maps that are a standard output of
\texttt{synfast}. Assuming a polarization angle and efficiency, we combine a
single detector's $T$, $Q$, and $U$ timestream into a single timestream. Using
simulated input maps of progressively higher resolution allows us to simulate
timestreams to arbitrary accuracy. Doing this, we find that using an Nside=2048
map produces negligible fractional differences from a still higher resolution
input map.
\subsection{Elliptical Gaussian Beam Convolution}
\label{sec:ellipconv}
Leakage from differential pointing is naturally handled in all the simulations
discussed above because each detector is allowed to
have its own pointing trajectory on the pre-smoothed input maps.
In studies of systematics where we wish to vary the simulated elliptical Gaussian
beam shape, we use multiple input maps which have each been pre-smoothed with
circular Gaussians of different widths. Convolution on the sphere is fast and
exact for any beam that is circularly symmetric \citep{wandelt01}.
To simulate beam widths that vary from detector
to detector, we use a perturbative method in which two or more Healpix maps of
bracketing widths are simultaneously read in and interpolated between at each
time step to approximate the timestream from a beam of intermediate width. Using
bracketing maps of closer and closer spacing allows simulation of differential
beamwidth to arbitrarily high accuracy, which we use to verify that our choice
of bracketing widths simulates leakage from beamwidth mismatch to sufficient
accuracy.
Elliptical beam convolution is handled by approximating elliptical beams as the
superposition of three or more circular sub-Gaussians of different widths, centers,
and amplitudes, the choice of which is a function of $p$, $c$ and $\sigma$ and is
predetermined from 2-d fits to elliptical Gaussians. Input maps pre-smoothed to
different circular Gaussian widths are read in and each is interpolated along
the sub-Gaussians' trajectories. The individual timestreams are then combined to
approximate the timestream from a detector with an elliptical Gaussian beam. The
amplitudes, widths, and relative centers of the sub-Gaussians are fixed, but the
orientation of the ellipse can vary along a scan trajectory according to the
beam's projected orientation. We have verified the accuracy of this approach
with special simulations using intrinsically flat input maps and explicit 2D
convolution. As with beamwidth, we can simulate elliptical beams to
arbitrarily high accuracy using superpositions of greater numbers of circular
Gaussians.
Defining ellipticity $e=(\sigma_{maj}^2-\sigma_{min}^2)/(\sigma_{maj}^2+\sigma_{min}^2)$
we find that our procedure, which uses three Gaussians, produces
timestreams from elliptical beams that are accurate for $e<0.15$.
\subsection{Arbitrary Beam Shape Convolution}
\label{sec:beamsimpipeline}
The preceding methods allow for nearly exact simulation of
elliptical Gaussian beams. We also allow for arbitrary beam shape convolution.
We perform arbitrary beam shape convolution
by forming a flat map projection of the input Healpix map, convolving this
projection directly with a 2D kernel, and interpolating off the flat map to
form simulated timestreams. We call these ``beam map simulations.'' Ordinarily,
such a brute force algorithm would be very computationally expensive when
simulating a large number of detectors observing over a long time period. For
\bicep2 we have considerably reduced the expense by exploiting the fact that
(1) the telescope's deck angle remains fixed during CMB scans,
(2) there is no sky rotation at the South Pole, and (3) \bicep2's scan pattern
is highly redundant. Thus, for a fixed deck angle,
each detector observes a given location on the sky with only one orientation,
and the convolution of the kernel with a flat sky map need only be performed
once per detector per each of the four deck angles.
This method suffers from distortion away from the center of the
projection. However, because the distortion is common to both members of a
detector pair, the difference signal is still predicted with high accuracy. We
test this by comparing the \textit{T}$\rightarrow$\textit{P}\ leakage simulated using the multiple Gaussian
approach described in Section~\ref{sec:ellipconv} (which, again, does not suffer from any flat sky
distortion effects and which we perform to high accuracy) to beam map
simulations that use as the convolution kernels elliptical Gaussians constructed
to reflect identical beam parameters. Any difference in the \textit{T}$\rightarrow$\textit{P}\ leakage from
the two methods is attributed to algorithmic limitations of the beam map
simulation procedure. We have verified that the method of flat sky beam
convolution is sufficient to accurately predict the level of leakage from all
modes of an elliptical Gaussian, both before and after deprojection. These
simulations make no assumptions of elliptical Gaussian beam structure, so this
test verifies that beam map simulations will accurately predict \textit{T}$\rightarrow$\textit{P}\ leakage
from arbitrary beam shape mismatch.
Deprojection is performed on these beam map simulations in the same way as in the
standard simulations. Therefore, the leakage templates suffer from no
corresponding distortion effects, and the main impact of projection distortion
in beam map simulations is to slightly degrade the ability of deprojection to
filter leaked power from the timestreams. This results in an artificial
``floor'' at $\simeq 10^{-5}~\mu$K$^2$ below which power
due to the mismatch of elliptical Gaussians will not deproject in a beam map
simulation. Beam map simulations thus always predict at least as much residual
contamination as is present in the real data.
We have developed the beam map simulation procedure so that we can use measured
beam maps as inputs. Because these empirical beam maps make no assumption of
elliptical Gaussian structure, their ability to reproduce the behavior of real
data spectra, both signal and jackknife, under
different deprojection options is powerful evidence against residual, unmodeled,
and undeprojected contamination from beam mismatch (see Section~\ref{sec:beamsim}).
\section{Jackknife tests}
\label{sec:jackknives}
\bicep2's most basic guard against systematics is jackknife tests
\citep{pryke09,chiang10}. As already described in Section~\ref{sec:instrsummary}, we
split the data into two subsets, form $T$, $Q$, and $U$ maps from each subset,
and difference the maps. Under the hypothesis that the observed signal is real
and ``on the sky,'' the difference map should be consistent with the
distribution of systematics-free signal-plus-noise simulations. If some or all of the observed signal is
from an instrumental systematic, then, depending on the type of hypothesized
systematic, the different halves of a split will contain either different amplitudes
or different spatial patterns of contamination. The \bicep2 jackknife tests
were discussed in Section~{VII.C} of the Results Paper. Here we review and give some
fuller details.
Different jackknives probe for different classes of systematics. Some jackknives
split the data according to the observing cycle, some according to detector pair
selection, and one according to a combination of both. Detector pair selection
jackknives are illustrated in Figure~\ref{fig:chjackfpmap}. Most systematics
will produce different contamination in the two halves for at least one of the
jackknife splits we form. The following is a description of \bicep2's
jackknives and the types of systematics that are expected to cause each to fail.
\begin{figure}
\begin{center}
\includegraphics[width=1\columnwidth]{chjack_fpmap.pdf}
\end{center}
\caption{ Map of the \bicep2\ focal plane projected onto the sky at
deck$=90^\circ$ illustrating detector pair selection jackknife splits. Dots
denote detector pairs that are coadded to form one half of the jackknife
split; X's denote detector pairs coadded to form the other half. All
functioning pairs are shown. Light gray symbols indicate pairs that are
excluded from the final map.}
\label{fig:chjackfpmap}
\end{figure}
\vspace{1em}
\begin{description}[nolistsep,itemindent=0cm,leftmargin=.5cm,before={\renewcommand\makelabel[1]{\normalfont ##1}}]
\item[\textnormal{\textit{Deck angle}}:] Splits data according to boresight orientation,
$68^\circ+113^\circ$ vs.\ $248^\circ+293^\circ$; highly sensitive to systematics that
change sign under a rotation of the instrument, such as beam mismatch with
dipolar symmetry (see Section~\ref{sec:dipolecancel}). Because of this, \bicep2's
differential pointing contaminates the deck jackknife more strongly than the
signal map (see Figure~\ref{fig:deprojperf}).
\item[\textnormal{\textit{Alternative deck}}:] Same as deck, but $68^\circ + 293^\circ$ vs.\ $113^\circ$ +
$248^\circ$; similar to the deck jack, probes contamination that varies with
boresight orientation.
\item[\textnormal{\textit{Temporal split}}:] Splits data into equal weight halves by date; sensitive to
any long-term drifting of instrument properties.
\item[\textnormal{\textit{Scan direction}}:] Splits data according to the telescope scanning direction,
left-going vs.\ right-going; sensitive to detector transfer function
mismatch. \textit{T}$\rightarrow$\textit{P}\ leakage from transfer function mismatch contaminates the scan
direction jackknife more strongly than the coadded map. Because it is the
jackknife with the lowest predicted residuals, it is also the jackknife most
sensitive to noise model errors.
\item[\textnormal{\textit{Azimuth}}:] Splits data according to interleaved 10~hr blocks of time
(phases) within the three-day observing cycle (phases B+E+H vs.\ C+F+I; see
Section~12.3 or Table 6 of the Instrument Paper for details). Because these phase
groups are offset from each other in azimuth, this jackknife probes azimuth
fixed contamination, such as would be expected from polarized ground pickup.
\item[\textnormal{\textit{Moon up/down}}:] Splits according to times when the moon is above vs.\ below
the horizon; sensitive to contamination due to the moon.
\item[\textnormal{\textit{Tile}}:] Splits data by detectors, tiles 1+3 vs.\ tiles 2+4; sensitive to
differences in detector properties, e.g. bandpass.
\item[\textnormal{\textit{Tile/deck}}:] Tiles 1/2 at deck $68^\circ$/$113^\circ$ + tiles 3/4 at deck
$248^\circ$/$293^\circ$ vs.\ tiles 1/2 at deck $248^\circ$/$293^\circ$ + tiles 3/4 at
deck $68^\circ$/$113^\circ$; sensitive to effects that are common between tiles.
(Rotating the receiver by $180^\circ$ places new tiles at a given projected
location on the sky. However, the physical orientations of tiles 1 and 2 as
installed in the focal plane are rotated $180^\circ$ from tiles 3 and 4, so that
the new tiles have the same projected orientation after rotation. Thus, the
regular deck jackknife does not directly probe for tile fixed effects that are
common among tiles.) Because \bicep2's instantaneous field of view is large
compared to the map area, this jackknife map has smaller useful coverage than
the other jackknives.
\item[\textnormal{\textit{Focal plane inner/outer}}:] Splits according to the inner 50\% of detectors
vs.\ the outer 50\% of detectors in the focal plane; sensitive to beam shape
mismatch that varies with distance from the center of the focal plane, as
would be expected of ellipticity induced by variable beam truncation in the
aperture plane.
\item[\textnormal{\textit{Tile top/bottom}}:] Splits according to top of each tile vs.\ bottom of each
tile, where the sense of top and bottom is defined with respect to the tile as
fabricated, not globally within the focal plane; sensitive to effects that
vary within an individual tile.
\item[\textnormal{\textit{Tile inner/outer}}:] Splits according to the inner 50\% vs.\ the outer 50\%
of detectors within a tile; sensitive to effects that vary within an
individual tile.
\item[\textnormal{\textit{Mux column}}:] Splits according to detector multiplexing column, even vs.\ odd;
sensitive to crosstalk contamination.
\item[\textnormal{\textit{Mux row}}:] Splits according to detector multiplexing row.
\item[\textnormal{\textit{Differential pointing best/worst}}:] Splits according to the 50\% of detector
pairs with the smallest differential pointing and the 50\% of detector pairs
with the greatest differential pointing. Like the deck and alt deck
jackknives, it is more sensitive to differential pointing contamination than
the signal maps.
\end{description}
\vspace{1em}
Table 1 of the Results Paper lists probability to exceed (PTE) values for four statistics,
computed separately for the $EE$, $BB$, and $EB$ spectra, for each of the above
14 jackknife spectra.
There are thus 168 PTE statistics but some of these are partially correlated.
There is one $BB$ or $EB$ PTE with a value $\leq 0.01$, the mux row
$BB$.
Of the 499 $\Lambda$CDM\ signal + noise simulations used in the main analysis
(which should reproduce the correlations), 306 realizations have one or more
$BB$ or $EB$ PTE $\leq 0.01$ so this is unsurprising.
The real data contain six $EE$ PTEs $\leq
0.01$.
Of the 499 simulations, 2 have 6 or more $EE$ PTEs $\leq 0.01$.
The Results Paper offers an explanation for the apparently anomalous number of
low $EE$ PTEs: because of the high signal-to-noise of \bicep2's $EE$
measurements, variation in the mean gain from detector pair to detector pair
results in failures of the detector selection jackknives shown in
Figure~\ref{fig:chjackfpmap}. The $BB$ detection is, of course, highly
significant as well, but the $EE$ signal-to-noise ratio, which is $\sim 500$ at
$\ell=100$, makes even the smallest absolute calibration difference between jackknife
halves impact the PTE, even though such absolute calibration errors do not imply
systematic contamination of the signal map.
We include this effect in 10 of the signal
simulations by multiplying each detector pair's data by the mean of its measured
absolute gain, $(g_A+g_B)/2$, shown in Figure~\ref{fig:abscal}
(see Section~\ref{sec:gainvar}). The difference of the $EE$ spectra with and without
gain variation is an estimate of the contaminating power, and is $\simeq
1\times10^{-3}~\mu$K$^2$ at $\ell=100$. Including this contaminating power
results in 9 of the 499 realizations having six or more $EE$ PTEs $\leq 0.01$.
Gain variation is not important for jackknives of the comparatively low signal-to-noise
$BB$ data.
\section{Deprojection Performance}
\label{sec:deprojperformance}
\begin{figure}[t]
\begin{center}
\begin{tabular}{c}
\includegraphics[width=1\columnwidth]{deproj_dp_resids.pdf}
\end{tabular}
\end{center}
\caption[example] { \label{fig:deprojperf} Points with error bars are
\bicep2's deck jackknife bandpowers with (red) no deprojection, (blue)
differential pointing deprojected with a \textit{WMAP7} V-band template, and (black)
differential pointing deprojected with a \textit{Planck} 143 GHz template, with error
bars taken as the standard deviation of $\Lambda$CDM\ plus instrumental noise
simulations that include \bicep2's measured differential pointing. The
solid lines are the corresponding simulated deck jackknife spectra, computed
as the mean of 50 noiseless simulations of $\Lambda$CDM\ $T$ and \bicep2's measured
differential pointing, deprojected with templates containing simulated
template noise. The dashed lines show the corresponding simulated non-jackknife, signal
$BB$ leakage. The dotted line shows a lensed $\Lambda$CDM\ + $r=0.2$ spectrum for
reference.}
\end{figure}
We characterize the performance of deprojection by specifying the residual
spurious power remaining in \bicep2's polarization power spectra after
deprojection. We split this characterization into two parts. First, we
approximate the beams as elliptical Gaussians and determine the residual
contamination from various mismatch modes using the simulations introduced
in Section~\ref{sec:ellipconv}.
This serves as a test of deprojection's fundamental limit.
Second, we use the beam map simulations described in Section~\ref{sec:beamsimpipeline}
to determine the actual residual contamination after deprojection, including
that from the portion of \bicep2's beams not described by elliptical Gaussians.
In this section, we deal only with the first characterization. The second is
described in Section~\ref{sec:beamsim}.
\subsection{Template Map Non-idealities}
We first consider how non-idealities in the deprojection template map limit the efficacy
of deprojection. By far, the most important non-ideality is simply statistical
noise in the deprojection template. We have deprojected \bicep2\ data with two different
templates --- a \textit{WMAP}7 V-band \citep{wmap10} and a \textit{Planck} HFI 143~GHz $T$ map
\citep{planckviii} --- which have different bandpasses and different noise
properties. We have also performed timestream simulations using the
measured elliptical Gaussian parameters discussed in Section~\ref{sec:beammeas}
and deprojected them with
templates containing simulated \textit{Planck} and \textit{WMAP} noise. (We describe the construction of simulated
template maps in Appendix~\ref{sec:practical}.)
These simulations predict that \bicep2's
differential pointing is by far the dominant source of contamination in the deck
jackknife, and the dominant source of contamination in the signal spectra prior
to deprojection. Furthermore,
as expected given the discussion
in Section~\ref{sec:dipolecancel} and the substantially coherent measured differential
pointing shown in Figure~\ref{fig:dipolequiver},
the deck jackknife spectrum is far more
contaminated by differential pointing than the signal spectrum.
We isolate the effect of differential pointing by simulating it separately from
other difference beam modes. Because we want to investigate the impact of
template map noise, we simulate \textit{T}$\rightarrow$\textit{P}\ leakage using noiseless realizations
of $\Lambda$CDM\ $T$ as input and noise added versions of those same maps,
downgraded to Nside=512, as the deprojection templates.
Figure~\ref{fig:deprojperf} shows the results as well as
real data for \bicep2's deck jackknife.
In these simulations, the efficacy of deprojection is
entirely determined by the level of noise in the template map.
The predicted
contamination in the signal spectrum after deprojection with either the \textit{WMAP}7 or
\textit{Planck} template (dashed blue and black lines)
is small compared to an $r=0.2$ IGW
spectrum at $\ell<150$. However, when deprojecting with the noisier \textit{WMAP}7
template, the \textit{T}$\rightarrow$\textit{P}\ leakage in the deck jackknife (solid blue line) is measurable and well
predicted by simulation.
Because the deck jackknife has much greater contamination than the
signal spectrum, it is a highly stringent test of contamination.
Our accurate prediction of residual contamination in the deck jackknife
is strong evidence against significant unmodeled leakage in the signal maps. In
\bicep2's main results, deprojection is performed with a \textit{Planck} 143 GHz
template, and \textit{T}$\rightarrow$\textit{P}\ leakage from differential pointing is negligible and
unmeasurable in even the deck jackknife.
We note that bandpass differences between \bicep2\ and the deprojection template
are not important. The \textit{WMAP} V-band template is centered at 60~GHz while the
\textit{Planck} template is centered at 143~GHz, much closer to \bicep2's central
frequency. In principle, the \textit{T}$\rightarrow$\textit{P}\ leakage at different frequencies is not the
same because of unpolarized foregrounds with non-CMB-like spectral dependencies.
The agreement of the data points and the solid lines in
Figure~\ref{fig:deprojperf} indicates that for even significant bandpass
differences, undeprojected leakage from foregrounds not present in the
deprojection templates is negligible. Foregrounds present in the deprojection
template that are fainter in \bicep2's band would be a source of unmodeled
template noise, which Figure~\ref{fig:deprojperf} indicates is also not an issue. We
have also simulated adding point sources to the template map that are not
present in simulated \bicep2\ maps, and this also has a negligible effect on
deprojection.
\subsection{Consistency With Beam Maps}
\label{sec:beammapconsist}
We confirm that deprojection filters contamination consistent with our measured
difference beams by comparing the differential beam parameters implied by the
deprojection fit coefficients of \bicep2's real data (calculated according to
Table~\ref{tab:deprojection}) to the independent measurements of the same
parameters described in Section~\ref{sec:beammeas}. Figure~\ref{fig:dpvsmeas} shows
the correlation of the deprojection derived differential beam parameters with the
beam-map-derived differential beam parameters. (Note that $\delta x$ and $\delta y$
are measured from beam maps, not from correlation of per-detector $T$ maps with
\textit{Planck} maps, as in Figure~\ref{fig:dipolequiver}, and are thus fully independent
of the deprojection coefficients, if somewhat lower signal-to-noise.)
The uncertainties of the beam-map-derived parameters are somewhat difficult to accurately estimate. However, the
scatter in the observed relation is consistent with the scatter on the
deprojection coefficients predicted from signal-plus-noise simulations,
indicating that noise in the CMB data dominates the scatter in
Figure~\ref{fig:dpvsmeas}.
\begin{figure}[t]
\begin{center}
\begin{tabular}{c}
\includegraphics[width=1\columnwidth]{real_dpcoeff_vs_meas.pdf}
\end{tabular}
\end{center}
\caption[example] { \label{fig:dpvsmeas} Differential beam parameters measured
from far-field beam maps (horizontal axis) and from template regression as
used in deprojection (vertical axis), shown as 2D histograms over detector
pairs. (Differential gains are determined from cross-correlation of
individual detector T maps with \textit{Planck}.) The solid line has a slope of 1 and
a $y$-intercept of 0. The dashed line has slope of 1 but has been offset
vertically by the bias in the recovered deprojection coefficients predicted
from simulation. The scatter and bias in the observed relation is broadly
consistent with that predicted from signal-plus-noise
simulations. }
\end{figure}
The significant bias visible in the plus-ellipticity deprojection coefficient
results from the inherent $TE$ correlation in $\Lambda$CDM\ cosmology, which ensures
some correlation between the true CMB polarization signal and the deprojection
templates. This bias does not impair the filtering of \textit{T}$\rightarrow$\textit{P}\ leakage, but it does
cause additional filtering of cosmological $E$-mode s (the effect on $B$-mode s is
negligible). The effect on both $E$-mode s and $B$-mode s is automatically accounted
for in the filter/beam suppression factors derived from simulations that apply
the same choice of deprojection (see Section~ VI.C of the Results Paper). We have
verified that the bias arises from
$\Lambda$CDM\ $TE$ correlation by observing that the bias disappears in simulations
with no $TE$ correlation.
Given good agreement between measured differential beam parameters and those
inferred from deprojection, we can choose to either deproject a given
differential mode or to subtract the contamination expected given our direct
measurements. Differential gain can, in principle, have a significant time
variable component, so we choose to deproject it. (We perform the deprojection
regression on approximately 9~hr chunks of data; see
Appendix~\ref{sec:practical} for details.)
Differential pointing is
measured with high signal-to-noise in beam maps and is expected to be constant
in time, but because it is \bicep2's largest source of \textit{T}$\rightarrow$\textit{P}\ leakage we
conservatively choose to deproject it to avoid any residual leakage arising from
noise in the calibration measurements. Since differential ellipticity
deprojection preferentially filters our $TE$ and $EE$ spectra, we choose to fix the
deprojection coefficients to the beam-map-derived values and subtract the
scaled deprojection templates from the data, rather than fitting the templates.
In the results of the beam map simulations described in Section~\ref{sec:beamsim},
we find this to be empirically equivalent to deprojecting ellipticity
The simulation of \bicep2's best-fit elliptical Gaussian beam shapes that include
all six differential modes demonstrates that \textit{T}$\rightarrow$\textit{P}\ leakage from pure
elliptical Gaussian mismatch can be cleaned to the $r\sim1\times10^{-4}$ level
with deprojection using a template with \textit{Planck} 143~GHz noise levels. At this level,
the component of \bicep2's beam mismatch not fit by the difference of elliptical
Gaussians is the dominant source of \textit{T}$\rightarrow$\textit{P}\ leakage.
\section{Systematics error budget}
\label{sec:syslevels}
\begin{figure*}[t]
\begin{center}
\begin{tabular}{c}
\includegraphics[width=2\columnwidth]{beammaps.pdf}
\end{tabular}
\end{center}
\caption[example] {\label{fig:beammap} Composite beam map for a representative
detector pair, showing the A and B beams. ($1~\mbox{dB}=10\log_{10}$.) The
coordinate system is centered on the mean pair centroid. The expected
crosstalk feature with $\sim-25$~dB amplitude is visible in the A detector on
the horizontal axis at a distance of $\sim+1.7^\circ$ from the beam
center. The first Airy ring is visible at a radius of $\sim1^\circ$. The
difference beam (not shown) is dominated by a dipole structure.}
\end{figure*}
Jackknife tests fail when the magnitude of contamination exceeds
the noise in the jackknife maps, which, in general, is comparable
to the noise in the signal maps. If the
contamination is uncorrelated in the two halves of the jackknife split, then
jackknife tests can place upper limits on possible contamination only as low as
the level of \bicep2's statistical uncertainty. We therefore rely on the
jackknife tests described in Section~\ref{sec:jackknives} primarily as a safeguard
against unknown and unmodeled systematics. Using special calibration data, we
constrain known possible systematics to much lower levels.
In this section, we use a few approaches to either directly determine or
place upper limits on the contamination from a given systematic. First, where a
systematic is strong enough relative to the sensitivity of calibration data, we
directly determine the $BB$ spectrum of the expected spurious signal using
simulations of the effect. Many of the calibration measurements are described in
the Instrument Paper, and are similar to those described in
\citet{takahashi10}. Second, where calibration data exist but the systematic
effect in question is not large enough to directly measure, we place upper
limits on the contamination given the sensitivity of the calibration
data. Third, in the absence of robust calibration data, we can determine the
level of a hypothesized systematic that would show an observable
effect in \bicep2's signal and jackknife spectra and set an upper limit this
way.
We quote the level of contamination from individual sources of systematics by
assigning a characteristic tensor/scalar ratio to the spurious $BB$\ power they
generate. We compute this characteristic $r$-value using the ``direct
likelihood'' method developed in \citet{barkats13} and used in Section~ XI.A of the
Results Paper. We first compute a weighted sum of bandpowers of the predicted
spurious signal. We use signal/variance weighting, with a signal equal to an
$r=0.1$ IGW spectrum and variance equal to the variance of bandpowers from
simulations of lensed-$\Lambda$CDM\ signal + instrument noise. The ratio of this
weighted sum (multiplied by $0.1$) to the identically weighted sum of a pure
$r=0.1$ IGW spectrum is the characteristic $r$-value of the contamination. (In
practice, the choice of fiducial $r$ makes no difference.)
Because this procedure strongly de-weights bandpowers above $\ell\simeq120$,
contamination at these multipoles will not be reflected in the quoted
$r$-values. Nonetheless, we plot systematics spectra
at $\ell<350$ and can therefore verify that systematics are small at all
scales presented in the main analysis.
\subsection{Undeprojected Residual Beam Mismatch}
\label{sec:beamsim}
In Section~\ref{sec:deprojection}, we described the deprojection algorithm that allows
us to filter out \textit{T}$\rightarrow$\textit{P}\ leakage from mismatched beams and in Section~\ref{sec:deprojperformance}
demonstrated that for idealized elliptical beams the residual
\textit{T}$\rightarrow$\textit{P}\ contamination after deprojection using the \textit{Planck} 143~GHz template map is
well below \bicep2's noise. Because deprojection, as parametrized, filters only
power corresponding to the modes of the difference of elliptical Gaussians,
the portion of any detector pair's difference beam not described by this model
creates residual, undeprojected contamination.
As described in Section~\ref{sec:beammeas} and in the Beams Paper, we have obtained high
signal-to-noise beam maps of every \bicep2\ detector. The source was observed
3 times each at 4 deck angles to produce a total of 12 individual
$8^\circ\times8^\circ$ beam maps for each detector. The central region of each
detector's beam map, at radius $r\leq1.2^\circ$, is covered by all 12
observations. This area contains $97\%$ of the total integrated beam power. The
regions of the beams at $r>1.2^\circ$ are not fully covered by observations
at a single deck angle. Beam map pixels at $r\leq3^\circ$ from the beam
center are observed at a minimum of two deck angles. Regions of the beam map at
$r>3^\circ$ are generally observed at a single deck angle.
We combine the available observations to form one composite beam map for
each detector. We do this in two ways: (1) we median filter the full beam maps
to produce $8^\circ\times8^\circ$ maps, and (2) we set to zero the portion of the
beam maps at $r>1.2^\circ$ and mean filter the observations. We refer to these
two composite maps as the (1) extended and (2) main beams. The median filter is
necessary for the outer regions of the beam maps because of
artifacts in some of the observations. The extended composite beam map for a
representative detector pair is shown in Figure~\ref{fig:beammap}.
We apply a gain mismatch by normalizing each detector's beam map to reflect the
differential gain measurements shown in Figure~\ref{fig:relgain}. (We normalize
each detector pair's two beam maps such that the mean gain is one and the
intra-pair ratio of the mean of the square root of the azimuthally averaged beam
window functions, $B_{\ell}$, in the multipole range $100<\ell<300$ equals the
ratio of the measured absolute gains. This procedure ensures we apply the
differential gain in simulation to the same multipole range as in which it was
measured.)
\subsubsection{Undeprojected Residual in Signal Maps}
We use these beam maps as inputs to the beam map simulation algorithm described
in Section~\ref{sec:beamsimpipeline} and compare the resulting \textit{T}$\rightarrow$\textit{P}\ leakage to the
real data.
The left panel of Figure~\ref{fig:beammapjacks} shows the predicted
$BB$ contamination from the main beam map simulations using different
deprojection options. The colored bands indicate the $\pm1\sigma$ uncertainty of
the predicted leakage, which is set by noise in the beam maps and the absolute
gain measurement uncertainty. The top right
panel of Figure~\ref{fig:beammapjacks} shows the change in simulated leakage
when applying deprojection as colored bands, as well as the observed change in
\bicep2's bandpowers under different choices of deprojection as points. Again,
the shaded bands indicate the $\pm1\sigma$ uncertainty of the beam map
simulations. The error bars on the points are the standard deviation of
bandpower differences from simulations that include lensed-$\Lambda$CDM\ signal and
instrumental noise. The details of the estimation of the leakage uncertainty
are given in Appendix~\ref{sec:beammapsimappendix}.
Figure~\ref{fig:beammapjacks} shows that deprojection of differential
pointing is absolutely necessary and differential gain is also
important.
Differential ellipticity is a smaller effect.
Once these deprojections are in effect the residual contamination
is seen to be very small.
\begin{figure*}[t]
\begin{center}
\begin{tabular}{c}
\includegraphics[width=2\columnwidth]{beammap_sim_jacks.pdf}
\end{tabular}
\end{center}
\caption[example] {\label{fig:beammapjacks} Left panel: $BB$ contamination
predicted from beam map simulations of \bicep2's measured main beams
(temperature only simulations using the Planck HFI 143 GHz $T$ map convolved
with measured, per-detector beam maps). The shaded bands indicate the
$1\sigma$ uncertainty of the contamination given the sensitivity of the beam
maps and gain mismatch measurements. The colors correspond to different choices of deprojection: (1) no
deprojection; (2) deprojection of differential pointing ($\delta x + \delta y$); (3)
deprojection of differential pointing and differential gain ($\delta x +
\delta y + \delta g$); and (4)
deprojection of differential pointing, differential gain, and differential
ellipticity ($\delta x + \delta y + \delta g + \delta p + \delta
c$). Right panels: Changes in bandpowers with
different deprojection choices for: (top) the $BB$ signal
spectrum, (middle) the $BB$ focal plane inner/outer jackknife, and
(bottom) the $TE$ tile
inner/outer jackknife.
The solid lines and shaded bands again indicate the mean and $1\sigma$
uncertainty of the predicted leakage given the sensitivity of the beam
maps and gain mismatch measurements. The points with error bars are the real data bandpower differences,
with error bars computed as the rms of \bicep2's standard,
lensed-$\Lambda$CDM\ signal plus instrumental noise simulation set. }
\end{figure*}
\subsubsection{Undeprojected Residual in Jackknife Maps}
We can further compare the action of deprojection on real data and simulations
for each of the jackknives described in Section~\ref{sec:jackknives}. As described
in Section~\ref{sec:cancellation}, some beam systematics undergo considerably less averaging
down due to incoherence across the focal plane and cancellation
due to instrument rotation in certain jackknifes.
In these cases, we can
investigate the behavior of deprojection in circumstances where it has to
``work harder'' than in the full signal map.
Examples of this are the focal plane inner/outer and tile inner/outer splits
illustrated in Figure~\ref{fig:chjackfpmap}. As seen in
Figure~\ref{fig:ellipquiver} \bicep2's beam ellipticities exhibit a dependence
on distance from the focal plane center
while the differential ellipticity is
strongest around the edges of individual tiles. The right center panel of
Figure~\ref{fig:beammapjacks} shows that the focal plane inner/outer jackknife
has a
much stronger response in $BB$ to differential ellipticity deprojection than the full signal
map, and that the degree of this response matches between real data and
simulations. Likewise, the bottom right panel shows that the tile inner/outer
jackknife responds as predicted in the $TE$ spectrum.
In general the simulated jackknife residuals match the real data for all the
jackknives, under all deprojection combinations. Even \textit{without}
differential ellipticity deprojection, the contamination of the $BB$ spectrum is
negligible, yet we still detect it in the jackknives that ought to be most
sensitive to it. These many additional tests build confidence that we
understand \textit{T}$\rightarrow$\textit{P}\ leakage from beam shape mismatch to an accuracy and precision
surpassing that required by our error budget.
\subsubsection{Undeprojected Residual Correction}
The simulated main beam leakage with differential gain, pointing and ellipticity
deprojection is robustly measured and is shown as the black line in the left
panel of Figure~\ref{fig:beammapjacks}. This leakage corresponds to
$r=1.1\times10^{-3}$ and is subtracted from the $BB$ bandpowers prior to fitting
$r$ in Section~ VIII.A of the Results Paper. The expected main beam contamination in the final
results is therefore zero.
The extended beam simulations are noisier than the main beam simulations and
the median filter makes statistics derived from them less robust. The predicted
extended beam leakage is consistent with zero and we adopt its $1\sigma$
uncertainty as the upper limit of possible remaining \textit{T}$\rightarrow$\textit{P}\ leakage from beam
shape mismatch after the main beam residual correction. Because
the extended beam maps include the main beam, this upper limit includes the
uncertainty of the residual leakage correction. Moreover, because the extended
beam maps include crosstalk beams, it includes \textit{T}$\rightarrow$\textit{P}\ leakage from multiplexer
crosstalk. The upper limit is shown in Figure~\ref{fig:sysfigs} and indicates
that beam shape mismatch contributes \textit{T}$\rightarrow$\textit{P}\ leakage corresponding to $r<3.0\times10^{-3}$.
\subsection{Further Consideration of Gain Mismatch}
\label{sec:gain}
Deprojection filters \textit{T}$\rightarrow$\textit{P}\ leakage from gain mismatch with such effectiveness
that the subtle choices of multipole ranges and normalization constants
described in Section~\ref{sec:beamsim} make
virtually no difference. We have simulated up to three times the level of
measured relative gain mismatch and found no change in the predicted residual
contamination after deprojection. The ``extended beam'' upper limit in
Figure~\ref{fig:sysfigs} includes contamination from gain mismatch.
Other lines of evidence against systematic contamination by gain mismatch are
the cross-spectrum of \bicep2 and \bicep1, and the passing of
jackknives. Because we calibrate the relative response of our detectors hourly
by executing elevation dips (the ``el nods'' described in Section~ 12.4 of the Instrument Paper)
and observing the large response from changing atmospheric loading, we expect
that a gain miscalibration will primarily be the result of intra-pair bandpass
mismatch coupling to differences between the color spectrum of the CMB and the
atmosphere at the South Pole \citep{bierman11}. As discussed
in Section~\ref{sec:monopole}, it is only coherent gain mismatch that will evade
\bicep2's jackknife tests. Because \bicep1 and \bicep2's bandpasses are
physically defined in very different ways (horn and mesh filter vs. antenna and
lumped-element filter), we expect that a coherent mismatch will not correlate
between the two experiments.
Also as discussed in Section~\ref{sec:monopole},
while coherent differential gain will not
contaminate jackknives in \bicep2, it will (1) contaminate jackknives in
\bicep1, and (2) contaminate the signal map \emph{differently} in \bicep1
because of the different layout of polarization angles. Thus, while the power
spectrum of contamination from uniform gain mismatch could be similar in \bicep1
and \bicep2\, correlation would not be expected. Thus, the
consistency of the \bicep1$\times$\bicep2\ cross-spectrum with the \bicep2\ auto
spectrum as presented in Figure~7 of the Results Paper is evidence against residual uniform
gain mismatch. Incoherent gain mismatch is still expected to contaminate pair
selection jackknives.
Additionally, a coherent gain mismatch
common to \bicep2\ and to the \textit{Keck Array}\ will not produce correlated power. In
\textit{Keck Array}\ maps from 2013 and after, a coherent gain mismatch will fully cancel
in signal maps as
well as contaminate $90^\circ$ split jackknives.
\subsection{Gain Variation}
\label{sec:gainvar}
\bicep2 applies a single absolute calibration to the final coadded
maps. Because the map coverage region is not the same for all detector pairs,
a variation of mean gain from pair to pair will cause
\textit{E}$\rightarrow$\textit{B}\ leakage, even if the intra-pair differential gain is zero.
The matrix-based map purification discussed in Section~ VI.B of the Results Paper
ensures that the \textit{E}$\rightarrow$\textit{B}\ leakage from timestream filtering and map apodization is
at a level corresponding to $r<10^{-4}$. We simulate \textit{E}$\rightarrow$\textit{B}\ leakage from gain
variation within the focal plane by applying the per-pair mean of the
absolute gains shown in Figure~\ref{fig:abscal} to
signal-only simulations containing unlensed $\Lambda$CDM\ $E$-mode power. The accuracy
of this procedure is limited by the
matrix purification, and so is an upper limit. We find that gain
variation within the focal plane contributes \textit{E}$\rightarrow$\textit{B}\ leakage
corresponding to $r < 5.3\times 10^{-5}$.
A separate issue is temporal gain variation. Temporal gain variation per
se is not a systematic (the full season coadded maps are calibrated against
\textit{Planck}), nor is static differential gain or temporal variation of
differential gain on timescales longer than $\sim9$~hr, the timescale over
which we perform the fit of the deprojection templates to the data (see
Appendix~\ref{sec:practical}). However, temporal variation of the differential gain on
timescales shorter than the 9~hr deprojection timescale will produce \textit{T}$\rightarrow$\textit{P}\ leakage that does not
fully deproject. We therefore reject $\sim1$~hr blocks of data (``scansets'')
from channels
whose el-nod-derived gains change by more than $30\%$ as measured at the
beginning and end of the scanset, and reject pairs whose ratio of
gains changes by more than $10\%$ (see Section~13.7 of the Instrument Paper). As
discussed in Appendix~\ref{sec:practical} and in Section~ IV.F of the Results Paper, the
deprojection timescale was chosen as a compromise between the desire for
robustness against temporal variation of sytematics (favoring shorter
timescales) and the desire to minimize unnecessary filtering of signal (favoring
longer timescales). We note that before this timescale was settled upon, the
power spectrum results using deprojection performed on hour-long timescales were
consistent with those using deprojection performed on 9~hr long timscales,
modulo the additional \textit{E}$\rightarrow$\textit{B}\ variance resulting from the more aggressive
filtering. We regard this as empirical evidence against the existence of leakage
from unknown temporal differential gain variation at relevant levels.
\subsection{Crosstalk}
\label{sec:crosstalk}
The leakage from the forms of crosstalk we expect in \bicep2
\citep{brevikthesis} is easily incorporated into our simulation pipeline. The
simulated timestreams from a detector's multiplexer neighbors are simply multiplied by
constants reflecting the level of crosstalk and added to the detector's
timestream. We have measured levels of crosstalk between channels in a variety
of ways: first, we use cosmic ray hits on the focal plane that induce large
changes in the signal on a given detector to map out the relative pickup on
other detectors, yielding a non-symmetric $N_{\mathrm{detector}}\times N_{\mathrm{detector}}$
matrix of crosstalk coefficients. Second, we determine crosstalk coefficients
for nearby detectors by cross-correlating individual detector CMB $T$ maps offset
by the known angular distance to the channel next to it in the multiplexing
ordering scheme. Third, we determine the crosstalk from nearby detectors by
fitting a 2D Gaussian to the secondary beams that are seen with high
signal-to-noise in individual detector beam maps, such as in
Figure~\ref{fig:beammap}. Fourth, we have extended the deprojection algorithm to
remove crosstalk leakage. (We do this by fitting the differential gain leakage
template of a detector pair's two multiplex neighbors to that detector pair;
averaging the coefficients over three years is a measure of the crosstalk
coefficient.)
Crosstalk \textit{T}$\rightarrow$\textit{P}\ leakage will partially cancel when coadding detectors within a
multiplexing column into maps \citep{sheehythesis}. Detector pairs that are
nearest neighbors in the multiplexing ordering scheme are second nearest
neighbors in the physical layout of the focal plane. Incrementing in multiplex
samples, every other detector along a physical row is first sampled, then the
interleaved detectors are sampled in the reverse physical direction. As a
consequence, the crosstalk induced \textit{T}$\rightarrow$\textit{P}\ leakage on two physically adjacent
detector pairs is equal in magnitude but opposite in sign when they are pointed
at the same location on the sky.
If the crosstalk coefficient is the same for all
detectors, the leakage almost fully cancels when adding the data from adjacent
pairs to form maps. Timestream simulations confirm that the cancellation
mechanism is highly effective as long as the average crosstalk is similar
between channels upstream in the multiplexing order and channels
downstream in the multiplexing order, which our various measurements indicate
is the case.
Direct simulations of crosstalk coefficients derived using all of the methods
described above predict similarly small levels of \textit{T}$\rightarrow$\textit{P}\ leakage. The least noisy
and most easily interpretable of these methods is the fitting to beam
maps. Simulation of the measured per-pair crosstalk, which has a median of
$\simeq0.3\%$, predicts leakage corresponding to $r \simeq 3.2\times10^{-3}$,
which we adopt as the predicted systematic contamination.
\subsection{Ghost Beams}
\label{sec:ghostbeams}
In addition to the $8^\circ\times8^\circ$ beam maps described
in Section~\ref{sec:beamsim}, we map the beam response out to radius $\sim20^\circ$ using a
bright non-thermal source. We observe a
small-amplitude ``ghost beam'' for each detector located at the position of that
detector's beam reflected across the boresight axis. These likely result from
reflections in the optics chain. The peak amplitude of these ghost beams is
small, $\simeq4\times10^{-4}$ relative to the main beams. We can detect them
because in the large beam maps we use a brighter microwave source than that used
for the main and extended beam maps. We fit and measure
the differential elliptical Gaussian parameters of these ghost beams, which are
generally different from those of the corresponding main beam. We directly
simulate the \textit{T}$\rightarrow$\textit{P}\ leakage from mismatched ghost beams by using the elliptical
Gaussian convolution approach described in Section~\ref{sec:ellipconv}. The predicted
leakage is small, corresponding to $r\simeq7.2\times10^{-6}$.
\subsection{Polarization Angles}
We divide the residual \textit{E}$\rightarrow$\textit{B}\ leakage from polarization angle miscalibration into
a systematic (fully coherent) and a random component.
\subsubsection{Systematic Polarization Angle Error}
\label{sec:syspol}
Section~VIII.B of the Results Paper describes \bicep2's
procedure for self-calibrating the overall polarization angle orientation of the
detectors, which removes the systematic component. Summarizing this procedure, we
find that, prior to calibration, the high-$\ell$ $TB$ and $EB$ spectra are
consistent with a coherent $-1.1^\circ$ polarization angle error and apply an equal and
opposite rotation to the polarization maps prior to computing power
spectra. Doing so filters the \textit{E}$\rightarrow$\textit{B}\ leakage from a systematic polarization angle
error. Given
the analytic expression for \textit{E}$\rightarrow$\textit{B}\ leakage found in Equation~5 of \citet{keating13} and
assuming a $\Lambda$CDM\ $EE$ spectrum, we then calculate the maximum possible residual
miscalibration by determining the coherent rotation at which \bicep2's $TB$ and $EB$
spectra would show significant non-zero power. For coherent angle errors $\ll 1$~rad,
the contamination of $BB$ scales quadratically with the angle error, while
contamination of $TB$ and $EB$ scales linearly. The $TB$ and $EB$ spectra are
therefore contaminated more strongly
than $BB$ for a given angle error, and the resulting
upper limit on $BB$ contamination is negligible. Given the sensitivity of
\bicep2's $TB$ and $EB$ spectra, a systematic
polarization angle rotation of $0.20^\circ$ produces a failure of the $EB$
$\chi$ statistic, which tests for coherently positive or negative residuals
(and is defined in Equation~8 of the Results paper) in $95\%$ of
\bicep2's signal-plus-noise simulations, which
limits the possible \textit{E}$\rightarrow$\textit{B}\ leakage to $r<4.0\times10^{-4}$.
\subsubsection{Random Polarization Angle Error}
Self-calibration removes the leakage from a systematic error in polarization
angle, but errors in relative polarization angles between detectors still
produce additional \textit{E}$\rightarrow$\textit{B}\ leakage. We measure detector polarization angles with a
dielectric sheet calibrator~\citep{takahashi10}. The measurements are described
in detail in Section~ 11.4 of the Instrument Paper. After accounting for the $-1.1^\circ$
systematic rotation, the difference between the measured and nominal
polarization angles is small. The distribution is approximately Gaussian, with
an rms of $0.14^\circ$. We have estimated that the precision of these measurements is
$\sim0.2^\circ$~\citep{aikinthesis}, so the relative misalignment of individual
detector polarization angles is not measured with high significance.
Leakage from random polarization angle errors
is easily simulated. We simply assume one set of per-detector polarization
angles in the simulation stage and use another in the map making stage. The
resulting $Q$ and $U$ maps contain $\Lambda$CDM\ $E$-mode power that has been rotated into
$B$-mode\ power. The difference between spectra estimated from maps made with the
``wrong'' polarization angles and the known, ``correct'' polarization angles is
the \textit{E}$\rightarrow$\textit{B}\ leakage from polarization angle error. Simulation of the
\textit{E}$\rightarrow$\textit{B}\ leakage from a random polarization angle error of $0.2^\circ$ rms predicts
contamination corresponding to $r\lesssim5.0\times10^{-5}$.
\subsection{Cross-polar Response}
In addition to the miscalibration of the polarization angle, there can be
higher order cross-polar response terms in the beam that give rise to
\textit{E}$\rightarrow$\textit{B}\ leakage. If a pair that is analyzed assuming it responds only to $Q$
polarization actually has some response to $U$, there will be polarization
rotation leading to \textit{E}$\rightarrow$\textit{B}\ leakage. Any $U$ response that is uniform across the
beam (i.e. a monopole) will be included in the polarization angle calibration.
However, non-uniform $U$ response cannot be fully removed by polarization angle
calibration. We have measured the beam patterns of response to $Q$ and $U$ for
\bicep2\ with a rotating polarized source in the far field.
After pair-differencing, the response to $U$ at any location is
$\lesssim0.8\%$ of the response to $Q$ at the peak.
The corresponding \textit{E}$\rightarrow$\textit{B}\ leakage is
at the level of $r\lesssim10^{-3}$. Boresight rotation and variation among
detectors would further reduce this effect, so this level is a conservative
upper limit. See the Beams Paper for further details.
\subsection{Thermal Instability}
\label{sec:thermalinstability}
As mentioned in Section~\ref{sec:instdesign} fluctuations
in the focal plane temperature will produce spurious polarization if the
response of detectors to a change in focal plane temperature differs within a
detector pair. Two NTD thermistors are located on the \bicep2\ focal plane and
are read out at the same rate as the detectors. Using the heaters on the focal
plane normally used for active thermal control, we directly measured
individual detector's response to a change in focal plane temperature by varying
the focal plane temperature over a range $\sim10$~mK.
We estimate the leakage from thermal fluctuations by replacing each detector's
timestream with the measured focal plane temperature multiplied by that
detector's thermal response. We then make maps exactly as for the real data, but
using these timestreams substituted for the real ones. (We co-add focal plane
temperature data from only the 2011 and 2012 seasons because the NTD thermistor
biases rendered focal plane temperature data from 2010 noisy.) This procedure
naturally includes the mitigation of leakage from ground subtraction
and averaging down across detectors. The polarization maps produced in
this manner are consistent with the readout noise of the NTD thermistors. The
$BB$ spectrum of these maps is a directly measured upper limit on leakage from
thermal drift in the focal plane and corresponds to $r<1.2\times10^{-5}$.
\subsection{Detector Transfer Functions}
The temporal response of \bicep2's detectors is very fast. Typical detector time
constants are $\tau\sim1$~ms, with a few detectors having $\tau=5-8$~ms. We
therefore do not deconvolve the detector response function from the time ordered
data. In principle, a mismatch of detector response results in \textit{T}$\rightarrow$\textit{P}\ leakage.
We have measured each detector's transfer function (the Fourier transform of the
temporal response) with high signal-to-noise --- details are in Section~10.6 of the
Instrument Paper. We simulate a conservative upper limit of the \textit{T}$\rightarrow$\textit{P}\ leakage
from transfer function mismatch by convolving simulated detector timestreams
with exponential response functions having $10\times$ the measured time
constants. This simulation predicts \textit{T}$\rightarrow$\textit{P}\ leakage at a level corresponding to
$r\simeq5.7\times10^{-4}$. We also verify from simulation that the scan
direction jackknife is contaminated by transfer function mismatch more strongly
than the signal spectra, making it a robust additional check against
leakage.
\subsection{Magnetic Pickup}
We do not attempt to directly simulate magnetic pickup in the SQUIDs. We
nonetheless have multiple lines of evidence disfavoring significant magnetic
contamination. First, and most importantly, the magnetic shielding employed by
\bicep2\ was found to suppress magnetic pickup from external sources by a factor
$\sim10^6$ (see Section~5.3 of the Instrument Paper for details).
Second, ground subtraction filtering exactly removes any signal that is constant
over hour-long timescales and fixed with respect to the telescope scan or to the
ground. Ground subtraction is performed on individual channels separately, so
detector to detector differences in magnetic response are accounted for. Such a
scan- or ground-fixed signal includes the Earth's magnetic field or any other
magnetic field that is fixed with respect to the telescope superstructure. Only
the slight misalignment in azimuth of the time-ordered points of corresponding
telescope scans would cause imperfect subtraction. Simulation of this effect
shows that ground subtraction filters scan synchronous signals to below
$r\lesssim1\times10^{-8}$.
Third, magnetic pickup varies from channel to channel (especially across
multiplexing columns) due to differences in shielding environment.
Investigation of channels with deliberately severed TES-SQUID links (dark
SQUIDS) and special calibration runs with detectors in the normal state do show
column-to-column differences in magnetic pickup. We therefore expect the Mux
column jackknife to be a moderately sensitive probe of magnetic
contamination. (Within a given column, multiplex neighbors show highly
correlated magnetic sensitivity, so that the Mux row jackknife, which splits the
data within a column by interleaved channels, is a very weak test of magnetic
pickup.)
Lastly, we note that \bicep1\ did not use SQUID readouts and thus had no
sensitivity to magnetic fields, so that a \bicep1$\times$\bicep2
cross-spectrum will show no contamination from magnetic pickup.
\subsection{Electromagnetic Interference}
\begin{figure}
\begin{center}
\includegraphics[width=1\columnwidth]{satcomcut.pdf}
\end{center}
\caption{EMI sensitivity parameter for all detector pairs included in \bicep2's
maps. The parameter is proportional to the contribution of possible EMI from a
given detector pair to \bicep2's polarization power spectra. The dashed line
indicates the cut threshold used in constructing the EMI sensitivity pair
exclusion test.}
\label{fig:satcomcut}
\end{figure}
After the completion of \bicep2\ observations, analysis of non-ground-subtracted
galactic maps revealed clear contamination during specific temporal periods
resulting from a satellite transmitter operating at the Amundsen-Scott South
Pole research station. The transmitter uplink operates in the S-band (2~GHz) for
approximately 7~hr per sidereal day. Details are given in Section~11.8 of the
Instrument
Paper.
A few factors limit the impact of electromagnetic interference (EMI) in CMB
observations. One, because the satellite uplink schedule is locked to sidereal
time, it so happened that the CMB field mapped by \bicep2\ was always in the opposite direction from
the transmitter when it was on. The opposite was true for the galactic field mapped
by \bicep2. Second, a small subset of detector pairs shows much stronger
differential sensitivity to the EMI than others, indicating that pair selection
jackknives should fail if EMI were contributing significant power. Third, the
EMI in these few pairs is visible in raw pair-difference timestreams prior to
ground subtraction so that we can study its phenomenology. We find that the EMI
is fixed in azimuth and constant in time, and that ground subtraction filters it
nearly perfectly. The only contamination that occurs is when the transmitter
turns on or off during a scanset, causing imperfect ground subtraction.
We have performed a pair exclusion test to test for EMI.
Figure~\ref{fig:satcomcut} shows an EMI sensitivity parameter for all
\bicep2\ detector pairs used in the final maps. The parameter is proportional to
the square of the level of EMI pickup seen in non-ground-subtracted
pair-difference maps of the Galactic field. The
contribution of a given pair's contamination to power spectra should thus
scale with this parameter.
Because a few pairs dominate possible contamination from EMI, a pair exclusion
test is more sensitive than a jackknife that splits based on the EMI
statistic. We performed the test by re-coadding the real data maps and 50
signal-plus-noise simulations, excluding the 18 most sensitive pairs. The change
in the resulting $BB$ bandpowers, $\Delta C_{\ell}^{BB} = C_{\ell}^{BB,\mathrm{cut}} -
C_{\ell}^{BB}$, is consistent with the slightly altered noise and weighting of
the new map and is statistically insignificant. In fact the first five bandpowers
shift slightly up when making the cut.
The $\chi$ statistic (also used in Section~\ref{sec:syspol})
has PTE~$\simeq0.05$. The ratio of the mean EMI sensitivity parameter
with and without the pair cut implies that the cut reduces any EMI contamination
that is present by $\simeq90\%$ in the polarization power spectra. Taking this
into account, we place an upper limit on contamination from EMI at
$r\lesssim1.7\times10^{-3}$ with 95\% confidence. At contamination greater than
this, the pair exclusion test we performed would have a $95\%$ likelihood of
producing statistically significant negative $\Delta C_{\ell}^{BB}$'s.
We also note that while the coupling of EMI is not fully known, it did not
appear to involve the detector antennas, most likely coupling directly to the TES
islands. The mechanism should therefore manifest differently or not at all in
\bicep1, which did not use TES technology. The contaminating power will
therefore not be present in a \bicep1$\times$\bicep2\ cross-spectrum.
\begin{deluxetable}{lc}[t]
\tablecolumns{2} \tablewidth{0pc} \tablecaption{Instrumental systematics
\label{tab:sysfinal}} \tablehead{\colhead{Systematic}
& \colhead{Characteristic $r$}}
\startdata
Crosstalk & $\simeq3.2\times10^{-3}$ \\
Beams (including gain mismatch) & $<3.0\times10^{-3}$ \\
EMI & $\lesssim1.7\times10^{-3}$ \\
Cross polar response & $\lesssim 10^{-3}$ \\
Eetector transfer functions & $<5.7\times10^{-4}$ \\
Systematic polarization angle error & $<4.0\times10^{-4}$ \\
Gain variation \textit{E}$\rightarrow$\textit{B} & $<5.3\times10^{-5}$ \\
Random polarization angle error & $\lesssim5.0\times10^{-5}$ \\
Thermal fluctuations & $<1.2\times10^{-5}$ \\
Ghost beams & $\simeq7.2\times10^{-6}$ \\
Scan synchronous contamination & $\lesssim1\times10^{-8}$ \\
\hline \\
Total & $\simeq(3.2-6.5)\times10^{-3}$
\enddata
\tablecomments{The comparable characteristic $r$ of \bicep2's statistical
uncertainty is $r=3.1\times10^{-2}$. }
\end{deluxetable}
\subsection{Overall Achieved Systematics Level}
\begin{figure}
\begin{center}
\includegraphics[width=1\columnwidth]{systlevels.pdf}
\end{center}
\caption{Estimated levels of systematics as compared to a
lensed-$\Lambda$CDM+$r=0.2$ spectrum. Solid lines indicate expected contamination.
Dashed lines indicate upper limits. All systematics are comparable to or
smaller than the extended beam mismatch upper limit, which is smaller than
\bicep2's statistical uncertainty. }
\label{fig:sysfigs}
\end{figure}
Figure~\ref{fig:sysfigs} shows the expected $BB$ contamination or upper limits
on contamination from the individual sources of systematics considered in this
section. Table~\ref{tab:sysfinal} summarizes the $r$-values quoted for them
above. To obtain a final constraint on instrumental systematics, we add the
values for systematics quoted as predicted values and add the upper limits in
quadrature --- or the upper limit divided by two for those that are $95\%$
confidence upper limits (EMI and systematic polarization angle error).
We note
that magnetic pickup, which is expected to be negligible, has not been included.
The total contamination and its uncertainty is almost completely dominated by
the expected \textit{T}$\rightarrow$\textit{P}\ leakage from crosstalk and the uncertainty on the residual
\textit{T}$\rightarrow$\textit{P}\ leakage from beam shape mismatch.
\section{Conclusions}
\label{sec:conclusions}
\bicep2's systematic control demonstrates the validity of our
experimental approach for high signal-to-noise CMB polarimetry. Instrumental
systematics are a negligible contributor to \bicep2's $BB$ auto spectrum. They
are also small compared to \bicep2's instrumental noise. Deprojection mitigates
\textit{T}$\rightarrow$\textit{P}\ leakage from beam mismatch to a level at least sufficient to detect
$r\simeq0.003$. Other calibration measurements allow us to limit additional
systematics to $r\lesssim0.006$. For comparison, \citet{bischoff13} claimed
a limit on instrumental systematics of $r<0.01$.
Cosmic variance limited measurements of CMB polarization promise to constrain
$\Lambda$CDM\ cosmology with greater precision than temperature data alone
\citep{snowmass,galli14}. They will require control of systematics similar to
\bicep2's. Leakage from constant fractional beam mismatch (not including
differential gain) scales with beam size, with leakage peaking near
the beam scale. Telescopes with larger apertures than \bicep2\ but similar
fractional mismatch will have \textit{T}$\rightarrow$\textit{P}\ leakage that peaks at correspondingly
smaller angular scales. We expect deprojection to be equally effective at
filtering \textit{T}$\rightarrow$\textit{P}\ leakage at higher multipoles, as the \textit{Planck} temperature maps
should be sufficiently low-noise. Even if they were not, it is possible to
deproject using an experiment's own temperature map, which should always have
sufficient sensitivity at the angular scales required. We do not take this
approach out of simplicity to avoid complications involved with map filtering.
In summary, \bicep2's proven systematics control demonstrates the power of
scanning, small aperture, pair differencing bolometric polarimeters that do not
use rotating half-wave plates or other polarization modulators. Our
experimental approach will maintain its usefulness as we continue characterizing
the detected $B$-mode\ signal.
\acknowledgements
\bicep2 was supported by the U.S. National Science Foundation under
grants ANT-0742818 and ANT-1044978 (Caltech/Harvard) and ANT-0742592
and ANT-1110087 (Chicago/Minnesota). The development of antenna-coupled
detector technology was supported by the JPL Research and Technology
Development Fund and grants 06-ARPA206-0040 and 10-SAT10-0017
from the NASA APRA and SAT programs. The development and testing of
focal planes were supported by the Gordon and Betty Moore Foundation
at Caltech. Readout electronics were supported by a Canada Foundation
for Innovation grant to UBC. The receiver development was supported
in part by a grant from the W. M. Keck Foundation. Partial support for
C. Sheehy was also provided by the
Kavli Institute for Cosmological Physics at the University of Chicago through
grant NSF PHY-1125897 and an endowment from the Kavli Foundation and its founder
Fred Kavli.
The computations in this paper were run on the Odyssey cluster
supported by the FAS Science Division Research Computing Group at
Harvard University. Tireless administrative support was provided by
Irene Coyle and Kathy Deniston.
We thank the staff of the U.S. Antarctic Program and in particular
the South Pole Station without whose help this research would not have been possible.
We thank all those who have contributed past efforts to the {\sc Bicep} /\textit{Keck Array}\
series of experiments, including the {\sc Bicep1}\ and \textit{Keck Array}\ teams,
as well as our colleagues on the {\sc Spider}\ team with whom
we coordinated receiver and detector development efforts at Caltech.
We dedicate this paper to the memory of Andrew Lange, whom we sorely miss.
|
1,314,259,993,641 | arxiv | \section{Introduction}
The formation of planets is inherently entangled with the formation and evolution of their natal protoplanetary disks. The physical conditions and chemical composition at the onset of planet formation determine the properties of the resulting planetary systems \citep[e.g.,][]{Armitage2011,Oberg2011b, Morbidelli2016}. The key question is then: at what stage of disk evolution do planets start to form?
The protoplanetary disks around Class II pre-main sequence stars were considered to be the starting point of the planet formation process. However, submillimeter surveys of those disks reveal that the mass reservoir available in Class II disks is much lower than the masses needed to explain the formation of the observed exoplanetary systems \citep{Andrews2007,Greaves2010, Williams2012,Najita2014, Manara2018}. Structures observed in the disks \citep[e.g.,][]{Marel2013a,Andrews2018, Long2019} are also evidence that planet formation is already underway in the Class II phase. One of the possible solutions to this conundrum is to move the onset of planet formation to the younger disks surrounding Class 0/I protostars \citep[< 0.5 Myr;][]{Dunham2014a}, where more material is available \citep{Andrews2007a, Greaves2011, Ansdell2017}.
There is other evidence for early planet formation. The distribution of different types of meteorites in our solar system can be explained by the formation of Jupiter's core in the first million years of the solar system \citep{Kruijer2014}. There is also evidence for dust growth in the earliest stages of disk formation \citep[e.g.,][]{Jorgensen2007,Kwon2009,Miotello2014,Harsono2018,Hsieh2019a}.
Another indication is provided by young sources with structures suggestive of ongoing planet formation \citep[e.g.,][]{,ALMA2015, Sheehan2018}. These all suggest that planet formation starts early in disks surrounding much younger Class 0 and Class I protostars rather than in Class II disks.
This begs the question: what is the amount of material available for planet formation in Class 0/I disks? \cite{Greaves2011}, in a study of a small sample of Class 0 disks known at the time, found that 20 - 2000 M$_\oplus$ dust mass is available in Class 0 disks around low-mass stars; they concluded that this is sufficient to form the most massive exoplanet systems found to date. An analysis of a sample of Class I disks in Taurus \citep{Andrews2013} combined with information about the occurrence of exoplanets led \cite{Najita2014} to conclude that Class I disks can explain the population of exoplanetary systems, contrary to Class II disks in the same region. A study of a larger sample of young disks extending to Class 0 protostars is needed to put constraints on planet formation timescales and efficiency.
In the first complete survey of Class 0/I protostars in a single cloud, Perseus, \cite{Tychoniec2018a} used Very Large Array (VLA) 9 mm observations at 75 au resolution to show that there is a declining trend in the dust masses from Class 0 to Class I disks. The median masses for the Class 0 and Class I phase ($\sim$250 and $\sim$100 M$_\oplus$, respectively) are explained by a significant fraction of the dust being converted into larger bodies already in the Class 0 phase. Moreover, they compared the results for Class 0/I disks in Perseus with Atacama Large Millimeter/submillimeter Array (ALMA) observations of several Class II regions which have mean dust masses in the range 5-15 M$_\oplus$ \citep{Ansdell2017}. This suggests that dust masses in the Class 0/I disks are an order of magnitude higher that those for Class II disks. Note, however, that the adopted dust mass absorption coefficient ($\kappa_{\nu}$ - dust opacity) varies in these studies.
ALMA observations by \citet{Tobin2020} in the Orion Molecular Cloud, based on the largest sample of protostars observed in a single region at sub-millimeter wavelengths (379 detections), found much lower mean dust masses for Class 0 and I disks than those in Perseus, 26 and 15 M$_\oplus$, respectively. Very low Class I mean dust disk mass (4 M$_\oplus$) were also reported in the Ophiuchus star-forming region
\citep{Williams2019}. Also in this case, different opacities assumed in those studies could contribute to the difference between the median masses measured.
Comparison of the VLA observations for Perseus
with other embedded disks surveys using ALMA is difficult because of
the different wavelength range of observations. The VLA obsercvations
at 9 mm can have a significant free-free emission contribution, which
could result in overestimating the actual flux coming from the dust
\citep[e.g.,][]{Choi2009}; although \cite{Tychoniec2018a} applied the correction for a free-free contribution using information from the C-band (4.1 and 6.4 cm) flux densities. On the other hand, the dust emission at those long
wavelengths is less likely to be optically thick than that
in the ALMA wavelength range \citep{Dunham2014b}. The way forward is to use observations of
young disks with VLA and ALMA in the same star-forming region, offering
a direct comparison of dust disk masses and determining if the difference in observing wavelengths can be the reason for the described
differences. Therefore, in this work we present ALMA observations of protostars in Perseus and compare them with our previous VLA data.
This work aims to compare the solid masses of the embedded (Class 0/I) disks with the masses of the exoplanetary systems observed to date to ultimately infer an efficiency of the planet formation. In Section 2, we describe the ALMA observations and data analysis. In Section 3, the integrated fluxes at 1 mm and 9 mm are compared, and dust masses are calculated based on those fluxes and then compared with other young and more mature dust disks observed with ALMA. In Section 4, we put the inferred masses in the context of known exoplanetary systems masses and planet formation models.
\section{Observations and analysis}
\subsection{Observations}
In this paper we analyze ALMA Band 6 continuum observations of 44 protostars in the Perseus molecular cloud. The data were obtained in September 2018 with a Cycle 5 program (2017.1.01693.S, PI: T. Hsieh). The absolute flux and bandpass calibrator was J0237+2848, and the phase calibrator was J0336+3218. Continuum images and spectral lines observed in this project are presented in \cite{Hsieh2019}. The continuum bandwidth was $\sim 1.85\,$GHz centered at 267.99 GHz (1.1 mm). The absolute flux calibration uncertainty is on the order of $\sim$ 30\%. The synthesized beam of the continuum observations in natural weighting is $0\farcs45 \times 0\farcs30$. The average spatial resolution of observations ($0\farcs38$) corresponds to 110 au (diameter) at the distance to Perseus (293$\pm 22$ pc; \citealt{Ortiz-Leon2018}). The typical rms value of the continuum images is $\sim$ 0.1 mJy beam$^{-1}$.
Additional data on 8 disks were obtained in a Cycle 5 program (2017.1.01078.S, PI: D. Segura-Cox). The continuum bandwidth was centered at 233.51 GHz (1.3 mm) with a total bandwidth of 2 GHz. The average synthesized beam of 0\farcs41$\times$0\farcs28 provides spatial resolution corresponding to 100 au at the distance of Perseus. The rms value of the images is $\sim$ 0.05 mJy beam$^{-1}$. The absolute flux and bandpass calibrator was J0510+1800 and the phase calibrator was J0336+3218. The accuracy of the flux calibration is on the order of $\sim$ 10\%. The measurement sets were self-calibrated and cleaned with the robust parameter 0.5.
We also use the flux densities of 25 disks published in \cite{Tobin2018} which were observed at 1.3 mm with a resolution of $0\farcs27\times0\farcs16$ and sensitivity of 0.14 mJy beam$^{-1}$. The flux and disk masses in \cite{Tobin2018} are measured using a Gaussian fit in the image domain to the compact component in the system without subtraction of an envelope component. Altogether we compile a sample of 77 Class 0 and Class I disks in Perseus observed with ALMA. In the following, when referring to ALMA data, we use 1 mm observations for short, but anywhere the wavelength is used to calculate properties of the source (e.g., disk mass) the exact value of the observed wavelength is used.
The VLA observations come from the VLA Nascent Disks and Multiplicity Survey (VANDAM) \citep{Tobin2015a, Tobin2016, Tychoniec2018}. The sample for the VANDAM survey was prepared based on unbiased infrared and submillimeter surveys of protostars in Perseus \citep{Enoch2009,Evans2009, Sadavoy2014}. Fluxes at 9.1 mm (Ka-band), obtained with 0.25\arcsec resolution from 100 Class 0 and I disks (including upper limits) were reported in \cite{Tobin2016}. \cite{Tychoniec2018} applied a correction for free-free emission, based on C-band (4.1 and 6.4 cm) observations. In that work, all sources with a Ka-band spectral index suggestive of emission not coming from dust ($\alpha \ll 2$) were marked as upper limits, and we use the same criteria here. We use the 9 mm fluxes corrected for the free-free emission for further analysis.
\subsection{Gaussian fitting}
Pre-ALMA surveys of embedded sources have found that disk masses are
typically only a small fraction of the total envelope mass in the
Class 0 phase (1-10\%), becoming more prominent as the system evolves
in the Class I phase (up to 60\%, e.g., \citealt{Jorgensen2009}). In the
much smaller ALMA beam, the envelope contamination is reduced
\citep[e.g.,][]{Crapsi2008}, but still needs to be corrected for \citep{Tobin2020}. Here both components, disk and envelope, are represented by Gaussians.
The CASA \citep{McMullin2007} v. 5.4.0 $\it imfit$ task was used to fit Gaussian profiles to the sources. After providing the initial guess, all parameters: position, flux, and shape of the Gaussian, were set free during the fit. All sources were inspected by eye to assess the number of necessary Gaussian components. In case of a single source without a noticeable contribution from the envelope, a single compact Gaussian with the size of the synthesized beam was provided as input to the $\it imfit$ task (Fig. \ref{fig:fit_examples}a). In cases where a contribution of the envelope by eye was significant, an additional broad Gaussian with a size of 3\arcsec was added to the initial guess parameters of the fitting (Fig. \ref{fig:fit_examples}b).
In two cases (Per-emb-4 and SVS13A2) it was necessary to fix the size of the Gaussian to the synthesized beam size for the fit to converge (Fig. \ref{fig:fit_examples}c). Two binary systems with separations below our resolution (Per-emb-2 and Per-emb-5) are treated as single systems with a common disk.
The flux density of the compact Gaussian is assumed to be that of the embedded disk. It is called `disk' here, even though no evidence for a Keplerian rotation pattern yet exists. We report the measured fluxes of the embedded disks in Table \ref{tab:table1}.
The broad component is used only to force the $\it imfit$ task to not fit extended emission without constraining the compact Gaussian size which would in turn overestimate the flux of the compact emission. It was necessary to add an envelope component to 31 sources out of 51 targets, specifically 20 Class 0 and 11 Class I sources. We assessed remaining 6 Class 0 sources and 20 Class I sources as not having significant contribution from their envelope.
The envelope flux remaining after subtracting the model of the disk component is measured as the flux in the area of the size of the FWHM of the disk in the residual image. This ratio of the envelope residual flux to the disk flux ranges from less than 1\% to usually below 30\%. In one case the source is dominated by the envelope emission (Per-emb-51; Fig. \ref{fig:fit_examples}d), but after the envelope component subtraction the residual flux is only $\sim\ 6\%$ of that of the disk (Table \ref{tab:table1}). The two sources with high values of the ratio - Per-emb-22-B and Per-emb-27B are heavily affected by the nearby binary component so the value is not reliable.
We stress that the remaining envelope fraction is not incorporated in the flux density of the disk component, and it is presented to show that fitting the envelope component is needed to exclude the contamination of the envelope from the disk. Fig. \ref{fig:fit_examples} shows that residuals are significantly reduced after removing the envelope contribution, and that without fitting the extended component some of this emission could contaminate the flux coming from the disk.
\section{Dust disk masses}
\subsection{Comparison of the integrated fluxes between 1 and 9 mm }
Measurements of the continuum emission at different wavelengths allow us to analyze the properties of the emitting material. First, it is important to verify that the fluxes at both 1 mm (ALMA) and 9 mm (VLA) have their origin in the same physical process. This is to confirm that the correction for contamination of the VLA observations by the free-free emission is accurate. In order to do so, we investigate the correlation between the flux densities at both wavelengths and the spectral index of the emission for each source.
The flux densities from the ALMA 1.1-1.3 mm observations are presented in Table \ref{tab:table1}. In Fig. \ref{fig:fluxes_ALMA_VLA} we compare the measurements with the VLA 9.1 mm observations \citep{Tobin2016, Tychoniec2018a}. There is a clear correlation with a close-to-linear slope (1.15 $\pm$ 0.10) obtained with the {\it lmfit} Python function. The fitting was performed excluding upper limits. The value of the slope indicates that all sources have a similar spectral index between ALMA and VLA wavelengths. Thus, the mechanism responsible for emission at both wavelength ranges is the same. Since it is generally accepted that the ALMA 1 mm emission is dominated by dust thermal emission, we can conclude that this is also the case for the 9 mm VLA observations. The sources with resolved emission at 9 mm can be modelled successfully with the disk \citep{Segura-Cox2016}.
\begin{figure}[h]
\centering
\includegraphics[width=0.95\linewidth,trim={0cm 0cm 0cm 2cm},clip]{plots/flux_VLA_ALMA_v3.png}
\caption{ALMA (1.1 and 1.3 mm) integrated fluxes plotted against VLA integrated fluxes at 9.1 mm. A total of 77 sources are plotted but only the 62 sources that are not upper limits are included in fitting the linear function. The best fit to the data is shown with the green line and has a slope of 1.15 $\pm$ 0.1. The Class 0 sources are shown in red and Class I sources in blue. Crosses mark sources that are unresolved binaries. }
\label{fig:fluxes_ALMA_VLA}
\end{figure}
The nature of this emission can be also investigated with the value of the spectral index between the two wavelengths. In the Rayleigh-Jeans approximation, the flux density $F_\nu$ changes with frequency $F_\nu \sim \nu^{\alpha}$, where $\alpha$ is the spectral index. The dust emissivity index $\beta$ defines the dependence of the dust opacity on the frequency $\kappa_{\nu} \sim \nu^{\beta}$. From the observed spectral index, the emissivity index can be derived accordingly:
\begin{equation}
\label{eq:beta}
\beta = (\alpha - 2)(1+\Delta)\,
\end{equation}
where $\Delta$ is the ratio of optically thick to optically thin emission \citep{Beckwith1990,Lommen2007}. It is often assumed that emission at millimeter wavelengths is optically thin, in which case $\Delta$ = 0 and then $\beta = \alpha - 2$.
The spectral index between the ALMA and VLA fluxes is calculated as
\begin{equation}
\label{eq:spectral_index}
\alpha_{\rm VLA/ALMA}=\frac{\ln (F_{\rm 1\ mm}/ F_{\rm 9\ mm})}{\rm ln(\rm 9\ mm/1\ mm)}.
\end{equation}
The mean spectral index obtained between ALMA and VLA is $\alpha \sim 2.4$, with a standard deviation for the sample of 0.5. This indicates that $\beta$ = 0.4 $\pm$ 0.5, which is lower than the typical ISM value, i.e. $\sim $1.8 for small grains \citep{Draine2006}. The index is also lower than 1, the value typically used for protoplanetary disks, specifically in our previous study of embedded disks in Perseus \citep{Tychoniec2018a}. If emission is optically thin, the low value of $\beta$ can point to dust growth as is commonly seen in Class II disks \citep[e.g.,][]{Natta2004a, Ricci2010, Testi2014}. There are other effects that could alter the value of the dust spectral index such as dust porosity \citep{Kataoka2014}, and grain composition \citep{Demyk2017b,Demyk2017a} but to explain $\beta < 1$ some grain growth is required \citep{Ysard2019}.
While the VLA 9 mm flux is unlikely to be optically thick, the ALMA 1 mm emission from young disks can be opaque. Optically thick emission at 1 mm would result in a lower spectral index value. If the indices obtained between 1 and 9 mm $\alpha_{\rm VLA/ALMA}$ are consistent with the Ka-band intraband indices $\alpha_{\rm VLA}$, it can be assumed that the emission at 1 mm is optically thin so that it is possible to measure the spectral index in a robust way. We calculate the VLA intraband spectral index, determined between the two sidebands of the Ka-band observations as follows:
\begin{equation}
\label{eq:spectral_index}
\alpha_{\rm VLA}=\frac{\ln (F_{\rm 8\ mm}/ F_{\rm 10\ mm})}{\rm ln(\rm 10\ mm/8\ mm)}.
\end{equation}
Fig. \ref{fig:robust_index} shows the range of the $\alpha_{\rm VLA/ALMA}$ and $\alpha_{\rm VLA}$ values measured.
For 23 sources out of 77 we found the $\alpha_{\rm VLA/ALMA}$ - $\alpha_{\rm VLA}$
$\leq$ 0.4, and therefore in reasonable agreement (see Fig. \ref{fig:SED}). For those sources, the emission at both 1 mm and 9 mm wavelengths is most likely optically thin, so the spectral index should provide information about the grain size. The mean spectral index of those sources is 2.5, which means that $\beta \sim 0.5$. This value confirms that significant dust growth is occurring in the observed disks. The spectral index calculated for the selected optically thin sample (0.5) is similar to that calculated for the full sample (0.4). We therefore proceed with assuming a value of $\beta=$ 0.5 for the further analysis, as an average value, which does not exclude that the 1 mm emission is optically thick. It is also likely that the spectral index varies with the radius of the disk due to optically thick emission close to the protostar and due to the grain growth further out \citep{Pinilla2012,vanTerwisga2018}, as well as a grain size that depends on radius \citep{Tazzari2016}. Our observed emission is largely unresolved and the measured spectral index is an average of those effects.
The emission at shorter wavelengths is more likely to be optically thick. With at least marginally resolved disks, we can obtain an estimate of the dust optical depth, because the extent of the emission allows to approximate the disk radius. We use the major axis deconvolved from the beam as the diameter of the disk. Then, we obtain optical depth as $\tau \sim \kappa_{\nu}\Sigma$, where $\kappa_{\nu}$ is dust opacity used to calculate the mass and $\Sigma$ is averaged surface density. Fig. \ref{fig:dustopacities} presents a distribution of calculated optical depths. For all the disks with available major axis value, we get $\tau < 0.4$ and in vast majortiy of cases $<0.1$.
Summarizing, we have identified dust thermal emission as the dominating physical process responsible for the emission at both 1 mm and 9 mm. What is more, from the sample of sources for which the emission is most likely optically thin, we calculate a spectral index value of 2.5, suggestive of significant grain growth already at these young stages.
\subsection{Disk mass measurements }
The continuum flux at millimeter wavelengths is commonly used as a proxy of the dust mass of the emitting region. Here we utilize the collected fluxes for the continuum flux in Perseus with VLA and ALMA to calculate the masses of the embedded disks. The key assumptions used in the calculation, temperature and dust opacity (dust mass absorption coefficient), are discussed. Then we proceed to compare the results with other disk surveys both at Class 0/I and at Class II phases.
From the integrated disk fluxes, the dust mass of the disk is calculated following the equation from \cite{Hildebrand1983}:
\begin{equation}
\label{eq:dustmass}
M=\frac{D^2F_\nu}{{\kappa}_\nu B_\nu(T_{\rm dust})}\,
\end{equation}
where $D$ is the distance to the source, $B_\nu$ is the Planck function for a temperature $T_{\rm dust}$ and $\kappa_\nu$ is the dust opacity with the assumption of optically thin emission. Temperature of the dust is set to 30 K, typical for dust in dense protostellar envelopes \citep{Whitney2003}, and disks are assumed to be isothermal. The same temperature is set for Class 0 and Class I disks. If the decrease of the temperature of dust from Class 0 to Class I is significant, the mass difference diminishes \citep[e.g.,][]{Andersen2019}. We consider two cases for the values of $\kappa_\nu$ at 1.3 and 9 mm.
First, since our aim is to compare the results with the Class II disk masses in the literature, most notably the Lupus star-forming region, we use $\kappa_{\rm 1.3\ mm} = 2.3 $~cm$^{2}$~g${^{-1}}$ as used in the determination of masses in \cite{Ansdell2016}. Fig. \ref{fig:ALMA_Per_Lup} (top panel) shows the cumulative distribution function (CDF) for Class II disks in Lupus and Class 0 and I disks in Perseus, all observed with ALMA. The CDF plot is prepared using the survival analysis with the {\it lifelines} package for Python \citep{DavidsonPilon2017}. The CDF plot describes the probability of finding the element of the sample above a certain value. Uncertainty of the cumulative distribution is inversely proportional to the size of the sample and 1$\sigma$ of the confidence interval is indicated as a vertical spread on the CDF plot. It takes into account the upper limits of the measurement, and the median is only reliable if the sample is complete. While the VLA observations sample is complete, the ALMA sample of disks is not, as we assemble $\sim 80\%$ of the total sample. Therefore the VLA median values and distributions are more reliable.
The median dust mass for young disks in Perseus measured with ALMA at 1 mm with the adopted opacity of $\kappa_{\rm 1.3\ mm} = 2.3 $~cm$^{2}$~g${^{-1}}$ is 47 M${_\oplus}$ and 12 M${_\oplus}$ for Class 0 and Class I disks, respectively (Fig. \ref{fig:ALMA_Per_Lup}, top panel). The median is taken from the value corresponding to the 0.5 probability on the CDF plot. The opacity value used here is likely close to the maximum value of the opacity at 1.3 mm \citep{Draine2006}. In an analysis of dust opacity value at 1.3 mm, \citet{Panic2008} find a range between 0.1 and 2 cm$^{2}$ g$^{-1}$. Therefore, the dust masses obtained with $\kappa_{\rm 1.3\ mm} = 2.3 $~cm$^{2}$~g${^{-1}}$ from \cite{Ansdell2016} should be considered as a lower limit to the disk masses in Perseus. Only if the grain composition is significantly different from the typical assumption, in particular if dust has a significant fraction of amorphous carbon, will the actual masses of the dust be lower by a factor of a few \citep{Birnstiel2018}, even when compared with the $\kappa_{\rm 1.3\ mm} = 2.3 $~cm$^{2}$~g${^{-1}}$ that we assume to provide a lower limit on the dust disk mass.
Regardless of the uncertainties, there is a clear evolutionary trend from Class 0 to Class II with disk masses decreasing with evolutionary phase. The median dust masses for disks in Perseus measured with ALMA of 47 $M{_\oplus}$ and 12 M${_\oplus}$ for Class 0 and Class I disks, respectively, are significantly higher than for Class II disks in Lupus which have a median mass of 3 M${_\oplus}$. We note that this value differs from the 15 M$_\oplus$ value reported in \cite{Ansdell2016}, since in that work the standard mean is calculated, contrary to the median taken from the CDF plot. Also, distances to Lupus disks have been updated with {\it Gaia} DR2 distances \citep{Gaia2018}. It should be noted that the dust temperature used in \cite{Ansdell2016} was 20 K, while we use 30 K, but the opacity value adopted to calculate the masses is the same. A lower temperature results in an increase of the total mass, based on Equation 4. Therefore if the temperature would be set to 20 K for the Perseus disks, the difference between Class 0/I and Class II disks would be even higher. Class 0/I disks are, however, expected to be warmer than Class II disks \citep{Harsono2015,vantHoff2020}.
As an alternative method, considering that ALMA fluxes can be optically thick, we use the VLA flux densities to estimate the disk masses. Here we adopt $\kappa_{9 {\rm mm}}= 0.28 \ {\rm cm^{2}\ g^{-1}}$ as provided by dust models of the DIANA project \citep{Woitke2016} that consider large grains up to 1 cm; we recall that significant grain growth is indicated in our data by the empirically measured value of $\beta = 0.5$. In \cite{Tychoniec2018a}, a value of $\kappa_{9 {\rm mm}}= 0.13 \ {\rm cm^{2}\ g^{-1}}$ was used, scaled from $\kappa_{1.3 {\rm mm}}= 0.9 \ {\rm cm^{2}\ g^{-1}}$ of \cite{Ossenkopf1994} using $\beta$ = 1. If $\beta$ = 0.5 is instead used to scale the opacity $\kappa_{1.3 {\rm mm}}$ to 9 mm, the value is consistent with that of DIANA.
The median masses measured from the 9 mm observations are 158 M${_\oplus}$ for Class 0 and 52 M${_\oplus}$ for Class I (Fig. \ref{fig:ALMA_Per_Lup}, bottom panel). Those masses are lower than the estimate provided in \cite{Tychoniec2018a} by a factor of two, which stems from the different opacity values used. Additionally the values quoted in \cite{Tychoniec2018a} are regular medians, taken from the sample of detected disks and the distance to the Perseus star-forming region has been revised from 235 to 293 pc \citep{Ortiz-Leon2018} which increases the estimate of the mass.
An important difference between the ALMA and VLA samples is that the VLA sample is complete, as it targeted all known protostars in Perseus \citep{Tobin2016}. Additionally, it is likely that the VLA flux densities are coming from optically thin emission, whereas the ALMA flux densities can become optically thick in the inner regions. We also use a refined model of the dust opacity of the DIANA project \citep{Woitke2016} including large grains. Therefore, the median masses reported with VLA (158 M${_\oplus}$ and 52 M${_\oplus}$ for Class 0 and Class I, respectively) can be considered more robust.
\begin{figure}[h]
\centering
\includegraphics[width=0.95\linewidth,trim={0cm 0cm 0cm 0cm},clip]{plots/ALMA_Per_Lup_v3.png}
\includegraphics[width=0.95\linewidth,trim={0cm 0cm 0cm 0cm},clip]{plots/VLA_Per_Lup_v3.png}
\caption{ Cumulative distribution plots of the dust disk masses. {\it Top:} Masses of the Perseus Class 0 and I disks measured with ALMA at 1 mm compared with the Lupus Class II disks measured with ALMA \citep{Ansdell2016}. The opacity value of $\kappa_{1.3 {\rm mm}}=2.3\ {\rm \ cm}^{2}\ {
\rm g^{-1}}$ is used to calculate the masses. The ALMA sample consist of 77 sources (38 Class 0 and 39 Class I) and the Lupus sample consist of 69 sources. {\it Bottom:} Masses of the Perseus Class 0 and I disks measured with VLA at 9 mm (red and blue, respectively), compared with the Lupus Class II disks measured with ALMA \citep{Ansdell2016}. The opacity value of $\kappa_{9 {\rm mm}}=0.28\ {\rm \ cm}^{2}\ {\rm g}^{-1}$ is used to calculate the VLA masses. The VLA sample consist of 100 sources (49 Class 0 and 51 Class I). Medians are indicated in the labels.}
\label{fig:ALMA_Per_Lup}
\end{figure}
\subsection{ALMA Class 0/I disk masses for different star-forming regions}
Recent ALMA observations of Orion and Ophiuchus reveal masses of embedded Class 0/I dust disks that are somewhat lower than those obtained for Perseus with the VLA \citep{Williams2019, Tobin2020}. Here we collect available ALMA observations for Perseus that use the same techniques as other embedded surveys. Such analysis can reveal the inherent differences between the different protostellar regions.
\begin{figure}[h]
\centering
\includegraphics[width=0.95\linewidth,trim={0 0cm 0 0cm},clip]{plots/CDF_Per_Ori_v2_30K.png}
\caption{Cumulative distribution plots for Perseus disk masses calculated from ALMA flux densities, and Orion disk masses from \citet{Tobin2020}, both calculated with the same opacity assumptions of $\kappa_{1.3 {\rm mm}}=0.9\ {\rm cm^{2}\ g^{-1}}$. The Perseus sample consist of 77 sources (38 Class 0 and 39 Class I), the Orion sample consist of 415 sources (133 Class 0 and 282). Medians are indicated in the labels.}
\label{fig:cdplots_per_ori}
\end{figure}
In Fig. \ref{fig:cdplots_per_ori} we show the cumulative distribution of disks observed with ALMA for Orion \citep{Tobin2020} and for Perseus. The Orion disks were targeted within the VANDAM survey of Orion protostars where 328 protostars were observed with ALMA Band 7 (0.87 mm) at 0\farcs1 (40 au) resolution. The sample in \cite{Tobin2020} is divided into Class 0, Class I and Flat spectrum sources. We incorporate the Flat Spectrum sources into Class I in the comparison. An opacity value of $\kappa_{0.87 {\rm mm}}=1.84\ {\rm cm^{2}\ g}^{-1}$ \citep{Ossenkopf1994} has been assumed to calculate the masses in the Orion survey. We use the same reference for opacity at 1.3 mm wavelengths, $\kappa_{1.3 {\rm mm}}=0.9\ {\rm cm^{2}\ g}^{-1}$, to calculate the masses from ALMA flux densities in Perseus to compare with the Orion sample. The median for Class 0 dust disk masses is still significantly lower in Orion, 67 M$_\oplus$ versus 131 M$_\oplus$ in Perseus, but is remarkably similar for Class I: 25 M$_\oplus$ versus 33 M$_\oplus$. Thus, using ALMA-measured flux densities and the same opacity assumption as \citet{Tobin2020}, we find that there are some inherent differences between the population of Class 0 disks in Perseus and Orion. Note that values calculated here are different than reported in \cite{Tobin2020} because the temperature of the dust was scaled with the luminosity in that work, while we use a constant $T$ = 30 K for a consistent comparison with our sample.
Differences in sound speed or initial core rotation can result in different disk masses \citep[e.g.,][]{Terebey1984,Visser2009}. The initial composition of grains could also affect the dust spectral index. The Orion Molecular Complex seems to show a higher fraction of the amorphous pyroxene than the typical ISM \citep{Poteet2012}. It is likely that such factors are resulting in different observed masses between Orion and Perseus. \cite{Tobin2020} noted that the 9 mm flux density distribution is similar between Perseus and Orion.
The low Class I median masses reported in Ophiuchus \citep[][median mass 3.8 M$_\oplus$]{Williams2019} are puzzling as it suggests that the problem of missing dust mass for planet formation extends from Class II to Class I disks. In our data, the median Class I disk mass median is 11 M$_\oplus$ for the same opacity assumption as in \cite{Williams2019}, a factor of 3 higher. The Ophiuchus sample does not include the entire population of Class I disks in Ophiuchus and may be contaminated with more evolved sources due to the high foreground extinction \citep{vanKempen2009a, McClure2010}. For this reason, we will not include it in the further analysis. Despite those caveats it is possible that the population of young disks in Ophiuchus is less massive than in Perseus and Orion.
\section{Exoplanetary systems and young disks - a comparison of their solid content}
Surveys of protoplanetary (Class II) disks around pre-main sequence stars reveal that dust masses of most disks are not sufficient to explain the inferred solid masses of exoplanetary systems \citep{Williams2012, Najita2014, Ansdell2017, Manara2018}. On the other hand, the results from the younger (Class 0/I) star-forming regions show that the dust reservoir available in younger disks is much higher than in the Class II phase \citep{Tychoniec2018a, Tobin2020}. Here, we aim to determine if the amount of dust available at the onset of planet formation (Class 0/I disks) agrees with the masses of the exoplanet systems observed for reasonable efficiencies of the planet formation process. Simply stated: are the masses of the embedded disks high enough to produce the observed population of the most massive exoplanet systems, or does the problem of the missing mass extend even to the youngest disks?
In this analysis we focus on the Perseus sample, the only complete sample that is available for Class 0/I protostars in a low-mass star-forming region. As such it guarantees that there is no bias towards the more massive disks. Perseus, however, may not be a representative star-forming region for the environment of our own Solar System \citep{Adams2010}. Also, because it is difficult to estimate the stellar mass of the Class 0/I sources, making a comparison of planets and disks around similar stellar types is challenging. Therefore, we include as well a comparison with the Orion disks. The Orion star-forming region contains more luminous protostars than Perseus; thus it might be more representative of the initial mass function. The other limitation is that we analyze mostly unresolved disks, hence the radial dust distribution in the disk is unknown.
\subsection{Exoplanet sample selection}
The exoplanet systems masses were obtained from the exoplanet.eu database \citep{Schneider2011}. From the catalog (updated 28.04.2020) we obtained 2074 exoplanets with provided value for the total mass, either a true mass, or a lower limit to the mass (M$\ \times\ sin(i)$). We do not filter for detection method, mass measurement method, or stellar type of the host star. The mass estimation method for the majority of planets with information on a mass detection method provided is a radial velocity method, which introduces a strong bias toward more massive exoplanets.
Indeed 1373 of the exoplanets in our analysis have a total (gas+dust) masses above 0.3 Jupiter masses (M$\rm_J$). There are 1062 systems with more than one planet where at least one is > 0.3 M$\rm_J$, and 173 systems with a single > 0.3 M$\rm_J$ planet.
Gaseous planets are expected to be less frequent than the rocky low-mass planets \citep[e.g.,][]{Mayor2011}. It is estimated that only 17-19\% of planetary systems would contain a planet more massive than 0.3 M$\rm_J$ within 20 au orbit \citep{Cumming2008}. Therefore, we use only systems containing at least one planet with a total mass of 0.3 M$\rm_J$ and normalize it to 18\% of the total population. This is done by setting the 18\% value of the CDF plot at the estimated solid fraction of the gaseous planet of 0.3 M$_{\rm J}$, which is 27.8 M$_{\oplus}$. We assume that 82\% of the systems have masses below that value. By doing so, we focus on the sample of gas giants and their solid material content.
In this work, we focus on a reliably estimated solid mass and its cumulative distribution for the most massive exoplanetary systems. By comparing their CDF to the total dust mass distribution from surveys of young disks we can answer the pivotal question: do the Class 0/I disks contain enough solids to explain the masses of those systems.
Our study focuses on the dust masses of disks. In order to compare the solid content between the disks and exoplanets, we calculate the solid content in exoplanets using the formula from \cite{Thorngren2016} for estimating the solid content in gaseous planets. This study is based on structural and thermal planetary evolution models relating the metallicity of a gas giant with the total mass of the planet. Importantly, the metals in those gas giants are assumed to be located not only in the core but also in the envelope of a planet. We combined the masses of planets orbiting the same star to retrieve the total dust mass of the system, resulting in 1235 systems in the analysis.
\begin{figure}[h]
\centering
\includegraphics[width=0.95\linewidth,trim={0cm 0cm 0cm 0cm},clip]{plots/CDF_VLA_cII_exo_v3.png}
\includegraphics[width=0.95\linewidth,trim={0cm 0cm 0cm 0cm},clip]{plots/CDF_VLA_cII_exo_v3_efficiency.png}
\caption{Cumulative distribution function of dust disk masses and solid content of exoplanets.
{\it Top:} Cumulative distribution function of dust masses for Class 0 (red) and Class I (blue) disks in Perseus and Class II disks (yellow) in Lupus measured with ALMA \citep{Ansdell2016}. In black, the masses of the exoplanet systems are normalized to the fraction of the gaseous planets \citep{Cumming2008}. Perseus disk masses calculated with $\kappa_{9 {\rm mm}}=0.28\ {\rm cm^{2}\ g^{-1}}$ from the VLA fluxes. Medians are indicated in the labels. {\it Bottom:} Zoom-in to the ranges where exoplanets are present. The color scale shows the efficiency needed for the planet formation for a given bin of the distribution.}
\label{fig:hist_exo+disks}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[width=0.95\linewidth,trim={0cm 0cm 0cm 0cm},clip]{plots/CDF_ALMA_cII_exo_v3.png}
\includegraphics[width=0.95\linewidth,trim={0cm 0cm 0cm 0cm},clip]{plots/CDF_ALMA_Orion_cII_exo_v3_30K.png}
\caption{Cumulative distribution function of dust masses for Class 0 and Class I disks in Perseus and Orion and Class II disks in Lupus measured with ALMA {\it Top:} Perseus disk masses calculated with $\kappa_{1.3 {\rm mm}}=2.3\ {\rm cm^{2}\ g^{-1}}$ from the ALMA fluxes. {\it Bottom: } Orion disk masses calculated with the $\kappa_{0.89 {\rm mm}}=1.3\ {\rm cm^{2}\ g^{-1}}$ \citep{Tobin2020}. Medians are indicated in the labels.}
\label{fig:exo_efficiency}
\end{figure}
\begin{figure*}[h]
\centering
\includegraphics[width=0.95\linewidth,trim={0cm 0cm 0cm 0cm},clip]{plots/exoplanet_scatter_v5.png}
\caption{Plot showing the distribution of masses of exoplanetary systems obtained from the exoplanet.eu catalog \citep{Schneider2011}, for the planets around main-sequence stars with the measured masses. Shaded areas mark the range of our best estimation of the dust disk masses in Perseus: Class 0 (red) and Class I (blue) calculated from the VLA fluxes with the opacity value of $\kappa_{9 {\rm mm}}=0.28\ {\rm cm^{2}\ g^{-1}}$. Medians of the distributions, 158 and 52 M$_\oplus$, for Class 0 and I, respectively, are indicated with the dashed lines. The median mass of the Class II disks in Lupus, 3 M$_\oplus$ \citep{Ansdell2016} is showed in yellow. The masses of the solids in exoplanetary systems are plotted against the stellar mass of the host star. All planets with available information on the mass are included in this plot, without introducing the 0.3 M$_{\rm J}$ threshold.}
\label{fig:exo_scatter}
\end{figure*}
\subsection{Comparison of young disk dust masses with the solids in exoplanetary systems}
In Fig. \ref{fig:hist_exo+disks} (top panel) we compare the sample of exoplanetary systems with dust masses of young Class 0 and Class I disks in Perseus with our best estimate of embedded disk dust masses in Perseus, i.e., with the complete VLA survey, using the DIANA opacities. In Fig. \ref{fig:hist_exo+disks} (bottom panel) we show efficiency of planet formation for a given bin of the disk and exoplanet distributions. The efficiency is calculated as a ratio of the total mass of the exoplanetary systems at the certain fraction of the cumulative distribution plot divided by the corresponding dust mass of the disk at the same value of the CDF plot. This calculation provides information on how much total dust disk mass will be converted to planets.
In order to reproduce the population of exoplanets with the top 18\% most massive disks in Perseus, the efficiency of planet formation would have to be on the order of 15\% for Class 0 and 34\% for Class I (Fig. \ref{fig:hist_exo+disks}). The average efficiency is measured by taking the mass at 10\% of the cumulative distribution plot (CDF plot is not well sampled for disk masses, and the 10\% value is the closest to the mean of the sample with data available). The Class 0/I disk masses in Perseus calculated from the VLA data at 9 mm for the refined value of the dust opacity suggest that on average there is enough mass available at those early phases to form the giant planet systems that we observe.
In Fig. \ref{fig:hist_exo+disks} (bottom panel), instead of the average value of the efficiency we attempt to measure the efficiency per each percentage level of the disk masses on the cumulative distribution plot . We note that this analysis has a higher uncertainty than the average value as the distribution of the exoplanets is uncertain. With known masses of a large number of giant planets and expected occurrences of such systems \citep{Cumming2008} we notice that the most massive exoplanets require efficiencies $\sim$ 30\%, stretching the requirements of some of the planet formation models (see Section 4.3). It is also possible that some of the most massive exoplanets or brown dwarfs present in the database do not follow the core accretion formation mechanisms, and excluding them would lower the requirement on efficiency.
If the underlying initial mass function of stars in Perseus is not representative of the stellar initial mass function, it could be that such exo-systems were produced from more massive disks than observed here. Another possibility is that young disks at the Class 0/I phase are still being replenished with material accreting from the envelope \citep{Hsieh2019}, making the effective material available for planet formation higher. It is also possible that the most massive systems are indeed producing the planets most efficiently.
In Fig. \ref{fig:exo_efficiency} (top panel)
the disk masses from the ALMA fluxes in Perseus using the opacity value from \citet{Ansdell2016} are presented. We remind that this is likely the maximum value of the dust opacity at those wavelengths \citep{Panic2008} and therefore the masses are lower limits. In the bottom panel of Fig. \ref{fig:exo_efficiency}, the Orion Class 0/I disk masses measured with ALMA 0.87 mm observations \citep{Tobin2020} are used.
The comparison of the exoplanet sample with the disk masses for the maximum value of the opacity used with the ALMA data (i.e., lower limit to the disk mass, Fig. \ref{fig:exo_efficiency}, top panel) shows that if those opacities were the correct ones, the efficiency of forming planets in the Class 0 phase would be on the order of 33\%. The efficiency of the Class I phase would be on the order of 72\%. It is in line with our expectations that for our most conservative estimates of disk mass, the efficiencies required for giant planet formation are high, whereas for our best estimate of the dust masses in the young disks, we achieve an average efficiency in agreement with models (see Section 4.3). It is also clear that use of dust opacities that result in an order of magnitude lower disk masses \citep{Birnstiel2018} do not provide dust masses that would be compatible with such models.
The Orion sample, which contains more luminous protostars and could be more representative of the IMF than Perseus, has comparable disk dust masses in Class I and much lower disk masses in Class 0. The efficiencies required to produce the exoplanet population from the Orion disks dust content, as measured with ALMA, are comparable with those measured for Perseus with VLA observations (Fig. \ref{fig:exo_efficiency} bottom panel): 16\% and 45\% for Class 0 and Class I, respectively.
Fig. \ref{fig:exo_scatter} presents a different visualization of the distribution of exoplanet systems dust masses compared to the range of disk masses observed in Class 0 and Class I disks in Perseus. The conclusions of our work show that for the complete sample of disks in Perseus and with a large sample of known exoplanets, there is enough solid material in Class 0 stage to explain the solid content in observed exoplanetary systems. This conclusion is consistent with \cite{Najita2014} and \cite{Greaves2011} but now with much more robust statistics.
In recent years several studies, specifically with the use of microlensing observations, estimate that nearly all stars can have at least 10 M{$_\oplus$} planet \citep{Cassan2012,Suzuki2016}. Very few cases and irreproducibility of microlensing observations suggest caution with extrapolating the results to all systems. It should be kept in mind that large population of wide-orbit planets could pose a challenge to efficiency of planet formation even in Class 0/I stage \citep{Najita2014}.
Actual timescales for the different phases of low-mass star formation are uncertain. Based on a statistical analysis of a population of protostars, it is estimated that the Class 0 evolutionary phase lasts for $\sim$0.1 Myr since the beginning of the collapse, and the Class I phase ends when the protostar is $\sim$ 0.5 Myr old \citep{Dunham2014}. More recent estimates of the half-lifes of the protostellar phases give Class 0 half-life values of $\sim$ 0.05 Myr and $\sim$ 0.08 Myr for Class I \citep{Kristensen2018}. Therefore our results indicate the start of the planet formation begins less than 0.1 Myr after the beginning of the cloud collapse. This is consistent with the ages of the oldest meteorites in our Solar system \citep{Connelly2012}.
\subsection{The context of planet formation models}
It is of worth to put our empirical constraints on the solid mass reservoir in the context of planet formation models. Broadly speaking, planets can either form bottom-up through the assembly of smaller building blocks, in the so-called \textit{core accretion} scenario, or top-down in the \textit{gravitational instability} scenario, via direct gravitational collapse of the disk material. In the latter case (see \citealt{Kratter2016} for a review), planets need not contain rocky cores and we may therefore have over-estimated the solid mass locked in planets. In addition, in this view planets form at the very beginning of the disk lifetime when the disk is gravitationally unstable, possibly at even earlier stages than we probe here. If this is the case, our observations do not put constraints on the mass budget required for planet formation since planets would already be formed in the disks we are considering.
Our results are instead relevant for the core accretion scenario. In this case, two large families of models can be defined, differing in the type of building block: planetesimal accretion \citep[e.g.,][]{Pollack1996} and pebble accretion \citep{Ormel2010,Lambrechts2012}.
If planets grow by accreting planetesimals, it should be kept in mind that our observations are sensitive only to the dust. Therefore, the efficiency we have defined in this paper should be intended as the product of two efficiencies: the efficiency of converting dust into planetesimals and the efficiency of converting planetesimals into planets. The latter is relatively well constrained from theory and observations. Indeed, the accretion of planetesimals is highly efficient in numerical models \citep[e.g.,][]{Alibert2013} and nearly all planetesimals are accreted into planets over Myr timescales. Observationally, constraints on the mass in planetesimals that are not locked into planets is set by the debris disk population \citep{Sibthorpe2018}. For Sun-like stars, the median planetesimal mass\footnote{As discussed in \citet{Wyatt2007}, this value is degenerate with the maximum planetesimal size; here we have assumed a diameter of 1000 km, and we note that the mass would be even lower if using smaller planetesimals.} is 3 M$_\oplus$, i.e. much smaller than the solid content of giant planets. In this context, the Solar System could be an exception because attempts at explaining its complex history (such as the well known Nice model, \citealt{Tsiganis2005}) require instead a much more massive (20-30 M$_\oplus$) population of planetesimals past the orbits of Uranus and Neptune that later evolved into the current Kuiper Belt. Even so, this mass is comparable to the mass in the cores of the giant planets, implying an efficiency of planet conversion from planetesimals $\gtrsim 50$\%.
Using this value, our observations place empirical constraints on the planetesimal formation efficiency. To satisfy our measurement of a total efficiency of $\sim 10$\%, a planetesimal formation efficiency of $\sim 20$\% would be needed. There is considerable uncertainty in planetesimal formation models \citep[e.g.,][]{Drazkowska2014, Drazkowska2018, Lenz2019}, but such an efficiency can in principle be reached by most of the expected conditions in the protoplanetary disks (see Fig. 9 of \citealt{Lenz2019} and Table 1 of \citealt{Drazkowska2014}). Thus, it is possible to explain the observed population of giant planets with the initial dust masses we report in this paper.
If instead planets grow by accreting pebbles, the growth rates can be significantly higher than in planetesimal accretion, but the formation efficiency is lower because most pebbles drift past the forming planets without being accreted \citep{Ormel2010,Ormel2017}. The efficiency of $\sim 10$\% reported here is among the highest that can be reached by pebble accretion \citep{Ormel2017} and it favours models where the disk is characterised by low turbulence ($\alpha < 10^{-3}$).
Assuming such a value for the turbulence, \citet{Bitsch2019} finds that in pebble accretion models that form giant planets the total planet formation efficiency is 5-15\%, in line with our findings. It is also suggestive that in their models giant planet formation requires initial dust masses larger than $200-300$ M$_\oplus$: note how this condition is satisfied for $\sim$ 20\% of the Class 0 dust mass distribution. Therefore, the mass constraints derived in this paper from the VLA data are also consistent with pebble accretion, provided that the turbulence in the disk is sufficiently low.
\section{Conclusions}
This work collects available ALMA and VLA data of a complete sample of Perseus young disks in the Class 0/I phase to provide robust estimates of the disk dust masses at the early phases of star and planet formation. The refined values are used to compare the inferred disk masses with the exoplanetary systems to obtain constraints on when exoplanets start to form.
A linear correlation is found between the fluxes obtained with VLA and ALMA, supporting the fact that thermal dust emission is responsible for the emission at both wavelengths. The value of the dust spectral index measured with ALMA and VLA observations is $\beta = 0.5 $, lower than the commonly used value of $\beta = 1 $, pointing to significant grain growth occurring already in the Class 0 and I phases. Therefore, compared with our previous study \citep{Tychoniec2018a} we recalculated the masses with the new dust opacity value that account for large grains. The best estimate of the median initial reservoir of dust mass available for planet formation in Perseus is 158 M$_\oplus$ and 52 M$_\oplus$ for Class 0 and Class I disks, respectively, derived from the VLA data.
Comparison of ALMA observations in Orion and Perseus shows that while disk masses in Class I disks agree well, Class 0 disks are more massive in Perseus than in Orion. This suggests that initial cloud conditions may lead to different masses of disks in the early phases.
Dust masses of disks measured with the VLA for Perseus are compared with the observed exoplanet systems.
If we assume that planet formation starts with the dust mass reservoir equal to the dust mass of Class I disks in Perseus, efficiency of $\sim$ 30\% is required to explain the currently observed systems with giant exoplanets. Lower efficiencies of $\sim$ 15\% on average are needed if the Class 0 disks are assumed as the starting point. We find strong evidence that there is enough dust mass in young disks to make planet formation possible already in the first $\sim$ 0.5 Myr of star formation. Given that low efficiencies are more in line with theoretical core accretion models, our results are most consistent with significant accumulation of material in larger bodies occurring already at the Class 0 phase.
\begin{acknowledgements}
{\L}T and EvD thank Dr. Yao Liu for discussions on dust opacities. {\L}T thanks Leon Trapman for discussions that helped in the presentation of results. G.R. acknowledges support from the Netherlands Organisation for Scientific Research (NWO, program number 016.Veni.192.233). JT acknowledges support from grant AST-1814762 from the National Science Foundation. The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc. This paper makes use of the following ALMA data: ADS/JAO.ALMA\#2017.1.01693.S, ADS/JAO.ALMA\#2017.1.01078.S, ADS/JAO.ALMA\#2015.1.00041.S, and ADS/JAO.ALMA\#2013.1.00031.S. ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada), MOST and ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO and NAOJ. Astrochemistry in Leiden is supported by the Netherlands Research School for Astronomy (NOVA), by a Royal Netherlands Academy of Arts and Sciences (KNAW) professor prize, and by the European Union A-ERC grant 291141 CHEMPLAN. This project has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 823823 (DUSTBUSTERS). This work was partly supported by the Deutsche Forschungs-Gemeinschaft (DFG, German Research Foundation) - Ref no. FOR 2634/1 TE 1024/1-1. This research made use of Astropy, a community-developed core Python package for Astronomy \citep{Astropy2013},
http://astropy.org); Matplotlib library \citep{Hunter2007}; NASA's Astrophysics Data System.
\end{acknowledgements}
|
1,314,259,993,642 | arxiv | \section{Introduction}\label{sec:Intro}
Let $\Gamma$ be a finitely generated group and $G$ be a connected semisimple Lie group. It is an interesting problem to select and study some connected components of the representation variety ${\rm Hom}(\Gamma,G)$ that consist of homomorphisms $\rho:\Gamma\to G$ that are well behaved and, ideally, reflect some interesting geometric properties of the group $\Gamma$. The best example of this framework is the case in which $\Gamma$ is the fundamental group $\Gamma_g$ of a closed surface of genus $g\geq 2$ and $G$ is ${\rm PSL}_2(\mathbb R)$. In this case the Teichm\"uller space arises as a component of ${\rm Hom}(\Gamma_g,{\rm PSL}_2(\mathbb R))/\!/{\rm PSL}_2(\mathbb R)$ that can be selected by means of a cohomological invariant \cite{Goldmanthesis}.
In the more general setting in which $G$ is any Hermitian Lie group, the so-called \emph{maximal representations} form a well studied union of connected components of the character variety ${\rm Hom}(\Gamma_g,G)/\!/G$ which generalize the Teichm\"uller component \cite{BGG,Toledo}. In analogy with holonomy representations of hyperbolizations, maximal representations can be characterized as those
representations that maximize an invariant, the \emph{Toledo invariant}, that can be defined in terms of bounded cohomology. Such representations are discrete and faithful, and give rise to interesting geometric structures \cite{Toledo, Anosov}. Maximal representations in Hermitian Lie groups were first studied by Toledo in \cite{Toledo-rep} where he proves that a maximal representation $\rho:\Gamma_g\to {\rm SU}(1,q)$ fixes a complex geodesic, and by Hernandez \cite{Hernandez} who studied maximal representations $\rho:\Gamma_g\to {\rm SU}(2,q)$ and showed that the image must stabilize a symmetric space associated to the group ${\rm SU}(2,2)$. In general any maximal representation stabilizes a tube-type subdomain \cite{Toledo}.
Despite this, a remarkable flexibility result holds for maximal representations $\rho$ of fundamental groups of surfaces: if the image of $\rho$ is a Hermitian Lie group of tube type, then $\rho$ admits a one parameter family of deformations consisting of Zariski dense representations \cite{Toledo, KP}.
An analogue of the Toledo invariant was defined by Burger and Iozzi in \cite{BIpreprint} for representations of a lattice $\Gamma$ in ${\rm SU}(1,p)$ with values in a Hermitian Lie group $G$. This allows to select a union of connected components of ${\rm Hom}(\Gamma,G)$ consisting of \emph{maximal representations}. These generalize maximal representations of fundamental groups of surfaces: the fundamental group of a surface is a lattice in ${\rm PU}(1,1)={\rm PSL}_2(\mathbb R)$. However, if $p$ is greater than one, a different behavior is expected: Goldman and Milson proved local rigidity for the standard embedding of $\Gamma$ in ${\rm SU}(1,q)$ \cite{GM}, and Corlette
proved that maximal representations of uniform complex hyperbolic lattices with values in ${\rm SU}(1,q)$ all come from the standard construction \cite{Corlette}.
The picture for rank one targets was completed independently by Koziarz and Maubon \cite{KMrank1} and by Burger and Iozzi \cite{BICartan}: any maximal representation of a lattice in ${\rm SU}(1,p)$ with values in ${\rm SU}(1,q)$ admits an equivariant totally geodesic holomorphic embedding $\H^p_\mathbb C\to \H^q_\mathbb C$.
Koziarz and Maubon generalized this result to the situation in which the target group is classical of rank 2 and the lattice is cocompact \cite{KM}\footnote{In his recent preprint Spinaci studies maximal representations of cocompact K\"ahler groups admitting an holomorphic equivariant map \cite{Spi}}. It is conjectured that every maximal representation of a complex hyperbolic lattice with target a Hermitian Lie group is superrigid, namely it extends, up to a representation of $\Gamma$ in the compact centralizer of the image, to a representation of the ambient group ${\rm SU}(1,p)$.
In this article we show that the conjecture holds for Zariski dense representations in ${\rm SU}(m,n)$, with $m$ different from $n$:
\begin{thm}\label{thm:Zariskisuperrigidity}
Let $\Gamma$ be a lattice in ${\rm SU}(1,p)$ with $p>1$. If $m$ is different from $n$, then every Zariski dense maximal representation of $\Gamma$ into ${\rm PU}(m,n)$ is the restriction of a representation of ${\rm SU}(1,p)$.
\end{thm}
This immediately implies the following:
\begin{cor}\label{cor:noZariskidense}
Let $\Gamma$ be a lattice in ${\rm SU}(1,p)$ with $p>1$. There are no Zariski dense maximal representations of $\Gamma$ into ${\rm SU}(m,n)$, if $1<m<n$.
\end{cor}
Exploiting results of \cite{tight}, and the classification of maximal representations between Hermitian Lie groups \cite{Ham1, Ham2, HamP}, we are able to use our main theorem to give a structure theorem for all maximal representations $\rho:\Gamma\to{\rm SU}(m,n)$.
\begin{thm}\label{thm:general}
Let $\rho:\Gamma\to {\rm SU}(m,n)$ be a maximal representation. Then the Zariski closure $L=\overline {\rho(\Gamma)}^Z$ splits as the product ${\rm SU}(1,p)\times L_t\times K$ where $L_t$ is a Hermitian Lie group of tube type without irreducible factors that are virtually isomorphic to ${\rm SU}(1,1)$, and $K$ is a compact subgroup of ${\rm SU}(m,n)$.
Moreover there exists an integer $k$ such that the inclusion of $L$ in ${\rm SU}(m,n)$ can be realized as
$$\Delta\times i\times {\rm Id}:L
\to {\rm SU}(1,p)^{m-k}\times{\rm SU}(k,k)\times K<{\rm SU}(m,n)$$
where $\Delta:{\rm SU}(1,p)\to {\rm SU}(1,p)^{m-k}$ is the diagonal embedding, $i: L_t\to {\rm SU}(k,k)$ is a tight holomorphic embedding and $K$ is contained in the compact centralizer of $\Delta\times i(L)$.
\end{thm}
It is possible to show that there are no tube-type factors in the Zariski closure of the image of $\rho$ by imposing some non-degeneracy hypothesis on the associated linear representation of $\Gamma$ into ${\rm GL}(\mathbb C^{m+n})$:
\begin{cor}\label{cor:general}
Let $\Gamma$ be a lattice in ${\rm SU}(1,p)$, with $p>1$ and let $ \rho$ be a maximal representation of $\Gamma$ into ${\rm SU}(m,n)$. Assume that the associated linear representation of $\Gamma$ on $\mathbb C^{n+m}$ has no invariant subspace on which the restriction of the Hermitian form has signature $(k,k)$ for some $k$. Then
\begin{enumerate}
\item $n\geq pm$,
\item $\rho$ is conjugate to $\overline \rho\times \chi_{\rho}$ where $\overline\rho$ is the restriction to $\Gamma$ of the diagonal embedding of $m$ copies of ${\rm SU}(1,p)$ in ${\rm SU}(m,n)$ and $\chi_\rho$ is a representation $\chi_{\rho}:\Gamma\to K$, where $K$ is a compact group.
\end{enumerate}
\end{cor}
Recently Klingler proved that all the representations of uniform complex hyperbolic lattices that satisfy a technical algebraic condition are locally rigid \cite{Klingler}. As a particular case his main theorem implies that if $\Gamma$ is a cocompact lattice in ${\rm SU}(1,p)$ and $\rho:\Gamma\to {\rm SU}(m,n)$ is obtained by restricting to $\Gamma$ the diagonal inclusion of ${\rm SU}(1,p)$ in ${\rm SU}(m,n)$, then $\rho$ is locally rigid. Since the invariant defining the maximality of a representation is constant on connected components of the representation variety, we get a new proof of Klingler's result in our specific case, and the generalization of this latter result to non-uniform lattices:
\begin{cor}\label{cor:local rigidity}
Let $\Gamma$ be a lattice in ${\rm SU}(1,p)$, with $p>1$, and let $\rho$ be the restriction to $\Gamma$ of the diagonal embedding of $m$ copies of ${\rm SU}(1,p)$ in ${\rm SU}(m,n)$. Then $\rho$ is locally rigid.
\end{cor}
Our proof of Theorem \ref{thm:Zariskisuperrigidity} is inspired by Margulis' beautiful proof of superrigidity for higher rank lattices: in order to show that a representation $\rho:\Gamma\to G$ extends to the group $H$ in which $\Gamma$ sits as a lattice, it is enough to exhibit a $\rho$-equivariant algebraic map $\phi:H/P\to G/L$ for some parabolic subgroups $P$ of $H$ and $L$ of $G$. The existence of measurable $\rho$-equivariant boundary maps $\phi: H/P\to G/L$ where $P<H$ is a minimal parabolic subgroup and $G$ is a linear algebraic group is by now well understood \cite{SUpq, Fur, BF}, and the crucial part in the proof of superrigidity for our representations is to show that such a measurable equivariant boundary map must indeed be algebraic.
In general not every representation of a complex hyperbolic lattice is superrigid: for example Livne constructed in his PhD dissertation a lattice in ${\rm SU}(1,2)$ that surjects onto a free group (cfr. \cite[Chapter 16]{DelMos}), moreover Mostow constructed examples of lattices $\Gamma_1,\Gamma_2$ in ${\rm SU}(1,2)$ admitting a surjection $\Gamma_1\twoheadrightarrow \Gamma_2$ with infinite kernel (cfr. \cite{Most, Tolmaps}).
These examples show that many representations of complex hyperbolic lattice do not extend to ${\rm SU}(1,2)$ and hence some additional information on the boundary map $\phi$ is needed in order to deduce its algebraicity.
We restrict our interest to maximal representations precisely to be able to gather some information on a measurable boundary map $\phi$. The maximality of a representation $\rho$ can be rephrased as a property of the induced pullback map $\rho^*:{\rm H}^2_{\rm cb}(G,\mathbb R)\to {\rm H}^2_{\rm b}(\Gamma,\mathbb R) $ in bounded cohomology. One of the advantages of bounded cohomology with respect to ordinary cohomology is that it can be isometrically computed from the complex of $\rm L^\infty$ functions on some suitable boundary of the group \cite{BMGAFA} and, in all geometric cases known so far \cite{BIAppendix}, the pullback map in bounded cohomology can be implemented using boundary maps. In particular we exploit results of \cite{Formula} and we show that the fact that the representation $\rho$ is maximal implies that a $\rho$-equivariant measurable boundary map must preserve some incidence structure on the boundary (this was proven in \cite{BICartan} in the case in which the image is of rank one).
To describe more precisely this incidence structure, recall that one of the key features of the complex hyperbolic space is the existence of complex geodesics tangent to any vector in $T^1\H_\mathbb C^p$: these are precisely the totally geodesic holomorphic embeddings of the Poincar\'e disc in $\H_\mathbb C^p$. The boundaries of these subspaces produce a family of circles in $\partial\mathbb H _{\mathbb C}^p$, the socalled \emph{chains}, that form an incidence structure that was first studied by Cartan in \cite{Cartan}.
Under many respects, the natural generalization to higher rank of the visual boundary of the complex hyperbolic space is the Shilov boundary of a Hermitian symmetric space and the generalization of a complex geodesic, when maximal representations are involved, is a maximal tube-type subdomain.
All these objects have an explicit linear description: it is well known that the boundary of the complex hyperbolic space can be identified with the set of isotropic lines in $\mathbb C^{p+1}$, and it is easy to check that a triple of lines $x,y,z$ is contained in a chain if and only if $\dim\,\<x,y,z\>=2$.
Similarly the Shilov boundary $\mathcal S_{m,n}$ of ${\rm SU}(m,n)$ can be described as the set of maximal isotropic subspaces of $\mathbb C^{m+n}$ and, again, a triple of transverse isotropic subspaces $x,y,z$ in $\mathcal S_{m,n}$ is contained in the boundary of a tube-type subdomain precisely when $\dim\,\<x,y,z\>=2m$. In such case we will say that $x,y,z$ are contained in an \emph{$m$-chain}.
As it turns out,
if $\rho:\Gamma\to {\rm SU}(m,n)$ is a maximal representation and $\phi:\partial\mathbb H _{\mathbb C}^p\to \mathcal S_{m,n}$ is a measurable $\rho$-equivariant boundary map,
then $\phi$ induces a map from the chain geometry of $\partial\mathbb H _{\mathbb C}^p$ to the geometry whose space is $\mathcal S_{m,n}$ and whose lines are the $m$-chains.
Therefore most of this paper is devoted to the study of these geometries. We generalize some results of Cartan \cite{Cartan} and Goldman \cite{Goldman} and this allows us to prove a strong rigidity result for measurable maps that preserve this geometry, that is a higher rank analogue of the main theorem of \cite{Cartan}:
\begin{thm}\label{thm:phirational}
Let $p>1$, $1<m<n$ and let $\phi:\partial\mathbb H _{\mathbb C}^p\to\mathcal S_{m,n}$ be a measurable map whose essential image is Zariski dense. Assume tha
, for almost every triple with $\dim\<x,y,z\>=2$, it holds $\dim\<\phi(x),\phi(y),\phi(z)\>=2m$. Then $\phi$ coincides almost everywhere with a rational map.
\end{thm}
\subsection*{Outline of the paper}
In Section \ref{sec:Toledo}, after recalling the relevant concepts about Hermitian symmetric spaces and continuous bounded cohomology, we prove that a measurable boundary map associated with a maximal representation induces a map between chain geometries. Sections \ref{sec:S_mn} to \ref{sec:reduction} are devoted to prove Theorem \ref{thm:phirational}: in Section \ref{sec:S_mn} we study the chain geometry of $\mathcal S_{m,n}$ and prove some properties of the incidence structure of chains; in Section \ref{sec:chain} we show that the restriction to almost every chain of a measurable boundary map associated with a maximal representation is rational; in Section \ref{sec:reduction} we show that this information is already enough to conclude. We finish the article with Section \ref{sec:rational}, where we prove all the remaining results announced in this introduction.
\section{Preliminaries}\label{sec:Toledo}
\subsection{Hermitian Symmetric spaces}
Let $G$ be a connected semisimple Lie group of noncompact type with finite center and let $K$ be a maximal compact subgroup. We will denote by $\mathcal X=G/K$ the associated symmetric space. Throughout this article we will be only interested in \emph{Hermitian} symmetric spaces, that is in those symmetric spaces that admit a $G$-invariant complex structure $J$. It is a classical fact \cite[Theorem III.2.6]{Koranyi} that these symmetric spaces admit a bounded domain realization, that means that they are biholomorphic to a bounded convex subspace of $\mathbb C^n$ on which $G$ acts via biholomorphisms. An Hermitian symmetric space is said to be \emph{of tube-type} if it is also biholomorphic to a domain of the form $V+i\Omega$ where $V$ is a real vector space and $\Omega\subset V$ is a proper convex open cone. Hermitian symmetric spaces were classified by Cartan \cite{Cartan-class}, and are the symmetric spaces associated to the exceptional Lie groups ${\rm E}_7(-25)$ and ${\rm E}_6(-14)$ together with 4 families of classical domains: the
ones
associated to ${\rm SU}(p,q)$, of type $I_{p,q}$ in the standard terminology\footnote{In Cartan's original terminology \cite{Cartan-class} the families $III_p$ and $IV_p$ are exchanged}, the ones associated to ${\rm SO}^*(2p)$, of type $II_{p}$, the symmetric spaces, $III_p$, of the groups ${\rm Sp}(2p,\mathbb R)$, and the symmetric spaces of the group $IV_p$ associated to ${\rm SO}_0(2,p)$. It is well known that the only spaces that are not of tube type are the symmetric space of ${\rm E}_6(-14)$ and the families $I_{p,q}$ with $q\neq p$ and $II_p$ with $p$ odd. It follows from the classification that any Hermitian symmetric space contains maximal tube-type subdomains, and those are all conjugate under the $G$-action, are isometrically and holomorphically embedded and have the same rank as the ambient symmetric space.
The $G$-action via biholomorphism on the bounded domain realization of $\mathcal X$ extends continuously on the topological boundary $\partial \mathcal X$. If the real rank of $G$ is greater than or equal to two, $\partial \mathcal X$ is not an homogeneous $G$-space, but contains a unique closed $G$-orbit, the \emph{ Shilov boundary} $\mathcal S_G$ of $\mathcal X$. If $\mathcal X$ is irreducible, the stabilizer of any point $s$ of $\mathcal S_G$ is a maximal parabolic subgroup of $G$. In the reducible case, if $\mathcal X=\mathcal X_1\times\ldots\times\mathcal X_n$ is the de Rham decomposition in irreducible factors whose isometry group is $G_i$, then $\mathcal S_G$ splits as the product $\mathcal S_{G_1}\times\ldots\times\mathcal S_{G_n}$ as well.
Moreover when $\mathcal Y$ is a maximal tube-type subdomain of $\mathcal X$, the Shilov boundary of $\mathcal Y$ embeds in the Shilov boundary of $\mathcal X$.
The diagonal action of $G$ on the pairs of points $(s_1,s_2)\in \mathcal S_G^2$ has a unique open orbit corresponding to pairs of opposite parabolic subgroups. Two points in $\mathcal S_G$ are \emph{transverse} if they belong to this open orbit. Whenever a pair $(s_1,s_2)$ of transverse points of $\mathcal S_G$ is fixed, there exists a unique maximal tube-type subdomain $\mathcal Y=G_T/K_T$ of $\mathcal X$ such that $s_i$ belongs to $\mathcal S_{G_T}$. In particular this implies that the Shilov boundaries of maximal tube-type subdomains define a rich incidence structure in $\mathcal S_G$.
Given three points in $\mathcal S_G$ there won't, in general, exist a maximal tube-type subdomain $\mathcal Y$ of $\mathcal X$ whose Shilov boundary contains all the three points. However it is possible to determine when this happens with the aid of the K\"ahler form. Recall that, since $\mathcal X$ is a Hermitian symmetric space, it is possible to define a differential two form via the formula
$$\o(X,Y)= g(X,JY)$$
where $g$ denotes the $G$-invariant Riemannian metric normalized so that its minimal holomorphic sectional curvature is $-1$, and $J$ is the complex structure of $\mathcal X$.
Since $\o$ is $G$-invariant, it is closed: this is true for every $G$-invariant differential form on a symmetric space. This implies that $\mathcal X$ is a K\"ahler manifold and $\o$ is its K\"ahler form.
Let $\mathcal X^{(3)}$ denote the triples of pairwise distinct points in $\mathcal X$ and let us consider the function
\begin{displaymath}\begin{array}{cccc} \b_{\mathcal X}:&\mathcal X^{(3)}&\to&\mathbb R\\ &(x,y,z)&\to& \frac 1{\pi }\int_{\Delta(x,y,z)}\o\end{array}\end{displaymath}
where we denote by $\Delta(x,y,z)$ any smooth geodesic triangle having $(x,y,z)$ as vertices. Since $\o$ is closed, Stokes theorem implies that $\b_{\mathcal X}$ is a well defined continuous $G$-invariant cocycle and it is proven in \cite{CO} that it extends continuously to the triples of pairwise transverse points in the Shilov boundary.
If a triple $(s_1,s_2,s_3)\in\mathcal S^3$ doesn't consist of pairwise transverse points, the limit of $\beta_\mathcal X(x_1^i,x_2^i,x_3^i)$ as $x_j^i$ approaches $s_j$ is not well defined, but Clerc proved that, restricting only to some preferred sequences (the one that converge \emph{radially} to $s_j$), it is possible to get a measurable extension of $\b_\mathcal X$ to the whole Shilov boundary. The obtained extension
$\b_\mathcal S:\mathcal S^{3}_G\to \mathbb R$ is called the \emph{Bergmann cocycle}\footnote{We choose the normalization of \cite{Clerc}, the normalization chosen in \cite{DT} is such that $\b_{DT}=\pi\cdot\b_{\mathcal S}$, the one of \cite{Toledo} is such that $\b_{BIW}=\frac{\beta_{\mathcal S}}{2}$} and it is a measurable strict cocycle.
The maximality of the Bergmann cocycle detects when a triple of points is contained in the Shilov boundary of a tube-type subdomain:
\begin{prop}\label{prop:bergmann}
\begin{enumerate}
\item $\beta_\mathcal S$ is a strict alternating $G$-invariant cocycle with values in $[-{\rm rk}\mathcal X,{\rm rk}\mathcal X]$,
\item If $\beta_\mathcal S(s_1,s_2,s_3)={\rm rk}\mathcal X$ then the triple $(s_1,s_2,s_3)$ is contained in the Shilov boundary of a tube-type subdomain.
\item The Bergmann cocycle is a complete invariant for the $G$ action on triples of pairwise transverse points contained in a tube type subdomain.
\end{enumerate}
\end{prop}
\begin{proof}
The first fact was proven in \cite{Clerc}, the second can be found in \cite[Proposition 5.6]{tight}, the third follows from the transitivity of the $G$-action on maximal tube type subdomains of $\mathcal S_G$ and \cite[Theorem 5.2]{CN}.
\end{proof}
We will call a triple $(s_1,s_2,s_3)$ in $\mathcal S_G^3$ satisfying $|\beta_\mathcal S(s_1,s_2,s_3)|={\rm rk} (\mathcal X)$ a \emph{maximal} triple.
In the case where $G$ is ${\rm SU}(1,p)$, that is a finite cover of the connected component of the identity in $\text{Isom} (\H^p_\mathbb C)$, the maximal tube-type subdomains are complex geodesics of $\H^p_\mathbb C$ and the Bergmann cocycle coincides with Cartan's angular invariant $c_p$ \cite[Section 7.1.4]{Goldman}. Following Cartan's notation we will denote by \emph{chains} the boundaries of the complex geodesics.
\subsection{Continuous (bounded) cohomology and maximal representations}
We introduce now the concepts we will need about continuous and continuous bounded cohomology, standard references are respectively \cite{BW} and \cite{Mon}. A quick introduction to the relevant aspects of continuous bounded cohomology can also be found in \cite{Formula}.
Throughout the section $G$ will be a locally compact second countable group, every finitely generated group fits in this class when endowed with the discrete topology. The \emph{continuous cohomology} of $G$ with real coefficients, ${\rm H}^n_{\rm c}(G,\mathbb R)$ is the cohomology of the complex $({\rm C}^n_{\rm c}(G,\mathbb R)^G,{\rm d})$ where
$${\rm C}^n_{\rm c}(G,\mathbb R)=\{f:G^{n+1}\to \mathbb R|\; f\text{ is a continuous function }\},$$
the invariants are taken with respect to the diagonal action,
and the differential ${\rm d}^n:{\rm C}_{\rm c}^{n}(G,\mathbb R)\to {\rm C}^{n+1}_{\rm c}(G,\mathbb R)$ is defined by the expression
$${\rm d}^nf(g_0,\ldots,g_{n+1})=\sum_{i=0}^{n+1}(-1)^if((g_0,\ldots,\hat g_i,\ldots, g_{n+1}).$$
Similarly the \emph{continuous bounded cohomology} ${\rm H}^n_{\rm cb}(G,\mathbb R)$ of $G$ is the cohomology of the subcomplex $({\rm C}^n_{\rm cb}(G,\mathbb R)^G,{\rm d})$ of $({\rm C}_{\rm c}^n(G,\mathbb R)^G, {\rm d})$ consisting of bounded functions.
The inclusion $i:{\rm C}^n_{\rm cb}(G,\mathbb R)^G\to {\rm C}_{\rm c}^n(G,\mathbb R)^G$ induces, in cohomology, the so-called \emph{comparison map} $c: {\rm H}^n_{\rm cb}(G,\mathbb R)\to{\rm H}^n_{\rm c}(G,\mathbb R)$.
The Banach norm on the cochain modules ${\rm C}^n_{\rm cb}(G,\mathbb R)$ defined by
$$\|f\|_\infty=\sup_{(g_0,\ldots, g_n)\in G^{n+1}}|f(g_0,\ldots,g_n)|$$
induces a seminorm on ${\rm H}^n_{\rm cb}(G,\mathbb R)$ that is usually referred to as the \emph{canonical seminorm} or \emph{Gromov's norm}.
Most of the results about continuous and continuous bounded cohomology are based on the functorial approach to the study of these cohomological theories that is classical in the case of continuous cohomology and was developed by Burger and Monod \cite{BMJEMS} in the setting of continuous bounded cohomology. This allows to show that the cohomology of many different complexes realizes canonically the given cohomological theory. Since we will only need applications of this machinery that are already present in the literature we will not describe it any further here and we refer instead to \cite{BW, Mon} for details on this nice subject.
A first notable application of this approach to continuous cohomology is van Est Theorem \cite{vanEst, Dupont} that realizes the continuous cohomology of a semisimple Lie group in terms of $G$-invariant differential forms on the associated symmetric space:
\begin{thm}[van Est]
Let $G$ be a semisimple Lie group without compact factors, then
$$ \Omega^n(\mathcal X,\mathbb R)^G\cong {\rm H}^n_{\rm c}(G,\mathbb R) .$$
Under this isomorphism the differential form $\omega$ corresponds to the class of the cocycle $c_\o$ defined by the formula
$$c_\o(g_0,\ldots,g_n)=\frac{1}{\pi}\int_{\Delta(g_0x,\ldots g_nx)}\o$$
for any fixed basepoint $x$ in $\mathcal X$.
\end{thm}
Let us now focus more specifically on the second bounded cohomology of a Hermitian Lie group $G$.
By van Est isomorphism the module ${\rm H}^2_{\rm c}(G,\mathbb R)$ is isomorphic to the vector space of the $G$-invariant differential 2-forms on $\mathcal X$ which are generated, as a real vector space, by the K\"ahler classes of the irreducible factors of the symmetric space $\mathcal X$. The class corresponding via van Est isomorphism to the K\"ahler class $\o$ of $\mathcal X$ is represented by the cocycle $c_\o(g_0,g_1,g_2)=\b_\mathcal X(g_0x,g_1x,g_2x)$ where $x\in\mathcal X$ is any fixed point.
It was proven in \cite{DT} for the irreducible classical domains and in \cite{CO} in the general case that the absolute value of the cocycle $c_\o$ is bounded by $\text{rk}(\mathcal X)$, hence the class $[c_\o]$ is in the image of the comparison map $c:{\rm H}^2_{\rm cb}(G,\mathbb R)\to {\rm H}^2_{\rm c}(G,\mathbb R)$.
Moreover, if $G$ is a connected semisimple Lie group
with finite center and without compact factors, the
comparison map $c$ is injective (hence an
isomorphism)
in degree 2 \cite{BMJEMS}. We will denote by $\k^b_G$ the \emph{bounded
K\"ahler class}, that is the class in ${\rm H}^2_{cb}(G,\mathbb R)$ satisfying $c(\k^b_G)=[c_\o]$. The Gromov norm of $\k^b_G$ can
be computed explicitly:
\begin{thm}[\cite{DT,CO,tight}]\label{thm:gromovnorm}
Let $G$ be a Hermitian Lie group with associated symmetric space $\mathcal X$ and let $\k^b_G$ be its bounded K\"ahler class. If $\|\cdot\|$ denotes the Gromov norm, then
$$\|\k^b_G\|=\text{\emph{rk}}(\mathcal X).$$
\end{thm}
Let now $M$ be a locally compact second countable topological group, $G$ a Lie group of Hermitian type, $\rho:M\to G$ a continuous homomorphism. The precomposition with $\rho$ at the cochain level induces a pullback map in bounded cohomology $\rho_b^*:{\rm H}^2_{\rm cb}(G,\mathbb R)\to {\rm H}^2_{\rm cb}(M,\mathbb R)$ that is norm non increasing.
\emph{Tight homomorphisms} were first defined in \cite{tight}, these are homomorphisms $\rho$ for which the pullback map is norm preserving, namely $\|\rho^*(\k_M^b)\|=\|\k_M^b\|$. In the same paper the following structure theorem is proven:
\begin{thm}[{\cite[Theorem 7.1]{tight}}]\label{thm:tight}
Let $L$ be a locally compact second countable group, $\mathbf G$ a connected algebraic group defined over $\mathbb R$ such that $G=\mathbf G(\mathbb R)^\circ$ is of Hermitian type. Suppose that $\rho:L\to G$ is a continuous tight homomorphism. Then
\begin{enumerate}
\item The Zariski closure $\mathbf H=\overline{\rho(L)}^Z$ is reductive.
\item The group $H=\mathbf H(\mathbb R)^\circ$ almost splits as a product $H_{nc}H_c$ where $H_c$ is compact and $H_{nc}$ is of Hermitian type.
\item If $\mathcal Y$ is the symmetric space associated to $H_{nc}$, then the inclusion of $\mathcal Y$ in $\mathcal X$ is totally geodesic and the Shilov boundary $\mathcal S_{H_{nc}}$ sits as a subspace of $\mathcal S_G$.
\end{enumerate}
\end{thm}
In the case when also $L$ is an Hermitian Lie group, tight homomorphisms can be completely classified: in fact it is possible to prove that, if $L$ has no simple factor locally isomorphic to ${\rm SU}(1,1)$, then the map $\rho$ is equivariant with an holomorphic map (see \cite{Ham2} for the case when $L$ is simple, and \cite{HamP} for the general case) and in particular the classification of \cite{Ham1} applies. In our setting this implies the following;
\begin{thm}\label{thm:tightol}
Let $i:L\to {\rm SU}(m,n)$ be a tight homomorphism, assume that no factor of $L$ is locally isomorphic to ${\rm SU}(1,1)$. Then each simple factor of $L$ is either isomorphic to ${\rm SU}(s,t)$ or is of tube type. Moreover if $L=L_t\times L_{nt}$ where $L_t$ is the product of all the irreducible factors of tube type, then there exists an orthogonal decomposition $\mathbb C^{m,n}=\mathbb C^{k,k}\oplus\mathbb C^{m-k,n-k}$ such that $L_{t}$ is included in ${\rm SU}(\mathbb C^{k,k})$ and $L_{nt}$ is included in ${\rm SU}(\mathbb C^{m-k,n-k})$.
\end{thm}
\begin{proof}
This can be found in \cite{HamP}.
\end{proof}
A key feature of bounded cohomology is that, whenever $\Gamma$ is a lattice in $G$, it is possible to construct a left inverse ${\rm T}_{\rm b}^\bullet:{\rm H}^\bullet_{\rm b}(\Gamma)\to{\rm H}^\bullet_{\rm cb}(G)$ of the restriction map.
Indeed the bounded cohomology of $\Gamma$ can be computed from the complex $({\rm C_{cb}^\bullet}(G,\mathbb R)^\Gamma,{\rm d})$ and the transfer map ${\rm T}_{\rm b}^\bullet$ can be defined on the cochain level by the formula
$${\rm T}^k_{\rm b}(c(g_0,\ldots,g_k))=\int_{\Gamma\backslash G}c(gg_0,\ldots,gg_k)\text{d}\mu(g)$$ where $\mu$ is the measure on $\Gamma\backslash G$ induced by the Haar measure of $G$ provided it is normalized to have total mass one.
It is worth remarking that when we consider instead continuous cohomology (without boundedness assumptions), a transfer map can be defined with the very same formula only for cocompact lattices, but the restriction map is in general not injective if the lattice is not cocompact.
Let us fix a representation $\rho:\Gamma\to G$. Since ${\rm H}^2_{\rm cb}({\rm SU}(1,p),\mathbb R)=\mathbb R\k^b_{{\rm SU}(1,p)}$, the class ${\rm T}^*_{\rm b}\rho^*(\k^b_G)$ is a scalar multiple of the K\"ahler class $\k^b_{{\rm SU}(1,p)}$ . The \emph{generalized Toledo invariant}\footnote{The original definition of the generalized Toledo invariant given in \cite{BIpreprint} used continuous cohomology only. However it is proven in \cite[Lemma 5.3]{MW} that the invariant that was originally defined in \cite{BIpreprint}, $i_\rho$ in the notation of that article, and the invariant we defined here, that there was denoted by $\rm t_{ b}(\rho)$, coincide.} of the representation $\rho$ is the number $i_\rho$ such that ${\rm T}^*_{\rm b}\rho^*(\k^b_G)= i_\rho\k^b_{{\rm SU}(1,p)}$. A consequence of Theorem \ref{thm:gromovnorm}, and the fact that the transfer map is norm non-increasing, is that $|i_\rho|\leq \text{rk}(\mathcal X)$.
\begin{defn}
A representation $\rho$ is \emph{maximal} if $|i_\rho|=\text{rk}(\mathcal X)$. Clearly maximal representation are in particular tight representations.
\end{defn}
The following lemma will be useful at the very end of the article, in the proof of Corollary \ref{cor:local rigidity}:
\begin{lem}\label{lem:Toledo constant}
The generalized Toledo invariant is constant on connected components of the representation variety.
\end{lem}
\begin{proof}
This is proven in \cite[Page 4]{BICartan}.
\end{proof}
\subsection{Boundary maps and maximal representations}
The existence of measurable boundary maps for Zariski dense homomorphisms in algebraic groups is by now classical:
\begin{prop}\cite[Proposition 7.2]{SUpq}\label{prop:boundary map}
Let $\Gamma$ be a lattice in ${\rm SU}(1,p)$, $G$ a Lie group of Hermitian type and let $\rho:\Gamma\to G$ be a Zariski dense representation. Then there exists a $\rho$-equivariant measurable map $\phi:\partial\mathbb H _{\mathbb C}^p\to\mathcal S_G$ such that, for almost every pair of points $x, y$ in $\partial\mathbb H _{\mathbb C}^p$, $\phi(x)$ and $\phi(y)$ are transverse.
\end{prop}
Let us now fix a measurable map $\phi:\partial\mathbb H _{\mathbb C}^p\to\mathcal S_{G}$, and define the \emph{essential Zariski closure} of $\phi$ to be the minimal Zariski closed subset $V$ of $\mathcal S_{G}$ such that $\mu(\phi^{-1}(V))=1$.
Such a set exists since the intersection of finitely many closed subset of full measure has full measure and $\mathcal S_{G}$ is an algebraic variety, in particular it is Noetherian.
We will say that a measurable boundary map $\phi$ is \emph{Zariski dense} if its essential Zariski closure is the whole $\mathcal S_{G}$.
\begin{prop}\label{prop:phiZariskidense}
Let $\rho$ be a Zariski dense representation, then $\phi$ is Zariski dense.
\end{prop}
\begin{proof}
Indeed let us assume by contradiction that the essential Zariski closure of $\phi(\partial\mathbb H _{\mathbb C}^p)$ is a proper Zariski closed subset $V$ of $\mathcal S_G$. The set $V$ is $\rho(\Gamma)$-invariant: indeed for every element $\gamma$ in $\Gamma$, we get $\mu(\phi^{-1}(\rho(\gamma)V))=\mu(\gamma\phi^{-1}(V))=1$, hence, in particular, $\rho(\gamma)V=V$ by minimality of $V$.
Let us now recall that the Shilov boundary $\mathcal S_G$ is an homogeneous space for $G$, and let us fix the preimage $W$ of $V$ under the projection map $G\to G/Q=\mathcal S_G$. $W$ is a proper Zariski closed subset of $G$, moreover if $g$ is any element in $W$, the Zariski dense subgroup $\rho(\Gamma)$ of $G$ is contained in $Wg^{-1}$ and this gives a contradiction.
\end{proof}
One of the advantages of bounded cohomology when proving rigidity statements is that the bounded cohomology of a group can be computed from a suitable boundary of the group itself, for example when $\Gamma$ is a lattice in ${\rm SU}(1,p)$, the complex $({\rm L}^\infty_{\text{alt}}((\partial\mathbb H _{\mathbb C}^p)^\bullet,\mathbb R)^\Gamma,d)$ realizes isometrically the bounded cohomology of $\Gamma$ \cite{BMGAFA}. Moreover, exploiting functoriality properties of bounded cohomology, one can implement the pullback via a measurable boundary map provided by Proposition \ref{prop:boundary map} thus getting the following result:
\begin{prop}[{\cite[Theorem 2.41]{Formula}} ]\label{prop:Formula}
Let $\Gamma$ be a lattice in ${\rm SU}(1,p)$ and let $G$ be a Hermitian Lie group. Let $\rho:\Gamma\to G$ be a representation, $\beta_\mathcal S:( \mathcal S_G)^{3}\to \mathbb R$ the Bergmann cocycle and $\phi:\partial\mathbb H _{\mathbb C}^p\to G/Q$ be a measurable $\rho$-equivariant boundary map. For almost every triple $(x,y,z)$ in $\partial\mathbb H _{\mathbb C}^p$, the formula
$$i_\rho c_p(x,y,z)=\int_{\Gamma\backslash {\rm SU}(1,p)}\beta_\mathcal S(\phi(gx),\phi(gy),\phi(gz)) {\rm d}\mu(g) $$ holds.
\end{prop}
We will now show that, since $\beta_\mathcal S$ is a strict $G$-invariant cocycle and ${\rm SU}(1,p)$ acts transitively on pairs of distinct points of $\partial\mathbb H _{\mathbb C}^p$, the equality holds for every triple of pairwise distinct points (this is an adaptation in our context of an argument due to Bucher: cfr. the proof of \cite[Proposition 3]{Mostow} in case $n=3$).
\begin{lem}\label{lem:everytriple}
The equality in Proposition \ref{prop:Formula} holds for every triple $(x,y,z)$ of pairwise distinct points.
\end{lem}
\begin{proof}
The formula of Proposition \ref{prop:Formula} is an equality between ${\rm SU}(1,p)$-invariant strict cocycles:
clearly this is true for the left-hand side, moreover the expression on the right-hand side is a strict cocycle since $\beta_\mathcal S$ is, and is ${\rm SU}(1,p)$ invariant since $\phi$ is $\rho$-equivariant and $\beta_\mathcal S$ is $G$-invariant.
Let us now fix a ${\rm SU}(1,p)$-invariant full measure set $\mathcal O\subseteq (\partial\mathbb H _{\mathbb C}^p)^{3}$ on which the equality holds.
Since $\mathcal O$ is of full measure, an application of Fubini's Theorem is that for almost every pair $(y_1,y_2)\in(\partial\mathbb H _{\mathbb C}^p)^2$ the set of points $z\in\partial\mathbb H _{\mathbb C}^p$ such that $(y_1,y_2,z)\in\mathcal O$ is of full measure. Let us fix a pair $(y_1,y_2)$ for which this holds and denote by $\mathcal W$ the set of points $z$ such that $(y_1,y_2,z)\in\mathcal O$.
Let us now fix a triple $(x_1,x_2,x_3)$ of points in $\partial\mathbb H _{\mathbb C}^p$.
Since the ${\rm SU}(1,p)$ action on $\partial\mathbb H _{\mathbb C}^p$ is transitive on pairs of distinct points, for every $i$ there exist an element $g_i$ such that $(x_i,x_{i+1})=(g_iy_1,g_iy_2)$.
Let us now fix a point $x_3$ in the full measure set $g_1\mathcal W\cap g_2\mathcal W\cap g_3\mathcal W$. Since $x_3$ is in $g_i\mathcal W$, we get that $g_i^{-1}x_3\in\mathcal W$, and hence $(x_i,x_{i+1},x_3)=g_i(y_1,y_2,g_i^{-1}{x_3})\in \mathcal O$.
In particular, computing the cocycle identity on the 4tuple $(x_0,x_1,x_2,x_3)$ we get that the identity of Proposition \ref{prop:Formula} holds for the triple $(x_0,x_1,x_2)$.
\end{proof}
\begin{cor}\label{cor:incidence preserved}
Let $\rho:\Gamma\to G$ be a maximal representation and let $\phi:\partial\mathbb H _{\mathbb C}^p\to\mathcal S_{G}$ be a $\rho$-equivariant measurable boundary map. Then for almost every maximal triple $(x,y,z)\in (\partial\mathbb H _{\mathbb C}^p)^3$, the triple $(\phi(x),\phi(y),\phi(z))$ is contained in the Shilov boundary of a tube-type subdomain and is a maximal triple.
\end{cor}
\begin{proof}
Let us fix a positively oriented triple $(x,y,z)$ of points on a chain. We know from Lemma \ref{lem:everytriple} that the equality
$$\int_{{\rm SU}(1,p)/\Gamma}\beta_\mathcal S(\phi(gx),\phi(gy),\phi(gz)) dg=\text{rk}(\mathcal X)$$
holds:
since $\rho$ is maximal, then $i_\rho={\rm rk} (\mathcal X)$, and since $(x,y,z)$ are on a chain then $c_p(x,y,z)=1$.
Since $\|\beta_\mathcal S\|_\infty={\rm rk}(\mathcal X)$,
it follows that $\beta_\mathcal S(\phi(gx),\phi(gy),\phi(gz))=\text{rk}(\mathcal X)$ for almost every $g$ in ${\rm SU}(1,p)$. By Proposition \ref{prop:bergmann}, this implies that for almost every $g\in {\rm SU}(1,p)$, the triple $(\phi(gx),\phi(gy),\phi(gz))$ is contained in the boundary of a tube-type subdomain. Since maximal triples in $\partial\mathbb H _{\mathbb C}^p$ form an ${\rm SU}(1,p)$-orbit, the fact that the result holds for almost every element $g$ implies that the result holds for almost every triple of positively oriented points in a chain. The same argument applies for negatively oriented triples.
\end{proof}
\begin{defn} A measurable map $\phi$ \emph{preserves the chain geometry} if, for almost every pair $x,y$ in $\partial\mathbb H _{\mathbb C}^p$, the images $\phi(x),\phi(y)$ are transverse subspaces and, for almost every maximal triple $(x,y,z)\in (\partial\mathbb H _{\mathbb C}^p)^3$, the triple $(\phi(x),\phi(y),\phi(z))$ is maximal.
\end{defn}
This amounts to saying that the map $\phi$ induces an almost everywhere defined morphism $(\phi,\hat \phi)$ from the geometry $\partial\mathbb H _{\mathbb C}^p\times\mathcal C$ whose points are points in $\partial\mathbb H _{\mathbb C}^p$ and whose lines are the chains, to the geometry $\mathcal S_G\times \mathcal T$ whose points are points in $\mathcal S_{G}$ and whose lines are the Shilov boundaries of maximal tube-type subdomains of $\mathcal S_{G}$. The morphism $(\phi,\hat\phi)$ has the property that it preserves the incidence structure almost everywhere.
Purpose of the next sections is to show that a measurable Zariski dense map $\phi:\partial\mathbb H _{\mathbb C}^p\to \mathcal S_{{\rm SU}(m,n)}$ that preserves the chain geometry coincides almost everywhere with an algebraic map.
\section{The Chain geometry of $\mathcal S_{m,n}$}\label{sec:S_mn}
For the rest of the paper we will restrict our attention to the Hermitian Lie group ${\rm SU}(m,n)$ consisting of complex matrices that preserve a non-degenerate Hermitian form of signature $(m,n)$ with $m\leq n$, and denote, for the sake of brevity, by $\mathcal S_{m,n}$ the Shilov boundary of ${\rm SU}(m,n)$ that was previously denoted by $\mathcal S_{{\rm SU}(m,n)}$. The purpose of this section is to understand some features of the incidence structure of the subsets of $\mathcal S_{m,n}$ that arise as Shilov boundaries of the maximal tube-type subdomains of the symmetric space associated to ${\rm SU}(m,n)$. For reasons that will be explained later we will call $m$-\emph{chains} such subsets. The main tool that we will introduce in our investigations is a projection map $\pi_x$, depending on the choice of a point $x\in \mathcal S_{m,n}$. The map $\pi_x$ associates to a point $y$ that is transverse to $x$ the uniquely determined $m$-chain that contains both $x$ and $y$. The central results of the section are Proposition \ref{prop:lifts of k-vertical
chains} and Proposition \ref{prop:errors}.
\subsection{A model for $\mathcal S_{m,n}$.}\label{sec:S_mn1}
Throughout the paper we will realize ${\rm SU}(m,n)$ as the subgroup of ${\rm SL}(m+n,\mathbb C)$ that preserves the Hermitian form $h$ represented with respect to the standard basis by the matrix $h=\bsm0&0&{\rm Id}_m\\0&-{\rm Id}_{n-m}&0\\{\rm Id}_m&0&0\esm$.
We will denote by ${\rm Gr}_m(\mathbb C^{m,n})$ the Grassmannian of $m$-dimensional subspaces of $\mathbb C^{m+n}$.
It is well known that the Shilov boundary $\mathcal S_{m,n}$ can be realized as the subset of ${\rm Gr}_m(\mathbb C^{m,n})$ consisting of subspaces that are isotropic for the form $h$. Both $\mathcal S_{m,n}$ and the action of ${\rm SU}(m,n)$ on it are real algebraic\footnote{More details can be found in \cite{Pthesis} and in \cite{SUpq} where it is also possible to find the description of a complex variety $\mathbf{\mathcal S_{m,n}}$ such that $\mathcal S_{m,n}=\mathbf{\mathcal S_{m,n}}(\mathbb R)$}.
It is classical and easy to verify that the unique open ${\rm SU}(m,n)$-orbit on $\mathcal S_{m,n}^2$ consists of pairs of points whose underlying vector spaces are \emph{transverse}. In particular we will often identify a point $x\in \mathcal S_{m,n}$ with its underlying vectorspace, and we will use the notation $x\pitchfork y$, that would be more suited for the linear setting, with the meaning that the pair $(x,y)$ is a pair of transverse points.
We will use the notation $\mathcal S_{m,n}^{(2)}$ for the set of pairs of transverse points, and we will denote the set of points in $\mathcal S_{m,n}$ that are transverse to a given point $x$ by
$$\mathcal S_{m,n}^x:=\{y\in\mathcal S_{m,n}|\;y\pitchfork x\}.$$
If $x$ and $y$ are transverse point, the linear span $\<x,y\>$ is a $2m$ dimensional subspace $V$ of $\mathbb C^{m+n}$ on which $h$ has signature $(m,m)$. The Shilov boundary of the maximal tube type subdomain containing $x$ and $y$ is the set
$$\mathcal S_{V}=\{z\in {\rm Gr}_m(V)|\;h|_z=0\}=\mathcal S_{m,n}\cap {\rm Gr}_m(V).$$
\begin{defn}
An $m$-\emph{chain} in $\mathcal S_{m,n}$ is a subspaces of the form $\mathcal S_V$ for some linear subspace $V$ of $\mathbb C^{m,n}$ on which $h$ restricts to an Hermitian form of signature $(m,m)$.
\end{defn}
Clearly any pair of transverse points $x,y$ in $\mathcal S_{m,n}$ uniquely determines a $m$-chain $\mathcal S_{\langle x,y\rangle}$ with the property that both $x$ and $y$ belong to $\mathcal S_{\langle x,y\rangle}$. We will denote by $T_{x,y}$ such a chain.
In the case $m=1$, that is $\mathcal X_{m,n}=\H^n_\mathbb C$, the 1-chains are boundaries of complex geodesics or \emph{chains} in Cartan's terminology. This is the reason why we chose to call the Shilov boundaries of maximal tube-type subdomains \emph{$m$-chains}. To be more consistent with Cartan's notation, we omit the 1, and simply call \emph{chains} the $1$-chains.
\subsection{The Heisenberg model $\mathcal H^{m,n}(x)$.}\label{ssec:Heisenberg}
We now want to give a model for the Zariski open subset of $\mathcal S_{m,n}$ consisting of points transverse to a given point $x$. The model we are introducing is sometimes referred to as (the boundary of) a Siegel domain of genus two and was studied, for example, by Koranyi and Wolf in \cite{KorWo}. In the case $m=1$ this model is described in \cite[Chapter 4]{Goldman} but our conventions here will be slightly different.
In the rest of the paper, for each complex matrix $X$ we will denote by $X^T$ the transpose of $X$, by $X^*=\overline X^T$ the transpose conjugate of $X$. If, moreover, $X$ is invertible we will denote by $X^{-1}$ the inverse of $X$ and by $X^{-*}$ the inverse of $X^*$. Moreover we will indicate a point $V$ in the Grassmannian ${\rm Gr}_m(\mathbb C^{m+n})$ with a $(n+m)\cdot m$ dimensional matrix: we will understand such a matrix as an ordered basis of the subspace $V$. Clearly two matrices $X,Y$ represent the same element in ${\rm Gr}_m(\mathbb C^{m+n})$ if and only if there exists a matrix $G\in {\rm GL}_m(\mathbb C)$ such that $X=YG$. A direct computation gives that a point $x\in {\rm Gr}_m(\mathbb C^{m+n})$ represented by the matrix $\left[\begin{smallmatrix} X_1\\X_2\\X_3\esm$ belongs to $\mathcal S_{m,n}$ if and only if $X_1^*X_3+X_3^*X_1-X_2^*X_2=0$ where $X_1$ and $X_3$ have $m$ rows and $X_2$ has $n-m$ rows.
Let us focus on the maximal isotropic subspace
$$v_\infty=\<e_i|\;1\leq i\leq m\>=\left[\begin{smallmatrix}{\rm Id}_m\\0\\0\esm\in \mathcal S_{m,n}.$$
The set of points $\mathcal S_{m,n}^{v_\infty}$ that are transverse to $v_\infty$ admit a basis of the form $\left[\begin{smallmatrix} X\\Y\\{\rm Id}_m\esm$ with $X^*+X-Y^*Y=0$. We will identify such a set with the linear space $M((n-m)\times m,\mathbb C)\times \mathfrak{u}(m)$ where $\mathfrak{u}(m)$ is the set of antiHermitian matrices. We use the symbol $\mathcal H^{m,n}(v_\infty)$ for such a linear space, that will be understood as parametrizing $\mathcal S_{m,n}^{v_\infty}$ via the map
$$\begin{array}{ccccc}
\mathcal H^{m,n}(v_\infty)=& M((n-m)\times m,\mathbb C)\times \mathfrak{u}(m)&\to &\mathcal S_{m,n}^{v_\infty}\\
&(X,Y)&\mapsto &\left[\begin{smallmatrix} Y+X^*X/2\\X\\{\rm Id}_m\esm.
\end{array}
$$
We refer to $\mathcal H^{m,n}(v_\infty)$ as the Heisenberg model. This is because, as we will now see, $\mathcal H^{m,n}(v_\infty)$ identifies with the generalized Heisenberg group that is the nilpotent radical of the stabilizer of $v_\infty$.
Let us denote by $Q$ the maximal parabolic subgroup of ${\rm SU}(\mathbb C^{m+n}, h)$ that is the stabilizer of $v_\infty$. It is easy to verify that
$$Q=\left\{\begin{bmatrix} A&B&E\\0&C&F\\0&0&A^{-*}\end{bmatrix}\hspace{-10pt}\begin{array}{l}_m\\_{n-m}\\_{m} \end{array} \left|
\begin{array}{l} A\in {\rm GL}_m(\mathbb C),\, C\in {\rm U}(n-m),\\A^{-1}B-F^*C=0 \{\rm E}^*A^{-*}+A^{-1}E-F^*F=0\\\det C\det A\det A^{-*}=1\end{array} \right\}. \right. $$
The group $ Q$ can be written as $L\ltimes N$ where
$$ L=\left\{ \left.\begin{bmatrix} A&0&0\\0&C&0\\0&0&A^{-*}\end{bmatrix} \right| \begin{array}{l}A\in {\rm GL}_m(\mathbb C), \\mathbb C\in {\rm U}(n-m)\\\det C\det A\det A^{-*}=1\end{array}\right\}$$
is reductive and
$$N=\left\{ \left.\begin{bmatrix} {\rm Id}&E^*&F\\0&{\rm Id}&E\\0&0&{\rm Id}\end{bmatrix}\right| F^*+F-E^*E=0\right\}$$
is nilpotent.
The group $L$ is the stabilizer of the two transverse points $v_\infty$ and $v_0=\<e_{n+1},\ldots,e_{n+m}\>$ of $\mathcal S_{m,n}$. If we denote by $a$ the determinant of the matrix $A$, an explicit isomorphism between ${\rm GL}_m(\mathbb C)\times {\rm SU}(n-m)$ and $L$ is given by:
$$\begin{array}{ccc}
{\rm GL}_m(\mathbb C)\times SU(n-m)&\to &L\\
(A,B)&\mapsto &\left[\begin{smallmatrix} A&0&0\\0&\overline aa^{-1}B&0\\0&0&A^{-*}\esm.
\end{array}
$$
Similarly the 2 step nilpotent group $N$ can be identified with $ M(n\times m,\mathbb C)\ltimes \mathfrak{u}(m)$:
$$\begin{array}{ccc}
M((n-m)\times m,\mathbb C)\ltimes \mathfrak{u}(m)&\to &N\\
(E,F)&\mapsto &\left[\begin{smallmatrix} {\rm Id}&E^*&F+EE^*/2\\0&{\rm Id}&E\\0&0&{\rm Id}\esm.
\end{array}
$$
It is particularly easy to describe the action of $L$ and $N$ on $\mathcal H^{m,n}(v_\infty)$: the group $N$ acts by left multiplication according to the group structure on $ M(n\times m,\mathbb C)\ltimes \mathfrak{u}(m)$
$$(E,F)\cdot(X,Y)=\left(E+X,F+Y+\frac{E^*X-X^*E}2\right)$$
and $L$ acts via right-left matrix multiplication on the first factor and conjugation on the second:
$$(A,B)\cdot(X,Y)=(\overline a a^{-1}BXA^{*},AYA^*).$$
\subsection{The projection $\pi_x$}\label{ssec:projpx}
We consider the projection on the first factor $\pi_{v_\infty}:\mathcal H^{m,n}(v_\infty)\to M((n-m)\times m,\mathbb C)$. Under the natural identification $\mathcal H^{m,n}(v_\infty)\cong N$, this projection corresponds to the group homomorphism whose kernel is the center $\mathfrak{u}(m)$ of $N$. Purpose of this section is to give a geometric interpretation of the quotient space $M((n-m)\times m,\mathbb C)$: it corresponds to a parametrization of the space of $m$-chains through the point $v_\infty$.
In order to make this statement more precise let us consider the set
$$ W_{v_\infty}=\{V\in {\rm Gr}_{2m}(\mathbb C^{m+n})|\; v_\infty< V,\; h|_{V} \text{ has signature } (m,m) \}.$$
The following lemma gives an explicit identification of $ W_{v_\infty}$ with the quotient space $M((n-m)\times m,\mathbb C)=N/_{\mathfrak{u}(m)}$:
\begin{lem}\label{lem:vchains}
There exists a bijection between $M((n-m)\times m,\mathbb C)$ and $ W_{v_\infty}$ defined by the formula
\begin{displaymath}\begin{array}{cccc} i:M((n-m)\times m,\mathbb C)&\to& W_{v_\infty}\\
A&\mapsto&\begin{bmatrix} A^*\\{\rm Id}\\0\end{bmatrix}^\bot.
\end{array}\end{displaymath}
\end{lem}
\begin{proof}
Let $V$ be a point in $W_{v_\infty}$. Then $V^\bot$ is a $(n-m)$ dimensional subspace of $\mathbb C^{m+n}$ that is contained in $v_\infty^\bot$. This implies that $V^\bot$ admits a basis of the form $\begin{bmatrix} A&B&0\end{bmatrix}^T$ where $A$ has $m$ rows and $n-m$ columns and $B$ is a square $n-m$ dimensional matrix. Since the restriction of $ h$ on $V$ has signature $(m,m)$, the restriction of $ h$ to $V^\bot$ is negative definite, in particular the matrix $B$ must be invertible. This implies that, up to changing the basis of $V^\bot$, we can assume that $B={\rm Id}_{n-m}$. This gives the desired bijection.
\end{proof}
$W_{v_\infty}$ parametrizes the $m$-chains containing the point $v_\infty$. We will call them \emph{vertical} chains: the intersection $T^{v_\infty}$ of a vertical chain $T$ with the Heisenberg model $\mathcal H_{m,n}(v_\infty)$ consists precisely of a fiber of the projection on the first factor in the Heisenberg model:
\begin{lem}\label{lem:vertical chains}
Let $T\subset \mathcal S_{m,n}$ be a vertical chain and $V$ be its associated linear subspace. If we denote by $p_T$ in $M((n-m)\times m,\mathbb C)$, the point $p_T=i^{-1}(V)$, we have:
\begin{enumerate}
\item for every $x$ in $T^{v_\infty}$, then $\pi_{v_\infty}(x)=p_T$,
\item $T^{v_\infty}=\pi_{v_\infty}^{-1}(p_T)$,
\item the center $ M$ of $N$ acts simply transitively on $T^{v_\infty}$.
\end{enumerate}
\end{lem}
\begin{proof}
(1) An element $w$ of $\mathcal H_{m,n}(v_\infty)$ with basis $\begin{bmatrix} X&Y& {\rm Id}_m\end{bmatrix}^T$ belongs to the chain $T$ if and only if $Y=p_T$: indeed the requirement that $V^\bot$ is contained in $w^\bot$ restates as
$$0=\begin{bmatrix} X^*&Y^*&{\rm Id}_m\end{bmatrix} \bpm0&0&{\rm Id}\\0&-{\rm Id}&0\\{\rm Id}&0&0\end{bmatrix}\begin{bmatrix} p_T^*\\{\rm Id}\\0\end{bmatrix}=p_T^*-Y^*.$$
This implies that for every $w\in T$, we have $\pi_{v_\infty}(w)=p_T$.
Viceversa if $\pi_{v_\infty}(w)=p_T$, then $w$ is contained in $V$ and this proves (2).
(3) The fact that $M$ acts simply transitively on $T^{v_\infty}$ is now obvious: indeed $M$ acts on the Heisenberg model by vertical translation stabilizing every vertical chain.
\end{proof}
The stabilizer $Q$ of $v_\infty$ naturally acts on the space $W_{v_\infty}$ and it is easy to deduce explicit formulae for this action from the formulae given in Section \ref{ssec:Heisenberg}.
In the sequel, when this will not cause confusion, we will identify $W_{v_\infty}$ with $M((n-m)\times m,\mathbb C)$ considering implicit the map $i^{-1}$.
It is worth remarking that everything we did so far doesn't really depend on the choice of the point $v_\infty$, and a map $\pi_x: \mathcal S_{m,n}^{\; x}\to W_x$ can be defined for every point $x\in \mathcal S_{m,n}$. We decided to stick to the point $v_\infty$, since the formulae in the explicit expressions are easier.
\subsection{Projections of chains}\label{ssec:otherchains}
We now want to understand what are the possible images under the map $\pi_{v_\infty}$ of other chains. We define the \emph{intersection index} of an $m$-chain $T$ with a point $x\in \mathcal S_{m,n}$ by
$$i_{x}(T)=\dim (x\cap V_T)$$
where $V_T$ is the $2m$ dimensional linear subspace of $\mathbb C^{m+n}$ associated to $T$.
Clearly $0\leq i_{v_\infty}(T)\leq m$, and $i_{v_{\infty}}(T)=m$ if and only if the chain $T$ is vertical. In general we will call \emph{$k$-vertical} a chain whose intersection index is $k$: with this notation vertical chains are $m$-vertical. Sometimes we will call \emph{horizontal} the chains that are 0-vertical (in particular each point in the chain is transverse to $v_\infty$).
In our investigations it will be precious to be able to relate different situations via the action of the group $ G={\rm SU}(\mathbb C^{m+n}, h)$, under this respect the following lemma will be fundamental:
\begin{lem}\label{lem:transitivity on k vertical chains}
For every $k\in\{0,\ldots m\}$ the group $ G$ acts transitively on
\begin{enumerate}
\item the pairs $(x,T)$ where $x\in \mathcal S_{m,n}$ is a point and $T$ is an $m$-chain with $i_x(T)=k$,
\item the triples $(x,y,T)$ where $x\pitchfork y$, $y\in T$ and $i_x(T)=k$.
\end{enumerate}
In particular the intersection index is a complete invariant of $m$-chains up to the ${\rm SU}(m,n)$-action.
\end{lem}
\begin{proof}
We will prove directly the second statement. By transitivity of the $G$-action on the set of transverse pairs we can assume that $x=v_\infty$, $y=v_0$, in particular this reduces the proof to showing that $L$ acts transitively on the set of chains through $v_0$. It is not hard to show that the orthogonal to such a chain $T$ has a basis of the form $\bpm0&{\rm Id}_{n-m}&Z_3\end{bmatrix}^T$ for some matrix $Z_3$: any vector contained in the orthogonal to $v_0$ has vanishing components in $v_\infty$, moreover, since the orthogonal to a chain is positive definite we can assume that the central block is the identity up to changing the basis. Moreover it holds that $m-{\rm rk} (Z_3)=i_{v_\infty}(T)$. The statement is now obvious.
\end{proof}
As explained at the beginning of the section we want to give a parametrization of a generic chain $T$ and study the restriction of $\pi_{v_\infty}$ to $T$. In view of Lemma \ref{lem:transitivity on k vertical chains}, it is enough to understand, for every $k$, the parametrization and the projection of a single $k$-vertical chain.
The $k$-vertical chain we will deal with is the chain with associated linear subspace
$$V_k=\<e_i,e_{j}+e_{m+j-k}+e_{n+j},v_0|\;1\leq i\leq k<j\leq m \>.$$
\begin{lem}
$T_k$ is the linear subspace associated to a $k$-vertical chain $T_k$.
\end{lem}
\begin{proof}
$V_k$ is a $2m$-dimensional subspace containing $v_0$. Moreover $V_k$ splits as the orthogonal direct sum
$$\begin{array}{rl}
V_k&=V_k^0\overset\bot \oplus V_k^1=\\
&=\<e_i,e_{n+i}|\;1\leq i\leq k\>\overset \bot\oplus\langle e_{j}+e_{m+j-k}+e_{n+j},e_{n+j}|\;k+1\leq j\leq m\rangle=\\
&=\<v_\infty\cap V_k,e_{n+i}|\;1\leq i\leq k\>\overset \bot\oplus\langle e_{j}+e_{m+j-k}+e_{n+j},e_{n+j}|\;k+1\leq j\leq m\rangle.
\end{array}
$$
Since $v_\infty\cap V_k=\langle e_1,\ldots, e_k\rangle$, we get that $i_{v_\infty}( T_k)$ is $k$. Since $ h|_{V_k^0}$ has signature $(k,k)$ and $ h|_{V_k^1}$ has signature $(m-k,m-k)$, we get that the restriction of $ h$ on $V_k$ has signature $(m,m)$ and this concludes the proof.
\end{proof}
\begin{lem}\label{lem:parametrization of k-vertical chains}
$\mathcal H_{m,n}(v_\infty)\cap T_k$ consists precisely of those subspaces of $\mathbb C^{m+n}$ that admit a basis of the form
$$\begin{array}{ccc}
\begin{bmatrix} \frac {E^*E}2 +C&E^*X\\
E&{\rm Id}+X\\
E &{\rm Id}+X\\
0&0\\
{\rm Id}_k & 0\\
0&{\rm Id}_{m-k}
\end{bmatrix}
& \text{ with } & \left\{\begin{array}{l} E\in M((m-k)\times k,\mathbb C)\\
X\in {\rm U}(m-k)\\
C\in \mathfrak u(k).
\end{array}\right.
\end{array}$$
The projection of $T_k$ is contained in an affine subspace of $M((n-m)\times m,\mathbb C)$ of dimension $m^2-km$, and consists of the points of $M((n-m)\times m,\mathbb C)$ that have expression
$\left[\begin{smallmatrix} E&{\rm Id}+X\\0&0\esm$ with $E\in M((m-k)\times k,\mathbb C)$ and $X\in {\rm U}(m-k)$.
\end{lem}
\begin{proof}
It is enough to check that the orthogonal to $V_k$ is
$$V_k^\bot=\< e_{m+j}+e_{n+j+k},e_{m+l}|\; 1\leq j\leq m-k<l\leq n-m\>.$$
This implies that any $m$-dimensional subspace $z$ of $V_k$, that is transverse to $v_\infty$, has a basis of the form
$$\begin{array}{ccc}
z=\begin{bmatrix} Z_{11} &Z_{12}\\
Z_{21}&Z_{22}\\
Z_{21} &Z_{22}\\
0&0\\
{\rm Id} & 0\\
0&{\rm Id}
\end{bmatrix}\hspace{-7pt}
\begin{array}{l}
_k\\_{m-k
\\_{m-k}\\_{n-2m+k}\\_{k}\\_{m-k}
\end{array}
& \begin{array}{l}\text{and the}\\\text{restriction }\\\text{of $ h$ to $z$ }\\\text{is zero}\\\text{if and only if }\end{array}&
\left\{\begin{array}{l} Z^*_{11}+Z_{11}=Z_{21}^*Z_{21}\;(11)\\
Z^*_{21}+Z_{12}=Z_{21}^*Z_{22}\; (12)\\
Z^*_{12}+Z_{21}=Z_{22}^*Z_{21}\; (21)\\
Z^*_{22}+Z_{22}=Z_{22}^*Z_{22}\;(22).
\end{array}\right.
\end{array}$$
Equation $(22)$ restates as $Z_{22}={\rm Id}+X$ for some $X\in {\rm U}(m-k)$:
indeed a square matrix $Z$ satisfies the equation $Z^*+Z=Z^*Z$, if and only if the equation $(Z-{\rm Id})^*(Z-{\rm Id})=Z^*Z-Z-Z^*+{\rm Id}={\rm Id}$ holds, which means that $Z-{\rm Id}$ belongs to ${\rm U}(m-k)$.
This concludes the proof of the first part of the lemma: the $(m-k)\times k$ matrix $Z_{21}$ can be chosen arbitrarily, Equation $(12)$ uniquely determines $Z_{12}$ in function of $Z_{21}$ and $Z_{22}$, and Equation $(11)$ determines the Hermitian part of $Z_{11}$ in function of $Z_{21}$, but is satisfied independently on the antiHermitian part of $Z_{11}$. This proves the first part of the lemma.
The second part is a direct consequence of the parametrization of $T_k^{v_\infty}$ we just gave, together with the identification of $W_{v_\infty}$ and $M((n-m)\times m,\mathbb C)$ given in Lemma \ref{lem:vchains}.
\end{proof}
\begin{defn}
We will call a subset of $ W_{x}$ that is the projection of a $k$-vertical chain a \emph{$(m,k)$-circle}.
\end{defn}
The reason for the name \emph{circle} is due to the fact that, in the case $(m,n)=(1,2)$ the projections of horizontal chains are Euclidean circles in $\mathbb C$. This fact was first observed and used by Cartan in \cite{Cartan}. In fact every Euclidean circle $C\subseteq \mathbb C^{p-1}$ is a circle in our generalized definition, namely is the projection of some 1-chain of $\partial\mathbb H _{\mathbb C}^p$. Indeed we know from Lemma \ref{lem:parametrization of k-vertical chains} that the Euclidean circle $(1+e^{it},0,\ldots,0)\in \mathbb C^{p-1}$ is the projection of the chain associated to the linear subspace $\<e_1+e_2,e_{p+1}\>$ of $\mathbb C^{p+1}$. Moreover the set of Euclidean circles is a homogeneous space under the group of Euclidean similarities of $\mathbb C^{p-1}$ and the group $Q=\mathrm{stab}(v_\infty)$ acts on $\mathbb C^{p-1}$ as the full group of Euclidean similarities.
In the general case it is important to record both the dimension of the $m$-chain that is projected and the dimension of the ${\rm U}(m-k)$ factor in the projection. This explains the notation.
We will call \emph{generalized circle} any subset of $M((n-m)\times m,\mathbb C)$ arising as a projection of an $m$-chain. In particular a generalized circle is an $(m,k)$-circle for some $k$.
The ultimate goal of this section is to understand the possible lifts of a given $(m,k)$-circle. We begin by analyzing the stabilizers in ${\rm SU}(\mathbb C^{m+n}, h)$ of some configurations:
\begin{lem}\label{lem:S_0}
The stabilizer in ${\rm SU}(\mathbb C^{m+n}, h)$ of the triple $(v_\infty,v_0,T_k)$ is the subgroup $S_0$ of $L\cong {\rm GL}_m(\mathbb C)\times{\rm SU}(n-m)$ consisting of pairs of the form
$$\begin{array}{ccc}
\left(
\begin{bmatrix} Y& X\\0&\overline y y^{-1}C_{11}\end{bmatrix}, \begin{bmatrix} C_{11}&0\\0&C_{22}\end{bmatrix}
\right)
& \text{with} &
\left\{
\begin{array}{l}
C_{11} \in {\rm U}(m-k)\\
C_{22}\in {\rm U}(n-2m+k)\\
X\in M(k\times(m-k),\mathbb C)\\
Y\in {\rm GL}_k(\mathbb C), \,y=\det(Y) .\end{array}\right.
\end{array}$$
\end{lem}
\begin{proof}
We determined in Section \ref{ssec:Heisenberg} that the stabilizer $ L$ in ${\rm SU}(\mathbb C^{m+n}, h)$ of the pair $(v_\infty,v_0)$ is isomorphic to ${\rm GL}_m(\mathbb C)\times {\rm SU}(n-m)$. The stabilizer of the triple $(v_\infty,v_0,T_k)$ is clearly contained in $ L$ and consists precisely of the elements of $ L$ stabilizing $V_k^\bot$.
In the proof of Lemma \ref{lem:parametrization of k-vertical chains} we saw that the subspace $V_k^\bot$ has a basis of the form $\bsm0\{\rm Id}_{n-m}\\X\esm$ where $X$ denotes the $m\times (n-m)$ matrix $\bsm0&0\{\rm Id}_{m-k}&0\esm$.
From the explicit expression of elements in $L$ we get
$$\begin{bmatrix} A&&\\&\overline a a^{-1}C&\\&&A^{-*}\end{bmatrix}\bpm0\\{\rm Id}\\X\end{bmatrix}=\bpm0\\\overline a a^{-1}C\\A^{-*}X\end{bmatrix}\cong\bpm0\\{\rm Id}\\a\overline a^{-1}A^{-*}XC^{-1}\end{bmatrix}.$$
In turn the requirement that $a\overline a^{-1}A^{-*}XC^{-1}=X$, that is $X=\overline a a^{-1}A^*XC$, implies, in the suitable block decomposition for the matrices, that
$$\begin{array}{c}
\overline a a^{-1}\begin{bmatrix} A_{11}^*&A_{21}^*\\A_{12}^*&A_{22}^*\end{bmatrix}\bpm0&0\\{\rm Id}_{m-k}&0\end{bmatrix}\begin{bmatrix} C_{11}&C_{12}\\mathbb C_{21}&C_{22}\end{bmatrix}=\\
\overline a a^{-1}\begin{bmatrix} A_{21}^*&0\\A_{22}^*&0\end{bmatrix}\begin{bmatrix} C_{11}&C_{12}\\mathbb C_{21}&C_{22}\end{bmatrix}=
\overline a a^{-1}\begin{bmatrix} A_{21}^*C_{11}&A_{21}^*C_{12}\\A_{22}^*C_{11}&A_{22}^*C_{12}\end{bmatrix}.
\end{array}$$
This implies that $A_{22}^{*}= a \overline a^{-1}C_{11}^{-1}$ and $C_{12}=A_{21}=0$.
Moreover since $C$ is unitary, also $C_{21}$ must be 0, and both $C_{22}$ and $C_{11}$ must be unitary.
This concludes the proof.
\end{proof}
Let us now denote by $o$ the point $o=\pi_{v_\infty}(v_0)=0$ in $W_{v_\infty}$ and by $C_k$ the $(m,k)$-circle that is the projection of $T_k$. We will denote by $S_1$ the stabilizer in $ Q$ of the pair $(o,C_k)$.
\begin{lem}\label{lem:S_1}
The stabilizer of the pair $(o,C_k)$ is the group
$$S_1=\mathrm{Stab}_{Q}(o,C_k)=M\rtimes S_0$$
where, as above, we denote by $M$ the center of the nilpotent radical $N$ of $Q$ and by $S_0$ the stabilizer in $Q$ of the pair $(v_0,T_k)$.
\end{lem}
\begin{proof}
Recall that any element in $Q$ can be uniquely written as a product $nl$ where $n$ is in $N$, and $l$ belongs to $L$, the Levi component of $ Q$.
Since any element in $S_1$ fixes, by assumption, the point $o=\pi_{v_\infty}(v_0)$ and since any element in $L$ fixes $o$, if $nl$ is in $S_1$ then $n(o)=o$ that, in turn, implies that $n$ belongs to $M$. Hence $S_1$ is of the form $M\rtimes S$ for some subgroup $S$ of $L$.
Let now $X$ be a point in $W_{v_\infty}=M((n-m)\times m,\mathbb C)$. The action of $(A,C)\in L$ on $W_{v_\infty}$ is $X\mapsto \overline a a^{-1}CXA^*$. We want to show that if $C_k$ is preserved then $(A,C)$ must belong to $S_0$. We have proven in Lemma \ref{lem:parametrization of k-vertical chains} that any point $z\in \pi_{v_\infty}(T_k)$ can be written as $\left[\begin{smallmatrix} E&{\rm Id}+Z\\0&0\esm$ for some matrices $E\in M((m-k)\times k,\mathbb C)$ and $Z\in {\rm U}(m-k)$. Explicit computations give that
$$\begin{array}{rl} \begin{bmatrix} E&{\rm Id}+Z\\0&0\end{bmatrix}&=\overline a a^{-1} \begin{bmatrix} C_{11}&C_{12}\\mathbb C_{21}&C_{22}\end{bmatrix}\begin{bmatrix} E&{\rm Id}+Z\\0&0\end{bmatrix}\begin{bmatrix} A_{11}^*&A_{21}^*\\A_{12}^*&A_{22}^*\end{bmatrix}=\\
&=\overline a a^{-1}\begin{bmatrix} C_{11}E&C_{11}({\rm Id}+Z)\\mathbb C_{21}E&C_{21}({\rm Id}+Z)\end{bmatrix}\begin{bmatrix} A_{11}^*&A_{21}^*\\A_{12}^*&A_{22}^*\end{bmatrix}.\end{array}$$
Since $A$ is invertible, $E$ is arbitrary and both ${\rm Id}$ and $-{\rm Id}$ are in ${\rm U}(m-k)$, the matrix $C_{21}$ must be zero. Hence $C$ must have the same block form of a genuine element of $S_0$. In particular $C_{11}$ is invertible.
Since $\overline a a^{-1}C_{11}(EA_{21}^*+({\rm Id}+Z)A_{22}^*)$ must be an element of ${\rm Id}+U(m-k)$ for every $E$, we get that $A_{21}^*$ must be zero.
The result now follows from Claim \ref{claim:fixing Um} below.
\end{proof}
\begin{claim}\label{claim:fixing Um}
Let $C\in {\rm U}(l)$ and $A\in {\rm GL}_l(\mathbb C)$ be matrices and let $\mathcal U$ denote the set
$$\mathcal U=\{{\rm Id}+X|\;X\in {\rm U}(l)\}\subset M(l\times l,\mathbb C).$$
If $C\mathcal U A^*=\mathcal U$ then $A=C$.
\end{claim}
\begin{proof}
Let us consider the birational map
$$\begin{array}{cccc}i:&M(l\times l,\mathbb C)&\to &M(l\times l,\mathbb C)\\ &X&\mapsto &X^{-1}\end{array}$$
that is defined on a Zariski open subset $\mathcal O$ of $M(l\times l,\mathbb C)$.
The image, under the involution $i$, of $\mathcal U$ is the set $$\mathcal L=\{W|\;{\rm Id}-W^*-W=0\}=\frac 12{\rm Id}+\mathfrak u(l).$$ Moreover $i( CXA^{*})=A^{-*}i(X)C^{-1}$, hence in order to show that the subgroup preserving $\mathcal U$ consists precisely of the pairs $(A,A)$, it is enough to check that the subgroup of ${\rm U}(l)\times {\rm GL}_l(\mathbb C)$ preserving $\mathcal L$ consists precisely of the pairs $(A,A)$ with $A\in {\rm U}(l)$.
This last statement amounts to show that the only matrix $B\in {\rm GL}_l(\mathbb C)$ such that ${\rm Id}-W^*B^{*}-BW=0$ for all $W\in \mathcal L$ is the identity itself.
Choosing $W$ to be $\frac 12 {\rm Id}$ we get that $B^*+B=2{\rm Id}$ hence in particular $B={\rm Id} +Z$ with $Z\in \mathfrak u(l)$. Since moreover $\mathcal L=\{\frac 12 {\rm Id}+M|\;M\in \mathfrak{u}(l)\}$ we have to show that if $ZM+M^*Z^*=ZM+MZ=0$ for all $M$ in $\mathfrak{u}(l)$ then $Z$ must be zero, and this can be easily seen, for example by choosing $M$ to be the matrix that is zero everywhere apart from the $l$-th diagonal entry where it is equal to $i$.
\end{proof}
We now have all the ingredients we need to prove the first crucial result of the section. Recall from Section \ref{sec:S_mn1} that every pair of transverse points $x,y$ in $\mathcal S_{m,n}$ uniquely determines an $m$-chain $T_{x,y}$ that is the unique chain that contains both $x$ and $y$.
\begin{prop}\label{prop:lifts of k-vertical chains}
Let $x\in \mathcal S_{m,n}$ be a point, $T$ be a chain with $i_x(T)=k$, $t\in T$ be a point, $y=\pi_{x}(t)\in W_{x}$. Then
\begin{enumerate}
\item $T$ is the unique lift of the $(m,k)$-circle $\pi_{x}(T)$ through the point $t$,
\item for any point $t_1$ in $T_{x,t}=\pi_x^{-1}(y)$ there exists a unique $m$ chain through $t_1$ which lifts $\pi_x(T)$.
\end{enumerate}
\end{prop}
\begin{proof}
(1) As a consequence of Lemma \ref{lem:transitivity on k vertical chains}, in order to prove the statements, we can assume that the triple $(x,t,T)$ is the triple $(v_\infty, v_0, T_k)$. Let $T'$ be another $m$-chain containing the point $t$ that lifts the $(m,k)$-circle $C_k$, a consequence of Lemma \ref{lem:transitivity on k vertical chains} is that there exists an element $g\in L$ such that $(v_\infty,v_0,T_k)=g(v_\infty,v_0,T')$. Moreover, since $\pi_{v_\infty}(T')=C_k$, we get that $g\in S_1$. But we know that $S_1\cap L=S_0$ and this proves that $T'=T_k$.
\newline (2) This is a consequence of the first part, together with the observation that $M$ acts transitively on the vertical chain $T_{v_0,v_\infty}$.
\end{proof}
We conclude the section by determining what are the lifts of a point $y$ that are contained in an $m$-chain $T$. For every $m$-chain $T$ we consider the subgroup
$$M_T=\mathrm{Stab}_{M}(T).$$
Clearly if $t$ is a lift of a point $y\in W_{v_\infty}$ that is contained in $T$, then all the orbit $M_T\cdot t$ consists of lifts of $y$ that are contained in $T$.
We want to show that also the other containment holds, namely that the lifts are precisely the $M_T$ orbit of any point.
\begin{lem}
For the chain $T_k$ we have $M_{T_k}=i(E_k)$ where
$$E_k=\{X\in \mathfrak u(m)|\; X_{ij}=0 \text { if } i>k \text{ or } j>k\}=\left\{\begin{bmatrix} X_1&0\\0&0\end{bmatrix}\Big|\;X_1\in\mathfrak{u}(k)\right\},$$
and $i:\mathfrak{u}(m)\to N$ is the inclusion of the center of the group.
\end{lem}
\begin{proof}
We already observed that the orthogonal to $V_k$ is $$V_k^\bot=\<e_{m+j}+e_{n+j+k},e_{l+m}|\;1\leq j\leq m-k< l\leq n-m\>.$$ Moreover an element of $M$ stabilizes $T_k$ if and only if it stabilizes $V_k^\bot$.
If now $m=\left[\begin{smallmatrix}{\rm Id}&0&E\\0&{\rm Id}&0\\0&0&{\rm Id}\esm$ is an element of $M$, then the image $m\cdot(e_{m+j}+e_{n+j+k})=\sum E_{ij}e_{j+k}+e_{m+j}+e_{n+j+k}$ that belongs to $V_k^\bot$ if and only if the $(j+k)$-th column of the matrix $E$ is zero. This implies that the subgroup of $M$ that fixes $V_k^\bot$ is contained in $i(E_k)$. Viceversa it is easy to check that $i(E_k)$ belongs to ${\rm SU}(V_k)$, in particular it preserves $T_k$.
\end{proof}
We denote by $Z_T$ the intersection of the linear subspace $V_T$ underlying $T$ with $v_\infty$:
$$Z_T=v_\infty\cap V_T.$$
In the standard case in which $T=T_k$ we will denote by $Z_k$ the subspace $Z_{T_k}$ which equals to the span of the first $k$ vectors of the standard basis of $\mathbb C^m$.
\begin{prop}\label{prop:errors}
Let $T$ be a $k$-vertical chain, then
\begin{enumerate}
\item If $g\in Q$ is such that $gT=T_k$, then $M_T=g^{-1}M_{T_k}g$.
\item For any point $x\in T$, we have $\pi_{v_\infty}^{-1}(\pi_{v_\infty}(x))\cap T=M_Tx$.
\item If $n\in N$, then $M_{nT}=M_T$.
\item If $a\in {\rm GL}(m)$ is such that $a(Z_T)=Z_k$, then $M_T=i(a^{-1}E_k a^{-*})$.
\end{enumerate}
\end{prop}
\begin{proof}
(1) This follows from the definition of $M_{T_k}$ and $M_T$ and the fact that $M$ is normal in $Q$.
\newline (2) Let us first consider the case $T=T_k$. In this case the statement is an easy consequence of the explicit parametrization of the chain $T_k$ we gave in Lemma \ref{lem:parametrization of k-vertical chains}: any two points in $T_k$ that have the same projection are in the same $M_{T_k}$ orbit. The general case is a consequence of the transitivity of $Q$ on $k$-vertical chains: let $g\in Q$ be such that $gT=T_k$ and let us denote by $y$ the point $gx$. Then we know that
$M_{T_k}y=\pi^{-1}_{v_\infty}(\pi_{v_\infty}(y))\cap T_k$. This implies that
$$\begin{array}{rl}
M_Tx&= g^{-1}M_{T_k}g x=g^{-1}(M_{T_k}y)=g^{-1}( \pi^{-1}_{v_\infty}(\pi_{v_\infty}(y))\cap T_k)=\\
&=g^{-1} \pi^{-1}_{v_\infty}(\pi_{v_\infty}(y))\cap g^{-1}T_k=\\
&=\pi^{-1}_{v_\infty}(\pi_{v_\infty}(x))\cap T.
\end{array}
$$
Where in the last equality we used that the $Q$ action on $\mathcal H_{m,n}(v_\infty)$ induces an action of $Q$ on $W_{v_\infty}$ so that the projection $\pi_{v_\infty}$ is $Q$ equivariant.
\newline (3) This is a consequence of the fact that $M$ is in the center of $N$: $M_{nT}=nM_Tn^{-1}=M_T$.
\newline (4) By (3) we can assume that $T$ is a chain through the point $v_0$: indeed there exists always an element $n\in N$ such that $nT$ contains $v_0$, moreover both $M_{nT}=M_T$ and $Z_{nT}=Z_T$ (the second assertion follows from the fact that any element in $N$ acts trivially on $v_\infty$).
Since $v_0\in T$ and we proved in Lemma \ref{lem:transitivity on k vertical chains} that $L$ is transitive on $k$-vertical chains through $v_0$, we get that there exists a pair $(C,A)\in {\rm U}(n-m)\times {\rm GL}_m(\mathbb C)$ such that, denoting by $g$ the corresponding element in $L$, we have $gT=T_k$. It follows from (1) that $M_T=g^{-1}M_{T_k} g$, in particular we have
$$\begin{bmatrix} A^{-1}&0&0\\0&C^{-1}&0\\0&0&A^*\end{bmatrix}\begin{bmatrix}{\rm Id}&0&E\\0&{\rm Id}&0\\0&0&{\rm Id}\end{bmatrix}\begin{bmatrix} A&0&0\\0&C&0\\0&0&A^{-*}\end{bmatrix}=\begin{bmatrix}{\rm Id}&0&A^{-1}EA^{-*}\\0&{\rm Id}&0\\0&0&{\rm Id}\end{bmatrix}$$
and hence the subgroup $M_T$ is the group $i(A^{-1}E_kA^{-*})$. Moreover, since $gT=T_k$ we have in particular that $gZ_T=Z_k$ and hence $A(Z_T)=Z_k$ if we consider $Z_T$ as a subspace of $v_\infty$.
In order to conclude the proof it is enough to check that for every $a\in {\rm GL}_m(\mathbb C)$ with $a(Z_T)=Z_k$ the subgroups $a^{-1}E_k a^{-*}$ coincide. Indeed it is enough to check that for every element $a\in {\rm GL}_m(\mathbb C)$ such that $a(Z_k)=Z_k$ then $a^{-1}E_k a^{-*}=E_k$. But if $a$ satisfies this hypothesis, the matrix $a^{-*}$ has the form $\left[\begin{smallmatrix} A_1&0\\A_2&A_3\esm$. In particular we can compute:
$$a^{-1}Xa^{-*}=\begin{bmatrix} A_1^*&A_2^*\\0&A_3^*\end{bmatrix}\begin{bmatrix} X_1&0\\0&0\end{bmatrix}\begin{bmatrix} A_1&0\\A_2&A_3\end{bmatrix}=\begin{bmatrix} A_1^*X_1A_1&0\\0&0\end{bmatrix}$$
and the latter matrix still belongs to $E_k$.
\end{proof}
\section{The restriction to a chain is rational}\label{sec:chain}
In this section we prove that the chain geometry defined in Section \ref{sec:S_mn} is rigid in the following sense:
\begin{thm}\label{thm:restriction rational}
Let $\phi:\partial\mathbb H _{\mathbb C}^p\to \mathcal S_{m,n}$ be a measurable, chain geometry preserving, Zariski dense map. Then for almost every chain $C$ in $\partial\mathbb H _{\mathbb C}^p$ the restriction $\phi|_{C}$ coincides almost everywhere with a rational map.
\end{thm}
Let us recall that, whenever a point $x\in \mathcal S_{m,n}$ is fixed, the center $M_x$ of the nilpotent radical $N_x$ of the stabilizer $Q_x$ of $x$ in ${\rm SU}(\mathbb C^{m+n}, h)$ acts on the Heisenberg model $\mathcal H_{m,n}(x)$. Moreover, for every $m$-chain $T$ containing the point $x$, the $M_x$ action is simply transitive on the Zariski open subset $T^x$ of $T$.
The picture above is true for both $\partial\mathbb H _{\mathbb C}^p\cong \mathcal S_{1,p}$ and $\mathcal S_{m,n}$ where, if $x\in \partial\mathbb H _{\mathbb C}^p$, the group $M_x$ can be identified with $\mathfrak{u}(1)$, and, if $\phi:\partial\mathbb H _{\mathbb C}^p\to \mathcal S_{m,n}$ is the boundary map, $M_{\phi(x)}\cong \mathfrak{u}(m)$.
The idea of the proof is to show that, for almost every point $x\in \partial\mathbb H _{\mathbb C}^p$, the boundary map is equivariant with respect to a measurable homomorphism $h: M_x\to M_{\phi(x)}$. Since such homomorphism must be algebraic, we get that the restriction of $\phi$ to almost every chain through $x$ must be algebraic.
In order to define the homomorphism $h$ we will prove first that a map $\phi$ satisfying our assumptions induces a measurable map $\phi_x:W_x\to W_{\phi(x)}$. Here $W_x$ can be identified with $\mathbb C^{p-1}$ and $W_{\phi(x)}$ can be identified with $M((n-m)\times m,\mathbb C)$, both these identifications are non canonical but we fix them once and forall. We will then use the map $\phi_x$ to define a cocycle $\alpha:M_x\times (\partial\mathbb H _{\mathbb C}^p)^x\to M_{\phi(x)}$ with respect to which $\phi$ is equivariant. We will then show that $\alpha$ is independent on the point $x$ and hence coincides almost everywhere with the desired homomorphism.
\subsection{First properties of chain preserving maps}
Recall from Section \ref{sec:Toledo} that a map $\phi$ is Zariski dense if the essential Zariski closure of $\phi(\partial\mathbb H _{\mathbb C}^p)$ is the whole $\mathcal S_{m,n}$, or, equivalently if the preimage under $\phi$ of any proper Zariski closed subset of $\mathcal S_{m,n}$ is not of full measure. Moreover, by definition, a measurable boundary map \emph{preserves the chain geometry} if the image under $\phi$ of almost every pair of distinct points is a pair of transverse points, and the image of almost every maximal triple $(x_0,x_1,x_2)$ in $(\partial\mathbb H _{\mathbb C}^p)^3$, is contained in an $m$-chain.
We will denote by $\mathcal T_1$ the set of chains in $\partial\mathbb H _{\mathbb C}^p$, and by $\mathcal T_m$ the set of $m$-chains of $\mathcal S_{m,n}$. The set $\mathcal T_1$ is a smooth manifold, indeed an open subset of the Grassmannian ${\rm Gr}_{2}(\mathbb C^{p+1})$, and we will endow $\mathcal T_1$ with its Lebesgue measure class.
The following lemma, an application in this context of Fubini's theorem, gives the first property of a chain geometry preserving map:
\begin{lem}\label{lem:1}
Let $\phi:\partial\mathbb H _{\mathbb C}^p\to \mathcal S_{m,n}$ be a chain geometry preserving map. For almost every chain $C\in\mathcal T_1$ there exists an $m$-chain $\hat\phi(C)\in \mathcal T_m$ such that, for almost every point $x$ in $C$, $\phi(x)\in\hat\phi(C)$.
\end{lem}
\begin{proof}
There is a bijection between the set $(\partial\mathbb H _{\mathbb C}^p)^{\{3\}}$ consisting of triples of distinct points on a chain and the set
$$\mathcal T_1^{\{3\}}=\{(C,x,y,z)|\;C\in \mathcal T_1\text{ and } (x,y,z)\in C^{(3)}\}.$$
In turn the projection onto the first factor endows the manifold $\mathcal T_1^{\{3\}}$ with the structure of a smooth bundle over $\mathcal T_1$. In particular Fubini's theorem implies that, for almost every chain $C\in \mathcal T_1$ and for almost every triple $(x,y,z)\in C^3$, the triple $(\phi(x),\phi(y),\phi(z))$ belongs to the same $m$-chain $\hat \phi(C)$.
Moreover $\hat \phi(C)$ has the desired properties again as a consequence of Fubini theorem.
\end{proof}
We can now use the fact that each pair of transverse points $a,b$ in $\mathcal S_{m,n}$ uniquely determines a chain $T_{a,b}$ to reformulate
Lemma \ref{lem:1} in the following way:
\begin{cor}\label{cor:map on chains}
Let $\phi:\partial\mathbb H _{\mathbb C}^p\to \mathcal S_{m,n}$ be a measurable chain preserving map. Then there exists a measurable map $\hat \phi:\mathcal T_1\to \mathcal T_m$ such that, for almost every pair $(x,T)\in \partial\mathbb H _{\mathbb C}^p\times \mathcal T_1$ with $x\in T$, then $\phi(x)\in \hat\phi(T)$.
\end{cor}
\begin{proof}
The only thing that we have to check is that the map $\hat \phi$ is measurable, but this follows from the fact that the map associating to a pair $(x,y)\in \mathcal S_{m,n}^{(2)}$ the $m$-chain $T_{x,y}$ is algebraic.
\end{proof}
Recall that, if $x$ is a point in $\partial\mathbb H _{\mathbb C}^p$, we denote by $W_x$ the set of chains through $x$. We use the identification of $W_x$ as subvariety of ${\rm Gr}_2(\mathbb C^{1,p})$ to endow the space $W_x$ with its Lebesgue measure class.
\begin{cor}\label{cor:point-chain}
For almost every $x\in\partial\mathbb H _{\mathbb C}^p$, almost every chain in $W_x$ satisfies Lemma \ref{lem:1}.
\end{cor}
\begin{proof}
It is again an application of Fubini's theorem. Let us indeed consider the manifold $\mathcal T^{\{1\}}=\{(C,x)|\;C\in\mathcal T,\,x\in C\}$ the projection on the first two factor $\mathcal T^{\{3\}}\to\mathcal T^{\{1\}}$ realizes the first manifold as a smooth bundle over the second with fiber $(\mathbb R\times\mathbb R)\backslash \Delta$. In particular for almost every pair in $\mathcal T^{\{1\}}$ the chain satisfies the assumption of Corollary \ref{cor:map on chains}. Since $\mathcal T^{\{1\}}$ is a bundle over $\partial\mathbb H _{\mathbb C}^p$ with fiber $W_x$ over $x$, the statement follows applying Fubini again.
\end{proof}
We will call a point $x$ that satisfies the hypotheses of Corollary \ref{cor:point-chain} \emph{generic} for the map $\phi$. Let us now fix, for the rest of the section, a point $x$ that is generic for the map $\phi$ and consider the diagram
$$\xymatrix{\mathcal H_{1,p}(x)\ar[r]^-\phi\ar[d]^{\pi_x}&\mathcal H_{m,n}(\phi(x))\ar[d]^{\pi_{\phi(x)}}\\W_x\ar@{.>}[r]^{\phi_x}&W_{\phi(x)}.}$$
\begin{lem}\label{lem:phix}
If $x$ is generic for $\phi$, there exists a measurable map $\phi_x$ such that the diagram commutes almost everywhere. Moreover $\phi_x$ induces a measurable map $\hat\phi_x$ from the set of circles of $W_x$ to the set of generalized circles of $W_{\phi(x)}$ such that, for almost every chain $T$, we have that $\hat\phi(T)$ is a lift of $\hat\phi_x(\pi_x(T))$.
\end{lem}
\begin{proof}
The fact that a map $\phi_x$ exists making the diagram commutative on a full measure set is a direct application of Corollary \ref{cor:point-chain}.
Since the set of horizontal chains in $\partial\mathbb H _{\mathbb C}^p$ is a smooth bundle over the set of Euclidean circles in $W_x\cong \mathbb C^{p-1}$, we have that, for almost every Euclidean circle $C$, the map $\hat \phi$ is defined on almost every chain $T$ with $\pi_x(T)=C$. Moreover a Fubini-type argument implies that, for almost every circle $C$, the diagram commutes when restricted to the preimage of $C$.
This implies that the projections $\hat\phi_x(C):=\pi_{\phi(x)}(\hat\phi(T_i))$ coincide for almost every lift $T_i$ of $C$ if $C$ satisfies the hypotheses of the previous paragraph, and this concludes the proof.
\end{proof}
\subsection{A measurable cocycle}\label{ssec:cocycle}
Recall that if $H,K$ are topological groups and $Y$ is a Borel $H$-space, then a map $\alpha:H\times Y\to K$ is a \emph{Borel cocycle} if it is a measurable map such that, for every $h_1,h_2$ in $H$ and for almost every $y\in Y$, it holds $\alpha(h_1h_2,y)=\alpha(h_1,h_2\cdot y)\alpha(h_2,y)$.
\begin{prop}\label{prop:cocycle}
Let $\phi$ be a measurable, chain preserving map $\phi:\partial\mathbb H _{\mathbb C}^p\to\mathcal S_{m,n}$. For almost every point $x$ in $\partial\mathbb H _{\mathbb C}^p$ there exists a measurable cocycle $\alpha: M_x\times (\partial\mathbb H _{\mathbb C}^p)^{x}\to M_{\phi(x)}$
such that $\phi$ is $\alpha$-equivariant.
\end{prop}
\begin{proof}
Let us fix a point $x$ generic for the map $\phi$. For almost every pair $(e,y)$ where $e\in M_x$ and $y\in\mathcal H_{1,p}(x)$, we have that the points $\phi(y)$ and $\phi(ey)$ are on the same vertical chain in $\mathcal H_{m,n}(\phi(x))$. In particular there exists an element $\alpha(e,y)\in M_{\phi(x)}$ such that $\alpha(e,y)\phi(y)=\phi(ey)$. We extend $\alpha$ by defining it to be 0 on pairs that do not satisfy this assumption. The function $\alpha$ is measurable since $\phi$ is measurable.
We now have to show that the map $\alpha$ we just defined is actually a cocycle.
In order to do this let us fix the set $\mathcal O$ of points $z$ for which $\hat\phi(T_{x,z})$ is $m$-vertical and $\phi(z)\in \hat \phi(T_{x,z})$. $\mathcal O$ has full measure as a consequence of Lemma \ref{lem:1}. Let us now fix two elements $e_1,e_2\in M_x$. For every element $z$ in the full measure set $\mathcal O\cap e_2^{-1}\mathcal O\cap e_1e_2^{-1}\mathcal O$, the three points $\phi(z),\phi(e_2z),\phi(e_1e_2z)$ belong to the same vertical $m$-chain, moreover, by definition of $\alpha$, we have
$$\begin{array}{rl}\alpha(e_1e_2,z)\phi(z)&=\phi(e_1(e_2z))\\&=\alpha(e_1,e_2z)\phi(e_2z)\\&=\alpha(e_1,e_2z)\alpha(e_2,z)\phi(z).\end{array}$$
The conclusion follows from the fact that the action of $M_{\phi(x)}$ on $\mathcal H_{m,n}(\phi(x))$ is simply transitive.
\end{proof}
\begin{prop}\label{prop:homom>algebraic}
Let us fix a point $x$. Assume that there exists a measurable function $\beta:M_x\times W_x\to M_{\phi(x)}$ such that for every $e\in M_x$, for almost every $T$ in $W_x$ and for almost every $z$ in $T$, the equality $\alpha(e,z)=\beta(e,T)$ holds. Then the restriction of the boundary map $\phi$ to almost every chain through the point $x$ is rational.
\end{prop}
\begin{proof}
We are assuming that for every $e$ in $M_x$ for almost every $T$ in $W_x$ and for almost every $z$ in $T$, the equality $\alpha(e,z)=\beta(e,T)$ holds. Fubini's Theorem then implies that for every $T$ in a full measure subset $\mathcal F$ of $W_x$, for almost every $e$ in $M_x$ and almost every $z$ in $T$ the equality $\alpha(e,z)=\beta(e,T)$ holds. In particular for every vertical chain $T$ in $\mathcal F$ and almost every pair $(e_1,e_2)$ in $M_x^2$ we have $\beta(e_1,T)\beta(e_2,T)=\beta(e_1e_2,T)$: it is in fact enough to chose $e_1$ and $e_2$ so that the equality of $\alpha(e_i,z)$ and $\beta(e_i,T)$ holds for almost every $z$ and compute the cocycle identity for $\alpha$ in a point $z$ that works both for $e_1$ and $e_2$.
It is classical that if $\pi:G\to J$ satisfies $\pi(xy)=\pi(x)\pi(y)$ for almost every pair $(x,y)$ in $G^2$, then $\pi$ coincides almost everywhere with an actual Borel homomorphism (cfr. \cite[Theorem B.2]{Zimmer}). In particular for every $T$ in $\mathcal F$, we can assume (up to modifying $\beta|_T$ on a zero measure subset) that the restriction of $\beta$ to $T$ is a measurable homomorphism $\beta_T:M_x\to M_{\phi(x)}$ and hence coincides almost everywhere with an algebraic map. Since the action of $M_x$ and $M_{\phi(x)}$ on each vertical chain is algebraic and simply transitive, we get that for almost every vertical chain $T$ the restriction of $\phi$ to $T$ is algebraic.
\end{proof}
The fact that we let $\beta$ depend on the vertical chain $T$ might be surprising, and it is probably possible to prove that the cocycle $\alpha$ coincides almost everywhere with an homomorphism that doesn't depend on the vertical chain $T$. However since it suffices to prove that the restriction of $\alpha$ to almost every vertical chain coincides almost everywhere with an homomorphism, and since this reduces the technicalities involved, we will restrict to this version.
The rest of the section is devoted to prove, using the chain geometry of $\mathcal S_{m,n}$, that the hypothesis of Proposition \ref{prop:homom>algebraic} is satisfied whenever $\phi$ is a Zariski dense, chain geometry preserving map and $m<n$. In the following proposition we deal with a preliminary easy case, in which the geometric picture behind the general proof should be clear.
\begin{prop}\label{prop:easy}
Let $\phi:\partial\mathbb H _{\mathbb C}^p\to \mathcal S_{m,n}$ be a measurable, Zariski dense, chain geometry preserving map, and let $n\geq 2m$. Then the restriction of $\phi$ to almost every chain coincides almost everywhere with a rational map.
\end{prop}
\begin{proof}
We want to apply Proposition \ref{prop:homom>algebraic} and show that the cocycle $\alpha:M_x\times \partial\mathbb H _{\mathbb C}^p\to M_{\phi(x)}$ only depends on the vertical chain a point belongs to. We consider the set $\mathcal F\subseteq W_x$ of chains $F$ such that
\begin{minipage}{.7\textwidth}
\begin{enumerate}
\item $\hat\phi(F)$ is an $m$-vertical chain, hence in particular $\phi_x(F)$ is defined,
\item for almost every circle $C$ containing the point $F\in W_x$, for almost every chain $T$ lifting $C$ the diagram of Lemma \ref{lem:phix} commutes almost everywhere.
\end{enumerate}
\end{minipage}
\hspace{.5cm}
\begin{minipage}{.3\textwidth}
\vspace{-.5cm}
\begin{tikzpicture}[scale=.8]
\draw (0,0) to (0,3);
\node [left] at (0,3){$F$};
\draw (1,1.5) circle [x radius=1.5, y radius=.5, rotate=15] ;
\node at (1,1.5) {$T$};
\node[above left] at (0,1.7) {$z$};
\filldraw (0,1.65) circle (1pt);
\end{tikzpicture}
\end{minipage}
It follows from the proof of Lemma \ref{lem:phix} that the set $\mathcal F$ is of full measure, moreover, we get, applying Fubini, that if $F$ is an element in $\mathcal F$, for almost every point $z$ in $F$ and almost every chain $T$ through $z$ the diagram of Lemma \ref{lem:phix} commutes almost everywhere when restricted to $T$. In particular, using Fubini again, this implies that for almost every point $w$ in $\partial\mathbb H _{\mathbb C}^p$ the diagram of Lemma \ref{lem:phix} commutes almost everywhere when restricted to the chain $T_{z,w}$.
\begin{minipage}{.56\textwidth}
Let us now fix a chain $F\in \mathcal F$ and denote by $\mathcal O$ the full measure set of points in $F$ for which that holds. For every element $e\in M_x$ we also consider the full measure set $\mathcal O_{e}=\mathcal O\cap e^{-1}\mathcal O$.
We claim that given two points $z_1,z_2\in \mathcal O_e$ the cocycle $\alpha(e,z_i)$ has the same value $\beta(e,F)$.
In fact let us consider the set $\mathcal A_{z_1,z_2,e}\subseteq\partial\mathbb H _{\mathbb C}^p$ consisting of points $w$ such that
\begin{enumerate}
\item $\phi (w)\in\hat \phi(T_{w,z_1})\cap \hat \phi(T_{w,z_2})$
\item $\phi (ew)\in\hat \phi(T_{ew,ez_1})\cap \hat \phi(T_{ew,ez_2})$
\item $\dim \<\phi(z_1),\phi(z_2),\phi(w)\>=3m$.
\end{enumerate}
\end{minipage}
\hspace{1.2cm}
\begin{minipage}{.2\textwidth}
\begin{tikzpicture}[scale=.8]
\draw (0,0) to (0,8);
\node [left] at (0,8){$F$};
\draw (1,1.5) circle [x radius=1.5, y radius=.5, rotate=25] ;
\node at (1,1.5) {$T_{z_1,w}$};
\node[left] at (0,1.5) {$z_1$};
\filldraw (0,1.5) circle (1pt);
\draw (.8,2.5) circle [x radius=1, y radius=.5, rotate=-25] ;
\node at (.8,2.5){$T_{z_2,w}$};
\node[ above right] at (1.7,2.3) {$w$};
\filldraw (1.7,2.3) circle (1pt);
\node[ above left] at (0,3.05) {$z_2$};
\filldraw (0,3.05) circle (1pt);
\draw (1,5) circle [x radius=1.5, y radius=.5, rotate=25] ;
\node at (2,4) {$T_{ez_1,ew}=eT_{z_1,w}$};
\node[left] at (0,5) {$ez_1$};
\filldraw (0,5) circle (1pt);
\draw (.8,6) circle [x radius=1, y radius=.5, rotate=-25] ;
\node at (1.8,7){$T_{ez_2,ew}=eT_{z_2,w}$};
\node[ above right] at (1.7,5.8) {$ew$};
\filldraw (1.7,5.8) circle (1pt);
\node[ above left] at (0,6.55) {$ez_2$};
\filldraw (0,6.55) circle (1pt);
\end{tikzpicture}
\end{minipage}
We claim that the set $\mathcal A_{z_1,z_2,e}$ is not empty. Indeed, by definition of $\mathcal O_e$, the set of points $w$ satisfying the first two assumption is of full measure. Moreover, since $n\geq 2m$, the set $\mathcal C$ of points in $\mathcal S_{m,n}$ such that $\dim \<\phi(z_1),\phi(z_2),\phi(w)\>< 3m$ is a proper Zariski closed subset of $\mathcal S_{m,n}$. Since the map $\phi$ is Zariski dense, the preimage of $\mathcal C$ cannot have full measure, and this implies that $\mathcal A_{z_1,z_2,e}$ has positive measure, in particular it contains at least one point. The third assumption on the point $w$ implies that the $m$-chain containing $\phi(w)$ and $\phi(z_i)$ is horizontal for $i=1,2$.
Let us fix a point $w\in \mathcal A_{z_1,z_2,e}$ and consider the $m$-chain $\hat \phi(eT_{w,z_i})$ for $i=1,2$. The $m$-chain $\hat \phi(eT_{w,z_i})$ is a lift of the $(m,0)$-circle $C_i=\pi_{\phi(x)}(\hat\phi(T_{w,z_i}))$ that contains both the points $\phi(e z_i)$ and $\phi(ew)$. In particular, since $\alpha(e,z_i)\hat \phi (T_{w,z_i})$ is a lift of $C_i$ containing $\phi(ez_i)$ we get that $ \alpha(e,z_i)\hat \phi (T_{w,z_i})=\hat \phi(eT_{w,z_i})$. Similarly we get that $ \alpha(e,w)\hat \phi (T_{w,z_i})=\hat \phi(eT_{w,z_i})$. This gives that $\alpha(e,z_i)^{-1} \alpha(e,w)\in M_{\hat\phi(T_{w,z_i})}$, but the latter group is the trivial group since we know that the chain $\hat\phi(T_{w,z_i})$ is 0-vertical. This implies that $\alpha(e,z_1)=\alpha(e,w)=\alpha(e,z_2)$.
\end{proof}
\subsection{Proof of Theorem \ref{thm:restriction rational}}
Let us now go back to the setting of Theorem \ref{thm:restriction rational}: we fix a measurable, chain geometry preserving, Zariski dense map $\phi:\partial\mathbb H _{\mathbb C}^p\to \mathcal S_{m,n}$, a generic point $x\in\partial\mathbb H _{\mathbb C}^p$ such that for almost every chain $t\in W_x$, for almost every point $y\in t$, $\phi(y)\in \hat \phi(t)$. We want to show that the measurable cocycle $\alpha:\partial\mathbb H _{\mathbb C}^p\backslash\{x\}\times \mathfrak u(1)\to\mathfrak u(m)$ coincides on almost every vertical chain with a measurable homomorphism. In particular it is enough to show that for almost every pair $z_1,z_2$ on a vertical chain in $\partial\mathbb H _{\mathbb C}^p$, the values $\alpha(e,z_1)$ and $\alpha(e,z_2)$ coincide. For a generic pair $(z_1,z_2)$ we have that the triple $(\phi(x),\phi(z_1),\phi(z_2))$ is contained in a tube type subdomain and hence we can compose the map $\phi$ with an element of the group ${\rm SU}(m,n)$ so that $\phi(x)=v_\infty$, $\phi(z_1)=v_0$ and $\phi(z_2)=v_d$, here and in the following we denote by $v_d$ the subspace with basis $\begin{bmatrix}{\rm Id}&0&d\end{bmatrix}^T$ for some
diagonal matrix $d$ with all entries equal to $\pm i$. In fact it is proven in \cite[Theorem 5.2]{
CN} that the Bergmann cocycle is a complete invariant for the ${\rm SU}(m,n)$ action on triples of pairwise transverse points in an $m$-chain, and varying the matrix $d$ one gets that $\beta_{\mathcal S}(v_0,v_\infty, v_d)$ achieves all possible values (cfr. Proposition \ref{prop:bergmann}).
For the rest of the section we restrict to the case $n<2m$, since the otherwise Theorem \ref{thm:restriction rational} follows from Proposition \ref{prop:easy} and denote by $l$ the integer $l=n-m$ and $k=2m-n$. In analogy with the proof of Proposition \ref{prop:easy} we denote by $\mathcal A_{z_1,z_2,e}$ to be the full measure subset of $\partial\mathbb H _{\mathbb C}^p$ consisting of points with
\begin{enumerate}
\item $\phi (w)\in\hat \phi(T_{w,z_1})\cap \hat \phi(T_{w,z_2}), $
\item $\phi (ew)\in\hat \phi(T_{ew,ez_1})\cap \hat \phi(T_{ew,ez_2}), $
\item$\dim \<v_0,v_d,\;\phi(w)\>=m+n.$
\end{enumerate}
We will consider the subset $\mathcal D_{v_0,v_d}$ of $\mathcal S_{m,n}$ defined by
$$\mathcal D_{v_0,v_d}=\{w\in\mathcal S_{m,n}|\; w \text{ is transverse to } v_0,v_d\text{ and } \<v_0,v_d\> \}.$$
It is easy to verify that $\mathcal D_{v_0,v_d}$ consists of points $w$ such that both chains $T_{v_0,w}$ and $T_{v_d,w}$ are well defined and $k$-vertical: indeed in our assumptions $w$ is transverse to $\<v_0,v_d\>=\<v_\infty,v_0\>$. In particular $\dim\<w,v_0,v_\infty\>=n+m$ and we get
$$n+m=\dim\<w,v_0\>+\dim v_\infty-\dim (v_\infty\cap \<v_0,w\>)$$
which implies that $i_{v_\infty}(T_{v_0,w})=3m-(n+m)=k$.
Our next goal is to associate to any point $w$ in $\mathcal D_{v_0,v_d}$ subgroups $E(w)$, $I(w)$ which, when we consider a point $w$ that is image of a point $z$ in $\mathcal A_{v_0,v_d,e}$, represent, respectively, the error allowed by the point $z$ for the cocycle $\alpha(e,v_0)$ and some information on the difference $\alpha(e,v_0)-\alpha(e,v_d)$ obtained applying the strategy of Proposition \ref{prop:easy} to the point $z$.
In order to define the subgroups properly, we use the identification $M_{v_\infty}=\mathfrak{u}(m)$ provided in Section \ref{sec:S_mn} and use the linear structure on $\mathfrak{u}(m)$. Moreover we will denote by $H $ the positive definite bilinear form on $\mathfrak{u}(m)$ given by $H(A,B)=\text{tr} A^*B$. We also denote by $\beta_{v_0} $ the map
$$\begin{array}{cccc}\beta_{v_0}:&\mathcal D_{v_0,v_d}\subseteq\mathcal S_{m,n}&\to&{\rm Gr}_k(v_\infty)\\&w&\to&\<v_0,w\>\cap v_\infty\end{array}$$
similarly we define $\beta_{v_d}$ so that $\beta_{v_d}(w)=\<v_d,w\>\cap v_\infty$.
We want to understand the subspaces on which the possible defect of the cocycle $\alpha$ to be an homomorphism are confined. Whenever two $k$-dimensional subspaces $Z_i$ of $\mathbb C^m$ are fixed we denote by $S(Z_1,Z_2)$ the subspace:
$$S(Z_1,Z_2)=\<z_1z_2^*-z_2z_1^*|\;z_i\in Z_i\><\mathfrak u(m).$$
The map $S$ is useful to define the error subgroup $E(w)$ associated to a point $w$ in $\mathcal D_{v_0,v_d}$:
$$E(w)=S(\beta_{v_0}(w),\beta_{v_0}(w))+S(\beta_{v_d}(w)\beta_{v_d}(w)).$$
For each point $z$ in the set $\mathcal A_{z_1,z_2,e}$ the error group $E(\phi(z))$ bounds the error of the cocycle $\alpha$:
\begin{lem}
For every point $z$ in $\mathcal A_{z_1,z_2,e}$ we get $\alpha(e,z_1)-\alpha(e,z_2)\in E(\phi(z)).$
\end{lem}
\begin{proof}By the assumption on $z$ we have that $\hat \phi(T_{ez,ez_1})$ and $\hat \phi(T_{z,z_1})$ project to the same $(m,k)$-circle, and in particular we get that
$$\alpha(e,z)-\alpha(e,z_1)\in M_{\hat\phi( T_{z,z_1})}.$$
In the same way one gets that
$$\alpha(e,z)-\alpha(e,z_2)\in M_{\hat\phi (T_{z,z_2})}.$$
It follows from Proposition \ref{prop:errors} that if $g_i\in{\rm GL}(m)$ is such that $g_1\beta_{v_0}(\phi(w))=\<e_1,\ldots, e_k\>$ (resp. $g_2\beta_{v_d}(\phi(w))=\<e_1,\ldots, e_k\>$) we have that $M_{\hat\phi (T_{w,z_i})}=i(g_i^{-1}E_kg_i^{-*})$, and this proves our first claim since it is easy to check, from the definition of the set $E(\phi(w))$ that $$E(\phi(w))=g_1^{-1}E_kg_1^{-*}+g_2^{-1}E_kg_2^{-*}.$$
\end{proof}
In particular it is enough to show that, for almost every pair $z_1,z_2$ on a vertical chain, the intersection $\bigcap_{z\in \mathcal A_{z_1,z_2,e}}E(\phi(z))=\{0\}$. In fact this would imply that the restriction of $\alpha$ to almost every chain essentially doesn't depend on the choice of the point, hence coincides with a measurable homomorphism. In order to do this we define another subgroup of $M_{v_\infty}$ associated to a point $w\in \mathcal D_{v_0,v_d}$. The information associated to $w$ will be
$$I(w)=S\left((\beta_{v_0}(w)^\bot, \beta_{v_d}(w)^\bot\right)$$
Here the orthogonals are considered with respect to the standard Hermitian form on $v_\infty=\mathbb C^m$. It is easy to verify that $I(w)$ is contained in the orthogonal to $E(w)$ with respect to the orthogonal form on $\mathfrak{u}(m)$ given by $H(A,B)=\text{tr}(A^*B)$.
We postpone the proof of the following technical lemma to the next section:
\begin{lem}\label{cor:aeE(z)cuts}
For every proper subspace $L$ of $\mathfrak u (m)$ the set $C(L)=\{z\in \mathcal D_{v_0,v_d}^{v_\infty}|\;I(z)\subseteq L\}$ is a proper Zariski closed subset of $\mathcal D_{v_0,v_d}^{v_\infty}\subseteq\mathcal S_{m,n}$.
\end{lem}
We now conclude the proof of Theorem \ref{thm:restriction rational} assuming Lemma \ref{cor:aeE(z)cuts}.
\begin{proof}[Proof of Theorem \ref{thm:restriction rational} ]
We choose $m^2$ points $w_1,\ldots,w_{m^2}$ in $\partial\mathbb H _{\mathbb C}^p$ such that $\<I(\phi(w))\>=\mathfrak{u}(m)$: we work by induction and assume that there exist $j$ points $w_1,\ldots, w_j$ with $\dim L_j=\dim\<I(\phi(w_i))|\;i\leq j\>\geq j$. If the set $L_j$
is equal to the whole $\mathfrak u(m)$ we are done. Otherwise it follows from Lemma \ref{cor:aeE(z)cuts} that the subset $C(L_j)$ of $\mathcal D_{v_0,v_d}^{v_\infty}$ is a proper Zariski closed subset of $\mathcal D_{v_0,v_d}^{v_\infty}$.
In particular, since $\phi$ is Zariski dense, its essential image cannot be contained in $C(L_j)\cup (\mathcal S_{m,n}\setminus\mathcal D_{v_0,v_d}^{v_\infty})$ that is a Zariski closed subset of $\mathcal S_{m,n}$. Hence we can find a point $w_{j+1}$ in the full measure set $\mathcal A_{z_1,z_2,e}$ such that $I(\phi(w_{j+1}))$ is not contained in $L_j$, and this implies that $L_{j+1}=\< I(\phi(w_i))|\;i\leq j+1\>$ strictly contains $L_j$, hence has dimension strictly bigger than $j$. This completes the proof of Theorem \ref{thm:restriction rational}.
\end{proof}
\subsection{Possible errors}\label{ssec:4.3}
A crucial step in the proof of Lemma \ref{cor:aeE(z)cuts} is to show that the map $\beta=\beta_{v_0}\times\beta_{v_d}:\mathcal D_{v_0,v_d}\subseteq\mathcal S_{m,n}\to{\rm Gr}_k(v_\infty)^2$ is surjective (cfr. Proposition \ref{prop:betasurj}). As a preparation for this result we give a parametrization of the image of $\mathcal D_{v_0,v_d}$ under the map $\pi_{v_0}\times\pi_{v_d}:\mathcal D_{v_0,v_d}\to W_{v_0}\times W_{v_d}$. It is easy to check that explicit parametrizations of $W_{v_0}$ and $W_{v_d}$ are given by
$$\begin{array}{ccc}\begin{array}{ccc}M(l\times m,\mathbb C)&\to&W_{v_0}\\A_0&\to&\begin{bmatrix} 0\\ {\rm Id}_l\\ A_0^*\end{bmatrix}^\bot\end{array}&&\begin{array}{ccc}M(l\times m,\mathbb C)&\to&W_{v_d}\\A_1&\to&\begin{bmatrix} A_1^*\\ {\rm Id}_l\\ dA_1^*\end{bmatrix}^\bot\end{array}\end{array}$$
Moreover a point $A_i$ in $W_{v_i}$ correspond to a $k$-vertical chain if ${\rm rk}(A_i)=m-k$.
This allows us to give an explicit description of the image:
\begin{lem}\label{lem:4.15}
Under the parametrizations above, the image of the map $\pi_{v_0}\times\pi_{v_d}:\mathcal D_{v_0,v_d}^{v_\infty}\to W_{v_0}\times W_{v_d}$ is the closed subset $\mathcal C_{v_0,v_d}$ of $M(l\times m,\mathbb C)\times M(l\times m,\mathbb C)$ defined by
$$\mathcal C_{v_0,v_d}=\left\{(A_0,A_1)\left|\begin{array}{l}A_1A_0^*\in{\rm Id}+{\rm U}(m)\\
A_0,A_1 \text{ have maximal rank}
\end{array}
\right.\right\}.$$
\end{lem}
\begin{proof}
We already observed that each point $w$ in $\mathcal D_{v_0,v_d}$ uniquely determines two $k$-vertical chains $T_{w,v_0}$, $T_{w,v_d}$. In particular for each pair $(A_0,A_1)$ in the image of $\pi_{v_0}\times\pi_{v_d}$ we have that $A_i$ has maximal rank. We will now show that $A_1A_0^*\in{\rm Id}+{\rm U}(m)$ if an only if the intersection of linear subspaces associated to the two $m$-chains contains a maximal isotropic subspace.
Two $m$-chains $T_0$, $T_1$ intersect in $\mathcal S_{m,n}$ if and only if the intersection $V_0\cap V_1$ of their underlying vector spaces $V_0$, $V_1$ contains a maximal isotropic subspace. In turn this is equivalent to the requirement that $(V_0\cap V_1)^\bot$ has signature $(0,l)$. Indeed, since $V_0^\bot$ has signature $(0,l)$ and is contained in $(V_0\cap V_1)^\bot$, we get that the signature of any subspace of $(V_0\cap V_1)^\bot$ is $(k_1,l+k_2)$ for some $k_1, k_2$. On the other hand if $V_0\cap V_1$ contains a maximal isotropic subspace $z$, then $(V_0\cap V_1)^\bot\subseteq z^\bot$ and the latter space has signature $(0,l)$. In particular the signature of $(V_0\cap V_1)^\bot$ would be $(0,l)$, and clearly the orthogonal of a subspace of signature $(0,l)$ contains a maximal isotropic subspace.
Since $(V_0\cap V_1)^\bot=\< V_0^\bot, V_1^\bot\>$, we are left to check that the requirement that signature of this latter subspace is $(0,l)$ is equivalent to the requirement that $A_1A_0^*$ belongs to ${\rm Id}+ U(l)$.
If now we pick a pair $(A_0,A_1)\in M(l\times m,\mathbb C)\times M(l\times m,\mathbb C)$ representing a pair of subspaces $(V_0,V_1)\in W_{v_0}\times W_{v_d}$ we have that the subspace $(V_0\cap V_1)^\bot$ is spanned by the columns of the matrix
$$\bpm0&A_1^*\\{\rm Id}&{\rm Id}\\A_0^*&dA_1^*\end{bmatrix}.$$
It is easy to compute the restriction of $h$ to the given generating system of $(V_0\cap V_1)^\bot$:
$$\begin{array}{c}\bpm0&{\rm Id}&A_0\\A_1&{\rm Id}&-A_1d\end{bmatrix}\begin{bmatrix}&&{\rm Id}\\&-{\rm Id}&\\{\rm Id}&&\end{bmatrix}\bpm0&A_1^*\\{\rm Id}&{\rm Id}\\A_0^*&dA_1^*\end{bmatrix}=\\
\bpm0&{\rm Id}&A_0\\A_1&{\rm Id}&-A_1d\end{bmatrix}\begin{bmatrix} A_0^*&dA_1^*\\-{\rm Id}&-{\rm Id}\\0&A_1^*\end{bmatrix}=\begin{bmatrix}-{\rm Id}&A_0A_1^*-{\rm Id}\\A_1A_0^*-{\rm Id}&-{\rm Id}\end{bmatrix}
\end{array}
$$
The latter matrix is negative semidefinite and has rank $l$ if and only if \begin{equation}\label{eqn:1}
A_0A_1^*A_1A_0^*-A_1A_0^*-A_0A_1^*=(A_0A_1^*-{\rm Id})(A_1A_0^*-{\rm Id})-{\rm Id}=0.
\end{equation}
In this case the restriction of $ h$ to $(V_0\cap V_1)^\bot$ has signature $(0,l)$.
The intersection $V_0\cap V_1$ contains maximal isotropic subspaces that are transverse to $v_0$ and $v_d$ if and only if the radical of $V_0\cap V_1$, which coincides with the radical of $(V_0\cap V_1)^\bot$, is transverse to both subspaces. It is easy to verify that this is always the case if $A_0$ and $A_1$ have maximal rank.
\end{proof}
We now turn to the analysis of the map
$$\begin{array}{cccc}\beta:&\mathcal D_{v_0,v_d}^{v_\infty}\subseteq\mathcal S_{m,n}&\to&{\rm Gr}_k(v_\infty)^2\\&z&\to&(\<v_0,z\>\cap v_\infty,\<v_d,z\>\cap v_\infty).\end{array}$$
\begin{prop}\label{prop:betasurj}
The map $\beta$ is surjective.
\end{prop}
\begin{proof}
If we denote by $\zeta$ the uniquely defined map with the property that $\beta=\zeta\circ(\pi_{v_0}\times\pi_{v_d})$, then it is easy to check that the map $\zeta$ has the following expression, with respect to the coordinates described above:
$$\begin{array}{cccc}\zeta:&W_{v_0}\times W_{v_d}&\to&{\rm Gr}_k(v_\infty)^2\\ &(A_0,A_1)&\mapsto &(\ker(A_0), \ker(A_1)).\end{array}$$
In order to conclude the proof it is enough to show that any pair $(V_0,V_1)$ of $k$-dimensional subspaces of $v_\infty$ can be realized as the kernels of a pair of matrices satisfying Equation \ref{eqn:1}.
We first consider the case in which the subspaces $V_0,V_1$ intersect trivially, of course this can only happen if $k\leq l$.
In this case there exists an element $g\in {\rm U}(m)$ such that $gV_0=\widetilde V_0=\<e_1\ldots,e_k\>$ and that $\widetilde V_1=gV_1$ is spanned by the columns of the matrix $\left[\begin{smallmatrix} B\\{\rm Id}_k\\0\esm$ where $B$ is a matrix in $M(k\times k,\mathbb C)$. Clearly $V_i$ is the kernel of $A_i$ if and only if $\widetilde V_i$ is the kernel of $\widetilde A_i=A_i g^{-1}$ and $A_ig^{-1}$ satisfies the equation \ref{eqn:1} if and only if $A_i$ does. In particular it is enough to exhibit matrices $\widetilde A_i$ whose kernel is $\widetilde V_i$.
Let us first notice that, for any matrix $B\in M(k\times k,\mathbb R)$ there exists a matrix $X\in {\rm GL}_k(\mathbb C)$ such that $XB$ is a diagonal matrix $D$ whose elements are only 0 or 1. Let us now consider the matrices
$$\begin{array}{cc}
\begin{array}{rcl}
\widetilde A_1^*=&\left[\begin{array}{ccc}0&2{\rm Id}&0\\0&0&2{\rm Id}\end{array}\right]&\hspace{-10pt}\begin{array}{l}_k\\_{l-k}\end{array}\\%&\begin{array}{ccc}^k &^k &\;^{l-k}\end{array}
\end{array}
&
\begin{array}{rcl}
\widetilde A_2^*=&\left[\begin{array}{ccc}X&D&0\\0&0&{\rm Id}\end{array}\right]&\hspace{-10pt}\begin{array}{l}_k\\_{l-k}\end{array}\\%&\begin{array}{ccc}^k &^k &\;^{l-k}\end{array}
\end{array}
\end{array}
$$
By construction $\widetilde V_1$ is the kernel of $\widetilde A_1$ and $\widetilde V_2$ is the kernel of $\widetilde A_2$, moreover we have that $\widetilde A_1\widetilde A_2^*$ satisfies Equation \ref{eqn:1}:
$$\widetilde A_1^*\widetilde A_2^=\bpm0&2{\rm Id}&0\\0&0&2{\rm Id}\end{bmatrix}\begin{bmatrix} X^*&0\\mathbb D^*&0\\0&{\rm Id}\end{bmatrix}=\bpm2D^*&0\\0&2{\rm Id}\end{bmatrix}\in {\rm Id}+U(m).$$ This implies that there is a point $z\in\mathcal S_{m,n}$ with $\beta(z)=(\widetilde V_0,\widetilde V_1)$.
The general case, in which the intersection of $V_i$ is not trivial, is analogous: we can assume, up to the ${\rm U}(m)$ action that $V_0\cap V_1=\<e_1,\ldots,e_s\>$ and we can restrict to the orthogonal to $V_0\cap V_1$ with respect to the standard Hermitian form.
\end{proof}
We can now prove Lemma \ref{cor:aeE(z)cuts}.
\begin{proof}[Proof of Lemma \ref{cor:aeE(z)cuts}]
The subspace $C(L)$ is Zariski closed since the subspaces of $\mathfrak u(m)$ that are contained in $L$ form a Zariski closed subset of the Grassmanian ${\rm Gr}(\mathfrak{u}(m))$ of the vector subspaces of $\mathfrak{u}(m)$, moreover the subspace $I(z)$ is obtained as the composition $I(z)=S\circ \beta$ of two regular maps.
In order to verify that $C(L)$ is a proper subset, unless $L=\mathfrak u(m)$, it is enough to verify that the subspaces of the form $S(Z_1^\bot,Z_2^\bot)$ with $Z_i$ transverse subspaces span the whole $\mathfrak u(m)$. Once this is proven, the result follows from the the surjectivity of $\beta$: since $\beta$ is surjective, the preimage of a proper Zariski closed subset is a proper Zariski closed subset. The fact that the span
$$\<z_1z_2^*-z_2z_1^*|\;z_1,z_2\in \mathbb C^{m} \text{ linearly independent}\>$$
is the whole $\mathfrak u(m)$ follows from the fact that every matrix of the form $iz_1z_1^*$ is in the span, since such a matrix can be obtained as the difference $z_1(z_2-iz_1)^*-(z_2-iz_1)z_1^*-(z_1z_2^*-z_2z_1^*)$.
\end{proof}
\section{The boundary map is rational}\label{sec:reduction}
In this section we will show that a Zariski dense, chain geometry preserving map $\phi:\partial\mathbb H _{\mathbb C}^p\to\mathcal S_{m,n}$ whose restriction to almost every chain is rational, coincides almost everywhere with a rational map.
Assume that the chain geometry preserving map $\phi$ is rational and let us fix a point $x$. Since the projection $\pi_{\phi(x)}:\mathcal S_{m,n}^{\phi(x)}\to W_{\phi(x)}$ is regular we get that the map $\phi_x:W_x\to W_{\phi(x)}$ induced by $\phi$ is rational as well. The first result of the section is that the converse holds, namely that if there exist enough many generic points $s_1,\ldots,s_l$ in $\partial\mathbb H _{\mathbb C}^p$ such that $\phi_{s_i}$ is rational, then the original map $\phi$ had to be rational as well.
In what follows we will denote by $l$ the smallest integer bigger than $1+m/(n-m)$.
\begin{lem}\label{lem:Oo}
There exists a Zariski open subset $\mathcal O\subset \mathcal S_{m,n}^l$, such that for any $(x_1,\ldots,x_l)$ in $\mathcal O$, there exists a Zariski open subset $\mathcal D_{x_1,\ldots,x_l}\subset \mathcal S_{m,n}$ such that for every $z\in \mathcal D_{x_1,\ldots,x_l}$ we have
$$\bigcap_{i=1}^l\<z,x_i\>=z.$$
\end{lem}
\begin{proof}
Let us consider the set $\mathcal F$ of $(l+1)$-tuples $(x_1,\ldots,x_l,z)$ in $\mathcal S_{m,n}^{l+1}$ with the property that $\bigcap_{i=1}^l\<z,x_i\>=z$. This is a Zariski open subset of $\mathcal S_{m,n}^{l+1}$: indeed, since $z$ is clearly contained in the intersection, the set $\mathcal F$ is defined by the equation $\dim\bigcap_{i=1}^l\<z,x_i\>\leq m$. In order to conclude the proof it is enough to show that for each $z\in \mathcal S_{m,n}$ the set of tuples $(x_1,\ldots,x_l)$ with the property that $(x_1,\ldots,x_l,z)\in\mathcal F$ is non empty: this implies that the set $\mathcal F$ is a non empty Zariski open subset, and in particular there must exist a Zariski open subset of $\mathcal S_{m,n}^l$ consisting of $l$-tuples $(x_1,\ldots,x_l)$ satisfying the hypothesis of the lemma.
Let us then fix a point $z\in \mathcal S_{m,n}$. We denote by $\mathcal A_{z}^k$ the set of $k$-tuples $x=(x_1,\ldots,x_k)$ in $\mathcal S_{m,n}^z$ such that $\dim\bigcap_{i=1}^k\<z,x_i\>=\max\{2m-(n-m)(k-1),m\}$. In order to conclude the proof it is enough to exhibit, for every $k$-tupla $x$ in $\mathcal A_z^k$ a non empty subset $\mathcal B$ of $\mathcal S_{m,n}$ such that $(x,b)\in\mathcal A_z^{k+1}$ for each $b$ in $\mathcal B$. If we denote by $V_k$ the subspace $\bigcap_{i=1}^k\<z,x_i\>$, that has, by our assumption on the tuple $x$, dimension $2m-(n-m)(k-1)$, we can take $\mathcal B$ to be the Zariski open subset
$$\mathcal B=\{x_{k+1}\in\mathcal S_{m,n}|x_{k+1}\pitchfork V_k, x_{k+1}\pitchfork z\}.$$
The set $\mathcal B$ is not empty since both transversality conditions are non-empty, Zariski open conditions, and for this choice we get
$$\begin{array}{rl}
\dim\bigcap_{i=1}^{k+1}\<z,x_i\>&=\dim(V_k\cap \<z,x_{k+1}\>)\\
&=\dim V_k+\dim \<z,x_{k+1}\>-\dim\<V_k,x_{k+1}\>=\\
&=2m-(n-m)(k-1)+2m\\&-\min\{m+n,\,2m-(n-m)(k-1)+m\}=\\
&=\max\{2m-(n-m)k,m\}.
\end{array}$$
\end{proof}
It is worth remarking that, if $n\geq 2m$, the set $\mathcal S_{m,n}^{(2)}$ is contained in $\mathcal O$ and for $x_1,x_2$ transverse the set $\mathcal D_{x_1,x_2}$ consists of the points $z$ that are transverse to $x_1,x_2$ and $\<x_1,x_2\>$. This is consistent with the notation in Section \ref{ssec:4.3}. In general we will assume (up to restricting $\mathcal D_{x_1,\ldots,x_l}$ to a smaller Zariski open subset) that each $z$ in $\mathcal D_{x_1,\ldots,x_l}$ is transverse to $x_i$ for each $i$.
\begin{lem}
Let $(x_1,\ldots,x_l)$ be an $l$-tuple of pairwise transverse points in the set $\mathcal O$ defined in Lemma \ref{lem:Oo}.
There exist a quasiprojective subset $\mathcal C_{x_1,\ldots,x_l}$ of $W_{x_1}\times\ldots\times W_{x_l}$ such that the map $\beta_{x_1,\ldots,x_l}=\pi_{x_1}\times\ldots\times\pi_{x_l}:\mathcal D_{x_1,\ldots,x_l}\to W_{x_1}\times\ldots\times W_{x_l}$ gives a birational isomorphism.
\end{lem}
\begin{proof}
We consider the set $\mathcal C'_{x_1,\ldots,x_l}$ consisting of tuples $(t_1,\ldots,t_l)$ with the property that the associated linear subspaces intersect in an $m$-dimensional isotropic subspace, and that $t_j-\pi_j(x_i)$ has maximal rank for every $i,j$. With this choice $\mathcal C'_{x_1,\ldots,x_l}$ is quasiprojective since the condition that the intersection has dimension at least $m$ and that the restriction of $h$ to the intersection is degenerate are closed condition (defined by polynomial), the condition that the intersection has dimension at most $m$ is an open condition. The set $\mathcal C_{x_1,\ldots,x_l}$ is the subset of $\mathcal C'_{x_1,\ldots,x_l}$ that is the image of $\beta_{x_1,\ldots,x_l}$.
The fact that the map $\beta_{x_1,\ldots,x_l}$ gives a birational isomorphism follows from the fact that a regular inverse to $\beta_{x_1,\ldots,x_l}$ is given by the algebraic map that associates to an $l$-tuple of points their unique intersection.
\end{proof}
We now have all the ingredients we need to prove the following
\begin{prop}
Let us assume that for almost every point $x\in\partial\mathbb H _{\mathbb C}^p$
the map $\phi_x$ coincides almost everywhere with a rational map. The same is true for $\phi$.
\end{prop}
\begin{proof}
Let us fix $l$ points $t_1,\ldots, t_l$ which are generic in the sense of Lemma \ref{lem:1}, which satisfy that $\phi_{t_i}$ coincides almost everywhere with a rational map, and with the additional property that the $l$-tupla $(\phi(t_1),\ldots,\phi(t_l))$ belongs to the set $\mathcal O$. We can find such points since the map $\phi$ is Zariski dense and the set $\mathcal O$ is Zariski open. Let us now
consider the diagram
$$\xymatrix{\partial\mathbb H _{\mathbb C}^p\setminus\{t_1,\ldots,t_l\}\ar[r]^{\phi}\ar[d]_{\pi_{t_1}\times\ldots\times\pi_{t_l}}&\mathcal D_{\phi(t_1),\ldots,\phi(t_l)}\subseteq \mathcal S_{m,n}\ar[d]^{\beta_{\phi(t_1),\ldots,\phi(t_l)}}\\ W_{t_1}\times\ldots\times W_{t_n}\ar[r]^-{\phi_{t_1}\times\ldots\times\phi_{t_n}}&\mathcal C_{\phi(t_1),\ldots,\phi(t_l)}}.$$
A consequence of Lemma \ref{lem:phix} and of the definition of the isomorphisms $\beta_{\phi(t_1),\ldots,\phi(t_l)}$ is that the diagram commutes almost everywhere. In particular, since the isomorphism $\beta_{\phi(t_1),\ldots\phi(t_l)}$ is birational, and $\pi_{t_1}\times\ldots\times\pi_{t_l}$ is rational, we get that $\phi$ coincides almost everywhere with a rational map.
\end{proof}
Let us now fix a point $x$ in $\partial\mathbb H _{\mathbb C}^p$, and identify the space $W_x$ with $\mathbb C^{p-1}$. We want to study the map $\phi_x:\mathbb C^{p-1}\to W_{\phi(x)}$.
It follows from Lemma \ref{lem:parametrization of k-vertical chains} restricted to the case $m=1$ that the projections of chains in $\partial\mathbb H _{\mathbb C}^p$ to $\mathbb C^{p-1}$ are Euclidean circles $C\subset \mathbb C^{p-1}$ (possibly collapsed to points).
\begin{lem}
If $x$ is generic in the sense of Lemma \ref{lem:1}, the restriction of $\phi_x$ to almost every Euclidean circle $C$ of $\mathbb C^{p-1}$ is rational.
\end{lem}
\begin{proof}
It follows from the explicit parametrization of a chain given in Lemma \ref{lem:parametrization of k-vertical chains} that, whenever a point $t$ in $\pi_x^{-1}(C)$ is fixed, the lift map $l:C\to T$ is algebraic, where $T$ is the unique lift of $C$ containing $t$.
In particular, if $T$ is a chain such that the restriction of $\phi$ to $T$ coincides almost everywhere with a rational map, the restriction of $\phi_x$ to $C=\pi_x(T)$ coincides almost everywhere with a rational map. We can now use a Fubini based argument to get that, for almost every circle $C$, the restriction of $\phi_x$ to $C$ is rational: for almost every chain $T$, the restriction to $T$ coincides almost everywhere with a rational map, the conclusion follows from the fact that the space of chains that do not contain $x$ is a full measure subset of the space of chains in $\partial\mathbb H _{\mathbb C}^p$ that forms a smooth bundle over the space of Euclidean circles in $\mathbb C^{p-1}$.
\end{proof}
An usual Fubini type argument implies now the following
\begin{cor}\label{cor:phiL}
For almost every complex affine line $\mathcal L\subseteq \mathbb C^{m}$, for almost every Euclidean circle $C$ contained in $\mathcal L$, the restriction of $\phi_x$ to $C$ is algebraic. Moreover the same is true for almost every point $p$ in $\mathcal L$ and almost every circle $C$ containing $p$.
\end{cor}
In order to conclude the proof we will apply many times the following well known lemma that allows to deduce that a map is rational provided that the restriction to sufficiently many subvarieties is rational. Given a map $\phi:A\times B\to C$ and given a point $a\in A$ we denote by $_a\phi:B\to C$ the map $_a\phi(b)=\phi(a,b)$ in the same way, if $b$ is a point in $B$, $^b\phi$ will the note the map $^b\phi(a)=\phi(a,b)$
\begin{lem}[{\cite[Theorem 3.4.4]{Zimmer}}]\label{lem:R^nxR^m}
Let $\phi:\mathbb R^{n+m}\to\mathbb R$ be a measurable function. Let us consider the splitting $\mathbb R^{n+m}=\mathbb R^n\times\mathbb R^m$. Assume that for almost every $a\in \mathbb R^n$ the function $_a\phi:\mathbb R^m\to\mathbb R$ coincides almost everywhere with a rational function and for almost every $b\in\mathbb R^m$ the function $^b\phi:\mathbb R^n\to\mathbb R$ coincides almost everywhere with a rational function, then $\phi$ coincides almost everywhere with a rational function.
\end{lem}
This easily gives that the restriction of $\phi_x$ to any complex affine line $\mathcal L$ in $\mathbb C^{p-1}$ coincides almost everywhere with a rational map:
\begin{lem}
For almost every affine complex line $\mathcal L\subset \mathbb C^{p-1}$, the restriction $\phi_x|_{\mathcal L}$ coincides almost everywhere with a rational map.
\end{lem}
\begin{proof}
Let us fix a line $\mathcal L$ satisfying the hypothesis of Corollary \ref{cor:phiL} and denote by $\phi_\mathcal L:\mathbb C\to W_{\phi(x)}$ the restriction of $\phi_x$ to $\mathcal L$, composed with a linear identification of $\mathcal L$ with the complex plane $\mathbb C$. By the second assertion of Corollary \ref{cor:phiL}, we can find a point $p\in \mathbb C$ such that for almost every Euclidean circle $C$ through $p$ the restriction of $\phi_{\mathcal L}$ to $C$ coincides almost everywhere with a rational map. Let us now consider the birational map $i_p:\mathbb C\to\mathbb C$ defined by $i_p(z)=(z-p)^{-1}$, and let us denote by $\psi_\mathcal L$ the composition $\psi_\mathcal L=\phi_\mathcal L\circ i_p^{-1}$. Since the image under $i_p$ of Euclidean circles through the point $p$ are precisely the affine real lines of $\mathbb C$ that do not contain $0$, we get that the restriction of $\psi_\mathcal L$ to almost every affine line coincides almost everywhere with a rational map. In particular a consequence of Lemma \ref{lem:R^nxR^m} is that the map $\psi_\mathcal L$ itself coincides almost everywhere with a
rational map. Since $\phi_
\mathcal L$ coincides almost everywhere with $\psi_\mathcal L\circ i_p$ we get that the same is true for the map $\phi_\mathcal L$ and this concludes the proof.
\end{proof}
Applying Lemma \ref{lem:R^nxR^m} again we deduce the following proposition:
\begin{prop}
Let $\phi:\partial\mathbb H _{\mathbb C}^p\to\mathcal S_{m,n}$ be a map with the property that for almost every chain $C$ the restriction of $\phi$ to $C$ coincides almost everywhere with a rational map. Then for almost every point $x\in\partial\mathbb H _{\mathbb C}^p$ the map $\phi_x$ coincides almost everywhere with a rational map.
\end{prop}
In turn this was the last missing ingredient to prove Theorem \ref{thm:phirational}
\section{Conclusion}\label{sec:rational}
The last step of Margulis' original proof of superrigidity involves showing that if a Zariski dense representation $\rho:\Gamma\to H$ of a lattice $\Gamma$ in the algebraic group $G$ admits an algebraic boundary map, then it extends to a representation of $G$ (cfr. \cite{Margulis} and \cite[Lemma 5.1.3]{Zimmer}). The same argument applies here to deduce our main theorem:
\begin{proof}[Proof of Theorem \ref{thm:Zariskisuperrigidity}]
Let $\rho:\Gamma\to {\rm PU}(m,n)$ be a Zariski dense maximal representation and let $\psi:\partial\mathbb H _{\mathbb C}^p\to\mathcal S_{m,n}$ be a measurable $\rho$-equivariant boundary map, that exists as a consequence of Proposition \ref{prop:boundary map} (the difference between ${\rm SU}(m,n)$ and ${\rm PU}(m,n)$ plays no role here, since the action of ${\rm SU}(m,n)$ on $\mathcal S_{m,n}$ factors through the projection to the adjoint form of the latter group). The essential image of $\psi$ is a Zariski dense subset of $\mathcal S_{m,n}$ as a consequence of Proposition \ref{prop:phiZariskidense}, moreover Corollary \ref{cor:incidence preserved} implies that $\psi$ preserves the chain geometry.
Since we proved that any measurable, Zariski dense, chain preserving boundary map $\psi$ coincides almost everywhere with a rational map (cfr. Theorem \ref{thm:phirational}), we get that there exists a $\rho$-equivariant rational map $\phi:\partial\mathbb H _{\mathbb C}^p\to\mathcal S_{m,n}$. The $\rho$-equivariance follows from the fact that $\phi$ coincides almost everywhere with $\psi$ that is $\rho$-equivariant. In particular, for every $\gamma$ in $\Gamma$ the set on which $ \phi(\gamma x)=\rho(\gamma)\phi(x)$ is a Zariski closed, full measure set, and hence is the whole $\partial\mathbb H _{\mathbb C}^p$.
Since $\phi$ is $\rho$-equivariant and rational it is actually regular: indeed the set of regular points for $\phi$ is a non-empty, Zariski open, $\Gamma$-equivariant subset of $\partial\mathbb H _{\mathbb C}^p$. Since, by Borel density \cite[Theorem 3.2.5]{Zimmer}, the lattice $\Gamma$ is Zariski dense in ${\rm SU}(1,p)$ and $\partial\mathbb H _{\mathbb C}^p$ is an homogeneous algebraic ${\rm SU}(1,p)$ space, the only $\Gamma$-invariant proper Zariski closed subset of $\partial\mathbb H _{\mathbb C}^p$ is the empty set, and this implies that the set of regular points of $\phi$ is the whole $\partial\mathbb H _{\mathbb C}^p$.
In the sequel it will be useful to deal with complex algebraic groups and complex varieties in order to exploit algebraic results based on Nullstellensatz. This is easily achieved by considering the complexification. We will denote by $G$ the algebraic group ${\rm SL}(p+1,\mathbb C)$ and by $H$ the group ${\rm PSL}(m+n,\mathbb C)$ endowed with the appropriate real structures so that ${\rm SU}(1,p)=G(\mathbb R)$ and ${\rm PU}(m,n)=H(\mathbb R)$.
Since $\partial\mathbb H _{\mathbb C}^p$ and $\mathcal S_{m,n}$ are homogeneous spaces that are projective varieties, there exist parabolic subgroups $P<{\rm SL}(p+1,\mathbb C)$ and $Q<{\rm PSL}(m+n,\mathbb C)$ such that $\partial\mathbb H _{\mathbb C}^p=(G/P)(\mathbb R)=G(\mathbb R)/(P\cap G(\mathbb R))$ and $\mathcal S_{m,n}=(H/Q)(\mathbb R)$.
The algebraic $\rho$-equivariant map $\phi:\partial\mathbb H _{\mathbb C}^p\to\mathcal S_{m,n}$ lifts to a map $\overline \phi:G(\mathbb R)\to \mathcal S_{m,n}$ and we can extend the latter map uniquely to an algebraic map $T:G\to H/Q$ using the fact that $G(\mathbb R)$ is Zariski dense in $G$. The extended map $T$ is $\rho$-equivariant since $G(\mathbb R)$ is Zariski dense in $G$: whenever an element $\gamma \in \Gamma$ is fixed, the set $\{g\in G|\; T(\gamma g)=\rho(\gamma)T(g)\}$ is Zariski closed and contains $G(\mathbb R)$.
Let us now focus on the graph of the representation $\rho:\Gamma\to H$ as a subset ${\rm Gr}(\rho)$ of the group $ G\times H$. Since $\rho$ is an homomorphism, ${\rm Gr}(\rho)$ is a subgroup of $G\times H$, hence its Zariski closure $\overline{{\rm Gr}(\rho)}^Z$ is an algebraic subgroup. The image under the first projection $\pi_1$ of $\overline{{\rm Gr}(\rho)}^Z$ is a closed subgroup of $G$: indeed the image of a rational morphism (over an algebraically closed field) contains an open subset of its closure, since in our case $\pi_1$ is a group homomorphism, its image is an open subgroup that is hence also closed. Moreover $\pi_1(\overline{{\rm Gr}(\rho)}^Z)$ contains $\Gamma$ that is Zariski dense in $G$ by Borel density, hence equals $G$.
We now want to use the existence of the algebraic map $T$ and the fact that $\rho(\Gamma)$ is Zariski dense in $H$ to show that $\overline{{\rm Gr}(\rho)}^Z$ is the graph of an homomorphism.
In fact it is enough to show that $\overline{{\rm Gr}(\rho)}^Z\cap (\{{\rm id}\}\times H)=({\rm id},{\rm id})$.
Let $({\rm id},f)$ be an element in $\overline{{\rm Gr}(\rho)}^Z\cap (\{{\rm id}\}\times H)$. Since $H$ is absolutely simple being an adjoint form of a simple Lie group, and $N=\bigcap_{h\in H}hQh^{-1}$ is a normal subgroup of $H$, it is enough to show that $f\in N$ or, equivalently, that $f$ fixes pointwise $H/Q$.
But, since $T$ is a regular map, and the actions of $G$ on itself and of $H$ on $H/Q$ are algebraic, we get that the stabilizer of the map $T$ under the $G\times H$- action,
$$\mathrm{Stab}_{G\times H}(T)=\{(g,h)|\;((g,h)\cdot T)(x)=h^{-1}T(gx)=T(x),\;\forall x\in G\},$$ is a Zariski closed subgroup of $G\times H$. Moreover $\mathrm{Stab}_{G\times H}(T)$ contains ${\rm Gr}(\rho)$ and hence also $\overline{{\rm Gr}(\rho)}^Z$. In particular $({\rm id},f)$ belongs to the stabilizer of $T$, hence the element $f$ of $H$ fixes the image of $T$ pointwise. Since the image of $T$ is $\rho(\Gamma)$-invariant, $\rho(\Gamma)$ is Zariski dense and the set of points in $H/Q$ that are fixed by $f$ is a closed subset, $f$ acts trivially on $H/Q$.
\end{proof}
We can now prove Theorem \ref{thm:general}:
\begin{proof}[Proof of Theorem \ref{thm:general}]
Let $\rho:\Gamma\to {\rm SU}(m,n)$ be a maximal representation and let $L$ be the Zariski closure of $\rho(\Gamma)$ in ${\rm SL}(m+n,\mathbb C)$. Here, as above, ${\rm SU}(m,n)=H(\mathbb R)$ with respect to a suitable real structure on $H={\rm SL}(m+n,\mathbb C)$. Since the representation $\rho$ is tight, we get, as a consequence of Theorem \ref{thm:tight}, that $L(\mathbb R)$ almost splits a product $L_{nc}\times L_{c}$ where $L_{nc}$ is a semisimple Hermitian Lie group tightly embedded in ${\rm SU}(m,n)$ and $K=L_c$ is a compact subgroup of ${\rm SU}(m,n)$.
Let us consider $L_1,\ldots, L_k$ the simple factors of $L_{nc}$, namely $L_{nc}$, being semisimple, almost splits as the product $L_1\times\ldots \times L_k$ where $L_k$ are simple Hermitian Lie groups. The first observation is that none of the groups $L_i$ can be virtually isomorphic to ${\rm SU}(1,1)$. In that case the composition of the representation $\rho$ with the projection $L_{nc}\to L_i$ would be a maximal representation of a complex hyperbolic lattice with values in a group that is virtually isomorphic to ${\rm SU}(1,1)$ and this is ruled out by \cite{BICartan}: indeed Burger and Iozzi prove, as the last step in their proof of \cite[Theorem 2]{BICartan}, that there are no maximal representations of complex hyperbolic lattices in ${\rm PU}(1,1)$.
This implies that the inclusion $i:L_{nc}\to {\rm SU}(m,n)$ fulfills the hypotheses of Theorem \ref{thm:tightol}. In particular it is enough to prove that each factor $L_i$ which is not of tube type is isomorphic to ${\rm SU}(1,p)$ and the composition of $\rho$ with the projection to $L_i$ is conjugate to the inclusion.
Since, by Corollary \ref{cor:noZariskidense}, there is no Zariski dense representation of $\Gamma$ in ${\rm SU}(m_i,n_i)$ if $1<m_i<n_i$, we get that $m_i=1$. Moreover, since the only Zariski dense tight representation of ${\rm SU}(1,p)$ in ${\rm SU}(1,q)$ is the identity map, we get that $n_i=p$ and the composition of $\rho$ with the projection to $L_i$ is conjugate to the inclusion.
This concludes the proof.
\end{proof}
\begin{proof}[Proof of Corollary \ref{cor:general}]
We know that the Zariski closure of the representation $\rho$ is contained in a subgroup of ${\rm SU}(m,n)$ isomorphic to ${\rm SU}(1,p)^t\times {\rm SU}(m-t,m-t)\times K$. The product $M={\rm SU}(1,p)^t\times {\rm SU}(m-t,m-t)$ corresponds to a splitting $\mathbb C^{m,n}=V_1\oplus\ldots\oplus V_t\oplus W\oplus Z$ where the restriction of $h$ to $V_i$ is non-degenerate and has signature $(1,p)$ and the restriction of $h$ to $W$ is non-degenerate and has signature $(m-t,m-t)$. The subspace $W$ is left invariant by $M$ hence also by $K$ (since $K$ commutes with $M$ and all the invariant subspaces for $M$ have different signature).
In particular the linear representation of $\Gamma$ associated with $\rho$ leaves invariant a subspace on which $h$ has signature $(k,k)$ for some $k$ greater than 1 unless there are no factors
of
tube-type in the decomposition of $L$. This latter case corresponds to standard embeddings.
\end{proof}
\begin{proof}[Proof of Corollary \ref{cor:local rigidity}]
Let us denote by $\rho_0:\Gamma\to {\rm SU}(m,n)$ the standard representation. Since by Lemma \ref{lem:Toledo constant} the generalized Toledo invariant is constant on components of the representation variety, we get that any other representation $\rho$ in the component of $\rho_0$ is maximal. By Theorem \ref{thm:general} this implies that $\overline{\rho(\Gamma)}^Z$ almost splits as a product $K\times L_t\times {\rm SU}(1,p)$, and is contained in a subgroup of ${\rm SU}(m,n)$ of the form ${\rm SU}(1,p)^t\times {\rm SU}(m-t,m-t)\times K$. If the group $L_t$ is trivial then $\rho$ is a standard embedding, and is hence conjugate to $\rho_0$ up to a character in the compact centralizer of the image of $\rho_0$. In particular this would imply that $\rho_0$ is locally rigid.
Let us then assume by contradiction that there are representations $\rho_i$ arbitrarily close to $\rho_0$ and with the property that the tube-type factor of the Zariski closure of $\rho_i(\Gamma)$ is non trivial. Up to modifying the representations $\rho_i$ we can assume that the compact factor $K$ in the Zariski closure of $\rho_i$ is trivial.
By Theorem \ref{thm:general} this implies that $\rho_i(\Gamma)$ is contained in a subgroup of ${\rm SU}(m,n)$ isomorphic to ${\rm SU}(m,pm-1)$, moreover we can assume, up to conjugate the representations $\rho_i$ in ${\rm SU}(m,n)$, that the Zariski closure of $\rho_i$ is contained in the same subgroup ${\rm SU}(m,pm-1)$ for every $i$. Since the representations whose image is contained in the subgroup ${\rm SU}(m,pm-1)$ is a closed subspace of ${\rm Hom}(\Gamma,G)/G$, we get that the image of $\rho_0$ is contained in ${\rm SU}(m,pm-1)$ and this is a contradiction, since the image of the diagonal embedding doesn't leave invariant any subspace on which the restriction of $h$ has signature $(m,pm-1)$.
\end{proof}
|
1,314,259,993,643 | arxiv |
\section{Introduction}
\label{sec:intro}
Synthesizing realistic views from varying viewpoints is essential for
interactions between humans and virtual environments, hence it is of key importance for many
virtual reality applications. The novel view synthesis task is especially relevant for indoor scenes, where it enables virtual navigation through buildings, tourist destinations, or game environments.
When scaling up such applications, it is preferable to minimize the amount of input data required to store and process, as well as its acquisition time and cost. In addition, a static scene requirement is increasingly difficult to fulfill for a longer capture duration. Thus, our goal is novel view synthesis at room-scale using few input views.
NeRF \cite{Mildenhall2020NeRFRS} represents the radiance field and density distribution of a scene as a multi-layer perceptron (MLP) and uses volume rendering to synthesize output images. This approach creates impressive, photo-realistic novel views of scenes with complex geometry and appearance.
Unfortunately, applying NeRF to real-world, room-scale scenes given only tens of images does not produce desirable results (\cref{fig:teaser}) for the following reason:
NeRF purely relies on RGB values to determine correspondences between input images. As a result, high visual quality is only achieved by NeRF when it is given enough images to overcome the inherent ambiguity of the correspondence problem.
Real-world indoor scenes have characteristics that further complicate the ambiguity challenge:
First, in contrast to an ``outside-in'' viewing scenario of images taken around a central object, views of rooms represent an ``inside-out'' viewing scenario, in which the same number of images will exhibit significantly less overlap with each other.
Second, indoor scenes often have large areas with minimal texture, such as white walls.
Third, real-world data often has inconsistent color values across views, e.g., due to white balance or lens shading artifacts.
These characteristics of indoor scenes are likewise challenging for SfM, leading to very sparse SfM reconstructions, often with severe outliers.
Our idea is to use this noisy and incomplete depth data
and from it produce a complete dense map alongside a per-pixel uncertainty estimate of those depths, thereby increasing its value for NeRF --- especially in textureless, rarely observed, or color-inconsistent areas.
We propose a method that guides the NeRF optimization with dense depth priors, without the need for additional depth input (e.g., from an RGB-D sensor) of the scene. Instead, we take advantage of the sparse reconstruction that is freely available as a byproduct of running SfM to compute camera pose parameters. Specifically, we complete the sparse depth maps with a network that estimates uncertainty along with depth. Taking uncertainty into account, we use the resulting dense depth to constrain the optimization and to guide the scene sampling. We evaluate the effectiveness of our method on complete rooms from the Matterport3D \cite{Chang2017Matterport3DLF} and ScanNet \cite{Dai2017ScanNetR3} datasets, using only a handful of input images. We show that our approach improves over recent and concurrent work that uses sparse depth from SfM or multi-view stereo (MVS) in NeRF \cite{wei2021nerfingmvs,Deng2021DepthsupervisedNF}.
In summary, we demonstrate that dense depth priors with uncertainty estimates enable novel view synthesis with NeRF on room-size scenes using only 18--36 images, enabled by the following contributions:
\begin{compactenum}
\item A data-efficient approach to novel view synthesis on real-world scenes at room-scale.
\item An approach to enhance noisy sparse depth input from SfM to support the NeRF optimization.
\item A technique for accounting for variable uncertainty when guiding NeRF with depth information.
\end{compactenum}
\section{Related Work}
\label{sec:related_work}
The ability to synthesize novel views of a scene from a set of observed images and corresponding camera viewpoints is necessary for enabling virtual experiences of real-world environments.
In situations where it is feasible to densely sample images of the scene,
novel viewpoints can be synthesized with simple light field interpolation~\cite{gortler96lumigraph,levoy96lightfield}.
However, when fewer observed views of the scene are available, it becomes increasingly necessary to use information about the scene's geometry to render new views. A common paradigm for geometry-based view synthesis is to use a triangle mesh representation of scene geometry to reproject observed images into each novel viewpoint and combine them using either heuristic~\cite{buehler2001unstructured,debevec1992modeling,wood2000surface} or learned~\cite{hedman2019deep,riegler2020free} blending algorithms. More recently, these mesh-based geometry models have been replaced by volumetric scene representations such as voxel grids~\cite{lombardi2019neuralvolumes} or multiplane images~\cite{flynn2019deepview,mildenhall2019llff,srinivasan2019pushing,zhou2018stereomag}.
NeRF~\cite{Mildenhall2020NeRFRS} popularized an approach that avoids the steep scaling properties of discrete voxel representations by representing a scene as a continuous volume, parameterized by a MLP that is optimized to minimize the loss of re-rendering all observed views of a scene. Since its introduction, NeRF has become the dominant scene representation for view synthesis, and many recent works are built on top of NeRF's neural volumetric model.
However, in situations where the scene is observed from very few sparsely-sampled viewpoints, NeRF's high capacity to model detailed geometry and appearance can
result in various artifacts, such as ``floaters'', i.e., artifacts caused by a flawed density distribution.
In this work, we directly address the few-input setting, proposing a strategy that takes advantage of sparse depth to constrain NeRF's scene geometry and improve rendering quality. This depth data is freely available as a byproduct of running SfM to compute camera poses from the input images (e.g., by using COLMAP~\cite{schoenberger2016sfm}).
Our method takes inspiration from techniques that generate complete dense depth maps from sparse depth inputs. These include classic techniques that fuse observed depths into a single 3D reconstruction, typically in the form of a truncated signed distance function~\cite{curlesslevoy,newcombe2016kinectfusion}, as well as more recent techniques that train deep networks to operate over the sparsely observed geometry in 3D~\cite{dai2018scancomplete,dai2020sgnn}. Although these methods are effective for dense 3D scene reconstruction, the resulting geometry is not ideal for view synthesis since its edges frequently do not align with edges in the observed images. Instead, we leverage recent work on 2D sparse depth completion that directly completes depth maps in image space~\cite{cheng2018cspn,Cheng2020LearningDW} and extend it to also predict uncertainty.
A few recent works have also proposed incorporating depth observations into NeRF reconstruction. NerfingMVS~\cite{wei2021nerfingmvs} uses depth from MVS to overfit a depth predictor to the scene. The resulting depth prior guides the NeRF sampling. In comparison, our method employs depth completion on the SfM depth and additionally employs a depth loss to supervise the geometry recovered by NeRF. This way, our novel views achieve significantly better color and depth quality in the few-input setting without relying on the computationally more expensive MVS preprocessing. Concurrent work on depth-supervised NeRF~\cite{Deng2021DepthsupervisedNF} directly uses sparse depth information from SfM in the NeRF optimization. To handle inaccuracy in the sparse reconstruction, the 3D points are weighted according to their reprojection error. In contrast, we learn dense depth priors with uncertainty to more effectively guide the optimization, leading to more detailed novel views, as well as more accurate geometry and higher robustness to SfM outliers.
\section{Method}
\label{sec:method}
\begin{figure*}[ht]
\centering
\includegraphics[width=\linewidth,trim={2.4cm 10.4cm 1.7cm 2.8cm},clip]{figures/figure_2.pdf}
\caption{Overview of our radiance field optimization pipeline. Given a small set of RGB images of a room, we run SfM to obtain camera parameters and a sparse reconstruction, from which a sparse depth map is rendered for each input view. A depth completion network predicts dense depth and standard deviation, which is used to focus the scene sampling on surfaces. The samples on a ray, its viewing direction and a per-camera latent code are input to the radiance field. The output color and density are integrated to obtain the pixel's color and the ray's expected termination depth. The radiance field is supervised using the input RGB and the depth completion output. }
\label{fig:pipeline}
\end{figure*}
Our method facilitates room-scale novel view synthesis from a small collection of RGB images $\{I_i\}^{N-1}_{i=0}$, $I_i \in [0, 1]^{H\times W\times3}$ (see \cref{fig:pipeline}). As a preprocessing step (e.g., using SfM), the camera pose $\mathbf{p}_i \in \mathbb{R}^6$, intrinsic parameters $K_i \in \mathbb{R}^{3\times3}$, and a sparse depth map $Z_i^{\mathrm{sparse}} \in [0, t_f]^{H\times W}$ are computed for each image. $0$ values of the sparse depth indicate invalid pixels, and $t_f$ is the far plane of the volume rendering.
Our approach builds upon NeRF \cite{Mildenhall2020NeRFRS}. Prior to the NeRF optimization, a network estimates depth with uncertainty from the sparse depth input (\cref{ssec:depth_completion}). We incorporate the resulting dense depth prior into the NeRF optimization by adding a depth constraint and a guided sampling approach (\cref{ssec:depth_supervision}).
\subsection{Depth Completion with Uncertainty}
\label{ssec:depth_completion}
\paragraph{Network Architecture} With the goal of completing sparse depth from SfM, two challenges presented by this input data play a key role in designing the depth prior network. First, sparse reconstructions are noisy and have outliers. As a consequence, dense depth predictions are expected to have spatially varying accuracy, which makes it crucial to know the uncertainty at a per-pixel level.
Second, the density of SfM point clouds varies significantly across space, depending on the number of image features.
E.g., SfM reconstructions from 18--20 images per ScanNet scene lead to sparse depth maps with 0.04\% valid pixels on average.
Hence, depth completion must be able to predict dense depth even from largely empty sparse depth maps.
In order to address the first challenge, we construct our depth prior network $D_{\theta_0}$ to predict dense depth maps $Z_i^{\mathrm{dense}} \in [0, t_f]^{H\times W}$ along with pixelwise standard deviations $S_i \in [0, \infty)^{H\times W}$ from the sparse depth maps:
\begin{equation}
\left[Z_i^{\mathrm{dense}}, S_i\right] = D_{\theta_0}\!\left(I_i, Z_i^{\mathrm{sparse}}\right)\, ,
\label{eq:depth_completion}
\end{equation}
where $D_{\theta_0}$ is a convolutional network with ResNet~\cite{He2016DeepRL} downsampling and skip connections to two upsampling branches to predict depth $Z_i^{\mathrm{dense}}$ and standard deviation $S_i$.
To address the second challenge of extremely sparse input depth, we employ a Convolutional Spatial Propagation Network (CSPN)~\cite{Cheng2020LearningDW} in each branch. This refinement block locally and iteratively applies a kernel with weights given by a learned affinity matrix. This refines the often blurry depth output to become more detailed and sharp. Equally important, this process spreads information to neighboring pixels; i.e., information propagates further with each iteration. An increased number of iterations in the depth and the uncertainty head prove helpful to handle very sparse input.
\paragraph{Network Training} Though we evaluate on RGB-only data using SfM, we train our model on RGB-D data from ScanNet~\cite{Dai2017ScanNetR3} and Matterport3D~\cite{Chang2017Matterport3DLF}.
To avoid both the effort of running SfM on a large dataset and the possibility of SfM failures in the training data, the sparse depth input is sampled from the range sensor depth.
As such, it is critical to subsample and perturb the dense depth from the sensor in a way that creates realistic sparse training depth to enable the network to generalize to real SfM input at test time.
Specifically, a SIFT feature extractor, e.g., from COLMAP~\cite{schoenberger2016sfm}, is used to determine locations where sparse depth points would exist in a SfM reconstruction. We sample the sensor depth at these points and perturb it with Gaussian noise $\mathcal{N}(0, s_{\mathrm{noise}}(z)^2)$, where the standard deviation $s_{\mathrm{noise}}$ increases with depth. The function $s_{\mathrm{noise}}(z)$ is determined by fitting a second-order polynomial to the depth deviation between the sparse SfM reconstructions and sensor depth.
Under the assumption that the output is normally distributed, we supervise the network by minimizing the negative log likelihood of a Gaussian:
\begin{equation}
\mathcal{L}_{\theta_0} = \frac{1}{n} \sum_{j=1}^{n}\left(\log(s_j^2) + \frac{\left(z_j - z_{\mathrm{sensor},j}\right)^2}{s_j^2}\right),
\label{eq:depth_completion_loss}
\end{equation}
where $z_j$ and $s_j$ are the predicted depth and standard deviation of pixel $j$, $z_{\mathrm{sensor},j}$ is the sensor depth at $j$, and $n$ is the number of valid pixels in the dense sensor depth map.
\subsection{Radiance Field with Dense Depth Priors}
\label{ssec:depth_supervision}
\paragraph{Scene Representation} Following NeRF \cite{Mildenhall2020NeRFRS}, we encode the radiance field of the scene into a MLP $F_{\theta_1}$ that predicts color $\mathbf{c} = [r, g, b]$ and volume density $\sigma$ from a position $\mathbf{x} \in \mathbb{R}^3$ and a unit-norm viewing direction $\mathbf{d} \in \mathbb{S}^2$. $\gamma$ applies a positional encoding with 9 frequencies on the position. Because our scenes are angularly undersampled, we minimize the capacity of our view-dependent network by omitting positional encoding for the viewing direction.
\begin{equation}
\left[\mathbf{c}, \sigma\right] = F_{\theta_1}\!\left(\gamma(\mathbf{x}), \mathbf{d}, \boldsymbol{\ell}_i\right).
\label{eq:mlp}
\end{equation}
As an additional input to $F_{\theta_1}$, we generate a per-image embedding vector $\boldsymbol{\ell}_i \in \mathbb{R}^e$ \cite{MartinBrualla2021NeRFIT}. This allows the network to compensate for view-specific phenomena such as inconsistent lighting or lens shading, which can cause severe artifacts in novel views, particularly with few input images.
\paragraph{Optimization with Depth Constraint} To optimize the radiance field, the color $\hat{\mathbf{C}}(\mathbf{r})$ of each pixel in the batch $R$ is computed by evaluating a discretized version of the volume rendering integral (\cref{eq:volume_rendering} \cite{Mildenhall2020NeRFRS}).
Specifically, a pixel determines a ray $\mathbf{r}(t) = \mathbf{o} + t\mathbf{d}$ whose origin is at the camera's center of projection $\mathbf{o}$. Rays are sampled along their traversal through the volume. For each sampling location $t_k \in [t_n, t_f]$ within the near and far planes, a query to $F_{\theta_1}$ provides the local color and density.
\begin{align}
\hat{\mathbf{C}}(\mathbf{r}) &= \sum_{k=1}^{K} w_k\mathbf{c}_k \, , \label{eq:volume_rendering} \\
\text{where} \quad w_k &= T_k\left(1 - \exp(-\sigma_k\delta_k)\right) \, ,
\label{eq:rendering_weight} \\
T_k &= \exp \left( -\sum_{k'=1}^{k} \sigma_{k'}\delta_{k'}\right)\, , \\
\delta_k &= t_{k+1} - t_k \, .
\end{align}
Besides the predicted color of a ray, a NeRF depth estimate $\hat{z}(\mathbf{r})$ and standard deviation $\hat{s}(\mathbf{r})$ are needed to supervise the radiance field according to the learned depth prior (\cref{ssec:depth_completion}). The NeRF depth estimate and standard deviation are computed from the rendering weights $w_k$:
\begin{align}
\hat{z}(\mathbf{r}) &= \sum_{k=1}^{K}w_kt_k \, , \quad
\hat{s}(\mathbf{r})^2 = \sum_{k=1}^{K}w_k(t_k - \hat{z}(\mathbf{r}))^2.
\end{align}
The network parameters $\theta_1$ are optimized using a loss function $\mathcal{L}_{\theta_1}$ composed of a mean squared error (MSE) term on the color output $\mathcal{L}_{\mathrm{color}}$ and a Gaussian negative log likelihood (GNLL) term on the depth output $\mathcal{L}_{\mathrm{depth}}$:
\begin{align}
\mathcal{L}_{\theta_1} &= \displaystyle\sum_{\mathbf{r}\in R}\Big(\mathcal{L}_{\mathrm{color}}(\mathbf{r}) + \lambda \mathcal{L}_{\mathrm{depth}}(\mathbf{r})\Big), \\
\mathcal{L}_{\mathrm{color}}(\mathbf{r}) &= \begin{Vmatrix} \hat{\mathbf{C}}(\mathbf{r}) - \mathbf{C}(\mathbf{r}) \end{Vmatrix}_2^2,
\end{align}
\begin{align}
\!\!\!\mathcal{L}_{\mathrm{depth}}(\mathbf{r}) = &\begin{cases} \log\left(\hat{s}(\mathbf{r})^2\right) + \frac{\left(\hat{z}(\mathbf{r}) - z(\mathbf{r})\right)^2}{\hat{s}(\mathbf{r})^2} &\!\! \text{if } P \text{ or } Q \\
\label{eq:depth_loss}
0 &\!\! \text{otherwise,} \end{cases} \\
\!\!\!\!\!\text{where} \quad P &= | \hat{z}(\mathbf{r}) - z(\mathbf{r}) | > s(\mathbf{r}) \, , \label{eq:conditionA}\\
\!\!\!\!\!Q &= \hat{s}(\mathbf{r}) > s(\mathbf{r}) \, . \label{eq:conditionB}
\end{align}
Here $z(\mathbf{r})$ and $s(\mathbf{r})$ are the target depth and standard deviation from the corresponding $Z_i^{\mathrm{dense}}$ and $S_i$.
The depth loss is applied to rays where at least one of the following conditions is true:
1) the difference between the predicted and target depth is greater than the target standard deviation \cref{eq:conditionA}, or
2) the predicted standard deviation is greater than the target standard deviation \cref{eq:conditionB}.
This way, the loss encourages NeRF to terminate rays within one standard deviation of the most certain surface observation in the depth prior.
At the same time, NeRF retains some freedom to allocate density to best minimize the color loss. The effectiveness of this depth loss in contrast to MSE is shown in the ablation study (\cref{ssec:ablation_studies}).
\paragraph{Depth-Guided Sampling} In addition to the depth loss function, the depth prior contains valuable signal to guide sampling along a ray. To render one pixel of a room-scale scene, we require the same number of MLP queries as the original NeRF; however, we replace the coarse network used for hierarchical sampling. During optimization, half of the samples are distributed between the near and far planes and the second half are drawn from the Gaussian distribution determined by the depth prior $\mathcal{N}(z(\mathbf{r}), s(\mathbf{r})^2)$.
At test time, when the depth is unknown, the first half of the samples are used to render an approximate depth $\hat{z}(\mathbf{r})$ and standard deviation $\hat{s}(\mathbf{r})$ that is then used to sample the second half according to $\mathcal{N}(\hat{z}(\mathbf{r}), \hat{s}(\mathbf{r})^2)$.
\section{Results}
\label{sec:experiments}
We evaluate our method with a baseline comparison (\cref{ssec:baseline_comparison}) and an ablation study (\cref{ssec:ablation_studies}) on the ScanNet~\cite{Dai2017ScanNetR3} and Matterport3D~\cite{Chang2017Matterport3DLF} datasets.
\subsection{Experimental Setup}
\paragraph{ScanNet}
We run COLMAP SfM~\cite{schoenberger2016sfm} to obtain camera parameters and sparse depth. Specifically, we run SfM on all images to obtain camera parameters. To ensure a clean split between train and test data, we withhold the test images when computing the point cloud used for rendering the sparse depth maps. On average, the resulting depth maps have 0.04\% valid pixels.
We use three sample scenes each with 18 to 20 train images and 8 test images.
This set of images results from excluding video frames with motion blur
while ensuring that surfaces are observed from at least one input view. Details are provided in \cref{ssec:datasets_scannet}.
\paragraph{Matterport3D}
Using RGB images from the PrimeSense camera, COLMAP SfM struggled to reconstruct complete rooms in Matterport3D, hence, we mimic sparse depth from SfM by sampling and perturbing the sensor depth as described for depth prior training in \cref{ssec:depth_completion}. Sparse depth maps rendered from a SfM point cloud are by nature 3D-consistent. While consistency in 3D is irrelevant for training 2D depth completion, it plays a critical role when optimizing a 3D scene representation with NeRF. Hence, we ensure 3D-consistent sparse depth on the scenes used for NeRF by projecting the sampled and perturbed 3D points to all other views. On average the resulting depth maps are 0.1\% complete.
The impact of the sparse depth density is studied in \cref{sec:sparse_depth_density}.
We evaluate three example rooms each with 24 to 36 train images and 8 test images.
\paragraph{NeRF Optimization}
We process rays in batches of 1024 and use the Adam optimizer~\cite{Kingma2015AdamAM} with learning rate 0.0005.
For fairness, all approaches in the ablation and baseline experiments are configured to use 256 MLP evaluations per pixel, independent of the used sampling approach. The radiance fields are optimized for 500k iterations.
Further NeRF and depth prior implementation details are available in \cref{sec:implementation_details}.
\paragraph{Evaluation Metrics}
For quantitative comparison, we compute the peak signal-to-noise ratio (PSNR), the Structural Similarity Index Measure (SSIM) \cite{Wang2004ImageQA} and the Learned Perceptual Image Patch Similarity (LPIPS) \cite{Zhang2018TheUE} on the RGB of novel views as well as the root-mean-square error (RMSE) on the expected ray termination depth of NeRF against the sensor depth in meters.
By comparing color values directly, PSNR has limited expressiveness, when the images of the scene have inconsistent color.
As shown in \cref{ssec:ablation_studies}, the latent codes used to represent view-specific appearance largely help to produce consistent colors across the scene. Still, the color of a rendered image will not necessarily be similar to that of the test view against which it is evaluated. To compensate for this difference, we report an additional PSNR value, which is computed after optimizing for the latent codes on the entire test views. We are unable to use the left/right image split evaluation procedure from NeRF-W \cite{MartinBrualla2021NeRFIT}, because appearance changes too drastically across the image, so these numbers should be considered an upper bound on performance.
This additional metric is listed in parentheses (\cref{tab:results_scannet,tab:results_matterport}) for all approaches that use a latent code.
All other metrics as well as all renderings in the paper are computed by setting the latent code to zero, given that the codes are unknown at test time.
\subsection{Depth Priors}
\cref{tab:depth_priors} shows the depth prior accuracy on the three ScanNet and three Matterport3D scenes used for NeRF. These scenes are part of the test sets during depth completion training.
\begin{table}[tb]
\centering
\small
\begin{tabular}{@{}lcc@{}}
\toprule
& \multicolumn{2}{c}{RMSE [m] $\downarrow$} \\
Dataset & Sparse depth & Dense depth \\
\midrule
ScanNet & 0.261 & 0.268 \\
Matterport3D & 0.041 & 0.135 \\
\bottomrule
\end{tabular}
\caption{Accuracy of depth priors.}
\vspace{-.3cm}
\label{tab:depth_priors}
\end{table}
The higher quality, generated sparse depth on Matterport3D leads to more accurate dense depth priors. However, the network interpolates the more noisy sparse depth from SfM on ScanNet without a relevant drop in accuracy.
\begin{figure*}[htbp
\centering
\begin{tabular}{@{\,\,\,\,}p{0.195\linewidth}@{\,\,}p{0.195\linewidth}@{\,\,}p{0.195\linewidth}@{\,\,}p{0.195\linewidth}@{\,\,}p{0.195\linewidth}@{\,\,\,\,}}
\multicolumn{5}{c}{
\begin{tikzpicture}[squarednode/.style={rectangle, draw=white, fill=white, very thin, minimum size=2mm, text opacity=1,fill opacity=0.5}]
\node[anchor=south west,inner sep=0] (image) at (0,0)
{\includegraphics[width=\linewidth]{figures/scannet_results.jpg}};
\begin{scope}[x=(image.south east),y=(image.north west)]
\bf
\small
\node[squarednode] at (0.9, 11/12) (a) {1};
\node[squarednode] at (0.9, 9/12) (b) {2};
\node[squarednode] at (0.9, 7/12) (c) {3};
\node[squarednode] at (0.9, 5/12) (d) {4};
\node[squarednode] at (0.9, 3/12) (e) {5};
\node[squarednode] at (0.9, 1/12) (f) {6};
\end{scope}
\end{tikzpicture}
} \\
\centering{NeRF \cite{Mildenhall2020NeRFRS}} & \centering{DS-NeRF \cite{Deng2021DepthsupervisedNF}} & \centering{NerfingMVS \cite{wei2021nerfingmvs}} & \centering{Ours} & \centering{Ground Truth} \\
\end{tabular}
\caption{Rendered RGB and depth error for test views from three ScanNet rooms next to the ground truth RGB and depth.}
\label{fig:results_scannet}
\end{figure*}
\begin{figure*}[b]
\centering
\begin{tabular}{@{\,\,\,\,}p{0.136\linewidth}@{\,\,}p{0.136\linewidth}@{\,\,}p{0.136\linewidth}@{\,\,}p{0.136\linewidth}@{\,\,}p{0.136\linewidth}@{\,\,}p{0.136\linewidth}@{\,\,}p{0.136\linewidth}@{\,\,\,\,}}
\multicolumn{7}{c}{
\begin{tikzpicture}[squarednode/.style={rectangle, draw=white, fill=white, very thin, minimum size=2mm, text opacity=1,fill opacity=0.5}]
\node[anchor=south west,inner sep=0] (image) at (0,0)
{\includegraphics[width=\linewidth]{figures/matterport_results.jpg}};
\begin{scope}[x=(image.south east),y=(image.north west)]
\bf
\small
\node[squarednode] at (13/14, 9/10) (a) {1};
\node[squarednode] at (13/14, 7/10) (b) {2};
\node[squarednode] at (13/14, 5/10) (c) {3};
\node[squarednode] at (13/14, 3/10) (d) {4};
\node[squarednode] at (13/14, 1/10) (e) {5};
\end{scope}
\end{tikzpicture}
} \\
\centering{NeRF \cite{Mildenhall2020NeRFRS}} & \centering{Ours w/o Completion} & \centering{Ours w/o Uncertainty} & \centering{Ours w/o GNLL} & \centering{Ours w/o Latent Code} & \centering{Ours} & \centering{Ground Truth} \\
\end{tabular}
\caption{Rendered RGB and depth error for test views from three Matterport3D rooms next to the ground truth RGB and depth.}
\label{fig:results_matterport}
\end{figure*}
\subsection{Baseline Comparison}
\label{ssec:baseline_comparison}
We compare our method to NeRF \cite{Mildenhall2020NeRFRS} as well as recent and concurrent work that equally uses sparse depth input in NeRF, namely Depth-supervised NeRF (DS-NeRF) \cite{Deng2021DepthsupervisedNF} and NerfingMVS \cite{wei2021nerfingmvs}. Since DS-NeRF and NerfingMVS rely on SfM and MVS depth, respectively, they are run on ScanNet. NeRF and our method are run on both datasets.
The quantitative results (\cref{tab:results_scannet}) show that our method outperforms the baselines in all metrics.
``Floaters'' are a common problem when applying NeRF approaches in a setting with few input views.
By using dense depth priors with uncertainty, our method strongly reduces these artifacts compared to the baselines (example 2 \cref{fig:results_scannet}).
This contributes to far more accurate depth output and greater detail in color, e.g., visible in the books and the door handle (example 3 \cref{fig:results_scannet}). We found that our method is more robust to outliers in the sparse depth input. E.g., erroneous SfM points in the area of the sofa back (example 5 \cref{fig:results_scannet}) cause much larger deficiencies in geometry and color of the other approaches. This suggests that dense depth priors with uncertainty focus the optimization on more certain and accurate views, while direct incorporation of sparse depth, as in DS-NeRF, is more error-prone.
Besides greater robustness to outliers, dense depth guides NeRF better at object boundaries that are not represented in the sparse depth input. This is observable in example 6 (\cref{fig:results_scannet}), where a part of the chair back is missing in DS-NeRF, while it is complete using our method.
\paragraphNoSpace{NerfingMVS Details}
The error map calculation in NerfingMVS fails when applied to an entire room as opposed to a local region, causing invalid sampling ranges. The issue is solved as detailed in \cref{sec:implementation_details_nerfingmvs}.
To improve the performance of this baseline, we train its depth predictor 10 epochs longer than was done in the paper.
Still, the depth priors on the ScanNet scenes remain at RMSE 0.379m. While our method uses only train images to compute the sparse depth input, NerfingMVS runs COLMAP MVS on train and test images together, which gives them an advantage.
\begin{table}[t]
\centering
\resizebox{\linewidth}{!}{
\begin{tabular}{@{}lc@{\,\,}c@{\,\,}c@{\,\,}c@{\,\,}c@{}}
\toprule
& & & & & Depth \\
Method & \multicolumn{2}{c}{PSNR$\uparrow$} & SSIM$\uparrow$ & LPIPS$\downarrow$ & RMSE $\downarrow$ \\
\midrule
NeRF \cite{Mildenhall2020NeRFRS} & 19.03 & & 0.670 & 0.398 & 1.163 \\
DS-NeRF \cite{Deng2021DepthsupervisedNF} & 20.85 & & 0.713 & 0.344 & 0.447 \\
NerfingMVS \cite{wei2021nerfingmvs} & 16.29 & & 0.626 & 0.502 & 0.482 \\
Ours w/o Completion & 20.43 & (22.10) & 0.707 & 0.366 & 0.526 \\
Ours w/o Uncertainty & 20.09 & (22.21) & 0.714 & 0.308 & 0.279 \\
Ours w/o GNLL & 20.80 & (22.23) & 0.733 & 0.312 & 0.275 \\
Ours w/o Latent Code & 20.87 & & 0.726 & \textbf{0.293} & 0.243 \\
Ours & \textbf{20.96} & (\textbf{22.30}) & \textbf{0.737} & 0.294 & \textbf{0.236} \\
\bottomrule
\end{tabular}
}
\caption{Quantitative results on ScanNet.
Parentheses indicate PSNRs obtained after optimizing a latent code, when applicable.}
\vspace{-.3cm}
\label{tab:results_scannet}
\end{table}
\subsection{Ablation Study}
\label{ssec:ablation_studies}
To verify the effectiveness of the added components, we conduct ablation experiments on the ScanNet and Matterport3D scenes.
The quantitative results (\cref{tab:results_scannet,tab:results_matterport}) show that the full version of our method achieves the best performance in image quality and depth estimates.
This is consistent with the qualitative results in \cref{fig:results_matterport}.
\paragraphNoSpace{Without Completion}
Omitting depth completion and supervising with sparse depth only leads to inaccurate depth and color due to ``floaters'' in areas without depth input.
Even in areas with sparse depth points, the results are less sharp than in versions that use completed depth.
\paragraphNoSpace{Without Uncertainty}
Removing uncertainty from the optimization causes problems in resolving inconsistency in overlapping areas of the 2D depth priors. This results in wrong edges in RGB and depth (example 2 \cref{fig:results_matterport}), duplication artifacts (example 4 \cref{fig:results_matterport}) or lacking detail, e.g., in the patterns on the back of the chair (example 1 \cref{fig:results_matterport}).
The quantitative results on ScanNet (\cref{tab:results_scannet}) show that considering uncertainty becomes even more important when using the lower quality sparse depth from SfM.
\paragraphNoSpace{Without GNLL}
In this experiment, we replace GNLL with MSE in our depth loss (\cref{eq:depth_loss}), and observe that MSE struggles to constrain density behind surfaces. The lack of sharp edges in the density distribution is most visible for novel views looking in tangential direction of a surface, e.g., looking into the corridor (example 3 \cref{fig:results_matterport}).
\paragraphNoSpace{Without Latent Code}
Omitting the latent code that models per-camera information, leads to incapability to produce smooth and consistent color output across the scene. When rendering novel views, the frustums of training images are clearly visible by causing severe shifts in color intensity (examples 2 and 3, \cref{fig:results_matterport}).
\begin{table}[tb]
\centering
\resizebox{\linewidth}{!}{
\begin{tabular}{@{}lc@{\,\,}c@{\,\,}c@{\,\,}c@{\,\,}c@{}}
\toprule
& & & & & Depth \\
Method & \multicolumn{2}{c}{PSNR$\uparrow$} & SSIM$\uparrow$ & LPIPS$\downarrow$ & RMSE $\downarrow$ \\
\midrule
NeRF \cite{Mildenhall2020NeRFRS} & 15.24 & & 0.531 & 0.610 & 1.362 \\
Ours w/o Completion & 16.90 & (18.84) & 0.615 & 0.521 & 0.427 \\
Ours w/o Uncertainty & 17.95 & (20.37) & 0.658 & 0.413 & 0.115 \\
Ours w/o GNLL & 18.00 & (20.65) & 0.669 & 0.423 & 0.133 \\
Ours w/o Latent Code & 17.42 & & 0.654 & 0.410 & \textbf{0.110} \\
Ours & \textbf{18.33} & (\textbf{20.82}) & \textbf{0.673} & \textbf{0.402} & 0.114 \\
\bottomrule
\end{tabular}
}
\caption{Quantitative results on Matterport3D, using the same format as \cref{tab:results_scannet}.}
\vspace{-.3cm}
\label{tab:results_matterport}
\end{table}
\subsection{Limitations}
Our method allows for a significant reduction in the number of input images for NeRF-based novel view synthesis while at the same time applying it to larger room-size scenes. However, other NeRF limitations
such as long optimization times and slow rendering remain.
As a consequence of
the drastic reduction in the number of input images, surfaces are typically not observed by more than two other views, hence view-dependent effects are limited.
While our approach optimizes NeRF given as few as 18 images, the depth prior network requires a larger training dataset.
Although these priors generalize well and only need to be trained once, it would be beneficial if the depth reconstruction could be also learned from a sparse setting.
\section{Conclusion}
We have presented a method for novel view synthesis using neural radiance fields (NeRF) that leverages dense depth priors, thus facilitating reconstructions with only 18 to 36 input images for a complete room.
By learning a depth prior that generalizes across scenes, our method takes advantage of depth information without requiring depth sensor input of the scene. Instead, the depth prior network relies on the sparse reconstruction, which is available for free after structure from motion (SfM) on the input images. With only a few input views available, we show that our dense depth priors with uncertainty effectively guide the NeRF optimization, thus leading to significantly higher image quality of novel views and more accurate depth estimates compared to other approaches using SfM or multi-view stereo output in NeRF. Overall, we believe that our method is an important step towards making NeRF reconstructions available in commodity settings.
\section*{Acknowledgements}
This project is funded by a TUM-IAS Rudolf Mößbauer Fellowship, the ERC Starting Grant Scan2CAD (804724), and the German Research Foundation (DFG) Grant Making Machine Learning on Static and Dynamic 3D Data Practical.
We thank Angela Dai for the video voice-over.
\section{Datasets}
\subsection{ScanNet \cite{Dai2017ScanNetR3}}
\label{ssec:datasets_scannet}
\paragraphNoSpace{Motion Blur Detection}
We consider motion blur when sampling a small subset of images to be used in NeRF: From each window of $n$ consecutive video frames the sharpest one is selected according to the following metric, where high values indicate sharpness: first, an image is converted to grayscale, then it is convolved with a discrete Laplacian kernel; finally, the variance is computed.
$n$ is set to 10 or 20, depending on how densely the video samples the scene.
\paragraphNoSpace{Train/Test Image Selection}
After removing images with severe motion blur, we consider the following criteria: 1) SfM successfully registers the set of images. 2) Surfaces to be reconstructed are observed from at least one input view.
In practice, images are removed if their content is visible by other images and the remaining set fulfils 1).
This way, 22\% of the train pixels are not observed by any other train view, 31\% are observed by one other, 47\% by two or more. Test views have on average 66\% overlap with their most overlapping train view.
\paragraphNoSpace{Image Resolution}
The image resolution is 468$\times$624 after downsampling and cropping dark borders from calibration.
\paragraphNoSpace{Test Scenes}
We ensure that the test scenes are complete, sufficiently large rooms. The following scenes are used for evaluation:
\begin{itemize}[noitemsep,topsep=0pt,parsep=0pt,partopsep=0pt]
\item scene0710\_00
\item scene0758\_00
\item scene0781\_00
\end{itemize}
\paragraphNoSpace{SfM Quality on Few Views}
\Cref{fig:sfm_error} shows the mean absolute error (MAE) of the SfM points against the sensor depth. It is computed on the 6291 points from the three ScanNet evaluation scenes. The maximal error is 5.85m. We do not filter the COLMAP SfM output, i.e., all points are projected to the corresponding input views and used as input to the depth completion.
\begin{figure}[htb]
\centering
\begin{tikzpicture}[squarednode/.style={rectangle, draw=white, fill=white, very thin, minimum size=2mm, text opacity=1,fill opacity=0,draw opacity=0}]
\node[anchor=south west,inner sep=0] (image) at (0,0)
{\includegraphics[width=\linewidth,trim={-0.1cm 0.3cm 0.0cm 0.2cm},clip]{figures/sfm_error.pdf}};
\begin{scope}[x=(image.south east),y=(image.north west)]
\footnotesize
\node[squarednode] at (0.5, -0.125) (a) {Depth $z$ [m]};
\node[squarednode, rotate=90] at (0.0, 0.5) (b) {MAE [m]};
\end{scope}
\end{tikzpicture}
\caption{SfM depth error on ScanNet.
}
\label{fig:sfm_error}
\end{figure}
\subsection{Matterport3D \cite{Chang2017Matterport3DLF}}
\paragraphNoSpace{Train/Test Image Selection}
Similar to ScanNet, it is ensured that surfaces are observed from at least one input view.
25\% of the train pixels are not observed by any other train view, 45\% are observed by one other, 30\% by two or more. Test views have on average 67\% overlap with their most overlapping train view.
\paragraphNoSpace{Image Resolution}
The image resolution is 504$\times$630 after downsampling and cropping dark borders from calibration.
\paragraphNoSpace{Test Scenes}
We avoid unbounded open space, which is challenging for NeRF approaches. The following scenes are used for evaluation:
\begin{itemize}[noitemsep,topsep=0pt,parsep=0pt,partopsep=0pt]
\item Region 5, house VzqfbhrpDEA
\item Region 2, house Vvot9Ly1tCj
\item Region 19, house Vvot9Ly1tCj
\end{itemize}
\section{Impact of Sparse Depth Density}
\label{sec:sparse_depth_density}
We investigate the impact of the sparse depth density on Matterport3D by decreasing it from 0.1\% to 0.05\% and 0.01\% (\cref{tab:results_sparse_depth_matterport}).
\begin{table}[tb]
\centering
\resizebox{\linewidth}{!}{
\begin{tabular}{@{}lc@{\,\,}c@{\,\,}c@{\,\,}c@{\,\,}c@{}}
\toprule
& Sparse depth & & & & Depth \\
Method & density & PSNR$\uparrow$ & SSIM$\uparrow$ & LPIPS$\downarrow$ & RMSE $\downarrow$ \\
\midrule
Ours w/o completion & 0.10\% & 16.90 & 0.615 & 0.521 & 0.427 \\
Ours & 0.10\% & \textbf{18.33} & \textbf{0.673} & \textbf{0.402} & \textbf{0.114} \\
Ours & 0.05\% & 18.10 & 0.662 & 0.414 & 0.136 \\
Ours & 0.01\% & 17.99 & 0.662 & 0.437 & 0.140 \\
\bottomrule
\end{tabular}
}
\caption{Impact of sparse depth density on Matterport3D. Depth RMSE is in meters.}
\label{tab:results_sparse_depth_matterport}
\end{table}
While reduced sparse depth lowers performance, it clearly shows that depth completion increases the value of very sparse depth input: With just one tenth of the sparse depth our method still performs better, than the version without completion. Despite 0.01\% being very sparse---just 32 points per image on average---we expect that using monocular depth estimation is challenging as view-consistent depth is needed.
\section{Implementation Details}
\label{sec:implementation_details}
\subsection{Our Method}
\paragraphNoSpace{Radiance Fields}
Our model architecture is based on NeRF~\cite{Mildenhall2020NeRFRS}. The encoded position $\gamma(\mathbf{x})$ is provided as input to the first of 8 layers as well as to the fifth, by concatenating it with the activations from the fourth layer. Layers 1--8 each have 256 neurons and ReLU activations. The output of layer 8 is passed through a single layer with softplus activation to produce density $\sigma$. The output of layer 8 is also passed through a 256-channel layer without activation, whose output is concatenated with the viewing direction $\mathbf{d}$ and the latent code $\boldsymbol{\ell}$. The concatenated vector is fed to a 128-channel layer with ReLU activation, before the final layer producing the color $\mathbf{c}$.
The latent codes ${\boldsymbol{\ell}}$ have a size of 4 on ScanNet and 16 on Matterport3D.
Due to the different characteristics of the depth input on the two datasets, a suitable depth loss weight $\lambda$ is determined for each approach and dataset and used across all scenes of the same dataset (\cref{tab:depth_loss_weight}).
\begin{table}[tb]
\centering
\small
\begin{tabular}{@{}lcc@{}}
\toprule
& ScanNet & Matterport3D \\
\midrule
Ours w/o Completion & 1.0 & 0.25 \\
Ours w/o Uncertainty & 0.001 & 0.007 \\
Ours w/o GNLL & 0.04 & 0.03 \\
Ours w/o Latent Code & 0.003 & 0.007 \\
Ours & 0.003 & 0.007 \\
\bottomrule
\end{tabular}
\caption{Depth loss weights $\lambda$.}
\vspace{-0.3cm}
\label{tab:depth_loss_weight}
\end{table}
\paragraphNoSpace{Depth Completion}
The depth completion network is based on the architecture from Cheng \etal \cite{Cheng2020LearningDW}. We use a ResNet-18~\cite{He2016DeepRL} encoder and add a second upsampling branch for uncertainty estimation. It equally consists of up-projection layers with skip connections to the same downsampling layers as the depth prediction branch. To increase performance on very sparse input depth, both branches use a CSPN module, configured to 48 iterations in the depth branch and 24 iterations in the standard deviation branch.
The depth completion network is trained at a lower resolution of 256$\times$320 on Matterport3D, and 240$\times$320 on ScanNet.
We use the Adam optimizer~\cite{Kingma2015AdamAM} with a learning rate of 0.0001 and a batch size of 8. We train for 50 epochs on Matterport3D and 12 epochs on ScanNet.
On Matterport3D 80 houses are used for training, 5 houses for validation, and 5 houses for testing. On ScanNet we use the provided data split. We ensure that the scenes used for NeRF are not included during training, and are instead in the test sets.
\subsection{NerfingMVS~\cite{wei2021nerfingmvs}}
\label{sec:implementation_details_nerfingmvs}
The error map calculation used by NerfingMVS was not sufficiently robust to by applied to entire rooms, so to improve this baseline's performance we adapted it as follows:
\paragraphNoSpace{Original Calculation} For each input view an error map is computed by projecting the 3D points according to the depth prior to all other views, where a depth reprojection error is computed and normalized with the projected depth.
The mean of the 4 smallest errors are used as values in the error map.
\paragraphNoSpace{Problem on Entire Rooms} When applying the computation on entire rooms as opposed to a local region, the projected 3D points from other views frequently lie behind the camera. As a result the computed mean is often negative.
Similarly, the computation of the near and far planes of the scenes is not suited for entire rooms, leading to a negative near plane in our case.
Negative near plane and negative error map content lead to invalid sampling ranges, where the far bound lies in front of the near bound. to address this, we set the near and far planes ($t_n$ and $t_f$) of each scene such that all depth prior values are contained.
In the error map calculation, we assign a maximal error $t_{f} - t_{n}$ for projected points that lie behind the camera. Afterwards, the error map values are still computed as the mean of the smallest 4 errors.
\subsection{DS-NeRF~\cite{Deng2021DepthsupervisedNF}}
We used the same positional encoding frequencies as described for our method in the main paper for this baseline, which improved its performance.
A depth loss weight of 0.1 was suitable for the ScanNet scenes.
\subsection{NeRF~\cite{Mildenhall2020NeRFRS}}
As in DS-NeRF, we used our own positional encoding frequencies for this baseline, which improved its performance.
|
1,314,259,993,644 | arxiv |
\section{Originality and Value}
This research presents a demand forecasting system of electronic components in manufacturing validated with real data. The contributions cover the areas of pre-processing, prediction and model selection and are suited for individuals with domain knowledge but limited understanding of machine learning methods. They are the following:
\begin{enumerate}
\item An industry case of demand prediction for a large manufacturer of electronics,
\item An evaluation of 14 different models for demand prediction of items with hierarchical dependencies,
\item An implementation of a method for demand forecasting based on matrix factorization,
\item A feature engineering technique that is both easy to implement and yields similar results to those obtained from using feature engineering requiring domain knowledge,
\item A methodology for model selection based on topological data analysis suited for large data sets in an industry-setting,
\item For reproducibility purposes, an implementation, and data set available for download\footnote{\url{https://github.com/rodrigorivera/icmla2019}}.
\end{enumerate}
\section{Problem Statement}\label{sec:problemStatement}
One of the world's largest manufacturers of electronics has to forecast demand for both its products and their respective individual components, amounting to millions of time series data to predict. Traditional forecasting techniques are here largely ineffective. Nevertheless, the manufacturer has to generate reliable estimates for its future demand over multiple periods.
\section{Research Abstract and Goals}\label{researchandgoals}
At the moment, there are more than \$12 trillion USD in inventory either stockpiled or in transit, amounting to 17\% of the world's Gross Domestic Product (GDP), \cite{doi:10.1080/00207543.2018.1524167}. An accurate demand forecasting is essential in the industry.
Nevertheless, imprecise demand planning is still pervasive. For new products, forecast errors are, on average 44-53\%, whereas, for improved products, it is 31\%, \cite{Kahn2000AnEI, jain2005}. Companies compensate for this inaccuracy gap through expensive operational measures such as trans-shipments \cite{simchi2008designing}. Nevertheless, retailers still experience out-of-stock (OOS) events with rates amounting to 8.3\% worldwide, \cite{gruen2002retail}.
\begin{figure}[!ht]
\begin{center}
\includegraphics[width=\columnwidth]{images/process_diagram.png}
\end{center}
\caption{Overview of the implemented methodology}
\label{fig:intro:system_overview}
\end{figure}
The objective of this research is to present three techniques for (1) data pre-processing, (2) prediction, and (3) model selection accessible to non-technical business experts and offering competitive results. They represent a cohesive system depicted in \autoref{fig:intro:system_overview}. The use of novel machine learning methods for this field is a promising area with little academic research and with insufficient efforts to expose practitioners to them, \cite{2017arXiv170905548R}. It is relevant to have robust methods accessible to broader audiences, \cite{Chase2013}. \cite{Fleisch2003} observed that for discrepancies as low as 2\%, it is worth investing in improving the accuracy of a forecast. \cite{wipro2013} goes as far as claiming that a 10\% reduction in OOS increases revenue of retailers by up to 0.5\%.
Nevertheless, companies struggle to hire adequate personnel to address these tasks, \cite{2017apec}. \cite{2018esade} reported that over 60\% of surveyed businesses are resorting to internal training to compensate for this. This work seeks to alleviate this situation by presenting an extensive comparison of methods, proposing a feature engineering technique well-suited for demand forecasting in manufacturing, evaluating a novel method based on matrix factorization, and proposing a technique for model selection that is both accurate as well as easy to communicate to decision-makers.
The research goal of this work is to propose a set of approaches for time series forecast that can be adopted by business practitioners. For this purpose, the study poses the questions: 1) What is state of the art in academic research of time series prediction with structures? 2) How does the proposed method differs from popular approaches applied to time series prediction tasks?
Two objectives achieve the research goal: a) To review the existing theory on time series prediction and especially on techniques for dynamic hierarchical structures; b) To make a performance comparison of the proposed technique.
The object of research is the balance between accessibility and precision of methods for time series in a massive data context within the industry.
The subject of the research is forecasting product demand using techniques for time series with hierarchies.
\section{Literature Review}\label{section:literaturereview}
Supply chain management (SCM) in general and demand forecasting, in particular, are fields that have commanded attention from different communities according to \cite{Attar2016}. A comprehensive treatment is available in the works of \cite{Chase2013} and \cite{Gilliand2015}. Sales forecasting is an essential part of the supply chain management. The forecasting community uses and trains quantitative methods of the statistical family of ARIMA, exponential smoothing models, and alike with historical data to forecast future points to improve the forecasting accuracy. However, \cite{Ahmed2010} argues that there have been few large scale comparative studies of machine learning models for regression or time series aimed at forecasting problems.
In the retail and manufacturing sectors, authors such as \cite{Tirkes2017} paid attention to the demand forecasting of edibles.
Similarly, \cite{Taylor2007} deals with time series characterized by a high volatility skewness to forecast daily sales for a supermarket chain at the point of sale.
\cite{Bianchi2017} experimented with recurrent neural networks for short term forecasting of real-valued time series. While \cite{Carbonneau2008} explored demand forecasting with incomplete information.
For the electronics manufacturing industry, \cite{Wan2016} introduced SVM regression to the supply chain of various producers.
Although SVM regression is a popular method for forecasting, not everyone has identified it as the most effective method. For example, \cite{Lu2012} presented a MARS model, and \cite{Yelland2010} proposed a Bayesian model.
Other manufacturing-centric sectors such as fashion have also delved into demand forecasting but to a different extent. \cite{Liu2013} claims that pure statistical methods are not yet commonplace in the fashion industry. It is preferred to make use of judgmental forecasts or a combination of quantitative and qualitative forecasts.
\section{Dataset}
The data consists of a data set of observations from an electronics manufacturer representing a subset of their total inventory. It contains the demand for 2562 different items with a length of $n = 45$. These items have varying amounts of required quantities, with many of them being requested sporadically, as seen in \autoref{fig:tda:orders per item:2}, and few of them being requested in large quantities.
\begin{figure}[!ht]
\begin{center}
\includegraphics[width=0.9\columnwidth]{images/third_dataset_nan.png}
\end{center}
\caption{Number of NaNs (zero orders) per item. X-axis: Item's ID, Y-axis: Number of zeroes}
\label{fig:tda:orders per item:2}
\end{figure}
\section{Diagonal feeding}
\label{sec:diagonal_feeding}
One of the contributions of this work is to introduce the practitioner to a data transformation technique, useful for multi-step structured forecasting from anticipatory data. It is part of the first step, 'Preprocess,' of the system introduced in \autoref{fig:intro:system_overview}. The main benefit of Diagonal Feeding is that it helps utilize the anticipatory nature of pre-orders' time-series data and makes forecasting the pre-order structure more streamlined.
It is made possible due to the data set containing information not only about the current demand but also on the volumes of pre-orders made in advance. Advance pre-orders are expectation-driven, naturally forward-looking, and known beforehand, as they reflect planning and anticipation of the market and economic environment at the end of the period when the order is to be fulfilled. At the same time, forecasting the pre-order structure over several next periods is of significant practical interest. It is reasonable to leverage the anticipatory information of the advance pre-orders, known by the present, for predicting the pre-order structure in the future by also taking into account the cross-correlations between the pre-orders.
Let $q_t^h$ be the volume of some item in the ``quantity'' field in the data set, $t$ corresponds to the ``delivery date'', and $h$ be the ``periods before delivery date''. The value $q_t^h$ denotes the total amount requested via {\it $h$ period advance pre-orders} to be delivered {\it by the end of period $t$}. The key property of the data set is that for every item, the value $q_t^h$ is effectively-known and available for use by the end of period $t - h$ -- the period when the $h$-ahead pre-orders were made. For example, $q_{t-1}^1$ is known at the end of $t-2$. It corresponds to the quantity requested at the end of $t-1$. That is the case due to the accumulation of pre-orders made by the end of $t-2$.
Since $q_t^h$ reflects expectations about the market conditions at $t$ and is known $h$ periods in advance, it seems reasonable to reorder the data set with respect to the period when they become known and reshape it to keep the pre-order structure. This makes predicting $q_t^h$ with $q_s^f$ data for $t - h > s - f$, which is either past ($q_{t-1}^h$) or anticipatory ($q_t^{h+1}$), more streamlined.
The proposed reshaping of the multivariate time series of a particular ``item'' is illustrated below. Since the quantity $q_t^h$ is known at time $t-h$, each diagonal $(q_{t+s+h}^h)_{h\geq 0}$ in the scheme above is {\it known} at $t+s$, $s\in \mathbb{Z}$; thus, potentially up to infinite periods:
\begin{small}
\begin{equation} \label{eq:diag_feed}
\begin{aligned}
\begin{pmatrix}
\colb{q_{t+0}^0} & \colb{q_{t+0}^1} & \colb{q_{t+0}^2} \\
\colr{q_{t+1}^0} & \colb{q_{t+1}^1} & \colb{q_{t+1}^2} \\
\colr{q_{t+2}^0} & \colr{q_{t+2}^1} & \colb{q_{t+2}^2} \\
\colr{q_{t+3}^0} & \colr{q_{t+3}^1} & \colr{q_{t+3}^2} \\
\end{pmatrix}
\rightarrow
\begin{pmatrix}
\colb{x_{t 0}} & \colb{x_{t 1}} & \colb{x_{t 2}} \\
\colr{y_{t 0}} & \colb{x_{t 3}} & \colb{x_{t 4}} \\
\colr{y_{t 1}} & \colr{y_{t 2}} & \colb{x_{t 5}} \\
\colr{y_{t 3}} & \colr{y_{t 4}} & \colr{y_{t 5}} \\
\end{pmatrix}.
\end{aligned}
\end{equation}
\end{small}
In \autoref{eq:diag_feed}, the target $\colr{y_t}$ is the output and represents the pre-order structure for the next $3$ periods beginning with $t+1$. The objective is to predict the lower diagonal of the matrix. It is done using $\colb{x_t}$ and its history as an input, i.e., past pre-order structure. Although, in principle, predicting the structure in $\colr{y_t}$ allows planning production volumes several months ahead, the most relevant targets for practical demand forecasting are on the largest diagonal of $\colr{y_t}$, since they are the earliest future volumes.
\section{Matrix Factorization}
\label{sec:matrix_factorization}
Matrix Factorization (MF) methods are used in a variety of applications such as recommender systems, signal processing, \cite{Weng2012}, computer vision, \cite{Chen2004}, and others. The second contribution of this work is adapting a method discussed in \cite{Rivera_2018} and \cite{yuetal2016} to demand forecasting in manufacturing.
Let $Y$ be $T\times n$ sparse or dense matrix of observations of $n$ objects spanning the period of $T$ time steps, i.e. each column $i=1,\,\ldots,\,n$ of $Y$ is a times series $y^{(i)} = (Y_{ti})_{t=1}^T$ related to the $i$-th object. The problem of factorizing a fully or partially observed $T\times n$ matrix $Y$ consists of finding $d$-dimensional factors $Z$ and the corresponding factor loadings $F$. It must be in the form of $T \times d$ and $d \times n$ matrices respectively. As such, their product $Z F$ most accurately recovers the observed $Y$, i.e. $Y_{ti} \approx \sum_{j=1}^d Z_{tj} F_{ji}$. This is usually achieved by solving the following optimization problem:
\begin{equation} \label{eq:general_mf}
\begin{aligned}
& \underset{F, Z}{\text{minimize}}
& & \tfrac1{2 \lvert \Omega\rvert}
\|\mathcal{P}_\Omega(Y - Z F)\|^2
+ \lambda_F \mathcal{R}_F(F)
+ \lambda_Z \mathcal{R}_Z(Z)
\,,
\end{aligned}
\end{equation}
where $\Omega\subset \{1..T\} \times \{1..n\}$ is the sparsity pattern of $Y$, $\mathcal{P}_\Omega$ zeroes out unobserved entries. The coefficients $\lambda_F$ and $\lambda_Z$ are non-negative regularization coefficients that govern the trade-off between the reconstruction error and the regularizing terms $\mathcal{R}_F$ and $\mathcal{R}_Z$. The latter depends on the particular desired properties of the factorization, typically in conjunction with a Ridge regression-type penalty ($\ell^2$ norm).
\section{Models}\label{sec:models}
The third contribution of this work is a large-scale study of various methods for demand forecasting. In the system presented in \autoref{fig:intro:system_overview}, they belong to the parts 'Training' and 'Testing.' In total, the assessment consists of fourteen different methods. They are 1) Adaboost, 2) ARIMAX, 3) ARIMA, 4) Bayesian Structural Time Series (BSTS), 5) Bayesian Structural Time Series with a Bayesian Classifier (BSTS Classifier), 6) Ensemble of Gradient Boosting (Ensemble), 7) Ridge regression (Ridge), 8) Kernel regression (Kernel), 9) Lasso, 10) Matrix Factorization from \autoref{sec:matrix_factorization} (MF), 11) Neural Network (NN), 12) Poisson regression (Poisson), 13) Random Forest (RF), 14) Support Vector Regression (SVR).
Each of them had as a target value three different options: a) Quantity (non-transformed), b) Log-transformed quantity, c) Min-Max transformed quantity. Additionally, Diagonal Feeding, presented in \autoref{sec:diagonal_feeding}, was evaluated for regression methods. Thus, one evaluates three settings: a) No Diagonal Feeding, b) Diagonal Feeding with an item by item training (One by One). In this case, a vector containing the input of a specific item is fed individually to a model, c) Diagonal Feeding fitting the model on the full data set (All Items). Here, one uses a matrix with the input from all items. In all three cases, one obtains an individual vector corresponding to a given item as an output. For a), extensive feature engineering is necessary, and the outcome was 360 features.
The specific features are documented in the code base provided\footnote{\url{https://github.com/rodrigorivera/icmla2019}}. The training set consisted of 37 periods, and the test set of 8. The Symmetric Mean Absolute Percent Error (SMAPE) serves to evaluate the performance of the models, and one defines it as $
\text{SMAPE} = \frac{200\%}{n} \sum_{t=1}^n \frac{|F_t-A_t|}{|A_t|+|F_t|}
$
with $F_t$ being the forecasted value and $A_t$ the actual value at time $t$ respectively. One can see the results of the experiment in \autoref{tab:models:2}. The table contains both the median and average SMAPE for all models, an average for models fit without Diagonal Feeding (DF), and a second average in the case where it was used. Further, the performance across models was uneven. The top 5 of models that achieved the lowest SMAPE for a given item were 1) Adaboost with 222 items, 2) Ensemble of Random Forests with 45, 3) BSTS with 42, 4) BSTS Classifier with 32 and 5) ARIMAX with 21 respectively.
\begin{small}
\begin{table}
\caption{Overview of results using mean SMAPE. Low values are better. 1:1: One by One. AI: All Items. MM: Min-Max. LT: Log-Transform. DF: Diagonal Feeding}
\begin{center}\label{tab:models:2}
\begin{tabulary}{\linewidth}{CCCC}
\toprule
Model & SMAPE & Model & SMAPE \\ \hline
Adaboost & 0,17 & Ridge 1:1 MM DF & 0,42 \\
Ensemble & 0,18 & Adaboost 1:1 LT DF & 0,43 \\
ARIMA & 0,27 & Kernel AI LT DF & 0,43 \\
Ridge & 0,3 & Ridge AI MM DF & 0,43 \\
SVR & 0,3 & Kernel 1:1 DF & 0,44 \\
ARIMAX & 0,32 & Adaboost 1:1 DF & 0,47 \\
RF 1:1 LT DF & 0,34 & Adaboost 1:1 MM DF & 0,47 \\
Poisson AI LT DF & 0,36 & Kernel 1:1 LT DF & 0,47 \\
Lasso AI DF & 0,37 & NN AI MM DF & 0,47 \\
Poisson 1:1 DF & 0,37 & MF & 0,5 \\
Poisson 1:1 LT DF & 0,37 & NN 1:1 MM DF & 0,52 \\
Poisson 1:1 MM DF & 0,37 & \textbf{AVERAGE ALL} & 0,53 \\
RF 1:1 DF & 0,37 & \textbf{AVERAGE DF} & 0,54 \\
RF 1:1 MM DF & 0,37 & SVR 1:1 LT DF & 0,55 \\
Ridge AI DF & 0,37 & Adaboost AI LT DF & 0,56 \\
Lasso 1:1 DF & 0,38 & SVR AI LT DF & 0,56 \\
NN AI LT DF & 0,38 & SVR 1:1 DF & 0,56 \\
RF AI LT DF & 0,38 & Lasso 1:1 MM DF & 0,6 \\
Ridge 1:1 DF & 0,38 & SVR 1:1 MM DF & 0,6 \\
Kernel AI DF & 0,39 & RF AI DF & 0,62 \\
Kernel AI MM DF & 0,39 & RF AI MM DF & 0,62 \\
Lasso 1:1 LT DF & 0,39 & NN 1:1 DF & 0,68 \\
Poisson AI DF & 0,4 & NN 1:1 LT DF & 0,74 \\
Poisson AI MM DF & 0,4 & SVR AI DF & 0,87 \\
Ridge 1:1 LT DF & 0,4 & BSTS & 0,93 \\
Lasso AI LT DF & 0,41 & BSTS classifier & 0,97 \\
Ridge & 0,41 & Adaboost AI DF & 1,1 \\
Ridge AI LT DF & 0,41 & Adaboost AI MM DF & 1,11 \\
\textbf{MEDIAN ALL} & 0,42 & NN AI DF & 1,13 \\
\textbf{AVERAGE NO DF} & 0,42 & Lasso AI MM DF & 1,15 \\
Kernel 1:1 MM DF & 0,42 & SVR AI MM DF & 1,76 \\
\bottomrule
\end{tabulary}
\end{center}
\end{table}
\end{small}
\section{TDA for Model Selection}\label{sec:tda}
In the system presented in \autoref{fig:intro:system_overview}, model selection is done with a method based on Topological Data Analysis. It represents the fourth contribution of this study. TDA is a new field that emerged from a combination of various statistical, computational, and topological methods during the first decade of the century. It allows us to find shape-like structures in the data and has proven to be a powerful exploratory approach for noisy and multi-dimensional data sets. For a detailed introduction, the reader is invited to consult \cite{chazal2017introduction}.
Two motivations lie behind this approach. First, in a production-setting with millions of time series to forecast, it is necessary in advance to decide on the appropriate model for a particular item in order to minimize computing costs and efforts. There are many periods with zero orders and peaks in demand. Second, SMAPE as the sole metric for decision-making can be misleading, especially if it is evaluated exclusively on the training set. For example in \autoref{fig:models:autoarima:1}, the best forecast using ARIMA is depicted. A relatively low SMAPE of 0,20 was obtained. Nevertheless, the model is only predicting the value at time $t+1$ using the value from time $t$.
\begin{figure}[!b]
\begin{center}
\includegraphics[width=0.9\columnwidth]{images/autoarima_50359_ff0.png}
\end{center}
\caption{Top forecast using ARIMA. X-axis: Period. Y-axis: Quantity. Blue color: Actual quantity. Orange color: Predicted quantity. SMAPE: 0,20}
\label{fig:models:autoarima:1}
\end{figure}
This research proposes a pipeline consisting of 8 steps to select a model. (A) For a subset of time series, in this case, 200, all possible models are fitted. For this experiment, one used only five models, see \autoref{fig:tda:results}. (B) On the test dataset and for the same items, one calculates SMAPE for each model. (C) For each time series, the best model is chosen based on SMAPE. The best model becomes a target. (D) One computes relevant features describing each time series, see \cite{christ2018time}. (F) A graph is constructed using the Mapper algorithm, see \cite{chazal2017introduction}. The Canberra distance, see \cite{lance1967mixed}, is used as a distance metric and the first principal component obtained from the Mapper algorithm as a lens.
(G) A graph partitioning algorithm, see \cite{slininger2013fiedlers}, is run recursively until reaching the lowest limit of data points per cluster. (H) One chooses the most frequently observed target (model) for all models within a cluster. (I) For a new time series, one can select the best model by running the K-nearest neighbors algorithm on the features obtained in point (D). For this experiment, one chooses seven features.
Based on the described pipeline, one obtains two clusters of nodes from the graph: a) AdaBoost, BSTS, BSTS classifier for 74\% of the time series, b) Poisson regression, and Random Forest for 26\% respectively. They are depicted in \autoref{fig:tda:results}. Using cross-validation for model selection, for 71\% of the time series, AdaBoost, BSTS, BSTS Classifier were the best choice. Hence, using only one graph, partitioning yields a small model selection error (6\%).
\begin{figure}[!b]
\begin{center}
\includegraphics[width=0.7\columnwidth]{images/final_1.png}
\end{center}
\caption{TDA pipeline for 5 models and 7 features with Canberra distance. Colors: Blue (BSTS), Orange (BTSTS classifier), Yellow (Poisson), Green (RF), Grey (Adaboost).}
\label{fig:tda:results}
\end{figure}
\section{Discussion \& Learnings}
\paragraph{On Diagonal Feeding}\label{sec:discussion:df}
The critical insight from the analysis of the data set through Diagonal Feeding is that the currently known one-period mostly determines the next period's gross total demand volume $q_{t+1}^0$ ahead pre-orders for the period ($q_{t+1}^1$).
The net-next period's volume, $\delta_{t+1}^0$, is the difference between $q_{t+1}^0$ and $q_{t+1}^1$. Viewed through Diagonal Feeding, it is mostly independent of the history of net pre-orders for the period $t+1$ and is thus less predictable from advance pre-order data, as indicated by the correlation analysis and the results of a grid search experiment. The apparent success of forecasting the $q_{t+1}^0$, especially in contrast to the other next period's pre-order volumes $q_{t+1+j}^j$ for $j \in \{1,2,3\}$, might be attributed to an observed high correlation of the one-period ahead pre-order volume $q_{t+1}^1$. Further, Diagonal Feeding delivers results comparable to those obtained doing extensive feature engineering. Along these lines, exploring different transformations of the target value is essential. For example, a Neural Network without a transformed quantity fitted on the full data set had a SMAPE of 1,13, with a log transformation, it was 0,38.
\paragraph{On Matrix Factorization}\label{sec:learnings:mf}
The contribution of this work concerning \cite{yuetal2016} is an implementation of MF with temporal regularization solving explicitly the following optimization problem (extended with graph similarity regularizer).
The major advantage is that in the high dimensional object mode $T \ll n$, it has fewer parameters ($T k + k n + k p$) to estimate than $p$-th order vector autoregression ($p n^2$), while retaining the power to capture the correlations among the time series in $Y$,~\cite{yuetal2016}. Nevertheless, criticism is twofold. First, the method is wasteful. Its most precise forecasts are one-step-ahead, since it relies on the ``dynamic'' forecasting method: the factor forecasts are computed based on the prior forecasts $\hat{Z}_{T+h\mid T,\, j} = \sum_{i=1}^p \phi_{ji} \hat{Z}_{T+h-i\mid T,\, j}$ with $\hat{Z}_{T+h-i\mid T,\, j} = Z_{T+h-i,\, j}$ for $i\geq h$. One attributes this deterioration of forecast accuracy to the accumulating forecast error inherent to this method. The secondary reason is that the $\ell^2$ and $\mathrm{AR}(p)$ regularizers jointly force stationary factor time series $(Z_{t,\,j})_{t=1}^T$, with the characteristic roots lying within the $\mathbb{C}$ unit disk. Therefore, the dynamic forecast, although capable of exhibiting complex dynamic patterns for high $p$, still has vanishing oscillations, eventually leveling to zero. The second shortcoming is that it is impossible to get the new latent factor values when one updates $Y$ with new data, other than re-estimating the factorization model. The key issue with re-estimation is that the re-estimated factors and loadings are not guaranteed to resemble the ones from the factorization before the data update.
Given these shortcomings, following guidelines for the application of the temporal regularized matrix factorization can be formulated. First, one should observe the experiments by \cite{yuetal2016} suggest that at least $25\%$ of the entries in $Y$ for an adequate reconstruction of the missing dynamics within the training set. Second, the structure of the $\mathrm{AR}(p)$ regularizer suggests that the factorization should not express extreme volatility. A comparison of the performance of this method with the second data set (non-sparse and moderately volatile) against the third one (highly sparse and volatile) supports this. Third, due to the dynamic nature of the factor forecasts, the best strategy is to compute one-step-ahead forecasts and re-estimate the factorization upon new data.
\paragraph{On the experiment}
The results from \autoref{tab:models:2} show that the best model was Adaboost with an SMAPE of 0,17. It was followed by the Ensemble of Random Forests with 0,18. Both performed significantly better than Arimax, the baseline used by the manufacturer, with 0,32. Worth highlighting are the results obtained by Diagonal Feeding. The best method using this transformation technique, a random forest with log-transform and fitted on the full data set, obtained 0,34. It was significantly better than an average consisting of methods trained on 360 features with a SMAPE of 0,42.
\paragraph{On the scope of the study}
The objective of this study was to improve the results obtained from the forecast method used by the manufacturer, ARIMAX. At the same time, it seeks to provide tools that demand planners at the electronics manufacturer can use without requiring extensive knowledge in computer science. In this study, ARIMAX showed good results using SMAPE as an error metric. However, looking at individual items, it only gave the best results for less than 10\% of the inventory. Besides, it showed that using Diagonal Feeding improves results without extensive feature engineering. From an academic perspective, this study filled a void. In the literature, there are no comprehensive studies on demand forecasting for manufacturers that practitioners can use as a reference.
\paragraph{On TDA for Model Selection}
The manufacturer's inventory consists of millions of components. Thus, proper and efficient model selection becomes essential. Model selection based on TDA produced fast and explainable results. It worked well even with a small number of data in comparison to the number of models, i.e., 200 time-series and five models.
To further validate this approach in an industry-setting, two comparisons were conducted between TDA and Dynamic Time Warping (DTW) with K-Means, see \cite{berndt1994using}. The first experiment consisted of 80 000 time series generated from the data set with added random noise. For TDA, it took less than 30 minutes on a standard commercial laptop, whereas DTW was not able to complete the process. A second experiment using DTW with K-Means was made under the same conditions described in \autoref{sec:tda}. It revealed that the first cluster consisting of AdaBoost, BSTS and BSTS classifier is the best suited for 69\% of the time series. For the second cluster containing Poisson regression and Random Forest, it was 31\%. Yet, it is still necessary to conduct experiments to
verify the purity of the cluster.
\section{Conclusion}
This work had as an objective to provide practitioners with a system for demand forecasting consisting of preprocessing, training, and prediction of a large number of models as well as model selection. As a preprocessing technique, Diagonal Feeding was introduced. It helps demand planners improve the accuracy of their methods whenever future delivery dates are known and without requiring domain knowledge or extensive feature engineering. For prediction and testing, a large study comparing over fourteen methods was presented. Also, it applied a method based on matrix factorization for demand forecasting.
Similarly, a model selection method based on TDA was presented. In an industry-setting, low error metrics such as SMAPE can be misleading. The trained model might be incapable of forecasting the actual demand. The methodology provided alleviates this and shows better results than similar techniques while being easy to communicate to stakeholders. As a further line of work, this study would like to point out two main directions. First, for matrix factorization, there is the need to improve it for sparse data as well as to be more computationally efficient. Second, for the model selection based on TDA, it is worth considering different approaches not based on graph partitioning. One example is clustering based on point clouds. In conclusion, there is a need to up-skill existing personnel, and researchers can contribute to close this gap. Given the significant demand for analytics talent in the years to come, one can expect that the academic community will focus their attention in this direction.
\bibliographystyle{IEEEtran}
|
1,314,259,993,645 | arxiv | \section{Introduction}
Random numbers are needed in various applications,
including cryptography \cite{Dav89}, stochastic optimization \cite{Aar89}, and
Monte Carlo methods \cite{Bin92}. Because of practical reasons
random numbers are usually produced by deterministic rules,
implemented as pseudorandom number generators. In spite
of their fully deterministic origin the quality of
pseudorandom numbers may often be good enough for many
applications.
To confirm the suitability of a given pseudorandom number generator
for practical use, it should be subjected to a rigourous test program
which reveals the strengths and weaknesses of the algorithm and,
in particular, its {\it implementation}. Recently, such an extensive
test program has been carried out by the present authors \cite{Vat93b}.
By performing a comparative evaluation using statistical, bit level
and visual tests we were able to assess the quality of a group of
random number generators which are commonly used in the applications
of physics.
One of the generators included in Ref. \cite{Vat93b} was RCARRY,
which uses the so called {\em ``subtract-and-borrow''} algorithm
which has been implemented by James \cite{Jam90}. In the tests,
RCARRY clearly displayed the poorest statistical properties of
the generators tested, suggesting possible problems in the
implementation. Supporting this, James has recently
reported \cite{Jam93} the observation of M. L\"uscher that the
original implementation of RCARRY may contain a small error, which
may adversely affect the quality of the random number sequence.
The purpose of the present work is to address this issue. To this end,
we present results of extensive statistical and bit level tests on
the corrected version of RCARRY, and compare the results to those
of Ref. \cite{Vat93b}. In addition, we test a slightly different version
of the RCARRY generator, which uses an {\it ``add-and-carry''} algorithm
based on the {\it addition} of a carry bit. We call this generator
ADCARRY. Our results reveal that there is very little
difference between the statistical properties of the original RCARRY
and its corrected version, as well as the ADCARRY generator.
All these generators display a relatively poor performance in two
of the gap tests presented here.
\section{Implementation of the Generators}
The three pseudorandom number generators tested in this work
are based on a lagged Fibonacci algorithm, which is augmented
by an occasional addition of a carry bit. The basic formula is:
\begin{equation}
X_{i} = (X_{i-24}\ \pm\ X_{i-10}\ \pm c) \mbox{ mod } b.
\end{equation}
The carry bit $c$ is zero if the sum is less than or equal to $b$,
and otherwise ``$c=1$ in the least significant bit position''
\cite{Jam90}. The choice for $b$ is $2^{24}$.
The period of the generator is about $2^{1407}$ \cite{Jam90}
and it produces random numbers distributed between [0,1).
Only the 24 most significant bits are guaranteed to be good.
The inclusion of the carry bit $c$ in the lagged Fibonacci algorithm
was done in order to improve its properties \cite{Mar90b}.
Recently, however, it has been shown \cite{Cou,Tez} that this
type of algorithms are in fact equivalent to linear congruential
generators with very large prime moduli. Consequently, they
inherit unfavourable lattice structures in higher dimensions.
The original implementation of Eq. (1) was done by James \cite{Jam90},
based on the ideas of Marsaglia {\em et al.} \cite{Mar90b}. It uses
the subtraction contained in Eq. (1). In this work, we shall denote
it by I1. The second generator I2 includes the suggested
correction of L\"uscher and James, who recommend replacing
line 13 of the code of Ref. \cite{Jam90}
\begin{tabular}{p{2cm} l}
& {\tt uni = seeds(i24) - seeds(j24) - carry,}
\end{tabular}
by
\begin{tabular}{p{2cm} l}
& {\tt uni = seeds(j24) - seeds(i24) - carry}.
\end{tabular}
The third generator ADCARRY (I3) uses the operation known as
``add-and-carry'', in which subtraction in Eq. (1) has been replaced by
addition. In this version the lines 13 - 15 of \cite{Jam90} are
rewritten as:
\begin{tabular}{p{2cm} l}
& {\tt uni = seeds(j24) + seeds(i24) + carry}\\
& {\tt if(uni.ge.1.) then}\\
& \mbox{\hspace{2cm}} {\tt uni = uni - 1.}
\end{tabular}
Otherwise, the implementation is identical to that of RCARRY
\cite{Lusch1,Lusch2}.
\section{Test methods}
Tests scrutinizing the quality of random numbers can be divided into
three main categories: statistical tests \cite{Knu81}, bit level tests
\cite{Alt88,Mar85,Vat93b}
for testing the properties of random numbers on binary level, and
visual tests \cite{Knu81} which may give some further qualitative
information on the statistical properties of random numbers.
A number of these tests were implemented and employed extensively in Ref.
\cite{Vat93b}. In this work, we have
repeated the same statistical tests for I2 and I3. They
are listed in Table 1, where the numbering refers to the
parameters of Ref. \cite{Vat93b}. From bit level tests, only the
$d$-tuple test \cite{Alt88,Mar85} was done since it was shown to be
sufficient. Finally, the random numbers were plotted in two
dimensions for purposes of visual inspection.
The test bench is described in detail in Ref. \cite{Vat93b}.
Description of the statistical tests can also be found in Ref. \cite{Knu81}.
In brief, the statistical accuracy of all the tests
was improved by utilizing a one way Kolmogorov - Smirnov test
\cite{Knu81} to a large number (1000 or more) of test statistics.
This approach has been realized earlier by L'Ecuyer \cite{Lec88}.
The final test variables are therefore the values $K^+$ and $K^-$ of
a Kolmogorov - Smirnov test statistic $K$ \cite{Knu81}. In each test
the generator was considered to fail the test if the observed descriptive
level $\delta = P(K \leq \{ K^+, K^-\} | H_0)$ was less than
0.05 or larger than 0.95.
\section{Results}
Results of the statistical tests for the descriptive levels
$\delta^+$ and $\delta^-$ are summarized in Table 2, where
the numbering refers to Table 1. In each test the chosen generator was
initialized with the seed 667790. In case a failure occurred, the
generator was subjected to another test starting from the final
state of the first test. If another immediate failure occurred,
the generator was tested for the third time starting from a new
state with an initial seed 14159 (from the decimals of $\pi$).
In Table 2, frames with thin lines indicate a single failure,
frames with double single lines two consequtive failures, and frames with
bold lines three consequtive failures in the corresponding tests.
The results of the original RCARRY ({\cal I1}) by James \cite{Jam90}
are shown on the left (from Ref. \cite{Vat93b}), whereas the results
of the corrected version ({\cal I2}) and ADCARRY ({\cal I3})
are at the center and on the right, respectively.
Based on the results, it is clear that the corrected version
of RCARRY (I2) using
arithmetic subtraction performs no significantly better than the
original RCARRY (I1). The main malady of RCARRY,
namely the clear failing of the gap tests 6 and 8 with parameters
$\alpha = 0, \beta = 0.05$ and $\alpha = 0.95, \beta = 1$ \cite{Knu81},
respectively, is still characteristic of I2. This signifies
the existence of local correlations in the vicinity of zero and one.
The same conclusion applies to ADCARRY as well, signaling basic
problems with these algorithms.
In the $d$-tuple test each implementation was tested two times and
the bits considered failed had two consequtive failures. The results
are shown in Table 3. In our notation, bit number one is the
most significant bit (excluding the sign bit). For the original
implementation of RCARRY (I1) only the 24 most significant bits
are guaranteed to be good, which the tests confirm \cite{Vat93b}.
The implementation I2 yields identical results, whereas
ADCARRY ({\cal I3}) gives only 22 good bits (see Fig. 3).
Finally, visual tests on bit level support the results above. In Figs.
1, 2 and 3 we show subsequent random numbers for I1, I2 and I3
in binary form on a $120 \times 120$ matrix, when only 24 most significant
bits are included. No clear correlations are visible,
except for the last two bits of ADCARRY where strong correlations are
apparent. No visual indications of the suggested \cite{Cou,Tez}
lattice structure were found in these generators.
\section{Summary and conclusions}
In this work, we have compared the results of detailed
statistical and bit level tests for three implementations of
random number algorithms using a lagged Fibonacci sum with the
addition of a carry bit. Results for RCARRY and its corrected
version show very little difference.
Also, a new generator ADCARRY
using purely additive arithmetics fares no better statistically, and
has two good bits less than RCARRY. Fortunately enough,
these bits are not among the most significant ones.
Overall, our results suggest
that the basic algorithm of Eq. (1) on which these generators are based
seems to lead to observable correlations. The persistent failure of this
class of generators in the gap tests may lead to problems in some
applications, e.g. in lattice simulations \cite{Luscher}.
Finally, we would like to emphasize the importance of extensive
testing such as presented here before using {\it any} new
pseudorandom number generator. Even a good algorithm can be corrupted
by a poor implementation, as we have previously demonstrated \cite{Vat93b}.
Hence, a good amount of
scepticism towards pseudorandom number generators without extensive
test results seems prudent. It should also be noted that even when
no statistical or bit level correlations are found, direct physical
tests of random number generators should be used to reveal possible
``hidden'' correlations \cite{Vat93c,Fer92}.
\clearpage
{\Large {\bf Acknowledgments}}
\medskip
We would like to thank Fred James and Martin L\"uscher
for fruitful correspondence.
The Centre for Scientific Computing Ltd., Tampere University
of Technology, and University of Helsinki
have provided ample computing resources. This research
has been supported by the Academy of Finland.
E-mail addresses:
{\tt
Ilpo.Vattulainen@csc.fi,
Kari.Kankaala@csc.fi,\\
jukkas@ee.tut.fi,
ala@phcu.helsinki.fi}
\pagebreak
|
1,314,259,993,646 | arxiv | \section{Introduction}
We are aiming to describe the response of thermo-visco-elastic material to applied external forces and the heat flux through the boundary. The system of equations capturing the displacement, temperature and visco-elastic strain of the body is
a consequence of physical principles such as balance of momentum and balance of energy, cf. \cite{GreenNaghdi, LandauLifshitz}, see also \cite{GKSG}. The equations are complemented by the constitutive relation for the Cauchy stress tensor and the constitutive equation for the evolution of the visco-elastic strain tensor.
Although we treat the case where the thermal expansion is negligible, but the changes of temperature affect the visco-elastic properties of the considered material. We shall observe it in the appearance of the temperature-dependent constitutive relation in the evolution equation for the visco-elastic strain tensor.
We assume that the body $\Omega \subset \mathbb{R}^3$ is an open bounded set with a $C^1$ boundary and moreover, the body is homogeneous in space. The material
undergoes
two kinds of deformations: elastic and visco-elastic. By the first type we understand the deformations which are reversible and the second ones are irreversible.
The problem is captured by the following system
\begin{equation}
\left\{
\begin{array}{rclr}
- \rm{div\,} \ten{T} &=& \vc{f} & \mbox{in } \Omega\times(0,T),
\\
\ten{T} &=& \ten{D}(\ten{\varepsilon}(\vc{u}) - \ten{\varepsilon}^{\bf p} ) & \mbox{in } \Omega\times(0,T),
\\
\ten{\varepsilon}^{\bf p}_t &=& \ten{G}(\theta,\ten{T}^d) & \mbox{in } \Omega\times(0,T),
\\
\theta_t - \Delta\theta &=& \ten{T}^d:\ten{G}(\theta,\ten{T}^d) & \mbox{in } \Omega\times(0,T),
\end{array}
\right.
\label{full_system_2}
\end{equation}
which describes the quasi-static evolution of the displacement of the material $\vc{u}:\Omega\times\mathbb{R}_+\rightarrow \mathbb{R}^3$, the temperature of the material $\theta:\Omega\times\mathbb{R}_+\rightarrow\mathbb{R}_+$ and the visco-elastic strain tensor $\ten{\varepsilon}^{\bf p}:\Omega\times\mathbb{R}_+\rightarrow \mathcal{S}^3_d$.
We denote by $\mathcal{S}^3$ the set of symmetric $3 \times 3$-matrices with real entries and by $\mathcal{S}^3_d$ a subset of $\mathcal{S}^3$ which contains traceless matrices. By $\ten{T}^d$ we mean the deviatoric part (traceless) of the tensor $\ten{T}$, i.e. $\ten{T}^d=\ten{T}-\frac{1}{3}tr(\ten{T})\ten{I}$, where $\ten{I}$ is the identity matrix from $\mathcal{S}^3$. Additionally, $\ten{\varepsilon}(\vc{u})$ denotes the symmetric part of the gradient of the displacement~$\vc{u}$, i.e. $\ten{\varepsilon}(\vc{u})=\frac{1}{2}(\nabla\vc{u} + \nabla^T\vc{u})$. The volume force is denoted by $\vc{f}:\Omega\times\mathbb{R}_+\rightarrow \mathbb{R}^3$.
The visco-elastic strain tensor is described by the evolutionary equation with prescribed constitutive function $\ten{G}(\cdot,\cdot)$.
The function $\ten{G}(\cdot,\cdot)$ is assumed to be monotone and to satisfy polynomial growth and coercivity conditions.
\begin{ass}
The function $\ten{G}(\theta,\ten{T}^d)$ is continuous with respect to $\theta$ and $\ten{T}^d$ and satisfies for $p\ge 2$
the following conditions:
\begin{itemize}
\item[a)] $(\ten{G}(\theta,\ten{T}^d_1)-\ten{G}(\theta,\ten{T}^d_2)):(\ten{T}^d_1-\ten{T}^d_2) \geq 0$, for all $\ten{T}_1^d,\ten{T}_2^d \in \mathcal{S}^3_d$ and $\theta\in\mathbb{R}_+$;
\item[b)] $|\ten{G}(\theta,\ten{T}^d)| \leq C(1 + |\ten{T}^d|)^{p-1}$, where $\ten{T}^d\in\mathcal{S}^3_d$, $\theta\in\mathbb{R}_+$;
\item[c)] $\ten{G}(\theta,\ten{T}^d):\ten{T}^d \geq \beta |\ten{T}^d|^p$, where $\ten{T}^d\in\mathcal{S}^3_d$, $\theta\in\mathbb{R}_+$,
\end{itemize}
\label{ass_G}
where $C$ and $\beta$ are positive constants, independent of the temperature $\theta$.
\end{ass}
We complete the considered problem by formulating the initial conditions
\begin{equation}
\left\{
\begin{array}{rcl}
\theta(x,0)&=&\theta_0(x),
\\
\ten{\varepsilon}^{\bf p}(x,0)&=&\ten{\varepsilon}^{\bf p}_0(x),
\end{array}
\right.
\label{init_0}
\end{equation}
in $\Omega$ and boundary conditions
\begin{equation}
\left\{
\begin{array}{rcl}
\vc{u}&=&\vc{g}, \\
\frac{\partial \theta}{\partial \vc{n}}&=&g_{\theta},
\end{array}
\right.
\label{boun_0}
\end{equation}
on $\partial\Omega\times(0,T)$.
The properties of the material under consideration determine the choice of the function $\ten{G}$.
Such a framework includes the classical
Norton-Hoff model, cf.~\cite{CheAl}, which we shall briefly discuss in a sequel.
There are various different relations considered,
e.g.
\begin{itemize}
\item
Bodner-Partom model \cite{Bartczak, MANA:MANA5, MMA:MMA802}:
\begin{equation}
\begin{split}
\ten{G}(\theta,\ten{T}^d) &= \mathcal{G}\left(\frac{\left\{|\ten{T}^d| + \beta(\theta) \right\}^+}{y} \right)\frac{\ten{T}^d}{|\ten{T}^d|}
,
\\
y_t & = \gamma(y) \mathcal{G}\left(\frac{|\ten{T}^d|}{y} \right) |\ten{T}^d| - A \delta(y),
\end{split}
\end{equation}
where $y : \Omega \times \mathbb{R}_+ \to \mathbb{R}_+$ describes the isotropic hardening of the metal, $\{\cdot\}^+$ stands for the positive part of $\{\cdot\}$, $\gamma : \mathbb{R}_+ \supset D (\gamma) \to \mathbb{R}_+$ and $\delta : \mathbb{R}_+ \supset D(\delta) \to \mathbb{R}_+$ are given functions and $A$ is a positive constant. Moreover, functions $\mathcal{G}(\cdot)$, $\gamma(\cdot)$, $\delta(\cdot)$ and $\beta(\cdot)$ fulfill some specific properties.
\item Mr\'{o}z model \cite{GKSG, brokate, Homberg200455}:
\begin{equation}
\ten{G}(\theta,\ten{T}^d) = g(\theta)\ten{T}^d,
\end{equation}
where $g:\mathbb{R}_+\to \mathbb{R}_+$ is a continuous function.
\item Prandtl-Reuss model with linear kinematic hardening \cite{ChR}
\begin{equation}
\begin{split}
\ten{\varepsilon}^{\bf p}_t &\in \partial I_{K(\theta)}(\ten{T} - \alpha \ten{\varepsilon}^{\bf p}),
\end{split}
\end{equation}
where $I_{K(\theta)}$ is the indicator function of the closed and convex subset $K(\theta) = \{\ten{T}\in\mathcal{S}^3: |\ten{T}^d|\leq k-\theta\}$ and $\alpha, k>0$ are material parameters. Furthermore, $\partial I_{K(\theta)}$ is a subdifferential of the function $I_{K(\theta)}$.
\end{itemize}
For further examples of constitutive relations (e.g. classical Maxwell model, models proposed by Chaboche, Hart, Miler, Bruhns and many others) we refer to \cite[Chapter 2.2]{Alber}.
Our motivation for current considerations were the results of
Alber and Chełmiński \cite{CheAl} and of H\"{o}mberg \cite{Homberg200455}.
In \cite{CheAl} the authors considered the quasi-static visco-elasticity\footnote{The authors used the notion {\it visco-plasticity} which is sometimes also applied in the literature to capture the appearance of irreversible deformations.}
models with Norton-Hoff constitutive function, namely of the power-law type
\begin{equation}
\ten{G} = c |\ten{T}|^{p-1} \ten{T}
\label{eq:norton-hoff}
\end{equation}
with $p>2$. The parameter $c$ was either assumed to be a positive constant or dependent on an additional relaxation parameter described by a separate equation.
The scheme of the proof in \cite{CheAl} was to formulate the problem in a way that it fits to the abstract theory of maximal monotone operators, cf.~\cite{Barbu}. In the current paper we include the thermal effects of the process through
the dependence of the constitutive function $\ten{G}$ on the temperature. This dependence obstructs following the same scheme and
requires different approach.
Furthermore, we assume that $\ten{G}(\theta,\cdot)$ depends only on the deviatoric part of the Cauchy stress tensor and its range is the set of traceless matrices. The last assumption, together with the fact that $\ten{\varepsilon}^{\bf p}_0(x)$ is traceless, provides that also $\ten{\varepsilon}^{\bf p}$ is traceless. Vanishing of the trace of the deformation tensor corresponds to preserving the volume of the material. Indeed, the volume change is associated only with the elastic response of the material, and the plastic response is essentially incompressible, cf.~\cite{Gurtin}. The dependence of $\ten{G}(\theta,\cdot)$ only on $\ten{T}^d$ is essential to maintain the coercivity of the model. Once we know that the range of $\ten{G}$ is
${\mathcal S}_d^3$, then even for the isothermal process, namely the case of $\ten{G}=\ten{G}(\ten{T}) $
we observe that $\ten{G}(\ten{T}):\ten{T}=\ten{G}(\ten{T}):\ten{T}^d$. Then e.g. taking as $\ten{T}$ the identity matrix we immediately see that $\ten{G}(\ten{I}):\ten{I}^d=0$.
Let us now comment on the technical consequences of this assumption. Contrary to the proof of Alber and Chełmiński, where they showed that $\ten{T}$ belongs to $L^p(0,T,L^p(\Omega,\mathcal{S}^3))$ for $p\geq 2$, the estimates conducted in the current situation provide only that $\ten{T}$ belongs to $L^2(0,T,L^2(\Omega,\mathcal{S}^3))$.
H\"{o}mberg in \cite{Homberg200455} considered more general physical phenomena including the electro-magnetic effects. The changes of temperature influenced the concentration of different phases of materials and
this dependence was prescribed by some general operator $\mathcal{P}[\cdot]$ having {\it good} properties.
Then the constitutive function describing the evolution of visco-elastic strain depends no more on the temperature, but on these concentrations. Moreover it is linear with respect to the deviatoric part of the Cauchy stress tensor, namely corresponds to the Mr\'{o}z model.
The similarities with our approach are related with the construction of the approximated problem, namely by the truncation of the terms which appear on the right-hand side of the heat equation and are only integrable. The method also follows the framework of Boccardo and Gallou\"et. Nevertheless, because of the different structure of the problem, H\"{o}mberg can show the strong convergence of the approximated sequence of the Cauchy stress tensor. For the concept of showing this strong convergence observe that in the case of linear Mr\'oz relation, and in fact also in the case of Norton-Hoff relation \eqref{eq:norton-hoff}, the stronger condition than monotonicity holds, namely the uniform monotonicity condition
$$(\ten{G}(\theta,\ten{T}^d_1)-\ten{G}(\theta,\ten{T}^d_2)):(\ten{T}^d_1-\ten{T}^d_2) \geq c | \ten{T}^d_1-\ten{T}^d_2|^p \mbox{ for all } \ten{T}_1^d,\ten{T}_2^d \in \mathcal{S}^3_d \mbox{ and }\theta\in\mathbb{R}_+.$$
For the proof see e.g.~\cite{maleknecas}.
The studies on the Mr\'oz model presented in~\cite{GKSG} essentially used the strong monotonicity of the function
$\ten{G}$ in the second variable. The existence proof used the methods developed in~\cite{GwSw2005, Sw2006, ChGw2007} arising from the tools of Young measures. In the present setting none of the assumptions of strong nor uniform monotonicity are needed. We only assume monotonicity of $\ten{G}$.
Following Bartczak \cite{Bartczak}, Chełmiński \cite{MMA:MMA802}, Chełmiński and Racke \cite{ChR}, Duvaut and J.L. Lions \cite{duvautLions}, Johnson \cite{johnson1,johnson2}, Ne\v{c}as and Hlav\'{a}\v{c}ek \cite{NH}, Suquet \cite{suquet1,suquet2,suquet3}, Temam \cite{temam1,temam2} and many others, we study the quasi-static evolution, i.e. the evolution, which is slow and we neglect the acceleration term in the equation for balance of momentum. Moreover, we consider the model with infinitesimal displacement. In a consequence, the dependence between the Cauchy stress tensor and the symmetric gradient of displacement is linear (generalized Hooke's law, for more details see \cite{NH} or \cite{rajagopal}).
Much of the approaches involve the models that are purely mechanical, namely concern
the theory of inelastic and infinitesimal deformations with the nonlinear inelastic constitutive relation of monotone type, however neglect all thermal influences, see~\cite{Alber} and also \cite{CHelG2,MANA:MANA5,MMA:MMA844, MMA:MMA802}. On the other hand,
the mathematical analysis of linear thermo-elasticity is also a classical, well understood topic, cf. \cite{JR}, contrary to an analysis of thermo-inelastic models. By the thermo-inelastic models we mean the systems consisting of balance of momentum for kind of inelastic deformation and the equation for an evolution of the temperature. In the equation for balance of momentum for inelastic deformation the stress is not proportional to the strain, i.e. there appear term which absorbs the mechanical energy. There are only some results for special models or for simplified models in the literature \cite{Bartczak, BR, ChR}.
If we introduce thermal effects into various purely mechanical models, then the right hand side of the heat equation
(the product $\ten{T}^d:\ten{G}(\theta,\ten{T}^d)$) turns out to be only an integrable function.
In such a case the standard energy methods fail and one needs to search for more delicate tools.
Using the Boccardo and Gallou\"{e}t \cite{Boccardo} approach to prove the existence of solutions to the heat equation the essential point is to use the truncation of the solution as a test function. This is however difficult to combine with a classical
Galerkin method as the truncation of a function may no longer be a linear combination of the functions from the Galerkin basis.
Therefore we appeal to non-standard energy methods, such as two-level Galerkin approximation, see also \cite{PhDBulicek,BFM,Bull}. The new difficulty which arises here is the construction of the appropriate basis for approximation of the strain tensor $\ten{\varepsilon}^{\bf p}$, for details see Appendix~\ref{B}.
All functions appearing in this paper are the functions of position $x$ and time $t$. We often omit the variables of the function and write $\vc{u}$ instead of $\vc{u}(x,t)$. All of the computation are conducted in Lagrangian coordinates. In view of the fact that the displacement is small, the stress tensor in the Lagrangian coordinates is approximated by the stress tensor in Eulerian coordinates. This is a standard way of considering the inelastic models, for more details see \cite[Chapter 13.2]{temammiranville}.
Before we formulate the definition of weak solutions and state the main result of the paper let us introduce the notation
$W^{1,p'}_{\vc{g}}(\Omega,\mathbb{R}^3):=\left\{\vc{u} \in W^{1,p'}(\Omega,\mathbb{R}^3): \vc{u}=\vc{g} \mbox{ on } \partial\Omega \right\}$.
\begin{defi
Let $p\geq 2$, $q<\frac{5}{4}$ and $s\in\mathbb{R}$ be large enough. The triple of functions
\begin{equation}
\begin{split}
\vc{u}&\in L^{p'}(0,T,W^{1,p'}_{\vc{g}}(\Omega,\mathbb{R}^3)
\\
\ten{T}&\in L^2(0,T,L^2(\Omega,\mathcal{S}^3))
\end{split}
\nonumber
\end{equation}
and
\begin{equation}
\theta\in L^q(0,T,W^{1,q}(\Omega))\cap C([0,T],W^{-s,2}(\Omega))
\nonumber
\end{equation}
is a weak solution to the system \eqref{full_system_2} if
\begin{equation}
\begin{split}
\int_0^T\int_{\Omega}\ten{T}:\nabla\vc{\varphi} \,{\rm{d}}x\,{\rm{d}}t
&= \int_0^T\int_{\Omega}\vc{f}\cdot \vc{\varphi} \,{\rm{d}}x\,{\rm{d}}t ,
\end{split}
\end{equation}
where
\begin{equation}
\ten{T}=\ten{D}(\ten{\varepsilon}(\vc{u}) - \ten{\varepsilon}^{\bf p}),
\end{equation}
and
\begin{equation}
\begin{split}
-\int_0^T\int_{\Omega} \theta\phi_t \,{\rm{d}}x\,{\rm{d}}t -
\int_{\Omega} \theta_0(x)\phi(0,x) \,{\rm{d}}x \qquad \qquad \qquad & \\ +
\int_0^T\int_{\Omega} \nabla\theta\cdot\nabla\phi \,{\rm{d}}x\,{\rm{d}}t -
\int_0^T\int_{\partial\Omega}g_{\theta}\phi \,{\rm{d}}x\,{\rm{d}}t &=
\int_0^T\int_{\Omega} \ten{T}^d:\ten{G}(\theta,\ten{T}^d)\phi \,{\rm{d}}x\,{\rm{d}}t,
\end{split}
\end{equation}
holds for every test function $\vc{\varphi}\in C^{\infty}([0,T],C^{\infty}_c(\Omega,\mathbb{R}^3))$ and $\phi\in C^{\infty}_c([-\infty,T),C^{\infty}(\Omega))$. Furthermore, the visco-elastic strain tensor can be recovered from the equation on its evolution, i.e.
\begin{equation}
\ten{\varepsilon}^{\bf p}(x,t) = \ten{\varepsilon}^{\bf p}_0(x) + \int_0^t \ten{G}(\theta(x,\tau),\ten{T}^d(x,\tau)) \,{\rm{d}}\tau,
\end{equation}
for a.e. $x\in\Omega$ and $t\in [0,T)$. Moreover, $\ten{\varepsilon}^{\bf p} \in W^{1,p'}(0,T,L^{p'}(\Omega,\mathcal{S}^3_d))$.
\end{defi}
\begin{tw}
Let $p\geq 2$ and let initial conditions satisfy $\theta_0 \in L^1(\Omega)$, $\ten{\varepsilon}^{\bf p}_0\in L^2(\Omega,\mathcal{S}^3_d)$, boundary conditions satisfy $\vc{g}\in L^p(0,T, W^{1-\frac{1}{p},p}(\partial\Omega,\mathbb{R}^3))$, $g_{\theta}\in L^2(0,T,L^2(\partial\Omega))$ and volume force $\vc{f}\in L^p(0,T,W^{-1,p}(\Omega,\mathbb{R}^3))$ and function $\ten{G}(\cdot,\cdot)$ satisfy the Assumption \ref{ass_G}. Then there exists a weak solution to system \eqref{full_system_2}.
\label{thm:main2}
\end{tw}
\begin{uwaga}
There is nothing about the uniqueness of solutions in Theorem \ref{thm:main2}. Using Boccardo and Gallou\"{e}t approach to the heat equation we obtain the existence of $\theta$ only in the space $L^q(0,T,W^{1,q}(\Omega))$ for all $q<\frac{5}{4}$, see the Appendix. The lack of uniqueness of the temperature implies the lack of uniqueness of the solution to the whole system.
In order to find the class of solutions providing both existence and uniqueness one should consider the renormalized solutions to the heat equation, see \cite{BlanchardMurat,Blanchard}. However, the existing theory concerns mostly the homogeneous Dirichlet boundary-value problems.
\end{uwaga}
The rest of the paper is organized as follows: Section \ref{druga} is mostly dedicated to physical aspects of the problem. Therefore in Section \ref{sec:model} we introduce the complete model and present the assumptions which brought us to the simplified setting. Then in Section \ref{comp} we concentrate on physical justification of the model after simplifications. Section \ref{sec:3} is only a technical part that prepares us to the proof of the main theorem, namely we transform the problem to a homogeneous boundary-value problem.
The whole
Section \ref{sec:proof} is devoted to the proof of Theorem \ref{thm:main2}. The subsequent subsections correspond to the steps of the proof such as existence of the approximate solutions, boundedness of the approximate solution and the behaviour of the energy of the system. Finally we pass to the limit in the Galerkin approximations.
We complete the introduction by introducing the notation.
As a result of integration $\int_{t_1}^{t_2}\frac{d g}{d t} dt$ we write $g|_{t_1}^{t_2} $ which is equal to $g(t_2) - g(t_1)$. Furthermore, we denote by $L^p(\Omega)$ standard Lebesgue spaces, for $k,m\in\mathbb{N}$ and $1\leq p,q \leq \infty$, by $W^{k,p}(\Omega)$ the Sobolev spaces, by $W^{\frac{m}{k},p}(\partial\Omega)$ the fractional order Sobolev space and by $L^p(0,T,L^q(\Omega))$ Bochner spaces, by $C(K)$ continuous functions on $K$, by $C_c^{\infty}(K)$ compactly supported smooth functions on $K$.
\section{The physical model. Motivations and simplifications.}\label{druga}
We will start the current section with formulating the full system describing the evolution of visco-elastic body including thermal effects. Subsequently we describe the assumptions that were made due to simplify the system and motivate considering equations \eqref{full_system_2}. The second part concerns the issue of thermodynamical completeness of the considered system. This part essentially follows \cite{GKSG}. However, since this is an important argument for choosing this model, we include the main steps for completeness. In the last subsection we include the technical step which allows to reduce the problem to homogeneous boundary-value problem.
\subsection{Origin of the model problem}
\label{sec:model}
Let us consider the system of equations in the bounded domain $\Omega\subset\mathbb{R}^3$ with a $C^1$ boundary $\partial \Omega$
\begin{align}
\varrho\vc{u}_{tt} - \rm{div\,} \ten{\sigma} &= \vc{f} & \mbox{in } \Omega\times(0,T),
\label{full_system1}
\\
\ten{\sigma} &= \ten{T} - \alpha(\theta-\theta_R)\ten{I} & \mbox{in } \Omega\times(0,T),
\label{full_system2}
\\
\ten{T} &= \ten{D}(\ten{\varepsilon}(\vc{u}) - \ten{\varepsilon}^{\bf p} ) & \mbox{in } \Omega\times(0,T),
\label{full_system3}
\\
\ten{\varepsilon}^{\bf p}_t &= \ten{G}(\theta,\ten{T}^d) & \mbox{in } \Omega\times(0,T),
\label{full_system4}
\\
\theta_t - \kappa\Delta\theta +\alpha(\theta-\theta_R)\rm{div\,} \vc{u}_t &= \ten{T}^d:\ten{G}(\theta,\ten{T}^d) + r & \mbox{in } \Omega\times(0,T).
\label{full_system5}
\end{align}
Derivation of this system was presented in \cite{GreenNaghdi}, \cite{GKSG} and \cite{LandauLifshitz}. The equation \eqref{full_system1} describes the balance of momentum. Equations \eqref{full_system2} and \eqref{full_system3} prescribe the constitutive relation for the Cauchy stress tensor and \eqref{full_system4} presents the constitutive relation for the evolution of the visco-elastic strain tensor. Finally, \eqref{full_system5} stands for the balance of energy.
The function $\ten{\sigma}:\Omega\times\mathbb{R}_+\rightarrow \mathcal{S}^3$ is the Cauchy stress tensor. The Cauchy stress tensor can be divided into two parts: mechanical and thermal. The mechanical part is $\ten{T}=\ten{D}(\ten{\varepsilon}(\vc{u}) - \ten{\varepsilon}^{\bf p})$, where the operator $\ten{D}:\mathcal{S}^3\rightarrow\mathcal{S}^3$ is linear, positively definite and bounded.
Assuming that $\Omega$ is a homogeneous material, the operator $\ten{D}$ is a four-index matrix, i.e. $\ten{D}=\left\{d_{i,j,k,l}\right\}_{i,j,k,l=1}^3$ and the following equalities hold
\begin{equation}
d_{i,j,k,l} = d_{j,i,k,l},
\quad
d_{i,j,k,l} = d_{i,j,l,k}
\quad
\mbox{and}
\quad
d_{i,j,k,l} = d_{k,l,i,j}
\quad
\forall i,j,k,l=1,2,3 .
\end{equation}
The evolution of the visco-elastic strain tensor $\ten{\varepsilon}^{\bf p}$ is governed by the constitutive relation $\ten{G}:\mathbb{R}_+\times\mathcal{S}^3_d\rightarrow\mathcal{S}^3_d$.
The visco-elastic strain tensor $\ten{\varepsilon}^{\bf p}=(\ten{\varepsilon}^{\bf p})^d$ is traceless if $\ten{\varepsilon}^{\bf p}_0$ is traceless. The temperature $\theta_R$ is the reference temperature.
The function $r:\Omega\times\mathbb{R}_+\to\mathbb{R}_+$ describes a given density of heat sources, $\kappa:\Omega\times\mathbb{R}_+\to\mathbb{R}_+$ is the material’s conductivity, which in the case of homogeneous materials is a positive constant, $\varrho$ is the constant density of the body.
Moreover, $\alpha$ describes the thermal expansion of the body. We will study the simplified situation, namely under the following assumptions
\begin{ass}
We consider only the problem with small inertial force, i.e. $\varrho\vc{u}_{tt}=0$.
\end{ass}
\begin{ass}
We assume that $\alpha=0$, i.e. the considered material is not subject to
the thermal expansion.
\end{ass}
The fact of neglecting the acceleration term implies that the system of equations may be supplemented only by the initial conditions
\eqref{init_0}.
Moreover, we complete the system with boundary conditions \eqref{boun_0}.
Using the Dirichlet condition for the displacement means that we control the shape of the body, and by using the Neumann condition for the temperature we control the flow of the energy through the boundary.
There are various simplifications that are proposed due to provide the mathematical analysis of the system.
In the linear thermoelasticity, the term connected with thermal expansion in the heat equation is approximated by a linear one, i.e. $\alpha(\theta - \theta_0 )\rm{div\,} \vc{u}_t\approx \alpha_0 \rm{div\,}{} \vc{u}_t$ with the argumentation that the temperature $\theta$ in the considered process is close to the reference temperature, cf. Bartczak \cite{Bartczak}, Chełmiński and Racke \cite{ChR}.
From the point of view of mathematical techniques used in the linear theory, such as e.g. linear semigroup theory, this approximation seems to be accurate. Unfortunately, in a consequence of this procedure one obtains the model which is not consistent with the physical principles.
Our simplification follows different way, we consider the case where no thermal expansion appears, hence $\alpha=0$. In the proceeding section we discuss in detail the issue of thermodynamical completeness of the system after the simplifications. Finally, we also formulate the last assumption.
\begin{ass}
We assume that there are no heat sources in the system, hence $r\equiv 0$.
The material's conductivity $\kappa$ is for simplicity equal to $1$.
\end{ass}
Taking into account the above conditions we obtain the considered system \eqref{full_system_2}.
\subsection{Thermodynamical completeness}\label{comp}
The purpose of the current section is to underline the physical advantages of the considered system. The assumptions used in the construction of the simplified model do not effect the loss of physical properties, i.e. the system \eqref{full_system_2} still conserves the energy, the temperature is positive and there exists a function of state, namely the entropy, which has a positive rate of production. We shall say that the system is thermodynamically complete if these properties are satisfied.
In \cite{GKSG} we showed the thermodynamical completeness of the system \eqref{full_system1}--\eqref{full_system5} in the case it is isolated, i.e. $\vc{f}=0$, we assume homogeneous boundary values and there are no heat sources ($r=0$).
All of the calculation in this section are formal.
\noindent
{\it Conservation of total energy}
\noindent
In the first step we intend to show that the global energy is preserved. Multiplying the first equation of system \eqref{full_system_2} by $\vc{u}_t$ and integrating over an arbitrary set $\mathcal{O}\subset\Omega$, we obtain
\begin{equation}
-
\int_{\mathcal{O}}{\rm{div\,}}\ten{T}\cdot\vc{u}_t \,{\rm{d}}x
= 0
\end{equation}
and hence
\begin{equation}
\int_{\mathcal{O}}\ten{T}:\nabla\vc{u}_t \,{\rm{d}}x
- \int_{\partial\mathcal{O}}\ten{T}\vc{n}\cdot\vc{u}_t \,{\rm{d}}s
= 0.
\label{rownanie_powyzej}
\end{equation}
We multiply the evolutionary equation for the visco-elastic strain by $\ten{T}$ and integrate over~$\mathcal{O}$. Subtracting this equation from \eqref{rownanie_powyzej} implies that
\begin{equation}
\begin{split}
\int_{\mathcal{O}}\big(\ten{T}:\nabla\vc{u}_t-
\ten{T}:\ten{\varepsilon}^{\bf p}_t\big)
\,{\rm{d}}x
- \int_{\partial\mathcal{O}}\ten{T}\vc{n}\cdot\vc{u}_t \,{\rm{d}}s
=
- \int_{\mathcal{O}}\ten{T}^d:\ten{G} \,{\rm{d}}x .
\end{split}
\end{equation}
Finally, using the symmetry of $\ten{T}$ we obtain
\begin{equation}
\begin{split}
\frac{1}{2}\frac{d}{dt}\int_{\mathcal{O}}\ten{T}:(\ten{\varepsilon}(\vc{u}) -\ten{\varepsilon}^{\bf p}) \,{\rm{d}}x
- \int_{\partial\mathcal{O}}\ten{T}\vc{n}\cdot\vc{u}_t \,{\rm{d}}s
=
- \int_{\mathcal{O}}\ten{T}^d:\ten{G} \,{\rm{d}}x .
\end{split}
\label{enrgia}
\end{equation}
Since the global energy of the set $\mathcal{O}$ is equal to $\mathcal{E}_{\mathcal{O}}(\tau)=\int_{\mathcal{O}}e(x,\tau)\,{\rm{d}}x $ and the density of the total energy is defined by $e(x,\tau)=\theta +\frac{1}{2}\ten{D}^{-1}\ten{T}:\ten{T} $, we obtain
\begin{equation}
\mathcal{E}_{\mathcal{O}}(t)
\int_{\mathcal{O}}\theta(t) \,{\rm{d}}x +
\frac{1}{2}\int_{\mathcal{O}}\ten{T}:(\ten{\varepsilon}(\vc{u})-\ten{\varepsilon}^{\bf p})(t) \,{\rm{d}}x .
\end{equation}
Consequently equation \eqref{enrgia} may be written in the following from
\begin{equation}
\begin{split}
\frac{d}{dt}\mathcal{E}_{\mathcal{O}}(t) & =
\frac{d}{dt}\int_{\mathcal{O}}\theta \,{\rm{d}}x
- \int_{\mathcal{O}}\ten{T}^d:\ten{G} \,{\rm{d}}x
+ \int_{\partial\mathcal{O}}\ten{T}\vc{n}\cdot\vc{u}_t \,{\rm{d}}s .
\end{split}
\end{equation}
Using \eqref{full_system_2}$_4$, we obtain
\begin{equation}
\begin{split}
\frac{d}{dt}\mathcal{E}_{\mathcal{O}}(t) &=
\int_{\mathcal{O}}\theta_t \,{\rm{d}}x
- \int_{\mathcal{O}}\theta_t \,{\rm{d}}x
+\int_{\mathcal{O}}\Delta\theta \,{\rm{d}}x
+ \int_{\partial\mathcal{O}}\ten{T}\vc{n}\cdot\vc{u}_t \,{\rm{d}}s
\\
& =
\int_{\partial\mathcal{O}}\big(\ten{T}\vc{u}_t +\nabla\theta\big)\cdot\vc{n} \,{\rm{d}}s .
\end{split}
\end{equation}
Zero external forces, homogeneous boundary conditions and no heat sources implies that $\vc{u}_t=0$ and $\nabla\theta\cdot\vc{n}=0$ on the boundary $\partial\Omega$. Therefore, the global energy $\mathcal{E}_{\Omega}$ is constant in time.
\noindent
{\it Positivity of the temperature}
\noindent
Let us assume that the initial temperature $\theta_0$ is positive.
The heat equation after simplifications has a form
\begin{equation}
\theta_t - \Delta\theta =
\ten{G}(\theta,\ten{T}^d):\ten{T}^d .
\label{eq:nee_heat}
\end{equation}
Hence, the assumptions on the function $\ten{G}(\cdot,\cdot)$ imply that the right hand side of \eqref{eq:nee_heat} is positive, namely
\begin{equation}
\theta_t - \Delta\theta \geq 0.
\end{equation}
When the initial and boundary conditions for the temperature are positive, then the temperature $\theta$ is positive.
\noindent
{\it Entropy inequality}
\noindent
Multiplying \eqref{eq:nee_heat} by $1/\theta$ and integrating over an arbitrary set $\mathcal{O}\subset\Omega$, we obtain
\begin{equation}
\begin{split}
\frac{d}{dt}\int_{\mathcal{O}}\ln\theta \,{\rm{d}}x
- \int_{\mathcal{O}}{\rm{div\,}}\frac{\nabla \theta}{\theta} \,{\rm{d}}x
- \int_{\mathcal{O}}\frac{|\nabla\theta|^2}{\theta^2} \,{\rm{d}}x
=
\int_{\mathcal{O}} \frac{\ten{G}(\theta,\ten{T}^d):\ten{T}^d}{\theta} \,{\rm{d}}x .
\end{split}
\nonumber
\end{equation}
Thus
\begin{equation}
\begin{split}
\frac{d}{dt}\int_{\mathcal{O}} \ln\theta
\,{\rm{d}}x
+ &\int_{\mathcal{O}}{\rm{div\,}}\Big(\frac{\vc{q}}{\theta}\Big) \,{\rm{d}}x
= \int_{\mathcal{O}} \frac{\ten{G}(\theta,\ten{T}^d):\ten{T}^d}{\theta} \,{\rm{d}}x
+ \int_{\mathcal{O}}\frac{|\nabla\theta|^2}{\theta^2} \,{\rm{d}}x .
\end{split}
\label{eq:ent_pow}
\end{equation}
By the properties of the function $\ten{G}(\cdot,\cdot)$ and positivity of $\theta$, the right hand side of \eqref{eq:ent_pow} is positive.
Therefore, an arbitrary choice of the domain $\mathcal{O}$ implies that the inequality holds
\begin{equation}
\Big( \ln\theta\Big)_t
+ {\rm{div\,}}\Big(\frac{\vc{q}}{\theta}\Big) \geq 0.
\end{equation}
The above relation is the so-called Clausius-Duhem inequality and it is one of the equivalent formulations of the second principle of thermodynamics. Hence, the homogeneous boundary conditions and the definition of the heat flux ($\vc{q}=-\nabla \theta$) implies that
\begin{equation}
\frac{d}{dt}\int_{\Omega} \ln\theta \geq 0 .
\end{equation}
Note that $\eta(\theta) = \ln\theta$ is one of the admissible entropies for system \eqref{full_system_2} what furnishes a formal justification for the thermodynamical completeness of the model. For the situation with linearization of the term $\alpha(\theta-\theta_R){\rm{div\,}} \vc{u}_t$ one can show that none of the thermodynamical principles is fulfilled.
\subsection{Transformation to a homogeneous boundary-value problem}
\label{sec:3}
Our aim is to reduce the problem to a homogeneous one.
For this purpose we are interested in a decoupled elastic systems and a heat equation. The first system is subject to the same external forces as problem \eqref{full_system_2} and both of the problems are complemented with the same boundary conditions as \eqref{full_system_2}.
Hence, given $\tilde{\theta}_0\in L^2(\Omega)$ we study
\begin{equation}
\left\{
\begin{array}{rcll}
-\rm{div\,} \tilde{\ten{T}} &=& \vc{f} & \mbox{in } \Omega\times (0,T), \\
\tilde{\ten{T}} &=& \ten{D}\ten{\varepsilon}(\tilde{\vc{u}}) & \mbox{in } \Omega\times (0,T), \\
\tilde{\vc{u}} &=& \vc{g} & \mbox{on } \partial\Omega\times (0,T),
\end{array}
\right.
\label{war_brz_u}
\end{equation}
and
\begin{equation}
\left\{
\begin{array}{rcll}
\tilde{\theta}_t -\Delta \tilde{\theta} &=& 0 & \mbox{in } \Omega\times (0,T), \\
\frac{\partial\tilde{\theta}}{\partial\vc{n}} &=& g_{\theta} & \mbox{on } \partial\Omega\times (0,T), \\
\tilde{\theta}(x,0) &=& \tilde{\theta}_0 & \mbox{in } \Omega.
\end{array}
\right.
\label{war_brz_t}
\end{equation}
\begin{lemat}
Let $\tilde{\theta}_0 \in L^2(\Omega)$, $\vc{g} \in L^p(0,T, W^{1-\frac{1}{p},p}(\partial\Omega,\mathbb{R}^3))$, $g_{\theta} \in L^2(0,T,L^2(\partial\Omega))$ and moreover $\vc{f}\in L^p(0,T,W^{-1,p}(\Omega,\mathbb{R}^3))$. Then there exists a solution to systems \eqref{war_brz_u} and \eqref{war_brz_t}. Additionally, the following estimates hold:
\begin{equation}
\begin{split}
\|\tilde{\vc{u}}\|_{L^p(0,T,W^{1,p}(\Omega))}
& \leq
C_1 \left(\|\vc{g}\|_{L^p(0,T, W^{1-\frac{1}{p},p}(\partial\Omega,\mathbb{R}^3))}+
\|\vc{f}\|_{L^p(0,T,W^{-1,p}(\Omega))} \right),
\\
\|\tilde{\theta}\|_{L^{\infty}(0,T,L^1(\Omega))} + \|\tilde{\theta}\|_{L^2(0,T,W^{1,2}(\Omega))}
& \leq
C_2 \left(\|g_{\theta}\|_{L^2(0,T,L^2(\partial\Omega))}+\|\tilde{\theta}_0\|_{L^2(\Omega)} \right).
\nonumber
\end{split}
\end{equation}
\label{wyrzucenie_war_brzeg}
Moreover, $\theta$ belongs to $C([0,T],L^2(\Omega))$.
\end{lemat}
\begin{uwaga}
From the trace theorem \cite[Chapter II]{Valent} there exist $\tilde{\vc{g}}\in L^{p}(0,T,W^{1,p}(\Omega,\mathbb{R}^3))$ such that $\tilde{\vc{g}}|_{\partial\Omega}=\vc{g}$. Then, finding the solution $\tilde {\vc{u}}$ to \eqref{war_brz_u} is equivalent to finding the solution $\tilde{\vc{u}}_1$ to the following problem
\begin{equation}
\left\{
\begin{array}{rcll}
-{\rm{div\,}} \ten{D}\ten{\varepsilon}(\tilde{\vc{u}}_1) &=& \vc{f} +{\rm{div\,}} \ten{D}\ten{\varepsilon}(\vc{\tilde{g}}) & \mbox{in } \Omega\times (0,T), \\
\tilde{\vc{u}}_1 &=& 0 & \mbox{on } \partial\Omega\times (0,T),
\end{array}
\right.
\label{war_brz_u_0}
\end{equation}
and $\tilde{\vc{u}} = \tilde{\vc{u}}_1 + \tilde{\vc{g}}$. Using \cite[Corollary 4.4]{Valent}, we obtain the estimates presented in Lemma \ref{wyrzucenie_war_brzeg}.
\end{uwaga}
Instead of finding $(\widehat{\vc u}, \widehat{\theta})-$ the solution to problem \eqref{full_system_2}-\eqref{init_0}-\eqref{boun_0}
we shall search for $(\vc{u}, \theta)$, where $\vc{u}=\widehat{\vc{u}}-\tilde{\vc{u}}$ and $\theta=\widehat{\theta}-\tilde{\theta}$ and $(\tilde{\vc{u}},\tilde{\theta})$ solve \eqref{war_brz_u} with ${\vc{g}}=0$ and \eqref{war_brz_t}. Furthermore, we get
\begin{equation}
\left\{
\begin{split}
- {\rm{div\,}} \ten{T} = - {\rm{div\,}} (\widehat{\ten{T}} - \tilde{\ten{T}}) & = 0 ,
\\
\ten{T} & = \ten{D}(\ten{\varepsilon}(\vc{u}) - \ten{\varepsilon}^{\bf p} ),
\\
\ten{\varepsilon}^{\bf p}_t & = \ten{G}(\widehat{\theta},\widehat{\ten{T}}^d)
\\
& = \ten{G}(\theta + \tilde{\theta},\ten{T}^d+\tilde{\ten{T}}^d),
\\
\theta_t - \Delta \theta = (\widehat{\theta} - \tilde{\theta})_t - \Delta(\widehat{\theta} - \tilde{\theta}) & = \widehat{\ten{T}}^d:\ten{G}(\theta + \tilde{\theta},\ten{T}^d+\tilde{\ten{T}}^d)
\\
& = \big(\ten{T}^d + \tilde{\ten{T}}^d\big):\ten{G}(\theta + \tilde{\theta},\ten{T}^d+\tilde{\ten{T}}^d).
\end{split}
\right.
\label{full_system_22a}
\end{equation}
Hence, we consider the problem
\begin{equation}
\left\{
\begin{split}
- {\rm{div\,}} \ten{T} & = 0 ,
\\
\ten{T} & = \ten{D}(\ten{\varepsilon}(\vc{u}) - \ten{\varepsilon}^{\bf p} ),
\\
\ten{\varepsilon}^{\bf p}_t & = \ten{G}(\theta + \tilde{\theta}, \ten{T}^d + \tilde{\ten{T}}^d),
\\
\theta_t - \Delta \theta & = \big(\ten{T}^d + \tilde{\ten{T}}^d\big):\ten{G}(\theta + \tilde{\theta}, \ten{T}^d + \tilde{\ten{T}}^d),
\end{split}
\right.
\label{full_system_22}
\end{equation}
with the initial and boundary conditions
\begin{equation}
\left\{
\begin{array}{rcll}
\vc{u} &=& 0 & \mbox{on } \partial\Omega\times (0,T), \\
\frac{\partial\theta}{\partial \vc{n}} &=& 0 & \mbox{on } \partial\Omega\times (0,T), \\
\theta(\cdot,0) &=& \widehat{\theta}_0 - \tilde{\theta}_0 \equiv \theta_0 & \mbox{in } \Omega, \\
\ten{\varepsilon}^{\bf p}(\cdot,0) &=& \ten{\varepsilon}^{\bf p}_0 & \mbox{in } \Omega,
\end{array}
\right.
\label{in_bou_cond}
\end{equation}
where $\widehat{\theta}_0$ is the given initial condition for the temperature and
$\tilde{\theta}_0$ is the initial condition for the system \eqref{war_brz_t}.
\begin{uwaga}
From the proof provided in Section \ref{sec:proof} it follows that the displacement $\ten{u}$, which is a solution to the homogeneous problem belongs to the space $C([0,T],L^{p'}(\Omega))$. However, in Theorem \ref{thm:main2}
the information on the continuity of the solution to the nonhomogeneous problem does not appear. This is the consequence of the the fact that $\tilde{\ten{u}}$ may fail to be continuous under the assumptions that we have for the volume force $\ten{f}$ and boundary data.
\end{uwaga}
\section{Proof of Theorem \ref{thm:main2}}\label{sec:proof}
\subsection{Approximate solutions}
\label{sec:4}
Let $k\in{\mathbb N}$ and $\mathcal{T}_k(\cdot)$ be a standard truncation operator
\begin{equation}
\mathcal{T}_k(x)=\left\{
\begin{split}
k \qquad & x> k \\
x \qquad & |x|\leq k \\
-k \qquad & x <-k.
\end{split}
\right.
\label{Tk}
\end{equation}
We are facing the problem of low regularity of the right hand side of the heat equation and the initial condition. Both functions are only integrable what enforces using some delicate methods, such as the approach of Boccardo and Gallou\"{e}t, cf. \cite{Boccardo}, for showing the existence of solutions. An essential step is testing the equation with the truncation of solution. However, this truncation need not to be a linear combination of basis functions. This is the reason why we use two level approximation, i.e. independent parameters of approximation in the displacement and temperature.
We pass to the limit, firstly with parameter $l$ corresponding to the dimension of the Galerkin basis for the temperature to get the sequence of infinite dimensional approximate solutions. Passing to the limit with parameter $k$ corresponding to the dimension of the Galerkin basis for the displacement requires closer attention.
We construct the approximated system using the Galerkin method. Consider the space $L^2(\Omega,\mathcal{S}^3)$ with a scalar product defined
\begin{equation}
(\ten{\xi},\ten{\eta})_{\ten{D}}:= \int_\Omega {\ten{D}}^\frac{1}{2}\ten{\xi}\cdot {\ten{D}}^\frac{1}{2}\ten{\eta} \,{\rm{d}}x
\quad\mbox{for }\ten{\xi},\ten{\eta}\in L^2(\Omega,\mathcal{S}^3)
\end{equation}
where ${\ten{D}}^\frac{1}{2}\circ{\ten{D}}^\frac{1}{2}=\ten{D}$.
Let $\{\vc{w}_i\}_{i=1}^{\infty}$ be the set of eigenfunctions of the operator $-\rm{div\,}\ten{D}\ten{\varepsilon}(\cdot)$ with the domain $W_0^{1,2}(\Omega,\mathbb{R}^3)$ and $\{ \lambda_i \}$ be the corresponding eigenvalues such that $\{\vc{w}_i\}$ is orthogonal in $W^{1,2}_0(\Omega,\mathbb{R}^3)$ with the inner product
\begin{equation}
( \vc{w}, \vc{v})_{W^{1,2}_0(\Omega)}=( \ten{\varepsilon}(\vc{w}), \ten{\varepsilon}(\vc{v}))_{\ten{D}}
\end{equation}
and orthonormal in $L^2(\Omega,\mathbb{R}^3)$.
Hence
\begin{equation}
\|\ten{\varepsilon}(\vc{w})\|^2_{\ten{D}}=( \ten{\varepsilon}(\vc{w}), \ten{\varepsilon}(\vc{v}))_{\ten{D}}.
\end{equation}
Using the eigenvalue problem for the operator $-\rm{div\,}\ten{D}\ten{\varepsilon}(\cdot)$ we obtain
\begin{equation}
\int_{\Omega}\ten{D}\ten{\varepsilon}(\vc{w}_i):\ten{\varepsilon}(\vc{w}_j) \,{\rm{d}}x = \lambda_i \int_{\Omega}\vc{w}_i\cdot\vc{w}_j \,{\rm{d}}x = 0
\end{equation}
Moreover, let $\{v_i\}_{i=1}^\infty$ be the set of eigenfunctions of the Laplace operator with the domain $W^{1,2}_n(\Omega)=\{ v\in W^{1,2}(\Omega):\quad \frac{\partial v}{\partial\vc{n}} = 0 \}$, let $\{\mu _i \}$ be the set of corresponding eigenvalues, let $\{v_i\}$ be orthogonal in $W^{1,2}_n(\Omega)$ and orthonormal in $L^2(\Omega)$. These two families of vectors shall be used to construct the finite dimensional approximations of the displacement and the temperature. To construct the basis for approximating the visco-elastic strain tensor we will proceed as follows.
Let us consider the symmetric gradients of first
$k$ functions from the basis $\{\ten{w}_i\}_{i=1}^{\infty}$. Due to the regularity of the eigenfunctions we observe that
$\ten{\varepsilon}(\ten{w}_i)$ are elements of $H^s(\Omega,\mathcal{S}^3)$, namely the fractional Sobolev space with a scalar product denoted by $\braket{\cdot,\cdot}_s$ and $s>\frac{3}{2}$.
Define now
\begin{equation}\label{Vk}
V_k:= (\mbox{span}\{\ten{\varepsilon}(\ten{w}_1),...,\ten{\varepsilon}(\ten{w}_k)\})^\bot,
\end{equation}
which is the orthogonal complement in $L^2(\Omega,\mathcal{S}^3)$ taken with respect to the scalar product $(\cdot,\cdot)_{\ten{D}}$ and also
\begin{equation}\label{Vks}
V_k^s:=V_k\cap H^s(\Omega,\mathcal{S}^3)
\end{equation}
Let $\{\ten{\zeta}^k_n\}_{n=1}^{\infty}$ denote the orthonormal basis of $V_k$, which is also an orthogonal basis of $V_k^s$, for more details see Appendix~\ref{B}.
For $k,l\in\mathbb{N}$,
we are ready to define
\begin{equation}
\begin{split}
\vc{u}_{k,l} & = \sum_{n=1}^k\alpha_{k,l}^n(t) \vc{w}_n,
\\
\theta_{k,l} & = \sum_{m=1}^l\beta_{k,l}^m(t) v_m,
\\
\ten{\varepsilon}^{\bf p}_{k,l} & = \sum_{n=1}^k\gamma_{k,l}^n(t) \ten{\varepsilon}(\vc{w}_n) +
\sum_{m=1}^l\delta_{k,l}^m(t) \ten{\zeta}_m^k,
\end{split}
\label{eq:postac}
\end{equation}
such that $\vc{u}_{k,l}$, $\ten{\varepsilon}^{\bf p}_{k,l}$ and $\theta_{k,l}$ solve the system of equations
\begin{equation}
\begin{array}{rll}
\int_{\Omega} \ten{T}_{k,l} : \ten{\varepsilon}(\vc{w}_n) \,{\rm{d}}x &= 0
& n=1,...,k ,
\\[1ex]
\ten{T}_{k,l} &= \ten{D}(\ten{\varepsilon}(\vc{u}_{k,l}) - \ten{\varepsilon}^{\bf p}_{k,l} ),
\\[1ex]
\int_{\Omega}(\ten{\varepsilon}^{\bf p}_{k,l})_t : \ten{D}\ten{\varepsilon}(\vc{w}_n) \,{\rm{d}}x &=
\int_{\Omega}\ten{G}(\theta_{k,l} + \tilde{\theta}, \ten{T}^d_{k,l} + \tilde{\ten{T}}^d ) : \ten{D}\ten{\varepsilon}(\vc{w}_n) \,{\rm{d}}x
& n=1,...,k ,
\\[1ex]
\int_{\Omega}(\ten{\varepsilon}^{\bf p}_{k,l})_t : \ten{D}\ten{\zeta}^k_m \,{\rm{d}}x &=
\int_{\Omega}\ten{G}(\theta_{k,l} + \tilde{\theta}, \ten{T}^d_{k,l} + \tilde{\ten{T}}^d ) : \ten{D}\ten{\zeta}^k_m \,{\rm{d}}x
& m=1,...,l ,
\\[1ex]
\int_{\Omega}(\theta_{k,l})_t v_m\,{\rm{d}}x + \int_{\Omega}\nabla\theta_{k,l}\cdot\nabla v_m \,{\rm{d}}x &
\\[1ex]
= \int_{\Omega} \mathcal{T}_k( (\ten{T}_{k,l}^d + \tilde{\ten{T}}^d ): & \ten{G}(\theta_{k,l} + \tilde{\theta}, \ten{T}^d_{k,l} + \tilde{\ten{T}}^d ) ) v_m \,{\rm{d}}x & m=1,...,l .
\end{array}
\label{app_system}
\end{equation}
for a.a. $t\in(0,T)$.
For each approximate equation we have the initial conditions in the following form
\begin{equation}
\left\{
\begin{array}{rclc}
\left( \theta_{k,l}(x,0), v_m\right) &=& \left( \mathcal{T}_k(\theta_0),v_m \right) & m=1,..,l, \\[1ex]
\left( \ten{\varepsilon}^{\bf p}_{k,l}(x,0), \ten{\varepsilon}(\vc{w}_n) \right)_{\ten{D}} &=& \left(\ten{\varepsilon}^{\bf p}_0, \ten{\varepsilon}(\vc{w}_n) \right)_{\ten{D}}
& n=1,..,k,
\\[1ex]
\left( \ten{\varepsilon}^{\bf p}_{k,l}(x,0), \ten{\zeta}_m^k) \right)_{\ten{D}} &=& \left(\ten{\varepsilon}^{\bf p}_0, \ten{\zeta}^k_m \right)_{\ten{D}}
& m=1,..,l,
\end{array}
\right.
\label{eq:warunki_pocz_app}
\end{equation}
where $\big(\cdot,\cdot\big)$ denotes the inner product in $L^2(\Omega)$ and $\big(\cdot,\cdot\big)_{\ten{D}}$ the inner product in $L^2(\Omega,\mathcal{S}^3)$.
Let us define
\begin{equation}
\begin{split}
\vc{\xi}_1(t) &=(\alpha_{k,l}^1(t),..., \alpha_{k,l}^k(t))^T, \\
\vc{\xi}_2(t) &= (\beta_{k,l}^1(t),...,\beta_{k,l}^l(t),\gamma_{k,l}^1(t),..., \gamma_{k,l}^k(t),\delta_{k,l}^1(t),...,\delta_{k,l}^l(t) )^T .
\end{split}
\nonumber
\end{equation}
The selection of the Galerkin bases and representation of the approximate solution \eqref{eq:postac} allows to notice that
\begin{equation}
\alpha_{k,l}^n(t) = \frac{1}{\lambda_n}
\gamma_{k,l}^n(t)\int_{\Omega}\ten{D}\ten{\varepsilon}(\vc{w}_n):\ten{\varepsilon}(\vc{w}_n) \,{\rm{d}}x = \gamma_{k,l}^n(t)
\\
\end{equation}
and hence we obtain
\begin{equation}
\left\{
\begin{split}
(\gamma_{k,l}^n(t))_t &=
\frac{1}{\lambda_n}
\int_{\Omega}\tilde{\ten{G}}(x,t,\vc{\xi}_1(t),\vc{\xi}_2(t)) : \ten{D}\ten{\varepsilon}(\vc{w}_n) \,{\rm{d}}x ,
\\
(\delta_{k,l}^m(t))_t &=
\int_{\Omega}\tilde{\ten{G}}(x,t,\vc{\xi}_1(t),\vc{\xi}_2(t)) : \ten{D}\ten{\zeta}_m^k\,{\rm{d}}x ,
\\
(\beta_{k,l}^m(t))_t &= \int_{\Omega} \mathcal{T}_k\Big( \big(( \ten{D}\sum_{n=1}^k\alpha_{k,l}^n\ten{\varepsilon}(\vc{w}_n) - \ten{D}(\sum_{n=1}^l\gamma_{k,l}^n(t) \ten{\varepsilon}(\vc{w}_n) +
\delta_{k,l}^n(t) \ten{\zeta}_n ))^d + \tilde{\ten{T}}^d \big)
\\
&\quad :\tilde{\ten{G}}(x,t,\vc{\xi}_1(t),\vc{\xi}_2(t)) \Big) v_m \,{\rm{d}}x + \mu_m \beta_{k,l}^m(t),
\end{split}
\right.
\label{app_system20}
\end{equation}
for $n=1,...,k$ and $m=1,...,l$, where
\begin{equation}
\begin{split}
& \quad \tilde{\ten{G}}(x,t,\vc{\xi}_1(t),\vc{\xi}_2(t))
\\
& := \ten{G}(\theta_{k,l} +\tilde{\theta},\ten{T}_{k,l}^d + \tilde{\ten{T}}^d)
\\
&=\ten{G}\Big(\sum_{j=1}^l \beta_{k,l}^j(t) v_j(x) + \tilde{\theta}, \Big(\ten{D}\sum_{j=1}^k \alpha_{k,l}^j(t)\ten{\varepsilon}(\vc{w}_j) - \ten{D}\sum_{j=1}^l \big(\gamma_{k,l}^j(t) \ten{\varepsilon}(\vc{w}_j) + \delta_{k,l}^j(t) \ten{\zeta}_j \big) \Big)^d + \tilde{\ten{T}}^d \Big)
\end{split}
\nonumber
\end{equation}
Hence
\begin{equation}
\left\{
\begin{split}
(\gamma_{k,l}^n(t))_t &=
\frac{1}{\lambda_n}
\int_{\Omega}\tilde{\ten{G}}(x,t,\vc{\xi}_1(t),\vc{\xi}_2(t)) : \ten{D}\ten{\varepsilon}(\vc{w}_n) \,{\rm{d}}x ,
\\
(\delta_{k,l}^m(t))_t &=
\int_{\Omega}\tilde{\ten{G}}(x,t,\vc{\xi}_1(t),\vc{\xi}_2(t)) : \ten{D}\ten{\zeta}_m^k\,{\rm{d}}x ,
\\
(\beta_{k,l}^m(t))_t &= \int_{\Omega} \mathcal{T}_k\Big( \big(( \ten{D}\sum_{n=1}^k\alpha_{k,l}^n\ten{\varepsilon}(\vc{w}_n) - \ten{D}(\sum_{n=1}^l\gamma_{k,l}^n(t) \ten{\varepsilon}(\vc{w}_n) +
\delta_{k,l}^n(t) \ten{\zeta}_n ))^d + \tilde{\ten{T}}^d \big)
\\
&\quad :\tilde{\ten{G}}(x,t,\vc{\xi}_1(t),\vc{\xi}_2(t)) \Big) v_m \,{\rm{d}}x + \mu_m \beta_{k,l}^m(t),
\end{split}
\right.
\label{app_system2}
\end{equation}
System \eqref{app_system2} with initial conditions \eqref{eq:warunki_pocz_app} can be equivalently written as the initial value problem
\begin{equation}\label{47}
\begin{split}
&\frac{d\vc{\xi}_2 }{dt} = \vc{F}(\vc{\xi}_1(t),\vc{\xi}_2(t),t),
\qquad
t\in [0,T),
\\
&\vc{\xi}_2(0) =\vc{\xi}_{2,0},
\end{split}
\end{equation}
where $\vc{\xi}_{2,0}$ is a vector of initial conditions obtained from \eqref{eq:warunki_pocz_app}. For $n\leq k$, we get $\alpha_{k,l}^n=\gamma_{k,l}^n$, hence $\vc{F}(\vc{\xi}_1(t),\vc{\xi}_2(t),t)$ can be treated as a function only of $\vc{\xi}_2(t)$, i.e. $\vc{F}(\vc{\xi}_1(t),\vc{\xi}_2(t),t)=\tilde{\vc{F}}(\vc{\xi}_2(t),t)$.
\begin{lemat}{(Existence of approximate solution)}
For initial condition satisfying $\ten{\varepsilon}^{\bf p}_0\in L^2(\Omega,\mathcal{S}^3_d)$ and $\theta_0\in L^1(\Omega)$ there exists an absolutely continuous in time solution to \eqref{47}.
\label{istnienie_przyblizone}
\end{lemat}
\begin{proof}
According to Carath\'eodory Theorem, see \cite[Theorem 3.4]{maleknecas} or \cite[Appendix $(61)$]{zeidlerB}, there exist unique absolutely continuous functions $\beta_{k,l}^m(t)$, $\gamma_{k,l}^n(t)$ and $\delta_{k,l}^m(t)$ for every $n \leq k$ and $m \leq l$ on some time interval $[0,t^*]$.
Moreover for every $n \leq k$ there exists a unique absolutely continuous function $\alpha_{k,l}^n(t)$
\end{proof}
\subsection{Boundedness of approximate solutions}
\label{sec:5}
In this section we show the uniform boundedness of approximate solutions. As the considered model describes the physical phenomena, then it is obvious that the total energy should be finite. The total energy of the system consists of potential energy and thermal energy.
\begin{defi
We say that $\mathcal{E}$
is the potential energy if
\begin{equation}
\mathcal{E}(\ten{\varepsilon}(\vc{u}),\ten{\varepsilon}^{\bf p}): = \frac{1}{2}\int_{\Omega}\ten{D}(\ten{\varepsilon}(\vc{u}) - \ten{\varepsilon}^{\bf p}):(\ten{\varepsilon}(\vc{u}) - \ten{\varepsilon}^{\bf p} ) \,{\rm{d}}x .
\nonumber
\end{equation}
\label{energia}
\end{defi}
\begin{lemat}
There exists a constant $C$ which
is uniform with respect to $k$ and $l$ such that
\begin{equation}
\sup_{t\in [0,T]} \mathcal{E}(\ten{\varepsilon}(\vc{u}_{k,l}) , \ten{\varepsilon}^{\bf p}_{k,l}) (t)
+ c \|\ten{T}_{k,l}^d + \tilde{\ten{T}}^d\|^p_{L^p(0,T,L^p(\Omega))}
\leq
C.
\end{equation}
\label{pom_2}
\end{lemat}
\begin{proof}
The potential energy is an absolutely continuous function and calculating the time derivative of $\mathcal{E}(t)$ we get for a.a. $t\in[0,T]$
\begin{equation}
\begin{split}
\frac{d}{dt} \mathcal{E}(\ten{\varepsilon}(\vc{u}_{k,l}) , \ten{\varepsilon}^{\bf p}_{k,l})
& =
\int_{\Omega}\ten{D}(\ten{\varepsilon}(\vc{u}_{k,l}) - \ten{\varepsilon}^{\bf p}_{k,l}):(\ten{\varepsilon}(\vc{u}_{k,l}))_t
\,{\rm{d}}x
\\
& \quad
-
\int_{\Omega}\ten{D}(\ten{\varepsilon}(\vc{u}_{k,l}) - \ten{\varepsilon}^{\bf p}_{k,l}): (\ten{\varepsilon}^{\bf p}_{k,l})_t \,{\rm{d}}x.
\end{split}
\label{pochodna}\end{equation}
In the first step we multiply \eqref{app_system}$_{(1)}$ by $\{(\alpha_{k,l}^n)_t\}$ for each $n\leq k$.
Summing over $n=1,...,k$
we obtain
\begin{equation}
\int_{\Omega}\ten{D}(\ten{\varepsilon}(\vc{u}_{k,l}) - \ten{\varepsilon}^{\bf p}_{k,l}): (\ten{\varepsilon}(\vc{u}_{k,l})_t \,{\rm{d}}x =
0.
\label{pierwsze_r}
\end{equation}
In the second step we multiply \eqref{app_system}$_{(4)}$ by
$\delta^m_{k,l}$ and
summing over
$m=1,...,l$, we obtain the identity, which is equivalent to
\begin{equation}
\int_{\Omega}(\ten{\varepsilon}^{\bf p}_{k,l})_t:\ten{T}_{k,l} \,{\rm{d}}x=
\int_{\Omega}\ten{G}(\tilde{\theta} + \theta_{k,l},\tilde{\ten{T}}^d + \ten{T}^d_{k,l}):\ten{T}_{k,l} \,{\rm{d}}x.
\label{drugie_r}
\end{equation}
Thus
\begin{equation}\label{ene}
\begin{split}
\frac{d}{dt} \mathcal{E}(\ten{\varepsilon}(\vc{u}_{k,l}) , \ten{\varepsilon}^{\bf p}_{k,l})
=
-
\int_{\Omega}\ten{G}(\tilde{\theta} + \theta_{k,l},\tilde{\ten{T}}^d + \ten{T}^d_{k,l}):\ten{T}^d_{k,l}\,{\rm{d}}x.
\end{split}
\end{equation}
Using Assumption 1c and the Young inequality we get
\begin{equation}
\begin{split}
\frac{d}{dt} \mathcal{E}(\ten{\varepsilon}(\vc{u}_{k,l}) , \ten{\varepsilon}^{\bf p}_{k,l})
& =
- \int_{\Omega} (\ten{T}_{k,l}^d+\tilde{\ten{T}}^d) : \ten{G}(\theta_{k,l}+\tilde{\theta},\ten{T}_{k,l}^d +\tilde{\ten{T}}^d) \,{\rm{d}}x
\\
& \quad
+ \int_{\Omega} \tilde{\ten{T}}^d : \ten{G}(\theta_{k,l}+\tilde{\theta},\ten{T}_{k,l}^d +\tilde{\ten{T}}^d) \,{\rm{d}}x
\\
&
\leq
- \beta \|\ten{T}_{k,l}^d + \tilde{\ten{T}}^d\|^p_{L^p(\Omega)}
+ \|\tilde{\ten{T}}^d\|_{L^p(\Omega)}\|\ten{G}(\theta_{k,l}+\tilde{\theta},\ten{T}_{k,l}^d +\tilde{\ten{T}}^d)\|_{L^{p'}(\Omega)}
\\
& \leq
- \beta \|\ten{T}_{k,l}^d + \tilde{\ten{T}}^d\|^p_{L^p(\Omega)}
+ c(\epsilon)\|\tilde{\ten{T}}^d\|_{L^p(\Omega)}^p + \epsilon\|\ten{G}(\theta_{k,l}+\tilde{\theta},\ten{T}_{k,l}^d +\tilde{\ten{T}}^d)\|_{L^{p'}(\Omega)}^{p'}
\nonumber
\end{split}\label{osz1}
\end{equation}
where $\epsilon=\frac{\beta}{2^{p+1}C}$, with a constant $C$ coming from Assumption 1b. Hence we estimate the last term as follows
\begin{equation}
\epsilon\|\ten{G}(\theta_{k,l}+\tilde{\theta},\ten{T}_{k,l}^d +\tilde{\ten{T}}^d)\|_{L^{p'}(\Omega)}^{p'} \le
\frac{\beta}{2}|\Omega|+\frac{\beta}{2} \|\ten{T}_{k,l}^d + \tilde{\ten{T}}^d\|^p_{L^p(\Omega)}.
\end{equation}
Finally, integrating over $(0,t)$, with $0\le t\le T$ we obtain
\begin{equation}\label{osz2}
\begin{split}
\mathcal{E}(\ten{\varepsilon}(\vc{u}_{k,l}) , \ten{\varepsilon}^{\bf p}_{k,l}) (t)
&+ \frac{\beta}{2} \|\ten{T}_{k,l}^d + \tilde{\ten{T}}^d\|^p_{L^p(0,T,L^p(\Omega))}
\\&\leq
c(\epsilon)\|\tilde{\ten{T}}^d\|^p_{L^p(0,T,L^p(\Omega))}
+ \mathcal{E}(\ten{\varepsilon}(\vc{u}_{k,l}) , \ten{\varepsilon}^{\bf p}_{k,l})(0) +\frac{\beta}{2} |\Omega|.
\end{split}\end{equation}
\end{proof}
\begin{uwaga}
From \eqref{osz2} we immediately observe that the sequence $\{\ten{T}_{k,l}^d\}$ is uniformly bounded in the space $L^p(0,T,L^p(\Omega,\mathcal{S}^3))$ with respect to $k$ and $l$.
Additionally, combining \eqref{osz1} and \eqref{osz2} we conclude the uniform boundedness of the sequence
$\{\ten{G}(\theta_{k,l}+\tilde{\theta},\ten{T}_{k,l}^d +\tilde{\ten{T}}^d)\}$ in the space $L^{p'}(0,T,L^{p'}(\Omega,\mathcal{S}^3))$ and hence the uniform boundedness of the sequence $\{(\ten{T}_{k,l}^d +\tilde{\ten{T}}^d):\ten{G}(\theta_{k,l}+\tilde{\theta},\ten{T}_{k,l}^d +\tilde{\ten{T}}^d)\}$ in $L^1(0,T,L^1(\Omega))$.
\label{wsp_ogr_T}
\end{uwaga}
\begin{uwaga}
The uniform boundedness of the potential energy implies that the sequence $\{\ten{T}_{k,l}\}$ is uniformly bounded in $L^{\infty}(0,T,L^2(\Omega,\mathcal{S}^3))$ and in particular in $L^2(0,T,L^2(\Omega,\mathcal{S}^3))$.
\end{uwaga}
\begin{lemat}
The sequence $\{(\ten{\varepsilon}^{\bf p}_{k,l})_t\}$ is uniformly bounded in $L^{p'}(0,T,(H^{s}(\Omega,\mathcal{S}^3))')$ with respect to $k$ and $l$.
\label{wsp_org_epa}
\end{lemat}
\begin{proof}
Let $P^l$ be a projection on ${\rm lin}\{\ten{\zeta}_1,\ldots,\ten{\zeta}_l\}$, $P^l(\ten{v}):=\sum_{i=1}^{l}(\ten{v},\ten{\zeta}_i)_{\ten{D}}\ten{\zeta}_i$, then $\|P^l\varphi\|_{H^s}\le\|\varphi\|_{H^s}$.
Let $P^k$ be a projection on ${\rm lin}\{\ten{\varepsilon}(\vc{w}_1),\ldots,\ten{\varepsilon}(\vc{w}_1)\}$,
$P^k(\ten{v}):=\sum_{i=1}^{k}(\ten{v},\ten{\varepsilon}(\vc{w}_i))_{\ten{D}}\ten{\varepsilon}(\vc{w}_i)$.
Since $P^k$ is the projection of a finite dimensional space, and the dimension of the space is independent of $l$, there exists a constant, also independent of $l$ such that
$\|P^k\varphi\|_{H^s}\le c\|\varphi\|_{H^s}$
Let $\varphi\in L^p(0,T,H^{s}(\Omega,\mathcal{S}^3))$ and we may estimate as follows
\begin{equation}
\begin{split}
\int_0^T |\langle (\ten{\varepsilon}^{\bf p}_{k,l})_t, \varphi\rangle |\,{\rm{d}}t &=
\int_0^T |\langle (\ten{\varepsilon}^{\bf p}_{k,l})_t, (P^k + P^l)\varphi\rangle |\,{\rm{d}}t
\\ &
\le\int_0^T |\langle (\ten{\varepsilon}^{\bf p}_{k,l})_t, P^k\varphi\rangle |\,{\rm{d}}t
+\int_0^T |\langle (\ten{\varepsilon}^{\bf p}_{k,l})_t, P^l\varphi\rangle |\,{\rm{d}}t ,
\end{split}
\end{equation}
where the equality results from orthogonality of subspaces $\mbox{lin}\{\ten{\varepsilon}(\vc{w}_1),\ldots,\ten{\varepsilon}(\vc{w}_k)\}$ and $\mbox{lin}\{\ten{\zeta}_1,\ldots, \ten{\zeta}_l\}$. Then
\begin{equation}
\begin{split}
\int_0^T |\langle (\ten{\varepsilon}^{\bf p}_{k,l})_t, \varphi\rangle |\,{\rm{d}}t &\le
\int_0^T |\int_\Omega
\ten{G}(\theta_{k,l}+\tilde{\theta},\ten{T}_{k,l}^d +\tilde{\ten{T}}^d) P^k\varphi\,{\rm{d}}x |\,{\rm{d}}t
\\ &
\quad +
\int_0^T |\int_\Omega
\ten{G}(\theta_{k,l}+\tilde{\theta},\ten{T}_{k,l}^d +\tilde{\ten{T}}^d) P^l\varphi\,{\rm{d}}x |\,{\rm{d}}t
\\ &\le
\int_0^T\|\ten{G}(\theta_{k,l}+\tilde{\theta},\ten{T}_{k,l}^d +\tilde{\ten{T}}^d)\|_{L^{p'}(\Omega)}
\|P^k\varphi\|_{L^{p}(\Omega)}\,{\rm{d}}t\\
&
\quad + \int_0^T\|\ten{G}(\theta_{k,l}+\tilde{\theta},\ten{T}_{k,l}^d +\tilde{\ten{T}}^d)\|_{L^{p'}(\Omega)}
\|P^{l}\varphi\|_{L^{p}(\Omega)}\,{\rm{d}}t
\\ &
\le \tilde c\int_0^T\|\ten{G}(\theta_{k,l}+\tilde{\theta},\ten{T}_{k,l}^d +\tilde{\ten{T}}^d)\|_{L^{p'}(\Omega)}
\|P^k\varphi\|_{H^{s}(\Omega)}\,{\rm{d}}t
\\ &
\quad + \tilde c \int_0^T\|\ten{G}(\theta_{k,l}+\tilde{\theta},\ten{T}_{k,l}^d +\tilde{\ten{T}}^d)\|_{L^{p'}(\Omega)}
\|P^{l}\varphi\|_{H^{s}(\Omega)}\,{\rm{d}}t
\\ &
\le c\tilde c\int_0^T\|\ten{G}(\theta_{k,l}+\tilde{\theta},\ten{T}_{k,l}^d +\tilde{\ten{T}}^d)\|_{L^{p'}(\Omega)}
\|\varphi\|_{H^{s}(\Omega)}\,{\rm{d}}t
\\ &
\quad + \tilde c\int_0^T\|\ten{G}(\theta_{k,l}+\tilde{\theta},\ten{T}_{k,l}^d +\tilde{\ten{T}}^d)\|_{L^{p'}(\Omega)}
\|\varphi\|_{H^{s}(\Omega)}\,{\rm{d}}t\\
&\le (1+c)\tilde c\|\ten{G}(\theta_{k,l}+\tilde{\theta},\ten{T}_{k,l}^d +\tilde{\ten{T}}^d)\|_{L^{p'}(0,T,L^{p'}(\Omega))}
\|\varphi\|_{L^p(0,T,H^{s}(\Omega))},\\
\end{split}\end{equation}
where $\tilde c$ is an optimal embedding constant of $H^s(\Omega,\mathcal{S}^3)\subset L^2(\Omega,\mathcal{S}^3)$.
Consequently, there exists $C>0$ such that
\begin{equation}
\sup_{\varphi\in L^p(0,T,H^{s}(\Omega))\atop
\|\varphi\|_{L^p(0,T,H^{s}(\Omega))}\le 1 }\int_0^T
|\langle (\ten{\varepsilon}^{\bf p}_{k,l})_t, \varphi\rangle |\,{\rm{d}}t\le C
\end{equation}
and hence sequence $\{(\ten{\varepsilon}^{\bf p}_{k,l})_t\}$ is uniformly bounded in
${L^{p'}(0,T,(H^{s}(\Omega,\mathcal{S}^3))')}$
\end{proof}
\begin{lemat}\label{LinftyL1}
The sequence $\{\theta_{k,l}\}$ is uniformly bounded in $L^\infty(0,T;L^1(\Omega))$ with respect to $k$ and $l$.
\end{lemat}
Since it can be immediately observed that
\begin{equation}
\sup_{0\leq t\leq T}\|\theta_{k,l}(t)\|_{L^1(\Omega)}\leq
C(1+\|\ten{T}_{k,l}^d + \tilde{\ten{T}}^d\|_{L^p(0,T,L^p(\Omega))})+
\|\theta_0\|_{L^1(\Omega)}
\nonumber
\end{equation}
and Lemma~\ref{pom_2} holds, we omit the details of the proof.
The lemma provides that the internal energy of $\Omega$ is finite at any time $t\in [0,T]$. It is possible to prove better estimates for the temperature, however they are uniform only with respect to $l$ and not with respect to $k$. We provide the details in the proceeding lemma.
\begin{lemat
There exists a constant $C$, depending on the domain $\Omega$ and the time interval $(0,T)$, such that for every $k\in\mathbb{N}$
\begin{equation}
\begin{split}
\sup_{0\leq t\leq T}&\|\theta_{k,l}(t)\|^2_{L^2(\Omega)} +
\|\theta_{k,l}\|^2_{L^2(0,T,W^{1,2}(\Omega))} +
\|(\theta_{k,l})_t\|^2_{L^2(0,T,W^{-1,2}(\Omega))}
\\
& \leq C\Big(\|\mathcal{T}_k\Big( (\ten{T}_{k,l}^d + \tilde{\ten{T}}^d ):\ten{G}(\theta_{k,l}+\tilde{\theta},\ten{T}_{k,l}^d+\tilde{\ten{T}}^d) \Big)\|^2_{L^2(0,T,L^2(\Omega))}+
\|\mathcal{T}_k(\theta_0)\|_{L^2(\Omega)}^2\Big).
\label{numer}
\end{split}
\end{equation}
\label{lm:7}
\end{lemat}
The proof follows from the standard tools for parabolic equations, see e.g. Evans \cite{Evans}.
\begin{uwaga}
The uniform boundedness of solutions (Lemma \ref{pom_2} and Lemma \ref{lm:7}) implies the global existence of approximate solutions, i.e. existence of solutions $\{\beta_{k,l}^m(t),\gamma_{k,l}^n(t),\delta_{k,l}^m(t)\}$ on the whole time interval $[0,T]$ for each $n=1,...,k$ and $m=1,...,l$. Moreover, there exist global solutions $\{\alpha_{k,l}^n(t)\}$ for all $n=1,...,k$.
\end{uwaga}
\subsection{Limit passage $l\to\infty$ and uniform estimates
}
\label{sec:7}
Before we pass to the limit let us multiply the system \eqref{app_system} by smooth time-dependent functions, integrate over $[0,T]$ and then
rewrite the system as follows
\begin{equation}\label{58}
\begin{split}
\int_0^T\int_{\Omega}\ten{T}_{k,l}:\nabla\vc{w}_n \varphi_1(t) \,{\rm{d}}x\,{\rm{d}}t
&= 0, \quad n=1,\dots, k
\end{split}
\end{equation}
\begin{equation}\label{59}\begin{split}
\int_0^T\int_{\Omega}(\ten{\varepsilon}^{\bf p}_{k,l})_t &: \ten{D}\ten{\varepsilon}(\vc{w}_n) \varphi_2(t)\,{\rm{d}}x\,{\rm{d}}t \\&=
\int_0^T\int_{\Omega}\ten{G}(\theta_{k,l} + \tilde{\theta}, \ten{T}^d_{k,l} + \tilde{\ten{T}}^d ) : \ten{D}\ten{\varepsilon}(\vc{w}_n)
\varphi_2(t)\,{\rm{d}}x\,{\rm{d}}t,
\quad n=1,...,k ,
\\
\int_0^T\int_{\Omega}(\ten{\varepsilon}^{\bf p}_{k,l})_t &: \ten{D}\ten{\zeta}^k_m \varphi_3(t)\,{\rm{d}}x\,{\rm{d}}t \\&=
\int_0^T\int_{\Omega}\ten{G}(\theta_{k,l} + \tilde{\theta}, \ten{T}^d_{k,l} + \tilde{\ten{T}}^d ) : \ten{D}\ten{\zeta}^k_m \varphi_3(t)\,{\rm{d}}x\,{\rm{d}}t,
\quad m=1,...,l ,
\end{split}\end{equation}
and for $m=1,\ldots,l$
\begin{equation}\label{60}
\begin{split}
&-\int_0^T\int_{\Omega} \theta_{k,l}\varphi_4'(t) v_m \,{\rm{d}}x\,{\rm{d}}t -
\int_{\Omega} \theta_0(x)\varphi_4(0) v_m \,{\rm{d}}x +
\int_0^T\int_{\Omega} \nabla\theta_{k,l}\cdot \varphi_4(t)\nabla v_m \,{\rm{d}}x\,{\rm{d}}t\\ & =
\int_0^T\int_{\Omega} \mathcal{T}_k\left((\ten{T}^d_{k,l}+\tilde{\ten{T}}^d):\ten{G}(\theta_{k,l}+\tilde{\theta},\ten{T}^d_{k,l}+\tilde{\ten{T}}^d)\right)\varphi_4(t)v_m \,{\rm{d}}x\,{\rm{d}}t,
\end{split}
\end{equation}
holds for every test functions ${\varphi}_1, {\varphi}_2, {\varphi}_3\in C^{\infty}([0,T])$ and $\varphi_4\in C^{\infty}_c([-\infty,T))$.
Firstly, we pass to the limit with $l \rightarrow \infty$ - the Galerkin approximation of temperature. From the previous section we get uniform boundedness with respect to $l$ for appropriate sequences. Then at least for a subsequence, but still denoted by the index $l$, we get the following convergences
\begin{equation}
\begin{array}{cl}
\ten{T}_{k,l}\rightharpoonup \ten{T}_k & \mbox{weakly in } L^2(0,T,L^2(\Omega,\mathcal{S}^3)),\\
\ten{T}^d_{k,l}\rightharpoonup \ten{T}^d_k & \mbox{weakly in } L^p(0,T,L^p(\Omega,\mathcal{S}^3_d)),\\
\ten{G}(\tilde{\theta} + \theta_{k,l},\tilde{\ten{T}}^d + \ten{T}_{k,l}^d)\rightharpoonup \ten{\chi}_k & \mbox{weakly in } L^{p'}(0,T,L^{p'}(\Omega,\mathcal{S}^3_d)), \\
\theta_{k,l}\rightharpoonup \theta_k & \mbox{weakly in } L^2(0,T,W^{1,2}(\Omega)),\\
\theta_{k,l}\rightarrow \theta_k & \mbox{a.e. in } \Omega \times (0,T),\\
(\ten{\varepsilon}^{\bf p}_{k,l})_t\rightharpoonup(\ten{\varepsilon}^{\bf p}_{k})_t& \mbox{weakly in }
L^{p'}(0,T,(H^s(\Omega,\mathcal{S}^3))').
\end{array}
\end{equation}
Passing now to the limit in \eqref{58}-\eqref{59} yields
\begin{equation}\label{limit1}
\begin{split}
\int_0^T\int_{\Omega}\ten{T}_{k}:\nabla\vc{w}_n \varphi_1(t) \,{\rm{d}}x\,{\rm{d}}t
&= 0, \quad n=1,\dots, k
\end{split}
\end{equation}
\begin{equation}\label{limit2}\begin{split}
\int_0^T\int_{\Omega}(\ten{\varepsilon}^{\bf p}_{k})_t : \ten{D}\ten{\varepsilon}(\vc{w}_n) \varphi_2(t)\,{\rm{d}}x\,{\rm{d}}t =
\int_0^T\int_{\Omega}\ten{\chi}_{k}: \ten{D}\ten{\varepsilon}(\vc{w}_n)
\varphi_2(t)\,{\rm{d}}x\,{\rm{d}}t,
\quad n=1,...,k ,
\\
\int_0^T\int_{\Omega}(\ten{\varepsilon}^{\bf p}_{k})_t : \ten{D}\ten{\zeta}^k_m \varphi_3(t)\,{\rm{d}}x\,{\rm{d}}t =
\int_0^T\int_{\Omega}\ten{\chi}_{k} : \ten{D}\ten{\zeta}^k_m \varphi_3(t)\,{\rm{d}}x\,{\rm{d}}t,
\quad m\in\mathbb{N},
\end{split}\end{equation}
holds for every test functions ${\varphi}_1, {\varphi}_2, {\varphi}_3\in C^{\infty}([0,T])$.
By the density of $\mbox{lin}\{\ten{\zeta}^k_m\}_{m=1}^\infty$ in $L^{p}(\Omega,\mathcal{S}^3)$ we conclude that
\begin{equation}\label{65}
\int_0^T\int_{\Omega}(\ten{\varepsilon}^{\bf p}_{k})_t : \ten{\varphi}\,{\rm{d}}x\,{\rm{d}}t =
\int_0^T\int_{\Omega}\ten{\chi}_{k} : \ten{\varphi}\,{\rm{d}}x\,{\rm{d}}t
\end{equation}
holds for all $ \ten{\varphi}\in C^\infty([0,T],L^{p}(\Omega,\mathcal{S}^3))$ and then also for all
$ \ten{\varphi}\in L^{p}(0,T;L^{p}(\Omega,\mathcal{S}^3))$.
In the rest of this section we identify the weak limit of the nonlinear term $\ten{\chi}_k$ and then show the convergence of
$$\int_0^T\int_\Omega\mathcal{T}_k\left(\ten{G}(\tilde{\theta} + \theta_{k,l},\tilde{\ten{T}}^d + \ten{T}_{k,l}^d):(\tilde{\ten{T}}^d+\ten{T}^d_{k,l})\right)\,{\rm{d}}x\,{\rm{d}}t$$ what shall allow to pass to the limit in \eqref{60}.
\begin{lemat}
The sequence $\{\ten{\varepsilon}^{\bf p}_{k}\}$ is uniformly bounded in $W^{1,p'}(0,T,L^{p'}(\Omega,\mathcal{S}^3))$ with respect to $k$.
\label{wsp_org_ep}
\end{lemat}
\begin{proof}
By Assumption 1b
and the fact that the constant $C$ is independent of temperature, we get
\begin{equation}
\ten{\varepsilon}^{\bf p}_{k}(x,t) = \ten{\varepsilon}^{\bf p}_{k}(x,0) + \int_0^t(\ten{\varepsilon}^{\bf p}_{k}(x,s))_s \,{\rm{d}}s .
\nonumber
\end{equation}
Hence
\begin{equation}
|\ten{\varepsilon}^{\bf p}_{k}|^{p'}(x,t) \leq c|\ten{\varepsilon}^{\bf p}_{k}|^{p'}(x,0) + ct^{1/p} \int_0^t|(\ten{\varepsilon}^{\bf p}_{k})_s|^{p'}(x,s) \,{\rm{d}}s
\nonumber
\end{equation}
and consequently
\begin{equation}
\begin{split}
\int_0^T\int_{\Omega}|\ten{\varepsilon}^{\bf p}_{k}|^{p'}(x,t) \,{\rm{d}}x\,{\rm{d}}t &\leq
c\int_0^T\int_{\Omega}|\ten{\varepsilon}^{\bf p}_{k}|^{p'}(x,0) \,{\rm{d}}x\,{\rm{d}}t + ct^{1/p} \int_0^T\int_{\Omega}\int_0^t|(\ten{\varepsilon}^{\bf p}_{k})_s|^{p'}(x,s)\,{\rm{d}}s \,{\rm{d}}x\,{\rm{d}}t
\\
&\leq C(T) (1 + \int_{\Omega}\int_0^T|\ten{G}(\theta_{k}+\tilde{\theta},\ten{T}_{k}^d +\tilde{\ten{T}}^d )|^{p'}) \,{\rm{d}}s \,{\rm{d}}x
\\
&\leq C(T)(1 + \int_0^T\int_{\Omega}|\ten{T}_{k}^d+\tilde{\ten{T}}^d|^p) \,{\rm{d}}x\,{\rm{d}}t .
\nonumber
\end{split}
\end{equation}
It follows from the previous lemma that the right hand side is uniformly bounded.
\end{proof}
\begin{lemat}
The sequence $\{\vc{u}_{k}\}$ is uniformly bounded in $L^{p'}(0,T,W^{1,p'}_0(\Omega,\mathbb{R}^3))$ with respect to $k$.
\label{wsp_org_u}
\end{lemat}
\begin{proof}
In view of Lemma \ref{pom_2} the sequence $\{\ten{T}_{k}\}$ is uniformly bounded in $L^2(0,T,L^2(\Omega,\mathcal{S}^3))$. Using the triangle inequality and boundedness of the operator $\ten{D}$ we obtain
\begin{equation}
|\ten{\varepsilon}(\vc{u}_{k})|^{p'}\leq c|\ten{\varepsilon}(\vc{u}_{k}) - \ten{\varepsilon}^{\bf p}_{k}|^{p'} + c|\ten{\varepsilon}^{\bf p}_{k}|^{p'}
\leq c|\ten{T}_{k}|^{p'} + c|\ten{\varepsilon}^{\bf p}_{k}|^{p'}.
\end{equation}
Integrating over $\Omega\times (0,T)$ and using that $1<p'\leq 2$ we get
\begin{equation}
\begin{split}
\int_0^T\int_{\Omega} |\ten{\varepsilon}(\vc{u}_{k})|^{p'} \,{\rm{d}}x\,{\rm{d}}t
& \leq c \int_0^T\int_{\Omega} |\ten{T}_{k}|^{p'} \,{\rm{d}}x\,{\rm{d}}t
+ c\int_0^T\int_{\Omega} |\ten{\varepsilon}^{\bf p}_{k}|^{p'} \,{\rm{d}}x\,{\rm{d}}t
\\
& \leq c \int_0^T\int_{\Omega} |\ten{T}_{k}|^{2} \,{\rm{d}}x\,{\rm{d}}t
+ c\int_0^T\int_{\Omega} |\ten{\varepsilon}^{\bf p}_{k}|^{p'} \,{\rm{d}}x\,{\rm{d}}t
\\
& \leq c \|\ten{T}_{k}\|^2_{L^2(0,T,L^2(\Omega))}
+ c\|\ten{\varepsilon}^{\bf p}_{k}\|^{p'}_{L^{p'}(0,T,L^{p'}(\Omega))} .
\end{split}
\end{equation}
The tensor $\ten{\varepsilon}(\vc{u}_{k})$ is the symmetric gradient of the displacement, thus using the Korn inequality
(cf.~\cite[Theorem 1.10]{maleknecas}) we conclude that the sequence $\{\vc{u}_{k}\}$ is uniformly bounded in $L^{p'}(0,T,W^{1,p'}_0(\Omega,\mathbb{R}^3))$.
\end{proof}
\begin{lemat}
The following inequality holds for the solution of approximate system
\begin{equation}
\limsup_{l\rightarrow\infty}\int_{0}^{t}\int_{\Omega}\ten{G}(\tilde{\theta} + \theta_{k,l},\tilde{\ten{T}}^d + \ten{T}^d_{k,l}):\ten{T}^d_{k,l} \,{\rm{d}}x\,{\rm{d}}t \leq
\int_{0}^{t}\int_{\Omega}\ten{\chi}_k:\ten{T}^d_k \,{\rm{d}}x\,{\rm{d}}t .
\label{teza-8}
\end{equation}
\label{lm:8}
\end{lemat}
\begin{proof}
For each $\mu>0, t_2\le T-\mu, s\ge0, $ let $\psi_\mu:\mathbb{R}_+\to\mathbb{R}_+$ be defined as follows
\begin{equation}\label{psi-mu}
\psi_{\mu,t_2}(s)=\left\{
\begin{array}{lcl}
1&{\rm for}&s\in[0,t_2),\\
-\frac{1}{\mu}s+\frac{1}{\mu}t_2+1&{\rm for}&s\in[t_2, t_2+\mu),\\
0&{\rm for}&s\ge t_2+\mu.
\end{array}\right.
\end{equation}
Next we shall use \eqref{ene} and multiply it by $\psi_{\mu,t_2}(t)$ and integrate over $(0,T)$
\begin{equation}\label{mu2}
\begin{split}
\int_{0}^{T}
\frac{d}{d\tau} \mathcal{E}(\ten{\varepsilon}(\vc{u}_{k,l}) , \ten{\varepsilon}^{\bf p}_{k,l}) \,\psi_{\mu,t_2}\,{\rm{d}}t
\
=
-
\int_0^T\int_{\Omega}\ten{G}(\tilde{\theta} + \theta_{k,l},\tilde{\ten{T}}^d + \ten{T}^d_{k,l}):\ten{T}^d_{k,l} \,\psi_{\mu,t_2}\,{\rm{d}}x\,{\rm{d}}t.
\end{split}
\end{equation}
Let us now integrate by parts the left hand side of \eqref{mu2}
\begin{equation}\label{mu3}
\begin{split}
\int_{0}^{T}
\frac{d}{d\tau} \mathcal{E}(\ten{\varepsilon}(\vc{u}_{k,l}) , \ten{\varepsilon}^{\bf p}_{k,l}) \,\psi_{\mu,t_2}\,{\rm{d}}t
=\frac{1}{\mu}\int_{t_2}^{t_2+\mu}\mathcal{E}(\ten{\varepsilon}(\vc{u}_{k,l}(t)) , \ten{\varepsilon}^{\bf p}_{k,l}(t)) \,{\rm{d}}t-
\mathcal{E}(\ten{\varepsilon}(\vc{u}_{k,l}(0)) , \ten{\varepsilon}^{\bf p}_{k,l}(0)).
\end{split}\end{equation}
Passing to the limit in \eqref{mu3} with $l\to\infty$ we obtain
\begin{equation}\label{mu4}
\begin{split}
\liminf\limits_{l\to\infty}\int_{0}^{T}
\frac{d}{d\tau}& \mathcal{E}(\ten{\varepsilon}(\vc{u}_{k,l}) , \ten{\varepsilon}^{\bf p}_{k,l}) \,\psi_{\mu,t_2}\,{\rm{d}}t\\
&=\liminf\limits_{l\to\infty}\frac{1}{\mu}\int_{t_2}^{t_2+\mu}\mathcal{E}(\ten{\varepsilon}(\vc{u}_{k,l}) , \ten{\varepsilon}^{\bf p}_{k,l}) \,{\rm{d}}t-
\lim\limits_{l\to\infty}\mathcal{E}(\ten{\varepsilon}(\vc{u}_{k,l}(0)) , \ten{\varepsilon}^{\bf p}_{k,l}(0))\\
&\ge \frac{1}{\mu}\int_{t_2}^{t_2+\mu}\mathcal{E}(\ten{\varepsilon}(\vc{u}_{k}(t)) , \ten{\varepsilon}^{\bf p}_{k}(t)) \,{\rm{d}}t-
\mathcal{E}(\ten{\varepsilon}(\vc{u}_{k}(0)) , \ten{\varepsilon}^{\bf p}_{k}(0))
\end{split}\end{equation}
Note that the last inequality holds due to the weak lower semicontinuity in
$L^2(0,T,L^2(\Omega;\mathcal{S}^3))$.
To complete the proof we choose in \eqref{limit1} the test functions
$\varphi_1(t)=((\alpha^n_k)_t*\eta_{\epsilon}\mathbf{1}_{(t_1,t_2)})*\eta_{\epsilon}$,
and in \eqref{65} $\ten{\varphi}=(\ten{T}_k^d*\eta_{\epsilon}\mathbf{1}_{(t_1,t_2)})*\eta_{\epsilon}$,
where $\eta_\epsilon$ is a standard mollifier and we mollify with respect to time. Thus we obtain
\begin{equation}
\begin{split}
\int_0^T\int_{\Omega} \ten{T}_{k} : \ten{\varepsilon}(((\alpha^n_k)_t*\eta_{\epsilon}\mathbf{1}_{(t_1,t_2)})*\eta_{\epsilon}\vc{w}_n) \,{\rm{d}}x &= 0,
\\
\int_0^T\int_{\Omega}(\ten{\varepsilon}^{\bf p}_{k})_t : (\ten{T}_k^d*\eta_{\epsilon}\mathbf{1}_{(t_1,t_2)})*\eta_{\epsilon} \,{\rm{d}}x
=
\int_0^T\int_{\Omega}\ten{\chi}_{k} : &(\ten{T}_k^d*\eta_{\epsilon}\mathbf{1}_{(t_1,t_2)})*\eta_{\epsilon}\,{\rm{d}}x ,
\end{split}
\label{app_system_n}
\end{equation}
for $n=1,...,k $. Summing \eqref{app_system_n}$_{(1)}$ over $n=1,...,k$ we obtain
\begin{equation}
\int_{t_1}^{t_2} \int_{\Omega}\ten{D}\left(\ten{\varepsilon}(\vc{u}_{k}) - \ten{\varepsilon}^{\bf p}_{k}\right)*\eta_{\epsilon}: (\ten{\varepsilon}(\vc{u}_{k})*\eta_{\epsilon})_t \,{\rm{d}}x\,{\rm{d}}t =
0.
\label{pierwsze_r1}
\end{equation}
and
\begin{equation}
\int_{t_1}^{t_2}\int_{\Omega}(\ten{\varepsilon}^{\bf p}_{k}*\eta_{\epsilon})_t:\ten{T}_{k}*\eta_{\epsilon} \,{\rm{d}}x\,{\rm{d}}t =
\int_{t_1}^{t_2}\int_{\Omega}\ten{\chi}_{k}*\eta_{\epsilon}:\ten{T}_{k}*\eta_{\epsilon} \,{\rm{d}}x\,{\rm{d}}t .
\label{drugie_r2}
\end{equation}
Products in \eqref{drugie_r2} are well defined, since for the matrices $\ten{A}\in\mathcal{S}^3_d$ and $\ten{B}\in\mathcal{S}^3$ the equivalence $\ten{A}:\ten{B}^d=\ten{A}:\ten{B}$ holds and the sequence $\{\ten{T}^d_{k}\}$ is uniformly bounded in $L^{p'}(0,T,L^{p'}(\Omega,\mathcal{S}^3_d))$.
\noindent
Passing with $\epsilon\to0$
we obtain the equality
\begin{equation}
\frac{1}{2}\int_{\Omega}\ten{D}(\ten{\varepsilon}(\vc{u}_k) - \ten{\varepsilon}^{\bf p}_k):(\ten{\varepsilon}(\vc{u}_k) - \ten{\varepsilon}^{\bf p}_k) \,{\rm{d}}x \Big|_{t_1}^{t_2}
= -
\int_{t_1}^{t_2}\int_{\Omega}\ten{\chi}_k:\ten{T}^d_k \,{\rm{d}}x\,{\rm{d}}t .
\label{granica_l}
\end{equation}
Since $\ten\varepsilon({\vc{u}_k}), \ten{\varepsilon}^{\bf p}_k\in C_{w}([0,T],L^2(\Omega,\mathcal{S}^3))$, then we may pass
with $t_1\to 0$ and conclude
\begin{equation}\label{gran}
\mathcal{E}(\ten{\varepsilon}(\vc{u}_{k}(t_2)) , \ten{\varepsilon}^{\bf p}_{k}(t_2)) -
\mathcal{E}(\ten{\varepsilon}(\vc{u}_{k}(0)) , \ten{\varepsilon}^{\bf p}_{k}(0))=-
\int_{0}^{t_2}\int_{\Omega}\ten{\chi}_k:\ten{T}^d_k \,{\rm{d}}x\,{\rm{d}}t .
\end{equation}
Multiplying \eqref{gran} by $\frac{1}{\mu}$ and integrating over the interval $(t_2, t_2+\mu)$ we get
\begin{equation}
\begin{split}
\frac{1}{\mu}\int_{t_2}^{t_2+\mu}\mathcal{E}(\ten{\varepsilon}(\vc{u}_{k}(t)) , \ten{\varepsilon}^{\bf p}_{k}(t)) \,{\rm{d}}t-
\mathcal{E}(\ten{\varepsilon}(\vc{u}_{k}(0)) , \ten{\varepsilon}^{\bf p}_{k}(0))=-
\frac{1}{\mu}\int_{t_2}^{t_2+\mu}\int_{0}^{\tau}\int_{\Omega}\ten{\chi}_k:\ten{T}^d_k \,{\rm{d}}x\,{\rm{d}}t \,{\rm{d}}\tau.
\end{split}\end{equation}
For brevity we denote $$F(s):=\int_{\Omega}\ten{\chi}_k:\ten{T}^d_k\,{\rm{d}}x$$ which is obviously in $L^1(0,T)$. Then we may apply the Fubini theorem
\begin{equation}\label{71}\begin{split}
\frac{1}{\mu}\int_{t_2}^{t_2+\mu}\int_0^\tau F(s) \,{\rm{d}}s\,{\rm{d}}\tau&=\frac{1}{\mu}\int_{\mathbb{R}^2}
\mathbf{1}_{\{0\le s\le \tau\}}(s)\mathbf{1}_{\{t_2\le \tau\le t_2+\mu\}}(\tau) F(s)\,{\rm{d}}s\,{\rm{d}}\tau\\
&=\frac{1}{\mu}\int_\mathbb{R} \left(\int_\mathbb{R}
\mathbf{1}_{\{0\le s\le\tau\}}(s)\mathbf{1}_{\{t_2\le \tau\le t_2+\mu\}} (\tau)\,{\rm{d}}\tau\right)F(s)\,{\rm{d}}s.
\end{split}
\end{equation}
The crucial observation is that
\begin{equation}\label{psi}
\psi_{\mu,t_2}(s)=\frac{1}{\mu}\int_\mathbb{R}
\mathbf{1}_{\{0\le t\le\tau\}}(t)\mathbf{1}_{\{t_2\le \tau\le t_2+\mu\}} (\tau)\,{\rm{d}}\tau.
\end{equation}
Hence using \eqref{mu2} and \eqref{mu4} we conclude
\begin{equation}
-\int_{0}^{T}\int_{\Omega}\ten{\chi}_k:\ten{T}^d_k \, \psi_{\mu,t_2}\,{\rm{d}}x\,{\rm{d}}t \le
\liminf\limits_{l\to\infty}\left(
-
\int_0^T\int_{\Omega}\ten{G}(\tilde{\theta} + \theta_{k,l},\tilde{\ten{T}}^d + \ten{T}^d_{k,l}):\ten{T}^d_{k,l} \,\psi_{\mu,t_2}\,{\rm{d}}x\,{\rm{d}}t \right)
\end{equation}
which is nothing else than
\begin{equation}
\limsup_{l\rightarrow\infty}\int_{0}^{T}\int_{\Omega}\ten{G}(\tilde{\theta} + \theta_{k,l},\tilde{\ten{T}}^d + \ten{T}^d_{k,l}):\ten{T}^d_{k,l} \ \psi_{\mu,t_2}\,{\rm{d}}x\,{\rm{d}}t \leq
\int_{0}^{T}\int_{\Omega}\ten{\chi}_k:\ten{T}^d_k \ \psi_{\mu,t_2}\,{\rm{d}}x\,{\rm{d}}t .
\label{jedna_nierownosc}
\end{equation}
Observe now that
\begin{equation}
\begin{split}
\limsup\limits_{l\to\infty}&\int_{0}^{t_2}\int_{\Omega}\ten{G}(\tilde{\theta} + \theta_{k,l},\tilde{\ten{T}}^d + \ten{T}^d_{k,l}):\ten{T}^d_{k,l} \,{\rm{d}}x\,{\rm{d}}t\\
&\le
\limsup\limits_{l\to\infty}\int_{0}^{t_2}\int_{\Omega}\ten{G}(\tilde{\theta} + \theta_{k,l},\tilde{\ten{T}}^d + \ten{T}^d_{k,l}):(\tilde{\ten{T}}^d + \ten{T}^d_{k,l}) \,{\rm{d}}x\,{\rm{d}}t\\
&-\lim\limits_{l\to\infty}\int_{0}^{t_2}\int_{\Omega}\ten{G}(\tilde{\theta} + \theta_{k,l},\tilde{\ten{T}}^d + \ten{T}^d_{k,l}):\tilde{\ten{T}}^d \,{\rm{d}}x\,{\rm{d}}t\\
&\le
\limsup\limits_{l\to\infty}\int_{0}^{t_2+\mu}\int_{\Omega}\ten{G}(\tilde{\theta} + \theta_{k,l},\tilde{\ten{T}}^d + \ten{T}^d_{k,l}):(\tilde{\ten{T}}^d + \ten{T}^d_{k,l})\psi_{\mu,t_2} \,{\rm{d}}x\,{\rm{d}}t\\
&-\lim\limits_{l\to\infty}\int_{0}^{t_2}\int_{\Omega}\ten{G}(\tilde{\theta} + \theta_{k,l},\tilde{\ten{T}}^d + \ten{T}^d_{k,l}):\tilde{\ten{T}}^d \,{\rm{d}}x\,{\rm{d}}t\\
&\le
\limsup_{l\rightarrow\infty}\int_{0}^{t_2+\mu}\int_{\Omega}\ten{G}(\tilde{\theta} + \theta_{k,l},\tilde{\ten{T}}^d + \ten{T}^d_{k,l}):\ten{T}^d_{k,l} \ \psi_{\mu,t_2}\,{\rm{d}}x\,{\rm{d}}t\\
&+\lim\limits_{l\to\infty}\int_{0}^{t_2+\mu}\int_{\Omega}\ten{G}(\tilde{\theta} + \theta_{k,l},\tilde{\ten{T}}^d + \ten{T}^d_{k,l}):\tilde{\ten{T}}^d \ \psi_{\mu,t_2}\,{\rm{d}}x\,{\rm{d}}t\\
&-\lim\limits_{l\to\infty}\int_{0}^{t_2}\int_{\Omega}\ten{G}(\tilde{\theta} + \theta_{k,l},\tilde{\ten{T}}^d + \ten{T}^d_{k,l}):\tilde{\ten{T}}^d \,{\rm{d}}x\,{\rm{d}}t\\
& \leq
\int_{0}^{t_2+\mu}\int_{\Omega}\ten{\chi}_k:\ten{T}^d_k \ \psi_{\mu,t_2}\,{\rm{d}}x\,{\rm{d}}t\\
&+\lim\limits_{l\to\infty}
\int_{t_2}^{t_2+\mu}\int_{\Omega}\ten{G}(\tilde{\theta} + \theta_{k,l},\tilde{\ten{T}}^d + \ten{T}^d_{k,l}):\tilde{\ten{T}}^d \psi_{\mu,t_2}\,{\rm{d}}x\,{\rm{d}}t\\
&=\int_{0}^{t_2+\mu}\int_{\Omega}\ten{\chi}_k:\ten{T}^d_k \ \psi_{\mu,t_2}\,{\rm{d}}x\,{\rm{d}}t +\lim\limits_{l\to\infty}\int_{t_2}^{t_2+\mu}\int_{\Omega}
\ten{G}(\tilde{\theta} + \theta_{k,l},\tilde{\ten{T}}^d + \ten{T}^d_{k,l}):\tilde{\ten{T}}^d_k\,{\rm{d}}x\,{\rm{d}}t
\end{split}\end{equation}
Passing with $\mu\to0$ yields \eqref{teza-8}. The proof is complete.
\end{proof}
To identify the weak limit~$\ten{\chi}_k$ we use the Minty-Browder trick. From the monotonicity of the function $\ten{G}(\cdot,\cdot)$ we obtain
\begin{equation}
\begin{split}
\int_{\Omega}\left(\ten{G}(\tilde{\theta} + \theta_{k,l},\tilde{\ten{T}}^d + \ten{T}^d_{k,l}) - \ten{G}(\tilde{\theta} + \theta_{k,l},\tilde{\ten{T}}^d + \ten{W}^d)\right) : &(\ten{T}_{k,l}^d - \ten{W}^d) \,{\rm{d}}x\geq 0
\\
&
\forall \ \ten{W}^d\in L^p(0,T,L^p(\Omega,\mathcal{S}^3)).
\label{eq:zal_G1}
\end{split}
\end{equation}
Hence
\begin{equation}
\begin{split}
\int_0^T\int_{\Omega} \ten{G}(\tilde{\theta} + \theta_{k,l},\tilde{\ten{T}}^d + \ten{T}^d_{k,l}):\ten{T}_{k,l}^d \,{\rm{d}}x\,{\rm{d}}t
- &\int_0^T\int_{\Omega} \ten{G}(\tilde{\theta} + \theta_{k,l},\tilde{\ten{T}}^d + \ten{T}^d_{k,l}):\ten{W}^d \,{\rm{d}}x\,{\rm{d}}t
\\
- \int_0^T\int_{\Omega} \ten{G}(\tilde{\theta} + \theta_{k,l},\tilde{\ten{T}}^d + \ten{W}^d) :\ten{T}_{k,l}^d \,{\rm{d}}x\,{\rm{d}}t
+ &\int_0^T\int_{\Omega} \ten{G}(\tilde{\theta} + \theta_{k,l},\tilde{\ten{T}}^d + \ten{W}^d) : \ten{W}^d \,{\rm{d}}x\,{\rm{d}}t\geq 0 .
\end{split}
\label{eq:44}
\end{equation}
The pointwise convergence of $\{\theta_{k,l}\}$ implies the pointwise convergence of $\{\ten{G}(\tilde{\theta} + \theta_{k,l},\tilde{\ten{T}}^d + \ten{W}^d)\}$. The function $|\tilde{\ten{T}}^d + \ten{W}^d|^{p-1}$ belongs to $L^{p'}(0,T,L^{p'}(\Omega))$, hence the sequence $\{\ten{G}(\tilde{\theta} + \theta_{k,l},\tilde{\ten{T}}^d +\ten{W}^d)\}$ is uniformly bounded in $L^{p'}(0,T,L^{p'}(\Omega,\mathcal{S}^3))$. Then, using the Lebesgue dominated convergence theorem
we obtain that $\ten{G}(\tilde{\theta} + \theta_{k,l},\tilde{\ten{T}}^d + \ten{W}^d)\rightarrow \ten{G}(\tilde{\theta} + \theta_k,\tilde{\ten{T}}^d +\ten{W}^d)$ in $L^{p'}(0,T, L^{p'}(\Omega,\mathcal{S}^3))$ for every $\ten{W}^d\in L^p(0,T,L^p(\Omega,\mathcal{S}^3))$.
Letting $l\to\infty$ in \eqref{eq:44}, we get
\begin{equation}
\int_0^T \int_{\Omega} \left(\ten{\chi}_k -\ten{G}(\tilde{\theta} + \theta_{k},\tilde{\ten{T}}^d + \ten{W}^d)\right):( \ten{T}_k^d - \ten{W}^d) \,{\rm{d}}x\,{\rm{d}}t \geq 0
\qquad \forall \ \ten{W}^d\in L^p(0,T,L^p(\Omega,\mathcal{S}^3)),
\end{equation}
and taking $\ten{W}^d = \ten{T}^d_k - \lambda \ten{U}^d$, where $\ten{U}^d\in L^p(0,T,L^p(\Omega,\mathcal{S}^3))$ and $\lambda>0$, then
\begin{equation}
\begin{split}
\int_0^T \int_{\Omega}\left( \ten{\chi}_k -\ten{G}(\tilde{\theta} + \theta_{k},\tilde{\ten{T}}^d + \ten{T}_k^d - \lambda \ten{U}^d)\right):( \lambda \ten{U}^d) \,{\rm{d}}x\,{\rm{d}}t \geq 0 \quad \forall \ \ten{U}^d\in L^p(0,T,L^p(\Omega,\mathcal{S}^3))
\end{split}
\end{equation}
hence
\begin{equation}
\begin{split}
\int_0^T \int_{\Omega} \left(\ten{\chi}_k -\ten{G}(\tilde{\theta} + \theta_{k},\tilde{\ten{T}}^d + \ten{T}_k^d - \lambda \ten{U}^d)\right): \ten{U}^d \,{\rm{d}}x\,{\rm{d}}t \geq 0 \quad \forall \ \ten{U}^d\in L^p(0,T,L^p(\Omega,\mathcal{S}^3)).
\end{split}
\end{equation}
Letting $\lambda\to0$ we obtain
\begin{equation}
\int_0^T \int_{\Omega} \left(\ten{\chi}_k -\ten{G}(\tilde{\theta} + \theta_{k},\tilde{\ten{T}}^d + \ten{T}_k^d )\right): \ten{U}^d \,{\rm{d}}x\,{\rm{d}}t \geq 0 \qquad \forall\ \ten{U}^d\in L^p(0,T,L^p(\Omega,\mathcal{S}^3)).
\end{equation}
Choosing now $\lambda<0$ we obtain the opposite inequality and hence
\begin{equation}
\int_0^T \int_{\Omega} \left(\ten{\chi}_k -\ten{G}(\tilde{\theta} + \theta_{k},\tilde{\ten{T}}^d + \ten{T}_k^d )\right): \ten{U}^d \,{\rm{d}}x\,{\rm{d}}t = 0 \qquad \forall\ \ten{U}^d\in L^p(0,T,L^p(\Omega,\mathcal{S}^3)).
\end{equation}
Thus
\begin{equation}
\ten{\chi}_k =\ten{G}(\tilde{\theta} + \theta_{k},\tilde{\ten{T}}^d + \ten{T}_k^d )\quad \mbox{a.e. in}\ (0,T)\times\Omega.
\label{82}\end{equation}
Consequently for every $k\in\mathbb{N}$
$$\ten{G}(\tilde{\theta} + \theta_{k,l},\tilde{\ten{T}}^d + \ten{T}_{k,l}^d)\rightharpoonup \ten{G}(\tilde{\theta} + \theta_k,\tilde{\ten{T}}^d + \ten{T}_k^d)\quad \mbox{in} \ L^{p'}(0,T,L^{p'}(\Omega,\mathcal{S}^3))$$
as $l\rightarrow \infty$.
\begin{lemat}
For each $k\in\mathbb{N}$ it holds
\begin{equation}\begin{split}
\lim\limits_{l\to\infty}\int_0^T\int_\Omega&\ten{G}(\tilde{\theta} + \theta_{k,l},\tilde{\ten{T}}^d + \ten{T}_{k,l}^d):(\tilde{\ten{T}}^d + \ten{T}_{k,l}^d )\,{\rm{d}}x\,{\rm{d}}t\\&
= \int_0^T\int_\Omega\ten{G}(\tilde{\theta} + \theta_k,\tilde{\ten{T}}^d + \ten{T}_k^d):(\tilde{\ten{T}}^d + \ten{T}_k^d)\,{\rm{d}}x\,{\rm{d}}t.
\end{split}\end{equation}
\end{lemat}
\begin{proof}
Using monotonicity of the function $\ten{G}(\cdot,\cdot)$
\begin{equation}
\begin{split}
0 &\leq
\int_0^T\int_{\Omega}
\left(
\ten{G}(\tilde{\theta} + \theta_{k,l},\tilde{\ten{T}}^d + \ten{T}_{k,l}^d) - \ten{G}(\tilde{\theta} + \theta_{k,l},\tilde{\ten{T}}^d + \ten{T}_{k}^d)\right):(\ten{T}_{k,l}^d - \ten{T}_k^d) \,{\rm{d}}x\,{\rm{d}}t
\\
&=\int_0^T\int_{\Omega}
\ten{G}(\tilde{\theta} + \theta_{k,l},\tilde{\ten{T}}^d + \ten{T}_{k,l}^d):(\ten{T}_{k,l}^d - \ten{T}_k^d) - \ten{G}(\tilde{\theta} + \theta_{k,l},\tilde{\ten{T}}^d + \ten{T}_{k}^d):(\ten{T}_{k,l}^d - \ten{T}_k^d) \,{\rm{d}}x\,{\rm{d}}t .
\end{split}
\label{eq:50}
\end{equation}
Passing with $l$ to $\infty$ we get that the second term from \eqref{eq:50} converges to zero. Furthermore, using Lemma \ref{lm:8}
\begin{equation}
\begin{split}
0 &\leq
\limsup_{l\rightarrow \infty}
\int_0^T\int_{\Omega}
\ten{G}(\tilde{\theta} + \theta_{k,l},\tilde{\ten{T}}^d + \ten{T}_{k,l}^d):(\ten{T}_{k,l}^d + \tilde{\ten{T}}^d - \tilde{\ten{T}}^d -\ten{T}_{k}^d) \,{\rm{d}}x\,{\rm{d}}t
\\
&=
\limsup_{l\rightarrow \infty}
\int_0^T\int_{\Omega}
\ten{G}(\tilde{\theta} + \theta_{k,l},\tilde{\ten{T}}^d + \ten{T}_{k,l}^d):(\tilde{\ten{T}}^d + \ten{T}_{k,l}^d) \,{\rm{d}}x\,{\rm{d}}t
\\
& \quad
- \lim_{l\rightarrow \infty}
\int_0^T\int_{\Omega}
\ten{G}(\tilde{\theta} + \theta_{k,l},\tilde{\ten{T}}^d + \ten{T}_{k,l}^d):(\tilde{\ten{T}}^d + \ten{T}_{k}^d) \,{\rm{d}}x\,{\rm{d}}t
\le 0 .
\end{split}
\end{equation}
Hence
\begin{equation}
0 = \lim_{l\rightarrow\infty}
\int_0^T\int_{\Omega}
\left(
\ten{G}(\tilde{\theta} + \theta_{k,l},\tilde{\ten{T}}^d + \ten{T}_{k,l}^d) - \ten{G}(\tilde{\theta} + \theta_{k,l},\tilde{\ten{T}}^d + \ten{T}_{k}^d)\right):(\ten{T}_{k,l}^d - \ten{T}_k^d) \,{\rm{d}}x\,{\rm{d}}t ,
\end{equation}
and
\begin{equation}
\begin{split}
\lim_{l\rightarrow \infty}
\int_0^T\int_{\Omega}
\ten{G}(\tilde{\theta} + \theta_{k,l},\tilde{\ten{T}}^d + \ten{T}_{k,l}^d):&(\tilde{\ten{T}}^d + \ten{T}_{k,l}^d) \,{\rm{d}}x\,{\rm{d}}t
\\&=
\int_0^T\int_{\Omega}
\ten{G}(\tilde{\theta} + \theta_{k},\tilde{\ten{T}}^d + \ten{T}_{k}^d):(\tilde{\ten{T}}^d + \ten{T}_{k}^d) \,{\rm{d}}x\,{\rm{d}}t ,
\end{split}
\nonumber
\end{equation}
which completes the proof.
\end{proof}
Hence now we can also pass to the limit in the heat equation, namely we obtain for all
$\phi\in C^\infty([0,T]\times\Omega)$
\begin{equation}
\begin{split}
-\int_0^T\int_{\Omega} \theta_k\phi_t \,{\rm{d}}x\,{\rm{d}}t -
\int_{\Omega} \theta_k(x,0)\phi(x,0) \,{\rm{d}}x +
\int_0^T\int_{\Omega} \nabla\theta_k\cdot\nabla\phi \,{\rm{d}}x\,{\rm{d}}t \\=
\int_0^T\int_{\Omega} \mathcal{T}_k\left((\ten{T}^d_{k}+\tilde{\ten{T}}^d):\ten{G}(\theta_{k}+\tilde{\theta},\ten{T}^d_{k}+\tilde{\ten{T}}^d)\right)\phi \,{\rm{d}}x\,{\rm{d}}t,
\\
\label{eq:after_limit_l_2}
\end{split}
\end{equation}
\subsection{Limit passage $k\to\infty$ }
\label{sec:8}
We start this section with considerations on the sequence of temperatures.
We are using the
result of Boccardo and Galllou\"{e}t \cite{Boccardo} for parabolic equation with only integrable data and Dirichlet boundary conditions.
Since our studies concern the problem with Neumann boundary conditions, we include the modification of their result in the Appendix A.
Consequently, we conclude for each $1<q<\frac{5}{4}$
\begin{equation}
\theta_k \rightharpoonup \theta \mbox{ weakly in } L^q(0,T,W^{1,q}(\Omega)). \\
\end{equation}
Moreover, the uniform estimates from the previous sections allow to conclude that at least for a subsequence the following holds
\begin{equation}
\begin{array}{cl}
\theta_k \rightarrow \theta & \mbox{ a.e. in }\Omega\times(0,T),\\
\vc{u}_k \rightharpoonup \vc{u} & \mbox{ weakly in } L^{p'}(0,T,W^{1,p'}_0(\Omega,\mathbb{R}^3)),\\
\ten{T}_k \rightharpoonup \ten{T} & \mbox{ weakly in } L^2(0,T,L^2(\Omega,\mathcal{S}^3)),\\
\ten{T}^d_k \rightharpoonup \ten{T}^d & \mbox{ weakly in } L^p(0,T,L^p(\Omega,\mathcal{S}^3_d)),\\
\ten{G}(\tilde{\theta} + \theta_k,\tilde{\ten{T}}^d +\ten{T}_k^d) \rightharpoonup \ten{\chi} & \mbox{ weakly in } L^{p'}(0,T,L^{p'}(\Omega,\mathcal{S}^3_d)), \\
(\ten{\varepsilon}^{\bf p}_k)_t \rightharpoonup (\ten{\varepsilon}^{\bf p})_t & \mbox{ weakly in } L^{p'}(0,T,L^{p'}(\Omega,\mathcal{S}^3_d)).
\end{array}
\end{equation}
Consequently, passing to the limit in \eqref{limit1}, \eqref{65} we obtain
\begin{equation}\label{limit1a}
\begin{split}
\int_0^T\int_{\Omega}\ten{T}:\nabla\vc{\varphi} \,{\rm{d}}x\,{\rm{d}}t =0
\end{split}
\end{equation}
\begin{equation}\label{65a}
\int_0^T\int_{\Omega}(\ten{\varepsilon}^{\bf p})_t : \ten{\psi}\,{\rm{d}}x\,{\rm{d}}t =
\int_0^T\int_{\Omega}\ten{\chi} : \ten{\psi}\,{\rm{d}}x\,{\rm{d}}t
\end{equation}
for all $ \ten{\varphi}\in C^\infty([0,T],L^2(\Omega,\mathcal{S}^3))$ and then also for all
$ \ten{\varphi}\in L^2(0,T;L^2(\Omega,\mathcal{S}^3))$ and for all $\ten{\psi}\in L^p(0,T;L^p(\Omega,\mathcal{S}^3))$
To characterize the limit $\ten{\chi}$ and pass to the limit in the heat equation we follow the similar lines as in the limit passage with $l\to\infty$.
\begin{lemat}
The following inequality holds for the solution of approximate systems.
\begin{equation}
\limsup_{k\rightarrow\infty}\int_{0}^{t_2}\int_{\Omega}\ten{G}(\tilde{\theta} + \theta_{k},\tilde{\ten{T}}^d + \ten{T}^d_{k}):\ten{T}^d_k \,{\rm{d}}x\,{\rm{d}}t \leq
\int_{0}^{t_2}\int_{\Omega}\ten{\chi}:\ten{T}^d \,{\rm{d}}x\,{\rm{d}}t .
\label{jedna_nierownosc_1}
\end{equation}
\end{lemat}
\begin{proof}
Due to \eqref{82} we can rewrite \eqref{gran} as follows
\begin{equation}\label{do-l}
\frac{d}{dt} \mathcal{E}(\ten{\varepsilon}(\vc{u}_{k}) , \ten{\varepsilon}^{\bf p}_{k})
=
-
\int_{\Omega}\ten{G}(\tilde{\theta} + \theta_{k},\tilde{\ten{T}}^d + \ten{T}^d_{k}):\ten{T}^d_{k}\,{\rm{d}}x.
\end{equation}
We multiply the above identity by $\psi_{\mu,t_2}$ given by formula \eqref{psi-mu} and integrate over $(0,T)$.
Passing to the limit $k\to\infty$ we proceed in the same manner as in the proof of Lemma~\ref{lm:8} and obtain
\begin{equation}\label{mu4a}
\begin{split}
\liminf\limits_{k\to\infty}\int_{0}^{T}
\frac{d}{d\tau}& \mathcal{E}(\ten{\varepsilon}(\vc{u}_{k}) , \ten{\varepsilon}^{\bf p}_{k}) \,\psi_{\mu,t_2}\,{\rm{d}}t\\
&=\liminf\limits_{k\to\infty}\frac{1}{\mu}\int_{t_2}^{t_2+\mu}\mathcal{E}(\ten{\varepsilon}(\vc{u}_{k}) , \ten{\varepsilon}^{\bf p}_{k}) \,{\rm{d}}t-
\lim\limits_{k\to\infty}\mathcal{E}(\ten{\varepsilon}(\vc{u}_{k}(0)) , \ten{\varepsilon}^{\bf p}_{k}(0))\\
&\ge \frac{1}{\mu}\int_{t_2}^{t_2+\mu}\mathcal{E}(\ten{\varepsilon}(\vc{u}_{k}(t)) , \ten{\varepsilon}^{\bf p}_{k}(t)) \,{\rm{d}}t-
\mathcal{E}(\ten{\varepsilon}(\vc{u}(0)) , \ten{\varepsilon}^{\bf p}(0)).
\end{split}\end{equation}
For the final step of the proof of the lemma we need to show that the energy equality holds. Contrary to the case of previous section, we cannot use the time derivative of the limit, namely $\vc{\varepsilon}(\vc{u})_t$ as the test function. Although we shall mollifty with respect to time, but the regularity with respect to space is not sufficient since possibly $p'< 2$. Therefore we proceed differently. We use an approximate sequence as a test function in the limit identity. Indeed, we take in \eqref{limit1a}
the test function $\vc{\varphi}=(\ten{\varepsilon}(\vc{u}_k)*\eta_\epsilon)_t\mathbf{1}_{(t_1,t_2)})*\eta_{\epsilon}$,
where again $\eta_\epsilon$ is a standard mollifier and we mollify with respect to time
\begin{equation}
\int_{t_1}^{t_2} \int_{\Omega}\ten{D}(\ten{\varepsilon}(\vc{u}) - \ten{\varepsilon}^{\bf p})*\eta_{\epsilon}: (\ten{\varepsilon}(\vc{u}_k)*\eta_{\epsilon})_t \,{\rm{d}}x\,{\rm{d}}t =
0.
\label{pierwsze_r2}
\end{equation}
Then we use the approximate equation \eqref{65} with a test function $\ten{\psi}=(\ten{T}_k^d*\eta_{\epsilon}\mathbf{1}_{(t_1,t_2)})*\eta_{\epsilon}$. In a consequence we obtain \eqref{drugie_r2}, which together with \eqref{82} yields
\begin{equation}
\int_{t_1}^{t_2}\int_{\Omega}(\ten{\varepsilon}^{\bf p}_{k}*\eta_{\epsilon})_t:\ten{T}*\eta_{\epsilon} \,{\rm{d}}x\,{\rm{d}}t =
\int_{t_1}^{t_2}\int_{\Omega}\ten{G}(\tilde{\theta}+\theta_k,\tilde{\ten{T}}^d+\ten{T}^d_k)*\eta_{\epsilon}:\ten{T}*\eta_{\epsilon} \,{\rm{d}}x\,{\rm{d}}t .
\label{drugie_r2a}
\end{equation}
Products in \eqref{drugie_r2a} are well defined, since for the matrices $\ten{A}\in\mathcal{S}^3_d$ and $\ten{B}\in\mathcal{S}^3$ the equivalence $\ten{A}:\ten{B}^d=\ten{A}:\ten{B}$ holds and tensor $\ten{T}^d$ belongs to $L^{p'}(0,T,L^{p'}(\Omega,\mathcal{S}^3))$.
Subtracting \eqref{drugie_r2a} from \eqref{pierwsze_r2} we get
\begin{equation}
\int_{t_1}^{t_2}\int_{\Omega}\ten{T}*\eta_{\epsilon}:(\ten{\varepsilon}(\vc{u}_k) - \ten{\varepsilon}^{\bf p}_k)_t*\eta_{\epsilon} \,{\rm{d}}x\,{\rm{d}}t=
-
\int_{t_1}^{t_2}\int_{\Omega}\ten{G}(\tilde{\theta}+\theta_k,\tilde{\ten{T}}^d + \ten{T}^d_k)*\eta_{\epsilon}:\ten{T}^d*\eta_{\epsilon} \,{\rm{d}}x\,{\rm{d}}t .
\label{granica_k_ptrzed}
\end{equation}
For every $\epsilon>0$ the sequence $\{(\ten{\varepsilon}(\vc{u}_k) - \ten{\varepsilon}^{\bf p}_k)_t*\eta_{\epsilon}\}$ belongs to $L^2(0,T,L^2(\Omega,\mathcal{S}^3))$ and is uniformly bounded in $L^2(0,T,L^2(\Omega,\mathcal{S}^3))$, hence we pass to the limit with $k\rightarrow\infty$ and we obtain
\begin{equation}
\int_{t_1}^{t_2}\int_{\Omega}\ten{T}*\eta_{\epsilon}:(\ten{\varepsilon}(\vc{u}) - \ten{\varepsilon}^{\bf p})_t*\eta_{\epsilon} \,{\rm{d}}x\,{\rm{d}}t=
-
\int_{t_1}^{t_2}\int_{\Omega}\ten{\chi}*\eta_{\epsilon}:\ten{T}^d*\eta_{\epsilon} \,{\rm{d}}x\,{\rm{d}}t .
\nonumber
\end{equation}
Using the properties of convolution we get
\begin{equation}
\int_{\Omega}\ten{T}*\eta_{\epsilon}:(\ten{\varepsilon}(\vc{u}) - \ten{\varepsilon}^{\bf p})*\eta_{\epsilon} \,{\rm{d}}x \Big|_{t_1}^{t_2}=
-
\int_{t_1}^{t_2}\int_{\Omega}\ten{\chi}*\eta_{\epsilon}:\ten{T}^d*\eta_{\epsilon}*\eta_{\delta} \,{\rm{d}}x\,{\rm{d}}t ,
\nonumber
\end{equation}
and finally passing to the limit with $\epsilon\rightarrow 0$ and then with $t_1\to0$
\begin{equation}
\int_{\Omega}\ten{D}(\ten{\varepsilon}(\vc{u}) - \ten{\varepsilon}^{\bf p}):(\ten{\varepsilon}(\vc{u}) - \ten{\varepsilon}^{\bf p}) \,{\rm{d}}x \Big|_{0}^{t_2}=
-
\int_{0}^{t_2}\int_{\Omega}\ten{\chi}:\ten{T}^d \,{\rm{d}}x\,{\rm{d}}t .
\label{granica_k}
\end{equation}
We multiply \eqref{granica_k} by $\frac{1}{\mu}$ and integrate over $(t_2,t_2+\mu)$ and proceed now in the same manner as in the proof of Lemma~\ref{lm:8} to complete the proof.
\end{proof}
Using the Minty-Browder trick to identify the weak limit $\ten{\chi}$ and the same argumentation as in the previous section, we obtain that
\begin{equation}
\ten{G}(\tilde{\theta} + \theta_k,\tilde{\ten{T}}^d + \ten{T}_k):(\tilde{\ten{T}}^d + \ten{T}^d_{k})\rightharpoonup \ten{G}(\tilde{\theta} + \theta,\tilde{\ten{T}} +\ten{T}):(\tilde{\ten{T}}^d + \ten{T}^d)
\quad\mbox{ in }L^1(0,T,L^1(\Omega)).
\end{equation}
Furthermore
\begin{equation}
\mathcal{T}_k\Big(\ten{G}(\tilde{\theta} + \theta_k,\tilde{\ten{T}}^d + \ten{T}_k):(\tilde{\ten{T}}^d + \ten{T}^d_{k})\Big)\rightharpoonup \ten{G}(\tilde{\theta} + \theta,\tilde{\ten{T}} +\ten{T}):(\tilde{\ten{T}}^d + \ten{T}^d)
\end{equation}
in $L^1(0,T,L^1(\Omega))$. Using convergences presented above we pass to the limit with $k\rightarrow \infty$ in the equations \eqref{limit1} and \eqref{eq:after_limit_l_2}, include the previously removed boundary and volume force term and obtain
\begin{equation}
\begin{split}
\int_0^T\int_{\Omega}\Big(\tilde{\ten{T}} + \ten{T} \Big):\nabla\vc{\varphi} \,{\rm{d}}x\,{\rm{d}}t
&= \int_0^T\int_{\Omega}\vc{f}\cdot \vc{\varphi} \,{\rm{d}}x\,{\rm{d}}t ,
\end{split}
\end{equation}
where
\begin{equation}
\ten{T}=\ten{D}(\ten{\varepsilon}(\vc{u}) - \ten{\varepsilon}^{\bf p} )
\qquad
\tilde{\ten{T}} = \ten{\varepsilon}(\tilde{\vc{u}}),
\end{equation}
and
\begin{equation}
\begin{split}
-\int_0^T\int_{\Omega} (\tilde{\theta} + \theta)\phi_t \,{\rm{d}}x\,{\rm{d}}t &-
\int_{\Omega} (\tilde{\theta}_0(x) + \theta_0(x))\phi(x,0) \,{\rm{d}}x
\\
+ \int_0^T\int_{\Omega} \nabla(\tilde{\theta} +\theta)\cdot\nabla\phi \,{\rm{d}}x\,{\rm{d}}t
&- \int_0^T\int_{\partial\Omega}g_{\theta}\phi \,{\rm{d}}s \,{\rm{d}}t
\\
&=
\int_0^T\int_{\Omega} (\tilde{\ten{T}}^d + \ten{T}^d):\ten{G}(\tilde{\theta}+\theta,\tilde{\ten{T}}^d+\ten{T}^d)\phi \,{\rm{d}}x\,{\rm{d}}t,
\end{split}
\end{equation}
and
\begin{equation}
\ten{\varepsilon}^{\bf p}(x,t) = \ten{\varepsilon}^{\bf p}_0(x) + \int_0^t \ten{G}(\tilde{\theta} + \theta,\tilde{\ten{T}}^d + \ten{T}^d)\,{\rm{d}}\tau ,
\end{equation}
what completes the proof of Theorem \ref{thm:main2}.
\begin{appendix}
\section{}
Let $\mathcal{T}_k(\cdot)$ be a standard truncation operator defined in \eqref{Tk}. In \cite{Boccardo}, the authors showed the existence of solutions for the heat equation with Dirichlet boundary conditions. The current section is devoted to the existence proof to the problem with Neumann boundary conditions. Two dimensional case was considered in \cite{1240/THESES}.
We consider the sequence of the heat equations with boundary and initial conditions and with the right hand side of equation in the form
\begin{equation}
f_k=\mathcal{T}_k\left((\tilde{\ten{T}}^d + \ten{T}_k^d) : \ten{G}(\tilde{\theta} + \theta_k,\tilde{\ten{T}}^d + \ten{T}_k^d)\right)
\end{equation}
which for every $k\in\mathbb{N}$ belongs to $L^2(0,T,L^2(\Omega))$ and moreover is uniformly bounded $\|f_k\|_{L^1(0,T,L^1(\Omega))}\leq C$ and $f_k\rightarrow f$ in $L^1(0,T,L^1(\Omega))$ as $k\to\infty$. Additionally, we have $\mathcal{T}_k(\theta_0)\in L^2(\Omega)$, $\|\mathcal{T}_k(\theta_0)\|_{L^1(\Omega)}\leq \|\theta_0\|_{L^1(\Omega)}$ and $\mathcal{T}_k(\theta_0)\rightarrow \theta_0$ in $L^1(\Omega)$. To simplify the notation in the remaining part of the Appendix we denote $\Omega\times(0,T)$ by $Q$.
Let us consider the following problem
\begin{equation}
\left\{
\begin{array}{cc}
(\theta_k)_t - \Delta\theta_k = f_k
& \mbox{ in } \Omega\times (0,T),
\\
\frac{\partial\theta_k}{\partial\vc{n}}=0
& \mbox{ in } \partial\Omega\times(0,T),
\\
\theta_k(\cdot,0)=\mathcal{T}_k(\theta_0)
& \mbox{ on } \quad \Omega .
\end{array}
\right.
\label{ukla_para_n}
\end{equation}
and its weak formulation
\begin{equation}
\begin{split}
\int_{0}^T \int_{\Omega}\theta_k\varphi_t \,{\rm{d}}x\,{\rm{d}}t +
\int_{0}^T\int_{\Omega}\nabla\theta_k\cdot\nabla\varphi \,{\rm{d}}x\,{\rm{d}}t
=
\int_{0}^T\int_{\Omega}f_k\varphi \,{\rm{d}}x\,{\rm{d}}t +\int_\Omega \theta_0\varphi(0)\,{\rm{d}}x,
\end{split}\label{slab}
\end{equation}
holding for all $\varphi\in L^q(0,T;W^{1,q}(\Omega))$.
\begin{lemat}
The sequence of approximate solutions to the heat equation \eqref{ukla_para_n} is uniformly bounded in the space $L^q(0,T,W^{1,q}(\Omega))$ for $q<\frac{2 (N + 1) - N}{N + 1}$ ($q<\frac{5}{4}$ in tree dimensional case $N = 3$).
\label{unif_boun_LqLq}
\end{lemat}
\begin{proof}
We define the special truncation function $\psi_m(\cdot)$ for every $m\in\mathbb{N}$:
\begin{equation}
\psi_m(s)=\left\{
\begin{array}{ccl}
1 & \mbox{if} & s\geq m+1, \\
s-m & \mbox{if} & m+1\geq s\geq m, \\
0 & \mbox{if} & |s|\leq m, \\
s+m & \mbox{if} & s\geq m+1, \\
-1 & \mbox{if} & s\leq -m-1. \\
\end{array}
\right.
\end{equation}
Using in \eqref{slab} the test function $\psi_m(\theta_k)$ we obtain
\begin{equation}
\begin{split}
\int_{0}^T \int_{\Omega}(\Psi_m(\theta_k))_t \,{\rm{d}}x\,{\rm{d}}t +
\int_{0}^T\int_{\Omega}\nabla\theta_k\cdot\nabla\psi_m(\theta_k) \,{\rm{d}}x\,{\rm{d}}t
=\int_{0}^T\int_{\Omega}f_k\psi_m(\theta_k) \,{\rm{d}}x\,{\rm{d}}t ,
\end{split}
\end{equation}
where $\Psi_m(s)=\int_0^s\psi_m(\sigma)d\sigma$. Thus
\begin{equation}
\begin{split}
\int_{\Omega}\Psi_m(\theta_k)(T) \,{\rm{d}}x +
\int_{0}^T\int_{\Omega}\nabla\theta_k\cdot\nabla\psi_m(\theta_k) \,{\rm{d}}x\,{\rm{d}}t =
\int_{0}^T\int_{\Omega}f_k\psi_m(\theta_k) \,{\rm{d}}x\,{\rm{d}}t +
\int_{\Omega}\Psi_m(\mathcal{T}_k(\theta_0)) \,{\rm{d}}x .
\nonumber
\end{split}
\end{equation}
The terms on the right side of the above equation can be estimated as follows
\begin{equation}
\begin{split}
\int_{0}^T\int_{\Omega}f_k\psi_m(\theta_k) \,{\rm{d}}x\,{\rm{d}}t & \leq
\|f\|_{L^1(0,T,L^1(\Omega))},
\nonumber \\
\int_{\Omega}\Psi_m(\mathcal{T}_k(\theta_0)) \,{\rm{d}}x & \leq
\|\theta_0\|_{L^1(\Omega)},
\nonumber
\end{split}
\end{equation}
for every $k,m\in\mathbb{N}$. Additionally, $\int_{\Omega}\Psi_m(\theta_k)(T) dx$ is nonnegative. Hence,
\begin{equation}
\begin{split}
\int_{B_m}|\nabla\theta_k|^2 \,{\rm{d}}x\,{\rm{d}}t =
\int_{0}^T\int_{\Omega}\nabla\theta_k\cdot\nabla\psi_m(\theta_k) \,{\rm{d}}x\,{\rm{d}}t \leq
\|f\|_{L^1(0,T,L^1(\Omega))} +
\|\theta_0\|_{L^1(\Omega)},
\nonumber
\end{split}
\end{equation}
where the set $B_m=\left\{ (x,t)\in \Omega\times(0,T): m\leq \theta_k(x,t)\leq m+1 \right\}$. Now let $q \leq \frac{2(N+1) - N}{N + 1}$ and $r = \frac{N+1}{N} q$ (in our case $q <\frac{5}{4}$ and $r=\frac{4}{3}q$). Using the H\"older inequality we obtain
\begin{equation}
\begin{split}
\int_{B_m}|\nabla \theta_k|^q \,{\rm{d}}x\,{\rm{d}}t
&\leq
\left(\int_{B_m}|\nabla\theta_k|^{q\frac{2}{q}} \,{\rm{d}}x\,{\rm{d}}t
\right)^{\frac{q}{2}}
\left(\int_{B_m}1^{\frac{2}{2-q}} \,{\rm{d}}x\,{\rm{d}}t \right)^{1-\frac{q}{2}}
\\
& \leq \left(\int_{B_m}|\nabla\theta_k|^2 \,{\rm{d}}x\,{\rm{d}}t \right)^{\frac{q}{2}}
\left(\int_{B_m} \,{\rm{d}}x\,{\rm{d}}t \right)^{1-\frac{q}{2}}
\\
& \leq
c_3 \left(\int_{B_m}\frac{|\theta_k|^r}{m^r} \,{\rm{d}}x\,{\rm{d}}t \right)^{1-\frac{q}{2}}
\\
& \leq
c_3 \left(\int_{B_m}|\theta_k|^r \,{\rm{d}}x\,{\rm{d}}t \right)^{1-\frac{q}{2}}\frac{1}{m^{\frac{r(2-q)}{2}}}
\\
& \leq
c_3 \left(\int_{B_m}|\theta_k|^r \,{\rm{d}}x\,{\rm{d}}t \right)^{1-\frac{q}{2}}\left(\frac{1}{m^{\frac{r(2-q)}{q}}}\right)^{\frac{q}{2}} .
\nonumber
\end{split}
\end{equation}
Then
\begin{equation}
\begin{split}
\int_Q|\nabla\theta_k|^q \,{\rm{d}}x\,{\rm{d}}t &\leq
c_4(n_0)+c_3\sum_{m=n_0}^{\infty}\left(\int_{B_m}|\theta_k|^r \,{\rm{d}}x\,{\rm{d}}t \right)^{1-\frac{q}{2}}\left(\frac{1}{m^{\frac{r(2-q)}{q}}}\right)^{\frac{q}{2}}
\\
& \leq
c_4(n_0)+c_3\left(\sum_{m=n_0}^{\infty}\int_{B_m}|\theta_k|^r \,{\rm{d}}x\,{\rm{d}}t \right)^{1-\frac{q}{2}}\left(\sum_{m=n_0}^{\infty}\frac{1}{m^{\frac{r(2-q)}{q}}}\right)^{\frac{q}{2}}
\\
& \leq
c_4(n_0)+c_3\left(\int_{Q}|\theta_k|^r \,{\rm{d}}x\,{\rm{d}}t \right)^{1-\frac{q}{2}}\left(\sum_{m=n_0}^{\infty}\frac{1}{m^{\frac{r(2-q)}{q}}}\right)^{\frac{q}{2}},
\end{split}
\label{eq:dobre_ograniczenie}
\end{equation}
where $c_4(n_0)=\int_{\{(x,t):|\theta_k(x,t)|\leq n_0\}}|\nabla \theta_k|^q \,{\rm{d}}x\,{\rm{d}}t $. Using the H\"older inequality we observe that $c_4(n_0)$ is bounded by the terms $\|f\|_{L^1(0,T,L^1(\Omega))}$, $\|u_0\|_{L^1(\Omega)}$ and the measure of the set $Q$. Furthermore, $\frac{r(2-q)}{q}>1$ and $\sum_{m=n_0}^{\infty}m^{-\frac{r(2-q)}{q}}$ is summable.
Using the interpolation inequality for $\|\theta_k\|_{L^q(\Omega)}$ we obtain
\begin{equation}
\begin{split}
\|\theta_k\|_{L^q(\Omega)}\leq
\|\theta_k\|_{L^1(\Omega)}^s
\|\theta_k\|_{L^{q^*}(\Omega)}^{1-s},
\end{split}
\label{eq:int_1}
\end{equation}
where $q^*=\frac{Nq}{N - q}$ ($=\frac{3q}{3 - q}$) and $\frac{1}{q}=\frac{s}{1}+\frac{1-s}{q^*}$. After simple calculations we get that $1-s=\frac{1 - q}{1 - q^*}\frac{q^*}{q}$ (and $0<s<1$). In Lemma \ref{LinftyL1} we showed that $\|\theta_k\|_{L^1(\Omega)}$ is uniformly bounded, hence
\begin{equation}
\begin{split}
\int_0^T\int_{\Omega}|\theta_k|^q \,{\rm{d}}x\,{\rm{d}}t \leq
C \int_0^T\|\theta_k\|_{L^{q^*}(\Omega)}^{(1-s)q} \,{\rm{d}}t \leq
C \int_0^T\|\theta_k\|_{L^{q^*}(\Omega)}^{\frac{1 - q}{1 - q^*}q^*} \,{\rm{d}}t .
\nonumber
\end{split}
\end{equation}
Using the H\"older inequality we obtain
\begin{equation}
\begin{split}
\int_0^T\int_{\Omega}|\theta_k|^q \,{\rm{d}}x\,{\rm{d}}t &\leq
C \int_0^T\|\theta_k\|_{L^{q^*}(\Omega)}^{\frac{1 - q}{1 - q^*}q^*} \,{\rm{d}}t
\\
& \leq
C \left(\int_0^T\|\theta_k\|_{L^{q^*}(\Omega)}^{\frac{1 - q}{1 - q^*}q^*\frac{q^* - 1}{q - 1}\frac{q}{q^*}} \,{\rm{d}}t \right)^{\frac{q - 1}{q^*-1}\frac{q^*}{q}}
\\
& =
C \left(\int_0^T\|\theta_k\|_{L^{q^*}(\Omega)}^q \,{\rm{d}}t \right)^{\frac{q - 1}{q^*-1}\frac{q^*}{q}}.
\nonumber
\end{split}
\end{equation}
Let us notice that the exponent $\frac{q - 1}{q^*-1}\frac{q^*}{q}=\frac{N(q-1)}{N(q-1)+q}<1$.
Using the interpolation inequality for $\|\theta_k\|_{L^r(\Omega)}$ we get
\begin{equation}
\begin{split}
\|\theta_k\|_{L^r(\Omega)}\leq
\|\theta_k\|^s_{L^1(\Omega)}
\|\theta_k\|^{1-s}_{L^{q^*}(\Omega)},
\label{eq:int_2}
\end{split}
\end{equation}
where $\frac{1}{r}=\frac{s}{1}+\frac{1-s}{q^*}$. The parameters $s$ are different in each of the interpolation inequalities \eqref{eq:int_1} and \eqref{eq:int_2}. Simple calculations yield that $1-s=\frac{1-r}{1-q^*}\frac{q^*}{r}$. By Lemma \ref{LinftyL1} we conclude that
\begin{equation}
\begin{split}
\|\theta_k\|^r_{L^r(0,T,L^r(\Omega))} & \leq
\int_0^T\|\theta_k\|^r_{L^r(\Omega)} \,{\rm{d}}t
\\
& \leq
\int_0^T \|\theta_k\|^{sr}_{L^1(\Omega)}\|\theta_k\|^{\frac{1-r}{1-q^*}\frac{q^*}{r}r}_{L^{q^*}(\Omega)} \,{\rm{d}}t
\\
& \leq
C \int_0^T \|\theta_k\|^q_{L^{q^*}(\Omega)} \,{\rm{d}}t =
C \|\theta_k\|^q_{L^q(0,T,L^{q^*}(\Omega))} .
\end{split}
\label{eq:drugie_dobre}
\end{equation}
The Sobolev embedding theorem implies that
\begin{equation}
\begin{split}
\|\theta_k\|^q_{L^q(0,T,L^{q^*}(\Omega))}=
\int_0^T \left( \int_{\Omega}|\theta_k|^{q^*} \,{\rm{d}}x \right)^{\frac{q}{q^*}} \,{\rm{d}}t \leq
C\left(\int_0^T\int_{\Omega}|\theta_k|^q \,{\rm{d}}x\,{\rm{d}}t + \int_0^T\int_{\Omega}|\nabla\theta_k|^q \,{\rm{d}}x\,{\rm{d}}t \right).
\nonumber
\end{split}
\end{equation}
Using the previous inequalities we obtain
\begin{equation}
\begin{split}
\|\theta_k\|^q_{L^q(0,T,L^{q^*}(\Omega))}
& \leq
C \|\theta_k\|_{L^q(0,T,L^{q^*}(\Omega))}^{\frac{q-1}{q^*-1}\frac{q^*}{q}}+
c_4(n_0)+D\left(\int_{Q}|\theta_k|^r \,{\rm{d}}x\,{\rm{d}}t \right)^{1-\frac{q}{2}}
\\
& \leq C \|\theta_k\|_{L^q(0,T,L^{q^*}(\Omega))}^{\frac{q-1}{q^*-1}\frac{q^*}{q}}+
c_4(n_0)+D\|\theta_k\|_{L^q(0,T,L^{q^*}(\Omega))}^{q\frac{2-q}{2}}
\nonumber
\end{split}
\end{equation}
and $\frac{q-1}{q^*-1}\frac{q^*}{q}<1$ and $q\frac{2-q}{2}<q$, so we have the uniform boundedness
\begin{equation}
\begin{split}
\|\theta_k\|^q_{L^q(0,T,L^{q^*}(\Omega))}
\leq
C,
\nonumber
\end{split}
\end{equation}
and from the previous inequalities we get the uniform boundedness of the sequence $\{\theta_k\}$ in the space
$L^q(0,T,L^{q^*}(\Omega))$. Using this uniform boundedness and inequalities \eqref{eq:dobre_ograniczenie} and \eqref{eq:drugie_dobre} we get the uniform boundedness of the sequence $\{\theta_k\}$ in the spaces $L^q(0,T,W^{1,q}(\Omega))$, which completes the proof.
\end{proof}
\begin{lemat}
The sequence $\{\nabla\theta_k\}$ converges strongly to $\nabla \theta$ in $L^1(0,T,L^1(\Omega))$.
\label{zbieznosc_nabla_teta}
\end{lemat}
\begin{proof}
Let $\varphi$ be such that, for $\varepsilon > 0$ fixed. Let us define a test function
\begin{equation}
\varphi(s) =
\left\{
\begin{array}{ll}
\varepsilon & s > \varepsilon,
\\
s & |s| \leq \varepsilon,
\\
-\varepsilon & s < -\varepsilon.
\end{array}
\right.
\end{equation}
Subtracting equation \eqref{ukla_para_n} with function on right side $f_n$ and $f_m$, and using the test function $\varphi(\theta_n -\theta_m)$ we obtain
\begin{equation}
\begin{split}
\int_{\Omega}\Phi(\theta_n - &\theta_m)(T) \,{\rm{d}}x +
\int_{D_{n,m,\varepsilon}}|\nabla(\theta_n-\theta_m)|^2 \,{\rm{d}}x\,{\rm{d}}t =
\nonumber \\
& \int_0^T\int_{\Omega}(f_n-f_m)\varphi(\theta_n-\theta_m) \,{\rm{d}}x\,{\rm{d}}t +
\int_{\Omega}\Phi(\mathcal{T}_n(\theta_0)- \mathcal{T}_m(\theta_0)) \,{\rm{d}}x ,
\nonumber
\end{split}
\end{equation}
where $\Phi(s)=\int_0^s\varphi(\tau)d\tau$ and $D_{n,m,\varepsilon}=\{ (x,t) \in \Omega\times (0,T): |\theta_n(x,t)-\theta_m(x,t)|\leq \varepsilon \}$. The sequence $\mathcal{T}_k(\theta_0)$ is convergent to $\theta_0$ in $L^1(\Omega)$, hence, we can find $n_0$ such that for every $n$, $m$ greater than $n_0$ we have $\int_{\Omega}\Phi(T_n(\theta_0)-T_m(\theta_0))<\varepsilon$. The function $\Phi$ is nonnegative and the right hand side of the equation above is bounded ($\|f_n\|_{L^1(0,T,L^1(\Omega))}\leq \|f\|_{L^1(0,T,L^1(\Omega))}=:B$), hence
\begin{equation}
\begin{split}
\int_{D_{n,m,\varepsilon}}|\nabla(\theta_n-\theta_m)|^2 \,{\rm{d}}x\,{\rm{d}}t \leq
2\varepsilon B+\varepsilon=(2 B+1 )\varepsilon.
\nonumber
\end{split}
\end{equation}
The H\"older inequality yields
\begin{equation}
\begin{split}
\int_{D_{n,m,\varepsilon}}|\nabla(\theta_n-\theta_m)| \,{\rm{d}}x\,{\rm{d}}t & \leq
\left(\int_{D_{n,m,\varepsilon}}|\nabla(\theta_n-\theta_m)|^2 \,{\rm{d}}x\,{\rm{d}}t \right)^{\frac{1}{2}}
\left(meas(D_{n,m,\varepsilon})\right)^{\frac{1}{2}}
\\
& \leq
C(2B+1)^{\frac{1}{2}}\varepsilon^{\frac{1}{2}}.
\nonumber
\end{split}
\end{equation}
Using the decomposition of $Q=D_{n,m,\varepsilon}\cup (Q\setminus D_{n,m,\varepsilon})$ we have to consider the integral over the second set.
\begin{equation}
\begin{split}
\int_{Q\setminus D_{n,m,\varepsilon}}|\nabla(\theta_n-\theta_m)| \,{\rm{d}}x\,{\rm{d}}t \leq
\left(\int_{Q\setminus D_{n,m,\varepsilon}}|\nabla(\theta_n-\theta_m)|^q \,{\rm{d}}x\,{\rm{d}}t \right)^{\frac{1}{q}}
\left(meas(Q\setminus D_{n,m,\varepsilon})\right)^{1-\frac{1}{q}}
\end{split}
\end{equation}
The first term on the right hand side is bounded, since the sequence $\{\theta_n\}$ is uniformly bounded in $L^q(0,T,W^{1,q}(\Omega))$. The sequence $\{\theta_n\}$ is a Cauchy sequence in $L^1(0,T,L^1(\Omega))$, so there exists $n_0$ such that for all $n,m>n_0$ occurs $\left(meas(Q\setminus D_{n,m,\varepsilon})\right)^{1-\frac{1}{q}}<\varepsilon$. Then from the previous inequalities we obtain
\begin{equation}
\begin{split}
\int_Q|\nabla \theta_n-\nabla \theta_m| \,{\rm{d}}x\,{\rm{d}}t & =
\int_{D_{n,m,\varepsilon}}|\nabla(\theta_n-\theta_m)| \,{\rm{d}}x\,{\rm{d}}t +
\int_{Q\setminus D_{n,m,\varepsilon}}|\nabla(\theta_n-\theta_m)| \,{\rm{d}}x\,{\rm{d}}t
\\
& \leq
c_1\varepsilon^{\frac{1}{2}}+c_2\varepsilon
\label{ciag_cau_szac}
\end{split}
\end{equation}
which implies that $\{\nabla\theta_n\}$ is a Cauchy sequence in $L^1(0,T,L^1(\Omega))$.
\end{proof}
\begin{lemat}{Aubin-Lions \cite[Lemma 7.7]{Roubicek}}
Let $V_1$, $V_2$ be Banach spaces, and $V_3$ be a metrizable Hausdorff locally convex space,
$V_1$ be separable and reflexive, $V_1\subset\subset V_2$ (a compact embedding), $V_2 \subset V_3$ (a continuous embedding), $1 < p <\infty$,
$1 \leq q \leq \infty$. Then $\{u: u\in L^p(0,T,V_1);u_t\in L^q(0,T,V_3 )\}\subset\subset L^p(0,T,V_2 )$ (a compact embedding).
\end{lemat}
From the uniform boundedness of the sequence $\{f_k\}$ in $L^1(0,T,L^1(\Omega))$ and from the uniform boundedness of the sequence $\{\theta_k\}$ in $L^q(0,T,W^{1,q}(\Omega))$ we obtain
that $\{(\theta_n)_t\}$ is a sequence bounded in the space $L^1(0,T,W^{-1,q}(\Omega))$.
Consequently the sequence $\{\theta_n\}$ is relatively compact in $L^1(0,T,L^1(\Omega))$. Due to Lemma \ref{unif_boun_LqLq} and Lemma \ref{zbieznosc_nabla_teta} we know that the sequence $\{\theta_n\}$ converges strongly to $\theta$ in $L^q(0,T,W^{1,q}(\Omega))$.
Moreover, for $s$ large enough $(\theta_k)_t$ converges strongly to $\theta_t$ in $L^1(0, T; W^{-1,s}(\Omega))$. Thus, $\theta_k$ converges strongly to $\theta$ in $C([0, T], W^{-1,s}(\Omega))$ and $\theta_k(\cdot, 0)$ converges to $\theta(\cdot, 0)$ in $W^{-1,s}(\Omega)$.
\begin{lemat}
For $q<\frac{2(N+1) - N}{N + 1}$ ($q<\frac{5}{4}$ when $N=3$) there exists $\theta\in L^q(0,T,W^{1,q}(\Omega))\cap C([0,T],W^{-s,2}(\Omega))$ - a solution to the system
\begin{equation}
\left\{
\begin{array}{cl}
\theta_t - \Delta\theta = f & \mbox{ in } \Omega\times (0,T),\\
\frac{\partial\theta}{\partial\vc{n}}=0 & \mbox{ on } \partial\Omega\times (0,T),\\
\theta (x,0) = \theta_0(x) & \mbox{ in } \Omega.
\end{array}
\right.
\end{equation}
\end{lemat}
\begin{proof}
Choosing in \eqref{slab} the test function $\psi\in C^{\infty}(\Omega\times [0,T))$ such that $\psi=0$ on $\Omega\times\{T\}$, we get
\begin{equation}
\begin{split}
\int_0^T\int_{\Omega}(\theta_n)_t\psi \,{\rm{d}}x\,{\rm{d}}t - \int_0^T\int_{\Omega}\Delta\theta_n\psi \,{\rm{d}}x\,{\rm{d}}t &= \int_0^T\int_{\Omega}f_n\psi \,{\rm{d}}x\,{\rm{d}}t .
\nonumber
\end{split}
\end{equation}
Then
\begin{equation}
\begin{split}
-\int_0^T\int_{\Omega}\theta_n\psi_t \,{\rm{d}}x\,{\rm{d}}t & +
\int_{\Omega}\theta_n\psi \,{\rm{d}}x \Big|_0^T
\\
& + \int_0^T\int_{\Omega}\nabla\theta_n\cdot\nabla\psi \,{\rm{d}}x\,{\rm{d}}t -
\int_0^T\int_{\partial\Omega}\frac{\partial\theta_n}{\partial\vc{n}}\psi \,{\rm{d}}x\,{\rm{d}}t
= \int_0^T\int_{\Omega}f_n\psi \,{\rm{d}}x\,{\rm{d}}t .
\nonumber
\end{split}
\end{equation}
And finally
\begin{equation}
\begin{split}
-\int_0^T\int_{\Omega}\theta_n\psi_t \,{\rm{d}}x\,{\rm{d}}t +
\int_0^T\int_{\Omega}\nabla\theta_n\cdot\nabla\psi \,{\rm{d}}x\,{\rm{d}}t
& = \int_0^T\int_{\Omega}f_n\psi \,{\rm{d}}x\,{\rm{d}}t +
\int_{\Omega}T_n(\theta_0)\psi \,{\rm{d}}x .
\nonumber
\end{split}
\end{equation}
Using the convergence of the temperatures' sequence we obtain
\begin{equation}
\begin{split}
-\int_0^T\int_{\Omega}\theta\psi_t +
\int_0^T\int_{\Omega}\nabla\theta\cdot\nabla\psi
= \int_0^T\int_{\Omega}f\psi +
\int_{\Omega}\theta_0\psi.
\nonumber
\end{split}
\end{equation}
\end{proof}
\section{}\label{B}
In the current section we present the construction of the basis used for approximation of the strain tensor. We adapt it for our particular case, however the idea follows the lines of \cite[Theorem 4.11]{maleknecas}.
The definitions of spaces $V_k$ and $V_k^s$ were introduced in Section~\ref{sec:4} by \eqref{Vk} and \eqref{Vks}.
Let us consider the following problem: find $\ten{\zeta}_i\in V^s_k$ and $\lambda_i\in\mathbb{R}$ such that
\begin{equation}
\braket{\ten{\zeta}_i,\ten{\Phi}}_s = \lambda_i (\ten{\zeta}_i,\ten{\Phi})_{\ten{D}} \qquad \forall\ \ten{\Phi}\in V^s_k.
\label{eq:war_wl}
\end{equation}
where by $\braket{\cdot,\cdot}_s$ we denote the scalar product in $H^s(\Omega, \mathcal{S}^3)$ and $(\cdot,\cdot)_{\ten{D}}$ is the previously defined scalar product in $L^2(\Omega,\mathcal{S}^3)$.
\begin{tw}
There exist a countable set of eigenvalues $\{\lambda_i\}_{i=1}^{\infty}$ and a corresponding family of eigenfunctions $\{\ten{\zeta}_i\}_{i=1}^{\infty}$ solving \eqref{eq:war_wl} such that
\begin{itemize}
\item $(\ten{\zeta}_i,\ten{\zeta}_j)_{\ten{D}} = \delta_{ij}$ for all $i,j\in\mathbb{N}$,
\item $1\leq \lambda_1 \leq \lambda_2\leq ... $ and $\lambda_i\to \infty$ as $i$ tends to $\infty$,
\item $\braket{\frac{\ten{\zeta}_i}{\sqrt{\lambda_i}},\frac{\ten{\zeta}_j}{\sqrt{\lambda_i}}}_s = \delta_{ij}$ for all $i,j\in\mathbb{N}$,
\item the set $\{\ten{\zeta}_i\}_{i=1}^{\infty}$ is a basis of $V^s_k$.
\item the set $\{\ten{\zeta}_i\}_{i=1}^{\infty}$ is a basis of $V_k$.
\end{itemize}
Moreover, let us define the subspace $H^N\equiv \mbox{span} \{\ten{\zeta}_1,..., \ten{\zeta}_N\}$ and projection $P^N:V^s_k \to H^N$ such that $P^N(\ten{V}) \equiv \sum_{i=1}^N(\ten{V},\ten{\zeta}_i)_{\ten{D}}\ten{\zeta}_i$, then we get
\begin{equation}
\|P^N\varphi\|_{H^s}\le\|\varphi\|_{H^s}
\end{equation}
\label{th:jos}
\end{tw}
\begin{proof}
Proof of Theorem \ref{th:jos} is divided into few steps.
\noindent
{\it Existence of $\ten{\zeta}_1$}
\noindent
Let us define
\begin{equation}
\frac{1}{\lambda_1} \equiv \sup_{\ten{V}\in V^s_k \atop \|\ten{v}\|_{H^s}\leq 1} (\ten{V},\ten{V})_{\ten{D}}.
\label{eq:definiecja_war_wl}
\end{equation}
Consequently, there exists a sequence $\{\ten{V}_i\}_{i=1}^{\infty}$ such that $(\ten{V}_i,\ten{V}_i)_{\ten{D}}\to \frac{1}{\lambda_1}$ as $i$ tends to $\infty$ and $\|\ten{V}_i\|_{H^s(\Omega)}=1$. Then, there exist a subsequence $\{\ten{V}_i\}_{i=1}^{\infty}$ (still denoted by i) and $\ten{\zeta}_1 \in V^s_k$ such that
\begin{equation}
\begin{split}
\ten{V}_i & \rightharpoonup \ten{\zeta}_1
\qquad \mbox{in } V^s_k,
\\
\ten{V}_i & \to \ten{\zeta}_1
\qquad \mbox{in } L^2(\Omega,\mathcal{S}^3).
\end{split}
\end{equation}
If $\|\ten{\zeta}_1\|_{H^s(\Omega)} < 1$, then let us define $\ten{\zeta}=\frac{\ten{\zeta}_1}{\|\ten{\zeta}_1\|_{H^s(\Omega)}}$ and then
\begin{equation}
\|\ten{\zeta}\|_{H^s(\Omega)} =1
\qquad
\mbox{ and }
\qquad
(\ten{\zeta},\ten{\zeta})_{\ten{D}} = \frac{(\ten{\zeta}_1,\ten{\zeta}_1)_{\ten{D}}}{\|\ten{\zeta}_1\|_{H^s(\Omega)}} > \frac{1}{\lambda_1},
\end{equation}
which is contrary with \eqref{eq:definiecja_war_wl} and it implies that $\|\ten{\zeta}_1\|_{H^s(\Omega)} = 1$. To finish the first step we show that $\ten{\zeta}_1$ is an eigenfunction. Let us take $\ten{H}\in V^s_k$ and define the function
\begin{equation}
\Phi(t) = \frac{(\ten{\zeta}_1 + t\ten{H},\ten{\zeta}_1 + t\ten{H})_{\ten{D}}}{\braket{\ten{\zeta}_1 + t\ten{H},\ten{\zeta}_1 + t\ten{H}}_s}.
\end{equation}
Calculating the derivative of function $\Phi(t)$, we obtain
\begin{equation}
\begin{split}
0 = \frac{d}{dt} \Phi(t) |_{t=0} & =
\frac{2(\ten{\zeta}_1,\ten{H})_{\ten{D}}\braket{\ten{\zeta}_1 ,\ten{\zeta}_1}_s - 2(\ten{\zeta}_1,\ten{\zeta}_1)_{\ten{D}}\braket{\ten{\zeta}_1,\ten{H}}_s}{((\ten{\zeta}_1 ,\ten{\zeta}_1 ))_s^2}
\\
& =
\frac{2(\ten{\zeta}_1,\ten{H})_{\ten{D}} - \frac{2}{\lambda_1}\braket{\ten{\zeta}_1,\ten{H}}_s}{\braket{\ten{\zeta}_1 ,\ten{\zeta}_1 }_s^2}
\end{split}
\end{equation}
and then
\begin{equation}
\lambda_1(\ten{\zeta}_1,\ten{H})_{\ten{D}} = \braket{\ten{\zeta}_1,\ten{H}}_s
\qquad
\forall\ \ten{H}\in V^s_k.
\end{equation}
\noindent
{\it Iterative construction}
\noindent
Assume that for $N\geq 1$ there exists the set of eigenvalues $\{\lambda_i\}_{i=1}^{N}$ and the set of corresponding eigenfunctions $\{\ten{\zeta}_i\}_{i=1}^{N}$. Let us define the space
\begin{equation}
W^N \equiv \{ \ten{V}\in V^s_k: \braket{\ten{V},\ten{\zeta}_i}_s =0, \quad i=1,...,N\}.
\end{equation}
Using the similar construction as in the previous step, we find the next eigenvalue and eigenfunction
\begin{equation}
(\ten{\zeta}_{N+1},\ten{\zeta}_{N+1})_{\ten{D}} = \sup_{\ten{V}\in W^N\atop \|\ten{V}\|_{H^s}=1} (\ten{V},\ten{V})_{\ten{D}} \equiv \frac{1}{\lambda_{N+1}}.
\end{equation}
Finally, we obtain
\begin{equation}
\begin{split}
&1\leq \lambda_1\leq \lambda_2\leq ...,
\\
&(\ten{\zeta}_i,\ten{\zeta}_j)_{\ten{D}}=0 \qquad \mbox{if } i\leq j,
\\
&\braket{\ten{\zeta}_i,\ten{\zeta}_j}_s=\delta_{ij}.
\end{split}
\end{equation}
\noindent
{\it Unboundedness of eigenvalues}
\noindent
Let us assume that the set of eigenvalues has a finite limit, i.e. $\lim_{i\to\infty}\lambda_i =\lambda <\infty$. Since $\|\ten{W}_i\|_{H^s}=1$, using subsequence if it is necessary, we get $\ten{W}_i \to \ten{W}$ in $L^2(\Omega,\mathcal{S}^3)$ as $i\to\infty$. Hence
\begin{equation}
\begin{split}
2 &= \braket{\ten{\zeta}_i,\ten{\zeta}_i}_s + \braket{\ten{\zeta}_j,\ten{\zeta}_j}_s
= \braket{\ten{\zeta}_i - \ten{\zeta}_j,\ten{\zeta}_i - \ten{\zeta}_j}_s
\\
& = \braket{\ten{\zeta}_i ,\ten{\zeta}_i - \ten{\zeta}_j}_s
- \braket{\ten{\zeta}_j,\ten{\zeta}_i - \ten{\zeta}_j}_s
\\
& = \lambda_i (\ten{\zeta}_i ,\ten{\zeta}_i - \ten{\zeta}_j)_{\ten{D}}
- \lambda_j(\ten{\zeta}_j,\ten{\zeta}_i - \ten{\zeta}_j)_{\ten{D}}.
\end{split}
\label{eq:sprz}
\end{equation}
Passing with $i,j$ to $\infty$ we obtain
\begin{equation}
\begin{split}
(\ten{\zeta}_i ,\ten{\zeta}_i - \ten{\zeta}_j)_{\ten{D}} \to 0,
\\
(\ten{\zeta}_j,\ten{\zeta}_i - \ten{\zeta}_j)_{\ten{D}} \to 0.
\end{split}
\label{eq:sprz2}
\end{equation}
Comparing \eqref{eq:sprz} and \eqref{eq:sprz2} we get the contradiction.
\noindent
{\it The Set $\{\lambda_i\}_{i=1}^{\infty}$ contains all eigenvalues} \\
Let us assume that there exists an eigenvalue $\lambda$ such that $\lambda \notin \{\lambda_i\}_{i=1}^{\infty}$. Let $\ten{W}$ be the corresponding eigenfunction to the eigenvalue $\lambda$ and
\begin{equation}
\braket{\ten{\zeta},\ten{\Phi} }_s=\lambda (\ten{\zeta},\ten{\Phi})_{\ten{D}}
\qquad
\ten{\Phi} \in V^s_k.
\end{equation}
Without loss of generality, $\|\ten{\zeta}\|_{H^s}=1$.
Moreover, there exists $i\in\mathbb{N}$ such that $\lambda_i < \lambda <\lambda_{i+1}$. Then, for all $k=1,...,i$
\begin{equation}
\begin{split}
\braket{\ten{\zeta}_k,\ten{\zeta} }_s = \lambda_k (\ten{\zeta}_k,\ten{\zeta} )_{\ten{D}},
\\
\braket{\ten{\zeta},\ten{\zeta}_k }_s = \lambda (\ten{\zeta},\ten{\zeta}_k )_{\ten{D}}.
\end{split}
\end{equation}
Hence, $(\ten{\zeta},\ten{\zeta}_k )_{\ten{D}}=0$ and therefore $\ten{\zeta}\in W^i$ and
\begin{equation}
(\ten{\zeta},\ten{\zeta})_{\ten{D}}=\frac{1}{\lambda}> \frac{1}{\lambda_i} = \sup_{\ten{V}\in W^N \atop \|\ten{V}\|_{s,2}=1} (\ten{V},\ten{V})_{\ten{D}},
\end{equation}
which is a contradiction.
\noindent
{\it The set $\{\ten{\zeta}_i\}_{i=1}^{\infty}$ is a basis in $V^s_k$}
\noindent
Let us define $X= \mbox{span}\{\ten{\zeta}_1,\ten{\zeta}_2,...\}$ and let us assume that $X\neq V^s_k$. Then, there exists $\ten{\Phi}\in V^s_k$ such that $\|\ten{\Phi}\|_{H^s(\Omega)} =1$ and $\braket{\ten{\Phi},\ten{\zeta}_i}_{s}=0$ for all $i\in\mathbb{N}$. Moreover, for all $i\in\mathbb{N}$
\begin{equation}
(\ten{\Phi},\ten{\Phi})_{\ten{D}} \leq \sup_{\ten{V}\in W^N \atop \|\ten{V}\|_{H^s}=1} (\ten{V},\ten{V})_{\ten{D}} = \frac{1}{\lambda_1},
\end{equation}
which implies that $\ten{\Phi}=\ten{0}$.
\noindent
{\it Renormalization of basis}
\noindent
To complete the proof we may renormalize the basis
\begin{equation}
\widehat{\ten{\zeta}_i}\equiv \frac{\ten{\zeta}_i}{\sqrt{\lambda_i}}.
\end{equation}
for all $i\in\mathbb{N}$.
\noindent
{\it The set $\{\ten{\zeta}_i\}_{i=1}^{\infty}$ is a basis in $V_k$}
\noindent
Observe that the space $V_k^s$ is dense in $V_k$ in $L^2(\Omega, \mathcal{S}^3)$ norm. For this purpose consider an element $\ten{\xi}$ of $V^k$. To show there exists a sequence $\ten{\xi}^n$ bounded in $V^k_s$ that converges to $\ten{\xi}$ recall that if $\ten{\xi}$ is in $L^2(\Omega,\mathcal{S}^3)$, then there exists an approximating sequence
$\overline{\ten{\xi}}^n$ in $H^s(\Omega,\mathcal{S}^3)$. Then the sequence $\ten{\xi}^n$ we construct as follows
$$\ten{\xi}^n:=\overline{\ten{\xi}}^n-P_k\overline{\ten{\xi}}^n,$$
where the projection $P^k$ was defined in the proof of Lemma~\ref{wsp_org_epa}. Then using the continuity of $P^k$ in $H^s(\Omega, \mathcal{S}^3)$ we immediately obtain that $\ten{\xi}^n$ is bounded in $H^s(\Omega, \mathcal{S}^3)$ and converges to $\ten{\xi}\in V_k$. Consequently, $\{\ten{\zeta}_i\}_{i=1}^{\infty}$ is also a basis in $V_k$.
\end{proof}
\end{appendix}
\bigskip
{\bf Acknowledgement} P.G. is a coordinator and F.K. is a PhD student in the International PhD Projects Programme of Foundation for Polish Science operated within the Innovative Economy Operational Programme 2007-2013 funded by European Regional Development Fund (PhD Programme: Mathematical Methods in Natural Sciences). PG was supported by the National Science Center, project no. 6085/B/H03/2011/40. FK was partially supported by grant NCN OPUS 2012/07/B/ST1/03306, A. \'S.-G. was supported by the grant IdP2011/000661.
\input{GwiazdaKlaweSwierczewska.bbl}
\bibliographystyle{plain}
\end{document} |
1,314,259,993,647 | arxiv | \section{Introduction}
Solar wind (SW) speed is one of the most important factors in the solar wind-magnetosphere interaction.
Long-term averages of SW speed are strongly modulated by the occurrence of high-speed streams (HSSs), which are known to originate from coronal holes \citep{Krieger_1973, Gosling_1976, Kojima_1990}.
The occurrence of HSSs at the Earth's orbit maximizes during the declining phase of the solar cycle, when high speed streams from equatorward extensions of polar coronal holes often reach low heliographic latitudes and the ecliptic plane \citep{Hakamada_1981}.
HSSs have a strong effect on geomagnetic activity especially at high latitudes and the auroral zone.
For example, the occurrence of magnetospheric substorms is strongly modulated by the occurrence of HSSs [\citeauthor{Tanskanen_2005}, \citeyear{Tanskanen_2005}, \citeyear{Tanskanen_2011}]. Recently, \citet{Lukianova_2012} showed that magnetic disturbances caused by HSSs can be seen even in annual means of the vertical magnetic field component (Z) on the polar cap and the horizontal magnetic field component (H) at auroral latitudes.
HSSs include enhanced Alfv\'en wave activity, which lead to a repeated occurrence of substorms \citep{Lyons_2009} and energetic particles \citep{Denton_2012}.
The HSS-related magnetic disturbances are mainly reflected in the westward auroral electrojet (WEJ), which is enhanced during substorms.
\citet{Lukianova_2012} found that the highest intensity of WEJ occurred in 2003, in the declining phase of the solar cycle (SC) 23, leading to a major reduction of H at auroral latitudes and strengthening of Z within the northern and southern polar caps.
\citet{Mursula_2015} exploited the strong correlation between the high-latitude magnetic disturbances and SW speed and reconstructed the annual means of SW speed from two magnetic stations (Godhavn/Qeqertarsuaq, GDH and Sodankyl\"a, SOD) with the longest series of high-latitude geomagnetic data since 1926 (GDH) and 1914 (SOD).
They thus covered most solar cycles of the so called Grand Modern Maximum (GMM) of solar activity, during which solar activity has been greater than centuries to millennia before \citep{Solanki_2000}.
\citet{Mursula_2015} found that high annual speeds occurred in the declining phase of all SCs 16–-23.
The estimated annual SW speed exceeded 500 km/s in 1930, 1941, 1951-1953, 1963, 1974, 1994 and 2003.
In the early 1950s, high HSS activity continued for three successive years, with the highest yearly activity, up to 570 km/s, found in the year 1952.
They noted that cycle 19, which marks the sunspot maximum period of the GMM, was preceded by exceptionally strong polar fields during the declining phase of cycle 18, which proves the $\Omega$-mechanism (conversion of poloidal fields to toroidal fields) of the solar dynamo theory \citep{Babcock_1961} for this period of very high solar activity.
The aim of this paper is to study the long-term evolution HSSs at monthly time resolution, using the same methodology as previously applied by \citet{Lukianova_2012} and \citet{Mursula_2015}.
This paper is organized as follows. In Section 2 we present the data and introduce the $\Delta{H}$ parameter.
In Section 3 we study the relation between monthly $\Delta{H}$ and SW speed.
In Section 4 we present monthly proxies of SW speed.
Discussion and conclusions are given in Section 5.
\section{Data}
We use hourly measurements of horizontal magnetic field at the Sodankyl\"a geophysical observatory (SOD; geographic latitude and longitude: $67.37^{\circ}$, $26.63^{\circ}$; geomagnetic latitude and longitude: $64^{\circ}$, $119^{\circ}$), located near the equatorward boundary of the auroral oval.
At Sodankyl\"a, recordings of the Earth's magnetic field vector have been made since 1914 (interrupted only during the World War II in 1945), forming the longest continuous high-latitude geomagnetic measurements available for more than 100 years.
Because of the proximity of SOD to the auroral electrojets, the largest perturbations occur in the geomagnetic horizontal H component which is directed to the magnetic north. During magnetospheric substorms, WEJ increases during the growth and expansion phases and decays back to quiet-time level during the recovery phase \citep{Akasofu_1964}.
The WEJ-related magnetic disturbances are mainly manifested in the midnight magnetic local time (MLT) sector and the eastward auroral electrojet in the late afternoon sector.
Accordingly, the amplitude of the daily curve varies in the range of approximately -500 to 500 nT.
At SOD, the 4-hour time interval of 20-23 UT (22-01 LT) is the most appropriate time to estimate the WEJ intensity.
We define the geomagnetic disturbance parameter $\Delta{H}$ as follows.
For each month we calculate the quiet-time level H(q) by averaging the night-time (20-23 UT) H values during the five quietest days. The quietest days have been calculated from the local K indices and they are available at the SGO database.
Then we calculate $\Delta{H}$ by calculating the difference between H(q) and the monthly average of all night-time (20-23 UT) H values.
($\Delta{H}$ = H(q) - H is positive since WEJ reduces H).
We also use the measured hourly SW speed (V) values of the OMNI data base (\texttt{http://omniweb.gsfc.nasa.gov/}) since 1964. To quantify the relationship between $\Delta{H}$ and SW speed we calculate a linear regression separately for different months. Because of numerous data gaps in the OMNI data base especially during 1980s and early 1990s, monthly means of SW speed cannot always be reliably calculated.
In the regressions we only use those values of $\Delta{H}$ and SW speed when both parameters have been measured simultaneously. Moreover, in order to have sufficient statistics for each month we neglect, when calculating the regressions, those months when the data coverage is less than 30\%.
Overall, there are 55 months (i.e., 9\%) neglected by this requirement.
The problem is worst in the early 1980s, when up to 8 months were neglected in the year 1984.
\section{Relationship between monthly solar wind speed and $\Delta{H}$}
Figure 1 shows, as an example, the correlation between monthly $\Delta{H}$ and SW speed for Januaries in 1964-2014.
The correlation is fairly linear and statistically significant (correlation coefficient is 0.76; zero correlation probability p = 0.0002 using a first order autoregressive (AR-1) noise model), and there are no actual outliers in Fig. 1.
Note that one station cannot always be exactly at the site of the maximum WEJ enhancement related to the substorm current wedge, which produces a source of error in Fig. 1.
Also the CMEs contribute to increasing scatter, especially in solar maximum years (see later).
Data gaps in SW measurements (to be discussed in detail in Section 4) are a source of significant scatter in Fig. 1.
Also one can see from Figure 1 that data points lie symmetrically around the regression line for the whole range of $\Delta{H}$.
This is better seen in the bottom panel of Fig. 1 which depicts the residuals of the regression, i.e., the differences between the measured and estimated monthly SW speeds.
The homoscedasticity of the residuals guarantees that the standard least squares fit performs well and can be reliably used to reconstruct the monthly means of SW speed. Table 1 gives the regression coefficients and correlation coefficients for similar fits for each month.
Figure 2 depicts the regression coefficients (slope and intercept) for each month. Figure 2 shows that while the intercept varies very little from month to month, the relative variation in the slope is much larger. The slope maximizes during mid-winter (Dec and Jan) and mid-summer (Jun), and minimizes around equinox months. Thus, the monthly average SW speed required to produce a given value of $\Delta{H}$ is largest during mid-winter and mid-summer, and the response at high latitudes ($\Delta{H}$) to SW speed is stronger (slope is lower) during equinoxes.
This is most likely related to the semiannual variation of geomagnetic activity, whose main drivers are the equinoctial mechanism \citep{Cliver_2000, Lyatsky_2001} related to the seasonally changing ionospheric conductivity and the Russell-McPherron mechanism \citep{Russell_1973, McPherron_2009}, where the solar equatorial magnetic field gets projected to the southward component in the GSM coordinate system.
Here we use the monthly regressions and related parameters to reconstruct the monthly SW speeds in the early 20th century.
Figure 3a shows the scatter plot of the actually measured SW speeds and estimated SW speeds (using the regression parameters in Table 1) for all months in 1964-2014.
(Here no requirement for data coverage is imposed).
One can see that the relation between the estimated and measured values is fairly linear and there are only two large outliers in the fit. (These two outliers, June and Feb 1982, were neglected when calculating the regressions).
Figure 3b shows the fit residuals $\delta$ = $V(measured)$ - $V(estimated)$ as a function of estimated SW speeds.
One can see that the majority of the residuals over almost the whole range of values are homoscedastically distributed.
Note that the data gaps in the solar wind measurements and from the non-HSS related SW effects in $\Delta{H}$ (mainly due to coronal mass ejections, CMEs) increase scatter and may increase the number of outliers.
We will discuss the data gaps in more detail in Section 4.
\section{Monthly solar wind speeds}
The above described monthly regressions between SW speed and $\Delta{H}$ are applied to reconstruct the monthly SW speeds during the last 100 years since 1914. Figure 4 depicts the monthly SW speed proxies estimated from $\Delta{H}$ in 1914-2014 together with their $\pm 1\sigma$ errors, and the measured monthly SW speeds in 1964-2014, separately for each month. (We have included in Figure 4 all available data for both the measured and proxy values). Figure 4 shows that, even at the monthly timescale, the $\Delta{H}$ based proxies represent the SW speed with reasonable accuracy.
The proxy covers the range of the measured SW speeds relatively well, except for some of the highest peaks and the lowest minima. There are some differences especially from 1980s until mid-l990s when there are numerous data gaps in the SW measurements, and the lower statistics increases random fluctuations.
Figure 5 shows the proxy and the measured monthly SW speeds in 1964-2014 for months with $30$\% data coverage condition imposed. Therefore Figure 5 includes more data gaps than Figure 4.
Here we have included only those hours when the two parameters had simultaneous measurements.
The bottom panel of Figure 5 depicts the monthly fractions of data gaps in SW measurements.
The data gaps practically end in 1995, since when the SW is continuously measured by the ACE (and later Wind) satellite. At this time the accuracy of the proxy values is also somewhat improved. The standard deviation of the difference between the estimated and measured values of SW speed is 31 km/s in 1995-2014 and 39 km/s in 1964-1994.
Despite the differences between the measured and the proxy values of monthly SW speeds, the highest peaks of each solar cycle tend to agree with each other. This is the case for solar cycle 20 (when the measured and the proxy peaks occur in April and March 1973), for cycle 22 (both in February 1994) and cycle 23 (both in June 2003). This comparison fails only for solar cycle 21, when the two peaks are in different years (April 1983 and Feb 1982). The largest number of data gaps take place in the early 1980s, in the declining phase of SC 21, when IMP-8 satellite was the only satellite measuring the solar wind.
However, we suspect that the difference between the measured and the proxy values especially during the high peaks in 1982 is not only due to the data gaps. These peaks occurred close to the maximum of solar cycle 21 and are most likely significantly affected by CMEs.
Note that CMEs tend to produce exceptionally high values for the proxy due to its other geo-effective factors. This is seen clearly in 1982--1983.
Figure 6 shows the reconstructed monthly SW speeds in 1914-2014 from Figure 4 as a single time series. Figure 6 shows that the highest peaks for each SC always occur during the declining phase of the solar cycle. The cycle peaks of monthly SW speeds of cycles 15-23 occurred in 1919, 1930, 1941, 1952, 1959, 1973, 1982, 1994 and 2003. These years are almost the same as the cycle peak years of the annual SW speeds from SOD (1918, 1930, 1941, 1952, 1959, 1974, 1984, 1994 and 2003; almost the same years were found in the GDH station as well) estimated by Mursula et al. (2015).
So, in 6 out of 9 cycles the cycle peaks for monthly and yearly peaks in the SW speed proxy occur in the same year. In two cycles they are found in successive years, when the high-speed stream forming the monthly maximum is very likely produced by the coronal hole which yields the annual speed maximum in the adjacent year.
This is the case for 1918-1919, when the monthly peak is found in May 1919 (520 km/s) while almost equally high monthly values were found already in 1918 (e.g., in December 520 km/s), which was the year of the most persistent yearly HSS activity.
Only for cycle 21, when three years 1982-1984 all have roughly equal annual SW speed values, the respective peaks have a difference of 2 years.
The fact that practically all of the highest monthly speeds are found during the same or adjacent years as the highest annual speeds implies that the most persistent coronal holes, which are responsible for the highest annual means of SW speed, are also the sources of the highest monthly SW speeds.
Persistent coronal holes can live for several months (up to one year), whence the highest monthly SW speed value in one calendar year can, actually, be produced by a persistent coronal hole extending to or from the adjacent year, where it forms the annual maximum occurrence of high-speed streams (maximum of annual solar wind speed).
These results imply that it is very unlikely that highest speed streams of roughly one-month (or solar rotation) duration would appear outside of the times when the most persistent coronal holes appear in the Sun.
These uniform results also strongly support the method used here to estimate the monthly SW speeds from the $\Delta{H}$ parameter.
\section{Discussion and conclusions}
In this paper we have utilized the longest available high-latitude measurement of the geomagnetic field made at Sodankyl\"a, Finland, and used a local night-time measure ($\Delta{H}$) of geomagnetic activity to estimate the strength of the westward auroral electrojet, which is a sensitive proxy of SW speed.
We have calculated linear regressions between $\Delta{H}$ and SW speed separately for all months.
Even at monthly timescale we find high correlations between the two parameters for all months, giving evidence that other factors in solar wind, especially the intensity of the interplanetary magnetic fields which is enhanced during CMEs [Richardson and Cane, 2012], have a significantly smaller effect for the monthly averages of $\Delta{H}$ at Sodankyl\"a. This supports the earlier studies which have shown the importance of the SW speed for substorm occurrence \citep{Tanskanen_2005} and high-latitude geomagnetic activity \citep{Finch_2008, Lukianova_2012, Holappa_2014}.
The relation between $\Delta{H}$ and SW speed shows a clear seasonal variation so that during equinoxes the coupling is stronger, i.e., a given SW speed value yields a higher value of $\Delta{H}$. This seasonal variation is most likely related to the semiannual variation of geomagnetic activity, mainly due to the equinoctial \citep{Cliver_2000} and Russell-McPherron effects \citep{Russell_1973, McPherron_2009}, which modulate the geoeffectiveness of HSSs. We take this seasonal variation into account by determining regression coefficients separately for different months.
Using the monthly regressions we have estimated the monthly means of the SW speed for the last 100 years (1914-2014). We find that the largest monthly SW speeds, i.e., the highest HSS-active months in each solar cycle occur in the declining phase of the cycle, in the years 1919, 1930, 1941, 1952, 1959, 1973, 1982, 1994 and 2003 for cycles 15-23, respectively.
This confirms the observation based on annual means that the most persistent high-speed streams occur during the declining phase of all cycles during the last century \citep{Mursula_2015}, and extends this observation to the shorter-living streams with duration of about one solar rotation.
Interestingly, for 8 out of 9 solar cycles studied (all except for cycle 21 when the statistics of SW measurement in 1980s was poor), the years with the highest monthly SW speeds of the respective cycle are the same or adjacent years to the peak years based on the annual SW speeds. This suggests that the most persistent coronal holes lasting for several months (solar rotations) are also the sources of the highest monthly (one solar rotation) SW speeds. Accordingly, no short-term coronal holes are found that would be large or effective enough to produce the highest monthly SW speed of any solar cycle.
In seven months (May-June 1930, February-March 1952, April-May 1994 and June 2003) the monthly mean SW speed based on the $\Delta{H}$ proxy exceeded 550 km/s.
All these months occur in years of the highest annual solar wind speed in the respective cycle (16, 18, 22 and 23).
This further supports the idea that coronal holes emitting fast SW speed indeed tend to be persistent and live longer than one solar rotation.
The temporal distribution of the highest SW speeds is interesting. We find strong HSS activity during the rather weak sunspot cycle 16, which compares with the HSS activity of the later, more active sunspot cycles.
Thus, although the mean level of solar wind speed slightly changes (increases) with the long-term evolution of sunspot activity, the occurrence of the highest SW speeds (i.e., coronal holes) does not follow them very closely.
\begin{acknowledgments}
We acknowledge the financial support by the Academy of Finland to the ReSoLVE Centre of Excellence (project no. 272157).
We thank the Sodankyl\"a Geophysical Observatory for providing the magnetic field data at (\texttt{http://www.sgo.fi/}).
The solar wind data were downloaded from the OMNI2 database (\texttt{http://omniweb.gsfc.nasa.gov/}).
\end{acknowledgments}
|
1,314,259,993,648 | arxiv | \section{\@startsection {section}{1}{\z@}{-1.5ex plus -.5ex
minus -.2ex}{1ex plus .2ex}{\large\bf}}
\makeatletter
\def\subsection{\@startsection {subsection}{1}{\z@}{-1.5ex plus -.5ex
minus -.2ex}{1ex plus .2ex}{\bf}}
\newcommand\Lm{\Lambda}
\newcommand{w}{w}
\newcommand{w_\cS}{w_{\mathcal S}}
\newcommand{{\mathsf H}}{{\mathsf H}}
\newcommand\E{{\mathbb E}}
\newcommand{{\mathbf{D}}}{{\mathbf{D}}}
\newcommand{{\mathbf{GUD}}}{{\mathbf{GUD}}}
\newcommand{{\mathbf{SUD}}}{{\mathbf{SUD}}}
\newcommand{{\mathbf{GD}}}{{\mathbf{GD}}}
\newcommand{{\mathbf{SD}}}{{\mathbf{SD}}}
\newcommand{{\bf{D}}}{{\bf{D}}}
\newcommand{{\mathsf{F}}}{{\mathsf{F}}}
\newcommand{{\mathbf{G}}}{{\mathbf{G}}}
\newcommand{{\mathbf{R}}}{{\mathbf{R}}}
\newcommand{{\hat D}}{{\hat D}}
\newcommand{{\pmb\pi}}{{\pmb\pi}}
\newcommand{{\bf 1}}{{\bf 1}}
\newcommand{{\pmb\tau}}{{\pmb\tau}}
\def\<{\langle}
\def\>{\rangle}
\newcommand\Tr{{\rm Tr}}
\newcommand{\tilde{\mathscr{A}}}{\tilde{\mathscr{A}}}
\newcommand{\tilde{\mathscr{B}}}{\tilde{\mathscr{B}}}
\newcommand{{\mathscr{A}}}{{\mathscr{A}}}
\newcommand{\tilde{\mathscr{G}}}{\tilde{\mathscr{G}}}
\newcommand{{\mathscr{F}}}{{\mathscr{F}}}
\newcommand{\ul{\mathscr{F}}}{\underline{\mathscr{F}}}
\newcommand{\ul{\mathscr{L}}}{\underline{\mathscr{L}}}
\newcommand{{\mathscr{G}}}{{\mathscr{G}}}
\newcommand{{\mathscr{H}}}{{\mathscr{H}}}
\newcommand{{\mathscr{L}}}{{\mathscr{L}}}
\newcommand{\ul{\mathscr{G}}}{\underline{\mathscr{G}}}
\newcommand{{\mathscr{R}}}{{\mathscr{R}}}
\newcommand{{\mathscr{B}}}{{\mathscr{B}}}
\DeclareMathOperator{\Spin}{Spin}
\newcommand{{L}}{{L}}
\newcommand{\bar{L}}{\bar{L}}
\newcommand{\omega}{\omega}
\usepackage{hyperref}
\newcommand{\email}{\email}
\newenvironment{keyword}{Keywords: \keywords}{}
\newcommand{\MSC}[1]{MSC #1:\subjclass}
\newcommand{\hspace{1ex}}{\hspace{1ex}}
\begin{document}
\title[Curtis-Tits groups and Phan groups]{\Large Realizations and properties of $3$-spherical Curtis-Tits Groups and Phan groups}
\author[R.~Blok]{Rieuwert J. Blok}
\email{blokr@member.ams.org}
\address{Department of Mathematics and Statistics\\
Bowling Green State University\\
Bowling Green, OH 43403\\
U.S.A.}
\author[C.~Hoffman]{Corneliu G. Hoffman}
\email{C.G.Hoffman@bham.ac.uk}
\address{University of Birmingham\\
Edgbaston, B15 2TT\\
U.K.}
\begin{abstract}
In this note we establish the existence of all Curtis-Tits groups and Phan groups with $3$-spherical diagram as classified in~\cite{BloHofShp2017} and investigate some of their geometric and group theoretic properties.
Whereas it is known that orientable Curtis-Tits groups with spherical or non-spherical and non-affine diagram are almost simple, we show that non-orientable Curtis-Tits groups are acylindrically hyperbolic and therefore have infinitely many infinite-index normal subgroups.
However, we also provide concrete examples of non-orientable Curtis-Tits groups whose quotients are finite simple groups of Lie type.
\end{abstract}
\maketitle
\begin{keyword}
Curtis-Tits groups, Phan groups, groups of Kac-Moody type, lattices, abstract simplicity.
\MSC[2010] 20G35 \hspace{1ex}
51E24%
\end{keyword}
\section{Introduction}
In~\cite{BloHofShp2017} Curtis-Tits amalgams and Phan amalgams over a finite field ${\mathbb F}_q$ with $3$-spherical diagram $\Gamma$ and weak system of fundamental root groups (Curtis-Tits case) or property (D) (Phan case) were completely classified, generalizing the result from~\cite{BloHof2014b} which covered the case of Curtis-Tits amalgams over a field of order $\ge 4$ with simply-laced
$3$-spherical diagram satisfying property (D).
In the present paper we shall generalize the results from~\cite{BloHof2016} showing that not only all Curtis-Tits amalgams but also all Phan amalgams classified in~\cite{BloHofShp2017} have non-trivial completions.
More precisely, using the notation of Definition~\ref{dfn:representative amalgams} we have the following.
\begin{mainthm}\label{mainthm:CT realization}
Every Curtis-Tits amalgam of the form ${\mathscr{G}}(\delta)$ has a non-trivial completion.
In particular, an arbitrary Curtis-Tits amalgam over ${\mathbb F}_q$ with $3$-spherical diagram having no $C_2(2)$-subdiagrams has a non-trivial completion if and only if it possesses a weak system of fundamental root groups.
\end{mainthm}
\noindent{\bf Proof}\hspace{7pt}
The first claim is the content of Theorems~\ref{thm:OCT realization}~and~\ref{thm:NOCT realization}.
We now recall that it was shown in~\cite{BloHofShp2017} that an arbitrary Curtis-Tits amalgam over ${\mathbb F}_q$ with $3$-spherical diagram possessing some non-trivial completion does have a weak system of fundamental root groups and is isomorphic to ${\mathscr{G}}(\delta)$ for some $\delta$, so the second claim follows from the first.
\rule{1ex}{1ex}
\begin{remark}\label{rem:CT non-universal}
Note that Curtis-Tits amalgams as defined here are all universal in the sense that the groups appearing in them are universal groups of Lie type. In~\cite{BloHof2016} we also derive existence criteria for Curtis-Tits type amalgams with $3$-spherical simply-laced diagrams that are not universal.
We expect that an analogous treatment of general $3$-spherical Curtis-Tits amalgams will yield a similar result, but we shall not work out the details here.
\end{remark}
\begin{mainthm}\label{mainthm:Phan realization}
Every Phan amalgam of the form ${\mathscr{G}}(\delta)$ has a non-trivial completion.
In particular, an arbitrary Phan amalgam over ${\mathbb F}_q$ with $3$-spherical diagram has a non-trivial completion if and only if it satisfies property (D).
\end{mainthm}
\noindent{\bf Proof}\hspace{7pt}
The first claim is the content of Part 2.~in~Lemma~\ref{lem:delta and deltabar}.
We now recall that it was shown in~\cite{BloHofShp2017} that an arbitrary Phan amalgam over ${\mathbb F}_q$ with $3$-spherical diagram possessing some non-trivial completion must satisfy property (D) and is isomorphic to ${\mathscr{G}}(\delta)$ for some $\delta$, so the second claim follows from the first.
\rule{1ex}{1ex}
\medskip
As before, the Curtis-Tits amalgams fall into two categories: Orientable Curtis-Tits groups are essentially groups of Kac-Moody type (Theorem~\ref{thm:OCT realization}). Non-orientable Curtis-Tits groups can be obtained as (central extensions of) subgroups of groups of Kac-Moody type fixed under an involution that interchanges positive and negative roots groups and permutes types non-trivially (Theorem~\ref{thm:NOCT realization}).
Note that the latter involutions are not Phan involutions.
Completions of Phan amalgams are obtained as subgroups of groups of Kac-Moody type fixed under a Phan involution (which does fix the types) (Lemmas~\ref{lem:fixed CT amalgam is Phan amalgam}~and~\ref{lem:delta and deltabar}).
We show (Proposition~\ref{prop:G is a lattice of bhKM+}) that the completions of non-orientable Curtis-Tits groups are lattices in the ambient group of Kac-Moody type, thus generalizing a result from~\cite{GraMuh2008}.
Now suppose $\Gamma$ is non-spherical and non-affine and that ${\rm k}$ is finite.
By~\cite{CapRem2009}, the corresponding orientable Curtis-Tits groups are almost simple (Corollary~\ref{cor:OCT almost simple}).
Note that the diagram of a non-orientable Curtis-Tits groups is either ${\widetilde{A}}_{n-1}$ or it is non-spherical and non-affine.
In the present note we use a result from~\cite{CapHum2015} to show that non-orientable Curtis-Tits groups over finite fields with $3$-spherical diagram different from ${\widetilde{A}}_{n-1}$ are acylindrically hyperbolic and in particular are not abstractly simple (Theorem~\ref{thm:non-orientable CT groups are not simple}).
In the case where $\Gamma$ has type ${\widetilde{A}}_{n-1}$ both orientable and non-orientable Curtis-Tits groups have interesting quotients~\cite{BloHof2014a}. In Subsection~\ref{subsec:examples} we also exhibit finite quotients of non-orientable Curtis-Tits groups with non-spherical and non-affine diagram.
Throughout the paper it is our intention to be as concrete and explicit as we can be.
\section{Curtis-Tits and Phan amalgams}\label{sec:amalgams}
\subsection{Amalgams of Curtis-Tits and Phan type}\label{subsec:CTP amalgams}
\begin{definition}\label{dfn: CTP structure}
Let $\Gamma=(I, E)$ be a Lie diagram.
A {\em Curtis-Tits (resp.~Phan) amalgam with Lie diagram $\Gamma$ over ${\mathbb F}_q$} is a collection
${\mathscr{G}}=\{{\mathbf{G}}_{i},{\mathbf{G}}_{i, j}, {\mathbf g}_{i,j} \mid i, j \in I\}$
such that
for every $i,j\in I$, ${\mathbf g}_{i,j}\colon {\mathbf{G}}_i\to{\mathbf{G}}_{i,j}$ is a homomorphism of groups and, setting
$\bar{\amgrpG}_i={\mathbf g}_{i,j}({\mathbf{G}}_i)$, the triple
$({\mathbf{G}}_{i,j}, \bar{\amgrpG}_i, \bar{\amgrpG}_j)$ is a Curtis-Tits / Phan standard pair of type $\Gamma_{i,j}(q^e)$, for some $e\ge 1$ as defined in~\cite{BloHofShp2017}.
Moreover $e=1$ is realized for some $i,j\in I$.
For any subset $K\subseteq I$ , we let
\begin{align*}
{\mathscr{G}}_K&=\{{\mathbf{G}}_{i},{\mathbf{G}}_{i, j}, {\mathbf g}_{i,j} \mid i, j \in K\}.
\end{align*}
A {\em completion} of ${\mathscr{G}}$ is a group ${{G}}$ together with a collection $\gamma_\bullet=\{\gamma_i,\gamma_{i,j}\colon i,j\in I\}$ of homomorphisms $\gamma_i\colon {\mathbf{G}}_i\to {{G}}$, and $\gamma_{i,j}\colon {\mathbf{G}}_{i,j}\to {{G}}$, whose images - often denoted ${{{G}}}_i=\gamma_i({\mathbf{G}}_i)$ - generate ${{G}}$, such that for any $i,j\in I$, $\gamma_{i,j}\mathbin{ \circ }{\mathbf g}_{i,j}=\gamma_i$.
The amalgam ${\mathscr{G}}$ is {\em non-collapsing} if it has a non-trivial completion.
As a convention, for any subgroup $\amgrp{H}\le {\mathbf{G}}_J$, let ${G}{H}=\gamma({\mathbf{H}})\le {{G}}$.
A completion $({\tilde{G}},{\tilde{\gamma}}_\bullet)$ is called {\em universal} if for any completion $({{G}},\gamma_\bullet)$ there is a unique surjective group homomorphism $\pi\colon {\tilde{G}}\to {{G}}$ such that $\gamma_\bullet=\pi\mathbin{ \circ }{\tilde{\gamma}}_\bullet$. A universal completion always exists and is unique, but it may be trivial.
\end{definition}
\begin{definition}\label{dfn:standard amalgam}
Let $\Gamma=(I, E)$ be a Lie diagram.
The {\em standard Curtis-Tits (resp.~Phan) amalgam with Lie diagram $\Gamma$ over ${\mathbb F}_q$} is the Curtis-Tits (resp.~Phan) amalgam
$\ul{\mathscr{G}}=\{{\mathbf{G}}_{i},{\mathbf{G}}_{i, j}, \ul{\mathbf g}_{i,j} \mid i, j \in I\}$ in which $\ul{\mathbf g}_{i,j}$ is the standard identification map as defined in~\cite{BloHofShp2017} for all $i,j\in I$.
\end{definition}
We now describe all Curtis-Tits and Phan amalgams arising from the classification results of~\cite{BloHofShp2017}.
To this end, first consider certain groups ${\mathbf{C}}_i$ of automorphisms of the vertex groups $\mathop{\flexbox{\rm SL}}\nolimits_2(q)$ (Curtis-Tits case) and $\SU_2(q)$ (Phan case). These are certain subgroups of the vertex groups of the Coefficient system ${\mathscr{A}}$ of~\cite{BloHofShp2017}.
\paragraph{${\mathbf{C}}_i$ Curtis-Tits case}
For any $i\in I$, there is some $e\in {\mathbb N}$ such that ${\mathbf{G}}_i=\mathop{\flexbox{\rm SL}}\nolimits_2(q^e)$ via the standard identification maps and ${\mathbf{C}}_i=Aut({\mathbb F}_{q^e})\times\langle \tau\rangle$ (with $\tau$ of order $2$).
Here $\alpha\in \mathop{\rm Aut}\nolimits({\mathbb F}_{q^e})$ acts as a Frobenius automorphism and $\tau$ acts as transpose-inverse.
\paragraph{${\mathbf{C}}_i$ Phan case}
For any $i\in I$, we have ${\mathbf{G}}_i=\SU_2(q)$ via the standard identification maps and ${\mathbf{C}}_i=\mathop{\rm Aut}\nolimits({\mathbb F}_{q^2})$.
Here $\alpha\in \mathop{\rm Aut}\nolimits({\mathbb F}_{q^2})$ acts as a Frobenius automorphism with respect to an orthonormal basis for the hermitian form.
Note that $\tau$ (transpose-inverse) acts as $\sigma\colon x\mapsto x^q$ ($x\in {\mathbb F}_{q^2}$) on $\SU_2(q)$.
\paragraph{The spanning tree for $\Gamma$}
As customary we view the Dynkin diagram $\Gamma$ as an oriented edge-labeled graph. Let $\ul{\liediag}$ denote the underlying undirected simple graph.
We now fix a spanning tree $T=(I,E(T))$ for $\ul{\liediag}$ and let $\edg\ul{\liediag}-\edg T=\{\{i_s,j_s\}\colon s=1,2,\ldots,r\}$ together with certain integers $\{e_s\colon s=1,2,\ldots,r\}$.
In the Phan case, any spanning tree suffices and $e_s=1$ for all $s$. In the Curtis-Tits case select $T$ such that (see~\cite{BloHofShp2017}):
\begin{enumerate}
\item\label{cond:A2} $({\mathbf{G}}_{\{i_s,j_s\}},{\mathbf g}_{i_s,j_s}({\mathbf{G}}_{i_s}),{\mathbf g}_{i_s,j_s}({\mathbf{G}}_{j_s}))$ has type $A_2(q^{e_s})$, where $e_s$ is some power of $2$.
\item\label{cond:minimal e} There is a loop $\Lambda_s$ containing $\{i_s,j_s\}$ such that
any vertex group of $\Lambda_s$ is isomorphic to $\mathop{\flexbox{\rm SL}}\nolimits_2(q^{e_s 2^l})$ for some $l\ge 0$.
\end{enumerate}
\begin{definition}\label{dfn:representative amalgams}
The main results of~\cite{BloHofShp2017} now says that the Curtis-Tits (resp.~Phan) amalgams over ${\mathbb F}_q$ with diagram $\Gamma$ are, up to type preserving isomorphism, in bijection with the set
\begin{align*}
{\mathbf{C}}=\prod_{s=1}^r{\mathbf{C}}_{i_s}.
\end{align*}
Under this bijection, the sequence $\delta=(\delta_s)_{s=1}^r$ corresponds to the amalgam
\begin{align}
{\mathscr{G}}(\delta)&=\{{\mathbf{G}}_{i},{\mathbf{G}}_{i, j}, {\mathbf g}^\delta_{i,j} \mid i, j \in I\} \label{eqn:amG^delta}
\end{align}
where ${\mathbf g}^\delta_{i,j}=\ul{\mathbf g}_{i,j}$ for all $i,j\in I$ except that
${\mathbf g}_{i_s,j_s}^\delta= \ul{\mathbf g}_{i_s,j_s}\mathbin{ \circ }\delta_s$ for all $s=1,2,\ldots,r$.
We shall call $\ul{\mathscr{G}}(\delta)$ {\em the amalgam representing $\delta$}.
We finish this discussion with some terminology applying only to Curtis-Tits amalgams. We say that $\ul{\mathscr{G}}(\delta)$ is {\em orientable} if $\delta_s\in \mathop{\rm Aut}\nolimits({\mathbb F}_q^{e_s})$ for all $s=1,\ldots,r$
(that is, $\delta$ does not involve any $\tau$) and {\em non-orientable} otherwise.
Note that we can interpret $\delta$ as the image of a homomorphism
\begin{align*}
\omega\colon \pi_1(\Gamma,0)& \to {\mathbf{C}}\\
[\Lambda_s]&\mapsto \delta_s
\end{align*}
where $0$ is some base vertex of $\Gamma$ and $[\Lambda_s]$ denotes the homotopy class of the loop $\Lambda_s$ above.
Consider the composition of $\omega$ and the natural projection map:
\begin{align}
\omega^*\colon \pi_1(\Gamma,0)\to \langle \tau\rangle\cong {\mathbb Z}/2{\mathbb Z}\label{eqn:classmap star}
\end{align}
Then ${\mathscr{G}}$ is orientable if and only if the image of the corresponding $\omega^*$ is trivial.
\end{definition}
From now on ${\mathscr{G}}=\ul{\mathscr{G}}(\delta)=\{{\mathbf{G}}_{i},{\mathbf{G}}_{i, j}, {\mathbf g}_{i,j} \mid i, j \in I\}$
for some $\delta$.
We shall only consider Curtis-Tits amalgams ${\mathscr{G}}$ over a finite field ${\mathbb F}_q$ possessing a weak system of fundamental root groups with $3$-spherical (but not spherical) diagram having no subdiagrams of type $C_2(2)$.
The latter condition was not necessary for classification, but it is necessary when considering completions, as it is necessary to satisfy condition (co) of~\cite{Mu1999}.
\section{Orientable Curtis-Tits groups are groups of Kac-Moody type}\label{sec:curtis-tits realizations}
\subsection{Realization of Orientable Curtis-Tits amalgams} Realization of all Curtis-Tits amalgams arising from the classification can be achieved along the lines of~\cite{BloHof2016,BloHof2014b}, where they were obtained for Curtis-Tits amalgams over a finite field ${\mathbb F}_q$ with $q\ge 4$ satisfying property (D) and having $3$-spherical simply-laced diagram. The present situation only requires us to modify the proof in certain places, and we will content ourselves with pointing out these differences.
We start by assuming that ${\mathscr{L}}=\{{\mathbf{L}}_i,{\mathbf{L}}_{i,j},{\mathbf l}_{i,j}\colon i\in I\}$ is an orientable Curtis-Tits amalgam over ${\mathbb F}_q$ with $3$-spherical diagram $\Lambda$ without $C_2(2)$-subdiagrams..
Following Tits~\cite{Ti1992} a {{\em group of Kac-Moody type}}{} is by definition a group with RGD such that a central quotient is the subgroup of $\mathop{\rm Aut}\nolimits(\Delta)$ generated by the root groups of an apartment in a Moufang twin-building {$\Delta$}{}. This central quotient will be called the associated {\em adjoint} group of Kac-Moody type. In this section we shall prove the following.
As a general reference for groups with root group datum we will use~\cite{CapRem2009a}. In particular, we assume that such a groups is generated by the root groups of the root group datum.
\begin{theorem} \label{thm:OCT realization}
The universal completion of ${\mathscr{L}}$ is a group of Kac-Moody type (and ${\mathscr{L}}$ is the Curtis-Tits amalgam for this group) if and only if ${\mathscr{L}}$ is orientable.
\eth
Theorem~\ref{thm:OCT realization} generalizes Corollary 1.2 of~\cite{BloHof2014b} and its proof follows the steps detailed in Section 5 of~\cite{BloHof2014b}.
So as not to repeat that proof nearly verbatim, we merely indicate how the more general assumptions of Theorem~\ref{thm:OCT realization} still yield the same result.
\medskip
\noindent{\bf Proof}\hspace{7pt}(of Theorem~\ref{thm:OCT realization})
Following Subsection~5.1 of~\cite{BloHof2014b} we consider a simply-connected locally split Kac-Moody group over ${\mathbb F}_q$ with diagram $\Lambda$ and consider the twin-building $\Delta=((\Delta_+,\delta_+),(\Delta_-,\delta_-),\delta_*)$ associated to its twin BN-pair $(B^+,N,B^-)$. Now as $\Lambda$ is $3$-spherical and has no $C_2(2)$ subdiagrams, it satisfies condition (co) of~\cite{MuRo1995} so the local structure determines the global structure. For the same reason $\Lambda$ satisfies condition (${\rm co}^*$) from~\cite{Cap2007} so that by the main result of that paper, we obtain the group as a central quotient of the universal completion of an amalgam
\begin{align*}
{\mathcal R}&=\{R_i,R_{i,j}, \rho_{i,j}\mid i,j\in I\},
\end{align*}
where $R_i=\langle X_i^+,X_i^-\rangle$, $R_{i,j}=\langle R_i,R_j\rangle$,
and $\{X_i^+\mid i\in I\}$ is a selection of positive root groups corresponding to a fundamental system of positive roots, and $ \rho_{i,j}\colon R_i \hookrightarrow R_{i,j}$ is given by inclusion of subgroups in $G$.
Now ${\mathcal R}$ is the desired Curtis-Tits amalgam. Note that the analog of Lemma 5.2 is not valid as for instance the center of $\mathop{\rm Sp}\nolimits_4(q)$ meets one of the rank $1$ Levi groups, but this does not affect the validity of the conclusion.
Note that ${\mathcal X}=\{X_i^+,X_i^-\colon i\in I\}$ is the weak system of fundamental root groups and the fact that we can select the $+$ signs to correspond to a fundamental system of positive roots means that the amalgam is orientable.
We now show that the universal completion of any orientable ${\mathscr{L}}$ has a central quotient that is the automorphism group of a twin-building following Subsection 5.3 of~\cite{BloHof2014b}. We now take the definition of a sound Moufang foundation as in~\cite{Mu1999}. As ${\mathscr{L}}$ is orientable, we obtain the rank-$2$ Moufang buildings $\Delta_{i,j}$ of type $\Lambda_{i,j}$ of the foundation ${\mathsf{F}}$ from the Curtis-Tits standard pairs $({\mathbf{L}}_{i,j},{\mathbf{L}}_i,{\mathbf{L}}_j)$ in ${\mathscr{L}}$ using the Borel subgroups $\{{\mathbf{B}}_{i,j}^+,{\mathbf{B}}_{i,j}^-\}$, which are uniquely determined by $\{{\mathbf{X}}_i^\pm,{\mathbf{X}}_j^\pm\}\subseteq {\mathcal X}$. We now select the chamber $C_{i,j}\in \Delta_{i,j}$ of the foundation to be those associated to ${\mathbf{B}}_{i,j}^+$. Also for each $i\in I$ we select an auxiliary chamber $C_i$ in the rank-$1$ building $\Delta_i$ associated to the BN pair $({\mathbf{B}}_i^+,N_i,{\mathbf{B}}_i^-)$. We define an inclusion map $\theta_i^j\colon \Delta_i\to \Delta_{i,j}$ induced by ${\mathbf l}_{i,j}$ and let the restriction maps of ${\mathsf{F}}$ be given by $\theta_{i,j}^{j,k}=\theta_j^k\mathbin{ \circ }(\theta_j^i)^{-1}$.
It is immediate that the $\theta_{i,j}^{j,k}$ satisfy the condition (Fo3) of a Moufang foundation.
Soundness of ${\mathsf{F}}$ follows from the fact that $\Lambda$ is $3$-spherical and the fact that the subamalgam of ${\mathscr{L}}$ of type $\Lambda_J$ is the Curtis-Tits amalgam associated to the corresponding spherical building.
The fact that the signature of ${\mathcal X}$ determines a twin-apartment is proved as in~\cite{BloHof2014b}.
The proof is completed exactly as in~\cite{BloHof2014b}: By~\cite{Mu1999}, the sound Moufang foundation can be integrated to a twin-building $\Delta$ whose automorphism group contains a non-trivial homomorphic image of the original Curtis-Tits amalgam, generated by the root groups associated to the roots intersecting the $E_2(c)$ for some chamber $c$. Universality gives a homomorphism from the universal completion ${\tilde{L}}$ of ${\mathscr{L}}$ to the subgroup $\mathop{\rm Aut}\nolimits(\Delta)^{\dagger}$ of $\mathop{\rm Aut}\nolimits(\Delta)$ generated by these root groups; the kernel of the action of ${\tilde{L}}$ on $\Delta$ must be central, as required.
\rule{1ex}{1ex}
\begin{remark}
In~\cite{BloHof2016, BloHof2014a} we obtained rather precise information on the particular central subgroups appearing in the kernel of the map from the universal completion of ${\mathscr{L}}$ to $\mathop{\rm Aut}\nolimits(\Delta)$. For the purposes of the present paper, it suffices to establish the existence of a completion for ${\mathscr{L}}$.
However, we expect that the techniques developed in~\cite{BloHof2016,BloHof2014a} can be used to handle this more general case, although the details will probably a bit more involved.
\end{remark}
In subsequent sections we shall prove that non-orientable Curtis-Tits (resp.~Phan) amalgams have non-trivial completions inside the subgroup of an orientable Curtis-Tits group fixed under a certain Cartan (resp.~Phan) involution.
\subsection{The twin-building $\Delta$ associated to ${\tilde{L}}$}\label{subsec:twin-building of amL}
Let $({\tilde{L}},{\tilde{\lambda}})$ be the universal completion of ${\mathscr{L}}$.
For future reference, we introduce the notation necessary to talk about the twin-building related to ${\tilde{L}}$.
Note that ${\tilde{L}}$ is a group of Kac-Moody type over ${\mathbb F}_q$ with diagram $\Lambda$.
Let $(W,S=\{s_i\colon i\in I\})$ the associated Coxeter system with root system $\Phi$.
Now ${\tilde{L}}$ is a group with a locally finite root group datum $\{{{U}}_\alpha\colon \alpha\in \Phi\}$ (namely ${{U}}_\alpha$ is finite for all $\alpha\in \Phi$ (see~\cite{CapRem2009,Cap2007}).
This means in particular that ${\tilde{L}}=\langle {{U}}_\alpha\colon \alpha\in \Pi\rangle$ (for a root base $\Pi$ of $\Phi$) has a twin $BN$-pair $(({{B}}^+,{{N}}), ({{B}}^-,{{N}}))$, where ${{B}}^\varepsilon={{D}}{{U}}^\varepsilon$, setting
\begin{align*}
{{U}}^\varepsilon&=\langle {{U}}_\alpha\colon \alpha\in \Phi^\varepsilon\rangle && (\varepsilon=+,-) \\
{{N}}&=\langle \mu(u)\colon u\in {{U}}_\alpha-\{1\}, \alpha\in \Pi\rangle,\\
{{D}}&=\bigcap_{\alpha\in \Phi}N_{{\tilde{L}}}({{U}}_\alpha).
\end{align*}
In fact this is the twin BN-pair giving rise to a twin-building $((\Delta_+,\delta_+),$ $ (\Delta_-,\delta_-),\delta_*)$.
As proved in~\cite{BloHofShp2017} the weak system of fundamental root groups can be selected so that for some fundamental system $\Pi=\{\alpha_i\colon i\in I\}$ of $\Phi$, and all $i\in I$, we have
${{U}}_{\alpha_i}=\lambda({\mathbf{X}}_i^+)$.
\section{Non-orientable Curtis-Tits groups are fixed groups of Cartan involutions}\label{sec:NO CT groups}
\subsection{The ambient orientable amalgam ${\mathscr{L}}$}
We shall now assume that ${\mathscr{G}}=\ul{\mathscr{G}}(\delta)=\{{\mathbf{G}}_{i},{\mathbf{G}}_{i, j}, {\mathbf g}_{i,j} \mid i, j \in I\}$
where $\delta=(\delta_s)_{s=1}^r\in \prod_{s=1}^r {\mathbf{C}}_{j_s}$ is non-orientable. This means that the map $\omega^*$ of~\eqref{eqn:classmap star} is surjective.
\begin{definition} {(The covering of diagrams $p\colon \Lambda\to\Gamma$)}
Consider the map $\omega^*\colon \pi(\Gamma,{0})\to \langle \tau\rangle$.
Its kernel is the fundamental group of a two-sheeted covering $p\colon \Lambda\to \Gamma$ sending some vertex $\hat{0}$ to $0$.
Since $\Gamma$ has no circuits of length $\le 3$, the quotient $\pi(\Gm,0)/\pi(\Lambda,\hat{0})=\langle\theta\rangle\cong {\mathbb Z}/2{\mathbb Z}$ acts as a group of deck transformations
commuting with $p$; in particular $p$ does not fix points or edges.
\end{definition}
We now lift ${\mathscr{G}}$ to a locally isomorphic amalgam ${\mathscr{L}}$ defined over $\Lambda$ and extend $\theta$ to ${\mathscr{L}}$.
\begin{definition}\label{dfn:hamG}
Let ${\mathscr{L}}=\{{\mathbf{L}}_{i},{\mathbf{L}}_{i, j}, \lambda_{i,j} \mid i, j \in \widehat{I}\}$ be the amalgam such that, for all $i,j\in \widehat{I}$,
\begin{enumerate}
\item[(${\mathscr{L}}$1)]
${\mathbf{L}}_i\mbox{ is a copy of }{\mathbf{G}}_{p(i)}$,
\item[(${\mathscr{L}}$2)]
$ {\mathbf{L}}_{i,j}\mbox{ is a copy of }
\begin{cases}
{\mathbf{G}}_{p(i),p(j)} &\mbox{if } \{i,j\}\in {E(\La)}, \\
{\mathbf{G}}_{p(i)}\times {\mathbf{G}}_{p(j)} & \mbox{else. } \\
\end{cases}$
\item[(${\mathscr{L}}$3)]
$ \lambda_{i,j}
=
\begin{cases}
\varphi_{p(i),p(j)} & \mbox{if } \{i,j\}\in {E(\La)},\\
\mbox{ canonical inclusion } & \mbox{else.}
\end{cases}$
\end{enumerate}
This means the following. Fix some $J\subseteq \widehat{I}$ with $1\le |J|\le 2$, and denote by $\pi\colon {\mathscr{L}}\to {\mathscr{G}}$ the homomorphism of amalgams induced by $p$.
That is, identifying ${\mathbf{G}}_{p(J)}$ with its copies ${\mathbf{L}}_J$ and ${\mathbf{L}}_{\theta(J)}$, we let the maps $\pi_J$ and $\pi_{\theta(J)}$ be the identity mappings.
Then, we have a commuting diagram of isomorphisms
\begin{align}\label{eqn:hamG covers amG}
\xymatrix{
x\in {\mathbf{L}}_J\ar[dr]^{\pi_J} \ar[rr]^\theta && {\mathbf{L}}_{\theta(J)} \ni x \ar[dl]^{\pi_{\theta(J)}}\\
& x\in {\mathbf{G}}_{p(J)} &
}
\end{align}
Thus, in~\eqref{eqn:hamG covers amG}, also $\theta$ is given by the identity mapping.
Then, by (${\mathscr{L}}$3) $\theta$ is an automorphism of ${\mathscr{L}}$. Also $\pi$ can be viewed as a $2$-covering of Curtis-Tits amalgams.
\end{definition}
Since $\pi$ induces an isomorphism on every vertex and edge group of ${\mathscr{L}}$ we have the following.
If $\delta_s\in\mathop{\rm Aut}\nolimits({\mathbb F}_{q^{2^{e_s}}})$, then the $p$-fiber over $\Lambda_s$ consists of two disjoint loops, and the subamalgams of ${\mathscr{L}}$ induced on each of these is isomorphic to the one induced on $\Lambda_s$, hence they correspond to $\delta_s$ as well.
Otherwise the $p$-fiber over $\Lambda_s$ is a single loop $\hat{\Lambda}_s$ doubly covering $\Lambda_s$ and it corresponds to $\delta_s^2\in\mathop{\rm Aut}\nolimits({\mathbb F}_{q^{2^{e_s}}})$.
Hence ${\mathscr{L}}$ is orientable. It follows from the classification theorem that it is isomorphic to some standard orientable Curtis-Tits amalgam.
\subsection{Realization of ${\mathscr{G}}$ in the twisted group of Kac-Moody type ${\mathbf{L}}^\theta$}
Since ${\mathscr{L}}$ is orientable, Theorem~\ref{thm:OCT realization} implies that ${\mathscr{L}}$ has a completion $({{L}},\lambda)$, where ${{L}}$ is a group of Kac-Moody type. Hence also the universal completion $({\tilde{L}},{\tilde{\lambda}})$ of ${\mathscr{L}}$ is not trivial.
For convenience, we shall replace ${\mathscr{L}}$ by its image in ${\tilde{L}}$.
Note that this image does not have to be isomorphic to ${\mathscr{L}}$, but when considering completions, there is no loss of generality.
By universality, the automorphism $\theta\colon {\mathscr{L}}\to{\mathscr{L}}$ induces a Cartan involution, also denoted $\theta$, on the universal completion ${\tilde{L}}$.
\begin{definition}\label{dfn:Cartan involution}
A {\em Cartan involution} of a group with twin- $BN$-pair $(({{B}}^+,{{N}}),({{B}}^-,{{N}}))$ is an automorphism $\theta$ satisfying
\begin{enumerate}
\item $\theta^2=\mathop{\rm id}\nolimits$
\item$({{B}}^+)^\theta={{B}}^-$,
\item $\theta$ normalizes the Weyl group $W={{N}}/{{B}}^-={{N}}/{{B}}^-$ inducing a graph automorphism of $\Gamma$ without fixed vertices or edges.
\end{enumerate}
\end{definition}
\noindent
Let ${\tilde{L}}^\theta$ be the fixed group of ${\tilde{L}}$ under $\theta$ and define the amalgam of fixed subgroups
\begin{align*}
{\mathscr{L}}^\theta & = \{{\mathbf l}^\theta_{i,j}\colon \langle {\mathbf{L}}_i,{\mathbf{L}}_{\theta(i)}\rangle^\theta\hookrightarrow\langle{\mathbf{L}}_{i,j}, {\mathbf{L}}_{\theta(i),\theta(j)}\rangle^\theta\colon i,j\in I\},
\end{align*}
where, for each $i,j\in I$, we consider the fixed subgroups
\begin{align*}
\langle {\mathbf{L}}_i,{\mathbf{L}}_{\theta(i)}\rangle^\theta &=\{x x^\theta\colon x\in {\mathbf{L}}_i\},\\
\langle {\mathbf{L}}_{i,j},{\mathbf{L}}_{\theta(i), \theta(j)}\rangle^\theta &=\{x x^\theta\colon x\in {\mathbf{L}}_{i,j}\}.
\end{align*}
There is a non-trivial surjective morphism ${\mathscr{G}}\to {\mathscr{L}}^\theta$.
In~\cite{BloHof2016} this is shown for the image of ${\mathscr{L}}^\theta$ in ${{L}}$.
However, the proof translates verbatim to obtain the result in the present more general setting.
Hence there is a non-trivial completion $({{G}},\gamma)$ of ${\mathscr{G}}$ with ${{G}}=\langle{\mathscr{L}}^\theta\rangle \le {\tilde{L}}^\theta$.
Thus, we have obtained the following.
\begin{theorem}\label{thm:NOCT realization}
Let ${\mathscr{G}}$ be a non-orientable Curtis-Tits amalgam over ${\mathbb F}_q$ with connected $3$-spherical Dynkin diagram $\Gamma$ having no subdiagrams of type $C_2(2)$. Then, there is a group of Kac-Moody type ${\tilde{L}}$ over ${\mathbb F}_q$ whose diagram is a two-sheeted covering of $\Gamma$ equipped with a Cartan involution $\theta$ such that the fixed group ${\tilde{L}}^\theta$ contains a non-trivial completion of ${\mathscr{G}}$.
\eth
\begin{remark}
Let $({\tilde{G}},{\tilde{\gamma}})$ be the universal completion of ${\mathscr{G}}$.
Then, there is a surjective homomorphism
\begin{align*}
\tilde{{{G}}}\twoheadrightarrow {{G}}\le {\tilde{L}}^\theta\le {\tilde{L}}.
\end{align*}
Let $\tilde{\pi}\colon{\tilde{L}}\to{{L}}$ be the map of completions of ${\mathscr{L}}$ given by universality.
Now ${\tilde{L}}$ is a central extension of ${{L}}$. If in fact ${{L}}={\tilde{L}}/Z({\tilde{L}})$, then $\theta$ induces an involution of ${{L}}$ so that $\tilde{\pi}$ sends ${\tilde{L}}^\theta$ to ${{L}}^\theta$. In this terminology,~\cite{BloHof2016} investigates the index $[{{L}}^\theta\colon\tilde{\pi}({{G}})]$ in great detail in the case of simply-laced diagrams.
\end{remark}
\section{Realization of Phan amalgams}\label{sec:Realization of Phan amalgams}
\subsection{Phan involutions}\label{subsec:phan involutions}
Recall the following definition of a Phan involution for groups with twin-root group datum.
\begin{definition}\label{dfn:orientable phan involution}
A {\em Phan involution} $\theta$ of a group ${{L}}$ with a twin root group datum (see Subsection~\ref{subsec:twin-building of amL}) is an automorphism of ${{L}}$ such that
\renewcommand{\theenumi}{\roman{enumi}}\begin{enumerate}
\item\label{Phan Inv i} $\theta^2=\mathop{\rm id}\nolimits$
\item\label{Phan Inv ii} $({{B}}^+)^\theta={{B}}^-$,
\item\label{Phan Inv iii} $\theta$ centralizes the Weyl group $W={{N}}/{{D}}$.
\end{enumerate}
\end{definition}
Now let ${\mathscr{G}}=\{{\mathbf g}_{i,j}\colon{\mathbf{G}}_i\to {\mathbf{G}}_{i,j}\colon i,j\in I\}$ be a (possibly non-orientable) Curtis-Tits amalgam over ${\mathbb F}_q$ with $3$-spherical diagram $\Gamma$.
Let $({\tilde{G}},{\tilde{\gamma}})$ denote its universal completion and let $({{G}},\gamma)$ be some completion
with canonical map $\tilde{\pi}\colon {\tilde{G}}\to{{G}}$.
\begin{definition}\label{dfn:phan involution of CT group}
A {\em Phan involution} of ${\mathscr{G}}$ is an automorphism $\theta=\{\theta_i,\theta_{i,j}\colon i,j\in I\}$ of ${\mathscr{G}}$ that induces a Phan involution on each group of ${\mathscr{G}}$.
\end{definition}
From now on let $\theta=\{\theta_i,\theta_{ij}\colon i,j\in I\}$ be a Phan involution of the Curtis-Tits amalgam ${\mathscr{G}}$.
As a source of examples, we have the following observation.
\begin{lemma}\label{lem:phan involutions inducing phan involutions on CT amalgam}
Suppose that ${\mathscr{L}}$ is the Curtis-Tits amalgam arising from the action of a group of Kac-Moody type ${{L}}$ on its twin-building $\Delta$ over ${\mathbb F}_q$ with $3$-spherical diagram $\Lambda$ and that
$\lambda\colon {\mathscr{L}}\to{{L}}$ is the completion map.
If $\theta$ is a Phan involution of ${{L}}$ preserving
${\mathscr{L}}$, then it induces a Phan involution on ${\mathscr{L}}$.
Conversely, a Phan involution $\theta$ of ${\mathscr{L}}$ induces a Phan involution on ${\tilde{L}}$.
\end{lemma}
\noindent{\bf Proof}\hspace{7pt}
First note that any automorphism of ${\mathscr{L}}$ induces a unique automorphism of its universal completion $({\tilde{L}},{\tilde{\lambda}})$ that preserves ${\mathscr{L}}$.
Conversely, any automorphism of ${{L}}$ which preserves ${\mathscr{L}}$, induces an automorphism of ${\mathscr{L}}$.
In both cases, if the original automorphism is an involution, then so is the induced one.
Let $\theta$ denote the automorphism of ${\mathscr{L}}$ as well as the induced automorphism of ${{L}}$ in the forward direction, and of ${\tilde{L}}$ in the backward direction.
As we saw in~Subsection~\ref{subsec:twin-building of amL} not only ${{L}}$, but also ${\tilde{L}}$ is a group with root group datum. So as not to overload notation, in both cases (directions) we shall denote the datum
$\{{{U}}_\alpha\colon \alpha\in \Phi\}$.
Note that when proving either direction, we can view ${\mathscr{L}}$ as a concrete amalgam for ${{L}}$ (resp.~${\tilde{L}}$) in the sense that for any $i,j\in I$, the connecting map
${\mathbf l}_{i,j}$ (resp.~$\tilde{\mathbf l}_{i,j}$) is just inclusion of subgroups
(In the backward direction, for purposes of completions, it is harmless to identify ${\mathscr{L}}$ with its image in ${\tilde{L}}$). In the terminology of~\cite[Section 4]{BloHofShp2017} this means that
if ${\mathcal X}=\{{\mathbf{X}}_i^+,{\mathbf{X}}_i^-\colon i\in I\}$ is the weak system of fundamental root groups of ${\mathscr{L}}$, then
for any $j\in I$, we have ${\mathbf{X}}_i^\varepsilon={\mathbf l}_{i,j}({\mathbf{X}}_i^\varepsilon)$ (resp.~ ${\mathbf{X}}_i^\varepsilon=\tilde{\mathbf l}_{i,j}({\mathbf{X}}_i^\varepsilon)$) ($\varepsilon=+,-$). By uniqueness of fundamental root groups in ${\mathbf{L}}_{i,j}$ whenever $\Lambda_{i,j}\ne A_1\times A_1$ and connectedness of $\Lambda$, we then have ${\mathcal X}=\{{{U}}_{\alpha_i},{{U}}_{-\alpha_i}\colon i\in I\}$. The significance here is that $\Pi=\{\alpha_i\colon i\in I\}$ is a root basis for $\Phi$, so that
\begin{align}
W\Pi=\Phi.\label{eqn:Chi^+ is a root basis for Phi}
\end{align}
Since in the forward (resp.~backward) direction the automorphism $\theta$ of ${{L}}$ (resp.~of~${\tilde{L}}$) preserves each group of ${\mathscr{L}}$, it follows that $\theta$ acts as
\begin{align}
{{U}}_{\alpha}^\theta&={{U}}_{-\alpha}\mbox{ for all }\alpha\in \Pi \label{eqn:theta swaps fundamental root groups}
\end{align}
{We now complete the proof of the forward direction.
Combining property~\eqref{Phan Inv iii} of the Phan involution $\theta$ on ${{L}}$, with~\eqref{eqn:Chi^+ is a root basis for Phi},~\eqref{eqn:theta swaps fundamental root groups} and applying the RGD axioms, we see that $\theta$ preserves the root group datum, while interchanging $\{{{U}}_\beta\colon \beta\in \Phi_J^+\}$ and $\{{{U}}_\beta\colon \beta\in \Phi^-_J\}$ for any $J\subseteq I$; in particular, $\theta$ preserves ${{N}}_J$ and ${{D}}$,
and centralizes $W_J={{N}}_J/{{D}}$.
Hence, $\theta_i$ and $\theta_{i,j}$ satisfy properties~\eqref{Phan Inv ii}~and~\eqref{Phan Inv iii} for each $i,j\in I$.
We now establish the backward direction in a similar manner using~\eqref{eqn:Chi^+ is a root basis for Phi}~and~\eqref{eqn:theta swaps fundamental root groups}. Thus, in order to establish that the automorphism $\theta$ of ${\tilde{L}}$ has property~\eqref{Phan Inv ii} it suffices to show that
it has property~\eqref{Phan Inv iii}. Recall that $N$ is generated by elements $\mu(u_i)$, for some $u_i\in {{U}}_{\alpha_i}$ with $\alpha_i\in \Pi$.
Let $(W,\{s_i\colon i\in I\})$ denote the Coxeter system where $s_i=\mu(u_i)D$.
Now note that $({\tilde{L}}_i,\{{{U}}_{\alpha_i},{{U}}_{-\alpha_i}\})$ is a group with root group datum since ${\tilde{L}}_i=\langle {{U}}_{\alpha_i},{{U}}_{-\alpha_i}\rangle_{{\tilde{L}}}$ (cf.~\cite[\S2.3]{CapRem2009a}).
Since $\theta_i$ satisfies property~\eqref{Phan Inv iii}, the element $\mu(u_i)^{\theta_i}\in {{U}}_{\alpha_i}u_i'{{U}}_{\alpha_i}$ (with $u_i'\in {{U}}_{-\alpha_i}$) must conjugate ${{U}}_\beta$ to ${{U}}_{s_i\beta}$ for all $\beta\in \Phi$; in particular it must do so for $\beta=\pm\alpha_i$.
However, in this standard root group datum of $\mathop{\flexbox{\rm SL}}\nolimits_2({\rm k})$ for some field ${\rm k}$, one verifies that this means that
$\mu(u_i)^{\theta_i}=\mu(u_i')$. Clearly $\mu(u_i)D=s_i=\mu_(u_i')D$. Thus $\theta$ centralizes $W$, as required.
\rule{1ex}{1ex}
}
\subsection{Fixed subamalgams}\label{subsec:fixed subamalgams}
\begin{definition}\label{dfn:fixed amalgam}
Let $\theta$ be a Phan involution of the Curtis-Tits amalgam ${\mathscr{G}}$. Define the fixed amalgam of ${\mathscr{G}}$ under $\theta$ as ${\mathscr{F}}=\{{\mathbf{F}}_{i,j},{\mathbf{F}}_i, {\mathbf f}_{i,j}\colon i,j\in I\}$,
where
\begin{align*}
{\mathbf{F}}_i & = {\mathbf{G}}_i^{\theta_i}, \\
{\mathbf{F}}_{i,j} & = {\mathbf{G}}_{i,j}^{\theta_{i,j}},\\
{\mathbf f}_{i,j} & = {\mathbf g}_{i,j}|_{{\mathbf{F}}_i}.
\end{align*}
We may therefore unambiguously write ${\mathscr{F}}={\mathscr{G}}^\theta$, ${\mathbf{F}}_J={\mathbf{G}}_J^\theta$ ($J\subseteq I$ with $0<|J|\le 2$) and denote the inclusion maps simply with ${\mathbf g}_{i,j}$ instead of ${\mathbf f}_{i,j}$.
\end{definition}
\begin{remark}\label{rem:orthogonal amalgams}
The case where $\theta$ induces $\tau$ on all groups of ${\mathscr{G}}=\ul{\mathscr{G}}$ is the one studied in~\cite{CapHum2015}.
For $A_n$ diagrams the situation is studied in detail in~\cite{Hoffman:2013aa}.
In the present paper we are mostly interested in the case where $\theta$ induces $\tau$ together with some field involution.
\end{remark}
\medskip
\noindent
\medskip
We now establish the relation between ${\mathscr{G}}$ and ${\mathscr{F}}$ in terms of the representative amalgams of Definition~\ref{dfn:representative amalgams}.
Let $\ul{\mathscr{G}}$ be the standard Curtis-Tits amalgam over ${\mathbb F}_{q^2}$ with connected $3$-spherical diagram $\Gamma$ (as in Definition~\ref{dfn:standard amalgam}), and assume that that for any $i,j\in I$,
$\Gamma_{i,j}\in \{A_1\times A_1, A_2,C_2/B_2\}$.
Note that since $\Gamma$ has no subdiagrams of type ${}^2\! {A}_3$, in fact we have that, for any $i,j\in I$
$({\mathbf{G}}_{i,j},\bar{\amgrpG}_i,\bar{\amgrpG}_j)$ is a Curtis-Tits standard pair of type $\Gamma_{i,j}(q^2)\in \{A_1(q^2)\times A_1(q^2), A_2(q^2),C_2(q^2)/B_2(q^2)\}$.
Fix some spanning tree $\Sigma\subseteq \Gamma$ and suppose that $\edg\Gamma-\edg\Sigma=\{\{i_s,j_s\}\colon s=1,2,\ldots,r\}$ so that $H_1(\Lambda,{\mathbb Z})\cong{\mathbb Z}^r$.
Let ${\mathscr{G}}=\ul{\mathscr{G}}(\delta)$ for some $\delta\in \prod_{s=1}^r{\mathbf{C}}_{i_s}$.
Note that ${\mathbf{C}}_{i_s}=\mathop{\rm Aut}\nolimits({\mathbb F}_{q^2})\times\langle\tau\rangle$ for all $s$.
We let $\theta=\sigma\mathbin{ \circ }\tau$, where we recall that $\sigma\colon x\mapsto x^q$ for $x\in {\mathbb F}_{q^2}$.
\begin{lemma}\label{lem:fixed CT amalgam is Phan amalgam}
The fixed amalgam ${\mathscr{F}}$ of ${\mathscr{G}}$ under $\theta$ is a Phan amalgam over ${\mathbb F}_q$ with diagram $\Gamma$.
\end{lemma}
\noindent{\bf Proof}\hspace{7pt}
We simply have to verify that if $({\mathbf{G}}_{i,j},\bar{\amgrpG}_i,\bar{\amgrpG}_j)$ is a Curtis-Tits standard pair of type $\Gamma_{i,j}(q^2)\in \{A_1(q^2)\times A_1(q^2), A_2(q^2),C_2(q^2)/B_2(q^2)\}$, then
$({\mathbf{F}}_{i,j}^{\theta_{ij}},\bar{\amgrpF}_i^{\theta_{i}},\bar{\amgrpF}_j^{\theta_{j}})$ is a Phan standard pair of type $\Gamma_{i,j}(q)\in \{A_1(q)\times A_1(q), A_2(q),C_2(q)/B_2(q)\}$.
Now note that $\theta$ is defined with respect to the standard basis used to define the Curtis-Tits standard pair.
Thus, the fixed group under $\theta$ is the intersection of the group in the Curtis-Tits standard pair and the unitary group preserving the hermitian form for which this standard basis is orthonormal.
In every case this is exactly the corresponding vertex or edge group of the Phan standard pair with the same diagram as the Curtis-Tits standard pair.
\rule{1ex}{1ex}
Let $\ul{\mathscr{F}}$ be the standard Phan amalgam with diagram $\Gamma$ over ${\mathbb F}_q$. Let ${\mathscr{F}}$ be the fixed amalgam of ${\mathscr{G}}$ under $\theta$. By Lemma~\ref{lem:fixed CT amalgam is Phan amalgam} ${\mathscr{F}}$ is a Phan amalgam. By the classification this means that ${\mathscr{F}}=\ul{\mathscr{F}}(\bar{\delta})$ for some $\bar{\delta}$. We now identify $\bar{\delta}$ given $\delta$.
\begin{lemma}\label{lem:delta and deltabar}
\begin{enumerate}
\item Suppose ${\mathscr{G}}=\ul{\mathscr{G}}(\delta)$ and ${\mathscr{F}}=\ul{\mathscr{F}}({\bar{\delta}})$ Then,
$\bar{\delta}$ is the image of $\delta=(\delta_{s})_{s=1}^r$ under the map
\begin{align}
\prod_{s=1}^r {\mathbf{C}}_{i_s} & \to \prod_{s=1}^r {\mathbf{C}}_{i_s}/\langle \theta_{i_s}\rangle.\label{eqn:classification restriction}
\end{align}
\item As a consequence, every Phan amalgam ${\mathscr{F}}$ appears as the fixed amalgam in $2^r$ pairwise non-isomorphic Curtis-Tits amalgams ${\mathscr{G}}$.
\end{enumerate}
\end{lemma}
\noindent{\bf Proof}\hspace{7pt}
Note that in the case of Curtis-Tits as well as Phan standard pairs, the standard identification maps, i.e.~the connecting maps for $\ul{\mathscr{G}}$ and $\ul{\mathscr{F}}$ are "identity maps''.
It follows that
\begin{align}
\ul{\mathscr{G}}^\theta&=\ul{\mathscr{F}}.\label{eqn:famGtheta=famF}
\end{align}
To conclude 1.~from~\eqref{eqn:famGtheta=famF} it suffices to note that if $\ul{\mathscr{F}}(\bar{\delta})$ is the fixed amalgam of $\ul{\mathscr{G}}(\delta)$, then
\begin{align*}
\ul{\mathbf g}_{i_s,j_s}|_{{\mathbf{G}}_{i_s}^\theta}\mathbin{ \circ } \bar{\delta}_{i_s}=\ul{\mathbf f}_{i_s,j_s}\mathbin{ \circ } \bar{\delta}_{i_s}={\mathbf f}_{i_s,j_s} & = {\mathbf g}_{i_s,j_s}|_{{\mathbf{G}}_{i_s}^\theta}=\ul{\mathbf g}_{i_s,j_s}|_{{\mathbf{G}}_{i_s}^\theta}\mathbin{ \circ } \delta_{i_s}|_{{\mathbf{G}}_{i_s}^\theta},
\end{align*}
and by restricting to ${{\mathbf{G}}_{i_s}^\theta}$, we pass from
${\mathbf{C}}_{i_s}=\mathop{\rm Aut}\nolimits({\mathbb F}_{q^2})\times \langle \tau\rangle$ to ${\mathbf{C}}_{i_s}/\langle\theta\rangle=\mathop{\rm Aut}\nolimits({\mathbb F}_{q^2})$ by identifying $\tau=\sigma$.
Part~2.~follows from the classification of Curtis-Tits and Phan amalgams combined with the observation that
the map~\eqref{eqn:classification restriction} has a kernel of order $2^r$.
\rule{1ex}{1ex}
\subsection{Completions of Phan amalgams}\label{subsec:completions of Phan amalgams}
Note that, if $({\tilde{G}},{\tilde{\gamma}})$ is non-trivial, then so is the image of ${\mathscr{F}}$ in ${\tilde{G}}$ under ${\tilde{\gamma}}$.
In particular, ${\mathscr{F}}$ also has a non-trivial completion contained in the fixed group ${\tilde{G}}^\theta$.
This proves the following:
\begin{proposition}\label{prop:Phan realization}
Let ${\mathscr{F}}$ be a Phan amalgam over ${\mathbb F}_q$ with connected $3$-spherical Dynkin diagram $\Gamma$ having no subdiagrams of type $C_2(2)$. Then, there is a Curtis-Tits amalgam ${\mathscr{G}}$ over ${\mathbb F}_{q^2}$ with diagram $\Gamma$ whose universal completion $({\tilde{G}},{\tilde{\gamma}})$ equipped with a
Phan involution $\theta$ such that its fixed group ${\tilde{G}}^\theta$ contains a non-trivial completion $({{F}},\phi)$ of ${\mathscr{F}}$.
\end{proposition}
\begin{corollary}\label{cor:Phan realization in KM group}
Let ${\mathscr{F}}$ be a Phan amalgam over ${\mathbb F}_q$ with connected $3$-spherical Dynkin diagram $\Gamma$ having no subdiagrams of type $C_2(2)$.
Then, there is a group of Kac-Moody type ${\tilde{L}}$ over ${\mathbb F}_{q^2}$ with diagram
$\Gamma$ equipped with a Phan involution $\theta$ such that the fixed group ${\tilde{L}}^\theta$ contains a non-trivial completion $({{F}},\phi)$ of ${\mathscr{F}}$.
\end{corollary}
We note that the completion $({{F}},\phi)$ in Proposition~\ref{prop:Phan realization}~and~Corollary~\ref{cor:Phan realization in KM group} can be a proper subgroup of ${\tilde{G}}^\theta$ and the completion does not have to be universal.
\section{Non-orientable Curtis-Tits groups are lattices}
\subsection{Cartan involutions of the twin-building $\Delta$}
We continue the notation from Section~\ref{sec:NO CT groups}.
Note that the properties of a Cartan involution can be reformulated in terms of the action on the building, as follows.
\begin{lemma}\label{lem:theta on Delta}
Let $\theta$ be a Cartan involution as in Definition~\ref{dfn:Cartan involution}. Then $\theta$ induces an automorphism, also denoted $\theta$, on $\Delta$ with the following properties
\begin{enumerate}
\item $\theta^2=\mathop{\rm id}\nolimits$,
\item $\theta(\Delta_+)= \Delta_-$ and, letting $c_\varepsilon$ be the chamber corresponding to ${{B}}_\varepsilon$ ($\varepsilon=+,-$), we have $c_-=c_+^\theta$ and $c_- \mathbin{\rm opp} c_+$,
\item $\theta$ preserves the twin-apartment $\Sigma(c_+,c_-)=(\Sigma_+(c_+,c_-),\Sigma_-(c_-,c_+))$ and
permutes the types in each fiber of $p\colon \Lambda\to \Gamma$.
\end{enumerate}
\end{lemma}
\subsection{The Coxeter groups $W$ and $W^\theta$}
Here we recall some definitions and results from~\cite{BloHof2014a} and indicate how one can prove these in the current more general setting of $3$-spherical diagrams, which are not necessarily simply-laced.
Let $(W,\{s_j\colon j\in \widehat{I}\})$ be the Coxeter system of type $\Lambda$ with $W=N/D$ and $s_j=\mu(u)D$ for $u\in U_{\alpha_j}$.
Let
$$\delta^\theta(W)=\{w\in W\mid \exists d_+\in \Delta_+\colon w=\delta_*(d_+,d_+^\theta)\}.$$
Let
$$\mathop{\rm Inv}\nolimits^\theta(W)=\{u\in W\mid u^\theta=u^{-1}\}.$$
and
$$W(\theta)=\{w(w^{-1})^\theta \mid w\in W\}.$$
\begin{lemma}\label{lem:tau does not preserve roots}
\label{lem:sws'<>w
\label{lem:delta^theta(W)}
$\mathop{\rm Inv}\nolimits^\theta(W)=\delta^\theta(W).$
More precisely, given any $u\in \mathop{\rm Inv}\nolimits^\theta(W)$ there exists a word $w\in W$ such that
$w(w^{-1})^\theta$ is a reduced expression for $u$.
[cf. Lemma 4.28 of ~\cite{BloHof2014a}]\label{lem:4.24BloHof}
\end{lemma}
This lemma is proved exactly as in~\cite{BloHof2014a} using Lemmas 4.24, 4.26, 4.27.
Note that the proof Lemma 4.24 relies on the fact that a connected spherical diagram does not admit an involution without fixed vertices or edges. As a consequence these results hold for any $3$-spherical diagram.
\begin{corollary}\label{cor:every d in theta apartment}
Let $d\in \Delta_\varepsilon$.
Then, there exists $(c,c^\theta)\in \Delta^\theta$ such that
$d\in \Sigma_+(c,c^\theta)$.
If the non-orientable Curtis-Tits amalgam ${\mathscr{L}}$ is defined over ${\mathbb F}_q$, there are at least $q^{\delta_+(c,d)}$ such chambers.
\end{corollary}
\noindent{\bf Proof}\hspace{7pt}
Let $u=\delta_*(d,d^\theta)$. We induct on $l(u)$.
If $u=1$, we are done.
Assume $l(u)>0$. From Lemma~\ref{lem:delta^theta(W)} we know that $u=w(w^{-1})^\theta$ is irreducible for some irreducible $w\in W$.
Pick any $s_i\in S$ such that $l(s_iw)=l(w)-1$ and let $e$ be a chamber $i$-adjacent to $d$.
Then, $\delta_*(e,e^\theta)=s_iw(w^{-1})^\theta s_{\theta(i)}$ is shorter by $2$, so by induction there exists
$c$ such that $e\in \Sigma(c,c^\theta)_+$.
Moreover, calling $\pi$ the $i$-panel on $d$ we have
$\proj^*_\pi(e^\theta)=d\in \Sigma(c,c^\theta)_+$. This in turn implies that if every panel has at least $q+1$ chambers, then there are at least $q$ choices for the pair $(e,e^\theta)$.
\rule{1ex}{1ex}
\begin{definition}
For each $u\in W^\delta$, define
\begin{align*}
\Delta_u^\theta&=\{d\in \Delta_+\colon \delta_*(d,d^\theta)=u\}.
\end{align*}
\end{definition}
\begin{corollary}\label{cor:comp transitive on De_u^theta}
The group ${{G}}$ acts transitively on the set $\Delta_u^\theta$ for each $u\in W^\delta$. Suppose that
the orientable Curtis-Tits amalgam ${\mathscr{L}}$ is defined over ${\mathbb F}_q$, let $d\in \Delta_u^\theta$ and suppose that $u=w(w^{-1})^\theta$ for some reduced $w\in W$.
Then, $|\mathop{\rm Stab}_{{G}}(d)|\ge q^{l(w)}$.
\end{corollary}
\noindent{\bf Proof}\hspace{7pt}
It was proved in~\cite{BloHof2016} that ${{G}}$ is transitive on the set $\Delta_1^\theta$.
The proof also applies in this case since all vertex groups are $\mathop{\flexbox{\rm SL}}\nolimits_2(q^e)$ acting on the panel as points of the projective line over ${\mathbb F}_{q^e}$ for some $e\ge 1$.
Given $d\in \Delta_u^\theta$, let $c\in \Delta_1^\theta$ be such that $d\in \Sigma_+(c,c^\theta)$.
Then, $d$ is the unique chamber in $\Sigma_+(c,c^\theta)$ with $\delta_+(c,d)=w$.
The transitivity of ${{G}}$ on $\Delta_u^\theta$ now follows.
In addition, by Corollary~\ref{cor:every d in theta apartment}, given $d$, there are $q^{l(w)}$ such chambers $c$, hence at least $q^{l(w)}$ elements in ${{G}}$ fixing $d$.
\rule{1ex}{1ex}
\subsection{The group $G$ is a lattice in $\bar{L}_+$}\label{subsec:lattice}
The action of the group ${{L}}$ on the positive building $\Delta_+$ turns it into a locally compact group with Haar measure $\mu$, and endowed with a metric $f_+$ as in~\cite{CapRem2009a,CapRem2009}.
Let $\bar{L}_+$ be the completion of ${{L}}$ with respect to this metric.
\begin{proposition}\label{prop:G is a lattice of bhKM+}
The group $G$ is a lattice in the group $\bar{L}_+$ for all $q\ge n$.
\end{proposition}
\noindent{\bf Proof}\hspace{7pt}
We follow the proof idea in~\cite{BloHof2014a} and~\cite{GraMuh2008}.
The group $\bar{L}_+$ is locally compact.
It follows from the fact that $U^+\cap {{G}}=\{1\}$ that ${{G}}$ acts discretely on $\Delta_+$ and {any two elements of ${{G}}$ have distance at least $2$ from each other in the metric $f_+$ defined there} (see also~\cite[\S 6]{CapRem2009a}).
We shall now show that ${{G}}$ has finite covolume in $\bar{L}_+$, that is ${{G}}\backslash \bar{L}_+$ has finite volume.
To see this we use~\cite[Proposition 1.4.2]{Bou2000} and instead show that the sum
\begin{align}\label{eqn:stabilizer series}
\sum_{d\in \Delta_+}\frac{1}{|\mathop{\rm Stab}_{{G}}(d)|}
\end{align}
converges.
By Corollary~\ref{cor:comp transitive on De_u^theta} we see that if $d\in \Delta_+$
then, $|\mathop{\rm Stab}_{{G}}(d)|\ge q^{l(w)}$.
It follows that~\eqref{eqn:stabilizer series} is dominated by the Poincar\'e series of the Coxeter group $W$ evaluated at $t=q^{-1}$:
\begin{align}\label{eqn:evaluated poincare series}
p_{(W,S)}(q^{-1})=\sum_{l\in {\mathbb N}}a_l(q^{-1})^l= \sum_{w\in W}\frac{1}{q^{l(w)}},
\end{align}
where $a_l$ is the number of elements in $W$ of length $l$ with respect to the generating set $S$.
Now since $S$ is a finite symmetric generating set for $W$, we have
\begin{align*}
\rho_{(W,S)}^{-1}=\omega(W,S)=\limsup_{l\to\infty}\sqrt[l]{a_l},
\end{align*}
where $\omega(W,S)=\limsup_{l\to\infty}\sqrt[l]{a_l}$ is the growth rate of $(W,S)$ and $\rho_{(W,S)}$ is the radius of convergence of $p_{(W,S)}(t)$ as a power series over ${\mathbb C}$.
Thus if $q\ge \omega(W,S)$, then~\eqref{eqn:evaluated poincare series} converges.
We now wish to maximise $\omega(W,S)$ over all $3$-spherical $(W,S)$ with fixed $n=|S|$. Recall the partial order $\preceq$ from ~\cite{Ter16} on the set of Coxeter systems.
If $(W,S)$ and $(W',S')$ are two Coxeter systems with $S$ and $S'$ finite and Coxeter matrices $M$ and $M'$, then, we write
$(W,S)\preceq (W',S')$ whenever there is an injective map $\varphi\colon S\hookrightarrow S'$ such that for all $r,s\in S$ we have $m_{r,s}\le m'_{\varphi(r),\varphi(s)}$.
By~\cite[Theorem A]{Ter16} we then have $\omega(W,S)\le \omega(W',S')$.
Now suppose that $(W',S')$ is a diagram dominating all $3$-spherical Coxeter systems with given $n=|S'|$ in the $\preceq$ order. Then~\eqref{eqn:evaluated poincare series} converges for all $q \ge \omega_{(W',S')}$.
For example, we can let $(W',S')$ be the Coxeter group whose diagram is the complete graph on $S'$ in which any two vertices are connected by a double edge.
To compute this series note that the set of spherical subsets of $\widehat{I}$ is ${\mathcal F}=\{J\subseteq \widehat{I}\colon |W'_J|<\infty\}=\{J\subseteq \widehat{I}\colon |J|\le 2\}$.
Then, by~\cite{St1968}, we have
\begin{align}
\frac{1}{p_{(W',S')}(t^{-1})}=\sum_{J\in {\mathcal F}}\frac{(-1)^{|J|}}{p_{(W'_J,S'_J)}(t)} = 1 -\frac{n}{p_1(t)}+{n\choose 2}\frac{1}{p_1(t)p_3(t)},
\end{align}
where, setting $p_m(t)=\sum_{i=0}^m t^i$, we have
\begin{align*}
p_{(W'_J,S'_J)}(t) & = \begin{cases}
1 & \mbox{ if }J=\emptyset\\
p_1(t) & \mbox{ if }|J|=1\\
p_1(t)p_3(t) & \mbox{ if } |J|=2.
\end{cases}
\end{align*}
It follows that
\begin{align*}
p_{(W',S')}(t)&=\frac{t^4}{t^4}\cdot \frac{2p_1(t^{-1})p_3(t^{-1})}{2p_1(t^{-1})p_3(t^{-1}) - 2np_3(t^{-1}) +n(n-1)}\\
&=\frac{2p_1(t)p_3(t)}{2p_1(t)p_3(t) - 2ntp_3(t) +t^4 n(n-1)}
\end{align*
The denominator equals
\begin{align*}
2(1-(n-1)t)(1+t+t^2+t^3)+t^4n(n-1)
\end{align*}
When $|z|\le \frac{1}{n}$ one verifies that the last term has norm at most $\frac{1}{n^2}$, whereas the other terms together have norm at least $\frac{1}{n}$, noting that $n\ge 3$. Therefore the Poincar\'e series converges at $q^{-1}$ for $q\ge n$.
Numerical evidence suggests that one cannot do much better as, e.g.~for $n=1000$, the norm of the smallest complex root is about $0.001001...$.
\rule{1ex}{1ex}
\section{Simplicity of Curtis-Tits groups}\label{sec:simplicity}
Combining Theorem~\ref{thm:OCT realization}, the Simplicity Theorem of~\cite{CapRem2009} and the observation that Curtis-Tits groups are perfect since they are generated by perfect subgroups, we have the following.
\begin{corollary}\label{cor:OCT almost simple}
Suppose ${\mathscr{L}}$ is an orientable Curtis-Tits amalgam over ${\mathbb F}_q$ with connected diagram $\Gamma$, which is $3$-spherical, but not spherical or affine. Then, its universal completion ${\tilde{L}}$ is almost simple.
\end{corollary}
By Theorem~\ref{thm:NOCT realization} (as well as~\cite{BloHofShp2017,BloHof2016}), any non-orientable Curtis-Tits amalgam ${\mathscr{G}}$ has a completion inside the centralizer in a group of Kac-Moody type ${{L}}$ of a Cartan involution $\theta$. In contrast to what happens in the orientable case, we now have the following.
\begin{theorem}\label{thm:non-orientable CT groups are not simple}
If ${{G}}$ is a non-orientable Curtis-Tits group over a finite field with irreducible non-spherical, non-affine diagram, then ${{G}}$ is acylindrically hyperbolic. In particular, it is not simple.
\eth
\noindent{\bf Proof}\hspace{7pt}
By Remark 3.7 of~\cite{CapHum2015} it suffices to note the following.
The group ${{L}}$ is a group of Kac-Moody type with non-spherical, non-affine diagram over a finite field ${\mathbb F}_q$.
The positive building associated to ${{L}}$, denoted $\Delta_+$ is a proper CAT(0) space and $\mathop{\rm Aut}\nolimits(\Delta_+)$ acts cocompactly on it.
Moreover, by~\cite[Theorem 1.1]{CapFuj2010} $\mathop{\rm Aut}\nolimits(\Delta_+)$ contains rank 1 elements since it is of irreducible, non-spherical and non-affine type.
By~\cite[Corollary 3.6]{CapHum2015} any lattice of $\mathop{\rm Aut}\nolimits(\Delta_+)$ is acylindrically hyperbolic and is therefore not simple.
Thus, the claim follows from Proposition~\ref{prop:G is a lattice of bhKM+}.
\rule{1ex}{1ex}
\subsection{Examples}\label{subsec:examples}
From the classification of Curtis-Tits amalgams it is clear that whenever the diagram is a tree, the amalgam is unique. In particular, the only Curtis-Tits groups with spherical diagram are the groups of Lie type.
The same holds for Curtis-Tits groups with affine diagram, other than ${\widetilde{A}}_{n-1}$.
In~\cite{BloHof2014a} all orientable and non-orientable Curtis-Tits groups with diagram ${\widetilde{A}}_{n-1}$ were described in terms of matrix groups and it was shown that all of them have interesting quotients (see also~\cite{BloHofVdo2012}).
Here we will give some examples of quotients arising from non-orientable Curtis-Tits groups with non-spherical and non-affine diagram.
We are particularly interested in finite quotients.
In a subsequent study of such quotients we will use the finite presentations for Curtis-Tits groups arising from the Curtis-Tits amalgam and combine this with relations in the vein of~\cite{BarMar15}.
As a simple example consider the non-orientable Curtis-Tits amalgam over ${\mathbb F}_2$ with the following diagram:
\begin{center}
\begin{tikzpicture}[scale=.5]
\node [label=below:$\tau$] (tau) at (5,-1) {};
\tikzstyle{every node} = [draw, line width = 1pt, shape=circle]
\node [label=below:$1$] (one) at (0,0) {};
\node [label=above:$2$] (two) at (4,2) {};
\node [label=below:$3$] (three) at (2,0) {};
\node [label=below:$4$] (four) at (4,0) {};
\node [label=below:$5$] (five) at (6,0) {};
\node [label=below:$6$] (six) at (8,0) {};
\foreach \from/\to in {one/three, three/four, four/five, five/six, two/four}
\path[draw, line width = 1pt] (\from) -- (\to);
\path[draw, line width = 1pt] (three) .. controls (5,-3) .. (six);
\end{tikzpicture}
\end{center}
Here the $\tau$ indicates that $\omega$ sends the loop $\Lambda_1=\{3,4,5,6,3\}$ to $\tau$.
Let ${\mathcal X}=\{{\mathbf{X}}_i^+,{\mathbf{X}}_i^-\colon i=1,2,\ldots,6\}$ and, for each $i$, let $x_i^+$, $x_i^-$, and $n_i$ be defined by
${\mathbf{X}}_i^+=\langle x_i^+\rangle$,
${\mathbf{X}}_i^-=\langle x_i^-\rangle$, and $n_i=x_i^+x_i^-x_i^+$.
Then, the universal completion has a presentation in terms of these generators. After adding the relation
\begin{align*}
(n_3n_4n_5n_6n_5n_4)^2=1,
\end{align*}
GAP~\cite{GAP4.8.5} was able to identify the quotient as the exceptional group $E_6(2)$.
Since the orientable amalgam with the same diagram is simple by Corollary~\ref{cor:OCT almost simple}, this example confirms the existence of non-spherical, non-affine non-orientable Curtis-Tits groups.
Note that since $|{\mathbb F}_2|<|S|=6$, this example is not covered by Proposition~\ref{prop:G is a lattice of bhKM+}.
We shall also study non-orientable Curtis-Tits amalgams with diagrams such as the following:
\begin{center}
\begin{tikzpicture}[scale=.5]
\node [label=below:$\tau$] (tau) at (4.25,1.5) {};
\tikzstyle{every node} = [draw, line width = 1pt, shape=circle]
\node [label= {[label distance=1ex] -90:\makebox(0,0){$1$}}] (two) at (0,0) {};
\node [label= {[label distance=1ex] -90:\makebox(0,0){$2$}}] (two) at (2.5,0) {};
\node [label= {[label distance=1ex] -90:\makebox(0,0){$k-1$}}] (enminusone) at (6,0) {};
\node [label= {[label distance=1ex] -90:\makebox(0,0){$k$}}] (en) at (8.5,0) {};
\node [label= {[label distance=1ex] -90:\makebox(0,0){$k+1$}}] (enplusone) at (11,0) {};
\node [label= {[label distance=1ex] -90:\makebox(0,0){$k+l-1$}}] (enpluskayminusone) at (14.5,0) {};
\node [label= {[label distance=1ex] -90:\makebox(0,0)[l]{$k+l$}}] (enpluskay) at (17,0) {};
\foreach \from/\to in {one/two, enminusone/en, en/enplusone, enpluskayminusone/enpluskay}
\path[draw, line width = 1pt] (\from) -- (\to);
\path[draw, line width = 1pt] (one) .. controls (2.5,3) and (6,3) .. (en);
\foreach \from/\to in {two/enminusone, enplusone/enpluskayminusone}
\path[draw, dashed, line width = 1pt] (\from) -- (\to);
\end{tikzpicture}
\end{center}
Here the $\tau$ indicates that $\omega$ sends the loop $\Lambda_1=\{1,2,\ldots,k,1\}$ to $\tau$.
Let $n_i$ ($i=1,2,\ldots,k+l$) be defined as above and consider the quotient of the universal completion of this amalgam over ${\mathbb F}_q$ ($q$ even) over the normal closure of the element
\begin{align*}
\left (n_1n_2\cdots n_{k-1}n_k n_{k-1}^{-1}\cdots n_2^{-1}\right)^2.
\end{align*}
Note here that the case $q=4$, $k=3$ and $l=1$, is covered by Proposition~\ref{prop:G is a lattice of bhKM+}, so that
the existence of infinitely many infinite-index normal subgroups is guaranteed via Theorem~\ref{thm:non-orientable CT groups are not simple}.
However, using GAP~\cite{GAP4.8.5} we were able to identify the following finite quotients.
For $k=3$ and $(q,l)= (2,1),(2,2),(2,3),(4,1),(4,2)$, the quotient is $\mathop{\flexbox{\rm PSL}}\nolimits_{k+l+1}(q)$.
For $k=4$, $l=1$, and $q=2,4$, the quotient is $\mathop{\flexbox{\rm P}\Omega}\nolimits^+(2(k+l),q)$.
|
1,314,259,993,649 | arxiv | \section{Introduction}
The phase diagram of quantum chromodynamics (QCD)
is being actively studied in heavy ion collision experiments
as well as theoretically. A form of matter with
remarkable properties~\cite{Muller:2008zzm}
has been observed in the Relativistic Heavy Ion Collider (RHIC)
experiments~\cite{Arsene:2004fa,Back:2004je,Adcox:2004mh,Adams:2005dq}.
It appears to be a strongly coupled
plasma of quarks and gluons (QGP), but no consensus on
a physical picture
that accounts for both equilibrium and non-equilibrium properties
has been reached yet.
On the other hand, below the short interval of temperatures
where the transition from the confined phase to the
QGP takes place~\cite{Aoki:2006br,Aoki:2009sc,Cheng:2007jq,Bazavov:2009zn},
it is widely believed that the most prominent degrees of freedom are the
ordinary hadrons. From this point of view,
the zeroth order approximation to the properties
of the system is to treat the hadrons as infinitely
narrow and non-interacting. We will refer to this approximation
as the hadron resonance gas model (HRG).
The HRG predictions were compared
with lattice QCD thermodynamics data
in~\cite{Karsch:2003vd,Cheng:2007jq}, and
lately they have been used to extrapolate certain results
to zero temperature~\cite{Bazavov:2009zn}.
The HRG is also the basis of the statistical model currently applied
to the analysis of hadron yields in
heavy ion collisions~\cite{Andronic:2008gu},
and recently the transport properties of a relativistic hadron gas
have been studied in detail~\cite{Demir:2008tr}.
Since any heavy ion reaction ends up in the low-temperature phase
of QCD, it is important to understand its properties in detail
in order to extract those of the high-temperature phase
with minimal uncertainty. In this Letter we study whether
the HRG model works in the absence of quarks, in other words
in the pure SU($N=3$) gauge theory, where the low-lying states
are glueballs. There are reasons to believe that
if the HRG model is to work at any
quark content of QCD, it is in the zero-flavor case.
Firstly, the mass gap in SU(3) gauge theory is very large,
$M_0/T_c\simeq 5.3$. As we shall see,
the thermodynamic properties up to quite close to $T_c$
are dominated by the states below the two-particle threshold,
which are exactly stable. Furthermore, because of their large mass,
neglecting their thermal width should be a good approximation.
Secondly, the scattering amplitudes between glueballs are
parametrically $1/N^2$ suppressed
while those between mesons are only $1/N$
suppressed~\cite{Witten:1979kh}.
This means that the glueballs should be free to a
better approximation than the hadrons of realistic QCD.
An additional motivation to study the thermodynamics of the
confined phase of SU(3) gauge theory is that it is a parameter-free
theory, simplifying the interpretation of its properties.
Its spectrum is known quite accurately up to the two-particle
threshold~\cite{Meyer:2004gx,Chen:2005mg}. By contrast,
in full QCD calculations, lattice data calculated
at pion masses larger than in Nature
are often compared out of necessity to the HRG model based on the
experimental spectrum~\cite{Cheng:2007jq,Bazavov:2009zn}.
Finally, calculations in the pure gauge theory
are at least two orders of magnitude faster, which allows us to
reach a high level of control of statistical and systematic errors;
in particular, we are able to perform calculations
in very large volumes.
\section{Lattice calculation}
We use Monte-Carlo simulations of the Wilson action
$S_{\rm g}= \frac{1}{g_0^2} \sum_{x,\mu,\nu} {\rm Tr\,}\{1-P_{\mu\nu}(x)\} $
for SU(3) gauge theory~\cite{Wilson:1974sk}, where $P_{\mu\nu}$ is the plaquette.
The lattice spacing is related to the bare coupling through
$g_0^2\sim 1/\log(1/a\Lambda)$.
We calculate the thermal expectation value of
$\theta\equiv T_{\mu\mu}$, the (anomalous)
trace of the energy-momentum tensor $T_{\mu\nu}$, and
of $\theta_{00}\equiv T_{00}-{\textstyle\frac{1}{4}} \theta$.
In the thermodynamic limit,
\begin{equation}
Ts = e+p = {\textstyle\frac{4}{3}} \<\theta_{00}\>_T,~~~~
e-3p= \<\theta\>_T - \<\theta\>_0.
\end{equation}
Here $e,p,s$ are respectively the energy density, pressure and entropy density.
The operator
$\theta_{00}=\frac{1}{2}(-{\bf E}^a\cdot{\bf E}^a+{\bf B}^a\cdot{\bf B}^a)$
requires no subtraction, because its vacuum expectation value vanishes.
The choice of of $\theta_{00}$ and $\theta$ as
independent linear combinations is convenient
because they both renormalize multiplicatively.
We use the `HYP-clover' discretization of the energy-momentum tensor
introduced in~\cite{Meyer:2007tm,Meyer:2007ed}.
The normalization of the $\theta_{00}$ operator differs from its
naive value by a factor that we parametrize as $Z(g_0)\chi(g_0)$.
The factor $Z(g_0)$ is taken from~\cite{Meyer:2007ic} and
rests on the results of~\cite{Engels:1999tk}; its accuracy is about one percent.
The factor $\chi(g_0)$ is obtained by calibrating
our discretization to the `bare plaquette' discretization in the
deconfined phase at $N_t=6$~\cite{Meyer:2007tm}.
We find, for $6/g_0^2$ between 5.90 and 6.41,
$\chi(g_0)=0.1306\cdot(6/g_0^2)-0.1865$ with an accuracy of half a percent.
For the lattice beta-function that renormalizes $\theta$, we use
the parametrization~\cite{Durr:2006ky} of the data in~\cite{Necco:2001xg}
and the same calibration method.
\begin{figure}
\centerline{\includegraphics[width=6.5 cm,angle=-90]{fv-detail2.ps}}
\caption{Finite volume effects on the entropy density
close to the deconfining temperature $T_c$.}
\label{fig:fv}
\end{figure}
Our results for the entropy density from $N_t=8$ and $N_t=12$
simulations are shown on Fig.~(\ref{fig:epp}). The displayed error
bars do not contain the uncertainty on the normalization factor,
which is much smaller and would introduce correlation between the points. This factor
varies by only $7\%$ over the displayed interval and so
to a first approximation amounts to an overall normalization of the curve.
Our data is about five times statistically more accurate than that of
previous thermodynamic studies~\cite{Boyd:1996bx,Namekawa:2001ih},
which were primarily focused on the deconfined phase.
Just as importantly, we kept the finite-spatial-volume effects under good
control, in particular very close to $T_c$.
Figure (\ref{fig:fv})
shows the size of finite-volume effects. For instance, at $0.985T_c$
the conventional choice $LT=4$ leads to an overestimate of the entropy
density by a factor three. The fact that the $N_t=12$ data fall on the
same smooth curve as the $N_t=8$ is strong evidence that discretization
errors are small. We parametrize the volume dependence empirically
by a $A+Be^{-cLT}$ curve, and use it to convert the $N_t=12$ data to $LT=8$.
At $0.929T_c$, there is no statistically significant
difference between $LT=6$ and 8 and we do not apply any correction.
It is the corrected $N_t=12$ data that is then displayed on Fig.~(\ref{fig:epp}).
In~\cite{Meyer:2009kn}, formulas for the leading finite-volume effects
on the thermodynamic potentials were derived in terms of the energy gap
of the theory defined on a $(1/T)\times L\times L$ spatial hypertorus.
Close to $T_c$, this gap corresponds to the mass of the ground state
flux loop winding around the cycle of length $1/T$.
If $\delta s(T,L) \equiv s(T,\infty)-s(T,L)$, the formula then reads
\begin{equation}
\delta s(T,L) = \frac{e^{-m(T) L}}{2\pi L}
\left[ m^2(T) + {\textstyle\frac{3}{2}}T\partial_T m^2(T)\right]\,.
\label{eq:s}
\end{equation}
Using the calculation of $m(T)$ described in the next section,
the predicted asymptotic approach to the
infinite-volume entropy density
for $0.985T_c$ is displayed on Fig.~(\ref{fig:fv}).
While the sign is correct,
the magnitude of the finite-volume effects
is not reproduced for $LT\leq 8$. We conclude that the
asymptotic approach to infinite volume sets in for
very large values of $LT$. Since $m(T)L$ is only about $4$ when $LT=6$,
it is not implausible that flux-loop states with high multiplicity
dominate the finite-volume effects at that box size.
\begin{figure}
\centerline{\includegraphics[width=6.5 cm,angle=-90]{tmass6.ps}}
\caption{The mass of the temporal flux loop as calculated from
Polyakov loop correlators, and the fit (\ref{eq:II}).
The $N_t>11$ data are from~\cite{Meyer:2004vr}, the
$N_t=5$ data from~\cite{Lucini:2005vg}.}
\label{fig:tmass4}
\end{figure}
Next we obtain the correlation length $\xi(T)$ of the order parameter
for the deconfining phase transition, the Polyakov loop.
The method consists in computing the two-point function
of zero-momentum operators, designed to have large overlaps
with the ground state flux loop, along a spatial direction.
We fit the lattice data for $m(T)\equiv1/\xi(T)$
displayed on Fig.~(\ref{fig:tmass4}) with the formula
\begin{equation}
{\textstyle\left(\frac{m(T)T}{T_c^2}\right)}^2 = a_0
- a_1 {\textstyle\left(\frac{T}{T_c}\right)}^2
- a_2 {\textstyle\left(\frac{T}{T_c}\right)}^4
\end{equation}
and find, either fitting $a_2$ or setting it to zero,
\begin{eqnarray}
&& a_0= 5.76(15),~ a_1 = 4.97(65), ~ a_2 = 0.55(54)
\label{eq:I}
\\
&& a_0= 5.90(9),~~ a_1 = 5.62(10),~~ a_2 = 0
\label{eq:II}
\end{eqnarray}
with in both cases a $\chi^2/$dof of about 0.3.
We remark that the $a_i$ are not far from
the Nambu-Goto string~\cite{Arvis:1983fp} values
$a_1=\frac{2\pi}{3}\frac{\sigma}{T_c^2}= 5.02(5)$~\cite{Lucini:2005vg}
and $a_2=0$ ($\sigma$ is the tension of the confining string).
We extract the `Hagedorn' temperature, defined
as in~\cite{Bringoltz:2005xx} by $m(T_h)=0$,
from the second fit,
\begin{equation}
T_h/T_c = 1.024(3).
\label{eq:Th}
\end{equation}
This extraction amounts to assuming mean-field exponents
near $T_h$ (it is not clear which universality class should be
used~\cite{Yaffe:1982qf}). The result is stable if
the fit interval is varied, and also if $a_2$ is fitted with
$a_0$ and $a_1$ constrained to the
known values of $(\sigma/T_c^2)^2$ and $\frac{2\pi}{3}\frac{\sigma}{T_c^2}$.
As a check on the normalization of the operators
$\theta_{00}$ and $\theta$,
we calculate the latent heat in two different ways. The latent heat
is the jump in energy density at $T_c$. Since the pressure
is continuous, we obtain it instead from the discontinuity
in entropy density or the `conformality measure' $e-3p$.
We obtain $s$ and $e-3p$ on either side of $T_c$
by extrapolating $LT=10$ data from the confined (deconfined)
phase towards $T_c$. The result is
\begin{equation}
\frac{\Delta s}{T_c^3} = 1.45(5)(5),~~
\frac{\Delta(e-3p)}{T_c^4} = 1.39(4)(5),
\end{equation}
where the first error is statistical and the second comes from
the uncertainty in the extrapolation (taken to be the difference
between a linear and quadratic fit).
The compatibility between these two estimates of $L_h/T_c^4$
is strong evidence that we control the normalization of
our operators. They are in good agreement with previous
calculations of the latent heat performed
on coarser lattices
\cite{Beinlich:1996xg,Lucini:2005vg}.
We have also verified more generally that the thermodynamic identity
$T\partial_T(s/T^3)=(1/T^3)\partial_T(e-3p)$
is satisfied within statistical errors.
\section{Interpretation}
In infinite volume the pressure associated with a single non-interacting,
relativistic particle species
of mass $M$ with $n_\sigma$ polarization states reads
\begin{equation}
p = \frac{n_\sigma}{2\pi^2}M^2\,T^2\sum_{n=1}^\infty \frac{1}{n^2} K_2(nM/T)
\end{equation}
where $K_2$ is a modified Bessel function.
By linearity, the knowledge of the glueball spectrum
leads to a simple prediction for the pressure and entropy density
$s=\frac{\partial p}{\partial T}$,
which is expected to become exact in the large-$N$ limit.
Since only the low-lying spectrum of glueballs is known, it
is useful to consider how the density of states might be extended above
the two-particle threshold $2M_0$, where $M_0$ is the mass of the
lightest (scalar) glueball.
The asymptotic closed bosonic string density of states in four dimensions
is given by~\cite{Zwiebach:2004tj}
\begin{equation}
\rho(M) = \frac{(2\pi)^3}{27\,T_h} \left(\frac{T_h}{M}\right)^4 e^{M/T_h}.
\label{eq:hagedorn}
\end{equation}
In the string theory, the Hagedorn temperature $T_h$ is related to the
string tension, $T_h^2=\frac{3\sigma}{2\pi}$, corresponding to
$T_h/T_c=1.069(5)$~\cite{Lucini:2003zr}. Below we use
this value as an alternative to the more direct determination (\ref{eq:Th}).
On Fig.~\ref{fig:epp}, we show the entropy contribution of the glueballs
lying below the two-particle threshold $2M_0$. The curve is just about consistent
with the smallest temperature lattice data point, but clearly fails to
reproduce the strong increase in entropy density as $T\to T_c$.
The figure also illustrates that the two lowest-lying states, the
scalar and tensor glueballs, account for about three quarters
of the stable glueballs' contribution.
We have used the continuum-extrapolated
lattice spectrum~\cite{Meyer:2004gx,Meyer:2008tr}.
Adding the Hagedorn spectrum contribution, Eq.~(\ref{eq:hagedorn})
with $T_h$ given by Eq.~(\ref{eq:Th}),
leads to the solid curve on Fig.~\ref{fig:epp}. It describes the direct
calculation of the entropy density surprisingly well, particularly
close to $T_c$. The curve tends to underestimate
somewhat the entropy density at the lower temperatures.
This is likely to be a cutoff effect. Indeed, at fixed $N_t$
lower temperatures correspond to a coarser lattice spacing,
and the scalar glueball mass in physical units is known
to be smaller on coarse lattices with the Wilson action~\cite{Lucini:2001ej}.
If we use the stable glueball spectrum calculated at $g_0^2=1$
instead of the continuum spectrum, the agreement of the non-interacting
glueball + Hagedorn spectrum with the lattice data at the lower
four temperatures is again excellent.
This difference provides an estimate for the size of lattice effects.
\begin{figure}
\centerline{\includegraphics[width=6.5 cm,angle=-90]{epp.ps}}
\caption{The entropy density in units of $T^3$ for $LT=8$.
We applied a (modest) volume-correction to the $N_t=12$ data.}
\label{fig:epp}
\end{figure}
To summarize, we have computed to high accuracy
the entropy of the confined phase of
QCD without quarks. The low-lying states
of the theory are therefore bound states
called glueballs, and their spectrum is well
determined~\cite{Meyer:2004gx,Chen:2005mg}.
If the size $N$ of the gauge group is increased, the interactions
of the glueballs are expected to be suppressed~\cite{Witten:1979kh}.
To what extent the glueballs really are weakly interacting at $N=3$
is not known precisely. Some evidence for the smallness of their low-energy
interactions was found some time ago
by looking at the finite-volume effects on their masses~\cite{Meyer:2004vr}.
But it seems unlikely that glueballs well above the two-particle
threshold would have a small decay width.
We have nevertheless compared the entropy density data to
the entropy density of a gas of non-interacting glueballs.
While restricting the spectral sum to the stable glueballs
leads to an underestimate by at least a factor two of the
entropy density near $T_c$, extending the spectral sum
with an exponential spectrum $\rho(M)\sim \exp(M/T_h)$,
suggested long ago by Hagedorn~\cite{Hagedorn:1965st}, leads to a prediction in excellent
agreement with the lattice data for the entropy density (Fig.~\ref{fig:epp}).
This is remarkable,
since the analytic form of the asymptotic spectrum
is completely predicted by free bosonic string theory,
including its overall normalization (Eq.~\ref{eq:hagedorn}). Therefore,
since we also separately computed the temperature (identified with $T_h$)
where the flux loop mass vanishes,
no parameter was fitted in the comparison with the thermodynamic data.
By contrast, the entropy density is not nearly as well described
if the Nambu-Goto value of $T_h$ is used, see Fig.~(\ref{fig:epp}).
The success of the non-interacting string density of states
in reproducing the entropy density
suggests that once the Hagedorn temperature has been determined
directly from the divergence of the flux-loop correlation length,
the residual effects of interactions on the thermodynamic potentials are small.
It may be that thermodynamic properties in general
are not strongly influenced by interactions when a large number
of states are contributing. A well-known example
is provided by the ${\cal N}=4$ super-Yang-Mills theory,
whose entropy density at very strong coupling
is only reduced by a factor 3/4 with respect to the
free theory~\cite{Gubser:1996de}.
In this interpretation,
the main effect of interactions among glueballs on thermodynamic properties
is to slightly shift the value of the Hagedorn
temperature $T_h$ with respect to its free-string value.
A possible mechanism is that the string tension
that effectively determines $T_h$ is an in-medium string tension
that is $\sim8\%$ lower than at $T=0$.
Returning to full QCD, our results lend support to the idea
that the hadron resonance gas model can largely account for the
thermodynamic properties of the low-temperature phase.
Whether the open string density of states
reproduces the entropy calculated on the lattice
can also be tested at quark masses not necessarily as light as in Nature
using a simple open string model~\cite{Selem:2006nd}.
\acknowledgments{
I thank B. Zwiebach for a discussion on the bosonic string
density of states. The simulations were done on the Blue Gene L rack
and the desktop machines of the
Laboratory for Nuclear Science at M.I.T.
This work was supported in part by
funds provided by the U.S. Department of Energy
under cooperative research agreement DE-FG02-94ER40818.
}
|
1,314,259,993,650 | arxiv | \section{Conclusion}
\label{sec:conclusion}
Our approach to discover multi-relational patterns with maximum entropy models
in a visual analytics tool is a significant step in formalizing a previously
unarticulated knowledge discovery problem and supporting its solution in an
interactive manner. We have primarily showcased results in intelligence
analysis; however, the theory and methods presented are applicable for analysis
of unstructured or discrete multi-relational data in general---such as for
biological knowledge discovery from text. The key requirement to apply our
methods is that the data should be transformed into our data model.
Some of the directions for future work include (i) obviating the need to mine
all biclusters prior to composition, (ii) improving the scalability of the
proposed models and framework to be able to deal with even larrger datasets,
(iii) enabling dynamic and flexible multi-relational schema generation to
support better sensemaking and hidden plot discovery, (iv) incorporating weights
on relationships to account for differing veracities and trustworthiness of
evidence. Ultimately, the key is to support more expressive forms of
human-in-the-loop knowledge discovery.
\section{Experiments}
\label{sec:exp}
We describe the experimental results over both synthetic and real datasets. For
real datasets, we focus primarily on datasets from the domain of intelligence
analysis. Through a case study, we demonstrate how the proposed maximum entropy
models embedded in our visual analytics approach
helps analysts to explore text datasets, such as used in intelligence analysis.
All experiments described in this section were conducted on a Xeon 2.4GHz
machine with 1TB memory. Performance results (for synthetic data)
were obtained by averaging over 10 independent runs.
\subsection{Results on Synthetic Data}
\label{sec:exp_syn}
To evaluate the runtime performance of the proposed maximum entropy models with
respect to the data characteristics, we generate synthetic datasets. Since we
focused on the runtime performance of the proposed models here, and the
multi-relational schema of the dataset will not affect how the proposed models
are inferred over the data matrix $D$, we will temporarily ignore the
multi-relational schema of the dataset in the synthetic data for now. The
synthetic datasets are parameterized as follows. The data matrix $D$ consists of
$N$ rows and $M$ columns, or entities, and $\beta$ denotes the density of the
data matrix $D$. For each entry in the data matrix $D$, we set its value to be
non-zero with probability $\beta$. For the binary case, the non-zero values would
naturally be one, and for the real-valued case, the non-zero values are generated
from a standard uniform distribution. In order to avoid the scenario that too
many rows or columns in $D$ contains only zeros, a non-zero value is placed
randomly in a row or column if it only contains zeros.
\begin{figure}[t]
\centering
\includegraphics[width=2.7in]{binary_train}\hfill
\includegraphics[width=2.7in]{real_train}
\caption{Time to infer the binary (left) and real-valued (right, Y-axis is
in log scale) maximum entropy model on synthetic
datasets. The error bars represent the standard deviation}\label{fig:model_infer}
\end{figure}
In our experiments, we explore data matrix $D$ sizes of
($N=1000, M=1000$),
($N=2000, M=2000$), and ($N=3000, M=3000$), and varied the density $\beta$ of the
data matrix $D$ from $0.01$ to $0.05$ in steps of $0.01$. To infer the
maximum entropy models, we use column margin and row margin tiles as the set of
constraint tiles for the proposed model (see Sect.~\ref{sec:model}). We first
investigate the time needed to infer the maximum entropy models.
Figure~\ref{fig:model_infer} shows the model inference time for the binary and
real-valued maximum entropy formulations. As expected, model inference increases with
dataset size and requires more time for the real-valued model.
Since the
real-valued maximum entropy model adopts the conjugate gradient method,
model inference time heavily depends
upon the structure of the given dataset, the number of constraint tiles, and how
fast the model converges to the optimal solution along the gradient direction.
For example, in our experiments we used the row and column margin tiles as the
constraints for the real-valued maximum entropy model, the dimension of the
gradient could be $2 (M + N)$ (that would be 4,000 dimension when $N=1000,
M=1000$ for our synthetic datasets).
Another interesting phenomenon we observed here is that as the density $\beta$
of the data matrix $D$ increases, the inference time required by the real-valued
maximum entropy model decreases. One explanation for this phenomenon is that
denser data matrices provide more information to the maximum entropy
model about the underlying data generation distribution through the constraint
tiles. This aids the model in rapidly learning the structure of the data space
and search for the optimal solution with fewer iterations of the conjugate gradient
algorithm.
\begin{figure}[t]
\centering
\includegraphics[width=2.7in]{binary_eval}\hfill
\includegraphics[width=2.7in]{real_eval}
\caption{Time to evaluate a set of tiles with the binary (left) and
real-valued (right) maximum entropy model on synthetic datasets. The set of
solid lines on the top represents the results of global score, and the set of
dash lines at the bottom represents the results of local score. The error
bars represent the standard deviation, and the Y-axis is in log
scale.}\label{fig:model_eval}
\end{figure}
We also measured the runtime performance of evaluating tile sets with the
proposed binary and real-valued maximum entropy models since the patterns
(biclusters or bicluster chains) whose qualities we would like to assess will
eventually be converted into a set of tiles in our framework. To be more
specific, we randomly generated a set of tiles over the synthetic data matrix,
and compared the time required to evaluate this tile set with both global score
and local score using converged binary and real-valued models, and
Figure~\ref{fig:model_eval} illustrates the results. As we can see from this
figure, in both binary and real-valued maximum entropy model, evaluating tile
sets using the global score requires more time than the local score, which is
expected since the global score requires a complete re-inference of the model.
The difference of runtime performance between global and local scores is significant
in the real-valued model due to this model inference step.
When applying the real-valued maximum entropy model in practical applications,
such as the one here necessitating real-time interaction, we can employ
an asynchronized model inference scheme, e.g.\
creating a daemon process to infer the model when the system is idle, and adopt
the local score to evaluate tile sets.
\subsection{Evaluation on Real Dataset: A Usage Scenario}
\label{sec:exp_real}
In this section, we walk through an intelligence analysis scenario to
demonstrate how \MaxEntBiSet, particularly incorporating the proposed
maximum entropy models for identifying surprising entity coalitions,
can support an analyst to discover a coordinated activity via visual analysis
of entity coalitions. For ease of description, we use a small
dataset, viz. \textsl{The Sign of the Crescent}
\cite{hughes2003discovery}, which contains $41$ fictional intelligence reports
regarding three coordinated terrorist plots in three US cities where each plot
involves a group of (at least four) suspicious people. In fact, $24$ of the
reports are relevant to the plots. We use LCM~\cite{uno2004efficient} to find
\textsl{closed} biclusters from the dataset with the \textsl{minimum support}
parameter set to 3, which assures that each bicluster has at least three
entities from one domain (e.g., people, location, date, etc.). This leads to 337
biclusters from 284 unique entities and 495 individual relationships (based on
entity co-occurrence in the reports).
In order to try to discover all the possible plots hidden in the
\textsl{Crescent} dataset, in \MaxEntBiSet, we set the threshold for the Jaccard
coefficient as 0.05, which is a loose constraint. This enables the model to
evaluate those neighboring biclusters that has a few entity overlaps with user
specified biclusters for assessment. Although \MaxEntBiSet fully supports
pattern evaluations with the real-valued \textsl{maximum entropy model}, we
observed that the model evaluation results of a given bicluster were similar
when using the binary and the real-valued \textsl{maximum entropy models} in our
experiments over the \textsl{Crescent} dataset. Thus, we only present the use
case study using the binary \textsl{maximum entropy model} here to demonstrate
the effectiveness of the proposed \MaxEntBiSet technique when assisting analysts
in conducting intelligence analysis tasks.
To illustrate the benefits of integrating the maximum entropy models
into visual analytic tools, in this intelligence analysis scenario, we
use BiSet~\cite{sun2015biset} as the baseline approach for comparison purposes.
Notice that BiSet does not has the capability of model evaluations, and thus it just
provides the \textsl{connection} oriented highlighting function for users to
manually explore entity coalitions. We begin our discussions with the use case
of BiSet, and then discuss the use case of \MaxEntBiSet.
In our scenario, suppose that Sarah is an
intelligence analyst. She is assigned a task to read intelligence reports and
identify potential terrorist threats and key persons from the \textsl{Crescent}
dataset. She opens BiSet, selects four identified domains (people, location,
phone number and date), and begins her analysis.
Figure~\ref{fig:biset-case-without-evaluation} demonstrates Sarah's key
analytical steps using BiSet. Figure~\ref{fig:biset-case-with-evaluation1} and
Figure~\ref{fig:biset-case-with-evaluation2} show the key steps of Sarah's
analytical process using \MaxEntBiSet.
\subsubsection{BiSet Use Case}
\begin{sidewaysfigure}
\centering
\includegraphics[width=0.9\textwidth]{biset_case_without_eva}
\caption{The process of finding a major threat plot with key steps. (1): Based
on \textsl{A. Ramazi}, finding that there are two similar bundles and two
cells. (2): One name and two bundles are highlighted when hovering the
mouse over \textsl{B. Dhaliwal}. (3): Three names and three bundles are
highlighted when exploring \textsl{F. Goba}. (4) Referring to the four
connected groups of useful entities for hypothesis generation.}
\label{fig:biset-case-without-evaluation}
\end{sidewaysfigure}
Sarah begins analysis by hovering over individual entities in the list of people's
names. BiSet highlights related bundles and entities, each time when she hovers
the mouse over an entity. Immediately she finds that \textsl{A. Ramazi} is
active in three bundles, which indicates that this person may be involved in
three coordinated activities. Sarah selects it
(Figure~\ref{fig:biset-case-without-evaluation} (1)) to focus on highlighted
entities of the three bundles. She finds that \textsl{A. Ramazi} is involved in
two cells with other five people. One is in Germany and the others may be more
broadly located in other four countries. \textsl{A. Ramazi} is the only person
connecting the two cells, and there are two overlapped subgroups of people
involved in the broader cell. Moreover, each subgroup has its unique person
(\textsl{B. Dhaliwal} and \textsl{F. Goba}).
Then Sarah decides to explore the two overlapped subgroups, because she aims to
know what brings the unique people to them. She checks \textsl{B. Dhaliwal}
first by hovering the mouse over it. After this, two bundles are highlighted.
Following edges from them, Sarah finds that they share two people's names and
three locations, but the bigger one (shown in
Figure~\ref{fig:biset-case-without-evaluation} (2)) is related to a new name
(\textsl{H. Pakes}). Then she examines \textsl{F. Goba} in the same way. This
time three bundles and three names are highlighted, and one name (\textsl{M.
Galab}) has a high frequency. This quickly catches Sarah's attention, so she
decides to temporarily pause the analytical branch of \textsl{B. Dhaliwal}, and
moves on with the branch of \textsl{F. Goba}. Sarah hovers the mouse over
\textsl{M. Galab} to check whether it leads to more information. However, it
turns out that no additional bundles or names are highlighted. Sarah realizes
that people potentially related with \textsl{M. Galab} have already been
highlighted in her current view. The bundle (shown in
Figure~\ref{fig:biset-case-without-evaluation} (3) as the black dot box in the
middle) reveals the people related with \textsl{M. Galab}, and all their
activities are in the US\@. With this bundle, Sarah acquires this key insight
revealed by a group of locations. The relations revealed in this bundle are
important, and Sarah infers that the three people (\textsl{M. Galab}, \textsl{Y.
Mosed} and \textsl{Z. al Shibh}) may work on something together in the US\@.
Thus, she decides to find more relevant information by following this tail
\cite{kang2009evaluating}.
Sarah selects the same bundle. BiSet highlights relevant bundles that
potentially form bicluster chains with the selected one. She finds that five
bundles, in the space between the location list and the phone number list, are
highlighted, and two bundles, in the space between the phone number list and the
date list, are highlighted. Relevant entities in lists are also highlighted.
In the two lists of newly highlighted bundles, Sarah finds that there are two
big ones (relatively longer in width shown in
Figure~\ref{fig:biset-case-without-evaluation} (4)) in each list. These two
bundles seem useful since they contain more relations. Sarah chooses to
investigate these first and tries to check how bundles from different
relationship lists are connected. For bundles between the location list and the
phone number list (from top to bottom), Sarah finds that the first bundle and
the third one share two locations (\textsl{Charlottesville} and
\textsl{Virginia}) with the selected bundle, and other highlighted bundles just
share one location with the selected one. Compared with the third bundle, the
first one is related with more locations that are not associated the selected
bundle. Sarah chooses to focus on information highly connected with the selected
bundle, rather than additional information. Thus, she considers the third bundle
a useful one. With the same strategy in another bicluster list, she finds that
the bigger bundle is more useful.
After this step, Sarah hides edges of other bundles with the right click menu to create
a clear view. Then her workspace shows that three bundles connected to each
other through two shared locations and three shared phone numbers. Sarah feels
that she has found a good number of relations, connecting four groups of
entities, which may reflect a suspicious activity. Therefore, she decides to
read relevant documents to find details of such connections and generate her
hypothesis.
The three connected bundles direct Sarah to eight reports, which are all
relevant to the plot. Sarah reads these reports by referring to the entities
with bright shading in the four connected groups (shown in
Figure~\ref{fig:biset-case-without-evaluation} (4)). The darker shading of an
entity indicates that it is shared more times. Sarah uses this information to help keep her
attention to more important entities in reports. After reading the reports, she
identifies a potential threat with four key persons as follows:
\begin{quote}
\textsl{F. Goba}, \textsl{M. Galab} and \textsl{Y. Mosed}, following the
commands from \textsl{A. Ramazi}, plan to attack \textsl{AMTRAK Train $19$} at
\textsl{9:00 am} on \textsl{April 30}.
\end{quote}
In this use case, Sarah has to manually check details about shared entities to
determine which biclusters are meaningful and useful because BiSet does not
provide the function of model based bicluster-chain evaluation. With just
\textsl{connection} oriented highlighting, Sarah has to verify many connected
biclusters to find potentially useful ones (e.g., finding \textsl{b} in
Figure~\ref{fig:biset-case-without-evaluation} (4) as a useful bicluster). This
limits her analysis strategy as stepwise search, and such search focuses on
checking the shared entities of investigated biclusters. Thus, it takes Sarah
significant effort to work at the
entity-level to finally identify a
meaningful bicluster chain.
\subsubsection{\MaxEntBiSet Use Case}
Similar to the previous case, Sarah begins analysis by hovering individual
entities in the list of people. \MaxEntBiSet highlights related bundles and
entities as she hovers the mouse over an entity. Immediately she finds that
\textsl{A. Ramazi} is active in three bundles
(Figure~\ref{fig:biset-case-with-evaluation1} (1)), which indicates that this
person is involved in three coordinated activities. Based on edges, Sarah finds
that two bundles are similar (see the black dotted box in
Figure~\ref{fig:biset-case-with-evaluation1} (1)) due to the number of their
shared entities. Thus, she decides to further investigate them.
With the right click menu on the two bundles, Sarah uses the \textsl{stepwise}
evaluation function, provided by \MaxEntBiSet, to find their neighboring bundles
that contain the most surprising information
(Figure~\ref{fig:biset-case-with-evaluation1} (2) and (3)). Based on evaluated
scores from the \textsl{maximum entropy model}, \MaxEntBiSet highlights their
most surprising neighboring bundles. She finds that the most surprising bundles
connected with the two investigated bundles are the same. This indicates that
the model-suggested most surprising bundle may be important and worthy of further
inspection, and so Sarah decides to find more relevant information from it.
\begin{sidewaysfigure}
\centering
\includegraphics[width=1\textwidth]{biset_case_with_eva1}
\caption{A process of finding one major threat plot with key steps. (1): Based
on \textsl{A. Ramazi}, finding that there are two similar bundles. (2):
Finding the most surprising bundle evaluated by the \textsl{Maximum
Entropy model} for one of the two similar bundles. (3): Finding the most
surprising bundle evaluated by the \textsl{maximum entropy model} for the
other one of the two similar bundles. (4). The most surprising
bicluster-chain suggested by the \textsl{maximum entropy model} for the
most surprising bundle identified in previous steps.}
\label{fig:biset-case-with-evaluation1}
\end{sidewaysfigure}
Sarah chooses the \textsl{full path} evaluation function on this
model-suggested bundle to find the most surprising bicluster-chain. \MaxEntBiSet
highlights the path (Figure~\ref{fig:biset-case-with-evaluation1} (4)) passing
through this bundle having the highest evaluation score from the
maximum entropy model. This provides four connected sets of entities
from all the selected domains (people, location, phone and date). Sarah feels
that she has discovered enough information for a story, so she checks entities
involved in this chain and reads documents from the three connected bundles. The
three bundles directs Sarah to nine reports in total, and eight of them are
relevant to each other. After reading these relevant reports, she identifies a
potential threat with four key persons as follows:
\begin{quote}
\textsl{F. Goba}, \textsl{M. Galab} and \textsl{Y. Mosed}, following the
commands from \textsl{A. Ramazi}, plan to attack \textsl{AMTRAK Train $19$} at
\textsl{9:00 am} on \textsl{April 30}.
\end{quote}
Sarah is satisfied with this finding and marks the bundles in this model
suggested chain as useful, using the right click menu. This informs the
integrated maximum entropy model in \MaxEntBiSet that the information
in these bundles has been known to the analyst, and so the model updates its
background information for further evaluations.
\begin{sidewaysfigure}
\centering
\includegraphics[width=1.02\textwidth]{biset_case_with_eva2}
\caption{A process of finding another key threat plot. (5): Finding the most
surprising bicluster-chain by the \textsl{maximum entropy model} from the
bundle marked in the black dotted box. (6): Requesting the
\textsl{stepwise} evaluation on the bundle \textsl{a} in (5). (7): Based
on the most surprising bundle shown in (6), requesting to find its most
surprising bicluster-chain. (8): Based on the shared entity, \textsl{B.
Dhaliwal}, requesting to find the most surprising bicluster-chain from the
bundle that includes \textsl{B. Dhaliwal} and \textsl{A. Ramazi}. The
chain from this step and that from the previous step merge together.}
\label{fig:biset-case-with-evaluation2}
\end{sidewaysfigure}
The content of one report, from the bundle in the middle of the surprising chain
(\textsl{a} in Figure~\ref{fig:biset-case-with-evaluation1} (4)), is irrelevant
to that of the other eight, but the entities extracted from this report are
connected with those in the identified threat. Thus, Sarah considers the
information in this report as potentially useful clues, which may lead to some
other threat plot(s). In order to check what new information it can bring in,
she uses the \textsl{full path} evaluation function on the bundle in the
middle of the surprising chain (\textsl{a} in
Figure~\ref{fig:biset-case-with-evaluation1} (4)). Based on this request,
\MaxEntBiSet highlights another chain
(Figure~\ref{fig:biset-case-with-evaluation2} (5)).
This newly highlighted chain has one new bundle (\textsl{a} in
Figure~\ref{fig:biset-case-with-evaluation2} (5)), and this chain merged with
previously suggested surprising chain (comparing
Figure~\ref{fig:biset-case-with-evaluation1} (4) with
Figure~\ref{fig:biset-case-with-evaluation2} (5)). By checking this newly
brought in bundle, Sarah finds that all its entities are different from those in
previously investigated bundles. In order to connect this new piece of
information with previously examined pieces, Sarah decides to use the
\textsl{stepwise} evaluation function on this bundle.
After this stage, \MaxEntBiSet highlights just one bundle
(Figure~\ref{fig:biset-case-with-evaluation2} (6)), which is the most surprising
one suggested by the model. From this bundle, Sarah finds that it includes the
person, \textsl{B. Dhaliwal}. This quickly catches her attention since she
remembers that \textsl{B. Dhaliwal} is connected with \textsl{A. Ramazi}
(Figure~\ref{fig:biset-case-with-evaluation1} (1)). Because of this connection,
Sarah decides to find more information from this bundle and another bundle that
includes \textsl{B. Dhaliwal} and \textsl{A. Ramazi} (the bundle on top in the
black dotted box in Figure~\ref{fig:biset-case-with-evaluation1} (1)), so she
requests the \textsl{full path} evaluation from them. Based on the request from
the newly highlighted bundle shown in
Figure~\ref{fig:biset-case-with-evaluation2} (6), \MaxEntBiSet highlights a new
bicluster-chain (Figure~\ref{fig:biset-case-with-evaluation2} (7)). Then based
on the evaluation request from the bundle including \textsl{B. Dhaliwal} and
\textsl{A. Ramazi}, \MaxEntBiSet highlights another chain. Sarah finds these two
chains merge together (Figure~\ref{fig:biset-case-with-evaluation2} (8)). The
two merged chains both include new pieces of information which connects with the
previous findings. Thus, Sarah decides to read the reports that are related to
these four bundles.
From the four bundles, in the document view of \MaxEntBiSet, Sarah finds in
total ten unique reports. Of the ten reports, six show evidences about a new
threat and three are those relevant to previously identified threat plot. Based
on the six reports, Sarah identifies the potential threat as:
\begin{quote}
\textsl{B. Dhaliwal} and \textsl{A. Ramazi} plan to attack the \textsl{New York
Stock Exchange} at \textsl{9:00 am} on \textsl{April 30}.
\end{quote}
Considering the connections between this plot and the previously identified one
(e.g., they share some people's names and date), Sarah also confirms that
\textsl{A. Ramazi} is the key person who coordinates the two planned attacks.
With the capability of model evaluations, in this use case, \MaxEntBiSet
effectively directs Sarah to discover potentially meaningful biclusters or
bicluster-chains. Using colors to visually indicate the model evaluation scores
in \MaxEntBiSet, Sarah can easily see the most surprising bicluster or
bicluster-chain, evaluated by the maximum entropy model. Compared with
the previous use case of BiSet, following the model-suggested biclusters or
chains saves Sarah significant time
in checking entity-level overlaps for meaningful
bicluster identification. In this use case, the maximum entropy model
shares the burden of Sarah for foraging information (e.g., finding potentially
useful biclusters or chains). Thus, compared with the first use case, Sarah can
spend more time and effort to synthesize the visualized structured information
for hypothesis generation.
\subsubsection{Comparison between BiSet and \MaxEntBiSet}
Both BiSet and \MaxEntBiSet can highlight entities and biclusters based on
connections, and visually present entities and biclusters (algorithmically
identified structured information) in an organized manner. However, compared
with BiSet, \MaxEntBiSet also enables the
highlighting entities and biclusters based
on identified surprising coalitions from the maximum entropy model.
Comparing the two cases discussed above, we find that \MaxEntBiSet better
supports the user's sensemaking process of exploring entity coalitions, than BiSet
does, from two key aspects: 1) efficiency and 2) exploring new analytical paths.
Compared with BiSet, \MaxEntBiSet more effectively directs users' attention to
potentially useful biclusters or bicluster-chains by visually prioritizing them
with colors based on their maximum entropy model evaluation scores. The model
evaluation function provided in \MaxEntBiSet eases the process for users to find
useful biclusters, particularly compared with manually entity overlap
investigation. For example, in the first use case, a user has to examine in
total 9 biclusters ($4$ in the left most bicluster list, $5$ in the middle
bicluster list and $2$ in the right most bicluster list as shown in
Figure~\ref{fig:biset-case-without-evaluation}), before she finally identifies a
meaningful bicluster-chain that covers the information of a potential threat.
However, in the second use case, \MaxEntBiSet directs the user to a
bicluster-chain after she investigates 3 biclusters (in the left most bicluster
list shown in Figure~\ref{fig:biset-case-with-evaluation1}). Although this chain
is slightly different from the manually identified one in the first use case, it
covers the same amount of information as the other one does. Thus, in the second
use case, \MaxEntBiSet saves the user from checking highlighted biclusters in
the other two lists, and effectively provides a useful bicluster-chain for users
to explore.
Based on the four user selected domains (visualized as a fixed schema), it is
hard to identify all three threat plots in the \textsl{Crescent} dataset because
not all pre-identified biclusters can be shown. However, from the two cases, we
can find that \MaxEntBiSet can direct users from one identified plot to a new
plot via a surprising bicluster-chain. However, when users manually forage
relevant information, it is not easy for them to make such transitions due to
cognitive tunneling \cite{thomas2001visual}. In the first use case, the key
bundle that can lead to a new plot is actually identified not as useful as the
one shown as \textsl{b} in Figure~\ref{fig:biset-case-without-evaluation}. Thus,
\MaxEntBiSet significantly aids in identifying coalitions of entities worthy of
further exploration.
\section{Introduction}
\label{sec:intro}
\begin{figure}[!t]
\centering
\includegraphics[width=1\textwidth]{intro}
\caption{Illustration of \MaxEntBiSet. (left) Discovery of coordinated
relationship chains is aided by regular incorporation of user feedback.
(right) Unaided algorithmic discovery of relationship chains leads to
long lists of patterns that might not lead to the desired answer.}\label{fig:intro}
\end{figure}
Unstructured exploration of large text datasets is a crucial problem in many
application domains, e.g., intelligence analysis, biomedical discovery, analysis
of legal briefs and opinions. The state-of-the-art today involves two broad
classes of techniques. Visual analytic tools, e.g.,
Jigsaw~\cite{stasko2008jigsaw}, support the exploration of relationships
extracted from large text datasets. While they promote human-in-the-loop
analysis, identifying promising leads to explore is left to the creativity of
the user. At the other end of the spectrum, text analysis techniques such as
storytelling~\cite{Hossain:2012:SEN:2339530.2339742} provide interesting
artifacts (e.g., stories, summaries) for analysis but are limited in their
ability to incorporate user input to steer the discovery process.
Our goal here is to realize an amalgamation of algorithmic and human-driven
techniques to support the discovery of coordinated relationship chains. A
coordinated relationship (also called a bicluster) is one in which a group of
entities are related to another group of entities via a common relation. It is
thus a generalization of a relationship instance. A chain of such coordinated
relationships enables us to bundle groups of entities across various domains and
relate them through a succession of individual relationships. The primary
artifact of interest are thus chains summarizing how entities in a document
collection are related. We introduce new maximum entropy (MaxEnt) models to
identify surprising chains of interest and rank them for inspection by the user.
In intelligence analysis, such chains can reveal how hitherto unconnected people
or places are related through a sequence of intermediaries. In biomedical
discovery, such chains can reveal how proteins involved in distinct pathways are
related through cross-talk via other proteins or signaling molecules. In legal
briefs, one can use chains to determine how rationale for court opinions vary
over the years and are buttressed by the precedence structure implicit in legal
history.
As shown in Fig.~\ref{fig:intro} (left), we envisage an interactive approach
wherein user feedback is woven at each stage and used to rank the most
interesting chains for further exploration. We will demonstrate through case
studies how such an approach gets users to their intended objectives faster than
a purely algorithmic approach (Fig.~\ref{fig:intro} (right)). The work
presented here is implemented in a system -- Maximum Entropy Relational Chain
ExploRer (\MaxEntBiSet) that uses a variety of visual exploration strategies and
algorithmic means to foster user exploration.
Our key contributions are:
\begin{enumerate}
\item \MaxEntBiSet is a marriage of two of our prior
works~\cite{Wu:2014:UPD:2664051.2664089, sun2015biset} but supercedes
the state-of-the-art in these papers in significant, orthogonal, ways. \MaxEntBiSet
is a significant improvement over the work presented
in~\cite{sun2015biset} because~\citet{sun2015biset} provides support for
only manual exploration of coordinated relationships. \MaxEntBiSet is a
significant improvement over the work presented
in~\cite{Wu:2014:UPD:2664051.2664089}
because~\citet{Wu:2014:UPD:2664051.2664089} only presents approaches to
rank chains involving a binary maximum entropy model whereas
\MaxEntBiSet introduces more general maximum entropy approaches for
real-valued data.
\item We present two path strategies (full path and stepwise) to help analyze
datasets. Using our proposed maximum entropy models, the full path strategy
discovers the most surprising bicluster chains from all possible chains
involving an analyst-selected bicluster. The stepwise strategy evaluates
biclusters neighboring a user-specified one, and prioritizes possible
connected information with the current pieces under investigation. Both strategies
directs analysts to reveal hidden plots involving surprising relational
patterns.
\item We describe new visual encodings and summary as well as detailed views to
support user-guided exploration of coordinated relationships in massive
datasets. Besides basic color codings (e.g., \textsl{connection-oriented}
highlighting presented in ~\cite{sun2015biset}), \MaxEntBiSet offers highlighting
mechanisms aimed at pointing out surprising information.
Enhanced with the proposed maximum entropy models, this highlighting capability not only
directs user's attention to important connected pieces of information, but
also visually prioritizes them in a usable manner.
\item We describe experimental results on both large, synthetic datasets (to
illustrate efficiency and effectiveness of our algorithms) and small, real
datasets (to illustrate how users can interactively explore a realistic text
dataset. In particular, we show how \MaxEntBiSet enables the user to more
quickly arrive at plots of interest than the traditional manual approach
described in~\cite{sun2015biset}.
\end{enumerate}
\section{Tile-based Maximum Entropy Model}
\label{sec:model}
Our problem statement is based on a notion of a multi-relational schema. In practice,
such multi-relational datasets are inferred from a transactional dataset (e.g., entities
discovered from a document collection, and then subsequently related by co-occurrence).
More specifically, we assume that our schema was generated from a transactional
data matrix $D$ (see Fig.~\ref{fig:mercer_arch}). This data matrix can be viewed
as a matrix of size $N$-by-$M$. We will introduce the method of obtaining a
schema from $D$ in Section~\ref{sec:score}. In this approach the columns of $D$
correspond to the entities of the schema. Hence, we will refer to the columns of
$D$ as entities.
\subsection{Maximum Entropy Model for Binary Data}
\label{sec:binary_maxent}
In this section, we will define the maximum entropy (MaxEnt) model for binary
data matrices using tiles as background knowledge---recall that a tile is a more
general notion than a bicluster. We will first introduce notation that will be
useful to understand the model derivation in the context of binary data. Then,
we will recall MaxEnt theory for modeling binary data given tiles as background
information, and finally, identify how we can fit the model to the data by
maximizing the likelihood.
\subsubsection{Notation for Tiles}
\label{sec:maxent:tile.notation}
Given a binary data matrix $D$ of size $N$-by-$M$ and a tile $T$, the frequency
of $T$ in $D$, $\mathit{fr}(T;D)$, is defined as
\begin{equation}
\label{eq:freq}
\mathit{fr}(T;D) = \frac{1}{|\sigma(T)|} \sum\limits_{(i,j) \in \sigma(T)} D(i,j) \quad .
\end{equation}
Here, $D(i,j)$ represents the entry $(i,j)$ in $D$, and $\sigma(T) = \{(i,j)
\mid i \in r(T), j \in c(T)\}$ denotes the cells covered by tile $T$ in data
$D$. Recall that a tile $T$ is called `exact' if the corresponding entries
$D(i,j)$ $\forall (i,j) \in \sigma(T)$ are all $1$ (resp.\ $0$), or in other
words, $\mathit{fr}(T;D) = 0$ or $\mathit{fr}(T;D) = 1$. Otherwise, it is called
a `noisy' tile.
Let $\mathcal{D}$ be the space of all the possible binary data matrices
of size $N$-by-$M$, and $p$ be the probability distribution defined over the
data matrix space $\mathcal{D}$. Then, the frequency of the tile $T$ with
respect to $p$ is
\begin{equation}
\label{eq:expected.freq}
\mathit{fr}(T;p) = \mathbb{E}\left[\mathit{fr}(T;D)\right] =
\sum\limits_{D \in \mathcal{D}} p(D)\mathit{fr}(T;D) \quad ,
\end{equation}
the expected frequency of tile $T$ under the data matrix probability distribution.
Combining these definitions, we have the following
lemma~\cite{Wu:2014:UPD:2664051.2664089}.
\begin{lemma
\label{lemma:1}
Given a dataset distribution $p$ and a tile $T$, the frequency of tile $T$ is
\begin{equation*}
\mathit{fr}(T;p) = \frac{1}{|\sigma(T)|} \sum\limits_{(i,j) \in \sigma(T)}
p\left(\left(i,j\right) = 1\right) \quad ,
\end{equation*}
where $p((i,j) = 1)$ represents the probability of a data matrix having 1 at
entry $(i,j)$ under the data matrix distribution $p$.
\end{lemma}
Lemma~\ref{lemma:1} is trivially proved by substituting $\mathit{fr}(T;D)$ in
Equation~\eqref{eq:expected.freq} with Equation~\eqref{eq:freq} and switching
the summations.
\subsubsection{Global MaxEnt Model from Tiles}
\label{sec:maxent:tile.global}
Here, we will construct a global statistical model based on tiles. Suppose we
are given a set of tiles $\mathcal{T}$, and each tile $T \in \mathcal{T}$ is
associated with a frequency $\gamma_{T}$---which typically can be trivially
obtained from the data. This tile set $\mathcal{T}$ provides information about
the data at hand, and we would like to infer a distribution $p$ over the space
of possible data matrices $\mathcal{D}$ that conforms with the information given
in $\mathcal{T}$. That is, we want to be able to determine how probable is a
data matrix $D \in \mathcal{D}$ given the tile set $\mathcal{T}$.
To derive a good statistical model, we take a principled approach and employ the
maximum entropy principle \cite{jaynes:59:maxent} from information theory.
Loosely speaking, the MaxEnt principle identifies the best distribution given
background knowledge as the unique distribution that represents the provided
background information but is maximally random otherwise. MaxEnt modeling has
recently become popular in data mining as a tool for identifying
\emph{subjective} interestingness of results with regard to background
knowledge~\cite{wang:06:summaxent,debie:11:dami,tatti:12:apples}.
To formally define a MaxEnt distribution, we first need to specify the
space of the probability distribution candidates. Here, these are all the
possible data matrix distributions that are consistent with the information
contained in the tile set $\mathcal{T}$. Hence, the data matrix distribution
space is defined as: $\mathcal{P} = \{p \mid \mathit{fr}(T;p) = \gamma_{T},
\forall T \in \mathcal{T}\}$. Among all these possible distributions, we choose
the distribution $p_{\mathcal{T}}^{*}$ that maximizes the entropy,
\begin{equation*}
p_{\mathcal{T}}^{*} = \arg \max_{p \in \mathcal{P}} H(p) \quad .
\end{equation*}
Here, $H(p)$ represents the entropy of the data matrix probability distribution
$p$, which is defined as
\begin{equation*}
H(p) = -\sum\limits_{D \in \mathcal{D}} p(D) \log p(D) \quad .
\end{equation*}
Next, to infer the MaxEnt distribution $p_{\mathcal{T}}^{*}$, we rely on a
classical theorem about how MaxEnt distributions can be factorized. In
particular, Theorem 3.1 in \cite{1975:Csiszar} states that for a given set of
testable statistics $\mathcal{T}$ (background knowledge, here a tile set), a
distribution $p_{\mathcal{T}}^{*}$ is the maximum entropy distribution if and
only if it can be written as
\begin{equation*}
p_{\mathcal{T}}^{*}(D) \propto \left\{
\begin{array}{ll}
\exp\big( \sum\limits_{T \in \mathcal{T}} \lambda_{T} \cdot |\sigma(T)|
\cdot \mathit{fr}(T;D)\big) & D \not\in \mathcal{Z} \\
0 & D \in \mathcal{Z}\quad , \\
\end{array} \right.
\end{equation*}
where $\lambda_{T}$ is a certain weight for $\mathit{fr}(T;D)$ and $\mathcal{Z}$
is a collection of data matrices such that $p(D) = 0$, for all $p \in
\mathcal{P}$.
\citet{debie:11:dami} formalized the MaxEnt model for a binary matrix $D$ given
row and column margins---also known as a \citet{rasch:60:probabilistic} model.
Here, we consider the more general scenario of binary data and tiles, for which
we additionally know \cite[Theorem 2 in][]{tatti:12:apples} that given a tile
set $\mathcal{T}$, with $\mathcal{T}(i,j) = \{T \in \mathcal{T} \mid (i,j) \in
\sigma(T)\}$, we can write the distribution $p_{\mathcal{T}}^{*}$ as
\begin{equation*}
p_{\mathcal{T}}^{*} = \prod\limits_{(i,j) \in D} p_{\mathcal{T}}^{*}((i,j) =
D(i,j)) \quad ,
\end{equation*}
where
\begin{equation*}
p_{\mathcal{T}}^{*}((i,j) = 1) = \frac{\exp\left(\sum_{T \in \mathcal{T}(i,j)}
\lambda_{T}\right)}{\exp\left(\sum_{T \in \mathcal{T}(i,j)}
\lambda_{T}\right) + 1} ~\text{or}~ 0,1 \quad .
\end{equation*}
This result allows us to factorize the MaxEnt distribution $p_{\mathcal{T}}^{*}$
of binary data matrices given background information in the form of a set of
tiles $\mathcal{T}$ into a product of Bernoulli random variables, each of which
represents a single entry in the data matrix $D$. We should emphasize here that
this model is different MaxEnt model than when we assume independence between
rows in the data matrix $D$~\cite[see,
e.g.,][]{tatti:06:computational,wang:06:summaxent,mampaey:12:mtv}. Here, for
example, in the special case where the given tiles are all exact ($\gamma_{T} =
0$ or $1$), the resulting MaxEnt distribution will have a very simple form:
\begin{equation*}
p_{\mathcal{T}}^{*}((i,j) = 1) = \left\{
\begin{array}{ll}
\gamma_{T} & \text{if } \exists T \in \mathcal{T} \text{ such that }
(i,j) \in \sigma(T)\\
\frac{1}{2} & \text{otherwise.} \\
\end{array} \right.
\end{equation*}
\subsubsection{Inferring the MaxEnt Distribution}
\label{sec:maxent:iter.scaling}
\begin{algorithm}[t]
\SetAlgoLined
\SetKwInOut{Input}{input}
\SetKwInOut{Output}{output}
\Input{a tile set $\mathcal{T}$, target frequencies
$\{\gamma_{T}\mid T \in \mathcal{T} \}$.}
\Output{maximum entropy distribution $p_{\mathcal{T}}^{*} \leftarrow p$.}
\BlankLine
$p \leftarrow$ a $N$-by-$M$ matrix with all values of $\frac{1}{2}$\;
\For{$T \in \mathcal{T}, \gamma_{T} = 0, 1$}{
$p(i,j) \leftarrow \gamma_{T}$, for all $(i,j) \in \sigma(T)$\;
}
\While{not converged}{
\For{$T \in \mathcal{T}, 0 < \gamma_{T} < 1$}{
find $x$ such that: $\mathit{fr}(T;p) = \sum_{(i,j) \in \sigma(T)}
\frac{x \cdot p(i,j)}{1 - (1 - x) \cdot p(i,j)}$\;
$p(i,j) \leftarrow \frac{x \cdot p(i,j)}{1 - (1 - x) \cdot p(i,j)}$, for all $(i,j) \in \sigma(T)$\;
}
}
\caption{Iterative Scaling Algorithm (binary dataset)} \label{alg:1}
\end{algorithm}
To discover the parameters of the Bernoulli random variable mentioned above, we
follow a standard approach and apply the well known Iterative Scaling (IS)
algorithm \cite{1972:iterative:scaling} to infer the tile based MaxEnt
distribution on binary data matrices. Algorithm~\ref{alg:1} illustrates the
details of this IS algorithm for binary data. Basically, for each tile $T
\in \mathcal{T}$, the algorithm updates the probability distribution $p$ such
that the expected frequency of $1$s under the distribution $p$ will match
the given frequency $\gamma_{T}$. Clearly, during this update we may change the
expected frequency for other tiles, and hence several iterations are needed
until the probability distribution $p$ converges. For a proof of convergence,
please refer to Theorem 3.2 in \cite{1975:Csiszar}. In practice, the algorithm
typically takes on the order of seconds to converge.
\subsection{Maximum Entropy Model for Real-valued Data}
\label{sec:real_maxent}
In this section, we introduce the MaxEnt model for real-valued data with tiles
as background knowledge. We first extend the concept of tiles from binary
transactional matrix to a real-valued transactional matrix. Then, we formulate
the global MaxEnt model over the real-valued transactional data, and finally, we
provide an efficient algorithm to infer the real-valued MaxEnt distribution.
\subsubsection{Notation for Tiles}
\label{sec:real_tiles}
As stated earlier, a document-entity transactional matrix $D$ usually contains
occurrence (count) information for each entity in every document of the corpus.
Count data is integer valued but without loss
of generality, the entries in the real-valued transactional matrix $D$ is
considered to be normalized into the range of $[0,1]$ (e.g.\ each entry of $D$
can be divided by the maximum entry of $D$).
A tile $T$ over a real-valued matrix $D$ is still defined as the tuple $T =
(r(T), c(T))$ which identifies a sub-matrix from $D$. Compared to the frequency
of a tile defined in the binary case, more descriptive statistical measures can
be defined for real-valued tiles. In our scenario, we choose the sum of the
values and sum of the squared values identified by a tile $T$, which are
represented by $f_m$ and $f_v$ respectively. More specifically, $f_m$ and $f_v$
are defined as follow:
\begin{align}
\label{eq:def_mean_var}
f_m(T \mid D) & = \sum_{\forall (i,j) \in \sigma(T)} D(i,j) \\
f_v(T \mid D) & = \sum_{\forall (i,j) \in \sigma(T)} D^2(i,j) \nonumber
\end{align}
\subsubsection{Global MaxEnt Model from Tiles}
\label{sec:maxent_real_tile}
A real-valued MaxEnt model was first proposed
by~\citet{real:value:maxent}. Given a set of real-valued tiles $\mathcal{T}$
where for every entry $(i,j)$ in the matrix $D$, there exists at least a tile $T
\in \mathcal{T}$ such that $(i,j) \in \sigma(T)$. Each tile $T \in \mathcal{T}$
is associated with $\tilde{f}_m(T)$ and $\tilde{f}_v(T)$ as its basic
statistics. Then, the probability distribution space of real-valued data
matrices can be defined as
$$
\mathcal{P} = \{p \mid \mathbb{E}_{p}[f_m(T \mid D)] = \tilde{f}_m(T),
\mathbb{E}_p[f_v(T \mid D)] = \tilde{f}_v(T), \forall T \in \mathcal{T} \}~.
$$
Here, $\tilde{f}_m$ and $\tilde{f}_v$ denote the empirical values of the
statistics associated with tiles, which can be computed from the given
real-valued data matrix, and $\mathbb{E}_{p}[\cdot]$ represents the expectation
with respect to the probability distribution $p$. Among all the candidate
distribution $p \in \mathcal{P}$, we choose the one that maximizes the entropy
according to:
$$
p^{*}_{\mathcal{T}} = \operatornamewithlimits{argmax}_{p \in \mathcal{P}} \left\{ -\oint\limits_{D} p(D)
\log p(D) dD \right\}~.
$$
To be more specific, inferring the MaxEnt distribution could be formulated as
the following optimization problem:
\begin{align}
\label{eq:maxent_optimize}
p^{*}_{\mathcal{T}} & = \operatornamewithlimits{argmax}_{p} \left\{ -\oint\limits_{D} p(D) \log p(D)
dD \right\} \\
\text{s.t.} \quad & \oint\limits_{D} p(D) f_{m}(T \mid D) d D =
\tilde{f}_{m}(T),~\forall T \in \mathcal{T} \nonumber \\
& \oint\limits_{D} p(D) f_{v}(T \mid D) d D = \tilde{f}_{v}(T),~\forall T \in
\mathcal{T} \nonumber \\
& \oint\limits_{D} p(D) d D = 1,~p(D) \geq 0 \nonumber
\end{align}
Since the optimization problem defined above is convex, by applying the approach
of Lagrange multipliers, we can derive that the MaxEnt distribution has the
following exponential form:
$$
p^{*}_{\mathcal{T}}(D) = \frac{1}{Z} \exp\left(-\sum_{T \in \mathcal{T}}
\lambda^{(m)}_{T} f_m(T \mid D) - \sum_{T \in \mathcal{T}} \lambda^{(v)}_{T}
f_v(T \mid D) \right)~.
$$
Substituting $f_m(T \mid D)$ and $f_v(T \mid D)$ with their definitions from
Equation~\eqref{eq:def_mean_var}, the MaxEnt distribution could be simplified
as:
\begin{align}
\label{eq:maxent_factorized}
p^{*}_{\mathcal{T}} & = \frac{1}{Z} \prod_{(i,j) \in D} \exp
\left(-\beta_{i,j} D^{2}(i,j) - \alpha_{i,j} D(i,j) \right) \\
& = \prod_{(i,j) \in D} p_{i,j}(D(i,j)) \nonumber
\end{align}
where
\begin{align*}
p_{i,j}(D(i,j)) & = \sqrt{\frac{\beta_{i,j}}{\pi}} \exp\left\{ -\frac{\left(
D(i,j) + \frac{\alpha_{i,j}}{2\beta_{i,j}}\right)^2}{1 / \beta_{i,j}}
\right\} \\
\alpha_{i,j} & = \sum_{\substack{(i,j) \in \sigma(T) \\ T \in \mathcal{T}}}
\lambda_{T}^{(m)}, \quad
\beta_{i,j} = \sum_{\substack{(i,j) \in \sigma(T) \\ T in \mathcal{T}}}
\lambda_{T}^{(v)} \nonumber
\end{align*}
Equation~\eqref{eq:maxent_factorized} indicates that the real-valued MaxEnt
distribution over the matrix $D$ could be factorized into the product of the
distributions of $D(i,j)$ where each $D(i,j)$ follows the Gaussian distribution:
$$
D(i,j) \sim \mathbb{N}\left( -\frac{\alpha_{i,j}}{2 \beta_{i,j}},
\frac{1}{2 \beta_{i,j}}\right)
$$
In addition, we can also compute the normalizing constant $Z$ in
Equation~\eqref{eq:maxent_factorized} as
\begin{align*}
Z & = \oint\limits_{D} \prod_{(i,j) \in D} \exp \left(-\beta_{i,j} D^2(i,j)
- \alpha_{i,j} D(i,j) \right) d D \\
& = \prod_{(i,j) \in D} \sqrt{\frac{\pi}{\beta_{i,j}}} \exp \left(
\frac{\alpha^{2}_{i,j}}{4 \beta_{i,j}}\right)
\end{align*}
\subsubsection{Inferring the MaxEnt Distribution}
\label{sec:infer_real}
\begin{algorithm}[t]
\SetAlgoLined
\SetKwInOut{Input}{input}
\SetKwInOut{Output}{output}
\SetKwFunction{updateAlphaBeta}{updateAlphaBeta}
\Input{a tile set $\mathcal{T}$, target tile statistics
$\{f_{m}(T \mid D), f_{v}(T \mid D) \mid T \in \mathcal{T} \}$.}
\Output{Maximum Entropy distribution $p_{\mathcal{T}}^{*}$ parameterized by
$\alpha_{i,j}$ and $\beta_{i,j}$.}
\BlankLine
Initialize $\lambda_{T}^{(m)}$ and $\lambda_{T}^{(v)}$ randomly $\forall T \in
\mathcal{T}$\;
$\boldsymbol{\lambda} \leftarrow [\lambda_T^{(m)}, \lambda_T^{(v)} \mid T \in
\mathcal{T}]$\;
\While{not converged}{
\updateAlphaBeta{$\boldsymbol{\lambda}$}\;
compute gradient using Equation~\eqref{eq:grad_m} and~\eqref{eq:grad_v}\;
perform a conjugate gradient update on $\boldsymbol{\lambda}$\;
}
\caption{MaxEnt model inference (real-valued dataset)} \label{alg:2}
\end{algorithm}
To infer the real-valued MaxEnt distribution, we need to estimate the
values of the model parameters $\lambda_{T}^{(m)}$ and $\lambda_{T}^{(v)}$.
We leverage the duality between maximum entropy and maximum likelihood
formulations~\cite{maxent-duality} by solving the following problem:
\begin{align*}
\max_{\boldsymbol{\lambda}}: \mathcal{L}(\boldsymbol{\lambda}) & = \log p(D)
= \sum_{T \in \mathcal{T}} \left( -\lambda_{T}^{(m)} \tilde{f}_{m}(T) -
\lambda_{T}^{(v)} \tilde{f}_{v}(T) \right) - \log Z \\
& = - \sum_{(i,j) \in D} \left[ \frac{1}{2} \log \left(
\frac{\pi}{\beta_{i,j}}\right) +
\frac{\alpha_{i,j}^{2}}{4\beta_{i,j}}\right] + \sum_{T \in \mathcal{T}}
\left( -\lambda_{T}^{(m)} \tilde{f}_{m}(T) - \lambda_{T}^{(v)}
\tilde{f}_{v}(T) \right) \\
\text{s.t.} \quad & \beta_{i,j} > 0, \quad \forall (i,j) \in D
\end{align*}
The above optimization problem is convex and can be solved
efficiently by state-of-the-art optimization algorithms. Here, we choose the
conjugate gradient method to solve this problem, where the gradient of the
objective function $\mathcal{L}(\boldsymbol{\lambda})$ is given by:
\begin{align}
\frac{\partial \mathcal{L}(\boldsymbol{\lambda})}{\partial
\lambda_{T}^{(m)}} & = - \sum_{(i,j) \in \sigma(T)} \left(
\frac{\alpha_{i,j}}{2 \beta_{i,j}} \right) - \tilde{f}_{m}(T)
\label{eq:grad_m} \\
\frac{\partial \mathcal{L}(\boldsymbol{\lambda})}{\partial
\lambda_{T}^{(v)}} & = \sum_{(i,j) \in \sigma(T)} \left(
\frac{1}{2\beta_{i,j}} + \frac{\alpha_{i,j}^{2}}{4 \beta_{i,j}^{2}}
\right) - \tilde{f}_{v}(T) \label{eq:grad_v}
\end{align}
\section{Preliminaries}
\label{sec:prelim}
Figure~\ref{fig:mercer_arch} illustrates the workflow in \MaxEntBiSet.
By taking the background information from the document-entity transactional
matrix, the \MaxEntBiSet system infers the maximum entropy model, which will be
described in detail in Section~\ref{sec:model}. From the document-entity matrix,
multiple entity-entity relations are extracted and surprisingness measure for
relational patterns is defined based on the MaxEnt model
(Section~\ref{sec:score}). By interacting with analysts, our visualization
interface displays the surprising relational patterns discovered from the
multiple entity-entity relations, and also provides analysts' feedback to the
MaxEnt model, which will in turn help to further discover additional surprising
patterns (Section~\ref{sec:biset}).
\begin{figure}[!t]
\centering
\includegraphics[width=1.0\textwidth]{architecture}
\caption{\MaxEntBiSet system workflow.}
\label{fig:mercer_arch}
\end{figure}
In this section, we introduce some preliminary concepts and notations that will
be useful to understand the \MaxEntBiSet system and the rest of this paper.
\paragraph{Multi-relational schema}
We assume that we are given $l$ domains or \emph{universes} which we will denote
throughout the paper by $U_i$, $i \in 1\ldots l$. An entity is a member of $U_i$
and an entity set $E_i$ is simply a subset of $U_i$. We use $R = R(U_i, U_j)$
to denote a binary relation between some $U_i$ and $U_j$. Given a set of
domains $\mathcal{U} = \{U_1, U_2,\ \ldots, U_l\}$ and a set of relations
$\mathcal{R} = \{R_1, R_2, \ \ldots, R_m\}$, a \emph{multi-relational schema}
$S(\mathcal{U}, \mathcal{R})$ is a connected bipartite graph whose vertex set is
given by $\mathcal{U} \cup \mathcal{R}$ and edge set is the collection of edges
each of which connects a relation $R_j$ in $\mathcal{R}$ and a domain $U_i$ in
$\mathcal{U}$ that the relation $R_j$ involves. In this paper, without loss of
generality, all vertices in $\mathcal{R}$ are assumed to have degree of two,
i.e., only binary relationships are considered. As is well known, ternary and
higher-order relations can be converted into sets of binary relationships. (No
such degree constraint exists for $\mathcal{U}$; a domain can participate in
many relationships.)
\paragraph{Tiles}
A \emph{tile} $T$, a notion introduced by~\citet{Geerts04tilingdatabases}, is
essentially a rectangle in a data matrix. Formally, it is defined as a tuple $T
= (r(T), c(T))$ where $r(T)$ is a set of row identifiers (e.g., row IDs) and
$c(T)$ is a set of column identifiers (e.g., column IDs) over the data matrix. In
this most general form, it imposes no constraints on values of the matrix
elements identified by a tile. So, each element in a tile could be any valid
value in the data matrix.
In the binary case, when all elements within a tile $T$ have the same value
(i.e., either all $1$s or all $0$s) we say it is an exact tile. Otherwise we
call it a noisy tile.
\paragraph{Biclusters}
As local patterns of interest over binary relations, we consider biclusters. A
\emph{bicluster}, denoted by $B = (E_i, E_j)$, on relation $R = R(U_i, U_j)$,
consists of two entity sets $E_i \subseteq U_i$ and $E_j \subseteq U_j$ such
that $E_i \times E_j \subseteq R$. As such a bicluster is a special case of an
exact tile, one in which all the elements are 1. Further, we say a bicluster $B
= (E_i, E_j)$ is {\it closed} if for every entity $e_i \in U_i \setminus E_i$,
there is some entity $e_j \in E_j$ such that $(e_i, e_j) \notin R$ and for every
entity $e_j \in U_j \setminus E_j$, there is some entity $e_i \in E_i$ such that
$(e_i, e_j) \notin R$. In other words, $E_i$ is maximal (w.r.t. $E_j$) so that
we cannot add more elements to $E_i$ without violating the premise of a
bicluster. If a pair of entities $e_i \in U_i, e_j \in U_j$ belongs to a
bicluster $B$, we denote this fact by $(e_i, e_j) \in B$.
\paragraph{Redescriptions}
Assume that we are given two biclusters $B = (E_i, E_j)$ and $C = (F_j, F_k)$,
where $E_i \subseteq U_i$, $E_j, F_j \subseteq U_j$, and $F_k \subseteq U_k$.
Note that $E_j$ and $F_j$ lie in the same domain. Assume that we are given a
threshold $0 \leq \varphi \leq 1$. We say that $B$ and $C$ are \emph{approximate
redescriptors} of each other, which we denote by $B \sim_{\varphi, j} C$ if the
Jaccard coefficient $\abs{E_j \cap F_j} / \abs{E_j \cup F_j} \geq \varphi$. The
threshold $\varphi$ is a user parameter, consequently we often drop $\varphi$
from the notation and write $B \sim_j C$. The index $j$ indicates the common
domain over which we should take the Jaccard coefficient. If this domain is
clear from the context we often drop $j$ from the notation.
If $B \sim_{1, j} C$, then we must have $E_j = F_j$ in which case we say that
$B$ is an \emph{exact redescription} of $C$.
This definition is a generalization of the definition given
by~\citet{Zaki:2005:RSU:1081870.1081912}, who define redescriptions for
itemsets over their mutual domain, transactions, such that the set $E_j$
consists of transactions containing itemset $E_i$ and the set $F_j$ consists
of transactions containing itemset $F_k$.
\paragraph{Bicluster Chains}
A \emph{bicluster chain} $C$ consists of an ordered set of biclusters $\{B_{1},
B_{2}, \ldots, B_{k}\}$ and an ordered bag of domain indices $\{j_1, j_2,
\ldots, j_{k - 1}\}$ such that for each pair of adjacent biclusters we have
$B_{i} \sim_{j_i} B_{i + 1}$. Note that this implicitly requires that two
adjacent biclusters share a common domain.
If a bicluster $B_{R_i}$ is a part of the bicluster chain $C$, we will represent
this by $B_{R_i} \in C$ in this paper.
\paragraph{Surprisingness}
In the knowledge discovery tasks studied here, the primary goal is to extract
novel, interesting, or unusual knowledge. That is, we aim to find results that
are highly informative with regard to what we already know---we are not so much
interested in what we already do know, or what we can trivially induce from such
knowledge.
To this end, we suppose a probability distribution $p$ that represents the
user's current beliefs about the data. When mining the data (e.g., for a
bicluster or chain), we can use $p$ to determine the likelihood of a result
under our current beliefs: if it is high, this indicates that we most likely
already know about it, and thus, reporting it to the user would provide little
new information. In contrast, if the likelihood of a result is very low, the
result is very surprising, thus potentially conveying new information. In
Section~\ref{sec:model}, we will discuss how to infer such a probability
distribution for both binary and real-valued data matrices.
\paragraph{Problem Statement}
Given a multi-relational dataset, a bicluster chain across multiple relations
describes a progression of entity coalitions. We are particularly interested in
chains that are surprising w.r.t.\ what we already know, as these could help to
uncover the plots hidden in the multi-relational dataset.
More formally, given a multi-relational dataset schema $\schema(\mathcal{U},
\mathcal{R})$, where $\mathcal{U} = \{U_1, U_2, \ldots, U_l\}$ and
$\mathcal{R} = \{R_1, R_2, \ldots, R_m\}$, we are interested in iteratively finding
non-redundant bicluster chains that are most surprising with regard to each
other and w.r.t.\ the background knowledge with the assistance of visual
analysis techniques.
\section{Related Work}
\label{sec:related}
In this section we survey related work. In particular, we discuss work related
with regard to mining surprising patterns, iterative data mining,
mining multi-relational datasets, finding plots in data, and bicluster
visualizations for data exploration.
\subsection{Mining Biclusters}
Mining biclusters is an extensively studied area of data mining, and many
algorithms for mining biclusters from varied data types have been
proposed, e.g.~\citep{Tibshirani99clusteringmethods, Califano00analysisof,
Segal01062001, Sheng27092003, Cheng:2000:BED:645635.660833, Zaki:1401887,
Uno:2005:LVC:1133905.1133916}. Bicluster mining, however, is not the primary aim
in this paper; instead it is only a component in our proposed framework.
Moreover, the above mentioned studies do not assess whether the mined clusters
are subjectively interesting. A comprehensive survey of biclustering algorithms
was given by \citet{Madeira:2004:BAB:1024308.1024313}.
\subsection{Mining Surprising Patterns}
There is, however, significant literature on mining
representative/succinct/sur-prising patterns
\citep[e.g.,][]{Kiernan:2009:CCS:1631162.1631169} as well as on explicit
summarization \citep[e.g.,][]{conf:sdm:DavisST09}. \citet{wang:06:summaxent}
summarized a collection of frequent patterns by means of a row-based MaxEnt
model, heuristically mining and adding the most significant itemsets in a
level-wise fashion. \citet{tatti:06:computational} showed that querying such a
model is PP-hard. \citet{mampaey:12:mtv} gave a convex heuristic, allowing more
efficient search for the most informative set of patterns.
\citet{debie:11:dami} formalized how to model a binary matrix by MaxEnt using
row and column margins as background knowledge, which allows efficient
calculation of probabilities per cell in the matrix. \citet{real:value:maxent}
first proposed a real-valued MaxEnt model for assessing patterns over
real-valued rectangular databases. These papers all focus on mining surprising
patterns from a single relation. They do not explore the multi-relational
scenario, and can hence not find connections among surprising patterns from
different relations---the problem we focus on.
\subsection{Iterative Data Mining}
Iterative data mining as we study was first proposed by
\citet{hanhijarvi:09:tell}. The general idea is to iteratively mine the result
that is most significant given our accumulated knowledge about the data. To
assess significance, they build upon the swap-randomization approach of
\citet{gionis:07:assessing} and evaluate empirical p-values. With the help of
real-valued MaxEnt model,~\citet{konto:13:numaxentit} proposed a subjective
interestingness measure called \textit{Information Ratio} to iteratively
identify and rank the interesting structures in real-valued data.
\citet{mampaey:12:mtv} and \citet{konto:13:numaxentit} show that ranking results
using a static MaxEnt model leads to redundancy in the top-ranked results, and
that iterative updating provides a principled approach for avoiding this type of
redundancy. \citet{tatti:12:apples} discussed comparing the informativeness of
results by different methods on the same data. They gave a proof-of-concept for
single binary relations, for which results naturally translate into tiles, and
gave a MaxEnt model in which tiles can be incorporated as background knowledge.
In this work we build upon this framework, translating bicluster chains (over
multiple relations) into tiles to measure surprisingness with regard to
background knowledge using a Maximum Entropy model.
\subsection{Multi-relational Mining}
Mining relational data is a rich research area~\citep{rdmbook} with a plethora
of approaches ranging from relational association
rules~\citep{Dehaspe:2001:DRA:567222.567232} to inductive logic programming
(ILP) \citep{ilp}. The idea of composing
redescriptions~\citep{Zaki:2005:RSU:1081870.1081912} and biclusters to form
patterns in multi-relational data was first proposed by
\citet{Jin:2008:CMM:1342320.1342322}. \citet{Cerf:2009:CPM:1497577.1497580}
introduced the \textsc{DataPeeler} algorithm to tackle the challenge of directly
discovering closed patterns from $n$-ary relations in multi-relational data.
Later, \citet{Cerf:2013:noise:tolerant} refined \textsc{DataPeeler} for finding
both closed and noise-tolerant patterns. These frameworks do not provide any
criterion for measuring subjective interestingness of the multi-relational
patterns. \citet{DBLP:conf/sdm/OjalaGGM10} studied randomization techniques for
multi-relational databases with the goal to evaluate the statistical
significance of database queries. \citet{spyropoulou:11:multirel} and
\citet{DBLP:SpyropoulouBB14} proposed to transform a multi-relational database
into a $K$-partite graph, and to mine maximal complete connected subset (MCCS)
patterns that are surprising with regard to a MaxEnt model based on the margins
of this data. \citet{Spyropoulou:local:patterns} extended this approach to
finding interesting local patterns in multi-relational data with $n$-ary
relationships. Bicluster chains and MCCS patterns both identify redescriptions
between relations, but whereas MCCS patterns by definition only identify exact
pair-wise redescriptions (completely connected subsets), bicluster chains also
allow for approximate redescriptions (incompletely connected subsets). All
except for the most simple bicluster chains our methods discovered in the
experiments of Section~\ref{sec:exp} include inexact redescriptions, and could
hence not be found under the MCCS paradigm. Besides that we consider two
different data models, another key difference is that we iteratively update our
MaxEnt model to include all patterns we mined so far.
\subsection{`Finding Plots'}
The key difference between finding plots, and finding biclusters or surprising
patterns is the notion of chaining patterns into a chain, or plot. Commercial
software such as {\it Palantir} provide significant graphic and visualization
capabilities to explore networks of connections but do not otherwise automate
the process of uncovering plots from document collections.
\citet{shahaf-guestrin-journal} studied the problem of summarizing a large
collection of news articles by finding a chain that represents the main events;
given either a start or end-point article, their goal is to find a chain of
intermediate articles that is maximally coherent. In contrast, in our setup we
know neither the start nor end points. Further, in intelligence analysis, it is
well known that plots are often loosely organized with no common all-connecting
thread, so coherence cannot be used as a driving criterion. Most importantly, we
consider data matrices where a row (or, document) may be so sparse or small
(e.g., 1-paragraph snippets) that it is difficult to calculate statistically
meaningful scores. Storytelling algorithms
\citep[e.g.,][]{Hossain:2012:SEN:2339530.2339742,storytelling-tkde,connectingpubmed}
are another related thread of research; they provide algorithmic ways to rank
connections between entities but do not focus on entity coalitions and how such
coalitions are maintained through multiple sources of evidence.
\citet{Wu:2014:UPD:2664051.2664089} proposed a framework to discover the plots
by detecting non-obvious coalitions of entities from multi-relational datasets
with Maximum Entropy principle and further support iterative, human-in-the-loop,
knowledge discovery. However, no visualization framework was developed to enable
analysts to be involved when discovering the surprising entity coaliations in
that work. Moreover, we also propose the full path and step-wise chain search
strategies and combine them together to help analyts to explore the data.
\subsection{Bicluster Visualizations}
Finally, we give an overview of work on bicluster visualization techniques.
Biclusters offer a usable and effective way to present coalitions among sets of
entities across multiple domains. Various visualizations have been proposed to
present biclusters for sensemaking of data in different fields. One typical
application domain of bicluster visualizations is bioinformatics, where
biclusters are visualized to help bioinformaticians to identify groups of genes
that have similar behavior under certain groups of conditions (e.g., BicAt
\cite{barkow2006bicat}, Bicluster viewer \cite{heinrich2011bicluster},
BicOverlapper 2.0 \cite{santamaria2014bicoverlapper}, BiGGEsTS
\cite{gonccalves2009biggests}, BiVoc \cite{grothaus2006automatic}, Expression
Profiler \cite{kapushesky2004expression}, GAP \cite{wu2010gap} and Furby
\cite{streit2014furby}). In addition, \citet{fiaux2013bixplorer} and
\citet{sun2014role} applied biclusters in Bixplorer, a visual analytics tool, to
support intelligence analysts for text analytics. Evaluations of these tools
show promising results, which indicates that using visualized bicluster to empower
data exploration is beneficial.
In order to systematically inform the design of bicluster visualizations, a
five-level design framework has been proposed \cite{sun2014five} and the key
design trade-off to visualize biclusters has been identified:
\textsl{Entity-centric} and \textsl{relationship-centric} \cite{sun2015biset}.
This design framework highlights five levels of relationships that underlie the
notions of biclusters and bicluster chains. The design trade-off suggests that
bicluster visualizations should visually represent both the membership of
entities and the overlap among biclusters in a human perceptible and usable
manner.
\iffalse
The relationships in this design framework are depicted using five keywords:
\textsl{entity}, \textsl{group}, \textsl{bicluster}, \textsl{chain} and
\textsl{schema}. \textsl{Entity}-level of relationships refer to relations
between two individual entities, while \textsl{group}-level of relationships are
those connecting one individual entity with a group of entities.
\textsl{Bicluster}-level and \textsl{chain}-level of relationships represent two
levels of coalitions between sets of entities: biclusters and bicluster-chains.
The latter is more complex than the former, because there are more involved sets
of entities in a bicluster-chain than those in a bicluster.
\textsl{Schema}-level of relationship indicates database-like patterns in a
dataset, which potentially reveals the overview of a dataset. Relations in
higher levels (e.g., \textsl{bicluster}-level and \textsl{chain}-level ) are
usually formed based on those in lower levels (e.g., \textsl{entity}-level and
\textsl{group}-level). Thus, lower levels of relations provide a critical
support for users to understand and further explore those in higher levels.
Three basic types of layouts can be applied to visualize biclusters: list,
matrix and node-link diagram. List organizes entities by domain and employs
visual links or entity highlights to show connections among entities (e.g.,
Jigsaw \cite{stasko2008jigsaw} and ConTour \cite{partl2014contour}). In a list
view, users have to visually follow links and mentally aggregate
entity-connections to manually find biclusters. Matrix placed entities of two
different domains in an orthogonal manner and use cells to indicate the
relations (e.g., Bicluster viewer \cite{heinrich2011bicluster} and BiVoc
\cite{grothaus2006automatic}). In a matrix, users have to permute (and sometimes
duplicate) rows and columns to visually place entities, belonging to the same
bicluster, close with each other. Node-link diagrams often apply certain layout
algorithms to organize entities and use edges to present relations among
entities (e.g., BicOverlapper 2.0 \cite{santamaria2014bicoverlapper}). To
visually show biclusters, other visualization techniques (e.g., Bubble sets
\cite{collins2009bubble}, KelpFusion \cite{meulemans2013kelpfusion} and LineSets
\cite{alper2011design}) have to be applied to a node-link diagram. These
techniques use additional visual elements (e.g., colored ribbon) to show the
membership of entities. Moreover, combining two or more basic layouts as a
hybrid layout can also be applied to visualize biclusters. Such hybrid layouts
usually use a list (e.g., Matchmaker \cite{lex2010comparative} and VisBrick
\cite{lex2011visbricks}) or a node-link diagram (e.g., Bixplorer
\cite{fiaux2013bixplorer}, Furby \cite{streit2014furby} and NodeTrix
\cite{henry2007nodetrix}) as the basic layout and replaces entities with
matrices (or other types of visualizations).
\fi
\section{Scoring Biclusters and Chains}
\label{sec:score}
We now turn our attention to using the above formalisms to help score our
patterns, viz., biclusters and bicluster chains. But before we do so, we need to
pay attention to the relational schema over which these patterns are inferred,
as this influences how patterns can be represented as tiles, in order to be
incorporated as knowledge in our maximum entropy models.
\subsection{Entity-Entity Relation Extraction}
\label{sec:data_model}
In this section, we describe the approach to construct a multi-relational schema
$\schema(\mathcal{U}, \mathcal{R})$ from a transaction data matrix $D$. Recall
that whenever an element $D(r, e_i)$ has a non-zero value (e.g.\ 1 in the binary
case or a fraction in the range of $[0,1]$ in the real-valued case), this
denotes that entity $e_i$ appears in row $r$ of $D$. As an example, when
considering text data, an entity would correspond to a word or concept, and a
row to a document in which this word occurs. (Thus, note that when considering
text data we currently model occurrences of entities at the granularity of
documents. Admittedly, this is a coarse modeling in contrast to modeling
occurrences at the level of sentences, but it suffices for our purposes.)
To extract entity-entity relations from transaction data matrix $D$, we utilize
the entity co-occurrence information. To be more specific, each binary relation
in $\mathcal{R}$ stores the entity co-occurrences in data matrix $D$ between two
entity domains, e.g.\ for each $R = R(U_i, U_j)$ in $\mathcal{R}$, $(e, f) \in
R$ for $e \in U_i$, $f \in U_j$, and $e$ and $f$ appear at least once together
in a row in $D$.
\subsection{Background Model Definition}
\label{sec:back_model}
Next, to discover non-trivial and interesting patterns, we need to incorporate
some basic information about the multi-relational schema $S(\mathcal{U},
\mathcal{R})$ into the model. For such basic background knowledge over $D$ we
use the column marginals and the row marginals for each entity domain. To this
end, following~\citet{Wu:2014:UPD:2664051.2664089} we construct a tile set
$\mathcal{T}_\mathit{col}$ consisting of a tile per column, a tile set
$\mathcal{T}_\mathit{row}$ consisting of a tile per row per entity domain, and a
tile set $\mathcal{T}_\mathit{dom}$ consisting of a tile per entity domain but
spanning all rows. Formally, we have
\begin{eqnarray*}
\mathcal{T}_\mathit{col} & = & \{(U_D, e) \mid e \in U, U \in
\mathcal{U} \setminus \{U_D\} \}, \\
\mathcal{T}_\mathit{row} & = & \{(r, U) \mid r \in U_D, U \in
\mathcal{U} \setminus \{U_D\} \}, \text{ and} \\
\mathcal{T}_\mathit{dom} & = & \{(U_D, U) \mid U \in \mathcal{U}
\setminus \{U_D\} \}.
\end{eqnarray*}
Here, $U_{D}$ represents the domain of all the documents in the dataset. We
refer to the combination of these three tile sets as the background tile set
$\mathcal{T}_\mathit{back} = \mathcal{T}_\mathit{row} \cup
\mathcal{T}_\mathit{col} \cup \mathcal{T}_\mathit{dom}$. Given the background
tiles $\mathcal{T}_\mathit{back}$, the background MaxEnt model $p_\mathit{back}$
can be inferred using iterative scaling (see
Sect.~\ref{sec:maxent:iter.scaling}) and the conjugate gradient method (see
Sect.~\ref{sec:infer_real}) for binary and real-valued cases, respectively.
\subsection{Quality Scores}
\label{sec:quality_score}
To assess the quality of a given bicluster $B$ with regard to our background
knowledge, we need to first convert it into tiles such that we can infer the
corresponding MaxEnt model. Below we specify how we do this conversion
for biclusters from entity-entity relations. For a given bicluster $B = (E_i,
E_j)$, we construct a tile set $\mathcal{T}_B$, consisting of
$\abs{E_i}\abs{E_j}$ tiles, as follows
\begin{equation}
\label{eq:biTiles}
\mathcal{T}_B = \{\left(\rows(X; D),\ X \right) \mid X = \{e_i,
e_j\} \mathrm{~with~} (e_i, e_j) \in B\}\quad,
\end{equation}
where $\rows(X; D)$ is the set of rows that contain $X$ in $D$, e.g.\ the
corresponding entries for $X$ in the matrix $D$ that have non-zero values.
To evaluate the quality of a bicluster chain $C$, for each bicluster $B \in C$,
we construct the set of tiles $\mathcal{T}_{B}$ as illustrated by
Equation~\eqref{eq:biTiles}, and the tile set that corresponds to a bicluster
chain $C$ is then $\mathcal{T}_\mathit{C} = \bigcup_{B \in \mathit{C}}
\mathcal{T}_{B}$.
Next, we describe the metrics that measure how much information a bicluster
$B$ (or the corresponding tile set $\mathcal{T}_{B}$) gives with regard to the
background model $p_{\mathit{back}}$. The global score is defined as follows:
\begin{align}
s_{\mathit{global}}(B) = \mathit{KL}(p_{B} || p_{\mathit{back}})~,
\label{eq:global}
\end{align}
where $p_{B}$ represents the MaxEnt distribution inferred over the background
tile set $\mathcal{T}_{\mathit{back}}$ and the tile set $\mathcal{T}_{B}$ for
the bicluster $B$.
For both of binary and real-valued MaxEnt model, the MaxEnt distribution $p(D)$
can be factorized as
$$
p(D) = \prod_{(i,j) \in D} p(D(i,j))~.$$
Thus, this global score can be written as:
\begin{align}
s_{\mathit{global}}(B) & = \oint\limits_{D} p_{B}(D) \log
\frac{p_{B}(D)}{p_{\mathit{back}}(D)} d D \nonumber \\
& = \oint\limits_{D} \prod_{(i,j) \in D} p_{B}(D(i,j)) \sum_{(i,j) \in D}
\log \frac{p_{B}(D(i,j))}{p_{\mathit{back}} (D(i,j))} d D \nonumber \\
& = \sum_{(i,j) \in D} \int_{-\infty}^{+\infty} p_{B}(D_{{i,j}}) \log
\frac{p_{B}(D(i,j))}{p_{\mathit{back}}(D(i,j))} d D(i,j) \nonumber \\
& = \sum_{(i,j) \in D} \mathit{KL}(p_{B}(D(i,j)) ||
p_{\mathit{back}}(D(i,j)))~.
\label{eq:global_real}
\end{align}
For the binary MaxEnt model, $D(i,j)$ follows the Bernoulli distribution
$$
D(i,j) \sim \mathit{Bernoulli}(q),~\text{where}~q = \frac{\exp\left(\sum_{T
\in \mathcal{T}(i,j)} \lambda_{T}\right)}{\exp\left(\sum_{T \in
\mathcal{T}(i,j)} \lambda_{T}\right) + 1}~,
$$
and the global score for binary MaxEnt model would be:
$$
s_{\mathit{global}}(B) = \sum_{(i,j) \in D} \left(q_{B} \log
\frac{q_B}{q_{\mathit{back}}} + (1 - q_B) \log \frac{1 - q_B}{1 -
q_{\mathit{back}}}\right)~.
$$
For the real-valued MaxEnt model, $D(i,j)$ follows the Gaussian distribution
$$
D(i,j) \sim \mathbb{N}\left(-\frac{\alpha_{i,j}}{2 \beta_{i,j}}, \frac{1}{2
\beta_{i,j}}\right)~.
$$
Given any two normal distribution $P_{\mathcal{N}_1} = \mathcal{N}(\mu_1,
\sigma_1^2)$ and $P_{\mathcal{N}_2} = \mathcal{N}(\mu_2, \sigma_2^2)$, we can
verify that the KL-divergence between these two normal distribution is:
\begin{align}
\mathit{KL}(P_{\mathcal{N}_1} || P_{\mathcal{N}_2}) = \log
\frac{\sigma_2}{\sigma_1} + \frac{\sigma_1^2 + {(\mu_1 - \mu_2)}^2}{2
\sigma_2^2} - \frac{1}{2}~.
\label{eq:kl_normal}
\end{align}
Combining Equation~\eqref{eq:global_real} and~\eqref{eq:kl_normal}, the global
score for the real-valued maximum entropy model is:
\begin{align}
s_{\mathit{global}} = \sum_{(i,j) \in D} \left[ \frac{1}{2} \log
\frac{\beta_{i,j}^{(B)}}{\beta_{i,j}^{(\mathit{back})}} +
\frac{\beta_{i,j}^{(\mathit{back})}}{2 \beta_{i,j}^{(B)}} +
\beta_{i,j}^{(\mathit{back})} {\left(
\frac{\alpha_{i,j}^{(\mathit{back})}}{2
\beta_{i,j}^{(\mathit{back})}} - \frac{\alpha_{i,j}^{(B)}}{2
\beta_{i,j}^{(B)}}\right)}^2 - \frac{1}{2} \right]
\label{eq:global_real_final}
\end{align}
However, using the global score defined above requires us to re-infer the MaxEnt model
for every candidate bicluster that needs to be evaluated, which could be
computationally expensive and thus not applicable to our
interactive mining sitting. Moreover, $s_{\mathit{global}}$ evaluates a candidate
globally, whereas typically most information is \textit{local}: at most a few entries
in the maximum entropy distribution will be affected by adding $B$ into the model.
Making use of this observation, to reduce the computational cost of candidate
bicluster evaluation, we define the score $s_{\mathit{local}}(B)$ that measures
the local surprisingness of a tile set as
\begin{align}
s_{\mathit{local}}(B) = - \sum_{T \in \mathit{T}_{B}} \sum_{(i,j) \in
\sigma(T)} \log p_{\mathit{back}}(D(i,j))~,
\label{eq:local}
\end{align}
where for both binary and real-valued MaxEnt model, $p_{\mathit{back}}(D(i,j))$
indicates the probability (or probability density) evaluated at the value
$D(i,j)$ under the current background MaxEnt model. Notice that although the
global and local scores are described using the notation of biclusters here,
they can also be directly adopted to assess the quality of bicluster chains
because fundamentally these scores are defined around the concept of tiles and
bicluster chains (and can thus be trivially converted to a set of tiles as described at
the beginning of this section).
\section{BiSet}
\section{\MaxEntBiSet}
\label{sec:biset}
\MaxEntBiSet is a visual analytics system, supported by the
maximum entropy model above,
to support interactive exploration of coordinated relationships
using biclusters. Coordinated relationships are groups of relations, connecting
sets of entities from different domains (e.g., people, location, organization,
etc.), which potentially indicate coalitions between these entities. \MaxEntBiSet
extends an existing bicluster visualization, viz. BiSet~\cite{sun2015biset}, by
incorporating MaxEnt models to support user exploration of entity coalitions
for sensemaking purposes. In this section, we first introduce BiSet, followed
by the enhancements that
\MaxEntBiSet provides.
\subsection{BiSet Technique Overview}
\iffalse
people and location, people and organization, location and organization, etc.)
is \textsl{formalized as biclusters}, which are algorithmically identified from
a dataset (e.g., using biclustering or closed itemset mining). Then based on the
order of user selected domains (e.g., people, location, organization and date),
BiSet generates the corresponding multi-relational schema and visually presents:
1) \textsl{entities in these selected domains}, and 2) \textsl{biclusters
between each two neighboring domains} (e.g., people and location, location and
organization, and organization and date).
\fi
\begin{figure}[!t]
\centering
\includegraphics[width=0.9\textwidth]{biset_bic}
\caption{Visual representations of a bicluster that includes three entities in \textsl{D1}
and three entities in \textsl{D2}. (A) displays all individual relationships between
the two sets of entities from the two domains, \textsl{D1} and \textsl{D2}.
(B) Realtionships are aggregated as an edge bundle that
represents a bicluster.}
\label{fig:biset-bic}
\end{figure}
The key idea is
that \textsl{BiSet visualizes the mined biclusters in
context as edge bundles} between sets of related entities. BiSet uses lists as
the basic layout to present entities and biclusters. Figure~\ref{fig:biset-bic}
shows an example of a visualized bicluster in BiSet. In
Figure~\ref{fig:biset-bic}, (A) shows all individual edges between related
entities and (B) presents the same bicluster as an edge bundle. BiSet enables
both ways to show the coalition of entities with two modes: \textsl{link mode}
and \textsl{bicluster mode}. \textsl{Link mode} displays the individual
connections among entities in a dataset, while \textsl{bicluster mode} offers a
more clear representation to show identified biclusters in the dataset. Based on
these visual representations, BiSet can visually show bicluster-chains as
connected edge bundles through their shared entities.
Figure~\ref{fig:biset-chain} shows four bicluster-chains (\textsl{b1} -
\textsl{b4}, \textsl{b2} - \textsl{b4}, \textsl{b2} - \textsl{b5} and
\textsl{b3} - \textsl{b5}) visualized using BiSet. Each of them consists of two
different biclusters including entities from three domains. The two biclusters
in each chain are visually connected through one or two shared entities. For
example, bicluster \textsl{b2} and \textsl{b4} are connected by entity
\textsl{e1} and \textsl{e2}. With edges, BiSet enables users to see members of
bicluster-chains and how these biclusters are connected. This potentially guides
users to interpret the coalition among sets of entities from multiple domains in
an organized manner (e.g., checking connected biclusters from left to right).
\begin{figure}[!t]
\centering
\includegraphics[width=0.9\textwidth]{biset_chain}
\caption{An example of four bicluster-chains (\textsl{b1} - \textsl{b4},
\textsl{b2} - \textsl{b4}, \textsl{b2} - \textsl{b5} and \textsl{b3} -
\textsl{b5}). These chains consist of entities from three domains,
\textsl{D1}, \textsl{D2} and \textsl{D3}. \textsl{b1} and \textsl{b4} are
connected through \textsl{e1}. \textsl{b2} and \textsl{b4} share \textsl{e1}
and \textsl{e2}. \textsl{b2} and \textsl{b5} are linked by \textsl{e3}.
\textsl{b3} and \textsl{b5} are connected by \textsl{e3} and \textsl{e5}.}
\label{fig:biset-chain}
\end{figure}
To support exploratory analysis, BiSet treats \textsl{edge bundles as first class
objects}, so users can directly manipulate them (e.g., drag and move) to
spatially organize them in meaningful ways. BiSet also offers automatic
ordering for entities and biclusters to help users organize them. For example,
entities can be ordered based on their frequency in a dataset and biclusters can
be ordered by size, i.e., the number of entities participating in a bicluster.
Moreover, BiSet can highlight bicluster-chains as users select their members.
This provides visual clues for users to follow in conducting their analysis.
\subsection{\MaxEntBiSet Visual Encoding}
\begin{figure}[!t]
\centering
\includegraphics[width=1\textwidth]{biset_visual}
\caption{Detailed visual encodings in \MaxEntBiSet. \textsl{1a}, \textsl{2a} and
\textsl{3a} depict the \textsl{normal} state of an entity, a bicluster and
edges, respectively. \textsl{1b}, \textsl{2b} and \textsl{3b} depict the
\textsl{connection-oriented} highlighting state of an entity, a bicluster and
edges, when users \textsl{select} bicluster \textsl{2c}, \textsl{hover} over entity \textsl{1f}
and \textsl{select} entity \textsl{1c}.
\textsl{1e}, \textsl{2d} and \textsl{3c} illustrate the
\textsl{surprisingness-oriented} highlighting state of an entity, a bicluster and edges. \textsl{1d}
demonstrates larger fonts of entities as users hover the mouse pointer over
their previously selected entity \textsl{1c}. Moreover, \textsl{2e} represents a
bicluster (in the \textsl{normal} state) with its edges chosen to be hidden by users.}
\label{fig:biset-visual}
\end{figure}
\subsubsection{Shape and Size}
In \MaxEntBiSet, entities and biclusters are represented as rectangles (e.g.,
\textsl{1a} and \textsl{2a} in Figure~\ref{fig:biset-visual}), and edges are
visualized as B\'ezier curves. We use B\'ezier curves because they can generate
more smooth edges, compared with polylines~\cite{Lambert:2010jg}. Rectangles
indicating entities are equal in length, while those representing biclusters are
not. \MaxEntBiSet applies a linear mapping function to determine the length of a bundle
based on the total number of its related entities. In a bicluster rectangle,
\MaxEntBiSet uses two colored regions (light green and light gray) to indicate the
proportion between its related entities in lists of both sides (left and right).
In an entity rectangle, a small rectangle is displayed on the left to indicate
its frequency in a dataset. The length of these rectangles is determined by the
frequency of the associated entities with a linear mapping function. These helps
users to visually discriminate entities from biclusters. Moreover, when users
hover over a selected entity or bicluster (e.g., entity \textsl{1c} and bicluster
\textsl{2c} in Figure~\ref{fig:biset-visual}), the font of its related entities
is enlarged (e.g., comparing \textsl{1d} with \textsl{1b} in
Figure~\ref{fig:biset-visual}). This helps users review relevant information
of their previous selections.
\subsubsection{Color Coding}
\MaxEntBiSet applies color coding to entities, biclusters and edges to indicate their
states and allows users to hide edges of biclusters to reduce visual clutter
(see \textsl{2e} in Figure \ref{fig:biset-visual}).
In \MaxEntBiSet, entities, biclusters and edges have two basic states:
\textsl{normal} and \textsl{highlighted}. The normal state is the default state
for entities, biclusters and edges. Examples of the normal state for them are
shown as \textsl{1a}, \textsl{2a} and \textsl{3a}, respectively, in
Figure~\ref{fig:biset-visual}. To encode surprisingness,
\MaxEntBiSet supports two types of highlighting states:
\textsl{connection oriented highlighting} (colored as \textsl{orange} in
Figure~\ref{fig:biset-visual}) and \textsl{surprisingness oriented highlighting}
(color as \textsl{red} in Figure~\ref{fig:biset-visual}), which encode two
levels of information: \textsl{the coalition of entities} and \textsl{the
surprisingness of the coalition}. The former indicates the linkage of entities,
emphasizing the connections between entities. The latter reveals the
model-evaluated surprisingness of different sets of entity coalitions. In
Figure~\ref{fig:biset-visual}, examples of connection-oriented highlighting
for entities, biclusters and edges are shown as \textsl{1b}, \textsl{2b}
and \textsl{3b}, respectively; while examples of surprisingness-oriented
highlighting are presented as \textsl{1e}, \textsl{2d} and
\textsl{3c}.
The connection oriented highlighting state is triggered as users hover or select
an entity or a bicluster. For example, when users hover the mouse pointer over the entity \textsl{1f},
its directly connected bicluster \textsl{2b} is highlighted and other entities
that belong to this bicluster are also highlighted. The surprisingness oriented
highlighting state is triggered by explicit user request of model evaluation.
For instance, in Figure~\ref{fig:biset-visual}, as users request to find the
most surprising chains with bicluster \textsl{2c} as the starting point,
\MaxEntBiSet highlights entities and biclusters in a chain that has the highest
score given by the proposed \textsl{Maximum Entropy model} (the approach to
discover such a chain will be described in Section~\ref{biset-model} below).
With our color codings, \MaxEntBiSet
empowers users to explore entity coalitions by directing them to computationally
identified surprising chains.
\subsection{Human-model Interaction}
\begin{figure}[!t]
\centering
\includegraphics[width=1\textwidth]{biset_user_model}
\caption{The human-model interaction flow in \MaxEntBiSet. Visual
representations in \MaxEntBiSet enable the interaction between users and
the proposed maximum entropy models.}\label{fig:biset-human-model}
\end{figure}
\MaxEntBiSet allows human-model interaction with visualizations to support
visual analytics of entity coalitions. To enable this capability, we incorporate
the proposed \textsl{Maximum Entropy models} into \MaxEntBiSet.
Figure~\ref{fig:biset-human-model} illustrates the human-model interaction flow
in \MaxEntBiSet. Visual representations in \MaxEntBiSet work as the bridge to
enable the interaction between users and the proposed models.
After inspecting the
visualized biclusters and bicluster-chains, users can explicitly request model
evaluations using right click menus on a bicluster. This further triggers the
maximum entropy model to evaluate either all paths passing through the
requested bicluster or its neighboring biclusters. Then, based on results of the
model evaluation, \MaxEntBiSet highlights the most surprising bicluster-chain
including the user requested bicluster or neighboring biclusters. We address
this with a detailed discussion in Section~\ref{biset-model}. Moreover, users
can mark highlighted bicluster(s), based on model evaluation, as useful one(s)
by using a right click menu on the bicluster(s). This implicitly evokes a model
update function, which informs the model that the information in a marked
bicluster has been known by users. Then the model updates its background
information to take the marked bicluster(s) into account and prepare for
further user requested evaluations. This human-model interaction flow in
\MaxEntBiSet enables the combination the human cognition with computations for the
exploration of entity coalitions.
\subsection{Model Evaluation Strategies}
\label{biset-model}
\MaxEntBiSet offers two strategies to evaluate bicluster-chains, using the proposed
maximum entropy models, based on explicit user requests: \textsl{full
path evaluation} and \textsl{stepwise evaluation}. Both ways require users to
explicitly specify a bicluster based on its visual information, e.g.\ size
of a bicluster, frequency of corresponding entities, etc., to initiate the
chain. The former evaluates all bicluster-chains passing through the bicluster
that users request for evaluation, while the latter evaluates neighboring
biclusters that satisfy a certain degree of overlap with the user-specified one.
\MaxEntBiSet enables users to explicitly issue an evaluation request from a
bicluster with a right click menu. From the menu, users can choose the desired
way of evaluation.
\subsubsection{Full Path Evaluation}
The \textsl{full path evaluation} in \MaxEntBiSet includes three key steps: 1) path
search, 2) path evaluation, and 3) path rank. In \MaxEntBiSet, a path, passing through
a bicluster, refers to a set of biclusters (e.g., \{\textsl{b2}, \textsl{b4}\}
in Figure~\ref{fig:biset-chain}), which can be connected through certain
entities to form a bicluster-chain. In the \textsl{path search} step, \MaxEntBiSet
finds all possible paths passing through the bicluster that users request for
evaluation. Similar to tree search, \MaxEntBiSet treats the user requested bicluster as
a root node and applies depth-first search to find all paths starting from this
bicluster. If the user requested bicluster is not from the left or right
most relation in the user specified multi-relational schema, \MaxEntBiSet
performs bidirectional search and then combines identified paths in the left and
those in the right together to obtain all paths going through this bicluster. Then
in the \textsl{path evaluation} step, \MaxEntBiSet converts each
bicluster-chain, found in the previous step, into a unique set of tiles following
the Equation~\eqref{eq:biTiles} in Section~\ref{sec:quality_score}, and applies
the maximum entropy models to score them. Finally, based on the score
from the model, in the \textsl{path rank} step, \MaxEntBiSet ranks these
bicluster-chains and visually highlights the one that has the highest score
(e.g., \{\textsl{2c}, \textsl{2d}\} in Figure~\ref{fig:biset-visual}). Thus,
with the \textsl{full path evaluation} in \MaxEntBiSet, users can get the most
surprising bicluster-chain for the bicluster requested for evaluation.
\subsubsection{Stepwise Evaluation}
\begin{figure}[!t]
\centering
\includegraphics[width=1\textwidth]{biset_stepwise}
\caption{Exampled results from the \textsl{stepwise} evaluation in
\MaxEntBiSet. (a) shows the bicluster selected by a user to initiate the
maximum
entropy model evaluation. (b) represents the most surprising bicluster in
the same bicluster list as the one requested for evaluation. (c) illustrates
the most surprising bicluster in another bicluster list.}
\label{fig:biset-step}
\end{figure}
The \textsl{stepwise evaluation} in \MaxEntBiSet examines neighboring biclusters for
the one that users request for evaluation. \textsl{Neighboring biclusters} for a
specific bicluster refers to those that can meet certain degree of overlaps,
with respect to participated entities, with a user requested bicluster.
\MaxEntBiSet uses the Jaccard coefficient to measure the degree of overlaps
between two biclusters with a default threshold set as $0.1$.
Thus, for a specific bicluster,
its potential neighboring biclusters are those sharing at least one domain
(e.g., people, location, date, etc.) with this one.
Similar to the \textsl{full path evaluation}, the \textsl{stepwise evaluation}
also has three key steps, including: 1) neighboring bicluster search, 2)
neighboring bicluster evaluation, and 3) neighboring bicluster coloring. Based
on a user specified bicluster for evaluation, \MaxEntBiSet first identifies its
neighboring biclusters using the Jaccard coefficient. Then, \MaxEntBiSet
converts the identified neighboring biclusters into different sets of tiles
following Equation~\eqref{eq:biTiles} and employs the maximum
entropy models to score them. Based on the model evaluation score, BiSet
applies a linear mapping function to assign the opacity value of
\textsl{surprisingness} oriented highlighting color to these biclusters. The
more red a color is, and the higher score this neighboring bicluster gets, which
indicates more surprising information. Figure~\ref{fig:biset-step} gives an
example of the \textsl{stepwise evaluation} in \MaxEntBiSet. In this example,
users request to evaluate a bicluster (see \textsl{a}), \MaxEntBiSet highlights
neighboring biclusters based on their model evaluation scores. Of these
highlighted biclusters, bicluster \textsl{b} shows the most surprising one in
the same bicluster list as that requested for evaluation, and bicluster
\textsl{c} is the most surprising bicluster in the adjacent bicluster list.
Although bicluster \textsl{b} here could not be used to extend the users
selected bicluster \textsl{a}, it has the potential to reveal entities related
to the bicluster \textsl{a} and the plots. Thus, we also take the most
surprising bicluster from the same relation of the users selected bicluster into
account. Such \textsl{stepwise evaluation} potentially enables to involve users
in the process of building a meaningful bicluster-chain. Each time after a
\textsl{stepwise evaluation}, users can investigate highlighted neighboring
biclusters, identify and then select useful one(s) for further exploration.
Users can iterate this process and build a bicluster-chain that is meaningful
for them.
\subsection{Bicluster based Evidence Retrieval}
\MaxEntBiSet allows users to review relevant documents directly from biclusters
with a right click menu. When investigating a bicluster, users can use a right
click menu to open a popup view where relevant documents are listed, shown in
Figure~\ref{fig:biset-doc}. This helps users review information relevant to
this bicluster and verify computationally identified coalitions of entities. This
document view is on top of the view for relationship exploration with
transparency, so users can simultaneously see both the visualized relationships
and corresponding documents. Moreover, after reading the documents, users can
quickly return to previous view by closing it.
\begin{figure}[!t]
\centering
\includegraphics[width=1\textwidth]{biset_doc}
\caption{Document view mode in \MaxEntBiSet. (A) depicts the bicluster ID, relevant document
ID(s) and associated entities. (B) shows the content of a document. (C) lists
all document IDs in a dataset with a search function.}
\label{fig:biset-doc}
\end{figure}
|
1,314,259,993,651 | arxiv | \section{Introduction}
The transverse momentum dependent (TMD) factorisation theorems for semi-inclusive deep inelastic scattering (SIDIS) and Drell-Yan type processes formulated in \cite{GarciaEchevarria:2011rb,Collins:2011zzd,Chiu:2012ir, Echevarria:2014rua} allow a consistent treatment of the rapidity divergences in the definition of spin (in)dependent TMD distributions. Within factorisation theorems TMD operators are self-contained defined objects, and can be considered individually by standard methods of quantum field theory without referring to a scattering process. TMD operators are intrinsically non-perturbative objects due to the infrared divergences, nevertheless in the limit of large-$q_t$ (or small $b_T$) a perturbative computation of matching coefficients of TMD distributions on the corresponding integrated functions can be performed. The unpolarised TMD distribution is the most studied case and it has been treated using different regularization schemes at the next-to-leading order (NLO) \cite{GarciaEchevarria:2011rb,Collins:2011zzd,Echevarria:2014rua} and the next-to-next-to-leading order (NNLO) \cite{Echevarria:2015usa,Echevarria:2016scs,Gehrmann:2014yya,Gehrmann:2012ze}. For polarised distribution such a program has been performed for helicity, transversity, pretzelosity and linearly polarised distribution at NLO \cite{Bacchetta:2013pqa,Echevarria:2015uaa,Gutierrez-Reyes:2017glx} and only for transversity and pretzelosity at NNLO \cite{Gutierrez-Reyes:2018iod}. In this work we present the TMDPDF for linearly polarised gluons at NNLO improving the status of the art on polarised TMD distributions. The phenomenological interest of gluons TMD distribution is supported on the fact that at high energies most of the scattering processes are triggered by gluons.
Gluon-gluon fusion is the main channel for Higgs production. The factorisation of this process in the infinite top-mas ($m_t$) limit and with $q_T\ll m_H$, where $q_T$ is the transverse momentum spectrum of the Higgs boson produced via gluon-gluon fusion and $m_H$ is its mass, has been demonstrated to follow the same pattern as in the Drell-Yan/vector boson case, and in this sense it has been reviewed in \cite{Echevarria:2015uaa}. We find intriguing the fact that the sign of linearly polarised gluon contribution can flip depending on the (pseudo) scalar nature of the Higgs boson \cite{Boer:2011kf, Boer:2013fca}. The possibility to test the parity of the Higgs relies heavily on the precision achievable experimentally, and a theoretical prediction which includes resummation at next-to-leading logarithmic (NLL) order has been done in \cite{Boer:2014tka}.
Another process where appears linearly polarised gluons is in the di-$J/\psi$ production, which leads a modulation $\cos\left(2\phi\right)\left(\cos\left(4\phi\right)\right)$ in the azimuthal angle due to single(double) gluon helicity flips \cite{Lansberg:2017dzg}.
\section{Matching Coefficients}
In this section we define the basic operators for the case of interesting following \cite{Gutierrez-Reyes:2017glx}, and show the matching coefficients, which are the main result of this work. The gluon TMD operator matrix element reads\footnote{We omit the transverse links necessary in singular gauges.}
\begin{equation}
\begin{aligned}
\Phi_{g\leftarrow h, \mu\nu}(x,\vec b)=&\langle P,S|\frac{1}{xp^+}\int \frac{d\lambda}{2\pi}e^{-ixp^+\lambda}
\bar T \left\{F_{+\mu}(\lambda n+\vec b)
\tilde W_{n}(\lambda n+\vec b)\right\}
\\ &\times
T\left\{ \tilde W_{n}^{\dagger}(0)
(\lambda,\vec b)F_{+\nu}(0)\right\}|P,S\rangle,
\end{aligned}
\label{def:TMD_OP_G}
\end{equation}
where $n$ is the lightlike vector and we use the standard notation for the lightcone components of vector $v^\mu=n^\mu v^-+\bar n^\mu v^++g_T^{\mu\nu}v_\nu$ (with $n^2=\bar n^2=0$, $n\cdot\bar n=1$, and $g_T^{\mu\nu}=g_{\mu\nu}-n^\mu \bar n^\nu-\bar n^\mu n^\nu$).
The Wilson lines $\tilde W$ are taken in the adjoint representation of the gauge group for the gluon case.
The hadron matrix elements of the TMD operators in eq.~(\ref{def:TMD_OP_G}) are decomposed in covariant Lorentz structures, the TMDPDF.
The decomposition of gluon operator in momentum space over all possible Lorentz variants can be found in \cite{Mulders:2000sh}.
Here we use the corresponding decomposition in impact parameter space which is more convenient when treating the formulation of the factorisation theorem.
The corresponding between decomposition in momentum and impact parameter space can be found in e.g. \cite{Boer:2011xd,Echevarria:2015uaa}.
In $b$-space we have
\begin{equation}
\begin{aligned}
\Phi_{g\leftarrow h}^{\mu\nu}(x,\vec b)=&
\frac{1}{2}\Big(-g_T^{\mu\nu}f_{1,g\leftarrow h} (x,\vec b)-
i\epsilon_T^{\mu\nu}S_Lg_{1L, g\leftarrow h} (x,\vec b)\\
&+2h_{1,g\leftarrow h}^{\perp} (x,\vec b)(\frac{g_T^{\mu\nu}}{2}+\frac{ b^\mu b^\nu}{\vec b^2})+...\Big),
\end{aligned}
\label{TMD_G_dec}
\end{equation}
where the vector $b^\mu$ is a 4-dimensional vector of the impact parameter ($b^+=b^-=0$ and $-b^2\equiv\vec{b}^2>0$). On the r.h.s of eq. ~(\ref{TMD_G_dec}) $f_1^g$ is the unpolarised gluon TMDPDF, $g_{1L}^g$ the helicity and finally we have $h_1^{\perp g}$ as the TMD for linearly polarised gluons, which is the object of the present work.
In eq. ~(\ref{TMD_G_dec}) we write only the TMD distribution that match the twist-2 integrated parton distribution function (PDF) and the twist-3 and higher parton distribution function are understood in the dots.
The small-$b$ operator product expansion (OPE) provides a relation between a TMD operators and collinear integrated operators which at lower twist reads
\begin{equation}
\Phi_{g\leftarrow h, \mu\nu}(x,\vec b)=\sum_f C_{g\leftarrow f, \mu\nu}^a(x,\vec b)\otimes \phi_{f\leftarrow h}^{a, {\rm{tw2}}}(x)+\dots
\label{OPE_G}
\end{equation}
where symbol $\otimes$ denotes the Mellin convolution in the variable $x$, the function $C(x,\vec{b})$ depend on $\vec{b}$ only logarithmically, $\phi^a\left(x\right)$ are collinear functions, the dots represent the power suppressed contributions and scale dependences are not shown.
at the lowest order of PDF, the function $\phi\left(x\right)$ are the formal limit of the TMD distribution $\Phi(x,\vec{0})$. The hadronic matrix elements of $\phi$ are the gluon/quark PDFs.
It is also convenient to extract to desired TMD using projectors and defining
\begin{eqnarray}
\Phi_{g\leftarrow h}^{[\Gamma]}=\Gamma^{\mu\nu}\Phi_{g\leftarrow h,\mu\nu}.
\end{eqnarray}
The projector for unpolarised gluons is
\begin{equation}
\Gamma^{\mu\nu}_{un}=\frac{g^{\mu\nu}_T}{2(1-\epsilon)}
\end{equation}
and for linearly polarised gluons we have
\begin{equation}
\Gamma^{\mu\nu}_{\ell in}=\left(g^{\mu\nu}_T-2 (1-\epsilon) \frac{b^\mu b^\nu}{b^2} \right)\frac{1}{2(1-2\epsilon)}.
\label{eq:l-proj}
\end{equation}
The small-$b$ matching of this distribution has been performed in \cite{Echevarria:2015uaa,Gutierrez-Reyes:2017glx} up to NLO.
The structure of rapidity divergences for the gluon TMD operators differs from the quark case only because of the colour factors. As for all TMDs, both ultraviolet (UV) and rapidity divergences are present in their perturbative calculation which are renormalized by appropriate renormalisation constant \cite{Echevarria:2016scs,Vladimirov:2017ksc}.
Hence, the renormalisation (or physical) TMDPDF depend on two scales (the UV renormalisation scale is denoted by
$\mu$ and the rapidity renormalisation scale is denoted by $\zeta$). The renormalised expression for gluon TMDPDF and renormalisation constan can be found in \cite{Echevarria:2016scs}.
In perturbation theory, the expression for the coefficient function can be presented as
\begin{equation}
\delta^L C_{g\leftarrow f'}(x,\mathbf{L}_\mu,\mathbf{l}_\zeta)=\sum_{n=0}^\infty a_s^n \sum_{k=0}^{n+1}\sum_{l=0}^n \mathbf{L}_\mu^k\, \mathbf{l}_\zeta^l \, \delta^L C^{(n;k,l)}_{g\leftarrow f'}(x),
\label{eq:pertTR}
\end{equation}
where $a_s=g^2/(4\pi)^2$. The coefficients $\delta^L C^{(n;k,l)}$ with $k+l>0$ are fixed order-by-order with the help of the renormalisation group equations and they can be found up to two loops in e.g. \cite{Echevarria:2015usa} as they are common to the unpolarised case.
The only non-trivial part up to two loops is so provided by $\delta^L C^{(2;0,0)}$, where
\begin{equation}
\begin{aligned}
\textcolor{black}{\delta^L C_{g\leftarrow g}^{(2;0,0)}(x)}& \textcolor{black}{=C_A^2\Big\{\frac{1}{x}(\frac{220}{9}+20(1-x)\zeta_2+16\ln x)-32 \frac{1-x}{x}{\rm Li}_3(1-x)} \\
&\textcolor{black}{-[16\frac{1-x}{x}+\frac{x}{2}(x+3)]{\rm Li}_2(x)} \\
&\textcolor{black}{+\frac{x+3}{4}[x{\rm Li}_3(x^2)+(1-x\ln x){\rm Li}_2(x^2)]} \\
&\textcolor{black}{+\frac{x+3}{2}\ln x[(1-x)\ln(1-x)+\ln(1+x)+\frac{x}{2}]} \\
&\textcolor{black}{+\frac{76}{3}\ln x+x(\frac{77}{6}+\frac{31}{36}x)-8\ln^2 x} \\
&\textcolor{black}{+\frac{x}{4}(x+3)(\zeta_2-\zeta_3)-\frac{1325}{36}\Big\}} \\
&\textcolor{black}{+C_F T_R n_f\Big\{8 \ln^2x-16\frac{(1-x)^3}{x} \Big\}} \\
&\textcolor{black}{+C_A T_R n_f\Big\{\frac{136}{9x} +\frac{16}{3}\ln x-\frac{8 x}{9}(x+3)-\frac{128}{9}\Big\},}
\end{aligned}
\label{C2gg}
\end{equation}
\begin{equation}
\begin{aligned}
\textcolor{black}{
\delta^L C^{(2;0,0)}_{g\leftarrow q}(x)}&\textcolor{black}{=C_F C_A \Big\{-16 \frac{1-x}{x}[{\rm Li}_2(x)+2{\rm Li}_3(1-x)]-\frac{40}{3}\frac{1-x}{x}\ln(1-x)}\\
&\textcolor{black}{-8\frac{1-x}{x}\ln^2(1-x)+(40+\frac{16}{x})\ln x-8\ln^2 x}\\
&\textcolor{black}{+\frac{1-x}{x}(\frac{88}{9}+20\zeta_2-8x)\Big\}}\\
&\textcolor{black}{+C_F^2\Big\{8(1-x)+4(\ln x-5)\ln x+8\frac{1-x}{x}[1+\ln(1-x)]\ln (1-x)\Big\}}\\
&\textcolor{black}{+\frac{32}{3} C_F T_R n_f\frac{1-x}{x}(\ln(1-x)+\frac{2}{3}).}
\end{aligned}
\label{C2gq}
\end{equation}
Note that we have no singularity for $x\rightarrow 1$.
\section{Conclusions}
We have provided a description of TMDPDF for linearly polarised at the same order of precision than TMDPDF for unpolarised gluons. We find out that TMDPDF for linearly polarised gluons is not divergent when x goes to 1, unlike for unpolarised gluons.
We have reviewed that the contribution of linearly polarised gluons to the Higgs cross section through gluon-gluon fusion is less than \%. This can be understood due to the fact that Higgs production takes place at electroweak energies scale, together to the non divergent behaviour of linearly polarised gluons when x $\rightarrow 1$ makes the contribution of unpolarized gluons to exceed linearly polarised gluons.
For quarkonium production, di-$J/\psi$ production is the most promising process to measure linearly polarised gluons effects, which effect can reach the 50\% of the corss section. Recently quarkonium production has been factorised \cite{Echevarria:2019ynx} which leads interesting opportunities to work in quarkonium production phenomenology.
\section*{Acknowledgements}
D.G.R., S.L.G. and I.S. are supported by the Spanish MECD grant FPA2016-75654-C2-2-P.
This project has received funding from the European Union Horizon 2020 research and innovation program under grant agreement No 824093 (STRONG-2020). S.L.G is supported by the Austrian Science Fund FWF under the Doctoral Program W1252-N27 Particles and Interactions.
\bibliographystyle{JHEP}
|
1,314,259,993,652 | arxiv | \section{Introduction}
This paper studies the design of revenue-optimal mechanism in the two-item, one-buyer, unit-demand setting. The solution to the problem is well known when the buyer's value is one-dimensional (\citet{Mye81}). The problem however becomes much harder when the buyer's value is multi-dimensional. Though many partial results are available in the literature, finding the general solution remains open in the two-item setting, be it with or without the unit-demand constraint.
In this paper, we consider the problem of optimal mechanism design in the two-item one-buyer unit-demand setting, when the valuations of the buyer are uniformly distributed in arbitrary rectangles in the positive quadrant having their left-bottom corners on the line $z_1=z_2$. Observe that this is a setting that occurs often in practice. As one example, consider a setting where two houses in a locality are sold. The seller is aware of a minimum and a maximum value for each house. Further, the buyer has a unit-demand, i.e., he can buy at most one of the houses, but submits his bids for both the houses. We consider that the buyer's valuations are uniform in the rectangle formed by those intervals. We compute the optimal mechanism for all cases when the minimum value for both the houses is the same. Another example is one where two sports team franchises in a sports league are sold to a potential buyer. The buyer needs at most one franchise, but submits his bids for both franchises.
\subsection{Prior work}
Consider the setting where the buyer is not restricted by the unit-demand constraint. \citet{DDT13,DDT14,DDT15} provided a solution when the buyer's valuation vector $z$ arises from a rich class of distribution functions each of which gives rise to a so-called ``well-formed canonical partition" of the support set of the distribution. The authors of these papers formulate this problem as an optimization problem, identify its dual as a problem of optimal transport, and exploit its solution to obtain a primal solution. \citet{GK14} computed the solution for the multi-item setting, but only when the valuations for each item are uniformly distributed in $[0,1]$. \citet{GK15} also provided closed form solutions in the two-item setting, when the distribution satisfies some sufficient conditions, by using a dual approach similar to \cite{DDT13,DDT15,DDT17}. In a companion paper \cite{TRN17a} (see also \cite{TRN16}), we used the same approach of solving the optimal transport problem as in \cite{DDT15} to obtain the solution when $z\sim\mbox{Unif}[c_1,c_1+b_1]\times[c_2,c_2+b_2]$ for arbitrary nonnegative values of $(c_1,c_2,b_1,b_2)$. The exact solution in the unrestricted setting has largely been computed using the dual approach designed in \cite{DDT15}.
The exact solution in the unit-demand setting, on the other hand, has been computed using various other methods. \citet{Pav11} obtained a solution both in the unrestricted setting and in the restricted setting of unit-demand constraint, when $z\sim\mbox{Unif}[c,c+1]^2$. The above paper used a marginal profit function $V$, whose properties are analogous to the virtual valuation function in \cite{Mye81}, to compute the exact solution. We thus call this method the {\em virtual valuation method}. The function however depends on the region of zero allocation, the exclusion region, and is thus not as straightforward to compute as the virtual valuation function in \cite{Mye81} for the single item case. \citet{Lev11} provided a solution for the unit-demand setting when the distribution is uniform in certain polygons aligned with the co-ordinate axes; the approach involves analyzing the utility function of the optimal mechanism at the edges of the polygon. \citet{KM16} identified the dual when the valuation space is convex and the space of allocations is restricted. They also solved examples when the allocations are restricted to satisfy either the unit-demand constraint or the deterministic constraint. Other than this lone example solved in \cite{KM16}, we are not aware of any work that computes the exact solution in the unit-demand setting using the duality approach.
There are interesting characterization results on optimal mechanisms in the unit-demand setting. \citet{WT14}, \cite{TW17} proved that when the distributions are uniform in any rectangle in the positive quadrant, the optimal mechanism is a {\em menu} with at most five items. However, the exact menus and associated allocations were left open. \citet{HH15} did a reverse mechanism design; they constructed a mechanism and identified conditions under which there exists a virtual valuation thereby establishing that the mechanism is optimal.
There has been some interest in finding approximately optimal solutions when the distribution of the buyer's valuations satisfies certain conditions. See \cite{Bhat10, BCKW10, BCKW15, CD11, CD15, CDW12a, CDW12b, CDW13, CDW16, CZ17, CHK07, CMS15, CM16}, \cite{DW11}, \cite{DW12}, \cite{Yao14} for relevant literature on approximate solutions. In this paper however we shall focus on exact solutions.
\subsection{Our contributions}
Our contributions are as follows:
\begin{enumerate}
\item[(i)] We identify the dual to the problem of optimal auction in the restricted unit-demand setting, using a result in \cite{KM16}\footnote{The dual to the problem of optimal auction was derived independently in the PhD thesis of the first author.}. We then argue that the computation of the dual measure in the unit-demand setting using the approach of optimal transport in \cite{DDT15} is intricate. Specifically, we consider three examples, $z\sim\mbox{Unif}[1.26,2.26]^2$, $z\sim\mbox{Unif}[1.5,2.5]^2$, and $z\sim\mbox{Unif}[0,1]\times[0,1.2]$, and show that the optimal dual variable differs significantly with variation in $c$, thus making it hard to discover the correct dual measure.
\item[(ii)] Motivated by the above, we explore the virtual valuation method in \cite{Pav11} and nontrivially extend this method to compute the exact solution when $z\sim\mbox{Unif}[c,c+b_1]\times[c,c+b_2]$, for arbitrary nonnegative values of $(c,b_1,b_2)$. We establish that the structure of the optimal mechanism falls within a class of five simple structures, each having at most five constant allocation regions. We also make some remarks on the general case $[c_1,c_1+b_1]\times[c_2,c_2+b_2]$.
\item[(iii)] To the best of our knowledge, our results appear to be the first to show the existence of optimal mechanisms with no region of exclusion (see Figures \ref{fig:e-ini} and \ref{fig:e'-ini}). The results in \citet{Arm96a} and \citet{PBBK14} assert that the optimal multi-dimensional mechanisms have a nontrivial exclusion region under some sufficient conditions on the distributions and the utility functions. \citet{Arm96a} assumes strict convexity of the support set, and \citet{PBBK14} assume strict concavity of the utility function in the allocations. Neither of these assumptions holds in our setting.
\end{enumerate}
In the literature, we already have qualitative results on the structure of optimal mechanism for distributions satisfying certain conditions. For instance, Pavlov \cite{Pav10} considered distributions with negative power rate, while Wang and Tang \cite{WT14} considered uniform distributions on arbitrary rectangles (which do have negative power rate). Our work considers uniform distributions with support set $[c,c+b_1]\times[c,c+b_2]$, a special case of the settings in \cite{Pav10} and \cite{WT14}. It follows that the optimal mechanisms in our setting can have allocations only of the form $(0,0)$, $(a,1-a)$ in accordance with Pavlov's result, and the menus can have at most five items in accordance with Wang and Tang's result.
Though our work is on a further special case, we are able to obtain finer results. We prove that the optimal mechanisms can only be one among the structures depicted in Figures \ref{fig:a-ini}--\ref{fig:e'-ini}. Our results bring out some unexpected structures such as those in Figures \ref{fig:e-ini} and \ref{fig:e'-ini}. Furthermore, our results are explicit in that we can compute the optimal mechanism for uniform distributions on any rectangle of the form $[c,c+b_1]\times[c,c+b_2]$.
The optimal mechanisms for various values of $(c,b_1,b_2)$ are mentioned in Theorem \ref{thm:consolidate}. The phase diagram in Figure \ref{fig:phase-diagram} represents how the structure of optimal mechanism changes when the values of $(c,b_1,b_2)$ change. We interpret the solutions and highlight their features as follows.
\begin{figure}[t]
\centering
\begin{tikzpicture}[scale=0.5,font=\normalsize,axis/.style={very thick, ->, >=stealth'}]
\draw [axis,thick,-] (0,0)--(0,12);
\node [right] at (6.5,-1) {$\frac{b_1}{b_2}$};
\draw [axis,thick,-] (0,0)--(12,0);
\node [above] at (-1,6.5) {$\frac{c}{b_2}$};
\draw [axis,thick,-] (12,0)--(12,12);
\draw [axis,thick,-] (0,12)--(12,12);
\node [left] at (-0.25,0) {$0$};
\draw [thin,-] (-0.25,2) -- (0.25,2);
\node [left] at (-0.25,2) {$2$};
\draw [thin,-] (-0.25,4) -- (0.25,4);
\node [left] at (-0.25,4) {$4$};
\draw [thin,-] (-0.25,6) -- (0.25,6);
\node [left] at (-0.25,6) {$6$};
\draw [thin,-] (-0.25,8) -- (0.25,8);
\node [left] at (-0.25,8) {$8$};
\draw [thin,-] (-0.25,10) -- (0.25,10);
\node [left] at (-0.25,10) {$10$};
\draw [thin,-] (-0.25,12) -- (0.25,12);
\node [left] at (-0.25,12) {$12$};
\node [below] at (0,-0.25) {$1$};
\draw [thin,-] (2,-0.25) -- (2,0.25);
\node [below] at (2,-0.25) {$1.2$};
\draw [thin,-] (4,-0.25) -- (4,0.25);
\node [below] at (4,-0.25) {$1.4$};
\draw [thin,-] (6,-0.25) -- (6,0.25);
\node [below] at (6,-0.25) {$1.6$};
\draw [thin,-] (8,-0.25) -- (8,0.25);
\node [below] at (8,-0.25) {$1.8$};
\draw [thin,-] (10,-0.25) -- (10,0.25);
\node [below] at (10,-0.25) {$2$};
\draw [thick,-] (0,1) to (12,1);
\draw [thick,-] (0,1) to (5,1.733) to (12,1.97);
\draw [thick,-] (0,1.372) to (5,1.733);
\draw [thick,-] (2.2,12) to[out=-60,in=170] (12,3.98);
\draw [thick,dotted] (5,0) to (5,12);
\node [right] at (5,11) {$b_1=(1.5)b_2$};
\draw [thick,->] (3,10) -- (2.2,8.75);
\node at (2.2,8.5) {asymptotic};
\node at (2.2,7.5) {to $b_1=b_2$};
\node at (9,3.8) {asymptotic};
\node at (9,2.8) {to $c=2b_2$};
\node at (10,1.4) {$c=b_2$};
\draw [thick,->] (8.5,2) -- (8.5,2.6);
\draw [thick,->] (8.5,5) -- (8.5,4.2);
\path[fill=red,opacity=.5] (0,1) to (12,1) to (12,0) to (0,0);
\path[fill=green,opacity=.5] (0,1) to (5,1.733) to (12,1.97) to (12,1) to (0,1);
\path[fill=blue,opacity=.5] (0,1) to (0,1.372) to (5,1.733) to (0,1);
\path[fill=yellow,opacity=.5] (0,1.372) to (0,12) to (2.2,12) to [out=-60,in=135] (5,8) to (5,1.733) to (0,1.372);
\path[fill=violet,opacity=.5] (5,8) to [out=-50,in=170] (12,3.98) to (12,1.97) to (5,1.733) to (5,8);
\path[fill=orange,opacity=.5] (2.2,12) to [out=-60,in=135] (5,8) to (5,12) to (2.2,12);
\path[fill=brown,opacity=.5] (5,12) to (5,8) to [out=-50,in=170] (12,3.98) to (12,12) to (5,12);
\draw [thick,-] (13,9.5) to (13,10.5);
\draw [thick,-] (13,10.5) to (14,10.5);
\draw [thick,-] (14,9.5) to (14,10.5);
\draw [thick,-] (13,9.5) to (14,9.5);
\path[fill=red,opacity=.5] (13,9.5) to (14,9.5) to (14,10.5) to (13,10.5) to (13,9.5);
\node [right] at (14,10) {Figure \ref{fig:a-ini}};
\draw [thick,-] (13,8) to (13,9);
\draw [thick,-] (13,9) to (14,9);
\draw [thick,-] (14,8) to (14,9);
\draw [thick,-] (13,8) to (14,8);
\path[fill=green,opacity=.5] (13,8) to (14,8) to (14,9) to (13,9) to (13,8);
\node [right] at (14,8.5) {Figure \ref{fig:b-ini}};
\draw [thick,-] (13,6.5) to (13,7.5);
\draw [thick,-] (13,7.5) to (14,7.5);
\draw [thick,-] (14,6.5) to (14,7.5);
\draw [thick,-] (13,6.5) to (14,6.5);
\path[fill=blue,opacity=.5] (13,6.5) to (14,6.5) to (14,7.5) to (13,7.5) to (13,7.5);
\node [right] at (14,7) {Figure \ref{fig:c-ini}};
\draw [thick,-] (13,5) to (13,6);
\draw [thick,-] (13,6) to (14,6);
\draw [thick,-] (14,5) to (14,6);
\draw [thick,-] (13,5) to (14,5);
\path[fill=yellow,opacity=.5] (13,5) to (14,5) to (14,6) to (13,6) to (13,5);
\node [right] at (14,5.5) {Figure \ref{fig:d-ini}};
\draw [thick,-] (13,3.5) to (13,4.5);
\draw [thick,-] (13,4.5) to (14,4.5);
\draw [thick,-] (14,3.5) to (14,4.5);
\draw [thick,-] (13,3.5) to (14,3.5);
\path[fill=orange,opacity=.5] (13,3.5) to (14,3.5) to (14,4.5) to (13,4.5) to (13,3.5);
\node [right] at (14,4) {Figure \ref{fig:e-ini}};
\draw [thick,-] (13,2) to (13,3);
\draw [thick,-] (13,3) to (14,3);
\draw [thick,-] (14,2) to (14,3);
\draw [thick,-] (13,2) to (14,2);
\path[fill=violet,opacity=.5] (13,2) to (14,2) to (14,3) to (13,3) to (13,2);
\node [right] at (14,2.5) {Figure \ref{fig:d'-ini}};
\draw [thick,-] (13,0.5) to (13,1.5);
\draw [thick,-] (13,1.5) to (14,1.5);
\draw [thick,-] (14,0.5) to (14,1.5);
\draw [thick,-] (13,0.5) to (14,0.5);
\path[fill=brown,opacity=.5] (13,0.5) to (14,0.5) to (14,1.5) to (13,1.5) to (13,0.5);
\node [right] at (14,1) {Figure \ref{fig:e'-ini}};
\end{tikzpicture}
\caption{A phase diagram of the optimal mechanism when $b_2\leq b_1\leq (2 .2)b_2$.}\label{fig:phase-diagram}
\end{figure}
\begin{itemize}
\item[$\bullet$] Beyond the exclusion (no sale) region, the allocation probabilities are the same for all $z$ falling in the same $45^\circ$ line (Theorem \ref{thm:pav-2}). Observe that this is in sharp contrast with the unrestricted setting, where the allocation probabilities are the same either for all $z$ falling in the same vertical line or the same horizontal line (see \cite[Fig.~1--3]{TRN16}). This is because, in the unit-demand case, the buyer demands at most one of the two items, and thus the seller decides the item to be sold based on the difference of valuations on the items\footnote{The item to be sold is decided based on the difference in valuations only for cases where $q_1 + q_2 = 1$ holds everywhere outside the exclusion region. It would be interesting to interpret the results for cases when $q_1 + q_2 < 1$ can occur outside the exclusion region, but this exploration is beyond the scope of this paper.}.
\item[$\bullet$] Consider the case when $c$ is low. The seller then knows that the buyer possibly could have very low valuations, and thus sets a high price $(c+\delta_i)$ to sell item $i$. Observe that this is a posted price mechanism with prices $c+\delta_1$ and $c+\delta_2$ for items $1$ and $2$ respectively (see Figure \ref{fig:a-ini}).
\item[$\bullet$] When $c$ increases, the seller now finds it optimal to set a second price over and above the first price $c+\delta_i$. He offers a lottery for the first price, and offers an individual item for the second and higher price (see Figures \ref{fig:b-ini} and \ref{fig:c-ini}).
\item[$\bullet$] When $c$ increases further, the seller sells item $i$ only when $z_i$ is very high compared to $z_{-i}$. In case the difference is not sufficiently high, then the seller finds it optimal to allocate randomly one or the other item (see Figures \ref{fig:d-ini} and \ref{fig:d'-ini}).
\item[$\bullet$] When $c$ is very high, the revenue gained by exclusion of certain valuations is always dominated by the revenue lost by it, and thus the seller finds no reason to withhold the items for any valuation profile\footnote{We refer the reader to Remark \ref{rem:armstrong} for a more precise explanation.}. So the optimal mechanism turns out to be a posted price mechanism with prices $c+\frac{b_1}{3}+\max(0,\frac{b_1}{6}-\frac{b_2}{4})$ and $c$ for items $1$ and $2$ respectively. In effect, it is a posted price mechanism with no exclusion region (see Figures \ref{fig:e-ini} and \ref{fig:e'-ini}).
\item[$\bullet$] Starting at $c=0$, consider moving the support set rectangle to infinity. Then, the optimal mechanism starts as a posted price mechanism with an exclusion region, and ends up again as a posted price mechanism but without an exclusion region. The other structures in Figures \ref{fig:b-ini}--\ref{fig:d-ini}, and Figure \ref{fig:d'-ini} are optimal for various intermediate values.
\end{itemize}
\begin{figure}[t]
\centering
\setlength\tabcolsep{0pt}
\begin{tabular}{cccccc}
\multicolumn{2}{c}{\subfloat[]{\label{fig:a-ini}\begin{tikzpicture}[scale=0.18,font=\footnotesize,axis/.style={very thick, -}]
\node at (-0.5,-1) {$(c,c)$};
\draw [axis,thick,-] (0,0)--(15,0);
\node at (12,-1) {$(c+b_1,c)$};
\draw [axis,thick,-] (0,0)--(0,12);
\node [above] at (1,12) {$(c,c+b_2)$};
\draw [axis,thick,-] (0,12)--(15,12);
\draw [axis,thick,-] (15,0)--(15,12);
\draw [axis,thick,-] (0,5)--(6.5,5);
\draw [axis,thick,-] (6.5,0)--(6.5,5);
\draw [thick,-] (6.5,5)--(13.5,12);
\node [above] at (3.5,1.5) {$(0,0)$};
\node [above] at (11.5,4.5) {$(1,0)$};
\node [above] at (3,8) {$(0,1)$};
\end{tikzpicture}}}&
\multicolumn{2}{c}{\subfloat[]{\label{fig:b-ini}\begin{tikzpicture}[scale=0.18,font=\footnotesize,axis/.style={very thick, -}]
\node at (-0.5,-1) {$(c,c)$};
\draw [axis,thick,-] (0,0)--(15,0);
\node at (12,-1) {$(c+b_1,c)$};
\draw [axis,thick,-] (0,0)--(0,12);
\node [above] at (1,12) {$(c,c+b_2)$};
\draw [axis,thick,-] (0,12)--(15,12);
\draw [axis,thick,-] (15,0)--(15,12);
\draw [axis,thick,-] (0,4)--(8,12);
\draw [axis,thick,-] (0,3.5)--(4,2.5);
\draw [axis,thick,-] (4,0)--(4,2.5);
\draw [thick,-] (4,2.5)--(13.5,12);
\node at (2,1.5) {$(0,0)$};
\node [above] at (11,4) {$(1,0)$};
\node [above] at (2,8) {$(0,1)$};
\node [rotate=45] at (6,7) {$(1-a_2,a_2)$};
\end{tikzpicture}}}&
\multicolumn{2}{c}{\subfloat[]{\label{fig:c-ini}\begin{tikzpicture}[scale=0.18,font=\footnotesize,axis/.style={very thick, -}]
\node at (-0.5,-1) {$(c,c)$};
\draw [axis,thick,-] (0,0)--(15,0);
\node at (12,-1) {$(c+b_1,c)$};
\draw [axis,thick,-] (0,0)--(0,12);
\node [above] at (1,12) {$(c,c+b_2)$};
\draw [axis,thick,-] (0,12)--(15,12);
\draw [axis,thick,-] (15,0)--(15,12);
\draw [axis,thick,-] (0,4)--(8,12);
\draw [axis,thick,-] (0,3.5)--(4,2.5);
\draw [axis,thick,-] (4,2.5)--(6,0);
\draw [axis,thick,-] (6.5,0)--(15,8.5);
\draw [thick,-] (4,2.5)--(13.5,12);
\node at (2,1.5) {$(0,0)$};
\node at (12,2) {$(1,0)$};
\node [above] at (2,8) {$(0,1)$};
\node [rotate=45] at (6,7) {$(1-a_2,a_2)$};
\node [rotate=45] at (8.5,4.5) {$(a_1,1-a_1)$};
\end{tikzpicture}}}\\
&\multicolumn{2}{c}{\subfloat[]{\label{fig:d-ini}\begin{tikzpicture}[scale=0.18,font=\footnotesize,axis/.style={very thick, -}]
\node at (-0.5,-1) {$(c,c)$};
\draw [axis,thick,-] (0,0)--(15,0);
\node at (12,-1) {$(c+b_1,c)$};
\draw [axis,thick,-] (0,0)--(0,12);
\node [above] at (1,12) {$(c,c+b_2)$};
\draw [axis,thick,-] (0,12)--(15,12);
\draw [axis,thick,-] (15,0)--(15,12);
\draw [axis,thick,-] (0,4)--(8,12);
\draw [axis,thick,-] (0,3.5)--(5,0);
\draw [thick,-] (5.5,0)--(15,9.5);
\node at (1.5,1.2) {\scriptsize$(0,0)$};
\node [above] at (12,2) {$(1,0)$};
\node [above] at (2,8) {$(0,1)$};
\node at (6.5,6) {$(1-a,a)$};
\end{tikzpicture}}}&
&\multicolumn{2}{c}{\subfloat[]{\label{fig:e-ini}\begin{tikzpicture}[scale=0.18,font=\footnotesize,axis/.style={very thick, -}]
\node at (-0.5,-1) {$(c,c)$};
\draw [axis,thick,-] (0,0)--(15,0);
\node at (12,-1) {$(c+b_1,c)$};
\draw [axis,thick,-] (0,0)--(0,12);
\node [above] at (1,12) {$(c,c+b_2)$};
\draw [axis,thick,-] (0,12)--(15,12);
\draw [axis,thick,-] (15,0)--(15,12);
\draw [thick,-] (5,0)--(15,10);
\node [above] at (12,3) {$(1,0)$};
\node [above] at (5,6) {$(0,1)$};
\end{tikzpicture}}}\\
&\multicolumn{2}{c}{\subfloat[]{\label{fig:d'-ini}\begin{tikzpicture}[scale=0.18,font=\small,axis/.style={very thick, -}]
\node at (-0.5,-1) {$(c,c)$};
\draw [axis,thick,-] (0,0)--(18,0);
\node at (16,-1) {$(c+b_1,c)$};
\draw [axis,thick,-] (0,0)--(0,12);
\node [above] at (1,12) {$(c,c+b_2)$};
\draw [axis,thick,-] (0,12)--(18,12);
\draw [axis,thick,-] (18,0)--(18,12);
\draw [axis,thick,-] (0,4)--(8,12);
\draw [axis,thick,-] (0,3.5)--(5,0);
\draw [thick,-] (5.5,0)--(17.5,12);
\node at (1.5,1.2) {\scriptsize$(0,0)$};
\node [above] at (14,3) {$(1,0)$};
\node [above] at (2,8) {$(0,1)$};
\node at (6.5,5.5) {$(1-a,a)$};
\end{tikzpicture}}}&
&\multicolumn{2}{c}{\subfloat[]{\label{fig:e'-ini}\begin{tikzpicture}[scale=0.18,font=\small,axis/.style={very thick, -}]
\node at (-0.5,-1) {$(c,c)$};
\draw [axis,thick,-] (0,0)--(18,0);
\node at (16,-1) {$(c+b_1,c)$};
\draw [axis,thick,-] (0,0)--(0,12);
\node [above] at (1,12) {$(c,c+b_2)$};
\draw [axis,thick,-] (0,12)--(18,12);
\draw [axis,thick,-] (18,0)--(18,12);
\draw [thick,-] (5,0)--(17,12);
\node [above] at (14,3) {$(1,0)$};
\node [above] at (5,6) {$(0,1)$};
\end{tikzpicture}}}
\end{tabular}
\caption{An illustration of all possible structures that an optimal mechanism can have.}\label{fig:all-structure-ini}
\end{figure}
\subsection{Our method}
Our method is as follows. We initially formulate the problem at hand (in the unit-demand setting) into an optimization problem, and compute its dual using a result in \cite{KM16}. The dual problem turns out to be an optimal transport problem that transfers mass from the support set $D$ to itself. Mass transfer must occur subject to the constraint that the difference between the mass densities transferred out of and transferred into the set convex-dominates a signed measure that depends only on the distribution of the valuations. The dual problem is similar to that in \cite{DDT15} for the unrestricted setting, but differs in the transportation cost.
The key challenge in solving the dual problem lies in constructing the ``shuffling measure'' that convex-dominates $0$, and in finding the location in the support set $D$ where the shuffling measure sits. The shuffling measure was always added at fixed locations in the unrestricted setting, and had a fixed structure for the uniform distribution of valuations over any rectangle in the positive quadrant (see \cite{TRN16}). In the unit-demand setting, however, we see that both the locations and the structures of the shuffling measure vary significantly for different values of $c$. There is as yet no clear understanding on how to construct the shuffling measure, and hence on how to compute the optimal solution via the dual method.
Motivated by the above, we explore the virtual valuation method used by \citet{Pav11}. \citet{Pav11} computed the optimal mechanism when the buyer's valuations are given by $z\sim\mbox{Unif}[c,c+1]^2$; the optimal mechanism was obtained only for distributions that are symmetric across the two items. When compared with the case of symmetric distributions, the case of asymmetric distributions poses the following challenge. The optimal mechanism is symmetric along a diagonal in the case of symmetric distributions. For asymmetric distributions, the mechanism must be computed over the larger region of the entire support set. The asymmetry leads to more parameters, more conditions to check for optimality, and a more complex variety of solutions determined, as we will soon see, by a larger number of polynomials. All these make the computation more difficult.
In this paper, we demonstrate how to compute the optimal mechanism for asymmetric distributions, when $z\sim\mbox{Unif}[c,c+b_1]\times[c,c+b_2]$. Specifically, we do the following.
\begin{itemize}
\item[$\bullet$] Taking cue from the result in \cite{WT14} that the optimal mechanism is a menu with at most five items, we first construct some possible menus, parametrized by at most four parameters.
\item[$\bullet$] We find the relation between the parameters using the sufficient conditions on the marginal profit function $V$. We show that the parameters can be computed by simultaneously solving at most two polynomials, each of degree at most $4$.
\item[$\bullet$] We then use continuity of the polynomials to prove that there exists a solution having desired values for all parameters. We then prove that the optimal mechanism has one of the five simple structures for arbitrary nonnegative values of $(c,b_1,b_2)$ (see Theorem \ref{thm:consolidate}).
\item[$\bullet$] We conjecture that the optimal mechanisms have a similar structure even when $z\in[c_1,c_1+b_1]\times[c_2,c_2+b_2]$ for all $(c_1,c_2,b_1,b_2)\geq 0$. We provide preliminary results to justify the conjecture (see Theorem \ref{thm:extension}).
\end{itemize}
Proofs of some case use Mathematica to verify certain algebraic inequalities. This is because (i) the parameters turn out to be solutions that simultaneously satisfy two polynomials of degree at most $4$; and (ii) the solutions are complicated functions of $(c,b_1,b_2)$ involving fifth roots and eighth roots of some expressions. Verifying that these expressions satisfy some bounds were automated via the Mathematica software. The results that use Mathematica have been marked with an asterisk in the statement of Theorem \ref{thm:consolidate}. We believe that all of these results can be proved in the strict mathematical sense; but we leave this for the future in the interest of timely dissemination of our conclusions and observations. The skeptical reader could proceed by interpreting the Mathematica-based conclusions as conjectures.
Our work thus provides insights into two well-known approaches to solve representative problems on optimal mechanisms in the multi-item setting, besides solving, in the process, one such problem for asymmetric distributions. Specifically, our work clarifies under what situations the duality approach is likely to work well, and the intrinsic difficulties in using that approach in some other settings. Furthermore, the special cases that we solve provide insights into various possible structures of the optimal mechanisms which, we feel, would act as a guideline to solve the problem of computing good menus in practical settings. We believe that our work is an important step towards understanding the applicability of the two different approaches, and a useful step addition to the growing canvas of canonical problems in multi-dimensional optimal auctions.
The rest of the paper is organized as follows. In Section 2, we first formulate an optimization problem under the unit-demand setting. We next compute its dual using a result in \cite{KM16}, and solve it for three representative examples of $(c,b_1,b_2)$. The main purpose behind these examples is to bring out the variety in structure, and therefore the difficulty in guessing and computing, the dual measure for more general settings. In Section 3, we nontrivially extend the {\em virtual valuation method\/} of \cite{Pav11} to provide a complete and explicit solution for the case of asymmetric distributions. In particular, we prove that the optimal mechanism has one of the five simple structures. In Section 4, we conjecture, with promising preliminary results, that the optimal mechanism when the valuations are uniformly distributed in an arbitrary rectangle $[c_1,c_1+b_1]\times[c_2,c_2+b_2]$ also has similar structures. In Section 5, we conclude the paper and provide some directions for future work.
\section{Exploring The Dual Approach}\label{sec:prelim}
Consider a two-item, single-buyer, unit-demand setting. The buyer's valuation is $z=(z_1,z_2)$ for the two items, sampled according to the joint density $f(z)=f_1(z_1)f_2(z_2)$, where $f_1(z_1)$ and $f_2(z_2)$ are marginal densities. The support set of $f$ is defined as $D:=\{z:f(z)>0\}$. Throughout the paper, we consider $D=[c,c+b_1]\times[c,c+b_2]$, where $(c,b_1,b_2)$ are nonnegative.
Our aim is to design a revenue-optimal mechanism. By the revelation principle \cite[Prop.~9.25]{NRTV07}, it suffices to focus only on direct mechanisms. Further, we focus on mechanisms where the buyer has a quasilinear utility. Specifically, we assume an allocation function $q:D\rightarrow\{(q_1,q_2):0\leq q_1,q_2,q_1+q_2\leq 1\}$ and a payment function $t:D \rightarrow \mathbb{R}_+$ that represent, respectively, the probabilities of allocation of the items to the buyer and the amount of transfer from the buyer to the seller. If the buyer's true valuation is $z$, and he reports $\hat{z}$, his realized (quasilinear) utility is $\hat{u}(z, \hat{z}) := z \cdot q(\hat{z}) - t(\hat{z})$, which is the expected value of the lottery he receives minus the payment.
A mechanism $(q,t)$ satisfies {\em incentive compatibility} (IC) when truth telling is a weakly dominant strategy for the buyer, i.e., $\hat{u}(z,z) \geq \hat{u}(z,\hat{z})$ for every $z,\hat{z}\in D$. In this case the buyer's realized utility is $u(z) := \hat{u}(z, z) = z\cdot q(z)-t(z)$. An incentive compatible mechanism satisfies {\em individual rationality} (IR) if the buyer is not worse off by participating in the mechanism, i.e., $u(z) \geq 0$ for every $z \in D$, with zero being the buyer's utility if he chooses not to participate.
The following result is well known:
\begin{theorem}\cite{Roc87}\/.\label{thm:Rochet}
A mechanism $(q,t)$, with $u(z) = z\cdot q(z)-t(z)$, is incentive compatible if and only if $u$ is continuous, convex and $\nabla u(z)=q(z)$ for a.e. $z\in D$.
\end{theorem}
An optimal mechanism is one that maximizes the expected revenue to the seller subject to incentive compatibility and individual rationality (\citet[p.~67]{VKBook}). By virtue of Theorem \ref{thm:Rochet}, an optimal mechanism solves the problem
\begin{alignat*}{2}
&\max_u \int_D(z\cdot\nabla u(z)-u(z))f(z)\,dz \\
\mbox{ subject to }\hspace*{.5in}
&(a)\,u\mbox{ continuous, convex},\nonumber\\
&(b)\,\nabla u(z)\in[0,1]^2,\nabla u(z)\cdot\mathbf{1}\in[0,1],\mbox{ a.e. }z \in D,\nonumber\\
&(c)\,u(z)\geq 0,\,\,\forall z\in D.\nonumber
\end{alignat*}
Using the arguments in \cite[Sec.~2.1]{DDT17}, we simplify the aforementioned problem as
\begin{align}\label{eqn:optim}
&\max_u\int_D(z\cdot\nabla u(z)-(u(z)-u(c,c)))f(z)\,dz \\
\mbox{ subject to }\hspace*{.5in}
&(a)\,u\mbox{ continuous, convex},\nonumber\\
&(b)\,\nabla u(z)\in[0,1]^2,\nabla u(z)\cdot\mathbf{1}\in[0,1],\mbox{ a.e. }z \in D.\nonumber
\end{align}
We now further simplify the objective function of the problem. Using integration by parts, the objective function can be written as $\int_D u(z)\mu(z)\,dz+\int_{\partial D}u(z)\mu_s(z)\,dz+u(c,c)\mu_p(c,c)$, where the functions $\mu$, $\mu_s$, and $\mu_p$ are defined as
\begin{align}\label{eqn:mu-measure}
\mu(z)&:=-z\cdot\nabla f(z)-3f(z),\,z\in D,\nonumber\\
\mu_s(z)&:=(z\cdot n(z))f(z),\,z\in\partial D,\\
\mu_p(z)&:=\delta_{\{(c,c)\}}(z).\nonumber
\end{align}
The vector $n(z)$ is the normal to the surface $\partial D$ at $z$ if it is defined, and $0$ otherwise (at corners). We regard $\mu$ as the density of a signed measure on the support set $D$ that is absolutely continuous with respect to (w.r.t.) the two-dimensional Lebesgue measure, and $\mu_s$ as the density of a signed measure on $\partial D$ that is absolutely continuous w.r.t. the surface Lebesgue measure. We shall refer to both Lebesgue measures as $dz$. We regard $\mu_p$ as a point measure of unit mass at the specified point. The notation $\delta$ denotes the Dirac-delta function. So $\mu_p(z)=1$ if $z=(c,c)$, and $0$ otherwise. By taking $u(z) = 1 ~ \forall z \in D$, we observe that
\begin{align}\label{eqn:mu-D}
&\int_D \mu(z)\,dz+\int_{\partial D}\mu_s(z)\,dz+\mu_p(c,c)\nonumber\\&\hspace*{.5in}=\int_D u(z)\mu(z)\,dz+\int_{\partial D}u(z)\mu_s(z)\,dz+u(c,c)\mu_p(c,c)\nonumber\\&\hspace*{.5in}=\int_D(z\cdot\nabla u(z)-u(z))f(z)\,dz+u(c,c)\nonumber\\&\hspace*{.5in}=\int_D(0-1)f(z)\,dz+u(c,c)=0.
\end{align}
We now define the measure $\bar{\mu}$, supported on set $D$, as
$$
\bar{\mu}(A):=\int_D\mathbf{1}_A(z)\mu(z)\,dz+\int_{\partial D}\mathbf{1}_A(z)\mu_s(z)\,dz+\mu_p(A\cap(c,c))
$$
for all measurable sets $A$. We thus observe that $\bar{\mu}(D)=0$. Observe that $\bar{\mu}$ is a signed Radon measure in $D$, and that the functions $\mu$ and $\mu_s$ are just the Radon-Nikodym derivatives of the respective components of $\bar{\mu}$ w.r.t. the two-dimensional and one-dimensional Lebesgue measures respectively. Based on the discussion in the paragraph after (\ref{eqn:optim}), the objective function of problem (\ref{eqn:optim}) can now be written as $\int_D u\,d\bar{\mu}$.
We now rewrite the constraint (b) in problem (\ref{eqn:optim}) as the following three constraints.
\begin{alignat*}{2}
&u(z_1,z_2)-u(z_1',z_2)\leq (z_1-z_1')_+,\, \forall z_1,z_1'\in D_1,\,\forall z_2\in D_2,\nonumber\\
&u(z_1,z_2)-u(z_1,z_2')\leq (z_2-z_2')_+,\, \forall z_1\in D_1,\,\forall z_2,z_2'\in D_2,\nonumber\\
&u(z_1,z_2)-u(z_1',z_2-z_1+z_1')\leq (z_1-z_1')_+,\, \forall z_1,z_1'\in D_1,\,\forall z_2\in D_2,\nonumber
\end{alignat*}
where $(\cdot)_+=\max(0,\cdot)$. Observe that these three constraints are equivalent to
$$
u(z)-u(z')\leq\max((z_1-z_1')_+,(z_2-z_2')_+),\,\forall z,z'\in D.
$$
So the optimization problem can now be written as
\begin{alignat}{2}\label{eqn:primal}
&\max_u \int_D u\,d\bar{\mu} \\
\mbox{ subject to }\hspace*{.5in}
&(a)\,u\mbox{ continuous, convex, increasing},\nonumber\\
&(b)\,u(z)-u(z')\leq\|z-z'\|_\infty,\,\forall z,z'\in D.\nonumber
\end{alignat}
Note that the objective function of the problem satisfies $\int_Dt(z)f(z)\,dz=\int_Du\,d\bar{\mu}$; thus the $\bar{\mu}$-measure can be interpreted as the marginal contribution of the utility $u$ to the revenue of the seller.
We now recall the definition of the convex ordering relation. A function $f$ is increasing if $z\geq z'$ component-wise implies $f(z)\geq f(z')$.
\begin{definition} (See for e.g., \cite{DDT14})
Let $\alpha$ and $\beta$ be measures defined on a set $D$. We say $\alpha$ {\em convex-dominates} $\beta$ ($\alpha\succeq_{cvx}\beta$) if $\int_Df\,d\alpha\geq\int_Df\,d\beta$ for all continuous, convex and increasing $f$.
\end{definition}
One can understand convex dominance as follows: A {\em risk-seeking} buyer\footnote{This is to be contrasted with second-order stochastic dominance which says that $\alpha$ second-order dominates $\beta$ (denoted as $\alpha\succeq_2\beta$) if a {\em risk-averse} buyer with an {\em increasing and concave} utility function prefers $\alpha$ to $\beta$. Mathematically, convex dominance and second-order stochastic dominance are related inversely under some conditions. More specifically, $\alpha\succeq_{cvx}\beta\Leftrightarrow\alpha\preceq_2\beta$ if (i) $D$ is a bounded rectangle in the positive orthant and (ii) $\int_D\|x\|_1\,d(\alpha-\beta)=0$ \cite[Lem. 8]{DDT14}}., with $u$ as his utility function (increasing and convex), will choose the lottery $\alpha$ over $\beta$ if $\alpha$ convex-dominates $\beta$.
The dual problem of (\ref{eqn:primal}) is found to be \cite[Thm.~3.1]{KM16}.
\begin{align}\label{eqn:dual}
&\min_\gamma\int_{D\times D}\|z-z'\|_\infty\,d\gamma(z,z')\\
\mbox{ subject to }\hspace*{.25in}
&(a)\,\gamma\in Radon_+(D\times D),\nonumber\\
&(b)\,\gamma(\cdot,D)=\gamma_1,\,\gamma(D,\cdot)=\gamma_2,\,\gamma_1-\gamma_2\succeq_{cvx}\bar{\mu}.\nonumber
\end{align}
By $\gamma\in Radon_+(D\times D)$, we mean that $\gamma$ is an unsigned Radon measure in $D\times D$. The dual is computed by using the following expressions in the statement of \cite[Thm.~3.1]{KM16}: (i) $l_S(z,z')=\|z-z'\|_\infty$, and (ii) $U(D,S)$ is the set of all utility functions that are continuous, convex, and increasing. We derive the weak duality result in \ref{app:new} to provide an understanding of how the dual arises and why $\gamma$ may be interpreted as prices for violating the primal constraint.
The next lemma gives a sufficient condition for strong duality.
\begin{lemma}\cite[Cor.~4.1]{KM16}\label{lem:compslack}
Let $u^*$ and $\gamma^*$ be feasible for the aforementioned primal (\ref{eqn:primal}) and dual (\ref{eqn:dual}) problems, respectively\/. Then the objective functions of (\ref{eqn:primal}) and (\ref{eqn:dual}) with $u=u^*$ and $\gamma=\gamma^*$ are equal if and only if (i) $\int_Du^*\,d(\gamma_1^*-\gamma_2^*)=\int_Du^*\,d\bar{\mu}$, and (ii) $u^*(z)-u^*(z')=\|z-z'\|_\infty$, hold $\gamma^*-$a.e.
\end{lemma}
We now present a few examples to indicate why it is hard to compute the optimal mechanism using this dual approach. We first compute the components of $\bar{\mu}$ (i.e., $\mu,\mu_s,\mu_p$), with $f(z)=\frac{1}{b_1b_2}$ for $z\in D=[c,c+b_1]\times[c,c+b_2]$, from (\ref{eqn:mu-measure}), as
\begin{align}
\mbox{(area density) } \, & \mu(z)=-3/(b_1b_2),\quad z \in D,\nonumber\\
\mbox{(line density) } \, & \mu_s(z)=\sum_{i=1}^2(-c\mathbf{1}(z_i=c)+(c+b_i)\mathbf{1}(z_i=c+b_i))/(b_1b_2),\nonumber\\&\hspace*{2.5in} z\in\partial D,\nonumber\\
\mbox{(point measure)} \, & \mu_p(z)=\delta_{\{(c,c)\}}(z).\label{eqn:bar-mu}
\end{align}
In the examples that we consider, we start by suggesting a certain mechanism, and prove that it is indeed the optimal mechanism by constructing a feasible $u$ and a feasible $\gamma$ that satisfy the complementary slackness constraints of Lemma \ref{lem:compslack}. While $u$ can be constructed easily from the allocations $q$, the construction of the transport variable $\gamma$ needs some work. This involves transporting mass from each point on the top and right boundaries of $D$ along the $45^\circ$ line containing the point. We shuffle the measure across the points on the boundary in case there is an excess or a deficit. The construction of the shuffling measure is the main challenge; it differs significantly across the examples we consider. We now fill in the details.
\subsection{Example 1: $z\sim\mbox{Unif }[1.26,2.26]^2$}
\begin{theorem}\label{thm:eg-1}\cite{Pav11}
Consider the case when $c=1.26$, and $b_1=b_2=1$. Then, the optimal mechanism is as depicted in Figure \ref{fig:illust-1}, with $\delta_1=\delta_2=20/63$ and $a_1=a_2=a=0.6615$.
\end{theorem}
\begin{proof}
\citet{Pav11} proved this via virtual valuations. We shall use the dual method. To prove this theorem, we must find a feasible $u$ and a feasible $\gamma$, and show that they satisfy the conditions of Lemma \ref{lem:compslack}. We define the allocation $q$ as given in Figure \ref{fig:illust-1}. The primal variable $u$ can be derived by fixing $u(c,c)=0$ and by using the allocation variable $q$, since $\nabla u=q$.
We now define functions $\alpha^{(1)},\beta^{(1)}:D\rightarrow\mathbb{R}$ as follows (see Figures \ref{fig:alpha} and \ref{fig:beta}).
\begin{figure}[H]
\centering
\begin{tabular}{cc}
\subfloat[]{\label{fig:illust-1}\begin{tikzpicture}[scale=0.28,font=\tiny,axis/.style={very thick, -}]
\node at (-1,-1) {$(c,c)$};
\draw [axis,thick,-] (0,0)--(16.5,0);
\node at (15.5,-1) {$(c+b_1,c)$};
\draw [axis,thick,-] (0,0)--(0,16.5);
\node [above] at (0,16.5) {$(c,c+b_2)$};
\draw [axis,thick,-] (0,16.5)--(16.5,16.5);
\draw [axis,thick,-] (16.5,0)--(16.5,16.5);
\draw [axis,thick,-] (5.5,0)--(16.5,11);
\node [below,rotate=90] at (16.5,11) {$(c+b_1,c+2b_2/3)$};
\draw [axis,thick,-] (0,5.5)--(11,16.5);
\node [above] at (11,16.5) {$(c+2b_1/3,c+b_2)$};
\draw [axis,thick,-] (0,4.5)--(3,3);
\draw [axis,thick,-] (4.5,0)--(3,3);
\draw [axis,thick,-] (3,3)--(16.5,16.5);
\node at (9,-1) {$(c+b_1/3,c)$};
\node at (3,-1) {$(c+\delta_1,c)$};
\node [rotate=90] at (-1,9) {$(c,c+b_2/3)$};
\node [rotate=90] at (-1,3) {$(c,c+\delta_2)$};
\foreach \Point in {(4.5,0),(5.5,0),(0,4.5),(0,5.5)}{
\node at \Point {\textbullet};}
\node at (1.7,1.5) {\small$(0,0)$};
\node at (13,3) {\small$(1,0)$};
\node at (3,13) {\small$(0,1)$};
\node [rotate=45] at (7.5,10.5) {\small$(1-a_2,a_2)$};
\node [rotate=45] at (10.5,7.5) {\small$(a_1,1-a_1)$};
\node at (3.2,6.5) {Slope$=$};
\node at (3,4.75) {$-\frac{1-a_2}{a_2}$};
\draw [thin,->] (0.25,4.25)--(2,6);
\node at (6,3.5) {Slope$=$};
\node at (5.4,1.75) {$\frac{-a_1}{1-a_1}$};
\draw [thin,->] (3.8,1.9)--(4.3,3.3);
\end{tikzpicture}}&
\subfloat[]{\label{fig:illust-2}\begin{tikzpicture}[scale=0.28,font=\tiny,axis/.style={very thick, -}]
\node at (-1,-1) {$(c,c)$};
\draw [axis,thick,-] (0,0)--(16.5,0);
\node at (15.5,-1) {$(c+b_1,c)$};
\draw [axis,thick,-] (0,0)--(0,16.5);
\node [above] at (0,16.5) {$(c,c+b_2)$};
\draw [axis,thick,-] (0,16.5)--(16.5,16.5);
\draw [axis,thick,-] (16.5,0)--(16.5,16.5);
\draw [axis,thick,-] (5.5,0)--(16.5,11);
\node [below,rotate=90] at (16.5,11) {$(c+b_1,c+2b_2/3)$};
\draw [axis,thick,-] (0,5.5)--(11,16.5);
\node [above] at (11,16.5) {$(c+2b_1/3,c+b_2)$};
\draw [axis,thick,-] (0,5)--(5,0);
\node at (9,-1) {$(c+b_1/3,c)$};
\node at (3,-1) {$(c+\delta_1',c)$};
\node [rotate=90] at (-1,9) {$(c,c+b_2/3)$};
\node [rotate=90] at (-1,3) {$(c,c+\delta_2')$};
\foreach \Point in {(5,0),(0,5)}{
\node at \Point {\textbullet};}
\node at (1.5,1.5) {\small$(0,0)$};
\node at (13,3) {\small$(1,0)$};
\node at (3,13) {\small$(0,1)$};
\node at (8,8) {\small$\left(\frac{1}{2},\frac{1}{2}\right)$};
\node at (4.25,3.5) {Slope$=-1$};
\end{tikzpicture}}
\end{tabular}
\caption{Optimal mechanism when $b_1=b_2=1$ and (a) $c=1.26$, (b) $c=1.5$.}\label{fig:illust}
\end{figure}
\begin{align}
\alpha^{(1)}(c+t,c+t'):&=\begin{cases}3t-1&(t,t')\in[0,2/3]\times\{1\},\\0&\mbox{otherwise}.\end{cases}\label{eqn:alpha}\\
\beta^{(1)}(c+t,c+t'):&=\begin{cases}3t-1&(t,t')\in[2/3,1-\delta_2]\times\{1\},\\3t+3a(1-t-\delta_2)-c-1&(t,t')\in[1-\delta_2,1]\times\{1\},\\0&\mbox{otherwise}.\end{cases}\label{eqn:beta}
\end{align}
The functions $\alpha^{(2)}$ and $\beta^{(2)}$ are defined similarly on the intervals $(\{c+1\}\times[c,c+2/3])$ and $(\{c+1\}\times[c+2/3,c+1])$ respectively. Observe that $\alpha^{(i)}$ and $\beta^{(i)}$ are densities (Radon-Nikodym derivatives) of measures that are absolutely continuous w.r.t. the surface Lebesgue measure. The measures themselves are denoted $\bar{\alpha}^{(i)}$ and $\bar{\beta}^{(i)}$, respectively.
We now construct the dual variable $\gamma$ as follows. First, let (i) $\gamma_1:=\gamma_1^Z+\gamma_1^{D\backslash Z}$, where $Z$ is the exclusion region; (ii) $\gamma_1^Z=\bar{\mu}^Z$, the $\bar{\mu}$ measure restricted to $Z$; and (iii) $\gamma_1^{D\backslash Z}=(\bar{\mu}^{D\backslash Z}+\sum_i(\bar{\alpha}^{(i)}+\bar{\beta}^{(i)}))_+$. So $\gamma_1$ is supported on $Z\cup([1.26,2.26]\times\{2.26\})\cup(\{2.26\}\times[1.26,2.26])$. We define $\gamma_1^s$ as the Radon-Nikodym derivative of $\gamma_1$ w.r.t. the surface Lebesgue measure. It is easy to see that $\gamma_1^s(z)=\mu_s(z)+\sum_i(\alpha^{(i)}(z)+\beta^{(i)}(z))$ when $z\in(Z\cap D)\cup([1.26,2.26]\times\{2.26\})\cup(\{2.26\}\times[1.26,2.26])$, and zero otherwise. We now specify a transition probability kernel $\gamma(\cdot~|~x)$ for all $x$ in the support of $\gamma_1$.
\begin{figure}[t]
\centering
\begin{tabular}{cc}
\subfloat[]{\label{fig:alpha}\begin{tikzpicture}[scale=0.2,font=\small,axis/.style={very thick, -}]
\draw [axis,thick,<->] (-1,7)--(17,7);
\node [right] at (17,6) {$t$};
\draw [axis,thick,<->] (0,0)--(0,15);
\node [above] at (-1,16) {$\alpha^{(1)}(1.26+t,2.26)$};
\draw [thick,-] (0,1)--(10.67,13);
\draw [thick,-] (10.67,13)--(10.67,7);
\node at (-0.5,6) {$0$};
\draw [thin,-] (5.33,6.75) -- (5.33,7.25);
\node [below,rotate=90] at (5.33,5.5) {$1/3$};
\draw [thin,-] (10.67,6.75) -- (10.67,7.25);
\node [below,rotate=90] at (10.67,5.5) {$2/3$};
\draw [thin,-] (16,6.75) -- (16,7.25);
\node [rotate=90] at (16,6.25) {$1$};
\draw [thin,-] (-0.25,1) -- (0.25,1);
\node at (-2,1) {$-1$};
\draw [thin,-] (-0.25,13) -- (0.25,13);
\node at (-2,13) {$1$};
\end{tikzpicture}}&
\subfloat[]{\label{fig:beta}\begin{tikzpicture}[scale=0.2,font=\small,axis/.style={very thick, -}]
\draw [axis,thick,<->] (-1,7)--(17,7);
\node [right] at (17,6) {$t$};
\draw [axis,thick,<->] (0,0)--(0,15);
\node [above] at (-1,16) {$\beta^{(1)}(1.26+t,2.26)$};
\draw [thick,-] (10.67,13)--(10.9,13.3);
\draw [thin,-] (10.9,13.3)--(10.9,5.72);
\draw [thick,-] (10.9,5.72)--(16,7.66);
\node at (-0.5,6) {$0$};
\draw [thin,-] (10.67,6.75) -- (10.67,7.25);
\node [rotate=90] at (10.3,5.5) {$2/3$};
\draw [thin,-] (10.9,6.75) -- (10.9,7.25);
\node [below,rotate=90] at (10.9,4.5) {$43/63$};
\draw [thin,-] (16,6.75) -- (16,7.25);
\node [rotate=90] at (16,6.25) {$1$};
\draw [thin,-] (-0.25,13) -- (0.25,13);
\node at (-2,11.67) {$1$};
\end{tikzpicture}}
\end{tabular}
\caption{(a) The measure $\alpha^{(1)}$. (b) The measure $\beta^{(1)}$.}
\end{figure}
\begin{enumerate}
\item[(a)] For $x\in Z$, we define $\gamma(y~|~x)=\delta_x(y)$. This is interpreted as no mass being transferred.
\item[(b)] For $x\in([1.26,2.26]\times\{2.26\})\cup(\{2.26\}\times[1.26,2.26])$, we define $\gamma(y~|~x)=(\mu(y)+\mu_s(y))_-/\gamma_1^s(x)$ if $y\in\{y\in D\backslash Z:y_1-y_2=x_1-x_2\}$, and zero otherwise. This is interpreted as a transfer of $\gamma_1^s(x)$ from the boundary point $x$ to (the $45^\circ$ line segment) $\{y\in D\backslash Z:y_1-y_2=x_1-x_2\}$, which has $x$ as one end-point.
\end{enumerate}
We then define $\gamma(F)=\int_{(x,y)\in F}\gamma_1(dx)\gamma(dy~|~x)$ for any measurable $F\in D\times D$. It is now easy to check that $\gamma_2^Z=\bar{\mu}^Z$, and $\gamma_2^{D\backslash Z}=(\bar{\mu}^{D\backslash Z}+\sum_i(\bar{\alpha}^{(i)}+\bar{\beta}^{(i)}))_-$. Thus we have $(\gamma_1-\gamma_2)^Z=0$, and $(\gamma_1-\gamma_2)^{D\backslash Z}=\bar{\mu}^{D\backslash Z}+\sum_i(\bar{\alpha}^{(i)}+\bar{\beta}^{(i)})$.
We now verify that $\gamma$ is feasible. Observe that the components of $\bar{\mu}^Z$ are positive only at the left-bottom corner of $D$ (i.e., at $(c,c)$) and negative elsewhere, and that $\bar{\mu}_+(Z)=1=\bar{\mu}_-(Z)$ (the second equality requires some calculations). So we have $\int_Z f\,d\bar{\mu}\leq 0$ for any increasing function $f$, and thus $\bar{\mu}^Z\preceq_{cvx}0=(\gamma_1-\gamma_2)^Z$. We next prove that $(\gamma_1-\gamma_2)^{D\backslash Z}\succeq_{cvx}\bar{\mu}^{D\backslash Z}$. Since $(\gamma_1-\gamma_2-\bar{\mu})^{D\backslash Z}=\sum_i(\bar{\alpha}^{(i)}+\bar{\beta}^{(i)})$, it suffices to prove that $\sum_i(\bar{\alpha}^{(i)}+\bar{\beta}^{(i)})\succeq_{cvx}0$. We do this in the next lemma.
\begin{lemma}\label{lem:cvx}
\begin{enumerate}
\item[(i)] The measure $\bar{\alpha}^{(1)}$ is such that $\bar{\alpha}^{(1)}([1.26,1.26+2/3]\times\{2.26\})=0$ and $\int_{1.26}^{1.26+2/3}(t-1.26)\,\bar{\alpha}^{(1)}(dt,2.26)\geq 0$. Hence for any $f$ constant on $[1.26,1.26+2/3]$, we have $\int_{1.26}^{1.26+2/3}f(t)\,d\bar{\alpha}^{(1)}(dt,2.26)=0$. Further, $\bar{\alpha}^{(1)}\succeq_{cvx}0$. A similar result holds for $\bar{\alpha}^{(2)}$.
\item[(ii)] $\bar{\beta}^{(1)}([1.26+2/3,2.26]\times\{2.26\})=0$ and $\int_{1.26+2/3}^{2.26}(t-1.26)\,\bar{\beta}^{(1)}(dt,2.26)=0$. Hence we have $\int_{1.26+2/3}^{2.26}f(t)\,\bar{\beta}^{(1)}(dt,2.26)=0$ for any affine $f$ on $[1.26+2/3,2.26]$. Further, $\bar{\beta}^{(1)}\succeq_{cvx}0$. A similar result holds for $\bar{\beta}^{(2)}$.
\end{enumerate}
\end{lemma}
\begin{proof}
See \ref{app:a}.\qed
\end{proof}
We have thus established that $\gamma_1-\gamma_2\succeq_{cvx}0$. We now verify if $u$ and $\gamma$ satisfy the conditions in Lemma \ref{lem:compslack}.
\begin{multline*}
\int_Du\,d(\gamma_1-\gamma_2)=\int_Zu\,d(\gamma_1-\gamma_2)+\int_{D\backslash Z}u\,d(\gamma_1-\gamma_2)\\=\int_{D\backslash Z}u\,d\left(\bar{\mu}+\sum_i(\bar{\alpha}^{(i)}+\bar{\beta}^{(i)})\right)=\int_{D\backslash Z}u\,d\bar{\mu}=\int_Du\,d\bar{\mu},
\end{multline*}
where the second equality holds because $(\gamma_1-\gamma_2)^Z=0$; the third equality holds because $u(z)$ is a constant when $z\in([1.26,1.26+2/3]\times\{2.26\})\cup(\{2.26\}\times[1.26,1.26+2/3])$, and $u(z)$ is affine when $z\in([1.26+2/3,2.26]\times\{2.26\})\cup(\{2.26\}\times[1.26+2/3,2.26])$; and the last equality holds because $u(z)=0$ when $z\in Z$. To see why $u(z)-u(z')=\|z-z'\|_\infty$ holds $\gamma$-a.e., it suffices to check this condition for those $(z,z')$ for which $\gamma(\cdot~|~z)$ is nonzero, as in the cases (a) and (b) above. For $z,z'$ in (a), $z=z'$ and hence $u(z)-u(z')=0$; in (b), $(z,z')$ lie on a $45^\circ$ line, and hence $u(z)-u(z')=(z_1-z_1')=(z_2-z_2')=\|z-z\|_\infty$. Thus $u(z)-u(z')=\|z-z'\|_\infty$ holds $\gamma$-a.e.\qed
\end{proof}
The dual measure $\gamma$ was defined so that the measure $\gamma_1-\gamma_2-\bar{\mu}$, called the shuffling measure, convex-dominates $0$. Our key challenge in computing the optimal mechanism lies in constructing the shuffling measure. In the next example, we use a significantly different shuffling measure.
\subsection{Example 2: $z\sim\mbox{Unif }[1.5,2.5]^2$}
\begin{theorem}\cite{Pav11}\label{thm:eg-2}
Consider the case when $c=1.5$, and $b_1=b_2=1$. Then, the optimal mechanism is as depicted in Figure \ref{fig:illust-2}, with $\delta_1'=\delta_2'=\sqrt{5/3}-1$.
\end{theorem}
We use the shuffling measure $\bar{\lambda}+\sum_i(\bar{\alpha}^{(i)}+\bar{\beta}^{(i)})$, defined as follows. We define $\alpha^{(i)}$ and $\beta^{(i)}$, the respective Radon-Nikodym derivatives of the measures $\bar{\alpha}^{(i)}$ and $\bar{\beta}^{(i)}$ w.r.t. the surface Lebesgue measure, as in (\ref{eqn:alpha}) and (\ref{eqn:beta}) respectively, but with $\delta_1=\delta_2=\frac{((3+\sqrt{33})/8)-1}{(27-3\sqrt{33})/32}>\delta_2'$ and $a=(27-3\sqrt{33})/32$. We define $\lambda:D\rightarrow\mathbb{R}$, the Radon-Nikodym derivative of the measure $\bar{\lambda}$ w.r.t. the surface Lebesgue measure, as follows (see Figure \ref{fig:measure}):
\begin{align}
\lambda(c&+(t-1+\delta_2)/2,c+\delta_2-(t-1+\delta_2)/2)\nonumber\\&=\lambda(c+\delta_2-(t-1+\delta_2)/2,c+(t-1+\delta_2)/2)\label{eqn:shuffle-eg-2}\\&=\begin{cases}3a(t-1+\delta_2)+c&t\in[1-\delta_2,1-\delta_2'],\\3t(a-1/2)+3/2(1-\delta_2')-3a(1-\delta_2)&t\in[1-\delta_2',1].\nonumber\end{cases}
\end{align}
$\lambda$ is defined to be $0$ at every other point in $D$. Observe that the function is defined on the line $z_1+z_2=2c+\delta_2$, and thus is symmetric about the line $z_1=z_2$.
\begin{figure}[t]
\centering
\begin{tikzpicture}[scale=0.2,font=\small,axis/.style={very thick, -}]
\draw [axis,thick,<->] (-1,6)--(16.5,6);
\node [right] at (16,5) {$t$};
\draw [axis,thick,<->] (0,-1)--(0,13);
\node [above] at (1,16.5) {\tiny L: $\lambda(c+(t-1+\delta_2)/2,c+\delta_2-(t-1+\delta_2)/2)$};
\draw [thick,-] (2,9)--(4,10);
\node [above] at (1,14.75) {\tiny R: $\lambda(c+\delta_1+(t-1+\delta_1)/2,c-(t-1+\delta_1)/2)$};
\draw [thin,-] (4,10)--(4,7);
\draw [thick,-] (4,7)--(8,3);
\draw [thick,-] (8,3)--(12,7);
\draw [thin,-] (12,7)--(12,10);
\draw [thick,-] (12,10)--(14,9);
\draw [thin,-] (2,5.75) -- (2,6.25);
\node [rotate=90] at (2,3) {$(1-\delta_2)$};
\draw [thin,-] (4,5.75) -- (4,6.25);
\node [rotate=90] at (4,3) {$(1-\delta_2')$};
\draw [thin,-] (8,5.75) -- (8,6.25);
\node [rotate=90] at (8,5.25) {$1$};
\draw [thin,-] (12,5.75) -- (12,6.25);
\node [rotate=90] at (12,3) {$(1-\delta_1')$};
\draw [thin,-] (14,5.75) -- (14,6.25);
\node [rotate=90] at (14,3) {$(1-\delta_1)$};
\draw [thin,-] (-0.25,9) -- (0.25,9);
\node at (-1,9) {$c$};
\draw [thin,dotted] (8,-1)--(8,14);
\node at (4,11.5) {Left};
\node at (12,11.5) {Right};
\end{tikzpicture}
\caption{The measure $\lambda$. The $y$-axis expressions for the left and the right portions of the graph are indicated using $L$ and $R$. The measure is symmetric because we have $\delta_1=\delta_2$.}\label{fig:measure}
\end{figure}
We construct the dual measure using $\bar{\lambda}+\sum_i(\bar{\alpha}^{(i)}+\bar{\beta}^{(i)})$ as the shuffling measure. Observe that the shuffling measure has a significantly different structure compared to Example-1. For a detailed proof of the theorem, we refer the reader to \ref{app:a}.
The results of Theorems \ref{thm:eg-1} and \ref{thm:eg-2} are parts of a more general result shown in \cite{Pav11}. Pavlov's proof uses a virtual valuation method, but our proofs use the dual approach. We now solve another example via the dual approach, going beyond those considered in \cite{Pav11}.
\subsection{Example 3: $z\sim\mbox{Unif }[0,1.2]\times[0,1]$}
\begin{theorem}\label{thm:eg-3}
Consider the case when $c=0$, $b_1=1.2$, and $b_2=1$. Then, the optimal mechanism is as in Figure \ref{fig:illust-3}, with $(\delta_1,\delta_2)$ simultaneously solving
\begin{align*}
&-3\delta_1\delta_2-c(\delta_1+\delta_2)+b_1b_2=0.\\
&-\frac{3}{2}\delta_2^2+2b_2\delta_2-\frac{b_2^2}{2}+(c-2b_2+3\delta_2)\delta_1=0.
\end{align*}
The values of $(\delta_1,\delta_2)$ can be solved numerically to be
$$
(\delta_1,\delta_2)\approx(0.678837,0.589243).
$$
\end{theorem}
We construct the shuffling measure $\bar{\alpha}+\bar{\alpha}^{(o)}+\bar{\alpha}^{(h)}$ as follows, using its respective Radon-Nikodym derivatives $\alpha$, $\alpha^{(o)}$, $\alpha^{(h)}$ w.r.t. the surface Lebesgue measure. The superscripts $(o)$ and $(h)$ stand for 'oblique' and 'horizontal'.
\begin{align}
\alpha(c+t,c+t'):&=\frac{1}{1.2}\begin{cases}3t-1&(t,t')\in[0,1-\delta_2]\times\{1\},\\3(1-\delta_2)-c-1&(t,t')\in[1-\delta_2,1+\delta^*]\times\{1\},\\0&\mbox{otherwise}.\end{cases}\label{eqn:shuffle-eg-3}\\
\alpha^{(o)}(c+t,c+t'):&=\frac{1}{1.2}\begin{cases}3t-1.2&(t,t')\in\{1.2\}\times[0,1.2-\delta_1],\\2(1.2)-3\delta_1&(t,t')\in\{1.2\}\times[1.2-\delta_1,1],\\0&\mbox{otherwise}.\end{cases}\label{eqn:shuffle-eg-3-o}\\
\alpha^{(h)}(c+t,c+t'):&=\frac{1}{1.2}\begin{cases}3(t-\delta_1+0.2)&(t,t')\in\{1.2\}\times[\delta_1-0.2,\delta_2],\\3(0.2-\delta^*)&(t,t')\in\{1.2\}\times[\delta_2,2/3],\\0&\mbox{otherwise}.\end{cases}\label{eqn:shuffle-eg-3-h}
\end{align}
We construct the dual measure using $\bar{\alpha}+\bar{\alpha}^{(o)}+\bar{\alpha}^{(h)}$ as the shuffling measure. For a detailed proof of Theorem \ref{thm:eg-3}, see \ref{app:a}. In point (d) of that proof, mass from certain points on the right-hand side boundary will be transferred to two line segments -- a $45^\circ$ line (oblique transfer via $\alpha^{(o)}$) and a horizontal line (via $\alpha^{(h)}$). Observe that the shuffling measure has a significantly different structure compared to Examples 1 and 2.
\begin{minipage}{.46\textwidth}
\begin{figure}[H]
\centering
\begin{tikzpicture}[scale=0.2,font=\tiny,axis/.style={very thick, -}]
\node at (-1,-1) {$(0,0)$};
\draw [axis,thick,-] (0,0)--(16.5,0);
\node [right] at (15,-1) {$(b_1,0)$};
\draw [axis,thick,-] (0,0)--(0,15);
\node [above] at (0,15) {$(0,b_2)$};
\draw [axis,thick,-] (0,15)--(16.5,15);
\draw [axis,thick,-] (16.5,0)--(16.5,15);
\draw [axis,thick,-] (0,8.8)--(9.5,8.8);
\node [rotate=90] at (-1,9) {$(0,\delta_2)$};
\draw [axis,thick,-] (9.5,0)--(9.5,8.8);
\node at (9.5,-1) {$(\delta_1,0)$};
\draw [thick,-] (9.5,8.8)--(15.7,15);
\draw [thick,dotted] (9.5,8.8)--(0.7,0);
\node [right] at (0.2,-1) {$(\delta^*,0)$};
\node at (14.4,16) {$(b_2+\delta^*,b_2)$};
\node [above] at (5,3) {\small$(0,0)$};
\node [above] at (13,6) {\small$(1,0)$};
\node [above] at (6,10) {\small$(0,1)$};
\end{tikzpicture}
\caption{Optimal mechanism when $c=0$, $b_1=1.2$, $b_2=1$.}\label{fig:illust-3}
\end{figure}
\end{minipage}
\begin{minipage}{.02\textwidth}
\hspace*{.02\textwidth}
\end{minipage}
\begin{minipage}{.46\textwidth}
\begin{figure}[H]
\centering
\begin{tikzpicture}[scale=0.2,font=\small,axis/.style={very thick, -}]
\draw [axis,thick,<->] (-1,7)--(17,7);
\node [right] at (17,6) {$t$};
\draw [axis,thick,<->] (0,0)--(0,15);
\node [above] at (-1,16) {$\left(\alpha^{(o)}+\alpha^{(h)}\right)(1.2,t)$};
\draw [thick,-] (0,1)--(7.7,8.17);
\draw [thick,-] (7.7,8.17)--(8.3,9.42);
\draw [thick,-] (8.3,9.42)--(9.4,10.47);
\draw [thick,-] (9.4,10.47)--(10.67,10.47);
\draw [thin,-] (10.67,10.47)--(10.67,8.8);
\draw [thick,-] (10.67,8.8)--(16,8.8);
\node at (-0.5,6) {$0$};
\draw [thin,-] (7.7,6.75) -- (7.7,7.25);
\node [rotate=90] at (7.15,4) {$\delta_1-0.2$};
\draw [thin,-] (8.3,6.75) -- (8.3,7.25);
\node [rotate=90] at (8.4,4) {$b_1-\delta_1$};
\draw [thin,-] (9.4,6.75) -- (9.4,7.25);
\node [rotate=90] at (9.75,5.5) {$\delta_2$};
\draw [thin,-] (10.67,6.75) -- (10.67,7.25);
\node [below,rotate=90] at (10.67,5.5) {$2/3$};
\draw [thin,-] (16,6.75) -- (16,7.25);
\node [rotate=90] at (16,6.25) {$1$};
\draw [thin,-] (-0.25,1) -- (0.25,1);
\node at (-2,1) {$-1.2$};
\end{tikzpicture}
\caption{The measure $\alpha^{(o)}+\alpha^{(h)}$, for $c=0$, $b_1=1.2$, $b_2=1$.}\label{fig:measure2}
\end{figure}
\end{minipage}
\vspace*{10pt}
We have computed the optimal mechanisms for three representative examples using the dual approach. The challenge in each of the examples was to construct the appropriate shuffling measure $\gamma_1-\gamma_2-\bar{\mu}$ that convex-dominates $0$. We now make some observations on the constructed shuffling measures.
\begin{enumerate}
\item[$\bullet$] The locations of the shuffling measure exhibit significant variations in our examples. For instance, the shuffling measure was non-zero only at the top boundary and the right boundary of $D$ in Theorems \ref{thm:eg-1} and \ref{thm:eg-3}, whereas, it was non-zero additionally on the line $z_1+z_2=2c+\delta_2$ in Theorem \ref{thm:eg-2}.
\item[$\bullet$] The structures of the shuffling measure also exhibit significant variations. The variations can be observed from the structures in Figures \ref{fig:measure} and \ref{fig:measure2}. This is in contrast to the unrestricted setting solved in \cite{TRN16}, where the shuffling measures were added at a fixed location and had a fixed structure.
\item[$\bullet$] In the case of $c=0, b_1=1.2, b_2=1$, the shuffling measure had to be constructed partly for a mass transfer along the $45^\circ$ line segment, and partly for a transfer along the horizontal line segment (see point (d) in the proof of Theorem \ref{thm:eg-3}, \ref{app:a}). The example thus had two shuffling measures: $\bar{\alpha}^{(o)}$ and $\bar{\alpha}^{(h)}$.
\end{enumerate}
The variability in the examples above makes it difficult for us to arrive at a general algorithmic method to construct shuffling measures, even for the restricted setting of uniform distributions. This motivates us to tackle the general problem using the virtual valuation method in \cite{Pav11}.
\section{Exploring The Virtual Valuation Method}
Recall that we consider the problem of optimal mechanism design in a two-item, one-buyer, unit-demand setting. In this section, we compute the optimal mechanism when the buyer's valuation $z\sim\mbox{Unif}[c,c+b_1]\times[c,c+b_2]$, using the virtual valuation method in \cite{Pav11}. We start with the following general result from \cite{Pav11}.
\begin{theorem}\cite[Prop.~1]{Pav11}\label{thm:pav-1}
If the distribution $f$ satisfies
$$
3f_1(z)f_2(z)+z_1f_1'(z)f_2(z)+z_2f_1(z)f_2'(z)\geq 0\,\forall z\in D,
$$
then the allocation function $q$ in the optimal mechanism is such that $q_1+q_2\in\{0,1\}$.
\end{theorem}
Thus, if $f$ satisfies the above sufficient condition, then for every $z\in D\backslash Z$, $q(z)$ satisfies $q_1(z)+q_2(z)=1$. Recall that $Z$ is the exclusion region. Observe that the sufficient condition in Theorem \ref{thm:pav-1} is clearly satisfied for the uniform distribution $\mbox{Unif}[c,c+b_1]\times[c,c+b_2]$. The utility of the buyer in $D\backslash Z$ can be written as $u(z)=(z_1-z_2)q_1(z)+z_2-t(z)$, where we have used $q_2=1-q_1$. Defining $\delta:=z_1-z_2$, we have $\delta\in[-b_2,b_1]$ for the case under consideration. The following theorem from \cite{Pav11} reduces the domains of $q$ and $t$ from two-dimensions to one-dimension.
\begin{theorem}\cite[Prop.~2]{Pav11}\label{thm:pav-2}
In the optimal mechanism, the allocations and the payments, $(q,t)$, can be rewritten so that they are a constant for every $\{z\in D\backslash Z:z_1-z_2=\delta\}$.
\end{theorem}
The theorem indicates that if $Z$ is fixed, then the domains of $(q,t)$ become one-dimensional in the region $D\backslash Z$; they can be written as $t(\delta)$ and $q_1(\delta)$, where $t:[-b_2,b_1]\rightarrow\mathbb{R}_+$, $q_1:[-b_2,b_1]\rightarrow[0,1]$, and $q_2=1-q_1$. As done in \cite{Pav11}, define $u_1:[-b_2,b_1]\rightarrow\mathbb{R}$, $u_1(\delta):=\delta q_1(\delta)-t(\delta)$, and define
\[
g(u_1(\delta),\delta):=\int_{\substack{z:z_1-z_2=\delta,\\u_1(\delta)+z_2>0}}f(z)\,dz.
\]
The function $g(u_1(\delta),\delta)$ resembles the marginal of $f$ along the $z_1-z_2$ axis, but for the fact that the marginal is computed by integrating only up to the point where $u_1(z_1-z_2)+z_2=0$. Call this point $z_2^*(\delta)$, and observe that $u(\delta+z_2^*(\delta),z_2^*(\delta))=u_1(\delta)+z_2^*(\delta)=0$. So $z_2=z_2^*(\delta)$ is the boundary point between the exclusion region $Z$, and the other regions. Further, $\{z:z_1-z_2=\delta,\,z_2<z_2^*(\delta)\}$ belongs to $Z$. So the function $g(u_1(\delta),\delta)$ is actually the marginal of $f$ in $D\backslash Z$, along the $z_1-z_2$ axis.
Consider the problem of maximizing the expected revenue subject to IC and IR constraints. The IC constraint, from \cite[Lem.~2]{Mye81}, can equivalently be written as (i) $q_1$ increasing, and (ii) $u_1(\delta)$ has the representation $u_1(\delta)=u_1(-b_2)+\int_{-b_2}^{\delta}q_1(\tilde{\delta})\,d\tilde{\delta}$ for every $\delta\in[-b_2,b_1]$. The optimal mechanism can thus be computed by solving the following optimization problem.
\begin{align}\label{eqn:optim-myerson}
&\max_{q_1(\cdot),u_1(\cdot)}\int_{-b_2}^{b_1}(\delta q_1(\delta)-u_1(\delta))g(u_1(\delta),\delta)\,d\delta\\
\mbox{subject to}\hspace*{0.5in}
&(a)\,q_1(\delta)\in[0,1]\,\forall\delta\in[-b_2,b_1];\,q_1\mbox{ increasing};\nonumber\\
&(b)\,u_1(\delta)=u_1(-b_2)+\int_{-b_2}^{\delta}q_1(\tilde{\delta})\,d\tilde{\delta}\,\forall\delta\in[-b_2,b_1].\nonumber
\end{align}
The IR constraint is already taken into account because the integral in the objective function of (\ref{eqn:optim-myerson}) is over $D\backslash Z$, i.e., where $u(z)\geq 0$.
Observe that the problem (\ref{eqn:optim-myerson}) is similar to the optimization problem in \cite[Lem.~3]{Mye81}. To solve the problem in a similar way, we now search for an equivalent of the virtual valuation function $\phi$ in our setting.
Applying integration by parts to the objective function of (\ref{eqn:optim-myerson}), we get $\int_{-b_2}^{b_1}\bar{V}(\delta)q_1(\delta)\,d\delta$, where the marginal profit function $\bar{V}:[-b_2,b_1]\rightarrow\mathbb{R}$ is defined as\footnote{We use the term marginal profit function, see \citet{Pav11}, based on the fact that $\bar{V}$ denotes the marginal contribution of allocation $q_1(\delta)$ to the profit of the seller.}
$$
\bar{V}(\delta):=\delta g(u_1(\delta),\delta)-\int_{\delta}^{b_1}g(u_1(\tilde{\delta}),\tilde{\delta})\,d\tilde{\delta}+\int_{\delta}^{b_1}(\tilde{\delta} q_1(\tilde{\delta})-u_1(\tilde{\delta}))\frac{\partial}{\partial u_1}g(u_1(\tilde{\delta}),\tilde{\delta})\,d\tilde{\delta}.
$$
Notice that in Myerson's setting, we have $g(u_1(\delta),\delta)=f(\delta)$, and thus $\bar{V}(\delta)=\delta f(\delta)-\int_{\delta}^{b_1}f(\tilde{\delta})\,d\tilde{\delta}=\phi(\delta)f(\delta)$. We thus expect $\bar{V}$ to have similar properties of $\phi$. The following result from \cite{Pav06} provides some ``ironing conditions'' on $\bar{V}$, similar to those on $\phi$ in Myerson's setting.
\begin{theorem}\cite[Lem.~3,~Prop.~5]{Pav06}\label{thm:Myerson}
A mechanism is optimal if and only if it satisfies the following conditions:
\begin{enumerate}
\item $q_1(\delta)$ is strictly increasing on $(\delta',\delta'')$ if and only if (iff) $\bar{V}(\delta)=0$ on this interval.
\item $q_1(\delta)=0$ for $\delta\in[\delta',\delta'']$ iff (a) $\delta'=-b_2$, (b) $\bar{V}(\delta'')=0$ unless $\delta''=b_1$, (c) $\int_{\delta'}^{\delta''}\bar{V}(\delta)\,d\delta=\underline{k}\leq 0$, and (d) $\int_{\delta'}^x\bar{V}(\delta)\,d\delta\geq\underline{k}$ for all $x\in[\delta',\delta'']$.
\item $q_1(\delta)=q\in(0,1)$ for $\delta\in[\delta',\delta'']$ iff (a) $\bar{V}(\delta')=0$ unless $\delta'=-b_2$, (b) $\bar{V}(\delta'')=0$ unless $\delta''=b_1$, (c) $\int_{\delta'}^{\delta''}\bar{V}(\delta)\,d\delta=0$, and (d) $\int_{\delta'}^x\bar{V}(\delta)\,d\delta\geq0$ for all $x\in[\delta',\delta'']$.
\item $q_1(\delta)=1$ for $\delta\in[\delta',\delta'']$ iff (a) $\bar{V}(\delta')=0$ unless $\delta'=-b_2$, (b) $\delta''=b_1$, (c) $\int_{\delta'}^{\delta''}\bar{V}(\delta)\,d\delta=\overline{k}\geq 0$, and (d) $\int_x^{\delta''}\bar{V}(\delta)\,d\delta\leq\overline{k}$ for all $x\in[\delta',\delta'']$.
\end{enumerate}
\end{theorem}
\begin{figure}[H]
\centering
\begin{tikzpicture}[scale=0.2,font=\small,axis/.style={very thick, -}]
\draw [axis,thick,->] (-1,0)--(17,0);
\node [right] at (17,-1) {$\delta$};
\draw [thin,-] (1,-0.5)--(1,0.5);
\node at (2,-1) {$-b_2$};
\draw [thin,-] (15,-0.5)--(15,0.5);
\node at (15,-1) {$b_1$};
\draw [axis,thick,->] (0,-1)--(0,8);
\node [above] at (-1,8) {$V_1(\delta)$};
\draw [axis,thick,-] (1,0) to[out=80,in=180] (3,4);
\draw [axis,thick,-] (3,4) to[out=0,in=180] (3.5,3.5);
\draw [axis,thick,-] (3.5,3.5) to[out=0,in=180] (5,7);
\draw [axis,thick,-] (5,7) to[out=0,in=180] (6,5.5);
\draw [axis,thick,-] (6,5.5) to[out=0,in=180] (7,7);
\draw [axis,thick,-] (7,7) to[out=0,in=180] (7.5,6.25);
\draw [axis,thick,-] (7.5,6.25) to[out=0,in=180] (8,6.75);
\draw [axis,thick,-] (8,6.75) to[out=0,in=180] (8.5,6);
\draw [axis,thick,-] (8.5,6) to[out=0,in=180] (9,7);
\draw [axis,thick,-] (9,7) to[out=0,in=180] (11,7);
\draw [axis,thick,-] (11,7) to[out=0,in=180] (13,4);
\draw [axis,thick,-] (13,4) to[out=0,in=180] (13.5,4.5);
\draw [axis,thick,-] (13.5,4.5) to[out=-10,in=100] (15,1);
\draw [thin,dotted] (1,7)--(15,7);
\node at (3,8) {A};
\draw [thin,dotted] (5,1)--(5,9);
\node at (6,8) {B};
\draw [thin,dotted] (7,1)--(7,9);
\node at (8,8) {C};
\draw [thin,dotted] (9,1)--(9,9);
\node at (10,8) {D};
\draw [thin,dotted] (11,1)--(11,9);
\node at (13,8) {E};
\node [right] at (20,8) {A: $q_1=0$};
\node [right] at (20,6) {B: $q_1=\bar{q}_1\in(0,1)$};
\node [right] at (20,4) {C: $q_1=\tilde{q}_1>\bar{q}_1$};
\node [right] at (20,2) {D: $q_1>\tilde{q}_1$ increasing};
\node [right] at (20,0) {E: $q_1=1$};
\end{tikzpicture}
\caption{Illustration of the conditions in Theorem \ref{thm:Myerson}.}\label{fig:my-illust-1}
\end{figure}
Define $V_1(\delta)=-\int_{-b_2}^{\delta}\bar{V}(\tilde{\delta})\,d\tilde{\delta}$. We now argue that the conditions in Theorem \ref{thm:Myerson} can be interpreted as conditions on $\delta$ where $V_1$ attains its global maximum. The theorem states that the mechanism is optimal if and only if the following conditions hold. Take $\delta'$ and $\delta''$ to be the left and right end points of an interval under consideration.
\begin{itemize}
\item Let $q_1(\delta)=0$ $\forall$ $\delta\in[\delta',\delta'']$. Then (a) $\delta'=-b_2$ and (b) $V_1(\delta)$ is maximized at $\delta''$ (see region A, Figure \ref{fig:my-illust-1}).
\item Let $q_1(\delta)=q\in(0,1)$ when $\delta\in[\delta',\delta'']$. Then $V_1(\delta)$ is maximized at both $\delta'$ and $\delta''$ (see regions B and C, Figure \ref{fig:my-illust-1}).
\item Let $q_1(\delta)$ be strictly increasing when $\delta\in[\delta',\delta'']$. Then $V_1(\delta)=\max_\delta V_1(\delta)$ for all $\delta\in[\delta',\delta'']$ (see region D, Figure \ref{fig:my-illust-1}).
\item Let $q_1(\delta)=1$ $\forall$ $\delta\in[\delta',\delta'']$. Then (a) $\delta''=b_1$ and (b) $V_1(\delta)$ is maximized at $\delta'$ (see region E, Figure \ref{fig:my-illust-1}).
\end{itemize}
Observe that the conditions mentioned above are a consequence of the conditions stated in Theorem \ref{thm:Myerson}. The conditions 2(c)--(d), 3(c)--(d), and 4(c)--(d), are representations that indicate that the global maximum must occur at certain end points of the interval. The value of $q_1$ changes only at those $\delta$ where $V_1$ attains its global maximum. We have a similar result in one-dimension, where the value of $q$ changes only at those $z$ where $-\int_0^{z}\phi(t)f(t)\,dt=z(1-F(z))$ is maximized \cite[p.~338]{NRTV07}.
Theorem \ref{thm:Myerson} and the above interpretation highlight the similarity between the virtual valuation functions $\phi$ and $\bar{V}$. The key difference between $\phi$ and $\bar{V}$ is that the former depends only on $f$, whereas the latter depends on $u_1(\delta)$ in addition, which is known only when the optimal mechanism is known. So the computation of $\bar{V}$ requires the knowledge of the mechanism itself. However, given a mechanism, we can use the theorem to determine if the mechanism is optimal or not.
We now simplify the computation of the marginal profit function. We define virtual valuation function $V:[-b_2,b_1]\rightarrow\mathbb{R}$ as $V(\delta):=\bar{\mu}(\{z:z_1-z_2\geq\delta\}\backslash Z)$ where $\bar{\mu}$ is as defined in Section \ref{sec:prelim}. We then have $\bar{\mu}(D)=0$ (see (\ref{eqn:mu-D})). The following lemma shows that $V$ is equal to the marginal profit function $\bar{V}$.
\begin{lemma}\label{lem:V-V'}
Let the allocation function $q$ be such that there exists a $u:D\rightarrow\mathbb{R}$ with $\nabla u=q$. Then, the functions $V$ and $\bar{V}$ are one and the same.
\end{lemma}
\begin{proof}
See \ref{app:b}.\qed
\end{proof}
This lemma could be understood as follows.
\begin{itemize}
\item Recall that the expected revenue equals $\int_{-b_2}^{b_1}\bar{V}(\delta)q_1(\delta)\,d\delta$. The expected revenue thus increases by $\bar{V}(\delta)$ for a differential increase in $q_1$ at $\delta$.
\item A differential increase in $q_1$ increases $u$ uniformly for all $\delta'\geq\delta$, since $q=\nabla u$.
\item From (\ref{eqn:primal}), we know that the expected revenue equals $\int_D u\,d\bar{\mu}$. So a uniform increase for all $\delta'\geq\delta$ increases the expected revenue by $\bar{\mu}(\{z:z_1-z_2\geq\delta\}\backslash Z)$.
\item Thus we have $\bar{V}(\delta)=\bar{\mu}(\{z:z_1-z_2\geq\delta\}\backslash Z)$.
\end{itemize}
Observe that the virtual valuation function $V$ can be computed if the exclusion region $Z$ is known. In the rest of the paper, we propose some structures for all possible values of $(c,b_1,b_2)\geq 0$, and then prove that the optimal mechanisms indeed have those structures, using Theorem \ref{thm:Myerson}.
\subsection{Optimal mechanisms for the uniform distribution on a rectangle}
Without loss of generality, we assume $b_1\geq b_2$. The following theorem asserts that the optimal mechanism falls within one of the structures depicted in Figures \ref{fig:a-ini}--\ref{fig:e'-ini}.
\begin{theorem}\label{thm:consolidate}
Consider $z\sim\mbox{Unif }[c,c+b_1]\times[c,c+b_2]$. The optimal mechanism in the unit-demand setting is described as follows.
\begin{multicols}{2}
\begin{enumerate}
\item Case $b_1\in[b_2,3b_2/2]$:
\begin{enumerate}
\item[(a) ] $c\in[0,b_2]$: Figure \ref{fig:a-ini}
\item[(b)$^*$] $c\in[b_2,\alpha_1]$: Figure \ref{fig:b-ini}
\item[(c)$^*$] $c\in[\alpha_1,\alpha_2]$: Figure \ref{fig:c-ini}
\item[(d)$^*$] $c\in[\alpha_2,\frac{27b_1^2b_2^2}{4(b_1^3-b_2^3)}]$: Figure \ref{fig:d-ini}
\item[(e) ] $c\geq\frac{27b_1^2b_2^2}{4(b_1^3-b_2^3)}$: Figure \ref{fig:e-ini}
\end{enumerate}
\item Case $b_1\geq 3b_2/2$:
\begin{enumerate}
\item[(a) ] $c\in[0,b_2]$: Figure \ref{fig:a-ini}
\item[(b)$^*$] $c\in[b_2,\beta]$: Figure \ref{fig:b-ini}
\item[(c) ] $c\in[\beta,\frac{216b_1^2b_2}{108b_1^2-108b_1b_2-5b_2^2}]$: Figure \ref{fig:d'-ini}
\item[(d) ] $c\geq\frac{216b_1^2b_2}{108b_1^2-108b_1b_2-5b_2^2}$: Figure \ref{fig:e'-ini}
\end{enumerate}
\end{enumerate}
\end{multicols}
The values of $\alpha_1$, $\alpha_2$ and $\beta$ are defined as follows.
\begin{itemize}
\item $c=\alpha_1$ is obtained by solving the following equations simultaneously for $(c,h,\delta^*)$.
\begin{align}
&3h^2/2+ch+2b_2\delta^*-b_1b_2+b_2^2/2=0.\label{eqn:fig-b-first}\\
27(c+h+\delta^*)&(b_2+\delta^*)^2-4(4b_2+3\delta^*)(3(h+\delta^*)/2+c)^2=0.\label{eqn:fig-b-second}\\
2b_1^3/27&-(c+h)h^2/2+b_2(\delta^*)^2-b_2\delta^*(b_1-b_2/2)=0.\label{eqn:fig-b-third}
\end{align}
\item $c=\alpha_2$ is the solution obtained by solving (\ref{eqn:fig-b-second}) and the following equations simultaneously for $(c,h,\delta^*)$.
\begin{align}
&(2b_1^3/27+b_2(\delta^*)^2-b_2\delta^*(b_1-b_2/2))(3h/2+c)^2\nonumber\\&\hspace*{1.5in}-(c+h)(2b_2\delta^*+b_2^2/2-b_1b_2)^2/2=0.\label{eqn:fig-c-second}\\
&2b_1b_2(b_2^2+4b_2\delta^*-2c(\delta^*+h)-3h(2\delta^*+h))\nonumber\\&\hspace*{0.5in}-(b_2^2+4b_2\delta^*-3\delta^*h)(b_2^2+4b_2\delta^*-2c\delta^*-3\delta^*h)=0.\label{eqn:fig-c-third}
\end{align}
\item $c=\beta\geq b_2$ solves
\begin{equation}\label{eqn:menu-2-bound}
72b_1^2b_2+144b_1b_2^2-90b_2^3+(-36b_1^2+84b_1b_2+399b_2^2)c-(96b_1+208b_2)c^2=0.
\end{equation}
\end{itemize}
\end{theorem}
\begin{remark}
The starred portions in the theorem statement indicate that we used Mathematica to verify certain inequalities in proving those parts.
\end{remark}
\begin{remark}
The values of $\alpha_1$ fall in the interval $[b_2,tb_2]$, where $t=3(37+3\sqrt{465})/176\approx 1.733379$. Similarly, the values of $\alpha_2\in[kb_2,tb_2]$ where $k\geq 1$ is the root of $32k^3-54k^2+19=0$ ($k\approx 1.37214$), and the values of $\beta\in[tb_2,2b_2)$. See Figure \ref{fig:phase-diagram}.
\end{remark}
The following is a pictorial representation of the results in Theorem \ref{thm:consolidate}. It depicts the regions in $(c,b_1,b_2)$ space at which each of the mechanisms depicted in Figures 2a--2g turns out to be optimal.
\begin{figure}[H]
\centering
\begin{minipage}{.45\textwidth}
\centering
\begin{tikzpicture}[scale=0.36,font=\small,axis/.style={very thick, ->, >=stealth'}]
\draw [axis,thick,->] (0,-1)--(0,13);
\node [right] at (11,-1) {$\frac{b_1}{b_2}$};
\draw [axis,thick,->] (-0.5,0)--(12,0);
\node [above] at (1,11) {$\frac{c}{b_2}$};
\node at (-1,0.25) {$0$};
\draw [thin,-] (-0.25,2) -- (0.25,2);
\node [left] at (-0.25,2) {$2$};
\draw [thin,-] (-0.25,4) -- (0.25,4);
\node [left] at (-0.25,4) {$4$};
\draw [thin,-] (-0.25,6) -- (0.25,6);
\node [left] at (-0.25,6) {$6$};
\draw [thin,-] (-0.25,8) -- (0.25,8);
\node [left] at (-0.25,8) {$8$};
\draw [thin,-] (-0.25,10) -- (0.25,10);
\node [left] at (-0.25,10) {$10$};
\draw [thin,-] (-0.25,12) -- (0.25,12);
\node [left] at (-0.25,12) {$12$};
\node at (0.5,-1) {$1$};
\draw [thin,-] (2,-0.25) -- (2,0.25);
\node [below] at (2,-0.25) {$1.2$};
\draw [thin,-] (4,-0.25) -- (4,0.25);
\node [below] at (4,-0.25) {$1.4$};
\draw [thin,-] (6,-0.25) -- (6,0.25);
\node [below] at (6,-0.25) {$1.6$};
\draw [thin,-] (8,-0.25) -- (8,0.25);
\node [below] at (8,-0.25) {$1.8$};
\draw [thin,-] (10,-0.25) -- (10,0.25);
\node [below] at (10,-0.25) {$2$};
\draw [thick,-] (0,1) to (12,1);
\draw [thick,-] (0,1) to (5,1.733) to (12,1.97);
\draw [thick,-] (0,1.372) to (5,1.733);
\draw [thick,-] (2.2,12) to[out=-60,in=170] (12,3.98);
\draw [thick,dotted] (5,0) to (5,12);
\node [right] at (5,11) {$b_1=(1.5)b_2$};
\node at (2.2,8.5) {asymptotic};
\node at (2.2,7.5) {to $b_1=b_2$};
\node at (9,3.8) {asymptotic};
\node at (9,2.8) {to $c=2b_2$};
\node at (10,1.4) {$c=b_2$};
\draw [thick,->] (8.5,2) -- (8.5,2.6);
\draw [thick,->] (8.5,5) -- (8.5,4.2);
\path[fill=gray!50,opacity=.5] (0,1) to (12,1) to (12,0) to (0,0);
\end{tikzpicture}
\end{minipage}
\begin{minipage}{0.025\textwidth}
\hspace*{0.025\textwidth}
\end{minipage}
\begin{minipage}{0.45\textwidth}
\centering
\begin{tikzpicture}[scale=0.25,font=\normalsize,axis/.style={very thick, -}]
\node [rotate=45] at (-1,0) {$c$};
\node [rotate=45] at (0,-1) {$c$};
\draw [axis,thick,-] (0,0)--(15,0);
\node [rotate=45] at (15,-1.2) {$c+b_1$};
\draw [axis,thick,-] (0,0)--(0,12);
\node [rotate=45] at (-1,12) {$c+b_2$};
\draw [axis,thick,-] (0,12)--(15,12);
\draw [axis,thick,-] (15,0)--(15,12);
\draw [axis,thick,-] (0,5)--(6.5,5);
\draw [thin,-] (-0.4,5)--(0.4,5);
\node [rotate=45] at (-2,4) {$c+\delta_2$};
\draw [axis,thick,-] (6.5,0)--(6.5,5);
\draw [thin,-] (6.5,-0.4)--(6.5,0.4);
\node [rotate=45] at (6.5,-1.8) {$c+\delta_1$};
\draw [thick,-] (6.5,5)--(13.5,12);
\draw [thick,dotted] (6.5,5)--(1.5,0);
\draw [thin,-] (1.5,-0.4)--(1.5,0.4);
\node [rotate=45] at (1,-1.8) {$c+\delta^*$};
\draw [thin,-] (13.5,11.6)--(13.4,12.4);
\node at (12.5,13) {$c+b_2+\delta^*$};
\node [above] at (3.5,1.5) {$(0,0)$};
\node [above] at (11.5,4.5) {$(1,0)$};
\node [above] at (3,8) {$(0,1)$};
\end{tikzpicture}
\end{minipage}
\caption{When $(c,b_1,b_2)$ falls in the shaded region in the left, the optimal mechanism is as depicted in the right. Item $1$ is offered for a price of $c+\delta_1$, and item $2$ is offered for a price of $c+\delta_2$.}\label{fig:a-new}
\end{figure}
\begin{figure}[H]
\centering
\begin{minipage}{.45\textwidth}
\centering
\begin{tikzpicture}[scale=0.36,font=\small,axis/.style={very thick, ->, >=stealth'}]
\draw [axis,thick,->] (0,-1)--(0,13);
\node [right] at (11,-1) {$\frac{b_1}{b_2}$};
\draw [axis,thick,->] (-0.5,0)--(12,0);
\node [above] at (1,11) {$\frac{c}{b_2}$};
\node at (-1,0.25) {$0$};
\draw [thin,-] (-0.25,2) -- (0.25,2);
\node [left] at (-0.25,2) {$2$};
\draw [thin,-] (-0.25,4) -- (0.25,4);
\node [left] at (-0.25,4) {$4$};
\draw [thin,-] (-0.25,6) -- (0.25,6);
\node [left] at (-0.25,6) {$6$};
\draw [thin,-] (-0.25,8) -- (0.25,8);
\node [left] at (-0.25,8) {$8$};
\draw [thin,-] (-0.25,10) -- (0.25,10);
\node [left] at (-0.25,10) {$10$};
\draw [thin,-] (-0.25,12) -- (0.25,12);
\node [left] at (-0.25,12) {$12$};
\node at (0.5,-1) {$1$};
\draw [thin,-] (2,-0.25) -- (2,0.25);
\node [below] at (2,-0.25) {$1.2$};
\draw [thin,-] (4,-0.25) -- (4,0.25);
\node [below] at (4,-0.25) {$1.4$};
\draw [thin,-] (6,-0.25) -- (6,0.25);
\node [below] at (6,-0.25) {$1.6$};
\draw [thin,-] (8,-0.25) -- (8,0.25);
\node [below] at (8,-0.25) {$1.8$};
\draw [thin,-] (10,-0.25) -- (10,0.25);
\node [below] at (10,-0.25) {$2$};
\draw [thick,-] (0,1) to (12,1);
\draw [thick,-] (0,1) to (5,1.733) to (12,1.97);
\draw [thick,-] (0,1.372) to (5,1.733);
\draw [thick,-] (2.2,12) to[out=-60,in=170] (12,3.98);
\draw [thick,dotted] (5,0) to (5,12);
\node [right] at (5,11) {$b_1=(1.5)b_2$};
\node at (2.2,8.5) {asymptotic};
\node at (2.2,7.5) {to $b_1=b_2$};
\node at (9,3.8) {asymptotic};
\node at (9,2.8) {to $c=2b_2$};
\node at (10,1.4) {$c=b_2$};
\draw [thick,->] (8.5,2) -- (8.5,2.6);
\draw [thick,->] (8.5,5) -- (8.5,4.2);
\path[fill=gray!50,opacity=.5] (0,1) to (5,1.733) to (12,1.97) to (12,1) to (0,1);
\end{tikzpicture}
\end{minipage}
\begin{minipage}{0.025\textwidth}
\hspace*{0.025\textwidth}
\end{minipage}
\begin{minipage}{0.45\textwidth}
\centering
\begin{tikzpicture}[scale=0.25,font=\normalsize,axis/.style={very thick, -}]
\node [rotate=45] at (-1,0) {$c$};
\node [rotate=45] at (0,-1) {$c$};
\draw [axis,thick,-] (0,0)--(15,0);
\node [rotate=45] at (15,-1.2) {$c+b_1$};
\draw [axis,thick,-] (0,0)--(0,12);
\node [rotate=45] at (-1,12) {$c+b_2$};
\draw [axis,thick,-] (0,12)--(15,12);
\draw [axis,thick,-] (15,0)--(15,12);
\draw [axis,thick,-] (0,4)--(8,12);
\draw [thin,-] (-0.4,4)--(0.4,4);
\node [rotate=45] at (-2,3.5) {$c+b_2/3$};
\draw [axis,thick,-] (0,3.5)--(4,2.5);
\draw [thin,-] (-0.4,3.5)--(0.4,3.5);
\node [rotate=45] at (-1.8,1.8) {$c+\delta_2$};
\draw [axis,thick,-] (4,0)--(4,2.5);
\draw [thin,-] (4,-0.4)--(4,0.4);
\node [rotate=45] at (3.4,-1.8) {$c+\delta_1$};
\draw [thin,<->] (4.5,0)--(4.5,2.5);
\node at (5,1) {$h$};
\draw [thick,-] (4,2.5)--(13.5,12);
\draw [thick,dotted] (4,2.5)--(1.5,0);
\draw [thin,-] (1.5,-0.4)--(1.5,0.4);
\node [rotate=45] at (1,-1.8) {$c+\delta^*$};
\draw [thin,-] (13.5,11.6)--(13.5,12.4);
\node at (14.5,13) {$c+b_2+\delta^*$};
\draw [thin,-] (8,11.6)--(8,12.4);
\node at (7.25,13) {\footnotesize$c+2b_2/3$};
\node at (2,1.5) {$(0,0)$};
\node [above] at (11,4) {$(1,0)$};
\node [above] at (2,8) {$(0,1)$};
\node [rotate=45] at (6,7) {$(1-a_2,a_2)$};
\end{tikzpicture}
\end{minipage}
\caption{When $(c,b_1,b_2)$ falls in the shaded region in the left, the optimal mechanism is as depicted in the right. Item $1$ is offered for a price of $c+\delta_1$, item $2$ is offered for a price of $c+b_2/3$, and a lottery with probabilities $(1-a_2,a_2)$ is offered for a price of $c+a_2\delta_2$.}\label{fig:b-new}
\end{figure}
\begin{figure}[H]
\centering
\begin{minipage}{.45\textwidth}
\centering
\begin{tikzpicture}[scale=0.36,font=\small,axis/.style={very thick, ->, >=stealth'}]
\draw [axis,thick,->] (0,-1)--(0,13);
\node [right] at (11,-1) {$\frac{b_1}{b_2}$};
\draw [axis,thick,->] (-0.5,0)--(12,0);
\node [above] at (1,11) {$\frac{c}{b_2}$};
\node at (-1,0.25) {$0$};
\draw [thin,-] (-0.25,2) -- (0.25,2);
\node [left] at (-0.25,2) {$2$};
\draw [thin,-] (-0.25,4) -- (0.25,4);
\node [left] at (-0.25,4) {$4$};
\draw [thin,-] (-0.25,6) -- (0.25,6);
\node [left] at (-0.25,6) {$6$};
\draw [thin,-] (-0.25,8) -- (0.25,8);
\node [left] at (-0.25,8) {$8$};
\draw [thin,-] (-0.25,10) -- (0.25,10);
\node [left] at (-0.25,10) {$10$};
\draw [thin,-] (-0.25,12) -- (0.25,12);
\node [left] at (-0.25,12) {$12$};
\node at (0.5,-1) {$1$};
\draw [thin,-] (2,-0.25) -- (2,0.25);
\node [below] at (2,-0.25) {$1.2$};
\draw [thin,-] (4,-0.25) -- (4,0.25);
\node [below] at (4,-0.25) {$1.4$};
\draw [thin,-] (6,-0.25) -- (6,0.25);
\node [below] at (6,-0.25) {$1.6$};
\draw [thin,-] (8,-0.25) -- (8,0.25);
\node [below] at (8,-0.25) {$1.8$};
\draw [thin,-] (10,-0.25) -- (10,0.25);
\node [below] at (10,-0.25) {$2$};
\draw [thick,-] (0,1) to (12,1);
\draw [thick,-] (0,1) to (5,1.733) to (12,1.97);
\draw [thick,-] (0,1.372) to (5,1.733);
\draw [thick,-] (2.2,12) to[out=-60,in=170] (12,3.98);
\draw [thick,dotted] (5,0) to (5,12);
\node [right] at (5,11) {$b_1=(1.5)b_2$};
\node at (2.2,8.5) {asymptotic};
\node at (2.2,7.5) {to $b_1=b_2$};
\node at (9,3.8) {asymptotic};
\node at (9,2.8) {to $c=2b_2$};
\node at (10,1.4) {$c=b_2$};
\draw [thick,->] (8.5,2) -- (8.5,2.6);
\draw [thick,->] (8.5,5) -- (8.5,4.2);
\path[fill=gray!50,opacity=.5] (0,1) to (0,1.372) to (5,1.733) to (0,1);
\end{tikzpicture}
\end{minipage}
\begin{minipage}{0.025\textwidth}
\hspace*{0.025\textwidth}
\end{minipage}
\begin{minipage}{0.45\textwidth}
\centering
\begin{tikzpicture}[scale=0.25,font=\normalsize,axis/.style={very thick, -}]
\node [rotate=45] at (-1,0) {$c$};
\node [rotate=45] at (0,-1) {$c$};
\draw [axis,thick,-] (0,0)--(15,0);
\node [rotate=45] at (15,-1.2) {$c+b_1$};
\draw [axis,thick,-] (0,0)--(0,12);
\node [rotate=45] at (-1,12) {$c+b_2$};
\draw [axis,thick,-] (0,12)--(15,12);
\draw [axis,thick,-] (15,0)--(15,12);
\draw [axis,thick,-] (0,4)--(8,12);
\draw [thin,-] (-0.4,4)--(0.4,4);
\node [rotate=45] at (-2,3.5) {$c+b_2/3$};
\draw [axis,thick,-] (0,3.5)--(4,2.5);
\draw [thin,-] (-0.4,3.5)--(0.4,3.5);
\node [rotate=45] at (-1.8,1.8) {$c+\delta_2$};
\draw [thin,<->] (4,0)--(4,2.5);
\node at (4.5,1) {$h$};
\draw [axis,thick,-] (4,2.5)--(6,0);
\draw [axis,thick,-] (6.5,0)--(15,8.5);
\draw [thin,-] (6,-0.4)--(6,0.4);
\node [rotate=45] at (4.6,-1.8) {$c+\delta_1$};
\draw [thin,-] (6.5,-0.4)--(6.5,0.4);
\node [rotate=45] at (6.5,-2) {$c+b_1/3$};
\draw [thin,-] (14.6,8.5)--(15.4,8.5);
\node [rotate=90] at (16,7.5) {$c+2b_1/3$};
\draw [thick,-] (4,2.5)--(13.5,12);
\draw [thick,dotted] (4,2.5)--(1.5,0);
\draw [thin,-] (1.5,-0.4)--(1.5,0.4);
\node [rotate=45] at (1,-1.8) {$c+\delta^*$};
\draw [thin,-] (13.5,11.6)--(13.5,12.4);
\node at (14.5,13) {$c+b_2+\delta^*$};
\draw [thin,-] (8,11.6)--(8,12.4);
\node at (7.25,13) {\footnotesize$c+2b_2/3$};
\node at (2,1.5) {$(0,0)$};
\node at (12,2) {$(1,0)$};
\node [above] at (2,8) {$(0,1)$};
\node [rotate=45] at (6,7) {$(1-a_2,a_2)$};
\node [rotate=45] at (8.5,4.5) {$(a_1,1-a_1)$};
\end{tikzpicture}
\end{minipage}
\caption{When $(c,b_1,b_2)$ falls in the shaded region in the left, the optimal mechanism is as depicted in the right. Item $1$ is offered for a price of $c+b_1/3$, item $2$ is offered for a price of $c+b_2/3$, a lottery with probabilities $(1-a_2,a_2)$ is offered for a price of $c+a_2\delta_2$, and a lottery with probabilities $(a_1,1-a_1)$ is offered for a price of $c+a_1\delta_1$.}\label{fig:c-new}
\end{figure}
\begin{remark}
The mechanisms depicted below in Figures \ref{fig:d-new} and \ref{fig:d'-new} differ only in that the line separating the regions with allocations $(1-a,a)$ and $(1,0)$ falls to the right of the line $z_1-z_2=b_1-b_2$ in the former, and to the left of it in the latter. These two structures meet at $b_1=3b_2/2$ when the line of separation exactly falls at $z_1-z_2=b_1-b_2$.
\end{remark}
\begin{figure}[H]
\centering
\begin{minipage}{.45\textwidth}
\centering
\begin{tikzpicture}[scale=0.36,font=\small,axis/.style={very thick, ->, >=stealth'}]
\draw [axis,thick,->] (0,-1)--(0,13);
\node [right] at (11,-1) {$\frac{b_1}{b_2}$};
\draw [axis,thick,->] (-0.5,0)--(12,0);
\node [above] at (1,11) {$\frac{c}{b_2}$};
\node at (-1,0.25) {$0$};
\draw [thin,-] (-0.25,2) -- (0.25,2);
\node [left] at (-0.25,2) {$2$};
\draw [thin,-] (-0.25,4) -- (0.25,4);
\node [left] at (-0.25,4) {$4$};
\draw [thin,-] (-0.25,6) -- (0.25,6);
\node [left] at (-0.25,6) {$6$};
\draw [thin,-] (-0.25,8) -- (0.25,8);
\node [left] at (-0.25,8) {$8$};
\draw [thin,-] (-0.25,10) -- (0.25,10);
\node [left] at (-0.25,10) {$10$};
\draw [thin,-] (-0.25,12) -- (0.25,12);
\node [left] at (-0.25,12) {$12$};
\node at (0.5,-1) {$1$};
\draw [thin,-] (2,-0.25) -- (2,0.25);
\node [below] at (2,-0.25) {$1.2$};
\draw [thin,-] (4,-0.25) -- (4,0.25);
\node [below] at (4,-0.25) {$1.4$};
\draw [thin,-] (6,-0.25) -- (6,0.25);
\node [below] at (6,-0.25) {$1.6$};
\draw [thin,-] (8,-0.25) -- (8,0.25);
\node [below] at (8,-0.25) {$1.8$};
\draw [thin,-] (10,-0.25) -- (10,0.25);
\node [below] at (10,-0.25) {$2$};
\draw [thick,-] (0,1) to (12,1);
\draw [thick,-] (0,1) to (5,1.733) to (12,1.97);
\draw [thick,-] (0,1.372) to (5,1.733);
\draw [thick,-] (2.2,12) to[out=-60,in=170] (12,3.98);
\draw [thick,dotted] (5,0) to (5,12);
\node [right] at (5,11) {$b_1=(1.5)b_2$};
\node at (2.2,8.5) {asymptotic};
\node at (2.2,7.5) {to $b_1=b_2$};
\node at (9,3.8) {asymptotic};
\node at (9,2.8) {to $c=2b_2$};
\node at (10,1.4) {$c=b_2$};
\draw [thick,->] (8.5,2) -- (8.5,2.6);
\draw [thick,->] (8.5,5) -- (8.5,4.2);
\path[fill=gray!50,opacity=.5] (0,1.372) to (0,12) to (2.2,12) to[out=-60,in=130] (5,8) to (5,1.733) to (0,1.372);
\end{tikzpicture}
\end{minipage}
\begin{minipage}{0.025\textwidth}
\hspace*{0.025\textwidth}
\end{minipage}
\begin{minipage}{0.45\textwidth}
\centering
\begin{tikzpicture}[scale=0.25,font=\normalsize,axis/.style={very thick, -}]
\node [rotate=45] at (-1,0) {$c$};
\node [rotate=45] at (0,-1) {$c$};
\draw [axis,thick,-] (0,0)--(15,0);
\node [rotate=45] at (15,-1.2) {$c+b_1$};
\draw [axis,thick,-] (0,0)--(0,12);
\node [rotate=45] at (-1,12) {$c+b_2$};
\draw [axis,thick,-] (0,12)--(15,12);
\draw [axis,thick,-] (15,0)--(15,12);
\draw [axis,thick,-] (0,4)--(8,12);
\draw [thin,-] (-0.4,4)--(0.4,4);
\node [rotate=45] at (-2,3.5) {$c+b_2/3$};
\draw [axis,thick,-] (0,3.5)--(5,0);
\draw [thin,-] (-0.4,3.5)--(0.4,3.5);
\node [rotate=45] at (-1.8,1.8) {$c+\delta_2$};
\draw [thin,-] (5,-0.4)--(5,0.4);
\node [rotate=45] at (3.5,-1.8) {$c+\delta_1$};
\draw [thick,-] (5.5,0)--(15,9.5);
\draw [thin,-] (5.5,-0.4)--(5.5,0.4);
\node [rotate=45] at (5.5,-2) {$c+b_1/3$};
\draw [thin,-] (14.6,9.5)--(15.4,9.5);
\node [rotate=90] at (16,8) {$c+2b_1/3$};
\draw [thin,-] (8,11.6)--(8,12.4);
\node at (7,13) {$c+2b_2/3$};
\node at (1.5,1.2) {$(0,0)$};
\node [above] at (12,2) {$(1,0)$};
\node [above] at (2,8) {$(0,1)$};
\node at (6.5,6) {$(1-a,a)$};
\end{tikzpicture}
\end{minipage}
\caption{When $(c,b_1,b_2)$ falls in the shaded region in the left, the optimal mechanism is as depicted in the right. Item $1$ is offered for a price of $c+b_1/3$, item $2$ is offered for a price of $c+b_2/3$, and a lottery with probabilities $(1-a,a)$ is offered for a price of $c+a\delta_2$.}\label{fig:d-new}
\end{figure}
\begin{figure}[H]
\centering
\begin{minipage}{.45\textwidth}
\centering
\begin{tikzpicture}[scale=0.36,font=\small,axis/.style={very thick, ->, >=stealth'}]
\draw [axis,thick,->] (0,-1)--(0,13);
\node [right] at (11,-1) {$\frac{b_1}{b_2}$};
\draw [axis,thick,->] (-0.5,0)--(12,0);
\node [above] at (1,11) {$\frac{c}{b_2}$};
\node at (-1,0.25) {$0$};
\draw [thin,-] (-0.25,2) -- (0.25,2);
\node [left] at (-0.25,2) {$2$};
\draw [thin,-] (-0.25,4) -- (0.25,4);
\node [left] at (-0.25,4) {$4$};
\draw [thin,-] (-0.25,6) -- (0.25,6);
\node [left] at (-0.25,6) {$6$};
\draw [thin,-] (-0.25,8) -- (0.25,8);
\node [left] at (-0.25,8) {$8$};
\draw [thin,-] (-0.25,10) -- (0.25,10);
\node [left] at (-0.25,10) {$10$};
\draw [thin,-] (-0.25,12) -- (0.25,12);
\node [left] at (-0.25,12) {$12$};
\node at (0.5,-1) {$1$};
\draw [thin,-] (2,-0.25) -- (2,0.25);
\node [below] at (2,-0.25) {$1.2$};
\draw [thin,-] (4,-0.25) -- (4,0.25);
\node [below] at (4,-0.25) {$1.4$};
\draw [thin,-] (6,-0.25) -- (6,0.25);
\node [below] at (6,-0.25) {$1.6$};
\draw [thin,-] (8,-0.25) -- (8,0.25);
\node [below] at (8,-0.25) {$1.8$};
\draw [thin,-] (10,-0.25) -- (10,0.25);
\node [below] at (10,-0.25) {$2$};
\draw [thick,-] (0,1) to (12,1);
\draw [thick,-] (0,1) to (5,1.733) to (12,1.97);
\draw [thick,-] (0,1.372) to (5,1.733);
\draw [thick,-] (2.2,12) to[out=-60,in=170] (12,3.98);
\draw [thick,dotted] (5,0) to (5,12);
\node [right] at (5,11) {$b_1=(1.5)b_2$};
\node at (2.2,8.5) {asymptotic};
\node at (2.2,7.5) {to $b_1=b_2$};
\node at (9,3.8) {asymptotic};
\node at (9,2.8) {to $c=2b_2$};
\node at (10,1.4) {$c=b_2$};
\draw [thick,->] (8.5,2) -- (8.5,2.6);
\draw [thick,->] (8.5,5) -- (8.5,4.2);
\path[fill=gray!50,opacity=.5] (5,8) to[out=-50,in=170] (12,3.98) to (12,1.97) to (5,1.733) to (5,8);
\end{tikzpicture}
\end{minipage}
\begin{minipage}{0.025\textwidth}
\hspace*{0.025\textwidth}
\end{minipage}
\begin{minipage}{0.45\textwidth}
\centering
\begin{tikzpicture}[scale=0.225,font=\normalsize,axis/.style={very thick, -}]
\node [rotate=45] at (-1,0) {$c$};
\node [rotate=45] at (0,-1) {$c$};
\draw [axis,thick,-] (0,0)--(18,0);
\node [rotate=45] at (18,-1) {$c+b_1$};
\draw [axis,thick,-] (0,0)--(0,12);
\node [rotate=45] at (-1,12) {$c+b_2$};
\draw [axis,thick,-] (0,12)--(18,12);
\draw [axis,thick,-] (18,0)--(18,12);
\draw [axis,thick,-] (0,4)--(8,12);
\draw [thin,-] (-0.4,4)--(0.4,4);
\node [rotate=45] at (-2,3.8) {$c+b_2/3$};
\draw [axis,thick,-] (0,3.5)--(5,0);
\draw [thin,-] (-0.4,3.5)--(0.4,3.5);
\node [rotate=45] at (-2,2) {$c+\delta_2$};
\draw [thin,-] (5,-0.4)--(5,0.4);
\node [rotate=45] at (3.8,-1.6) {$c+\delta_1$};
\draw [thick,-] (5.5,0)--(17.5,12);
\draw [thin,-] (5.5,-0.4)--(5.5,0.4);
\node [rotate=45] at (5.3,-2.7) {$c+\frac{b_1}{2}-\frac{b_2}{4}$};
\draw [thin,-] (17.5,11.6)--(17.5,12.4);
\node at (16,13.1) {$c+\frac{b_1}{2}+\frac{3b_2}{4}$};
\draw [thin,-] (8,11.6)--(8,12.4);
\node at (8,13) {\footnotesize$c+2b_2/3$};
\node at (1.5,1.2) {$(0,0)$};
\node [above] at (14,3) {$(1,0)$};
\node [above] at (2,8) {$(0,1)$};
\node at (6.5,5.5) {$(1-a,a)$};
\end{tikzpicture}
\end{minipage}
\caption{When $(c,b_1,b_2)$ falls in the shaded region in the left, the optimal mechanism is as depicted in the right. Item $1$ is offered for a price of $c+b_1/2-b_2/4$, item $2$ is offered for a price of $c+b_2/3$, and a lottery with probabilities $(1-a,a)$ is offered for a price of $c+a\delta_2$.}\label{fig:d'-new}
\end{figure}
\begin{remark}
Observe that the mechanisms depicted in Figures \ref{fig:b-new}, \ref{fig:c-new}, \ref{fig:d-new}, and \ref{fig:d'-new} meet at $b_1=3b_2/2$, $c=tb_2$. They meet because at this $(c,b_1,b_2)$, the parameter $h$ (in Figures \ref{fig:b-new} and \ref{fig:c-new}) becomes $0$, and $\delta^*=\delta_1=b_1/2-b_2/4=b_1/3=b_1-b_2$.
\end{remark}
\begin{remark}
The mechanisms depicted below in Figures \ref{fig:e-new} and \ref{fig:e'-new} differ only in that the line separating the regions with allocations $(0,1)$ and $(1,0)$ falls to the right of the line $z_1-z_2=b_1-b_2$ in the former, and to the left of it in the latter. These two structures meet at $b_1=3b_2/2$ when the line of separation exactly falls at $z_1-z_2=b_1-b_2$.
\end{remark}
\begin{figure}[H]
\centering
\begin{minipage}{.45\textwidth}
\centering
\begin{tikzpicture}[scale=0.36,font=\small,axis/.style={very thick, ->, >=stealth'}]
\draw [axis,thick,->] (0,-1)--(0,13);
\node [right] at (11,-1) {$\frac{b_1}{b_2}$};
\draw [axis,thick,->] (-0.5,0)--(12,0);
\node [above] at (1,11) {$\frac{c}{b_2}$};
\node at (-1,0.25) {$0$};
\draw [thin,-] (-0.25,2) -- (0.25,2);
\node [left] at (-0.25,2) {$2$};
\draw [thin,-] (-0.25,4) -- (0.25,4);
\node [left] at (-0.25,4) {$4$};
\draw [thin,-] (-0.25,6) -- (0.25,6);
\node [left] at (-0.25,6) {$6$};
\draw [thin,-] (-0.25,8) -- (0.25,8);
\node [left] at (-0.25,8) {$8$};
\draw [thin,-] (-0.25,10) -- (0.25,10);
\node [left] at (-0.25,10) {$10$};
\draw [thin,-] (-0.25,12) -- (0.25,12);
\node [left] at (-0.25,12) {$12$};
\node at (0.5,-1) {$1$};
\draw [thin,-] (2,-0.25) -- (2,0.25);
\node [below] at (2,-0.25) {$1.2$};
\draw [thin,-] (4,-0.25) -- (4,0.25);
\node [below] at (4,-0.25) {$1.4$};
\draw [thin,-] (6,-0.25) -- (6,0.25);
\node [below] at (6,-0.25) {$1.6$};
\draw [thin,-] (8,-0.25) -- (8,0.25);
\node [below] at (8,-0.25) {$1.8$};
\draw [thin,-] (10,-0.25) -- (10,0.25);
\node [below] at (10,-0.25) {$2$};
\draw [thick,-] (0,1) to (12,1);
\draw [thick,-] (0,1) to (5,1.733) to (12,1.97);
\draw [thick,-] (0,1.372) to (5,1.733);
\draw [thick,-] (2.2,12) to[out=-60,in=170] (12,3.98);
\draw [thick,dotted] (5,0) to (5,12);
\node [right] at (5,11) {$b_1=(1.5)b_2$};
\node at (2.2,8.5) {asymptotic};
\node at (2.2,7.5) {to $b_1=b_2$};
\node at (9,3.8) {asymptotic};
\node at (9,2.8) {to $c=2b_2$};
\node at (10,1.4) {$c=b_2$};
\draw [thick,->] (8.5,2) -- (8.5,2.6);
\draw [thick,->] (8.5,5) -- (8.5,4.2);
\path[fill=gray!50,opacity=.5] (2.2,12) to[out=-60,in=130] (5,8) to (5,12) to (2.2,12);
\end{tikzpicture}
\end{minipage}
\begin{minipage}{0.025\textwidth}
\hspace*{0.025\textwidth}
\end{minipage}
\begin{minipage}{0.45\textwidth}
\centering
\begin{tikzpicture}[scale=0.25,font=\normalsize,axis/.style={very thick, -}]
\node [rotate=45] at (-1,0) {$c$};
\node [rotate=45] at (0,-1) {$c$};
\draw [axis,thick,-] (0,0)--(15,0);
\node [rotate=45] at (15,-1.2) {$c+b_1$};
\draw [axis,thick,-] (0,0)--(0,12);
\node [rotate=45] at (-1,12) {$c+b_2$};
\draw [axis,thick,-] (0,12)--(15,12);
\draw [axis,thick,-] (15,0)--(15,12);
\draw [thick,-] (5,0)--(15,10);
\draw [thin,-] (5,-0.4)--(5,0.4);
\node [rotate=45] at (5,-2) {$c+b_1/3$};
\draw [thin,-] (14.6,10)--(15.4,10);
\node [rotate=90] at (16,9) {$c+2b_1/3$};
\node [above] at (12,3) {$(1,0)$};
\node [above] at (5,6) {$(0,1)$};
\end{tikzpicture}
\end{minipage}
\caption{When $(c,b_1,b_2)$ falls in the shaded region in the left, the optimal mechanism is as depicted in the right. Item $1$ is offered for a price of $c+b_1/3$, and item $2$ is offered for a price of $c$.}\label{fig:e-new}
\end{figure}
\begin{figure}[H]
\centering
\begin{minipage}{.45\textwidth}
\centering
\begin{tikzpicture}[scale=0.36,font=\small,axis/.style={very thick, ->, >=stealth'}]
\draw [axis,thick,->] (0,-1)--(0,13);
\node [right] at (11,-1) {$\frac{b_1}{b_2}$};
\draw [axis,thick,->] (-0.5,0)--(12,0);
\node [above] at (1,11) {$\frac{c}{b_2}$};
\node at (-1,0.25) {$0$};
\draw [thin,-] (-0.25,2) -- (0.25,2);
\node [left] at (-0.25,2) {$2$};
\draw [thin,-] (-0.25,4) -- (0.25,4);
\node [left] at (-0.25,4) {$4$};
\draw [thin,-] (-0.25,6) -- (0.25,6);
\node [left] at (-0.25,6) {$6$};
\draw [thin,-] (-0.25,8) -- (0.25,8);
\node [left] at (-0.25,8) {$8$};
\draw [thin,-] (-0.25,10) -- (0.25,10);
\node [left] at (-0.25,10) {$10$};
\draw [thin,-] (-0.25,12) -- (0.25,12);
\node [left] at (-0.25,12) {$12$};
\node at (0.5,-1) {$1$};
\draw [thin,-] (2,-0.25) -- (2,0.25);
\node [below] at (2,-0.25) {$1.2$};
\draw [thin,-] (4,-0.25) -- (4,0.25);
\node [below] at (4,-0.25) {$1.4$};
\draw [thin,-] (6,-0.25) -- (6,0.25);
\node [below] at (6,-0.25) {$1.6$};
\draw [thin,-] (8,-0.25) -- (8,0.25);
\node [below] at (8,-0.25) {$1.8$};
\draw [thin,-] (10,-0.25) -- (10,0.25);
\node [below] at (10,-0.25) {$2$};
\draw [thick,-] (0,1) to (12,1);
\draw [thick,-] (0,1) to (5,1.733) to (12,1.97);
\draw [thick,-] (0,1.372) to (5,1.733);
\draw [thick,-] (2.2,12) to[out=-60,in=170] (12,3.98);
\draw [thick,dotted] (5,0) to (5,12);
\node [right] at (5,11) {$b_1=(1.5)b_2$};
\node at (2.2,8.5) {asymptotic};
\node at (2.2,7.5) {to $b_1=b_2$};
\node at (9,3.8) {asymptotic};
\node at (9,2.8) {to $c=2b_2$};
\node at (10,1.4) {$c=b_2$};
\draw [thick,->] (8.5,2) -- (8.5,2.6);
\draw [thick,->] (8.5,5) -- (8.5,4.2);
\path[fill=gray!50,opacity=.5] (5,12) to (5,8) to[out=-50,in=170] (12,3.98) to (12,12) to (5,12);
\end{tikzpicture}
\end{minipage}
\begin{minipage}{0.025\textwidth}
\hspace*{0.025\textwidth}
\end{minipage}
\begin{minipage}{0.45\textwidth}
\centering
\begin{tikzpicture}[scale=0.225,font=\normalsize,axis/.style={very thick, -}]
\node [rotate=45] at (-1,0) {$c$};
\node [rotate=45] at (0,-1) {$c$};
\draw [axis,thick,-] (0,0)--(18,0);
\node [rotate=45] at (18,-1) {$c+b_1$};
\draw [axis,thick,-] (0,0)--(0,12);
\node [rotate=45] at (-1,12) {$c+b_2$};
\draw [axis,thick,-] (0,12)--(18,12);
\draw [axis,thick,-] (18,0)--(18,12);
\draw [thick,-] (5,0)--(17,12);
\draw [thin,-] (5,-0.4)--(5,0.4);
\node [rotate=45] at (4.8,-2.7) {$c+\frac{b_1}{2}-\frac{b_2}{4}$};
\draw [thin,-] (17,11.6)--(17,12.4);
\node at (16,13.1) {$c+\frac{b_1}{2}+\frac{3b_2}{4}$};
\node [above] at (14,3) {$(1,0)$};
\node [above] at (5,6) {$(0,1)$};
\end{tikzpicture}
\end{minipage}
\caption{When $(c,b_1,b_2)$ falls in the shaded region in the left, the optimal mechanism is as depicted in the right. Item $1$ is offered for a price of $c+b_1/2-b_2/4$, and item $2$ is offered for a price of $c$.}\label{fig:e'-new}
\end{figure}
\begin{remark}
The mechanisms depicted in Figures \ref{fig:d-new}, \ref{fig:d'-new}, \ref{fig:e-new}, and \ref{fig:e'-new} meet at $b_1=3b_2/2$, $c=(243/38)b_2$. They meet because at this $(c,b_1,b_2)$, the parameter $a$ (in Figures \ref{fig:d-new} and \ref{fig:d'-new}) becomes $0$, and $b_1/2-b_2/4=b_1/3=b_1-b_2$.
\end{remark}
\begin{remark}\label{rem:armstrong}
The mechanisms in Figures \ref{fig:e-new} and \ref{fig:e'-new} show an interesting result -- the existence of an optimal multi-dimensional mechanism without an exclusion region. An intuitive explanation for the absence of exclusion region in Figure \ref{fig:e'-new} is as follows. Consider the case where the seller offers each allocation with a small increase in price, say $\epsilon$. The seller then loses a revenue of $c$ from the valuations $\{z:u(z)\leq\epsilon\}$, and gains an extra revenue of $\epsilon$ from the valuations $\{z:u(z)\geq\epsilon\}$. The mechanism will have no exclusion region when the loss dominates the gain. Observe that the expected loss in revenue is \begin{eqnarray*}
c\cdot Pr(\{z:u(z)\leq\epsilon\}) & = & \frac{c}{b_1b_2}(\epsilon(b_1/2-b_2/4+\epsilon))+\frac{(b_1/2-b_2/4)}{b_1b_2}\frac{\epsilon^2}{2}\\
& \approx & \frac{c}{b_1b_2}(\epsilon(b_1/2-b_2/4)),
\end{eqnarray*}
and that the expected gain in revenue is $$\epsilon\cdot Pr(\{u(z)\geq\epsilon\})=\epsilon\cdot(1-Pr(\{u(z)\leq\epsilon\}))\approx\epsilon.$$ The loss dominates the gain when $c \geq \frac{4b_1b_2}{2b_1-b_2}$. (The actual threshold will depend on more precise calculations than our order estimates.) Observe that both the loss and the gain are of the order of $\epsilon$, which explains the possibility of the loss dominating the gain at very high values of $c$. Figure \ref{fig:e-new} has no exclusion region due to a similar reason.
\end{remark}
\begin{remark}\label{rem:role}
The notations $\delta_1$, $\delta_2$, and $\delta^*$, used in various mechanism depictions, can be understood as follows. (i) The first transition from $q=(0,0)$ on the bottom boundary of $D$ occurs at $\delta=\delta_1$. (ii) Similarly, the first transition on the left boundary of $D$ occurs at $\delta=-\delta_2$. (iii) The final transition of $q$ on the top/right boundary of $D$ (in mechanisms depicted in Figures \ref{fig:a-new}--\ref{fig:c-new}) occurs at $\delta=\delta^*$.
\end{remark}
For a summarizing phase diagram see Figure \ref{fig:phase-diagram}. To see a portrayal of all possible structures that an optimal mechanism can take, see Figures \ref{fig:a-ini}--\ref{fig:e'-ini}.
We now proceed to prove Theorem \ref{thm:consolidate}. We consider every structure separately, and go through the following steps in order to prove that the optimal mechanism has the specific structure.
\begin{enumerate}
\item[\bf{Step 1:}] We compute the virtual valuation function $V(\delta)$ for every $\delta\in[-b_2,b_1]$.
\item[\bf{Step 2:}] We find the relation between the variables of interest, ($\delta_1$, $\delta_2$, $\delta^*$, $h$, $a_1$, $a_2$), using the equality conditions in Theorem \ref{thm:Myerson}.
\item[\bf{Step 3:}] We prove that the solution that satisfies the relations obtained in Step 2 are indeed meaningful, by evaluating bounds for the variables of interest.
\item[\bf{Step 4:}] We verify that all the inequality conditions of Theorem \ref{thm:Myerson} hold. The bounds evaluated in Step 3 are crucially used in this process of verification.
\end{enumerate}
We now proceed to prove parts 1(a) and 2(a) of Theorem \ref{thm:consolidate}.
\begin{theorem}\label{thm:menu-1}
Let $c\in[0,b_2]$. Then the optimal mechanism is as depicted in Figure \ref{fig:a-ini} (see also Figure \ref{fig:a-new}). The values of $\delta_1$ and $\delta_2$ are computed by solving the following equations simultaneously.
\begin{align}
&-3\delta_1\delta_2-c(\delta_1+\delta_2)+b_1b_2=0.\label{eqn:fig-a-first}\\
&-\frac{3}{2}\delta_2^2+2b_2\delta_2-\frac{b_2^2}{2}+(c-2b_2+3\delta_2)\delta_1=0.\label{eqn:fig-a-second}
\end{align}
\end{theorem}
\begin{proof}
{\bf Step 1:} We compute the virtual valuation function for the mechanism depicted in Figure \ref{fig:a-new}. Since $\bar{\mu}(D)=0$, we compute $V$ using the formula
\begin{equation}\label{eqn:V-alternate}
V(\delta)=-\bar{\mu}(\{z:z_1-z_2<\delta\}\cup Z).
\end{equation}
\begin{equation}\label{eqn:V-fig-a}
V(\delta)=\frac{1}{b_1b_2}\begin{cases}\bar{\mu}(Z)+\frac{3}{2}\delta^2+2b_2\delta+\frac{b_2^2}{2}&\delta\in[-b_2,-\delta_2]\\V(-\delta_2)-(c-2b_2+3\delta_2)(\delta+\delta_2)&\delta\in[-\delta_2,\delta^*]\\V(\delta^*)-(c-2b_2)(\delta-\delta^*)+\frac{3}{2}((\delta_1-\delta)^2-\delta_2^2)&\delta\in[\delta^*,b']\\V(b')-(c-2b_1+3\delta_1)(\delta-b_1+b_2)&\delta\in[b',\delta_1]\\-\frac{3}{2}\delta^2+2b_1\delta-\frac{b_1^2}{2}&\delta\in[\delta_1,b_1]\end{cases}
\end{equation}
where $b_1-b_2$ is denoted as $b'$. For ease of notation, we drop the factor $\frac{1}{b_1b_2}$ in the rest of the paper.
{\bf Step 2:} The mechanism has three unknowns: $\delta^*$, $\delta_1$, and $\delta_2$. Observe that the line between the points $(c+b_2+\delta^*,c+b_2)$ and $(c+\delta^*,c)$ passes through $(c+\delta_1,c+\delta_2)$. So we have $\delta^*=\delta_1-\delta_2$.
We now proceed to compute $\delta_1$ and $\delta_2$. We do so by equating $\bar{\mu}(Z)=0$ and $V(\delta^*)=0$. The latter follows from Theorem \ref{thm:Myerson} because $q_1=0$ for $\delta\in[-b_2,\delta^*]$. We thus obtain equations (\ref{eqn:fig-a-first}) and (\ref{eqn:fig-a-second}).
{\bf Step 3:} We now show that there exists a meaningful solution $(\delta_1,\delta_2)$ that simultaneously solves (\ref{eqn:fig-a-first}) and (\ref{eqn:fig-a-second}). Specifically, we show that there exists a $(\delta_1,\delta_2)\in[\frac{b_1}{2}-\frac{b_2}{6},\frac{2b_1-c}{3}]\times[\frac{b_2}{3},\frac{2b_2-c}{3}]$ as a simultaneous solution to (\ref{eqn:fig-a-first}) and (\ref{eqn:fig-a-second}). To show this, we do the following.
\begin{itemize}
\item We first define $\delta_1|_{\delta_2=x}$ to be the value of $\delta_1$ that satisfies (\ref{eqn:fig-a-first}) when $\delta_2=x$ and $\delta_2|_{\delta_1=x}$ to be the value of $\delta_2$ that satisfies (\ref{eqn:fig-a-first}) when $\delta_1=x$. We then show that there exists a $(\delta_1,\delta_2)\in[\frac{b_1}{2}-\frac{b_2}{6},\frac{2b_1-c}{3}]\times[\frac{b_2}{3},\frac{2b_2-c}{3}]$ satisfying (\ref{eqn:fig-a-first}). We do this by showing that (a) $\delta_1|_{\delta_2=x}$ is continuous in $x$, (b) $\delta_1|_{\delta_2=\frac{b_2}{3}}\geq\frac{b_1}{2}-\frac{b_2}{6}$, and (c) $\delta_1|_{\delta_2=\frac{2b_2-c}{3}}\leq\frac{2b_1-c}{3}$. We further show that in addition to continuity, $\delta_1|_{\delta_2=x}$ is also monotone; it decreases as $x$ increases.
\item It now suffices to show that the entry and the exit points of the curve $(\delta_1|_{\delta_2=x},x)$ in the rectangle $[\frac{b_1}{2}-\frac{b_2}{6},\frac{2b_1-c}{3}]\times[\frac{b_2}{3},\frac{2b_2-c}{3}]$ changes sign when substituted on the left-hand side of (\ref{eqn:fig-a-second}). The possible entry points are $(\frac{b_1}{2}-\frac{b_2}{6},\delta_2|_{\delta_1=\frac{b_1}{2}-\frac{b_2}{6}})$ and $(\delta_1|_{\delta_2=\frac{2b_2-c}{3}},\frac{2b_2-c}{3})$; we substitute the entry points on left-hand side of (\ref{eqn:fig-a-second}) and show that the expression is nonnegative in both cases. Similarly, the possible exit points are $(\delta_1|_{\delta_2=\frac{b_2}{3}},\frac{b_2}{3})$ and $(\frac{2b_1-c}{3},\delta_2|_{\delta_1=\frac{2b_1-c}{3}})$; we substitute the exit points on left-hand side of (\ref{eqn:fig-a-second}) and show that the expression is nonpositive in both cases.
\end{itemize}
We now fill in the details. We have $\delta_1|_{\delta_2}=\frac{b_1b_2-c\delta_2}{3\delta_2+c}$ and $\delta_2|_{\delta_1}=\frac{b_1b_2-c\delta_1}{3\delta_1+c}$ from (\ref{eqn:fig-a-first}). It is clear that $\delta_1|_{\delta_2=x}$ is continuous, and also monotonically decreases in $x$. We now verify that $\delta_1|_{\delta_2=\frac{b_2}{3}}\geq\frac{b_1}{2}-\frac{b_2}{6}$; indeed,
$$
\frac{b_1b_2-cb_2/3}{c+b_2}\geq\frac{b_1b_2-cb_2/3}{2b_2}\geq\frac{b_1b_2-b_2^2/3}{2b_2}=\frac{b_1}{2}-\frac{b_2}{6},
$$
where both the inequalities hold because $c\leq b_2$. We now verify that $\delta_1|_{\delta_2=\frac{2b_2-c}{3}}\leq\frac{2b_1-c}{3}$:
$$
\frac{b_1b_2-c(2b_2-c)/3}{2b_2}\leq\frac{4b_1b_2/3-2b_2c/3}{2b_2}=\frac{2b_1-c}{3},
$$
where the inequality $c^2\leq b_1b_2$ holds because of $c\leq b_2\leq b_1$.
We now consider the points $(\delta_1|_{\delta_2=\frac{2b_2-c}{3}},\frac{2b_2-c}{3})$ and $(\delta_1|_{\delta_2=\frac{b_2}{3}},\frac{b_2}{3})$. Substituting $\delta_1=\frac{b_1b_2-c\delta_2}{c+3\delta_2}$ in (\ref{eqn:fig-a-second}), we obtain
\begin{equation}\label{eqn:fig-a-delta_2}
-\frac{9}{2}\delta_2^3+\delta_2^2(6b_2-\frac{9}{2}c)+\delta_2(4b_2 c-c^2-\frac{3}{2}b_2^2+3b_1b_2)-\frac{1}{2}b_2^2c+b_1b_2c-2b_1b_2^2=0.
\end{equation}
When $\delta_2=\frac{2b_2-c}{3}$, the left-hand side of (\ref{eqn:fig-a-delta_2}) equals $\frac{1}{3}b_2(b_2^2-c^2)\geq 0$, and when $\delta_2=\frac{b_2}{3}$, it equals $-b_2(b_1-c/3)(b_2-c)\leq 0$.
We now consider the points $(\frac{2b_1-c}{3},\delta_2|_{\delta_1=\frac{2b_1-c}{3}})$ and $(\frac{b_1}{2}-\frac{b_2}{6},\delta_2|_{\delta_1=\frac{b_1}{2}-\frac{b_2}{6}})$. Substituting $\delta_2=\frac{b_1b_2-c\delta_1}{3\delta_1+c}$ in (\ref{eqn:fig-a-second}), we obtain
\begin{multline}\label{eqn:fig-a-delta_1}
-\frac{3}{2}b_1^2b_2^2+2b_1b_2^2c-\frac{1}{2}b_2^2c^2+(6b_1b_2^2+6b_1b_2c-3b_2^2c-4b_2c^2+c^3)\delta_1\\+(9b_1b_2-\frac{9}{2}
b_2^2-18b_2c+\frac{3}{2}c^2)\delta_1^2-18b_2\delta_1^3=0.
\end{multline}
When $\delta_1=\frac{2b_1-c}{3}$, the left-hand side of (\ref{eqn:fig-a-delta_1}) equals $\frac{1}{6}(-8b_1^3b_2+3b_1^2b_2^2+4b_1^2c^2+2b_1b_2c^2-c^4)$. We claim that this expression is negative for $b_1\geq b_2$, $c\in[0,b_2]$. Observe that its derivative with respect to $c$ satisfies $4c(b_1(2b_2+b_2)-c^2)\geq 0$ for all $c\in[0,b_2]$, and thus the expression attains its maximum when $c=b_2$. At $c=b_2$, the expression equals $b_2(b_1-b_2)(-8b_1^2-b_1b_2+b_2^2)$ which clearly is nonpositive when $b_1\geq b_2$. We have proved our claim.
Now when $\delta_1=\frac{b_1}{2}-\frac{b_2}{6}$, the left-hand side of (\ref{eqn:fig-a-delta_1}) equals
\begin{eqnarray*}
\frac{1}{24}(b_2-c)(27b_1^2b_2-18b_1b_2^2-b_2^3+(42b_1b_2-9b_1^2-b_2^2)c+4(b_2-3b_1)c^2) \\
=\frac{1}{24}(b_2-c)(A_0+A_1c+A_2c^2).
\end{eqnarray*}
Observe that we have a quadratic expression in $c$, with $A_2$ being negative. So to prove that this quadratic expression is nonnegative for $c\in[0,b_2]$, it suffices to prove that it is nonnegative at $c=0$ and $c=b_2$. At $c=0$, the expression equals $27b_1^2b_2-18b_1b_2^2-b_2^3\geq 0$ for $b_1\geq b_2$, and at $c=b_2$, it equals $18b_1^2b_2+12b_1b_2^2+2b_2^3\geq 0$.
We have thus shown that there exists a solution $(\delta_1\,\delta_2)\in[\frac{b_1}{2}-\frac{b_2}{6},\frac{2b_1-c}{3}]\times[\frac{b_2}{3},\frac{2b_2-c}{3}]$ that simultaneously solves (\ref{eqn:fig-a-first}) and (\ref{eqn:fig-a-second}), for every $c\in[0,b_2]$ and $b_1\geq b_2$.
{\bf Step 4:} We now proceed to prove parts (c) and (d) in Theorem \ref{thm:Myerson}(2) and \ref{thm:Myerson}(4). Observe that the proof is complete if we prove that $V(\delta)\leq 0$ when $\delta\in[-b_2,\delta^*]$, and $V(\delta)\geq 0$ when $\delta\in[\delta^*,b_1]$. We now compute $V'(\delta)$ for almost every $\delta\in[-b_2,b_1]$.
\begin{equation}\label{eqn:menu-1-V'}
V'(\delta)=\begin{cases}3\delta+2b_2&\delta\in(-b_2,-\delta_2)\\-(c-2b_2+3\delta_2)&\delta\in(-\delta_2,\delta^*]\\-(c-2b_2)-3(\delta_1-\delta)&\delta\in[\delta^*,b_1-b_2)\\-(c-2b_1+3\delta_1)&\delta\in(b_1-b_2,\delta_1)\\-3\delta+2b_1&\delta\in(\delta_1,b_1).\end{cases}
\end{equation}
Observe that $V'(\delta)$ is negative when $\delta\in[-b_2,-\frac{2b_2}{3}]$, and positive when $\delta\in[-\frac{2b_2}{3},\delta^*]$ (follows because $\delta_2\leq\frac{2b_2-c}{3})$. We also have $V(-b_2)=V(\delta^*)=0$. So $V(\delta)=V(-b_2)+\int_{-b_2}^{\delta}V'(\tilde{\delta})\,d\tilde{\delta}\leq 0$ for all $\delta\in[-b_2,\delta^*]$, and hence $\int_{-b_2}^{\delta^*}V(\delta)\,d\delta\leq 0$, and $\int_{-b_2}^xV(\delta)\,d\delta\geq\int_{-b_2}^{\delta^*}V(\delta)\,d\delta$ for all $x\in[-b_2,\delta^*]$.
We now claim that $V'(\delta)$ is positive when $\delta\in[\delta^*,\frac{2b_1}{3}]$, and negative when $\delta\in[\frac{2b_1}{3},b_1]$. Observe that $V'(\delta)$ is continuous at $\delta=\delta^*$, and that it increases in the interval $[\delta^*,b_1-b_2]$. So $V'(\delta)\geq 0$ when $\delta\in[\delta^*,b_1-b_2]$. Also, $V'(\delta)\geq 0$ when $\delta\in[b_1-b_2,\delta_1]$ because $\delta_1\leq\frac{2b_1-c}{3}$. That $V'(\delta)$ is positive when $\delta\in[\delta_1,\frac{2b_1}{3}]$, and negative when $\delta\in[\frac{2b_1}{3},b_1]$ is obvious. We have proved our claim.
Since we also have $V(\delta^*)=V(b_1)=0$, it follows that $V(\delta)=V(\delta^*)+\int_{\delta^*}^{\delta}V'(\tilde{\delta})\,d\tilde{\delta}\geq 0$ for all $\delta\in[\delta^*,b_1]$. So we have $\int_{\delta^*}^{b_1}V(\delta)\,d\delta\geq 0$ and $\int_x^{b_1}V(\delta)\,d\delta\leq\int_{\delta^*}^{b_1}V(\delta)\,d\delta$ for all $x\in[\delta^*,b_1]$.\qed
\end{proof}
With the above theorem, we have completely solved the $c\leq b_2$ case. We now analyze the case at which the transition occurs. At $c=b_2$, when we solve (\ref{eqn:fig-a-first}) and (\ref{eqn:fig-a-second}) simultaneously, we obtain $\delta_2=\frac{b_2}{3}=\frac{2b_2-c}{3}$ and $\delta_1=\frac{b_1}{2}-\frac{b_2}{6}$. When $c>b_2$, the left-hand side of (\ref{eqn:fig-a-delta_2}) still continues to change sign at $\delta_2=\frac{b_2}{3}$ and $\delta_2=\frac{2b_2-c}{3}$, but since $\frac{b_2}{3}>\frac{2b_2-c}{3}$, the solution $\delta_2$ now belongs to the interval $[\frac{2b_2-c}{3},\frac{b_2}{3}]$. We thus have (i) $V(-\frac{b_2}{3})=0=V(\delta^*)$, and (ii) $V'(\delta)\geq 0$ when $\delta\in[-\frac{2b_2}{3},-\delta_2]$ and $V'(\delta)\leq 0$ when $\delta\in[-\delta_2,\delta^*]$. These both imply that $V(\delta)\geq 0$ when $\delta\in[-\frac{b_2}{3},\delta^*]$. So the minimum of $\int_{-b_2}^{x}V(\delta)\,d\delta$ can never occur at $x=\delta^*$, causing the condition in part (d) of Theorem \ref{thm:Myerson}(2) to fail.
At $c=b_2$, a transition occurs from the structure depicted in Figure \ref{fig:a-ini} to that in Figure \ref{fig:b-ini}. We now proceed to prove the optimality of the structure in \ref{fig:b-ini}, i.e., parts 1(b) and 2(b) in Theorem \ref{thm:consolidate}.
\begin{theorem}\label{thm:menu-2}
Let $c\in[b_2,\beta]$ if $b_1\geq 3b_2/2$ and let $c\in[b_2,\alpha_1]$ if $b_1\in[b_2,3b_2/2]$ with $\alpha_1$ and $\beta$ as defined in Theorem \ref{thm:consolidate}. Then, the optimal mechanism is as depicted in Figure \ref{fig:b-ini} (see also Figure \ref{fig:b-new}). The values of $h$ and $\delta^*$ are obtained by solving (\ref{eqn:fig-b-first}) and (\ref{eqn:fig-b-second}) simultaneously, and the values of $(\delta_1,\delta_2)$ are given by
$$
(\delta_1,\delta_2)=\left(h+\delta^*,\frac{b_1b_2-(3h/2+c)(h+\delta^*)}{3/2(h+\delta^*)+c}\right).
$$
The probability of allocation $a_2$ is given by $a_2=\frac{h+\delta^*}{\delta_2+\delta^*}$.
\end{theorem}
\begin{proof}
{\bf Step 1:} We compute the virtual valuation function for the mechanism depicted in Figure \ref{fig:b-new}.
$$
V(\delta)=\begin{cases}V(-\delta_2)-(c-2b_2+3\delta_2)(\delta+\delta_2)+\frac{3}{2}\frac{\delta_2-h}{\delta_2+\delta^*}(\delta+\delta_2)^2&\delta\in[-\delta_2,\delta^*]\\V(\delta^*)-(c-2b_2)(\delta-\delta^*)+\frac{3}{2}((\delta_1-\delta)^2-h^2)&\delta\in[\delta^*,b']\end{cases}
$$
where $b_1-b_2$ is denoted by $b'$. The expression for $V(\delta)$ when $\delta\in[-b_2,-\delta_2]\cup[b_1-b_2,b_1]$ remains the same as in (\ref{eqn:V-fig-a}).
{\bf Step 2:} The mechanism has five parameters: $h$, $\delta^*$, $\delta_1$, $\delta_2$, and $a_2$. Observe that the $45^\circ$ line segment joining the points $(c+b_2+\delta^*,c+b_2)$ and $(c+\delta^*,c)$ passes through $(c+\delta_1,c+h)$. So we have $\delta_1=h+\delta^*$. Since $q=\nabla u$, a conservative field, we must have the slope of the line separating $(0,0)$ and $(1-a_2,a_2)$ allocation regions satisfying $-\frac{1-a_2}{a_2}=\frac{h-\delta_2}{h+\delta^*}$. This yields $a_2=\frac{h+\delta^*}{\delta_2+\delta^*}$.
We now proceed to compute $h$, $\delta_2$ and $\delta^*$. We do so by equating $\bar{\mu}(Z)=0$, $V(\delta^*)=0$, and $\int_{-\frac{b_2}{3}}^{\delta^*}V(\delta)\,d\delta=0$. The latter two conditions follow from Theorem \ref{thm:Myerson} 3(b) and 3(c) because $q_1(\delta)=1-a_2\in(0,1)$ for $\delta\in[-\frac{b_2}{3},\delta^*]$. We then have the following implications.
\begin{equation}
\bar{\mu}(Z)=0\Rightarrow-\frac{3}{2}(h+\delta^*)(h+\delta_2)-c(\delta_2+h+\delta^*)+b_1b_2=0.\label{eqn:fig-b-delta2}
\end{equation}
From (\ref{eqn:V-alternate}), we see that $V(\delta^*)$ is the negative of $\bar{\mu}$ measure of the nonconvex pentagon bound by $(c,c)$, $(c,c+b_2)$, $(c+b_2+\delta^*,c+b_2)$, $(c+\delta_1,c+h)$, and $(c+\delta_1,c)$. Thus
\begin{align}
V(\delta^*)=0&\Rightarrow-\frac{3}{2}h^2-ch-\frac{3}{2}b_2(b_2+2\delta^*)+b_2(b_2+\delta^*)+b_1b_2=0\label{eqn:fig-b-h-initial}\\&\Rightarrow h=\frac{-c+\sqrt{c^2+3b_2(2b_1-b_2-4\delta^*)}}{3}.\label{eqn:fig-b-h}
\end{align}
Next,
\begin{align}
&\int_{-\frac{b_2}{3}}^{\delta^*}V(\delta)\,d\delta=0\Rightarrow\int_{-\frac{b_2}{3}}^{-\delta_2}V(\delta)\,d\delta+\int_{-\delta_2}^{\delta^*}V(\delta)\,d\delta=0\nonumber\\&\Rightarrow b_2(\delta_2^2-b_2^2/9)+\frac{1}{2}(b_2^3/27-\delta_2^3)+b_2^2/2(b_2/3-\delta_2)\nonumber\\&\hspace*{.1in}-(2b_2\delta_2-3\delta_2^2/2-b_2^2/2)(\delta^*+\delta_2)-(c-2b_2+2\delta_2+h)(\delta^*+\delta_2)^2/2=0\nonumber\\&\Rightarrow\frac{1}{54}(4b_2+3\delta^*)(b_2+3\delta^*)^2-\frac{(c+h+\delta^*)}{2}(\delta^*+\delta_2)^2=0.\label{eqn:fig-b-last}
\end{align}
The values of $h$, $\delta^*$, and $\delta_2$ can be obtained by solving (\ref{eqn:fig-b-delta2}), (\ref{eqn:fig-b-h}), and (\ref{eqn:fig-b-last}) simultaneously. We now proceed to prove that $(h,\delta^*)$ can be obtained by solving (\ref{eqn:fig-b-first}) and (\ref{eqn:fig-b-second}) simultaneously. From (\ref{eqn:fig-b-h-initial}), we get
\begin{equation}\label{eqn:fig-b-first-replica}
3h^2/2+ch+2b_2\delta^*-b_1b_2+b_2^2/2=0
\end{equation}
which is (\ref{eqn:fig-b-first}). We next find an expression for $\delta_2+\delta^*$. Rearranging (\ref{eqn:fig-b-delta2}), we get
\begin{equation}\label{eqn:fig-b-delta2-clear}
\delta_2=\frac{b_1b_2-(3h/2+c)(h+\delta^*)}{3/2(h+\delta^*)+c}=\frac{2b_2\delta^*+b_2^2/2-\delta^*(3h/2+c)}{3/2(h+\delta^*)+c}
\end{equation}
where we have used (\ref{eqn:fig-b-first-replica}). Thus
$$
\delta_2+\delta^*=\frac{(b_2+3\delta^*)(b_2+\delta^*)/2}{3/2(h+\delta^*)+c}.
$$
Plugging this into (\ref{eqn:fig-b-last}), we eliminate $\delta_2$, and obtain
\begin{equation}\label{eqn:fig-b-second-replica}
27(c+h+\delta^*)(b_2+\delta^*)^2-4(4b_2+3\delta^*)(3(h+\delta^*)/2+c)^2=0
\end{equation}
which is (\ref{eqn:fig-b-second}). It is thus clear that $(h,\delta^*)$ can be obtained by simultaneously solving (\ref{eqn:fig-b-first}) and (\ref{eqn:fig-b-second}).
{\bf Step 3:} We now prove that a meaningful solution that satisfies (\ref{eqn:fig-b-first-replica}) and (\ref{eqn:fig-b-second-replica}) exists, by evaluating the bounds of the variables $h$, $\delta^*$, and $\delta_2$ . In Step 3a, we prove the bounds on $(h,\delta^*)$ when $b_1\geq 3b_2/2$. In Step 3b, we prove the bounds on $(h,\delta^*)$ when $b_1\in[b_2, 3b_2/2]$. In Step 3c, we prove the bounds on $\delta_2$ for all $b_1$.
{\bf Step 3a:} Consider the case when $b_1\geq 3b_2/2$. We consider a pair of $(\delta^*,h)$ values that satisfy (\ref{eqn:fig-b-first-replica}) as the end points, and prove that the expression on the left-hand side of (\ref{eqn:fig-b-second-replica}) changes sign at those end points. Given that $h$ is a decreasing function of $\delta^*$ (see (\ref{eqn:fig-b-h})), this suffices to show the bounds of $(\delta^*,h)$.
We claim that when $c\in[0,\beta]$, there exists a $(\delta^*,h)\in[\frac{c^2+6b_1b_2-7b_2^2}{12b_2},\frac{b_1}{2}-\frac{b_2}{4}]\times[0,\frac{2b_2-c}{3}]$ that solves (\ref{eqn:fig-b-first-replica}) and (\ref{eqn:fig-b-second-replica}) simultaneously. Observe that $h$ is a decreasing function of $\delta^*$ (see (\ref{eqn:fig-b-h})), and that the pairs $(\delta^*,h)=(\frac{b_1}{2}-\frac{b_2}{4},0)$ and $(\delta^*,h)=(\frac{c^2+6b_1b_2-7b_2^2}{12b_2},\frac{2b_2-c}{3})$ satisfy (\ref{eqn:fig-b-first-replica}). The choice $h=\frac{2b_2-c}{3}$ will be motivated later. It suffices now to indicate that it is to satisfy condition 3(d) of Theorem \ref{thm:Myerson}. We now prove that the left-hand side of (\ref{eqn:fig-b-second-replica}) has opposite signs at these pairs of $(\delta^*,h)$. Substituting $(\delta^*,h)=(\frac{c^2+6b_1b_2-7b_2^2}{12b_2},\frac{2b_2-c}{3})$, we obtain
\begin{equation}\label{eqn:fig-b-lower-bound}
-\frac{(c-b_2)(6b_1b_2^2+b_2^3+6b_1b_2c+9b_2^2c+b_2c^2+c^3)}{4b_2}\leq 0
\end{equation}
for every $c\geq b_2$. Substituting $(\delta^*,h)=(\frac{b_1}{2}-\frac{b_2}{4},0)$, we obtain
$$
\frac{1}{16}(72b_1^2b_2+144b_1b_2^2-90b_2^3+(-36b_1^2+84b_1b_2+399b_2^2)c-(96b_1+208b_2)c^2)
$$
which is nonnegative for every $c\in[0,\beta]$. So by continuity of (\ref{eqn:fig-b-second-replica}), there exists a $(\delta^*,h)$ in the rectangle $[\frac{c^2+6b_1b_2-7b_2^2}{12b_2},\frac{b_1}{2}-\frac{b_2}{4}]\times[0,\frac{2b_2-c}{3}]$, and by the continuity of (\ref{eqn:fig-b-first-replica}), the pair $(\delta^*,h)$ also satisfies (\ref{eqn:fig-b-first-replica}). We have thus proved our claim.
{\bf Step 3b:} Consider the case when $b_1\in[b_2,3b_2/2]$. We claim that there exists a $(\delta^*,h)\in[\frac{c^2+6b_1b_2-7b_2^2}{12b_2},b_1-b_2]\times[\frac{-c+\sqrt{c^2+3b_2(3b_2-2b_1)}}{3},\frac{2b_2-c}{3}]$ simultaneously solving (\ref{eqn:fig-b-first-replica}) and (\ref{eqn:fig-b-second-replica}). As before, substitution of $(\delta^*,h)=(\frac{c^2+6b_1b_2-7b_2^2}{12b_2},\frac{2b_2-c}{3})$ yields (\ref{eqn:fig-b-lower-bound}). We now substitute the other pair of $(\delta^*,h)$ on the left-hand side of (\ref{eqn:fig-b-second-replica}), and obtain
\begin{multline}\label{eqn:menu-2-upper}
9b_1^2\left(3b_1-3b_2+2c+\sqrt{9b_2^2-6b_1b_2+c^2}\right)\\-(3b_1+b_2)\left(3b_1-3b_2+c+\sqrt{9b_2^2-6b_1b_2+c^2}\right)^2.
\end{multline}
We now show that this expression is nonnegative for every $b_1\in[b_2,3b_2/2]$, $c\in[b_2,\alpha_1]$. We do so by the following steps: (a) We first differentiate the expression with respect to $c$ and show that the differential is nonpositive; (b) We then evaluate the expression at $c=2(t-1)(b_1-b_2)+b_2$ (recall from Remark 2 that $t=3(37+3\sqrt{465})/176$) and show that it is nonnegative; and (c) We finally show that $\alpha_1\leq 2(t-1)(b_1-b_2)+b_2$.
We now differentiate the expression w.r.t. $c$. Fix $v=\sqrt{9b_2^2-6b_1b_2+c^2}$. When $b_1\in[b_2,3b_2/2]$ and $c\geq b_2$, we have
\begin{align*}
\mbox{(i) }&v=\sqrt{9b_2^2-6b_1b_2+c^2}\geq\sqrt{9b_2^2-6(3b_2/2)b_2+c^2}=c,\\
\mbox{(ii) }&v=\sqrt{9b_2^2-6b_1b_2+c^2}\leq\sqrt{9b_2^2-6(b_2)b_2+c^2}=\sqrt{3b_2^2+c^2}\leq 2c.
\end{align*}
So we have $c\leq v\leq 2c$. Differentiating (\ref{eqn:menu-2-upper}) with respect to $c$, we have
\begin{align*}
&18b_1^2+\frac{9b_1^2c}{v}-2(3b_1+b_2)(-3b_2+3b_1+c+v)(1+c/v)\\&=\frac{18b_1^2v+9b_1^2c-2(3b_1+b_2)(c+v)^2-(18b_1^2+2(-6b_1b_2-3b_2^2)(c+v))}{v}\\&=\frac{-9b_1^2c+2(c+v)(3b_2(2b_1+b_2)-(3b_1+b_2)(c+v))}{v}\\&=\frac{-9b_1^2c+2(c+v)((2b_1+b_2)(2b_2-c-v)+b_2(2b_1+b_2)-b_1(c+v))}{v}\\&\leq\frac{-9b_1^2c+2(c+v)b_2^2}{v}\leq\frac{-9b_1^2c+6cb_2^2}{v}\leq 0
\end{align*}
where the first inequality follows from $c+v\geq 2c\geq 2b_2$, the second inequality from $c+v\leq 3c$, and the third inequality from $b_2\leq b_1$.
We now proceed to evaluate the expression at $c=2(t-1)(b_1-b_2)+b_2$. Substituting $c=2(t-1)(b_1-b_2)+b_2$ in (\ref{eqn:menu-2-upper}), we now verify if
\begin{multline*}
\frac{15(117\sqrt{465}-4189)b_1^3+13(13417-225\sqrt{465})b_1^2b_2}{1936}\\-\frac{(70269-981\sqrt{465})b_1b_2^2+9(5021-21\sqrt{465})b_2^3}{1936}\\+\left(\frac{-(201+27\sqrt{465})b_1^2+(134+18\sqrt{465})b_1b_2+(111+9\sqrt{465})b_2^2}{44}\right)\\\sqrt{9b_2^2-6b_1b_2+\left(2\left(\frac{3(37+3\sqrt{465})}{176}-1\right)(b_1-b_2)+b_2\right)^2}\geq 0
\end{multline*}
Writing the above expression as $X+Y\sqrt{Z}$, we note that (i) $X\leq 0$ when $b_1\in b_2[1,1.03873]$, and $X\geq 0$ when $b_1\in b_2[1.03873,1.5]$; (ii) $Y\geq 0$ when $b_1\in b_2[1,1.04088]$, and $Y\leq 0$ when $b_1\in b_2[1.04088,1.5]$. So we now verify if $X^2-Y^2Z\leq 0$ when $b_1\in b_2[1,1.03873]$, and if $X^2-Y^2Z\geq 0$ when $b_1\in b_2[1.04088,1.5]$. That $X+Y\sqrt{Z}\geq 0$ when $b_1\in b_2[1.03873,1.04088]$ is clear since both $X$ and $Y$ are positive in that interval. Evaluating $X^2-Y^2Z$, we have
\begin{multline*}
\frac{9}{42592}(b_1-b_2)(3b_2-2b_1)((20196\sqrt{465}-447876)b_2^4\\+(108900\sqrt{465}-2234628)b_1b_2^3+(32337\sqrt{465}-952857)b_1^2b_2^2\\+(4841141-276237\sqrt{465})b_1^3b_2+(140940\sqrt{465}-1820460)b_1^4)
\end{multline*}
which is negative when $b_1\in b_2[1,1.03977]$ and positive when $b_1\in b_2[1.03977,1.5]$. We have thus shown that the expression in (\ref{eqn:menu-2-bound}) is nonnegative when $b_1\in[b_2,3b_2/2]$, $b_2\leq c\leq 2(t-1)(b_1-b_2)+b_2$. That $\alpha_1\leq 2(t-1)(b_1-b_2)+b_2$ is shown via Mathematica (see \ref{app:c.1}(4)).
{\bf Step 3c:} For both the cases, we now claim that $\delta_2\in[\frac{2b_2-c}{3},\frac{b_2}{3}]$. To prove the claim, we do the following.
\begin{itemize}
\item We show the upper bound $\delta_2\leq\frac{b_2}{3}$ via Mathematica (see \ref{app:c.1}(2)).
\item We next show the lower bound. Since $\delta_2=(b_1b_2-(3h/2+c)(h+\delta^*))/(3(h+\delta^*)/2+c)$ decreases with $(h+\delta^*)$, we first find the upper bound on $(h+\delta^*)$.
\item We then substitute this obtained upper bound on $(h+\delta^*)$ and simplify, resulting in the lower bound $\delta_2\geq\frac{2b_2-c}{3}$.
\end{itemize}
We now fill in the details. To find the upper bound on $\delta_1=h+\delta^*$, we first show that $\delta_1$, as a function of $\delta^*$, decreases with increase in $\delta^*$. Differentiating the expression for $\delta_1=(h+\delta^*)$ with $h$ as in (\ref{eqn:fig-b-h}), we get $1-\frac{2b_2}{\sqrt{c^2+3b_2(2b_1-b_2-4\delta^*)}}$ which is nonpositive for $\delta^*\geq(c^2+6b_1b_2-7b_2^2)/(12b_2)$. But this is exactly the lower bound that we computed for $\delta^*$. The highest value of $\delta_1$ thus occurs at $(h,\delta^*)=(\frac{2b_2-c}{3},\frac{c^2+6b_1b_2-7b_2^2}{12b_2})$. Using these expressions, we get $\delta_1=(h+\delta^*)\leq\frac{c^2+6b_1b_2+b_2^2-4b_2c}{12b_2}$.
We now substitute the end points of $h+\delta^*$ in (\ref{eqn:fig-b-delta2-clear}), to evaluate the lower bound of $\delta_2$.
\begin{align*}
\delta_2&=\frac{b_1b_2-(3h/2+c)(h+\delta^*)}{3(h+\delta^*)/2+c}\\&\geq\frac{b_1b_2-(b_2+c/2)(c^2+6b_1b_2+b_2^2-4b_2c)/(12b_2)}{(c^2+6b_1b_2+b_2^2+4b_2c)/(8b_2)}\\&=\frac{2b_2-c}{3}+\frac{4b_2(c^2-b_2^2)}{3(c^2+6b_1b_2+b_2^2+4b_2c)}\\&\geq\frac{2b_2-c}{3}
\end{align*}
where the first inequality occurs from the upper bound $h\leq (2b_2-c)/3$ and the above upper bound on $(h+\delta^*)$, and the second inequality from $c\geq b_2$. We have thus shown the lower bound. We have also shown that the probability of allocation $a_2=\frac{h+\delta^*}{\delta_2+\delta^*}\leq 1$, since $\delta_2\geq\frac{2b_2-c}{3}\geq h$.
{\bf Step 4:} We now proceed to prove parts (c) and (d) of Theorem \ref{thm:Myerson} (2)--(4). The expression for $V'(\delta)$ is the same as in the proof of Theorem \ref{thm:menu-1}, except in $[-\delta_2,\delta^*]$, where it is given by
\begin{equation}\label{eqn:menu-2-V'}
V'(\delta)=-(c-2b_2+3\delta_2)+3\frac{\delta_2-h}{\delta_2+\delta^*}(\delta+\delta_2),\forall\delta\in(-\delta_2,\delta^*].
\end{equation}
From (\ref{eqn:menu-1-V'}), observe that $V'(\delta)$ is negative when $\delta\in[-b_2,-\frac{2b_2}{3}]$ and positive when $\delta\in[-\frac{2b_2}{3},-\frac{b_2}{3}]$. We also have from (\ref{eqn:V-fig-a}) that $V(-b_2)=V(-\frac{b_2}{3})=0$. So $V(\delta)=V(-b_2)+\int_{-b_2}^{\delta}V'(\tilde{\delta})\,d\tilde{\delta}\leq 0$ for all $\delta\in[-b_2,-\frac{b_2}{3}]$. It follows that $\int_{-b_2}^{-\frac{b_2}{3}}V(\delta)\,d\delta\leq 0$, and that $\int_{-b_2}^xV(\delta)\,d\delta\geq\int_{-b_2}^{-\frac{b_2}{3}}V(\delta)\,d\delta$ for all $x\in[-b_2,-\frac{b_2}{3}]$. Thus condition (2) of Theorem \ref{thm:Myerson} is verified.
We now prove that $\int_{-\frac{b_2}{3}}^{x}V(\delta)\,d\delta\geq 0$ for every $x\in[\frac{b_2}{3},\delta^*]$. Observe that $V'(\delta)$ is positive when $\delta\in[-\frac{b_2}{3},-\delta_2]$, negative when $\delta\in[-\delta_2,l_2]$ for some $l_2\in[-\delta_2,\delta^*]$, and positive when $\delta\in[l_2,\delta^*]$. These statements follow from (i) $\delta_2\geq\frac{2b_2-c}{3}$, (ii) $V'(\delta)$ increasing in the interval $[-\delta_2,\delta^*]$, and (iii) $h\leq\frac{2b_2-c}{3}$, all of which can be obtained from (\ref{eqn:menu-2-V'}). We also have $V(-\frac{b_2}{3})=V(\delta^*)=\int_{-\frac{b_2}{3}}^{\delta^*}V(\delta)\,d\delta=0$, which we used to derive the parameters $h$, $\delta_2$, and $\delta^*$. It follows that $\int_{-\frac{b_2}{3}}^xV(\delta)\,d\delta\geq 0$ for all $x\in[-\frac{b_2}{3},\delta^*]$. Thus condition (3) of Theorem \ref{thm:Myerson} is verified.
The proof that the conditions of Theorem \ref{thm:Myerson} (4) are satisfied trace the same steps as in the proof of Theorem \ref{thm:menu-1}, provided $\delta_1\leq\frac{2b_1-c}{3}$. If $\delta_1>\frac{2b_1-c}{3}$, then $V'(\delta)$ is no more positive in the interval $[b_1-b_2,\delta_1]$. We consider two cases.
Let $b_1\geq 3b_2/2$. Then we claim that $V(\delta)\geq 0$ holds for all $\delta\in[\delta^*,b_1]$, even when $V'(\delta)\leq 0$ for $\delta\in[b_1-b_2,\delta_1]$. Observe that (i) $V(\delta)=\frac{1}{2}(3\delta-b_1)(b_1-\delta)\geq 0$ for all $\delta\in[\max(b_1-b_2,\frac{b_1}{3}),b_1]$, and (ii) $\delta_1\geq b_1-b_2\geq\frac{b_1}{3}$, when $b_1\geq 3b_2/2$. So, $V(\delta)\geq 0$ for all $\delta\in[b_1-b_2,b_1]$. Now $V(\delta)\geq 0$ also holds in the interval $\delta\in[\delta^*,b_1-b_2]$ since $V'(\delta)\geq 0$ in that interval (see the discussion following (\ref{eqn:menu-1-V'})), and since $V(\delta^*)=0$. We have proved our claim.
We now consider the case when $b_1\in[b_2,3b_2/2]$. $V(\delta)$ could possibly be negative at some values of $\delta$. We now evaluate $\int_{\delta^*}^{\frac{b_1}{3}}V(\delta)\,d\delta$:
\begin{align*}
&\int_{\delta^*}^{\frac{b_1}{3}}V(\delta)\,d\delta\\&=\int_{\delta^*}^{b_1-b_2}V(\delta)\,d\delta+\int_{b_1-b_2}^{\delta_1}V(\delta)\,d\delta+\int_{\delta_1}^{\frac{b_1}{3}}V(\delta)\,d\delta\\&=-\frac{2}{27}b_1^3-b_2(\delta^*)^2+b_2\delta^*(b_1-b_2/2)+b_1b_2h-\frac{b_2^2h}{2}-2b_2h\delta^*-\frac{ch^2}{2}-h^3\\&=-\frac{2}{27}b_1^3+\frac{(c+h)}{2}h^2-b_2(\delta^*)^2+b_2\delta^*(b_1-b_2/2)+hV(\delta^*)
\end{align*}
where $V(\delta^*)$ is obtained from (\ref{eqn:fig-b-h-initial}). The last expression is the same as (\ref{eqn:fig-b-third}), since $V(\delta^*)=0$. From Mathematica, (\ref{eqn:fig-b-third}) is nonnegative for all $c\in[b_2,\alpha_1]$ (see
\ref{app:c.1}(3)). Since $\int_{\frac{b_1}{3}}^{b_1}V(\delta)\,d\delta=\frac{2}{27}b_1^3\geq 0$, we have $\int_{\delta^*}^{b_1}V(\delta)\,d\delta\geq 0$. This verifies condition 4(c) of Theorem \ref{thm:Myerson}.
Observe that $V'(\delta)\leq 0$ only when $\delta\in[b_1-b_2,\delta_1]$. Also, $V(\delta^*)=0=V(\frac{b_1}{3})$. So $V(\delta)$ can be negative only when $\delta$ is in some subset of $[b_1-b_2,\frac{b_1}{3}]$, say in the interval $[l_1,\frac{b_1}{3}]$. Observe that the integral $\int_x^{b_1}V(\delta)\,d\delta$ thus attains its maximum either at $\delta^*$ or at $\frac{b_1}{3}$. But we just evaluated $\int_{\delta^*}^{\frac{b_1}{3}}V(\delta)\,d\delta\geq 0$, and so the maximum cannot be at $x=\frac{b_1}{3}$. Thus we have $\int_x^{b_1}V(\delta)\,d\delta\leq\int_{\delta^*}^{b_1}V(\delta)\,d\delta$ for all $x\in[\delta^*,b_1]$. Hence the result.\qed
\end{proof}
Observe that at $c=\alpha_1$, we have $\int_{\delta^*}^{\frac{b_1}{3}}V(\delta)\,d\delta=0$. When $c>\alpha_1$, the quantity turns negative, causing the condition in Theorem \ref{thm:Myerson}(4d) to fail. A transition occurs from the structure depicted in Figure \ref{fig:b-ini} to that depicted in Figure \ref{fig:c-ini}. We now proceed to prove the optimality of the structure in Figure \ref{fig:c-ini}, i.e., part 1(c) of Theorem \ref{thm:consolidate}.
\begin{theorem}\label{thm:menu-3}
Consider the case when $b_1\in[b_2,3b_2/2]$, and $c\in[\alpha_1,\alpha_2]$, where $\alpha_1$ and $\alpha_2$ are as defined as in Theorem \ref{thm:consolidate}. Then the optimal mechanism is as depicted in Figure \ref{fig:c-ini} (see also Figure \ref{fig:c-new}). The values of $h$ and $\delta^*$ are found by solving (\ref{eqn:fig-b-second}) and (\ref{eqn:fig-c-second}) simultaneously, and the values of $\delta_1$ and $\delta_2$ are given by
$$
(\delta_1,\delta_2)=\left(\delta^*+\frac{b_1b_2-2b_2\delta^*-b_2^2/2}{3h/2+c},\frac{b_1b_2-(3h/2+c)\delta_1}{3/2(h+\delta^*)+c}\right).
$$
The values of $a_1$ and $a_2$ are given by $(a_1,a_2)=\left(\frac{h}{\delta_1-\delta^*},\frac{h+\delta^*}{\delta_2+\delta^*}\right)$.
\end{theorem}
\begin{proof}
See \ref{app:b}. This too relies on Mathematica for verification of certain inequalities.\qed
\end{proof}
Consider $b_1\in[b_2,3b_2/2]$. The proof (in \ref{app:b}) indicates that at $c=\alpha_2$, we have $a_1+a_2=1$, and that when $c>\alpha_2$, we have $a_1+a_2<1$. This causes the monotonicity of $q_1$ to fail (recall that $q_1$ increasing is one of the constraints of Problem (\ref{eqn:optim-myerson})). Further, when $a_1+a_2=1$, the slope of the line segment joining $(c,c+\delta_2)$, $(c+h+\delta^*,c+h)$, and the slope of the line segment joining $(c+h+\delta^*,c+h)$, $(c+\delta_1,c)$, are equal, i.e., $-\frac{1-a_2}{a_2}=-\frac{a_1}{1-a_1}$. The two line segments thus turn into a single line segment that joins $(c,c+\delta_2)$, $(c+\delta_1,c)$. A transition thus occurs from the structure depicted in Figure \ref{fig:c-ini} to that in Figure \ref{fig:d-ini}, with $a_2=1-a_1=a$.
Consider $b_1\geq 3b_2/2$. At $c=\beta$, we have $h=0$. Thus a transition occurs from the structure depicted in Figure \ref{fig:b-ini} to that in Figure \ref{fig:d'-ini}.
We now proceed to prove the optimality of the structures depicted in Figures \ref{fig:d-ini}--\ref{fig:e'-ini}, i.e., parts 1(d)--(e) and 2(c)--2(d) of Theorem \ref{thm:consolidate}.
\begin{theorem}\label{thm:menu-4,5}
\begin{enumerate}
\item[(i)] Consider the case when $c\in[\beta,\frac{216b_1^2b_2}{108b_1^2-108b_1b_2-5b_2^2}]$, and $b_1\geq 3b_2/2$, where $\beta$ is as defined Theorem \ref{thm:consolidate}. Then the optimal mechanism is as depicted in Figure \ref{fig:d'-ini} (see also Figure \ref{fig:d'-new}). The values of $\delta_1$ and $\delta_2$ are computed by solving the following equations simultaneously.
\begin{align*}
&-\frac{3}{2}\delta_1\delta_2-c(\delta_1+\delta_2)+b_1b_2=0.\\&-\frac{2}{27}b_2^3+\frac{1}{2}\delta_1\delta_2(\delta_2-\delta_1)+\frac{c}{2}(\delta_2^2-\delta_1^2)+\frac{1}{16}b_2(2b_1-b_2)^2=0.
\end{align*}
The value of $a$ is given by $a=\frac{\delta_1}{\delta_1+\delta_2}$. If $c\geq\frac{216b_1^2b_2}{108b_1^2-108b_1b_2-5b_2^2}$, then the optimal mechanism is as depicted in Figure \ref{fig:e'-ini} (see also Figure \ref{fig:e'-new}).
\item[(ii)] Consider the case when $b_1\in[b_2,3b_2/2]$, and $c\in[\alpha_2,\frac{27b_1^2b_2^2}{4(b_1^3-b_2^3)}]$, where $c=\alpha_2$ is as defined in Theorem \ref{thm:consolidate}. Then the optimal mechanism is as depicted in Figure \ref{fig:d-ini} (see also Figure \ref{fig:d-new}). The values of $\delta_1$ and $\delta_2$ are computed by solving the following equations simultaneously.
\begin{align*}
&-\frac{3}{2}\delta_1\delta_2-c(\delta_1+\delta_2)+b_1b_2=0.\\&\frac{2}{27}(b_1^3-b_2^3)+\frac{1}{2}\delta_1\delta_2(\delta_2-\delta_1)+\frac{c}{2}(\delta_2^2-\delta_1^2)=0.
\end{align*}
The value of $a$ is given by $a=\frac{\delta_1}{\delta_1+\delta_2}$. If $c\geq\frac{27b_1^2b_2^2}{4(b_1^3-b_2^3)}$, then the optimal mechanism is as depicted in Figure \ref{fig:e-ini} (see also Figure \ref{fig:e-new}).
\end{enumerate}
\end{theorem}
\begin{proof}
See \ref{app:b}.\qed
\end{proof}
\section{On Extending to Uniform Distributions on General Rectangles}
We have computed the optimal mechanism in the two-item unit-demand setting when $z\sim\mbox{Unif}[c,c+b_1]\times[c,c+b_2]$ for every nonnegative $(c,b_1,b_2)$. Our computation used the method based on the virtual valuation function designed in \cite{Pav11}. We can now ask if there is a generalization of this method for more general distributions, specifically for uniform distributions on rectangles $[c_1,c_1+b_1]\times[c_2,c_2+b_2]$, when $c_1\ne c_2$. We conjecture that the optimal mechanisms would have structures similar to the five structures as in the case of $c_1=c_2$. We now report some promising preliminary results that support this conjecture.
\begin{theorem}\label{thm:extension}
Consider the case when $b_1\geq b_2$. Let $c_2\geq 0$, $c_1\geq c_2$, and $2c_1-c_2\leq b_2$. Then, the optimal mechanism is as depicted in Figure \ref{fig:a-ini} (see also Figure \ref{fig:a-new}). The values of $\delta_1$ and $\delta_2$ are computed by solving the following equations simultaneously.
\begin{align*}
&-3\delta_1\delta_2-c_2\delta_1-c_1\delta_2+b_1b_2=0.\\
&-\frac{3}{2}\delta_2^2+2b_2\delta_2-\frac{b_2^2}{2}-d(b_2-\delta_2)+(c_2-2b_2+3\delta_2)\delta_1=0.
\end{align*}
\end{theorem}
\begin{proof}
See \ref{app:b}. The proof traces the same steps as in the proof of Theorem \ref{thm:menu-1}.\qed
\end{proof}
\section{Conclusion and Future Work}
We solved the problem of computing the optimal mechanism for the two-item one-buyer unit-demand setting, when the buyer's valuation $z\sim\mbox{Unif}[c,c+b_1]\times[c,c+b_2]$ for arbitrary nonnegative values of $(c,b_1,b_2)$. Our results show that a wide range of structures arise out of different values of $c$. When the buyer guarantees that his valuations for the items are at least $c$, the seller offers different menus based on the guaranteed minimum $c$ and the upper bounds $c+b_i$, $i = 1,2$.
Taking a cue from the solution method in the unrestricted setting \cite{TRN16}, we initially attempted to solve the problem using the duality approach in \cite{DDT15}, but constructing a dual measure in the unit-demand setting turned out to be intricate. We then used the virtual valuation method used in \cite{Pav11} to compute the solution. We now characterize the pros and cons of these approaches.
The duality approach could not be pursued systematically because the construction of a shuffling measure that both convex-dominates $0$ and spans over more than one line segment appears to be difficult. Observe that in both Examples 2 and 3, there exists some constant allocation region that is a part of both the top boundary and the right boundary of $D$. So the shuffling measure had to be constructed so that it spans over two line segments connected at the top-right corner of $D$. To get around this issue, we had to construct (i) a shuffling measure on the line $z_1+z_2=2c+\delta_2$ in Example 2, and (ii) a shuffling measure that transfers mass horizontally in Example 3. The problem of constructing a ``generalized'' shuffling measure that both convex-dominates $0$ and also spans over two segments, thereby rendering the dual approach practical, is a possible direction for future work.
The virtual valuation method on the other hand, did not pose any issue when constant allocation regions span over the top-right corner. The approach provides a generalized procedure to verify if a menu at hand is optimal or not, under the (only) constraint that the distribution satisfies the negative power rate condition (stated in Theorem \ref{thm:pav-1}). So unlike the duality approach, we cannot use this approach to solve the problem for general distributions. But our results for $z\sim\mbox{Unif}[c,c+b_1]\times[c,c+b_2]$ and the extension to general rectangles suggest that this approach can be used to solve the problem of computing the optimal mechanism for all distributions satisfying the negative power rate condition. The key challenge in solving these problems is to find the exclusion region $Z$ for arbitrary distributions, so that we can use Theorem \ref{thm:Myerson} to verify if the menu is optimal or not. Coming up with a generalized procedure to compute $Z$ is a possible direction for future work.
Our proofs used Mathematica to verify certain algebraic inequalities that turn out to be complicated functions of $(c,b_1,b_2)$ involving fifth roots and eighth roots of some expressions. This leads us to the following questions. From a rather abstract perspective, Pavlov's sufficient conditions lead to the identification of a family of polynomial equalities and inequalities in the variables $(h,\delta^*,\delta_1,\delta_2)$ in Figures \ref{fig:a-new}--\ref{fig:e'-new}, indexed by the parameters $(c,b_1,b_2)$. In a nutshell, our work is a careful analysis of the solution space, denoted $L_{c,b_1,b_2}$, associated with the polynomial equalities and inequalities. We argued that $L_{c,b_1,b_2}$ is nonempty for every parameter $(c,b_1,b_2)$. We also captured the transitions of $L_{c,b_1,b_2}$ as the parameters vary. Can this view provide a more systematic procedure to solve the case of uniform distribution on any rectangle in the positive quadrant, or more generally, the case of any distribution of valuations on the positive quadrant? Alternatively, can the procedure of this paper (both existence of solutions and capture of transitions) be automated on Mathematica or other similar tool? These are some computation related problems that might be of interest to the computer scientists.
\section*{Acknowledgements}
This work was supported by the Defence Research and Development Organisation [Grant no. DRDO0667] under the DRDO-IISc Frontiers Research Programme. The first author thanks Prof. G. Pavlov for a very informative discussion.
|
1,314,259,993,653 | arxiv | \section*{Introduction}
{
Characterizing exoplanets atmospheres has recently become within reach. Nowadays, a significant number of atmospheric measurements have been acquired on a dozen of exoplanets. Unfortunately, none of those measurements were done with a dedicated instrument. Although { researchers have} made the best use of available telescopes, the observations still suffer from large error bars, from possible instrumental noise~\citep{Hansen2014}, are averaged over large bins of frequency, and measurements at different wavelength are usually made at different times. The construction of a reliable spectrum is therefore a difficult task. Few unambiguous molecular detections have been claimed and most of the physical characterizations are qualitative rather than quantitative. Better data are needed. The future of exoplanet characterization should be based on high signal-to-noise, spectrally resolved observations with a large spectral coverage accessible in a single observation.
A mission with those capabilities was proposed to the ESA Cosmic Vision program in 2014. With its large spectral coverage ($4-16\, \micro\meter$), high spectral resolution ($\lambda/\Delta\lambda>300$ below $5\,\micro\meter$ and $\lambda/\Delta\lambda>30$ above $5\,\micro\meter$), and $1.5\,\meter$ mirror, EChO (the Exoplanet Characterization Observatory) is an ideal instrument to characterize exoplanets atmospheres~\citep[see][for more technical details about the mission]{Tinetti2012}. Although it was not selected in 2014, it should serve as a baseline for future missions with similar goals. The following review is based on the expected capabilities of EChO but is also relevant for any future mission with similar characteristics. In the following, the term \emph{EChO} should therefore be understood as \emph{an EChO-class mission}.
We will now review why a mission such as EChO will be a decisive step toward understanding exoplanets atmospheres and atmospheric physics in general.}
\section*{On the large diversity of observable exoplanets atmospheres}
\begin{figure}[h!]
\begin{minipage}[c]{0.48\linewidth}
\includegraphics[width=1\linewidth, trim=25 10 50 25]{./TauSync2.pdf}
\centering
\caption{Tidal synchronization timescale based on~\citet{Guillot1996} for all known exoplanets with a measured mass and radius in function of their orbital period for a dissipation factor $Q=6\times10^{5}$, typical for hot Jupiters~\citep{Ferraz-Mello2013} and an initial rotation rate equal to Jupiter's one. Planets in the shaded area are likely to be tidally locked.}
\label{fig::PlotSync}
\end{minipage}
\hfill
\begin{minipage}[c]{0.48\linewidth}
\centering
\includegraphics[width=0.85\linewidth, trim=25 10 50 25]{./TeqG.pdf}
\caption{Equilibrium temperature (assuming zero albedo) of exoplanets with a measured mass and radius. Planets are color-coded by their gravity. The blue (red) line is the equilibrium temperature for a planet orbiting a M5 (A5) type star. Planets with an orbital period smaller than $\approx 10$ days are likely to have a rotation period equal to their orbital period (see Fig.~\ref{fig::PlotSync}).}
\label{fig::PlotTeq}
\end{minipage}
\end{figure}
Most EChO targets -- and the ones for which the best observations will be available -- are planets orbiting close to their host star. Tidal interactions should force them toward a tidally locked state~\citep{Lubow1997,Guillot2002} where their rotation period is the same as their revolution period (see Fig.~\ref{fig::PlotSync}). A whole range of atmospheric constraints is obtainable for those close-in, tidally locked planets because \emph{we know which hemisphere is facing us at any orbital phase}. Monitoring the star-planet system during its whole orbit, one can obtain longitudinal information on the planet's brightness distribution~\citep{Knutson2008}. During the ingress and egress of the secondary eclipse, the technique of eclipse mapping~\citep{Majeau2012,DeWit2012} can constrain the horizontal (both longitudinal and latitudinal) brightness distribution of the planet's dayside. { Finally, the frequency dependence of the thermal flux emitted by the planet and of the stellar flux filtered { through} the planet atmosphere during transit depends principally on the temperature profile, the atmospheric composition and their variations with depth~\citep[]{Barstow2013,deWit2013}.Thus, with a high enough signal-to-noise ratio and a large enough spectral coverage}, the spectral resolution of transmission and emission spectra can translate into vertical resolution of the temperature and composition of the atmosphere. Combining those techniques, EChO will provide a three dimensional vision of numerous close-in planets.
Hundreds of close-in transiting planets with very different gravities and orbital periods are already known and more will be discovered and confirmed before the launch of the mission. Although, for a given star, the irradiation is only function of the distance to the star, the large diversity in exoplanets stellar hosts ensure a good coverage of the rotation period / equilibrium temperature parameter space. { As seen in Fig.~\ref{fig::PlotTeq}, the irradiation temperature can vary by a factor $4$ (corresponding to a factor $256$ for the irradiation flux) between planets with similar rotation period but orbiting different stellar types. Planet gravity, for its part, varies by more than two orders of magnitude among known planets, ranging from $\approx 2.5$ to $\approx500\meter\per\second\squared$. The sample of planets EChO will observe thus covers a large area in the irradiation / rotation / planet gravity parameter space, three of the main parameters shaping the atmospheric circulation.}
Thermal structure, composition and atmospheric circulation are essential characteristics of planetary atmospheres. They affect each other via the different mechanisms described in Fig.~\ref{fig::Triangle}. The thermal structure sets the chemical equilibrium whereas the composition determines the atmospheric opacities, controlling the radiative transfer and thus the temperature. The atmospheric circulation is driven by the temperature contrasts. It transports heat and material, { which} shapes the temperature and composition both horizontally and vertically. Finally, the presence of ionized material directly affects the circulation via the Lorentz forces. The spatial variation of the temperature and composition, together with their departure from equilibrium are thus signatures of the atmospheric circulation.
EChO can observe hundreds of exoplanets atmospheres with a high spectral resolution and an exquisite photometric precision. It can obtain a full exoplanet spectrum in one observation and will be able to observe periodically a given target. Such a mission is essential to determine the spatio-temporal variability of exoplanets atmospheres and understand their diversity in terms of composition, thermal structure and dynamics. Hereafter we list several key scientific questions concerning { the thermal structure and atmospheric dynamics of} gas giant atmospheres that EChO's observations will help to solve. { Questions related to atmospheric chemical composition are treated in a separate article.}
\begin{figure}[h]
\centering
\includegraphics[width=0.7\linewidth]{./Triangle.png}
\captionsetup{justification=centering}
\caption{Schematic view of the main atmospheric characteristics and how they affect each other.}
\label{fig::Triangle}
\end{figure}
\section*{Key questions in atmospheric structure and dynamics to be addressed by an EChO-class mission}
\subsection{What is the longitudinal structure of the temperature
in hot Jupiter atmospheres, and how does it depend on depth?}
High-quality lightcurves---as obtainable from EChO for a wide range of
close-in planets---will allow longitudinal maps of brightness
temperature to be derived. This will allow the
longitudinal locations of hot and cold spots, among other features, to
be identified; observations at many wavelengths will allow the
depth-dependence to be determined in the range $\sim$0.001--10
bar. Spitzer observations of several hot Jupiters, including HD
189733b~\citep{Knutson2007,Knutson2009,Knutson2012}, Ups And b~\citep{Crossfield2010}, and WASP-43b~\citep{Stevenson2014} indicate that the hottest regions are displaced eastward of
the substellar point by tens of degrees of longitude or more (see Figs.~\ref{fig::Knutson1} and~\ref{fig::Knutson2}). This
phenomenon was predicted and has now been reproduced in a
wide range of three-dimensional circulation models under conditions
appropriate to benchmark hot Jupiters such as HD 189733b and HD
209458b \citep{Showman2002,Cooper2005,Showman2008,Showman2009,Menou2009,Dobbs-Dixon2008,Dobbs-Dixon2010,Rauscher2010,Rauscher2012b,Heng2011a,Heng2011,Perna2012}. In these models, the eastward
displacement results from advection by an eastward ``superrotating''
jet stream at the equator. Theory shows that, on tidally locked
planets, such superrotation is the natural result of the day-night
heating pattern, which leads to planetary-scale waves that pump
angular momentum to low latitudes~\citep{Showman2011}. Nevertheless, current predictions---yet to be tested---suggest
that the longitudinal offset of the hotspot should scale inversely
with incident stellar flux~\citep{Showman2011,Perna2012,Showman2013}. The extent to which such longitudinal
offsets are prevalent on hot Jupiters---and their dependence on
incident stellar flux, planetary rotation rate, atmospheric
composition, and other factors---remains unknown. { Recent
magnetohydrodynamic calculations that properly represent the full
coupling of the dynamics to the magnetic field furthermore suggest that,
under particularly hot conditions, a westward equatorial jet can sometimes
emerge~\citep{Rogers2014,Rogers2014a}, potentially
leading to a {\it westward} hot spot offset in these cases.}
EChO can address
this question with a broad census, { determining the amplitude and
sign of the offset under a broad range of conditions,} and map the depth dependence of
these features.
\begin{figure}[h!]
\begin{minipage}[c]{0.48\linewidth}
\includegraphics[height=8.5cm]{./fig1.pdf}
\centering
\caption{Thermal phase curve of HD189733 observed with the IRAC instrument on the Spitzer Space Telescope at 8 microns by~\citet{Knutson2007}. In the top panel, the transit (orbital phase 0) and secondary eclipse of the planet orbital phase 0.5) are visible. In the bottom panel, the increase of flux between the transit and the secondary eclipse is due to the planet phase: before and after the transit the planet shows its cold and thus dark nightside whereas before and after the secondary eclipse it shows its warm, and thus luminous, dayside.{\it Reprinted by permission from Macmillan Publishers Ltd. Nature Copyright 2007.} }
\label{fig::Knutson1}
\end{minipage}
\hfill
\begin{minipage}[c]{0.48\linewidth}
\centering
\includegraphics[height=4.5cm, trim=0 0 40 0]{./fig3a.pdf}
\includegraphics[height=4cm]{./fig3b.pdf}
\caption{Longitudinal temperature map of the planet HD189733b retrieved from the phase curve observation depicted in the previous figure~\citep[from]{Knutson2007}. The shift of the hottest point of the planet east of the substellar point is attributed to fast eastward equatorial winds~\citep{Showman2009}.{\it Reprinted by permission from Macmillan Publishers Ltd. Nature Copyright 2007.}
\vspace{1.8cm}}
\label{fig::Knutson2}
\end{minipage}
\end{figure}
\subsection{What sets the day-night temperature contrast? How
does it vary with depth (wavelength) and among different planets?
What is the mechanism that controls the day-night temperature
contrast on tidally locked planets?}
Current lightcurve observations have allowed the day-night brightness
temperature contrast to be determined for over a dozen hot Jupiters.
These observations suggest a trend wherein cooler planets exhibit
modest fractional day-night temperature contrasts whereas hotter
planets exhibit near-unity fractional day-night temperature variations~\citep{Cowan2011,Perna2012,Perez-Becker2013a}. { As emphasized by
Perez-Becker \& Showman 2013,} the details of this trend place strong constraints on the
mechanisms that maintain the day-night temperature differences on hot
Jupiters (e.g., on the relative roles of horizontal advection, vertical advection, wave
propagation, and radiative cooling) and on the conditions under which frictional drag and ohmic drag become important~\citep{Li2010,Rauscher2012a,Rauscher2013, Showman2013}. Current observations exist at
only a few broadband wavelengths, and full spectral information as obtainable
from EChO would provide significant information on how the
transition from small to large fractional day-night flux difference
depends on wavelength, and in turn how this transition depends
on depth in the atmosphere.
{
\subsection{What physical mechanisms determine the vertical temperature profile at the terminator of the planet ?}
The terminator of close-in, tidally locked planets is extremely interesting but very complex. It is located at the middle of the largest temperature gradients and where the fastest winds are present. Hydrodynamics shocks might be present~\citep{Heng2012a}. Scattering should become important due to the grazing path of the stellar rays~\citep{Fortney2005b}. Condensation of numerous species is expected to take place close to the terminator, depositing latent heat and increasing even more the importance of scattering. From the combined effects of the dynamics and the condensation processes, a significant differences in the cloud coverage between the western and the eastern atmospheric limbs is expected~\citep{Iro2005}. Whether the ions produced in the hot dayside recombine before or after crossing the terminator will influence the strength of the magnetic forces acting on the fluid. At low pressures, non local thermodynamic equilibrium (LTE) effects should also play a major role~\citep{Barman2002}.
\begin{figure}[h]
\includegraphics[width=0.495\linewidth,clip]{./Temp-HD189-Limb-b.pdf}
\includegraphics[width=0.495\linewidth,clip]{./Temp-HD209-Limb-b.pdf}
\caption{Pressure-temperature profile at the terminator of the hot Jupiter HD189733b (left) and HD209458b (right). Data are retrieved from the sodium absorption line of the planet observed during transit by~\citet{Vidal-Madjar2011} and~\citet{Huitson2012}. For HD 209458b, the pressure scale is based on the detection of the Rayleigh scattering by $\rm H_{\rm 2}$. For HD 189733b, the pressure scale is model dependent: it is determined assuming that the top of the cloud deck is at $10^{-4}\,\bbar$. The red line is obtained from the grid of 1D numerical models used in~\citet{Parmentier2014b}. The green lines are all the limb temperature profiles predicted by the 3D model (SPARC/MIT GCM). The difference between the 1D and the 3D temperature profiles is mainly due to the advection of heat by the atmospheric circulation. At pressures lower than $10^{-5}\,\bbar$, non-LTE effects, not taken into account in the models become important~\citet{Barman2002}. Figure adapted from~\citet{Huitson2012}.}
\label{fig::Huitson2012}
\end{figure}
The temperature at the terminator of a planet can be retrieved from the slope of the spectral features apparent in the transit spectrum~\citep{LecavelierDesEtangs2008}. From the absorption feature of the Sodium D line, the temperature profile at the terminator of HD 189733b~\citep{Vidal-Madjar2011} and HD 209458b~\citep{Huitson2012} have been retrieved. As shown in Figure~\ref{fig::Huitson2012}, the retrieved temperatures in the upper atmosphere of HD 189733b and HD 209458b are larger than predicted by current LTE models. They are nonetheless consistent with observations of hot hydrogen in the upper atmosphere of HD 209458b by~\citet{Ballester2007} and necessary to explain the extended atmosphere observed in both planets~\citep{Vidal-Madjar2003,LecavelierDesEtangs2010}. At higher pressures, the temperature of HD 209458b is unexpectedly low and cannot be explained by current 1D and 3D models. Those low temperatures are however consistent with the condensation of sodium at low pressures as shown by~\citet{Sing2008}.
EChO will accurately determine the mean temperature profile at the terminator of a wide range of planets from their transit spectrum. It will disentangle the contributions of the dynamical, chemical and radiative processes shaping the temperature profile at the terminator. For the brightest targets, it will observe the differences between the ingress and the egress of the transit, shedding light on the differences in temperature, chemical composition and cloud coverage between the western and the eastern atmospheric limbs.
}
\subsection{What is the latitudinal structure of the temperature
in hot Jupiter atmospheres?}
The high and low latitudes of a planet differ by the amount of irradiation they receive and by the strength of the Coriolis forces. As a result, in hot Jupiters atmospheric models, the circulation patterns change from a deep super-rotating jet at the equator to a day-to-night circulation at the poles~\citep{Showman2013}. Chemical composition and cloud coverage could follow this trend and be significantly different between the poles and the equator~\citep[see][and Fig.~\ref{fig::Clouds} hereafter]{Parmentier2013}. The secondary eclipse of an exoplanet yields latitudinal information about the temperature structure of its atmosphere. During a secondary eclipse, the planet disappears behind its host star. For non-zero impact parameter, the disappearance and appearance of the planet happen by slices that are tilted with respect to the north/south direction. The ingress and egress of an exoplanet's secondary eclipse can thus allow the construction of full two-dimensional maps of the dayside hemisphere ~\citep{Majeau2012,DeWit2012}, in opposition to phase curves that lead to longitudinal maps only. Furthermore, as each wavelength probes different optical depth of the dayside atmosphere, multi-wavelength observations, as the ones EChO will provide, can allow tri-dimensional maps of the atmosphere. As an example, the eclipse mapping of HD\, 189733b using Spitzer 8 microns data constrains its hot spot to low latitudes and provides independent confirmation of its eastward shift relative to the substellar point~\citep{Majeau2012,DeWit2012}.
Based on the technique developed by~\citet{DeWit2012} we present in Fig.~\ref{fig::maps} the map retrieval of a synthetic version of the hot Jupiter HD\,189733b\footnote{ { We use EChO's noise model introduced in \cite{Barstow2013}. In particular, we use a telescope effective area of $1.13$ square meter, a detector quantum efficiency of $0.7$, a duty-cycle of $0.8$, and an optical throughput of 0.378 from 2.5 to 5 $\mu$m, relevant for this simulation showed in Fig.~\ref{fig::maps}}} with a hypothetical hot spot with a temperature contrast of $\Delta T/T\approx30\%$ located { in} the northern hemisphere. Such a hot spot in a given spectral bin could be formed by the presence of patchy clouds (see Fig.~\ref{fig::Clouds}) or chemical differences between the poles and the equator. With one secondary eclipse, EChO will detect the presence of latitudinal asymmetry in the planet's brightness distribution. With $\sim$10 (resp. $\sim$100) secondary eclipses, the temperature contrast will be measured with a precision of $300\,\rm{K}$ (resp. $100\,\rm{K}$) and the latitudinal location of the hot-spot will be known with a precision of $10\degree$ (resp. $3.5\degree$). This observations will be available in different spectral intervals, with a spectral resolution of $\approx20$, for the most favorable targets.
\begin{figure}[h!]
\includegraphics[width=1\linewidth]{./retrievied_maps_Temperature.pdf}
\centering
\caption{ { Simulated retrieval of dayside brightness temperature
patterns using ingress/egress mapping for a hypothetical case
where a large thermal hotspot resides in the high northern
latitudes of the dayside. Planetary and stellar parameters of
HD 189733b are adopted. The top left map depicts the synthetic
data. The top right, bottom left, and bottom right shows the
ability of ingress/egress mapping to recover the temperature
structure of the synthetic data with 1, 10, and 100 secondary
eclipses observed by EChO, respectively, in a spectral bin of
resolution $20$.}}
\label{fig::maps}
\end{figure}
\subsection{How common are clouds, what are they made of, and
what is their spatial distribution?} The atmospheres of many hot and warm
Jupiters have temperatures that cross the condensation curves for
various refractory materials, suggesting that cloud formation may be
an important process on some of those planets. Transmission spectra
indicate that HD 189733b and perhaps HD 209458b exhibit haze-dominated
atmospheres~\citep{Pont2013,Deming2013}. This may also be
true for the super-Earth GJ 1214b~\citep[e.g.][]{Bean2011,Berta2012,Morley2013} and GJ 3470b~\citep{Crossfield2013,Nascimbeni2013}. Given the cold conditions on the
nightsides of typical hot Jupiters, many chemical species should
condense on the nightside. Three-dimensional circulation models including condensable tracers~\citep{Parmentier2013}
indicate that complex spatial
distributions of clouds---on both the dayside and nightside---can
result from such nightside condensation (see Fig.~\ref{fig::Clouds}).
Multi-wavelength lightcurves obtained by EChO will provide
major constraints not only on the chemical composition and thermal
structure but on the existence and properties of clouds in gas giant's atmospheres. Phase curves in the visible frequency range will provide insight on the longitudinal variation in albedo along the planet, which could be a strong signature of inhomogeneous cloud coverage on the planet atmosphere~\citep{Demory2013,Heng2013}. By monitoring planets with widely different equilibrium temperatures, EChO is expected to characterize the transition from cloudy to cloudless atmospheres and the change in the dominant condensable species with equilibrium temperature, from silicate clouds at high temperatures to water clouds in temperate planets.
\begin{figure}[h!]
\includegraphics[width=1\linewidth]{./Parmentier2013.pdf}
\centering
\caption{Spatio-temporal variability of tracer particles (color) and winds (arrows) representing clouds in a hot-Jupiter model of~\citet{Parmentier2013}. The particles efficiently trace the main circulation patterns of the atmosphere.}
\label{fig::Clouds}
\end{figure}
{
\subsection{How common are stratospheres, and what determines their distribution and properties ?}
Thermal inversions are a natural consequence of visible/UV absorption of the incident star light high in the atmosphere. For an isolated planetary atmosphere in hydrostatic equilibrium and
no local energy sources, the atmospheric temperature decreases
with pressure. In planetary atmospheres irradiated by their
host star, strong optical/UV absorbers in the upper layers can intercept part of the incident star light.
With such a local heating, a zone where the temperature increases with decreasing pressure can form. Most solar system planets have temperature inversions in their atmospheres. In Earth atmosphere it is caused by ozone, which is a strong absorber in the UV~\citep{Chamberlain1987}. In Jupiter, it is mainly caused by the strong absorption in the visible by hazes resulting from methane photochemistery.
The compounds producing thermal inversions in solar system atmospheres do not survive the high temperatures of hottest hot Jupiters. Nevertheless, it has been proposed that thermal inversions in the $\sim1\,\milli\bbar--1\,\bbar$ level could form in the atmosphere of very hot Jupiters due to the strong absorption of the incident stellar radiation in the visible by gaseous titanium oxide~\citep{Hubeny2003}, a compound present in brown dwarfs with similar atmospheric temperatures~\citep{Kirkpatrick2005}. The so-called TiO-hypothesis differentiates between planets hot enough to have gaseous TiO and thus a thermal inversion and planets too cold to have gaseous TiO and thus without thermal inversion~\citep{Fortney2008}. Evidence for the presence of a thermal inversion have been claimed for several planets. Most of these claims were based on the ratio between the $3.6\,\micro\meter$ and the $4.5\,\micro\meter$ thermal fluxes observed with the Spitzer space telescope~\citep{Knutson2008,Burrows2008}. Assuming that water is the main absorber at those wavelengths, a higher flux at $4.5\,\micro\meter$ than at $3.6\,\micro\meter$ can be interpreted as an emission band, created by an inverted temperature profile whereas a smaller flux at $4.5\,\micro\meter$ than at $3.6\,\micro\meter$ can be interpreted as an absorption feature, resulting from a non-inverted temperature profile. Up to now, most of the claims did not survive a more exhaustive analysis that included a large range of possible atmospheric chemical composition and temperature profiles~\citep{Madhusudhan2010}. In current data there is thus no strong evidence for a thermal inversion but it is not ruled out either~\citep{Hansen2014}.
Given the apparent lack of large thermal inversions and strong observational signatures of TiO in the transit spectrum of several planets~\citep[e.g.][]{Desert2008,Huitson2013,Sing2013}, many authors challenged the TiO hypothesis. Condensation in the deep atmosphere~\citet{Showman2009,Spiegel2009} or in the nightside of the planet~\citep{Parmentier2013} could deplete TiO from the dayside atmosphere.~\citet{Knutson2010} noted that TiO could be destroyed by the strong stellar FUV flux, implying that only planets orbiting low activity stars could have an inversion.~\citet{Madhusudhan2011} showed that atmospheres with a carbon to oxygen ratio higher than one should have a reduced TiO abundance, making them unable to maintain a thermal inversion.~\citet{Zahnle2009} and ~\citet{Pont2013} proposed that absorption by hazes instead of TiO could be responsible for the thermal inversion whereas~\citet{Menou2012} showed that ohmic dissipation could also lead to an inverted temperature profile.
EChO will perform a broad census of which hot Jupiters exhibit
a thermal inversion and which do not, and will determine to which extent the presence of thermal inversions correlates with incident stellar flux,
stellar activity, atmospheric composition, day/night temperature gradients and other parameters.
Because EChO will obtain full IR spectra from which absorption and emission
features can be well identified, the determination of whether
a planet exhibits a stratosphere---and the pressure range of any
stratosphere---will be much more robust than possible with existing
Spitzer and groundbased data. Moreover,
spectral features seen in transit and secondary eclipse will provide
strong constraints on the specific chemical absorber that allows
for the existence of stratospheres.
}
\subsection{What are the main dynamical regimes and what determines
the shift from one to another?}
Hot Neptunes and Jupiters span an enormous range of incident stellar
fluxes, orbital parameters, masses, surface gravities, and rotation
rates, among other parameters. Not surprisingly, then, theory and
numerical simulations suggest that such planets exhibit several
fundamentally different circulation regimes depending on these
parameters. Most circulation models to date have emphasized the
benchmark hot Jupiters HD 189733b and HD 209458b~\citep{Showman2002,Cooper2005,Showman2008,Showman2009,Menou2009,Dobbs-Dixon2008,Dobbs-Dixon2010,Thrastarson2010,Thrastarson2011,Rauscher2010,Rauscher2012b,Lewis2010,Heng2011,Heng2011a,Perna2012,Miller-RicciKempton2012,Parmentier2013}. These models tend to produce several broad zonal
(east-west) jets including a fast superrotating equatorial jet, and
day-night temperature differences of hundreds of Kelvin at photospheric
levels. Nevertheless, recent theoretical explorations of wider
parameter spaces suggest that at extremely large stellar fluxes, the
fractional day-night temperature differences increases and the
longitudinal offset of hot spots decreases~\citep{Perna2012,Perez-Becker2013a}. This shift is also accompanied by a
shift from a circulation dominated by zonal (east-west) jets at
moderate stellar flux to a circulation dominated by day-to-night flow
at extreme stellar flux~\citep{Showman2013}. At orbital separations
beyond those typically identified with hot Jupiters ($>0.1\,$AU),
models suggest that the eastward equatorial jet will give way to a
circulation exhibiting one or more eastward jets in the midlatitudes
of each hemisphere generated by baroclinic instability---a pattern
more reminiscent of Earth or Jupiter~\citep{Showman2012}. { The spatial variation of temperature, clouds, and chemical composition can efficiently trace the atmospheric circulation patterns~\citep{Parmentier2013}. By determining those spatial variations for a wide range of planetary conditions, EChO will determine the main circulation regimes of exoplanets atmospheres.}
\subsection{What is the role of magnetic coupling in the circulation
of hot exoplanets?}
Several authors have suggested that, at the extreme temperatures
achieved on the most highly irradiated hot Jupiters, thermal
ionization may allow a coupling of the atmosphere to the planet's
magnetic field, causing the Lorentz force to become dynamically
important~\citep{Perna2010,Perna2010a,Rauscher2013}. This
could lead to qualitative changes in the day-night temperature
difference and the geometry and speed of the global wind pattern
relative to an otherwise similar planet without such coupling~\citep{Batygin2013}.
Dynamical coupling to the magnetic field could even allow
feedbacks that influence the existence and amplitude of a dayside
stratosphere~\citep{Menou2012}. Moreover, such coupling could lead
to Ohmic dissipation, with possible implications for the planet's
long-term evolution~\citep{Batygin2010,Perna2010a,Huang2012,Wu2013}. The sensitivity of the magnetic effects to the ionisation rate -- given by the composition and the temperature profile -- will allow EChO to identify their role in the hottest planets.
\subsection{Are hot Jupiters temporally variable, and if so,
what is the nature and distribution of the variability?}
Atmospheres of planets in the solar system are turbulent, leading to
temporal fluctuations on a wide range of space and time scales. This
question is also a crucial one for hot Jupiters, especially because
the temporal behavior of any variability contains telltale clues about
the atmospheric state that would be hard to obtain using other
techniques. A variety of searches for variability have taken place
over the years, so far without any firm detections of variability.
Using Spitzer observations of { seven secondary-eclipses} of HD 189733b,~\citet{Agol2010} demonstrated an upper limit of 2.7\% of the variability of
the secondary-eclipse depth at 8 $\mu$m. Most
3D circulation models of typical hot Jupiters exhibit relatively
steady circulation patterns; for example, circulation models coupled
to radiative transfer predict variability in the secondary-eclipse
depth of $\sim$1\% in the Spitzer IRAC bandpasses~\citep{Showman2009}. Nevertheless, some circulation models predict
high-amplitude variability of up to 10\% or more at global scales~\citep{Cho2003,Cho2008,Rauscher2007}. The amplitude and temporal spectrum of variability have much
to tell about the basic atmospheric structure. Periods of
variability are likely to be linked to the periods for dynamical instabilities in the atmosphere.
In turn, these fundamental periods are influenced by the structure of the circulation's basic state including the stratification (e.g., the Brunt-Vaisala frequency), the vertical shear of the horizontal wind, and other parameters. As a dedicated mission, EChO will be able to observe systematically \emph{all} transits and secondary eclipses of a given planet for a given amount of time and shed light on the different timescale and on the amplitude of the variability of a handfull of hot Jupiters. This way, EChO will allow insights into the dynamics not obtainable in any other way.
\subsection{What are the conditions in the deep, usually unobservable atmosphere ?}
\begin{SCfigure*}[][h]
\includegraphics[clip=true,width=0.6\linewidth]{./Agundez2014.pdf}
\centering
\caption{Abundance of methane in the equatorial plane of HD 189733b predicted by the pseudo-2D chemical model of~\citet{Agundez2014}. The x-axis represents longitude with respect to the substellar point. In the dayside the abundance in the $10^{-5}-0.1\,\bbar$ pressure range is quenched to the abundance at the $0.1\,\bbar$ level. At lower pressures photochemistery becomes important and the abundance drops. the nightside abundance is quenched to the dayside one due to the horizontal advection by an eastward jet.}
\label{fig::Agundez2014}
\end{SCfigure*}
EChO observations can be used to detect and infer the atmospheric
abundances of major molecules potentially including H$_2$O, CO,
CH$_4$, CO$_2$, and various other trace and/or disequilibrium species~\citep[see][]{Barstow2013}.
To these extent that these species exhibit chemical interactions with
short timescales, they may exhibit spatially variable
three-dimensional distributions (e.g., differing dayside and nightside
abundances). Any detected spatial variations or homogeneity in such chemical species
across the planet would thus provide important constraints on the
dynamics.
Several species, including CO and CH$_4$, are predicted to have long
interconversion timescales, implying that they will be chemically
``quenched'' in the observable atmosphere at constant abundances that
should vary little from one side of the planet to the other~\citep[see Figure~\ref{fig::Agundez2014} and also][]{Cooper2006,Moses2011,Agundez2014}. The quench level---above which
the abundances are in disequilibrium and below which they are
approximately in equilibrium--is predicted to be at $\sim$0.1--10 bars
pressure on typical hot Jupiters~\citep{Cooper2006,Agundez2014} and
even deeper for cooler planets.
Interestingly, this can be deeper than directly probed by thermal
emission measurements (which sense pressures less than $\sim$10 bar).
Because the quenched abundances depend
on the atmospheric vertical mixing rate, this implies that precise
measurements of the CO and CH$_4$ abundances will place constraints
on the dynamical mixing rates at pressures deeper than can be
directly sensed. These insights on the dynamics via chemistry will
thus be highly complementary to insights obtained on the dynamics
from light curves and ingress/egress mapping. Moreover, they will give constrains on the deep atmosphere, a fundamental zone for understanding the interior and evolution of gas giant planets~\citep{Guillot2002}. On cooler planets,
quenching in the N$_2$/NH$_3$ system can provide analogous insights.
\subsection{Why are some hot Jupiters inflated?}
Transit observations show that many hot Jupiters have radii larger
than can be expected from standard evolution models~\citep[see the review by][]{Guillot2005}. The best way of explaining these
radii is that some hot Jupiters experience an interior heat source
(not accounted for in ``classical'' evolution models) that maintains
a large interior entropy and thereby planetary radius. Several
explanations have been put forward for this missing energy source,
including tidal dissipation ~\citep[e.g.][]{Bodenheimer2001},
mechanical energy transported downward into the interior by the atmosphere~\citep{Guillot2002}, suppression of convective heat loss
in the interior as a result of compositional layering~\citep{Chabrier2007,Leconte2012},
and Ohmic dissipation associated with ionized atmospheric winds
~\citep{Batygin2010,Perna2010,Huang2012,Wu2013}. However, the amount of extra-heating needed to keep hot Jupiters inflated is strongly affected by the ability of the atmosphere to transport the energy from the deep interior to the outer space~\citep{Guillot2011}. The efficiency of this transport is tied to the deep atmospheric temperature and it's spatial variations~\citep{Rauscher2014}. { The deep temperature is partly determined by the ability of the upper atmosphere to absorb and re-emit the incoming stellar irradiation~\citep{Parmentier2014a,Parmentier2014b}. EChO will determine the chemical composition and the thermal profile of the observable atmosphere. This will restrict the range of possible thermal structures for the deep atmosphere, providing better constraints on the strength of the unknown mechanism inflating hot Jupiters}.
\subsection{How does the circulation respond to seasonal and
extreme forcing?}
Several transiting hot Jupiters, including HD 80606b, HAT-P-2b, and
HD 17156 have orbital eccentricities exceeding 0.5, which imply
that these planets receive an order of magnitude or more stellar
flux at apoapse than at periapse. This extreme time-variable heating
may have significant effects on the atmospheric circulation~\citep{Kataria2013}. As the planet goes back and forth between apoapse and periapse, EChO will provide a unique opportunity to see the atmosphere heating up and cooling down at different wavelength, measuring its global thermal inertia~\citep{Lewis2013} and how it varies with depth. Then, those heating and cooling rates can be used to better understand the atmospheric dynamics of planets on a circular orbits, where this measurement is not possible.
\section*{Conclusion}
EChO is a dedicated instrument to observe exoplanets atmospheres proposed to the European Space Agency. { Although it was not selected in 2014, its exquisite photometric precision and high spectroscopic resolution over a wide spectral range make it the archetype of a future space-mission dedicated to the spectroscopic characterization of exoplanets in tight orbit.} Such a future mission will perform a broad survey of exoplanets atmospheres, exploring a large range of stellar irradiation, rotation period and planetary gravity, three parameters that determine the main dynamical regimes of planetary atmospheres. It will provide a deeper understanding of some benchmark planets, characterizing their three-dimensional thermal, chemical and compositional structure and their variation with time, opening the field of climate study to exoplanets.
\begin{multicols}{2}
|
1,314,259,993,654 | arxiv | \section{Introduction}
Higgs production via gluon fusion is one of the most
important LHC processes. Its computation at higher orders requires
renormalization and factorization to cancel UV and IR divergences. The
renormalization is less trivial than the one of standard QCD processes
due to the required renormalization of non-renormalizable
operators. The virtual corrections have been computed in conventional dimensional
regularization ({\scshape cdr})~\cite{Harlander:2000mg,Moch:2005tm,Baikov:2009bg,
Gehrmann:2010ue,Gehrmann:2010tu};
the required theory of operator renormalization in {\scshape cdr}\ has been developed
in Ref.~\cite{Spiridonov:1984br}, based on general work in
Refs.~\cite{KlubergStern:1974rs,Joglekar:1975nu}.
In the past years, several alternative regularization schemes have
been developed. Purely four-dimensional schemes such as implicit
regularization \cite{Cherchiglia:2010yd,Ferreira:2011cv} and
{\scshape fdr}~\cite{Pittau:2012zd} have been proposed and
used to compute processes of practical interest such as
$H\to\gamma\gamma$ \cite{Cherchiglia:2012zp,Donati:2013iya} and $H\to gg$
\cite{Pittau:2013qla}.
The present paper is devoted to regularization
by dimensional reduction ({\scshape dred}) \cite{Siegel:1979wq}
and the related four-dimensional helicity ({\scshape fdh}) scheme \cite{BernZviKosower:1992}.
Both schemes are actually the same regarding UV renormalization,
but they differ in the treatment of external partons related to IR
divergences.\footnote{Parts of the literature, e.\,g.
Refs.~\cite{Kunszt:1993sd, Catani:2000ef, Catani:1996pk}
used the term DR/dimensional reduction for what is called {\scshape fdh}\ here.
}
There has been significant progress
in the understanding of {\scshape fdh}\ and {\scshape dred}:
the equivalence to {\scshape cdr}\ \cite{Jack:1993ws,Jack:1994bn},
mathematical consistency and the quantum action principle
\cite{Stockinger:2005gx},
infrared factorization \cite{Signer:2005,Signer:2008va} have been established ---
these results solved several problems that had been reported earlier,
related to violation of unitarity \cite{vanDamme:1984ig}, Siegel's
inconsistency \cite{Siegel:1980qs}, and the factorization problem of
\cite{Beenakker:1989, Smith:2005}. In addition, explicit multi-loop calculations
have been carried out \cite{Harlander:2006rj,Harlander:2006xq,Kant:2010tf,
Kilgore:2011ta,Kilgore:2012tb}.
More recently, the multi-loop IR divergence structure of {\scshape fdh}\ and {\scshape dred}\
amplitudes has been studied in Ref.~\cite{Gnendiger:2014nxa}.
It has been shown that IR divergences in {\scshape fdh}\ and {\scshape dred}\ can be
described by a generalization of the {\scshape cdr}\ formulas given in
Refs.~\cite{Becher:2009qa,Becher:2009cu,Gardi:2009zv,Magnea:2012pk}.
The description involves IR anomalous dimensions $\gamma^i$ for each
parton type $i$.
In Ref.~\cite{Gnendiger:2014nxa} they have been computed for the cases
of quarks and gluons by comparing the general IR factorization formulas
with explicit results for the quark and gluon form factor.
In {\scshape fdh}\ and {\scshape dred}, however, the
gluon can be decomposed into a $D$-dimensional gluon ${\hat{g}}$ and
$(4-D)$ additional degrees of freedom, so-called $\epsilon$-scalars
${\tilde{g}}$. In {\scshape dred}, $\epsilon$-scalars also appear as external states.
The present paper is devoted to a detailed two-loop computation of the
amplitude $H\to gg$ in {\scshape fdh}\ and {\scshape dred}. In {\scshape dred}, this involves the
computations of $H\to {\hat{g}}\ghat$ and
$H\to{\tilde{g}}\gtilde$, since the external gluons can either be
gauge fields or $\epsilon$-scalars. The {\scshape fdh}\ result is identical to
the one for $H\to {\hat{g}}\ghat$ and has already been given in
Ref.~\cite{Gnendiger:2014nxa}, but we will provide further details
here.
This detailed computation is of interest for two
reasons: First, it provides the basis for obtaining the remaining IR
anomalous dimension for $\epsilon$-scalars at the two-loop
level. Second, it provides an example of the required renormalization
in {\scshape fdh}\ and {\scshape dred}, including operator renormalization and operator
mixing. The difficulty of renormalization in {\scshape fdh}\ and {\scshape dred},
particularly in connection with $H\to gg$, has been pointed out
e.\,g. in Refs.~\cite{Kilgore:2012tb, Anastasiou:2008rm}.
The outline of the paper is as follows: Section \ref{sec:setup} gives a brief
description of the regularization schemes and of the
relevant Lagrangian and operators.
It ends with a detailed list
of the required ingredients of the calculation.
Apart from the actual two-loop computation and ordinary parameter and field
renormalization that are described in Sections \ref{sec:two-loop} and
\ref{sec:fieldparameterrenormalization}, respectively,
the main difficulty lies in the renormalization and
mixing of the operators generating $H\to gg$. This is discussed in
general in Section \ref{sec:operatorrenormalization},
and specific two-loop results are presented in
Section \ref{sec:CT2Lb}.
Section \ref{sec:onshell_results} then provides the final results for the on-shell
amplitudes for $H\to {\hat{g}}\ghat$ and $H\to{\tilde{g}}\gtilde$.
The appendix contains details on our projection operators and gives Feynman
rules for the different operator insertions.
\section{Regularization schemes and $H\to gg$}
\label{sec:setup}
It is useful to distinguish the following regularization schemes
\cite{Signer:2008va}:
conventional dimensional regularization ({\scshape cdr}),
the 't Hooft-Veltman ({\scshape hv}) scheme,
the four-dimensional helicity ({\scshape fdh}) scheme, and
dimensional reduction ({\scshape dred}). In all these schemes, momenta are treated in
$D=4-2\epsilon$ dimensions (the associated space is denoted by $QDS$ with
metric tensor
${\hat{g}}^{\mu\nu}$). In order to define the schemes, one also needs an
additional quasi-4-dimensional space ($Q4S$, metric $g^{\mu\nu}$) and the
original 4-dimensional space ($4S$, metric ${\bar{g}}^{\mu\nu}$).
The treatment of gluons in the four schemes is given in Tab.~\ref{tab:RSs}.
In the table,
``internal'' gluons are defined as either virtual gluons that are part of a
one-particle irreducible loop diagram or, for real correction
diagrams, gluons in the initial or final state that are collinear or
soft. ``External gluons'' are defined as all other gluons.
\begin{table}
\begin{center}
\begin{tabular}{l|cccc}
&{\scshape cdr}&{\scshape hv}&{\scshape fdh}&{\scshape dred}\\
\hline
internal gluon&${\hat{g}}^{\mu\nu}$&${\hat{g}}^{\mu\nu}$&
$g^{\mu\nu}$&$g^{\mu\nu}$\\
external gluon&${\hat{g}}^{\mu\nu}$&${\bar{g}}^{\mu\nu}$&
${\bar{g}}^{\mu\nu}$&$g^{\mu\nu}$
\end{tabular}
\end{center}
\caption{
Treatment of internal and external gluons in the four different
regularization schemes,
i.e.\ prescription which metric tensor has to be used in propagator
numerators and polarization sums.
\label{tab:RSs}
}
\end{table}
Mathematical consistency and $D$-dimensional gauge invariance require
that $Q4S\supset QDS\supset 4S$ and forbid to identify
$g^{\mu\nu}$ and ${\bar{g}}^{\mu\nu}$. Details can be found
in Refs.\ \cite{Stockinger:2005gx,Signer:2008va,Gnendiger:2014nxa}.
The most important relations for the present paper are
\begin{align}
g^{\mu\nu}&={\hat{g}}^{\mu\nu}+{\tilde{g}}^{\mu\nu},&
{\hat{g}}^{\mu\rho}{\tilde{g}}_{\rho}{}^\nu&=0,&
{\hat{g}}^{\mu\rho}\bar{g}_{\rho}{}^\nu&=\bar{g}^{\mu\nu},&
{\hat{g}}^{\mu\nu}{\hat{g}}_{\mu\nu}&=D,&
{\tilde{g}}^{\mu\nu}{\tilde{g}}_{\mu\nu}&=N_\epsilon,&
\end{align}
where a complementary $2\epsilon$-dimensional metric tensor
${\tilde{g}}^{\mu\nu}$ has been introduced. With these metric tensors we
can decompose a quasi-4-dimensional gluon field $A^\mu$ as
\begin{align}
A^\mu&={\hat{g}}^{\mu\nu}A_\nu+{\tilde{g}}^{\mu\nu}A_\nu=\hat{A}^\mu+\tilde{A}^\mu
\end{align}
into a $D$-dimensional gauge field $\hat{A}^\mu$ and an associated
$\epsilon$-scalar field $\tilde{A}^\mu$ with
multiplicity~$N_\epsilon=2\epsilon$.\,\footnote{%
In many applications of {\scshape fdh}\ the dimensionality of $Q4S$ is left as a
variable $D_s$, which is eventually set to $D_s=4$. The multiplicity
of $\epsilon$-scalars is then $N_\epsilon=D_s-D$.}
Correspondingly, there are two types of
particles in the regularized theory: $D$-dimensional gluons ${\hat{g}}$
and $\epsilon$-scalars ${\tilde{g}}$. The unregularized external gluons
$\bar{g}$ of {\scshape fdh}\ are a part of ${\hat{g}}$.
The regularized Lagrangian of massless QCD then reads
\begin{subequations}
\begin{align}
{\cal{L}}_{QCD,\text{ regularized}} &= -\frac{1}{4}{\hat{F}}^{\mu\nu}_a{\hat{F}}_{\mu\nu,a}
-\frac{1}{2\xi}(\partial^\mu{\hat{A}}_{\mu,a})^2
+i\,\overline{\psi}\hat{\slashed{D}}\psi
+\partial^\mu\overline{c}_{a}{\hat{D}}_\mu c_{a}+\cal{L}_\epsilon,\\[15pt]
%
\cal{L}_\epsilon &=
-\frac{1}{2}(\hat{D}^\mu{\tilde{A}}^\nu)_a(\hat{D}_\mu{\tilde{A}}_\nu)_{a}
-g_{e}\,\overline{\psi}{\mathrlap{\not{\phantom{A}}}\tilde{A}}\psi
-\frac{1}{4!}\left(g_{4\epsilon}^2\right)^{\alpha\beta\gamma\delta}_{abcd}
{\tilde{A}}_{\alpha,a}{\tilde{A}}_{\beta ,b}
{\tilde{A}}_{\gamma,c}{\tilde{A}}_{\delta,d}.
\label{LQCDregularized}
\end{align}
\end{subequations}
Here, ${\hat{F}}^{\mu\nu}$ and ${\hat{D}}^\mu=\partial^\mu+ig_{s}{\hat{A}}^\mu$ denote the
non-abelian field strength tensor and the covariant derivative in $D$ dimensions;
$\psi$ and $c$ are the quark and ghost fields.
In Eq.~(\ref{LQCDregularized}) the coupling of $\epsilon$-scalars to (anti-)quarks is given by
the evanescent Yukawa-like coupling~$g_{e}$. This could in principle be
set equal to the strong coupling~$g_{s}$. But, since both couplings renormalize differently
this would only hold at tree-level and for one particular renormalization
scale~\cite{Jack:1993ws};
the same is true for the quartic $\epsilon$-scalar coupling $g_{4\epsilon}$.
In Eq.~(\ref{LQCDregularized}) we introduce an abbreviation that includes the appearing
Lorentz and color structure:
$\left(g_{4\epsilon}^2\right)^{\alpha\beta\gamma\delta}_{abcd}\mathrel{\mathop:}=
g_{4\epsilon}^2(f_{abe}f_{cde}
{\tilde{g}}^{\alpha\gamma}{\tilde{g}}^{\beta\delta}+\text{perm.})$, where ``perm.''
denotes the 5 permutations arising from symmetrization in the multi-indices
$(a,\alpha)\dots(c,\gamma)$.
In the following we use all couplings in the form
$\alpha_i=\frac{g_i^2}{4\pi}$ with $i = s, e, 4\epsilon$.
The process $H\to gg$ is generated by an effective Lagrangian which
arises from integrating out the top quark in the Standard Model. In
{\scshape cdr}\ it contains only the term
$-\frac{1}{4}\lambda H {\hat{F}}^{\mu\nu}_a{\hat{F}}_{\mu\nu, a}$.
In {\scshape fdh}\ and {\scshape dred}\ one again has to distinguish several gauge invariant
structures containing either $D$-dimensional gluons or
$\epsilon$-scalars. The effective Lagrangian can be written as
\begin{align}
{\cal L}_{\text{eff}} &=
\lambda HO_1+\lambda_\epsilon H\tilde{O}_1+
\sum_i\lambda_{4\epsilon,i}H\tilde{O}_{4\epsilon,i},
\label{Leff}
\end{align}
with
\begin{subequations}
\begin{align}
O_1 &=-\frac{1}{4} \hat{F}^{\mu\nu}_a \hat{F}_{\mu\nu,a},\\
{\tilde{O}}_1 &=
-\frac{1}{2}(\hat{D}^\mu{\tilde{A}}^\nu)_a(\hat{D}_\mu{\tilde{A}}_\nu)_a.
\end{align}
\end{subequations}
${\tilde{O}}_{4\epsilon,i}$ denote operators involving
products of four $\epsilon$-scalars.
Such operators are not important in the present paper and will
not be given explicitly.
Like for $\alpha_s,\alpha_e$ and $\alpha_{4\epsilon}$, the couplings $\lambda$ and
$\lambda_\epsilon$ can be set equal at tree-level, but they
renormalize differently and have different $\beta$ functions.
Our final goal is the calculation of the two-loop form factors for
gluons and $\epsilon$-scalars. This requires the
on-shell calculation of the 3-point function
$\Gamma_{H A^\mu A^\nu}(q,-p,-r)$. All momenta are
defined as incoming, so $q=p+r$. The 3-point function can be separated
into $\Gamma_{H {\hat{A}}^\mu {\hat{A}}^\nu}$ and $\Gamma_{H {\tilde{A}}^\mu
{\tilde{A}}^\nu}$, corresponding to the amplitudes for $H\to {\hat{g}}\ghat$ and
$H\to{\tilde{g}}\gtilde$, respectively. In {\scshape dred}, both on-shell amplitudes
are needed according to Tab.\ \ref{tab:RSs}. In {\scshape fdh}, only $H\to
\bar{g}\bar{g}$ is needed, which however is identical to
$H\to{\hat{g}}\ghat$ and will not be discussed seperately.
The on-shell calculation requires the knowledge of the two-loop
renormalization constants $\delta Z_\lambda^{\text{2L}}$ and
$\delta Z_{\lambda_\epsilon}^{\text{2L}}$.
These in turn can be obtained from an off-shell calculation of
$\Gamma_{H A^\mu A^\nu}$. Projectors extracting the required
renormalization constants from the off-shell Green functions
and precisely defining the gluon and $\epsilon$-scalar form factors are
given in appendix~\ref{sec:appendix_A}.
We have now all ingredients to discuss the classes of Feynman diagrams
that contribute to $\Gamma_{H A^\mu A^\nu}$ in {\scshape fdh}\ and {\scshape dred}:
\begin{enumerate}
\item Genuine two-loop diagrams $\Gamma_{H A^\mu A^\nu}^{\text{2L}}$.
Some remarks concerning the calculation are presented in
Sec.~\ref{sec:two-loop}.
\item Counterterm diagrams $\Gamma_{H A^\mu A^\nu}^{\text{1LCT,a}}$
and $\Gamma_{H A^\mu A^\nu}^{\text{2LCT,a}}$
arising from one- and two-loop renormalization
of the fields, the gauge parameter $\xi$, and of the couplings
$\alpha_s$, $\alpha_e$, and $\alpha_{4\epsilon}$.
The required renormalization constants are
presented in Sec.~\ref{sec:fieldparameterrenormalization}.
\item Counterterm diagrams $\Gamma_{H A^\mu A^\nu}^{\text{1LCT,b}}$
arising from one-loop renormalization of the effective Lagrangian
(\ref{Leff})
at the one-loop level, which includes the renormalization of
$\lambda$ and $\lambda_{\epsilon}$. This is a major complication and will be presented in
Sec.~\ref{sec:operatorrenormalization}.
\item Overall two-loop counterterm diagrams $\Gamma_{H A^\mu A^\nu}^{\text{2LCT,b}}$
arising from the two-loop renormalization of the effective
Lagrangian (\ref{Leff}), equivalently from the
renormalization constants
$\delta Z_{\lambda}^{\text{2L}}$ and $\delta Z_{\lambda_\epsilon}^{\text{2L}}$.
These renormalization constants are generally defined by the requirement
that the appropriate off-shell Green functions are UV finite after
renormalization. For the case of $\delta Z_\lambda$, an elegant
alternative determination is possible~\cite{Spiridonov:1984br}, but that
method fails for $\delta Z_{\lambda_\epsilon}$. The results for
$\delta Z_{\lambda}^{\text{2L}}$ and
$\delta Z_{\lambda_\epsilon}^{\text{2L}}$ are
presented in Sec.~\ref{sec:CT2Lb}.
\end{enumerate}
\section{Genuine two-loop diagrams}
\label{sec:two-loop}
\begin{figure}[t]
\begin{center}
\scalebox{.70}{
\begin{picture}(135,90)(0, 0)
\Vertex(25,45){2}
\Vertex(62.5,62.5){2}
\Vertex(62.5,27.5){2}
\Vertex(100,80){2}
\Vertex(100,10){2}
\DashLine(0,45)(25,45){2}
\Gluon(25,45)(62.5,62.5){4}{3}
\Gluon(25,45)(62.5,27.5){4}{3}
\ArrowLine(100,10)(100,80)
\ArrowLine(100,80)(62.5,62.5)
\ArrowLine(62.5,62.5)(62.5,27.5)
\ArrowLine(62.5,27.5)(100,10)
\Gluon(100,80)(130,80){4}{3}
\Gluon(100,10)(130,10){4}{3}
\Text(10,50)[b]{\scalebox{1.42}{$H$}}
\end{picture}
\quad
\begin{picture}(135,90)(0, 0)
\Vertex(25,45){2}
\Vertex(62.5,62.5){2}
\Vertex(62.5,27.5){2}
\Vertex(100,80){2}
\Vertex(100,10){2}
\DashLine(0,45)(25,45){2}
\DashLine(62.5,62.5)(25,45){4}
\DashLine(25,45)(62.5,27.5){4}
\ArrowLine(100,10)(100,80)
\ArrowLine(100,80)(62.5,62.5)
\ArrowLine(62.5,62.5)(62.5,27.5)
\ArrowLine(62.5,27.5)(100,10)
\DashLine(100,80)(130,80){4}
\DashLine(130,10)(100,10){4}
\Text(10,50)[b]{\scalebox{1.42}{$H$}}
\end{picture}
\quad
\begin{picture}(135,90)(0,0)
\Vertex(30,45){2}
\Vertex(60,45){2}
\Vertex(90,45){2}
\DashCArc(45,45)(15,0,180){3}
\DashCArc(45,45)(15,180,360){3}
\GlueArc(75,45)(15,0,180){4}{4}
\GlueArc(75,45)(15,180,360){4}{4}
\DashLine(0,45)(30,45){2}
\Gluon(90,45)(130,70){4}{4}
\Gluon(90,45)(130,20){4}{4}
\Text(10,50)[b]{\scalebox{1.42}{$H$}}
\end{picture}
\quad
\begin{picture}(135,90)(0, 0)
\Vertex(30,45){2}
\Vertex(60,45){2}
\Vertex(90,45){2}
\DashCArc(45,45)(15,0,180){3}
\DashCArc(45,45)(15,180,360){3}
\DashCArc(75,45)(15,0,180){3}
\DashCArc(75,45)(15,180,360){3}
\DashLine(0,45)(30,45){2}
\DashLine(90,45)(130,70){4}
\DashLine(90,45)(130,20){4}
\Text(10,50)[b]{\scalebox{1.42}{$H$}}
\end{picture}
}
\caption{\label{fig:twoloopdiagrams}
Sample two-loop diagrams for the process $H\to {\hat{g}}\ghat$ and
$H\to{\tilde{g}}\gtilde$ in {\scshape dred}.
$\epsilon$-scalars are denoted by dashed lines. The appearing coupling
combinations from left to right are $\lambda\alpha_s^2$, $\lambda_\epsilon
\alpha_e^2$, $\lambda_\epsilon\alpha_s^2$, $\lambda_\epsilon\alpha_{4\epsilon}^2$.
}
\end{center}
\end{figure}
As mentioned above the Green function $\Gamma_{H A^\mu A^\nu}$ can be separated into
$\Gamma_{H {\hat{A}}^\mu {\hat{A}}^\nu}$ and $\Gamma_{H {\tilde{A}}^\mu {\tilde{A}}^\nu}$,
corresponding to $H\to {\hat{g}}\ghat$ and $H\to{\tilde{g}}\gtilde$.
Examples for genuine two-loop diagrams with either external gluons or $\epsilon$-scalars
are shown in Fig.~\ref{fig:twoloopdiagrams}.
All loop calculations have been performed using the following setup: the generation of diagrams
and analytical expressions is done with the Mathematica package FeynArts~\cite{Hahn:2000kx};
to cope with the extended Lorentz structure in $Q4S$ we use a modified version of TRACER~\cite{Jamin:1991dp};
all planar on-shell integrals are reduced and evaluated with an inplementation of an
in-house algorithm that is based on integration-by-parts methods and the
Laporta-algorithm~\cite{Laporta:2001dd};
all non-planar and off-shell integrals are reduced and evaluated with the packages
FIRE~\cite{Smirnov:2008iw} and FIESTA~\cite{Smirnov:2008py}.
\section{Parameter and field renormalization in FDH and DRED}
\label{sec:fieldparameterrenormalization}
\begin{figure}[t]
\begin{center}
\scalebox{.70}{
\begin{picture}(135,90)(0,0)
\Vertex(25,45){2}
\Text(63,55)[b]{\scalebox{2}{\ding{53}}}
\Vertex(62.5,62.5){2}
\Vertex(62.5,27.5){2}
\DashLine(0,45)(25,45){2}
\Gluon(25,45)(62.5,62.5){4}{4}
\Gluon(25,45)(62.5,27.5){4}{4}
\DashLine(62.5,62.5)(100,80){4}
\DashLine(62.5,27.5)(100,10){4}
\DashLine(62.5,27.5)(62.5,62.5){4}
\Text(10,50)[b]{\scalebox{1.42}{$H$}}
\end{picture}
\quad
\begin{picture}(135,90)(0,0)
\Vertex(25,45){2}
\Text(63,55)[b]{\scalebox{2}{\ding{53}}}
\Vertex(62.5,62.5){2}
\Vertex(62.5,27.5){2}
\DashLine(0,45)(25,45){2}
\ArrowLine(62.5,62.5)(25,45)
\ArrowLine(25,45)(62.5,27.5)
\ArrowLine(62.5,27.5)(62.5,62.5)
\DashLine(62.5,62.5)(100,80){4}
\DashLine(62.5,27.5)(100,10){4}
\Text(10,50)[b]{\scalebox{1.42}{$H$}}
\end{picture}
\quad
\begin{picture}(135,90)(0,0)
\Vertex(30,45){2}
\Vertex(70,45){2}
\Text(70,38)[b]{\scalebox{2}{\ding{53}}}
\DashCArc(50,45)(20,0,180){4}
\DashCArc(50,45)(20,180,360){4}
\DashLine(0,45)(30,45){2}
\DashLine(70,45)(100,10){4}
\DashLine(70,45)(100,90){4}
\Text(10,50)[b]{\scalebox{1.42}{$H$}}
\end{picture}
\quad
\begin{picture}(135,90)(0,0)
\Vertex(25,45){2}
\Text(43,47)[b]{\scalebox{2}{\ding{53}}}
\Vertex(62.5,62.5){2}
\Vertex(62.5,27.5){2}
\DashLine(0,45)(25,45){2}
\Gluon(25,45)(62.5,62.5){4}{4}
\Gluon(25,45)(62.5,27.5){4}{4}
\DashLine(62.5,62.5)(100,80){4}
\DashLine(62.5,27.5)(100,10){4}
\DashLine(62.5,27.5)(62.5,62.5){4}
\Text(10,50)[b]{\scalebox{1.42}{$H$}}
\end{picture}
}
\caption{\label{fig:CTa}
Sample one-loop counterterm diagrams originating from the renormalization
of the couplings $\alpha_s, \alpha_e$, $\alpha_{4\epsilon}$, and of the gauge
parameter $\xi$, respectively.
}
\end{center}
\end{figure}
We now consider the counterterm contributions
$\Gamma_{H A^\mu A^\nu}^{\text{1LCT,a}}$
and $\Gamma_{H A^\mu A^\nu}^{\text{2LCT,a}}$. They are given by
diagrams exemplified in Fig.~\ref{fig:CTa}, where the counterterm
insertions are generated by the usual multiplicative QCD renormalization
of the couplings and fields present in Eq.~(\ref{LQCDregularized}).
In the following we present the values of the required $\beta$ functions
and anomalous dimensions, which govern the renormalization constants.
\subsection{$\beta$ functions}
The renormalization of the couplings $\alpha_s, \alpha_e$, and $\alpha_{4\epsilon}$ is done
by replacing the bare couplings with the renormalized ones.
As renormalization scheme we choose a modified version of the $\overline{\text{MS}}$ scheme:
like in Ref.~\cite{Gnendiger:2014nxa} we treat the
multiplicity $N_\epsilon$ of the $\epsilon$-scalars as an initially arbitrary quantity
and subtract divergences of the form $\left(\frac{N_\epsilon}{\epsilon}\right)^n$.
As a consequence, the corresponding $\beta$ functions depend on $N_\epsilon$:
$\overline{\beta}^{i}\equiv\mu^2\frac{\text{d}}{\text{d}\mu^2}\left(\frac{\alpha_i}{4\pi}\right)=
\overline{\beta}^{i}(\alpha_s,\alpha_e,\alpha_{4\epsilon},N_\epsilon)$,
with $i = s, e, 4\epsilon$.
They are given in Refs.~\cite{Kilgore:2012tb, Gnendiger:2014nxa} and read:
\begin{subequations}
\begin{align}
\begin{split}
\overline{\beta}^{s
= &
-\Big(\frac{\alpha_s}{4\pi}\Big)^2 \Bigg[C_A \left(\frac{11}{3}-\frac{N_\epsilon}{6}\right)-\frac{2}{3}N_F\Bigg]\\
&-\Big(\frac{\alpha_s}{4\pi}\Big)^3 \Bigg[C_A^2 \left(\frac{34}{3}-\frac{7}{3}N_\epsilon\right)-\frac{10}{3}C_A N_F-2 C_F N_F\Bigg]\\
&-\Big(\frac{\alpha_s}{4\pi}\Big)^2 \Big(\frac{\alpha_e}{4\pi}\Big) \Bigg[C_F N_FN_\epsilon\Bigg]+\mathcal{O}(\alpha^4),
\end{split}
\\[15pt]
\begin{split}
\overline{\beta}^{e
= &
-\Big(\frac{\alpha_s}{4\pi}\Big) \Big(\frac{\alpha_e}{4\pi}\Big)\,6\,C_F
-\Big(\frac{\alpha_e}{4\pi}\Big)^2 \Bigg[C_A (2-N_\epsilon)+C_F (-4+N_\epsilon)-N_F\Bigg]
+\mathcal{O}(\alpha^3).
\end{split}
\end{align}
\end{subequations}
The renormalization of the quartic coupling
$\left(\alpha_{4\epsilon}\right)^{\alpha\beta\gamma\delta}_{abcd}$
is more complicated since the tree-level color structure,
$f_{abe}f_{cde}$, is not preserved under renormalization \cite{Jack:1993ws}.
In the case of an SU(3) gauge group one therefore has to introduce three quartic couplings,
$\alpha_{4\epsilon,i}$ with $i=1,2,3$, each of them related to one specific color structure in a
basis of color space. Examples for such a basis are given e.\,g.
in Refs.~\cite{Harlander:2006rj, Harlander:2006xq}.
In the present case of $H\to g g$ the renormalization constant for $\alpha_{4\epsilon}$ only
appears in diagrams like the third of Fig.~\ref{fig:CTa}. Hence, only the following
contracted $\beta$ function is needed:
\begin{align}
\begin{split}
(\overline{\beta}^{4\epsilon})^{\alpha\beta\gamma\delta}_{abcd}\,
\delta_{ab}^{\phantom{\beta}}\,
{\tilde{g}}_{\alpha\beta}^{\phantom{\beta}}
=\Bigg\{
&\Big(\frac{\alpha_s}{4\pi}\Big)^2 C_A^2 (9+6\,N_\epsilon)\phantom{\bigg]
+\Big(\frac{\alpha_s}{4\pi}\Big) \Big(\frac{\alpha_{4\epsilon}}{4\pi}\Big) C_A^2\,(1-N_\epsilon)\,12\phantom{\bigg]}\\&
+\Big(\frac{\alpha_e}{4\pi}\Big)^2 \Big[C_A N_F (4-2\,N_\epsilon)+C_F N_F(-8-4\,N_\epsilon)\Big]\phantom{\bigg]}\\&
+\Big(\frac{\alpha_e}{4\pi}\Big) \Big(\frac{\alpha_{4\epsilon}}{4\pi}\Big) C_A N_F\,(1-N_\epsilon)(-4)\phantom{\bigg]}\\&
+\Big(\frac{\alpha_{4\epsilon}}{4\pi}\Big)^2 C_A^2\,(1-N_\epsilon)(-7-2\,N_\epsilon)
\Bigg\}\,\delta_{cd}\,{\tilde{g}}^{\gamma\delta}+\mathcal{O}(\alpha^3).\phantom{\bigg]}
\end{split}
\end{align}
This result is obtained from a direct off-shell calculation.
It agrees with a general result from \cite{Luo:2002ti}.
\subsection{Anomalous dimensions}
For the off-shell calculation of $\Gamma_{HA^\mu A^\nu}$ also
renormalization of the fields and of the gauge parameter $\xi$ is needed.
The renormalization of $\xi$ is fixed by the requirement that the gauge
fixing term does not renormalize: $\xi\to Z_{{\hat{A}}}\xi$.
The anomalous dimensions $\gamma_i=\mu^2\frac{d}{d\mu^2}\text{ln}\,Z_i$
of gluon and $\epsilon$-scalar fields are obtained from a direct off-shell
calculation of the respective two-loop self energies.
Their values up to two-loop level read:
\begin{subequations}
\begin{align}
\notag
\gamma_{\hat{A}}=&
-\Big(\frac{\alpha_s}{4\pi}\Big)\Bigg[C_A\left(\frac{13}{6}-\frac{\xi}{2}-\frac{N_\epsilon}{6}\right)-\frac{2}{3}N_F\Bigg]
\\ \notag
&-\Big(\frac{\alpha_s}{4\pi}\Big)^2 \Bigg[C_A^2 \left(\frac{59}{8}-\frac{11}{8}\xi-\frac{\xi^2}{4}-\frac{15}{8}N_\epsilon\right)
-\frac{5}{2} C_A N_F-2\, C_F N_F\Bigg]\\
&-\Big(\frac{\alpha_s}{4\pi}\Big) \Big(\frac{\alpha_e}{4\pi}\Big) C_F N_F N_\epsilon
+\mathcal{O}(\alpha^3),\phantom{\Bigg]}
\label{eq:anomalousDimGlu}
\\[15pt] \notag
\gamma_{{\tilde{A}}}=&
-\Big(\frac{\alpha_s}{4\pi}\Big)C_A(3-\xi)
-\Big(\frac{\alpha_e}{4\pi}\Big)\Big[-N_F\Big]
\\ \notag
&-\Big(\frac{\alpha_s}{4\pi}\Big)^2 \Bigg[C_A^2 \left(\frac{61}{6}-2\xi-\frac{\xi^2}{4}-\frac{11}{12}N_\epsilon\right)-\frac{5}{3} C_A N_F\Bigg]
-\Big(\frac{\alpha_s}{4\pi}\Big) \Big(\frac{\alpha_e}{4\pi}\Big)\Big[-5\,C_F N_F\Big]
\\ \notag
&-\Big(\frac{\alpha_e}{4\pi}\Big)^2 \Bigg[C_A N_F\left(-1+\frac{N_\epsilon}{2}\right)+C_F N_F\left(2+\frac{N_\epsilon}{2}\right)\Bigg]
-\Big(\frac{\alpha_{4\epsilon}}{4\pi}\Big)^2 C_A^2\,(1-N_\epsilon)\,\frac{3}{4}\\
\phantom{\Bigg]}&+\mathcal{O}(\alpha^3).
\label{eq:anomalousDimEps}
\end{align}
\end{subequations}
Setting $N_\epsilon$ and $\alpha_e$ to zero in Eq.~(\ref{eq:anomalousDimGlu}) yields the well-known
gluon anomalous dimension in {\scshape cdr}, see e.\,g.~\cite{Larin:1993tp}.
The value of $\gamma_{\tilde{A}}$ agrees with the general result for the
anomalous dimension of a scalar field \cite{Luo:2002ti},
confirming the point of view that $\epsilon$-scalars behave like ordinary
scalar fields with multiplicity $N_\epsilon$.
\section{Operator renormalization and mixing in FDH and DRED}
\label{sec:operatorrenormalization}
The second type of counterterm contributions, denoted by
$\Gamma_{H A^\mu A^\nu}^{\text{1LCT,b}}$ and
$\Gamma_{H A^\mu A^\nu}^{\text{2lCT,b}}$,
originates from the necessary renormalization
of the effective Lagrangian (\ref{Leff}), equivalently of the operators
$O_1$ and ${\tilde{O}}_1$.
One major difficulty is that multiplicative renormalization of the parameters
$\lambda$ and $\lambda_\epsilon$ is not sufficient since the operators
mix with further operators.
We will show that the full operator mixing involving gauge non-invariant
operators has to be taken into account. The renormalization constants
cannot be predicted from known QCD renormalization constants but need to
be determined from an off-shell calculation.
The general theory of operator mixing in gauge theories and the
classification of gauge invariant and gauge
non-invariant operators has been developed long
ago~\cite{KlubergStern:1974rs,Joglekar:1975nu,Deans:1978wn}.
In the following we briefly describe operator mixing in the much simpler case of
{\scshape cdr}\ and then explain the cases of {\scshape fdh}\ and {\scshape dred}, which involve further operators.
\subsection{Operators in CDR}
In {\scshape cdr}, a useful basis of scalar dimension-4 operators, which is closed under
renormalization, is given in Ref.~\cite{Spiridonov:1984br}:
\begin{subequations}
\label{eq:O1to5}
\begin{align}
O_1&= -\frac{1}{4} \hat{F}^{\mu\nu}_a \hat{F}_{\mu\nu, a}^{\phantom{\mu}},
\phantom{\frac{1}{1}}
\\*
O_2&= 0,
\phantom{\frac{1}{1}}\\*
O_3&= \frac{i}{2}\,\overline{\psi}\,\overleftrightarrow{\slashed{D}}\,\psi,
\phantom{\frac{1}{1}}\\*
O_4&= {\hat{A}}^{\nu}_a({\hat{D}}^{\mu}{\hat{F}}_{\mu\nu})_a
-g_{s}\overline{\psi}{\mathrlap{\not{\phantom{A}}}\hat{A}}\psi
-(\partial^{\mu}\overline{c}_a)(\partial_{\mu}c_{a}),
\phantom{\frac{1}{1}}\\*
O_5&= (D^\mu \partial_\mu \overline{c})_a c_{a}.
\phantom{\frac{1}{1}}
\end{align}
\end{subequations}
Operator $O_{1}$ is gauge invariant and related to coupling renormalization;
$O_2=m\overline{\psi}\psi$ in Ref.~\cite{Spiridonov:1984br} and corresponds
to the fermion mass renormalization; we set $m=0$.
All other operators are constrained by BRS invariance and
Slavnov-Taylor identities \cite{KlubergStern:1974rs,Joglekar:1975nu};
operators $O_4$ and $O_5$ are not gauge invariant.
The basis is chosen such that $O_{3}$, $O_{4}$ and $O_{5}$
are related to field renormalization of
$\psi$, $\hat{A}^\mu$ and $c$, respectively.
In particular, the first two terms of
$O_4$ are generated by applying the functional derivative
\begin{align}
{\hat{A}}_a^\nu(x)\frac{\delta}{\delta{\hat{A}}_a^\nu(x)}
\end{align}
on the gauge invariant part of the QCD action; the remaining term is
then required by BRS invariance and the non-renormalization of the
gauge fixing term.\footnote{%
See Refs.~\cite{Joglekar:1975nu,Deans:1978wn} for more details; the
full operator $O_4$ can be obtained from evaluating $W
Y_{{\hat{A}}^\nu_a}{\hat{A}}^\nu_a+W (\partial^\nu\overline{c}_a)A_{\nu,a}$,
where $W$ is the linearized Slavnov-Taylor operator and
$Y_{{\hat{A}}^\nu_a}$ is the source of the BRS transformation of ${\hat{A}}^\nu_a$ in
the functional integral. Since $W$ is nilpotent, this definition shows
that $O_4$ is compatible with BRS invariance and the Slavnov-Taylor
identity and can appear in the operator mixing.}
The operators renormalize as
\begin{align}
O_i &\to Z_{ij}O_{j,\text{bare}},
\end{align}
where $O_{j,\text{bare}}$ arises from $O_j$ by replacing all
parameters and fields by the respective bare quantities.
Following an elegant proof in Ref.~\cite{Spiridonov:1984br} the
nontrivial {\scshape cdr}\ renormalization matrix $Z_{ij}$ can be written in the form
\begin{align}
Z_{ij} &= \delta_{ij}+\mathbbmsl{D}_i\,\text{ln}\mathbbmsl{Z}_{j}.
\label{SpiridonovResult1}
\end{align}
Here, $\mathbbmsl{D}_i$ are derivatives with respect to parameters and $\mathbbmsl{Z}_j$
are combinations of ordinary QCD renormalization constants. As a
result, in particular the renormalization of $Z_{11}$ is given by
\begin{align}
Z_{11} &= 1+\alpha_s\frac{\partial}{\partial \alpha_s}\ln Z_{\alpha_s},
\label{SpiridonovResult2}
\end{align}
with the multiplicative renormalization constant of $\alpha_s$,
$Z_{\alpha_s}$. In this way the renormalization of the parameter
$\lambda$ in the {\scshape cdr}\ version of ${\cal L}_{\text{eff}}$ is related to the
renormalization of~$\alpha_s$.
\subsection{Operators in FDH and DRED}
In {\scshape fdh}\ and {\scshape dred}, the basis of operators needs to contain additional
terms involving $\epsilon$-scalars. We use a basis constructed
analogously to Eqs.~\eqref{eq:O1to5} from gauge invariant operators and
operators corresponding to field
renormalization. Then there are two kinds of changes: there are
modifications of the operators $O_{3}$ and $O_4$, and there are
additional basis elements. The new basis operators correspond
to the $\epsilon$-scalar kinetic term, ${\tilde{O}}_1$, to the new
parameters $\alpha_e$ and $\alpha_{4\epsilon}$, ${\tilde{O}}_3$ and
${\tilde{O}}_{4\epsilon,i}$,
and to the field renormalization of ${\tilde{A}}^\mu$, ${\tilde{O}}_4$. The
notation is chosen such that in all cases $O_j$ and ${\tilde{O}}_j$ have
a similar structure:
\begin{subequations}
\label{eq:O1toO5DRED}
\begin{align}
O_1&= -\frac{1}{4} \hat{F}^{\mu\nu}_a \hat{F}_{\mu\nu, a}^{\phantom{\mu}},
\phantom{\frac{1}{\hat{A}}}
\\
O_2&= 0,
\phantom{\frac{1}{\hat{A}}}
\\
O_3&= \frac{i}{2}\,\overline{\psi}\,\overleftrightarrow{\slashed{D}}\,\psi
-g_{e} \overline{\psi} {{\mathrlap{\not{\phantom{A}}}\tilde{A}}} \psi,
\phantom{\frac{1}{\hat{A}}}
\\
O_4&= {\hat{A}}^{\nu}_a({\hat{D}}^{\mu}{\hat{F}}_{\mu\nu})_a
+g_{s} f_{abc}(\partial^\mu{\tilde{A}}^{\nu}_{a}){\hat{A}}_{\mu, b}{\tilde{A}}_{\nu, c}
-g_{s}\overline{\psi}{{\mathrlap{\not{\phantom{A}}}\hat{A}}}\psi-\left(\partial^\mu \overline{c}_a\right)\left(\partial_\mu c_a\right),
\phantom{\frac{1}{\hat{A}}}
\\
O_5&= ({\hat{D}}^\mu \partial_\mu \overline{c}_a)c_a,
\phantom{\frac{1}{\hat{A}}}
\\[15pt]
{\tilde{O}}_{1}&= -\frac{1}{2}(\hat{D}^\mu{\tilde{A}}^\nu)_a(\hat{D}_\mu{\tilde{A}}_\nu)_a,
\phantom{\frac{1}{\hat{A}}}
\\
{\tilde{O}}_{3}&= g_{e} \overline{\psi} {\mathrlap{\not{\phantom{A}}}\tilde{A}} \psi,
\phantom{\frac{1}{\hat{A}}}
\\
{\tilde{O}}_{4}&= {\tilde{A}}^\nu_a ({\hat{D}}^\mu {\hat{D}}_\mu{\tilde{A}}_\nu)_a,
\phantom{\frac{1}{\hat{A}}}
\\
{\tilde{O}}_{4\epsilon,i} & = {\cal O}({\tilde{A}}^4).
\phantom{\frac{1}{\hat{A}}}
\end{align}
\end{subequations}
Since we consider massless QCD there is no $\epsilon$-scalar mass term.
Like in Eq.\ (\ref{Leff}), operators involving four
$\epsilon$-scalars are not needed explicitly.
This set of operators differs in a crucial way from the
{\scshape cdr}\ case. The difference between operators ${\tilde{O}}_1$ and ${\tilde{O}}_4$
is related to the total derivative $\Box{\tilde{A}}^\mu{\tilde{A}}_\mu$.
Hence, the basis for space-time
integrated operators (zero-momentum insertions) does not coincide with
the one for non-integrated operators (non-vanishing momentum
insertions).
As discussed by Spiridonov in Ref.\ \cite{Spiridonov:1984br}, in such a case
his method cannot be used. Therefore, in {\scshape fdh}\ and {\scshape dred}\ it is
not possible to derive
complete results for the operator mixing analogous to
Eqs.~(\ref{SpiridonovResult1}) and (\ref{SpiridonovResult2}).
This implies two difficulties: First, the two-loop renormalization of
${\tilde{O}}_1$ and the corresponding parameter $\lambda_\epsilon$ cannot
be obtained from a priori
known two-loop QCD renormalization constants but need to be determined
from an explicit two-loop off-shell calculation. Second, the off-shell
Green functions get contributions from unphysical, gauge non-invariant
operators, so the full operator mixing needs to be taken into account.
We have carried out the explicit one-loop calculations to obtain all
required one-loop results for $Z_{1j}$ and $Z_{\tilde{1}j}$.
The results are
\begin{subequations}
\label{eq:dZij}
\begin{align}
\delta Z_{11}^{\text{1L}} &=\left(\frac{\alpha_s}{4\pi} \right)
\left[\Big(-\frac{11}{3}+\frac{N_\epsilon}{6}\,\Big)C_A+\frac{2}{3}N_F\right]\frac{1}{\epsilon},
\label{Z11}
\\*
\delta Z_{\tilde{1}1}^{\text{1L}} &= 0,\phantom{\frac{\alpha_s}{4\pi}}
\label{Zt11}
\\
\delta Z_{1\tilde{1}}^{\text{1L}} &= 0,\phantom{\frac{\alpha_s}{4\pi}}
\label{Z11t}
\\*
\delta Z_{\tilde{1}\tilde{1}}^{\text{1L}} &=\bigg[
\Big(\frac{\alpha_s}{4\pi}\Big)(-3)C_A
+\Big(\frac{\alpha_e}{4\pi}\Big) N_F
-\Big(\frac{\alpha_{4\epsilon}}{4\pi} \Big)(1-N_\epsilon)C_A
\bigg]\frac{1}{\epsilon},
\label{Z1t1t}
\\
\delta Z_{13}^{\text{1L}} &= 0,\phantom{\frac{\alpha_s}{4\pi}}
\\*
\delta Z_{\tilde{1}3}^{\text{1L}} &= \left(\frac{\alpha_e}{4\pi} \right)
\frac{N_\epsilon}{2}C_F\frac{1}{\epsilon},
\\
\delta Z_{14}^{\text{1L}} &= \left(\frac{\alpha_s}{4\pi} \right)
\frac{3}{4}C_A\frac{1}{\epsilon},
\\*
\delta Z_{\tilde{1}4}^{\text{1L}} &= 0,\phantom{\frac{\alpha_s}{4\pi}}
\\
\delta Z_{1\tilde{4}}^{\text{1L}} &= \left(\frac{\alpha_s}{4\pi} \right)
\Big(-\frac{3}{2}\Big)C_A\frac{1}{\epsilon},
\\*
\delta Z_{\tilde{1}\tilde{4}}^{\text{1L}} &= \left(\frac{\alpha_s}{4\pi} \right)
\frac{1}{2}(3-\xi)C_A\frac{1}{\epsilon},
\\
\delta Z_{15}^{\text{1L}} &= 0,\phantom{\frac{\alpha_s}{4\pi}}
\\*
\delta Z_{\tilde{1}5}^{\text{1L}} &= 0.\phantom{\frac{\alpha_s}{4\pi}}
\end{align}
\end{subequations}
Renormalization constants involving operators ${\tilde{O}}_3$ or
${\tilde{O}}_{4\epsilon,i}$ are not needed for the calculations in the
present paper.
The renormalization constants (\ref{Z11})-(\ref{Z1t1t}) agree with those
given in Ref.~\cite{Gnendiger:2014nxa}. The only gauge-dependent quantity is
$Z_{\tilde{1}\tilde{4}}^{\text{1L}}$. This is due to the fact that operator
${\tilde{O}}_4$ is related to the field renormalization of the $\epsilon$-scalars.
In all other renormalization constants related to field renormalization the
gauge-dependent parts incidentally cancel out.
With these results the bare effective Lagrangian can be written as
\begin{align}
{\cal L}_{\text{eff}}^{\text{bare}} &=
H \sum_{j}\left(
\lambda\, Z_{1j}O_{j,\text{bare}}
+
\lambda_\epsilon\, Z_{\tilde{1}j}O_{j,\text{bare}}^{\phantom{I}}
\right),
\end{align}
where the sum runs over all operators in Eqs.~\eqref{eq:O1toO5DRED}.
Sometimes it is useful to write this using multiplicative renormalization
constants for $\lambda$ and $\lambda_\epsilon$ as
\begin{align}
{\cal L}_{\text{eff}}^{\text{bare}} &=
Z_{\lambda} \lambda H O_{1,\text{bare}}
+
Z_{\lambda_\epsilon} \lambda_\epsilon H O_{\tilde{1},\text{bare}}
+\ldots,
\label{Zlambda}
\end{align}
suppressing operators not present at tree level, such that
$\lambda Z_{\lambda} =
\lambda Z_{11} +
\lambda_\epsilon Z_{\tilde{1}1} $ and similar for
$Z_{\lambda_\epsilon}$.
The one-loop counterterm effective Lagrangian involving the
renormalization constants of Eqs.~\eqref{eq:dZij} is
then given by
\begin{align}
{\cal L}_{\text{eff}}^{\text{1LCT}} &=
H\sum_{j}\left(
\lambda\,\delta Z_{1j}^{\text{1L}}O_{j}^{\phantom{\text{1L}}}
+
\lambda_{\epsilon}\,\delta Z_{\tilde{1}j}^{\text{1L}}
O_{j\phantom{\tilde{1}}}^{\phantom{\text{1L}}}
\right).
\end{align}
We have now all ingredients for the one-loop counterterm diagrams $\Gamma_{H
A^\mu A^\nu}^{\text{1LCT,b}}$
relevant for the computation of
$H\to gg$, where the gluons are either $D$-dimensional gauge fields or
$\epsilon$-scalars. These counterterm contributions arise from
one-loop counterterm diagrams with one insertion of
${\cal L}_{\text{eff}}^{\text{1LCT}}$. Sample diagrams are given in
Fig.\ \ref{fig:operatorRenSample}. They show insertions of
operators $O_3$, $O_4$, ${\tilde{O}}_4$ and $O_5$.
Feynman rules for operator insertions are given in
appendix~\ref{sec:appendix_B}.
\begin{figure}[t]
\begin{center}
\scalebox{.70}{
\begin{picture}(135,90)(0,0)
\Vertex(25,45){2}
\Text(25,37.5)[b]{\scalebox{2}{\ding{53}}}
\Vertex(62.5,62.5){2}
\Vertex(62.5,27.5){2}
\DashLine(0,45)(25,45){2}
\ArrowLine(62.5,62.5)(25,45)
\ArrowLine(25,45)(62.5,27.5)
\ArrowLine(62.5,27.5)(62.5,62.5)
\DashLine(62.5,62.5)(100,80){4}
\DashLine(62.5,27.5)(100,10){4}
\Text(5,50)[b]{\scalebox{1.42}{$O_3$}}
\end{picture}
\quad
\begin{picture}(135,90)(0,0)
\Vertex(30,45){2}
\Text(30,37.5)[b]{\scalebox{2}{\ding{53}}}
\Vertex(60,65){2}
\ArrowArc(47.5,52)(18,40,200)
\ArrowArc(43,59)(18,240,15)
\DashLine(0,45)(30,45){2}
\DashLine(30,45)(100,10){4}
\DashLine(60,65)(100,90){4}
\Text(10,50)[b]{\scalebox{1.42}{$O_3$}}
\end{picture}
}
\quad
\scalebox{.70}{
\begin{picture}(135,90)(0,0)
\Vertex(25,45){2}
\Text(25,37.5)[b]{\scalebox{2}{\ding{53}}}
\Vertex(62.5,62.5){2}
\Vertex(62.5,27.5){2}
\DashLine(0,45)(25,45){2}
\Gluon(25,45)(62.5,62.5){4}{4}
\Gluon(25,45)(62.5,27.5){4}{4}
\DashLine(62.5,62.5)(100,80){4}
\DashLine(62.5,27.5)(100,10){4}
\DashLine(62.5,27.5)(62.5,62.5){4}
\Text(5,50)[b]{\scalebox{1.42}{$O_4$}}
\end{picture}
\quad
\begin{picture}(135,90)(0,0)
\Vertex(30,45){2}
\Text(30,37.5)[b]{\scalebox{2}{\ding{53}}}
\Vertex(60,65){2}
\GlueArc(47.5,52)(18,40,200){4}{4}
\DashCArc(43,59)(18.5,240,15){4}
\DashLine(0,45)(30,45){2}
\DashLine(30,45)(100,10){4}
\DashLine(60,65)(100,90){4}
\Text(10,50)[b]{\scalebox{1.42}{$O_4$}}
\end{picture}
}
\\[15pt]
\scalebox{.70}{
\begin{picture}(135,90)(0,0)
\Vertex(25,45){2}
\Text(25,37.5)[b]{\scalebox{2}{\ding{53}}}
\Vertex(62.5,62.5){2}
\Vertex(62.5,27.5){2}
\DashLine(0,45)(25,45){2}
\DashLine(62.5,62.5)(25,45){4}
\DashLine(25,45)(62.5,27.5){4}
\Gluon(62.5,27.5)(62.5,62.5){4}{4}
\DashLine(62.5,62.5)(100,80){4}
\DashLine(62.5,27.5)(100,10){4}
\Text(5,50)[b]{\scalebox{1.42}{${\tilde{O}}_4$}}
\end{picture}
\quad
\begin{picture}(135,90)(0,0)
\Vertex(30,45){2}
\Text(30,37.5)[b]{\scalebox{2}{\ding{53}}}
\Vertex(60,65){2}
\GlueArc(47.5,52)(18,40,200){4}{4}
\DashCArc(43,59)(18.5,240,15){4}
\DashLine(0,45)(30,45){2}
\DashLine(30,45)(100,10){4}
\DashLine(60,65)(100,90){4}
\Text(10,50)[b]{\scalebox{1.42}{${\tilde{O}}_4$}}
\end{picture}
}
\quad
\scalebox{.70}{
\begin{picture}(135,90)(0,0)
\Vertex(25,45){2}
\Text(25,37.5)[b]{\scalebox{2}{\ding{53}}}
\Vertex(62.5,62.5){2}
\Vertex(62.5,27.5){2}
\DashLine(0,45)(25,45){2}
\DashArrowLine(62.5,62.5)(25,45){2}
\DashArrowLine(25,45)(62.5,27.5){2}
\DashArrowLine(62.5,27.5)(62.5,62.5){2}
\Gluon(62.5,62.5)(100,80){4}{4}
\Gluon(62.5,27.5)(100,10){4}{4}
\Text(5,50)[b]{\scalebox{1.42}{$O_5$}}
\end{picture}
\quad
\begin{picture}(135,90)(0,0)
\Vertex(30,45){2}
\Text(30,37.5)[b]{\scalebox{2}{\ding{53}}}
\Vertex(60,65){2}
\DashArrowArc(47.5,52)(18,40,200){2}
\DashArrowArc(43,59)(18.5,240,15){2}
\DashLine(0,45)(30,45){2}
\Gluon(60,65)(100,90){4}{4}
\Gluon(100,10)(30,45){4}{8}
\Text(10,50)[b]{\scalebox{1.42}{$O_5$}}
\end{picture}
}
\caption{\label{fig:operatorRenSample}
Sample one-loop counterterm diagrams originating from operators
$O_3$, $O_4$, ${\tilde{O}}_4$ and $O_5$.
}
\end{center}
\end{figure}
The calculation shows that all these operators
generate non-vanishing contributions to
$\Gamma_{H A^\mu A^\nu}^{\text{1LCT,b}}$.
However, in the extraction of the form factors and two-loop
renormalization constants to be discussed in the next section
there are cancellations, and $O_4$ is the only new operator which
contributes.
\section{Two-loop renormalization constants of $\lambda$ and $\lambda_\epsilon$}
\label{sec:CT2Lb}
Putting together the results from the previous three sections it is possible to calculate the
two-loop renormalization constants $\delta Z^{\text{2L}}_{\lambda}$
and $\delta Z^{\text{2L}}_{\lambda_{\epsilon}}$ appearing in Eq.~(\ref{Zlambda}).
They can be obtained from a complete off-shell two-loop calculation and the requirement
that the corresponding Green-functions are UV finite after renormalization:
\begin{align}
\Gamma_{H A^\mu A^\nu}^{\text{2L}}
+\Gamma_{H A^\mu A^\nu}^{\text{1LCT,a}}
+\Gamma_{H A^\mu A^\nu}^{\text{2LCT,a}}
+\Gamma_{H A^\mu A^\nu}^{\text{1LCT,b}}
+\Gamma_{H A^\mu A^\nu}^{\text{2LCT,b}}\bigg|_{\text{UV div.}}^{\text{off-shell}}= 0.
\label{deltaLambda}
\end{align}
All ingredients except the last term are computed in the
previous sections, and Eq.~\eqref{deltaLambda} is then used to extract $\delta
Z^{\text{2L}}_{\lambda}$ and $\delta Z^{\text{2L}}_{\lambda_{\epsilon}}$.
The result for $\delta Z^{\text{2L}}_{\lambda}$ is:
\begin{align}
\begin{split}
\delta Z^{\text{2L}}_{\lambda} = &
\Big(\frac{\alpha_s}{4\pi}\Big)^2 \Bigg\{
C_A^2 \Bigg[
\frac{\frac{121}{9}-\frac{11}{9}N_\epsilon+\frac{N_\epsilon^2}{36}}{\epsilon^2}
+\frac{-\frac{34}{3}+\frac{7}{3}N_\epsilon}{\epsilon}
\Bigg]
\\&\quad\quad\quad\quad
+C_A N_F \Bigg[
\frac{-\frac{44}{9}+\frac{2}{9}N_\epsilon}{\epsilon^2}
+\frac{10}{3\epsilon }
\Bigg]
+C_F N_F\frac{2}{\epsilon }+N_F^2\frac{4}{9 \epsilon ^2}\Bigg\}\\
&+\Big(\frac{\alpha_s}{4\pi}\Big) \Big(\frac{\alpha_e}{4\pi}\Big)
C_F N_F\frac{\left(-1-\frac{\lambda_{\epsilon}}{\lambda }\right)N_\epsilon}{2 \epsilon }.
\label{eq:dZlambda}
\end{split}
\end{align}
Since the
off-shell calculations have been done numerically with the help of
FIESTA~\cite{Smirnov:2008py} the analytical expressions have been
obtained by rounding to a least
common denominator. The numerical uncertainty is less than
$\frac{1}{72}$ for the terms of the order
$\mathcal{O}(\epsilon^{-2})$ and $\frac{1}{6}$ for the terms of the
order $\mathcal{O}(\epsilon^{-1})$.
Result (\ref{eq:dZlambda}) is not new; it agrees with Ref.~\cite{Gnendiger:2014nxa},
where it has been obtained using Spiridonov's method. The recalculation serves as a test of the
setup and the results given in the previous sections. At the same time
a comparison with Ref.~\cite{Gnendiger:2014nxa} confirms that
Eq.~(\ref{eq:dZlambda}) is actually exactly correct, in spite of
numerical uncertainties.
In the same way, we obtain the renormalization constant $\delta Z^{\text{2L}}_{\lambda_{\epsilon}}$:
\begin{align}
\notag
\delta Z^{\text{2L}}_{\lambda_{\epsilon}} =
& \Big(\frac{\alpha_s}{4\pi}\Big)^2\Bigg\{
C_A^2 \Bigg[
\frac{\frac{49}{4}+\frac{5}{4}N_\epsilon}{\epsilon ^2}
+\frac{-\frac{113}{24}+\frac{71}{24}N_\epsilon+\frac{\lambda}{\lambda_{\epsilon}}\Big(2-\frac{N_\epsilon}{2}\Big)}{\epsilon }
\Bigg]
+C_A N_F \Bigg[
-\frac{1}{\epsilon^2}
+\frac{\frac{5}{6}-2\frac{\lambda }{\lambda_{\epsilon}}}{\epsilon}
\Bigg]
\Bigg\}
\\ \notag
&+\Big(\frac{\alpha_s}{4\pi}\Big) \Big(\frac{\alpha_e}{4\pi}\Big) \Bigg\{
C_A N_F \Bigg[
-\frac{3}{\epsilon ^2}
+\frac{\frac{3}{2}+3\frac{\lambda }{\lambda_{\epsilon}}}{\epsilon }
\Bigg]
+C_F N_F \Bigg[
-\frac{3}{\epsilon ^2}
+\frac{\frac{5}{2}-3\frac{\lambda }{\lambda_{\epsilon}}}{\epsilon }
\Bigg]\Bigg\}
\\ \notag
&+\Big(\frac{\alpha_s}{4\pi}\Big) \Big(\frac{\alpha_{4\epsilon}}{4\pi}\Big)
C_A^2\,(1-N_\epsilon)\Bigg[
\frac{6}{\epsilon ^2}
+\frac{-4-3\frac{\lambda}{\lambda_{\epsilon}}}{\epsilon }
\Bigg]
\\ \notag
&+\Big(\frac{\alpha_e}{4\pi}\Big)^2\Bigg\{
C_A N_F\frac{-\frac{3}{2}+\frac{3}{4}N_\epsilon}{\epsilon }
+C_F N_F \Bigg[
-\frac{3N_\epsilon}{2\,\epsilon ^2}
+\frac{3-\frac{7}{4}N_\epsilon}{\epsilon }
\Bigg]
+\frac{N_F^2}{
\epsilon ^2}
\Bigg\}
\\ \notag
&+\Big(\frac{\alpha_e}{4\pi}\Big) \Big(\frac{\alpha_{4\epsilon}}{4\pi}\Big)
C_A N_F\,(1-N_\epsilon)\Bigg[
-\frac{2}{\epsilon^2}
+\frac{3}{2\epsilon}
\Bigg]
\\
&+\Big(\frac{\alpha_{4\epsilon}}{4\pi}\Big)^2 C_A^2\,(1-N_\epsilon) \Bigg[
\frac{-\frac{5}{4}-N_\epsilon}{\epsilon^2}
+\frac{15}{8\epsilon}
\Bigg].
\label{eq:dZlambdaeps}
\end{align}
Compared to Eq.~(\ref{eq:dZlambda}) this result is more complicated and includes all combinations
of the three couplings $\alpha_s, \alpha_e$ and $\alpha_{4\epsilon}$.
This result is new; as described in
Sec.\ \ref{sec:operatorrenormalization} it cannot be obtained using
Spiridonov's method. The numerical uncertainty is less than
$\frac{1}{48}$ for all terms. A forthcoming comparison with a prediction of the
infrared structure of $H\to {\tilde{g}}\gtilde$ will confirm that
expression (\ref{eq:dZlambdaeps}) is exactly correct \cite{IRstructure}.
\section{UV renormalized form factors of gluons and $\epsilon$-scalars}
\label{sec:onshell_results}
Now that all renormalization constants are known it is possible
to calculate the two-loop form factors of gluons and $\epsilon$-scalars
in the {\scshape fdh}\ and {\scshape dred}\ scheme.
We present the results in two ways: First, we give results with
independent couplings needed to determine the IR anomalous dimensions
of gluons and $\epsilon$-scalars; second, we give simplified
results, where all couplings are set equal. These can be viewed as the
final results for the UV renormalized but IR regularized form factors.
We give them including higher orders in the $\epsilon$-expansion.
\subsection{Results for independent couplings}
\label{sec:onshell_results_complete}
The UV renormalized but IR divergent form factor for $H\to {\hat{g}}\ghat$
in {\scshape dred}\ is given at the one-loop and two-loop level by
\begin{align}
\notag
\bar{F}^{\text{1L}}_{{\hat{g}}} & (\alpha_s,\lambda_{\epsilon}/\lambda,N_\epsilon)
\\ \notag
=&\Big(\frac{\alpha_s}{4\pi}\Big)
\Bigg\{C_A \Bigg[
-\frac{2}{\epsilon ^2}
+\frac{-\frac{11}{3}+\frac{N_\epsilon}{6}}{\epsilon }
+\frac{\pi ^2}{6}+\frac{\lambda_{\epsilon}}{\lambda }N_\epsilon
+\epsilon \Big(-2+\frac{14}{3}\zeta (3)+3\frac{\lambda_{\epsilon}}{\lambda }N_\epsilon\Big
\Bigg]
+\frac{2 N_F}{3 \epsilon }\Bigg\}
\\&+\mathcal{O}(\epsilon^2),\phantom{\Bigg\}}
\label{eq:gluonFF1}
\\[15pt]
\notag
\bar{F}^{\text{2L}}_{{\hat{g}}} & (\alpha_s,\alpha_e,\lambda_{\epsilon}/\lambda,N_\epsilon)
\\ \notag
=&\Big(\frac{\alpha_s}{4\pi}\Big)^2 \Bigg\{C_A^2 \Bigg[
\frac{2}{\epsilon ^4}
+\frac{\frac{77}{6}-\frac{7}{12}N_\epsilon}{\epsilon^3}
+\frac{\frac{175}{18}-\frac{\pi ^2}{6}-N_\epsilon\Big(1+2\frac{\lambda_{\epsilon}}{\lambda}\Big)+\frac{N_\epsilon^2}{36}}{\epsilon ^2}
\\ \notag
&\quad\quad\quad\quad\quad\quad
+\frac{-\frac{238}{27}-\frac{11}{36}\pi ^2-\frac{25}{3}\zeta (3)
+N_\epsilon\Big(\frac{49}{27}+\frac{\pi^2}{72}-\frac{29}{3}\frac{\lambda_{\epsilon}}{\lambda }\Big)
+\frac{1}{6}\frac{\lambda_{\epsilon}}{\lambda }N_\epsilon^2}{\epsilon
\Bigg]
\\ \notag
&\quad\quad\quad\quad
+C_A N_F \Bigg[
-\frac{7}{3 \epsilon ^3}
+\frac{-\frac{13}{3}+\frac{2}{9}N_\epsilon}{\epsilon ^2}
+\frac{\frac{64}{27}+\frac{\pi ^2}{18}+\frac{2}{3}\frac{\lambda_{\epsilon}}{\lambda }N_\epsilon}{\epsilon
\Bigg]
+C_F N_
\frac{1}{\epsilon
+\frac{4 N_F^2}{9 \epsilon ^2}\Bigg\}
\\
&-\Big(\frac{\alpha_s}{4\pi}\Big)\Big(\frac{\alpha_e}{4\pi}\Big
C_F N_
\frac{N_\epsilon}{2\epsilon
+\mathcal{O}(\epsilon^0).\phantom{\Bigg\}}
\label{eq:gluonFF2}
\end{align}
As mentioned in the beginning the ${\hat{g}}$ form factor in {\scshape dred}\ is
identical to the gluon form factor in {\scshape fdh}, and Eq.~(\ref{eq:gluonFF2})
agrees with the result given in Ref.~\cite{Gnendiger:2014nxa}.
Since there are no external $\epsilon$-scalars in diagrams related to the gluon
form factor internal $\epsilon$-scalars have to be part of a closed $\epsilon$-scalar loop
or have to couple to a closed fermion loop.
Hence, the effective coupling $\lambda_{\epsilon}$ always appears together with at
least one power of $N_\epsilon$ in Eqs.~(\ref{eq:gluonFF1}) and (\ref{eq:gluonFF2}).
The $\epsilon$-scalar form factor for $H\to{\tilde{g}}\gtilde$ in
{\scshape dred}\ is given by
\begin{align}
\notag
\bar{F}^{\text{1L}}_{{\tilde{g}}} & (\alpha_s,\alpha_e,\alpha_{4\epsilon},\lambda/\lambda_{\epsilon},N_\epsilon)
\\ \notag
=&\Big(\frac{\alpha_s}{4\pi}\Big)
C_A \Bigg[
-\frac{2}{\epsilon ^2}
-\frac{4}{\epsilon }
-2+\frac{\pi ^2}{6}+2\frac{\lambda }{\lambda_{\epsilon}}
+\epsilon \Big(-4+\frac{\pi^2}{12}+\frac{14}{3}\zeta (3)+4\frac{\lambda}{\lambda_{\epsilon}}\Big)
\Bigg]\\
&+\Big(\frac{\alpha_e}{4\pi}\Big)\frac{N_F}{\epsilon }
+\Big(\frac{\alpha_{4\epsilon}}{4\pi}\Big)
C_A\,(1-N_\epsilon)\Bigg[
+\epsilon\Big(4-\frac{\pi ^2}{12}
\Big)
\Bigg
+\mathcal{O}(\epsilon^2),\phantom{\Bigg\}}
\\[15pt] \notag
\bar{F}^{\text{2L}}_{{\tilde{g}}} & (\alpha_s,\alpha_e,\alpha_{4\epsilon},\lambda/\lambda_{\epsilon},N_\epsilon)
\\ \notag
=&\Big(\frac{\alpha_s}{4\pi}\Big)^2 \Bigg\{C_A^2 \Bigg[
\frac{2}{\epsilon ^4}
+\frac{\frac{27}{2}-\frac{N_\epsilon}{4}}{\epsilon^3}
+\frac{\frac{281}{18}-\frac{\pi ^2}{6}-\frac{N_\epsilon}{9}-4\frac{\lambda }{\lambda_{\epsilon}}}{\epsilon^2}
\\ \notag
&\quad\quad\quad\quad\quad\quad
+\frac{\frac{469}{216}-\frac{5}{12}\pi ^2-\frac{25}{3}\zeta (3)+N_\epsilon\Big(\frac{233}{216}+\frac{\pi^2}{24}\Big)-16\frac{\lambda }{\lambda_{\epsilon}}}{\epsilon
\Bigg]
\\ \notag
&\quad\quad\quad\quad+C_A N_F \Bigg[
-\frac{1}{\epsilon ^3}-\frac{7}{9 \epsilon ^2}
+\frac{\frac{113}{54}+\frac{\pi ^2}{6}}{\epsilon }
\Bigg]\Bigg\}
\\ \notag
&+\Big(\frac{\alpha_s}{4\pi}\Big)\Big(\frac{\alpha_e}{4\pi}\Big) \Bigg\{C_A N_F \Bigg[
-\frac{2}{\epsilon ^3}-\frac{4}{\epsilon^2}
+\frac{-2-\frac{\pi ^2}{6}+2\frac{\lambda }{\lambda_{\epsilon}}}{\epsilon}
\Bigg]
+C_F N_F \Bigg[
-\frac{3}{\epsilon ^2}+\frac{5}{2 \epsilon}
\Bigg]\Bigg\}
\\ \notag
&+\Big(\frac{\alpha_s}{4\pi}\Big)\Big(\frac{\alpha_{4\epsilon}}{4\pi}\Big) C_A^2\,(1-N_\epsilon)\Bigg[
-\frac{4}{\epsilon ^2}
+\frac{-16+\frac{\pi ^2}{6}}{\epsilon}
\Bigg]
\\ \notag
&+\Big(\frac{\alpha_e}{4\pi}\Big)^2 \Bigg\{C_A N_F \Bigg[
\frac{-1+\frac{N_\epsilon}{2}}{\epsilon ^2}
+\frac{\frac{1}{2}-\frac{N_\epsilon}{4}}{\epsilon}
\Bigg]
+C_F N_F \Bigg[
\frac{2-\frac{N_\epsilon}{2}}{\epsilon^2}
+\frac{-1-\frac{N_\epsilon}{4}}{\epsilon}
\Bigg]
+\frac{N_F^2}{\epsilon ^2}\Bigg\}
\\ \notag
&+\Big(\frac{\alpha_e}{4\pi}\Big)\Big(\frac{\alpha_{4\epsilon}}{4\pi}\Big)C_A N_F\,(1-N_\epsilon)\
\frac{2}{\epsilon}\phantom{\Bigg\}}
\\
&+\Big(\frac{\alpha_{4\epsilon}}{4\pi}\Big)^2 C_A^
(1-N_\epsilon)\frac{-3}{8\epsilon}
+\mathcal{O}(\epsilon^0).\phantom{\Bigg\}}
\end{align}
Compared to Eqs.~(\ref{eq:gluonFF1}) and (\ref{eq:gluonFF2}) the result with external
$\epsilon$-scalars is more complicated and includes all combinations of the couplings
$\alpha_s, \alpha_e$ and $\alpha_{4\epsilon}$.
In this result, like in all previous results, the evanescent coupling $\alpha_e$
appears always together with at least one power of $N_F$ and the quartic coupling
$\alpha_{4\epsilon}$ is always accompanied by a factor $(1-N_\epsilon)$.
\subsection{Results for equal couplings}
During the renormalization process the couplings $\alpha_s$, $\alpha_e$, $\alpha_{4\epsilon}$
and $\lambda$, $\lambda_{\epsilon}$ have to be distinguished.
After renormalization they can be set equal, giving a simpler form of
the final result.\footnote{%
If the results of Sec.~\ref{sec:onshell_results_complete} were not desired
for independent couplings, the genuine two-loop diagrams could have been computed
in a simpler way, with all couplings set equal from the beginning --- this is
what is done in many applications of {\scshape fdh}\ and {\scshape dred}\ in the literature.}
The results for $N_\epsilon=2\epsilon$ at the one(two)-loop level up
to order $\mathcal{O}(\epsilon^4)$ ($\mathcal{O}(\epsilon^2)$) then read:
\begin{align}
\notag
\bar{F}^{\text{1L}}_{{\hat{g}}} = \ & \Big(\frac{\alpha_s}{4\pi}\Big)
\Bigg\{
C_A \Bigg[
-\frac{2}{\epsilon ^2}
-\frac{11}{3 \epsilon }
+\frac{1}{3}
+\frac{\pi ^2}{6}
+\epsilon\frac{14}{3}\zeta (3)
+\epsilon ^2\frac{47}{720}\pi ^4
\\&\quad\quad\quad\quad\quad\notag
+\epsilon^3 \Big(
\frac{62}{5}\zeta (5)
-\frac{7}{18}\pi ^2 \zeta (3)
\Big)
+\epsilon^4 \Big(
\frac{949}{60480}\pi ^6
-\frac{49}{9}\zeta(3)^2
\Big)
\Bigg]
\\ & \quad\quad\quad
+\frac{2 N_F}{3 \epsilon}
\Bigg\}
+\mathcal{O}(\epsilon^5),
\\[15pt] \notag
\bar{F}^{\text{2L}}_{{\hat{g}}}
= \ &\Big(\frac{\alpha_s}{4\pi}\Big)^2
\Bigg\{
C_A^2 \Bigg[
\frac{2}{\epsilon ^4}
+\frac{77}{6 \epsilon ^3}
+\frac{
\frac{77}{9}
-\frac{\pi ^2}{6}
}{\epsilon ^2}
+\frac{
-\frac{400}{27}
-\frac{11}{36}\pi ^2
-\frac{25}{3}\zeta (3)
}{\epsilon }
\\&\quad\quad\quad\quad\quad\quad\notag
+\frac{5711}{162}
+\frac{17}{9}\pi ^2
-33 \zeta (3)
-\frac{7}{60}\pi ^4
\phantom{\Bigg\}}
\\&\quad\quad\quad\quad\quad\quad\notag
+\epsilon\Bigg(
\frac{189767}{972}
+\frac{65}{27}\pi ^2
-\frac{1058}{27}\zeta (3)
-\frac{1111}{2160}\pi ^4
+\frac{71}{5}\zeta (5)
+\frac{23}{18}\pi ^2 \zeta (3)
\Bigg)
\\&\quad\quad\quad\quad\quad\quad\notag
+\epsilon^2\Bigg(
\frac{4972715}{5832}
-\frac{233}{324}\pi ^2
-\frac{26404}{81}\zeta (3)
-\frac{307}{360}\pi ^4
-\frac{341}{5}\zeta (5)
\\&\quad\quad\quad\quad\quad\quad\quad\quad\quad\notag
+\frac{257}{1680}\pi ^6
-\frac{11}{54}\pi ^2 \zeta (3)
+\frac{901}{9}\zeta (3)^2
\Bigg)
\Bigg]
\\&\quad\ \,\notag
+C_A N_F \Bigg[
-\frac{7}{3 \epsilon ^3}
-\frac{13}{3 \epsilon ^2}
+\frac{
\frac{76}{27}
+\frac{\pi ^2}{18}
}{\epsilon}
-\frac{916}{81}
-\frac{5}{18}\pi^2
-2 \zeta (3)
\\&\quad\quad\quad\quad\quad\quad\notag
+\epsilon\Bigg(
-\frac{14603}{243}
-\frac{8}{27}\pi ^2
-\frac{604}{27}\zeta (3)
-\frac{59}{1080}\pi ^4
\Bigg)
\\&\quad\quad\quad\quad\quad\quad\notag
+\epsilon^2\Bigg(
-\frac{366023}{1458}
+\frac{127}{162}\pi ^2
-\frac{4448}{81}\zeta (3)
-\frac{257}{648}\pi ^4
-\frac{98}{5}\zeta (5)
+\frac{61}{27}\pi ^2 \zeta (3)
\Bigg)
\Bigg]
\\&\quad\ \,\notag
+C_F N_F \Bigg[
\frac{1}{\epsilon }
-\frac{73}{6}
+8 \zeta (3)
+\epsilon\Bigg(
-\frac{2045}{36}
+\frac{7}{18}\pi ^2
+\frac{92}{3}\zeta (3)
+\frac{4}{27}\pi ^4
\Bigg)
\\&\quad\quad\quad\quad\quad\quad\notag
+\epsilon ^2\Bigg(
-\frac{53269}{216}
+\frac{263}{108}\pi ^2
+\frac{1232}{9}\zeta (3)
+\frac{46}{81}\pi ^4
+32 \zeta (5)
-\frac{20}{9}\pi ^2 \zeta (3)
\Bigg)
\Bigg]
\\&\quad\quad\quad\quad
+\frac{4 N_F^2}{9 \epsilon ^2}
\Bigg\}
+\mathcal{O}(\epsilon^3),
\\[15pt] \notag
\bar{F}^{\text{1L}}_{{\tilde{g}}} = \ &
\Big(\frac{\alpha_s}{4\pi}\Big)\Bigg\{
C_A \Bigg[
-\frac{2}{\epsilon ^2}
-\frac{4}{\epsilon }
+2
+\frac{\pi ^2}{6}
+\epsilon\frac{14}{3}\zeta (3)
+\epsilon ^2\frac{47}{720}\pi ^4
\\&\quad\quad\quad\quad\quad\, \notag
+\epsilon^3 \Big(
\frac{62}{5}\zeta (5)
-\frac{7}{18}\pi ^2 \zeta (3)
\Big)
+\epsilon^4 \Big(
\frac{949}{60480}\pi ^6
-\frac{49}{9}\zeta(3)^2
\Big)
\Bigg]
\\&\quad\quad\quad
+\frac{N_F}{\epsilon}
\Bigg\}
+\mathcal{O}(\epsilon^5),
\\[15pt] \notag
\bar{F}^{\text{2L}}_{{\tilde{g}}} = \ &
\Big(\frac{\alpha_s}{4\pi}\Big)^2\Bigg\{
C_A^2 \Bigg[
\frac{2}{\epsilon ^4}
+\frac{27}{2 \epsilon ^3}
+\frac{
\frac{64}{9}
-\frac{\pi ^2}{6}
}{\epsilon ^2}
+\frac{
-\frac{1211}{54}
-\frac{\pi ^2}{4}
-\frac{25}{3}\zeta (3)
}{\epsilon }
\\&\quad\quad\quad\quad\quad\notag
+\frac{6052}{81}
+\frac{263}{108}\pi ^2
-\frac{323}{9}\zeta (3)
-\frac{7}{60}\pi ^4
\\&\quad\quad\quad\quad\quad\notag
+\epsilon\Bigg(
\frac{263363}{972}
+\frac{1489}{324}\pi ^2
-\frac{1655}{27}\zeta (3)
-\frac{67}{120}\pi ^4
+\frac{71}{5}\zeta (5)
+\frac{23}{18}\pi ^2 \zeta (3)
\Bigg)
\\&\quad\quad\quad\quad\quad\notag
+\epsilon ^2\Bigg(
\frac{6457043}{5832}
+\frac{6803}{972}\pi ^2
-\frac{34459}{81}\zeta (3)
-\frac{15221}{12960}\pi ^4
-\frac{235}{3}\zeta (5)
\\&\quad\quad\quad\quad\quad\quad\quad\quad\notag
+\frac{257}{1680}\pi ^6
-\frac{16}{27}\pi ^2 \zeta (3)
+\frac{901}{9}\zeta (3)^2
\Bigg)
\Bigg]
\\&\notag
+C_A N_F \Bigg[
-\frac{3}{\epsilon^3}
-\frac{52}{9 \epsilon^2}
+\frac{151}{27 \epsilon }
-\frac{1925}{162}
-\frac{25}{54}\pi ^2
-\frac{28}{9}\zeta (3)
\\&\quad\quad\quad\quad\ \,\notag
+\epsilon\Bigg(
-\frac{10538}{243}
-\frac{46}{81}\pi ^2
-\frac{922}{27}\zeta (3)
-\frac{61}{720}\pi ^4
\Bigg)
\\&\quad\quad\quad\quad\ \,\notag
+\epsilon ^2\Bigg(
-\frac{291065}{1458}
+\frac{419}{486}\pi ^2
-\frac{8678}{81}\zeta (3)
-\frac{3971}{6480}\pi ^4
\\&\quad\quad\quad\quad\quad\quad\quad\ \,\notag
-\frac{382}{15}\zeta (5)
+\frac{203}{54}\pi ^2 \zeta (3)
\Bigg)
\Bigg]
\\&\notag
+C_F N_F \Bigg[
-\frac{1}{\epsilon ^2}
+\frac{1}{2 \epsilon }
-41
-\frac{\pi ^2}{3}
+12 \zeta (3)
\\&\quad\quad\quad\quad\ \ \notag
+\epsilon\Bigg(
-\frac{669}{4}
-\frac{3}{2}\pi ^2
+\frac{196}{3}\zeta (3)
+\frac{2}{9}\pi ^4
\Bigg)
\\&\quad\quad\quad\quad\ \ \notag
+\epsilon^2\Bigg(
-\frac{4607}{8}
-\frac{61}{12}\pi ^2
+\frac{868}{3}\zeta (3)
+\frac{67}{60}\pi ^4
+48 \zeta (5)
-\frac{10}{3}\pi ^2 \zeta (3)
\Bigg)
\Bigg]
\\&
+\frac{N_F^2}{\epsilon ^2}
\Bigg\}
+\mathcal{O}(\epsilon^3).
\end{align}
\section{Conclusions}
\label{sec:conclusions}
We have computed the $H\to gg$ amplitudes
at the two-loop level in the {\scshape fdh}\ and {\scshape dred}\ scheme and presented the
$\overline{\text{MS}}$ renormalized on-shell results up to the order $\epsilon^2$. In
{\scshape dred}, this involves two different amplitudes for $H\to {\hat{g}}\ghat$
and $H\to {\tilde{g}}\gtilde$ with external gluons/$\epsilon$-scalars.
The computation is motivated because it contains key elements which
constitute important building blocks for further computations, and
because it is essential for the complete understanding of the
infrared divergence structure of {\scshape fdh}\ and {\scshape dred}\ amplitudes.
The renormalization procedure has been described in detail. It is less trivial
than in many QCD calculations in {\scshape cdr}, since not only the strong
coupling needs to be renormalized but also evanescent couplings of the
$\epsilon$-scalar. The computation provides a further example of
the well-known fact that regardless of whether {\scshape fdh}\ or {\scshape dred}\ is used,
the evanescent couplings have to be renormalized independently.
Further, the renormalization of the effective dimension-5 operators
involves mixing with new, $\epsilon$-scalar dependent operators.
A suitable basis of operators has been provided. One unavoidable fact is
that the extended operator space contains operators which are total
derivatives. As a result the required operator mixing renormalization
constants cannot be obtained in the same elegant way of
Ref.~\cite{Spiridonov:1984br} as in {\scshape cdr}.
Instead, they had to be obtained from explicit one- and two-loop
off-shell calculations.
The results for the UV renormalized but infrared divergent form factors
can also be used to complete the study of the general infrared divergence
structure of two-loop amplitudes in {\scshape fdh}\ and {\scshape dred}, begun in
Ref.~\cite{Gnendiger:2014nxa,Kilgore:2012tb}. From general principles it is known
that all infrared divergences can be expressed in terms of cusp and
parton anomalous dimensions. The results of the present paper allow
to extract the final missing two-loop anomalous dimension for
external $\epsilon$-scalars. This extraction, together with further
checks and results, will be presented in a forthcoming paper
\cite{IRstructure}, where the infrared structure will also be investigated by
a SCET approach.
\section{Appendix}
\subsection{Projectors and form factors of gluons and $\epsilon$-scalars}
\label{sec:appendix_A}
According to its Lorentz structure the on-shell Green-function
$\Gamma_{H{\hat{A}}^\mu{\hat{A}}^\nu}^{\text{on-shell}}$ can be represented as
\begin{align}
\Gamma_{H{\hat{A}}^\mu{\hat{A}}^\nu}^{\text{on-shell}} =
a\,(p\cdot r)\,{\hat{g}}^{\mu\nu}+ b\,p^\nu r^\mu + c\,p^\mu r^\nu + d\,p^\mu p^\nu + e\,r^\mu r^\nu,
\end{align}
where the coefficients $a\dots e$ are momentum-dependent quantities,
and coefficient $a$ is the gluon form factor.
Due to QCD Ward-identities the relation $a=-b$ holds, see e.\,g. Ref.~\cite{Harlander:2000mg}.
Accordingly, the on-shell Green-function $\Gamma_{H{\tilde{A}}^\mu{\tilde{A}}^\nu}^{\text{on-shell}}$
with external $\epsilon$-scalars can be represented as
\begin{align}
\Gamma_{H{\tilde{A}}^\mu{\tilde{A}}^\nu}^{\text{on-shell}} =
f\,(p\cdot r)\,{\tilde{g}}^{\mu\nu},
\end{align}
where we refer to $f$ as $\epsilon$-scalar form factor.
All coefficients of the covariant decomposition can be extracted with
appropriate projection operators that are given below.
In the off-shell case the UV divergence structure of
$\Gamma_{H{\hat{A}}^\mu{\hat{A}}^\nu}$
can be represented in a more specific way as
\begin{align}
\Gamma_{H{\hat{A}}^\mu{\hat{A}}^\nu}\Big|^{\text{off-shell}}_{\text{UV div.}} =
\left[A+A'\,\frac{p^2+r^2}{(p\cdot r)}\right](p\cdot r)\,{\hat{g}}^{\mu\nu}
+ B\,p^\nu r^\mu + C\,p^\mu r^\nu + D\,p^\mu p^\nu + E\,r^\mu r^\nu,
\end{align}
where the coefficients $A\dots E$ are now momentum-independent.
Since these divergences can be absorbed by counterterms corresponding
to operators $O_1$ and $O_4$ the relation $A=-B$ again holds,
see e.\,g. Feynman rules (\ref{frO1a}) and (\ref{frO4a}).
Due to this there are two possibilities of extracting coefficient $A$,
which corresponds to the desired renormalization constant
$\delta Z^{\text{2L}}_{\lambda}$:
The first one is to extract the coefficient of $(p\cdot r)\,{\hat{g}}^{\mu\nu}$
and neglect terms $\propto p^2, r^2$;
the second is to extract coefficient $-B$.
We checked explicitly that the relations $a=-b$ and $A=-B$
hold throughout the paper.
Again, the covariant decomposition with external $\epsilon$-scalars is much simpler and reads:
\begin{align}
\Gamma_{H{\tilde{A}}^\mu{\tilde{A}}^\nu}\Big|^{\text{off-shell}}_{\text{UV div.}} =
\left[F+F'\,\frac{p^2+r^2}{(p\cdot r)}\right](p\cdot r)\,{\tilde{g}}^{\mu\nu}.
\end{align}
The desired coefficient for the computation of
$\delta Z^{\text{2L}}_{\lambda_{\epsilon}}$ is $F$.
Accordingly, we extract the coefficient of $(p\cdot r)\,{\tilde{g}}^{\mu\nu}$ and neglect terms
$\propto p^2, r^2$.
The corresponding projection operators are:
\begin{subequations}
\begin{align}
\begin{split}
P^{\mu\nu}_{g,(p\cdot r){\hat{g}}^{\mu\nu}} &= \Big\{
{\hat{g}}^{\mu\nu}\left[(p\cdot r)^2-p^2r^2\right]\\
&\quad\quad-(p^\nu r^\mu+p^\mu r^\nu)(p\cdot r)\\
&\quad\quad+p^\mu p^\nu r^2+r^\mu r^\nu p^2\Big\}
\frac{1}{(D-2)(p\cdot r)\left[(p\cdot r)^2-p^2r^2\right]},
\end{split}\\
%
\begin{split}
P^{\mu\nu}_{g,p^\nu r^\mu} &=\Big\{
{\hat{g}}^{\mu\nu}\,(p\cdot r)\left[p^2 r^2-(p\cdot r)^2\right]\\
&\quad\quad+p^\nu r^\mu\left[(p\cdot r)^2+p^2 r^2 (D-2)\right]
+p^\mu r^\nu\,(p\cdot r)^2\,(D-1)\\
&\quad\quad+(p^\mu p^\nu r^2+r^\mu r^\nu p^2)(p\cdot r)(1-D)\Big\}
\frac{1}{(D-2)\left[(p\cdot r)^2-p^2r^2\right]^2},
\end{split}\\
%
P^{\mu\nu}_{{\tilde{g}},(p\cdot r){\tilde{g}}^{\mu\nu}} &=
\frac{{\tilde{g}}^{\mu\nu}}{N_\epsilon(p\cdot r)}.
\end{align}
\end{subequations}
\subsection{Feynman rules}
\label{sec:appendix_B}
In the following we give Feynman rules according to operators
$O_1, {\tilde{O}}_1, O_4$ and ${\tilde{O}}_4$
that are needed for the renormalization in the {\scshape fdh}\ and {\scshape dred}\ scheme.
Feynman rules including four $\epsilon$-scalars
are not relevant in this paper and are not given explicitly.
\begin{align}
\intertext{$\bullet$ Feynman rules according to the Lagrangian term $\lambda H O_1$:}
\scalebox{.6}{
\begin{picture}(200,0)(0,50)
\Vertex(35,50){2}
\DashLine(0,50)(35,50){2}
\Gluon(35,50)(100,100){5}{6}
\Gluon(35,50)(100,0){5}{6}
\LongArrow(80,96)(60,80)
\LongArrow(80,6)(60,21)
\Text(63, 100)[c]{\scalebox{1.67}{$k_2$}}
\Text(63, 5 )[c]{\scalebox{1.67}{$k_1$}}
\Text( 25, 70)[c]{\scalebox{1.67}{$O_1$}}
\Text(-15, 50)[c]{\scalebox{1.67}{$H$}}
\Text(115, -5)[c]{\scalebox{1.67}{${\hat{A}}^{\alpha}_{a}$}}
\Text(115,105)[c]{\scalebox{1.67}{${\hat{A}}^{\beta }_{b}$}}
\label{frO1a}
\end{picture} }
&=\quad i\lambda\,\Big[(k_1\cdot k_2)\,{\hat{g}}^{\,\alpha\beta}-k_1^{\,\beta}\,k_2^{\,\alpha}\Big]\,\delta^{ab}
\phantom{\begin{aligned}\bigg|\\ \bigg|\\ \bigg|\end{aligned}}
\\
\scalebox{.6}{
\begin{picture}(200,0)(0,50)
\Vertex(35,50){2}
\DashLine(0,50)(35,50){2}
\Gluon(35,50)(100,100){5}{5}
\Gluon(35,50)(100,50){5}{4}
\Gluon(35,50)(100,0){5}{5}
\LongArrow(80,96)(60,80)
\LongArrow(80,6)(60,21)
\LongArrow(80,60)(60,60)
\Text( 63,100)[c]{\scalebox{1.67}{$k_3$}}
\Text( 95, 65)[c]{\scalebox{1.67}{$k_2$}}
\Text( 63, 5)[c]{\scalebox{1.67}{$k_1$}}
\Text( 25, 70)[c]{\scalebox{1.67}{$O_1$}}
\Text(-15,50 )[c]{\scalebox{1.67}{$H$}}
\Text(115, -5)[c]{\scalebox{1.67}{${\hat{A}}^{\alpha}_{a}$}}
\Text(115, 50)[c]{\scalebox{1.67}{${\hat{A}}^{\beta }_{b}$}}
\Text(115,105)[c]{\scalebox{1.67}{${\hat{A}}^{\gamma}_{c}$}}
\end{picture} }
&=\quad-\lambda\,g_{s} f^{abc}\times\left[
\begin{aligned}
& {\hat{g}}^{\alpha\beta }\left(k_1-k_2\right)^{\gamma} \\ &
+ {\hat{g}}^{\beta\gamma }\left(k_2-k_3\right)^{\alpha} \\ &
+ {\hat{g}}^{\gamma\alpha}\left(k_3-k_1\right)^{\beta}
\end{aligned}
\right]
\phantom{\begin{aligned}\bigg|\\ \bigg|\\ \bigg|\end{aligned}}
\\
\scalebox{.6}{
\begin{picture}(200,0)(0,50)
\Vertex(35,50){2}
\DashLine(0,50)(35,50){2}
\Gluon(35,50)(100,100){5}{5}
\Gluon(35,50)(100, 66){5}{5}
\Gluon(35,50)(100, 33){5}{5}
\Gluon(35,50)(100, 0){5}{5}
\Text( 25, 70)[c]{\scalebox{1.67}{$O_1$}}
\Text(-15, 50)[c]{\scalebox{1.67}{$H$}}
\Text(115, -6)[c]{\scalebox{1.67}{${\hat{A}}^{\alpha}_{a}$}}
\Text(115, 31)[c]{\scalebox{1.67}{${\hat{A}}^{\beta }_{b}$}}
\Text(115, 68)[c]{\scalebox{1.67}{${\hat{A}}^{\gamma}_{c}$}}
\Text(115,105)[c]{\scalebox{1.67}{${\hat{A}}^{\delta}_{d}$}}
\end{picture} }
&=\quad -i\lambda\, g_{s}^2\times\left[\begin{aligned}
& {\hat{g}}^{\alpha\beta }{\hat{g}}^{\gamma\delta}\left(f^{ace}f^{bde}+f^{ade}f^{bce}\right)\\
&+{\hat{g}}^{\alpha\gamma}{\hat{g}}^{\beta\delta }\left(f^{abe}f^{cde}-f^{ade}f^{bce}\right)\\
&-{\hat{g}}^{\alpha\delta}{\hat{g}}^{\beta\gamma }\left(f^{abe}f^{cde}+f^{ace}f^{bde}\right)
\end{aligned}\right]
\phantom{\begin{aligned}\bigg|\\ \bigg|\\ \bigg|\end{aligned}}
%
%
%
%
%
\intertext{$\bullet$ Feynman rules according to the Lagrangian term $\lambda_{\epsilon} H {\tilde{O}}_1$}
\scalebox{.6}{
\begin{picture}(200,0)(0,50)
\Vertex(35,50){2}
\DashLine(0,50)(35,50){2}
\DashLine(35,50)(100,100){4}
\DashLine(100,0)(35,50){4}
\LongArrow(80,96)(60,80)
\LongArrow(80,6)(60,21)
\Text(63, 100)[c]{\scalebox{1.67}{$k_2$}}
\Text(63, 5 )[c]{\scalebox{1.67}{$k_1$}}
\Text( 25, 70)[c]{\scalebox{1.67}{${\tilde{O}}_1$}}
\Text(-15,50 )[c]{\scalebox{1.67}{$H$}}
\Text(115, -5)[c]{\scalebox{1.67}{${\tilde{A}}^{\,\alpha}_{a}$}}
\Text(115,105)[c]{\scalebox{1.67}{${\tilde{A}}^{\,\beta }_{b}$}}
\end{picture} }
&=\quad i\lambda_{\epsilon}\,\Big[(k_1\cdot k_2)\,{\tilde{g}}^{\,\alpha\beta}\Big]\,\delta^{ab}
\phantom{\begin{aligned}\bigg|\\ \bigg|\\ \bigg|\end{aligned}}
\\
\scalebox{.6}{
\begin{picture}(200,0)(0,50)
\Vertex(35,50){2}
\DashLine(0,50)(35,50){2}
\Gluon(35,50)(100,100){5}{5}
\DashLine(35,50)(100,50){4}
\DashLine(100,0)(35,50){4}
\LongArrow(80,96)(60,80)
\LongArrow(80,6)(60,21)
\LongArrow(80,60)(60,60)
\Text( 63,100)[c]{\scalebox{1.67}{$k_3$}}
\Text( 95, 65)[c]{\scalebox{1.67}{$k_2$}}
\Text( 63, 5)[c]{\scalebox{1.67}{$k_1$}}
\Text( 25, 70)[c]{\scalebox{1.67}{${\tilde{O}}_1$}}
\Text(-15,50 )[c]{\scalebox{1.67}{$H$}}
\Text(115, -5)[c]{\scalebox{1.67}{${\tilde{A}}^{\alpha}_{a}$}}
\Text(115, 50)[c]{\scalebox{1.67}{${\tilde{A}}^{\beta }_{b}$}}
\Text(115,105)[c]{\scalebox{1.67}{${\hat{A}}^{\gamma}_{c}$}}
\end{picture} }
&=\quad -\lambda_{\epsilon}\,g_{s} f^{abc}\,{\tilde{g}}^{\,\alpha\beta}\,(k_1-k_2)^{\gamma}
\phantom{\begin{aligned}\bigg|\\ \bigg|\\ \bigg|\end{aligned}}
\\
\scalebox{.6}{
\begin{picture}(200,0)(0,50)
\Vertex(35,50){2}
\DashLine(0,50)(35,50){2}
\Gluon(35,50)(100,100){5}{5}
\Gluon(35,50)(100, 66){5}{5}
\DashLine(35,50)(100, 33){4}
\DashLine(35,50)(100, 0){4}
\Text( 25, 70)[c]{\scalebox{1.67}{${\tilde{O}}_1$}}
\Text(-15, 50)[c]{\scalebox{1.67}{$H$}}
\Text(115, -6)[c]{\scalebox{1.67}{${\tilde{A}}^{\alpha}_{a}$}}
\Text(115, 31)[c]{\scalebox{1.67}{${\tilde{A}}^{\beta }_{b}$}}
\Text(115, 68)[c]{\scalebox{1.67}{${\hat{A}}^{\gamma}_{c}$}}
\Text(115,105)[c]{\scalebox{1.67}{${\hat{A}}^{\delta}_{d}$}}
\end{picture} }
&=\quad -i\lambda_{\epsilon}\,g_{s}^2\, {\tilde{g}}^{\,\alpha\beta}{\hat{g}}^{\gamma\delta}\left(f^{ace}f^{bde}+f^{ade}f^{bce}\right)
\phantom{\begin{aligned}\bigg|\\ \bigg|\\ \bigg|\end{aligned}}
%
%
%
%
%
\intertext{$\bullet$ Feynman rules according to the Lagrangian term $H O_4$:}
\scalebox{.6}{
\begin{picture}(200,0)(0,50)
\Vertex(35,50){2}
\DashLine(0,50)(35,50){2}
\Gluon(35,50)(100,100){5}{5}
\Gluon(35,50)(100,0){5}{5}
\LongArrow(80,96)(60,80)
\LongArrow(80,6)(60,21)
\Text(63, 100)[c]{\scalebox{1.67}{$k_2$}}
\Text(63, 5 )[c]{\scalebox{1.67}{$k_1$}}
\Text( 25, 70)[c]{\scalebox{1.67}{$O_4$}}
\Text(-15, 50)[c]{\scalebox{1.67}{$H$}}
\Text(115, -5)[c]{\scalebox{1.67}{${\hat{A}}^{\alpha}_{a}$}}
\Text(115,105)[c]{\scalebox{1.67}{${\hat{A}}^{\beta }_{b}$}}
\label{frO4a}
\end{picture} }
&=\quad -i\,\Big[\left(k_1^2 + k_2^2\right){\hat{g}}^{\,\alpha\beta}
-\left(k_1^{\,\alpha}\,k_1^{\,\beta}+k_2^{\,\alpha}\,k_2^{\,\beta}\right)\Big]\,\delta^{ab}
\phantom{\begin{aligned}\bigg|\\ \bigg|\\ \bigg|\end{aligned}}
\\
\scalebox{.6}{
\begin{picture}(200,0)(0,50)
\Vertex(35,50){2}
\DashLine(0,50)(35,50){2}
\Gluon(35,50)(100,100){5}{5}
\Gluon(35,50)(100,50){5}{5}
\Gluon(35,50)(100,0){5}{5}
\LongArrow(80,96)(60,80)
\LongArrow(80,6)(60,21)
\LongArrow(80,60)(60,60)
\Text( 63,100)[c]{\scalebox{1.67}{$k_3$}}
\Text( 95, 65)[c]{\scalebox{1.67}{$k_2$}}
\Text( 63, 5)[c]{\scalebox{1.67}{$k_1$}}
\Text( 25, 70)[c]{\scalebox{1.67}{$O_4$}}
\Text(-15, 50)[c]{\scalebox{1.67}{$H$}}
\Text(115, -5)[c]{\scalebox{1.67}{${\hat{A}}^{\alpha}_{a}$}}
\Text(115, 50)[c]{\scalebox{1.67}{${\hat{A}}^{\beta }_{b}$}}
\Text(115,105)[c]{\scalebox{1.67}{${\hat{A}}^{\gamma}_{c}$}}
\end{picture} }
&=\quad3\,g_{s} f^{abc}\times\left[
\begin{aligned}
& {\hat{g}}^{\alpha\beta }\left(k_1-k_2\right)^{\gamma} \\ &
+ {\hat{g}}^{\beta\gamma }\left(k_2-k_3\right)^{\alpha} \\ &
+ {\hat{g}}^{\gamma\alpha}\left(k_3-k_1\right)^{\beta}
\end{aligned}
\right]
\phantom{\begin{aligned}\bigg|\\ \bigg|\\ \bigg|\end{aligned}}
\\
\scalebox{.6}{
\begin{picture}(200,0)(0,50)
\Vertex(35,50){2}
\DashLine(0,50)(35,50){2}
\Gluon(35,50)(100,100){5}{5}
\DashLine(35,50)(100,50){4}
\DashLine(100,0)(35,50){4}
\LongArrow(80,6)(60,21)
\LongArrow(80,60)(60,60)
\Text( 95, 65)[c]{\scalebox{1.67}{$k_2$}}
\Text( 63, 5)[c]{\scalebox{1.67}{$k_1$}}
\Text( 25, 70)[c]{\scalebox{1.67}{$O_4$}}
\Text(-15, 50)[c]{\scalebox{1.67}{$H$}}
\Text(115, -5)[c]{\scalebox{1.67}{${\tilde{A}}^{\alpha}_{a}$}}
\Text(115, 50)[c]{\scalebox{1.67}{${\tilde{A}}^{\beta }_{b}$}}
\Text(115,105)[c]{\scalebox{1.67}{${\hat{A}}^{\gamma}_{c}$}}
\end{picture} }
&=\quad -g_{s} f^{abc}\,{\tilde{g}}^{\,\alpha\beta}\,(k_1-k_2)^{\gamma}
\phantom{\begin{aligned}\bigg|\\ \bigg|\\ \bigg|\end{aligned}}
\\
\scalebox{.6}{
\begin{picture}(200,0)(0,50)
\Vertex(35,50){2}
\DashLine(0,50)(35,50){2}
\ArrowLine(35,50)(100,100)
\ArrowLine(100,50)(35,50)
\Gluon(35, 50)(100,0){5}{5}
\Text( 25, 70)[c]{\scalebox{1.67}{$O_4$}}
\Text(-15, 50)[c]{\scalebox{1.67}{$H$}}
\Text(115, -5)[c]{\scalebox{1.67}{${\tilde{A}}^{\alpha}_{a}$}}
\Text(115, 50)[c]{\scalebox{1.67}{$q_j$}}
\Text(115,105)[c]{\scalebox{1.67}{$\overline{q}_i$}}
\end{picture} }
&=\quad -ig_{s}\,\hat{\gamma}^{\alpha}\left(T^a\right)_{ij}
\phantom{\begin{aligned}\bigg|\\ \bigg|\\ \bigg|\end{aligned}}
\\
\scalebox{.6}{
\begin{picture}(200,0)(0,50)
\Vertex(35,50){2}
\DashLine(0,50)(35,50){2}
\DashArrowLine(35,50)(100,100){2}
\DashArrowLine(100,0)(35,50){2}
\LongArrow(80,96)(60,80)
\LongArrow(80,6)(60,21)
\Text(63, 100)[c]{\scalebox{1.67}{$k_2$}}
\Text(63, 5 )[c]{\scalebox{1.67}{$k_1$}}
\Text( 25, 70)[c]{\scalebox{1.67}{$O_4$}}
\Text(-15, 50)[c]{\scalebox{1.67}{$H$}}
\Text(115, -5)[c]{\scalebox{1.67}{$c_a$}}
\Text(115,105)[c]{\scalebox{1.67}{$\overline{c}_b$}}
\end{picture} }
&=\quad i\,(k_1\cdot k_2)\,\delta_{ab}
\phantom{\begin{aligned}\bigg|\\ \bigg|\\ \bigg|\end{aligned}}
%
%
%
%
%
\intertext{$\bullet$ Feynman rules according to the Lagrangian term $H {\tilde{O}}_4$:}
\scalebox{.6}{
\begin{picture}(200,0)(0,50)
\Vertex(35,50){2}
\DashLine(0,50)(35,50){2}
\DashLine(35,50)(100,100){4}
\DashLine(35,50)(100,0){4}
\LongArrow(80,96)(60,80)
\LongArrow(80,6)(60,21)
\Text(63, 100)[c]{\scalebox{1.67}{$k_2$}}
\Text(63, 5 )[c]{\scalebox{1.67}{$k_1$}}
\Text( 25, 70)[c]{\scalebox{1.67}{${\tilde{O}}_4$}}
\Text(-15, 50)[c]{\scalebox{1.67}{$H$}}
\Text(115, -5)[c]{\scalebox{1.67}{${\hat{A}}^{\alpha}_{a}$}}
\Text(115,105)[c]{\scalebox{1.67}{${\hat{A}}^{\beta }_{b}$}}
\end{picture} }
&=\quad -i\left(k_1^2 + k_2^2\right){\tilde{g}}^{\,\alpha\beta}\,\delta^{ab}
\phantom{\begin{aligned}\bigg|\\ \bigg|\\ \bigg|\end{aligned}}
\\
\scalebox{.6}{
\begin{picture}(200,0)(0,50)
\Vertex(35,50){2}
\DashLine(0,50)(35,50){2}
\Gluon(35,50)(100,100){5}{5}
\DashLine(35,50)(100,50){4}
\DashLine(100,0)(35,50){4}
\LongArrow(80,6)(60,21)
\LongArrow(80,60)(60,60)
\Text( 95, 65)[c]{\scalebox{1.67}{$k_2$}}
\Text( 63, 5)[c]{\scalebox{1.67}{$k_1$}}
\Text( 25, 70)[c]{\scalebox{1.67}{${\tilde{O}}_4$}}
\Text(-15, 50)[c]{\scalebox{1.67}{$H$}}
\Text(115, -5)[c]{\scalebox{1.67}{${\tilde{A}}^{\alpha}_{a}$}}
\Text(115, 50)[c]{\scalebox{1.67}{${\tilde{A}}^{\beta }_{b}$}}
\Text(115,105)[c]{\scalebox{1.67}{${\hat{A}}^{\gamma}_{c}$}}
\end{picture} }
&=\quad -\,2\,g_{s} f^{abc}\,{\tilde{g}}^{\,\alpha\beta}\,(k_1-k_2)^{\gamma}
\phantom{\begin{aligned}\bigg|\\ \bigg|\\ \bigg|\end{aligned}}
\end{align}
\end{appendix}
\subsection*{Acknowledgments}
We are grateful to M.\ Steinhauser and W.\ Kilgore for
useful discussions. We acknowledge financial support from the DFG
grant STO/876/3-1.
A. Visconti is supported by the Swiss National Science
Foundation (SNF) under contract 200021-144252.
\input{98_appendix}
|
1,314,259,993,655 | arxiv |
\section{Conclusion}
\label{sec:conclusion}
The automatic seizure detectors using frequency domain features of EEG signals and traditional machine learning techniques such as k-nearest neighbor and Support Vector Machine (SVM) have been extensively researched in literature \cite{kNN,nonlinear} in the past. Recently, deep neural network techniques have gained attention enabling seizure prediction from pre-ictal EEG patterns with high sensitivity and low FAR \cite{LSTM}. Since the contribution of different channels in seizure detection or prediction performance is patient specific and depends on the seizure onset zone, these models cannot
be generally used for every one and require subject-specific sensor selection and optimization techniques to reduce the complexity \cite{sensor_selection}. In addition, wearing an EEG cap is not feasible for a long-term monitoring of patients leading a normal daily life.
The autonomic nervous system dysfunction induced by epileptic brain activity is easier to detect using extracerebral sensors \cite{discharge}. In particular, heart rate variability (HRV), accelerometry and Electrodermal Activity (EDA) have gained attention for measuring the autonomic dysfunction. \cite{EMG,ACC}The frequency domain features such as Very Low Frequency (VLF)(0.0033-0.04~Hz), Low Frequency (LF)(0.04-0.15~Hz), and High Frequency (HF) (0.15-0.4~Hz) measure the Sympathetic (SNS) and parasympathetic (PNS) nervous balance \cite{heart-brain}. However, recent studies have suggested the LF/HF ratio is a controversial measure of sympatho/vagal balance. Since SNS and PNS activities are nor linearly counterparts, decreasing activity in one does not suggest an increase in the other. In fact, both SNS and PNS are contributing to LF power, whereas HF power is associated with PNS activity \cite{LF-HF}.
The Catecholamines released as anticonvulsant during epileptic seizures affect the
blood circulation and vascular function. The hemodynamics induced by seizures can
be measured by PPG sensors. In this study, we investigated six different
morphological features in PPG and ECG signal recorded from 12 epileptic subjects. A
total of 102 hours and 30 minutes of ictal and inter-ictal data was recorded. The
extracted features include heart rate, crest time, maximum velocity time, pulse
transmit time, pulse amplitude, and the first principle coefficient derived from
the second derivative of PPG pulse shape. Among these features, the crest time, the
maximum velocity time, and the pulse transmit time are influenced by the heart rate. In order to eliminate the effect of heart rate, these feature were normalized to pulse width.
All the investigated features showed significant variation in majority of recorded
seizures (refer to Table \ref{table:sig}). Except for patient number 11, who had a few seizures in which the pulse-transmit time was significantly increased, a consistent pattern across all seizures/subjects was observed. The reduction in pulse amplitude and the increase in normalized crest time suggests an increase in vascular resistance and hypovolemia in limbs induced by vasoconstriction. This is the first time a comprehensive analysis on PPG morphology along with ECG data induced by epilepsy is performed.
\section{Data Collection and Analysis }
\label{sec:data_analysis}
\subsection{Data Collection}
A total of 12 subjects (8 males and 4 females with a mean age of
33.64 $\pm$ 13.3) were recruited among the patients with refractory
epilepsy who underwent long-term extra-cranial monitoring admitted to
University of California Irvine Medical Center (UCIMC). The informed
consents were obtained before the start of the monitoring as required
by the Institutional Review Boards of University of California,
Irvine. The 20-channel surface EEG data and one-lead ECG data (Lead II) were
continuously recorded
as a standard of care procedure with sampling
rate of 500~Hz (The Nihon Kohden JE-921, paired with the QI-123A LAN converter). Our study also involved collection of PPG data recorded simultaneously by Empatica E4 with 64~Hz of sampling
frequency. The PPG data collected the blood pulse from the left ankle
of the subjects. The subjects were laying on a bed while being
monitored to reduce the motion artifact. The seizure onset time and
foci were extracted by a clinical neurophysiologist based on the
revision of the EEG data. A total of 60 seizures were recorded where 3 seizures were withdrawn from the analysis due to extensive noise in PPG data.
Table \ref{table:subjects} represents the subjects' clinical information
including the age and the seizure onset zone.
\begin{table*}[t]
\caption{Subjects' Clinical Characteristics}
\centering
\begin{tabular}{c c c c c c c c}
\hline
\hline
Patient&
Sex&
Age& Seizure type & Origin & Number of seizures & Medication & hour:min \\
\hline
1 &Male &27 &Partial &Unknown &1 & Clonazepam, Depakote &2:0' \\
2 &Female &26 &Generalized &- &1 & Topiramate, Lamictal&1:30'\\
3 &Male &27 &Partial &Left Frontal, Left Temporal &17 & Keppra, Vimpat &19:0'\\
4 &Male &23 &Partial &Right Frontal, Right Temporal &7 & Trileptal&23:21' \\
5 &Male &58 &Partial &Left Temporal &1 & Keppra, Levetricateram &5:0'\\
6 &Female &50 &Partial &Right Temporal &12 & Clonazepam, Levetricateram&16:0'\\
7 &Male &25 &Partial &Right Frontal, &5 & Trileptal, Zonisamide&10:41' \\
8 &Female &33 &Partial & Left Temporal, Left Parietal &1 & Phenytoin &4:0'\\
9 &Male &27 &Partial & Right Temporal, &1 & Lamictal, Clonazepam, Depakote, Mirtazapine &4:0'\\
10 &Male &25 &Partial & Left Occipital, Left Parietal &1 & Zonisamide&4:0' \\
11 &Male &59 &Partial & Left Temporal &9 & Keppra, Topiramate &21:0'\\
12 &Female &21 &Partial & Left Frontal &1 & Zonisamide, Phenytoin&2:0' \\
\hline
\end{tabular}
\label{table:subjects}
\end{table*}
\begin{table*}[t]
\caption{Collected Seizures}
\centering
\begin{tabular}{c c c c c c c c c c c c c c c c c c c c c}
\hline
\hline
Seizure Number & S1 & S2 & S3 & S4 & S5& S6 & S7 & S8& S9 & S10 &S11 & S12& S13 & S14 & S15 & S16 & S17 &S18 & S19\\
\hline
Patient number & P1 & P2& P3 & P3& P3 & P3 & P3 & P3 & P3 & P3 & P3 & P3 & P3 & P3 & P3 & P3 & P3 & P3 & P3 \\
Duration(sec) &110& 70 & 60 & 78 &117 & 113 & 42 & 46 & 56 &40 & 63 &175 &37 &102 &81 & 297 &19 & 85 & 59 \\
\hline
\hline \\
Seizure Number & S20& S21 & S22 & S23 & S24 & S25 & S26 & S27& S28 & S29 & S30& S31 & S32 & S33 & S34 & S35 &S36 &S37 & S38
\\
\hline
Patient number & P4 &P4& P4 &P4& P4 &P4&P4 &P5 &P6 & P6 & P6 & P6 & P6 & P6 & P6 &P6& P6 & P6 & P6\\
Duration(sec) & 39 & 84 & 93 & 71 & 98 & 104 & 61 & 79 & 94 & 91 & 201 & 156 & 49 & 42 & 35 & 35 & 36 & 47 &48 \\
\hline
\hline \\
Seizure Number & S39 &S40& S41 & S42 &S43 &S44 &S45 &S46 & S47 & S48 & S49 & S50 & S51 &S52 & S53 & S54 &S55 &S56 &S57\\
\hline
Patient number & P6&P7& P7 &P7&P7 &P7 & P8 & P9 & P10 & P11 & P11& P11 & P11& P11 & P11& P11 & P11 & P11 &P12\\
Duration(sec) & 59 & 36& 31 & 39 & 32 & 39& 28 & 265 & 13 & 54& 44 &48 &71 &45 & 37&61 &29 &38 &72 \\
\hline
\hline\\
\end{tabular}
\label{table:seizures}
\end{table*}
\subsubsection{Timing Synchronization}
Since the EEG/ECG data and the PPG data are recorded from separate devices,
the time offset and the drift noise present in the clocks of the two devices
will cause timing error in recording events. In order to synchronize the events
in Nihon Kohden EEG recording machine and the Empatica E4 PPG recording device,
we designed an apparatus that generates time stamps twice during each recording
session. The time stamp is a sequence of 10 square pulses with a pulse width of
1 second and duty cycle of 50\%. The Nihon Kohden JE-921 has an analog input
box that enables the collection of an external analog signal \cite{NK}. The
designed synchronizer feeds the analog input port of Nihon Kohden with the
sequence square pulses (5 volts of pulse amplitude). At the same time, the E4 device is placed in front of a green LED which is fed with the same sequence of pulses. As a result, the LED blinks with a frequency of 1 Hz and duty cycle of 50\%. The photo detector embedded in E4 captures the LED flashes as the time stamp. The time stamp is generated two times, once initially before attaching the E4 to the subject's left ankle and once again at the end after detaching it from the subject's body. \FIG{fig:synch} shows the block diagram of the synchronization apparatus. Once the recording session is finished, the timing is tuned by maximally aligning the time stamps captured by E4 photo detector and Nihon Kohden analog input box.
\begin{figure}
\centerline{\includegraphics[trim=90 20 50 20,width=140pt]{synch.eps}}
\caption{Synchronization apparatus}
\label{fig:synch}
\end{figure}
\label{subsub:analysis}
\begin{figure*}
\centering
\begin{subfigure}{0.3\textwidth}
\includegraphics[trim=0 10 0 20,width=\textwidth]{seiz_hr.eps}
\caption{}
\end{subfigure}
\centering
\begin{subfigure}{0.3\textwidth}
\includegraphics[trim=0 10 0 20,width=\textwidth]{seiz_MV.eps}
\caption{}
\end{subfigure}
\centering
\begin{subfigure}{0.3\textwidth}
\includegraphics[trim=0 10 0 20,width=\textwidth]{seiz_NCT.eps}
\caption{}
\end{subfigure}
\centering
\begin{subfigure}{0.3\textwidth}
\includegraphics[trim=0 10 0 0,width=\textwidth]{seiz_pa.eps}
\caption{}
\end{subfigure}
\centering
\begin{subfigure}{0.3\textwidth}
\includegraphics[trim=0 10 0 0,width=\textwidth]{seiz_PCA1.eps}
\caption{}
\end{subfigure}
\centering
\begin{subfigure}{0.3\textwidth}
\includegraphics[trim=0 10 0 0,width=\textwidth]{seiz_PTT.eps}
\caption{}
\end{subfigure}
\caption{Significant changes in z-scores proceeded by epileptic seizure in the first
recorded seizure from Patient 6. (a) $z_{\text{HR}}$,(b) $z_{t_{\text{MV}}}$, (c)
$z_{t_{\text{NCT}}}$, (d) $z_{\text{PA}}$,(e) $z_{\text{PCA1}}$, (f)
$z_{\text{PTT}}$.}
\label{fig:onesiez
\end{figure*}
\subsection{Preprocessing}
\label{subsec:preprocessing}
After synchronizing the PPG and EEG data, a neurologist marked the seizure onset and offset times based on the EEG waveform and video recordings. A total of 60 seizures were recognized among 12 subjects. Next, both PPG and EEG data were clipped so that it includes the seizure incidents as well as 2 hours of baseline before the seizure onset and 2 hours of data after the seizure offset. Thus, there is at least one seizure occurring in every 4 hours of the data. In case there are multiple seizures occurring close to each other(within less than 2 hours), the data is clipped such that it includes 2 hours of baseline before the first seizure and 2 hours of baseline after the last seizure. The clipped data was preprocessed to reject motion artifact and used for analysis in this study.
\subsubsection{Artifact Rejection}
\label{subsub:artifactRejection}
The PPG waveform is susceptible to motion artifact and can be completely obscured by noise. In order to reject noisy PPG pulses, the criteria introduced in \cite{segment_artifact} is employed. A clean PPG pulse should consist of a monotonically increasingly systolic upstroke followed by a diastolic descent. According to \cite{segment_artifact} the diastolic fall-off in a clean PPG pulse should not include more than two distinct notches. The PPG pulses with two, one, or zero dicrotic notches are considered clean. The artifactual parts are rejected in two phases. First, the algorithm proposed in \cite{segment_artifact}
automatically marks the PPG troughs and detects the artifacts. In the second phase, an expert visually inspects the PPG data and manually removes any missed artifact and rejects the false positives. After the noise rejection, two seizures from subject 8 and one seizure from subject 11 were eliminated due to excessive amount of noise.
Table \ref{table:subjects} shows the duration of the clipped data for each specific subject. Table \ref{table:seizures} demonstrates the duration of the seizures being used for analysis. A total of 57 seizures were used for analysis.
\subsubsection{Feature Analysis}
The morphological features were extracted from PPG data as defined in Section
\ref{sec:material}. Each feature forms a time series derived from the PPG pulses over time, where the individual samples belong to the feature values from each specific pulse. In order to compare the changes in
the features in time, each time series is segmented into non overlapping windows of 5-minutes length. Next,
the z-score of each feature sample is derived separately for every 5-min segment
based on the standard deviation and the mean value from its previous segment. For
instance, let us denote the mean and the standard deviation for the heart rate in
the $k^{th}$ segment as $\mu_{(\text{HR})}^k$ and
$\sigma_{(\text{HR})}^k$. For one individual detected PPG pulse inside
$k^{\text{th}}$ segment, the heart rate is denoted as $\text{HR}^{k}[i]$ implying
that the $i^{\text{th}}$ pulse belongs to the $k^{\text{th}}$ segment. Next the z-score for HR feature in this pulse is derived as follows:
\begin{equation}\label{tvalue}
z_{\text{HR}}^k[i]=\frac{\text{HR}^{k}[i]-\mu_{(\text{HR})}^{(k-1)}}{\sigma_{(\text{HR})}^{(k-1)}},
\end{equation}
where $\mu_{(\text{HR})}^{(k-1)}$ and $\sigma_{(\text{HR})}^{(k-1)}$ are the mean
and standard deviation from the previous segment. Normalizing the feature values based on the previous mean and standard deviation signifies the changes of these parameters over time. Let us denote the z-scores derived accordingly for all the hemodynamic-related features by $z_{\text{PA}}^{k}[i]$, $z_{t_{\text{NMVT}}}^{k}[i]$,
$z_{t_{\text{NCT}}}^{k}[i]$, $z_{\text{PCA1}}^{k}[i]$, and $z_{\text{PTT}}^{k}[i]$. \FIG{fig:onesiez} depicts the variations of
z-scores derived for the hemodynamic-related features in the first seizure of subject 6. According to \FIG{fig:onesiez}, the heart rate, normalized crest time, and normalized maximum velocity time experience a significant increment after the seizure onsets, while the pulse amplitude, pulse transmit time, and the first principle coefficient encounter significant decrements. In order to be able to see the patterns of changes in all the recorded seizures, \FIG{fig:hr_pdf} demonstrates the z-scores derived for all the features.
Comparing the z-scores for all the features pre- and post seizure shows a consistent pattern of changes across all the seizures. In \FIG{fig:hr_pdf}, the middle black lines placed at zero time is when the seizure onsets happen. The negative
time values on x-axis are the 5-minute segments prior to seizures and the positive time
values demonstrate the 5-minute post seizure onset segments. The y-axis
shows the seizure numbers and the color codes represent z-score values between
$-4$ and 4, where the highlighted parts mean a higher z-score. In addition, the probability
distribution functions (pdf) of the derived z-scores for pre- and post-seizures are also depicted. The
blue diagram represents the z-scores derived using the data in the 5-minute
pre-seizure periods for all the 57 seizures. The orange diagrams depict the pdfs
of z-scores for all 57 post seizure onset features. In order to quantify the significance of the variations, we adopted one-way Analysis of variance (ANOVA) with the z-scores from interictal as the baseline and a 5-minutes of data after the seizure onset. Table \ref{table:sig} shows the percentage of seizures with significant changes after the seizure onset as well as the percentages of significant increases and decreases.
\begin{figure*}
\centering
\begin{subfigure}{0.4\textwidth}
\includegraphics[trim=0 0 0 20,width=\textwidth]{hr_pdf.eps}
\caption{}
\end{subfigure}
\begin{subfigure}{0.4\textwidth}
\includegraphics[trim=0 0 0 20,width=\textwidth]{sdrr_pdf.eps}
\caption{}
\end{subfigure}
\begin{subfigure}{0.4\textwidth}
\includegraphics[trim=0 0 0 0,width=\textwidth]{rmssd_pdf.eps}
\caption{}
\end{subfigure}
\begin{subfigure}{0.4\textwidth}
\includegraphics[trim=0 0 0 0,width=\textwidth]{nn50_pdf.eps}
\caption{}
\end{subfigure}
\begin{subfigure}{0.4\textwidth}
\includegraphics[trim=0 0 0 0,width=\textwidth]{lf_pdf.eps}
\caption{}
\end{subfigure}
\begin{subfigure}{0.4\textwidth}
\includegraphics[trim=0 0 0 0,width=\textwidth]{hf_pdf.eps}
\caption{}
\end{subfigure}
\begin{subfigure}{0.4\textwidth}
\includegraphics[trim=0 0 0 0,width=\textwidth]{lfhf_pdf.eps}
\caption{}
\end{subfigure}
\centering
\begin{subfigure}{0.4\textwidth}
\includegraphics[trim=0 0 0 0,width=\textwidth]{NMV_pdf.eps}
\caption{}
\end{subfigure}
\begin{subfigure}{0.4\textwidth}
\includegraphics[trim=0 0 0 0,width=\textwidth]{NCT_pdf.eps}
\caption{}
\end{subfigure}
\begin{subfigure}{0.4\textwidth}
\includegraphics[trim=0 0 0 0,width=\textwidth]{PA_pdf.eps}
\caption{}
\end{subfigure}
\begin{subfigure}{0.4\textwidth}
\includegraphics[trim=0 0 0 0,width=\textwidth]{PCA1_pdf.eps}
\caption{}
\end{subfigure}
\begin{subfigure}{0.4\textwidth}
\includegraphics[trim=0 0 0 0,width=\textwidth]{PPT_pdf.eps}
\caption{}
\end{subfigure}
\caption{The z-score values depicted in color for features derived from PPG signal. The x-axis represents the time for 5 minutes before and after the seizure onsets. The y-axis enumerates the seizures across all the subjects with total of 57 seizures. (a)$z_{HR}$, (b) $z_{SDNN}$, (c) $z_{RMSSD}$, (d) $z_{NN50}, $(e) $ z_{LF}$, (f) $z_{HF}$, (g)$z_{LF/HF}$, (h) $z_{t_{NMV}}$, (i) $z_{t_{NCT}}$, (j) $z_{PA}$, (k) $z_{PCA1}$, (l) $z_{PTT}$}.
\label{fig:hr_pdf}
\end{figure*}
\begin{table*}[t]
\caption{Percentage of total recorded seizures showing significant changes ($p<0.01$) }
\centering
\begin{tabular}{c c c c c c c c c c c c c}
\hline
\hline
Feature & HR &SDNN&RMSSD&NN50&LF&HF&LF/HF& $t_{\text{NMV}}$ & $t_{\text{NCT}}$ & \id{PA} & \id{PCA1}& \id{PTT} \\
\hline
Percentage of Significant Seizures & 94.6\%&92.8\% &94.6\%&89.4\%&52.2\%&42.1\%& 52.2\%& 84.2\%& 84.2\% & 87.7\%& 66.6\% & 68.7\% \\
Percentage of Significant Increase & 94.6\% &92.8\%&94.6\%&89.4\%&50.8\%&42.1\%& 0\%& 84.2\%& 84.2\% & 0\% & 0\% &8.7\%\\
Percentage of Significant Decrease & 0\% &0\%&0\%&0\%& 1.4\%\%&0\% & 52.2\%&0\%&0\% & 87.7\% & 66.6\% &60\%\\
\hline
\hline \\
\end{tabular}
\label{table:sig}
\end{table*}
\section{Introduction}
\label{sec:introduction}
\IEEEPARstart{E}{pilepsy} is a chronic disease affecting more than
50
million people world wide \cite{demog}. In majority of
patients, the seizures are controlled using medication; however,
one-third of
the patients still do not respond to treatments and
continue to have
seizures. Although epilepsy is considered a
neurological disorder, several studies have reported autonomic
imbalance and cardiovascular dysfunction during ictal and postictal
period \cite{heart of epi}. In fact, Sudden Unexpected Death in
Epilepsy (SUDEP) has been attributed to cardiovascular and pulmonary
dysfunction induced by uncontrolled seizures \cite{SUDEP}.
Recently,
epilepsy is considered as a set of coexisting comorbids,
where
Electroencephalography (EEG) abnormal waves can be preceded by other
extracerebral
manifestations \cite{autonomic}.\\
\indent A
personal diary of seizures plays a key role in clinical
diagnosis
and research. However, keeping a report of the seizures is
not
trivial, especially for patients who have seizures during sleep
\cite{record}. In addition, since the patient may lose consciousness
during a seizure, timely intervention by a caregiver may be
required.
In order to have an accurate record of seizures and
identify the overlapping symptoms, new seizure detecting devices have
adopted multimodal data fusion techniques incorporating small sensors
such as
electrodermal\cite{EDA,EDA-seizure}, accelorometer \cite{ACC}, electromyogram \cite{EMG}, and electrocardiogram (ECG)
\cite{devices,automatic_detection,ECG_nocturnal,ECG_singlelead,ECG_feedback}. The use of multimodal sensors
facilitates automatic seizure detection without using bulky and
uncomfortable EEG caps. However, the performance of these techniques are dependent on the seizure type \cite{automatic_detection}. Specifically electrodermal, accelorometer, and electromyogram sensors are used for generalized tonic-clonic seizures and their performance degrades for partial seizures.\\
\indent
Ictal tachycardia is the most prevalent autonomic imbalance
manifestation observed in 82\% of seizures \cite{ictal-tachycardia}.
Ictal tachycardia
happens as a result of increase in sympathetic
tone. However, in less
than 5\% of cases, parasympathetic activity
can also predominate,
leading to bradycardia
\cite{ictal-bradycardia}. Studies suggest a consistent and
stereotyped
progression of autonomic dynamics leading to heart rate
variability (HRV) during and even before the
seizure onset \cite{model}.
The HRV due to autonomic imbalance has been
exploited for seizure prediction a few seconds before the seizure
onset \cite{prediction}. In general, the HRV-based seizure detectors perform poorly in terms of high false alarm rates and low sensitives compared to EEG-based detectors \cite{devices}. Besides heart rate variability, other less
prevalent cardiac conduction
abnormalities are inconsistent with
respect to age, laterization,
and seizure foci. Abnormalities such
as atrial fibrillation, bundle
branch block, atrial premature
depolarization, asystole, and
ST-segment elevation have also been
reported in total of less than
14\% of investigated cases and were not used for seizure detection
\cite{heart of epi}.\\
\indent In addition to cardiac conduction
abnormalities, the excess
release of catecholamines during seizure
will influence the vascular
function and hemodynamics.
Catecholamines such as norepinephrine are types of
neurotransmitters released to act as
anticonvulsant in ictal phase
\cite{norepinephrine}. These
neurotransmitters are responsible for a
series of autonomic responses
such as blood flow manipulation. The
blood perfusion is controlled by
constriction and dilation of the
vessels performed by muscle cells
present in the blood vessel wall.
These cells are abundant with alpha
adrenegric receptors that are
the targets of neurotransmitters such as
norepinephrine. The release
of norepinephrine calls off the blood from
non-vital organs leading
to a reduction in skin perfusion in limbs. The skin
vasoconstriction
due to release of catecholamines can be measured by
photoplethysmogram (PPG) \cite{ppg_hemody}. PPG is an optical sensor
composed of a light emitting diode and a photo detector. The
reflection of the light from skin captures the volumetric changes of
the blood pulse. Conventionally, PPG signal was used to measure
blood
oxygenation level and heart rate. Recent studies have shown
the
morphology of PPG signal contains valuable information about
the
cardiovascular function, autonomic nervous system (ANS) and its
related
hemodynamics \cite{arousal,ppg_hemody,stressPPG}. The blood pressure and vascular compliance affect the morphological features of a PPG pulse. In addition, the HRV features conventionally derived from ECG signal can still be derived using PPG signal and the results of seizure detection is comparable with ECG signal \cite{MPDIsensor}.\\
\section{Seizure Detection}
\label{sec:ML}
The entire data set consists of 102 hours and 30 minutes of recording among which a total of 1 hour and 9 minutes is in ictal phase. In order to quantify the performance, we adopted three metrics of sensitivity, Positive Predictive Value (PPV), and False Alarm Rate (FAR).
The sensitivity is defined as:
\begin{equation}\label{sens}
\id{Sensitivity}=\frac{\id{TP}}{\id{TP}+\id{FN}},
\end{equation}
where $\id{TP}$ and $\id{FN}$ are true positives and false negatives, respectively. The FAR is the number of erroneously detected seizures per hour. The PPV is the ratio of true positives over all the issued alarms and is defined as:
\begin{equation}\label{ppv}
\id{PPV}=\frac{\id{TP}}{\id{TP}+\id{FP}},
\end{equation}
where $\id{FP}$ is the false positives rate.
In order to avoid overfitting, we randomly excluded a continuous segment of 2-hours inside the total of 102 hours 30 minutes data as the testing data. The remaining data is used for training the model. This random selection was repeated 20 times and the model was trained using the obtained training set. The overall performance is obtained by averaging over the sensitivity, PPV, and FAR from every repetition. We adopted a two layer LSTM neural network architecture. LSTM is able to capture long-term dependencies which helps identifying the temporal progression of PPG features in epilepsy.
\subsection{LSTM Architecture}
The goal of automatic seizure detector is to classify windows of input data into
two labels of seizure and non-seizure. The stream of z-scores for each feature
forms a time series. A sliding window method was employed to subsegment the time
series into windows of 60 samples. \FIG{fig:LSTM} shows
the proposed architecture for the LSTM model. The LSTM architecture is composed of LSTM cells where each cell is fed with the hidden state and the cell state from the previous cells in time. Our proposed LSTM architecture contains two layers of 60 cells. A $20\%$ dropout was applied to the first layer to avoid over fitting. The Adam optimizer was used for training the architecture. \FIG{fig:LSTM} (a) shows the equations governing each cell, where $\text{h[n]}$ and $\text{c[n]}$ are the hidden state and the cell state corresponding to time $n$. $\boldsymbol{W}$ and $\boldsymbol{b}$ are the weight matrices and the bias vectors. The hidden state is dependent on the input vector, the previous hidden state and the cell state. The vector $\boldsymbol{i}[n]$ decides whether to use the input and the past hidden state to update the cell state $\boldsymbol{c}[n]$. The vector $\boldsymbol{f}[n]$ decides wether to use the past cell state $\boldsymbol{c}[n-1]$ to update the cell state. The equations governing the LSTM cells are as follows:
\begin{align}\label{equ:lstm}
&\boldsymbol{f}[n]=\sigma\big(\boldsymbol{W}_f\big[\boldsymbol{h}[n-1],\boldsymbol{x}[n]\big]+\boldsymbol{b}_f\big)\nonumber\\
&\boldsymbol{i}[n]=\sigma\big(\boldsymbol{W}_i\big[\boldsymbol{h}[n-1],\boldsymbol{x}[n]\big]+\boldsymbol{b}_i\big)\nonumber\\
&\boldsymbol{\tilde{c}}[n]=tanh\big(\boldsymbol{W}_c\big[\boldsymbol{h}[n-1],\boldsymbol{x}[n]\big]+\boldsymbol{b}_c\big)\nonumber\\
&\boldsymbol{o}[n]=\sigma\big(\boldsymbol{W}_o\big[\boldsymbol{h}[n-1],\boldsymbol{x}[n]\big]+\boldsymbol{b}_o\big)\nonumber\\
&\boldsymbol{c}[n]=\boldsymbol{f}[n]\odot \boldsymbol{c}[n-1]+ \boldsymbol{i}[n]\odot \tilde{\boldsymbol{c}[n]}\nonumber\\
&\boldsymbol{h}[n]=\boldsymbol{o}[n]\odot tanh\big(\boldsymbol{c}[n]\big),
\end{align}
where $\odot$ is the element-wise multiplication. \\
\subsection{HRV-related versus hemodynamic-related features}
\indent Conventional seizure detection algorithms, derive the HRV-based features from ECG signal and use them for seizure detection. In this work, the HRV-related features were extracted from PPG signal, i.e. $\text{HR}$, $\text{SDNN}$, $\text{RMSSD}$, $\text{NN50}$, $\text{LF}_{norm}$, $\text{HF}_{norm}$, and $\text{LF/HF}$. In addition to the HRV-related features, 5 other PPG features were studied which are known to be related to vascular compliance and hemodynamics, i.e. $t_{NMV}$, $t_{NCT}$, $t_{PA}$, $PCA1$, $PTT$. As it Despite the HRV-related features, all the hemodynamic-related features show consistent patterns of ictal change across majority of the seizures. In order to be able to measure how these hemodynamic-related features are contributing to the improvement of seizure detection, we trained the proposed LSTM architecture twice. First using the 7 HRV-related features, the dimensions of the input vector and the hidden state and cell state for the first layer were 7. The dimensions of the hidden state and the cell state vectors for the second layer were chosen to be 5. The performance results of the trained HRV-based LSTM is brought in table \ref{table:result}. For brevity, this detector is called LSTM7.\\
\indent Next, the 5 hemodynamic-related features were appended to the input vector, making the input vectors of 12 features in time. The hemodynamic-based LSTM is denoted by LSTM12 in table \ref{table:result}.T he dimensions of the hidden states and the cell states for the first and the second layer were 12 and 5 respectively. As shown in table \ref{table:result}, although our proposed seizure detector is subject independent, our HRV-based detector LSTM7 is showing an improvement in FAR and PPV compared to \cite{ECG_feedback,ECG_singlelead}. The LSTM7 results performance is comparable to \cite{ECG_nocturnal}, considering the fact that the data in \cite{ECG_nocturnal} is nocturnal, while our data is both nocturnal and diurnal. Adding the 5 hemodynamic-related features in LSTM12 improves the sensitivity to $92\%$. In addition , the FAR rate have improved from 0.91 to 0.52 which is a $\text{42}\%$ improvement.
\begin{table*}[t]
\caption{Results}
\centering
\begin{tabular}{c c c c c c c}
\hline
\hline
work & sensitivity & PPV &FAR & type of seizure & time & subject dependent \\
\hline
\cite{ECG_nocturnal} & 77.6\%\% & 30.7\%& 0.33 per hour & GTC + partial& nocturnal & Y \\
\cite{ECG_singlelead}& 81.9\% & 7.9\%&1.97 per hour & GTC+partial & nocturnal+diurnal &Y\\
\cite{ECG_feedback}& 77.1\% & 3.25\%&1.24 per hour & GTC+partial & nocturnal+diurnal &Y\\
\cite{MPDIsensor}& 77.1\% & 3.25\%&1.24 per hour & GTC+partial & nocturnal+diurnal &Y\\
LSTM based on HRV (LSTM7) &82\%&26\%&0.91 per hour>C+partial&nocturnal+dirunal&N\\
LSTM based on PPG morphological &92\%&43\%&0.52 per hour>C+partial&nocturnal+dirunal&N\\
features (LASTM12)&&&&&&\\
\hline
\hline \\
\end{tabular}
\label{table:result}
\end{table*}
\begin{figure}[!t]
\begin{subfigure}{0.5\textwidth}
\includegraphics[trim=0 0 0 0,width=220pt]{LSTM_cell.eps}
\caption{}
\end{subfigure}
\begin{subfigure}{0.45\textwidth}
\includegraphics[trim=0 0 0 0,width=220pt]{LSTM.eps}
\caption{}
\end{subfigure}
\caption{The LSTM architecture.}
\label{fig:LSTM}
\end{figure}
\section{Methods and Material}
\label{sec:material}
PPG is an optical sensor capturing the skin blood flow using a light
emitting diode (LED) and a photo detector. The LED illuminates the
skin, and the backscattered light from the surface of the skin changes
with respect to blood volume pulse and the blood color. The captured
signal represents the arterial pressure as the blood is ejected from
the left ventricle, circulates through the vessels, and finally goes
back to the right atrium. The PPG signal has a distinct shape that can be
separated into anacrotic and catacrotic phases as shown in
\FIG{fig:ppg}.
The anacrotic phase consists of the systolic upstroke corresponding to the to acceleration of the aortic blood flow as a result of left ventricular blood ejection. The ascending slope depends on several parameters such as the left ventricular ejection pressure, arterial peripheral resistance, and the arterial compliance and elasticity \cite{arterial_waveform}. The upstroke reaches the maximum arterial pressure point called the systolic point. People with less arterial compliance and elasticity tend to have higher systolic peaks.
The catacrotic phase consists of the systolic decline, the dicrotic
notch, and the diastolic run-off. The systolic decline happens when
the left ventricular contraction is about to end. The efflux of blood
from arteries to veins back to the heart is faster than the influx of
the blood from the heart left ventricle. The dicrotic notch happens
when the aortic valve closes. There is a sudden increase in arterial
pressure after the dicrotic notch. In fact, after the closure of the
aortic valve, the blood volume has less space to move and it can only
travel toward the peripheral arteries, causing a secondary peak in
arterial pressure in diastolic phase due to resistance of peripheral
arteries. The dicrotic notch is more slurred in people with a higher
arterial compliance. Finally, there is a gradual drop in the arterial
pressure in diastolic phase as the blood goes back to the right atrium.
\begin{figure}[!t]
\centerline{\includegraphics[trim=90 20 50 20,width=180pt]{ppg2.eps}}
\caption{PPG pulse.}
\label{fig:ppg}
\end{figure}
\begin{figure}[!t]
\centerline{\includegraphics[trim=90 20 50 20,width=180pt]{derivative3.eps}}
\caption{PPG features.}
\label{fig:derppg}
\end{figure}
\subsection{Data Analysis}
In this section the investigated morphological features derived from ECG and PPG data is described along with their physiological interpretation. The temporal and spectral features related to HRV, i.e. $\text{HR}$, $\text{SDNN}$, $\text{RMSSD}$, $\text{NN50}$, $\text{LF}$, $\text{HF}$, and $\text{LF/HF}$, were conventionally derived from ECG signal and are called HRV-related features hence forth. There are 5 other features related to hemodynamics and vascular compliance derived from PPG morphology, i.e. ${t}_{\text{NMV}}$, ${t}_{\text{NCT}}$, $\text{PA}$, $\text{PCA1}$, and $\text{PTT}$. For brevity, these 5 features are called hemodynamic-related features.
\subsubsection{Heart Rate (HR)}
Heart rate is estimated using the time between troughs of consecutive
PPG pulses. Representing the pulse width by $t_{\text{SS}}$ as shown in
\FIG{fig:derppg}, heart rate is defined as the reciprocal of the pulse
width as:
\begin{equation}\label{hr}
\text{HR}=\frac{1}{t_{\text{SS}}}.
\end{equation}
In addition to heart rate, there are other temporal and spectral parameters extended from heart rate which have been conventionally used as bio-markers of ANS dynamics and can be extracted from PPG signal as follows \cite{MPDIsensor}:
\begin{itemize}
\item \textbf{SDNN}:
SDNN is the standard deviation of pulse widths and is defined as:
\begin{equation}\label{sdnn}
\text{SDNN}=\mathbb{E}\{(\text{t}_{SS}-\mu_{\text{t}_{SS}})^2\},
\end{equation}
where $\mathbb{E}\{.\}$ is the expectation denotation and $\mu_{t_{SS}}$ is the mean value of the pulse widths.
\item \textbf{RMSSD}:
RMSSD is the root mean square of the expected value of squared differences of successive pulse widths and is defined as:
\begin{equation}\label{rmssd}
\text{RMSSD}=\sqrt{\mathbb{E}\{(\text{t}_{SS}[i+1]-\text{t}_{SS}[i])^2\}},
\end{equation}
where $t_{SS}[i]$ represents the $i^{th}$ pulse width.
\item \textbf{NN50}:
NN50 is the number of successive pulses with more than $50 msec$ difference in their widths.
\item {$\boldsymbol{\text{LF}_{norm}}$}:
$LF_{norm}$ is the power spectral density of heart rate in low frequency band (0.04-0.15 Hz) and normalized to the total spectral power of heart rate.
\item $\boldsymbol{\text{HF}_{norm}}$:
$\text{HF}_{norm}$ is the power spectral density of heart rate in high frequency band (0.15-0.4 Hz) and normalized to the total spectral power of heart rate.
\item $\boldsymbol{\text{LF/HF}}$:
$\text{LF/HF}$ is the ratio of power spectral density of heart rate between the low and high frequency band.
\end{itemize}
\subsubsection{Pulse Amplitude (PA)}
PA is defined as the height of the PPG signal, which is measured by the vertical distance between the diastolic trough to the systolic peak of the next pulse as shown in \FIG{fig:ppg}. PA is directly related to cardiac volume stroke, vascular distensibility, and vascular resistance \cite{ppg_morphology}. In case of hypovolemia and dehydration, the left ventricular volume stroke is small and the PA goes low. In addition, in case of vasoconstriction in peripheral arteries where the vascular resistance of peripheral arteries goes high, a reduction in the PA is observed \cite{PTT,diameter}. PA is also related to arterial compliance and elasticity. Compliance is the ability of blood vessel wall to distend in response to changes in blood pressure. People with lower arterial compliance tend to have a higher PA \cite{ppg_morphology}.
\subsubsection{Normalized Crest Time ($t_{\text{NCT}}$)} The crest time is the
time between the start point of systolic upstroke (denoted by S in
\FIG{fig:derppg}) and the systolic peak (denoted by P in \FIG{fig:derppg}) and is
represented by $t_{\text{SP}}$. Disregarding the fact that $t_{\text{SP}}$ is
expected to reduce due to tachycardia, normalizing the crest time with respect to
pulse width reveals an increase in ictal and post-ictal phase. In fact, regardless
of ictal reduction in crest time, the increase in normalized crest time
($T_{\text{NCT}}$) yields information about vasoconstriction and vascular resistance in limbs during seizure. The normalized crest time is defined as:
\begin{equation}\label{NCT}
t_{\text{NCT}}=\frac{t_{\text{SP}}}{t_{\text{SS}}}.
\end{equation}
\subsubsection{Normalized Pulse Transmit Time ($\text{PTT}$)}
Pulse transmit time is the time it takes from the onset of the left ventricle
depolarization to the time the blood reaches the peripheral arteries and is
measured as the interval between the R peak in ECG signal and the next PPG trough
as shown by $t_{\text{RS}}$ in \FIG{fig:derppg} \cite{PEP}. The pulse transmit time is related to pulse wave velocity or the speed of blood flow in arteries. The pulse wave velocity itself is a function of blood density, arterial dimension properties such as vessel thickness and arterial diameter, and blood pressure \cite{PTT}. The normalized pulse transmit time is defined as follows:
\begin{equation}\label{NPTT}
\text{PTT}=\frac{t_{\text{RS}}}{t_{\text{SS}}}.
\end{equation}
\subsubsection{Maximum Velocity and its Normalized Time ($t_{\text{NMV}}$ )} The point of maximum upstroke slope in systolic phase denoted by peak $a$ in the first derivative waveform represents the maximum velocity in PPG pulse \FIG{fig:derppg}. This maximum slope depends on the blood viscosity, arterial pressure and vascular resistance. In addition the normalized maximum slope time is defined as:
\begin{equation}\label{NST}
t_{\text{NMV}}=\frac{t_{Sa}}{t_{\text{SS}}}.
\end{equation}
\subsubsection{Principle Component from the Second Derivative of PPG (PCA1)} The second derivative of the PPG pulse shape (SDP) is a
measure of blood flow acceleration in vessels. The fiducial points of the second derivative
waveform are related to arterial stiffness and arterial pressure
\cite{elghandi2012}. In this study, the raw PPG data is segmented into PPG pulses.
Each PPG pulse starts from the diastolic trough and ends in the diastolic trough
of the next pulse as shown in \FIG{fig:derppg}. After calculating the second derivative of each pulse shape, the dominant shape of
the second derivative of the pulses in inter-ictal phase (baseline) is
derived using principle component analysis (PCA). All the PPG pulses occurring
before 15 minutes prior to the seizure onset and after 5 minutes post-seizure
offset are considered as baseline \cite{prediction}. The shapes of the SDP in
ictal phases are compared to the principle components of the
baseline using subspace projection. In order to be able to apply PCA to the second derivative of PPG pulse shape, the data samples are interpolated such that the PPG pulses have the same number of samples denoted by
$N$.
Let us represent the $k^{\text{th}}$ pulse as a $N \times 1$ vector $\boldsymbol{b}_k$ and its second derivative as the baseline acceleration vector $\boldsymbol{b}''_k$. Stacking all the baseline acceleration vectors, the $N\times K$ baseline acceleration matrix $\boldsymbol{B}$ is formed as:
\begin{equation}\label{B}
\boldsymbol{B}=[\boldsymbol{b}''_1, \boldsymbol{b}''_2, ...,\boldsymbol{b}''_K]_{N\times K},
\end{equation}
where $K$ is the total number of baseline pulses.
An eigenvalue decomposition of its covariance is as follows;
\begin{equation}\label{eigen}
\frac{1}{K}\boldsymbol{B}\boldsymbol{B}^T=\boldsymbol{\Psi}\boldsymbol{\Lambda}\boldsymbol{\Psi}^T,
\end{equation}
where $\boldsymbol{B}^T$ is the transpose of $\boldsymbol{B}$ and $\boldsymbol{\Psi}=[\boldsymbol{\psi}_1,\boldsymbol{\psi}_2,...,\boldsymbol{\psi}_N]$ is the $N\times N$ matrix of eigenvectors with $\boldsymbol{\psi}_i$ being the $i^{th}$ eigenvector. $\boldsymbol{\Lambda}$ is an $N \times N$ diagonal matrix of eigenvalues. The eigenvectors corresponding to the major eigenvalues contain the dominant shape of the baseline acceleration waveform. In order to compare the ictal SDP shape with baseline, we adopted a subspace projection approach as explained in the following. Let us represent the ictal PPG pulses and their second derivatives by $N \times 1$ vectors ${\boldsymbol{b}}_\ell$ and ${\boldsymbol{b}}''_{\ell}$, where $\ell$ is the index of ictal pulse. In order to quantify the deviation of each acceleration waveform from the baseline, the subspace projection of ${\boldsymbol{b}}''_{\ell}$ with the subspace spanned by the principle components in \EQN{eigen} is derived as follows:
\begin{equation}\label{correlation}
\text{PCA1}[\ell]=\frac{{\boldsymbol{\psi}_1}^T{\boldsymbol{b}}''_{\ell}{{\boldsymbol{b}}''_{\ell}}^T\boldsymbol{\psi}_1}{{{\boldsymbol{b}}''_{\ell}}^T{\boldsymbol{b}}''_{\ell}},
\end{equation}
where $\boldsymbol{\psi}_1$ is the first principle component of baseline SDP and $\ell$ is the index of the ictal pulse.
|
1,314,259,993,656 | arxiv | \section{INTRODUCTION}
With the increasing applications of visual perception methods~\cite{bruls2018mark,valada2017adapnet,kumar2021omnidet,zou2022real} in robotics and autonomous driving, the robustness of these methods has become more and more important in their visual system. However, the performance of current vision methods significantly degrades in rainy weather in autonomous driving scenarios, since raindrops on the windshield or camera lens cause inevitable visual obstruction, as mentioned in Porav et al.~\cite{porav2019can}. Therefore, removing waterdrops from videos on rainy days is highly important for self-driving cars and various robot applications.
Although many researchers~\cite{fu2017clearing,zhang2019image,chen2021hinet,li2019heavy,chen2019gated,wang2020model,yi2021structure,zamir2021multi,chen2021robust,wang2021rain,chen2020pmhld,wang2020model,jiang2020multi} propose dedicated frameworks to remove rain streaks, video waterdrop removal receive much less attention and is still an open question. Since there exists a significant geometric gap between waterdrops and rain streaks, the dedicated methods proposed to remove rain streaks cannot perform well on the waterdrop removal task as mentioned in~\cite{quan2021removing}. As shown in Fig.~\ref{fig:1}, directly applying a state-of-the-art rain streak removal method~\cite{Zamir2021Restormer} to remove waterdrops will not yield satisfactory results. Instead of being like the line shape of rain streaks, waterdrops tend to be ellipse scattered on windshields. And each waterdrop usually occupies a larger area than a rain streak, thus being more difficult to deal with.
To tackle the waterdrop removal problem, some researchers~\cite{eigen2013restoring,qian2018attentive,quan2019deep,shi2021stereo,hao2019learning,porav2019can} propose specialized frameworks to remove waterdrops from a single image. However, they still cannot handle complex driving videos, which are very common in Autonomous Driving. On the one hand, their datasets are usually collected by sprinkling waterdrops on the glass, thus there is a large domain gap between these data and real driving scenes. On the other hand, their methods lack temporal information utilization, which is necessary for video tasks.
The lack of paired real-world training data for video waterdrop removal limits the performance of learning-based methods in the real world. It is almost impossible to collect perfectly aligned driving videos with and without waterdrops.
To address this issue, we propose a large-scale synthetic video waterdrop dataset with the paired data for training. To promote our proposed method to generalize to real driving scenes, we adopt a cross-modality training strategy that jointly trains our model on our large-scale synthetic video dataset and a small-scale real-world image dataset proposed by~\cite{qian2018attentive}.
We also propose a spatio-temporal fusion-based framework to restore the background information under the regions occupied by sparse waterdrops and even the streaks of waterdrops, as shown in Fig.~\ref{fig:1}. With the dedicated framework and cross-modality strategy, the proposed method achieves the best-performing video waterdrop removal in real driving scenes.
\begin{figure}[t!]
\centering
\begin{tabular}{@{}c@{\hspace{1mm}}c@{\hspace{1mm}}c@{}}
\includegraphics[width=0.16\textwidth]{figs/introduction/fig3_291_rainy_frame.jpg} & \includegraphics[width=0.16\textwidth]{figs/introduction/fig3_291_rainy_frame_restormer.jpg} & \includegraphics[width=0.16\textwidth]{figs/introduction/fig3_291_fake_clean_frame.jpg} \\
Input (sparse case) & Restormer~\cite{Zamir2021Restormer} & Our Result\\
\includegraphics[width=0.16\textwidth]{figs/introduction/fig3_100_rainy_frame.jpg} & \includegraphics[width=0.16\textwidth]{figs/introduction/fig3_100_rainy_frame_restormer.jpg} & \includegraphics[width=0.16\textwidth]{figs/introduction/fig3_100_fake_clean_frame.jpg}\\
Input (streak case) & Restormer~\cite{Zamir2021Restormer} & Our Result\\
\end{tabular}
\vspace{-2mm}
\caption{Real-world driving videos with waterdrops.~\label{fig:1}}
\vspace{-6mm}
\end{figure}
Our contributions can be summarized as follows.
\begin{enumerate}
\item We propose a dedicated framework for video waterdrop removal in complex driving scenes. Our framework is based on spatio-temporal fusion that uses a self-attention mechanism to exploit both spatial and temporal information for restoring clean video frames.\\
\vspace{-2mm}
\item We are the first to propose a large-scale synthetic video dataset for waterdrop removal task. We design a video waterdrop synthesis algorithm based on Hao~\textit{et al}.~\cite{hao2019learning} to generate synthetic data.\\
\vspace{-2mm}
\item To tackle the domain gap between the synthetic data and real driving scenes, we propose a cross-modality strategy for jointly training the proposed method on our synthetic video data and the real-world image data~\cite{qian2018attentive}.\\
\vspace{-2mm}
\item The extensive evaluations demonstrate that our waterdrop removal method significantly outperforms previous works on our synthetic dataset and real driving scenes quantitatively and qualitatively.
\end{enumerate}
\vspace{-2mm}
\section{Related Work}
\subsection{Single-Image Methods}
Although there are many methods focusing on image deraining, they mainly attempt to remove rain streaks instead of waterdrops from images. To remove watedrops from images, some learning-based methods have been proposed in recent years. Eigen~\textit{et al}.~\cite{eigen2013restoring} propose the first CNN-based method to remove the waterdrops from a single degraded image. Due to the over-shallow network architecture design, their method shows poor removal performance for large and dense waterdrops. Qian~\textit{et al}.~\cite{qian2018attentive} propose a generative adversarial network (GAN) based method. Their method generates an attention map for each input image. However, with the simple concatenation of the input image and its corresponding attention map, their method still only focuses on the local spatial information instead of the global. Quan~\textit{et al}.~\cite{quan2019deep} propose a shape-driven attention and channel re-calibration to locate and process waterdrops. However, this method is still limited by the local attention mechanism that does not sufficiently explore the long-range but helpful information from the whole single image. Hao~\textit{et al}.~\cite{hao2019learning} propose a waterdrop synthesis algorithm for the single image. With their synthetic image dataset, they train a deep network for waterdrop detection and removal. However, their method shows the poor performance in real driving scenes. Quan~\textit{et al}.~\cite{quan2021removing} propose to utilize the neural architecture search method~\cite{liu2018darts} to generate a complementary network to tackle rain streak removal and waterdrop removal jointly. But their method is still too weak to remove waterdrops from real driving scenes.
\subsection{Multi-Image Methods}
\label{section:related_work_2}
Due to limited clues in a single image for clean background information recovery, some researchers~\cite{you2013adherent,shi2021stereo,liu2020learning,alletto2019adherent} propose to take multiple degraded images as input to reconstruct the clean ones. You~\textit{et al}.~\cite{you2013adherent} propose to calculate the dense motion change to detect waterdrops in each frame. However, their method needs to retrieve similar but clean information over nearly one hundred frames to restore the regions under waterdrops. Shi~\textit{et al}.~\cite{shi2021stereo} propose a dedicated framework for the stereo waterdrop removal task. Their method relies heavily on the disparity map in each stereo pair, which is not available in monocular driving videos. A coarse-to-fine method is proposed by Liu~\textit{et al}.~\cite{liu2020learning} to tackle obstruction removals, such as reflection removal and fence removal. Their method can be extended to waterdrop removal task. Based on the flow estimation, they propose to warp nearby frames to provide helpful information for recovery. Most recently, Alletto~\textit{et al}.~\cite{alletto2019adherent} propose a spatio-temporal de-raining model to tackle single-image and video waterdrop removal. They also propose a waterdrop synthesis algorithm to simulate driving scenes during rain. Although the abovementioned two methods can remove sparse waterdrops from multiple images, they fail to estimate a correct optical flow when there are numerous waterdrops over a sequence of frames. Considering the bottleneck of flow estimation, we propose to utilize a temporal attention block that directly provides effective information from nearby frames to restore clean background information.
\vspace{-1mm}
\section{Approach}
\vspace{-1mm}
\begin{figure*}[t]
\centering
\includegraphics[width=\textwidth]{figs/illustration/illustration_sz.pdf}
\vspace{-6mm}
\caption{Overview of the proposed method. We propose a framework with spatial-temporal fusion, based on a self-attention mechanism \cite{vaswani2017attention}, to reconstruct each feature across spatial and temporal dimensions, respectively. Meanwhile, we also adopt a cross-modality strategy to train our model on our proposed synthetic video data and the real-world image data~\cite{qian2018attentive} jointly.~\label{fig:ill.1}}
\vspace{-6mm}
\end{figure*}
Given a sequence of frames with waterdrops $\left \{\tilde{F}_{t}\right\}_{t=1}^{T}$, where $T$ is the sequence length, our goal is to remove the waterdrops in these frames and recover the clean ones $\left \{ \hat{F}_{t}\right \}_{t=1}^{T}$.
To tackle waterdrop removal, we consider two types of occlusions: the partial occlusion and the complete occlusion, as shown in Fig.~\ref{fig:2}. There is still some meaningful background information covered by the waterdrop in the partial occlusion, while the information in the complete occlusion is totally lost for background restoration. To address this problem, we propose a \textit{pixel attention block} to pixel-wisely re-weight the intermediate feature of each input frame. The goal of the pixel attention block is to enhance the background information in the partial occlusion and suppress the meaningless information in the complete occlusion.
After re-weighting each pixel in the feature, we need to refine the pixels in partially occluded regions and fill completely occluded regions with meaningful values. We treat this processing as an inpainting task~\cite{yu2018generative,zeng2020learning}. Inspired by~\cite{zeng2020learning}, we adopt a self-attention mechanism to refine and fill regions degraded by waterdrops. Unlike previous solely CNN-based works which are limited to exploiting local information, we utilize the self-attention mechanism to restore background information by fusing global information across the whole spatial dimension. Based on such a self-attention mechanism, we propose a \textit{spatial attention block} to refine re-weighted features from the pixel attention block.
Although the \textit{spatial attention block} can restore most regions in features, some regions that are still degraded can be restored by utilizing features from nearby frames. Hence, we propose a \textit{temporal attention block} to fully exploit the valid information from nearby frames. Similarly, but differently to the self-attention mechanism in the \textit{spatial attention block}, the \textit{temporal attention block} refines multiple features simultaneously by fusing the global information across the temporal dimension. Through the above spatio-temporal fusion, the proposed method can reconstruct cleaned frames accurately.
\begin{figure}[t]
\centering
\begin{minipage}[t]{0.235\textwidth}
\centering
\includegraphics[width=\textwidth]{figs/approach/partial_occlusion.jpg}
\subcaption{Partial occlusion}
\end{minipage}
\begin{minipage}[t]{0.235\textwidth}
\centering
\includegraphics[width=\textwidth]{figs/approach/complete_occlusion.jpg}
\subcaption{Complete occlusion}
\end{minipage}
\caption{Two types of occlusions caused by waterdrops. Examples are from the dataset~\cite{qian2018attentive}. ~\label{fig:2}}
\vspace{-6mm}
\end{figure}
\subsection{Pixel Attention Block (PAB)}
Given a sequence of frames within waterdrops $\left \{\tilde{F}_{t} \right \}_{t=1}^{T}, \tilde{F}_{t}\in \mathbb{R}^{H\times W\times 3}$, we first feed them into the encoder and obtain a sequence of features $\left \{ f_{t} \right \}_{t=1}^{T}, f_{t}\in \mathbb{R}^{h\times w\times c}$, where $h=\frac{H}{4}$, $w=\frac{W}{4}$ and $c$ denotes the channel number. After encoding, each feature is fed into the pixel attention block to obtain its corresponding pixel-wise confidence map $a_t$, as shown in Fig.~\ref{fig:ill.2}:
\begin{equation}
a_t = \sigma(\Theta_{PA}(f_{t})),
\end{equation}
where $1\leq t\leq T$, $\sigma$ denotes the sigmoid activation. To obtain a pixel-wise confidence map for each feature, we adopt a similar block in SENet~\cite{hu2018squeeze} but we discard the global average pooling layer and replace the fully connected layer with the convolutional layer $\Theta_{PA}$. Since each waterdrop only occupies a narrow region across the spatial dimension in each feature channel, this encourages us to re-weight each pixel by a pixel-wise confidence map $a_t$ instead of a common weight as in~\cite{hu2018squeeze}:
\begin{equation}
f^{'}_{t} = a_t\odot f_{t},
\end{equation}
where $\odot$ denotes a pixel-wise multiplication operation.
During re-weighting, the value of each element in the pixel-wise confidence map is expected to be $0$ for the complete occlusion, $1$ for the clean background region.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.4\textwidth]{figs/illustration/pixel_attn.pdf}
\vspace{-2mm}
\caption{The detail of the pixel attention block in the proposed method.~\label{fig:ill.2}}
\vspace{-6mm}
\end{figure}
\subsection{Spatio-Temporal Fusion}
\subsubsection{Spatial Attention Block (SAB)}
After re-weighting, we need to refine the pixels in the partial occlusion and fill the complete occlusion with meaningful pixel values. To address this, we utilize a self-attention mechanism~\cite{vaswani2017attention} to reconstruct each feature by fusing patches with a predicted attention map:
\begin{equation}
\begin{split}
q_{t} = \Psi _{q}(f^{'}_{t}),\quad k_{t} = \Psi _{k}(f^{'}_{t}),\quad v_{t} = \Psi _{v}(f^{'}_{t}),
\end{split}
\end{equation}
where $\Psi _{q}(\cdot)$, $\Psi _{k}(\cdot)$ and $\Psi _{v}(\cdot)$ denote the feature embedding for query, key and value, which consists of the 2D convolutional layers with $1\times 1$ kernel size. \
We extract $N=h \times w$ patches with size $1\times 1\times c$ from the query, key and value features, respectively. To calculate an attention map, we reshape each patch of query and key features into a 1-D vector, so the similarity between each query patch $p^{q}$ and each key patch $p^{k}$ is
\begin{equation}
s_{i,j} = \frac{p^{q}_{i}\cdot (p^{k}_{j})^{T}}{\sqrt{1\times 1\times c}},
\end{equation}
where $1\leq i, j \leq N$, $p^{q}_{i}$ and $p^{k}_{i}$ denote the $i$-th query patch and $j$-th key patch, respectively, $\cdot$ denotes a matrix multiplication operation. As it is mentioned in~\cite{vaswani2017attention}, the similarity value normalized by the dimension of each vector can avoid a small gradient caused by a subsequent softmax function. Finally, the attention map is
\begin{equation}
a_{i,j} = \frac{\mathrm{exp}(s_{i,j})}{\sum_{n=1}^{N}\mathrm{exp}(s_{i,n})}.
\end{equation}
With the attention map, the output $p^{o}_{i}$ for each query patch $p^{q}_{i}$ is a weighted fusion of corresponding value patches $p^{v}$:
\begin{equation}
p^{o}_{i} = \sum_{j=1}^{N}a_{i,j}p^{v}_{j}.
\end{equation}
After receiving these output patches, we reshape them into the size $h\times w\times c$ to obtain the spatially refined feature.
\subsubsection{Temporal Attention Block (TAB)}
In the temporal attention block, we extract multi-scale patches for queries, keys, and values from multiple input frames features as in~\cite{zeng2020learning}. For the waterdrop removal task, the large patches are marvelous for semantic-level reconstruction, and the small ones encourage texture-level reconstruction. For the trade-off between the computation cost and recovering performance, we split each feature into two parts with size $h\times w\times \frac{c}{2}$ and extract different-size patches ($2\times 2\times \frac{c}{2}$ and $8\times 8\times \frac{c}{2}$) from them. After fusing patches with a self-attention mechanism, we reshape fused patches into $h\times w\times c$ features to receive temporally refined features.
After the spatio-temporal fusion, we feed each temporally refined feature into the decoder to obtain cleaned frames $\left \{ \hat{F}_{t} \right \}_{t=1}^{T}$.
\begin{figure*}[t]
\centering
\includegraphics[width=\linewidth]{figs/experiments/comparison_sz.pdf}
\vspace{-2mm}
\caption{Qualitative comparison on real driving scenes collected from the Internet. The results show that our method can remove various-shaped waterdrops in multiple kinds of weather. The last column shows the proposed method still presents a satisfying performance in the extremely challenging case.~\label{fig:4}}
\vspace{-7mm}
\end{figure*}
\subsection{Training}
\subsubsection{Cross-modality Training Strategy}
Although the proposed method only trained on our synthetic dataset can generalize well to real driving scenes, training it jointly on our synthetic video data and the real-world image data~\cite{qian2018attentive} as shown in Fig.~\ref{fig:ill.1} improves the generalization performance of our method. As mentioned in~\cite{qian2018attentive}, the dataset~\cite{qian2018attentive} only consists of degraded images with corresponding clean images, and there is a slight spatial misalignment in each image pair, so we need to utilize these image data carefully.
\subsubsection{Loss Functions for Frame Sequences}
\paragraph{Mask Loss} In the pixel attention block, to encourage the network to predict the pixel-wise attention map precisely as much as possible, we feed the predicted pixel-wise attention map into the mask decoder to receive a waterdrop mask for each input frame, the mask loss is
\begin{equation}
\mathcal L^{F}_{mask} = \frac{1}{T}\sum_{t=1}^{T}BCE(\hat{M}_{t}, M_{t}),
\end{equation}
where $BCE(\cdot)$ denotes binary cross entropy loss function, $\hat{M}_{t}$ denotes the predicted mask and $M_{t}$ denotes the ground-truth mask from our synthetic dataset.
\paragraph{Reconstruction Loss} For the final cleaned frame reconstruction, we add a pixel-wise loss between the network outputs and the ground truths:
\begin{eqnarray}
\mathcal L^{F}_{recons} &=& \frac{1}{T}\sum_{t=1}^{T}\left \| \hat{F}_{t} - F_{t} \right \|^{2}_{2},
\end{eqnarray}
where $\tilde{F}_{t}$ denotes the cleaned frame output.\\
\paragraph{Temporal Loss} To guarantee the temporal consistency among network outputs, we adopt the conditional video discriminator \cite{wang2018video} to calculate the temporal loss (TL) for network outputs:
\begin{equation}
\mathcal L^{F}_{TL} = E_{\widehat{x}\sim P_{\mathbf{\tilde{F}_{T}}}(\widehat{x})}[logD_{V}(\widehat{x})],
\end{equation}
where $P(\cdot)$ denotes the data distribution, $\mathbf{\tilde{F}_{T}}$ denotes the concatenation of all the network outputs, $D_{V}$ denotes the discriminator whose loss function is
\begin{equation}
\begin{split}
\mathcal L_{D} = E_{x\sim P_{\mathbf{F_{T}}(x)}}[logD_{V}(x)]+E_{\widehat{x}\sim P_{\mathbf{\tilde{F}_{T}}(\widehat{x})}}[1-logD_{V}(\widehat{x})],
\end{split}
\end{equation}
where $\mathbf{F_{T}}$ denotes the concatenation of all the ground-truth frames from our synthetic dataset.\\
The total loss functions for each frame sequence are concluded as follows:
\begin{equation}
\mathcal L^{F}_{total} = \lambda ^{F}_{1} \cdot \mathcal L^{F}_{mask}++\lambda ^{F}_{2} \cdot \mathcal L^{F}_{recons} +\lambda ^{F}_{3} \cdot \mathcal L^{F}_{TL},
\end{equation}
where the weights for different losses are set as $\lambda ^{F}_{1}=10$, $\lambda ^{F}_{2}=25$, and $\lambda ^{F}_{3}=5$.
\subsubsection{Loss Functions for Images}
Considering the spatial misalignment between the input image $\tilde{I}$ and its clean ground truth $I$, we only adopt a feature matching loss between the network output and the ground truth:
\begin{equation}
\mathcal L^{I}_{feat} =\sum_{l} \lambda _{l}\left \|\Phi_{l}(\hat{I})-\Phi_{l}(I)\right \|_{1},
\end{equation}
where $\hat{I}$ denotes the network output, $\Phi_{l}$ denotes the layer $l$ in the VGG-16 network~\cite{johnson2016perceptual} where we select the layers $\mathrm{conv1}\_2$, $\mathrm{conv2}\_2$, $\mathrm{conv3}\_2$, $\mathrm{conv4}\_2$ and $\mathrm{conv5}\_2$, $\left \{\lambda _{l}\right \}$ denotes the weights for the layers we select. Besides, since there is no need to exploit the temporal information for image reconstruction, we discard the temporal attention block during attention-based fusion.
\vspace{-2mm}
\section{Synthetic Video Data Generation}
Due to the lack of paired data for multi-image/video methods training, we propose a synthetic waterdrop dataset for driving scenes. To the best of our knowledge, there are only three image waterdrop synthesis works~\cite{hao2019learning,alletto2019adherent,porav2019can} which cannot be utilized to generate waterdrops for videos directly. This encourages us to extend such algorithms to video waterdrop synthesis by considering the followings:
\begin{enumerate}
\item For a driving video, each waterdrop remains at the same position over a sequence of frames. With the wind or the intense movement of the car, there will be a tiny shift in each waterdrop.\\
\vspace{-2mm}
\item With the evaporation of waterdrops, some completely occluded regions turn to be partially occluded along a sequence of frames.
\end{enumerate}
Based on these observations, we design a video waterdrop synthesis algorithm based on~\cite{hao2019learning}. For a sequence of clean frames, we generate 150 to 400 waterdrops in the first frame. For each next frame, we add a shift along a random direction to each waterdrop. Meanwhile, we linearly enlarge the blur kernel size to make waterdrops progressively blurry over frames, which can simulate the evaporation of waterdrops. During synthesizing, we set the length of each sequence as 5, the shift value as 1 pixel, and the blur kernel size from 3 to 20 pixels.
To collect clean driving videos for data synthesis, we choose the DR(eye)VE Dataset~\cite{dreyeve2018} which contains 51 videos with multiple scenes and weathers, varying from the morning, evening, night, sunny, cloudy, countryside, downtown, and highway.
To summarize, we propose a large-scale synthetic dataset with numerous triplets $(F, \tilde{F}, M)$. In each triplet, there are one clean frame $F$, one frame $\tilde{F}$ with synthetic waterdrops and one binary mask $M$ for waterdrops. Totally, we have 67500 triplets from 45 videos for training and 600 triplets from 6 videos for testing.
\vspace{-2mm}
\section{Experiments}
\subsection{Implementation Details}
We train the network by minimizing $(\mathcal L^{F}_{total}+\mathcal L^{I}_{feat})$ on synthetic data and real data jointly. To be specific, for every 1000 training iterations, we train the network on synthetic data for 900 iterations while on real-world data for 100 iterations. We set the proportion as $9:1$, the best one we select from $5:1$, $7:1$, $9:1$ and $11:1$ by evaluations. Totally, we train the network for $75000$ iterations on both datasets using the ADAM optimizer~\cite{DBLP:journals/corr/KingmaB14} with $lr=0.0001$ and $(\beta _{1}, \beta _{2})=(0.9, 0.999)$. We set the length $T$ of each frame sequence as 5, and channel number $c$ as 256 during training.
\vspace{-2mm}
\subsection{Comparison to State-of-the-Art}
As abovementioned in Sec.~\ref{section:related_work_2}, there is only one video waterdrop removal method proposed by Alletto~\textit{et al}.~\cite{alletto2019adherent}. However, the lack of source code and real-scene evaluations impedes us from making a fair comparison with them. To evaluate our model and make fair comparisons, we select the most recent and the most competitive methods for comparisons:
\begin{enumerate}
\item AttentGAN~\cite{qian2018attentive}, Quan~\textit{et al}.~\cite{quan2019deep}, CCN~\cite{quan2021removing} are learning-based methods for single-image waterdrop removal.
\item Vid2Vid~\cite{wang2018video} is a recent and typical video-to-video translation method that is regarded as the spatio-temporal baseline in evaluation.
\item LSTO~\cite{liu2020learning} is a method for multi-image obstruction removal. This method can be extended to video waterdrop removal.
\end{enumerate}
\begin{table}[t!]
\centering
\tabcolsep=0.03cm
\caption{Quantitative comparison on our proposed synthetic dataset. ~\label{table:score}}
\vspace{-2mm}
\setlength\arrayrulewidth{1.0pt}
\resizebox{1\linewidth}{!}{
\begin{tabular}{cccc}
\toprule
\multicolumn{1}{p{1.5cm}}{}
&\multicolumn{1}{p{2cm}}{\centering CCN~\cite{quan2021removing}}
&\multicolumn{1}{p{2cm}}{\centering Vid2Vid~\cite{wang2018video}}
&\multicolumn{1}{p{1.5cm}}{\centering \textbf{Ours}} \\
\cmidrule(lr){2-4}
PSNR $\uparrow$ & 26.2878 & 27.5029 & \textbf{29.5789} \\
MS-SSIM $\uparrow$ & 0.9220 & 0.9439 & \textbf{0.9627} \\
LPIPS $\downarrow$ & 0.1896 & 0.1231 & \textbf{0.0848} \\
$E_{warp}$~\cite{lei2020dvp} $\downarrow$ & 0.0828 & 0.0811 & \textbf{0.0799} \\
\bottomrule
\end{tabular}}
\vspace{-4mm}
\end{table}
\subsubsection{Qualitative Evaluation}
We present some qualitative comparisons in Fig.~\ref{fig:4}, evaluated on real driving scenes which are collected from the Internet (without ground truth). For various waterdrop cases, image methods show poor performance on video waterdrop removal, especially CCN~\cite{quan2021removing}, since the network architecture dedicated to the single-image task is too weak to handle such complex driving scenes. Although Vid2Vid~\cite{wang2018video} shows satisfying removing performance on the sparse and small waterdrops as shown in Fig.~\ref{fig:4} column 4, this CNN-based method fails to address streak case with large waterdrops and it cannot restore the background information under the complete occlusions, as shown in Fig.~\ref{fig:4} column 1. LSTO~\cite{liu2020learning} can remove most waterdrops in real driving scenes, while this multi-image method relies heavily on flow estimation, which may fail when there are many waterdrops along a sequence of frames. Furthermore, this method is too time-consuming, which costs 60 seconds for each frame of size $960\times 512$, while our method only costs 0.5 seconds for the same one. Based on our pixel attention block and spatio-temporal fusion, the proposed method shows outstanding waterdrop removal performance in sparse cases and streak cases. Even under dark circumstances and heavy rain, the proposed method still presents satisfying results in such extreme cases, as shown in Fig.~\ref{fig:4} columns 2 and 5.
\subsubsection{Quantitative Evaluation}
\paragraph{PSNR and MS-SSIM} Due to the lack of paired data of real driving scenes, we evaluate the different methods on the testing set of our synthetic dataset. Since the lack of training codes for other methods, we only compare the proposed method with CCN~\cite{quan2021removing} and Vid2Vid~\cite{wang2018video}, which are re-trained on the proposed dataset with the same cross-modality training strategy. As shown in Table~\ref{table:score}, we compute the PSNR and MS-SSIM~\cite{wang2003multiscale} between the result cleaned frames of different methods and ground-truth clean frames. The proposed method shows strong performance over previous works on synthetic data.
\paragraph{User Study} We also conduct a user study for the results of different methods on real driving scenes. For each evaluation, we compare our method to the other five methods. Each user is presented with a driving-scene frame degraded by waterdrops and six cleaned frames from different methods. The user needs to choose the cleaned one which has better visual quality. There are 25 driving-scene frames presented in the comparisons and 30 users in this user study. The results are shown in Table \ref{table:user study}. There is nearly $77.84$ percent of users prefer our results, which means our method outperforms others significantly.\\
\begin{table}[t!]
\centering
\tabcolsep=0.03cm
\caption{User study for video waterdrop removal task.~\label{table:user study}}
\vspace{-2mm}
{
\setlength\arrayrulewidth{1.0pt}
\resizebox{1\linewidth}{!}{
\begin{tabular}{ccccccc}
\toprule
\multicolumn{1}{p{1.5cm}}{\centering AttentGAN\\~\cite{qian2018attentive}}
&\multicolumn{1}{p{1.5cm}}{\centering Quan~\textit{et al}.\\~\cite{quan2019deep}}
&\multicolumn{1}{p{1cm}}{\centering CCN\\~\cite{quan2021removing}}
&\multicolumn{1}{p{1cm}}{\centering Vid2Vid\\~\cite{wang2018video}}
&\multicolumn{1}{p{1cm}}{\centering LSTO\\~\cite{liu2020learning}}
&\multicolumn{1}{p{1cm}}{\centering Ours} \\
\cmidrule(lr){1-6}
4.54$\%$ & 10.23$\%$ & 0$\%$ & 6.82$\%$ & 0.56$\%$ & \textbf{77.84$\%$} \\
\bottomrule
\end{tabular}}}
\end{table}
\begin{table}[t]
\centering
\tabcolsep=0.03cm
\caption{Ablation study on our proposed synthetic dataset. The best and second-best scores are indicated in \textbf{\textcolor{red}{red}} and \textbf{\textcolor{blue}{blue}}.~\label{table:ablation}}
\vspace{-2mm}
{
\setlength\arrayrulewidth{1.0pt}
\resizebox{\linewidth}{!}{
\begin{tabular}{cccccc}
\toprule
\multicolumn{1}{p{1.cm}}{}
&\multicolumn{1}{p{1.5cm}}{\centering Ours-noPAB}
&\multicolumn{1}{p{1.5cm}}{\centering Ours-noSAB}
&\multicolumn{1}{p{1.5cm}}{\centering Ours-noTAB}
&\multicolumn{1}{p{1.6cm}}{\centering Ours-noCMS}
&\multicolumn{1}{p{1.4cm}}{\centering \textbf{Ours (full)}}\\
\cmidrule(lr){2-6}
PSNR $\uparrow$ & 27.6397 & 28.2355 & 28.0135 & \textbf{\textcolor{red}{30.1052}} & \textbf{\textcolor{blue}{29.5789}} \\
MS-SSIM $\uparrow$ & 0.9456 & 0.9530 & 0.9492 & \textbf{\textcolor{blue}{0.9625}} & \textbf{\textcolor{red}{0.9627}} \\
LPIPS $\downarrow$ & 0.1209 & 0.1318 & 0.1363 & \textbf{\textcolor{red}{0.0740}} & \textbf{\textcolor{blue}{0.0848}} \\
$E_{warp}$~\cite{lei2020dvp} $\downarrow$ & 0.0815 & 0.0810 & 0.0831 & \textbf{\textcolor{blue}{0.0800}} & \textbf{\textcolor{red}{0.0799}} \\
\bottomrule
\end{tabular}}}
\vspace{-1mm}
\end{table}
\begin{figure}[!ht]
\centering
\begin{minipage}[t]{.15\textwidth}
\centering
\includegraphics[width=\textwidth]{figs/experiments/ablation/input/79_rainy_frame.jpg}
\subcaption{Input Frame}
\end{minipage}
\begin{minipage}[t]{.15\textwidth}
\centering
\includegraphics[width=\textwidth]{figs/experiments/ablation/wo_trans/39_fake_clean_frame_final.jpg}
\subcaption{Ours-noCMS}
\end{minipage}
\begin{minipage}[t]{.15\textwidth}
\centering
\includegraphics[width=\textwidth]{figs/experiments/ablation/ours/79_fake_clean_frame.jpg}
\subcaption{Ours (full)}
\end{minipage}
\caption{Qualitative comparison on real driving scenes for our ablation study.~\label{fig:6}}
\vspace{-4mm}
\end{figure}
\paragraph{Temporal Consistency} To evaluate the quality of results across the temporal dimension, we adopt a similar evaluation metric proposed by Lei~\textit{et al}.~\cite{lei2020dvp}. For each cleaned frame $\tilde{F}_{t}$, we calculate the warping error with $\tilde{F}_{t-1}$, $\tilde{F}_{t-3}$ and $\tilde{F}_{t-5}$ for considering multi-scale temporal consistency. As shown in Table~\ref{table:ablation}, our model without the temporal attention block suffers heavily from temporal inconsistency, which means the temporal attention block not only restores the regions with the clues from nearby frames but also keeps the network outputs temporally consistent.
\vspace{-2mm}
\subsection{Ablation Study}
To better analyze each block or cross-modality training strategy that contributes to the final performance of our network, we remove or replace each block one by one and re-train each modified network on the same datasets. As shown in Table.~\ref{table:ablation}, each attention block shows a positive effect for removing and recovering performance. Besides, although the model trained without cross-modality strategy (Ours-noCMS), which is only trained on our synthetic dataset, performs the best in quantitative evaluation in Table~\ref{table:ablation}, our full model shows better performance on real driving scenes, as shown in Fig.~\ref{fig:6}. The cleaned frame maintains the original color in good condition with our cross-modality training strategy, and some tiny waterdrops disappear.
\vspace{-1mm}
\section{Conclusion}
\vspace{-1mm}
We present a new method with spatio-temporal fusion for video waterdrop removal. To train our proposed method, we build a large-scale synthetic waterdrop dataset for complex driving scenes. With an appropriate cross-modality training strategy, the proposed method shows an impressive waterdrop removal performance in real driving scenes. Furthermore, we are the first one to compare the proposed method with others on numerous synthetic data and real driving data. Both quantitative and qualitative evaluations show the proposed method is better than others.
{\small
\bibliographystyle{IEEEtran}
|
1,314,259,993,657 | arxiv |
\section{Introduction}\label{sec:introduction}
Modelling temporal series of data is important in many different domains, including disciplines as diverse as hydrology or economics, but also to monitor and understand human behaviours from wireless sensor networks in smart environments \cite{Sun15,Galiana14,LeBorgne07}.
Typically, different processes require different models to interpret and forecast new sensor data.
The nature of the process, the amount of required data and the extend of the forecasting determine the kind of model finally chosen.
Temporal models should be able to capture the frequencies of important event occurrences -- e.g. the routine activities performed by a home-monitoring system for the elderly.
Methods for frequency analysis (i.e. Fourier transform) can reveal periodic patterns in the sensor data but, if occurring only within specific time intervals, they fail to determine when these periodicities start and end. Moreover, short events that manifest in localized peaks of the sensor signal are difficult to be captured by standard Fourier analysis, unless a large number of frequency components are considered.
And even with a high frequency resolution, temporal information -- i.e. when those peaks are happening -- is lost in Fourier analysis.
In this paper, therefore, we propose a new wavelet-based method that is suitable for modelling sparse periodic and/or very short events in sensor data. Wavelet analysis indeed has the advantage that it simultanously provides temporal and frequency information of a signal with very little loss of information, and it is therefore more powerful than Fourier analysis in capturing and forecasting sensor data in many real-world applications.
One of these, Active \& Assisted Living~(AAL), is an important application area where good temporal representations of events can enable the implementation of many useful well-being services~\cite{Bellotto17}.
To this end, our wavelet-based temporal model can be used to identify patterns of human activity from smart-home sensors and detect anomalies in the occurrence of typical daily routines.
The latter, indeed, have a significant temporal component, which is often periodic, but with occasional variations and very short-term events (e.g.~repeatedly opening/closing the fridge in the morning, but only on weekdays).
In particular, we adopt the {\em anomaly} definition in~\cite{Fernandez-Carmona2017}, which considers the amount of motion in specific locations as a normalized entropy beyond some given thresholds.
Note that the term ``motion'' is used in a broad sense to include the activation of various binary sensors, such as passive infrared (PIR) motion detectors or contact sensors on doors, cupboards, etc. We also refer to this type of motion in the environment as {\em activity level}.
\begin{figure}[t]
\centering
\includegraphics[width=\columnwidth]{functionalDiagram4.pdf}
\caption{Wavelet-based anomaly detection system. The expected and actual normalized entropies from wavelet-based temporal models and sensor data, respectively, are compared by a HMLN-based inference module that contains expert rules.\label{fig:system_structure} }
\end{figure}
In this work, we apply our wavelet-based representation of human activities to a new anomaly detection system for AAL (see Fig.~\ref{fig:system_structure}).
In particular, given a set of smart-home binary sensors (i.e. motion detectors and contact sensors), we build accurate temporal models to represent and forecast their expected output.
Then, using an entropy-based method~\cite{Fernandez-Carmona2017}, we estimate the current and expected levels of human activity. These two are finally compared by an original inference system based on an Hybrid Markov Logic Network~(HMLN)~\cite{Wang2008}, to detect potential anomalies.
The paper includes three main contributions:
\begin{itemize}
\item First, we propose a novel technique for temporal modelling of (long-term) human activities based on wavelet transforms.
Among its possible applications, this wavelet-based temporal model enables the forecasting of smart-sensor signals for the detection of potential anomalies, i.e. human activities that deviate significantly from the norm.
A software implementation of this temporal modelling tool is made publicly available.
\item Second, we describe a new automatic system for anomaly detection that uses a HMLN to combine three sources of information about human activities, namely i)~actual entropy level from smart-home sensors, ii)~expected entropy from wavelet-based temporal models, and iii)~expert knowledge in the form of logic rules.
\item Finally, we present extensive experimental results based on two large datasets, one previously recorded in an office environment~\cite{Fernandez-Carmona2017} and a new one from a real elderly home, which we also made publicly available. These datasets were recorded in MongoDB format~\cite{Chodorow2013} for easy access and re-usability by the scientific community.
\end{itemize}
The remainder of the paper is organized as follows.
Sec.~\ref{sec:rel_work}~reviews state-of-the-art methods for temporal modelling and anomaly detection with smart-home sensors, including relevant public datasets.
Sec.~\ref{sec:temporal_model}~briefly introduces the wavelet transform and describes the respective temporal models of sensor data.
Sec.~\ref{sec:entropy}~explains the entropy-based method used to represent human activity levels in smart-home scenarios.
Sec.~\ref{sec:mln} describes the design of the HMLN-based inference systems and its expert rules to analyse and detect anomalies in human activities.
Sec.~\ref{sec:arch}~illustrates the architecture and practical implementation of the anomaly detection system.
Sec.~\ref{sec:experiments}~presents datasets and experiments to validate the effectiveness of the temporal models and the anomaly detection in an office and AAL scenarios.
Finally, Sec.~\ref{sec:conclusions} discusses advantages and disadvantages of the proposed approach, suggesting directions for future work in this area.
\section{Related work}\label{sec:rel_work}
A reliable temporal model of human activities can benefit many smart-home and robotics applications for AAL~\cite{coppola16}.
Such model could help an automated system understand the current scenario and plan opportune interventions, for example by sending a mobile service robot to a human user when it is more likely to be actually helpful.
Temporal modelling is widely used to detect regular patterns in data.
From time series analysis, a relevant tool is the \textit{autoregressive integrated moving average model} (ARIMA) and its derivations \cite{Chen10}, including the stationary process case described by the \textit{autoregressive moving average} (ARMA) model.
The main problem with these models is that they may become unstable \cite{Zou2004} or are only suitable for relatively short temporal windows or known temporal trends \cite{Xie2018}.
Other non-linear techniques, such as Gaussian Processes~\cite{Povinelli04}, could theoretically achieve the full reconstruction of signals from mixture models.
Similarly, Ghassemi \& Deisenroth~\cite{Ghassemi14} use periodic Gaussian Processes for long-term forecasting.
In~\cite{Ihler06}, Poisson processes are used instead as probabilistic models to recognize patterns and, in combination with Markov Chains, to identify anomalies in the data.
These models are typically robust against model instabilities, but they require heavy computational processes.
A technique called FreMEn~(Frequency Map Enhancement) has recently been proposed for spatio-temporal representations of robot environments in long-term scenarios~\cite{fremen14}.
It uses Fourier analysis to extract periodicities in sensor data, in combination with a Bernoulli distribution or Poisson processes~\cite{Jovan16} to represent binary information states.
FreMEn is a simple yet effective modelling tool, but it is not suitable to describe sparse or very short events.
Wavelet-based methods have been used for temporal modelling in many different fields such as drought or price forecasting~\cite{Maity16,Conejo2005}, passenger flow prediction~\cite{Sun15}, human motion analysis \cite{Ayrulu2011} or iris recognition \cite{Majumder2013}.
Since wavelets contain both frequency and time domain information, they are particularly suitable to represent sparse non-stationary signals.
Some temporal models are specifically tailored to the specific sensor or data source.
For example, \cite{Aran2016,Steen2013}~proposed spatio-temporal models of motion detectors in which an anomaly is seen as a significant deviation from the typical sensor response.
Although relatively simple, this approach is very sensitive to potential misplacements or faults of the deployed sensors
Alternative activity and temporal models were proposed by~\cite{Okeyo2014} using 4D-fluents (i.e. logic predicates that depend on time) to add a temporal layer on the top of an underlying description logic.
\cite{Soulas2015},~instead, proposed an Extended Episode Discovery model that defines habits in terms of length, frequency and periodicity for offline processing.
In~\cite{Chahuara2016}, the authors compare three sequential activity models -- \gls{hmm}, \gls{crf} and sequential \gls{mln} -- where feature vectors were generated during fixed-time windows for on-line processing. These sequential activity models offer a straightforward approach to anomaly detection, which is not addressed in those works though.
Typically, anomaly detection systems are designed for the specific sensor(s) used.
Depending on the input data, approaches may vary greatly.
Wearable activity trackers like the one proposed by~\cite{Godfrey2010}, for example, provide rich and continuous motion and pose information without requiring any additional preprocessing. But wearables can be forgotten, misplaced or misused by volunteers, leading to false anomalies in the datasets.
Automated video sequence-based analysis, instead, does not require explicit user intervention.
However, extra effort is needed to extract meaningful information from the input sequences.
For example, Xu~et~al.~\cite{Xu2016} used a multiple one-class \gls{svm} models to predict anomaly scores, while Leyva~et~al.\cite{Leyva2017} used Markov Chains to detect abnormal events on a video stream.
Compared to camera-based systems, smart-home sensors offer a cheaper alternative for anomaly detection~\cite{Fernandez-Carmona2017}.
Markov Logic Networks~(MLNs) are both a modelling \cite{Li2019,Alen2012} and inference \cite{Jiang2017,Liu2017} tool, often used for their flexibility to define
rich models.
They are able to perform inferences using imprecise or incomplete inputs, useful to deal with sensor faults and network errors.
In addition, they can blend both sensor data and expert logic rules within a probabilistic framework for robust inference in real time applications~\cite{Sztyler2018}.
Compared to the SVM and HMM-based system, the advantage of using MLNs for anomaly detection is that they require a smaller amount of sensor
data to build their models and that they better handle uncertain information~\cite{Gayathri2015}.
SVM have been successfully combined with deep learning (DL) techniques for anomaly detection and achieved promising results in high dimensional problems~\cite{Erfani2016}, but without exploiting the available temporal information.
The HMLN proposed in this paper combines wavelet-based temporal models and expert rules, mixing for the first time discrete and continuous predicates, to infer about potential anomalies. These expert rules allow also to overcome the lack of data otherwise required to train DL-based methods.
Public datasets with labelled sensor data are important to test and compare different algorithms.
The dataset hosted by Tim~van~Kasteren\footnote{\url{https://sites.google.com/site/tim0306/datasets}}~\cite{Kasteren10} offers a collection of compressed Matlab files with several recordings of binary sensors (e.g. open/closed doors; pressure mats; motion detectors).
The Center for Advanced Studies in Adaptive Systems~(CASAS) also provides an extensive collection of datasets\footnote{\url{http://casas.wsu.edu/datasets/}} for activity recognition, in which every entry has a different format, usually a compressed text or binary file.
The Smart project~\cite{Barker12}, even if focused on energetic sustainability and consumption management,
created a wide collection of datasets from real houses, including smart-home sensors\footnote{\url{http://traces.cs.umass.edu/index.php/Smart/Smart}}.
All these datasets contain non-standard, plain text or binary files which are difficult to handle by other researchers, especially if of large size.
They lack of an standarized format and access mechanisms, suitable for systematic data processing in big data.
To our knowledge, there are no smart-home datasets based on such standardised and easily manageable formats.
Our new dataset, instead, was created by storing raw data in a MongoDB database.
This approach provides an accessible, platform- and application-independent format readily available for other research in our paper's application area and beyond.
\section{Wavelet-based Temporal Forecasting}\label{sec:temporal_model}
In this section we present a novel approach to forecast sensor data for human activity monitoring using a wavelet-based temporal model. We start with a brief description of the discrete wavelet transform algorithm, and then we explain how to tune and use this algorithm for building our temporal model of the sensor data.
Standard Fourier analysis is useful for the frequency decomposition of signals, but it does not keep important time information.
That is, we know which frequency components are present in a signal, but not {\em when} they are present.
In addition, signal discontinuities are poorly represented by Fourier transform, since its basis is non-local.
This is known as the Gibb's phenomenon~\cite{Hewitt79}.
Wavelets provide an alternative representation that overcomes the limitations of Fourier analysis.
They decompose signals into individual components, which maintain both frequency and time information.
Also, they can effectively represent and provide localized information about discontinuities.
These advantages (i.e. time-frequency and discontinuity representations) are very important to handle the non-periodic and often ``spiky'' nature of real-world sensor data, especially in the context of activity monitoring.
\subsection{Discrete Wavelet Transform}\label{sec:dwt}
A discrete wavelet transform (DWT) is a sampled wavelet transform applicable to digital signals.
Let us consider a discrete time signal $x[n]$ in the $L^2$ space, with finite energy and defined in the interval ${[0,N-1]}$ with a sampling frequency $f_s$.
This signal can be represented using the following orthogonal decomposition:
\begin{eqnarray}
L^{2} &=& V_{0}\oplus W_{0} \label{eq:orthogonal_}
\end{eqnarray}
\noindent where $W_0$ is the orthogonal complement of subspace $V_0$ inside $L^{2}$.
The subspace $V_0$ can be further subdivided into two orthogonal subspaces $V_{1}\oplus W_{1} \nonumber$, and so recursively:
\begin{eqnarray}
V_{j} &=& V_{j+1}\oplus W_{j+1} \text{~~~~with } j = 0, 1, \ldots, Q \nonumber \\
\\
L^{2} &=& V_{Q}\oplus W_{Q}\oplus W_{Q-1}\oplus W_{Q-2}\dots \oplus W_{0} \nonumber
\end{eqnarray}
\noindent defines the $Q$-level decomposition of the $L^2$ space.
The subspace $V_{Q}$ maintains the time domain properties of the signal, whereas the
subspaces $W_{0...Q}$ preserve its properties in the frequency domain.
These time and frequency subspaces are generated by the following function families:
\begin{eqnarray} \label{eq:function_families}
V_{j} &=& \operatorname {span} (\phi _{j,k}: k \in \mathbb{Z} ), \nonumber \\
&& {\text{ where }}\phi _{j,k}[n]=2^{-j/2}\phi [2^{-j}n-k] \nonumber \\ \\
W_{j} &=& \operatorname {span} (\psi _{j,k}: k\in \mathbb{Z} ), \nonumber \\
&&{\text{ where }}\psi _{j,k}[n]=2^{-j/2}\psi [2^{-j}n-k]~. \nonumber
\end{eqnarray}
The scaling functions~$\phi _{j,k}$ are weighted and displaced versions of a ``father wavelet'' function $\phi[n]$.
They can also be obtained iteratively re-scaling a previous one.
The parameter $j$ determines the scale and magnitude of the corresponding scaling function, keeping the energy constant.
As a result, ~$\phi _{j,k}$ is only defined in the interval ${[0,\frac{N}{2^j}-1]}$ .
Values of $j$ close to infinity will turn the scaling function into a delta function, whereas the opposite will lead to an almost constant (and low) value.
Finally, the parameter $k$ determines the time displacement of the wavelet.
Similarly, every wavelet function $\psi _{j,k}[n]$ is built scaling and displacing a ``mother wavelet'' $\psi[n]$, or recursively.
However, they are related to the higher frequency components of $x[n]$ instead of its average trends.
Any function $x[n]$ belonging to $L^2$ can then be represented by the following linear combination of $Q+1$ subspaces:
\begin{eqnarray}
x[n] &=& \sum_{k=0}^{\frac{N}{2^Q}-1} c_{Q,k}\phi_{Q,k}[n] + \sum_{j=1}^{Q}\sum_{k=0}^{\frac{N}{2^j}-1} d_{j,k}\psi_{j,k}[n]
\end{eqnarray}
\noindent where the averaging coefficients $c_{Q,k}$ and detail coefficients $d_{j,k}$ are obtained using the following inner products:
\begin{eqnarray} \label{eq:inner_products}
c_{Q,k} &=& \langle x[n],\phi _{Q,k}[n] \rangle \nonumber \\
\\
d_{j,k} &=& \langle x[n],\psi _{j,k}[n] \rangle. \nonumber
\end{eqnarray}
This set of coefficients $C$ and the original wavelets are all we require to perform the inverse discrete wavelet transform (IDWT):
\begin{eqnarray} \label{eq:coeff_set}
C &=& \{ {c}_{Q,k}, {d}_{j,k} | k = [0, ... \frac{N}{2^Q}-1], j = [1,... Q] \} \nonumber \\
x[n] &=& IDWT(C,\phi,\psi)~. \nonumber
\end{eqnarray}
Using a geometric analogy, this process can be seen as a change of basis.
For example, Fig.~\ref{fig:vectorAnalogy}(a) shows a signal $x[n]$ in the vector space defined by the function family ${S_{j} = \operatorname {span} (\delta[n-k] : k \in \mathbb{Z})}$.
The DWT of $x[n]$, shown in Fig.~\ref{fig:vectorAnalogy}(b), is a representation of the same signal but with a different basis defined by the functions in (\ref{eq:function_families}). The inner products in (\ref{eq:inner_products}) are then used to obtain the coordinates in the new basis.
\begin{figure}
\centering
\includegraphics[width=0.65\columnwidth]{vectorAnalogy.pdf}
\caption{Geometric analogy for wavelet transform.\label{fig:vectorAnalogy}}
\end{figure}
The DWT is usually performed by a bank of equivalent filters~\cite{Vetterli92}, as depicted in Fig.~\ref{fig:bankFilter}.
The input signal is processed by a series of low- and high-pass filters $g[n]$ and $h[n]$, respectively, and then subsampled to obtain the averaging and detail coefficients.
The figure shows also how each group of coefficients is related to a specific range of frequencies.
Fig.~\ref{fig:frequencies}, instead, illustrates the frequency bands corresponding to the function families $V_j$ and $W_j$. Here, the detail coefficients ($d_{j,k}$) concentrate on higher frequency bands depending on the decomposition level, while the averaging ones ($c_{Q,k}$) belong to the narrow low-frequency band.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{bankFilter.pdf}
\caption{Bank of filters configuration for discrete wavelet transform (DWT).\label{fig:bankFilter}}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{frequencies.pdf}
\caption{Discrete frequency bands of the decomposition' subspaces.\label{fig:frequencies}}
\end{figure}
Using wavelets, we can study a signal using different frequency resolutions at once.
Fig.~\ref{fig:scalogram} shows the scalogram of a signal generated by an infrared motion detector, installed in an office environment, over a period of 24h using the dataset from \cite{Fernandez-Carmona2017}.
In this representation, the $x$ axis shows the temporal displacement $k$, while the $y$ axis indicates the scale (or period) $j$ of the DWT.
Higher scale values of the scalogram correspond to higher frequencies of the signal, although with reduced temporal resolution.
In the figure, we can see several peaks representing sudden spikes of the sensor data, repeated throughout the day, localized at certain temporal instants.
The vertical bar on the left shows also the average energy per scale (or period) $j$ of the DWT, which can be interpreted as a discrete Fourier transform of the original signal.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{2016-09-09-WW-Lounge-scalogram.png}
\caption{Example of wavelet scalogram for an office's motion detector.}
\label{fig:scalogram}
\end{figure}
\subsection{Parameter Selection for Wavelet Transforms}\label{sec:param_select}
In order to fully describe a DWT, we need to define its mother wavelet and decomposition level.
The mother wavelet is usually chosen through quantitative or qualitative approaches.
The former favour wavelets that are visually similar to the decomposed signals.
The latter instead optimize specific parameters such as number of components to describe the signal, fidelity of the reconstructed signal, denoising capabilities of the chosen wavelet, etc.
In order to obtain the best possible fidelity, in our model we use a \gls{MSE} criterion.
Originally proposed by~\cite{Phinyomark09}, this criterion chooses the mother wavelet that minimizes the error on the reconstructed signal.
The decomposition level is limited by the length of the signal and by the chosen wavelet.
Looking at the bank of filters implementation in Fig.~\ref{fig:bankFilter}, we can see that every decomposition level halves the length of the signal.
A practical rule is to stop the decomposition before the signal becomes shorter than the length of the low pass filter $g[n]$.
Let be $L$ the length of the filter $g[n]$ and $N$ the length of the signal. The maximum decomposition level is the following:
\begin{eqnarray}
\max Q = \log_2 \left (\frac{N}{L-1} +1 \right )~.
\end{eqnarray}
\noindent However, reaching the maximum decomposition level is not always necessary.
In~\cite{Lei13}, for example, the authors proposed a method to choose the decomposition level based on the sparseness (i.e. number of zero-elements) of the signal.
The same will be applied to our model to obtain a compact representation of the sensor data.
Another relevant parameter is the coefficient thresholding level.
The number of coefficients obtained from the wavelet transform is initially equal to the length of the discrete input.
Some of these coefficients, however, carry very little information, especially if the mother wavelet is optimal.
We can discard coefficients below a thresholding level, and still reconstruct the original signal good approximation:
\begin{eqnarray} \label{eq:c}
\hat{C} = \{ c_i \in {c}_{Q,k}, {d}_{j,k} \mid (c_i > \tau) \wedge IDWT(\hat{C},\phi,\psi) \approx x[n] \}~. \nonumber
\end{eqnarray}
This approach is commonly used in imaged processing to remove noise and perform lossless compression~\cite{Fathi12}.
Here we will use a \textit{statistical threshold}, originally proposed by~\cite{Nashat16}, that preserves some statistics on the compressed signal.
In practice, we will use the set of coefficients $\hat{C}$ above a certain threshold~$\tau$ that still allows a lossless reconstruction of the signal.
All the remaining coefficients, below the selected threshold, will be removed from our sensor data model.
\subsection{Sensor Data Modelling and Forecasting}
\label{sec:sensor_model}
After introducing the wavelet transform and its parameters,
we can use them to model smart-home sensors and to forecast their data.
Our model is an efficient representation of a generic temporal signal,
similar to some compression techniques commonly used in image processing~\cite{Fathi12}.
Let us consider the signal $x[n]$ generated by a smart-home sensor over time.
The sampling frequency of the sensor data is $f_s$.
Our training model signal is transformed into the wavelet domain using a 1-level DWT decomposition.
Since the input data is relatively sparse (i.e. mostly containing localized activation peaks),
a higher decomposition level would not bring any particular advantage to the resulting wavelet transform.
We then threshold the wavelet coefficients and keep a significantly smaller number of them, while maintaining a low Root Mean Square Error~(RMSE).
We can finally reconstruct the signal using this small subset of coefficients and the inverse wavelet transform (IDWT).
Our wavelet-based model $\mathcal{M}$ is therefore described by this subset~$\hat{C}$ of coefficients,
a mother wavelet~$\phi$, the decomposition level~$Q$, a coefficient threshold~$\tau$, the number of samples~$N$, the sampling frequency~$f_s$, and the time reference~$t_0$:
\begin{eqnarray}
\label{eq:model}
\mathcal{M} = \{ \hat{C}, \phi, Q, \tau, N, f_s, t_0 \}~.
\end{eqnarray}
Once this model is available, it is possible to represent the sensor output at a future time instant $t_f$.
The model in~(\ref{eq:model}) assumes that the sensor output has periodicity $N$ starting from time $t_0$.
The index $n_i$ of the sensor data sample at time $t_f$ is therefore given by the following equation:
\begin{equation}
n_{i} = \lceil (t_f - t_0) * f_s \rceil \mod N
\end{equation}
and the actual sensor data sample can then be obtained from the reconstructed signal $\hat{x}[n]$ as follows:
\begin{equation}
\label{eq:prediction}
\hat{x}[n_i] = IDWT(\hat{C},\phi,\psi)[n_i]~.
\end{equation}
In Sec.~\ref{sec:results:wavelets:modelling} we will describe an empirical method to determine the parameters of this model, including the most suitable mother wavelet and the thresholding level of the coefficients.
\section{Entropy-based Activity Representation}\label{sec:entropy}
\subsection{Normalized Entropy}
The metric used in our system to describe anomalous situations is based on the concept of entropy~$H$ of a (discrete) probabilistic distribution~$P(x)$, as defined in information theory.
Entropy is invariant to probability permutations and it describes the overall information contained in the distribution as follows:
\begin{equation}
H = - \Sigma_{x} P(x) \log_2 P(x)~. \label{eq:entropy}
\end{equation}
Highly probable events carry little information, and therefore reduce the entropy.
On the other hand, uniform probability distributions are characterised by high levels of entropy,
denoting situations with significant amount of information (i.e. high uncertainty).
We normalize $H$ using the maximum entropy of a discrete uniform distribution.
Such entropy is given by the logarithm of the total number of possible outcomes.
Therefore, our normalized entropy $\hat{H}$ for a probability distribution with entropy $H$ and $R$ possible outcomes is defined as follows:
\begin{equation}
\hat{H} = \frac{H}{\log_2 R}~. \label{eq:rel_entropy}
\end{equation}
In an environment monitored by $R$ sensors, the above quantity defines a metric to measure the amount of information that is associated to the events $x$ detected by the sensors.
A method to determine the probability $P$ of an ``activity'' event from a motion detector was proposed in~\cite{Fernandez-Carmona2017} and it is described in the next section, extended to the general case of binary smart-home sensors.
\subsection{Activity Levels}
We consider a network of $R$ binary smart-home sensors (e.g. motion detectors, contact sensors, etc.) distributed around different rooms, areas, or objects of interest in an indoor environment (e.g. office and apartment in Fig.~\ref{fig:setup_lcas} and Fig.~\ref{fig:setup_lace} of the experiments). We want to model the probabilities of human activities associated to those sensors and compute the normalized entropy for the whole environment.
In this case, the activity probability can be obtained observing the sensor's output during a fixed time interval (i.e. 30 seconds).
For example, motion detectors trigger an event whenever something moves within their detection field, while contact sensors can check whether doors have been opened or closed.
From this, it is possible to observe for how long such activity was detected by sensor~$s_i$, that is, the amount of time~$T(s_i)$ that the sensor was ``on''.
Under the assumption that there are no overlapping sensors (i.e. each sensor covers a different room, area, or object), we can define the probability $P(s_i)$ of an activity detected by $s_i$:
\begin{equation}
P(s_i)=\frac{T(s_i)}{\sum_{j = 1}^R T(s_j)}~. \label{eq:prob}
\end{equation}
\begin{figure}[tb]
\centering
\includegraphics[width=\columnwidth]{FDDandEntropy1.pdf}
\caption{Example of probability distribution and normalized entropy of motion activities in five different rooms. Mean and standard deviation in case (A) are different when the 'Bedroom' and 'Entrance' probabilities are swapped in (B). The total entropy for the whole environment remains the same instead. }
\label{fig:fddp_entropy}
\end{figure}
The distribution of these $R$ probabilities provides some information about the current activity level in the environment,
but it is not a good metric on its own to determine whether such activity should be considered ``normal'' or not.
For example, the distribution depends on the order of the considered sensors, and a simple permutation of different sensor probabilities would change the distribution's mean and standard deviation.
This is illustrated by the example in Fig.~\ref{fig:fddp_entropy}: after the activity probabilities of two motion detectors in different rooms are swapped, the mean $\mu$ and the standard deviation $\sigma$ of the distribution change significantly, whereas the total (normalized) entropy remains unaffected. The latter will be used therefore to represent the activity level in the environment as input for our anomaly detection system.
\section{Anomaly Detection}\label{sec:mln}
Markov Logic Networks can be used to combine different sources of information for probabilistic inference.
In this paper, we use both smart-home motion sensors and their wavelet-based models to analyse the difference between {\em actual} and {\em expected} entropy, respectively, of the environment.
The first one represents the current activity level, whereas the second one represents the most likely one.
These entropy values, together with direct sensor inputs and expert rules, provide the necessary information for our MLN to detect anomalous situations, as shown also in Fig.~\ref{fig:system_structure}.
\subsection{Hybrid Markov Logic Networks}
MLNs combine both probabilistic and logical reasoning~\cite{Richardson2006}.
Briefly, a MLN consists of a set of weighted first-order logic formulas or clauses.
The latter include the following elements:
\begin{itemize}
\item \emph{constants}, which are possible objects in the domain of interest;
\item \emph{variables}, describing a set of objects in that domain;
\item \emph{functions}, mapping relations between different objects;
\item \emph{predicates}, defining logical attributes or relationships over the domain's elements, which can be combined into more complex formulas using logical connectors.
\end{itemize}
Functions, variables and constants are called \emph{terms}.
If they do not contain variables, they are \emph{ground terms}.
A predicate that contains only ground terms is a \emph{ground predicate}.
When a logical value is assigned to all grounded predicates in a network, we have a \textit{possible world}.
Using evidences, MLNs can produce Markov networks
that describe the probability of all possible combinations of grounded clauses.
We can then perform inference on these Markov networks, usually by using approximate methods such as MC-SAT~\cite{Wang2008}.
Besides discrete evidence value, it is also possible to consider continuous ones using an extension called Hybrid Markov Logic Network (HMLN)~\cite{Wang2008}.
Thanks to the latter, we can thus consider predicates based on continuous variables that contain our entropy values of the activity levels.
\subsection{Wavelet Model as Prior for HMLN}
The wavelet-based sensor data model defined in Sec.~\ref{sec:sensor_model} can be used to predict the expected output of a particular sensor based on historical data.
From the expected output of all the sensors, it is also possible to compute the normalized entropy $\hat{H}_W$ that represents the {\em expected activity level} for the whole environment (see Fig.~\ref{fig:system_structure}).
The entropy $\hat{H}$ from all the real sensors represents instead the current activity level.
These two activity levels, current and expected, are compared by the following HMLN to determine whether an anomalous situation is occurring.
We define two clauses to combine our sources of information: one to check whether the current entropy is above a certain threshold, and the other to compare current versus expected entropy.
The occurrence of one or both conditions indicates a potentially anomalous situation at time $t_i$, captured by the predicate ${\tt IsStatisticAnomaly}$:
\begin{equation}
\hat{H}(t_i) \geq \hat{H}^* \Rightarrow {\tt IsStatisticalAnomaly}(t_i) \nonumber\\
\end{equation}
\begin{equation} \label{eq:stat_anomaly}
\hat{H}(t_i) > \hat{H}_W(t_i) \Rightarrow {\tt IsStatisticalAnomaly}(t_i)~.
\end{equation}
Here $\hat{H}^*$ is the 90\% of $\hat{H}$. This threshold was first suggested in~\cite{Goldberg2015} as a statistically meaningful indicator of anomaly.
The predicate in~(\ref{eq:stat_anomaly}) and its clauses are represented by the blue connected nodes in Fig.~\ref{fig:mln},
which shows the graph of a grounded HMLN at time~$t_i$.
An advantage of MLNs is that they can combine different logical rules.
This allows us to include additional expert rules that describe ``inappropriate behaviours''.
For AAL applications, such rules could be provided by clinicians or professional carers and adapted to the specific person being monitored.
For example, typical behaviours that are cause for concern in case of people with cognitive impairments include wandering and repetitive actions~\cite{Cubit07}.
In our system these can be monitored by means of motion detector and contact sensors on doors and appliances.
Their outputs determine the state of the predicate ${\tt IsActionAnomaly}$, which is implemented in our HMLN as follows (see also yellow nodes in Fig.~\ref{fig:mln}):
\begin{equation}
{\tt TimeActive}(t_i, Door) > t_0 \Rightarrow {\tt IsActionAnomaly}(t_i) \nonumber \\
\end{equation}
\begin{equation} \label{eq:action_anomaly}
{\tt IsActive}(t_i, Motion) \wedge (t_i\!\subset\!T_{rest})\!\Rightarrow\!{\tt IsActionAnomaly}(t_i)
\end{equation}
where $Door$ is a contact sensor, $Motion$ is a motion detector, $t_0$ is the minimum time of a door left open for considering it an anomaly, and $T_{rest}$ is the resting time interval suggested by some human expert (e.g. 11:00~P.M. to 7:00~A.M.).
The two types of anomaly are finally combined by the following ${\tt IsAnomaly}$ predicate (central node in Fig.~\ref{fig:mln}):
\begin{eqnarray} \label{eq:anomaly}
{\tt IsStatisticAnomaly}(t_i) \vee {\tt IsActionAnomaly}(t_i) \nonumber\\
\Rightarrow {\tt IsAnomaly}(t_i)~.
\end{eqnarray}
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{mln-graph.pdf}
\caption{Grounded Markov network from the predicates in~(\ref{eq:stat_anomaly}), (\ref{eq:action_anomaly}) and~(\ref{eq:anomaly}). The blue nodes capture statistical differences between current and expected activity levels. The yellow nodes instead implement ad-hoc expert rules. }
\label{fig:mln}
\end{figure}
In Sec.~\ref{sec:results:anomaly} we will evaluate the anomaly detection with and without the contribution of the expert rules in~(\ref{eq:action_anomaly}) to better understand the contribution of the wavelet- and entropy-based methods.
\section{System Implementation}\label{sec:arch}
The solutions described in the previous sections have been implemented in ENRICHME\footnote{\url{http://www.enrichme.eu}}, a research project integrating ambient intelligence and robotics to provide AAL services for elderly people wit mild cognitive impairments~\cite{Bellotto17}.
The ENRICHME system monitors the activity of these people at home, exchanging information between a network of smart-home sensors, a mobile robot and an auxiliary Ambient Intelligence Server~(AIS) (see Fig.~\ref{fig:data_sources}).
The latter consists of an embedded PC, located at home, which acts as a multiprotocol gateway, collecting and forwarding the information shared wirelessly between robot and smart-home sensors for monitoring human motion, doors/cupboards use, and energy consumption.
The sensor network is based on the Z-Wave communication protocol and uses the OpenHAB middleware\footnote{\url{http://www.openhab.org}}, which supports a wide range of different smart-home technologies with a uniform interface,
decoupling sensor information from specific smart-home protocols and manufacturers~\cite{Smirek2014}.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{global-setup.pdf}
\caption{Smart-home sensors integration in ENRICHME.\label{fig:data_sources}}
\end{figure}
The embedded PC for data recording and processing is an Intel NUC i7-5557U~CPU @~3.10GHz with 8~GB of RAM, running Linux OS Ubuntu 14.04 64~bits (see Fig.~\ref{fig:sensor_2:nuc}).
The smart-home sensors are commercial Z-Wave wireless devices produced by the Fibar Group\footnote{\url{http://www.fibaro.com}} (see Fig.~\ref{fig:sensor_2:sensors}).
These sensors are small, easily deployable, widely available and have a long battery life.
\begin{figure}
\begin{subfigure}{\columnwidth}
\centering
\includegraphics[width=0.5\columnwidth]{big_nuc.png}
\caption{AIS computer.\label{fig:sensor_2:nuc}}
\end{subfigure}
\begin{subfigure}{\columnwidth}
\centering
\includegraphics[width=0.2\columnwidth]{big_presence.png}
\includegraphics[width=0.2\columnwidth]{plug.png}
\includegraphics[width=0.2\columnwidth,angle=90]{big_door.png}
\caption{Domotic sensors.\label{fig:sensor_2:sensors}}
\end{subfigure}
\caption{Smart home devices in ENRICHME.\label{fig:sensor_2}}
\end{figure}
The anomaly detection system is implemented as a Robot Operating System\footnote{\url{http://www.ros.org}}~(ROS) module making use of efficient MLN libraries for online inference~\cite{Fernandez16}.
ROS provides a common framework for information exchange between AIS and robot, so that the latter can easily access the results of the HLMN inference engine.
The HMLN can be queried using evidence provided by any ROS source, including the actual and expected house entropies obtained from the sensors and the wavelet models, respectively.
The output of the inference process is also available to any other node on the ROS network, for example to trigger a specific robot behaviour or alert a remote telecare system.
\section{Experiments}\label{sec:experiments}
The performance of our proposed solutions were evaluated using real data recorded from different scenarios.
In this sections, we will first describe two different datasets: one already presented in~\cite{Fernandez-Carmona2017} and one newly recorded.
Then, we will use them to evaluate the forecasting capabilities of our wavelet sensor model compared to another similar tool in the literature.
Based on these wavelet models, we will calculate the expected entropy levels of the testing environments and finally demonstrate their use as priors for anomaly detection.
\subsection{Sensor Datasets}\label{sec:results:dataset}
All the datasets were recorded using MongoDB, an open-source cross-platform document-oriented database.
MongoDB is a NoSQL database program, using JSON-like documents with schemas.
Compared to traditional log and spreadsheet files, this storage approach offers better data management and manipulation, which is particularly important for long-term datasets like ours.
MongoDB provides also efficient and flexible querying methods, so we can easily retrieve any data interval, sensor set, or even combine data from other sources~\cite{Niemueller2012}.
The first dataset was collected in an office environment (L-CAS dataset~\cite{Fernandez-Carmona2017}) including: a lounge with sofas and a coffee table; a kitchenette with various appliances and cupboards for storing and preparing food; an entrance and a workshop area.
This dataset contains data from ten different physical devices, which provided six different types of sensor data readings:
humidity, temperature, light, energy consumption, motion, and binary contact (for door activation).
The sensors were located in five different locations, and their data recorded every 30~seconds, generating more than 400,000 data entries in total.
More than ten people were working in the L-CAS premises during the recording.
The sensors were mostly concentrated in places where a rich set of activities were typically performed (entering, exiting, eating, drinking, resting, etc.).
Fig.~\ref{fig:setup_lcas} illustrates our sensors' deployment and approximate area coverage.
The dataset is split in two parts:
the first one, used for training, includes sensor data continuously recorded for three months and a half; the second one includes one week of data used for testing.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{dataSet_red.pdf}
\caption{L-CAS dataset environment.\label{fig:setup_lcas}}
\end{figure}
\begin{figure}
\centering
\begin{subfigure}{\columnwidth}
\includegraphics[width=\columnwidth]{LACE-sensors.pdf}
\caption{LACE apartment's layout.\\~\\\label{fig:lace_layout}}
\end{subfigure}
\begin{subfigure}{\columnwidth}
\includegraphics[width=\columnwidth]{livingroom_perspective.jpg}
\caption{Living room of the LACE apartment.\label{fig:lace_flat}}
\end{subfigure}
\caption{ENRICHME dataset environment.\label{fig:setup_lace}}
\end{figure}
{\renewcommand{\arraystretch}{1.2
\setlength\tabcolsep{1.5pt}
\begin{table*}
\centering
\scriptsize
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
Dataset &Sensors &Locations &\parbox{1.2cm}{\centering Total num.\\ entries} &\parbox{1.5cm}{\centering Data\\ types} &People in dataset & Duration (days) \\ \hline
\multirow{2}{*}{L-CAS} &\multirow{2}{*}{ \parbox{4.7cm}{ \centering Motion, Binary Contact, Humidity, Light, Energy Consumption, Temperature }} &\multirow{2}{*}{\parbox{4.2cm}{ \centering Entrance, Fridge, Kitchen, Lounge, Workshop } } &\multirow{2}{*}{492,441} &\multirow{2}{*}{\parbox{1.5cm}{\centering Binary, Float, Integer}} &\multirow{2}{*}{12} &{104} \\ \cline{7-7}
& & & & & &{7} \\ \hline
\multirow{2}{*}{ ENRICHME} &\multirow{2}{*}{ \parbox{4.7cm}{\centering Motion, Binary Contact, Light, Energy Consumption, Temperature } } &\multirow{2}{*}{\parbox{4.2cm}{ \centering Entrance, Fridge, Kitchen, Bathroom, Bedroom, Livingroom, TV}}&\multirow{2}{*}{33,838} &\multirow{2}{*}{\parbox{1.5cm}{\centering Binary, Float, Integer}} &\multirow{2}{*}{2 } &\multirow{2}{*}{31} \\
& & & & & & \\ \hline
\end{tabular}
\\~\\
\caption{Dataset entries summary.\label{tab:datasets}}
\end{table*}
\setlength\tabcolsep{6pt}
}
The new dataset was recorded in the apartment of an elderly couple within the residential facilities of LACE Housing\footnote{\url{http://lacehousing.org}} as part of the ENRICHME project.
It contains one month of sensor data with five types of readings (temperature, light, energy consumption, motion and door activation), corresponding to approximately 33,000 entries in total. %
The sensors covered most of the apartment area, recording data from the entrance, the kitchen, the living room, the main bedroom and the bathroom.
Fig.~\ref{fig:setup_lace} illustrates the approximate sensors' position and area coverage.
The first three weeks of the dataset were used for training, while the last week for testing.
Table~\ref{tab:datasets} summarizes the locations, the sensors and the general characteristics of the recorded datasets.
\footnote{Datasets are publicly available at LCAS website: ENRICHME \url{https://lcas.lincoln.ac.uk/wp/lace-house-domotic-sensors-dataset/} and LCAS
\url{https://lcas.lincoln.ac.uk/wp/research/data-sets-software/l-cas-domotic-sensors-dataset/} }
\subsection{Performance of Wavelet-based Models}\label{sec:results:wavelets}
In the following subsections, we first present an empirical method to select the best parameters of our wavelet-based sensor model (Sec.~\ref{sec:sensor_model}), and then use the latter to predict the expected sensor output in our datasets.
Our wavelet-based sensor modelling system is available \footnote{\url{https://github.com/LCAS/wtfacts}} as a ROS action server.
This sofware allows creation, management and querying multiple binary models.
\subsubsection{Model Parameters Selection} \label{sec:results:wavelets:modelling}
A key step for the compact representation of sensor data with our new model is the selection of the mother wavelet.
There are several methods to do this,
but in general it is common practice to choose a wavelet that better describes a signal through minimization of a given parameter.
Here we propose to minimizes the RMSE of the reconstructed signal.
We tested a set of wavelets from four different families, analysing different motion detection sequences as input signals from the sensors used to estimate the activity level.
We compared the Daubechies, Haar, Biorthogonal and Reverse Biorthogonal wavelet families by accurately reconstructing signals with a large number of coefficients (i.e. low threshold~$\tau$).
The signals were one-month long sequences from the L-CAS dataset,
transformed using a 1-level DWT decomposition.
We used the smallest coefficient threshold that produced a non perfect reconstruction in all variants.
Among the reconstructed sequences, the ones using the Reverse Biorthogonal family produced the lowest RMSE, when compared to the original signals.
Fig.~\ref{fig:rbio_fam_comparison} shows different RMSE values using wavelets from the Reverse Biorthogonal family.
The best performance was obtained using the \textit{rbio3.1} wavelet, with small differences from other wavelets of the same family (i.e. $0.25\% < $~RMSE~$ < 0.46\%$).
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{RMSEinRBIOfamily.pdf}
\caption{Minimum RMSE of reconstructed signals using different Reverse Biorthogonal wavelets.\label{fig:rbio_fam_comparison}}
\end{figure}
After selecting the mother wavelet, the second step is to choose a threshold level for the coefficients.
As anticipated in Sec.~\ref{sec:param_select}, this threshold determines a subset of meaningful coefficients, which should be as few as possible but also enough to reconstruct the original signal with good approximation.
Because we are dealing with binary sensors, using a subset of coefficients introduces an error in the reconstruction, since the inverse transform generates non-binary values.
We therefore discretize the reconstructed signal into a binary one, and measure the RMSE between the latter and the original signal.
This process can be observed in Fig.~\ref{fig:prediction:fremen}, where the green subplot is the (non-binary) inverse transform of the original signal (red subplot), and the yellow one is the final binary prediction.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{2016-07-18-WW-Lounge-probs.png}
\caption{Example of FreMEn and Wavelet predictions for the Lounge motion detector in the L-CAS dataset.\label{fig:prediction:fremen}}
\end{figure}
Fig.~\ref{fig:rmse_numCoeffs_vs_threshold} illustrates the trade-off between the fidelity of our wavelet model representation (in terms of RMSE) and its size (as number of coefficients) for the Entrance motion detector in the L-CAS dataset.
The blue line shows the decreasing number of coefficients as the threshold increases.
The red line shows instead the increasing reconstruction error for the same threshold increase.
We therefore chose the lowest threshold ($\tau = 0.54$) across all the wavelet sensor models, allowing full reconstruction of all the signals in the training dataset.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{CoefsMSE-Rbio31.pdf}
\caption{Coefficient thresholding effect on the signal reconstruction error for a particular wavelet and sensor. The abscissa is in logarithmic scale.\label{fig:rmse_numCoeffs_vs_threshold}}
\end{figure}
We can thus reconstruct these signals using a small subset of coefficients and the inverse transform, discretized into binary values to obtain the original sensor output.
Our wavelet-based model $\mathcal{M}$ in (\ref{eq:model}) is described therefore by the subset of non-thresholded coefficients {$\hat{C}$},
the mother wavelet {$\phi =$ \textit{rbio3.1}},
the decomposition level {$Q = 1$},
the threshold {$\tau = 0.54$},
the number of samples {$N = 89200$},
the sampling frequency {$f_s = 1/30$~Hz.} and
the time reference {$t_0 = 1510012800$~s} (POSIX time).
\subsubsection{Model Training and Forecasting}\label{sec:results:wavelets:forecasting}
We divided our datasets (see Table~\ref{tab:datasets}) into two folds: one for training and one for testing the prediction.
In the L-CAS dataset, we used the first three months of sensor data for training and then one week for testing.
The ENRICHME dataset had a smaller number of entries, so we used three weeks for training and one week for testing.
In order to evaluate the prediction quality of our wavelet sensor model, we compared it to another tool called Frequency Map Enhancement (FreMEn)~\cite{fremen14}, which was originally developed for robotics applications but then applied also to smart-home sensors~\cite{coppola16}.
FreMEn is a method that allows to model periodic changes of the environment using Fourier-based spectral analysis.
It considers the probability of the environment's state to be a function of time, represented by a (compressed) combination of harmonic components.
The problem of Fourier-based methods though is that they are usually not suitable for describing sparse (i.e. non periodic) or very short events, at least not without considering a very large number of harmonics, which is impractical for many applications.
In these experiment we aim to demonstrate how the wavelet-based model overcomes some of those limitations in delivering more reliable sensor predictions.
To start with, Tab.~\ref{tab:stats_LCAS} presents some statistics of the predictions in the L-CAS dataset.
For all the considered metrics, we can see that our new wavelet model clearly outperforms the frequency-based one. In particular, the wavelet model performs much better in terms of accuracy.
Tab.~\ref{tab:stats_ENRICHME} presents also some results on the ENRICHME dataset.
In this case, the precision of FreMEn is slightly higher than our wavelet model, probably due to the periodic nature of the activities in the considered scenario.
FreMEn indeed captures all the most relevant frequency components, so the predicted activations can be very precise (i.e. high number of true positives).
However, for the recall, which considers the correct predictions over the total number of real activations, we can observe a significant improvement of the wavelet models compared to FreMEn, since the latter is not able to predict some of the sensor activations.
This improvement is further confirmed by the F1 score and the accuracy, also shown in the same table.
{
\renewcommand{\arraystretch}{1.4}
\setlength\tabcolsep{1.5pt}
\begin{table}
\centering
\scriptsize
\begin{tabular}{|c|c|c|c|c|c|c|}
\cline{3-7}
\multicolumn{2}{c|}{}&\multicolumn{4}{c|}{ Presence detectors} & \multirow{3}{*}{\makecell[cc]{Sensor \\Average}}\\
\cline{1-6}
\multicolumn{2}{|c|}{\makecell[cc]{Sensor \\Location}}&Entry&Kitchen&Lounge&Workshop& \\ \hline
\rowcolor{Gray}
\multirow{2}{*}{ \cellcolor{White} \makecell[cc]{ \cellcolor{White} Precision \\ \cellcolor{White} (\%)}}&W&\textbf{63.0}& \textbf{65.3}& \textbf{61.0}& \textbf{53.5} & \textbf{63.3} \\ \cline{2-7}
&F& 48.4& 50.3& 44.9& 37.9 & 46.3 \\ \hline
\rowcolor{Gray}
\multirow{2}{*}{ \cellcolor{White} \makecell[cc]{ \cellcolor{White} Recall \\ \cellcolor{White} (\%)}}&W& 57.1& \textbf{69.8}& \textbf{62.4}& \textbf{51.5}& \textbf{63.0} \\ \cline{2-7}
&F&\textbf{70.8}& 66.5& 42.1& 47.7 & 60.6 \\ \hline
\rowcolor{Gray}
\multirow{2}{*}{ \cellcolor{White} \makecell[cc]{ \cellcolor{White} Accuracy \\ \cellcolor{White} (\%)}}&W&\textbf{88.3}&\textbf{80.2}&\textbf{87.4}&\textbf{97.6} & \textbf{88.4} \\ \cline{2-7}
&F& 59.8& 58.1& 70.5& 72.1 & 65.1 \\ \hline
\rowcolor{Gray}
\multirow{2}{*}{ \cellcolor{White} \makecell[cc]{ \cellcolor{White} F1 score\\ \cellcolor{White} (\%)}}&W&\textbf{59.9}&\textbf{67.5}&\textbf{61.7}&\textbf{52.5} & \textbf{63.2} \\ \cline{2-7}
&F& 57.5& 57.2& 43.4& 42.2 & 52.5 \\ \hline
\end{tabular}
\\~\\
\caption{Comparison between predictions from Wavelet (W) and FreMEn (F) models in the L-CAS dataset.
\label{tab:stats_LCAS}}
\end{table}
}
{
\renewcommand{\arraystretch}{1.4}
\setlength\tabcolsep{1.5pt}
\setlength\tabcolsep{1.5pt}
\begin{table}
\centering
\scriptsize
\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
\cline{3-9}
\multicolumn{2}{c|}{}&\multicolumn{4}{c|}{\makecell[cc]{Presence detectors}}&\multicolumn{2}{c|}{\makecell[cc]{Door sensors}} & \multirow{3}{*}{\makecell[cc]{Sensor \\Average}} \\ \cline{1-8}
\multicolumn{2}{|c|}{\makecell[cc]{Sensor \\Location}}&Bathroom &Bedroom &Kitchen &Living room &Entrance &Fridge & \\ \hline
\rowcolor{Gray}
\multirow{2}{*}{ \cellcolor{White} \makecell[cc]{ \cellcolor{White} Pre.\\ \cellcolor{White} (\%)}}&W&94.0 & 93.9& 89.8& 87.3 &99.6 & 99.6& 94.0\\ \cline{2-9}
&F&\textbf{95.8}&\textbf{95.6}&\textbf{92.8}&\textbf{89.5}&99.7&99.6& \textbf{95.5}\\ \hline
\rowcolor{Gray}
\multirow{2}{*}{ \cellcolor{White} \makecell[cc]{ \cellcolor{White} Rec.\\ \cellcolor{White} (\%)}}&W&\textbf{93.3}&\textbf{93.8}&\textbf{90.1}&\textbf{87.3}&\textbf{99.6}&\textbf{99.4}& \textbf{93.9}\\ \cline{2-9}
&F& 90.8& 84.2& 66.5& 59.0& 81.3& 83.1& 77.5\\ \hline
\rowcolor{Gray}
\multirow{2}{*}{ \cellcolor{White} \makecell[cc]{ \cellcolor{White} Acc.\\ \cellcolor{White} (\%)}}&W& \textbf{88.1}& \textbf{88.6}& \textbf{82.1}& \textbf{77.9}& \textbf{99.3}& \textbf{99.1}& \textbf{89.2}\\ \cline{2-9}
&F& 87.7& 81.5& 65.5& 58.4& 81.1& 82.9& 76.2\\ \hline
\rowcolor{Gray}
\multirow{2}{*}{ \cellcolor{White} \makecell[cc]{ \cellcolor{White} F1\\ \cellcolor{White} (\%)}}&W& \textbf{93.6}& \textbf{93.9}& \textbf{90.0}& \textbf{87.3}& \textbf{99.6}& \textbf{99.5}& \textbf{94.0}\\ \cline{2-9}
&F& 93.3& 89.5& 77.5& 71.1& 89.6& 90.6& 85.3\\ \hline
\end{tabular}
\\~\\
\caption{Comparison between predictions from Wavelet (W) and FreMEn (F) models in the ENRICHME dataset.
\label{tab:stats_ENRICHME}}
\end{table}
}
The wavelet model can also capture very short peaks of sensor signal. Fig.~\ref{fig:prediction:fremen} illustrates the real temporal evolution of a sensor (red),
the activation probability and prediction computed by FreMEn (blue and purple, respectively),
the output of our wavelet model (green) and its binarized version (yellow).
Due to the limitations of the frequency-only representation, FreMEn fails to reproduce the original sensor data, whereas our wavelet model provides a reasonably good approximation of it.
The improvement can be further appreciated in Fig.~\ref{fig:models}, where the FreMEn and wavelet models of the same sensor are compared over a week-time period, showing that the average daily activation of the sensor is better predicted by our model.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{WW-Lounge-weekly-probs.png}
\caption{Average activation of the Lounge motion detector (L-CAS dataset) over a week from real sensor data (top), FreMEn model and wavelet model.
\label{fig:models}}
\end{figure}
\subsection{Performance of Activity Representation}\label{sec:results:fremen}
In the following sub-sections we illustrate the performance of our system to represent human activities using the normalized entropy method in Sec.~\ref{sec:entropy} and comparing the expected levels of activity to the actual ones.
\subsubsection{Real vs. Predicted Entropies}
We compared the entropies of human activity predicted by our wavelet model with the actual ones computed on both datasets.
We used three popular metrics to measure the statistical similarity between these two entropies: RMSE, correlation coefficient, and explained variance.
Table~\ref{tab:stats_entropy} illustrates the good performance of our solution in predicting the entropy of human activities, showing better results than a FreMEn-based approach.
We can also see that the entropy predicted by our wavelet model is slightly better for the ENRICHME dataset compared to the L-CAS dataset (i.e. lower RMSE; higher correlation and explained variance).
However, for both cases, our results confirms that real and predicted entropies are reasonably similar and, therefore, that the wavelet-based model is suitable to forecast the level of activity in the environment.
\begin{table}
\begin{tabular}{|c|c|c|c|c|}
\hline
Dataset & Model & \begin{tabular}[c]{@{}c@{}}RMSE\\ (\%)\end{tabular} & \begin{tabular}[c]{@{}c@{}}Correlation\\ (\%)\end{tabular} & \begin{tabular}[c]{@{}c@{}}Explained \\Variance (\%)\end{tabular} \\ \hline
\rowcolor{Gray}
\multirow{2}{*}{ \cellcolor{White} L-CAS} & W & 23.1 & 68.0 & 36.2 \\ \cline{2-5}
& F & 25.7 & 60.5 & 16.2 \\ \hline
\rowcolor{Gray}
\multirow{2}{*}{ \cellcolor{White} ENRICHME} & W & 20.2 & 74.2 & 51.6 \\ \cline{2-5}
& F & 21.1 & 65.2 & 27.6 \\ \hline
\end{tabular}
\\~\\
\caption{Measures of similarity between real and predicted entropies of human activity in the L-CAS and ENRICHME datasets using Wavelet (W) and FreMEn (F) models.
\label{tab:stats_entropy}
}
\end{table}
\subsubsection{Examples of Activity Forecasting}
\begin{figure*}
\centering
\begin{subfigure}[b]{\columnwidth}
\includegraphics[width=\textwidth]{2017-11-02-LH-entropy-signal-predicition.png}
\caption{}
\label{fig:entropy_comparison:lh}
\end{subfigure}
\begin{subfigure}[b]{\columnwidth}
\includegraphics[width=\textwidth]{2016-08-26-WW-entropy-signal-prediction.png}
\caption{}
\label{fig:entropy_comparison:ww}
\end{subfigure}
\caption{Examples of real vs. predicted entropies computed on the ENRICHME (a) and L-CAS (b) datasets during a 5-hours interval. \label{fig:entropy_comparison}} %
\end{figure*}
As explained in Sec.~\ref{sec:entropy}, human activities can be represented by the normalized entropy of the environment. Fig.~\ref{fig:entropy_comparison} illustrates two examples of such entropy calculated from the real sensors and predicted by our wavelet-based model.
In particular, the red graph shows the real normalized house entropy (as percentage) based on the available sensor setups.
The blue graph is the predicted entropy at the same time, using the wavelet models of our sensors.
Fig.~\ref{fig:entropy_comparison:lh} is based on the ENRICHME dataset, collected in the relatively quiet apartment of an elderly couple.
The figure refers to a typical morning of the two residents.
The predicted entropy of their activities differs from the real one of less than 10\%, with only two significant exceptions:
in the morning, around 10:00, the activity's level was higher than expected (about 20\% error between real and predicted entropies); a little later, around 11:30, the real activity's entropy decreased sharply a few minutes after the usual time (still about 20\% error).
These differences between real and predicted data, however, are understandable under normal variations of the resident's schedule, which cannot be predicted by our model.
It is worth to notice that the latter is able to predict a very sharp transition, where the activity's entropy goes from high to no activity at all.
This shows the capability of our system to consider high-frequency elements in its wavelet-based model.
Fig.~\ref{fig:entropy_comparison:ww} refers instead to the activity of a non-typical Friday afternoon in the L-CAS offices.
The real entropy (red) shows that it was a particularly busy day, with a high activity level for most of the time.
However, a significant decrease of the entropy between 18:00 and 19:00, when most of the researchers left the office, is followed by another increase between 19:00 and 20:00, when some people came back.
The activity remained then relatively high for the rest of the evening, which was unusual.
The entropy's prediction (blue) is able to capture several important trends of the activity levels, including
a few small negative peaks between 17:00 and 18:00 hours,
which are probably due to some researchers leaving the office, and the sharp decrease around 18:00 hours, when most of them left.
Our model captures also some of the evening activities and the entropy's increase between 19:00 and 20:00.
Although after this time there a significant difference between real and predicted entropies, due to the unusual presence of people on a Friday night,
the general trends of the activity's entropy are correctly captured by our prediction system.
\subsection{Performance of Anomaly Detection}\label{sec:results:anomaly}
In this final set of experiments we compare our HMLN for anomaly detection (Sec.~\ref{sec:mln} and~\ref{sec:arch}),
which integrates wavelet and entropy-based activity priors, to other existing approaches.
The normalized entropy computed by our system can be used indeed as a time series for unsupervised anomaly detection.
Here we evaluate our HMLN-based anomaly detector against two state-of-the-art unsupervised methods from a previous statistical framework~\cite{Fairbanks2013}.
We consider in particular the following anomaly detectors\footnote{See the open-source framework for real-time anomaly detection -- \url{https://github.com/MentatInnovations/datastream.io}}:
\begin{itemize}
\item Gaussian1D -- A frequentist anomaly detection method that assumes the intput data is gaussian, searching for low likelihood values.
\item LOFEstimator -- This method relies on local deviations of the density of a given sample with respect to its neighbors. It is local in the sense that the anomaly score depends on how isolated the object is from the surrounding neighborhood.
\end{itemize}
To compare our HLMN to the above methods, we first count the number of anomalies that each detector has in common with the other two.
The results are summarized in Table~\ref{tab:stats_anom:LCAS} and~\ref{tab:stats_anom:ENRICHME} for the L-CAS and ENRICHME datasets, respectively.
For a fair comparison, the tables include also a variant of our method (HMLN*) that does not implement any expert rule, but considers only statistical anomalies based on activity entropy.
We can see that all the anomalies reported by the HMLN* with no rules are also reported by the original HMLN, but not the opposite, as expected.
The results show also that our HMLN approach shares a significant number of detections with the other two statistical methods.
In particular, our solutions enable a more balanced detector that captures a reasonable number of anomalies from both Gaussian1D and LOFEstimator.
\begin{table}
\centering
\scriptsize
\begin{tabular}{|p{1.3cm}|c|c|c|p{0.8cm}|}
\cline{2-5}
\multicolumn{1}{c|}{} & Gaussian1D & LOFDetector & HMLN & HMLN* \\ \hline
Gaussian1D & 100 & 14.3 & 8.4 & 8.1 \\ \hline
LOFDetector & 2.1 & 100 & 19.5 & 19.2 \\ \hline
HMLN & 2.0 & 31.3 & 100 & 98.8 \\ \hline
HMLN* & 1.9 & 31.3 & 100 & 100 \\ \hline
\end{tabular}
\\~\\
\caption{Percentage of anomalies detected by a particular method (row) that are also detected by another one (column) in the L-CAS dataset.
\label{tab:stats_anom:LCAS}}
\end{table}
\begin{table}
\centering
\scriptsize
\begin{tabular}{|p{1.3cm}|c|c|c|p{0.8cm}|}
\cline{2-5}
\multicolumn{1}{c|}{} & Gaussian1D & LOFDetector & HMLN & HMLN* \\ \hline
Gaussian1D & 100 & 25.0 & 77.4 & 64.8 \\ \hline
LOFDetector & 14.0 & 100 & 20.0 & 16.7 \\ \hline
HMLN & 38 & 17.5 & 100 & 84.0 \\ \hline
HMLN* & 37.8 & 17.4 & 100 & 100 \\ \hline
\end{tabular}
\\~\\
\caption{Percentage of anomalies detected by a particular method (row) that are also detected by another one (column) in the ENRICHME dataset.
\label{tab:stats_anom:ENRICHME}}
\end{table}
\begin{figure}
\centering
\begin{subfigure}[b]{0.9\columnwidth}
\includegraphics[width=\textwidth]{F1_HMLNx.pdf}
\caption{}
\label{fig:stats_anom:GREC_HMLNx}
\end{subfigure}
\begin{subfigure}[b]{0.9\columnwidth}
\includegraphics[width=\textwidth]{F1_HMLN.pdf}
\caption{}
\label{fig:stats_anom:GREC_HMLN}
\end{subfigure}
\caption{F1 score comparison for the considered methods using our HMLN anomaly detector (a)~without and~(b) with expert rules.\label{fig:stats_anom:GREC}}
\end{figure}
To identify the best one among these detection systems, but lacking a consistent and reliable annotation of true anomalies,
we used the method proposed by Lamiroy \& Sun~\cite{Lamiroy2011} to estimate precision and recall, and from these compute the F1 score. Although not accurate in absolute terms, this approach has been shown to be useful for ranking different binary classifiers in absence of ground-truth.
Fig.~\ref{fig:stats_anom:GREC} summarizes our results for the two datasets.
In particular, if no expert rules are considered (HMLN*, Fig.~\ref{fig:stats_anom:GREC_HMLNx}), our approach performs always better than the other two methods.
If the rules are taken into account though (HMLN, Fig.~\ref{fig:stats_anom:GREC_HMLN}), the relative performance of our anomaly detector increases for the ENRICHME dataset, but decreases for the L-CAS one. The reason of such change is that our expert rules were specifically designed for the AAL scenarios in the former dataset. This shows indeed that it is possible to 'tune' the sensibility of our anomaly detection system in case additional expert knowledge is available, which is a desired feature in many applications.
\section{Conclusions and Future Work}\label{sec:conclusions}
This paper presented a new approach for wavelet-based temporal modelling of smart binary sensors, which we used to forecast levels of human activity in dynamic indoor environments.
We also proposed an original application of HMLNs combining real and predicted entropies of human activity with expert rules to detect potential anomalies.
Our solutions have been evaluated using two large public datasets, one of which newly collected from a real elderly home, to demonstrate their effectiveness.
Although the proposed wavelet temporal model can be applied to any arbitrary signal,
our current implementation focused only on binary sensor data, partly because it simplifies the subsequent entropy-based representation of human activities.
It remains to be studied how analog smart sensors (e.g.~light, temperature) can also be integrated and exploited by our system.
Finally, despite the flexibility of HMLNs, there are still limitations in the way logic rules are formulated and their weights learned, which requires particular attention and fine tuning to guarantee the convergence of the training process. Also, the time required by the latter grows exponentially with the number and complexity of the rules, which can be a problem in case a richer spectrum of human activities and sensor data is considered.
Possible alternatives combining deep neural networks and symbolic representations, like Logic Tensor Networks~\cite{Serafini2016}, could potentially overcome some of these problems and enable more powerful inference systems for anomaly detection.
\section*{Acknowledgment}
The research leading to these results has received funding from the EC H2020 Programme under grant agreement No.~643691, ENRICHME.
\ifCLASSOPTIONcaptionsoff
\newpage
\fi
\bibliographystyle{IEEEtran}
|
1,314,259,993,658 | arxiv | \section{The weak law of the excluded middle}
Let ${\sf IPC}$ denote the intuitionistic propositional calculus.
The weak law of the excluded middle ({w.l.e.m.}\ for short) is the principle
\begin{equation} \label{wlem}
\neg p \vee \neg\neg p.
\end{equation}
We view this as an axiom {\em schema\/}, in which we can substitute any
formula for the variable $p$.
Consider the logic ${\sf IPC} + \neg p \vee \neg\neg p$, that is, the
closure under deductions and substitutions of ${\sf IPC}$ and the {w.l.e.m.}\
The logic ${\sf IPC} + \neg p \vee \neg\neg p$ has been studied
extensively, and is known in the literature under various names.
It has been called
\begin{itemize}
\item {\em the logic of the weak law of the excluded middle\/} by Jankov,
\item {\em Jankov logic\/} by various Russian authors,
\item {\em De Morgan logic\/} by various American authors,
\item {\em testability logic\/} by some others, and
\item ${\sf KC}$ by still many others.
\end{itemize}
The term {\em principle of testability\/} for $\neg p \vee \neg\neg p$ goes
back to Brouwer himself.
In \cite[p80]{Brouwer1964} he writes (our comment in brackets):
\begin{quote}
``Another corollary of the simple principle of the
excluded third [i.e.\ $\tau\vee\neg \tau$] is the
{\em simple principle of testability},
saying that every assignment $\tau$
of a property to a mathematical entity can be {\em tested\/},
i.e.\ proved to be either non-contradictory [$\neg\neg\tau$]
or absurd [$\neg\tau$].''
\end{quote}
Apparently the name ${\sf KC}$ derives from
Dummett and Lemmon \cite{DummettLemmon}, who used $\sf LC$ to denote the
``linear calculus'', and K alphabetically follows~L,
hence~${\sf KC}$.
In this paper we will study the following sequence $\{\varphi_{k}\}_{k \ge 1}$
of formulas generalizing the {w.l.e.m.}:
\begin{definition}
Let $\varphi_1 = \neg p \vee \neg\neg p$, and
for every $k>1$ define
\begin{equation} \label{eq:phik}
\varphi_k =
\bigvee_{i\neq j} \big(\neg p_i \rightarrow \neg p_j\big) \vee
\neg(\neg p_1\wedge\ldots \wedge \neg p_k)
\end{equation}
(where $1 \le i,j \le k$).
\end{definition}
Notice that the formula $\varphi_1$ can be seen as a special case of $\varphi_k$:
indeed, $\varphi_k$ is equivalent over ${\sf IPC}$ to
\begin{equation}
\neg p_1 \vee \ldots \vee \neg p_k \vee
\bigvee_{i\neq j} \big(\neg p_i \rightarrow \neg p_j\big) \vee
\neg(\neg p_1\wedge\ldots \wedge \neg p_k)
\end{equation}
because $\neg p_i$ implies $\neg p_j \rightarrow \neg p_i$ in ${\sf IPC}$.
Then $\varphi_1$ is the special case $k=1$.
Also note that ${\sf IPC}$ proves $\varphi_k \rightarrow \varphi_{k+1}$ for every $k\geq 1$.
This follows for example from Theorem~\ref{thm:char}, or from
Theorem~\ref{thm:algebraic} below.
Below, we will study the logics ${\sf IPC} + \varphi_k$, which again is the
deductive closure of ${\sf IPC}$ and the axiom schema $\varphi_k$.
In particular ${\sf IPC} + \varphi_k$ proves any substitution instance of~$\varphi_k$.
\section{Kripke semantics}
In this section we characterize the formulas $\varphi_k$
in \eqref{eq:phik} in terms of Kripke frames,
and relate them to a class of formulas introduced by
Smorynski~\cite{Smorynski}.
We briefly recall some elementary notions about Kripke semantics.
For unexplained terminology about Kripke frames and models we refer
the reader to~\cite{ChagrovZakharyaschev} or~\cite[p67]{Gabbay}.
A \emph{Kripke frame\/} $\langle K, R \rangle$ is a nonempty set $K$,
partially ordered by an \emph{accessibility relation\/} $R$.
Throughout this paper, we will work with Kripke frames
that have a {\em root\/}, that is,
a least element with respect to~$R$,
though this is not standardly part of the definition.
As usual, we distinguish between models and frames:
A \emph{Kripke model} $\langle K,R,V\rangle$ is a Kripke frame
together with a {\em valuation\/} $V$, that associates with
every variable $p$ a set $V(p)\subseteq K$, such that if
$x\in V(p)$ and $xRy$ then $y\in V(p)$ for every $x$ and~$y$.
Now the forcing relation $x \Vdash\varphi$, with $x\in K$ and $\varphi$
a formula, is defined by
\begin{itemize}
\item $x \Vdash p$ if $x\in V(p)$;
\item $x \Vdash \phi \wedge \psi$ if and only if $x \Vdash \phi $
and $x \Vdash \psi$;
\item $x \Vdash \phi \vee \psi$ if and only if $x \Vdash \phi $
or $x \Vdash \psi$;
\item
$x \Vdash \phi \rightarrow \psi$ if and only if for every $y$ with $x\, R y$, if
$y \Vdash \phi $ then $y \Vdash \psi$;
\item
$x \Vdash \neg \phi$ if and only if there is no $y$ with $x\, R y$ and $y \Vdash \phi $.
\end{itemize}
A formula $\phi$ \emph{holds\/} in a frame $K$, denoted by $K \models \phi$,
if $K \Vdash \phi$ (meaning that $x \Vdash \phi$ for every $x \in K$), for
every valuation $V$ on the frame. A logic ${\sf L}$ is {\em complete with
respect to\/}, or {\em characterizes\/}, a class of frames $\mathcal{K}$ if a formula
is derivable in ${\sf L}$ if and only if it holds on every frame in $\mathcal{K}$.
\begin{definition}
A Kripke frame with accessibility relation $R$ has {\em topwidth\/} $k$ if it
has $k$ maximal nodes $x_1,\ldots,x_k$ such that for every $y\in K$ there is
an $i$ with $y R x_i$
\end{definition}
Following Jankov~\cite{Jankov}, Gabbay~\cite[p67]{Gabbay} showed that the logic
${\sf IPC}+ \neg p \vee \neg\neg p$ is complete with respect
to the class of Kripke frames of topwidth~1.
Smorynski~\cite{Smorynski} introduced, for every $k\ge 1$, the formula
\begin{equation}\label{Smor}
\sigma_{k}=
\bigwedge_{0\leq i < j \leq k} \neg\big(\neg p_i \wedge \neg p_j\big)
\rightarrow
\bigvee_{0 \le i \le k} \Big(\neg p_i \rightarrow \bigvee_{j\neq i} \neg p_j\Big)
\end{equation}
and showed that the logic ${\sf IPC}+\sigma_{k}$ characterizes
the class of Kripke frames of topwidth at most $k$ (henceforth
we refer to this result as Smorynski's Completeness Theorem).
In particular,
${\sf IPC}$ proves that $\sigma_k \rightarrow \sigma_{k+1}$
and ${\sf IPC}+\sigma_{1}$ coincides with the logic of the {w.l.e.m.}.
Note that $\varphi_k$ has $k$ variables and $\sigma_k$ has $k+1$.
The relation between these formulas is sorted out below.
We now turn to a characterization of the formulas $\varphi_k$ in~(\ref{eq:phik})
in terms of Kripke frames. We start with some preliminaries about canonical
models. For more on canonical models we refer to~\cite{ChagrovZakharyaschev}.
The \emph{canonical model\/} $K$ of a logic ${\sf L}$ containing ${\sf IPC}$ consists
of tableaux, that is, pairs $(\Gamma, \Delta)$ of sets of formulas,
satisfying the following properties:\footnote{%
Gabbay~\cite{Gabbay} uses saturated sets of formulas to define the canonical
model, which is similar but different.}
\begin{enumerate}[(i)]
\item \label{Kpr1} $(\Gamma, \Delta)$ is {\em consistent\/} with ${\sf L}$,
meaning that for no $\phi_1, \ldots \varphi_n\in \Delta$, $\Gamma$
proves $\varphi_1\vee\ldots\vee \varphi_n$ over ${\sf L}$,
\item \label{Kpr2}
$(\Gamma, \Delta)$ is {\em maximal\/} in the sense that
$\Gamma\cup\Delta$ is the set of all formulas.
\end{enumerate}
The accessibility relation $R$ in the canonical model is
defined by
$$
(\Gamma,\Delta)\, R \,(\Gamma',\Delta') \Longleftrightarrow
\Gamma\subseteq \Gamma' \Longleftrightarrow \Delta \supseteq \Delta'.
$$
This defines the canonical frame, and to make it into
a model it is defined that every atomic formula in $\Gamma$
is forced in the node $(\Gamma,\Delta)$.
It is a basic property of $K$ that for every node
$(\Gamma, \Delta)$ and every formula $\varphi$,
$$
(\Gamma, \Delta) \Vdash \varphi \Longleftrightarrow \varphi\in\Gamma.
$$
Note that it follows from properties (\ref{Kpr1}) and (\ref{Kpr2})
that $\Gamma$ is closed under ${\sf L}$-provability.
\begin{lemma} \label{lem:compdir1}
Suppose $K$ is a Kripke frame of topwidth $n+1$ in which
$\varphi_k$ does not hold.
Then $\binom{n}{\lfloor n/2\rfloor} \geq k$.
\end{lemma}
\begin{proof}
Under the assumptions,
we prove that the power set $\P(\{1,\ldots,n\})$ has an antichain of
size~$k$. The lemma then follows from Sperner's Theorem,
(\cite{Sperner1928}; cf.\ also \cite{Proofs}) stating
that $\binom{n}{\lfloor
n/2\rfloor}$ is the greatest number $k$ for which there is an antichain
of $k$ pairwise incomparable subsets of $\{1,\ldots, n\}$.
Since there is a model on the frame $K$ that falsifies $\varphi_k$, there must
be a maximal node in which $\neg p_1\wedge\ldots\wedge \neg p_k$ holds. This
leaves $n$ nodes to falsify all implications $\neg p_i \rightarrow \neg p_j$
with $i\neq j$. Label these nodes by $1,\ldots,n$. Let $S_i \subseteq
\{1,\ldots,n\}$ be the set of nodes where $p_i$ holds, with $i = 1,\ldots,
k$. Then the sets $S_i$ form an antichain since for every pair $i\neq j$
there is a node that falsifies $\neg p_i \rightarrow \neg p_j$, hence in
which $p_i$ and $\neg p_j$ hold.
\end{proof}
\begin{lemma} \label{lem:compdir2}
Suppose $(\Gamma_1,\Delta_1),\ldots,(\Gamma_n,\Delta_n)$ are distinct
maximal nodes in the canonical model of ${\sf L}$. Then for every $S\subseteq
\{1,\ldots,n\}$ there is a formula $A$ such that $A \in \Gamma_j$ if and only
if $j \in S$.
\end{lemma}
\begin{proof}
By maximality, the $\Gamma_i$ are pairwise $\subseteq$-incomparable, hence
for every $i \ne j$, there is a formula $A_{i,j}\in \Gamma_i - \Gamma_j$.
Hence, taking, $A_i=\bigwedge_{j \ne i} A_{i,j}$,
for every $i$, it is easy to see that $(\Gamma_i, \Delta_i)\Vdash A_i
\rightarrow \neg A_j$
for every $i\neq j$. Now let $A = \bigvee_{j \in S} A_j$.
\end{proof}
\begin{theorem} \label{thm:char}
${\sf IPC} + \varphi_k$ is complete with respect to
the class of Kripke frames of topwidth at most~$n$,
where $n$ is minimal such that
$$
\binom{n}{\lfloor n/2\rfloor} \geq k.
$$
\end{theorem}
\begin{proof}
For the right-to-left implication,
suppose $K$ is a frame of topwidth $m+1\le n$
in which $\varphi_k$ does not hold. Then by Lemma~\ref{lem:compdir1},
$\binom{m}{\lfloor m/2\rfloor} \geq k$,
hence by minimality of $n$ we have $m\geq n$, a contradiction.
Hence any frame of topwidth $l\leq n$ satisfies $\varphi_k$.
For the converse direction, we have to show that if $\varphi$ is a formula that
${\sf IPC}+\varphi_k$ does not prove, then there is a Kripke frame of topwidth at
most~$n$,
where $n$ and $k$ are related as in the statement of the theorem, in which
$\varphi$ does not hold, i.e.\ there is a model on this frame on which $\varphi$
does not hold. We show that a part of the canonical model of ${\sf IPC}+\varphi_k$
has this property.
Now if $\varphi$ is not provable in ${\sf IPC}+\varphi_k$, then
its negation is consistent, hence $\neg \varphi$ is forced
at some node $t=(\Gamma, \Delta)$ of the canonical model,
and $\varphi$ does not hold in $t$.
Let $K^t$ denote the part of $K$ that is $R$-reachable from $t$.
We prove that $K^t$ has the required property.
First we note that every node in $K$ is below an $R$-maximal
one: every path in $K$ has an upper bound (by taking unions on the
first coordinate and intersections on the second),
hence an application of Zorn's lemma gives a maximal element
above any node in $K$.
We now show that $K^t$ has at most $n$ $R$-maximal nodes.
Suppose for a contradiction that there exist at least $n+1$
distinct maximal nodes
\[
(\Gamma_1,\Delta_1),\ldots,(\Gamma_{n+1},\Delta_{n+1}).
\]
Since $\binom{n}{\lfloor n/2\rfloor} \geq k$ there is an
antichain $S_1,\ldots, S_k$ in $\P(\{1,\ldots,n\})$ of size~$k$.
For every $S_i$, with the help of Lemma~\ref{lem:compdir2}
choose a formula $A_i$ such that
\begin{equation} \label{Ai}
A_i \in \Gamma_j \Longleftrightarrow j\in S_i
\end{equation}
and such that $A_i \notin \Gamma_{n+1}$.
Note that by maximality it follows from \eqref{Ai} that
$$
\neg A_i \in \Gamma_j
\Longleftrightarrow A_i \notin \Gamma_j
\Longleftrightarrow j\notin S_i.
$$
But now we can prove that $\varphi_k$ is not forced in $t$:
First $t \not \Vdash \neg(\neg A_1\wedge\ldots\wedge \neg A_k)$
because
$(\Gamma_{n+1},\Delta_{n+1}) \Vdash \neg A_1\wedge\ldots\wedge\neg A_k$
by choice of $A_i$.
Also $t \not \Vdash \neg A_i \rightarrow \neg A_{i'}$ for
every $i\neq i'$ with $i,i'\leq k$. Namely,
the elements $S_i$ and $S_{i'}$ of the antichain are incomparable,
hence $j\in S_{i'} - S_i$ for some $j\in\{1,\ldots,n\}$.
Thus, by definition of $A_i$, we have
$A_{i'}\in \Gamma_j$ and $\neg A_i \in \Gamma_j$, and hence
$(\Gamma_j,\Delta_j)\Vdash \neg A_i \wedge A_{i'}$.
So we see that $t$ does not force the formula
$\varphi_k(A_1,\ldots,A_k)$ obtained from $\varphi_k$ by
substituting $A_i$ for every variable $p_i$.
But then it follows that $t\not\Vdash \varphi_k$, for if
$t\Vdash \varphi_k$ then $t$ would also force $\varphi_k(A_1,\ldots,A_k)$
because we work over the logic ${\sf IPC}+\varphi_k$,
which by definition proves every substitution instance of~$\varphi_k$.
\end{proof}
A logic ${\sf L}$ is called {\em canonical\/} if every formula of ${\sf L}$ holds in the
canonical frame of ${\sf L}$. Note that the proof of Theorem~\ref{thm:char} shows
that the logics of $\varphi_k$ are canonical in this sense.
Following \cite[p69]{Gabbay}, a condition $F$ on a partially ordered set
$\langle K,R,0 \rangle$ with least element $0$, is \emph{absolute} if it can
be formulated in higher order language (with symbols for $R, 0, =$), and for
every $\langle K,R,0\rangle$ satisfying $F$, there exists a finite $K_0
\subseteq K$ such that for every $K'$, with $K_0 \subseteq K' \subseteq K$,
we have that also $\langle K', R\mbox{\raisebox{.5mm}{$\upharpoonright$}} K', 0\rangle$ satisfies $F$. It is
known, see e.g.~Gabbay~\cite[p69]{Gabbay}, that if ${\sf L}$ is an intermediate
logic which characterizes a class of Kripke frames, consisting of exactly the
frames satisfying an absolute condition $F$, then ${\sf L}$ also characterizes
the class of finite Kripke frames satisfying~$F$. An intermediate logic
${\sf L}$ is said to have the \emph{finite model property}, if for every $\phi$
with $\phi\notin {\sf L}$, there exists a finite Kripke model which does not
satisfy $\phi$. By a classical theorem of Harrop (\cite{Harrop1958}; see also
\cite[p.~266]{Gabbay}), if an intermediate logic ${\sf L}$ has the finite model
property and is finitely axiomatizable, then ${\sf L}$ is decidable. Therefore
we have:
\begin{theorem}\label{thm:fmp}
Each ${\sf IPC} + \phi_k$ is complete with respect to the class of
{\em finite\/} Kripke frames with topwidth at most $n$, where $n$
is least such that $\binom{n}{\lfloor n/2\rfloor} \geq k$.
Moreover, ${\sf IPC} + \phi_k$ is decidable.
\end{theorem}
\begin{proof}
The claim follows by the above quoted remark and the fact the
condition of being a Kripke frame with topwidth at most $n$, and $n$ least
such that $\binom{n}{\lfloor n/2\rfloor} \geq k$, is absolute.
\end{proof}
Finally, we have the following additional characterization of
${\sf IPC}+\varphi_{k}$:
\begin{corollary}\label{cor:characterization}
${\sf IPC}+\phi_{k}={\sf IPC}+\sigma_{n}$,
for all $n$ and $k$ such that $n$ is minimal with
$\binom{n}{\lfloor n/2\rfloor} \geq k$.
\end{corollary}
\begin{proof}
This follows from Theorem~\ref{thm:char} and Smorynski's Completeness Theorem.
\end{proof}
Notice that the sequence of logics ${\sf IPC}+\phi_{k}$ is decreasing, but not
strictly decreasing, with respect to inclusion. Namely, if $k_{1} < k_{2}$
and $n$ is the least such that $\binom{n}{\lfloor n/2\rfloor} \geq k_{1}$,
but $n$ is also the least such that $\binom{n}{\lfloor n/2\rfloor} \geq
k_{2}$, then
\[
{\sf IPC}+\varphi_{k_{1}}={\sf IPC}+\varphi_{k_{2}}={\sf IPC}+\sigma_{n}.
\]
\section{Algebraic semantics} \label{sec:Brouwer}
A \emph{Brouwer algebra} is an algebra $\langle L, +, \times, \rightarrow,
\neg, 0,1\rangle$ where $\langle L, +, \times, 0,1\rangle$ is a bounded
distributive lattice (with $+$ and $\times$ denoting the operations of $\sup$
and $\inf$, respectively)
and $\rightarrow$ is a binary operation satisfying
\begin{equation}\label{eqn:arrow}
b \le a + c \Leftrightarrow a\rightarrow b \le c,
\end{equation}
or, equivalently,
\[
a\rightarrow b= \text{least }\{c: b \le a +c\},
\]
and $\neg$ is the unary operation, given by $\neg a= a \rightarrow 1$. A
Brouwer algebra $L$ \emph{satisfies} a propositional formula $\sigma$
(denoted by $L \models \sigma$) if whatever substitution of elements of $L$
in place of the propositional variables of $\sigma$ (interpreting the
connectives $\lor$, $\wedge$, $\rightarrow$, $\neg$ with the operations
$\times$, $+$, $\rightarrow$, $\neg$, respectively) yields the element~$0$.
(Note that this definition of truth is dual to that in a Heyting algebra; see
also the remarks on Heyting algebras below.) Let
$$
{\rm Th}(L)=\{\sigma: L\models \sigma\}.
$$
It is well known that ${\sf IPC} \subseteq {\rm Th}(L)$, for every Brouwer algebra $L$.
An intermediate logic ${\sf L}$ is \emph{complete with respect to} a class
of Brouwer algebras, if for every formula $\sigma$, ${\sf L}$ derives
$\sigma$ if and only if every algebra in the class satisfies~$\sigma$.
Recall that in a distributive lattice $L$, we have that an element $a \in L$
is join-irreducible if and only if $a \le x + y$ implies $a \le x$ or $a \le
y$, for every $x,y \in L$. Thus if $L$ is a Brouwer algebra, $b \in L$ with
$b=\sum X$, where $X$ consists of join-irreducible elements, then for every
$a \in L$,
\begin{equation}\label{arrow}
a \rightarrow b= \sum \{x \in X: x \not \le a\}:
\end{equation}
This follows from the fact that $b \le a + y$, where $y= \sum \{x \in X: x
\not \le a\}$, and by join-irreducibility of each element of $X$, we have
that $x \le c$ for every $c$ such that $b \le a + c$ and every $x \in X$ such
that $x \nleq a$. Thus $y$ is the least such that $b \le a + y$.
Finally, if $X$ is an antichain of join-irreducible elements
in a distributive lattice, and $I, J \subseteq X$ are finite sets, then
\begin{equation}\label{incomp}
\sum I \le \sum J \Leftrightarrow I \subseteq J.
\end{equation}
Recall the following well-known construction (see \cite{Fitting}) which
associates with every Kripke frame a Brouwer algebra, whose
identities coincide with the formulas that hold in the frame. Let $K$ be a
given Kripke frame, with accessibility relation $R$:
a subset $A \subseteq K$ is \emph{open}, if for every
$x,y\in K$ we have that $x \in A$ and $x R y$ then $y \in A$.
Let ${\sf Op}(A)$ be the collection of open subsets of $K$.
\begin{lemma}[\cite{Fitting}]\label{lem:Fitting}
The distributive lattice ${\sf Alg}(K)=\langle {\sf Op}(K), +, \times, \rightarrow 0,
1\rangle$ is a Brouwer algebra, where $A+B=A\cap B$, $A \times B=A \cup B$,
$A\rightarrow B=\{x\in K: (\forall y \in K)
[x R y \wedge y \in A \Rightarrow y \in B]\}$,
$0=K$, and $1=\emptyset$. Moreover
$$
\{\phi: K \models \phi\}=\{\phi: {\sf Alg}(K)\models \phi\}.
$$
\end{lemma}
\begin{proof}
See \cite{Fitting}. In fact, the theorem in \cite{Fitting} is formulated in
terms of Heyting algebras. Recall that $L$ is a Heyting algebra if the dual
$L^{{\rm op}}$ is a Brouwer algebra. If $L$ is a Heyting algebra, we write $L
\models^H \sigma$, if $L^{{\rm op}} \models \sigma$. In \cite{Fitting} it is shown
that the collection of open sets together with the operations $+=\cup$,
$\times=\cap$, $0=\emptyset$, $1=K$, and
$$
A\rightarrow B=\bigset{x\in K: (\forall y \in K)
[x R y \wedge y \in A \Rightarrow y \in B]},
$$
is a Heyting algebra which satisfies the same formulas as $K$. To prove our
result, given a frame $K$, apply Fitting's theorem to get a Heyting algebra,
and then take its dual: the claim then follows from the obvious fact that the
formulas satisfied (under $\models$) by a Brouwer algebra are the same as the
ones satisfied (under $\models^H$) by its dual Heyting algebra.
\end{proof}
Conversely, given a Brouwer algebra $L$ with meet-irreducible $0$,
let $I(L)$ be the collection of prime ideals of $L$,
which becomes a Kripke frame ${\sf Kr}(L)=\langle I(L), \subseteq \rangle$.
(Note that ${\sf Kr}(L)$ satisfies our assumption that all Kripke frames
have a root, since $0\in L$ is meet-irreducible, so that $\{0\}$ is
a prime ideal.)
\begin{lemma}\cite{Ono-Kripke}\label{lem:Ono}
For every Brouwer algebra $L$, we have
$$
\{\phi: L \models \phi\}\subseteq \{\phi: {\sf Kr}(L) \models \phi\}.
$$
Moreover, equality holds if $L$ is finite.
\end{lemma}
\begin{proof}
See \cite{Ono-Kripke}. Again, a few words may be spent on the proof, since
\cite{Ono-Kripke} uses Heyting algebras instead of Brouwer algebras. So,
suppose we are given a Brouwer algebra $L$, take its dual $L^{{\rm op}}$, which is
a Heyting algebra, and then use \cite{Ono-Kripke} to conclude that $\langle
F(L^{{\rm op}}), \subseteq\rangle$ (where $F(L^{{\rm op}})$ is the collection of prime
filters of $L^{{\rm op}}$) is a Kripke frame $K$ that satisfies $\{\phi: L^{{\rm op}}
\models^H \phi\}\subseteq \{\phi: K \models \phi\}$, with equality if
$L^{{\rm op}}$ is finite. The claim then follows from the fact that $\{\phi: L^{{\rm op}}
\models^H \phi\}=\{\phi: L \models \phi\}$, and $F(L^{{\rm op}})$ is order
isomorphic to $I(L)$ under $\subseteq$, as easily follows from recalling that
in a distributive lattice $L$, for every $X\subseteq L$, $X$ is a prime
filter if and only if $L-X$ is a prime ideal.
\end{proof}
Theorem~\ref{thm:fmp} has the following algebraic counterpart:
\begin{theorem} \label{thm:algebraic}
${\sf IPC} + \varphi_k$ is complete with respect to the class of all finite Brouwer
algebras $L$ with meet-irreducible $0$ and at most $n$ coatoms, where $n$ is
minimal such that $ \binom{n}{\lfloor n/2\rfloor} \geq k$.
\end{theorem}
\begin{proof}
The proof follows from Theorem~\ref{thm:char},
Lemma~\ref{lem:Fitting}, Lemma~\ref{lem:Ono},
together with the following observations:
\begin{enumerate}
\item If $K$ has topwidth $n$, then ${\sf Alg}(K)$ has $n$ coatoms: indeed,
for every maximal element $x$ in the frame, the singleton $\{x\}$ is
open, and this is clearly a coatom in ${\sf Alg}(K)$; moreover the coatoms
in ${\sf Alg}(K)$ are all of this form.
\item If a finite Brouwer algebra $L$ has $n$ coatoms, then ${\sf Kr}(L)$ is
of topwidth $n$:
indeed, in a finite Brouwer algebra $L$, the ideals
generated by the coatoms are prime and contain all other prime
ideals, generated by meet-irreducible elements. In other words the
coatoms correspond exactly to the maximal elements in ${\sf Kr}(L)$.
\end{enumerate}
Finally, notice that, for every Kripke frame $K$,
${\sf Alg}(K)$ has meet-irreducible~$0$, since the Kripke frames
in this paper always have a least element.
\end{proof}
For finite Brouwer algebras, we may also describe the completeness property
in terms of join-irreducible elements joining to the greatest element $1$.
\begin{definition}
For every $n$, let $\mathfrak{B}_n$ denote the class of Brouwer algebras in
which the top element is the join of some antichain of $n$ join-irreducible
elements.
\end{definition}
\noindent
Notice that in any distributive lattice, if $\sum X=\sum Y$, where $X, Y$
are finite antichains of join-irreducible elements, then it follows from
(\ref{incomp}) that $X=Y$. Thus, in a finite distributive lattice $L$, or
more generally in a distributive lattice $L$ having the finite descending
chain condition (see e.g.~\cite[Theorem~III.2.2]{Balbes-Dwinger:Book}) each
element is the join of a unique antichain of join-irreducibles,
and thus $L$ belongs to $\mathfrak{B}_n$, for a unique $n$.
\begin{lemma}\label{lem:coatoms-joinirr}
If $L$ is a finite Brouwer algebra, then $L$ has exactly $n$
coatoms if and only if $L \in \mathfrak{B}_n$.
\end{lemma}
\begin{proof}
Suppose that $L \in \mathfrak{B}_n$ is finite, and let $b_1, \ldots, b_n$ be
the antichain of $n$ join-irreducible elements such that $1=\sum_{i=1}^n
b_i$. For every $i$, let $\hat{b_i}=\sum_{j\ne i}b_j$. We claim that each
$\hat{b_i}$ is a coatom. Indeed $\hat{b_i}<1$, as $b_i \nleq \hat{b_i}$;
moreover, assume that $\hat{b_i} \le b$, and let $b=\sum X$ where $X$ is an
antichain of join-irreducible elements. (Here we use that $L$ is finite.) By
join irreducibility, we have
\[
\{b_j: j \ne i\} \subseteq X \subseteq \{b_j: 1 \le j \le n\}
\]
thus either $\hat{b_i}=b$ or $b=1$. It follows that $L$ has at least $n$
coatoms. On the other hand, suppose that $L$ has also a coatom
$a\notin \{\hat{b_i}: 1\le i \le n\}$. Then for every~$i$,
$\hat{b_i}+ a=1$, thus $b_i \le \hat{b_i}+ a$,
hence by join irreducibility, $b_i \le a$.
This implies that $\sum_i b_i \le a$, hence $a=1$, a contradiction.
Conversely, suppose that $L$ is a finite Brouwer algebra that has $n$
coatoms. Since $L$ is finite, there exists $m$ such that
$L\in\mathfrak{B}_m$. On the other hand, the above argument shows that $m=n$,
so that $L \in \mathfrak{B}_n$.
\end{proof}
Let $\mathfrak{B}_n^\bot$ be the subclass of $\mathfrak{B}_n$, consisting of
the algebras with meet-irreducible $0$. It follows:
\begin{corollary}\label{cor:fin-char-II}
${\sf IPC} + \varphi_k$ is complete with respect to the class of finite
Brouwer algebras $\mathfrak{B}_n^\bot$, where $n$ is minimal such
that $ \binom{n}{\lfloor n/2\rfloor} \geq k$.
\end{corollary}
\begin{proof}
Immediate from Theorem~\ref{thm:algebraic}, and Lemma~\ref{lem:coatoms-joinirr}.
\end{proof}
Finally, we prove Theorem~\ref{thm:validity} below, which holds also of
Brouwer algebras that are not necessarily finite. We need a preliminary
lemma, which illustrates the range of $\neg$ in a Brouwer algebra from
$\mathfrak{B}_n$.
\begin{lemma} \label{lemman}
Let $L\in \mathfrak{B}_n$, and let $b_1, \ldots, b_n$ be an antichain of
join-ir\-red\-uc\-ible elements such that $1=b_1+ \cdots +b_n$. Then every
negation $\neg a$ in $L$ is of the form $\neg a = \Join_{i\in I} b_i$ for
some subset $I\subseteq\{1,\ldots, n\}$ (where, of course, $\neg a=0$ if
$I=\emptyset$).
In particular, $\neg b_{i}= \Join_{j \ne i}b_{j}$.
\end{lemma}
\begin{proof}
By (\ref{arrow}) we have $\neg a= \sum_{i \in I} b_{i}$,
where $I=\{i: b_i \not\leq a\}$.
\end{proof}
\begin{theorem}\label{thm:validity}
Let $\binom{n}{\lfloor n/2\rfloor} = k$. Then the following hold:
\begin{enumerate}[\rm (i)]
\item If $L \in \mathfrak{B}_{m}$ and $m \le n$, then
$L\models \varphi_{k}$;
\item if $L \in \mathfrak{B}_{m}^{\bot}$ and $m >n$ then $L\not\models
\varphi_{k}$.
\end{enumerate}
\end{theorem}
\begin{proof}
(i) Let $k$ and $n$ be as in the statement of the theorem. Let $L \in
\mathfrak{B}_{m}$, $m \le n$, with $b_{1}, \ldots, b_{m}$ join-irreducible
elements that join to $1$. In order to show that $\varphi_k$ holds in $L$, we
take any sequence $a_i$ of $k$ elements in $L$ and show that $\varphi_k$
evaluates to $0$ for $p_i=a_i$. If there are $i\neq j$ such that $\neg a_i$
and $\neg a_j$ are comparable then the first clause of $\varphi_k$ is satisfied.
So suppose that all $\neg a_i$ are pairwise incomparable. We have to show
that then the last clause of $\varphi_k$ is satisfied, i.e.\ that $\neg(\neg a_1
+ \ldots + \neg a_k) = 0$, or equivalently, $\sum_{i=1}^k \neg a_i =1$. By
Lemma~\ref{lemman} every $\neg a$ is of the form $\neg a = \Join_{i\in I}
b_i$. Note that $\Join_{i\in I} b_i \leq \Join_{j\in J} b_j$ if and only if
$I\subseteq J$, as follows from (\ref{incomp}). So to the $k$ incomparable
negations $\neg a_i$ corresponds a collection of $k$ pairwise
$\subseteq$-incomparable subsets of $\{1,\ldots, m\}$. Sperner's Theorem says
that $\binom{m}{\lfloor m/2\rfloor}$ is the maximum number $k$ for which
there is such an antichain of $k$ pairwise incomparable subsets of
$\{1,\ldots, m\}$. Hence because $\binom{m}{\lfloor m/2\rfloor} \leq k$, the
collection corresponding to the $\neg a_i$ covers all of $\{1,\ldots, m\}$,
and in particular
$$
\sum_{i=1}^k \neg a_i = \sum_{i=1}^m b_i =1,
$$
which is what we had to prove.
(ii)
Suppose that $L \in \mathfrak{B}_{m}^{\bot}$, with $m>n$: let
\[
I=\{b_1, \ldots, b_n, b_{n+1}, \ldots, b_{m}\}
\]
be an antichain of join-irreducible elements such that in $L$ we have
$1=\sum_{1 \le i \le m}b_{i}$. By Sperner's Theorem take a collection of $k$
incomparable subsets $\{I_i: 1\le i \le k\}$ of $\{1,\ldots, n\}$. For every
$i=1, \ldots, k$ choose $a_i$ so that $\neg a_i = \Join_{j\in I_i} b_j$. (The
proof of Lemma~\ref{lemman} shows how to achieve this: take $a_i=\sum_{j
\notin I_i}b_j$.) Then the negations $\neg a_i$ are incomparable because the
sets $I_i$ form an antichain, and hence the first clause of $\varphi_k$ is
nonzero (as $0$ is meet-irreducible in $L$). We also have
\[
\sum_{i=1}^{k} \neg a_i = \sum_{\substack{1\le i \le k\\ j\in I_{i}}} b_j \neq 1
\]
(because no $b_{j}$, with $j>n$, is included), hence
$\neg(\Join_{i=1}^k \neg a_i)\neq 0$ and the second clause of $\varphi_k$ is
also nonzero. So $\phi_k$ does not evaluate to $0$ in $L$, since in this
algebra, $0$ is meet-irreducible.
\end{proof}
\section{An application to the Medvedev lattice }\label{sec:addendum}
This section is an addendum to~\cite{SorbiTerwijn}. We thank Paul Shafer
\cite{Shafer} for pointing out some inaccuracies in that paper. In
\cite{SorbiTerwijn} logics of the form ${\rm Th}({\mathfrak M}/\mathbf{A})$ are studied,
where ${\mathfrak M}$ is the Medvedev lattice, $\mathbf{A}\in {\mathfrak M}$, and ${\mathfrak M}/\mathbf{A}$
is the initial segment of ${\mathfrak M}$ consisting of all $\mathbf{B}\in {\mathfrak M}$ such that
$\mathbf{B}\le \mathbf{A}$. The Medvedev lattice arises from the following
reducibility on subsets of $\omega^\omega$ (also called \emph{mass
problems}): if $\mathcal{A}, \mathcal{B}$ are mass problems, then
$\mathcal{A}\le \mathcal{B}$, if there is an oracle Turing machine which,
when given as oracle any function $g \in \mathcal{B}$, computes a function
$f\in \mathcal{A}$. The \emph{Medvedev degrees}, or simply, \emph{M-degrees},
are the equivalence classes of mass problems under the equivalence relation
generated by $\le$. The collection of all M-degrees constitutes a bounded
distributive lattice, called the \emph{Medvedev lattice}, which turns out to
be in fact a Brouwer algebra, i.e.\ it is equipped with a suitable operation
$\rightarrow$, satisfying \eqref{eqn:arrow}. Hence every factor of the form
${\mathfrak M}/\mathbf{A}$ is itself a Brouwer algebra, being closed under
$\rightarrow$, with $\neg$ given by $\neg \mathbf{B}=\mathbf{B} \rightarrow
\mathbf{A}$. In the following we use the notation from~\cite{SorbiTerwijn},
to which the reader is also referred for more details and information about
the Medvedev lattice and intermediate propositional logics.
In order to show that there are infinitely many logics of the form
${\rm Th}({\mathfrak M}/\mathbf{A})$, in~\cite{SorbiTerwijn} a sequence of M-degrees
$\mathbf{B}_n$, $n\in\omega$, is introduced. In Corollary 5.8 of
\cite{SorbiTerwijn} it is claimed that the logics ${\rm Th}({\mathfrak M}/\mathbf{B}_n)$ are
all different but no detailed proof of this is given. Below we prove that
indeed these logics are all different from each other. In particular for any
$f\in\omega^\omega$ consider the mass problem
$$
\mathcal{B}_f = \bigset{g\in\omega^\omega : g\not\leq_T f }:
$$
then the Medvedev degree $\mathbf{B}_f$ of $\mathcal{B}_f$ is join-irreducible,
\cite{Sorbi:Brouwer}.
Recall that the top element $1$ of ${\mathfrak M}/\mathbf{B}_n$ is the join
$$
\mathbf{B}_n=\mathbf{B}_{f_1}+ \ldots + \mathbf{B}_{f_n}
$$
where $\bigset{f_i: i \in \omega}$ is a collection of functions whose Turing
degrees are pairwise incomparable. In particular, the top element of
${\mathfrak M}/\mathbf{B}_1$ is join-irreducible and the top elements of all other
factors ${\mathfrak M}/\mathbf{B}_n$ are not. Hence ${\rm Th}({\mathfrak M}/\mathbf{B}_1)$ can be
distinguished from all the other theories by the formula \eqref{wlem}.
Namely, the {w.l.e.m.}\ holds in a factor ${\mathfrak M}/\mathbf{A}$ if and only if
$\mathbf{A}$ is join-irreducible, cf.\ \cite{Sorbi:Quotient}.
We recall that the least element of $\mathfrak{M}$, and thus of every
factor ${\mathfrak M}/\mathbf{A}$, is meet-irreducible.
Hence ${\mathfrak M}/\mathbf{B}_{n}\in \mathfrak{B}_n^{\bot}$.
(This is in fact enough for the proof below.)
\begin{corollary} \label{cor}
If $m\ne n$ then ${\rm Th}({\mathfrak M}/\mathbf{B}_{m}) \ne {\rm Th}({\mathfrak M}/\mathbf{B}_{n})$.
\end{corollary}
\begin{proof}
Assume $n<m$, and let $k=\binom{n}{\lfloor n/2\rfloor}$. Since
${\mathfrak M}/\mathbf{B}_{n}\in \mathfrak{B}_n^{\bot}$, by Theorem~\ref{thm:validity},
we have that $\varphi_{k} \in {\rm Th}({\mathfrak M}/\mathbf{B}_{n})$, but $\varphi_{k} \notin
{\rm Th}({\mathfrak M}/\mathbf{B}_{m})$. Notice also that by
Corollary~\ref{cor:characterization}, we can now also conclude that
$\sigma_{n} \in {\rm Th}({\mathfrak M}/\mathbf{B}_{n})$, but $\sigma_{n} \notin
{\rm Th}({\mathfrak M}/\mathbf{B}_{m})$
\end{proof}
\section{Acknowledgements}
Thanks to Paul Shafer for his comments on the paper \cite{SorbiTerwijn}. We
thank Lev Beklemishev for remarks about ${\sf KC}$, Wim Veldman for the reference
to Brouwer, and Rosalie Iemhoff for general discussions about ${\sf IPC}$.
|
1,314,259,993,659 | arxiv | \section{Introduction}
Survival analysis, also known as time-to-event analysis aims to predict the first time of the occurrence of a stochastic event, conditioned on a set of features. An example in the case of medical data is the time of death or a graft failure after an operation. In cases where the time of event for many samples is missing because the event wasn't observed, this can be framed as a particular type of semi-supervised learning where part of the target values are referred to as right-censored. Formally we can say that for some examples we do not have the time of event $T$, but rather a time $T_0$ (censoring time) such that we know $T > T_0$. The classical approach to survival analysis is the Cox proportional hazards model \cite{cox1972regression} that takes into account censored samples. Ranking approaches \cite{steck2008ranking} are also a way to take these censored samples into account by incorporating them into the training using pairwise ranking loss where although the exact time of event is not known the pairwise relationship with respect to a censoring date is known for event occurring before the censored event. We would like to predict the probability distribution of an event as it will help in treatment planning. For example, determining if the risk of kidney graft rejection is constant or peaked after some time.
In this study, we propose to use the Wasserstein metric to have a model predict the probability distribution of the event time. This approach not only provides an interpretable prediction but allows us to impute the distribution of censored samples given global survival statistics with the non-parametric Kaplan Meier estimate. Our intuition is that training with the KM estimate provides a richer signal during training than a rank loss would provide. Also, we find that this approach directly optimizes the C-index \cite{harrell1982evaluating} which is the most common evaluation metric for ranking survival models. We compared our proposed loss with a set of common ranking-specific losses on several reference survival datasets.
\section{Survival data}
In what follows, we will use the following notations.
Let $\mathbf{x}^{(i)}$ be the feature vector of the $i$-th example and let $\mathbf{y}^{(i)}_t$ take value 1 if event $i$ happened at time $t$ and 0 otherwise. Moreover, let $\hat{\mathbf{y}}^{(i)}_t$ be the estimated probability of event $i$ happening at time $t$ and let $t^{(i)}$ be the (scalar) actual time of event $i$. We denote by $\mathbf{z}^{(i)}_t$ and $\hat{\mathbf{z}}^{(i)}_t$ the true and estimated cumulative probability distribution of $y$. Namely, $\mathbf{z}^{(i)}_{t_0} = \sum_{t < t_0} \mathbf{y}^{(i)}_t$. Finally, let $c^{(i)}$ be 1 if example $i$ is observed (non-censored) and 0 otherwise.
\subsection{Ties and censored data}
Survival datasets describe medical events that can have a low temporal resolution (time scale) causing ties between patients. A given \emph{unique time} (at a given resolution, e.g., one day) can correspond to multiple events. Such events are \emph{tied} and that would imply that more precise predictions are not relevant. However, they must be given special attention in constructing loss functions.
As mentioned earlier, another characteristic of survival data is that they are right-censored. We can still use these examples by only comparing with patients that had an event before the date of censorship or by imputing the event time based on statistics over the data.
\subsection{Metric of evaluation}
The concordance index or C-index \cite{harrell1982evaluating} is the standard evaluation metric for survival data. It corresponds to the normalized Kendall tau metric between the true and predicted distribution \cite{kendall1938new}. It can be seen as a generalization of the Area Under the Receiver Operating Characteristic Curve (AUROC) that can handle right-censored data \cite{steck2008ranking}.
We define an \emph{acceptable} pair as one for which we are sure the first event occurs before the second. These are the pairs for which the first element is non-censored, and for which the censoring or event time of the second element is strictly greater than the first. Let $\mathcal{A}$ be the set of acceptable pairs.
Then, the C-index to be maximized can be written as:%
$$
\frac{1}{|\mathcal{A}|} \sum_{\scriptscriptstyle (\mathbf{x}^{(i)}, \mathbf{x}^{(j)}) \in \mathcal{A}} \Scale[0.8]{ \mathds{1}\Big(f(\mathbf{x}^{(i)}) < f(\mathbf{x}^{(j)})\Big) + \frac{1}{2}\mathds{1}\Big(f(\mathbf{x}^{(i)}) = f(\mathbf{x}^{(j)})\Big)}.
$$
\vspace{-10pt}
\section{Loss functions for censored data}
In this section, we present loss functions in the context of survival prediction for censored data. We divide these loss functions into three categories: partial likelihood methods, rank methods, and our classification method based on a Wasserstein metric (WM).
\subsection{Cox Model}
Cox introduced a general conditional log-likelihood to fit survival models, in which the probability of observations is maximized \cite{cox1972regression}. It was demonstrated by \cite{steck2008ranking} that maximizing the Cox's partial likelihood is approximately equivalent to maximizing the C-index. We present the general formula, with a real-valued score prediction function $f_\theta$ estimating the probability of the event at a particular time, given input features $\mathbf{x}^{(i)}$. Denoting the predicted score $f_\mathbf{\theta}(\mathbf{x}^{(i)})$ the loss is:
\begin{equation*}
\ell (\theta) = \sum_{i:c^{(i)} = 1} \Big(\log f_\theta(\mathbf{x}^{(i)}) - \log \sum_{j : t^{(j)} \geq t^{(i)}} f_\theta(\mathbf{x}^{(j)}) \Big).
\end{equation*}%
We also consider a variant of this loss, Efron's approximation \cite{efron1977efficiency} that commonly improves performance when there are many tied event times.
In our experiments, the Cox variant refers to a multi-layer perceptron (MLP) $f_\theta$ trained with the normal Cox loss or with Efron's approximation loss, as in \cite{katzman2016deep, luck2017deep}.
\subsection{Ranking losses}
Many methods attempt to directly predict the rank of the different examples. This is done by learning the following objective
$$
\argmax_{\mathbf{\theta}} \frac{1}{|\mathcal{A}|} \sum_{(\mathbf{x}^{(i)}, \mathbf{x}^{(j)}) \in \mathcal{A}} \phi(f_{\mathbf{\theta}}(\mathbf{x}^{(i)}) - f_{\mathbf{\theta}}(\mathbf{x}^{(j)}))
$$%
where $\phi(z)$ is a function that relaxes the non-differentiable $\mathds{1}$ of the C-index~\cite{steck2008ranking}.
We evaluated the functions used in~\cite{steck2008ranking}, Ranking SVM ~\cite{herbrich2000large}, Rankboost~\cite{freund2003efficient} and RankNet~\cite{burges2005learning}. These functions have been shown in~\cite{kalbfleisch1978non} to correspond to lower bounds on the C-index. We use $\sigma$ to denote the Sigmoid function $z \rightarrow \frac{1}{1+\exp(-z)}$.
\subsection{Wasserstein metric}
While there have to our knowledge been no previous attempts to use the Wasserstein metric on survival data or ranking problems,~\cite{frogner2015learning} used a Wasserstein loss for image classification and tag prediction.~\cite{hou2016squared} and ~\cite{beckham2017unimodal} apply a Wasserstein metric for the more restrictive case of ordinal classification. Recently,~\cite{mena2018learning} used the Sinkhorn algorithm, which is commonly used in optimal transport applications, as an analogy to the Softmax for permutations.
The WM is the minimum cost to transport the mass from one probability distribution to another. In the case of distributions of discrete supports (histograms of class probabilities), this is computed by moving probability mass from one class to another, according to the ground distance matrix specifying the cost to transport probability mass to and from different classes. Thus, the WM takes advantage of knowledge of the structure of the space of values considered, e.g., the 1-dimensional real-valued time axis, so that some errors (e.g. between neighboring events) are appropriately penalized less than others.
The WM is particularly adapted to a survival context. We denote $p_r$ the true data distribution, and $p_\theta$ the distribution estimated by the model. We write $\Pi$ the set of joint distributions $p(\cdot, \cdot)$ with left and right marginals $p_\theta$ and $p_r$ respectively. Given an example $\mathbf{x}$ and corresponding real time of event $T$, we can write:%
$$
W(p_\theta, p_r) = \inf_{p(\cdot, \cdot | \mathbf{x}) \in \Pi} \mathbb{E}_{T_1, T_2 \sim p(\cdot, \cdot | \mathbf{x})} \big[d(T_1, T_2) \big]
$$%
As $p_r$ is a Dirac, we have that:%
$$
\mathbb{E}_{T_1, T_2 \sim p(\cdot, \cdot | \mathbf{x})} \big[d(T_1, T_2) \big] = \mathbb{E}_{T_1 \sim p(\cdot, T | \mathbf{x})} \big[d(T_1, T) \big]
$$%
In all that follows, $d(T_1, T_2)$ is chosen to be proportional to the number of train set elements having events between $T_1$ and $T_2$. The term is therefore $\mathbb{E}_{T \sim p_\theta(\cdot, T | \mathbf{x})} \big[1-\text{Cindex} \big]$.
\subsubsection{Use as a learning objective}
\cite{levina2001earth} notes that under certain conditions satisfied in the case of ordinal classification, the WM takes the following expression:%
$$
\text{WM}(p, q) = \Big(\frac{1}{T}\Big)^{1/l} || CDF(p) - CDF(q) ||_l,
$$
where $T$ is the size of the Softmax layer and $CDF(.)$ is a function that returns the cumulative density function of its input density. Here, $p$ and $q$ are two probability distributions with discrete supports. We use $l=1.5$ in our experiments. We write $f_\theta (\mathbf{x}^{(i)}) = \hat{\mathbf{z}}^{(i)}_\theta$ to highlight the dependency on $\theta$. The objective can be written as:%
$$
\argmin_\theta \frac{1}{T}\sum_i ||\hat{\mathbf{z}}^{(i)}_\theta - \mathbf{z}^{(i)} ||^l.
$$
\subsubsection{Imputing missing values for classification}
In order to allow the WM objective to lead to good training, we have imputed the CDF of the censored data with $1. - KM$, where $KM$ is the Kaplan-Meier non parametric estimate of the survival distribution function computed on the training set (see Figure~\ref{model}). With the KM estimator, the survival distribution function $S(t)$ is estimated as a step function, where the value at time $t_{i}$ is calculated as follows:
\begin{align*}
\hat{S}(t_{i}) = \hat{S}(t_{i-1})(1-d_{i}/n_{i}),
\end{align*}
with $d_{i}$ denoting the number of events at $t_{i}$ and $n_{i}$ the number of patients alive just before $t_{i}$.
\begin{figure}[h]
\centering
\includegraphics[width=0.4\textwidth]{WassersteinSurvivalOverview}
\caption{An overview of the proposed distribution matching loss. In the case that a sample is censored the KM estimate is used to impute the probability that should be assigned for that event.}
\label{model}
\end{figure}
\section{Experiments}
\subsection{Datasets}
We assess the presented models on a variety of publicly available datasets. The characteristics of these datasets are summarized in Table \ref{tab:datasets}.
\begin{table}[!ht]
\resizebox{\columnwidth}{!}{%
\centering
\begin{tabular}{ l r r r r r }
\toprule
Datasets & \thead{Nb.\\samples} & \thead{Nb. ($\%$) \\censored} & \thead{Nb. ($\%$) \\unique times } & \thead{Nb. \\features }\\
\midrule
SUPPORT2 & 9105 & 2904 (32.2) & 1724 (19.1) & 98 \\
AIDS3 & 3985 & 2223 (55.8) & 1506 (37.8) & 19 \\
COLON & 929 & 477 (51.3) & 780 (84.0) & 48 \\
\bottomrule
\end{tabular}}
\caption{Characteristics of the datasets used in our evaluation. The datasets have different numbers of samples, percentage of censored, and tied patients. The features are typically continuous or discrete clinical attributes.}
\label{tab:datasets}
\end{table}
\vspace*{-40pt}
\textbf{SUPPORT2}\footnote{available at \protect\url{http://biostat.mc.vanderbilt.edu/wiki/Main/DataSets}} records the survival time for patients of the SUPPORT study.
\textbf{AIDS3}\footnote{available at \url{https://vincentarelbundock.github.io/Rdatasets/datasets.html}\label{first_footnote}} corresponds to the Australian AIDS Survival Data.
\textbf{COLON}\footref{first_footnote} consists of data from the first successful trials of adjuvant chemotherapy for colon cancer. We considered death as a target event for our study.
\subsection{Data pre-processing}
We used a one-hot encoding for categorical features, and unit scaling for continuous features. For features with missing values, we added an indicator function for the absence of a value.
We performed 5 fold cross validation and kept 20\% of the train set as a validation set. The prediction performance was reported as mean $\pm$ standard error of the C-index over the 5 folds. Early stopping was performed on the validation C-index.
We used a multi-layer perceptron (3 layers with 100 units each) with ReLU activation functions where applicable, and used Dropout~\cite{hinton2012improving}, Batch Normalization~\cite{ioffe2015batch} and L2 regularization on the weights. We used the Adam optimizer. For the ranking and log-likelihood methods the output was a single unit with a linear activation function. For the methods requiring a prediction of output times, we used a Softmax function. Our code was written in PyTorch~\cite{paszke2017automatic}
We perform a grid-search for each split independently for the L2 regularization coefficient on the weight and the learning rate. We add a small constant (1 for Support2 and Aids3, 10 for Colon) to the distance between bins before normalizing. For colon we used a bin size of 2 days, and 1 day for the other two datasets.
\subsection{Comparison of different ranking methods}
We study the impact of the different loss functions in Table \ref{tab:results}. We study how the standard Cox model performs in comparison to ranking and classification losses.
\begin{table}[!ht]
\resizebox{\columnwidth}{!}{%
\centering
\begin{tabular}{ l l l l l l }
\toprule
Loss Type& Variant & SUPPORT2 & AIDS3 & COLON\\
\midrule
Partial likelihood & Cox & 84.90$\pm$0.63 & 54.84$\pm$0.82 & \textbf{64.66}$\pm$\textbf{0.44}\\
Partial likelihood & Cox Efron's & 84.91$\pm$0.60 & 54.03$\pm$1.21 & 63.08$\pm$0.93 \\
Ranking & $\sigma(z)$ & \textbf{85.53}$\pm$\textbf{0.56} & 55.35$\pm$1.19 & 64.22$\pm$0.61\\
Ranking & Log-sigmoid & 85.44$\pm$0.57 & 55.28$\pm$1.29 & 63.36$\pm$0.52 \\
Ranking & $(z-1)_+$ & 84.96$\pm$0.56 & 55.41$\pm$1.20 & 63.98$\pm$1.12 \\
Ranking & $1 - \exp(-z)$ & 85.35$\pm$0.58 & 55.73$\pm$0.93 & 61.96$\pm$0.91 \\
Classification & WM (ours) & 85.33$\pm$0.52 & \textbf{56.03}$\pm$\textbf{1.01} & 64.32$\pm$0.39 \\
\bottomrule
\end{tabular}}
\caption{Performance scores of the different methods. The table reports the C-index mean $\pm$ standard error over the 5 fold. For each dataset, the best model in terms of mean score is highlighted in bold. We draw the readers attention to the classification losses which are among the losses that give the best results.}
\label{tab:results}
\end{table}
\subsection{Impact of using censored data}
The purpose of this section is to explore how censoring is informative and demonstrate that we should not just ignore/process away censoring.
We compare three methods to account for censored data. We first completely removed censored examples from the training set (no censored data). We also considered the time of censoring to correspond to an actual event occurrence (transforming each example censored at time $t$ into the same example with an event occurring at time $t$) (death at censoring). Finally, we also listed results for the standard approach (with censored data). In the case of WM, the censored times are imputed with the ($1 - KM$) curve.
We run this experiment on the SUPPORT2 dataset for the three best methods of each category as it is the largest public dataset we have : Cox Efron's, $\sigma(z)$ and our methods WM. The results are presented in Table~\ref{table_censored}.
\begin{table}[h!]
\begin{center}
\resizebox{\columnwidth}{!}{%
\begin{tabular}{@{}cccc@{}}
\toprule
Method & WM & Ranking & Cox \\
\midrule
No censored data & 83.31$\pm$0.51 & 83.40$\pm$0.52 & 82.34$\pm$0.49 \\
Death at censoring & 82.34$\pm$0.58 & 81.97$\pm$0.67 & 80.67$\pm$0.55 \\
With censored data & \textbf{85.33}$\pm$\textbf{0.52} & \textbf{85.53}$\pm$\textbf{0.56} & \textbf{84.91}$\pm$\textbf{0.60} \\
\bottomrule
\end{tabular}}
\end{center}
\caption{We explore how the three categories of methods are impacted by adding censored data. The table reports the C-index mean $\pm$ standard error over the 5 fold. For "Death at censoring", we set the death event as the censored time. It is clear that censored data contains information that we can use to make better predictions.}
\label{table_censored}
\end{table}
\subsection{Exploring the impact of censored data}
In order to determine how much of an improvement we can obtain from incorporating censored data we can vary the composition of samples that are censored in the training data, while keeping the validation and test sets the same. In Figure \ref{varycensored} we show the evolution of the C-index with different percentages of censoring of the training set in the SUPPORT2 dataset.
\begin{figure}[h]
\centering
\includegraphics[width=0.5\textwidth]{increased_cens.png}
\caption{Here we study how the composition of censored and uncensored patients during training impacts the C-index mean $\pm$ standard error over the 5 fold in the SUPPORT2 dataset. The validation and test sets are fixed and the training set has censored patients introduced by marking patients as censored at random. The plot starts at 30\% because the dataset has that many censored patients by default. We find that the WM classification loss is robust to the introduction of censored data.}
\label{varycensored}
\end{figure}
\section{Conclusion}
We proposed a new method for learning to rank survival data. Experiments on the different datasets show that our models trained with the WM loss gives accurate predictions compared to the more classical losses of the Cox model and ranking loss functions, which directly approximate a lower bound of the C-index. While not always state of the art, our method is always among the best results for each dataset.
We also find that this approach allows the method to tolerate a high percentage of censored samples and continue to predict well given results consistently in the same range of the best methods. Also, we demonstrate that our method can be seen as directly optimizing the expected C-index which is the most common evaluation metric for ranking survival models. Moreover, our results demonstrate that imputing the values with the KM curve for the missing times in a classification framework can increase the resulting C-index.
\section*{Acknowledgements}
We thank Christopher Pal and Christopher Beckham for their input on the project. This work is partially funded by a grant from the U.S. National Science Foundation Graduate Research Fellowship Program (grant number: DGE-1356104) and the Institut de valorisation des donnees (IVADO). This work utilized the supercomputing facilities managed by the Montreal Institute for Learning Algorithms, NSERC, Compute Canada, and Calcul Quebec.
\newpage
|
1,314,259,993,660 | arxiv | \section{\bf #1}}}
\newcommand{\SUBSECTION}[1]{\bigskip{\large\subsection{\bf #1}}}
\newcommand{\SUBSUBSECTION}[1]{\bigskip{\large\subsubsection{\bf #1}}}
\begin{titlepage}
\begin{center}
\vspace*{2cm}
{\large \bf Classical Electromagnetism as a Consequence of Coulomb's Law,
Special Relativity and Hamilton's Principle and its Relationship to Quantum Electrodynamics
\footnote{This paper is dedicated to the memory of Valentine Telegdi}}
\vspace*{1.5cm}
\end{center}
\begin{center}
{\bf J.H.Field }
\end{center}
\begin{center}
{
D\'{e}partement de Physique Nucl\'{e}aire et Corpusculaire
Universit\'{e} de Gen\`{e}ve . 24, quai Ernest-Ansermet
CH-1211 Gen\`{e}ve 4.}
\end{center}
\begin{center}
{e-mail; john.field@cern.ch}
\end{center}
\vspace*{2cm}
\begin{abstract}
It is demonstrated how all the mechanical equations
of Classical Electromagnetism (CEM) may be derived from only Coulomb's inverse square force law,
special relativity and Hamilton's Principle. The instantaneous nature of the
Coulomb force in the centre-of-mass frame of two interacting charged objects,
mediated by the exchange of space-like virtual photons, is predicted by QED.
The interaction Lagrangian of QED is shown to be identical, in the appropriate
limit, to the potential energy term in the Lorentz-invariant Lagrangian of CEM.
A comparison is made with the Feynman-Wheeler action-at-a-distance formulation
of CEM.
\end{abstract}
\vspace*{1cm}
{\it Keywords}; Special Relativity, Classical Electrodynamics.
\newline
\vspace*{1cm}
PACS 03.30+p 03.50.De
\end{titlepage}
\SECTION{\bf{Introduction}}
At the beginning of Book III of the Principia~\cite{Principia} Newton introduced
four `Rules of Reasoning in Philosophy'. The first of them was:
\par {\tt We are to admit no more causes of natural things than such as are
both \newline true and sufficient to explain their appearences.}
\par It is still a salutary exercise to apply this simple principle to any domain of
science. What are fundamental and truly important in the scientific description
of phenomena are those concepts that cannot be discarded without destroying the
predictive power of the theory. The most powerful, the best, scientific theory
is that which describes the widest possible range of natural phenomena in terms
of the minimum number of essential (i.e. non-discardable) concepts. It is the
aim of the present paper to apply this precept of Newton to Classical Electromagnetism
(CEM). The relation of CEM to Quantum Electrodynamics (QED), in the attempt
to obtain a deeper physical understanding of the former, will also be discussed.
\par From the work of Coulomb, Amp\`{e}re and Faraday on, the basic phenomena
of CEM, i.e. what are actually observed in experiments, are the forces between
electric charges at rest or in motion, or the dynamical consequences of such
forces. The force between two static charges is given by Coulomb's inverse
square law. This law will taken as a postulate in the following, but no other
dynamical concept or theoretical construction will be introduced as an
independent hypothesis in order to build up the theory. Later, it will be
seen that, in QED, this law is a necessary consequence of the existence of,
and exchange of, space-like virtual photons between the electric charges.
\par It will be assumed throughout that the system of interacting electric
charges is a conservative one, in Classical Mechanics, and so may be
described by a Lagrangian that is a function of the coordinates and velocities
of the charges, but does not depend explicitly on the time. Calculating
the Action from the Lagrangian of the system and applying Hamilton's
Principle that the Action be an extremum with respect to variation of
the space-time trajectories of the charges, yields, in the well-known
manner, the Lagrange equations that provide a complete dynamical description
of the system~\cite{Golds1}.
\par It is further required that the physical description be consistent
with Special Relativity. For this, the Lagrangian must be a
Lorentz scalar. To introduce the method to be used to construct the
Lagrangian, which is likely to be familiar only to particle physicists,
I quote a passage taken from some lecture notes by R.Hagedorn~\cite{Hagedorn} on
relativistic kinematics dating from some four decades ago:
\par {\tt If a question is of such a nature that its answer will be always the
\newline same, no matter in which Lorentz system one starts, then it is
possible to formulate the answer entirely with the help of those invariants
which one \newline can build with the available four vectors. One then finds the
answer in a particular Lorentz system which one can choose freely and in
such a way \newline that the answer there is obvious and most easy. One looks then
how the \newline invariants appear in this particular system, expresses the answer
to the \newline problem by these invariants and one has found at the same time aleady the
\newline general answer... It is worthwhile to devote some thinking to this method
of calculation until one has completely understood that there is really
no jugglery or guesswork in it and that it is absolutely safe.}
\par It is important to stress the last sentence in this passage in relation
to the word `true' in Newton's philosophical precept quoted above. Just the method
outlined above was used to derive the Bargmann-Michel-Telegdi (BMT) equation for spin motion in
arbitary magnetic and electric fields~\cite{BMT}.
\par It will be demonstrated in the following that it is sufficient to apply
Hagedorn's programme to the simplest possible non-trivial
electrodynamical system that may be considered: two mutually interacting
electric charges, in order to derive all the mechanical equations of CEM, as well
as Maxwell's equations, with Coulomb's inverse square law as the only dynamical
hypothesis. The 'mechanical' equations comprise the relativistic generalisation of the Biot and Savart Law,
the Lorentz force equation and those describing electromagnetic induction effects with
uniformly moving source currents and test charges~\footnote{Not included are induction effects related
to AC currents, where source charges are accelerated. Although described, in an identical manner, by the
Faraday-Lenz Law, as non-accelerated charges, real as well as virtual photons must be taken
into account, at the fundamental level, in this case. For uniformly moving charges, no real photons are created.}
\par An aspect that is not touched upon in the above programme is radiation.
In this case a fundamental classical description of the phenomenon,
in the sense of Newton's precept, is not possible and Quantum Mechanics
must be invoked. In the language of QED, the existence of real photons
as well as the virtual photons responsible for the
Coulomb force, must be admitted. Indeed, extra degrees of freedom must
be added to the Lagrangian to describe the propagation of real photons
and their interaction with electric charges. Also the corresponding potentials and fields
are retarded, not instantaneous. A brief comment is made in the
concluding section on the relation of Maxwell's equations to radiation phenomena;
however, no detailed comparison with QED is attempted.
\par It is also assumed throughout the paper that the effects of gravitation,
that is of the curvature of space-time, on the interaction between the
charged physical objects considered, may be neglected.
\SECTION{\bf{Lorentz Invariant Lagrangian for Two Mutually Interacting
Electrically Charged Objects}}
Two physical objects O$_1$ and O$_2$ of masses $m_1$ and $m_2$ and electric charges
$q_1$ and $q_2$, respectively, are assumed to be in spatial proximity, far from
all other electric charges, so that they interact electromagnetically, but
are subjected to no external forces. The
spatial positions of O$_1$ and O$_2$ are specified, relative to their common
center of energy, by the vectors $\vec{r}_1$ and $\vec{r}_2$ respectively.
The spatial distance separating the two objects in their common center-of-mass (CM)
frame: $r_{12} = r_{21}$ is given by the modulus of the vectors $\vec{r}_{12}$,
$\vec{r}_{21}$ where:
\begin{equation}
\vec{r}_{12} = - \vec{r}_{21} = \vec{r}_1-\vec{r}_2
\end{equation}
\par The non-relativistic (NR) Lagrangian describing the motion of the objects O$_1$ and O$_2$
in their overall CM frame is~\cite{GoldsteinNRL}\footnote{Gaussian electromagnetic units are used.}
\begin{equation}
L_{NR}( \vec{r}_1,\vec{v}_1;\vec{r}_2,\vec{v}_2) \equiv T_1+T_2-V
= \frac{1}{2}m_1 v_1^2+ \frac{1}{2}m_2 v_2^2 -\frac{q_1q_2}{r_{12}}
\end{equation}
$T_1$, $v_1$ ($T_2$, $v_2$) are the kinetic energies and velocities, respectively of O$_1$ (O$_2$) and $V$
is the potential energy of the system.
A Lorentz-invariant Lagrangian describing the system O$_1$, O$_2$ will now be constructed in such a way
that it reduces to Eqn(2.2) in the non-relativistic limit. The Lagrangian must be a Lorentz scalar
constructed from the 4-vectors\footnote{ From translational invariance,
the interaction between the objects does not depend upon the absolute positions of the
objects, but only on their relative spatial separation: $|\vec{x}_1- \vec{x}_2|$. Therefore the
dependence of the Lagrangian on the independent 4-vector $x_1+x_2$ may be neglected.}:
$x_1-x_2$, $u_1$ and $u_2$ that completely specify the spatial
and kinematical configuration of the interacting system. Here $\vec{x}_1 = \vec{r}_1$,
$\vec{x}_2 = \vec{r}_2$ and the `4-vector velocity', $u$, is defined as:
\begin{equation}
u \equiv \frac{d x}{d \tau} = \gamma \frac{d~}{dt}(ct;\vec{x}) = (\gamma c ; \gamma \vec{v})
\end{equation}
where $\tau$ is the proper time of the object, $\gamma \equiv 1/\sqrt{1-\beta^2}$ and $\beta \equiv v/c$.
In general, the Lagrangian may
depend on the following six Lorentz invariants, constructed from the relevant 4-vectors:
\[ (x_1-x_2)^2,~~~~ u_1 \cdot (x_1-x_2),~~~~ u_2 \cdot (x_1-x_2),~~~~ u_1^2,~~~~ u_2^2,~~~~ u_1 \cdot u_2 \]
so that the Lagrangian may be written as:
\begin{equation}
L(x_1,u_1;x_2,u_2) =\alpha_0+ \alpha_1 (x_1-x_2)^2+ \alpha_2 u_1 \cdot (x_1-x_2) + \alpha_3 u_2 \cdot (x_1-x_2)
+ \alpha_4 u_1^2 + \alpha_5 u_2^2 + \alpha_6 u_1 \cdot u_2
\end{equation}
where the coefficients $\alpha_0$-$\alpha_6$ are Lorentz-scalars that may also be, in general,
arbitary functions of the six Lorentz invariants listed above.
Taking the NR limit:
\[ u_1 \rightarrow (c; \vec{v}_1),~~~ u_2 \rightarrow (c; \vec{v}_2)\]
gives\footnote{Note that the term containing $\vec{v}_1 \cdot \vec{v}_2$ vanishes in the NR limit
where terms of O( $\beta_1 \beta_2$) are neglected:
$u_1 \cdot u_2 \rightarrow c^2(1-\vec{v}_1 \cdot \vec{v}_2/c^2)= c^2 + O( \beta_1 \beta_2)$.}:
\begin{equation}
L(x_1,u_1;x_2,u_2) = \alpha_0 - \alpha_1 r_{12}^2 - \alpha_2 \vec{v}_1 \cdot \vec{r}_{12}-
\alpha_3 \vec{v}_2 \cdot \vec{r}_{12}
- \alpha_4 v_1^2 - \alpha_5 v_2^2 + ( \alpha_4 + \alpha_5 +\alpha_6) c^2
\end{equation}
where a time-like metric is chosen for 4-vector products.
Note that $\vec{x}_1$ and $\vec{x}_2$ are defined at the same time, $t$,
in the CM frame of
O$_1$ and O$_2$, so that $t_1=t_2 = t$ in the 4-vectors $x_1$ and $x_2$. Thus the Coulomb interaction
is assumed to be instantaneous in the CM frame. As discussed in Section 6 below, such behaviour is
a prediction of QED.
Consistency between Eqns(2.2) and (2.5) requires that\footnote{The symmetry of the Lagrangian
with respect to the labels 1,2 requires that the term $\alpha_6 u_1 \cdot u_2$ be
identified with the potential energy term in (2.2).}:
\begin{equation}
\alpha_1 = \alpha_2 = \alpha_3 = 0, ~~\alpha_4 = -\frac{m_1^2}{2}, ~~\alpha_5 = -\frac{m_2^2}{2},
~~ \alpha_6 = -\frac{q_1 q_2}{c^2 r_{12}}, ~~ \alpha_0 + (\alpha_4 + \alpha_5)c^2 = 0~~~
\end{equation}
The choice $\alpha_0 = c^2(m_1^2+m_2^2)/2$ satisfies the last condition in (2.6) and yields for
the Lorentz-scalar Lagrangian:
\begin{equation}
L(x_1,u_1;x_2,u_2) = -\frac{m_1 u_1^2}{2} -\frac{m_2 u_2^2}{2} - \frac{j_1 \cdot j_2}{c^2 r_{12}}
\end{equation}
Where the current 4-vectors: $j_1 \equiv q_1 u_1$ and $j_2 \equiv q_2 u_2$ have been introduced.
This Lagrangian may be written in a manifestly Lorentz-invariant manner by noting that:
\[ x_1-x_2 = (0;\vec{x}_1-\vec{x}_2) = (0; \vec{r}_{12}) \]
so that $r_{12} = \sqrt{-(x_1-x_2)^2}$ and
\begin{equation}
L(x_1,u_1;x_2,u_2) = -\frac{m_1 u_1^2}{2} -\frac{m_2 u_2^2}{2}
- \frac{j_1 \cdot j_2}{c^2 \sqrt{-(x_1-x_2)^2}}
\end{equation}
The Lagrangian (2.7), when substituted into the covariant Lagrange equations derived from
Hamilton's Principle~\cite{Golds1}:
\begin{equation}
\frac{d~}{d \tau}\left(\frac{\partial L}{\partial u_i^{\mu}}\right)
-\frac{\partial L}{\partial x_i^{\mu}} = 0~~~(i=1,2;~\mu = 0,1,2,3) :
\end{equation}
is shown in the following Sections to enable all the
concepts and equations of CEM concerning inter-charge forces, in the absence of radiation,
to be derived without introducing any further postulate.
Note that, since the Lagrangian (2.7) is a Lorentz scalar, it provides a description
of the motion of O$_1$ and O$_2$ in any inertial reference frame.
\SECTION{\bf{The 4-vector Potential, Electric and Magnetic Fields, the Lorentz Force Equation
and the Biot and Savart Law}}
Considering only the motion of O$_1$, introducing the `4-vector potential', $A_2$, according
to the definition:
\begin{equation}
A_2 \equiv \frac{j_2}{c r_{12}}
\end{equation}
the well-known~\cite{Golds2} Lorentz-invariant Lagrangian describing the motion of the
object O$_1$ in the `electromagnetic field created by the object O$_2$':
\begin{equation}
L(x_1,u_1) = -\frac{m_1 u_1^2}{2} - \frac{1}{c} q_1 u_1 \cdot A_2
\end{equation}
is recovered. In the same way, the motion of O$_2$ in the
`electromagnetic field created by the object O$_1$' is given by the invariant Lagrangian:
\begin{equation}
L(x_2,u_2) = -\frac{m_2 u_2^2}{2} - \frac{1}{c} q_2 u_2 \cdot A_1
\end{equation}
where:
\begin{equation}
A_1 \equiv \frac{j_1}{c r_{12}}
\end{equation}
To now introduce the concepts of distinct `electric' and `magnetic' fields it is
sufficient to consider only the motion of O$_1$. To simplify the equations the
labels `1' and `2' will be dropped in Eqn(3.2) and the following notation is used
for spatial partial derivatives:
\begin{equation}
\partial_i = -\partial^i \equiv \frac{\partial~}{\partial x^i} \equiv \nabla_i~~~(i=1,2,3)
\end{equation}
The Lagrangian (3.2) is now introduced into the Lagrange equations (2.9). Considering the
1 spatial components of the 4-vectors, the first term on the LHS of Eqn(2.9) is:
\begin{equation}
\frac{d~}{d \tau}\left(\frac{\partial L}{\partial u^1}\right) = \frac{d~}{d \tau}
(mu^1+\frac{q}{c} A^1) = \gamma(m \frac{d u^1}{dt}+\frac{q}{c}\frac{d A^1}{dt})
\end{equation}
and the second is:
\begin{equation}
-\frac{\partial L}{\partial x^1} = -\frac{q}{c} u \cdot(\partial^1 A)
\end{equation}
Combining Eqns(2.9), (3.6) and (3.7)and transposing:
\begin{equation}
\gamma m \frac{d u^1}{dt} = \gamma \frac{d p^1}{dt} = \frac{q}{c}[ u \cdot(\partial^1 A)
-\gamma\frac{d A^1}{dt}]
\end{equation}
where the `energy-momentum 4-vector' $p \equiv m u$ has been introduced. Substituting the
Euler formula for the total time derivative\footnote{The implict time dependence of $A^1$ in the first
term on the right side of (3.9) arises from the instantaneous motion of the `source' O$_2$,
whereas the remaining terms
describe the variation of $A^1$ due to the motion of O$_1$.}:
\begin{equation}
\frac{d A^1}{dt} = \frac{\partial A^1}{\partial t} - v^1 \partial^1 A^1 - v^2 \partial^2 A^1
- v^3 \partial^3 A^1
\end{equation}
into (3.8), writing out explicitly the 4-vector product $ u \cdot(\partial^1 A)$, and cancelling a common
factor $\gamma$ from each term, gives:
\begin{equation}
\frac{d p^1}{dt} = \frac{q}{c}\left[ c \partial^1 A^0- \frac{\partial A^1}{\partial t}
+v^2(\partial^2 A^1- \partial^1 A^2) -v^3(\partial^1 A^3- \partial^3 A^1) \right]
\end{equation}
Introducing now 3-vector `electric' and `magnetic' fields, $E^i$ and $B^i$ respectively,
according to the definitions:
\begin{equation}
E^i \equiv \partial^i A^0- \frac{1}{c}\frac{\partial A^i}{\partial t} = \partial^i A^0- \partial^0 A^i
\end{equation}
and
\begin{equation}
B^k \equiv -\epsilon_{ijk}(\partial^i A^j- \partial^j A^i) = (\vec{\nabla} \times \vec{A})^k
\end{equation}
where $\epsilon_{ijk}$ is the alternating tensor equal to $+1(-1)$ when $ijk$ is an
even (odd) permutation of 123, and zero otherwise,
enables Eqn(3.10) to be written as the compact expression:
\begin{equation}
\frac{d p^1}{dt} = q \left[ E^1 + \frac{1}{c}(\vec{v} \times \vec{B})^1 \right]
\end{equation}
which is the 1 component of the Lorentz force equation. The 2 and 3 components
are derived by cyclic permutations of the indices 1,2,3 in Eqn(3.10), yielding finally the
3-vector Lorentz force equation:
\begin{equation}
\frac{d\vec{p}}{dt} = q \left[\vec{E} + \frac{\vec{v}}{c} \times \vec{B} \right]
\end{equation}
The concepts of `electric' and `magnetic' fields have therefore appeared naturally
as a means to simplify the Lorentz force equation (3.10). However, the RHS of this equation
is completely defined, via Eqn(3.1), by the 4-vector current $j_2$, the spatial separation $r_{12}$
of O$_1$ and O$_2$ and the 3-velocity of O$_1$, so that the 4-vector potential $A$ may be
eliminated from the Lorentz force equation. Substituting the definition of $A$ from Eqn(3.1)
into Eqns(3.11) and (3.12), and restoring the labels of quantities associated with O$_2$,
gives\footnote{Note that the partial time derivative in (3.11) implies
that $\vec{x}_1$ but not $\vec{x}_2$ is held constant. The implicit time variation of $A^1$
in (3.11) then has contributions from both $\vec{j_2}$ and $\vec{x}_2$ which yield,
respectively, the last two terms on the right side of (3.15).}:
\begin{equation}
\vec{E} = \frac{j_2^0 \vec{r}}{c r^3} -\frac{1}{c^2 r}\frac{d \vec{j_2}}{d t}
-\frac{\vec{j_2}}{c^2}\frac{(\vec{r} \cdot \vec{v_2})}{r^3}
\end{equation}
\begin{equation}
\vec{B} = \frac{q_2 \gamma_2 (\vec{v_2} \times \vec{r})}{c r^3} = \frac{\vec{j_2} \times \vec{r}}{c r^3}
\end{equation}
where $\vec{r} \equiv \vec{r}_{12}$.
Eqn(3.16) is the relativistic generalisation of the Biot and Savart Law. It differs
from the usual CEM formula by a factor $\gamma_2$. Note that the electric field is,
in general, non-radial. The non-radial part of the field, associated with the last term on the
right side of (3.15), originates in the second term on the right side of (3.11). This is the
electric field that is associated with the time variation of the magnetic field in the
Faraday-Lenz Law. For the case of a source charge in uniform motion in the $x$-direction,
with velocity $v_2$, the electric and magnetic fields given by (3.15) and (3.16) at the field
point $\vec{r} = \hat{\imath} \cos \psi + \hat{\jmath} \sin \psi$ are:
\begin{eqnarray}
\vec{E} & = & \frac{q}{r^2}\left[\frac{\hat{\imath} \cos \psi}{\gamma_2}+ \gamma_2 \hat{\jmath} \sin \psi \right] \\
\vec{B} & = & \frac{\vec{v}_2 \times \vec{E}}{c}
\end{eqnarray}
where $\hat{\imath}$ and $\hat{\jmath}$ are unit vectors in the $x$- and $y$-directions. These equations
may be compared with the pre-relativistic Heaviside~\cite{Heaviside} formulae for this case:
\begin{eqnarray}
\vec{E}(H) & = & \frac{q \vec{r}}{r^3 \gamma_2^2(1- \beta_2^2 \sin^2 \psi)^{\frac{3}{2}}} \\
\vec{B}(H) & = & \frac{\vec{v}_2 \times \vec{E}(H)}{c}
\end{eqnarray}
The fields $\vec{E}(H)$ and $\vec{B}(H)$ are also the `present time' fields as derived~\cite{PP} from the
retarded Li\'{e}nard-Wiechert potentials~\cite{LW}. By considering a simple two-charge `magnet', in a
particular spatial configuration, either
in motion or at rest, it has been shown~\cite{JHFEMI} that the radial electric field of (3.19) predicts a vanishing
induction effect for a moving magnet and stationary test charge. In the same configuration
(3.17) predicts the same induction force on the test charge as the Faraday-Lenz Law. The Heaviside
formulae are therefore valid only to first order in $\beta$, in which case the predictions
of (3.19) and (3.20) are the same as those of (3.17) and (3.18). It is interesting to recall that
just this problem, of induction in different frames of reference, was discussed in the Introduction
of Einstein's 1905 special relativity paper~\cite{Ein1}.
\par Substitution of (3.15) and
(3.16) into (3.14) and restoring the labels associated with O$_1$ yields the `fieldless'
Lorentz force equations\footnote{The right sides of these equations
are `forces' according to the relativistic generalisation of
Newton's Second Law. In fact, however, the force concept does not appear at
any place in their derivation. Also the relativistic 3-momentum $\vec{p} = \gamma \vec{\beta}m c$
appears naturally in the equations as a necessary consequence of the initial postulates.
For an interesting recent discussion
of the force concept in modern physics see~\cite{Wilczek}.} for two, discrete, mutually electromagnetically interacting, physical objects:
\begin{eqnarray}
\frac{d\vec{p_1}}{dt} & = & \frac{q_1}{c}\left[\frac{ j_2^0\vec{r} + \vec{\beta}_1 \times
(\vec{j_2} \times \vec{r})}{r^3} -\frac{1}{c r}\frac{d \vec{j_2}}{d t}-\vec{j_2}
\frac{(\vec{r} \cdot \vec{\beta}_2)}{r^3}
\right] \\
\frac{d\vec{p_2}}{dt} & = & -\frac{q_2}{c}\left[\frac{ j_1^0\vec{r} +\vec{\beta}_2 \times
(\vec{j_1} \times \vec{r})}{r^3}+\frac{1}{c r}\frac{d \vec{j_1
}}{d t} -\vec{j_1}
\frac{(\vec{r} \cdot \vec{\beta}_1)}{r^3}
\right]
\end{eqnarray}
It may be thought that the terms $\simeq 1/r$ should be assocated with radiative procesees
(see Section 7 below) but they are in fact of particle-kinetic nature. Since $\vec{j} = (q/m)\vec{p}$
the two differential equations are coupled via the $d\vec{j}/dt$ terms on the right sides of each.
The solution of these equations for the case of circular Keplerian orbits has been derived~\cite{JHFRSKO}.
One result obtained is the relativistiic generalisation of Kepler's Third Law of planetary motion
for this case:
\begin{equation}
\tau^2 = \frac{(2 \pi)^2 {\cal E}^* \left[1 -\frac{(q_1 q_2)^2}{m_1 m_2 c^4 r^2}\right] r^3}
{|q_1||q_2|(1+\beta_1 \beta_2)}
\end{equation}
where
\begin{equation}
{\cal E}^* \equiv \frac{{\cal E}_1^*{\cal E}_2^*}{{\cal E}_1^*+{\cal E}_2^*}
\end{equation}
and
\begin{eqnarray}
{\cal E}_1^* & \equiv & \frac{\gamma_1 m_1 c^2}{\gamma_2-\frac{|q_1||q_2| \gamma_1}{m_2 c^2 r}} \\
{\cal E}_2^* & \equiv & \frac{\gamma_2 m_2 c^2}{\gamma_1-\frac{|q_1||q_2| \gamma_2}{m_1 c^2 r}}
\end{eqnarray}
Eqn(3.23) gives the period, $\tau$, of two objects of mass $m_1$ and $m_2$ with (opposite)
electric charges $q_1$ and $q_2$, in circular orbits around their common center of energy,
separated by the distance $r$. The $d \vec{j}/d t$ terms in (3.21) and (3.22) give the
terms $\simeq 1/r$ in the denominators on the right sides of (3.25) and (3.26).
These terms effectively modify the masses of the objects due to the electromagnetic
interaction.
\par It is also demonstrated in Ref.\cite{JHFRSKO} that stable, circular, Keplerian
orbits are impossible under the retarded forces generated by Li\'{e}nard-Wiechert potentials.
\par Considering now the time components of the 4-vectors in (2.11), the first term on the LHS
is:
\begin{equation}
\frac{d~}{d \tau}\left(\frac{\partial L}{\partial u^0}\right) =
\gamma(-m \frac{d u^0}{dt}-\frac{q}{c}\frac{d A^0}{dt})
\end{equation}
while the second is:
\begin{equation}
-\frac{\partial L}{\partial x^0} = \frac{q}{c} u \cdot(\partial^0 A)
= \frac{q}{c} u \cdot \left(\frac{1}{c}\frac{\partial A}{\partial t}\right)
\end{equation}
Substituting (3.27) and (3.28) into (2.9) and rearranging gives:
\begin{equation}
\gamma \frac{d {\cal E}}{d t}=\frac{q}{c}\left[ u \cdot \left(\frac{1}{c}\frac{\partial A}{\partial t}\right)
-\gamma\frac{q}{c}\frac{d A^0}{dt}\right]
\end{equation}
where $ {\cal E} \equiv m u^0 c$ is the relativistic energy of O$_1$. Using the Euler formula
(3.9) to express $d A_0/d t$ in terms of partial derivatives, and writing out the different
terms in the 4-vector scalar products, the terms $\partial A^0/ \partial t$ are seen to
cancel. Dividing out the factor $\gamma$ on both sides of the equation then gives the
result:
\begin{equation}
\frac{d {\cal E}}{d t} = q[v_1(\partial^1 A^0-\partial^0 A^1)+v_2(\partial^2 A^0-\partial^0 A^2)
+v_3(\partial^3 A^0-\partial^0 A^3)] = q \vec{v} \cdot \vec{E}
\end{equation}
where $\vec{E}$ is the electric field defined in (3.11). Restoring now the labels of O$_1$ and
O$_2$ gives the `fieldless' equations for the time derivatives of their relativistic energies:
\begin{eqnarray}
\frac{d{\cal E}_1}{dt} & = & q_1 \left[ j_2^0 \frac{\vec{\beta_1} \cdot \vec{r}}{r^3}
-\frac{1}{c r} \vec{\beta_1} \cdot \frac{d \vec{j_2}}{d t}-
\frac{(\vec{\beta_1} \cdot \vec{j_2})(\vec{r} \cdot \vec{\beta_2})}{r^3}
\right] \\
\frac{d{\cal E}_2}{dt} & = & -q_2 \left[ j_1^0 \frac{\vec{\beta_2} \cdot \vec{r}}{r^3}
+\frac{1}{c r} \vec{\beta_2} \cdot \frac{d \vec{j_1}}{d t}+
\frac{(\vec{\beta_2} \cdot \vec{j_1})(\vec{r} \cdot \vec{\beta_1})}{r^3}
\right]
\end{eqnarray}
The equations (3.21),(3.22) and (3.31),(3.32) give a complete description of
the purely mechanical aspects of CEM (that is, neglecting radiative effects) for two massive, electrically
charged, objects
interacting mutually through electromagnetic forces.
\par The Lagrangian (2.7) is readily generalised to describe the mutual electromagnetic interactions of
an arbitary number of charged objects:
\begin{equation}
L(x_1,u_1;x_2,u_2; ...,x_n,u_n) = -\frac{1}{2} \sum_{i=1}^{n} m_i u_i^2 -
\frac{1}{c^2} \sum_{i>j}q_i q_j \frac{u_i \cdot u_j}{r_{ij}}
\end{equation}
Here $r_{ij} = |\vec{r}_i-\vec{r}_j|$ where $\vec{r}_i$ and $\vec{r}_j$ specify the positions
of O$_i$ and O$_j$, respectively, relative to the centre-of-energy on the $n$ interacting objects.
Note that, as all these distances are specified at a fixed time in the overall CM frame
of the objects, the $r_{ij}$ are Lorentz invariant quantities, similar to $r_{12}$ in Eqn(2.7).
See also~\cite{JHF1} for a general discussion of such invariant length intervals. The Lagrangian
describing the motion of the object $i$ `in the electromagnetic field of' the remaining $n-1$
objects may be derived from Eqn(3.33):
\begin{equation}
L(x_i,u_i) = -\frac{m_i u_i^2}{2} - \frac{1}{c} q_i u_i \cdot A(n-1)
\end{equation}
where
\begin{equation}
A(n-1) \equiv \sum_{j \ne i}^{n}\frac{q_j u_j}{r_{ij}}
= \sum_{j \ne i}^{n}\frac{j_j}{r_{ij}}
\end{equation}
This equation embodies the classical superposition principle for the electromagnetic
4-vector potential, and hence, via the linear equations (3.11) and (3.12),
that for the electric and magnetic fields.
\SECTION{\bf{Derivation of Maxwell's Equations}}
Writing out explicitly the spatial components of the quantity $\vec{\nabla} \cdot \vec{B}$ using
the definition of $\vec{B}$, Eqn(3.12):
\begin{eqnarray}
\partial^1 B^1 & = & \partial^1 \partial^3 A^2 - \partial^1 \partial^2 A^3 \\
\partial^2 B^2 & = & \partial^2 \partial^1 A^3 - \partial^2 \partial^3 A^1 \\
\partial^3 B^3 & = & \partial^3 \partial^2 A^1 - \partial^3 \partial^1 A^2
\end{eqnarray}
it follows, since $\partial^i\partial^j=\partial^j\partial^i~~(i,j=1,2,3)$ that, on summing
Eqns(4.1), (4.2) and (4.3),
\begin{equation}
\vec{\nabla} \cdot \vec{B} = -(\partial^1 B^1+ \partial^2 B^2+ \partial^3 B^3) = 0
\end{equation}
which is the magnetostatic Maxwell equation. Since $\vec{B} \equiv \vec{\nabla} \times \vec{A}$, (4.4) can also
be seen to follow from the 3-vector identity $\vec{a} \cdot (\vec{a} \times \vec{b}) \equiv 0$ for arbitary
$\vec{a}$ and $\vec{b}$.
\par The Faraday-Lenz Law follows directly from the defining equations Eqn(3.11), (3.12)
of the electric and magnetic fields. Taking the curl of both sides of the 3-vector form
of Eqn(3.11) with $\vec{\nabla}$ gives:
\begin{equation}
\vec{\nabla} \times \vec{E} =-\vec{\nabla} \times (\vec{\nabla} A^0)
- \frac{\partial~}{\partial t}(\vec{\nabla} \times \vec{A})
\end{equation}
Since $\vec{\nabla} \times (\vec{\nabla} \phi) = {\rm curl} ({\rm div} \phi) = 0$ for an arbitary scalar $\phi$,
the first term on the RHS of Eqn(4.5) vanishes. Subsituting the 3-vector
form of Eqn(3.12) in the second term on the RHS of Eqn(4.5) then yields the
Faraday-Lenz Law:
\begin{equation}
\vec{\nabla} \times \vec{E} = - \frac{1}{c} \frac{\partial \vec{B}}{\partial t}
\end{equation}
\par The electrostatic Maxwell equation:
\begin{equation}
\vec{\nabla} \cdot \vec{E} = 4 \pi J^0
\end{equation}
is a well-known consequence of the inverse square law for a `static' electric field
defined by only the first term on the RHS of Eqn(3.11) and Gauss' theorem~\cite{Jackson1}.
The 4-vector current density: $J \equiv (c\rho; \vec{J})$, the 0 component of which appears in
Eqn(4.7), is related to the currents, $j_i$, of elementary charges $q_i$ by the relation:
\begin{equation}
J = \frac{1}{V_R}\sum_{i \subset R} j_i
\end{equation}
where $V_R$ is the volume of a spatial region $R$. Hence $\rho = J^0/c$ is, in the
non-relativistic limit where $\gamma \simeq 1$, the average
spatial density of electric charge in the region $R$. Conservation of electric charge
requires that:
\begin{equation}
\frac{\partial \rho}{\partial t} +\vec{\nabla} \cdot \vec{J} = 0
\end{equation}
This continuity equation may be simply derived from the properties of the
4-vector product:
\begin{equation}
\partial \cdot j_i = \partial^0 j_i^0 -\sum_{k=1}^3\partial^k j_i^k = q_i \left[
c\frac{\partial \gamma_i}{\partial t}+\vec{\nabla} \cdot ( \gamma_i \vec{v_i}) \right]
\end{equation}
In the rest frame of the object
O$_1$, $ \gamma_i-1 = | \vec{v_i}| = 0$, so that $ \partial \cdot j_i = 0$. Since
$ \partial \cdot j_i$ is a Lorentz invariant this quantity then vanishes in all
inertial refererence frames.
Taking the scalar product of $\partial$ and $J$ gives:
\begin{equation}
\partial \cdot J = \frac{\partial \rho}{\partial t} +\vec{\nabla} \cdot \vec{J}
= \frac{1}{V_R}\sum_{i \subset R} \partial \cdot j_i = 0
\end{equation}
Which is just the continuity equation (4.9).
It can be seen that the conservation of electric charge is a consequence of its
Lorentz-scalar nature, i.e. the charge $q_i$ in Eqn(4.10) does not depend on the
frame in which $\vec{v_i}$ is evaluated. Indeed, the definition $j_i \equiv q_i u_i$
implies that $j_i \cdot j_i = q_i^2 u_i \cdot u_i = c^2 q_i^2$, so that $q_i^2$
is manifestly Lorentz invariant,
in precise analogy with the mass of an object: $p_i \cdot p_i = m_i^2 u_i \cdot u_i = c^2 m_i^2$.
Both $j_i$ and $p_i$ are proportional to the 4-vector velocity $u_i$.
\par A relation similar to (4.9) is:
\begin{equation}
\frac{1}{c}\frac{\partial A^0}{\partial t}+\vec{\nabla} \cdot \vec{A} = 0
\end{equation}
the so-called `Lorenz Condition'\footnote{ Not `Lorentz Condition', as found in many text
books. See Reference~\cite{JackOkun}.}, which may also be written more simply as $\partial \cdot A = 0$.
This relation is, in the present approach, not, as in conventional discussions of CEM,
the result of a particular choice of gauge in the definition
of $\vec{A}$, but an identity following from the definition of $A$ in Eqn(3.1). In fact as is easily
shown:
\begin{equation}
\vec{\nabla} \cdot \vec{A} = -\frac{\vec{j} \cdot \vec{r}}{c r^3} = -\frac{1}{c}\frac{\partial A^0}{\partial t}
\end{equation}
Here the derivatives in $\vec{\nabla}$ are with respect to the `field point' $\vec{x}_1$ in contrast with
those in $\vec{\nabla}$ in Eqns(4.9)-(4.11), which are with respect to the spatial coordinate
$\vec{x}_2$ of the object O$_2$ associated with the current $\vec{j}$. The partial time derivative
in (4.12) is defined for $\vec{x}_1$ constant. The time variation of $A^0$ is then due solely
to the time dependence of $\vec{x}_2$, which leads to the second member of (4.13). Eqn(4.12) shows that the
4-vector potential, like the current and energy-momentum 4-vectors corresponds to a conserved
(Lorentz invariant) quantity: $A \cdot A = q^2 /r^2$ \footnote{$r$ is the manifestly
invariant quantity $\sqrt{-(x_1-x_2)^2}$ that appears in eqn(2.10) above.}.
So both $j$ and $A$ differ only by
Lorentz invariant multiplicative factors from $p$ and $u$:
\begin{equation}
c^2 = u \cdot u = \frac{p \cdot p }{m^2} = \frac{j \cdot j}{q^2} = c^2 r^2 \frac{A \cdot A}{q^2}
\end{equation}
The relation (4.12) is found to be important in an interpretation of the
electrodynmamic Maxwell equation,
(4.20) below, as a description of radiation phenomena (creation of real photons). This point will be briefly
discussed in Section 7.
\par The electrodynamic Maxwell equation (Amp\`{e}re's Law, including Maxwell's `displacement
current') is derived immediately on writing the electrostatic Maxwell equation
(4.7) in a covariant form. The latter then appears as an equation for the 0 component of
a 4-vector. The corresponding spatial components, written down simply by inspection,
are Amp\`{e}re's Law.
Writing Eqn(4.7) in 4-vector notation, and introducing also the `non-static' component
of the electric field, given by the second term on the RHS of Eqn(3.11), gives:
\begin{equation}
(\sum_{i=1}^3 -\partial^i \partial^i) A^0 - \partial^0(\sum_{i=1}^3 -\partial^i A^i) = 4 \pi J^0
\end{equation}
Adding to Eqn(4.15) the identity:
\[\partial^0 \partial^0 A^0 -\partial^0 \partial^0 A^0 = 0 \]
gives:
\begin{equation}
(\partial \cdot \partial) A^0 -\partial^0(\partial \cdot A) = 4 \pi J^0
\end{equation}
Since the coefficients of $A^0$ and $-\partial^0$ are Lorentz scalars, the corresponding
$i$th spatial component of the 4-vector $J$, is from the manifest covariance of Eqn(4.16),
given by the equation:
\begin{equation}
(\partial \cdot \partial) A^i -\partial^i(\partial \cdot A) = 4 \pi J^i
\end{equation}
This is Amp\`{e}re's Law in 4-vector notation. In order to recover the more familiar 3-vector
equation, the 4-vector potential must be eliminated in favour of the electric and
magnetic fields defined in Eqns(3.11) and (3.12) respectively. To do this, consider
the contribution of the spatial parts (SP) of the 4-vector products on the LHS of Eqn(4.17)
to $J^1$. This gives:
\begin{eqnarray}
4 \pi J^1(SP) & = & - \sum_{i=1}^3(\partial^i)^2 A^1+\partial^1 \sum_{i=1}^3(\partial^i A^i)
= \sum_{i=1}^3\partial^i(\partial^1 A^i-\partial^i A^1) \nonumber \\
& = & (\partial^1)^2A^1 +\partial^2 \partial^1 A^2+ \partial^3 \partial^1 A^3
- (\partial^1)^2A^1 -(\partial^2)^2A^1 -(\partial^3)^2A^1 \nonumber \\
& = & = -\partial^2(\partial^2 A^1 -\partial^1 A^2) + \partial^3(\partial^1 A^3 -\partial^3 A^1)
\nonumber \\
& = & - \partial^2 B^3 + \partial^3 B^2 = (\vec{\nabla} \times \vec{B})^1
\end{eqnarray}
where, in the fourth line the definition, Eqn(3.12), of the magnetic field has been used.
The contribution of the temporal parts (TP) of the 4-vector products on the LHS of Eqn(4.17)
to $J^1$ is:
\begin{equation}
4 \pi J^1(TP) = (\partial^0)^2 A^1-\partial^1 \partial^0 A^0 = \partial^0 (\partial^0 A^1
-\partial^1 A^0) = -\frac{1}{c} \frac{\partial E^1}{\partial t}
\end{equation}
Adding the spatial and temporal contributions to $J^1$ from Eqns(4.18) and (4.19) gives the 1
component of the electrodynamic Maxwell equation:
\begin{equation}
\vec{\nabla} \times \vec{B} -\frac{1}{c} \frac{\partial \vec{E}}{\partial t} = 4 \pi \vec{J}
\end{equation}
The 2 and 3 components are obtained by cyclic permutation of the indices 1,2,3 in \newline
Eqns(4.18) and (4.19). This derivation of Amp\`{e}re's Law, starting from the electrostatic Maxwell
equation, (4.7) has been previously given by Schwartz~\cite{Schwartz}, and, independently, by
the present author in Reference~\cite{JHF2}, where it was noted that Eqn(4.17) may be derived
from Eqn(4.16) using space-time exchange symmetry invariance.
\SECTION{\bf{Fundamental Concepts and Different Levels of Mathematical Abstraction}}
Equations (3.21),(3.22),(3.31) and (3.32) show that the dynamics of any system of mutually
interacting electrically charged
objects is completely specified by their masses, electric charges and 4-vector positions
and velocities. Other useful and important concepts of CEM such as the 4-vector potential
and electric and magnetic fields are completely specified, in terms of the geometrical and
kinematical configuration of the charged objects by Eqn(3.1) for $A^{\mu}$, Eqns(3.1)
and (3.11) for $\vec{E}$ and
Eqns(3.1) and (3.12) for $\vec{B}$. Historically, of course, Faraday arrived at the concepts
of electric and magnetic fields in complete ignorance of the existence of elementary
electric charges or of Special Relativity. With our present-day understanding of both the existence
of the former and the necessary constraints provided by the latter, it can be seen that both the
4-vector potential and electric and magnetic fields are, in fact, only convenient mathematical
abstractions. The 4-vector potential is at a first level of abstraction. The phenomenologically
most useful concepts of CEM, the electric and magnetic fields are, in turn, completely
specified by $A^{\mu}$ and so are at a second level of abstraction from the fundamental
and irreducible concepts (charged, interacting, physical objects) of the theory.
\par Indeed, there is yet a third level of abstraction, the tensor $F^{\mu \nu}$ of the
electromagnetic field defined as:
\begin{equation}
F^{\mu \nu} = \partial^{\mu} A^{\nu} - \partial^{\nu} A^{\mu}
\end{equation}
This description was introduced by Einstein in his original paper on General Relativity~\cite{Einstein1}
in analogy with the tensor $G^{\mu \nu}$ of the classical gravitational field. It has the merit
of enabling the electrostatic and electrodynamic Maxwell equations to be written as a single compact
equation\footnote{ The covariant operator $\partial_{\nu}$ is introduced by multiplying the
contravariant operator $\partial^{\mu}$ by the metric tensor:
$\partial_{\nu} = g_{\nu \mu} \partial^{\mu}$ where $ g_{\nu \mu}= 0$ for $\nu \ne \mu$ and
$ g_{\mu \mu} =(1,-1,-1,-1)$. Repeated upper and lower indices: $\nu$,$\mu$ are summed
over 0,1,2,3.}:
\begin{equation}
\partial_{\nu} F^{\mu \nu} = 4 \pi J^{\mu}
\end{equation}
As in the case of the introduction of electric and magnetic fields into the covariant Lorentz
force equation (3.10) to obtain the 3-vector version (3.14), a cumbersome equation is reduced
to an elegant one, at the the cost of introducing a higher level of mathematical abstraction.
Viewed, however in the light of the strict criteria of Newton's precept, $A^{\mu}$, $\vec{E}$,
$\vec{B}$ and $F^{\mu \nu}$, (although in the case of $\vec{E}$ and $\vec{B}$ extremely
useful phenomenologically) are certainly not `sufficient' to explain, in any fundamental
manner, the phenomena of CEM. On the contrary, as shown above, Coulomb's law and Special
Relativity, given, of course, the {\it a priori} existence of charged physical objects,
do provide such a fundamental description, in which the `fields' of electromagnetism
appear naturally by mathematical substitution. If all that was known of CEM was Eqn(5.2),
it is hard to see any logical path to derive from it the Lorentz Force, Biot and Savart
and Faraday-Lenz Laws that actually describe the results of laboratory experiments in CEM.
However these laws, Eqn(5.2) and the magnetostatic Maxwell equation
(4.4) are all necessary
consequences of Coulomb's Law, Special Relativity and Hamilton's Principle. The higher
the level of mathematical abstraction, the more elegant the electrodynamic formulae
appear to be, but the further removed they become from the physical realities of
the subject.
\par Although Einstein spent some decades of his life in the unsuccessful attempt
to realise a unifying synthesis between the classical field tensors
$F^{\mu \nu}$ and $G^{\mu \nu}$ he still made a clear distinction between physical
reality and mathematical abstraction\cite{Einstein2}:
\par {\tt We have seen, indeed, that in a more
complete analysis the energy tensor\newline can be regarded only as a provisional means of representing
matter. In \newline reality, matter consists of electrically charged particles, and is to be \newline
regarded itself as a part, in fact the principle part, of the \newline electromagnetic field.}
\par In fact, electrically charged particles and real and virtual photons (which are also particles)
are the true irreducible concepts of CEM. These are not the `principle part' of the
electromagnetic field, but rather {\it replace it}
in the most fundamental description of the phenomena of CEM.
\par Since the only dynamical postulate in CEM is Coulomb's Law, the only way to obtain
a deeper physical understanding is by a deeper understanding of this Law.
Indeed, as will be discussed in the following Section, this does seem to be possible by considering
the particle aspects of the microscopic underlying QED process, which is basically M\o ller scattering:
$e^-e^- \rightarrow e^-e^-$.
\SECTION{\bf{Quantum Electrodynamical Foundations of Classical Electromagnetism}}
If the electrodynamical force is transmitted by particle exchange, and it is assumed that the
magnitude of the force is proportional to the number of interacting particles, which are emitted
isotropically by the source, the inverse square law follows from spatial geometry and
conservation of the number of particles\footnote{A similar physical reasoning was followed
by Kepler in his attempts to understand the gravitational force. As, however, the agents of
force were constrained to propagate in the plane of a planetary orbit, rather than in
three spatial dimensions, a $1/r$ force law was predicted~\cite{Kepler}. Also, as a consequence
of Kepler's Aristotelian understanding of dynamics, the force was conjectured to sweep
the planets around the Sun in the transverse direction, rather than diverting them
radially from their natural rectilinear motion, as in Newtonian dynamics.}.
However, in the Coulomb interaction the exchanged
particle is a virtual, not a real, photon. This means that it cannot always be considered to move in
a particular direction in space-time. It will be shown below, however, that the Fourier transform
of the momentum-space virtual photon propagator does yield a space-time propagator with the
$1/r$ dependence of the Coulomb potential, which corresponds, in the classical limit,
to an inverse square force law. It is also shown that, in the CM frame of the interacting charged particles, this
interaction is instantaneous, as assumed in the derivation of the classical Lagrangian (2.7).
According to QED, the Biot and Savart and
Lorentz Force Laws are the classical limit of M\o ller scattering for very large numbers of
electrons at very large spatial separations. Conversely, M\o ller scattering is the
quantum limit of the Biot and Savart and Lorentz Force Laws when each current contains
a single electron and the spatial separation of the currents is very small. The fundamental
quantum mechanical laws governing M\o ller scattering do not change when many electrons,
with macroscopic spatial separations, participate in the observed physical phenomenon.
A more fundamental understanding of CEM is therefore provided, not by any kind of
field concept, but by properly taking into account the existence of virtual
photons, just as an analysis in terms of real photon production
is mandatory for a fundamental description of the radiative processes
of CEM, a subject beyond the scope of the present paper.
\par The invariant QED amplitude for M\o ller scattering by the exchange of
a single virtual photon\footnote{Actually there are two such amplitudes
related by exchange of the identical final state electrons. In the
present case, where the classical limit of CEM is under discussion,
it suffices to consider only the amplitude given by (6.1) in the
limit $q^2 \rightarrow 0$. The contribution of the
second amplitude is negligible in this limit.} is given by the expression~\cite{HM1}
\footnote{Here units with $\hbar = c = 1$ are assumed.}:
\begin{equation}
T_{fi} = -i \int \frac{{\cal J}^A(x_A) \cdot {\cal J}^B(x_A)}{q^2} d^4x_A
\end{equation}
The corresponding Feynman and momentum-space diagrams are shown in Fig.1.
The virtual photon is exchanged between the 4-vector currents ${\cal J}^A$
and ${\cal J}^B$ defined in terms of plane-wave solutions, $u_i$,~$u_f$ of the Dirac
equation:
\begin{equation}
{\cal J}^A_{\mu} \equiv - e \overline{u}_f^A \gamma_{\mu} u_i^A \exp[i(p_f^A-p_i^A)\cdot x_A]
\end{equation}
where $p_i^A$ and $p_f^A$ are the energy-momentum 4-vectors of the incoming
and scattered electron, respectively, that emit a virtual photon at the space-time
point $x_A$ and $-e$ is the electron charge. The overall centre-of-mass frame (Fig1b) is a Breit frame for the virtual
photon, i.e. the latter has vanishing energy:
\begin{equation}
q^{A0} = p_i^{A0}- p_f^{A0} = - q^{B0} = p_f^{B0}- p_i^{B0} = 0
\end{equation}
Thus, in this frame, the invariant amplitude may be written:
\begin{equation}
T_{fi} = i \int \frac{{\cal J}^A(x_A) \cdot {\cal J}^B(x_A)}{|\vec{q}|^2} d^4x_A
\end{equation}
As shown in the Appendix, use of the Fourier transform:
\begin{equation}
\frac{1}{|\vec{q}|^2} = \frac{1}{4 \pi} \int \frac{d^3 x e^{i\vec{q} \cdot \vec{x}}}{|\vec{x}|}
\end{equation}
enables the invariant amplitude to be written as the space-time integral:
\begin{equation}
T_{fi} = \frac{i}{4 \pi} \int dt_A \int d^3 x_A \int d^3 x_B \frac{{\cal J}^A(\vec{x}_A,t_A) \cdot
{\cal J}^B(\vec{x}_B,t_A)}{|\vec{x}_B-\vec{x}_A|}
\end{equation}
It can be seen that the integrand in Eqn(6.6) has exactly the same $j \cdot j/r$ structure as the
potential energy term in the invariant CEM Lagrangian (2.7). Indeed this is to be expected in the Feynman
Path Integral (FPI) formulation of quantum mechanics~\cite{Feyn1}.
The physical meaning of Eqn(6.6) is that
the total amplitude is given by integration over all spatial positions: $\vec{x}_A(t_A)$, $\vec{x}_B(t_A)$
at time $t_A$, and all times $t_A$, of emission and absorption of a single virtual photon
\footnote{Thus the simple momentum-space propagator $1/q^2$ of
Eqn(6.1) is equivalent, in space-time, to the exchange of an infinity of virtual
photons emitted and absorbed at different spatial positions and times. All these virtual photons however
have, according to Eqn(6.6), infinite velocity.} in the scattering
process: $e^-e^- \rightarrow e^-e^-$. Since the virtual photon is not observed, this is just a
manifestation of quantum mechanial superposition: a sum of different probability amplitudes with the same
initial and final states.
Notice that the virtual photon propagates with infinite velocity between the spatial positions
$\vec{x}_A$, $\vec{x}_B$ so that the ambiguity in the direction of propagation of the
space-like virtual
photon (see Fig.1b and c) has no relevance. Thus QED predicts that virtual photons
produce instantaneous `action at a distance' in the overall centre-of-mass frame of M\o ller scattering.
This is also implicit in the discussion of CEM in Sections 2 and 3 above, since all forces are
defined at a fixed time in the CM frame of the interacting charges. The meaning of the
retarded Li\'{e}nard-Wiechert~\cite{LW} potentials and `causality' in relation to the instantaneous
forces transmitted by space-like virtual photons is discussed in the concluding section
of this paper.
\begin{figure}[htbp]
\begin{center}
\hspace*{-0.5cm}\mbox{
\epsfysize15.0cm\epsffile{emagf1c.eps}}
\caption{{\sl a) Feynman diagram for M\o ller scattering: $e^+e^- \rightarrow e^+e^-$, by exchange of a single
space-like virtual photon. b), c) show the possible momentum space diagrams for M\o ller scattering
in the CM frame. In b)[c)] the virtual photon transfers momentum from the current ${\cal J}^A$ [${\cal J}^B$]
to ${\cal J}^B$ [${\cal J}^A$]. These are equivalent descriptions. In both cases the energy of the virtual
photon vanishes and it has infinite velocity.}}
\label{fig-fig1}
\end{center}
\end{figure}
\par To examine more closely the connection between Eqn(6.6) and the FPI formalism, consider the general
FPI expression for a transition amplitude~\cite{Feyn1}:
\begin{equation}
T_{fi}^{FPI} \equiv \langle \chi(t_f)|\psi(t_i) \rangle = \int_{{\rm paths}}
\chi^{\ast}(x_f,t_f) e^{iS} \psi(x_i,t_i){\cal D} x \equiv
\int_{{\rm paths}} \langle f|e^{iS}|i \rangle {\cal D} x
\end{equation}
where the Action, $S$, is given by the time integral of the {\it classical} Lagrangian, $L$, of the
quantum system under consideration:
\begin{equation}
S = \int_{t_i}^{t_f} L(x,\dot{x}) dt \end{equation}
(here the upper dot denotes time differentiation) and
\begin{equation}
{\cal D} x \equiv {\rm Lim}~(\epsilon \rightarrow 0)~ \frac{d x_0}{A} \frac{d x_1}{A}... \frac{d x_{j-1}}{A}
\frac{d x_{j}}{A}
\end{equation}
where $x_0$,$x_1$,... denote sucessive positions along the path, each separated by a small, fixed, time
interval $\epsilon$. Also $dx_j \equiv x_j - x_{j-1}$. $A$ is a normalistation constant that depends
upon $\epsilon$. In the case of present interest, M\o ller scattering, the one dimensional FPI (6.7),
with a single particle, is generalised to three spatial dimensions and two particles
with the label $p = A, B$, $x \rightarrow (x_p^1,x_p^2,x_p^3)$, corresponding to the two electrons which
scatter from each other (see Fig.1).
In this case, (6.7) is generalised to~\cite{FHPI}:
\begin{equation}
T_{fi}^{FPI} = \int_{{\rm paths}} \langle f|e^{iS}|i \rangle\prod_{p=A,B} \prod_{j = 1}^3 {\cal D} x_p^j(t)
\end{equation}
and (6.8) to
\begin{equation}
S = \int_{t_i}^{t_f} L(\vec{x}_A, \dot{\vec{x}}_A;\vec{x}_B, \dot{\vec{x}}_B)dt
\end{equation}
where $i$ and $f$ are the initial and final states of the M\o ller scattering process.
Assuming that the transition $f \rightarrow i$ is caused by a small term $S_{int}$ in the action where
$S = S_0+S_{int}$ and $\langle f|S_0| i\rangle = 0$, enables (6.10) to be written as:
\begin{eqnarray}
T_{fi}^{FPI} & = & \int_{{\rm paths}} \langle f|e^{iS_{int}}|i \rangle \prod_{p=A,B} \prod_{j = 1}^3 {\cal D} x_p^j(t)
\nonumber \\
& = & \int_{{\rm paths}} \langle f|1 + iS_{int} +\frac{(iS_{int})^2}{2!}+...)
|i \rangle \prod_{p=A,B} \prod_{j = 1}^3 {\cal D} x_p^j(t) \nonumber \\
& = & i \int_{{\rm paths}}\langle f|S_{int}| i \rangle
\prod_{p=A,B} \prod_{j = 1}^3 {\cal D} x_p^j(t) + O(S_{int}^2)\nonumber \\
& \simeq & i \int dt \int_{{\rm paths}}\langle f|L_{int}| i \rangle
\prod_{p=A,B} \prod_{j = 1}^3 {\cal D} x_p^j(t) \nonumber \\
&\propto & i \int dt_A \int d^3 x_A \int d^3 x_B\langle f| L_{int} | i \rangle
\end{eqnarray}
In the last line the formal differentials ${\cal D} x^i_p(t)$ for arbitary space-time paths
$x^i_p(t)$ are replaced by those corresponding
to the electrons A and B in the M\o ller scattering process that, in the classical limit, propagate
along straight-line paths so that\footnote{Different choices of $\epsilon$ correspond to different values
of $j$ in Eqn(6.9), for a given value of $\Delta x = x_j -x_0$. In the case of a straight line
path the value of the limit in (6.9) is independent of the value of $\epsilon$. In particular, the
choice $\epsilon = \Delta x/v$ is possible. This yields Eqn(6.13).}:
\begin{equation}
\prod_{p=A,B} \prod_{j = 1}^3 {\cal D} x_p^j(t) \propto d^3x_A d^3x_B
\end{equation}
where the normalisation constant in (6.9) has been dropped, since only the proportionality of the
matrix elements (6.6) and (6.12) is under investigation.
Comparison of Eqns(6.6) and (6.12) gives:
\begin{equation}
\langle f| L_{int} | i \rangle \propto \frac{{\cal J}^A(\vec{x}_A,t_A) \cdot
{\cal J}^B(\vec{x}_B,t_A)}{4 \pi|\vec{x}_B-\vec{x}_A|}
\end{equation}
In order to compare Eqn(6.14) with the potential
energy term in Eqn(2.7) which has the same 4-vector structure as Eqn(6.14), the classical
limit of the QED transition currents ${\cal J}^A$ and ${\cal J}^B$, where the momentum carried
by the virtual photon vanishes, must be considered. For this it is convenient to use the Gordon
Identity~\cite{IZ1} for the spinor product appearing in Eqn(6.2):
\begin{equation}
\tilde{{\cal J}}^{\mu} \equiv -e \overline{u}_f \gamma^{\mu} u_i =
\frac{-e}{2m} \overline{u}_f \left[ (p_f + p_i)^{\mu} + i \sigma^{\mu \nu}(p_f- p_i)_{\nu}\right] u_i
\end{equation}
where
\[ \sigma^{\mu \nu} \equiv \frac{1}{2}(\gamma^{\mu}\gamma^{\nu}-\gamma^{\nu}\gamma^{\mu}) \]
In the overall centre-of-mass frame in the limit of vanishing virtual photon momentum:
$(p_f- p_i)_{\nu} \rightarrow 0$, $(p_f+ p_i)^{\mu} \rightarrow 2 p^{\mu}$, and
$ \overline{u}_f u_i \rightarrow \overline{u}_i u_i = 2m$~\cite{HM2}. Thus, in this classical
limit Eqn(6.2) gives:
\begin{equation}
\tilde{{\cal J}}^{\mu}_{class} = -e 2 p^{\mu} = 2 Q u^{\mu} = 2 j^{\mu}
\end{equation}
where $Q = -e$. Therefore, up to a multiplicative constant, the classical limit of the QED transition
current $\tilde{{\cal J}}^{\mu}$ is identical to the CEM current $Q u^{\mu}$ introduced in
Section 2 above, and also, up to a constant multiplicative factor, the classical limit
of the matrix element of the QED interaction Lagrangian $\langle f| L_{int} | i \rangle$
is equal to the
potential energy term in the CEM Lagrangian (2.7). There is thus a
seamless transition from QED to CEM.
\par Hamilton's Principle of classical mechanics is the $h \rightarrow 0$ limit
of Feynman's path integral formulation of quantum mechanics. So it may be said that the third postulate in the
derivation, from first principles, of CEM presented in this paper is not
really an independent premise, but rather a prediction of quantum mechanics. To show this, it is
necessary to consider the behaviour of the fundamental FPI formula (6.7)
for a transition amplitude in the classical limit. The action S in this formula is a functional of the different
space-time paths $x(t)$. Writing explicitly the dependence on Planck's constant, gives
a multiplicative factor $\exp(iS[x(t)]/\hbar)$ in the transition amplitude. If the paths
$x(t)$ are chosen such that the variation of $S$ is large in comparison to $\hbar$,
this factor will exhibit rapid phase oscillations and give a negligible contribution
to the transition amplitude. If, however, the paths are chosen in such a way that
$S$ is near to an extremum with respect to their variation, $S$ will change
only very slowly from path to path, so that the contributions of different paths
have have almost the same phase, resulting in a large contribution to the
scattering amplitude. The classical limit corresponds to $\hbar \rightarrow 0$,
where only the path giving the extremum of $S$ contributes. This path is just
the classical trajectory as defined by Hamilton's Principle. This argument,
that may be called the `Stationary Phase Principle', was first given by
Dirac in 1934~\cite{Dirac1}(see also Reference~\cite{Dirac2}) and was an
important motivation for Feynman's space-time reformulation of the
principles of quantum mechanics~\cite{Feyn1,FHPI}.
\par At this point it can be truthfully said that there is
`nothing left to explain' for an understanding of the fundamental physics
of CEM, given the laws of special relativity and quantum mechanics.
The irreducible physical concepts are electrically charged physical objects
and space-like virtual photons. Coulomb's Law is a consequence of the
exchange of the latter between the former. Hamilton's Principle is naturally given by
the classical limit of the FPI formulation of quantum mechanics.
\par It is interesting, in the light of this `complete understanding' that quantum mechanics
and relativity provide about CEM, to consider two further quotations
from the {\it Principia}. The first is taken from the `General Scholium'~\cite{NP1}.
After describing the inverse-square law of the gravitational force Newton states:
\par {\tt But hitherto I have not been able to discover the cause of these \newline properties of gravity
from phenomena, and I frame no hypothesis, for \newline whatever is not derived from the
phenomena is to be called a hypothesis, and hypotheses, whether metaphysical
or physical, whether of occult qualities or mechanical, have no place in experimental
philosophy.}
\par The second is from the `Author's Preface to The Reader'~\cite{NP2}
\par {\tt I wish we could derive the rest of the phenomena of Nature by the same
\newline kind of reasoning from mechanical principles, for I am induced by many reasons
to suspect that they may all depend upon certain forces by which the particles of bodies, by some
causes hitherto unknown, are either mutually \newline impelled towards one another
and cohere in regular figures or are repelled and recede from one another.}
\par Now, at the beginning of the 21st century, Newton's wish to understand,
at a deeper level, the forces of nature, has been granted,
at least for the case of electromagnetic ones. What was needed was not
`the same kind of reasoning from mechanical principles' that Newton considered
but the discovery of relativity and quantum mechanics. The cause `hitherto unknown'
of the electromagnetic force is the exchange of space-like virtual photons according to the
known laws of QED.
\par Regrettably science is, at the time of this writing, riddled by many `hypotheses'
of the type referred to in the first of the above quotations. One such hypothesis, that
has persisted through much of the 19th century and all of the 20th is that: `No physical
influence can propagate faster than the speed of light'. This is contradicted by the
arguments given above and, as discussed in the following section, also by the results of
some recent experiments.
\SECTION{\bf{Discussion and Outlook}}
The starting point and aims of the present paper are very close to those of Feynman
and Wheeler when they attempted, in the early 1940's, to reformulate CEM in terms of direct
inter-charge interactions without the {\it a priori} introduction of any
electromagnetic field concept. In this way the infinite self-energy terms associated
with the electric field of a point charge are eliminated. As Feynman put it~\cite{Feyn2}:
\par{\tt You see then that my general plan was to first solve the classical \newline problem,
to get rid of the infinite self-energies in the classical theory, and to hope that when
I made a quantum theory of it everything would be\newline just fine.}
\par Feynman and Wheeler had a project to write three papers on the subject~\cite{Feyn3}.
The first of these papers was to be a study of the classical limit of the
quantum theory of radiation.
Feynman had yet to formulate his space-time version of QED, and this paper
was never written. In the remaining two papers~\cite{FW1,FW2} it was proposed to introduce
direct interparticle action by including the effects of both retarded and `advanced' potentials
as well as an array of `absorbers'. As suggested by Dirac~\cite{Dirac3} half of the difference
between the retarded
potential of an accelerated charge and of the `advanced' potential from the absorbers correctly
predicts the known radiative damping force of CEM. The second paper,~\cite{FW2}, developed
further this theory by exploiting the Fokker action principle formulation of
action-at-a-distance in CEM~\cite{Fokker}. As stated in the introduction of
this paper, a description was being sought that was:
\begin{itemize}
\item[(a)] {\tt well defined}
\item[(b)] {\tt economical in postulates}
\item[(c)] {\tt in agreement with experience}
\end{itemize}
that is, in other words, in accordance with Newton's first `Rule of Reasoning in Philosophy' quoted above.
However, in order to reproduce the known results of CEM by such a theory
`advanced' potentials had to be introduced. This immediately gives an apparent breakdown
of causality and the logical distinction between `past', `present' and `future'.
As concisely stated by
Feynman and Wheeler themselves~\cite{FW2}:
\par {\tt The apparent conflict with causality begins with the thought: if the \newline present motion of
$a$ is affected by the future motion of $b$, then the \newline observation of $a$ attributes a certain
inevitabilty to the motion of $b$. Is not this conclusion in conflict with our recognised ability
to influence \newline the future motion of $b$?}
\par Feynman and Wheeler then gave a rather artificial example (which the present writer finds unconvincing)
that was claimed to resolve this causal paradox.
\par In fact, Feynman and Wheeler were compelled
to introduce `advanced' potentials because they were assuming, as did also Fokker and earlier
authors attempting to formulate theories of direct interparticle action in CEM, that causality
meant that no physical influence could be transmitted faster than the speed of light in
vacuum. This definition of `causality' seems to have been introduced into physics by
C.F.Gauss in 1845~\cite{Gauss}. Somewhat later, C.Neumann proposed~\cite{Neumann} that the
electric potential responsible for interparticle forces should be transmitted, not
at the speed of light, but instantaneously, like the gravitational force in Newton's
theory. As shown above, this is indeed how, in QED, space-like virtual photons
transmit the electromagnetic force between charged objects in their common
CM frame. These two hypotheses will be refered to below below
as `Gaussian' and `Neumann' Causality. The fundamental Lagrangian of CEM describing the interaction
of charged objects, in any inertial frame, is the simple expression Eqn(2.7) above, not
the conjectured, and much more complicated, Fokker action that embodies Gaussian Causality.
It is important to
stress that the instantaneous action-at-a-distance, of Neumann Causality, which is just the
limit of Gaussian Causality as $c \rightarrow \infty$, unlike an `advanced' potential,
poses no logical problem of the influence of the future on the present, as
succinctly stated by Feynman and Wheeler in the above quotation.
\par Gaussian Causality has been an unstated (and unquestioned) axiom of physics since the advent
of Special Relativity
a century ago. The speed
of light is certainly the limiting velocity of any physical object described
by a time-like energy-momentum 4-vector. However Einstein at the time when he
invented special relativity, and Feynman himself, at the time of his collaboration
with Wheeler, were not aware of the concept of the `virtual' particles. The latter,
associated with the space-time propagators introduced into
QED by Feynman and Stueckelberg, may be described
by space-like energy-momentum 4-vectors. The instantaneous action at a distance of the virtual
photons in M\o ller scattering described by the invariant amplitude in Eqn(6.6) above,
can be simply understood from the relativistic kinematics of such virtual particles.
The relativistic velocity
$\beta = v/c$ of a particle in terms of its 3-momentum $\vec{p}$ and 4-momentum $p$ is,
in general, given by the expression:
\begin{equation}
\beta = \frac{|\vec{p}|}{\sqrt{\vec{p}^2 + p \cdot p}}
\end{equation}
Thus space-like virtual particles, for which, by definition: $ p \cdot p < 0$, are tachyons.
For the case of the virtual photons exchanged in the center-of-mass-system of M\o ller
scattering (Figs1b and 1c): $ p \cdot p = -\vec{p}^2$, since $p^0 =0$, and so $\beta$ is infinite,
consistent with the space-time description in Eqn(6.6).
\par That the Feynman propagator for a {\it massive} particle
violates Gaussian Causality was pointed out by Feynman himself in his first
QED paper~\cite{Feyn4} and later discussed by him in considerable detail~\cite{Feyn5}.
This fact is also sometimes mentioned in books on Quantum Field Theory, that otherwise make the
contradictory claim that, in general, quantum field operators commute for space-like
separations, so that, in consequence, no physical influence can propagate faster than the
speed of light\footnote{ For example in Reference~\cite{IZ1} it is stated, in
connection with the commutation relation for a pair of scalar fields (Eqn(3.55) of
~\cite{IZ1}) that: `Measurements at space time separated points do not interfere
as a consquence of locality and causality', whereas in the discussion of
the Feynman propagator $G_F(x)$ in Section 1.3.1 it is stated that:
`While the previous Green functions were zero outside the light cone this is not the case for
$G_F(x)$ which has an exponential tail at negative $x^2$.' The $G_F(x)$ discussed
here is that corresponding to a classical field , but the corresponding quantum
propagator, $S_F(x)$, has a similar property~\cite{Feyn4,Feyn5}.}
The space time propagator of a massive particle was shown by Feynman to be, in
general, a Hankel function of the second kind~\cite{Feyn4}. For an on-shell particle, or a virtual
particle propagating over a large proper time interval, $ \Delta \tau $,
the propagator has a simpler
functional dependence $\simeq \exp(-i m \Delta \tau )$ where $m$ is the pole mass of the particle
and the proper time interval is defined by the relations:
\[ \Delta \tau \equiv \sqrt{\Delta t^2-\Delta x^2}~~~{\rm for}~~~\Delta t^2 \ge \Delta x^2 \]
\[ \Delta \tau \equiv -i\sqrt{\Delta x^2-\Delta t^2}~~~{\rm for}~~~\Delta x^2 > \Delta t^2 \]
For space-like separations: $\Delta x^2 > \Delta t^2$ appropriate for the virtual photons
mediating the Coulomb force, $ \Delta \tau$ is imaginary. This would imply an exponentially
damped range of the associated force for the exchange of a massive particle\footnote{For example
the Yukawa force due to the exchange of virtual pions in nuclear physics.}. Since, however,
the pole mass of the photon vanishes, no such damping occurs for the exchange of virtual
photons. The corresponding force law is then the same as for the exchange of real (`on-shell')
particles, that is, inverse square.
\par It is instructive to compare Feynman's own discussion of the virtual photon propagator
in space-time~\cite{Feyn6} to the related one of the invariant amplitude for M\o ller
scattering in Section 6 above. Feynman writes out explicitly the 4-vector product in
Eqn(6.1) to obtain:
\begin{equation}
T_{fi} = -i \int \frac{({\cal J}^{A0}{\cal J}^{B0}-{\cal J}^{A1}{\cal J}^{B1}
-{\cal J}^{A2}{\cal J}^{B2} -{\cal J}^{A3}{\cal J}^{B3})}{q^2} d^4x_A
\end{equation}
Conservation of the current ${\cal J}$ gives the condition:
\begin{equation}
q \cdot {\cal J} = q^0 {\cal J}^0-|\vec{q}| {\cal J}^3 = 0
\end{equation}
where the 3 axis has been chosen parallel to $\vec{q}$. Use of (7.3) to eliminate $ {\cal J}^{A3}$
and $ {\cal J}^{B3}$
enables (7.2) to be written as:
\begin{equation}
T_{fi} = i \int\left[ \frac{({\cal J}^{A1}{\cal J}^{B1}
+{\cal J}^{A2}{\cal J}^{B2})}{q^2}+\frac{{\cal J}^{A0}{\cal J}^{B0}}{\vec{q}^2}\right] d^4x_A
\end{equation}
Feynman then performs a Fourier transform of $(\vec{q})^{-2}$ using Eqn(6.5) to obtain, for the last
term in the large square bracket of Eqn(7.4) an equation similar to (6.6) above, but with
the replacement: ${\cal J}^{A} \cdot {\cal J}^{B} \rightarrow {\cal J}^{A0}{\cal J}^{B0}$.
The instantaneous nature of the Coulomb interaction in this term is noted, but it is also implied
that the contribution of the transverse polarisation modes:
$({\cal J}^{A1}{\cal J}^{B1} +{\cal J}^{A2}{\cal J}^{B2})/q^2$ is not instantaneous. Feynman stated:
\par {\tt The total interaction which includes the interaction of transverse \newline photons then gives
rise to the retarded interaction.}
\par This statement is not true when $ T_{fi}$ is evaluated in the CM frame.
In this case: $q^2 = -\vec{q}^2$, Eqn(6.6) results and the whole interaction of the virtual
photon is instantaneous.
\begin{figure}[htbp]
\begin{center}
\hspace*{-0.5cm}\mbox{
\epsfysize15.0cm\epsffile{emagf2c.eps}}
\caption{{\sl Momentum space diagrams for M\o ller scattering of ultra-relativistic electrons
by $\pi/4$ radians in the CM frame, as in Fig 1 b) and c), as viewed by different
observers. In a)[b)] the observer is moving parallel to $\vec{p}_i^B$ [$\vec{p}_i^A$]
with velocity 3c/5 relative to the CM frame. Momentum conservation requires that
in a) the virtual photon propagates from ${\cal J}^A$ to ${\cal J}^B$ so that
$t_B>t_A$, whereas in b) the photon propagates from ${\cal J}^B$ to ${\cal J}^A$ and
$t_A>t_B$, where $t_A$ and $t_B$ are the effective times of emission or absorption of the photon
by the currents ${\cal J}^A$ and ${\cal J}^B$. In both cases the effective velocity $v = pc^2/E$
of the photon is superluminal: v = 1.044c.}}
\label{fig-fig2}
\end{center}
\end{figure}
\par It is amusing to note that a faint `ghost' of Wheeler and Feynman's `advanced' and
`retarded' potentials subsists in the momentum space diagrams Fig.1b and 1c.
The two kinematically distinct situations (i) a virtual photon with momentum $\vec{q}^A$
propagates from current A to current B (Fig1b) and (ii) a virtual photon with momentum
$\vec{q}^B = -\vec{q}^A$ propagates from current B to current A (Fig1c) are completely equivalent
descriptions of the scattering process in the CM frame where $t_A = t_B$. As shown in Fig 2, however,
this is no longer the the case if the scattering process is observed in a different inertial
frame. In Fig 2a the observer is moving with relativistic velocity $\beta = 3/5$ parallel to
the direction of $\vec{p}_i^B$ in the CM frame. Thus in the observer's proper frame,
$|\vec{p}_i^B|$ is halved and $|\vec{p}_i^A|$ doubled. In Fig2b, the observer moves with the
same velocity relative to the CM frame, parallel to $\vec{p}_i^A$. In both cases it follows
from momentum conservation that there is no possible ambiguity between the momentum space
configurations shown in Figs 2a and 2b. In Fig 2a the virtual photon must propagate from current A to B
and so $t_B > t_A$, and in Fig 2b from current B to A
so that $t_A > t_B$. Assuming that the electrons are ultrarelativistic, $E \simeq |\vec{p}|$, and that
the electrons scatter through $\pi/4$ rad in the CM frame, as shown in Fig 1, the relativistic velocity
of the virtual photon in the observer's frame is $\beta = 1.044$ for both cases shown in Fig 2.
Thus the causal description of scattering processes in momentum space is, in general, frame dependent,
being ambiguous only in
the CM frame\footnote{It must not be forgotten, however, that the configurations shown in Fig.2 are
in momentum space, not space-time. The different time ordering of `events' in different
frames that seems apparent on comparing Fig.2a and Fig.2b must therefore be treated
with caution. In fact, as discussed previously, there is not, in space-time, the exchange
of single virtual photons with fixed 4-momenta, as seen in Figs 1 and 2, but rather the
sum over an infinite number of amplitudes corresponding to exchanges of virtual photons
between all space-time points occupied by the trajectories of the scattered particles,
as in Eqn(6.6). In the CM frame all such photons have infinite velocity.}. Because of this
ambiguity, in the kinematical configuration of Fig 1b, the virtual
photon $\gamma_A$ can be considered as the limit as $t_B-t_A \rightarrow 0$ of an `retarded' interaction
from A as seen by B, whereas in Fig 1c $\gamma_B$ corresponds to the
limit as $t_B-t_A \rightarrow 0$ of an `advanced' interaction produced by B that interacts with A
\footnote{This corresponds to time increasing from left to right in the momentum space diagrams
of Fig.1b and 1c, in the same way as in Fig.2a or the Feynman diagram in Fig.1a.}.
Since the two descriptions are equivalent, the effect is the same as the $t_B-t_A \rightarrow 0$
limit of half the sum of the retarded interaction produced by the current B and the
`advanced' interaction produced by the current A. This is the `ghost' of Feynman
and Wheeler's advanced and retarded potentials mentioned above.
The current A (B) behaves as the `absorber' for the
interactions of the current B (A). Unlike in Feynman and Wheeler's formulation however there is no radiation
and therefore no `radiation resistance'. The photons responsible for the intercharge
interaction are purely virtual.
\par As often emphasised by Feynman~\cite{Feyn7}, QED is based on only three elementary amplitudes
describing, respectively, the propagation of electrons or photons from one space-time point to
another and the amplitude for an electron to absorb or emit a photon. The latter is proportional
to the classical electric charge of the electron. Since only kinematics, and not the coupling
constant of QED, changes when virtual photons are replaced by real ones it should not be surprising
if the various field concepts introduced to describe the effect of the virtual photons that generate
intercharge forces should also be able to provide a description of the observed effects of
the creation and absorption of real photons. As will now be shown, this is indeed the case.
\par A clear distinction should be made however, at the outset, between the fields so far discussed in the present
paper, representing the effects of virtual photon exchange, and the related fields
denoted here as $A_{rad}$, $\vec{E}_{rad}$ and $\vec{B}_{rad}$ that provide a description of
physical systems comprised
of large numbers of real photons\footnote{This distinction is usually not
made in text books on CEM}. As shown in a recent paper by the present author~\cite{JHF3},
a complex representation of these radiation fields may be identified, in the limit of very low photon
density, with the quantum wavefunction of a single real photon\footnote{This wavefunction occurs for example,
in the construction of invariant amplitudes of all processes in which real photons are
created or destroyed. The related problem of 'non localisability' of photons is also discussed
in Reference~\cite{JHF3}.}
\par The electrodynamic Maxwell equation (4.20) as written above therefore describes only
the effects of virtual photon exchange. All fields and currents are defined at some unique
time in the CM frame of the interacting charges. The solutions of this equation, $\vec{E}$,~$\vec{B}$ are given
by Eqns(3.11), (3.12) respectively and (3.1). To arrive at a description of real photons it is
convenient to express the electrodynamic Amp\`{e}re Law of Eqn(4.20) uniquely in terms of the 3-vector potential
by using the Lorenz Condition (4.12) to eliminate the scalar potential $A^0$.
The result of this simple exercise in 3-vector algebra, which may be found in
any text-book on CEM, is:
\begin{equation}
-\nabla^2\vec{A}_{rad} +\frac{1}{c^2} \frac{\partial^2 \vec{A}_{rad}}{\partial t^2}
= 4 \pi \vec{j}_{rad}
\end{equation}
The `radiation' suffix has been added to $\vec{A}$ and $\vec{j}$ to distinguish them
from the quantities $\vec{A}$ and $\vec{j}$ defined in Eqns(3.1) and (4.8) since the
latter are not solutions of Eqn(7.5) unless $c$ is infinite.
Similarly by using the Lorenz condition to eliminate $ \vec{A}$ in favour
of $A^0$ the inhomogeneous D'Alembert equation for the scalar potential may be
derived:
\begin{equation}
-\nabla^2 A^0_{rad} +\frac{1}{c^2} \frac{\partial^2 A^0_{rad}}{\partial t^2}
= 4 \pi j^0_{rad}
\end{equation}
As shown for example in Reference~\cite{Jackson}, the solutions of Eqn(7.5) and (7.6) are similar to (3.1)
except that they are retarded in time:
\begin{equation}
\vec{A}_{rad}(t) = \left\{\frac{\vec{j}}{c(r-\frac{\vec{v} \cdot \vec{r}}{c})}\right\}
_{t-\frac{r}{c}}
\end{equation}
\begin{equation}
A^0_{rad}(t) = \left\{\frac{j^0}{c(r-\frac{\vec{v} \cdot \vec{r}}{c})}\right\}
_{t-\frac{r}{c}}
\end{equation}
where the large curly bracket indicates that $\vec{j}$ and $r$ are evaluated at the retarded time
$t-r/c$. It follows that $ \vec{A}_{rad}(t)$ and the associated
electromagnetic fields $\vec{E}_{rad}(t)$ and $\vec{B}_{rad}(t)$ describe some physical
effect produced by the source current at time $t-r/c$, i.e. that propagates from the
source to the point of observation with velocity $c$. In reality, the energy-momentum
flux, associated with the corresponding `electromagnetic wave' produced by the source,
consists of a very large number of real photons whose energy distribution depends on the
acceleration of the source at their moment of emission. Thus the solutions
(7.7) and (7.8) imply the existence of massless physical objects (`photons')~\cite{LLB,JHF4},
created by the source current. As discussed in Reference~\cite{JHF3}, comparison of the
known properties of both photons and the classical electromagnetic waves associated with the
fields $A_{rad}$, $\vec{E}_{rad}$ and $\vec{B}_{rad}$ enables many fundamental concepts
of quantum mechanics to be understood in a simple way.
\par Text books and papers on CEM do not usually make the above distinction between the fields
$\vec{E}$ and $\vec{B}$, describing the mechanical forces acting on charges, and
$\vec{E}_{rad}$ and $\vec{B}_{rad}$ that provide the classical description of radiation
phenomena, employing identical symbols for both types of fields. An important exception to this
is the work of Reference~\cite{CSR}. In this paper,
the instantaneous nature of the interactions mediated by the $\vec{E}$ and $\vec{B}$ fields,
derived in the previous section from QED, is conjectured. These fields are solutions of
the Maxwell equations: (4.4), (4.6), (4.7) and (4.20). Different, retarded, fields,
solutions of the D'Alembert equation, equivalent to $\vec{E}_{rad}$ and $\vec{B}_{rad}$,
denoted as $\vec{E^*}$ and $\vec{B^*}$ were also introduced. The application of the
Poynting vector and spatial energy density formulae uniquely to the fields
$\vec{E^*}$ and $\vec{B^*}$ was pointed out. However, instead of the formulae
(7.7) and (7.8) above, only `sourceless' solutions of the homogeeous D'Alembert equation
were considered. Also it was proposed, instead of the formulae (3.15) and (3.16) above,
to define $\vec{E}$ and $\vec{B}$ as the standard `present time' Li\'{e}nard and Wichert
formulae\footnote{ See, for example, Reference~\cite{PP}.} which, for a uniformly moving charge,
are actually equivalent to {\it retarded} fields. The discussion of Reference~\cite{CSR}.
was carried out entirely at the level of classical fields, considered as solutions of
partial differential equations with certain boundary conditions. No identification of
$\vec{E}$ and $\vec{B}$ with the exchange of virtual photons and $\vec{E^*}$ and $\vec{B^*}$
as the classical description of real photons was made. The suggestion that $\vec{E}$ and $\vec{B}$
should be associated with exchange of virtual photons `not subject to causal limitations'
has, however, been made in a recent paper~\cite{ALK}
\par The electric and magnetic fields derived from the Li\'{e}nard and Wiechert potentials
(7.7) and (7.8) contain terms with both $1/r^2$ and $1/r$ dependencies. Both fields are
retarded, but conventionally only the latter are associated with radiative effects (the fields
$\vec{E}_{rad}$ and $\vec{B}_{rad}$) in CEM. It is interesting to note that there
is now mounting experimental evidence~\cite{Walker,KMSTCM}, that the fields $\simeq~1/r^2$
are instantaneous and not retarded, and so should be associated with the force fields
$\vec{E}$ and $\vec{B}$ mediated by virtual photon exchange. Particularly convincing are the
results shown in Reference~\cite{KMSTCM} where the temporal dependence of near- and far-magnetic
fields were investigated by measuring electromagnetic induction at different distances from
a circular antenna. Figure 8 of~\cite{KMSTCM}apparently shows clear evidence for the instantaneous
nature of the $1/r^2$ `bound fields' (i.e. fields associated with virtual photon exchange). This suggests
that the retarded $1/r^2$ solutions of (7.5) and (7.6) should be discarded as unphysical,
whereas the retarded $1/r$ solutions describing correctly the `far-field' in the
experiment~\cite{KMSTCM} do give the correct classical description of the radiation of real
photons. There seems now to be therefore experimental evidence for electromagnetic fields
respecting both
Neumann causality (the force fields $\vec{E}$ and $\vec{B}$) as well as Gaussian causality
(the radiation fields $\vec{E}_{rad}$ and $\vec{B}_{rad}$).
\par Maxwell's original discovery of electromagnetic waves~\cite{Maxwell} was based on an equation
similar to (7.5) for components of the electromagnetic fields, but without any source term, which
is just the well-known classical Wave Equation in three spatial dimensions
Although this procedure leads, in a heuristic manner, to the concept of
`electromagnetic waves' propagating at speed $c$, with vast practical, political and
sociological consequences, it can be seen, with hindsight, to have been a mistake from
the viewpoint of fundamental physics. In fact, if the current vanishes, so, by definition,
do all the fields whether instantaneous as in Eqn(3.1) or retarded as in Eqn(7.5). If all the
fields vanish there can evidently be no `waves'.
The result of this mistake was many decades of fruitless work by Maxwell and others
to invent a medium (the luminiferous aether) in which such `sourceless' waves might
propagate and whose properties would predict the value of $c$. Now it is understood that
the energy density $(\vec{E}_{rad}^2+\vec{B}_{rad}^2)/8\pi$ of a plane `electromagnetic wave'
is simply that of the beam of real photons of which it actually consists~\cite{JHF3}.
\par The existence of photons, massless particles with constant velocity c, is predicted
by Eqn(7.5) that necessarily follows from Eqns(4.12) and (4.20). These in turn may be
derived from the Lagrangian (2.7) and Hamilton's Principle. It is then interesting to ask
where the constant `c' was introduced into the derivation. The answer is Eqn(2.2), the definition
of 4-vector velocity. The same formula contains, implicitly, the information that a massless
particle has the constant velocity, c , that is used to identify the 'electromagnetic wave',
with velocity c predicted by Eqns(7.5) and (7.6), with the propagation of the massless real photons
produced by the source.
\par The only dynamical assumption in the derivation of CEM presented above is Coulomb's Law.
If it is explained in QED as an effect due to virtual photon exchange, it also seems to require
via Eqns(7.5) and (7.6), the existence of real, massless, photons. Although clearly of interest,
the further study of the relationship between CEM and QED for radiative processes is, as stated
earlier, beyond the scope of the present paper.
\par In conclusion, the results obtained in the present paper are compared with those of the
similarly motivated project of Feynman and Wheeler. The latter made the following general comments
on their approach~\cite{FW2}:
\par {\tt (1)~ There is no such concept as ``the'' field, an independent entity
\newline with degrees of freedom
of its own. }
\par {\tt (2)~There is no action of an elementary charge upon itself and
\newline consequently no problem of
an infinity in the energy of the electromagnetic\newline field.}
\par {\tt (3)~The symmetry between past and future in the prescription of the
\newline fields not a mere
logical possibility, as in the usual theory, but a
\newline postulational requirement.}
\par The statements (1) and (2) remain true in the approach described in Sections 2-4 above. However
the writer' opinion is that the `infinite self energy' problem of CEM is really an artifact
of the possibly unphysical concept of a `point charge' rather than a shortcoming of the
classical electromagnetic field concept {\it per se}. That being said, it remains true that
the virtual photons interacting with a given charge are produced by other charges so there
is no way for the charge to `interact with itself'. If the energy of the `electromagentic field'
is identified with that of the exchanged virtual photons in the CM frame, it vanishes, so, there
is, as in (2) above, certainly no self energy problem. However, the statements (1)and (2) are only
applicable to the `force' fields introduced in Eqns(3.1), (3.10) and (3.12) above, that may be
denoted as $A_{for}$, $\vec{E}_{for}$ and $\vec{B}_{for}$ to distingish them from the
`radiation' fields describing real photons. It is important to reiterate that the
definitions and physical meanings of these two types of fields are quite distinct. The
quantity: $(\vec{E}_{for}^2+\vec{B}_{for}^2)/8\pi$ does not correctly describe the energy
density of the electromagnetic field associated with virtual photons, and, in contradiction to (1),
extra degrees of freedom must be added to the Lagrangian to correctly describe real photons.
No distinction was made between real and virtual photons by Feynman and Wheeler.
In the approach of the present paper, point (3) with its introduction of acausal `advanced'
potentials is no longer valid. It was a consequence of Feynman and Wheeler's taking Gaussian
Causality as an axiom. The latter is true, as shown by Eqn(7.7) and (7.8), for any interaction
transmitted by real photons (i.e. for the fields $A_{rad}$, $\vec{E}_{rad}$ and $\vec{B}_{rad}$)
but not, as shown in Section 5 above, for the force fields describing the effects of
the exchange of space-like virtual photons. These are always tachyonic (as in Fig.2) and
may be instantaneous (as in Fig1b and c) but do not, unlike `advanced potentials',
violate causality. Feynman and Wheeler's mistake, the same as that of many
previous authors, was to try to describe the physical
effects of virtual photon exchange by fields respecting Gaussian, instead of Neumann, Causality.
\par It is instructive to compare the discussion of CEM in the present paper with that
of Reference~\cite{LL}, which also takes as fundamental physical assumptions, in constructing
the theory, special relativity and Hamilton's Principle. However in Reference~\cite{LL},
the existence of the 4-vector potental $A$ and the relativistic Lagrangian equivalent
to (3.2) above are both
postulated {\it a priori}. This procedure is justified by the statement~\cite{LL1}:
\par {\tt The assertions which follow should be regarded as being, to a certain \newline extent,
the consequence of experimental data. The form of the action for \newline a particle in an
electromagnetic field cannot be fixed on the basis of \newline general considerations alone
(such as, for example the requirement of \newline relativistic invariance). }
\par This is true, as far as it goes, but fails to take account of either the constructive
principle put forward in the quotation from Hagedorn cited above, or the known essential physics
of the problem embodied in the inverse-square force law between charges in the static limit.
As demonstrated in Section 2 above, the assumption of this law, together with the classical
definition of potential energy and relativistic invariance is in fact sufficient to derive
just the Lagrangian that is assumed {\it a priori} in~\cite{LL}. The derivations of the
Lorentz force equation and the covariant definitions of electric and magnetic fields
(3.11) and (3.12) given in~\cite{LL} are identical to those presented above, as are also the
derivations of the magnetostatic Maxwell equation and the Faraday-Lenz law. In Chapter 3 of
~\cite{LL} there is a lengthy discussion of the motion of particles in magnetic fields. However
at this point the magnetic field is a purely abstract mathematical concept. How it may be obtained
from its sources $-$ charges in motion $-$ has still not been even mentioned! Only after the
electrostatic and electrodynamic Maxwell equations have been derived in Chapter 4 from the principle
of least action, by treating the electromagnetic fields as `co-ordinates', is the relation between
fields and
their sources established. Coulomb's law is then derived at the begining of Chapter 5 (page 100!)
from the Poisson equation. In contrast, in the present paper, Coulomb's law (and hence the
Poisson equation) is assumed at the outset, and the electromagnetic Maxwell equation is
derived, simply by inspection, from the covariant form of Poisson's equation. At this point identical
results have been obtained from the same essential input (the Lagrangian (3.2)) by the present
paper and~\cite{LL}. However the present writer feels that there are enormous pedagogical advantages,
(especially in view of the crucial role of Coulomb's law in QED, discussed above) to start the
discussion with the vital experimental fact $-$ the inverse-square force law $-$ rather than
to derive it after 100 pages of complicated mathematics, as is done in~\cite{LL}. Also, in~\cite{LL}
no distinction is made between $A_{for}$, $\vec{E}_{for}$, $\vec{B}_{for}$ and $A_{rad}$, $\vec{E}_{rad}$,
$\vec{B}_{rad}$. All fields
are assumed to be derived from the same, non-relativistic, retarded, Li\'{e}nard and Wichert potentials.
\par Finally the approach of the present paper may be compared with that of another recent paper by
the present author~\cite{JHF1} in which the Lorentz Force Law, magnetic field concept and
the Faraday-Lenz Law are
derived from a different set of postulates. The electrostatic definition of the electric
field $\vec{E}_{stat} = -\vec{\nabla} V$ is first generalised to the covariant form of Eqn(3.11)
above by imposing space-time exchange symmetry invariance~\cite{JHF2}. The magnetic field concept
and the Lorentz Force Law are then shown to follow from the covariance of Eqn(3.11), and the
derivation of the Faraday-Lenz law is identical to that given above.
Neither Coulomb's Law nor Hamilton's Principle were invoked in this case, demonstrating the robustness
of some essential formulae of CEM to the choice of axioms for their derivation. Another example of this
is provided by Reference~\cite{LL} where Coulomb's law is derived from the principle of least
action and the relativistic Lagrangian (3.2), as initial postulates.
\par{ \bf Acknowledgement}
\par I thank B.Echenard and P.Enders for their comments on this paper. Pertinent and constructive
critical comments by an anonymous referee have enabled me to simplify, or improve, the presentation
in several places. They are gratefully acknowledged.
\newpage
{\bf Appendix}
\par Factoring out the space-time dependent factor in the transition current ${\cal J}_{\mu}$ according
to the definition
\begin{eqnarray}
{\cal J}_{\mu} & = & -e \overline{u} \gamma_{\mu} u \exp[i(p_f-p_i) \cdot x]
\nonumber \\
& \equiv & \tilde{{\cal J}}_{\mu} \exp[i(p_f-p_i) \cdot x]~~~~~~~(A1) \nonumber
\end{eqnarray}
enables the invariant amplitude $T_{fi}$ of Eqn(6.4) to be written as:
\begin{eqnarray}
T_{fi} & = & i \int dt_A \int d^3x_A \frac{\tilde{{\cal J}}^A e^{i(p_f^A-p_i^A) \cdot x_A}\cdot
\tilde{{\cal J}}^B e^{i(p_f^B-p_i^B) \cdot x_A}}{|\vec{q}^2|} \nonumber \\
& = & i \int dt_A \int d^3 x_A \frac{\tilde{{\cal J}}^A e^{i(p_f^{A0}-p_i^{A0})t_A}\cdot
\tilde{{\cal J}}^B e^{i(p_f^{B0}-p_i^{B0}) t_A}}
{|\vec{q}^2|}~~~~~~~(A2) \nonumber
\end{eqnarray}
since, from momentum conservation:
\[ \vec{p}_f^A-\vec{p}_i^A = - (\vec{p}_f^B-\vec{p}_i^B)~~~~~~~(A3) \]
Using now Eqn(6.5) gives
\[ T_{fi} = i \int dt_A \int d^3x_A
\tilde{{\cal J}}^A e^{i(p_f^{A0}-p_i^{A0})t_A}\cdot
\tilde{{\cal J}}^B e^{i(p_f^{B0}-p_i^{B0})t_A}
\int \frac{e^{ i\vec{q} \cdot \vec{x}}d^3x}{4 \pi |\vec{x}|}~~~~~~~~(A4) \]
Making the change of variables:
\[ \vec{x} = \vec{x}_B- \vec{x}_A,~~~d^3x = d^3x_B \]
and noting that $\vec{q} = \vec{p}_i^A- \vec{p}_f^A$ gives, from Eqn(A4):
\[ T_{fi} = i \int dt_A \int d^3x_A
\tilde{{\cal J}}^A e^{i(p_f^{A0}-p_i^{A0})t_A}\cdot
\tilde{{\cal J}}^B e^{i(p_f^{B0}-p_i^{B0})t_A}
\int \frac{e^{ i(\vec{p}_f^A- \vec{p}_i^A) \cdot (\vec{x}_B -\vec{x}_A) }d^3x_B}
{4 \pi |\vec{x}_B- \vec{x}_A|}~~~~~~~~(A5) \]
Now
\[(\vec{p}_f^A- \vec{p}_i^A) \cdot (\vec{x}_B -\vec{x}_A) = -(\vec{p}_f^B- \vec{p}_i^B) \cdot \vec{x}_B
-(\vec{p}_f^A- \vec{p}_i^A) \cdot \vec{x}_A~~~~~~~~(A6) \]
where Eqn(A3) has been used.
Substituting (A6) into (A5), yields Eqn(6.6) of
the text.
\pagebreak
|
1,314,259,993,661 | arxiv |
\section{Introduction}
Doppler surveys for exoplanets are monitoring almost every bright,
chromospherically inactive solar type star (late F, G and early
K types) and a fraction of the known M~dwarfs within 30 parsecs. These
successful programs have discovered about 400 exoplanets ranging
in mass from a few times that of the Earth up to the lower mass limit
for brown dwarfs (i.e., about 12 \mbox{$M_{\rm Jup}$}). This ensemble of exoplanets
has revealed correlations between both the metallicity and mass of the
host star with the occurrance rate of exoplanets \citep{sim04, fv05,
lm07, j07}.
The occurrence rate of gas giant planets around M~dwarfs appears to be lower than
for F, G and early K type stars. \citet{e03} detected only one planetary companion
in a sample of 100 M dwarf primaries. \citet{j07} carried out a statistical
analysis and showed that late K and early M~dwarf stars have a $1.8 \pm 1.0$\%
occurrence rate for Jovian planets compared to $4.2 \pm 0.7$\% for solar-mass
stars and $8.9 \pm 2.9$\% for higher mass subgiants.
However, the protoplanetary disks of M~dwarfs may still be a robust environment
for the formation of lower mass planets. \citet{f09} find that 30\% of
the twenty known planets with $\msini < 0.1 \mbox{$M_{\rm Jup}$}$ orbit M dwarf stars.
Consistent with this result, microlensing surveys
\citep{sumi09} find far more planets with masses in the
range between a few to 20 \mbox{$M_\oplus$}.
These empirical results are compatible with core accretion
models of planet formation. \citet{il05} predict that gas accretion will
be quenched by gap formation in disks with smaller aspect ratios; the low
surface density in the protoplanetary disks of M~dwarfs implies a low
formation probability for gas giants because massive cores don't form
before gas depletion. \citet{l04} also surmise that a significant
population of ``failed Jupiters'' with cores of a few Earth masses should exist
around M~dwarfs.
M~dwarfs constitute the majority of stars populating our galaxy.
Among the $\sim 150$ stars within 8 parsecs, about 120 are early-type
M~dwarfs, while only 15 are G dwarfs.
Low mass stars are attractive targets for searches of terrestrial mass
planets within the habitable zone where liquid water might be present
\citep{t07, g07}. The habitable zone is closer to
low luminosity stars and the shorter period planets induce larger,
more easily detected reflex velocities.
Yet, fewer than ten percent of late K and early M dwarf stars within 30 parsec
are being monitored by Doppler surveys \citep{m09, b06, e03}. These
stars are challenging targets because they are intrinsically faint
at optical wavelengths. Indeed, many late type M~dwarfs beyond ten parsecs
are only now being detected; recently, the
SUPERBLINK survey \citep{l05, ls05} identified a few thousand M~dwarfs closer
than 30 parsecs,
opening up the possibility for larger scale surveys of these stars.
\section{A New M and K dwarf Survey}
To learn more about the rate of exoplanet occurrence for M~dwarf stars
and mass and stellar metallicity dependences, we have started a survey of 1600 late K and
early M dwarfs. The target stars are drawn from the SUPERBLINK survey, which
provides an all-sky census of stars with proper motion larger than
40 $mas\ yr^{-1}$ down to a magnitude limit $R\approx19.5$. Main sequence
stars are identified based on their location in a reduced proper motion
diagram. A fraction of the stars are listed in the Hipparcos catalog; for
those stars we calculate the distance from their Hipparcos parallax. All
other stars have their distances estimated based on the empirical $V,V-J$
color-magnitude relationship from \citet{l05}. We select as probable
late K and M dwarfs all stars with optical-to-infrared color $V-J > 2.75$,
which roughly corresponds to absolute visual magnitudes $M_V > 8.5$ and
masses $M < 0.6 M_{\odot}$. We further narrow down our sample to stars with
photometric distances $d < 50$ pc and visual magnitudes $V < 12.5$. The
low proper motion limit and short distance range minimizes the kinematic
bias, and the sample is expected to be complete for stars with transverse
velocities $v_T>9.5\ \mbox{km s$^{-1}$}$ ($v_T>5.7\ \mbox{km s$^{-1}$}$ for the 30 pc subsample),
which effectively samples the local disk population over the full range
of typical stellar ages, as the selection even includes stars from nearby
young moving groups \citet{ls09}. Of the final sample of 1600 stars,
40\% are listed in the Hipparcos catalog.
We carry out low resolution spectroscopic screening of all candidates before
they are observed at Keck using the Mark III spectrograph at the 1.3-m
McGraw-Hill Telescope at teh MDM Observatory at Kitt Peak. These observations
are used to confirm the spectral type and main sequence status before
beginning observations at Keck.
Modeled after the N2K program \citep{f05}, our program uses a quick-look
strategy to obtain three or four nearly consecutive observations of target stars
to flag short-period candidates. We then continue to obtain approximately
log-spaced time series observations to detect longer period planets. Here,
we present the first planet to emerge from this program.
\section{HIP 79431}
\subsection{Stellar Characteristics}
HIP~79431 is an M3V star with apparent magnitudes of $V = 11.34$,
$J = 7.56$, $H = 6.86$, $K = 6.59$ and color \bv\ = 1.486.
The {\it Hipparcos} parallax \citep{esa97, vanleeuwen07}
of $69.46 \pm 3.12$ mas yields a distance of
14.4 parsecs and we use this to calculate the absolute magnitudes
$M_V = 10.47$, $M_J = 6.69$, $M_H = 5.99$, $M_K = 5.72$. The proper
motions $\mu_{RA} = 28.31\ {\rm mas\ yr^{-1}}$, $\mu_{Dec} = -208.36\ {\rm mas\ yr^{-1}}$
measured by the {\it Hipparcos} mission \citep{esa97} and our measured absolute
radial velocity of -4.7 \mbox{km s$^{-1}$}\ correspond to the small space velocities
of $U = +8.5\ \mbox{km s$^{-1}$}$, $V = +2.1\ \mbox{km s$^{-1}$} $, $W = -4.8\ \mbox{km s$^{-1}$}$.
\citet{d00} have established mass-luminosity calibrations for
low mass stars and find low dispersion relations for $M_J$,
$M_H$ and $M_K$. The average of these three infrared mass-luminosity
relations yield an average stellar mass of 0.49 \mbox{$M_{\odot}$}\ for HIP~79431.
It is challenging to assess metallicity from M dwarf spectra,
largely because of uncertainties in the molecular line data.
\citet{bo05} first derived a photometric calibration for M~dwarf metallicities
and this work has been updated by \citet{ja09}. Both groups used spectral
synthesis modeling to determine \mbox{\rm [Fe/H]}\ for bright F and G type stars
in binary systems with M dwarf companions
and assigned the metallicity of the primary star to the
M dwarf companion. Based on the star's height above the $M_K$ vs $V-K$
main sequence, the calibration of \citet{ja09} yields a metallicity of
$\mbox{\rm [Fe/H]} = +0.4 \pm 0.1$.
The effective temperature for M~dwarfs is usually derived from
a color-temperature relation. The K-band temperature relation
calibrated by \citet{c08} is insensitive to metallicity and M dwarf
fluxes peak near the infrared $K$ band. The \citet{c08} relation
has a scatter of only 19 K and yields \mbox{$T_{\rm eff}$} = 3191 K for
HIP~79431. This is similar to the effective temperature of 3236 K
that can be derived with the less robust \bv\ calibration of
\citet{vf05} for FGK type stars.
Because the continuum flux near the \ion{Ca}{2} H \& K\ lines is so low
in M~dwarfs it is typical to see strong emission from the core of the
\ion{Ca}{2} H \& K\ lines. \citet{if09} measure $S_{HK} = 1.15 \pm 0.06$ in this star.
The \mbox{$\log R^\prime_{\rm HK}$}, activity-based rotation period and ages are not calibrated for stars
with \bv\ greater than 1.0, however \citet{if09} find that this value of
$S_{HK}$ is in the 50th percentile for stars of this color, so the
star does not have an unusually high level of chromospheric activity for
a dwarf star with this \bv\ color.
The stellar characteristics are summarized in Table 1.
\subsection{Doppler Observations and Keplerian Fit}
HIP~79431 was added to the Keck program in April 2009 as part of our exoplanet
survey of low mass stars. We obtained 13 Doppler measurements
of HIP~79431 over six months using the HIRES spectrometer \citep{v94}.
Exposure times of 600 seconds for this $V=11.3$ star
yielded a \mbox{\rm signal-to-noise ratio}\ of just under 100.
Our Doppler analysis makes use of an iodine absorption cell in the
light path before the entrance slit of the HIRES spectrometer. The iodine
absorption lines in each program observation are used to model the
wavelength scale and the instrumental profile of the telescope and
spectrometer optics for each observation \citep{m92, b96}. The typical
Doppler precision for this faint star is about 3 \mbox{m s$^{-1}$}. Based on the \mbox{rms}\ scatter
of other stars on our program, we have empirically derived an additional
error which may arise from astrophysical noise sources or systematic
uncertainties in our data. Figure 1 shows time series Doppler observations
for stars with constant radial velocities and \bv\ colors similar to HIP~79431.
The typical \mbox{rms}\ scatter is about 2.5 \mbox{m s$^{-1}$}. We adopt 2.5 \mbox{m s$^{-1}$}\ as representative
of the combined radial velocity jitter and systematic errors for late type
stars and add this in quadrature with our formal single measurement errors
when fitting and plotting our data.
The observation dates, radial velocity data and measurement
uncertainties for HIP~79431 are listed in Table 2. In this Table,
the measurement uncertainties are not increased by the 2.5 \mbox{m s$^{-1}$}\ jitter term.
The radial velocities exhibit an unambiguous signal. The radial velocities
were fit with a Keplerian model and the uncertainties in the orbital parameters
were derived using a bootstrap Monte Carlo analysis with 1000 realizations
of the data \citep{ma05} in the following way: first, the theoretical velocities
from the best-fit Keplerian model are subtracted, then the residual velocities
are randomized, added back to the theoretical velocities and refit. This
analysis captures the full range of uncertainties, including systematic errors,
however, it will also scramble a signal from any additional planets, so it is
only appropriate for good \rchisq\ fit models.
The Keplerian fit yields an orbital period of $111.7 \pm 0.7$ days and a
semi-velocity amplitude of $149.5\ \pm 2.5\ \mbox{m s$^{-1}$}$. The stellar mass of 0.49 \mbox{$M_{\odot}$}\
implies a companion mass with \msini\ = 2.1 \mbox{$M_{\rm Jup}$}\ and a semi-major axis of 0.35 AU.
The parameters for the Keplerian orbit are listed in Table 3.
The Doppler velocities are plotted in Figure 2 with a solid line indicating
the best fit Keplerian model. In this plot, the single measurement errors have
been increased by adding a modest 2.5 \mbox{m s$^{-1}$}\ for stellar jitter and systematic
errors. The \mbox{rms}\ of the residuals to our Keplerian model is
3.9 \mbox{m s$^{-1}$}\ and the \rchisq\ fit is 0.84, demonstrating that the fit is consistent
with our single measurement errors and the added assumed jitter of 2.5 \mbox{m s$^{-1}$}.
As such a good fit implies, there is no evidence for additional planets.
\section{Planet or Brown Dwarf?}
Because the Doppler technique derives \msini\ and not the total mass,
this leaves open the question of whether this is actually a stellar
binary system in a nearly face-on orbit. HIP~79431 is only 14.4~pc
away and a stellar companion on a nearly face-on orbit would induce
motion around the barycenter of order 10~mas. The re-analysis of the
{\it Hipparcos} astrometry is consistent (goodness-of-fit parameter of
0.15) with the proper motion of a single star \citep{vanleeuwen07}.
We used the intermediate astrometric data \citep{esa97} to place an
upper limit on the companion mass as a function of confidence level
(one minus the false alarm probability FAP). The abscissa residuals to
the {\it Hipparcos} parallax and proper motion solution span about 10 orbits
and are plotted in Figure 3. There is no power in a Lomb-Scargle periodogram
near the orbital period of 111 days in these data. Assuming gaussian
statistics, a companion more massive than 0.3 $M_{\odot}$ can be ruled
out with 90\% confidence (Figure 4), but, the {\it Hipparcos} data by
themselves cannot eliminate the possibility of a late-type M or brown
dwarf companion. We can, however, rule out the possibility that the
location of HIP~79431 above the main sequence is due to a nearly
equally-luminous companion and therefore conclude that this location is consistent
with the high stellar metallicity \citep{ja09}.
Even a brown dwarf mass would require an improbable nearly
face-on orbit. If we take the lower limit for stellar mass as 70 \mbox{$M_{\rm Jup}$}\
then the maximum inclination $i$ for our system to surpass the stellar mass threshold
is given by:
\begin{equation}
i_{70\ Jup} = \arcsin{\msini \over 70\ \mbox{$M_{\rm Jup}$}},
\end{equation}
The probability that this particular inclination was observed can
be calculated under the assumption that the orientation of orbits
is random as:
\begin{equation}
P(star) = \cos(0) - \cos(i_{70\ Jup}) = 4 \times 10^{-4},
\end{equation}
\noindent
where $i$ is the orbital inclination from Eqn (1).
We can likewise calculate the probability that HIP~79431b falls
in the brown dwarf range. This time, the inclination limit is far
less severe since the planet only needs to have a true mass
of 12 \mbox{$M_{\rm Jup}$}:
\begin{equation}
i_{12\ Jup} = \arcsin{\msini \over 12 \mbox{$M_{\rm Jup}$}}
\end{equation}
\begin{equation}
P(brown\ dwarf) = \cos(i_{70\ Jup}) - \cos(i_{12\ Jup}) = 0.015
\end{equation}
So, in order for HIP~79431b to be a stellar companion, an inclination
of less than $2^\circ$ from face-on is required and that orientation
has a probability of only 0.04\%. If HIP~79431b is really a brown
dwarf companion, the orbital inclination must be between $2^\circ$ -
$10^\circ$; the probability of this orientation is 1.5\%.
\section{Transit Ephemeris}
Because HIP~79431b has a semi-major axis of 0.36 AU, the planet has a
relatively low transit probability. The planet also resides in
a moderately eccentric orbit that brings it as close as 0.25 AU to the
host star at periastron. For the orbit of HIP~79431b,
$\omega = 288 \pm 3$ so a primary transit would occur when the planet
is close to apastron. This further reduces the probability
of a primary transit and we calculate a transit probability of only 0.5\%.
In this system, the secondary eclipse is somewhat more likely; without
prior knowledge of a primary transit, we calculate a probability of
0.85\% for the secondary eclipse.
If a transit were to occur, the transit depth would be remarkable.
A planet with a mass of 2.1 \mbox{$M_{\rm Jup}$}\ is supported by electron
degeneracy and has a radius that is only a factor of four times smaller
than the M dwarf host. We calculate a prospective primary transit
depth of 98 millimags and transit
duration of almost 7 hours. The probability for a secondary eclipse
is about double that for the primary transit, however the photometric
decrement would only be about 0.004 millimags.
Prospective transit times were predicted from the best-fit Keplerian parameters.
The Thiele-Innes constants were used to project
the orbit onto the plane of the sky and to determine the true anomaly
corresponding to the central transit time, $T_c$. The time of the next
$T_c$ is calculated to be 2010 Feb 21 at UT 18:17:23:92 with an uncertainty
of 1.6 hours. The uncertainty in $T_c$ was determined by fitting the 1000
trials of Keplerian fits (used to derive uncertainties in the orbital parameters).
For each trial, the central transit time, ingress and egress were
calculated and the standard deviation provided an estimate of the uncertainty.
Future transit times can be estimated by rolling forward an
integral number of orbital periods, however uncertainty in $T_c$ increases
with time from our last Doppler measurement.
\section{Discussion}
As part of a new Doppler quick-look survey for late K and early M~dwarfs,
we have detected a planet with \msini\ = 2.1 \mbox{$M_{\rm Jup}$}\ orbiting the
M3V star, HIP~79431. The host star appears to be metal-rich with a
photometrically-determined \mbox{\rm [Fe/H]} = +0.4.
Massive planets around late type stars are easy detections.
However, the sample of M~dwarf stars with planets is relatively small;
only six have detected gas giant planets ($\msini > 0.5 \mbox{$M_{\rm Jup}$}$).
The remarkable system of planets orbiting GJ~876 (M4V) has three planets and two
of these are gas giants in a 2:1 mean motion resonance; GJ~876b
has \msini\ = 1.935 \mbox{$M_{\rm Jup}$}\ and orbital period of about 30 d
and GJ~876c has \msini\ = 0.619 \mbox{$M_{\rm Jup}$}\ with an orbital period
of 60 d \citep{m01, m98, d98}. GJ~849 (M3.5V) has a planet
with \msini = 0.82 \mbox{$M_{\rm Jup}$}\ in a 5-year orbit with significant
residual velocities that now exhibit curvature, consistent with an additional planet
in a wider orbit.
GJ~317 (M3.5V) has a planet
with \msini\ = 1.17 \mbox{$M_{\rm Jup}$}\ with a period of about 1.9 years and residual velocities
consistent with Jupiter-mass planet in an orbit of more than seven years \citep{j07}.
GJ~832 (M1.5V) has a Jupiter-like planet in a 9.4-year orbit \citep{b09}.
GJ~179 (M3.5V) has a Jupiter-mass planet in a 6.3 year orbit \citep{how09}.
GJ~649 (M1.5V) has a Saturn-mass planet in a 1.6 year orbit \citep{j09}.
It is curious that half of the known giant planets orbiting low
mass M~dwarf stars reside in relatively wide orbits. \citet{kk09} find
that grain growth time scales are limited by photo evaporation with
more rapid loss of disks in high mass stars. They note that the stellar mass-dependent
disk dispersal timescales may account for the fact that planets orbiting higher-mass
stars have wider orbits, since the disk may disperse more rapidly and halt
inward migration. Perhaps lower surface density creates an analogous migration-hindering
environment for low mass stars. Or, perhaps the protoplanetary disk
of M stars forms fewer planets so that planet-planet scattering events are less
common. With less competition for raw materials in the disk, it might be possible
for single or well-spaced giant planets to dominate at wide separations.
This is also a domain where gravitational instability could play a role {boss06}.
HIP~79431b is one of the most massive planets detected in orbit around an M dwarf
star. The calculated probability that this is actually
an equal mass stellar companion is vanishingly small: $4 \times 10^{-4}$. The
The {\it Hipparcos} astrometric data do not rule out the
possibility that this is a low mass star or brown dwarf companion, but
we calculate only a 1.5\% probability that HIP~79431b has an orbital
inclination less than $10^\circ$ and a true mass greater than 12 \mbox{$M_{\rm Jup}$}.
Statistically, this leaves a 98.5\% probability that HIP~79431b is a
planet with a mass between 2.1 and 12 \mbox{$M_{\rm Jup}$}.
The Keplerian motion induced by a single planet can explain our data,
although much less massive planets may exist on interior or exterior
orbits. The excellent agreement of the single planet Keplerian planet solution
with the data limit the mass of any second planet in the system. We
performed a linear perturbation analysis around the one-planet
solution and found that we can rule out planets more
massive than $M_3 \sin i$ of 12 or 15 Earths near the 3:1 or 2:1
interior resonances, respectively.
Although Doppler surveys show that massive companions to M~dwarf
stars are rare, evidence for a few Jovian-mass planets candidates has emerged
from microlensing surveys that sample wider orbits.
(Because low mass stars dominate the galactic population, they are
common lenses for background stars.)
\citet{g02} place an upper limit and find that fewer than
33\% of M~dwarfs have Jupiter-mass companions between 1.5 and 4 AU.
Nevertheless, only a small number
of Jovian-mass planets have been detected in microlensing surveys
\citep{b04, u05, g08, d09}.
\acknowledgements
We gratefully acknowledge the dedication and support of the Keck
Observatory staff, especially Grant Hill and Scott Dahm for support
of HIRES and Greg Wirth for support of remote observing.
DAF acknowledges research support from NASA grant
NNX08AF42G as well as NASA support through the Keck PI data analysis
Fund. EG acknowledges support by NSF grant AST0908419.
AWH gratefully acknowledges support from a Townes
Postdoctoral Fellowship at the U.C. Berkeley Space Sciences
Laboratory.
Data presented herein were obtained at the W. M. Keck
Observatory from telescope time allocated to the National Aeronautics
and Space Administration through the agency's scientific
partnership with the California Institute of Technology and the
University of California. We also thank the TAC of the University
of Hawaii for providing time for this project.
The Observatory was made possible by
the generous financial support of the W. M. Keck Foundation.
The authors wish to recognize and acknowledge the very significant
cultural role and reverence that the summit of Mauna Kea has always
had within the indigenous Hawaiian community. We are most fortunate
to have the opportunity to conduct observations from this mountain.
This research has made use of the SIMBAD database, operated
at CDS, Strasbourg, France, and of NASA's Astrophysics Data System
Bibliographic Services.
\clearpage
\input{bib.tex}
\clearpage
\begin{figure}
\plotone{f1.eps}
\figcaption{Doppler measurements of four M~dwarf stars with
constant radial velocities. These stars have an \mbox{rms}\ scatter of
about 2.5 \mbox{m s$^{-1}$}\ and represent velocity precision typical of
late type stars on our program. }
\end{figure}
\begin{figure}
\plotone{f2.eps}
\figcaption{Time series radial velocities from Keck
Observatory are plotted for HIP~79431 with
2.5 \mbox{m s$^{-1}$}\ of expected velocity jitter added in
quadrature with the single measurement uncertainties.
The Keplerian model is overplotted with
an orbital period of 111 days, velocity amplitude
of 149.5 \mbox{m s$^{-1}$}\ and eccentricity, $e = 0.29$.
With these parameters and the stellar mass of 0.49 \mbox{$M_{\odot}$},
we derive a planet mass, \msini\ = 2.1 \mbox{$M_{\rm Jup}$}\ and semi-major
axis of 0.36 AU. }
\end{figure}
\begin{figure}
\plotone{f3.eps}
\figcaption{Astrometric residuals (unbinned) from the {\it Hipparcos} intermediate
data after fitting for parallax and proper motion.}
\end{figure}
\begin{figure}
\plotone{f4.eps}
\figcaption{False alarm probability (one minus the confidence level) of the companion exceeding a given mass given the {\it Hipparcos} abscissa residuals for HIP~79431}
\end{figure}
\clearpage
\begin{deluxetable}{ll}
\tablenum{1}
\tablecaption{Stellar Parameters: HIP~79431}
\tablewidth{0pt}
\tablehead{\colhead{Parameter} & \colhead{} \\
}
\startdata
$V$ & 11.34 \\
$M_V$ & 10.47 \\
$M_K$ & 5.72 \\
\bv & 1.486 \\
Spectral Type & M3V \\
Distance [pc] & 14.4 \\
\mbox{$T_{\rm eff}$} & 3191 (1000 \\
\mbox{\rm [Fe/H]} & +0.4 (0.1) \\
$M_{star}$ & 0.49 (0.05) \\
\mbox{$S_{\rm HK}$} & 1.15 (0.06) \\
Radial Velocity [\mbox{km s$^{-1}$}] & -4.7 \\
\enddata
\end{deluxetable}
\begin{deluxetable}{rrcc}
\tablenum{2}
\tablecaption{Radial Velocities for HIP~79431}
\tablewidth{0pt}
\tablehead{
\colhead{} &
\colhead{RV} &
\colhead{$\sigma_{\rm RV}$} \\
\colhead{JD-2440000} &
\colhead{(\mbox{m s$^{-1}$})} &
\colhead{(\mbox{m s$^{-1}$})} \\
}
\startdata
14928.07496 & -63.11 & 2.73 \\
14929.11611 & -70.45 & 2.84 \\
14984.87477 & 103.92 & 2.68 \\
14986.96264 & 125.53 & 3.14 \\
14987.94866 & 140.47 & 3.10 \\
15028.96005 & -7.06 & 3.88 \\
15041.89538 & -74.88 & 3.21 \\
15042.92741 & -81.75 & 3.55 \\
15048.75289 & -104.55 & 2.80 \\
15074.72733 & -140.47 & 2.84 \\
15078.74506 & -119.78 & 2.90 \\
15084.73424 & -64.40 & 2.97 \\
15106.73408 & 139.86 & 3.52 \\
\enddata
\end{deluxetable}
\clearpage
\begin{deluxetable}{ll}
\tablenum{3}
\tablecaption{Orbital Parameters for HIP~79431b}
\tablewidth{0pt}
\tablehead{\colhead{Parameter} & \colhead{} \\
}
\startdata
Period (days) & 111.7 (0.7) \\
${\rm T}_{\rm p}$ (JD) & 2454980.3 (1.2) \\
$\omega$ (deg) & 287.4 (3.2) \\
Eccentricity & 0.29 (0.02) \\
K$_1$ (\mbox{m s$^{-1}$}) & 149.5 (2.5) \\
$a_{rel}$ (AU) & 0.36 \\
$M\sin i$ (M$_{Jup}$) & 2.1 \\
${\rm Nobs}$ & 13 \\
RMS (\mbox{m s$^{-1}$}) & 3.9 \\
\rchisq\ & 0.84 \\
\enddata
\end{deluxetable}
\clearpage
\end{document}
|
1,314,259,993,662 | arxiv | \section{Introduction.}
\bigskip
Aluffi \cite{A04} introduced a class of algebras which are intermediate between
the symmetric algebra and the Rees algebra of an ideal in order to define the characteristic
cycle of a hypersurface parallel to the conormal cycle in intersection theory.
These algebras were investigated by Nejad and Simis \cite{NS11}, who called them Aluffi algebras. At the end of his paper, Aluffi observed that it would be computationally desirable to up-grade his methods to more general schemes. A first step in the direction of Aluffi's proposed up-grade is a good notion of the Rees algebra of a module, such as the one described by Simis, Ulrich, and Vasconcelos in \cite{SUV03}. A second step in the direction of Aluffi's proposed up-grade is a good notion of the Aluffi algebra of a module, such as the one the one introduced by Ramos and Simis in \cite{RS15}.
Ramos and Simis compute the Aluffi algebra of the module of derivations of the homogeneous coordinate ring of a smooth projective hypersurface. In other contexts this module is also called the module of tangent vector fields or the differential idealizer or the module of logarithmic
derivations. As part of the Ramos-Simis program, it is necessary to understand the homological nature of the ideal $J=\operatorname{Pf}_4({\mathbf X})+I_1({\mathbf t}{\mathbf X})$, where ${\mathbf X}$ is a generic alternating matrix and ${\mathbf t}$ is a generic row vector. Simis told us that he and Ramos conjectured that $J$ is a Gorenstein ideal of height two more than the height of $\operatorname{Pf}_4({\mathbf X})$. The purpose of this paper is to prove the Ramos-Simis conjecture.
Let $R_0$ be an arbitrary commutative Noetherian ring, $\goth f$ be an integer with $4\le \goth f$, $$\mathcal R
=R_0[\{x_{i,j}\mid 1\le i<j\le \goth f\}\cup \{t_i\mid 1\le i\le \goth f\}]$$ be
a polynomial ring in $\binom{\goth f}2+\goth f$ indeterminates, $\mathbf X$ be the be the $\goth f\times \goth f$ alternating matrix with with $x_{i,j}$ in position (row $i$, column $j$) for $i<j$, $\mathbf t$ be the $1\times \goth f$ matrix with $t_j$ in column $j$, $I$ be the ideal $\operatorname{Pf}_4(\mathbf{X})$ which is generated
by the set of Pfaffians of the principal $4\times 4$ submatrices of $\mathbf X$,
$K$ be the ideal $
I_1(\mathbf {tX})$, which is
generated by the entries of the product of $\mathbf t$ times $\mathbf X$, and
$J$ be the ideal $I+K$ of $\mathcal R$.
The main result in the paper, Theorem~\ref{main-Theorem}, is that $J$ is a perfect Gorenstein ideal in $\mathcal R$ of grade $\binom{\goth f-2}2+2$. In particular, if $R_0$ is a Gorenstein ring, then $\mathcal R/J$ is a Gorenstein ring. Some consequences of the main result are contained in Corollary~\ref{corollary} where it is shown that $\mathcal R/J$ is a domain, or a normal ring, or a unique factorization domain if and only if the base ring $R_0$ has the same property.
The main ingredient in the proof of Theorem~\ref{main-Theorem} takes place over the polynomial ring $$R
=R_0[\{x_{i,j}\mid 1\le i<j\le \goth f\}].$$ We prove in Lemma~\ref{main-Dream-Lemma} that ``the column space of $\mathbf X$, calculated mod $I$'', which is equal to the submodule
\begin{equation}\label{column}\textstyle \{{\mathbf X}\theta\in (\frac RI)^\goth f\mid \theta\in (\frac RI)^\goth f\} \quad\text{of $(\frac RI)^\goth f$,}\end{equation}
is a perfect $R$-module of projective dimension $\binom{\goth f-2}2$.
If $R_0$ is a Cohen-Macaulay domain, then we prove, in Observation~\ref{yet-to-come}.\ref{yet.d}, that
the module of (\ref{column}) is a self-dual maximal Cohen-Macaulay $R/I$-module of rank two;
and we prove, see Remark~\ref{R4}.\ref{here it is},
that the ideal $J(\mathcal R/I)$, which is the central object in this paper, is a
Bourbaki ideal for $\mathcal R\otimes_R\text{(\ref{column})}$; and therefore, homological properties of (\ref{column}) are
inherited by $\mathcal R/J$. For more discussion about Bourbaki ideals, see, for example, \cite{A66, M80, BHU87, SUV03}.
The ideal $I$ is the ideal of ``quadratic relations'' which define the homogeneous coordinate ring of the image of the Pl\"ucker embedding of the Grassmannian $\operatorname{Gr}(2,\goth f)$ into projective space $\mathbb P^{\binom{\goth f} 2-1}$. Properties of the Schubert subvarieties of $\operatorname{Gr}(2,\goth f)$ play a crucial role in our proof of the properties of the module
(\ref{column}).
The ideal $I_1(\mathbf{tX})$ has already been studied. If $\goth f$ is odd, then
$I_1(\mathbf{tX})$ is a type two almost complete intersection ideal introduced by Huneke and Ulrich in \cite{HU85} and further studied in \cite{K95}. If $\goth f$ is even, then $I_1(\mathbf{tX})$ is a mixed ideal; its unmixed part is $I_1(\mathbf{tX})+\operatorname{Pf}_{\goth f}({\mathbf X})$; see, for example, \cite{KPU-Ann}. This ideal is a deviation two, grade $\goth f-1$ Gorenstein ideal also introduced in \cite{HU85} and further studied in \cite{K86,S91,K92}.
\bigskip
\section{Notation, conventions, and preliminary results.}\label{Prelims}
\bigskip
\begin{chunk}Let $M$ and $N$ be modules over a commutative Noetherian ring $R$. Whenever the meaning is unambiguous, we write $M^*$, $M\otimes N$, $\Hom(M,N)$, and $\bigwedge^i M$ in place of $\Hom_R(M,R)$, ${M\otimes_R N}$, $\Hom_R(M,N)$, and $\bigwedge_R^i M$, respectively.
\end{chunk}
\begin{chunk} An element $x$ of a ring $R$ is {\em regular} on the $R$-module $M$ if $x$ is a non-zero-divisor on $M$. In other words, if $xm=0$ for some element $m\in M$, then $m=0$. \end{chunk}
\begin{chunk}If $x$ is a non-nilpotent element of a commutative Noetherian ring $R$, then {\em the localization of $R$ at $x$}, denoted $R_x$, is the ring $S^{-1}R$ where $S$ is the set $\{1,x,x^2,x^3,\dots\}$. If $x$ is a regular element of $R$, then we use the notation $R_x$ and $R[x^{-1}]$ interchangeably.
\end{chunk}
\begin{chunk}We denote the ring of integers by $\mathbb Z$.
\end{chunk}
\subsection{Perfection}\label{Terminology}
\bigskip
\begin{chunk}\label{perfection} Let $R$ be a Noetherian ring, $I$ be a proper ideal of $R$, and $M$ be a non-zero finitely generated $R$-module. \begin{enumerate}[\rm(a)]
\item The {\it grade} of $I$ is the length of a maximal regular sequence on $R$ which is
contained in $I$. (If $R$ is Cohen-Macaulay, then the grade
of $I$ is equal to the height of $I$.)
\item The $R$-module $M$ is called {\it perfect} if the grade of the annihilator of $M$ (denoted $\operatorname{ann}_R M$) is equal to the projective dimension of $M$ (denoted $\operatorname{pd}_R M$). The inequality \begin{equation}\label{auto}\operatorname{grade}(\operatorname{ann}_R M)\le \operatorname{pd}_R M\end{equation} holds automatically if $M\not=0$.
\item\label{constant-pd} If $M$ is a perfect $R$-module, then
$$\operatorname{pd}_{R_P}M_P=\operatorname{grade} \operatorname{ann}_{R_P}M_P=\operatorname{grade}\operatorname{ann}_RM$$ for all prime ideals $P$ in the support of $M$. (See, for example, \cite[Prop.~16.17]{BV}.)
\item \label{perfection.d} If $R$ is a polynomial ring over a field or over the ring of integers and $M$ is a finitely generated graded $R$-module, then
$M$ is a perfect $R$-module if and only if $M$ is a Cohen-Macaulay $R$-module. (This is not the full story. For more information, see, for example, \cite[Prop.~16.19]{BV} or \cite[Thm.~2.1.5]{BH}.)
\item\label{gor} The ideal $I$ in $R$ is called a {\it perfect ideal} if $R/I$ is a perfect $R$-module.
A perfect ideal $I$ of grade $g$ is a {\it Gorenstein ideal}
if $\operatorname{Ext}^g_R(R/I,R)$
is a cyclic R-module.
\end{enumerate}
\end{chunk}
The concept of perfection is particularly useful because of the ``Persistence of Perfection Principle'', which is also known as the ``transfer of perfection''; see \cite[Prop. 6.14]{Ho75} or \cite[Thm. 3.5]{BV}.
\begin{theorem}\label{PoPP}
Let $R\to S$ be a homomorphism of Noetherian rings, $M$ be a perfect $R$-module, and $\mathbb P$ be a resolution of $M$ by projective $R$-modules. If $S\otimes_RM\neq 0$ and $$\operatorname{grade} (\operatorname{ann} M)\le \operatorname{grade} (\operatorname{ann}(S\otimes_RM)),$$
then $S\otimes_RM$ is a perfect $S$-module with $\operatorname{pd}_S(S\otimes_RM)=\operatorname{pd}_RM$ and $S\otimes_R\mathbb P$ is a resolution of $S\otimes_RM$ by projective $S$-modules.
\end{theorem}
\subsection{Multilinear algebra.}
\begin{chunk} \label{87Not1}
Many of our calculations are made in a coordinate-free manner. If the calculation is coordinate free, then the signs take care of themselves.
In particular, when working with Pfaffians, we prefer to use elements of an exterior algebra rather than to define and keep track of sign conventions which mimic operations that take place in an exterior algebra.
Let $R$ be a commutative Noetherian ring and $F$ be a free module of finite rank $\goth f$ over
$R$.
We make much use of
the exterior algebras $\bigwedge^{\bullet} F$ and $\bigwedge^{\bullet} F^*$,
the fact that
$\bigwedge^{\bullet} F$ and $\bigwedge^{\bullet} F^*$ are modules over one another, and the fact that the even part of an exterior algebra comes equipped with a divided power structure.
The
rules for a divided power algebra are recorded in \cite[section 7]{GL} or \cite[Appendix 2]{Ei95}.
(In practice these rules say that $w^{
(a)}$ behaves like $w^
a/(a!)$ would behave if $a!$ were
a unit in $R$.)
\end{chunk}
\begin{chunk}\label{2.3} We recall some of the properties of the divided power structure on the subalgebra $\bigwedge^{2\bullet}F$ of the exterior algebra $\bigwedge^{\bullet}F$. Suppose that $e_1,\dots,e_{\goth f}$ is a basis for the free $R$-module $F$ and $$f_2=\sum_{1\le i_1<i_2\le \goth f}a_{i_1,i_2} \ e_{i_1}\wedge e_{i_2}$$ is an element of $\bigwedge^2F$, for some $a_{i_1,i_2}$ in $R$.
Let $A$ be the $\goth f\times \goth f$ alternating matrix with
$$A_{i,j}=\begin{cases} a_{i,j},&\text{if $i<j$,}\\0,&\text{if $i=j$, and}\\-a_{i,j},&\text{if $j<i$}.\end{cases}$$
For each positive integer $\ell$, the $\ell$-th divided power of $f_2$ is
$$f_2^{(\ell)}=\sum\limits_I A_I e_I\in\textstyle\bigwedge^{2\ell}F,$$ where the $2\ell$-tuple $I=(i_1,\dots,i_{2\ell})$ roams over all increasing sequences of integers with $1\le i_1$ and $i_{2\ell}\le \goth f$, $e_I=e_{i_1}\wedge \dots \wedge e_{i_{2\ell}}$, and $A_I$ is the Pfaffian of the submatrix of $A$ which consists of rows and columns $\{i_1,\dots,i_{2\ell}\}$, in the given order. Furthermore, $\bigwedge^{2\bullet}F$ is a DG$\Gamma$-module over $\bigwedge^{\bullet}F^*$. In particular, if $\tau\in F^*$ and $v_1,\dots,v_s$ are homogeneous elements of $\bigwedge^{2\bullet}F$, then
\begin{equation}\label{Gamma}\tau\left(v_1^{(\ell_1)}\wedge \dots\wedge v_{s}^{(\ell_s)}\right)=
\sum\limits_{j=1}^s\tau(v_j)\wedge v_1^{(\ell_1)}\wedge \dots\wedge
v_j^{(\ell_j-1)}
\wedge \dots\wedge v_{s}^{(\ell_s)}.\end{equation} For more details see, for example,
\cite[Appendix~A2.4]{Ei95} or \cite[Appendix and Sect. 2]{BE77}.
\end{chunk}
The following fact about the interaction of the module structures of
$\bigwedge^\bullet F $ on $\bigwedge^\bullet F^* $ and $\bigwedge^\bullet F^* $ on $\bigwedge^\bullet F $
is well known; see \cite[section 1]{BE75} and \cite[Appendix]{BE77}.
\begin{proposition} \label{A3} Let $F$ be a free module of finite rank
over a commutative
Noetherian ring $R$. If $f_1\in
F$, $f_p\in \textstyle \bigwedge^{p}F$, and $\phi_q\in\textstyle \bigwedge^{q}(F^{*})$, then
$$ (f_1(\phi_q))(f_p)=f_1\wedge (\phi_q(f_p))+(-1)^{1+q}\phi_q(f_1\wedge f_p).$$
\end{proposition}
\vskip-24pt\qed
\medskip The following fact is important for our purposes. We prove it carefully in order to illustrate some of the ideas contained in \ref{87Not1} and \ref{2.3}.
\begin{observation} \label{doo-8.2} Let $R$ be a commutative Noetherian ring, $F$ be
a free $R$-module of finite rank, and $f_2$ be an element of
$\bigwedge^{2}F$.
\begin{enumerate}[\rm(a)]\item \label{doo-8.2.a} If
$\phi_3\in \bigwedge^3F^*$, then
$[f_2(\phi_3)](f_2)=\phi_3(f_2^{(2)})$.
\item\label{doo-8.2.b} If $\phi_1$, $\phi_1'$, and $\phi_1''$ are in $F^*$, then
$$f_2(\phi_1\wedge \phi_1'\wedge \phi_1'')=f_2(\phi_1\wedge \phi_1')\cdot \phi_1''-f_2(\phi_1\wedge \phi_1'')\cdot \phi_1'+f_2(\phi_1'\wedge \phi_1'')\cdot \phi_1.$$\end{enumerate}
\end{observation}\begin{proof}
We
prove (\ref{doo-8.2.a}) by showing that the two elements
$[f_2(\phi_3)](f_2)$ and $\phi_3(f_2^{(2)})$
of $F$ are equal
by showing that $\phi_1\Big([f_2(\phi_3)](f_2)\Big) =\phi_1\Big(\phi_3(f_2^{(2)})\Big)$ for
every element $\phi_1$ of $F^*$.
Observe that
$$\begin{array}{lllllllll}\phi_1([f_2(\phi_3)](f_2))&=&-[f_2(\phi_3)][\phi_1(f_2)]
&=&-[\phi_1(f_2)][f_2(\phi_3)]&=&-[\phi_1(f_2)\wedge f_2](\phi_3)\vspace{5pt}\\&=&-[\phi_1(f_2^{(2)})](\phi_3)&=&-\phi_3[\phi_1(f_2^{(2)})]&=&\phantom{-}\phi_1[\phi_3(f_2^{(2)})].\end{array}$$
The first and last equalities hold because $\bigwedge^{\bullet}F$ is a module over the graded-commutative ring $\bigwedge^{\bullet}F^*$. The second and fifth equalities
follow from the fact that the module actions of $\bigwedge^\bullet F^*$ on $ \bigwedge^\bullet F$ and $\bigwedge^\bullet F$ on $\bigwedge^\bullet F^*$ are compatible in the sense that \begin{equation}\label{compat} \phi_i(f_i)=f_i(\phi_i)\text{ for }\phi_i\in \textstyle\bigwedge^iF^*\text{ and } f_i\in \textstyle\bigwedge^iF.\end{equation} The third equality
is a consequence of the module action of $\bigwedge^\bullet F$ on $\bigwedge^\bullet F^*$. The fourth equality
is explained in
(\ref{Gamma}).
The proof of (\ref{doo-8.2.b}) is similar.
\end{proof}
\subsection{The set up.}
\bigskip
\begin{chunk}We set up the data in a coordinate-free manner in \ref{data2} and \ref{Not2}; a version with coordinates is given in \ref{R2}.
The critical calculation, Lemma~\ref{main-Dream-Lemma}, involves ``$x_{i,j}$'s'', but not ``$t_i$'s''; the ambient ring for this calculation is called $R$. The information about $R$ is given in \ref{data2}.\ref{data2-one}, \ref{Not2}.\ref{Not2.a}, and \ref{R2}.\ref{R2.z}.
The main result in the paper, Theorem~\ref{main-Theorem}, involves both ``$x_{i,j}$'s'' and ``$t_i$'s''; the ambient ring for this result is called $\mathcal R$. The ring $\mathcal R$ is an extension of $R$; the extra information about $\mathcal R$ is given in \ref{data2}.\ref{data2-both}, \ref{Not2}.\ref{Not2.b}, and \ref{R2}.\ref{R2.a} \end{chunk}
\begin{data}{\label{data2}} Let $\goth f$ be a positive integer, $R_0$ a commutative Noetherian ring,
and $V$ be a free $R_0$-module of rank $\goth f$.
\begin{enumerate}[\rm(a)]
\item\label{data2-one} Let $R=\bigoplus_{i=0}^\infty R_i$ be the standard graded polynomial ring
$$\textstyle R=\operatorname{Sym}_\bullet^{R_0}(\bigwedge_{R_0}^2V^*)$$and $F$ be the free $R$-module $F=R\otimes_{R_0} V$. Consider the $R$-module homomorphism
$$\textstyle\xi\in \Hom_R(\bigwedge^2_R F^*, R)=\bigwedge_R^2F,$$ which is given as the composition
$$\xi:\textstyle\bigwedge^2_R F^*=R\otimes_{R_0} \bigwedge^2_{R_0} V^*=R\otimes R_1\xrightarrow{{\rm multiplication}}R.$$
\item\label{data2-both} View $\bigwedge_{R_0}^2V^*\oplus V$ as a bi-graded free $R_0$-module where each element of $\bigwedge_{R_0}^2V^*$ has degree $(1,0)$ and each element of $V$ has degree $(0,1)$. Let $\mathcal R$ be the bi-graded polynomial ring
$$\textstyle \mathcal R=\operatorname{Sym}_\bullet^{R_0}(\bigwedge_{R_0}^2V^*\oplus V)$$and $\mathcal F$ be the free $\mathcal R$-module $\mathcal F=\mathcal R\otimes_{R_0} V$. Consider the $\mathcal R$-module homomorphism
$$\textstyle
\tau\in \Hom_\mathcal R(\mathcal F, \mathcal R)=\mathcal F^*$$ which is given as the composition
$$\mathcal F=\mathcal R\otimes_{R_0} V=\mathcal R\otimes \mathcal R_{\ 0,1}\xrightarrow{{\rm multiplication}}\mathcal R.$$
\item There is a natural inclusion map $\xymatrix{R\ar@{^(->}[r]&\mathcal R}$ and a natural projection map $\xymatrix{\mathcal R\ar@{->>}[r]& R}$. The $\mathcal R$-module $\mathcal F$ of (\ref{data2-both}) is also equal to $\mathcal F=\mathcal R\otimes_R F$; furthermore, the element $\xi\in\bigwedge_R^2F$ of (\ref{data2-one}) is also equal to the element $$\textstyle \xi=1\otimes \xi\text{ of }\bigwedge_{\mathcal R}^2\mathcal F=\mathcal R\otimes_R\bigwedge^2_RF.$$
\end{enumerate}\end{data}
\begin{notation}{\label{Not2}} Adopt Data \ref{data2}.
\begin{enumerate}[\rm(a)]\item\label{Not2.a}Let\begin{enumerate}[\rm(i)]\item $I$ be the ideal
$
I=\operatorname{im} (\xi^{(2)}:\textstyle\bigwedge^4F^*\to R)$, of $R$,
\item $A$ be the ring $R/I$, \item\label{overline} $\overline{\phantom{x}}$ be the functor $A\otimes_R-$, and \item\label{Not2.a.iii} $N$ be the cokernel of the map $\overline{d_1}:\bigwedge^3\overline{F}^*\to \overline{F}^*$ where $d_1:\bigwedge^3F^*\to F^*$ is the map $d_1(\phi_3)=\xi(\phi_3)$, for $\phi_3\in \bigwedge^3F^*$.
\end{enumerate}
\item\label{Not2.b}Let \begin{enumerate}[\rm(i)]\item $K$ and $J$ be the ideals
\begin{align*}K&=\operatorname{im} (\tau(\xi):\mathcal F^*\to \mathcal R),\quad\text{and}\\
J&=I\mathcal R+K\end{align*} of $\mathcal R$,\item $\mathcal A$ be the ring $\mathcal R\otimes_R A$, and\item $\mathcal N$ be the $\mathcal R$-module $\mathcal R\otimes_R N$.
\end{enumerate}\end{enumerate}
\end{notation}
\begin{remark}{\label{R2}} Adopt Data~\ref{data2} and Notation~\ref{Not2}. If one picks dual bases $e_1,\dots,e_\goth f$ for $V$ and $e_1^*,\dots,e_\goth f^*$ for $V^*$, lets $x_{i,j}$ represent $e_j^*\wedge e_i^*\in \bigwedge^2_{R_0}V^*=\mathcal R_{\ 1,0}$, for $1\le i<j\le \goth f$, and lets $t_i$ represent $e_i\in V=\mathcal R_{\ 0,1}$, for $1\le i\le \goth f$, then the following statements hold.\begin{enumerate} [\rm(a)]
\item\label{R2.z} The standard graded polynomial ring $R$ is $R=R_0[\{x_{i,j}\mid 1\le i<j\le \goth f\}]$. Furthermore,
\begin{enumerate}[\rm(i)]\item\label{R2.b} the element $\xi$ of $\bigwedge ^2F$ is $\xi=\sum\limits_{i<j} x_{i,j}\ e_i\wedge e_j$,
\item\label{R2.e} the matrix for the $R$-module homomorphism $d_0:F^*\to F$, with $d_0(\phi_1)=\phi_1(\xi)$,
with respect to the bases $\{e_j^*\}$ and $\{e_i\}$, is $-{\mathbf X}$, where ${\mathbf X}$ is the generic $\goth f\times \goth f$ alternating matrix
whose entry in position (row $i$, column $j$) is$$\begin{cases} x_{i,j}&\text{if $i<j$}\\0&\text{if $i=j$}\\-x_{j,i}&\text{if $j<i$},\end{cases}$$
and
\item\label{R2.g} the ideal $I$ of Notation \ref{Not2} is equal to $\operatorname{Pf}_4(\mathbf{X})$, which is the ideal of $R$ generated by the set of Pfaffians of the principal $4\times 4$ submatrices of $\mathbf X$.
\end{enumerate}
\item\label{R2.a} The bi-graded polynomial ring $\mathcal R$ is $\mathcal
R=R_0[\{x_{i,j}\mid 1\le i<j\le \goth f\}\cup\{t_i\mid 1\le i\le \goth f\}]$.
Furthermore, \begin{enumerate}[\rm(i)]
\item\label{R2.c} the element $\tau$ of $\mathcal F^*$ is $\tau=\sum_i t_ie_i^*$ ,
\item\label{R2.d} the matrix for $\tau:\mathcal F\to \mathcal R$ with respect to the basis $\{e_i\}$ for $\mathcal F$ is the row vector $${{\mathbf t}=[t_1,\dots, t_\goth f]},$$
\item\label{R2.f} the element $\tau(\xi)$ in $\mathcal F$ is an $\mathcal R$-module homomorphism $\mathcal F^*\to \mathcal R$ and the matrix for this homomorphism, with respect to the basis $\{e_i^*\}$ is the row vector ${\mathbf t}{\mathbf X}$, and
\item\label{R2.h} the ideal $K$ of Notation \ref{Not2} is equal to $
I_1(\mathbf {tX})$, which is the ideal of $\mathcal R$ generated by the entries of the product of $\mathbf t$ times $\mathbf X$, and
\item\label{R2.i} the ideal $J$ of Notation \ref{Not2} is equal to $\operatorname{Pf}_4(\mathbf {X})\cdot \mathcal R+
I_1(\mathbf {tX})$.
\end{enumerate} \end{enumerate}\end{remark}
\begin{remark}\label{promise}Adopt the language of \ref{data2}.\ref{data2-one} and \ref{Not2}.\ref{Not2.a}.
The following maps appear often in the paper:
\begin{equation}\label{pre-cplx}\textstyle \bigwedge^3F^*\xrightarrow{d_1} F^*\xrightarrow{d_0}F \xrightarrow{\delta_1}\bigwedge^3F,\end{equation}with
$d_1(\phi_3)=\xi(\phi_3)$, $d_0(\phi_1)=\phi_1(\xi)$, and
$\delta_1(f_1)=f_1\wedge \xi$, for $\phi_3\in \bigwedge^3F^*$, $\phi_1\in F^*$, and $f_1\in F$. Use Observation~\ref{doo-8.2}.\ref{doo-8.2.a} and (\ref{Gamma}) to see that
$$(d_0\circ d_1)(\phi_3)=[\xi(\phi_3)](\xi)=\phi_3(\xi^{(2)})\quad\text{and}\quad
(\delta_1\circ d_0)(\phi_1)=[\phi_1(\xi)]\wedge \xi=\phi_1(\xi^{(2)});$$so, in particular
$A\otimes_R (\text{\rm\ref{pre-cplx}})$ is a complex. In (\ref{prove-3.2}) we prove that a modification of ${A\otimes_R\text{(\ref{pre-cplx})}}$ is exact and in Observation~\ref{yet-to-come} we prove that $A\otimes_R\text{(\ref{pre-cplx})}$ is exact.
If one uses the notation Remark~\ref{R2}.\ref{R2.z}, then the matrix for $d_0$ is $-\mathbf X$, the matrix for $d_1$ has $\goth f$ rows and $\binom{\goth f}3$ columns and the column corresponding to
$e_k^*\wedge e_j^*\wedge e_i^*$, for $1\le i<j<k\le \goth f$, is
\setcounter{MaxMatrixCols}{20}
$$\bmatrix 0&\dots&0&x_{j,k}&0&\dots&0&-x_{i,k}&0&\dots&0&x_{i,j}&0&\dots&0\endbmatrix^{\rm T},$$where the non-zero entries appear in positions $i$, $j$, and $k$, respectively; see Observation~\ref{doo-8.2}.\ref{doo-8.2.b} and Remark~\ref{R2}.\ref{R2.b}. (We use $M^{\rm T}$ to represent the transpose of the matrix $M$.) The matrix for $\delta_1$ is the transpose of the matrix for $d_1$.
\end{remark}
\section{The main ingredient.}\label{dream}
In this section we prove the following result.
\begin{lemma}\label{main-Dream-Lemma}
Adopt Data~{\rm\ref{data2}.\ref{data2-one}} and Notation~{\rm\ref{Not2}.\ref{Not2.a}} with $4\le \goth f$.
If the base ring $R_0$ is an arbitrary commutative Noetherian ring, then the $R$-module $N$ is perfect of projective dimension $\binom {\goth f-2}2$.
\end{lemma}
In Observation~\ref{yet-to-come} we show that the module $N$ of Lemma~\ref{main-Dream-Lemma} is isomorphic to the module of (\ref{column}).
\begin{remark}
The assertion of Lemma~\ref{main-Dream-Lemma} does not hold for $\goth f=3$. Indeed, in the language of Remark~\ref{R2}, $N$, which is resolved by
$$0\to R\xrightarrow{\bmatrix x_{2,3}&-x_{1,3}&x_{1,2}\endbmatrix^{\rm T}}R^3,$$ is not a perfect $R$-module and has projective dimension one, which is not equal to $\binom {\goth f-2}2$. (We use $M^{\rm T}$ to represent the transpose of the matrix $M$.)
\end{remark}
It is convenient to let
\begin{align*}&\text{
$A'$ be the ring $A/(x_{1,2}\ ,\ x_{2,3}\ ,\ x_{1,3})$},
\end{align*}
in the language of Remark~\ref{R2}.\ref{R2.z}. Our proof of Lemma~\ref{main-Dream-Lemma} is given in \ref{prove-dream} at the end of the section; it depends on Lemma~\ref{exact-seq} and on information about the rings $A$ and $A'$ which is contained in Lemma~\ref{KL}.
\begin{lemma}\label{exact-seq}
Adopt Data~{\rm\ref{data2}.\ref{data2-one}} and Notation~{\rm\ref{Not2}.\ref{Not2.a}} with $3\le \goth f$.
If the base ring $R_0$ is a commutative Noetherian domain, then
there is an exact sequence of $A$-modules{\rm:}
\begin{equation}\label{exact-seq.1}0\to N\to A^3\to A\to A'\to 0.\end{equation}
In particular, if $(0)$ is the zero ideal of $A$, then $N_{(0)}$ is isomorphic to $A_{(0)}\oplus A_{(0)}$.
\end{lemma}
\begin{remarks}\begin{enumerate}[\rm(a)]
\item A strengthened version of Lemma~\ref{exact-seq} may be found in Proposition~\ref{improved}.
\item Lemma~{\rm\ref{exact-seq}} does hold when $\goth f=3$; indeed,
(\ref{exact-seq.1}) becomes
$$0\to \frac{R^3}{\left(\bmatrix x_{2,3}\\-x_{1,3}\\x_{1,2}\endbmatrix\right)}\xrightarrow{\bmatrix 0&x_{1,2}&x_{1,3}\\-x_{1,2}&0&x_{2,3}\\-x_{1,3}&-x_{2,3}&0\endbmatrix} R^3\xrightarrow{\bmatrix x_{2,3}&-x_{1,3}&x_{1,2}\endbmatrix} R\to R_0\to 0,
$$which is exact.\end{enumerate}
\end{remarks}
\bigskip
The proof of Lemma~\ref{exact-seq} is given in \ref{prove-3.2}.
\begin{definition}\label{I-KL}Adopt the language of \ref{data2}.\ref{data2-one}, \ref{Not2}.\ref{Not2.a} and \ref{R2}.\ref{R2.z}. For each integer $\lambda$, between $1$ and $\goth f-1$, let $I_{\lambda}$ be the ideal $$I_\lambda=I+(\{x_{i,j}\mid 1\le i<j\le \lambda\})$$ of $R$. \end{definition}
\begin{example}\label{3.4}Retain the notation of Definition~{\rm\ref{I-KL}}. The ideal $I_1$ is equal to $I$ (because the empty set generates the zero ideal) and the ideal $I_{\goth f-1}$ is equal to $(\{x_{i,j}\mid 1\le i<j\le \goth f-1\})$ (because $I$ is contained in the ideal $(\{x_{i,j}\mid 1\le i<j\le \goth f-1\})$. In particular, $A=R/I_1$ and $A'=R/I_3$.\end{example}
\begin{lemma}\label{KL} Adopt the language of {\rm \ref{data2}.\ref{data2-one}}, {\rm\ref{Not2}.\ref{Not2.a}}, {\rm \ref{R2}.\ref{R2.z}}, and {\rm\ref{I-KL}}.
Let $\lambda$ be an integer between $1$ and $\goth f-1$.
\begin{enumerate}[\rm(a)]\item\label{KL.a} If the base ring $R_0$ is an arbitrary commutative Noetherian ring, then $I_\lambda$ is a perfect ideal in $R$ of grade $\textstyle\binom{\goth f-2}2+\lambda-1$. In particular, if $4\le \goth f$, then $\operatorname{grade} I_3=\operatorname{grade} I_1+2$.
\item\label{KL.b} If the base ring $R_0$ is a commutative Noetherian domain, then $I_\lambda$ is a prime ideal.
\item\label{KL.c} If the base ring $R_0$ is an arbitrary commutative Noetherian ring, then $I$ is a Gorenstein ideal in the sense of {\rm\ref{perfection}.\ref{gor}}. In particular, if $R_0$ is a Gorenstein ring, then $R/I$ is a Gorenstein ring.
\end{enumerate}
\end{lemma}
\begin{remark}\label{require} The ``in particular assertion'' in (\ref{KL.a}) would be false if $\goth f$ were equal to $3$; because, in this case, $I_1$, which is equal to $(0)$, has grade $0$, and $I_3$, which is equal to $(x_{1,2}\ ,\ x_{1,3}\ ,\ x_{2,3})$, has grade $3$. Of course, the parameter $\lambda$, which is assumed to be at most $\goth f-1$, is not permitted to be $3$, when $\goth f=3$.
\end{remark}
\begin{proof}(\ref{KL.a},\ref{KL.b}) The ideal $I_\lambda$ is equal to the ideal $\operatorname{Pf}(X;\lambda;\lambda)$ of \cite{KL80}. The assertion follows from \cite[Thm.~12]{KL80}. The statement of \cite[Thm.~12]{KL80} only considers the case where $R_0$ is a domain; however, as soon as one knows that $I_\lambda$ is a perfect ideal when $R_0$ is equal to the ring of integers and when $R_0$ is equal to a field, then $I_\lambda$ built with $R_0=\mathbb Z$ is a generically perfect ideal and consequently $I_\lambda$ built over an arbitrary commutative Noetherian $R_0$ is a perfect ideal; see, for example \cite[Prop.~3.2 and Thm.~ 3.3]{BV}.
\medskip\noindent(\ref{KL.c}) A proof that $R/I$ is a Gorenstein ring whenever $R_0$ is Gorenstein is given in \cite[Thm.~17]{KL80}. A more explicit statement and proof of this result is given in \cite[Corollary]{A79}. In particular, when $R_0$
is equal to the ring of integers, then there exists a resolution $\mathbb F$ of $R/I$ by free $R$-modules which has the property that the length of $\mathbb F$ is $\binom{\goth f-2}2$ and the free module of $\mathbb F$ in position $\binom{\goth f-2}2$ has rank one. Now let $R_0$ be an arbitrary commutative Noetherian ring. We explained in the proof of (\ref{KL.a}) and (\ref{KL.b}) that $I$ is a perfect ideal in $R$. The ``Persistence of Perfection Principle'', Theorem~\ref{PoPP}, now guarantees that the back Betti number in a resolution of $R/I$ by free $R$-modules is one; and therefore, $I$ is a Gorenstein ideal in the sense of \ref{perfection}.\ref{gor}.
\end{proof}
\begin{remark}An alternate phrasing of the proof of Lemma~\ref{KL}, parts (\ref{KL.a}) and (\ref{KL.b}), (but really the same argument in a different form) involves the Grassmannian $\operatorname{Gr}(2,\goth f)$ of rank $2$ free summands of the rank $\goth f$ free $R_0$-module $V$. The ideal $I$ is the ideal of ``quadratic relations'' which define the homogeneous coordinate ring of the image of the Pl\"ucker embedding of $\operatorname{Gr}(2,\goth f)$ into $\mathbb P(\bigwedge^2 V)$. The ideal $I_\lambda$ defines the homogeneous coordinate ring of the Pl\"ucker embedding of the Schubert subvariety $\Omega(\goth f-\lambda,\goth f)$ of $\operatorname{Gr}(2,\goth f)$. The Schubert subvariety ${\Omega(\goth f-\lambda,\goth f)}$ consists of all $W$ in $\operatorname{Gr}(2,\goth f)$ such that $i\le \operatorname{rank} (W\cap V_i)$ for the flag $V_1\subsetneq V_2$ where $V_1$ is the the summand of $V$ with basis $e_{\lambda+1},\dots,e_\goth f$ and $V_2=V$. The original proofs that the homogeneous coordinate rings of the Schubert subvarieties of the Grassmannian are Cohen-Macaulay domains are \cite[Thm.~3.1$^*$, (3.10), Cor.~4.2]{H73}, \cite[Thm.~1]{L72}, and \cite[Thm.~II.4.1 and Thm.~III.4.1]{M72}. A version which contains many details is \cite[Thm.~1.4, the bottom of page 52, Cor.~5.18, Thm.~6.3]{BV}.
\end{remark}
One consequence of Lemma~\ref{KL} is that $I$ is grade unmixed. This fact facilitates the identification of regular elements in $A$. Corollary~\ref{KL-consq} and its style of proof are used in the proof of Corollary~\ref{corollary}.\ref{cor-a}.
\begin{corollary}\label{KL-consq} Adopt the language of {\rm \ref{data2}.\ref{data2-one}}, {\rm\ref{Not2}.\ref{Not2.a}}, and {\rm \ref{R2}.\ref{R2.z}}. If the base ring $R_0$ is an arbitrary commutative Noetherian ring, then $x_{1,2}\ ,\ x_{1,3}$ is a regular sequence on $A$.
\end{corollary}
\begin{proof}
Every associated prime $P$ of $R/I$ in $R$ has grade $PR_P=\binom{\goth f-2}2$.
Lemma~\ref{KL}.\ref{KL.a} assures that
$I_2$, which equals $(I\ ,\ x_{1,2})$, is a perfect ideal of grade $\binom{\goth f-2}2+1$ in $R$; hence $x_{1,2}$ is not in any associated prime of $R/I$ (that is, $x_{1,2}$ is regular on $R/I$) and every associated prime $P$ of $R/(I\ ,\ x_{1,2})$ in $R$ has grade $PR_P=\binom{\goth f-2}2+1$. We prove that $x_{1,3}$ is regular on $R/(I\ ,\ x_{1,2})$ by showing that $\textstyle\binom{\goth f-2}2+2\le \operatorname{grade} PR_P$ for all primes $P$ of $R$ which contain $(I\ ,\ x_{1,2}\ ,\ x_{1,3})$. Let $P$ be such a prime. Consider the Pfaffian
$$
x_{1,2}x_{3,j}-x_{1,3}x_{2,j}+x_{1,j}x_{2,3}\in I\subseteq P.$$Thus, $x_{2,3}x_{1,j}$ is in $P$ for $3\le j\le \goth f$. It follows that either $I_3$, which is $(I\ ,\ x_{1,2}\ ,\ x_{1,3}\ ,\ x_{2,3})$, is contained in $P$ or $(I\ ,\ x_{1,2}\ ,\ x_{1,3}\ ,\ \dots\ ,\ x_{1,\goth f})\subseteq P$. Lemma~\ref{KL}.\ref{KL.a} ensures that $I_3$ has grade ${\binom{\goth f-2}2+2}$. The ideal $(I\ ,\ x_{1,2}\ ,\ x_{1,3}\ ,\ \dots\ ,\ x_{1,\goth f})$ is equal to $\operatorname{Pf}_4(\mathbf X')$ plus an ideal generated by $\goth f-1$ indeterminates, where $\mathbf X'$ is $\mathbf X$ with row and column $1$ deleted. Thus $(I\ ,\ x_{1,2}\ ,\ x_{1,3}\ ,\ \dots\ ,\ x_{1,\goth f})$ has grade
$$\textstyle \binom{\goth f-3}2+\goth f-1=\binom{\goth f-2}2+ 2.$$ In either event, $\binom {\goth f-2}2+2\le \operatorname{grade} P$ and the proof is complete.
\end{proof}
\begin{chunk}\label{prove-3.2} {\bf Proof of Lemma \ref{exact-seq}.}
We prove that
\begin{equation}\label{key}\textstyle \bigwedge^3\overline{F}^*\xrightarrow{\ \ \overline{d_1}\ \ }\overline{F}^*
\xrightarrow{\ \ d_0'\ \ }
A^3\xrightarrow{\ \ \rho\ \ }A\to A'\to 0 \end{equation}is an exact sequence of $A$-modules, where
$d_1:\bigwedge^3F^*\to F^*$ is $d_1(\phi_3)=\xi(\phi_3)$, as given in
Notation~\ref{Not2}.\ref{Not2.a.iii} and Remark~\ref{promise},
$d_0'$ is the composition
$$\textstyle \overline{F}^*\xrightarrow{\ \ \overline{d_0}\ \ }\overline{F}=\bigoplus\limits_{i=1}^\goth f Ae_i
\xrightarrow{\ \ \text{projection}\ \ }\bigoplus\limits_{i=1}^3 Ae_i,
$$ where $d_0:F^*\to F$ is
$d_0(\phi_1)=\phi_1(\xi)$ as described in Remark~\ref{R2}.\ref{R2.e} and Remark~\ref{promise}, and $\rho$
is given by the matrix \begin{equation}\label{delta1}\rho=\bmatrix x_{2,3}&-x_{1,3}&x_{1,2}\endbmatrix.\end{equation} (The basis $e_1,\dots,e_\goth f$ for $F$ is introduced in Remark~\ref{R2}.)
Once we show that (\ref{key}) is an exact sequence, then the proof is complete. Indeed,
\begin{align}&N=\operatorname{coker} \overline{d_1}\cong \operatorname{im} d_0'=\ker \rho;
\quad\text{hence,}\label{3.10.3}\\
&0\to N\to
A^3\xrightarrow{\ \ \rho\ \ }A\to A'\to 0\notag\end{align} is exact, as claimed in (\ref{exact-seq.1}).
We first show that (\ref{key}) is a complex. To show that $d_0'\circ \overline{d_1}=0$ it suffices to show that the image of $d_0\circ d_1$ is contained in $I\cdot F$ and this was done in Remark~\ref{promise}. The matrix for $\rho$ is given in (\ref{delta1}) and the matrix
\begin{equation}\label{d0'}d_0'=-\bmatrix 0&x_{1,2}&x_{1,3}&x_{1,4}&\dots&x_{1,\goth f}\\
-x_{1,2}&0&x_{2,3}&x_{2,4}&\dots&x_{2,\goth f}\\
-x_{1,3}&-x_{2,3}&0&x_{3,4}&\dots&x_{3,\goth f}\endbmatrix
\end{equation} for
$d_0'$
may be read
from the discussion in Remark~\ref{promise}.
It is clear that $\rho\circ d_0'=0$ and that the complex (\ref{exact-seq.1}) is exact at $A$ and $A'$. We next show that (\ref{exact-seq.1}) is exact at $A^3$.
Suppose
$$\alpha=\bmatrix a_1&a_2&a_3\endbmatrix^{\rm T}$$ is an element of $A^3$ with
$\rho(\alpha)=0$ in $A$. (We use $M^{\rm T}$ to represent the transpose of the matrix $M$.)
In other words, $x_{2,3}a_1-x_{1,3}a_2+x_{1,2}a_3=0$ in $A$. In particular,
$$x_{2,3}a_1\in (x_{1,2}\ ,\ x_{1,3})A\subseteq (x_{1,2}\ ,\ x_{1,3}\ ,\ \dots\ ,\ x_{1,\goth f})A.$$ The ideal $(x_{1,2}\ ,\ x_{1,3}\ ,\ \dots\ ,\ x_{1,\goth f})A$ of $A$ is prime; indeed,
$$(x_{1,2}\ ,\ x_{1,3}\ ,\ \dots\ ,\ x_{1,\goth f})+I=(x_{1,2}\ ,\ x_{1,3}\ ,\dots\ ,\ x_{1,\goth f})+\operatorname{Pf}_4(\mathbf X'),$$ where ${\mathbf X}'$ is the matrix $\mathbf X$ of Remark~\ref{R2}.\ref{R2.e} with row one and column one deleted. The matrix $\mathbf X'$ is a generic alternating matrix which does not involve the variables $x_{1,2}\ ,\ \dots\ ,\ x_{1,\goth f}$; so \cite[Thm.~12]{KL80} guarantees that $\operatorname{Pf}_4(\mathbf X')$ is prime; see, for example Lemma~\ref{KL}.
The product $x_{2,3}a_1$ is in the prime ideal $(x_{1,2}\ ,\ \dots\ ,\ x_{1,\goth f})A$ and $x_{2,3}\notin(x_{1,2}\ ,\ \dots\ ,\ x_{1,\goth f})A$; thus, $a_1\in (x_{1,2}\ ,\ \dots\ ,\ x_{1,\goth f})A$ and a quick glance at (\ref{d0'}) shows that there is an element $\overline{\phi_1}$ in $\overline{F}$ such that
$$\alpha -d_0'(\overline{\phi_1})=\bmatrix 0&a_2'&a_3'\endbmatrix^{\rm T},$$
for some $a_2'$ and $a_3'$ in $A$. The equation $-x_{1,3}a_2'+x_{1,2}a_3'=0$ in $A$ shows that $x_{1,3}a_2'$ is an element of the prime ideal
$(x_{1,2})A=I_2A$; see Lemma~\ref{KL}. Hence, $a_2'$ is in $(x_{1,2})A$ and
a further modification $\alpha -d_0'(\overline{\phi_1})$ by a boundary which only involves the first column of $d_0'$ yields an element of the kernel of $\rho$ of the form $\bmatrix 0&0&a_3''\endbmatrix^{\rm T}$. The element $a_3''$ is zero because $A$ is a domain; and therefore, $\alpha\in \operatorname{im} d_0'$.
The argument that (\ref{exact-seq.1}) is exact at $\overline{F}^*$ is very similar to the preceding argument. Suppose $\alpha=[a_1\ ,\ \dots\ ,\ a_{\goth f}]^{\rm T}$ is an element of $\ker d_0'$. The third row of the equation $d_0'\alpha=0$ yields that $x_{3,\goth f}a_f$ is an element of the prime ideal $I_{\goth f-1}A$, in the language of Definition~\ref{I-KL} and Lemma~\ref{KL}; but $x_{3,\goth f}\notin I_{\goth f-1}A$; so $a_{\goth f}\in I_{\goth f-1}$. On the other hand, for each $x_{i,j}\in I_{\goth f-1}$, $$\overline{d_1}(e_\goth f^*\wedge e_j^*\wedge e_i^*)=x_{j,\goth f}e_i^*-x_{i,\goth f}e_j^*+x_{i,j}e_\goth f^*;$$ hence there is an element $\overline{\phi_3}\in \bigwedge^3\overline{F}^*$ so that $$\alpha -\overline {d_1}(\overline{\phi_3})=[a_1'\ ,\ \dots\ ,\ a_{\goth f-1}'\ ,\ 0]^{\rm T}.$$ The third row of the equation $d_0'(\alpha -\overline {d_1}(\overline{\phi_3}))=0$ yields that $x_{3,\goth f-1}a_{\goth f-1}'\in I_{\goth f-2}A$.
Use elements of the form $\overline{d_1}(e_{\goth f-1}^*\wedge e_j^*\wedge e_i^*)$ to remove $a_{\goth f-1}'$ (while keeping $0$ in the bottom position).
Continue in this manner to find $\overline{\phi_3}^{\,\dagger}\in \bigwedge^3\overline{F}^*$ so that $$\alpha -\overline {d_1}(\overline{\phi_3}^{\,\dagger})=[a_1^\dagger\ ,\ a_2^\dagger\ ,\ a_3^\dagger\ ,\ 0\ ,\ \dots,0]^{\rm T}.$$The second equation of
$$\bmatrix 0&x_{1,2}&x_{1,3}\\-x_{1,2}&0&x_{2,3}\\-x_{1,3}&-x_{2,3}&0\endbmatrix \bmatrix a_1^\dagger\\[3pt]a_2^\dagger\\[3pt]a_3^\dagger\endbmatrix =d_0'(\alpha -\overline {d_1}(\overline{\phi_3}^\dagger)=0$$ yields $a_3^\dagger\in (x_{1,2})A$; hence there exists $\overline{\phi_3}^{\,\ddag}\in \bigwedge^3\overline{F}^*$, so that $$\alpha -\overline {d_1}(\overline{\phi_3}^{\,\ddag})=[a_1^\ddag\ ,\ a_2^\ddag\ ,\ 0\ ,\ \dots\ ,\ 0]^{\rm T}.$$Now one sees that $x_{1,2}a_1^\ddag=x_{1,2}a_2^\ddag=0$ in the domain $A$; hence $a_1^\ddag=a_2^\ddag=0$, $\alpha$ is a boundary, (\ref{exact-seq.1}) is exact.
The final assertion, that $N$ has rank two as an $A$-module, is an immediate consequence of the exactness of (\ref{exact-seq.1}). Indeed, $A$ is a domain (see \cite[Thm.~12]{KL80} or Lemma~\ref{KL}) and $A'_{(0)}=0$.
\qed
\end{chunk}
\begin{chunk}\label{prove-dream}{\bf{Proof of Lemma \ref{main-Dream-Lemma}.}} The module $N$, built over an arbitrary ring $R_0$, is obtained from the module $N$, built over the ring of integers $\mathbb Z$, by way of the base change $R_0\otimes_{\mathbb Z}-$. According to the theory of generic perfection (see, for example \cite[Prop.~3.2 and Thm.~3.3]{BV}) in order to prove that $N$, built over an arbitrary ring $R_0$, is a perfect $R$-module, it suffices to prove that $N$ is a perfect $R$-module
when $R_0=\mathbb Z$ and when $R_0$ is a field. Fix one of these choices for $R_0$ and consider the exact sequence of Lemma~\ref{exact-seq}.
It was observed in Example~\ref{3.4} that $A=R/I_1$ and $A'=R/I_3$; consequently, Lemma~\ref{KL}.\ref{KL.a} guarantees that $A$ and $A'$ are perfect $R$-modules and $\operatorname{pd}_R A'=\operatorname{pd}_R A+2$. (This is where the hypothesis $4\le \goth f$ is required; see Remark~\ref{require}.)
Let $P$ be a prime ideal of $R$ which is in the support of $N$. Lemma~\ref{exact-seq} shows that the module $N$ embeds into a free $A$-module; hence, $P$ is in the support of $A$ and $A_P$ is a Cohen-Macaulay ring. The localization $A'_P$ is either zero or
a Cohen-Macaulay ring with $\dim A'_P=\dim A_P-2$. In either event,
we apply the usual argument about the growth of depth in an exact sequence
(see, for example, \cite[Prop.~1.2.9]{BH}), to
the localization of the exact sequence (\ref{exact-seq.1}) at $P$ in order to conclude that $\operatorname{depth} A_P\le \operatorname{depth} N_P$. At this point the inequalities
\begin{equation}\label{3.11.0}\operatorname{depth} N_P\le \dim N_P\le_* \dim A_P=\operatorname{depth} A_P\le \operatorname{depth} N_P\end{equation} all hold; consequently, equality holds throughout. (The inequality labeled * holds because $N_P$ is an $A_P$-module.) Thus, $N_P$ is a Cohen-Macaulay $R_P$-module and
\begin{equation}\label{just-above}\textstyle\operatorname{pd}_{R_P}N_P=\operatorname{pd}_{R_P}A_P=\operatorname{pd}_RA=\binom {\goth f-2}2.\end{equation}
(The first equality is a consequence of the Auslander-Buchsbaum theorem; the second equality is explained in \ref{perfection}.\ref{constant-pd}; and the third equality is a consequence of Lemma~\ref{KL}.) Thus, $N$ is a perfect $R$-module of projective dimension $\binom{\goth f-2}2$ (see \ref{perfection}.\ref{perfection.d}, if necessary) and the proof is complete.
\qed\end{chunk}
Section~\ref{main} is concerned with the ring $\mathcal R$ of Data~{\rm\ref{data2}} and Notation~{\rm\ref{Not2}}. The ring $\mathcal R$ is a polynomial ring over $R$ and the $\mathcal R$-modules $\mathcal N=\mathcal R\otimes_RN$,
$\mathcal A=\mathcal R\otimes_RA$, and $\mathcal F=\mathcal R\otimes_RF$ are obtained from the corresponding $R$-modules by way of a base change. It is convenient to record the results of the present section in the language of the future section.
\begin{corollary}\label{carry forward}
Adopt Data~{\rm\ref{data2}} and Notation~{\rm\ref{Not2}} with $4\le \goth f$.
If the base ring $R_0$ is an arbitrary commutative Noetherian ring, then the $\mathcal R$-modules $\mathcal N$ and $\mathcal A$ are perfect of projective dimension $\binom {\goth f-2}2${$;$} furthermore $I\mathcal R$ is a Gorenstein ideal.
\end{corollary}
\begin{proof} Apply Lemmas~\ref{main-Dream-Lemma} and \ref{KL}. \end{proof}
We close this section by redeeming
assorted
promises. Assertion~(\ref{yet.a}) was promised in
Remark~\ref{promise}.
Assertion~(\ref{yet.b}) was promised in the introduction when we claimed that Section~\ref{dream} is about the image of $\overline{d_0}$; however, until this point, it appears that Section~\ref{dream} is about $N$, which is the cokernel of $\overline{d_1}$.
The homological properties of $N$, which are listed in (\ref{yet.c}) and (\ref{yet.d}), were also promised in the introduction.
\begin{observation}\label{yet-to-come}
Adopt the language of {\rm \ref{data2}.\ref{data2-one}}, {\rm\ref{Not2}.\ref{Not2.a}}, and {\rm \ref{R2}.\ref{R2.z}}. Assume that $R_0$ is a domain.
\begin{enumerate}[\rm(a)]
\item\label{yet.a} The complex $A\otimes_R\text{\rm(\ref{pre-cplx})}$ is exact. \item\label{yet.b} The module $N$ {\rm (}of Lemma {\rm\ref{main-Dream-Lemma}} and elsewhere{\rm)} is isomorphic to the module of {\rm (\ref{column})}.
\item \label{yet.c} The $A$-module $N$ is self-dual.
\item \label{yet.d} If $R_0$ is a Cohen-Macaulay domain, then
$N$ is a self-dual maximal Cohen-Macaulay $A$-module of rank two.
\item \label{yet.e}
If $R_0$ is a Gorenstein domain, and
$$\mathbb X:\quad \cdots \xrightarrow{d_4}\mathbb X_3\xrightarrow{d_3}\mathbb X_2\xrightarrow{d_2}
\textstyle \bigwedge^3\overline{F}^*\xrightarrow{\overline{d_1}} \overline{F}^*
$$
is a resolution of $N$ by free $A$-modules, then
$$\textstyle \mathbb Y:\quad \cdots \xrightarrow{d_4}\mathbb X_3\xrightarrow{d_3}\mathbb X_2\xrightarrow{d_2}\bigwedge^3\overline{F}^*\xrightarrow{\overline{d_1}} \overline{F}^*\xrightarrow{\overline{d_0}}\overline{F} \xrightarrow{\overline{\delta_1}}\bigwedge^3\overline{F}\xrightarrow{d_2^*}\mathbb X_2^*\xrightarrow{d_3^*}\mathbb X_3^*\xrightarrow{d_4^*}\cdots$$
is a self-dual totally acyclic complex. {\rm(}In other words, $\operatorname{H}_\bullet(\mathbb Y)=\operatorname{H}_{\bullet}(\mathbb Y^*)=0$ and, after making the appropriate shift, $\mathbb Y^*$ is isomorphic to $\mathbb Y$.{\rm)}
\end{enumerate}\end{observation}
\begin{proof} (\ref{yet.a}) We are supposed to prove that the complex \begin{equation}\label{name-me}\textstyle \bigwedge^3\overline{F}^*\xrightarrow{\overline{d_1}} \overline{F}^*\xrightarrow{\overline{d_0}}\overline{F} \xrightarrow{\overline{\delta_1}}\bigwedge^3\overline{F}\end{equation} is exact. (Recall from \ref{Not2}.\ref{overline} that $\overline{\phantom{x}}$ is the functor $A\otimes_R-$.) We showed in (\ref{key}) that $$\textstyle \bigwedge^3\overline{F}^*\xrightarrow{\overline{d_1}} \overline{F}^*\xrightarrow{\operatorname{projection}\circ\overline{d_0}}A^3$$is exact. It follows that
$$\operatorname{im} \overline{d_1}\subseteq \ker \overline{d_0}\subseteq \ker(\operatorname{projection}\circ\overline{d_0})= \operatorname{im} \overline{d_1}$$ and (\ref{name-me}) is exact at $\overline{F}^*$.
We now prove that (\ref{name-me}) is exact at $\overline{F}$. Let $f_1=\sum_{i=1}^\goth f a_ie_i$ be in $\ker \overline{\delta_1}$, with $a_i\in A_i$ and $e_1,\dots, e_n$ a basis for $\overline{F}$. Use the coefficient of $e_1\wedge e_i\wedge e_j$ in $0=\overline{\delta_1}(f_1)$ in order to see that
$$x_{i,j}a_1\in(x_{1,i}\ ,\ x_{1,j})\subseteq (x_{1,2}\ ,\ x_{1,3}\ ,\ \dots\ ,\ x_{1,\goth f})$$
for all $i$ and $j$ with $2\le i<j\le \goth f$. The ideal $(x_{1,2}\ ,\ x_{1,3}\ ,\ \dots\ ,\ x_{1,\goth f})$ of $A$ is prime (indeed, $A/(x_{1,2}\ ,\ x_{1,3}\ ,\ \dots\ ,\ x_{1,\goth f})$ is the domain defined by ``$\operatorname{Pf}_4$'' of a smaller generic matrix) and $x_{i,j}$ is not in $(x_{1,2}\ ,\ x_{1,3}\ ,\ \dots\ ,\ x_{1,\goth f})$. Therefore, $a_1\in (x_{1,2}\ ,\ x_{1,3}\ ,\ \dots\ ,\ x_{1,\goth f})$ and there is an element $\phi_1\in \overline{F}^*$ with $f_1^\dagger=f_1-\overline{d_0}(\phi_1)=\sum_{i=2}^\goth f a_i^\dagger e_i$. (Recall that $-\mathbf X$ is the matrix for $d_0$.) The coefficient of $e_1\wedge e_2\wedge e_3$ in $0=\overline{\delta_1}(f_1^\dagger)$ shows $a_2^\dagger x_{1,3}$ is in the prime ideal $(x_{1,2})$; hence, $a_2^\dagger\in (x_{1,2})$ and one may use the first column of $\mathbf X$ to remove $a_2^\dagger$ without damaging $a_1^\dagger=0$. In other words, there exists $\phi_1^\ddag\in \overline{F}^*$ with $f_1^\ddag=f_1-d_0(\phi_1^\ddag)=\sum_{i=3}^\goth f a_ie_i$. The coefficient of $e_1\wedge e_2\wedge e_j$ in $0=\overline{\delta_1}(f_1^\ddag)$ shows that $x_{1,2}a_j^\ddag=0$ for $3\le j\le \goth f$. Hence, $a_j^\ddag=0$ for $3\le j\le \goth f$, $f_1$ is a boundary in (\ref{name-me}), and (\ref{name-me}) is exact.
\medskip(\ref{yet.b}) Apply (\ref{yet.a}) to see that $N=\operatorname{coker} \overline{d_1}\cong \operatorname{im} \overline{d_0}={\rm (\ref{column})}$.
\medskip(\ref{yet.c}) The definition $N=\operatorname{coker} \overline{d_1}$ guarantees that
$\textstyle\bigwedge^3\overline{F}^*\xrightarrow{\overline{d_1}} \overline{F}^*
\to N\to 0$ is exact. Apply $\Hom_A(-,A)$ to learn that
$$
0\to N^*\to \overline{F}^{**}\xrightarrow{\overline{d_1}^*} \textstyle\bigwedge^3\overline{F}^{**}$$is exact. It is easy to see that
$\overline{F}^{**}\xrightarrow{\overline{d_1}^*} \bigwedge^3\overline{F}^{**}$
is isomorphic to
$\overline{F}\xrightarrow{\overline{\delta_1}} \bigwedge^3\overline{F}$.
Assertion (\ref{yet.a}) now gives that $N\cong \ker \overline{\delta_1}\cong \ker \overline{d_1}^*\cong N^*$.
\medskip(\ref{yet.d}) Lemma~\ref{main-Dream-Lemma}, especially (\ref{3.11.0}) ensures that $N$ is a maximal Cohen-Macaulay $A$-module. The rank of $N$ is calculated in Lemma~\ref{exact-seq}. The self-duality of $N$ is established in (\ref{yet.c}).
\medskip(\ref{yet.e}) It follows from local duality (or the Auslander-Bridger formula, see, for example, \cite[Thms.~1.4.8 and 1.4.9]{Ch00}) that the maximal Cohen-Macaulay module $N$ over the Gorenstein ring $A$ satisfies $\operatorname{Ext}^i_A(N,A)=0$ for all positive $i$. So $\mathbb X\to N\to 0$ and $0\to N^*\to \mathbb X^*$ are both acyclic. The complexes $\mathbb X$ and $\mathbb X^*$ may be patched together at $N\cong N^*$ to form the totally acyclic complex $\mathbb Y$.
\end{proof}
\bigskip
\section{The main result.}\label{main}
\bigskip
The main result of the paper is Theorem~\ref{main-Theorem} where we prove that $J$ is a perfect Gorenstein ideal of grade $\binom{\goth f-2}2+2$. We estimate the grade of $J$ in Lemma~\ref{*.enough} and we use the exact sequence (\ref{*.claim.1}) to estimate the projective dimension of $\mathcal R/J$.
\begin{lemma}\label{*.enough}Adopt the language of {\rm\ref{data2}} and
{\rm\ref{Not2}} with $3\le \goth f$. If the base ring $R_0$ is
an arbitrary commutative Noetherian ring, then the height of the ideal $J$ satisfies the inequality $$\textstyle \binom{\goth f-2}2+2\le \operatorname{ht} J.$$
\end{lemma}
\begin{remark}\label{4.2} The assertion of Lemma~\ref{*.enough} is false when $\goth f=2$ because in this case $J$ equals $(t_1x_{1,2}\ ,\ t_2x_{1,2})$, which has height $1$; see Remark~\ref{R4}.\ref{funny} for a continuation of this example. On the other hand, Lemma~\ref{*.enough} does hold when $\goth f=3$; indeed, in this case, $J$ is the ideal generated by the maximal minors of the generic matrix
$$\bmatrix t_1&t_2&t_3\\x_{2,3}&-x_{1,3}&x_{1,2}\endbmatrix;$$see Remark~\ref{R4}.\ref{funny+} for a continuation of this example.
\end{remark}
\begin{proof} It suffices to replace $R_0$ with $R_0/p$ for some minimal prime ideal $p$ in $R_0$ and to prove the result when $R_0$ is a domain.
We use the language of Remark~\ref{R2} and view $J$ as the ideal $\operatorname{Pf}_4(\mathbf {X})+
I_1(\mathbf {tX})$ in the ring $\mathcal R=R_0[\{x_{i,j}\},\{t_i\}]$. Let $P$ be a prime ideal of $\mathcal R$ which contains $J$. We show $$\textstyle\binom{\goth f-2}2+2\le \operatorname{ht} P.$$
If $t_1\in P$, then $I'=\operatorname{Pf}_4(\mathbf{X})+(t_1)$ is a prime ideal of height $\binom{\goth f-2}2+1$ which is contained in $P$; furthermore, the first entry of $\mathbf {tX}$ is a non-zero element of $P\setminus I'$. Thus,
$\binom{\goth f-2}2+2\le \operatorname{ht} P$.
If $t_1\notin P$, then let
${\mathbf X}'$ be $\mathbf X$ with the first column removed,
${\mathbf X}''$ be $\mathbf X$ with the first row and first column removed,
and $I''$ be the ideal $\operatorname{Pf}_4({\mathbf X}'')$.
Observe that $I''$ is a
prime ideal of height $\binom{\goth f-3}2$ (this is where we use the hypothesis that $3\le \goth f$); $I''$ is contained in $P$;
and the entries of $\mathbf {tX}'$ form a regular sequence on $\mathcal R_{\ t_1}/I''\mathcal R_{\ t_1}$ in $P\mathcal R_{\ t_1}$. It follows that
$$\textstyle
\binom{\goth f-2}2+2=\binom{\goth f-3}2+\goth f-1\le \operatorname{ht} P \mathcal R_{\ t_1}=\operatorname{ht} P.
\vspace{-18pt}$$
\end{proof}
\begin{proposition}\label{*.claim}Adopt the language of {\rm\ref{data2}} and {\rm\ref{Not2}}. If $2\le \goth f$ and $R_0$ is a Cohen-Macaulay domain,
then there is an exact sequence of $\mathcal A$-modules{\rm:}
\begin{equation}\label{*.claim.1}0\to \mathcal A\xrightarrow{\tau}
\mathcal N\xrightarrow{\tau(\xi)}\mathcal A\to \mathcal R/ J\to 0.\end{equation} The map
$\tau:\mathcal A\to \mathcal N$ sends the element $1$ of $\mathcal A$ to the class of $\tau$ in $$\textstyle \mathcal N
=
\mathcal A\otimes_{\mathcal R}\operatorname{coker}\big(\xi:\bigwedge^3\mathcal F^*\to \mathcal F^*\big).$$ If $\phi_1$ is in $\mathcal F^*$, then the map $\tau(\xi):
\mathcal N\to \mathcal A$ sends the class of $\phi_1$ in $
\mathcal N$ to the class of $[\tau(\xi)](\phi_1)$ in $\mathcal A=\mathcal R/(I\cdot \mathcal R)$. The map $\mathcal A\to \mathcal R/ J$ is the natural quotient map $$\mathcal A=\mathcal R/(I\cdot \mathcal R)\to \mathcal R/(I\cdot \mathcal R+ K)=\mathcal R/J.$$
\end{proposition}
\begin{remarks}\label{R4}\begin{enumerate}[\rm(a)]\item After we prove Theorem~\ref{main-Theorem}, we are able to improve Proposition~\ref{*.claim}. In the improved version, $R_0$ is allowed to be an arbitrary commutative Noetherian ring. See Proposition~\ref{improved}.
\item\label{here it is} The exact sequence $0\to \mathcal A\to \mathcal N\to J\mathcal A \to 0$, which is a consequence of (\ref{*.claim.1}), exhibits $J\mathcal A$ as a Bourbaki ideal of $\mathcal N$, in the sense of \cite{A66, M80, BHU87, SUV03}.
\item The map $\tau({\xi})$ of (\ref{*.claim.1}) is well-defined.
Indeed,
if $\phi_3\in \bigwedge^3\mathcal F^*$, then $\xi(\phi_3)$ represents $0$ in $\mathcal N$ and
$[\tau(\xi)](\xi(\phi_3))$, which is equal to $\xi^{(2)}(\phi_3\wedge \tau)$ by (\ref{Gamma}) and (\ref{compat}), is equal to $0$ in $\mathcal A$.
\item It is not difficult to see that (\ref{*.claim.1}) is a complex of $\mathcal A$-modules.
\item \label{funny}If $\goth f=2$, then $\mathcal R=\mathcal A$ and, in the language of Remark~\ref{R2}, the complex (\ref{*.claim.1}) is
$$0\to\mathcal R \xrightarrow{\bmatrix t_1\\t_2\endbmatrix}\mathcal R^2\xrightarrow{\bmatrix -t_2x_{1,2}&t_1x_{1,2}\endbmatrix}\mathcal R\to \mathcal R/(t_1x_{1,2}\ ,\ t_2x_{1,2})\to 0,$$ which is exact, see Remark~\ref{4.2}
\item \label{funny+}If $\goth f=3$, then $\mathcal R=\mathcal A$ and, in the language of Remark~\ref{R2}, the complex (\ref{*.claim.1}) is
\begin{align*}0&\to\mathcal R \xrightarrow{\bmatrix t_1\\t_2\\t_3\endbmatrix}\frac{\mathcal R^3}{\left(\bmatrix x_{2,3}\\-x_{1,3}\\x_{1,2}\endbmatrix\right)}\xrightarrow{\bmatrix -t_2x_{1,2}-t_3x_{1,3}&t_1x_{1,2}-t_3x_{2,3}&t_{1}x_{1,3}+t_2x_{2,3}\endbmatrix}\mathcal R\\&\to \mathcal R/(-t_2x_{1,2}-t_3x_{1,3}\ ,\ t_1x_{1,2}-t_3x_{2,3}\ ,\ t_{1}x_{1,3}+t_2x_{2,3})\to 0,\end{align*} which is exact; see Remark~\ref{4.2}\end{enumerate}
\end{remarks}
\medskip\noindent
Observation
\ref{*.10.9}
and Lemma \ref{*.critical} are used in the proof of Proposition~\ref{*.claim}, which is given in \ref{*.proof-of-claim}.
\begin{observation}\label{*.10.9} Retain the hypotheses of Proposition~{\rm\ref{*.claim}}. The complex {\rm(\ref{*.claim.1})} is exact at $\mathcal R/J$ and at both copies of $\mathcal A$.\end{observation}
\begin{proof} It is clear that (\ref{*.claim.1}) is exact at $\mathcal R/J$ and at the right hand $\mathcal A$. We prove that (\ref{*.claim.1}) is exact at the left hand $\mathcal A$. Let $r\in \mathcal R$ with $r\cdot \tau\equiv\xi(\phi_3)\mod I\mathcal F$ for some $\phi_3\in \bigwedge^3 \mathcal F^*$. Apply $r\tau$ to $\xi$ and use Observation~\ref{doo-8.2}.\ref{doo-8.2.a} to learn that
$$r\cdot \tau(\xi)\equiv[\xi(\phi_3)](\xi)\equiv\phi_3(\xi^{(2)})\in I\mathcal F.$$It follows that $r\cdot K\subseteq I$. The ideal $I$ is prime and degree considerations show that $K\not\subseteq I$. It follows that $r\in I$. Thus, $\tau: \mathcal A\to \mathcal N$ is an injection.
\end{proof}
\begin{lemma}\label{*.critical}Adopt the language of {\rm\ref{data2}} and {\rm\ref{Not2}}.
Let $\phi_1,\phi_1'$ be elements of $\mathcal F^*$ with the property that the element $\phi_1\wedge \phi_1'$ is part of a basis for $\mathcal F^*$ and let $x$ be the element $\xi(\phi_1\wedge \phi_1')$ of $\mathcal R$. Then the following statements hold. \begin{enumerate}[\rm(a)] \item\label{*.critical.a} If the base ring $R_0$ is a commutative Noetherian domain, then the localization {\rm(\ref{*.claim.1})}$_x$ of the complex {\rm(\ref{*.claim.1})} at $x$ is isomorphic to
\begin{align*}0\to \mathcal A_{\, x}\xrightarrow {\bmatrix -[\tau(\xi)](\phi_1')\\\phantom{-}[\tau(\xi)](\phi_1)\endbmatrix} \mathcal A_{\,x} \oplus \mathcal A_{\,x} &\xrightarrow {\bmatrix [\tau(\xi)](\phi_1)&[\tau(\xi)](\phi_1')\endbmatrix} \mathcal A_{\,x}\\ &\longrightarrow \frac{\mathcal A_{\,x}}{([\tau(\xi)](\phi_1)\ ,\ [\tau(\xi)](\phi_1'))\mathcal A_{\,x}} \to 0.\end{align*}
\item\label{*.critical.b}
If $R_0$ is a Cohen-Macaulay domain, then
the localization {\rm(\ref{*.claim.1})}$_x$ is exact.\end{enumerate}\end{lemma}
\begin{remark-no-advance} Once we prove Theorem~\ref{main-Theorem}, then a much stronger version of Lemma~\ref{*.critical}
is also true, see Proposition~\ref{improved}.
\end{remark-no-advance}
\begin{proof} (\ref{*.critical.a}) The element $x$ in $\mathcal R$ is a non-zero element of $\mathcal R_{\ (1,0)}$. The ideal $I\cdot \mathcal R$ of $\mathcal R$ is a prime ideal generated by elements of $\mathcal R_{\ (2,0)}$; hence $x$ is a non-zero-divisor in $\mathcal A=\mathcal R/(I\cdot \mathcal R)$. Consider the map
\begin{equation}\label{*.natural-map}\mathcal A_{\,x}\oplus \mathcal A_{\,x}\longrightarrow
\mathcal N_{\ \,x},\end{equation}which sends $\bmatrix a_1&a_2\endbmatrix^{\rm T}$ to the class of $a_1 \phi_1+a_2\phi_1'$. This map is onto because, if $\phi_1''\in \mathcal F$, then the equation
\begin{equation}\label{*.because}0=\xi(\phi_1\wedge \phi_1'\wedge \phi_1'')=x\cdot\phi_1''-\xi(\phi_1\wedge \phi_1'')\cdot \phi_1'+\xi(\phi_1'\wedge \phi_1'')\cdot \phi_1\end{equation}
holds in $\mathcal N$ (see Observation \ref{doo-8.2}.\ref{doo-8.2.b}); and therefore the class of $\phi_1''$ in $\mathcal N_{\ \,x}$ is in the image of the map (\ref{*.natural-map}). Let $(0)$ be the prime ideal $(0)$ in the domain $\mathcal A$ and $L$ be the kernel of (\ref{*.natural-map}). We know from Lemma~\ref{exact-seq} that $\mathcal N_{\ \,(0)}=\mathcal A_{\,(0)}\oplus \mathcal A_{\,(0)}$; hence $L_{(0)}=0$. On the other hand, $L$ is a submodule of a free $\mathcal A_{\, x}$-module and $\mathcal A_{\,x}$ is a domain; thus, $L=0$ and (\ref{*.natural-map}) is an isomorphism.
Apply (\ref{*.because}), with $\tau$ in place of $\phi_1''$, to see that the composition
$$\mathcal A_{\,x}\xrightarrow{x}\mathcal A_{\,x}\xrightarrow {\tau} \mathcal N_{\ \,x}$$ sends $1\in \mathcal A_{\,x}$ to
$$x\tau=[\tau(\xi)](\phi_1)\cdot \phi_1'-[\tau(\xi)](\phi_1')\cdot \phi_1$$
in $\mathcal N_{\ \,x}$; and therefore, the composition
$$\mathcal A_{\,x}\xrightarrow{x}\mathcal A_{\,x}\xrightarrow {\tau} \mathcal N_{\ \,x}\xrightarrow {\text{(\ref{*.natural-map})}^{-1}}\mathcal A_{\,x}\oplus \mathcal A_{\,x}$$ sends $1\in \mathcal A_{\,x}$ to
$$\bmatrix -[\tau(\xi)](\phi_1')\\\phantom{-}[\tau(\xi)](\phi_1)\endbmatrix\in \mathcal A_{\,x}\oplus \mathcal A_{\,x}.$$ It is clear that the composition
$$\mathcal A_{\,x}\oplus \mathcal A_{\,x}\xrightarrow{\text{(\ref{*.natural-map})}} \mathcal N_{\ \,x}\xrightarrow{\tau(\xi)} \mathcal A_{\,x}$$ sends
$$\bmatrix 1\\0\endbmatrix \mapsto [\tau(\xi)](\phi_1)\quad \text{and} \quad \bmatrix 0\\1\endbmatrix \mapsto [\tau(\xi)](\phi_1').$$ This completes the proof of (\ref{*.critical.a}).
\medskip\noindent(\ref{*.critical.b}) We know from (\ref{*.critical.a}) that the ideal $J\mathcal A_{\,x}$ is generated by $[\tau(\xi)](\phi_1)$ and $[\tau(\xi)](\phi_1')$ and we know from Lemma~\ref{*.enough} that $2\le \operatorname{ht}(J\mathcal A)$. The ring $\mathcal A$ is Cohen-Macaulay; so,
$$2\le \operatorname{ht}(J\mathcal A)=\operatorname{grade} J\mathcal A\le \operatorname{grade} J\mathcal A_{\,x}.$$It follows that {\rm(\ref{*.claim.1})}$_x$, which, according to (\ref{*.critical.a}), is isomorphic to the augmented Koszul complex on the generating set $\{[\tau(\xi)](\phi_1)\ ,\ [\tau(\xi)](\phi_1')\}$ of $J\mathcal A_{\,x}$, is exact.
\end{proof}
\begin{chunk}\label{*.proof-of-claim}{\bf The proof of Proposition~\ref{*.claim}.} In light of Remark~\ref{R4}.\ref{funny}, we may assume that ${4\le \goth f}$.
We know from Observation \ref{*.10.9} that (\ref{*.claim.1}) is a complex of $\mathcal A$-modules which is exact everywhere except possibly at $\mathcal N$. Let $\operatorname{H}$ be the homology of (\ref{*.claim.1}) at $\mathcal N$. We argue by contradiction. Assume that $\operatorname{H}\neq 0$. Let $P$ be an associated prime of $\operatorname{H}$.
Lemma~\ref{*.critical} shows
that $\operatorname{H}_x=0$ for every $x$ in $\mathcal R$ of the form
\begin{equation}\label{form}\text{$x=\xi(\phi_1\wedge \phi_1')$ where $\phi_1$ and $\phi_1'$ are in $\mathcal F^*$ with $\phi_1\wedge \phi_1'$ part of a basis for $\textstyle\bigwedge^2\mathcal F^*$.}\end{equation} The fact that $\operatorname{H}_x=0$ and $\operatorname{H}_P\neq 0$ forces $x$ to be an element of $P$. The $R_0$-module $\mathcal R_{(1,0)}$ is generated by elements $x$ of the form (\ref{form}); therefore, $\mathcal R_{\ (1,0)}\subseteq P$.
Consider the complex (\ref{*.claim.1}). Let $B$ be the image of $\tau:\mathcal A\to \mathcal N$ and $Z$ be the kernel of $\tau(\xi):\mathcal N\to \mathcal A$. Combine the exact sequences
\begin{align*}&0\to \mathcal A\to B\to 0&&\text{from Observation~\ref{*.10.9},}\quad\text{and}\\
&0\to B\to Z\to \operatorname{H}\to 0\end{align*}
in order to obtain
the exact sequence \begin{equation}\label{*.ses}0\to \mathcal A\to Z\to \operatorname{H}\to 0.\end{equation} The $\mathcal R$-modules $\mathcal A$ and $\mathcal N$ are both perfect
and their annihilators have grade $\binom{\goth f-2}2$; see Corollary~\ref{carry forward}.
The ring $\mathcal R$ is Cohen-Macaulay; so,
$\mathcal A_{\,P}$ and $\mathcal N_{\ \,P}$ are both Cohen-Macaulay $\mathcal R_{\ P}$-modules with
$$\operatorname{depth} \mathcal N_{\ \,P}=\dim \mathcal N_{\ \,P}=\dim \mathcal A_{\,P}=\operatorname{depth} \mathcal A_{\,P};$$ and this common number is equal to $\dim \mathcal R_{\ P}-\binom{\goth f-2}2$. Furthermore, the ideal $(\mathcal R_{\ 1,0})$ of $\mathcal R$, which is prime of height $\binom{\goth f}2$, is contained in $P$.
It follows that
$$\textstyle 2\le \binom{\goth f}2-\binom{\goth f-2}2\le \dim \mathcal A_{\,P}.$$(The left most inequality holds because $3\le \goth f$.)
The module $Z_P$ is a non-zero submodule of $\mathcal N_{\ \,P}$; so $1\le \operatorname{depth} Z_P$. We have chosen $P$ with $\operatorname{H}_P\neq 0$ and $\operatorname{depth} \operatorname{H}_P=0$. The usual argument about the growth of depth in a short exact sequence shows that the exact sequence
$$0\to \mathcal A_{\,P}\to Z_P\to \operatorname{H}_P\to 0,$$ which is obtained by localizing the short exact sequence (\ref{*.ses}) at $P$,
is impossible; see, for example, \cite[Prop.~1.2.9]{BH}. This contradiction establishes the result. \qed
\end{chunk}
\begin{theorem}\label{main-Theorem}Adopt the language of {\rm\ref{data2}} and {\rm \ref{Not2}}. If $4\le \goth f$ and $R_0$ is an arbitrary commutative Noetherian ring, then $J$ is a perfect Gorenstein ideal of $\mathcal R$ of grade $\binom{\goth f-2}2+2$. In particular, if $R_0$ is a Gorenstein ring, then $\mathcal R/ J$ is a Gorenstein ring. \end{theorem}
\begin{proof} We employ the theory of generic perfection as described at the beginning of \ref{prove-dream}. It suffices to prove the result when $R_0$ is equal to the ring of integers and when $R_0$ is a field. In particular, we may assume that $R_0$ is a Cohen-Macaulay domain. Proposition~\ref{*.claim} guarantees that there exists an exact sequence of $\mathcal R$-modules
$$0\to \mathcal A\to \mathcal N\to \mathcal A\to \mathcal R/J\to 0$$
and Corollary~\ref{carry forward} ensures that $\mathcal A$ and $\mathcal N$ have free resolutions of length $\binom{\goth f-2}2$; furthermore, the back Betti number in the resolution of $\mathcal A$ is one. Resolve $\mathcal A$ and $\mathcal N$ and form the iterated mapping cone in order to find a free resolution of $\mathcal R/J$ of length $\binom{\goth f-2}2+2$. The back Betti number in the resolution of $\mathcal R/J$ is one. We see that
$$\textstyle \binom{\goth f-2}2+2\le \operatorname{grade} J\le \operatorname{pd}_{\mathcal R}\mathcal R/J\le \binom{\goth f-2}2+2.$$(The first inequality is Lemma~\ref{*.enough} and the second inequality is (\ref{auto}).) Thus, equality holds throughout and the proof is complete.
\end{proof}
\section{Consequences of the main result.}\label{consequences}
In this section, especially in Corollary~\ref{corollary}, we prove some consequences of the fact that $J$ is a perfect ideal in $\mathcal R$.
We begin by identifying some relations on the generators of $J$. These relations are used in the proof of Corollary~\ref{corollary}.\ref{cor-b} that $(\mathcal R/J)_{x_{i,j}}$ is a polynomial ring over $R_0[x_{i,j}\ ,\ x_{i,j}^{-1}]$.
\begin{definition}\label{5.1}Adopt the language of {\rm\ref{data2}} and {\rm\ref{Not2}}. Define the maps and modules
$$\mathbb E_2\xrightarrow{D_2}\mathbb E_1\xrightarrow{D_1}\mathbb E_0$$
by \begingroup\allowdisplaybreaks\begin{align*}&\mathbb E_2= \begin{matrix}\bigwedge^3\mathcal F^*
\\\oplus\\ \ker\left(\mathcal F^*\otimes \bigwedge^5\mathcal F^*\xrightarrow{\operatorname{multiplication}}\bigwedge^6\mathcal F^*\right),
\\\oplus\\\bigwedge^3\mathcal F^*\otimes \bigwedge^3 \mathcal F^*\end{matrix}\quad\mathbb E_1= \begin{matrix}\mathcal F^*\\\oplus\\\bigwedge^4\mathcal F^*\end{matrix},\quad\mathbb E_0=\mathcal R,\\
&D_2\left(\bmatrix \phi_3\\0\\\phi_3'\otimes \phi_3'' \endbmatrix\right)=\bmatrix \xi(\phi_3)\hfill\\\tau\wedge \phi_3+\xi(\phi_3')\wedge \phi_3''-\phi_3'\wedge \xi(\phi_3'')\endbmatrix,\\&D_1\left(\bmatrix \phi_1\\\phi_4 \endbmatrix\right)=[\tau(\xi)](\phi_1)+\xi^{(2)}(\phi_4),\end{align*}\endgroup and the middle component of $D_2$ is induced by the map $F^*\otimes \bigwedge^5\mathcal F^*\to \bigwedge^4\mathcal F^*$ which sends $\phi_1\otimes \phi_5$ to $[\phi_1(\xi)](\phi_5)$.
\end{definition}
\begin{observation}\label{D}The maps and modules of Definition~{\rm\ref{5.1}} form a complex and the image of $D_1$ is the ideal $J$ of {\rm\ref{Not2}}.\end{observation}
\begin{proof}We verify that $D_1\circ D_2=0$. We use (\ref{Gamma}), (\ref{compat}), Observation~\ref{doo-8.2}.\ref{doo-8.2.a}, and the module action of $\bigwedge^{\bullet}\mathcal F$ and $\bigwedge^{\bullet}\mathcal F^*$ on one another to compute
\begingroup\allowdisplaybreaks\begin{align*}(D_1\circ D_2)(\phi_3)&=D_1\left(\bmatrix \xi(\phi_3)\\\tau\wedge \phi_3\endbmatrix\right)=[\tau(\xi)](\xi(\phi_3))+\xi^{(2)}(\tau\wedge \phi_3)\\&=
[\tau(\xi^{(2)})](\phi_3)-(\phi_3\wedge \tau)(\xi^{(2)})=0, \\
(D_1\circ D_2)(\sum_i\phi_{1,i}\otimes \phi_{5,i})&=\sum_i D_1([\phi_{1,i}(\xi)](\phi_{5,i}))=\sum_i \xi^{(2)}([\phi_{1,i}(\xi)](\phi_{5,i}))\\
&=\sum_i [\phi_{1,i}(\xi^{(3)})](\phi_{5,i})=(\sum_i\phi_{5,i} \wedge \phi_{1,i})(\xi^{(3)})=0, \quad\text{and}\\
(D_1\circ D_2)(\phi_3'\otimes \phi_3'')&=D_1\big(\xi(\phi_3')\wedge \phi_3''-\phi_3'\wedge \xi(\phi_3'')\big)=\xi^{(2)}\big(\xi(\phi_3')\wedge \phi_3''-\phi_3'\wedge \xi(\phi_3'')\big)\\&=\xi^{(2)}\big(\xi(\phi_3')\wedge \phi_3''\big)-\xi^{(2)}\big(\phi_3'\wedge \xi(\phi_3'')\big).
\\\intertext{Furthermore, we compute}
\xi^{(2)}\big(\xi(\phi_3')\wedge \phi_3''\big)&=-[\phi_3''\wedge \xi(\phi_3')](\xi^{(2)})=-\phi_3''\big([\xi(\phi_3')](\xi^{(2)})\big)=-\phi_3''\big([\xi(\phi_3')](\xi)\wedge \xi\big)\\&=-\phi_3''\big(
\phi_3'(\xi^{(2)})\wedge \xi\big)=-[\phi_3'(\xi^{(2)})](\xi(\phi_3''))=
-[\xi(\phi_3'')\wedge \phi_3'](\xi^{(2)})\\
&=[\phi_3'\wedge \xi(\phi_3'')](\xi^{(2)})=\xi^{(2)}[\phi_3'\wedge \xi(\phi_3'')];
\end{align*}\endgroup
and therefore, $(D_1\circ D_2)(\phi_3'\otimes \phi_3'')=0$.
\end{proof}
\begin{corollary}\label{corollary}Adopt the language of {\rm\ref{data2}}, {\rm \ref{Not2}}, and {\rm \ref{R2}}. Assume that $4\le \goth f$ and $R_0$ is an arbitrary commutative Noetherian ring. The following statements hold.
\begin{enumerate}[\rm(a)]
\item\label{cor-a} The elements $x_{1,2}\ ,\ x_{1,3}$ form a regular sequence
on $\mathcal R/J$.
\item\label{cor-b} For each pair $i,j$ with $1\le i<j\le \goth f$, the localization of the ring $\mathcal R/J$ at the element $x_{i,j}$ is isomorphic to a polynomial ring over $R_0[x_{i,j}\ ,\ x_{i,j}^{-1}]$.
\item\label{cor-c} The ring $\mathcal R/J$ is a domain if and only if $R_0$ is a domain.
\item\label{cor-c.5} If $R_0$ is a domain, then $(x_{1,2})\mathcal R/J$ is a prime ideal in $\mathcal R/J$.
\item\label{cor-d} The ring $\mathcal R/J$ is normal if and only if $R_0$ is normal.
\item \label{cor-e} If $R_0$ is a normal domain, then the divisor class group of $R_0$ is isomorphic to the divisor class group of $\mathcal R/J$. In particular, $R_0$ is a unique factorization domain if and only if $\mathcal R/J$ is a unique factorization domain. \end{enumerate}
\end{corollary}
\begin{proof}(\ref{cor-a})
We employ the method of proof that is described in Corollary~\ref{KL-consq}. It suffices to show that
\begin{align*}
&\textstyle \binom{\goth f-2}2+3\le \operatorname{grade} PR_P&&\text{for all $P\in \operatorname{Spec} \mathcal R$ with $J+(x_{1,2})\subseteq P$}&&\text{situation 1, and}\\
&\textstyle\binom{\goth f-2}2+4\le \operatorname{grade} PR_P&&\text{for all $P\in \operatorname{Spec} \mathcal R$ with $J+(x_{1,2}\ ,\ x_{1,3})$}\subseteq P&&\text{situation 2}\\
\end{align*}
Fix a prime $P$ from situation 1 or situation 2.
There are two cases. Assume first that $t_{\goth f}\notin P$. The ring $\mathcal R_{\ P}$ is a localization of $\mathcal R_{\ t_f}$ and $\mathcal R_{\ t_f}$ is equal to the polynomial ring $$(R_0[t_1\ ,\ \dots\ ,\ t_\goth f\ ,\ t_\goth f^{-1},\{x_{i,j}\mid 1\le i<j\le \goth f-1\}])[(\mathbf{tX})_1\ ,\ \dots\ ,\ (\mathbf{tX})_{\goth f-1}].$$
Let $\mathbf X'$ represent $\mathbf X$ with row and column $\goth f$ deleted. Apply Corollary ~\ref{KL-consq}. In situation 1,
the ideal $(x_{1,2},J)\mathcal R_{\ t_\goth f}$ contains the $\operatorname{grade} \binom{\goth f-3}2+1$ ideal $(x_{1,2}\ ,\ \operatorname{Pf}_{4}(\mathbf X')$ of \begin{equation}\label{ring}R_0[t_1\ ,\ \dots\ ,\ t_\goth f\ ,\ t_\goth f^{-1}\ ,\ \{x_{i,j}\mid 1\le i<j\le \goth f-1\}]\end{equation} as well as the $\goth f-1$ indeterminates $(\mathbf{tX})_1,\dots,(\mathbf{tX})_{\goth f-1}$. Thus,
$$\textstyle \binom{\goth f-2}2+3=\Big(\binom{\goth f-3}2+1\Big)+(\goth f-1)\le \operatorname{grade}(x_{1,2}\ ,\ J)\mathcal R_{\ t_\goth f}.$$Similarly, in situation 2, Corollary ~\ref{KL-consq} guarantees that
$$\textstyle \binom{\goth f-3}2+2\le \operatorname{grade} (x_{1,2}\ ,\ x_{1,3}\ ,\ \operatorname{Pf}_{4}(\mathbf X'))\cdot \text{(\ref{ring})};$$so, $\binom{\goth f-2}2+4\le \operatorname{grade} (x_{1,2}\ ,\ x_{1,3}\ ,\ J)\mathcal R_{\ t_{\goth f}}$.
The same argument works if $t_{\goth f-1}\notin P$. The second case is $t_{\goth f}$ and $t_{\goth f-1}$ are both in $P$. In this case, Corollary~\ref{KL-consq} yields
\begin{align*} &(t_{\goth f}\ ,\ t_{\goth f-1})+\operatorname{Pf}_4(\mathbf X)+(x_{1,2})\subseteq P&&\textstyle\text{and $\binom{\goth f-2}2+3\le \operatorname{grade} P$}&&\text{in situation 1, and}\\
&(t_{\goth f}\ ,\ t_{\goth f-1})+\operatorname{Pf}_4(\mathbf X)+(x_{1,2}\ ,\ x_{1,3})\subseteq P&&\textstyle\text{and $\binom{\goth f-2}2+4\le \operatorname{grade} P$}&&\text{in situation 2}
.\end{align*}
\medskip\noindent(\ref{cor-b}) It is notationally convenient
to
prove the result for $(i,j)=(1,2)$.
Let $S_1$ and $S_2$ be the following subsets of $\mathcal R$:
\begin{align}S_1=&\{x_{i,j}|1\le i\le 2\ ,\ 3\le j\le \goth f\}\cup \{t_j\mid 3\le j\le \goth f\}\quad\text{and} \label{Ssub1}\\S_2=&\textstyle\{x_{1,2}x_{i,j}-x_{1,i}x_{2,j}+x_{1,j}x_{2,i}\mid 3\le i<j\le \goth f\}\notag\\&\cup \Big\{x_{1,2}t_2+\sum\limits_{j=3}^{\goth f} x_{1,j}t_j\ ,\ x_{1,2}t_1-\sum\limits_{j=3}^{\goth f} x_{2,j}t_j\Big\}.\notag\end{align}
\noindent Notice that \begin{enumerate}[\rm(A)]
\item\label{cor-A} $S_1\cup S_2$ is a set of indeterminates over the ring $R_0[x_{1,2}\ ,\ x_{1,2}^{-1}]$,
\item\label{cor-B} $\big(R_0[x_{1,2}\ ,\ x_{1,2}^{-1}]\big)[S_1\cup S_2]=\mathcal R[x_{1,2}^{-1}]$, and
\item\label{cor-C} $J\mathcal R[x_{1,2}^{-1}]=(S_2)\mathcal R[x_{1,2}^{-1}]$. \end{enumerate}
Assertions (\ref{cor-A}) and (\ref{cor-B}) are obvious.
Once (\ref{cor-C}) is established, we will have shown that
\begin{equation}\label{VARS} \begin{array}{l}\text{$(\mathcal R/J)_{x_{1,2}}$ is the polynomial ring }R_0[x_{1,2}\ ,\ x_{1,2}^{-1}][S_1]\text{ over $R_0[x_{1,2}\ ,\ x_{1,2}^{-1}]$}\vspace{5pt}\\\text{for $S_1$ given in (\ref{Ssub1})}.\end{array}\end{equation}
We now prove (\ref{cor-C}). Observe first that $S_2\subset J$. Indeed, in the language of Observation~\ref{D} and Remark~\ref{R2}, the ideal $S_2\mathcal R$ is the image, under $D_1$, of the submodule $$\textstyle W=\mathcal R e_1^*\oplus \mathcal R e_2^*\oplus
\mathcal R (e_1^*\wedge e_2^*)\wedge \bigwedge^2 \mathcal F^*$$ of $\mathbb E_1$. We show that
\begin{equation}\label{near}
\begin{array}{rcl}
x_{1,2} \mathcal F^* &\subseteq& W+\operatorname{im} D_2\\
x_{1,2} \mathcal R(e_1^*\ ,\ e_2^*)\wedge \bigwedge^3\mathcal F^* &\subseteq& W+\operatorname{im} D_2,\quad \text{and}\\
x_{1,2}\bigwedge^4\mathcal F^*&\subseteq& W+\mathcal R(e_1^*\ ,\ e_2^*)\wedge \bigwedge^3\mathcal F^*+\operatorname{im} D_2. \end{array}\end{equation}
Once (\ref{near}) is established, then iteration of (\ref{near}) gives $x_{1,2}^2\mathbb E_1\subseteq W+\operatorname{im} D_2$; hence, $x_{1,2}^2J$ is contained in $S_2 \mathcal R$ and (\ref{cor-C}) holds.
If $\phi_1\in \mathcal F^*$, then use Observation~\ref{doo-8.2}.\ref{doo-8.2.b} to see that
\begin{align*}x_{1,2}\phi_1&=\xi(e_2^*\wedge e_1^*)\cdot \phi_1=\xi(\phi_1\wedge e_2^*\wedge e_1^*)+\xi(\phi_1\wedge e_1^*)\cdot e_2^*-\xi(\phi_1\wedge e_2^*)\cdot e_1^*\\
&=D_2(\phi_1\wedge e_2^*\wedge e_1^*)-\tau\wedge \phi_1\wedge e_2^*\wedge e_1^*+\xi(\phi_1\wedge e_1^*)\cdot e_2^*-\xi(\phi_1\wedge e_2^*)\cdot e_1^*
\in W+\operatorname{im} D_2.\end{align*}If $\phi_3\in \bigwedge^3\mathcal F^*$, then
\begin{align*}&x_{1,2}e_1^*\wedge \phi_3=\xi(e_2^*\wedge e_1^*)\cdot e_1^*\wedge \phi_3
=[e_1^*(\xi)](e_2^*\wedge e_1^*\wedge \phi_3)+\text{an element of $W$}\\
=&D_2(e_1^*\otimes e_2^*\wedge e_1^*\wedge \phi_3)+\text{an element of $W$}\in W+\operatorname{im} D_2.
\end{align*}The calculation $x_{1,2}e_2^*\wedge \phi_3\in W+\operatorname{im} D_2$ is similar.
\noindent
If $\phi_1\in \mathcal F^*$ and $\phi_3\in \bigwedge^3\mathcal F^*$, then \begingroup\allowdisplaybreaks
\begin{align*}{}&x_{1,2}\phi_1\wedge\phi_3=\xi(e_2^*\wedge e_1^*)\cdot \phi_1\wedge\phi_3\\
{}=&\xi(\phi_1\wedge e_2^*\wedge e_1^*)\wedge \phi_3
+\xi(\phi_1\wedge e_1^*)\cdot e_2^*\wedge\phi_3
-\xi(\phi_1\wedge e_2^*)\cdot e_1^*\wedge\phi_3\\
{}=&D_2((\phi_1\wedge e_2^*\wedge e_1^*)\otimes \phi_3)
+\phi_1\wedge e_2^*\wedge e_1^*\wedge \xi(\phi_3)
+ \text{an element of $\mathcal R(e_1^*\ ,\ e_2^*)\wedge \textstyle\bigwedge^3\mathcal F^*$}\\
{}\in&W+\mathcal R(e_1^*\ ,\ e_2^*)\wedge \textstyle\bigwedge^3\mathcal F^*+\operatorname{im} D_2.\end{align*}\endgroup This completes the proof of (\ref{near}) and hence the proof of (\ref{cor-b}).
\medskip\noindent(\ref{cor-c}) Apply (\ref{cor-a}) and then (\ref{cor-b}) to see that $\mathcal R/J$ is a domain if and only if $(\mathcal R/J)_{x_{1,2}}$ is a domain if and only if $R_0$ is a domain.
\medskip\noindent(\ref{cor-c.5}) Suppose $\alpha$ and $\beta$ are elements of $\mathcal R/J$ with $\alpha\beta\in (x_{1,2})\cdot \mathcal R/J$. We know from (\ref{cor-b}) that $(x_{1,2})\cdot (\mathcal R/J)_{x_{1,3}}$ is a prime ideal; so, one of the elements $\alpha$ or $\beta$ (say, $\alpha$) is in $(x_{1,2})\cdot (\mathcal R/J)_{x_{1,3}}$. It follows that $x_{1,3}^s\alpha\in (x_{1,2})\cdot \mathcal R/J$, for some $s$. Apply (\ref{cor-a}) to see that $\alpha$ is in $(x_{1,2})\cdot \mathcal R/J$.
\medskip\noindent(\ref{cor-d}) ($\Leftarrow$) We apply the Serre criteria for normality in order to prove that $\mathcal R/J$ is normal. It suffices to prove that $(\mathcal R/J)_P$ is normal for all primes $P$ with $\operatorname{depth}(\mathcal R/J)_P\le 1$. If $\operatorname{depth} (\mathcal R/J)_P\le 1$, then (\ref{cor-a}) guarantees that at least one of the elements $x_{1,2}$ or $x_{1,3}$ is not in $P$. Thus, we know from (\ref{cor-b}) that $(\mathcal R/J)_P$ is a localization of a polynomial ring over $R_0$; hence, $(\mathcal R/J)_P$ is a normal domain.
\medskip\noindent(\ref{cor-d}) ($\Rightarrow$) The hypothesis that $\mathcal R/J$ is normal guarantees that $\mathcal R/J$ is reduced; and therefore, $R_0$ is reduced. The localization $(\mathcal R/J)_{x_{1,2}}$ is also normal. Recall from
(\ref{VARS}) that $(\mathcal R/J)_{x_{1,2}}$ is equal to $T[x_{1,2}^{-1}]$ where $T$ is the polynomial ring $R_0[x_{1,2}\ ,\ S_1]$ and $S_1$ is the list of indeterminates given in (\ref{Ssub1}).
Apply Lemma~\ref{BV}, with $y=x_{1,2}$, to conclude that $T$ is normal. Now a standard argument yields that $R_0$ is also normal.
\medskip\noindent(\ref{cor-e}) Avramov's proof \cite{A79} that $R/\operatorname{Pf}_{2t}(\mathbf X)$ is a unique factorization domain may be applied without change. In other words, there are isomorphisms of the following divisor class groups:
$$\xymatrix{\operatorname{Cl}(\mathcal R/J)\ar[r]^(.45){\alpha}& \operatorname{Cl}((\mathcal R/J)_{x_{1,2}})\ar[r]^(.48){\beta}& \operatorname{Cl}(R_0[S_1\ ,\ x_{1,2}^{-1}])
\ar@{<-}[r]^(.54){\gamma}
&
\operatorname{Cl}(R_0[S_1])\ar@{<-}[r]^(.54){\delta}&
\operatorname{Cl}(R_0).}$$ The element $x_{1,2}$ generates a prime ideal in $\mathcal R/J$ by (\ref{cor-c.5}); so the isomorphism $\alpha$ is Nagata's Lemma \cite[Cor.~7.3]{F73}. We proved in (\ref{VARS}) that $(\mathcal R/J)_{x_{1,2}}$ is equal to the polynomial ring $R_0[S_1\ ,\ x_{1,2}^{-1}]$, where $S_1$ is the list of indeterminates given in (\ref{Ssub1}); so the isomorphism $\beta$ is the identity map. The isomorphism $\gamma$ is again Nagata's Lemma and the isomorphism $\delta$ is Gauss' Lemma \cite[Thm.~8.1]{F73}.
\end{proof}
We have used the following normality criterion which appears as \cite[Lemma~16.24]{BV}. The result follows quickly from Serre's normality criterion. \begin{lemma}\label{BV} Let $T$ be a Noetherian ring, and $y$ be a regular element of $T$ such that
$T/T y$ is reduced and $T[y^{-1}]$ is a normal ring. Then $T$ is a normal ring.
\end{lemma}
Now that we know that $J$ is a perfect ideal, we are able to improve some of the results that we used in order to prove that $J$ is perfect. Notice that there are no hypotheses on the ring $R_0$.
\begin{proposition}\label{improved} Adopt the language of {\rm\ref{data2}}, {\rm\ref{Not2}} and {\rm\ref{R2}}. Let $R_0$ be an arbitrary commutative Noetherian ring. \begin{enumerate}[\rm(a)]
\item\label{improved.a}
The maps and modules of {\rm(\ref{*.claim.1})} form an exact sequence.
\item\label{improved.b}
The maps and modules of {\rm(\ref{exact-seq.1})} form an exact sequence.
\item\label{improved.X} The ideal $J\mathcal A_{\,x_{1,2}}$ is generated by the regular sequence $(\mathbf{tX})_1$, $(\mathbf{tX})_2$.
\item\label{improved.Y} The element $x_{1,2}$ of $R$ is regular on both $A$ and $N$ and $N_{x_{1,2}}\cong A_{x_{1,2}}\oplus A_{x_{1,2}}$.
\end{enumerate}
\end{proposition}
\begin{proof} (\ref{improved.a}) Let $\mathcal A$, $\mathcal N$, $\mathcal R$, and $J$ be the relevant modules built over $R_0$ and $\mathcal A_{\,\mathbb Z}$, $\mathcal N_{\ \,\mathbb Z}$, $\mathcal R_{\ \mathbb Z}$, and $J_{\mathbb Z}$ be the relevant modules built over ${\mathbb Z}$. We have shown in Proposition~\ref{*.claim} that
\begin{equation}\label{star}0\to \mathcal A_{\, \mathbb Z}\xrightarrow{\tau}
\mathcal N_{\ \,\mathbb Z}\xrightarrow{\tau(\xi)}\mathcal A_{\,\mathbb Z}\to \mathcal R_{\ \mathbb Z}/ J_{\mathbb Z}\to 0\end{equation} is an exact sequence. We know from Corollary~\ref{carry forward} and Theorem~\ref{main-Theorem} that $\mathcal A_{\,\mathbb Z}$, $\mathcal N_{\,\ \mathbb Z}$, and $\mathcal R_{\ \mathbb Z}/J_{\mathbb Z}$ are generically perfect $\mathbb Z[X]$-modules in the sense of \cite[Prop.~3.2 and Thm.~3.3]{BV}; and so, in particular, these modules are flat $\mathbb Z$-modules. Apply $R_0\otimes_{\mathbb Z}-$ to the constituent short exact sequences of (\ref{star}) in order to learn that $\operatorname{Tor}_1^{\mathbb Z}(R_0,J_{\mathbb Z}\mathcal A_{\mathbb Z})=0$ and $R_0\otimes_{\mathbb Z}\text{(\ref{star})}$, which is isomorphic to (\ref{*.claim.1}), is exact.
\medskip \noindent(\ref{improved.b}) The
proof from (\ref{improved.a}) also works for (\ref{improved.b}) because the $\mathbb Z[X]$-modules $A$ and $A'$, built over $\mathbb Z$, are also generically perfect, see Lemma~\ref{KL}.
\medskip \noindent(\ref{improved.X}) The proof of Corollary~\ref{corollary}.\ref{cor-b} shows that $\mathcal A_{\,x_{1,2}}$ is equal to the polynomial ring $$R_0[x_{1,2}\ ,\ x_{1,2}^{-1}][S_1\ ,\ (\mathbf{tX})_1\ ,\ (\mathbf{tX})_2],$$ where $S_1$ is the list of indeterminates given in (\ref{Ssub1}); furthermore, $J\mathcal A_{\,x_{1,2}}$ is generated by the two variables $(\mathbf{tX})_1$ and $(\mathbf{tX})_2$.
\medskip \noindent(\ref{improved.Y}) We saw in Corollary~\ref{KL-consq} that $x_{1,2}$ is regular on $A$. Recall from Lemmas~\ref{main-Dream-Lemma} and \ref{KL} that the ring $A$ and the $A$-module $N$ are perfect $R$-modules, and their annihilators (as $R$-modules) have the same grade. It follows that $\operatorname{Ass} N\subseteq \operatorname{Ass} A$ and that $x_{1,2}$ is also regular on $N$. The final assertion is obtained by localizing (\ref{exact-seq.1}), which is exact by (\ref{improved.b}), at $x_{1,2}$.
\end{proof}
\bigskip
\section{Remarks and questions.}\label{further}
\bigskip
The definition of $N$, as given in Notation~\ref{Not2}.\ref{Not2.a.iii}, is that $$\textstyle N=\frac RI\otimes_R\operatorname{coker}(d_1:\bigwedge^3F^*\to F^*).$$ However, if $2$ is a unit in $R_0$, then the next result shows that it is not necessary to apply the functor $\frac RI\otimes_R-$.
\begin{observation}\label{two}Adopt the language of {\rm\ref{data2}.\ref{data2-one}}, {\rm\ref{Not2}.\ref{Not2.a}} and {\rm(\ref{pre-cplx})}. If $2$ is a unit in $R_0$, then
$$\textstyle \operatorname{coker}(d_1:\bigwedge^3F^*\to F^*)$$
is an $R/I$ module; so, in particular $N=\operatorname{coker}(d_1:\bigwedge^3F^*\to F^*)$.
\end{observation}
\begin{proof}If $\phi_4\in \bigwedge^4F^*$ and $\phi_1\in F^*$, then
\begin{align*}\xi^{(2)}(\phi_4)\cdot \phi_1&= [\phi_1(\xi^{(2)})](\phi_4)+\xi^{(2)}(\phi_1\wedge \phi_4)&&\text{Proposition~\ref{A3}}\\
&= [\phi_1(\xi)\wedge \xi](\phi_4)+{\textstyle\frac 12}\xi(\xi(\phi_1\wedge \phi_4))&&\text{(\ref{Gamma})}\\
&=\xi\Big ([\phi_1(\xi)](\phi_4)+{\textstyle\frac 12}\xi(\phi_1\wedge \phi_4)\Big),\end{align*} which represents $0$ in $N$. \end{proof}
\begin{remarks}Adopt the language of {\rm\ref{data2}} and {\rm \ref{Not2}}. \begin{enumerate}[\rm(a)] \item The hypothesis ``$2$ is a unit in $R_0$'' is essential in Observation~\ref{two}. For example, if $R_0$ is the field $\mathbb Z/(2)$, then
$$\bmatrix 0&0&0&0&x_{1,2}x_{3,4}-x_{1,3}x_{2,4}+x_{1,4}x_{2,3}\endbmatrix^{\rm T}$$is zero in $N$, but is not in image of $d_1$. So, in particular, if $R_0$ is a field, then the first Betti number of $N$, as a module over $R$, depends on the characteristic of $R_0$, even when $\goth f=5$. We recall that the first Betti number of $A$, as a module, over $R$ depends on the characteristic $R_0$, but not until $\goth f=8$; see, for example, \cite{K-I,K-II,H95}.
\item\label{R5.b} Assume $R_0$ is a field. Suppose that $\mathbb F:\ \dots \to F_i\to \dots$ and $\mathbb G:\ \dots \to G_i\to \dots$ are minimal homogeneous resolutions of $A$ and $N$ by free $R$-modules with $F_i=\bigoplus R(-j)^{\beta_{i,j}}$ and $G_i=\bigoplus R(-j)^{\gamma_{i,j}}$. Then the proof of Theorem~\ref{main-Theorem} shows that the minimal bi-homogeneous resolution of $\mathcal R/J$ by free $\mathcal R$-modules is $\mathbb L:\ \dots \to L_i\to \dots$, with
$$L_i= \bigoplus \mathcal R(-j-1,-2)^{\beta_{i-2,j}}\oplus \bigoplus \mathcal R(-j-1,-1)^{\gamma_{i-1,j}}\oplus
\bigoplus \mathcal R(-j,0)^{\beta_{i,j}}.$$
Indeed, the iterated mapping cone
associated to
$$\xymatrix{ \mathcal R(-1,-2)\otimes_R \mathbb F[-2]\ar[r]&\mathcal R(-1,-2)\otimes_R A \ar@{>->}[d]^{\tau}\\
\mathcal R(-1,-1)\otimes_R \mathbb G[-1]\ar[r]&\mathcal R(-1,-1)\otimes_R N \ar[d]^{\tau(\xi)} \\
\mathcal R\otimes_R \mathbb F\ar[r]&\mathcal R\otimes_R A\ar@{->>}[d]\\&\mathcal R/J}$$
is a bi-homogeneous resolution of $\mathcal R/J$ and consideration of the $t$-degree shows that this resolution is minimal.
\item Retain the language of (\ref{R5.b}). If the characteristic of $R_0$ is zero, then the resolution $\mathbb F$ is given in Theorem 6.4.1 and Exercises 31--33 on page 222 in \cite{W03}. Can the geometric method of \cite{W03} also be used to obtain the minimal homogeneous equivariant resolution of $N$ by free $R$-modules?
\item One consequence of (\ref{R5.b}) is that the minimal homogeneous resolution $\mathbb G$ of $N$ by free $R$-modules is self-dual. Is this fact obvious for
some other reason?
\item Is the resolution $\mathbb X$ of $N$ by free $A$-modules from Observation~{\rm\ref{yet-to-come}.\ref{yet.e}} a linear complex?
\end{enumerate}\end{remarks}
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.